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+{"text":"\\section{Introduction}\n\nMost currently popular models for the formation of galaxies and larger\nstructures postulate that growth occurs hierarchically through\ngravitational clustering in such a way that small objects form first \nand then aggregate into larger systems (Frenk et al. 1988, Carlberg\nand Couchman 1989, Kauffman and White 1993, Lacey and Cole 1993). The \nobservational evidence seems to support these models since it is\npossible to trace the influence of merging both on galaxies and on\nclusters of galaxies. We can classify mergers roughly according to the\nmasses of the objects involved: (1) major mergers and (2) minor\nmergers. The first involve galaxies of comparable mass and are often  \ninvoked as a mechanism to form elliptical galaxies\nfrom spiral systems (Toomre 1977, Negroponte and White 1983, Schweizer\n1990, Barnes 1992, Silk and Wyse 1993). Minor mergers involve a giant \ngalaxy accreting a small satellite. The Large Magellanic Cloud and our\nown Galaxy are a clear example of such an ongoing minor merger. In\nspite of a general consensus that minor mergers may drive internal \nevolution in galaxies, the consequences of these events are still not\nclear (Quinn and Goodman 1986, Quinn, Hernquist and Fullagar\n1993, Walker, Mihos and Hernquist 1994 and, Huang and Carlberg 1997; \nhereafter QG, QHF, WMH and HC, respectively).\\\\\n\nAn important constraint on the effects of minor mergers was raised\nby Ostriker (1990) and T\\'oth and Ostriker (1992, hereafter TO).\nIn a high density Universe dominated by cold dark matter (CDM) about \n$80$ per cent of the dark haloes have undergone a merger in the past $5$\nbillion years which increased their mass by 10  per cent or more\n(Frenk et al. 1988, Kauffman and White 1993, Navarro, Frenk and White \n1994). TO argued that such a merger rate is too high to be compatible\nwith the observed thinness and coldness of discs in spiral galaxies. \nUsing a semi-analytic treatment of the problem, they derived an \nuncomfortably low upper limit for the mass that could be accreted by \na disc like that of the Milky Way -- no more than $4$ per cent of the \npresent mass within the solar circle could have been accreted during \nthe last $5$ billion years, given the observed local values of Toomre's \nstability Q-parameter and of disc scale-height. They argued that this\nconstraint favours a low density universe (perhaps with a cosmological\nconstant) in which the expected merger rate is low.\\\\\n\nSeveral complementary lines of investigation have been used to\nevaluate the argument proposed by TO. One approach\nconcentrates on understanding the later stages of merging and the\ndynamical evolution of disc structure.\nSemi-restricted and full N-body simulations\nhave been employed to explore how infalling satellites perturb \ndiscs (QG, TO, QHF, WMH, HC). The most recent work by WMH and\nHC used full N-body simulations and incorporated a key\ningredient in the accretion process, the responsiveness of the\nhalo. A possible weakness of all these studies is that they \nfocussed on satellites on nearly\ncircular orbits. For instance, WMH  followed their satellite only\nafter it was already within $21$ kpc of the galaxy centre; this choice\nwas imposed by their decision to use a very large number of particles \n(500,000) in order to reduce numerical noise and to delay the\ngrowth of a bar in their model disc. They found that accretion of a \nsatellite with $10$ per cent of the disc mass was already enough\nto provoke a 60 percent thickening of the stellar disc \nat the solar circle. In contrast, HC found that satellites with\nmasses between $10$-$30$ per cent of the disc mass could be put on\nnear-circular orbits at about $10$ disc half-mass radii,\nand would be sufficiently disrupted by tides before interacting\nstrongly with the disc that their effects on it are quite small.\nThe present paper is a continuation and extension of this line of\nresearch.\\\\\n\nA complementary approach concentrates on evaluating the rate at which\nsatellites are accreted as a function of mass and orbital parameters.\nObservation-based arguments can be used to\nestimate the current merger rate either for dark haloes or for \ngalaxies. Thus, signs of disturbance like tidal tails and shells \ncan be considered signs of a recent merger and thus allow the merger \nrate of luminous galaxies to be obtained. From a sample of\n$4000$ galaxies, 10 were identified by Toomre (1977) as results of a\nrecent merger. From this he was able to derive a lower limit of\n$0.005$ Gyr$^{-1}$ for the current merger rate. A higher merger rate \nof about $0.04$ Gyr$^{-1}$ was found by Carlberg, Pritchet \\& Infante\n(1994) based on the observed numbers of close galaxy pairs and\nthe assumption is that pairs will merge if their closest approach\ndistance and relative velocity are less than their characteristic \nradius and their internal velocity dispersion, respectively (Aarseth\nand Fall 1980). On larger scales it is clear that substructure is a \ncommon feature of clusters of galaxies, suggesting that many of them \nhave formed recently by the merging of several smaller systems\n(Dressler and Shectman 1988; Jones and Forman 1992; Richstone, Loeb \nand Turner 1992). Using another version of the TO argument, the\nobserved amount of substructure in clusters can give an estimate of\nthe current merging rate and so of the cosmic density parameter\n$\\Omega$ (Richstone, Loeb and Turner 1992; Lacey and Cole 1993;\nKauffman and White 1993; Evrard et al. 1994). Neither of these\narguments, however, can give the satellite accretion rates needed to\nevaluate the TO argument.\\\\\n\nSatellite accretion rates can be estimated from theoretical arguments\nbased either on the Press-Schechter model for hierarchical clustering\n(Lacey and Cole 1993) or on high resolution simulations. For example,\nNavarro, Frenk and White (1994, 1995) have carried out a series of\nsimulations of the formation of galaxy-satellite systems in an \n$\\Omega = 1$ CDM universe; their results suggest that the existence of\nthin discs may, perhaps, be reconciled with such a model, because\ndiscs are less efficient at accreting material than are their\nsurrounding dark haloes. In these simulations fewer than 30 per cent\nof the discs grew by more than 10 per cent in mass over the last 5 \nGyr, whereas about $80$ of the haloes grew by this much or more.\nThe distributions of orbital orientation and orbital eccentricity of\ntheir satellites were essentially uniform. Thus,\nLacey and Cole's (1993) suggestion that the TO constraint might be\navoided if satellites are primarily on near-circular orbits does not\nseem to be viable in a realistic hierarchical clustering model. Note\nthat while the simulations of Navarro et al. (1995) give useful \nindications about the rates\nof satellite accretion onto discs, their resolution is too low to\ngive information about how this accretion affects disc structure.\\\\\n\nIn the present paper we address the heating of the\nstellar disc by infalling satellites following the first approach \ndiscussed above. We consider satellites with a variety of internal \nstructures and on orbits with a variety of initial orientations and\neccentricities. We follow the evolution of the systems by using\nfull N-body simulations of all the components. Our paper is organized as\nfollows: section 2 contains brief descriptions of the models we adopt\nfor our primary and satellite galaxies, of our numerical methods, and\nof the parameters of the set of simulations we have carried out. \nSection 3 studies how the disc is heated and thickened by\ninfalling satellites and a comparison with TO's results is given in\nsection 4. We discuss the tilting and warping of the disc resulting\nfrom such events in section 5. Section 6 concentrates on the\ndisruption of the satellite and the evolution of its orbit. We show,\nin section 7, that the latter can be well reproduced by Chandrasekhar's local\nformulation of dynamical friction. In section 8 we replace our\n``live'' halo by a rigid one to demonstrate how halo response affects\nthe accretion and disc heating processes. Finally, in section 9, we\nsummarize our main conclusions.\\\\\n\n\n\n\\begin{table}\n\\begin{center}\n\\centering\n\\label {symbol1}\n\\caption{ Galactic parameters.}\n\\begin{tabular}{|l|c|l|} \\hline \n      \t&  Symbol &  Value \\\\ \\hline \\hline\nDisc: \t&            &              \t\t\t \t\\\\\n\t&  $N_D$     & $40\\,960$\t\t\t\t\t\\\\\n      \t&  $M_D$     & $ 5.6 \\times 10^{10}$ M$_\\odot$ \t\t\\\\\n      \t&  $R_D$     & $ 3.5$ kpc \\\\\n\t&  $z_o$     & $ 700 $ pc\\\\\n\t&  $Q_\\odot $ & $1.5$ \t\t\t\t\t\\\\\n\t&  $R_\\odot $ & $8.5$ kpc \\\\\n\t&$\\epsilon_D$& $175$ pc \\\\ \\hline\nBulge:  &\t     &\t\t\t\t\t\t\\\\\n\t&  $N_B$     & $4\\,096$\t\t\t\t\t\\\\\n\t&  $M_B$     & $1.87 \\times 10^{10}$ M$_\\odot$\t\t\\\\\n\t&  $a^*$     & $525$ pc\t\t\\\\\n\t&$\\epsilon_B$& $175$ pc \\\\ \\hline\nHalo:\t& \t     & \t\t\t\t\t\t\\\\\n\t& $N_H$\t     & $171\\,752$\t\t\t\t\t\\\\\n\t& $M_H$      & $7.84 \\times 10^{11}$ M$_\\odot$ \\\\\n\t& $\\gamma$   & $3.5$ kpc \\\\\n\t& $r_{cut}$  & $84$ kpc \\\\\n\t&$\\epsilon_H$& $175$ pc \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\medskip\n$^*$ The bulge half-mass radius is $1.27$ kpc. $N_D$, $N_B$ and $N_H$\ncorrespond to the number of particles used for each galaxy component. \nThe system of units is such that $G=M_D=R_D=1$.\n\n\\end{table}\n\n\\section{Numerical Preliminaries}\n\nIn this section we describe briefly the idealised models which we\nadopt to describe the primary disc galaxy and the satellites which\nmerge with it. We also describe the numerical tools employed to\nfollow the dynamical evolution of the primary\/satellite system, and\nthe parameters which define the specific set of simulations which we\nhave carried out.\n\n\\subsection{The Primary Galaxy Model}\n\nWe use the methods of Hernquist (1993) to set up a self-consistent \nN-body realization of a galaxy model consisting of three\ncomponents: a disc, a bulge and a halo.  A detailed description of \nthe technique can be found in Hernquist's paper. The density\ndistributions of the three components are: \\\\ \n\n\\begin{equation}\n\\rho_D(R,z)\\,=\\,{{M_D}\\over{4 \\pi R_D^2 z_o}} \\exp(-R\/R_D)\n\\hbox{ sech}^2(z\/z_o),\n\\end{equation}\n\n\\begin{equation}\n\\rho_B(r)\\,=\\,{{M_B}\\over{2 \\pi}}{{a}\\over{r(a+r)^3}},\n\\end{equation}\n\n\\begin{equation}\n\\rho_H(r)\\,=\\,{{M_H \\alpha}\\over{2 \\pi^{3\/2}r_{cut}}}{{\\exp\n(-r^2\/r_{cut}^2)}\\over{r^2+\\gamma^2}}.\n\\end{equation}\n\nHere, $M_D$, $M_B$ and $M_H$ correspond to the masses of the disc, the\nbulge and the halo, repectively. $R_D$ and $z_o$ are the radial and vertical\nscale lengths of the disc. $a$ defines the scale length of the bulge\nand corresponds to a half-mass radius of $a(1+\\sqrt{2})$ (Hernquist\n1990). Finally, $\\gamma$ and $r_{cut}$ are the core and cut-off radii\nfor the halo and $\\alpha$ is a normalisation constant. Notice that we\nassume both the bulge and the halo to be spherical.\\\\ \n\nThe velocities are derived from the Jeans equations (e.g. Binney and\nTremaine 1987). Isotropic gaussians are assumed for the halo and\nbulge velocity distributions. For the disc, the square of the radial\nvelocity dispersion is taken to be proportional to the surface density\nof the disc, $\\sigma_R^2 \\propto \\exp(- R_D\/R)$ (Lewis and Freeman\n1989) and the vertical component of the velocity ellipsoid is\ndetermined from $\\sigma_z^2=\\pi G \\Sigma (R) z_o$ in agreement with an\nisothermal sheet (Spitzer 1942). The azimuthal component is obtained\nfrom the epicyclic approximation, $\\sigma_\\phi^2=\\sigma_R^2\n\\kappa^2\/(4 \\Omega^2)$. Finally, the constant of proportionality is\ndetermined by fixing the value of Toomre's stability Q-parameter to a\ngiven value. We select $Q_\\odot = Q( R_\\odot) =1.5$ at the Solar radius. \\\\\n\nWe have chosen a system of units such that $U_m=M_D=1$, $U_l=R_D=1$ and\n$G=1$. For a disc mass of $5.6 \\times 10^{10}$ M$_\\odot$ and a disc radial \nscale length of $3.5$ kpc (Bahcall, Schmidt and Soneira 1983) the units of \ntime and velocity are $1.3 \\times 10^7$ yr and $262$ kms$^{-1}$, \nrespectively.  The half-mass radius of the disc is $\\sim 1.7\\,R_D$ with \na rotation period at this radius of about $13$ time units. The model can \nbe easily scaled through the following expressions for the time and velocity \nunits:\n\n\\begin{equation}\nU_t\\,=\\,4.709\\times 10^{11}\\,\\Big{(}{U_l^3\\over U_m}\\Big{)}^{1\/2}\\, \\hbox{yr}\n\\end{equation}\n \n\\begin{equation}\nU_v\\,=\\,2.076 \\times 10^{-3}\\,\\Big{(}{U_m\\over U_l}\\Big{)}^{1\/2}\\,\\hbox{kms}^{-1}\n\\end{equation}\n \n\\noindent where $U_m$ and $U_l$ are given in solar masses and \nkpc, respectively. In Table 1 we summarize the values of the \nparameters that define our primary galaxy model. \\\\\n\nWe should notice that \n(1) our halo is probably too small to be realistic.\nStudies of satellites in the Local Group and around external\ngalaxies show that galactic haloes extend to radii beyond $200$ kpc\nwith masses exceeding $2 \\times 10^{12}$ M$_\\odot$ (Zaritsky et al.\n1989, Zaritsky and White 1994). However, our halo is consistent with\nthe largest velocities observed for halo stars in the solar neighborhood\n(Carney and Lathman 1987) and should be massive enough to give\nrealistic orbital velocities for eccentric satellite orbits. (2) Our \nhalo is maybe too concentrated. Persic, Salucci and Stel (1996) argue for a \nhalo core radius of about $1-2\\,R_{opt}$ where $R_{opt}=3.2\\,R_D$ is the \noptical radius. However, a model with such a halo will be prone to \nform a bar which is an undesirable additional source of disc heating and \nto prevent its growth it will be necessary to increase the bulge \ncontribution. The values of the parameters listed in Table 1 guarantee \nstability against bar formation of the disc galaxy model in isolation (Vel\\'azquez \nand White, in preparation). \\\\\n \nThe rotation curve of our galaxy model (solid lines) is shown in figure 1. \nFor comparison, two other rotation curves are displayed for different bulge \nmasses but with the same disc and halo. The rotation curve for our galaxy \nmodel at the optical radius is $V_C(R_{opt})\\approx 243$ kms$^{-1}$ in model units while the \nmodels with a bulge mass of $M_B=0.2\\,M_D\\,(=1.12 \\times 10^{10}\\,\\hbox{M}_\\odot)$ and $M_B=2\/3\\,M_D\\,\n(=3.73 \\times 10^{10}\\,\\hbox{M}_\\odot)$ have values of \n$V_C(R_{opt})\\approx 236$ kms$^{-1}$ and $V_C(R_{opt})\\approx 257$ kms$^{-1}$, respectively. We can \nobserve that, at difference of the sample of rotation curves given by \nPersic et al. (1996), the rotation curve of our model shows a steeper rise \nin the inner region which is consistent with the observed nuclear rotation curves of galaxies in the CO-line emission and with the optical rotation \ncurves for large bright galaxies (Sofue et al. 1997, Courteau 1997).  Furthermore, it peaks near $R_{opt}$ in agreement with the results of \nCourteau (1997). \\\\\n\n\n\\begin{figure}\n\\plotone{Fig01.ps}\n\\caption{ The rotation curve for our disc galaxy model \n(solid lines) has a value of $V_C(R_{opt})\\approx 243$ kms$^{-1}$ where \n$R_{opt}=3.2\\,R_D$. Dotted-lines and dashed-lines \ncorrespond to rotation curves for bulge masses of $0.2\\,M_D$ and $2\/3\\,M_D$ \nwith values of $V_C(R_{opt})\\approx 236$ kms$^{-1}$ and $V_C(R_{opt})\\approx 257$ kms$^{-1}$, \nrespectively. For clarity the contribution of the disc and halo are not \nshown. The arrow indicates the Solar position.}\n\\label{Fig. 1}\n\\end{figure}\n\nFinally, we should mention that for this galaxy model we have used a \ntotal number of  $216\\,808$ particles. Most of these are in the halo \ncomponent $(171\\,752)$ where large numbers are required to reduce \nheating of the disc by two-body relaxation effects. \\\\\n\n\n\\begin{table}\n\\begin{center}\n\\centering\n\\label {symbol2}\n\\caption{ Satellite models.}\n\\begin{tabular}{|l|c|l|} \\hline\n Model  &  Symbol    & Value \\\\ \\hline \\hline\n S1:\t&            &              \t\t\t \t\\\\\n      \t&  $M_S$     & $ 5.60 \\times 10^9$ M$_\\odot$ \t\t\\\\\n      \t&  $r_c$     & $1$ kpc \\\\\n\t&  $c$       & $0.8$ \\\\  \n\t&  $\\rho_c$  & $0.52$ M$_\\odot$\/pc$^3$ \\\\ \n\t&  $\\sigma_c$& $52$ kms$^{-1}$\t\\\\\t\\hline\n S2:    &\t     &\t\t\t\t\t\t\\\\\n      \t&  $M_S$     & $ 5.60 \\times 10^9$ M$_\\odot$ \t\t\\\\\n      \t&  $r_c$     & $ 500$ pc \\\\\n\t&  $c$       & $ 1.1$\t\t\t\t    \t\\\\ \n\t&  $\\rho_c$  & $0.84$ M$_\\odot$\/pc$^3$ \\\\ \n\t&  $\\sigma_c$& $60$ kms$^{-1}$ \\\\ \\hline\n S3:    &\t     &\t\t\t\t\t\t\\\\\n      \t&  $M_S$     & $ 1.12 \\times 10^{10}$ M$_\\odot$\t\\\\\n      \t&  $r_c$     & $ 875$ pc \\\\\n\t&  $c$       & $ 1$\t\t\t\t    \t\\\\ \n\t&  $\\rho_c$  & $ 1.36 $ M$_\\odot$\/pc$^3$ \\\\ \n\t&  $\\sigma_c$& $ 71$ kms$^{-1}$ \\\\\t\t\\hline\n\\end{tabular}\n\\end{center}\n\\medskip\n$c$ defines the concentration of the satellite model and is given by\n$\\log_{10}(r_{t}\/r_c)$ where $r_c$ and $r_t$ are, respectively, the\ncore and tidal radii of a King model. $M_S$, $\\rho_c$ and $\\sigma_c$ denote \nthe satellite mass, the central density and the central one-dimensional \nvelocity dispersion, respectively.\n\\end{table}\n\n\n\\begin{figure}\n\\plotone{Fig02.ps}\n\\caption{ Rotation curve for our satellite models.}\n\\label{Fig. 2}\n\\end{figure}\n\n\\subsection{ Satellite Models}\n\nThe satellites are represented by self-consistent King models (King\n1966) which provide a reasonable fit to early-type and nucleated dwarf\ngalaxies (Vader and Chaboyer 1994). These models are a sequence of\ntruncated isothermal spheres parametrized by a concentration $c \\equiv\n\\log_{10}(r_t\/r_c)$, where $r_t$ and $r_c$ are the so-called tidal and\ncore radii, respectively. To specify completely a satellite model we\nprovide the satellite mass, $M_S$, its concentration and its tidal radius. \nThe latter is estimated from the density contrast\n$\\rho_S(r_t)\/\\overline{\\rho}_G(r_a) \\sim 3 $ at the apocentric\ndistance, $r_a$, of the orbit where the satellite is initially\nplaced. The parameters that characterize our satellite \nmodels, including the central density and the central one-dimensional \nvelocity dispersion, are listed in Table 2 which are within the observed \nvalues (Vader and Chaboyer 1994; Binggeli and Cameron 1991; Bender, Burstein \nand Faber 1992). The rotation curve for each satellite model is shown in \nfigure 2. We can notice that our satellite models are more concentrated than \nDD154-type dwarf galaxies (Moore 1994). In all cases, the \nsatellite consists of $8192$ particles. \\\\\n\n\\subsection{ Numerical Methods and Orbital Parameters}\n\nTo follow the evolution of the galaxy-satellite system we use a tree\nalgorithm with a tolerance parameter of \n$\\theta_{\\hbox{{\\footnotesize tol}}}=0.75$ and\nan integration timestep of $1.3 \\times 10^6$ yrs. Forces between particles are\ncomputed including the quadrupole components (see Barnes and Hut (1986)\nand Hernquist (1987) for details of this code). With these values, the\nconservation of global energy and angular momentum is better than $1$ \nper cent for all our models.\\\\ \n\nSince our initial galaxy and satellite models are not in a perfect\nequilibrium we allow them to relax separately before starting an\ninteraction simulation. The disc galaxy is relaxed for 20 time units, \nabout one and a half rotation periods at its half-mass radius, \nwhile the satellite\nis allowed to evolve for $40$ time units to reduce initial transient\neffects. These two configurations are then superposed to create\nour initial conditions. We have performed a set\nof experiments varying the parameters that are likely to\ninfluence the result of a galaxy-satellite interaction, for example, the\n`circularity' of the orbit, the angle between the angular momentum of\nthe satellite and the disc, and the satellite structure. In Table 3\nwe list the parameters\nof these simulations. Here, the `circularity' of the orbit has been\ndefined as $\\epsilon_J \\equiv  J\/J_C(E)$, where $J$ is the\nangular momentum of the satellite and $J_C(E)$ is the corresponding\nangular momentum for a circular orbit of the same energy $E$ as the \nsatellite's orbit. Also, we\nhave followed the subsequent evolution of the galaxy model in\nisolation (our control model) to distinguish effects produced by\ntwo-body encounters from those provoked by the accretion of the\nsatellite. The discussion of the following sections all refers to\nsatellites on orbits which actually intersect the disc. \\\\\n\n\n\\begin{table}\n\\begin{center}\n\\centering\n\\label {symbol3}\n\\caption{ Simulations.}\n\\begin{tabular}{|l|c|c|c|c|c|} \\hline\n Name  &  Sat. model &  $\\theta_i$ & $\\epsilon_J$ & $r_p\/R_D$ & $r_a\/R_D$  \\\\ \\hline \\hline\nG1S1    &    S1\t     &  $45^{o}$  &   $0.33$   &  $1.5$ & $16.86$ \\\\\nG1S2    &    S1\t     &  $0^{o}$   &   $0.55$   &  $3$   & $15.71$ \\\\\nG1S3    &    S1\t     &  $45^{o}$  &   $0.55$   &  $3$   & $15.71$ \\\\\nG1S4    &    S1\t     &  $90^{o}$  &   $0.55$   &  $3$   & $15.71$ \\\\\nG1S5    &    S1\t     &  $135^{o}$ &   $0.55$   &  $3$   & $15.71$ \\\\ \nG1S6    &    S1\t     &  $180^{o}$ &   $0.55$   &  $3$   & $15.71$ \\\\\nG1S7    &    S1\t     &  $0^{o}$   &   $0.82$   &  $6$   & $13.29$ \\\\\nG1S8    &    S1\t     &  $45^{o}$  &   $0.82$   &  $6$   & $13.29$ \\\\\nG1S9    &    S2\t     &  $0^{o}$   &   $0.55$   &  $3$   & $15.71$ \\\\\nG1S10   &    S2\t     &  $45^{o}$  &   $0.55$   &  $3$   & $15.71$ \\\\\nG1S11   &    S2\t     &  $90^{o}$  &   $0.55$   &  $3$   & $15.71$ \\\\\nG1S12   &    S2\t     &  $135^{o}$ &   $0.55$   &  $3$   & $15.71$ \\\\\nG1S13   &    S2\t     &  $180^{o}$ &   $0.55$   &  $3$   & $15.71$ \\\\\nG1S14   &    S3\t     &  $45^{o}$  &   $0.55$   &  $3$   & $15.71$ \\\\ \nG1S15   &    S3\t     &  $135^{o}$ &   $0.55$   &  $3$   & $15.71$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\medskip\n$\\theta_i$ refers to the angle between the initial angular momentum of\nthe satellite and the initial angular momentum of the disc. $\\epsilon_J$ \ndefines the circularity of the orbit, $r_p$ and $r_a$ correspond to\nthe initial pericentric and apocentric radii of the orbit, respectively. \n\\end{table}\n\n\n\n\n\\begin{figure*}\n\\vskip 16cm\n\\caption{ The evolution of the disc for model G1S10. A fourth of the \ntotal number of disc particles has been plotted. The satellite has \nbeen tidally disrupted at time 252. By this time, the disc has been \nslightly tilted and shows an asymmetric configuration (see text for \na discussion of some of these issues).}\n\\label{Fig. 3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\vskip 16cm\n\\caption{ The disc evolution for the retrograde counterpart of \nfigure 3 (see model G1S12 in Table 3). The disc looks thicker \nthan it really is because of tilting.}\n\\label{Fig. 4}\n\\end{figure*}\n\n\n\\begin{figure}\n\\plotone{Fig05.ps}\n\\caption{ The kinematical properties of the disc in our\nmodel G1S10. Solid lines give the kinematics of the disc in this \nmodel after $3.3$ Gyr, while short-dashed lines show corresponding \nquantities for our control model. Dotted lines show the \ndifference between the two, and so the part of the changes due purely \nto the interaction. In all plots the vertical long-dashed line indicates \nthe position at $R_\\odot$.}\n\\label{Fig. 5}\n\\end{figure}\n\n\\section{ Disc Heating and Thickening }\n\nIn figures 3 and 4 we show the evolution of the disc of our models \nG1S10 and G1S12 (see Table 3). At the end of these simulations the \nsatellite has been completely disrupted and the disc component has \nbeen altered in three different ways: (1) it is hotter and thicker; \n(2) it is no longer axisymmetric; and (3) it is tilted and warped. \nWe discuss some of these effects in this and the following sections. \\\\\n\n\n\\begin{figure}\n\\plotone{Fig06.ps}\n\\caption{ The response of the disc to the infall of\na satellite with an initial mass of $0.2\\,M_D$. The satellite at this\ntime ($2.34$ Gyr) has been completely destroyed. The solid lines\ncorrespond to a satellite following a prograde orbit with initial\nangle of $\\theta_i = 45^o$, (G1S14) while the long-dashed lines\ncorrespond to its retrograde counterpart (G1S15) with $\\theta_i =\n135^o$. For reference, the\nkinematical structure in our isolated control model is also\nshown (short-dashed lines). The kinematical changes for model G1S14 \nare indicated by dotted lines and for model G1S15 by dashed-dotted lines. \nThe vertical line again indicates the position at the Solar radius.}\n\\label{Fig. 6}\n\\end{figure}\n\nTo determine the kinematics of the disc we rotate to axes aligned with\nthe principal\naxes of the disc inertia tensor and then average particle properties in\nconcentric cylindrical annuli. These averages are a\ncrude measure of the disc kinematics since warping and\nazimuthal asymmetry are not taken into account. The heating and thickening of the disc can be described by the\nchanges of the velocity dispersions $(\\Delta \\sigma_R, \\Delta\n\\sigma_\\phi, \\Delta \\sigma_z)$ and by the increase of the vertical\nscale length, $\\Delta z_o$. Effects due to two-body relaxation can\nbe subtracted by comparing the final state to that of the control model.\nThe increase of Toomre's stability parameter, $Q$, is also shown.\nWe compute the vertical scale length in each annulus \nusing the following definition $z_o (R) \\equiv <z^2>^{1\/2}$. \\\\ \n\nIn figure 5 we show the disc kinematical properties of our model G1S10 (solid lines)\ntogether with the corresponding properties of the control model (short-dashed lines). It is evident \nfrom this figure that thickening of the stellar disc\ndoes not occur uniformly at all radii; regions beyond $R_\\odot$\nare much more susceptible to damage by the infalling satellite than\nthe inner disc. Given the complexity of the\nfinal disc structure we found it convenient to sample the kinematics at\nthree representative radii: near the centre (averaged out to $\\sim 2$ kpc), at $R_\\odot$, and at $14$ kpc. This is sufficient to provide us\nwith a global view of the heating and thickening. In Table 4 we give \nthe final disc structure at these radii for all our simulations listed in \nTable 3\\footnote{Satellite particles are not considered in computing the \nkinematical properties of the disc because, in most of the cases, their orbits \ndo not fall into the disc plane and so they would be separeted from disc \nparticles by their kinematics. Particles from satellites on prograde \ncoplanar orbits would be more difficult to distinguish, however, in these \nsimulations they form a disc-like structure with a hole of about $3-4$ kpc \nin the central region (see also Barnes 1996).}. For example, the velocity \nellipsoid of the disc in model  \n(G1S10) grows by $(18,12,8)$ kms$^{-1}$, $(12,10,5)$ kms$^{-1}$ and\n$(19,14,8)$ kms$^{-1}$ (from inside to outside) resulting in \nincreases of\nToomre's $Q$ parameter by about $0.4$, $0.8$ and, $2.2$. Before the accretion event $Q$ was roughly constant between $3$ kpc  \nand $9$ kpc with a value of $\\sim 1.5$ and after\nsatellite accretion it has risen to $Q \\sim 2$ between $3$ kpc and $7$ kpc. \nThe vertical scale length in this same case increases by $100$ pc, $275$ pc \nand $500$ pc, representing  a thickening by about $14$, $39$\nand, $71$ per cent; while for its retrograde counterpart (model G1S11) is of $21$, $21$ and $43$ per cent. Disc thickening is even lower \nfor model G1S3 being of $7$, $14$ and $43$ per cent (see entries in Table 4). \nIn some of the models \n(G1S5, G1S10, G1S12 and G1S14) satellite accretion induces \nadditional heating and thickening by exciting a bar-like instability \nin the inner $5$ kpc. \\\\ \n\nThree main trends are observed in our simulations: (1) a mighty\ncorrelation between the change in disc structure and the relative\norientation of disc and satellite angular momenta. We find\nthat, in general, the disc is much more susceptible to damage by a satellite\non a prograde orbit than by its retrograde counterpart. This suggests\na resonant coupling between satellite orbit and disc. This difference\nis most clearly seen by comparing the entries in Table 4 for \nthe coplanar prograde case (model \nG1S9) and the corresponding polar case (model G1S11). In figure 6, we \nillustrate these effects using our two simulations with massive \nsatellites (G1S14 and G1S15 in Table\n3). Notice that for the retrograde case the disc is less strongly\naffected at all radii. (2) In general, the planar\ncomponents of the velocity ellipsoid\nrespond more strongly than the vertical component to the\naccretion of the satellite. Indeed, the ratio of the two planar components\nis fixed by epicyclic theory. The degree of anisotropy \nis correlated with both the inclination and the sense of motion of the\norbit, being greatest for prograde coplanar orbits.\nAs a result Toomre's stability parameter $Q$ responds more\nsensitively than the vertical scale length to the\naccretion event. This is largely a consequence of the fact (addressed\nin section 5) that\nvertical perturbations tend to tilt the disc rather than to heat it.\n(3) For our model G1S3 the effects of accretion on the disc are\nquite modest, but a more massive and compact satellite can have a very\nlarge effect (see entries of model G1S3 and G1S14 in Table 4). \\\\ \n\n\n\\begin{table*}\n\\begin{center}\n\\centering\n\\caption{Disc kinematical changes.}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|} \\hline\n\\multicolumn{1}{|c|}{} & \\multicolumn{3}{|c|}{$R_{c}^{*}$} & \\multicolumn{3}{|c|}{$R_\\odot$} & \\multicolumn{3}{|c|}{$R_{4}^{\\dag}$} \\\\ \\hline\nModel & $\\scriptstyle{(\\Delta \\sigma_R,\\Delta \\sigma_\\phi,\\Delta \\sigma_z)}$ & $\\scriptstyle{\\Delta Q}$ & $\\scriptstyle{\\Delta z_o}$ & $\\scriptstyle{(\\Delta \\sigma_R,\\Delta \\sigma_\\phi,\\Delta \\sigma_z)}$ & $\\scriptstyle{\\Delta Q}$ & $\\scriptstyle{\\Delta z_o}$ & $\\scriptstyle{(\\Delta \\sigma_R,\\Delta \\sigma_\\phi,\\Delta \\sigma_z)}$ & $\\scriptstyle{\\Delta Q}$ & $\\scriptstyle{\\Delta z_o}$ \\\\ \n\t  &    (kms$^{-1}$) &        &    (pc)& (kms$^{-1}$)&        &  (pc)\n& (kms$^{-1}$) &       &  (pc) \\\\ \\hline \\hline\n{G1S1}    &    $(9,6,4)$  & $0.3$  &  {$50$}  & $(5,3,2)$  & $0.5$   &  {$150$}  & $(10,7,4)$   & $1.2$ & $225$ \t\\\\\n{G1S2}    &    $(16,8,4)$ & $0.4$   &  {$75$}  & $(16,10,4)$ & $1.1$  &  {$200$}  & $(32,21,6)$ & $2.6$ & $300$ \t\\\\\n&    [$(16,7,5)$] & [$0.5$]   &  [{$100$}]  & [$(31,12,4)$] & [$1.4$]  &  [{$200$}]  & [$(47,26,9)$] & [$2.9$] & [$450$]     \\\\\n{G1S3}    &    $(10,7,5)$  & $0.3$   &  {$50$}  & $(5,3,2)$  & $0.5$   &  {$100$}  & $(11,6,5)$ & $1.4$  & $300$    \t\\\\\n{G1S4}    &    $(12,6,4)$  & $0.5$  &  {$100$}  & $(4,3,2)$   & $0.4$  &  {$100$}  & $(6,5,4)$  & $0.9$ & $250$  \t\\\\\n{G1S5$^{\\ddag}$}    &    $(24,16,8)$   & $0.6$  &  {$200$}  & $(4,2,3)$  & $0.5$   &  {$100$}  & $(6,6,4)$  & $1.0$ & $150$  \t\\\\ \n{G1S6}    &    $(6,5,7)$  & $0.3$ &  {$75$}  & $(5,4,3)$  & $0.4$ &  {$100$}  & $(10,7,5)$ & $1.7$ & $275$  \t\\\\\n{G1S7}    &    $(38,18,8)$  & $0.4$ &  {$100$}  & $(16,12,5)$  & $1.2$ &  {$250$}  & $(27,14,7)$ & $3.4$ & $450$  \t\\\\\n    &    [$(38,17,9)$]  & [$0.4$] &  [{$100$}]  & [$(15,12,6)$]  & [$1.1$] &  [{$250$}]  & [$(29,16,8)$] & [$3.4$] & [$500$] \\\\\n{G1S8}    &    $(10,10,6)$  & $0.3$  &  {$50$}  & $(10,7,4)$  & $0.8$  &  {$150$}  & $(17,12,6)$ & $2.2$ & $350$  \t\\\\\n{G1S9}    &    $(19,13,11)$  & $0.5$ &  {$100$}  & $(22,15,7)$  & $1.5$ &  {$300$}  & $(37,23,9)$  & $2.6$ & $550$  \t\\\\\n       &   [$(19,13,11)$]  & [$0.5$] &  [{$100$}]  & [$(25,15,8)$]  & [$1.6$] &  [{$350$}]  & [$(43,28,12)$]  & [$2.9$] & [$675$]  \t\\\\\n{G1S10$^{\\ddag}$}    &    $(18,12,8)$  & $0.4$ &  {$100$}  & $(12,10,5)$  & $0.8$ &  {$275$}  & $(19,14,8)$  & $2.2$ & $500$  \t\\\\\n{G1S11}   &    $(11,10,6)$  & $0.4$ &  {$100$}  & $(7,4,4)$ & $0.5$ &  {$150$}  & $(11,8,5)$ & $1.0$ & $300$  \t\\\\\n{G1S12$^{\\ddag}$}    &    $(17,11,11)$  & $0.8$ &  {$150$}  & $(4,4,4)$ & $0.4$ &  {$150$}  & $(10,6,6)$ & $1.0$ & $300$  \t\\\\ \n{G1S13}    &    $(7,8,4)$  & $0.5$ &  {$75$}  & $(12,10,7)$ & $0.8$ &  {$250$}  & $(19,15,10)$ & $2.4$ & $700$  \t\\\\\n{G1S14$^{\\ddag}$}    &    $(32,20,12)$  & $0.6$ &  {$150$}  & $(22,15,12)$ & $1.2$ &  {$550$}  & $(27,18,16)$ & $2.9$ & $800$  \t\\\\\n{G1S15}    &    $(20,12,8)$  & $0.6$ &  {$100$}  & $(11,9,6)$ & $0.8$ &  {$300$}  & $(21,12,10)$ & $2.2$ & $525$  \t\\\\ \\hline\n\n\\end{tabular}\n\\end{center}\n\\medskip\nSymbols $*$ and $\\dag$ denote quantities at the centre and $4\\,R_D$,\nrespectively. Models forming a bar are indicated by the symbol\n$\\ddag$. Quantities in brackets are computed by taking into account satellite \nparticles.\n\\end{table*}\n\n\\subsection{ Disc Heating and the Bulge\/Disc Mass Ratio}\n\nTo address the effect of the bulge in the heating process of the stellar \ndisc we have repeated a few of our simulations but with two additional \ngalactic models. One of them has a bulge with a mass of $M_B=0.2\\,M_D$ and the other with a mass of $M_B=2\/3\\,M_D$. We did not pursue a larger reduction of \nthe bulge mass since it requires a huge number of particles $(> 5\\times 10^5)$ to keep stable the disc against bar formation during the \ndisruption times in our simulations; this would be prohibitive (e.g. Walker et al. 1996). In these models both the disc and halo are \nthe same as in our primary galaxy model of section 2. In figure 1 we \nshow the rotation curves for these new galactic models.\\\\\n\nIn Table 5 we summarize the disc kinematical changes for these new experiments. \nPrefix G2 and G3 refer to the galactic model with the less and more \nmassive bulge, respectively. We should mention that model `G2' develops a \nbar in isolation at time $\\sim 3.2$ Gyr inside $4$ kpc. Comparing the entries \nin tables 4 and 5 at the Solar radius we can see that the disc has \nbeen thickened by $39$, $39$ and $21$ per cent for models G1S10, G2S10 \nand G3S10, respectively, while for their retrograde counterparts (G1S12, G2S12 and G3S12) the corresponding values are only of $21$, $29$ and \n$14$ per cent. This suggest that a very \nmassive bulge would be more efficient in reducing the heating and \nthickening of the stellar disc. This effect can be more clearly appreciated in \nour models with the more massive satellite (models G1S14, G1S15, G3S14 \nand G3S15). Thus, the difference between the thickening found \nby Walker et al. (1996) ($\\sim 60$\\%) and that in our simulations can \nbe attributed \nto the fact that in the former case the satellite interacts strongly with \nthe disc because of the orbit chosen and because of the bulgeless galaxy \nmodel. The latter forms a bar during the early stages of the accretion. \\\\   \n\n\n\\begin{table*}\n\\begin{center}\n\\centering\n\\caption{Disc kinematical changes.}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|} \\hline\n\\multicolumn{1}{|c|}{} & \\multicolumn{3}{|c|}{$R_{c}^{*}$} & \\multicolumn{3}{|c|}{$R_\\odot$} & \\multicolumn{3}{|c|}{$R_{4}^{\\dag}$} \\\\ \\hline\nModel & $\\scriptstyle{(\\Delta \\sigma_R,\\Delta \\sigma_\\phi,\\Delta \\sigma_z)}$ & $\\scriptstyle{\\Delta Q}$ & $\\scriptstyle{\\Delta z_o}$ & $\\scriptstyle{(\\Delta \\sigma_R,\\Delta \\sigma_\\phi,\\Delta \\sigma_z)}$ & $\\scriptstyle{\\Delta Q}$ & $\\scriptstyle{\\Delta z_o}$ & $\\scriptstyle{(\\Delta \\sigma_R,\\Delta \\sigma_\\phi,\\Delta \\sigma_z)}$ & $\\scriptstyle{\\Delta Q}$ & $\\scriptstyle{\\Delta z_o}$ \\\\ \n\t  &    (kms$^{-1}$) &        &    (pc)& (kms$^{-1}$)&        &  (pc)\n& (kms$^{-1}$) &       &  (pc) \\\\ \\hline \\hline\n{G2S10$^{\\ddag}$}    &    $(16,11,10)$  & $0.7$  &  {$75$}  & $(12,7,7)$  & $0.8$   &  {$275$}  & $(18,11,9)$   & $1.3$ & $600$ \t\\\\\n{G2S12$^{\\ddag}$}    &    $(9,10,5)$ & $0.3$   &  {$100$}  & $(7,5,4)$ & $0.7$  &  {$200$}  & $(11,6,9)$ & $0.7$ & $400$ \t\\\\ \\hline\n{G3S3}    &    $(5,6,6)$  & $0.6$   &  {$50$}  & $(4,3,3)$  & $0.4$   &  {$100$}  & $(7,6,4)$ & $0.8$  & $200$    \t\\\\\n{G3S10}    &    $(9,6,5)$   & $0.8$  &  {$100$}  & $(7,5,5)$  & $0.5$   &  {$150$}  & $(16,10,7)$  & $1.5$ & $400$  \t\\\\ \n{G3S12}    &    $(6,5,3)$  & $0.7$ &  {$50$}  & $(4,3,2)$  & $0.4$ &  {$100$}  & $(6,6,5)$ & $0.7$ & $275$  \t\\\\\n{G3S14}    &    $(16,6,9)$  & $0.8$  &  {$75$}  & $(10,8,8)$  & $0.8$  &  {$325$}  & $(26,14,13)$ & $2.5$ & $750$  \t\\\\\n{G3S15}    &    $(10,8,4)$  & $0.5$  &  {$50$}  & $(6,4,5)$  & $0.5$  &  {$200$}  & $(13,9,8)$ & $1.4$ & $425$  \t\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\medskip\nSymbols as in Table 4.\n\\end{table*}\n\n\n\\begin{figure*}\n\\ppplotone{Fig07.ps}\n\\caption {Disc kinematical changes in the Solar neighborhood. Here, we\nshow the changes in the parameter $Q$ (top-left), in the vertical\nscale length (top-right), and in the vertical (bottom-left) and radial\n(bottom-right) velocity dispersions as resulting by the accretion of a\nsatellite of mass $M_S$. Solid and dashed lines represent the\nkinematical changes as predicted by TO's formulae while starred \nsymbols represent the results from all our simulations (see text for details).}\n\\label{Fig. 7}\n\\end{figure*}\n\n\\section{ Comparison with TO's Results}\n\n\nIt is interesting to compare the size of the effects we find with\nthose predicted analytically by TO. They give formulae for the changes\nin the disc vertical scale length, velocity dispersions and stability\nparameter $Q$ (their equations 2.16, 2.21, 2.25 and 2.26). These can be\nused to estimate the quantities we list in Table 4 and 5. We apply\nthese formulae by inserting the properties of the unperturbed disc (as\nmeasured in our control model) at the Solar radius and the satellite\ntotal mass, and by averaging over the orbital inclination of the\nsatellite. In figure 7,\nwe show the resulting disc changes as predicted by TO\n(solid and dashed lines for two extreme cases) together with \nthose found in our simulations (denoted by starred symbols). The solid \nlines correspond to a vertical ($\\epsilon_\\perp$) and \nepicyclic ($\\epsilon_\\parallel$) efficiency heating assuming circular \norbits with random inclinations for the \nvelocity distribution of the satellites while the dashed lines refer to \nan isotropic Maxwellian distribution (see TO, Appendix A). Clearly, the \nanalytical predictions tend to\noverestimate the disruptive effects of satellite accretion by a\nnon-negligible factor; typically about $2-3$. These\ndifferences are more marked for the more massive satellite. Notice\nthat for models G1S2, G1S7 and G1S9 the change in\nthe stability parameter is closer to that predicted by TO. This is\nbecause the satellite in these models (which follows a prograde orbit\nnear the disc plane) induces substantial deviations in $\\Sigma_D(R)$. These\ndeviations were taken into account in determining the change in $Q$\nfor figure 7. If we ignore such variations,\nas TO did, then $\\Delta Q$ is entirely determined by $\\Delta\n\\sigma_R$; the difference is then more dramatic since\nthe change in $Q$ is reduced to $0.8$, $0.7$ and $1.0$ for models G1S2, \nG1S7 and G1S9, respectively. \\\\\n\nThe fact that we find substantially weaker effects than those\npredicted by TO suggests that their limits on the accretion rate of\nsatellites (based on the thinness of observed discs, and their\nsusceptibility to spiral instabilities) are likely to be too strict.\nThere are some additional effects which may result in our own results\nstill being a significant overestimate of the extent of the damage.\n\\begin{itemize}\n\\item The stellar dynamical heating of the disc in our \nsimulations may be partially compensated by dissipative effects in a\ngaseous component. N-body\/Hydrodynamical simulations carried out by Mihos and\nHernquist (1994) and the recent work by HC reveal that gas cooling can make\nan important contribution to sustaining spiral structure in galaxies.\n\\item Our simulations begin with the satellite already\nquite close to the disc and have relatively low mass halos. They are\nthus likely to underestimate disruptive effects on the satellite\nbefore it begins to interact strongly with the disc (see HC).\nProgress on this point will require larger simulations and properly\nrealistic ``cosmological'' initial conditions. Currently available \nresults suggest that discs may accrete material less efficiently\nthan haloes (NFW94, NFW95). \n\\item Damaging effects on disc structure will be\ndiminished whenever the satellite is less concentrated than those we\nhave considered and so can be disrupted before encountering the disc\n(again see HC).  Although simulations from\ncosmological initial conditions show that the distribution of orbital \neccentricities should be almost uniform so that a significant\nfraction of satellites reach the disc essentially unperturbed on their first\npassage, such\nhigh velocity, near-radial encounters cause less damage than\nprograde encounters at lower velocities.\n\\end{itemize}\nIt is interesting that the properties of the most strongly perturbed\ndiscs in our simulations do seem to agree with those of the thick disc\ncomponent at the solar radius. This component has a velocity ellipsoid\nof about $(63,42,38)$ kms$^{-1}$, a vertical scale length of $1$ kpc\nand an asymmetric drift of about $25$ kms$^{-1}$ (Freeman\n1996). Similar structure can be seen in the\ngalaxy NGC $4565$ for which the vertical scale length is $\\approx \\,\n1.5$ kpc (Morrison 1996). \\\\\n\nThe difference between TO's analytical results and those of our\nsimulations may be due to one or a combination of the following\nfactors: \n\\begin{itemize}\n\\item A self-gravitating satellite has the capacity to absorb part of\nits orbital energy which can be carried away by the stripped\nmaterial. In this way, the deposition of energy onto the disc is diminished. \n\\item Another important ingredient in the galaxy-satellite interaction\nis the responsiveness of the halo. This regulates the transfer of\nangular momentum between the different components and may weaken the\nresonant response of the disc to the infalling satellite.\n\\item In addition, TO do not make any distinction between satellites\non prograde and retrograde orbits and this has important consequences \nfor the survival and evolution of the disc (see section 3 and 5).\n\\item Another controversial point in TO's derivation is their\nassumption that the deposition of a satellite's orbital energy in the\ndisc is a local process. They argued that the overall energy change in\nthe disc would be the same if the excitation of modes\nin the disc were taken into account, but the formation of spirals and\nwarps reveals that {\\sl global} modes are excited by\nthe infalling satellite before it crosses the main disc.\n\\end{itemize}\n\n\n\\begin{figure}\n\\plotone{Fig08.ps}\n\\caption{Evolution of the vertical kinetic energy of the\ndisc for several of our models.\nThe solid line represents the vertical heating\nproduced by numerical relaxation. The starred symbols refer to the\nvertical kinetic energy as measured from a frame that coincides with\nthe main axes of the inertia tensor while the open circles show its\nevolution in the initial reference frame.} \n\\label{Fig. 8}\n\\end{figure}\n\n\\section{ Disc Tilting and Warping}\n\nAs the satellite sinks to the galactic centre it transfers energy and\nangular momentum to its surroundings (QHF, HC). Part of this energy,\nas we saw in section 3, goes to the disc in random vertical motions\ngiving rise to the disc thickening. There is also, however, a coherent\nresponse of the disc to the satellite accretion which is associated\nwith the tilting observed in figures 3 and 4. Looking in figure 8 we can also\nappreciate the effect of the orbital type, compactness and mass of the \nsatellite and the mass of the bulge on the evolution of the vertical \nkinetic energy. The solid line corresponds to the\nvertical heating of the disc component coming from numerical\nrelaxation while the starred symbols refer to the vertical kinetic\nenergy as measured in a reference frame that coincides with the main\naxes of the inertia tensor and the open circles correspond to the\nvertical kinetic energy in the original reference frame. We can appreciate \nsome general trends: (i) In the prograde models most of the energy \ngoes to thickening \nthe disc which is best illustrated by models G1S2, G1S9 and G1S14. \nIn contrast, the \nretrograde models clearly show a coherent response of the disc (e.g. G1S12 and \nG1S15). Thus, a smaller fraction of the energy goes to thickening the disc in \nthis case which tell us that the disc is more robust to mergers with satellites \non retrograde orbits than on prograde ones. Notice that for satellites on \npolar and retrograde coplanar orbits all the energy goes to the disc, \nhowever, the \nthickening is lower than in their prograde counterparts suggesting that the \nvertical kinetic energy is distributed in the satellite remnants and \nhalo particles. (ii) As pointed out in section 3, a more massive bulge \nmay reduce the vertical heating of the disc which is more dramatic in models \nG1S10, G2S10 and G3S10 but this is not the case in their \nretrograde counterparts for which the coherent response of the disc seems to \nbe reduced (e.g. models G1S12, G2S12 and G3S12). \\\\\n\n\n\n\\begin{figure*}\n\\vskip 18cm\n\\caption{Representation of the disc by a set of rings equally spaced\nin the polar radius. Each ring has been displaced to the centre\ndefined by the centre of mass of the particles contained in it and\nthen rotated to the principal axes of its inertia tensor. The size of\nthe boxes are $56 \\times 56 \\times 3.5$ in kpc so the plotted distortions \nfrom planarity exaggerate the true distortions. The mean radii of the\nrings shown in this figure are: $4.37$, $7.88$, $11.38$, $14.87$, and\n$18.35$ kpc.}\n\\label{Fig. 9}\n\\end{figure*}\n\nIn the absence of a satellite there is a small exchange of angular\nmomentum between the disc and the halo resulting in a small tilt of the\ndisc, less than $3^o$ over $\\sim 4.5$ Gyr. When a satellite is\nintroduced in the galaxy, the transfer of angular momentum changes\ndramatically. By the end of the simulation, the disc of model G1S14 \nhas been tilted by $\\sim 11^o$ while the angle $\\theta_i$ (for both bound\nand unbound satellite particles) has fallen by $\\sim 26^o$ from \nits initial value. For model G1S15, the disc has been tilted by $\\sim\n15^o$, which, as noted above, is larger than for the corresponding\nprograde case. In this last model, the angle $\\theta_i$ remained\nalmost unchanged which tell us about the importance of the\ncoupling between disc stars and the satellite orbit. However, in both\ncases about $45$ per cent of the total initial angular momentum of the\nsatellite remains in the satellite debris, the rest going to the\ndisc and the halo. Less massive satellites experience less dramatic\nevolution since the sinking rate and, hence, \nthe loss of angular momentum is proportional to the square of the\nsatellite mass. Thus, in our models G1S3 and G1S5 about 61\\% and 68\\%,\nrespectively, of the angular momentum remains in the satellite \ndebris. Also the disc tilting is only about $6^o$ and $8^o$ in these\ncases. \\\\\n\nOn the other hand, observations of edge-on disc galaxies show that a\nconsiderable fraction of them have a conspicuous feature in the form of\nan `integral sign', commonly known as a warp. Most of these structures\ndevelop far beyond the optical disc and are very extended (Briggs\n1990). A typical example is our own Galaxy for which the HI warp,\ndetected at $21\\,$cm, is revealed beyond the solar circle  (Henderson\net al. 1982). However, HI warps are observable mainly where the\noptical disc ends and the detection of stellar warps is harder. A\nsystemic attempt was made by S\\'anchez-Saavedra, Battaner and Florido\n(1990) by selecting a sample of $82$ edge-on disc galaxies from the\nPalomar Survey. They were able to identify $23$ cases of stellar warps\nin that sample. This study suggests that stellar warps are also a\ncommon feature, but their origin remains a puzzle.\\\\ \n\nFrom a theoretical perspective, several scenarios have been proposed to\ntry to explain the origin of the warping of disc galaxies (an\nexcellent review on the subject is found in Binney 1992). Among the\nmost frequently invoked is a cosmological origin reflecting the time when\nthe galaxy was built up. Current cosmological simulations suggest that\ndark haloes are highly flattened with mean axis ratios of $<c\/a> \\sim\n0.5$ and $<b\/a> \\sim 0.7$ (Dubinski and Carlberg 1991). As a\nconsequence, the disc angular momentum may not be in\nalignment with one of the main halo axes which may then induce \na warping mode in the disc. This tilting mode was studied\nby Sparke and Casertano (1988) for the case of a disc embedded in a\nflattened rigid halo; they found that it is consistent with observed \nwarps. However, recent self-consistent N-body simulations\ncarried out by Dubinski and Kuijken (1995) have shown that the\nresponse of the halo can not be ignored, since dynamical\nfriction can damp the warps in much less than a Hubble time. Nelson and\nTremaine (1995) used perturbation theory to arrive at a similar\nconclusion, suggesting that, in general, a primordial origin of\ngalactic warps may be ruled out.\\\\ \n\nAnother mechanism to excite and maintain warps resorts to the accretion\nof satellites (Binney 1992) which has been highlighted in a recent\npaper by Weinberg (1996). This last author has shown the importance\nof the response of the halo (assumed spherical) of our Galaxy to the\ninteraction with the LMC, and its subsequent effect on the formation of\nthe disc's HI warp. Since in all our simulations the halo is assumed \nspherical we can address the relevance of such events in the formation\nof stellar warps. For this, we proceeded by rotating our disc along\nthe main axes of the inertia tensor of the particles located inside\n$3\\,R_D$ and representing it by a system of initially concentric\nrings equally spaced and each containing a sufficient number of\nparticles. This done, each ring is displaced to a new centre\ndetermined by the centre of mass of the particles within it. Finally,\nwe compute the inertia tensor for each ring and \\mbox{rotate} to its principal \naxes. Figure 9 shows the result of this process for our isolated\ngalaxy model `G1' (upper panel) and for our models G1S3, G1S8 and G1S14,\nrespectively. The size of the each box is $56 \\times 56 \\times 3.5$ in\nkpc. We can see that model G1S3 produces no \ndistinguishable warp while a\nmore massive satellite (model G1S14) in the same orbit or a \nsatellite on\na more nearly circular orbit (model G1S8)\nproduces bigger changes. However, it is important to remark that even\nin the most favorable case the departure of the warp from the disc\nplane is less than $1.75$ kpc in the outer parts which \ncorrespond to an angle less than $7^o$.\\\\\n\n\n\\section{ Satellite Sinking and Disruption Times}\n\nIn order to verify the reliability of the decay rate and disruption\ntime of the satellites in our simulations we built up a self-consistent\nmodel called 'NEW' similar to our model G1S3 but with only a fourth\nof the number of particles in each of the components present in the\nmodel. The mass evolution and\ndisruption rate of the satellite for these models are shown in figure\n10(a)-(b) where the open circles represent model G1S3 and\nthe open squares joined by solid lines correspond to the 'NEW'\nmodel. Notice that the agreement is good with differences in the\ndisruption process of the satellite being more evident in the last\nstages of its evolution.\\\\ \n\n\n\\begin{figure}\n\\plotone{Fig10.ps}\n\\caption{(a) This figure shows the mass of the satellite as a function\nof time. In both models G1S3 (open circles) and NEW we obtained similar \ntimes for the total disruption of the satellite which occurs about $2.6-2.8$\nGyr. (b) The satellite position is presented as a function of\ntime. Notice that the satellite is completely destroyed before it\nreachs the centre of the galaxy. }\n\\label{Fig. 10}\n\\end{figure}\n\n\n\\begin{figure}\n\\plotone{Fig11.ps}\n\\caption{The disruption times of the satellites for most of our\nsimulations. Most of the damage to the satellites occurs at pericentre \nsince the tidal force due to the galactic potential is strongest there.}\n\\label{Fig. 11}\n\\end{figure}\n\n\n\\begin{figure}\n\\plotone{Fig12.ps}\n\\caption{The orbital decay rate for the satellites \nin most of our simulations.}\n\\label{Fig. 12}\n\\end{figure}\n\nAs the satellite is being accreted by the host galaxy, there are two main \nphysical \\mbox{mechanisms} that regulate its evolution: the tidal\ninteraction and dynamical friction. Figures 11 and 12 show the effects\nof these mechanisms in the overall evolution of the satellites for most of \nour simulations. These  effects are illustrated by\nour model G1S1. Notice that the biggest `jumps' in the mass evolution\nof the satellite correspond to its passage through pericentre. This is\neasily understood since at that position the tidal interaction of the\nsatellite with the host galaxy is the strongest since the\ngradient of the galactic gravitational force reachs a maximimum at\npericentre. Comparison of the different models yields the following\nconclusions: \n(i) for a given circularity and satellite model, the disruption of the\nsatellite occurs faster for prograde orbits than for retrograde\nones being more evident for coplanar orbits. Furthermore, just from \nthe mass loss and disruption total times, it is\ndifficult to distinguish between polar and retrograde orbits. In the\ncase of prograde orbits, the satellites suffer a faster disruption and\nsinking if they follow orbits which are closer to coplanar. \n(ii) For a given orbit, the decay rate and disruption of our satellite \nwith an initial mass of $0.2\\,M_D$ are faster than for the satellite with an \ninitial mass of $0.1\\,M_D$ since the decay rate is proportional to the\nsatellite mass. (iii) For a given satellite mass and orbit a\nmore compact satellite survives for a longer time than a less\ncompact one since its binding energy is greater. (iv) Finally,\nalthough we do not have a large number of simulations, we note that, \nfor a given satellite, the disruption time scales roughly as \n$\\epsilon_J^{1\/3}$.\\\\\n\n\\section{ Chandrasekhar's Dynamical Friction Formula}\n\nIn this section we briefly address the reliability of Chandrasekhar's\ndynamical friction formula (Chandrasekhar 1960) for describing the decay\nrate of the satellites in our N-body simulations. To carry\nout this study, we built up a galaxy model similar to our fully\nself-consistent one but replacing the self-gravitating halo by a rigid\npotential which is free to move in response to the particle\ndistribution of the other components. For a Maxwellian distribution of\nvelocities with a dispersion velocity $\\sigma(r)$ we have that the\ndynamical friction on the satellite is given by (Binney and Tremaine\n1987): \n\n\\begin{equation}\n{\\bf F}_{df}\\,=\\,-{{4\\pi \\ln \\Lambda G^2 M_S^2 \\rho_H(r)}\\over{v_S^3}}[\\hbox{ erf}(X)-{{2 X}\\over{\\sqrt{\\pi}}}\\hbox{ e}^{-X^2}]\\,{\\bf v}_S,\n\\end{equation} \n\n\\noindent where $X \\equiv {v}_S\/(\\sqrt{2}\\sigma(r))$ and $\\ln \\Lambda$ is the\nCoulomb logarithm. The basic underlying assumption in the derivation\nof this equation \nis that a point particle of mass $M_S$ moves with a velocity ${\\bf\nv}_S$ in an homogeneous and infinite background of lighter\nparticles whose self-gravity is completely ignored. As this point\nparticle travels across the background it deflects particles\n(gravitational focusing) producing a density enhancement (a wake)\nbehind it which is responsible for the drag force expressed in equation\n(6) (Mulder 1983). Obviously, the manifestation of this force requires\na halo made of independent particles and hence, it will be not present in a\nrigid model. Since we want to check the reliability of equation (6) to\ndescribe the sinking rate in our self-consistent simulations we have\nintroduced it `by hand' in the case of the rigid halo models to\nemulate a drag force acting on the satellite. This will allow us to\ndetermine the importance of the self-gravity of the background particles of \nthe halo particles to the sinking process. We should remark that the\ndisc was kept alive and, hence, `disc friction' is included. \\\\ \n\n\n\n\\begin{figure}\n\\plotone{Fig13.ps}\n\\caption{Decay and disruption rates for a few of our models. Open\ncircles corresponds to our fully self-consistent simulations while\nsolid lines represent the same model but with a rigid halo free to\nmove instead. Dynamical friction has been computed using\nChandrasekhar's relation.} \n\\label{Fig. 13}\n\\end{figure}\n\nIn addition to the study of the orbital decay of satellites within\ntheir host galaxies (e.g. QG, QHF) dynamical friction has been invoked to \nexplain the evolution of cD galaxies in clusters \nof galaxies (Ostriker and Tremaine 1975, White 1976) as well as \nthe formation of galactic nuclei (Tremaine, Ostriker and\nSpitzer 1975). Despite numerous studies, there is\nstill disagreement about the applicability of Chandrasekhar's relation in \nsuch situations and the relevance of its inherent local character. \nThus, Lin and Tremaine (1983) using a semi-restricted N-body code found\nthat equation (6) provides accurate decay rates for satellites\nfollowing circular orbits for a large part of the parameter space\n(satellite mass, satellite size, and number density and mass of halo\nstars) suggesting that the self-gravity of the halo stars is\nunimportant. In contrast, White (1983) using fully self-consistent N-body\ncalculations based on a spherical harmonics expansion found that a\nlocal description of the dynamical friction was not enough to\ndetermine the sinking times in satellites. However, Bontekoe and van\nAlbada (1987) reached a conclusion more in agreement with Lin and\nTremaine (1987) since they found that higher order terms of the\nharmonic expansion had no discernible effect on their results. In an\nattempt to clarify this situation, Zaritsky and White (1988) carried\nout an exhaustive study using several codes and found that the\nself-gravity of the halo is not important for the sinking rate of the\nsatellite. Also, Hernquist and Weinberg (1989) used self-consistent\nand semi-restricted N-body algorithms to study this problem and to try\nto understand the discrepancy. They concluded that orbital decay is\nstrongly suppressed if self-gravity of the response is taken into\naccount in disagreement with Zaritsky and White (1988). Since most of\nthe previous work was restricted to satellites following circular\norbits, S\\'eguin and Dupraz (1994, 1996) employed a semi-restricted\nmethod to determine the applicability of Chandrasekhar's relation to\nhead-on encounters. They also conclude that the global response of the\ngalaxy cannot be ignored. \\\\ \n\n\\begin{table*}\n\\begin{center}\n\\centering\n\\caption{Disc kinematical changes. Rigid haloes}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|} \\hline\n& \\multicolumn{3}{|c|}{$R_{c}^{*}$} & \\multicolumn{3}{|c|}{$R_\\bullet$} & \\multicolumn{3}{|c|}{$R_{4}^{\\dag}$} \\\\ \\hline\n{Model} & $\\scriptstyle{(\\Delta \\sigma_R,\\Delta \\sigma_\\phi,\\Delta \\sigma_z)}$ & $\\scriptstyle{\\Delta Q}$ & $\\scriptstyle{\\Delta z_o}$ & $\\scriptstyle{(\\Delta \\sigma_R,\\Delta \\sigma_\\phi,\\Delta \\sigma_z)}$ & $\\scriptstyle{\\Delta Q}$ & $\\scriptstyle{\\Delta z_o}$ & $\\scriptstyle{(\\Delta \\sigma_R,\\Delta \\sigma_\\phi,\\Delta \\sigma_z)}$ & $\\scriptstyle{\\Delta Q}$ & $\\scriptstyle{\\Delta z_o}$ \\\\ \\hline \\hline\nR-G1S1    &    $(9,6,4)$  & $0.4$  &  $50$  & $(5,4,5)$  & $0.4$   &  $200$  & $(10,9,8)$   & $2.6$ & $550$ \t\\\\\nR-G1S3    &    $(9,6,5)$ & $0.3$   &  $50$  & $(5,4,5)$ & $0.4$  &  $150$  & $(13,8,10)$ & $2.2$ & $750$ \t\\\\\nR-G1S8    &    $(9,8,6)$  & $0.3$   &  $50$  & $(10,5,5)$  & $0.6$   &  $250$  & $(20,15,16)$ & $2.8$  & $1000$    \t\\\\\nR-G1S14    &    $(27,16,13)$  & $0.5$  &  $150$  & $(26,16,30)$   & $1.8$  &  $1200$  & $(30,23,28)$  & $3.4$ & $1700$ \t\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\medskip\nAs in Table 4, symbols $*$ and $\\dag$ denote quantities at the centre\nand $4\\,R_D$, respectively. \n\\end{table*}\n\nDue to this confusion about the role played by the global response of\nthe halo in the sinking process of the satellite we have repeated four\nof our simulations (models G1S1, G1S3, G1S8 and G1S14) but replacing the\nself-consistent halo by a rigid one which is free to move. With these new\nsimulations we try to cover several situations (different\neccentricities and satellite masses) to see whether the expression\n(6) deviates from the results in our self-consistent N-body\ncalculations and from a satellite that does not follow a circular\norbit as is claimed by S\\'eguin and Dupraz (1994, 1996). Figure 13\nshows the sinking and disruption times for the satellites in the new\nsimulations. Open squares represent the fully self-consistent\nsimulations while solid lines correspond to the model with rigid\nhaloes. To quantify the sinking and distruption rates we have used\nequation (6) with $\\rho_H(r)=\\rho(r)$ representing the local halo\ndensity at the satellite position and the satellite mass $M_S=M_S(t)$\nas the particles that remain bound. ${\\bf F}_{df}$ is applied to all\nthe bound particles. Notice that the agreement is quite\nremarkable and it suggests that a local description, as is implicit\nin Chandrasekhar's formula, is adequate to determine the sinking\ntimes of satellites whenever the satellite is immersed in the halo provided \nwe choose the right value for $\\Lambda $ and we define the satellite\nmass $M_S$ as the total mass of bound particles. The values for\n$\\Lambda $ that appear in figure 13 were estimated initially from the \nexpression $\\Lambda=p_{max}\/p_{min}$ \\footnote{Here, $p_{max}$ and\n$p_{min}$ are the maximum and minimum impact parameter, respectively,\nand are defined by $p_{max}\\equiv <R> \\simeq GM^2\/(2|W|)$ and\n$p_{min}\\equiv GM_S\/\\sigma_H^2(r_p)$ where $M$, $|W|$, $M_S$, \n$\\sigma_H(r_p)$ are the total mass of the spiral galaxy, the total\npotential energy of the spiral, the satellite's initial mass and the\none-dimensional dispersion velocity of the halo evaluated at the\npericentric radius $r_p$, respectively.} and afterwards were\nfine-tuned to get the `right' values. We must point out that to\ndetermine the dependence of $\\Lambda$ on the orbital parameters and\nthe satellite structure further work is necessary.\\\\\n\n\n\\begin{figure}\n\\plotone{Fig14.ps}\n\\caption{Disc kinematics for model R-GS3 (solid\nlines). Dashed lines represent the kinematics of the equivalent\nisolated galaxy model while dotted lines correspond to the changes\nsuffered by the disc as a consequence of the satellite accretion.}\n\\label{Fig. 14}\n\\end{figure}\n\n\\section{ Rigid Haloes and Disc Heating and Thickening}\n\nWe have carried out a detailed analysis of the simulations involving rigid\nhaloes in order to address the importance of the\nself-consistent \nrepresentation of the haloes in the accretion of a satellite. In\nfigure 14 we show the resulting disc kinematics (solid lines) for\nthe case equivalent to our model G1S3 but with a rigid halo\n(model R-G1S3). The kinematics of the corresponding isolated control\nmodel is displayed in the same figure\n(dashed lines) and the kinematical changes induced by the accretion\nevent are indicated by dotted lines. From comparison of figures 5 and\n14 we can notice that: (i) the disc thickening due purely to numerical\nrelaxation has been diminished by the introduction of a rigid halo;\nin an isolated model the disc heating is dominated by encounters\nbetween disc and halo particles. (ii)\nAfter completion of the accretion event, the radial and azimuthal\nchanges of the velocity ellipsoid induced by the satellite are\nroughly similar to those found in model G1S3. Thus, limits on\naccreted mass based only on the value of $\\Delta Q$ ($\\propto \\Delta\n\\sigma_R$) and ignoring any change of $\\Sigma_D(R)$ and any gas\ncooling will be essentially the same even if the spherical halo is  \nrepresented by a rigid potential rather than by a distribution of\nparticles provided the decay and disruption rates of the satellite are\nconsidered adequately (in our case by equation 6). (iii) However, the\nresponse of the vertical structure\nof the disc seems to be strongly coupled to the responsiveness of the\nhalo. For example, for\nmodel R-G1S3 we found that the change of the vertical scale length at\n$14$ kpc is about a factor of $2$ bigger than in model G1S3 \nwhile at the solar radius the factor is $\\sim 1.5$. This difference is\nsubstantial despite the fact that we have mimicked the satellite's\norbital energy loss to the rigid halo through equation 6. This may be\nbecause the disc develops some other\ninstabilities, as satellite accretion proceeds, which can be damped\nby a self-gravitating halo. (iv) As the\nsatellite becomes more massive, the necessity of a self-consistent\ntreatment of the halo is more evident; see figure 15\nwhich is the equivalent of our model G1S14 (compare it with figure\n6). The kinematics of this set of simulations using \nrigid haloes is summarized in Table 6.\\\\ \n\n\n\\begin{figure}\n\\plotone{Fig15.ps}\n\\caption{The same as figure 14 but for a satellite \nwith $0.2$ M$_D$. }\n\\label{Fig. 15}\n\\end{figure}\n\n\\section{ Conclusions}\n\nWe can summarize our main conclusions as follows:\n\n\\begin{itemize}\n\\item A comparison of our results with those obtained using TO's\nformulae shows that the mass limits they derived are too strict. In\ngeneral, TO tend to\noverestimate the damaging effects of satellite accretion by a\nfactor of about $2-3$ at the Solar radius. The\ndamaging effects in our simulations may also be an overestimation since\nwe ignore any contribution of gas cooling during the accretion process. The\norigin of the discrepancy between TO's predictions and our results may\nlie in the fact that: (i) their analysis ignores the coherent response of\nthe disc and its interaction with the halo. (ii) Their assumption that\nthe satellite's orbital\nenergy is deposited locally in the disc is clearly unrealistic. (iii)\nA fully self-consistent treatment of the dynamics is needed to get reliable\nresults. A rigid halo (with dynamical friction introduced through\nequation (6) leads to a larger increase of the vertical scale length of\nthe disc by a factor of $1.5-2$. A self-consistent treatment is more\nimportant for more massive satellites.\n\n\\item The damaging\neffects (heating and thickening) on the disc are very different for \nsatellites on prograde and\nretrograde orbits. A satellite on a prograde orbit tends to heat the\nstellar disc while its retrograde counterpart induces a coherent\nresponse of the disc in a form of a tilt. A massive satellite on a \nretrograde orbit may be accreted by a spiral galaxy without destroying its\ndisc. Furthermore, a massive central bulge may reduce the vertical heating \nof the disc for prograde orbits (but weaker for retrograde) \nwhile it may slightly diminish the tilting of the disc for retrograde \nones (but not for prograde).\n\n\\item A satellite as massive as $0.1\\,M_D$ and moving in a roughly\nelongated orbit (e.g. our model G1S3) does not produce a\nstrong stellar warp. The\nmost noticeable case of warp formation occurs in our model G1S14 where\nwe see a departure from the disc plane of less than $7^o$ in the outer\nregions (at about $15$ kpc). \n\n\\item Chandrasekhar's dynamical \nfriction formula gives remarkably good estimates for the sinking and\ndisruption rates of satellites in a variety of situations provided\na suitable value is chosen for the Coulomb logarithm and the\nsatellite mass is taken to be the mass still bound to\nthe satellite at each moment.\n\n\\end{itemize}\n\n\\section{ACKNOWLEDGMENTS}\n\nWe thank L. Hernquist for provide us with his algorithms. We also thanks \nan anonymous referee for his helpful comments to improve this paper. HV \nacknowledges useful conversations with L. Aguilar and S. Levine. HV \nthanks CONACyT (Consejo Nacional de Ciencia y Tecnolog\\'{\\i}a) of M\\'exico \nfor a studentship and the Max-Planck f\\\"ur Astrophysik for a Stipendium for \nthe fulfillment of this research. Most of the simulations were run at the \nMax-Planck Society's Computer Centre at Garching, Germany. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n The beams of different kind of waves, such as electromagnetic or sonic, experience diffraction and broaden as they propagate in a homogeneous medium. This spreading of energy in space diminishes the wave amplitude of the beam in the axis along propagation, unless the spreading is balanced by some focusing mechanism. Still, it is possible to prevent this fundamental wave propagation property and create non-diverging beams. Among the beam patterns without divergence the most popular are the linear Bessel beams \\cite{Drunin87}, and the nonlinear solitonic or self-trapped beams \\cite{Stegeman99}. Recently, another method for creating linear non-diverging beam was proposed, for waves propagating in a periodic medium. Such beams have been named self-collimated beams, and were first proposed for light beams in photonic crystals \\cite{Kosaka1999}, and later extended to other type of waves. The phenomenon of self-collimation has attracted much attention, as a technique to propagate optical, acoustical or even matter wave beams at long distances without a sensible loss of amplitude. Self-collimation is highly sensitive to the frequency: the divergence of beams can be reduced or even suppressed only at particular frequencies, those presenting particular dispersion characteristics, namely flat regions in the isofrequency contours \\cite{Kosaka1999}. The size of the self-collimated beam is also limited by the extension of such flat region in angular space. \n Self-collimation of low amplitude (linear), monochromatic acoustic waves has been demonstrated in 2D \\cite{Espinosa2007} and 3D \\cite{Soliveres2009} sonic crystals. More recently, the simultaneous self-collimation of two beams of different frequencies was also demonstrated experimentally \\cite{Soliveres2011}. These results show that the conditions for self-collimation can be achieved also for non-monochromatic beams; in particular the case of the superposition of beams of one frequency and its second harmonic was considered in \\cite{Soliveres2011}. The latter results are valid in linear regime; actually in \\cite{Soliveres2011} both frequency components were present in the input beam, and the corresponding beams propagated in the crystal without nonlinear interaction between them. \n \n In the linear case, the propagation of light and sound beams obey similar equations, and similar propagation characteristics are expected. The similarities between photonic and sonic crystals are well established \\cite{Miyashita05}, and have motivated many studies, where analogous effects in both systems have been investigated. The analogy, however, breaks for high amplitude waves, where nonlinear effects appear. For example, second and higher harmonic generation processes may be essentially different in optics and acoustics. One reason is the absence of intrinsic dispersion for acoustic waves propagating in homogeneous media. Nonlinear acoustical waves in nondispersive media as homogeneous fluids, eventually generate shock waves, which are not observed in optics. Also, the type and strength of nonlinearity may be different. While most common optical nonlinearities are cubic (kerr-type), in fluids and homogeneous solids, quadratic nonlinearity is dominant in acoustics. Even, nontraditional acoustic nonlinearities (power-law, hysteretic,...) are typical of some complex or microstructured acoustic media. In this sense, nonlinear effects of acoustic waves in periodic media, and in particular the self-collimation problem considered here are not a direct extension of the same effects in the optical case. Furthermore, the propagation of nonlinear acoustic beams in sonic crystals has never been addressed before.\n\nThe basic effect in nonlinear acoustics is harmonic generation \\citep{Hamilton2008}. It is known that efficient harmonic generation is only possible under fulfilment of phase matching conditions. For acoustic waves in fluids, this condition is rather natural, being always fulfilled for all harmonics due to the absence of dispersion, however in optics this requires special materials and special phase matching techniques \\cite{Boyd2003}. \n\nAcoustic harmonic generation has been studied in a variety of highly-dispersive nonlinear media, as bubbly liquids, or acoustic waveguides \\cite{Hamilton2008}, and weakly dispersive media as elastic plates \\cite{DeLima2003, Muller2010}, nonlinear porous-elastic media \\cite{Donskoy1997} and in granular media \\cite{Legland2012}. It has been proven as a useful effect in different  applications, as material characterization \\cite{Hirsekorn1994,Zheng1999}, ultrasound imaging and echography, \\cite{Humphrey2000}, biological tissue characterization \\cite{Law1981} and other medical ultrasound applications. \n\n\nThe purpose of this paper is to study nonlinear propagation of high intensity sound beams in periodic media, and in particular to demonstrate the formation of nonlinear  self-collimated acoustic beams, and discuss the conditions under which this process occurs with maximal efficiency. A sonic crystal is designed, by using an iterative method, to fulfil the three conditions for optimal energy transfer between harmonics: flatness of isofrequency contour for each harmonic, phase matching and large overlap between distributions of the interacting Bloch modes. The predictions are checked by FDTD simulations of the nonlinear problem, that demonstrate the efficient generation of fundamental and second harmonic acoustic narrow beams. \n\n\\section{Nonlinear sound beam propagation model}\n\nSeveral models can be used to describe nonlinear sound wave propagation though a fluid medium, with different levels of accuracy. An accurate description, when thermal and viscous effects are negligible, follows from the conservation laws of mass and momentum, can be written, respectively, in a Eulerian form \\cite{Hamilton2008}: \n\\begin{eqnarray}\n\t\\frac{{\\partial \\rho }}{{\\partial t}} =  - \\nabla  \\cdot \\left( {\\rho {\\bf{v}}} \\right),\\label{eq_mass}\\\\\n\t\\rho \\left( {\\frac{{\\partial {\\bf{v}}}}{{\\partial t}} + {\\bf{v}} \\cdot \\nabla {\\bf{v}}} \\right) =  - \\nabla p,\\label{eq_momentum}\n\\end{eqnarray}\nwhere $\\textbf{v}$ is the particle velocity vector, $p$ is the acoustic pressure, $\\rho$ is the total density field that can be expressed as $\\rho = \\rho_0 + \\rho'$, where $\\rho _0$ the ambient fluid density and $\\rho'$ is the acoustic density. The system is closed by the equation of state of the fluid, that under our assumptions is a pressure-density relation, $p=p(\\rho)$. A commonly used expression is obtained after Taylor expansion, keeping nonlinear terms up to second order. Then \n\\begin{equation}\\label{eq_state}\np = c_0^2 \\rho + \\frac{c_0^2}{\\rho_0}\\frac{B}{2A}\\rho^{2},\n\\end{equation}\nwhere $B\/A$ is the nonlinear parameter of the medium (which is known for most of materials, see e.g. \\citep{Naugolnykh1998}) and $c_0$ the sound speed in the medium.\nNote that Eqs. (\\ref{eq_mass}) and (\\ref{eq_momentum}) also contain nonlinearities related to (1) mass and momentum advection , or (2) geometrical nonlinearities. However, for the second harmonic generation they are of minor importance compared with the nonlinear terms in the equation of state, Eq. (\\ref{eq_state}). \n\n The above formulation of nonlinear propagation problem remains valid when the propagating medium is inhomogeneous, including the case of sonic crystals where inhomogeneity is periodically distributed in space. In such case, the medium parameters $c_0$ and $\\rho _0$ are space-dependent, represented by periodic functions. To our knowledge, the propagation of acoustic beams in periodic media has been only studied in the linear regime, and the corresponding nonlinear problem is addressed here for the first time.\n \n\\begin{figure}[t]\n\t\\includegraphics[width=8cm]{Figure1a.eps}\\\\\n\t\\includegraphics[width=4.1cm]{Figure1b.eps}\n\t\\includegraphics[width=4.1cm]{Figure1c.eps}\n\n\t\\caption{Top: Band structure (left) of a square lattice of rigid cylinders with $r = 0.11 a$, where $a$ is the lattice constant, immersed in water (right). Red and blue lines mark the bands (2nd and 8th) for which simultaneous self-collimation for both fundamental and second harmonic searched, along $\\Gamma-\\mathrm{X}$ direction. Bottom: Isofrequency contours for the 2nd (left) and 8th bands (right).Points denote the wavevectors for both waves, lying on flat segments respectively}\n\t\\label{FigDiagram}\n\\end{figure}\n\n\n\\section{Self-collimation of intense acoustic beams}\n\nWe consider a narrow, intense acoustic beam incident on a 2D sonic crystal made of cylindrical scatterers of radius $r$ embedded in a fluid, arranged in a square-lattice with lattice constant $a$. The corresponding filling fractio is $f=\\pi (r\/a)^2$. The beam width is roughly $6$ lattice periods. For the sake of simplicity the scatterers are considered perfectly rigid (the sound field is totally reflected from the wall of scatterer). Assuming water as a host fluid, the material parameters are $\\rho _0 = 1000$ kg\/m$^3$, $c_0 = 1490$ m\/s.\n\nThe special conditions required for a sound beam to propagate without diffraction are presented in this section. The problem of self-collimation has been already discussed for linear, monofrequency \\cite{Perez2007,Espinosa2007} and bi-frequency \\cite{Soliveres2011} beams. Since a nonlinear beam is composed by a fundamental frequency component and its high frequency harmonics, self-collimation of the nonlinear beam requires self-collimation of its constituent frequency components. We remind that in self-collimation regime the sonic beam does not spread diffractively because Bloch wave vectors lying on the flat segment of the spatial dispersion curve have equal longitudinal components and thus do not dephase mutually in propagation. In general, this flatness of the dispersion curve appears at a particular frequency, but as shown in \\cite{Soliveres2011} it can be also obtained for a wave and its second harmonic regarded they propagate in different propagation bands. \n\nTo illustrate this case, we show in Fig. \\ref{FigDiagram} the dispersion diagram of the sonic crystal for small-amplitude excitations, obtained using the Plane Wave Expansion (PWE) method on a linearized version of Eqs. (\\ref{eq_mass})-(\\ref{eq_state}). The conventional form of the band diagram is represented on the trajectory along the principal directions of the crystal, $\\Gamma-\\mathrm{X}-\\mathrm{M}-\\Gamma$, which are the boundary of the irreducible Brillouin zone (BZ). Figure~\\ref{FigDiagram} (top) shows the dispersion diagram for $\\Gamma-\\mathrm{X}$ direction. The red line in Fig.~\\ref{FigDiagram} denotes the 2nd propagation band. The fundamental (driving) field lies on this band. Its frequency $\\Omega$ is chosen such that the corresponding isofrequency contour contains flat regions. The blue line in Fig.~\\ref{FigDiagram} denotes the 8th propagation band. The second harmonic frequency, $2\\Omega$, lies in this band for the particular crystal parameters considered.  \nAs shown in \\cite{Soliveres2011}, for a given crystal, it is possible to choose the fundamental frequency such that the isofrequency contours for both frequencies present flat regions. \nIn our particular crystal, this happens when $\\Omega$ = 0.125 [these are dimensionless frequencies, related to physical frequencies $\\omega$ as  $\\omega=\\Omega (2 \\pi c_0\/a)]$. Similarly, a normalized Bloch wavevector is defined as $\\mathrm{K}=\\mathrm{k}_x(a\/\\pi)$\n\nSuch condition is necessary to achieve self-collimated propagation of the nonlinear beam.     \nIn order to obtain an efficient generation of the second harmonic, together with simultaneous self-collimation for both waves two additional geometric conditions must be fulfilled, related to the wavenumber and spatial shape of the interacting beams. \nThese conditions have been discussed for photonic crystals \\cite{Nistor2008, Nistor2010}. \n\n\\subsection{Phase matching conditions}\n\n It is well known from nonlinear optics \\cite {Boyd2003} that a proper phase relationship between the fundamental and second harmonic waves must be satisfied for an efficient nonlinear frequency conversion along the propagation direction. In a dispersive medium, the wavenumbers of first harmonics do not combine to result precisely in wavevector of second harmonics, and a phase mismatch $\\Delta \\mathrm{K}=2\\mathrm{K}(\\Omega)-\\mathrm{K}(2\\Omega)$ occurs. As a consequence, the second harmonic field is limited in amplitude: it does not grow linearly but oscillates in propagation, with a characteristic period given by the coherence length $l_c=\\frac{\\pi}{\\Delta \\mathrm{k}}=\\frac{a}{\\Delta \\mathrm{K}}$ \\citep{Boyd2003}.  \n \n The conversion efficiency into second harmonics generally is smaller in optics than in acoustics, because of the inherent material dispersion for light waves (absent for sound waves in fluids), that causes the fundamental and second harmonic waves to travel along the crystal with different phase velocities. Thus, the presence of the scatterers is the only important source of dispersion in the acoustic case.  \n \n\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics [width=8cm]{Figure2.eps}\n\t\\caption{Dispersion curves involved in simultaneous self-collimation for the fundamental (red line) at the 2nd band and second harmonic (blue line) at the 8th band. The \"doubled\" dispersion curve is represented (dashed red line) to identify phase matching of harmonics. The intersection denotes the frequency presenting phase matching. A closest view (inset) shows that for the self-collimated second harmonic, two solutions (modes A and B, with distinct $\\mathrm{K}$) are found, phase and non-phase matched, respectively}\n\t\\label{SHmatch}\n\\end{figure}\n\nPhase matching corresponds to $\\Delta \\mathrm{K} = 0$. Figure ~\\ref{SHmatch} shows that it can be actually achieved for the pair of frequencies where self-collimation occurs, as follows from the previous analysis. There, we represent the dispersion branches involved in self-collimation, along $\\Gamma-\\mathrm{X}$ direction, as in Fig.~\\ref{FigDiagram}. Fundamental and second harmonic modes correspond to the crossings of the dotted horizontal lines with the corresponding dispersion branches. For a given fundamental frequency $\\Omega$, in order to check the fulfillment of the phase matching condition, the curve corresponding to the ''double'' of the 2nd band dispersion curve, $2\\Omega(2\\mathrm{K})$, has been represented in Fig.~\\ref{SHmatch} as a red dashed line. Phase matching  is satisfied at the intersection between this curve with the corresponding curve at the 8th band. This corresponds to the mode labeled A in Fig.~\\ref{SHmatch}, which is phase-matched with the fundamental mode.\n\nNote that due to the concavity of the 8th band, at frequency 2$\\Omega$ a second mode labelled with B in Figure.~\\ref{SHmatch} can be also excited. This solution presents a large phase mismatch with the fundamental mode, and its contribution to the second harmonic field is negligible. \n\nThe simultaneous fulfillment of both conditions is obtained by an iterative procedure, which implies a re-design of the crystal parameters. The procedure is as follows: we start from a pair of frequencies $(\\Omega,2\\Omega)$ showing self-collimation for a given crystal parameters. Around this doublet, we seek the closest pair of frequencies $\\Omega'=\\Omega+\\delta \\Omega$ and 2$\\Omega'$ showing phase-matching. Then the isofrequency curves (Fig.~\\ref{FigDiagram}) are again calculated in order to evaluate the deviation of flatness in the isofrequency contours. The sonic crystal parameters are then modified, e.g. by a slight variation of the filling fraction, in order to tune the dispersion relations to get again self-collimation conditions. The process is repeated again until both conditions (flatness and phase matching) are simultaneously satisfied. We note that, despite this is out of the scope of this paper, optimization techniques as genetic algorithms can be applied here in order to find an optimal structure.\n\n\\subsection{Nonlinear coupling of Bloch modes}\n\n\\label{sec:Nonlinearcoupling}\nEfficient energy transfer between harmonics requires also a strong mode coupling, which depends on the spatial overlapping between the two interacting waves. For plane waves in a homogeneous medium a perfect spatial nonlinear coupling between first and second harmonic is assured, since mode overlapping is maximal. The propagation eigenmodes in a periodic medium are Bloch waves, whose amplitudes are spatially modulated and does not necessarily overlap. If two modes do not overlap in space, the energy transfer is less efficient even if they are phase matched. The amount of energy transfer can be estimated evaluating the spatial overlap between the envelopes of the corresponding Bloch modes. Let $B_{1}$ and $B_{2}$ be the spatial envelopes of the Bloch modes of the fundamental and and second harmonic waves, respectively. The nonlinear coupling coefficient is calculated as the cross-correlation between the functions $B_{1}^{2}$ and $B_{2}$ normalized in such a way that unity would correspond to the perfect matching of the modes \\cite{Nistor2008}. We define the coupling coefficient as\n\\begin{equation}\n\\kappa = \\frac{| \\int_{M}  B_{1}^{2}\\,B_{2}^{*}  \\mathrm{d}r |} { \\sqrt{ \\int_{C} \\, | B_{1}^{4}|\\,\\mathrm{d}r \\, \\int_{C} \\, | B_{2}|^{2}\\,\\mathrm{d}r }}\n\\end{equation}\nwhere the upper integral is calculated in the nonlinear medium from one unit cell, while the lower integrals are taken over the entire unit cell. \nTo calculate $B_{1}$ and $B_{2}$ we solve the eigenvalue problem for the pressure field by means of the PWE method, which converts the differential equation to an infinite matrix eigenvalue problem that can be truncated and solved numerically. For that, we follow the same procedure as in \\cite{Perez2007} however the problem is solved inversely, i.e. for a given frequency, the corresponding wave vectors, satisfying the phase matching condition, are obtained. Then $B_{1}$ and $B_{2}$ are obtained as the eigenvectors corresponding to fundamental and second harmonic frequencies, respectively. In Fig.~\\ref{BlochMode} we plot the spatial distributions of $B_{1}$ and $B_{2}$, respectively, for the selected final design where a coupling coefficient of $\\kappa = 0.85$ is obtained. This value is of the same order as the coupling in homogeneous media, $\\kappa = 1$, and therefore sufficiently large for an efficient harmonic generation.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=4.1cm]{Figure3a.eps}\n\t\\includegraphics[width=4.1cm]{Figure3b.eps}\n\t\\caption{Spatial distribution of the pressure field for the Bloch modes of the findamental (left) and the second harmonic (right) waves. The coupling coefficient is estimated to be  $\\kappa = 0.85$.}\n\t\\label{BlochMode}\n\\end{figure}\n\n\n\\subsection{Numerical simulation}\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=4cm]{Figure4a.eps}\n\t\\includegraphics[width=4cm]{Figure4b.eps}\\\\\n\t\\includegraphics[width=4cm]{Figure4c.eps}\n\t\\includegraphics[width=4cm]{Figure4d.eps}\\\\\n\t\\includegraphics[width=4cm]{Figure4e.eps}\n\t\\includegraphics[width=4cm]{Figure4f.eps}\n\t\\caption{Pressure distributions obtained by FDTD simulations. Normalized intensity cross section of fundamental (a) and second harmonic (b) at $x=80a$ propagating in the crystal (continuous line) and in a homogeneous (water) medium (dashed lines). Beam spatial distribution for simultaneously self-collimated harmonics in the sonic crystal (c-d) and in homogeneous fluid (e-f). Pressures are normalized to the maximum pressure.}\n\t\\label{figFDTD}\n\\end{figure}\n\nA full-wave nonlinear simulation was performed, using the FDTD method, to validate the efficiency of the second harmonic generation in the proposed structure. The crystal parameters, obtained after the iterative procedure describe above, are as in Figs. (\\ref{FigDiagram}) and (\\ref{SHmatch}). The source is a plane piston with a width of $6a$, located near the crystal, and radiating a harmonic wave with normalized frequency $\\Omega$ = 0.125. In order to minimize numerical dispersion a computational grid with $N_{\\lambda}=45$ elements per wavelength was used, and a Courant-Friedrich-Levy number of $S=0.95$. In Fig.~\\ref{figFDTD} we present the numerically obtained spatial distributions of the fundamental and its nonlinearly generated second harmonic. As predicted, both beams are nearly collimated. For comparison, the beam spatial distribution calculated for an homogeneous material (removing the crystal) are represented in Fig.~\\ref{figFDTD}~(e-f), where the diffractive broadening of the beams is visible. Transversal intensity distributions are shown in Fig.~\\ref{figFDTD}~(a-b) for a distance $80a$, where beam widths are compared with the reference beams in the homogeneous medium, broadened by diffraction. \n\nThe pressure amplitudes along the beam axis are shown in Fig.~\\ref{FWSHout} for each harmonic. Here, the analytic solution for a nonlinear plane wave propagating in a homogeneous (nondispersive) medium is plotted for reference (dashed lines) \\cite{Naugolnykh1998}, given by $p_n\/p_0=2 J_n(n\\sigma)\/n\\sigma$, where $J_n$ is the Bessel funcion of order $n$, $\\sigma=(\\varepsilon \\omega p_0\/\\rho_0 c_0^3) x$ is coordinate normalized to the shock formation distance, $p_0$ is the pressure at the source and $\\beta=1+ B\/2A$ the nonlinearity parameter. Such analytical solution is valid in the pre-shock region $\\sigma<1$. The growth rate of the self-collimated second harmonic beam propagating in the crystal matches well the growth rate of a plane wave in a homogeneous medium in such preshock region, which is a consequence of the weak divergence of the beam and the high degree of phase matching. Also, the second harmonic field can reach even higher amplitudes than those corresponding to nondispersive media (where harmonics decay beyond the shock formation distance). The latter effect can be understood in terms of phase mismatch of higher harmonics: in homogeneous media all harmonics are phase matched while in the crystal only the second harmonic is phase matched. In Fig. (\\ref{FWSHout}) the third harmonic is also plotted, where its small contribution to the beam is evident. The phase mismatch in third and higher harmonics decrease the energy flow into these components. Finally, the amplitude in Fig. (\\ref{FWSHout}) decays because non-perfect conditions for self-collimations, that makes the beam to start diverging after a long distance, or non-perfect phase matching, which results in a beating period with long coherence length.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=8cm]{Figure5.eps}\n\t\\centering\n\t\\caption{Normalized field amplitude along the acoustical axis $y=0$, for the fundamental, second and third harmonic beams. The dashed lines represent the analytical solutions for harmonic evolution on a plane wave propagating in a nondispersive medium.}\n\t\\label{FWSHout}\n\\end{figure}\n\n\n\\section{Conclusions and Remarks}\n\nWe have demonstrated the possibility of efficient second harmonic generation of sound in a sonic crystal, by means of the formation of narrow, weakly diverging nonlinear acoustic beams. Three conditions must be simultaneously present for a efficient second harmonic generation, which are: 1) simultaneous self-collimation, 2) phase mathing and 3) high spatial coupling of interacting harmonics. The use of simultaneous self-collimation regime limits the diffraction of both harmonic beams, maintaining the amplitude at the axis and therefore the nonlinear interaction. Under ideal conditions (no divergence and losses), the decrease of the first harmonic beam is mainly attributed to the energy transfer to second and higher harmonics. The sonic crystal parameters can be chosen to fulfill phase matching with the second harmonic, maximizing second harmonic generation due to synchronous cumulative interaction. Finally, the spatial coupling (overlapping) between interacting modes is also analyzed by calculating a nonlinear coupling coefficient. It is shown that its value ($\\kappa=0.85$ for the case studied) is not far from the ideal case, revealing a strong spatial overlap between both Bloch modes that leads to high energy transfer. \n\n\n The study show that linear dispersion characteristics (band structures, isofrequency contours) can be used to predict the behaviour of nonlinear beams propagating in periodic media. This opens the possibility of extending the study of nonlinear sound beam propagation in sonic crystals to other cases of interest. For example, crystals with higher filling factors present full frequency bandgaps, that may be used to filter out the propagation of selected higher harmonics. In this sense, sonic crystals can be a way to control the spectrum of intense acoustic waves, using the strong dispersion properties introduced by the periodicity.  \n\n\n\\bibliographystyle{apsrev}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nAn important feature of finite temperature QCD is the generation of an\nelectric and magnetic screening mass which play an important role in\ncontrolling\nthe infrared behaviour of the theory. The electric screening mass leads to\na Debye\nscreened static quark-antiquark potential and is given for SU(N) \nwith $N_{f}$ quark flavours, and for vanishing chemical potential and \nquark mass, in\nleading order\nperturbation theory by $m^{2}_{el} = \\frac{g^{2}}{3}(N+N_{f}\/2)T^{2}$ \n\\cite{kala}.\nRenormalization group improved perturbation\ntheory\ntells us that the effective coupling is a function of the temperature, and\ndecreases\nwith increasing temperature. This suggests that at sufficiently high\ntemperatures,\nabove the deconfining phase transition, the screening mass may be computed in\nperturbation theory. Because of the singular\ninfrared behaviour of the perturbative series, however, the computation\nof the next to leading order contribution requires a resummation of infrared\ndivergent diagrams which turns out to be sensitive to the magnetic\nscreening mass \\cite{rebh}.\nThis mass vanishes in lowest order perturbation theory and is expected to be\nof ${\\cal O}(g^{2}T)$. The coefficient multiplying $g^{2}T$ turns out however\nto be incalculable \\cite{lind}.\nMaking use\nof an improved perturbation theory proposed by Braaten and Pisarski \n\\cite{braa}, which\nresums hard thermal loops, and of a gauge invariant\ndefinition of the\nelectric screening mass \\cite{kobe}, Rebhan \nhas calculated the ${\\cal O}(g^{3}T^{2})$\ncorrections to the non-abelian screening mass-squared and has shown that\nnext to leading order contributions give rise to an enhancement \\cite{rebh}.\n\nThe lattice formulation of QCD allows one to\ndetermine the electric screening mass non-perturbatively.\nThe\nscreening mass is extracted from correlators of Polyakov loops\n[6--9], or\nfrom the long distance behaviour of the gluon propagator \n\\cite{hell}. For small\nquark-antiquark\nseparations lattice perturbation theory for the Polyakov loop correlation\nfunction\nis expected to describe the\nMonte Carlo data, since\nfor a finite lattice volume one is not confronted with the infrared\nproblems encountered in thermal perturbation theory. This has been checked\nin \\cite{pete} for the SU(3) gauge theory\nby taking careful account of finite size effects and, in particular,\nof the zero momentum modes which do not allow\none to take the thermodynamical limit for fixed coupling, as one would do in\nstandard perturbation theory. \nFor larger quark-antiquark separations, beyond the ``perturbative horizon'',\nthe colour averaged potential is expected to have a Debye screened form.\nMonte Carlo simulations confirm this screening picture [6--9].\nIn the case of pure SU(2) and SU(3) gauge theories the electric screening\nmass, when determined from Polyakov loop correlation functions, is found to\nbe about $10\\%$ larger \\cite{gao,irba} \nthan the leading order perturbative result \nif the temperature dependent coupling constant is \ndetermined from Polyakov loop correlators in the perturbative region.\nAs was pointed out by Rebhan, \\cite{rebh}, such an enhancement could also\nbe expected if next to leading order corrections to the continuum screening\nmass are taken into account through resummed perturbation theory. A\nquantitative comparison with the results obtained in the above simulations\nis however very difficult and has, to our knowledge, not been carried out\nso far. In contrast to the work of ref.\\ \\cite{gao,irba}, the electric\nscreening mass as extracted in ref.\\ \\cite{hell} from the gluon propagator\nin the Landau gauge was found to deviate strongly form the leading order\nperturbative result. \n\nIn comparing the Monte Carlo data for the electric screening mass with\nleading order, or resummed, perturbation theory it is important to have\nan estimate of the size of lattice artefacts to be expected from a finite\nlattice spacing.\nTo obtain such an estimate\nwe compute the electric screening\nmass in one-loop order on the lattice, at finite temperature and\nchemical potential, in the infinite volume\nlimit, and compare it with the continuum. \nFor the case of QED with {\\it naive} fermions the screening mass has been \ncalculated by Pietig \\cite{piet}.\nThe screening mass is calculated\nfrom the zero-momentum\nlimit of the 44-component of the vacuum polarization tensor\nevaluated for vanishing Matsubara\nfrequency.\nIn one-loop order this definition of the screening mass is gauge invariant,\nand consistent\nwith the more general gauge invariant definition given in ref.\\ \\cite{kobe},\nwhere\nthe screening mass is determined from the position of the\npole\nof the gluon propagator for vanishing Matsubara frequency.\n\nThe paper is organized as follows: In the following section we calculate the\nelectric screening mass for QCD in one-loop order for\nthe case of Wilson fermions. The Feynman\nrules and frequency summation formulae required for the computation are\nrelegated\nto two Appendices. As we shall see, the resulting integral expression has a\nvery transparent form. In section 3\nwe then evaluate the momentum integrals for the\nscreening mass numerically and compare the results with the\ncontinuum. We show that the lattice artefacts due to a finite\nlattice spacing give rise\nto an enhancement of the screening mass as compared to the continuum. We\ndiscuss the magnitude of this enhancement as a function of \nthe temperature and chemical potential\nfor lattices with different number of lattice sites in the\ntemporal direction which can be implemented in numerical simulations.\nMost of the enhancement\nis found to be due to the fermion\nloop contribution.\nSection\n4 contains a summary of our results.\n\\section{Electric Screening Mass in One Loop Order}\nIn this section we compute the electric screening mass in lattice QCD to\none loop order\nfrom the zero momentum limit of the 44-component of the vacuum polarization\ntensor evaluated for vanishing Matsubara frequency. The Feynman diagrams\ncontributing\nin this order are shown in fig.\\ 1. While diagrams (a,c,d,e) have a\ncontinuum analog, the\nremaining diagrams, required by gauge invariance on the lattice, do not\npossess a\ncontinuum counterpart.\n\\begin{figure}[ht]\n\\leavevmode\n\\centering\n\\epsfxsize10cm \\epsffile{loops.ps}\n\\caption{Feynman diagrams contributing to the vacuum polarization in\n  one loop order on the lattice.}\n\\end{figure}\nThe finite temperature, finite chemical potential\nlattice Feynman\nrules are obtained from the $T=\\mu =0$ rules by replacing the fourth\ncomponent of\nthe fermion and boson momenta by $\\hat{\\omega}^{-}_{\\ell} +i\\hat{\\mu}$ and\n$\\hat{\\omega}^{+}_{\\ell}$,\nrespectively, where  $\\hat{\\omega}^{+}_{\\ell} = \\frac{2\\pi}{\\hat{\\beta}}\\ell\n\\ (\\hat{\\omega}^{-}_{\\ell} = \\frac{(2\\ell+1)\\pi}{\\hat{\\beta}})$, with\n$\\ell \\in Z$, \nare the Matsubara\nfrequencies\nfor bosons (fermions), and $\\hat{\\beta}$ is the inverse temperature.\nQuantities with a\n\"hat\" are always understood to be measured in lattice units. Furthermore,\nintegrals over the\nfourth component of momenta at zero temperature are replaced at finite\ntemperature by\nsums over Matsubara frequencies in the interval $[-\\frac{\\hat{\\beta}}{2},\n\\frac{\\hat{\\beta}}{2}-1]$, \nwhere we have taken $\\hat{\\beta}$ to be even. The\nFeynman rules\nare collected in Appendix A. The relevant formulae for carrying out the sums\nover Matsubara frequencies are derived in Appendix B.\n\nThe contributions of diagrams (a-g) to the vacuum polarization tensor are\ndiagonal\nin colour space,\n\\begin{equation}\n\\hat{\\Pi}^{(\\beta,\\mu)AB}_{\\mu\\nu}(\\hat{\\omega}^{+}_{\\ell},\\vec{\\hat{k}}) =\n\\delta_{AB}\\hat{\\Pi}^{(\\beta,\\mu)}_{\\mu\\nu}(\\hat{\\omega}^{+}_{\\ell},\\vec{\\hat{\nk}})\\ . \\label{diag:col}\n\\end{equation}\nThe electric screening mass (in lattice units) is then defined by\n\\begin{equation}\n\\hat{m}_{el}^{2} = \\lim_{\\vec{\\hat{k}}\\rightarrow 0}\n\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})\\ . \n\\end{equation}\nIn the following we first consider the contributions to\n$\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})$ coming from the fermion\nloops, i.e.,\ndiagrams (a) and (b).\\\\ \\\\\n{\\it i) Contribution of diagram (a)}\\\\ \n\nA straight forward application of the finite temperature, finite chemical\npotential\nlattice Feynman rules yields\n\\alpheqn\n\\begin{equation}\n\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(a)}\n=-\\frac{N_{f}}{2}g^{2}\\frac{1}{\\hat{\\beta}}\n \\sum^{\\frac{\\hat{\\beta}}{2}-1}_{\\ell=-\\frac{\\hat{\\beta}}{\n 2}}\\int^{\\pi}_{-\\pi}\\frac{d^{3}\\hat{p}}{(2\\pi)^{3}}f^{(a)}\n(e^{i(\\hat{\\omega}_{\\ell}^{-}+i\\hat{\\mu})};\\vec{\\hat{p}},\n \\vec{\\hat{k}})\\ , \\label{pi:a}\n\\end{equation}\nwhere\n\\begin{equation}\nf^{(a)}(z;\\vec{\\hat{p}},\\vec{\\hat{k}})=\n\\frac{2(z^{4}+1)-2\\eta(z^{3}+z)+4\\xi{\\cal G}z^{2}}{\\Pi^{4}_{i=1}(z-z_i)}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\eta &=& \\frac{1}{[1+\\hat{M}(\\vec{\\hat{p}})]} + \\frac{1}{[1+\\hat{M}\n (\\vec{\\hat{p}}+\\vec{\\hat{k}})]} \\ , \\\\\n\\xi &=& \\frac{1}{[1+\\hat{M}(\\vec{\\hat{p}})][1+\\hat{M}(\\vec{\\hat{p}}+\n \\vec{\\hat{k}})]} \\ , \\\\\n{\\cal G} &=& 1+\\sum_{j}\\sin\\hat{p}_{j}\\sin(\\hat{p}+\\hat{k})_{j} \\ , \\\\\n\\hat{M}(\\vec{\\hat{p}}) &=& \\hat{m} +2 \\sum_{j} \\sin^{2}\\frac{\\hat{p}_{j}}{2}\\ ,\n\\end{eqnarray}\n\\reseteqn\nand $N_{f}$ is the number of quark-flavours.\nThe position of the poles of $f^{(a)}(z;\\vec{\\hat{p}},\\vec{\\hat{k}})$ \nare given by\n\\alpheqn\n\\begin{eqnarray}\nz_{1} &=& e^{\\phi};\\ \\ \\ z_{2}=e^{-\\phi} \\ , \\nonumber \\\\\nz_{3} &=& e^{\\psi};\\ \\ \\ z_{4}=e^{-\\psi} \\ ,\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\phi &=& \\tilde{\\cal E}(\\vec{\\hat{p}}) \\ , \\nonumber \\\\\n\\psi &=& \\tilde{\\cal E}(\\vec{\\hat{p}}+\\vec{\\hat{k}})\\ ,\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\tilde{\\cal E}(\\vec{\\hat{q}}) = \\ln\\left[ K(\\vec{\\hat{q}}) +\n \\sqrt{K^{2}(\\vec{\\hat{q}})-1}\\, \\right]\n = \\mbox{arcosh} K(\\vec{\\hat{q}})\n\\end{equation}\nand\n\\begin{equation}\nK(\\vec{\\hat{q}}) = 1 + \\frac{\\bar{E}^{2}(\\vec{\\hat{q}})}{2[1+\n \\hat{M}(\\vec{\\hat{q}})]}\\ , \\ \\\n\\bar{E}(\\vec{\\hat{q}}) = \\sqrt{\\sum_{j} \\sin^{2} \\hat{q}_{j} +\n \\hat{M}^{2}(\\vec{\\hat{q}})}\\ .\n\\end{equation}\n\\reseteqn\nNote that\n\\begin{equation}\n\\phi \\stackrel{\\vec{\\hat{p}} \\rightarrow\n-\\vec{\\hat{p}}-\\vec{\\hat{k}}}{\\longleftrightarrow} \\psi \\ , \\label{subst}\n\\end{equation}\nwhile $\\eta$, $\\xi$ and ${\\cal G}$ are invariant under the\ntransformation $\\vec{p} \\rightarrow -\\vec{p}-\\vec{k}$. \nThis will be important further\nbelow.\nThe frequency sum can be performed making use of eq.\\ (\\ref{sum:fermg}) \nderived in\nAppendix B. One\nfinds that\n\\alpheqn\n\\begin{eqnarray}\n\\frac{1}{\\hat{\\beta}}\\sum^{\\frac{\\hat{\\beta}}{2}-1}_\n{\\ell=-\\frac{\\hat{\\beta}}{2}}f^{(a)}\n (e^{i(\\hat{\\omega}_{\\ell}^{-}+i\\hat{\\mu})};\n\\vec{\\hat{p}},\\vec{\\hat{k}}) &=& 2 + h(\\phi,\\psi,\\eta,\\xi,{\\cal G})\n\\left[ \\frac{1}{e^{\\hat{\\beta}(\\phi+\\hat{\\mu})}+1} -\n \\frac{1}{e^{-\\hat{\\beta}(\\phi-\\hat{\\mu})}+1} \\right] \\nonumber \\\\\n& & \\mbox{} + h(\\psi,\\phi,\\eta,\\xi,{\\cal G})\n\\left[ \\frac{1}{e^{\\hat{\\beta}(\\psi+\\hat{\\mu})}+1} -\n \\frac{1}{e^{-\\hat{\\beta}(\\psi-\\hat{\\mu})}+1} \\right]\\ , \n \\nonumber \\\\ \\label{eq:aa}\n\\end{eqnarray}\nwhere\n\\begin{equation}\nh(\\phi,\\psi,\\eta,\\xi,{\\cal G})= \\frac{\\cosh 2\\phi-\\eta\\cosh\\phi\n+\\xi{\\cal G}}{\\sinh\\phi(\\cosh\\phi-\\cosh\\psi)}\\ . \\label{eq:ab}\n\\end{equation}\n\\reseteqn\nTo obtain $\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0;\\vec{\\hat{k}})_{(a)}$, \nwe must integrate\nthis expression over $\\vec{\\hat{p}}$, with $\\hat{p}_{j}\n\\in [-\\pi,\\pi]$. Noting that $\\eta$, $\\xi$ and\n${\\cal G}$ are invariant under the transformation $\\vec{\\hat{p}}\n\\rightarrow -\\vec{\\hat{p}}-\\vec{\\hat{k}}$,\nand making use of (\\ref{subst}), \nas well as of the fact that the integrand in (\\ref{pi:a})\nis a periodic function in $\\hat{p}_{j}$ and $\\hat{k}_{j}$, we\ncan combine the last two contributions on the r.h.s. of (\\ref{eq:aa}) \nand obtain\n\\alpheqn\n\\begin{eqnarray}\n\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat k})_{(a)} &=&\n N_{f}g^{2}\\int^{\\pi}_{-\\pi}\\frac{d^3\\hat{p}}{(2\\pi)^{3}}\n [h(\\phi,\\psi,\\eta,\\xi,\n {\\cal G})-1] \\nonumber \\\\\n& &\\mbox{} -N_{f}g^{2}\\int^{\\pi}_{-\\pi}\\frac{d^3\\hat{p}}{(2\\pi)^{3}}\n h(\\phi,\\psi,\n \\eta,\\xi,{\\cal G})[\\hat{\\eta}_{FD}(\\phi)+\\bar{\\hat{\\eta}}_{FD}\n (\\phi)]\\ . \\label{erg:a}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\hat{\\eta}_{FD}(\\phi)=\\frac{1}{e^{\\hat{\\beta}(\\phi-\\hat{\\mu})}+1}\\ ,\\ \\\n\\bar{\\hat{\\eta}}_{FD}(\\phi)=\\frac{1}{e^{\\hat{\\beta}\n (\\phi+\\hat{\\mu})}+1}\\ ,\n\\end{equation}\n\\reseteqn\nare the lattice Fermi-Dirac distribution functions\nfor particles and antiparticles.\\\\ \\\\\n{\\it ii) Contribution of diagram (b)} \\\\\n\nWe next compute the contribution to $\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\n\\vec{\\hat{k}})$ of the Feynman diagram (b) depicted in Fig.\\ 1. \nThis diagram has\nno analog in the continuum and is given by\n\\alpheqn\n\\begin{equation}\n\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(b)} = - N_{f} g^{2}\n \\frac{1}{\\hat{\\beta}}\n\\sum^{\\frac{\\hat{\\beta}}{2}-1}_{\\ell=\\frac{\\hat{\\beta}}{2}}\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}} f^{(b)}(e^{i(\\hat{\\omega}_{\\ell}^{-}+i\n \\hat{\\mu})},\\vec{\\hat{p}} )\n\\end{equation}\nwhere\n\\begin{eqnarray}\n f^{(b)}(z;\\vec{\\hat{p}}) &=& -\\frac{z^2-2\\rho z+1}{(z-z_1)(z-z_2)}\\ , \\\\\n \\rho &=& \\frac{1}{1+\\hat{M}(\\vec{\\hat{p}})}\\ ,\n\\end{eqnarray}\n\\reseteqn\nand where $z_{1}$ and $z_{2}$ have been defined\nin (2.4a-d). Making again use of the frequency summation formula\n(\\ref{sum:fermg}), one verifies that\n\\begin{eqnarray}\n\\lefteqn{\n \\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(b)}=-N_{f}g^{2}\n \\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}} \\left(\\coth \\phi- \\frac{\\rho}{\\sinh\n \\phi}-1 \\right) } \\nonumber \\\\\n& & \\mbox{} + N_{f} g^{2} \\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}} \\left(\\coth \\phi- \\frac{\\rho}{\\sinh\n \\phi} \\right)[\\hat{\\eta}_{FD}(\\phi)\n +\\bar{\\hat{\\eta}}_{FD}(\\phi)]\\ .\n\\end{eqnarray}\nCombining this expression with (\\ref{erg:a}) one finds that\n\\alpheqn\n\\begin{equation}\n\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})\n =\\hat{\\Pi}^{(vac)}_{44}(0,\\vec{\\hat{k}}) +\n N_{f} g^{2} \\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}}H(\\phi,\\psi,\\rho,\\eta,\\xi,{\\cal G})\n [ \\hat{\\eta}_{FD}(\\phi)+\\bar{\\hat{\\eta}}_{FD}(\\phi) ]\\ , \\label{eq:bc}\n\\end{equation}\nwhere\n\\begin{equation}\n H(\\phi,\\psi,\\rho,\\eta,\\xi,{\\cal G}) = \\coth \\phi - \\frac{\\rho}{\\sinh\\phi}-\n  h(\\phi,\\psi,\\eta,\\xi,{\\cal G})\n\\end{equation}\nand\n\\begin{equation}\n \\hat{\\Pi}^{(vac)}_{44}(0,\\vec{\\hat{k}}) = -N_{f}g^{2} \\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}}H(\\phi,\\psi,\\rho,\\eta,\\xi,{\\cal G}\n ) \\label{eq:ba}\n\\end{equation}\n\\reseteqn\nis the $T = \\mu = 0$ contribution.\nAs we now show, $\\hat{\\Pi}^{(vac)}_{44}(0,\\vec{\\hat{k}})$ vanishes in the limit\n$\\vec{\\hat{k}} \\rightarrow 0$, and hence does not contribute to\nthe screening mass.\n\nConsider the function $h(\\phi,\\psi,\\eta,\\xi,{\\cal G})$ defined in\n(\\ref{eq:ab}). It is singular for $\\vec{\\hat{k}} \\rightarrow 0$, \nsince in this\nlimit $\\psi \\rightarrow \\phi$. \nThe singularity is however integrable.\\footnote{We\nfollow here\nand in the following a technique used in ref. \\cite{piet}, where the author\nhas calculated the screening mass for naive\nfermions in lattice QED to one loop order}\nThis can be seen as follows. Since according to (\\ref{subst}), \nand the statement\nfollowing it\n\\begin{equation}\n h(\\phi,\\psi,\\eta,\\xi,{\\cal G}) \n \\stackrel{\n \\vec{\\hat{p}}\\rightarrow -\\vec{\\hat{p}}-\\vec{\\hat{k}}}{\\longrightarrow}\n h(\\psi,\\phi,\\eta,\\xi,{\\cal G})\n\\end{equation}\nwe can also write (\\ref{eq:ba}) in the form\n\\alpheqn\n\\begin{eqnarray}\n \\hat{\\Pi}^{(vac)}_{44}(0,\\vec{\\hat{k}})\n&=& -N_{f}g^{2}\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}}\\left(\\coth\\phi-\\frac{\\rho}{\n \\sinh\\phi}\\right) \\nonumber \\\\\n& &\\mbox{}+\\frac{1}{2}N_{f}g^{2}\\int^{\\pi}_{-\\pi}\\frac{d^{3}\\hat{p}}{\n (2\\pi)^{3}}\n \\tilde{h}(\\phi,\\psi,\\eta,\\xi,{\\cal G}) \\ , \\label{eq:bb}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n \\tilde{h}(\\phi,\\psi,\\eta,\\xi,{\\cal G})=h(\\phi,\\psi,\\eta,\\xi,{\\cal G})\n  +h(\\psi,\\phi,\\eta,\\xi,{\\cal G}) \\ . \\label{eq:bd}\n\\end{equation}\n\\reseteqn\nAlthough each term on the rhs of (\\ref{eq:bd}) is singular for \n$\\vec{\\hat{k}} \\rightarrow 0$\n$(\\psi \\rightarrow \\phi)$, the sum possesses a finite\nlimit.\nThus setting $\\psi=\\phi+\\epsilon$ and taking the limit \n$\\vec{\\hat{k}} \\rightarrow 0$\n($\\epsilon \\rightarrow 0$),\none verifies that\n\\begin{equation}\n \\lim_{\\vec{\\hat{k}} \\rightarrow 0}\\tilde{h}(\\phi,\\psi,\\eta,\\xi,{\\cal G})\n = 2\\left( \\coth\\phi-\\frac{\\rho}{\\sinh\\phi} \\right) \\ . \\label{eq:be}\n\\end{equation}\nFrom (\\ref{eq:bb}) we therefore conclude that\n\\[ \\lim_{\\vec{\\hat k}\\to 0}\\hat\\Pi^{(vac)}_{44}(0,\\vec{\\hat k})\n=0 \\ . \\]\nThis result is not unexpected, since for vanishing temperature and chemical\npotential it is\nwell known in the continuum formulation, that Lorentz and gauge invariance\nprotects the gluon from acquiring a mass. The screening mass is therefore\ndetermined by the\nfinite temperature (f.T.), finite chemical potential contribution, given by the\nintegral in (\\ref{eq:bc}). By making again use of the fact that\n$\\phi\\leftrightarrow\\psi$, when $\\vec{\\hat{p}} \\rightarrow -\n\\vec{\\hat{p}}-\\vec{\\hat{k}}$, while $\\eta$, $\\xi$ and ${\\cal G}$\nremain invariant under this change of variables, we can write this\ncontribution in\nthe form\n\\begin{eqnarray}\n \\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{f.T.} &=& N_{f}g^{2}\n \\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}}\\left( \\coth\\phi-\\frac{\\rho}{\n \\sinh\\phi} \\right) [ \\hat{\\eta}_{FD}(\\phi)+\\bar{\\hat{\\eta}}_{FD}\n(\\phi) ] \\nonumber \\\\\n& &\\mbox{} -\\frac{1}{2}N_{f}g^{2}\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}} \\left\\{ h(\\phi,\\psi,\\eta,\\xi,{\\cal G})\n[ \\hat{\\eta}_{FD}(\\phi)+\\bar{\\hat{\\eta}}_{FD}(\\phi) ]\\right. \\nonumber \\\\\n& &\\mbox{}+ \\left. h(\\psi,\\phi,\\eta,\\xi,{\\cal G})[ \\hat{\\eta}_{FD}(\\psi)\n+\\bar{\\hat{\\eta}}_{FD}(\\psi)] \\right\\} \\ . \\label{pi:fft}\n\\end{eqnarray}\nConsider the second integral. It can be rewritten as follows\n\\[ \\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}} \\{ \\tilde{h}(\\phi,\\psi,\\eta,\\xi,{\\cal G})\n[\\hat{\\eta}_{FD}(\\psi)+\\bar{\\hat{\\eta}}_{FD}(\\psi)]+\n h(\\phi,\\psi,\\eta,\\xi,{\\cal G})\n \\bigtriangleup\\hat{\\eta}_{FD}(\\phi,\\psi) \\} \\ , \\]\nwhere $\\tilde{h}$ has been defined in (\\ref{eq:bd}), and\n\\[ \\bigtriangleup\\hat{\\eta}_{FD}(\\phi,\\psi) =\n [\\hat{\\eta}_{FD}(\\phi)-\\hat{\\eta}_{FD}(\\psi)]+\n [\\bar{\\hat{\\eta}}_{FD}(\\phi)-\\bar{\\hat{\\eta}}_{FD}(\\psi)]\\ . \\]\nAccording to (\\ref{eq:be}), $\\tilde{h}$ approaches a finite limit for\n$\\vec{\\hat{k}} \\rightarrow 0$. \nUpon making the change of variables \n$\\vec{\\hat{p}} \\rightarrow -\\vec{\\hat{p}}-\\vec{\\hat{k}}$\nthe contribution proportional to $\\tilde{h}$ is seen to be\ncancelled by\nthe first integral in (\\ref{pi:fft}). We therefore conclude that\n\\begin{equation}\n \\lim_{\\vec{\\hat{k}} \\rightarrow 0}\\hat{\\Pi}^{(\\beta,\\mu)}_{44}\n (0,\\vec{\\hat{k}})=\n -\\frac{1}{2}N_{f}g^{2}\\lim_{\\vec{\\hat{k}}\\rightarrow 0}\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}}\n h(\\phi,\\psi,\\eta,\\xi,{\\cal G})\\bigtriangleup\\hat{\\eta}_{FD}\n (\\phi,\\psi)\\ .\n\\end{equation}\nWe have now dropped the subscript \"f.T.\", since in this\nlimit only (\\ref{pi:fft}) contributes to the screening mass. To calculate\nthis limit we proceed as before and set $\\psi=\\phi+\\epsilon$.\nOne then verifies that for $\\epsilon \\rightarrow 0$ \n(or $\\vec{\\hat{k}} \\rightarrow 0$) the\nfermionic\ncontribution to the screening mass squared is given by\n\\begin{equation}\n [\\hat{m}_{el}^{2}(\\hat{\\beta},\\hat{\\mu},\\hat{\nm})]_{ferm}=N_{f}g^{2}\\hat{\\beta}\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{p}}{(2\\pi)^{3}}\n \\left\\{ \\frac{e^{\\hat{\\beta}(\\phi+\\hat{\\mu})}}{\n [e^{\\hat{\\beta}(\\phi+\\hat{\\mu})}+1]^2}+\n \\frac{e^{\\hat{\\beta}(\\phi-\\hat{\\mu})}}{\n [e^{\\hat{\\beta}(\\phi-\\hat{\\mu})}+1]^2}\\right\\} \\ . \\label{mel:fermg}\n\\end{equation}\nIn the continuum limit the electric screening mass is given by\n\\begin{equation}\n m^{2}_{el} = \\lim_{a\\rightarrow 0}\\frac{1}{a^{2}} \\hat{m}_{el}^{2} \n  (\\frac{\\beta}{a},\\mu a,ma)\\ . \\label{mel:lim}\n\\end{equation}\nFor Wilson fermions, only momenta $\\vec{\\hat{p}}$ in the immediate\nneighbourhood of\n$\\vec{\\hat{p}} = 0$ contribute to the integral (\\ref{mel:fermg}) for \n$\\hat{\\beta} \\rightarrow \\infty$,\n$\\hat{\\beta}\\hat{\\mu} = \\beta\\mu$, $\\hat{\\beta}\\hat{m} = \\beta m$ fixed.\nBut in this\nlimit $\\hat{\\beta}\\phi(\\vec{\\hat{p}}) \\rightarrow \n\\beta\\sqrt{\\vec{p}^{\\, 2}+m^{2}}$.\nIntroducing in (\\ref{mel:fermg}) the dimensioned\nmomenta $\\vec{p} = \\vec{\\hat{p}}\/a$ as new integration variables, with $a$\nthe lattice\nspacing, one then verifies, after performing\nthe angular integration, and a partial integration that\n\\begin{equation}\n [m^{2}_{el}]_{ferm}=N_{f}\\frac{g^{2}}{2\\pi^{2}}\\int^{\\infty}_{0}dp\n \\frac{2p^{2}+m^{2}}{\\sqrt{p^{2}+m^{2}}}\\left[\\eta_{FD}(E,\\mu) +\n \\bar{\\eta}_{FD}(E,\\mu)\\right]\\ ,\n\\end{equation}\nwhere $E = \\sqrt{\\vec{p}^{\\, 2}+m^{2}}$, and\n\\begin{equation}\n \\eta_{FD}(E,\\mu) = \\frac{1}{e^{\\beta (E-\\mu)}+1}\\ ; \\ \\\n \\bar{\\eta}_{FD}(E,\\mu) = \\frac{1}{e^{\\beta (E+\\mu)}+1} \n\\end{equation}\nare the Fermi-Dirac distribution functions for particles and antiparticles.\n\nWe next consider the contribution to the electric screening mass arising\nfrom diagrams\n(c-g) in fig.\\ 1.\nThey only involve sums over Matsubara frequencies\nof the bosonic type. Since in the continuum limit the only dimensioned\nscale is the temperature, their contribution to the screening mass will be\nof the form $const \\times g T$. For finite lattice spacing, however, the\ntemperature dependence will be modified by lattice artefacts. In\nthe following we first consider diagrams (c-e) which have an analog\nin the continuum.\\\\ \\\\\n{\\it iii) Contribution of diagram (c)}\\\\\n\nUsing the lattice Feynman rules given in Appendix A one finds after some\nalgebra that\nthis diagram contributes as follows to\n$\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})$,\ndefined in (\\ref{diag:col}),\n\\alpheqn\n\\begin{equation}\n\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(c)}=\\frac{3}{2}\n g^{2}\\frac{1}{\\hat{\\beta}}\\sum^{\\frac{\\hat{\\beta}}{2}-1}_{\\ell=\n -\\frac{\\hat{\\beta}}{2}} \\int^{\\pi}_{-\\pi}\\frac{d^{3}\\hat{q}}{\n (2\\pi)^{3}}f^{(c)}(e^{i\\hat{\\omega}^{+}_{\\ell}};\\vec{\\hat{q}},\n \\vec{\\hat{k}})\\ ,\n\\end{equation}\nwhere\n\\begin{equation}\n f^{(c)}(z;\\vec{\\hat{q}},\\vec{\\hat{k}})=\\frac{a(\\vec{\\hat{k}})(z^{2}-1)^{2}-b\n (\\vec{\\hat{q}},\\vec{\\hat{k}})z(z+1)^{2}}{\n \\Pi^{4}_{i=1}[z-\\bar{z}_i]} \\label{eq:ca}\n\\end{equation}\nand\n\\begin{eqnarray}\n a(\\vec{\\hat{k}}) &=& \\sum_{j}\\cos^{2}\\frac{\\hat{k}_{j}}{2}\\ , \\\\\n b(\\vec{\\hat{q}},\\vec{\\hat{k}}) &=& \\frac{1}{4} \\left[ \\sum_{j}\n \\widetilde{(\\hat{q}-\\hat{k})}^{2}_{j} +\n \\sum_{j}\\widetilde{(\\hat{q}+2\\hat{k})}^{2}_{j} \\right] \\ .\n\\end{eqnarray}\n\\reseteqn\nHere $\\tilde{\\hat{p}}$ is generically defined by\n$\\tilde{\\hat{p}}_{\\mu}=2\\sin\\frac{\\hat{p}_{\\mu}}{2}$.\nThe zeros of the denominator in (\\ref{eq:ca}) are located at\n\\alpheqn\n\\begin{eqnarray}\n \\bar{z}_{1} &=& e^{\\tilde{\\phi}}\\ ;\\ \\bar{z}_{2}=e^{-\\tilde{\\phi}}\\ ,\n  \\nonumber \\\\\n \\bar{z}_{3} &=& e^{\\tilde{\\psi}}\\ ;\\ \\bar{z}_{4}=e^{-\\tilde{\\psi}}\\ ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n \\tilde{\\phi} &=& \\mbox{arcosh} H(\\vec{\\hat{q}})\\ , \\nonumber \\\\\n \\tilde{\\psi} &=& \\mbox{arcosh} H(\\vec{\\hat{q}}+\\vec{\\hat{k}})\\ , \\\\\n H(\\vec{\\hat{p}}) &=& 1+2\\sum_{j}\\sin^{2}\\frac{\\hat{p}_{j}}{2}\\ . \\nonumber\n\\end{eqnarray}\n\\reseteqn\nThe frequency sum can be calculated by making use of \n(\\ref{sum:bosg}). After some\nstraight forward algebra one finds that\n\\alpheqn\n\\begin{eqnarray}\n\\lefteqn{\n \\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(c)}= 6g^2\n \\int^{\\pi}_{-\\pi}\\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\n h(\\tilde{\\phi},\\tilde{\\psi},\n a,b)\\hat{\\eta}_{BE}(\\tilde{\\phi}) } \\nonumber \\\\\n& &\\mbox{}+\\frac{3}{2} g^{2} \\left\\{ a(\\vec{\\hat{k}})+\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}[h(\\tilde{\\phi},\\tilde{\\psi},a,b)+h\n (\\tilde{\\psi},\\tilde{\\phi},a,b)] \\right\\}\\ , \\label{eq:cb}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n h(\\tilde{\\phi},\\tilde{\\psi},a,b)=\\frac{-a\\, \\sinh^{2}\\tilde{\\phi}\n +\\frac{1}{2} b[\\cosh\\tilde{\\phi}+1]}{\\sinh\\tilde{\\phi}\n [\\cosh\\tilde{\\phi}-\\cosh\\tilde{\\psi}]} \\label{eq:cc}\n\\end{equation}\nand\n\\begin{equation}\n \\hat{\\eta}_{BE}(\\tilde{\\phi}) = \\frac{1}{e^{\\hat{\\beta}\\tilde{\\phi}}-1}\n\\end{equation}\n\\reseteqn\nis the lattice version of the Bose-Einstein distribution function.\n\nIn obtaining this result we have made use of the fact that\n\\begin{equation}\n \\tilde{\\phi} \\stackrel{\\vec{\\hat{q}} \\rightarrow\n -\\vec{\\hat{q}}-\\vec{\\hat{k}}}{\\longleftrightarrow} \\tilde{\\psi} \\label{substb}\n\\end{equation}\nwhile $a(\\vec{\\hat{k}})$ and $b(\\vec{\\hat{q}},\\vec{\\hat{k}})$ are invariant\nunder the\ntransformation \n$\\vec{\\hat{q}} \\rightarrow -\\vec{\\hat{q}}-\\vec{\\hat{k}}$. Note that the\nfunction\n(\\ref{eq:cc}) is singular for $\\vec{\\hat{k}} \\rightarrow 0$,\nsince in this limit $\\tilde{\\phi} \\rightarrow \\tilde{\\psi}$. \nThe singularity is however\nintegrable as can be seen by making use of (\\ref{substb}) to write\n(\\ref{eq:cb}) in the form\n\\alpheqn\n\\begin{eqnarray}\n\\lefteqn{\n \\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(c)}=3g^2\n \\int^{\\pi}_{-\\pi}\\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\n h(\\tilde{\\phi},\\tilde{\\psi},a,b)\n \\bigtriangleup\\hat{\\eta}_{BE}(\\tilde{\\phi},\\tilde{\\psi}) } \\\\\n& &\\mbox{}+\\frac{3}{2}g^{2} \\left\\{ a(\\vec{\\hat{k}})+\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}[h(\\tilde{\\phi},\\tilde{\\psi},a,b)+\n h(\\tilde{\\psi},\n \\tilde{\\phi},a,b)][1+2\\hat{\\eta}_{BE}(\\tilde{\\phi})] \\right\\}\\ , \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation}\n \\bigtriangleup\\hat{\\eta}_{BE}(\\tilde{\\phi},\\tilde{\\psi}) = \n \\hat{\\eta}_{BE}(\\tilde{\\phi})\n -\\hat{\\eta}_{BE}(\\tilde{\\psi}) \\ . \\label{def:delbe}\n\\end{equation}\n\\reseteqn\nThe limit $\\vec{\\hat{k}} \\rightarrow 0$ \ncan now be easily taken and one obtains the\nfollowing contribution to the electric screening mass\n\\begin{eqnarray}\n (\\hat{m}^{2}_{el})_{(c)} &=& \\frac{3}{2} g^{2} \\left\\{ 3-\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}} \\left[\n 3\\coth\\tilde{\\phi}+\\frac{1}{2\\sinh\n \\tilde{\\phi}} \\right] [1+2\\hat{\\eta}_{BE}(\\tilde{\\phi})] \\right\\} \\nonumber \\\\\n& &\\mbox{}+\\frac{15}{2} g^{2}\\hat{\\beta}\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\n \\frac{e^{\\hat{\\beta}\\tilde{\\phi}}}{[e^{\\hat{\\beta}\\tilde{\\phi}}-1]^{2}}\n \\ . \\label{mel:c}\n\\end{eqnarray} \\pagebreak \\\\\n{\\it iv) Contribution of diagram (d)}\\\\\n\nThis diagram involves the 4-gluon vertex, which consists of types of terms\ndiffering in the colour structure: terms involving the structure constants\n$f_{ABC}$,\nand terms involving the completely symmetric colour couplings $d_{ABC}$. We\ndenote\nthe corresponding contributions to $\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(\\hat{k})$\nby  $[\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(\\hat{k})]_{[f]}$ and\n$[\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(\\hat{k})]_{[d]}$,\nrespectively.\nConsider first $[\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(d)}]_{[f]}$.\nAfter some algebra one finds that\n\\alpheqn\n\\begin{equation}\n \\left[\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(d)}\\right]_{[f]}=\n \\frac{3}{4} g^{2}\\frac{1}{\\hat{\\beta}}\\sum^{\\frac{\\hat{\\beta}}{2}-1}_{\n \\ell=-\\frac{\\hat{\\beta}}{2}}\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\n \\left[ f^{(d)}(e^{i\\hat{\\omega}^{+}_{\\ell}};\\hat{q},\\hat{k})\\right]_{\n [f]}\\ , \\label{eq:da}\n\\end{equation}\nwhere\n\\begin{equation}\n \\left[ f^{(d)}(z;\\vec{\\hat{q}},\\vec{\\hat{k}})\\right]_{[f]}=\n \\frac{-c(\\vec{\\hat{k}})(z^{2}+1)+d(\\vec{\\hat{k}})(z-1)^{2}+\n P(\\vec{\\hat{q}},\\vec{\\hat{k}})z}{\n [z-\\bar{z}_{1}][z-\\bar{z}_{2}]}\n\\end{equation}\nand\n\\begin{eqnarray}\n c(\\vec{\\hat{k}}) &=& 1+2\\sum_{j}\\cos \\hat{k}_{j} \\ , \\nonumber \\\\\n d(\\vec{\\hat{k}}) &=& 1-\\frac{1}{3}\\sum_{j} \\tilde{\\hat{k}}_{j}^{2} \\ , \\\\\n P(\\vec{\\hat{q}},\\vec{\\hat{k}}) &=& 2+\\frac{1}{6}\\sum_{j}\n [\\widetilde{(\\hat{q}+\\hat{k})}_{j}^{2} +\n \\widetilde{(\\hat{q}-\\hat{k})}_{j}^{2}] \\ . \\nonumber\n\\end{eqnarray}\n\\reseteqn\nPerforming the frequency sum in (\\ref{eq:da}) one obtains\n\\begin{eqnarray}\n\\lefteqn{\n \\left[ \\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(d)} \\right]_{[f]}=\n \\frac{3}{4} g^{2} \\left\\{ -c(\\vec{\\hat{k}})+\n d(\\vec{\\hat{k}}) \\right. } \\nonumber \\\\\n& &\\mbox{}+\\int^{\\pi}_{-\\pi}\\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\n \\left. \\left[ (c-d)\n \\coth\\tilde{\\phi}+(d-\\frac{1}{2}P)\\frac{1}{\\sinh\\tilde{\\phi}} \\right]\n [1+2\\hat{\\eta}_{BE}(\\tilde{\\phi})] \\right\\} \\ .\n\\end{eqnarray}\nTaking the limit $\\vec{\\hat{k}} \\rightarrow 0$ one finds the following\ncontribution to\nthe screening mass\n\\begin{equation}\n (\\hat{m}^{2}_{el})^{(d)}_{[f]}=\\frac{1}{2}g^{2} \\left\\{ -9+\\frac{1}{\n 2}\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}} \\left[ 17\\coth\n \\tilde{\\phi}+\\frac{1}{\\sinh\n \\tilde{\\phi}} \\right] [1+2\\hat{\\eta}_{BE}(\\tilde{\\phi})] \\right\\}\n \\ . \\label{mel:df}\n\\end{equation}\nNext consider the contribution \n$[\\hat{\\Pi}_{44}^{(\\beta,\\mu)}(0,\\vec{\\hat{k}})_{(d)}]_{[d]}$.\nIt is given by\n\\alpheqn\n\\begin{equation}\n \\left[\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{\n (d)}\\right]_{[d]}=-\\frac{1}{2}\n g^{2}\\frac{1}{\\hat{\\beta}}\\sum^{\\frac{\\hat{\\beta}}{2}-1}_{\\ell=-\n \\frac{\\hat{\\beta}}{2}}\\int^{\\pi}_{-\\pi}\\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\n \\left[ f^{(c)}(e^{i\\hat{\\omega}^{+}_{\\ell}};\\vec{\\hat{q}},\\vec{\\hat{k}})\n \\right]_{[d]}\\ ,\n\\end{equation}\nwhere\n\\begin{eqnarray}\n \\left[ f^{(d)}(z;\\vec{\\hat{q}},\\vec{\\hat{k}})\\right]_{[d]} &=&\\frac{1}{12}\n \\left(\\frac{20}{3}+d(A)\\right) \\frac{ K(\\vec{\\hat{k}})(z-1)^{2}-L(\\vec{\\hat{\n q}},\\vec{\\hat{k}})z}{\n [z-\\bar{z}_{1}][z-\\bar{z}_{2}]}\\ , \\\\\n K(\\vec{\\hat{k}}) &=& \\sum_{j}\\tilde{\\hat{k}}^{2}_{j}\\ , \\\\\n L(\\vec{\\hat{q}},\\vec{\\hat{k}}) &=& \\sum_{j}\\tilde{\\hat{q}}_{j}^{2}\n  \\tilde{\\hat{k}}^{2}_{j}\\ ,\n\\end{eqnarray}\nand where $d(A)$ is defined by\n\\begin{equation}\n \\sum^{8}_{E,F=1}[2d_{AFE} d_{BFE}+d_{FFE}d_{ABE}]=d(A)\\delta_{AB}\\ .\n\\end{equation}\n\\reseteqn\nPerforming the frequency sum one finds\n\\begin{eqnarray*}\n\\lefteqn{\n \\left[\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}}\n )_{(d)}\\right]_{[d]} = g^{2}\n \\frac{1}{24}\\left(\\frac{20}{3}+d(A)\\right) } \\\\\n& &\\times\\left\\{ -K+\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\\left[ K \\coth\\tilde{\\phi}-\n \\frac{ K+\\frac{1}{2} L}{\\sinh\\tilde{\\phi}} \\right]\n [1+\\hat{\\eta}_{BE}\n (\\tilde{\\phi})] \\right\\}\\ .\n\\end{eqnarray*}\nSince $K(\\vec{\\hat{k}})$ and $L(\\vec{\\hat{q}},\\vec{\\hat{k}})$ vanish for\n$\\vec{\\hat{k}} \\rightarrow 0$, it does not contribute\nto the screening mass, i.e.,\n\\begin{equation}\n [(\\hat{m}_{el})_{(d)}]_{[d]} = 0 \\ . \\label{mel:dd}\n\\end{equation}\\\\\n{\\it v) Contribution of diagram (e)}\\\\\n\nThe only other diagram possessing a continuum analog is the ghost\nloop shown\nin fig.\\ 1e. Its contribution is given by\n\\alpheqn\n\\begin{equation}\n\\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(e)} = \\frac{3}{2}\n g^{2}\\frac{1}{\\hat{\\beta}}\n \\sum^{\\frac{\\hat{\\beta}}{2}-1}_{\\ell=-\n \\frac{\\hat{\\beta}}{2}}\\int^{\\pi}_{-\\pi}\\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\n f^{(d)}(e^{i\\hat{\\omega}^{+}_{\\ell}};\\vec{\\hat{q}},\\vec{\\hat{k}})\\ ,\n\\end{equation}\nwhere\n\\begin{equation}\n f^{(e)}(z;\\vec{\\hat{q}},\\vec{\\hat{k}}) = -\\frac{1}{2}\n \\frac{(z^{2}-1)^{2}}{\\Pi^{4}_{i=1}[z-\\bar{z}_{i}]} \\ .\n\\end{equation}\n\\reseteqn\nPerforming the frequency sum one finds that\n\\alpheqn\n\\begin{eqnarray}\n \\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(e)} &=& \\frac{3}{4}g^{2}\n \\left\\{ -1+\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{\n (2\\pi)^{3}}[g(\\tilde{\\phi},\\tilde{\\psi})+g(\\tilde{\\psi},\\tilde{\\phi})]\n \\right. \\nonumber \\\\\n& &\\mbox{}+ \\left. 4\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}g(\\tilde{\\phi},\\tilde{\\psi})\n \\hat{\\eta}_{BE}(\\tilde{\\phi}) \\right\\} \\ , \\label{eq:ea}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n g(\\tilde{\\phi},\\tilde{\\psi}) = \\frac{\\sinh \\tilde{\\phi}}{\\cosh\\tilde{\\phi} -\n \\cosh\\tilde{\\psi}}\\ .\n\\end{equation}\n\\reseteqn\nThis function is again singular for $\\vec{\\hat{k}} \\rightarrow 0$. \nTo compute the\nlimit we\nproceed as discussed earlier and write (\\ref{eq:ea}) in the form\n\\begin{eqnarray*}\n \\hat{\\Pi}^{(\\beta,\\mu)}_{44}(0,\\vec{\\hat{k}})_{(e)} &=& \\frac{3}{4}g^{2}\n \\left\\{  -1+\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}[g(\\tilde{\\phi},\\tilde{\\psi})+\n g(\\tilde{\\psi},\\tilde{\\phi})]\n [1+2\\hat{\\eta}_{BE}(\\tilde{\\phi})]\\right. \\\\\n& &\\mbox{}+ \\left. 2\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{\n (2\\pi)^{3}}g(\\tilde{\\phi},\\tilde{\\psi})\\bigtriangleup\n \\hat{\\eta}_{BE}(\\tilde{\\phi},\n \\tilde{\\psi}) \\right\\}\\ ,\n\\end{eqnarray*}\nwhere $\\bigtriangleup\\hat{\\eta}_{BE}(\\tilde{\\phi},\\tilde{\\psi})$ \nhas been defined in (\\ref{def:delbe}).\nTaking the limit $\\vec{\\hat{k}} \\rightarrow 0$ \none obtains\n\\begin{eqnarray}\n(\\hat{m}^{2}_{el})_{(e)} &=& \\frac{3}{4}g^{2}\\left\\{ -1 +\n \\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}[1+2\\hat{\\eta}_{BE}(\\tilde{\\phi})]\n \\coth\\tilde{\\phi} \\right. \\nonumber \\\\\n& &\\mbox{} \\left. -2\\hat{\\beta}\\int^{\\pi}_{-\\pi}\n \\frac{d^{3}\\hat{q}}{\n (2\\pi)^{3}}\\frac{e^{\\hat{\\beta}\\tilde{\\phi}}}{[e^{\\hat{\\beta}\\tilde{\\phi}}-1\n ]^{2}} \\right\\} \\ . \\label{mel:e}\n\\end{eqnarray}\nCombining the results (\\ref{mel:c}), (\\ref{mel:df}), (\\ref{mel:dd}) \nand (\\ref{mel:e}), we therefore\nfind that\nthose diagrams possessing\na continuum analog yield the following contribution to the electric screening\nmass in the gluonic sector\n\\begin{eqnarray}\n(\\hat{m}^{2}_{el})_{(c+d+e)} &=& g^{2}\\left\\{ -\\frac{3}{4}\n +\\frac{1}{2}\\int_{-\\pi}^{\\pi}\\frac{d^{3}\\hat{q}}{(2\\pi)^{3}} \n \\left( \\coth\\tilde{\\phi} -\n \\frac{1}{\\sinh\\tilde{\\phi}}\\right)[1+2\\hat{\\eta}_{BE}(\\tilde{\\phi})]\n \\right\\} \\nonumber \\\\\n& &\\mbox{}+6g^{2}\\hat{\\beta}\\int_{-\\pi}^{\\pi}\\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\n \\frac{e^{\\hat{\\beta}\\tilde{\\phi}}}{[e^{\\hat{\\beta}\\tilde{\\phi}}-1]^{2}}\n \\ . \\label{mel:cde}\n\\end{eqnarray}\n\nThe computation of the remaining contributions arising from diagrams (f)\nand (g),\nwhich are a consequence of the lattice regularization, is\nstraight forward. One finds that they cancel the first term in\n(\\ref{mel:cde}).\nHence the gluonic sector (G) contributes as follows to the screening\nmass\n\\begin{equation}\n [\\hat{m}_{el}^{2}(\\hat{\\beta},\\hat{\\mu},\\hat{m})]_{G}=\n 6g^{2}\\hat{\\beta}\\int^{\\pi}_{-\\pi}\\frac{d^{3}\\hat{q}}{(2\\pi)^{3}}\n \\frac{e^{\\hat{\\beta}\\tilde{\\phi}}}{[e^{\\hat{\\beta}\\tilde{\\phi}}-1]^{2}}\\ .\n\\end{equation}\nIn the continuum limit (\\ref{mel:lim}) the corresponding expression for the\n(dimensioned)\nscreening mass squared is given by\n\\begin{eqnarray*}\n (m^{2}_{el})_{G} &=& \\lim_{a\\rightarrow 0}6g^{2}\\beta\\int^{\\frac{\\pi}{a}}_{\n -\\frac{\\pi}{a}}\n \\frac{d^{3}q}{(2\\pi)^{3}}\\frac{e^{\\frac{1}{a}\\beta\\tilde{\\phi}\n (a\\vec{q}\\, )}}{\n [e^{\\frac{1}{a}\\beta\\tilde{\\phi}(a\\vec{q}\\, )}-1]^{2}} \\\\\n&=& \\frac{3}{\\pi^{2}}g^{2}\\beta\\int^{\\infty}_{0} dq\\, q^{2} \n \\frac{e^{\\beta q}}{[e^{\\beta q} -1]^{2}}\\ ,\n\\end{eqnarray*}\nwhere $q = |\\vec{q}\\, |$.\nAfter a partial integration this expression takes the form\n\\[ (m^{2}_{el})_{G} = \\frac{6}{\\pi^{2}}g^{2} T^{2}\\int^{\\infty}_{0} dx \n \\frac{x}{e^{x}-1}\\ . \\]\nMaking use of\n\\[ \\int^{\\infty}_{0} dx \\frac{x^{\\alpha-1}}{e^x-1} =\n \\Gamma(\\alpha)\\zeta(\\alpha)\\ , \\ \\ \\alpha > 1 \\ , \\]\nwhere $\\Gamma(\\alpha)$ is the Euler Gamma function, and $\\zeta(\\alpha)$ the\nRiemann\nZeta-function, we recover the well known result:\\ \n$(m^{2}_{el})_{G} = g^{2}T^{2}$.\n\\section{Lattice Artefacts in the Screening Mass}\n\\setcounter{equation}{0}\nIn this section we compare the one loop result for the electric screening mass\non the lattice with the continuum. This will provide us with an estimate of the\nmagnitude of the lattice artefacts to be expected in numerical simulations.\nThe numerical data we present is for two mass-degenerate quarks.\n\nOn a lattice the temperature can be varied by either keeping the lattice\nspacing fixed and varying the number $N_{\\tau}$ of temporal lattice sites, or\nby varying the lattice spacing (or equivalently the coupling) keeping\n$N_{\\tau}$ fixed. For fixed lattice spacing the dependence of the screening\nmass on the temperature, fermion mass and chemical potential is given by\n(see eq. (\\ref{mel:lim}))\n\\begin{equation}\n [m_{el}(T,m,\\mu,a)]_{latt} = \\frac{1}{a}\\hat{m}_{el}(\\frac{1}{Ta},\\mu a,\n ma)\n\\end{equation}\n\\begin{figure}[htb]\n\\leavevmode\n\\centering\n\\epsfxsize10cm \\epsffile{art2.eps} \n\\caption{Dependence of $[m_{el}]_{latt}\/[m_{el}]_{cont}$ on $\\frac{T}{m}$\n  for $\\mu=0$ and different lattice spacings measured in units of $m^{-1}$.\n  Open squares (filled stars) correspond to lattices with $N_{\\tau}=8$\n  ($N_{\\tau}=16$).}\n\\end{figure}\nIf the lattice expression is to\napproximate the continuum, then the lattice spacing must be small compared\nto all\nphysical length scales in the problem. Hence we must have that\n$a \\ll \\frac{1}{T}$,\n$a \\ll \\frac{1}{m}$ and $a \\ll \\frac{1}{\\mu}$. We therefore expect that\nfor temperatures $T \\ll \\frac{1}{a}$ the continuum is well approximated for\n$ma \\ll 1$ and\n$\\mu a \\ll 1$. \nThis is shown in figs. 2 and 3 where we have plotted\n$[m_{el}]_{latt}\/[m_{el}]_{cont}$ as a function of $T\/m$ at $\\mu = 0$ and\n$\\mu\/m =1.5$\nfor various lattice spacings measured\nin units of $m^{-1}$. For $\\mu = 0$ and $ma \\equiv \\tilde{a} \\in [0, .3]$,\nthe deviation of this ratio from\nunity is seen to be at most\n$1.75\\%$ for $\\frac{T}{m} \\leq \\frac{1}{16\\tilde{a}}$. The rhs of this\ninequality is the temperature associated with a lattice with 16 sites\nin the temporal direction. \nFor $\\frac{T}{m} \\approx \\frac{1}{8\\tilde{a}}$ the deviation is already\n$7.5\\%$.\nThe endpoint of the curves\nfor $\\tilde{a} = 0.05$, $0.1$ and $0.3$ correspond to the minimal number\nof temporal lattice\nsites, i.e. $N_{\\tau} =2$, and the open squares (filled stars) to\n$N_{\\tau} =8$ ($N_{\\tau} =16$).\n\\begin{figure}[htb]\n\\leavevmode\n\\centering\n\\epsfxsize10cm \\epsffile{art3.eps}\n\\caption{Dependence of $[m_{el}]_{latt}\/[m_{el}]_{cont}$ on $\\frac{T}{m}$\n  for $\\frac{\\mu}{m}=1.5$ and different lattice spacings measured in units \n  of $m^{-1}$.\n  Open squares (filled stars) correspond to lattices with $N_{\\tau}=8$\n  ($N_{\\tau}=16$). The dashed-dotted (dashed) lines show the temperature\n  dependence for a fixed number of temporal lattice sites, $N_{\\tau}=8$\n  ($N_{\\tau}=16$). }\n\\end{figure}\n\nFor $\\mu\/m = 1.5$ we must also ensure that $\\mu a \\ll 1$. We therefore\nexpect that the allowed range of lattice spacings for achieving an accuracy\nof $2\\%$ for $T \\leq \\frac{1}{16 a}$ is now restricted to a smaller interval.\nFig.\\ 3 shows that the continuum screening mass is well approximated for\n$\\tilde{a} < 0.1$ in this temperature range.\n\nIn lattice simulations one is interested in determining the electric screening\nmass above the deconfining phase transition. It is\nextracted from correlators of Polyakov loops [6--9],\nor from the long distance behaviour of\nthe gluon propagator \\cite{hell}. \nIn these simulations the number of lattice sites\n$N_{\\tau} \\equiv \\hat{\\beta}$ is\nfixed. The electric screening mass in physical units, divided by\nthe temperature, is then given by\n\\begin{equation}\n [m_{el}]_{latt}\/T = N_{\\tau} \\hat{m}_{el}(N_{\\tau},(\\mu\/T)N^{-1}_{\\tau},\n  (m\/T)N^{-1}_{\\tau})\\ ,\n\\end{equation}\nwhile in the continuum limit ($N_{\\tau} \\rightarrow \\infty$) \nthis ratio is just a\nfunction of\n$m\/T$ and $\\mu\/T$.\nThe above conditions for approximating the continuuum now read\n(a) $N_{\\tau} \\gg 1$; (b) $ma = (\\frac{m}{T})\\frac{1}{N_{\\tau}} \\ll 1$;\n(c) $\\mu a = (\\frac{\\mu}{T})\\frac{1}{N_{\\tau}} \\ll 1$.\nWe therefore expect that for fixed $N_{\\tau}$ the continuum is best\napproximated for high temperatures. \nFor $\\frac{\\mu}{m}=1.5$ this is shown in fig.\\ 3, where\nthe temperature dependence of $[m_{el}]_{latt}\/[m_{el}]_{cont}$ for\n$N_{\\tau}=8$ and $N_{\\tau}=16$ is given by the dash-dotted and dashed curves\nrespectively. The strong deviation of this ratio at\nlow temperatures is due to the fermion loop contribution. \n\\begin{figure}[htb]\n\\leavevmode\n\\centering\n\\epsfxsize10cm \\epsffile{art4.eps} \\\\\n\\caption{Dependence of the pure gluonic contribution to\n  $[m_{el}]_{latt}\/[m_{el}]_{cont}$ on the number of lattice sites $N_{\\tau}$.\n  The solid line interpolates between different numbers of lattice sites,\n  $N_{\\tau}$.}\n\\end{figure}\nThis is evident\nfrom fig.\\ 4, where we have plotted \n$[m_{el}]_{latt}\/[m_{el}]_{cont}$ for the pure SU(3) gauge theory for various\nvalues of $N_{\\tau}$. This ratio only depends on the number of lattice sites,\n$N_{\\tau}$. The solid line interpolates between different numbers of\ntemporal lattice sites.  \nThe deviation from the continuum is seen to be small already\nfor $N_{\\tau}=8$. For $N_{\\tau}=8$, and $N_{\\tau}=16$, it is about $2\\%$,\nand $0.4\\%$, respectively.\n\n\\begin{figure}[htb]\n\\leavevmode\n\\centering\n\\begin{tabular}{cc}\n\\epsfxsize7.5cm \\epsffile{art5.eps} & \n\\epsfxsize7.5cm \\epsffile{art6.eps} \\\\\n\\end{tabular}\n\\caption{Dependence of $[m_{el}]_{latt}\/[m_{el}]_{cont}$ on $\\frac{m}{T}$\n  and $\\frac{\\mu}{T}$ for (a) $N_{\\tau}=8$, and (b) $N_{\\tau}=16$.}\n\\end{figure}\nFinally, in fig.\\ 5\nwe have plotted the ratio $[m_{el}]_{latt}\/[m_{el}]_{cont}$ as a function\nof $\\frac{m}{T}$ and $\\frac{\\mu}{T}$ for \n$N_{\\tau} = 8, 16$. In the parameter\nrange\nconsidered the above inequalities are well satisfied and the\ndeviation of this ratio from unity is seen to be at most\n$1.7\\%$ for $N_{\\tau} = 16$,\nand $10\\%$ for $N_{\\tau} = 8$.\nThe range of values $m\/T$ and $\\mu\/T$ for which the continuum is\nwell approximated will of course increase with increasing $N_{\\tau}$.\n\\section{Conclusions}\n\\setcounter{equation}{0}\nIn this paper we have have computed the electric screening mass for\nWilson fermions in the infinite volume limit for lattice QCD at finite\ntemperature and chemical potential in one\nloop order. The expression we obtained had a very transparent structure in\nwhich the artefacts arising from a finite lattice spacing were concentrated\nin two functions which in the naive continuum limit reduced to the\non shell energies of a free quark and gluon. We then studied the\ndependence of the lattice\nscreening mass on the temperature and chemical potential for fixed values of\nthe lattice spacing. It was found that lattice artefacts\ngive rise to an enhancement of the screening mass. For $\\mu =0$\nand lattice spacings $a < .3m^{-1}$ the deviation\nof $[m_{el}]_{latt}\/[m_{el}]_{cont}$ from the continuum was found to\nbe less than $1.75\\%$ for $\\frac{T}{m} \\leq \\frac{1}{16\\tilde{a}}$,\nwhere $\\tilde{a}=ma$. For\n$\\frac{\\mu}{m} = 1.5$ a substantially smaller lattice spacing was\nrequired to approximate the continuum. Most of the deviation was found to\nbe due to the fermion loop contribution.\n\nSince in numerical simulations the temperature dependence of the screening\nmass is extracted from a given lattice by varying the coupling\n(lattice spacing), we have also studied the dependence of the lattice\nscreening mass on the temperature and chemical potential for \nfixed $N_{\\tau}$. It\nwas found that for $N_{\\tau} =16$, and temperatures larger than the fermion\nmass and chemical potential, the continuum screening mass was approximated\nto within $1.75\\%$. The corresponding\ndeviation for $N_{\\tau} = 8$ was found to be\nat most $10\\%$ and to be due to the fermion loop contribution.\nIn the pure SU(3) gauge theory the continuum\nwas already approximated to $2\\%$ for only 8 lattice sites in\nthe temporal direction.\n\nOur analysis was carried out for infinite lattice volume.\nFor finite spatial volume the momentum spectrum becomes\ndiscrete and zero momentum modes must be treated separately in a\nperturbative expansion. In comparing the data for the electric\nscreening mass obtained in Monte Carlo simulations with continuum\nperturbation theory, finite volume effects need to be\nincluded, while finite lattice spacing effects may, as our analysis\nsuggests, be negligible\nfor those couplings and lattice sizes at which the simulations have been\nperformed.\\\\ \\\\\n\\begin{center} {\\bf ACKNOWLEDGEMENTS} \\end{center}\nWe are very grateful to T. Reisz for several discussions and constructive\ncomments.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe investigate algorithmic computability of a particular class of coalitional games (cooperative games), \ncalled \\emph{simple games} (voting games).  \nOne can think of simple games as representing voting methods; alternatively,\nas representing ``manuals'' or ``contracts.'' \nWe give a characterization of computable simple games; it implies that a computable simple game uses\ninformation about only finitely many players, but how much information it uses depends on each coalition.\nWe also apply the characterization to the theory of the core.\nFor the latter application, we extend Nakamura's \ntheorem (\\citeyear{nakamura79}) regarding the core of simple games to the framework\nwhere not all subsets of players are deemed to be a coalition.\n\n\\subsection{Computability analysis of social choice}\n\nMost of the paper (except the part on the theory of the core) \ncan be viewed as a contribution to the foundations of \n\\emph{computability analysis of social choice}, \nwhich studies algorithmic properties of social decision-making.  \nThis literature includes \\citet{kelly88}, \n\\citet{lewis88}, \\citet{bartholdi-tt89vs,bartholdi-tt89cd},\nand \\citet{mihara97et,mihara99jme,mihara04mss},\nwho study issues in social choice using \\emph{recursion theory} \n(the theory of computability and complexity).\\footnote\nThese works, which are mainly concerned with the complexity of rules or cooperative games in themselves,\ncan be distinguished from the closely related studies of the complexity of \\emph{solutions} for cooperative games, \nsuch as \\citet{deng-p94} and \\citet{fang-zcd02}; %\nthey are also distinguished from the studies, such as \\citet{takamiya-t0701}, of the complexity of deciding\nwhether a given cooperative game has a certain property.\n(More generally, applications of recursion \ntheory to economic theory and game theory include\n\\citet{spear89}, \\citet{canning92}, \\citet{anderlini-f94},\n\\citet{anderlini-s95},  \\citet{prasad97},\n\\citet{richter-w99jme, richter-w99et}, and \\citet{evans-t01}.\nSee also \\citet{lipman95} and \\citet{rubinstein98} for surveys of the literature on \nbounded rationality in these areas.)}\n\nThe importance of computability in social choice theory would be unarguable.\nFirst, the use of the language by social choice theorists suggests the importance.\nFor example, Arrow defined a social welfare function to be \na ``\\emph{process or rule}'' which, for each profile of \nindividual preferences, ``states'' a corresponding social \npreference \\citep[p.~23]{arrow63}, and called the function a \n``\\emph{procedure}'' \\citep[p.~2]{arrow63}.  Indeed, he later \nwrote \\citep[p.~S398]{arrow86} in a slightly different context, ``The next \nstep in analysis, I would conjecture, is a more consistent assumption of \n\\emph{computability} in the formulation of economic hypotheses'' (emphasis added).\nSecond, there is a normative reason.  Algorithmic \nsocial choice rules specify the procedures in such a way that the same \nresults are obtained irrespective of who carries out a computation, leaving \nno room for personal judgments.  In this sense, \ncomputability of social choice rules formalizes the notion of ``due \nprocess.''\\footnote\n\\citet{richter-w99jme} give reasons for studying \ncomputability-based economic theories from the viewpoints of \nbounded rationality, computational economics, and complexity analysis.\nThese  reasons partially apply to studying computable rules in social choice.}\n\n\\subsection{Simple games with countably many players}\\label{Intro2}\n\nSimple games have been central to the study of social choice \n\\citep[e.g.,][]{banks95,austensmith-b99,peleg02hbscw}.\n\\emph{Simple games}  on an algebra of coalitions of players assign either 0 or~1 to each coalition\n(member of the algebra).\nIn the setting of players who face a yes\/no question, \na coalition intuitively describes those players who vote yes.\nA simple game is characterized by its \\emph{winning} coalitions---those assigned the value~1.\n(The other coalitions are \\emph{losing}.)\nWinning coalitions are understood to be those coalitions whose unanimous votes are decisive.\n\n\nWhen there are only \\emph{finitely} many players, we can construct a finite table listing \nall winning coalitions.  Computability is automatically satisfied, since such a table gives an \nalgorithm for computing the game.\nThe same argument does not hold when there are \\emph{infinitely} many players.\nIndeed, some simple games are noncomputable,\nsince there are uncountably many simple games but only countably many computable ones\n (because each computable game is associated with an algorithm).\n \nTaking the ``fixed population'' approach,\\footnote\nThere are two typical approaches to introducing infinite population to a social choice model.\nIn the ``variable population'' approach, \nplayers are potentially infinite, but each problem (or society) involves only finitely many\nplayers.  Indeed, well-known schemes, such as simple majority rule, \nunanimity rule, and the Condorcet and the Borda rules, are all algorithms that \napply to problems of any finite size.\n\\citet{kelly88} adopts this approach, giving examples of noncomputable social choice rules.\nIn the ``fixed population'' approach, which we adopt, each problem involves the whole set of \ninfinitely many players.  This approach dates back to \\citet{downs57}, who\nconsider continuous voter distributions.\nThe paper by \\citet{banks-dl06} is a recent example of this approach to political theory.}  %\nwe consider a fixed infinite set of players in this study of simple games.\nRoughly speaking, a simple game is \\emph{computable} if there is a Turing program (finite algorithm) that can decide\nfrom a description (by integer) of each coalition whether it is winning or losing.\nTo be more precise, we have to be more specific about what we mean by a ``description'' of a coalition.\nThis suggests the following:\nFirst, since each member of a coalition should be describable in words (in English), it is natural to assume that the\nset $N$ of (the names of) players is countable, say, $N=\\mathbb{N}=\\{0,1,2, \\dots \\}$.\nSecond, since one can describe only countably many coalitions, we have to restrict coalitions.\nFinite or cofinite coalitions can be described by listing their members or nonmembers completely.\nBut restricting coalitions to these excludes too many coalitions of interest---such as the set of even numbers.\nA natural solution is to describe coalitions by a Turing program that can decide for the name of each player whether\nshe is in the coalition.  Since each Turing program has its code number (G\\\"{o}del number),\nthe coalitions describable in this manner are describable by an integer, as desired.\nOur notion of computability (\\emph{$\\delta$-computability}) focuses on this\nclass of coalitions---\\emph{recursive} coalitions---as well as the method (characteristic index) of describing them.\n\nA fixed population of countably many \\emph{players} arises not only in voting but in other contexts, \nsuch as a special class of multi-criterion decision making---depending on how we interpret a ``player'':\n\\begin{description}\n\\item [Simulating future generations] \nOne may consider countably many \\emph{players} (people) extending into the indefinite future.\n\n\\item [Uncertainty]\nOne may consider finitely many \\emph{persons} facing countably many states of the world~\\citep{mihara97et}:\neach \\emph{player} can be interpreted as a particular \\emph{person} in a particular \\emph{state}.\nThe decision has to be made before a state is realized and identified.\n(This idea is formalized by \\citet{gomberg-mt05}, who introduce\n``$n$-period coalition space,'' where $n$ is the number of persons.)\n\n\\item [Team management]\nPutting the right people (and equipment) in the right places is basic to team management.\\footnote{In line with much of \ncooperative game theory, we put aside the important problems of economics of organization, such as coordinating the\nactivities of the team members by giving the right incentives.}\nTo ensure ``due process'' (which is sometimes called for), can a manager of a company write \na ``manual'' (computable simple game\\footnote{Like \\citet{anderlini-f94}, who view contracts as algorithms, we view ``manuals'' as algorithms.  They derive contract incompleteness through computability analysis.}) elaborating the conditions that a team must meet?\n\nFix a particular task such as operating an exclusive agency of the company.%\n\\footnote{Extension to finitely or countably many tasks is straightforward.\nRedefine a \\emph{team} as consisting of members, equipment, \\emph{and} tasks.\nThen introduce a player for each task.  \nSince a task can be regarded as a negative input, it will be more natural to assign\n$0$ to those tasks undertaken and $1$ to those not undertaken (think of the monotonicity condition).}\nA \\emph{team} consists of members (people) and equipment.\nThe manager's job is to organize or give a licence to a team that satisfactorily performs the task.\nEach member (or equipment) is described by attributes such as skills, position, availability at a particular time and place (in case of equipment such as a computer, the attributes may be the kind of operating system, the combination of software that may run at the same time, as well as hardware and network specifications).\nEach such \\emph{attribute} can be thought of as a particular yes\/no question, \nand there are countably many such questions.\\footnote{According to a certain approach \\citep[e.g.,][]{gilboa90}\nto modeling scientific inquiry, \na ``state'' is an infinite sequence of $0$'s and $1$'s (answers to countably many questions) \nand a ``theory'' is a Turing program describing a state.\nThis team management example is inspired by this approach in philosophy of science.\nIf we go beyond the realm of social choice, we can indeed find many other interpretations\nhaving a structure similar to this example, such as elaborating the conditions for a certain medicine to take the \ndesired effects and deciding whether a certain act is legal or not.}\n\n\nHere, each \\emph{player} can be interpreted as a particular attribute of a particular member (or equipment).%\n\\footnote{From the viewpoint of ``due process,'' it would be reasonable to define a simple game not for the \nset of (the names of) members but for the set of attributes.  \n(This is particularly important where games cannot meet anonymity.) \nConsidering characteristic games for the set of attributes \n(``skills'') can make it easy to express certain allocation problems and give solutions to them \n\\citep[see][]{yokoo-05aaai}.}\nIn other words, each \\emph{coalition} is identified with a $0$-$1$ ``matrix'' of \nfinitely many rows (each row specifying a member) and \ncountably many columns (each column specifying a particular attribute).\\footnote{Since different \nquestions may be interrelated,\nsome ``matrix'' may not make much sense.  One might thus want to restrict admissible ``matrices.''  \nThis point is not crucial to our discussion, provided that there are infinitely many \nadmissible ``matrices'' consisting of infinitely many $0$'s and infinitely many $1$'s (in such cases\ncharacteristic indices are the only reasonable way of naming coalitions).}\n\n\\end{description}\n\n\n\\subsection{Overview of the results}\n\n\nAdopting the above notion of computability for simple games, \n\\citet{mihara04mss} gives a sufficient condition and necessary conditions \nfor computability.  The sufficient condition \\citep[Proposition~5]{mihara04mss} is intuitively plausible: \nsimple games with a \\emph{finite carrier} (such games are in effect finite, ignoring all except finitely many,\nfixed players' votes) are computable.\nA necessary condition \\citep[Corollary~10]{mihara04mss} in the paper seems to exclude ``nice'' \n(in the voting context) infinite games:\ncomputable simple games have both finite winning coalitions and cofinite losing coalitions.\nHe leaves open the questions (i)~whether there exists a computable simple game that has no finite carrier and\n(ii)~whether there exists a noncomputable simple game that has both finite winning coalitions and cofinite \nlosing coalitions. \nThe first of these questions is particularly important since if the answer were no, then\nonly the games that are in effect finite would be computable, a rather uninteresting result.\nThe answers to these questions (i) and~(ii) are affirmative.  \nWe construct examples in Section~\\ref{examples} to show their existence.\nThe construction of these examples depends in essential ways on Proposition~\\ref{cutprop}\n(which gives a necessary condition for a simple game to be computable) or on\nthe easier direction of Theorem~\\ref{delta0det} (which gives a sufficient condition).\nIn contrast, the results in \\citet{mihara04mss} are not useful enough for \nus to construct such examples.\n\n\nTheorem~\\ref{delta0det} gives a necessary and sufficient condition for simple games to be computable.\nThe condition roughly states that ``finitely many, unnecessarily fixed players matter.''  \n\nTo explain the condition, let us introduce the notion of a ``determining string.''   \nGiven a coalition~$S$, its \\emph{$k$-initial segment} is the string of $0$'s and $1$'s of length $k$\nwhose $j$th element (counting from zero) is $1$ if $j\\in S$ and is $0$~if $j\\notin S$.  \nFor example, if $S=\\{0,2,4\\}$, \nits $0$-initial segment, $1$-initial segment, \\ldots, $8$-initial segment, \\ldots are, respectively, the empty string,\nthe string~$1$, the string $10$, the string~$101$, the string $1010$, the string $10101$,\nthe string $101010$, the string $1010100$, the string $10101000$, \\ldots.\nWe say that a (finite) string $\\tau$ is \\emph{winning determining} if any coalition~$G$ extending $\\tau$\n(i.e., $\\tau$ is an initial segment of~$G$) is winning.  We define \\emph{losing determining} strings similarly.\n\nThe necessary and sufficient condition for computability according to Theorem~\\ref{delta0det}\nis the following: there are computably listable sets \n$T_0$ of losing determining strings and $T_1$ of winning determining strings such that \nany coalition has an initial segment in one of these sets.  \nIn the above example, the condition \nimplies that at least one string from among the empty string, $1$, $10$, \\ldots, $10101000$, \n\\ldots is in $T_0$ or in $T_1$---say, $1010$ is in $T_1$.\nThen any coalition of which $0$ and $2$ are members but $1$ or $3$ is not, is winning.\nIn this sense, \\emph{one can determine whether a coalition is winning or losing\nby examining only finitely many players' membership}.  \nIn general, however, \\emph{one cannot do so by picking finitely many players before a coalition is given}.\n\nTheorem~\\ref{delta0det} has an interesting implication for the nature of  ``manuals'' or ``contracts,'' \nif we regard them as being composed of computable simple games\n (e.g., the team management example in Section~\\ref{Intro2}).\nConsider how many ``criteria''  (players; e.g., member-attribute pairs) are needed for a ``manual'' to determine \nwhether a given ``situation'' (coalition; e.g., team) is ``acceptable'' (winning; e.g., satisfactorily performs a given task).\n\\emph{While increasingly complex situations may require increasingly many criteria, \nno situation (however complex) requires infinitely many criteria.}\nThe conditions  (such as ``infinitely many of the prime-numbered criteria must be met'') \nbased on infinitely many criteria are ruled out.\n\nThe proof of Theorem~\\ref{delta0det} uses the \\emph{recursion theorem}.\nIt involves much more intricate arguments of recursion theory\nthan those in \\citet{mihara04mss} giving only a partial characterization of the computable \ngames.\\footnote{Theorem~\\ref{delta0det} can also be derived from results in \\citet{kreisel-ls59}\nand \\citet{ceitin59}.  See Remark~\\ref{kreisel59}.}\n\nA natural characterization result might relate computability to well-known properties of simple games, \nsuch as monotonicity, properness, strongness, and nonweakness.\nUnfortunately, we are not likely to obtain such a result: as we clarify in a companion paper~\\citep{kumabe-m06csg64}, \ncomputability is ``unrelated to'' the four properties just mentioned.\n\n\nThe earlier results~\\citep{mihara04mss} are easily obtained from Theorem~\\ref{delta0det}.\nFor example, if a computable game has a winning coalition, then,\nan initial segment of that coalition is winning determining, implying that (Proposition~\\ref{d0neg1cor}) \nthe game has a finite winning coalition and a cofinite winning coalition.\nWe give simple proofs to some of these results in Section~\\ref{applications}.\nIn particular, Proposition~\\ref{nonfinanonymous} strengthens the earlier result \\citep[Corollary~12]{mihara04mss} \nthat computable games violate anonymity.\\footnote\nDetailed studies of anonymous rules based on infinite simple games include\n\\citet{mihara97scw}, \\citet{fey04}, and \\citet{gomberg-mt05}.}\n\n\\subsection{Application to the theory of the core}\n\nMost cooperative game theorists are more interested in the properties of a \\emph{solution} (or value) \nfor games than in the properties of a game itself.\nIn this sense, Section~\\ref{core} deals with more interesting applications of Theorem~\\ref{delta0det}.\n(Most of the section is of independent interest, and can be read without a knowledge of recursion theory.)\n\nTheorem~\\ref{nakamura-thm} is our main contribution to the study of \nacyclic preference aggregation rules in the spirit of \nNakamura's theorem (\\citeyear{nakamura79}) on the core of simple games.\\footnote\n\\citet{banks95}, \\citet{truchon95}, and \\citet{andjiga-m00} are recent contributions\nto this literature.\n(Earlier papers on acyclic rules can be found in \\citet{truchon95} \nand \\citet{austensmith-b99}.)\nMost works in this literature (including those just mentioned) consider finite sets of players.\n\\citet{nakamura79} considers arbitrary (possibly infinite) sets of players and the algebra of all subsets of players.\nIn contrast, we consider arbitrary sets of players and \\emph{arbitrary algebras} of coalitions.}\n\n\nCombining a simple game with a set of alternatives and a profile of individual preferences, we define a \n\\emph{simple game with (ordinal) preferences}.\nNakamura's theorem (\\citeyear{nakamura79}) gives a necessary and sufficient condition for a simple game with\npreferences to have a nonempty core for all profiles: \nthe number of alternatives is below a certain number, called the \\emph{Nakamura number} of the simple game.\nWe extend (Theorem~\\ref{nakamura-thm}) Nakamura's theorem to the framework \nwhere simple games are defined on an arbitrary algebra of coalitions\n(so that not all subsets of players are coalitions).\n\\emph{It turns out that our proof for the generalized result is more elementary than Nakamura's original proof;\nthe latter is more complex than need be.}\n\nSince computable (nonweak) simple games have a finite winning coalition, \nwe can easily prove that they have a finite Nakamura number (Corollary~\\ref{nakamura-finite}).\nTheorem~\\ref{nakamura-thm} in turn implies (Corollary~\\ref{core-nakamura}) that \nif a game is computable, \nthe number of alternatives that the set of players can deal with rationally is restricted by this number.\nWe conclude Section~\\ref{core} with Proposition~\\ref{filter-nakamura}, which suggests the fundamental \ndifficulty of obtaining computable aggregation rules in Arrow's setting (\\citeyear{arrow63}),\neven after relaxing the transitivity requirement for (weak) social preferences.\\footnote\n\\citet{mihara97et,mihara99jme} studies computable aggregation rules \\emph{without} relaxing the\ntransitivity requirement; these papers build on \\citet{armstrong80,armstrong85}, who generalizes\n\\citet{kirman-s72}.}\n\n\n\n\\section{Framework}\n\n\\subsection{Simple games}\\label{notions}\n\nLet $N=\\mathbb{N}=\\{0,1,2, \\dots \\}$ be a countable set of (the names of) \nplayers.  Any \\textbf{recursive} (algorithmically decidable) \nsubset of~$N$ is called a \\textbf{(recursive) coalition}.\n\nIntuitively, a simple game describes in a crude manner the power \ndistribution among \\emph{observable} (or describable) subsets of players.  Since the cognitive \nability of a human (or machine) is limited, it is not natural to \nassume that all subsets of players are observable when there are \ninfinitely many players.  We therefore assume that only \n\\textbf{recursive} subsets are \nobservable.  This is a natural assumption in the present context, \nwhere algorithmic properties of simple games are investigated.  \nAccording to \\emph{Church's thesis} \\citep[see][]{soare87,odifreddi92}, the recursive coalitions \nare the sets of players for which there is an algorithm that can decide for \nthe name of each player whether she is in the set.\\footnote{\\citet{soare87} and \\citet{odifreddi92}\ngive a more precise definition of \\emph{recursive sets} as well as detailed discussion of recursion theory.\n\\citet{mihara97et,mihara99jme} contain short reviews of recursion theory.}\nNote that \\textbf{the class~$\\mathrm{REC}$ of recursive \ncoalitions} forms a \\textbf{Boolean algebra}; that is, it includes $N$ \nand is closed under union, intersection, and complementation.\n(We assume that observable coalitions are recursive, not just r.e.\\ (\\emph{recursively enumerable}).  \n\\citet[Remarks~1 and 16]{mihara04mss} gives three reasons: \nnonrecursive r.e.\\ sets are observable in a very limited sense; \nthe r.e.\\  sets do not form a Boolean algebra;\nno satisfactory notion of computability can be defined if a simple game is \ndefined on the domain of all r.e.\\ sets.)\n\n\nFormally, a \\textbf{(simple) game} is a collection~$\\omega\\subseteq\\mathrm{REC}$ of (recursive) coalitions.\nWe will be explicit when we require that $N\\in \\omega$.\nThe coalitions in $\\omega$ are said to be \\textbf{winning}.  \nThe coalitions not in $\\omega$ are said to be \\textbf{losing}. \nOne can regard a simple game as a function from~REC to $\\{0,1\\}$, assigning the value 1 or 0 to each \ncoalition, depending on whether it is winning or losing.\n\nWe introduce from the theory of cooperative games a few basic \nnotions of simple games~\\citep{peleg02hbscw,weber94}.\\footnote{The desirability of these properties depends, \nof course, on the context.  Consider the team management example in Section~\\ref{Intro2}, for example.\nMonotonicity makes sense, but may be too optimistic (adding a member may turn an acceptable team into an\nunacceptable one).  Properness may be irrelevant or even undesirable \n(ensuring that a given task can be performed by \ntwo non-overlapping teams may be important from the viewpoint of reliability).\nThis observation does not diminish the contribution of the main theorem (Theorem~\\ref{delta0det}),\nwhich does not refer to these properties.\nIn fact, one can show~\\citep{kumabe-m06csg64}\n that computability is ``unrelated to'' monotonicity, properness, strongness, and weakness.}\nA simple game $\\omega$ is said to be \n\\textbf{monotonic} if for all coalitions $S$ and $T$, the \nconditions $S\\in \\omega$ and $T\\supseteq S$ imply $T\\in\\omega$.  \n$\\omega$ is \\textbf{proper} if for all recursive coalitions~$S$, \n$S\\in\\omega$ implies $S^c:=N\\setminus S\\notin\\omega$.  $\\omega$ is \n\\textbf{strong} if for all coalitions~$S$, $S\\notin\\omega$ \nimplies $S^c\\in\\omega$.  $\\omega$ is \\textbf{weak} if \n$\\omega=\\emptyset$ or\nthe intersection~$\\bigcap\\omega=\\bigcap_{S\\in\\omega}S$ of the winning coalitions is nonempty.  \nThe members of $\\bigcap\\omega$ are called \\textbf{veto players}; they \nare the players that belong to all winning coalitions.  \n(The set $\\bigcap\\omega$ of veto players may or may not be observable.)\n$\\omega$ is \\textbf{dictatorial} if there exists some~$i_0$ (called a \n\\textbf{dictator}) in~$N$ such that $\\omega=\\{\\,S\\in\\mathrm{REC}: i_0\\in \nS\\,\\}$.  Note that a dictator is a veto player, but a veto player is \nnot necessarily a dictator.\n\nWe say that a simple game $\\omega$ is \\textbf{finitely anonymous} if for any finite permutation $\\pi: N\\to N$ \n(which permutes only finitely many players) and for \nany coalition $S$, we have \n$S\\in \\omega \\iff \\pi(S) \\in \\omega$.  \nIn particular, finitely anonymous games treat any two coalitions with the same finite number of players equally.\nFinite anonymity is a notion much weaker than \nthe version of anonymity that allows any (measurable) permutation $\\pi: N\\to N$.\nFor example, free ultrafilters (nondictatorial ultrafilters) defined below are finitely anonymous.  \n\nA \\textbf{carrier} of a simple game~$\\omega$ is a coalition $S\\subset N$ \nsuch that\n\\[ \nT\\in\\omega \\iff S\\cap T\\in \\omega\n\\]\nfor all coalitions~$T$.\nWe observe that if $S$ is a carrier, then so is any coalition $S'\\supseteq S$.\n\nFinally, we introduce a few notions from the theory of Boolean \nalgebras~\\citep{koppelberg89}; they can be regarded as properties of \nsimple games.  \nA monotonic simple game~$\\omega$ satisfying\n$N\\in \\omega$ and $\\emptyset\\notin\\omega$\nis called a \\textbf{prefilter} if it has the finite intersection property:\nif $\\omega'\\subseteq \\omega$ is finite, then $\\bigcap\\omega'\\neq \\emptyset$.\nIntuitively, a prefilter consists of ``large'' coalitions.\nA prefilter is \\textbf{free} if and only if it is nonweak (i.e., it has no veto players).\nA free prefilter does not contain any finite coalitions (Lemma~\\ref{nakamura-ceiling}).\nA prefilter~$\\omega$ is a \\textbf{filter} if it is closed with respect to finite intersection: \nif $S$, $S'\\in\\omega$, then $S\\cap S'\\in \\omega$.\nThe \\textbf{principal filter generated by $S$} is\n$\\omega=\\{T\\in \\mathrm{REC}: S\\subseteq T\\}$.  It is a typical example of a filter that is\nnot free; it has a carrier, namely, $S$.  A filter is a \\textbf{principal filter} if it is \nthe principal filter generated by some~$S$.\nA filter~$\\omega$ is called an \\textbf{ultrafilter} if it is a \nstrong simple game.  If $\\omega$ is an ultrafilter, then $S\\cup \nS'\\in\\omega$ implies that $S\\in\\omega$ or $S'\\in\\omega$.\nAn ultrafilter is free if and only if it is not dictatorial.\n\n\\subsection{An indicator for simple games}\\label{indicators}\n\nTo define the notion of computability for simple games, we introduce below \nan indicator for them.  In order to do that, \nwe first represent each recursive coalition by a characteristic index ($\\Delta_0$-index).\nHere, a number $e$~is a \\textbf{characteristic index} for a coalition~$S$\nif $\\varphi_e$ (the partial function computed by the Turing program with code number~$e$)\nis the characteristic function for~$S$. \nIntuitively, a characteristic index for a coalition describes\nthe coalition by a Turing program that can decide its membership.\nThe indicator then assigns the value 0 or 1 to each \nnumber representing a coalition, depending on whether the \ncoalition is winning or losing.  When a number does not represent a \nrecursive coalition, the value is undefined.\n\nGiven a simple game $\\omega$, its \\textbf{$\\delta$-indicator} is the partial \nfunction~$\\delta_\\omega$ on~$\\mathbb{N}$ defined by\n\\begin{equation}\n\t\\label{d:eq}\n\t\\delta_\\omega(e)=\\left\\{\n\t    \\begin{array}{ll}\n\t\t1 & \\mbox{if $e$ is a characteristic index for a recursive\n\tset in $\\omega$},  \\\\\n\t\t0 & \\mbox{if $e$ is a characteristic index for a recursive\n\tset not in $\\omega$},  \\\\\n\t\t\\uparrow & \\mbox{if $e$ is not a characteristic\n\tindex for any recursive set}.\n\t\\end{array}\n\t\\right.\n\\end{equation}\nNote that $\\delta_\\omega$ is well-defined since each $e\\in\\mathbb{N}$ can be a \ncharacteristic index for at most one set.\n\n\\subsection{The computability notion}\n\\label{comp:notions}\n\nWe now introduce the notion of \\emph{$\\delta$-computable} simple games.\nWe start by giving a scenario or intuition underlying the notion of $\\delta$-computability.\nA number (characteristic index) representing a coalition\n(equivalently, a Turing program that can decide the membership of a coalition)\nis presented by an inquirer to the aggregator (planner), \nwho will compute whether the coalition is winning or not.\nThough there is no effective (algorithmic) procedure to decide whether a number given by the \ninquirer is legitimate (i.e., represents some recursive coalition),\na human can often check manually (non-algorithmically) if such a number is a legitimate representation.\nWe assume that the inquirer gives the aggregator only those indices that he has checked and proved its legitimacy.\nThis assumption is justified if we assume that the aggregator always demands such proofs.\nThe aggregator, however, cannot know a priori which indices will possibly be presented to her.\n(There are, of course, indices unlikely to be used by humans.  \nBut the aggregator cannot a priori rule out some of the indices.)\nSo, \\emph{the aggregator should be ready to compute whenever a legitimate representation\n is presented to her}.\\footnote\nAn alternative notion of computability might use a ``multiple-choice format,''\n in which the aggregator gives possible indices that the inquirer can choose from. \nUnfortunately, such a ``multiple-choice format'' would not work as one might wish\n\\citep[Appendix~A.1]{kumabe-m07csgcc}.} \nThis intuition justifies the following condition of computability.\\footnote\n\\citet{mihara04mss} also proposes a stronger condition, \\emph{$\\sigma$-computability}.\nWe discard that condition since it is too strong a notion of computability (Proposition~3 of that paper;\nfor example, even \\emph{dictatorial} games are not $\\sigma$-computable).} %\n\n\\begin{description} \\item[$\\delta$-computability] $\\delta_\\omega$ has \nan extension to a partial recursive function.  \n\\end{description}\n\nInstead of, say, $\\delta$-computability, one might want to require the indicator~$\\delta_\\omega$ \\emph{itself} \n(or its extension that gives a number different from $0$ or~$1$ whenever $\\delta_\\omega(e)$ is undefined)\nto be partial recursive \\citep[Appendix~A]{mihara04mss}.  \nSuch a condition cannot be satisfied, however, since the domain of~$\\delta_\\omega$ is\nnot r.e. \\citep[Lemma~2]{mihara97et}.\n\n\\section{A Characterization Result}\\label{mainresult}\n\n\n\\subsection{Determining strings}\n\nThe next lemma states that for any coalition~$S$ of a \n$\\delta$-computable simple game, \nthere is a cutting number~$k$ such \nthat any \\emph{finite} coalition~$G$ having the same $k$-players \nas~$S$ (that is, $G$ and $S$ are equal if players~$i\\geq k$ are \nignored) is winning (losing) if $S$ is winning (losing).  Note that \nif $k$ is such a cutting number, then so is any $k'$ greater than~$k$.\n\n\\medskip\n\n\\textbf{Notation}.  We identify a natural number~$k$ with the finite \nset $\\{0,1,2,\\ldots,k-1\\}$, which is an initial segment of~$\\mathbb{N}$.  \nGiven a coalition $S\\subseteq N$, we write $S\\cap k$ to represent the \ncoalition $\\{i\\in S: i<k\\}$ consisting of the members of $S$ whose \nname is less than~$k$.  \nWe call $S\\cap k$ the \\textbf{$k$-initial segment of $S$}, and view it \neither as a subset of~$\\mathbb{N}$ or as the string $S[k]$ of length~$k$ of 0's and 1's \n(representing the restriction of its characteristic function to \n$\\{0,1,2,\\ldots,k-1\\}$).  \nFor example, if $S=\\{0, 2, 4\\}$, we have \n$S[7]=1010100=\\varphi_e(0)\\varphi_e(1) \\cdots \\varphi_e(6)$,\nwhere $e$ is a characteristic index for~$S$.\nNote that if $G$ is a coalition and $G\\cap \nk=S\\cap k$ (that is, $G$ and $S$ are equal if players~$i\\geq \nk$ are ignored), the characteristic function of~$G$ extends the $k$-initial \nsegment (viewed as a string of 0's and 1's) of~$S$.\n\n\\begin{lemma} \\label{cutlemma}\nLet $\\omega$ be a $\\delta$-computable simple game.  If $S\\in\\omega$, \nthen there is an initial segment $k\\geq 0$ of~$\\mathbb{N}$ such that for any \n\\emph{finite}~$G\\in \\mathrm{REC}$, if $G\\cap k=S\\cap k$, then \n$G\\in\\omega$.  Similarly, if $S\\notin\\omega$, then there is an initial \nsegment $k\\geq 0$ of~$\\mathbb{N}$ such that for any \\emph{finite}~$G\\in \n\\mathrm{REC}$, if $G\\cap k=S\\cap k$, then $G\\notin\\omega$.\n\\end{lemma}\n\n\\begin{proof}\nLet $S\\in\\omega$ and assume for a contradiction that there is no such \ninitial segment~$k$.  Then, for each initial segment~$k$ of~$\\mathbb{N}$, there \nis a finite coalition $G_k$ such that $G_k\\cap k=S\\cap k$ and \n$G_k\\notin\\omega$.  Note that we can find such $G_k$ recursively \n(algorithmically) in~$k$ since it is finite.\n\nLet $K$ be a nonrecursive r.e.\\ set such as $\\{\\,e:e\\in W_e\\,\\}$.  \nSince $K$ is r.e., there is a recursive set $R\\subseteq\\mathbb{N}\\times\\mathbb{N}$ \nsuch that $e\\in K\\Leftrightarrow \\exists z R(e,z)$.  Define \n$g(e,u)=\\mu y\\leq u\\ R(e,y)$ (i.e., the least $y\\leq u$ such that \n$R(e,y)$) if such $y$ exists, and $g(e,u)=0$ otherwise.  Then $g$~is \nrecursive.\n\nUsing the Parameter Theorem, define a recursive function~$f$ by\\footnote\nIntuitively, given $e$, we define $\\varphi_{f(e)}(u)$ at each step~$u$ as follows:\nLet $u':=\\mu y\\ R(e,y)$ if $e\\in K$;  $u':=+\\infty$ otherwise.\nTry to find $z \\leq u$ such that $R(e, z)$. \nIf $u$ is small (i.e., $u<u'$), there is no such $z$ (the first and the second cases);\nfor those~$u$,  $\\varphi_{f(e)}$ takes the same value as the characteristic function for~$S$.\nIf $u$ is large enough (i.e., $u\\geq u'=g(e,u)$, which is true only if $e\\in K$),\nthere is such $z$ (the third and the fourth cases);\nfor those~$u$, $\\varphi_{f(e)}$ takes the same value as the characteristic function for a certain\nlosing coalition~$G_k$ defined above (namely, $k=u'$).\nNote that the construction of $G_k$ ensures that the characteristic functions for $S$ and \nfor $G_k$ take the same values for small $u$ as well.}\n\\begin{eqnarray*}\n\\varphi_{f(e)}(u)=1 & & \\textrm{if $\\neg\\exists z\\leq u\\ R(e,z)$ and \n$u\\in S$},\\\\\n\\varphi_{f(e)}(u)=0 & & \\textrm{if $\\neg\\exists z\\leq u\\ R(e,z)$ and \n$u\\not\\in S$},\\\\\n\\varphi_{f(e)}(u)=1 & & \\textrm{if $\\exists z\\leq u\\ R(e,z)$ and $u\\in \nG_{g(e,u)}$, and}\\\\\n\\varphi_{f(e)}(u)=0 & & \\textrm{otherwise}.\n\\end{eqnarray*}\n\nNow, on the one hand, $e\\in K$ implies that $f(e)$ is a characteristic \nindex for $G_{u'}\\notin\\omega$ for some~$u'$.  (\\emph{Details}: Given \n$e\\in K$, let $u'=\\mu y\\ R(e,y)$, which is well-defined since $\\exists \nz R(e,z)$.  Then $\\varphi_{f(e)}(u)=1$ iff (i)~$u<u'$ and $u\\in S$ \n[that is, $u<u'$ and $u\\in G_{u'}$] or (ii)~$u\\geq u'$ and [since \n$g(e,u)=u'$ in this case] $u\\in G_{g(e,u)}=G_{u'}$.  Thus \n$\\varphi_{f(e)}(u)=1$ iff $u\\in G_{u'}$.)  Hence \n$\\delta_{\\omega}(f(e))=0$.  On the other hand, $e\\notin K$ implies \nthat $f(e)$ is a characteristic index for $S\\in\\omega$.  Hence \n$\\delta_{\\omega}(f(e))=1$.\n\nSince $\\delta_{\\omega}$ has an extension to a p.r.\\ function (because \n$\\omega$ is $\\delta$-computable), the last paragraph implies that $K$ \nis recursive.  This is a contradiction.\n\n\\medskip\n\nTo prove the last half of the lemma, note that the set-theoretic \ndifference $\\hat{\\omega}=\\mathrm{REC}-\\omega$ is also a \n$\\delta$-computable simple game.  Let $S\\notin\\omega$.  Then the first \nhalf applies to $\\hat{\\omega}$ and $S\\in\\hat{\\omega}$.  Since \n$G\\in\\hat{\\omega}$ iff $G\\notin\\omega$, the desired result \nfollows.\\end{proof}\n\nIn fact, the coalition~$G$ in Lemma~\\ref{cutlemma} need not be finite.  \nBefore stating an extension (Proposition~\\ref{cutprop}) of Lemma~\\ref{cutlemma}, we introduce\nthe notion of \\emph{determining strings}:\n\n\\begin{definition}\nConsider a simple game.  A string $\\tau$ (of 0's and 1's) of \nlength~$k\\geq 0$ is said to be \\textbf{determining} if either any \ncoalition $G\\in\\mathrm{REC}$ extending $\\tau$ (in the sense that $\\tau$ is an \ninitial segment of $G$, i.e., $G\\cap k=\\tau$) is winning or any \ncoalition $G\\in\\mathrm{REC}$ extending $\\tau$ is losing.  A string $\\tau$ is \nsaid to be \\textbf{determining for finite coalitions} if either any \nfinite coalition~$G$ extending $\\tau$ is winning or any finite \ncoalition~$G$ extending $\\tau$ is losing.  A string is  \n\\textbf{nondetermining} if it is not determining.\n\\end{definition}\n\nProposition~\\ref{cutprop} below states that for $\\delta$-computable simple \ngames, (the characteristic function for) every coalition~$S$ has an \ninitial segment $S\\cap k$ that is determining.  (The number~$k-1$ may \nbe greater than the greatest element, if any, of~$S$):\n\n\\begin{prop} \\label{cutprop}\nSuppose that a $\\delta$-computable simple game is given.  \n\\textup{(i)}~If a coalition $S$ is winning, then there is an initial segment $k\\geq 0$ \nof~$\\mathbb{N}$ such that for \\emph{any} (finite or infinite) coalition~$G$, \nif $G\\cap k=S\\cap k$, then $G$ is winning.  \n\\textup{(ii)}~If $S$ is losing, then there is an initial segment $k\\geq 0$ of~$\\mathbb{N}$ such that for \n\\emph{any} coalition~$G$, if $G\\cap k=S\\cap k$, then $G$ is losing.\n\\textup{(iii)}~If $S\\cap k$ is an initial segment that is determining \n\\emph{for finite coalitions}, then $S\\cap k$ is an initial segment that is determining.\n\\end{prop}\n\n\\begin{proof} \nIt suffices to prove~(i).  As a byproduct, we obtain~(iii).\n\nSuppose $S\\in\\omega$, where \n$\\omega$ is a $\\delta$-computable simple game.  Then by the first half \nof Lemma~\\ref{cutlemma}, there is $k\\geq 0$ such that (a)~for any \nfinite $G'$, if $G'\\cap k=S\\cap k$, then $G'\\in\\omega$.\n\nTo obtain a contradiction, suppose that there is $G\\notin\\omega$ such \nthat (b)~$G\\cap k=S\\cap k$.  By the last half of Lemma~\\ref{cutlemma}, \nthere is $k'\\geq 0$ such that (c)~for any finite $G'$, if $G'\\cap \nk'=G\\cap k'$, then $G'\\notin\\omega$.  Without loss of generality, \nassume $k'\\geq k$.\n\nConsider $G'=G\\cap k'$, which is finite.  Then, on the one hand, since \n$G'\\cap k'=G\\cap k'$, we get $G'\\notin\\omega$ by (c).  On the other \nhand, since $k'\\geq k$, we get $G'\\cap k=G\\cap k=S\\cap k$ (the last \nequality by~(b)).  Then (a)~implies that $G'\\in\\omega$.  This is a \ncontradiction.\\end{proof}\n\n\\subsection{Characterization of computable games}\n\nThe next theorem characterizes $\\delta$-computable simple games in \nterms of sets of determining strings.  Roughly speaking,\nfinitely many players determine whether a coalition is winning or losing.\nThough we cannot tell in advance which finite set of players determines that, \nwe can list such sets in an effective manner.\n\nNote that $T_0\\cup T_1$ in the theorem does not necessarily contain all determining strings.\n(The $\\Longrightarrow$ direction can actually be strengthened:  \nwe can find \\emph{recursive}, not just r.e., sets $T_0$ and $T_1$ satisfying the conditions.\nWe do not prove the strengthened result, since we will not use it.)\n\n\\begin{theorem}\\label{delta0det}\nA simple game~$\\omega$ is $\\delta$-computable if and only if there are \nan r.e.\\ set~$T_0$ of losing determining strings and an r.e.\\ \nset~$T_1$ of winning determining strings such that (the characteristic \nfunction for) any coalition has an initial segment in $T_0$ or in \n$T_1$.\\end{theorem}\n\n\\begin{remark}\\label{kreisel59}\nWe can derive Theorem~\\ref{delta0det} from a result in \\citet{kreisel-ls59}\nand \\citet{ceitin59}.  \nIn the working paper \\citep[Appendix A.2]{kumabe-m07csgcc}, \nfollowing the terminology of \\citet{odifreddi92},\nwe outline such a proof, which is based on a topological argument.\nIn this paper, we give a different, more self-contained proof.\nThe fact that the proof uses the recursion theorem should also be of some interest.\\end{remark}\n\n\\begin{proof} \n($\\Longleftarrow$).  We give an algorithm that can decide for each \ncoalition whether it is winning or not: Given is a characteristic \nindex~$e$ of a coalition~$S$.  Generate the elements of $T_0$ and \n$T_1$; we can do that effectively since these sets are r.e. Wait \nuntil an initial segment of~$S$ is generated.  (Since a characteristic \nindex is given, we can decide whether a string generated is an initial \nsegment of~$S$.)  If the initial segment is in~$T_0$, then $S$~is \nlosing; if it is in~$T_1$, then $S$~is winning.\n\n\\bigskip\n\n($\\Longrightarrow$).  Suppose $\\omega$~is $\\delta$-computable.  Let \n$\\delta'$ be a p.r.\\ extension of $\\delta_{\\omega}$; such a $\\delta'$ \nexists since $\\omega$~is $\\delta$-computable.\n\n\n\\emph{Overview.} \n\nFrom Proposition~\\ref{cutprop}, our goal is to  \\emph{effectively enumerate}  \na determining initial segment~$S\\cap k$ of each losing coalition~$S$ in~$T_0$ \nand that of each winning coalition in $T_1$.\n\nWe will define a certain recursive function~$y(e)$ in Step~1.  \nIn Step~2, we will first define the sets $T_i$, where $i\\in\\{0,1\\}$, as the \ncollection of certain strings (of 0's and 1's) of length~$k(e)$ (to be \ndefined) for those $e\\in\\mathbb{N}$ satisfying $\\delta'(y(e))=i$.  \nIn particular, $T_0\\cup T_1$~includes, for each \ncharacteristic index~$e$, the $k(e)$-initial segment of the recursive \ncoalition indexed by~$e$.  We will then show that $T_0$ \nand~$T_1$ satisfy the conditions stated.\n\n\nWe use the following \\textbf{notation}.  We write $\\varphi_{e,s}(x)=y$ if $x$, \n$y$, $e<s$ and $y$ is the output of $\\varphi_e(x)$ in less than $s$ \nsteps of the $e$th Turing program~\\citep[p.~16]{soare87}.  \nIn particular, if $e$th Turing program does not give an output for $x$ in $\\leq s$ steps, \nthen $\\varphi_{e,s}(x)$ is undefined.\nWe fix a Turing program for~$\\delta'$ and denote by $\\delta'_s(y)$ the \ncomputation of $\\delta'(y)$ up to step~$s$ of the program.\n\n\\bigskip \n\n\\emph{Step~1.  Defining a recursive function~$y(e)$.}\n\nWe define a recursive function~$f(e,y)$ in Step~1.1.  \nIn Step~1.2, we apply a variant of the Recursion Theorem to \n$f(e,y)$ and obtain~$y(e)$.\n\n\\medskip\n\n\\emph{Step~1.1.  Defining a recursive function $f(e,y)$.}\n\nDefine an r.e.\\ set $Q_0\\subseteq\\mathbb{N}$ by\n$y\\in Q_0$ iff there exists~$s$ such that\n$\\delta'_{s}(y)=0$ or $\\delta'_{s}(y)=1$.\nDefine a p.r.\\ function \n\\[\ns_0(y)=\\mu s [\\delta'_{s}(y)\\in\\{0,1\\}],\n\\]\nwhich converges for $y\\in Q_0$.  \n\nFix a recursive set~$\\mathbf{F}$ of characteristic indices for finite sets \nsuch that each finite set has at least one characteristic index \nin~$\\mathbf{F}$.  (An example of $\\mathbf{F}$ is the set consisting of the code \nnumbers (G\\\"odel numbers) of the Turing programs of a particular \nform.)  For $s\\in\\mathbb{N}$, let $\\mathbf{F}_s=\\mathbf{F}\\cap s$ be the finite set of numbers \n$e<s$ in~$\\mathbf{F}$.\n\n\nDefine a set $Q_1\\subseteq\\mathbb{N}\\times\\mathbb{N}$ by $(e,y)\\in Q_1$ iff (i)~$y\\in \nQ_0$ and (ii)~there exist $s'\\geq s_0:=s_0(y)$ and $e'\\in\\mathbf{F}_{s'}$ such \nthat (ii.a)~$\\delta'_{s'}(e')=1-\\delta'_{s_0}(y)$ and that \n(ii.b)~$\\varphi_{e',s'}$ is an extension of $\\varphi_{e,s_0-1}$.\\footnote{\\label{note:extension\nCondition~(ii.b), written $\\varphi_{e',s'}\\supseteq\\varphi_{e,s_0-1}$,\nmeans that if $\\varphi_{e,s_0-1}(z)=u$, then $\\varphi_{e',s'}(z)=u$.\nIn particular, if $\\varphi_{e,s_0-1}(z)$ is undefined, then $\\varphi_{e',s'}(z)$ can take any value or no value.\nIt is possible that $\\varphi_{e,s_0-1}$ is undefined (has a ``hole'') for $z$ but defined for some $z'>z$.} %\nNote that if $(e,y)\\in Q_1$, then \n$s_0=s_0(y)$ is defined and $\\delta'_{s_0}(y)\\in \\{0,1\\}$.  We can \neasily check that $Q_1$ is r.e. Given $(e,y)\\in Q_1$, let $s_1$ be the \nleast $s'\\geq s_0$ such that conditions (ii.a) and (ii.b) hold for \nsome $e'\\in\\mathbf{F}_{s'}$.  Let $e_0$ be the least $e'\\in\\mathbf{F}_{s'}$ such that \nconditions (ii.a) and (ii.b) hold for $s'=s_1$.  We can view $e_0$ as \na p.r.\\ function $e_0(e,y)$, which converges for $(e,y)\\in Q_1$.\n\nDefine a partial function~$\\psi$ by \n\\[\n\\psi(e,y,z) =\\left\\{ \n\t\\begin{array}{ll}\n\t\t\\varphi_{e_0}(z) & \\mbox{if $y\\in Q_0$ and $(e,y)\\in Q_1$},\\\\\n\t\t\\varphi_{e,s_0-1}(z) & \\mbox{if $y\\in Q_0$ and $(e,y)\\notin Q_1$},\\\\\n\t\t\\varphi_e(z) & \\mbox{if $y\\notin Q_0$}.\n\t\\end{array}\n\t\\right.\n\\]\n\n\\begin{lemma}\n$\\psi$ is p.r.\n\\end{lemma}\n\n\\begin{subproof}\nWe show there is a sequence of p.r.\\ functions~$\\psi^s$\nsuch that $\\psi=\\bigcup_s \\psi^s$.  We then apply the Graph Theorem to conclude $\\psi$ is p.r.\n\n\\medskip\n\nFor each $s\\in \\mathbb{N}$, define a recursive set $Q_0^s\\subseteq\\mathbb{N}$ by\n$y\\in Q_0^s$ iff there exists~$s'\\leq s$ such that\n$\\delta'_{s'}(y)=0$ or $\\delta'_{s'}(y)=1$.  We have $y\\in Q_0$ iff $y\\in Q_0^s$ for some~$s$.\nNote that if $y\\in Q_0^s$, then $s\\ge s_0=s_0(y)$.\n\nFor each $s\\in \\mathbb{N}$, define a recursive set $Q_1^s\\subseteq\\mathbb{N}\\times\\mathbb{N}$ by \n$(e,y)\\in Q_1^s$ iff (i)~$y\\in Q_0^s$ and \n(ii)~there exist $s'$ such that $s_0:=s_0(y)\\le s' \\le s$ and $e'\\in\\mathbf{F}_{s'}$ such \nthat (ii.a)~$\\delta'_{s'}(e')=1-\\delta'_{s_0}(y)$ and that \n(ii.b)~$\\varphi_{e',s'}$ is an extension of $\\varphi_{e,s_0-1}$.  \n(Conditions (ii.a) and~(ii.b) are the same as those in the definition of~$Q_1$.)\nWe have $(e,y)\\in Q_1$ iff $(e,y)\\in Q_1^s$ for some $s$.\nNote that if $(e,y)\\in Q_1^s$, then $s_0\\le s_1 \\le s$.\n\nFor each $s\\in \\mathbb{N}$, define a p.r.\\ function~$\\psi^s$ by \n\\[\n\\psi^s(e,y,z) =\\left\\{ \n\t\\begin{array}{ll}\n\t\t\\varphi_{e_0,s}(z) & \\mbox{if $y\\in Q_0^s$ and $(e,y)\\in Q_1^s$},\\\\\n\t\t\\varphi_{e,s_0-1}(z) & \\mbox{if $y\\in Q_0^s$ and $(e,y)\\notin Q_1^s$},\\\\\n\t\t\\varphi_{e,s}(z) & \\mbox{if $y\\notin Q_0^s$}.\n\t\\end{array}\n\t\\right.\n\\]\n\n\\emph{We claim that $\\bigcup_s \\psi^s$ is a partial function \n(i.e., $\\bigcup_s \\psi^s(e,y,z)$ does not take more than one value)\nand that $\\psi=\\bigcup_s \\psi^s$}:\n\n\\begin{itemize}\n\\item Suppose $y\\notin Q_0$.  Then $y\\notin Q_0^s$ for any~$s$.  So, for all $s$, $\\psi^s(e,y,z)=\\varphi_{e,s}(z)$.\nHence $\\bigcup_s \\psi^s(e,y,z)=\\varphi_{e}(z)=\\psi(e,y,z)$ as desired.\n\n\\item Suppose ($y\\in Q_0$ and) $(e,y)\\in Q_1$.  Then $s_0$, $s_1$, and $e_0$ are defined and $s_1\\ge s_0$.\nIf $s<s_0$, then since $y\\notin Q_0^s$, we have $\\psi^s(e,y,z)=\\varphi_{e,s}(z)$.\nIf $s_0\\le s<s_1$, then since $y\\in Q_0^s$ and $(e,y)\\notin Q_1^s$, we have \n$\\psi^s(e,y,z)=\\varphi_{e,s_0-1}(z)$.\nIf $s_1\\le s$, then since $y\\in Q_0^s$ and $(e,y)\\in Q_1^s$, we have \n$\\psi^s(e,y,z)=\\varphi_{e_0,s}(z)$.\nHence $\\bigcup_s \\psi^s(e,y,z)=\\varphi_{e,s_0-1}(z)\\cup (\\bigcup_{s\\ge s_1}\\varphi_{e_0,s}(z))$.\nThe definition of $s_1$ implies that when $s_1\\le s$, \n$\\varphi_{e,s_0-1} \\subseteq \\varphi_{e_0,s_1}\\subseteq \\varphi_{e_0,s}$.\nThus $\\bigcup_s \\psi^s(e,y,z)=\\bigcup_{s\\ge s_1}\\varphi_{e_0,s}(z)=\n\\varphi_{e_0}(z)=\\psi(e,y,z)$ as desired.\n\n\\item Suppose $y\\in Q_0$ and $(e,y)\\notin Q_1$.  Then $s_0$ is defined.\nIf $s<s_0$, then since $y\\notin Q_0^s$, we have $\\psi^s(e,y,z)=\\varphi_{e,s}(z)$.\nIf $s_0\\le s$, then since $y\\in Q_0^s$ and $(e,y)\\notin Q_1^s$, we have \n$\\psi^s(e,y,z)=\\varphi_{e,s_0-1}(z)$.\nHence $\\bigcup_s \\psi^s(e,y,z)=\\varphi_{e,s_0-1}(z)=\\psi(e,y,z)$ as desired.\n\\end{itemize}\n\nDefine a partial function~$\\hat{\\psi}$ by $\\hat{\\psi}(s,e,y,z)=\\psi^s(e,y,z)$.\nThen from the construction of $\\psi^s$, $\\hat{\\psi}$ is p.r.\\ by Church's Thesis.\nBy the Graph Theorem, the graph of~$\\hat{\\psi}$ is r.e.\n\n\\emph{We claim that $\\psi=\\bigcup_s \\psi^s$ is p.r.}  \nBy the Graph Theorem it suffices to show that its graph is r.e.  We have \n$(e,y,z,u)\\in \\psi  \\Longleftrightarrow  \\exists s\\ (e,y,z,u)\\in \\psi^s \n \\Longleftrightarrow \\exists s\\ (s,e,y,z,u)\\in \\hat{\\psi}$.\nSince the graph of~$\\hat{\\psi}$ is r.e., it follows that the graph of~$\\psi$ is r.e.\\end{subproof}\n\n\\bigskip\n\nSince $\\psi$ is p.r., there is a recursive function~$f(e,y)$ such that\n$\\varphi_{f(e,y)}(z)=\\psi(e,y,z)$ by the Parameter Theorem.\n\n\\medskip\n\n\\emph{Step~1.2.  Applying the Recursion Theorem to obtain $y(e)$.}\n\nSince $f(e,y)$ is recursive, by the Recursion Theorem with Parameters \\citep[p.~37]{soare87} \nthere is a recursive function~$y(e)$ such that $\\varphi_{y(e)}=\\varphi_{f(e,y(e))}$.  So, we have\n$\\varphi_{y(e)}(z)=\\psi(e,y(e),z)$.\n\n\\emph{We claim that $y=y(e)$ cannot meet the first case ($y\\in Q_0$ and $(e,y)\\in Q_1$)\nin the definition of $\\psi$}.  Suppose $y(e)\\in Q_0$ and $(e,y(e))\\in Q_1$.\nSince $\\varphi_{y(e)}(z)=\\psi(e,y(e),z)$, by the definition of~$\\psi$ we have\non the one hand $\\varphi_{y(e)}=\\varphi_{e_0}$.\nBy~(ii.a) of the definition of~$Q_1$ and by the definition of~$e_0$, we have\non the other hand $\\delta'(e_0)=1-\\delta'(y(e)) \\neq \\delta'(y(e))$.\nThis contradicts the fact that $\\delta'$ extends the $\\delta$-indicator~$\\delta_\\omega$.\n\nTherefore, we can express $\\varphi_{y(e)}(z)=\\psi(e,y(e),z)$ as follows:\\footnote\nRoughly speaking, the first case in the definition of~$\\psi$ corresponds to the existence of an\nindex $e'$ for a finite set extending $\\varphi_{e,s_0-1}$ such that \n$\\delta'(e')$ is different from $\\delta'(y(e))$.\nObserve that $y(e)$ is defined so that \nthis case (as just shown) as well as  the third case (Step 2.3 below) in the definition of~$\\psi$\nwill not occur in (\\ref{varphi-y}) if $e$ is a characteristic index.\nIt follows that if $e$ is a characteristic index, then \nall finite coalitions extending $\\varphi_{e,s_0-1}$ have the same winning\/losing status\n(the $\\delta'$-value of their indices is the same as $\\delta'(y(e))$).\nThus $\\varphi_{e,s_0-1}$ is like a determining string---except that it may fail to be a string because of ``holes'' as discussed in footnote~\\ref{note:extension}.}\n\\begin{equation} \\label{varphi-y}\n\\varphi_{y(e)}(z) =\\left\\{ \n\t\\begin{array}{ll}\n\t\t\\varphi_{e,s_0-1}(z) & \\mbox{if $y(e)\\in Q_0$ ($(e,y(e))\\notin Q_1$ implied)},\\\\\n\t\t\\varphi_e(z) & \\mbox{if $y(e)\\notin Q_0$}.\n\t\\end{array}\n\t\\right.\n\\end{equation}\n\n\\bigskip\n\n\\emph{Step~2  Defining $T_0$ and~$T_1$ and verifying the conditions.}\n\nFor $i\\in \\{0,1\\}$, let $T_i$ be the collection of all the strings~$\\tau$ of length~$k(e):=s_0-1$\n(where $s_0=s_0(y(e))$) that extends $\\varphi_{e,s_0-1}$ for all those~$e$ such that $\\delta'(y(e))=i$.\n(Note that $\\delta'(y(e))\\in \\{0,1\\}$ iff $y(e)\\in Q_0$.)\nWe show $T_0$ and $T_1$ satisfy the conditions.\n\n\n\\medskip\n\n\\emph{Step~2.1.  $T_0$ and $T_1$ are r.e.}\n\nThis is obvious since $s_0$, $\\delta'$, and $y$ are p.r.\n(In other words, for each $e$,  first find whether $\\delta'(y(e))\\in \\{0,1\\}$. If not, we do not enumerate\nany segment in $T_0$ or $T_1$.  If  $\\delta'(y(e))=i \\in \\{0,1\\}$, then we\nhave  corresponding strings whose length is effectively obtained. So we enumerate\nthem in $T_i$.  This procedure ensures that $T_0$ and $T_1$ are r.e.)\n\n\\medskip\n\n\\emph{Step~2.2.  $T_0$ and $T_1$ consist of determining strings.}\n\nWe show that $T_0$ consists of losing determining strings.  \nWe can show that $T_1$ consists of winning determining strings in a similar way.\n\nSuppose $\\delta'(y(e))=0$.  Then $y(e)\\in Q_0$.  Let $s_0=s_0(y(e))$ and $k(e)=s_0-1$.\nSince $(e,y(e))\\notin Q_1$ by (\\ref{varphi-y}), \nthere is no $e'\\in \\mathbf{F}$ such that $\\delta'(e')=1-\\delta'(y(e))=1$ and that \n$\\varphi_{e'}$ is an extension of $\\varphi_{e,s_0-1}$.  \n(Note that $\\delta'(e')\\in \\{0,1\\}$ if $e'\\in \\mathbf{F}$.)\nHence any finite coalition that extends $\\varphi_{e,s_0-1}$ is losing.\n\nTherefore,  all  strings~$\\tau$ in $T_0$\n(i.e., all the finite strings of length~$k(e)$ that extend $\\varphi_{e,s_0-1}$ \nfor some~$e$ such that $\\delta'(y(e))=0$) are losing determining for \\emph{finite} coalitions.\nBy Proposition~\\ref{cutprop}~(iii), all strings~$\\tau$ in~$T_0$ are losing determining strings.\n\n\\medskip\n\n\\emph{Step~2.3.  Any coalition has an initial segment in $T_0\\cup T_1$.}\n\nLet $S$ be a coalition, which is recursive.  Pick a characteristic index $e$  for~$S$.\nWe first show that $\\delta'(y(e))\\in \\{0,1\\}$ (i.e., $y(e)\\in Q_0$).\nSuppose $y(e)\\notin Q_0$.  By (\\ref{varphi-y}),\nwe have $\\varphi_{y(e)}=\\varphi_e$.  So $y(e)$~is a characteristic index for~$S$.\nHence  $\\delta'(y(e))\\in \\{0,1\\}$.  That is, $y(e)\\in Q_0$, which is a contradiction.\n\nBy the definitions of $T_0$ and $T_1$, since $\\delta'(y(e))\\in \\{0,1\\}$,\nall the strings of length $k(e)=s_0-1$ extending $\\varphi_{e,s_0-1}$\nare in $T_0\\cup T_1$.   In particular, since the $k(e)$-initial segment $S\\cap k(e)$ of \nthe characteristic function $\\varphi_e$ for~$S$ extends $\\varphi_{e,s_0-1}$,\nthe initial segment $S\\cap k(e)$ is in~$T_0\\cup T_1$.\\end{proof}\n\n\\section{Applications: Finite Carriers, Finite Winning Coalitions, Prefilters, and Nonanonymity}\n\\label{applications}\n\nTheorem~\\ref{delta0det} is a powerful theorem.\nWe can obtain as its corollaries some of the results in \\citet{mihara04mss}.\n\n\\subsection{Finite carriers}\n\nThe following proposition asserts that games that are essentially finite satisfy \n$\\delta$-computability, as might be expected.  \nWe give here a proof that uses the characterization theorem.\n\n\\begin{prop}[Mihara, 2004, Proposition~5] \\label{d0pos}\n\tSuppose that a simple game~$\\omega$ has a finite carrier.\n\tThen $\\omega$ is $\\delta$-computable.\n\\end{prop}\n\n\\begin{proof}\nSuppose that $\\omega$ has a finite carrier~$S$.\nLet $k=\\max S+1$ (we let $k=0$ if $S=\\emptyset$).\nLet \\[\nT_1=\\{\\tau: \\textrm{$\\tau$ is a string of length~$k$ and $\\tau\\in\\omega$} \\}\n\\]\n(where $\\tau\\in\\omega$ means that \nthe set $\\{i<k: \\tau(i)=1\\}$ represented by $\\tau$, viewed as a characteristic function,\nis in $\\omega$)\nand $T_0=\\{\\tau: \\textrm{$\\tau$ is a string of length~$k$ and $\\tau\\notin\\omega$} \\}$.\nWe verify the conditions of Theorem~\\ref{delta0det}.\n\nSince  $T_0$ and $T_1$ are finite, they are r.e.  \nSince $T_0\\cup T_1$ consists of all strings of length~$k$, any coalition has an initial segment in it.\n\nWe show that $T_1$ consists of winning determining strings.\n(We can show that $T_0$ consists of losing determining strings in a similar way.)\nSuppose $G\\cap k=\\tau\\in T_1$.  It suffices to show that $G\\in \\omega$.\nBy the definition of $T_1$, we have $\\tau\\in \\omega$.  \nThis implies $\\tau\\cap S\\in \\omega$ since $S$ is a carrier.\nSince $\\tau\\cap S=(G\\cap k)\\cap S=G\\cap(k\\cap S)=G\\cap S$, it follows that\n$G\\cap S\\in \\omega$.  Since $S$ is a carrier, we get $G\\in\\omega$.\\end{proof}\n\nIn particular, \\emph{if a simple game~$\\omega$ is dictatorial, then $\\omega$ \nis $\\delta$-computable}.  Indeed, the coalition \nconsisting of the dictator is a finite carrier for the dictatorial \ngame~$\\omega$.\n\n\n\\subsection{Finite winning coalitions}\\label{sec:finwin}\n\nNote in Proposition~\\ref{d0pos} that if a game has a finite carrier~$S$ and $N$ is winning,\nthen there exists a finite winning coalition, namely $S=N\\cap S$.  \nWhen there does not exist a finite \nwinning coalition, it is a corollary of the following negative result---itself \na corollary of Theorem~\\ref{delta0det}---that the computability condition is violated.\n\n\\begin{prop}[Mihara, 2004, Proposition~6]\\label{d0neg1}\n\tSuppose that a simple game~$\\omega$ has an infinite winning \n\tcoalition~$S\\in\\omega$ such that for each $k\\in\\mathbb{N}$, its $k$-initial \n\tsegment $S\\cap k$ is losing.  Then $\\omega$ is not $\\delta$-computable.\n\\end{prop}\n\n\\begin{proof}\nSuppose that  $\\omega$ is $\\delta$-computable.  Since $S$ is winning, \nby Lemma~\\ref{cutlemma} (or by Proposition~\\ref{cutprop} or by Theorem~\\ref{delta0det})\nthere is some $k\\in \\mathbb{N}$ such that $G=S\\cap k$ is winning.\nThis contradicts the assumption of the proposition.\\end{proof}\n\nTheorem~\\ref{delta0det} immediately gives the following extension of \nCorollary~7 of \\citet{mihara04mss}.  It gives a useful criterion for \nchecking computability of simple games.  \nHere, a \\emph{cofinite} set is the complement of a finite set.\n\n\\begin{prop}\\label{d0neg1cor}\n\tSuppose that a $\\delta$-computable simple game has a winning coalition.\n\tThen, it has both finite winning coalitions and cofinite winning coalitions.\n\\end{prop}\n\nWe also prove a result that is close to Proposition~\\ref{d0neg1}.\n\n\\begin{prop}[Mihara, 2004, Proposition~8] \\label{d0neg2} \nSuppose that $\\emptyset\\notin\\omega$.  Suppose that the simple \ngame~$\\omega$ has an infinite coalition~$S\\in\\omega$ such that for \neach $k\\in\\mathbb{N}$, its difference $S\\setminus k=\\{\\,s\\in S: s\\geq k\\,\\}$ from the \ninitial segment is winning.  Then $\\omega$ is not  $\\delta$-computable.\n\\end{prop}\n\n\\begin{proof}\nSuppose $\\omega$ is  $\\delta$-computable.\nSince $\\emptyset\\notin\\omega$, there is a losing determining string\n$\\tau=00\\cdots0$ of length~$k$ by Theorem~\\ref{delta0det}.  \nBy assumption, $S\\setminus k$ is winning.\nBut $(S\\setminus k)\\cap k=\\tau$ and that $\\tau$ is a losing determining string imply that\n$S\\setminus k$ is losing, which is a contradiction.\\end{proof}\n\nAgain, the following proposition gives a useful criterion for \nchecking computability of simple games.\n\n\\begin{prop}\\label{d0neg2cor} \n\tSuppose that a $\\delta$-computable simple game has a losing coalition.\n\tThen, it has both finite losing coalitions and cofinite losing coalitions.\n\\end{prop}\n\n\\subsection{Prefilters, filters, and ultrafilters}\n\nFrom the propositions in Section~\\ref{sec:finwin}, examples of a noncomputable simple game are easy to come by.\n\n\\begin{example}\\label{q-compl}\nFor any $q$ satisfying $0< q \\leq \\infty$, let $\\omega$ be the \\textbf{$q$-complement rule} defined as follows:  \n$S\\in \\omega$ if and only if $\\#(N\\setminus S)< q$.  For example, if $q=1$, then the $q$-complement rule is the unanimous game, consisting of $N$ alone.  If $q=\\infty$, then the game consists of cofinite coalitions \n(the complements of finite coalitions).  Proposition~\\ref{d0neg1cor}\nimplies that \\emph{$q$-complement rules are not $\\delta$-computable}, since they have no finite winning coalitions.\nAny $q$-complement rule is a prefilter and it is a monotonic, proper, nonstrong, and anonymous simple game.  \nIf $q>1$, it is nonweak, but any finite intersection of winning coalitions is nonempty\n (i.e., has an infinite Nakamura number, to be defined in Section~\\ref{core}).\nNote that if $1<q<\\infty$, then the $q$-complement rule is not a filter since it is not closed with respect to finite intersection.\\end{example}\n\nExample~\\ref{q-compl} gives examples of a prefilter that is not a filter.  \nIt also gives two examples of a filter that is not an ultrafilter:\nthe unanimous game is a principal filter and \nthe game consisting of all cofinite coalitions is a nonprincipal filter.\n\\citet{mihara01scw} gives a constructive example of an ultrafilter.\n\nSome prefilters are computable, but that is true only if they have a veto player:\naccording to Proposition~\\ref{filter-nakamura} below, if a prefilter\nis $\\delta$-computable, then it is weak.\n \nIf $\\omega$ is a filter, then it is $\\delta$-computable\nif and only if it is has a finite carrier~\\citep[Corollary~11]{mihara04mss}.\nIn particular, the principal filter $\\omega=\\{T\\in \\mathrm{REC}: S\\subseteq T\\}$\n generated by $S$ has a carrier, namely, $S$.\nSo, if $S$ is finite, the principal filter is computable.  \nIf $S$ is infinite, it is noncomputable, since it does not have a finite winning coalition.\nFor example, the principal filter generated by $S=2\\mathbb{N}:=\\{0, 2, 4, \\ldots \\}$ is a monotonic,\nproper, nonstrong, weak, and noncomputable simple game.\n\nIf $\\omega$ is a nonprincipal ultrafilter, it  is not $\\delta$-computable by Proposition~\\ref{d0neg1cor}, \nsince it has no finite winning coalitions (or no cofinite losing coalitions).\nIt is a monotonic, proper, strong, and nonweak noncomputable simple game.\nIn fact, an ultrafilter is $\\delta$-computable if and only if it is dictatorial~\\citep[Lemma~4]{mihara97et}.\n\n\n\\subsection{Nonanonymity}\n\nAs an application of the characterization result in Section~\\ref{mainresult},  we show that \n$\\delta$-computable simple games violate finite anonymity, a weak notion of anonymity.\nProposition~\\ref{nonfinanonymous} below strengthens \nan earlier result \\citep[Corollary~12]{mihara04mss} about computable games.\nThe latter result assumes proper, monotonic, $\\delta$-computable simple games, \ninstead of just $\\delta$-computable simple games.\n\n\\begin{prop}\\label{nonfinanonymous}\nSuppose that $N\\in\\omega$ and $\\emptyset\\notin\\omega$. \nIf the simple game $\\omega$ is $\\delta$-computable, then it is not finitely anonymous.\n\\end{prop}\n\n\\begin{proof}\nLet $\\omega$ be a  finitely anonymous $\\delta$-computable simple game such that $N\\in\\omega$ and $\\emptyset\\notin\\omega$. \nSince $N\\in\\omega$, there is an initial segment $k:=\\{0, 1, \\ldots, k-1\\}=11\\cdots1$ (string of 1's of length $k$),\nby Lemma~\\ref{cutlemma} (or by Proposition~\\ref{cutprop} or by Theorem~\\ref{delta0det}).\nSince $\\emptyset\\notin\\omega$, there is a losing determining string \n$\\tau=00\\cdots0$ of length~$k'$ by Theorem~\\ref{delta0det}.\nThen the concatenation $\\tau*k=00\\cdots011\\cdots1$ of $\\tau$ and $k$, viewed as a set,\n has the same number of elements as $k$.\nSince coalitions $\\tau*k$ and $k$ are finite and have the same number of elements, they should be treated equally\nby the finitely anonymous $\\omega$.  But $\\tau*k$  is losing and $k$ is winning.\\end{proof}\n\n\n\\section{The Number of Alternatives and the Core}\\label{core}\n\nIn this section, we apply Theorem~\\ref{delta0det} to a social choice problem.\nWe show (Corollary~\\ref{core-nakamura}) \nthat computability of a simple game entails a restriction on the number of alternatives\nthat the set of players (with the coalition structure described by the simple game) can deal with rationally.\n\nFor that purpose, we define the notion of a \\emph{simple game with (ordinal) preferences},\na combination of a simple game and a set of alternatives and individual preferences.\nAfter defining the \\emph{core} for simple games with preferences,\nwe extend (Theorem~\\ref{nakamura-thm}) Nakamura's theorem (\\citeyear{nakamura79})\n about the nonemptyness of the core:\nthe core of a simple game with preferences is always (i.e., for all profiles of preferences) nonempty \nif and only if the number of alternatives is finite and below a certain critical number, \ncalled the \\emph{Nakamura number} of the simple game.\nWe need to make this extension since what we call a ``simple game'' is not generally \n what is called a ``simple game'' in \\citet{nakamura79}.\n\nWe show (Corollary~\\ref{nakamura-finite}) that the Nakamura number of a nonweak simple game is finite \nif it is computable, though (Proposition~\\ref{nakamura-any}) there is no upper bound for the set\n of the Nakamura numbers of such games.\\footnote\n\\citet{kumabe-m07nc} study the relations between the Nakamura number and \ncomputable simple games having various properties.}\nIt follows from Theorem~\\ref{nakamura-thm} that (Corollary~\\ref{core-nakamura}) in order for a set of alternatives to always have a maximal element given a nonweak, computable game,\nthe number of alternatives must be restricted.\nIn contrast, \\emph{some} noncomputable \n(and nonweak) simple games do not have such a restriction (Proposition~\\ref{filter-nakamura}),\nand in fact have some nice properties.\nThese results have implications for social choice theory; \nwe suggest its connection with the study of Arrow's Theorem (\\citeyear{arrow63}).\n\n\\subsection{Framework}\n\nLet $N'$ be an arbitrary nonempty set of players and \n$\\mathcal{B}\\subseteq 2^{N'}$ an arbitrary Boolean algebra of  subsets (called ``coalitions'' in this section) of~$N'$. \nA $\\mathcal{B}$-\\textbf{simple game} $\\omega$ is a subcollection of~$\\mathcal{B}$ such that $\\emptyset\\notin \\omega$.\nThe elements of $\\omega$ are said to be winning, and the other elements in~$\\mathcal{B}$ \nare losing, as before.\nOur ``simple game'' is a $\\mathcal{B}$-simple game with $N'=\\mathbb{N}$ and $\\mathcal{B}=\\mathrm{REC}$, if it does not contain~$\\emptyset$.\nNakamura's ``simple game'' (\\citeyear{nakamura79}) is one with $\\mathcal{B}=2^{N'}$.\nThe properties (such as monotonicity and weakness, defined in Section~\\ref{notions}) for simple games are \nredefined for $\\mathcal{B}$-simple games in an obvious way.\n\nLet $X$ be a (finite or infinite) set of \\textbf{alternatives}, with cardinal number $\\#X\\geq 2$.\nLet $\\mathcal{A}$ be the set of \\textbf{(strict) preferences}, i.e., \n\\emph{acyclic} (for any finite set $\\{x_1, x_2, \\ldots, x_m\\}\\subseteq X$,\nif $x_1 \\succ x_2$, \\ldots, $x_{m-1} \\succ x_m$, then $x_m \\not\\succ x_1$;\nin particular, $\\succ$ is asymmetric and irreflexive) binary relations~$\\succ$ on $X$. \n(If $\\succ$ is acyclic, we can show that the relation~$\\succeq$, defined by \n$x\\succeq y \\Leftrightarrow y\\not\\succ x$, is complete, i.e., reflexive and total.)\nA \\textbf{($\\mathcal{B}$-measurable) profile} is a list \n$\\mathbf{p}=\\mbox{$(\\pref)_{i\\in N'}$} \\in\\mathcal{A}^{N'}$ of {individual preferences} $\\succ_i^\\p$\nsuch that $\\{\\,i\\in N':{x\\pref y}\\,\\} \\in\\mathcal{B}$\nfor all~$x$, $y\\in X$.  Denote by $\\A^{N'}_\\B$ the set of all profiles. \n\nA \\textbf{$\\mathcal{B}$-simple game with (ordinal) preferences} is a list $(\\omega, X, \\mathbf{p})$ of\na $\\mathcal{B}$-simple game~$\\omega\\subseteq\\mathcal{B}$, a set $X$ of alternatives, and\na profile $\\mathbf{p}=\\mbox{$(\\pref)_{i\\in N'}$}\\in \\A^{N'}_\\B$.\nGiven the $\\mathcal{B}$-simple game with preferences, \nwe define the dominance relation~$\\succ^\\mathbf{p}_\\omega$ by $x\\succ^\\mathbf{p}_\\omega y$ if and only if there is a winning coalition\n$S\\in\\omega$ such that $x\\succ_i^\\p y$ for all $i\\in S$.\\footnote\nIn this definition, $\\{\\,i\\in N':{x\\pref y}\\,\\}$ need not be winning since we do not assume $\\omega$ is monotonic.\n\\citet{andjiga-m00} study Nakamura's theorem,\nadopting the notion of dominance that requires the above coalition to be winning.} %\nThe \\textbf{core} $C(\\omega, X, \\mathbf{p})$ of the $\\mathcal{B}$-simple game with preferences is the set of undominated alternatives:\n\\[\nC(\\omega, X, \\mathbf{p})=\\{x\\in X: \\textrm{$\\not\\!\\exists y\\in X$ such that $y\\succ^\\mathbf{p}_\\omega x$}\\}.\n\\]\n\nA \\textbf{(preference) aggregation rule}  is\na map $\\succ \\colon \\mathbf{p}\\mapsto \\, \\succ^\\mathbf{p}$ from profiles~$\\mathbf{p}$ of preferences \nto binary relations (social preferences)~$\\succ^\\mathbf{p}$ \non the set of alternatives.  For example, the mapping $\\succ_\\omega$ \nfrom profiles $\\mathbf{p}\\in\\A^{N'}_\\B$ of acyclic preferences to dominance relations $\\succ^\\mathbf{p}_\\omega$ \nis an aggregation rule.\nWe typically restrict individual and social preferences to those binary relations $\\succ$ on $X$\nthat are asymmetric (i.e., complete $\\succeq$) and \neither (i)~acyclic or (ii)~transitive (i.e., quasi-transitive $\\succeq$) or\n(iii)~negatively transitive (i.e., transitive $\\succeq$).\nAn aggregation rule is often referred to as a \\emph{social welfare function} when individual preferences\nand social preferences are restricted to the asymmetric, negatively transitive relations.\n\n\\subsection{Nakamura's theorem and its consequences}\n\n\n\\citet{nakamura79} gives a necessary condition for a $2^{N'}$-simple game with preferences\nto have a nonempty core for any profile~$\\mathbf{p}$, which is also sufficient if the set $X$ of alternatives is finite.  \nTo state Nakamura's theorem, we define the \\textbf{Nakamura number} $\\nu(\\omega)$ of a $\\mathcal{B}$-simple game~$\\omega$ to be the size of the smallest collection of winning coalitions having empty intersection\n\\[\n\\nu(\\omega)=\\min \\{\\#\\omega': \\textrm{$\\omega' \\subseteq \\omega $ and $\\bigcap\\omega'=\\emptyset$} \\}\n\\]\nif $\\bigcap\\omega=\\emptyset$ (i.e., $\\omega$ is nonweak); otherwise, set $\\nu(\\omega)=\\#(2^X)>\\#X$.\n\nThe following useful lemma \\citep[Lemma~2.1]{nakamura79} states that the Nakamura number of\na $\\mathcal{B}$-simple game cannot exceed the size of a winning coalition by more than one.\n\n\\begin{lemma}\\label{nakamura-ceiling}\nLet $\\omega$ be a nonweak $\\mathcal{B}$-simple game.  \nThen $\\nu(\\omega)\\leq \\min \\{\\#S: S\\in\\omega\\}+1$.\n\\end{lemma}\n\n\\begin{proof}\nChoose a coalition $S\\in \\omega$ such that $\\#S=\\min \\{\\#S: S\\in\\omega\\}$.\nSince $\\bigcap\\omega=\\emptyset$, for each $i\\in S$, there is some $S^i\\in \\omega$ with $i\\notin S^i$.\nSo, $S\\cap (\\bigcap_{i\\in S} S^i)=\\emptyset$.  Therefore, $\\nu(\\omega)\\leq \\#S+1$.\\end{proof}\n\nIt is easy to prove \\citep[Corollary~2.2]{nakamura79} that the Nakamura number of a nonweak $\\mathcal{B}$-simple game is at \nmost equal to the cardinal number~$\\#N$ of the set of players and that this maximum is attainable if $\\mathcal{B}$ \ncontains all finite coalitions.\nIn fact, one can easily construct a computable, nonweak simple game with any given Nakamura number:\n\n\\begin{prop}\\label{nakamura-any}\nFor any integer $k\\geq 2$,  there exists a $\\delta$-computable, nonweak simple game~$\\omega$\nwith Nakamura number $\\nu(\\omega)=k$.\n\\end{prop}\n\n\\begin{proof}\nGiven an integer $k\\geq 2$, \nlet $S=\\{0,1, \\ldots, k-1\\}$ be a carrier and define $T\\in \\omega$ iff $\\#(S\\cap T) \\geq k-1$.  \nThen $\\nu(\\omega)=k$.\\end{proof}\n\nSince computable, nonweak simple games have winning coalitions, \nit has \\emph{finite} winning coalitions by Proposition~\\ref{d0neg1cor}.\nAn immediate corollary of Lemma~\\ref{nakamura-ceiling} is the following:\n\\begin{cor}\\label{nakamura-finite}\nLet $\\omega$ be a $\\delta$-computable, nonweak simple game.  \nThen its Nakamura number~$\\nu(\\omega)$ is finite.\n\\end{cor}\n\n\\citet{nakamura79} proves the following theorem for $\\mathcal{B}=2^{N'}$:\n\n\\begin{theorem} \\label{nakamura-thm}\nLet $\\mathcal{B}$ be a Boolean algebra of sets of $N'$.\nSuppose that $\\emptyset \\notin \\omega$ and $\\omega\\neq \\emptyset$.\nThen the core $C(\\omega, X, \\mathbf{p})$ of a $\\mathcal{B}$-simple game $(\\omega, X, \\mathbf{p})$ with preferences is nonempty for all (measurable) profiles $\\mathbf{p}\\in \\A^{N'}_\\B$\nif and only if $X$ is finite and $\\#X<\\nu(\\omega)$.\\end{theorem}\n\n\\begin{remark}\nAt first glance, Nakamura's proof \\citep[Theorem~2.3]{nakamura79} of the necessary condition~$\\#X<\\nu(\\omega)$,\ndoes not appear to generalize to an arbitrary Boolean algebra~$\\mathcal{B}$:\nhe constructs certain coalitions from winning coalitions by taking possibly\n\\emph{infinite unions and intersections}, as well as complements;\na difficulty is that the resulting set of players may not belong to the Boolean algebra~$\\mathcal{B}$.\nHowever, it turns out that once we make use of the other necessary condition \n(disregarded by Nakamura) that $X$~is finite, \nwe only need to consider \\emph{finite} unions and intersections, and his proof actually works.\nSince accessible proofs are readily available in the literature \\citep[e.g.,][Theorem~3.2]{austensmith-b99}\nfor $\\mathcal{B}=2^{N'}$ and finite sets~$N'$ of players, we choose to relegate the proof to\nthe working paper \\citep[Appendix~A.3]{kumabe-m07csgcc}. \nUnlike others', our proof treats the \\emph{measurability} condition ($\\mathbf{p}\\in \\A^{N'}_\\B$) particularly carefully.\\end{remark}\n\nIt follows from Theorem~\\ref{nakamura-thm} that if a $\\mathcal{B}$-simple game $\\omega$ is \\emph{weak} \n(and satisfies $\\emptyset \\notin \\omega$ and $\\omega\\neq \\emptyset$), \nthen the core $C(\\omega, X, \\mathbf{p})$ is nonempty for all profiles $\\mathbf{p}\\in \\A^{N'}_\\B$ \nif and only if $X$~is finite.\nThe more interesting case is where $\\omega$ is nonweak.\nCombined with Corollary~\\ref{nakamura-finite}, Theorem~\\ref{nakamura-thm} has a consequence\nfor nonweak, computable simple games:\n\n\\begin{cor}\\label{core-nakamura}\nLet $\\omega$ be a $\\delta$-computable, nonweak simple game satisfying\n$\\emptyset \\notin \\omega$.\nThen there exists a finite number $\\nu$ (the Nakamura number~$\\nu(\\omega)$) such that\nthe core $C(\\omega, X, \\mathbf{p})$ is nonempty for all profiles $\\mathbf{p}\\in \\mathcal{A}^{N}_\\mathrm{REC}$\nif and only if $\\#X<\\nu$.\\end{cor}\n\nIf we drop the computability condition, the above conclusion no longer holds.\nAn example of $\\omega$ that has no such restriction on the size of the set~$X$ of alternatives\nis a nonweak prefilter (e.g., the $q$-complement rule of Example~\\ref{q-compl}, for $q>1$),\nwhich has an infinite Nakamura number.\n\nIn fact, we can say more, if we shift our attention from the core---the set of undominated \nalternatives with respect to the dominance relation~$\\succ^\\mathbf{p}_\\omega$---to the \ndominance relation itself. \n(The proof of the following proposition is in the working paper \\citep[Appendix~A.4]{kumabe-m07csgcc}.)\n\n\\begin{prop}\\label{filter-nakamura}\nLet $\\omega$ be a nonweak simple game\nsatisfying $\\emptyset \\notin \\omega$.\n\\textup{(i)}~$\\omega$ cannot be a $\\delta$-computable prefilter.\n\\textup{(ii)}~If $\\omega$ is $\\delta$-computable; then $\\nu(\\omega)$ is finite, and\n$\\succ^\\mathbf{p}_\\omega$  is acyclic for all $\\mathbf{p}\\in\\mathcal{A}^N_\\mathrm{REC}$ if and only if $\\#X<\\nu(\\omega)$.\n\\textup{(iii)}~If $\\omega$ is a prefilter,\nthen $\\succ^\\mathbf{p}_\\omega$  is acyclic for all $\\mathbf{p}\\in\\mathcal{A}^N_\\mathrm{REC}$, regardless \nof the cardinal number $\\#X$ of~$X$.\\end{prop}\n\n\n\nWe can strengthen the acyclicity of the dominance relation~$\\succ^\\mathbf{p}_\\omega$ in statement~(iii)\nof Proposition~\\ref{filter-nakamura} by replacing the statement with one of the following:\n(iv)~if $\\omega$ is a \\emph{filter},\nthen $\\succ^\\mathbf{p}_\\omega$  is transitive for all $\\mathbf{p}$ such that all individuals have transitive preferences~$\\succ_i^\\p$;\n(v)~if $\\omega$ is an \\emph{ultrafilter},\nthen $\\succ^\\mathbf{p}_\\omega$  is asymmetric and negatively transitive for all $\\mathbf{p}$ \nsuch that all individuals have asymmetric, negatively transitive preferences~$\\succ_i^\\p$.\nIn fact, statements (iii), (iv), and (v) each gives an aggregation rule $\\succ_\\omega\\colon \\mathbf{p}\\mapsto \\, \\succ^\\mathbf{p}$ that satisfies Arrow's conditions of ``Unanimity'' and ``Independence of irrelevant alternatives.''\nThese results are immediate from the relevant definitions \n(\\citet[Proposition~3.2]{armstrong80} gives a proof).\nAccording to Arrow's Theorem (\\citeyear{arrow63}), however, if the set $N$ of players were \nreplaced by a \\emph{finite} set, then social welfare functions given by statement~(v) \nwould be dictatorial (and $\\omega$ would be weak).\n\nIn an attempt to escape from Arrow's impossibility, many authors have investigated the consequences of\nrelaxing the rationality requirement (negative transitivity of $\\succ^\\mathbf{p}_\\omega$)\nfor social preferences.\nIn view of the close connection \\citep[Theorems 2.6 and 2.7]{austensmith-b99} \nbetween the rationality properties of an aggregation rule and preflters \n\\citep[also][]{kirman-s72,armstrong80,armstrong85},\nProposition~\\ref{filter-nakamura} has a significant implication for this investigation.\n\n\n\\section{Examples}\\label{examples}\n\nPropositions~\\ref{d0pos}, \\ref{d0neg1cor}, and \\ref{d0neg2cor} show that the class of computable games\n(i)~includes the class of games that have finite carriers and \n(ii)~is included in the class of games that have both finite winning coalitions and cofinite losing coalitions.\nIn this section, we construct examples showing that these inclusions are strict.\n\nWe can find such examples without sacrificing the voting-theoretically desirable properties of simple games.\nWe pursue this task thoroughly in a companion paper~\\citep{kumabe-m06csg64}.\nThe \\emph{noncomputable} simple game example in Section \\ref{ex:noncomp} \nthat has both finite winning coalitions and cofinite losing coalitions is a sample of that work.\nIt is monotonic, proper, strong, and nonweak.\nExamples of a \\emph{computable} simple game that is monotonic, proper, strong, nonweak, \nand has no finite carrier is given in \\citet{kumabe-m06csg64,kumabe-m07nc}.\n\n\n\\subsection{A noncomputable game with finite winning coalitions} \n\\label{ex:noncomp}\n\nWe exhibit here a noncomputable simple game\nthat is monotonic, proper, strong, nonweak, \nand have both finite winning coalitions and cofinite losing coalitions.\nIt shows in particular that the class of computable games is strictly smaller than the class \n of games that have both finite winning coalitions and cofinite losing coalitions.\nIn this respect, the game is unlike nonweak prefilters\n(such as the $q$-complement rules in Example~\\ref{q-compl}); \nthose examples do not have any finite winning coalitions.\nFurthermore, unlike nonprincipal ultrafilters---which are also \nmonotonic, proper, strong, and nonweak noncomputable simple games---the \ngame is nonweak in a stronger sense: it violates the finite intersection property.\n\n\nLet $A=N\\setminus \\{0\\}=\\{1,2,3,\\ldots\\}$.  We define the simple game~$\\omega$ as follows:\nAny coalition except $A^c=\\{0\\}$ extending the string 1 of length~1 (i.e., any coalition containing~0)  is winning; \nany coalition except $A$ extending the string 0  is losing; $A$ is winning and $A^c$ is losing.\nIn other words, for all $S\\in \\mathrm{REC}$,\n\\[ \nS\\in \\omega \\iff [\\textrm{$S=A$ or ($0\\in S$ \\& $S\\neq A^c$)}].\n\\]\n\n\\begin{remark}\nThe reader familiar with the notion of repeated games (or binary rooted trees) may find the following visualization helpful.\nThink of the extensive form of an infinitely repeated game played by you, \nwith the stage game consisting of two moves 0 and~1.  \nIf you choose~1 in the first stage, you will win unless you keep choosing 0 indefinitely thereafter; \nif you choose~0 in the first stage, you will lose unless you keep choosing 1 indefinitely thereafter.\nNow, you ``represent'' a certain coalition and play 1 in stage $i$ if $i$ is in the coalition; \nyou play~0 in that stage otherwise.\nThen the coalition that you represent is winning if you win; it is losing if you lose.\n\\end{remark}\n\n\\begin{lemma}\n$\\omega$ is not $\\delta$-computable.\n\\end{lemma}\n\nThe following proof demonstrates the power of Theorem~\\ref{delta0det},\nalthough its full force is not used  (Proposition~\\ref{cutprop} suffices).\nProposition~\\ref{d0neg1}, which appeared earlier in \\citet{mihara04mss}, does not have this power.\n\n\\bigskip\n\n\\begin{proof}\nIf $\\omega$ is $\\delta$-computable, then by Theorem~\\ref{delta0det} (or by Proposition~\\ref{cutprop}),\n$A$ has an initial segment $A\\cap k$ that is a winning determining string.\nBut $A\\cap k$ itself is not winning (though it extends the string trivially).\n\\end{proof}\n\n\\begin{lemma}\n$\\omega$ has both finite winning coalitions and cofinite losing coalitions.\n\\end{lemma}\n\n\\begin{proof}\nFor instance, $\\{0,1\\}$ is finite and winning.\n$N\\setminus \\{0,1\\}=\\{2, 3, 4, \\ldots\\}$ is cofinite and losing.\n\\end{proof}\n\n\\begin{lemma}\n$\\omega$ is monotonic.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $S\\in \\omega$ and $S\\subsetneq T$.  There are two possibilities.\nIf $S=A$, then $T=N$, and we have $N\\in \\omega$ by the definition of $\\omega$.\nOtherwise, $S$ contains 0 and some other number $i$.  The same is true of $T$, \nimplying that $T\\in \\omega$.\n\\end{proof}\n\n\\begin{lemma}\n$\\omega$ is proper and strong.\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to show that $S^c\\in\\omega \\iff S\\notin\\omega$.\nFrom the definition of $\\omega$, we have\n\\begin{eqnarray*}\nS \\notin\\omega & \\iff & \\textrm{$S\\neq A$ \\& ($0\\notin S$ or $S=A^c$)} \\\\\n\t\t\t\t\t\t& \\iff & \\textrm{$S^c \\neq A^c$ \\& ($0\\in S^c$ or $S^c= A$)} \\\\\n\t\t\t\t\t\t& \\iff & \\textrm{($0\\in S^c$  \\& $S^c \\neq A^c$) or $S^c= A$} \\\\\n\t\t\t\t\t\t& \\iff & S^c\\in\\omega.\n\\end{eqnarray*}\\end{proof}\n\n\\begin{lemma}\n$\\omega$ is not a prefilter.  In particular,  it is not weak.  \n\\end{lemma}\n\n\\begin{proof}\nWe show that the intersection of some finite family of winning coalitions is empty.\nThe coalitions $\\{0,1\\}$, $\\{0,2\\}$, and~$A$ form such a family.\n(Incidentally, this shows that the Nakamura number of $\\omega$ is three, since $\\omega$ is proper.)\\end{proof}\n \n\n\n\\subsection{A computable game without a finite carrier}\\label{ex:nocarrier}\n\n\nWe exhibit here a computable simple game that does not have a finite carrier. \nIt shows that the class of computable games is strictly larger than \nthe class of games that have finite carriers.\n\nOur approach is to construct r.e.\\ (in fact, recursive) sets $T_0$ and $T_1$ \nof determining strings (of 0's and 1's) satisfying the conditions of Theorem~\\ref{delta0det} \n(the full force of the theorem is not needed; the easier direction suffices).\nWe first give a condition that any string in $T_0\\cup T_1$ must satisfy.  \nWe then specify each of $T_0$ and $T_1$, and construct the simple game by means of these sets.\nWe conclude that the game is computable by checking (Lemmas~\\ref{ex:nocarrier-rec}, \\ref{ex:nocarrier-string1},\nand \\ref{ex:nocarrier-det}) \nthat $T_0$ and $T_1$ satisfy the conditions of the theorem.\nFinally, we show (Lemma~\\ref{ex:nocarrier-none}) that the game does not have a finite carrier.\n\n\\bigskip\n\nLet $\\{k_s\\}_{s=0}^\\infty$ be an effective listing (recursive enumeration) of the members of \nthe r.e.\\ set $\\{k : \\varphi_k(k)\\in \\{0,1\\}\\}$, \nwhere $\\varphi_k(\\cdot)$ is the $k$th p.r.\\ function of one variable.\nWe can assume that all elements $k_s$ are distinct.\n(Such a listing $\\{k_s\\}_{s=0}^\\infty$ exists by the Listing Theorem~\\citep[Theorem~II.1.8 and Exercise II.1.20]{soare87}.)\nThus, \n\\[ \\mathrm{CRec} \\subset \\{k : \\varphi_k(k)\\in \\{0,1\\}\\} = \\{k_0, k_1, k_2, \\ldots\\}, \\]\nwhere $\\mathrm{CRec}$ is the set of characteristic indices for recursive sets.\n\nLet $l_{0}=k_0+1$, and for $s>0$, let $l_{s}=\\max \\{l_{s-1}, k_{s}+1\\}$.\nWe have $l_s\\geq l_{s-1}$ (that is, $\\{l_s\\}$ is an nondecreasing sequence of numbers) and \n$l_s>k_s$ for each $s$.  Note also that $l_s\\geq l_{s-1}>k_{s-1}$, and $l_s\\geq l_{s-2}>k_{s-2}$, etc.\\ \nimply that $l_s> k_s$, $k_{s-1}$, $k_{s-2}$, \\ldots.\n\nFor each $s$, let $F_s$ be the set of strings $\\alpha=\\alpha(0)\\alpha(1)\\cdots\\alpha(l_s-1)$ \n(the *'s denoting the concatenation are omitted)\nof length $l_s$ such that \n\\begin{equation} \\label{ex:nocarrier1}\n\\textrm{$\\alpha(k_{s})=\\varphi_{k_{s}}(k_{s})$ and for each $s'<s$,  $\\alpha(k_{s'})=1-\\varphi_{k_{s'}}(k_{s'})$.}\n\\end{equation}\nNote that (\\ref{ex:nocarrier1}) imposes no constraints on  $\\alpha(k_{s'})$ for $s'>s$ and \nno constraints on $\\alpha(k)$ for $k\\notin\\{k_0,k_1,k_2, \\ldots\\}$, \nwhile it imposes real constraints for $s'\\leq s$,\nsince $|\\alpha|=l_s> k_{s'}$ for such $s'$.\nWe observe that if $\\alpha\\in F_s\\cap F_{s'}$, then $s=s'$.\n \nLet $F=\\bigcup_{s}F_s$.  ($F$ will be the union of $T_0$ and $T_1$ defined below.)\nWe claim that for any two distinct elements $\\alpha$ and $\\beta$ in $F$\nwe have neither $\\alpha\\subseteq \\beta$ ($\\alpha$ is an initial segment of~$\\beta$)\nnor $\\beta\\subseteq\\alpha$ \n(i.e., there is $k< \\min\\{|\\alpha|,|\\beta|\\}$ such that $\\alpha(k)\\neq \\beta(k)$).\n To see this, let $|\\alpha|\\leq |\\beta|$, without loss of generality.  \n If $\\alpha$ and $\\beta$ have the same length, then the \n conclusion follows since otherwise they become identical strings.\n If $l_s=|\\alpha|< |\\beta|=l_{s'}$, then $s<s'$ and by (\\ref{ex:nocarrier1}),\n $\\alpha(k_{s})=\\varphi_{k_{s}}(k_{s})$ on the one hand, but \n$\\beta(k_{s})=1-\\varphi_{k_{s}}(k_{s})$ on the other hand.  So $\\alpha(k_{s})\\neq \\beta(k_{s})$.\n\nThe game $\\omega$ will be constructed from the sets $T_0$ and $T_1$ of strings defined as follows:\n\\begin{eqnarray*}\n\\alpha\\in T_0 & \\iff & \\exists s \\, \\textrm{[$\\alpha\\in F_s$ and $\\alpha(k_{s})=\\varphi_{k_{s}}(k_{s})=0$]} \\\\\n\\alpha\\in T_1 & \\iff & \\exists s \\, \\textrm{[$\\alpha\\in F_s$ and $\\alpha(k_{s})=\\varphi_{k_{s}}(k_{s})=1$].} \n\\end{eqnarray*}\nWe observe that $T_0\\cup T_1=F$ and $T_0\\cap T_1=\\emptyset$.\n\n\\emph{Define $\\omega$ by $S\\in \\omega$ if and only if $S$ has an initial segment in $T_1$}.\nLemmas~~\\ref{ex:nocarrier-rec}, \\ref{ex:nocarrier-string1}\nand \\ref{ex:nocarrier-det} establish computability of $\\omega$\nby way of Theorem~\\ref{delta0det}.\n\n\\begin{lemma} \\label{ex:nocarrier-rec}\n$T_0$ and $T_1$ are recursive.\n\\end{lemma}\n\n\\begin{proof}\nWe prove that $T_0$ is recursive; the proof for $T_1$ is similar.\nWe give an algorithm that can decide for each given string whether it is in $T_0$ or not.\n\nTo decide whether a string $\\sigma$ is in $T_0$, generate $k_0$, $k_1$, $k_2$, \\ldots,\ncompute $l_0$, $l_1$, $l_2$, \\ldots, and determine $F_0$, $F_1$, $F_2$, \\ldots\nuntil we find the least $s$ such that $l_s\\geq |\\sigma|$.\nIf $l_s > |\\sigma|$, then $\\sigma\\notin F_s$.\nSince $l_s$ is nondecreasing in $s$ and $F_s$ consists of strings of length~$l_s$, \nit follows that $\\sigma\\notin F$, implying $\\sigma\\notin T_0$.\n\nIf $l_s= |\\sigma|$, then check whether $\\sigma \\in F_s$; this can be \ndone since the values of $\\varphi_{k_{s'}}(k_{s'})$ for $s'\\leq s$ in (\\ref{ex:nocarrier1})\nare available and $F_s$ determined by time $s$.\nIf $\\sigma\\notin F_s$ and $l_{s+1}>l_s$, then $\\sigma \\notin T_0$ as before.\nOtherwise check whether $\\sigma \\in F_{s+1}$.  \nIf $\\sigma \\notin F_{s+1}$ and $l_{s+2}>l_{s+1}=l_s$, then $\\sigma \\notin T_0$ as before.\nRepeating this process, we either get $\\sigma\\in F_{s'}$ for some $s'$ or \n$\\sigma\\notin F_{s'}$ for all $s'\\in \\{s': l_{s'}=l_s\\}$.\nIn the latter case, we have $\\sigma \\notin T_0$.\nIn the former case, if $\\sigma(k_{s'})=\\varphi_{k_{s'}}(k_{s'})=1$, \nthen $\\sigma \\in T_1$ by the definition of $T_1$; hence it is not in $T_0$.\nOtherwise $\\sigma(k_{s'})=\\varphi_{k_{s'}}(k_{s'})=0$, \nand we have $\\sigma \\in T_0$.\\end{proof}\n\n\\begin{lemma}\\label{ex:nocarrier-string1}\n$T_1$ consists only of winning determining strings for $\\omega$;\n$T_0$ consists only of losing determining strings for $\\omega$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\alpha\\in T_1$.  If a coalition~$S$ extends $\\alpha$, then by the definition of~$\\omega$,\n$S$~is winning.  This proves that $\\alpha$~is a winning determining string.\n\nLet $\\alpha\\in T_0$.  Suppose a coalition~$S$ extends $\\alpha\\in T_0\\subset T_0\\cup T_1=F$.\nIf $\\beta\\in F$ and $\\beta\\neq \\alpha$, \nwe have, as shown before, $\\alpha \\not \\subseteq \\beta$ and $\\beta\\not \\subseteq\\alpha$,\nwhich implies that $S$~does not extend~$\\beta$.  So, in particular, $S$ does not extend any\nstring in~$T_1$.  It follows from the definition of~$\\omega$ that $S$~is losing.\nThis proves that $\\alpha$~is a losing determining string.\\end{proof}\n\n\\begin{lemma} \\label{ex:nocarrier-string}\nFor each $s$, any string $\\alpha$ of length $l_s$ such that $\\alpha(k_s)=\\varphi_{k_s}(k_s)$ \nextends a string in $\\bigcup_{t\\leq s}F_t$.\n\\end{lemma}\n\n\\begin{proof}\nWe proceed by induction on $s$.\nLet $\\alpha$ be a string of length $l_s$ such that $\\alpha(k_s)=\\varphi_{k_s}(k_s)$. \nIf $s=0$, we have $\\alpha\\in F_0$; hence the lemma holds for $s=0$. \nSuppose the lemma holds for $s'<s$. If for some $s'<s$, $\\alpha(k_{s'})=\\varphi_{k_{s'}}(k_{s'})$,\nthen by the induction hypothesis, the $l_{s'}$-initial segment $\\alpha \\cap l_{s'}$ of $\\alpha$\nextends a string in $\\bigcup_{t\\leq s'}F_t$. So $\\alpha$  extends a string in $\\bigcup_{t\\leq s}F_t$.  \nOtherwise,  we have for each $s'<s$,  $\\alpha(k_{s'})=1-\\varphi_{k_{s'}}(k_{s'})$. \nThen by (\\ref{ex:nocarrier1}), $\\alpha\\in F_s\\subset \\bigcup_{t\\leq s}F_t$.\\end{proof}\n\n\\begin{lemma} \\label{ex:nocarrier-det}\nAny coalition $S\\in\\mathrm{REC}$ has an initial segment in $T_0$ or $T_1$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $\\varphi_k$ is the characteristic function for a recursive coalition~$S$. \nThen $k\\in\\{k_0,k_1,k_2, \\ldots\\}$ since this set contains the set $\\mathrm{CRec}$ of characteristic indices.\nSo $k=k_s$ for some $s$.  Consider the initial segment $S\\cap l_s$.  It extends a string in\n$\\bigcup_{t\\leq s}F_t$ by Lemma~\\ref{ex:nocarrier-string}.  The conclusion follows \nsince $\\bigcup_{t\\leq s}F_t \\subset F=T_0\\cup T_1$.\\end{proof}\n\n\\begin{lemma} \\label{ex:nocarrier-none}\n$\\omega$ does not have a finite carrier.\n\\end{lemma}\n\n\\begin{proof}\nWe will construct a set~$A$ such that for infinitely many $l$,\nthe $l$-initial segment $A\\cap l$ has an extension (as a string) that is winning \nand for infinitely many $l'$, $A\\cap l'$ has an extension that is losing.  \nThis implies that $A\\cap l$ is not a carrier of~$\\omega$ for any such~$l$.\nSo no subset of $A\\cap l$ is a carrier.  Since there are arbitrarily large such~$l$, \nthis proves that $\\omega$ has no finite carrier.\n\nLet $A$ be a set such that for each $k_t$, $A(k_t)=1-\\varphi_{k_t}(k_t)$. \nFor any $s'>0$ and $i \\in \\{0,1\\}$, there is an $s>s'$ such that $k_s> l_{s'}$ and $\\varphi_{k_s}(k_s)=i$.\n\nFor a temporarily chosen $s'$, fix $i$ and fix such $s$.  Then choose the greatest $s'$ satisfying these conditions.\nSince $l_s>k_s>l_{s'}$, there is a string $\\alpha$ of length~$l_s$ extending (as a string) $A\\cap l_{s'}$ \nsuch that $\\alpha\\in F_s$.  \nSince $\\alpha(k_s)=\\varphi_{k_s}(k_s)=i$, we have $\\alpha\\in T_i$.  \n\nThere are infinitely many such $s$, so there are infinitely many such $s'$. \nIt follows that for infinitely many $l_{s'}$, the initial segment $A\\cap l_{s'}$ is a substring \nof some  string~$\\alpha$ in $T_1$ (by Lemma~\\ref{ex:nocarrier-string1}, $\\alpha$ is winning in this case), \nand for infinitely many $l_{s'}$, $A\\cap l_{s'}$ is a substring of some (losing) string~$\\alpha$ in $T_0$.\\end{proof}\n\n\\section*{Acknowledgements}\n\nWe would like to thank an anonymous referee for useful suggestions.  %\nThis is an outcome of a  long-term collaboration started in 1998.  \nIt could not have been produced without a lack of grants for shorter-term projects.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSuppose $X$ is a compact oriented manifold acted on by a compact\nconnected Lie group $K$ of dimension $s$; one may\nthen define the equivariant\ncohomology $\\hk(X)$.\nThroughout this paper we shall consider only cohomology\nwith complex coefficients.\nIf $X$ is a symplectic manifold\nwith symplectic form $\\om$ and the action of $K$ is Hamiltonian\n(in other words, there is a moment map $\\mu: X \\to \\lieks$), then we\nmay form the symplectic quotient $\\xred = \\mu^{-1}(0)\/K$. The restriction\nmap $i_0: X \\to \\mu^{-1}(0)$ gives a ring homomorphism\n$i_0^*: \\hk(X) \\to \\hk (\\mu^{-1}(0) ) $. Using Morse theory and the gradient\nflow of the function $|\\mu|^2: X \\to {\\Bbb R }$, it is proved\nin  \\cite{Ki1}\nthat the map\n$i_0^*$ is surjective.\n\nSuppose in addition that $0$ is a {\\em regular value} of the\nmoment map $\\mu$. This assumption is equivalent to the assumption\nthat the stabilizer $K_x$ of $x$ under\n the action of $K$ on\n$X$ is {\\em finite } for every $x \\in \\mu^{-1} (0)$, and\nit implies that $\\xred$ is an orbifold, or $V$-manifold, which inherits\na symplectic form $\\omega_0$ from the symplectic form\n$\\omega$ on $X$. In this situation there is a canonical isomorphism\n$\\pi_0^*: H^*(\\mu^{-1}(0)\/K) \\to \\hk (\\zloc)$.\\footnote{This\nisomorphism is induced by the map $\\pi_0: \\zloc \\times_K EK \\to \\zloc\/K$.\nRecall that we are only considering cohomology with complex coefficients.}\nHence we have a surjective ring homomorphism\n\\begin{equation} \\label{0.1}\n\\kappa_0 = (\\pi_0^*)^{-1} \\circ i_0^*: \\hk(X) \\to\nH^*(\\xred). \\end{equation}\n\n\nHenceforth, if $\\eta \\in \\hk(X)$ we shall denote $\\kappa_0(\\eta)$ by\n$\\eta_0$.\nPrevious work \\cite{Brion,Ki2} on determining the ring structure\nof $\\hk(X)$ has presented methods which in some situations\npermit the direct determination of the kernel of the\nmap $\\kappa_0$, and hence  of generators and relations\nin $H^*(\\xred)$ in terms of generators\nand relations in $\\hk(X)$.  (Note that  the generators of $\\hk(X)$\ngive generators of $H^*(\\xred)$ via the surjective  map $\\kappa_0$, and also that\ngenerators of $H^*(BK)$ together with extensions to\n$\\hk(X)$ of generators of $H^*(X)$ give generators of\n$\\hk(X)$ because the spectral sequence\nof the fibration\n$X \\times_K EK \\to BK$ degenerates \\cite{Ki1}.)\n Here we present\nan alternative approach to determining the ring structure of\n$H^*(\\xred)$ when $0$ is a regular value of $\\mu$, which complements\nthe results obtained by directly studying the kernel\nof $\\kappa_0$. Our approach is based on the observation that since\n$H^*(\\xred) $ satisfies Poincar\\'e duality, a class\n$\\eta \\in \\hk(X)$ is in the kernel of $\\kappa_0$ if and only if for all\n$\\zeta \\in \\hk(X)$ we have\n\\begin{equation} \\eta_0  \\zeta_0 [\\xred]\n= (\\eta \\zeta)_0 [\\xred] = 0 . \\end{equation}\nHence to determine the kernel of $\\kappa_0$ (in other words the\nrelations in the ring $H^*(\\xred)$) given\nthe ring structure of $\\hk(X)$, it suffices to know\nthe {\\em intersection pairings}, in other words the evaluations\non the fundamental class $[\\xred]$ of all possible classes $\n\\xi_0 = \\kappa_0 (\\xi)$.\nIn principle the intersection pairings thus determine generators\nand relations for the cohomology ring $H^*(\\xred)$, given\ngenerators and relations for $\\hk(X)$.\n\n\n There is a natural\npushforward map $\\pisk: \\hk(X) \\to \\hk $ $ = \\hk({\\rm pt} )$\n$ \\cong S(\\lieks)^K$, where we have identified\n$\\hk$ with the space of $K$-invariant polynomials on the\nLie algebra $\\liek$. This map can be thought of as integration\nover $X$ and will sometimes be denoted by\n$\\int_X$.\nIf $T$ is a compact {\\em abelian} group\n(i.e. a torus) and $\\zeta \\in \\hht(X)$,\nthere is a formula\\footnote{Atiyah and Bott\n\\cite{abmm} give\na cohomological proof of this formula, which was first proved by\nBerline and Vergne \\cite{BV1}.} (the {\\em abelian localization\ntheorem})\nfor $\\pist \\zeta$ in terms of\nthe restriction of $\\zeta$ to the components of the\nfixed point set for the action of $T$. In particular, for a general\ncompact Lie group $K$ with maximal torus $T$ there\nis a canonical map\n$\\tau_X: \\hk(X) \\to \\hht(X)$, and we may apply the\nabelian localization theorem to $\\tau_X (\\zeta)$ where\n$\\zeta \\in \\hk(X)$.\n\n\n\n\n\n\n\n\n\n\nIn terms of the components $F$ of the fixed point set\nof $T$ on $X$, we obtain a\nformula (the residue formula, Theorem \\ref{t8.1})\nfor the evaluation\n of a class  $\\eta_0 \\in H^*(\\xred) $ on the\nfundamental class $[\\xred]$, when $\\eta_0$ comes\nfrom a class $\\eta \\in \\hk(X)$. There are two main\ningredients in the proof\nof Theorem \\ref{t8.1}. One is the abelian localization\ntheorem \\cite{abmm,BV1},  while\nthe other is an equivariant  normal\nform for $\\om$  in a neighbourhood of $\\zloc$,\ngiven in \\cite{STP}\nas a consequence of the coisotropic embedding theorem.\nThe result is the following:\n\n\\noindent{\\bf Theorem 8.1} {\\em\nLet $\\eta \\in \\hk(X)$ induce $\\eta_0 \\in H^*(\\xred)$.\nThen we have\n$$\n\\eta_0 e^{i\\omega_0}  [\\xred]  =\n \\frac{(-1)^{n_+} }{(2 \\pi)^{s-l}  |W|  \\,{\\rm vol}\\, (T) }\n\\treso\n\\Biggl (     \\: \\nusym^2 (\\psi)\n \\sum_{F \\in {\\mbox{$\\cal F$}}} e^{i \\mu_T(F) (\\psi) }\n\\int_F \\frac{i_F^* (\\eta(\\psi) e^{i \\omega} )  }{e_F(\\mar  \\psi) }\n[d \\psi] \\Biggr ). $$\nIn this formula,  $n_+$ is the number of\npositive roots of $K$,  and\n$\\nusym(\\psi) = \\prod_{\\gamma > 0} \\gamma(\\psi)$ is the product of the\n positive roots, while\n${\\mbox{$\\cal F$}}$  is the set of components of the fixed point\nset of the maximal torus $T$ on $X$. If $F \\in\n{\\mbox{$\\cal F$}}$ then $i_F$ is the inclusion of $F$ in $ X$ and\n$e_F$ is the equivariant Euler class of the normal\nbundle to $F$ in $X$.}\n\n\\noindent Here, via the {\\em Cartan model},\nthe class $\\tau_X ( \\eta )\\in \\hht(X)$\nhas been identified with a family of differential\nforms $\\eta (\\psi)$ on $X$\nparametrized by $\\psi \\in \\liet$. The definition   of the\nresidue map $\\treso$ (whose domain is a suitable class of\nmeromorphic differential forms on\n$\\liet \\otimes {\\Bbb C }$)\n will be given in Section\n8 (Definition \\ref{d8.5n}). It is a linear map, but\nin order to apply it to the individual terms\nin the statement of Theorem 8.1 some\nchoices must be made. The choices do not affect the\nresidue of the whole sum. When $\\liet$ has dimension one the\nformula becomes\n\n\n\n\n\n$$\n\\eta_0 e^{i\\omega_0}  [\\xred]  =\n -\\frac{1}{2}\n{\\rm Res}_0\n\\Biggl ( \\psi^2      \\:\n \\sum_{F \\in {\\mbox{$\\cal F$}}_+} e^{i \\mu_T(F) (\\psi) }\n\\int_F \\frac{i_F^* (\\eta(\\psi) e^{i \\omega} )  }{e_F(\\mar  \\psi) }\n \\Biggr ), $$\nwhere ${\\rm Res}_0$ denotes the coefficient of\n$1\/\\psi$, and ${\\mbox{$\\cal F$}}_+$ is the subset of the\nfixed point set of $T = U(1)$ consisting of those\ncomponents $F$ of the $T$ fixed point set for which\n$\\mu_T(F) > 0 $.\n\n\nWe note that if $\\dim \\eta_0 = \\dim \\xred$ then the\nleft hand side of the equation in Theorem \\ref{t8.1}\nis just $\\eta_0 [\\xred]$. More generally one may obtain\na formula for $\\eta_0[\\xred]$ by replacing the\nsymplectic form $\\omega$ by $\\delta \\omega$\n(where $\\delta > 0 $ is a small parameter), and taking\nthe limit as $\\delta \\to 0$. This  has the\neffect of replacing the moment map $\\mu$ by $\\delta \\mu$.\nIn the limit $\\delta \\to 0$, the Residue Formula (Theorem \\ref{t8.1})\nbecomes a sum of terms corresponding to the components $F$ of the\nfixed point set, where the term corresponding to $F$ is (up to a constant)\nthe residue (in the sense of Section 8) of\n$\\nusym^2(\\psi) \\int_F {i_F^* (\\eta(\\psi) )  }\/{e_F(\\mar  \\psi) } $, and\nthe only role played by the symplectic form and the moment\nmap is in determining which\n $F$\ngive a nonzero\ncontribution to the residue of the sum and the signs with which\nindividual terms enter.\n\n\n\n\n\n\n\n\nResults for the case when $K = S^1$, which are\nrelated to our Theorem \\ref{t8.1}, may\nbe found in the papers of Kalkman \\cite{Kalk} and\nWu \\cite{wu}.\n\n\nWitten in Section 2 of \\cite{tdg} gives a related result, the\n{\\em nonabelian localization theorem}, which also interprets\nevaluations $\\eta_0[\\xred]$ of classes on the\nfundamental class $[\\xred]$ in terms of appropriate\ndata on $X$.  For $\\epsilon > 0$ and $\\zeta \\in \\hk(X)$,\nhe defines\\footnote{The normalization of the measure in\n$\\ie(\\zeta)$ will be described at the beginning of\nSection 3. As above, $\\pis: \\hk(X) \\to \\hk \\cong S(\\lieks)^K$ is the\nnatural pushforward map, where $S(\\lieks)^K$ is the space of\n$K$-invariant polynomials on $\\liek$. }\n\\begin{equation} \\label{1.1}\n\\ie(\\zeta) = \\frac{1}{(2 \\pi i)^s  \\,{\\rm vol}\\,  K}\n\\intk [d \\phi] e^{- \\epsilon \\inpr{\\phi, \\phi}\/2 }\n\\pisk \\zeta (\\phi) \\end{equation}\n(where $\\inpr{\\cdot,\\cdot} $ is a fixed invariant inner\nproduct on $\\liek$, which we shall use throughout to identify\n$\\lieks$ with $\\liek$)\n and expresses it as a sum\nof local contributions.\n\n\n\n\nWitten's theorem tells us  that just as $\\pisk \\zeta$ would have\ncontributions  from\nthe components of the fixed point set of $K$ if $K $ were\nabelian, the quantity $\\ie(\\zeta)$ (if $K$ is not necessarily\nabelian) reduces to a sum of integrals\nlocalized around the critical set of the function\n$\\rho = |\\mu|^2,$\ni.e. the set of points $x$ where $(d |\\mu|^2)_x = 0$.\n(Of course $d |\\mu|^2 = 2\\inpr{\\mu, d \\mu}$, so the\nfixed point set of the $K$ action, where $d \\mu = 0$,\nis a subset of the critical set of $d|\\mu|^2$.)\nMore precisely the critical set of $\\rho = |\\mu|^2$\ncan be expressed as a disjoint union of closed subsets\n$C_\\beta$ of $X$ indexed by a finite subset\n${\\mbox{$\\cal B$}}$ of the Lie algebra $\\liet$ of the maximal\ntorus $T$ of $K$ which is explicitly\nknown in terms of the moment map\n$\\mu_T$ for the action of $T$ on $X$ \\cite{Ki1}. If\n$\\beta \\in {\\mbox{$\\cal B$}}$ then the critical subset\n$C_\\beta$ is of the form $C_\\beta = K (Z_\\beta \\cap\n\\mu^{-1}(\\beta))$ where $Z_\\beta$ is a union of\nconnected components of the fixed point set of the\nsubtorus of $T$ generated by $\\beta$. The subset\n$\\mu^{-1}(0)$ on which $\\rho = |\\mu|^2$ takes its\nminimum value is $C_0$.\nThere is a natural map\\footnote{This map\nis induced by the projection ${\\rm pr}:\\zloc \\to {\\rm pt}.$}\n${\\rm pr}^*: \\hk \\to \\hk(\\zloc)$\n  so that the distinguished  class $f(\\phi) = -\\inpr{\\phi, \\phi}\/2$\nin $H^4_K$\ngives rise to a distinguished class\n$\\Theta \\in H^4(\\zloc\/K) \\cong H^4_K(\\zloc).$\nWitten's result can then be expressed in the form\n\\begin{theorem} \\label{t1.1}\n$$ \\ie(\\zeta) = \\zeta_0e^{\\epsilon \\Theta}  [\\xred]  + \\sum_{\\beta \\in\n{\\mbox{$\\cal B$}} - \\{0\\} }  \\int_{U_\\beta}\n{\\zeta'}_\\beta. $$\nHere,\nthe $U_\\beta$ are\nopen neighbourhoods in $X$ of\n the nonminimal critical subsets $C_\\beta$ of the function\n$\\rho$.\nThe $\\zeb$ are\ncertain differential forms on $U_\\beta$ obtained from $\\zeta$.\n\\end{theorem}\n\nIn the special case $\\zeta = \\eta \\exp \\iins \\bar{\\om} $ (where $\\bar{\\om}(\\phi)\n = \\om + \\evab{\\phi}{ \\mu} $ is the standard extension of the symplectic\nform $\\om$\nto an element of $H^2_K(X)$, and\n$\\eta$ has polynomial dependence\\footnote{The equivariant\ncohomology $\\hk(X) $ is defined to consist of classes which have\npolynomial dependence on the generators of $\\hk$, but we shall\nalso make use of formal classes such as $\\exp  \\iins \\bom$ which\nare formal power series in these generators.}\non the generators of $\\hk$),\n Witten's results give us   the following\n estimate\non the growth of the terms $\\int_{U_\\beta} \\zeb$\nas $\\epsilon \\to 0$:\n\\begin{theorem} \\label{t1.2}\nSuppose $\\zeta = \\eta \\exp i \\bar{\\om}$ for some $\\eta \\in \\hk(X)$. If\n$\\beta \\in {\\mbox{$\\cal B$}} - \\{0\\}$ then\n$\\int_{U_\\beta} \\zeb = e^{- \\rho_\\beta\/{2 \\epsilon} } \\:\nh_\\beta(\\epsilon)$, where $\\rho_\\beta = |\\beta|^2$\n is the value of $|\\mu|^2$\non the critical set $C_\\beta$ and $|h_\\beta(\\epsilon)|$ is bounded by a polynomial\nin $\\epsilon^{-1}$.\n\\end{theorem}\n\nThus one should think of $\\epsilon > 0$ as a small parameter,\nand one may use the asymptotics of the integral $\\ie$\nover $X$ to calculate the intersection pairings\n$\\eta_0 e^{\\epsilon \\Theta} e^{i\\om_0} [\\xred],$ since the  terms in\nTheorem \\ref{t1.2} corresponding to the other critical subsets\nof $\\rho$ vanish exponentially fast as $\\epsilon \\to 0$.\nNotice that when $\\zeta = \\exp i\\bom$, the vanishing of\n$\\mu$ on $\\zloc$ means that $\\zeta_0 = \\exp i\\om_0$, where\n$\\omega_0$ is the symplectic form induced\nby $\\omega$ on $\\xred = \\mu^{-1}(0)\/K$.\n\n\n\n\n\nIn this paper we shall give a proof of a variant of\nTheorems \\ref{t1.1} and \\ref{t1.2}, for the case\n$\\zeta = \\eta \\exp i \\bom  $ where $\\eta \\in \\hk(X)$.\n Before outlining our proof, it will\nbe useful to briefly recall Witten's argument.\nWitten introduces a $K$-invariant 1-form $\\lambda$ on\n$X$, and shows that $\\ie(\\zeta) = \\ie(\\zeta \\exp s D \\lambda)$,\nwhere $D $ is the differential in equivariant\ncohomology and $s \\in {\\Bbb R }^+$. He then does the\nintegral over $\\phi \\in \\liek$ and shows that in the limit\nas $s \\to \\infty$,\nthis\nintegral vanishes  over any region of\n$X$  where $\\lambda(\\va)  \\ne 0$ for at least one of the   vector fields\n$\\va, j = 1, \\dots, s$ given by the infinitesimal action of a basis of\n$\\liek$ on $X$ indexed by $j$.\n Thus, after integrating\nover $\\phi \\in \\liek$, the limit as $s \\to \\infty$\nof $\\ie(\\zeta)$ reduces to a sum of contributions\nfrom sets where $\\lambda(\\va) = 0$ for all the $\\va$.\n\nIn our case, when $X$ is a symplectic manifold and the action\nof $K$ is Hamiltonian, Witten chooses $\\lambda(Y) = d|\\mu|^2(JY)$,\nwhere $J$ is a $K$-invariant almost complex structure on\n$X$. Thus $\\lambda(\\va)(x) = 0$ for all $j$\nif and only if $(d|\\mu|^2)_x = 0$,\nso  $\\ie(\\zeta)$ reduces to a sum of contributions from the critical sets\nof $\\rho = |\\mu|^2$.\nFurther, he obtains the contribution from $ \\zloc$\nas $e^{\\epsilon \\Theta} \\zeta_0 [\\xred] $. If $\\zeta = \\eta e^{\\iins \\bom}  $\nhe also obtains the estimates in Theorem \\ref{t1.2}\non the contributions from the neighbourhoods $U_\\beta$.\n\nIn general,\nthe contributions to the localization theorem depend\n on the choice of $\\lambda$. In the symplectic\ncase, with $\\lambda = J d |\\mu|^2$, the contribution\nfrom $\\mu^{-1}(0)$ is canonical but the contributions\nfrom the other critical sets $C_\\beta$ depend in principle\non the choice of $J$. Further, the properties of these\nother terms are difficult to study. Ideally they\nshould reduce to integrals over the critical sets\n$C_\\beta$, and indeed when proving  Theorem \\ref{t1.2} Witten\nmakes   the assumption (before (2.52))  that the $C_\\beta$\nare nondegenerate critical manifolds in the sense of Bott\n\\cite{bnd}.\nIn general the $C_\\beta$ are not manifolds; and\neven when  they are manifolds, they are not necessarily nondegenerate. They\n satisfy only a weaker condition called  {\\em minimal\ndegeneracy} \\cite{Ki1}.\\footnote{However, minimal degeneracy\nmay be sufficient for Witten's argument.} We shall treat the\nintegrals over neighbourhoods of the $C_\\beta$ in a future paper.\n\n\nIn the case when $X$ is a symplectic manifold and\n$\\zeta = \\eta \\exp \\bom$ for any $\\eta \\in\n\\hk(X)$,\nwe have been able to use our methods\nto  prove a variant\nof Theorems \\ref{t1.1} and \\ref{t1.2}\n(see Theorems \\ref{t4.1}, \\ref{t4.3} and \\ref{t7.1} below)\nwhich\nbypasses these analytical difficulties and reduces the result\nto fairly well known results on Hamiltonian group actions on\nsymplectic manifolds. We assume that $0$ is\na regular value of $\\mu$, or equivalently\nthat $K$ acts on $\\mu^{-1}(0)$ with finite\nstabilizers.\\footnote{Witten assumes that $K$ acts\nfreely on $\\mu^{-1}(0)$.}  By treating the pushforward\n$\\pisk \\zeta$ as a function on $\\liek$, we\nmay use the  abelian\nlocalization  formula \\cite{abmm,BV1} for the\npushforward  in equivariant cohomology of\ntorus actions to find an explicit expression for\n$\\pisk \\zeta$ as a function on $\\liek$.\nThus, analytical problems relating to\nintegrals  over neighbourhoods of $C_\\beta$ are circumvented,\nand localization reduces to studying the image of the moment\nmap and the pushforward of the symplectic or Liouville\nmeasure under the moment map.\\footnote{If $K$ is\nabelian, this pushforward measure  is equal  (at $\\phi \\in \\liek$)\nto Lebesgue measure multiplied by  a function which gives\nthe symplectic volume of the reduced space\n$\\mu^{-1}(\\phi)\/K$; this function is sometimes called the\n Duistermaat-Heckman polynomial \\cite{DH}.}\nSeen in this light, the nonabelian localization\ntheorem is a consequence of the same results that\nunderlie the residue formula:\nthe  abelian localization formula for torus\n actions \\cite{abmm,BV1} and the normal\nform for $\\om$ in a neighbourhood of $\\zloc$.\n\n\nWe now summarize the key steps in our proof. Having replaced\nintegrals over $X$ by integrals over $\\liek$,\nwe observe that in turn these may be replaced by integrals over the\nLie algebra\n$\\liet$ of the maximal torus. Then, applying properties of\nthe Fourier transform, we rewrite $\\ie$ as the\nintegral over\n$\\liets$ of a Gaussian $\\gtsoe(y) $ $\\sim e^{- |y|^2\/(2 \\epsilon) } $\n multiplied by a\nfunction $Q = D_\\nusym R$ where $R$ is piecewise\npolynomial and $D_\\nusym$ is a differential operator on $\\liets$:\n\\begin{equation} \\label{0.2}\\ie =i^{-s} \\tintt \\widetilde{\\gsoe} (y) Q(y),  \\end{equation}\nwhere $s$ is the dimension of $K$.\n The function $Q$ is obtained by combining the\nabelian localization theorem (Theorem \\ref{t2.1})\nwith a result\n(Proposition \\ref{p3.5}) on Fourier transforms of a certain class of\nfunctions which arise in the\n formula for the pushforward.\n\nThe\nfunction $Q$ is smooth in a neighbourhood of the origin when\n$0$ is a regular value of $\\mu$:\nthus there is a polynomial $Q_0\n = D_\\nusym R_0 $ which is equal to $Q$ near\n$0$. It turns out that the cohomological expression\n$\\eeth[\\xred]$ is obtained as the integral over $\\liets$\nof a Gaussian multiplied not\nby $Q$ but by the polynomial $Q_0$:\n\\begin{equation} \\eeth [\\xred] = i^{-s}  \\tintt \\widetilde{\\gsoe} (y) Q_0(y). \\end{equation}\nThis result follows from a normal form for $\\om$ near\n$\\zloc$.\\footnote{This\nnormal form is a key tool in the original proof \\cite{DH} of\nthe Duistermaat-Heckman theorem; this theorem\nmotivated  the proof by Atiyah and Bott  \\cite{abmm} of the\nabelian localization theorem.}\n\nTo obtain our analogue of Witten's  estimate (Theorem\n\\ref{t1.2}) for the asymptotics of\n$\\ie - \\eeth [\\xred]$ as $\\epsilon \\to 0$,\nwe then write\n\\begin{equation}  \\ie - \\eeth [\\xred] =\ni^{-s} \\tintt \\widetilde{\\gsoe}(y)\nD_\\nusym (R - R_0) (y). \\end{equation}\nHere, $R - R_0$ is piecewise polynomial and\nsupported {\\em away} from $0$.\nBy studying\nthe minimum distances from $0$\nin the support of $R - R_0$\nwe obtain an estimate (Theorem \\ref{t4.1}) similar to\nWitten's estimate (Theorem \\ref{t1.2}). In our estimate,\nthe terms in the sum are indexed by\nthe set ${\\mbox{$\\cal B$}} - \\{0\\}$; however, our estimate is\nweaker than Witten's estimate since some of the subsets\n$C_\\beta$ indexed by\n$\\beta \\in {\\mbox{$\\cal B$}} - \\{0\\}$ (which a priori\ncontribute to our sum\\footnote{In a future  paper\nwe hope to prove that the nonzero contributions\nto our estimate (Theorem \\ref{t4.1}) come only\nfrom those $|\\beta|^2$ which are nonzero critical values\nof $|\\mu|^2$.}\n)\nmay be empty in which case $\\rho_\\beta\n= |\\beta|^2$ may not be a critical value\nof $|\\mu|^2$.\n\n\n\nTo summarize, the following related quantities appear in this paper:\n\\begin{enumerate}\n\\item The cohomological quantity $\\eta_0 e^{i \\om_0}\n[\\xred]$.\n\\item The integral ${\\mbox{$\\cal I$}}^\\epsilon $ (\\ref{1.1}) coming from\nthe pushforward of an equivariant cohomology class\n$\\eta \\in \\hk(X)$  to $\\hk$.\n\\item Sums of terms of the form\n$$\\int_F \\frac{i_F^* \\eta}{e_F}$$\nwhere\n$F$ is a connected component of\nthe  fixed point set  of the maximal torus $T$ acting\non $X$.\nSuch sums appear after mapping $\\eta \\in\n\\hk(X)$ into $\\hht(X) $ and then applying the abelian\nlocalization theorem.\n\\end{enumerate}\nWitten's work relates (1) and (2), while our\nTheorem \\ref{t8.1} relates (1) and (3).\n\n\n\n\n\n\n\n\n\nThis paper is organized as follows. Section 2 contains\nbackground material on equivariant cohomology\nand the  abelian localization formula.\nIn Section 3 we collect a number of preliminary results\nwhich we use in Section 4 to reduce our integral\n$\\ie$ to an integral over $\\liets$ of a piecewise\npolynomial function multiplied by a Gaussian.\nSection 4 also contains the statement of two of\nour main results,\nTheorems \\ref{t4.1} and \\ref{t4.3}; Theorem\n\\ref{t4.3} is proved in Section 5, and Theorem \\ref{t4.1}\nin Section 6. In Section 7, Theorems \\ref{t4.1}\nand \\ref{t4.3} (which are for the case $\\zeta = \\exp \\bom$)\nare extended to the case $\\zeta = \\eta \\exp \\bom $ for\n$\\eta \\in \\hk(X)$: the result is\nTheorem \\ref{t7.1}.  Finally, in Section 8 we\nprove the  residue\n formula (Theorem \\ref{t8.1}) for  the evaluation of cohomology\nclasses from $\\hk(X)$ on the fundamental class of $\\xred$, and\nin Section 9 we\napply it when $K = SU(2)$ to specific examples.\nThis formula may be related to an unpublished\nformula due to Donaldson.\n\nIn  future papers we shall treat the case when\n$\\xred$ is singular using intersection homology; we shall also\napply the nonabelian localization formula to moduli spaces of\nbundles over Riemann surfaces regarded as finite\ndimensional symplectic quotients, in singular as\nwell as nonsingular cases.\n\n\\noindent{\\em Acknowledgement:}\nWe are most grateful to H. Duistermaat and M. Vergne for\nhelpful suggestions and  careful readings of the paper.\n\nIn addition, one\n of us (L.C.J.) wishes\nto thank E. Lerman and E. Prato for explaining their work,\nand also to thank E. Witten for discussions about\nthe nonabelian localization formula while the paper\n\\cite{tdg} was being written.\n\n\n\n\n\n{ \\setcounter{equation}{0} }\n\\section{Equivariant cohomology and pushforwards}\n\nIn this section we recall the localization formula for torus\nactions (Theorem\n\\ref{t2.1}) and express it in a form convenient\nfor our later use (Lemma \\ref{l2.2}).\n\nLet $X$ be a compact manifold equipped with the action of\na compact Lie group $K$ of dimension $s$ with\nmaximal torus $T$ of dimension $l$. We denote the Lie algebras\nof $K$ and $T$ by $\\liek$ and $\\liet$ respectively, and\nthe Weyl group by $W$.\nWe assume an  invariant inner product\n$\\inpr{\\cdot, \\cdot} $ on  $\\liek$ has\nbeen chosen (for example, the Killing form): we shall use\nthis to identify $\\liek$ with its dual. The orthocomplement\nof $\\liet$ in $\\liek$ will be denoted $\\lietp$.\n\nThroughout this paper all cohomology groups are assumed to have\ncoefficients in\nthe field ${\\Bbb C }$.\nThe $K$-equivariant cohomology of a point is $\\hk = H^*(BK)$,\nand similarly the $T$-equivariant cohomology is $\\hht\n= H^*(BT)$. We  identify $\\hk $  with\n$S(\\liek^*)^K$, the $K$-invariant polynomial functions on $\\liek$,\nand $\\hht$ with $S(\\liets)$.\nHence we have a bijective map  (obtained from the restriction from\n$\\lieks$ to $\\liets$) which identifies $\\hk$ with the subset of\n$\\hht$ fixed by the action of the Weyl group $W$:\n\\begin{equation} \\label{2.1}\n\\hk \\cong S(\\lieks)^K\n\\cong S(\\liets)^W \\subseteq S(\\liets) \\cong \\hht \\end{equation}\nThis  natural map $\\hk \\to \\hht$ will be denoted $\\tau$ or\n$\\tau_X$.\nWe shall use the symbol $\\phi$ to denote a point in\n$\\liek$, and $\\psi$ to denote a\npoint in $\\liet$. For $f \\in \\hk$ we shall write\n$f = f (\\phi)$ as a function of $\\phi$.\n\nThe $K$-equivariant cohomology of $X$ is the cohomology\nof a certain chain complex (see, for instance, Chapter 7 of\n\\cite{BGV} or Section 5\nof \\cite{MQ}; the construction is due to Cartan\n\\cite{cartan}) which\ncan be expressed  as\n\\begin{equation} \\label{2.0}\n\\Om^*_K(X) = \\Bigl ( S(\\lieks) \\otimes \\Om^*(X)   \\Bigr )^K \\end{equation}\n(where $\\Om^*(X)$ denotes   differential forms on $X$).\nAn element in $\\Om^*_K(X)$ may be thought of as a\n$K$-equivariant polynomial function\nfrom $\\liek$ to $\\Om^*(X)$.\nFor\n $ \\alpha \\in \\Om^*(X)$ and $f \\in S(\\lieks)$,\nwe write $(\\alpha \\otimes f) (\\phi) = f(\\phi) \\alpha$.\nIn this notation, the differential $D$ on the\ncomplex $\\Om^*_K(X)$ is then defined  by\\footnote{This definition and\nthe definition (\\ref{2.0''}) of the extension $\\bom(\\phi)\n= \\omega +  \\mu(\\phi)$\nof the\nsymplectic form $\\omega$ to an equivariant cohomology\nclass are different from the conventions\nused by Witten \\cite{tdg}: a factor $\\phi$ appears in the\nour definitions where $i \\phi$ appears in Witten's definition.\nIn other words Witten's definition is\n$D(\\alpha \\otimes f) (\\phi) =\nf(\\phi) (d \\alpha -  i \\iota_{ \\tilde{\\phi} } \\alpha)$\nand $\\bom(\\phi)\n= \\omega +  i \\mu(\\phi)$.\nWitten makes this substitution\nso that the oscillatory\nintegral $ \\int_X \\exp (\\om + i \\evab{\\phi}{ \\mu} ) $   will appear\nas the integral of an equivariant cohomology class.}\n\\begin{equation} \\label{2.0'}\nD(\\alpha \\otimes f) (\\phi) =\nf(\\phi) (d \\alpha - \\iota_{ \\tilde{\\phi} } \\alpha)\n=  \\, f (\\phi)  d \\alpha\n-  \\sum_{j= 1}^s \\phi_j f(\\phi)  \\: \\iota_{\\va} \\alpha    . \\end{equation}\nHere, $\\tilde{\\phi}$ is the vector\nfield on $X$ given by the action\nof $\\phi \\in \\liek$, and  $\\iota_{\\tilde{\\phi} }$ is the interior\nproduct with the vector field $\\tilde{\\phi}$.\nWe have  introduced an orthonormal basis $\\{ \\hat{e}^j,\n\\: j = 1, \\dots, s \\} $ for $\\liek$, and the $\\phi_j$ $ \\in \\lieks$\nare\nsimply the coordinate functions $\\phi_j = \\inpr{\\hat{e}^j, \\phi},\n $  while the $\\va$ are the vector fields on $X$ generated by\nthe action of $\\hat{e}^j$. The $\\phi_j $ are assigned\ndegree $2$, so that the differential $D$ increases degrees by $1$.\n\n\nOne may define the pushforward\n$\\pis^K: \\hk(X) \\to \\hk$, which\ncorresponds to integration over the fibre\nof the map $X \\times_K EK \\to BK $ (see Section 2\nof \\cite{abmm}).  The pushforward  satisfies\n$\\pis^T = \\tau \\circ \\pisk$. Because of this identification,\nwe shall usually simply write $\\pis$ for $\\pis^K$ or\n$\\pis^T$.\nA localization formula for $\\pist$ was given by\nBerline and Vergne in \\cite{BV1}; a more topological\nproof of this formula is given in Section 3 of\n\\cite{abmm}.\n\\begin{theorem} \\label{t2.1} {\\bf \\cite{BV1} }\nIf $\\si \\in \\hht(X)$ and $\\psi \\in \\liet$ then\n$$(\\pist \\si) (\\psi) = \\sum_{F  \\in {\\mbox{$\\cal F$}}}\n\\int_F\n\\frac{i_F^* \\si(\\psi)  }{e_F( \\psi) }. $$\nHere we sum over the set ${\\mbox{$\\cal F$}}$ of components $F$ of the fixed point\nset of $T$, and $e_F $ is the $T$-equivariant\nEuler class of the normal bundle of $F$; this Euler class\nis an element of  $H^*_T(F) \\cong H^*(F) \\otimes \\hht$,\nas is $i_F^* \\si$. The map $i_F: F \\to X$ is the inclusion map.\nThe right hand side of the above expression is to be interpreted as\na rational function of $\\psi$.\n\\end{theorem}\n\n\nWe shall now prove a lemma about the image of the pushforward,\nwhich will be applied in Section 4.\n\\begin{lemma}\\label{l2.2}\nIf $\\si \\in \\hht(X) $ then $(\\pist \\si)(\\psi)$ is a sum of terms\n\\begin{equation} \\label{2.m0} (\\pist \\si)(\\psi) =\n\\sum_{F \\in {\\mbox{$\\cal F$}}, \\;\\alpha \\in {\\mbox{$\\cal A$}}_F}\\tau_{F,\\alpha}  \\end{equation}\nsuch that each term $\\tau_{F,\\alpha}$  is of the form\n\\begin{equation} \\label{2.m1}\n\\tau_{F,\\alpha} =\n\\frac{\\int_F c_{F, \\alpha}(\\psi)  }{ \\efo(\\mar \\psi) \\prod_j\n\\beta_{F,j}(\\mar \\psi)^{n_{F,j}(\\alpha)} } \\end{equation}\nfor some component $F$  of the fixed point set of the\n$T$ action. Here, the $\\beta_{F,j}$ are the weights of\nthe $T$ action on the normal bundle $\\nu_F$, and\n$\\efo (\\mar \\psi) = \\prod_j \\beta_{F,j}(\\mar \\psi)$ is the product of all the\nweights, while $n_{F,j}(\\alpha)$ are some nonnegative integers. The class\n$c_{F,\\alpha}$ is in $H^*(F) \\otimes H^*_T$, and is equal to\n$i_F^* \\si \\in H^* (F)  \\otimes H^*_T$ times some\ncharacteristic\nclasses of subbundles of $\\nu_F$.\n\\end{lemma}\n\n\\Proof\nThe normal bundle $\\nu_F$ to $F$ decomposes as a direct sum of\nweight spaces  $\\nu_F  =\n\\oplus_{j = 1}^r \\nu_F^{(j)}$, on each of which\n$T$ acts with weight $\\bfj$. All these weights must be nonzero.\nBy passing to a split manifold if necessary (see section 21 of\n\\cite{BT}), we may assume without loss of generality\nthat the subbundle on which $T$ acts with a given weight decomposes\ninto a direct sum of $T$-invariant real subbundles\nof rank $2$.\nIn other words, we may assume that the $\n\\nu_F^{(j)}$ are rank $2$ real bundles, and  the\n$T$ action enables one to   identify them  in a\nstandard way with complex line bundles.\n\nThen the equivariant\nEuler class $e_F(\\psi)$ is given for $\\psi \\in \\liet$\n by\n\\begin{equation} \\label{2.3}\ne_F(\\psi) = \\prod_{j = 1}^r  \\Bigl (c_1(\\nu_F^{(j)}) + \\beta_{F,j}(\\psi)\n\\Bigr  ). \\end{equation}\nThus we have\n$$\n\\frac{1}{e_F (\\mar \\psi)}  = \\frac{1}{\\efo(\\mar \\psi) } \\,\n\\prod_j (1 + \\frac{c_1 (\\nu_F^{(j)} )}{\\beta_{F,j}(\\mar \\psi) } )^{-1} $$\n\\begin{equation}  \\label{2.4}\n= \\frac{1}{\\efo(\\mar \\psi) } \\: \\prod_j \\sum_{r_j \\ge 0 }\n\\: \\, (-1)^{r_j} \\Bigl ( \\frac{c_1 (\\nu_F^{(j)} )}\n{\\beta_{F,j}(\\mar \\psi) } \\Bigr )^{r_j}\n. \\end{equation}\nHere, $c_1(\\nu_F^{(j)} ) \\in H^2(F)$, so that\n$c_1(\\nu_F^{(j)} )\/\\beta_{F,j}(\\mar \\psi)$ is {\\em nilpotent} and the inverse\nmakes sense in $H^*(F) \\otimes {\\Bbb C }(\\psi_1, \\dots, \\psi_l)$,\nwhere ${\\Bbb C } (\\psi_1, \\dots, \\psi_l)$ denotes the complex\nvalued rational functions\non $\\liet$.\n$\\square$\n\n\nLet us now assume that $X$  is a symplectic manifold\nand the action of $K$ is Hamiltonian with moment map\n$\\mu: X \\to \\lieks$. Denote by $\\mu_T $\nthe moment map for the action of $T$ given by  the composition\nof $\\mu$ with the restriction map $\\lieks \\to  \\liets$.\nWe shall be interested in one particular (formal)\nequivariant cohomology\nclass $\\si $, defined by\n\\begin{equation} \\label{2.0''}\\si(\\phi) =\n\\exp  \\iins \\bom(\\phi), \\phantom{bbbbb} \\bom(\\phi)\n =  \\omega +   \\mu( \\phi ) .  \\end{equation}\n For this\nclass the localization formula gives\n\\begin{equation} \\label{2.2}\n (\\pist \\si) (\\psi) = \\sum_F  \\rf(\\psi), \\phantom{bbbbb}\n\\rf (\\psi) =  \\int_F \\frac{  e^{i \\mu_T(F)(\\psi )\n } e^\\om  } {e_F(\\mar \\psi) }. \\end{equation}\n(This formula does not require the fixed point set\nof $T$ to consist of\n isolated fixed points.)\n\n\\noindent{\\em Remark:} For any\n$\\eta \\in \\hk(X) $ the function $\\pisk \\eta \\in \\hk$ is\n a polynomial on $\\liek$, and in particular is  smooth. However,\n$\\sigma = e^{\\iins \\bom} $ does not have\npolynomial dependence on $\\phi$.\nAlthough it  is not immediately obvious\nfrom the formula (\\ref{2.2}), the\nfunction $\\pisk (\\eta e^{\\iins \\bom}) $ is still  a\n{ smooth}\nfunction on $\\liek$ (for any $\\eta \\in \\hk(X)$ represented\nby an element $\\tilde{\\eta} \\in \\Om^*_K (X)$): this\nfollows from its description as\n$$ \\pisk (\\eta e^{\\iins \\bom}) (\\phi) = \\int_{x \\in X} e^{i \\om}\n \\tilde{\\eta} (\\phi)\ne^{i \\mu(x) (\\phi) }. $$\n\n\n\n{ \\setcounter{equation}{0} }\n\\section{Preliminaries}\n\nThis section contains results which will be applied in the\nnext section to reduce the integral $\\ie$ to an integral over $\\liets$\nof a Gaussian multiplied by a piecewise polynomial function. The\nfirst, Lemma \\ref{l3.1}, reduces integrals over $\\liek$ to\nintegrals over $\\liet$. Lemma \\ref{l3.2} enables\nus to replace the $L^2$ inner product of two functions by\nthe $L^2$ inner product of their Fourier transforms. Lemma\n\\ref{l3.3} relates Fourier transforms on $\\liek$ to\nFourier transforms on $\\liet$. Finally Proposition \\ref{p3.5}\ndescribes  certain functions\nwhose Fourier transforms are the terms\nappearing\nin the  localization formula (\\ref{2.4}).\n\nWe would like to study a certain integral that arises out\nof equivariant cohomology:\n\\begin{equation} \\label{3.1}\n{\\mbox{$\\cal I$}}^\\epsilon = \\frac{1}{(2 \\pi i )^s \\,{\\rm vol}\\,  K} \\int_{\\phi \\in \\liek}\n[d\\phi] \\, e^{- \\epsilon \\inpr {\\phi, \\phi } \/2 }  \\int_X \\si(\\mar \\phi). \\end{equation}\nHere, $\\si \\in \\Om^*_K(X) $\n(see (\\ref{2.0})); we are mainly interested in the\nclass $\\si$ defined by (\\ref{2.0''}).\nAlso,\n $\\epsilon > 0 $ and we shall consider the behaviour of ${\\mbox{$\\cal I$}}^\\epsilon$\nas $\\epsilon \\to 0^+$.\nThe measure $[d \\phi]$ is a measure on $\\liek$ which\ncorresponds to a choice of invariant metric on $\\liek$\n(for instance, the metric given by the Killing form):  such\na metric induces a  volume form on $K$, and\n$ \\,{\\rm vol}\\,  K$ is the integral of this volume  form over $K$. Thus\n$[d \\phi]\/ \\,{\\rm vol}\\,  K$ is independent of the choice of metric on\n$\\liek$. The metric also gives a  measure  $[d \\psi] $\non $\\liet$ and a volume form on $T$: it is\nimplicit in our notation that the measures  on $T$ and $\\liet$\ncome from the same invariant metric as those on $K$ and $\\liek$.\n\nIt will be convenient to recast integrals over $\\liek$ in terms\nof integrals over $\\liet$. For this we use\n  a function $\\nusym: \\liet \\to {\\Bbb R }$,\nsatisfying $\\nusym(w \\psi) = (\\det w) \\nusym(\\psi)$ for\nall elements $w$ of the Weyl group  $ W$, and\ndefined by\n\\begin{equation} \\label{3.2}\n\\nusym(\\psi) = \\prod_{\\gamma > 0} \\gamma(\\psi), \\end{equation}\nwhere $\\gamma$ runs over the positive roots.\nUsing the inner product to identify $\\liet$ with\n$\\liets$, $\\nusym$ also defines a function $\\liets \\to {\\Bbb R }$.\nWe  have\n\\begin{lemma} \\label{l3.1}[Weyl Integration Formula]\nIf $f: \\liek \\to {\\Bbb R }$ is $K$-invariant, then\n$$\\int_{\\phi \\in \\liek} f (\\phi) [d \\phi]\n= \\ck^{-1}  \\intt f (\\psi) \\nusym(\\psi)^2\n[d \\psi], $$\nwhere $s$ and $l$ are the dimensions of $K$ and $T$, and\n$ \\ck =  { \\wn  \\,{\\rm vol}\\,  T}\/ \\,{\\rm vol}\\,  K $.\n\\end{lemma}\n\\Proof There is an orthonormal basis $\\{X_\\gamma, Y_\\gamma | $\n$\\gamma $ a positive root$\\}$ for $\\lietp$ such that\n$$[X_\\gamma, \\psi] =  \\gamma(\\psi) Y_\\gamma, $$\n$$ [Y_\\gamma, \\psi] = -  \\gamma(\\psi) X_\\gamma$$\nfor all $\\psi \\in \\liet$. The Riemannian volume form of the\ncoadjoint orbit\nthrough $\\psi \\in \\liet \\cong \\liets$ (with the metric\non the orbit pulled back from the metric on $\\lieks$\ninduced by the inner product $\\inpr{\\cdot, \\cdot}$)\nevaluated on the tangent vectors $[X_\\gamma, \\psi]$ and $[ Y_\\gamma, \\psi]$\nis thus $ \\prod_{\\gamma> 0} \\gamma(\\psi)^2$, while the volume\nform\nof the homogeneous space $K\/T$ (induced by\nthe chosen metric on $\\liek$) evaluated on the tangent vectors\ncorresponding to\n$X_\\gamma, Y_\\gamma \\in \\liek $ is $1$. Hence\n the Riemannian volume of the  orbit\nthrough $\\psi \\in \\liet$ is $ \\nusym(\\psi)^2$ times the\nvolume of the homogeneous space $K\/T$.\n$\\square$\n\n\\newcommand{\\lamax}{\\Lambda^{\\rm max} }\n\\newcommand{\\dist}{{\\mbox{$\\cal D$}}'}\n\nIt will be convenient also to work with the Fourier transform. Given\n$f: \\liek \\to {\\Bbb R }$ we define $F_K f: \\lieks \\to {\\Bbb R }$,\n$F_T f: \\liets \\to {\\Bbb R }$\nby\n\\begin{equation} \\label{3.3}\n(F_K f) (z) = \\frac{1}{(2 \\pi)^{s\/2} } \\intk f (\\phi)\ne^{-i \\evab{\\phi}{z} }  \\, [d \\phi], \\end{equation}\n\\begin{equation} \\label{3.4}\n(F_T f) (y) = \\frac{1}{(2 \\pi)^{l\/2} } \\intt f (\\psi)\ne^{-i \\evab{\\psi}{y} } \\, [d \\psi]. \\end{equation}\nMore invariantly, the Fourier transform\nis defined on a vector space $V$  of dimension $n$ with\ndual space $V^*$ as a map\n$F: \\Omega^{\\rm max}(V) \\to \\Omega^{\\rm max} (V^*)$\nwhere\n$\\Omega^{\\rm max}  (V) = \\Lambda^{\\rm max} (V^*) \\otimes \\dist (V)$,\n$\\Lambda^{\\rm max}(V^*) $\n  is the top exterior power of\n$V^*$ and $\\dist (V) $ are the tempered distributions\non $V$ (see \\cite{hor}).\nIndeed for $z \\in V^*$, $f \\in \\dist(V) $ and $u \\in \\lamax(V^*)$\nwe define\n\\begin{equation} \\Bigl ( F (u \\otimes f )\\Bigr ) (z) = \\frac{v}{(2 \\pi)^{n\/2}}\n\\int_{\\phi \\in V}  f(\\phi) e^{- i z(\\phi)}  u  , \\end{equation}\nwhere the element\n$v\\in \\lamax (V) $ satisfies $u(v)  = 1 $ under the\nnatural pairing $\\lamax(V^*) \\cong \\Bigl (\\lamax (V) \\Bigr)^*.$\n(The normalization has been chosen so that\n$$ \\tilde{f} (\\phi) = F (F \\tilde{f}) (- \\phi)  $$\nfor any $\\tilde{f} \\in \\Omega^{\\rm max}(V)$.)\nFor notational convenience\nwe shall often ignore this subtlety\nand identify $\\liek, \\liet$ with $\\lieks, \\liets$\nunder the invariant inner product $\\inpr{\\cdot, \\cdot}$:\nwe shall also suppress the exterior powers of\n$\\liek$ and $\\liet$ and write $\\fk: \\dist(\\liek) \\to\n\\dist (\\lieks)$, $\\ft: \\dist(\\liet) \\to \\dist(\\liets)$.\nFurther, although we shall work with functions whose definition\ndepends on the choice of the element  $[d \\phi] \\in \\lamax(\\lieks)$\n(associated to the inner product), some of our   end results\ndo not depend on this choice\\footnote{The statement of\nTheorem \\ref{t8.1},\nfor instance, does not depend on\nthe invariant\ninner product $\\inpr{\\cdot, \\cdot}$.}\nand the use  of such\nfunctions is just  a notational convenience.\n\n\nA fundamental property of the Fourier transform\nis that it preserves the $L^2$ inner product:\n\\begin{lemma} \\label{l3.2} {\\bf [Parseval's Theorem]}(\\cite{hor},\nSection 7.1)\nIf $f:  \\liek \\to  {\\Bbb C }$  is a tempered distribution\nand $g: \\liek \\to {\\Bbb C }$ is a Schwartz function then\n$F_K f: \\liek \\to {\\Bbb C }$ is also a tempered distribution and\n$F_K g: \\liek \\to {\\Bbb C }$ a Schwartz function, and we have\n$$ \\intk \\overline{g(\\phi)}\n f (\\phi) [d \\phi] = \\tintk \\overline{(F_K g)(z) }\n(F_K f)(z) \\, [d z], $$\n$$ \\intt \\overline{g(\\psi)}\n f (\\psi) [d \\psi] = \\tintt \\overline{(F_Tg)(y) }\n(F_T f)(y) \\, [d y]. $$\n\\end{lemma}\nWe note also that if\n$g_\\epsilon: \\liek \\to {\\Bbb R }$ is the\nGaussian defined by $g_\\epsilon(\\phi) =  e^{-\\epsilon \\inpr{\\phi, \\phi}\/2  }\n$, then\n\\begin{equation} \\label{3.5} (F_K g_\\epsilon) (z) = \\frac{1}{\\epsilon^{s\/2} } e^{- \\inpr{z,z}\n\/{2 \\epsilon}\n} = \\frac{1}{\\epsilon^{s\/2} }  \\gsoe(z),  \\phantom{bbbbb}\n(F_T g_\\epsilon  ) (y) = \\frac{1}{\\epsilon^{l\/2} } e^{- \\inpr{y,y} \/{2 \\epsilon}\n} =  \\frac{1}{\\epsilon^{l\/2} } \\gsoe(y). \\end{equation}\n\nWe have also that\n\\begin{lemma} \\label{l3.2'} The symplectic volume form\n$d \\Om^S_\\phi$ at a point $\\phi$ in the\norbit $K \\cdot \\psi$\n  through $\\psi \\in \\liek$\nis related to the Riemannian volume form\n$d \\Om^R_\\phi$ (induced by\nthe metric on $\\liek$)\nby\n$$d \\Om^R_\\phi =  \\nusym(\\psi) d \\Om^S_\\phi.$$\n\\end{lemma}\n\\Proof The symplectic form is\n  $K$-invariant\nand is given at the point $\\psi \\in  \\liet_+$  in the orbit\n$K \\cdot \\psi$\n(for $\\xi, \\eta \\in \\lietp$ giving rise to tangent\nvectors $[\\xi,\\psi], $ $[\\eta, \\psi]$ to the orbit) by\n\\begin{equation} \\label{3.7}\n\\om([\\xi,\\psi], [ \\eta,\\psi]) = \\inpr{[ \\xi,\\psi], \\eta }\n= \\inpr{\\psi, [ \\eta,\\xi] }.  \\end{equation}\nIn the notation of Lemma \\ref{l3.1}, the symplectic\nvolume form evaluated on the tangent vectors\n$[ X_\\gamma,\\psi], $ $[ Y_\\gamma,\\psi]$ is given by\n$\\prod_{\\gamma > 0} \\om ([ X_\\gamma,\\psi],  [ Y_\\gamma,\\psi] ).$\nBut $\\om ([X_\\gamma,\\psi],  [ Y_\\gamma,\\psi] ) =  \\inpr{[ X_\\gamma,\\psi], Y_\\gamma} $\n$ =  \\gamma(\\psi)$, from which (comparing with the proof of\nLemma \\ref{l3.1}) the Lemma follows. $\\square$\n\nWe shall use this Lemma to prove the following\nLemma  relating Fourier\ntransforms on $\\liek$ to those on $\\liet$:\n\\begin{lemma} \\label{l3.3}\nLet $f \\in \\dist(\\liek)$ be $K$-invariant,\nand let $\\nusym$ be defined\nby (\\ref{3.2}). Then\n$$\\ft(\\nusym f) = \\nusym \\fk(f)$$\nas distributions on $\\liet$.\n\\end{lemma}\n\\Proof $$(\\fk f) (z) = \\frac{1}{(2 \\pi)^{s\/2} } \\intk e^{-i\n\\evab{\\phi}{z} } \\, f(\\phi) [d \\phi] $$\n\\begin{equation} \\label{3.6}\n= \\frac{1} {(2 \\pi)^{s\/2} } \\int_{\\psi \\in \\liet_+}\n\\int [d \\psi] f(\\psi) \\: \\int_{\\phi \\in K \\cdot \\psi}\ne^{- i \\evab{\\phi}{z} } d \\Om^R_\\phi \\end{equation}\n(where $\\liet_+$ denotes the fundamental Weyl chamber).\n\nWe have from Lemma \\ref{l3.2'} that\n$d \\Om^R_\\phi =  \\nusym(\\psi) d \\Om^S_\\phi$.\nThus we have\n\\begin{equation} \\label{3.8}(F_K f)(z) = (2 \\pi)^{-s\/2} \\int_{\\psi \\in \\liet_+}\n[d \\psi] f (\\psi) \\nusym(\\psi) \\int_{\\phi \\in K \\cdot \\psi}\ne^{- i \\evab{\\phi}{ z} }  d \\Om^S_\\phi. \\end{equation}\n\nNow the integral over the coadjoint orbit may be computed\nby the Duistermaat-Heckman theorem \\cite{DH} applied to the\naction of $T$ on the orbit $K \\cdot \\psi$\n(or equivalently as a consequence\nof the\nabelian localization theorem, Theorem \\ref{t2.1}): we have\n(see e.g. \\cite{BGV} Theorem 7.24)\n\\begin{equation} \\label{3.9}\n\\int_{\\phi \\in K \\cdot \\psi} e^{- i \\evab{\\phi}{ z} }\n\\, d \\Om^S_\\phi =\n\\frac{(2 \\pi)^{(s-l)\/2} } {\\nusym(z) } \\sum_{w \\in W} e^{- i \\evab{w \\psi}{ z}\n}\n(\\det w). \\end{equation}\nThis is a well known formula due originally to Harish-Chandra\n(\\cite{Harish}, Lemma 15).\n(Notice that the  $z$ in (\\ref{3.9}) plays the role of the\n$\\psi$ in Theorem \\ref{t2.1}, while the $\\psi$ in\n(\\ref{3.9}) specifies the orbit.)\nNow since $\\nusym(w \\psi) = (\\det w) \\nusym(\\psi), $ we may replace\nthe integral over $\\liet_+$ by an integral over $\\liet$: in other\nwords we have\n\\begin{equation} \\label{3.10}\n\\nusym(z) (F_K f) (z) = (2 \\pi)^{- l\/2} \\int_{\\psi \\in \\liet}\n[d \\psi] \\, f(\\psi) \\nusym(\\psi) \\eminevb{\\psi}{ z} = F_T(\\nusym f) (z).\n\\phantom{bbbbb} \\square \\end{equation}\n\n\\newcommand{\\ims}{i^{-s} }\n\nApplying this to the Gaussian\n$\\gse(\\psi) =  e^{-\\epsilon \\inpr{\\psi, \\psi}\/2  }  $ we have\n\\begin{corollary} \\label{c3.4}\n$$\nF_T(\\gse \\nusym) (y) = \\frac{1}{\\epsilon^{s\/2} } \\nusym(y) e^{- \\inpr{y,y} \/{2 \\epsilon} }\n= \\frac{1}{\\epsilon^{s\/2} } \\nusym(y)  \\gsoe(y).  $$\n\\end{corollary}\n\nWe shall also need\nthe following result which occurs\nin the work of Guillemin, Lerman, Prato\nand Sternberg\n concerning Fourier transforms of\na class of functions on\n$\\liets$. This result will be applied to functions\nappearing in the abelian localization formula (\\ref{2.4}).\n\\begin{prop} \\label{p3.5}\n{\\bf (a)} (see \\cite{JGP},  section 3.2 of \\cite{GLS},\nand \\cite{GP})\nDefine $h(y) = H_{\\bar{\\beta} }(y + \\tau)$ for some\n$\\tau \\in \\liet$,\nwhere $H_{\\bar{\\beta} } (y) =  \\,{\\rm vol}\\,   \\{ (s_1, \\dots,\ns_N): s_i \\ge 0,  \\phantom{a}  y = \\sum_j s_j \\beta_j \\}$ for\nsome $N$-tuple   $\\bar{\\beta} = $\n$\\{ \\beta_1, \\dots, \\beta_N \\}$ , $\\beta_j \\in \\liets$,\nsuch that\nthe $\\beta_j$ all lie in the interior of\n some half-space of $\\liets$.\nThus $H_{\\bar{\\beta} } $ is a piecewise polynomial\nfunction supported on the cone $C_\\barb =\n\\{ \\sum_j s_j \\beta_j \\, | \\, s_j \\ge 0 \\}. $\nThen the Fourier transform of $h$ is given for\n$\\psi$ in the complement of the hyperplanes\n$\\{ \\psi \\in \\liet| {\\beta_j}({ \\psi}) = 0 \\}$ by\nthe formula\n\\begin{equation} \\label{3.10'} F_T h(\\psi) = \\frac{\\epinev{\\tau}{ \\psi} }\n{i^N \\prod_{j = 1}^N \\beta_j (\\mar \\psi) }.  \\end{equation}\n\n\\noindent{\\bf(b)} (see Section 2 of \\cite{JGP}\nand (2.15) of \\cite{GP}) The function $H_{\\bar{\\beta}} (y) $ is also given\nas\n$$H_{\\bar{\\beta}} (y) = H_{\\beta_1} * H_{\\beta_2} * \\dots\n* H_{\\beta_r}, $$\nwhere for $\\beta \\in {\\rm Hom}({\\Bbb R }^l,{\\Bbb R })$ we have  $H_\\beta = (i_\\beta)_* dt$,\ni.e. $H_\\beta$ is the pushforward of the Euclidean\nmeasure $dt$ on ${\\Bbb R }^+$ under the map $i_\\beta: {\\Bbb R }^+ \\to {\\Bbb R }^l$\ngiven by $i_\\beta(t) = \\beta t$.  Here, $*$ denotes convolution.\n\n\\noindent{\\bf (c)} (see Section 2 of \\cite{JGP}) The function\n$H_\\barb $ satisfies the differential equation\n$$ \\prod_{j = 1}^N \\beta_j (\\partial\/\\partial y) H_\\barb(y) =\n\\delta_0(y) $$\nwhere $\\delta_0$ is the Dirac delta distribution.\n\n\\noindent{\\bf (d)} (see Proposition 2.6 of \\cite{JGP})\nThe function $H_\\barb$ is smooth at any\n$y \\in U_\\barb$, where $U_\\barb$ are the\npoints in $\\liets$  which are not in any cone\nspanned by a subset of $\\{\\beta_1, \\dots, \\beta_N \\} $ containing\nfewer than $l$ elements.\n\n\n\\end{prop}\n\n{ \\setcounter{equation}{0} }\n\\section{Reduction to a piecewise polynomial function}\n\n\tIn this section we apply the results stated in\nSection 3 to reduce $\\ie$ to the form given in Proposition \\ref{p4.2},\nas the integral over $y \\in \\liets$ of a\nGaussian $\\gtsoe(y) = \\gsoe(y)\/((2 \\pi)^s  \\wn  \\,{\\rm vol}\\, ( T) \\epsilon^{s\/2} ) $\ntimes a function\n$Q(y)$ which is $D_\\nusym R(y)$\nwhere $D_\\nusym$ is a differential operator\nand $R$ is a piecewise polynomial function.\nAs a byproduct we obtain also a generalization of\nTheorem 2.16 of \\cite{GP}: we may relax the hypothesis of \\cite{GP}\nthat\nthe action of the maximal torus $T$ have isolated fixed points.\n\n Two of our main results related to Witten's work in\n\\cite{tdg}\nare Theorem \\ref{t4.1} and \\ref{t4.3}: these are stated in this section.\nThe  proof of Theorem \\ref{t4.3} will be given in Section 5. It\ntells us  that the cohomological contribution $\\eeth[\\xred]$\ngiven in Witten's Theorem \\ref{t1.2} for the zero locus\nof the moment map is given by the integral\nover $y \\in \\liets$ of $\\gtsoe(y)$ times a polynomial\n$Q_0(y)$ which is equal to $Q$ near $y = 0$.\nTheorem \\ref{t4.1} is our version of the asymptotic\nestimates given in Witten's Theorem \\ref{t1.1}, and will be\nproved in Section 6 below. Theorem \\ref{t7.1} in Section 7\nextends Theorems \\ref{t4.1} and \\ref{t4.3} to more general\nequivariant cohomology classes.\n\n\n\nWe assume throughout the rest of the\npaper that $K$ acts on $X$ in a Hamiltonian fashion,\nand that $0$ is a regular value of the moment map\n$\\mu$ for the $K$ action.\nThis is equivalent to the assumption that\n $K$ acts on $\\mu^{-1}(0)$ with finite stabilizers, and it\n implies that $\\mu^{-1}(0)$ is a smooth manifold.\n Under these\nhypotheses, the space $\\xred = \\mu^{-1}(0)\/K$ is a $V$-manifold or\norbifold\n(see \\cite{kaw}) and $P = \\mu^{-1}(0) \\to\n\\mu^{-1}(0)\/K $ is a $V$-bundle: we  have a class\n$\\Theta \\in H^4 (\\xred) $ which represents the class\n$-\\inpr{\\phi, \\phi}\/2 \\in H^4_K(\\mu^{-1}(0) ) $\n$\\cong H^4(\\zloc\/K)$,\nand which is a four-dimensional characteristic class\nof the bundle $P \\to \\xred$.\n\nIn \\cite{Ki1} it is proved that the set of\ncritical points of the function $\\rho = |\\mu|^2: X \\to {\\Bbb R }$\nis a disjoint union of closed subsets $C_\\beta$ in\n$X$ indexed by a finite subset ${\\mbox{$\\cal B$}}$ of $\\liet$.\nIn fact if $\\liet_+$ is a fixed positive Weyl\nchamber for $K$ in $\\liet$ then $\\beta \\in {\\mbox{$\\cal B$}}$ if and only\nif $\\beta \\in \\liet_+$ and $\\beta$ is the closest point to $0$\nof the convex hull in $\\liet$ of some nonempty\nsubset of the finite set $\\{ \\mu_T(F): F \\in {\\mbox{$\\cal F$}}\\}$,\ni.e. the image under $\\mu_T$ of the set of fixed\npoints of $T$ in $X$. Moreover if $\\beta \\in {\\mbox{$\\cal B$}}$\nthen $$C_\\beta = K(Z_\\beta \\cap \\mu^{-1}(\\beta))$$\nwhere $Z_\\beta$ is the union of those connected components\nof the set of critical points of the function $\\mu_\\beta$\ndefined by $\\mu_\\beta (x) = \\eva{\\mu(x)}{  \\beta} $\non which $\\mu_\\beta$ takes the value $|\\beta|^2$. Note that\n$C_0 = \\mu^{-1}(0) $ and in general the value\ntaken by the function $\\rho = |\\mu|^2$\non the critical set $C_\\beta$ is just $|\\beta|^2$.\n\n\n\n\nWe shall prove the following version\nof Witten's nonabelian localization theorem, for\nthe integral $\\ie$  defined in (\\ref{3.1}) with\nthe\nclass $\\si  = e^{\\iins \\bom} $ defined by (\\ref{2.0''}):\n\\begin{theorem}\n\\label{t4.1}  For each $\\beta \\in {\\mbox{$\\cal B$}}$ let $\\rho_\\beta = |\\beta|^2$\n(this is   the  critical value\nof the function $\\rho = |\\mu|^2: X \\to {\\Bbb R }$ on the critical\nset $C_\\beta$ when this set is nonempty).\nThen\nthere exist functions $h_\\beta: {\\Bbb R }^+ \\to {\\Bbb R } $  such that\nfor some $N_\\beta \\ge 0$,\n$\\epsilon^{N_\\beta} h_\\beta(\\epsilon)$ remains bounded as $\\epsilon \\to 0^+$, and\nfor which\n$$|\\ie  - e^{\\epsilon \\Theta} e^\\om [\\xred]  | \\le\n\\sum_{\\beta \\in {\\mbox{$\\cal B$}} - \\{0\\} }\n e^{- \\rho_\\beta  \/{2 \\epsilon}  } \\, h_\\beta(\\epsilon). $$\n\n\\end{theorem}\n\n\\noindent{\\em Remark:} The estimate given in Theorem\n\\ref{t4.1} is {\\em weaker} than Witten's estimate\n(Theorem 1.2) since $|\\beta|^2 $ is not in fact a\ncritical value of $|\\mu|^2 $ when $C_\\beta$ is\nempty.\nNevertheless in many interesting cases all the $C_\\beta$\nare nonempty, and our estimate then coincides with Witten's.\n\n\nTo prove this result, we shall rewrite $\\ie$ using Lemma\n\\ref{l3.1}:\n\\begin{equation} \\label{4.1} \\ie = \\frac{1}{(2 \\pi i)^s  \\wn  \\,{\\rm vol}\\, (T) }\n \\: \\intt [d \\psi] \\, \\Bigl (\n\\gse(\\psi) \\: \\nusym(\\psi) \\Bigr ) \\:\n\\Bigl ( \\: \\nusym(\\psi) (\\pist \\sigma ) (\\psi) \\Bigr ). \\end{equation}\nNow we apply Lemma \\ref{l3.2} to get\n\\begin{equation} \\label{4.2}\n\\ie = \\frac{1}{(2 \\pi i)^s \\wn  \\,{\\rm vol}\\, (T) }\n\\tintt [dy] \\Bigl ( F_T (g_\\epsilon \\nusym) (y) \\Bigr )\n\\Bigl ( F_T (\\nusym \\, \\pist \\si ) (y) \\Bigr ). \\end{equation}\nApplying Corollary \\ref{c3.4} we have\n\\begin{equation} \\label{4.3}\n\\ie = \\frac{1}{(2 \\pi i)^s \\wn  \\,{\\rm vol}\\, (T) \\epsilon^{s\/2}  }\n\\tintt [dy] \\nusym(y) e^{- \\inpr{y,y}\/{2 \\epsilon} } \\,\nF_T ( \\nusym \\pist \\si) (y). \\end{equation}\n\n\\newcommand{\\bwd}{\\lasub}\n\\newcommand{\\cf}{C_F}\n\\newcommand{\\cfl}{C_{F,\\lasub } }\n\\newcommand{\\dcf}{\\check{C}_{F,\\lasub} }\n\n\nFollowing Guillemin, Lerman and Sternberg \\cite{JGP},\nwe may  use the abelian localization formula (Theorem\n\\ref{t2.1})\nto give a formula for $F_T ( \\pist \\si)$ where $\\si = e^{\\iins \\bom}$,\nand from it obtain a formula\nfor $F_T (\\nusym \\pist \\si)$.\nIn terms of the notation of  Lemma \\ref{l2.2}, we choose\na component $\\bwd$ of the set $\\cap_{F,j}  \\Bigl \\{\n\\psi \\in \\liet:$ $ \\bfj(\\psi) \\ne 0 \\Bigr \\}, $\nwhere $\\beta_{F, j}$ are the weights of the action\nof $T$ on the normal bundle to a component $F$ of the\nfixed point set.  Thus $\\bwd$\nis a cone in $\\liet$. If we denote by $\\cf = \\cfl$\nthe component of $\\cap_j \\{ \\psi \\in \\liet:\n$ $ \\bfj (\\psi) \\ne 0 \\} $ containing $\\bwd$, then\n$\\bwd = \\cap_F \\cfl$. Also, $\\bfj \\in \\liets$ lies\nin the dual cone $\\dcf$ of $\\cfl$: indeed, this dual\ncone is simply the cone $\\dcf = \\{\n\\sum_j s_j \\bfj: s_j \\ge 0 \\}. $\nWe then define\n$\\sigma_\\fj = {\\rm sign} \\bfj(\\xi) $ for any\n$\\xi \\in \\bwd$, and $\\bfjw =\n\\sigma_\\fj \\bfj$. Then we set $k_F(\\alpha) =\n\\sum_{j, \\sigma_\\fj = - 1} n_\\fj(\\alpha)$.\n\nWe define a function  $\\hh: \\liets \\to {\\Bbb C }$\nby\n\\begin{equation} \\label{4.g}\n\\hh(y) = \\sum_{F \\in {\\mbox{$\\cal F$}}} \\sum_{\\alpha  \\in {\\mbox{$\\cal A$}}_F}\n(-1)^{k_F(\\alpha) } H_{\\bar{\\gamma_F}(\\alpha)  } (-y + \\mu_T(F) )\n\\int_F (e^{i\\om} \\tilde{c}_{F,\\alpha}) . \\end{equation}\nHere as before ${\\mbox{$\\cal F$}}$ is the set of components\n$F$ of the fixed point set of $T$ and\nthe ${\\mbox{$\\cal A$}}_F$ are the indexing sets which appeared\nin Lemma \\ref{l2.2}. If $\\alpha \\in {\\mbox{$\\cal A$}}_F$ then\n$\\bar{\\gamma_F}(\\alpha)$ consists of the elements\n$\\bfjw$ where each $\\bfjw$ appears with multiplicity\n$n_\\fj(\\alpha) $.\nThen\n$H_{\\bar{\\gamma_F}(\\alpha) } $ is  as defined in Proposition \\ref{p3.5}.\nThe $\\tilde{c}_{F,\\alpha} \\in H^*(F) $ are related to\nthe $c_{F,\\alpha} $  in (\\ref{2.m1}), in that\n\\begin{equation} \\label{4.004} c_{F, \\alpha}(\\psi)  = e^{i\\om}  e^{i \\eva{\\mu_T(F)}{ \\psi} }\n\\tilde{c}_{F,\\alpha}.  \\end{equation}\n\n\nWe then have the following Theorem, which in the case\nwhen the action of $T$ has isolated fixed points is the main\ntheorem\nof Section 3 of the  paper \\cite{JGP}\nof Guillemin, Lerman and Sternberg.\nFor the most part our proof is a direct extension\nof the proof given in that paper; the major difference is in\nthe use of the abelian localization theorem rather than\nstationary phase.\n\n\n\\begin{theorem} \\label{t4.1'} The (piecewise polynomial)\nfunction\n$\\hh$ given in (\\ref{4.g}) is identical\nto  the distribution $ \\ggh  = F_T ( \\pist e^{\\iins \\bom}) $.\n\\end{theorem}\n\\Proof We first\n apply the abelian localization formula (\\ref{2.2})\nto $\\pist \\si$.\n Then\nthe formula obtained from Lemma \\ref{l2.2}  for\n$\\pist \\si$ is\n\\begin{equation} \\label{4.3p} \\pist \\si =\n\\sum_{F \\in {\\mbox{$\\cal F$}}} \\rf, \\phantom{bbbbb} \\rf = \\sum_{\\alpha \\in {\\mbox{$\\cal A$}}_F} \\tau_{F, \\alpha} , \\end{equation}\n\\begin{equation} \\label{4.03p}\\tau_{F, \\alpha} =\n (-1)^{k_F(\\alpha) }  \\frac{ \\epinevb{\\psi}{ \\mu_T(F)}\n\\int_F (e^{\\iins\\om} \\tilde{c}_{F,\\alpha} ) }\n{\\prod_j \\Bigl ( \\bfjw (\\mar \\psi) \\Bigr )^{{n}_\\fj(\\alpha)} } , \\end{equation}\nwhere $\\mu_T $, the moment map for the $T$ action, is simply\nthe projection of $\\mu$ onto $\\liet$\nand the  $ \\bfj  $ are the weights of the $T$ action. Recall that\nthe quantity $\\bfjw$ is  $\\bfj$   if $\\bfj (\\xi) > 0 $\nand $- \\bfj$ if $\\bfj (\\xi) < 0 $.\nThe class $\\tilde{c}_{F, \\alpha} \\in H^*(F)$ is equal to\nsome characteristic classes of\nsubbundles of the normal bundle $\\nu_F$; it is independent\nof $\\psi$.\n\nNotice that each  $\\bfjw$ $\\in \\liets$\nlies in the half space\n$\\{ y  \\in \\liets \\, | y( \\xi) > 0 \\}$, for any $\\xi \\in \\bwd$.\nThe expression  (\\ref{4.03p}) is hence\nof the form  appearing on the right hand side\nof (\\ref{3.10'}), up to multiplication  by a\nfactor independent of $\\psi$.\nThe conclusion of the proof goes, as in Section 3 of \\cite{JGP},\nby applying a lemma about distributions (see Appendix A of \\cite{GP}):\n\\begin{lemma} \\label{l4} Suppose $\\gggh$ and $\\hhh$ are\ntwo tempered distributions on ${\\Bbb R }^l$ such that:\n\\begin{enumerate}\n\\item $F_T \\gggh - F_T \\hhh$  is supported on a finite union\nof hyperplanes.\n\\item There is a half space $\\{ y \\, | \\, \\inpr{y,\\zeta} > k_0 \\}$\ncontaining the support of $\\hhh - \\gggh$.\n\\end{enumerate}\nThen $\\gggh = \\hhh$.\n\\end{lemma}\nHere we apply the lemma to $\\hhh$ as given in\n(\\ref{4.g}) and $\\gggh = F_T ( \\pist e^{\\iins \\bom} ) $.\nThe first hypothesis  is satisfied  because we know from Proposition\n\\ref{p3.5} that $F_T \\hh$  is given by the formula (\\ref{4.3p})\non the complement of the hyperplanes\n$\\{ \\psi \\, | \\, \\bfjw ( \\psi)  = 0 \\}$; but this\nis just the formula for $F_T \\ggh = \\pist e^{\\iins \\bom}$.\nFurther, $\\hh$  is supported in a half space since all the\nweights $\\bfjw$ satisfy $\\bfjw(\\xi) > 0$ for any $\\xi \\in \\bwd$, while\nthe support of $\\ggh$ is\ncontained in the compact set $\\mu_T(X)$  (see Section 5 below).\nTherefore the support of $H-G$ is contained\nin a half space of the form $\\{ y: \\inpr{y,\\xi} > k_0 \\}$\nfor some $k_0$.\nThis completes the proof of Theorem \\ref{t4.1'}. $\\square$\n\n\n\n\nDefine\n\\begin{equation} \\label{4.3pp}\nR(y) =  F_T ( \\pist \\si) (y), \\end{equation}\nwhere $\\si = \\exp \\iins \\bom$.\nThen\n (\\ref{4.3}) and (\\ref{4.g}) give us  the following\n\\begin{prop} \\label{p4.2}\nThe function  $R$ is a piecewise polynomial function supported on\ncones each of which has apex at $\\mu_T(F)$ for some component\n$F$ of the fixed point set of $T$.\nLet $Q$ be the distribution defined by\n$$ Q(y) = \\nusym(y) D_\\nusym R(y) $$\nwhere\nthe differential operator $D_\\nusym$ is given by\n$$D_\\nusym = \\prod_{\\gamma> 0} (i \\gamma(\\partial\/\\partial y) )$$\nand $\\gamma$ runs over the positive roots (cf. (\\ref{3.2})).\nThen\n\\begin{equation} \\label{4.3ppp}\n\\ie = \\frac{1}{(2 \\pi i )^s \\wn  \\,{\\rm vol}\\, ( T) \\epsilon^{s\/2} }\n\\tintt [dy] e^{- \\inpr{y,y}\/{2 \\epsilon}  }  Q(y). \\end{equation}\n\\end{prop}\n\\Proof It only remains\nto note that  $\\ft (\\nusym \\pist \\si) = D_\\nusym \\ft (\\pist \\si), $\nwhere $D_\\nusym$ is defined above. $\\square$\n\n\\noindent{\\em Remark:} The formulas for $Q(y)$ obtained\nfrom (\\ref{4.3p}) will in general be different for\ndifferent choices of $\\bwd$.\n\n\n\nIn addition, certain formulas simplify if we impose\nthe additional assumption that at any point $x$ in a component\n$F$ of the fixed point set of the $T$ action,\nthe orthocomplement $\\lietp$ of $\\liet$ in $\\liek$\ninjects into the tangent space $T_x X$ under the infinitesimal\naction of $K$: in other words, that the stabilizer ${\\rm Stab} (x)$\nof $x$ is such that ${\\rm Stab}(x)\/T$ is a finite group.\nUnder this  additional hypothesis,\nwe may indeed prove a somewhat stronger result.\n Notice that by Lemma \\ref{l2.2},\neach term (corresponding to a component $F$ in the fixed point\nset of $T$) in the localization\nformula for $\\pist \\si$ has a factor $\\efo(\\mar \\psi)$ in the denominator.\nNow\nfor $x \\in F$, the fibre $(\\nu_F)_x$ over $x$  of the normal bundle\nto $F$\nwill contain $ \\lietp \\cdot x$, the image of $\\lietp$\nunder the infinitesimal action of $K$. Under the additional assumption\nthat $\\lietp$ injects into $T_x X$,\nthe set of weights for $\\nu_F$ contains for each root $\\gamma > 0$\neither the root $\\gamma$ or the root $-  \\gamma$: in other words,\n$\\efo (\\mar \\psi)$ is divisible by $\\nusym(\\psi)$. Thus from Lemma\n\\ref{l2.2} we obtain\nthe formula for\n$\\nusym(\\psi) \\pist \\si$:\n\\begin{equation} \\label{4.3pinj}\n\\nusym(\\psi) \\pist \\si = \\sum_{F \\in {\\mbox{$\\cal F$}}, \\alpha \\in {\\mbox{$\\cal A$}}_F}\n\\ttfa, \\end{equation}\n\\begin{equation} \\label{4.03pinj} \\ttfa =\n (-1)^{k_F(\\alpha) }  \\frac{ \\epinevb{\\psi}{ \\mu_T(F)}\n }\n{\\prod_j \\Bigl ( \\bfjw (\\mar \\psi) \\Bigr )^{\\tilde{n}_\\fj(\\alpha)} }\n\\int_F ( e^{i \\om} \\tilde{c}_{F,\\alpha} ) . \\end{equation}\nHere,  the notation is as in (\\ref{4.3p})\nand (\\ref{4.03p}) except that\n$\\tilde{n}_\\fj(\\alpha) = n_\\fj(\\alpha)  $ if $\\bfj$ is not a\nroot, while   $\\tilde{n}_\\fj(\\alpha) = n_\\fj(\\alpha) - 1$ if $\\bfj$ is a root.\n\n\n\n We may then use the abelian localization formula\nto give a formula for $F_T (\\nusym \\pist \\si)$ where $\\si = e^{\\iins \\bom}$.\nAs in (\\ref{4.g}),\nwe define a function $\\hh^\\nusym(y) $ for $y \\in \\liets$\nby\n\\begin{equation} \\label{4.ginj}\n\\hh^\\nusym(y) = \\sum_{F \\in {\\mbox{$\\cal F$}}} \\sum_{\\alpha  \\in {\\mbox{$\\cal A$}}_F}\n(-1)^{k_F(\\alpha) } H_{\\hat{\\gamma_F}(\\alpha)  } (-y +\\mu_T(F) )\n\\int_F e^\\omega \\tilde{c}_{F,\\alpha} . \\end{equation}\nThe notation is as in ({\\ref{4.g}) except that\neach $\\bfjw$ appears in $\\hat{\\gamma}_F(\\alpha)$ with multiplicity\n$n_\\fj(\\alpha)  $ if it is not a root and $n_\\fj(\\alpha) - 1 $ if it is a\nroot. The function\n$H_{\\hat{\\gamma_F}(\\alpha) } $ is then as defined in Proposition \\ref{p3.5}.\n\n\n\nIn the case\nwhen the action of $T$ has isolated fixed points, the theorem\n\\ref{t4.1'inj} below\nis the main\nresult\n(Theorem 2.16) of Part I  of the paper \\cite{GP} of Guillemin\nand Prato. For the most part our proof translates directly from\nthe proof given in that paper, except that we use\nthe abelian localization theorem in place of\nstationary phase.\n\n\n\\begin{theorem} \\label{t4.1'inj}\nSuppose\nthat $\\lietp$ injects into $T_x X$ for all fixed points\n$x$ of the action of $T$.\nThen the distribution\n$\\hh^\\nusym$ given in (\\ref{4.ginj}) is identical\nto  the distribution $ \\ggh  = F_T (\\nusym \\pist e^{\\iins \\bom}) $.\n\\end{theorem}\n\\Proof This theorem is proved in exactly\nthe same way as   Theorem \\ref{t4.1'}.\nThe conclusion of the proof goes, as in the\nproof of Theorem 2.16 of \\cite{GP}, by applying\nLemma \\ref{l4} directly to\n$\\hh^\\nusym$ as given in\n(\\ref{4.ginj}) and $\\gggh = F_T (\\nusym \\pist e^{\\iins \\bom} ) $.\n $\\square$\n\n\n\n\n\nIn particular we have the following\n\\begin{prop} \\label{p4.2inj}\nSuppose\nthat $\\lietp$ injects into $T_x X$ for all fixed points\n$x$ of the action of $T$.\nDefine\n\\begin{equation} \\label{4.3ppinj}\nQ(y) = \\nusym(y) F_T (\\nusym \\pist \\si) (y), \\end{equation}\nso that\n\\begin{equation} \\label{4.3pppinj}\n\\ie = \\frac{1}{(2 \\pi i )^s  \\wn  \\,{\\rm vol}\\, ( T) \\epsilon^{s\/2} }\n\\tintt [dy] e^{- \\inpr{y,y}\/{2 \\epsilon}  }  Q(y). \\end{equation}\nThen $Q$ is a piecewise polynomial function supported on\ncones each of which has apex at $\\mu_T(F)$ for some component\n$F$ of the fixed point set of $T$.\n\\end{prop}\n\n\n\nWe now drop the hypothesis that $\\lietp$ injects into $T_x X$\nat the fixed points $x$ of the action of $T$, and\nreturn to the general situation described in\nProposition \\ref{p4.2}.\nIt will follow from (\\ref{5.6}) that $Q$ is smooth near\n$y = 0$: thus in particular there is a polynomial $Q_0$ which\nis equal to $Q$ near $y = 0$.  Of course $Q_0 = D_\\nusym R_0$ where\n$R_0$ is the polynomial which is equal to $\\ft (\\pist \\si)$ near\n$y = 0$.\nIn the next section we shall provide an alternative\ndescription of $Q_0$ and prove\n\\begin{theorem}\n\\label{t4.3}\n$$\n\\ie_0 \\;\\: {\\stackrel{ {\\rm def} }{=} } \\;\\:\n\\frac{1}{ (2 \\pi i)^s   \\wn  \\,{\\rm vol}\\, ( T) \\epsilon^{s\/2} }\n \\tintt [dy] e^{- \\inpr{y,y}\/{2 \\epsilon} }\n\\: Q_0(y)  = e^{\\epsilon \\Theta} e^{i \\om_0} [\\xred]. $$\n\\end{theorem}\nThis tells us that the contribution to\n$\\ie$ from $\\mu^{-1}(0)\/K$ is obtained by integrating\n$Q_0$ rather than $Q$ (weighted by the Gaussian $\\tilde{g}_{\\epsilon^{-1}}$\ndefined in the first paragraph of this Section)\nover $y \\in \\liets$.\n\n\n\n\n{ \\setcounter{equation}{0} }\n\\section{The proof of Theorem 4.7}\n\nThis section gives the proof of Theorem \\ref{t4.3}, which\nidentifies $\\eeth [\\xred]$ with the integral of\na Gaussian $\\gtsoe$ times a polynomial $Q_0$.\nThe key step in the proof is\nthe well known result Proposition  \\ref{p5.2} below, which\ngives a normal form for the symplectic form, the $K$ action\nand the moment map in a neighbourhood ${\\mbox{$\\cal O$}}$ of $\\zloc$.\nWe first recast the distribution $Q$ in terms of an integral\nover $X$ (Proposition \\ref{p5.1}), so that\n$Q(y)$ is given by the integral over $X$ of a distribution\nsupported where $\\mu$ takes the value $y$. (This step occurs\nalso in the  proof \\cite{DH} of the Duistermaat-Heckman theorem.)\nHence, for sufficiently small $y$, this distribution\nis supported in ${\\mbox{$\\cal O$}}$ and we may do the integral over\n$X$ to obtain the value of the polynomial $Q_0 $ which\nis equal to $Q $ near $0$.\nThis turns out to be given by an integral over $\\xred$ involving\nthe symplectic form and the curvature of a bundle\nover $\\xred$ (see (\\ref{5.6})).\nFinally, we multiply $Q_0$\nby the Gaussian $\\gtsoe$ and integrate over $\\liet$\nto see that the result is\n $\\eeth [\\xred]$.\n\n\n\\begin{prop} \\label{p5.1}\n$Q(y) = \\nusym^2(y) (2 \\pi)^{s\/2} \\int_{x \\in X}\ne^{i \\om} \\delta (y - \\mu(x) ) $,\nwhere $\\delta$ denotes the (Dirac) delta distribution.\n\\end{prop}\n\\Proof We have by Lemma \\ref{l3.3} that\n$$Q = \\nusym F_T (\\nusym \\pist \\si ) =\n\\nusym^2 F_K(\\pisk \\si) ,$$\nso that\n$$ Q(y) = \\frac{\\nusym^2 (y) }{(2 \\pi)^{s\/2} }\n \\intk  [d \\phi] \\, e^{-i \\evab{\\phi}{ y} }\n\\int_{x \\in X} e^{i \\om} e^{ i \\eva{\\mu(x)}{ \\phi}  } \\, $$\n \\begin{equation} \\label{5.1}\n= \\nusym^2 (y) (2 \\pi)^{s\/2} \\int_X e^{i \\om} \\delta(\\mu - y).  \\square \\end{equation}\n\nWe would like to study this  for $|y| < h $ for sufficiently\nsmall $h > 0$. Now there is a neighbourhood of $\\mu^{-1}(0)$\non which the symplectic form is given in a standard way\nrelated to the symplectic form $\\om_0$ on $\\xred$: this\nfollows from the coisotropic embedding theorem\n(see   sections 39-41 of \\cite{STP}).\n\\begin{prop} \\label{p5.2} {\\bf (Gotay\\cite{gotay}, Guillemin-Sternberg\n\\cite{STP}, Marle \\cite{marle})}\nAssume $0$ is a regular value of $\\mu$ (so that $\\mu^{-1}(0)  $\nis a smooth manifold and  $K$ acts on $\\mu^{-1}(0) $\nwith finite stabilizers).\nThen there is a neighbourhood ${\\mbox{$\\cal O$}} \\cong \\mu^{-1}(0) \\times\n\\{ z \\in \\lieks, |z| \\le h \\} $\n$\\subseteq \\mu^{-1}(0) \\times \\lieks$  of $\\mu^{-1} (0)$ on which the\nsymplectic form is given as follows. Let\n$P \\;\\: {\\stackrel{ {\\rm def} }{=} } \\;\\: \\mu^{-1}(0) \\stackrel{q}{\\to} \\xred $\nbe the orbifold  principal $K$-bundle given by\nthe projection map $q: \\zloc \\to \\zloc\/K$,\n and let $\\theta$ $ \\in \\Om^1(P) \\otimes \\liek$\nbe a connection\nfor it. Let $\\omr$   denote the induced symplectic form on\n$\\xred$, in other\nwords $q^* \\om_0 = i_0^* \\om$.\nThen if we define a 1-form\n$\\tau$ on ${\\mbox{$\\cal O$}}\\subset P \\times \\lieks$ by\n$\\tau_{p,z}  = z(\\theta)$ (for $p \\in P$ and $z \\in \\lieks$),\n  the symplectic form on ${\\mbox{$\\cal O$}}$\nis given by\n\\begin{equation} \\label{5.2} \\om = q^* \\omr + d \\tau.  \\end{equation}\nFurther,  the moment map on ${\\mbox{$\\cal O$}} $\nis given by $\\mu (p, z) = z$.\n\\end{prop}\n\n\\noindent{\\em Proof of Theorem \\ref{t4.3}:}~\nWe assume for simplicity of notation that $K$\nacts freely on $\\mu^{-1}(0)$, but all of the following\nmay be transferred to the case when $K$ acts with finite\nstabilizers by introducing $V$-manifolds or orbifolds (see \\cite{kaw}).\nIn other words, we work locally on  finite covers\nof subsets of $\\mu^{-1}(0)$ and $\\mu^{-1}(0)\/K$, where the\ncovering group is the stabilizer of the $K$ action at a\npoint $x \\in \\mu^{-1}(0)$.\n\n\nWhen  $|y| < h$ and $h$ is sufficiently small,\nthe distribution $\\delta(\\mu(x)  - y)$\nis supported in ${\\mbox{$\\cal O$}}$, so we may compute\n$Q_0(y)$ from (\\ref{5.1}) by restricting to ${\\mbox{$\\cal O$}}$.\nWe have\n\\begin{equation} \\label{5.3}\nQ_0(y) = (2 \\pi)^{s\/2}  \\nusym^2(y)  \\int_{(p, z') \\in P \\times \\lieks}\ne^{i \\om} \\delta(y - z')  \\end{equation}\n\\begin{equation} \\label{5.4}\n \\phantom{a}  = (2 \\pi)^{s\/2} \\nusym^2(y)\n\\int_{(p, z') \\in P \\times \\lieks}\n\\exp i(q^* \\om_0 + \\evab{d \\theta}{ z'})  \\exp  i\\evab{\\theta}{ dz'} \\delta(y - z')\n\\end{equation}\nNow the term in  $\\exp  i \\evab{\\theta}{ dz'} $ which contributes\nto the integral (\\ref{5.4}) is $i^s \\Om \\,[dz'] $\nwhere $[dz'] $ is the volume form on $\\liek$\n(since all factors $d z'_1 \\dots {dz'}_l$ must appear in\norder to get a contribution to the integral).\nHere, $\\Om  = \\prod_{j = 1}^s \\theta^j$ (for $j$ indexing an\northonormal basis of $\\liek$ and $\\theta^j$ the\ncorresponding components of the connection\n$\\theta$) is a form integrating to $ \\,{\\rm vol}\\, (K)$\nover each fibre of $P \\to \\xred$.\n\nDoing the integral over $z' \\in \\lieks$, we get\n\\begin{equation} \\label{5.5}\nQ_0(y) = i^s \\nusym^2(y) (2 \\pi)^{s\/2}\n\\: \\int_P \\exp\ti (q^* \\omr + \\evab{d \\theta +\n[\\theta, \\theta]\/2 }{ y }\n ) \\: \\Om \\end{equation}\n\\begin{equation} \\label{5.6}\n\\phantom{bbbbb} = i^s  \\nusym^2 (y) (2 \\pi)^{s\/2}  \\: \\int_{\\zloc} \\exp\ni\n(q^* \\omr +  \\evab{F_\\theta }{ y } ) \\, \\Om.\n\\end{equation}\nHere, $F_\\theta = d \\theta +  {\\frac{1}{2} } [\\theta, \\theta]$ is the curvature\nassociated to the connection $\\theta$;\nwe may introduce the term $[\\theta, \\theta]$ into the\nexponential in (\\ref{5.5}) since\nthe additional factors $\\theta$  will give zero under\nthe wedge product with $\\Om$. Formula (\\ref{5.6})\nshows that $Q_0$ is a polynomial in $y$.\n\nNow we were interested in\n$$\\ie_0 = \\frac{1}{(2 \\pi i )^s   \\wn  \\,{\\rm vol}\\, ( T) \\epsilon^{s\/2} }\n\\tintt [dy] e^{- \\inpr{y,y}\/{2 \\epsilon} } Q_0(y) $$\n\\begin{equation} \\label{5.7}\n= \\frac{1 }\n{ (2 \\pi)^{s\/2}  \\wn    \\,{\\rm vol}\\, ( T) \\epsilon^{s\/2} }\n\\tintt [dy] \\nusym^2 (y) \\:  e^{- \\inpr{y,y}\/{2 \\epsilon} } \\:\n\\int_{P} \\exp (q^* \\omr + \\evab{F_\\theta}{ y} ) \\, \\Om, \\end{equation}\n\\begin{equation} \\label{5.9}\n= \\frac{1}{ (2 \\pi \\epsilon)^{s\/2}   \\,{\\rm vol}\\, (K)  } \\int_{z \\in \\lieks}\n\\, [dz] \\, e^{- \\inpr{z,z}\/{2\\epsilon}  } \\: \\int_{P}\n\\exp (q^* \\omr + \\evab{F_\\theta}{z} ) \\, \\Om, \\end{equation}\nwhere the last step uses Lemma \\ref{l3.1} and the\nfact that $\\int_{P} \\exp (q^* \\omr + \\evab{F_\\theta}{ z} )  \\Om$\nis an invariant function of $z$.\n\n\n\n\nWe now regard $F_\\theta$ as a formal parameter and complete the\nsquare to do the integral over $z$: we have\n(identifying $z(F_\\theta)$ with $\\inpr{F_\\theta, z}$  using the invariant\ninner product $\\inpr{\\cdot, \\cdot}$)\n\\begin{equation} \\label{5.10}\n\\int_{z \\in \\lieks} [dz] e^{- \\inpr{z,z}\/{2 \\epsilon} }\n\\: \\exp \\inpr{F_\\theta, z}\n =  (2 \\pi \\epsilon)^{s\/2} \\exp \\epsilon \\inpr{F_\\theta, F_\\theta}\/2. \\end{equation}\nBut $\\inpr{F_\\theta, F_\\theta}\/2$ is just the class\n$\\pi^* \\Theta$ on $P$, for $\\Theta \\in H^4 (\\xred)$.\nHence we obtain (integrating over the fibre of\n$P \\to \\xred$ and using the fact that the\nintegral of $\\Om$ over the fibre is $ \\,{\\rm vol}\\, (K)$)\n\\begin{equation} \\label{5.11}\n\\ie_0 = \\int_{\\xred} \\exp i\\om_0 \\: \\exp \\epsilon \\Theta, \\end{equation}\ncompleting the proof of Theorem \\ref{t4.3}. $\\square$\n\n{ \\setcounter{equation}{0} }\n\\section{The proof of  Theorem 4.1 }\n\nIn this section we complete the proof of Theorem\n\\ref{t4.1}. This is done by observing\nthat $\\ie - \\eeth[\\xred]$ is of the\nform $\\ims \\int_{\\liet} \\gtsoe (Q - Q_0) = $\n$\\int_{\\liet} \\gtsoe D_\\nusym ( R - R_0)$ =\n$\\int_{\\liet} (D_\\nusym^* \\gtsoe) (R-R_0)$, where\n$R - R_0$ is piecewise polynomial and\nsupported away from $0$. (Here,\n$D_\\nusym^* = (-1)^{(s-l)\/2} D_\\nusym.$)\nThe results of  \\cite{Ki1}\nestablish that the   distance\nof any point of Supp($Q - Q_0$) from $0$\nis at least $|\\beta|$ for some nonzero $\\beta$\nin the indexing set ${\\mbox{$\\cal B$}}$ defined in Section 4.\nHence we obtain the estimates in Theorem \\ref{t4.1}.\n\nIn fact the function $R - R_0$ is known explicitly\nin terms of the values of $\\mu_T(F)$ (where $F$ are\nthe components of the fixed point set), the\nintegrals over $F$ of characteristic classes of subbundles\nof the normal bundle $\\nu_F$, and the weights of\nthe action of $T$ on $\\nu_F$ (see (\\ref{2.4}) and\nProposition\\ \\ref{p3.5}). The function $R - R_0$ is\npolynomial on polyhedral regions of $\\liet$,\nso that the quantity  $\\ie -  \\eeth [\\xred]$ can in principle\nbe computed from the integral of a polynomial times a Gaussian\nover these polyhedral regions.\nWe shall study these integrals in another paper and\nrelate them to the cohomology of the higher strata\nin the stratification of $X$ according to the gradient\nflow of $|\\mu|^2$ given in \\cite{Ki1}.\n\n\n\n\n\n\n\n\n\n\n\nWe now   examine $\\ie - \\ie_0$ and prove Theorem \\ref{t4.1}.\nRecall from section 4 that the indexing\nset ${\\mbox{$\\cal B$}}$ of the critical\nsets $C_\\beta$ for the\nfunction $\\rho = |\\mu|^2$ is\n ${\\mbox{$\\cal B$}} = \\liet_+ \\cap W{\\mbox{$\\cal B$}}$ where\n$W {\\mbox{$\\cal B$}} = \\{ w \\beta: \\; \\beta \\in {\\mbox{$\\cal B$}}, \\phantom{a} \nw \\in W \\}$ is the\nset of closest points to $0$\nof convex hulls of nonempty subsets\nof the set $\\{ \\mu_T (F): F \\in {\\mbox{$\\cal F$}}\\}$\nof images under $\\mu_T$ of the connected\ncomponents of the fixed point set of $T$ in $X$.\nWe shall refer to $\\{ \\mu_T (F): F \\in {\\mbox{$\\cal F$}}\\}$  as the\nset of {\\em weights}\\footnote{The motivation for this is\nthat if $X$ is a nonsingular subvariety of complex\nprojective space ${ \\Bbb P}_n$ and $T$ acts\non $X$ via a linear action on ${\\Bbb C }^{n+1} $\nthen each $\\mu_T(F)$ is a weight of this action\nwhen appropriate identifications are made.}  associated to $X$ equipped with\nthe\naction of $T$.\n\nLet ${\\mbox{$\\cal J$}}$ denote the locus\n$${\\mbox{$\\cal J$}} = \\{ y \\in \\liets  : \\, Q \\phantom{a} \n\\mbox{is not smooth at} \\phantom{a}  y\\}. $$\nThen we have\n\\begin{prop}\n${\\mbox{$\\cal J$}} \\subset \\caljab, $ where\n$\\caljab = \\{ y \\in \\liets : \\, \\ft\n(\\pis e^{\\iins \\bom})  $ is not smooth at $y \\}. $\n\\end{prop}\n\\Proof $Q = \\nusym \\ft (\\nusym \\pis e^{\\iins \\bom})$\n$ = \\nusym D_\\nusym\n\\ft(\\pis e^{\\iins \\bom})$.\nHence if $\\ft (\\pis e^{\\iins \\bom})$ is smooth at $y$ then so is\n$Q$. $\\square$\n\nNow it follows from \\cite{JGP} (Section 5) that\n$\\caljab = \\cup_{\\gamma \\in \\Gamma} \\mu_T (V_\\gamma)$ where\n$V_\\gamma$ is a component of the fixed point set of a one parameter subgroup\n$T_\\gamma$ of $T$ and $\\Gamma$ indexes all such one parameter subgroups and\ncomponents of their fixed point sets.\nLet $$D = \\cap \\{ D_\\beta: \\;  \\beta \\in W{\\mbox{$\\cal B$}} - \\{0\\}\n   \\}$$\nwhere $D_\\beta$ denotes the open half-space\n$$ D_\\beta = \\{ y \\in \\liets: \\;  y(\\beta)\n<  |\\beta|^2 \\}. $$\nNote that if $\\beta \\in {\\mbox{$\\cal B$}} -  \\{0\\}$\nthen $D_\\beta$ contains $0$ and its boundary\nis the hyperplane\n$$H_\\beta = \\{ y \\in \\liets: \\; y(\\beta) = |\\beta|^2 \\}.$$\n\\begin{lemma} \\label{l6.2}The support of $Q - Q_0$ is contained in\nthe complement of $D$ (or equivalently\n$Q = Q_0$ on $D$).\n\\end{lemma}\n\\Proof\nSuppose $V_\\gamma$ is a component of the fixed point set of a one\nparameter subgroup $T_\\gamma$.\nBy the Atiyah-Guillemin-Sternberg convexity theorem\n\\cite{aam,gsconv},\n$\\mu_T(V_\\gamma)$ is the convex hull of some subset of the\nweights. Hence the closest point to $0$ in\n$\\mu_T(V_\\gamma)$ is in $W {\\mbox{$\\cal B$}}$.\nNow either this closest point is $0$,\nor else the closest point is a point\n$\\beta \\in\nW{\\mbox{$\\cal B$}} - \\{0\\}$ (in which case $\\mu_T(V_\\gamma) \\subset \\liet - D_\\beta\n\\subset \\liet - D$).\n\nNow if $x$ is a point in Supp($Q - Q_0$) then the ray from\n$0$ to $x$ must pass through at least one point in ${\\mbox{$\\cal J$}}$: hence\nit suffices to prove ${\\mbox{$\\cal J$}} \\subset \\liet - D$ since $D$ is\nthe intersection of a number of\nopen half spaces all of which contain $0$.\n\nBut ${\\mbox{$\\cal J$}} \\subset \\cup_\\gamma \\mu_T (V_\\gamma)$ and every\npoint in $\\mu_T(V_\\gamma)$ is in $\\liet - D$ unless $0$ is\nin $\\mu_T(V_\\gamma)$. Moreover if $x \\in \\mu_T(V_\\gamma) \\cap {\\mbox{$\\cal J$}}$\nand $0 \\in \\mu_T(V_\\gamma)$, we may consider the function\n$Q - Q_0$ restricted to a small neighbourhood of the ray\nfrom $0$ through $x$. This ray lies in the hyperplane\n$\\tilde{H}_\\gamma$ which\nis the orthocomplement in\n$\\liet$ to the Lie algebra $\\liet_\\gamma$\nof $T_\\gamma$. (Since the component of\n$\\mu_T$ in the direction of $\\liet_\\gamma$ is\nconstant along $V_\\gamma$ and since\n$0 \\in \\mu_T(V_\\gamma)$, it follows\nthat $\\mu_T(V_\\gamma)$ is contained in\n$\\tilde{H}_\\gamma$.)  Since\n$Q - Q_0$ is identically zero near $0$ but not near $x$,\nthe ray from $0$ to $x$ must contain a point $x'$ in\n${\\mbox{$\\cal J$}}$ which is contained in some $\\mu_T(V_{\\gamma'})$\nwith $t_\\gamma \\ne t_{\\gamma'}. $ If $0 \\notin\n\\mu_T(V_{\\gamma'})$\n  then $\\mu_T(V_{\\gamma'} ) \\subset \\liet - D$\nand so $x \\in \\liet - D$. If $0 \\in\n\\mu_T(V_{\\gamma'}) $ then we simply\nrepeat the argument, considering the restriction\nof $Q - Q_0$ to a neigbourhood\nof $\\tilde{H}_\\gamma \\cap \\tilde{H}_{\\gamma'}$.\nSince $0 \\notin {\\mbox{$\\cal J$}}$, after finitely many repetitions of\nthis argument we get the required result.\n Hence the Lemma is proved. $\\square$\n\n\n\n\n\n\n\n\n\n\nTo complete the proof of Theorem \\ref{t4.3},\nwe then use Lemma \\ref{l6.2} to express $\\ie - \\ie_0 $ as\n\\begin{equation} \\label{6.3}\n\\ie - \\ie_0\n =  \\frac{1}{(2 \\pi i)^s |W|    \\,{\\rm vol}\\,  T \\epsilon^{s\/2} }\n\\int_{\\liets - D} [dy] (Q - Q_0) e^{- |y|^2\/{2 \\epsilon} } . \\end{equation}\nDenote by $C$ the set\n$\\{ y \\in \\liets  - D  \\: : \\: |Q(y) - Q_0(y) | \\le 1 \\}. $\nThen $$(2 \\pi)^s |W|  \\,{\\rm vol}\\, (T) \\epsilon^{s\/2}\n|\\ie - \\ie_0| \\le  \\int_C [dy]\\stg +\n\\int_{\\liets - D}[dy]  |Q - Q_0|^2 \\stg. $$\nIf $b $ is the minimum value of $|\\beta | $\nover all $\\beta \\in {\\mbox{$\\cal B$}} - \\{0\\}$, then\n$$\\int_C [dy]\\stg  \\le \\int_{|y| \\ge b } [dy] \\stg\n\\le e^{- b^2\/{2 \\epsilon} } q(\\epsilon), $$\nwhere $q(\\epsilon) $ is a polynomial in $\\epsilon^ {\\frac{1}{2} } $.\n\nFurther, denote by $p$ the function  $|Q - Q_0|^2$.\nThen\n\\begin{equation} \\label{6.4}\\int_{\\liets - D} [dy] p(y) \\stg\n  \\le \\sum_{\\beta \\in W{\\mbox{$\\cal B$}} - \\{0\\}}\n\\int_{y \\in D_\\beta}[dy] \\stg p(y). \\end{equation}\nFor each $\\beta \\in W {\\mbox{$\\cal B$}} - \\{0\\} $ one can now decompose\n$y \\in \\liets$ into $y = w_0 \\hatb + w, w \\in \\beta^\\perp$, $w_0 \\in {\\Bbb R }$\n(where $\\hatb = \\beta\/|\\beta|$).\nHence each of\nthe integrals (\\ref{6.4})  is of the form\n$$ \\int_{w_0 \\ge |\\beta| } \\int_{w \\in \\beta^\\perp}\ne^{- w_0^2\/{2 \\epsilon} } e^{ - |w|^2\/{2 \\epsilon} } p(w_0, w), $$\nand this is clearly bounded by $e^{- |\\beta|^2\/{2 \\epsilon} } $ times\na polynomial in $\\epsilon$. This completes the proof of Theorem 4.1.\n$\\square$\n\n\n\n\n\n\n\n\n{ \\setcounter{equation}{0} }\n\\section{Extension of Theorems 4.1 and 4.7 to other classes}\n\nIn this section we extend Theorems \\ref{t4.1} and \\ref{t4.3}\nto equivariant cohomology classes of the form $\\zeta = \\eta e^{\\iins \\bom} $\nwhere $\\eta \\in \\hk(X)$.\n More precisely\nwe shall show the following\n\\begin{theorem} \\label{t7.1}\n Suppose $\\eta \\in \\hk(X)$\nand suppose\nthat\n$i_0^* \\eta \\in \\hk (\\zloc)$ is represented by $\\eta_0 \\in H^*(\\zloc\/K)$\n(where $i_0: \\zloc \\to X$ is the inclusion map). Then\n\n{\\bf(a)} We have that\n$$ \\eta_0 e^{\\epsilon \\Theta}  e^{i \\om_0}\n[\\xred]  = \\frac{1}{(2 \\pi i )^s |W|  \\,{\\rm vol}\\, (T) \\epsilon^{s\/2} }\n\\tintt [dy] e^{- |y|^2\/{2 \\epsilon} } \\,  Q_0^\\eta (y), $$\nwhere $Q^\\eta(y) = \\nusym(y) \\ft (\\nusym \\pist (\\eta \\exp \\iins \\bom) ) $\nand $Q_0^\\eta(y) $ is a polynomial\nwhich is equal to $Q^\\eta(y) $ near $y = 0$.\n\n{\\bf (b)}\nLet\n$ \\rho_\\beta = |\\beta|^2$ be  the\nvalue of the function $|\\mu|^2: X \\to {\\Bbb R }$ on the\ncritical set $C_\\beta$.   Then\nthere exist functions $h_\\beta: {\\Bbb R }^+ \\to {\\Bbb R } $  such that\nfor some $N_\\beta \\ge 0$,\n$\\epsilon^{N_\\beta} h_\\beta(\\epsilon)$ remains bounded as $\\epsilon \\to 0^+$, and\nfor which\n$$\\Bigl | \\,  \\frac{1}{(2 \\pi i )^s  \\,{\\rm vol}\\, (K) } \\intk  [d \\phi]\ne^{-\\epsilon |\\phi|^2\/2} \\pisk \\eta e^{\\iins \\bom}\n - \\eta_0 e^{\\epsilon \\Theta} e^{i \\om_0}  [\\xred]  \\,  \\Bigr | \\le\n\\sum_{\\beta \\in {\\mbox{$\\cal B$}} - \\{0\\} }\n e^{- \\rho_\\beta  \/{2 \\epsilon}  } \\, h_\\beta(\\epsilon). $$\n\n{\\bf (c)} Suppose $\\eta$ is represented by\n$\\tilde{\\eta} = \\sum_J \\eta_J \\phi^J $ $ \\in \\Om^*_K(X)$\nfor\n$\\eta_J \\in \\Om^*(X)$. Then $Q^\\eta$ is of the form\n$Q^\\eta (y) = \\sum_J D_J R_J(y)$, where $D_J$ are\ndifferential operators on $\\liets$ and $R_J$ are piecewise\npolynomial functions on $\\liets$.\n\\end{theorem}\n\\noindent{\\em Proof of } {\\bf(a)}: We examine the function $ \\qeta(y) $ near\n$y = 0$. We have from Lemma \\ref{l3.3} and the paragraph before\nTheorem \\ref{t2.1} that\n$$ \\nusym \\ft (\\nusym \\pist (\\eta e^{\\iins \\bom} ) ) (y)\n = \\Bigl (\\nusym^2 \\fk  \\pisk (\\eta e^{\\iins \\bom} ) \\Bigr )  (y)$$\n$$ = \\frac{\\nusym^2 (y)}{(2 \\pi)^{s\/2} }  \\intk [d \\phi] \\int_{x \\in X }\ne^{- i \\eva{y}{ \\phi} } e^{i \\om} e^{ i \\eva{\\mu(x)}{ \\phi} } \\eta (\\phi). $$\nNow since $\\eta (\\phi) = \\sum_I \\phi^I \\eta_I$ for $\n\\eta_I \\in \\Om^*(X)$  (where the $I$ are multi-indices), we may\ndefine for  any $x \\in X$ a distribution ${\\mbox{$\\cal S$}}_x$ with\nvalues in $\\Lambda^* T^*_x X$ as follows: for any $y \\in \\liets$ we have\n$${\\mbox{$\\cal S$}}_x(y) = \\intk [d \\phi] \\eminev{y}{ \\phi }   e^{i \\om}\n\\epinev{\\mu(x)}{ \\phi} \\sum_I \\eta_I \\phi^I $$\n\\begin{equation}  = \\sum_I (i \\partial\/\\partial y)^I \\intk [d \\phi]\n\\epinev{ (\\mu(x) - y)}{ \\phi} e^{i \\om} \\eta_I \\end{equation}\n$$ = (2 \\pi)^s\n\\sum_I (i \\partial\/\\partial y)^I \\delta(\\mu(x) - y) e^{i \\om} \\eta_I. $$\nThus the distribution ${\\mbox{$\\cal S$}}_x(y)$, viewed as\na distribution $  {\\mbox{$\\cal S$}} (x,y)   $ on $X \\times \\liets$,\n is supported on $\\{ (x , y) \\in X \\times \\liets   \\, | \\,\n\\mu(x) = y \\}. $ Hence for sufficiently small $y$,\n${\\mbox{$\\cal S$}}(x,y) $ (viewed now\nas a function of $x$)   is supported on $ {\\mbox{$\\cal O$}}$ (in the notation\nof Section 5) and we find that\n\\begin{equation} \\qeta(y)  = \\frac{\\nusym^2(y)}{\\htps}\n\\int_{x \\in X} {\\mbox{$\\cal S$}}(x, y) \\end{equation}\n$$ = \\frac{\\nusym^2(y)}{ (2 \\pi)^{s\/2} }  \\intk [d \\phi]  \\int_{P \\times\n\\lieks}\n\\epinev{(\\mu(x)  - y)}{ \\phi}  e^{i \\om} \\eta(\\phi)  . $$\n\nNow consider the restriction of $\\eta$ to $\\hk(P \\times \\lieks) \\cong\n\\hk(P)$ (where $P = \\mu^{-1}(0)$ as in Section\n5). Recall that  there exists $\\eta_0 \\in\n\\Om^*(P\/K)$ such that $\\eta - \\pi^* q^*\\eta_0 = D \\gamma$ for some\n$\\gamma$, where $D$ is the equivariant cohomology differential on\n$P \\times \\lieks$ and $\\pi: P \\times \\lieks \\to P  \\times \\{0\\}$ and\n $q: P \\to \\to P\/K$ are the\nprojection maps.\\footnote{This is because the map\n$i \\circ \\pi: P \\times \\lieks \\to P \\times \\lieks$ is homotopic\nto the identity by a homotopy through equivariant maps, where\n $i: P \\times \\{ 0 \\} \\to\nP \\times \\lieks$ is the inclusion map. Hence\n$i$ induces an isomorphism $i^*: \\hk (P \\times \\lieks)\n\\to \\hk(P \\times \\{0\\}). $}\n We then have that\n\\begin{equation} \\label{7.d3}\nQ^\\eta (y) - \\frac{\\nusym^2(y)}{(2 \\pi)^{s\/2} } \\intk [d \\phi]\n\\int_{x \\in\nP \\times \\lieks} \\epinev{(\\mu(x) - y)}{\\phi} e^{i \\om} \\pi^* q^*\\eta_0  \\end{equation}\n$$ \\phantom{a}  \\phantom{a}  = \\frac{\\nusym^2(y)}{(2 \\pi)^{s\/2} } \\intk [d \\phi]\n\\int_{x \\in P \\times \\lieks} \\epinev{(\\mu(x) - y)}{ \\phi} e^{i \\om}\nD \\gamma \\;\\: {\\stackrel{ {\\rm def} }{=} } \\;\\: \\frac{\\nusym^2(y)}{(2 \\pi)^{s\/2}}  \\bigtriangleup. $$\nBut also\n$$ \\epinev{(\\mu(x) - y)}{ \\phi} e^{i \\om} D \\gamma = D (\n\\epinev{(\\mu(x) - y)}{ \\phi}\ne^{i \\om} \\gamma ) $$\n(since $D \\phi_j = 0 $  and $D \\bom = 0$). Hence\n$\\bigtriangleup = \\intk [ d \\phi] \\int_{P \\times \\lieks}\nd (\\epinev{(\\mu(x) - y)}{ \\phi} e^{i \\om} \\gamma) $ (since for any\ndifferential form $\\Psi$, the term\n$\\iota_{\\tilde{\\phi}}  \\Psi$ in $D \\Psi$\ncannot contain differential forms of top degree in $x$).\nUsing Stokes' Theorem and replacing $P \\times \\lieks$ by\n$P \\times B(\\lieks) $ where $B(\\lieks)$ is a large ball in\n$\\lieks$ with boundary $S(\\lieks)$, we have that\n$$ \\bigtriangleup =\n\\intk [d \\phi] \\int_{P \\times S(\\lieks) }\n\\epinev{(\\mu(x) - y)}{ \\phi} e^{i \\om} \\sum_I \\gamma_I \\phi^I$$\nfor $\\gamma_I \\in \\Om^*(P \\times \\lieks)$. Thus we have\n$ \\bigtriangleup = \\sum_I  (i \\partial\/\\partial y)^I S_I(y) $ where\n$$ S_I (y) =\n\\intk [d \\phi] \\int_{x \\in P \\times S(\\lieks) }\n\\epinev{(\\mu(x) - y)}{ \\phi} e^{i \\om} \\gamma_I. $$\nNow we do the integral\nover $\\phi$ to obtain\n$$ S_I(y)  = (2 \\pi)^{s}  \\int_{x \\in P \\times S(\\lieks)} \\delta (\\mu(x) - y)\ne^{i \\om} \\gamma_I. $$\nThis is zero since the delta distribution is supported\noff $S(\\lieks)$ (recall that $\\mu(p, z) = z$ for\n$(p, z) \\in P \\times \\lieks$). Hence we have that\n$\\bigtriangleup = 0$, and so\n\\begin{equation}  \\qeta(y) = \\frac{\\nusym^2(y)}{(2 \\pi)^{s\/2} } \\intk [d \\phi]\n\\int_{(p, z) \\in P \\times \\lieks} \\epinev{(z - y)}{ \\phi}\ne^{i \\om} \\pi^* \\eta_0, \\end{equation}\nand the rest of the proof is exactly the same as the proof of\nTheorem \\ref{t4.3} which was for the case $\\eta_0 = 1$.\nIn particular the analogue of (\\ref{5.6}) is\n\\begin{equation} \\label{5.6ext}\nQ^\\eta_0(y)  = i^s \\nusym^2 (y) (2 \\pi)^{s\/2}  \\: \\int_{\\zloc}\n\\eta_0 \\exp\n(q^*\\omr +  \\evab{F_\\theta}{  y } ) \\, \\Om \\, ;\n\\end{equation} this equation\nshows that $Q_0^\\eta$ is a polynomial (and in\nparticular smooth) in $y$.\n\n\\noindent{\\em Proof of }{\\bf(b):}   This  is a direct extension of the\nproof of Theorem \\ref{t4.1}, with $Q$ and $Q_0$ replaced by\n$\\qeta$ and $Q^\\eta_0$.\n\n\\noindent{\\em Proof of }{\\bf (c):}  Since $\\eta = \\sum_J \\eta_J \\phi^J$,\nthe  abelian localization formula for\n$\\pis (\\eta e^{\\iins \\bom})$ yields\n\\begin{equation} \\label{7.6n}\n\\nusym(\\phi) \\pis (\\eta e^{\\iins \\bom}) (\\phi) =  \\nusym(\\phi)\n\\sum_{F \\in {\\mbox{$\\cal F$}}} \\rfe(\\phi), \\phantom{bbbbb}\n\\rfe(\\phi) =  \\int_F \\frac{i_F^* \\eta( \\phi) e^{i \\bom(\\phi)} }{e_F(\\phi)},\n\\end{equation} $$ \\phantom{bbbbb} =\n\\sum_J \\phi^J\ne^ {i \\eva{\\mu_T(F)}{ \\phi} }\n \\int_F \\frac{i_F^* \\eta_J e^{i \\om}\n  }{e_F(\\mar \\phi) } $$\n$$ \\phantom{a}  = \\nusym(\\phi) \\sum_J \\phi^J\n\\sum_{F \\in {\\mbox{$\\cal F$}}, \\alpha \\in {\\mbox{$\\cal A$}}_F} \\tfaj(\\phi) , $$\n\\begin{equation} \\label{7.06}\\mbox{where} \\phantom{a}  \\phantom{a}   \\tfaj(\\phi) =\n (-1)^{k_F(\\alpha) }\n\\int_F i_F^* \\eta_J\ne^{i \\om}\n\\tilde{c}_{F,\\alpha}\n   \\frac{ \\epinev{ \\mu_T(F)}{ \\phi }   }\n{\\prod_j \\Bigl ( \\bfjw (\\mar \\phi) \\Bigr )^{{n}_\\fj(\\alpha)} }\n  \\end{equation}\n(and the $\\tilde{c}_{F,\\alpha}$ are as in\n(\\ref{4.004})).\nHere, as in Section 4, we may form a distribution\n\\begin{equation} \\label{7.07}H(y) = D_\\nusym   \\tilde{H},\n\\phantom{a}  \\tilde{H} = \\sum_J (i \\partial\/\\partial y)^J\n\\sum_{F \\in {\\mbox{$\\cal F$}}, \\alpha \\in {\\mbox{$\\cal A$}}_F} (-1)^{k_F(\\alpha)}\n\\Bigl ( \\int_F  i_F^* \\eta_J\ne^{i \\om}\n\\tilde{c}_{F,\\alpha} \\Bigr )  H_{\\bar{\\gamma}_F(\\alpha) } ( \\mu_T(F) - y ), \\end{equation}\nwhere the piecewise polynomial function\n$H_{\\bar{\\beta} }$ is as in Proposition \\ref{p3.5}. Thus $\\tilde{H}$ is of\nthe form $\\tilde{H}(y) = \\sum_J D_J R_J (y)$ where\nthe $R_J(y)$ are piecewise polynomial and\nthe $D_J$ are differential operators.\n\nWe also define the distribution\n\\begin{equation} \\label{7.07'}\nG(y) =  \\ft( \\nusym \\pis \\eta e^{\\iins \\bom}) . \\end{equation}\nThus $\\nusym(y) G(y) = Q^\\eta (y)$.\n\nThen we may show\nthat  the distributions $G$ and $H$ are identical.\nFor we apply Lemma \\ref{l4} as before. We find by Proposition\n\\ref{p3.5}(a) that $(\\ft G)(\\psi) $ and $(\\ft H)(\\psi)  $ are identical\noff the hyperplanes $\\bfj( \\psi) = 0$, so the first\nhypothesis of Lemma \\ref{l4} is satisfied. Moreover,\n$H$ is supported in a half space and\n\\begin{equation} G(y) = \\nusym(y) \\fk (\\pis \\sum_J \\eta_J \\phi^J e^{\\iins \\bom} ) (y) \\end{equation}\n$$ = (2 \\pi)^s \\nusym(y) \\sum_J (i \\partial\/\\partial y)^J \\int_{x \\in X}\n\\eta_J  \\delta(\\mu(x) - y) e^{i \\om}, $$\nso $G$ is supported in the compact set $\\mu(X) \\cap \\liets$ and hence\n$H - G$ is supported in a half space. Thus the second\nhypothesis of Lemma \\ref{l4} is also satisfied and\nwe may conclude that $G = H$, which completes the proof.\n$\\square$\n\n\n\n\n\n\n\n\n\n\n\n\n{ \\setcounter{equation}{0} }\n\\section{Relations in the cohomology ring of symplectic quotients}\nIn this section we shall prove a  formula\nfor the evaluation of cohomology\nclasses from $\\hk(X) $ on the fundamental\nclass of $\\xred$, and apply\nit to study the cohomology\nring $H^*(\\xred)$ in two examples.\n\n\\newcommand{\\gamp}{\\Gamma(P) }\n\\newcommand{\\npl}{ {n_+} }\n\nWe shall prove the following theorem:\n\n\\begin{theorem}\\label{t8.1}\nLet $\\eta \\in \\hk(X)$ induce $\\eta_0 \\in H^*(\\xred)$. Then we have\n\\begin{equation} \\label{8.00}\n\\eta_0  e^{i \\om_0} [\\xred]  =\n\\frac{(-1)^\\npl }{ (2 \\pi)^{s-l} |W|  \\,{\\rm vol}\\, (T) }\n\\treso\n\\Biggl (     \\: \\nusym^2 (\\psi)\n \\sum_{F \\in {\\mbox{$\\cal F$}}} \\rfe(\\psi) [d \\psi] \\Biggr ), \\end{equation}\nwhere\n$$ \\phantom{a}  \\rfe(\\psi) =  e^{i \\mu_T(F) (\\psi) }\n\\int_F \\frac{i_F^* (\\eta(\\psi) e^{i \\omega})  }{e_F(\\mar  \\psi) }\n . $$\n\\end{theorem}\nHere $s$ and  $l$ are the dimensions of $K$ and\nits  maximal torus  $T$,\nand $\\nusym(\\psi) = \\prod_{\\gamma > 0 }\n\\gamma(\\psi)$ where $\\gamma$ runs over the positive roots of $K$;\nthe number of positive roots\n$(s - l)\/2$  is denoted by $\\npl$.\nIf $F \\in {\\mbox{$\\cal F$}}$ (where ${\\mbox{$\\cal F$}}$ denotes the set\nof components of the $T$ fixed point set) then\n$i_F: F \\to X$ is the inclusion of $F$ in $X$ and\n$e_F $ is the equivariant Euler class of the\nnormal bundle to $F$ in $X$.\nThe quantity $\\treso (\\Omega)$\nwill be defined\nbelow (Definition \\ref{d8.5n}).\nThe definition  of $\\treso (\\Omega)$\n will depend on the choice of a cone $\\lasub$, a test function\n$\\testf$, and\n a ray in $\\liets$ specified by\na parameter  $ \\zray \\in \\liets$ -- but in fact if the form $\\Omega$\nis sufficiently well behaved\n(as is the case in (\\ref{8.00}))  then the\nquantity $\\treso (\\Omega) $ will turn out to be\nindependent of these choices. (See  Propositions\n\\ref{p8.6n}, \\ref{p8.7n} and \\ref{p8.8n}.)\n\n\n\n\n\nIn the case of $K = SU(2) $  the result of\nTheorem  \\ref{t8.1} is as follows:\n\n\n\n\n\\begin{corollary} \\label{c8.2}\nLet $K = SU(2)$, and let $\\eta \\in \\hk(X)$ induce\n$\\eta_0 \\in H^*(\\xred)$.\nThen\nthe cohomology class $\\eta_0   e^{i \\om_0}  $ evaluated on\nthe fundamental class of $\\xred$ is given by the\nfollowing formula:\n$$ \\eta_0  e^{i \\om_0}  [\\xred] =\n-\\frac{ 1}{2} {\\rm Res}_0\n\\Biggl (  \\psi^2\n\\sum_{F  \\in {\\mbox{$\\cal F$}}_+ } \\rfe(\\psi) \\Biggr ), \\phantom{a} \n{\\rm where} \\phantom{a}    \\rfe (\\psi) =\ne^{i \\mu_T(F)(\\psi) }\n\\int_F\n\\frac{i_F^* \\eta(\\psi) e^{i \\omega}   }{e_F( \\psi) }.\n$$ Here,\n${\\rm Res_0}$ denotes the coefficient of $1\/\\psi$, and\n${\\mbox{$\\cal F$}}_+$ is the subset of the fixed point set of $T = U(1)$\nconsisting of those components $F$ of the $T$ fixed point set\nfor which $\\mu_T(F) > 0 $.\n\\end{corollary}\nAn important  special case (cf. Witten \\cite{tdg}, Section 2.4) is as follows:\n\\begin{corollary} \\label{c8.00}\nLet $\\eta \\in \\hk(X)$ induce $\\eta_0 \\in H^*(\\xred)$,\nand let $\\Theta \\in H^4(\\xred)$ be induced by the\npolynomial function $- \\inpr{\\phi, \\phi}\/2$\nof $\\phi$, regarded as an element of $H^4_K$.\nThen if $\\epsilon > 0$ we have\n\\begin{equation} \\label{8.010}\n\\eta_0 e^{\\epsilon \\Theta} e^{i \\om_0} [\\xred]  =\n \\frac{(-1)^\\npl }{(2 \\pi)^{(s-l)} |W|  \\,{\\rm vol}\\, (T) }\n\\sum_{m \\ge 0} \\frac{1}{m!} \\treso\n\\Biggl (    (- \\epsilon |\\psi|^2\/2)^m  \\: \\nusym^2 (\\psi)\n \\sum_{F \\in {\\mbox{$\\cal F$}}}  \\rfe(\\psi) [d \\psi] \\Biggr ), \\end{equation}\n$$  \\phantom{bbbbb} {\\rm where} \\phantom{bbbbb}  \\rfe (\\psi) = e^{i \\mu_T(F) (\\psi) }\n\\int_F \\frac{i_F^* (\\eta(\\psi) e^{i \\om})  }{e_F(\\mar  \\psi) }\n. $$\n\\end{corollary}\n\n\nWe  now give a general definition (Definition\n\\ref{d8.5n}) of $\\ress h [d \\psi]$\nwhen $h $ is a meromorphic function on an open\nsubset of $\\liet \\otimes {\\Bbb C }$ satisfying\ncertain growth conditions at infinity. For the restricted class of\nmeromorphic forms of the form\n$e^{i \\lambda(\\psi)} [d \\psi]\/\\prod_j \\beta_j (\\psi), $ Definition \\ref{d8.5n}\nimplies\nthe existence of an explicit procedure for computing these residues\nby successive contour integrations, which is outlined in\nProposition \\ref{p8.4}.\n\nOur definition of the residue will be based on the following\nwell-known result (\\cite{hor}, Theorem 7.4.2 and Remark following\nTheorem 7.4.3):\n\\begin{prop} \\label{cty}\n(i) Suppose $u$ is a distribution on $\\liets$. Then\nthe set $\\gu = \\{ \\xi \\in \\liet: \\: e^{(\\cdot,  \\xi) } u $\nis a tempered distribution$\\}$ is convex. (Here, $(\\cdot, \\cdot)$\ndenotes the pairing between $\\liet$ and $\\liets$.)\n\n\\noindent (ii) If the interior $\\guo$ of $\\gu$ is nonempty,\nthen there is an analytic function $\\hat{u} $ in\n$\\liet + i \\guo$ such that the Fourier transform\nof $e^{(\\cdot,  \\xi) } u $ is $\\hat{u} (\\cdot + i \\xi)$\nfor all $\\xi \\in \\guo$.\n\n\\noindent (iii) For every compact subset $M$ of $\\guo$\nthere is an estimate\n\\begin{equation}  \\label{bds} | \\hat{u} (\\zeta) | \\le C (1 + |\\zeta|)^N, \\phantom{a} \n{\\rm Im} (\\zeta) \\in M. \\end{equation}\n\n\\noindent (iv) Conversely if $\\Gamma$ is an open convex set\nin $\\liet$ and $h$ is a holomorphic function on $\\liet + i \\Gamma$\nwith bounds of the form (\\ref{bds}) for every compact\n$M \\subset \\Gamma$, then there is a distribution\n$u$ on $\\liets$  such that $e^{(\\cdot, \\xi)} u $ is a tempered distribution\nand has Fourier transform\n$h (\\cdot + i \\xi)$ for all $\\xi \\in \\Gamma$.\n\n\\noindent (v)  Finally, if $u$ itself is tempered then the Fourier transform\n$\\hat{u}$\n is the limit  (in the space ${\\mbox{$\\cal S$}}'$\nof tempered distributions)\nof the distribution  $ \\psi \\mapsto\n\\hat{u} (\\psi + i t \\theta)$\nas $t \\to 0^+$, for any $\\theta \\in \\guo$.\n\\end{prop}\n\n\n\n\\begin{definition} \\label{d8.5n}\nLet $\\lasub$ be a (proper) cone in $\\liet$. Let $h$ be a holomorphic\nfunction on $\\liet - i {\\rm Int} (\\Lambda) \\subseteq \\liet \\otimes {\\Bbb C } $\n  such that\nfor any compact subset $M$ of $\\liet - i {\\rm Int} (\\lasub) $ there\nis   an integer $N \\ge 0 $ and a  constant $C$ such that\n$|h (\\zeta)| \\le C (1 + |\\zeta|)^N$\n for  all   $\\zeta \\in M$.\nLet $\\testf: \\lieks \\to {\\Bbb R }$ be a smooth invariant function with\ncompact support and strictly positive  in some  neighbourhood of $0$,\n and let $\\htestf =\n\\fk \\testf: $ $\\liek\n\\to {\\Bbb C }$ be its Fourier transform. Let $\\htestfe(\\phi) =\n\\htestf (\\epsilon \\phi)$ so that $(\\fk \\htestfe)(z) = \\testfe (z) =\n\\epsilon^{-s} \\testf (z\/\\epsilon). $ Assume $\\htestf(0) = 1. $\nThen we define\n\\begin{equation} \\label{8.1n} \\reslch  (h [d \\psi])\n = \\lim_{\\epsilon \\to 0^+} \\frac{1}{(2 \\pi i )^l }\n\\int_{\\psi \\in \\liet - i \\xi} \\htestf (\\epsilon \\psi) h (\\psi) [ d\\psi] \\end{equation}\nwhere $\\xi$ is any element of $\\Lambda$.\n \\end{definition}\n\nBy the Paley-Wiener theorem (\\cite{hor}, Theorem 7.3.1), for any\nfixed $\\xi \\in \\liek$ the function $\\htestf (\\psi - i \\xi) =\n\\fk(e^{- \\inpr{\\xi, \\cdot} } \\testf)(\\psi) $ is rapidly decreasing\nsince $\\fk \\htestf = \\testf$ is smooth and compactly supported. Hence\nthe integral (\\ref{8.1n}) converges. Now the function $\\htestf$ extends\nto a holomorphic function on $\\liek \\otimes {\\Bbb C }$ and in\nparticular on $\\liet \\otimes {\\Bbb C }$ (Proposition \\ref{cty}), and\nby assumption  $h$ extends\nto a holomorphic function on\n$\\liet - i \\, {\\rm Int} (\\lasub)$;\n hence, by Cauchy's theorem, the integral\n(\\ref{8.1n}) is  independent of $\\xi \\in {\\rm Int}  (\\lasub)$.\n\n\nThe independence of $\\reslch (\\Omega)$ of the\nchoices of $\\lasub$ and $\\testf$ when $\\Omega$ is sufficiently\nwell behaved is established by the next results.\n\\begin{prop} \\label{p8.6n}\n Let $h: \\liek \\to {\\Bbb C }$ be a\n$K$-invariant function. Assume that\n$\\fk h$ is compactly supported; it then follows that $h: \\liek \\to {\\Bbb C }$\nis smooth (\\cite{hor}, Lemma 7.1.3) and extends to a\nholomorphic function on $\\liek \\otimes {\\Bbb C }$ (Proposition \\ref{cty}).\n Then $\\reslch ( h [d \\psi]  ) $\nis independent of the cone $\\lasub$.\n\\end{prop}\n\n\n\\Proof As above, define $\\testfe (z) = \\epsilon^{-s} \\testf (z\/\\epsilon),$\nso that $\\htestfe (\\phi) = \\htestf(\\epsilon \\phi). $\nThe function $h$ extends in particular to a holomorphic\nfunction on $\\liet \\otimes {\\Bbb C }$, and by the remarks after\nDefinition \\ref{d8.5n}, the function $\\htestfe$ also extends to a\nholomorphic function on $\\liet \\otimes {\\Bbb C }$.\nHence Cauchy's theorem shows that\nfor any choice of the cone $\\lasub$,\n\\begin{equation} \\label{8.4n}\n\\reslch ( h(\\psi) \\dps ) =  \\lim_{\\epsilon \\to 0^+}\n\\frac{1}{(2 \\pi i)^l}\n \\intt \\htestf (\\epsilon\\psi) h(\\psi) [d \\psi]. \\phantom{bbbbb} \\square    \\end{equation}\n\\noindent{\\em Remark:} If $h$ is as in the statement of\nProposition \\ref{p8.6n},  then $\\nusym^2 h$ also satisfies the hypotheses\nof the Proposition, so\n$\\reslch(\\nusym^2 h [d \\psi] ) $ is also independent of the cone\n$\\lasub$.\n\n\nThe following is a consequence of Proposition \\ref{cty}:\n\n\n\\begin{prop} \\label{p8.7n}\n Let $u: \\liets \\to {\\Bbb C }$ be a distribution,\nand assume the set $\\gu$ defined in Proposition \\ref{cty} contains\n$- {\\rm Int} (\\Lambda)$. Thus $h = \\ft u$ is a holomorphic\nfunction on $\\liet - i {\\rm Int} \\Lambda$, satisfying the\nhypotheses in Definition \\ref{d8.5n}. Suppose in addition\nthat $u$ is smooth at $0$. Then\n$\\reslch (h \\dps) $ is independent of the test function $\\chi$, and\nequals $i^{-l} u(0)\/(2 \\pi)^{l\/2} $.\n\\end{prop}\n\n\nWe shall be dealing with functions $h: \\liek \\to {\\Bbb C }$ whose Fourier\ntransforms are smooth at $0$ but which are sums of other\nfunctions not all of whose Fourier transforms need be smooth\nat $0$. For this reason we must introduce a small generic parameter\n$\\zray \\in \\liets$ where all the functions in this sum are smooth.\nMore precisely we make the following\n\\begin{definition} \\label{d8.7n} Let $\\lasub, $ $\\testf$ and $h$ be as in\nDefinition \\ref{d8.5n}. Let $\\zray \\in \\liets$ be such that the\ndistribution $\\ft h$ is smooth on the ray $t \\zray$ for  $t \\in (0, \\delta)$\nfor some $\\delta > 0 $,\nand suppose $(\\ft h)(t \\rho) $ tends to a well defined limit\nas $t \\to 0^+$. Then we\ndefine\n\\begin{equation} \\label{8.6n}  \\resrlch (h  \\dps)\n= \\lim_{t \\to 0^+} \\reslch \\Bigl ( h (\\psi )\ne^{i t \\zray (\\psi) }  \\dps \\Bigr ). \\end{equation}\n\\end{definition}\nUnder these hypotheses, $\\resrlch ( h \\dps) $ is independent of $\\testf$\n(by Proposition \\ref{p8.7n}), but it may depend on the\nray $\\{ t \\zray: t \\in {\\Bbb R }^+ \\}. $ However\nwe have  by Proposition \\ref{p8.7n}\n\\begin{prop} \\label{p8.8n} Suppose $\\ft h $ is smooth at $0$. Then\nthe quantity $\\resrlch ( h \\dps)$ satisfies\n$\\resrlch (h \\dps) = \\reslch ( h \\dps) $.\n\\end{prop}\n\n\\noindent{\\em Remark:} In the light of Propositions \\ref{p8.6n},\n\\ref{p8.7n} and \\ref{p8.8n}, it makes sense to write\n${\\rm Res} (\\Omega) $ for $\\resrlch (\\Omega) $ when\n$\\Omega = \\nusym^2 h \\dps $ for  a\n$K$-invariant function $h: \\liek \\to {\\Bbb C }$ for which $\\fk h$ is compactly\nsupported and $\\ft (\\nusym^2 h)$ is smooth at $0$. In the proof of\nTheorem \\ref{t8.1} which we are about to give, we shall check\nthe validity\nof these hypotheses for the form $\\Omega$ which appears in\nthe statement of the Theorem.\n\nWe now give the proof of Theorem \\ref{t8.1}. We shall first show\n\n\n\n\\begin{prop} \\label{p7.6}\n\n{\\noindent (a)} The distribution\n$\\fk(\\pisk \\eta e^{i \\bom} ) (\\zp) $  defined for\n$\\zp \\in \\liek$ is represented by a smooth\nfunction for $\\zp$ in a sufficiently small neighbourhood of $0$.\n\n{\\noindent (b) }  We have\n\\begin{equation}  \\label{7.001} \\eta_0   e^{i \\om_0}\n[\\xred]  = \\frac{1}{(2 \\pi)^{s\/2}i^s\n \\,{\\rm vol}\\, (K) } F_K (\\pis \\eta e^{\\iins \\bom}) (0),  \\end{equation}\n\n\\begin{equation} \\label{7.002} \\phantom{ \\eta_0   e^{i \\om}\n[\\xred]}   = \\frac{(2 \\pi)^{l\/2} }{(2 \\pi)^{s}\n|W|  \\,{\\rm vol}\\, (T)i^s } F_T (\\nusym^2 \\pis \\eta e^{\\iins \\bom}) (0). \\end{equation}\n\\end{prop}\n\n\\noindent{\\em Proof of (a):}\n To evaluate $\\fk(\\pis \\eta e^{i \\bom} ) (0)$, we introduce\na test function $\\testf: \\lieks \\to {\\Bbb R }^+$ which is smooth and\nof compact support, and for which $(\\fk \\testf) (0) = 1\/(2 \\pi)^{s\/2}. $\nWe define $\\testfe(z) = \\epsilon^{-s} \\testf(z\/\\epsilon) $; as $\\epsilon \\to 0 $,\nthe functions $\\testfe $ tend to the Dirac delta distribution on $\\lieks$\n(in the space ${\\mbox{$\\cal D$}}'$ of distributions on $\\lieks$).\nThen we have\n\\begin{equation} (\\fk \\testfe) (\\phi) = (\\fk \\testf) (\\epsilon \\phi). \\end{equation}\nNow to evaluate $\\fk (\\pis \\eta e^{i \\bom} )  (\\zp)$, we integrate it against\nthe sequence of test functions $\\testfe$:\n\\begin{equation} \\fk ( \\pis \\eta e^{i \\bom} ) (\\zp) = \\lim_{\\epsilon \\to 0^+}\ni^s  \\je (\\zp) \\end{equation}\nwhere\n\\begin{equation} i^s \\je (\\zp)= \\tintk [d z] \\fk (\\pis \\eta\ne^{i \\bom} ) (z) \\chi_\\epsilon (z - \\zp)  \\end{equation}\n$$ \\phantom{bbbbb} = \\intk [d \\phi] \\int_{x \\in X} \\eta(\\phi) e^{i \\om}\ne^{i (\\mu(x) - \\zp)  (\\phi) }  \\htestfe(\\phi) $$\n(by Parseval's Theorem).\nNow because $\\testfe$ is smooth and of compact support, the Paley-Wiener\nTheorem (Theorem 7.3.1 of \\cite{hor}) implies that $\\htestfe$ is rapidly\ndecreasing. So we may use Fubini's theorem to interchange the order\nof integration and get\n\\begin{equation} \\label{7.t1} i^s \\je(\\zp)  = \\int_{x \\in X } e^{i \\om} \\int_{ \\phi \\in\n\\liek}\n[d \\phi] \\eta(\\phi) \\htestfe(\\phi) e^{i (\\mu(x) - \\zp) (\\phi)} \\end{equation}\n\\begin{equation} \\label{7.t2} \\phantom{bbbbb} = (2 \\pi)^{s\/2}\n\\int_{x \\in X} e^{i \\om} \\eta( - i \\partial\/\\partial z) \\testfe(z)|_{z =\n\\mu(x) - \\zp}. \\end{equation}\nAs $\\epsilon \\to 0$, $\\testfe$ is supported on an arbitrarily\nsmall neighbourhood of $\\mu^{-1}(0)$. Thus, because of\nProposition  \\ref{p5.2},\nthe integral may be replaced by an integral over $P \\times \\lieks$:\n\\begin{equation} i^s \\je(z') = (2 \\pi)^{s\/2} \\int_{(p,z) \\in P \\times \\lieks}\ne^{i \\om} \\eta( - i \\frac{ \\partial }{\\partial z}) \\testfe(z - \\zp)\n\\end{equation}\nNow by the same argument as given in the proof of Theorem \\ref{t7.1}(a),\nthere is $\\eta_0 \\in \\Omega^*(P\/K) $ such that\n\\begin{equation} \\eta - \\pi^* \\eta_0 = D \\gamma \\phantom{a}  \\mbox{for some $\\gamma$, } \\end{equation}\nwhere $D$ is the equivariant cohomology differential on $P \\times \\lieks$\nand $\\pi: P \\times \\lieks \\to P \\to P\/K$ is the projection map.\nBy the argument given after (\\ref{7.d3}), we have\n\\begin{equation} i^s \\je(\\zp)  - (2 \\pi)^{s\/2} \\int_{(p,z) \\in P \\times \\lieks}\ne^{i \\om} (\\pi^* \\eta_0) \\testfe(z - \\zp)  \\end{equation}\n$$ \\phantom{bbbbb} = \\int_{(p,z) \\in P \\times \\lieks} e^{i \\om}\n\\int_{ \\phi \\in \\liek} [d \\phi] D \\gamma(\\phi) \\htestfe(\\phi)\ne^{i (z - \\zp)( \\phi) } $$\n$$ = \\int_{ \\phi \\in \\liek} [d \\phi] \\htestfe( \\phi)\n\\int_{ (p,z) \\in P \\times \\lieks } e^{i \\om} D \\gamma (\\phi)\ne^{i (z - \\zp) (\\phi) } \\;\\: {\\stackrel{ {\\rm def} }{=} } \\;\\: \\deleps. $$\nBut\n\\begin{equation} e^{i \\om} e^{i (z-\\zp)(\\phi) }\nD \\gamma = D (e^{i \\om} e^{i (z - \\zp) (\\phi) }\n\\gamma(\\phi) ). \\end{equation}\nHence\n\\begin{equation} \\deleps = \\intk [d \\phi] \\htestfe (\\phi)\n\\int_{(p,z) \\in  P \\times \\lieks}\nd ( e^{i \\om} e^{i (z - \\zp) (\\phi) } \\gamma(\\phi) ) \\end{equation}\n(since for  any differential form $\\Psi$, the term\n$\\iota_{\\tilde{\\phi}}  \\Psi$ in $D \\Psi$\ncannot contain differential forms of top degree in $x$).\nReplacing $P \\times \\lieks$ by\n$P \\times B_R(\\lieks)$ where $B_R (\\lieks)$ is a large ball in $\\lieks$\nwith boundary $S_R (\\lieks)$, we have that\n\\begin{equation} \\deleps = \\lim_{R \\to \\infty}\n\\intk [d \\phi] \\htestfe( \\phi)\n\\int_{(p,z) \\in P \\times B_R (\\lieks) } d \\Bigl (\ne^{i \\om} e^{i (z - \\zp) (\\phi) }\n\\gamma( \\phi)    \\Bigr ) . \\end{equation}\nWe then interchange the order of integration to get\n\\begin{equation} \\deleps =  \\lim_{R \\to \\infty}\n\\int_{(p,z) \\in P \\times B_R (\\lieks) } d \\left (\ne^{i \\om} \\intk [d \\phi] \\htestfe ( \\phi) e^{i (z - \\zp) (\\phi) }\n\\gamma( \\phi) \\right ) \\end{equation}\n$$ \\phantom{bbbbb} =  \\lim_{R \\to \\infty}\n\\int_{(p,z) \\in P \\times B_R(\\lieks)  } d\n\\Bigl (e^{i \\om}\n\\gamma( \\frac{\\partial}{\\partial z}) \\testfe (z - \\zp) \\Bigr  ) . $$\n$$ \\phantom{bbbbb} =  \\lim_{R \\to \\infty}\n\\int_{(p,z) \\in P \\times S_R(\\lieks)  }\n\\Bigl (e^{i \\om}\n\\gamma( \\frac{\\partial}{\\partial z}) \\testfe (z - \\zp) \\Bigr  )\n\\phantom{a}  \\mbox{by Stokes' Theorem}. $$\nThis limit\nequals $0$ for sufficiently\nsmall $\\zp$ since $\\testfe$ is\ncompactly supported on $\\lieks$.  Hence $\\deleps = 0 $.\n\nFinally we obtain using the expression for $\\omega$\ngiven in Proposition \\ref{p5.2}\n\\begin{equation} i^s \\je(\\zp)  = (2 \\pi)^{s\/2} \\int_{(p,z) \\in P \\times \\lieks} e^{i \\om}\n(\\pi^* \\eta_0) \\testfe (z - \\zp), \\end{equation}\n\\begin{equation} \\phantom{bbbbb} = ( 2 \\pi)^{s\/2} \\int_{ (p,z) \\in P \\times \\lieks}\ne^{i \\pi^* \\om_0} (\\pi^* \\eta_0) e^{i \\theta(dz)} e^{i d \\theta(z) }\n\\testfe (z - \\zp). \\end{equation}\nThus we have as in Section 5\n\\begin{equation} \\label{8.26}\n \\lim_{\\epsilon \\to 0 } \\je (\\zp) = ( 2 \\pi)^{s\/2} \\int_{(p,z) \\in\nP \\times \\lieks} e^{i \\pi^* \\om_0} (\\pi^* \\eta_0)\n\\Om [dz] e^{i F_\\theta (z)} \\delta(z - \\zp) \\end{equation}\n$$ = ( 2\\pi)^{s\/2} \\int_P e^{i \\pi^* \\om_0}\n (\\pi^* \\eta_0)\ne^{i F_\\theta(\\zp) }\n\\Om $$\nwhere $\\Omega$ is the differential form introduced after (\\ref{5.4}).\nThis shows in particular that $(\\fk  \\rbare)(\\zp)$\n (where $\\rbare (\\phi) = \\pis (\\eta e^{i \\bom})( \\phi) $)\nis a\npolynomial in $\\zp$ for small $\\zp$, and hence is smooth  in\n$\\zp$ for $\\zp$ sufficiently close to $0$. This completes the\nproof of (a).\n\nWhen $\\zp = 0 $,  equation (\\ref{8.26})\nbecomes\n$$\\lim_{\\epsilon \\to 0}\n\\je (0) =\ni^{-s} F_K (\\pis \\eta e^{\\iins \\bom})(0) = {(2 \\pi)^{s\/2}    \\,{\\rm vol}\\, (K) }\n\\eta_0 e^{i \\om_0} [\\xred], $$\nwhich proves (\\ref{7.001}).\nUsing Lemma \\ref{l3.1} again, we have for any\n$K$-invariant function $f$ on $\\liek$ that\n\\begin{equation} \\label{7.003}\n\\frac{(2 \\pi)^{s\/2} }{  \\,{\\rm vol}\\, (K)  }\n(F_K f) (0) =\n\\frac{( 2 \\pi)^{l\/2} }{|W|  \\,{\\rm vol}\\, (T) }\nF_T(f \\nusym^2) (0). \\end{equation}\nCombining this with (\\ref{7.001}) we obtain\n$$\n\\eta_0 e^{i \\om_0}[\\xred]\n   = \\frac{(2 \\pi)^{l\/2} }{(2 \\pi)^{s}\n|W|  \\,{\\rm vol}\\, (T) i^s  } F_T (\\nusym^2 \\pis \\eta e^{\\iins \\bom}) (0), $$\nwhich is (\\ref{7.002}). $\\square$\n\nTheorem \\ref{t8.1} follows from Proposition \\ref{p7.6} and\nProposition \\ref{p8.7n}\nby applying\nTheorem \\ref{t2.1}  to decompose $r^{{\\eta}}  =\n\\pis (\\eta e^{i \\bom} ) $ as a sum\nof meromorphic functions $\\rfe$ on $\\liet \\otimes {\\Bbb C } $ corresponding\nto the components $F$ of the fixed point set of $T$.  We now\ncomplete the proof of this theorem.\n\n\\noindent{\\em Proof of Theorem \\ref{t8.1}:} As in (\\ref{7.6n}),\nthe abelian localization formula yields\n$ r^{{\\eta} } (\\psi) = \\sum_{F \\in {\\mbox{$\\cal F$}}}  \\rfe (\\psi), $\nwhere\n$$ \\rfe(\\psi) = e^{i \\mu_T(F) (\\psi) } \\int_F\n\\frac{i_F^* \\eta(\\psi) e^{i \\om} } {e_F(\\psi) }. $$\nNow the distribution $(\\fk r^{{\\eta} } ) (\\phi) $ is represented\nby a smooth function near $0$ (Proposition \\ref{p7.6} (a));\nalso, the distribution $\\ft(\\nusym^2 r^{{\\eta} } ) $\n$ = D_{\\nusym} \\ft (\\nusym \\rbare) $ is represented\nby a smooth function near $0$ since $\\ft (\\nusym r^{{\\eta} } )  =\n\\nusym \\fk r^{{\\eta} } $ (Lemma\n\\ref{l3.3}) and $\\fk r^{{\\eta} } $ is smooth near $0$ (Proposition\n\\ref{p7.6}(a)).\nWe choose $ \\zray \\in \\liets$ so that the distribution\n$\\ft \\rfe$ is smooth along the ray $t \\zray, t \\in (0, \\delta) $, for all\n$F$ and sufficiently small $\\delta > 0 $, and that this distribution\ntends to a well defined limit as $t \\to 0^+$:\nthis is possible because the $\\rfe (\\psi)$ are sums of\nterms of the form $e^{i \\mu_T(F) (\\psi) } \/ \\prod_j \\bfj(\\psi)^{n_j} $\nso their Fourier transforms $\\ft \\rfe$ are piecewise polynomial\nfunctions of the form $H_\\barb (y) $ (see Proposition \\ref{p3.5}).\nThese functions are smooth on the set $U_\\barb$ consisting of all points\n$y$ where $y$ is not in the cone spanned by any subset of the\n$\\bfjw$ containing less than $l$ elements. Thus\n$$\\lim_{t \\to 0^+} (2 \\pi)^{-l\/2}  i^{-l}   \\ft \\rfe(t \\zray) =\n \\resrlch (\\rfe \\dps) $$ by\nDefinition \\ref{d8.5n} and Proposition \\ref{p8.7n}. It follows that\n\\begin{equation} \\label{8.7n}\n(2 \\pi)^{-l\/2} i^{-l}  \\ft (\\nusym^2 \\rbare) (0) =\n\\resrlch (\\nusym^2 \\rbare \\dps ) \\end{equation}\n\\begin{equation}  \\label{8.8n} \\phantom{bbbbb}  = \\sum_{F \\in {\\mbox{$\\cal F$}}} \\resrlch (\\nusym^2 \\rfe).  \\end{equation}\nThe residue in (\\ref{8.7n}) is independent of $\\testf$, $\\lasub$ and $\\zray$\nby Propositions \\ref{p8.6n}, \\ref{p8.7n} and\n\\ref{p8.8n}. The residues in (\\ref{8.8n}) are independent of $\\testf$ by\nProposition \\ref{p8.7n}, but they may depend on $\\zray$\nand $\\lasub$.\n\nTo conclude the proof of Theorem \\ref{t8.1} we note that (\\ref{7.002})\ngives\n\\begin{equation}\n  \\eta_0   e^{i \\om}\n[\\xred]   = \\frac{(2 \\pi)^{l\/2} }{(2 \\pi)^{s}\n|W|  \\,{\\rm vol}\\, (T)i^s } F_T (\\nusym^2 \\pis \\eta e^{\\iins \\bom}) (0),  \\end{equation}\n$$   = \\frac{i^l  }{(2 \\pi)^{s-l}\n|W|  \\,{\\rm vol}\\, (T)i^s }  \\reso(\\rbare(\\psi)    \\nusym^2(\\psi)   [ d\\psi]) \\phantom{a} \n\\mbox{(by Proposition \\ref{p8.7n})} $$\n$$  = \\frac{-1)^\\npl }{ (2 \\pi)^{s-l}  |W|   \\,{\\rm vol}\\, (T) }   \\reso\n\\Biggl (  \\sum_{F \\in {\\mbox{$\\cal F$}}} \\rfe(\\psi)  \\nusym^2 (\\psi) [d \\psi]\n\\Biggr ) $$\nas claimed. $\\square$\n\n\\noindent{\\em Proof of Corollary 8.2:} In a normalization\nwhere $ \\,{\\rm vol}\\, (T) = 1$, the factor $\\nusym(\\psi) $ becomes $2 \\pi \\psi$.\nThis gives\n$$ \\frac{1}{(2 \\pi)^{s-l}}\\treso\n\\Biggl (     \\: \\nusym^2 (\\psi)\n \\sum_{F \\in {\\mbox{$\\cal F$}}} \\rfe(\\psi) \\dps \\Biggr )\n= \\reslch  (  \\psi^2\n \\sum_{F \\in {\\mbox{$\\cal F$}}} \\rfe(\\psi) [d \\psi] ). $$\nEach term $\\rfe (\\psi) $ is a sum of terms of the form\n$\\tau_\\alpha (\\psi) = c_{\\alpha} \\psi^{-n_\\alpha}  e^{i \\mu(F) \\psi}  $ for some constants\n$c_\\alpha$ and integers $n_\\alpha$.\nBy (\\ref{8.1n}), the residue is given by\n\\begin{equation} \\label{8.001} \\reslch (h \\dps) =\n\\lim_{ \\epsilon \\to 0^+ }\n\\frac{1} {2 \\pi i} \\int_{ \\psi \\in {\\Bbb R } - i \\xi}\n\\htestf (\\epsilon \\psi) h(\\psi) \\dps, \\end{equation}\nwhere we choose $\\xi$ to be in the cone $\\lasub = {\\Bbb R }^+$.\nProposition \\ref{cty} (i), (ii) shows that\nthe function $\\htestf: {\\Bbb R } \\to {\\Bbb C }$ extends to an entire\nfunction on ${\\Bbb C }$.\n\n\n\nWe may now  decompose the integral\n in (\\ref{8.001}) into terms corresponding to the $\\tau_\\alpha$. If\n$n_\\alpha > 0 $, we\ncomplete each such integral over ${\\Bbb R } - i \\xi$ to a\ncontour integral by adding a semicircular curve  at infinity, which\nis in the upper half plane if $\\mu(F) > 0 $ and in the lower half\nplane if $\\mu(F) < 0 $. This choice of contour is  made so that\nthe function\n$\\tau_\\alpha (\\psi) $  is bounded on the added\ncontours, so the added semicircular curves  do not contribute to\nthe integral. Since only the contours corresponding to values\nof $F$ for which $\\mu(F) > 0 $  enclose the pole at $0$, application\n of Cauchy's residue formula  now gives the result.\nA similar argument establishes that the terms  $\\tau_\\alpha $ for which\n$n_\\alpha \\le  0 $ contribute $0$ to the sum.\\footnote{The\nformula obtained by choosing $\\lasub = {\\Bbb R }^-$ is in fact\nequivalent to the formula we have\nobtained using the choice $\\lasub = {\\Bbb R }^+$.\n This can be seen directly from the Weyl invariance\nof the function $\\rbare$, where the action of the Weyl group takes\n$\\psi$ to $- \\psi$ and so converts terms with $\\mu(F) > 0  $ to\nterms with $\\mu(F) < 0 $.}\n$\\square$\n\n\n\n\n\\noindent{\\em Remarks:}\n\n (a) The quantity $\\resrlch  (\\nusym^2 \\rfe \\dps ) $ depends on\nthe cone $\\lasub$ for each $F$;\nhowever, it follows from Proposition\n\\ref{p8.6n} that the sum $\\sum_{F \\in {\\mbox{$\\cal F$}}}\n\\resrlch  ( \\nusym^2 \\rfe \\dps ) $ is independent of $\\lasub$. }\n\n\\noindent (b) Let ${\\mbox{$\\cal F$}}_\\lasub$ be the set of those $F \\in {\\mbox{$\\cal F$}}$ for which\n$\\mu_T(F)$ lies in the cone $\\dcf =\n\\{ \\sum_j s_j \\bfjw: s_j \\ge 0 \\} $ (defined in  Section 4) spanned\nby the $\\bfjw$. Then by\nProposition \\ref{p8.4} (iii), $\\resrlch ( \\nusym^2 \\rfe ) = 0 $ if\n $F \\notin {\\mbox{$\\cal F$}}_\\lasub$,\nso in fact $\\resrlch (\\nusym^2  \\rbare \\dps)  =\n\\resrlch \\sum_{F \\in {\\mbox{$\\cal F$}}_\\lasub} ( \\nusym^2 \\rfe  \\dps ) $.\n\n\n\\noindent (c)  Finally, it follows from Proposition \\ref{p8.4} that\nif we replace the symplectic form\n$\\omega$ and the moment map $\\mu$ by $\\delta \\omega$ and $\\delta \\mu$\n(where $\\delta > 0 $) and then let $\\delta $ tend to $0$, we obtain\nan expression where $\\mu$ and $\\omega $ appear only in determining the\nset ${\\mbox{$\\cal F$}}_\\lasub$ indexing terms which yield a nonzero contribution.\n\n\nWe now restrict ourselves to the special case when $\\Omega$ is of the\nform $$\\Omega_\\lambda(\\psi)  = e^{i \\lambda(\\psi)}\n[d \\psi]\/\\prod_{j = 1}^N \\beta_j (\\psi).$$\nIf $\\beta_j \\in \\lasub $ then the distribution\n$\\resrlch (\\Omega_\\lambda) $ is just (up to multiplication by a\nconstant) the piecewise polynomial function $H_\\barb(\\lambda)$ from Proposition\n\\ref{p3.5}.\nWe shall now give  a Proposition\n which gives a list of properties satisfied\nby the residues $\\reso (\\Oma{\\lambda})$: these properties in fact characterize the\nresidues uniquely and enable one to compute them.\n\n\n\\newcommand{\\pee}{P}\n\\begin{prop} \\label{p8.4} Let $\\xi \\in \\liet$ and\nsuppose\n $\\beta_1, \\dots, \\beta_N \\in \\liets$ are\nall in the dual cone of a cone $\\lasub \\in\n\\liet$.\nDenote by $\\pee: \\liet \\to {\\Bbb R } $ the\nfunction $\\pee(\\psi) = \\prod_j \\beta_j(\\psi)$.\nSuppose  $\\lambda \\in U_\\barb \\subset \\liets$ (see\nProposition \\ref{p3.5}), and\ndefine $\\Oma{\\lambda}(\\psi) =\ne^{i \\lambda(\\psi)}[d \\psi]\/\\pee(\\psi) $.\n\n\n\nThen we have\n\\begin{description}\n\\item[(i)]\n $\\reso (\\psi_k \\psi^J \\Oma{\\lambda})\n= (- i \\partial\/\\partial \\lambda_k) \\reso (\\psi^J \\Oma{\\lambda}).\n$\n\\item[(ii)] $(2 \\pi i)^l \\reso (\\Oma{\\lambda})\n=  i^N H_\\barb (\\lambda)$, where $H_\\barb$ is\nthe distribution given in Proposition \\ref{p3.5}. (Recall we have assumed\nthat\n$\\beta_j $ is in the dual cone of\n$ \\lasub $ for all $j$.)\n\\item[(iii)] $\\reso (\\Oma{\\lambda}) = 0 $ unless $\\lambda$ is in the cone\n$C_\\barb$ spanned\nby the $\\beta_j$.\n\\item[(iv)] $$\\lim_{s \\to 0^+} \\reso \\Bigl (\n\\Oma{s \\lambda} \\psi^J\n\\Bigr ) = 0$$\nunless $ N  - |J| =  l$.\n\\item[(v)]\n$$\\lim_{s \\to 0^+} \\reso \\Bigl (\n\\Oma{s \\lambda} \\psi^J\n\\Bigr ) = 0 $$\nif the monomials $\\beta_j$ do not span $\\liets$.\n\\item[(vi)] If\n$\\beta_1, \\dots, \\beta_l$ span $\\liets$ and\n$\\lambda = \\sum_j \\lambda^j \\beta_j$\nwith all $\\lambda^j > 0$,\n then\n$$ \\lim_{s \\to 0^+} \\reso\n\\Bigl ( \\frac{ e^{i s \\lambda(\\psi) } [d \\psi] }{\\beta_1(\\psi) \\dots\n\\beta_l(\\psi)\n}   \\Bigr )\n= \\frac{1}{\\det \\barb } ,  $$\nwhere $\\det \\barb $ is the determinant of the $l $ by $l$\nmatrix whose columns are the coordinates of $\\beta_1, \\dots,\n\\beta_l$ written in terms of any orthonormal\nbasis of $\\liet$.\n\\item[(vii)] $$ \\reso  \\Bigl (\n\\frac{e^{i \\lambda(\\psi) }  \\psi^J   }\n{\\pee(\\psi)}\n[d \\psi]\n\\Bigr  ) = \\sum_{m \\ge 0} \\lim_{s \\to 0^+}\n\\reso \\Bigl (  \\frac{ (i \\lambda(\\psi))^m e^{i s \\lambda(\\psi) } \\psi^J  }\n{m! \\pee(\\psi)}\n [d \\psi]\\Bigr ). $$\n\n\\end{description}\n\\end{prop}\n\n\\noindent{\\em Remark:} The limits in Proposition\n\\ref{p8.4} are not part of the\ndefinition of the residue map: rather these limits\nare described in order to specify a procedure\nfor computing the piecewise polynomial function\n$H_\\barb (\\lambda) = (2 \\pi i)^l i^{-N}\n\\reso (\\Om_\\lambda). $ (See the example below.)\nProposition \\ref{p8.4} (ii)\nidentifies\n$H_\\barb (\\lambda)$ with  an integral over $\\liet$,\nwhich may be completed to an appropriate\ncontour integral: the choice of contour is\ndetermined by the value of $\\lambda$,\nand requires $\\lambda$ to be a nonzero point in\n$U_\\barb$. Proposition \\ref{p8.4}\n(vii) says that one may compute $H_\\barb$ by\nexpanding the numerator $e^{i \\lambda(\\psi)}$ in\na power series, but only provided one keeps\na factor $e^{is \\lambda(\\psi)}$ in the\nintegrand (for small $s > 0$) in order to\nspecify the contour. The limits in\nProposition \\ref{p8.4} (iv)-(vii) exist because\naccording to Proposition \\ref{p8.4}\n(i) and (ii), they specify limits\nof derivatives of polynomials on subdomains of\n$U_\\barb$, as one approaches the point $0$\nin the boundary of $U_\\barb$ along the fixed\ndirection $s \\lambda$ as $s \\to 0$ in ${\\Bbb R }^+$.\n\n\n\\noindent{\\em Proof of  (i):} This\nfollows directly from Definition \\ref{d8.5n}.\n\n\n\n\\noindent{\\em Proof of (ii):}\nThis follows because $(2 \\pi i)^l i^{- N} \\reso(\\Oma{\\lambda})$\nis the fundamental solution $E(\\lambda)$ of the differential\nequation $P(\\partial\/\\partial \\lambda)E(\\lambda) = \\delta_0$\nwith support in a  half space containing the $\\beta_j$.\n(See\n\\cite{abg} Theorem 4.1 or  \\cite{hor2} Theorem 12.5.1.)\nBut this fundamental solution is given\nby $H_\\barb$ (see Proposition  \\ref{p3.5}(c)).\n\n\n\n\n\\noindent{\\em Proof of (iii):}\nThis is an immediate consequence of (i) and (ii), in view\nof Proposition  \\ref{p3.5}(a).\\footnote{Alternatively there is the\nfollowing direct argument, which was pointed out to us\nby J.J. Duistermaat. We recall that\n$$\n \\reso (\\Omega_\\lambda) = \\frac{1}{(2 \\pi i)^l} \\intt\n\\frac{ e^{i \\lambda(\\psi - i \\xi) }}{P(\\psi - i \\xi) }  \\, [d \\psi], $$\nwhich is defined and independent of $\\xi$ for  $\\xi \\in (C_\\barb)^* $\n(see the discussion after\nDefinition \\ref{d8.5n}). Hence we may replace $\\xi $ by $t \\xi $\nfor any $t \\in {\\Bbb R }^+$:\n\\begin{equation} \\label{resrec}\n \\reso (\\Omega_\\lambda) = \\frac{1}{(2 \\pi i)^l} \\intt\n\\frac{ e^{i \\lambda(\\psi - i t\\xi) }}{P(\\psi - i t \\xi) }  \\, [d \\psi].  \\end{equation}\nTaking the limit as $t \\to \\infty$, we\nsee that $\\reso (\\Omega_\\lambda) = 0 $ if $ \\lambda(\\xi) < 0 $,\nbecause of the factor $e^{  t \\lambda(\\xi) } $ that appears\nin the numerator of (\\ref{resrec}). Since this holds\nfor all $\\xi \\in (C_\\barb)^*$,\n$\\reso (\\Omega_\\lambda)$ is only nonzero when $\\lambda(\\xi) \\ge 0 $\nfor all $\\xi \\in (C_\\barb)^*$, in other words when\n$\\lambda \\in C_\\barb$.}\n\n\n\\noindent{\\em Proof of (iv):} By (i),\n$$\n\\limpl \\reso \\Bigl ( \\Oma{s \\lambda_0} \\psi^J\n \\Bigr ) = \\limpl (- i \\partial\/\\partial \\lambda)^J\nH_\\barb (\\lambda)|_{\\lambda = s\\lambda_0}, $$\nbut $H_\\barb$ is a homogeneous piecewise polynomial function\nof degree $N-l$, so the conclusion holds for $|J| > N - l.$\nIf $|J| < N - l$, we find that $(\\partial\/\\partial \\lambda)^J\nH_\\barb(\\lambda)$ is homogeneous of order $N - l - |J|$ (on any\nopen  subset of\n$\\liets$ where $H_\\barb$ is smooth). Hence it\nis of order $s^{N - l - |J|} $ at $\\lambda = s \\lambda_0$ as $s \\to 0^+$,\nand the conclusion also holds in this case.\n\n\\noindent{\\em Proof of (v):} By (ii), we know that\n$\\reso \\Bigl ( \\Oma{\\lambda}\n \\Bigr )\n = 0$ for $\\lambda$ in a neighbourhood of $s \\lambda_0$ (since $s \\lambda_0$\nis not in the support of $H_\\barb$). Applying\n(i), $\\reso  \\Bigl (\n\\Oma{\\lambda} \\psi^J\n\\Bigr ) $\nmust also be zero.\n\n\\noindent{\\em Proof of (vi):} If $\\beta = \\{\\beta_1, \\dots,\n\\beta_l \\} $ span $\\liets$, and $\\lambda = \\sum_j \\lambda^j \\beta_j$\nwith all $\\lambda^j > 0$, we have\n$$\\reso \\Bigl ( \\Oma{\\lambda} \\Bigr )  = \\frac{1}{(2 \\pi i)^l}\n\\intt \\frac{ [d \\psi] e^{i \\sum_j \\lambda^j (\\psi_j - i  \\vt_j) } }\n{\\prod_{k = 1}^l (\\psi_k - i \\vt_k ) } , $$\nwhere the $\\vt_k = \\beta_k (\\vt) > 0$.\nBut $ [d \\psi]  = d \\psi_1 \\dots d \\psi_l\/(\\det \\barb)$,\nwhere $\\psi_j = \\beta_j(\\psi)$.\nSince the integrals over $\\psi_1, \\dots, \\psi_l$ may be\ncompleted to integrals over semicircular contours\n$C_+(R) $  in the\nupper half plane, for each of which the contour integral\nmay be evaluated by the Residue Theorem to give the\ncontribution $ 2 \\pi i$, we obtain the result.\n\n\n\\noindent{\\em Proof of (vii):} We have\n$$ \\int_{\\psi + i \\vt \\in {\\Bbb R }^l} \\frac{  e^{i \\lambda(\\psi)} }{P(i\\psi)} [d \\psi]\n= (2 \\pi i)^l i^{-N} \\reso(\\Oma{\\lambda}) =\nH_\\barb (\\lambda). $$\nAlso, by (i),\n$$\\sum_{m_1, \\dots,\nm_l \\ge 0}  \\frac{(2 \\pi i)^l i^{-N} }{m_1!\\dots m_l! }\n\\limpl\n\\reso \\Bigl (\n(i \\lambda^1\\psi_1)^{m_1}\n\\dots (i \\lambda^l\\psi_l)^{m_l} \\Oma{s \\lambda_0} \\Bigr )  $$\n$$ = \\sum_{m_1, \\dots,\nm_l \\ge 0}  \\frac{(2 \\pi i)^l i^{-N}\n(\\lambda^1)^{m_1}  \\dots\n(\\lambda^l)^{m_l}}\n{m_1!\\dots m_l! }\n\\limpl ( \\partial\/\\partial \\lambda^1)^{m_1} \\dots\n( \\partial\/\\partial \\lambda^l)^{m_l}  \\reso (\\Oma{\\lambda} )\n|_{\\lambda = s \\lambda_0} $$\n$$ = \\sum_{m_1, \\dots,\nm_l \\ge 0}  \\frac{\n(\\lambda^1)^{m_1}  \\dots\n(\\lambda^l)^{m_l}}\n{m_1!\\dots m_l! }\n\\limpl ( \\partial\/\\partial \\lambda^1)^{m_1} \\dots\n( \\partial\/\\partial \\lambda^l)^{m_l}  H_\\barb(\\lambda)\n|_{\\lambda = s \\lambda_0} $$\nwhich is equal to $H_\\barb(\\lambda)$ since $H_\\barb$ is\na polynomial on certain conical subregions\nof $\\liets$, and there is such\na subregion containing the ray $\\lambda = s \\lambda_0$.$\\square$\n\n\n\n\\noindent{\\underline{ Example.}}\nIn the following  simple example, the explicit formula\n for $H_\\barb$ follows immediately from the definition of $H_\\barb$ in\nProposition \\ref{p3.5}(a). The example is included to show how\nthis result may alternatively be derived by successive contour\nintegrations.\n\nSuppose $l = 2$ and\n$N = 3$, and $\\beta_1(\\psi) = \\psi_1$, $\\beta_2(\\psi) = \\psi_2$,\n$\\beta_3(\\psi) = \\psi_1 + \\psi_2$. We compute\n\\begin{equation} \\label{10.6}{\\mbox{$\\cal R$}} = \\reso \\Bigl ( \\frac{e^{i \\lambda(\\psi)} }\n{\\psi_1 \\psi_2 (\\psi_1 + \\psi_2)} \\Bigr ) \\end{equation}\nwhere $\\lambda(\\psi) = \\lambda^1 \\psi_1 + \\lambda^2 \\psi_2. $\nWe assume $\\lambda^1, \\lambda^2 > 0$, and $\\vt_1, \\vt_2 > 0 $.\nThe quantity (\\ref{10.6}) is given by\n\\begin{equation} {\\mbox{$\\cal R$}}  =\n\\frac{1}{(2 \\pi i)^2}\n\\int_{\\psi_2 + i \\vt_2 \\in {\\Bbb R }} d \\psi_2 \\frac{e^{i \\lambda^2 \\psi_2}}{\\psi_2}\n\\int_{\\psi_1+ i \\vt_1\\in {\\Bbb R }}\n \\frac{e^{i \\lambda^1 \\psi_1}}{(\\psi_1 + \\psi_2)\\psi_1} . \\end{equation}\n(This integral in fact gives the Duistermaat-Heckman\npolynomial $F_T (\\pist e^{\\iins \\bom}) (\\lambda)$ near $\\mu_T(F) $ where\n$F$ is\n  a fixed point of the action of\n$T$ on $X$,  when $X$ is a\ncoadjoint orbit of $SU(3)$ and $T\\cong (S^1)^2$ is the\nmaximal torus : see \\cite{JGP}.)\nWe compute this by first integrating over $\\psi_1$: since $\\lambda^1 > 0 $,\nthe integral may be completed to a contour integral over a semicircular\ncontour in the upper half plane. We obtain contributions\nfrom the two residues $\\psi_1 = 0 $ and $\\psi_1 = - \\psi_2$.\nHence we have\n\\begin{equation} {\\mbox{$\\cal R$}} =\n\\frac{1}{(2 \\pi i)}\n\\int_{\\psi_2 + i \\vt_2 \\in {\\Bbb R }} d \\psi_2 \\frac{e^{i \\lambda^2 \\psi_2}}{\\psi_2^2}\n-\\frac{1}{(2 \\pi i)}\n\\int_{\\psi_2 + i \\vt_2 \\in {\\Bbb R }} d \\psi_2 \\frac{e^{i (\\lambda^2-\n\\lambda^1) \\psi_2}}{\\psi_2^2} . \\end{equation}\nSince $\\lambda^2 > 0 $, the first of these integrals may\nbe completed to a contour integral over a semicircular\ncontour in the upper half plane, and the residue\nat $0$ yields the value $i \\lambda^2$. If $\\lambda^2 - \\lambda^1 > 0 $, the second\nintegral likewise yields $- i (\\lambda^2 - \\lambda^1)$.\nHowever if $\\lambda^2 - \\lambda^1 < 0 $ the second integral\nis instead equal to a contour integral over a semicircular\ncontour in the {\\em lower} half plane, which\ndoes not enclose the pole at $0$, and hence the second integral gives $0$.\nThus we have\n\\begin{equation} \\label{9.009}(2 \\pi i)^2 \\reso(\\Oma{\\lambda}) = \\cases{i \\lambda^2, &\n$\\lambda^1 > \\lambda^2$ \\cr\ni \\lambda^1, &\n$\\lambda^2 > \\lambda^1$. \\cr } \\end{equation}\n\nAccording to (iv) and (vii), the quantity ${\\mbox{$\\cal R$}}$ is\nalso given by\n\\begin{equation} {\\mbox{$\\cal R$}} = \\limpl \\int_{\\psi+ i \\vt\\in {\\Bbb R }^2}\n\\frac{ e^{i s (\\lambda^1 \\psi_1 + \\lambda^2 \\psi_2)} (i \\lambda^1 \\psi_1\n+ i \\lambda^2 \\psi_2)[d \\psi]}{\\psi_1 \\psi_2 (\\psi_1 + \\psi_2)} \\end{equation}\n$$ =\n  \\limpl (i \\lambda^1) \\int_{\\psi+ i \\vt\\in {\\Bbb R }^2}\n\\frac{ e^{i s [\\lambda^1 (\\psi_1 + \\psi_2) +  (\\lambda^2 - \\lambda^1)\\psi_2]}\n[d \\psi] }{(\\psi_1 + \\psi_2) \\psi_2}\n+\n  \\limpl (i \\lambda^2) \\int_{\\psi+ i \\vt\\in {\\Bbb R }^2}\n\\frac{ e^{i s [(\\lambda^1 - \\lambda^2)\\psi_1 +  \\lambda^2 (\\psi_1 + \\psi_2)]}\n[d \\psi] }{\\psi_1 (\\psi_1 + \\psi_2) }\n. $$\nThis clearly gives the result  (\\ref{9.009}).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n{ \\setcounter{equation}{0} }\n\\section{Examples}\n\nIn this section we shall show\nin the case $K = SU(2)$  how Corollary  \\ref{c8.2} may\nbe used to prove relations in the cohomology\nring $H^*(\\xred)$ for two specific $X$.\nThese $X$ are the examples treated at the end of\nSection 6 of \\cite{Ki2}. There, all the relations\nin the cohomology ring are determined, which is equivalent\nto exhibiting\nall the   vanishing intersection\npairings. We shall show how the results\nof the present paper may be used to show these are indeed vanishing\n  intersection pairings, although we shall not\nbe able to rederive the result that there are no others.\n\n\n\\noindent\\underline{\\em Example 1: $X = ({\\Bbb P}_1)^N, $  $N$ odd.}\nConsider the action of $K = SU(2)$ on the space\n$X = ( {\\Bbb P } _1)^N$ of ordered $N$-tuples\nof points on the complex projective line\n$ {\\Bbb P } _1$, defined by the $N$th tensor\npower of the standard representation of $K$ on\n${\\Bbb C }^2$.\nEquivalently when $ {\\Bbb P } _1$ is identified with\nthe unit sphere $S^2$ in ${\\Bbb R }^3$ then $K$ acts\non $X = (S^2)^N$ by rotations of the sphere. When\nthe dual of the Lie algebra of $K$ is identified\nsuitably with ${\\Bbb R }^3$ then the moment map\n$\\mu$ is given (up to a constant scalar\nfactor depending on the conventions used)\nby $$\\mu(x_1, \\dots, x_N)  = x_1 + \\dots + x_N$$\nfor $x_1 , \\dots, x_N \\in S^2$. We assume\nthat $0$ is a regular value for $\\mu$; this happens\nif and only if there is no $N$-tuple\nin $\\mu^{-1}(0)$ containing a pair of antipodal\npoints in $S^2$ each with multiplicity $N\/2$,\nand so $0$ is a regular value if and only\nif $N$ is odd.\n\nIn order to apply Corollary \\ref{c8.2} we note that\nthe fixed points of the action of the standard\nmaximal torus $T$ of $K$ are the $N$-tuples\n$(x_1, \\dots, x_N)$ of points in\n$ {\\Bbb P } _1$ such that each\n$x_j$ is either $0$ or $\\infty$. We shall index these\nby sequences $n = (n_1, \\dots, n_N)$  where\n$n_j = + 1$ if $x_j = 0$ and $n_j = - 1$\nif $x_j = \\infty$.\nDenote by $e_n$ the fixed point indexed\nby $n$. Then\n$$ \\mu_T(e_n) = \\sum_{j = 1}^N n_j $$\nand the weights of the action of $T$\nat $e_n$ are just $\\{ n_1, \\dots, n_N\\}$.\nHence the sign of the product of weights\nat $e_n$ is $\\prod_j n_j$ and its\nabsolute value is $1$.\n\nThe cohomology ring $H^*(X)$ has $N$\ngenerators $\\xi_1, \\dots, \\xi_N$ say, of degree two,\nsatisfying $\\xi_j^2  = 0$\nfor $1 \\le j \\le N$. The equivariant cohomology\nring $H^*_T(X)$ with respect to the torus $T$\nhas generators\n$\\xi_1, \\dots, \\xi_N$ and $\\alpha$ of\ndegree two subject to the relations\n$$(\\xi_j)^2 = \\alpha^2$$\nfor $1 \\le j \\le N$. The Weyl group action sends $\\alpha$\nto $-\\alpha$ so $\\hk(X)$ has generators\n$\\xi_1, \\dots, \\xi_n, \\alpha^2$ subject to the\nsame relations.\n\nAccording to the last example of section 6 of\n\\cite{Ki2}, the kernel of the map $\\hk(X) \\to H^*(\\xred)$\nis spanned by elements of the form\n\\begin{equation} \\label{8.beta} (1\/\\alpha) \\Bigl ( q(\n\\xi_1, \\dots, \\xi_N, \\alpha) \\prod_{i \\in Q} (\\xi_i + \\alpha)\n- q(\n\\xi_1, \\dots, \\xi_N, -\\alpha) \\prod_{i \\in Q} (\\xi_i - \\alpha) \\Bigr )  \\end{equation}\nfor some $Q \\subset \\{ 1, \\dots, N\\} $  containing\nat least $(N+1)\/2$ elements\nand some polynomial $q$ in $N + 1$ variables\nwith complex coefficients.\nWe can  use Corollary \\ref{c8.2}\nto give an alternative proof that the evaluation against the\nfundamental class $[\\xred]$ of the image in\n$H^*(\\xred)$ of any element\nof this form of degree $N-3$ is zero. This amounts\nto showing that\n\\begin{equation} {\\rm Res}_0 \\frac{1}{\\psi^{N-1} }\n\\sum_{\\stackrel{\\indd_j = \\pm 1,}{ \\sum_j \\indd_j > 0} } \\Bigl ( \\prod_j\n\\indd_j  \\Bigr )\n\\Bigl \\{ q(\\psi \\indd_1, \\dots, \\psi \\indd_N, \\psi) \\prod_{i \\in Q}\n\\psi(\\indd_i + 1)\n- q(\\psi \\indd_1, \\dots, \\psi \\indd_N, -\\psi) \\prod_{i \\in Q}\n\\psi(\\indd_i - 1) \\Bigr \\} \\end{equation}\nis zero when $q$ is homogeneous of degree $N - 2 - |Q|$.\nIn other words it amounts to showing\nthat $\\dell = 0$ for any $\\dell$ of the form\n\\begin{equation} \\dell =\n\\sum_{\\stackrel{\\indd_j = \\pm 1,}{ \\sum_j \\indd_j > 0}}\n\\Bigl ( \\prod_j \\indd_j  \\Bigr )\n\\Bigl \\{\nq( \\indd_1, \\dots,  \\indd_N, 1) \\prod_{i \\in Q}\n(\\indd_i + 1)\n- q( \\indd_1, \\dots,  \\indd_N, -1) \\prod_{i \\in Q}\n(\\indd_i - 1) \\Bigr \\} \\end{equation}\nwhere $q$ is homogeneous of degree $N - 2 - |Q|$.\nLet us assume without loss of generality that\n$q (\\xi_1, \\dots, \\xi_N, \\alpha) = \\prod_i \\xi_i^{r_i} \\alpha^p$\nwhere $p + \\sum_i r_i + |Q| = N-2$.\nThus $p - 1 = |Q| + \\sum_i r_i $ (mod $2$) since\n$N$ is odd. Hence we have that\n\\begin{equation} \\dell =\n \\sum_{\\stackrel{\\indd_j = \\pm 1,}  {\\sum_j \\indd_j > 0}\n}\n\\prod_j  (\\indd_j^{r_j + 1} ) \\Bigl \\{\n\\prod_{i \\in Q} (\\indd_i + 1) +\n(-1)^{|Q|} \\prod_k (-1)^{r_k} \\prod_{i \\in Q} (\\indd_i - 1)  \\:\n\\Bigr \\}  \\end{equation}\n\\begin{equation} \\phantom{a}  = \\sum_{\\stackrel{\\indd_j = \\pm 1,}  {\\sum_j \\indd_j > 0}\n}\n\\prod_j  (\\indd_j^{r_j + 1} )\n\\prod_{k \\in Q} (\\indd_k + 1)\n-\n \\sum_{\\stackrel{\\Indd_j = \\pm 1,}{ \\sum_j \\Indd_j < 0} }\n\\prod_j  (\\Indd_j^{r_j + 1} )\n\\prod_{k \\in Q} (\\Indd_k + 1)  \\end{equation}\nwhere we have introduced $\\Indd_j = - \\indd_j$.\n\nNow the second sum vanishes, for if $\\Indd_j = 1$\nfor all $j \\in Q$ then we must have $\\sum_j \\Indd_j > 0$\nsince $|Q|> N\/2$.\nHence we are reduced to proving the vanishing of\n$$ \\sum_{  \\indd\\in \\Gamma}\n\\prod_j \\indd_j^{r_j + 1} $$\nwhere\n$$\\Gamma = \\{ \\indd \\: | \\: \\sum_j \\indd_j > 0, \\;\n\\indd_j = 1 \\; \\mbox{for $j \\in Q$} \\}. $$\nHence we have to prove the vanishing\nof $$\\dell = \\sum_{\\indd_j = \\pm 1, j \\notin Q}\n\\prod_{j \\in S} n_j$$\nwhere $S$ is the set $\\{ j \\notin Q \\, | \\,\nr_j = 0 \\pmod{2} \\}. $\nThe sum thus vanishes by cancellation in pairs provided $S$ is\nnonempty. However if $S$ were empty then\n$r_j = 1 \\pmod{2}$ for all $j \\notin Q$,\nso $r_j \\ge 1$ for all $j \\notin Q$,\nwhich is impossible since $\\sum_j r_j + |Q| \\le N-2$.\nThis proves the desired result.\n\n\n\\noindent\\underline{\\em Example 2: $X =  {\\Bbb P } _N$, $N$ odd.}\n A closely related example\nis given by the action of $K = SU(2)$ on the complex projective\nspace\n$X = {\\Bbb P}^N $  defined by the $N$th symmetric\npower of the standard representation of $K$ on\n${\\Bbb C }^2$. Equivalently we can identify $X$ with\nthe space of\nof {\\em unordered}\n$N$-tuples of points in  the complex\nprojective line $ {\\Bbb P } _1$ or the sphere $S^2$, and\nthen $K$ acts by rotations as in example $1$.\nWe take the symplectic form $\\om$ on $X$\nto be the Fubini-Study form on $ {\\Bbb P } _N$.\nThe moment map is given  by the composition\nof the restriction map ${\\bf u}(N+1)^* \\to \\lieks$\nwith the map\n$\\mu:  {\\Bbb P } _N \\to {\\bf u}(N+1)^* $ defined for\n$a \\in {\\bf u}(N+1)$ by\n$$ \\inpr{\\mu(x) , a} = (2 \\pi i |x^*|^2 )^{-1}\n\\bar{x^*}^t a x^*, $$\nwhere $x^* = (x^*_0, \\dots, x^*_N)$ is\nany point\nin ${\\Bbb C }^{N+1}$  lying over the point $x  \\in  {\\Bbb P } _N$.\nThe restriction of this moment map\nto $\\liets$ is\n$\\mu_T (x) = (2 \\pi i |x^*|^2 )^{-1} )\n\\sum_{j = 0}^N (N - 2j)|x^*_j|^2  $.\nAgain we assume that $N$ is odd in order\nto ensure that $0$ is a regular value of\nthe moment map $\\mu$.\n\nAgain the fixed points of the action of $T$ are\nthe $N$-tuples of points in $ {\\Bbb P } _1$ consisting\nentirely of copies of $0$ and $\\infty$. Equivalently\nthey are the points\n$e_0 = [1, 0, \\dots, 0]$,\n$e_1 = [0, 1, \\dots, 0]$,\n$\\dots, e_N = [0, \\dots, 0, 1]$ of\n$ {\\Bbb P } _N$. The image of $e_k$ under $\\mu_T$ is\n$\\mu_T(e_k) = N - 2 k = \\mu_k$ say. Since\n$\\diag (t, t^{-1} ) \\in T$ acts on $ {\\Bbb P } _N$\nby sending $[x_0, \\dots, x_j, \\dots, x_N]$\nto $[t^{-N} x_0, \\dots, t^{2j-N} x_j, \\dots, t^N x_N]$\nthe weights at $e_k$ are\n$$\\{ 2 (j - k): \\: 0 \\le j \\le N, \\; j \\ne k \\}$$\nThe number of negative weights at $e_k$ is equal to $k$\n(modulo $2$), and the absolute value of the product\nof weights at $e_k$ is\n$v_k = \\prod_{j \\ne k } |j - k| = 2^N k! (N-k)! $\n\n\n\nThe cohomology ring $H^*( {\\Bbb P } _N)$ is generated\nby $\\xi$ of degree two subject to the relation\n$\\xi^{N+1} = 0$.\nThe equivariant cohomology ring $H^*_T ( {\\Bbb P } _N)$\nis generated by $\\xi$ and $\\alpha$ of degree two\nsubject to the\nrelation $\\prod_{0 \\le j \\le N} (\\xi - (2j-N) \\alpha) = 0, $\nand the equivariant cohomology ring $\\hk( {\\Bbb P } _N)$\nis generated by $\\xi$ and $\\alpha^2$ subject to the\nsame relation.\n\nAccording to section 6 of \\cite{Ki2} the kernel\nof the natural map $\\hk(X) \\to H^*(\\xred)$\nis generated as an ideal in $\\hk(X) $ by $P_+(\\xi, \\alpha)$\nand $P_-(\\xi,\n\\alpha)\/\\alpha$ where\n$$P(\\xi, \\alpha) = \\prod_{k > N\/2} (\\xi + \\mu_k \\alpha) $$\nand\n$$P_\\pm(\\xi, \\alpha) = P (\\xi, \\alpha) \\pm P(\\xi, - \\alpha). $$\n(Note that  $P_+ (\\xi, \\alpha)$ and $P_-(\\xi, \\alpha)\/\\alpha$\nare actually polynomials in $\\xi$ and $\\alpha^2$.)\nWe would like to check that the evaluation against the\nfundamental class $[\\xred]$\nof the image of $R_+(\\xi, \\alpha^2) P_+(\\xi, \\alpha) $\nand $R_-(\\xi, \\alpha^2)  P_-(\\xi, \\alpha)\/\\alpha$ in\n$H^*(\\xred)$ is zero for any $R_\\pm(\\xi, \\alpha^2)$\n$\\in \\hk(X)$ of the appropriate degree.\n\nNow we have from the abelian fixed point formula that  for any\n$S(\\gnx, \\gna^2)$,\n$$\\Pi^+_*  (S(\\gnx, \\gna^2) ) =\n\\sum_{k < N\/2}  (-1)^k\n\\frac{S(\\mu_k \\psi, \\psi^2) }{\\veee_k  }\n\\psi^{-N}. $$\n(Here, if  $\\zeta \\in \\hk(X)$, the notation $\\Pi^+(\\zeta)$\nmeans  the portion of the abelian formula (\\ref{2.1})\nfor $\\pis(\\zeta)$\ncorresponding to fixed points $F$ for which $\\mu_T(F) > 0$.)\nTo evaluate this on the fundamental class of $\\xred$\nwe must then find the term of degree $-1$ in\n$\\nusym^2(\\psi) \\Pi^+_* (S(\\gnx, \\gna^2 ) )(\\psi) $, or in other words\nthe term of degree $N - 3$ in\n$\\sum_{k < N\/2}(-1)^k  S(\\mu_k \\psi, \\psi^2) \/(\\veee_k )\n$.\nHaving found the term of degree $N-3$ in $\\psi$, we evaluate\nit at $\\psi = 1$ to get the residue.\nIn the case when $S(\\mu_k \\psi, \\psi^2) = R_+(\\mu_k \\psi, \\psi^2)\nP_+(\\mu_k \\psi, \\psi) $ or\n$S(\\mu_k \\psi, \\psi^2) = R_-(\\mu_k \\psi, \\psi^2)\nP_-(\\mu_k \\psi, \\psi)\/\\psi $\nis homogeneous of  degree $N - 3$ in $\\psi$, we need to show that\n$$ \\sum_{k = 0 }^{(N-1)\/2}   \\:  (-1)^k\n\\frac{1}{\\veee_k } \\Bigl ( \\,\nR_\\pm (\\mu_k, 1) \\prod_{j \\ge N\/2} (\\mu_k + \\mu_j)\n\\pm  R_\\pm(\\mu_k, -1) \\prod_{j \\ge N\/2} (\\mu_k - \\mu_j) \\: \\Bigr )  = 0. $$\nSince $\\mu_k = - \\mu_{N-k} $, we have $\\prod_{j \\ge N\/2}\n (\\mu_k + \\mu_j ) = 0$\nfor all $k < N\/2$, so we just have to prove the vanishing of\n$$\\sum_{k \\le  N\/2}  \\: (-1)^k\n\\frac{1}{ \\veee_k }\nR(\\mu_k) \\prod_{j \\ge N\/2} (\\mu_k - \\mu_j)\n $$\nfor every polynomial $R$ of degree at most $(N-1)\/2 - 2$,\nor without loss of generality the vanishing of\n$$ \\sum_{k \\le N\/2} (\\mu_k)^s \\frac{(-1)^k}{ \\prod_{l \\ne k}\n|l - k| } \\prod_{j \\ge N\/2} (\\mu_k - \\mu_j), $$\nwhere $s \\le (N-1)\/2 - 2$.\nSince $ \\prod_{l \\ne k}\n|l - k| = k! (N-k)!$ and $\\mu_k - \\mu_j = 2(j - k), $\nwe have that\n$$ \\prod_{j \\ge N\/2} |\\mu_k - \\mu_j | = 2^{(N+1)\/2} \\frac{(N - k)!}\n{( (N-1)\/2 - k )! } $$\nDefine $r = (N-1)\/2$.\nWe need to show the vanishing of\n$$\\sum_{k = 0}^r (N-2k)^s \\frac{(-1)^k (N-k)!}{k! (N-k)! (r - k)! }\n$$ for $s \\le r - 2$.\nIt then suffices to show the vanishing of\n$$\\sum_{k = 0}^r k^s (-1)^k \\colvec{r}{k} , \\phantom{bbbbb} \\mbox{\n$s \\le r - 2$.}$$\nThis follows since one may expand $(1- e^\\lambda)^r$ as a power series\nin $e^\\lambda$ using the binomial theorem: we have\n$$ (1 - e^\\lambda)^r = \\sum_{j= 0}^r (-1)^j \\colvec{r}{j}\n\\exp j \\lambda \\: = \\sum_{s \\ge 0}  \\lambda^s\/s!\n\\sum_{j= 0}^r (-1)^j j^s \\colvec{r}{j}\n,  $$\nbut the terms in this expansion corresponding to $s < r$ must vanish\nsince $1 - e^\\lambda  = \\lambda h(\\lambda)$ for some  function\n$h$ of $\\lambda$ which is analytic at $\\lambda = 0$.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nAs a fundamental task for video understanding, video action recognition has drawn much attention from the community and industry~\\cite{tran2015learning, carreira2017quo, tran2018closer, feichtenhofer2019slowfast}. Unlike image-related tasks, networks for video-related tasks are normally easier to overfit due to the complexity of the tasks~\\cite{tran2017convnet, tran2015learning, kataoka2020would}. The common practice is to firstly pre-train the network on large-scale datasets (\\eg, Kinetics~\\cite{carreira2019short} of up to $650,000$ video clips) and then finetune on downstream small datasets to obtain better performance~\\cite{feichtenhofer2019slowfast,qian2021spatiotemporal,pan2021videomoco,han2020self}.\n\nHowever, since annotating large-scale video datasets is time-consuming and expensive, training models on a large dataset collected with complete annotations is impeded. To utilize large-scale datasets with acceptable costs, some researchers have turned to designing semi-supervised learning models which have good generalization ability with limited annotations~\\cite{jing2021videossl,singh2021semi,zou2021learning,xiong2021multiview}. As pseudo-label based methods (\\eg, FixMatch~\\cite{sohn2020fixmatch} and MixMatch~\\cite{ berthelot2019mixmatch}) have shown outstanding performance on semi-supervised image classification, most previous video-based methods are heavily built on them to utilize the unlabeled data. Although these preliminary attempts have obtained acceptable results, most methods ~\\cite{jing2021videossl,zou2021learning} are just taking video clips as `images' in 3D without further consideration of the video properties. \n\n\nVideos are significantly different from images, and the key differences are the temporal information span in multiple frames and the inherent multimodal property. The temporal information refers to the motion signal between frames, and usually the features of contiguous frames from the same video change smoothly. \\zzrevise{The multimodal consistency refers to the features of multiple modalities in the same video clip should be consistent since they are encoding the same content.}{The multimodal consistency expects the features extracted from the same video clip to be consistent, as they encode identical content.} Without special designs to specifically focus on temporal information and multimodal consistency, the potential of semi-supervised action recognition is not fully unleashed.\n\n\nSome previous studies\\cite{wang2016temporal, zhao2018recognize} introduce temporal gradient\\footnote{The difference between two RGB video frames with a short interval.} as an additional modality to better utilize the temporal information encoded in videos as it is rich in motion signals. The temporal gradient can be formulated as:\n$TG = x_t^{RGB} - x_{t+n}^{RGB}$,\nwhere $x$ represents a video, $t$ denotes the frame index and $n$ denotes the interval for calculating temporal gradient. \n\n\nInspired by these studies, we made a trial with temporal gradient under the semi-supervised settings and found that a much better performance could be generated when the input frames in RGB are replaced with temporal gradient. As shown in~\\Cref{fig:motivation}, the Top-1 accuracy of temporal gradient is $\\sim$25\\% higher than using the RGB as input on the UCF-101 dataset with only $20$\\% of labeled data for training.\n\n\n\nWhy is the temporal gradient so much better than the RGB frames when the training data are limited? We hypothesize that the key is in the detailed and fine-grained motion signals encoded in the temporal gradient. The gradient along the temporal dimension is color-invariant and explicitly encodes the representative motion information of the actions in the video. This helps models generalize much easier when the labels are extremely limited. Therefore, in this paper, we propose to train a semi-supervised action recognition RGB based model to mimic both the fine-grained and high-level features from the temporal gradient. \n\n\nWe start from FixMatch~\\cite{sohn2020fixmatch}, a typical pseudo-label based semi-supervised model, as the baseline framework. However, without any further constraints in the feature level, pseudo-label based methods perform poorly in the case of very limited labels, as many generated pseudo-labels are inaccurate.\nTherefore, we propose two constraints to help the model extract temporal information in video with multiple modalities and improve the consistency between the multimodal representations. To leverage the detailed and fine-grained motion signals from temporal gradient, we propose a knowledge distillation strategy using block-wise dense alignment. It helps the student RGB model learn from the teacher temporal gradient model efficiently and effectively. To further improve high-level representation space across different modalities, we perform contrastive learning between the features from RGB and temporal gradient sequences to enforce the high-order similarity. Given the two constraints at the feature level, our proposed model is able to achieve much better performance.\n\nUnlike the existing methods, our model has two unique advantages. First, our model requires no additional computation or parameters for inference. In the training, we distill the knowledge from temporal gradient to the RGB-based network; in the testing, only the RGB model is required. Second, our model is simple, yet effective. We conducted experiments on multiple public action recognition benchmarks including UCF-101, HMDB-51, and Kinetics-400. Our proposed method significantly outperforms all the state-of-the-art methods by a large margin.\n\n\n\\section{Related Work}\n\\noindent\\textbf{Semi-supervised learning in images.} The semi-supervised image classification task has been well studied and many methods have been proposed including Pseudo-Label~\\cite{lee2013pseudo},  S4L~\\cite{zhai2019s4l}, MeanTeacher~\\cite{tarvainen2017mean}, MixMatch~\\cite{berthelot2019mixmatch}, UDA~\\cite{xie2020unsupervised},  FixMatch~\\cite{sohn2020fixmatch}, UPS~\\cite{rizve2020defense}, etc. The Pseudo-Label~\\cite{lee2013pseudo} is an early method which uses the confidence (softmax probabilities) of the unlabeled data as labels and to train the network jointly with a small ratio labeled data and much more unlabeled data. Many improved versions of Pseudo-Label have been proposed while the key is to improve the quality of the labels~\\cite{sohn2020fixmatch, rizve2020defense}. Following a  state-of-the-art method on image classification---FixMatch~\\cite{sohn2020fixmatch}, many FixMatch-alike methods  achieve the state-of-the-art performance on many other tasks including detection~\\cite{wang20213dioumatch}, segmentation~\\cite{zou2020pseudoseg}, etc. Although these methods achieve remarkable performance on image-based tasks, some recent studies show that the performances are not satisfying when directly applying these methods to video semi-supervised tasks~\\cite{jing2021videossl, singh2021semi}.\n\n\\smallskip\\noindent\\textbf{Semi-supervised learning in videos.} Although there have been a few semi-supervised video action recognition methods~\\cite{jing2021videossl,singh2021semi,zou2021learning, xiong2021multiview} proposed, most of them directly apply the image-based methods to videos with less focus on the temporal dynamics of videos. VideoSSL~\\cite{jing2021videossl} made the first attempt to build a benchmark for the video semi-supervised learning task by training the network with ImageNet pre-trained models, which explicitly guides the model to learn the appearance information in each video. It also shows that the existing image-based methods (\\eg Pseudo-Label~\\cite{lee2013pseudo}, Mean-Teacher~\\cite{tarvainen2017mean}) have inferior performance on video semi-supervised benchmarks. TCL~\\cite{singh2021semi} is a recently proposed method that jointly optimizes the network by employing a self-supervised auxiliary task and a group contrastive learning. By using multimodal data, MvPL~\\cite{xiong2021multiview} achieved the state-of-the-art performance by sharing the same model with different input modalities (RGB, temporal gradient, and optical flow) and generating pseudo labels with the ``confidence'' of multiple modalities. Compared with these methods, our method specifically focuses on learning the temporal information from Temporal Gradient with our proposed constraints and significantly outperforms the state-of-the-art methods on multiple public benchmarks.\n\n\\smallskip\\noindent\\textbf{Multimodal Video Feature Learning.} Videos could be viewed from different modalities while each modality encodes information from a unique perspective. For example, a video in general RGB couples both spatial and temporal information, the temporal gradient is color invariant which mainly encodes the difference between frames, and the optical flow explicitly encodes the motion information for each pixel. The features from different modalities are normally complementary to each other, and therefore the feature fusions are normally performed for better performance. The pioneer work is the Two-Stream~\\cite{simonyan2014two, feichtenhofer2016convolutional} model which fuses features from both RGB video clips and optical flow clips. With the complementary information from different modalities, the multimodal network is able to achieve better performance~\\cite{simonyan2014two, feichtenhofer2016convolutional, wang2015towards,wang2020makes,alwassel2019self}. However, there are additional computation and latency during inference. Unlike the normal multimodal feature fusion model, our model distills the motion-related representation from the temporal gradient to the base RGB model, while only the base model and RGB frames are needed during the inference stage. Moreover, our model outperforms the teacher model with only RGB as input during the inference.\n\n\\smallskip\\noindent\\textbf{Contrastive learning.} Contrastive learning methods have achieved remarkable performance on downstream image classification~\\cite{tian2020contrastive, chen2020simple,misra2020self, he2020momentum,grill2020bootstrap,caron2020unsupervised}. The key idea is that the representation can be learned by minimizing the distance of features of positive pairs (two views of the same data sample) and maximizing the distance of features of negative pairs (two different data samples). Recently, many researchers proposed to use temporal contrastive learning for video self-supervised learning~\\cite{han2020self, feichtenhofer2021large,qian2021spatiotemporal,pan2021videomoco,huang2021self}. In this paper, to better utilize the unlabeled data for semi-supervised action recognition, we propose to use the cross-modal contrastive loss to enforce the consistency of features from RGB clips and temporal gradient clips. We demonstrate that the cross-modal contrastive method is very effective for the proposed semi-supervised learning.\n\n\\section{Method}\nThe objective of our method is to improve the performance on the semi-supervised action recognition task by introducing and utilizing an effective view of videos: Temporal Gradient. The overview of our proposed framework is shown in \\Cref{fig:overview}, which consists of three main components: (1) the FixMatch framework with weak-strong augmentation strategy to generate better pseudo-labels for unlabeled data (2) cross-modal dense feature alignment between the features from RGB clips and TG clips for network to learn the fine-grained motion signals, and (3) cross-modal contrastive learning to learn high-level consistency feature across RGB and TG clips. The formulation for each component is introduced in the following subsections.\n\n\\subsection{FixMatch}\n\nConsidering a multi-class classification problem, we denote $\\mathcal{X} = \\{(x_i, y_i)\\}_{i=1}^{N_l}$ as the \\emph{labeled} training set, where $x_i \\in \\mathcal{R}^{T\\times H\\times W\\times 3}$ is the $i$-th sampled video clip, $y_i$ is the corresponding one-hot ground truth label, and $N_l$ is the number of data points in the labeled set.\nSimilarly, we denote $\\mathcal{U} = \\{x_j\\}_{j=1}^{N_u}$ as the \\emph{unlabeled} set, where $N_u$ is the number of data points in the unlabeled set. We use $f_{\\theta}$ to denote a classification model with trainable parameters $\\theta$. We use $\\alpha(\\cdot)$ to represent the weak (standard) augmentation (\\ie, random horizontal flip, random scaling, and random crop in video action recognition), and $\\mathcal{A}(\\cdot)$ to represent the strong data augmentation strategies (i.e., Randaugment~\\cite{cubuk2020randaugment}).\n\nThe network $f_{\\theta}$ is optimized with each video clip consisting of $T$ frames as $x_i$. \nFor a mini-batch of \\emph{labeled} data $\\{(x_i, y_i)\\}_{i=1}^{B_l}$, the network is optimized by minimizing the cross-entropy loss $\\mathcal{L}_l$ as\n\\begin{equation}\n\\small\n    \\mathcal{L}_l = -\\frac{1}{B_l} \\sum_{i=1}^{B_l} \n    y_i \\log f_{\\theta}(\\alpha(x_i)),\n    \\label{eq:loss_l}\n\\end{equation} \nwhere $B_l$ is the number of labeled samples in a batch.\n\nFor a mini-batch of \\emph{unlabeled} data $\\{x_j\\}_{j=1}^{B_u}$, FixMatch enforces the model to produce consistent predictions of the same unlabeled data sample with different extent of augmentations. Specifically, pseudo labels $\\hat{y}$ for the unlabeled data are usually generated via confidence thresholding as: \n\\begin{equation}\n\\label{eq:pseudo_label}\n\\small\n    \\mathcal{C} = \\{x_j | \\text{max} f_{\\theta}(\\alpha(x_j)) \\geq \\gamma\\},\n\\end{equation}\nwhere $\\gamma$ denotes a pre-defined threshold and $\\mathcal{C}$ is the confident example set from a mini-batch. The confident predictions $f_{\\theta}(\\alpha(x_j))$ in the set $\\mathcal{C}$ are then transformed into one-hot labels $\\hat{y}_j$ by taking the \\textit{argmax} operation. Then a cross-entropy loss $\\mathcal{L}_u$ will be optimized over the samples in $\\mathcal{C}$ and its generated one-hot labels as: \n\\begin{equation}\n\\small\n    \\mathcal{L}_u = -\\frac{1}{B_u} \\sum_{x_j \\in \\mathcal{C}}\n    \\hat{y}_j \\log f_{\\theta}(\\mathcal{A}(x_j)),\n    \\label{eq:loss_u}\n\\end{equation}\nwhere $B_u$ is the number of unlabeled samples in a batch.\n\nWith the loss over both labeled and unlabeled data, the entire FixMatch is optimized with the objective function as: \n\\begin{equation}\n\\small\n    \\mathcal{L}_{fm} = \\mathcal{L}_l + \\mathcal{L}_u.\n\\end{equation}\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{figures\/main.pdf}\n\\caption{\\textbf{An overview of our proposed framework.} Our method consists of two parallel models with different input modalities (\\ie, RGB and TG) of video clips. The entire framework is jointly optimized with (1) two parallel FixMatch frameworks with pseudo-labeling, (2) cross-modal dense feature alignment, and (3) cross-modal contrastive learning. }\n\\label{fig:overview}\n\\vspace{-2mm}\n\\end{figure*}\n\\subsection{Parallel Framework for Temporal Gradient}\n\\label{sec:parallel}\n\n\nTemporal gradient (TG) $(\\frac{\\partial V}{\\partial t})$ between two RGB frames in a video encodes the appearance change and corresponds to the temporal information that changes dynamically. Therefore, the response is accentuated by the moving objects, especially the boundaries. FixMatch\\cite{sohn2020fixmatch} is originally designed for the image classification task and pays little attention to the temporal information of videos, therefore, we extend it to jointly train with RGB and TG to explicitly focus more on capturing the temporal information. To avoid additional computation and delays for processing temporal gradients during model inference on unseen videos, we propose to distill fine-grained motion signals from TG to RGB without introducing extra input or parameters for inference.\n\n\nThe RGB and temporal gradient information are complementary with each other. The RGB encodes spatial and temporal information in a general way, while the temporal gradient has a focus on the motion signals, as illustrated in \\Cref{fig:motivation}. Therefore, for each video clip, the predictions from both RGB network and TG network are averaged and then used to generate the pseudo labels. In this way, the fused pseudo-label generation is reformulated as:\n\\begin{equation}\n\\label{eq:pseudo_label_avg}\n\\small\n    \\mathcal{C} = \\{x_j | \\text{max} (\\frac{f_{\\theta_{R}}(\\alpha(x_j^{RGB})) + f_{\\theta_{T}}(\\alpha(x_j^{TG}))}{2} )  \\geq \\gamma\\}.\n\\end{equation}\n\n\nHaving access to both features from RGB and TG, the quality of the fused pseudo labels are more accurate than the predictions from each model alone, and a more detailed ablation study is provided in \\Cref{sec:Abla}. The fused pseudo labels will be jointly used with unlabeled data to train both the TG and RGB model. For the temporal gradient model, the training objective is also a summation of \\Cref{eq:loss_l} and \\Cref{eq:loss_u} but for TG.\n\\begin{equation}\n\\small\n    \\mathcal{L}_{fm}^{TG} = \\mathcal{L}_{l}^{TG} + \\lambda_u \\mathcal{L}_u^{TG}.\n\\end{equation}\n\n\n\\subsection{Cross-modal Dense Feature Alignment}\n\\label{sec:alignemnt}\n\nTo learn detailed fine-grained motions from temporal gradient, we propose to distill the knowledge from temporal gradient model to the RGB model. The similarities between the features from both temporal gradient and RGB clips are minimized by the cross-model dense feature alignment module as: \n\\begin{equation}\n\\label{eq:alignment obj}\n\\small\n\\min \\left[\\mathcal{D}\\left(\\mathcal{F}^{RGB}_i, \\mathcal{F}^{TG}_i\\right)\\right],\n\\end{equation} where $\\mathcal{F}^{RGB}_i , \\mathcal{F}^{TG}_i\\in \\mathbb{R}^{C_i \\times T_i \\times H_i \\times W_i}$ denote the output features of the $i$-th block in the RGB and TG models, and $\\mathcal{D}$ represents a pairwise function evaluating the representation differences. There are many choices for $\\mathcal{D}$ and we experiment with three different functions: L1, L2 and Cosine Similarity losses (shown in \\Cref{eq:alignment func}, where $\\|\\cdot\\|_{1}$ and $\\|\\cdot\\|_{2}$ are $\\ell_{1}\/\\ell_{2}$-norm). A more detailed discussion is provided in \\Cref{sec:Abla}.\n\\begin{align}\n\\label{eq:alignment func}\n\\small\n\\begin{split}\n\\mathcal{D}_{L1}\\left(\\mathcal{F}_{1}, \\mathcal{F}_{2}\\right)&={\\left\\|\\mathcal{F}_{1}-\\mathcal{F}_{2}\\right\\|_{1}},\n\\\\\n\\mathcal{D}_{L2}\\left(\\mathcal{F}_{1}, \\mathcal{F}_{2}\\right)&={\\left\\|\\mathcal{F}_{1}-\\mathcal{F}_{2}\\right\\|_{2}},\n\\\\\n\\mathcal{D}_{cos}\\left(\\mathcal{F}_{1}, \\mathcal{F}_{2}\\right)&=-\\frac{\\mathcal{F}_{1}}{\\left\\|\\mathcal{F}_{1}\\right\\|_{2}} \\cdot \\frac{\\mathcal{F}_{2}}{\\left\\|\\mathcal{F}_{2}\\right\\|_{2}}.\n\\end{split}\n\\end{align}\n\n\nAn key setting in our online knowledge distillation method is the stop-gradient $(stopgrad)$ operation on the temporal gradient side, which means the teacher model would not receive any gradient from the alignment loss. This helps the TG model avoid degeneration by the alignment with the RGB student model. As shown in \\Cref{eq:alignment loss}, the alignment loss term for learning fine-grained motion features is:\n\\begin{equation}\n\\label{eq:alignment loss}\n\\small\n\\mathcal{L}_{kd} = \\left[\\mathcal{D}\\left(\\mathcal{F}^{RGB}_i, stopgrad(\\mathcal{F}^{TG}_i)\\right)\\right].\n\\end{equation}\n\n\\subsection{Cross-modal Contrastive Learning}\n\\label{sec:contrastive}\n\nThe dense feature alignment explicitly enables the RGB network to mimic the fine-grained motion signals from temporal gradient. We hypothesize that the global high-level representations across different modalities are also valuable and crucial. Therefore, cross-modal contrastive learning is employed as another module to discover the mutual information that coexists in both TG and RGB clips.\n Following the principle of SimCLR\\cite{chen2020simple} and CMC\\cite{tian2020contrastive}, we form the contrastive learning with positive pairs and negative pairs. Specifically, we consider the two modalities of the same video clip as a positive pair $\\{k^{+}\\}$ and the two modalities of different video clips as negative pairs $\\{k^{-}\\}$. The learning objective is to maximize the similarity of positive pairs and minimize the similarity of negative ones. We adopt \\mbox{InfoNCE} loss \\cite{oord2018representation} as the objective function over the features extracted from RGB and TG:\n\\begin{equation}\\label{eq:infonce} \t\\small\n\t\\mathcal{L}_{clr} = -\\log{\\frac{ {\\sum_{k \\in \\{k^+\\}}} \\exp\\left({\\mathrm{sim}(q, k) \/ \\tau}\\right)}{ {\\sum_{k \\in \\{k^+, k^-\\}}} {\\exp\\left({\\mathrm{sim}(q, k) \/ \\tau}\\right)}   }},\n\\end{equation} with $\\tau$ being a temperature hyper-parameter for scaling.\nAll embeddings are $\\ell_2$  normalized and dot product (cosine) similarity is used to compare them $\\mathrm{sim}(q, k) = q^\\top k \/ \\lVert q\\rVert \\lVert k\\rVert$.\n\t\nIt is worth noting that this cross-modal contrastive learning directly uses all weakly augmented samples of the two modalities ($\\alpha(x^{RGB\/TG}_i)$) in the FixMatch, including both labeled (the labels are not used) and unlabeled data. Therefore, there is no additional computation for the data loading and preprocessing.\n\n\n\\smallskip\\noindent\\textbf{Total Loss}:\nOur entire model based is jointly trained with cross-entropy loss over labeled data,cross-entropy loss over the unlabeled data with pseudo-labels, the dense alignment over both labeled and unlabeled data, and the cross-modal contrastive loss over both labeled and unlabeled data. Overall, the final objective function of our method is:\n\\begin{equation}\n\\small\n    \\mathcal{L}_{total} =w_{fm} (\\mathcal{L}_{fm}^{RGB} + \\mathcal{L}_{fm}^{TG}) + w_{kd}\\mathcal{L}_{kd} + w_{clr}\\mathcal{L}_{clr}.\n\\end{equation}\n\n\n\\begin{table*}[htbp]\n\t\t\\centering\n\t\t\\tablestyle{6pt}{1.1}\n\t\t\\small\n        \\begin{tabular}{cc|cc|cc|cc|cc|cc}\n        \\shline\n         && \\multicolumn{4}{c|}{Kinetics-400} & \\multicolumn{4}{c|}{UCF-101} & \\multicolumn{2}{c}{HMDB-51}\n        \\\\ \\hline\n        &            & \\multicolumn{2}{c|}{1\\%} & \\multicolumn{2}{c|}{10\\%} & \\multicolumn{2}{c|}{10\\%}                     & \\multicolumn{2}{c|}{20\\%}\n                             & \\multicolumn{2}{c}{50\\%}\n                             \\\\ \\shline\n        Alignment & Contrast                           & Top-1         & Top-5          & Top-1          & Top-5          & Top-1                     & Top-5                     & Top-1                     & Top-5\n        & Top-1                     & Top-5   \n        \\\\ \\shline\n        \\xmark       &     \\xmark              & 5.4          & 17.0          & 40.2          & 65.4          & {38.4} & {64.8} & {54.1} & {78.1} \n        & 37.8 &\t68.6\n        \\\\\n        \\cmark & \\xmark & 9.4          & 25.5          & 43.5          & 68.8          & 60.4                     & 84.4                     & 74.6                     & 91.7  &47.3 &74.8                   \\\\\n        \\xmark & \\cmark & 5.2 & 23.1 & 42.6 & 67.4 & 58.0                     & 82.5                     & 68.6         & 89.2\n        & 46.1 & 73.8\n        \\\\\n        \\cmark & \\cmark &\\textbf{9.8} & \\textbf{26.0} & \\textbf{43.8} & \\textbf{69.2} & \\textbf{62.4}            & \\textbf{84.9}            & \\textbf{76.1}            & \\textbf{92.1}\n        &  \\textbf{48.4}\t& \\textbf{75.9}\n        \\\\   \n        \\shline \n        \\end{tabular}\n\t\t\\centering\n\t\t\\caption{\\textbf{Effectiveness of the cross-modal alignment and contrastive learning.} The results are evaluated on the validation sets. The first row shows the results of the FixMatch baseline model without any proposed modules.}\n\t\t\\label{tab:abla_major}\n\t\t\\vspace{-4mm}\n\t\\end{table*}\n\t\n\t\n\n\n\n\\section{Experimental Results}\n\\subsection{Datasets and Evaluation}\n\\noindent\\textbf{Datasets.} Following previous state-of-the-art semi-supervised video action recognition methods~\\cite{jing2021videossl,zou2021learning, xiong2021multiview}, we evaluate our method on three public action recognition benchmarks: UCF-101~\\cite{soomro2012ucf101}, HMDB-51~\\cite{kuehne2011hmdb}, and Kinetics-400~\\cite{kay2017kinetics}. UCF-101 is a widely used dataset which consists of $13,320$ videos belonging to $101$ classes. HMDB-51 is a smaller dataset which consists of $6,766$ videos with $51$ classes. For UCF-101 and HMDB-51, we follow the data splits that released by VideoSSL\\cite{jing2021videossl}. The Kinetics-400 dataset is a large-scale dataset\nconsisting of $\\sim$235k training videos and $\\sim$20k validation videos belonging to 400 classes. For Kinetics-400, we follow the most recent state-of-the-art method MvPL~\\cite{xiong2021multiview} by forming two balanced labeled subsets by randomly sampling 6 and 60 videos per class for 1\\% and 10\\% settings.\n\n\\smallskip\\noindent\\textbf{Evaluation.}\nWe report Top-1 accuracy for major comparisons and Top-5 accuracy for some ablation studies.\n\n\\subsection{Implementation Details}\n\\label{sec:implementation}\n\\noindent\\textbf{Network architecture.} For a fair comparison with the state-of-the-art methods~\\cite{jing2021videossl, xiong2021multiview}, the FixMatch~\\cite{sohn2020fixmatch} framework is used as the backbone model while the 3D ResNet-18~\\cite{tran2018closer, he2016deep} is adopted as feature extractors for both RGB and TG (\\Cref{sec:parallel}) modalities. For each feature extractor, two individual contrastive heads with 3-layer non-linear MLP architecture are added for the cross-modal contrastive learning (\\Cref{sec:contrastive}).\n\n\\smallskip\\noindent\\textbf{Video augmentations.} There are two types of data augmentations: weak augmentation and strong augmentation. For the weak augmentation, the random horizontal flipping, random scaling, and random cropping following \\cite{zou2021learning}. To be specific, given a video clip, we firstly resize the video making the short side be $256$, and then a randomly resized crop operation is performed. The cropped clips are then resized to 224$\\times$224 pixels and flipped horizontally with a $50$\\% probability. For strong augmentation, the RandAugment~\\cite{cubuk2020randaugment} is chosen which randomly selects a small set of transformations from a large augmentation pools (\\eg, rotation, color inversion, translation, contrast adjustment, etc.) for each sample and then performs the selected data augmentation over the samples. It is worth noting that both the teacher (TG) and student (RGB) share the same weak augmentation (\\ie, the inputs are identically cropped in the same area and both flipped or not). This provides a direct positional information matching, which plays a crucial role for the dense alignment in \\Cref{sec:parallel}. \n\n\n\\smallskip\\noindent\\textbf{Training details.} All experiments are done with an initial learning rate of 0.2 on 8 GPUs by following the settings in~\\cite{feichtenhofer2019slowfast,zou2021learning, xiong2021multiview} using the cosine learning rate decaying scheduler~\\cite{loshchilov2016sgdr} and also a linear warm-up strategy \\cite{goyal2017accurate}. We use momentum of 0.9 and weight decay of 10$^\\text{-4}$. Dropout \\cite{srivastava2014dropout} of 0.5 is used before the final classifier layer to reduce the over-fitting. Following~\\cite{zou2021learning}, each mini-batch consists of 5 labeled data clips and 5 unlabeled data clips, while each input clip consists of 8 frames with a sampling stride of 8, which covers 64 frames of the raw video. We consistently train our models with 180 and 360 epochs for all experiments on UCF-101 and HMDB-51, while 45 (1\\%) and 90 (10\\%) epochs are trained for Kinetics-400. More training details are provided in the supplementary material. For the pseudo-label threshold, we follow~\\cite{xiong2021multiview} which sets it to 0.3 for getting more training samples.\nFor the loss weights, $w_{fm}$ is set to 0.5 while $w_{kd}$ and $w_{clr}$ are set to 1.\n\n\\smallskip\\noindent\\textbf{Inference.} \nFollowing the recent state-of-the-art methods \\cite{feichtenhofer2019slowfast, zou2021learning, xiong2021multiview}, 10 clips are uniformly sampled for each video along its temporal axis and each clip is taken 3 crops of 256$\\times$256. A total of 3x10 crops are evaluated for each video.\n\n\n\n\n\\subsection{Effectiveness of the Cross-modal Dense Alignment and Contrastive Learning}\n\n\n\nWe begin with a direct comparison to examine our hypothesis:\n\\emph{multimodal constraints on local and global features can serve as two complementary extensions to existing semi-supervised methods} (FixMatch~\\cite{sohn2020fixmatch} as the baseline).\nTo this end, our dense alignment (\\Cref{sec:alignemnt}) is devised to regularize the local features, and our contrastive loss (\\Cref{sec:contrastive}) is developed to distinguish global features. \nFor a fair comparison, we have ablated four experimental settings (detailed in \\Cref{tab:abla_major}): (1) none, (2) alignment-only, (3) contrast-only, and (4) both.\nKinetics-400, UCF-101, and HMDB-51 with different labeled data ratios (\\ie, 1\\%, 10\\%, 20\\%, and 50\\%) are used to ensure the generalizability of the following observations. \\textit{First}, FixMatch (none) exhibits acceptable but worse performance than its three counterparts, suggesting that pseudo labeling only is inadequate when using very limited labeled data. \n\\textit{Second}, dense alignment significantly elevates the performance (more than contrast-only), indicating that the fine-grained motion signal across multimodal plays an essential role in semi-supervised action recognition. \\textit{Third}, introducing contrastive loss across RGB and TG modalities improves Top-1\/Top-5 accuracy, revealing that global consistency in different modalities is advantageous.\n\\textit{Finally}, dense alignment and contrastive loss enforce the model learning from complementary perspectives because implementing both on top of FixMatch surpasses either one of them.\nWe hope that our discovery on multimodal constraints can shed new light on semi-supervised action recognition in video analysis.\n\n\\zzrevise{}{\n\\smallskip\\noindent\\textbf{Overfitting is alleviated.}  \\Cref{tab:overfitting} (Suppl.) presents a significant accuracy gap between the training and testing set, showing that FixMatch severely overfits to the training set. Our method effectively reduces the gap by imposing additional regularization on models with RGB as input.\n}\n\n\n\\begin{table*}[!t]\n\\centering\n\n\\begin{threeparttable}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{lccccccccccccc}\n\\toprule\n & w\/ ImageNet           &          & \\multicolumn{2}{c}{Kinetics-400} &           & \\multicolumn{4}{c}{UCF-101}                                   &  & \\multicolumn{3}{c}{HMDB-51}                   \\\\\n\\cmidrule{4-5}\n\\cmidrule{7-10}\n\\cmidrule{12-14}\nMethod                                           & distillation & Backbone    & 1\\%              & 10\\%              &           & 5\\%             & 10\\%            & 20\\%            & 50\\%            &  & 40\\%            & 50\\%            & 60\\%            \\\\ \\midrule\nPseudo-Label~\\cite{lee2013pseudo} (ICMLW 2013)   & \\xmark       & R3D-18      & 6.3            & -               &           & 17.6          & 24.7          & 37.0          & 47.5          &  & 27.3          & 32.4          & 33.5          \\\\\nMeanTeacher~\\cite{tarvainen2017mean} (NIPS 2017) & \\xmark       & R3D-18      & 6.8            & 19.5            &           & 17.5          & 25.6          & 36.3          & 45.8          &  & 27.2          & 30.4          & 32.2          \\\\\nS4L~\\cite{zhai2019s4l} (ICCV 2019)               & \\xmark       & R3D-18      & 6.3            & -               &           & 22.7          & 29.1          & 37.7          & 47.9          &  & 29.8          & 31.0          & 35.6          \\\\\nUPS~\\cite{rizve2020defense} (ICLR 2021)          & \\xmark       & R3D-18      & -              & -               &           & -             & -             & 39.4          & 50.2          &  & -             & -             & -             \\\\ \\midrule\nVideoSSL~\\cite{jing2021videossl} (WACV 2021)     & \\cmark       & R3D-18      & -              & 33.8            &           & 32.4          & 42.0          & 48.7          & 54.3          &  & 32.7          & 36.2          & 37.0          \\\\\nTCL~\\cite{singh2021semi} (CVPR 2021)             & \\xmark       & R3D(TSM)-18 & 7.7            & -               &           & -             & -             & -             & -             &  & -             & -             & -             \\\\\nActorCutMix~\\cite{zou2021learning} (arXiv 2021)  & \\xmark       & R(2+1)D-34  & -              & -               &           & 27.0          & 40.2          & 51.7          & 59.9          &  & 32.9          & 38.2          & 38.9          \\\\\nMvPL*~\\cite{xiong2021multiview} (arXiv 2021)     & \\xmark       & R3D-18      & 5.0            & 36.9            &           & 41.2          & 55.5          & 64.7          & 65.6          &  & 30.5          & 33.9          & 35.8          \\\\\n\\textbf{Ours}                                    & \\xmark       & R3D-18      & \\textbf{9.8}   & \\textbf{43.8}   & \\textbf{} & \\textbf{44.8} & \\textbf{62.4} & \\textbf{76.1} & \\textbf{79.3} &  & \\textbf{46.5} & \\textbf{48.4} & \\textbf{49.7}\n\n\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\begin{tablenotes}\n      \\small\n      \\item * indicates the method is reimplemented by ourselves. The input modalities are RGB and TG.\n    \\end{tablenotes}\n\\end{threeparttable}\n\\caption{\\textbf{Comparison with the state-of-the-arts methods.}\nThe results are reported with Top-1 accuracy (\\%) on the validation sets.\nThe best performance of each setting is in \\textbf{bold}.\n}\n\\vspace{-5mm}\n\\label{tab:main_all}\n\\end{table*}\n\\subsection{Comparison with State-of-the-art Methods}\n\nTo demonstrate the capability and potential of our proposed method, we compared with the most recent state-of-the-art methods for the semi-supervised action recognition task on public datasets including Kinetics-400, UCF-101 and HMDB-51. As shown in \\Cref{tab:main_all}, we mainly compare with two types of methods including image-based methods~\\cite{lee2013pseudo, tarvainen2017mean, zhai2019s4l} which were originally designed for image classification and then simply adopted to video tasks and video based methods ~\\cite{jing2021videossl, singh2021semi, zou2021learning, xiong2021multiview} which were specifically designed for video action recognition task. \n\n\n\\smallskip\\noindent\\textbf{Comparison with image-based methods.}\nThe first three rows in \\Cref{tab:main_all} show the results of image-based methods including \\ie, Pseudo-Label\\cite{lee2013pseudo}, MeanTeacher\\cite{tarvainen2017mean} and S4L\\cite{zhai2019s4l}. In general, the results of all the three image-based methods across over the three datasets with all different labeled percentages are much lower than the results of all video-based based methods. This confirms that it is necessary to propose methods specifically designed based on the video temporal and multimodal attributes.\n\n\n\\smallskip\\noindent\\textbf{Comparison with video-based methods.} The overall performance of the video-based performance are much higher. VideoSSL surpasses all the image-based methods by using Imagenet pre-trained model to guide the learning, and TCL~\\cite{singh2021semi} use self-supervised learning task as auxiliary task and the group contrastive for the video semi-supervised learning. Both the ActorCutMix~\\cite{zou2021learning} and MvPL~\\cite{xiong2021multiview} are adapted from FixMatch~\\cite{sohn2020fixmatch}. Benefited by our proposed cross-modal dense alignment and cross-modal contrastive, our method outperforms all these methods by a significant margin on three datasets under all the experimental settings (different ratio of labels).\n\n\n\n\\subsection{Ablation Studies}\n\\label{sec:Abla}\n\nTo understand the impact of each part of the design in our method, we conduct extensive ablation studies on the UCF-101 dataset with 20\\% labeled setting.\n\n\n\\begin{figure}[t]\n\t\\centering\n    \\hspace*{-0.5cm} \n\t\\includegraphics[width=1\\columnwidth]{figures\/compare_slowfast_tg.pdf}\n\t\\caption{\\textbf{Visualization of the slow and fast temporal gradient.} Slow temporal gradient contains a more noisy background of the shooting environment while fast temporal gradient focuses more on the activity-related moving objects.} \n\t\\label{fig:compare_slowfast_tg}\n\t\\vspace{-5mm}\n\\end{figure}\n\n\\smallskip\\noindent\\textbf{Fast temporal gradient is better.}\nTemporal gradient (TG) is calculated by differing two RGB frames and the stride of them could be small or large to generate either fast or slow TG. To delve deeper into the effect of different strides, we conduct experiments with fast TG (calculation stride = 1) and slow TG (calculation stride = 7), and the results are shown in \\Cref{tab:abla_fast}. The first group compare the performance with the baseline FixMatch framework with different modalities of data as input. The results confirm that both the slow TG and fast TG perform much better than RGB (more than 25\\% higher), and also demonstrate that the Fast TG is better than the slower TG for the semi-supervised setting. The second group of \\Cref{tab:abla_fast} compares the final performance of our model with different temporal gradients. When the pseudo-labels are generated by the fast TG, the model beats the performance with slow TG with a large margin (74.1\\% vs. 68.2\\%). To figure out the reason why the performance of fast TG is much higher than slower TG, we visualized the two types of temporal gradient for three video clips and the visualization are shown in \\Cref{fig:compare_slowfast_tg}. The comparison shows that the slow TG has much noisy background information especially when the cameras have significant movement while the fast temporal gradient information focuses more on the boundary of the fast moving objects (\\eg, people, balls). Both the quantitative and qualitative results \nverify the advantages of the fast TG over the slower TG for semi-supervised action recognition. \n\n\\begin{table*}[t]\n    \\captionsetup[subtable]{font=normalsize}\n\n\t\\begin{subtable}[t]{0.25\\textwidth}\n\t\t\\centering\n\t\t\\tablestyle{2pt}{1.02}\n\t\t\\small\n\t\t\\begin{tabular}{cc|c}\nStudent & Teacher & Top-1          \\\\ \\shline\nRGB     & -       & 52.9          \\\\\nSlow TG & -       & 67.3          \\\\\nFast TG & -       & \\textbf{68.3} \\\\ \\hline\nRGB     & Slow TG  & 68.2          \\\\\nRGB     & Fast TG  & \\textbf{74.1}\n\\end{tabular}\n\t\t\\caption{\\textbf{Fast temporal gradient is better.} \n\t}\n\t\t\\label{tab:abla_fast}\n\t\\end{subtable}\n\t\\begin{subtable}[t]{0.25\\textwidth}\n\t\t\\tablestyle{2pt}{1.02}\n\t\t\\small\n\t\t\\begin{tabular}{c|cc}\n     Align. Loss  & Top-1 & Top-5 \\\\ \\shline\n     \n- & 54.1 & 78.1 \\\\ \\hline\nL1     & 74.0 & 91.3 \\\\\nL2     & 74.4 & 91.4 \\\\\nCosine & \\textbf{74.6} & \\textbf{91.7}\n\\end{tabular}\n\\vspace{6pt}\n\t\t\\caption{\\textbf{Dense alignment functions.}}\n\t\t\\label{tab:abla_alignloss}\n\t\\end{subtable}\n\t\\begin{subtable}[t]{0.2\\textwidth}\n\t\t\\tablestyle{2pt}{1.02}\n\t\t\\small\n\t\t\\begin{tabular}{c|cc}\n        Stopgrad & Top-1          & Top-5          \\\\ \\shline\n        \\xmark        & 60.0          & 84.4          \\\\\n        \\cmark        & \\textbf{74.6} & \\textbf{91.7}\n        \\end{tabular}\n        \\vspace{17pt}\n\t\t\\caption{\\textbf{Stop gradient in knowledge distillation.}}\n\t\t\\label{tab:abla_stopgrad}\n\t\\end{subtable}\n\t\\begin{subtable}[t]{0.3\\textwidth}\n\t\t\\tablestyle{2pt}{1.02}\n\t\t\\small\n\t\t\\begin{tabular}{c|cc}\n        Pseudo-label Metric & Top-1          & Top-5          \\\\ \\shline\n        RGB                 & 73.6          & 91.0          \\\\\n        TG                  & 74.1          & 91.3          \\\\\n        Self          & 72.8          & 91.6          \\\\\n        Average         & \\textbf{74.6} & \\textbf{91.7}\n        \\end{tabular}\n        \\vspace{6pt}\n\t\t\\caption{\\textbf{Metrics for the pseudo-labels.}}\n\t\t\\label{tab:abla_pl_metric}\n\t\\end{subtable}\n\t\n\t\n\t\n\t\\begin{subtable}[t]{0.35\\textwidth}\n\t\t\\centering\n\t\t\\tablestyle{2pt}{1.02}\n\t\t\\small\n\t\t\\begin{tabular}{cccc|cc}\n\\multicolumn{4}{c|}{Aligned Block Index} & \\multicolumn{2}{c}{Accuracy}  \\\\ \\hline\n1st     & 2nd     & 3rd     & 4th & Top-1          & Top-5          \\\\ \\shline\n\\xmark  & \\xmark  & \\xmark  & \\cmark  & 71.4          & 90.2          \\\\\n\\xmark  & \\xmark  & \\cmark  & \\cmark  & 74.0          & 91.4          \\\\\n\\xmark  & \\cmark  & \\cmark  & \\cmark  & 74.4          & \\textbf{91.8} \\\\\n\\cmark  & \\cmark  & \\cmark  & \\cmark  & \\textbf{74.6} & 91.7         \n\\end{tabular}\n\t\t\\caption{\\textbf{Align them in block-wise.} }\n\t\t\\label{tab:abla_blockwise}\n\t\\end{subtable}\n\t\\begin{subtable}[t]{0.3\\textwidth}\n\t\\centering\n\t\t\\tablestyle{2pt}{1.02}\n\t\t\\small\n\t\t\\begin{tabular}{l|cc}\n                   & Top-1          & Top-5          \\\\ \\shline\n    Plain               & 71.1          & 90.0          \\\\\n    + LR warm-up          & 71.9          & 91.1          \\\\\n    + Sup. warm-up & 74.1          & 91.0          \\\\\n    + PreciseBN         & \\textbf{74.6} & \\textbf{91.7}\n    \\end{tabular}\n    \\vspace{5pt}\n    \\caption{\\textbf{The crucial training tricks.} \n\t}\n\t\\label{tab:abla_tricks}\n\t\\end{subtable}\n\t\\begin{subtable}[t]{0.35\\textwidth}\n\t\\centering\n\t\t\\tablestyle{2pt}{1.02}\n\t\t\\small\n\t\t\\begin{tabular}{c|cc}\n        Tempature $\\tau$ & Top-1          & Top-5 \\\\ \\shline\n        0.1              & 74.8          & 91.8 \\\\\n        0.2              & 75.2          & \\textbf{92.4} \\\\\n        0.5              & \\textbf{76.1} & 92.1 \\\\\n        1.0              & 74.3          & 92.2\n        \\end{tabular}\n        \\vspace{5pt}\n        \\caption{\\textbf{Ablation on contrastive temperature.}}\n        \\label{tab:abla_temperature}\n\t\\end{subtable}\n\t\n\t\n\t\\caption{\\textbf{Ablation studies} on UCF101 split-1 under 20\\% semi-supervised setting  (only use 20\\% labeled data). The results are reported with Top-1 and Top-5 accuracy on the validation set. Backbone: 3D ResNet-18~\\cite{he2016deep,tran2018closer}, each input clip consists of 8 frames sampled from a single video with the inter-frame interval of 8. Except for the study (a), all the other results are evaluated with PreciseBN. Except for the study (g), all the other experiments are without cross-modal contrastive learning for better comparisons.}\n\t\\vspace{-2mm}\n\t\\vspace{-5pt}\n\t\\label{tab:ablations}\n\\end{table*}\n\n\\smallskip\\noindent\\textbf{The choice of alignment functions.} As discussed in \\Cref{sec:alignemnt}, there are many possible choices for the alignment loss function as long as it can effectively enforce the similarity between the two features. Here we studied the performance of three different alignment functions including L1, L2 and Cosine Similarity loss. As shown in \\Cref{fig:compare_slowfast_tg} (b), all the three loss functions in alignment achieve high performance while Cosine Similarity (74.6\\%) outperforms the other two functions (74.0\\% \\& 74.4\\%). A possible explanation is that L1 and L2 have more strict constraints on the scale of two representations, while the Cosine Similarity loss focuses on the vector orientation (\\eg, L1 and L2 losses of $\\Vec{v_1}$=(10,10,10) and $\\Vec{v_2}$=(1,1,1) are large while the Cosine Similarity loss is 0). Although TG is normalized to the 0-255 range during training, there is still a gap in the scales between the representations of RGB and TG. A strict constraint like L1 or L2 would have negative effects on the model for learning motion features.\n\n\\smallskip\\noindent\\textbf{Stop gradient in knowledge distillation.} The stop-gradient operation on the TG side stated in \\Cref{sec:alignemnt} is one of the keys to the successful knowledge distillation with dense alignment. However, as the student RGB has much appearance information which TG does not have, directly training  with the dense alignment strategy would make the teacher TG model degenerate greatly and hard to focus on extracting the fine-grained motion features. The stop-gradient avoids the fine-grained motion-related representations in TG model can be disturbed by the RGB model. As shown in \\Cref{tab:abla_stopgrad}, there is a 14.6\\% performance drop on Top-1 accuracy (60.0\\% vs. 74.6\\%) when stop gradient is taken off. \n\n\n\\smallskip\\noindent\\textbf{How to generate pseudo-labels?} There are multiple ways to generate pseudo-labels since our model takes two input modalities. We compare the performance of four settings: 1) use the prediction from RGB model as pseudo-labels, 2) use the prediction from TG model as pseudo-labels, 3) each model uses the probabilities of its self-modality, and 4) fuse the results from both RGB and TG as pseudo-labels. \\Cref{tab:abla_pl_metric} shows that the fused pseudo-labels are more reliable and achieve the best performance benefiting from comprehensive information of both RGB and TB.\n\n\\smallskip\\noindent\\textbf{Dense alignment in block-wise.} An intuitive question about our knowledge distillation framework is that which block or blocks should be densely aligned. Therefore, we conduct this ablation study by adding dense alignment to different positions (\\ie, blocks) and the results are shown in \\Cref{tab:abla_blockwise}. As the common practice of previous knowledge distillation methods \\cite{hinton2015distilling,thoker2019cross,stroud2020d3d} is to align the high-level features of the last layers. Therefore, we start to add the dense alignment module over the features from the last (4-th) block (ResNet basic block) and then experiment with more blocks. Their performances are consistently improved when more blocks are densely aligned and the best Top-1 accuracy is achieved with all blocks aligned. Compared with the baseline, our block-wise dense alignment strategy gains a considerable improvement of 20.5\\% (54.1\\% to 74.6\\%) which demonstrates that fine-grained motion signals are better at semi-supervised model generalization.\n\n\\smallskip\\noindent\\textbf{Crucial training tricks.} Through the extensive experiments, we identified several training tricks which are essential to lead to the high performance. \\Cref{tab:abla_tricks} shows the impact of learning rate warm-up \\cite{goyal2017accurate}, supervised warm-up\\cite{xiong2021multiview} and PreciseBN\\cite{wu2021rethinking}. All the three tricks could make a decent improvement, while the supervised warm-up (training with only the labeled data at the first several epochs) is the most effective one which gains an improvement of 2.7\\% (71.9\\% to 74.6\\%). This shows that the supervised warm-up can alleviate the cold-start issue that low-quality pseudo-labels would be generated at the beginning. The performance of the semi-supervised learning model could easily have large variations~\\cite{singh2021semi,oliver2018realistic,wang20213dioumatch}. These three tricks can solidly improve the performance while in the meantime make the training more stable.\n\n\\smallskip\\noindent\\textbf{Contrastive temperature.}\nAn appropriate temperature is important to the good performance of contrastive learning~\\cite{chen2020simple}, we ablate the contrastive loss temperature in \\Cref{eq:infonce}. As shown in \\Cref{tab:abla_temperature}, a modest temperature (e.g., 0.2 or 0.5) could help the proposed cross-modal contrastive learning work better while a large (1.0) or small (0.1) temperature is not that optimal.\n\n\n\n\n\\section{Conclusion}\n\n\nThis paper has presented a novel semi-supervised learning method which introduces temporal gradient for the fruitful motion-related information and extra representation consistency crossing multiple modalities. Our proposed method uses the block-wise dense alignment strategy and cross-modal contrastive learning. Without additional computation or delay during inference, our method substantially outperforms all prior methods while achieving state-of-the-art performance on UCF-101, HMDB-51, and Kinetics-400 datasets with all the experimented settings (different labeled ratios). In the future, we plan to study the effectiveness of temporal gradient on other video-based tasks and to automatically search or generate powerful modalities.\n\n\\smallskip\\noindent\\textbf{Acknowledgments:} This work was supported by the National Science Foundation under Grant No. NSF-1763705.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{INTRO}\n\nCondensation and fragmentation are basic and widely-studied concepts of Bose-Einstein condensate\nemanating from the properties of the reduced one-particle density matrix \\cite{RDM1,RDM2,RDM3,RDM4,RDM5}.\nThe bosons are said to be condensed if there is a single macroscopic eigenvalue of the\nreduced one-particle density matrix \\cite{PEN_ONS} and fragmented if there\nare two or more such macroscopic eigenvalues \\cite{FRAG_REV}. \nThese eigenvalues are commonly called natural occupation numbers and\nthe respective eigenfunctions of the reduced one-particle density matrix are referred to as natural orbitals.\nFragmentation of Bose-Einstein condensate has been investigated, e.g.,\nin \\cite{FRAG1,FRAG2,FRAG3,FRAG4,FRAG5,FRAG6,FRAG7,FRAG8,FRAG9,FRAG10,FRAG11,FRAG12,\nFRAG13,FRAG14,FRAG15,FRAG16,FRAG17,FRAG18,FRAG19,FRAG20}.\n\nThe condensation and especially fragmentation of\nthe reduced two-particle density matrix of interacting identical bosons is less studied, see, e.g., \\cite{RDMs_FRAG}.\nHere, the analysis of the reduced two-particle density matrix\nwould determine whether pairs of bosons are condensed or fragmented.\nThe respective eigenfunctions of the reduced two-particle density matrix\nare often called natural geminals.\nWe note that natural geminals in electronic systems have long been explored, see, e.g.,\n\\cite{GEM1,GEM2,GEM3,GEM4,GEM5,GEM6,GEM7,GEM8,GEM9,GEM10,GEM11,GEM12}.\n\nConsider now a mixture of two kinds of identical bosons,\nwhich are labeled species $1$ and species $2$.\nMixtures of Bose-Einstein condensate is a highly investigated topic, see, e.g., \\cite{MIX1,MIX2,MIX3,MIX4,MIX5,MIX6,MIX7,MIX8,MIX9,MIX10,MIX11,MIX12,MIX13,\nMIX14,MIX15,MIX16,MIX17,MIX18,MIX19,MIX20,MIX21,MIX22,MIX23,MIX24,MIX25,MIX26,\nMIX27,MIX28,MIX29,MIX30}.  \nOne may ask,\njust like for single-species bosons,\nabout condensation or fragmentation of each of the species\nand how, for instance,\none species is affected by the presence of the other species and vice versa.\nTo answer this question the intra-species reduced one-particle density matrices\nof species $1$ and $2$ are required, i.e.,\nanalyzing the intra-species occupation numbers and natural orbitals.\nFollowing the above line,\none could also investigate fragmentation of higher-order intra-species reduced density matrices in the mixture.\nFor instance, to investigate whether pairs of identical bosons,\nof either species $1$ or species $2$, are fragmented,\ndiagonalizing of the intra-species reduced two-particle density matrices is needed.\nSummarizing, fragmentation of identical bosons and its manifestation\nin higher-order reduced density matrices stem from the properties of intra-species quantities.\n\nBut, a mixture of Bose-Einstein condensates\noffers a degree-of-freedom or many-particle construction\nwhich do not exist for single-species bosons, namely, inter-species reduced density matrices.\nNow, if fragmentation of identical bosons and pairs is defined as the macroscopic occupation\nof respective eigenvalues following the diagonalization of intra-species reduced density matrices,\nwe may analogously define fragmentation of distinguishable bosons' pairs as\nmacroscopic occupation of the eigenvalues of the inter-species reduced two-particle density matrix.\nObviously, the later is the lowest-order inter-species quantity,\nsince at least one particle of each species is needed to build an\ninter-species entity.\n\nThe above discussion defines the goals of the present work which are:\n(i) To investigate fragmentation of pairs of identical bosons and establish fragmentation of pairs of \ndistinguishable bosons in a mixture of Bose-Einstein condensates;\n(ii) To construct the respective natural geminals of the mixture, for identical pairs and for distinguishable pairs;\n(iii) To show that fragmentation of distinguishable bosons' pairs in the mixture persists with\nhigher-order inter-species reduced density matrices;\n(iv) To construct the Schmidt decomposition of the mixture's wavefunction and discuss\nsome of its properties at the limit of an infinite-number of particles where the mixture is $100\\%$ condensed;\nand\n(v) Achieving the first four goals analytically, using an exactly solvable model.\n\nTo this end we recruit the harmonic-interaction model for mixtures\n\\cite{HIM_MIX1,HIM_MIX_ENTANGLE,HIM_MIX2,HIM_MIX_RDM,HIM_MIX_VAR,HIM_MIX_FLOQUET},\nor, more precisely here, a symmetric version of which \\cite{HIM_MIX_CP}.\nThe harmonic-interaction model for single-species bosons (and fermions)\nhas been used extensively in the literature including for investigating properties of Bose-Einstein condensates\n\\cite{HIM_RDM,HIM_DIAG1,HIM_DIAG2,HIM_JCP,HIM_SCH,HIM1,HIM2,HIM3,HIM4,HIM5,HIM6,HIM7,HIM8,HIM9,HIM10}.\nIn our work\nwe build on results obtained and techniques used \nfor the reduced density matrices of single-species bosons within the harmonic-interaction model\n\\cite{HIM_RDM,HIM_DIAG1,HIM_DIAG2,HIM_JCP,HIM_SCH},\nand, among others, generalize and extend them for \nthe intra-species and particularly the inter-species reduced density matrices of mixtures \\cite{HIM_MIX_RDM}.\n\nThe structure of the paper is as follows.\nIn Sec.~\\ref{PAIR} we construct and investigate fragmentation of\nintra-species and inter-species pair functions in the mixture.\nIn Sec.~\\ref{MORE} we extend the results and explore fragmentation of\npairs of distinguishable pairs.\nFurthermore, a complementary result for the Schmidt decomposition\nof the mixture's wavefucntion at the limit of an infinite number of particles is offered.\nIn Sec.~\\ref{SUM_OUT} a summary of the results and an outlook of some prospected research topics are provided. \nFinally, appendix \\ref{APP} collects for comparison with the mixture \nthe details of fragmentation of bosons and pairs in the single-species system.\n\n\\section{Intra-species and inter-species natural pair functions}\\label{PAIR}\n\n\\subsection{The symmetric two-species harmonic-interaction model}\\label{PAIR_1}\n\nWe consider a mixture of two Bose-Einstein condensates\ndescribed by the Hamiltonian of the symmetric two-species harmonic-interaction model\n\\cite{HIM_MIX_CP,HIM_MIX_RDM}:\n\\begin{eqnarray}\\label{HIM_MIX}\n& & \\hat H(x_1,\\ldots,x_N,y_1,\\ldots,x_N) =\n\\sum_{j=1}^{N} \\left( -\\frac{1}{2m}\\frac{\\partial^2}{\\partial x_j^2} + \\frac{1}{2}m\\omega^2 x_j^2 \\right)\n+ \\lambda \\sum_{1\\le j <k}^{N} \\left(x_j-x_k\\right)^2 + \\nonumber \\\\ \n& & +\n\\sum_{j=1}^{N} \\left( -\\frac{1}{2m}\\frac{\\partial^2}{\\partial y_j^2} + \\frac{1}{2}m\\omega^2 y_j^2 \\right)\n+ \\lambda \\sum_{1\\le j <k}^{N} \\left(y_j-y_k\\right)^2 + \\lambda_{12} \\sum_{j=1}^{N} \\sum_{k=1}^{N} \\left(x_j-y_k\\right)^2. \\\n\\end{eqnarray}\nThere are $N$ bosons of type $1$, $N$ bosons of type $2$,\nand the mass of each boson is $m$.\n$\\lambda$ is the intra-species interaction strength, either between two bosons of type $1$ or two bosons of type $2$,\nand $\\lambda_{12}$ is the inter-species interaction strength between type $1$ and type $2$ bosons.\nDimensionality plays no role in the present work hence we work in one spatial dimensions.\n$\\hbar=1$ is used throughout.\nEmploying Jacoby coordinates for the mixture and translating back to the laboratory frame,\nthe $2N$-boson wavefunction and corresponding many-particle density matrix are given by\n\\begin{subequations}\\label{HIM_MIX_WF_DEN_MAT}\n\\begin{eqnarray}\\label{HIM_MIX_WF_DEN_MAT1}\n& &\n\\Psi(x_1,\\ldots,x_N,y_1,\\ldots,y_N) =\n\\left(\\frac{m\\Omega}{\\pi}\\right)^{\\frac{N-1}{2}}\n\\left(\\frac{M_{12}\\Omega_{12}}{\\pi}\\right)^{\\frac{1}{4}}\n\\left(\\frac{M\\omega}{\\pi}\\right)^{\\frac{1}{4}} \\times \\nonumber \\\\\n& &\n\\times e^{-\\frac{\\alpha}{2}\\sum_{j=1}^N \\left(x_j^2 + y_j^2\\right) - \\beta \\sum_{1\\le j < k}^N \\left(x_jx_k+y_jy_k\\right)\n+ \\gamma \\sum_{j=1}^N\\sum_{k=1}^N x_jy_k}, \\\n\\end{eqnarray}\n\\begin{eqnarray}\\label{HIM_MIX_WF_DEN_MAT2}\n& &\n\\Psi(x_1,\\ldots,x_N,y_1,\\ldots,y_N)\\Psi^\\ast(x'_1,\\ldots,x'_N,y'_1,\\ldots,y'_N) = \n\\left(\\frac{m\\Omega}{\\pi}\\right)^{N-1}\n\\left(\\frac{M_{12}\\Omega_{12}}{\\pi}\\right)^{\\frac{1}{2}}\n\\left(\\frac{M\\omega}{\\pi}\\right)^{\\frac{1}{2}} \\times \\nonumber \\\\\n& &\n\\times\ne^{-\\frac{\\alpha}{2}\\sum_{j=1}^N \\left(x_j^2 + {x'_j}^2 +  y_j^2 + {y'_j}^2\\right) -\n\\beta \\sum_{1\\le j < k}^N \\left(x_jx_k+x'_jx'_k+y_jy_k+y'_jy'_k\\right)\n+ \\gamma \\sum_{j=1}^N\\sum_{k=1}^N \\left(x_jy_k+x'_jy'_k\\right)}, \\\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\\label{HIM_MIX_WF_DEN_MAT3}\n& &\n\\Omega = \\sqrt{\\omega^2+\\frac{2N}{m}\\left(\\lambda+\\lambda_{12}\\right)}, \\qquad\n\\Omega_{12} = \\sqrt{\\omega^2+\\frac{4N}{m}\\lambda_{12}}, \\nonumber \\\\\n& &\n\\alpha = m\\Omega + \\beta, \\qquad\n\\beta = \\frac{m}{2N}\\left(\\Omega_{12}+\\omega-2\\Omega\\right), \\qquad\n\\gamma=\\frac{m}{2N}\\left(\\Omega_{12}-\\omega\\right), \\\n\\end{eqnarray}\n\\end{subequations}\nand the relative center-of-mass $M_{12}=\\frac{m}{2N}$ and center-of-mass $M=2mN$ masses.\nThe wavefunction and similarly the many-particle density of the mixture \ndepend on two dressed frequencies, $\\Omega$ and $\\Omega_{12}$,\nand consist\nof three parts:\nOne-body part with coefficient $\\alpha$,\nintra-species two-body coupling with coefficient $\\beta$,\nand inter-species two-body coupling with coefficient $\\gamma$.\nWhereas $\\alpha$ and $\\beta$ depend on the intra-species and inter-species interactions,\n$\\gamma$ depends on the inter-species interaction only.\n\nAnother issue worth mentioning is the stability of the mixture.\nFor a stable, i.e., bound,\nmixture both dressed frequencies $\\Omega$ and $\\Omega_{12}$ must be positive.\nThis implies the conditions $\\lambda+\\lambda_{12} > - \\frac{m\\omega^2}{2N}$ and\n$\\lambda_{12} > - \\frac{m\\omega^2}{4N}$, respectively, on the interactions.\nIn other words, the inter-species interaction $\\lambda_{12}$ is bound from below,\nimplying that the mutual repulsion between the two species cannot be too strong,\nbut is not bound from above,\nmeaning that the mutual attraction between the two species can be unlimitedly strong.\nFurthermore,\nthe intra-species interaction $\\lambda$ can take\nany value as long as the inter-species interaction is sufficiently attractive, i.e.,\n$\\lambda > - \\frac{m\\omega^2}{2N} - \\lambda_{12}$.\nWe shall return to the dressed\nfrequencies $\\Omega$ and $\\Omega_{12}$ below.\n\n\\subsection{Intra-species natural pair functions}\\label{PAIR_2}\n\nThe intra-species reduced density matrices are defined when all bosons of the other type are integrated out.\nWe concentrate in what follows on the reduced one-particle\nand in particular the two-particle density matrices of species $1$,\n\\begin{eqnarray}\\label{RDMs_1_2}\n& & \\rho_1^{(1)}(x,x') = N \\int dx_2 \\cdots dx_N dy_1 \\cdots dy_N \n\\Psi(x,x_2,\\ldots,x_N,y_1,\\ldots,y_N) \\times \\nonumber \\\\\n& &\n\\times \\Psi^\\ast(x',x_2,\\ldots,x_N,y_1,\\ldots,y_N), \\nonumber \\\\\n& & \\rho_1^{(2)}(x_1,x_2,x'_1,x'_2) = N(N-1) \\int dx_3 \\cdots dx_N dy_1 \\cdots dy_N \n\\Psi(x_1,x_2,x_3,\\ldots,x_N,y_1,\\ldots,y_N) \\times \\nonumber \\\\\n& &\n\\times \\Psi^\\ast(x'_1,x'_2,x_3,\\ldots,x_N,y_1,\\ldots,y_N). \\\n\\end{eqnarray}\nIn a symmetric mixture, the corresponding reduced density matrices of species $2$, \n$\\rho_2^{(1)}(y,y')$ and\n$\\rho_2^{(2)}(y_1,y_2,y'_1,y'_2)$,\nare the same and need not be repeated.\n\nThe reduction of the many-particle density (\\ref{HIM_MIX_WF_DEN_MAT2})\nto its finite-order reduced density matrices is\nsomewhat lengthly and given in \\cite{HIM_MIX_RDM}.\nWe start from the final expression for the intra-species reduced one-particle density matrix\nwhich is given by\n\\begin{eqnarray}\\label{1_RDM}\n& & \\rho_1^{(1)}(x,x') = N \\left(\\frac{\\alpha+C_{1,0}}{\\pi}\\right)^{\\frac{1}{2}}\ne^{-\\frac{\\alpha}{2}\\left(x^2+{x'}^2\\right)} \ne^{- \\frac{1}{4} C_{1,0} \\left(x+x'\\right)^2} = \\nonumber \\\\\n& &\n= N \\left(\\frac{\\alpha+C_{1,0}}{\\pi}\\right)^{\\frac{1}{2}}\ne^{-\\frac{\\alpha+\\frac{C_{1,0}}{2}}{2}\\left(x^2+{x'}^2\\right)} \ne^{- \\frac{1}{2} C_{1,0} xx'}, \\nonumber \\\\\n& &\n\\alpha + C_{1,0} =\n(\\alpha-\\beta)\n\\frac{\\left[(\\alpha-\\beta) + N\\beta\\right]^2 - N^2\\gamma^2}\n{\\left[(\\alpha-\\beta) + N\\beta\\right]\\left[(\\alpha-\\beta) + (N-1)\\beta\\right] - N(N-1)\\gamma^2}.\n\\end{eqnarray}\nThe coefficient $C_{1,0}$ governs the properties of the intra-species reduced one-particle density matrix\nand reminds one that all bosons of type $2$ and all but a single boson of type $1$ are integrated out.\nAs might be expected, $\\rho_1^{(1)}(x,x')$\ndepends on the three parts of the many-boson wavefunction,\ni.e., on the $\\alpha$, $\\beta$, and $\\gamma$ terms (\\ref{HIM_MIX_WF_DEN_MAT1}).\nIn the absence of inter-species interaction, i.e., for $\\gamma=0$,\nthe coefficient $C_{1,0}$\nboils down to that of the single-species harmonic-interaction model,\nsee appendix \\ref{APP} for further discussion.\n\nJust as for the case of single-species bosons \\cite{HIM_JCP,HIM_SCH},\nthe intra-species reduced one-particle density matrix (\\ref{1_RDM}) can be diagonalized using Mehler's formula.\nMehler's formula can be written as follows\n\\begin{eqnarray}\\label{MEHLER}\n& & \\left[\\frac{(1-\\rho)s}{(1+\\rho)\\pi}\\right]^{\\frac{1}{2}}\ne^{-\\frac{1}{2}\\frac{(1+\\rho^2)s}{1-\\rho^2}\\left(x^2+{x'}^2\\right)} \\,\ne^{+\\frac{2\\rho s}{1-\\rho^2}xx'}\n= \\nonumber \\\\\n& & \n\\quad = \\sum_{n=0}^\\infty (1-\\rho)\\rho^n \\frac{1}{\\sqrt{2^n n!}} \\left(\\frac{s}{\\pi}\\right)^{\\frac{1}{4}} H_n(\\sqrt{s}x) e^{-\\frac{1}{2}s x^2}\n\\frac{1}{\\sqrt{2^n n!}} \\left(\\frac{s}{\\pi}\\right)^{\\frac{1}{4}} H_n(\\sqrt{s}x') e^{-\\frac{1}{2}s {x'}^2}, \\\n\\end{eqnarray}\nwith $s>0$ and, generally, for intra-species and inter-species\nreduced density matrices as well as later on for\nSchmidt decomposition of the wavefunction,\n$1 > \\rho \\ge 0$.\n$H_n$ are the Hermite polynomials.\n\nComparing the structure of the intra-species reduced\none-particle density matrix $\\rho_1^{(1)}(x,x')$\nwith that of Mehler's formula one readily has\n\\begin{eqnarray}\\label{S_1_1}\n& & s_1^{(1)} = \\sqrt{\\alpha\\left(\\alpha+C_{1,0}\\right)} =\n\\sqrt{\\frac{\\alpha(\\alpha-\\beta)\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]^2 - N^2\\gamma^2\\right\\}}\n{\\left[(\\alpha-\\beta) + N\\beta\\right]\\left[(\\alpha-\\beta) + (N-1)\\beta\\right] - N(N-1)\\gamma^2}},\n\\nonumber \\\\ \\nonumber \\\\\n& & \\rho_1^{(1)} = \\frac{\\alpha - s_1^{(1)}}{\\alpha + s_1^{(1)}} =\n\\frac{\\sqrt{\\frac{\\alpha\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]\\left[(\\alpha-\\beta) + (N-1)\\beta\\right] - N(N-1)\\gamma^2\\right\\}}{(\\alpha-\\beta)\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]^2 - N^2\\gamma^2\\right\\}}}-1}\n{\\sqrt{\\frac{\\alpha\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]\\left[(\\alpha-\\beta) + (N-1)\\beta\\right] - N(N-1)\\gamma^2\\right\\}}{(\\alpha-\\beta)\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]^2 - N^2\\gamma^2\\right\\}}}+1},\n\\nonumber \\\\\n& & 1 - \\rho_1^{(1)} = \\frac{2s_1^{(1)}}{\\alpha + s_1^{(1)}}. \\\n\\end{eqnarray}\nHere, $1 - \\rho_1^{(1)}$ is the condensate fraction of species $1$ (and of species $2$),\ni.e., the fraction of condensed bosons,\nand $\\rho_1^{(1)}$ is the depleted fraction,\nnamely, the fraction of bosons residing outside the lowest, condensed mode.\n$s_1^{(1)}$ is the scaling, or effective frequency, of the intra-species\nnatural orbitals.\nThe condensate fraction, depleted fraction, and scaling of the natural orbitals\nare all given in closed form as a function of the number of bosons $N$, and the intra-species $\\lambda$\nand inter-species $\\lambda_{12}$ interaction strengths.\nA specific application of the general expressions\n(\\ref{S_1_1}) for the mixture appears below.\n\nFor the intra-species two-particle reduced density matrix we have:\n\\begin{eqnarray}\\label{2_RDM}\n& &\n\\rho_1^{(2)}(x_1,x_2,x'_1,x'_2) = \nN(N-1) \\left(\\frac{\\alpha+C_{1,0}}{\\pi}\\right)^{\\frac{1}{2}} \\left(\\frac{\\alpha+C_{2,0}}{\\pi}\\right)^{\\frac{1}{2}} \\times \\nonumber \\\\\n& & \n\\times e^{-\\frac{\\alpha}{2}\\left(x_1^2+x_2^2+{x'_1}^2+{x'_2}^2\\right)} e^{-\\beta\\left(x_1x_2 + x'_1x'_2\\right)}\ne^{-\\frac{1}{4} C_{2,0} \\left(x_1+x_2+x'_1+x'_2\\right)^2}, \\nonumber \\\\\n& &\n\\alpha+\\beta+2C_{2,0} =\n(\\alpha-\\beta)\n\\frac{\\left[(\\alpha-\\beta) + N\\beta\\right]^2 - N^2\\gamma^2}\n{\\left[(\\alpha-\\beta) + N\\beta\\right]\\left[(\\alpha-\\beta) + (N-2)\\beta\\right] - N(N-2)\\gamma^2}, \\\n\\end{eqnarray}\nwhere $C_{1,0}$ is the coefficient of the intra-species reduced one-particle density matrix (\\ref{1_RDM})\nand the combination $\\left(\\alpha+\\beta+2C_{2,0}\\right)$ would appear shortly after.\n\nTo obtain the natural geminals of $\\rho_1^{(2)}(x_1,x_2,x'_1,x'_2)$ we \ndefine the variables $q_1 = \\frac{1}{\\sqrt{2}}\\left(x_1+x_2\\right)$, $q_2 = \\frac{1}{\\sqrt{2}}\\left(x_1-x_2\\right)$\nand $q'_1 = \\frac{1}{\\sqrt{2}}\\left(x'_1+x'_2\\right)$, $q'_2 = \\frac{1}{\\sqrt{2}}\\left(x'_1-x'_2\\right)$,\ni.e., the center-of-mass and relative coordinate of\ntwo identical bosons.\nWith this rotation of coordinates we have for the different terms in (\\ref{2_RDM}): \n\\begin{eqnarray}\\label{TRAS1}\n& & x_1^2+x_2^2+{x'_1}^2+{x'_2}^2 = q_1^2 + {q'_1}^2 + q_2^2 + {q'_2}^2, \\nonumber \\\\\n& & x_1x_2+x'_1x'_2 = \\frac{1}{2}\\left(q_1^2 + {q'_1}^2 - q_2^2 - {q'_2}^2\\right), \\nonumber \\\\\n& & \\left(x_1+x_2+x'_1+x'_2\\right)^2 = 2 \\left(q_1^2 + {q'_1}^2 + 2q_1q'_1\\right). \\\n\\end{eqnarray} \nConsequently, one readily finds the diagonal form\n\\begin{eqnarray}\\label{2_RDM_DIAG}\n& &\n\\rho_1^{(2)}(q_1,q'_1,q_2,q'_2) = \nN(N-1)\n\\left(\\frac{\\alpha-\\beta}{\\pi}\\right)^{\\frac{1}{2}}\ne^{-\\frac{\\alpha-\\beta}{2}\\left(q_2^2 + {q'_2}^2\\right)} \\times \\nonumber \\\\\n& & \\times\n\\left(\\frac{\\alpha+\\beta+2C_{2,0}}{\\pi}\\right)^{\\frac{1}{2}}\ne^{-\\frac{\\alpha+\\beta+C_{2,0}}{2}\\left(q_1^2 + {q'_1}^2\\right)}\ne^{-C_{2,0} q_1q'_1},\n\\end{eqnarray}\nwhere\nthe normalization coefficients before and after diagonalization are, of course, equal and satisfy\n$\\left(\\alpha+C_{1,0}\\right)\\left(\\alpha+C_{2,0}\\right)=\\left(\\alpha-\\beta\\right)\\left(\\alpha+\\beta+2C_{2,0}\\right)$.\n\nAfter the transformation (\\ref{TRAS1}),\nthe first term of $\\rho_1^{(2)}(q_1,q'_1,q_2,q'_2)$ is separable as a function of $q_2$ and $q'_2$ whereas,\nusing Mehler's formula onto the variables $q_1$ and $q'_1$,\nthe second term can be diagonalized. \nThus, comparing the second term in (\\ref{2_RDM_DIAG}) and Eq.~(\\ref{MEHLER}) we find\n\\begin{eqnarray}\\label{S_1_2}\n& & s_1^{(2)} = \\sqrt{\\left(\\alpha+\\beta\\right)\\left(\\alpha+\\beta+2C_{2,0}\\right)} =\n\\sqrt{\\frac{(\\alpha^2-\\beta^2)\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]^2 - N^2\\gamma^2\\right\\}}\n{\\left[(\\alpha-\\beta) + N\\beta\\right]\\left[(\\alpha-\\beta) + (N-2)\\beta\\right] - N(N-2)\\gamma^2}},\n\\nonumber \\\\ \\nonumber \\\\\n& & \\rho_1^{(2)} = \\frac{\\left(\\alpha + \\beta\\right) - s_1^{(2)}}{\\left(\\alpha +\\beta\\right) + s_1^{(2)}} =\n\\frac{\\sqrt{\\frac{\\left(\\alpha+\\beta\\right)\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]\\left[(\\alpha-\\beta) + (N-2)\\beta\\right] - N(N-2)\\gamma^2\\right\\}}{(\\alpha-\\beta)\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]^2 - N^2\\gamma^2\\right\\}}}-1}\n{\\sqrt{\\frac{\\left(\\alpha+\\beta\\right)\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]\\left[(\\alpha-\\beta) + (N-2)\\beta\\right] - N(N-2)\\gamma^2\\right\\}}{(\\alpha-\\beta)\\left\\{\\left[(\\alpha-\\beta) + N\\beta\\right]^2 - N^2\\gamma^2\\right\\}}}+1},\n\\nonumber \\\\\n& & 1 - \\rho_1^{(2)} = \\frac{2s_1^{(2)}}{\\alpha + s_1^{(2)}}. \\\n\\end{eqnarray}\nWith expressions (\\ref{S_1_2}),\nthe decomposition of the intra-species reduced two-particle density matrix in terms of its natural\ngeminals is explicitly given by\n\\begin{eqnarray}\\label{2_RDM_NAT_GEM}\n& &\n\\rho_1^{(2)}(x_1,x_2,x'_1,x'_2) = \nN(N-1) \\sum_{n=0}^\\infty \\left(1-\\rho_1^{(2)}\\right)\\left(\\rho_1^{(2)}\\right)^n\n\\Phi^{(2)}_{1,n}(x_1,x_2) \\Phi^{(2),\\ast}_{1,n}(x'_1,x'_2), \\nonumber \\\\\n& & \\Phi^{(2)}_{1,n}(x_1,x_2) =\n\\frac{1}{\\sqrt{2^n n!}}\n\\left(\\frac{s_1^{(2)}}{\\pi}\\right)^{\\frac{1}{4}}\nH_n\\left(\\sqrt{\\frac{s_1^{(2)}}{2}}\\left(x_1+x_2\\right)\\right)\ne^{-\\frac{1}{4}s_1^{(2)}\\left(x_1+x_2\\right)^2} \\times \\nonumber \\\\\n& & \\times\n\\left(\\frac{\\alpha-\\beta}{\\pi}\\right)^{\\frac{1}{4}}\ne^{-\\frac{1}{4}\\left(\\alpha-\\beta\\right)\\left(x_1-x_2\\right)^2}.\\\n\\end{eqnarray}\nEquation (\\ref{2_RDM_NAT_GEM}) is a general result on\nthe inter-species natural geminals of the mixture.\nTogether with (\\ref{S_1_2}) they imply\nthat\n$1 - \\rho_2^{(1)}$ is the fraction of condensed pairs of species $1$ (and of species $2$),\n$\\rho_1^{(2)}$ is the fraction of depleted pairs,\ni.e., the fraction of pairs residing outside the lowest, condensed natural geminal,\nand \n$s_1^{(2)}$ is the scaling, or effective frequency, of the intra-species\nnatural pair functions.\nThe intra-species natural geminals along with their condensate and depleted fractions\nare prescribed as explicit functions of the number of bosons $N$, and the intra-species $\\lambda$\nand inter-species $\\lambda_{12}$ interactions.\nA specific application of the general decomposition\n(\\ref{S_1_2}),\n(\\ref{2_RDM_NAT_GEM})\nto natural geminals of the mixture is provided below.\nFinally, we point out that \nthe generalization to higher-order intra-species reduced density matrices and corresponding natural\nfunctions follows the above pattern\nand will not be discussed further here.\n\nLet us work out an explicit application\nwhere we shall find and analyze fragmentation of identical bosons' pairs.\nConsider the specific scenario where $\\lambda+\\lambda_{12}=0$,\ni.e., that the intra-species interaction is inverse to and `compensates'\nthe effect of the inter-species interaction on each of the species\nin the manner that the intra-species frequency is that of non-interacting particles,\n$\\Omega=m\\omega$.\nThen, the coefficients of the three parts of the wavefunction simplify and one has\n$\\alpha=m\\omega+\\beta = m\\omega \\left[1 + \\frac{1}{2N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]$\nand\n$\\beta=\\gamma=\\frac{m}{2N}\\left(\\Omega_{12}-\\omega\\right)$.\nConsequently, expressions (\\ref{S_1_1}) and (\\ref{S_1_2}) simplify and the intra-species reduced\none-particle and two-particle density matrices can be evaluated further.\nThus we readily find\n\\begin{subequations}\\label{S_INTRA_EXAM}\n\\begin{eqnarray}\\label{S_1_1_EXAM}\n& & s_1^{(1)} = m\\omega \\sqrt{\\frac{1 + \\frac{1}{2N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)}\n{1 + \\frac{1}{2N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}},\n\\nonumber \\\\ \\nonumber \\\\\n& & \\rho_1^{(1)} = \\frac{\\sqrt{\\left[1 + \\frac{1}{2N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{1}{2N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}-1}\n{\\sqrt{\\left[1 + \\frac{1}{2N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{1}{2N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}+1},\n\\nonumber \\\\\n& & 1 - \\rho_1^{(1)} = \\frac{2}\n{\\sqrt{\\left[1 + \\frac{1}{2N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{1}{2N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}+1}\\\n\\end{eqnarray}\nfor the intra-species reduced one-particle density matrix,\nwhere $\\alpha+C_{1,0} = m\\omega \\frac{1}{1 + \\frac{1}{2N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}$ is used,\nand\n\\begin{eqnarray}\\label{S_1_2_EXAM}\n& & s_1^{(2)} = m\\omega \\sqrt{\\frac{1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)}\n{1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}},\n\\nonumber \\\\ \\nonumber \\\\\n& & \\rho_1^{(2)} = \\frac{\\sqrt{\\left[1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}-1}\n{\\sqrt{\\left[1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}+1},\n\\nonumber \\\\\n& & 1 - \\rho_1^{(2)} = \\frac{2}\n{\\sqrt{\\left[1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}+1} \\\n\\end{eqnarray}\n\\end{subequations}\nfor the intra-species reduced two-particle density matrix,\nwhere \n$\\alpha+\\beta = m\\omega \\left[1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]$ and\n$\\alpha+\\beta+2C_{2,0} = m\\omega \\frac{1}{1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}$ are utilized.\nWe see that fragmentation of identical pairs and bosons\nis governed by the ratio $\\frac{\\Omega_{12}}{\\omega}$ and its inverse $\\frac{\\omega}{\\Omega_{12}}$,\nmeaning that it takes place both at the attractive and repulsive sectors of interactions.\nMoreover, the condensed and depleted fractions\nof the pairs and bosons are symmetric to interchanging $\\frac{\\Omega_{12}}{\\omega}$ and \n$\\frac{\\omega}{\\Omega_{12}}$, see discussion below.\n\nLet us analyze explicitly macroscopic fragmentation of geminals, i.e.,\nwhen there is macroscopic occupation of more than a single\nintra-species natural pair function of\\break\\hfill $\\rho_1^{(2)}(x_1,x_2,x'_1,x'_2)$.\nAs a reference, we also refer to the corresponding and standardly defined\nmacroscopic fragmentation of the intra-species natural orbitals of $\\rho_1^{(1)}(x,x')$. \nThe structure of the eigenvalues,\nemanating from Mehler's formula and its applicability to the various reduced density matrices,\nsuggests that, say, the `middle' value $\\rho=1-\\rho=\\frac{1}{2}$, i.e., when the condensed and depleted fractions are equal,\nis a convenient manifestation of macroscopic fragmentation.\nIndeed, for this value the first few natural occupation fractions $(1-\\rho)\\rho^n$, $n=0,1,2,3,4,\\ldots$ are\n\\begin{equation}\\label{rho_half}\n\\frac{1}{2}, \\quad \\frac{1}{4}, \\quad \\frac{1}{8}, \\quad \\frac{1}{16}, \\quad \\frac{1}{32}, \\ldots,\n\\end{equation}\nnamely,\nthere is $50\\%$ occupation of the first natural geminal,\n$25\\%$ occupation of the second,\n$12.5\\%$ of the third,\n$6.25\\%$ of the fourth,\n$3.125\\%$ of the fifth,\nand so on.\nFor brevity,\nwe refer to the fragmentation values in (\\ref{rho_half}) as $50\\%$ fragmentation. \n\nNow, one can compute for which ratio $\\frac{\\Omega_{12}}{\\omega}$,\nor, equivalently, for which inter-species interaction\n$\\lambda_{12} = \\frac{m\\omega^2}{4N}\\left[\\left(\\frac{\\Omega_{12}}{\\omega}\\right)^2-1\\right]$,\nthe intra-species reduced\ntwo-particle and one-particle density matrices are macroscopically fragmented as in (\\ref{rho_half}).\nThus, solving (\\ref{S_1_1_EXAM}) for $50\\%$ natural-orbital fragmentation we find\n\\begin{eqnarray}\\label{FRAG_INTRA_1}\n \\rho_1^{(1)}=\\frac{1}{2} \\quad\n\\Longrightarrow \\quad\n\\frac{\\Omega_{12}}{\\omega} =\n\\left(1 + \\frac{8N^2}{N-\\frac{1}{2}}\\right) \\pm \\sqrt{\\left(1 + \\frac{8N^2}{N-\\frac{1}{2}}\\right)^2 - 1},\n\\end{eqnarray}\nand working out (\\ref{S_1_2_EXAM}) for $50\\%$ natural-geminal fragmentation we get\n\\begin{eqnarray}\\label{FRAG_INTRA_2}\n \\rho_1^{(2)}=\\frac{1}{2} \\quad\n\\Longrightarrow \\quad\n\\frac{\\Omega_{12}}{\\omega} = \n\\sqrt{1+\\frac{4N}{m\\omega^2}\\lambda_{12}} = \n\\left(1 + \\frac{4N^2}{N-1}\\right) \\pm \\sqrt{\\left(1 + \\frac{4N^2}{N-1}\\right)^2 - 1}.\n\\end{eqnarray}\nThere are two `reciprocate' solutions for both\nthe natural geminals and natural orbitals:\nWe see that $50\\%$ fragmentation occurs for strong attractions,\ni.e., when $\\frac{\\Omega_{12}}{\\omega}$ is large,\nor near the border of stability for repulsions, that is when $\\frac{\\Omega_{12}}{\\omega}$ is close to zero.\nAlso, to achieve the same degree of $50\\%$ with a larger number $N$ of species $1$ (and species $2$) bosons,\na stronger attraction or repulsion is needed.\nFinally, comparing natural-geminal with natural-orbital fragmentation at the same $50\\%$ value,\none sees from (\\ref{FRAG_INTRA_2}) and (\\ref{FRAG_INTRA_1}) that slightly weaker\ninteractions, attractions or repulsions, are needed for the former.\n\nIt is also useful to register the one-particle and two-particle densities, i.e.,\nthe diagonal parts $\\rho_1^{(1)}(x)=\\rho_1^{(1)}(x,x'=x)$ and $\\rho_1^{(2)}(x_1,x_2)=\\rho_1^{(2)}(x_1,x_2,x'_1=x_1,x'_2=x_2)$,\nwhich read\n\\begin{eqnarray}\\label{DENSITIES_1_2_EXAM}\n& & \\rho_1^{(1)}(x) =\nN \\left(\\frac{\\alpha+C_{1,0}}{\\pi}\\right)^{\\frac{1}{2}} e^{-\\left(\\alpha+C_{1,0}\\right)x^2} = \nN \\left(\\frac{m\\omega}{\\pi\\left[1 + \\frac{1}{2N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}\\right)^{\\frac{1}{2}}\ne^{-\\frac{m\\omega}{1 + \\frac{1}{2N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}x^2},\n\\nonumber \\\\\n& & \\rho_1^{(2)}(x_1,x_2) = \nN(N-1)\n\\left(\\frac{\\alpha-\\beta}{\\pi}\\right)^{\\frac{1}{2}}\ne^{-\\frac{\\alpha-\\beta}{2}\\left(x_1-x_2\\right)^2}\n\\left(\\frac{\\alpha+\\beta+2C_{2,0}}{\\pi}\\right)^{\\frac{1}{2}}\ne^{-\\frac{\\alpha+\\beta+2C_{2,0}}{2}\\left(x_1+x_2\\right)^2} = \\nonumber \\\\\n& &\n= N(N-1)\n\\left(\\frac{m\\omega}{\\pi}\\right)^{\\frac{1}{2}}\ne^{-\\frac{m\\omega}{2}\\left(x_1-x_2\\right)^2}\n\\left(\\frac{m\\omega}{\\pi\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}\\right)^{\\frac{1}{2}}\ne^{-\\frac{m\\omega}{2\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}\\left(x_1+x_2\\right)^2}.\n\\end{eqnarray}\nFrom the densities (\\ref{DENSITIES_1_2_EXAM}) we can infer a measure for the size\nof identical pairs' and bosons' clouds using the widths of the respective Gaussian functions therein.\nThus, we have\n\\begin{subequations}\\label{WIDTH_INTRA}\n\\begin{eqnarray}\\label{WIDTH_INTRA_GEN}\n& &\n\\sigma_{1,x}^{(1)} = \\sqrt{\\frac{1 + \\frac{1}{2N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}{2m\\omega}}, \\nonumber \\\\\n& &\n\\sigma_{1,\\frac{x_1+x_2}{\\sqrt{2}}}^{(2)} = \\sqrt{\\frac{1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}{2m\\omega}},\n\\quad\n\\sigma_{1,\\frac{x_1-x_2}{\\sqrt{2}}}^{(2)} = \\sqrt{\\frac{1}{2m\\omega}}. \\\n\\end{eqnarray}\nTo assess the combined impact of the intra-species and inter-species interactions\natop the fragmentation of the reduced density matrices,\nit is useful to compute the sizes (\\ref{WIDTH_INTRA_GEN}) for large inter-species attractions or \ninter-species repulsions at the border of stability.\nOne finds, respectively,\n\\begin{eqnarray}\\label{WIDTH_INTRA_LIM}\n& & \\lim_{\\frac{\\Omega_{12}}{\\omega} \\to \\infty} \\sigma_{1,x}^{(1)} = \n\\sqrt{\\frac{1 - \\frac{1}{2N}}{2m\\omega}}, \\qquad\n\\sigma_{1,x}^{(1)} \\longrightarrow \\infty \\quad \\mathrm{for} \\quad \\frac{\\Omega_{12}}{\\omega} \\to 0^+, \\nonumber \\\\\n& & \\lim_{\\frac{\\Omega_{12}}{\\omega} \\to \\infty} \\sigma_{1,\\frac{x_1+x_2}{\\sqrt{2}}}^{(2)} = \n\\sqrt{\\frac{1 - \\frac{1}{N}}{2m\\omega}}, \\qquad\n\\sigma_{1,\\frac{x_1+x_2}{\\sqrt{2}}}^{(2)} \\longrightarrow \\infty \\quad \\mathrm{for} \\quad \\frac{\\Omega_{12}}{\\omega} \\to 0^+,\n\\\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\sigma_{1,\\frac{x_1-x_2}{\\sqrt{2}}}^{(2)}$ is independent of the interactions.\nInterestingly, the size of the densities \nfor strong inter-species attractions, which is accompanied by strong intra-species repulsions because $\\lambda+\\lambda_{12}=0$,\nsaturates at about the trap's size and does not depend on the strengths of interactions.\nIn other words, a high degree of fragmentation is possible in the mixture without shrinking of the density due to strong inter-species\nattractive interaction or expansion of the intra-species densities due to strong intra-species repulsive interaction. \nFor the sake of comparative analysis, it is instructive to make contact with\nfragmentation of single-species bosons in the harmonic-interaction model,\nsee appendix \\ref{APP}.\n\n\\subsection{Inter-species natural pair functions}\\label{PAIR_3}\n\nAs mentioned above,\nin a mixture of two types of identical bosons there are other kinds of pairs, namely,\npairs of distinguishable particles.\nIf we are to examine the lowest-order inter-species reduced density matrix,\nwe can ask regarding distinguishable pairs questions analogous to those asked concerning identical pairs.\nThe purpose of this subsection is to\nderive the relevant tools and answer such questions.\n\nThe inter-species reduced two-particle density matrix,\ni.e., the lowest-oder inter-species quantity, is defined from the all-particle density as\n\\begin{eqnarray}\\label{2_RDM_12_SYM_GEN}\n& & \\rho_{12}^{(2)}(x,x',y,y') = N^2 \\int dx_2 \\cdots dx_N dy_2 \\cdots dy_N \n\\Psi(x,x_2,\\ldots,x_N,y,y_2,\\ldots,y_N) \\times \\nonumber \\\\\n& &\n\\times \\Psi^\\ast(x',x_2,\\ldots,x_N,y',y_2,\\ldots,y_N). \\\n\\end{eqnarray}\nFor the harmonic-interaction model of the symmetric mixture it\ncan be computed analytically and, starting from (\\ref{HIM_MIX_WF_DEN_MAT2}),\nis given by \\cite{HIM_MIX_RDM}\n\\begin{subequations}\\label{2_RDM_12_SYM}\n\\begin{eqnarray}\\label{2_RDM_12_SYM_DEN}\n& & \\rho_{12}^{(2)}(x,x',y,y') = N^2\n\\left[\\frac{(\\alpha_1+C_{1,1})^2-D_{1,1}^2}{\\pi^2}\\right]^\\frac{1}{2} \\times \\nonumber \\\\\n& & \\times e^{-\\frac{\\alpha_1}{2} \\left(x^2+{x'}^2+y^2+{y'}^2\\right)}\ne^{-\\frac{1}{4}C_{1,1}\\left[\\left(x+x'\\right)^2+\\left(y+y'\\right)^2\\right]}\ne^{+\\frac{1}{2}D_{1,1} \\left(x+x'\\right)\\left(y+y'\\right)} \ne^{+\\frac{1}{2}D'_{1,1} \\left(x-x'\\right)\\left(y-y'\\right)}, \\\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{2_RDM_12_SYM_COEFF}\n& & \\alpha + C_{1,1} \\mp D_{1,1} =\n(\\alpha-\\beta)\n\\frac{(\\alpha-\\beta) + N(\\beta\\mp \\gamma)}\n{(\\alpha-\\beta) + (N-1)(\\beta\\mp \\gamma)},\n\\nonumber \\\\\n& & D'_{1,1} = \\gamma.\n\\end{eqnarray}\n\\end{subequations}\nWe see that the structure of the inter-species reduced two-particle density matrix\nis more involved than that of the intra-species reduced two-particle density matrix\nas well as that of the product of the two, species $1$ and species $2$ intra-species reduced one-particle density matrices.\nNonetheless, it can be diagonalized.\n\nTo diagonalize\n$\\rho_{12}^{(2)}(x,x',y,y')$\none must couple and make linear combinations of\ncoordinates associated with distinguishable bosons.\nDefining $u = \\frac{1}{\\sqrt{2}}\\left(x+y\\right)$, $v = \\frac{1}{\\sqrt{2}}\\left(x-y\\right)$\nand \n$u'= \\frac{1}{\\sqrt{2}}\\left(x'+y'\\right)$, $v'= \\frac{1}{\\sqrt{2}}\\left(x'-y'\\right)$\nwe have for the different terms in (\\ref{2_RDM_12_SYM}):\n\\begin{eqnarray}\\label{TRAS3}\n& & x^2+y^2+{x'}^2+{y'}^2 = u^2 + {u'}^2 + v^2 + {v'}^2, \\\\\n& & \\left(x+x'\\right)^2+\\left(y+y'\\right)^2 = \\left(u+u'\\right)^2 + \\left(v+v'\\right)^2 =\nu^2 + {u'}^2 + v^2 + {v'}^2 + 2\\left(uu'+vv'\\right), \\nonumber \\\\\n& & \\left(x \\pm x'\\right)\\left(y \\pm y'\\right) = \\frac{1}{2}\\left[\\left(u\\pm u'\\right)^2 - \\left(v\\pm v'\\right)^2\\right] =\n\\frac{1}{2}\\left(u^2 + {u'}^2 - v^2 - {v'}^2\\right) \\pm \\left(uu'-vv'\\right). \\nonumber \\\n\\end{eqnarray}\nConsequently, we readily find the decomposition\n\\begin{eqnarray}\\label{2_RDM_12_SYM_DIAG}\n& &\n\\rho_{12}^{(2)}(u,u',v,v') = N^2\n\\left(\\frac{\\alpha_1+C_{1,1}-D_{1,1}}{\\pi}\\right)^\\frac{1}{2}\ne^{-\\frac{\\alpha_1+\\frac{C_{1,1}}{2}-\\frac{D_{1,1}+D'_{1,1}}{2}}{2}\\left(u^2 + {u'}^2\\right)}\ne^{-\\frac{1}{2}\\left[C_{1,1}-\\left(D_{1,1}-D'_{1,1}\\right)\\right]uu'} \\times \\nonumber \\\\\n& & \\times \n\\left(\\frac{\\alpha_1+C_{1,1}+D_{1,1}}{\\pi}\\right)^\\frac{1}{2}\ne^{-\\frac{\\alpha_1+\\frac{C_{1,1}}{2}+\\frac{D_{1,1}+D'_{1,1}}{2}}{2}\\left(v^2 + {v'}^2\\right)}\ne^{-\\frac{1}{2}\\left[C_{1,1}+\\left(D_{1,1}-D'_{1,1}\\right)\\right]vv'},\n\\end{eqnarray}\nwhere the normalizations after and before diagonalization are, of course, equal.\nAs might be expected,\nsince the structure of $\\rho_{12}^{(2)}(x,x',y,y')$ is more involved than that\nof $\\rho_1^{(2)}(x_1,x_2,x'_1,x'_2)$ the diagonalization of the former is more intricate.\nFortunately, we can do that using the application of Mehler's formula twice,\non the appropriately-constructed inter-species `mixed' coordinates $u, u'$ and $v, v'$.\nWe thus get\n\\begin{eqnarray}\\label{S_12_2}\n& & s_{12,\\pm}^{(2)} = \\sqrt{\\left(\\alpha \\mp D'_{1,1}\\right)\\left(\\alpha+C_{1,1} \\mp D_{1,1}\\right)} =\n\\sqrt{(\\alpha\\mp \\gamma)(\\alpha-\\beta)\n\\frac{(\\alpha-\\beta) + N(\\beta\\mp \\gamma)}\n{(\\alpha-\\beta) + (N-1)(\\beta\\mp \\gamma)}},\n\\nonumber \\\\\n& & \\rho_{12,\\pm}^{(2)} = \\frac{\\left(\\alpha \\mp D'_{1,1}\\right) - s_{12,\\pm}^{(2)}}{\\left(\\alpha \\mp D'_{1,1}\\right) + s_{12,\\pm}^{(2)}} =\n\\frac{\\frac{(\\alpha\\mp \\gamma)\\left[(\\alpha-\\beta) + (N-1)(\\beta\\mp \\gamma)\\right]}{(\\alpha-\\beta)\\left[(\\alpha-\\beta) + N(\\beta\\mp \\gamma)\\right]}-1}{\\frac{(\\alpha\\mp \\gamma)\\left[(\\alpha-\\beta) + (N-1)(\\beta\\mp \\gamma)\\right]}{(\\alpha-\\beta)\\left[(\\alpha-\\beta) + N(\\beta\\mp \\gamma)\\right]}+1},\n\\nonumber \\\\\n& & 1 - \\rho_{12,\\pm}^{(2)} = \\frac{2s_{12,\\pm}^{(2)}}{\\left(\\alpha \\mp D'_{1,1}\\right) + s_{12,\\pm}^{(2)}}, \\\n\\end{eqnarray}\nwhere the ``$+$'' terms quantify the fragmentation in the $u,u'$ part of the intra-species reduced two-particle density matrix and\nthe ``$-$'' terms quantify the fragmentation in the $v,v'$ part of the intra-species reduced two-particle density matrix,\nalso see below.\nEquation (\\ref{S_12_2}) is one of the main results of the present work and bears\na clear and appealing physical meaning,\nthat pairs made of distinguishable bosons can be fragmented,\nand that this fragmentation is governed\nby the center-of-mass and \nby a relative coordinate of distinguishable bosons.\nWe shall return to this point in what follows.\n\nWe can now prescribe the decomposition of the\ninter-species reduced two-particle density matrix\nto its distinguishable natural pair functions which is given by\n\\begin{eqnarray}\\label{12_2_RDM_NAT_DIS_GEM}\n& &\n\\rho_{12}^{(2)}(x,x',y,y') = \\nonumber \\\\\n& & = N^2 \n\\sum_{n_+=0}^\\infty \\sum_{n_-=0}^\\infty\n\\left(1-\\rho_{12,+}^{(2)}\\right)\n\\left(1-\\rho_{12,-}^{(2)}\\right)\n\\left(\\rho_{12,+}^{(2)}\\right)^{\\!n_+}\n\\left(\\rho_{12,-}^{(2)}\\right)^{\\!n_-}\n\\Phi^{(2)}_{12,n_+,n_-}(x,y) \\Phi^{(2),\\ast}_{12,n_+,n_-}(x',y'), \\nonumber \\\\\n& & \\Phi^{(2)}_{12,n_+,n_-}(x,y) =\n\\frac{1}{\\sqrt{2^{n_+} {n_+}!}}\n\\left(\\frac{s_{12,+}^{(2)}}{\\pi}\\right)^{\\frac{1}{4}}\nH_{n_+}\\left(\\sqrt{\\frac{s_{12,+}^{(2)}}{2}}\\left(x+y\\right)\\right)\ne^{-\\frac{1}{4}s_{12,+}^{(2)}\\left(x+y\\right)^2} \\times \\nonumber \\\\\n& & \\times\n\\frac{1}{\\sqrt{2^{n_-} {n_-}!}}\n\\left(\\frac{s_{12,-}^{(2)}}{\\pi}\\right)^{\\frac{1}{4}}\nH_{n_-}\\left(\\sqrt{\\frac{s_{12,-}^{(2)}}{2}}\\left(x-y\\right)\\right)\ne^{-\\frac{1}{4}s_{12,-}^{(2)}\\left(x-y\\right)^2}.\\\n\\end{eqnarray}\nAll in all,\n(\\ref{12_2_RDM_NAT_DIS_GEM}) implies\nthat the distinguishable-pair `condensed fraction' is given by\\break\\hfill\n$\\left(1-\\rho_{12,+}^{(2)}\\right)\\left(1-\\rho_{12,-}^{(2)}\\right)$ \nand the respective depleted fraction by\n$1-\\left(1-\\rho_{12,+}^{(2)}\\right)\\left(1-\\rho_{12,-}^{(2)}\\right)=\n\\rho_{12,+}^{(2)}+\\rho_{12,-}^{(2)}-\\rho_{12,+}^{(2)}\\rho_{12,-}^{(2)}$. \nEach of the inter-species `mixed' coordinates $\\frac{x\\pm y}{\\sqrt{2}}$\ncarry its own scaling, $s_{12,\\pm}^{(2)}$.\nThe distinguishable natural geminals \n$ \\Phi^{(2)}_{12,n_+,n_-}(x,y)$\nare, needless to say, orthonormal to each other. \n\nWe proceed now for an application.\nWe considered above the specific case of $\\lambda+\\lambda_{12}=0$\nwhich leads to \n$\\Omega=m\\omega$,\n$\\alpha=m\\omega \\left[1 + \\frac{1}{2N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]$, and\n$\\beta=\\gamma=\\frac{m}{2N}\\left(\\Omega_{12}-\\omega\\right)$.\nTo evaluate $\\rho_{12}^{(2)}(x,x',y,y')$\nwe also need the combinations\n$\\left(\\alpha + C_{1,1} - D_{1,1}\\right) = m \\omega$\nand\n$\\left(\\alpha - D'_{1,1}\\right) = m\\omega$ for the ``$+$'' branch\nas well as\n$\\left(\\alpha + C_{1,1} + D_{1,1}\\right) =\nm\\omega \\frac{1}{1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}$\nand\n$\\left(\\alpha + D'_{1,1}\\right) =\nm\\omega \\left[1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]$\nfor the ``$-$'' branch.\n\nThus, expressions (\\ref{S_12_2}) can readily be evaluated and the following picture\nof inter-species fragmentation is found:\n\\begin{subequations}\\label{S2_INTER_EXAM}\n\\begin{eqnarray}\\label{S2_12_+_EXAM}\n& & s_{12,+}^{(2)} = m\\omega,\n\\qquad\n\\rho_{12,+}^{(2)} = 0,\n\\qquad\n1 - \\rho_{12,+}^{(2)} = 1, \\\n\\end{eqnarray}\nindicating that there is no contribution to fragmentation from the symmetric `mixed' coordinate $u, u'$.\nOn the other end,\n\\begin{eqnarray}\\label{S2_12_-_EXAM}\n& & s_{12,-}^{(2)} = m\\omega \\sqrt{\\frac{1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)}\n{1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}},\n\\nonumber \\\\ \\nonumber \\\\\n& & \\rho_{12,-}^{(2)} = \\frac{\\sqrt{\\left[1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}-1}\n{\\sqrt{\\left[1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}+1},\n\\nonumber \\\\\n& & 1 - \\rho_{12,-}^{(2)} = \\frac{2}\n{\\sqrt{\\left[1 + \\frac{1}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}+1}, \\\n\\end{eqnarray}\n\\end{subequations}\nnamely, that the fragmentation fully originates from the asymmetric `mixed' coordinate $v, v'$.\nWe conclude that,\nwhereas fragmentation of identical pairs is associated with their center-of-mass coordinate,\nfragmentation of distinguishable pairs is linked, in this explicit case, only with a relative coordinate between\ntwo distinguishable bosons. \nInterestingly,\nthe degree of intra-species and inter-species pair fragmentation is the same\nin the specific case considered,\ndespite pertaining to different parts of the mixtures' many-boson wavefunction.\nFurthermore,\nthere are different numbers of pairs:\n$\\frac{N}{2}$ intra-species identical pairs (for each of the species)\nand $N$ inter-species pairs of distinguishable bosons.\n\nNow, one can compute the ratio $\\frac{\\Omega_{12}}{\\omega}=\\sqrt{1+\\frac{4N}{m\\omega^2}\\lambda_{12}}$\nfor which the inter-species reduced two-particle density matrix is $50\\%$ fragmented as in (\\ref{rho_half}).\nSince $\\rho_{12,+}^{(2)}=0$ does not contribute,\nthe only contribution to fragmentation comes from\n$\\rho_{12,-}^{(2)}$.\nThus, solving (\\ref{S2_12_-_EXAM}) for $50\\%$ distinguishable-pair-function fragmentation we obtain\n\\begin{eqnarray}\\label{FRAG_INTER_2}\n\\rho_{12,-}^{(2)}=\\frac{1}{2} \\quad\n\\Longrightarrow \\quad\n\\frac{\\Omega_{12}}{\\omega} =\n\\left(1 + \\frac{4N^2}{N-1}\\right) \\pm \\sqrt{\\left(1 + \\frac{4N^2}{N-1}\\right)^2 - 1}.\n\\end{eqnarray}\nAs above, there are two `reciprocate' solutions,\none for strong inter-species attraction and the second close to the border of stability for intermediate-strength\ninter-species repulsion.\nWe remind that since $\\lambda+\\lambda_{12}=0$ in our example,\nthe respective intra-species interaction is opposite in sign.\nAlso, to achieve the same degree of $50\\%$ fragmentation\nwith a larger number $N$ of\ndistinguishable pairs,\na stronger inter-species attraction or repulsion is needed.\nFurthermore, as discussed above,\ncomparing distinguishable-pair and identical-pair fragmentation at the same $50\\%$ value\nin this example,\none sees from (\\ref{FRAG_INTER_2}) and (\\ref{FRAG_INTRA_2}) that the same interaction\nis needed.\n\nFinally, we prescribe the inter-species two-particle density,\nnamely,\nthe diagonal part $\\rho_{12}^{(2)}(x,y)=\\rho_{12}^{(2)}(x,x'=x,y,y'=y)$,\nwhich is given by\n\\begin{eqnarray}\\label{DENSITY_2_12_EXAM}\n& &\n\\rho_{12}^{(2)}(x,y) = N^2\n\\left(\\frac{\\alpha_1+C_{1,1}-D_{1,1}}{\\pi}\\right)^\\frac{1}{2}\ne^{-\\frac{\\alpha_1+C_{1,1}-D_{1,1}}{2}\\left(x+y\\right)^2}\n\\times \\nonumber \\\\\n& & \\times \n\\left(\\frac{\\alpha_1+C_{1,1}+D_{1,1}}{\\pi}\\right)^\\frac{1}{2}\ne^{-\\frac{\\alpha_1+C_{1,1}+D_{1,1}}{2}\\left(x-y\\right)^2} = \\nonumber \\\\\n& &\n= N^2\n\\left(\\frac{m\\omega}{\\pi}\\right)^{\\frac{1}{2}}\ne^{-\\frac{m\\omega}{2}\\left(x+y\\right)^2}\n\\left(\\frac{m\\omega}{\\pi\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}\\right)^{\\frac{1}{2}}\ne^{-\\frac{m\\omega}{2\\left[1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}\\left(x-y\\right)^2}.\n\\end{eqnarray}\nNext, the size of the distinguishable pairs' cloud can be assessed \nfrom the density (\\ref{DENSITY_2_12_EXAM})\nusing the widths of the respective Gaussian functions.\nAccordingly, we find\n\\begin{subequations}\\label{WIDTH_INTER}\n\\begin{eqnarray}\\label{WIDTH_INTER_GEN}\n& & \\sigma_{12,\\frac{x+y}{\\sqrt{2}}}^{(2)} = \\sqrt{\\frac{1}{2m\\omega}}, \\quad\n\\sigma_{12,\\frac{x-y}{\\sqrt{2}}}^{(2)} = \\sqrt{\\frac{1 + \\frac{1}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}{2m\\omega}}. \\\n\\end{eqnarray}\nTo show the combined effect of the intra-species and inter-species interactions\naccompanying\nfragmentation of $\\rho_{12}^{(2)}(x,x',y,y')$,\nit is useful to compute the sizes (\\ref{WIDTH_INTER_GEN}) for large inter-species attractions or \ninter-species repulsions at the border of stability.\nWe obtain, respectively,\n\\begin{eqnarray}\\label{WIDTH_INTER_LIM}\n& & \\lim_{\\frac{\\Omega_{12}}{\\omega} \\to \\infty} \\sigma_{12,\\frac{x-y}{\\sqrt{2}}}^{(2)} = \n\\sqrt{\\frac{1 - \\frac{1}{N}}{2m\\omega}}, \\qquad\n\\sigma_{12,\\frac{x-y}{\\sqrt{2}}}^{(2)} \\longrightarrow \\infty \\quad \\mathrm{for} \\quad \\frac{\\Omega_{12}}{\\omega} \\to 0^+,\n\\\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\sigma_{12,\\frac{x+y}{\\sqrt{2}}}^{(2)}$ is independent of the interactions.\nWe see that the size of the inter-species density\nsaturates as well at about the trap's size and does not depend on the strengths of interactions\nin the limit of strong inter-species attractions.\nAnalogously to identical pairs,\na strong fragmentation of distinguishable pairs\nis possible in the mixture without shrinking of the inter-species \ndensity due to strong inter-species\nattractive interaction.\nAt the other end,\nwhen the inter-species repulsion is close to the border of stability,\nthe inter-species density expands boundlessly.\nSummarizing, inter-species fragmentation is governed by the ratio $\\frac{\\Omega_{12}}{\\omega}$\nand takes place both at the attractive and repulsive sectors of interactions.\nFor the sake of analysis, we compared the results for inter-species pair fragmentation with intra-species pair fragmentation\nand discussed the similarity and differences between the respective two-particle densities\n(\\ref{DENSITY_2_12_EXAM}) and (\\ref{DENSITIES_1_2_EXAM}).\n\n\\section{Pair of distinguishable pairs and Schmidt decomposition of the wavefunction}\\label{MORE}\n\nFollowing the results of the previous section\non fragmentation of distinguishable pairs,\nthere are two questions that warrant answers.\nThe first is whether inter-species fragmentation persists beyond distinguishable pairs,\nsay, to pairs of distinguishable pairs?\nIn as much as single-species and intra-species fragmentations\ntake place at the lowest-level reduced one-particle density matrix,\nand persist at higher-level single-species reduced density matrices,\nwe wish to establish the result of inter-species fragmentation at the level of\nhigher-order reduced density matrices.\nAfter all, the reduced two-particle density matrix is the lowest-order inter-species one.\nThe second question deals with the nature of the inter-species coordinates governing fragmentation.\nAt the level of distinguishable-pair fragmentation,\ni.e., within the inter-species reduced two-particle density matrix,\none cannot unambiguously tell whether the relative center-of-mass\ncoordinate of the two species is involved or whether other relative inter-species coordinates govern fragmentation.\nThis is because for a pair of distinguishable particles one cannot\ndistinguish between the two types of coordinates.\n\nAs seen in the previous section,\nthe inter-species reduced two-particle density matrix is more intricate\nthan the intra-species ones, \nand consequently its diagonalization is more involved.\nWe derive now the inter-species reduced four-particle density matrix\nand examine which `normal coordinates' govern its diagonalization.\nThen, the natural four-particle functions are obtained explicitly and investigated.\n\nFinally and as a complementary result of the techniques\nused for inter-species fragmentation,\nwe carry the connection between inter-species and intra-species center-of-mass coordinates,\nin conjunction with the usage of Mehler's formula within a mixture, further.\nThis is done\nby constructing the Schmidt decomposition of the mixture's wavefunction\nand discussing consequences of this decomposition at the limit of an infinite number of particles.\n\n\\subsection{Inter-species fragmentation in higher-order reduced density matrices}\\label{MORE_FRAG}\n\nThe inter-species reduced four-particle density matrix is defined as\n\\begin{eqnarray}\\label{4_RDM_12_SYM_GEN}\n& & \\rho_{12}^{(4)}(x_1,x_2,x'_1,x'_2,y_1,y_2,y'_1,y'_2) = N^2 \\left(N-1\\right)^2\n\\int dx_3 \\cdots dx_N dy_3 \\cdots dy_N \\times \\nonumber \\\\\n& & \\times \\Psi(x_1,x_2,x_3,\\ldots,x_N,y_1,y_2,y_3,\\ldots,y_N)\n\\Psi^\\ast(x'_1,x'_2,x_3,\\ldots,x_N,y'_1,y'_2,y_3,\\ldots,y_N). \\\n\\end{eqnarray}\nNote that here we only treat the four-particle quantity with two identical bosons per each species.\nIntegrating the harmonic interaction-model for symmetric mixtures we find the final expression explicitly\n\\begin{subequations}\\label{4_RDM_12_SYM}\n\\begin{eqnarray}\\label{4_RDM_12_SYM_DEN}\n& & \\rho_{12}^{(4)}(x_1,x_2,x'_1,x'_2,y_1,y_2,y'_1,y'_2) = N^2 \\left(N-1\\right)^2\n\\left[\\frac{(\\alpha+C_{1,1})^2-D_{1,1}^2}{\\pi^2}\\right]^\\frac{1}{2}\n\\left[\\frac{(\\alpha+C_{2,2})^2-D_{2,2}^2}{\\pi^2}\\right]^\\frac{1}{2} \\times \\nonumber \\\\\n& & \\times e^{-\\frac{\\alpha}{2} \\left(x_1^2+x_2^2+{x'_1}^2+{x'_2}^2+y_1^2+y_2^2+{y'_1}^2+{y'_2}^2\\right)}\ne^{-\\beta\\left(x_1x_2+x'_1x'_2+y_1y_2+y'_1y'_2\\right)} \\times \\nonumber \\\\\n& & \\times\ne^{-\\frac{1}{4}C_{2,2}\\left[\\left(x_1+x_2+x'_1+x'_2\\right)^2+\\left(y_1+y_2+y'_1+y'_2\\right)^2\\right]} \\times \\nonumber \\\\\n& & \\times e^{+\\frac{1}{2}D_{2,2} \\left(x_1+x_2+x'_1+x'_2\\right)\\left(y_1+y_2+y'_1+y'_2\\right)} \ne^{+\\frac{1}{2}D'_{2,2} \\left(x_1+x_2-x'_1-x'_2\\right)\\left(y_1+y_2-y'_1-y'_2\\right)}, \\\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{4_RDM_12_SYM_COEFF}\n& & \\alpha + \\beta +2\\left(C_{2,2} \\mp D_{2,2}\\right) =\n(\\alpha-\\beta)\n\\frac{(\\alpha-\\beta) + N(\\beta\\mp \\gamma)}\n{(\\alpha-\\beta) + (N-2)(\\beta\\mp \\gamma)},\n\\nonumber \\\\\n& & D'_{2,2} = \\gamma,\n\\end{eqnarray}\n\\end{subequations}\nand $\\alpha+C_{1,1}\\mp D_{1,1}$ are given in (\\ref{2_RDM_12_SYM_COEFF}).\nThe combinations of parameters $\\alpha + \\beta +2\\left(C_{2,2} \\mp D_{2,2}\\right)$\nwould appear below shortly.\n\nTo diagonalize \n$\\rho_{12}^{(4)}(x_1,x_2,x'_1,x'_2,y_1,y_2,y'_1,y'_2)$\nwe need to mix and rotate the coordinates of the two species\ninto new coordinates appropriately.\nThus, defining the new coordinates as the center-of-mass, relative center-of-mass,\nand relative coordinates of two identical pairs, one for each of the species,\n$u_1 = \\frac{1}{2}\\left[\\left(x_1+x_2\\right)+\\left(y_1+y_2\\right)\\right]$,\n$v_1 = \\frac{1}{2}\\left[\\left(x_1+x_2\\right)-\\left(y_1+y_2\\right)\\right]$,\n$u_2 = \\frac{1}{\\sqrt{2}}\\left(x_1-x_2\\right)$,\n$v_2 = \\frac{1}{\\sqrt{2}}\\left(y_1-y_2\\right)$\nand\n$u'_1 = \\frac{1}{2}\\left[\\left(x'_1+x'_2\\right)\\right.+\\left.\\left(y'_1+y'_2\\right)\\right]$,\n$v'_1 = \\frac{1}{2}\\left[\\left(x'_1+x'_2\\right)-\\left(y'_1+y'_2\\right)\\right]$,\n$u'_2 = \\frac{1}{\\sqrt{2}}\\left(x'_1-x'_2\\right)$,\n$v'_2 = \\frac{1}{\\sqrt{2}}\\left(y'_1-y'_2\\right)$,\nwe have for the different terms in (\\ref{4_RDM_12_SYM_DEN}): \n\\begin{eqnarray}\\label{TRAS4}\n& & x_1^2+x_2^2+y_1^2+y_2^2+{x'_1}^2+{x'_2}^2+{y'_1}^2+{y'_2}^2 =\nu_1^2+{u'_1}^2+v_1^2+{v'_1}^2+u_2^2+{u'_2}^2+v_2^2+{v'_2}^2, \\nonumber \\\\\n& & x_1x_2+x'_1x'_2+y_1y_2+y'_1y'_2 =\n\\frac{1}{2}\\left(u_1^2+{u'_1}^2+v_1^2+{v'_1}^2-u_2^2-{u'_2}^2-v_2^2-{v'_2}^2\\right),  \\nonumber \\\\\n& & \\left(x_1+x_2+x'_1+x'_2\\right)^2+\\left(y_1+y_2+y'_1+y'_2\\right)^2 =\n2\\left[\\left(u_1+u'_1\\right)^2 + \\left(v_1+v'_1\\right)^2\\right], \\nonumber \\\\\n& & \\left[\\left(x_1+x_2\\right)\\pm\\left(x'_1+x'_2\\right)\\right]\\left[\\left(y_1+y_2\\right)\\pm\\left(y'_1+y'_2\\right)\\right] = \n\\left(u_1 \\pm u'_1\\right)^2 - \\left(v_1 \\pm v'_1\\right)^2. \\\n\\end{eqnarray}\nRelations (\\ref{TRAS4}) imply that\none could equally define inter-species linear combinations of the relative coordinates,\nsince\n$u_2^2+v_2^2 = \\left[\\frac{\\left(x_1-x_2\\right)+\\left(y_1-y_2\\right)}{2}\\right]^2 + \n\\left[\\frac{\\left(x_1-x_2\\right)-\\left(y_1-y_2\\right)}{2}\\right]^2$ and\n${u'_2}^2+{v'_2}^2 = \\left[\\frac{\\left(x'_1-x'_2\\right)+\\left(y'_1-y'_2\\right)}{2}\\right]^2 + \n\\left[\\frac{\\left(x'_1-x'_2\\right)-\\left(y'_1-y'_2\\right)}{2}\\right]^2$.\nWe\nchose the former combinations.\n\nPlugging (\\ref{TRAS4}) into (\\ref{4_RDM_12_SYM}) we readily find for the transformed\ninter-species reduced four-particle density matrix\n\\begin{eqnarray}\\label{4_RDM_12_SYM_DIAG}\n& &\n\\rho_{12}^{(4)}(u_1,u'_1,v_1,v'_1,u_2,u'_2,v_2,v'_2) = N^2 \\left(N-1\\right)^2\n\\times \\nonumber \\\\\n& & \\times \n\\left(\\frac{\\alpha-\\beta}{\\pi}\\right)^{\\frac{1}{2}} e^{-\\frac{\\alpha-\\beta}{2}\\left(u_2^2 + {u'_2}^2\\right)}\n\\left(\\frac{\\alpha-\\beta}{\\pi}\\right)^{\\frac{1}{2}} e^{-\\frac{\\alpha-\\beta}{2}\\left(v_2^2 + {v'_2}^2\\right)} \\times \\nonumber \\\\\n& & \\times \n\\left[\\frac{\\alpha+\\beta+2\\left(C_{2,2}-D_{2,2}\\right)}{\\pi}\\right]^\\frac{1}{2}\ne^{-\\frac{\\alpha+\\beta+C_{2,2}-\\left(D_{2,2}+D'_{2,2}\\right)}{2}\\left(u_1^2 + {u'_1}^2\\right)}\ne^{-\\left[C_{2,2}-\\left(D_{2,2}-D'_{2,2}\\right)\\right]u_1u'_1} \\times \\nonumber \\\\\n& & \\times \n\\left[\\frac{\\alpha+\\beta+2\\left(C_{2,2}+D_{2,2}\\right)}{\\pi}\\right]^\\frac{1}{2}\ne^{-\\frac{\\alpha+\\beta+C_{2,2}+\\left(D_{2,2}+D'_{2,2}\\right)}{2}\\left(v_1^2 + {v'_1}^2\\right)}\ne^{-\\left[C_{2,2}+\\left(D_{2,2}-D'_{2,2}\\right)\\right]v_1v'_1}, \\\n\\end{eqnarray}\nwhere\nthe normalization coefficients before and after diagonalization are, naturally, equal and fulfill\n$\\left[\\alpha+\\left(C_{1,1}\\mp D_{1,1}\\right)\\right]\\left[\\alpha+\\left(C_{2,2}\\mp D_{2,2}\\right)\\right]=\n\\left(\\alpha-\\beta\\right)\\left[\\alpha+\\beta+2\\left(C_{2,2}\\pm D_{2,2}\\right)\\right]$.\n\nAs can be seen in (\\ref{4_RDM_12_SYM_DIAG}) and (\\ref{2_RDM_12_SYM_DIAG}),\nthe similarities and differences between the structures\nof $\\rho_{12}^{(4)}(u_1,u'_1,v_1,v'_1,u_2,u'_2,v_2,v'_2)$ and $\\rho_{12}^{(2)}(u,u',v,v')$ \nclarify the issue of which coordinates are coupled and identify the coordinates that are not.\nIn particular, just like for the two-particle quantity,\nwe can apply Mehler's formula twice,\non the appropriately-constructed inter-species `mixed coordinates' $u_1, u'_1$ and $v_1, v'_1$,\nto diagonalize the inter-species reduced four-particle density matrix.\nWhen this is done one obtains\n\\begin{eqnarray}\\label{S_12_4}\n& & s_{12,\\pm}^{(4)} = \\sqrt{\\left(\\alpha+\\beta\\mp 2D'_{2,2}\\right)\n\\left[\\alpha+\\beta+2\\left(C_{2,2} \\mp D_{2,2}\\right)\\right]} = \\nonumber \\\\\n& & = \\sqrt{(\\alpha+\\beta\\mp 2\\gamma)(\\alpha-\\beta)\n\\frac{(\\alpha-\\beta) + N(\\beta\\mp \\gamma)}\n{(\\alpha-\\beta) + (N-2)(\\beta\\mp \\gamma)}},\n\\nonumber \\\\\n& & \\rho_{12,\\pm}^{(4)} = \\frac{\\left(\\alpha+\\beta \\mp 2D'_{2,2}\\right) - s_{12,\\pm}^{(4)}}{\\left(\\alpha+\\beta \\mp 2D'_{2,2}\\right) + s_{12,\\pm}^{(4)}} =\n\\frac{\\frac{(\\alpha+\\beta\\mp 2\\gamma)\\left[(\\alpha-\\beta) + (N-2)(\\beta\\mp \\gamma)\\right]}{(\\alpha-\\beta)\\left[(\\alpha-\\beta) + N(\\beta\\mp \\gamma)\\right]}-1}\n{\\frac{(\\alpha+\\beta\\mp 2\\gamma)\\left[(\\alpha-\\beta) + (N-2)(\\beta\\mp \\gamma)\\right]}{(\\alpha-\\beta)\\left[(\\alpha-\\beta) + N(\\beta\\mp \\gamma)\\right]}+1},\n\\nonumber \\\\\n& & 1 - \\rho_{12,\\pm}^{(4)} = \\frac{2s_{12,\\pm}^{(4)}}{\\left(\\alpha+\\beta \\mp 2D'_{2,2}\\right) + s_{12,\\pm}^{(4)}}, \\\n\\end{eqnarray}\nwhere the ``$+$'' terms quantify the fragmentation in the $u_1, u'_1$ part of the inter-species reduced four-particle density matrix and\nthe ``$-$'' terms determine\nthe fragmentation in the $v_1, v'_1$ part of the inter-species reduced four-particle density matrix.\nAs found and shown in (\\ref{4_RDM_12_SYM_DIAG}),\nthere is no fragmentation due to the relative-coordinate parts $u_2, u'_2$ and $v_2, v'_2$.\nEquation (\\ref{S_12_4}) adds to the main results of the present work and bears\na transparent and appealing physical meaning:\nIn the mixture inter-species fragmentation is quantified by\nthe eigenvalues obtained from Mehler's formula when the latter\nis applied to the mixture's center-of-mass and relative center-of-mass coordinates\nof distinguishable pairs of pairs.\nExtensions to larger distinguishable aggregates of species $1$ and species $2$ identical bosons in the mixture is possible along the above lines,\nand will not be pursued further here.\n\nWe can now prescribe the decomposition of the intra-species reduced four-particle density matrix,\nin as much as the reduced two-particle density matrix was decomposed, \ninto its natural four-particle functions made of distinguishable particles which is given by\n\\begin{eqnarray}\\label{12_4_RDM_NAT_DIS_GEM}\n& &\n\\rho_{12}^{(4)}(x_1,x_2,x'_1,x'_2,y_1,y_2,y'_1,y'_2) = N^2(N-1)^2 \n\\sum_{n_+=0}^\\infty \\sum_{n_-=0}^\\infty\n\\left(1-\\rho_{12,+}^{(4)}\\right)\n\\left(1-\\rho_{12,-}^{(4)}\\right) \\times \\nonumber \\\\\n& & \\times \\left(\\rho_{12,+}^{(4)}\\right)^{\\!n_+}\n\\left(\\rho_{12,-}^{(4)}\\right)^{\\!n_-}\n\\Phi^{(4)}_{12,n_+,n_-}(x_1,x_2,y_1,y_2) \\Phi^{(4),\\ast}_{12,n_+,n_-}(x'_1,x'_2,y'_1,y'_2), \\nonumber \\\\\n& & \\Phi^{(4)}_{12,n_+,n_-}(x_1,x_2,y_1,y_2) = \\nonumber \\\\\n& & =\n\\frac{1}{\\sqrt{2^{n_+} {n_+}!}}\n\\left(\\frac{s_{12,+}^{(4)}}{\\pi}\\right)^{\\frac{1}{4}}\nH_{n_+}\\left(\\frac{\\sqrt{s_{12,+}^{(4)}}}{2}\\left[\\left(x_1+x_2\\right)+\\left(y_1+y_2\\right)\\right]\\right)\ne^{-\\frac{1}{8}s_{12,+}^{(4)}\\left[\\left(x_1+x_2\\right)+\\left(y_1+y_2\\right)\\right]^2} \\times \\nonumber \\\\\n& & \\times\n\\frac{1}{\\sqrt{2^{n_-} {n_-}!}}\n\\left(\\frac{s_{12,-}^{(4)}}{\\pi}\\right)^{\\frac{1}{4}}\nH_{n_-}\\left(\\frac{\\sqrt{s_{12,-}^{(4)}}}{2}\\left[\\left(x_1+x_2\\right)-\\left(y_1+y_2\\right)\\right]\\right)\ne^{-\\frac{1}{8}s_{12,-}^{(4)}\\left[\\left(x_1+x_2\\right)-\\left(y_1+y_2\\right)\\right]^2} \\times \\nonumber \\\\\n& & \n\\times \\left(\\frac{\\alpha-\\beta}{\\pi}\\right)^{\\frac{1}{4}}\ne^{-\\frac{\\alpha-\\beta}{4}\\left(x_1-x_2\\right)^2}\n\\left(\\frac{\\alpha-\\beta}{\\pi}\\right)^{\\frac{1}{4}}\ne^{-\\frac{\\alpha-\\beta}{4}\\left(y_1-y_2\\right)^2}. \\\n\\end{eqnarray}\nEquation (\\ref{12_4_RDM_NAT_DIS_GEM}) means\nthat the pair-of-distinguishable-pairs `condensed fraction' is given by\nthe product\n$\\left(1-\\rho_{12,+}^{(4)}\\right)\\left(1-\\rho_{12,-}^{(4)}\\right)$ \nand the respective depleted fraction is\\break\\hfill\n$1-\\left(1-\\rho_{12,+}^{(4)}\\right)\\left(1-\\rho_{12,-}^{(4)}\\right)=\n\\rho_{12,+}^{(4)}+\\rho_{12,-}^{(4)}-\\rho_{12,+}^{(4)}\\rho_{12,-}^{(4)}$.\nThe center-of-mass and relative center-of-mass coordinates of the two pairs,\n$\\frac{\\left(x_1+x_2\\right)\\pm\\left(y_1+y_2\\right)}{2}$,\ncarry the respective scalings $s_{12,\\pm}^{(4)}$.\nThe natural four-particle functions $\\Phi^{(4)}_{12,n_+,n_-}(x_1,x_2,y_1,y_2)$\nare enumerated by the two quantum numbers $n_+$, $n_-$ and\nare obviously orthonormal to each other. \n\nWe proceed now to examine fragmentation in this higher-order inter-species reduced density matrix.\nWe investigate as above the specific case of $\\lambda+\\lambda_{12}=0$.\nTo compute $\\rho_{12}^{(4)}(x_1,x_2,x'_1,x'_2,y_1,y_2,y'_1,y'_2)$\nwe require the quantities\n$\\left[\\alpha+\\beta+2\\left(C_{2,2} - D_{2,2}\\right)\\right] = m \\omega$\nand\n$\\left(\\alpha+\\beta-2D'_{2,2}\\right) = m\\omega$\nfor the ``$+$'' branch as well as\n$\\left[\\alpha+\\beta+2\\left(C_{2,2} + D_{2,2}\\right)\\right] =\nm\\omega \\frac{1}{1 + \\frac{2}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}$\nand\n$\\left(\\alpha+\\beta+2D'_{2,2}\\right) =\nm\\omega \\left[1 + \\frac{2}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]$\nfor the ``$-$'' branch.\n\nNow, expressions (\\ref{S_12_4}) can readily be evaluated and the following picture\nof higher-order inter-species fragmentation is found:\n\\begin{subequations}\\label{S4_INTER_EXAM}\n\\begin{eqnarray}\\label{S4_12_+_EXAM}\n& & s_{12,+}^{(4)} = m\\omega,\n\\qquad\n\\rho_{12,+}^{(4)} = 0,\n\\qquad\n1 - \\rho_{12,+}^{(4)} = 1, \\\n\\end{eqnarray}\nindicating that there is no contribution to fragmentation from the center-of-mass `mixed coordinate' $u_1, u'_1$.\nThis is additional to the no contribution to fragmentation coming from the relative coordinates\n$u_2, u'_2$ and $v_2, v'_2$,\nsee (\\ref{4_RDM_12_SYM_DIAG}).\nFor the relative center-of-mass `mixed coordinate' $v_1, v'_1$,\non the other end,\none finds\n\\begin{eqnarray}\\label{S4_12_-_EXAM}\n& & s_{12,-}^{(4)} = m\\omega \\sqrt{\\frac{1 + \\frac{2}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)}\n{1 + \\frac{2}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}},\n\\nonumber \\\\ \\nonumber \\\\\n& & \\rho_{12,-}^{(4)} = \\frac{\\sqrt{\\left[1 + \\frac{2}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{2}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}-1}\n{\\sqrt{\\left[1 + \\frac{2}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{2}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}+1},\n\\nonumber \\\\\n& & 1 - \\rho_{12,-}^{(4)} = \\frac{2}\n{\\sqrt{\\left[1 + \\frac{2}{N}\\left(\\frac{\\Omega_{12}}{\\omega}-1\\right)\\right]\n\\left[1 + \\frac{2}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}+1}, \\\n\\end{eqnarray}\n\\end{subequations}\nnamely, that the fragmentation of pairs of distinguishable pairs \nfully originates from the relative center-of-mass `mixed coordinate' $v_1, v'_1$.\nWe see that also the higher-order inter-species fragmentation is governed by the ratio $\\frac{\\Omega_{12}}{\\omega}$\nand takes place both at the attractive and repulsive sectors of interactions.\nConcluding, higher-order inter-species fragmentation is proved.\n\nNow, one can compute the ratio $\\frac{\\Omega_{12}}{\\omega}=\\sqrt{1+\\frac{4N}{m\\omega^2}\\lambda_{12}}$\nfor which the inter-species reduced four-particle density matrix is $50\\%$ fragmented as in (\\ref{rho_half}).\nSince $\\rho_{12,+}^{(4)}=0$ does not contribute in this specific case,\nthe only contribution to fragmentation comes from\n$\\rho_{12,-}^{(4)}$.\nThus, solving (\\ref{S4_12_-_EXAM}) for $50\\%$ distinguishable-four-particle-function fragmentation we obtain\n\\begin{eqnarray}\\label{FRAG_INTER_4}\n\\rho_{12,-}^{(4)}=\\frac{1}{2} \\quad\n\\Longrightarrow \\quad\n\\frac{\\Omega_{12}}{\\omega} =\n\\left(1 + \\frac{2N^2}{N-2}\\right) \\pm \\sqrt{\\left(1 + \\frac{2N^2}{N-2}\\right)^2 - 1}.\n\\end{eqnarray}\nAs for distinguishable pairs,\nthere are two `reciprocate' solutions,\none for strong attractions and the second close to the border of stability for repulsions.\nAlso, to achieve the same degree of $50\\%$\nwith a larger number $\\frac{N}{2}$ of distinguishable four-boson aggregates,\na stronger attraction or repulsion is needed.\nFurthermore,\ncomparing distinguishable-four-boson and distinguishable-two-boson fragmentation at the same $50\\%$ value,\none sees from (\\ref{FRAG_INTER_4}) and (\\ref{FRAG_INTER_2}) that \nslightly weaker interactions, attractions or repulsions, are needed for the former.\nThis behavior of fragmentation of increasing orders of inter-species reduced density matrices\nis analogous to and generalizes\nthat of intra-species and single-species reduced density matrices,\nsee the previous section and the appendix, respectively.\n\nFinally, we present for completeness\nthe inter-species four-particle density,\ni.e., the diagonal part\n$\\rho_{12}^{(4)}(x_1,x_2,y_1,y_2)=\n\\rho_{12}^{(4)}(x_1,x_2,x'_1=x_1,x'_2=x_2,y_1,y_2,y'_1=y_1,y'_2=y_2)$,\nwhich is given by\n\\begin{eqnarray}\\label{DENSITY_4_12_EXAM}\n& &\n\\rho_{12}^{(4)}(x_1,x_2,y_1,y_2) = N^2 \\left(N-1\\right)^2\n\\left(\\frac{\\alpha-\\beta}{\\pi}\\right)\ne^{-\\frac{\\alpha-\\beta}{2}\\left(x_1-x_2\\right)^2}\ne^{-\\frac{\\alpha-\\beta}{2}\\left(y_1-y_2\\right)^2} \\times \\nonumber \\\\\n& & \\times \n\\left[\\frac{\\alpha+\\beta+2\\left(C_{2,2}-D_{2,2}\\right)}{\\pi}\\right]^\\frac{1}{2}\ne^{-\\frac{\\alpha+\\beta+2\\left(C_{2,2}-D_{2,2}\\right)}{4}\\left[\\left(x_1+x_2\\right)+\\left(y_1+y_2\\right)\\right]^2}\n\\times \\nonumber \\\\\n& & \\times \n\\left[\\frac{\\alpha+\\beta+2\\left(C_{2,2}+D_{2,2}\\right)}{\\pi}\\right]^\\frac{1}{2}\ne^{-\\frac{\\alpha+\\beta+2\\left(C_{2,2}+D_{2,2}\\right)}{4}\\left[\\left(x_1+x_2\\right)-\\left(y_1+y_2\\right)\\right]^2} = \\nonumber \\\\\n& & \n= N^2 \\left(N-1\\right)^2 \\left(\\frac{m\\omega}{\\pi}\\right)^{\\frac{3}{2}}\ne^{-\\frac{m\\omega}{2}\\left(x_1-x_2\\right)^2} e^{-\\frac{m\\omega}{2}\\left(y_1-y_2\\right)^2} \ne^{-\\frac{m\\omega}{4}\\left[\\left(x_1+x_2\\right)+\\left(y_1+y_2\\right)\\right]^2} \\times \\nonumber \\\\\n& & \\times \\left(\\frac{m\\omega}{\\pi\\left[1 + \\frac{2}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}\\right)^{\\frac{1}{2}}\ne^{-\\frac{m\\omega}{4\\left[1 + \\frac{2}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)\\right]}\n\\left[\\left(x_1+x_2\\right)-\\left(y_1+y_2\\right)\\right]^2}.\n\\end{eqnarray}\nTo proceed,\nthe size of the distinguishable four-boson cloud can be estimated \nfrom the widths of the respective Gaussian functions in the density (\\ref{DENSITY_4_12_EXAM}).\nThus, we obtain\n\\begin{subequations}\\label{WIDTH_4_INTER}\n\\begin{eqnarray}\\label{WIDTH_4_INTER_GEN}\n& & \\sigma_{12,\\frac{x_1-x_2}{\\sqrt{2}}}^{(4)} = \\sqrt{\\frac{1}{2m\\omega}}, \\quad\n\\sigma_{12,\\frac{y_1-y_2}{\\sqrt{2}}}^{(4)} = \\sqrt{\\frac{1}{2m\\omega}}, \\quad\n\\sigma_{12,\\frac{\\left(x_1+x_2\\right)+\\left(y_1+y_2\\right)}{2}}^{(4)} =\n\\sqrt{\\frac{1}{2m\\omega}}, \\nonumber \\\\\n& & \n\\sigma_{12,\\frac{\\left(x_1+x_2\\right)-\\left(y_1+y_2\\right)}{2}}^{(4)} = \\sqrt{\\frac{1 + \\frac{2}{N}\\left(\\frac{\\omega}{\\Omega_{12}}-1\\right)}{2m\\omega}}. \\\n\\end{eqnarray}\nTo show the combined effect of the inter-species and intra-species interactions\naccompanying\nfragmentation of $\\rho_{12}^{(4)}(x_1,x_2,x'_1,x'_2,y_1,y_2,y'_1,y'_2)$,\nit is instrumental to compute the sizes (\\ref{WIDTH_4_INTER_GEN}) for large inter-species attractions or \ninter-species repulsions at the border of stability.\nWe obtain, respectively,\n\\begin{eqnarray}\\label{WIDTH_4_INTER_LIM}\n& & \\!\\!\\!\\!\\!\\!\\!\\! \\lim_{\\frac{\\Omega_{12}}{\\omega} \\to \\infty} \\sigma_{12,\\frac{\\left(x_1+x_2\\right)-\\left(y_1+y_2\\right)}{2}}^{(4)} = \n\\sqrt{\\frac{1 - \\frac{2}{N}}{2m\\omega}}, \\qquad\n\\sigma_{12,\\frac{\\left(x_1+x_2\\right)-\\left(y_1+y_2\\right)}{2}}^{(4)}\n\\longrightarrow \\infty \\quad \\mathrm{for} \\quad \\frac{\\Omega_{12}}{\\omega} \\to 0^+,\n\\\n\\end{eqnarray}\n\\end{subequations}\nwhere\n$\\sigma_{12,\\frac{x_1-x_2}{\\sqrt{2}}}^{(4)}$,\n$\\sigma_{12,\\frac{y_1-y_2}{\\sqrt{2}}}^{(4)}$, and\n$\\sigma_{12,\\frac{\\left(x_1+x_2\\right)+\\left(y_1+y_2\\right)}{2}}^{(4)}$\ndo not depend on the interactions.\nWe see that the size of the inter-species four-boson density\nsaturates as well at about the trap's size and does not depend on the strengths of interactions\nin the limit of strong inter-species attractions.\nAs for the pair of distinguishable bosons,\na strong fragmentation is possible in the mixture with\nhardy any shrinking of the density in comparison with the bare trap\ndue to the condition $\\lambda+\\lambda_{12}=0$,\ni.e., that strong inter-species attractive interaction is accompanied by \nstrong intra-species repulsion of equal magnitude.\nSummarizing, inter-species fragmentation in higher-order reduced density matrices\nis also governed by the ratio $\\frac{\\Omega_{12}}{\\omega}$\nand takes place both at the attractive and repulsive sectors of interactions.\n\n\\subsection{Inter-species entanglement and the limit of an infinite number of particles}\\label{MORE_Schmidt}\n\nIn the previous sections the reduced density matrices \nfor identical and distinguishable pairs of bosons were diagonalized and\nthe intra-species and inter-species fragmentations explored.\nBoth kinds of fragmentations are critical phenomena in the sense that,\ngoing to the limit of an infinite number of particles while keeping the interaction parameters \n(products of the number of particles times the interaction strengths) constant,\nthe respective reduced density matrix per particle becomes $100\\%$ condensed \\cite{HIM_MIX_RDM}.\nThis can be easily obtained from the leading natural eigenvalues of the natural functions\nexplicitly obtained above,\nsee the general (\\ref{S_1_1}), (\\ref{S_1_2}), (\\ref{S_12_2}), (\\ref{S_12_4})\nand specific (\\ref{S_INTRA_EXAM}), (\\ref{S2_INTER_EXAM}),  (\\ref{S4_INTER_EXAM}) expressions,\nwhich are all equal to $1$ in this limit.\n\nIn the present, concluding subsection we touch upon a property of the mixture which does not diminish at the limit\nof an infinite number of particles.\nClassifying properties of Bose-Einstein condensates and their mixtures\nat the limit of an infinite number of particles,\nand especially when many-body and mean-field theories do not coincide,\nis an active field of research,\nwhere variances and the overlap between the many-body and mean-field\nwavefunctions are discussed elsewhere,\nsee \\cite{HIM_MIX_VAR,HIM_MIX_FLOQUET,HIM_MIX_CP,INF1,INF2,INF3,INF4,INF5,INF6,INF7,INF8,INF9}.\nHere, combining the techniques used in the previous sections,\nwe apply Mehler's formula to perform the Schmidt decomposition of the wavefunction.\n\nLet us examine the mixture's wavefunction,\nfor which the coordinates of the two species are coupled to each other owing to the inter-species interaction,\nsee the last term in (\\ref{HIM_MIX_WF_DEN_MAT1}).\nTo remind, the wavefunction is obtained by representing the Hamiltonian (\\ref{HIM_MIX}) with the mixture's Jacoby coordinates,\nfor which it is fully diagonalized, and translating back to the laboratory frame.\nTo decouple the coordinates of each species, in the sense of prescribing the Schmidt decomposition of the wavefunction,\nit is useful to go `half a step' backward,\nand express (\\ref{HIM_MIX_WF_DEN_MAT1}) using the individual species' Jacoby coordinates.\n\nThe Jacoby coordinates of each species are given by\n\\begin{eqnarray}\\label{JACOBY_12}\n& & X_k = \\frac{1}{\\sqrt{k(k+1)}} \\sum_{j=1}^k \\left(x_{k+1}-x_j\\right), \\quad 1\\le k \\le N-1, \\qquad\nX_{N} = \\frac{1}{\\sqrt{N}} \\sum_{j=1}^{N} x_j, \\nonumber \\\\\n& & Y_k = \\frac{1}{\\sqrt{k(k+1)}} \\sum_{j=1}^k \\left(y_{k+1}-y_j\\right), \\quad 1\\le k \\le N-1, \\qquad\nY_{N} = \\pm \\frac{1}{\\sqrt{N}} \\sum_{j=1}^{N} y_j, \\\n\\end{eqnarray}\nwhere, for the derivation given below,\nit is useful to distinguish between the two cases for the definition of, say, $Y_N$:\nThe plus sign is assigned to positive $\\gamma$, namely,\nto attractive inter-species interactions for which $\\Omega_{12} > \\omega$,\nand the minus sign is assigned to negative $\\gamma$, i.e.,\nto repulsive inter-species interactions where $\\Omega_{12} < \\omega$.\n\nFor the symmetric mixture,\ngiven the above Jacobi coordinates of each species, Eq.~(\\ref{JACOBY_12}),\nthe wavefunction reads\n\\begin{subequations}\\label{WF_MIX}\n\\begin{eqnarray}\\label{WF_MIX_JACOBY}\n& & \\Psi(X_1,\\ldots,X_{N},Y_1,\\ldots,Y_{N}) =\n\\left(\\frac{m\\Omega}{\\pi}\\right)^{\\frac{N-1}{2}}\n\\left(\\frac{M_{12}\\Omega_{12}}{\\pi}\\right)^{\\frac{1}{4}}\n\\left(\\frac{M\\omega}{\\pi}\\right)^{\\frac{1}{4}} \\times \\nonumber \\\\\n& & \\times e^{-\\frac{1}{2} m \\Omega \\sum_{k=1}^{N-1} \\left(X_k^2 + Y_k^2\\right)}\ne^{-\\frac{\\frac{1}{2}m\\left(\\Omega_{12}+\\omega\\right)}{2}\\left(X_{N}^2 + Y_{N}^2\\right)}\ne^{\\pm\\frac{1}{2}m\\left(\\Omega_{12}-\\omega\\right)X_{N}Y_{N}}. \\\n\\end{eqnarray}\nIndeed, all relative coordinates are decoupled and the only coupling due to the inter-species interaction\nis between the center-of-mass\n$X_N$ of species $1$ bosons\nand the center-of-mass\n$Y_N$ of species $2$ bosons.\nConsequently, applying Mehler's formula to the intra-species\ncenter-of-mass Jacoby coordinates $X_{N}$ and $Y_{N}$\nthe Schmidt decomposition of (\\ref{WF_MIX_JACOBY}) is readily performed and given by\n\\begin{eqnarray}\\label{WF_MIX_SCHMIDT}\n& & \\Psi(X_1,\\ldots,X_{N},Y_1,\\ldots,Y_{N})\n= \\sum_{n=0}^{\\infty} \\sqrt{1-\\rho^2_{SD}} \\rho_{SD}^n \\Phi_{1,n}(X_1,\\ldots,X_N) \\Phi_{2,n}(Y_1,\\ldots,Y_N), \\nonumber \\\\\n& & \\Phi_{1,n}(X_1,\\ldots,X_N) = \\left(\\frac{m\\Omega}{\\pi}\\right)^{\\frac{N-1}{4}}\ne^{-\\frac{1}{2} m \\Omega \\sum_{k=1}^{N-1} X_k^2}\n\\frac{1}{\\sqrt{2^n n!}} \\left(\\frac{s_{SD}}{\\pi}\\right)^{\\frac{1}{4}} H_n\\left(\\sqrt{s_{SD}}X_N\\right) e^{-\\frac{1}{2}s_{SD} X_N^2}, \\nonumber \\\\\n& & \\Phi_{2,n}(Y_1,\\ldots,Y_N) = \\left(\\frac{m\\Omega}{\\pi}\\right)^{\\frac{N-1}{4}}\ne^{-\\frac{1}{2} m \\Omega \\sum_{k=1}^{N-1} Y_k^2}\n\\frac{1}{\\sqrt{2^n n!}} \\left(\\frac{s_{SD}}{\\pi}\\right)^{\\frac{1}{4}} H_n\\left(\\sqrt{s_{SD}}Y_N\\right) e^{-\\frac{1}{2}s_{SD} Y_N^2}, \\nonumber \\\\\n& & \\sqrt{1-\\rho^2_{SD}} =\n\\frac{2\\sqrt{\\frac{\\Omega_{12}}{\\omega}}}{1+\\frac{\\Omega_{12}}{\\omega}}, \\qquad\n\\rho_{SD} =\n\\frac{\\left(\\frac{\\Omega_{12}}{\\omega}\\right)^{\\pm 1}-1}{\\left(\\frac{\\Omega_{12}}{\\omega}\\right)^{\\pm 1}+1}, \\qquad\ns_{SD} = m\\sqrt{\\omega\\Omega_{12}}.\n\\\n\\end{eqnarray}\n\\end{subequations}\nWe remind that the plus sign is for attraction and the minus for repulsion,\nwhich is what guarantees that $\\rho_{SD}$ \nand consequently the Schmidt coefficients $\\sqrt{1-\\rho^2_{SD}} \\rho_{SD}^n$, $n=0,1,2,3,\\ldots$\nare always positive.\n$s_{SD}$ defines the inverse width of the individual species' center-of-mass Gaussians in\nthe Schmidt basis $\\Phi_{1,n}(X_1,\\ldots,X_N)$ and $\\Phi_{2,n}(Y_1,\\ldots,Y_N)$. \n\nLet us concisely discuss properties of the Schmidt decomposition of the mixture, Eq.~(\\ref{WF_MIX_SCHMIDT}). \nClearly and interestingly,\nthe Schmidt coefficients are independent of the intra-species dressed frequency $\\Omega$,\nwhich only appears in conjunction with intra-species relative coordinates, \ni.e., the Schmidt coefficients depend solely on the\ninter-species interaction.\nFurthermore,\nthere is a kind of symmetry between respective attractive and repulsive inter-species interactions,\nas one gets the\nsame Schmidt coefficients for the inter-species frequency $\\frac{\\Omega_{12}}{\\omega}=\\sqrt{1+\\frac{4N\\lambda_{12}}{m\\omega^2}}$ and\ninverse frequency $\\frac{\\omega}{\\Omega_{12}}=\\frac{1}{\\sqrt{1+\\frac{4N\\lambda_{12}}{m\\omega^2}}}$.\n\nLast but not least,\nthe same Schmidt coefficients are obtained when the product\nof the number of bosons in each species times the inter-species interaction strength, $N\\lambda_{12}$, is held fixed,\nand $N$ is increased to infinity.\nIn other words,\nwhereas identical and distinguishable bosons, pairs, four-particle aggregates, etc.\nare $100\\%$ condensed at the limit of an infinite number of particles, \ni.e., the leading eigenvalue of \nall finite-order intra-species and inter-species reduced density matrices per particle is $1$,\nthe mixture's wavefunction exhibits a fixed amount of entanglement at the infinite-particle-number limit.\nThis is a good place to bring the present study to an end. \n\n\\subsection{Summary and Outlook}\\label{SUM_OUT}\n\nThe present work aims at developing and combining concepts from quantum theory\nof many-particle systems with novel results on the physics\nof trapped mixtures of Bose-Einstein condensates.\nThe notions of natural orbitals and\nnatural geminals are fundamental to many-particle systems\nmade of identical particles.\nThese natural functions entail the diagonalization of the reduced one-particle and two-particle density matrices, respectively.\nIn a mixture of two kinds of identical particles,\nhere explicitly two types of bosons,\nthere are, naturally, identical bosons and pairs made of indistinguishable bosons of either species.\nTo find their natural orbitals and natural geminals,\nthe construction and subsequent diagonalization of\nrespective intra-species reduced density matrices is in need.\nIn the mixture there are, additionally, pairs made of distinguishable bosons.\nAnalogously, their theoretical description would require\nassembling, diagonalizing, and analyzing the inter-species\nreduced two-particle density matrix. \nIn the present work we have investigated pairs made of identical or distinguishable\nbosons in a mixture of Bose-Einstein condensates,\ncovering both the structure of the respective natural pair functions,\non the more formal theoretical side,\nand the exploration of pairs' fragmentation.\nLike identical bosons,\nwhich can, depending on whether the reduced one-particle density matrix has\none or more macroscopic eigenvalues,\nbe condensed or fragmented,\nso do pairs of bosons.\nWe showed in the present work that,\nin the mixture, both pairs made of identical bosons\nand pairs consisting of distinguishable bosons\ncan be condensed and more so fragmented.\n\nTo tackle the above and other questions,\nwe employed a solvable model,\nthe symmetric harmonic-interaction model for mixtures.\nThe natural geminals for pairs made of identical or distinguishable bosons\nwere explicitly contracted as a function of the inter-species and intra-species interactions.\nThis was done\nby diagonalizing the corresponding intra-species and inter-species reduced two-particle density matrices\nusing applications of Mehler's formula on appropriately-constructed linear combinations of intra-species and inter-species coordinates.\nHere, the role of the mixture's center-of-mass and relative center-of-mass coordinates was identified and explained.\nThe structure of identical and distinguishable pairs in the mixture was discussed,\nand a generalization to pairs of distinguishable pairs using the inter-species reduced four-body density matrix was made.\nA particular case, where attractive and repulsive inter-species and intra-species interactions are opposite in magnitude,\nhas been worked out explicitly.\nFragmentation of bosons, pairs, and pairs of pairs in the mixture has been proven,\nand the size of the respective densities analyzed.\nLast but not least,\nas a complementary investigation,\nthe exact Schmidt decomposition of the mixture's wavefunction was performed.\nThe entanglement between the two species was shown to be governed by the coupling of their individual center-of-mass coordinates\nand, consequently, not to vanish at the limit of an infinite number of particles where\nany finite-order intra-species and inter-species reduced density matrix per particle is 100\\% condensed.\n\nThe present investigations suggest several directions for further developments.\nWe have treated the symmetric mixture\nand an anticipated extension to\ngeneric trapped mixtures, with different numbers of bosons, masses, and interaction strengths for each species, would be in place.\nIn what capacity the fragmentation of identical pairs in the different species can be made to differ,\nand to what extent the fragmentation of distinguishable pairs would become more complex? \nDo the center-of-mass and relative center-of-mass coordinates keep their\nrole in the diagonalization of inter-species reduced density matrices for a generic mixture?\nIt also makes sense, in a generic mixture,\nto investigate fragmentation of\naggregates with unequal numbers of bosons from each species,\nlike, for instance, the analysis of inter-species reduced three-particle density matrices.\nAnother extension foreseen is to mixtures with more species and, if feasible,\nto generic multi-species mixtures where, e.g., one species could serve as a bridge between two baths.\nFinally, one could forecast that the topic of Bose-Einstein condensates and mixtures in the limit of an infinite number of particles\nwould be enriched by exploring the Schmidt decomposition of the wavefunction.\nRecall that at the infinite-particle-number limit any finite-order intra-species and inter-species reduced\ndensity matrix per particle is $100\\%$ condensed.\nHere, observables' variances and wavefunctions' overlaps\nhave deepened our understanding of the\ndifferences between many-body and mean-field theories of Bose-Einstein condensates and mixtures\nat limit of an infinite number of particles,\nbut are properties already defined for single-species bosons.\nThe Schmidt decomposition, on the other hand, is a property that\nenters the topic of the infinite-particle-number limit starting, obviously,\nonly from a two-species mixture.\nAll of which paves the way for further intriguing investigations to come.\n    \n\\section*{Acknowledgements}\n\nThis research was supported by the Israel Science Foundation \n(Grant No. 1516\/19). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\n\\subsection{Objective}\n\nDesigning computing environments consuming a limited amount of energy while achieving computationally complex tasks is an objective of utmost importance, especially in distributed systems involving a large number of computing entities. In this paper, we aim at designing energy-efficient algorithms for the standard LOCAL and CONGEST models of distributed computing in networks~\\cite{peleg2000}. Both models assume a network modeled as an $n$-node graph $G=(V,E)$, where each node is provided with an identifier, i.e., an integer that is unique in the network, which can be stored on $O(\\log n)$ bits. All nodes are assumed to run the same algorithm, and computation proceeds as a series of synchronous rounds (all nodes start simultaneously at round~1). During a round, every node sends a message to each of its neighbors, receives the messages sent by its neighbors, and performs some individual computation. The two models LOCAL and CONGEST differ only in the amount of information that can be exchanged between nodes at each round. \n\nThe LOCAL model does not bound the size of the messages, whereas the CONGEST model allows only messages of size $O(\\log n)$ bits. Initially, every node~$v\\in V$ knows solely its identifier~$\\mathsf{id}(v)$, an upper bound of the number~$n$ of nodes, which is assumed to be polynomial in~$n$ and to be the same for all nodes, plus possibly some input bit-string $x(v)$ depending on the task to be solved by the nodes. In this paper, we denote by $N$ the maximum between the largest identifier and the upper bound on~$n$ given to all nodes. Hence $N=O(\\mbox{poly}(n))$, and is supposed to be known by all nodes. After a certain number of rounds, every node outputs a bit-string~$y(v)$, where the correctness of the collection of outputs $y=\\{y(v):v\\in V\\}$ is defined with respect to the specification of the task to be solved, and may depend on the collection of inputs $x=\\{x(v):v\\in V\\}$ given to the nodes, as well as on the graph~$G$ (but not on the identifiers assigned to the nodes, nor on the upper bound~$N$). \n\n\\paragraph{Activation complexity.} \n\nWe measure the energy consumption of an algorithm~$A$ by counting how many times each node and each edge is activated during the execution of the algorithm. More specifically, a node~$v$ (resp., an edge~$e$) is said to be \\emph{active} at a given round~$r$ if $v$ is sending a message to at least one of its neighbors at round~$r$ (resp., if a message traverses $e$ at round~$r$). The \\emph{node-activation} and the \\emph{edge-activation} of an algorithm~$A$ running in a graph $G=(V,E)$ are respectively  defined as \n\\[\n\\mathsf{nact}(A):=\\max_{v\\in V}\\#\\mbox{activation}(v),\n\\;\\; \\mbox{and} \\;\\;\n\\mathsf{eact}(A):=\\max_{e\\in E}\\#\\mbox{activation}(e),\n\\]\nwhere $\\#\\mbox{activation}(v)$ (resp., $\\#\\mbox{activation}(e)$) denotes the number of rounds during which node~$v$ (resp., edge~$e$) is active along the execution of the algorithm~$A$.\nBy definition, we have that, in any graph of maximum degree~$\\Delta$, \n\\begin{equation}\\label{eq:ineq-activation}\n\\mathsf{eact}(A)\\leq 2\\cdot \\mathsf{nact}(A), ~\\textrm{ and }~ \\mathsf{nact}(A) \\leq \\Delta\\cdot\\mathsf{eact}(A).\n\\end{equation}\n\n\\paragraph{Objective.} \n\nOur goal is to design \\emph{frugal} algorithms, that is, algorithms with \\emph{constant} node-activation, or to the least \\emph{constant} edge-activation, independent of the number~$n$ of nodes and of the number~$m$ of edges. Indeed, such algorithms can be viewed as consuming the least possible energy for solving a given task. Moreover, even if the energy requirement for solving the task naturally grows with the number of components (nodes or edges) of the network, it grows \\emph{linearly} with this number whenever using frugal algorithms. We refer to \\emph{node-frugality} or \\emph{edge-frugality} depending on whether we focus on node-activation or edge-activation, respectively. \n\n\\subsection{Our Results}\n\nWe first show that every Turing-computable problem\\footnote{A problem is Turing-computable if there exists a Turing machine that, given any graph with identifiers and inputs assigned to the nodes, computes the output of each node in the graph.} can thus be solved by a node-frugal algorithm in the LOCAL model as well as in the CONGEST model. It follows from Eq.~\\ref{eq:ineq-activation} that every Turing-computable problem can be solved by an edge-frugal algorithm in both models. In other words, every problem can be solved by an energy-efficient distributed algorithm. One important question remains: what is the round complexity of frugal algorithms? \n\nIn the LOCAL model, our node-frugal algorithms run in $O(\\mbox{poly}(n))$ rounds. However, they may run in exponentially many rounds in the CONGEST model. We show that this cannot be avoided. Indeed, even if many symmetry-breaking problems such as computing a maximal-independent set ({\\sc mis}) and computing a $(\\Delta+1)$-coloring can be solved by a node-frugal algorithm performing in $O(\\mbox{poly}(n))$ rounds, we show that there exist problems (e.g., deciding $C_4$-freeness or deciding the presence of symmetries in the graph) that cannot be solved in $O(\\mbox{poly}(n))$ rounds in the CONGEST model by any edge-frugal algorithm. \n\nFinally, we discuss the relation between node-activation complexity and edge-activation complexity. We show that the bounds given by Eq.~\\ref{eq:ineq-activation} are essentially the best that can be achieved in general. Precisely, we identify a problem, namely \\textsc{Depth First Pointer Chasing}{} (\\textsc{dfpc}), which has edge-activation complexity $O(1)$ for all graphs with an algorithm running in $O(\\mbox{poly}(n))$ rounds in the CONGEST model, but satisfying that, for every $\\Delta =O\\left(\\frac{n^{1\/4}}{\\sqrt{\\log n}}\\right)$, its node-activation complexity in graphs with maximum degree $\\Delta$ is $\\Omega(\\Delta)$ whenever solved by an algorithm bounded to run in $O(\\mbox{poly}(n))$ rounds in the CONGEST model. In particular, \\textsc{Depth First Pointer Chasing}{} has constant edge-activation complexity but node-activation complexity $\\tilde{\\Omega}(n^{1\/4})$ in the CONGEST model (for $O(\\mbox{poly}(n))$-round algorithms). \n\nOur main results are summarized in Table~\\ref{tab:summary}. \n\n\\begin{table}\n\\begin{center} \n\\begin{tabular}{l|l|l|l|}\n& \\hspace{1cm} \\textbf{Awakeness}\n& \\hspace{.1cm} \\textbf{Node-Activation}\n& \\hspace{.3cm} \\textbf{Edge-Activation} \\\\\n\\hline\nLOCAL \n& $\\bullet \\;\\forall \\Pi, \\Pi \\in O(\\log n)$ with  \n& $\\bullet \\;\\forall \\Pi, \\Pi \\in O(1)$ with \n& $\\bullet \\; \\forall \\Pi, \\Pi \\in O(1)$ with\\\\\n& $\\;\\; O(\\mbox{poly}(n))$ rounds \\cite{BarenboimM21} \n& $\\;\\; O(\\mbox{poly}(n))$ rounds \n& $\\;\\; O(\\mbox{poly}(n))$ rounds\\\\\n& $\\bullet \\; \\mbox{\\sc st}\\in \\Omega(\\log n)$ \\cite{BarenboimM21} \n& & \\\\\n\\hline \nCONGEST \n& $\\bullet \\; \\mbox{\\sc mis}\\in O(\\mbox{polyloglog}(n))$  \n& $\\bullet \\; \\forall \\Pi, \\Pi\\in O(1)$ \n& $\\bullet \\; \\forall \\Pi, \\Pi \\in O(1)$\\\\\n& \\;\\; with $O(\\mbox{polylog}(n))$  \n& $\\bullet \\;\\mbox{poly}(n)$ rounds \n& $\\bullet \\; \\mbox{poly}(n) $ rounds \\\\\n& \\;\\; rounds \\cite{DufoulonMP22} (randomized)\n& $\\;\\; \\Rightarrow \\exists \\Pi \\in \\Omega(\\mbox{poly}(n)) $ \n& $\\;\\; \\Rightarrow \\exists \\Pi \\in \\Omega(\\mbox{poly}(n)) $ \\\\\n& $\\bullet \\; \\mbox{\\sc mst} \\in O(\\log n)$\n& $\\bullet \\;\\mbox{poly}(n)$ rounds \n& $\\bullet \\;\\mbox{\\sc dfpc}\\in O(1)$ with \\\\\n& \\;\\; with $O(\\mbox{poly}(n))$\n& $\\;\\; \\Rightarrow \\mbox{\\sc dfpc}\\in \\tilde{\\Omega}(n^{1\/4})$\n& $\\; \\; O(\\mbox{poly}(n))$ rounds \\\\\n& \\;\\; rounds \\cite{AugustineMP22}\n&\n& $\\bullet \\;\\Pi\\in \\mbox{FO and}\\; \\Delta=O(1) $\\\\\n&\n&\n& $\\;\\; \\Rightarrow \\Pi\\in O(1) \\; \\mbox{with}$ \\\\\n&\n&\n& $\\;\\; O(\\mbox{poly}(n))$ rounds~\\cite{GrumbachW09} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\sl Summary of our results where, for a problem $\\Pi$, $\\Pi\\in O(f(n))$ means that the corresponding complexity of $\\Pi$ is $O(f(n))$ (same shortcut for $\\Omega$).}\n\\label{tab:summary}\n\\end{table}\n\n\\paragraph{Our Techniques.} \n\nOur upper bounds are mostly based on similar types of upper bounds techniques used in the sleeping model~\\cite{BarenboimM21,ChatterjeeGP20} (cf. Section~\\ref{subsec:related-work}), based on constructing spanning trees along with\ngathered and broadcasted information. However, the models considered in this paper do not suffer from the same limitations as the sleeping model (cf. Section~\\ref{sec:preliminaries}), and thus one can achieve activation complexity $O(1)$ in scenarios where the sleeping model limits the awake complexity to $\\Omega(\\log n)$. \n\nOur lower bounds for CONGEST are based on reductions from 2-party communication complexity. However, as opposed to the standard CONGEST model in which the simulation of a distributed algorithm by two players is straightforward (each player performs the rounds sequentially, one by one, and exchanges the messages sent across the cut between the two subsets of nodes handled by the players at each round), the simulation of distributed algorithms in which only subsets of nodes are active at various rounds requires more care. This is especially the case when the simulation must not only control the amount of information exchanged between these players, but also the number of communication steps performed by the two players. Indeed, there are 2-party communication complexity problems that are hard for $k$ steps, but trivial for $k+1$ steps~\\cite{NisanW93}, and some of our lower bounds rely on this fact. \n\n\n\n\\subsection{Related Work}\n\\label{subsec:related-work}\n\nThe study of frugal algorithms has been initiated in~\\cite{GrumbachW09}, which focuses on the edge-frugality in the CONGEST model. It is shown that for {\\emph{bounded-degree graphs}}, any problem expressible in first-order logic (e.g., $C_4$-freeness) can be solved by an edge-frugal algorithm running in $O(\\mbox{poly}(n))$ rounds in the CONGEST model. This also holds for planar graphs with no bounds on the maximum degree, whenever the nodes are provided with their local combinatorial embedding. Our results show that these statements cannot be extended to arbitrary graphs as we prove that any algorithm solving $C_4$-freeness in $O(\\mbox{poly}(n))$ rounds in the CONGEST model has edge-activation $\\tilde{\\Omega}(\\sqrt{n})$. \n\nMore generally, the study of energy-efficient algorithms in the context of distributed computing in networks has been previously considered in the framework of the \\emph{sleeping} model, introduced in~\\cite{ChatterjeeGP20}. This model assumes that nodes can be in two states: \\emph{awake} and \\emph{asleep}. A node in the awake state performs as in the LOCAL and CONGEST models, but may also decide to fall asleep, for a prescribed amount of rounds, controlled by each node, and depending on the algorithm executed at the nodes. A sleeping node is totally inactive in the sense that it does not send messages, it cannot receive messages (i.e., if a message is sent to a sleeping node by an awake neighbor, then the message is lost), and it is computationally idle  (apart from counting rounds). The main measure of interest in the sleeping model is the \\emph{awake complexity}, defined as the maximum, taken over all nodes, of the number of rounds at which each node is awake during the execution of the algorithm. \n\nIn the LOCAL model, it is known~\\cite{BarenboimM21} that all problems have awake complexity $O(\\log n)$, using algorithms running in $O(\\mbox{poly}(n))$ rounds. This bound is tight in the sense that there are problems (e.g., spanning tree construction) with awake complexity $\\Omega(\\log n)$ \\cite{BarenboimM21,ChangDHHLP18}. \n\nIn the CONGEST model, It was first shown~\\cite{ChatterjeeGP20} that {\\sc mis} has constant \\emph{average} awake complexity, thanks to a \\emph{randomized} algorithm running in $O(\\mbox{polylog}(n))$ rounds. The round complexity was improved in~\\cite{GhaffariP22} with a \\emph{randomized} algorithm running in $O(\\log n)$ rounds. The (worst-case) awake complexity of {\\sc mis} was proved to be $O(\\log\\log n)$ using a  \\emph{randomized} Monte-Carlo algorithm running in $O(\\mbox{poly}(n))$ rounds~\\cite{DufoulonMP22}. This (randomized) round complexity can even be reduced to $O(\\log^3n \\cdot \\log\\log n \\cdot \\log^\\star n)$, to the cost of slightly increasing the awake complexity to $O(\\log\\log n \\cdot \\log^\\star n)$. {\\sc mst} has also been considered, and it was proved~\\cite{AugustineMP22} that its (worst-case) awake complexity is $O(\\log n)$ thanks to a (deterministic) algorithm running in $O(\\mbox{poly}(n))$ rounds. The upper bound on the awake complexity of {\\sc mst} is tight, thank to the lower bound for spanning tree ({\\sc st}) in~\\cite{BarenboimM21}. \n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nIn this section, we illustrate the difference between the standard LOCAL and CONGEST models, their sleeping variants, and our node- and edge-activation variants. Fig.~\\ref{fig:1}(a) displays the automaton corresponding to the behavior of a node in the standard models. A node is either \\emph{active}~(A) or \\emph{terminated}~(T). At each clock tick (i.e., round) a node is subject to message events corresponding to sending and receiving messages to\/from neighbors. A node remains active until it terminates. \n\n\\begin{figure}[!h]\n    \\centering\n    \\scalebox{0.8}{\n    \\input{Figure1.tex}}\n    \\caption{(a) Classical model (b) Sleeping model, (c) Activation model.  }\n    \\label{fig:1}\n\\end{figure}\n\n\nFig.~\\ref{fig:1}(b) displays the automaton corresponding to the behavior of a node in the sleeping variant. In this variant, a node can also be in a \\emph{passive} (P) state. In this state, the clock event can either leave the node passive, or awake the node, which then moves back to the active state.  \n\nFinally, Fig.~\\ref{fig:1}(c) displays the automaton corresponding to the behavior of a node in our  activation variants. It differs from the sleeping variant in that a passive node is also subject to message events, which can leave the node passive, but may also move the node to the active state.  In particular, a node does not need to be active for receiving messages, and incoming messages may not trigger an immediate response from the node (e.g., forwarding information). Instead, a node can remain passive while collecting information from each of its neighbors, and eventually react by becoming active. \n\n\\paragraph{Example 1: Broadcast.} \n\nAssume that one node of the $n$-node cycle~$C_n$ has a token to be broadcast to all the nodes. Initially, all nodes are active. However, all nodes but the one with the token become immediately passive when the clock ticks for entering the second round. The node with the token sends the token to one of its neighbors, and becomes passive at the next clock tick. Upon reception of the token, a passive node becomes active, forwards the token, and terminates. When the source node receives the token back, it becomes active, and terminates. The node-activation complexity of broadcast is therefore~$O(1)$, whereas it is known that broadcasting has awake complexity $\\Omega(\\log n)$ in the sleeping model~\\cite{BarenboimM21}. \n\n\\paragraph{Example 2: At-least-one-leader.} \n\nAssume that each node of the cycle~$C_n$ has an input-bit specifying whether the node is leader or not, and the nodes must collectively check that there is at least one leader. Every leader broadcasts a token, outputs accept, and terminates. Non-leader nodes become passive immediately after the beginning of the algorithm, and start waiting for $N$~rounds (recall that $N$ is an upper bound on the number~$n$ of nodes). Whenever the ``sleep'' of a (passive) non-leader is interrupted by the reception of a token, it becomes active, forwards the token, outputs accept, and terminates. After $N$ rounds, a passive node that has not been ``awaken'' by a token becomes active, outputs reject, and terminates. This guarantees that there is at least one leader if and only if all nodes accept. The node-activation complexity of this algorithm is $O(1)$, while the awake complexity of at-least-one-leader is $\\Omega(\\log n)$ in the sleeping model, by reduction to broadcast. \n\n\\medskip \n\nThe following observation holds for LOCAL and CONGEST, by noticing that every algorithm for the sleeping model can be implemented with no overheads in terms of node-activation. \n\n\n\\begin{observation}\\label{obs:sleep-vs-activation}\nIn $n$-node graphs, every algorithm with awake complexity~$a(n)$ and round complexity~$r(n)$ has node-activation complexity~$a(n)$ and round complexity~$r(n)$. \n\\end{observation}\n\nIt follows from Observation~\\ref{obs:sleep-vs-activation} that all upper bound results for the awake complexity directly transfer to the node-activation complexity. However, as we shall show in this paper, in contrast to the sleeping model in which some problems (e.g., spanning tree) have awake complexity $\\Omega(\\log n)$, even in the LOCAL model, all problems admit a frugal algorithm in the CONGEST model, i.e., an algorithm with node-activation~$O(1)$. \n\n\\begin{definition}\nA LOCAL or CONGEST algorithm is \\emph{node-frugal} (resp., \\emph{edge-frugal}) if the activation of every node (resp., edge) is upper-bounded by a constant independent of the graph, and of the identifiers and inputs given to the nodes. \n\\end{definition}\n\n\\section{Universality of Frugal Algorithms}\n\nIn this section we show that every Turing-computable problem can be solved by frugal algorithms, both in the LOCAL and  CONGEST models. Thanks to Eq.~\\ref{eq:ineq-activation}, it is sufficient to prove that this holds for node-frugality.\n\n\\begin{lemma} \\label{lemm:leader}\nThere exists a CONGEST algorithm electing a leader, and constructing a BFS tree rooted at the leader, with node-activation complexity~$O(1)$, and performing in $O(N^2) = O(\\mbox{\\rm poly}(n))$ rounds.\n\\end{lemma}\n\n\\begin{proof}\nThe algorithm elects as leader the node with smallest identifier, and initiates a breadth-first search from that node. At every node~$v$, the protocol performs as follows. \n\n\\begin{itemize}\n\\item If $v$ has received no messages until round $\\mathsf{id}(v) \\cdot N$, then $v$ elects itself as leader, and starts a BFS by sending message $(\\mathsf{id}(v),0)$ to all its neighbors. Locally, $v$~sets its parent in the BFS tree to \\(\\bot\\), and the distance to the root to~$0$.\n\n\\item Otherwise, let $r$ be the first round at which vertex~$v$  receives a message. Such a message is of type $(\\mathsf{id}(u),d)$ where $u$ is the neighbor of $v$ which sent the message to~$v$, and $d$ is the distance from $u$ to the leader in the graph. Node $v$ sets its parent in the BFS tree to $\\mathsf{id}(u)$, its distance to the root to $d+1$, and, at round $r+1$, it sends the message $(\\mathsf{id}(v),d+1)$ to all its neighbors. (If $v$ receives several messages at round~$r$, from different neighbors, then $v$ selects the messages coming from the neighobors with smallest identifier).\n\\end{itemize}\n\nThe node $v$ with smallest identifier is indeed the node initiating the BFS, as the whole BFS is constructed between\nrounds ${\\mathsf{id}(v)\\cdot N}$ and ${\\mathsf{id}(v)\\cdot N + N - 1}$, and $N\\geq n$. The algorithm terminates at round at most~$O(N^2)$. \n\\qed\n\\end{proof}\n\nAn instance of a problem is a triple $(G,\\mathsf{id},x)$ where $G=(V,E)$ is an $n$-node graph, $\\mathsf{id}:V\\to [1,N]$ with $N=O(\\mbox{poly}(n))$, and $x:V\\to [1,\\nu]$ is the input assignment to the nodes. Note that the input range $\\nu$ may depend on~$n$, and even be exponential in~$n$, even for classical problems, e.g., whenever weights assigned to the edges are part of the input. A solution to a graph problem is an output assignment $y:V\\to [1,\\mu]$, and the correctness of~$y$ depends on $G$ and $x$ only, with respect to the specification of the problem. We assume that $\\mu$ and $\\nu$ are initially known to the nodes, as it is the case for, e.g., {\\sc mst}, in which the weights of the edges can be encoded on $O(\\log n)$ bits.\n\n\\begin{theorem}\\label{theo:all-in-LOCAL}\nEvery Turing-computable problem\nhas a LOCAL algorithm with $O(1)$ node-activation complexity, and running in  $O(N^2) = O(\\mbox{\\rm poly}(n))$ rounds.\n\\end{theorem}\n\n\\begin{proof}\nOnce the BFS tree~$T$ of Lemma \\ref{lemm:leader} is constructed, the root can (1)~gather the whole instance $(G,\\mathsf{id},x)$, (2)~compute a solution~$y$, and (3)~broadcast~$y$ to all nodes. Specifically, every leaf~$v$ of~$T$ sends the set \n\\[\nE(v)=\\big\\{\\{(\\mathsf{id}(v),x(v)),(\\mathsf{id}(w),x(w))\\}:w\\in N(v)\\big\\}\n\\]\nto its parent in~$T$. An internal node~$v$ waits for receiving a set of edges $S(u)$ from each of its children~$u$ in~$T$, and then forwards the set \n\\[\nS(v)=E(v)\\cup(\\cup_{u\\in \\mbox{\\tiny child}(v)}S(u))\n\\]\nto its parent. Each node of $T$ is activated once during this phase, and thus the node-activation complexity of gathering is~1. Broadcasting the solution~$y$ from the leader to all the nodes is achieved along the edges of~$T$, again with node-activation~1.\n\\qed\n\\end{proof}\n\nThe algorithm used in the proof of Theorem~\\ref{theo:all-in-LOCAL} cannot be implemented in CONGEST due to the size of the messages, which may require each node to be activated more than a constant number of times. To keep the node-activation constant, we increased the round complexity of the algorithm. \n\n\\begin{lemma}\\label{lem:simulation-local-congest}\nEvery node-frugal algorithm $\\mathcal{A}$ performing in $R$ rounds in the LOCAL model with messages of size at most~$M$ bits can be implemented by a node-frugal algorithm $\\mathcal{B}$ performing in $R\\,2^M$ rounds in the CONGEST model.\n\\end{lemma} \n\n\\begin{proof}\nLet $v$ be a node sending a message~$m$ through an incident edge~$e$ at round~$r$ of~$\\mathcal{A}$. Then, in~$\\mathcal{B}$, $v$ sends one ``beep'' through edge~$e$ at round $r\\,2^M+t$ where $t$ is lexicographic rank of $m$ among the at most $2^M$ messages generated by~$\\mathcal{A}$. \n\\qed\n\\end{proof}\n\n\\begin{theorem}\\label{theo:all-in-CONGEST}\nEvery Turing-computable problem\nhas a CONGEST algorithm with $O(1)$ node-activation complexity, and \n running in \n$2^{\\mbox{\\rm\\scriptsize poly}(n)+O((\\nu+\\mu)\\log n)}$ rounds for inputs in the range $[1,\\nu]$ and outputs in the range $[1,\\mu]$. \n\\end{theorem}\n\n\\begin{proof}\nThe algorithm used in the proof of Theorem~\\ref{theo:all-in-LOCAL} used messages of size at most $2N^2+\\nu\\log N$ bits during the gathering phase, and of size at most $\\mu\\log N$ bits during the broadcast phase. The result follows from Lemma~\\ref{lem:simulation-local-congest}.\\qed\n\\end{proof}\n\n\nOf course, there are many problems that can be solved in the CONGEST model by a frugal algorithm much faster than the bound from Theorem~\\ref{theo:all-in-CONGEST}. This is typically the case of all problems that can be solved by a sequential greedy algorithm visiting the nodes in arbitrary order, and producing a solution at the currently visited node based only on the partial solution in the neighborhood of the node. Examples of such problems are {\\sc mis}, $\\Delta + 1$-coloring, etc. We call such problem \\emph{sequential-greedy}. \n\n\\begin{theorem}\nEvery sequential-greedy problem whose solution at every node can be encoded on $O(\\log n)$ bits has a node-frugal CONGEST algorithm running in $O(N) = O({\\mbox{\\rm poly}(n)})$ rounds.\n\\end{theorem}\n\n\\begin{proof}\nEvery node $v \\in V$ generates its output  at round~$\\mathsf{id}(v)$ according to its current knowledge about its neighborhood, and sends this output to all its neighbors. \n\\qed\n\\end{proof}\n\n\\section{Limits of CONGEST Algorithms with Polynomially Many Rounds}\n\nGiven a graph \\(G = (V,E)\\) such that \\(V\\) is partitioned in two sets \\(V_A, V_B\\), the set of edges with one endpoint in \\(V_A\\) and the other in \\(V_B\\) is called the \\emph{cut}. We denote by \\(e(V_A,V_B)\\) the number of edges in the cut, and by \\(n(V_A,V_B)\\) the number of nodes incident to an edge of the cut.  Consider the situation where there are two players, namely Alice and Bob. We say that a player controls a node \\(v\\) if it knows all its incident edges and its input. For a CONGEST algorithm \\(\\mathcal{A}\\), we denote \\(\\mathcal{A}(\\mathcal{I})\\) the output of \\(\\mathcal{A}\\) on input \\(\\mathcal{I} = (G,\\mathsf{id}, x)\\). We denote \\(R_\\mathcal{A}(n)\\) the round complexity of \\(\\mathcal{A}\\) on inputs of size \\(n\\).\n\n\\begin{lemma}[Simulation lemma]\\label{lem:simu}\n    Let  \\(\\mathcal{A}\\) be an algorithm in the CONGEST model, let \\(\\mathcal{I} = (G,\\mathsf{id},x)\\) be an input for $\\mathcal{A}$, and let \\(V_A, V_B\\) be a partition of \\(V(G)\\).  Suppose that Alice controls all the nodes in \\(V_A\\), and Bob controls all the nodes in \\(V_B\\).  Then, there exists a communication protocol \\(\\mathcal{P}\\) between Alice and Bob with at most \\(2\\cdot\\min(n(V_A,V_B)\\cdot \\mathsf{nact}(\\mathcal{A}), e(V_A,V_B)\\cdot \\mathsf{eact}(\\mathcal{A}))\\) rounds and using total communication \\(\\mathcal{O}(\\min(n(V_A,V_B)\\cdot \\mathsf{nact}(\\mathcal{A}), e(V_A,V_B)\\cdot \\mathsf{eact}(\\mathcal{A}))\\cdot \\log n \\cdot \\log(R_{\\mathcal{A}}(n))\\), such that each player computes the value of \\(\\mathcal{A}(\\mathcal{I})\\) at all nodes he or she controls. \n\\end{lemma}\n\n\\begin{proof}\n\n\nIn protocol \\(\\mathcal{P}\\), Alice and Bob simulate the rounds of algorithm \\(\\mathcal{A}\\)  in all the nodes they control. The simulation run in phases. Each phase is used to simulate up to a certain number of rounds \\(t\\) of algorithm \\(\\mathcal{A}\\), and takes two rounds of protocol \\(\\mathcal{P}\\) (one round for Alice, and one round for Bob). By simulating \\(\\mathcal{A}\\) up to \\(t\\) rounds, we mean that Alice and Bob know all the states of all the nodes they control, on every round up to round \\(t\\). \n\nIn the first phase, players start simulating \\(\\mathcal{A}\\) from the initial state. Let us suppose that both Alice and Bob have already executed \\(p\\geq 0\\) phases, meaning that they had correctly simulated \\(\\mathcal{A}\\) up to round \\(t = t(p)\\geq 0\\). Let us explain phase \\(p+1\\) (see also Figure~\\ref{fig:simulationphase}). \n\n\n\\begin{figure}[!h]\n    \\centering\n    \\input{Figure3.tex}\n    \\caption{Illustration of one phase of the simulation protocol. Assuming that the players agree on the simulation of  algorithm \\(\\mathcal{A}\\) up to round \\(t\\), each player runs an oblivious simulation at the nodes they control. In the example of the figure, the next message corresponds to a node controlled by Bob, who sends a message to a node in \\(V_A\\) at round \\(r_b\\). The oblivious simulation of Alice is not aware of this message, and incorrectly considers that a message is sent from \\(V_A\\) to \\(V_B\\) at round \\(r_a > r_b\\). Using the communication rounds in this phase, the players agree that the message of Bob was correct. Thus  the simulation is correct up to round \\(r_b\\), for both players.   }\n    \\label{fig:simulationphase}\n\\end{figure}\n\n\nStarting from round \\(t\\), Alice runs an \\emph{oblivious simulation} of algorithm \\(\\mathcal{A}\\) over all nodes that she controls. By oblivious, we mean that Alice assumes that no node of \\(V_B\\) communicates a message to a node in \\(V_A\\) in any round at least \\(t\\). The oblivious simulation of Alice stops in one of the following two possible scenarios:\n\n\\begin{itemize}\n\\item[(1)] All nodes that she controls either terminate or enter into a passive state that quits only on an incoming message from \\(V_B\\). \n\\item[(2)] The simulation reaches a round \\(r_a\\) where a message is sent from a node in \\(V_A\\) to a node in \\(V_B\\).\n\\end{itemize}\n\nAt the same time, Bob runs and oblivious simulation of \\(\\mathcal{A}\\) starting from round \\(t\\) (i.e. assuming that no node of \\(V_A\\) sends a message to a node in \\(V_B\\) in any round at least \\(t\\)). The oblivious simulation of Bob stops in one of the same two  scenarios analogous to the ones  above. In this case, we call \\(r_b\\) the round reached by Bob in his version of scenario~(2). \n\nAt the beginning of a phase, it is the turn of Alice to speak.  Once the oblivious simulation of Alice stops, she is ready to send a message to Bob. If the simulation stops in the  scenario (1), Alice sends a message \"\\emph{scenario 1}\" to Bob. Otherwise, Alice sends \\(r_a\\) together with all the messages sent from nodes in \\(V_A\\) to nodes in \\(V_B\\) at round \\(r_a\\), to Bob. When Bob receives the message from Alice, one of the following situations holds:\\\\\n\n\\noindent{Case 1:} the oblivious simulation of both Alice and Bob stopped in the first scenario. In this case, since \\(\\mathcal{A}\\) is correct, there are no deadlocks. Therefore, all vertices of \\(G\\) reached a terminal state, meaning that the oblivious simulation of both players was in fact a real simulation of \\(\\mathcal{A}\\), and the obtained states are the output states. Therefore, Bob sends a message to Alice indicating that the simulation is finished, and indeed Alice and Bob have correctly computed the output of \\(\\mathcal{A}\\) for all the nodes they control. \\\\\n\n\\noindent{Case 2:} the oblivious simulation of Alice stopped in scenario (1), and the one of Bob stopped in the scenario (2). In this case, Bob infers that his oblivious simulation was correct. He sends  \\(r_b\\) and all the messages communicated in round \\(r_b\\) through the cut to Alice. When Alice receives the message of Bob, she updates the state of the nodes she controls up to round \\(r_b\\). It follows that both players have correctly simulated algorithm \\(\\mathcal{A}\\) up to round \\(r_b > t\\).\\\\ \n\n\\noindent{Case 3:} the oblivious simulation of Alice stopped in scenario (2), and the one of Bob stopped in scenario (1). In this case, Bob infres that the simulation of Alice was correct up to round \\(r_a\\). He sends a message to Alice indicating that she has correctly simulated \\(\\mathcal{A}\\) up to round \\(r_a\\), and he updates the states of all the nodes he controls up to round \\(r_a\\). It follows that both players have correctly simulated  \\(\\mathcal{A}\\) up to round \\(r_a > t\\). \\\\\n\n\\noindent{Case 4:} the oblivious simulation of both players stopped in scenario (2), and \\(r_a > r_b\\). Bob infers that his oblivious simulation was correct up to \\(r_b\\), and that the one of Alice was not correct after round \\(r_b\\). Then, the players act in the same way as   described in Case 2. Thus, both players have correctly simulated \\(\\mathcal{A}\\) up to round~\\(r_b\\).\\\\\n\n\\noindent{Case 5:} the oblivious simulation of both players stopped in scenario (2), and \\(r_b > r_a\\). Bob infers that his oblivious simulation was incorrect after round \\(r_a\\), and that the one of Alice was correct up to round \\(r_a\\). Then, the players act in the same way as described in Case 3. Thus, both players have correctly simulated \\(\\mathcal{A}\\) up to round~\\(r_a\\). \\\\\n\n\\noindent{Case 6:} the oblivious simulation of both players stopped in scenario (2), and \\(r_b = r_a\\). Bob assumes that both oblivious simulations were correct. He sends \\(r_b\\) together with all the messages communicated from his nodes at round \\(r_b\\) through the cut. Then, he updates the states of all the nodes he controls up to round \\(r_b\\). When Alice receives the message from Bob, she updates the states of the nodes she controls up to round \\(r_b\\).\nIt follows that both players have correctly simulated  \\(\\mathcal{A}\\) up to round \\(r_b > t\\). \\\\\n\nObserve that, except when the algorithm terminates, on each phase of the protocol, at least one node controlled by Alice or Bob is activated. Since the number of rounds of \\(\\mathcal{P}\\) is twice the number of phases, we deduce that the total number of rounds is at most\n\\[ 2\\cdot \\min(n(V_A,V_B)\\cdot\\mathsf{nact}(\\mathcal{A}), e(V_A,V_B)\\cdot \\mathsf{eact}(\\mathcal{A})).  \\]\nMoreover, on each round of $\\mathcal{P}$,  the players communicate \\(O(\\log(R_{\\mathcal{A}}(n)) \\cdot \\log n \\cdot e(V_A,V_B))\\) bits. As a consequence, the total communication cost of $\\mathcal{P}$ is \\[O(\\log(R_{\\mathcal{A}}(n)) \\cdot \\log n \\cdot e(V_A,V_B)) \\cdot \\min(n(V_A,V_B)\\cdot \\mathsf{nact}(\\mathcal{A}), e(V_A,V_B)\\cdot \\mathsf{eact}(\\mathcal{A})) ),\\]\nwhich completes the proof. \\qed \\end{proof}\n\nWe use the simulation lemma to show that there are problems that cannot be solved by a frugal algorithm in a polynomial number of rounds. In problem \\textsc{C4-free\\-ness}{}, all nodes of the input graph $G$ must accept if $G$ has no cycle of 4 vertices, and at least one node must reject if such a cycle exists. Observe that this problem is expressible in first-order logic, in particular it has en edge-frugal algorithm with a polynomial number of rounds in graphs of bounded degree~\\cite{GrumbachW09}. We show that, in graphs of unbounded degree, this does not hold anymore.\nWe shall also consider problem \\textsc{Symmetry}{}, where the input is a graph $G$ with $2n$ nodes indexed from $1$ to $2n$, and with a unique edge  $\\{1,n+1\\}$ between $G_A = G[\\{1,\\dots,n\\}]$ and $G_B = G[\\{n+1,\\dots,2n\\}]$. Our lower bounds holds even if  every node is identified by its index. All nodes must output \\emph{accept} if the function $f:\\{1,\\dots,n\\} \\to \\{n+1,\\dots,2n\\}$ defined by $f(x)=x+n$ is an isomorphism from $G_A$ to $G_B$, otherwise at least one node must output \\emph{reject}.\n\n The proof of the following theorem is based on classic reductions from communication complexity problems \\textsc{Equality} and \\textsc{Set Disjointness} (see, e.g., \\cite{KushilevitzNisan}), combined with Lemma~\\ref{lem:simu}.\n\n \n\\begin{theorem}\\label{th:limits}\nAny CONGEST algorithm solving \\textsc{Symmetry}{} (resp., \\textsc{C4-free\\-ness}{}) in polynomially many rounds has node-activation and edge-activation at least $\\Omega\\left(\\frac{n^2}{\\log^2 n}\\right)$ (resp., $\\Omega\\left(\\frac{\\sqrt{n}}{\\log^2 n}\\right)$).\n\\end{theorem}\n\n\n\\begin{proof}\nIn problem \\textsc{Equality}, two players Alice and Bob have a boolean vector of size $k$, $x_A$ for Alice and $x_B$ for Bob. Their goal is to answer \\emph{true} if $x_A = x_B$,  and \\emph{false} otherwise. The communication complexity of this problem is known to be~$\\Theta(k)$~\\cite{KushilevitzNisan}.\nLet $k = n^2$. We can interpret $x_A$ and $x_B$ as the adjacency matrix of two graphs $G_A$ and $G_B$ in an instance of $\\textsc{Symmetry}$. It is a mere technicality to \"shift\" $G_B$ as if its vertices were indexed from $1$ to~$n$, such that \\textsc{Symmetry}{} is true for $G$ iff $x_A = x_B$. Moreover, Alice can construct $G_A$ from its input $x_A$, and Bob can construct $G_B$ from $x_B$. Both can simulate the unique edge joining the two graphs in $G$. Therefore, by Lemma~\\ref{lem:simu} applied to $G$, if Alice controls the vertices of $G_A$, and Bob controls the vertices of $G_B$, then any CONGEST algorithm $\\mathcal{A}$ solving \\textsc{Symmetry}{} in polynomially many rounds yields a two-party protocol for \\textsc{Equality} on $n^2$ bits. Since graphs $G_A$ and $G_B$ are linked by a unique edge, the total communication of the protocol is $O(\\mathsf{eact}(\\mathcal{A}) \\cdot \\log^2 n)$ and  $O(\\mathsf{nact}(\\mathcal{A})\\cdot \\log^2 n)$. The result follows.\n\n\\medskip \n\nIn  \\textsc{Set Disjointness}, each of the two players Alice and Bob has a Boolean vector of size $k$, $x_A$ for Alice, and $x_B$ for Bob. Their goal is to answer \\emph{true} if there is no index $i \\in [k]$ such that both $x_A[i]$ and $x_B[i]$ are true (in which case, $x_A$ and $x_B$ are disjoint),  and \\emph{false} otherwise. The communication complexity of this problem is known to be~$\\Theta(k)$~\\cite{KushilevitzNisan}.\nWe use the technique in~\\cite{DruckerKO14} to construct an  instance $G$ for $C_4$ freeness, with a small cut, from two Boolean vectors $x_A, x_B$ of size $k = \\Theta(n^{3\/2})$. Consider a $C_4$-free $n$-vertex graph~$H$ with a maximum number of edges. Such a graph has $k = \\Theta(n^{3\/2})$ edges, as recalled in~\\cite{DruckerKO14}. We can consider the edges $E(H)$ as indexed from $1$ to~$k$, and $V(H)$ as $[n]$. Let now $x_A$ and $x_B$ be two Boolean vectors of size~$k$. These vectors can be interpreted as edge subsets $E(x_A)$ and $E(x_B)$ of~$H$, in the sense that the edge indexed $i$ in $E(H)$ appears in $E(x_A)$ (resp. $E(x_B)$) iff $x_A[i]$ (resp. $x_B[i]$) is true. Graph $G$ is constructed to have $2n$ vertices, formed by two sub-graphs $G_A = G[\\{1,\\dots, n\\}]$ and $G_B = G[\\{n+1,\\dots, 2n\\}]$. \nThe edges of $E(G_A)$ are exactly the ones of $E(x_A)$. Similarly, the edges of $E(G_B)$ correspond to $E(x_A)$, modulo the fact that the vertex indexes are shifted by~$n$, i.e., for each edge $\\{u,v\\} \\in E(x_B)$, we add edge  $\\{u+n,v+n\\}$ to~$G_B$. Moreover we add a perfect matching to $G$, between $V(G_A)$ and $V(G_B)$, by adding all edges $\\{i,i+n\\}$, for all $i \\in [n]$. Note that $G$ is $C_4$-free if and only if vectors $x_A$ and $x_B$ are disjoint. Indeed, since $G_A, G_B$ are isomorphic to sub-graphs of $H$, they are $C_4$-free. Thus any $C_4$ of $G$ must contain two vertices in $G_A$ and two in $G_B$, in which case the corresponding edges in $G_A$ and $G_B$ designate the same bit of $x_A$ and $x_B$ respectively.\nMoreover Alice and Bob can construct $G_A$ and $G_B$, as well as the edges in the matching, from their respective inputs $x_A$ and~$x_B$.  Therefore, thanks to Lemma~\\ref{lem:simu}, a CONGEST algorithm $\\mathcal{A}$ for \\textsc{C4-free\\-ness}{} running in a polynomial number of rounds can be used to design a protocol $\\mathcal{P}$ solving \\textsc{Set Disjointness} on $k = \\Theta(n^{3\/2})$ bits, where Alice controls $V(G_A)$ and Bob controls $V(G_B)$. The communication complexity of the protocol is $O(\\mathsf{eact}(\\mathcal{A})\\cdot n\\cdot \\log^2 n)$, and $O(\\mathsf{nact}(\\mathcal{A}) \\cdot n\\cdot \\log^2 n)$, since the cut between $G_A$ and $G_B$ is a matching. The result follows. \n\\qed    \n\\end{proof}\n\n\n\\section{Node versus Edge Activation}\n\nIn this section we exhibit a problem that admits an edge-frugal CONGEST algorithm running in a polynomial number of rounds,  for which any algorithm running in a polynomial number of rounds has large node-activation complexity.\n\nWe proceed by reduction from a two-party communication complexity problem. However, unlike the previous section, we are now also interested in the number of rounds of the two-party protocols. We consider protocols in which the two players Alice and Bob do not communicate simultaneously. For such a protocol~$\\mathcal{P}$, a \\emph{round} is defined as a maximal contiguous sequence of messages emitted by a same player. We denote by $R(\\mathcal{P})$ the number of rounds of~$\\mathcal{P}$. \n\nLet \\(G\\) be a graph, and \\(S\\) be a subset of nodes of~$G$. We denote by \\(\\partial S\\) the number of vertices in \\(S\\) with a neighbor in \\(V\\setminus S\\). \n\n\\begin{lemma}[Round-Efficient Simulation lemma]\\label{lem:simu2}\n    Let  \\(\\mathcal{A}\\) be an algorithm in the CONGEST model, let \\(\\mathcal{I} = (G,\\mathsf{id},x)\\) be an input for~$\\mathcal{A}$, and let \\(V_A, V_B\\) be a partition of~\\(V(G)\\).  Let us assume that Alice controls all the nodes in~\\(V_A\\), and Bob controls all the nodes in \\(V_B\\), and both players know the value of \\(\\mathsf{nact}(\\mathcal{A})\\).  Then, there exists a communication protocol \\(\\mathcal{P}\\) between Alice and Bob such that, in at most \\(\\min(\\partial V_A, \\partial V_B)\\cdot \\mathsf{nact}(\\mathcal{A})\\) rounds, and using total communication \\(O(((\\partial(V_A) + \\partial(V_B)) \\cdot \\mathsf{nact}(\\mathcal{A})))^2 \\cdot \\log n \\cdot \\log R_{\\mathcal{A}}(n))  \\) bits, each player computes the value of \\(\\mathcal{A}(\\mathcal{I})\\) at all the nodes he or she controls. \n\\end{lemma}\n\n\\begin{proof}\nIn protocol \\(\\mathcal{P}\\), Alice and Bob simulate the rounds of algorithm \\(\\mathcal{A}\\)  at all the nodes each player controls. Without loss of generality, we assume that algorithm \\(\\mathcal{A}\\) satisfies that the nodes send messages at different rounds, by merely multiplying by \\(N\\) the number of rounds. \n\nInitially,  Alice runs a oblivious simulation of \\(\\mathcal{A}\\) that stops when every node in \\(V_A\\) either has terminated, or entered into the passive state that it may leave only after having received a message from a node in~\\(V_B\\) (this corresponds to what we call the first scenario in the proof of Lemma~\\ref{lem:simu}). Then, Alice sends to Bob the integer \\(t_1 = 0\\), and the set \\(M^1_A\\) of all messages sent from nodes in \\(V_A\\) to nodes in \\(V_B\\) in the communication rounds that she simulated, together with their corresponding timestamps. If the number of messages communicated by Alice exceeds \\(\\mathsf{nact}(\\mathcal{A})\\cdot \\partial A\\), we trim the list up to this threshold.\n\nLet us suppose that the protocol \\(\\mathcal{P}\\) has run for \\(p\\) rounds, and let us assume that  it is the turn of Bob to speak at round \\(p+1\\) --- the case where Alice speaks at round \\(p+1\\) can be treated in the same way. Moreover, we assume that \\(\\mathcal{P}\\) satisfies the following two conditions:\n\\begin{enumerate}\n\\item At round \\(p\\), Alice sents an integer \\(t_p\\geq 0\\), and a list of timestamped messages \\(M^p_A\\) corresponding to messages sent from nodes in \\(V_A\\) to nodes in \\(V_B\\) in an oblivious simulation of \\(\\mathcal{A}\\) starting from a round \\(t_p\\). \n\\item Bob had correctly simulated \\(\\mathcal{A}\\) at all the nodes he controls, up to round \\(t_p\\). \n\\end{enumerate}\n\n\n\n\\begin{figure}[!h]\n    \\centering\n    \\input{Figure4.tex}\n    \\caption{Illustration of the round-efficient simulation protocol for algorithm \\(\\mathcal{A}\\). After round \\(p\\), Alice has correctly simulated the algorithm up to round \\(t_p\\). It is the turn of Bob to speak in round \\(p+1\\). In round \\(p\\), Alice sent to Bob the set of messages \\(M^p_A\\), obtained from an oblivious simulation of \\(\\mathcal{A}\\) starting from \\(t_p\\). Only the first three messages are correct, since at round \\(t_{p+1}\\) Bob communicates a message to Alice. Then, Bob runs an oblivious simulation of \\(\\mathcal{A}\\) starting from \\(t_{p+1}\\), and communicates all the messages sent from nodes \\(V_B\\) to nodes in \\(V_A\\).  In this case the two first messages are correct. }\n    \\label{fig:efficientsimulationround}\n\\end{figure}\n\n\nWe now describe round \\(p+1\\) (see also Figure~\\ref{fig:efficientsimulationround}). \nBob initiates a simulation of \\(\\mathcal{A}\\) at all the nodes he controls. However, this simulation is \\emph{not} oblivious. Specifically, Bob simulates \\(\\mathcal{A}\\) from round \\(t_p\\) taking into account all the messages sent from nodes in \\(V_A\\) to nodes in \\(V_B\\), as listed in the messages~\\(M^{p}_A\\). The simulation stops when Bob reaches a round \\(t_{p+1}>t_p\\) at which a node in \\(V_B\\) sends a message to a node in \\(V_A\\). Observe that, up to round \\(t_{p+1}\\), the oblivious simulation of Alice was correct. At this point, Bob initiates an oblivious simulation of \\(\\mathcal{A}\\) at all the nodes he controls, starting from \\(t_{p+1}\\). Finally, Bob sends to Alice \\(t_{p+1}\\), and the list \\(M^{p+1}_B\\) of all timestamped messages sent from nodes in \\(V_B\\) to nodes in \\(V_A\\) resulting from the oblivious simulation of the nodes he controls during rounds at least \\(t_{p+1}\\). Using this information, Alice infers that her simulation was correct up to round \\(t_{p+1}\\), and she starts the next round for protocol~$\\mathcal{P}$. \n\nThe simulation carries on until one of the two players runs an oblivious simulation in which all the nodes he or she controls terminate, and no messages were sent through the cut in at any intermediate round. In this case, this player sends a message \"\\emph{finish}\" to the other player, and both infer that their current simulations are correct. As  a consequence, each player has correctly computed the output of \\(\\mathcal{A}\\) at all the nodes he or she controls.\n\nAt every communication round during which Alice speaks, at least one vertex of \\(V_A\\) which has a neighbor in \\(V_B\\) is activated. Therefore, the number of rounds of Alice is at most \\(\\partial V_A \\cdot \\mathsf{nact}(\\mathcal{A})\\). By the same argument, we have that the number of rounds of Bob is at most \\(\\partial V_B \\cdot \\mathsf{nact}(\\mathcal{A})\\). It follows that \n\\[\nR(\\mathcal{P}) = \\min(\\partial V_A,\\partial V_B) \\cdot \\mathsf{nact}(\\mathcal{A}).\n\\]\nAt each communication round, Alice sends at most \\(\\partial(V_A) \\cdot \\mathsf{nact}(\\mathcal{A})\\) timestamped messages, which can be encoded using \\(O(\\partial(V_A) \\cdot \\mathsf{nact}(\\mathcal{A})) \\cdot \\log n \\cdot \\log R_{\\mathcal{A}}(n)) \\) bits. Similarly, Bob sends \\(O(\\partial(V_B) \\cdot \\mathsf{nact}(\\mathcal{A})) \\cdot \\log n \\cdot \\log R_{\\mathcal{A}}(n)) \\) bits. It follows that\n\\[\nC(\\mathcal{P}) = O(((\\partial(V_A) + \\partial(V_B)) \\cdot \\mathsf{nact}(\\mathcal{A})))^2 \\cdot \\log n \\cdot \\log R_{\\mathcal{A}}(n)),\n\\]\nwhich completes the proof. \n\\qed\n\\end{proof}\n\nIn order to separate the node-activation complexity from the edge-activation complexity, we consider a problem called \\textsc{Depth First Pointer Chasing}, and we show that this problem can be solved by an edge-frugal CONGEST algorithm running in $O(\\mbox{poly}(n))$ rounds, whereas the node-activation complexity of any algorithm running in $O(\\mbox{poly}(n))$ rounds for this problem is $\\Omega(\\Delta)$, for any $\\Delta \\in O(\\frac{\\sqrt{n}}{\\log n})$. The lower bound is proved thanks to the Round-Efficient Simulation Lemma (Lemma~\\ref{lem:simu}), by reduction from the two-party communication complexity problem \\textsc{Pointer Chasing}{}, for which too few rounds imply large communication complexity~\\cite{NisanW93}.\n\nIn the \\textsc{Depth First Pointer Chasing}, each node $v$ of the graph is given as input its index $\\mbox{DFS}(v)\\in [n]$ in a depth-first search ordering (as usual we denote $[n]=\\{1,\\dots,n\\}$). Moreover the vertex indexed~$i$ is given a function $f_i:[n] \\to [n]$, and the root (i.e., the node indexed~1) is given a value $x \\in [n]$ as part of its input. The goal is to compute the value of $f_n \\circ f_{n-1} \\circ \\dots \\circ f_1(x)$ at the root. \n\n\\begin{lemma}\\label{lem:algoforDFPC}\nThere exists an edge-frugal CONGEST algorithm for problem \\textsc{Depth First Pointer Chasing}, with polynomial number of rounds.\n\\end{lemma}\n\n\n\\begin{proof}\nThe lemma is established using an algorithm that essentially traverses the DFS tree encoded by the indices of the nodes, and performs the due partial computation of the function at every node, that is, the node with index~$i$ computes $f_i \\circ f_{i-1} \\dots f_1 (x)$, and forwards the result to the node with index~$i+1$.\n\nAt round 1, each node $v$ transmits its depth-first search index $\\mbox{DFS}(v)$ to its neighbors. Therefore, after this round, every node knows its parent, and its children in the DFS tree. Then the algorithm merely forwards messages of type $m(i) = f_i \\circ f_{i-1} \\dots f_1 (x)$, corresponding to iterated computations for increasing values~$i$, along the DFS tree, using the DFS ordering. That is, for any node $v$, let $\\mbox{MaxDFS}(v)$ denote the maximum DFS index appearing in the subtree of the DFS tree rooted at $v$. We will not explicitly compute this quantity but it will ease the notations. At some round, vertex $v$ of DFS index $i$ will receive a message $m(i-1)$ from its parent (of index $i-1$). Then node $v$ will be in charge of computing message $m(\\mbox{MaxDFS}(v))$, by ``calling'' its children in the tree, and sending this message back to its parent. In this process, each edge in the subtree rooted at $v$ is activated twice.\n\nThe vertex of DFS index~1 initiates the process at round~2, sending $f_1(x)$ to its child of DFS index~$2$. Any other node~$v$ waits until it receives a message from its parent, at a round that we denote~$r(v)$. This  message is precisely $m(i-1) = f_{i-1} \\circ f_{i-2} \\dots f_1 (x)$, for $i = \\mbox{DFS}(v)$. Then $v$ computes message \n$m(i) = f_{i} \\circ f_{i-1} \\dots f_1 (x)$ using its local function $f_i$. If it has no children, then it sends  this message $m(i)$ to its parent at round $r(v)+1$. Assume now that $v$ has $j$ children in the DFS tree, denoted  $u_1,u_2,\\dots,u_j$, sorted by increasing DFS index. Observe that, by definition of DFS trees, $\\mbox{DFS}(u_k) = \\mbox{MaxDFS}(u_{k-1})+1$ for each $k \\in \\{2,\\dots,j\\}$. \nNode~$v$ will be activated $j$ times, once for each edge $\\{v,u_k\\}$, $1 \\leq k \\leq j$, as follows. \nAt round $r(v)+1$ (right after receiving the message from its parent), $v$~sends message $m(i)$ to its child $u_1$, then it awaits until round $r^1(v)$ when it gets back a message from $u_1$. \n\nThe process is repeated for $k=2,\\dots,j$: at round $r^{k-1}(v)+1$, node $v$ sends the message $m(\\mbox{DFS}(u_{k})-1)$ received from $u_{k-1}$ to $u_k$, and waits until it gets back  a message from $u_k$, at round $r^k(v)$. Note that if $k<j$ then this message is $m(\\mbox{DFS}(u_{k+1})-1)$, and if $k=j$ then this message is $m(\\mbox{MaxDFS}(v))$. At round $r^j(v)+1$, after having received messages from all its children, $v$~backtracks message $m(\\mbox{MaxDFS}(v))$ to its parent. If $v$ is the root, then the process stops.\n\nThe process terminates in $O(n)$ rounds, and, except for the first round,  every edge of the DFS tree is activated twice: first, going downwards, from the root towards the leaves, and, second, going upwards. At the end, the root obtains the requested message $m(n) = f_{n} \\circ f_{n-1} \\dots f_1 (x)$.\n\\qed\n\\end{proof}\n\n\\medbreak\n\nLet us recall the \\textsc{Pointer Chasing}{} problem as defined in~\\cite{NisanW93}.  \nAlice is given a function $f_A : [n] \\to [n]$, and a number $x_0 \\in [n]$. Bob is given function $f_B : [n] \\to [n]$. Both players have a parameter $k \\in [n]$. Note that the size of the input given to each player is $\\Theta(n \\log n)$ bits.\nThe goal is to compute $(f_A \\circ f_B)^k(x_0)$, i.e., $k$ successive iterations of $f_A \\circ f_B$ applied to~$x_0$. We give a slightly simplified version of the result in~\\cite{NisanW93}.\n\n\\begin{lemma}[Nissan and Wigderson \\cite{NisanW93}]\n\\label{lem:PointerChasing}\nAny two-party protocol for \\textsc{Pointer Chasing}{} using less than $2k$ rounds has communication complexity $\\Omega(n - k \\log n)$.\n\\end{lemma}\n\nWe have now all ingredients for proving the main result of this section. \n\n\\begin{theorem}\\label{thm:NodeEdgeSep}\nFor every $\\Delta \\in O\\left(\\frac{n^{1\/4}}{\\sqrt{\\log n}}\\right)$, every CONGEST algorithm solving  \\textsc{Depth First Pointer Chasing}{} in graphs of maximum degree $\\Delta$ with polynomialy many rounds has node-activation complexity~$\\Omega(\\Delta)$.\n\\end{theorem}\n\n\\begin{proof}\nLet $k$ be the parameter of \\textsc{Pointer Chasing}{} that will be fixed later. The lower bound is established for this specific parameter~$k$. Let us consider an arbitrary instance of \\textsc{Pointer Chasing} $f_A, f_B:[n] \\to [n]$, and $x_0 \\in [n]$, with parameter~$k$. \nWe reduce that instance to a particular instance of \\textsc{Depth First Pointer Chasing} (see Fig.~\\ref{fig:DFPC}). \n\n\\begin{figure}[!h]\n    \\centering\n    \\input{Figure2.tex}\n    \\caption{Reduction from \\textsc{Pointer Chasing} to \\textsc{Depth First Pointer Chasing}.}\n    \\label{fig:DFPC}\n\\end{figure}\n\nThe graph is a tree $T$ on $n$ vertices, composed of a path $(v_1,\\dots,v_{n-2k})$, and $2k$ leaves $v_{n-2k+1}, \\dots, v_n$, all adjacent to $v_{n-2k}$. Node $v_1$ is called the root, and node $v_{n-2k}$ is said central. Note that the ordering obtained by taking $\\mbox{DFS}(v_i)=i$ is a depth-first search of~$T$, rooted at $v_1$. \nThe root $v_1$ is given value $x_0$ as input. If $i \\leq n - 2k$, then function~$f_i$ is merely the identity function~$f$ (i.e., $f(x)=x$ for all~$x$). \nFor every $j \\in [k]$, let $a_j  = v_{n-k+2j-1}$, and $b_j = v_{n-k+2j}$. All nodes $b_j$ get as input the function~$f_B$, and all nodes $a_j$ get the function~$f_A$. Observe that the output of \\textsc{Depth First Pointer Chasing} on this instance is precisely the same as the output of the initial instance of \\textsc{Pointer Chasing}. Indeed,  $f_{n-2k} \\circ f_{n-2k - 1} \\circ \\cdots \\circ f_1$ is the identity function, and  the sequence $f_n \\circ f_{n-1} \\circ \\cdots \\circ f_{n-2k+2} \\circ f_{n-2k+1}$ alternates nodes of ``type'' $a_j$ with nodes of ``type'' $b_j$, for decreasing values of $j \\in [k]$, and thus corresponds to $f_A \\circ f_B \\circ \\cdots \\circ f_A \\circ f_B$, where the pair $f_A \\circ f_B$ is repeated $k$ times, exactly as in problem \\textsc{Pointer Chasing}.\n\nWe can now apply Round-Efficient Simulation Lemma. Let \nAlice control all vertices $a_j$, for all $j \\in [k]$, and vertices $v_1,\\dots,v_{n-2k}$. Let Bob control vertices $b_j$, for all $j \\in [k]$. See Fig.~\\ref{fig:DFPC}. Note that Alice and Bob can construct the subgraph that they control, based only on their input in the considered \\textsc{Pointer Chasing}{} instance, and they both now value $k$. \n\n\\begin{claim}\nIf there exists a CONGEST algorithm $\\mathcal{A}$ for \\textsc{Depth First Pointer Chasing}{} on $n$-node graphs performing in $R_{\\mathcal{A}}$ rounds with node-activation smaller than~$2k$, then \\textsc{Pointer Chasing} can be solved by a two-party protocol $\\mathcal{P}$ in less than $2k$ rounds, with communication complexity $O(k^4 \\log n \\log R_{\\mathcal{A}})$ bits.\n\\end{claim}\n\nThe claim directly follows from Lemma~\\ref{lem:simu2}. Indeed, by construction, $\\partial V_A = 1$ and $\\partial V_B = k$. Since we assumed $\\mathsf{nact}(\\mathcal{A}) < 2k$, the two-way protocol $\\mathcal{P}$ provided by Lemma~\\ref{lem:simu2} solves the  \\textsc{Pointer Chasing}{} instance in less than $2k$ rounds, and uses $O(k^4 \\log n \\log R_{\\mathcal{A}})$ bits.  \n\nBy Lemma~\\ref{lem:PointerChasing}, we must have $k^4 \\log n \\log R_{\\mathcal{A}} \\in \\Omega(n - k \\log n)$. Therefore, if our CONGEST algorithm $\\mathcal{A}$ has polynomially many rounds, we must have  $k \\in \\Omega\\left(\\frac{n^{1\/4}}{\\sqrt{\\log n}}\\right)$. Since our graph has maximum degree $\\Delta = 2k+1$, the conclusion follows. \n\\qed    \n\\end{proof}\n\n\\section{Conclusion}\n\nIn this paper, we have mostly focused on the round complexity of (deterministic) \\emph{frugal} algorithms solving general graph problems in the LOCAL or CONGEST model. It might be interesting to consider specific classical problems. As far as ``local problems'' are concerned, i.e., for locally checkable labeling (LCL) problems, we have shown that MIS and $(\\Delta+1)$-coloring admit frugal algorithms with polynomial round complexities. It is easy to see, using the same arguments, that problems such as maximal matching share the same properties. It is however not clear that the same holds for $(2\\Delta-1)$-edge coloring. \n\n\\begin{openprob}\nIs there a (node or edge) frugal algorithm solving $(2\\Delta-1)$-edge-coloring with round complexity $O(\\mbox{\\rm poly}(n))$ in the CONGEST model? \n\\end{openprob}\n\nIn fact, it would be desirable to design frugal algorithms with sub-polynomial round complexities for LCL problems in general. In particular: \n\n\\begin{openprob}\nIs there a (node or edge) frugal algorithm solving {\\sc mis} or $(\\Delta+1)$-coloring with round complexity $O(\\mbox{\\rm polylog}(n))$ in the LOCAL model? \n\\end{openprob}\n\nThe same type of questions can be asked for global problems. In particular, it is known that MST has no ``awake frugal'' algorithms, as MST has awake complexity $\\Omega(\\log n)$, even in the LOCAL model. In contrast, frugal algorithms for MST do exist as far as node-activation complexity is concerned. The issue is about the round complexities of such algorithms. \n\n\\begin{openprob}\nIs there a (node or edge) frugal algorithm solving {\\sc mst} with round complexity $O(\\mbox{\\rm poly}(n))$ in the CONGEST model? \n\\end{openprob}\n\nAnother intriguing global problem is depth-first search ({\\sc dfs}), say starting from an identified node. This can be performed by an edge-frugal algorithm performing in a linear number of rounds in CONGEST. However, it is not clear whether the same can be achieved by a node-frugal algorithm. \n\n\\begin{openprob}\nIs there a node-frugal algorithm solving {\\sc dfs} with round complexity $O(\\mbox{\\rm poly}(n))$ in the CONGEST model? \n\\end{openprob}\n\nFinally, we have restricted our analysis to \\emph{deterministic} algorithms, and it might obviously be worth considering \\emph{randomized} frugal algorithms as well.  \n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Background}\nThe anomalous Hall effect~\\cite{Nagaosa2010} (AHE) refers to a nonzero transverse voltage in zero applied magnetic field. Its underlying physical mechanisms are an enduring topic of research, due both to the insight provided into the effects of electronic structure and impurity scattering on conduction and to the potential spintronic applications of materials hosting a large AHE. \nSpecifically, the AHE shares its microscopic origins with the spin Hall effect (SHE), which generates a pure spin current essential for semiconductor spintronics, including the spin-transfer torque manipulation of spin valves, leading to intense experimental interest in materials hosting large AHE as a gateway to large SHE.~\\cite{Sinova2015}\n\nThe measured Hall resistivity is the sum of two contributions: the ordinary Hall effect (OHE), a potential difference across the sample due to carrier deflection by the Lorentz force, and the anomalous Hall effect (AHE), which is proportional to the magnetization $M_z(H)$ and stems from a combination of asymmetric scattering and band structure effects unique to magnetically ordered materials.\\footnote{While the AHE was long understood to be present exclusively in ferromagnets, recent results indicate that certain noncollinear antiferromagnets also host a substantial AHE~[A. K. Nayak \\emph{et al.}, Sci. Adv. \\textbf{2}, e1501870 (2016)].}\nWe write this phenomenologically as,~\\cite{Pugh1930}\n\\begin{equation} \\label{eqHall}\n    \\rho_{xy}(H) = \\rho_{xy}^{\\mathrm{OH}}(H) + \\rho_{xy}^{\\mathrm{AH}}(H) = R_0 H + R_S M_z(H),\n\\end{equation}\nwhere $M_z(H)$ is the $z$-component of the magnetization measured with the field $H$ perpendicular to the plane of the sample, and $R_0$ and $R_S$ are the ordinary and anomalous Hall coefficients, respectively. $\\rho_{xy}^{\\mathrm{OH}}$ contributes a linear in $H$ ``background'' whose slope shares the sign of the carriers in the sample, while $\\rho_{xy}^{\\mathrm{AH}}$ adopts the shape of $M_z(H)$. \n\nAs understanding of the AHE deepened, analytical focus shifted from the anomalous Hall resistivity $\\rho_{xy}^{\\mathrm{AH}}$ to the more fundamental anomalous Hall conductivity (AHC) $\\sigma_{xy}^{\\mathrm{AH}}$, which when $\\rho_{xy}^{\\mathrm{AH}}$ is small relative to the longitudinal conductivity $\\rho_{xx}$ is given by $\\sigma_{xy}^{\\mathrm{AH}} \\approx \\rho_{xy}^{\\mathrm{AH}}\/\\rho_{xx}^2$. The AHC originates from a combination of three mechanisms: skew-scattering, side-jump scattering, and the intrinsic mechanism. \nSkew-scattering refers to asymmetric scattering of spin-polarized carriers by spin-orbit-coupled impurities;~\\cite{Smit1955} while side-jump scattering refers to the transverse velocity deflection caused by the opposite electric fields experienced by spin-polarized carriers upon approaching and leaving a spin-orbit-coupled impurity.~\\cite{Berger1970} \nThe intrinsic contribution is proportional to the integral over occupied states of the Berry curvature $\\Omega(\\vec{k})$, which can be written as a summation over eigenstates:~\\cite{Xiao2010} \n\\begin{equation} \\label{eq:BerryCurvature}\n    \\Omega(\\vec{k}) = i \\sum_{n \\neq n'}^{} \\frac{\\bra{n}\\grad_{\\vec{k}}\\hat{H}\\dyad{n'}\\grad_{\\vec{k}}\\hat{H}\\ket{n}} {(\\varepsilon_n - \\varepsilon_{n'})^2}.\n\\end{equation}\nEq.~\\ref{eq:BerryCurvature} emphasizes that this intrinsic contribution to the AHC depends only on the topology of the band structure and not on the details of scattering in the material.~\\cite{Jungwirth2002, Onoda2002} \n\nThe relative importance of these three mechanisms has long been disputed, but some clarity has come from a unified theory, originally proposed by Onoda~\\cite{Onoda2006} and elaborated upon by Liu,~\\cite{Liu2011} which separates systems into three regimes based on longitudinal conductivity $\\sigma_{xx} = \\rho_{xx}^{-1}$: a high conductivity ``clean'' limit where $\\sigma_{xx} \\gtrsim 10^6 \\,(\\mathrm{\\Omega cm})^{-1}$ and $\\sigma_{xy}^{\\mathrm{AH}} \\propto \\sigma_{xx}$, indicating that the AHC is dominated by skew-scattering; an intermediate ``good metal'' regime where $10^4 \\lesssim \\sigma_{xx} \\lesssim 10^6 \\,(\\mathrm{\\Omega\\; cm})^{-1}$ and $\\sigma_{xy}^{\\mathrm{AH}}$ is independent of $\\sigma_{xx}$, indicating that the intrinsic mechanism dominates the AHC; and a low conductivity regime where $\\sigma_{xx} \\lesssim 10^4 \\,(\\mathrm{\\Omega\\; cm})^{-1}$, which was less well understood. \nAt the time of Onoda's work, this last regime was described empirically as having $\\sigma_{xy}^{\\mathrm{AH}} \\sim \\sigma_{xx}^{\\gamma}$, with $\\gamma$ ranging between $1.6-1.8$; Onoda recovered the scaling $\\gamma = 1.6$ in the hopping regime, and Liu found $1.33 \\leq \\gamma \\leq 1.76$ by studying multiple hopping processes which typically occur deep in the low conductivity regime ($\\sigma_{xx} \\lesssim 10^2 \\,(\\mathrm{\\Omega\\; cm})^{-1}$), demonstrating that the microscopic origin of the AHE determines its scaling in this insulating regime. \n\nRecent experimental results from Karel \\textit{et al.}~\\cite{Karel2016} explored the high conductivity edge of the low conductivity regime, employing an empirical scaling argument based on the intrinsic AHC in \\textit{crystalline} Fe to suggest that the AHC in the low conductivity metal amorphous iron-silicon (\\textit{a}-Fe$_x$Si$_{1-x}$\\xspace) is dominated by the intrinsic mechanism. \nThis conclusion is intially surprising, because perturbative expressions such as Eq.~\\ref{eq:BerryCurvature} indicate that the intrinsic anomalous Hall effect is maximized when the spin-orbit interaction splits band dispersions to create anti-crossing points with a small gap near the Fermi level, which resonantly enhances the integral in question~\\cite{Onoda2006} by enhancing the momentum-space Berry curvature at these points. \nHowever, full-band calculations on crystalline materials have called into question assumptions based on perturbative treatments of anti-crossings near the Fermi level;~\\cite{Sahin2015} in addition, for an amorphous system, $\\vec{k}$ is no longer a good quantum number due to the absence of lattice periodicity. We will instead introduce the energy-resolved density of Berry curvature~\\cite{Sahin2015, Guo2008} for an amorphous system and show that it supplants a typical Brillouin-zone integral to allow a more general expression of the electronic origins of the intrinsic AHC.\nIn a quite distinct but relevant approach, Marrazzo and Resta~\\cite{Marrazzo2017a} have reformulated the expression for the intrinsic AHC in terms of real-space wavefunctions, enabling the calculation of this contribution in systems where $\\vec{k}$ is not a good quantum number, including structurally disordered systems such as \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace. \n\nHere we study amorphous iron-germanium (\\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace), which promises similarly tunable magnetization and resistivity to \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace, but accompanied by spin-orbit coupling (SOC) large enough to fundamentally alter both the electronic structure and the magnetic texture in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace by amplifying the antisymmetric Dzyaloshinskii-Moriya interaction (DMI).\nWhile recent investigation in the Fe-Ge system has focused on the  magnetic skyrmion lattice in single crystals and epitaxial films of cubic B20 FeGe,~\\cite{Huang2012, Porter2014, Gallagher2016a} even prior to the skyrmion discovery, the Fe-Ge system had established a record of varied spin configurations that depend intimately on structure and composition: hexagonal B35 FeGe hosts a conical structure at low temperature, which gives way to a collinear antiferromagnet at higher temperatures; monoclinic FeGe is also a collinear antiferromagnet; tetragonal FeGe$_2$ hosts a collinear antiferromagnet at low temperatures which transitions to a noncollinear antiferromagnet at higher temperatures; finally, both hexagonal and cubic Fe$_3$Ge are ferromagnetic. \nAmorphous Fe-Ge films have previously been fabricated;~\\cite{Suran1976, Daver1977, Terzieff1979, Randhawa1981, Lorentz1984} these films showed ferromagnetism above a critical Fe concentration $x \\approx 0.4$, and in one case preliminary indications of noncollinear spin textures were observed in Lorentz transmission electron micrographs.~\\cite{Randhawa1981}\nHowever, as much of this work predates the discovery of the Berry phase and the ensuing resurgence of interest in the anomalous Hall effect, magnetotransport in this material has yet to be thoroughly explored.\n\nIn this work we conduct experimental and computational investigations of the low temperature ($T=2$ K) magnetization and magnetotransport including Hall resistivity in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace  with $0.38 \\leq x \\leq 0.61$, contextualizing our findings via the isoelectronic \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace system to highlight broader trends among amorphous ferromagnets. \nUltimately, we show that \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace shares considerable physics with \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace and, unlike \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace, with its crystalline counterparts. \nWe supplement the empirical scaling argument with a new theoretical paradigm for the intrinsic anomalous Hall conductivity in aperiodic materials, based on the energy-resolved density of Berry curvature, and demonstrate how this model and our data together corroborate the intrinsic origin of the anomalous Hall effect in the low conductivity regime. \nFinally, we comment on the implications of our results for future work on other spin- and orbitronic phenomena arising from the Berry curvature.\n\n\n\n\n\\section{Methods}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[height = 1.75in]{structuralEDS.pdf}\n\\caption{Structural characterization of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace thin films.  Transmission electron diffractogram (a) taken with a 200nm probe on $x = 0.61$ shows the diffuse ring characteristic of an amorphous material. High resolution energy dispersive X-ray spectrometry images with 20 nm scale bar show nanoscale elemental homogeneity in the distribution of (b) Fe and (c) Ge for $x = 0.56$. The pixellation in these images is the sub-nm data resolution limit.}\n\\label{fStruct}\n\\end{figure*}\n\n\n\\subsection{Thin Film Growth and Characterization}\nFilms of amorphous iron-germanium (65-80 nm  thickness) were deposited onto commercially available amorphous silicon nitride substrates ($5000 \\mathrm{\\AA}$ LPCVD Si$_3$N$_4$\/$300 \\mathrm{\\AA}$ SiO$_2$\/Si $\\hkl<1 0 0>$ wafer) by co-evaporation of iron from an electron beam source and germanium from an effusion cell in a chamber with base pressure below $8 \\times 10^{-9}$ Torr.  \nGrowth rates were monitored using separate quartz crystal microbalances and were, for each deposition, maintained at a constant value  between $0.23$ \\AA\/s and $0.27$ \\AA\/s. Immediately following film growth, a capping layer (3-5 nm thickness) which prevents sample oxidation was deposited \\emph{in situ} by sublimation of Al$_2$O$_3$ from an electron beam source.\n\n\n\nFilm thickness was measured in a KLA-Tencor Alpha-Step IQ surface profiler. Compositions and atomic number densities were determined from Rutherford backscattering spectra measured at the Pelletron at Lawrence Berkeley National Laboratory; a spectrum corresponding to each measurement was simulated and iterated to fit the measured spectrum to obtain the sample composition and density. \nSample amorphicity was verified using x-ray and electron diffraction [Fig.~\\ref{fStruct}(a)], while high-resolution energy dispersive X-ray spectrometry [Fig.~\\ref{fStruct}(b) and (c)] demonstrates the films' nanoscale elemental homogeneity. \nTaken together, these data rule out the possibility of precipitation, percolation, or nanocrystallization of the \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace films.\nFurthermore, the amorphous quality of \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace films grown by identical techniques has been previously shown using high-resolution cross-sectional transmission electron microscopy.~\\cite{Gray2011} \nGiven that silicon and germanium are both well-known glass formers, it is unsurprising that both \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace and \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace readily condense into an amorphous phase in the studied range of $x$.\n\n\nMagnetization and magnetoelectrical transport were measured in a Quantum Design Magnetic Property Measurement System (MPMS XL) equipped with a 7 T superconducting magnet.  The reciprocating sample option (RSO) was used for all magnetization measurements, with four distinct SQUID voltage curves averaged for each magnetic moment data point. To isolate the magnetic signal from the film, a diamagnetic background due to the substrate was calculated from the sample mass, measured in a microbalance, and substrate susceptibility, known from previous magnetization measurements of virgin substrates. This calculated background was used to verify the accuracy of a linear fit to the high-field regions of a $T = 300$ K $M(H)$ measurement on each sample, and a straight line with the slope of this fit was subtracted from the total measured magnetization. \n\nLongitudinal and transverse resistivities were obtained using the van der Pauw method,~\\cite{VanderPauw1958} in which each rectangular film (side length $3-5$ mm) is mounted onto a specialized MPMS sample rod, with four evenly spaced pointlike (diameter $< 0.5$ mm) electrical contacts attached to the film perimeter by indium soldering. Individual four-point resistances used to calculate the film resistivity were measured using standard AC lock-in techniques, with currents of amplitude $2\\; \\mu$A and frequency 16 Hz. The transverse and longitudinal resistivities were then calculated following the procedures detailed by van der Pauw, with the aid of a numerical solver to find roots of the transcendental equation defining the longitudinal resistivity.\n\n\\subsection{Density Functional Theory Calculations}\nDensity functional theory (DFT) calculations were performed using the projector augmented wave (PAW) method~\\cite{Blochl1994, Kresse1999} and a plane wave basis set, as implemented in the Vienna \\textit{ab initio} simulation package (VASP);~\\cite{Kresse1993, Kresse1996} an energy cutoff of 350 eV was used for the plane wave basis expansion.\nThe exchange-correlation interactions were described by the generalized-gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional,~\\cite{Perdew1996} taking spin polarization into account.\nTwo separate sets of amorphous Fe$_x$Ge$_{1-x}$ structures were simulated, with one set using a 64-atom supercell (with 29, 31, 35, and 39 Fe atoms) and the other using a 128-atom supercell (with 58, 61, 69, and 78 Fe atoms), corresponding to $x \\simeq 0.45$, 0.48, 0.54, and 0.61, matching the experimental compositions. The similarity of the results of these calculations indicates that both supercell sizes capture the patterns of amorphous systems. \n\nFor each value of $x$ and each supercell size, 20 different independent amorphous structures were simulated using \\textit{ab initio} molecular dynamics (AIMD).\nInitial configurations were created by randomly substituting Fe and Ge atoms onto zincblende crystal structure sites. \nTo randomize the atomic structural positions, the systems underwent a melting step (2000 K for the 64-atom systems; 3000 K followed by a 4 ps anneal to ensure a fully molten state for the 128-atom systems) and a quenching step (2000 K to 200 K at $3 \\times 10^{14}$ K\/s for the 64-atom systems; 3000 K to 300 K at $4.5 \\times 10^{14}$ K\/s for the 128-atom systems); the 64-atom systems underwent a subsequent annealing step (200 K for 4.5 ps) in a canonical ensemble. \nThe structures were then further relaxed using the conjugate gradient method until the forces on each atom were less than 0.01 eV\/\\AA. Only the $\\Gamma$-point was used to sample the Brillouin zone during the melting, quenching, and annealing processes; however, a $3 \\times 3 \\times 3$ Monkhorst-Pack (MP) $k$-point mesh was used for the atomic geometry relaxation of all structures. \nThis mesh was also used for the 128-atom electronic structure calculation, while the 64-atom electronic structure calculation employed a $6 \\times 6 \\times 6$ MP $k$-point mesh.\n\n\n\\begin{figure}\n    \\includegraphics{Fig1_20191005.pdf}\n    \\caption{$M(H)$ curves for films with $0.38 \\leq x \\leq 0.61$ measured at $T = 2$ K with $H$ parallel to the film plane.  A temperature independent diamagnetic background corresponding to the combined magnetization of each sample's substrate and capping layer as measured at $T=300$ K has been subtracted from each measurement. Inset: expanded view of the $M(H)$ curves with $x < 0.50$, each normalized to its saturation magnetization as defined in the text and overlaid with Brillouin functions corresponding to $S = 3\/2$ and $S = 1\/2$. Data points are connected as a guide to the eye.}\n    \\label{f1}\n\\end{figure}\n\n\n\\section{Results}\n\\subsection{Magnetization}\n\nExperimental magnetization curves at $T = 2$ K with the applied field $H$ in the plane of the film are shown in Fig.~\\ref{f1} for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace with $x=0.38$, 0.45, 0.48, 0.54, and 0.61. \nWhile the M(H) curves for $x > 0.50$ primarily exhibit a squareness and absence of hysteresis associated with an otherwise unremarkable soft ferromagnet, we note the curvature at $H < 2500$~Oe. For lower $x$ ($x = 0.48$ and 0.45) this curvature is dramatically exaggerated and the curve takes on an S-shape while also developing a small hysteresis at low fields. \nThe low-$H$ curvature at high $x$ and S-shaped curves at lower $x$ hint at non-collinear spin textures in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace ($0.45 \\leq x \\leq 0.61$). Such spin textures have since been observed directly in this system via resonant X-ray scattering and Lorentz transmission electron microscopy; detailed results will be reported elsewhere.~\\cite{Chen2019,Streubel2019}\n\n\n\n\\begin{figure}\n    \\includegraphics{Fig2ab_withDFT_natomic_20190429.pdf}\n    \\caption{Comparison of magnetic and structural properties of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace (blue symbols) and \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace (green symbols) indicates that both materials exhibit a Stoner model-type itinerant ferromagnetism. The purple $\\mathbf{\\times}$ marks $T = 5$ K measurements of B20 FeGe for comparison. (a) Magnetization per Fe atom (filled circles) increases roughly linearly as a function of Fe concentration $x$ for both \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace and \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace. MD-DFT calculations of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace using a 64-atom supercell (open blue squares) yield the same values of $M_S$ as those using a 128-atom supercell (open blue diamonds); these are discussed further in the text. (b) Total atomic number density increases slightly with $x$; solid lines are linear fits to experimental data and dotted lines are extrapolations of those fits shown as guides to the eye. \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace MD-DFT calculations and $T = 2$ K experimental data are from Ref.~\\citenum{Karel2014a}, and $T = 5$ K B20 FeGe experimental data is from Ref.~\\citenum{Gallagher2016a}.}\n    \\label{f2}\n\\end{figure}\n\n\nAlthough the $M(H)$ loops for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace are qualitatively similar to \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace~\\cite{Karel2016} for the same $x$, the volume magnetization is slightly greater in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace and the S-shape of the magnetization in the low-$x$ samples is exaggerated, indicating a more complex spin structure in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace that persists to higher fields.\nThe measured and calculated values of the $T=2$~K saturation magnetization per Fe atom of \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace and \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace are compared quantitatively in Fig.~\\ref{f2}(a), where the data and calculations for \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace are from Ref.~\\citenum{Karel2014a}. \nThe \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace saturation magnetization was obtained for high $x$ (reasonably square loops in Fig.~\\ref{f1}) by extrapolating the $M(H)$ curve for $H \\geq 4$ T to $H = 0$; this extrapolation yields almost exactly $M_S = M(H = 5\\mathrm{\\,T})$ so the $H = 5$ T value of $M$ was chosen as $M_S$ for the low $x$ measurements (S-shaped loops in Fig.~\\ref{f1}). \nThe enhanced $M_S$ in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace compared to \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace is supported by the presence of sharper peaks in the calculated spin-resolved density of states in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace, which will be further discussed later in the paper.\n\n\n\\begin{figure*}\n    \\includegraphics{rhoOfT_and_sigma0_20191005.pdf}\n    \\caption{Longitudinal charge transport properties of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace (blue lines and hexagons), shown with \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace data from Ref.~\\citenum{Karel2016} (green triangles) for comparison. (a) Longitudinal resistivity $\\rho_{xx}$ as a function of temperature $T$ exhibits a negative temperature coefficient of resistance for all compositions $x$, and decreases with increasing $x$ at all $T$. Individual data points are too closely spaced to be distinguishable, so a solid line is shown instead. (b) Residual longitudinal conductivity $\\sigma_{xx,0}$ measured at $T = 4 \\,\\mathrm{K}$ as a function of Fe concentration $x$ for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace (solid blue hexagons) and \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace (solid green triangles). Spline fits to the data (dashed lines) are shown as a guide to the eye. (c) Carrier concentration $n_h$ obtained from Hall effect measurements at $T = 2$ K for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace (open blue hexagons) and \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace (open green triangles). Spline fits to the data (dashed lines) are shown as a guide to the eye. }\n    \\label{fRho}\n\\end{figure*}\n\n\\begin{figure}\n    \\includegraphics{MR_allX_20191005.pdf}\n    \\caption{Normalized magnetoresistance with $H$ oriented out of the film plane for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace. Data is symmetrized according to $\\rho(H) = \\frac{1}{2}[\\rho(+H) + \\rho(-H)]$ to remove contact asymmetry effects; data points are connected as a guide to the eye. Poor signal-to-noise is a side effect of the measurement geometry, which was chosen to optimize the Hall voltage signal. (a) Composition dependence of magnetoresistance at $T = 2$ K shows stronger negative magnetoresistance for lower $x$. (b) Temperature dependence of magnetoresistance for $x = 0.48$ shows negative magnetoresistance to $T = 40$ K, disappearing by $T = 80$ K.}\n    \\label{fMR}\n\\end{figure}\n\n\nFig.~\\ref{f2}(b) shows measured and calculated values of the atomic number density $n_{\\mathrm{total}}$\\xspace for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace and \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace; $n_{\\mathrm{total}}$\\xspace for crystalline B20 FeGe is also shown for comparison. \nThe increased size of the Ge atom compared to the Si atom is responsible for the 15-20\\% reduction in $n_{\\mathrm{total}}$\\xspace of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace compared to that of \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace.\nSince Fe structures itself more compactly than either Si or Ge, $n_{\\mathrm{total}}$\\xspace unsurprisingly increases with increasing $x$ for both systems; DFT calculations reproduce this trend in good qualitative agreement with experiment for both \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace and \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace. \nCursory extrapolations of the measured trend indicate that, for this $x$ regime, $n_{\\mathrm{total}}$\\xspace for both systems approaches the atomic number density of bulk Fe ($n_{\\mathrm{Fe}}$, orange arrow on right vertical axis) as $x$ approaches 1. \nHowever, neither extrapolation recovers its group IV element atomic number density (arrows on left vertical axis: green for $n_{\\mathrm{Si}}$ and blue for $n_{\\mathrm{Ge}}$) as $x$ approaches 0, suggesting the existence of one or more polyamorphous transitions in both systems.\n\n\n\n\n\n\n\\subsection{Magnetotransport}\nLongitudinal charge transport measurements are shown in Figs.~\\ref{fRho} and~\\ref{fMR}, while transverse measurements are shown in Fig.~\\ref{f3}; we first examine the $H = 0$ longitudinal transport properties. \nFig.~\\ref{fRho}(a) shows the temperature dependence of the longitudinal resistivity $\\rho_{xx}$, or simply $\\rho$, for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace with $0.40 \\leq x \\leq 0.65$, with higher $x$ values in darker colors (solid lines are shown because the individual data points are too closely spaced to be distinguishable). \nThe temperature dependence of $\\rho$ for each $x$ is relatively weak compared to the composition dependence of $\\rho$ across the range of $x$ studied. \nBoth the weak temperature dependence of the resistivity and the negative sign of the temperature coefficient of resistance $\\alpha$ ($\\alpha \\equiv \\rho^{-1} \\, \\partial \\rho \/ \\partial T$) are consistent with the temperature-independent scattering length $\\ell \\sim a$ of an amorphous metal (where $a$ is the average interatomic spacing); this temperature-independent $\\ell$ leads to the condition $\\rho \\gtrsim 150 \\, \\mu\\Omega\\cdot\\mathrm{cm}$ for amorphous metals, which is apparent in our measurements. \nThe systematic increase in $\\rho$ with decreasing $x$ reflects the decrease in carrier concentration with decreasing $x$, which will be further discussed next.\n\n\\begin{figure}\n    \\includegraphics{Fig3_20191005.pdf}\n    \\caption{Measured Hall resistivity $\\rho_{xy}$ as a function of applied magnetic field at $T = 2 \\mathrm{K}$ for various Fe concentrations $x$ of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace shows a dominant anomalous Hall effect with a very large magnitude for all $x$. Inset: normalized anomalous Hall resistivity $\\rho_{xy}^{\\mathrm{AH}} (H) \\equiv \\rho_{xy} (H) - R_0 H$ (green) overlaid with normalized measured magnetization (orange), showing excellent agreement and further corroborating the weak ferromagnetism in the $x=0.45$ and $0.48$ films. Data points are connected as a guide to the eye.}\n    \\label{f3}\n\\end{figure}\n\n\nFig.~\\ref{fRho}(b) shows the residual conductivity $\\sigma_{xx,0}(T = 4\\,\\mathrm{K})$ while Fig.~\\ref{fRho}(c) shows the carrier concentration for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace compared to \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace (\\textit{a}-Fe$_x$Si$_{1-x}$\\xspace data from Ref.~\\citenum{Karel2016}).\nThe carrier concentration was obtained from magnetotransport measurements carried out at $T = 2$ K (Fig.~\\ref{f3}) with the samples set up in the van der Pauw geometry. \nThe anomalous Hall effect dominates $\\rho_{xy}$ for all four samples; the additional positive slope due to the ordinary Hall effect (indicative of hole carriers) is apparent for the $x=0.45$ and $0.48$ samples, but is hardly visible at higher $x$ so the density of holes $n_h$ was extracted from $\\rho_{xy}(H)$ by iteratively fitting the parameters $R_0$ and $R_S$ to Eq.~\\ref{eqHall} in Python, using measured $M_z(H)$ data for each sample. \n$R_0$ is related to the carrier concentration by $|R_0| = (ne)^{-1}$, where $n$ is the carrier concentration and $e$ is the electronic charge; our fits yield the values of hole concentration $n_h$ shown by open blue hexagons in Fig.~\\ref{fRho}(c). \nComparing our results to the measured (for $0.43 \\leq x \\leq 0.48)$ and extrapolated (for $x > 0.48$) values of $n_h$ for \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace (open green triangles) confirms that the higher carrier concentration in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace is responsible for its higher $\\sigma_{xx,0}$ for any $x$, represented by filled blue hexagons in Fig.~\\ref{fRho}(b), compared to \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace (filled green triangles).\n\nThe magnetoresistance (MR) ratio, defined as $\\mathrm{MR}(H) \\equiv [\\rho(H) - \\rho(0)]\/\\rho(0)$, was also measured for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace and is shown in Fig.~\\ref{fMR}(a) at $T = 2$ K with $H$ oriented perpendicular to the film plane. \nAs expected for ferromagnets, the MR is negative,~\\cite{Spaldin2010} and the weakly ferromagnetic $x = 0.45$ and 0.48 samples show a larger negative magnetoresistance due to increased suppression of spin-disorder scattering by an applied magnetic field. \nFig.~\\ref{fMR}(b) shows that, in contrast to the positive MR of \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace for $T \\geq 16$ K,~\\cite{Karel2016} this significantly enhanced negative contribution to the MR of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace dominates up to temperatures between 40 K and 80 K, resulting in a net negative MR for $x = 0.48$ over a wide temperature range.\nWe note that, while still small, the magnitude of the $T = 2$ K MR in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace is more than 200 times greater than the effect measured in the same geometry (H perpendicular to the film and therefore also perpendicular to the current) in \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace for comparable Fe concentrations, despite similar $M(H)$. \nSince we have shown our samples to be structurally and chemically homogeneous (Fig.~\\ref{fStruct}), we attribute this enhanced negative MR in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace both to a stronger coupling between carriers and local moments and to a greater reduction in the disorder seen by the carriers. \nThe latter effect reinforces our observation, based on the $M(H)$ curves, that \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace hosts a more complicated spin texture than \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace. \n\n\n\n\n\n\n   \n   \n   \n   \n   \n\n\n\nWe isolate the anomalous component of the Hall resistivity by subtracting the OHE from the total $\\rho_{xy}$, and show the result for $x=0.45$ in the inset to Fig.~\\ref{f3} normalized by its $H=5$ T value and overlaid with the correspondingly normalized $M_z(H)$ to emphasize the unmistakable magnetic origin of this Hall signal, even in our most weakly ferromagnetic sample. \nThis inset also exemplifies the absence of additional measurable contributions to the Hall effect, such as a topological Hall effect that has been observed in several systems as a consequence of the adiabatic change in the carrier phase by a noncollinear spin texture. \nWhile the samples shown here likely host local noncollinear spin textures due to the Dzyaloshinskii-Moriya interaction in the Fe-Ge system, the absence of global chirality in the amorphous structure reduces any net signal from localized topological contributions to the Hall resistivity below the sensitivity of our setup. \n\n\n\\begin{figure}\n    \\includegraphics{Fig4_20190614.pdf}\n    \\caption{Scaling plot of the anomalous Hall conductivity $\\sigma_{xy}^{\\mathrm{AH}}$ reveals the origin of the anomalous Hall effect in \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace and \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace across different regimes of longitudinal conductivity $\\sigma_{xx}$. (a) Normalizing $\\sigma_{xy}^{\\mathrm{AH}}$ by the saturation magnetization $M_z$ yields a power law dependence that deviates from the empirical $\\sigma_{xy}^{\\mathrm{AH}} \\sim \\sigma_{xx}^{1.6-1.8}$. (b) Normalizing further by a free-electron-type dependence on the carrier concentration, $n_h^{2\/3}$, yields an essentially constant value for both \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace and \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace. Dashed lines are average values of $\\sigma_{xy}^{\\mathrm{AH}}\/(M_z n_h^{2\/3})$ for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace (blue) and \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace (green). Amorphous and epitaxial Fe-Si data from Ref.~\\citenum{Karel2016}; B20 FeGe data from Ref.~\\citenum{Gallagher2016a}.}\n    \\label{f4}\n\\end{figure}\n\n\n\n\n\n\n\\section{Discussion}\n\\subsection{Magnetization}\nThe onset of magnetic order at $T = 2$ K in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace occurs between $x = 0.45$ and $x = 0.38$, in agreement with previous work.~\\cite{Suran1976} The inset of Fig.~\\ref{f1} confirms this by comparing the magnetization of the $x = 0.38,\\,0.45,$ and $0.48$ films to Brillouin functions of ideal paramagnets with $S = 1\/2$ and $S = 3\/2$; the paramagnetic $x=0.38$ curve falls squarely between the Brillouin functions while no single Brillouin function adequately models both the low- and high-field behavior of the $x=0.45$ and $x=0.48$ curves. The steep slope and finite hysteresis of their magnetization at low fields suggests ferromagnetism, but the magnetization continues to increase without saturating to fields above $H = 7$ T, leading to the conclusion that \\textit{a}-Fe$_{0.45}$Ge$_{0.55}$ and \\textit{a}-Fe$_{0.48}$Ge$_{0.52}$ are only weakly ferromagnetic.\nNotably, the Ising model from which the Brillouin functions arise does not capture the profusion of spin textures that form across the crystalline Fe-Ge system thanks to the competition between Heisenberg and DM interactions in different lattice configurations, and the non-square features of the hysteresis loops noted previously indicate that this competition persists in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace.\n\nFor both \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace and \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace, the magnetization per Fe atom increases with increasing Fe concentration $x$; however, for any given $x$, \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace has a significantly higher  magnetization per Fe atom than \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace, despite having only slightly higher magnetization per unit volume, due to the different atomic densities of these two alloys at all $x$ [Fig.~\\ref{f2}(b)]. \nIn contrast to the Fe-Si system, in which the magnetization of the crystalline phase is an order of magnitude (or more) less than than that of \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace,~\\cite{Karel2014a} the measured magnetization per Fe atom of B20 FeGe at $T = 5$ K from Ref.~\\citenum{Gallagher2016a} aligns very well with our experimental data for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace; taken together with existing evidence that the local atomic environment strongly influences the magnetic state of amorphous transition metal germanides and silicides, this suggests that the local atomic environment in our amorphous films resembles that of B20 FeGe.\nThe DFT calculations of magnetization in \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace~\\cite{Karel2014a} and \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace [open diamonds and squares in Fig.~\\ref{f2}(a)] reproduce this experimental trend as well as the relative values of $M$ between both systems, but yield excellent quantitative agreement with experiment only in \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace. \nThe discrepancy between calculated and measured magnetization in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace is likely due to a noncollinear spin texture, which suppresses the net magnetization and is not computationally taken into account because of the difficulty of calculating the spin texture of an amorphous system using DFT. \n\n\\begin{figure}\n    \\includegraphics{DoScomparison_20191005.pdf}\n    \\caption{Calculated density of states (DOS) near the Fermi level for the compositions $x$ of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace studied experimentally (blue curves), and for \\textit{a}-Fe$_{0.65}$Si$_{0.35}$ (green dashed curve, from Ref.~\\citenum{Karel2014a}), all based on a 128-atom unit cell. The peaks of the majority spin in the DOS are narrowed and sharpened by substituting the larger Ge atom for the Si atom for a fixed $x$, as can be seen from comparing the DOS of \\textit{a}-Fe$_{0.65}$Si$_{0.35}$ to that of \\textit{a}-Fe$_{0.61}$Ge$_{0.39}$, explaining the enhanced magnetization in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace.}\n    \\label{fDoS}\n\\end{figure}\n\n\n\nThe increased size of the Ge atom relative to the Si atom enhances the magnetization of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace relative to \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace [Fig.~\\ref{f2}(a)].\nA local-moment picture does not fully capture the physics in these systems, so we turn to the calculated densities of states (DOS) of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace ($0.45 \\leq x \\leq 0.61$)and a-Fe$_{0.65}$Si$_{0.35}$~\\cite{Karel2014a} shown in Fig.~\\ref{fDoS} for a more complete explanation of the role of Ge and Si atomic size. \nIntuitively, substituting the larger, isoelectronic Ge atom for Si in a \\emph{crystalline} system will increase the lattice constant, shrink the Brillouin zone, and compress the energy bands in $\\vec{k}$-space, thereby narrowing and sharpening the features in the density of states . \nFor our \\emph{amorphous} systems, $\\vec{k}$ is no longer a good quantum number so Brillouin zones and energy bands are not meaningful; however, the average interatomic spacing and the density of states make no reference to $\\vec{k}$ and so provide a meaningful extension of our crystalline intuition into amorphous systems.\nThe reduced atomic number density corresponds to an increased average interatomic spacing in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace compared to \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace. \nThis leads to a sharper DOS peak and a stronger spin split near the Fermi level, as shown in Fig.~\\ref{fDoS}, which explains the enhanced magnetization, as well as its pronounced $x$-dependence, in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace when considered in the framework of a simple Stoner band model of itinerant ferromagnetism.~\\cite{Stoner1938} We therefore conclude that the scattering from the disorder associated with the amorphous structure is sufficiently weak that it does not substantially redistribute the majority and minority DOS of the Fe-$3d$ electrons,  so the exchange interaction remains strong enough to spin-split the density of states and yield a net magnetization.\n\nFurthermore, the density of states in Fig.~\\ref{fDoS} allows us to approximate the spin polarization in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace; for example, we count states at the Fermi energy $\\varepsilon_F$ for \\textit{a}-Fe$_{0.61}$Ge$_{0.39}$ and estimate a 37\\% spin polarization, nearly identical to the calculated spin polarization for \\textit{a}-Fe$_{0.65}$Si$_{0.35}$.~\\cite{Karel2014a} \nThe low-temperature spin polarization of \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace with $0.58 \\leq x \\leq 0.68$ has been measured using point-contact Andreev reflection spectroscopy and was found to peak at almost 70\\% for $x = 0.65$,~\\cite{Karel2018} roughly double the prediction from the calculated DOS; this discrepancy was there attributed to the small size of the supercell used in the calculations. The result in \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace  leads us to suspect a similarly large spin polarization exists in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace.\n\n\\subsection{Anomalous Hall Effect}\nThese films fall at the high conductivity edge of the low conductivity regime, with $10^3 < \\sigma_{xx} < 10^4 \\,(\\mathrm{\\Omega\\; cm})^{-1}$ and finite $\\rho_{xx}$ as $T \\to 0$, so hopping conduction does not apply. To apply a standard empirical scaling argument to the AHC in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace, an appropriate normalization is essential; we consider two factors to enable comparison of our results to the scaling of the AHC in \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace.~\\cite{Karel2016}\nFirst, in changing the Fe concentration $x$, $M_z$ is also modified. \nSince $\\sigma_{xy}^{\\mathrm{AH}} \\propto \\rho_{xy}^{\\mathrm{AH}} \\propto M_z$, changing $x$ alters $\\sigma_{xy}^{\\mathrm{AH}}$ directly due to the dependence on $M_z$; we remove this dependence by dividing $\\sigma_{xy}^{\\mathrm{AH}}$ by $M_z$ and obtain the data shown in Fig.~\\ref{f4}(a).\nThese data can be fit by $\\sigma_{xy}^{\\mathrm{AH}} \\propto \\sigma_{xx}^{1.1}$, similar to what was obtained for \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace but inconsistent with the theoretical predictions of Onoda~\\cite{Onoda2006} and Liu.~\\cite{Liu2011} \nAdditionally, since the unified theory is expressed in terms of the carrier lifetime $\\tau$ and not the conductivity $\\sigma_{xx}$, we must account for the effect of changing $x$ on $\\tau$. Hence, we consider a simple free-electron-type model which gives $n_h^{2\/3}$ as our second normalization factor, using the values of $n_h$ shown in Fig.~\\ref{fRho}(b). This normalization yields the scaling plot in Fig.~\\ref{f4}(b), in which $\\sigma_{xy}^{\\mathrm{AH}}\/M_z n_h^{2\/3}$ is essentially independent of $\\sigma_{xx}$. Both scaling plots in Fig.~\\ref{f4} also show the appropriately normalized anomalous Hall conductivity in \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace from Ref.~\\citenum{Karel2016} as a point of comparison; we attribute the quantitative similarity in the normalized AHC of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace and \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace to the spin-orbit coupling introduced near $\\varepsilon_F$ by the Fe $d$-states in both systems, since the spin-orbit coupling due to the Ge (or Si) $p$-states is diluted over a wider energy range.\n\n\\begin{figure}\n    \\includegraphics{AnomalousHallConductivity_theoryVsExp_20190429.pdf}\n    \\caption{Comparison of calculated \\emph{intrinsic} anomalous Hall conductivity from 128-atom MD-DFT (turquoise diamonds) to experimentally measured \\emph{total} anomalous Hall conductivity (blue dots). The main axis shows the AHC normalized by the saturation magnetization $M_z$, where experimental values have been normalized by the experimentally measured $M_z$ and theoretical values have been normalized by the theoretically calculated $M_z$ for consistency. Unnormalized values of the measured total and calculated intrinsic AHC are shown in the inset. Spline fits to data (dashed lines) are shown as a guide to the eye.}\n    \\label{f:AHCtheoryexpt}\n\\end{figure}\n\nThe fact that $\\sigma_{xy}^{\\mathrm{AH}}\/M_z n_h^{2\/3}$ is independent of $\\sigma_{xx}$ implies that either the side-jump or the intrinsic mechanism, or some combination of the two, is responsible for the anomalous Hall conductivity. \nIn the case of \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace, these two contributions were qualitatively deconvolved by comparison to measurements on crystalline Fe$_x$Si$_{1-x}$, which were compared to calculations of the intrinsic and side-jump mechanisms in the AHC of bcc Fe.~\\cite{Karel2016} \nTo more rigorously separate the contributions of these two mechanisms in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace, we compare \\textit{ab initio} values of the \\textit{intrinsic} anomalous Hall conductivity calculated from our spin-orbit-coupled 128-atom MD-DFT, which are normalized by the calculated magnetization and shown in Fig.~\\ref{f:AHCtheoryexpt}, to experimental values of the \\textit{total} anomalous Hall conductivity, normalized by the measured magnetization; this comparison shows excellent agreement in the trend of the calculated and experimental values, suggesting that the measured AHC consists of a large intrinsic component, which varies with composition and is partially offset by a composition-independent side-jump component. \nWhile the intrinsic AHC is generally expressed as the integral of the Berry curvature taken over all occupied bands, this result independently verifies the outcome of our scaling argument and unambiguously shows that the intrinsic mechanism persists in systems whose band structure is ill-defined, further validating the real-space formulation of the AHC in Ref.~\\citenum{Marrazzo2017a}. \n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{DensityOfCurvature_1005.pdf}\n    \\caption{Density of Berry curvature as a function of chemical potential calculated from 128-atom MD-DFT for \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace with $x = 0.45$, 0.48, 0.54 and 0.61. }\n    \\label{f:DOC}\n\\end{figure}\n\nOur spin-orbit coupled DFT calculations provide computational foundations and a theoretical paradigm shift for understanding the \\emph{intrinsic} anomalous Hall conductivity in an amorphous material. The intrinsic AHC is the integral over occupied states of the Berry curvature; however, as $\\vec{k}$ is not a good quantum number in an amorphous system, expressing the Berry curvature as $\\Omega(\\vec{k})$ is meaningless here.\nInstead, we construct a framework appropriate to our system based on the \\textit{density of Berry curvature}:~\\cite{Guo2008, Sahin2015}\n\\begin{equation}\n    \\rho_{\\mathrm{DOC}}(\\varepsilon) = \\sum_{\\vec{k}}^{} \\Omega(\\vec{k}) \\delta(\\varepsilon_{\\vec{k}} - \\varepsilon)\n\\end{equation}\nwhich corresponds to the Berry curvature within an energy range bounded by $\\varepsilon$ and $\\varepsilon + d\\varepsilon$ integrated throughout the Brillouin zone. \nThis can, in principle, be computed in an amorphous material without reference to $\\vec{k}$-space by adding together the partial densities of states for local orbital states with spin-orbit correlation parallel and antiparallel, which are energetically offset by the self-consistently determined exchange energy.\nIf the energy scales of the disorder potentials that scatter states from momentum $\\vec{k}$ to $\\vec{k}'$ are relatively weak compared to the energy scale spanned by the features in $\\rho_{\\mathrm{DOC}}(\\varepsilon)$, then the density of Berry curvature would be approximately preserved even as momentum fails to remain a good quantum number in an amorphous system.\n\nFig.~\\ref{f:DOC} shows the density of Berry curvature as a function of chemical potential for the compositions $x$ of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace studied in this work; integrating the density of curvature over occupied energy states yields the intrinsic AHC shown in Fig.~\\ref{f:AHCtheoryexpt}, in the same way that integrating the spin-resolved density of states over occupied energy states yields the magnetization.\nFig.~\\ref{f:DOC} further indicates that the energy scale of the density of Berry curvature is on the order of 5 eV, an energy scale set in part by the spin-orbit interaction-induced splitting of parallel and antiparallel spin-orbit correlated states and in part by the exchange energy in our system. \nAs long as the disorder potentials are typically smaller than 5 eV our conclusions about the AHC would remain similar in both the clean and disordered systems; otherwise, the spin-orbit correlations that give rise to the AHC would be lost.\nThis energy scale lends confidence that the calculated density of curvature, which imposes an artificial periodicity on the simulated 128-atom amorphous supercell, is a suitable approximation for that of the fully amorphous material: the similarity between the simulated and measured atomic number densities [Fig.~\\ref{f2}(b)] indicates that the supercell accurately reproduces the structural disorder, and hence the disorder potentials, of the actual amorphous structure.\n\n   \n   \n   \n   \n\n\n\n\n\n\n\\section{Conclusion}\nIn summary, we presented experimental and computational studies of the magnetic and transport properties of amorphous Fe$_x$Ge$_{1-x}$ ($0.45 \\leq x \\leq 0.61$) thin films, including the first measurement of the anomalous Hall effect in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace. \nIts magnetization is well explained by a Stoner band model above the onset of magnetic order around $x=0.4$, with the larger Ge atom distorting the density of states and causing enhanced magnetization at all $x$ in \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace compared to \\textit{a}-Fe$_x$Si$_{1-x}$\\xspace.\nThe resistivity of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace agrees with established work on amorphous metals, and indicates that a metal-insulator transition occurs at lower $x$ than the onset of ferromagnetism. \nThe anomalous Hall conductivity of \\textit{a}-Fe$_x$Ge$_{1-x}$\\xspace is independent of the longitudinal conductivity when appropriately normalized by $M_z n_h^{2\/3}$ to account for the role of changing the Fe concentration on the magnetization and carrier lifetime; our DFT calculations refine this scaling argument and indicate the AHC is comprised of a dominant intrinsic component whose magnitude is influenced by the Fe concentration, and of an opposing composition-independent side-jump component.\nThe calculated density of Berry curvature shows that the AHC in this amorphous material is indeed intrinsic; moreover, because the density of curvature can be calculated from either the spin-orbit correlations of local orbital states in a disordered material or the band structure in a crystalline one, it provides a novel and versatile Stoner-esque model for understanding the electronic origins of the intrinsic AHC in systems possessing and lacking long-range order.\nSince the spin and orbital Hall conductivities arise from mathematically analogous Berry curvature-like quantities, we anticipate that the straightforward extension of this model to those phenomena will spur experimental investigation of the spin and orbital Hall effects in amorphous materials, portending future application of such materials as spin and orbital torque generators in next-generation devices.\n\n\\begin{acknowledgements}\nThe authors thank C. \\c{S}ahin, J. Karel, S. Mack, and N. D. Reynolds for illuminating discussions.\nThis work was primarily funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05-CH11231 (NEMM program MSMAG). \nWork for 128-atom DFT calculations was supported by DOE-BES (Grant No. DE-FG02-05-ER46237). Additionally, BHZ acknowledges support from the Basic Research Program of China (2015CB921400).\nSupport for the density of Berry curvature framework for the AHC was provided by the Center for Emergent Materials, an NSF MRSEC under Award No. DMR-1420451.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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We considered\nseveral lattice spacings, with the DBW2 and Iwasaki gauge actions and\ndifferent sea quark masses on a volume of $16^3\\times32$ and a fifth\ndimension of size 8.  All data were generated on the QCDOC machines.\nDespite large residual massses and a limited number of sea quark mass\nvalues with which to perform chiral extrapolations, our results are in\nreasonable agreement with experiment and scale within errors. Finite size\neffects on most ensembles appear to be small.\n}\n\n\\PoS{PoS(LAT2005)098}\n\\ShortTitle{Baryons in 2+1 flavour domain wall QCD}\n\\FullConference{XXIIIrd International Symposium on Lattice Field Theory\n\\\\Trinity College, Dublin, Ireland\\\\25-30 July 2005}\n\\begin{document}\n\n\n\n\n\n\n\\section{Introduction}\nThe calculation of many quantities from lattice QCD with quenched, or\nheavy dynamical, quarks has produced results which, although in\nqualitative agreement with experiment, have uncontrolled sources of\nerror. To progress beyond this, calculations with light dynamical\nquarks are necessary. Using fermions which satisfy the Ginsparg-Wilson\n(GW) relation is advantageous, as the theory is well defined within\nbroad parameter ranges. It has lattice versions of the symmetries of\ncontinuum QCD, and the correct flavour content. Mixing of operators\nwith different chirality is suppressed, renormalisation is simplified,\nand a continuum-like chiral perturbation theory can be used for the\nextrapolation of quantities to the chiral limit. Domain wall fermions\n(DWF) satisfy these requirements. The QCDOC machine and the RHMC\nalgorithm have allowed for the first time calculations using DWF with\n$2+1$ flavours of dynamical quarks.\n\nBaryon physics is an area rich in\nphenomenology. Unanswered questions as to the nature of some of the\nexcited states, the decay of the proton predicted by some Grand\nUnified and super~symmetric models, and the determination of \nmatrix elements related to the structure functions and the neutron electric\ndipole moment can, in principle, be determined by lattice QCD\ncalculations. These calculations are very computationally challenging,\nas they need both light sea quarks and large volumes, as well as fermions\nwith the correct symmetry properties. In this work we report on a study\nof the lowest lying states in the baryon spectrum, $\\{N , \\Delta ,\n\\Omega , N^{\\star}\\}$, on ensembles of configurations produced by\nQCDOC. It is important for any calculation claiming to be ``full QCD''\nto be able to reproduce this spectrum.  These initial ensembles were\nproduced primarily for a search of parameter space to guide larger\nproduction runs.  As such, they have a small volume, too small\ncertainly for excited states such as the $N^{\\star}$, but this can be\nused, eventually in combination with the productions runs on larger\nvolumes to try and estimate the size of the finite volume effects on\nthe remaining spectrum.\n\nThe DWF action is given in~\\cite{Furman:1994ky} with\nthe Pauli-Villars field in~\\cite{Vranas:1997da} for the dynamical\nsimulation. The gauge fields were generated with renormalisation group\n(RG) improved actions, as follows:\n\\begin{equation}\n  S_G=-\\frac{\\beta}{3}\\left[ (1-8) c_1 \\sum_{x,\\mu\\nu} P(x)_{\\mu\\nu} +\n  c_1 \\sum_{x,\\mu\\neq\\nu} R(x)_{\\mu\\nu}\\right]\n\\end{equation}\nwith either $c_1=-1.4069$ for the DBW2 action~\\cite{deForcrand:1999bi}\nor $c_1=-0.331$ for the Iwasaki action~\\cite{Iwasaki:1984cj}. It has\nbeen noted in the quenched DWF calculations~\\cite{Blum:2000kn} that\nthese actions reduce the chiral symmetry breaking resulting from the\nfinite fifth dimension. The mechanism and size of chiral symmetry\nbreaking on these ensembles is studied in detail in~\\cite{PeterBoyle}.\n\nIn the DWF formalism, the 4D quark fields are constructed\nfrom left and right projections of the 5D fermion fields\non the boundaries. With a finite fifth dimension, there is still\nan overlap between these left- and right-handed fields, which manifests\nitself as an additive mass renormalisation, known as the residual mass.\nThis can be determined directly from the axial Ward-Takahashi identity\n\\begin{eqnarray}\n  \\partial_\\mu A_\\mu(x)& = & 2am_fP(x)+ 2 J_5(x)\\\\\\nonumber\n               & \\approx & 2 (am_f + am_{\\rm res})P(x)\n\\end{eqnarray}\nwhere $J_5(x)$ is the point-split current constructed from fields at the \nmid-point of the fifth dimension and $P(x)$ the pseudoscalar density.\n\n\n\\section{Details of the calculation}\nThe ensembles were all generated on QCDOC machines, using the exact\nRHMC algorithm~\\cite{Kennedy:1998cu,MikeClark}, with a volume of\n$16^3\\times32$ and a fifth dimension of size $L_S=8$ and are detailed in\nTable~\\ref{tab:Ensembles}. The integrated autocorrelation times are\nestimated to be $\\order{50}$ for mesonic correlators. To maximise\nstatistics, the correlation functions were measured every 5\ntrajectories, and then binned so that the separation between\nindependent measurements is 100 trajectories. The correlators were\ncomputed from up to four sources on different time-planes with as many as\nthree different smeared sources.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cccccccc}\n$\\beta$ & $am_l\/am_s$ & $r_0\/a$ & $a^{-1}$(GeV) & $L$(fm) & $Lm_P$ & $m_P\/m_V$ & $\\#$traj \\\\\\hline\n$0.72$ & 0.04\/0.04 & & & & $7.7(1)$ & $0.692(5)$ & 3400\\\\\n$0.72$ & 0.02\/0.04 & \\raisebox{1.5ex}[0pt]{$4.3(1)$} &\\raisebox{1.5ex}[0pt]{$1.7(1)$} & \\raisebox{1.5ex}[0pt]{$1.9(1)$} & $6.0(1)$ & $0.589(3)$ & 6000 \\\\\\hline\n$0.764$ & 0.04\/0.04 & & & &$6.7(1)$ & $0.699(4)$ & 5750\\\\\n$0.764$ & 0.02\/0.04 & \\raisebox{1.5ex}[0pt]{$5.1(2)$} &\\raisebox{1.5ex}[0pt]{$2.0(1)$} & \\raisebox{1.5ex}[0pt]{$1.6(1)$} & $5.1(1)$ & $0.619(4)$ & 3000 \\\\\\hline\n$2.13$ & 0.04\/0.04 & & & & $7.5(1)$ & $0.700(8)$ & 3600\\\\\n$2.13$ & 0.02\/0.04 & \\raisebox{1.5ex}[0pt]{$4.6(2)$} &\\raisebox{1.5ex}[0pt]{$1.8(1)$} & \\raisebox{1.5ex}[0pt]{$1.8(1)$} & $5.8(1)$ & $0.615(5)$ & 3600 \\\\\\hline\n$2.2$ & 0.04\/0.04 & & & & $6.8(1)$ & $0.726(2)$ & 4500\\\\\n$2.2$ & 0.02\/0.04 & \\raisebox{1.5ex}[0pt]{$5.3(1)$} &\\raisebox{1.5ex}[0pt]{$2.1(1)$} & \\raisebox{1.5ex}[0pt]{$1.5(1)$}& $5.1(1)$ & $0.667(8)$ & 3200 \\\\\\hline\n\\end{tabular}\n\\caption{Properties of the ensembles used in this study. The value of $r_0\/a$ was determined in~\\cite{KoichiHashimoto}.\nThe lattice spacing, and thus the volume are set by choosing $r_0=0.5$fm. Trajectories are of length 0.5}\n\\label{tab:Ensembles}\n\\end{center}\n\\end{table}\n\nThe masses of the pseudoscalar and vector mesons were determined by simultaneous\nfits to the local and smeared correlators with two exponentials, the ground state and the\nexcited state. Similarly $m_{\\rm res}$ was determined by a simultaneous fit of a plateau to \na ratio of correlators, with both smeared and local sources.\n\nThe standard baryon interpolating operator is given by\n\\begin{equation}\n  \\Omega(x)=\\epsilon_{ijk}\\left[\\psi_i(x) C \\Gamma \\psi_j(x)\\right]\\psi_k(x)\n\\end{equation}\nFor the $I=\\{\\frac{1}{2},\\frac{3}{2}\\}$ baryons,\n$\\Gamma=\\{\\gamma_5,\\gamma_5\\gamma_k\\}$.  Another operator, which\nprojects onto the negative parity $I=\\frac{1}{2}$ state, with\n$\\Gamma=1$ was also used.  For baryon correlators in a finite box\nwith periodic boundary conditions, the backward propagating state is\nthe negative parity partner, that is\n\\begin{equation}\n  C_B(t) = A_+e^{-m_+t} + A_-e^{-m_-(T-t)}\n  \\label{eqn:baryonCorr}\n\\end{equation}\nFor the $I=\\frac{1}{2}$ baryon, the masses of the positive and negative\nparity states were determined by a simultaneous fit to equation\n(\\ref{eqn:baryonCorr}) using the standard operator, and a single\nexponential to the negative parity correlator. This is shown in\nFigure~\\ref{fig:effMass}(a).  Typically this was computed for the\nsmeared correlator only, as the local correlator had a poor\nsignal. \n\nThe mass of the $I=\\frac{3}{2}$ baryon was determined from a single\nexponential fit to the smeared correlator, as the signal for the\nexcited, negative parity partner was poor, as might be expected on\nrelatively small numbers of configurations.\n\n\\begin{figure}\n\\begin{center}\n  \\epsfig{file=Nstar_m0.02.eps,height=0.49\\textwidth,width=0.49\\textwidth}\n  \\epsfig{file=EdinburghPlot.eps,height=0.49\\textwidth,width=0.49\\textwidth}\n  \\caption{(a) The effective mass of $I=\\frac{1}{2}$ baryon correlator on the DBW2\n    $\\beta=0.72$, $am_l\/am_s=0.02\/0.04$ ensemble. The red symbol shows\n    the data for the positive parity state, the closed blue squares are\n    the time-reversed backward propagating negative parity state data and\n    the open blue squares are the negative parity data from the second correlator.\n    (b) The Edinburgh plot for the different ensembles.}\n  \\label{fig:effMass}\n\\end{center}\n\\end{figure}\n\n\\section{Preliminary Results}\nShown in Figure~\\ref{fig:effMass}(b) is the Edinburgh plot. It is\nreassuring that, even at relatively coarse lattice spacing, with a\nsmall fifth dimension and consequently moderate chiral\nsymmetry breaking, the data follows the phenomenological curve very well. \nThe only exception is the lightest Iwasaki $\\beta=2.2$ datum. Naively, one\nmight expect this to be a finite size effect, especially, examining \nTable~\\ref{tab:Ensembles}, given that the lattice spacing\nand thus the box size are rather small. However, the value of $Lm_P$ is not\nsignificantly smaller than the other data sets and, critically, $Lm_P>4$, suggesting\nthat the box is big enough as measured by the pseudoscalar meson. \n\nFor this particular ensemble the signal for the vector meson mass is\nnot good. In particular, the effective mass plot has a poor plateau,\nand a stable fit can only be achieved for the lowest region. The net\neffect is for a rather low vector meson mass. It is probable that this\nis due simply to low statistics. A low estimate of the vector mass\nwould cause this datum in the Edinburgh plot to be shifted up and to the\nright, which could be mistaken for a finite volume effect increasing\nthe mass of the nucleon.\n\nThe quark mass was defined as\n\\begin{equation}\n  am_q=am_f + am_{\\rm res}(m_f)\n\\end{equation}\nwith the chiral limit at $am_q=0$. With two sea quark masses,\nonly a crude chiral extrapolation could be attempted, {\\em i.e.} drawing a\nstraight line through the two data points. The strange quark mass, (for the\n$\\Omega$ baryon) was set from the kaon mass.\n\n\\begin{figure}\n\\begin{center}\n  \\epsfig{file=scaling.eps,height=0.49\\textwidth,width=0.49\\textwidth}\n  \\caption{Scaling of the baryon spectrum with lattice spacing squared.\n\t   The symbols denote the following: Closed circles $N$, open circles\n\t   $N^{\\star}$, squares $\\Delta$, diamonds $\\Omega$. Black symbols denote\n           experiment, red DBW2 ensembles, blue Iwasaki ensembles.\n\t   The value of $r_0=0.5$fm was chosen, to give an indication of \n\t   the experimental spectrum in these units. The $N^{\\star}$ measured\n\t   on the $\\beta=2.13$ ensemble (the furthest right open blue circle) has\n\t   been offset to the left for clarity.}\n  \\label{fig:Scaling}\n\\end{center}\n\\end{figure}\n\nShown in Figure~\\ref{fig:Scaling} is the dependence of the baryon\nspectrum, in dimensionless units, on the lattice spacing. A continuum\nextrapolation cannot be attempted with these ensembles. However, the\ndata for the ground states, $\\{N,\\Delta ,\\Omega\\}$, shows reasonable\nscaling, albeit with large errors.  These large uncertainties are due\nto the crude nature of the chiral extrapolation. The determination of\n$r_0$ is well defined for each ensemble, and thus it is a good\nquantity to use to examine scaling behaviour. As the value of $r_0$ in\nthe continuum is unknown, rather than setting the absolute scale with\n$r_0$, a better strategy is to predict dimensionless ratios of\nphysical quantities in the continuum, while using $r_0$ to just\nexamine the scaling behaviour.\n\nThe negative parity partner of the nucleon, the $N^\\star$ is expected\nto become degenerate with the nucleon in a small enough box. This\neffect can be clearly seen from Figure~\\ref{fig:Scaling}; the\n$N^\\star$ mass drops dramatically as the volume is reduced. This\nsuggests that finite size effects may also be beginning to affect the\nground states for the ensembles at finest lattice spacing. These\nfinite size effects would tend increase the mass of the ground\nstates. The slight upward tendancy in the scaling plot as the lattice\nspacing decreases is consistent with the finite size effects spoiling\notherwise very good scaling, or a small scaling violation for the\nnucleon mass.\n\n\\section{Conclusions}\nWe have determined the spectrum of the lowest lying baryon states for several\nensembles with two different gauge actions. The QCDOC machines and the RHMC algorithm\nhave made $2+1$ flavours of DWF ensembles possible for the first time. With limited\nstatistics, and only two different sea quark masses, we qualitatively reproduce the\nexperimental spectrum. There is limited evidence that finite size effects may be \ninfluencing the ground state baryons on the smaller volumes. Both the\nEdinburgh plot, and the scaling analysis suggest that a programme of baryon\nphysics on larger volumes and at lighter quark mass will yield very interesting\nresults.\n\n\\section{Acknowledgements}\nWe thank Saul Cohen, Sam Li and Meifeng Lin for help generating the datasets used\nin this work. We thank Dong Chen, Norman Christ, Saul Cohen, Calin\nCristian, Zhihua Dong, Alan Gara, Andrew Jackson, Chulwoo Jung,\nChanghoan Kim, Ludmila Levkova, Xiaodong Liao, Guofeng Liu, Robert\nMawhinney, Shigemi Ohta, Konstantin Petrov and Tilo Wettig for\ndeveloping with us the QCDOC machine and its software. This development\n and the resulting computer equipment used in this calculation were funded\n by the U.S. DOE grant DE-FG02-92ER40699, PPARC JIF grant PPA\/J\/S\/1998\/00756\n and by RIKEN. This work was supported by PPARC grant PPA\/G\/O\/2002\/00465.\n\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{Introduction}\n\nSolar flares have been studied in detail both observationally and theoretically ever since their discovery by \\citet{Ca859}. Although increasingly more sophisticated instrumentation provides ever more detailed data, we still lack the basic understanding of many processes at work in a solar flare. \n\nThe common flare picture as deduced from hard X-ray (HXR) observations features an HXR source in the corona~\\citep[coronal or loop-top source,][]{Fr71,Hu78}, and two or more HXR sources (footpoints) in the chromosphere \\citep{Hoy81}. These sources are thought to be due to bremsstrahlung emission produced by fast electrons accelerated somewhere above the loop. If we assume that a single particle beam creates both coronal and footpoint emission, the most basic model would involve thin target emission at the top of the coronal loop and thick target emission from the footpoints, which both produce characteristic spectra. \n\\citet{Wh95} developed a more sophisticated model (intermediate thin-thick target or ITTT model) to fit observations by Yohkoh. They based their model on observations by \\citet{Fe94}, who found high column densities at the loop top, which might act as a thick target below a certain electron energy. In the ITTT model, the shape of the coronal and footpoint non-thermal spectra and the relation between them, observed by Yohkoh, can be explained. The column density in the coronal source determines a critical energy level for the electrons. Electrons that have an energy below this critical energy are stopped in the coronal region. Consequently, the distribution of electron energies measured at the footpoints is depleted in low energy electrons. If the column density is high, the coronal source may act as a thick target to electrons of energies as high as 60 keV, which would leave almost no footpoint emission. Observational evidence for such coronal thick targets were found in RHESSI observations \\citep[eg.][]{Ve04}.\n\nLess extreme cases, flares with one or more footpoints, have frequently been observed by RHESSI.\nTo study the spectral time evolution of individual sources, five well-observed events were analyzed by \\citet{Ba06} who focused on the differences between the spectral indices of coronal and footpoint spectra. They found that in two of those events, the differences at specific times as well as the time-averaged difference was significantly larger than two, ruling out a simple thin-thick target interpretation. In \\citet{Ba07}, the spectra of the five events were compared with the predictions of the ITTT model. The authors exploited the order of magnitude improvement in spectral resolution of RHESSI over the 4-point Yohkoh spectra and showed that most RHESSI observations could not be explained by the ITTT model. \n\n\\citet{Ba07} proposed that by considering non-collisional energy loss inside the loop this inconsistency could be resolved.  \nA possible mechanism that causes non-collisional energy loss is an electric field. Accelerating electrons out of the coronal source region drives a return current to maintain charge neutrality in the whole loop. For finite conductivity, Ohm's law implies that an electric field must be present. The beam electrons lose energy because of work expended in moving inside the electric potential. This produces a change in the shape of the electron spectrum at the footpoints. The formation and evolution of these return currents were studied by various authors \\cite[e.g.][]{Kn77,SS84,La89,Oo90}. \\citet{Zh06} proposed that return currents could explain the high energy break observed in flare HXR spectra. \nMost studies have, however, been theoretical proposals or numerical simulations, based on standard flare values, that do not attempt to explain or reproduce true solar flare observations. \n\n\\citet{Ba07} compared RHESSI spectra to the ITTT-model of \\citet{Wh95}, demonstrating that the qualitative shape and relations between coronal and footpoint-spectrum often do not agree with the model predictions. In this study, we take an additional step by completing a quantitative analysis of the relation linking coronal and footpoint spectra in the context of the thin-thick target model; we demonstrate that, in some cases, electric fields related to return currents can indeed explain the relation between coronal and footpoint spectra.\n\nIn Sect.~\\ref{Theory} we summarize the basic physical concepts applied in the paper. Section~\\ref{evdescription} provides a brief overview of the analyzed events and a description of the spectral analysis. In Sect. \\ref{Method}, we describe our calculation of the energy loss required to reproduce the observed footpoint spectrum, constrained by the coronal emission. Our results are presented in Sect.~\\ref{results}. In Sect. \\ref{retcurrnefield}, we link those results to the concept of return currents. \n\n\\section{Thin and Thick target emission} \\label{Theory}\n\nTwo types of bremsstrahlung emission are distinguished. If the electrons pass a target without losing a significant amount of energy, the corresponding emission is referred to as thin target \\citep{Da73}. This situation is expected to occur in coronal regions when electrons pass through a target of insufficient column density to stop them. If the electrons are fully stopped inside the target, the resultant emission is called thick target emission \\citep{Br71}. This is the case for the dense chromospheric material at the footpoints. \n\nFor an input power-law electron distribution of the shape $F(E)=A_{E}E^{-\\delta}$, the non-relativistic bremsstrahlung theory predicts power-law photon spectra \n\\[I(\\epsilon)\\propto \\epsilon^{-\\gamma}\\quad\\mbox{where}\\quad\\left\\{\\begin{array}{l} \\gamma=\\delta+1\\quad\\mbox{in the thin target case} \\\\ \\gamma=\\delta-1\\quad\\mbox{in the thick target case} \\end{array} \\right. \\] \nThe observable distinction between the two emission mechanisms is a difference $\\Delta \\gamma$ of value 2 in their observed photon spectral indices. \n\nAssuming a final column density in the coronal source, the coronal source spectrum is a thick target at low energies and a thin target at high energies with a break at some critical energy. The footpoint spectrum is depleted at low energies, as low energy electrons do not reach the chromosphere. An illustration of this can be found in \\citet{Wh95} or \\citet{Ba07}.\n\nIn the events that we analyze here, a thermal component is present in all observations. Observation of the non-thermal emission is therefore only possible at photon energies higher than 15 keV.  For these energies we can assume that the coronal source is a pure thin target and the footpoints are a pure thick target.\n\n\n\\section{Event description and spectral analysis} \\label{evdescription}\nThe two events that we analyze were described in detail by \\citet{Ba06, Ba07}. They were selected because of the significant differences between the footpoint and coronal non-thermal spectral index.  The first event occurred on 24 October 2003 around 02:00 UT (GOES M7.7), the second on 13 July 2005 around 14:15 UT (GOES M5.1). Both events occurred close to the limb but were not occulted; two footpoints were therefore fully observed in both cases. RHESSI light curves from the time of main emission are shown in Fig.~\\ref{2flimages}. An EIT image of the 24 October 2003 event and a GOES SXI image for the event of 13 July 2005 are presented for orientation. The contours of the coronal source and the footpoints from RHESSI images are overlaid. \n\\begin{figure*} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig1.eps}}\n\\caption {\\textit{Top}: RHESSI light curves in the 6-12 and 25-50 keV energy band. The analyzed time interval is indicated by the \\textit{vertical bar}. \\textit{Bottom}: SOHO\/EIT image at 24 October 2003 02:47:31 (\\textit{left}), GOES SXI image at 13 July 2005 14:19:04 (\\textit{right}). The 30 \\%, 50 \\% and 70 \\% contours from RHESSI Pixon images of the coronal source at 12-16 keV (\\textit{solid contours}) and 20-24 keV (\\textit{dashed contours}, mainly non-thermal emission) are overlaid along with the 50\\% contour of the footpoint sources at 25-50 keV.}\n\\label{2flimages}\n\\end{figure*}\n\n\n\\subsection{Spectral fitting and analysis} \\label{fitting}\n\nThe two events were analyzed using imaging spectroscopy with the PIXON  algorithm \\citep{Me96, Hur02}. Images were made in a 30 second time interval during which the flux was sufficiently high for good images but pile-up was low. The image times are given in Table~\\ref{fitpartable} and shown in Fig.~\\ref{2flimages}. The spectra of the footpoints and the coronal source were measured and fitted. The regions of interest from which the spectra were computed were chosen to be a circle around the coronal source and a polygon around the footpoints to include all of the emission at all energies. The effects of this method of region selection are discussed in \\cite{Ba05}. As a simplification, the footpoints were treated as one region and the spectrum was fitted with a single power-law. In the presence of the footpoints, the non-thermal emission in coronal sources is difficult to observe. Therefore, two methods of fitting the coronal source were used. First, a thermal component was fitted to the spectrum at low energies and a single power-law to the energies higher than about 25 keV. As a second method, the full sun thermal spectrum was fitted. As shown in \\cite{Ba05}, the thermal emission observed in full sun spectra is mostly coronal emission. We therefore used the thermal full sun fit as an approximation to the coronal thermal component and completed a power-law fit at the higher energies, while the thermal emission was fixed. This supports the idea that non-thermal emission exists in the coronal source and provides an estimate of the accuracy of the non-thermal coronal fit. The energy ranges for the fits were 8-36 keV for the coronal source and 24-80 keV for the footpoints. All fit parameters are provided in Table~\\ref{fitpartable}.\n\nIn the thin-thick target model an electron beam is assumed to be injected into the center of the coronal source.\nThe column depth that the electrons travel through in the corona is then $\\Delta N=n_e\\cdot l$, where the path length \\textit{l} is half the coronal source length. \nFrom RHESSI images in the 10-12 keV band, the source area \\textit{A} was measured to be the 50\\% contour of the maximum emission. We approximate the source volume to be $V=A^{3\/2}$ and the path length to be $l=\\sqrt{A}\/2$. Using the observed emission measure EM, the particle density is computed to be $n_e=\\sqrt{EM\/V}$ which corresponds to a column depth of $\\Delta N=\\sqrt{EM}A^{-1\/4}\/2$ expressed in observable terms.\nA volume filling factor of  1 was assumed for the computation of the density, which will be improved in Sect.~\\ref{expfemissionloss}.\nThe emission measures were taken from the spectral fits to the coronal source and to full sun spectra. Additionally, temperatures and emission measures observed by GOES were included. This provides a range for the emission measures, temperatures, and column depths, and an estimate of their uncertainty.  \\\\\n\n\n\\section{Method}\\label{Method}\n\nStarting from the assumption that the observed coronal spectrum at high photon energies is caused by thin target emission, we compute the electron distribution and therefore the expected footpoint photon spectrum. \nThis is completed via the following steps.\n\\begin{enumerate}\n\\item We assume that the observed coronal photon spectrum can be fitted by a power law:\n\\begin{equation}\n F^{cs}_{obs}(\\epsilon)=A_{\\epsilon}^{cs}\\epsilon^{-\\gamma^{cs}},\n\\end{equation} where A$_{\\epsilon}^{cs}$ is the normalization and $\\gamma^{cs}$ the photon spectral index.\n\\item Using thin target emission, the injected electron spectrum F(E) is then proportional to  \\label{itemelsp}\n\\begin{equation} \\label{bla}\nF(E)=A_E E^{-\\delta}\\sim\\frac{A_{\\epsilon}^{cs}}{\\Delta N}E^{-\\gamma^{cs}+1}\n\\end{equation} where $\\Delta N$ is the column depth the electrons travel through inside the coronal target \\citep{Da73}. \n\\item The expected thick target emission $F_{exp}^{fp}(\\epsilon)$ caused by this electron distribution in the footpoints can be computed as follows \\citep{Br71}:\n\\begin{equation}\n F^{fp}_{exp}(\\epsilon)=A_{\\epsilon,exp}^{fp}\\epsilon^{-\\gamma^{fp}_{exp}}\\sim\\frac{A_{\\epsilon}^{cs}}{\\Delta N}\\epsilon^{-(\\gamma^{cs}-2)}\n\\end{equation}\nThe superscripts fp and cs denote the footpoint and coronal source values, respectively.\n\\item The normalization and spectral index of $F^{fp}_{exp}(\\epsilon)$ is compared to the observed footpoint spectrum $F^{fp}_{obs}(\\epsilon)$.\n\\end{enumerate}\nIn thin-thick target models, the difference in spectral index $\\Delta \\gamma=|\\gamma^{cs}-\\gamma^{fp}|$ is 2. As the observed difference is larger than 2 in the selected events, a mechanism has to be found that causes the electron spectrum to harden while the beam passes down the loop. We present a mechanism that assumes an electric field which causes electrons to lose the energy $\\mathcal{E}_{loss}$ independently of the initial electron energy. The resulting spectrum is flatter, although not strictly a power-law function anymore (Fig.~\\ref{elsp}a). The deviation becomes substantial below 2 $\\mathcal{E}_{loss}$.\n\n\nThe necessary energy loss is determined as follows:\n\\begin{enumerate}\n\\item We start with the coronal electron distribution as found from Point~\\ref{itemelsp} in the above list.\n\\item By assuming a thin target, the electron distribution leaves the coronal source and propagates down the loop. A constant energy loss $\\mathcal{E}_{loss}$ is subtracted from the electron energies as the energy loss is independent of the electron energy.\n\\item We compute the expected thick target photon spectrum $F^{fp}_{exp}(\\epsilon)$ from this altered electron spectrum.\n\\item A power law is fitted to  $F^{fp}_{exp}(\\epsilon)$. The fitted energy range is 30-80~keV. This is the range for which footpoint emission is typically observed. \n\\end{enumerate}\nThe relation between the energy loss experienced  and the corresponding photon spectral index of the best-fit power law function depends on the initial electron spectral index $\\delta$ and the energy loss $\\mathcal{E}_{loss}$. It is equivalent to the elementary charge times the electric potential between the coronal source and the footpoints. If the initial electron spectral index is 8 for instance, the thick target photon spectral index without energy loss is 7. With increasing energy loss, this value decreases rapidly. The effect is less pronounced when the initial electron spectrum is harder. This is shown in Fig.~\\ref{elsp}b for several values of $\\delta$ and $\\mathcal{E}_{loss}$. Using the curves in this figure, we can easily determine the energy loss that causes an electron spectrum of spectral index $\\delta$ to result in a fitted photon spectral index $\\gamma^{fp}$. \n\n\\begin{figure*} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig2.eps}}\n\\caption {a) Normalized electron power-law spectrum (\\textit{solid}) and altered spectrum due to a constant energy loss of 30 keV (\\textit{dashed}). b) Relation between loss energy $\\mathcal{E}_{loss}$ and fitted thick target photon power-law spectral index $\\gamma^{fp}$ for initial electron spectral index $\\delta=3,4,5,6,7,8$ in the accelerator.}\n\\label{elsp}\n\\end{figure*}\n\n\n\n\n\n\n\\begin{table*}\n\n\\begin{minipage}[t][]{2\\columnwidth}\n\n\n\\caption{Overview of main event properties and fit parameters }\n\\renewcommand{\\footnoterule}{} \n\\begin{tabular}{l|lll|lll}  \n\\hline \\hline\nTime interval &\\multicolumn{3}{c}{24 October 2003 02:48:20-02:48:50 }& \\multicolumn{3}{c}{13 July 2005 14:15:00-14:15:30 }\\\\\n\\hline\nArea [cm$^2$] \/ Volume [cm$^3$]&\\multicolumn{3}{c}{$7.9\\cdot 10^{18}$ \/ $2.2\\cdot 10^{28}$}&\\multicolumn{3}{c}{$1.7\\cdot 10^{18}$ \/ $2.2\\cdot 10^{27}$} \\\\\n\\hline\n&full sun&imspec& GOES&full sun&imspec&GOES\\\\\n\\hline\nTemperature [MK]\\footnote{ Thermal parameters for three different measuring methods (RHESSI full sun fit, RHESSI imaging spectroscopy, GOES). See also comment in Sect~\\ref{obsspectra}. }&21.6&23.2&15.4&23.8&22.3&18.1\\\\\nEmission measure [$10^{49}$cm$^{-3}$]&0.98&0.44&3.4&0.22&0.22&0.47\\\\\nElectron density [$10^{10}$cm$^{-3}$]&2.1&1.4&3.9&3.2&3.2&4.6\\\\\nColumn density [$10^{19}$cm$^{-2}$]&2.9&2.0&5.5&2.1&2.1&3.0 \\\\\n\\hline\n&footpoints&cs fit 1&cs fit2&footpoints&cs fit 1&cs fit 2\\\\\n$\\gamma$\\footnote{ Two different values (cs fit 1, cs fit 2) for the non-thermal coronal fit, distinguishing the two fitting methods used (compare Sect~\\ref{fitting}).}&2.6&6.2&6.1&2.9&5.1&5.6 \\\\\n$F_{50}[\\mathrm{photons\\, cm^{-2}s^{-1}keV^{-1}}]$&2.0&0.07&0.07&0.88&0.06&0.05 \\\\\n\n\\hline\n\\end{tabular}\n\\label{fitpartable}\n\\end{minipage}\n\n\\end{table*}\n\n\n\n\n\\section{Results}\\label{results}\n\n\\subsection{Observed spectra} \\label{obsspectra}\n\nFigure~\\ref{spectra} shows the observed spectra overlaid with the spectral fits. As indicated in Sect. \\ref{fitting}, the thermal fits differ slightly from each-other. The main reason why the fits do not agree is the wider energy binning adopted by imaging spectroscopy. With this binning, the atomic lines are not resolved, contrary to full sun spectroscopy.\n\\begin{figure*} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig3.eps}}\n\\caption {Coronal and footpoint source spectra overlaid with the according fits. \\textit{Dots} are the measured footpoint spectrum, the \\textit{dashed-dotted} line indicates the fit to this spectrum in the range 30-80 keV. \\textit{Squares} indicate the observed coronal source spectrum. The \\textit{solid} lines provide the thermal and non-thermal fits as found from imaging spectroscopy. The \\textit{dotted} lines give the thermal fit to the full sun spectrum and the resulting non-thermal fit in imaging spectroscopy.}\n\\label{spectra}\n\\end{figure*}\n\nUsing the different fitting methods as an estimate of the uncertainty, an average difference in spectral index of $\\Delta \\gamma = 3.55 \\pm 0.07$ for the event of 24 October 2003 and $\\Delta \\gamma = 2.45 \\pm 0.35$ for the event of 13 July 2005 is found between the coronal source and footpoints. \n\\subsection{Expected footpoint emission and energy loss} \\label{expfemissionloss}\nAs described in Sect.~\\ref{Method}, we computed the electron distribution from the coronal source photon spectrum and the expected thick target emission (footpoint spectrum) caused by this electron distribution. Figure~\\ref{efieldspec} shows the measured spectra, the expected footpoint spectrum from a pure thick target, and the footpoint spectrum when introducing energy loss. \n\nTo compute the electron flux, the coronal column density is required. As given in Table~\\ref{fitpartable}, three different values of the column density were estimated. Using those, we are able to reproduce a range of possible electron spectra (Table \\ref{beamvalues}) and, therefore footpoint spectra. The confidence range of the footpoint spectra is indicated by a light gray (green) area in Fig.~\\ref{efieldspec}. \n\n\n\\begin{figure*} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig4.eps}}\n\\caption {Observed (fitted) power-laws of the non-thermal coronal source and footpoints (\\textit{solid}). The \\textit{light-gray (green)} area indicates the range of expected footpoint spectra without energy loss. The \\textit{dark-gray (red)} area marks the range of expected footpoint spectra when energy loss is applied to the electrons to find the same spectral index as the observed footpoint spectrum. The \\textit{dash-dotted} lines give the expected spectra at the footpoints if the only transport effect was Coulomb collisions of the beam electrons (cf. Sect.~\\ref{collisions}).}\n\\label{efieldspec}\n\\end{figure*}\n\n\nThe derived energy loss depends on the fitted coronal and footpoint spectra. For the two different coronal fitting methods, a range of loss energies $\\mathcal{E}_{loss}=[58.0,59.4]$ keV  is found for the event of 24 October 2003 and $\\mathcal{E}_{loss}=[8.7,26]$ keV for the event of 13 July 2005. The normalization of the new spectrum depends on the initial electron distribution. The initial electron distribution is computed according to Eq.~(\\ref{bla}). It depends on the column depth. If the column depth is lower, more electrons are needed to produce the same X-ray intensity. The resulting range of possible spectra is shaded in dark-gray in Fig.~\\ref{efieldspec}. As shown in the figure, the footpoint spectrum with energy loss reproduces well the observed footpoint spectrum for the event of 13 July 2005. \n\nDuring the event of 24 October 2003, the predicted footpoint spectrum is, however, an order of magnitude less intense than observed. In the context of the ITTT model, this implies that the electron flux density emanating from the coronal region is higher than predicted. The discrepancy may be explained by density inhomogeneities in the coronal source resulting in a smaller effective column density. The electron flux is underestimated if the non-thermal X-ray emission originates in regions that are less dense than average. In the following we therefore assume that the coronal source has an inhomogeneous density; this is represented by dense regions with filling factor smaller than 1 for thermal emission and, for the non-thermal coronal source in the 24 October 2003 event, a density that is lower by an order of magnitude. The observations do not allow to determine the filling factor.\nAs can be seen from Fig.~\\ref{2flimages}, the coronal source in the event of 13 July 2005 is very compact, while the source in the event of 24 October 2003 is more extended, showing isolated intense regions. This supports the assumption that the density is inhomogeneous in the 24 October 2003 coronal source and the true column density in the X-ray emitting plasma might be smaller than deduced from the measurements. Since $F_{exp}^{fp}(\\epsilon)$ is proportional to $1\/\\Delta N$, an effective column density of an order of magnitude less than the observed could produce the observed footpoint spectrum. \nIn the computations presented in Sect.~\\ref{retcurrnefield}, we assume an effective column density $\\Delta N_{eff} = \\Delta N\/14$ for the event of 24 October 2003.\n\n\n\\section{Return current and electric field} \\label{retcurrnefield}\nIn the above analysis, we assumed that the electrons experienced a constant energy loss while streaming down the loop. We now demonstrate that this energy loss could be caused by an electric potential in the loop that drives a return current. \nThere was much controversy surrounding the precise physical mechanism that generates the return current \\citep[eg.][]{Kn77, SS84, Oo90}. The basic scenario is the following: We assume that the electrons are accelerated in the coronal source region. When a beam of accelerated electrons, which is not balanced by an equal beam of ions, leaves this region, a return current prohibits charge build-up and the induction of a beam-associated magnetic field. In the return current, thermal electrons move towards the coronal source. Since their velocity is relatively small, they collide with background ions and cause resistivity. Ohm's law then implies the presence of an electric field in the downward direction.  \nThe return current density $j_{ret}$ can be derived from the equation of motion for the background electrons \\citep{Bebook}.\n\\begin{equation}\n\\frac{\\partial \\vec{v}}{\\partial t}+(\\vec{v}\\cdot \\triangledown)\\vec{v}=-\\frac{e}{m}\\vec{E_{ind}}-\\frac{e}{mc}(\\vec{v}\\times \\vec{B})-\\nu_{e,i}\\vec{v}\n\\end{equation}\nwhere $\\vec{E_{ind}}$ is the electric field induced by the return current, $\\vec{v}$ is the mean velocity of the electrons that represent the return current, and $\\nu_{e,i}$ is the electron-ion collision frequency. Using $\\vec{j_{ret}}=-en\\vec{v}$, this expression can be written as\n\\begin{equation} \\label{fullohm}\n\\left(\\frac{\\partial}{\\partial t}+\\nu_{e,i}\\right)\\vec{j_{ret}}=\\frac{(\\omega_p^e)^2}{4\\pi}\\vec{E_{ind}}+\\frac{e}{mc}[\\vec{j_{ret}}\\times (\\vec{B_0}+\\vec{B_{ind}})],\n\\end{equation}\nwhere $B_0$ is the guiding magnetic field and $B_{ind}$ the field induced by the beam. We neglect the last term on the right side of Eq.~\\ref{fullohm} by assuming that the beam and return currents are anti-parallel, oriented along the guiding magnetic field, and that the perpendicular component of $B_{ind}$ vanishes. We then obtain\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial t}+\\nu_{e,i}\\right)\\vec{j_{ret}}=\\frac{(\\omega_p^e)^2}{4\\pi}\\vec{E_{ind}}.\n\\end{equation}\n\nSince we consider a fixed time interval that is far longer than the collision time, we assume a steady state, neglecting the time derivative of the return current. The equation then takes the form of the classical Ohm's law:\n\\begin{equation}\n\\vec{j_{ret}}=\\frac{(\\omega_p^e)^2}{4\\pi \\nu_{e,i}}\\vec{E_{ind}}=\\sigma \\vec{E_{ind}}.\n\\end{equation}\n\n\nWe estimate whether the energy loss computed in Sect.~\\ref{results} is caused by this electric field.\nFrom the observations, we estimated the electron loss energy $\\mathcal{E}_{loss}$ that the electrons experience in the loop (Sect.~\\ref{expfemissionloss}). Assuming this loss is caused by the induced electric field $E_{ind}$ and across the distance \\textit{s} from the coronal source to the footpoints (i.e. half the loop length), we compute the electric field to be\n\\begin{equation}\nE_{ind}=\\frac{\\mathcal{E}_{loss}}{e\\cdot s}.\n\\end{equation}\n\nUsing Spitzer conductivity~\\citep{Spbook}, the term for the return current is related to the observed loss energy by:\n\\begin{equation} \\label{returneq}\nj_{ret}=6.9\\cdot 10^6T_{loop}^{3\/2}\\frac{\\mathcal{E}_{loss}}{e\\cdot s}\\quad \\mathrm{[statamp\/cm^2]},\n\\end{equation}\nwhere $T_{loop}$ is the temperature in the loop.\n\nOn the other hand, the beam current density can be written as\n\\begin{equation} \\label{jbeam}\nj_{beam}=\\frac{F_{tot}(E)}{A_{fp}}\\cdot e \\quad \\mathrm{[statamp\/cm^2]},\n\\end{equation}\nwhere $A_{fp}$ is the total footpoint area. The total electron flux per second $F^{tot}(E)$ is computed from the observed electron spectrum as follows: Let the electron spectrum be $F(E)=A_eE^{-\\delta}$. The total flux of streaming electrons per second above a cutoff energy $E_{cut}$ is then:\n\\begin{equation} \nF_{tot}(E)=\\int_{E_{cut}}^{\\infty}F(E) \\mathrm{d}E=\\frac{A_e}{\\delta-1}E_{cut}^{-(\\delta-1)}.\n\\end{equation}\n\nIn a steady state, the relation \n\\begin{equation}\nj_{beam}=j_{ret}\n\\end{equation}\n is valid. \\\\\n\nComparing the beam current as described in Eq.~(\\ref{jbeam}) with the return current from the observed energy loss according to Eq.~(\\ref{returneq}), we test whether the assumption of Spitzer conductivity holds.\n\n\n\\subsection{Results} \\label{retresults}\nTable \\ref{beamvalues} presents the relevant physical parameters necessary for the derivation of the beam- and return currents.\nFor Spitzer conductivity, the loop temperature $T_{loop}$ is required. Its is expected to have a value between the coronal source temperature and the footpoint temperature (see Table~\\ref{beamvalues}). As a first assumption, a mean temperature of $T_{loop}=15$ MK is chosen. The loop length is evaluated from RHESSI images, approximating the distance between the sources from the centroid positions and assuming a symmetrical loop structure. This provides a typical half loop length of $4\\cdot 10^9$ cm. \nThe footpoint area is measured from the 50\\% contour in RHESSI images in the 25-50 keV energy range, yielding a total footpoint area of $\\approx (6-7)\\cdot 10^{17}$ cm$^2$. \nThe beam current density depends critically on the electron cut off energy $E_{cut}$. We use a value of 20 keV. This gives an approximate lower limit to the total amount of streaming electrons.  \n\nUsing the presented observations, Eq.(\\ref{jbeam}) and by assuming Spitzer conductivity (Eq. \\ref{returneq}), the return current results to be of an order of magnitude higher than the beam current. This contradicts the assumptions of a steady state, and is also unphysical. \n\\begin{table*}\n\\caption{Values used for the computation of the beam and return currents and computed currents. } \n\\begin{tabular}{lrr}  \n\\hline \\hline\nEvent & 24 October 2003 & 13 July 2005 \\\\\n\\hline\nAssumed loop temperature $T_{loop}$ [MK] & 15 & 15 \\\\\n1\/2 Loop length s [cm] & $3.2\\cdot 10^9$&$4.3\\cdot 10^9$     \\\\\nElectron flux $F(E)$ [s$^{-1}$ keV$^{-1}$]&6.33$\\cdot 10^{41}E^{-5.2}$-1.5$\\cdot 10d^{42}E^{-5.2}$&1.7$\\cdot10^{40}E^{-4.1}-8.4\\cdot 10^{40}E^{-4.6}$\\\\\nElectron cutoff energy [keV] & 20 & 20  \\\\\nTotal footpoint area [cm$^2$] & $7.2\\cdot10^{17}$& $6.2\\cdot10^{17}$ \\\\\n$E_{loss}$ [keV] & 58-59.4 & 8.7-26 \\\\\nElectric field strength [statvolt\/cm] &$(6-6.3) \\cdot 10^{-8}$&$(6.7-20.3)\\cdot 10^{-9}$ \\\\\n\\hline\n$j_{ret}$[$statamp\/cm^2$]&$(2.4-2.5) \\cdot 10^{10}$& $(2.7-8.1)\\cdot 10^9$\\\\\n$j_{beam}$ [$statamp\/cm^2$]&$(5.1-14.4) \\cdot 10^9$&$(1.9-3.6) \\cdot 10^8$ \\\\\n\\hline\n\\end{tabular}\n\n\\label{beamvalues}\n\\end{table*}\n\n\n\\section{Discussion} \\label{discussion}\n\n\\subsection{Instability}\nIn Sect.~\\ref{retresults}, we assumed Spitzer conductivity when computing the return current which produced the unphysical result of $j_{ret}>j_{beam}$. Using Eq.~(\\ref{returneq}), the loop temperature required to maintain equality between the return current and beam current ($j_{ret}=j_{beam}$) can be computed. In the 24 October 2003 event, the loop temperature $T_{loop}$ would need to be smaller than 10 MK; in the 13 July 2005 event, $T_{loop}$ should be less than 3.9 MK. Such low loop temperatures are highly unlikely.\nHowever, it is possible that the return current is unstable to wave growth. For an extended discussion of instabilities in parallel electric currents, see e.g. \\citet{Bebook}. Instability causes an enhanced effective collision frequency of electrons in the return current and therefore a lower effective conductivity. The ion cyclotron instability develops if the drift velocity of the beam particles $V_d$ exceeds the thermal ion velocity $v_{th}^{ion}$ as follows: \n\n\\begin{equation} \\label{instcond}\nV_d\\ge15\\frac{T_i}{T_e}v_{th}^{ion}\n\\end{equation}\nwith $v_{th}^{ion}=\\sqrt{\\frac{k_BT_i}{m_i}}$ and $T_e$ and $T_i$ being the electron and ion temperatures, respectively. \n\nWe assume a steady state for which $j_{beam}=j_{ret}=n_e eV_d$, where $V_d$ is the mean drift velocity of the electrons constituting the return current. We therefore express $V_d$ as\n\\begin{equation}\nV_d=\\frac{j_{beam}}{n_e e}\n\\end{equation}\nand substitute this expression and that for $v_{th}^{ion}$ in Eq. (\\ref{instcond}). Assuming $T_e = T_i=T_{loop}$ and solving Eq.~(\\ref{instcond}) for $T_{loop}$ the instability condition holds\n\\begin{equation} \\label{tcond}\nT_{loop}\\le 2.3\\cdot 10^8 \\left(\\frac{j_{beam}}{n_e}\\right)^2 \\quad \\mathrm{[K]}.\n\\end{equation}\n\nSince the loop temperature and density are not known exactly, \nthis relation is illustrated in Fig.~\\ref{instab} for several values of $n_e$ and $T_{loop}$ typical in flare loops. For the values of $j_{beam}$ found for the observations of the two events, we find that instability occurs in the 24 October 2003 event for all values of $n_e$ and $T_{loop}$ in Fig.~\\ref{instab}.\nFor the 13 July 2005 flare, three distinct regions in the diagram can be found. The solid line indicates the relation of Eq. (\\ref{tcond}). Below this line, the return current is unstable. At high densities and low temperatures (lower right), the return current is stable and $j_{beam}=j_{ret}$ with Spitzer conductivity. The range of beam currents $j_{beam}$ deduced from the data allows for loop temperatures $< 3.9$ MK. \nIn the upper right quadrangle, Spitzer conductivity would imply $j_{ret}>j_{beam}$, which is unphysical. If the loop was in this parameter range, the current instability would be most likely saturated and $T_e>T_i$. This would shift the instability threshold in Fig.~\\ref{instab} to the right. Further, a loop in the state presented by the uppermost part of the figure (temperature above 10 MK, high density) would be detectable by the RHESSI satellite even in the presence of the coronal source. Since no loop emission is observed, we conclude that the loop is either less dense, cooler or both. The values in the upper right quadrangle are therefore unlikely. \n\n\n\\begin{figure} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig5.eps}}\n\\caption {Region of instability in the density\/temperature space for the event of 13 July 2005. The \\textit{grey} region indicates the densities and temperatures $T_{loop}$ for which the return current is unstable (Eq.~\\ref{instcond}). In the lower right part, the return current is stable and $j_{beam}=j_{ret}$ with Spitzer conductivity. The current in the event of 24 October 2003 is unstable for all values of density and temperature in the Figure.}\n\\label{instab}\n\\end{figure}\n\n\n\\subsection{Low energy electron cutoff}\nIn the above computations, a value of 20 keV for the electron cutoff energy $E_{cut}$ was assumed. This value is within the range for which the thermal and non-thermal components of the spectrum intersect. Values around 20 keV or higher are also supported by detailed studies of the exact determination of low-energy cutoffs \\citep[eg.][]{Sa05,Ve05,Sui07}. What if the cutoff energy were substantially lower than 20 keV? A cutoff energy as low as 10 keV would increase the total electron flux and therefore the beam current by an order of magnitude, leading to $j_{ret}\\approx j_{beam}$ for Spitzer conductivity. Conductivity could then not be reduced significantly by wave turbulence, and instability would be marginal. If the low energy cutoff were even lower than 10 keV, we would find that $j_{ret}< j_{beam}$. This could not be explained in terms of the model used here.\n\\subsection{Source inhomogeneity and filling factor} \\label{filling}\nIn the above paragraphs, it was demonstrated that the energy loss for the electrons due to an electric field could resolve the inconsistency in the difference between footpoint and coronal source spectral indices (Sect. \\ref{results}). For the event of the 24 October 2003, this produces footpoint emission that is lower than the observed emission (Fig. \\ref{efieldspec}). Images show that the coronal source in this event is not compact, but extended with brighter and darker regions. It is therefore possible that the standard density estimate, which favors high density regions, produces higher densities than average and that the effective column density of the regions, where the largest part of the non-thermal emission originates, is lower. This would provide a higher expected footpoint emission, in closer agreement with observations. \n\nAs mentioned in Sect.~\\ref{fitting}, a filling factor of 1 was used for the computation of the column density. A filling factor smaller than 1 would lead to even higher densities of the SXR emitting plasma. However, this would not affect the lower effective column density of the regions, where the HXR emission originates when assuming source inhomogeneity. \n\n\\subsection{Collisions and other possible scenarios} \\label{collisions}\nWhile return currents may not be the only means of attaining non-collisional energy loss, they are the most obvious and best studied. However, other scenarios are conceivable, which could produce a harder footpoint spectrum (or a softer coronal spectrum). \nIn the model presented here, collisional energy loss of beam electrons is neglected. This is a valid assumption for the following reasons: If collisions of the beam electrons in the loop were to play an important role, significant HXR emission should originate in the loop. Within the dynamical range limitations of RHESSI, this is not the case. However, GOES SXI and SOHO\/EIT images imply that the loop is filled with hot material. To study possible effects of collisions, we compared the change in the electron spectrum and the resulting footpoint spectrum for collisional energy loss and energy loss due to the electric field. The change in the electron spectrum due to collisions depends on the column depth through which the beam passes and was computed by \\citet{Le81} and \\citet{Br75}. We assume a column depth derived from the density in the loop times half the loop length. Assuming the same density as in the coronal source, we derive an upper limit to the collisional effects. The expected footpoint spectrum from purely collisional losses is indicated in Fig.~\\ref{efieldspec} as dash-dotted lines. Collisions affect the low energetic electrons most where a significant change in the spectrum is found. At the higher energies observed in this study, the spectrum does not change significantly. The neglect of collisional effects is therefore justified.\n\\citet{Br08} showed that in certain cases, emission from non-thermal recombination can be important, generating a coronal spectrum that is steeper than expected by the thin-target model. This could also produce a difference in the spectral index that is larger than two. Acceleration over an extended region \\citep[as proposed by][]{Xu08} could alter the electron distribution at the footpoints. If the distribution was harder at the edge of the region, a spectral index difference larger than 2 would result. A thorough comparison of such models with observations may be the scope of future work. \n\\section{Conclusions} \\label{conclusions}\nThe spectral relations between coronal and footpoint HXR-sources provide information about electron transport processes in the coronal loop between the coronal source and the footpoints. Most models neglect these processes in the prediction of the shape and quantitative differences between the source spectra. As shown by \\citet{Ba07}, the observations of some solar flares do not fit the predictions of such models, in particular the intermediate thin-thick target model by \\citet{Wh95}: there is a discrepancy concerning the difference in coronal and footpoint spectral indices, which is expected to be 2. \n\nWe have analyzed the two out of five events that display a spectral index difference larger than two in more detail. Such a behavior can be attributed to energy loss during transport that is not proportional to electron energy, but $\\mathcal{E}_{loss}\/E$ is larger at low energies. Such an energy loss causes the footpoint spectrum to flatten, which increases the difference in spectral indices. Two loss mechanisms come to mind immediately: Coulomb collisions and an electric potential. Figure 4 demonstrates that the assumption of an electric potential reproduces the observations more accurately.\n\nIn one of the two events, there remains a discrepancy between the observed and expected footpoint emission, such that the electron flux at the footpoints is larger than predicted. This flux was estimated from the observed non-thermal HXR (photon) flux and the observed thermal emission of the coronal source. We attribute the discrepancy to propagation or acceleration in low density plasma, which also heats the adjacent high-density regions. \n\nThe energy loss can therefore be explained by an electric field in the loop associated to the return current, which builds up as a reaction to the electrons streaming down the loop and the associated beam current. In a steady state ($j_{beam}=j_{ret}$), the return current is unstable to wave growth in one event for all realistic temperature and density parameters in the loop. The kinetic current instability drives a wave turbulence that enhances the electric resistivity by many orders of magnitude. This anomalous resistivity in turn significantly enhances the electric field. In the event of 13 July 2005, the return current may be stable if the loop density is high and the temperature is low, and Spitzer conductivity is applied. Both cases (out of five) present strong evidence for a return current in flares for the first time.\n\nTransport effects by return currents constitute a considerable energy input by Ohmic heating into the loop outside the acceleration region. It may be observable in EUV. Comprehensive MHD modeling including the coronal source, the footpoints, and the region in-between, may be the goal of future theoretical work.\n\n\n\n\n\n\\begin{acknowledgements}\nRHESSI data analysis at ETH Z\\\"urich is supported by ETH grant TH-1\/04-2 and the Swiss National Science Foundation (grant 20-105366). \nWe thank S\\\"am Krucker for helpful comments and discussions.\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe ability to design nanostructures which accurately self-assemble from \nsimple units is central to the goal of engineering objects and machines \non the nanoscale. \nWithout self-assembly, structures must be laboriously constructed in a step \nby step fashion.  Double-stranded DNA (dsDNA) has the ideal properties for \na nanoscale building block,\\cite{Seeman2003,Pitchiaya2006} \nwith structural length scales determined by the \nseparation of base pairs, the helical pitch and its persistence length \n(approximately 0.33\\,nm, 3.4\\,nm (Ref.\\ \\onlinecite{Saenger1984}) and \n50\\,nm,\\cite{Hagerman1988} respectively). \nOver these distances, dsDNA acts as an almost rigid \nrod and so it is capable of forming well-defined three dimensional structures.\n\nIt is the selectivity of base pairing between single strands, however, that makes DNA ideal for controlled self-assembly. \nBy designing sections of different strands to be complementary, a certain configuration of a system of oligonucleotides can be specified as the global minimum of the energy landscape. \nIn this way the target structure (usually consisting of branched double helices) can be `programmed' into the sequences. \nThis approach was initially demonstrated for a four-armed junction by the Seeman\ngroup in 1983.\\cite{Kallenbach83}\nSuch junctions and more rigid double crossover motifs\\cite{Fu93}\ncan then be used to create two-dimensional lattices.\\cite{Winfree98,Malo2005}\nYan \\it et al. \\rm  \\cite{Yan2003} have also constructed ribbons and two dimensional lattices from larger four-armed structures, each arm consisting of a junction of four strands.  \nFurthermore, using Rothemund's DNA ``origami'' approach an almost arbitrary \nvariety of two-dimensional shapes can be created.\\cite{Rothemund06}\n\nProgress in forming three-dimensional DNA nanostructures was initially much \nslower. The Seeman goup managed to synthesize a DNA cube\\cite{Chen91} and \na truncated octahedron,\\cite{Zhang94} but only after a long series of steps and \nwith a low final yield.\nMore recently, approaches have been developed that allow polyhedral cages, \nsuch as tetrahedra,\\cite{Goodman2005} trigonal bipyramids,\\cite{Erben07} \noctahedra,\\cite{Shih04,Anderson08} dodecahedra and truncated icosahedra,\n\\cite{He2008} to been obtained in high yields simply by cooling \nsolutions of appropriately designed oligonucleotides from high temperature. \nAdditional structures have also been produced using pre-assembled modular building blocks incorporating other organic molecules.\\cite{Aldaye2007,Zimmermann08} \n\nIn designing strand sequences, it is important to minimize the stability of competing structures with respect to the stability of the target configuration. \nIn addition, if systems can be designed to follow certain routes through configuration space---for example, by the hierarchical assembly of simple motifs \\cite{Pistol2006}---the target can potentially be reached more efficiently. \nA standard approach to hierarchical assembly, such as that described by \nHe {\\it et al.},\\cite{He2008}\ninvolves choosing sequences so that bonds between different pairs of oligonucleotides become stable at different temperatures. \nThis allows certain motifs to form in isolation at high temperatures before bonding to each other as the solution is cooled. \nAn alternative, elegant system for programming assembly pathways has been proposed by Yin \\it et al. \\rm \\cite{Yin2008}, which relies on the metastability of single stranded loop structures and the possibility of catalyzing their interactions using other oligonucleotides.\n\nGiven these recent experimental advances in creating DNA nanostructures, it would be useful to have\ntheoretical models that allow further insights into the self-assembly process.\nIn particular, a successful model would be able to provide information on the \nformation pathways and free energy landscape associated with the self-assembly, \nand as such would be of use to experimentalists wishing to consider increasingly more complex designs. \nAtomistic simulations of DNA would offer potentially the most spatially-detailed descriptions of the self-assembly.  \nHowever, they are computationally very expensive, and are generally restricted to time scales that are \ntoo short to study self-assembly.\\cite{Cheatham2004}\n\nStatistical approaches such as that of Poland and Scheraga \\cite{Poland1970} and the nearest-neighbour model \\cite{Everaers2007} use simple expressions for the free energy of helix and random coil states to obtain equilibrium results for the bonding of two strands. \nWhilst the parameters in these models can be tuned to give very accurate correspondence with experimental data,\\cite{SantaLucia1998,SantaLucia2004} they give no information on the dynamics and formation pathways and hence are only useful for ensuring that the target structure has significantly lower free energy than competing configurations. \nFurthermore, any description purely based on secondary structure (i.e.\\ which bases are paired) is inherently incapable of accounting for topological effects such as linking of looped structures.\\cite{Bois2005} \n\nCoarse-grained or minimal models offer a compromise  between detail and computational simplicity, and are well  suited to the study of hybridization of oligonucleotides.  The aim of these models is to be capable of describing  both the thermodynamic and kinetic behaviour of systems, a vital feature if kinetic metastability is inherent in assembly pathways.\\cite{Yin2008} In developing such minimal models the approach is usually to retain just those physical features of the system that are essential to the behaviour that is of interest. \n\nDauxois, Peyrard and Bishop models,\\cite{Dauxois1993} and modified versions such as that proposed by Buyukdagli, Sanrey, and Joyeux,\\cite{Buyukdagli2005} \nconstitute the simplest class of dynamical models. \nAlthough these are dynamic models in the sense that the energy is a function of the separation between each base, the nucleotides are constrained to move in one dimension. This lack of conformational freedom means that these models are \nincapable of capturing the nuances of the self-assembly from single-stranded DNA (ssDNA).\n\nRecently, models have been proposed which capture the helicity of dsDNA using two \\cite{Drukker2001} or three \\cite{Knotts2007}  interaction sites per nucleotide. \nThese models, however, are optimized for studying deviations from the ideal double-stranded state, and so have not been used to examine self-assembly.\nAlthough they have been used to study the thermal denaturation of dsDNA, \nit is essential for our purposes to be able to simulate the assembly of a \nstructure from ssDNA as it is this process that will reveal the kinetic traps \nand free energy landscape associated with the formation of a particular \nDNA nanostructure.  \n\nSimpler,  linear models, also with two interaction sites per nucleotide, have been used to investigate duplex hybridization,\\cite{Araque2006} hairpin formation \\cite{Sales-Pardo2005} and gelation of colloids functionalized with oligonucleotides.\\cite{Starr2006} \nThese models all use two interaction sites to represent one nucleotide, with backbone sites linked to each other to represent the sugar-phosphate chain, and interaction sites which represent the bases. This work investigates the possibility of extending the use of such coarse-grained models to study the self-assembly of nanostructures that involve multiple strands forming branched duplexes. We hypothesize that the self-assembly properties of DNA are dominated by the fact that ssDNA is a semi-flexible polymer with selective attractive interactions. \nWe introduce an extremely simple model, similar to that of Starr and Sciortino,\nto test this hypothesis. \\cite{Starr2006}  \nThis simplicity enables us to explore the thermodynamics and kinetics of self-assembly in the model in great depth, and hence examine the feasibility of simulating nanostructure formation through minimal models.\n\nWe first describe the model in Section \\ref{methods}, then examine its success in reproducing the general features of hybridization in Section \\ref{Duplex Formation}. \nNext in Section \\ref{Holliday Junction}, we apply it to the formation of a Holliday junction, a simple nanostructure consisting of a four-armed cross.\\cite{Malo2005}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.4cm]{Fig1.eps}\n\\end{center}\n\\caption{(Colour online) A schematic representation of the model. \nThe thick lines represent the rigid backbone monomer units and the large circles the repulsive Lennard-Jones interactions at their centres.\nThe smaller, darker circles represent the bases. \nThe panels illustrate the definitions of (a) the bending angle between two units ($\\theta$), and (b) the torsional angle ($\\phi$) which is found after the monomers have been rotated to lie parallel.}\n\\label{model picture}\n\\end{figure}\n\n\\section{Methods}\n\\label{methods}\n\\subsection{Model}\n\\label{model}\n\nWe introduce an off-lattice model inspired by that which Starr and Sciortino \nused to study the gelation of four-armed DNA dendrimers.\\cite{Starr2006} \nAs our aim is to reproduce the basic physics with as simple a \nmodel as possible, we neglect contributions to the interactions\ndue to base stacking, and the charge and asymmetry of the phosphate backbone. \nWe do not attempt to include the detailed geometrical structure \nof DNA, but instead\nrepresent the oligonucleotides as a chain of monomer units,\neach corresponding to one nucleotide (Fig.\\ \\ref{model picture}). \nA monomer consists of a rod (chosen to be rigid for simplicity) of length $l$ \nwith a repulsive backbone interaction site at the centre of the rod.\nIn addition, each unit has a bonding interaction site (or base) at a distance \nof $0.3\\,l$ from the backbone site \n(perpendicular to the rod). Each monomer is also assigned a base type \n(A,G,C,T) to model the selective nature of bonding. In this model we only \nconsider bonds between the complementary pairs A-T and G-C.\n\n\nWe do not explicitly include any solvent molecules in our simulations, but instead use effective potentials\nto describe the interactions between the DNA.\nSites interact through shifted-force \nLennard-Jones (LJ) potentials, where, as well as truncating and shifting the \npotential, an extra term is included to ensure the force goes smoothly to zero \nat the cutoff $r_c$. \nFor $r < r_{c}$\n \\begin{equation} \nV_{\\rm sf}(r)=V_{\\rm LJ }(r) - V_{\\rm LJ}(r_{c}) - \n              (r-r_{c})\\left.\\frac{dV_{\\rm{LJ}}}{dr}\\right|_{r=r_c}\n \\label{potential} \n \\end{equation} \nwhere\n \\begin{equation} \n V_{\\rm LJ}(r) = 4\\epsilon \\left [\\left({\\sigma \\over r} \\right) ^{12} - \\left( {\\sigma \\over r} \\right) ^{6} \\right], \\label{LJ} \n\\end{equation} \nand $V_{\\rm sf}(r) = 0$ for $r\\ge r_{c}$. \nBackbone sites (except adjacent units on the same strand) interact through \nEq.\\ (\\ref{potential}) with $\\sigma = l$ and $r_{c} = 2^{1\/6}\\sigma$. \nThis purely repulsive interaction models the steric repulsion between strands. \nBonding sites (again excluding adjacent units on a strand) interact via \nEq.\\ (\\ref{potential}) with $\\sigma = 0.35\\,l$ and  $r_{c} = 2.5\\,\\sigma$ for complementary bases (to allow for attraction) and $r_{\\rm c} = 2^{1\/6}\\sigma$ for all other pairings. \nThe depth of the resulting potential well between complementary bases, \n$\\epsilon^{\\rm eff}_{\\rm base}$, is $0.396\\epsilon$. In what follows we will measure\nthe temperature in terms of a reduced temperature, \n$T^*=k_B T\/\\epsilon^{\\rm eff}_{\\rm base}$. \nThe above choice of parameters ensures that the attractive interaction between complementary bases is largely shielded by backbone repulsion. Monomers therefore bond selectively and can only bond strongly to one other monomer at a given instant. \nThese are the key features of Watson-Crick base pairing that make DNA so useful for self-assembly.\n\nThe model also includes potentials between consecutive monomers associated with bending and twisting the strand: \n\\begin{equation} \nV_{\\rm bend} = \\begin{cases}\n                k_1 (1-\\cos(\\theta))& \n                        \\text{if $\\theta < {3\\pi\\over4}$},\\\\ \n\t\t\\infty& \\text{otherwise}\n\t\t\t\\end{cases}\n\\label{bend} \n\\end{equation}\nand \n\\begin{equation}\nV_{\\rm twist} = k_2 (1-\\cos(\\phi)).\n\\label{twist}\n\\end{equation}\nWe define $\\theta$ as the angle between the vectors along adjacent monomer rods. As previously mentioned, consecutive backbone sites do not interact via LJ potentials. \nInstead, a hard cutoff is introduced in Eq.\\ (\\ref{bend}) to reflect the fact that an oligonucleotide cannot double back on itself. \n$\\phi$ is taken as the angle between adjacent backbone to bonding site vectors after the monomers have been rotated to lie parallel (Fig.\\ \\ref{model picture}). \nFor simplicity, we choose the torsional potential to have a minimum at $\\phi=0$. Thus, neither ssDNA or dsDNA\nwill be helical in our model.\n\n$k_1$ is chosen to be $0.1\\,\\epsilon$ to give a persistence length, $l_{\\rm ps}=3.149\\,l$ at a reduced temperature of $T^*=0.09677$ for ssDNA. \nWe obtain this result by simulating a single strand 70 bases in length, and using the definition:\\cite{Cifra2004}\n\\begin{equation}\nl_{\\rm{ps}} = \\frac{\\langle {\\bf{L} \\cdot \\bf{l_{\\rm 1}}}\\rangle} {\\langle l_{\\rm 1} \\rangle},\n\\end{equation} \nwhere $\\bf{L}$ is the end to end vector of the strand, $\\bf{l_{\\rm 1}}$ is the vector associated with the first monomer and $\\langle \\rangle$ indicates a thermal average. \nTaking $l$ to be 6.3{\\AA}, $T^*=0.09677$ is mapped to $24^\\circ$C and so the model is consistent with experimental data for ssDNA in 0.445M NaCl solution.\\cite{Murphy2004,lengthscale} $k_2$ is chosen to be $0.4\\,\\epsilon$.\n\nIn neglecting the geometrical structure of a double helix we do not accurately represent certain types of bonding. \n`Bulged' bonding occurs when consecutive bases in one of the strands attach to non-consecutive bases in the other strand. \n`Internal loops' consist of stretches of non-complementary bases (either symmetric or asymmetric in the number of bases involved in each strand). \n`Hairpins' result when a single strand doubles back and bonds to itself. The details of these motifs are complicated but an empirical description of their thermodynamic properties is given in Ref.\\ \\onlinecite{SantaLucia2004}. \nImportantly, they are generally penalized due to the disruption of the geometry of DNA in a way which is not well reproduced by our model. \nIn the case of the short strands we consider, these motifs will only play a small role as the strands are not specifically intended to have stable structures of these forms. \nIn fact, as the base sequences we use were designed to form the Holliday junction, the possibility of forming these motifs at relevant temperatures was deliberately avoided.\\cite{MaloThesis}\n\nFor simplicity we therefore include only two alterations to the model. Firstly, we define `kinked states' as those for which the number of unpaired bases between two bonding pairs on either side of a duplex is not equal (including asymmetric loops and bulges). We impose an infinite energy penalty on the formation of these kinked states if the total number of intermediate bases is less than six. Secondly, we treat complementary units within six bases of each other on the same strand as non-complementary, but allow all other hairpins without penalty. \n\nIt should be also noted that this model neglects the directional asymmetry of the sugar-phosphate backbone. Therefore,\nparallel as well as anti-parallel bonding is possible in our model, whereas parallel bonding does \nnot occur in experiment.    \n\n\\subsection{Monte Carlo Simulation}\n\\label{Monte Carlo Simulation}\n\nIn a fully-atomistic model of DNA the natural way to simulate its dynamics would be to use molecular dynamics. However, the best way to simulate the dynamics\nin a coarse-grained model is an important, but not fully resolved, question, and\none that will depend on the nature of the model. Clearly, for the current model\nstandard molecular dynamics is inappropriate as it will lead to ballistic \nmotion of the strands between collisions because of the absence of explicit \nsolvent particles, whereas DNA in solution undergoes diffusive Brownian motion. \nAn alternative approach is to use Metropolis Monte Carlo (MC) \nalgorithm\\cite{Metropolis1953,Frenkel2001} where the moves \nare restricted to be local, as it has been argued that this can provide a \nreasonable approximation to the dynamics.\\cite{Kikuchi91,Berthier07,Tiana07} \nThis is the approach that we use here to simulate the dynamics of \nself-assembly of DNA duplexes and Holliday junctions. \nIn particular, the local MC moves that we use are translation and \nrotation of whole strands and bending of a strand about a particular monomer,\nthus ensuring that the strands undergo an approximation to diffusive Brownian \nmotion in the simulations.\nTherefore we expect the MC simulations, which are all initiated with free single strands, to mimic the real self-assembly processes in our model. It is \nimportant to note that this will include, as well as successful assembly into the target structure, kinetic trapping in non-equilibrium\nconfigurations and that when the latter occurs this reflects the inefficiency of\nthe self-assembly under those conditions.\n\n\nAlthough a true measure of time is impossible in Monte Carlo simulations, an approximate time scale for diffusion-limited processes can be found by comparing the diffusive properties of objects to experiment. \nBy measuring the diffusion of isolated strands, and assuming diffusion coefficients comparable to those of double strands and hairpin loops of similar length,\\cite{Lapham1997} we conclude that one step per strand corresponds to a time scale of approximately 2\\,ps. \nThus, our model allows our systems to be studied on millisecond time scales. \n\nAt the end of the above MC simulations, our systems will not necessarily have reached equilibrium, \nboth because the energy barriers to escape from misbonded configurations can be difficult to \novercome at low temperature and the low rate of association at higher \ntemperatures.\nTherefore, as a comparison we also compute\nthe equilibrium thermodynamic properties of our systems using umbrella \nsampling.\\cite{Torrie1977,Frenkel2001} \nFormally, we can write the thermal average of a function $B(\\bf{r}^N)$ in the canonical ensemble as:\n\\begin{equation}\n\\langle{B}\\rangle = \\frac\n   {\\int{\\frac{B}{W(Q)} \\left[{W(Q) \\exp(-V\/\\rm{k_B}\\it T)}\\right]} d\\bf{r}^{N}} \n   {\\int{\\frac{1}{W(Q)}\\left[{W(Q) \\exp(-V\/\\rm{k_B} \\it T)}\\right]} d\\bf{r}^{N}} \n\\label{US avg}\n\\end{equation}\nwhere $Q=Q(\\bf{r}^N)$ is an order parameter or reaction coordinate and $V=V(\\bf{r}^N)$ is the potential energy. \nWe are free to choose $W(Q)$, and by taking the term in square brackets as the weighting of states and keeping statistics for $B\/W$ and $1\/W$ at each step we can find $\\langle{B}\\rangle$. \nIn standard Metropolis MC, $W=1$, but by choosing $W(Q)$ in such a way that those states \nwith intermediate values of $Q$ are visited more frequently, the effective free energy barrier between \n(meta)stable states can be lowered allowing the system to pass easily between the free energy minima, and equilibrium to be reached.\n\nTo ensure that each value of $Q$ is equally likely to be sampled in an \numbrella sampling simulation, one would choose $W(Q)=\\exp(\\beta A(Q))$,\nwhere $A(Q)$ is the free energy as a function of the order parameter. \nTo achieve this, however, would require knowledge of $A(Q)$. \nInstead, there are standard methods to \nconstruct $W(Q)$ iteratively, but for the current examples it was possible to\nconstruct $W(Q)$ manually, because of the relative simplicity of the free\nenergy profiles.\n\nTo a first approximation the interaction between fully bonded structures is negligible. \nTherefore, in the umbrella sampling simulations we consider systems containing the minimum number of strands \nrequired to form a given object (two for a duplex and four for a Holliday junction). We then use the relative weight of bound and free states to extrapolate the expected fractional concentrations for larger systems.\\cite{Ouldridgeunpub}\nThe natural choice for the order parameter $Q$ is the number of correct bonds, where \ntwo monomers are defined to be bonded if their energy of interaction is negative.\n\n\\section{Results}\n\\subsection{Duplex Formation}\n\\label{Duplex Formation}\n\nWe test the model by analysing the duplex bonding of two different complementary strands. \nWe simulate systems of ten oligonucleotides, initially not bonded, in a periodic cell with a concentration of $5.49\\times10^{-5}$ molecules\\,$l^{-3}$\n(or $3.65\\times10^{-4}$\\,M). \nWe separately consider strands consisting of 7 and 13 monomers, which correspond to two of the arms of the Holliday junction studied experimentally by Malo {\\it et al.}\\cite{MaloThesis,Malo2005} and which we consider in Section \\ref{Holliday Junction}:\n\\begin{equation}\n\\text{7 bases} \\begin{cases}\n\t\t\t\t\t\\text{G-A-G-T-T-A-G}\\\\\n\t\t\t\t\t\\text{C-T-A-A-C-T-C}\n\t\t\t\t\t\\end{cases}\n\\label{7 bases}\t\t\t\t\t\n\\end{equation}\n\\begin{equation}\t\t\t\t\t\n\\text{13 bases} \\begin{cases}\n\t\t\t\t\t\\text{G-C-G-A-T-G-A-G-C-A-G-G-A}\\\\\n\t\t\t\t\t\\text{T-C-C-T-G-C-T-C-A-T-C-G-C}\n\t\t\t\t\t\\end{cases}\n\\label{13 bases}\n\\end{equation}\nwhere we have listed strands in the 5'--3' sense for consistency with the literature. \nThe yields of correctly-bonded and misbonded structures at the end of the simulations are depicted in Figure \n\\ref{7 & 13 mers} as a function of temperature.\nWe also display the predicted equilibrium fraction of correctly bonded strands for both systems obtained using umbrella sampling. \nFor convenience, we define a correctly bonded structure to have more than 70\\% of the bonds of the complete duplex and no bonds to other strands. Any other structure is recorded as `misbonded'.\n\n\\begin{figure}\n\\includegraphics[width=6.2cm,angle=-90]{Fig2.eps}\n\\caption{(Colour online) Yields of correctly-formed duplexes and misbonded configurations at the end \nof our MC simulations (lines with data points, as labelled) compared to the equilibrium probability of \nthe strands adopting the correct structure as obtained by umbrella sampling.\nThe solid and dashed lines represent results for strands with 7 and 13 monomers, respectively.\nThe MC results are averages over ten runs of length $3\\times10^8$ steps per strand with ten strands in \nthe simulation cell. }\n\\label{7 & 13 mers}\n \\end{figure}\n\nFigure \\ref{7 & 13 mers} shows a maximum in the yield as a function of temperature. \nSuch behaviour is typical of self-assembling systems \n\\cite{Brooks06,Hagan2006,Wilber2007,Rapaport08,Whitelam08} and reflects the \nthermodynamic and dynamic constraints on the self-assembly process. \nFirstly, the yield is zero at high temperature where only ssDNA is stable, and rises just below the expected \nequilibrium value as the temperature is decreased, \nthe deviation arising due to the large number of steps required to reach equilibrium. \nAt low temperatures, the yield falls away due to the presence of kinetic traps which are now stable with respect to isolated strands, as evidenced by the rise in `misbonded structures' in Figure \\ref{7 & 13 mers}. \nThus, there is a non-monotonic dependency of yield on temperature and an optimum region for successful assembly, which corresponds to the region where only the desired structure is stable against thermal fluctuations. \nFigure \\ref{5 13mers} is a snapshot from near the end of a simulation in this regime. \nIt should be noted that neglecting helicity has the effect of increasing the flexibility of dsDNA in the direction \nperpendicular to the plane of bonding \n(a helix cannot bend in any direction without disturbing its internal structure whereas a `ladder' can). \n\n\\begin{figure}\n\\includegraphics[width=8.4cm]{Fig3.eps}\n\\caption{(Colour online) Snapshot of a fully-assembled configuration in a MC simulation of\nten 13-base strands  at $T^*=0.0971$.\nIn this image the colour of the backbone indicates the type of strand: red for G-C-G-A-T-G-A-G-C-A-G-G-A and grey for its complement.\nBackbone sites are indicated by the large spheres, and bases by the small, blue spheres.}\n\\label{5 13mers}\n\\end{figure}\n\nThe heat capacity obtained from umbrella sampling of pair formation is shown in Figure \\ref{thermo_duplex}(a). \nThe heat capacity peaks indicate a transition from single strands to a duplex. \nAs the formation of duplexes is essentially \na chemical equilibrium between monomers and clusters of a definite size \n(in this case two)\nthe width of the peaks will remain finite as the number of strands is increased. \nThe transition does, however, become increasingly narrow as the DNA strands become longer, as is evident from comparing the \nheat capacity peaks for the 7-mer and 13-mers.\n\n\\begin{figure}\n\\includegraphics[width=8.4cm]{Fig4.eps}\n\\caption{Thermodynamics for the formation of a single duplex. \n(a) Heat capacity curves for 7- and 13-base systems, as labelled.\n(b) Free energy profile associated with the formation of a 13 base pair duplex at different temperatures \n(as labelled), where $Q$ represents the number of correctly formed bonds in the duplex.}\n\\label{thermo_duplex}\n\\end{figure} \n\nFigures \\ref{thermo_duplex}(b) shows the free energy profile, $F(Q)$, for the \nformation of a duplex. \nThe initial peak at low $Q$ is accounted for by the entropic cost of bringing two strands together. \nOnce bonds are formed, however, adding extra bonds costs much less entropy whilst providing a significant decrease in energy, explaining the monotonic decrease in $F(Q)$ beyond $Q=2$. \nThe rise between $Q=1$ and $Q=2$ is partly due to the fact that in order to form two bonds between strands the relative orientation of strands must be specified whereas this is not true for $Q=1$: \nhence there is an additional entropy penalty to the formation of the second bond. \nIn addition, there exist structures with only one correct bond that are \nstabilized by additional incorrect bonds and these misbonded configurations \nalso contribute to $F(1)$. \nThe constant gradient above $Q=2$ indicates that the energetic gain and entropic cost of forming an extra bond are approximately constant at a given temperature, which is consistent with the assumptions underlying nearest-neighbour models of DNA melting.\\cite{Everaers2007}\n\n\\begin{figure}\n\\includegraphics[width=6.2cm,angle=-90]{Fig5.eps}\n\\caption{(Colour online) The bulk equilibrium probability of strands being in \na correct duplex extrapolated from our umbrella sampling simulations (solid lines) compared to the predictions of an empirical two-state model (dashed lines).\nResults are presented for strands with 7 and 13 bases, as labelled.\nFor the two-state model, as well as the results for the sequences\nin Eqs.\\ (\\ref{7 bases}) and (\\ref{13 bases}) (lines (a) and (d)),\nsequences corresponding to the other two arms of the Holliday junction\n(Figure \\ref{HJ schematic}) are considered.\nOnly one line is shown for the umbrella sampling results, because A-T and G-C\nhave the same binding energy in our model.\nTemperatures in our model are converted to  ${\\rm^o}$C using the same mapping given in Section II A.}\n\\label{2 state}\n\\end{figure} \n\n In Figure \\ref{2 state}, we compare our melting curves to those predicted by a simple two-state model,\\cite{Everaers2007} using the same mapping of the reduced temperature as in Section \\ref{model}. In the two-state model the molar concentrations of product (AB) and reactants (A,B) are given by the equilibrium relation: \n\\begin{equation}\n\\frac{\\left[{\\rm AB }\\right]}{\\left[{\\rm A} \\right] \\left[{\\rm B}\\right]} = \\exp{\\left({\\frac{-\\Delta H_0 + T \\Delta S_0}{\\rm{k_B}\\it T}}\\right)},\n\\label{two state model}\n\\end{equation}\nwhere $\\Delta H_0$ and $\\Delta S_0$ are assumed to be constants which depend only on the strand sequences and the salt concentration (we take $\\rm{[Na^+}] = 0.445\\rm{M}$ as in Section \\ref{model}). \nWe use the enthalpy and entropy changes of duplex formation calculated by ``HyTher\",\\cite{Hyther} a program that estimates these values using the ``unified oligonucleotide nearest neighbour parameters\".\\cite{SantaLucia1998, Peyret1999} \nThe authors claim that the thermodynamic parameters predicted by HyTher give the melting temperature $T_{\\rm m}$ (the temperature at which the fraction of bonded strands is 1\/2)\nof a duplex to within a standard error of $\\pm2.2^{\\rm{o}}\\rm{C}$.\\cite{SantaLucia1998}\nFigure \\ref{2 state} shows that our system reflects the melting temperatures predicted by the two-state model with reasonable accuracy, excepting sequence dependent effects which are not included in our model, because the interaction energies \nbetween A-T and C-G complementary base pairs have for simplicity been taken to be the same. The widths of transitions are seen to be of the same order, but slightly larger for our model. \nThis feature, which is typical of coarse-grained models,\\cite{Knotts2007} indicates that the degree of entropy loss on hybridization is too small in our model, \nand is due to a failure to accurately incorporate all degrees of freedom which become frozen on hybridization. \nHowever, the agreement is sufficiently good that the basic features of physical DNA assembly should be reproducible.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=15cm]{Fig6.eps}\n\\end{center}\n\\caption{(Colour online) Snapshots illustrating four stages in the process of displacement. \n(a) A third strand binds to a misbonded pair. \n(b) The third strand is prevented from forming a complete duplex by the misbond. \n(c) Thermal fluctuations cause bonds in the misbonded structure to break and be replaced by the correct duplex. \n(d) The misbonded strand is displaced and the correct duplex is formed.}\n\\label{displacement}\n\\end{figure*}\n\nA further satisfying feature of the model is that `displacement' was observed on several occasions. This process, during which a misbonded pair of strands is broken up by a third strand, is illustrated in Figure \\ref{displacement}. The third strand is able to bond to the pair, as some bases are free in the misbonded structure. \nThermal fluctuations allow the new strand to bond to sites previously involved in misbonding, in a process known as `branch migration'. Eventually one of the misbonded strands is completely displaced, leaving a correct duplex and an isolated single strand. This behaviour is observed in real DNA systems, and is the driving mechanism of some nanomachines \\cite{Yurke2000} and DNA catalyzed reactions.\\cite{Yin2008, Zhang2007}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.4cm]{Fig7.eps}\n\\end{center}\n\\caption{(Colour online) A schematic diagram showing the sequences of the strands used in our Holliday junction simulations, and the alternative bound states that are possible: (a) the square planar configuration and (b) the $\\chi$-stacked form.}\n\\label{HJ schematic}\n\\end{figure}\n\n\\subsection{Holliday Junction}\n\\label{Holliday Junction}\nEncouraged by the above results,\nwe next apply the model to the formation of a Holliday junction. \nHolliday junctions consist of four single strands which bind to form a four-armed cross. \nIn our case we consider a Holliday junction with two long arms (13 bases long) and two short arms (7 bases long). \nWe use the experimental base ordering of Malo {\\it et al.}\\cite{MaloThesis} with the `sticky ends' removed. \n(These sticky ends consist of six unpaired bases on the end of arms and their purpose is to allow the Holliday junctions to bond together to form a lattice). \nThe sequences of the four DNA strands and schematic diagrams of the \npossible junctions that they can form are shown in Figure \\ref{HJ schematic}.\n\nInitially we studied a system of 20 strands (five of each type) that has the potential to form five separate junctions. \nWe use a concentration of $1.56 \\times 10^{-5}$ molecules\\,$l^{-3}$\n(which corresponds to $1.04 \\times 10^{-4}$M). \nThe results are displayed in Figure \\ref{20mers}. \n\n\\begin{figure}\n\\includegraphics[width=6.0cm,angle=-90]{Fig8.eps}\n\\caption{(Colour online) \nA comparison of the kinetics and thermodynamics for a system of 20 strands that \ncan potentially form five Holliday junctions, where the MC simulations\nare initiated from a purely single-stranded configuration. \nThe MC results (lines with data points) are the final yield of Holliday junctions, and the fraction of strands involved in a correctly-formed long ($\\alpha$) or \nshort ($\\beta$) arm, or in misbonding, as labelled.\nThe results are averages over five runs of length $10^{9}$ steps per strand.\nFor comparison, the equilibrium probabilities of being in a $\\alpha$-bonded dimer\nand a Holliday junction are also plotted, along with the equilibrium probability of \nbeing in a $\\beta$-bonded dimer if the longer arms are not allowed to hybridize.\n}\n\\label{20mers}\n\\end{figure}\n\nThe results are as expected for the bonding of the longer arms (which we now describe as `$\\alpha$-bonding'). \nThe yield again displays the characteristic non-monotonic dependence on temperature. \nWe obtain very few complete junctions, however, which is due to two effects. \nFirstly, each simulation is performed at constant temperature, which means the hierarchical route to assembly is less favoured than when the system is cooled, as in the experiments.\\cite{MaloThesis,Malo2005}\nWhen the system is gradually cooled, Figure \\ref{20mers} suggests that at around $T_{\\rm m}(\\alpha) =0.111$ we would expect to find a region in which only $\\alpha$-bonded dimers were stable with respect to ssDNA. \nIf the cooling was sufficiently slow on the timescale of bonding, all strands would form $\\alpha$-structures at around $T_{\\rm m}(\\alpha)$. \nAt lower temperatures, when the Holliday junction becomes stable with respect to the $\\alpha$-bonded dimers, many competing minima would then be inaccessible to the system as they would require the disassociation of stable $\\alpha$-bonded pairs. \nThe free energy landscape of two $\\alpha$-structures forming a Holliday junction is consequentially much simpler than that of four single strands forming a junction at a given temperature.  \nTherefore, one expects the yield for self-assembly at constant temperature to be \nlower than when the system is cooled, because there is only a relatively narrow \ntemperature window between where the Holliday junction becomes stable and misbonded\nconfigurations start to appear. Indeed, the shorter arms are only marginally more \nstable than some competing minima, as evidenced by \nthe rise in misbonded structures in Fig.\\ \\ref{20mers} at temperature just below \nwhere $\\beta$-bonded structures first appear.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.4cm]{Fig9.eps}\n\\end{center}\n\\caption{(Colour online) Snapshot showing five Holliday junctions formed at $T=0.0842$ after \n$5.67 \\times 10^8$ MC steps per strand. Again, the backbone colour indicates strand type (1: red, 2: grey,\n 3: orange, 4: yellow) where numbers refer to Figure \\ref{HJ schematic}}\n\\label{5 HJs}\n\\end{figure}\n\nSecondly, even in the temperature range where Holliday junction formation is \nnot hindered by the formation of misbonded configurations, the yield is low\nbecause the Metropolis MC algorithm artificially reduces the diffusion of bound \npairs, and hence the likelihood that two pairs of $\\alpha$-bonded strands come together to form a junction is also reduced.\nThis is because the acceptance probability of trial moves for bonded strands is much lower than for isolated strands \\cite{Luijten2006}, due to the energy penalty associated with trying to move a bound pair apart. \n\n\\begin{figure}\n\\includegraphics[width=6.0cm,angle=-90]{Fig10.eps}\n\\caption{(Colour online) The yields of Holliday junctions (HJ) and misbonded configurations \nfor MC simulations, where the initial configuration was a pair of \n$\\alpha$-bonded dimers. For comparison, \nthe equilibrium probabilities of being in a $\\alpha$-bonded dimer \nand a Holliday junction are also plotted.\nThe results are averages over five runs of length \n$7.5 \\times 10^8$ steps per strand.\n}\n\\label{HJ thermo}\n\\end{figure}\n\nInterestingly, examination of the equilibrium lines in Figure \\ref{20mers} shows that the Holliday junctions are actually stable at a higher temperature than the individual shorter arms. \nThis is because the total loss of entropy when two $\\alpha$-bonded dimers bind together is considerably less than that for two short arms in isolation (as fewer translational degrees of freedom are lost), whereas the energy change is comparable. \nThus, there is a small temperature window at $T^*\\approx 0.1$ where hierarchical assembly can occur at constant temperature as the short arms are only stable once $\\alpha$-bonding has taken place. \nHowever, due to the deficiencies in the MC simulations mentioned above, the yield\nof Holliday junctions in this region is practically zero. Instead, the maximum\nyield of Holliday junctions occurs at lower temperatures where non-hierarchical\npathways that proceed by the addition of single strands become feasible.\n\nThe above simulations were only able to successfully model the first stage of \nthe Holliday junction assembly, namely the formation of $\\alpha$-bonded dimers. \nTo probe the second stage of assembly, we must first make two modifications to \nour simulation approach to overcome the two deficiencies mentioned above. \nFirstly, we study systems initially consisting of pairs of $\\alpha$-bonded strands, which we assume have successfully formed at some higher temperature---this is \nreasonable given the results of our earlier simulations.\nSecondly, we also include simple local cluster moves in addition to those which move only one strand, i.e.\\ translations, rotations and bending of pairs of $\\alpha$-bonded strands. \nWith these changes incorporated, we simulate the same system for $7.5 \\times 10^8$ steps per strand at a range of temperatures below $T_{\\rm m}(\\alpha)$. \nIt should be noted that due to a change in the size of typical moves, one move per strand now corresponds to approximately 10ps.\n\nWe find that Holliday junctions form over a wide range of intermediate temperatures, whilst kinetic traps at low temperature lead to incomplete bonding and consequently to the possibility of forming large clusters. A typical result from the high-yield regime is shown in Figure \\ref{5 HJs}. \nAs fully-bonded Holliday junctions are essentially inert, \nit is reasonable to analyse their assembly behaviour by considering only one \njunction.  The smaller system size has the effect of increasing the assembly rate, because the strands have less distance to diffuse, but does not affect the basic assembly mechanism. \nWe therefore simulated systems consisting of two $\\alpha$-bonded pairs with the same concentration as above. \n\nWe also introduced some modifications to the the umbrella sampling scheme in order\nto more efficiently compute the thermodynamics of the second-stage of \nHolliday junction formation. \nAs well as cluster moves, we also introduced a `tethering' component in the \nweighting function $W(Q)$. \nWe introduce a length $r_{\\rm min}$ that corresponds to the shortest distance between any pair of backbone sites on different strands. \nWe then split $Q=0$ into two regions: we weight those states with  $r_{\\rm min} < 3l$ with $W=1$ but for  $r_{\\rm min} \\geq 3l$ we use $W=0.1$. \nThis enables us to increase the rate of transitions between $Q=0$ and 1, and reduces the time spent \nsimply simulating the diffusion of $\\alpha$-bonded dimers waiting for a collision to occur.\n\nThe MC results are plotted in Figure \\ref{HJ thermo} along with \nthe equilibrium results obtained from umbrella sampling.\nWith the cluster moves in place, we now see a high yield of Holliday junctions\nand a broad maximum in the yield as a function of temperature.\nThe hierarchical pathway has the effect suggested earlier. \nNamely, the temperature window over which correct formation can occur is \nvastly increased, as the most significant competing minima are inaccessible because their formation would require dissociation of the $\\alpha$-bonded pairs. \n\nThe model is therefore consistent with the experimentally observed hierarchical assembly of Holliday junctions as the system is cooled.\\cite{Malo2005, MaloThesis} \nIt should be noted, however, that the junctions in our model usually form in the `square planar' as opposed to the `$\\chi$-stacked' shape (Figure \\ref{HJ schematic}) that is observed under normal experimental conditions. \nThe preference for one structure is a subtle consequence of the concentration of cations and the precise helical geometry of DNA.\\cite{Ortiz-Lombardia1999} \nThis level of detail is not included in our coarse-grained model, so it is not surprising that it cannot reproduce the preference for $\\chi$-stacked structures.\nMoreover, it is relatively easy to see why in our model, which forms \n`ladders' rather than helices, a planar geometry is preferred for the junction.\n\n\\begin{figure}\n\\includegraphics[width=8.4cm]{Fig11.eps}\n\\caption{Thermodynamics for the formation of a single Holliday junction.\n(a) Heat capacity curves for two $\\alpha$-bonded dimers forming a Holliday junction (HJ) \nand four strands forming two $\\alpha$-bonded dimers, as labelled.\n(b) Free energy profiles for the formation of a Holliday junction from two $\\alpha$-bonded dimers \nat different temperatures, as labelled. \n$Q$ represents the total number of correct bonds in the short arms of the junction.\n}\n\\label{thermo_HJ}\n\\end{figure}\n\nSome of the equilibrium thermodynamic properties associated with the formation\nof a Holliday junction are shown in Figure \\ref{thermo_HJ}. \nIn particular, Fig.\\ \\ref{thermo_HJ}(b) shows the free energy profile for the \nformation of a Holliday junction from two $\\alpha$-bonded pairs. \nThe initial peak and subsequent drop is very similar to that for the duplexes and can be accounted for in the same way. \nHowever, the formation of the two arms is not like the zipping up of a \n14-base duplex, because there is much more relative freedom of movement for \nthe bases on either side of the $\\alpha$-bonded sections in the dimers\nthan for consecutive bases on single-stranded DNA.\nThus, there is a rise between $M=7$ and $M=8$ that is a result of the entropy \npenalty of bringing together the two ends to make the second short arm. \nWe note that the penalty is much smaller than the initial cost of bringing the two $\\alpha$-bonded pairs together, and as a result, \nthe value of $T_{\\rm{m}}$ for the junction is higher than for the short arms \nin isolation, as noted earlier.\n\nAn interesting feature of Figure \\ref{thermo_HJ}(b) is the plateau between $Q=6$ and $Q=7$. \nIn general, when two $\\alpha$-bonded pairs meet to form one short arm, there is an entropic penalty associated with the excluded volume that the remaining bases in the $\\alpha$-structures represent to each other. \nThis excluded volume is a large fraction of the total available space if one complete short arm is formed, so that there are no free monomers between the short arm and the $\\alpha$-bonded sections. As a consequence, there is not the usual free energy benefit from forming the final bond in the short arm (the one closest to the centre of the Holliday junction), as the excluded volume penalty is large and those states that are allowed involve distortion of the backbones and bonds near the centre of the Holliday junction. \nAlthough the details of this free energy penalty and the other features in \nFig.\\ \\ref{thermo_HJ}(b) will depend on the exact geometry of the system, \nwe expect the calculated free energy profile to be representative of that \nfor real DNA.\n\nIt is possible to extend the two-state model discussed in Section \\ref{Duplex Formation} to the formation of a Holliday junction by considering the concentrations of all four isolated strands, the two $\\alpha$-bonded intermediates and the junction itself. \nWe assume Eq.\\ (\\ref{two state model}) holds for every possible transition, use the same thermodynamic parameters as before and apply conservation of total strand number. To estimate $\\Delta H_0$ and $\\Delta S_0$  associated with the formation of a Holliday junction, we construct a single strand by linking the ends of the oligonucleotides together with four non-bonding bases. The thermodynamic parameters associated with the folding of this structure are predicted by ``UNAFold''.\\cite{Markham2008} \nThe correction for the fact that our strands are not connected by loops is discussed by Zuker.\\cite{Zuker2003} \nThis leaves five simultaneous equations (assuming perfect stochiometry) which can be solved numerically.\n\n\\begin{figure}\n\\includegraphics[width=6.2cm,angle=-90]{Fig12.eps}\n\\caption{(Colour online) Bulk equilibrium probability of strands being in a Holliday junction (HJ) or\nan $\\alpha$-bonded dimer computed by umbrella sampling (solid lines) and by \nthe extended two-state model (dashed lines), where lines (a) and (b) represent the two possible $\\alpha$-bonded dimers.\n}\n\\label{etsm HJ}\n\\end{figure}\n\nFigure \\ref{etsm HJ} compares this extended two-state model (ETSM) with the bulk thermodynamics predicted by umbrella sampling (using the same temperature scaling as before). \nETSM predictions for both stages of Holliday junction formation agree well with our results, which again supports our hypothesis that much of the physics of self-assembly can be reproduced by a simple coarse-grained model. The extra width of the transitions in our model occurs for the same reasons as mentioned in Section \\ref{Duplex Formation} when discussing Fig.\\ \\ref{2 state}. \n\n\\begin{figure}\n\\includegraphics[width=6.1cm,angle=-90]{Fig13.eps}\n\\caption{(Colour online) Simulation results for the badly-designed Holliday \njunction of Section \\ref{negative}, where the initial configuration was a pair \nof $\\alpha$-bonded dimers. The lines with data points give the yield of \ncorrectly-formed junctions and misbonded configurations, as labelled.\nFor comparison the solid line gives the equilibrium probability that the \nwell-designed sequences of Section \\ref{Holliday Junction} adopt a \nHolliday junction.}\n\\label{thermo neg}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.4cm]{Fig14.eps}\n\\end{center}\n\\caption{(Colour online) Example of a competing minimum for a badly designed Holliday junction. The snapshot is \ntaken from a simulation at $T^*=0.0936$. \n}\n\\label{misbond neg}\n\\end{figure}\n\n\\subsection{Negative Design}\n\\label{negative}\nThe hierarchical pathway for the formation of the Holliday junction is one \naspect of the sequence design that aids the formation of the correct structure.\nThe experimental base ordering of the Holliday junction, however, was also chosen to minimize the number of competing structures---a typical example of `negative design'.\\cite{Doye04b} \nWe illustrate the importance of such negative design by considering a badly-designed junction, where the complementary seven-base sections consist of just one base type each. \nWe simulate a system of two $\\alpha$-bonded pairs with the short arms so modified \nunder the same conditions as for Fig.\\ \\ref{HJ thermo} and \nthe results are shown in Figure \\ref{thermo neg}. \nAlthough there is some probability of forming the correct junction, the simulations\nare dominated by misbonded junctions, such as the one depicted in Figure \\ref{misbond neg}.\nAlthough these competing structures are energetically less stable than the target\njunction because of the presence of unpaired bases at the `dangling' ends,\nthey are readily accessible, because the likelihood that the first bonds formed\nbetween two $\\alpha$-bonded pairs are in the same registry as the target structure  \nis low. The yield of the correct Holliday junctions will then depend upon how\nreadily the system is able escape from these malformed junctions. Clearly,\nthis process is slow on the time scales of the current simulations, and is \nalso likely to hinder the location of the target structure in experiment.\n\n\\section{Discussion}\nIn this paper we have introduced a simple coarse-grained model of DNA in order to test the feasibility of modeling the self-assembly of DNA nanostructure by \nMonte Carlo simulations. Any such model involves a trade-off between detail and \ncomputational simplicity, and here we deliberately chose to keep the model as simple\nas possible in order to give us the best chance of being able to probe the time\nscales relevant to self-assembly. The model involves just two interaction sites\nper nucleotide. \n\nThe results from our model are very encouraging. Firstly, we have shown that\nusing our model it is feasible to model the self-assembly of both DNA duplexes\nand a Holliday junction. The latter represents, to the best of our knowledge, the\nfirst example of the simulation of the self-assembly of a DNA structure beyond a \nduplex. Secondly, the model succeeds in reproducing many of the known thermodynamic\nand dynamic features of this self-assembly. \nFor example, the equilibrium melting curves agree well with those predicted by \nthe nearest-neighbour two-state model,\\cite{SantaLucia1998} which is known\nto predict melting temperatures very accurately. \nThe model is also able to capture important dynamical \nphenomena such as displacement.\n\nThirdly, by analysing the thermodynamic and dynamic constraints on assembly, we \nhave been able to gain some important physical insights into the nature of DNA\nself-assembly and how to control it. For example, the optimal conditions for \nself-assembly are in the temperature range just below the melting temperature of the\nthe target structure, where this structure is the only one stable with respect to \nthe precursors, be they ssDNA or some intermediate in a hierarchical assembly \npathway.  At lower temperatures, misbonded configurations can be formed that act \nas kinetic traps and reduce the assembly yield.\nSimilar trade-offs between the thermodynamic driving force and \nkinetic accessibility have been previously seen in a variety of self-assembling\nsystems,\\cite{Brooks06,Hagan2006,Wilber2007,Rapaport08,Whitelam08} \nand also give rise to a maximum in the yield near to and below\nthe temperature at which the target structure becomes stable.\n\nWe have also seen how hierarchical self-assembly through cooling can be a \nparticularly useful strategy to aid self-assembly, because the formation of stable\nintermediates at higher temperatures simplifies the free energy landscape for the\nassembly of the next stage in the hierarchy by reducing the number of misbonded \nconfigurations available to the system. This simplification of the energy\nlandscape is likely to be a general feature of hierarchical self-assembly.\n\nThus, our results have confirmed the utility of using coarse-grained DNA models to study the self-assembly of DNA nanostructures, and supported our hypothesis that much of the physics can be explained by describing DNA as a semi-flexible polymer with selective attractive interactions. The model's success in forming junctions in reasonable computational time suggests that it will be possible to develop further models that have an increased level of detail, but which can still access the time scales relevant to self-assembly.\n\nThe model has also highlighted some features which it would be advantageous to include in such models. For example, greater accuracy in the details of oligonucleotide geometry, particularly the helicity of dsDNA, would allow features such as the characteristically long persistence length of hybridized strands to be reproduced and give the appropriate degree of rigidity to simulated nanostructures. Such improvement might also allow more complicated motifs to be accounted for, such as the preference for $\\chi$-stacked Holliday junctions that the current model could not reproduce. \n\nIt should be noted that if one is to introduce helicity in a physically\nreasonable way it should also allow for ssDNA to undergo a stacking transition \nto a helical form. \nThis transition may play a significant role in the thermodynamics and kinetics of self-assembly.\\cite{Holbrook1999}\nPreviously proposed coarse-grained DNA models that incorporate helicity have not been designed to accurately reproduce this feature. \nIncorporating extra degrees of freedom which are relevant to the stacking transition, such as the rotation of the base with respect to the sugar-base bond, may also help to increase the entropy change on hybridization and hence make the transition narrower as required.\n\nThe approximation to diffusive dynamics provided by the local move Metropolis Monte Carlo algorithm could also be improved. \nCurrently the `local' moves involve displacing, rotating or bending entire strands or pairs of strands---these effectively constitute cluster moves of groups of strongly bound nucleotides, and result in slow relaxation and translation times within bound structures. \nMore realistic dynamics may be achievable by considering trial moves of individual nucleotides, and incorporating cluster moves in a more systematic fashion, such as in the `virtual move' MC algorithm proposed by Whitelam and Geissler.\\cite{Whitelam2007,Whitelam08} \n\nOne potential issue with any coarse-graining is how it preserves the different\ntime scales in a system.\nIn Section \\ref{Monte Carlo Simulation} we assigned an approximate mapping \nbetween the number of Monte Carlo steps and physical time based upon comparison \nof diffusion coefficients. There are, however, other important time scales in the \nsystem, such as the time scale for the internal dynamics of an isolated strand and \nthe time scale over which the `zipping-up' of two strands occurs after a bond has been formed. \nComparisons of experimental diffusion coefficients \\cite{Lapham1997} and \nmelting and bubble formation from molecular dynamics \nsimulations \\cite{Drukker2001,Knotts2007} suggest a large separation in time scale \nbetween diffusion-limited processes and those that rely on the dynamics of \nindividual nucleotides. \nEncouragingly, we observe a similar time scale separation in our model:\nzipping-up and thermal relaxation of isolated strands occur over times scales \nshorter than $10^5$ steps per strand, whereas association typically required on the \norder of $10^7$ to $10^8$ steps per strand near the melting temperature \n(corresponding to tens or hundreds of microseconds). Furthermore, we would\nargue that it is this time scale separation, and not the precise ratios of the\nrelevant rate constants, that it is important to reproduce in self-assembly \nsimulations.\n\nWe should also note that the mapping of the diffusion constants between the model\nand experiment will not necessarily ensure that the rate of association is \naccurate in our model, because although the frequency of collisions in our model \nshould be correct, \nthere is also the contribution\nto the association rate from the probability that a collision will lead to \nsuccessful association. That we can reproduce the thermodynamics of the DNA melting transitions implies that the rates of association and disassociation have the right\nratio, but not that they necessarily have the correct absolute value. \nFor example, it is conceivable that helicity (both in dsDNA and possibly in ssDNA),\nwhich is not included in the current model, will influence the likelihood that a \ncollision is successful.\n\n\n\\begin{acknowledgments}\nThe authors are grateful for financial support from the EPSRC \nand the Royal Society. We also wish to acknowledge helpful discussions\nwith Jonathan Malo, John Santalucia Jr and Michael Zuker.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nLet $M$ be a closed oriented smooth $d$-manifold. Let $D: H_*(M)\n\\xrightarrow{\\cong} H^{d-*}(M)$ be the Poincar\\'e duality map. Following a practice in string topology, we shift the homology grading downward by $d$ and let $\\mathbb H_{-*}(M)=H_{d-*}(M)$. The Poincar\\'e duality now takes the form $D:\\mathbb H_{-*}(M) \\xrightarrow{\\cong} H^{*}(M)$. For a homology element $a$, let $|a|$ denote its \n$\\mathbb H_*$-grading of $a$. \n\nThe intersection product $\\cdot$ in homology is defined as the Poincar\\'e dual of the cup product. Namely, for $a,b\\in \\mathbb H_*(M)$, $D(a\\cdot b)=D(a)\\cup D(b)$. If $\\alpha\\in H^*(M)$ is dual to $a$, then $\\alpha\\cap b=a\\cdot b$, and its Poincar\\'e dual is $\\alpha\\cup D(b)$. Thus, through Poincar\\'e duality, the intersection product, the cap product, and the cup product are all the same. In particular, the cap product and the intersection product commute:\n\\begin{equation}\n\\alpha\\cap (b\\cdot c)=(\\alpha\\cap b)\\cdot c\n=(-1)^{|\\alpha||b|}b\\cdot(\\alpha\\cap c). \n\\end{equation}\nIn fact, the direct sum $H^*(M)\\oplus \\mathbb H_*(M)$ can be made into a graded commutative associative algebra with unit, given by $1\\in H^0(M)$, using the cap and the cup product.\n\nFor an infinite dimensional manifold $N$, there is no longer\nPoincar\\'e duality, and geometric intersections of finite dimensional\ncycles are all trivial. However, cap products can still be nontrivial\nand the homology $H_*(N)$ is a module over the cohomology ring\n$H^*(N)$.\n\nWhen the infinite dimensional manifold $N$ is a free loop space $LM$\nof continuous maps from the circle $S^1=\\mathbb R\/\\mathbb Z$ to $M$,\nthe homology $\\mathbb H_*(LM)=H_{*+d}(LM)$ has a great deal more structure. As before, $|a|$ denotes the $\\mathbb H_*$-grading of a homology element $a$ of $LM$. Chas and\nSullivan \\cite{CS} showed that $\\mathbb H_*(LM)$ has a degree preserving associative graded\ncommutative product $\\cdot$ called the loop product, a\nLie bracket $\\{\\ ,\\ \\}$ of degree $1$ called the loop bracket\ncompatible with the loop product, and the BV operator $\\Delta$ of\ndegree $1$ coming from the homology $S^1$ action. These structures turn\n$\\mathbb H_*(LM)$ into a Batalin-Vilkovisky (BV) algebra. The purpose of this\npaper is to clarify the interplay between the cap product with cohomology elements and the BV structure in $\\mathbb H_*(LM)$.\n\n\n\nLet $p: LM \\rightarrow M$ be the base point map\n$p(\\gamma)=\\gamma(0)$ for $\\gamma\\in LM$. For a cohomology class\n$\\alpha\\in H^*(M)$ in the base manifold, its pull-back $p^*(\\alpha)\\in\nH^*(LM)$ is also denoted by $\\alpha$. Let $\\Delta: S^1\\times LM\n\\longrightarrow LM$ be the $S^1$-action map. This map induces a degree $1$ map $\\Delta$ in homology given by $\\Delta a=\\Delta_*([S^1]\\times a)$\nfor $a\\in \\mathbb H_*(LM)$. For a cohomology class $\\beta\\in H^*(LM)$, the formula \n$\\Delta^*(\\beta)=1\\times\\beta+\\{S^1\\}\\times\\Delta\\beta$ defines a degree $-1$ map $\\Delta$ in cohomology, where $\\{S^1\\}$ is the fundamental cohomology class. Although we use the same notation $\\Delta$ in three different but closely related situations, what is meant by $\\Delta$ should be clear in the context.  \n\n\\begin{Theorem A} Let $b,c\\in \\mathbb H_*(LM)$. \nThe cap product with $\\alpha\\in H^*(M)$ graded commutes with the loop product. Namely \n\\begin{equation}\n\\alpha\\cap(b\\cdot c)=(\\alpha\\cap b)\\cdot\nc=(-1)^{|\\alpha||b|}b\\cdot(\\alpha\\cap c).\n\\end{equation}\n\nFor $\\alpha\\in H^*(M)$, the cap product with $\\Delta\\alpha\\in H^*(LM)$\nacts as a derivation on the loop product and the\nloop bracket\\textup{:}\n\\begin{align}\n(\\Delta\\alpha)\\cap(b\\cdot c)=(\\Delta\\alpha\\cap b)\\cdot c+\n(-1)^{(|\\alpha|-1)|b|}b\\cdot (\\Delta\\alpha\\cap c), \\\\\n(\\Delta\\alpha)\\cap\\{b,c\\}=\\{\\Delta\\alpha\\cap b,c\\}+\n(-1)^{|\\alpha|-1)(|b|+1)}\\{b,\\Delta\\alpha\\cap c\\}.\n\\end{align}\n\nThe operator $\\Delta$ acts as a derivation on the cap product. Namely, for $\\alpha\\in H^*(M)$ and $b\\in \\mathbb H_*(LM)$.\n\\begin{equation}\n\\Delta(\\alpha\\cap b)=\\Delta\\alpha\\cap b + (-1)^{|\\alpha|} \\alpha\\cap\n\\Delta b.\n\\end{equation}\n\\end{Theorem A}\n \nWe recall that in the BV algebra $\\mathbb H_*(LM)$, the following identities\nare valid for $a,b,c\\in \\mathbb H_*(LM)$ \\cite{CS}:\n\\begin{gather}\n\\Delta(a\\cdot b)=(\\Delta a)\\cdot b + (-1)^{|a|}a\\cdot \\Delta b +\n(-1)^{|a|}\\{a,b\\} \n\\tag{BV identity} \\\\ \n\\{a, b\\cdot c\\} =\\{a,b\\}\\cdot c + (-1)^{|b|(|a|+1)}b\\cdot\\{a,c\\} \n\\tag{Poisson identity} \\\\ \na\\cdot b=(-1)^{|a||b|}b\\cdot a,\\qquad\n\\{a,b\\}=-(-1)^{(|a|+1)(|b|+1)}\\{b,a\\}\n\\tag{Commutativity} \\\\\n\\{a,\\{b,c\\}\\}=\\{\\{a,b\\},c\\}+(-1)^{(|a|+1)(|b|+1)}\\{b,\\{a,c\\}\\}\n\\tag{Jacobi identity}\n\\end{gather} \nHere, $\\deg a\\cdot b=|a|+|b|, \\deg \\Delta a=|a|+1$, and $\\deg\\{a,b\\}=|a|+|b|+1$. \n\nWe can extend the loop product and the loop bracket in $\\mathbb H_*(LM)$ to\ninclude $H^*(M)$ in the following way. For $\\alpha\\in H^*(M)$ and\n$b\\in \\mathbb H_*(LM)$, we define the loop product and the loop bracket of \n$\\alpha$ and $b$ by \n\\begin{equation}\n\\alpha\\cdot b=\\alpha\\cap b, \\qquad\n\\{\\alpha,b\\}=(-1)^{|\\alpha|}(\\Delta\\alpha)\\cap b.\n\\end{equation} \n\nFurthermore, the BV structure in $\\mathbb H_*(LM)$ can be extended to the direct sum $A_*=H^*(M)\\oplus \\mathbb H_*(LM)$ by defining the BV operator $\\boldsymbol\\Delta$ on $A_*$ to be trivial on $H^*(M)$ and to be the usual homological $S^1$ action $\\Delta$ on $\\mathbb H_*(LM)$. Here in $A_*$, elements in $H^k(M)$ are regarded as having homological degree $-k$. \n\n\\begin{Theorem B} The direct sum $H^*(M)\\oplus \\mathbb H_*(LM)$ has a structure of a BV algebra. In particular, for $\\alpha\\in H^*(M)$ and $b,c\\in \\mathbb H_*(LM)$, the following form of\nPoisson identity and the Jacobi identity hold\\textup{:}\n\\begin{gather} \n\\begin{split}\n\\{\\alpha\\cdot b,c\\}&=\\alpha\\cdot\\{b,c\\}+\n(-1)^{|b|(|c|+1)}\\{\\alpha,c\\}\\cdot b \\\\ &=\\alpha\\cdot\\{b,c\\} +\n(-1)^{|\\alpha||b|} b\\cdot\\{\\alpha,c\\},\n\\end{split} \\\\\n\\{\\alpha,\\{b,c\\}\\}=\\{\\{\\alpha,b\\},c\\} +\n(-1)^{(|\\alpha|+1)(|b|+1)}\\{b,\\{\\alpha,c\\}\\}.\n\\end{gather}\n\\end{Theorem B}\n\nAll the other possible forms of Poisson and Jacobi identities are\nalso valid, and the above two identities are the most nontrivial\nones. These identities are proved by using standard properties of the\ncap product and the BV identity above in $\\mathbb H_*(LM)$ relating the BV operator $\\Delta$ and the loop bracket $\\{\\,,\\,\\}$, but without using Poisson identities nor Jacobi identities in the BV algebra $\\mathbb H_*(LM)$.\n\nThe above identities may seem rather surprising, but they become\ntransparent once we prove the following result.\n\n\\begin{Theorem C} For $\\alpha\\in H^*(M)$, \nlet $a=\\alpha\\cap [M]\\in \\mathbb H_*(M)$ be its Poincar\\'e dual. Then for $b\\in \\mathbb H_*(LM)$,\n\\begin{equation}\\label{cap}\n\\alpha\\cap b=a\\cdot b,\\qquad (-1)^{|\\alpha|}\\Delta\\alpha\\cap\nb=\\{a,b\\}.\n\\end{equation} \nMore generally, for cohomology elements $\\alpha_0,\\alpha_1,\\dotsc\n\\alpha_r\\in H^*(M)$, let $a_0, a_1,\\dotsc a_r\\in \\mathbb H_*(M)$ be their Poincar\\'e duals. Then for $b\\in \\mathbb H_*(LM)$, we have\n\\begin{equation}\\label{composition of derivations}\n(\\alpha_0\\cup \\Delta\\alpha_1\\cup\\dotsm \\cup \\Delta\\alpha_r)\\cap b\n=(-1)^{|a_1|+\\dotsb+|a_r|}a_0\\cdot\\{a_1,\\{a_2,\\dotsc \\{a_r,b\\}\\dotsb \\}\\}.\n\\end{equation}\n\\end{Theorem C}\n\nSince the cohomology $H^*(M)$ and the homology $\\mathbb H_*(M)$ are isomorphic through Poincar\\'e duality and $\\mathbb H_*(M)$ is a subring of $\\mathbb H_*(LM)$, the first formula in \\eqref{cap} is not surprising. However, the main difference between $H^*(M)$ and $\\mathbb H_*(M)$ in our context is that the homology $S^1$ action $\\Delta$ is trivial on $\\mathbb H_*(M)\\subset \\mathbb H_*(LM)$, although cohomology $S^1$ action $\\Delta$ is nontrivial on $H^*(M)$ and is related to loop bracket as in \\eqref{cap}.  \n\nTheorem A and Theorem C describes the cap product action of the cohomology $H^*(LM)$ on the BV algebra $\\mathbb H_*(LM)$ for most elements in $H^*(LM)$. For example, for $\\alpha\\in H^*(M)$, the cap product with $\\Delta\\alpha$ is a derivation on the loop algebra $\\mathbb H_*(LM)$ given by a loop bracket, and consequently the cap product with a cup product $\\Delta\\alpha_1\\cup\\cdots\\cup\\Delta\\alpha_r$ acts on the loop algebra as a composition of derivations, which is equal to a composition of loop brackets, according to \\eqref{composition of derivations}. If $H^*(LM)$ is generated by elements $\\alpha$ and $\\Delta\\alpha$ for $\\alpha\\in H^*(M)$ (for example, this is the case when $H^*(M)$ is an exterior algebra, see Remark \\ref{exterior algebra}), then Theorem C gives a complete description of the cap product with arbitrary elements in $H^*(LM)$ in terms of the BV algebra structure in $\\mathbb H_*(LM)$. However, $H^*(LM)$ is general bigger than the subalgebra generated by $H^*(M)$ and $\\Delta H^*(M)$. \n\nSince $\\mathbb H_*(LM)$ is a BV algebra, in view of Theorem C, the validity of Theorem B may seem obvious. However, in the proof of Theorem B, we only used standard properties of the cap product and the BV identity. In fact, Theorem B gives an alternate elementary and purely homotopy theoretic proof of Poisson and Jacobi identities in $\\mathbb H_*(LM)$, when at least one of the elements $a,b,c$ are in $\\mathbb H_*(M)$. Similarly, Theorem C gives a purely homotopy theoretic interpretation of the loop product and the loop bracket if one of the elements are in $\\mathbb H_*(M)$.\n\nOur interest in cap products in string topology comes from an\nintuitive geometric picture that cohomology classes in $LM$ are dual\nto finite codimension submanifolds of $LM$ consisting of certain loop\nconfigurations. We can consider configurations of loops\nintersecting in particular ways (for example, two loops having their base points in common), or we can consider a family of loops\nintersecting transversally with submanifolds of $M$ at\ncertain points of loops. In a given family of loops, taking the cap product with a cohomology class selects a subfamily of a certain loop configuration, which are ready for certain loop interactions. In this context, roughly speaking, composition of two interactions of loops correspond to the cup product of corresponding cohomology classes.\n\nThe organization of this paper is as follows. In section 2, we\ndescribe a geometric problem of describing certain family of  intersection configuration of loops in terms of cap products. This gives a geometric motivation for the remainder of the paper. In section 3, we review the loop product in $\\mathbb H_*(LM)$ in detail from the point of view of the intersection product in $\\mathbb H_*(M)$. Here we pay careful attention to signs. In particular, we give a homotopy theoretic proof of graded commutativity in the BV algebra $\\mathbb H_*(LM)$, which turned out to be not so trivial. In section 4, we prove compatibility\nrelations between the cap product and the BV algebra structure, and\nprove Theorems A and B. In the last section, we prove Theorem C.\n\nWe thank the referee for numerous suggestions which lead to clarification and improvement of exposition. \n\n\n\n\n\n\\section{Cap products and intersections of loops}\n\nLet $A_1,A_2,\\dotsc A_r$ and $B_1,B_2,\\dotsc B_s$ be oriented closed\nsubmanifolds of $M^d$. Let $F\\subset LM$ be a compact family of\nloops. We consider the following question. \n\n\\smallskip\n\n\\textbf{Question} : Fix $r$ points $0\\le t_1^*,t_2^*,\\dotsc,t_r^*\\le\n1$ in $S^1=\\mathbb R\/\\mathbb Z$. Describe the homology class of the subset $I$ of the compact family $F$\nconsisting of loops $\\gamma$ in $F$ such that $\\gamma$ intersects\nsubmanifolds $A_1,\\dotsc A_r$ at time $t_1^*,\\dotsc t_r^*$ and\nintersects $B_1,\\dotsc B_s$ at some unspecified time.\n\n\\smallskip\n\nThis subset $I\\subset F$ can be described as follows. We consider the following diagram of an evaluation map and a projection map: \n\\begin{equation} \\label{eval}\n\\begin{CD}\n\\overset{s}{\\overbrace{(S^1\\times\\dotsb\\times S^1)}}\\times LM @>{e}>>\n  \\overset{r}{\\overbrace{M\\times \\dotsb\\times M}}\\times\n\\overset{s}{\\overbrace{M\\times\\dotsb\\times M}}  \\\\\n@V{\\pi_2}VV @.   \\\\\nLM  @. \n\\end{CD}\n\\end{equation}\ngiven by $e\\bigl((t_1,\\dotsc t_s),\\gamma\\bigr)\n=\\bigl(\\gamma(t_1^*),\\dotsc \\gamma(t_r^*), \\gamma(t_1), \\dotsc\n\\gamma(t_s)\\bigr)$. Then the pull-back set\n$e^{-1}(\\prod_iA_i\\times\\prod_jB_j)$ is a closed \nsubset of $S^1\\times\\dotsb\\times S^1\\times LM$. Let\n\\begin{equation*}\n\\tilde{I}=e^{-1}(\\prod_iA_i\\times \\prod_jB_j)\\cap \n(S^1\\times\\dotsb\\times S^1\\times F). \n\\end{equation*} \nThe set $I$ in\nquestion is given by $I=\\pi_2(\\tilde{I})$. We want to understand this set $I$ homologically, including multiplicity. Although $e^{-1}(\\prod_iA_i\\times\\prod_jB_j)$ is infinite dimensional, it has finite codimension in $(S^1)^r\\times LM$. So we work cohomologically.\n\nLet $\\alpha_i, \\beta_j\\in H^*(M)$ be cohomology classes dual to $[A_i], [B_j]$ for $1\\le i\\le r$ and $1\\le j\\le s$. Then the subset $e^{-1}(\\prod_iA_i\\times \\prod_jB_j)$ is dual to the cohomology class  $e^*(\\prod_i\\alpha_i\\times\\prod_j\\beta_j)\\in H^*((S^1)^s\\times LM)$. Suppose the family $F$ is parametrized by a closed oriented manifold $K$ by an onto map $\\lambda:K \\longrightarrow F$ and let $b=\\lambda_*([K])\\in \\mathbb H_*(LM)$ be the homology class of $F$ in $LM$. Then the homology class of $\\tilde{I}$ in $(S^1)^s\\times LM$ is given by \n\\begin{equation}\n[\\tilde{I}]=e^*(\\prod_i\\alpha_i\\times\\prod_j\\beta_j)\\cap ([S^1\\times\\dotsb\\times S^1]\\times b). \n\\end{equation} \nNote that the homology class $(\\pi_2)_*([\\tilde{I}])$ represents the homology class of $I$ with multiplicity. \n\n\\begin{proposition}\\label{loop intersection} With the above notation, $(\\pi_2)_*([\\tilde{I}])$ is given by the following formula in terms of the cap product or in terms of the BV structure\\textup{:}\n\\begin{equation}\n\\begin{aligned}\n(\\pi_2)_*([\\tilde{I}])&=(-1)^{\\sum_jj|\\beta_j|-s}\n\\bigl(\\alpha_1\\dotsm\\alpha_r(\\Delta\\beta_1)\\dotsm(\\Delta\\beta_s)\\bigr)\n\\cap b \\\\\n&=(-1)^{\\sum_jj|\\beta_j|-s}[A_1]\\cdots[A_s]\\cdot\\{[B_1],\\{\\cdots\\{[B_s],b\\}\\cdots\\}\n\\in \\mathbb H_*(LM). \n\\end{aligned}\n\\end{equation}\n\\end{proposition} \n\\begin{proof} The evaluation map $e$ in \\eqref{eval} is \ngiven by the following composition.\n\\begin{multline*} \n\\overset{s}{\\overbrace{S^1\\times\\dotsb\\times S^1}}\\times LM \n\\xrightarrow{1\\times\\phi} \n(S^1\\times\\dotsb\\times S^1) \\times\n\\overset{r+s}{\\overbrace{LM\\times\\dotsb\\times LM}} \\\\\n\\xrightarrow{T} \n\\overset{r}{\\overbrace{LM\\times\\dotsb\\times LM}}\\times\n\\overset{s}{\\overbrace{(S^1\\times LM)\\times\\dotsb\\times(S^1\\times LM)}}  \\\\\n\\xrightarrow{1^r\\times\\Delta^s}\n(LM\\times\\dotsb\\times LM) \\times\n(LM\\times\\dotsb\\times LM)\n\\xrightarrow{p^r\\times p^s} \n(M\\times\\dotsb\\times M) \\times\n(M\\times\\dotsb\\times M),\n\\end{multline*} \nwhere $\\phi$ is a diagonal map, $T$ moves $S^1$ factors.  Since we\napply $(\\pi_2)_*$ later, we only need terms in $e^*(\\prod A_i\\times\n\\prod B_j)$ containing the factor\n$\\{S^1\\}\\times\\dotsb\\times\\{S^1\\}$. Since\n$\\Delta^*p^*(\\beta_j)=1\\times p^*(\\beta_j)\n+\\{S^1\\}\\times\\Delta\\beta_j$ for $1\\le j\\le s$, following the above\ndecomposition of $e$, we have\n\\begin{equation*}\ne^*(\\alpha_1\\times\\dotsb\\times\\alpha_r\\times\\beta_1\\times\\dotsb\\times\\beta_s)\n=\\varepsilon\n\\{S^1\\}^s\\times \\bigl(\\alpha_1\\dotsm\\alpha_r(\\Delta\\beta_1)\\dotsm\n(\\Delta\\beta_s)\\bigr) + \\text{ other terms},\n\\end{equation*}\nwhere the sign $\\varepsilon$ is given by $\\varepsilon=\n(-1)^{\\sum_{\\ell=1}^s(s-\\ell)(|\\beta_{\\ell}|-1)\n+s\\sum_{\\ell=1}^r|\\alpha_{\\ell}|}$.  Thus, taking the cap product with\n$[S^1]^s\\times b$ and applying $(\\pi_2)_*$, we get\n\\begin{multline*}\n{\\pi_2}_*\n\\bigl(e^*(\\alpha_1\\times\\dotsb\\times\\alpha_r\\times\n\\beta_1\\times\\dotsb\\times\\beta_s)\n\\cap ([S^1]\\times\\dotsb\\times[S^1]\\times b)\\bigr) \\\\\n=(-1)^{\\sum_{\\ell=1}^{s}\\ell|\\beta_{\\ell}|-s}\n\\alpha_1\\dotsm\\alpha_r(\\Delta\\beta_1)\\dotsm(\\Delta\\beta_s)\\cap b.\n\\end{multline*}\nThe second equality follows from the formula \\eqref{composition of derivations}. \n\\end{proof}\n\n\\begin{remark} In the diagram \\eqref{eval}, \nin terms of cohomology transfer ${\\pi_2}^!$ we have \n\\begin{equation}\n{\\pi_2}^!\ne^*(\\alpha_1\\times\\dotsb\\times\\alpha_r\\times\\beta_1\\times\\dotsb\\times\\beta_s)\n=\\pm\n\\alpha_1\\dotsm\\alpha_r(\\Delta\\beta_1)\\dotsm(\\Delta\\beta_s), \n\\end{equation}\nwhere ${\\pi_2}^!(\\alpha)\\cap\nb=(-1)^{s|\\alpha|}{\\pi_2}_*\\bigl(\\alpha\\cap{\\pi_2}_!(b)\\bigr)$ for any\n$\\alpha\\in H^*((S^1)^s\\times LM)$ and $b\\in \\mathbb H_*(LM)$. Here\n${\\pi_2}_!(b)=[S^1]^s\\times b$.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The intersection product and the loop product}\n\n\n\n\n\n\n\nLet $M$ be a closed oriented smooth $d$-manifold.  The loop product in\n$\\mathbb H_*(LM)$ was discovered by Chas and Sullivan \\cite{CS}, in terms of\ntransversal chains. Later, Cohen and Jones \\cite{CJ} gave a homotopy\ntheoretic description of the loop product. The loop product is a\nhybrid of the intersection product in $\\mathbb H_*(M)$ and the Pontrjagin\nproduct in the homology of the based loop spaces $H_*(\\Omega M)$. In\nthis section, we review and prove some properties of the loop product\nin preparation for the next section. Our treatment of the loop product\nfollows \\cite{CJ}. However, we will be precise with signs and give a\nhomotopy theoretic proof of the graded commutativity of the loop\nproduct, which \\cite{CJ} did not include. For the Frobenius compatibility formula with careful discussion of signs, see \\cite{T2}. For homotopy theoretic deduction of the BV identity, see \\cite{T3}. \n\nFor our purpose, the free loop space $LM$ is the space of {\\it continuous} maps from $S^1=\\mathbb R\/\\mathbb Z$ to $M$. Our discussion is homotopy theoretic and does not require smoothness of loops, although we do need smoothness of $M$ which is enough to allows us to have tubular neighborhoods for certain submanifolds in the space of continuous loops. Recall that the space $LM$ of continuous loops can be given a structure of a smooth manifold. See the discussion before Definition \\ref{definition of loop product}. \n\nLet $p: LM \\longrightarrow M$ be the base\npoint map given by $p(\\gamma)=\\gamma(0)$. Let $s: M \\longrightarrow\nLM$ be the constant loop map given by $s(x)=c_x$, where $c_x$ is the\nconstant loop at $x\\in M$. Since $p_*\\circ s_*=1$, $\\mathbb H_*(M)$ is\ncontained in $\\mathbb H_*(LM)$ through $s_*$ and we often regard $\\mathbb H_*(M)$ as a subset of $\\mathbb H_*(LM)$.\n\nWe start with a discussion on the intersection ring $\\mathbb H_*(M)$ and later we compare it with the loop homology algebra $\\mathbb H_*(LM)$. An exposition on intersection products in homology of manifolds can be found on Dold's book \\cite{D}, Chapter VIII, \\S13. Our sign convention (which follows Milnor \\cite{M}) is slightly different from Dold's. We give a fairly detailed discussion of the intersection ring $\\mathbb H_*(M)$ because the\ndiscussion for the loop homology algebra goes almost in parallel, and\nbecause our choice of the sign for the loop product comes from and is\ncompatible with the intersection product in $\\mathbb H_*(M)$. Compare formulas \\eqref{intersection product} and \\eqref{loop product}.\n\nThose who are familiar with intersection product and loop products can skip this section after checking Definition \\ref{definition of loop product}. \n\nLet $D: \\mathbb H_*(M) \\xrightarrow{\\cong} H^{d-*}(M)$ be the Poincar\\'e duality map such that $D(a)\\cap[M]=a$ for $a\\in \\mathbb H_*(M)$.  We discuss two ways to define\nintersection product in $\\mathbb H_*(M)$. The first method is the official one\nand we simply define the intersection product as the Poincar\\'e dual\nof the cohomology cup product. Thus, $D(a\\cdot b)=D(a)\\cup D(b)$ for\n$a,b\\in \\mathbb H_*(M)$. For example, we have $a\\cdot\nb=(-1)^{|a||b|}b\\cdot a$.\n\nThe second method uses the transfer map induced from the diagonal map\n$\\phi: M \\longrightarrow M\\times M$. Let $\\nu$ be the normal bundle to\n$\\phi(M)$ in $M\\times M$, and we orient $\\nu$ by $\\nu\\oplus\n\\phi_*(TM)\\cong T(M\\times M)|_{\\phi(M)}$. Let $u'\\in\nH^d(\\phi(M)^{\\nu})$ be the Thom class of $\\nu$. Let $N$ be a closed\ntubular neighborhood of $\\phi(M)$ in $M\\times M$ so that $D(\\nu)\\cong\nN$, where $D(\\nu)$ is the associated closed disc bundle of $\\nu$. Let\n$\\pi: N \\longrightarrow M$ be the projection map. Then the above Thom\nclass can be thought of as $u'\\in \\tilde{H}^d(N\/\\partial N)$, and we\nhave the following commutative diagram, where $c: M\\times M\n\\longrightarrow N\/\\partial N$ is the Thom collapse map, and $\\iota_N$\nand $j$ are obvious maps.\n\\begin{equation}\n\\begin{CD} \nH^d(N, N-\\phi(M)) @>{\\cong}>> H^d(N,\\partial N)\\ni u' \\\\\n@A{\\cong}A{\\iota_N^*}A   @VV{c^*}V   \\\\\nu''\\in H^d\\bigl(M\\times M, M\\times M-\\phi(M)\\bigr) @>{j^*}>>  H^d(M\\times M)\\ni u\n\\end{CD}\n\\end{equation} \nLet $u''\\in H^d\\bigl(M\\times M, M\\times M-\\phi(M)\\bigr)$ and $u\\in\nH^d(M\\times M)$ be the classes corresponding to the Thom class. We\nhave $u=c^*(u')=j^*(u'')$. This class $u$ is characterized by the\nproperty $u\\cap[M\\times M]=\\phi_*([M])$, and $\\phi^*(u)=e_M\\in H^d(M)$\nis the Euler class of $M$. See for example section 11 of \\cite{M}. The\ntransfer map $\\phi_!$ is defined as the following composition:\n\\begin{equation}\n\\phi_!: H_*(M\\times M) \\xrightarrow{c_*} \\tilde{H}_*(N\/\\partial N)\n\\xrightarrow[\\cong]{u'\\cap(\\ )} H_{*-d}(N) \\xrightarrow[\\cong]{\\pi_*}\nH_{*-d}(M).\n\\end{equation} \n\nFor a homology element $a$, let $|a|'$ denote its regular homology degree of $a$, so that we have $a\\in H_{|a|}(M)$ and $|a|'=|a|+d$. \n\n\\begin{proposition}\\label{properties of phi} Suppose $M$ is a connected \noriented closed $d$-manifold with a base point $x_0$. The transfer map\n$\\phi_!: H_*(M\\times M) \\longrightarrow H_{*-d}(M)$ satisfies the\nfollowing properties. For $a,b\\in H_*(M)$,\n\\begin{align}\n\\phi_*\\phi_!(a\\times b)&=u\\cap(a\\times b) \\\\ \\phi_!\\phi_*(a\\times\nb)&=\\chi(M)[x_0]\n\\end{align}\nFor $\\alpha\\in H^*(M\\times M)$ and $b,c\\in H_*(M)$, we have \n\\begin{equation}\n\\phi_!\\bigl(\\alpha\\cap(b\\times c)\\bigr)\n=(-1)^{d|\\alpha|}\\phi^*(\\alpha)\\cap\\phi_!(b\\times c).\n\\end{equation} \nThe intersection product and the transfer map coincide up to a sign. \n\\begin{equation}\\label{intersection product}\na\\cdot b=(-1)^{d(|a|'-d)}\\phi_!(a\\times b).\n\\end{equation}\n\\end{proposition} \n\\begin{proof} For the first identity, we consider the following \ncommutative diagram, where $M^2$ denotes $M\\times M$. \n\\begin{equation*}\n\\begin{CD}\nH_*(M^2) @>{c_*}>> H_*(N,\\partial N) @>{u'\\cap (\\ )}>{\\cong}> H_{*-d}(N) \n@>{\\pi_*}>{\\cong}>  H_{*-d}(M)  \\\\\n@|   @V{\\cong}V{{\\iota_N}_*}V  @VV{{\\iota_N}_*}V   @VV{\\phi_*}V   \\\\\nH_*(M^2)  @>{j_*}>> H_*\\bigl(M^2, M^2-\\phi(M)\\bigr) \n@>{u''\\cap(\\ )}>{\\cong}>  H_{*-d}(M^2) @=  H_{*-d}(M^2)\n\\end{CD}\\end{equation*}\nThe commutativity implies that for $a,b\\in H_*(M)$, we have $\\phi_*\\phi_!\n=u''\\cap j_*(a\\times b)=j^*(u'')\\cap(a\\times b)=u\\cap(a\\times b)$. \n\nTo check the second formula, we first compute\n$\\phi_*\\phi_!\\phi_*([M])$. By the first formula,\n$\\phi_*\\phi_!\\phi_*([M])=u\\cap\\phi_*([M])=\\phi_*(\\phi^*(u)\\cap[M])$. Since\n$\\phi^*(u)$ is the Euler class $e_M$, this is equal to\n$\\phi_*(e_M\\cap[M])=\\chi(M)[(x_0,x_0)]$. Since $M$ is assumed to be\nconnected, $\\phi_*$ is an isomorphism in $H_0$. Hence\n$\\phi_!\\phi_*([M])=\\chi(M)[x_0]\\in H_0(M)$.\n\nFor the next formula, we examine the following commutative diagram. \n\\begin{equation*}\n\\begin{CD} \nH_*(M^2) @>{c_*}>> H_*(N,\\partial N) @>{u'\\cap(\\ )}>{\\cong}> H_{*-d}(N)\n@<{\\iota_*'}<{\\cong}< H_{*-d}(M) \\\\ \n@V{\\alpha\\cap(\\ )}VV\n@V{\\iota_N^*(\\alpha)\\cap(\\ )}VV @V{\\iota_N^*(\\alpha)\\cap(\\ )}VV\n@V{\\iota^*(\\alpha)}VV \\\\ \nH_{*-|\\alpha|}(M^2) @>{c_*}>>\nH_{*-|\\alpha|}(N,\\partial N) @>{u'\\cap(\\ )}>{\\cong}> H_{*-d-|\\alpha|}(N)\n@<{\\iota_*'}<{\\cong}< H_{*-d-|\\alpha|}(M)\n\\end{CD}\n\\end{equation*}\nwhere $\\iota': M\\rightarrow N$ is an inclusion map and\n$\\iota_*'=(\\pi_*)^{-1}$. The middle square commutes up to\n$(-1)^{|\\alpha|d}$. The commutativity of this diagram immediately\nimplies that $\\iota^*(\\alpha)\\cap\\phi_!(a\\times b)\n=(-1)^{|\\alpha|d}\\phi_!\\bigl(\\alpha\\cap(a\\times b)\\bigr)$.\n\nFor the last identity, we apply $\\phi_*$ on both sides and\ncompare. Since $a\\cdot b=\\phi^*(D(a)\\times D(b))\\cap[M]$, we have\n\\begin{align*} \n\\phi_*(a\\cdot b)&=\\bigl(D(a)\\times D(b)\\bigr)\\cap \\phi_*([M]) \\\\\n&=\\bigl(D(a)\\times D(b)\\bigr)\\cap \\bigl(u\\cap[M]\\bigr) \\\\\n&=(-1)^{d(|a|'-d)}u\\cap(a\\times b)=(-1)^{d(|a|'-d)}\\phi_*\\phi_!(a\\times b).\n\\end{align*} \nSince $\\phi_*$ is injective, we have $a\\cdot\nb=(-1)^{d(|a|'-d)}\\phi_!(a\\times b)$.\n\\end{proof} \n\nThese two intersection products, one defined using the Poincar\\'e\nduality, and the other using Pontrjagin Thom construction, differ only\nin signs. However, the formulas for graded commutativity take\ndifferent forms.\n\\begin{align}\na\\cdot b&=(-1)^{(d-|a|')(d-|b|')}b\\cdot a \\\\\n\\phi_!(a\\times b)&=(-1)^{|a|'|b|'+d}\\phi_!(b\\times a)\n\\end{align}\nThe sign $(-1)^d$ in the second formula above comes from the fact that\nthe Thom class $u\\in H^d(M\\times M)$ satisfies $T^*(u)=(-1)^du$, where\n$T$ is the switching map of factors.\n\nNext we turn to the loop product in $H_*(LM)$. We consider the\nfollowing diagram.\n\\begin{equation}\n\\begin{CD}\nLM\\times LM @<{j}<< LM\\times_M LM  @>{\\iota}>> LM  \\\\\n@V{p\\times p}VV    @V{q}VV        @.  \\\\\nM\\times M @<{\\phi}<<  M @.  \n\\end{CD}\n\\end{equation}\nwhere $LM\\times_M LM=(p\\times p)^{-1}(\\phi(M))$ consists of pairs of\nloops $(\\gamma,\\eta)$ with the same base point, and\n$\\iota(\\gamma,\\eta)=\\gamma\\cdot\\eta$ is the product of composable\nloops. Let $\\widetilde{N}=(p\\times p)^{-1}(N)$ and let $\\tilde{c}:\nLM\\times LM \\longrightarrow \\widetilde{N}\/\\partial\\widetilde{N}$ be the Thom collapse map. Let $\\tilde{\\pi}: \\widetilde{N} \\longrightarrow\nLM\\times_MLM$ be a projection map defined as follows. For\n$(\\gamma,\\eta)\\in\\widetilde{N}$, let their base points be $(x,y)\\in\nN$. Let $\\pi(x,y)=(z,z)\\in \\phi(M)$. Since $N\\cong D(\\nu)$ \nhas a bundle structure,\nlet $\\ell(t)=(\\ell_1(t),\\ell_2(t))$ be the straight ray in the fiber\nover $(z,z)$ from $(z,z)$ to $(x,y)$. Then let\n$\\tilde{\\pi}\\bigl((\\gamma,\\eta)\\bigr)=(\\ell_1\\cdot\\gamma\\cdot\\ell_1^{-1},\n\\ell_2\\cdot\\eta\\cdot\\ell_2^{-1})$. By considering $\\ell_{[t,1]}$, we\nsee that $\\tilde{\\pi}$ is a deformation retraction. \n\nIn fact, more is true. Stacey (\\cite{St}, Proposition 5.3) showed that when $L_{\\text{smooth}}M$ is the space of {\\it smooth} loops, $\\widetilde{N}$ has an actual structure of a tubular neighborhood of $LM\\times_MLM$ inside of $LM\\times LM$ equipped with a diffeomorphism  $p^*\\bigl(D(\\nu)\\bigr)\\cong \\widetilde{N}$. His proof only uses the smoothness of $M$ and exactly the same proof applies to the space $LM$ \nof {\\it continuous} loops and $\\widetilde{N}$ still has the structure of a tubular neighborhood and we again have a diffeomorphism $p^*\\bigl(D(\\nu)\\bigr)\\cong \\widetilde{N}$ between spaces of continuous loops. But we do not need this much here. \n\nLet $\\tilde{u}'=(p\\times p)^*(u')\\in\n\\tilde{H}^d(\\widetilde{N}\/\\partial\\widetilde{N})$, and $\\tilde{u}=(p\\times p)^*(u)\\in H^d(LM\\times LM)$ be pull-backs of Thom classes. Define the\ntransfer map $j_!$ by the following composition of maps.\n\\begin{equation} \nj_!: H_*(LM\\times LM) \\xrightarrow{\\tilde{c}_*} \n\\tilde{H}_*(\\widetilde{N}\/\\partial\\widetilde{N})\n\\xrightarrow[\\cong]{\\tilde{u}'\\cap(\\ )}  H_{*-d}(\\widetilde{N}) \n\\xrightarrow[\\cong]{\\tilde{\\pi}_*}\nH_{*-d}(LM\\times_MLM).\n\\end{equation}\nThe tubular neighborhood structure of $\\widetilde{N}$ implies that the middle map is a genuine Thom isomorphism. \n\n\\begin{definition}\\label{definition of loop product} \nLet $M$ be a closed oriented $d$-manifold. For $a,b\\in \\mathbb H_*(LM)$, their loop product, denoted by $a\\cdot b$, is defined by\n\\begin{equation}\\label{loop product} \na\\cdot b=(-1)^{d(|a|'-d)}\\iota_*j_!(a\\times b)\n=(-1)^{d|a|}\\iota_*j_!(a\\times b). \n\\end{equation}\n\\end{definition} \n\nThe sign $(-1)^{d(|a|'-d)}$ appears in \\cite{CJY} in the commutative\ndiagram (1-7). We include this sign explicitly in the definition of\nthe loop product for at least three reasons. The most trivial reason\nis that on the left hand side, the dot representing the loop product\nis between $a$ and $b$. On the right hand side, $j_!$ of degree $-d$\nrepresenting the loop product is in front of $a$. Switching $a$ and\n$j_!$ gives the sign $(-1)^{d|a|'}$. The other part of the sign\n$(-1)^d$ comes from our choice of orientation of $\\nu$ and ensures\nthat $s_*([M])\\in \\mathbb H_0(LM)$, with the $+$ sign, is the unit of the loop\nproduct. We quickly verify the correctness of the sign.\n\\begin{lemma} \nThe element $s_*([M])\\in\\mathbb H_0(LM)$ is the unit of the loop\nproduct. Namely for any $a\\in \\mathbb H_*(LM)$,\n\\begin{equation}\ns_*([M])\\cdot a=a\\cdot s_*([M])=a.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} We consider the following diagram. \n\\begin{equation*}\n\\begin{CD}\n@. LM @= LM  \\\\\n@. @A{\\pi_2}AA @| @. \\\\\nM\\times M @<{1\\times p}<< M\\times LM  @<{j'}<< M\\times_M LM @= LM  \\\\\n@| @V{s\\times 1}VV @V{s\\times_M1}VV @|  \\\\\nM\\times M  @<{p\\times p}<< LM\\times LM  @<{j}<< LM\\times_M LM @>{\\iota}>> LM \n\\end{CD}\n\\end{equation*} \nIn the induced homology diagram with transfers $j_!$ and $j'_!$, the\nbottom middle square commutes because transfers are defined using Thom\nclasses pulled back from the same Thom class $u$ of the base\nmanifold. Thus,\n\\begin{equation*}\ns_*([M])\\cdot a=\\iota_*j_!(s\\times 1)_*([M]\\times a)=j'_!([M]\\times a). \n\\end{equation*}\nHere, since $[M]$ has degree $d$, the sign in \\eqref{loop product} is $+1$. Since $\\pi_2\\circ j'=1$, the identity on $LM$,\n\\begin{equation*}\nj'_!([M]\\times a)={\\pi_2}_*j'_*j'_!([M]\\times a)\n={\\pi_2}_*\\bigl((1\\times p)^*(u)\\cap([M]\\times a)\\bigr).\n\\end{equation*}\nDue to the way $\\nu$ is oriented, the Thom class $u$ is of the form\n$u=\\{M\\}\\times 1+\\dotsb+(-1)^d1\\times\\{M\\}$. Hence\n${\\pi_2}_*\\bigl((1\\times p)^*(u)\\cap([M]\\times\na)\\bigr)={\\pi_2}_*([x_0]\\times a+\\dotsm)=a$.\n\nThe other identity $a\\cdot s_*([M])=a$ can be proved similarly. This\ncompletes the proof.\n\\end{proof} \n\n\nThe second reason is that this choice of sign for the loop product is\nthe same sign appearing in the formula for the intersection product\ndefined in terms of the transfer map \\eqref{intersection\nproduct}. This makes the loop product compatible with the intersection\nproduct in the following sense.\n\n\\begin{proposition}\nBoth of the following maps are algebra maps preserving units between\nthe loop algebra $\\mathbb H_*(LM)$ and the intersection ring $\\mathbb H_*(M)$.\n\\begin{equation}\np_*: \\mathbb H_*(LM) \\longrightarrow \\mathbb H_*(M),\\qquad \ns_*: \\mathbb H_*(M) \\longrightarrow\n\\mathbb H_*(LM).\n\\end{equation}\n\\end{proposition} \n\\begin{proof} The proof is more or less straightforward, \nbut we discuss it briefly. We consider the following diagram.\n\\begin{equation*}\n\\begin{CD}\nLM\\times LM @<{j}<< LM\\times_M LM @>{\\iota}>> LM \\\\ @V{p\\times p}VV\n@V{p}VV @V{p}VV \\\\ M\\times M @<{\\phi}<< M @= M \\\\ @V{s\\times s}VV\n@V{s}VV @V{s}VV \\\\ LM\\times LM @<{j}<< LM\\times_M LM @>{\\iota}>> LM\n\\end{CD}\\end{equation*}\nSince the Thom classes for embeddings $j$ and $\\phi$ are compatible\nvia $(p\\times p)^*$, the induced homology diagram with transfers\n$j_!$ and $\\phi_!$ is commutative. Then by diagram chasing, we can\neasily check that $p_*$ and $s_*$ preserve products because of the\nsame signs appearing in \\eqref{intersection product} and \n\\eqref{loop product}. \n\\end{proof} \n\nThe third reason of the sign for the loop product is that it gives the\ncorrect graded commutativity, as given in \\cite{CS} proved in terms of chains. We discuss a homotopy theoretic proof of graded commutativity because \\cite{CJ} did not include it, and because the homotopy theoretic proof itself is not so trivial with careful treatment of transfers and signs. Contrast the present homotopy theoretic proof with the simple geometric proof given in \\cite{CS}. \n\n\\begin{proposition} \nFor $a,b\\in \\mathbb H_*(LM)$, the following graded commutativity relation holds\\textup{:}\n\\begin{equation} \na\\cdot b=(-1)^{(|a|'-d)(|b|'-d)}b\\cdot a=(-1)^{|a||b|}b\\cdot a. \n\\end{equation}\n\\end{proposition} \n\\begin{proof}\nWe consider the following commutative diagram, where $R_{\\frac12}$ is\nthe rotation of loops by $\\frac12$, that is,\n$R_{\\frac12}(\\gamma)(t)=\\gamma(t+\\frac12)$.\n\\begin{equation*}\n\\begin{CD}\nLM\\times LM @<{j}<< LM\\times_M LM @>{\\iota}>> LM \\\\\n@V{T}VV    @V{T}VV  @V{R_{\\frac12}}VV   \\\\\nLM\\times LM @<{j}<< LM\\times_M LM @>{\\iota}>> LM\n\\end{CD}\n\\end{equation*}\nSince $R_{\\frac12}$ is homotopic to the identity map, we have\n${R_{\\frac12}}_*=1$. Hence\n\\begin{equation*}\na\\cdot b=(-1)^{d(|a|'-d)}\\iota_*j_!(a\\times b)\n=(-1)^{d(|a|'-d)}\\iota_*T_*j_!(a\\times b).\n\\end{equation*} \nNext we show that the induced homology square with transfer $j_!$, we\nhave $T_*j_!=(-1)^dj_!T_*$. Since the left square in the above diagram\ncommutes on space level, we have that $T_*j_!$ and $j_!T_*$ coincides\nup to a sign. To determine this sign, we compose $j_*$ on the left of\nthese maps and compare. Since the homology square with induced\nhomology maps commute,\n\\begin{equation*}\nj_*T_*j_!(a\\times b)=T_*j_*j_!(a\\times\nb)=T_*\\bigl(\\tilde{u}\\cap(a\\times b)\\bigr).\n\\end{equation*}\nOn the other hand, \n\\begin{equation*}\nj_*j_!T_*(a\\times b)=\\tilde{u}\\cap T_*(a\\times b)\n=T_*\\bigl(T^*(\\tilde{u})\\cap(a\\times b)\\bigr). \n\\end{equation*}\nWe compare $T^*(\\tilde{u})$ and $\\tilde{u}$. Since $\\tilde{u}=(p\\times\np)^*(u)$, we have $T^*(\\tilde{u})=(p\\times p)^*T^*(u)$. Since $u$ is\ncharacterized by the property $u\\cap[M\\times M]=\\phi_*([M])$ and\n$T\\circ \\phi=\\phi$, we have\n\\begin{equation*}\n\\phi_*([M])=T_*\\phi_*([M])=T^*(u)\\cap T_*([M\\times\nM])=T^*(u)\\cap(-1)^d[M\\times M].\n\\end{equation*}\nThus $T^*(u)=(-1)^du$. Hence $T^*(\\tilde{u})=(-1)^d\\tilde{u}$. In view\nof the above two identities, this implies that\n$j_*T_*j_!=(-1)^dj_*j_!T_*$, or $T_*j_!=(-1)^dj_!T_*$. \n\nContinuing our computation,\n\\begin{equation*} \na\\cdot b=(-1)^{d|a|'}\\iota_*j_!T_*(a\\times b)\n=(-1)^{|a|'|b|'+d|\\alpha|}\\iota_*j_!(b\\times a)=(-1)^{(|a|'-d)(|b|'-d)}b\\cdot a.\n\\end{equation*}\nThis completes the homotopy theoretic proof of commutativity formula. \n\\end{proof}\n\n\\begin{remark}\nIf we let $\\mu=\\iota_*j_!:\\mathbb H_*(LM)\\otimes \\mathbb H_*(LM) \\longrightarrow \\mathbb H_*(LM)$, then using the method in \\cite{T2}, we can show that the associativity of $\\mu$ takes the form $\\mu\\circ(1\\otimes \\mu)=(-1)^d\\mu\\circ(\\mu\\otimes 1)$. With our choice of the sign for the loop product in Definition \\ref{definition of loop product}, we can get rid of the above sign and we have a usual associativity relation $(a\\cdot b)\\cdot c=a\\cdot(b\\cdot c)$ for the loop product without any signs for $a,b,c\\in\\mathbb H_*(LM)$. This is yet another reason of our choice of the sign in the definition of the loop product. \n\\end{remark}\n\n\nThe transfer map $j_!$ enjoys the following properties similar to\nthose satisfies by $\\phi_!$ as given in Proposition~\\ref{properties of\nphi}. The proof is similar, and we omit it.\n\n\\begin{proposition}\nFor $a,b\\in \\mathbb H_*(LM)$ and $\\alpha\\in H^*(LM\\times LM)$, the following\nformulas are valid.\n\\begin{align}\nj_*j_!(a\\times b)&=\\tilde{u}\\cap(a\\times b) \\label{j_!1}\\\\\nj_!\\bigl(\\alpha\\cap(a\\times b)\\bigr)&=\n(-1)^{d|\\alpha|}j^*(\\alpha)\\cap j_!(b\\times c) \\label{j_!2}\n\\end{align}\n\\end{proposition}\n\nThe second formula says that $j_!$ is a $H^*(LM\\times LM)$-module map. \n\n\n\n\n\n\n\n\n\n\n\\section{Cap products and extended BV algebra structure}\n\n\n\n\n\n\nWe examine compatibility of the cap product with the various\nstructures in the BV-algebra $\\mathbb{H}_*(LM)=H_{*+d}(LM)$.\n\nWe recall that a BV-algebra $A_*$ is an associative graded commutative\nalgebra equipped with a degree $1$ Lie bracket $\\{\\ ,\\ \\}$ and a\ndegree $1$ operator $\\Delta$ satisfying the following relations for\n$a,b,c\\in A_*$:\n\\begin{gather}\n\\Delta(a\\cdot b)=(\\Delta a)\\cdot b + (-1)^{|a|}a\\cdot \\Delta b +\n(-1)^{|a|}\\{a,b\\} \n\\tag{BV identity} \\\\ \n\\{a, b\\cdot c\\} =\\{a,b\\}\\cdot c + (-1)^{|b|(|a|+1)}b\\cdot\\{a,c\\} \n\\tag{Poisson identity} \\\\ \na\\cdot b=(-1)^{|a||b|}b\\cdot a,\\qquad \n\\{a,b\\}=-(-1)^{(|a|+1)(|b|+1)}\\{b,a\\}\n\\tag{Commutativity} \\\\\n\\{a,\\{b,c\\}\\}=\\{\\{a,b\\},c\\}+(-1)^{(|a|+1)(|b|+1)}\\{b,\\{a,c\\}\\}\n\\tag{Jacobi identity}\n\\end{gather} \nHere, degrees of elements are given by $\\Delta a\\in A_{|a|+1}, a\\cdot b\\in A_{|a|+|b|}$, and $\\{a,b\\}\\in A_{|a|+|b|+1}$. \nOne way to view these relations is to consider operators $D_a$ and\n$M_a$ acting on $A_*$ for each $a\\in A_*$ given by $D_a(b)=\\{a,b\\}$ and\n$M_a(b)=a\\cdot b$. Let $[x,y]=xy-(-1)^{|x||y|}yx$ be the graded\ncommutator of operators. Then the Poisson identity and the Jacobi\nidentity take the following forms:\n\\begin{equation}\n[D_a,M_b]=M_{\\{a,b\\}},\\qquad [D_a,D_b]=D_{\\{a,b\\}},\n\\end{equation}\nwhere degrees of operators are $|D_a|=|a|+1$ and $|M_b|=|b|$. \n\nOne nice context to understand BV identity is in the context of odd symplectic geometry (\\cite{G}, \\S2), where BV operator $\\Delta$ appears as a mixed second order odd differential operator, and BV identity can be simply understood as Leipnitz rule in differential calculus. This context actually arises in loop homology. In \\cite{T1}, we explicitly computed the BV structure of $\\mathbb H_*(LM)$ for the Lie group $\\text{SU}(n+1)$ and complex Stiefel manifolds. There, the BV operator $\\Delta$ is given by second order mixed odd differential operator as above, and $\\mathbb H_*(LM)$ is interpreted as the space of polynomial functions on the odd symplectic vector space. \n\nThe fact that the loop algebra $\\mathbb{H}_*(LM)$ is a\nBV-algebra was proved in \\cite{CS}. Note that the above BV relations\nare satisfied with respect to $\\mathbb{H}_*$-grading, rather than the\nusual homology grading. \nThe same is true for compatibility relations\nwith cap products. \n\nFirst we discuss cohomological $S^1$ action operator $\\Delta$ on $H^*(LM)$. Let $\\Delta:\nS^1\\times LM \\longrightarrow LM$ be the $S^1$ action map given by\n$\\Delta(t,\\gamma)=\\gamma_t$, where $\\gamma_t(s)=\\gamma(s+t)$ for\n$s,t\\in S^1=\\mathbb{R}\/\\mathbb{Z}$. The degree $-1$ operator\n$\\Delta: H^*(LM) \\longrightarrow H^{*-1}(LM)$ is defined by the\nfollowing formula for $\\alpha\\in H^*(LM)$:\n\\begin{equation}\n\\Delta^*(\\alpha)=1\\times\\alpha + \\{S^1\\}\\times \\Delta\\alpha\n\\end{equation}\nwhere $\\{S^1\\}$ is the fundamental cohomology class of $S^1$. The\nhomological $S^1$ action $\\Delta$ is not a derivation with respect to the loop product and the deviation from being a derivation is given \nby the loop bracket. However, the\ncohomology $S^1$-operator $\\Delta$ is a derivation with respect to the cup product.\n\n\\begin{proposition}\nThe cohomology $S^1$-operator $\\Delta$ satisfies $\\Delta^2=0$, and it acts as a derivation on the cohomology ring $H^*(LM)$. That is, for\n$\\alpha,\\beta\\in H^*(LM)$,\n\\begin{equation}\\label{delta and cup}\n\\Delta(\\alpha\\cup\\beta)=(\\Delta\\alpha)\\cup\\beta +\n(-1)^{|\\alpha|}\\alpha\\cup\\Delta\\beta.\n\\end{equation}\n\\end{proposition}\n\\begin{proof} The property $\\Delta^2=0$ is straightforward \nusing the following diagram\n\\begin{equation*}\n\\begin{CD}\nS^1\\times S^1 \\times LM @>{1\\times\\Delta}>> S^1\\times LM \\\\ @V{\\mu\\times 1}VV\n@V{\\Delta}VV \\\\ S^1\\times LM @>{\\Delta}>> LM\n\\end{CD}\n\\end{equation*}\nComparing both sides of $(1\\times\\Delta)^*\\Delta^*(\\alpha)=(\\mu\\times\n1)^*\\Delta^*(\\alpha)$, we obtain $\\Delta^2(\\alpha)=0$.\n\nFor the derivation property, we consider the following diagram. \n\\begin{equation*}\n\\begin{CD}\nS^1\\times LM @>{\\phi\\times\\phi}>>  (S^1\\times S^1)\\times (LM\\times LM) \n@>{1\\times T\\times 1}>>  (S^1\\times LM)\\times (S^1\\times LM)  \\\\\n@V{\\Delta}VV    @.   @V{\\Delta\\times\\Delta}VV    \\\\\nLM @>{\\phi}>> LM\\times LM @= LM\\times LM \n\\end{CD}\n\\end{equation*} \nOn the one hand, $\\Delta^*\\phi^*(\\alpha\\times\n\\beta)=\\Delta^*(\\alpha\\cup\\beta) =1\\times(\\alpha\\cup\\beta) +\n\\{S^1\\}\\times \\Delta(\\alpha\\cup\\beta)$. On the other hand,\n\\begin{equation*}\n(\\phi\\times\\phi)^*(1\\times T\\times 1)^*(\\Delta\\times\n\\Delta)^*(\\alpha\\times \\beta) =1\\times(\\alpha\\cup\\beta) +\n(-1)^{|\\alpha|}\\{S^1\\}\\times \\bigl(\\alpha\\cup\\Delta\\beta \n+ \\Delta\\alpha\\cup\\beta\\bigr).\n\\end{equation*}\nComparing the above two identities, we obtain the derivation formula.\n\\end{proof} \n\nWe can regard the cohomology ring $H^*(LM)$ together with cohomological $S^1$ action $\\Delta$ as a BV algebra with trivial bracket product. \n\nNow we show that the cap product is compatible with the loop product\nin the BV-algebra $\\mathbb H_*(LM)$. The following theorem describes the behavior of the cap product with those elements in the subalgebra of $H^*(LM)$ generated by $H^*(M)$ and $\\Delta\\bigl(H^*(M)\\bigr)$.\n\n\\begin{theorem} Let $\\alpha\\in H^*(M)$ and $b,c\\in \\mathbb H_*(LM)$. \nThe cap product with $p^*(\\alpha)$ behaves\nassociatively and graded commutatively with respect to the loop\nproduct. Namely\n\\begin{equation}\\label{cap and loop product} \np^*(\\alpha)\\cap(b\\cdot c)=(p^*(\\alpha)\\cap b)\\cdot c\n=(-1)^{|\\alpha||b|}b\\cdot(p^*(\\alpha)\\cap c).\n\\end{equation}\n\nThe cap product with \n$\\Delta\\bigl(p^*(\\alpha)\\bigr)$ is a derivation on the loop product. Namely, \n\\begin{equation}\\label{cap derivation} \n\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap(b\\cdot c)\n=\\bigl(\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b\\bigr)\\cdot c +\n(-1)^{(|\\alpha|-1)|b|}b\\cdot\\bigl(\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap\nc\\bigr).\n\\end{equation}\n\\end{theorem} \n\\begin{proof}\nFor the first formula, we consider the following diagram, \nwhere $\\pi_i$ is the projection onto the $i$th factor for $i=1,2$.  \n\\begin{equation*}\n\\begin{CD} \nLM @<{\\pi_i}<<  LM\\times LM @<{j}<< LM\\times_M LM  @>{\\iota}>> LM  \\\\\n@V{p}VV  @V{p\\times p}VV  @V{q}VV   @V{p}VV \\\\\nM @<{\\pi_i}<< M\\times M  @<{\\phi}<< M @= M \n\\end{CD}\n\\end{equation*} \nSince $p^*(\\alpha)\\cap(b\\cdot c)\n=(-1)^{d|b|}\\iota_*\\bigl(\\iota^*p^*(\\alpha)\\cap j_!(b\\times c)\\bigr)$, \nwe need to understand $\\iota^*p^*(\\alpha)$. From the above commutative \ndiagram, we have $\\iota^*p^*(\\alpha)=j^*\\pi_i^*p^*(\\alpha)$, which is\nequal to either $j^*(p^*(\\alpha)\\times 1)$ or $j^*(1\\times\np^*(\\alpha))$. In the first case, continuting our computation using \n\\eqref{j_!2}, we have \n\\begin{align*}\np^*(\\alpha)\\cap(b\\cdot c)&=\n(-1)^{d|b|}\\iota_*\\bigl(j^*(p^*(\\alpha)\\times 1)\n\\cap j_!(b\\times c)\\bigr) \\\\\n&=(-1)^{d|b|+d|\\alpha|}\\iota_*j_!\n\\bigl((p^*(\\alpha)\\times 1)\\cap(b\\times c)\\bigr) \\\\\n&=(-1)^{d|b|+d|\\alpha|}\\iota_*j_!\n\\bigl((p^*(\\alpha)\\cap b)\\times c\\bigr)  \\\\\n&=(p^*(\\alpha)\\cap b)\\cdot c.\n\\end{align*}\nSimilarly, using $\\iota^*p^*(\\alpha)\n=j^*\\bigl(1\\times p^*(\\alpha)\\bigr)$, we get the other identity. This\nproves (1). \n\nFor (2), we first note that the element\n$\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap(b\\cdot c)$ is equal to \n\\begin{equation*}\n\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap (-1)^{d|b|}\\iota_*j_!(b\\times c)\n=(-1)^{d|b|}\\iota_*\\bigl(\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)\n\\cap j_!(b\\times c)\\bigr).\n\\end{equation*}\nThus, we need to understand the element\n$\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)$. We need some notations. Let\n$I=I_1\\cup I_2$, where $I_1=[0,\\frac12]$ and $I_2=[\\frac12,1]$, and\nset $S_i^1=I_i\/\\partial I_i$ for $i=1,2$. Let $r: S^1=I\/\\partial I \n\\longrightarrow I\/\\{0,\\frac12,1\\}=S^1_1\\vee S^1_2$ be an\nidentification map, and let $\\iota_i: S^1_i \\longrightarrow S^1_1\\vee\nS^1_2$ be the inclusion map into the $i$th wedge summand. We examine\nthe following diagram \n\\begin{equation*}\n\\begin{CD}\nS^1\\times(LM\\underset{M}{\\times}LM)  @>{r\\times 1}>> \n(S^1_1\\vee S^1_2)\\times\n(LM\\underset{M}{\\times}LM) @<<< \\{0\\}\\times (LM\\underset{M}{\\times}LM)  \\\\\n@V{1\\times\\iota}VV   @V{e'}VV   @V{\\iota}VV   \\\\\nS^1\\times LM  @>{e}>> M @<{p}<<  LM \n\\end{CD}\n\\end{equation*} \nwhere $e=p\\circ\\Delta$ is the evaluation map for $S^1\\times LM$, and\nthe other evaluation map $e'$ is given by \n\\begin{equation*}\ne'(t,\\gamma,\\eta)=\n\\begin{cases}\n\\gamma(2t)& 0\\le t\\le\\frac12,\\\\\n\\eta(2t-1)& \\frac12\\le t\\le 1.\n\\end{cases}\n\\end{equation*}\nFor $\\alpha\\in H^*(M)$, we let \n\\begin{equation*}\n{e'}^*(\\alpha)=1\\times \\iota^*p^*(\\alpha) \n+\\{s^1_1\\}\\times \\Delta_1(\\alpha) \n+\\{S^1_2\\}\\times \\Delta_2(\\alpha)\n\\end{equation*}\nfor some $\\Delta_i(\\alpha)\\in H^*(LM\\times_M LM)$ for $i=1,2$. The\nfirst term in the right hand side is identified using the right square\nof the above commutative diagram. Since $r^*(\\{S^1_i\\})=\\{S^1\\}$ for\n$i=1,2$,  \n\\begin{equation*}\n(r\\times 1)^*{e'}^*(\\alpha)=1\\times \\iota^*p^*(\\alpha) + \n\\{S^1\\}\\times \\bigl(\\Delta_1(\\alpha)+\\Delta_2(\\alpha)\\bigr). \n\\end{equation*}\nThe commutativity of the left square implies that this must be equal to \n\\begin{equation*}\n(1\\times\\iota)^*\\Delta^*p^*(\\alpha)\n=1\\times \\iota^*p^*(\\alpha)+\\{S^1\\}\\times \n\\iota^*\\Delta\\bigl(p^*(\\alpha)\\bigr). \n\\end{equation*}\nHence we have \n\\begin{equation*}\n\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)\n=\\Delta_1(\\alpha)+\\Delta_2(\\alpha)\\in H_*(LM\\times_M LM). \n\\end{equation*}\nTo understand elements $\\Delta_i(\\alpha)$, we consider the following\ncommutative diagram, where $\\ell_1(t)=2t$ for $0\\le t\\le\\frac12$ and \n$\\ell_2(t)=2t-1$ for $\\frac12\\le t\\le 1$. \n\\begin{equation*}\n\\begin{CD}\nS^1_i\\times(LM\\times_M LM) @>{\\ell_i\\times j}>> S^1\\times(LM\\times LM)\n@>{1\\times\\pi_i}>> S^1\\times LM \\\\\n@V{\\iota_i\\times 1}VV @. @V{\\Delta}VV \\\\\n(S^1_1\\vee S^1_2)\\times(LM\\times_M LM) @>{e'}>> M  @<{p}<<  LM \n\\end{CD} \n\\end{equation*} \nOn the one hand, $(\\iota_1\\times 1)^*{e'}^*(\\alpha)\n=1\\times\\iota^*p^*(\\alpha)+ \\{S^1_1\\}\\times\\Delta_1(\\alpha)$. On the\nother hand, \n\\begin{equation*}\n(\\ell_1\\times j)^*(1\\times \\pi_1)^*\\Delta^*p^*(\\alpha)=\n1\\times j^*\\bigl(p^*(\\alpha)\\times 1\\bigr)\n+\\{S^1_1\\}\\times j^*\\bigl(\\Delta(p^*(\\alpha))\\times 1\\bigr).\n\\end{equation*}\nBy the commutativity of the diagram, we\nget $\\Delta_1(\\alpha)=j^*\\bigl(\\Delta(p^*(\\alpha))\\times\n1\\bigr)$. Similarly, $i=2$ case implies $\\Delta_2(\\alpha)\n=j^*\\bigl(1\\times\\Delta(p^*(\\alpha))\\bigr)$. Hence we finally obtain \n\\begin{equation*}\n\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)\n=j^*\\bigl(\\Delta(p^*(\\alpha))\\times 1 + 1\\times\n\\Delta(p^*(\\alpha))\\bigr).\n\\end{equation*}\nWith this identification of $\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)$\nas $j^*$ of some other element, we can continue our initial\ncomputation. \n\\begin{equation*}\n\\begin{split}\n\\Delta\\bigl(&p^*(\\alpha)\\bigr)\\cap(b\\cdot c)\n=(-1)^{d|b|}\\iota_*\\bigl(j^*\\bigl(\\Delta(p^*(\\alpha))\\times 1 + \n1\\times \\Delta\\bigl(p^*(\\alpha)\\bigr)\\bigr)\\cap j_!(b\\times c)\\bigr)\n\\\\\n&=(-1)^{d|b|+(|\\alpha|-1)d} \n\\iota_*j_!\\Bigl(\\bigl(\\Delta(p^*(\\alpha))\\times 1 \n+1\\times \\Delta\\bigl(p^*(\\alpha)\\bigr)\\bigr)\\cap(b\\times c)\\Bigr) \\\\\n&=(-1)^{d(|\\alpha|+|b|-1)}\\iota_*j_!\\Bigl(\n\\bigl(\\Delta(p^*(\\alpha))\\cap b\\bigr)\\times c +\n(-1)^{(|b|+d)(|\\alpha|-1)}b\\times \n\\bigl(\\Delta(p^*(\\alpha))\\cap c\\bigr)\\Bigr) \\\\\n&=\\bigl(\\Delta(p^*(\\alpha))\\cap b\\bigr)\\cdot c +\n(-1)^{(|\\alpha|-1)|b|}b\\cdot\\bigl(\\Delta(p^*(\\alpha))\\cap c\\bigr).\n\\end{split}\n\\end{equation*}\nThis completes the proof of the derivation property of the cap product\nwith respect to the loop product. \n\\end{proof} \n\nNext we describe the relation between the cap product and the BV\noperator in homology and cohomology. \n\n\\begin{proposition} For $\\alpha\\in H^*(LM)$ and $b\\in \\mathbb H_*(LM)$, the BV-operator $\\Delta$ satisfies \n\\begin{equation}\\label{delta and cap}\n\\Delta(\\alpha\\cap b)=(\\Delta\\alpha)\\cap b+(-1)^{|\\alpha|}\\alpha\\cap\n\\Delta b.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nOn the one hand, the $S^1$-action map $\\Delta: S^1\\times LM\n\\longrightarrow LM$ satisfies \n\\begin{equation*}\n\\Delta_*\\bigl(\\Delta^*(\\alpha)\\cap([S^1]\\times b)\\bigr)\n=\\alpha\\cap\\Delta_*([S^1]\\times b)\n=\\alpha\\cap\\Delta b.\n\\end{equation*}\nOn the other hand, since $\\Delta^*(\\alpha)=1\\times\\alpha +\n\\{S^1\\}\\times \\Delta\\alpha$, we have \n\\begin{equation*}\n\\begin{split}\n\\Delta_*\\bigl(\\Delta^*(\\alpha)\\cap([S^1]\\times b)\\bigr)\n&=\\Delta_*\\bigl(\n(-1)^{|\\alpha|}[S^1]\\times(\\alpha\\cap b) +\n(-1)^{|\\alpha|-1}[pt]\\times(\\Delta\\alpha\\cap b)\\bigr)  \\\\\n&=(-1)^{|\\alpha|}\\Delta(\\alpha\\cap b) + \n(-1)^{|\\alpha|-1}\\Delta\\alpha\\cap b. \n\\end{split}\n\\end{equation*}\nComparing the above two formulas, we obtain \n$\\Delta(\\alpha\\cap b)=\\Delta\\alpha\\cap b + \n(-1)^{|\\alpha|}\\alpha\\cap\\Delta b$. \n\\end{proof} \nSince homology BV operator $\\Delta$ on $\\mathbb H_*(LM)$ acts trivially on $\\mathbb H_*(M)$, the\nfollowing corollary is immediate. \n\\begin{corollary}\nFor $\\alpha\\in H^*(M)$, the cap product of $\\Delta\\alpha$ with\n$\\mathbb H_*(M)\\subset \\mathbb H_*(LM)$ is trivial.\n\\end{corollary} \n\\begin{proof} For $b\\in\\mathbb H_*(M)$, the operator $\\Delta$ acts trivially on both $\\alpha\\cap b$ and $b$. Hence formula \\eqref{delta and cap} implies $(\\Delta\\alpha)\\cap b=0$. \n\\end{proof} \n\nNext, we discuss a behavior of the cap product with respect to the\nloop bracket.\n\n\\begin{theorem}  The cap product with $\\Delta\\bigl(p^*(\\alpha)\\bigr)$ \nis a derivation on the loop bracket. Namely, for $\\alpha\\in H^*(M)$\nand $b,c\\in \\mathbb H_*(LM)$,\n\\begin{equation}\\label{cap-loop bracket}\n\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap\\{b,c\\}\n=\\{\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b, c\\} +\n(-1)^{(|\\alpha|-1)(|b|-1)}\\{b, \\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap\nc\\}.\n\\end{equation}\n\\end{theorem} \n\\begin{proof}  Our proof is computational using previous results. \nWe use the BV identity as the definition of the loop bracket. Thus,\n\\begin{equation*}\n\\{b,c\\}=(-1)^{|b|}\\Delta(b\\cdot c)-(-1)^{|b|}(\\Delta b)\\cdot c \n-b\\cdot \\Delta c.\n\\end{equation*} \nWe compute the right hand side of \\eqref{cap-loop bracket}. \nFor simplicity, we write $\\Delta\\alpha$\nfor $\\Delta\\bigl(p^*(\\alpha)\\bigr)$.  Each term in the right hand side\nof \\eqref{cap-loop bracket} gives\n\\begin{gather*}\n\\!\\!\\!\\!\\!\\!\\!\\{\\Delta\\alpha\\cap b, c\\}\n=(-1)^{|b|-|\\alpha|+1}\\Delta\\bigl((\\Delta\\alpha\\cap b)\\cdot c\\bigr)\n-(-1)^{|b|}(\\Delta\\alpha\\cap \\Delta b)\\cdot c \n-(\\Delta\\alpha\\cap b)\\cdot\\Delta c, \\\\\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\{b,\\Delta\\alpha\\cap c\\}\n=(-1)^{|b|}\\Delta\\bigl(b\\cdot(\\Delta\\alpha\\cap c)\\bigr)\n-(-1)^{|b|}\\Delta b\\cdot(\\Delta\\alpha\\cap c) \n-(-1)^{|\\alpha|-1}b\\cdot(\\Delta\\alpha\\cap\\Delta c),\n\\end{gather*}\nHere we used \\eqref{delta and cap} for the second term in the first\nidentity and in the third term in the second identity. Combining these\nformulas, we get\n\\begin{multline*} \n\\{\\Delta\\alpha\\cap b, c\\}+ (-1)^{(|\\alpha|-1)(|b|+1)}\n\\{b,\\Delta\\alpha\\cap c\\}  \\\\\n=\\bigl[ \n(-1)^{|b|-|\\alpha|+1}\\Delta\\bigl((\\Delta\\alpha\\cap b)\\cdot c\\bigr)\n+(-1)^{|b|+(|\\alpha|-1)(|b|+1)}\n\\Delta\\bigl(b\\cdot(\\Delta\\alpha\\cap c)\\bigr)\\bigr]  \\\\\n-\\bigl[(-1)^{|b|}(\\Delta\\alpha\\cap\\Delta b)\\cdot c + \n(-1)^{(|\\alpha|-1)(|b|+1)+|b|}\\Delta b\\cdot(\\Delta\\alpha\\cap c)\\bigr] \\\\\n-\\bigl[(\\Delta\\alpha\\cap b)\\cdot \\Delta c +\n(-1)^{(|\\alpha|-1)(|b|+1)+|\\alpha|-1}b\\cdot(\\Delta\\alpha\\cap \\Delta c)\\bigr].\n\\end{multline*}\nUsing the derivation formula for $\\Delta\\alpha\\cap(\\ )$ with respect\nto the loop product \\eqref{cap derivation}, three pairs of terms above\nbecome\n\\begin{multline*}\n(-1)^{|b|-|\\alpha|+1}\\Delta\\bigl(\\Delta\\alpha\\cap(b\\cdot c)\\bigr)\n-(-1)^{|b|}\\Delta\\alpha\\cap(\\Delta b\\cdot c)\n-\\Delta\\alpha\\cap(b\\cdot\\Delta c) \\\\\n=\\Delta\\alpha\\cap\\bigl[(-1)^{|b|}\\Delta(b\\cdot c)\n-(-1)^{|b|}\\Delta b\\cdot c -b\\cdot\\Delta c\\bigr] = \\Delta\\alpha\\cap\\{b,c\\}.\n\\end{multline*}\nThis completes the proof of the derivation formula for the loop bracket. \n\\end{proof} \n\nRecall that in the BV algebra $\\mathbb H_*(LM)$, for every $a\\in \\mathbb H_*(LM)$ the\noperation $\\{a,\\ \\cdot \\ \\}$ of taking the loop bracket with $a$ is a\nderivation with respect to both the loop product and the loop bracket,\nin view of the Poisson identity and the Jacobi identity. Since we have\nproved that the cap product with $\\Delta p^*(\\alpha)$ for $\\alpha\\in\nH^*(M)$ is a derivation with respect to both the loop product and the\nloop bracket, we wonder if we can extend the BV structure in $\\mathbb H_*(LM)$ to a BV structure in $H^*(M)\\oplus \\mathbb H_*(LM)$. Indeed this is possible by extending the loop product and the loop bracket to elements in $H^*(M)$ as follows.\n\\begin{definition}\nFor $\\alpha,\\beta\\in H^*(M)$ and $b\\in \\mathbb H_*(LM)$, we define their loop product and loop bracket by\n\\begin{equation}\n\\begin{gathered}\n\\alpha\\cdot b=\\alpha\\cap b,\\qquad \n\\{\\alpha,b\\}=(-1)^{|\\alpha|}(\\Delta\\alpha)\\cap b,\\\\\n\\alpha\\cdot\\beta=\\alpha\\cup\\beta, \\qquad \\{\\alpha,\\beta\\}=0.\n\\end{gathered}\n\\end{equation}\nThis defines an associative graded commutative loop product by \\eqref{cap and loop product}, and a bracket product on $H^*(M)\\oplus\\mathbb H_*(LM)$. \n\\end{definition}\nNote that this loop product on $H^*(M)\\oplus\\mathbb H_*(LM)$ reduces to the ring structure on $H^*(M)\\oplus \\mathbb H_*(M)$ mentioned in the introduction. \n\nWith this definition, Poisson identities and Jacobi identities are\nstill valid in $H^*(M)\\oplus \\mathbb H_*(LM)$.\n\n\\begin{theorem} Let $\\alpha,\\beta\\in H^*(M)$, and let $b,c\\in \\mathbb H_*(LM)$. \n\n\\noindent\\textup{(I)} The following Poisson identities are valid in $H^*(M)\\oplus\\mathbb H_*(LM)$\\textup{:} \n\\begin{align}\n\\{\\alpha,\\beta\\cdot c\\}\n&=\\{\\alpha,\\beta\\}\\cdot c+ \n(-1)^{|\\beta|(|\\alpha|-1)}\\beta\\cdot\\{\\alpha,c\\} \\label{eq1}\\\\\n\\{\\alpha\\beta,c\\}&=\\alpha\\cdot\\{\\beta,c\\}\n+(-1)^{|\\alpha||\\beta|}\\beta\\cdot\\{\\alpha,c\\} \\label{eq2}\\\\\n\\{\\alpha,b\\cdot c\\}&=\\{\\alpha,b\\}\\cdot c\n+(-1)^{(|b|-d)(|\\alpha|-1)}b\\cdot\\{\\alpha,c\\} \\label{eq3}\\\\\n\\{\\alpha\\cdot b,c\\}&=\\alpha\\cdot\\{b,c\\}+\n(-1)^{|\\alpha|(|b|-d)}b\\cdot\\{\\alpha,c\\}. \\label{eq4} \n\\end{align}\n\n\\noindent\\textup{(II)} The following Jacobi identities are valid in $H^*(M)\\oplus\\mathbb H_*(LM)$\\textup{:}\n\\begin{align}\n\\{\\alpha,\\{\\beta,c\\}\\}&=\\{\\{\\alpha,\\beta\\},c\\}+\n(-1)^{(|\\alpha|-1)(|\\beta|-1)}\\{\\beta,\\{\\alpha,c\\}\\} \\label{eq5} \\\\\n\\{\\alpha,\\{b,c\\}\\}&=\\{\\{\\alpha,b\\},c\\}+\n(-1)^{(|\\alpha|-1)(|b|-d+1)} \\{b,\\{\\alpha,c\\}\\}.  \\label{eq6}\n\\end{align}\n\\end{theorem}\n\\begin{proof}  If we unravel definitions, we see that \\eqref{eq1} \nand \\eqref{eq5} are really the same as the graded commutativity of the\ncup product of the following form\n\\begin{align*}\n(\\Delta\\alpha)\\cap(b\\cap c)&=\n(-1)^{|\\beta|(|\\alpha|-1)}\\beta\\cap(\\Delta\\alpha\\cap c), \\\\\n(\\Delta\\alpha)\\cap(\\Delta\\beta\\cap c)&=(-1)^{(|\\alpha|-1)(|\\beta|-1)}\n(\\Delta\\beta)\\cap\\bigl((\\Delta\\alpha)\\cap c\\bigr).\n\\end{align*}\nthe identity \\eqref{eq2} is equivalent to the derivation formula \\eqref{delta and cup} of\nthe cohomology $S^1$ action operator with respect to the cup product.\n\\begin{equation*}\n\\Delta(\\alpha\\cup \\beta)=(\\Delta\\alpha)\\cup \\beta + (-1)^{|\\alpha|}\n\\alpha\\cup (\\Delta \\beta).\n\\end{equation*}\nThe identity \\eqref{eq3} says that $\\Delta\\alpha\\cap (\\ )$ is a\nderivation with respect to the loop product, and the identity\n\\eqref{eq6} says that $\\Delta\\alpha\\cap(\\ )$ is a derivation with\nrespect to the loop bracket. We have already verified both of these\ncases. Thus, what remains to be checked is formula \\eqref{eq4}, which\nsays\n\\begin{equation*}\n\\{\\alpha\\cap b,c\\}=\\alpha\\cap\\{b,c\\}+\n(-1)^{|\\alpha||b|+|\\alpha|}b\\cdot(\\Delta\\alpha\\cap c).\n\\end{equation*}\nUsing the BV identity, the derivation formula \\eqref{delta and cap} \nof the BV operator with\nrespect to the cap product, and properties of $\\alpha\\cap(\\ )$ and\n$\\Delta\\alpha\\cap(\\ )$, we can prove this identity as follows.\n\\begin{multline*}\n(-1)^{|b|-|\\alpha|}\\{\\alpha\\cap b,c\\}\n=\\Delta\\bigl((\\alpha\\cap b)\\cdot c\\bigr)-\\Delta(\\alpha\\cap b)\\cdot c\n-(-1)^{|b|-|\\alpha|}(\\alpha\\cap b)\\cdot\\Delta c \\\\\n=\\Delta\\bigl(\\alpha\\cap(b\\cdot c)\\bigr)\n-(\\Delta\\alpha\\cap b+(-1)^{|\\alpha|}\\alpha\\cap\\Delta b)\\cdot c\n-(-1)^{|b|-|\\alpha|}\\alpha\\cap(b\\cdot\\Delta c) \\\\\n=(\\Delta\\alpha)\\cap(b\\cdot c)\n-(\\Delta\\alpha\\cap b)\\cdot c\n+(-1)^{|\\alpha|}\\alpha\\cap\\Delta(b\\cdot c) \\\\\n-(-1)^{|\\alpha|}\\alpha\\cap(\\Delta b\\cdot c)\n-(-1)^{|b|-|\\alpha|}\\alpha\\cap(b\\cdot\\Delta c) \\\\\n=(-1)^{(|\\alpha|-1)|b|}b\\cdot(\\Delta\\alpha\\cap c)\n+(-1)^{|\\alpha|+|b|}\\alpha\\cap\\{b,c\\}.\n\\end{multline*}\nCanceling some signs, we get the desired formula. This completes the proof. \n\\end{proof}\n\nOther Poisson and Jacobi identities with cohomology elements in the second argument formally follow from above identities by making following definitions for $\\alpha\\in H^*(M)$ and $b\\in\\mathbb H_*(LM)$: \n\\begin{equation*}\nb\\cdot\\alpha=(-1)^{|\\alpha||b|}\\alpha\\cdot b,\\qquad \n\\{b,\\alpha\\}=-(-1)^{(|\\alpha|+1)(|b|+1)}\\{\\alpha,b\\}.\n\\end{equation*}\n\nFor $\\alpha\\in H^*(M)$ we showed that $\\Delta\\alpha\\cap(\\ )$ is a\nderivation for both the loop product and the loop bracket, and\n$\\alpha\\cap(\\ )$ is graded commutative and associative with respect to\nthe loop product.  What is the behavior\nof $\\alpha\\cap(\\ )$ is with respect to the loop bracket? \nFormula \\eqref{eq4} says that $\\alpha\\cap(\\ \\cdot\\ )$ on loop bracket  is not a derivation or graded commutativity: it is a Poisson identity!\n\nPoisson identities and Jacobi identities we have just proved in $A_*=H^*(M)\\oplus\\mathbb H_*(LM)$ show that $A_*$ is a Gerstenhaber algebra. In fact, $A_*$ can be formally turned into a BV algebra by defining a BV operator $\\boldsymbol\\Delta$ on $A_*$ to be trivial on $H^*(M)$ and to be the usual one on $\\mathbb H_*(LM)$ coming from the homological $S^1$ action. \n\n\\begin{corollary}\\label{BV structure on direct sum} \nThe direct sum $A_*=H^*(M)\\oplus\\mathbb H_*(LM)$ has the structure of a BV algebra. \n\\end{corollary}\n\\begin{proof} Since $\\mathbb H_*(LM)$ is a BV algebra and since we have already verified Poisson identities and Jacobi identities in $A_*$, we only have to verify BV identities in $A_*$. For $\\alpha,\\beta\\in H^*(M)$, an identity \n\\begin{equation*}\n\\boldsymbol\\Delta(\\alpha\\cup\\beta)=(\\boldsymbol\\Delta\\alpha)\\cup\\beta\n+(-1)^{|\\alpha|}\\alpha\\cup(\\boldsymbol\\Delta\\beta)\n+(-1)^{|\\alpha|}\\{\\alpha,\\beta\\}\n\\end{equation*}\nis trivially satisfied since all terms are zero by definition of BV operator $\\boldsymbol\\Delta$ and the loop bracket on $H^*(M)\\subset A_*$. \n\nNext, let $\\alpha\\in H^*(M)$ and $b\\in\\mathbb H_*(LM)$. Since the BV operator $\\boldsymbol\\Delta$ on $A_*$ acts trivially on $H^*(M)$, an identity \n\\begin{equation*}\n\\boldsymbol\\Delta(\\alpha\\cap b)=(\\boldsymbol\\Delta\\alpha)\\cap b\n+(-1)^{|\\alpha|}\\alpha\\cap(\\boldsymbol\\Delta b)\n+(-1)^{|\\alpha|}\\{\\alpha,b\\}\n\\end{equation*}\nis really a restatement of the derivative formula of the homology $S^1$ action operator $\\Delta$ on cap product: $\\Delta(\\alpha\\cap b)=(-1)^{|\\alpha|}\\alpha\\cap(\\Delta b)+(\\Delta\\alpha)\\cap b$ in formula \\eqref{delta and cap}. \n\\end{proof}\n\nIn connection with the above Corollary, we can ask whether $H^*(LM)\\oplus \\mathbb H_*(LM)$ has a structure of a BV algebra. Of course, $H^*(LM)$ together with the cohomological $S^1$ action operator $\\Delta$, which is a derivation, is a BV algebra with trivial bracket product. Thus, as a direct sum of BV algebras, $H^*(LM)\\oplus\\mathbb H_*(LM)$ is a BV algebra, although products between $H^*(LM)$ and $\\mathbb H_*(LM)$ are trivial. More meaningful question would be to ask whether the direct sum $H^*(LM)\\oplus\\mathbb H_*(LM)$ has a BV algebra structure extending the one on $A_*$ described in Corollary \\ref{BV structure on direct sum}. If we want to use the cap product as an extension of the loop product, the answer is no. This is because the cap product with an arbitrary element $\\alpha\\in H^*(LM)$ does not behave associatively with respect to the loop product in $\\mathbb H_*(LM)$: if $\\alpha$ is of the form $\\alpha=\\Delta\\beta$ for some $\\beta\\in H^*(M)$, then $\\alpha\\cap(\\ \\cdot\\ )$ acts as a derivation on loop product in $\\mathbb H_*(LM)$ due to \\eqref{cap derivation} and does not satisfy associativity.  \n\n\\begin{remark} In the course of our investigation, we noticed the \nfollowing curious identity, which is in some sense symmetric in three\nvariables, for $\\alpha\\in H^*(M)$ and $b,c\\in\\mathbb H_*(LM)$. \n\\begin{equation}\n\\begin{split}\n\\{\\alpha,b\\cdot c\\}+(-1)^{|b|}\\alpha\\cdot\\{b,c\\}\n&=\\{\\alpha,b\\}\\cdot c+(-1)^{|b|}\\{\\alpha\\cdot b,c\\} \\\\\n&=(-1)^{(|\\alpha|+1)|b|}\\bigl(b\\cdot\\{\\alpha,c\\}\n+(-1)^{|\\alpha|}\\{b,\\alpha\\cdot c\\}\\bigr).\n\\end{split}\n\\end{equation}\nThis identity is easily proved using Poisson identities. But we wonder\nthe meaning of this symmetry.\n\\end{remark} \n\n\n\n\n\n\n\n\n\n\\section{Cap products in terms of BV algebra structure} \n\n\n\n\n\n\nIn the previous section, we showed that the BV algebra structure in $\\mathbb H_*(LM)$ can be extended to the BV algebra structure in $H^*(M)\\oplus \\mathbb H_*(LM)$ by proving Poisson identities and Jacobi identities. This may be a bit surprising. But this turns out to be very natural through Poincar\\'e duality in the following way. For $a\\in \\mathbb H_*(M)$, we denote the element $s_*(a)\\in \\mathbb H_*(LM)$ by $a$, where $s:M\\to LM$ is the inclusion map.  \n\n\\begin{theorem} For $a\\in \\mathbb H_*(M)$, let $\\alpha=D(a)\\in H^*(M)$ be its Poincar\\'e dual. Then for any $b\\in \\mathbb H_*(LM)$, the following identities hold. \n\\begin{equation}\np^*(\\alpha)\\cap b=a\\cdot b,\\qquad \n(-1)^{|\\alpha|}\\Delta \\bigl(p^*(\\alpha)\\bigr)\\cap b=\\{a,b\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof} Let $1=s_*([M])\\in \\mathbb H_0(LM)$ be the unit of the loop product. Since $p^*(\\alpha)\\cap b=p^*(\\alpha)\\cap(1\\cdot b)=\n\\bigl(p^*(\\alpha)\\cap1\\bigr)\\cdot b$ by \\eqref{cap and loop product}, and since \n\\begin{equation*}\np^*(\\alpha)\\cap1=p^*(\\alpha)\\cap s_*([M])=s_*\\bigl(s^*p^*(\\alpha)\\cap[M]\\bigr)\n=s_*(\\alpha\\cap[M])=a,\n\\end{equation*}\nwe have $p^*(\\alpha)\\cap b=a\\cdot b$. This proves the first identity. \n\nFor the second identity, in the BV identity\n\\begin{equation*}\n(-1)^{|a|}\\{a,b\\}=\\Delta(a\\cdot b)-(\\Delta a)\\cdot b\n-(-1)^{|a|}a\\cdot\\Delta b,\n\\end{equation*}\nthe first term in the right hand side gives \n\\begin{equation*}\n\\Delta(a\\cdot b)=\\Delta(p^*(\\alpha)\\cap\nb)=\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b\n+(-1)^{|\\alpha|}p^*(\\alpha)\\cap\\Delta b\n\\end{equation*}\nin view of the first identity we just proved and the derivation\nproperty of the homological $A^1$ action operator on cap products. Here\n$p^*(\\alpha)\\cap\\Delta b=a\\cdot\\Delta b$. Since $a\\in \\mathbb H_*(M)$ is a\nhomology class of constant loops, we have $\\Delta a=0$. Thus,\n\\begin{equation*}\n(-1)^{|a|}\\{a,b\\}=\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b\n+(-1)^{|\\alpha|}a\\cdot\\Delta b-(-1)^{|a|}a\\cdot\\Delta b\n=\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b,\n\\end{equation*}\nsince $|\\alpha|=-|a|$. Thus, $\\{a,b\\}=(-1)^{|\\alpha|}\n\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b$. This completes the proof. \n\\end{proof}\n\nIn view of this theorem, since $\\mathbb H_*(LM)$ is already a BV algebra, the\nPoisson identities and Jacobi identities we proved in section 4 may\nseem obvious. However, what we did in section 4 is that we gave a\n\\emph{new and elementary homotopy theoretic proof} of Poisson\nidentities and Jacobi identities using only basic properties of the\ncap product and the BV identity, when at least one of the elements is\nfrom $\\mathbb H_*(M)$.\n\nThe above theorem shows that loop products and loop brackets with elements in $\\mathbb H_*(M)$ can be written as cap products with cohomology elements in $LM$. Thus, compositions of loop products and loop brackets with elements in $\\mathbb H_*(M)$ corresponds to a cap product with the product of corresponding cohomology classes in $H^*(LM)$. Namely,\n\\begin{corollary}\nLet $a_0, a_1, \\dotsc, a_r\\in \\mathbb H_*(M)$, and let\n$\\alpha_0,\\alpha_1,\\dotsc\\alpha_r\\in H^*(M)$ be their Poincar\\'e\nduals. Then for $b\\in \\mathbb H_*(LM)$,\n\\begin{equation}\na_0\\cdot\\{a_1,\\{a_2,\\dotsc\\{a_r,b\\}\\dotsb\\}\\}\n=(-1)^{|a_1|+\\dotsb+|a_r|}\n\\bigl[\\alpha_0(\\Delta\\alpha_1)(\\Delta\\alpha_2)\n\\dotsm(\\Delta\\alpha_r)\\bigr]\\cap b.\n\\end{equation}\n\\end{corollary}\n\nIn section 2, we considered a problem of intersections of loops with\nsubmanifolds in certain configurations, and we saw that the homology\nclass of the intersections of interest can be given by a cap product\nwith cohomology cup products of the above form (Proposition \\ref{loop\nintersection}). The above corollary computes this homology class in\nterms of BV structure in $\\mathbb H_*(LM)$ using the homology classes of these\nsubmanifolds.\n\n\\begin{remark} \\label{exterior algebra}\nIn general, elements $\\alpha,\\Delta\\alpha$ for $\\alpha\\in H^*(M)$ do not generate the entire cohomology ring $H^*(LM)$. However, if $H^*(M;\\mathbb{Q})=\\Lambda_{\\mathbb{Q}}(\\alpha_1,\\alpha_2,\\dotsc\\alpha_r)$ is an exterior algebra, over $\\mathbb{Q}$, then using minimal models or spectral sequences, we have\n\\begin{equation}\nH^*(LM;\\mathbb{Q})=\\Lambda_{\\mathbb{Q}}(\\alpha_1,\\alpha_2,\\dotsc\\alpha_r)\\otimes\n\\mathbb{Q}[\\Delta\\alpha_1,\\Delta\\alpha_2,\\dotsc\\Delta\\alpha_r],\n\\end{equation}\nand thus we have the complete description of the cap products with any elements in $H^*(LM;\\mathbb{Q})$ in terms of the BV structure in $\\mathbb H_*(LM;\\mathbb{Q})$. \n\\end{remark}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNASA's \\textit{Kepler} mission has been continuously monitoring more than\n150\\,000 stars for the past 4 years, searching for transiting exoplanets\n\\citep{Borucki-Kepler}.\nThe unprecedented quality of the photometric light curves delivered by\n\\textit{Kepler} makes them also very well suited to study stellar variability in\ngeneral. Automated light curve classification techniques with the goal to\nrecognize \nand identify the many variable stars hidden in the \\textit{Kepler} database have\nbeen developed. The application of these methods to the\npublic \\textit{Kepler} Q1 data is described in \\cite{Debosscher:2011}. There,\nthe authors paid special attention to the detection of pulsating stars \nin eclipsing binary systems. These systems are relatively rare, and especially\ninteresting for asteroseismic studies \\citep[see\ne.g.][]{Maceroni:2009,Welsh:2011}. By modelling the orbital dynamics of the\nbinary, using photometric time series complemented with spectroscopic follow-up\nobservations, we can\nobtain accurate constraints on the masses and radii of the pulsating stars.\nThese\nconstraints are needed for asteroseismic modelling, and are difficult to obtain\notherwise. \n\nNumerous candidate pulsating binaries were identified in the\n\\textit{Kepler} data, and spectroscopic follow-up is ongoing. In this work, we\npresent the results obtained for KIC 11285625 (BD+48 2812), which turns out to be an\neclipsing binary system containing a $\\gamma$ Dor pulsator. The KIC (\\textit{Kepler} Input Catalog) lists the following properties for this target: $V = 10.143$ \\,mag,  $T_{\\rm eff}$ = 6882 K, $\\log g$ = 3.753,\n$R$ = 2.61 $\\unit{R_{\\sun}}$ and $[Fe\/H]$ = -0.127.\nSpectroscopic follow-up revealed it to be a double-lined binary (Section \\ref{rv}). Currently, only a few $\\gamma$ Dor pulsators in double-lined spectroscopic binaries are known, making their analysis very relevant for asteroseismology. \\cite{Maceroni:2013} studied a $\\gamma$ Dor pulsators in an eccentric binary system, observed by CoRoT. Here, we are dealing with a non-eccentric system with a longer orbital period. The longer time span and the higher photometric precision of the \\textit{Kepler} observations (almost a factor 6) allowed us to study the pulsation spectrum with significantly increased frequency resolution and down to lower amplitudes.\n\nA combined\nanalysis of the \\textit{Kepler} light curve and spectroscopic radial velocities\nallowed us to obtain a good binary model for KIC 11285625, resulting in accurate\nestimates of the masses and radii of both components (Section \\ref{binmod}). \nThis binary model was also\nused to disentangle the pulsations from the orbital variability in the\n\\textit{Kepler} light curve in an iterative way. In this paper, we describe\nprocedures to\nperform this task in an automated way. The low signal-to-noise composite\nspectra used for the determination of the radial velocities were used to obtain\nhigher signal-to-noise mean spectra of the components, by means of\nspectral disentangling. These spectra were then used to obtain fundamental\nparameters of the stars (Section \\ref{disentangling}). \nThe resulting pulsation signal of the primary is analysed in detail in Section \\ref{puls-spec}.\nThere we discuss the global characteristics of the frequency spectrum, we list\nthe dominant frequencies and their amplitudes detected by means of prewhitening,\nand search for signs of rotational splitting.  \n  \n\n\\section{\\textit{Kepler} data}\n\nKIC 11285625 has been almost continuously observed by \\textit{Kepler}; data are\navailable for observing quarters Q0-Q10. We only used long cadence data in this work (with a time resolution of 29.4 minutes), since short cadence data is only available for three quarters and is not needed for our purposes. During quarter Q4, one of the CCD modules failed, the reason why part of the Q4 data are missing. No Q8 data could be observed either, since\nthe target was positioned on the same broken CCD module during that quarter.\nThe \\textit{Kepler} spacecraft needs to make rolls every 3 months (for\ncontinuous illumination of its solar arrays), causing targets to fall on\ndifferent CCD modules depending on the observing quarter. \nGiven the different nature of the CCDs, and the different aperture masks used,\nthis caused some issues with the data reduction. The average flux level of the\nlight curve for KIC 11285625 varies significantly \nbetween quarters, and for some, instrumental trends are visible. The top\npanel of Fig. \\ref{LC-all-quarters} plots all the observed\ndatasets, showing the quarter-to-quarter\nvariations. Merging the quarters correctly is not\ntrivial, since the trends have to be removed for each quarter separately, and \nthe data have to be shifted so that all quarters are at the same average level\n(see below).\nOften, polynomials are used to remove the trends, but it is difficult to\ndetermine a reasonable order for the polynomial. This is especially the case\nwhen large amplitude variability, at time scales comparable to the total time\nspan of the data, is present in the light curve.\n\nFortunately, pixel target files are available for all observed quarters for\nKIC 11285625, allowing us to do the light curve extraction based on custom aperture masks. We can define a custom aperture mask, determining\nwhich pixels to include or not. It turned out that the standard aperture mask, used by\nthe data reduction pipeline, was not optimal for all quarters. \nThis is clearly visible in the top panel of Fig. \\ref{LC-all-quarters} for\nquarters Q3 and Q7. The clear upward trends and smaller variability amplitudes\ncompared to the other quarters are caused by a suboptimal aperture mask. The\ntrends can be explained by a small drift of the star on the CCD, changing the\namount of stellar flux included in the aperture during the quarter.\nChecking the target pixel files for those quarters revealed that pixels with\nsignificant flux contribution were not included in the mask. Fig. \\ref{mask}\nshows the \\textit{Kepler} aperture from the automated pipeline, and a single\ntarget pixel image obtained during quarter Q3. As can be seen, the\n\\textit{Kepler} aperture misses a pixel\nwith significant flux contribution (the lowest blue coloured pixel).   \nAdding this pixel to the aperture mask effectively removed the trend in the\nlight curve and increased the variability amplitude to the level of the other\nquarters.\n\nAn automated method was developed to optimize the aperture mask for each\nquarter, with the goal to maximize the signal-to-noise ratio (S\/N) in the\nFourier amplitude spectrum. \nThe standard mask provided with the target pixel files is used as a starting\npoint. The\nmethod then loops over each pixel in the images outside of the original mask\ndelivered by the \\textit{Kepler} pipeline. For each of those pixels, a new light\ncurve is constructed by adding the flux values of the pixel\nto the summed flux of the pixels within the original \\textit{Kepler} mask. The\namplitude spectrum of the resulting light curve is then computed and the S\/N of\nthe highest peak (in this case, the main pulsation frequency of the star) is\ndetermined.\nIf the addition of the pixel increases the S\/N (by a user specified amount), it\nwill be added to the final new light curve once each pixel has been analysed\nthis way. The method also avoids adding pixels containing significant flux of\nneighbouring contaminating targets, since these will normally decrease\nthe S\/N of the signal coming from the main target. It is also possible to detect\ncontaminating pixels within the original \\textit{Kepler} mask, using exactly the\nsame method, but now by excluding one pixel at a time from the original mask \nand checking the resulting S\/N of the new light curves.\n\nAfter determining a new optimal aperture mask for each quarter, the resulting\nlight curves still showed some small trends and offsets, but they were easily\ncorrected using second order polynomials.  \nSpecial care is needed however when shifting quarters to the same level, since\nthe average value of the light curve might be ill determined, especially\nwhen large amplitude non-sinusoidal variability is present at time scales\nsimilar\nto the duration of an observing quarter.\nIn our case, we want to make the average out-of-eclipse brightness match\nbetween different quarters, and not the global average of the quarters, since\nthe latter is shifted due to the presence of the eclipses. \nTherefore, we cut out the eclipses first and interpolated the data points in the\nresulting gaps using cubic splines. This was done only to determine the\npolynomial coefficients, which were then used to subtract \nthe trends from the original light curves (including the eclipses). We first\ntransformed the fluxes into magnitudes for each quarter separately, prior to\ntrend removal. The binary modelling code we describe further needs magnitudes\nas input, but the conversion to magnitude also\nresolves any potential quarter-to-quarter variability amplitude changes caused, e.g., by a difference in CCD gains (or any other instrumental effect changing the\nflux values in a linear way).        \n\nThe lower panel of Fig. \\ref{LC-all-quarters} shows the resulting light curve\nafter application of our detrending procedure using pixel target files.\n\n\n\n\\begin{figure*}\n \\centering\n  \n\\includegraphics[width=14cm,angle=270,scale=0.7]{kplr11285625-all-quarters.ps}\n\\caption{Combination of observing quarters Q1-Q10 for KIC 11285625. The top\npanel shows the `raw'\nSAP (simple aperture photometry) fluxes, as they were delivered, while the lower\npanel shows the resulting combined dataset after optimal mask selection and detrending.}\n\\label{LC-all-quarters}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n \\centering\n   \\includegraphics[width=14cm,angle=0,scale=1.0]{aperture.ps}\n\\caption{\\textit{Kepler} aperture (left) and a single target pixel image (right)\nfor KIC 11285625, quarter Q3. Red colours in the pixel image indicate the lowest flux levels, white colours indicate the highest flux levels.}\n\\label{mask}\n\\end{figure*}\n\n\n\n\n\\section{Spectroscopic follow-up and RV determination}\n\\label{rv}\nSpectroscopic follow-up observations were obtained with the HERMES Echelle\nspectrograph at the Mercator telescope on La Palma (see \\cite{HERMES} for a detailed\ndescription of the instrument). \nIn total, 63 spectra with good orbital phase coverage were observed, with\nS\/N in \\textit{V} in the range 40-70. These spectra revealed the\ndouble-lined nature of the spectroscopic binary. \n\nRadial velocities were derived with the HERMES reduction pipeline, using the\ncross-correlation technique \nwith an F0 spectral mask. The choice for this mask was based on the F0 spectral\ntype given by SIMBAD and the effective temperature listed in the KIC (Kepler\nInput Catalogue). We also tried additional spectral masks, given the\ndouble-lined\nnature of the binary, but we did not obtain better results in terms of scatter\non the radial velocity points. Due to the numerous metal lines in the spectrum,\nthe cross-correlation technique worked very well, despite the relatively low S\/N\nspectra. The upper panel of  Fig. \\ref{rv-lc} shows the radial velocity measurements obtained for\nboth components of KIC 11285625.\nThe black circles correspond to the primary and the red circles to the\nsecondary. From the scatter on the radial velocity measurements, we conclude\nthat the primary component is pulsating. A Keplerian model was fitted for both\ncomponents (shown also in the upper panel of Fig. \\ref{rv-lc} ), resulting in the orbital parameters listed in Table \\ref{final-par}.\n\nUncertainties were estimated using a Monte-Carlo perturbation approach. \nAlthough the orbital period of the system can be a free parameter in the fitting\nprocedure for the Keplerian model, we fixed it to the much more accurate value obtained from the\n\\textit{Kepler} light curve, given its longer time span (see Section \\ref{binmod}). The quality\nof the fit obtained this way (as judged from the $\\chi^2$ values) is significantly better compared to the case where\nthe orbital period is left as a free parameter.\n\n\n\n\n\\section{Binary model}\n   \n\\label{binmod}\nWe used the combined \\textit{Kepler} Q1-Q10 data to obtain a binary model,\nwhich, combined with the results from the spectroscopic analysis, provided us\nwith accurate estimates of the main astrophysical properties\nof both components. Given that we are dealing with a detached binary with no or only\nlimited distortion of both components, we used JKTEBOP, written by\nJ. Southworth \\citep[see][]{Southworth:JKTEBOP1,Southworth:JKTEBOP2}.\nThis code is based on the EBOP code, originally developed by Paul B. Etzel\n\\citep[see][]{Etzel:EBOP,Popper:EBOP}. JKTEBOP has the advantage of being very\nstable, fast, and it is applicable to large datasets with thousands of\nmeasurements, such as the \\textit{Kepler} light curves. \nMoreover, it can easily be scripted (essential in our approach) and includes\nuseful error analysis options such as Monte Carlo and bootstrapping methods.\\\\\n\nSince we are dealing with a light curve containing both orbital variability\n(eclipses) and pulsations, we had to disentangle both phenomena in order to\nobtain a reliable binary model. \nIn the amplitude spectrum of the \\textit{Kepler} light curve (see Fig.\n\\ref{puls-residuals}), the $\\gamma$ Dor type pulsations have numerous significant peaks in\nthe range 0-0.7 $\\unit{d^{-1}}$, with clear repeating patterns up to around 4 $\\unit{d^{-1}}$. In the same region of the amplitude spectrum, we\nalso find peaks \ncorresponding to the orbital variability: a comb-like pattern of harmonics of\nthe orbital frequency ($f_\\mathrm{orb}$, $2f_\\mathrm{orb}$, $3f_\\mathrm{orb}$,...). Moreover, the\nmain pulsation frequency of 0.567 $\\unit{d^{-1}}$ is close to $6f_\\mathrm{orb}$ (0.556\n$\\unit{d^{-1}}$ ), though the peaks are clearly separated, given an estimated\nfrequency resolution\nof $ 1\/T \\approx0.0012\\, d^{-1}$, with T the total time span \nof the combined \\textit{Kepler} Q1-Q10 data. This near coincidence complicated the disentangling\nof both types of variability in the light curve, unlike the case where the\norbital variability is well separated from the \npulsations in frequency domain (e.g. typical for a $\\delta$ Sct pulsator in a\nlong-period binary). We used an iterative procedure, consisting of an\nalternation\nof binary modelling with JKTEBOP, and prewhitening of \nthe remaining variability (pulsations) after removal of the binary model. This\nprocedure can be done in an automated way, and the number of iterations can be\nchosen. The idea is that we gradually improve both the binary model and the\nresidual pulsation spectrum at the same time. In each step of the procedure, we\nused the entire Q1-Q10 dataset without any rebinning. \nOur iterative method is similar to the one described in \\cite{Maceroni:2013} and\nconsists of the following steps: \n\\begin{enumerate}\n \\item Remove the eclipses from the original \\textit{Kepler} light curve and\ninterpolate the resulting gaps using cubic splines.\n \\item Derive a first estimate of the pulsation spectrum by means of iterative\nprewhitening of the light curve without eclipses.\n \\item Remove the pulsation model derived in the previous step from the original\n\\textit{Kepler} light curve.\n \\item Find the best fitting binary model to the residuals using JKTEBOP.\n \\item Remove this binary model from the original \\textit{Kepler} light curve\n(dividing by the model when working in flux, or subtracting the model when\nworking in magnitudes).\n \\item Perform frequency analysis on the residuals, which delivers a new\nestimate of the pulsation spectrum.  \n \\item Subtract the pulsation model obtained in the previous step from the\noriginal \\textit{Kepler} light curve. \n \\item Model the residuals (an improved estimate of the orbital variability)\nwith JKTEBOP and repeat the procedure starting from step five. \n\\end{enumerate}\n\nThe procedure is then stopped when convergence is obtained: the $\\chi^{2}$\nvalue of the binary model no longer decreases significantly.\nIn practice, convergence is obtained after just a few iterations, at least for\nKIC 11285625. The procedure can be run automatically,\nprovided that the user has a good initial guess for the orbital parameters.\nThe computation time is dominated by the prewhitening\nstep, which requires the repeated calculation of amplitude spectra. In our case,\nthe complete procedure required a few hours on a single desktop CPU. The top panel of Fig. \\ref{puls-residuals} shows part of the original \\textit{Kepler} data,\nwith the disentangled pulsation contribution overplotted in red, the lower\npanel shows the corresponding amplitude spectrum. In Fig.\n\\ref{rv-lc}, the Keplerian model fit to the HERMES RV data, the binary model\nfit to the \\textit{Kepler} data (pulsation part removed) and the residual light\ncurve are shown together, phased with the orbital period (using the\nzero-point $T_\\mathrm{0}$ = 2454953.751335 d, corresponding to a time of minimum\nof the primary eclipse). The slightly larger scatter in the residuals at the\ningress\nand egress of the primary eclipse is caused by the fact that the\nocculted surface of the pulsating primary is non-uniform and changing over\ntime, due to the non-radial pulsations.\n\nWe also investigated a different iterative approach, where we started by fitting\na binary model directly to the original \\textit{Kepler} light curve, instead of\nfirst removing the pulsations. This way, we do not remove information from the\nlight curve by cutting the eclipses, and we do not need to interpolate the data.\nAlthough the iterative procedure also converged quickly using this approach, the\nfinal binary model was not accurate. Removing the model from the original\n\\textit{Kepler} light curve introduced systematic offsets during the eclipses.\nThe reason is that the initial binary model obtained from the original light\ncurve is inaccurate due to the large amplitude and non-sinusoidal nature of the\npulsations, causing the mean light level between eclipses to be badly defined. \nThe approach starting from the light curve with the pulsations removed prior to\nbinary modelling provided much better results, as judged from the\n$\\chi^{2}$ values and visual inspection of the \nresiduals during the eclipses.\\\\\n\nA linear limb-darkening law was used for both stars, with coefficients\nobtained from \\cite{Prsa:2011}\\footnote{\\url{http:\/\/astro4.ast.villanova.edu\/aprsa\/?q=node\/8}}. \nThese coefficients have been computed for a grid of {$T_{\\rm eff}$},\n{$\\log g$} and [M\/H] values, taking into account \\textit{Kepler's} transmission,\nCCD quantum efficiency and optics. We estimated them using the KIC\n(Kepler Input Catalogue) parameters for a first\niteration, but later adjusted them using our obtained values for {$T_{\\rm eff}$},\n{$\\log g$} and [M\/H] from the combination of binary modelling and spectroscopic\nanalysis (see Section \\ref{disentangling}). We did not take gravity darkening and refection effects into account, given that both components are well separated, and are not significantly deformed by rapid rotation or binarity (the oblateness values returned by JKTEBOP are very small).\n\n\nDuring the iterative procedure, special care was paid to the following\nissues, since they all influence the quality of the final binary and\npulsation models:\n\n\n\\begin{itemize}\n \\item Removal of the pulsations from the original light curve: here, we first\ndetermined all the significant pulsation frequencies (or, more general:\nfrequencies most likely not caused by the orbital motion) from the light curve\nwith the\neclipses removed. Only frequencies with an amplitude signal-to-noise ratio (S\/N)\nabove four are considered. The noise level in the amplitude spectrum is\ndetermined from the region 20- 24 d$^{-1}$, where no significant peaks are\npresent. The often used procedure of computing the noise level in a region\naround the peak of interest would not provide reliable S\/N estimates in our\ncase, given the high density of significant peaks at low frequencies. \nWe also used false-alarm probabilities as an additional significance test,\nwith very similar results regarding the number of significant frequencies. \n\n\\item Initial parameters of the binary model: when running JKTEBOP, initial\nparameters have to be provided for the binary model. These are then refined\nusing non-linear optimization techniques (e.g. Levenberg-Marquardt). \nAlthough the optimization procedure is stable and converges fast, we have no\nguarantee that the global best solution is obtained. If the initial parameters\nare too far off from their true values, the procedure can end up in a local\nminimum. Therefore, we did some exploratory analysis first, to find a good set\nof initial \nparameters, aided by the constraints obtained from the radial velocity data. The\ninitial parameters were also refined: after completion of the first\niterative procedure, the final parameters were used as initial values for a new\nrun, etc. This also confirmed the stability of the solution, although it does\nnot guarantee that the overall best solution has been found.  \n\\end{itemize}\n\nError analysis of the final binary model was done using the Monte-Carlo method\nimplemented in JKTEBOP. Here, the input light curve (with the pulsations\nremoved) is perturbed by adding Gaussian noise with standard deviation estimated\nfrom the residuals, and the binary model is recomputed. This procedure is\nrepeated typically 10000 times, to obtain confidence intervals for the obtained\nparameter values.\nThe final parameters from the combined spectroscopic and photometric analysis,\nand their estimated uncertainties (1$\\sigma$), are listed in Table \\ref{final-par}.\n\nGiven the long time span and excellent time sampling of the light curve, we also checked for the presence of eclipse time variations (e.g. due to the presence of a third body).\nWe used two different methods to check for deviations of pure periodicity of the eclipse times. The first method consists of computing the amplitude spectrum of each observing quarter \nof the light curve separately, and comparing the peaks caused by the binary signal (the comb of harmonics of the orbital frequency). We could not detect any change in orbital frequency this way.\nMoreover, the orbital peaks in the amplitude spectrum of the entire light curve also do not show any broadening or significant sidelobes (indicative of frequency changes), compared to the amplitude spectrum of the purely periodic light curve of our best binary model computed with JKTEBOP. The second method consists of determining the times of minima for each eclipse individually by fitting a parabola to the bottom of each eclipse and determining the minimum. We then compared those times with the predicted values using our best value of the orbital period (O-C diagram). This was done both for the original light curve and the light curve with pulsations removed. In the first case, we found indications of periodic shifts of the eclipse times, but these are clearly linked to the pulsation signal in the light curve, which also affects the eclipses. No periodic shifts or trends could be found when analysing the light curve with pulsations removed, and we conclude that we do not detect eclipse time variations.\n\n\\begin{table}\n\\tiny\n\\renewcommand{\\tabcolsep}{0.4mm}\n\\center\n\\caption{Orbital and physical parameters for both components of KIC 11285625,\nobtained from the combined spectroscopic and photometric analysis.}\n \\begin{tabular}{lcc}\n \\hline\n &System&\\\\\n \\hline\n Orbital period  $P$ (days)&  10.790492 $\\pm$ 0.000003  &\\\\\n Eccentricity $e$ &$0.005\\pm0.003$ &\\\\\n Longitude of periastron $\\omega$ (\\degr)&90.103 $\\pm$ 0.006&\\\\\n Inclination $i$ (\\degr) & 85.32 $\\pm$ 0.02 &\\\\\n Semi-major axis $a$ ($\\unit{R_{\\sun}}$) &28.8 $\\pm$ 0.1&\\\\\n Light ratio $L_1\/L_2$ &0.38 $\\pm$ 0.01&\\\\\n System RV $\\gamma$ (km $\\unit{s^{-1}}$)&-11.7 $\\pm$ 0.2&\\\\\n $T_\\mathrm{0}$ (days)\\tablefootmark{a}&2454953.751335 $\\pm$ 0.000014 \\\\\n \\hline  \n &Primary&Secondary\\\\\n \\hline\n Mass $M$ ($\\unit{M_{\\sun}}$)&$1.543\\pm0.013$ &$1.200\\pm0.016$\\\\\n Radius $R$ ($\\unit{R_{\\sun}}$) & 2.123 $\\pm$ 0.010& 1.472 $\\pm$ 0.014\\\\\n log $g$ &3.973 $\\pm$ 0.006 & 4.18 $\\pm$ 0.01 \\\\\n\\hline\n\\end{tabular}\n\\tablefoottext{a}{Reference time of minimum of a primary eclipse.}\n\\label{final-par}\n\\end{table}\n\n\n\n\\begin{figure}\n \\centering\n   \\includegraphics[width=14cm,angle=0,scale=0.66]{rv-lc-model-plots-new.ps}\n\\caption{Upper panel: phased HERMES radial velocity data (open circles) and Keplerian model fits (lines) for both components (black: primary, red: secondary). Middle panel: phased \\textit{Kepler} data (pulsations removed) with the binary model overplotted in red.\nLower panel: phased residuals of the \\textit{Kepler} data, after removal of both the pulsations and the binary model. }\n\\label{rv-lc}\n\\end{figure}\n\n\n\\begin{figure*}\n \\centering\n  \n\\includegraphics[width=14cm,angle=270,scale=0.7]{kplr11285625-puls-ampspec-comparison.ps}  \n\\caption{Upper panel: resulting light curve after iterative removal of the\norbital variability (in red), with the original light curve shown for comparison\n(in black). Lower panel: Amplitude spectrum of the original\n\\textit{Kepler} light curve (in black), and the \namplitude spectrum of the pulsation `residuals' (in red) after iterative\nremoval of the orbital variability.}\n\\label{puls-residuals}\n\\end{figure*}\n\n\n\\section{Spectral disentangling}\n\\label{disentangling}\n\nTo derive the fundamental parameters  $T_{\\rm eff}$, $\\log g$, $[M\/H]$, etc. of the $\\gamma$ Dor pulsator from\nthe high\nresolution HERMES spectra, we first needed to separate the contributions\n of both stars in the measured composite spectra. Given the similar spectral types of the components, and the large number of metal lines in the spectra, \nwe could apply the technique of spectral disentangling to accomplish this\n\\citep{Simon:1994,Hadrava:1995}. Here,\nwe used the FDBinary code \\footnote{http:\/\/sail.zpf.fer.hr\/fdbinary\/}\n\\citep{Ilijic:2004} which is based on Hadrava's Fourier approach\n\\citep{Hadrava:1995}. The overall procedures used to determine the orbital\nparameters and reconstruct the spectra of the component stars from time series\nof observed composite spectra of a spectroscopic double-lined eclipsing binary\nhave been described extensively in \\cite{Hensberge:2000} and\n\\cite{Pavlovski:2005}. \n\nThe user has to provide good initial guesses and confidence intervals for the\norbital parameters, otherwise the method might not converge towards the correct\nsolution. Luckily, we had very good\ninitial parameter values available from the Keplerian orbital fit to the radial\nvelocity data, as was described in Section \\ref{rv}. The final orbital\nparameters\nturned out to be in excellent agreement with the initial values derived from\nspectroscopy. \n\nSpectral disentangling methods have the advantage that they enable us to\ndetermine\nthe orbital parameters of the system in an independent way (although good\ninitial estimates are necessary), and that the resulting component spectra have\na higher signal-to-noise ratio than the individual original composite spectra:\nS\/N $\\sim$ $\\sqrt{N}$, with N the number of composite spectra used for\ndisentangling. The increase in S\/N is illustrated in\nFig. \\ref{disentangled-primary}, where a single observed spectrum (corrected for\nDoppler shift) is compared to the disentangled spectrum for the primary\ncomponent. Renormalization of the disentangled spectra was done using\nthe light factors obtained from the binary modelling of the \\textit{Kepler}\nlight\ncurve. \n\n\\begin{figure}\n\n \\centering\n\n\\includegraphics[width=14cm,angle=270,scale=0.5]{disentangled-spectrum.ps}\n\\caption{Comparison of a single observed composite spectrum (black circles) with\nthe disentangled spectrum of the primary component (red lines), illustrating\nthe significant increase in S\/N that is obtained.}\n\\label{disentangled-primary}\n\\end{figure}\n\nFor the spectrum analysis of both components of KIC\\,11285625, we\nuse the GSSP code \\citep[Grid Search in Stellar Parameters,][]{Tkachenko2012} that finds the optimum\nvalues of $T_{\\rm eff}$, $\\log g$, $\\xi$, $[M\/H]$, and $v\\sin{i}$\\ from the minimum\nin $\\chi^2$ obtained from a comparison of the observed spectrum with\nthe synthetic ones computed from all possible combinations of the\nabove mentioned parameters. The errors of measurement (1$\\sigma$\nconfidence level) are calculated from the $\\chi^2$ statistics, using the\nprojections of the hypersurface of the $\\chi^2$ from all grid points\nof all parameters in question. In this way, the estimated error bars include\nany possible model-inherent correlations between the parameters but do not take\ninto account imperfections of the model (such as incorrect atomic data, non-LTE\neffects, etc.) and\/or continuum normalization. A\ndetailed description of the method and its application to the\nspectra of  \\textit{Kepler} $\\beta$\\,Cep and SPB candidate stars as well as\n$\\delta$\\,Sct and $\\gamma$\\,Dor candidate stars are given in\n\\citet{Lehmann2011} and \\citet{Tkachenko2012}, respectively.\n\nFor the calculation of synthetic spectra, we used the LTE-based\ncode SynthV \\citep{Tsymbal1996} which allows the computation of the\nspectra based on individual elemental abundances. The code uses\ncalculated atmosphere models which have been computed with the\nmost recent, parallelised version of the LLmodels program\n\\citep{Shulyak2004}. Both programs make use of the VALD database\n\\citep{Kupka2000} for a selection of atomic spectral lines.\nThe main limitation of the LLmodels code is that the models are\nwell suited for early and intermediate spectral type stars, but\nnot for very hot and cool stars where non-LTE effects or\nabsorption in molecular bands may become relevant, respectively.\n\n\n\\begin{table}\n\\caption{Fundamental parameters of both components of KIC\\,11285625.}\\label{Table: Fundamental parameters}\n\\begin{tabular}{lll}\n\\hline\\hline\n\\multicolumn{1}{c}{Parameter\\rule{0pt}{9pt}} & Primary & \\multicolumn{1}{c}{Secondary}\\\\\n\\hline\n$T_{\\rm eff}$\\,(K)\\rule{0pt}{11pt} &$6960\\pm 100$ & $7195\\pm200$\\\\\n$\\log g$\\,(fixed)\\rule{0pt}{11pt} & 3.97 & 4.18\\\\\n$\\xi$\\,(km\\,s$^{-1}$)\\rule{0pt}{11pt} & $0.95\\pm0.30$ & $0.09\\pm0.25$\\\\\n$v\\sin{i}$\\,(km\\,s$^{-1}$)\\rule{0pt}{11pt} & $14.2\\pm1.5$ & $8.4\\pm1.5$\\\\\n$[M\/H]$\\,(dex)\\rule{0pt}{11pt} & $-0.49\\pm0.15$ & $-0.37\\pm0.3$\\\\\n\\hline\n\\end{tabular}\n\\tablefoot{The temperature of the secondary is not reliable, as discussed in the text.}\n\\end{table}\n\nGiven that KIC\\,11285625 is an eclipsing, double-lined (SB2)\nspectroscopic binary for which unprecedented quality (\\textit{Kepler})\nphotometry is available, the masses and the radii of both components\nwere determined with very high precision. Having those two\nparameters, we evaluated surface gravities of the two stars with\nfar better precision than one would expect from the spectroscopic\nanalysis given that the S\/N of our spectra varies between 40 and 70,\ndepending on the weather conditions on the night when the\nobservations were taken. Thus, we fixed $\\log g$\\ for both\ncomponents to their photometric values (3.97 and 4.18 for the\nprimary and secondary, respectively) and adjusted the effective\ntemperature $T_{\\rm eff}$, micro-turbulent velocity $\\xi$, projected\nrotational velocity $v\\sin{i}$, and overall metallicity [M\/H] for both\nstars based on their disentangled spectra.\nGiven that the contribution of the primary component to the total\nlight of the system is significantly larger than that of the\nsecondary (72\\% compared to 28\\%) and that its decomposed spectrum\nis consequently better defined and is of higher quality than that of\nthe secondary, we were also able to evaluate individual abundances\nfor this star besides the fundamental atmospheric parameters.\nTable~\\ref{Table: Fundamental parameters} lists the fundamental\nparameters of the two stars whereas Table~\\ref {Table: Individual abundances} summarizes the results\nof chemical composition analysis for the primary component. The overall metallicities of the two\nstars agree within the quoted errors, but the derived temperature for the\nsecondary is not reliable, since we find \nit to be about 200~K hotter than the primary. From the relative eclipse depths,\nwe estimate the temperature of the secondary to be about 6400~K. The cause of\nthis temperature discrepancy is the poor quality of the disentangled spectrum of\nthe secondary, given its smaller light contribution. Normalization errors in\nthe spectra can easily translate into temperature errors of several hundred\nKelvin. Note that the listed uncertainties for the spectroscopic temperatures do not take normalization errors into account.\n\n\nFigure~\\ref{Figure: HR diagram} shows the position of the primary component in\nthe $\\log(T_{\\rm eff})$-$\\log g$\\ diagram, with respect to the observational  $\\delta$~Sct\\  (solid\nlines) and $\\gamma$~Dor\\ (dashed lines) instability strips as given by\n\\cite{Rodriguez:2001} and \\cite{Handler:DSCUT}, respectively. The primary falls\ninto the $\\gamma$~Dor\\ instability strip, meaning that pure g-modes are expected to be\nexcited in its interior.\n\n\\begin{table}\n\\tabcolsep 2.2mm\\caption{Atmospheric chemical composition of the\nprimary component of KIC\\,11285625.}\n\\label{Table: Individual abundances}\n\\begin{tabular}{llllll}\n\\hline\\hline\n\\multicolumn{1}{c}{Element\\rule{0pt}{9pt}} & Value & \\multicolumn{1}{c}{Sun} & \\multicolumn{1}{c}{Element\\rule{0pt}{9pt}} & Value & \\multicolumn{1}{c}{Sun}\\\\\n\\hline\nFe\\rule{0pt}{11pt} & --0.58\\,(15) & --4.59 & Mg & --0.65\\,(23) & --4.51\\\\\nTi\\rule{0pt}{11pt} & --0.40\\,(20) & --7.14 & Ni & --0.53\\,(20) & --5.81\\\\\nCa\\rule{0pt}{11pt} & --0.41\\,(27) & --5.73 & Cr & --0.35\\,(27) & --6.40\\\\\nMn\\rule{0pt}{11pt} & --0.50\\,(35) & --6.65 & C & --0.28\\,(40) & --3.65\\\\\nSc\\rule{0pt}{11pt} & --0.31\\,(40) & --8.99 & Si & --0.68\\,(40) & --4.53\\\\\nY\\rule{0pt}{11pt} & --0.22\\,(45) & --9.83 &  &  & \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{All values are in dex and on a\nrelative scale (compared to the Sun). Label ``Sun'' refers to the\nsolar composition given by \\citet{Grevesse2007}. Error bars\n(1-$\\sigma$ level) are given in parentheses in terms of last digits.}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[scale=0.9,clip=]{K11285625_Teff_logg.eps}\n\\caption{{\\small Location of the primary of KIC\\,11285625 in the\n$\\log(T_{\\rm eff})$-$\\log g$\\ diagram. The $\\gamma$~Dor\\ and the red edge of the $\\delta$~Sct\\\nobservational instability strips\nare represented by the dashed and solid lines, correspondingly. According to\nthe photometric $T_{\\rm eff}$\\ and $\\log g$, the secondary would be located in the lower\nright corner of the diagram, outside both instability strips. }} \\label{Figure:\nHR diagram}\n\\end{figure}\n\n\n\\section{Pulsation spectrum}\n\\label{puls-spec}\n\nFig. \\ref{as-puls-global} shows the amplitude spectrum of the \\textit{Kepler}\nlight curve with the binary model removed, in the region 0-2 d$^{-1}$, where\nmost of the dominant pulsation frequencies are found. Clearly visible\nare the three groups of peaks around 0.557, 1.124 and 1.684 {d$^{-1}$}. Some of the\nfrequencies found in the groups around 1.124 and\n1.684 {d$^{-1}$} are harmonics of frequencies present in the group around\n0.557 {d$^{-1}$}. These harmonics are caused by the\nnon-linear nature of the pulsations, and have been observed for many pulsators\nobserved by CoRoT and  \\textit{Kepler}, along almost the entire main-sequence\n\\citep{Degroote:2009,Poretti:2011,Breger:2011,Balona:2012}, and in $\\gamma$~Dor\\ stars in\nparticular \\citep{Tkachenko:2013}. Closer\ninspection of the\nmain frequency groups revealed that they consist of several closely spaced\npeaks,\nalmost\nequally spaced with a frequency of $\\sim$ 0.010 {d$^{-1}$}. A likely explanation is\namplitude modulation of the pulsation signal on a timescale of $\\sim$ 100\ndays. Mathematically, the effect of amplitude modulation can be\ndescribed as follows: imagine a simplified case where a single periodic\nnon-linear\npulsation signal is being modulated with a general periodic function. We can\nwrite this signal as a product of two sums of sines, where the number of terms\n(harmonics) in each sum depends on how non-linear (non-sinusoidal) the signals\nare: \n\\begin{equation}\n\\sum_{i=1}^{N_\\mathrm{mod}} a_\\mathrm{i} \\sin{[2\\pi f_\\mathrm{mod}\ni t + \\phi^\\mathrm{mod}_\\mathrm{i}]} \\sum_{j=1}^{N_\\mathrm{puls}} b_\\mathrm{j} \\sin{[2\\pi f_\\mathrm{puls}\nj t + \\phi^\\mathrm{puls}_\\mathrm{j}]} ,\n\\end{equation}\nwith $f_\\mathrm{mod}$ the modulation frequency and $f_\\mathrm{puls}$ the pulsation frequency.\nThis product can be rewritten as a sum, using Simpson's rule:\n\\begin{eqnarray}\n\\sum_{i=1}^{N_\\mathrm{mod}} \\sum_{j=1}^{N_\\mathrm{puls}} \\frac{a_\\mathrm{i} b_\\mathrm{j}}{2}(\\cos{[2\\pi\n(f_\\mathrm{puls} j-f_\\mathrm{mod} i)t+\\phi^\\mathrm{puls}_\\mathrm{j}-\\phi^\\mathrm{mod}_\\mathrm{i}]}-\\\\\n\\cos{[2\\pi (f_\\mathrm{puls} j+f_\\mathrm{mod} i)t+\\phi^\\mathrm{puls}_\\mathrm{j}+\\phi^\\mathrm{mod}_\\mathrm{i}]}). \\nonumber\n\\end{eqnarray}\nIn the Fourier transform of this signal, we will thus see peaks at\nfrequencies which are linear combinations of the pulsation frequency and the\nmodulation frequency, where the number of combinations depends on how\nnon-linear\nboth signals are. Most of the observed substructure in the amplitude spectrum\ncan be explained by amplitude modulation of a pulsation signal with\n$f_\\mathrm{mod}\\sim$ 0.010 {d$^{-1}$}. A more detailed description of amplitude\nand frequency modulation in light curves can be found in \\cite{Benko:2011}.\nSince the period of the amplitude modulation is close to the length of a\n\\textit{Kepler} observing quarter, we checked for a possible instrumental\norigin. A very strong argument against this, is the fact that the modulation is\nnot present in the eclipse signal of the light curve, but is only affecting the\npulsation peaks. \n\nTo study the pulsation signal in detail, we performed a complete frequency\nanalysis using an iterative prewhitening procedure. The Lomb-Scargle\nperiodogram was used in combination with false-alarm probabilities to detect the\nsignificant frequencies present in the light curve. Prewhitening of the\nfrequencies was performed using linear least-squares fitting with non-linear\nrefinement. In total, hundreds of formally significant frequencies were\ndetected. We stress here that formal significance does not imply that a\nfrequency has a physical interpretation, e.g. in the sense of all being\nindependent pulsation modes. In fact, most of them are not independent at all;\nmany combination\nfrequencies and harmonics are present and many peaks arise from the fact that\nwe are performing Fourier analysis on a signal that is not strictly periodic.\nTable \\ref{freqs} lists the 50\nmost significant frequencies, together with their amplitude, and estimated S\/N.\nThe noise level used to calculate the S\/N was estimated from the average\namplitude in the frequency range between 20 and 24 {d$^{-1}$}. The last column\nindicates possible combination frequencies and harmonics. The lowest order\ncombination is always listed, but for some frequencies, these can be\nwritten equivalently as a different higher order combination of more dominant\nfrequencies (they are listed in brackets). A complete list of frequencies\nis available online in electronic format at the CDS \\footnote{Centre de\nDonn\\'{e}es astronomiques de Strasbourg,\n  http:\/\/cdsweb.u-strasbg.fr\/}.\nClearly, many frequencies can\nbe explained by low-order linear combinations of the three most dominant\nfrequencies. However, given the amplitude modulation,\nthese dominant frequencies are likely not independent (corresponding to\ndifferent\npulsation modes). Instead, they are probably combination frequencies of the\nreal\npulsation frequency and (harmonics of) the modulation frequency $f_\\mathrm{mod}$\n$\\sim$ 0.010 {d$^{-1}$}. \nLooking back at the RV data for the pulsating component,\nwe expect the larger scatter in comparison to the secondary to be caused by the\npulsations. After subtraction of the Keplerian orbit fit, we performed\nfrequency analysis on the residuals, and found 2 significant\npeaks corresponding to frequencies detected in the \\textit{Kepler} data\n(f1 and f7 in Table \\ref{freqs}). The amplitude spectra of the RV residuals for\nboth components are shown in Fig. \\ref{ampspec-rv-res}. \n\nFrom spectroscopy, we found v sin i = 14.2 $\\pm$ 1.5 km\/s for the pulsating\nprimary component and v sin i = 8.4 $\\pm$ 1.5 km\/s for the secondary. Assuming that the rotation axes are perpendicular to the orbital plane, and using the orbital and stellar\nparameters listed in Table \\ref{final-par}, this corresponds to a rotation\nperiod of about 7.5 $\\pm$ 1.3 d for the primary and 8.8 $\\pm$ 1.9 d for the\nsecondary. These values suggest super-synchronous rotation, but on their own do\nnot provide sufficient evidence, given the uncertainties (1$\\sigma$), especially\nnot for the secondary.\n\nWe analysed the pulsation signal to\ndetect signs of rotational splitting of the pulsation frequencies in two\ndifferent ways:  by computing the autocorrelation function of the\namplitude spectrum and by using the list of detected significant frequencies.\nThe autocorrelation function between 0 and 0.7 {d$^{-1}$} is\nshown in Fig. \\ref{autocorrelation}. Clearly, the amplitude spectrum is\nself-similar for many different frequency shifts, as is evidenced by the\ncomplex structure of the autocorrelation function. Although the\nautocorrelation function is dominated by the highest peaks in the\namplitude spectrum and their higher harmonics, we can relate some smaller peaks\nto the properties of the system. For example, the peaks indicated with\nthe red arrows correspond to the orbital frequency and the rotational frequency of the primary as derived from spectroscopy (at 0.0926 and 0.1333 {d$^{-1}$} respectively). \n\nNext, we searched the list of significant frequencies for any possible frequency\ndifferences occurring several times (given the frequency resolution). This way,\nsplittings of lower amplitude peaks can be detected more easily, which is not the\ncase when using the autocorrelation function. Since the number of significant\nfrequencies is so large, several frequency differences occur multiple times\npurely by chance (this was checked by using a list of randomly generated\nfrequencies), so one must be careful when interpreting the results\n\\citep[see e.g.][]{Papics:2012}.\nIn our search for splittings, we used frequency lists of different lengths,\nranging from the first 50 to a maximum of several hundred significant\nfrequencies (down to a S\/N of 10). We used a rather strict cutoff value of\n0.0001 {d$^{-1}$} to accept frequency differences\nas being equal ( 0.1\/T, with T the total time span of the light curve). Next, we ordered all possible frequency differences in increasing\norder of occurrence. This always resulted in the same differences showing up in\nthe top of the lists. Table \\ref{freqdiffs} shows the most abundant differences detected in a conservative list of `only' 400 frequencies (down to a S\/N of about 50). \nClearly, many of these differences are related to the\namplitude modulation in the light curve (as discussed above), with values around\n0.010 {d$^{-1}$} (suspected modulation frequency) and 0.020 {d$^{-1}$} (twice the\nmodulation frequency) occurring often. Values around 0.567 {d$^{-1}$} are related to\nthe presence of higher harmonics of the pulsation frequencies. \nIn the autocorrelation function, clear peaks are present around the orbital\nfrequency, while the orbital frequency itself only\nshows up in the list of frequency differences when going down to S\/N values\naround 10. Most likely, this is caused by small residuals of the binary signal\n(eclipses) in the light curve. \nWe also find values very close to the rotation\nfrequency from spectroscopy and values in between the rotation frequency and the orbital frequency. They are also visible as the group of peaks in the\nautocorrelation function around 0.1 {d$^{-1}$}. We interpret these as the result of\nrotational splitting. The values are compatible with the spectroscopic results\nobtained for the primary, and hence provide stronger evidence for\nsuper-synchronous rotation. We cannot unambiguously identify one unique\nrotational splitting, but rather a range of possible values. The difference in\nmeasured splitting values across the spectrum suggests non-rigid rotation in\nthe interior of the primary \\citep[see e. g.][]{Aerts:2003, Aerts:book,Dziembowski:2008}.   \n                                               \n \nIn Fig. \\ref{as-puls-global}, the structures in the amplitude spectrum caused by\nrotational splitting are indicated by means of red arrows. Clearly, the pattern\nis repeated for the higher harmonics of the pulsation frequencies. \n\nFinally, the second-most dominant peak in\nthe amplitude spectrum almost coincides with the sixth harmonic of the orbital\nfrequency. This suggests tidally affected\npulsation, although this is not expected for non-eccentric systems.\nMoreover, the frequencies do not match perfectly, and the second-most dominant\npeak is likely not an independent pulsation mode, but caused by the amplitude\nmodulation of the main oscillation frequency.\n\n\\begin{figure*}\n \\centering\n   \\includegraphics[width=14cm,angle=270,scale=0.7]{as-puls-global.ps}\n\\caption{Part of the amplitude spectrum (below 2 {d$^{-1}$}) of the \\textit{Kepler}\nlight curve with the binary\nmodel removed. The red arrows indicate the splitting of groups of peaks caused\nby rotation with a  splitting value around 0.1 {d$^{-1}$}.}\n\\label{as-puls-global}\n\\end{figure*}\n\n\n\\begin{figure}\n \\centering\n   \\includegraphics[width=14cm,angle=270,scale=0.5]{autocorrelation.ps}\n\\caption{Autocorrelation of the amplitude spectrum, after removal of the binary\nmodel.}\n\\label{autocorrelation}\n\\end{figure}\n\n\n\n\n\\begin{table*}\n\\center\n\\caption{Dominant fifty frequencies with their amplitude and S\/N, detected in the\n\\textit{Kepler} light curve with the binary model removed, using the\nprewhitening technique as explained in the text.}\n \\begin{tabular}{c|c|c|c|c}\nNumber&Frequency (d$^{-1}$) & Amplitude (mmag)& S\/N&Combination\\\\\n\\hline\n 1       &0.5673 &17.487&720.1&-\\\\\n 2       &0.5575 &13.482&656.0&-\\\\\n 3       &0.5685 &8.378 &460.2&-\\\\\n 4       &1.1250 &7.468 &450.2&f1+f2\\\\\n 5       &0.5467 &6.559 &400.4&2f2-f3\\\\\n 6       &1.1359 &5.462 &355.8&f1+f3\\\\\n 7       &0.6594 &4.704 &323.4&-\\\\\n 8       &1.1141 &4.466 &311.7&f1+f5 (f1+2f2-f3)\\\\\n 9       &0.5563 &4.124 &296.7&f4-f3 (f1+f2-f3)\\\\\n 10      &0.4348 &3.192 &237.7&-\\\\\n 11      &0.4403 &3.288 &246.5&-\\\\\n 12      &1.6936 &2.842 &215.5&f3+f4 (f1+f2+f3)\\\\\n 13      &0.5666 &2.547 &195.9&-\\\\\n 14      &0.8985 &2.286 &178.8&-\\\\\n 15      &1.1041 &2.123 &168.1&f2+f5 (3f2-f3)\\\\\n 16      &0.3029 &2.080 &164.6&2f10-f13\\\\\n 17      &0.4483 &2.014 &161.4&-\\\\\n 18      &1.6925 &2.009 &163.2&f1+f4 (2f1+f2)\\\\\n 19      &1.6716 &1.969 &159.8&f4+f5 (f1+3f2-f3)\\\\\n 20      &0.5357 &1.963 &156.4&2f5-f2 (3f2-2f3)\\\\\n 21      &0.2365 &1.864 &150.7&2f1-f14\\\\\n 22      &1.1258 &1.847 &150.2&f2+f3\\\\\n 23      &0.4777 &1.831 &150.8&2f3-f7\\\\\n 24      &0.4337 &1.780 &148.4&2f5-f7 \\\\\n 25      &0.5784 &1.770 &147.7&f4-f5 (f1-f2+f3)\\\\\n 26      &1.2268 &1.766 &149.6&f1+f7\\\\\n 27      &1.6816 &1.738 &146.2&2f1+f5 (2f1+2f2-f3)\\\\\n 28      &0.4656 &1.712 &145.9&f4-f7 (f1+f2-f7)\\\\\n 29      &0.1272 &1.729 &149.5&f1-f11\\\\\n 30      &1.0023 &1.643 &142.2&f1+f10\\\\\n 31      &0.9979 &1.667 &146.7&f2+f11\\\\\n 32      &0.5582 &1.612 &143.8&f22-f1 (f2+f3-f1)\\\\\n 33      &0.6583 &1.539 &138.1&2f5-f10\\\\\n 34      &1.1346 &1.514 &135.7&2f1\\\\\n 35      &0.2477 &1.456 &131.7&2f7-2f20\\\\\n 36      &0.9804 &1.398 &127.6&f5+f24 (3f5-f7)\\\\\n 37      &1.2167 &1.399 &127.5&f2+f7\\\\\n 38      &1.7031 &1.388 &126.5&f1+f3\\\\\n 39      &0.4115 &1.363 &124.1&f7-f35 (2f20-f7)\\\\\n 40      &2.2609 &1.326 &122.2&f4+f6 (2f1+f2+f3)\\\\\n 41      &0.5257 &1.308 &120.3&2f5-f1\\\\\n 42      &0.5458 &1.267 &116.8&f8-f3 (f1+2f2-2f3)\\\\\n 43      &0.4200 &1.272 &117.2&2f7-f14\\\\\n 44      &0.0909 &1.240 &113.9&f7-f3\\\\\n 45      &0.5677 &1.220 &112.9&f1\\\\\n 46      &1.7043 &1.209 &113.8&f3+f6 (f1+2f3)\\\\\n 47      &0.6720 &1.201 &114.5&2f9-f11\\\\\n 48      &1.1368 &1.194 &114.4&2f3\\\\\n 49      &1.1470 &1.160 &113.0&2f6-f4 (f1-f2+2f3)\\\\\n 50      &0.2579 &1.159 &113.3&f4-2f24\\\\\n\\hline\n\n\n\\end{tabular}\n\\tablefoot{Typical uncertainties on the frequency values are\n$\\sim$ $10^{-3}$ d$^{-1}$ (using the Rayleigh criterion), and $\\sim 5 \\times\n10^{-3}$ mmag for the amplitudes. The last column indicates possible\ncombination frequencies and harmonics.}\n\\label{freqs}\n\\end{table*}\n\n\n\\begin{table*}\n\\center\n\\caption{Frequency differences and corresponding periods (in order of decreasing occurrence), detected in the list of the first 400 frequencies obtained using iterative prewhitening.}\n \\begin{tabular}{c|c|c}\nFrequency difference (d$^{-1}$)&  Period (d)&Remarks\\\\\n\\hline\n       0.5674  &      1.7624 & harmonics of pulsation frequencies\\\\ \n       0.0108  &    92.5926 & amplitude modulation\\\\\n       0.5675  &    1.7621   & harmonics of pulsation frequencies\\\\\n       0.1244   &     8.0386  & rotational splitting? \\\\ \n       0.5575   &     1.7937 &  harmonics of pulsation frequencies  \\\\\n       0.1133   &     8.8261  &  rotational splitting? \\\\\n       0.5685   &     1.7590   & harmonics of pulsation frequencies  \\\\ \n       0.5672   &     1.7630  & harmonics of pulsation frequencies \\\\\n       0.1010   &     9.9010   & rotational splitting? \\\\\n       0.0210  &     47.6190  &  amplitude modulation\\\\ \n       0.5577  &      1.7931 & harmonics of pulsation frequencies  \\\\\n       0.3132  &      3.1928  &  \\\\\n       0.2492  &      4.0128   &  \\\\ \n       0.0110  &     90.9091  &  amplitude modulation \\\\\n       0.0284  &     35.2113   &  \\\\\n       0.2267  &      4.4111    &  \\\\ \n       0.3318  &      3.0139   &  \\\\\n       0.0109  &     91.7431    & amplitude modulation \\\\\n       0.0070  &    142.8571  & \\\\ \n       0.3353  &      2.9824 &  \\\\\n       ...         &     ... &\\\\\n\\hline                       \n\\end{tabular}\n\\label{freqdiffs}\n\\end{table*}\n\n\n\n\\begin{figure}\n \\centering\n   \\includegraphics[width=14cm,angle=270,scale=0.5]{ampspec-rv-res.ps}\n\\caption{Amplitude spectrum of the radial velocity residuals for both\ncomponents, after subtraction of the best Keplerian orbit fit. The two highest\npeaks for the primary (indicated) correspond to frequencies detected in the\n\\textit{Kepler} light curve.}\n\\label{ampspec-rv-res}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusions}\n\nWe have obtained accurate system parameters and astrophysical\nproperties for KIC 11285625, a double-lined eclipsing binary\nsystem with a $\\gamma$ Dor pulsator discovered by the \\textit{Kepler} space\nmission. The excellent \\textit{Kepler} data with a total time span of almost\n1000 days have been analysed together with high resolution HERMES spectra.\nThe individual composite spectra could not be used to derive fundamental\nparameters such as\n{$T_{\\rm eff}$} and {$\\log g$}, given their insufficient S\/N. This was achieved after using\nthe spectral disentangling technique for both components. \n\nAn iterative automated method was developed to separate the orbital variability\nin the \\textit{Kepler} light curve from the variability due to the pulsations of\nthe primary. The fact that the orbital frequency and its overtones are located \nin the same frequency range as the pulsation frequencies, made a simple\nseparation technique (such as a filter in the frequency domain) insufficient.\nWe plan to develop this technique further and apply it to other binary systems\nin the \\textit{Kepler} database.\n\nAfter removal of the best binary model, we studied the residual pulsation\nsignal in detail, and found indications for rotational splitting of the pulsation\nfrequencies, compatible with super-synchronous and non-rigid internal rotation.\nA detailed asteroseismic analysis of the $\\gamma$ Dor pulsator and comparison\nwith theoretical models can now be attempted on the basis of this observational work, which\nconstitutes an excellent starting point for stellar modelling of a $\\gamma$~Dor\\\nstar. A concrete interpretation of the detected amplitude modulation must await a much longer \\textit{ Kepler} light curve, given the relatively long modulation period.\n\n\\begin{acknowledgements}\nThe research leading to these results has received funding from the European\nResearch Council under the European Community's Seventh Framework Programme\n(FP7\/2007--2013)\/ERC grant agreement n$^\\circ$227224 (PROSPERITY), from the\nResearch Council of K.U.Leuven (GOA\/2008\/04), and from the Belgian federal\nscience\npolicy office (C90309: CoRoT Data Exploitation); A. Tkachenko and P. Degroote\nare postdoctoral fellows of the Fund for Scientific Research (FWO), Flanders,\nBelgium. \nFunding for the \\textit{Kepler} Discovery mission is provided by NASA's Science\nMission Directorate.\nSome of the data presented in this paper were obtained from the\nMultimission Archive at the Space Telescope Science Institute (MAST). STScI is\noperated by the Association of Universities for Research in Astronomy, Inc.,\nunder NASA contract NAS5-26555. Support for MAST for non-HST data is provided by\nthe NASA Office of Space Science via grant NNX09AF08G and by other grants and\ncontracts.\nThis research has made use of the SIMBAD database,\noperated at CDS, Strasbourg, France.\nWe would like to express our special thanks to the numerous people who helped\nmake the \\textit{Kepler} mission possible.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminary results}\n\nIt is well known from elementary calculus that an arbitrary\npolynomial $f$ with complex coefficients (complex polynomial) of\ndegree $n \\in \\mathbb{N}$\n$$f(z)= a_0 z^{n}+ a_1 z^{n-1}+ \\dots + a_{n-1} z + a_n, \\ a_0\\neq\n0,\\eqno(1)$$\nhaving a root $\\lambda \\in \\mathbb{C}$ of multiplicity $\\mu, \\ 1\\le\n\\mu\\le n$,  shares it with each of its derivatives  up to order $\\mu-1$,\nbut $f^{(\\mu)}(\\lambda)\\neq 0$. When $\\lambda$ is a unique root of\n$f$, it has the  form  $f(z)=a(z-\\lambda)^n$, $\\mu=n$ and $\\lambda$ is\nthe same root of each derivative of $f$ up to order $n-1$. We will\ncall such a polynomial as a trivial polynomial.  Obviously, as it\nfollows from fundamental theorem of algebra, $f$ has at least two\ndistinct roots, i.e. a polynomial of degree $n$ is non-trivial, if\nand only if its maximum multiplicity of roots $r$ does not exceed\n$n-1$.\n\n\nIn 2001 Casas- Alvero \\cite{CA} conjectured that an arbitrary polynomial $f$  degree $n\n\\ge 1$ with complex coefficients  is of the form   $f(z)=\na(z-b)^n, a, b \\in \\mathbb{C}$,    if   and \\ only   if $f$  shares a root with each of its derivatives $f^{(1)}, f^{(2)},\n\\dots, f^{(n-1)}.$\n\nWe will call a possible non-trivial polynomial,  which has a common root with each of its non-constant derivatives   as the CA-polynomial. The conjecture says that there  exist no CA-polynomials.  The problem is still open. However, it is proved for small degrees, for infinitely many degrees, for instance, for all powers $n$,   when $n$ is a prime   (see in \\cite{Drai}, \\cite{Graf}, \\cite{Pols} ).  We observe that such a kind of  CA-polynomial of degree $n \\ge 2$  cannot have all distinct roots since at least one root is common   with its  first derivative.  Therefore it has a multiplicity at least 2 and a maximum of possible distinct  roots is $n-1$.\n\n\n\n Our main goal here is to derive necessary and sufficient conditions for an arbitrary polynomial (1) to be trivial.  For example, solving a simple differential equation of the first order, we easily prove that a polynomial is trivial, if and only if it is divisible by its first derivative.  In the sequel we establish other criteria, which will guarantee that an arbitrary polynomial has a unique joint root.\n\n Without loss of generality one can assume in the sequel that $f$ is a monic polynomial of degree $n$, i.e. $a_0=1$ in (1).  Generally, it has $k$ distinct roots $\\lambda_j$  of multiplicities $r_j, \\   j= 1,  \\dots,  k, 1\\le k\\le n$ such that\n %\n$$ r_1+ r_2+ \\dots r_{k} = n\\eqno(2).$$\n %\n By $r$ we will denote the maximum of multiplicities (2), $r= \\hbox{max}_{1\\le j\\le k}  (r_j)$,  $r_0= \\hbox{min}_{1\\le j\\le k}  (r_j)$ and by  $  \\xi^{(m)}_\\nu, \\  \\nu = 1,\\dots, n-m$  zeros of $m$-th derivative $f^{(m)}, \\ m=1,\\dots, n-1.$\n For further needs we specify zeros of the $n-1$-th and $n-2$-th derivatives, denoting them by $\\xi^{(n-1)}_1=z_{n-1}$ and $\\xi^{(n-2)}_2=z_{n-2}$, respectively. It is easy to find another zero of the $n-2$-th derivative, which is equal to $\\xi^{(n-2)}_1= 2z_{n-1} -z_{n-2} $.  When  zeros $ z_{n-1},\\  z_{n-2}$ are real we write, correspondingly,   $ x_{n-1},\\  x_{n-2}$.    The value $z_{n-1}$ is called the centroid. It is a center of gravity of roots and by Gauss-Lucas theorem it is contained in the convex hull of all non-constant polynomial derivatives (see details in \\cite{Rah}).\n\n The paper is structured as follows: In Section 2 we study properties of the Abel-Goncharov interpolation polynomials, including integral and series representations and upper bounds.  Section 3 deals with the Sz.-Nagy type identities and Obreshkov-Chebotarev type inequalities for roots of polynomials and   their derivatives. As applications  new criteria are found for an arbitrary polynomial with only real roots to be trivial.   Section 4 is devoted to the Laguerre type inequalities for polynomials with only real roots to localize their zeros. The final Section 5 contains applications of these results towards  solution of the  Casas-Alvero  conjecture and its particular cases.\n\n\n \\section{Abel-Goncharov polynomials, their upper bounds and  integral and genetic sum's representations}\n\nWe begin, choosing a sequence of complex numbers (repeated terms are permitted)\n$z_0, z_1, z_2, \\dots, z_{n-1}, n \\in \\mathbb{N}$, where $z_0 \\in \\{\\lambda_1, \\lambda_2, \\dots, \\lambda_k\\},\\\nz_m \\in  \\{\\xi^{(m)}_1,\\  \\xi^{(m)}_2, \\dots, \\xi^{(m)}_{n-m}\\}, \\  m=1,2,\\dots, n-1$, satisfying conditions $f^{(m)}(z_m) =0, \\\nm =0, 1,\\dots,  \\ n-1$ and, clearly $f^{(n)} (z)= n!$.     Then we represent  $f(z)$ in the form\n\n$$\n    f(z)= z^n + P_{n-1} (z),\\eqno(3)\n$$\nwhere $P_{n-1} (z)$ is a polynomial of degree at most $n-1$. To determine  $P_{n-1} (z)$  we  differentiate the latter equality $m$ times,  and we  calculate the corresponding derivatives in $z_m$ to obtain\n$$\n P_{n-1}^{(m)} (z_m) =  -  \\frac{n!}{ (n-m)!}  z_m ^{n- m}  ,   \\quad  m= 0,1, \\dots, n-1.\\eqno(4)\n$$\nBut this is the known Abel-Goncharov interpolation problem (see \\cite{Evgrafov}) and the polynomial $P_{n-1}(z)$ can be uniquely determined via the linear system (4) of $n$ equations with $n$ unknowns and triangular\nmatrix with non-zero determinant.   So, following \\cite{Evgrafov}, we derive\n\n$$\n     P_{n-1} (z) = -  \\sum_{k=0}^{n-1}  \\frac{n!}{ (n- k)!}  z_k ^{n- k} G_k(z) , \\eqno(5)\n$$\nwhere $G_k(z),  k=0, 1,\\dots, n-1$ is the system of the Abel-Goncharov polynomials\n \\cite{Evgrafov}, \\cite{Levinson1},  \\cite{Levinson2}.    On the other hand it is known that\n\n$$\n     G_n(z)= z^n  -  \\sum_{k=0}^{n-1}  \\frac{n!}{ (n- k)!}  z_k ^{n- k}  G_k(z).\\eqno(6)\n$$\nThus comparing with (3),  we  find that\n$$\nG_n(z)\\equiv  G_n\\left(z, z_0, z_1, z_2,\\dots, z_{n-1}\\right)=f(z),\n$$\nand\n$$\n G_n\\left(\\lambda_j, z_0, z_1, z_2,\\dots, z_{n-1}\\right) = f(\\lambda_j)= 0,   \\quad\nj= 1,2, \\dots,  k.\n$$\nPlainly, one can make a relationship of possible CA-polynomials with the corresponding Abel-Goncharov polynomials, fixing a sequence $\\{z_m\\}_0^{n-1}$ such that\n$$z_m \\in \\{\\lambda_1, \\lambda_2, \\dots, \\lambda_k\\}, \\  m=0,1,\\dots,\\ n-1.$$\n\nFurther, It is known \\cite{Evgrafov} that the Abel-Goncharov polynomial can\nbe represented as a multiple integral in the complex plane\n$$\nG_n(z)=   n! \\int_{z_0}^z  \\int_{z_1}^{s_{1}} \\dots\n\\int_{z_{n-1}}^{s_{n-1}} d s_{n} \\dots d s_{1}.\\eqno (6)\n$$\nMoreover, making  simple changes  of variables in (6),  it can be verified that $G_n(z)$ is\n a homogeneous function of degree $n$ (cf. \\cite{Levinson1}). Therefore\n$$G_n(\\alpha z)= G_n\\left(\\alpha z, \\alpha z_0, \\alpha z_1,\\dots, \\alpha z_{n-1}\\right)\n = \\alpha^n G_n(z), \\ \\alpha \\neq 0.\\eqno(7)$$\nThe following Goncharov upper bound holds for  $G_n$ (see \\cite{Gon}, \\cite{Evgrafov}, \\cite{Levinson1}, \\cite{Ibra})\n$$\n\\left| G_n(z) \\right|  \\le  \\left( |z-z_0| + \\sum_{s=0}^{n-2} \\left\n|z_{s+1}- z_s\\right| \\right)^n.\\eqno(8)\n$$\n  Let us represent the Abel-Goncharov polynomials $G_n(z)$ in a different  way.  To do this, we will use the following representation of the Gauss hypergeometric function given by relation (2.2.6.1) in \\cite{Prud}, namely\n$$\\int_a^b (z-a)^{\\alpha-1} (b-z)^{\\beta-1}(z + c)^\\gamma dz =  (b-a)^{\\alpha+\\beta-1} (a+c)^\\gamma\nB(\\alpha,\\beta) {}_2F_1 \\left(\\alpha, -\\gamma; \\alpha +\\beta;\n\\frac{a-b}{ a+c} \\right),\\eqno(9)$$ where $\\alpha, \\beta, \\gamma $\nare positive integers,  $a, b, c \\in \\mathbb{C}$ and\n$B(\\alpha,\\beta)$ is the Euler beta-function. So, our goal  will be a\nrepresentation of the Abel-Goncharov polynomials  in terms of the\nso-called genetic sums considered, for instance, in \\cite{Apteka}.\nMoreover, this will drive us to a  sharper upper bound for these  polynomials, improving  the Goncharov  bound (8).\n Indeed,   $G_1(z)= z-z_0$. When $n \\ge 2$, we employ  multiple integral (6),  and appealing to\n representation (9), we obtain recursively\n$$\nG_n(z)=   n! \\int_{z_0}^z  \\int_{z_1}^{s_{1}} \\dots \\int_{z_{n-2}}^{s_{n-2}} \\  (s_{n-1} - z_{n-1})\n d s_{n-1} \\dots d s_{1}\n$$\n$$=   n! (z_{n-2} - z_{n-1})  \\int_{z_0}^z  \\int_{z_1}^{s_{1}} \\dots \\int_{z_{n-3}}^{s_{n-3}} \\  (s_{n-2} - z_{n-2})\n {}_2F_1 \\left(1, -1 ;  \\ 2; \\   \\frac{ z_{n-2} - s_{n-2}}{ z_{n-2} -  z_{n-1} } \\right) d s_{n-2} \\dots d s_{1} $$\n$$=  n!   \\sum_{j_1=0} ^ 1\\frac{(-1)_{j_1}(-1)^{j_1} }{(2)_{j_1}}  (z_{n-2} - z_{n-1})^{1-j_1} \\int_{z_0}^z\n\\int_{z_1}^{s_{1}} \\dots \\int_{z_{n-3}}^{s_{n-3}} \\  (s_{n-2} - z_{n-2})^{1+j_1}  d s_{n-2} \\dots d s_{1} .$$\nHence, employing  properties of the Pochhammer symbol and repeating this process, we find\n$$\nG_n(z)=  n!   \\sum_{j_1=0} ^ 1\\frac{(z_{n-2} - z_{n-1})^{1-j_1}  }{(2)_{j_1} (1-j_1)!}   \\int_{z_0}^z  \\int_{z_1}^{s_{1}}\n\\dots \\int_{z_{n-3}}^{s_{n-3}} \\  (s_{n-2} - z_{n-2})^{1+j_1}  d s_{n-2} \\dots d s_{1} $$\n$$= n!   \\sum_{j_1=0} ^ 1  \\sum_{j_2=0} ^{1+j_1} \\frac{(z_{n-2} - z_{n-1})^{1-j_1} (z_{n-3} - z_{n-2})^{1+j_1-j_2}  }\n{(2)_{j_2} (1-j_1)!(1+j_1 -j_2)!}   \\int_{z_0}^z  \\int_{z_1}^{s_{1}} \\dots \\int_{z_{n-4}}^{s_{n-4}} \\\n(s_{n-3} - z_{n-3})^{1+j_2}  d s_{n-3} \\dots d s_{1} .$$\nContinuing to calculate  iterated integrals with the use of (9), we\narrive  finally  at the following genetic sum's representation of\nthe Abel-Goncharov polynomials ($j_0 = j_n=0,\\   z_{-1}\\equiv z $)\n$$\nG_n(z)=   n!   \\sum_{j_1=0} ^ 1  \\sum_{j_2=0} ^{1+j_1} \\dots  \\sum_{j_{n-1}=0} ^{1+j_{n-2}}  \\\n \\prod_{s=0}^{n-1} \\frac{ (z_{n-2-s}- z_{n-1-s} )^ {1+ j_s- j_{s+1}} }{ (1+ j_s- j_{s+1})!}.\\eqno(10) $$\nAnalogously, we derive the genetic  sum's representation for  the $m$-th derivative\n$G_{n}^{(m)}(z)$, namely  ($j_0  =0 $)\n $$\nG_n^{(m)} (z)=   n!   \\sum_{j_1=0} ^ 1  \\sum_{j_2=0} ^{1+j_1} \\dots  \\sum_{j_{n-1-m }=0} ^{1 +j_{n-2-m}}\n  \\frac{ (z - z_{m} )^ {1 + j_{n-1-m}} }{ (1 + j_{n-1-m})!} \\\n \\prod_{s=0}^{n-2-m} \\frac{ (z_{n-2-s}- z_{n-1-s} )^ {1 + j_s- j_{s+1}} }{ (1+ j_s- j_{s+1} )!},\\eqno(11) $$\nwhere $m= 0,1,\\dots, n-1$.\n\nMeanwhile,  the Taylor expansions of $ G_n^{(m)} (z)$ in the neighborhood of points  $z_m$ give the formulas\n$$G_n^{(m)} (z)  = \\frac{n!}{(n- m)!} (z-z_m)^{n-m}  + \\frac {G_n^{(n-1)} (z_m)}{(n-m-1)!} (z-z_m)^{n-m- 1} + \\dots\n+  G_n^{(1+m)} (z_m)  (z-z_m),\\eqno(12)$$\nwhere $m= 0,1,\\dots, n-1$.  Thus comparing coefficients in front of $(z-z_m)^s, \\  s= 1, \\dots , n- m-1$ in (11) and (12), we find the values of derivatives $G_n^{(s+ m)} (z_m)$ in terms  $z_m, z_{m+1}, \\dots, z_{n-1}$. Precisely, we obtain   ($j_0  =0 $)\n$$G_n^{(s+m)} (z_m)   =   n!   \\sum_{j_1=0} ^ 1  \\sum_{j_2=0} ^{1+j_1} \\dots\n\\sum_{j_{n-2-m }=0} ^{1 +j_{n-3-m}}   \\frac{ (z_m - z_{m+1} )^ {2 + j_{n-2-m}-s} }{ (2 + j_{n- 2-m}-s)!} \\\n \\prod_{l=0}^{n-3-m} \\frac{ (z_{n-2-l}- z_{n-1-l} )^ {1 + j_l- j_{l+1}} }{ (1+ j_l- j_{l+1} )!},\\eqno(13)$$\nwhere $s=1,2,\\dots, n-m, \\  m=0,1,\\dots, n-1$.\n\nFinally, in this section,  we will establish a  sharper  upper bound for the\nAbel-Goncharov polynomials.    We have\n\n{\\bf Theorem 1}. {\\it Let $z, z_0, z_1, z_2,\\dots, z_{n-1} \\in\n\\mathbb{C},\\ n \\ge 1$. The following  upper bound holds for the\nAbel-Goncharov polynomials\n$$| G_n\\left(z, z_0, z_1, z_2,\\dots, z_{n-1}\\right)| \\le\n\\sum_{k_0=0}^1  \\sum_{k_1=0} ^{2-k_0} \\dots  \\sum_{k_{n-2}=0} ^{n-1\n- k_0-k_1-\\dots- k_{n-3}}  \\ {n! \\choose k_0! k_1! \\dots k_{n-2}! \\\n(n - k_0-k_1-\\dots- k_{n-2})! }$$\n$$\\times \\prod_{s=0}^{n-1} |z_{n-2-s}- z_{n-1-s}|^ {k_s},\\eqno(14) $$\nwhere $ z_{-1}\\equiv z$ and\n$${n! \\choose l_0! l_1! \\dots l_m!} = \\frac{n!}{l_0! l_1! \\dots\nl_m!}, \\ l_0+l_1+\\dots +l_m= n$$ are multinomial coefficients. This\nbound is sharper than the Goncharov upper bound $(8)$.}\n\n\\begin{proof} In fact, making simple substitutions $k_s= 1+j_s- j_{s+1}, \\ s=0,1,\\dots, n-1,  j_0=j_n=0$\nand writing identity (10) for the Abel-Goncharov polynomials (6), we\nestimate their  absolute value, coming out immediately with inequality\n(14). Furthermore, appealing to the multinomial theorem, we estimate\nthe right-hand side of (14) in the following way\n$$\\sum_{k_0=0}^1  \\sum_{k_1=0} ^{2-k_0} \\dots  \\sum_{k_{n-2}=0} ^{n-\n1- k_0-k_1-\\dots- k_{n-3}}  \\ {n! \\choose k_0! k_1! \\dots k_{n-2}\\\n(n - k_0-k_1-\\dots- k_{n-2})! } \\prod_{s=0}^{n-1} |z_{n-2-s}-\nz_{n-1-s}|^ {k_s}$$$$\\le \\sum_{l_0+l_1+\\dots+ l_{n-1} = n} \\ {n!\n\\choose  l_0! l_1!\\dots l_{n-1}!} \\prod_{s=0}^{n-1} |z_{n-2-s}-\nz_{n-1-s}|^{l_s}$$\n$$=  \\left(\\sum_{m=0}^{n-1} \\left |z_{m-1}- z_{m}\\right|\\right)^n,$$\nwhere the summation now  is taken over all combinations of\nnonnegative integer indices $l_0$ through $l_{n-1}$ such that the\nsum of all $l_j$ is $n$. Thus it  yields (8) and completes the\nproof.\n\n\\end{proof}\n\n\n\n\\section{Sz.-Nagy type identities for roots of polynomials and their derivatives}\n\nIn this section we prove  Sz.-Nagy type identities \\cite{Rah} for zeros of monic polynomials with complex coefficients and their derivatives.  All   notations of  roots and their multiplicities given in Section 1 are involved.\n\n    We begin with\n\n{\\bf  Lemma 1.}  {\\it Let $f$ be  a monic polynomial of degree $n \\ge 2$ with complex coefficients,  $m= 0,1,\\dots, n-2$ and  $z \\in \\mathbb{C}$. Then the following Sz.-Nagy type identities, which are related to the roots of $f$ and its $m$-th derivative,  hold }\n\n$$z_{n-1} - z  = {1\\over n} \\sum_{j=1}^k r_{j}(\\lambda_j- z) =   {1\\over n-m} \\sum_{j=1}^{n-m}  (\\xi^{(m)}_j - z),\\eqno(15)$$\n\n$$   (z_{n-1} - z_{n-2})^2={1\\over n(n-1)} \\left[ \\sum_{j=1}^k r_{j}(\\lambda_j- z)^2-  n (z_{n-1} -  z)^2\\right]\n=  {1\\over (n-m)(n-m -1)}$$$$\\times  \\left[ \\sum_{j=1}^{n-m}\n(\\xi^{(m)}_j - z)^2 -  (n-m) (z_{n-1} -  z)^2\\right],\\eqno(16)$$\n\n$$   (z_{n-1} - z_{n-2})^2={1\\over n^2(n-1)} \\sum_{1\\le j <  s\\le k} r_{j}r_{s}(\\lambda_j- \\lambda_s)^2\n=  {1\\over (n-m)^2(n-m -1)} \\sum_{1\\le j < s \\le n-m} (\\xi^{(m)}_j -\n\\xi^{(m)}_s)^2.\\eqno(17)$$\n\n\n\n\\begin{proof}  In fact,  the first Vi\\'{e}te  formula (see \\cite{Rah}) says that the coefficient $a_1$ ($a_0=1$) in (1) is equal to\n$$- a_1= r_{1}\\lambda_1  +  r_{2} \\lambda_{2} +   \\dots +  r_{k}\\lambda_{k}.$$\nOn the other hand, differentiating (1) $n-1$ times, we find $z_{n-1}\n= - a_1\/ n$.  Thus minding (2) we prove the first equality in (15). The second\nequality can be done similarly, using the properties of centroid,\nwhich is differentiation invariant, see, for instance, in\n\\cite{Rah}.  In order to establish the first equality in (16), we\ncall formula (11) to find\n$$\\frac{f^{(n-2)} (z)}{(n-2)!} =  \\frac{n(n-1)}{2} (z - z_{n-2}) (z + z_{n-2}- 2 z_{n-1}).\\eqno(18)$$\nMoreover,  as a consequence of the second Vi\\'{e}te formula,  the\ncoefficient $a_2$ in (1), which  equals\n$$a_2=   \\frac{f^{(n-2)} ( z)}{(n-2)!} -  \\frac{n(n-1)}{2} z^2 + n(n-1) z_{n-1}z\\eqno(19)$$\ncan be expressed as follows\n$$a_2=   {1\\over 2} \\left(\\sum_{j=1}^k r_{j} \\lambda_j\\right)^2 -  {1\\over 2} \\sum_{j=1} ^k r_j\\lambda_j^2.\\eqno(20) $$\nHence letting $z=z_{n-2}$ in (18),  and taking into account (15) with $z=0$,\nwe deduce\n\n$$2a_2=   n^2z^2_{n-1}  -   \\sum_{j=1} ^k r_{j} \\lambda_j^2 =  2 n(n-1) z_{n-1}z_{n-2}  -  n(n-1) z_{n-2}^2 . $$\nTherefore,   using again (15) and (2), we easily come out with the\nfirst equality in (16). The second one can be prove in the same\nmanner, involving roots of derivatives. Finally, we prove the first\nequality in (17). Concerning the second equality,  see Lemma 6.1.5\nin \\cite{Rah}. Indeed, calling the first equality in (16),  letting\n$z= z_{n-1}$ and employing (15), we derive\n\n$$  n^2(n-1)(z_{n-1} - z_{n-2})^2= n \\sum_{j=1}^k r_{j}\\lambda^2_j +\n\\left(\\sum_{s=1}^k r_{s}\\lambda_s\\right)^2 - 2\\left(\\sum_{s=1}^k\nr_{s}\\lambda_s\\right)\\left(\\sum_{j=1}^k r_{j}\\lambda_j\\right)$$\n$$= n \\sum_{j=1}^k r_{j}\\lambda^2_j - \\sum_{s=1}^k r^2_{j}\\lambda^2_j -\n2\\sum_{1\\le j <  s\\le k} r_{j}r_{s}\\lambda_j\\lambda_s=  \\sum_{1\\le j\n<  s\\le k} r_{j}r_{s}(\\lambda_j- \\lambda_s)^2.$$\n\n\\end{proof}\n\nThe following result gives an identity, which is associated with  zeros of a monic\npolynomial and common zeros  of its derivatives. Precisely, we have\n\n\n{\\bf Lemma 2.} {\\it Let $f$ be a monic polynomial of exact degree $n\\ge 2$, having $k$ distinct  roots of multiplicities $(2)$. Let  $z_{n-1}=\\lambda_1$ be a common root of $f$ of multiplicity $r_1$ with  the unique root of its $n-1$-th\nderivative. Let also $z_m= \\xi_{n-m}^{(m)}= \\lambda_{k_m}$ be a common root of $f$ of multiplicity $r_{k_m}$ and\nits $m$-th derivative, $m \\in \\{1,2,\\dots,  n-2\\}$. Then,  involving other roots of $f^{(m)}$,\nthe following identity holds}\n\n$$\\left[\\frac{n-m-2}{(n-m)^2} + \\frac{r_{k_m}+r_1-n}{n (n-1)}\\right]\\sum_{s=1}^{n-m-1} (z_m- \\xi^{(m)}_s)^2\n+ \\frac{n-m-2}{(n-m)^2}\\sum_{1\\le s < t \\le n-m-1} (\\xi^{(m)}_s- \\xi^{(m)}_t)^2\n$$$$= \\frac{(n-m)^2 r_{k_m}- (n-r_1)(n-m+2)}{n (n-1)} (z_m- z_{n-1})^2$$$$+ \\\n\\frac{2}{n (n-1)}\\sum_{j\\neq 1, k_m} r_j \\sum_{1 \\le s < t \\le\nn-m-1} (\\lambda_j- \\xi^{(m)}_s)(\\lambda_j- \\xi^{(m)}_t).\\eqno(21)$$\n\n\\begin{proof}  We begin, appealing to  (15) and letting  $z=0$.  We get\n$$\\sum_{s=1}^{n-m} \\xi^{(m)}_s= (n-m) z_{n-1}, \\quad   \\xi_{n-m}^{(m)}=z_m.\\eqno(22) $$\nHence via identities  (17) with $z=z_m$ we write the chain of equalities\n$$\\sum_{1\\le s < t \\le n-m} (\\xi^{(m)}_s- \\xi^{(m)}_t)^2=\n\\frac{(n-m-1)(n-m)^2}{n (n-1)} r_{k_m}(z_m- z_{n-1})^2 +\n\\frac{(n-m-1)(n-m)^2}{n (n-1)}\\sum_{j\\neq 1, k_m} r_j (\\lambda_j -\nz_{n-1})^2$$\n$$=\\frac{(n-m-1)(n-m)^2}{n (n-1)} r_{k_m}(z_m- z_{n-1})^2 +\n\\frac{n-m-1}{n (n-1)}\\sum_{j\\neq 1, k_m} r_j \\left(\\lambda_j - z_m+\n\\sum_{s=1}^{n-m-1} (\\lambda_j- \\xi^{(m)}_s) \\right)^2$$\n$$=\\frac{(n-m-1)(n-m)^2}{n (n-1)} r_{k_m}(z_m- z_{n-1})^2 +\n\\frac{n-m-1}{n (n-1)}\\left[\\sum_{j\\neq 1, k_m} r_j (\\lambda_j -\nz_m)^2 +  \\sum_{j\\neq 1, k_m} r_j \\left(\\sum_{s=1}^{n-m-1}\n(\\lambda_j- \\xi^{(m)}_s) \\right)^2\\right.$$\n$$\\left.+ 2 \\sum_{j\\neq 1, k_m} r_j \\sum_{s=1}^{n-m-1} (\\lambda_j -\nz_m)(\\lambda_j- \\xi^{(m)}_s)\\right]= \\frac{(n-m-1)(n-m)^2}{n (n-1)}\nr_{k_m}(z_m- z_{n-1})^2$$\n$$+ \\frac{n-m-1}{n (n-1)}\\left[(2(n-m)-1) \\sum_{j\\neq 1, k_m} r_j (\\lambda_j -\nz_m)^2 +  \\sum_{j\\neq 1, k_m} r_j \\left(\\sum_{s=1}^{n-m-1}\n(\\lambda_j- \\xi^{(m)}_s) \\right)^2\\right.$$\n$$\\left.+ 2 \\sum_{j\\neq 1, k_m} r_j \\sum_{s=1}^{n-m-1} (\\lambda_j -\nz_m)(z_m - \\xi^{(m)}_s)\\right]= \\frac{(n-m-1)(n-m)^2}{n (n-1)}\nr_{k_m}(z_m- z_{n-1})^2$$\n$$+ \\frac{n-m-1}{n (n-1)}\\left[(2(n-m)-1) \\sum_{j\\neq 1, k_m} r_j (\\lambda_j -\nz_m)^2 +  \\sum_{j\\neq 1, k_m} r_j \\left(\\sum_{s=1}^{n-m-1}\n(\\lambda_j- \\xi^{(m)}_s) \\right)^2\\right.$$\n$$\\left. - 2 (n-m)(n-r_1)(z_m-z_{n-1})^2 \\right]= \\frac{(n-m-1)}{n (n-1)}\n\\left((n-m)^2 r_{k_m}- n+r_1\\right) (z_m- z_{n-1})^2$$\n$$+ (n-m-1)(3(n-m)-2)(z_{n-1}- z_{n-2})^2  +  \\frac{(n-m-1)(n-r_1)}{n (n-1)}\\sum_{s=1}^{n-m-1}\n(z_{n-1}- \\xi^{(m)}_s)^2$$$$-    \\frac{r_{k_m}(n-m-1)}{n\n(n-1)}\\sum_{s=1}^{n-m-1} (z_{m}- \\xi^{(m)}_s)^2+ 2 \\ \\frac{n-m-1}{n\n(n-1)}\\sum_{j\\neq 1, k_m} r_j \\sum_{1 \\le s < t \\le n-m-1}\n(\\lambda_j- \\xi^{(m)}_s)(\\lambda_j- \\xi^{(m)}_t).$$ Applying again\n (17), (22), we split the right-hand side of the latter\nequality in (17) in two parts, selecting the root $z_m$. Thus in the\nsame manner after straightforward calculations it becomes\n$$\\left[\\frac{n-m-2}{(n-m)^2} + \\frac{r_{k_m}+r_1-n}{n (n-1)}\\right]\\sum_{s=1}^{n-m-1} (z_m- \\xi^{(m)}_s)^2\n+ \\frac{n-m-2}{(n-m)^2}\\sum_{1\\le s < t \\le n-m-1} (\\xi^{(m)}_s-\n\\xi^{(m)}_t)^2\n$$$$= \\frac{(n-m)^2 r_{k_m}- (n-r_1)(n-m+2)}{n (n-1)} (z_m- z_{n-1})^2+ \\\n\\frac{2}{n (n-1)}\\sum_{j\\neq 1, k_m} r_j \\sum_{1 \\le s < t \\le\nn-m-1} (\\lambda_j- \\xi^{(m)}_s)(\\lambda_j- \\xi^{(m)}_t),$$\ncompleting the proof of Lemma 2.\n\n\\end{proof}\n\n{\\bf Remark 1}.  It is easy to verify identity (21) for the least case $m=n-2$, when double sums are\nempty and $\\xi^{(n-2)}_1= 2z_{n-1} -z_{n-2} $ (see above).\n\n{\\bf  Corollary 1.}  {\\it A polynomial with only real roots of degree $n\\ge 2$ is trivial, if and only if its  $n-2$-th derivative has a double root}.\n\n\\begin{proof}  Indeed, necessity is obvious.  To prove sufficiency we see that since the $n-2$-th derivative  has a double real root $x_{n-2}$,  it  is equal to the root $x_{n-1}$ of the $n-1$-th derivative. Therefore letting in (16)\n$z= x_{n-1}$, we find that its left-hand side becomes zero and,\ncorrespondingly, all squares in the right-hand side are zeros. This\ngives a conclusion that all roots are equal to $x_{n-1}$.\n\n\\end{proof}\n\n{\\bf  Corollary 2.}  {\\it Let $f$ be an arbitrary  polynomial of\ndegree $n \\ge 3$ with at least two distinct roots, whose $n-2$-th\nderivative has a double root. Then it contains at least one  complex\nroot}.\n\n\\begin{proof} In fact, if all roots are real it is trivial via Corollary 1.\n\n\\end{proof}\n\n Evidently, each derivative up to $f^{(r-1)}$ of a  polynomial $f$ with only real roots,  where $r$ is the maximum of multiplicities of roots  shares a root with $f$. Moreover, since  via the Rolle theorem all roots of $f^{(m)}, \\  m=r,r+1,\\dots, n-1$ are simple,  we have that a possible common root with $f$ is simple too (we note,  that a number of common roots does not exceed $k-2$, because  minimal and  maximal  roots cannot be zeros of $f^{(m)},\\ m\\ge r$).  This circumstance gives an immediate\n\n {\\bf Corollary 3.} {\\it There exists no non-trivial polynomial with only real roots, having two distinct zeros  and sharing a root  with at least one of its derivatives, whose order exceeds $r-1,\\  r= \\hbox{max}_{1\\le j\\le k}  (r_j)$. }\n\n\\begin{proof} Indeed, in the case of existence of such a polynomial,  these two distinct roots cannot be within zeros  of any derivative $f^{(m)},\\  m > r$ owing to the Rolle theorem.  Moreover, if any of two roots  is in common with roots of  $f^{(r)}$, its multiplicity is greater than $r$, which is impossible.\n\n\\end{proof}\n\nWe extend Corollary 3 on three distinct real roots.  Precisely, it drives to\n\n {\\bf Corollary 4.} {\\it There exists no non-trivial polynomial $f$ of degree $n \\ge 3$ with only real roots, having three distinct zeros  and sharing a root  with its $n-2$-th and $n-1$-th derivatives. }\n\n\\begin{proof}  Let such a polynomial exist. Calling its roots  $\\lambda_1= x_{n-1}, \\   \\lambda_2= x_{n-2}$ and $\\lambda_3$ of multiplicities  $r_1, \\ r_2, \\ r_3$, respectively. Hence employing identities (16), we write for this case\n$$  (n^2- n- r_2) (x_{n-1} - x_{n-2})^2=  r_{3}(\\lambda_3 - x_{n-1}) ^2.$$\nIn the meantime, making square of both sides of the first equality in (15) for this case after simple modifications , we obtain\n$$ r_2^2(x_{n-1} - x_{n-2})^2=  r_{3}^2 (\\lambda_3 - x_{n-1}) ^2.$$\nHence, comparing with the  previous equality, we come out with the relation\n$$ (n^2- n- r_2) r_3=  r_2^2.$$\nBut $n=r_1+r_2+r_3,\\ r_j \\ge 1, j=1,2,3.$  Consequently,\n$$r_2^2 \\ge  n(n-1)- r_2 >  (n-1)^2- r_2 \\ge (r_1+r_2)^2-r_2 \\ge r_2^2+ r_2+ r_1^2 > r_2^2,$$\nwhich is impossible.\n\\end{proof}\n\n{\\bf Remark 2.}  If we omit the condition for $f$ to have a common root with the $n-2$-th derivative in Corollary 4, it becomes false. In fact,  this circumstance can be shown  by the counterexample $f(x)= x^3-x.$\n\nThe following result deals with the case of 4 distinct roots.  We have,\n\n{\\bf Corollary 5.} {\\it There exists no non-trivial polynomial $f$ of degree $n \\ge 4$ with only real roots, having four  distinct zeros  and sharing a root  with its $n-2$-th and $n-1$-th derivatives. }\n\n\\begin{proof}  Similarly to the previous corollary, we assume the existence of such a polynomial and call  its  roots\n$\\lambda_1= x_{n-1}, \\   \\lambda_2= x_{n-2}$ and $\\lambda_3, \\lambda_4$ of multiplicities $r_j, \\  j=1,2,3,4$, respectively.\nHence the first  identity  in (16)  yields\n$$  (n^2- n- r_2) (x_{n-1} - x_{n-2})^2=  r_{3}(\\lambda_3 - x_{n-1}) ^2+   r_{4}(\\lambda_4 - x_{n-1}) ^2 .\\eqno(23)$$\nMeanwhile, using  the first equality in (15) for this case, we derive in a similar manner\n$$ r_2^2(x_{n-1} - x_{n-2})^2=  r_{3}^2 (\\lambda_3 - x_{n-1}) ^2+ r_{4}^2 (\\lambda_4 - x_{n-1}) ^2+\n2 r_{3}r_4 (\\lambda_3 - x_{n-1}) (\\lambda_4 - x_{n-1}).$$\nThus,  after straightforward calculations,  we come out with the quadratic equation\n$$Ay^2+ By+ C=0$$\nin variable $y=  (\\lambda_3 - x_{n-1}) \/  (\\lambda_4 - x_{n-1})$ with coefficients $A= r_3r_2^2- r_3^2(n^2-n-r_2),\\\nB= - 2 r_3r_4 (n^2-n-r_2),\\ C= r_4r_2^2- r_4^2(n^2-n-r_2).$   But, it is easy to verify that $B^2-4AC >0.$ Therefore the quadratic\nequation has two distinct real roots.  Writing $\\lambda_3 - x_{n-1}= y (\\lambda_4 - x_{n-1})$ and substituting into (23), we obtain\n$$ (n^2- n- r_2) (x_{n-1} - x_{n-2})^2= ( r_{3}y^2+ r_4) (\\lambda_4 - x_{n-1})^2.$$\nAt the same time, since $y\\neq 0$,  we have $  \\lambda_4 - x_{n-1}=\ny^{-1}  (\\lambda_3 - x_{n-1})$ and\n$$y^2 (n^2- n- r_2) (x_{n-1} - x_{n-2})^2= ( r_{3}y^2+ r_4) (\\lambda_3 - x_{n-1})^2.$$\nHence,\n$$\\lambda_4= x_{n-1}  \\pm \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}|,$$\n$$\\lambda_3= x_{n-1}  \\pm |y| \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}}\\   |x_{n-1} - x_{n-2}|.$$\nConsequently,\n$$\\lambda_4- \\lambda_3 =  \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}| ( 1-|y|)=\n - \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}| ( 1+|y|)$$\n $$=  \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}| ( 1+|y|)=\n \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}| (|y|-1),$$\nwhich are possible only in the case $x_{n-1}=x_{n-2},$\\  $\\lambda_3=\\lambda_4$. Thus we get a contradiction with Corollary 1 and complete the proof.\n\\end{proof}\n\nIn the same manner we prove\n\n{\\bf Corollary 6.} {\\it There exists no non-trivial polynomial $f$ of degree $n \\ge 5$ with only real roots, having five   distinct zeros  and sharing  roots  with its $n-2$-th and $n-1$-th derivatives. }\n\n\\begin{proof}  Assuming its existence, it has  the roots  $\\lambda_1= x_{n-1}, \\   \\lambda_2= x_{n-2}$,  $\\lambda_3= 2x_{n-1}- x_{n-2}, \\  \\lambda_4$ and $\\lambda_5$ of multiplicities $r_j, \\  j=1,2,3,4, 5$, respectively. Hence\n$$  (n^2- n- r_2-r_3) (x_{n-1} - x_{n-2})^2=  r_{4}(\\lambda_4 - x_{n-1}) ^2+   r_{5}(\\lambda_5 - x_{n-1}) ^2 .$$\nTherefore  using similar ideas as in the proof of Corollary 5, we come out again to the contradiction.\n\n\\end{proof}\n\nFor general number of distinct zeros we establish the following\n\n{\\bf Corollary 7.} {\\it There exists no non-trivial polynomial $f$ of degree $n$ with only real roots, having $k \\ge 2$   distinct zeros of multiplicities $(2)$ $r_j,\\  j=1,\\dots, k$ and  among them all roots of $f^{(m)}$ for some  $m$, satisfying   the relations\n$$ r \\le m <  {1\\over 2}\\left(1-{1\\over r_0}\\right)(n-1),\\eqno(24)$$\nwhere $r,\\  r_0$ are maximum and minimum multiplicities of roots of $f$.}\n\n\\begin{proof}  In fact, as a consequence of (16) we have the identity \n$${(n-m)(n-m -1)\\over n(n-1)}  \\sum_{j=1}^k r_{j}(\\lambda_j- x_{n-1})^2 =  \n \\sum_{j=1}^{n-m} (\\xi^{(m)}_j - x_{n-1})^2\\eqno(25)$$\nfor some $m$,  satisfying condition (24).  Hence, since $m\\ge r$,  it has $n-m \\le k-2$ and $\\xi^{(m)}_j= \\lambda_{m_j}, \\  m_j \\in \\{1,\\dots, k\\}, \\  j= 1,\\dots, n-m$  are simple roots of $f^{(m)}$.  Thus  we find \n$$ \\sum_{j=1}^{n-m} \\left[ r_{m_j} {(n-m)(n-m -1)\\over n(n-1)} -1\\right]  (\\lambda_{m_j}- x_{n-1})^2 + \n{(n-m)(n-m -1)\\over n(n-1)}   \\sum_{j=n-m+1}^k r_{m_j}(\\lambda_{m_j} - x_{n-1})^2 =  0.$$\nBut, owing to condition (24) \n$$r_{m_j} {(n-m)(n-m -1)\\over n(n-1)} -1 \\ge  r_{0} {(n-m)(n-m -1)\\over n(n-1)} -1 \\ge 0, \\ j=  1,\\dots, n-m.$$\nIndeed, we have from the latter inequality \n$$m \\le n- {1\\over 2} - \\sqrt{\\frac{n^2-n}{r_0} + {1\\over 4}}$$\nand, in turn,\n$$ n- {1\\over 2} - \\sqrt{\\frac{n^2-n}{r_0} + {1\\over 4}}=  \\frac{2 (1-r_0^{-1}) (n^2-n)}\n{ 2n- 1  + \\sqrt{4(n^2-n)r^{-1}_0 + 1}}\\ge \\frac{ (1-r_0^{-1}) (n^2-n)}\n{ 2n- 1} >  {1\\over 2}\\left(1-{1\\over r_0}\\right)(n-1).$$\nTherefore  $\\lambda_j= x_{n-1}, \\ j=1,\\dots, k$ and this contradicts to the fact that all roots are distinct.  \n\\end{proof}\n\n\n\nFinally, in this section, we will employ identities (17) to prove an analog of the Obreshkov- Chebotarev theorem for multiple roots (see \\cite{Rah}, Theorem 6.4.3), involving estimates for smallest and largest of distances between consecutive zeros of polynomials and their derivatives.  Namely, it has\n\n{\\bf Theorem 2.} {\\it  Let $f$ be a polynomial of degree $n >  2$ with only real zeros. Denote the largest and the smallest of the distances between consecutive zeros of $f$ by $\\Delta$ and $\\delta$, respectively.  Denoting the corresponding quantities associated with $f^{(m)}, \\ m=1,2,\\dots,\\  n-2$ by  $\\Delta^{(m)}$ and $\\delta^{(m)}$, the following inequalities take place\n\n$$\\delta^{(m)}  \\le \\Delta \\  {rk \\over n} \\  \\sqrt{  \\frac{k^2-1}{ (n-m+1)(n-1)}},\\eqno(26)$$\n$$\\delta \\  {r_0 k \\over n} \\  \\sqrt{  \\frac{k^2-1}{ (n-m+1)(n-1)}}\\le  \\Delta^{(m)} ,\\eqno(27)$$\n$$\\delta \\  {r_0 k \\over 2 n} \\  \\sqrt{  \\frac{k^2-1}{ 3 (n-1)}}\\le |x_{n-1} - x_{n-2}| \\le\n \\Delta \\  {rk \\over 2 n} \\  \\sqrt{  \\frac{k^2-1}{ 3(n-1)}},\\eqno(28)$$\nwhere $r_0,\\  r$ are minimum and maximum multiplicities of roots of $f$, respectively,  and $k \\ge 2$ is a number of distinct roots.}\n\n\\begin{proof}  Following similar ideas as in the proof of Theorem 6.4.3 in \\cite{Rah},  we assume distinct roots of $f$ in the increasing order and roots of its  $m$-th derivative in the non-decreasing order, and taking the second identity in (17), we deduce\n$$ {[\\delta^{(m)}]^2 \\over (n-m)^2(n-m -1)} \\sum_{1\\le j < s \\le n-m} (s-j)^2 \\le {[\\Delta r]^2  \\over n^2(n-1)} \\sum_{1\\le j < s \\le k} (s-j)^2.$$\nHence, minding the value of the sum\n$$ \\sum_{1\\le j < s \\le q} (s-t)^2 = {1\\over 12} q^2(q^2-1),$$\nafter simple manipulations we arrive at the inequality (26).  In the same manner (cf. \\cite{Rah}) we establish inequalities (27), (28), basing Sz.-Nagy type identities (17). \n\\end{proof}\n\n\n\n\\section{Laguerre's  type inequalities }\n\nIn 1880 Laguerre proved his famous theorem for polynomials with only\nreal roots, which provides their localization with  upper and lower\nbounds (see details in \\cite{Rah}).  Precisely,  we have the following\nLaguerre inequalities\n$$x_{n-1}-  (n-1)\\left|x_{n-1}- x_{n-2}\\right| \\le w_j \\le  x_{n-1} + (n-1)\\left|x_{n-1}- x_{n-2}\\right|, \\\n j=1,\\dots, n,$$\nwhere $w_j$ are  roots of the polynomial $f$ of degree $n$ and $x_{n-1}, \\  x_{n-2}$ are roots of  $f^{(n-1)},\\ f^{(n-2)}$, respectively.   First  we prove an analog of the Laguerre inequalities  for multiple roots.\n\n{\\bf  Lemma 3.}  {\\it Let $f$ be a polynomial with only real roots of degree  $n \\in \\mathbb{N}$,  having $k$  distinct roots\n $\\lambda_j, \\ j=1,\\ \\dots, k$ of multiplicities $(2)$ and $x_{n-1}, \\  x_{n-2}$ be roots of  $f^{(n-1)},\\ f^{(n-2)}$, respectively.\n  Then the following Laguerre   type inequalities  hold}\n$$x_{n-1}-  \\sqrt{\\frac{(n-r_j)(n-m-1)}{r_j-m} }\\left|x_{n-1}- x_{n-2}\\right| \\le \\lambda_j \\le  x_{n-1} +\n\\sqrt{\\frac{(n-r_j)(n-m-1)}{r_j-m} }\\left|x_{n-1}- x_{n-2}\\right|,\\eqno(29)$$\nwhere $ j=1,\\dots, k, \\  m= 0,1,\\dots, r_j-1.$\n\n\\begin{proof} In fact, appealing to the  Sz.-Nagy type identities (15), (16) and the Cauchy -Schwarz inequality, we find\n$$   (x_{n-1} - x_{n-2})^2={1\\over (n-m)(n-m-1)} \\left[ \\sum_{s=1}^{n-m} (\\xi^{(m)}_s-  \\lambda_j)^2-  (n-m) (x_{n-1} -\n\\lambda_j)^2\\right]$$$$ \\ge  {1\\over (n-m)(n-m-1)} \\left[\n\\frac{1}{n-r_j} \\left( \\sum_{s=1}^{n-m} (\\xi^{(m)}_s-\n\\lambda_j)\\right)^2-  (n-m) (x_{n-1} - \\lambda_j)^2\\right]$$$$=\n\\frac{r_j-m} {(n-r_j)(n-m-1)} \\left(x_{n-1}- \\lambda_j\\right)^2, \\\nm= 0,1,\\dots, r_j-1,$$ which yields (29).\n\n\\end{proof}\n\nAs a corollary we improve the Laguerre inequality (28) for multiple roots.\n\n\n{\\bf Corollary 8.} {\\it  Let $f$ be a polynomial with only real roots of degree  $n \\in \\mathbb{N}$.\nThen the multiple zero $\\lambda_j$ of multiplicity $r_j\\ge 1, \\   j=1,\\dots, k$ lies in the interval}\n$$\\left[ x_{n-1}-  \\sqrt{\\left(\\frac{n}{r_j}-1\\right)(n-1) }\\left|x_{n-1}- x_{n-2}\\right|, \\quad   x_{n-1} +\n\\sqrt{\\left(\\frac{n}{r_j}-1\\right)(n-1) }\\left|x_{n-1}- x_{n-2}\\right| \\right].\\eqno(30)$$\n\n\n\\begin{proof}  Indeed, the fraction $\\frac{(n-r_j)(n-m-1)}{r_j-m}$ attains its minimum value, letting $m=0$ in (29).\n\\end{proof}\n\n{\\bf Remark 3.}  When all roots are simple, the latter interval\ncoincides with the one generated by (28).\n\nA localization of roots of the  $m$-th derivative $f^{(m)}, \\ m=0,1,\\dots,  n-2$ is given by\n\n{\\bf  Lemma 4.}  {\\it Roots of the $m$-th derivative $f^{(m)}, \\ m=0,1,\\dots,  n-2$ satisfy the following Laguerre   type inequalities}\n$$x_{n-1}-  (n-m-1)\\left| x_{n-1}- x_{n-2}\\right| \\le \\xi^{(m)}_\\nu \\le\n x_{n-1} +  (n-m-1)\\left| x_{n-1}- x_{n-2}\\right|,\\eqno(31)$$\nwhere   $\\nu=1,\\dots, n-m.$\n\n\\begin{proof} Similarly to the proof of Lemma 3, we employ  the  Sz.-Nagy type identities (15), (16) and the Cauchy -Schwarz inequality to deduce\n$$   (x_{n-1} - x_{n-2})^2={1\\over (n-m)(n-m-1)} \\left[ \\sum_{s=1}^{n-m} (\\xi^{(m)}_s-  \\xi^{(m)}_\\nu)^2-  (n-m) (x_{n-1} -\n\\xi^{(m)}_\\nu)^2\\right]$$$$ \\ge  {1\\over (n-m)(n-m-1)} \\left[\n\\frac{1}{n-m-1} \\left( \\sum_{s=1}^{n-m} (\\xi^{(m)}_s-\n\\xi^{(m)}_\\nu)\\right)^2-  (n-m) (x_{n-1} -\n\\xi^{(m)}_\\nu)^2\\right]$$$$= \\frac{1}{(n-m-1)^2} \\left(x_{n-1}-\n\\xi^{(m)}_\\nu\\right)^2, \\  m= 0,1,\\dots, n-2.$$ Thus we come out\nwith (31) and complete the proof.\n\n\\end{proof}\n\nWhen $x_{n-1}=\\lambda_1$ be in common with $f$ of multiplicity $r_1$, we have\n\n{\\bf  Lemma 5.}  {\\it Let $f$ be a polynomial with only real roots\nof degree $n \\ge 2$ and $x_{n-1}=\\lambda_1$ be a  common zero with\n$f$ of multiplicity $r_1$, having $k \\ge 2$  distinct roots $\\lambda_j$ of multiplicities\n$r_j, j=1,\\dots, k$. Then the following Laguerre type inequalities\nhold}\n$$x_{n-1}-   \\sqrt{\\left({1\\over r_s}- {1\\over n-r_1}\\right) (n^2-n)}\\left| x_{n-1}- x_{n-2}\\right| \\le \\lambda_s \\le  x_{n-1}\n $$$$+  \\sqrt{\\left({1\\over r_s}- {1\\over n-r_1}\\right) (n^2-n)}\\left| x_{n-1}- x_{n-2}\\right|,\\eqno(32)$$\nwhere   $s=2,\\dots, k.$\n\n\\begin{proof} In the same manner  we involve   the  first Sz.-Nagy type identity in  (15) with $z=  \\lambda_s$,  which can be written in the form\n$$(n-r_1) (x_{n-1} - \\lambda_s)  =  \\sum_{j=2}^k r_{j}(\\lambda_j- \\lambda_s).$$\nHence squaring  both sides of the latter equality and appealing to the Cauchy -Schwarz inequality, we derive by virtue of  (16)\n$$(n-r_1)^2 (x_{n-1} - \\lambda_s)^2  = \\left( \\sum_{j=2}^k r_{j}(\\lambda_j- \\lambda_s)\\right)^2$$\n$$\\le (n-r_1-r_s)\\sum_{j=2}^k r_{j}(\\lambda_j- \\lambda_s)^2=  (n-r_1-r_s)\\left[ (n^2-n) ( x_{n-1}- x_{n-2})^2 +\n(n-r_1)(x_{n-1}- \\lambda_s)^2\\right].$$\nThus after simple calculations we easily arrive at (32).\n\n\\end{proof}\n\n\n{\\bf Remark 4}. Inequalities (27) are sharper than the corresponding relations, generated by interval (30).\n\n\nThe following result gives a Laguerre type localization for common roots of a possible CA-polynomial with only real roots and its $m$-th derivative.\n\n\n{\\bf Lemma 6.} {\\it     Let $f$ be a CA-polynomial of degree $n \\ge\n2$ with only real distinct zeros of multiplicities $(2)$,   including common\nroots $x_{n-1}=\\lambda_1$ of its $n-1$-th derivative and $x_m$ of\nits $m$-th derivative,   $m=  r, r+1, \\dots, n-2$, where  $r=\n\\hbox{max}_{1\\le j\\le k} (r_j)$. Then the following Laguerre type\ninequality holds\n$$ \\frac{n-r_1- r_{k_m}}{(n-r_1)^2}\n\\left(n^2-r_1+ (n-r_1)(n-m) (n-m-2)\\right) (x_{n-1}-  x_{n-2})^2 \\ge  (x_{n-1}- x_{m})^2,\\eqno(33)$$\nwhere $x_{n-2}$ is a root of $f^{(n-2)}$ and $r_{k_m}$ is the multiplicity of  $x_m$ as a root of $f$}.\n\n\\begin{proof}   Appealing  again to  Sz.-Nagy's type  identities (15), (16) with $z=x_m$, inequality (31)  and the Cauchy-Schwarz inequality, we  find\n$$   (x_{n-1} - x_{n-2})^2={1\\over n(n-1)} \\left[ \\sum_{j=2}^k r_{j}(\\lambda_j- x_m)^2-  (n-r_1) (x_{n-1} -\nx_m)^2\\right]$$$$ \\ge {1\\over n(n-1)} \\left[ \\sum_{j=2}^k\nr_{j}(\\lambda_j- x_m)^2- (n-r_1)(n-m-1)^2(x_{n-1}-  x_{n-2})^2\\right]$$\n$$ \\ge {1\\over n(n-1)} \\left[ {1\\over n-r_1- r_{j_m}} \\left(\\sum_{j=2}^k\nr_{j}(\\lambda_j- x_m)\\right)^2 -  (n-r_1)(n-m-1)^2(x_{n-1}-  x_{n-2})^2\\right]$$\n$$= {n-r_1\\over n(n-1)} \\left[ {n-r_1\\over n-r_1- r_{j_m}} (x_{n-1}- x_m)^2 - (n-m-1)^2(x_{n-1}-  x_{n-2})^2\\right].$$\nHence, making straightforward calculations,  we  derive (33), completing the proof of Lemma 6.\n\n\\end{proof}\n\n\nLet us denote by $d,\\ d^{(m)},\\  D, D^{(m)}$ the following values\n$$   d=  \\hbox{min}_{2\\le j\\le k}  |\\lambda_j-   x_{n-1}|, \\quad\n  d^{(m)}=  \\hbox{min}_{1\\le j\\le n-m}   |\\xi^{(m)}_j -   x_{n-1}|,\\eqno(34)$$\n$$   D=  \\hbox{max}_{2\\le j\\le k}  |\\lambda_j-   x_{n-1}|, \\quad\n  D^{(m)}=  \\hbox{max}_{1\\le j\\le n-m}   |\\xi^{(m)}_j -   x_{n-1}|,\\eqno(35)$$\nand by\n$$\\hbox{span} (f) = \\lambda^*- \\lambda_*,$$\nwhere\n$$  \\lambda^*= \\hbox{max}_{1\\le j\\le k} (\\lambda_j), \\quad     \\lambda_*= \\hbox{min}_{1\\le j\\le k} (\\lambda_j)$$\nare roots of $f$ of multiplicities $r^*, \\ r_*$, respectively.   It has the properties  $ D^{(m+1)}  \\le  D^{(m)}\\le D$ and\n  (cf. \\cite{Rah})  $\\hbox{span}(f^{(m+1)}) \\le \\hbox{span}(f^{(m)}) \\le \\hbox{span}(f)$, where $\\hbox{span}(f^{(m)})$\n   is the span of the $m$-th derivative.  Moreover, the strict inequalities $ D^{(m)}  <  D$,\\    $\\hbox{span}(f^{(m)}) < \\hbox{span}(f)$   hold when $m$ is sufficiently large.\n\n  {\\bf Lemma 7}. {\\it Let $x_{n-1}=\\lambda_1, \\  x_{n-2}=\\lambda_2$ be  common roots of $f$  with its $n-1$-th, $n-2$-th derivatives, respectively,  of multiplicities $r_1, r_2$ as roots of $f$,  and the maximum distance $D$ (see $(35)$) be attained at the root $\\lambda_{s_0}, \\ s_0 \\in \\{ 3,\\dots, k\\},\\ k \\ge 3$   of $f$ of multiplicity $r_{s_0}$.   Then the following inequalities hold }\n  %\n    $$\\sqrt{\\frac{n^2-n- r_2}{n-r_1-r_2}}\\  \\left|x_{n-1} - x_{n-2}\\right|   \\le D \\le \\sqrt{\\frac{n^2-n-r_2}{r_{s_0}}}\n     \\left|x_{n-1} - x_{n-2}\\right|,\\eqno(36)$$\n %\n  $${1\\over 2} \\sqrt{ {r_{s_0} \\over 3(n-r_1) }\\left(5 + \\frac{ r_2} {n^2-n-r_2}\\right)}\\hbox{span}(f) \\le D \\le \\sqrt{{1\\over n-r_1} \\left[ n-r_1- {r_{s_0} \\over 4}\\left(5 + \\frac{ r_2} {n^2-n-r_2}\\right)\\right] } \\hbox{span}(f).\\eqno(37)$$\n \n\\begin{proof}  In order to establish (36), we employ identities (16) and under condition of the lemma we write \n$$ (n^2-n- r_2)(x_{n-1} -   x_{n-2})^2 = \\sum_{j=3}^k r_{j}(\\lambda_j- x_{n-1})^2 \\le (n-r_1-r_2) D^2.$$\nSince $n > r_1+r_2$ and $x_{n-2}\\neq \\lambda_{s_0}$ (otherwise $f$ is trivial, because equalities $x_{n-2}= \\lambda_{s_0}= \\lambda^*$ or  $x_{n-2}= \\lambda_{s_0}= \\lambda_*$ mean that the maximum multiplicity $r > n-2$,  and we appeal to Corollary 3),  we come up with the lower  bound (36) for $D$.  The lower bound comes immediately from the estimate \n$$(n^2-n- r_2)(x_{n-1} -   x_{n-2})^2 = \\sum_{j=3}^k r_{j}(\\lambda_j- x_{n-1})^2    \\ge  r_{s_0} D^2.$$\nNow,  since $2D \\ge  \\hbox{span}(f)$, we find from (36)\n$$\\hbox{span}(f) \\le 2 \\sqrt{\\frac{n^2-n-r_2}{r_{s_0}}} \\  \\left|x_{n-1} - x_{n-2}\\right|.$$\nHence,  since  $D = \\hbox{max} \\left(|\\lambda^*- x_{n-1}|, \\ |\\lambda_*- x_{n-1}|\\right)$, the\n$n-2$-th derivative has roots $x_{n-2}$ and $2x_{n-1}- x_{n-2}$ and\n$\\hbox{span}(f)= D + \\Lambda$, where $\\Lambda = \\hbox{min}\n\\left(|\\lambda^*- x_{n-1}|, \\ |\\lambda_*- x_{n-1}|\\right)$, we\nappeal to the first equality in (16), letting $z= \\lambda_{s_0}$ and writing it in the form\n$$ (n-r_1)(x_{n-1} -  \\lambda_{s_0})^2 = \\sum_{j=2}^k r_{j}(\\lambda_j- \\lambda_{s_0})^2 - n(n-1)(x_{n-1} -\nx_{n-2})^2.$$\nTherefore,\n$$(n-r_1)D^2 \\le \\left[ n-r_1- {5\\over 4} r_{s_0} - \\frac{r_{s_0} r_2} {4(n^2-n-r_2)}\\right]  [\\hbox{span}(f)]^2$$ \nand we establish the upper bound (37) for $D$. On the other hand $\\hbox{span}(f)= D+ \\Lambda$. So,\n$$D^2 \\le \\left(1 -   {r_{s_0} \\over 4(n-r_1) }\\left(5 + \\frac{ r_2} {n^2-n-r_2}\\right)\\ \\right) \\left(D^2 + \\Lambda^2 +\n2D\\Lambda\\right)$$ and we easily come out with the lower bound (37)\nfor $D$, completing the proof of Lemma 7.\n\n\\end{proof}\n\n{\\bf Lemma 8}. {\\it Let $x_{n-1}=\\lambda_1, x_{n-2}=\\lambda_2$ be\ncommon roots of $f$ with its $n-1$-th, $n-2$-th   derivatives of\nmultiplicities $r_1, r_2,\\ r_1+r_2 < n$, respectively. Then  we\n have the following lower bound for $\\hbox{span}(f)$}\n  %\n  $$\\hbox{span}(f)\\ge \\sqrt{\\frac{n^2-r_1}{n-r_1-r_2}}\\ |x_{n-1}-x_{n-2}|.\\eqno(38)$$\n\n\\begin{proof} Indeed,  identities  (16) with $z=x_{n-2}$ yield\n$$(n^2-r_1)(x_{n-1}-x_{n-2})^2=\\sum_{j=3}^k r_{j}(\\lambda_j- x_{n-2})^2$$\nand we derive\n$$(n^2-r_1)(x_{n-1}-x_{n-2})^2\\le (n-r_1-r_2)[\\hbox{span}(f)]^2,$$\nwhich implies (38).\n\\end{proof}\n\nNext, we establish an analog of Lemma 5 for roots of derivatives.   Precisely, it has\n\n {\\bf Lemma 9}. {\\it Let  $x_{n-1},  \\  x_{n-2}$ be roots of the $n-1$-, $n-2$-th   derivatives of $f$, respectively.  Then\n %\n $$D^{(m)} \\ge \\sqrt{n-m-1}\\  |x_{n-1}-x_{n-2}|,\\eqno(39)$$\n where $m \\in \\{r, r+1,\\dots, n-2 \\}, \\   r= \\hbox{max}_{1\\le j\\le k} (r_j).$  Besides, if $x_{n-1}$ is a root of $f^{(m)}$, then\n we have a stronger inequality $$D^{(m)} \\ge \\sqrt{n-m}\\  |x_{n-1}-x_{n-2}|.\\eqno(40)$$\nMoreover,\n  %\n  $$2\\  D^{(m)} \\ge \\hbox{span}(f^{(m)}) \\ge   \\frac{n-m}{n-m-1}\\   D^{(m)}.\\eqno(41)$$\n  and if $x_{n-1}$ is a  root of  $f^{(m)}$,  it becomes\n   $$2\\  D^{(m)} \\ge \\hbox{span}(f^{(m)}) \\ge  \\sqrt{ \\frac{(n-m)(n-m-1)+1}{(n-m-1)(n-m-2)}}\\   D^{(m)},\\eqno(42)$$\nwhere  $m \\in \\{r, r+1,\\dots, n-3 \\}.$}\n\\begin{proof}  In fact,  since (see (16))\n$$(n-m)(n-m-1)(x_{n-1} - x_{n-2})^2= \\sum_{j=1}^{n-m} (\\xi^{(m)}_{j} -  x_{n-1})^2 \\le (n-m) \\left[ D^{(m)}\\right]^2,$$\nwe get (39). Analogously, we immediately come out with (40), when $x_{n-1}$ is a root of $f^{(m)}$, because one element of the sum of squares is zero.  In order to prove (41), we appeal again to (16), letting  $z= \\xi^{(m)}_{s_0}, \\ s_0 \\in \\{1,2, \\dots,  n-m\\}$,  $m \\in \\{r, r+1,\\dots, n-2 \\}, \\   r= \\hbox{max}_{1\\le j\\le k} (r_j)$, which is a root of  $f^{(m)}$, where the maximum $D^{(m)}$ is attained.  Hence owing to Laguerre type inequality (31)\n$$(n-m)\\left[D^{(m)}\\right]^2\\le (n- m-1) [\\hbox{span}(f^{(m)})]^2 - \\frac{n-m}{n-m-1} \\left[D^{(m)}\\right]^2,$$\nwhich drives to  the lower bound for $ \\hbox{span}(f^{(m)})$ in   (41).   The upper bound is straightforward\nsince $x_{n-1}$ belongs to the smallest interval containing roots of $f^{(m)}$.  In the same manner we establish (42),  since in this case\n$$(n-m-1)\\left[D^{(m)}\\right]^2\\le (n- m-2) [\\hbox{span}(f^{(m)})]^2 - \\frac{n-m}{n-m-1} \\left[D^{(m)}\\right]^2.$$\n\n\\end{proof}\n\n{\\bf Remark 5}.  The case $m=n-2$ gives  equalities in (39), (41).  Letting the same value of $m$ in (40), we easily get\na contradiction, which means that the only trivial polynomial is within polynomials with only real roots,\nwhose derivatives $f^{(n-2)}, \\  f^{(n-1)}$ have a common root (see Corollary 1).\n\n\n\n\\section{Applications to the Casas- Alvero conjecture}\n\nIn this final section we will discuss properties of possible CA-polynomials,  which share roots with each of their non-constant derivatives. We will investigate particular cases of the Casas-Alvero conjecture, especially for polynomials with only real roots, showing when it holds true or, possibly, is false. \n\nWe begin with \n\n\n{\\bf Proposition 1}. {\\it The Casas-Alvero conjecture holds true, if and only if  it is true for common roots $\\{z_\\nu \\}_0^{n-1}$ lying in the unit circle.}\n\n\\begin{proof}   The necessity is trivial. Let's l prove the sufficiency. Let the conjecture be true for common roots $\\{z_\\nu \\}_0^{n-1}$ of a complex polynomial $f$ and its non-constant derivatives, which lie in the unit circle.  Associating with $f$ an Abel-Goncharov polynomial $G_n$ (6),    one can choose an arbitrary $\\alpha  >0$ such that $\\ |z_\\nu| < \\alpha^{-1}, \\   \\nu=0, 1, \\dots , n-1. $  Hence owing to  (7)\n$$ f (\\alpha z_\\nu) = G_n\\left(\\alpha z_0, \\alpha z_\\nu, \\alpha z_1,\\dots, \\alpha z_{n-1}\\right)\n= \\alpha^n G_n(  z_\\nu) = \\alpha^n f (  z_\\nu)=  0,\\  \\nu=0, 1,\\dots , n-1,$$\nand\n$$f^{(\\nu)} _n(\\alpha z ) =  n! {d^{\\nu}\\over d z^{\\nu}}\n\\int_{\\alpha z_0}^{\\alpha z}  \\int_{\\alpha z_1}^{s_{1}} \\dots \\int_{\\alpha z_{n-1}}^{s_{n-1}} d s_{n} \\dots d s_{1}\n=n!  \\alpha ^\\nu \\int_{\\alpha z_\\nu}^{\\alpha z}  \\int_{\\alpha z_{\\nu+1}}^{s_{\\nu+ 1}} \\dots \\int_{\\alpha z_{n-1}}^{s_{n-1}} d s_{n} \\dots d s_{\\nu+1},$$\nwe find $f^{(\\nu)} _n(\\alpha z_\\nu ) =0$.   Hence  $\\alpha z_\\nu, \\  \\nu=0, 1,\\dots , n-1$ are common\nroots of  $\\nu$-th derivatives  $f^{(\\nu)}$ and $f$,  lying in the unit circle.   Consequently,  since via assumption the\nCasas-Alvero conjecture is true when common roots are inside the unit circle, we have that  $f$ is trivial  and $z_0=z_1=\\dots = z_{n-1} = a$ is a unique joint root of $f$ of the multiplicity $n$.  Proposition 1  is proved.\n\\end{proof}\n\n\n\nThe following lemma will be useful in the sequel.\n\n{\\bf Lemma 10}. {\\it Let  $f$ be a  CA-polynomial with only real roots of degree $n \\ge 2$ and  $\\{x_\\nu\n\\}_{0}^{n-1}$ be a sequence of common  roots of $f$ and the corresponding derivatives $f^{(\\nu)}$.  \nLet  $f^{(s+\\nu)} (x_\\nu) \\ge 0, \\  s =1,2,\\dots, n-\\nu-1$ and $\\nu=0,1,\\dots, n-1$. Then $x_\\nu$ is a\nmaximal root of the derivative  $f^{(\\nu)}$.}\n\n\\begin{proof}  In fact,  the proof is an immediate consequence of  expansion (12), where we let $G_n(x)=f(x)$.  Indeed,  $f^{(\\nu)} (x_\\nu) =0,   \\nu=0,1,\\dots, n-1$ and when $ x > x_\\nu$   we have from (12) $f^{(\\nu)} (x) > 0,   \\nu=0,1,\\dots, n-1$.   So,  this  means that there is no roots, which are bigger than $x_\\nu$. This completes  the proof of Lemma 10.\n\\end{proof}\n\n\n{\\bf Proposition  2}. {\\it  Under conditions of Lemma 10  the Casas-Alvero conjecture holds true for polynomials with only real roots.}\n\n\\begin{proof}  We will show that under conditions of Lemma 10 there exists no CA-polynomial $f$ with only real roots.  Indeed, assuming its existence,  we find via conditions of  the lemma that the root $x_0$ is a maximal zero  of $f(x)$. This means that $x_0 \\ge x_1$. On the other hand, classical Rolle's theorem states that between zeros $x_0, \\  x_1$\nin the case $x_0 > x_1$ there exists at least one zero of the derivative $f^{(1)} (x)$, say $\\xi_1^{(1)}$,\n which is bigger than $x_1$.  But this is impossible because $x_1$ is a maximal zero of the first derivative.\n  Thus $x_0=x_1\\ge x_2$.   Then   between $x_1$ and $x_2$ in the case $x_1 > x_2$ there exists a zero $\\xi_2^{(1)}$\n   of the first derivative such that $x_1>  \\xi_2^{(1)}  > x_2$.  Hence between $x_1$ and   $\\xi_2^{(1)}$ there exists at least one     zero of the second derivative, which is bigger than $x_2$. But this is impossible, since $x_2$ is a maximal zero of $f^{(2)} (x)$.      Therefore   $x_0=x_1=x_2$.   Continuing this process we observe that the sequence  $\\{x_\\nu \\}_0^{n-1}$ is stationary and $f$ has a unique joint root, which contradicts the definition of the CA-polynomial. \n\\end{proof}\n\n{\\bf Corollary 9}. {\\it   There exists no CA-polynomial $f$ with only real roots, having non-increasing sequence $\\{x_\\nu \\}_0^{n-1}$ of roots in common with $f$ and its non-constant derivatives.}\n\n\\begin{proof}    Obviously,  via (13) $f^{(s+\\nu)} (x_\\nu) \\ge 0, \\  s=1,2,\\dots, n-\\nu-1$ and conditions of Lemma 10 are satisfied.\n\\end{proof}\n\n\n{\\bf Corollary 10}. {\\it   There exists no CA-polynomial $f$ with only real roots, such that each $x_\\nu$ in the  sequence $\\{x_\\nu \\}_0^{n-1}$  is a maximal root of the derivative  $f^{(\\nu)} (x),  \\  \\nu=0,1,\\dots, n-1$.}\n\n\\begin{proof}    The proof is similar to the proof of Proposition  2.\n\\end{proof}\n\nAn immediate consequence of Corollaries 3,4,5 is \n\n{\\bf Corollary 11}. {\\it   The CA-polynomial, if any,   with only real roots has at least 5 distinct zeros. }\n\n\n\n Let us  denote by $l(m)$ the  number of distinct roots of the $m$-th derivative $f^{(m)},\\ m=0, 1,\\dots, n-2$,   which are in common with $f$ and different from $\\lambda_1=x_{n-1}$, which is a common root with $f^{(n-1)}$,   i.e.  the $m$-th derivative $f^{(m)}$ has $l(m)$ common roots with $f$\n\n$$\\lambda_{j_1}, \\dots,  \\lambda_{j_{l(m)}}  \\subseteq  \\{ \\lambda_2, \\lambda_3, \\dots, \\  \\lambda_k\\} $$\nof multiplicities\n$$r_{j_1}, \\dots,  r_{j_{l(m)}}  \\subseteq  \\{ r_2, r_3, \\dots, \\  r_k\\} .$$\nFor instance,  $l(0)= k-1, \\   l(1)= k-1-s$, where $s$ is a number\nof simple roots of $f$. So, we see that $n-m \\ge l(m) \\ge 0$ and\nsince $f$ is a  CA-polynomial,  $l(m)=0$ if and only if\n$x_{n-1}=\\lambda_1$ is the only common root of $f$ with $f^{(m)}$.\n\n\n\n{\\bf Lemma  11}. {\\it There exists no  CA-polynomial with only real roots, having the property\n$l(m)= l(m+1)=0$ for some $m \\in \\{r, r+1, \\dots,n-2\\},$ where $r= \\hbox{max}_{1\\le j\\le k} (r_j)$.}\n\n\\begin{proof} In fact, as we saw above, since all roots are real,  it follows that all roots of $f^{(m)}, \\  m \\ge r$ are simple, which contradicts equalities $l(m)= l(m+1)=0$.   Indeed, the latter equalities yield that $x_{n-1}$ is a multiple root of $f^{(m)}$. Therefore $r\\ge r_1 >  m+1\\ge r+1$, which is impossible.\n\\end{proof}\n\nFurther, as in Lemma 7 we involve the root  $\\lambda_{s_0}$ of multiplicity $r_{s_0}$,  and $D= |\\lambda_{s_0}- x_{n-1}|$ (see (35)). Thus $\\lambda_{s_0}= \\lambda_*$ or $\\lambda_{s_0}= \\lambda^*$ and, correspondingly,  $r_{s_0}=r_*$ or $r_{s_0}=r^*$. Hence, calling  Sz.-Nagy identities (15), we  let  $z= x_{n-1}$ and assume without loss of generality that $\\lambda_{s_0}= \\lambda^*$. Then  we obtain for  $m \\ge r$\n$$r_*(x_{n-1}- \\lambda_*) = r^*D + \\sum_{j=2, \\ r_j\\neq r_*,\\  r^*}^{k} r_j(\\lambda_j -  x_{n-1}) \\ge  r^*D - D^{(m)}\n \\sum_{s=1}^{l(m)} r_{j_s} -   D^{(m+1)} \\sum_{s=1}^{l(m+1)} r_{l_s} $$\n$$- \\left(n-r_1-  r^*- r_*-  \\sum_{s=1}^{l(m)} r_{j_s}-  \\sum_{s=1}^{l(m+1)} r_{l_s}\\right)D.$$\nBut $x_{n-1}- \\lambda_*= \\hbox{span}(f)- D$.  Therefore,\n$$r_*  \\hbox{span}(f) + \\left(n-r_1- 2(  r^*+ r_*)\\right) D \\ge (D- D^{(m)})  \\sum_{s=1}^{l(m)} r_{j_s} +\n(D- D^{(m+1)})  \\sum_{s=1}^{l(m+1)} r_{l_s}.$$\nThe right-hand side of the latter inequality is, obviously,  greater or equal to $r_0 \\left(l(m)+ l(m+1)\\right) (D- D^{(m)}) $, where $1 \\le r_0= \\hbox{min}_{1\\le j\\le k} (r_j)$.  Moreover,  since $ \\hbox{span}(f) \\le 2D$,  the left-hand side does not exceed $\\left(n-r_1\\right) D-  r^* \\hbox{span}(f)$. Thus  we come out with the inequality\n$$r_0 \\left(l(m)+ l(m+1)\\right) (D- D^{(m)}) \\le \\left(n-r_1\\right) D-  r^* \\hbox{span}(f)$$\nor since $D- D^{(m)} > 0$ \\ ($m \\ge r$), it becomes\n$$l(m)+ l(m+1)\\le \\frac{\\left(n-r_1\\right) D-  r^* \\hbox{span}(f)}{r_0(D- D^{(m)}) }.\\eqno(43)$$\nMeanwhile, appealing to (16), we get similarly \n$$n(n-1) ( x_{n-1}- x_{n-2})^2 = r^*D^2 + r_* (\\lambda_*- x_{n-1})^2 +  \\sum_{j=2, \\ r_j\\neq r_*,\\  r^*}^{k}\n r_{j}(\\lambda_j-  x_{n-1})^2 $$$$\n\\le  r^*D^2 + r_*\\left(\\hbox{span}(f)- D\\right)^2+  \\left[D^{(m)}\\right]^2 \\sum_{s=1}^{l(m)} r_{j_s} +   \\left[D^{(m+1)}\\right]^2 \\sum_{s=1}^{l(m+1)} r_{l_s}$$$$+ \\left(n-r_1-  r^*- r_*-  \\sum_{s=1}^{l(m)} r_{j_s}-\n \\sum_{s=1}^{l(m+1)} r_{l_s}\\right)D^2.$$\n %\n Therefore, analogously to (43), we arrive at the inequality\n %\n $$l(m)+ l(m+1)\\le   \\frac{(n-r_1)D^2 + r_* \\left[\\hbox{span}(f)\\right]^2 - n(n-1) ( x_{n-1}- x_{n-2})^2 -2D r_*\\  \\hbox{span}(f)}{r_0(D^2- \\left[D^{(m)}\\right]^2) }.$$\n{\\bf Proposition 3}. {\\it There exists no CA- polynomial with only real roots of degree $n$ such that }\n$$  \\hbox{span}(f) >  \\left(r^*\\right)^{-1} \\left[ (n-r_1-r_0)D + r_0 D^{(m)}\\right],\\   m\\ge r.\\eqno(44)$$\n\\begin{proof}  Under condition (44), the right-hand side of (43) is less than one.  Thus $l(m)= l(m+1)=0$ and Lemma 11 completes the proof.\n\\end{proof}\nLet $m=n-2$.   Then since $l(n-1)=0$, inequality (43) becomes\n$$l(n-2) \\le  \\frac{ (n-r_1)D -  r^* \\hbox{span}(f)}{r_0(D-   \\left|x_{n-1}- x_{n-2}\\right|)}.\\eqno(45)$$\n{\\bf Proposition 4}. {\\it There exists no CA- polynomial with only real roots of degree $n$ such that }\n$$D <  \\left[r^* \\  \\sqrt{ \\frac{n^2-r_1}{n-r_1-r_2}} -r_0\\right] \\frac{ \\left|x_{n-1}- x_{n-2}\\right|}{n-r_1-r_0}.\\eqno(46)$$\n\n\\begin{proof}  Indeed, employing the lower bound (38) for $ \\hbox{span}(f)$, we find that under condition (46) the right-hand side of (45) is strictly less than one.  Consequently,  $l(n-2)=0$ and owing to Corollary 1 $f$ is trivial. If the maximum of multiplicities $r > n-2$,    $f$ has at most 2 distinct zeros and  it is trivial via Corollary 3.\n\\end{proof}\n\nFinally, we prove \n\n{\\bf Proposition 5}. {\\it Let  CA- polynomial with only real roots exist.  Then it has the property\n$$  \\frac{d}{D} \\le \\sqrt{\\frac{2(n-m-1)}{2(k-1)-1}},\\eqno(47)$$\nwhere $d, D$ are defined by $(34), (35)$, respectively,  and $m,\\ m+1$ belong to the interval $\\left[r, \\   {1\\over 2}\\left(1-{1\\over r_0}\\right)(n-1) \\right)$.}\n\\begin{proof}  Since $m,\\ m+1$ are chosen from  the interval $\\left[r, \\   {1\\over 2}\\left(1-{1\\over r_0}\\right)(n-1) \\right)$,  condition (24) holds for these values. Hence assuming the existence of the CA-polynomial, we return to the Sz.-Nagy type identity (25) to  have the estimate \n$$0 \\ge  l(m)  \\left( r_{0} {(n-m)(n-m -1)\\over n(n-1)} -1\\right) d^2 +  \\left(k-1-l(m)\\right) d^2- (n-m-l(m)) D^2$$\n$$\\ge  (k-1) d^2- (n-m) D^2  + l(m) (D^2-d^2).$$ \nWriting the same inequality for $m+1$\n$$0 \\ge  (k-1) d^2- (n-m-1) D^2  + l(m+1) (D^2-d^2)$$ \nand adding two inequalities, we find\n$$0 \\ge  2 (k-1) d^2- (2(n-m) -1) D^2  +(l(m)+  l(m+1)) (D^2-d^2),$$\nwhich means\n$$l(m)+  l(m+1) \\le \\frac{(2(n-m) -1) D^2 - 2 (k-1) d^2}{D^2-d^2}.$$\nSo, for the existence of the CA-polynomial it is necessary that the right-hand side of the latter inequality is more or equal to 1.  Thus we come out with condition (47) and complete the proof. \n\\end{proof}\n\n\n\n\n{\\bf  Acknowledgment}.   The present investigation was supported, in\npart, by the \"Centro de Matem{\\'a}tica\" of the University of Porto.\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA standard model (SM) like Higgs boson \\cite{higgs} (denoted as H(125) in this paper) was discovered at the Large Hadron Collider (LHC) in 2012.\nIn order to test the SM and discover possible physics beyond the SM (BSM),\nit is crucial to measure the Higgs Yukawa couplings and Higgs self couplings at the LHC and future high energy colliders.\nIn the first run of the LHC, the CMS and ATLAS has constrained $h\\bar t t$ Yukawa coupling indirectly through the global fit,\nwith a precision of  20$\\%$ and  30$\\%$  respectively \\cite{Khachatryan:2014jba,a}.\nWith 300\/fb, Yukawa couplings will be measured up to 23$\\%$, 13$\\%$  and 14$\\%$ for $h\\bar b b$, $h\\tau^+ \\tau^-$ and $h\\bar t t$\nrespectively \\cite{Peskin:2012we}.\nIt was also proposed to measure the top Yukawa coupling via the associated Higgs boson production with a single top quark\n\\cite{Barger:2009ky,Ellis:2013yxa,Biswas:2012bd,Biswas:2013xva,Farina:2012xp,Englert:2014pja,Chang:2014rfa}.\n The Higgs self coupling can be measured up to $50\\%$ at the LHC with 300\/fb \\cite{Peskin:2012we}.\nThere are extensive studies on measuring anomalous  triple Higgs coupling directly at the LHC \\cite{Baur:2002rb,Baur:2002qd,Baur:2003gp,Dolan:2012rv,Baglio:2012np} and\nfuture electron-positron colliders  \\cite{Baer:2013cma,Asner:2013psa}.\n\n\n\nFor the future high-luminosity electron-positron colliders, it is proposed to measure\nthe Higgs self coupling up to $28\\%$ for $\\sqrt{s_{e^+e^-}}=$ 240 GeV under the model-dependent assumption that only the Higgs self coupling is modified \\cite{McCullough:2013rea} .\nThe  precision of Higgs self coupling  can only be reached based on the precisely measured cross section of ZH associated production  up to $0.4\\%$ \\cite{Gomez-Ceballos:2013zzn}.\n Entering $e^+e^- \\rightarrow ZH$ via loops, the  triple Higgs coupling will be possibly polluted heavily by other anomalous couplings,\n and among them the dominant one is the $h-Z-Z$ coupling which appears even at tree-level.\n The first run results of LHC shows that the HVV couplings including $h-Z-Z$ coupling are consistent with those in the SM \\cite{Khachatryan:2014jba,a}.\n The Higgs-top coupling contributes to the process\n$e^+e^- \\rightarrow ZH$ via loops and is potentially important for triple Higgs coupling extraction.\nActually the full one-loop correction to $e^+e^- \\rightarrow ZH$ in the SM was calculated about two decades ago \\cite{Kniehl:1991hk,Denner:1992bc,Fleischer:1982af,Denner:1991ue}.\nIn this paper we will focus on the anomalous Higgs-top coupling, especially its effects on the extraction of  triple Higgs coupling.\n\nThis paper is arranged as following. In section II, we estimate the deviation of the cross section for the process $e^+e^- \\rightarrow ZH$ arising from anomalous Higgs-top coupling,\n and compare to that from triple Higgs coupling. In section III, we explore how to measure CP-violated Higgs-top coupling via the forward-backward asymmetry $A_{FB}$ for\nthe process $e^+e^- \\rightarrow ZH$. The last section contains our conclusion and discussion.\n\n\n\n\\section{ Pollution from Higgs-Top anomalous Coupling}\n\nIn the SM, the process $e^+e^- \\rightarrow ZH$ occurs at tree level and the Feynman diagram is shown in Fig. \\ref{fig1}.\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehz.pdf}\n\\caption{Feyman diagram at tree-level for the process $e^+ e^- \\rightarrow Zh$}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehz3h2.pdf}\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehz3h.pdf}\n\\caption{Feynman diagram containing the anomalous $3h$ coupling, depicted as the black dot, at one-loop level for the process $e^+ e^- \\rightarrow Zh$. }\n\\label{fig2}\n\\end{figure}\nIn order to measure the triple Higgs coupling, one way is to produce the Higgs pair, provided that the center of mass energy of $e^+e^-$ is high enough via $ e^+e^- \\rightarrow HHZ$ or\n$ e^+e^- \\rightarrow HH \\nu \\bar \\nu$ \\cite{Levy:2015fva}. For such processes, the cross sections are notorious small.\nHigh energy and high luminosity are both required. Another way to measure the\ntriple Higgs coupling is via the virtual effects which are shown in Fig. \\ref{fig2}.\nThe capacity of measuring triple Higgs has been estimated by ref \\cite{McCullough:2013rea}. For completeness we recalculate the analytical result for\n$$\\delta_{\\sigma}\\equiv \\frac{\\Delta \\sigma}{\\sigma}=\\frac{\\sigma_{\\delta_h \\ne 0}-\\sigma_{\\delta_h = 0}}{\\sigma_{\\delta_h = 0}}$$ from the triple Higgs coupling\n$C_{SM}(1+ \\delta_h) H H H=-3i \\frac{m^2_h}{v}(1+ \\delta_h) H H H $ as\n\\begin{equation}\n\\begin{split}\n  \\delta_{\\sigma}(3h) =\\frac{3 \\alpha m_h^2 \\delta_h} {16 \\pi \\beta c_w^2 s_w^2 m_z^2}\n                    Re\\Big[&2\\rho \\Big (C_1(m_h^2)+C_{11}(m_h^2)+C_{12}(m_h^2) \\Big)\\\\\n                    & -\\beta \\Big (B_0-4 C_{00}(m_h^2)+4 m_z^2C_0(m_h^2)+3m_h^2 B'_0 \\Big ) \\Big],\n\\end{split}\n\\end{equation}\nwhere $\\delta_h=0$ corresponds to the case in the SM.\nHere $$ \\beta=m_h^4-2 m_h^2 (m_z^2+s)+m_z^4+10 m_z^2 s+s^2,$$\n$$ \\rho=(m_h^2-m_z^2-s)\\left((m_h-m_z)^2-s\\right)\\left((m_h+m_z)^2-s\\right)$$\nThe definition of the one loop scalar functions B, C etc. can be found in Ref. \\cite{Ellis:2007qk} and\n$ C_0(m_h^2)=C_0(m_h^2,m_z^2,s,m_h^2,m_h^2,m_z^2)$,\n$ C(m_h^2)=C(m_h^2,s,m_z^2,m_h^2,m_h^2,m_z^2)$,\n$ B_0=B_0(m_h^2,m_h^2,m_h^2)$,\n$ B'_0=\\frac{\\partial B_0}{\\partial p^2}\\bigg|_{p^2=m_h^2}$.\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehzhtt1.pdf}\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehzhtt2.pdf}\n\\caption{Feynman diagram containing the anomalous $h\\bar t t$ coupling, depicted as the black dot, at one-loop level for the process $e^+ e^- \\rightarrow Zh$.}\n\\label{fig3}\n\\end{figure}\nIn this paper we will calculate the contributions from Higgs-top coupling which are shown in Fig. \\ref{fig3}\n\\footnote{In fact,the contributions from $Z\/\\gamma-H$ bubble transition diagrams are zero.}. The Higgs-top coupling can be parameterized as\n$$\nC_{SM}(1+ \\delta_t) H \\bar t t=-i \\frac{m_t}{v}(1+ \\delta_t) H \\bar t t,\n$$\nwhere $\\delta_t=0$ corresponds to the case in the SM.\n\nThe analytical results can be written as\n\\begin{equation}\n \\begin{split}\n &\\delta_{\\sigma}(htt)\\\\\n  =&-\\frac{4 N_c\\alpha m_t^2 (s-m_z^2) v_1 v_2 \\delta_t}{3 \\pi s \\beta m_z^2 (v_1^2+a_1^2)}Re\\Big[\\beta \\Big (B_0(m_h^2)-4 C_{00}(m_t^2) \\Big)\\\\\n  &-2 \\rho \\Big(C_1(m_t^2)+C_{11}(m_t^2)+C_{12}(m_t^2) \\Big)-6 m_z^2 s \\big(s+m_z^2-m_h^2 \\big ) C_0(m_t^2)\\Big] \\\\\n                       &+\\frac{N_c \\alpha m_t^2 \\delta_t}{\\pi c_w^2 s_w^2 m_z^2 \\beta}  Re\\Big[ \\beta \\Big( 2 \\big (v_2^2+a_2^2 \\big )\\big(B_0(m_h^2)- 4 C_{00}(m_t^2)\\big) +2 a_2^2 \\big (B_0(s)+B_0(m_z^2) \\big) \\Big)  \\\\\n                        &-\\rho\\Big( 4 \\big (v_2^2+a_2^2\\big )\\big (C_1(m_t^2)+C_{11}(m_t^2)+C_{12}(m_t^2)\\big)+2a_2^2 C_2(m_t^2)  \\Big)\\\\\n                        &+\\Big(\\big (v_2^2+a_2^2 \\big) \\big((4 m_t^2- m_z^2-s) \\beta - \\rho \\big) +\\big (v_2^2-a_2^2 \\big) \\big (m_h^2-4 m_t^2 \\big ) \\beta \\Big) C_0(m_t^2)  \\Big]\\\\\n                      &+\\frac{N_c\\alpha m_t^2 \\delta_t}{4 \\pi c_w^2 s_w^2 m_z^2}Re\\Big[-B_0(m_h^2)+\\big(4 m_t^2- m_h^2 \\big) B'_0(m_h^2)\\Big]\n \\end{split}\n \\label{eq2}\n\\end{equation}\nIn Eq. (\\ref{eq2}), the first\/second\/third terms are from the contributions of the diagram with photon propagator\/Z boson propagator\/the counter term of ZZH vertex, respectively.\nHere\n$ \\alpha = \\frac{e^2}{4\\pi}$,\n$N_c=3$,\n$ v_1=-\\frac{1}{4}+s_w^2$,\n$ a_1=\\frac{1}{4}$,\n$ v_2=\\frac{1}{4}-\\frac{2}{3}s_w^2$,\n$ a_2=-\\frac{1}{4}$,\n$ C_0(m_t^2)=C_0(m_h^2,m_z^2,s,m_t^2,m_t^2,m_t^2)$,\n$ C(m_t^2)=C(m_h^2,s,m_z^2,m_t^2,m_t^2,m_t^2)$,\n$ B_0(m_h^2)=B_0(m_h^2,m_t^2,m_t^2)$,\n$ B_0(m_z^2)=B_0(m_z^2,m_t^2,m_t^2)$,\n$ B_0(s)=B_0(s,m_t^2,m_t^2)$.\n\n\n\nWe use LoopTools \\cite{Hahn:1998yk} to do the scalar integral for different  c.m. energies.\nIn Fig. \\ref{fig4}, we show the deviation of cross section arising from $\\delta_t$ and  $\\delta_h$ as a function of $\\sqrt{s}_{e^+ e^-}$.\nSeveral numerical results for the typical c.m. energy are\n\\begin{equation}\n \\delta_\\sigma^{240,350,400,500}=1.45,0.27,0.05,-0.19\\times \\delta_h \\%\n\\end{equation}\n \\begin{equation}\n \\delta_\\sigma^{240,350,400,500}=-0.49,1.38,2.14,2.12\\times \\delta_{t} \\%\n\\end{equation}\n\\begin{figure}[!htbp]\n\n\\includegraphics[width=0.3\\textwidth]{htthhh.pdf}\n\\caption{ Relative correction $\\delta_\\sigma$  due to anomalous $h\\bar{t}t$-coupling $\\delta_t$ (red)\n         and anomalous triple Higgs coupling $\\delta_h$ (blue), as a function of the $e^+ e^-$ center-of-mass (c.m.) energy from 220 GeV to 500 GeV. Note that the precision of\n         relative correction can reach $0.4\\%$ for high luminosity $e^+ e^-$ colliders. }\n\\label{fig4}\n\\end{figure}\n\nThe figures show that the behavior for $\\delta_t$ and $\\delta_h$ is opposite. At low energy end, the relative correction $\\delta_\\sigma$ happen to be dominant by $\\delta_h$, on\nthe contrary for the high energy end, the $\\delta_\\sigma$ arising from anomalous Higgs-top coupling can't be neglected.\nFor the proposed collider of Circular Electron-Positron Collider with $\\sqrt{s}_{e^+e^-} \\simeq 240$ GeV, the extraction\nof triple Higgs coupling is  polluted by Higgs-top coupling. For the International Linear Collider with option of high energy,\nthe pollution from Higgs-top coupling must be taken into account.\n\n\\section{ Measuring CP-violated Higgs-Top Coupling}\n\nThough the newly discovered Higgs boson H(125) is SM-like, it does not exclude the possibility that H(125) is CP mixing state. As emphasized by \\cite{zhu,Mao:2014oya} that\nCP spontaneously broken \\cite{Lee} may be closely related to the lightness of the H(125). In fact, current measurements are insensitive to the mixing, especially for\nH decaying into gauge bosons since the CP violation usually entering the couplings via loops.\n\nIn this paper we parameterize the CP violation through\n$$\nC_{SM} H\\left(1+\\delta_t +i \\delta_a \\gamma_5\\right)=-i \\frac{m_t}{v} H\\left(1+\\delta_t +i \\delta_a \\gamma_5\\right).\n$$\nIndirect constraints on $\\delta_t$ and $\\delta_a$ at the LHC have been studied in \\cite{Ellis:2013yxa}.\nAt the 68\\% CL  the allowed region for ($1+\\delta_t$ , $\\delta_a$) is a crescent with apex close to the SM point(1,0) \\cite{Ellis:2013yxa}.\nThe parameter space close to the SM point, namely $\\delta_t \\rightarrow 0$ and $\\delta_a \\rightarrow 0$ is allowed. At the same time, the parameter space with\nboth non-zero $\\delta_t,  \\delta_a$ is also allowed. In fact, it is quite challenging for LHC to completely exclude the latter case via the indirect method.\nOn the contrary, based on the last section analysis, the cross section deviation depends only on $\\delta_t$ but not $\\delta_a$. This point will be made clear below.  Therefore\nit is important to explore the method to measure the $\\delta_a$ at electron-positron collider.\n\nThe analytical results for the differential cross section arising from $\\delta_a$ can be written as\n\\begin{equation}\n \\begin{split}\n &\\frac{1}{\\delta_a} \\frac{d\\sigma}{d cos\\alpha}\\\\\n =&\\frac{32 N_c a_1 m_t^2 \\pi \\alpha^3 cos\\alpha \\sqrt{\\left((m_h-m_z)^2-s\\right) \\left((m_h+m_z)^2-s\\right)}}{c^4_w s^4_w \\left(m_z^2-s\\right)}\\\\\n &Im\\Big[\\frac{1}{3} v_2  C_0(m_t^2)\n   +\\frac{ s}{c_w^2 s_w^2 \\left(m_z^2-s\\right)}  v_1\n\\big((v_2^2+a_2^2) C_0(m_t^2)+2 a_2^2 C_2(m_t^2)\\big)\\Big]\n \\end{split}\n\\end{equation}\nHere $cos\\alpha$ is the angle between the momentum of the electron and the Z boson. The differential cross section is  proportional to $cos\\alpha$, which is\ndue to the term $\\varepsilon_{\\mu \\nu \\rho \\lambda}  \\varepsilon^{\\mu \\nu \\alpha \\beta}  p_2^{\\rho}  p_1^{\\lambda}  k_{1\\alpha}  k_{2\\beta}$\nwhere $p_1$ $p_2$ are the momentum of electron and positron and $k_1$ $k_2$ are the momentum of Higgs and Z. Another critical requirement for non-vanishing contribution\nto the differential cross section\nis that there should be imaginary part from top loops. This requires that the $\\sqrt{s}_{e^+e^-}$ must be great than $2 m_t$.\n\n\nIt is obvious that the CP-odd contributions to the total cross section is zero.\nIn order to show the different contributions from $\\delta_t$ and $\\delta_a$ respectively, we plot the normalized differential cross sections for several $\\sqrt{s}_{e^+e^-}$\nand set the corresponding parameter $\\delta_t$ or $\\delta_a$ equal to 1.\n\\begin{figure}[!htbp]\n\\includegraphics[width=0.5\\textwidth]{total4.pdf}\n\\caption{Differential scattering cross section as a function of the scattering angle with $\\sqrt{s}=240GeV$(orange),$350GeV$(red),$400GeV$(green),$500GeV$(blue).\nAnd solid\/dashed lines stand for the contributions\n from  $\\delta_t$\/$\\delta_a$ respectively. }\n\\label{6}\n\\end{figure}\nFrom the  figure, it is quite clear that the differential cross sections arising from $\\delta_t$ are symmetric and anti-symmetric from $\\delta_a$.\nFor $\\sqrt{s}=240GeV$, the contribution from $\\delta_a$ is zero because there is no imaginary part of $ C_0(m_t^2)$.\nWhen $\\sqrt{s}_{e^+e^-} > 2 m_t$ there are nonzero contributions from $\\delta_a$ as expected.\n\nIn order to gauge the forward-backward asymmetry, we introduce\n$$\nA_{FB} \\equiv \\frac{\\int_{0}^1 d\\cos \\alpha \\frac{d\\sigma}{d\\cos\\alpha}  -  \\int_{-1}^0 d\\cos \\alpha \\frac{d\\sigma}{d\\cos\\alpha} }{\\sigma_{tot} }\n$$\n\nIn Fig.  \\ref{fig9}, we plot $A_{FB}$ as a function of $\\sqrt{s}_{e^+e^-}$ with $\\delta_a=1$ for polarized and unpolarized electron\/positron beam.\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=0.4\\textwidth]{afbsum.pdf}\n\\caption{  $A_{FB}$ as a function of $\\sqrt{s}_{e^+e^-}$ from $220GeV$ to $500GeV$ for polarized or unpolarized electron\/positron beams.\nThe black line, blue dashed and red dashed seperately\ncorrespond to  unpolarized electron\/positron beams,$e_R^+ e_L^-$ and $e_L^+ e_R^-$ polarizations.}\n\\label{fig9}\n\\end{figure}\n\n\nFrom the figure we can see that the asymmetry can reach $0.7 \\%$ for $\\sqrt{s}_{e^+e^-}$. Such precision is comparable to that of cross section measurement.\nIt seems that the high luminosity collider is necessary.\n\n\n\n\\section{Conclusion and discussion}\n\nIn this paper, we explore the Higgs-top anomalous coupling pollution to the extraction of Higgs self coupling via precisely measuring cross section of $e^+e^- \\rightarrow ZH$.\nThe important conclusion is that the pollution is small for the $\\sqrt{s}_{e^+e^-} =240$ GeV, but can be sizable for higher energy collider.\nThe contributions to total cross section from Higgs-top CP-odd coupling is vanishing, while such interaction can\n be scrutinized via forward-backward asymmetry for $\\sqrt{s}_{e^+e^-}$ greater than $2 m_t$.\n\n\n\\section*{Acknowledgement}\n\n We would like to thank Shao Long Chen , Gang Li and Pengfei Yin for the useful discussions. This work was supported in part by the Natural Science Foundation\n of China (Nos. 11135003 and 11375014).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{E:introduction}\n\nWormholes are handles or tunnels in spacetime\nconnecting widely separated regions of our\nUniverse or different universes altogether.\nWhile there had been some forerunners,\nmacroscopic traversable wormholes were first\ndiscussed in detail by Morris and Thorne\n\\cite{MT88} in 1988.  A few years later,\nSung-Won Kim \\cite{Kim} proposed the\npossible existence of an evolving wormhole\nin the context of the\nFriedmann-Lemaitre-Robertson-Walker (FLRW)\ncosmological model by assuming that the\nmatter content can be divided into two\nparts, the cosmological part that depends on\ntime only and the wormhole part that depends\non space only.  The discussion was later\nexpanded by Cataldo et al. \\cite{mC13}.\n\nThe purpose of this paper is to study the\nrelationship between wormholes inspired by\nnoncommutative geometry and the Kim model.\nThe noncommutative-geometry background\nsimultaneously affects both the wormhole\nconstruction and the cosmological part of\nthe solution.  This result differs\nsignificantly from the outcomes in Refs.\n\\cite{Kim} and \\cite{mC13}.\n\nRegarding the strategy, Ref. \\cite{MT88}\nconcentrates mainly on the wormhole\ngeometry by specifying the metric\ncoefficients.  This strategy requires a\nsearch for matter or fields that can produce\nthe energy-momentum tensor needed to sustain\nthe wormhole.  Here it needs to be emphasized\nthat we are able to satisfy the geometric\nrequirements from the physical properties.\nThe result is an evolving zero-tidal force\nwormhole solution; it is restricted to the\ncurvature parameters $k=0$ and $k=-1$,\ncorresponding to an open Universe.\n\nViewed from a broader perspective, it has\nalready been shown that noncommutative\ngeometry, which is an offshoot of string\ntheory, can account for the flat galactic\nrotation curves \\cite{pK17, fR12}, but\nunder certain conditions, noncommutative\ngeometry can also support traversable\nwormholes \\cite{KG14, pK18, pK16, pK15,\n Jamil14, FKRI12}.\n\nThis paper is organized as follows: Sec.\n\\ref{S:structure} briefly recalls the\nstructure of wormholes and the basic\nfeatures of noncommutative geometry.\nSec. \\ref{S:Kim} continues with the\nSung-Won Kim model. Here the discussion\nis necessarily more detailed, partly\nin the interest of completeness, but mainly\nto allow the inclusion of a more general\nform of the Einstein field equations.\nThese are subsequently used in Sec. \\ref\n{S:special} to obtain a wormhole solution\nthat does not depend on the separation\nof the matter content.  In Sec. \\ref{S:nc}\nwe derive a wormhole solution from the\nnoncommutative-geometry background,\nfollowed by a discussion of the null\nenergy condition in Sec. \\ref {S:violation}.\nSec. \\ref{S:comparison} features a\n comparison to an earlier solution.\n In Sec. \\ref{S:conclusion}, we conclude.\n\n\n\\section{Wormhole structure and\n   noncommutative geometry}\\label{S:structure}\n\nMorris and Thorne \\cite{MT88} proposed the\nfollowing static and spherically symmetric\nline element for a wormhole spacetime:\n\\begin{equation}\\label{E:line1}\nds^{2}=-e^{2\\Phi(r)}dt^{2}+\\frac{dr^2}{1-b(r)\/r}\n+r^{2}(d\\theta^{2}+\\text{sin}^{2}\\theta\\,\nd\\phi^{2}),\n\\end{equation}\nusing units in which $c=G=1$.  Here $b=b(r)$\nis called the \\emph{shape function} and\n$\\Phi=\\Phi(r)$ is called the \\emph{redshift\nfunction}, which must be everywhere finite\nto avoid an event horizon.  For the shape\nfunction we must have $b(r_0)=r_0$, where\n$r=r_0$ is the radius of the \\emph{throat}\nof the wormhole.  The wormhole spacetime\nshould be asymptotically flat, i.e.,\n$\\text{lim}_{r\\rightarrow \\infty}\\Phi(r)\n=0$ and  $\\text{lim}_{r\\rightarrow \\infty}\nb(r)\/r=0$.  An important requirement is the\n\\emph{flare-out condition} at the throat:\n$b'(r_0)<1$, while $b(r)<r$ near the\nthroat.  The flare-out condition can only\nbe met by violating the null energy\ncondition (NEC), which states that\n\\begin{equation}\n  T_{\\alpha\\beta}k^{\\alpha}k^{\\beta}\\ge 0\n\\end{equation}\nfor all null vectors $k^{\\alpha}$, where\n$T_{\\alpha\\beta}$ is the energy-momentum\ntensor.  Matter that violates the NEC is\ncalled ``exotic\" in Ref. \\cite{MT88}.  In\nparticular, for the outgoing null vector\n$(1,1,0,0)$, the violation has the form\n\\begin{equation}\n   T_{\\alpha\\beta}k^{\\alpha}k^{\\beta}=\n   \\rho +P^r<0.\n\\end{equation}\nHere $T^t_{\\phantom{tt}t}=-\\rho$ is the energy\ndensity, $T^r_{\\phantom{rr}r}= P^r$ is the\nradial pressure, and\n$T^\\theta_{\\phantom{\\theta\\theta}\\theta}=\nT^\\phi_{\\phantom{\\phi\\phi}\\phi}=P^t$ is\nthe lateral pressure.\n\nReturning now to the noncommutative-geometry\nbackground mentioned earlier, we need to\nrecall that, as an offshoot of string theory,\nnoncommutative geometry replaces point-like\nparticles by smeared objects.  (For a detailed\ndiscussion, see Refs. \\cite {SS03, NSS06,\nNS10}.)  As a result, spacetime can be\nencoded in the commutator\n$[\\textbf{x}^{\\mu},\\textbf{x}^{\\nu}]\n=i\\theta^{\\mu\\nu}$, where $\\theta^{\\mu\\nu}$ is\nan antisymmetric matrix that determines the\nfundamental cell discretization of spacetime\nin the same way that Planck's constant\ndiscretizes phase space \\cite{NSS06}.  An\ninteresting and effective way to model the\nsmearing effect, discussed in Refs.\n\\cite{pK15,LL12, NM08}, is to assume that\nthe energy density of the static,\nspherically symmetric, smeared, and\nparticle-like gravitational source is\ngiven by\n\\begin{equation}\\label{E:rho}\n  \\rho(r)=\\frac{\\mu\\sqrt{\\beta}}\n     {\\pi^2(r^2+\\beta)^2},\n\\end{equation}\nwhich can be interpreted to mean that the\ngravitational source causes the mass $\\mu$\nof a particle to be diffused throughout the\nregion of linear dimension $\\sqrt{\\beta}$\ndue to the uncertainty; so $\\sqrt{\\beta}$\nhas units of length.\n(Ref. \\cite{NSS06} uses a Gaussian\ndistribution instead of Eq. (\\ref{E:rho})\nto represent $\\rho$.)  Eq. (\\ref{E:rho})\nleads to the mass distribution\n\\begin{equation}\\label{E:source}\n   \\int^r_04\\pi(r')^2\\rho(r')dr'=\n   \\frac{2M\\sqrt{\\beta}}{\\pi}\n   \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n   \\frac{r}{\\sqrt{\\beta}}-\n   \\frac{r}{r^2+\\beta}\\right),\n\\end{equation}\nwhere $M$ is now the total mass of the source.\n\nAccording to Ref. \\cite{NSS06}, noncommutative\ngeometry is one of the basic properties of\nspacetime that does not depend on particular\nfeatures such as curvature.  Moreover, since\nthe noncommutative effects can be implemented\nby modifying only the energy-momentum tensor,\nthere is no need to change the Einstein tensor\nin the field equations.  As a result, the\nlength scales need not be microscopic.\n\n\\section{The Sung-Won Kim model}\\label{S:Kim}\n\nThe Sung-Won Kim cosmological model with a\ntraversable wormhole is given by \\cite{Kim}\n\\begin{equation}\\label{E:line2}\nds^{2}=-e^{2\\Phi(r)}dt^{2}+[R(t)]^2\\left[\n\\frac{dr^2}{1-kr^2-b(r)\/r}\n+r^{2}(d\\theta^{2}+\\text{sin}^{2}\\theta\\,\nd\\phi^{2})\\right],\n\\end{equation}\nwhere $R(t)$ is the scale factor of the\nUniverse and $k$ is the sign of the\ncurvature of spacetime, i.e., $k=+1$,\n0, or $-1$.   (So while $b=b(r)$ is still\nthe shape function, it is subject to\nconditions that are different from those\nof a Morris-Thorne wormhole.)  The\nEinstein field equations in Ref. \\cite\n{Kim} are based on Eq. (\\ref{E:line2}),\nbut it is subsequently assumed that\n$\\Phi(r)\\equiv 0$ to be consistent with\nthe FLRW model.\n\nThe discussion of the Sung-Won Kim model\nis continued and elaborated on in Ref.\n\\cite{mC13}.  Unfortunately, the field\nequations in Refs. \\cite{Kim} and\n\\cite{mC13} do not agree and need to be\nrederived.  Here we follow Ref. \\cite{Kim}\nand base the calculations on line element\n(\\ref{E:line2}).  The results are\n\\begin{equation}\\label{E:Einstein1}\n   8\\pi\\rho(r,t)=3\\left(\\frac{\\dot{R}}{R}\\right)^2\n   e^{-2\\Phi(r)}+\\frac{3k}{R^2}+\\frac{b'(r)}{R^2r^2},\n\\end{equation}\n\\begin{equation}\\label{E:Einstein2}\n   8\\pi P^r(r,t)=-2\\frac{\\ddot{R}}{R}e^{-2\\Phi(r)}\n   -\\left(\\frac{\\dot{R}}{R}\\right)^2e^{-2\\Phi(r)}\n   -\\frac{k}{R^2}-\\frac{b(r)}{R^2r^3}+\\frac{2}{R^2r}\n   \\Phi'(r)\\left(1-kr^2-\\frac{b(r)}{r}\\right),\n\\end{equation}\n\\begin{multline}\\label{E:Einstein3}\n   8\\pi P^t(r,t)=-2\\frac{\\ddot{R}}{R}e^{-2\\Phi(r)}\n   -\\left(\\frac{\\dot{R}}{R}\\right)^2e^{-2\\Phi(r)}\n   -\\frac{k}{R^2}+\\frac{b(r)-rb'(r)}{2R^2r^3}\\\\\n   +\\frac{1}{R^2}\\left[(\\Phi'(r))^2\\left(1-kr^2-\n   \\frac{b(r)}{r}\\right)+\\Phi''(r)\\left(1-kr^2-\n   \\frac{b(r)}{r}\\right)\\right.\\\\\n   \\left.-\\frac{1}{2}\\Phi'(r)\\left(2kr\n   +\\frac{rb'(r)-b(r)}{r^2}\\right)\\right]\n   +\\frac{1}{R^2r}\\Phi'(r)\\left(1-kr^2\n   -\\frac{b(r)}{r}\\right),\n\\end{multline}\n\\begin{equation}\\label{E:Einstein4}\n  8\\pi T_{01}=\\frac{\\dot{R}}{R^2}\n  e^{-\\Phi(r)}\\Phi'(r)\n  \\left(1-kr^2-\\frac{b(r)}{r}\\right)^{1\/2},\n\\end{equation}\nwhere $T_{01}$ is the outward energy flow.\n(The prime and overdots denote the derivatives\nwith respect to $r$ and $t$, respectively.)\n\nIf $\\Phi(r)\\equiv 0$, the results agree with\nthose in Ref. \\cite{mC13} (omitting $\\Lambda$,\nthe cosmological constant).  For convenience,\nthese will now be restated:\n\\begin{equation}\\label{E:E1}\n   8\\pi\\rho(r,t)=3\\left(\\frac{\\dot{R}}{R}\\right)^2\n   +\\frac{3k}{R^2}+\\frac{b'}{R^2r^2},\n\\end{equation}\n\\begin{equation}\\label{E:E2}\n   8\\pi P^r(r,t)=-2\\frac{\\ddot{R}}{R}-\n   \\left(\\frac{\\dot{R}}{R}\\right)^2\n   -\\frac{k}{R^2}-\\frac{b}{R^2r^3},\n\\end{equation}\n\\begin{equation}\\label{E:E3}\n   8\\pi P^t(r,t)=-2\\frac{\\ddot{R}}{R}\n   -\\left(\\frac{\\dot{R}}{R}\\right)^2\n   -\\frac{k}{R^2}+\\frac{b-rb'}\n   {2R^2r^3}.\n\\end{equation}\n\nThe next step depends on a key insight,\ndue to Sung-Won Kim \\cite{Kim}, that\nallows the separation of the Einstein\nfield equations into two parts, namely,\nthe following ansatz for the matter\nparts:\n\\begin{equation}\\label{E:Kim1}\n   R^2(t)\\rho(r,t)=R^2(t)\\rho_c(t)\n       +\\rho_w(r),\n\\end{equation}\n\\begin{equation}\\label{E:Kim2}\n   R^2(t)P^r(r,t)=R^2(t)P_c(t)\n      +P^r_w(r),\n\\end{equation}\n\\begin{equation}\\label{E:Kim3}\n   R^2(t)P^t(r,t)=R^2(t)P_c(t)\n      +P^t_w(r).\n\\end{equation}\nThe subscripts $c$ and $w$ refer,\nrespectively, to the cosmological\nand wormhole parts.  So $P_c$\nnecessarily represents the isotropic\npressure.\n\nThe ansatz now allows us to separate\nEqs. (\\ref{E:E1})-(\\ref{E:E3}) into\ntwo parts, the left side being a\nfunction of $t$ and the right side\na function of $r$, also carried out in\nRef. \\cite{mC13}:\n\\begin{equation}\\label{E:separate1}\n   R^2\\left[8\\pi\\rho_c-3\\left(\\frac{\\dot{R}}\n   {R}\\right)^2-\\frac{3k}{R^2}\\right]=\n   \\frac{b'}{r^2}-8\\pi\\rho_w=l,\n\\end{equation}\n\\begin{equation}\\label{E:separate2}\n  R^2\\left[8\\pi P_c+2\\frac{\\ddot{R}}{R}\n  +\\left(\\frac{\\dot{R}}{R}\\right)^2\n  +\\frac{k}{R^2}\\right]=-\\frac{b}{r^3}\n  -8\\pi P^r_w=m,\n\\end{equation}\n\\begin{equation}\\label{E:separate3}\n   R^2\\left[8\\pi P_c+2\\frac{\\ddot{R}}{R}\n   +\\left(\\frac{\\dot{R}}{R}\\right)^2\n   +\\frac{k}{R^2}\\right]=\n   \\frac{b-rb'}{2r^3}-8\\pi P^t_w=m,\n\\end{equation}\nwhere $l$ and $m$ are constants.  The\nreason is that a function of $t$ cannot\nbe equal to a function of $r$ for all\n$t$ and $r$ unless they are equal to\nsome constant.  In Eqs. (\\ref{E:separate2})\nand (\\ref{E:separate3}), the constants\nare the same since the cosmological parts\nare equal.  The constants $l$ and $m$ may\nbe taken as arbitrary.\n\nReturning now to Eqs.\n(\\ref{E:Einstein1})-(\\ref{E:Einstein4}),\nrecall that we obtained the more general\nform of the field equations by using\nline element (\\ref{E:line2}), as\nsuggested in Ref. \\cite{Kim}.  It now\nbecomes apparent, however, that the\nseparation in Eqs.\n(\\ref{E:separate1})-(\\ref{E:separate3})\ncannot be carried out by means of Eqs.\n(\\ref{E:Kim1})-(\\ref{E:Kim3}) unless\nwe assume that $\\Phi(r)\\equiv 0$, which\ntakes us back to the FLRW model.\n\n\\section{The noncommutative wormhole}\n   \\label{S:nc}\nTo obtain a wormhole solution, we will\nconsider the special case $l=-3m$,\nfollowing Ref. \\cite{mC13}.  Eqs.\n(\\ref{E:separate1})-(\\ref{E:separate3})\nthen yield\n\\begin{equation}\\label{E:R1}\n  3\\left(\\frac{\\dot{R}}{R}\\right)^2\n   +\\frac{3k}{R^2}-\\frac{3m}{R^2}\n   =8\\pi\\rho_c\n\\end{equation}\nand\n\\begin{equation}\\label{E:R2}\n   -2\\frac{\\ddot{R}}{R}-\n   \\left(\\frac{\\dot{R}}{R}\\right)^2\n   -\\frac{k}{R^2}+\\frac{m}{R^2}\n   =8\\pi P_c\n\\end{equation}\nfor the cosmological part, while the\nwormhole part is given by\n\\begin{equation}\\label{E:W1}\n   \\frac{b'}{r^2}-8\\pi\\rho_w=-3m,\n\\end{equation}\n\\begin{equation}\\label{E:W2}\n   -\n\\frac{b}{r^3}-8\\pi P^r_w=m,\n\\end{equation}\n\\begin{equation}\\label{E:W3}\n   \\frac{b-rb'}{2r^3}-8\\pi P^t_w=m.\n\\end{equation}\n\nThe wormhole solution can be obtained\nfrom Eqs. (\\ref{E:W1})-(\\ref{E:W3}) by\nmaking use of Eq. (\\ref{E:rho}).  It\nremains to be seen what restrictions\nwill be placed on the solutions due to\nthe cosmological part and to the\nnecessary violation of the NEC.  But\nfor now we have from Eq. (\\ref{E:W1})\nand Eq. (\\ref{E:rho}) that\n\\begin{equation}\n    b'(r)=8\\pi\\frac{\\mu\\sqrt{\\beta}r^2}\n    {\\pi^2(r^2+\\beta)^2}-3mr^2.\n\\end{equation}\nIntegrating and using the condition $b(r_0)\n=r_0$, we obtain\n\\begin{multline}\\label{E:shape2}\n  b(r)=\\frac{4M\\sqrt{\\beta}}{\\pi}\n  \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n  \\frac{r}{\\sqrt{\\beta}}-\\frac{r}{r^2+\\beta}\n  \\right)-mr^3\\\\\n  -\\frac{4M\\sqrt{\\beta}}{\\pi}\n  \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n  \\frac{r_0}{\\sqrt{\\beta}}-\\frac{r_0}{r_0^2\n  +\\beta}\\right)+mr_0^3+r_0,\n\\end{multline}\nwhere $M$ is the mass of the wormhole.\nIt now becomes apparent that the resulting\nspacetime is not asymptotically flat.  The\nnormal procedure is to cut off the wormhole\nmaterial at some $r=a$ and then join the\nstructure to an external Schwarzschild\nspacetime. We will see in the next section,\nhowever, that the need to violate the NEC\n($\\rho +P^r<0$) requires a slight\nmodification of the shape function,\nresulting in the required asymptotic\nflatness.\n\n\\section{Violating the NEC}\\label{S:violation}\nTo check the violation of the NEC\n($\\rho +P^r<0$) for the wormhole, we\nlet $R\\equiv \\text{constant}$\nand obtain from Eqs. (\\ref{E:E1}) and\n(\\ref{E:E2}),\n\\begin{equation}\n   b-rb'-2kr^3>0.\n\\end{equation}\nAt $r=r_0$, we therefore get\n\\begin{equation}\n   r_0-r_0\\frac{8\\pi\\mu\\sqrt{\\beta}r_0^2}\n   {\\pi^2(r_0^2+\\beta)^2}+3mr_0^3\n   -2kr_0^3>0.\n\\end{equation}\nSince $\\sqrt{\\beta}$ is extremely small, we\nactually have\n\\begin{equation}\\label{E:NEC}\n   r_0+(3m-2k)r_0^3\\gtrsim 0.\n\\end{equation}\nTo check this condition, we need to return\nto Ref. \\cite{NSS06} for some additional\nobservations.  The relationship between\nthe radial pressure and energy density\nis given by\n\\begin{equation}\\label{E:EoS}\n   P^r=-\\rho.\n\\end{equation}\nThe reason is that the source is a\nself-gravitating droplet of anisotropic\nfluid of density $\\rho$ and the radial\npressure is needed to prevent the\ncollapse back to the matter point.  In\naddition, the  lateral pressure is\ngiven by\n\\begin{equation}\\label{E:tr1}\n   P^t=-\\rho-\\frac{r}{2}\n     \\frac{\\partial\\rho}{\\partial r}.\n\\end{equation}\nSince the length scales can be\nmacroscopic, we can retain Eq.\n(\\ref{E:EoS}) and then use Eq.\n(\\ref{E:tr1}) to write\n\\begin{equation}\\label{E:tr2}\n    P^t=-\\rho-\\frac{r}{2}\n     \\frac{\\partial\\rho}{\\partial r}\n     =P^r+\\frac{2\\mu r^2\\sqrt{\\beta}}\n     {\\pi^2(r^2+\\beta)^3}\n\\end{equation}\nby Eq. (\\ref{E:rho}).  So on larger\nscales, we have $P^r=P^t$.  Since the\npressure becomes isotropic, we can\nassume the equation of state to be\n$P_c=-\\rho_c$.  Substituting in Eqs.\n(\\ref{E:R1}) and (\\ref{E:R2}), we get\n\\begin{equation*}\n   -2\\frac{\\ddot{R}}{R}\n   +2\\left(\\frac{\\dot{R}}{R}\\right)^2\n   +\\frac{2k}{R^2}-\\frac{2m}{R^2}=0.\n\\end{equation*}\nThis equation can be rewritten as\n\\begin{equation}\n   3\\frac{\\ddot{R}}{R}\n   -3\\left(\\frac{\\dot{R}}{R}\\right)^2=\n   \\frac{3k}{R^2}-\\frac{3m}{R^2}.\n\\end{equation}\nSubtracting the Friedmann equations\n\\[\n   3\\frac{\\ddot{R}}{R}=-4\\pi(\\rho_c\n      +3P_c)\n\\]\nand\n\\[\n   3\\left(\\frac{\\dot{R}}{R}\\right)^2\n   =8\\pi\\rho_c-\\frac{3k}{R^2}\n\\]\nnow yields\n\\begin{equation}\n   \\frac{3k}{R^2}-\\frac{3m}{R^2}=\n   -4\\pi(\\rho_c+3P_c)-8\\pi\\rho_c\n   +\\frac{3k}{R^2}.\n\\end{equation}\nSo if $P_c=-\\rho_c$, we obtain\n\\begin{equation}\n   m=0, \\text{independently of}\\quad k.\n\\end{equation}\nApplied to Eq. (\\ref{E:NEC}), the NEC\nis violated if\n\\begin{equation}\n   k=0\\quad \\text{or}\\quad k=-1.\n\\end{equation}\nThese conditions correspond to an open\nUniverse.\n\nTo summarize, we employed basic physical\nprinciples to derive the following\nzero-tidal force solution:\n\\begin{equation}\n   \\Phi(r)\\equiv 0\n\\end{equation}\nand (since $m=0$)\n\\begin{multline}\\label{E:shape1}\n  b(r)=\\frac{4M\\sqrt{\\beta}}{\\pi}\n  \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n  \\frac{r}{\\sqrt{\\beta}}-\\frac{r}{r^2+\\beta}\n  \\right)\\\\\n  -\\frac{4M\\sqrt{\\beta}}{\\pi}\n  \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n  \\frac{r_0}{\\sqrt{\\beta}}-\\frac{r_0}{r_0^2\n  +\\beta}\\right)+r_0.\n\\end{multline}\nThe slowly evolving wormhole solution is\nrestricted to the values $k=0$ and $k=-1$\nto ensure that the NEC is violated.  The\nwormhole spacetime is asymptotically flat.\n\n\\section{The special case $k=0$}\n    \\label{S:special}\nFor completeness let us briefly consider\na wormhole solution that does not depend\non the separation of the Einstein field\nequations.  We can combine Eqs.\n(\\ref{E:E1}) and (\\ref{E:E2})\nto obtain\n\\begin{equation}\n   8\\pi r^3R^2\\left[\\rho(r,t)+P^r(r,t)\n   \\right]=2r^3(\\dot{R}^2-R\\ddot{R})\n   +2r^3k+rb'(r)-b(r).\n\\end{equation}\nIf we now let $k=0$, then Eq.\n(\\ref{E:line2}) represents an evolving\nMorris-Thorne wormhole with the usual\nshape function $b=b(r)$.  The NEC is\nviolated at the throat $r=r_0$ for all\n$t$ whenever\n\\begin{equation}\\label{E:NEC1}\n   8\\pi r_0^3R^2\\left[\\rho(r_0,t)-\n   P^r(r_0,t)\\right]=2r_0^3(\\dot{R^2}\n   -R\\ddot{R})+r_0b'(r_0)-b(r_0)<0.\n\\end{equation}\nIf the Universe is indeed accelerating,\nthen the term $-R\\ddot{R}$ eventually\nbecomes dominant due to the\never-increasing $R$.  So for\nsufficiently large $R$, the NEC is\nviolated, thereby fulfilling a key\nrequirement for the existence of\nwormholes.  (Inflating Lorentzian\nwormholes are discussed in Ref.\n\\cite{tR93}.)\n\nRecalling that the radial tension $\\tau$\nis the negative of $P^r$, Inequality\n(\\ref{E:NEC1}) can be written (since\n$b(r_0)=r_0$)\n\\begin{equation}\\label{E:NEC2}\n   8\\pi r_0^2R^2\\left[\\tau(r_0)-\\rho(r_0)\n   \\right]=2r_0^2(-\\dot{R^2}+R\\ddot{R})\n   -b'(r_0)+1>0.\n\\end{equation}\nIf $R(t)\\equiv 1$, this reduces to the\nstatic Morris-Thorne wormhole; so if\n$b'(r_0)<1$, then $\\tau(r_0)>\\rho(r_0)$,\nrequiring exotic matter.  In Inequality\n(\\ref{E:NEC2}), however, $\\tau(r_0)>\n\\rho(r_0)$ could result from the\ndominant term $R\\ddot{R}$.  In that\ncase, the NEC is violated without\nrequiring exotic matter  for the\nconstruction of the wormhole itself.\n\n\\section{Comparison to an earlier solution}\n    \\label{S:comparison}\nA wormhole solution inspired by\nnoncommutative geometry had already been\nconsidered in Ref. \\cite{pK15}.  The\nEinstein field equation $\\rho(r)=\nb'(r)\/(8\\pi r^2)$, together with Eq.\n(\\ref{E:rho}), leads directly to the\nstatic solution, Eq. (\\ref{E:shape1}).\n(Here it is understood that $k=0$,\nbut $R(t)$ could be retained.)\nUnfortunately, this simple approach\nleaves the redshift function\nundetermined.  The desirability of\nzero tidal forces then suggested the\nassumption $\\Phi(r)\\equiv 0$ in Ref.\n\\cite{pK15}.  It is shown in Ref.\n\\cite{pK09}, however, that this\nassumption causes a Morris-Thorne\nwormhole to be incompatible with the\nFord-Roman constraints from quantum\nfield theory.  Given the\nnoncommutative-geometry background,\nrather than the purely classical setting\nin Ref. \\cite{MT88}, this objection\ndoes not apply directly.\n\nIt is interesting to note that in the\npresent paper, the zero-tidal force\nsolution is built into the Sung-Won\nKim model and does not require any\nadditional considerations.\n\n\\section{Conclusion}\\label{S:conclusion}\nMorris-Thorne wormholes typically require\na reverse strategy for their theoretical\nconstruction: specify the geometric\nrequirements and then manufacture or search\nthe Universe for matter or fields to obtain\nthe required energy-momentum tensor.  One\nof the goals in this paper is to obtain a\ncomplete wormhole solution from certain\nphysical principles.  To this end, we assume\na noncommutative-geometry background, as in\nprevious studies, but we also depend on a\ncosmological model due to Sung-Won Kim that\nis based on the FLRW model with a\ntraversable wormhole.  The basic assumption\nis that the matter content can be divided\ninto two parts, a cosmological part that\ndepends only on $t$ and a wormhole part\nthat depends only on the radial coordinate\n$r$.  The result is a complete zero-tidal\nforce solution; it is restricted, however,\nto the values $k=0$ and $k=-1$, corresponding\nto an open Universe.  This conclusion is\nconsistent with the special case $k=0$\ndiscussed in Sections \\ref{S:special}\nand \\ref{S:comparison}.\n\n\nThe wormhole is slowly evolving due to the\nscale factor $R(t)$ and, critically, the\nnoncommutative-geometry background not\nonly produces the wormhole solution, it\nalso affects in a direct manner the\ncosmological part of the solution.  This\nconclusion differs significantly from\nthose in Refs. \\cite {Kim} and \\cite{mC13}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe inequalit\n\\begin{equation}\nf\\left( \\frac{a+b}{2}\\right) \\leq \\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right)\ndx\\leq \\frac{f\\left( a\\right) +f\\left( b\\right) }{2}  \\label{h}\n\\end{equation\nwhich holds for all convex functions $f:[a,b]\\rightarrow \n\\mathbb{R}\n$, is known in the literature as Hermite-Hadamard's inequality.\n\nIn \\cite{G}, Toader defined $m-$convexity as the following:\n\n\\begin{definition}\nThe function $f:[0,b]\\rightarrow \n\\mathbb{R}\n,$ $b>0$, is said to be $m-$convex where $m\\in \\lbrack 0,1],$ if we hav\n\\begin{equation*}\nf(tx+m(1-t)y)\\leq tf(x)+m(1-t)f(y)\n\\end{equation*\nfor all $x,y\\in \\lbrack 0,b]$ and $t\\in \\lbrack 0,1].$ We say that $f$ is \nm- $concave if $\\left( -f\\right) $ is $m-$convex.\n\\end{definition}\n\nIn \\cite{D}, Dragomir proved the following theorem.\n\nLet $f:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $m-$convex function with $m\\in \\left( 0,1\\right] $ and $0\\leq a<b.$\nIf $f\\in L_{1}\\left[ a,b\\right] ,$ then the following inequalities hold\n\\begin{eqnarray}\nf\\left( \\frac{a+b}{2}\\right) &\\leq &\\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left(\nx\\right) +mf\\left( \\frac{x}{m}\\right) }{2}dx  \\label{h1} \\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{f\\left( a\\right) +mf\\left( \\dfrac{a}{m\n\\right) }{2}+m\\frac{f\\left( \\frac{b}{m}\\right) +mf\\left( \\dfrac{b}{m^{2}\n\\right) }{2}\\right] .  \\notag\n\\end{eqnarray}\n\nIn \\cite{DI} and \\cite{DPP}, the authors connect together some disparate\nthreads through a Hermite-Hadamard motif. The first of these threads is the\nunifying concept of a $g-$convex dominated function. Similarly, in \\cite{KOS\n, Kavurmac\\i\\ et al. introduced the following class of functions and then\nproved a theorem for this class of functions related to (\\ref{h1}).\n\n\\begin{definition}\nLet $g:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n$ be a given $m-$convex function on the interval $\\left[ 0,b\\right] $. The\nreal function $f:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n$ is called $\\left( g,m\\right) -$convex dominated on $\\left[ 0,b\\right] $ if\nthe following condition is satisfie\n\\begin{eqnarray*}\n&&\\left\\vert \\lambda f(x)+m(1-\\lambda )f(y)-f\\left( \\lambda x+m\\left(\n1-\\lambda \\right) y\\right) \\right\\vert \\\\\n&\\leq &\\lambda g(x)+m(1-\\lambda )g(y)-g\\left( \\lambda x+m\\left( 1-\\lambda\n\\right) y\\right)\n\\end{eqnarray*\nfor all $x,y\\in \\left[ 0,b\\right] $, $\\lambda \\in \\left[ 0,1\\right] $ and \nm\\in \\left[ 0,1\\right] .$\n\\end{definition}\n\n\\begin{theorem}\n\\label{a} Let $g:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $m-$convex function with $m\\in \\left( 0,1\\right] $. $f:\\left[\n0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ is $\\left( g,m\\right) -$convex dominated \\ mapping and $0\\leq a<b.$ If \nf\\in L_{1}\\left[ a,b\\right] ,$ then one has the inequalities\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left( x\\right) +mf\\left( \\frac\nx}{m}\\right) }{2}dx-f\\left( \\frac{a+b}{2}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{1}{b-a}\\int_{a}^{b}\\frac{g\\left( x\\right) +mg\\left( \\frac{x}{m\n\\right) }{2}dx-g\\left( \\frac{a+b}{2}\\right)\n\\end{eqnarray*\nan\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{1}{2}\\left[ \\frac{f\\left( a\\right) +mf\\left( \\dfrac{a}{m\n\\right) }{2}+m\\frac{f\\left( \\frac{b}{m}\\right) +mf\\left( \\dfrac{b}{m^{2}\n\\right) }{2}\\right] -\\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left( x\\right)\n+mf\\left( \\frac{x}{m}\\right) }{2}dx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{g\\left( a\\right) +mg\\left( \\dfrac{a}{m\n\\right) }{2}+m\\frac{g\\left( \\frac{b}{m}\\right) +mg\\left( \\dfrac{b}{m^{2}\n\\right) }{2}\\right] -\\frac{1}{b-a}\\int_{a}^{b}\\frac{g\\left( x\\right)\n+mg\\left( \\frac{x}{m}\\right) }{2}dx.\n\\end{eqnarray*}\n\\end{theorem}\n\nIn \\cite{MIH}, definition of $\\left( \\alpha ,m\\right) -$convexity was\nintroduced by Mihe\\c{s}an as the following.\n\n\\begin{definition}\n\\label{d1} The function $f:[0,b]\\rightarrow \n\\mathbb{R}\n,$ $b>0$, is said to be $\\left( \\alpha ,m\\right) -$convex, where $\\left(\n\\alpha ,m\\right) \\in \\lbrack 0,1]^{2},$ if we hav\n\\begin{equation*}\nf(tx+m(1-t)y)\\leq t^{\\alpha }f(x)+m(1-t^{\\alpha })f(y)\n\\end{equation*\nfor all $x,y\\in \\lbrack 0,b]$ and $t\\in \\lbrack 0,1].$\n\\end{definition}\n\nDenote by $K_{m}^{\\alpha }\\left( b\\right) $ the class of all $\\left( \\alpha\n,m\\right) -$convex functions on $\\left[ 0,b\\right] $ for which $f\\left(\n0\\right) \\leq 0.$ If we take $\\left( \\alpha ,m\\right) =\\left\\{ \\left(\n0,0\\right) ,\\left( \\alpha ,0\\right) ,\\left( 1,0\\right) ,\\left( 1,m\\right)\n,\\left( 1,1\\right) ,\\left( \\alpha ,1\\right) \\right\\} ,$ it can be easily\nseen that $\\left( \\alpha ,m\\right) -$convexity reduces to increasing: \n\\alpha -$starshaped, starshaped, $m-$convex, convex and $\\alpha -$convex,\nrespectively.\n\nIn \\cite{SSOR}, Set et al. proved the following Hadamard type inequalities\nfor $\\left( \\alpha ,m\\right) -$convex functions.\n\n\\begin{theorem}\nLet $f:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) -$convex function with $\\left( \\alpha\n,m\\right) \\in \\left( 0,1\\right] ^{2}.$ If \\ $0\\leq a<b<\\infty $ and $f\\in $ \nL_{1}\\left[ a,b\\right] \\cap L_{1}\\left[ \\frac{a}{m},\\frac{b}{m}\\right] ,$\nthen the following inequality holds\n\\begin{equation}\nf\\left( \\frac{a+b}{2}\\right) \\leq \\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left(\nx\\right) +m\\left( 2^{\\alpha }-1\\right) f\\left( \\frac{x}{m}\\right) }\n2^{\\alpha }}dx.  \\label{h2}\n\\end{equation}\n\\end{theorem}\n\n\\begin{theorem}\nLet $f:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) -$convex function with $\\left( \\alpha\n,m\\right) \\in \\left( 0,1\\right] ^{2}.$ If \\ $0\\leq a<b<\\infty $ and $f\\in $ \nL_{1}\\left[ a,b\\right] ,$ then the following inequality holds\n\\begin{equation}\n\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\leq \\frac{1}{2}\\left[ \\frac\nf\\left( a\\right) +f\\left( b\\right) +m\\alpha f\\left( \\frac{a}{m}\\right)\n+m\\alpha f\\left( \\frac{b}{m}\\right) }{\\alpha +1}\\right] .  \\label{h3}\n\\end{equation}\n\\end{theorem}\n\nFor the recent results based on the above definition see the papers \\cit\n{BOP}, \\cite{BPR}, \\cite{OKS}, \\cite{OAK} and \\cite{SSO}.\n\nIn \\cite{OSA}, the power mean $M_{r}(x,y;\\lambda )$ of order $r$ of positive\nnumbers $x,y$ is defined b\n\\begin{equation*}\nM_{r}(x,y;\\lambda )=\\left\\{ \n\\begin{array}{cc}\n\\left( \\lambda x^{r}+\\left( 1-\\lambda \\right) y^{r}\\right) ^{\\frac{1}{r}}, & \nr\\neq 0 \\\\ \nx^{\\lambda }y^{1-\\lambda }, & r=0\n\\end{array\n\\right. \n\\end{equation*\nA positive function $f$ is $r-$convex on $[a,b]$ if for all $x,y\\in \\lbrack\na,b]$ and $\\lambda \\in \\lbrack 0,1]\n\\begin{equation}\nf(\\lambda x+(1-\\lambda )y)\\leq M_{r}(f\\left( x\\right) ,f\\left( y\\right)\n;\\lambda ).  \\label{h4}\n\\end{equation\nThe generalized logarithmic mean of order $r$ of positive numbers $x,y$ is\ndefined b\n\\begin{equation}\nL_{r}\\left( x,y\\right) =\\left\\{ \n\\begin{array}{cc}\n\\frac{r}{r+1}\\frac{x^{r+1}-y^{r+1}}{x^{r}-y^{r}}, & r\\neq 0,1,x\\neq y \\\\ \n&  \\\\ \n\\frac{x-y}{\\ln x-\\ln y}, & r=0,x\\neq y \\\\ \n&  \\\\ \nxy\\frac{\\ln x-\\ln y}{x-y}, & r=-1,x\\neq y \\\\ \n&  \\\\ \nx, & x=\n\\end{array\n\\right.   \\label{L}\n\\end{equation\nIn \\cite{GPP}, the following theorem was proved by Gill et al. for $r-\nconvex functions.\n\n\\begin{theorem}\nSuppose $f$ is a positive $r-$convex function on $\\left[ a,b\\right] .$ Then \n\\begin{equation}\n\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\leq L_{r}\\left( f\\left(\na\\right) ,f\\left( b\\right) \\right) .  \\label{h5}\n\\end{equation\nIf $f$ is a positive $r-$concave function, then the inequality is reversed.\n\\end{theorem}\n\nIn the following sections our main results are given: We establish several\nnew convex dominated functions and then we obtain new Hadamard type\ninequalities.\n\n\\section{$\\left( g-\\left( \\protect\\alpha ,m\\right) \\right) $-convex\ndominated functions}\n\n\\begin{definition}\n\\label{d2} Let $g:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n,$ $b>0$ be a given $\\left( \\alpha ,m\\right) $-convex function on the\ninterval $\\left[ 0,b\\right] $. The real function $f:\\left[ 0,b\\right]\n\\rightarrow \n\\mathbb{R}\n$ is called $\\left( g-\\left( \\alpha ,m\\right) \\right) $-convex dominated on \n\\left[ 0,b\\right] $ if the following condition is satisfie\n\\begin{eqnarray}\n&&\\left\\vert \\lambda ^{\\alpha }f(x)+m(1-\\lambda ^{\\alpha })f(y)-f\\left(\n\\lambda x+m\\left( 1-\\lambda \\right) y\\right) \\right\\vert  \\label{h6} \\\\\n&\\leq &\\lambda ^{\\alpha }g(x)+m(1-\\lambda ^{\\alpha })g(y)-g\\left( \\lambda\nx+m\\left( 1-\\lambda \\right) y\\right)  \\notag\n\\end{eqnarray\nfor all $x,y\\in \\left[ 0,b\\right] $, $\\lambda \\in \\left[ 0,1\\right] $ and \n\\left( \\alpha ,m\\right) \\in \\left[ 0,1\\right] ^{2}.$\n\\end{definition}\n\nThe next simple characterisation of $\\left( \\alpha ,m\\right) $-convex\ndominated functions holds.\n\n\\begin{lemma}\n\\label{l1} Let $g:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) $-convex function on the interval $\\left[\n0,b\\right] $ and the function $f:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n.$ The following statements are equivalent:\n\\end{lemma}\n\n\\begin{enumerate}\n\\item $f$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) $-convex dominated on \n$\\left[ 0,b\\right] .$\n\n\\item The mappings $g-f$ and $g+f$ are $\\left( \\alpha ,m\\right) $-convex\nfunctions on $\\left[ 0,b\\right] .$\n\n\\item There exist two $\\left( \\alpha ,m\\right) $-convex mappings $h,k$\ndefined on $\\left[ 0,b\\right] $ such tha\n\\begin{equation*}\n\\begin{array}{ccc}\nf=\\frac{1}{2}\\left( h-k\\right) & \\text{and} & g=\\frac{1}{2}\\left( h+k\\right\n\\end{array\n.\n\\end{equation*}\n\\end{enumerate}\n\n\\begin{proof}\n1$\\Longleftrightarrow $2 The condition (\\ref{h6}) is equivalent t\n\\begin{eqnarray*}\n&&g\\left( \\lambda x+m\\left( 1-\\lambda \\right) y\\right) -\\lambda ^{\\alpha\n}g(x)-m(1-\\lambda ^{\\alpha })g(y) \\\\\n&\\leq &\\lambda ^{\\alpha }f(x)+m(1-\\lambda ^{\\alpha })f(y)-f\\left( \\lambda\nx+m\\left( 1-\\lambda \\right) y\\right)  \\\\\n&\\leq &\\lambda ^{\\alpha }g(x)+m(1-\\lambda ^{\\alpha })g(y)-g\\left( \\lambda\nx+m\\left( 1-\\lambda \\right) y\\right) \n\\end{eqnarray*\nfor all $x,y\\in I$, $\\lambda \\in \\left[ 0,1\\right] $ and $\\left( \\alpha\n,m\\right) \\in \\left[ 0,1\\right] ^{2}.$ The two inequalities may be\nrearranged a\n\\begin{equation*}\n\\left( g+f\\right) \\left( \\lambda x+m\\left( 1-\\lambda \\right) y\\right) \\leq\n\\lambda ^{\\alpha }\\left( g+f\\right) (x)+m(1-\\lambda ^{\\alpha })\\left(\ng+f\\right) (y)\n\\end{equation*\nan\n\\begin{equation*}\n\\left( g-f\\right) \\left( \\lambda x+m\\left( 1-\\lambda \\right) y\\right) \\leq\n\\lambda ^{\\alpha }\\left( g-f\\right) (x)+m(1-\\lambda ^{\\alpha })\\left(\ng-f\\right) (y)\n\\end{equation*\nwhich are equivalent to the $\\left( \\alpha ,m\\right) $-convexity of $g+f$\nand $g-f,$ respectively.\n\n2$\\Longleftrightarrow $3 We define the mappings $f,g$ as $f=\\frac{1}{2\n\\left( h-k\\right) $ and $g=\\frac{1}{2}\\left( h+k\\right) $. Then, if we sum\nand subtract $f,g,$ respectively, we have $g+f=h$ and $g-f=k.$ By the\ncondition 2 of Lemma 1, the mappings $g-f$ and $g+f$ are $\\left( \\alpha\n,m\\right) $-convex on $\\left[ 0,b\\right] ,$ so $h,k$ are $\\left( \\alpha\n,m\\right) $-convex mappings too.\n\\end{proof}\n\n\\begin{theorem}\n\\label{t1} Let $g:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) -$convex function with $\\left( \\alpha\n,m\\right) \\in \\left( 0,1\\right] ^{2}$. $f:\\left[ 0,\\infty \\right)\n\\rightarrow \n\\mathbb{R}\n$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated mapping\nand $0\\leq a<b.$ If $f\\in L_{1}\\left[ a,b\\right] \\cap L_{1}\\left[ \\frac{a}{m\n,\\frac{b}{m}\\right] ,$ then the first inequality holds\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left( x\\right) +m\\left(\n2^{\\alpha }-1\\right) f\\left( \\frac{x}{m}\\right) }{2^{\\alpha }}dx-f\\left( \n\\frac{a+b}{2}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{1}{b-a}\\int_{a}^{b}\\frac{g\\left( x\\right) +m\\left( 2^{\\alpha\n}-1\\right) g\\left( \\frac{x}{m}\\right) }{2^{\\alpha }}dx-g\\left( \\frac{a+b}{2\n\\right)\n\\end{eqnarray*\nand if $f\\in L_{1}\\left[ a,b\\right] $ then the second inequality holds: \n\\begin{eqnarray*}\n&&\\left\\vert \\frac{1}{2}\\left[ \\frac{f\\left( a\\right) +mf\\left( \\dfrac{a}{m\n\\right) }{\\alpha +1}+m\\alpha \\frac{f\\left( \\frac{b}{m}\\right) +mf\\left( \n\\dfrac{b}{m^{2}}\\right) }{\\alpha +1}\\right] -\\frac{1}{b-a}\\int_{a}^{b}\\frac\nf\\left( x\\right) +mf\\left( \\frac{x}{m}\\right) }{2}dx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{g\\left( a\\right) +mg\\left( \\dfrac{a}{m\n\\right) }{\\alpha +1}+m\\alpha \\frac{g\\left( \\frac{b}{m}\\right) +mg\\left( \n\\dfrac{b}{m^{2}}\\right) }{\\alpha +1}\\right] -\\frac{1}{b-a}\\int_{a}^{b}\\frac\ng\\left( x\\right) +mg\\left( \\frac{x}{m}\\right) }{2}dx.\n\\end{eqnarray*}\n\\end{theorem}\n\n\\begin{proof}\nBy Definition \\ref{d2} with $\\lambda =\\frac{1}{2}$, as the mapping $f$ is \n\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated function, we\nhave tha\n\\begin{equation*}\n\\left\\vert \\frac{f\\left( x\\right) +m\\left( 2^{\\alpha }-1\\right) f\\left(\ny\\right) }{2^{\\alpha }}-f\\left( \\frac{x+my}{2}\\right) \\right\\vert \\leq \\frac\ng\\left( x\\right) +m\\left( 2^{\\alpha }-1\\right) g\\left( y\\right) }{2^{\\alpha \n}-g\\left( \\frac{x+my}{2}\\right)\n\\end{equation*\nfor all $x,y\\in \\left[ 0,\\infty \\right) $ and $\\left( \\alpha ,m\\right) \\in\n\\left( 0,1\\right] ^{2}.$ If we choose $x=ta+(1-t)b,$ $y=\\left( 1-t\\right) \n\\frac{a}{m}+t\\frac{b}{m}$ and $t\\in \\left[ 0,1\\right] ,$ then we ge\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{f\\left( ta+(1-t)b\\right) +m\\left( 2^{\\alpha }-1\\right)\nf\\left( \\frac{\\left( 1-t\\right) a+tb}{m}\\right) }{2^{\\alpha }}-f\\left( \\frac\na+b}{2}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{g\\left( ta+(1-t)b\\right) +m\\left( 2^{\\alpha }-1\\right) g\\left( \n\\frac{\\left( 1-t\\right) a+tb}{2}\\right) }{2^{\\alpha }}-g\\left( \\frac{a+b}{2\n\\right) .\n\\end{eqnarray*\nIntegrating over $t$ on $\\left[ 0,1\\right] $ we deduce tha\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{\\int_{0}^{1}f\\left( ta+(1-t)b\\right) dt+m\\left( 2^{\\alpha\n}-1\\right) \\int_{0}^{1}f\\left( \\frac{\\left( 1-t\\right) a+tb}{m}\\right) dt}\n2^{\\alpha }}-f\\left( \\frac{a+b}{2}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{\\int_{0}^{1}g\\left( ta+(1-t)b\\right) dt+m\\left( 2^{\\alpha\n}-1\\right) \\int_{0}^{1}g\\left( \\frac{\\left( 1-t\\right) a+tb}{m}\\right) dt}\n2^{\\alpha }}-g\\left( \\frac{a+b}{2}\\right)\n\\end{eqnarray*\nand so the first inequality is proved.\n\nSince $f$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated\nfunction, we hav\n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }f\\left( x\\right) +m(1-t^{\\alpha })f\\left( y\\right)\n-f\\left( tx+m(1-t)y\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }g\\left( x\\right) +m(1-t^{\\alpha })g\\left( y\\right)\n-g\\left( tx+m(1-t)y\\right) ,\\text{ for all }x,y>0\n\\end{eqnarray*\nwhich gives for $x=a$ and $y=\\frac{b}{m}\n\\begin{eqnarray}\n&&\\left\\vert t^{\\alpha }f\\left( a\\right) +m(1-t^{\\alpha })f\\left( \\frac{b}{m\n\\right) -f\\left( ta+m(1-t)\\frac{b}{m}\\right) \\right\\vert  \\label{h7} \\\\\n&&  \\notag \\\\\n&\\leq &t^{\\alpha }g\\left( a\\right) +m(1-t^{\\alpha })g\\left( \\frac{b}{m\n\\right) -g\\left( ta+m(1-t)\\frac{b}{m}\\right)  \\notag\n\\end{eqnarray\nand for $x=\\frac{a}{m}$, $y=\\frac{b}{m^{2}}$ and then multiply with $m$ \n\\begin{eqnarray}\n&&\\left\\vert mtf\\left( \\frac{a}{m}\\right) +m^{2}(1-t)f\\left( \\frac{b}{m^{2}\n\\right) -mf\\left( t\\frac{a}{m}+(1-t)\\frac{b}{m}\\right) \\right\\vert\n\\label{h8} \\\\\n&&  \\notag \\\\\n&\\leq &mtg\\left( \\frac{a}{m}\\right) +m^{2}(1-t)g\\left( \\frac{b}{m^{2}\n\\right) -mg\\left( t\\frac{a}{m}+(1-t)\\frac{b}{m}\\right)  \\notag\n\\end{eqnarray\nfor all $t\\in \\left[ 0,1\\right] .$ By properties of modulus, if we add the\ninequalities in $\\left( \\text{\\ref{h7}}\\right) $ and $\\left( \\text{\\ref{h8}\n\\right) $, we get \n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }\\left[ f\\left( a\\right) +mf\\left( \\frac{a}{m}\\right)\n\\right] +m(1-t^{\\alpha })\\left[ f\\left( \\frac{b}{m}\\right) +mf\\left( \\frac{\n}{m^{2}}\\right) \\right] \\right. \\\\\n&& \\\\\n&&-\\left. \\left[ f\\left( ta+m(1-t)\\frac{b}{m}\\right) +mf\\left( t\\frac{a}{m\n+(1-t)\\frac{b}{m}\\right) \\right] \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }\\left[ g\\left( a\\right) +mg\\left( \\frac{a}{m}\\right)\n\\right] +m(1-t^{\\alpha })\\left[ g\\left( \\frac{b}{m}\\right) +mg\\left( \\frac{\n}{m^{2}}\\right) \\right] \\\\\n&& \\\\\n&&-\\left[ g\\left( ta+m(1-t)\\frac{b}{m}\\right) +mg\\left( t\\frac{a}{m}+(1-t\n\\frac{b}{m}\\right) \\right] .\n\\end{eqnarray*\nThus, integrating over $t$ on $\\left[ 0,1\\right] $ we obtain the second\ninequality. The proof is completed.\n\\end{proof}\n\n\\begin{remark}\nIf we choose $\\alpha =1$ in Theorem \\ref{t1}, we get two inequalities of\nHermite-Hadamard type for functions that are $\\left( g,m\\right) -$convex\ndominated in Theorem \\ref{a}.\n\\end{remark}\n\n\\begin{theorem}\n\\label{t2} Let $g:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) -$convex function with $\\left( \\alpha\n,m\\right) \\in \\left( 0,1\\right] ^{2}$. $f:\\left[ 0,\\infty \\right)\n\\rightarrow \n\\mathbb{R}\n$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated \\ mapping\nand $0\\leq a<b.$ If $f\\in L_{1}\\left[ a,b\\right] ,$ then the following\ninequality holds\n\\begin{eqnarray}\n&&\\left\\vert \\frac{1}{2}\\left[ \\frac{f\\left( a\\right) +f\\left( b\\right)\n+m\\alpha f\\left( \\frac{a}{m}\\right) +m\\alpha f\\left( \\frac{b}{m}\\right) }\n\\alpha +1}\\right] -\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert\n\\label{h9} \\\\\n&&  \\notag \\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{g\\left( a\\right) +g\\left( b\\right) +m\\alpha\ng\\left( \\frac{a}{m}\\right) +m\\alpha g\\left( \\frac{b}{m}\\right) }{\\alpha +1\n\\right] -\\frac{1}{b-a}\\int_{a}^{b}g\\left( x\\right) dx  \\notag\n\\end{eqnarray}\n\\end{theorem}\n\n\\begin{proof}\nSince $f$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated\nfunction, we hav\n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }f\\left( a\\right) +m(1-t^{\\alpha })f\\left( \\frac{b}{m\n\\right) -f\\left( ta+m(1-t)\\frac{b}{m}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }g\\left( a\\right) +m(1-t^{\\alpha })g\\left( \\frac{b}{m\n\\right) -g\\left( ta+m(1-t)\\frac{b}{m}\\right)\n\\end{eqnarray*\nan\n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }f\\left( b\\right) +m(1-t^{\\alpha })f\\left( \\frac{a}{m\n\\right) -f\\left( tb+m(1-t)\\frac{a}{m}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }g\\left( b\\right) +m(1-t^{\\alpha })g\\left( \\frac{a}{m\n\\right) -g\\left( tb+m(1-t)\\frac{a}{m}\\right)\n\\end{eqnarray*\nfor all $t\\in \\left[ 0,1\\right] .$ Adding the above inequalities, we ge\n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }\\left[ f\\left( a\\right) +f\\left( b\\right) \\right]\n+m(1-t^{\\alpha })\\left[ f\\left( \\frac{a}{m}\\right) +f\\left( \\frac{b}{m\n\\right) \\right] -f\\left( ta+m(1-t)\\frac{b}{m}\\right) -f\\left( tb+m(1-t)\\frac\na}{m}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }\\left[ g\\left( a\\right) +g\\left( b\\right) \\right]\n+m(1-t^{\\alpha })\\left[ g\\left( \\frac{a}{m}\\right) +g\\left( \\frac{b}{m\n\\right) \\right] -g\\left( ta+m(1-t)\\frac{b}{m}\\right) -g\\left( tb+m(1-t)\\frac\na}{m}\\right) .\n\\end{eqnarray*\nIntegrating over $t\\in \\left[ 0,1\\right] $ and then by dividing the\nresulting inequality with $2,$ we get the desired result. The proof is\ncompleted.\n\nAnother proof can be done as the following.\n\nSince $f$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated,\nwe have by Lemma \\ref{l1} that $g+f$ and $g-f$ are $\\left( \\alpha ,m\\right)\n- $convex on $\\left[ 0,b\\right] ,$ and so by the Hadamard's type inequality\nfor $\\left( \\alpha ,m\\right) -$convex functions in $\\left( \\text{\\ref{h3}\n\\right) \n\\begin{eqnarray}\n&&\\frac{1}{b-a}\\int_{a}^{b}\\left( g+f\\right) \\left( x\\right) dx  \\label{h10}\n\\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{\\left( g+f\\right) \\left( a\\right) +\\left(\ng+f\\right) \\left( b\\right) +m\\alpha \\left( g+f\\right) \\left( \\frac{a}{m\n\\right) +m\\alpha \\left( g+f\\right) \\left( \\frac{b}{m}\\right) }{\\alpha +1\n\\right]  \\notag\n\\end{eqnarray\nan\n\\begin{eqnarray}\n&&\\frac{1}{b-a}\\int_{a}^{b}\\left( g-f\\right) \\left( x\\right) dx  \\label{h11}\n\\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{\\left( g-f\\right) \\left( a\\right) +\\left(\ng-f\\right) \\left( b\\right) +m\\alpha \\left( g-f\\right) \\left( \\frac{a}{m\n\\right) +m\\alpha \\left( g-f\\right) \\left( \\frac{b}{m}\\right) }{\\alpha +1\n\\right]  \\notag\n\\end{eqnarray\nBy using the inequalities in $\\left( \\text{\\ref{h10}}\\right) $ and $\\left( \n\\text{\\ref{h11}}\\right) $, we get the inequality in $\\left( \\text{\\ref{h9}\n\\right) $.\n\\end{proof}\n\n\\section{$\\left( g,r\\right) -$convex dominated functions}\n\n\\begin{definition}\n\\label{d3} Let positive function $g:\\left[ a,b\\right] \\rightarrow \n\\mathbb{R}\n$ be a given $r-$convex function on $\\left[ a,b\\right] $. The real function \nf:\\left[ a,b\\right] \\rightarrow \n\\mathbb{R}\n$ is called $\\left( g,r\\right) -$convex dominated on $\\left[ a,b\\right] $ if\nthe following condition is satisfied\n\\begin{eqnarray*}\n&&\\left\\vert M_{r}(f\\left( x\\right) ,f\\left( y\\right) ;\\lambda )-f\\left(\n\\lambda x+\\left( 1-\\lambda \\right) y\\right) \\right\\vert  \\\\\n&\\leq &M_{r}(g\\left( x\\right) ,g\\left( y\\right) ;\\lambda )-g\\left( \\lambda\nx+\\left( 1-\\lambda \\right) y\\right) \n\\end{eqnarray*\nfor all $x,y\\in \\left[ a,b\\right] $ and $\\lambda \\in \\left[ 0,1\\right] .$\n\\end{definition}\n\n\\begin{theorem}\nLet positive function $g:\\left[ a,b\\right] \\rightarrow \n\\mathbb{R}\n$ be an $r-$convex function on $\\left[ a,b\\right] $. $f:\\left[ a,b\\right]\n\\rightarrow \n\\mathbb{R}\n$ is $\\left( g,r\\right) -$convex dominated\\ mapping and $0\\leq a<b.$ If \nf\\in L_{1}\\left[ a,b\\right] ,$ then the following inequality holds\n\\begin{equation*}\n\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{1}\nb-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\leq L_{r}\\left( g\\left(\na\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a}\\int_{a}^{b}g\\left(\nx\\right) dx\n\\end{equation*\nfor all $x,y\\in I$, $\\lambda \\in \\left[ 0,1\\right] $ and $L_{r}\\left(\nf\\left( a\\right) ,f\\left( b\\right) \\right) $ as in (\\ref{L}).\n\\end{theorem}\n\n\\begin{proof}\nBy the Definition \\ref{d3} with $r=0,\\ f\\left( a\\right) \\neq f\\left(\nb\\right) ,$ we have \n\\begin{eqnarray*}\n&&\\left\\vert f^{\\lambda }\\left( a\\right) f^{1-\\lambda }\\left( b\\right)\n-f\\left( \\lambda a+\\left( 1-\\lambda \\right) b\\right) \\right\\vert \\\\\n&\\leq &g^{\\lambda }\\left( a\\right) g^{1-\\lambda }\\left( b\\right) -g\\left(\n\\lambda a+\\left( 1-\\lambda \\right) b\\right) .\n\\end{eqnarray*\nIntegrating the above inequality over $\\lambda $ on $\\left[ 0,1\\right] ,$ we\nhave \n\\begin{eqnarray*}\n&&\\left\\vert f\\left( b\\right) \\int_{0}^{1}\\left[ \\frac{f\\left( a\\right) }\nf\\left( b\\right) }\\right] ^{\\lambda }d\\lambda -\\int_{0}^{1}f\\left( \\lambda\na+\\left( 1-\\lambda \\right) b\\right) d\\lambda \\right\\vert \\\\\n&& \\\\\n&\\leq &g\\left( b\\right) \\int_{0}^{1}\\left[ \\frac{g\\left( a\\right) }{g\\left(\nb\\right) }\\right] ^{\\lambda }d\\lambda -\\int_{0}^{1}g\\left( \\lambda a+\\left(\n1-\\lambda \\right) b\\right) d\\lambda .\n\\end{eqnarray*\nBy a simple calculation we hav\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{f\\left( b\\right) -f\\left( a\\right) }{\\ln f\\left( b\\right)\n-\\ln f\\left( a\\right) }-\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right)\ndx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{g\\left( b\\right) -g\\left( a\\right) }{\\ln g\\left( b\\right) -\\ln\ng\\left( a\\right) }-\\frac{1}{b-a}\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nThe above inequality can written a\n\\begin{equation*}\n\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{1}\nb-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\leq L_{r}\\left( g\\left(\na\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a}\\int_{a}^{b}g\\left(\nx\\right) dx.\n\\end{equation*\nFor $r=0,\\ f\\left( a\\right) =f\\left( b\\right) ,$ we have with the same\ndevelopmen\n\\begin{eqnarray*}\n&&\\left\\vert f\\left( a\\right) -f\\left( \\lambda a+\\left( 1-\\lambda \\right)\nb\\right) \\right\\vert \\\\\n&\\leq &g\\left( a\\right) -g\\left( \\lambda a+\\left( 1-\\lambda \\right) b\\right)\n\\end{eqnarray*\nand this inequality can be written a\n\\begin{equation*}\n\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{1}\nb-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\leq L_{r}\\left( g\\left(\na\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a}\\int_{a}^{b}g\\left(\nx\\right) dx.\n\\end{equation*\nBy the Definition \\ref{d3} with $r\\neq 0,-1,\\ f\\left( a\\right) \\neq f\\left(\nb\\right) ,$ we hav\n\\begin{eqnarray*}\n&&\\left\\vert \\left( \\lambda f^{r}\\left( a\\right) +\\left( 1-\\lambda \\right)\nf^{r}\\left( b\\right) \\right) ^{\\frac{1}{r}}-f\\left( \\lambda a+\\left(\n1-\\lambda \\right) b\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\left( \\lambda g^{r}\\left( a\\right) +\\left( 1-\\lambda \\right)\ng^{r}\\left( b\\right) \\right) ^{\\frac{1}{r}}-g\\left( \\lambda a+\\left(\n1-\\lambda \\right) b\\right) .\n\\end{eqnarray*\nIntegrating the above inequality over $\\lambda $ on $\\left[ 0,1\\right] ,$ we\nhave \n\\begin{eqnarray*}\n&&\\left\\vert \\frac{r}{r+1}\\frac{f^{r+1}\\left( a\\right) -f^{r+1}\\left(\nb\\right) }{f^{r}\\left( a\\right) -f^{r}\\left( b\\right) }-\\frac{1}{b-a\n\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{r}{r+1}\\frac{g^{r+1}\\left( a\\right) -g^{r+1}\\left( b\\right) }\ng^{r}\\left( a\\right) -g^{r}\\left( b\\right) }-\\frac{1}{b-a\n\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nThe above inequality can be written a\n\\begin{eqnarray*}\n&&\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{\n}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\\\\n&\\leq &L_{r}\\left( g\\left( a\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a\n\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nFor$\\ r\\neq 0$ and $f\\left( a\\right) =f\\left( b\\right) ,$ we have similarl\n\\begin{eqnarray*}\n&&\\left\\vert \\left( f^{r}\\left( a\\right) \\right) ^{\\frac{1}{r}}-f\\left(\n\\lambda a+\\left( 1-\\lambda \\right) b\\right) \\right\\vert \\\\\n&\\leq &\\left( g^{r}\\left( a\\right) \\right) ^{\\frac{1}{r}}-g\\left( \\lambda\na+\\left( 1-\\lambda \\right) b\\right) .\n\\end{eqnarray*\nThen integrating the above inequality over $\\lambda $ on $\\left[ 0,1\\right]\n, $ we hav\n\\begin{eqnarray*}\n&&\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{\n}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\\\\n&\\leq &L_{r}\\left( g\\left( a\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a\n\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nFinally, let $r=-1.$ For$\\ f\\left( a\\right) \\neq f\\left( b\\right) $ we have\nagai\n\\begin{eqnarray*}\n&&\\left\\vert \\left( \\lambda f^{-1}\\left( a\\right) +\\left( 1-\\lambda \\right)\nf^{-1}\\left( b\\right) \\right) ^{-1}-f\\left( \\lambda a+\\left( 1-\\lambda\n\\right) b\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\left( \\lambda g^{-1}\\left( a\\right) +\\left( 1-\\lambda \\right)\ng^{-1}\\left( b\\right) \\right) ^{-1}-g\\left( \\lambda a+\\left( 1-\\lambda\n\\right) b\\right) .\n\\end{eqnarray*\nIntegrating the above inequality over $\\lambda $ on $\\left[ 0,1\\right] ,$ we\nhave \n\\begin{eqnarray*}\n&&\\left\\vert \\frac{f\\left( a\\right) f\\left( b\\right) }{f\\left( b\\right)\n-f\\left( a\\right) }\\int_{\\frac{1}{f\\left( a\\right) }}^{\\frac{1}{f\\left(\nb\\right) }}\\lambda ^{-1}d\\lambda -\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right)\ndx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{g\\left( a\\right) g\\left( b\\right) }{g\\left( b\\right) -g\\left(\na\\right) }\\int_{\\frac{1}{f\\left( a\\right) }}^{\\frac{1}{f\\left( b\\right) \n}\\lambda ^{-1}d\\lambda -\\frac{1}{b-a}\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nThe above inequality can be written a\n\\begin{eqnarray*}\n&&\\left\\vert L_{-1}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{\n}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\\\\n&\\leq &L_{-1}\\left( g\\left( a\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a\n\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nThe proof is completed.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPopulation estimation methods have been widely used in ecology to study the development of species over the last several decades~\\cite{krebs1989ecological}. Existing population estimation methods focus on open populations, where births, deaths, and migrations are taken into account, or closed population methods, where the population is assumed to remain static over the population estimation study. In this work, we focus on closed population methods and study how their respective population estimates can be used to evaluate an evaluation set of generated samples, given a reference set of real samples. In this work, samples are either contextualized word or sentence embeddings.\nWe study two mark-recapture methods, namely the Petersen~\\cite{petersen} and the Schnabel~\\cite{schnabel} estimators, where samples are captured and marked, released, and then recaptured. The number of marked, recaptured, and captured samples can then be used to estimate the population size. We additionally study one maximum-likelihood method, Program CAPTURE\n~\\cite{capture}, which uses the number of marked or unique samples over multiple captures to estimate the population size.\n\nAccurate evaluation of generated data is essential to correctly measure in what degrees we can improve the overall generation process. Depending on the use case, single-valued metrics may suffice to assess specific conditional language generation tasks, such as machine translation and text summarization, where we are interested in evaluating the similarities of a generated translation or summary to a specific reference translation or summary, respectively. \nOn the other hand, on unconditional language generation, for example, it may be useful to have separate measures for the diversity and quality of the generated set, enabling the identification of possible shortcomings of our generation system and try to fix it accordingly. This has been an active area of generative models research, specifically generative adversarial networks~\\cite{gans}, where several works have focused on stimulating diversity while maintaining the overall sample quality~\\cite{veegan,pacgan,dropoutgan,microbatchgan,sauder2020best}. Hence, depending on the context, a single-valued or double-valued metric may be more desirable.\n\nMark-Evaluate (ME) is a family of 3 novel language evaluation methods based on the above population estimation methods: ME$_\\text{Petersen}$ and ME$_\\text{CAPTURE}$ retrieve a single-valued metric to assess an evaluation set, while ME$_\\text{Schnabel}$ returns a double-valued metric, separately measuring the quality and diversity of the evaluation set.\nOur main contributions can be listed as follows:\n\\begin{enumerate*}[label=(\\roman*)]\n    \\item Proposal of 3 novel language metrics (Section~\\ref{sec:mark_evaluate}) that are sensitive to mode collapse (Section~\\ref{sec:mode_collapse}) and quality detriment (Section~\\ref{sec:swap}) and show a high correlation to human evaluation on challenging text generation tasks, such as unconditional language generation (Section~\\ref{sec:language_generation}), machine translation (Section~\\ref{sec:machine_translation}) and text summarization (Section~\\ref{sec:text_summarization}).\n    \\item In-depth study of the language assessment capability of popular existing metrics, \\textit{i.e.} FID~\\cite{fid}, PRD~\\cite{prd} and IMPAR~\\cite{impar}, primarily used to evaluate image generation in the past.\n    \\item Usage of contextual information and different levels of granularity to assess language, by using either contextualized word (Sections~\\ref{sec:machine_translation} and ~\\ref{sec:text_summarization}) or sentence embeddings (Sections~\\ref{sec:synthetic_experiments}, \\ref{sec:language_generation}) derived from BERT~\\cite{bert}.\n    \\item Code for the reproducibility of the results will be publicly available.\n\\end{enumerate*}\n\n\\section{Related work}\n\nWhile acknowledging the importance of traditional evaluation metrics, such as BLEU~\\cite{bleu}, ROUGE~\\cite{rouge} and METEOR~\\cite{meteor}, we will focus on the new trend of unsupervised methods that use embedding representations of pre-trained models to assess a set of evaluation samples. Our family of methods analyzes the data manifold to assess the evaluation set by using \\textit{k-nearest neighbors} to determine the capture volume. Several methods have been recently proposed to assess data generation using topological information, however, they were primarily intended to assess image generation~\\cite{prd,khrulkov2018geometry,impar,fti}.\nIn this work, we investigate the performance of such methods in the text domain, analyzing their behavior on synthetic experiments and their correlation with human evaluation.\n\nPrecision and recall for distributions (PRD) was proposed by~\\newcite{prd} and uses of \\textit{k-means}~\\cite{kmeans_2} to build histograms of the discrete reference and evaluation distributions over the clusters' centers. The evaluation distribution is then assessed in terms of relative probability densities. Precision is obtained by calculating the probability of an evaluation sample falling within the reference distribution's support. On the other hand, recall is retrieved by calculating the probability of a reference sample falling within the evaluation distribution's support.\n\n\\newcite{impar} suggested several improvements to the above method, which we call improved precision and recall (IMPAR). First, instead of \\textit{k-means}, they proposed to use \\textit{k-nearest neighbors} to approximate the reference and evaluation manifolds by building a hypersphere around each sample to its $k$-th nearest neighbor. Second, they simplify PRD's notions of precision and recall, by calculating the probability of an evaluation sample to fall within at least one reference sample's hypersphere, and vice-versa, respectively. The proposed manifold approximations by the usage of hyperspheres present a simple, yet effective way of representing the reference and evaluation manifold in an explicit, non-parametric way. We build upon this idea and use identical hyperspheres to determine the capture volume used by our different estimators to estimate the population size.\n\nFr\u00e9chet Inception Distance or FID~\\cite{fid} is a widely used single-valued metric that assesses data similarity by calculating the distance between the reference and evaluation distributions. Even though originally proposed for the image domain, \\newcite{semeniuta2018accurate} adapted FID to evaluate text generation by getting vector representations from InferSent~\\cite{infersent}, instead of Inception-V3~\\cite{inception}. Even though this metric precedes PRD and IMPAR, FID is still commonly used to assess generative models in the image and text domain.\n\nAs previously mentioned, we study both the usage of sentence embeddings, derived from SBERT~\\cite{sbert}, as well as contextualized word embeddings from BERT~\\cite{sbert}, which have been recently shown to improve language assessment, both in a supervised~\\cite{bertr,bleurt} and unsupervised manner~\\cite{moverscore,bertscore}.\nMore specifically, BERTScore measures precision and recall from a reference and evaluation text by calculating the required transport of each word of a given text to the most semantically similar word in the other text. On the other hand, MoverScore measures the semantic distance between two texts by calculating the minimum transport required between the reference and evaluation texts. \nThese metrics are ideal for conditional text generation, such as machine translation and text summarization, where an evaluation text should match a given reference text.\n\n\\section{Mark-Evaluate}\n\\label{sec:mark_evaluate}\n\nIn this work, we consider population estimation methods for closed populations, where the true population size remains constant throughout the estimation study. In our use case, our population consists of two sets, namely a reference set $S_r$ and an evaluation set $S_e$. The true population size ($P$) is then known \\textit{a priori} and represents the total number of samples in the two sets: $P=|S_r|+|S_e|$. Given an estimated population size ($\\widehat{P}$) from one of the used estimators, we measure the accuracy loss ($A$) as follows:\n\n\\begin{equation}\n\\label{eq:accuracy}\nA(P,\\widehat{P}) = \\text{max}\\Big(\\Big|\\frac{\\widehat{P}-P}{P}\\Big|,1\\Big),\n\\end{equation}\n\nwith a low accuracy loss, \\textit{i.e.} $A(P,\\widehat{P})\\approx 0$, representing a good population estimate, and a high accuracy loss, \\textit{i.e.} $A(P,\\widehat{P})\\approx 1$, otherwise.\nOur population estimation methods assume all samples to have an equal chance of capture, which is influenced by our capture volumes: hyperspheres that reaches each reference or evaluation sample's $k$-th nearest reference or evaluation neighbor, respectively.\nHence, if evaluation samples tend not to be inside any reference sample's hypersphere and vice-versa, the population estimate will likely be poor due to the lack of captured samples in the estimation study.\n\nOur methods can be separated into three categories: single marking and recapture (ME$_\\text{Petersen}$), multiple markings and recaptures (ME$_\\text{Schnabel}$) and multiple markings and captures (ME$_\\text{CAPTURE}$). Let us consider two sets of samples $S$ and $S'$, where each sample is a contextualized word embedding or a sentence embedding derived from BERT, depending on the task. \nFigure~\\ref{fig:mark_recapture} illustrates our family of methods. \n\n\\begin{figure}\n\\caption{Each sample $s \\in S$ and $s' \\in S'$ is represented as a blue and red circle, respectively. Marked samples are represented by filled circles. In this illustration, hyperspheres reach to each sample's nearest neighbor of the same set, \\textit{i.e.} $K=1$.\nFor ME$_\\text{Petersen}$ and ME$_\\text{Schnabel}$, we first capture and mark all samples inside any hypersphere of $s$. Then, for ME$_\\text{Petersen}$, we count the number of marked samples or recaptures inside any hypersphere of $s'$. For ME$_\\text{Schnabel}$, we perform a similar process iteratively, marking and recapturing samples inside the hypersphere of each $s'$, resulting in all samples being marked in the end.\nOn the other hand, ME$_\\text{CAPTURE}$ captures and marks samples inside each hypersphere of $s$ and $s'$.\n}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Figures\/method.pdf}\n\\label{fig:mark_recapture}\n\\end{figure}\n\nAdapting \\newcite{impar}'s formulations, we define a binary function $f$ that returns whether a sample $s' \\in S'$ lays inside any capture volume or hypersphere of a sample $s \\in S$:\n\n\\begin{equation}\n    f(s', S) = \n    \\begin{cases}\n        1, \\quad \\text{if} \\quad ||s'-s||_2 \\leq ||s - \\text{NN}_k(s, S)[-1]||_2 \\quad \\text{for at least one} \\quad s \\in S\\\\\n        0, \\quad \\text{otherwise,}\\\\\n    \\end{cases}\n\\end{equation}\n\nwhere NN$_k(s, S)$ returns an ordered set containing $s$ and its $k$-nearest neighbors in the set $S$, in ascending order of Euclidean distances to $s$. Hence, NN$_k(s, S)[-1]$ represents the $k$'th nearest neighbor of $s$.\nWe may refer to individual samples in $S$ and $S'$ as $\\{s_1,\\ldots,s_{|S|}\\}$ and $\\{s'_1,\\ldots,s'_{|S'|}\\}$, respectively.\n\nThe Petersen estimator~\\cite{petersen}, relies on a single marking step and a single recapture step. It merely assumes that the ratio of marked samples ($M$) in the marking step and the population size ($P$) is equivalent to the ratio of recaptured samples ($R$) and captured samples ($C$) in the recapture step. The population size estimate ($\\widehat{P}_\\text{Petersen}$) is then calculated as follows:\n\n\\begin{equation}\n\\label{eq:petersen}\n    \\widehat{P}_\\text{Petersen}(S, S') = \\dfrac{C(S, S') M(S, S')}{R(S, S')}.\n\\end{equation}\n\nDuring the marking step, we mark all samples inside at least one hypersphere of $s$: $M(S, S') = |S| + \\sum\\limits_{s' \\in S'} f(s', S)$. During the recapture step, we do the opposite, marking all samples inside at least one hypersphere of $s'$: $C(S, S') = |S'| + \\sum\\limits_{s \\in S} f(s, S')$. Additionally, in the recapture step, we count the number of captured samples that are already marked from the marking step, \\textit{i.e.} the recaptured samples. This corresponds to the number of samples in $S'$ inside at least one hypersphere of $s$ as well as the number of samples in $S$ inside at least one hypersphere of $s'$': $R(S, S') = \\sum\\limits_{s' \\in S'} f(s', S) + \\sum\\limits_{s \\in S} f(s, S')$.\n\n\\begin{comment}\nME$_\\text{Petersen}$, which returns a single-valued metric, can be defined as:\n\n\\begin{equation}\n    \\text{ME}_\\text{Petersen}(S,S') = 1 - \\text{max}\\Bigg(A\\Big(P,\\widehat{P}_\\text{Petersen}(S_r,S_e)\\Big),1\\Bigg).\n\\end{equation}\n\\end{comment}\n\nThe Petersen estimator was extended by \\newcite{schnabel} to incorporate multiple markings and recaptures. The population size estimate ($\\widehat{P}_\\text{Schnabel}$) is calculated from $T$ consecutive Petersen estimates:\n\n\\begin{equation}\n\\label{eq:schnabel}\n    \\widehat{P}_\\text{Schnabel}(S, S') = \\dfrac{C_T(S, S')M_T(S, S')}{R_T(S, S')},\n\\end{equation}\n\nThe set of marked samples at each iteration $t \\in \\{1,\\ldots,T\\}$, can be defined recursively as:\n\\begin{equation}\n    M(t, S, S') = \n    \\begin{cases}\n        S \\cup \\{s' \\in S'| f(s', S) = 1\\}, &\\text{if }t = 1,\\\\\n        S \\cup S', &\\text{if }t = T,\\\\\n        M(1,S,S') \\cup \\Bigg(\\bigcup\\limits_{i=1}^{t-1}\\text{NN}_k(s'_i, S')\\Bigg), &\\text{otherwise}.\n    \\end{cases}\n\\end{equation}\n\nME$_\\text{Schnabel}$'s first marking step is identical to ME$_\\text{Petersen}$'s single marking step ($M(1,S,S')=M(S,S')$), with all samples in $S$ as well as samples in $S'$ that are inside at least one hypersphere of $s$ being marked. By the final marking step, all samples will be marked since we iterated through all of them: $M_T(S,S') = |M(T,S,S')|$. For the other iterations, $1<t<T$, samples in $S'$ that are captured, \\textit{i.e.} are $k$-nearest neighbors of the $s'$ being iterated, but are not yet marked, are added to the marked set.\n\nAfter all recapture steps, which excludes the first marking step, the number of captured samples will be the number of samples in $S'$ and their respective $k$'th nearest neighbors as well as samples in $S$ that are inside the hypersphere of each $s'$: $C_T = (K+1)*|S'| + \\sum\\limits_{s' \\in S'} \\sum\\limits_{s \\in S} f(s, \\text{NN}_k(s', S'))$.\nSince all samples in $S$ have been marked in the first marking step, the number of total recaptures is the number of samples in $S$ inside the hypersphere of each $s'$ as well as the number of $k$-nearest neighbors of the iterated $s'$ that have already been marked: $R_T(S,S') = \\sum\\limits_{i = 1}^{|S'|} \\sum\\limits_{j = 1}^{|S|} \\Big(f(s_j, \\text{NN}_k(s'_i, S')) + |M(i,S,S') \\cap \\text{NN}_k(s'_i, S')|\\Big)$. \n\n\\begin{comment}\nME$_\\text{Schnabel}$ is thus defined as:\n\n\\begin{equation}\n    \\text{ME}_\\text{Schnabel}(S,S') = 1 - \\text{max}\\Bigg(A\\Big(P,\\widehat{P}_\\text{Schnabel}(S,S')\\Big), 1\\Bigg).\n\\end{equation}\n\\end{comment}\n\n\\begin{comment}\nAlternatively, \\cite{schumacher} proposed to use linear regression to estimate the number of samples marked prior to each time step and the proportion of samples marked in the current time step. At the last time step $T$, the population size can be estimated as follows:\n\n\n\\begin{equation}\n\\label{eq:schumacher}\n    \\widehat{P}_\\text{Schumacher}(S, S') = \\dfrac{C_T(S,S')M_T(S,S')^2}{R_T(S,S')M_T(S,S')},\n\\end{equation}\n\\end{comment}\n\n\\begin{comment}\nwith ME$_\\text{Schumacher}$ defined as:\n\n\\begin{equation}\n    \\text{ME}_\\text{Schumacher}(S,S') = 1 - \\text{max}\\Bigg(A\\Big(P,\\widehat{P}_\\text{Schumacher}(S,S')\\Big), 1\\Bigg).\n\\end{equation}\n\\end{comment}\n\nBoth ME$_\\text{Petersen}$ and ME$_\\text{Schnabel}$ are mark-recapture methods since they rely on marking and recapturing information to estimate the population size. We further used a maximum log-likelihood method: the model null from Program CAPTURE~\\cite{capture}. By considering the total number of marked samples ($M_T$) and the total number of captures ($C_{total}$) over $T$ iterations, with $T=|S \\cup S'|$, we iterate through several provisional population estimates ($P_\\text{CAPTURE} \\in \\mathbb{N}_{\\geq M}$) and compute their log-likelihood:\n\n\\begin{equation}\n\\label{eq:capture}\n\\begin{split}\n    L_n(P_\\text{CAPTURE};S,S') = \\ln\\Bigg(\\dfrac{P_\\text{CAPTURE}!}{(P_\\text{CAPTURE}-M_T(S,S'))!}\\Bigg) + C_{total}(S,S') \\times ln\\Big(C_{total}(S,S')\\Big)\\\\ + \\Big(TP_\\text{CAPTURE}-C_{total}(S,S')\\Big) \\times \\ln\\Big(TP_\\text{CAPTURE}-C_{total}(S,S')\\Big) - (TP_\\text{CAPTURE})\\ln(TP_\\text{CAPTURE}).\n\\end{split}\n\\end{equation}\n\nThe total number of captures corresponds to the number of samples in $S$ and $S'$ and their respective neighbors, as well as the number of samples in $S$ inside the hypersphere of a given $s'$ and vice-versa:\n$C_{total}(S,S') = \\sum\\limits_{s \\in S} \\sum\\limits_{s' \\in S'} ( f(s', \\text{NN}_k(s, S)) + |\\text{NN}_k(s, S)| ) + \\sum\\limits_{s' \\in S'} \\sum\\limits_{s \\in S} ( f(s, \\text{NN}_k(s', S')) + |\\text{NN}_k(s', S')|)$. \nThe final population estimate ($\\widehat{P}_\\text{CAPTURE}$) is then the estimate that maximizes Equation~\\ref{eq:capture}: \n\n\\begin{equation}\n\\label{eq:capture_2}\n\\widehat{P}_\\text{CAPTURE}(S,S') = \\argmax\\limits_{P_\\text{CAPTURE}}\\widehat{L_n}(P_\\text{CAPTURE};S,S').\n\\end{equation}\n\n\\begin{comment}\nME$_\\text{CAPTURE}$ is defined as:\n\n\\begin{equation}\n    \\text{ME}_\\text{CAPTURE}(S,S') = 1 - \\text{max}\\Bigg(A\\Big(P,\\widehat{P}_\\text{CAPTURE}(S,S')\\Big)\\Bigg).\n\\end{equation}\n\\end{comment}\n\nOur family of methods uses the accuracy loss of each estimator to compute their scores as follows:\n\n\\begin{equation}\n\\label{eq:mark_evaluate}\n    \\text{ME}_\\text{\\{Petersen, Schnabel, CAPTURE\\}}(S,S') = 1 - A\\Big(P,\\widehat{P}_\\text{\\{Petersen, Schnabel, CAPTURE\\}}(S,S')\\Big).\n\\end{equation}\n\nNote that, due to its iterative nature, ME$_\\text{Schnable}$ may be used to separately assess the quality and diversity of an evaluation set $S_e$ given a reference set $S_r$. More specifically, quality may be calculated by ME$_\\text{Schnable}(S_r, S_e)$, whereas diversity may be measured by ME$_\\text{Schnable}(S_e, S_r)$. On the other hand, ME$_\\text{Petersen}$ and ME$_\\text{CAPTURE}$ are single-valued metrics, since ME$_\\text{Petersen}(S_r,S_e)$ = ME$_\\text{Petersen}(S_e,S_r)$ and ME$_\\text{CAPTURE}(S_r,S_e)$ = ME$_\\text{CAPTURE}(S_e,S_r)$.\nWe refer to the Appendix for theoretical discussions. \n\nTo study the effects of different capture volumes, determined by different $K$, we used SBERT to get the sentence embeddings of 10k training sentences from MNLI~\\cite{williams2017broad} as the reference set, and 10k validation sentences as the evaluation set. Results are shown in Figure~\\ref{fig:num_k}, with $K \\in \\{1,\\ldots, 40\\}$. We observe that as $K$ increases, the population size estimated by all estimators converges to the true population size. In turn, the scores of our family of methods also converge to their maximum value of 1.\n\n\\begin{figure} \n\\caption{Effects of using different capture volumes, \\textit{i.e.} changing the number of neighbors ($K$), in the population size estimated of each estimator (left) and our respective method's score (right).}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{Figures\/num_k.pdf}\n\\label{fig:num_k}\n\\end{figure}\n\n\n\\begin{comment}\n\\begin{equation}\n  \\text{ME}(S_r,S_e) =\n    \\begin{cases}\n        \\text{ME}_\\text{Petersen}(S_r,S_e) =  \\text{ME}_\\text{Petersen}(S_e,S_r) = 1 - A(\\widehat{P}_\\text{Petersen}(S_r,S_e)),&\\text{(single)}\\\\\n        \\text{ME}_\\text{CAPTURE}(S_r,S_e) = \\text{ME}_\\text{CAPTURE}(S_e,S_r) = 1 - A(\\widehat{P}_\\text{CAPTURE}(S_r,S_e)),&\\text{(single)}\\\\\n        \\text{ME}_\\text{Schnabel}(S_r,S_e) = 1 - A(\\widehat{P}_\\text{Schnabel}(S_r,S_e)),&\\text{(precision)}\\\\\n        \\text{ME}_\\text{Schnabel}(S_e,S_r) = 1 - A(\\widehat{P}_\\text{Schnabel}(S_e,S_r)),&\\text{(recall)}\\\\\n        \\text{ME}_\\text{Schumacher}(S_r,S_e) = 1 - A(\\widehat{P}_\\text{Schumacher}(S_r,S_e)),&\\text{(precision)}\\\\\n        \\text{ME}_\\text{Schumacher}(S_e,S_r) = 1 - A(\\widehat{P}_\\text{Schumacher}(S_e,S_r)).&\\text{(recall)}\\\\\n    \\end{cases}\n\\end{equation}\n\\end{comment}\n\n\\section{Synthetic experiments}\n\\label{sec:synthetic_experiments}\n\n\\begin{figure}\n\\caption{Mode collapse experiment where sentences from certain topics are dropped from the evaluation set. Quality (full lines) is expected to remain constant while diversity (dotted lines) is expected to drop as mode collapse aggravates.}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{Figures\/mode_collapse.pdf}\n\\label{fig:mode_collapse}\n\\end{figure}\n\nTo simulate drops in quality and diversity, we used the MNLI dataset~\\cite{williams2017broad}, which consists of 433k sentence pairs annotated with one out of 5 possible topics. \nSince we were interested in the sentence-pair information for these experiments, we treated each sentence independently. Our reference set consists of sentences from the training set, whereas our evaluation set has sentences from the validation set. We kept the size of the reference and evaluation sets equal throughout our experiments to reduce possible method instabilities regarding sample size. We follow the experiments in \\newcite{semeniuta2018accurate} and simulate diversity loss by dropping sentences from certain topics (Section~\\ref{sec:mode_collapse}), whereas quality detriment is induced by swapping the words of each sentence (Section~\\ref{sec:swap}). For both experiments, we use SBERT embeddings from BERT-base pre-trained on SNLI\n~\\cite{snli} and MNLI~\\cite{williams2017broad} datasets ( \\textit{'bert-base-nli-mean-tokens'~\\footnote{https:\/\/github.com\/UKPLab\/sentence-transformers}}).\n\n\\subsection{Mode collapse}\n\\label{sec:mode_collapse}\n\nTo evaluate mode collapse, we dropped the sentences from specific topics from the evaluation set, containing sentences from all the available topics. Thus, the evaluation set only contains sentences from a subset of topics. What differs at each step is the number of topics included in the evaluation set: for example, dropping one topic means that the evaluation set only contains samples from the rest of the four available topics. The reference set remained unaltered throughout this process. We used 4k reference and 4k evaluation samples throughout this experiment.\n\nWe expect quality assessments to remain constant and diversity assessments to drop as fewer topics are represented in the evaluation set. For single-metric methods, we expect a detriment of the overall score throughout the mode dropping process. Results are presented in Figure~\\ref{fig:mode_collapse}, where we observe that our family of methods displays the expected behavior. IMPAR and FID also show expected performance (note that higher FID is worse since it represents a distance from the reference and evaluation distributions). On the other hand, PRD's quality assessment or precision drops significantly as mode collapse aggravates, which is not expected since the quality of the evaluation set is not affected in this experiment.\nWe further observe that all methods show high sensitivity when only one topic is represented in the evaluation set. For example, when 4 topics are dropped, IMPAR's quality assessment shifts by $\\approx 0.4$, while ME$_\\text{Schnabel}$ and ME$_\\text{Schnabel}$ shifts by $\\approx 0.5$. Hence, both methods show similar variance, despite the visualization contrast originated from different y-scales.\n\n\\subsection{Word swap}\n\\label{sec:swap}\n\nTo evaluate quality detriment, we swapped the words of each sentence in the evaluation set with a certain swap probability. Similarly to the mode collapse experiment, the reference set remains constant throughout this study. We used 10k reference and 10k evaluation samples. \n\\newcite{semeniuta2018accurate} showed that sentence embeddings derived from InferSent~\\cite{infersent} and Transformers~\\cite{vaswani2017attention} models were unable to detect similar quality perturbations. However, we observe that BERT embeddings can capture such quality detriment, observed by the variance of the scores of all the tested methods.\n\nFor this experiment, precision is expected to drop, while recall should remain constant. The overall score of single-valued metrics should deteriorate as the swap probability increases. Results are presented in Figure~\\ref{fig:swap}. Both the quality and diversity assessments of ME$_\\text{Schnabel}$ show the expected behavior, similarly to our single-metrics and FID. On the other hand, the recall or diversity assessment of both PRD and IMPAR drops unexpectedly. Moreover, IMPAR's quality or precision does not drop as significantly at higher swap probabilities, which is not desirable. \n\n\\begin{figure} \n\\caption{Swap experiment where words from each sentence in the evaluation set are swapped with a certain swap probability. Quality (full lines) is expected to drop while diversity (dotted lines) is expected to remain constant as the swap probability increases.}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{Figures\/swap.pdf}\n\\label{fig:swap}\n\\end{figure}\n\n\\section{Language generation}\n\\label{sec:language_generation}\n\nWe further assessed the text generated by ten different language generation models presented in \\newcite{eval_all}. The models include a traditional language model and several types of autoencoders, namely variational, adversarial, adversarially regularized, and plain autoencoders. We used the human ratings assigned to each model's fluency presented in their work to study the correlation of our family of methods and other tested metrics to human evaluation. \nReverse and forward cross-entropy, \\textit{i.e.} Reverse CE and Forward CE, have been commonly used to assess text generation in the past~\\cite{eval_all,semeniuta2018accurate,lm_ppl}. \nThe reported Reverse CE and Forward CE results were taken from \\newcite{eval_all}, obtained by training a language model on English Gigaword~\\cite{napoles2012annotated}.\nWe refer to \\newcite{eval_all} for additional details.\n\n\\begin{comment}\n\\begin{table} \n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\centering\n\\begin{adjustbox}{width=1.0\\textwidth}\n\\begin{tabular}{ccccccccc}\n\\toprule\n{\\bf Corr.} & Reverse CE & {FID} & {PRD} & {IMPAR} & {\\bf ME$_\\text{Schnabel}$} & {\\bf ME$_\\text{Schumacher}$} & {\\bf ME$_\\text{Petersen}$} & {\\bf ME$_\\text{CAPTURE}$} \\\\\n\\midrule\n\\bf $r$ & 0.440 & 0.902 & 0.830 (K=5) & 0.745 (K=2) & \\bf 0.917 (K=1) & \\bf \\underline{0.919} (K=1) & 0.872 (K=1) & \\textbf{0.902 (K=1)}\\\\\n\\bf $k$ & 0.333 & 0.867 & 0.822 (K=15) & 0.778 (K=2) & \\bf \\underline{0.911} (K=1) & \\bf \\underline{0.911} (K=1) & \\bf \\underline{0.911} (K=1) & \\bf 0.867 (K=1)\\\\\n\\bf $p$ & 0.491 & 0.964 & 0.939 (K=15) & 0.903 (K=2)  & \\bf \\underline{0.976} (K=1) & \\bf \\underline{0.976} (K=1) & \\bf \\underline{0.976} (K=1) & \\bf 0.964 (K=1)\\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{(best K, SBERT (Large)) Pearson, Kendall and Spearman correlations to system-level human evaluation of the different methods. Best values for each embedding type are underlined. Correlations where our methods outperform or match the other methods' performance are in bold. We observe that all contextual-based embeddings methods outperform the previous baselines on Absolute Pearson correlation. Moreover, contextual-based embeddings show higher correlation than non-contextual embeddings on Kendall. Absolute Pearson\/Kendall values for FID. Pearson scores for CAPTURE are with len(ref)+len(eval) occasions, whereas kendall and spearman scores are with 2 occasions.}\n\\label{tab:human_eval_correlations_best}\n\\end{table}\n\\end{comment}\n\n\\begin{table}[b] \n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\centering\n\\begin{tabular}{ccccccccc}\n\\toprule\n{\\bf Correlations} & Forward CE & Reverse CE & {FID} & {PRD} & {IMPAR} & {\\bf ME$_\\text{Schnabel}$} & {\\bf ME$_\\text{Petersen}$} & {\\bf ME$_\\text{CAPTURE}$} \\\\\n\\midrule\n\\bf Pearson $r$ & 0.606 & 0.440 & 0.902 & 0.830 & 0.745 & \\bf 0.917 & 0.872 & \\textbf{0.902}\\\\\n\\bf Kendall $k$ & 0.556 & 0.333 & 0.867 & 0.822 & 0.778 & \\bf \\underline{0.911} & \\bf \\underline{0.911} & \\bf 0.867\\\\\n\\bf Spearman $p$ & 0.697 & 0.491 & 0.964 & 0.939 & 0.903  & \\bf \\underline{0.976} & \\bf \\underline{0.976} & \\bf 0.964\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{Absolute correlations to human evaluation regarding the fluency of 10 different models with the best $K$. Best scores for each correlation are underlined. Bold values represent the correlations where our methods outperform or match all of the other methods' performance.\n}\n\\label{tab:human_eval_correlations_best}\n\\end{table}\n\nWe used SBERT embeddings from BERT-large trained on SNLI and MNLI datasets (\\textit{'bert-large-nli-mean-tokens'}) since they achieved the best-reported performance in\n~\\newcite{sbert}. Note that, since human evaluation is only related to each model's fluency, we only report the quality assessment scores for ME$_\\text{Schnabel}$, PRD, and IMPAR.\nFor our family of methods, as well as PRD and IMPAR, we iterate through $K$ values until correlation drops and present the results with the best $K$ of each method. Table~\\ref{tab:human_eval_correlations_best} shows the Pearson $p$, Kendall $k$, and Spearman $p$ correlations to human evaluation. \n\nOverall, our family of methods achieves the highest correlations to human evaluation. Note that despite being outperformed by FID, ME$_\\text{Petersen}$ still outperforms PRD and IMPAR across all correlations. Additional results with default $K$ for our family of methods, PRD, and IMPAR, as well as a comparison with InferSent and different SBERT embeddings, are provided in the Appendix.\n\n\\section{Contextualized word embeddings}\n\\label{sec:word_embeddings_mt}\n\nWe will now shift our focus to conditional language generation under finer-grained representations, \\textit{i.e.} contextualized word embeddings. We used embeddings from BERT-base fine-tuned on MNLI, identically to~\\newcite{moverscore}. For a fair comparison, we used the same embedding representations for all the methods in the following experiments. Due to the likely imbalance of reference and evaluation samples, we only report the quality assessment or precision of double-valued metrics. Similarly to Section\n~\\ref{sec:language_generation}, we report the results with the best $K$. Additional results with default $K$ can be found in the Appendix.\n\nUsing the information of the last layers of BERT has been shown to help in several downstream tasks~\\cite{liu_last_layers}. This has also been shown for language assessment, observed by, for example, the fact that the best performing layers of BERTScore are often latter layers~\\cite{bertscore}. MoverScore extends this thinking and aggregates the representations of the last five layers of BERT with $p$-means.\nFor our methods, instead of aggregating or routing this information, we use the vector representation from the last five layers for each specific word. Thus, each word has five representations, defined as five samples, in our scheme. See Figure~\\ref{fig:word_embeddings} for an illustration of this process. This also allows us to produce a better population estimate in the end due to the increase of the sample size.\n\n\\begin{figure} \n\\caption{When using contextualized word embeddings in our family of methods, each word is represented by five samples corresponding to the embeddings of the last five layers of BERT. We used the same procedure as Mark-Evaluate for PRD and IMPAR. On the other hand, MoverScore uses p-means to aggregate the information of the last five embeddings, and BERTScore uses the embeddings of a given layer for each word.}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{Figures\/word_embeddings.pdf}\n\\label{fig:word_embeddings}\n\\end{figure}\n\n\\subsection{Machine translation}\n\\label{sec:machine_translation}\n\nWe start by assessing system-level machine translations from the WMT17 metrics task~\\cite{wmt17}. We evaluated the different methods on the five language pairs provided by \\newcite{moverscore}'s implementation \\footnote{https:\/\/github.com\/AIPHES\/emnlp19-moverscore}. Namely, we assess translations from Czech (cs), German (de), Russian (ru), Turkish (tr), and Chinese (zh) to English (en). Each language pair has around 3k reference with the respective evaluation translations from multiple systems (the number of systems for each language pair varies).\n\n\\begin{comment}\n\\begin{table*}\n  \\centering\n  \\begin{adjustbox}{width=1.0\\textwidth}\n  \\begin{tabular}{cccccccccccccccccc}\n    \\hline\n    \\textbf{Translations} & BERTScore & MoverScore & PRD & IMPAR & \\textbf{ME$_\\text{Schnabel}$} & \\textbf{ME$_\\text{Schumacher}$} & \\textbf{ME$_\\text{Petersen}$} & \\textbf{ME$_\\text{CAPTURE}$}\\\\\n    \\hline\n    \\bf cs-en ($r$) & 0.966 & 0.983 & \\underline{0.992} & 0.987 (K=2) & 0.989 (K=2) & 0.990 (K=2) & 0.988 (K=2) & 0.987 (K=2)\\\\ \\hline\n    \\bf de-en ($r$) & 0.859 & 0.920 & 0.769 & 0.934 (K=3) & \\bf 0.944 (K=2) & \\bf 0.946 (K=2) & \\bf \\underline{0.953} (K=1) & \\bf \\underline{0.953} (K=1)\\\\ \\hline\n    \\bf ru-en ($r$) & 0.868 & 0.921 & \\underline{0.933} & 0.896 (K=2) & 0.902 (K=2) & 0.900 (K=2) & 0.908 (K=2) & 0.908 (K=2)\\\\ \\hline\n    \\bf tr-en ($r$) & 0.938 & 0.931 & 0.935 & 0.959 (K=6) & \\bf \\underline{0.970} (K=6) & \\bf 0.969 (K=6) & \\bf 0.960 (K=2) & \\bf 0.959 (K=2)\\\\ \\hline\n    \\bf zh-en ($r$) & 0.894 & 0.943 & 0.889 & 0.933 (K=1) & \\bf 0.957 (K=1) & \\bf \\underline{0.959} (K=2) & 0.936 (K=1) & 0.936 (K=1)\\\\ \\hline\n    \\bf Average ($r$) & 0.905 & 0.940 & 0.904 & 0.942 & \\bf 0.952 & \\bf \\underline{0.953} & \\bf 0.949 & \\bf 0.949\\\\ \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n  \\caption{System-level results for the WMT17 metrics task with best $K$. The best correlation of each language pair is underlined. Correlations where our methods outperform or match all of the other methods' are highlighted in bold.\n  }\n  \\label{tab:mt_word_best}\n\\end{table*}\n\\end{comment}\n\n\\begin{table*}\n  \\centering\n  \\begin{adjustbox}{width=1.0\\textwidth}\n  \\begin{tabular}{ccccccccccccccccc}\n    \\hline\n    \\textbf{Translations} & BERTScore & MoverScore & PRD & IMPAR & \\textbf{ME$_\\text{Schnabel}$}  & \\textbf{ME$_\\text{Petersen}$} & \\textbf{ME$_\\text{CAPTURE}$}\\\\\n    \\hline\n    \\bf cs-en ($r$) & 0.966 & 0.983 & \\underline{0.992} & 0.987 & 0.989 & 0.988 & 0.987\\\\ \\hline\n    \\bf de-en ($r$) & 0.859 & 0.920 & 0.769 & 0.934 & \\bf 0.944 & \\bf \\underline{0.953} & \\bf \\underline{0.953}\\\\ \\hline\n    \\bf ru-en ($r$) & 0.868 & 0.921 & \\underline{0.933} & 0.896 & 0.902 & 0.908 & 0.908\\\\ \\hline\n    \\bf tr-en ($r$) & 0.938 & 0.931 & 0.935 & 0.959 & \\bf \\underline{0.970} & \\bf 0.960 & \\bf 0.959\\\\ \\hline\n    \\bf zh-en ($r$) & 0.894 & 0.943 & 0.889 & 0.933 & \\bf \\underline{0.957} & 0.936 & 0.936 \\\\ \\hline\n    \\bf Average ($r$) & 0.905 & 0.940 & 0.904 & 0.942 & \\bf \\underline{0.952} & \\bf 0.949 & \\bf 0.949\\\\ \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n  \\caption{Pearson correlations on the WMT17 metrics task. The best correlation of each language pair is underlined, while correlations of our methods that outperform or match any other method are in bold.\n  }\n  \\label{tab:mt_word_best}\n\\end{table*}\n\nPearson ($r$) correlations with human evaluation are presented in Table~\\ref{tab:mt_word_best}. Our family of metrics outperforms all the rest in several language pair translations. Moreover, our metrics show the highest correlation to human evaluation when considering the average correlation across all language pairs. BERTScore results were calculated using the embeddings from the last fifth layer of the aforementioned BERT model. \n\n\\subsection{Text summarization}\n\\label{sec:text_summarization}\n\nWe further assessed text summarization with the TAC-2009 dataset\\footnote{http:\/\/tac.nist.gov\/}, consisting of news articles from ten different topics, with four reference summaries and fifty-five evaluation summaries from summarization systems per article. We evaluate each evaluation summary independently, performing a summary-level evaluation. Two scores were assigned to each evaluation summary: the \\textit{pyramid score}, which evaluates the semantic similarity between the reference and evaluation summaries, and the \\textit{responsiveness score}, that measures the overall quality of the evaluation summary in terms of grammar and content.\n\n\nTable~\\ref{tab:human_eval_text_summarization} shows the Kendall ($k$), Pearson ($r$), and Spearman ($p$) correlation to human evaluation for each score. Considering Kendall and Spearman correlations, our family of methods outperforms all the rest on responsiveness score. Moreover, ME$_\\text{Petersen}$ and ME$_\\text{CAPTURE}$ outperform all methods on the above correlations on the pyramid score. \nConsidering Pearson correlation, our family of methods outperforms PRD, and at least one of our metrics consistently outperforms IMPAR on both scores. We hypothesize that the lower Pearson correlations of our metrics could be explained by the instability of the population estimation process due to the low amount of samples, \\textit{i.e. }reference and evaluation words.\n\n\\begin{comment}\n\\begin{table} \n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\centering\n\\begin{adjustbox}{width=1.0\\textwidth}\n\\begin{tabular}{cccccccccc}\n\\toprule\n\\bf Metrics & BERTScore & MoverScore & PRD & IMPAR & \\bf ME$_\\text{Schnabel}$ & \\bf ME$_\\text{Schumacher}$ & \\bf ME$_\\text{Petersen}$ & \\bf ME$_\\text{CAPTURE}$\\\\\n\\midrule\n\\bf Responsiveness ($k$) & - & 0.482 & 0.398 & 0.481 (K=5) & \\textbf{0.483} (K=4) & \\textbf{0.483} (K=4) & \\underline{\\textbf{0.487}} (K=4) & \\textbf{0.484} (K=5) \\\\\n\\bf Responsiveness ($r$) & 0.739 & \\underline{0.754} & 0.564 & 0.743 (K=5) & 0.739 (K=3) & 0.740 (K=3) & 0.683 (K=1) & 0.747 (K=4) \\\\\n\\bf Responsiveness ($p$) & 0.580 & 0.594 & 0.501 & 0.594 (K=5) & \\textbf{0.595} (K=4) & \\textbf{0.596} (K=4) & \\underline{\\textbf{0.598}} (K=5) & \\textbf{0.596} (K=7) \\\\\n\\midrule\n\\bf Pyramid ($k$) & - & 0.550 & 0.444 & 0.541 (K=5) & 0.548 (K=4) & \\textbf{0.550} (K=4) & \\textbf{0.555} (K=9) & \\underline{\\textbf{0.565}} (K=7)\\\\\n\\bf Pyramid ($r$) & 0.823 & \\underline{0.831} & 0.658 & 0.804 (K=7) & 0.813 (K=4) & 0.814 (K=4) & 0.770 (K=3) & 0.808 (K=4) \\\\\n\\bf Pyramid ($p$) & 0.703 & 0.701 & 0.588 & 0.693 (K=5) & 0.698 (K=4) & 0.700 (K=4) & \\textbf{0.704} (K=5) & \\underline{\\textbf{0.718}} (K=11)\\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{Summary-level correlations to human evaluation on TAC 2009. Bold values represent correlations from our proposed metrics that outperform or math all the other methods. The best correlations for each score are underlined. BERTScore results were taken from~\\cite{moverscore}\n}\n\\label{tab:human_eval_text_summarization}\n\\end{table}\n\\end{comment}\n\n\\begin{table} \n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\centering\n\\begin{tabular}{ccccccccc}\n\\toprule\n\\bf Metrics & BERTScore & MoverScore & PRD & IMPAR & \\bf ME$_\\text{Schnabel}$ & \\bf ME$_\\text{Petersen}$ & \\bf ME$_\\text{CAPTURE}$\\\\\n\\midrule\n\\bf Responsiveness ($k$) & - & 0.482 & 0.398 & 0.481 & \\textbf{0.483} & \\underline{\\textbf{0.487}} & \\textbf{0.484} \\\\\n\\bf Responsiveness ($r$) & 0.739 & \\underline{0.754} & 0.564 & 0.743 & 0.739 & 0.683 & 0.747\\\\\n\\bf Responsiveness ($p$) & 0.580 & 0.594 & 0.501 & 0.594 & \\textbf{0.595} & \\underline{\\textbf{0.598}} & \\textbf{0.596} \\\\\n\\midrule\n\\bf Pyramid ($k$) & - & 0.550 & 0.444 & 0.541 & 0.548 & \\textbf{0.555} & \\underline{\\textbf{0.565}}\\\\\n\\bf Pyramid ($r$) & 0.823 & \\underline{0.831} & 0.658 & 0.804 & 0.813 & 0.770 & 0.808 \\\\\n\\bf Pyramid ($p$) & 0.703 & 0.701 & 0.588 & 0.693 & 0.698 & \\textbf{0.704} & \\underline{\\textbf{0.718}}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{Summary-level correlations to human evaluation on TAC 2009. Bold values represent correlations from our proposed metrics that outperform or match all the other methods. The best correlations of each score are underlined. BERTScore results were taken from~\\newcite{moverscore}\n}\n\\label{tab:human_eval_text_summarization}\n\\end{table}\n\n\n\\section{Conclusion}\n\nIn this work, we present a family of methods derived from popular population size estimators that have been widely used in ecology in the past several decades. We show that our family of methods is able to assess language systems under different representations effectively, \\textit{i.e.} using contextualized word and sentence embeddings. Our methods show a high correlation to human evaluation on challenging language generation tasks as well as the desired sensitivity to detect mode collapse and quality detriment.\n\nIn the future, we would like to evaluate our family of metrics on image generation tasks, reinforcing the general applicability of our methods. Moreover, we plan to extend our family of methods to also cover popular open populations estimation methods, where the population size may vary over time. In the end, we hope that combining the information from closed and open population methods will improve the overall assessment of language systems, further fostering the adoption of ecology methods in NLP.\n\n\\bibliographystyle{coling}\n\n\\section{Credits}\n\nThis document has been adapted from the instructions for  \nCOLING-2018 proceedings compiled by Xiaodan Zhu and Zhiyuan Liu,\nwhich are, in turn, based on\nthe instructions for\nCOLING-2016 proceedings compiled by Hitoshi Isahara and Masao Utiyama,\nwhich are, in turn, based on\nthe instructions for\nCOLING-2014 proceedings compiled by Joachim Wagner, Liadh Kelly\nand Lorraine Goeuriot,\nwhich are, in turn, based on the instructions for earlier ACL proceedings,\nincluding \nthose for ACL-2014 by Alexander Koller and Yusuke Miyao,\nthose for ACL-2012 by Maggie Li and Michael\nWhite, those for ACL-2010 by Jing-Shing Chang and Philipp Koehn,\nthose for ACL-2008 by Johanna D. 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+{"text":"\n\\section*{Acknowledgement}\n\n\n\n\n\\section{Introduction}\\label{intro}\n\nFinite Element Analysis (FEA), which plays a critical role in modern engineering design and quality evaluations, provides numerical solutions for boundary value problems and simulates the distribution of physical fields with high precision \\cite{axelsson2001finite}. However, FEA generally requires high-performance computational resources and has an expensive time cost which becomes a bottleneck in the present manufacturing industry, especially for advanced time-sensitive techniques: inverse modeling, agile manufacturing, generative design system, etc. In order to solve this problem, many researchers have employed machine learning technologies to regress the distribution of physical fields to obtain instant simulation results while compromising the accuracy. Some researchers employed conventional machine learning methods (Support Vector Regression-SVG \\cite{capuano2019smart}, Artificial Neural Network - ANN \\cite{tamaddon2020data}) to approximate the physical field distribution and solve the boundary value problem. However, most methods are for specific boundary conditions and require extra training ahead of the application. This limits their capability of serving as a general FEA surrogate model, and their ability to solve abstract decision-making problems.\n\nRecently, with the rapid development of Deep Learning (DL) techniques, some researchers employed Convolutional Neural Network (CNN) as the backbone for solving physical field prediction problems, and found out that CNN based surrogate models cannot only solve basic boundary value problems (e.g., solid mechanics \\cite{nie2020stress, kantzos2019design, khadilkar2019deep}, fluid dynamics \\cite{kutz2017deep, guo2016convolutional, zhang2018application}, etc.) but also perform efficiently on some abstract design problems based on predicted physical field, for instance, topology optimization \\cite{lee2020cnn, nie2021topologygan,zhang2019deep,banga20183d}. However, CNN-based surrogate models still have inevitable disadvantages. Firstly, CNN can only accept input tensor with fixed input size, which becomes an obstacle for input tensor encoding, especially for some complex FEA conditions. Also, since CNN can only perform calculations on Euclidean coordinates, most CNN-based surrogate models employ square-shaped or cube-shaped meshes (2D quadrilateral or 3D hexahedron meshes with uniform side lengths) that require homogeneous grid generation on the input FEA model. The grid resolution sometimes cannot reach a sufficiently high level since the CNN usually cannot handle a large tensor input with hundreds of meshes along X\/Y\/Z axes due to the limitation of modern computation resources. Lastly, most FEA run on structured meshes (e.g., triangular meshes, or polygon meshes with different side lengths) that cannot directly fit into the CNN input tensor. Some researchers have presented alternate solutions to handle this problem, which is illustrated in Fig.~\\ref{fig:0_1_FemCompare}. We provide a von Mises stress distribution prediction for a 2D cantilever beam based on simulation results from ABAQUS \\cite{abaqus2011abaqus}. The beam is made of A36 steel with Young's modulus 200 GPa and Poisson's ratio of 0.32 and has an elliptical hole inside. Its left edge is fixed and an external force 1000N is exerted on the center of its right edge. The ground truth simulation result with triangular meshes (with an approximate global mesh size of 1.0 mm) is shown in Fig.~\\ref{fig:0_1_FemCompare}. One type of method for using CNN to regress the physical field is to approximate the original input model's geometry with a high-resolution voxelized representation, shown in \"Simulation results with quadrilateral meshes\" in Fig.~\\ref{fig:0_1_FemCompare} \\cite{nie2020stress}. However, the resolution is still limited by computational resources, and the simulation result can significantly change since the change in model shape can largely influence the global boundary condition, shown in Fig.~\\ref{fig:0_1_FemCompare}. The other type of method usually uses some post-process tasks, for instance, averaging the physical fields to fit into the square\/cube-shaped meshes. This will still lead to extra approximation error for the output results, shown in \"Average the results\" in Fig.~\\ref{fig:0_1_FemCompare}. All of those defects call for a more efficient DL surrogate model for FEA. The Graph Neural Network (GNN) comes into sight.\n\n\\begin{figure}[htp]\n\\vspace{-0.12in}\n\\centering\n\t\\includegraphics[keepaspectratio,width=0.7\\textwidth]{fig\/0_1_FemCompare.png}\n\t\\caption{Comparison between different FEM simulation and prediction results}\n\t\\label{fig:0_1_FemCompare}\n\t\\vspace{-0.12in}\n\\end{figure}\n\n\nThe GNN is a generalized version of CNN for non-Euclidean graphs, which has been widely applied to abstract decision-making and data-regression problems on different graph data structures, for instance: point cloud, triangular meshes, and molecular structures \\cite{alet2019graph, wang2020kalibre, belbute2020combining, guo2020semi, ogoke2020graph,  pfaff2020learning, sanchez2020learning, gilmer2017neural}. Since the structured meshes can be naturally embedded by an undirected graph $G$, many researchers investigated GNN-based surrogate models to solving FEA regression problems, show in Table~\\ref{tab:0_1_Compare}. It is assumed that an undirected graph $G=(V, E)$ represents the embedded structured meshes in an FEA model. Then, generally, $V={1, \\cdots, N}$ can be the set of N graph vertices (or graph nodes) denoting N geometrical node points in FEA while $ E \\subseteq V \\times V$ is the set of graph links (or graph edges) representing the connection of neighborhood geometrical nodes. Graph vertex $v$ contains features $f_v$ encoding the properties of a geometrical node point (e.g., coordinates, material properties, boundary conditions). Graph links generally employ adjacent matrix $E \\in [0,1]^{N \\times N}$ to describe the connection between geometrical node points. Nowadays, most applications employed GNN layers with Message Passing Neural Network (MPNN) framework \\cite{gilmer2017neural} whose updated message $m_v^{t+1}$ of graph vertex $v$ at layer $t+1$ can be represented by:\n\n\\begin{equation}\nm_v^{t+1} = \\sum_{\\omega \\in N(v)}^{} M_t (h_v^{t}, h_\\omega^{t}, e_{v\\omega})\n\\label{mpnn1}\n\\end{equation}\n\\begin{equation}\nh_v^{t+1}=U_t(h_v^{t}, m_v^{t+1})\n\\label{mpnn2}\n\\end{equation}\n\\begin{equation}\n\\hat{p}=R( \\left \\{ h_v^{T}|v \\in G \\right \\})\n\\label{mpnn3}\n\\end{equation} \n\\newline Where $M_t$, $U_t$ and $R$ are message function, vertex update function and readout function, respectively; $h$ represents the hidden state in GNN layers (e.g., $h_v^{t}$ is the hidden state of $v$ at layer $t$); $N(v)$ denotes the neighbors of $v$ in graph $G$; $e_{v\\omega}$ represents the graph link features; $\\hat{p}$ is the final prediction results. From the Eq.~(\\ref{mpnn2}), it can be seen that GNN layers can only pass the information of the embedded feature to its neighbor vertices. If the distance on a graph for passing an important message is relatively long, with current graph embedding approaches, more layers of GNN are needed. For instance, this can happen when GNN is employed to pass the information from the element with boundary condition information (external force or fixture, e.g., element marked by \"Element with force information\" in Fig.~\\ref{fig:0_1_FemCompare}) to the element whose stress field is required to be solved (\"Target element\" in Fig.~\\ref{fig:0_1_FemCompare}). One of the shortest message propagation paths from the boundary element to the target element is marked with red shown in Fig.~\\ref{fig:0_1_FemCompare}, which requires several layers of GNN to pass the information. However, deeply stacking the GNN layers can suffer from over-smoothing problems which cause indistinguishable graph-vertex embeddings and make GNN eventually not predict anything \\cite{li2020deepergcn, yang2020revisiting}. In general, this problem restricts most GNN models to around 3 layers with only short-range graph-vertex interactions and limited message passing distance.\n\n\\input{tab\/0_1_tab}\n\nGiven the limitations of GNN, from Table~\\ref{tab:0_1_Compare}, it can be seen that the GNN is effective on two types of physical field regression problems: The first type is the problems which input a small-scale graph data into GNN with generally no more than 100 graph vertices, and usually requires data pre-processing or model simplification \\cite{alet2019graph, wang2020kalibre}; the other type can solve the large-scale-graph problem, however, has only short-range graph-vertex interactions. This means the graph distance for the propagation of boundary information is relatively short since the elements embedding boundary information is relatively close to the target element whose physical field is required to be solved. For instance, Pfaff et al. \\cite{pfaff2020learning} developed an efficient GNN model for dynamic simulations based on triangular meshes. However, since the dataset contained sequential physical fields (displacement field, velocity field, stress field, etc.) and the input physical field for the current frame (or simulation step) was derived from the previous frame, the boundary condition information can slowly propagate from the boundary elements to the elements inside the model frame by frame. Then, the physical field of the target element was only determined by its surrounding elements that embed boundary information, which only requires the short-range graph-vertex interaction. The same happens when the regression model works on sequential physical field predictions(e.g., \\cite{sanchez2020learning}). Ogoke et al. \\cite{ogoke2020graph} predicted drag force on a large graph using velocity fields , but only elements around the streamlined object contributed to the total force. Although some researchers successfully developed efficient GNN based prediction model on large graphs, their applications were still limited. Belbute-Peres et al. \\cite{ belbute2020combining} developed a GNN based fluid dynamics prediction model on large graphs by interpolating PDE solver's results, which had a relatively short-range graph distance for the propagation of boundary-condition information since the result from the PDE solver carried the model's boundary condition. Guo et al. \\cite{ guo2020semi} developed a semi-supervised GNN model to predict optimized architected material structure on a large graph, but the selected training dataset was chosen from inside of the material which embedded boundary conditions. Also, the training needed to be conducted before each application in \\cite{ guo2020semi}.\n\nFrom above, it can be seen that there's no ready-to-use generalized machine learning approach for solving structured-mesh-based boundary value problems, especially when long-range graph-vertex interactions are involved. FEA generally requires structured meshes to accurately approximate the boundary of the input model geometry. It usually provides finer meshes to the model's delicate structures to improve the computation accuracy and employ coarse meshes to save the computational resource. The CNN-based FEA surrogate models have difficulty approximating the structured-mesh-based results, which becomes a bottleneck for the efficiency of related algorithms. The GNN-based approach shows its unique advantage of regressing structured-mesh-based results, however, a precise FEA model usually requires fine meshing to provide a high-resolution result, which can largely increase the graph size and therefore extends the propagation distance of the boundary information. This requires a deep GNN model and usually can cause the oversmoothing problem that is difficult to solve. Also, when GNN is applied to solve the static boundary value problems (different from the dynamic problems using the sequential physical field as its input with multiple frames\\cite{pfaff2020learning, sanchez2020learning}), the target element needs to gather all the boundary condition information in one frame (or simulation step) and then provides the prediction result. For instance, in the cantilever beam problem, the target element stress field is determined by all the elements with the boundary information (fixture, model's geometrical edges, external forces, shown in Fig.~\\ref{fig:0_1_FemCompare}), and this cannot be solved by any of the current approaches. Therefore, in order to solve these problems and provide a potential general DL-based surrogate regression model, we focus on the most fundamental solid mechanics problem - cantilever beam problem and develop a Boundary Oriented Graph Embedding (BOGE) approach. With the novel BOGE method, the GNN model works effectively on high-resolution triangular-mesh-based FEA simulation and predicts the accurate distribution of von Miss stress. The topological optimization predicted by GNN shows the capability of  our model to conduct abstract decision-making results based on the model's boundary condition. In this paper, Section \\ref{data} introduces our novel data representation which provides both multi-hop local adjacent information and shortcuts for global boundary information. Section \\ref{dataset} describes the details of our dataset preparation and the parametric designs for the FEA simulation results. Section \\ref{train} shows the model's training results that \\textcolor{blue}{achieves the regression with MSE of 0.011706 (2.41\\% MAPE) for stress field prediction and 0.002735 MSE (with 1.58\\% elements having error larger than 0.01) for topological optimization.} The outperformance of our model's prediction accuracy as well as its irreplaceable properties on regression of FEA results shows our models' un-substitutable significance on fast FEA simulations. We summarize the advantages of our GNN based surrogate model as follows:\n\\begin{itemize}[leftmargin=*]\n\\input{0_highlights}\n\\end{itemize}\n\n\n\n\\section{Data representation}\\label{data}\n\nThe physical field distribution is generally derived by resolving partial differential equations under the given boundary condition. For the stress analysis of the cantilever beam problem, denote the cantilever beam's geometry as $\\Omega$. Then, the global boundary condition $\\partial \\Omega$ of the problem can be expressed by:\n\n\\begin{equation}\n\\partial \\Omega = S_u + S_p\n\\label{bc}\n\\end{equation}\n\\newline Where $S_u$ and $S_p$ respectively represent the displacement (essential) and force\/stress (natural) boundary condition. In an FEA model, the algorithm usually employs discretized approach to decompose the input geometry $\\Omega$ into finite elements so that the complex boundary value problem can be simplified into solving the partial differential equation for each element. For the linear elasticity problem on isotropic materials, the canonical form of the relationship between mesh nodal displacement $\\mathbf{U}$ and the global boundary condition $\\mathbf{F}$ can be expressed by:\n\n\\begin{equation}\n\\mathbf{K}\\mathbf{U}=\\mathbf{F}\n\\label{hooke}\n\\end{equation}\n\\newline Where $\\mathbf{K}$ is the global stiffness matrix. Here, the global stiffness matrix $\\mathbf{K}$ is assembled by the elemental stiffness matrices $\\mathbf{k_e}$, which can be represented by an integration function of the element area $A$:\n\n\\begin{equation}\n\\mathbf{k_e} = \\int_{A}^{}\\mathbf{B_e^T}\\mathbf{C_e}\\mathbf{B_e} \\textup{d}A\n\\label{ke}\n\\end{equation}\n\\newline Where $\\mathbf{B_e}$ is the displacement differentiation matrix obtained by differentiation of displacements expressed through shape functions and nodal displacements; $\\mathbf{C_e}$ is the elasticity matrix constructed by Young's modulus ($E$) and Poisson's ratio ($\\nu$). Then, the nodal displacement field $\\mathbf{U}$ can be solved. Based on $\\mathbf{U}$, the stress field $\\mathbf{\\sigma} = [\\sigma_x, \\sigma_y, \\tau_{xy}, \\tau_{yx}]^T$ can be expressed by:\n\n\\begin{equation}\n\\mathbf{\\sigma} = \\mathbf{C_g}\\mathbf{B}\\mathbf{u}\n\\label{stress}\n\\end{equation}\n\\newline Where $\\mathbf{B}$ is the assembly of $\\mathbf{B_e}$ matrices; the block-diagonal matrix $\\mathbf{C_g}$ is constructed from elemental elasticity tensor $\\mathbf{C}$ that can be derived by $\\mathbf{C_e}$ \\cite{nie2020stress}. From the equation, it can be seen that the element nodal stress state and the nodal displacement are not only determined by the global boundary condition $\\partial \\Omega$ but also influenced by local boundaries determined by its neighbor elements due to the assembly of $\\mathbf{K}$. Specifically, if a GNN model is applied to solve the cantilever beam problem, the target element requires both long-range global boundary condition information and the short-range neighbor elements' properties to generate its stress field. Also, in a 2D cantilever-beam FEA simulation, three key items need to be fully defined to determine the final simulation solution: global boundary condition ($\\partial \\Omega$), predefined elements for the model geometry ($\\Omega$), and material properties. This section mainly introduces the BOGE approach which can comprehensively feed that information from the FEA model to the GNN.\n\n\\subsection{Graph vertex embedding method}\\label{chs}\n\nMost GNN-based FEA surrogate models embed mesh nodes as graph vertices for physical field simulation \\cite{belbute2020combining, guo2020semi, ogoke2020graph, pfaff2020learning, sanchez2020learning}. However, boundary conditions can be cast on mesh edges or mesh bodies, which is not convenient to be represented using element nodes. Also, it's natural to encode the material property inside the mesh body rather than mesh nodes, especially for the conditions when material property changes spatially, for instance, composite materials and non-homogeneous material. The mesh nodes are between the mesh bodies, which is hard to embed the change of mesh body properties. In order to solve this problem, according to the convention from the previous research \\cite{pfaff2020learning}, the mesh edge, as well as the mesh body and the mesh node, are considered as graph vertices. However, compared with the condition when only element nodes are considered, embedding each mesh node, mesh edge, and mesh body as separate graph vertices will expand the graph size 3 times for triangular-mesh-based simulations, which can run out of the computational resources or lead to an extremely long message passing distance for the propagation of boundary conditions. Therefore, we combine the properties of the mesh node, mesh edge, and mesh body into one graph vertex and embed the overall mesh features into the graph vertex to provide a compressed graph representation.\n\nThe format of our graph embedding features $f_v$ is illustrated in Fig.~\\ref{fig:1_1_Encoding}. The first several channels encode the center of the mesh which provides the approximate position information of the mesh body. Material properties follow those channels which provide mesh body properties. After that, we encode the geometrical properties of mesh edges using the edge's normal vector and its distance to the mesh center to fully define the edge's geometrical position. Each edge's geometrical property follows the edge's physical property that encodes the global boundary condition of the simulation model. Following the edge properties, additional point-based information can be added to the data representation, for instance, the external force cast on the mesh.\n\n\\begin{figure}[htp]\n\\vspace{-0.12in}\n\\centering\n\t\\includegraphics[keepaspectratio,width=0.7\\textwidth]{fig\/1_1_Encoding.png}\n\t\\caption{Encoding approach for triangular mesh}\n\t\\label{fig:1_1_Encoding}\n\t\\vspace{-0.12in}\n\\end{figure}\n\nIn Fig.~\\ref{fig:1_1_Encoding}, we provide an example of our data representation for solving the cantilever beam problem (details of the problem definition are explained in Sec.~\\ref{dataset}). For a 2D cantilever beam problem based on triangular-mesh simulation, the first two channels encode the mesh center $(x_e, y_e)$,  which is calculated by the average of the coordinates for all the mesh nodes. The following body property for a stress analysis simulation can be Young's modulus ($E$) and Poisson's ratio ($\\nu$) of the mesh material. After that, three mesh edges with their distances from the center ($d_k$) and their normal vectors ($(x_{\\vec{n}k}, y_{\\vec{n}k})$) are encoded (where k=1, 2, 3 representing each edge of the mesh). Following each edge geometry, the boundary condition is encoded by its boundary state ($S_{uk}$). We set the boundary state as 1 when the edge is on the outer contour of the input model geometry, while $S_{uk}=2$ for edges on fixtures. After that, the data representation encodes the external 2D force ($S_p$) by attaching its relative position ($(x_{F_{ext}}, y_{F_{ext}})$, calculated according to the mesh center) and its magnitude ($(F_{ext-x}, F_{ext-y})$) to the graph vertex features. Meshes without the external force need to be padded with zero to meet the requirement of graph embedding. Then, all the necessary information from a triangular mesh has been embedded into the graph vertex.\n\n\\subsection{Adjacency information}\\label{adj}\n\nThe GNN requires the input of graph links $E$ to figure out the message passing direction, which is encoded by the adjacency matrix. Conventionally, the mesh-based graph embedding is bounded by the geometry of the input mesh model  ($\\Omega$), which only considers the graph links between the mesh and its neighboring meshes. However, in order to shorten the message passing distance to use shallow layers of GNN to regress the physical fields, we provide shortcuts for both the global boundary condition and the local neighboring meshes.\n\nMost researchers focus on one-hop graph vertices, which are the meshes straightly connecting to the target mesh. Denote the distance between the target element and its neighbor element on the graph as $l_e$, then we represent the graph links between one-hop elements and the target element as $l_e=0$, shown in Fig.~\\ref{fig:1_2_Shortcut} with red graph links. However, in order to provide more local information to the target mesh and smooth the physical-field results, we also consider two-hop ($l_e=1$) and three-hop ($l_e=2$) graph links (marked with green and blue graph links respectively in Fig.~\\ref{fig:1_2_Shortcut}) and encode those to the adjacency matrix. Adding information about meshes' local connectivity assists to provide more details of local boundary conditions, whose performance is presented in Sec.~\\ref{train}.\n\n\\begin{figure}[htp]\n\\vspace{-0.12in}\n\\centering\n\t\\includegraphics[keepaspectratio,width=0.7\\textwidth]{fig\/1_2_Shortcut.png}\n\t\\caption{Encoding for graph connectivity and shortcuts}\n\t\\label{fig:1_2_Shortcut}\n\t\\vspace{-0.12in}\n\\end{figure}\n\nThen, we provide mesh connectivity between the target mesh and all the meshes that have the information of global boundary conditions ($\\partial \\Omega$) in order to provide the shortcut for solving the large-scale graph problem. In the cantilever beam problem, all the meshes with information of the model's geometrical edges, fixtures, and external forces, are connected to every internal mesh inside the beam, which is illustrated in Fig.~\\ref{fig:1_2_Shortcut}. With these graph linking shortcuts, the GNN serves for collecting weighted boundary conditions from the entire simulation model, other than only passing the mesh information to its neighborhood meshes with short-range distance. Therefore, the shallow-layer GNN structure can still regress high-resolution structured-mesh-based simulation results. The performance of this improvement is shown in detail in Sec.~\\ref{train}.\n\nIn this section, we present the BOGE approach to embed triangular-mesh-based FEA problems into the graph structure. This concept can be generalized to other polygon-mesh simulations with more mesh-edge information. Padding the channels with zeros to keep the uniform graph-vertex feature size for all the meshes, data representation can encode simulation with mixed type of structured mesh (e.g., triangular meshes and quadrilateral meshes) and will incur only a small amount of extra computational resources since GNN generally employs sparse tensor production during training and computing. Our BOGE approach largely compresses the graph size and provides shortcuts for the boundary conditions, which can improve the long-range graph-vertex interactions in boundary value problems. It is compatible with most MPNN-based GNN structures and can be generalized to solve other boundary value problems.\n\\section{Dataset preparation}\\label{dataset}\n\nIn order to validate the efficiency of this approach, we focus on two aspects of the cantilever beam problem: stress field prediction and topology optimization. We summarized all the cantilever beam models provided in \\cite{nie2020stress} and created a parameterized synthetic FEA simulation dataset. According to \\cite{zhang2018featurenet, peddireddy2021identifying}, we provide uniform random values to provide design parameters for cantilever beams to improve the repeatability and portability of the dataset. Details on dataset preparation are explained in this section.\n\n\\subsection{Stress field simulation}\\label{stress}\n\nAccording to \\cite{nie2020stress}, a balanced FEA simulation dataset with 9 types of cantilever beam models is summarized and shown in Fig.~\\ref{fig:3_1_BeamDataset} which includes all the beam geometries from \\cite{nie2020stress}. All the design parameters for the cantilever beams geometry are generated uniformly-randomly according to the range presented in Fig.~\\ref{fig:3_1_BeamDataset} with a minimum increment of 0.1mm. In the case of beam structures with holes, certain constraints have been considered to prevent thin walls with a thickness less than 1mm. The fixtures ($S_u=0$) in this dataset are always on the left edge of the cantilever beam while the external force ($S_p$) is exerted on a random position selected from the right edge of the beam. The magnitude and the angle of the force are uniform-randomly generated with a minimum increment of 100N and $\\frac{\\pi}{6}$ respectively, ranging from 100N to 1000N and from 0 to $2\\pi$. Although the generated cantilevered structures cannot represent all the beam structures, the selected ones represent the most common structures in the mechanical areas \\cite{nie2020stress}. Also, following the similar parametric design pattern, other features can be considered and added into the dataset in the future according to previous synthetic CAD design researches \\cite{zhang2018featurenet, peddireddy2021identifying}. \n\n\\begin{figure}[htp]\n\\vspace{-0.12in}\n\\centering\n\t\\includegraphics[keepaspectratio,width=0.7\\textwidth]{fig\/3_1_BeamDataset.png}\n\t\\caption{Dataset design parameters}\n\t\\label{fig:3_1_BeamDataset}\n\t\\vspace{-0.12in}\n\\end{figure}\n\nThe ground truth of the stress field distribution is calculated by a commercial FEA software - ABAQUS \\cite{abaqus2011abaqus}. The chosen material for the simulations is the A36 steel with a Young's modulus of 200 GPa and a Poisson's ratio of 0.32. The process of triangular tessellation for the cantilever beam structures is carried out by ABAQUS built-in functions with the approximate global mesh size of 1.0 mm. The settings for deviation factor and minimum size factor are both chosen to be the default value 0.1mm to guarantee a fine triangular-mesh approximation. ABAQUS conducts the FEA algorithm explained in Sec.~\\ref{data} to calculate the displacement field and the stress field for the cantilever beams. We choose von Mises stress ($\\sigma_{\\textup{vm}}$) as the target stress field for the final regression, which can be represented by:\n\n\\begin{equation}\n\\sigma_{\\textup{vm}} = \n\\sqrt{\\sigma^2_{x} + \\sigma^2_{y} -\n\\sigma_{x} \\sigma_{y} +\n3 \\, \\tau^2_{xy} }\n\\label{vm}\n\\end{equation}\n\nWe encode the triangular meshes with their boundary condition into the graph according to the BOGE approach explained in Sec.~\\ref{data}. The external force has been normalized in vertex features and the material properties have not been included since only one material is considered. Table~\\ref{tab:2_1_Dataset} provides details regarding the generated dataset.\n\n\\subsection{Topology optimization}\\label{topo}\n\nBased on the simulated stress field, we calculate topological optimization results to validate the abstract decision-making capability of our approach. ABAQUS employs density-based Simple Isotropic Material with Penalization (SIMP) method to optimize the cantilever beam structure. The optimization process works toward minimizing the compliance function $C(z)$ which is the sum of the strain energy of all the elements:\n\n\\begin{equation}\n\\left.\\begin{matrix}\nx_\\textup{min} & :C(z)=\\mathbf{U^{T}}\\mathbf{K}\\mathbf{U} & =\\sum_{e=1}^{N} (z_e)^{p} \\mathbf{u_e^{T}}\\mathbf{k_e}\\mathbf{u_e} \\\\ \n\\textup{s.t.} & \\frac{V(z)}{V_0}=V_f & \\\\ \n & \\mathbf{K}\\mathbf{U}=\\mathbf{F} & \\\\ \n & 0 < z_{min} & \n\\end{matrix}\\right\\}\n\\label{simp}\n\\end{equation}\n\\newline Where $V(z)$ and $V_0$ represent the optimized material volume and the model's design domain volume; $V_f$ is the volume friction, which has been chosen as 0.3 for the final optimization target. Variable $z$ represents the element material density that varies from $z_{min}$ (void if 0, but generally nonzero to avoid singularity) to 1 (the full solid); $p$ is the penalization power whose value is usually 3. \\cite{lee2020cnn, nie2021topologygan} Using ABAQUS built-in optimization algorithm, we set the maximum design cycle as 25 and solve the SIMP problem. The optimized cantilever beams have a final $V_f\\in (0.29,0.3]$ which satisfies the required design target. The optimized $z_e$ for each element has been exported as the regression ground truth for GNN to be trained.\n\nWe finish all the dataset preparation tasks on a personal computer with an Intel i7-10700k CPU. The data generation algorithm runs in less than 10 threads with the total CPU occupation less than 90\\%, which has not influenced the evaluation of the computation efficiency of ABAQUS. The average computation time for the stress prediction is around 11.5s while that for the topological optimization is around 489s. We provide a balanced dataset with 45k simulation results (5k simulations for each class from Table~\\ref{fig:3_1_BeamDataset} with 3.5k\/0.75k\/0.75k simulations for training\/validating\/testing subsets) for both stress field distribution predictions and topological optimizations. Another balanced dataset with 180k simulation results (20k simulations for each class from Table~\\ref{fig:3_1_BeamDataset} with 14k\/3k\/3k simulations for training\/validating\/testing subsets) for stress field predictions only is generated to investigate the influence of the dataset size. \n\n\\input{tab\/2_1_tab}\n\nThe parameterized synthetic dataset can be generalized to other fields related to the FEA surrogate DL model. The synthetic dataset sometimes has an expensive time cost that can become a bottleneck for our approach to be a general-purpose surrogate model. However, the dataset prepared by physical experiments can also serve the training task. The stress fields can be measured by installing strain gauges on real cantilever beams on the area whose position corresponds to that of the predefined meshes in the simulation. Multiple data can be generated within a short time when the external forces change. In this condition, the physical experiment can serve as a fast PDE solver which assists to generate sufficient data for the surrogate model. All of these ideas contribute to the realization of a generalized model with high-precision predictions under various working conditions.\n\n\n\\section{Neural Network Training and Results}\\label{train}\n\nIn order to present the efficiency of the BOGE approach, we compare its performance with the conventional graph embedding methods (without any graph shortcuts) on different state of art GNN structures (GCN \\cite{kipf2016semi}, GAT \\cite{velivckovic2017graph}, UNet \\cite{gao2019graph}, DeepGCN \\cite{li2019deepgcns, li2020deepergcn}). We choose the MSE (Mean Square Error) function as the loss function and the evaluation metric to quantify our prediction performance \\cite{nie2020stress, koeppe2020intelligent}. The predicted outputs \u2013 the magnitude of von Mises stress $\\sigma_\\textup{vm}$ and the volume density $z_e$, are all 1D tensors which can be written as $\\hat{p}=(\\hat{p}_1, \\hat{p}_1,..., \\hat{p}_n)$, while the ground truth results can be represented by $p=(p_1, p_2, ...,p_n)$. The MSE can be represented as:\n\n\\begin{equation}\n\\mathrm{MSE}=\\frac{1}{n}\\sum_{i=1}^{n}(p_i-\\hat{p}_i)^2\n\\label{mse}\n\\end{equation}\n\nAccording to \\cite{koeppe2020intelligent}, we also employ the MAPE (Mean Absolute Percentage Error) to present the error rate in percentage, which can be expressed by:\n\n\\begin{equation}\n\\mathrm{MAPE}=\\frac{100}{n}\\sum_{i=1}^{n}|\\frac{p_i-\\hat{p}_i}{p_i+\\varepsilon}|\n\\label{mape}\n\\end{equation}\n\\newline Where we set $\\varepsilon=0.01$ to avoid computation error when the ground truth result is zero or extremely small.\n\nAll the codes have been written in Pytorch (shared in Github\\footnote{\\url{https:\/\/github.com\/stxyfu\/Stress\\_GNN}}) with Pytorch Geometric library \\cite{fey2019fast} and run on an NVIDIA RTX 3090 GPU and an Intel i7-10700k CPU. For all training experiments, the ADAM optimization algorithm has been employed to adjust the hyperparameters in the model. The training batch size is set to 8 to make maximal use of GPU memory and provide a precise training gradient while the learning rate is 0.01 obtained through the grid search to provide the best performance of the final GNN models. 500 training epochs have been employed for each experiment to ensure the GNNs are all fully trained with suitable prediction accuracy. The entire training process for the selected GNN models (Table~\\ref{tab:3_3_stress} and Table~\\ref{tab:3_4_topo}) takes around 125 hours on 45k datasets and 493 hours on the 150k dataset. The training time can be largely reduced if multiple high-performance GPUs are employed. \n\n\\subsection{Stress field prediction with conventional graph embedding}\\label{deepGNN}\n\nWe first investigate the regression capability of GNNs with conventional graph embedding methods on the 45k stress field prediction dataset, shown in Table~\\ref{tab:2_1_Dataset}. Conventional graph embedding methods only consider one-hop graph links (with $l_e = 0$ in our condition) whose adjacency matrix does not contain any shortcuts for boundary elements. Using this graph embedding method, shallow layer GNN (\"GCN(3 layers)\" in Table~\\ref{tab:3_1_results}) cannot pass the boundary information to all the internal elements which cannot perform any regression at the end of the training. Fig.~\\ref{fig:ConvRes}(\\subref{fig:Conv3GCN}) illustrates that, with the 3-layer GCN, the boundary information only propagates through limited meshes near the model's geometrical boundary and cannot reach the internal elements inside the material. The deep-layer GNN can save the weighted features passed from other elements in its hidden state $h$ according to Eq.~(\\ref{mpnn1}) and provides a relatively long-range message passing. According to our statistics, the largest graph distance between two structured meshes is around $l_e=250$ which requires at least 8 layers of GNN for general MPNN layers. However, deep GNN usually incurs oversmoothing that makes the embedded features similar and prevents the further regression process. For example, in Fig.~\\ref{fig:ConvRes}(\\subref{fig:Conv8GCN}), the 8-layer GCN (\"GCN(8 layers)\" in Table~\\ref{tab:3_1_results}) cannot predict anything but only outputs uniform values after 500 training epochs, which illustrates this extreme oversmoothing phenomenon. In order to solve this problem, researchers have employed different methods to mitigate the influence of oversmoothing and realized a few state-of-art GNN structures with deep GNN layers. From those methods, the most popular anti-oversmoothing techniques are DeepGCN \\cite{li2019deepgcns, li2020deepergcn}, attention layers(e.g., GAT  \\cite{velivckovic2017graph}) and DropEdge \\cite{rong2019dropedge}. Some other GNN structures, though have been developed for other purposes, can also reach deep layers with sufficient accuracy, for instance, the g-U-net \\cite{gao2019graph}. However, most of these methods mainly work on classification problems. For the regression problem in our condition, extra validations are needed. Therefore, we have tested those models using the conventional graph embedding approach and shown those results in Table~\\ref{tab:3_1_results} and Fig.~\\ref{fig:ConvRes}.\n\n\n\\begin{table}[!h]\n\\centering\n\\includegraphics[keepaspectratio, width=0.7\\columnwidth]{fig\/3_2_Convention.png}\n\\input{tab\/3_1_tab}\n\\caption{Training results with different GNN structures and graph embedding methods (training\/validating\/testing accuracy are presented without regularization)}\n\\label{tab:3_1_results}\n\\end{table}\n\nDeepGCN employs the GEN (GENeralized Graph Convolution layers \\cite{li2020deepergcn}), layerNorm \\cite{li2019deepgcns} and ResNet structure \\cite{he2016deep}, making GNN sufficiently robust against oversmoothing and can reach 56 layers with high classification accuracy. However, 8-layers DeepGCN (\"DeepGCN(8 layers)\" in Table~\\ref{tab:3_1_results}) with the ResGCN backbone \\cite{li2019deepgcns} still cannot reach a high accuracy and performs some oversmoothing shown in Fig.~\\ref{fig:ConvRes}(\\subref{fig:Conv8DeepGCN}). GAT employs the attention mechanism that provides weighted attention coefficients to various graph features to ensure that GNN can pass the important information through deep layers. We employ the same DeepGCN backbone, but substitute GAT (with 4 attention heads) for GEN and try to improve the prediction accuracy, shown as \"GAT\" in Table~\\ref{tab:3_1_results}. According to our tests, the combination of GAT and the DeepGCN backbone performs better than simply stacking GAT layers on our dataset. However, the training result still appears to be incapable of obtaining acceptable results and shows an extremely oversmoothed result, shown in Fig.~\\ref{fig:ConvRes}(\\subref{fig:Conv8GAT}). The g-U-Net can encode and decode both high-level features and local features which can reach 5 layers in \\cite{gao2019graph}. However, each graph pooling layer drops both graph vertices (mesh elements) and graph links. When only one-hop graph links ($l_e=0$) are considered, with a very sparse adjacency matrix, stacking pooling layers in the g-U-net makes the adjacent matrix too sparse, so that only a few graph links remain in high-level feature scope, which cannot pass ample information to other elements through GNN and ruin the g-U-net's prediction capability. Also, a deeper g-U-net can still incur the extreme oversmoothing problem shown in Fig.~\\ref{fig:ConvRes}(\\subref{fig:ConvgUnet}) (with the training result shown as \"gUnet\" in Table~\\ref{tab:3_1_results}). Other anti-oversmoothing techniques like DropEdge can ruin the integrity of the graph, which is not suitable for our problem as we have tested. The overall training results show that deep-layer GNNs with the conventional graph embedding approach cannot simply regress the boundary value problem with long-range graph-vertex interactions.\n\n\n\n\\input{fig_results_conv\/fig_results_conv}\n\n\n\\subsection{Stress field prediction with BOGE}\\label{stress}\n\nThe BOGE approach shows its unique benefits for regressing the results of large-scale boundary value problems with simple shallow-layer GNNs. By providing shortcuts for both global boundary conditions and local information of neighboring elements, GNNs with BOGE bypass the message passing limitation of the MPNN framework, and all the internal elements can directly get the boundary condition information to solve the boundary value problem. We have employed the 3-layers DeepGCN and BOGE with $l_e=1,2,3$ graph links to validate its performance on the 45k stress field prediction dataset. The results are shown in Table~\\ref{tab:3_3_stress}. \n\nCompared with the predicted results from \"DeepGCN(3 layers)\" in Table~\\ref{tab:3_1_results}, BOGE with $l_e=0$ graph links (model \\#1 in Table~\\ref{tab:3_3_stress}) largely improves the prediction accuracy and reaches 0.016383 testing accuracy with MAPE of 2.85\\%, which is accurate enough to serve as an efficient FEA surrogate prediction model. The testing results has been compared for BOGE with different adjacency matrices that encodes local graph links with $l_e=1,2$. From Table~\\ref{tab:3_3_stress}, it can be seen that additional local mesh information cannot improve the prediction accuracy. The testing accuracy for $l_e=1$ (model \\#2 in Table~\\ref{tab:3_3_stress}) drops to 0.021737 with MAPE of 3.08\\% while that for $l_e=2$ (model \\#3 in Table~\\ref{tab:3_3_stress}) drops to 0.027302 with MAPE of 3.44\\%. More local graph edges provide worse prediction accuracy, which can be caused by message congestion \\cite{loukas2019graph} since more elements enlarge the size of the message passing at each training epoch. Also, the adjacency matrix with more graph links can lead to a denser tensor product which takes more computational resources. Therefore, we only employ BOGE with $l_e=0$ for the rest of the training task to provide a satisfying prediction result.\n\n\\begin{table}[!h]\n\\centering\n\\includegraphics[keepaspectratio, width=\\columnwidth]{fig\/3_3_StressGNN.png}\n\\caption{Stress prediction results (notations for neural network layers are explained in Table~\\ref{tab:3_1_results}; training\/validating\/testing accuracy are presented without regularization)}\n\\label{tab:3_3_stress}\n\\end{table}\n\nWe also consider the effect of the dataset size. Another training experiment with 180k simulation dataset (\"Stress prediction (180k)\" in Table~\\ref{tab:2_1_Dataset}) is generated by the same 3-layer DeepGCN structure (model \\#4 in Table~\\ref{tab:3_3_stress}). The testing accuracy increases to 0.011706 which shows a better training performance, but only slightly improves the MAPE that reaches 2.41\\%. This indicates that while the larger training dataset provides better results, a proper smaller dataset with around 45k data can still provide appropriate training results. Some of the predicted results from model \\#4 in Table~\\ref{tab:3_3_stress} are presented in Fig.~\\ref{fig:StressResults} which illustrates the effectiveness of the BOGE approach. The average prediction time for the GNN model (model \\#4 in Table~\\ref{tab:3_3_stress}) with GPU is 0.012ms which is far less than the ABAQUS computation time (around 11.5s in Table~\\ref{tab:2_1_Dataset}). Though the computation time for the GNN model does not include the time for data loading, input checking, and other unknown processes running in ABAQUS, the large difference of the order of magnitude validates the efficiency of our GNN surrogate model. The training loss and the validation\/testing accuracy for the stress predictions is shown in Fig.~\\ref{fig:loss}(\\subref{fig:StressLoss})\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.4\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{fig\/stressloss.png}\n  \\caption{Stress prediction}\n  \\label{fig:StressLoss}\n\\end{subfigure}%\n\\begin{subfigure}{0.4\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{fig\/topoloss.png}\n  \\caption{Topology optimization prediction}\n  \\label{fig:TopoLoss}\n\\end{subfigure}\n\\caption{Training loss, validation accuracy, and testing accuracy}\n\\label{fig:loss}\n\\end{figure}\n\n\\input{fig_stress\/fig_stress}\n\n\\subsection{Topology optimization prediction}\\label{stress}\n\nUsing the same method, we regress the topology optimization results to investigate the BOGE's potential on solving abstract decision-making problems. Using the same DeepGCN structure, we employ the same graph embedding inputs to regress the volume density $z_e$. Since topology optimization usually happens after the stress field simulation, we also consider the condition when the stress field information is added to the features of graph vertices. In this condition, the stress field information $\\mathbf{\\sigma} = [\\sigma_x, \\sigma_y, \\tau_{xy}]^T$ is attached to the graph-vertex features in BOGE approach. The training data employs BOGE with $l_e=1$ on 45k topology optimization dataset (\"Topology optimization (45k)\" in Table~\\ref{tab:2_1_Dataset}), and its training results are shown in Table~\\ref{tab:3_4_topo}. The dropout layers have been added after each layer with the dropout rate of 0.1 to provide a better prediction result.\n\n\\begin{table}[!h]\n\\centering\n\\includegraphics[keepaspectratio, width=\\columnwidth]{fig\/3_4_TopoGNN.png}\n\\caption{Topology optimization prediction results (notations for neural network layers are explained in Table~\\ref{tab:3_1_results}; training\/validating\/testing accuracy are presented without regularization)}\n\\label{tab:3_4_topo}\n\\end{table}\n\n\\begin{table}[!h]\n\\centering\n\\includegraphics[keepaspectratio, width=\\columnwidth]{fig\/3_5_TopoGNN.png}\n\\caption{Analyses for topology optimization prediction results (training\/validating\/testing accuracy are presented without regularization)}\n\\label{tab:3_5_acc}\n\\end{table}\n\nFrom the training results, it can be seen that BOGE without encoding the stress information (model \\#1 in Table~\\ref{tab:3_4_topo}) reaches 0.002735 testing accuracy, which is better than the best testing MSE - 0.059943 in \\cite{nie2021topologygan} with similar cantilever beam settings and the comparable number of meshes. However, the MAPE is relatively large and reaches 22.53\\% in testing accuracy. It is due to two effects. First, the range or the average value for the volume density in the topological optimization is relatively small. For stress prediction shown in Table~\\ref{tab:2_1_Dataset}, the average von Mises stress is around 2MPa while the maximum stress magnitude can reach 74.09MPa, which means the error divided by its ground truth value can be relatively small for the MAPE. Assume one element with the stress of 2MPa (which is the average stress) has the prediction error of 0.1MPa, the MAPE for this element is only 5\\%. However, from Table~\\ref{tab:2_1_Dataset}, the average volume density is around 0.29 whereas the maximum volume density is 1.0, which means that the same error can contribute largely to the final MAPE. For example, if the element with 0.29 volume density has the prediction error of 0.1, the MAPE for this element can be 34\\%. This explains one of the reasons that some researchers haven't employed the MAPE for evaluating topology optimization results \\cite{nie2021topologygan, zhang2019deep, banga20183d}. Further, the large MAPE is generated because some outliers around the edge of the optimized beam contribute to most of the errors. Assume that the resolution of the volume density is 0.01, which is also the $\\varepsilon$ in Eq.~(\\ref{mape}). From model \\#1 in Table~\\ref{tab:3_5_acc}, when only meshes with its ground-truth volume density larger than $\\varepsilon$ are considered (${p|p_i>0.01}$), the MAPE for those meshes is only 9.70\\%. Also, for each testing graph (or simulation model), around 22\\% - 23\\% of its meshes have the volume density larger than 0.01 for its ground truth value, but only 1.58\\% of the total meshes contribute to the errors larger than $\\varepsilon$. Some prediction results from model \\#1 in Table~\\ref{tab:3_4_topo} are shown in Fig.~\\ref{fig:TopoResults}. It can be seen that those 1.58\\% of meshes mostly are located primarily on the edges of the optimized beams. Since the boundary of the optimized beams usually has a volume density around $\\varepsilon=0.01$ as its ground truth value, any small prediction error can be largely amplified by the MAPE. However, regardless of the MAPE, the predicted volume density shown in Fig.~\\ref{fig:TopoResults} and its testing accuracy are sufficient to validate the BOGE's advantage in regressing complex decision-making problems. \n\nWith the stress field information, BOGE shows a worse prediction result, which reaches the MSE of 0.003378 for testing accuracy (model \\#2 in Table~\\ref{tab:3_4_topo}). Also, more outliers (6.00\\% from model \\#2 in Table~\\ref{tab:3_5_acc}) are found in this condition. Adding additional features to the graph vertex can lead to message congestion\\cite{loukas2019graph} and oversquashing\\cite{alon2020bottleneck}. Therefore, unlike the CNN-based surrogate model\\cite{nie2021topologygan}, GNN models are sensitive to the size of the input tensor and cannot be simply enhanced by adding extra input features. Balancing the input feature, hidden layer features, and the GNN layers can provide better results according to the GNN's property.\n\nThe topology optimization prediction shows BOGE's capability of making abstract decisions which benefits the smart design technology. The average prediction time for the topology-optimization GNN model (model \\#1 in Table~\\ref{tab:3_4_topo}) on GPU is around 0.011ms, similar to that of GNN stress field prediction, but far less than the ABAQUS computation time (around 489s in Table~\\ref{tab:2_1_Dataset}). This is because the SIMP method requires several iterations of computations that take much longer than the stress prediction. However, the GNN surrogate model can directly output the optimized result which largely saves the time cost. The overall performance of BOGE demonstrates its irreplaceable advantage in regressing the physical field on structured elements.The training loss and the validation\/testing accuracy for the stress predictions is shown in Fig.~\\ref{fig:loss}(\\subref{fig:TopoLoss})\n\n\\input{fig_topo\/fig_topo}\n\n\\section{Conclusion}\\label{conclusion}\n\nWe develop a Boundary Oriented Graph Embedding (BOGE) approach for the GNN-based FEA surrogate model, especially for regressing triangular-mesh-based FEA simulation results. The BOGE approach bypasses the limitations of the MPNN framework and provides shortcuts for both global boundary information and local multi-hop elements, which allows shallow-layer GNN to regress boundary value problems with long-range graph-vertex interactions. Working on solving the cantilever beam problem, the BOGE approach with 3-layer DeepGCN model \\textcolor{blue}{achieves the regression with MSE of 0.011706 (2.41\\% MAPE) for stress field prediction and 0.002735 MSE (with 1.58\\% elements having error larger than 0.01) for topological optimization.}, which validates the BOGE's efficiency in complex decision-making problems. The model has the potential to be applied to most physical-field fitting problems with any type of polygon meshes or mixed types of meshes. This study has potential limitations. The GNN performance is sensitive to the input-tensor shape and requires more theoretical analysis to investigate appropriate methods to improve the structure of GNN. Future work can focus on modifying GNN structures to obtain more accurate regression results and applying the BOGE approach to other boundary value problems.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and main result}\\label{sec1}\n\n\\noindent\nWe study H\\\"older regularity up to the boundary for the weak solutions of the Dirichlet problem\n\\begin{equation}\\label{dir}\n\\begin{cases}\n(-\\Delta)_p^s u=f & \\text{in $\\Omega$} \\\\\nu=0 & \\text{in $\\Omega^c$}.\n\\end{cases}\n\\end{equation}\nHere $\\Omega\\subset{\\mathbb R}^N$ ($N>1$) is a bounded domain with a $C^{1,1}$ boundary $\\partial\\Omega$, $\\Omega^c={\\mathbb R}^N\\setminus\\Omega$, $s\\in(0,1)$ and $p\\in(1,\\infty)$ are real numbers and $f\\in L^\\infty(\\Omega)$. The $s$-fractional $p$-Laplacian operator is the gradient of the functional\n\\[J(u):=\\frac{1}{p}\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\\, dx\\, dy,\\]\ndefined on\n\\[W^{s,p}_0(\\Omega):=\\{u\\in L^p({\\mathbb R}^N): J(u)<\\infty, \\,\\, u=0 \\text{ in $\\Omega^c$}\\},\\]\nwhich is a Banach space with respect to the norm $J(u)^{1\/p}$. Under suitable smoothness conditions on $u$ the operator can be written as\n\\begin{equation*}\n(- \\Delta)_p^s\\, u(x) = 2 \\lim_{\\varepsilon \\searrow 0} \\int_{B_\\varepsilon^c(x)} \\frac{|u(x) - u(y)|^{p-2}\\, (u(x) - u(y))}{|x - y|^{N+sp}}\\, dy, \\quad x \\in {\\mathbb R}^N.\n\\end{equation*}\nA weak solution $u\\in W^{s,p}_0(\\Omega)$ of problem \\eqref{dir} satisfies,\nfor every $\\varphi\\in W^{s,p}_0(\\Omega)$, \n\\[\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}} \\,dx\\,dy=\\int_\\Omega f(x)\\varphi(x) \\,dx.\\]\nProblem \\eqref{dir} is thus well posed and, in the case $p=2$, it corresponds to an inhomogeneous fractional Laplacian equation with Dirichlet boundary condition.  \nFor the sake of completeness we recall that in the literature the fractional Laplacian is often defined by\n\\begin{equation*}\n \\langle (-\\Delta)^su,\\varphi\\rangle\n  = \\frac{c(N,s)}{2}  \\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))(\\varphi(x)-\\varphi(y))}{|x-y|^{N+2s}} \\,dx\\,dy,\\quad \\varphi \\in  W^{s,2}_0(\\Omega),\n\\end{equation*}\nwhere $c(N,s) = s 2^{2s} \\, \\Gamma((N+2s)\/2) \/ (\\pi^{N\/2} \\Gamma(1-s))$, in order to be coherent with the\nFourier definition of $(-\\Delta)^s$ (see Remark 3.11 of \\cite{CS1}). We point out that, in the current literature, there are several\nnotions of fractional Laplacian, all of which agree when the problems\nare set on the whole ${\\mathbb R}^N$, but some of them disagree in a bounded\ndomain.  We refer the reader to~\\cite{SV2} for a discussion on the comparison\nbetween the integral fractional laplacian and the regional (or spectral) notion\nobtained by taking the $s$-powers of the Laplacian operator $-\\Delta$ with zero Dirichlet\nboundary conditions.\n\\vskip2pt\n\\noindent\nIn the case $p\\neq 2$, problem \\eqref{dir} is a non-local and non-linear one. \nIts leading term $(-\\Delta)^s_p$ is furthermore degenerate when $p>2$ and singular when $1<p<2$. Determining sufficiently good regularity estimates {\\em up to the boundary} is not only relevant by itself, but it also has useful applications in obtaining multiplicity results for more general non-linear and non-local equations, such as those investigated in \\cite{ILPS} in the framework of topological methods and Morse theory. To this regard, \nthis contribution provides a first step in order to obtain the results of \\cite{IMS} in the general case $p\\neq 2$.\n\\vskip2pt\n\\noindent\nThe regularity up to the boundary of fractional problems in the case $p=2$ is now rather \nwell understood, even when more general kernels and nonlinearities are considered.  Using a viscosity solution approach, the model linear case gives regularity for fully non-linear equations which are ``uniformly elliptic\" in a suitable sense.\nRegarding the viscosity approach to fully non-linear, elliptic non-local equation, see \\cite{CafSil1,CafSil2} for interior regularity theory with smooth kernels, and \\cite{Serra} for rough kernels; regarding boundary regularity, see \\cite{RS3} for nearly optimal results and a detailed discussion on the delicate role that the kernel's regularity class plays in such problems.\n\\vskip2pt\n\\noindent\nEquation \\eqref{dir}, however, does not fall in the category of non-local non-linear equations treated in the aforementioned works. This is not surprising, due to the degenerate\/singular nature of the nonlinearity, and the $s$-fractional $p$-Laplacian is the non-local analogue of a degenerate\/singular non-linear {\\em divergence form} equation, rather than of a uniformly elliptic fully non-linear one. Local H\\\"older continuity has been addressed in \\cite{DKP1,DKP2} using methods {\\em \\'a la} De Giorgi, and in \\cite{Ling} with a Krylov-Safanov approach for $p>1\/(1-s)$. In \\cite{BCF} the fully non-linear approach is used to study the non-local analogue of the $p$-Laplacian equation in non-divergence form\n\\[ \\Delta u +(p-2)\\frac{\\nabla u}{|\\nabla u|}D^2u\\frac{\\nabla u}{|\\nabla u|}=0,\\]\narising from non-local `tug of war' games. Interior $C^{1,\\alpha}$ estimates and H\\\"older continuity up to the boundary is proved under rather general assumptions.\n\\vskip2pt\n\\noindent\nOur main result is the following:\n\n\\begin{theorem}\\label{main}\nThere exist $\\alpha\\in(0,s]$ and $C_\\Omega>0$, depending only on $N$, $p$, $s$, with $C_\\Omega$ also depending on $\\Omega$, such that, for all weak solution $u\\in W^{s,p}_0(\\Omega)$ of problem \\eqref{dir}, $u\\in C^\\alpha(\\overline\\Omega)$ and\n\\begin{equation}\\label{thm57tesi}\n\\|u\\|_{C^\\alpha(\\overline\\Omega)}\\le C_\\Omega\\|f\\|_{L^\\infty(\\Omega)}^\\frac{1}{p-1}.\n\\end{equation}\n\\end{theorem}\n\n\\noindent\nNotice that, regarding regularity {\\em up to the boundary}, one cannot expect more than $s$-H\\\"older continuity due to explicit examples (see Section \\ref{sec3} below). On the other hand, the optimal H\\\"older exponent up to the boundary seems to be $s$ for any $p>1$, while we prove $C^\\alpha$ regularity for an unspecified small $\\alpha$, the issue being a lack of higher (at least $C^s$) regularity results in the interior of the domain.\n\\vskip2pt\n\\noindent\nLet us describe the strategy to prove Theorem \\ref{main}. We choose to use the notion of weak rather than viscosity solution, since we feel that the equation is more naturally seen as a variational one. However, we will frequently use barrier arguments, rather than De Giorgi-Nash-Moser techniques. Indeed, the proof of Theorem \\ref{main} is performed in the spirit of Krylov's approach to boundary regularity, see \\cite{krylov}, and uses two main ingredients:\n\\begin{itemize}\n\\item[$(a)$]\na uniform H\\\"older control (see Theorem \\ref{estid}) on how $u$ reaches its boundary values, which amounts to \n\\begin{equation}\\label{deltasintro}\n|u(x)|\\leq C\\|f\\|_\\infty^{\\frac{1}{p-1}}{\\rm dist}^s(x,\\Omega^c);\n\\end{equation}\n\\item[$(b)$]\na local regularity estimate (see Theorem \\ref{osc}) in terms of quantities which may blow up in general when reaching \nthe boundary, but remain bounded for functions satisfying \\eqref{deltasintro}.\n\\end{itemize}\nPoint $(a)$ is obtained through a barrier argument, and stems from the fact that  $(-\\Delta)^s_p (x_+)^s=0$  in the half line ${\\mathbb R}_+$. Notice that for $p\\neq 2$ we do not have at our disposal the fractional Kelvin transform, and the concrete calculus of the $s$-fractional $p$-Laplacian even on smooth functions is a prohibitive task, in general. \nThus constructing upper barriers can be quite technical, and is done as following:\n\\begin{itemize}\n\\item\nConsider $u_N(x)=(x_N)_+^s$: explicit calculus shows that $(-\\Delta)^s_p u_N=0$ in the half-space ${\\mathbb R}^N_+$. We locally deform the half-space to $\\Omega^c$ by a  diffeomorphism $\\Phi$ close to the identity, and obtain a function $u_N\\circ \\Phi$ with small $s$-fractional $p$-Laplacian in a small ball $\\hat B$ centered at a point of $\\partial\\Omega$.\n\\item\nThe resulting function $u_N\\circ \\Phi$ can be controlled in $\\hat B\\cap \\Omega$ by distance-like functions from the boundary, and we can modify it to globalize the controls, while keeping the smallness of $(-\\Delta)^s_p(u_N\\circ\\Phi)$ in $\\hat B\\cap \\Omega$.\n\\item\nWe exploit the non-local nature of the equation to add a fixed {\\em positive} quantity to $(-\\Delta)^s_p(u_N\\circ\\Phi)$ in $\\hat B\\cap \\Omega$, by truncation away from $\\hat B$. Since $(-\\Delta)^s_p(u_N\\circ\\Phi)$ is arbitrarily small, its truncation has therefore $s$-fractional $p$-Laplacian bounded from below by a positive constant in $\\hat B\\cap \\Omega$, and provides the local upper barrier.\n\\end{itemize}\nPoint $(b)$ is a generalization, in the whole range $p>1$, to non-homogeneous equations of Theorem 1.2 from \\cite{DKP1}, and it could be deduced in the case $p>2-s\/N$ using the results of \\cite{KMS} and in the case $p>1\/(1-s)$ using \\cite{Ling}. However we choose to prove it with a different approach. Much in the spirit of \\cite{S}, rather than considering the non-locality of the equation as an additional technical difficulty to the implementation of the De Giorgi-Moser regularity theory, we use it at our advantage to construct a more elementary proof. It should be noted that we do not employ Caccioppoli-like inequalities, or estimates on $\\log u$ (which are the elementary counterpart of John-Nirenberg's lemma). Actually we don't even need a Poincar\\'e or Sobolev inequality, which are usually looked at as basic tools for (variational) regularity theory. This feature seems typical of the non-local framework and it should be noted that the proof doesn't  seem to immediately ``pass to the limit to local equations'' as the obtained estimates blow up for  $s\\to 1$. \n\\vskip2pt\n\\noindent\nRegarding possible developments and generalizations, a first remark regards the choice of the kernel in the non-local operator\n\\[L(u)={\\rm PV}\\int_{{\\mathbb R}^N}|u(x)-u(y)|^{p-2}(u(x)-u(y))K(x,y)\\, dy.\\]\nRegarding interior regularity, a bound from above and below in terms of the model kernel $|x-y|^{-N-ps}$ seems to suffice to obtain H\\\"older regularity, due to the results of \\cite{DKP1,KMS}. For non-local, fully non-linear, uniformly elliptic equation, higher interior regularity (up to $C^{2, \\alpha}$) is proved in \\cite{CafSil1,CafSil2,Serra} when the kernel satisfies additional structural and regularity assumption, but no such result is known for the $s$-fractional $p$-Laplacian.  \nRegarding regularity up to the boundary things are more subtle. In the uniformly elliptic case ($p=2$), the optimal regularity is $C^s(\\overline{\\Omega})$ due to the results of \\cite{RS3}, but only for a subclass of rough symmetric kernels arising from stable L\\'evy processes, of the form\n\\[K(x, y)=H(x-y),\\quad H(z)=\\frac{a\\big(z\/|z|\\big)}{|z|^{N+2s}}, \\quad 0<\\lambda\\leq a\\leq \\Lambda.\\] \nCounterexamples show that this is the largest kernel's class where to expect such regularity up to the boundary. However, for any $p>1$, one still expects $C^\\alpha(\\overline{\\Omega})$ regularity for arbitrarily rough symmetric kernels, for a small $\\alpha<s$.\n\\vskip2pt\n\\noindent\nAnother point of interest is the H\\\"older regularity up to the boundary of $u\/{\\rm dist}^s(x,\\Omega^c)$, when $(-\\Delta)^s_p u$ is bounded in $\\overline{\\Omega}$. This is proven in \\cite{RS} for the fractional Laplacian, and in \\cite{RS3} for the L\\'evy stable fully non-linear, uniformly elliptic non-local equations. While undoubtedly being relevant in light of the applications depicted in \\cite{ILPS}, we do not treat this problem here.\n\\vskip2pt\n\\noindent\nThe structure of the paper is as follows:\n\\begin{itemize}\n\\item In Section \\ref{sec2} we mainly discuss the relationship between weak and strong (i.e., in a suitable principal value sense) solutions of \\eqref{dir}. In doing so we clarify how barrier arguments (which are more suited to viscosity solutions) can be applied in the framework of weak solutions of non-linear non-local problems.\n\\item In Section \\ref{sec3} we study the $s$-fractional $p$-Laplacian of distance-related functions, and consider their stability with respect to local diffeomorphisms of the domain.\n\\item In Section \\ref{sec4} we construct some upper barriers, derive $L^\\infty$-bounds for solutions of \\eqref{dir} and prove estimate \\eqref{deltasintro}.\n\\item In Section \\ref{sec5} we tackle the local regularity through a weak Harnack inequality. Then we couple it with \\eqref{deltasintro} to prove Theorem \\ref{main}.\n\\end{itemize}\n\\vskip2pt\n\\noindent\nA short description of the result obtained in the present paper can be found in \\cite{IMS-RLM}.\n\n\\section{Preliminaries}\\label{sec2}\n\n\\subsection{Notations and function spaces}\n\n\\noindent\nGiven a subset $A\\subseteq{\\mathbb R}^N$ we will set $A^c={\\mathbb R}^N\\setminus A$ and for $A, B\\subseteq{\\mathbb R}^N$,\n\\[{\\rm dist}(A, B)=\\inf_{x\\in A,\\,y\\in B}|x-y|, \\quad \\delta_A(x)={\\rm dist}(x, A^c),\\]\n\\[{\\rm dist}_H(A, B)= \\max\\Big\\{\\sup_{x\\in A}{\\rm dist}(x, B), \\sup_{y\\in B}{\\rm dist}(y, A)\\Big\\}.\\]\nFor all $x\\in{\\mathbb R}^N$, $r>0$ we denote by $B_r(x)$, $\\overline B_r(x)$, and $\\partial B_r(x)$, respectively, the open ball, the closed ball and the sphere centered at $x$ with radius $r$. When the center is not specified, we will understand that it's the origin, e.g. $B_1=B_1(0)$. For all measurable $A\\subset{\\mathbb R}^N$ we denote by $|A|$ the $N$-dimensional Lebesgue measure of $A$. If $u$ is a measurable function and $A$ is a measurable subset of ${\\mathbb R}^N$, we will set for brevity\n\\[\\inf_A u=\\essinf_A u,\\quad \\sup_A u=\\essinf_A u.\\]\nFor all measurable $u:{\\mathbb R}^N\\to{\\mathbb R}$ we define\n\\[[u]_{s,p}=\\Big(\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}} \\,dx\\,dy\\Big)^{\\frac{1}{p}},\\]\n\\[\\|u\\|_{W^{s,p}(\\Omega)}=\\|u\\|_{L^p(\\Omega)}+\\Big(\\int_{\\Omega\\times\\Omega}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\\, dx\\, dy\\Big)^{\\frac{1}{p}}\\]\nand we will consider the following spaces (see \\cite{DPV} for details):\n\\begin{align*}\n&W^{s,p}(\\Omega)=\\big\\{u\\in L^p(\\Omega):\\|u\\|_{W^{s,p}(\\Omega)}<\\infty\\big\\}, \\\\\n&W^{s,p}_0(\\Omega) =\\big\\{u\\in W^{s,p}({\\mathbb R}^N):\\,u=0 \\ \\text{in $\\Omega^c$}\\big\\},\\\\\n&W^{-s,p'}(\\Omega)=(W^{s,p}_0(\\Omega))^*,\n\\end{align*}\nwhere the last one is the Banach dual, whose pairing with $W^{s,p}_0(\\Omega)$ will be denoted by $\\langle\\cdot,\\cdot\\rangle_{s,p,\\Omega}$.\nWe will extensively make use of the following space:\n\\begin{definition}\\label{defwtilde}\nLet $\\Omega\\subseteq {\\mathbb R}^N$ be bounded. We set\n\\[\\widetilde{W}^{s,p}(\\Omega):=\\Big\\{u\\in L^p_{\\rm loc}({\\mathbb R}^N):\\,\\exists\\,U\\Supset\\Omega \\ \\text{s.t.}\\,\\|u\\|_{W^{s,p}(U)}+\\int_{{\\mathbb R}^N}\\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx<\\infty\\Big\\}.\\]\nIf $\\Omega$ is unbounded, we set\n\\[\\widetilde{W}^{s,p}_{\\rm loc}(\\Omega):=\\big\\{u\\in L^p_{\\rm loc}({\\mathbb R}^N):\\,u\\in \\widetilde{W}^{s,p}(\\Omega')\\, \\text{for any bounded $\\Omega'\\subseteq\\Omega$}\\big\\}.\\]\n\\end{definition}\n\n\\noindent\nWe notice that the condition \n\\[\\int_{{\\mathbb R}^N}\\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx<\\infty\\]\nholds if  $u\\in L^\\infty({\\mathbb R}^N)$ or $[u]_{C^s({\\mathbb R}^N)}<\\infty$. The spaces $\\widetilde{W}^{s,p}(\\Omega)$, $\\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ can be endowed with a topological vector space structure as inductive limit, but we will not use it.\nFor all $\\alpha\\in(0,1]$ and all measurable $u:\\overline\\Omega\\to{\\mathbb R}$ we set\n\\[[u]_{C^\\alpha(\\overline\\Omega)}=\\sup_{x,y\\in\\overline\\Omega,\\,x\\neq y}\\frac{|u(x)-u(y)|}{|x-y|^\\alpha},\\]\n\\[C^\\alpha(\\overline\\Omega)=\\big\\{u\\in C(\\overline\\Omega):\\,[u]_{C^\\alpha(\\overline\\Omega)}<\\infty\\big\\},\\]\nthe latter being a Banach space under the norm $\\|u\\|_{C^\\alpha(\\overline\\Omega)}=\\|u\\|_{L^\\infty(\\overline\\Omega)}+[u]_{C^\\alpha(\\overline\\Omega)}$. A similar definition is given for $C^{1,\\alpha}(\\overline\\Omega)$. When no misunderstanding is possible, we set for all measurable $D\\subset{\\mathbb R}^N$, $x\\in D$, and all measurable $\\psi:D\\times D\\to{\\mathbb R}$\n\\[{\\rm PV}\\int_{D}\\psi(x, y)\\,dy=\\lim_{\\varepsilon\\to 0^+}\\int_{D\\setminus B_\\varepsilon(x)}\\psi(x, y)\\,dy.\\]\nFor all measurable $u:{\\mathbb R}^N\\to{\\mathbb R}$ we recall that the {\\em non-local tail} centered at $x\\in{\\mathbb R}^N$ with radius $R>0$, introduced in \\cite{DKP1}, is defined as\n\\begin{equation}\\label{deftail}\n{\\rm Tail}(u;x,R)=\\Big(R^{ps}\\int_{B_R^c(x)}\\frac{|u(y)|^{p-1}}{|x-y|^{N+ps}}\\,dy\\Big)^\\frac{1}{p-1}.\n\\end{equation}\nWe will also set ${\\rm Tail}(u; 0, R)={\\rm Tail}(u; R)$. Unless otherwise stated, the numbers $p>1$ and $s\\in(0,1)$ will be fixed as the order of summability and the order of differentiability. By a {\\em universal} constant we mean a constant $C=C(N,p,s)$. This dependence will always be omitted, even when other dependencies are present, in which case they are the only ones explicitly stated: for example $C_\\Omega$ will denote a constant depending on $N, p, s$, and $\\Omega$. During chains of inequalities, universal constants will be denoted by the same letter $C$ even if their numerical value may change from line to line. The same treatment will be used for constants which retain their dependencies from line to line. When needed, we will denote a specific universal constant with a number, e.g.\\ $C_1$, $C_2$ {\\em et cetera}.\n\n\\subsection{Some elementary inequalities} \n\nFor all $a\\in{\\mathbb R}$, $q>0$, we set\n\\[a^{q}=|a|^{q-1}a.\\]\nThis notation has great advantages in readability and, for future reference, we recall here some more or less known elementary inequalities about the function $a\\mapsto a^q$. We will provide a sketch of proof for the less frequent ones.\n\\vskip2pt\n\\noindent\nWe begin with the well known inequalities\n\\begin{equation}\\label{in3}\n(a+b)^q\\le 2^{q-1}(a^q+b^q) \\quad  a,b\\ge 0,\\,q\\ge 1;\n\\end{equation}\n\\begin{equation}\\label{in2}\n(a+b)^q\\le a^q+b^q \\quad  a,b\\ge 0,\\,q\\in(0,1];\n\\end{equation}\n\\begin{equation}\\label{in4}\n|a^q-b^q|\\le q(|a|^{q-1}+|b|^{q-1})|a-b| \\quad a,b\\in{\\mathbb R},\\,q\\ge 1,\n\\end{equation}\nthe last one being a trivial consequence of Taylor's formula. We will also use\n\\begin{equation}\\label{in6}\na^q-(a-b)^q\\le C_M\\max\\{b,b^q\\} \\quad  |a|\\le M,\\,b\\ge 0,\\,q>0,\n\\end{equation}\nwhich follows immediately from \\eqref{in2} if $q\\in (0,1]$. If $q>1$ we can prove it distinguishing the cases $b\\leq M$, where we use \\eqref{in4}, and the case $b\\geq M$, where we use $a^q-(a-b)^q\\leq M^q+2M^q\\leq 3b^q$.\nWe now prove\n\\begin{equation}\\label{in7}\n(a+b)^q-a^q\\le\\theta a^q+C_\\theta b^q \\quad  a,b\\ge 0,\\,q\\ge 1,\\,C_\\theta\\to\\infty \\ \\text{as} \\ \\theta\\to 0^+.\n\\end{equation}\nLetting  $C_q= 1$ if $q\\leq 1$ and $C_q=2^{q-1}$ if $q\\geq 1$, \\eqref{in3} and \\eqref{in2} can be written as\n\\[(a+b)^q\\le C_q(a^q+b^q) \\quad  a,b\\ge 0,\\,q>0.\\]\nNow \\eqref{in7} can be proved using Taylor's formula and Young's inequality:\n\\begin{align*}\n(a+b)^q-a^q &\\leq C_q(a^{q-1}+b^{q-1})b= (\\theta q'a)^{q-1} \\frac{C_q b}{(\\theta q')^{q-1}}+C_qb^q\\\\\n&\\leq\\theta a^q+\\frac{1}{q}\\Big(\\frac{C_q}{(\\theta q')^{q-1}}\\Big)^q b^q +C_qb^q.\n\\end{align*}\nWe prove the  following inequality:\n\\begin{equation}\\label{in1}\na^{q}-(a-b)^{q}\\ge 2^{1-q}b^{q} \\quad a\\in{\\mathbb R},\\,b\\ge 0,\\,q\\ge 1.\n\\end{equation}\nWe can suppose $b>0$ and consider the function\n\\[f(t)=t^{q}-(t-b)^{q},\\quad f'(t)=q(|t|^{q-1}-|t-b|^{q-1}).\\]\nTherefore $f$ is positive, increasing for $t>b$ and decreasing for $t<-b$ and thus it's coercive. Since $f'(t)=0$ if and only if $t=b\/2$, its global minimum is $f(b\/2)=2^{1-q}b^{q}$.\n\\vskip2pt\n\\noindent\nFinally, we will use the following inequality, holding for all $A,B\\subset{\\mathbb R}^N$ with $A$ bounded and ${\\rm dist}(A, B^c)=d>0$:\n\\begin{equation}\\label{lkj}\n|x-y|\\geq C(A, B)(1+|y|), \\quad x\\in A,\\,y\\in B^c.\n\\end{equation}\n\n\\subsection{Weak and strong solutions}\n\n\\noindent\nWe compare in the following different notions of solutions for equations driven by $(-\\Delta)^s_p$.\n\n\\begin{definition}\nLet $\\Omega$ be bounded, $u\\in \\widetilde{W}^{s,p}(\\Omega)$ and $f\\in W^{-s,p'}(\\Omega)$. We say that $u$ is a {\\em weak solution} of $(-\\Delta)^s_p u=f$ in $\\Omega$ if for all $\\varphi\\in W^{s,p}_0(\\Omega)$\n\\[\n\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy=\\langle f, \\varphi\\rangle_{s,p,\\Omega}\n\\]\nIf $\\Omega$ is unbounded, we say that $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ solves $(-\\Delta)^s_p u=f$ (with $f\\in W^{-s, p'}(\\Omega)$) weakly in $\\Omega$ if it does so in any bounded open set $\\Omega'\\subseteq\\Omega$.\n\\end{definition}\n\n\\noindent\nThe inequality $(-\\Delta)^s_p u\\leq f$ weakly in $\\Omega$ will mean that\n\\[\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy\\le\\langle f, \\varphi\\rangle_{s,p,\\Omega}\\]\nfor all $\\varphi\\in W^{s,p}_0(\\Omega)$, $\\varphi\\ge 0$, and similarly for $(-\\Delta)^s_p u\\ge f$. Noticing that $\\pm K\\in W^{-s,p'}(\\Omega)$ for any $K>0$ and any bounded $\\Omega$, by $|(-\\Delta)^s_p u|\\le K$ weakly in $\\Omega$ we mean that both $-K\\le(-\\Delta)^s_p u\\le K$ weakly in $\\Omega$.\n\\vskip2pt\n\\noindent\nIn the following proposition we will prove that $(-\\Delta)^s_p u\\in W^{-s,p'}(\\Omega)$ if $u\\in\\widetilde{W}^{s,p}(\\Omega)$, which implies that the previous definition makes sense.\n\n\\begin{lemma}\n\\label{remws}\nLet $\\Omega$ be bounded and $u\\in \\widetilde{W}^{s, p}(\\Omega)$. Then the functional \n\\[W^{s,p}_0(\\Omega)\\ni \\varphi\\mapsto ( u, \\varphi):= \\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}} \\,dx\\,dy\\]\nis finite and belongs to $W^{-s,p'}(\\Omega)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $U\\Supset\\Omega$ be such that\n\\begin{equation}\\label{Uwtilde}\n\\|u\\|_{W^{s,p}(U)}+\\int_{{\\mathbb R}^N}\\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx<\\infty,\n\\end{equation}\nand write\n\\begin{equation}\\label{<>}\n\\begin{split}\n(u, \\varphi)&=\\int_{U\\times U}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}} \\,dx\\,dy\\\\\n&\\quad +\\int_{U\\times U^c}\\frac{(u(x)-u(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}} \\,dx\\,dy\\\\\n&\\quad -\\int_{U^c\\times U}\\frac{(u(x)-u(y))^{p-1}\\varphi(y)}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&=\\int_{U\\times U}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}} \\,dx\\,dy\\\\\n&\\quad +2\\int_{\\Omega\\times U^c}\\frac{(u(x)-u(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}} \\,dx\\,dy,\n\\end{split}\n\\end{equation}\nsince ${\\rm supp}(\\varphi)\\subset\\overline\\Omega$.\nThe integral in $U\\times U$ is finite and continuous with respect to strong convergence of $\\varphi\\in W^{s,p}_0(\\Omega)$ since $u\\in W^{s,p}(U)$. For the second term, observe that for a.e.\\ $x\\in \\Omega$ it holds\n\\begin{equation}\\label{hhh}\n\\begin{split}\n&\\int_{U^c}\\frac{|u(x)-u(y)|^{p-1}}{|x-y|^{N+ps}}\\,dy\\\\\n&\\leq C\\Big(|u(x)|^{p-1}\\int_{U^c}\\frac{1}{|x-y|^{N+ps}}\\,dy+\\int_{U^c}\\frac{|u(y)|^{p-1}}{(|x-y|)^{N+ps}} \\,dy\\Big)\\\\\n&\\leq C\\Big(|u(x)|^{p-1}+\\int_{{\\mathbb R}^N}\\frac{|u(y)|^{p-1}}{(1+|y|)^{N+ps}} \\,dy\\Big),\n\\end{split}\n\\end{equation}\nwhere we used \\eqref{lkj}\nwith $A=\\Omega$ and $B=U$. The right hand side of \\eqref{hhh} belongs to $L^{p'}(\\Omega)$ since $\\Omega$ is bounded and $u\\in L^p(\\Omega)$. Thus the second term in \\eqref{<>} is continuous with respect to $L^p(\\Omega)$-convergence of $\\varphi$. Therefore it is also continuous in $W^{s,p}_0(\\Omega)$.\n\\end{proof}\n \n\\begin{definition}[Point-wise and strong solutions]\nLet $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ and $f:\\Omega\\to {\\mathbb R}$ be measurable. We say that $u$ is an {\\em a.e.\\ point-wise} solution of $(-\\Delta)^s_p u=f$ in $\\Omega$ if for a.a.\\ Lebesgue point $x\\in \\Omega$ of $u$  it holds\n\\begin{equation}\n\\label{psl-strong.1}\n2\\, {\\rm PV}\\int_{{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\, dy=f(x).\n\\end{equation}\nMoreover, for $f\\in L^1_{\\rm loc}(\\Omega)$ we say that $u$ is a {\\em strong} solution of $(-\\Delta)^s_p u=f$ if\n\\begin{equation}\n\\label{psl-strongg}\n2\\int_{B_\\varepsilon^c(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\, dy\\to f\\quad \\text{strongly in $L^1_{\\rm loc}(\\Omega)$, as $\\varepsilon\\to 0^+$.}\n\\end{equation}\n\\end{definition}\n\n\\noindent\nSimilar definitions are given for sub- and supersolutions. \n\\vskip2pt\n\\noindent\nNow we prove that a strong solution is also a weak solution. First, we introduce a more general result, which will be used in the following: we denote by ${\\bf D}$ the diagonal of ${\\mathbb R}^N\\times{\\mathbb R}^N$.\n\n\\begin{lemma}\\label{symmset}\nLet $u\\in\\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$. For all $\\varepsilon>0$ let $A_\\varepsilon\\subset{\\mathbb R}^N\\times{\\mathbb R}^N$ be a neighborhood of ${\\bf D}$ which satisfies \n\\begin{enumroman}\n\\item\\label{symmset1}\n$(x,y)\\in A_\\varepsilon$ for all $(y,x)\\in A_\\varepsilon$;\n\\item\\label{symmset2}\n${\\rm dist}_H(A_\\varepsilon, {\\bf D})\\to 0$ as $\\varepsilon\\to 0^+$.\n\\end{enumroman}\nFor all $x\\in{\\mathbb R}^N$ we set $A_\\varepsilon(x)=\\{y\\in{\\mathbb R}^N:(x,y)\\in A_\\varepsilon\\}$ and\n\\[g_\\varepsilon(x)=\\int_{A_\\varepsilon^c(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy.\\]\nIf $2g_\\varepsilon\\to f$ in $L^1_{\\rm loc}(\\Omega)$, then $u$ is a weak solution of $(-\\Delta)^s_p u=f$ in $\\Omega$.\n\\end{lemma}\n\\begin{proof}\nWe can suppose that $\\Omega$ is bounded and let $U\\Supset\\Omega$ be such that \\eqref{Uwtilde} holds for $u$, fix $\\varphi\\in C^\\infty_c(\\Omega)$ and let $K={\\rm supp}(\\varphi)$. First we prove that $g_\\varepsilon\\in L^1(K)$. For all $x\\in K$ there exists $\\rho>0$ such that $B_\\rho(x)\\subset A_\\varepsilon(x)$, and by a covering argument we may choose $\\rho$ independent of $x$ (while $\\rho$ depends on $\\varepsilon$). Moreover, for all $x\\in K$ and $y\\in A_\\varepsilon^c(x)$ we have $|x-y|\\ge C(1+|y|)$ (see \\eqref{lkj}). So we can compute\n\\begin{align*}\n\\int_K|g_\\varepsilon(x)|\\,dx &\\le C\\int_K\\int_{A_\\varepsilon^c(x)}\\frac{|u(x)|^{p-1}}{|x-y|^{N+ps}}\\,dy\\,dx+C\\int_K\\int_{A_\\varepsilon^c(x)}\\frac{|u(y)|^{p-1}}{|x-y|^{N+ps}}\\,dy\\,dx \\\\\n&\\le C\\int_K|u(x)|^{p-1}\\,dx\\int_{B_\\rho^c}\\frac{1}{|z|^{N+ps}}\\,dz+C\\int_K\\int_{A^c_\\varepsilon(x)}\\frac{|u(y)|^{p-1}}{(1+|y|)^{N+ps}}\\,dy\\\\\n&\\le C_\\varepsilon\\int_U|u(x)|^{p-1}\\,dx+C|K|\\int_{{\\mathbb R}^N}\\frac{|u(y)|^{p-1}}{(1+|y|)^{N+ps}}\\,dy < \\infty.\n\\end{align*}\nLemma \\ref{remws} shows that \n\\[\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\in L^1({\\mathbb R}^N\\times {\\mathbb R}^N)\\]\nand thus, through \\ref{symmset1}, \\ref{symmset2}, and Fubini's theorem we have\n\\begin{align*}\n&\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&\\underset{(ii)}{=} \\lim_{\\varepsilon\\to 0^+}\\int_{A_\\varepsilon^c}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dy\\,dx\\\\\n&\\underset{(i)}{=}\\lim_{\\varepsilon\\to 0^+}2\\int_K\\int_{A_\\varepsilon^c(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\varphi(x)\\,dx\\,dy\\\\\n&=\\lim_{\\varepsilon\\to 0^+}2\\int_Kg_\\varepsilon(x)\\varphi(x)\\,dx.\n\\end{align*}\nSince $2g_\\varepsilon\\to f$ in $L^1(K)$, the density of $C^\\infty_c(\\Omega)$ in $W^{s,p}_0(\\Omega)$ and Lemma \\ref{remws} give the assertion.\n\\end{proof}\n\n\\begin{remark}\nAs the proof shows, it suffices to assume that the convergence in \\eqref{psl-strongg} be in $L^1_{\\rm loc}(\\Omega)$ weakly. We deliberately choose to assume strong $L^1_{\\rm loc}$-convergence since in all subsequent applications this is enough.\n\\end{remark}\n\n\\begin{corollary}\n\\label{simplw}\nLet $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ be a strong solution of $(-\\Delta)^s_p u=f$ in $\\Omega$, with $f\\in L^1_{\\rm loc}(\\Omega)$. Then $u$ is a weak solution of $(-\\Delta)^s_p u=f$ in $\\Omega$.\n\\end{corollary}\n\\begin{proof}\nIt follows from Lemma \\ref{symmset} with $A_\\varepsilon=\\{(x,y)\\in{\\mathbb R}^N\\times{\\mathbb R}^N:\\,|x-y|<\\varepsilon\\}.$\n\\end{proof}\n\n\\subsection{Some basic properties of $(-\\Delta)^s_p$}\n\nThe following result describes a fundamental non-local feature of $(-\\Delta)^s_p$.\n\\begin{lemma}[Non-local behavior of $(-\\Delta)^s_p$]\n\\label{psadd}\nSuppose $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ solves $(-\\Delta)^s_p u=f$ weakly, strongly or point-wisely in $\\Omega$ for some $f\\in L^1_{\\rm loc}(\\Omega)$. Let $v\\in L^1_{\\rm loc}({\\mathbb R}^N)$ be such that \n\\begin{equation}\n\\label{hyppertv}\n{\\rm dist}({\\rm supp}(v),\\Omega)>0,\\quad \\int_{\\Omega^c}\\frac{|v(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx<\\infty,\n\\end{equation}\nand define for a.e.\\ Lebesgue point $x\\in \\Omega$ of $u$\n\\[h(x)=2\\int_{{\\rm supp}(v)}\\frac{(u(x)-u(y)-v(y))^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy.\\]\nThen $u+v\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ and it solves $(-\\Delta)^s_p(u+v)=f+h$ weakly, strongly or pointwisely respectively in $\\Omega$.\n\\end{lemma}\n\n\\begin{proof}\nAs usual, it suffices to consider the case $\\Omega$ bounded, and we first prove that $u+v\\in \\widetilde{W}^{s,p}(\\Omega)$. \nLet $K={\\rm supp}(v)$ and $U$ be such that \\eqref{Uwtilde} holds for $u$, and suppose without loss of generality that $\\Omega\\Subset U\\Subset K^c$. Clearly $u+v=u$ in $U$, and thus it belongs to $W^{s,p}(U)$. Moreover\n\\[\\int_{{\\mathbb R}^N}\\frac{|u(x)+v(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx\\leq C\\Big(\\int_{{\\mathbb R}^N}\\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx+\\int_{K}\\frac{|v(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx\\Big),\\]\nand the last term is finite due to \\eqref{hyppertv}.\nWith a similar estimate, we see that the integral defining $h$ is finite (due also to \\eqref{hyppertv} and  \\eqref{lkj}).\nConsider now the case where $(-\\Delta)^s_p u=f$ weakly.\nChoose $\\varphi\\in C^\\infty_c(\\Omega)$ and compute\n\\begin{align*}\n&\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)+v(x)-u(y)-v(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&=\\int_{\\Omega\\times \\Omega}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&\\quad + \\int_{\\Omega\\times \\Omega^c}\\frac{(u(x)-u(y)-v(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&\\quad -\\int_{\\Omega^c\\times \\Omega}\\frac{(u(x)+v(x)-u(y))^{p-1}\\varphi(y)}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&= \\int_{{\\mathbb R}^N\\times {\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&\\quad -\\int_{\\Omega \\times \\Omega^c}\\frac{(u(x)-u(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&\\quad +\\int_{\\Omega^c\\times \\Omega}\\frac{(u(x)-u(y))^{p-1}\\varphi(y)}{|x-y|^{N+ps}}\\,dx\\,dy+2\\int_{\\Omega\\times \\Omega^c}\\frac{(u(x)-u(y)-v(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&= \\int_\\Omega f(x)\\varphi(x)\\,dx+2\\int_{\\Omega\\times \\Omega^c}\\frac{(u(x)-u(y)-v(y))^{p-1}-(u(x)-u(y)))^{p-1}}{|x-y|^{N+ps}}\\varphi(x)\\,dx\\,dy \\\\\n&= \\int_\\Omega(f(x)+h(x))\\varphi\\,dx,\n\\end{align*}\nwhere in the end we have used Fubini's theorem. The density of $C^\\infty_c(\\Omega)$ in $W^{s,p}_0(\\Omega)$ allows to conclude.\n\\vskip2pt\n\\noindent\nSuppose now that $(-\\Delta)^s_p u=f$ strongly or pointwisely in $\\Omega$. Let for $x\\in V\\Subset\\Omega$ and $\\varepsilon<{\\rm dist}(V, \\Omega^c)$\n\\[g_\\varepsilon(x)=\\int_{B^c_\\varepsilon(x)}\\frac{(u(x)+v(x)-u(y)-v(y))^{p-1}}{|x-y|^{N+ps}}\\,dy.\\] \nUsing \\eqref{hyppertv} we get\n\\begin{align*}\n&g_\\varepsilon(x)=\\int_{\\Omega\\setminus B_\\varepsilon(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy+\\int_{\\Omega^c}\\frac{(u(x)-u(y)-v(y))^{p-1}}{|x-y|^{N+ps}}\\,dy\\\\\n&= \\int_{B_\\varepsilon^c(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy+\\int_{K}\\frac{(u(x)-u(y)-v(y))^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy.\n\\end{align*}\nTaking the limit for $\\varepsilon\\to 0^+$ gives the claim in the pointwise case. To show that $(-\\Delta)^s_p (u+v)=f+h$ strongly it suffices to show that\n\\[x\\mapsto \\int_{K}\\frac{(u(x)-u(y)-v(y))^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy\\]\nbelongs to $L^1(K)$, which can be  done proceeding as in \\eqref{hhh} and using \\eqref{hyppertv} for the term involving $v$. \n\\end{proof}\n\n\\noindent\nWe also recall the well known homogeneity, scaling, and rotational invariance properties of $(-\\Delta)^s_p$. For all $\\rho>0$, $M\\in O_N$ (the orthogonal group), $v$ measurable, $\\Omega\\subseteq{\\mathbb R}^N$, set\n\\begin{align*}\nv_\\rho(x)& =v(\\rho x),\\quad \\rho^{-1}\\Omega=\\{x\/\\rho:x\\in \\Omega\\},  \\\\\nv_M(x)&=v(Mx),\\quad M^{-1}\\Omega =\\{M^{-1}x:x\\in \\Omega\\}.\n\\end{align*}\n\\begin{lemma}\\label{hs}\nLet $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ satisfy $(-\\Delta)^s_p u=f$ weakly in $\\Omega$ for some $f\\in L^{1}_{\\rm loc}(\\Omega)$. Then we have\n\\begin{enumroman}\n\\item\\label{hs.1} for all $h>0$, $(-\\Delta)^s_p (hu)=h^{p-1}f$ weakly in $\\Omega$;\n\\item\\label{hs.2} for all $\\rho>0$, $u_\\rho\\in \\widetilde{W}^{s,p}(\\rho^{-1}\\Omega)$ and $(-\\Delta)^s_p u_\\rho=\\rho^{ps}f_\\rho$ weakly in $\\rho^{-1}\\Omega$;\n\\item\\label{hs.3} for all $M\\in O_N$, $u_M\\in \\widetilde{W}^{s,p}(M^{-1}\\Omega)$ and  $(-\\Delta)^s_p u_M=f_M$ weakly in $M^{-1}\\Omega$.\n\\end{enumroman}\n\\end{lemma}\n\n\\noindent\nFinally, from Lemma 9 of \\cite{LL} we have the following comparison principle for $(-\\Delta)^s_p$.\n\n\\begin{proposition}[Comparison Principle]\\label{comp}\nLet $\\Omega$ be bounded, $u,v\\in \\widetilde{W}^{s,p}(\\Omega)$ satisfy $u\\le v$ in $\\Omega^c$ and, for all $\\varphi\\in W^{s,p}_0(\\Omega)$, $\\varphi\\ge 0$ in $\\Omega$,\n\\begin{align*}\n&\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&\\le \\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(v(x)-v(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy.\n\\end{align*}\nThen $u\\le v$ in $\\Omega$.\n\\end{proposition}\n\n\\begin{proof}\nThe proof follows by the arguments of \\cite{LL}. It is sufficient to know that both sides \nof the inequality are finite and $(u-v)_+\\in W^{s,p}_0(\\Omega)$, which is used there as a test function.\nBy Lemma \\ref{remws}, both sides are finite. We claim that $w:=(u-v)_+\\in W^{s,p}_0(\\Omega)$. Let $U\\Supset\\Omega$ be as in Definition \\ref{defwtilde} for both $u$ and $v$. We split the Gagliardo norm in ${\\mathbb R}^N$ as\n\\begin{align*}\n&\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|w(x)-w(y)|^p}{|x-y|^{N+ps}}\\, dx\\, dy\\\\\n&= \\int_{U\\times U}\\frac{|w(x)-w(y)|^p}{|x-y|^{N+ps}}\\, dx\\, dy+2\\int_{\\Omega\\times U^c}\\frac{|w(x)|^p}{|x-y|^{N+ps}}\\, dx\\, dy\n\\end{align*}\nwhere we used that $w=0$ in $\\Omega^c$ by assumption.\nThe first term is bounded since $u, v\\in W^{s,p}(U)$, which is a lattice. The second term is non-singular since ${\\rm dist}(\\Omega, U^c)>0$ and using  \\eqref{lkj}  we get\n\\begin{align*}\n\\int_{\\Omega\\times U^c}\\frac{|w(x)|^p}{|x-y|^{N+ps}}\\,dx\\,dy\n&\\leq C_{\\Omega, U}\n\\int_\\Omega (|u(x)|^{p}+ |v(x)|^{p})\\, dx\\int_{{\\mathbb R}^N}\\frac{1}{(1+|y|)^{N+ps}}\\, dy \\\\\n& \\leq C_{\\Omega, U}\\int_\\Omega (|u(x)|^{p}+ |v(x)|^{p})\\, dx,\n\\end{align*}\nwhich proves the claim. \n\\end{proof}\n\n\\subsection{$(-\\Delta)^s_p$ on smooth functions}\n\nNext we show that in the class of sufficiently smooth functions, the $s$-fractional $p$-Laplacian exists strongly (and thus weakly) and is locally bounded. First we recall the following definition of $(-\\Delta)^s_p$, equivalent to \\eqref{psl-strong.1} (by a simple change of variable):\n\\begin{equation}\\label{psl-strong}\n(-\\Delta)^s_p u(x) = {\\rm PV}\\int_{{\\mathbb R}^N}\\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\,dz.\n\\end{equation}\nOur first lemma displays an estimate which allows us to remove the singularity at $0$, when $u$ is smooth enough:\n\n\\begin{lemma}\\label{zero}\nIf $u\\in C^{1,\\gamma}_{\\rm loc}(\\Omega)$, $\\gamma\\in [0,1]$, and $K\\subset\\Omega$ is compact, then there exist $C_{K, u},R_K>0$ \nsuch that for all $x\\in K$, $z\\in B_{R_K}$\n\\begin{equation*}\n\\big|(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}\\big| \n\\leq \n\\begin{cases}\n C_{K, u}|z|^{\\gamma+p-1}  & \\text{if $p\\geq 2$}\\\\\n C_{K, u}|z|^{(\\gamma+1)(p-1)} & \\text{if $p< 2$}.\n\\end{cases}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $K$ is compact, we can find $R_K>0$ such that\n\\[\\Omega_K:=\\{x\\in{\\mathbb R}^N:\\,{\\rm dist}(x,K)\\le R_K\\}\\subset\\Omega.\\]\nConsider first the case $p\\geq 2$. Since $u\\in C^{1,\\gamma}(\\Omega_K)$, for all $x\\in K$, $z\\in B_{R_K}$ there exist $\\tau_1,\\tau_2\\in[0,1]$ with\n\\begin{align*}\n&\\big|(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}\\big| \\\\\n&= \\big|(Du(x+\\tau_1 z)\\cdot z)^{p-1}-(Du(x-\\tau_2 z)\\cdot z)^{p-1}\\big| \\\\\n&\\le (p-1)\\sup_{B_{R_K}(x)}|Du|^{p-2}|z|^{p-2}\n\\big|\\big(Du(x+\\tau_1 z)-Du(x-\\tau_2 z)\\big)\\cdot z\\big| \\\\\n&\\le C\\|Du\\|_{C^{0,\\gamma}(\\Omega_{K})}^{p-1}|z|^{\\gamma+p-1}.\n\\end{align*}\nIf $1<p<2$ then $t\\mapsto t^{p-1}$ is globally $(p-1)$-H\\\"older continuous and in this case we directly have, with the same notation as before,\n\\begin{align*}\n&\\big|(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}\\big| \\\\\n&\\le C\\big|Du(x+\\tau_1 z)\\cdot z-Du(x-\\tau_2 z)\\cdot z\\big|^{p-1}\\\\\n&\\le C\\|Du\\|_{C^{0,\\gamma}(\\Omega_{K})}^{p-1}|z|^{\\gamma(p-1)}|z|^{p-1},\n\\end{align*}\nwhich concludes the proof.\n\\end{proof}\n\n\\noindent\nThe following result shows sufficient conditions to write $(-\\Delta)^s_p u$ as a locally bounded function:\n\n\\begin{proposition}[$(-\\Delta)^s_p$ on $C^{1,\\gamma}$ functions]\\label{weak-strong}\nSuppose $\\Omega$ is bounded, $u\\in \\widetilde{W}^{s,p}(\\Omega)\\cap C^{1,\\gamma}_{\\rm loc}(\\Omega)$, with $\\gamma\\in [0,1]$ such that\n\\begin{equation}\\label{assumptionsgamma}\n\\gamma>\n\\begin{cases}\n1-p(1-s) & \\text{if $p\\ge 2$} \\\\\n\\displaystyle\\frac{1-p(1-s)}{p-1} & \\text{if $p<2$}.\n\\end{cases}\n\\end{equation}\nThen $(-\\Delta)^s_p u=f$ strongly in $\\Omega$ for some $f\\in L^\\infty_{\\rm loc}(\\Omega)$\n\\end{proposition}\n\\begin{proof}\nLet $U$ be as in Definition \\ref{defwtilde} for $u$, fix a compact set $K\\subset\\Omega$ and let $R_K,C_K>0$ be as in Lemma \\ref{zero}. Define, for $x\\in K$, $\\varepsilon>0$, \n\\begin{align*}\ng_\\varepsilon(x)&:=\\int_{B_\\varepsilon^c}\\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\, dz\\\\\n&=2\\int_{B_\\varepsilon^c}\\frac{(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\, dz.\n\\end{align*}\nWe claim that $g_\\varepsilon$ converges as $\\varepsilon\\to 0^+$ in a dominated way to some $f\\in L^\\infty(K)$. We split the integral in one for $z\\in B_{R_K}$ and one over $B_{R_K}^c$. For the first one, the previous lemma gives\n\\[\\Big|\\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\Big| \\le \\frac{C_{K, u}}{|z|^{N+ps-\\sigma}},\\]\nwhere $\\sigma=\\gamma+p-1$ if $p\\geq 2$ and $\\sigma=(\\gamma+1)(p-1)$ if $1<p<2$. Notice that, in both cases, we have $ps-\\sigma<0$. Due to assumptions \\eqref{assumptionsgamma}, the integral is thus non-singular, and it holds\n\\[\\lim_{\\varepsilon\\to 0}\\int_{B_{R_K}\\setminus B_\\varepsilon}\\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\, dz=:f_1(x),\\]\n\\[\\Big|\\int_{B_{R_K}\\setminus B_\\varepsilon}\\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\, dz\\Big|\\leq \\int_{B_{R_K}}\\frac{C_{K, u}}{|z|^{N+ps-\\sigma}}\\, dz,\\]\nwhich is a bound independent of $x\\in K$ and $\\varepsilon>0$. For the integral over $z\\in B^c_{R_K}$ we have, as in \\eqref{hhh},\n\\begin{align*}\n|f_2(x)|&:=\\Big|2\\int_{B^c_{R_K}}\\frac{(u(x)-u(x+z))^{p-1}}{|z|^{N+ps}}\\, dz\\Big|\\\\\n&\\leq  C_{K, U}\\Big(\\|u\\|_{L^\\infty(K)}^{p-1}+\\int_{{\\mathbb R}^N}\\frac{|u(y)|^{p-1}}{(1+|y|)^{N+ps}}\\, dy\\Big).\n\\end{align*}\nGathering togheter the two estimates, we get\n\\[|g_\\varepsilon(x)|\\leq C_{K, u, U}\\quad \\text{$\\forall x\\in K$, $\\varepsilon>0$},\\]\n\\[\\lim_{\\varepsilon\\to 0^+}g_\\varepsilon(x)= f_1(x)+f_2(x)\\quad \\forall x\\in K,\\]\nand thus by the dominated convergence theorem $g_\\varepsilon\\to f_1+f_2$ in $L^1(K)$. \n\\end{proof}\n\n\\begin{remark}\nIt is useful to outline the dependence of $\\|(-\\Delta)^s_p u\\|_\\infty$ on $s$ in the previous proposition. Suppose, to fix ideas, that $p\\geq 2$ and $u\\in C^\\infty_c({\\mathbb R}^N)$, so that the domain $\\Omega$ has no role. Then, following the proof, we can find a constant $c_N$ depending only on $N$ such that\n\\[\\|(-\\Delta)^s_p u\\|_{\\infty}\\leq c_N\\frac{\\|u\\|_{C^2({\\mathbb R}^N)}^{p-1}}{1-s}.\\]\nThis is in accordance with the well known fact that $(1-s)(-\\Delta)^s_p \\to -\\Delta_p$ as $s\\to 1^-$ (see e.g.\\ \\cite{ponce}).\n\\end{remark}\n\n\\begin{remark}\n\\label{remarkL}\nConsider the class of functions\n\\[{\\mathcal L}(\\Omega)=\\{u\\in \\widetilde{W}^{s,p}(\\Omega):(-\\Delta)^s_p u=f\\,\\,\\text{strongly for some $f\\in L^\\infty_{\\rm loc}(\\Omega)$}\\}.\\]\nThe previous theorem asserts that if $p\\geq 2$, then $C^2(\\Omega)\\subseteq {\\mathcal L}(\\Omega)$. However, if $1<p<2$, it may be difficult  to find smooth functions (e.g., smooth cut-offs) belonging to ${\\mathcal L}(\\Omega)$, since the second condition in \\eqref{assumptionsgamma} coupled with $\\gamma\\leq 1$ forces $s<2(p-1)\/p$. One may think that this is just a technical limit of the proof, or that requiring higher regularity than $C^2$ could solve the issue. Unfortunately, due to the singular nature of the operator for $1<p<2$, this is not the case: there are smooth functions $u$ such that $(-\\Delta)^s_p u$ (in the strong sense) cannot be pointwise bounded. Consider for example $u(x)=x^2\\eta(x)$, where $\\eta\\in C^\\infty_c({\\mathbb R})$ and $\\eta=1$ on $[-1,1]$. Calculating $(-\\Delta)^s_p u(0)$ as a principal value  gives\n\\[|(-\\Delta)^s_p u(0)|<\\infty\\quad \\Leftrightarrow\\quad s<2\\frac{p-1}{p}.\\]\n\\end{remark}\n\n\\section{Distance functions}\\label{sec3}\n\n\\noindent\nIn this section we produce some explicit solutions for $(-\\Delta)^s_p$ in special domains. Then we prove that $(-\\Delta)^s_p\\delta^s$ is bounded in a neighborhood of $\\partial\\Omega$ (here we define $\\delta=\\delta_\\Omega$ as in Section \\ref{sec2}). We begin by getting a solution of $(-\\Delta)^s_p u=0$ on the half-line ${\\mathbb R}_+$.\n\n\\begin{lemma}\\label{sol1d}\nSet $u_1(x)=x_+^s$ for all $x\\in{\\mathbb R}$. Then $u_1\\in \\widetilde{W}^{s,p}_{\\rm loc}({\\mathbb R})$ and it solves $(-\\Delta)^s_p u_1=0$ strongly and weakly in ${\\mathbb R}_+$.\n\\end{lemma}\n\\begin{proof}\nLet $K\\subseteq (\\rho, \\rho^{-1})$ for some $\\rho\\in(0,1)$. For $x\\in K$, $\\varepsilon>0$ consider the function\n\\begin{equation}\n\\label{defg}\ng^{(1)}_\\varepsilon(x)=\\int_{B_\\varepsilon^c(x)}\\frac{(x^s-y_+^s)^{p-1}}{|x-y|^{1+ps}}\\, dy.\n\\end{equation}\nWe claim that $g^{(1)}_\\varepsilon\\to 0$ uniformly on $K$, as $\\varepsilon\\to 0^+$. Note that for any $\\varepsilon<x$ it holds\n\\[0<x-\\varepsilon<x+\\varepsilon<\\frac{x^2}{x-\\varepsilon}.\\]\nWe split the integral accordingly, namely\n\\begin{align*}\ng^{(1)}_\\varepsilon(x)&=\\int_{-\\infty}^0\\frac{(x^s-y_+^s)^{p-1}}{|x-y|^{1+ps}}\\, dy\n+\\int_{x+\\varepsilon}^{\\frac{x^2}{x-\\varepsilon}}\\frac{(x^s-y_+^s)^{p-1}}{|x-y|^{1+ps}}\\, dy \\\\\n&\\quad +\\Big(\\int_0^{x-\\varepsilon}\\frac{(x^s-y_+^s)^{p-1}}{|x-y|^{1+ps}}\\, dy+\\int_{\\frac{x^2}{x-\\varepsilon}}^{\\infty}\\frac{(x^s-\ny_+^s)^{p-1}}{|x-y|^{1+ps}}\\, dy\\Big) \\\\\n&= I_1(x)+I_2(x, \\varepsilon)+I_3(x, \\varepsilon).\n\\end{align*}\nLet us estimate the three terms separately. It holds\n\\[I_1(x)=\\int_{-\\infty}^0\\frac{(x^s-y_+^s)^{p-1}}{|x-y|^{1+ps}}\\, dy=\\frac{x^{-s}}{ps}.\\]\nRegarding the integral between $x+\\varepsilon$ and $\\frac{x^2}{x-\\varepsilon}$, since $\\xi\\mapsto \\xi^s$ is globally $s$-H\\\"older, we have\n\\[|I_2(x, \\varepsilon)|\\leq C\\int_{x+\\varepsilon}^{\\frac{x^2}{x-\\varepsilon}}\\frac{|x-y|^{s(p-1)}}{|x-y|^{1+ps}}\\, dy=\\frac{Cx^{-s}}{s}\\frac{x^s-(x-\\varepsilon)^s}{\\varepsilon^s}.\\]\nFinally,\n\\begin{align*}\nI_3(x, \\varepsilon)&=\\frac{x^{s(p-1)}}{x^{1+ps}}\\Big(\\int_0^{x-\\varepsilon}\\frac{(1-(y\/x)^s)^{p-1}}{(1-y\/x)^{1+ps}}\\, dy-\\int_{x^2\/(x-\\varepsilon)}^{\\infty}\\frac{((y\/x)^s-1)^{p-1}}{(y\/x-1)^{1+ps}}\\, dy\\Big)\\\\\n&\\!\\!\\!\\!\\!\\underset{t=\\frac{y}{x}=\\xi}{=} x^{-s}\\Big(\\int_0^{1-\\varepsilon\/x}\\frac{(1-t^s)^{p-1}}{(1-t)^{1+ps}}\\, dt-\\int_{(1-\\varepsilon\/x)^{-1}}^{\\infty}\\frac{(\\xi^s-1)^{p-1}}{(\\xi-1)^{1+ps}}\\, d\\xi\\Big)\\\\\n&\\!\\!\\!\\!\\underset{\\xi=t^{-1}}{=}x^{-s}\\Big(\\int_0^{1-\\varepsilon\/x}\\frac{(1-t^s)^{p-1}}{(1-t)^{1+ps}}\\, dt-\\int_0^{1-\\varepsilon\/x}\\frac{(t^{-s}-1)^{p-1}}{(t^{-1}-1)^{1+ps}}\\, \\frac{dt}{t^2}\\Big)\\\\\n&=x^{-s}\\int_0^{1-\\varepsilon\/x}\\frac{(1-t^s)^{p-1}}{(1-t)^{1+ps}}(1-t^{s-1})\\, dt\\\\\n&=x^{-s}\\Big[\\frac{1}{ps}\\frac{(1-t^s)^p}{(1-t)^{ps}}\\Big]_0^{1-\\varepsilon\/x}=\\frac{x^{-s}}{ps}\\Big(\\Big(\\frac{x^s-(x-\\varepsilon)^s}{\\varepsilon^s}\\Big)^p-1\\Big).\n\\end{align*}\nLet for $\\varepsilon<x$\n\\begin{equation}\n\\label{defpsi}\n\\psi(x, \\varepsilon)=\\frac{x^s-(x-\\varepsilon)^s}{\\varepsilon^s},\n\\end{equation}\nand notice that from the subadditivity of $x\\mapsto x^s$ we get $\\psi(x, \\varepsilon)\\leq 1$. Gathering together the three previous estimates we get\n\\begin{equation}\n\\label{estg}\n|g^{(1)}_\\varepsilon(x)|\\leq Cx^{-s}(\\psi(x, \\varepsilon)+\\psi^p(x, \\varepsilon))\\leq  Cx^{-s}\\psi(x, \\varepsilon) \\quad \\forall x>\\varepsilon>0,\n\\end{equation}\nwhere $C$ is a universal constant.\nSince  $\\psi(x, \\varepsilon)\\to 0$ uniformly on $[\\rho, \\rho^{-1}]\\supseteq K$, as $\\varepsilon\\to 0^+$, the claim follows. Finally we prove that $u_1\\in \\widetilde{W}^{s,p}(a,b)$ for any $a<0<b$. We have\n\\begin{align*}\n&\\int_{[a,b]\\times[a,b]}\\frac{|u_1(x)-u_1(y)|^p}{|x-y|^{1+ps}}\\, dx\\, dy\\\\\n&\\,\\,\\,\\,=2\\int_0^b\\int_0^x\\frac{|x^s-y^s|^p}{|x-y|^{1+ps}}\\, dy\\, dx+2\\int_0^bx^{sp}\\int_a^0\\frac{1}{|x-y|^{1+ps}}\\, dy\\, dx\\\\\n&\\underset{t=y\/x}{=}2\\int_0^b\\int_0^1\\frac{|1-t^s|^p}{|1-t|^{1+ps}}\\, dt\\, dx+\\frac{2}{ps}\\int_0^bx^{sp}\\Big(\\frac{1}{x^{ps}}-\\frac{1}{|x-a|^{ps}}\\Big)\\, dx,\n\\end{align*}\nwhich is readily checked to be finite. The assertion follows through Lemma \\ref{remws} and Corollary \\ref{simplw}.\n\\end{proof}\n\n\\noindent\nNow we study the solution $u(x)=u_1(x_N)$ in the half-space\n\\[{\\mathbb R}^N_+=\\{x\\in{\\mathbb R}^N:\\,x_N>0\\}.\\]\n\n\\begin{lemma}\\label{solnd}\nSet for any $A\\in GL_N$ and $x\\in {\\mathbb R}^N_+$,\n\\[g_{\\varepsilon}(x, A)=\\int_{B_\\varepsilon^c}\\frac{(u_1(x_N)-u_1(x_N+z_N))^{p-1}}{|Az|^{N+ps}}\\, dz\\]\nand $u(x)=u_1(x_N)$.\nThen $g_\\varepsilon\\to 0$ uniformly in any compact $K\\subseteq {\\mathbb R}^N_+\\times GL_N$ and $u\\in \\widetilde{W}^{s,p}_{\\rm loc}({\\mathbb R}^N)$ solves $(-\\Delta)^s_p u=0$ strongly and weakly in ${\\mathbb R}^N_+$.\n\\end{lemma}\n\\begin{proof}\nIt suffices to prove the statement for $K=H\\times H'$, where $H\\subseteq {\\mathbb R}^N_+$ and $H'\\subseteq GL_N$ are compact (recall that $GL_N$ is open in ${\\mathbb R}^{N^2}$). \nTo estimate $g_{\\varepsilon}$ we use elliptic coordinates. A consequence of the singular value decomposition is that $A S^{N-1}$ is an ellipsoid whose semiaxes are the singular values of $A$, and thus its diameter is $2\\|A\\|_2$, where the latter is the spectral norm of $A$.\nThe corresponding elliptic coordinates are uniquely defined by\n\\[y=\\rho\\omega, \\quad \\omega\\in AS^{N-1},\\quad \\rho>0,\\]\nfor $y\\in {\\mathbb R}^N\\setminus\\{0\\}$. It holds $dy=\\rho^{N-1}d\\omega\\, d\\rho$ where $d\\omega$ is the surface element of $A S^{N-1}$. Setting \n\\[e_A:=A^{-1}e_N,\\quad E_A:=\\{x\\in {\\mathbb R}^N:\\,x\\cdot e_A\\geq 0\\},\\]\nwe compute, through the change of variable $z=A^{-1}y$,\n\\begin{align*}\n&g_{\\varepsilon}(x, A)\\\\\n&=\\int_{B^c_\\varepsilon}\\frac{(u(x)-u(x+z))^{p-1}}{|Az|^{N+ps}}\\, dz=|{\\rm det} A|^{-1}\\int_{AB^c_\\varepsilon}\\frac{(u(x)-u(x+A^{-1}y))^{p-1}}{|y|^{N+ps}}\\, dy\\\\\n&=\\int_{AS^{N-1}}\\frac{1}{|{\\rm det} A||\\omega|^{N+ps}}\\int_\\varepsilon^{\\infty}\\frac{(u_1(x_N) - u_1(x_N+\\omega\\cdot e_A\\rho))^{p-1}}{\\rho^{1+ps}}\\, d\\rho\\, d\\omega\\\\\n&=\\int_{AS^{N-1}\\cap E_A}\\!\\!\\frac{|\\omega\\cdot e_A|^{1+ps}}{|{\\rm det} A||\\omega|^{N+ps}}\\int_{(-\\varepsilon, \\varepsilon)^c}\\frac{(u_1(x_N)-u_1(x_N+\\omega\\cdot e_A\\rho))^{p-1}}{|\\omega\\cdot e_A\\rho|^{1+ps}}\\, d (\\omega\\cdot e_A\\rho)\\, d\\omega\\\\\n&=\\int_{AS^{N-1}\\cap E_A}\\frac{|\\omega\\cdot e_A|^{1+ps}}{|{\\rm det} A||\\omega|^{N+ps}}\\,g^{(1)}_{\\omega\\cdot e_A\\varepsilon}(x_N) \\, d\\omega,\n\\end{align*}\nwhere $g^{(1)}_{\\omega\\cdot e_A\\varepsilon}$ is defined as in \\eqref{defg}. Since $|\\omega\\cdot e_A|\\leq \\|A\\|_2\\|A^{-1}\\|_2$, the condition\n\\[\\omega\\cdot e_A\\varepsilon<x_N\\]\nholds for $\\varepsilon\\leq \\bar \\varepsilon$ where $\\bar\\varepsilon$ depends only on $H$ and $H'$. For any such $\\varepsilon$ we can apply \\eqref{estg} to obtain \n\\[|g_{\\varepsilon}(x, A)|\\leq Cx_N^{-s}\\int_{AS^{N-1}\\cap E_A}\\frac{|\\omega\\cdot e_A|^{1+ps}}{|{\\rm det} A||\\omega|^{N+ps}}\\psi(x_N, \\omega\\cdot e_A\\varepsilon)\\, d\\omega,\\]\nwhere $\\psi$ is defined in \\eqref{defpsi}. Since $\\xi\\mapsto \\xi^s$ is concave for $0<s<1$, we have\n\\[s(x_N-t)^{s-1}t\\ge x_N^s-(x_N-t)^s,\\]\nand being $1>s>0$ it follows\n\\[\\frac{\\partial \\psi(x_N, t)}{\\partial t}=s\\frac{(x_N-t)^{s-1}t-x_N^s+(x_N-t)^s}{t^{1+s}}\\geq 0, \\quad \\text{for $0< t\\le x_N$}.\\]\nTherefore $\\psi(x_N, t)$ is non-decreasing in $t$, thus we get\n\\begin{align*}\n|g_{\\varepsilon}(x, A)|&\\leq Cx_N^{-s}\\psi(x_N, \\|A\\|_2\\|A^{-1}\\|_2\\varepsilon)\\int_{AS^{N-1}}\\frac{|\\omega\\cdot e_A|^{1+ps}}{|{\\rm det} A||\\omega|^{N+ps}}\\, d\\omega\\\\\n&\\leq Cx_N^{-s}\\psi(x_N, \\|A\\|_2\\|A^{-1}\\|_2\\varepsilon)\\int_{S^{N-1}}\\frac{|\\omega\\cdot e_N|^{1+ps}}{|A\\omega|^{N+ps}}\\, d\\omega\\\\\n&\\leq Cx_N^{-s}\\psi(x_N, \\|A\\|_2\\|A^{-1}\\|_2\\varepsilon)\\|A^{-1}\\|_2^{N+ps}.\n\\end{align*}\nNow $ \\|A\\|_2$ and $\\|A^{-1}\\|_2$ are bounded on $H'$ from below and above, as well as $x_N$ on $H$, and the uniform convergence follows. As in the previous proof, it is readily checked that $u\\in \\widetilde{W}^{s,p}(V)$ for any bounded $V$, and the second statement follows as before. \n\\end{proof}\n\n\\begin{remark}\\label{rotate}\nDue to rotational invariance, Lemma \\ref{solnd} easily extends to any half-space\n\\[H_e=\\{x\\in{\\mathbb R}^N:\\,x\\cdot e\\ge 0\\} \\quad (e\\in S^{N-1}),\\]\nsimply considering the solution $u(x)=(x\\cdot e)_+^s$.\n\\end{remark}\n\n\\noindent\nThe following lemma gives a control on the behaviour of $(-\\Delta)^s_p (x_N)^s_+$ under a smooth change of variables.\n\\begin{lemma}[Change of variables]\n\\label{lemmadiffeo}\nLet $\\Phi$ be a $C^{1,1}$ diffeomorphism of ${\\mathbb R}^N$ such that $\\Phi=I$ in $B_r^c$, $r>0$. Then the function $v(x)=(\\Phi^{-1}(x)\\cdot e_N)_+^s$ belongs to $\\widetilde{W}^{s,p}_{\\rm loc}({\\mathbb R}^N)$ and is a weak solution of $(-\\Delta)^s_p v=f$ in $\\Phi({\\mathbb R}^N_+)$, with\n\\begin{equation}\n\\label{stimaf}\n\\|f\\|_\\infty\\leq C(\\|D\\Phi\\|_\\infty, \\|D\\Phi^{-1}\\|_\\infty, r)\\|D^2\\Phi\\|_\\infty.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst we recall that, since $D\\Phi$ is globally Lipschitz in ${\\mathbb R}^N$ with constant $L>0$, \nthen $D^2\\Phi(x)$ exists in the classical sense for a.a. $x\\in{\\mathbb R}^N$, and $\\|D^2\\Phi\\|_{L^\\infty({\\mathbb R}^N)}\\le L$. Let $J_\\Phi(\\cdot)=|{\\rm det}\\,D\\Phi(\\cdot)|$, $u_1(t)=t_+^s$. Due to Lemma \\ref{symmset}, applied with $A_\\varepsilon=\\{|\\Phi^{-1}(x)-\\Phi^{-1}(y)|<\\varepsilon\\}$ it suffices to show that\n\\[g_\\varepsilon(x)=\\int_{\\{|\\Phi^{-1}(x)-\\Phi^{-1}(y)|\\ge\\varepsilon\\}}\\frac{(v(x)-v(y))^{p-1}}{|x-y|^{N+ps}}\\, dy\\]\nconverges in $L^1(K)$ for any compact $K\\subseteq \\Phi({\\mathbb R}^N_+)$. Changing variables $x=\\Phi(X)$, this is equivalent to claiming that\n\\begin{equation}\n\\label{claimgepsilon}\nX\\mapsto \\int_{B_\\varepsilon^c(X)}\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\n\\end{equation}\nconverges as $\\varepsilon\\to 0$ in $L^1_{\\rm loc}({\\mathbb R}^N_+)$. To prove this claim, we write\n\\begin{equation}\\label{diffeo1}\n\\begin{split}\ng_\\varepsilon(x) &= \\int_{B_\\varepsilon^c(X)}\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|D\\Phi(X)(X-Y)|^{N+ps}}h(X, Y)\\, dY\\\\\n&\\quad +\\int_{B_\\varepsilon^c(X)}J_\\Phi(X)\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|D\\Phi(X)(X-Y)|^{N+ps}}\\, dY,\n\\end{split}\n\\end{equation}\nwhere \n\\[h(X, Y)=\\frac{|D\\Phi(X)(X-Y)|^{N+ps}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}} J_\\Phi(Y)-J_\\Phi(X), \\quad X\\neq Y.\\]\nWe will now prove the following estimate, from which convergence of \\eqref{claimgepsilon} will follow:\n\\begin{equation}\n\\label{diffeo2}\n|h(X, Y)|\\leq C_\\Phi\\|D^2\\Phi\\|_\\infty\\min\\{|X-Y|, 1\\},\n\\end{equation}\nwhere $C_\\Phi$ depends on $N$, $p$, $s$ as well as on $\\|D\\Phi\\|_\\infty$, $\\|D\\Phi^{-1}\\|_\\infty$ and $r$. Write\n\\begin{align*}\nh(X, Y) &=\\frac{|D\\Phi(X)(X-Y)|^{N+ps}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}} (J_\\Phi(Y)-J_\\Phi(X))\\\\\n&\\quad +J_\\Phi(X)\\Big(\\frac{|D\\Phi(X)(X-Y)|^{N+ps}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}} -1\\Big)\\\\\n&=:J_1+J_2.\n\\end{align*}\nFirst observe that using Taylor formula yields\n\\[\\frac{|D\\Phi(X)(X-Y)|}{|\\Phi(X)-\\Phi(Y)|}\\leq C\\|D\\Phi\\|_\\infty\\|D\\Phi^{-1}\\|_\\infty,\\]\ntherefore\n\\[|J_1| \\le \\tilde{C}\\|D^2\\Phi\\|_{L^\\infty({\\mathbb R}^N)}|X-Y|.\\]\nTo estimate $J_2$, we note that the mapping $t\\mapsto t^{(N+ps)\/2}$ is smooth in a neighborhood of $1$ and that\n\\[\\lim_{Y\\to X}\\frac{|D\\Phi(X)(X- Y)|^2}{|\\Phi(X)-\\Phi(Y)|^2}=1,\\]\nhence\n\\begin{equation}\n\\label{j2}\n|J_2|\\leq C_\\Phi\\Big(\\frac{|D\\Phi(X)(X-Y)|^{2}}{|\\Phi(X)-\\Phi(Y)|^{2}} -1\\Big).\n\\end{equation}\nBesides, for all $Y\\in{\\mathbb R}^N$ there exist $\\tau_1,\\ldots,\\tau_N\\in[0,1]$ such that\n\\[\\Phi^i(X)-\\Phi^i(Y)=D\\Phi^i(\\tau_i X+(1-\\tau_i)Y)\\cdot(X-Y), \\quad i=1,\\ldots,N,\\]\nwhere $\\Phi^i$ denotes the $i$-th component of $\\Phi$. So we have (still allowing $C_\\Phi>0$ to depend on $\\|D\\Phi\\|_{L^\\infty({\\mathbb R}^N)}$)\n\\begin{align*}\n&\\big||\\Phi(X)-\\Phi(Y)|^2-|D\\Phi(X)(X-Y)|^2\\big| \\\\\n&= \\big|(\\Phi(X)-\\Phi(Y)+D\\Phi(X)(X-Y))\\cdot(\\Phi(X)-\\Phi(Y)-D\\Phi(X)(X-Y))\\big| \\\\\n&\\le C_\\Phi|X-Y|\\sum_{i=1}^N|\\Phi^i(X)-\\Phi^i(Y)-D\\Phi^i(X)(X-Y)| \\\\\n&\\le C_\\Phi|X-Y|^2\\sum_{i=1}^N|D\\Phi^i(\\tau_i X+(1-\\tau_i)Y)-D\\Phi^i(X)| \\\\\n&\\le C_\\Phi\\|D^2\\Phi\\|_\\infty|X-Y|^3.\n\\end{align*}\nInserting into \\eqref{j2} we obtain\n\\begin{equation*}\n|J_2| \\le C_\\Phi\\frac{\\big||D\\Phi(X)(X-Y)|^2-|\\Phi(X)-\\Phi(Y)|^2\\big|}{|\\Phi(X)-\\Phi(Y)|^2} \n\\le C_\\Phi\\|D^2\\Phi\\|_\\infty|X-Y|,\n\\end{equation*}\nwhich yields\n\\[|h(X, Y)|\\leq C_\\Phi\\|D^2\\Phi\\|_{L^\\infty({\\mathbb R}^N)}|X-Y|,\\quad \\text{for all $X, Y\\in {\\mathbb R}^N$},\\]\nand thus \\eqref{diffeo2} for $|X-Y|\\leq 2r$. Assume now $|X-Y|>2r$, then at least one of $X$, $Y$ lies in $\\overline B_r^c$. Clearly, if $X,Y\\in\\overline B_r^c$, then $h(X,Y)=0$. If $X\\in\\overline B_r$, $Y\\in\\overline B_r^c$, then for any $1\\le i\\le N$ we define a mapping $\\eta_i\\in C^{1,1}([0,1])$ by setting\n\\[\\eta_i(t)=\\Phi^i(X+t(Y-X)).\\]\nIt is readily checked that $|\\eta''_i|\\leq C\\|D^2\\Phi\\|_\\infty|X-Y|^2$ for a.e.\\ $t\\in (0,1)$.\nMoreover, if $t\\geq 2r\/|X-Y|$ then $X+t(Y-X)\\in B_r^c$, and since $\\Phi=I$ outside $B_r$ it holds \n\\[\\eta_i(t)=(X+t(Y-X))\\cdot e_i\\quad \\text{for $t\\geq \\frac{2r}{|X-Y|}$}.\\]\nTherefore $\\eta_i''(t)\\equiv 0$ for $t\\geq 2r\/|X-Y|$ and applying the Taylor formula with integral remainder we have\n\\begin{align*}\n&|\\Phi^i(Y)-\\Phi^i(X)+D\\Phi^i(X)(X-Y)|\\\\\n&=|\\eta_i(1)-\\eta_i(0)-\\eta'_i(0)|\\leq \\int_0^1|\\eta''_i(t)|(1-t)\\,dt\\\\\n&\\leq \\int_0^{2r\/|X-Y|}|\\eta''_i(t)|(1-t)\\,dt\\le C_\\Phi\\|D^2\\Phi\\|_\\infty|X-Y|.\n\\end{align*}\nSo we have\n\\begin{align*}\n&|h(X,Y)|\\\\\n&\\le \\Big|\\frac{|D\\Phi(X)(X-Y)|^{N+ps}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}-1\\Big|+|1-J_\\Phi(X)| \\\\\n&\\le C_\\Phi\\Big|\\frac{|D\\Phi(X)(X-Y)|^2-|\\Phi(X)-\\Phi(Y)|^2}{|\\Phi(X)-\\Phi(Y)|^2}\\Big|+C_\\Phi\\|D^2\\Phi\\|_\\infty \\\\\n&\\le C_\\Phi\\frac{\\big|D\\Phi(X)(X-Y)+\\Phi(X)-\\Phi(Y)\\big|}{|\\Phi(X)-\\Phi(Y)|^2}\\\\\n&\\quad \\cdot\\big|D\\Phi(X)(X-Y)-\\Phi(X)+\\Phi(Y)\\big|+C_\\Phi\\|D^2\\Phi\\|_\\infty \\\\\n&\\le \\frac{C_\\Phi}{|X-Y|}\\sum_{i=1}^N\\big|D\\Phi^i(X)(X-Y)-\\Phi^i(X)+\\Phi^i(Y)\\big|+C_\\Phi\\|D^2\\Phi\\|_\\infty \\\\\n&\\le C_\\Phi\\|D^2\\Phi\\|_\\infty.\n\\end{align*}\nIf $X\\in\\overline B_r^c$, $Y\\in\\overline B_r$, we argue in a similar way. Thus \\eqref{diffeo2} is achieved for all $X,Y\\in{\\mathbb R}^N$.\n\\vskip2pt\n\\noindent\nLet us go back to \\eqref{diffeo1}. The first integral can be estimated as follows:\n\\begin{equation}\n\\label{hsf}\n\\begin{split}\n&\\int_{B_\\varepsilon^c(X)}\\Big|\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|D\\Phi(X)(X-Y)|^{N+ps}}h(X, Y)\\Big|\\, dY \\\\\n&\\leq C_\\Phi\\|D^2\\Phi\\|_\\infty\\int_{B_\\varepsilon^c(X)}\\frac{\\min\\{|X-Y|,1\\}}{|X-Y|^{N+s}}\\, dY\\\\\n&\\leq C_\\Phi\\|D^2\\Phi\\|_\\infty\\Big(\\int_\\varepsilon^1\\frac{1}{t^{s}}\\, dt+\\int_1^{\\infty}\\frac{1}{t^{1+s}}\\, dt\\Big)\\\\\n\\noalign{\\vskip2pt}\n&\\leq C_\\Phi\\|D^2\\Phi\\|_\\infty(\\varepsilon^{1-s}+1).\n\\end{split}\n\\end{equation}\nThe second integral in \\eqref{diffeo1} vanishes for $\\varepsilon\\to 0$, and is estimated through Lemma \\ref{solnd}: since $D\\Phi({\\mathbb R}^N)$ is a compact subset of $GL_N$, the integral vanishes uniformly in any compact $\\Phi^{-1}(K)\\subseteq {\\mathbb R}^N_+$, and therefore uniformly in any compact $K \\subseteq \\Phi({\\mathbb R}^N_+)$.  Lemma \\ref{symmset} thus gives that $(-\\Delta)^s_p v= f$ weakly in any open bounded $U\\subseteq \\Phi({\\mathbb R}^N_+)$, \nwhere \n\\[f(x):=2\\lim_{\\varepsilon\\to 0}g_\\varepsilon(x).\\]\nTaking the limit for $\\varepsilon \\to 0$ in estimate \\eqref{hsf}  gives \\eqref{stimaf}. \n\\end{proof}\n\n\\noindent\nFinally, we consider a general bounded domain $\\Omega$ with a $C^{1,1}$ boundary. First we recall some geometrical properties, which can be found e.g. in \\cite{AKSZ} (see figure \\ref{geometry}):\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=1.5]\n\\clip (-3.7,-0.1) rectangle (3.4,4.1);\n\\shade [left color=lightgray, right color=white, shading=axis, shading angle=180] (-3.7,0.3) to  [out=45, in=180] (0,2) to [out=0, in=200] (3.4,2) -- (3.4,4) -- (-3.7,4);\n\\draw (-3.4, 3) node[right]{$\\Omega$};\n\\draw[thick] (-3.7,0.3) to  [out=45, in=180] (0,2) to [out=0, in=200] (3.4,2);\t\t\n\\draw (0,1)  node[right]{$x_2$} circle (1cm);\n\\filldraw (0,1) circle (0.5pt);\n\\draw (1.3,0.5) node {$B_{\\rho}(x_2)$};\n\n\\draw (1.3,2.5) node{$B_{\\rho}(x_1)$};\n\\draw (0,3)  node[right]{$x_1$} circle (1cm);\n\\filldraw (0,3) circle (0.7pt);\n\n\\filldraw (0,2) circle (0.5pt);\n\\draw (0,2)  node[below right]{$x_0$};\n\n\\draw[thick] (0,2) -- (0, 3);\n\n\\end{tikzpicture}\n\\caption{The interior and exterior balls at $x_0\\in \\partial\\Omega$. For all $x\\in [x_0, x_1]$ it holds $\\delta(x)=|x-x_0|$.}\n\\label{geometry}\n\\end{figure}\n\n\\begin{lemma}\\label{geo1}\nLet $\\Omega\\subset{\\mathbb R}^N$ be a bounded domain with a $C^{1,1}$ boundary $\\partial\\Omega$. Then, there exists $\\rho>0$ such that for all $x_0\\in\\partial\\Omega$ there exist $x_1,x_2\\in{\\mathbb R}^N$ on the normal line to $\\partial\\Omega$ at $x_0$, with the following properties:\n\\begin{enumroman}\n\\item $B_{\\rho}(x_1)\\subset\\Omega$, $B_\\rho(x_2)\\subset\\Omega^c$;\n\\item $\\overline B_\\rho(x_1)\\cap\\overline B_\\rho(x_2)=\\{x_0\\}$;\n\\item $\\delta(x)=|x-x_0|$ for all $x\\in[x_0,x_1]$.\n\\end{enumroman}\n\\end{lemma}\n\n\\noindent\nAs a byproduct, we prove that $(-\\Delta)^s_p\\delta^s$ is bounded in a neighborhood of the boundary.\n\n\\begin{theorem}\\label{deltas}\nLet $\\Omega\\subset{\\mathbb R}^N$ be a bounded domain with a $C^{1,1}$ boundary. There exists $\\rho=\\rho(N, p, s, \\Omega)$ such that $(-\\Delta)_p^s\\delta^s=f$ weakly in \n\\[\\Omega_\\rho:=\\{x\\in \\Omega:\\delta(x)<\\rho\\},\\]\nfor some $f\\in L^\\infty(\\Omega_\\rho)$.\n\\end{theorem}\n\\begin{proof}\nSuppose that $\\rho$ is smaller than the one given in Lemma \\ref{geo1}. We choose a finite covering of $\\Omega_{\\rho}$ made of balls of radius $2\\rho$ and center $x_i\\in \\partial\\Omega$. Using a partition of unity, it suffices to prove the statement in any set $\\Omega \\cap B_{2\\rho}(x_i)$. To do so, we flatten the boundary near the point $x_i$, which we can suppose without loss of generality to be the origin. Choosing a smaller $\\rho$ (depending only on the geometry of $\\partial\\Omega$) if necessary, there exists a diffeomorphism $\\Phi\\in C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)$, $\\Phi(X)=x$ such that $\\Phi=I$ in $B_{4\\rho}^c$ and\n\\begin{equation}\n\\label{hasd}\n\\Omega\\cap B_{2\\rho}\\Subset \\Phi(B_{3\\rho}\\cap {\\mathbb R}^N_+),\\quad \\delta(\\Phi(X))=(X_N)_+,\\quad \\forall X\\in B_{3\\rho}.\n\\end{equation}\nWe claim that \n\\[g_\\varepsilon(x)=\\int_{\\{|\\Phi^{-1}(x)-\\Phi^{-1}(y)|\\geq \\varepsilon\\}}\\frac{(\\delta^{s}(x)-\\delta^s(y))^{p-1}}{|x-y|^{N+ps}}\\, dy\\to f(x)\n\\quad\\text{in $L^1_{\\rm loc}(\\Omega\\cap B_{2\\rho})$}.\\]\nWe change variables setting $X=\\Phi^{-1}(x)$, noting that $X\\in B_{3\\rho}\\cap {\\mathbb R}^N_+$ for any $x\\in \\Omega\\cap B_{2\\rho}$, and compute\n\\begin{align*}\ng_\\varepsilon(x)&= \\int_{\\{|X-Y|\\geq \\varepsilon\\}}\\frac{(\\delta^{s}(\\Phi(X))-\\delta^s(\\Phi(Y)))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&=\\int_{B_\\varepsilon^c(X)\\cap B_{3\\rho}}\\frac{(\\delta^{s}(\\Phi(X))-\\delta^s(\\Phi(Y)))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&+\\int_{B_{3\\rho}^c}\\frac{(\\delta^{s}(\\Phi(X))-\\delta^s(\\Phi(Y)))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&=\\int_{B_\\varepsilon^c(X)}\\frac{ (u_1(X_N)-u_1(Y_N))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&\\quad +\\int_{B_{3\\rho}^c}\\frac{(\\delta^{s}(\\Phi(X))-\\delta^s(\\Phi(Y)))^{p-1}-(u_1(X_N)-u_1(Y_N))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&= f_{1,\\varepsilon}(X)+f_2(X),\n\\end{align*}\nfor sufficiently small $\\varepsilon$, where we used the fact that\n\\[\\delta^s(\\Phi(Z))=u_1(Z_N)\\quad \\text{for all $Z\\in B_{3\\rho}$}\\]\nthanks to \\eqref{hasd}.\nClearly $f_2\\circ \\Phi^{-1}\\in L^1(\\Omega\\cap B_{2\\rho})$, and to estimate its $L^\\infty$-norm we observe that, due to \\eqref{hasd}, \n\\[{\\rm dist}(\\Phi^{-1}(\\Omega\\cap B_{2\\rho}), B_{3\\rho}^c)>\\theta_{\\Phi, \\rho}>0.\\]\nThen, using the $s$-H\\\"older regularity of $\\delta^s\\circ\\Phi$ and $u_1$, and recalling that $\\Phi^{-1}\\in {\\rm Lip}({\\mathbb R}^N)$ and \\eqref{lkj}, we obtain\n\\begin{align*}\n|f_2(X)|&\\leq C_{\\Phi, \\rho}\\int_{B^c_{3\\rho}}\\frac{|X-Y|^{s(p-1)}}{|X-Y|^{N+ps}}\\, dY\\\\\n&\\leq C_{\\Phi, \\rho}\\int_{{\\mathbb R}^N}\\frac{1}{(1+|Y|)^{N+s}}\\, dY\\\\\n&\\leq C_{\\Phi, \\rho},\\quad \\forall X\\in \\Phi^{-1}(\\Omega\\cap B_{2\\rho}).\n\\end{align*}\nRegarding $f_{1,\\varepsilon}$, it coincides with the $g_\\varepsilon$ of \\eqref{diffeo1}. Therefore claim \\eqref{claimgepsilon} of Lemma \\ref{lemmadiffeo} shows that the limit \n\\[f_1(X):=\\lim_{\\varepsilon\\to 0}\\int_{B_\\varepsilon^c(X)}\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\]\nholds in $L^1_{\\rm loc}({\\mathbb R}^N_+)$, and $\\|f_1\\|_{\\infty}\\leq C_{\\Phi, \\rho}$. Therefore $g_\\varepsilon\\to f_1\\circ \\Phi^{-1}+f_2\\circ\\Phi^{-1}$ in $L^1_{\\rm loc}(\\Omega\\cap B_{2\\rho})$, and both are bounded. Lemma \\ref{symmset} finally gives the conclusion.\n\\end{proof}\n\n\\section{Barriers}\\label{sec4}\n\n\\noindent\nIn this section we provide some barrier-type functions and prove {\\em a priori} bounds for the bounded weak solutions of problem \\eqref{dir}. We begin by considering the simple problem\n\\begin{equation}\\label{dir1}\n\\begin{cases}\n(-\\Delta)^s_p v=1 & \\text{in $B_1$} \\\\\nv=0 & \\text{in $B_1^c$.}\n\\end{cases}\n\\end{equation}\nThe following lemma displays some properties of the solution of \\eqref{dir1}:\n\n\\begin{lemma}\\label{cupola}\nLet $v\\in W^{s,p}_0(B_1)$ be a weak solution of \\eqref{dir1}. Then, $v\\in L^\\infty({\\mathbb R}^N)$ is unique, radially non-increasing, and for all $r\\in(0,1)$ it holds $\\inf_{B_r}v>0$.\n\\end{lemma}\n\\begin{proof}\nFirst we prove uniqueness. Let the functional $J:W^{s,p}_0(B_1)\\to{\\mathbb R}$ be defined by\n\\[J(u)=\\frac{1}{p}\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\\,dx\\,dy-\\int_{B_1}u(x)\\,dx.\\]\n$J$ is strictly convex and coercive, hence it admits a unique global minimizer $v\\in W^{s,p}_0(B_1)$, which is the only weak solution of \\eqref{dir1}. By Lemma \\ref{hs} \\ref{hs.3} we see that $v$ is radially symmetric, that is, $v(x)=\\psi(|x|)$ for all $x\\in{\\mathbb R}^N$, where $\\psi:{\\mathbb R}_+\\to{\\mathbb R}_+$ is a mapping vanishing in $[1,\\infty)$. Let $v^\\#$ be the symmetric non-increasing rearrangement of $v$. By the fractional P\\'olya-Szeg\\\"o inequality (see Theorem 3 of \\cite{B}) we have $J(v^\\#)\\le J(v)$, so by uniqueness $v=v^\\#$, that is, $\\psi$ is non-increasing and continuous from the right in ${\\mathbb R}_+$. Now let\n\\[r_0=\\inf\\{r\\in(0,1]:\\,\\psi(r)=0\\}.\\]\nClearly $r_0\\in(0,1]$. Arguing by contradiction, assume $r_0\\in(0,1)$. Then $v\\in W^{s,p}_0(B_{r_0})$ and \nit solves weakly\n\\[\\begin{cases}\n(-\\Delta)^s_p v=1 & \\text{in $B_{r_0}$} \\\\\nv=0 & \\text{in $B^c_{r_0}$.}\n\\end{cases}\\]\nReasoning as above and using uniqueness and Lemma \\ref{hs} \\ref{hs.2}, we see that $v(x)=r_0^{-ps}v(r_0^{ps}x)$ in $B_{r_0}$, so\n\\[\\psi(r_0^2)=r_0^{ps}\\psi(r_0)=0,\\]\nwith $r_0^2<r_0$, against the definition of $r_0$. So, for all $r\\in(0,1)$ we have\n\\[\\inf_{B_r} v = \\psi(r) >0.\\]\nFinally, we prove that $v\\in L^\\infty({\\mathbb R}^N)$. Let $w\\in C^s({\\mathbb R}^N)\\cap \\widetilde{W}^{s,p}(B_1)$ be defined by\n\\[w(x)=\\min\\{(2-x_N)_+^s,5^s\\}.\\]\nNotice that $w(x)=(2-x_N)_+^s=u_1(2-x_N)$ for all $x\\in B_2$. Thus we can apply Lemma \\ref{psadd} in $B_{3\/2}$ to \n\\[w(x)=u_1(2-x_N)-(u_1(2-x_N)-5^s)_+\\]\nto get, by Lemma \\ref{solnd} \n\\begin{align*}\n(-\\Delta)^s_p w(x) &= 2\\int_{\\{y_N\\le-3\\}}\\frac{((2-x_N)_+^s-5^s)^{p-1}-((2-x_N)_+^s-(2-y_N)_+^s)^{p-1}}{|x-y|^{N+ps}}\\,dy\\\\\n&=: I(x)\n\\end{align*}\nweakly in $B_1$. The function $I:\\bar B_1\\to{\\mathbb R}$ is continuous and positive, so there exists $\\alpha>0$ such that\n\\[(-\\Delta)^s_p w(x)\\ge\\alpha \\ \\text{weakly in $B_1$.}\\]\nWe set $\\tilde w=\\alpha^{-1\/(p-1)}w$, so we have\n\\[\\begin{cases}\n(-\\Delta)^s_p v=1\\le(-\\Delta)^s_p\\tilde w & \\text{weakly in $B_1$} \\\\\nv=0\\le\\tilde w & \\text{in $B^c_1$,}\n\\end{cases}\\]\nand Proposition \\ref{comp} yelds\n\\[0\\le v \\le\\tilde w \\le \\frac{5^s}{\\alpha^\\frac{1}{p-1}},\\quad \\text{in ${\\mathbb R}^N$},\\]\nso $v\\in L^\\infty({\\mathbb R}^N)$, concluding the proof.\n\\end{proof}\n\n\\noindent\nNext we introduce {\\em a priori} bounds for functions with bounded fractional $p$-Laplacian.\n\n\\begin{corollary}[$L^\\infty$-bound]\\label{apb}\nLet $u\\in W^{s,p}_0(\\Omega)$ satisfy $|(-\\Delta)^s_p u|\\le K$ weakly in $\\Omega$ for some $K>0$. Then\n\\[\\|u\\|_\\infty\\le (C_d K)^\\frac{1}{p-1},\\]\nfor some $C_d=C(N,p,s, d)$, $d={\\rm diam}(\\Omega)$.\n\\end{corollary}\n\\begin{proof}\nLet $v\\in W^{s,p}_0(B_1)$ be as in Lemma \\ref{cupola}, $x_0$ such that $\\Omega\\Subset B_d(x_0)$, and set\n\\[\\tilde v(x)=(Kd^{ps})^\\frac{1}{p-1}v\\Big(\\frac{x-x_0}{d}\\Big).\\]\nBy Lemma \\ref{hs} \\ref{hs.1}, \\ref{hs.2} we have weakly\n\\[\\begin{cases}\n(-\\Delta)^s_p u\\le K=(-\\Delta)^s_p\\tilde v & \\text{in $\\Omega$} \\\\\nu=0\\le\\tilde v & \\text{in $\\Omega^c$},\n\\end{cases}\\]\nwhich, by Proposition \\ref{comp}, implies $u\\le\\tilde v$ in ${\\mathbb R}^N$. A similar argument, applied to $-u$, gives the lower bound.\n\\end{proof}\n\n\\noindent\nWe can now produce (local) upper barriers on the complements of balls.\n\n\\begin{lemma}[Local upper barrier]\\label{moon}\nThere exist $w\\in C^s({\\mathbb R}^N)$, and universal $r>0$, $a\\in (0,1)$, $c>1$ with\n\\[\\begin{cases}\n(-\\Delta)^s_p w\\ge a & \\text{weakly in $B_r(e_N)\\setminus\\overline B_1$} \\\\\nc^{-1}(|x|-1)_+^s\\le w(x)\\le c(|x|-1)_+^s & \\text{in ${\\mathbb R}^N$}.\n\\end{cases}\\]\n\\end{lemma}\n\\begin{proof}\nBy translation, rotation invariance and scaling (Lemma \\ref{hs}), it suffices to prove the statement for any fixed ball of radius $R>2$, at any fixed point $\\bar x_R$ of its boundary. To fix ideas, we set $\\tilde x_R=(0,-(R^2-4)^{1\/2})$ and $\\bar x_R=\\tilde x_R+Re_N$, so that $B_R(\\tilde x_R)$ intersects the hyperplane ${\\mathbb R}^{N-1}\\times\\{0\\}$ in the $(N-1)$-ball $\\{|x'|<2\\}$ (we use the notation $x=(x',x_N)\\in{\\mathbb R}^{N-1}\\times{\\mathbb R}$).\n\\vskip2pt\n\\noindent\nIn the following we will choose $R$ large enough, depending only on $N, p, s$. If $R>2$, we can find $\\varphi\\in C^{1,1}({\\mathbb R}^{N-1})$ such that $\\|\\varphi\\|_{C^{1,1}({\\mathbb R}^{N-1})}\\le C\/R$ and\n\\[\\varphi(x')=\\big((R^2-|x'|^2)^{1\/2}-(R^2-4)^{1\/2}\\big)_+ \\ \\text{for all $|x'|\\in[0,1]\\cup[3,\\infty)$.}\\]\nWe set\n\\[U_+=\\{x\\in{\\mathbb R}^N:\\,\\varphi(x')<x_N\\}\\]\n(see figure \\ref{figball}).\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=0.8]\n\\clip (-6, -4.65) rectangle (6, 1.7);\n\n\\shade [left color=lightgray, right color=white, shading=axis, shading angle=180] (-6, 0) -- (-3, 0) .. controls (-2.55, 0) .. node[above right=3pt]{$\\varphi$} (-1, 0.3164)  arc (101.537:78.463:5)  .. controls (2.55,0) .. (3,0) -- (6,0) -- (6,1.7) -- (-6, 1.7) -- (-6,0);\n\\draw[very thick] (-6, 0) -- (-3, 0) .. controls (-2.55, 0) ..  (-1, 0.3164)  arc (101.537:78.463:5)  .. controls (2.55,0) .. (3,0) -- (6,0);\n\\draw[->] (-6, 0) -- (6, 0);\n\\draw[->] (0, -5) -- (0, 1.7) ;\n\\draw (0, -4.582) circle (5cm);\n\\filldraw (0, -4.582) circle (1.2pt);\n\\draw (0, -4.582) node[above left]{$\\widetilde{x}_R$} -- node[right=1pt]{$R$} (2,0)\n\n\\filldraw (2, 0) circle (1.2pt);\n\\draw (2.1, -0.4) node{\\small $2$};\n\n\\draw (0, 0.4174) circle (1cm);\n\\draw (1.4,1.45) node{$B_1(\\bar x_R)$};\n\n\\draw (-5, 1) node{$U_+$};\n\\filldraw (0, 0.4174)  circle (1.2pt);\n\\draw  (0, 0.4174) node[above right]{$\\bar x_R$};\n\n\\end{tikzpicture}\n\\caption{The balls $B_R(\\widetilde{x}_R)$ and $B_1(\\bar x_R)$. The thick line is the graph of $\\varphi$, whose epigraph is $U_+$.}\n\\label{figball}\n\\end{figure}\nWe claim that for any sufficiently large $R$ there exists a diffeomorphism $\\Phi\\in C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)$ such that $\\Phi(0)=\\bar x_R$, $\\Phi=I$ in $B^c_4$, and\n\\begin{equation}\n\\label{propPhi}\n\\|\\Phi-I\\|_{C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)}+\\|\\Phi^{-1}-I\\|_{C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)}\\le\\frac{C}{R}, \\ \\Phi({\\mathbb R}^N_+)=U_+.\n\\end{equation}\nIndeed, let $\\eta\\in C^2({\\mathbb R})$ satisfy $\\eta\\in [0,1]$, $\\eta(0)=1$, ${\\rm supp}\\,  \\eta\\subseteq (-1,1)$. Set for all $X=(X',X_N)\\in{\\mathbb R}^{N-1}\\times{\\mathbb R}$\n\\[\\Phi(X)=X+\\varphi(X')\\eta(X_N)e_N.\\]\nThen, for sufficiently large $R$, $\\Phi\\in C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)$ is a bijection since $\\Phi(X_1)=\\Phi(X_2)$ implies $X_1'=X_2'$, and the map $t\\mapsto t+\\varphi(X')\\eta(t)$ is increasing whenever\n\\[\\sup_{X\\in {\\mathbb R}^N}\\varphi(X')|\\eta'(X_N)|=\\frac{4\\sup_{{\\mathbb R}}|\\eta'|}{R+\\sqrt{R^2-4}}< 1.\\]\nIts inverse mapping satisfies\n\\begin{equation}\\label{moon2}\n\\Phi^{-1}(x)=x-\\varphi(x')\\eta(\\Phi^{-1}(x)\\cdot e_N)e_N \\ \\text{for all $x\\in{\\mathbb R}^N$,}\n\\end{equation}\nbesides $\\Phi(0)=\\bar x_R$. Moreover, for all $X\\in B^c_4$ we have either $|X'|\\ge 3$ or $|X_N|\\ge 1$, in both cases $\\Phi(X)=X$. The $C^{1,1}$-bounds on $\\varphi$, $\\eta$ and \\eqref{moon2} yield the required $C^{1,1}$-bounds on $\\Phi-I$ and $\\Phi^{-1}-I$. Finally, reasoning as above, the monotonicity of $t\\mapsto t+\\varphi(X')\\eta(t)$ implies that $\\Phi({\\mathbb R}^N_+)=U_+$, and \\eqref{propPhi} is proved.\n\\vskip2pt\n\\noindent\nLet $v_1(x)=u_1(\\Phi^{-1}(x)\\cdot e_N)$. Lemma \\ref{lemmadiffeo} ensures that $v_1\\in \\widetilde W^{s,p}_{{\\rm loc}}({\\mathbb R}^N)$ and\n\\begin{equation}\n\\label{pslv1}\n(-\\Delta)^s_p v_1=f\\quad \\text{weakly in $U_+$, with $\\|f\\|_\\infty\\leq C\/R$}.\n\\end{equation}\nDefine \n\\[v(x)=\\min\\{v_1(x), 5^s\\},\\]\nwhich belongs to $\\widetilde{W}^{s,p}(B_4)$. From $\\Phi=I$ in $B_4^c$ we infer $\\Phi^{-1}(B_4)=B_4$ and thus \n\\[v_1(x)=u_1(x_N)\\quad\\text{in $B_4^c$},\\quad v_1\\leq 4^s\\quad\\text{in $B_4$}. \\]\nHence\n\\[v_1(x)-v(x)= (x_N)_+^s-5^s\\quad\\text{in $\\{x_N\\geq 5\\}$},\\quad v_1-v=0\\quad\\text{in $\\{x_N\\leq 5\\}\\Supset B_4$}.\\]\nThus the function $v-v_1$ satisfies conditions \\eqref{hyppertv} in $B_4$, and Lemma \\ref{psadd} provides weakly in $B_4$\n\\[(-\\Delta)^s_p v=(-\\Delta)^s_p (v_1+(v-v_1))=f+g,\\]\nwhere \n\\begin{align*}\ng(x)&=2\\int_{B_4^c}\\frac{(v_1(x)-v(y))^{p-1}-(v_1(x)-v_1(y))^{p-1}}{|x-y|^{N+ps}}\\, dy\\\\\n&\\geq 2\\int_{\\{y_N\\geq 5\\}}\\frac{((x_N)_+^s-5^s)^{p-1}-((x_N)_+^s-(y_N)_+^s)^{p-1}}{|x-y|^{N+ps}}\\, dy\n\\end{align*}\nfor any $x\\in B_4$. As in the proof of Lemma \\ref{cupola}, there is a universal $\\alpha>0$ such that $g(x)\\geq \\alpha$ for all $x\\in B_4$, and therefore using \\eqref{pslv1} we have\n\\[(-\\Delta)^s_p v\\geq f+g\\geq \\alpha - \\frac{C}{R}\\quad \\text{weakly in $U_+\\cap B_4$}.\\]\nTaking  $R$ big enough we thus find $B_2(\\bar x_R)\\Subset B_4$ and \n\\begin{equation}\n\\label{moon6}\n(-\\Delta)^s_p v\\geq \\frac{\\alpha}{2}>0,\\quad \\text{weakly in  $U_+\\cap B_2(\\bar x_R)$}.\n\\end{equation}\nSet for all $x\\in{\\mathbb R}^N$\n\\[d_R(x)=(|x-\\tilde x_R|-R)_+.\\]\nWe can estimate $v$ by multiples of $d_R^s$ {\\em globally} from above but only {\\em locally} from below. Indeed, since $v=0$ in $U_+^c$, $B_R(\\tilde x_R)\\subset U^c_+$, and $v\\in C^s({\\mathbb R}^N)$, there exists $\\tilde c>1$ such that\n\\begin{equation}\\label{moon7}\nv(x)\\leq \\tilde c\\, {\\rm dist}(x,U^c_+)^s\\leq \\tilde c\\, d_R^s(x), \\,\\,\\,\\,\\, \\text{for all $x\\in {\\mathbb R}^N$.}\n\\end{equation}\nOn the other hand, for all $x\\in B_{1}(\\bar x_R)$ it holds either $x\\in B_1(\\bar x_R)\\setminus U_+\\subseteq B_R(\\tilde x_R)$, in which case $d_R^s(x)=0=\\tilde c v(x)$, or $x\\in B_1(\\bar x_R)\\cap U_+\\subseteq B_R^c(\\tilde x_R)$. In the latter case let $X=(X', X_N)$ be such that $x=\\Phi(X)$, $Z=(X', 0)$ and $z=\\Phi(Z)$. It holds $|X'|\\leq 1$ and by the construction of $\\Phi$, it follows that $z\\in \\partial B_R(\\tilde x_R)$, therefore\n\\[d_R^s(x)\\leq |x-z|^s\\le\\tilde c|X-Z|^s=\\tilde cX_N^s=\\tilde cv(x).\\]\nThus we have (taking $\\tilde c>1$ bigger if necessary)\n\\begin{equation}\\label{moon8}\nv\\ge\\frac{1}{\\tilde c}d_R^s \\ \\text{in $B_{1}(\\bar x_R)$.}\n\\end{equation}\nWe aim at extending \\eqref{moon8} to the whole ${\\mathbb R}^N$, while retaining \\eqref{moon6} and \\eqref{moon7}. For any $\\varepsilon\\in(0,1\/\\tilde c)$ set\n\\[v_\\varepsilon=\\max\\{v,\\varepsilon d_R^s\\}.\\]\nClearly $v_\\varepsilon$ satisfies estimates like \\eqref{moon7} and \\eqref{moon8} in ${\\mathbb R}^N$ with a constant $\\tilde c_\\varepsilon=\\max\\{\\tilde c+\\varepsilon,\\varepsilon^{-1}\\}$. Besides $v\\le v_\\varepsilon\\le v+\\varepsilon d_R^s$ in ${\\mathbb R}^N$, being $\\varepsilon<1\/\\tilde{c}$, $v_\\varepsilon-v=0$ in $B_1(\\bar x_R)$. So, by \\eqref{moon6}, Lemma \\ref{psadd} and \\eqref{in6} (with $M=5^s$ and $q=p-1$)\n\\begin{align*}\n(-\\Delta)^s_p v_\\varepsilon(x)\n&= (-\\Delta)^s_p v(x)-2\\int_{B^c_{1\/2}(\\bar x_R)}\\frac{(v(x)-v(y))^{p-1}-(v(x)-v_\\varepsilon(y))^{p-1}}{|x-y|^{N+ps}}\\,dy \\\\\n&\\ge \\frac{\\alpha}{2}-C\\int_{B^c_{1}(\\bar x_R)}\\frac{\\max\\{\\varepsilon d_R^s(y),(\\varepsilon d_R^s(y))^{p-1}\\}}{|\\bar x_R-y|^{N+ps}}\\,dy \\\\\n&\\ge \\frac{\\alpha}{2}-CJ(\\varepsilon)\n\\end{align*}\nweakly in $B_{1\/2}(\\bar x_R)\\cap U_+$ (in the end we have used the inequality $|x-y|\\ge 1\/2|\\bar x_R-y|$ for all $x\\in B_{1\/2}(\\bar x_R)$, $y\\in B^c_{1}(\\bar x_R)$). Notice that $J(\\varepsilon)\\to 0$ as $\\varepsilon\\to 0^+$ independently of $x$, thus, for $\\varepsilon>0$ small enough we have\n\\[(-\\Delta)^s_p v_\\varepsilon(x)\\ge\\frac{\\alpha}{4}>0 \\ \\text{weakly in $B_{1\/2}(\\bar x_R)\\setminus B_R(\\tilde x_R)$.}\\]\nTo obtain the barrier of the thesis, we set $w(x)=v_\\varepsilon(\\tilde{x}_R+Rx)$ and using Lemma \\ref{hs} we reach the conclusion for $r=1\/(2R)$, $a=\\alpha\/(4R^{ps})$, $c=R^s\\max\\{\\tilde c+\\varepsilon,\\varepsilon^{-1}\\}$.\n\\end{proof}\n\n\\noindent\nFinally, we prove that any bounded weak solution of \\eqref{dir} can be estimated by means of a multiple of $\\delta^s$.\n\n\\begin{theorem}\\label{estid}\nLet $u\\in W^{s,p}_0(\\Omega)$ satisfy $|(-\\Delta)^s_p u|\\le K$ weakly in $\\Omega$ for some $K>0$. Then\n\\begin{equation}\n\\label{thm44tesi}\n|u|\\le (C_\\Omega K)^\\frac{1}{p-1}\\delta^s \\quad \\text{a.e.\\ in $\\Omega$,}\n\\end{equation}\nfor some $C_\\Omega=C(N,p,s,\\Omega)$.\n\\end{theorem}\n\\begin{proof}\nConsidering $u\/K^{1\/(p-1)}$ and using homogeneity, we can prove \\eqref{thm44tesi} in the case $K=1$. Thanks to Corollary \\ref{apb} we may focus on a neighborhood of $\\partial\\Omega$. Let $\\rho>0$ be as in Lemma \\ref{geo1}, and let $r\\in (0,1)$ be defined in Lemma \\ref{moon}. Set\n\\[U=\\Big\\{x\\in\\Omega:\\,\\delta(x)<r\\frac{\\rho}{2}\\Big\\},\\]\n$\\bar x\\in U$ and $x_0=\\Pi(\\bar x)\\in \\partial\\Omega$ its point of minimal distance from $\\Omega^c$.\nThere exists two balls $B_{\\rho\/2}(x_1)$ and $B_{\\rho}(x_2)$ exteriorly tangent to $\\partial\\Omega$ at $x_0$, and (by scaling and translating the supersolution of the previous Lemma \\ref{moon}) a function $w\\in C^s$ such that\n\\begin{equation}\\label{estid2}\n(-\\Delta)^s_p w\\ge a  \\quad \\text{weakly in $B_{r\\rho\/2}(x_0)\\setminus B_{\\rho\/2}(x_1)$}\n\\end{equation}\nand\n\\begin{equation}\n\\label{estid3}\nc^{-1}d^s(x)\\le w(x)\\le cd^s(x)  \\quad \\text{in ${\\mathbb R}^N$}.\n\\end{equation}\nwhere we have set\n\\[d(x)={\\rm dist}(x,B^c_{\\rho\/2}(x_1)).\\]\nNotice that the constants in \\eqref{estid2} \\eqref{estid3} depend only on $\\rho$, $N$, $p$ and $s$, and we will suppose henceforth that $a, r, c^{-1}\\in (0,1)$.\nBy Lemma \\ref{geo1} it holds\n\\begin{equation}\\label{estid2b}\nd(\\bar x)=\\delta(\\bar x)=|\\bar x-x_0|,\n\\end{equation}\nmoreover \n\\[d(x)\\ge \\theta>0,\\quad \\text{in  $B_{\\rho}^c(x_2)\\setminus B_{r\\rho\/2}(x_0)$}.\\]\nfor a constant $\\theta$ which depends only on $\\rho$ and $r$ (and thus on $\\Omega$ alone).\nSince $\\Omega\\subseteq B_{\\rho}^c(x_2)$, the latter inequality together with \\eqref{estid3} provides\n\\begin{equation}\n\\label{estid4}\nw\\geq c^{-1}\\theta^s=:\\alpha>0,\\quad \\text{in $\\Omega\\setminus B_{r\\rho\/2}(x_0)$}.\n\\end{equation}\nWe define the open set\n\\[V=\\Omega\\cap B_{r\\frac{\\rho}{2}}(x_0)\\subseteq B_{r\\frac{\\rho}{2}}(x_0)\\setminus B_{\\rho\/2}(x_1),\\]\nwhere we will apply the comparison principle. Suppose without loss of generality that in \\eqref{estid4} $\\alpha\\in (0,1)$ and let $C_d>1$ be as in Corollary \\ref{apb}. Set\n\\[M=\\frac{1}{\\alpha}\\Big(\\frac{C_d}{a}\\Big)^\\frac{1}{p-1}, \\quad \\bar w=Mw.\\]\nBy \\eqref{estid2} and $C_d\/\\alpha^{p-1}\\geq 1$ we have\n\\[(-\\Delta)^s_p \\bar w=M^{p-1}(-\\Delta)^s_p w\\geq \\frac{C_d}{\\alpha^{p-1}}\\geq 1\\geq (-\\Delta)^s_p u,\\quad \\text{weakly in $V$}.\\]\nMoreover $u=0\\leq \\bar w$ in $\\Omega^c$, while \\eqref{estid4}, $a<1$ and Corollary \\ref{apb} give \n\\[\\bar w\\geq  M\\alpha=\\Big(\\frac{C_d}{a}\\Big)^\\frac{1}{p-1}\\geq \\sup_{\\Omega} u,\\quad \\text{in $\\Omega\\setminus B_{r\\rho\/2}(x_0)$}.\\] \nTherefore  $\\bar w\\geq u$ in the whole $V^c$, and Proposition \\ref{comp} together with \\eqref{estid3} yelds\n\\[u(x)\\leq \\bar w(x)\\leq cMd^s(x)\\quad \\text{for a.e.\\ $x\\in {\\mathbb R}^N$}.\\]\nRecalling \\eqref{estid2b} we get\n\\[u(\\bar x)\\leq cMd^s(\\bar x)=cM\\delta^s(\\bar x)\\quad \\text{for all $\\bar x=x_0-tn_{x_0}$, $t\\in\\Big[0, r\\frac{\\rho}{2}\\Big]$},\\]\nwhere $n_{x_0}$ is the exterior normal to $\\partial\\Omega$ at $x_0$, which gives the thesis since $cM$ depends only on $N, p, s$, $\\rho$, $r$, and $\\Omega$. A similar argument applied to $-u$ yields the lower bound.\n\\end{proof}\n\n\\section{H\\\"older regularity}\\label{sec5}\n\n\\noindent\nIn this section we will obtain the H\\\"older regularity of solutions.\n\n\\subsection{Interior H\\\"older regularity}\n\n\\noindent\nWe now study the behavior of a weak supersolution in a ball, proving a weak Harnack inequality. Then we will obtain an estimate of the oscillation of a bounded weak solution in a ball (this can be interpreted as a first interior H\\\"older regularity result). All balls are meant to be centered at $0$, as translation invariance of $(-\\Delta)^s_p$ allows to extend the results to balls centered at any point.\n\\vskip2pt\n\\noindent\nWe begin with a curious Jensen-type inequality:\n\n\\begin{lemma}\\label{jensen}\nLet $E\\subset{\\mathbb R}^N$ be a set of finite measure and let $u\\in L^1(E)$ satisfy\n\\[\\Xint-_E u\\,dx=1.\\]\nThen, for all $r\\ge 1$ and $\\lambda\\ge 0$, it holds\n\\[\\Xint-_E(u^r-\\lambda^r)^\\frac{1}{r}\\,dx\\ge 1-2^\\frac{r-1}{r}\\lambda.\\]\n\\end{lemma}\n\\begin{proof}\nAvoiding trivial cases, we assume $r>1$ and $\\lambda>0$. Set, for all $t\\in{\\mathbb R},$\n\\[g(t)=(t^r-\\lambda^r)^\\frac{1}{r}.\\]\nThen, for all $t\\in{\\mathbb R}\\setminus\\{0,\\lambda\\}$, we have\n\\[g'(t)=|t^r-\\lambda^r|^\\frac{1-r}{r}|t|^{r-1}.\\]\nIn particular, $t_\\lambda=2^{-1\/r}\\lambda$ is the only solution of $g'(t)=1$. Elementary calculus shows that for all $t\\in{\\mathbb R}$\n\\[g(t)\\ge g(t_\\lambda)+g'(t_\\lambda)(t-t_\\lambda)=t-2t_\\lambda.\\]\nSo we have\n\\[\\Xint-_E (g\\circ u)\\,dx\\ge\\Xint-_E(u-2t_\\lambda)\\,dx=1-2^\\frac{r-1}{r}\\lambda,\\]\nwhich concludes the proof.\n\\end{proof}\n\n\\noindent\nNow we prove a weak Harnack-type inequality for non-negative supersolutions:\n\n\\begin{theorem}[Weak Harnack inequality]\\label{harnack}\nThere exist universal $\\sigma\\in(0,1)$, $\\bar C>0$ with the following property: if $u\\in \\widetilde{W}^{s,p}(B_{R\/3})$ satisfies weakly\n\\[\\begin{cases}\n(-\\Delta)^s_p u\\ge -K & \\text{in $B_{R\/3}$} \\\\\nu\\ge 0 & \\text{in ${\\mathbb R}^N$,}\n\\end{cases}\\]\nfor some $K\\ge 0$, then\n\\[\\inf_{B_{R\/4}}u\\ge\\sigma\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}-\\bar C(KR^{ps})^\\frac{1}{p-1}.\\]\n\\end{theorem}\n\\begin{proof}\nWe first consider the case $p\\ge 2$. Let $\\varphi\\in C^\\infty({\\mathbb R}^N)$ be such that $0\\le\\varphi\\le 1$ in ${\\mathbb R}^N$, $\\varphi=1$ in $B_{3\/4}$, $\\varphi=0$ in $B^c_1$, and by Proposition \\ref{weak-strong} $|(-\\Delta)^s_p\\varphi|\\le C_1$ weakly in $B_1$. We rescale by setting $\\varphi_R(x)=\\varphi(3x\/R)$, so $\\varphi_R\\in C^\\infty({\\mathbb R}^N)$, $0\\le\\varphi_R\\le 1$ in ${\\mathbb R}^N$, $\\varphi_R=1$ in $B_{R\/4}$, $\\varphi_R=0$ in $B^c_{R\/3}$, and $|(-\\Delta)^s_p\\varphi_R|\\le C_1R^{-ps}$ weakly in $B_{R\/3}$ (taking $C_1$ bigger).\n\\vskip2pt\n\\noindent\nFor all $\\sigma\\in(0,1)$ we set\n\\[L=\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}, \\quad w=\\sigma L\\varphi_R+\\chi_{B_R\\setminus B_{R\/2}} u\\]\n(see figure \\ref{w-barr}).\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=1.8]\n\\draw[->] (-3, 0) -- (3, 0)\n\\draw[->] (0, -0.5) -- (0, 2)\n\n\\draw[thick, rounded corners=3pt]  (-0.68, 0) -- (-0.6, 0) -- (-0.54, 0.4) -- node[above=2pt, fill=white, inner sep=2pt]{$\\sigma L\\varphi$} (0.54, 0.4) -- (0.6, 0) -- (0.68,0);\n\\draw (-0.47, 0) node[below]{\\small $\\frac{R}{4}$} -- (-0.47, 0.4);\n\n\\draw[thick] (-2,0) node[below]{\\small $R$} -- (-2, 0.7) parabola bend (-1.8, 1) (-1.6,0.5) parabola bend (-1.45, 0.9) (-1.4, 0.4) parabola bend (-1.2, 0.8) (-1,0.6) -- (-1,0) node[below]{\\small $\\frac{R}{2}$} -- (-0.65, 0) ;\n\\draw (-1.5,1.4) node{$\\chi_{B_R\\setminus B_{R\/2}}u$};\n\\draw[thick] (0.66,0) -- (1,0) -- (1,0.9) parabola bend (1.1, 1.25)  (1.38,0.3) parabola bend (1.43, 0.6) (1.5,0.2) parabola bend (1.6,1) (1.78,0.4) parabola bend (1.85, 0.8) (2,0.6) -- (2,0);\n\\draw (1.5,1.4) node{$\\chi_{B_R\\setminus B_{R\/2}}u$};\n\n\\end{tikzpicture}\n\\caption{The lower barrier $w$.}\n\\label{w-barr}\n\\end{figure}\nSo $w\\in \\widetilde{W}^{s,p}(B_{R\/3})$  and by Lemma \\ref{psadd} and \\eqref{in1} we have, weakly in $B_{R\/3}$,\n\\begin{align*}\n&(-\\Delta)^s_p w(x)\\\\\n&= (-\\Delta)^s_p(\\sigma L\\varphi_R)(x)+2\\int_{B_R\\setminus B_{R\/2}}\\frac{(\\sigma L\\varphi_R(x)-u(y))^{p-1}-(\\sigma L\\varphi_R(x))^{p-1}}{|x-y|^{N+ps}} \\, dy\\\\\n&\\le \\frac{C_1(\\sigma L)^{p-1}}{R^{ps}}-2^{3-p}\\int_{B_R\\setminus B_{R\/2}}\\frac{u^{p-1}(y)}{|x-y|^{N+ps}}\\,dy \\\\\n&\\le \\frac{C_1(\\sigma L)^{p-1}}{R^{ps}}-\\frac{C_2L^{p-1}}{R^{ps}}.\n\\end{align*}\nIf we assume\n\\[\\sigma<\\min\\Big\\{1,\\Big(\\frac{C_2}{2C_1}\\Big)^\\frac{1}{p-1}\\Big\\},\\]\nwe get the upper estimate\n\\begin{equation}\\label{hk1}\n(-\\Delta)^s_p w(x)\\le-\\frac{C_2L^{p-1}}{2R^{ps}} \\quad  \\text{weakly in $B_{R\/3}$.}\n\\end{equation}\nWe set $\\bar C=(2\/C_2)^{1\/(p-1)}$ and distinguish two cases:\n\\begin{itemize}\n\\item if $L\\le\\bar C(KR^{ps})^{1\/(p-1)}$, then clearly\n\\[\\inf_{B_{R\/4}}u\\ge 0\\ge\\sigma L-\\bar C(KR^{ps})^\\frac{1}{p-1};\\]\n\\item if $L>\\bar C(KR^{ps})^{1\/(p-1)}$, then we use \\eqref{hk1} to have \n\\[\\begin{cases}\n(-\\Delta)^s_p w\\le -K\\le (-\\Delta)^s_p u & \\text{weakly in $B_{R\/3}$} \\\\\nw=\\chi_{B_R\\setminus B_{R\/2}} u\\le u & \\text{in $B^c_{R\/3}$,}\n\\end{cases}\\]\nwhich by Proposition \\ref{comp} implies $w\\le u$ in ${\\mathbb R}^N$, in particular\n\\[\\inf_{B_{R\/4}}u\\ge \\sigma L.\\]\n\\end{itemize}\nIn any case we have\n\\[\\inf_{B_{R\/4}}u \\ge \\sigma L-\\bar C(KR^{ps})^\\frac{1}{p-1},\\]\nwhich is the conclusion.\n\\vskip2pt\n\\noindent\nNow we consider the case $p\\in(1,2)$. Due to Remark \\ref{remarkL}, in this case we cannot use the cut-off function $\\varphi$ as before to construct the barrier $w$. We use instead the weak solution $v$ of \\eqref{dir1} introduced in Lemma \\ref{cupola}, recalling that $\\inf_{B_{3\/4}}v>0$, and we set\n\\[\\varphi_R(x)=\\Big(\\inf_{B_{3\/4}}v\\Big)^{-1}v\\Big(\\frac{3x}{R}\\Big),\\] \nso that $0\\le\\varphi_R\\le\\alpha$ (for some universal $\\alpha>0$) in ${\\mathbb R}^N$, $\\varphi_R\\ge 1$ in $B_{R\/4}$, $\\varphi_R=0$ in $B^c_{R\/3}$, and $(-\\Delta)^s_p\\varphi_R= C_1R^{-ps}$ weakly in $B_{R\/3}$. Accordingly, to obtain the estimate \\eqref{hk1} we apply Lemma \\ref{jensen} to the function $(u\/L)^{p-1}$ with $E=B_R\\setminus B_{R\/2}$, $r=1\/(p-1)$, and $\\lambda=(\\sigma\\varphi_R(x))^{p-1}$, so that\n\\[\\Xint-_{B_R\\setminus B_{R\/2}}\\Big(\\frac{u(y)}{L}-\\sigma\\varphi_{R}(x)\\Big)^{p-1}\\,dy\\ge 1-2^{2-p}(\\sigma\\varphi_R(x))^{p-1},\\]\nfor a.e.\\ $x\\in B_{R\/3}$. This, in turn, implies that for a.e.\\ $x\\in B_{R\/3}$\n\\begin{align*}\n&2\\int_{B_R\\setminus B_{R\/2}}\\frac{(\\sigma L\\varphi_R(x)-u(y))^{p-1}-(\\sigma L\\varphi_R(x))^{p-1}}{|x-y|^{N+ps}}\\, dy\\\\ \n&\\le \\frac{C_2}{R^{ps}}\\Xint-_{B_R\\setminus B_{R\/2}}(\\sigma L\\varphi_R(x)-u(y))^{p-1}\\,dy \\\\\n&\\le \\frac{C_2}{R^{ps}}\\big(2^{2-p}(\\sigma L\\varphi_R(x))^{p-1}-L^{p-1}\\big) \\\\\n&\\le -\\frac{C_2L^{p-1}}{2R^{ps}},\n\\end{align*}\nwhere we have chosen $\\sigma<2^{\\frac{p-3}{p-1}}\\alpha^{-1}$. Then, by taking $\\sigma$ even smaller if necessary, we get \\eqref{hk1} and the rest of the proof follows {\\em verbatim}.\n\\end{proof}\n\n\\noindent\nWe need to extend Theorem \\ref{harnack} to supersolutions which are only non-negative in a ball. To do so, we introduce a tail term defined as in \\eqref{deftail}:\n\n\\begin{lemma}\\label{harnackloc}\nThere exist $\\sigma\\in(0,1)$, $\\tilde C>0$, and for all $\\varepsilon>0$ a constant $C_\\varepsilon>0$ with the following property: if $u\\in \\widetilde{W}^{s,p}(B_{R\/3})$ satisfies weakly\n\\[\\begin{cases}\n(-\\Delta)^s_p u\\ge -K & \\text{in $B_{R\/3}$} \\\\\nu\\ge 0 & \\text{in $B_R$,}\n\\end{cases}\\]\nfor some $K\\geq 0$, then\n\\begin{equation}\n\\label{hloctesi}\n\\inf_{B_{R\/4}}u\\ge\\sigma\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}-\\tilde C(KR^{ps})^\\frac{1}{p-1}-C_\\varepsilon{\\rm Tail}(u_-;R)-\\varepsilon\\sup_{B_R}u.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst we consider the case $p\\ge 2$. We apply Lemma \\ref{psadd} to the functions $u$ and $v=u_-$, so that $u+v=u_+$, and $\\Omega=B_{R\/3}$: we have in a weak sense in $B_{R\/3}$\n\\begin{align*}\n&(-\\Delta)^s_p u_+(x) \\\\\n&= (-\\Delta)^s_p u(x)+2\\int_{B^c_{R\/3}}\\frac{(u(x)-u(y)-u_-(y))^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy \\\\\n&\\ge -K+2\\int_{\\{u<0\\}}\\frac{u(x)^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy \\\\\n&\\ge -K+C\\int_{\\{u<0\\}}\\frac{u(x)^{p-1}-(u(x)-u(y))^{p-1}}{|y|^{N+ps}}\\,dy,\n\\end{align*}\nwhere in the end we have used that $|x-y|\\ge 2\/3|y|$. By \\eqref{in7}, for any $\\theta>0$ there exists $C_\\theta>0$ such that weakly in $B_{R\/3}$\n\\begin{align*}\n(-\\Delta)^s_p u_+(x) &\\ge -K-\\theta\\big(\\sup_{B_R}u\\big)^{p-1}\\int_{B^c_R}\\frac{1}{|y|^{N+ps}}\\,dy-\\frac{C_\\theta}{R^{ps}}{\\rm Tail}(u_-;R)^{p-1} \\\\\n&\\ge -K-\\frac{C\\theta}{R^{ps}}\\big(\\sup_{B_R}u\\big)^{p-1}-\\frac{C_\\theta}{R^{ps}}{\\rm Tail}(u_-;R)^{p-1} =: -\\tilde K.\n\\end{align*}\nNow, by applying Theorem \\ref{harnack} to $u_+$ we have for any $\\varepsilon>0$ and $\\theta<(\\varepsilon\/\\bar C)^{p-1}$,\n\\begin{align*}\n\\inf_{B_{R\/4}}u &\\ge \\sigma\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}-\\bar C(\\tilde KR^{ps})^\\frac{1}{p-1} \\\\\n&\\ge \\sigma\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}-\\tilde C(KR^{ps})^\\frac{1}{p-1}-C_\\varepsilon{\\rm Tail}(u_-;R)-\\varepsilon\\sup_{B_R}u\n\\end{align*}\nfor some universal constant $\\tilde C>0$ and a convenient $C_\\varepsilon>0$ depending also on $\\varepsilon$.\n\\vskip2pt\n\\noindent\nNow we turn to the case $p\\in(1,2)$. The argument in this case is in fact easier, as by \\eqref{in2} we have\n\\[\\int_{\\{u<0\\}}\\frac{u(x)^{p-1}-(u(x)-u(y))^{p-1}}{|y|^{N+ps}}\\,dy\\le\\frac{1}{R^{ps}}{\\rm Tail}(u_-;R)^{p-1}\\quad \\text{for a.e.\\ $x\\in B_{R\/3}$},\\]\nthen we argue as above using \\eqref{in3} instead of \\eqref{in1} when required.\n\\end{proof}\n\n\\noindent\nClearly, symmetric versions of Theorem \\ref{harnack} and Lemma \\ref{harnackloc} also hold. Now we use the above results to produce an estimate of the oscillation of a bounded function $u$ such that $(-\\Delta)^s_p u$ is locally bounded. We set for all $R>0$, $x_0\\in {\\mathbb R}^N$\n\\[Q(u; x_0, R) = \\|u\\|_{L^\\infty(B_R(x_0))}+{\\rm Tail}(u; x_0, R),\\quad Q(u; R)=Q(u; 0, R).\\]\nOur result is as follows:\n\n\\begin{theorem}\\label{osc}\nThere exist universal $\\alpha\\in(0,1)$, $C>0$ with the following property: if $u\\in \\widetilde{W}^{s,p}(B_{R_0})\\cap L^\\infty(B_{R_0})$ satisfies $|(-\\Delta)^s_p u|\\le K$ weakly in $B_{R_0}$ for some $R_0>0$, then for all $r\\in(0,R_0)$\n\\[\\underset{B_r}{\\rm osc}\\,u\\le C\\big[(KR_0^{ps})^\\frac{1}{p-1}+Q(u;R_0)\\big]\\frac{r^\\alpha}{R_0^\\alpha}.\\]\n\\end{theorem}\n\\begin{proof}\nFirst we consider the case $p\\ge 2$. For all integer $j\\ge 0$ we set $R_j=R_0\/4^j$, $B_j=B_{R_j}$, and $\\frac{1}{2} B_j=B_{R_j\/2}$. We put forward the following\n\\vskip2pt\n\\noindent\n{\\em Claim.} There exist a universal $\\alpha\\in(0,1)$ and a real $\\lambda>0$ (depending on all the data), a non-decreasing sequence $(m_j)$, and a non-increasing sequence $(M_j)$, such that for all $j\\ge 0$\n\\[m_j\\le\\inf_{B_j}u\\le\\sup_{B_j}u\\le M_j, \\quad  M_j-m_j=\\lambda R_j^\\alpha.\\]\nWe argue by induction on $j$. Step zero: we set $M_0=\\sup_{B_0}u$ and $m_0=M_0-\\lambda R_0^\\alpha$, where $\\lambda>0$ satisfies\n\\begin{equation}\\label{io1}\n\\lambda\\ge \\frac{2\\|u\\|_{L^\\infty(B_0)}}{R_0^\\alpha},\n\\end{equation}\nwhich clearly implies\n\\[\\inf_{B_0}u\\ge m_0.\\]\nInductive step: assume that sequences $(m_j)$, $(M_j)$ are constructed up to the index $j$. Then\n\\begin{equation}\\label{iox}\n\\begin{split}\nM_j-m_j &= \\Xint-_{B_j\\setminus\\frac{1}{2} B_j}(M_j-u)\\,dx+\\Xint-_{B_j\\setminus\\frac{1}{2}B_j}(u-m_j)\\,dx \\\\\n&\\le \\Big(\\Xint-_{B_j\\setminus\\frac{1}{2}B_j}(M_j-u)^{p-1}\\,dx\\Big)^\\frac{1}{p-1}+\\Big(\\Xint-_{B_j\\setminus\\frac{1}{2} B_j}(u-m_j)^{p-1}\\,dx\\Big)^\\frac{1}{p-1}.\n\\end{split}\n\\end{equation}\nLet $\\sigma\\in(0,1)$, $\\tilde C>0$ be as in Lemma \\ref{harnackloc}, and multiply the previous inequality by $\\sigma$ to obtain, via \\eqref{hloctesi}, \n\\begin{equation}\n\\label{ioxx}\n\\begin{split}\n\\sigma(M_j-m_j) &\\le \\inf_{B_{j+1}}(M_j-u)+\\inf_{B_{j+1}}(u-m_j)+2\\tilde C(KR_0^{ps})^\\frac{1}{p-1} \\\\\n&\\quad + C_\\varepsilon\\big[{\\rm Tail}((M_j-u)_-;R_j)+{\\rm Tail}((u-m_j)_-;R_j)\\big]\\\\\n&\\quad +\\varepsilon\\Big[\\sup_{B_j}(M_j-u)+\\sup_{B_j}(u-m_j)\\Big].\n\\end{split}\n\\end{equation}\nSetting universally $\\varepsilon=\\sigma\/4$, $C=\\max\\{2\\tilde C,C_\\varepsilon\\}$ and rearranging, we have\n\\begin{equation}\\label{io2}\n\\begin{split}\n\\underset{B_{j+1}}{\\rm osc}\\,u &\\le \\Big(1-\\frac{\\sigma}{2}\\Big)(M_j-m_j)\\\\\n&\\quad +C\\big[(KR_0^{ps})^\\frac{1}{p-1}+{\\rm Tail}((M_j-u)_-;R_j)+{\\rm Tail}((u-m_j)_-;R_j)\\big].\n\\end{split}\n\\end{equation}\nNow we provide an estimate of both non-local tails, firstly noting that\n\\begin{equation}\\label{io3}\n\\begin{split}\n{\\rm Tail}((u-m_j)_-;R_j)^{p-1}&=R_j^{ps}\\sum_{k=0}^{j-1}\\int_{B_k\\setminus B_{k+1}}\\frac{(u(y)-m_j)_-^{p-1}}{|y|^{N+ps}}\\,dy\\\\\n&\\quad +R_j^{ps}\\int_{B^c_0}\\frac{(u(y)-m_j)_-^{p-1}}{|y|^{N+ps}}\\,dy.\n\\end{split}\n\\end{equation}\nWe consider the first term: by the inductive hypothesis, for all $0\\le k\\le j-1$ we have in $B_k\\setminus B_{k+1}$\n\\[(u-m_j)_-\\le m_j-m_k \\le (m_j-M_j)+(M_k-m_k) = \\lambda(R_k^\\alpha-R_j^\\alpha),\\]\nhence\n\\begin{align*}\n\\sum_{k=0}^{j-1}\\int_{B_k\\setminus B_{k+1}}\\frac{(u(y)-m_j)_-^{p-1}}{|y|^{N+ps}}\\,dy &\\le \\lambda^{p-1}R_j^{\\alpha(p-1)}\\sum_{k=0}^{j-1}\\int_{B_k\\setminus B_{k+1}}\\frac{(4^{\\alpha(j-k)}-1)^{p-1}}{|y|^{N+ps}}\\,dy \\\\\n&\\le C\\lambda^{p-1} S(\\alpha)R_j^{\\alpha(p-1)-ps},\n\\end{align*}\nwhere we have set for all $\\alpha\\in(0,1)$\n\\[S(\\alpha)=\\sum_{h=1}^\\infty\\frac{(4^{\\alpha h}-1)^{p-1}}{4^{psh}},\\]\nnoting that $S(\\alpha)\\to 0$ as $\\alpha\\to 0^+$. Regarding the second term, by the inductive hypothesis we have\n\\[m_j\\le\\inf_{B_j}u\\le\\sup_{B_j}u\\le\\|u\\|_{L^\\infty(B_0)},\\]\nhence\n\\[\\int_{B^c_0}\\frac{(u(y)-m_j)_-^{p-1}}{|y|^{N+ps}}\\,dy\\le \\int_{B^c_0}\\frac{(\\|u\\|_{L^\\infty(B_0)}+|u(y)|)^{p-1}}{|y|^{N+ps}}\\,dy \\le \\frac{CQ(u; R_0)^{p-1}}{R_0^{ps}}.\\]\nChoosing $\\alpha<ps\/(p-1)$ and plugging the above inequalities in \\eqref{io3}, we get\n\\begin{align*}\n{\\rm Tail}((u-m_j)_-;R_j) &\\le C\\Big[\\lambda^{p-1}S(\\alpha)R_j^{\\alpha(p-1)}+\\frac{Q(u; R_0)^{p-1}R_j^{ps}}{R_0^{ps}}\\Big]^\\frac{1}{p-1} \\\\\n&\\le C\\Big[\\lambda S(\\alpha)^\\frac{1}{p-1}+\\frac{Q(u; R_0)}{R_0^\\alpha}\\Big]R_j^\\alpha.\n\\end{align*}\nAn analogous estimate holds for ${\\rm Tail}((M_j-u)_-;R_j)$, so from \\eqref{io2} we have\n\\begin{equation}\\label{io4}\n\\begin{split}\n\\underset{B_{j+1}}{\\rm osc}\\,u &\\le \\Big(1-\\frac{\\sigma}{2}\\Big)\\lambda R_j^\\alpha+C\\Big[(KR_0^{ps})^\\frac{1}{p-1}+\\lambda S(\\alpha)^\\frac{1}{p-1}R_j^\\alpha+\\frac{Q(u; R_0)R_j^\\alpha}{R_0^\\alpha}\\Big] \\\\\n&\\le 4^\\alpha\\Big[\\Big(1-\\frac{\\sigma}{2}\\Big)+CS(\\alpha)^\\frac{1}{p-1}\\Big]\\lambda R_{j+1}^\\alpha\\\\\n&\\quad +4^\\alpha C\\Big[K^\\frac{1}{p-1}R_0^{\\frac{ps}{p-1}-\\alpha}+\\frac{Q(u; R_0)}{R_0^\\alpha}\\Big]R_{j+1}^\\alpha.\n\\end{split}\n\\end{equation}\nNow we choose $\\alpha\\in(0,ps\/(p-1))$ universally such that\n\\[4^\\alpha\\Big[\\Big(1-\\frac{\\sigma}{2}\\Big)+CS(\\alpha)^\\frac{1}{p-1}\\Big]\\le 1-\\frac{\\sigma}{4},\\]\nand we set\n\\begin{equation}\\label{io5}\n\\lambda=\\frac{4^{\\alpha+1}}{\\sigma} C\\Big[K^\\frac{1}{p-1}R_0^{\\frac{ps}{p-1}-\\alpha}+\\frac{Q(u; R_0)}{R_0^\\alpha}\\Big],\n\\end{equation}\nwhich implies \\eqref{io1} as $4^{\\alpha+1} C\/\\sigma>2$.\nSo, \\eqref{io4} forces\n\\[\\underset{B_{j+1}}{\\rm osc}\\,u\\le \\lambda R_{j+1}^\\alpha.\\]\nWe may pick $m_{j+1}$, $M_{j+1}$ such that\n\\[m_j\\le m_{j+1}\\le\\inf_{B_{j+1}}u\\le\\sup_{B_{j+1}}u\\le M_{j+1}\\le M_j, \\ M_{j+1}-m_{j+1}=\\lambda R_{j+1}^\\alpha,\\]\nwhich completes the induction and proves the claim.\n\\vskip2pt\n\\noindent\nNow fix $r\\in(0,R_0)$ and find an integer $j\\ge 0$ such that $R_{j+1}\\le r<R_j$, thus $R_j\\leq 4r$. Hence, by the claim and \\eqref{io5}, we have\n\\[\\underset{B_r}{\\rm osc}\\,u \\le \\underset{B_j}{\\rm osc}\\,u \\le \\lambda R_j^\\alpha \\le C[(KR_0^{ps})^\\frac{1}{p-1}+Q(u; R_0)\\big]\\frac{r^\\alpha}{R_0^\\alpha},\\]\nwhich concludes the argument.\n\\vskip2pt\n\\noindent\nNow we consider the case $p\\in(1,2)$. The only major difference is in \\eqref{io2}: instead of \\eqref{iox} we use the inductive hypothesis to see that \n\\[M_j-u \\le (M_j-m_j)^{2-p}(M_j-u)^{p-1},\\quad \\text{in $B_j$},\\]\nand similarly for $u-m_j$. Hence\n\\[M_j-m_j\\le(M_j-m_j)^{2-p}\\Big[\\Xint-_{B_j\\setminus\\frac{1}{2}B_j}(M_j-u)^{p-1}\\,dx+\\Xint-_{B_j\\setminus\\frac{1}{2}B_j}(u-m_j)^{p-1}\\,dx\\Big],\\]\nwhich in turn implies through \\eqref{in3}\n\\begin{align*}\nM_j-m_j&\\le\\Big[\\Xint-_{B_j\\setminus\\frac{1}{2}B_j}(M_j-u)^{p-1}\\,dx+\\Xint-_{B_j\\setminus\\frac{1}{2}B_j}(u-m_j)^{p-1}\\,dx\\Big]^\\frac{1}{p-1}\\\\\n&\\leq 2^{\\frac{2-p}{p-1}}\\Big[\\Big(\\Xint-_{B_j\\setminus\\frac{1}{2} B_j}(M_j-u)^{p-1}\\,dx\\Big)^{\\frac{1}{p-1}}+\\Big(\\Xint-_{B_j\\setminus\\frac{1}{2} B_j}(u-m_j)^{p-1}\\,dx\\Big)^{\\frac{1}{p-1}}\\Big].\n\\end{align*}\nMultiplying by $\\sigma\/2^{\\frac{2-p}{p-1}}$ and applying Lemma \\ref{harnackloc} we obtain \\eqref{ioxx} with $\\tilde\\sigma=\\sigma\/2^{\\frac{2-p}{p-1}}$, and the proof follows {\\em verbatim}.\n\\end{proof}\n\n\\noindent\nThe next corollary of Theorem \\ref{osc} follows from standard arguments.\n\n\\begin{corollary}\n\\label{corlocalpha}\nThere exists universal $C>0$ and $\\alpha\\in (0,1)$ with the following property: for all $u\\in \\widetilde{W}^{s,p}(B_{2R_0}(x_0))\\cap L^\\infty(B_{2R_0}(x_0))$ \nsuch that $|(-\\Delta)^s_p u|\\le K$ weakly in $B_{2R_0}(x_0)$,\n\\begin{equation}\n\\label{tesicorloc}\n[u]_{C^\\alpha(B_{R_0}(x_0))}\\le C\\big[(KR_0^{ps})^\\frac{1}{p-1}+Q(u; x_0, 2R_0)\\big]R_0^{-\\alpha}.\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nGiven $x, y$ in $B_{R_0}(x_0)$, let $r=|x-y|$. It suffices to apply Theorem \\ref{osc} to the ball $B_{R_0}(x)\\subseteq B_{2R_0}(x_0)$. Clearly $\\|u\\|_{L^\\infty(B_{R_0}(x))}\\leq \\|u\\|_{L^\\infty(B_{2R_0}(x_0))}$ and \n\\begin{align*}\n&{\\rm Tail}(u; x, R_0)^{p-1}\\\\\n&=R_0^{ps}\\int_{B_{R_0}^c(x)}\\frac{|u(y)|^{p-1}}{|x-y|^{N+ps}}\\, dy\\\\\n&\\leq CR_0^{ps}\\Big[\\int_{B_{2R_0}(x_0)\\setminus B_{R_0}(x)}\\frac{\\|u\\|_{L^\\infty(B_{2R_0}(x_0))}^{p-1}}{|x-y|^{N+ps}}\\, dy+\\int_{B_{2R_0}^c(x_0)}\\frac{|u(y)|^{p-1}}{|x-y|^{N+ps}}\\, dy\\Big]\\\\\n&\\leq C\\|u\\|_{L^\\infty(B_{2R_0}(x_0))}^{p-1}+CR_0^{ps}\\int_{B_{2R_0}^c(x_0)}\\frac{|u(y)|^{p-1}}{|x_0-y|^{N+ps}}\\, dy\n\\end{align*}\nfor a universal $C$, where as usual we used $|x-y|\\geq |x_0-y|\/2$ for  $y\\in B^c_{2R_0}(x_0)$, $x\\in B_{R_0}(x_0)$.\nThis implies that\n\\[Q(u; x, R_0)\\leq CQ(u; x_0, 2R_0),\\]\nand thus the desired estimate on the H\\\"older seminorm.\n\\end{proof}\n\n\\subsection{Global H\\\"older regularity}\n\nWe finally prove the stated H\\\"older regularity result up to the boundary.\n\\vskip4pt\n\\noindent\n\\textbf{Proof of Theorem \\ref{main}.} We set $K=\\|f\\|_{L^\\infty(\\Omega)}$. Corollary \\ref{apb} already provides the desired estimate for the $\\sup$-norm, namely\n\\[\\|u\\|_{L^\\infty(\\Omega)}\\le CK^\\frac{1}{p-1},\\]\nso we can focus on the H\\\"older seminorm.\n\\vskip2pt\n\\noindent\nLet $\\alpha$ be the one given in Corollary \\ref{corlocalpha}. We can assume $\\alpha\\in(0,s]$. Through a covering argument, \\eqref{tesicorloc} implies that $u\\in C^{\\alpha}_{\\rm loc}(\\overline\\Omega')$ for all $\\Omega'\\Subset\\Omega$, with a bound of the form\n\\[\\|u\\|_{C^\\alpha(\\overline\\Omega')}\\leq C_{\\Omega'}K^{\\frac{1}{p-1}},\\quad C_{\\Omega'}=C(N, p, s, \\Omega, \\Omega').\\]\nHence it suffices to prove \\eqref{thm57tesi} in the closure of a fixed $\\rho$-neighbourhood of $\\partial\\Omega$. We will suppose that $\\rho>0$ is so small (depending only on $\\Omega$) that Lemma \\ref{geo1} holds, and thus the metric projection \n\\[\\Pi:V\\to \\partial\\Omega, \\quad \\Pi(x)=\\underset{y\\in \\Omega^c}{\\rm Argmin}|x-y|\\]\nis well defined on $V:=\\{x\\in\\overline\\Omega:\\delta(x)\\leq \\rho\\}$. We claim that \n\\begin{equation}\n\\label{claimfin}\n[u]_{C^\\alpha(B_{r\/2}(x))}\\leq C_\\Omega K^{\\frac{1}{p-1}},\\quad \\text{for all $x\\in V$ and $r=\\delta(x)$} \n\\end{equation}\nfor some constant $C_\\Omega=C(N, p, s, \\Omega)$, independent on $x\\in V$. We recall \\eqref{tesicorloc}, which in the present case rephrases (up to a universal constant) as\n\\[[u]_{C^\\alpha(B_{r\/2}(x))}\\le C\\big[(Kr^{ps})^\\frac{1}{p-1}+\\|u\\|_{L^\\infty(B_r(x))}+{\\rm Tail}(u;x,r)\\big]r^{-\\alpha}.\\]\nTo prove \\eqref{claimfin}, it suffices to bound the three terms on the right hand side of the above inequality. The first one it trivially dealt with since $\\alpha\\leq s\\leq  ps\/(p-1)$, and thus\n\\[r^{-\\alpha}(Kr^{ps})^\\frac{1}{p-1}\\leq K^{\\frac{1}{p-1}}\\rho^{\\frac{ps}{p-1}-\\alpha}.\\]\nFor the second one we use Theorem \\ref{estid} and $\\alpha\\leq s$ to get\n\\[\\|u\\|_{L^\\infty(B_{r}(x))}\\leq CK^{\\frac{1}{p-1}}(\\delta(x)+r)^s\\leq CK^{\\frac{1}{p-1}}\\rho^{s-\\alpha}r^\\alpha,\\]\nand thus the claimed bound. Similarly for the last term we employ again \\eqref{thm44tesi}, together with \n\\[\\delta(y)\\leq |y-\\Pi(x)|\\leq |y-x|+|x-\\Pi(x)|\\leq |y-x|+r\\leq 2|x-y|,\\quad \\forall y\\in B_r^c(x),\\]\nto get\n\\begin{align*}\n{\\rm Tail}(u;x,r)^{p-1} &\\leq CKr^{ps}\\int_{B_{r}^c(x)}\\frac{\\delta^{s(p-1)}(y)}{|x-y|^{N+ps}}\\, dy\\\\\n&\\leq CKr^{ps}\\int_{B_{r}^c(x)}\\frac{|x-y|^{s(p-1)}}{|x-y|^{N+ps}}\\, dy\\\\\n&\\leq CKr^{s(p-1)}.\n\\end{align*}\nAgain due to $\\alpha\\leq s$ we obtain the claimed bound, and the proof of \\eqref{claimfin} is completed.\nTo prove the theorem, pick $x, y\\in V$ and suppose without loss of generality that $|x-\\Pi(x)|\\geq|y-\\Pi(y)|$. Two cases may occur:\n\\begin{itemize}\n\\item either $2|x-y|< |x-\\Pi(x)|$, in which case we set $r=\\delta(x)$ and apply \\eqref{claimfin} in $B_{r\/2}(x)$, to obtain \n\\[|u(x)-u(y)|\\leq C K^{\\frac{1}{p-1}}|x-y|^\\alpha;\\]\n\\item or $2|x-y|\\geq |x-\\Pi(x)|\\geq |y-\\Pi(y)|$, in which case \\eqref{thm44tesi} ensures\n\\begin{align*}\n|u(x)-u(y)|&\\leq |u(x)|+|u(y)|\\leq C K^{\\frac{1}{p-1}}(\\delta^s(x)+\\delta^s(y))\\\\\n&=C K^{\\frac{1}{p-1}}(|x-\\Pi(x)|^s+|y-\\Pi(y)|^s)\\\\\n&\\leq C K^{\\frac{1}{p-1}}|x-y|^s\\\\\n&\\leq C K^{\\frac{1}{p-1}}\\rho^{s-\\alpha}|x-y|^\\alpha.\n\\end{align*}\n\\end{itemize}\nThus in both cases the $\\alpha$-H\\\"older seminorm is bounded in $V$ and the proof is completed. \\qed\n\n\\begin{remark}\n\\label{remalpha}\nAs the proofs above show, interior regularity (Theorem \\ref{osc}) forces in particular $\\alpha<ps\/(p-1)$, while in order to control the behavior of weak solutions near the boundary we need the more restrictive bound $\\alpha\\le s$. Anyway, our H\\\"older exponent remains not explicitly determined.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe dynamics of the Universe is shaped by the gravity force. The\nunderstanding of gravity is thus the key to unveil the nature of the\nUniverse and its cosmological evolution. After almost one century from\nits original formulation, Einstein's General Relativity (GR) is still\nthe dominant theory to describe gravity. However, despite its lasting\nexperimental successes, some fundamental open issues are motivating\nchallenging efforts aimed at searching for possible expansions or\nradically alternative models \\citep{amendola2013}. First of all, GR is\nnot a quantum field theory and cannot be reconciled with the\nprinciples of quantum mechanics. Secondly, an increasingly large\namount of astronomical observations cannot be explained by GR and\nbaryonic matter alone, requiring the introduction of a never observed\nform of dark matter \\citep[DM,][]{zwicky1937}. Finally, independent\ncosmological observations give indisputable evidences for an\naccelerated expansion of the Universe, requiring the introduction of\nanother dark component, generally dubbed dark energy (DE), when\ninterpreted in the GR framework \\citep{riess1998,\n  perlmutter1999}. Whether GR provides a reliable description of the\nlarge scale structure of the Universe, and thus the latter is\ndominated by unknown dark components, or, on the contrary, such a\ngravity theory has to be corrected, or even abandoned, is one of the\nkey questions of modern physics and cosmology.\n\nOne of the most effective ways to test the gravity theory on large\nscales, that is where DM and DE arise, is to exploit the apparent\nanisotropies observed in galaxy maps, the so-called redshift-space\ndistortions \\citep[RSD,][]{kaiser1987, hamilton1998}.  Indeed, by\nmodelling redshift-space clustering anisotropies it is possible to\nconstrain the linear growth rate of cosmic structures, $f\\equiv\nd\\log\\delta\/d\\log a$, where $a$ is the dimensionless scale factor and\n$\\delta$ the linear fractional density perturbation, provided that the\ngalaxy bias is known.  The first RSD measurements were exploited\nprimarily to estimate the mean matter density, $\\Omega_{\\rm M}$\\,, that can be\nderived directly from $f(z)$ when a gravity model is assumed\n\\citep[][]{peacock2001, hawkins2003}. Later on, \\citet{guzzo2008} and\n\\citet{zhang2008} showed that RSD can be effectively used to\ndiscriminate between DE and modified gravity scenarios. RSD thus\nstarted to be considered as a powerful probe of gravity, and several\napplications to galaxy redshift surveys followed rapidly, both in the\nlocal Universe and at larger redshifts, up to $z\\sim1$, such as 6dFGS\n\\citep{beutler2012}, SDSS \\citep{samushia2012, chuang2013,\n  chuang2013b}, WiggleZ \\citep{blake2012, contreras2013}, VIPERS\n\\citep{delatorre2013b}, and BOSS \\citep{tojeiro2012, reid2012,\n  reid2014}.\n\nAll the above measurements have been obtained from large galaxy\nredshift surveys. First tentative studies are starting to consider\nalso different tracers, but results are still strongly affected by\nstatistical uncertainties, due to the paucity of the catalogues\nused. Thanks to the wealth of observational data now available, or\nexpected in the next future, for instance from the ESA Euclid\nmission\\footnote{http:\/\/www.euclid-ec.org} \\citep{laureijs2011}, the\nNASA Wide Field Infrared Space Telescope (WFIRST)\nmission\\footnote{http:\/\/wfirst.gsfc.nasa.gov} \\citep{spergel2013}, and\nthe extended Roentgen Survey with an Imaging Telescope Array (eROSITA)\nsatellite mission\\footnote{http:\/\/www.mpe.mpg.de\/eROSITA}\n\\citep{merloni2012}, it will soon become possible to apply these\nmethodologies on both cluster and Active Galactic Nuclei (AGN)\ncatalogues. Galaxy clusters are the biggest collapsed structures in\nthe Universe. They appear more strongly biased than galaxies, that is\ntheir clustering signal is higher. Moreover, they are relatively\nunaffected by non-linear dynamics on small scales. This could help in\nmodelling redshift-space distortions in their clustering pattern, as\nthe effect of small scale incoherent motions, that generate the\nso-called fingers-of-God pattern, is substantially less severe\nrelative to the galaxy clustering case, improving also the\ncosmological constraints that can be extracted from baryon acoustic\noscillations \\citep{veropalumbo2014}. The main drawback is the low\ndensity of galaxy cluster samples and the fact that these sources can\nbe reliably detected only at relatively small redshifts. On the other\nhand, AGN can be detected up to very large distances. Powered by\naccreting supermassive black holes hosted at their centres, their\nextreme luminosities make them optimal tracers to investigate the\nlargest scales of the Universe, and thus its cosmological evolution\n\\citep[see e.g.][and references therein]{marulli2008, marulli2009,\n  bonoli2009}.\n\nThe aim of this paper is to investigate the {\\em accuracy} of the\nso-called {\\em dispersion model} (that will be described in\n\\S\\ref{sec:RSD}) as a function of comoving scales and sample\nselection. To achieve this goal, we exploit realistic mock samples of\ngalaxies, galaxy clusters and AGN extracted from the {\\em Magneticum}\nhydrodynamic simulation, at six snapshots in the range $0.2\\leq\nz\\leq2$. We adopt a similar methodology as in \\citet{bianchi2012},\nthough substantially extending that analysis. Specifically, i) in line\nwith the majority of recent studies, we investigate the systematic\nerrors of $f\\sigma_8$, instead of $\\beta$, ii) we consider different\nkinds of realistic mock tracers, instead of just DM haloes, iii) we\nanalyse the dependency of the errors as a function of redshift, and\niv) as a function of different sample selections.  In agreement with\nprevious investigations \\citep[see e.g.][]{bianchi2012, mohammad2016},\nwe find that the rough modelisation of non-linear dynamics provided by\nthe dispersion model introduces systematic errors on $f\\sigma_8$\nmeasurements, that depend non trivially on the redshift and bias of\nthe tracers. On the other hand, as we will demonstrate, it is possible\nto substantially reduce the systematic errors on $f\\sigma_8$ provided\nthat the fit is restricted in a proper range of scales, that depends\non sample selection. An alternative investigation of the impact of the\ngalaxy sample selection function on the ability to test GR with RSD\nhas been carried out recently by \\citet{hearin2015}, analysing the\nsmall-scale effects of the assembly bias in velocity space.\n\nThe software that we implemented to perform all the analyses presented\nin this paper can be freely downloaded at this link:\nhttp:\/\/apps.difa.unibo.it\/files\/people\/federico.marulli3\/. It consists\nof a set of C++ libraries that can be used for several astronomical\ncalculations, in particular to measure the two-point correlation\nfunction and to model RSD \\citep[{\\small\n    CosmoBolognaLib},][]{marulli2016}. We also provide the full\ndocumentation for these libraries, and some example codes that explain\nhow to use them.\n\nThe paper is organised as follows. In \\S\\ref{sec:magneticum} we\ndescribe the {\\em Magneticum} simulation used to construct mock\nsamples of galaxies, clusters and AGN, whose main properties are\nreported in Appendix \\ref{app:samples}. In \\S\\ref{sec:RSD} and\n\\S\\ref{sec:methodology} we present the formalism used to model RSD and\nto measure the two-point correlation function in redshift-space mock\ncatalogues, respectively. The results of our analyses are presented in\n\\S\\ref{sec:results}. In \\S\\ref{sec:conclusions} we summarise our\nconclusions. Finally, in the appendices \\ref{app:random} and\n\\ref{app:errors} we discuss the tests performed to investigate the\nrobustness of the methods applied in our analysis.\n\n\n\n\n\\section{The {\\em Magneticum} simulation}\n\\label{sec:magneticum}\n\nThe most direct way to investigate systematic errors on linear growth\nrate measurements from RSD is to exploit large numerical simulations,\nand to apply on them the same methodologies used for real data. As our\ngoal is to study the effects of different sample selections, we make\nuse of realistic mock catalogues of different cosmic\ntracers. Specifically, we analyse galaxy, cluster and AGN mock\ncatalogues extracted from the hydrodynamic {\\em Magneticum}\nsimulation\\footnote{www.magneticum.org} \\citep{dolag2016}.\n\nThis simulation is based on the parallel cosmological TreePM-SPH code\n{\\small P-GADGET3} (\\citealp{springel2005gadget2}). The code uses an\nentropy-conserving formulation of smoothed-particle hydrodynamics\n(SPH) \\citep{springel2002} and follows the gas using a low-viscosity\nSPH scheme to properly track turbulence \\citep{dolag2005}.  It also\nallows a treatment of radiative cooling, heating from a uniform\ntime-dependent ultraviolet background and star formation with the\nassociated feedback processes. The latter is based on a sub-resolution\nmodel for the multiphase structure of the interstellar medium (ISM)\n\\citep{springel2003b}.\n\nRadiative cooling rates are computed by following the same procedure\npresented by \\citet{wiersma2009}. We account for the presence of the\ncosmic microwave background and of ultraviolet\/X-ray background\nradiation from quasars and galaxies, as computed by\n\\citet{haardt2001}. The contributions to cooling from each one of the\nfollowing 11 elements, H, He, C, N, O, Ne, Mg, Si, S, Ca, Fe, have\nbeen pre-computed using the publicly available {\\small CLOUDY}\nphotoionization code \\citep{ferland1998} for an optically thin gas in\nphotoionization equilibrium.\n\nIn the multiphase model for star-formation \\citep{springel2003}, the\nISM is treated as a two-phase medium where clouds of cold gas form\nfrom cooling of hot gas and are embedded in the hot gas phase assuming\npressure equilibrium whenever gas particles are above a given\nthreshold density. The hot gas within the multiphase model is heated\nby supernovae and can evaporate the cold clouds. A certain fraction of\nmassive stars (10 per cent) is assumed to explode as supernovae type\nII (SNII). The released energy by SNII ($10^{51}$~erg) is modelled to\ntrigger galactic winds with a mass loading rate being proportional to\nthe star formation rate to obtain a resulting wind velocity of\n$v_{\\mathrm{wind}} = 350$ km\/s.  Our simulation also includes a\ndetailed model of chemical evolution according to\n\\citet{tornatore2007}. Metals are produced by SNII, by supernovae type\nIa (SNIa) and by intermediate and low-mass stars in the asymptotic\ngiant branch (AGB). Metals and energy are released by stars of\ndifferent mass by properly accounting for mass-dependent lifetimes,\nwith a lifetime function according to \\citet{padovani1993}, the\nmetallicity-dependent stellar yields by \\citet{woosley1995} for SNII,\nthe yields by \\citet{vandenhoek1997} for AGB stars and the yields by\n\\citet{thielemann2003} for SNIa. Stars of different mass are initially\ndistributed according to a Chabrier initial mass function\n\\citep{chabrier2003}.\n\nMost importantly, the {\\em Magneticum} simulation also includes a\nprescription for black hole growth and for feedback from AGN.  As for\nstar formation, the accretion onto black holes and the associated\nfeedback adopts a sub-resolution model. Black holes are represented by\ncollisionless ``sink particles'' that can grow in mass by accreting\ngas from their environments, or by merging with other black holes.\nOur implementation is based on the model presented in\n\\citet{springel2005} and \\citet{dimatteo2005}, including the same\nmodifications as in the study of \\citet{fabjan2010} and some new,\nminor changes, as described in \\citet{hirschmann2014}.\n\nThe black hole (BH) gas accretion rate, $\\dot{M}_{\\rm BH}$, is\nestimated by using the Bondi-Hoyle-Lyttleton approximation\n\\citep{hoyle1939, bondi1944, bondi1952}:\n\\begin{equation}\n  \\dot{M}_{\\rm BH} = \\frac{4 \\pi G^2 M_{\\rm BH}^2 f_{\\rm boost}\n    \\rho}{(c_s^2 + v^2)^{3\/2}},\n  \\label{Bondi}\n\\end{equation}\nwhere $\\rho$ and $c_s$ are the density and the sound speed of the\nsurrounding (ISM) gas, respectively, $f_{\\rm boost}$ is a boost\nfactor for the density which is typically set to $100$ and $v$ is\nthe velocity of the black hole relative to the surrounding gas. The\nblack hole accretion is always limited to the Eddington rate. The\nradiated bolometric luminosity, $L_{\\rm bol}$, is related to the\nblack hole accretion rate by\n\\begin{equation}\n  L_{\\rm bol} = \\epsilon_{\\rm r} \\dot{M}_{\\rm BH} c^2,\n\\end{equation} \nwhere $\\epsilon_{\\rm r}$ is the radiative efficiency, for which we\nadopt a fixed value of $0.1$, standardly assumed for a radiatively\nefficient accretion disk onto a non-rapidly spinning black hole\naccording to \\citet{shakura1973} \\citep[see\n  also][]{springel2005gadget2, dimatteo2005}.\n\nThe simulation covers a cosmological volume with periodic boundary\nconditions initially occupied by an equal number of $1526^3$ gas and\nDM particles, with relative masses that reflect the global baryon\nfraction, $\\Omega_{\\rm b}\/$$\\Omega_{\\rm M}$\\,. The box side is $896$ $h^{-1}\\,\\mbox{Mpc}$\\,.  The\ncosmological model adopted is a spatially flat $\\Lambda$CDM universe\nwith matter density $\\Omega_{\\rm M}$\\,$=0.272$, baryon density $\\Omega_{\\rm\n  b}=0.0456$, power spectrum normalisation $\\sigma_8=0.809$, and\nHubble constant $H_0=70.4 {\\rm km\\, s^{-1} Mpc^{-1}}$, chosen to match\nthe seven-year Wilkinson Microwave Anisotropy Probe \\citep[WMPA7,\n][]{komatsu2011}.  More details on the {\\em Magneticum} simulation and\nthe derived mock catalogues can be found in \\citet{hirschmann2014,\n  saro2014, bocquet2015}.\n\nFirstly, we will analyse large samples of mock galaxies, clusters and\nAGN selected in stellar mass, cluster mass and BH mass,\nrespectively. Then, we will consider several subsamples with different\nselections, as reported in Tables \\ref{tab:table1}, \\ref{tab:table2}\nand \\ref{tab:table3} (see \\S\\ref{sec:results}).\n\n\n\n\n\\section{Redshift-space distortions}\n\\label{sec:RSD}\n\nTo construct redshift-space mock catalogues, we adopt the same\nmethodology used in \\citet{bianchi2012} and \\citet{marulli2011,\n  marulli2012a, marulli2012b}. We provide here a brief description. We\nconsider a local virtual observer at $z=0$, and place the centre of\neach snapshot of the {\\em Magneticum} simulation at a comoving\ndistance, $D_{\\rm c}$, corresponding to its redshift, that is:\n\\begin{equation}\nD_{\\rm c}(z)=c\\int_0^z\\frac{dz_c'}{H(z_c')} \\; ,\n\\label{eq:distance}\n\\end{equation} \nwhere $c$ is the speed of light, and the Hubble expansion rate is:\n\\begin{multline}\nH(z) = H_0\\left[\\Omega_{\\rm M}(1+z)^3+\\Omega_{\\rm\n    k}(1+z)^2\\right. \\\\ \\left. +\\Omega_{\\rm DE}\n  \\exp\\left(3\\int_0^z\\frac{1+w(z)}{1+z}\\right)\\right]^{0.5} \\; ,\n\\label{eq:HubbleFull}\n\\end{multline}\nwhere $\\Omega_{\\rm k}=1-\\Omega_{\\rm M}-\\Omega_{\\rm DE}$, $w(z)$ is the\nDE equation of state, and the contribution of radiation is assumed\nnegligible. In our case: $\\Omega_{\\rm k}=1$, $w(z)=1$, so $\\Omega_{\\rm\n  DE}\\equiv\\Omega_{\\Lambda}=1-\\Omega_{\\rm M}$ and the Hubble expansion\nrate reduces to:\n\\begin{equation}\nH(z) = H_0\\left[\\Omega_{\\rm M}(1+z)^3+\\Omega_{\\Lambda}\\right] \\; .\n\\label{eq:Hubble}\n\\end{equation}\n\nTo estimate the distribution of mock sources in redshift-space, we\nthen transform the comoving coordinates of each object into angular\npositions and observed redshifts. The latter are computed as follows:\n\\begin{equation}\nz_{\\rm obs}=z_{\\rm c}+\\frac{v_\\parallel}{c}(1+z_{\\rm c}) \\; ,\n\\label{eq:redshift}\n\\end{equation}\nwhere $z_{\\rm c}$ is the {\\em cosmological} redshift due to the Hubble\nrecession velocity at the comoving distance of the object and\n$v_\\parallel$ is the line-of-sight component of its centre of mass\nvelocity. In this analysis we do not include either errors in the\nobserved redshift, or geometric distortions caused by an incorrect\nassumption of the background cosmology \\citep[see][for more\n  details]{marulli2012b}.\n\nIn the linear regime, the velocity field can be determined from the\ndensity field, and the amplitude of RSD is proportional to the\nparameter $\\beta$, defined as follows:\n\\begin{equation}\n  \\beta(z) \\equiv \\frac{f(z)}{b(z)} \\; ,\n\\label{eq:beta}\n\\end{equation}\nwhere the linear growth rate, $f$, can be approximated in most\ncosmological frameworks as:\n\\begin{equation}\n  f(z) \\simeq \\Omega_{\\rm M}^{\\gamma}(z) \\; ,\n\\label{eq:fs8}\n\\end{equation}\nwith $\\gamma\\simeq0.545$ in $\\Lambda$CDM. The linear bias factor can\nbe estimated as:\n\\begin{equation}\n  b(z) = \\langle \\sqrt{\\frac{\\xi(r;z)}{\\xi_{\\rm DM}(r;z)}} \\rangle \\;\n  ,\n\\label{eq:bias}\n\\end{equation}\nwhere the mean is computed at sufficiently large scales at which\nnon-linear effects can be neglected.\n\nIn the linear regime, the redshift-space two-point correlation\nfunction can be written as follows:\n\\begin{equation} \n\\xi^{\\rm lin}(s,\\mu) =\n\\xi_0(s)P_0(\\mu)+\\xi_2(s)P_2(\\mu)+\\xi_4(s)P_4(\\mu) \\; ,\n\\label{eq:ximodellin}\n\\end{equation}\nwhere $\\mu\\equiv\\cos\\theta=s_\\parallel\/s$ is the cosine of the angle\nbetween the separation vector and the line of sight,\n$s=\\sqrt{s_\\perp^2+s_\\parallel^2}$, and $P_l$ are the Legendre\npolynomials \\citep{kaiser1987, lilje1989, mcgill1990, hamilton1992,\n  fisher1994}. Eq.~(\\ref{eq:ximodellin}) is derived in the\ndistant-observer approximation, that is reasonable at the scales\nconsidered in this analysis. The multipoles of $\\xi(s_\\perp,s_\\parallel)$\\, are:\n\\begin{subequations}\n  \\begin{align}\n    \\xi_0(s) & = \\left(1+ \\frac{2}{3}\\beta + \\frac{1}{5}\\beta^2 \\right)\n    \\cdot \\xi(r)\n    \\label{eq:xi0_1} \\\\ \n    & =  \\left[ (b\\sigma_8)^2 + \\frac{2}{3} f\\sigma_8 \\cdot b\\sigma_8 +\n      \\frac{1}{5}(f\\sigma_8)^2 \\right] \\cdot \\frac{\\xi_{\\rm DM}(r)}{\\sigma_8^2} \\; ,\n    \\label{eq:xi0_2} \n  \\end{align}\n\\end{subequations}\n\n\\begin{subequations} \n  \\begin{align}\n    \\xi_2(s) &  = \\left(\\frac{4}{3}\\beta +\n    \\frac{4}{7}\\beta^2\\right)\\left[\\xi(r)-\\overline{\\xi}(r)\\right] \n    \\label{eq:xi2_1} \\\\ \n    & = \\left[\\frac{4}{3}f\\sigma_8 \\cdot b\\sigma_8 +\n      \\frac{4}{7}(f\\sigma_8)^2\\right]\\left[\\frac{\\xi_{\\rm\n          DM}(r)}{\\sigma_8^2}-\\frac{\\overline{\\xi}_{\\rm\n          DM}(r)}{\\sigma_8^2}\\right] \\; ,\n    \\label{eq:xi2_2} \n  \\end{align}\n\\end{subequations}\n\n\\begin{subequations} \n  \\begin{align}\n    \\xi_4(s) & = \\frac{8}{35}\\beta^2\\left[\\xi(r) +\n      \\frac{5}{2}\\overline{\\xi}(r)\n      -\\frac{7}{2}\\overline{\\overline{\\xi}}(r)\\right] \n    \\label{eq:xi4_1} \\\\ \n    & = \\frac{8}{35}(f\\sigma_8)^2\\left[ \\frac{\\xi_{\\rm\n          DM}(r)}{\\sigma_8^2} + \\frac{5}{2}\\frac{\\overline{\\xi}_{\\rm\n          DM}(r)}{\\sigma_8^2}\n      -\\frac{7}{2}\\frac{\\overline{\\overline{\\xi}}_{\\rm DM}(r)}{\\sigma_8^2}\n      \\right] \\; ,\n    \\label{eq:xi4_2} \n  \\end{align}\n\\end{subequations}\nwhere $\\xi(r)$ and $\\xi_{\\rm DM}(r)$ are the real-space {\\em\n  undistorted} correlation functions of tracers and DM, respectively,\nwhereas the {\\em barred} functions are:\n\\begin{equation} \n\\overline{\\xi}_{\\rm DM}(r) \\equiv \\frac{3}{r^3}\\int^r_0dr'\\xi_{\\rm\n  DM}(r')r'{^2} \\; ,\n\\label{eq:xi_}\n\\end{equation}\n\\begin{equation}\n\\overline{\\overline{\\xi}}_{\\rm DM}(r) \\equiv \\frac{5}{r^5}\\int^r_0dr'\\xi_{\\rm DM}(r')r'{^4} \\; .\n\\label{eq:xi__} \n\\end{equation}\nEqs.~(\\ref{eq:xi0_2}), (\\ref{eq:xi2_2}) and (\\ref{eq:xi4_2}) are\nderived from Eqs.~(\\ref{eq:xi0_1}), (\\ref{eq:xi2_1}) and\n(\\ref{eq:xi4_1}) respectively, using Eq.~(\\ref{eq:beta}). In this\nanalysis, $\\xi_{\\rm DM}(r)$ is estimated by Fourier transforming the\nlinear power spectrum computed with the software {\\small CAMB}\n\\citep{lewis2002} for the cosmological model here considered. We could\nalternatively measure the DM correlation function directly from the\nsnapshots of the simulation. However, by using {\\small CAMB} we can\nsubstantially reduce the computational time, we get a smooth model\nwith no errors due to measurements and, more importantly, we can\nclosely mimic the analysis we would have done with real data.\n\nEq.~(\\ref{eq:ximodellin}) is an effective description of RSD only at\nlarge scales, where non-linear effects are negligible \\citep[but see\n  e.g.][for a more accurate modelling]{scoccimarro2004, taruya2010,\n  seljak2011, wang2014}.  An empirical model that can account for both\nlinear and non-linear dynamics is the so-called dispersion model\n\\citep{peacock1996, peebles1980, davis1983}, that describes the\nredshift-space correlation function as a convolution of the\nlinearly-distorted correlation with the distribution function of\npairwise velocities, $f(v)$:\n\\begin{equation} \n \\xi(s_\\perp, s_\\parallel) = \\int^{\\infty}_{-\\infty}dv\n f(v)\\xi\\left(s_\\perp, s_\\parallel - \\frac{v(1+z)}{H(z)}\\right)_{\\rm\n   lin} \\; ,\n\\label{eq:ximodel}\n\\end{equation}\nwhere the pairwise velocity $v$ is expressed in physical coordinates.\n\nSince we are not considering redshift errors, we use the exponential\nform for $f(v)$ \\citep{marulli2012b}, namely:\n\\begin{equation}\nf(v)=\\frac{1}{\\sigma_{12}\\sqrt{2}}\n\\exp\\left(-\\frac{\\sqrt{2}|v|}{\\sigma_{12}}\\right)\n\\label{eq:fvexp} \n\\end{equation}\n\\citep{davis1983, fisher1994, zurek1994}.  The quantity $\\sigma_{12}$\ncan be interpreted as the dispersion in the pairwise random peculiar\nvelocities, and is assumed to be independent of pair separations. \n\nThe dispersion model given by\nEqs.~(\\ref{eq:ximodellin})-(\\ref{eq:fvexp}) depends on three free\nquantities, $f\\sigma_8$, $b\\sigma_8$ and $\\sigma_{12}$ (since\n$\\xi_{\\rm DM} \\propto \\sigma_8^2$), and on the reference background\ncosmology used both to convert angles and redshifts into distances and\nto estimate the real-space DM two-point correlation function.  This\nmodel has been widely used in the past years to estimate the linear\ngrowth rate both in configuration space, that is by modelling $\\xi(s_\\perp,s_\\parallel)$\\, or\nits multipoles \\citep[e.g.][]{peacock2001, hawkins2003, guzzo2008,\n  ross2007, cabre2009a, cabre2009b, contreras2013} and in Fourier\nspace, by modelling the power spectrum \\citep[e.g.][]{percival2004,\n  tegmark2004, blake2011c}. In this analysis, we investigate the\naccuracy of the dispersion model in configuration space, while\nalternative RSD models will be analysed in an upcoming paper.\n\n\n\n\n\\section{Methodology}\n\\label{sec:methodology}\n\nThe aim of this work is to test widely-used statistical methodos to\nmodel RSD and to extract constraints on the linear growth rate. Thus,\nwe will use methodologies that can be applied directly to real\ndata. On the other hand, we will not consider any specific mock\ncatalogue, in order to keep our analysis general, that is not\nrestricted to any specific real survey.  To measure the two-point\ncorrelation functions of our mock samples we make use of the\n\\citet{landy1993} estimator:\n\\begin{equation}\n\\xi(r)=\\frac{OO(r)-2OR(r)+RR(r)}{RR(r)} \\; ,\n\\label{eqn:landy}\n\\end{equation}\nwhere $OO(r)$, $OR(r)$ and $RR(r)$ are the fractions of\nobject--object, object--random and random--random pairs, with spatial\nseparation $r$, in the range $[r-\\delta r\/2, r+\\delta r\/2]$, where\n$\\delta r$ is the bin size.  The random catalogues are constructed to\nbe three times larger than the associated mock samples. As we are\nconsidering mock catalogues in cubic boxes, with no geometrical\nselection effects and periodic conditions, we could easily estimate\nthe two-point correlation function directly from the density field, or\ncomputing the random counts analytically. Nevertheless, we prefer to\nuse the \\citet{landy1993} estimator to mimic more closely the analysis\non real data, as stressed before. In any case, this choice does not\nimpact the results of our analysis. Moreover, as we verified, the\nnumber of random objects used in this work is large enough to have no\nsignificant effects on our measurements (see Appendix\n\\ref{app:random}).  We compute the correlation functions up to a\nmaximum scale of $r=50$ $h^{-1}\\,\\mbox{Mpc}$\\,, both in the parallel and perpendicular\ndirections. As we have verified with a sub-set of mocks, going to\nhigher scales does not change our results, as the constraints on\n$f\\sigma_8$ are mainly determined by the RSD signal at smaller scales.\n\nTo model RSD and extract constraints on $f\\sigma_8$, we exploit the\nmodel given by Eqs.~(\\ref{eq:ximodellin})-(\\ref{eq:fvexp}). We\nconsider two case studies. The first one consists in modelling only\nthe monopole of the redshift-space two-point correlation function at\nlarge scales, via Eq.~(\\ref{eq:xi0_2}), that is in the Kaiser limit,\nassuming that the non-linear effects can be neglected at these\nscales. In this case, the $b\\sigma_8$ factor has to be fixed a\npriori. When analysing real catalogues, this factor can be determined\neither directly with the deprojection technique \\citep[see\n  e.g.][]{marulli2012b}, or from the three-point correlation function\n\\citep[see e.g.][]{moresco2014}, or by combining clustering and\nlensing measurements \\citep[see e.g.][]{sereno2015}. To minimise the\nuncertainties and systematics possibly present in the above methods,\nwe derive $b\\sigma_8$ directly from the real-space two-point\ncorrelation functions of our mock tracers, through\nEq.~(\\ref{eq:bias}), while the DM correlation is obtained by Fourier\ntransforming the linear {\\small CAMB} power spectrum, fixing\n$\\sigma_8$ at the value of the {\\em Magneticum} simulation.\n\nThe second method considered in this work consists in fully exploiting\nthe redshift-space two-dimensional anisotropic correlation function\n$\\xi(s_\\perp,s_\\parallel)$\\,. An alternative approach would be to use the multipole moments\nof the correlation function. The advantage is to reduce the number of\nobservables, thus helping in the computation of the covariance matrix\n\\citep[e.g.][]{delatorre2013b, petracca2016}. However, we are not\nmimicking here any real redshift survey. Thus, a detailed treatment of\nthe statistical errors is not necessary for the purposes of the\npresent paper, that focuses instead on the systematic uncertainties\ncaused by a poor modelisation of non-linear effects. Therefore, we\nwill use just diagonal covariance matrices, whose elements are\nestimated analytically from Poisson statistics. As discussed in\n\\citet{bianchi2012}, the effect on growth rate measurements caused by\nassuming negligible off-diagonal elements in the covariance matrix is\nsmall (of the order of only a few percent in that analysis), thanks to\nthe large volumes of the mock samples with respect to the scales used\nfor the parameter estimations. Appendix \\ref{app:errors} provides some\nfurther tests, using the mock samples analysed in this work, that\nappear in overall agreement with the \\citet{bianchi2012} conclusions\n\\citep[see also][]{petracca2016}. Moreover, to investigate the\naccuracy of the dispersion model as a function of scales, it is more\nconvenient to exploit $\\xi(s_\\perp,s_\\parallel)$\\,. Indeed, following this approach, we can\nexplore the full $r_\\parallel-r_\\perp$ plane, searching for the\noptimal region to be used to minimise model uncertainties.\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{xi_ratio_z0.2.ps}\n\\caption{The ratio between the redshift-space and real-space two-point\n  spherically averaged correlation functions, $\\xi(s)\/\\xi(r)$, at\n  $z=0.2$, for galaxies with $\\log(M_{\\rm STAR} [h^{-1}\\,{\\mbox\n      M_\\odot}])>10$ ({\\em top panel}), clusters with $\\log(M_{500}\n  [h^{-1}\\,{\\mbox M_\\odot}])>13$ ({\\em central panel}), and AGN with\n  $\\log(M_{\\rm BH} [h^{-1}\\,{\\mbox M_\\odot}])>6.3$ ({\\em bottom\n    panel}). The redshift-space correlation functions have been\n  measured directly from the simulation, while the real-space ones\n  have been either measured from the simulation ({\\em red dots}), or\n  derived from the {\\small CAMB} power spectrum ({\\em blue\n    squares}). The error bars represent the statistical noise as\n  prescribed by \\citet{mo1992}. The black solid lines show the\n  expected values of $\\xi(s)\/\\xi(r)$ at large scales predicted by\n  Eq.~(\\ref{eq:xi0_2}). The dotted blue and dashed red lines are the\n  best-fit ratios estimated from the blue and red points,\n  respectively.}\n\\label{fig:xi_ratio1}\n\\end{figure}\n\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nIn this section, we present our main results obtained from mock\nsamples of three different tracers -- galaxies, clusters and AGN --\nusing the two approaches described in \\S\\ref{sec:methodology}. For\neach catalogue, we analyse six snapshots corresponding to the\nredshifts $z=\\{0.2, 0.52, 0.72, 1, 1.5, 2\\}$. Moreover, at each\nredshift we consider different sample selections. In total, we analyse\n$270$ mock catalogues, whose main properties, including the number of\nobjects in each sample, are reported in Tables \\ref{tab:table1},\n\\ref{tab:table2} and \\ref{tab:table3}, and in Fig.~\\ref{fig:nz} in\nAppendix~\\ref{app:samples}.\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{fsigma8_RSD4.ps}\n\\caption{{\\em Top panel}: the best-fit values of $f(z)\\sigma_8(z)$ for\n  galaxies with $\\log(M_{\\rm STAR} [h^{-1}\\,{\\mbox M_\\odot}])>10$\n  ({\\em red dots}), clusters with $\\log(M_{500} [h^{-1}\\,{\\mbox\n      M_\\odot}])>13$ ({\\em blue squares}), and AGN with $\\log(M_{\\rm\n    BH} [h^{-1}\\,{\\mbox M_\\odot}])>6.3$ ({\\em green diamonds}),\n  obtained with the method described in \\S\\ref{subsec:resI}. The black\n  line shows the function $\\Omega_{\\rm M}(z)^{0.545}\\cdot\\sigma_8(z)$,\n  where $\\Omega_{\\rm M}(z)$ and $\\sigma_8(z)$ are the known values of\n  the simulation. {\\em Bottom panel}: the percentage systematic errors\n  on $f(z)\\sigma_8(z)$, $[(f\\sigma_8)^{\\rm\n      measured}$-$(f\\sigma_8)^{\\rm simulation}]\/(f\\sigma_8)^{\\rm\n    simulation}\\cdot100$, that is the percentage differences between\n  the points and the black line shown in the top panel. The error bars\n  have been estimated with Eq.~(\\ref{eq:bianchi}). The values reported\n  have been slightly shifted for visual clarity. To guide the eyes,\n  the light and grey shaded areas highlight the $5\\%$ and $10\\%$ error\n  regions, respectively.}\n\\label{fig:fsigma8}\n\\end{figure}\n\n\n\n\n\\subsection{The linear growth rate from the clustering monopole}\n\\label{subsec:resI}\n\nWe start analysing the spherically averaged two-point correlation\nfunction -- the clustering monopole -- at large linear scales. The aim\nof this exercise is to investigate the accuracy of the RSD model\nin the simplest case possible, that is in the so-called Kaiser limit.\n\nFig.~\\ref{fig:xi_ratio1} shows the ratio between the redshift-space\nand real-space two-point correlation functions,\n$\\xi(s)\/\\xi(r)$. Specifically, we show here the results obtained at\n$z=0.2$, for galaxies with $\\log(M_{\\rm STAR} [h^{-1}\\,{\\mbox\n    M_\\odot}])>10$, clusters with $\\log(M_{500} [h^{-1}\\,{\\mbox\n    M_\\odot}])>13$, and AGN with $\\log(M_{\\rm BH} [h^{-1}\\,{\\mbox\n    M_\\odot}])>6.3$, as reported by the labels. These correspond to\nthe upper left samples reported in Tables \\ref{tab:table1},\n\\ref{tab:table2} and \\ref{tab:table3}. Results obtained from the other\nmock samples are similar, but more scattered due to the lower\ndensities.  The redshift-space correlation functions have been\nmeasured directly from the simulation, through the procedure described\nin \\S\\ref{sec:methodology}. The real-space correlation functions have\nbeen estimated in two different ways. Either they are measured\ndirectly from the simulation, or they are derived from the linear\n{\\small CAMB} power spectrum, and assuming a linear bias factor. The\ndifferences at small scales, $r\\lesssim5$ $h^{-1}\\,\\mbox{Mpc}$\\,, between the ratios\ncomputed with the measured (blue squares) and {\\small CAMB} (red dots)\nreal-space correlation functions are due to non-linear effects. We\nverified that using the non-linear {\\small CAMB} power spectrum, via\nthe {\\small HALOFIT} routine \\citep{smith2003}, does not fully remove\nthe discrepancy. Nevertheless, as we model here the large scale\nclustering, this has no effects on our results. On the other hand, the\nsmall discrepancies at scales $r\\gtrsim40$ $h^{-1}\\,\\mbox{Mpc}$\\, can introduce\nsystematics, that however result smaller than the estimated\nuncertainties. The error bars in Fig.~\\ref{fig:xi_ratio1} show the\nstatistical Poisson noise \\citep{mo1992}. Scales larger than $60$\n$h^{-1}\\,\\mbox{Mpc}$\\,, not shown in the plot, are too noisy to affect the fit. More\nspecifically, we verified that a convenient scale range to get robust\nresults is $10<r[h^{-1}\\,\\mbox{Mpc}]<50$. The black solid lines show\nthe expected values of $\\xi(s)\/\\xi(r)$ at large scales, as predicted\nby Eq.~(\\ref{eq:xi0_2}).\n\nBy modelling the clustering ratio, $\\xi(s)\/\\xi(r)$, via\nEq.~(\\ref{eq:xi0_1}), it is possible to estimate the factor\n$\\beta$. We do this by measuring both $\\xi(s)$ and $\\xi(r)$ from the\nsimulation. The result is shown by the dotted blue lines in\nFig.~\\ref{fig:xi_ratio1}. Instead, to estimate $f\\sigma_8$ from the\nmonopole of the two-point correlation function, we use\nEq.~(\\ref{eq:xi0_2}). In this case the factor $b\\sigma_8$ has to be\nfixed. This requires to estimate the real-space two-point correlation\nfunction of the DM. As described above, we obtain the latter by\nFourier transforming the {\\small CAMB} power spectrum. Moreover, to\nestimate the mean linear bias, a range of scales has to be chosen in\nEq.~(\\ref{eq:bias}). We compute the linear bias in the range\n$\\bar{r}<r[$$h^{-1}\\,\\mbox{Mpc}$\\,$]<50$, where $\\bar{r}$ is a free parameter that sets\nthe minimum scale beyond which the bias is fairly\nscale-independent. Such a minimum scale depends on redshift and sample\nselection, and we estimate it for each sample considered. The dashed\nred lines are the best-fit ratios estimated with this method.\n\n\\begin{figure}\n  \\includegraphics[width=0.48\\textwidth]{iso_model_galaxies.ps}\n  \\caption{The iso-correlation contours of the redshift-space\n    two-point correlation function, corresponding to the values\n    $\\xi(s_\\perp,s_\\parallel)$\\,$=[0.05, 0.1, 0.2, 0.4, 1, 3]$, for galaxies with\n    $\\log(M_{\\rm STAR} [h^{-1}\\,{\\mbox M_\\odot}])>10$, at $z=0.2$\n    (black contours). The dot-dashed green and solid red contours show\n    the best-fit model given by Eq.~(\\ref{eq:ximodel}), with the\n    real-space correlation function $\\xi(r)$ measured from the\n    simulation, and estimated from the {\\small CAMB} power spectrum,\n    respectively. The blue dashed contours show the linear best-fit\n    model given by Eq.~(\\ref{eq:ximodellin}) with the {\\small CAMB}\n    real-space correlation function.}\n  \\label{fig:iso_gal}\n\\end{figure}\n\n\\begin{figure}\n  \\includegraphics[width=0.48\\textwidth]{fsigma8_RSD5_gal.ps}\n  \\includegraphics[width=0.48\\textwidth]{best_range_gal.ps}\n  \\caption{{\\em Top panel of the upper window}: the best-fit values of\n    $f(z)\\sigma_8(z)$ for galaxies with $\\log(M_{\\rm STAR}\n    [h^{-1}\\,{\\mbox M_\\odot}])>10$. The open and solid blue squares\n    show the values obtained by fitting the data with the models given\n    by Eqs.~(\\ref{eq:ximodellin}) and (\\ref{eq:ximodel}),\n    respectively, in the scale ranges\n    $3<r_\\perp[h^{-1}\\,\\mbox{Mpc}]<35$ and\n    $3<r_\\parallel[h^{-1}\\,\\mbox{Mpc}]<35$. The open and solid red\n    dots have been obtained with the method described in\n    \\S\\ref{subsubsec:galaxies}, fitting the data with the models given\n    by Eqs.~(\\ref{eq:ximodellin}) and (\\ref{eq:ximodel}),\n    respectively. The black line shows the function $\\Omega_{\\rm\n      M}(z)^{0.545}\\cdot\\sigma_8(z)$, where $\\Omega_{\\rm M}(z)$ and\n    $\\sigma_8(z)$ are the known values of the simulation. For\n    comparison, the grey dots show a set of recent observational\n    measurements from galaxy surveys (see\n    \\S\\ref{subsubsec:galaxies}). {\\em Bottom panel of the upper\n      window}: the percentage systematic errors on $f(z)\\sigma_8(z)$,\n    defined as $[(f\\sigma_8)^{\\rm measured}$-$(f\\sigma_8)^{\\rm\n        simulation}]\/(f\\sigma_8)^{\\rm simulation}\\cdot100$, that is\n    the percentage differences between the points and the black line\n    shown in the top panel. To guide the eyes, the light and grey\n    shaded areas highlight the $5\\%$ and $10\\%$ error regions. {\\em\n      Lower window}: best-fit values of $r_\\perp^{\\rm min}$ and\n    $r_\\parallel^{\\rm max}$ of the method described in\n    \\S\\ref{subsubsec:galaxies}.  The grey areas show the regions\n    explored by the method.  }\n  \\label{fig:fs8_gal}\n\\end{figure}\n\nAs it can be seen by comparing red and blue lines, the two methods\nprovide concordant results, both of them in agreement with the\nexpectations. Indeed, the real-space correlation function provided by\n{\\small CAMB} and assuming a linear bias is in reasonable agreement\nwith the one measured directly from the simulation.\n\nThe constraints on $f\\sigma_8$ as a function of redshift are shown in\nthe upper panel of Fig.~\\ref{fig:fsigma8}. As in\nFig.~\\ref{fig:xi_ratio1}, we show the results obtained for galaxies\nwith $\\log(M_{\\rm STAR} [h^{-1}\\,{\\mbox M_\\odot}])>10$, clusters with\n$\\log(M_{500} [h^{-1}\\,{\\mbox M_\\odot}])>13$, and AGN with\n$\\log(M_{\\rm BH} [h^{-1}\\,{\\mbox M_\\odot}])>6.3$. The black line shows\nthe function $\\Omega_{\\rm M}(z)^{0.545}\\cdot\\sigma_8(z)$, where\n$\\Omega_{\\rm M}(z)$ and $\\sigma_8(z)$ are the values of the {\\em\n  Magneticum} simulation.  In the lower panel we show the percentage\nsystematic errors on $f\\sigma_8$, defined as $[(f\\sigma_8)^{\\rm\n    measured}$-$(f\\sigma_8)^{\\rm simulation}]\/(f\\sigma_8)^{\\rm\n  simulation}\\cdot100$. The error bars have been estimated by\npropagating on $f\\sigma_8$ the $\\beta$ errors provided by the scaling\nformula presented in \\citet{bianchi2012}, that gives the statistical\nerrors as a function of bias, $b$, volume, $V$, and density, $n$:\n\\begin{equation}\n\\frac{\\delta\\beta}{\\beta}\\simeq\nCb^{0.7}V^{-0.5}\\exp\\left(\\frac{n_0}{b^2n}\\right) \\; ,\n\\label{eq:bianchi} \n\\end{equation}\nwhere $n_0=1.7\\cdot10^{-4}\\,h^{3}\\,\\mbox{Mpc}^{-3}$ and\n$C=4.9\\cdot10^2 \\,h^{-1.5}\\,\\mbox{Mpc}^{1.5}$. In this case, these\nerror bars have to be considered just as lower limits, as the scaling\nformula has been calibrated based on fits of the full two-dimensional\nanisotropic correlation function $\\xi(s_\\perp,s_\\parallel)$\\,. Moreover, they have been\ncomputed in different scale ranges and with Friends-of-Friends DM\nhaloes, differently from this analysis, where we consider tracers\nhosted in DM sub-haloes. Using the full covariance matrix in this\nstatistical analysis does not change our conclusions, as shown in\nAppendix \\ref{app:errors}.\n\n\\begin{figure}\n  \\includegraphics[width=0.48\\textwidth]{err_fsigma8_RSD5_gal_sel.ps}\n  \\caption{The percentage systematic errors on $f(z)\\sigma_8(z)$ for\n    galaxies, $[(f\\sigma_8)^{\\rm measured}$-$(f\\sigma_8)^{\\rm\n        simulation}]\/(f\\sigma_8)^{\\rm simulation}\\cdot100$, as a\n    function of redshift and sample selections, as indicated by the\n    labels. $S1-S5$ refer to the five selection thresholds reported in\n    Tables \\ref{tab:table1}, \\ref{tab:table2} and \\ref{tab:table3},\n    for the different tracers and properties used for the selection.}\n  \\label{fig:errfs8_gal}\n\\end{figure}\n\nAs demonstrated by Figs.~\\ref{fig:xi_ratio1} and \\ref{fig:fsigma8}, it\nis indeed possible to get almost unbiased constraints on $f\\sigma_8$\nfrom the monopole of the two-point correlation function, at all\nredshifts considered, and for both galaxies, clusters and AGN,\nprovided that the linear bias is estimated at sufficiently large\nscales. The advantage of this method is the minimal number of free\nparameters necessary for the modelisation. However, the linear bias\nhas to be assumed, or measured from other probes.\n\n\n\n\n\\subsection{The linear growth rate from the anisotropic 2D clustering}\n\\label{subsec:resII}\n\nTo extract the full information from the redshift-space two-point\ncorrelation function, the anisotropic correlation $\\xi(s_\\perp,s_\\parallel)$\\, or,\nalternatively, all the relevant multipoles have to be modelled. As\ndiscussed in \\S\\ref{sec:methodology}, we consider the first\napproach. Differently from the analysis of \\S\\ref{subsec:resI}, here\nwe can jointly constrain the two terms $f\\sigma_8$ and $b\\sigma_8$. In\nthis section we investigate the accuracy of the constraints on\n$f\\sigma_8$ provided by the dispersion model, as a function of sample\nselection, with $b\\sigma_8$ and $\\sigma_{12}$ as free parameters.\n\n\n\n\n\\subsubsection{Galaxies}\n\\label{subsubsec:galaxies}\n\nWe start presenting our results for the galaxy mock samples. The black\nlines of Fig.~\\ref{fig:iso_gal} show the iso-correlation contours of\nthe redshift-space two-point correlation function, $\\xi(s_\\perp,s_\\parallel)$\\,, for galaxies\nwith $\\log(M_{\\rm STAR} [h^{-1}\\,{\\mbox M_\\odot}])>10$, at\n$z=0.2$. The other lines are the best-fit models obtained with three\ndifferent methods. The dot-dashed green lines are obtained by using\nthe full non-linear dispersion model given by Eq.~(\\ref{eq:ximodel}),\nwith the real-space correlation function $\\xi(r)$ measured directly\nfrom the simulation. The blue dashed and red solid contours are\nobtained with $\\xi(r)$ estimated from the {\\small CAMB} power\nspectrum, and by fitting the linear model given by\nEq.~(\\ref{eq:ximodellin}) and the non-linear one by\nEq.~(\\ref{eq:ximodel}), respectively. The fitting is done in the scale\nranges $3<r_\\perp[h^{-1}\\,\\mbox{Mpc}]<35$ and\n$3<r_\\parallel[h^{-1}\\,\\mbox{Mpc}]<35$.\n\nAs one can see, while all the three models provide a good description\nof the anisotropic clustering at scales larger than $\\sim5$ $h^{-1}\\,\\mbox{Mpc}$\\,,\nnone of them can reproduce accurately the fingers-of-God pattern at\nsmall scales. This is expected for the blue contours, obtained without\nmodelling the non-linear dynamics. For the other two cases, the\ndiscrepancy is due to the not sufficiently accurate description of\nnon-linearities provided by the dispersion model. We verified that the\nminimum scales considered for the fit have a negligible impact on the\nfingers-of-God shape, that is we find the same results even\nconsidering scales smaller that 3 $h^{-1}\\,\\mbox{Mpc}$\\, in the fit. The green contours\nappear in better agreement with the measurements at small scales, due\nto the fact that the real-space correlation function is measured from\nthe mock catalogue\\footnote{The model corresponding to the green\n  contours provides constraints on $\\beta$, not on $f\\sigma_8$. We\n  show it here only to highlight the differences at small scales\n  between the dispersion model estimated with $\\xi(r)$ measured from\n  the simulation, and the one with $\\xi(r)$ from the linear {\\small\n    CAMB} power spectrum, assuming a linear scale-independent bias.}.\n\n\\begin{figure}\n  \\includegraphics[width=0.48\\textwidth]{iso_model_cl.ps}\n  \\caption{The iso-correlation contours of the redshift-space\n    two-point correlation function, corresponding to the values\n    $\\xi(s_\\perp,s_\\parallel)$\\,$=[0.2, 0.4, 1, 3]$, for clusters with $\\log(M_{500}\n         [h^{-1}\\,{\\mbox M_\\odot}])>13$, at $z=0.2$ (black\n         contours). The other contours are as in\n         Fig.~\\ref{fig:iso_gal}.}\n  \\label{fig:iso_cl}\n\\end{figure}\n\n\\begin{figure}\n  \\includegraphics[width=0.48\\textwidth]{fsigma8_RSD5_cl.ps}\n  \\includegraphics[width=0.48\\textwidth]{best_range_cl.ps}\n  \\caption{The best-fit values of $f(z)\\sigma_8(z)$ for clusters with\n    $\\log(M_{500} [h^{-1}\\,{\\mbox M_\\odot}])>13$. All the symbols are\n    as in Fig.~\\ref{fig:fs8_gal}.}\n  \\label{fig:fs8_cl}\n\\end{figure}\n\n\\begin{figure}  \n  \\includegraphics[width=0.48\\textwidth]{err_fsigma8_RSD5_cl_sel.ps}\n  \\caption{The percentage systematic errors on $f(z)\\sigma_8(z)$ for\n    clusters, as a function of redshift and sample selections. All\n    symbols are as in Fig.~\\ref{fig:errfs8_gal}.}\n  \\label{fig:errfs8_cl}\n\\end{figure}\n\nThe best-fit values of $f\\sigma_8$ as a function of redshift are shown\nin the top panel of the upper window of Fig.~\\ref{fig:fs8_gal}. The\nopen and solid blue squares show the values obtained by fitting the\ndata with the models given by Eqs.~(\\ref{eq:ximodellin}) and\n(\\ref{eq:ximodel}), respectively, that is they correspond to the blue\nand red contours of Fig.~\\ref{fig:iso_gal} (at $z=0.2$). The bottom\npanel of the upper window shows the percentage systematic errors on\n$f\\sigma_8$, that is $[(f\\sigma_8)^{\\rm measured}$-$(f\\sigma_8)^{\\rm\n    simulation}]\/(f\\sigma_8)^{\\rm simulation}\\cdot100$.\n\nAs it can be seen, the best-fit values of $f\\sigma_8$ result strongly\nbiased with respect to the true ones, with a systematic error that\ndepends on the redshift. At $z=0.2$, we get an underestimation of\n$\\sim10$ ($20$) $\\%$ with the non-linear (linear) dispersion model, at\n$z=1$ the error reduces to $\\sim5$ ($8$)$\\%$, while at $z=2$ we get an\noverestimation of $\\sim20\\%$, with both linear and non-linear\nmodelling. This result is fairly in agreement with what found by\n\\citet{bianchi2012} at $z=1$, although the two analyses are not\ndirectly comparable, as \\citet{bianchi2012} considered a sample of\nFriends-of-Friends DM haloes, slightly affected by fingers-of-God. As\nexpected, the convolution given by Eq.~(\\ref{eq:ximodel}) reduces the\nsystematic error on $f\\sigma_8$, especially at low redshifts. However,\nit does not totally remove the discrepancies, in agreement with\nprevious findings \\citep[e.g.][and references\n  therein]{mohammad2016}. For comparison, the grey dots of\nFig.~\\ref{fig:fs8_gal} show a set of recent observational measurements\nof $f\\sigma_8$ from large galaxy surveys: 6dFGS at $z=0.067$\n\\citep{beutler2012}; SDSS(DR7) Luminous Red Galaxies at $z=0.25, 0.37$\n\\citep{samushia2012} from scales lower than $60$ and $200$ $h^{-1}\\,\\mbox{Mpc}$\\,; BOSS\nat $z=0.3, 0.57, 0.6$ \\citep{tojeiro2012, reid2012}; WiggleZ at\n$z=0.44, 0.6, 0.73$ \\citep{blake2012}; VIPERS at $z=0.8$\n\\citep{delatorre2013b}. These results have been obtained using\ndifferent RSD models, most of them more accurate than the dispersion\nmodel considered in this work. Nevertheless, many of these\nmeasurements underestimate $f\\sigma_8$ with respect to GR+$\\Lambda$CDM\npredictions \\citep[see e.g.][]{macaulay2013}.  In line with our\nfindings, this could be explained, at least partially, by model\nuncertainties still present in the more sophisticated RSD models\nconsidered.\n\nThe direct way to reduce these systematics is to improve the\nmodelisation of RSD at non-linear scales \\citep[see\n  e.g.][]{scoccimarro2004, taruya2010, seljak2011, wang2014,\n  delatorre2013b}. We explore here a different approach, investigating\nthe dependency of the systematic error on the comoving scales\nconsidered in the analysis. In the forthcoming plots, we show the\nresults obtained by repeating our fitting procedure for different\nvalues of the minimum perpendicular separation and the maximum\nparallel separation used in the fit, that is $r_\\perp^{\\rm min}$ and\n$r_\\parallel^{\\rm max}$. By increasing the value of $r_\\perp^{\\rm\n  min}$ we can cut the region more affected by fingers-of-God\nanisotropies, while by reducing $r_\\parallel^{\\rm max}$ we avoid the\nregion more affected by shot noise. As we verified, changing also\n$r_\\perp^{\\rm max}$ and $r_\\parallel^{\\rm min}$, or adopting different\nscale selection criteria, do not affect significantly the results.\nThe aim here is to search for optimal regions in this plane to\npossibly get unbiased constraints. As non-linear dynamics impact on\ndifferent scales for different redshifts and biases of the tracers, we\nexpect that the best values of $r_\\perp^{\\rm min}$ and\n$r_\\parallel^{\\rm max}$, that is the ones that minimise systematics,\nwill be different for different sample selections.\n\nThe results of this analysis are shown in Fig.~\\ref{fig:fs8_gal} with\nopen and solid red dots, obtained by fitting the data with the model\ngiven by Eqs.~(\\ref{eq:ximodellin}) and (\\ref{eq:ximodel}),\nrespectively, that is by considering either the linear or the\nnon-linear RSD model.  We explore the ranges $5<r_\\perp^{\\rm\n  min}[h^{-1}\\,\\mbox{Mpc}]<20$ and $15<r_\\parallel^{\\rm\n  max}[h^{-1}\\,\\mbox{Mpc}]<35$, highlighted by the grey areas in the\nbottom panels of the lower window of Fig.~\\ref{fig:fs8_gal}.\nSpecifically, we consider a 10x10 grid in the $r_\\perp^{\\rm min}$,\n$r_\\parallel^{\\rm max}$ plane. The best-fit values of $r_\\perp^{\\rm\n  min}$ and $r_\\parallel^{\\rm max}$, shown in the bottom panels of the\nlower window, are the ones that minimise the systematic error on\n$f\\sigma_8$. The ranges considered are large enough to get systematic\nerrors on $f\\sigma_8$ lower than $\\sim 5\\%$. Indeed, for proper values\nof $r_\\perp^{\\rm min}$ and $r_\\parallel^{\\rm max}$, it is possible to\nsignificantly reduce the systematic error on $f\\sigma_8$ at all\nredshifts, without having to improve the treatment of\nnon-linearities. The error bars have been estimated using the scaling\nformula given by Eq.~\\ref{eq:bianchi}.\n\nBoth the linear and non-linear dispersion models provide similar\nresults when $r_\\perp^{\\rm min}$ and $r_\\parallel^{\\rm max}$ are kept\nfree. In other words, it seems more convenient simply to not consider\nthe scales where non-linear effects have a non-negligible impact,\nrather than to try modelling them with the empirical description\nprovided by Eq.~(\\ref{eq:ximodel}).\n\nThese results show that i) it is possible to reduce significantly the\nsystematics in RSD constraints, even with the dispersion model, by\ncutting the $r_\\perp^{\\rm min}$, $r_\\parallel^{\\rm max}$ plane\nconveniently, and ii) the {\\em optimal} $r_\\perp^{\\rm min}$,\n$r_\\parallel^{\\rm max}$ range depends on sample selection, thus it\ncannot be just fixed in multi-tracer analyses. As it can be seen in\nFig.~\\ref{fig:fs8_gal}, we do not find any clear trend of\n$r_\\perp^{\\rm min}$ and $r_\\parallel^{\\rm max}$ as a function of\nredshift, as it would be expected if the systematics were caused by\nnon-linearities. This is possibly caused by the not sufficiently large\nvolumes considered. The analysis of larger data sets, that we defer to\na future work, will hopefully clarify this point.\n\n\\begin{figure}\n  \\includegraphics[width=0.48\\textwidth]{iso_model_agn.ps}\n  \\caption{The iso-correlation contours of the redshift-space\n    two-point correlation function, corresponding to the values\n    $\\xi(s_\\perp,s_\\parallel)$\\,$=[0.1, 0.2, 0.4, 1, 3]$, for AGN with $\\log(M_{\\rm BH}\n         [h^{-1}\\,{\\mbox M_\\odot}])>6.3$, at $z=0.2$ (black\n         contours). The other contours show the best-fit models, as\n         in Fig.~\\ref{fig:iso_gal}.}\n  \\label{fig:iso_agn}\n\\end{figure}\n\n\\begin{figure}\n  \\includegraphics[width=0.48\\textwidth]{fsigma8_RSD5_agn.ps}\n  \\includegraphics[width=0.48\\textwidth]{best_range_agn.ps}\n  \\caption{The best-fit values of $f(z)\\sigma_8(z)$ for AGN with\n    $\\log(M_{\\rm BH} [h^{-1}\\,{\\mbox M_\\odot}])>6.3$. All the symbols\n    are as in Fig.~\\ref{fig:fs8_gal}.}\n  \\label{fig:fs8_agn}\n\\end{figure}\n\n\\begin{figure}  \n  \\includegraphics[width=0.48\\textwidth]{err_fsigma8_RSD5_agn_sel.ps}\n  \\caption{The percentage systematic errors on $f(z)\\sigma_8(z)$ for\n    AGN, as a function of redshift and sample selections. All symbols\n    are as in Fig.~\\ref{fig:errfs8_gal}.}\n  \\label{fig:errfs8_agn}\n\\end{figure}\n\nFor completeness, we then apply our method to several subsamples with\ndifferent selections. Specifically, we consider galaxy catalogues\nselected in stellar mass, $M_{\\rm STAR}$, g-band absolute magnitude,\n$M_g$, and star formation rate, SFR. The selection thresholds\nconsidered and the number of galaxies in each sample are reported in\nTable \\ref{tab:table1}. The percentage systematic errors on\n$f\\sigma_8$ are reported in Fig.~\\ref{fig:errfs8_gal}, as a function\nof redshift and sample selections, as indicated by the labels. The\nresult is quite remarkable: it is indeed possible to get almost\nunbiased constraints on $f\\sigma_8$ with the dispersion model,\nindependent of sample selections, provided that $r_\\perp^{\\rm min}$\nand $r_\\parallel^{\\rm max}$ are chosen conveniently. For the largest\ngalaxy sample shown in Fig.~\\ref{fig:fsigma8}, the preferred value of\n$r_\\perp^{\\rm min}$ is $\\sim 15$ $h^{-1}\\,\\mbox{Mpc}$\\, at low and high redshifts, and\n$\\sim 5$ $h^{-1}\\,\\mbox{Mpc}$\\, around $z=1$, while $r_\\parallel^{\\rm max}$ is $\\sim\n15$ $h^{-1}\\,\\mbox{Mpc}$\\, at low redshifts, and increases up to $\\sim 35$ $h^{-1}\\,\\mbox{Mpc}$\\, at\n$z=2$. Again, we do not find any clear trend of $r_\\perp^{\\rm min}$\nand $r_\\parallel^{\\rm max}$ as a function of redshift or sample bias.\n\nOverall, all of these results show that our modelling of RSD is mostly\nsensitive to the lower fitting cut-off, depending on how the tracers\nrelate to the underlying mass. Indeed, the effect of the different\nselections considered here is just to select subsamples of haloes of\ndifferent masses (hence bias) hosting the observed galaxies. As noted\nabove, this has to be considered in multi-tracer analyses, where a\nsingle sample selection might introduce systematics.\n\n\n\n\n\\subsubsection{Galaxy groups and clusters}\n\\label{subsubsec:clusters}\n\nIn this section, we perform the same analysis presented in\n\\S\\ref{subsubsec:galaxies} on the mock samples of galaxy groups and\nclusters. To simplify the discussion, in the following we will use the\nterm {\\em clusters} to refer to all of these objects, though the less\nmassive ones should be considered as galaxy groups, or just haloes,\nfrom an observational perspective (see Table \\ref{tab:table2}).\n\nFig.~\\ref{fig:iso_cl} shows the iso-correlation contours of the\nredshift-space two-point correlation function of clusters with\n$\\log(M_{500} [h^{-1}\\,{\\mbox M_\\odot}])>13$, at $z=0.2$. The other\ncontours are the best-fit models, as in Fig.~\\ref{fig:iso_gal}. As it\ncan be seen, the fingers-of-God anisotropies are almost absent,\ndifferently from the galaxy correlation function shown in\nFig.~\\ref{fig:iso_gal}, due to the lower values of non-linear motions\nof clusters at small scales. This makes the convolution of\nEq.~(\\ref{eq:ximodel}) negligible, as it can be seen comparing the\ndashed blue and solid red lines. Interestingly, when the real-space\ncorrelation function $\\xi(r)$ is measured from the mocks (green\ncontours), the dispersion model fails to match the small-scale\nclustering shape. This is primarily caused by the paucity of the\ncluster sample. Due to the low number of cluster pairs at small\nseparations, the small-scale clustering cannot be estimated\naccurately, thus introducing systematics in the model. On the\ncontrary, when $\\xi(r)$ is estimated from {\\small CAMB}, the\nsmall-scale clustering shape can be accurately described. For visual\nclarity, we have shown here only the iso-correlation contours down to\n$\\xi(s_\\perp,s_\\parallel)$\\,$=0.2$, as for lower values they appear too scattered.\n\nFig.~\\ref{fig:fs8_cl} shows the best-fit values of $f\\sigma_8$ as a\nfunction of redshift. In contrast with what we found with galaxies,\nthe constraints on the linear growth rate obtained in the scale ranges\n$3<r_\\perp[h^{-1}\\,\\mbox{Mpc}]<35$ and\n$3<r_\\parallel[h^{-1}\\,\\mbox{Mpc}]<35$ (blue squares) result severely\noverestimated at all redshifts, independent of using the linear or the\nnon-linear dispersion model. To investigate how these systematics\ndepend on the scale range considered in the fit, we repeat the same\nanalysis described in \\S\\ref{subsubsec:galaxies}, keeping free the\nparameters $r_\\perp^{\\rm min}$ and $r_\\parallel^{\\rm max}$. The result\nis shown by the red dots in Fig.~\\ref{fig:fs8_cl}. As one can see, we\nare able to substantially reduce the systematics on $f\\sigma_8$, at\nall redshifts. This is obtained by cutting the small perpendicular\nseparations, limiting the analysis at $r_\\perp\\gtrsim20$ $h^{-1}\\,\\mbox{Mpc}$\\,, as\nshown in the lower window of Fig.~\\ref{fig:fs8_cl}.\n\nThese results appear in overall agreement with what previously found\nby \\citet{bianchi2012}, where systematic errors become positive for DM\nhalo masses larger than $10^{13}h^{-1}\\,{\\mbox M_\\odot}$, that is for\ngroup masses and above. This represents a nice confirmation, not\nrelated to the specific simulation or halo\/cluster selection, but only\nto the dynamics of haloes of large masses.\n\nThen, we consider several cluster subsamples selected in mass,\n$M_{500}$, temperature, $T_{500}$, and luminosity, $L_{500}$, whose\nmain properties are reported in Table\n\\ref{tab:table2}\\footnote{$M_{500}$, $T_{500}$ and $L_{500}$ are,\n  respectively, the mass, the temperature and the luminosity enclosed\n  within a sphere of radius $r_{500c}$, in which the mean matter\n  density is equal to $500$ times the critical density $\\rho_{\\rm\n    crit}(z) = 3H^2(z)\/8\\pi G$, where $H(z)$ is the Hubble\n  parameter.}.  Fig.~\\ref{fig:errfs8_cl} shows the percentage\nsystematic errors on $f\\sigma_8$ as a function of redshift and sample\nselections. For the largest samples analysed we do not get any\nsignificant bias on $f\\sigma_8$, similarly to what we found with\ngalaxies. However, when selecting too few objects, the results appear\nquite scattered, though without systematic trends. Indeed, the\nadvantages resulting from the low fingers-of-God anisotropies\ncompensate with the paucity of cluster pairs at small\nscales. Summarising, it seems possible to get unbiased constraints on\nthe linear growth rate from the RSD of galaxy clusters, but the\ncluster sample has to be sufficiently numerous.\n\n\n\n\n\\subsubsection{AGN}\n\\label{subsubsec:AGN}\n\nFinally, we analyse the AGN mock samples.  The results are shown in\nFigs.~\\ref{fig:iso_agn}, \\ref{fig:fs8_agn} and \\ref{fig:errfs8_agn},\nthat are the analogous of the plots presented in the previous two\nsections for galaxies and clusters. As expected, we find similar\nresults to the ones found with galaxies. Differently from the cluster\ncase, the fingers-of-God are clearly evident in the AGN correlation\nfunction, and not accurately described by the dispersion model, as it\ncan be seen in Fig.~\\ref{fig:iso_agn}. Nevertheless, the convolution\nof Eq.~(\\ref{eq:ximodel}) significantly helps in reducing the\nsystematic error on $f\\sigma_8$, even more than what we found when\nanalysing the galaxy sample, as evident by comparing\nFigs.~\\ref{fig:iso_gal} and \\ref{fig:iso_agn}. Also in this case, it\nis possible to get almost unbiased constraints on $f\\sigma_8$ at all\nredshifts and for the different selections considered, provided that\nthe analysis is restricted at $r_\\perp^{\\rm min}\\gtrsim15$ $h^{-1}\\,\\mbox{Mpc}$\\, and\n$r_\\parallel^{\\rm max}\\lesssim15$ $h^{-1}\\,\\mbox{Mpc}$\\,. Specifically, we considered\nhere AGN catalogues selected in BH mass, $M_{\\rm BH}$, bolometric\nluminosity, $L_{\\rm bol}$, and Eddington factor, $f_{\\rm Edd}$, as\nreported in Table \\ref{tab:table3}.  Only the AGN samples with the\nlower number of objects analysed here (cyan points in\nFig.~\\ref{fig:errfs8_agn}) are not large enough to provide robust\nconstraints on $f\\sigma_8$.\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe results presented in this paper extend the previous work by\n\\citet{bianchi2012}, who analysed mock samples of DM haloes at\n$z=1$. One of the main differences with respect to that work is that\nhere we used galaxy, cluster and AGN simulated catalogues, that are\nmore directly comparable to observational samples. Moreover, we\ninvestigated the systematic errors of $f\\sigma_8$, instead of $\\beta$,\nand extended the analysis at different redshifts and sample\nselections. The use of these simulated objects as {\\em test particles}\nprovides a better treatment of non-linearities in small-scale\ndynamics, that translates to a more realistic description of the\nfingers-of-God in the redshift-space galaxy and AGN clustering, as it\ncan be seen in Figs.~\\ref{fig:iso_gal} and \\ref{fig:iso_agn}. On the\nother hand, the selected cluster samples are more similar to the DM\nhalo samples analysed by \\citet{bianchi2012}, as expected, with\nfingers-of-God anisotropies almost absent, as shown in\nFig.~\\ref{fig:iso_cl}. Nevertheless, the {\\small Magneticum} samples\nprovide cluster properties that allowed us to apply more realistic\nselections.\n\nOne of the main results found by \\citet{bianchi2012} is that, when\nusing the dispersion model, the systematic errors on the linear growth\nrate are positive for DM haloes with masses lower than $\\sim10^{13}\nh^{-1}\\,{\\mbox M_\\odot}$, and negative otherwise \\citep[see\n  also][]{okumura2011, marulli2012b}. Our findings reinforce these\nconclusions. Indeed, regardless of the specific selections considered,\nthe overall trends of our systematic errors depend ultimately on the\nmass of the DM haloes hosting the selected tracers.\n\nMoreover, extending the \\citet{bianchi2012} analysis, we investigated\nhow the systematic errors depend on the comoving scales\nconsidered. Our main finding is that it is possible to substantially\nreduce the systematic errors at all redshifts and selections by\nrestricting the analysis in a proper subregion of the\n$r_\\perp-r_\\parallel$ plane. The latter depends on the properties of\nthe selected objects, specifically on the mass of the host halo, due\nto the difficulties in modelling the non-Gaussian nature of the\nvelocity probability density function of the tracers.\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nClustering anisotropies in redshift-space are one of the most\neffective probes to test Einstein's General Relativity on large\nscales. Though only galaxy samples have been considered so far for\nthese analyses, other astronomical tracers, such as clusters and AGN,\nwill be used in the next future, aimed at maximising the dynamic and\nredshift ranges where to constrain the linear growth rate of cosmic\nstructures. In particular, clusters of galaxies can be efficiently\nused to probe the low redshift Universe. Their large bias and low\nvelocity dispersion at small scales make them optimal tracers for\nlarge scale structure analyses \\citep{sereno2015, veropalumbo2014,\n  veropalumbo2016}. On the other hand, AGN samples can be exploited to\npush the GR test on larger redshifts.\n\nIn this work, we made use of both galaxy, cluster and AGN mock samples\nextracted from the {\\em Magneticum}, a cosmological hydrodinamic\nsimulation of the standard $\\Lambda$CDM Universe, to investigate the\naccuracy of the widely used dispersion model as a function of scale\nand sample selection. Instead of analysing the clustering multipoles,\nwe preferred to extract $f\\sigma_8$ constraints from the monopole\nonly, or from the full 2D anisotropic correlation $\\xi(s_\\perp,s_\\parallel)$\\,. This allowed\nus to explore the $r_\\perp-r_\\parallel$ plane, aimed at finding the\noptimal range of scales to minimise systematic uncertainties.\n\nThe main results of this analysis are the following.\n\n\\begin{itemize}\n\n\\item It is possible to get almost unbiased constraints on $f\\sigma_8$\n  by modelling the large-scale monopole of the two-point correlation\n  function with the Kaiser limit of the dispersion model\n  (Eq.~\\ref{eq:xi0_2}), provided that the linear bias factor\n  $b\\sigma_8$ is known. The latter has to be estimated at sufficiently\n  large scales, to avoid non-linearities. With real data, the linear\n  bias can be estimated either with the deprojection technique, or\n  from the three-point correlation function, or by combining\n  clustering and lensing measurements (see the discussion in\n  \\S\\ref{sec:methodology}).\n\n\\item When higher multipoles are considered, $f\\sigma_8$ and\n  $b\\sigma_8$ can be jointly constrained, and the statistical error on\n  $f\\sigma_8$ is substantially reduced \\citep[see\n  e.g.][]{delatorre2013b, petracca2016}. However, systematic errors\n  can arise if non-linearities are not treated accurately, such as\n  with the dispersion model\n  (Eqs.~\\ref{eq:ximodellin}-\\ref{eq:fvexp}). On the other hand, the\n  analysis shown in this paper demonstrates that it is still possible\n  to reduce the systematic errors on $f\\sigma_8$ if the range of\n  scales used to model RSD is chosen appropriately. The latter depends\n  on the properties of the tracers. We do not find however any clear\n  prescription to relate these scale ranges to the sample properties.\n\n\\item If the linear growth rate is estimated from the RSD of either\n  galaxies or AGN, the value of $f\\sigma_8$ will be underestimated at\n  $z\\lesssim1$, and overestimated at larger redshifts, when the fit is\n  performed in the fixed range of scales $3<r_\\perp,\n  r_\\parallel[h^{-1}\\,\\mbox{Mpc}]<35$. With $r_\\perp^{\\rm min}\\sim15$\n  $h^{-1}\\,\\mbox{Mpc}$\\, and $r_\\parallel^{\\rm max}\\sim20$ $h^{-1}\\,\\mbox{Mpc}$\\,, we get almost\n  unbiased constraints. However, the proper values of $r_\\perp^{\\rm\n    min}$ and $r_\\parallel^{\\rm max}$ depend on sample selection, and\n  should be fixed differently for each sample analysed, calibrating\n  the analysis using specific mock catalogues.\n\n\\item If the RSD of galaxy groups and clusters are modelled in the\n  range of scales $3<r_\\perp, r_\\parallel[h^{-1}\\,\\mbox{Mpc}]<35$, the\n  value of $f\\sigma_8$ results severely overestimated at all\n  redshifts. The results at $z=1$ appear in overall agreement with\n  what found by \\citet{bianchi2012}.  The systematic error can be\n  reduced if the fit is performed at larger scales, as expected due to\n  the low number of cluster pairs at small separations. Specifically,\n  with $r_\\perp^{\\rm min}\\sim20$ $h^{-1}\\,\\mbox{Mpc}$\\,, we get unbiased constraints,\n  but only for the densest group samples. The paucity of the most\n  massive cluster samples does not allow us to obtain robust results.\n\n\\end{itemize}\n\nTo summarise, as the dispersion model cannot accurately describe the\nnon-linear motions at small scales, the constraints on $f\\sigma_8$ can\nbe biased. At the present time, it is difficult to include model\nuncertainties in this kind of analyses, and even more to improve the\nRSD modelling at small scales \\citep[see e.g.][and references\n  therein]{delatorre2012, bianchi2015}. Instead of trying to improve\nthe model, in this analysis we investigated how the systematic errors\non the linear growth rate caused by model uncertainties depend on the\ncomoving scales considered.  As we have shown, in order to reduce\nthese systematics it is necessary to restrict the analysis in a proper\nsubregion of the $r_\\perp-r_\\parallel$ plane. More specifically, we\nfound that it is enough to choose a proper value for $r_\\perp^{\\rm\n  min}$ and $r_\\parallel^{\\rm max}$. Indeed, as we verified, changing\nalso $r_\\perp^{\\rm max}$ and $r_\\parallel^{\\rm min}$ does not affect\nsignificantly the results.\n\nOverall, we find that, in order to reduce the systematics in the\n$f\\sigma_8$ measurements, it is more convenient simply to not consider\nthe small scales where non-linear effects distort significantly the\nclustering shape, rather than to try modelling them with the empirical\ndescription provided by the dispersion model given by\nEq.~(\\ref{eq:ximodel}).  The drawback of this approach is to\ninevitably increase the statistic error on $f\\sigma_8$. Moreover, the\n$r_\\perp-r_\\parallel$ region to be selected depends on the properties\nof the tracers, and should be determined using suitable mock\ncatalogues. On the other hand, without a sufficiently accurate\nmodelisation of non-linear dynamics at small scales, this approach is\nthe only one possible to get unbiased constraints.\n\nThis can have a non-negligible impact on multiple tracer analyses\n\\citep[see e.g.][and references therein]{mcdonald2009, blake2013,\n  marin2015} if the RSD are modelled at small non-linear scales with\nthe dispersion model. To avoid different systematic errors on the\nlinear growth rate estimated from the small-scale RSD of different\ntracers, the $r_\\perp-r_\\parallel$ plane analysed should be different\nfor each sample. On the other hand, in multiple tracer techniques that\nrequire to analyse different samples at the same scales, it is\nnecessary to limit the analysis at large enough scales where\nsystematic errors are lower than statistical errors for all the\ndifferent tracers.  And the same when joining together different\nredshifts, where RSD constraints can be differently biased.\n\n\\begin{figure}\n  \\includegraphics[width=0.49\\textwidth]{nz.ps}\n  \\caption{The number of mock galaxies ({\\em red dots}), clusters\n    ({\\em blue squares}) and AGN ({\\em green diamonds}) as a function\n    of redshift, for the three selections: Gs1, Cl1, Af1 (see Tables\n    \\ref{tab:table1}, \\ref{tab:table2} and \\ref{tab:table3}).}\n  \\label{fig:nz}\n\\end{figure}\n\n\n\n\\section*{Acknowledgments}\n\nWe acknowledge the support from the grants ASI\/INAF n.I\/023\/12\/0\n`Attivit\\`a relative alla fase B2\/C per la missione Euclid' and MIUR\nPRIN 2010-2011 `The dark Universe and the cosmic evolution of baryons:\nfrom current surveys to Euclid'. L.M. acknowledges financial\ncontributions from contracts and PRIN INAF 2012 `The Universe in the\nbox: multiscale simulations of cosmic structure'. K.D. acknowledges\nthe support by the DFG Cluster of Excellence `Origin and Structure of\nthe Universe'. We are especially grateful for the support by\nM.~Petkova through the Computational Center for Particle and\nAstrophysics (C$^2$PAP). Computations have been performed at the\n`Leibniz-Rechenzentrum' with CPU time assigned to the Project\n`pr86re', as well as at the `Rechenzentrum der Max-Planck-\nGesellschaft' at the `Max-Planck-Institut f\\\"ur Plasmaphysik' with CPU\ntime assigned to the `Max-Planck-Institut f\\\"ur Astrophysik'.\nInformation on the {\\it Magneticum Pathfinder} project is available at\nhttp:\/\/www.magneticum.org. Finally, we thank the referee for several\nuseful suggestions that improved the presentation of our results.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHigh energy diffractive scattering can be well described by the\nPomeron exchange mechanism \\cite{pom1,pom2,pom3,pom4,pom5}. Within\nthe framework of quantum chromodynamics(QCD), the\nBalitskii-Fadin-Kuraev-Lipatov(BFKL) Pomeron\n\\cite{bfkl1,bfkl2,bfkl3}, the so-called hard Pomeron, is viewed as\nthe Green's function of two interacting Reggeized gluons with vacuum\nquantum number, which is evaluated by resuming the leading energy\nlogarithms in perturbative QCD(pQCD) applicable regime, e.g., all\n$(\\alpha_s \\text{ln} s)^n $ terms in leading logarithmic\napproximation(LLA) and all $\\alpha_s (\\alpha_s \\text{ln} s)^n $\nterms in the next-to-leading approximation(NLA). People noticed that\nthe virtual photon-photon scattering is an ideal process\n\\cite{virpho1,virpho2} to testify the BFKL predictions and hence the\npQCD calculation reliability, which is highly expected in the study\nof strong interaction physics. The cross section\n$\\sigma_{tot}^{\\gamma^{*} \\gamma^{*}}$ is calculable in the BFKL\napproach with relatively high precision and can be realized in\nexperiment through a measurement of the reaction $e^{+}\ne^{-}\\rightarrow e^{+} e^{-} + X$ by tagging the outgoing leptons at\nhigh energy electron-positron colliders, like LEP and more\noptimistically CEPC \\cite{cepc} or ILC \\cite{ilc} in the future.\nMoreover, the recently proposed electron-ion colliders like EIC\n\\cite{eic} and EicC \\cite{eicc} may also be good places to study the\nsingle diffractive process and hence on small-x physics at lower\nenergies.\n\nTill now, the leading order(LO) BFKL predictions for $e^{+}\ne^{-}\\rightarrow e^{+} e^{-} + X$ process have been confronted to\nthe LEP data \\cite{lep1,lep2,lep3}, in which the experimental\nmeasurement is found above the two-gluon-exchange model result,\nwhile below the leading order BFKL Pomeron prediction. Now that the\nnext-to-leading order(NLO) corrections to BFKL kernel is ready and\nnumerically big and negative \\cite{bfkl3,nlb2}, the higher order\ncorrections may lower the the BFKL Pomeron exchange calculation and\napproach to the experimental measurement. However, in the NLO BFKL\ncalculations, there is still a task remaining, calculating the NLO\ncorrections to the coupling of the BFKL Pomeron and the external\nphotons, which is called photon impact factor\\cite{im1,im2,im3}.\n\nThe photon impact factor can be obtained by calculating the\ndiscontinuity of the process $\\gamma^{*}+~reggon \\longrightarrow\n\\gamma^{*}+~reggon$ process in light of the optical theorem, which\ntells that the discontinuity is proportional to the scattering\namplitude, $\\gamma^{*}+~reggeon\\longrightarrow Q\\bar{Q}$, squared.\nThe reggon, is identified with the elementary t-channel gluons. The\nNLO corrections to the photon impact factor with massless internal\nquarks were given in Ref.\\cite{qiao}, while here the $Q(\\bar{Q})$\nare massive quarks, charm or bottom quarks. In the NLO calculation,\nin analogous to Ref.\\cite{qiao}, we calculate the NLO corrections to\n$\\gamma^{*}+~reggeon\\longrightarrow Q\\bar{Q}$ amplitude on for\nexample left-hand side of the discontinuity line, and the leading\n$Q\\bar{Q}g$ intermediate state on both sides of the discontinuity\nline. The calculation of NLO QCD correction hence proceeds in three\nsteps: (i) calculate the NLO corrections to the\n$\\gamma^{*}+~reggeon\\longrightarrow Q\\bar{Q}$ vertex. (ii) calculate\nthe leading order vertex $\\gamma^{*}+~reggeon \\longrightarrow\nQ\\bar{Q}g$, and (iii) carry out the integration over the phase space\nof the intermediate states. In fact, here the calculation procedure is similar to the massless case performed in Ref.\\cite{qiao}. In this paper, as in \\cite{qiao}, we give the results of\nthe first step, in which the vertex is extracted from the scattering\nprocess $\\gamma^{*}+~q\\longrightarrow Q\\bar{Q}+q$ in the high energy\nlimit.\n\n\\section{Technical preliminaries}\\label{II}\n\n\\begin{figure}[!hbtp]\n\\centering\n\\includegraphics[scale=0.8]{kinematics.eps}%\n\\caption{\\small Kinematics of the process $\\gamma^* + q \\to Q\\bar{Q}\n+ q$. } \\label{kinematics}\n\\end{figure}\n\n\nThe kinematics of the $\\gamma^* + q \\to Q\\bar{Q} + q$ process is\nillustrated Fig.\\ref{kinematics}, where $q$ and $p$ are the\n4-momenta of photon and incident quark respectively. In the\ncalculation, $\\varepsilon$ represents for the polarization\nvector of the photon, $s$ for the collision energy of the $\\gamma^*\nq$ system, and $m$ for the mass of the heavy quark. The Lorentz\ninvariants appeared include $s=(q+p)^2$, $Q^2=-q^2$, $t_{a}=k^2$,\n$t_{b}=(q-k-r)^2$, $M^2=(q+r)^2$, $t=r^2$, and $x= Q^2\/2p\\cdot q$,\nthe Bjorken scaling variable.\n\nThe momenta $k$ and $r$ can be decomposed in the Sudakov form, i.e,\n\n\\begin{eqnarray}\nk &=&\\alpha q^\\prime+\\beta p+k_{\\bot}\\ , \\\\\nr &=& \\frac{t}{s} q^\\prime -\\frac{t_{a}+t_{b}-2m^2}{s} p + r_{\\bot}\\\n,\n\\end{eqnarray}\nwith $q^\\prime =q+xp$, ${q^{\\prime}}^2=p^2=0$ and $\\beta s\n=\\frac{k_{\\bot}^2+m^2}{1-\\alpha}-Q^2$.\n\n\\begin{figure}[tb]\n  \\begin{center}\n    \\epsfig{file=graphs,width=6in}\n  \\end{center}\n  \\caption{Feynman diagrams for the process $\\gamma^* + q \\to Q\\bar{Q} +\n    q$.}\n  \\label{graphs}\n\\end{figure}\n\nThe typical Feynman diagrams for NLO corrections are shown in\nFig.\\ref{graphs}. Of these diagrams, except Fig.\\ref{graphs}.14, the\nupper part quark-antiquark exchange diagrams are implied. For the\ncolor octet t-channel configuration, the sum of all diagrams has to\nbe antisymmetric if we interchange quark and antiquark: $k\n\\rightarrow q-k-r$, $\\lambda\\rightarrow\\lambda^{\\prime}$, where\n$\\lambda$, $\\lambda^{\\prime}$ are the helicities of the quark and\nantiquark respectively. In particular, the \"box\" graph shown in Fig. 3.14\nhas to be antisymmetric by itself. Throughout the calculation,\nFeynman gauge is employed, and for the t-channel gluons the metric\ntensor is decomposed into \\begin{eqnarray}\ng_{\\mu\\nu}=\\frac{2}{s}(p_{\\mu}q^{\\prime}_{\\nu}+p_{\\nu}q^{\\prime}_{\\mu})\n+g^{\\bot}_{\\mu\\nu}\\end{eqnarray} as usual. Note, in practical calculations, the\ntransverse term is not taken into account, whose contributions are\nsuppressed by powers of $s$. We use the helicity formalism as well,\nand then our results can be expressed in\nterms of the following matrix elements similar to \\cite{qiao}, i.e.,\n\\begin{eqnarray}\n\\label{HTdef} H_{T}^a&=& \\bar{u}(k+r, \\lambda) \\not\\!{p} \\not\\!{k}\n\\not\\!{\\varepsilon}\\;\n\\lambda^av(q-k,\\lambda') \\; , \\\\\n\\label{HEdef} H_{\\varepsilon}^a&=& \\bar{u}(k+r, \\lambda)\n\\not\\!{\\varepsilon}\\;  \\lambda^a\nv(q-k,\\lambda') \\; , \\\\\n\\label{HKdef} H_{k}^a&=& \\bar{u}(k+r, \\lambda) \\not\\!{k}\\; \\lambda^a\nv(q-k,\\lambda')\n\\; , \\\\\n\\label{HPdef} H_{p}^a&=& \\bar{u}(k+r, \\lambda) \\not\\!{p}\\; \\lambda^a\nv(q-k,\\lambda')\n\\; , \\\\\n\\label{HKEdef} H_{k\\varepsilon}^a&=& \\bar{u}(k+r, \\lambda)\n\\not\\!{k}\\not\\!{\\varepsilon}\\; \\lambda^a v(q-k,\\lambda')\n\\; , \\\\\n\\label{HPEdef} H_{p\\varepsilon}^a&=& \\bar{u}(k+r, \\lambda)\n\\not\\!{p}\\not\\!{\\varepsilon}\\; \\lambda^a v(q-k,\\lambda')\n\\; , \\\\\n\\label{HPKdef} H_{pk}^a&=& \\bar{u}(k+r, \\lambda)\n\\not\\!{p}\\not\\!{k}\\; \\lambda^a v(q-k,\\lambda')\n\\; , \\\\\n\\label{HMdef} H_{m}^a&=& \\bar{u}(k+r, \\lambda) \\; \\lambda^a\nv(q-k,\\lambda'). \\end{eqnarray} Here $\\lambda^a$ is the generators of the\ncolor group. Note that the last four helicity matrix elements do not exist in \\cite{qiao},\nwhich always come up with the factor of $m$.\n\nTake high energy limit in the calculation, where\n\\begin{eqnarray} t,\nQ^2, t_a, t_b, M^2, m^2 \\ll s,\n\\end{eqnarray}\nand do not impose any restrictions on the remaining invariants, we then obtain the\nfollowing amplitude\n\\begin{eqnarray} \\label{eq:16} T = g^2\\;T^{(0)} + g^4\\;T^{(1)},\n\\end{eqnarray}\nwith\n\\begin{eqnarray} T^{(0)} = \\Gamma_{\\gamma^* \\to q\\bar{q}}^{(0),a}\\; \\frac{2s}{t}\n\\; \\Gamma_{qq}^{(0),a},\n\\end{eqnarray} and \\begin{eqnarray} \\Gamma_{qq}^{(0),a}&=&\n\\frac{1}{s} \\bar{u}(p-r, \\,\\lambda^{\\prime}) \\not\\!{q}' \\lambda^a\nu(p,\\,\\lambda)\\ . \\end{eqnarray}\nThe only unknown piece of the NLO amplitude is\n$T^{(1)}$, which corresponds to the processes in Fig.\\ref{graphs}.1 - \\ref{graphs}.14\nand will be calculated analytically in the following.\n\n\\section{Calculation methods}\n\nThe method we use in our calculation will be briefly described\nin the following before presenting the explicit results.\n\nIn our calculation the MATHEMATICA package FeynArts \\cite{feynarts}\nis applied to generate the Feynman diagrams and amplitudes that are\nrelevant to our process. We use FeynCalc \\cite{feyncalc} to\ncalculate and simplify the amplitudes. Color in the t-channel is\nprojected onto the antisymmetric octet as done in \\cite{qiao}. After\nsimplifying the Dirac matrix and employing the Dirac equation of motion, we\nexpress the amplitudes as the combination of helicity matrix\nelements and Passarino-Veltman integrals. For integrals that contain\ndivergences, we separate the divergence part from the finite one.\nThe high energy limit is taken through out our calculation, i.e. $t,\nQ^2, t_a, t_b, M^2, m^2 \\ll s$.\n\nIn this work an extra scale $m$ exists in comparison to \\cite{qiao}, and\nit is too lengthy and redundancy to turn all the Passarino-Veltman\nintegrals into scalar integrals or logarithms and dilogarithms\nfunctions. For amplitudes that do not contain divergences and\nhigh energy scale $s$, we express them as the combination of\nPassarino-Veltman integrals and helicity matrix elements. The\nnumerical values of these integrals will be evaluated\nby Looptools \\cite{looptools}. For the pentagon diagram\nFig.\\ref{graphs}.13, the reduction method given in\n\\cite{denner} is employed to reduce the corresponding integrals to box integrals.\nIn the end we find our result agrees with that in \\cite{qiao} when taking the $m\\rightarrow 0$ limit.\n\n\\section{Analytic Results}\n\nThe NLO amplitudes in $T^{(1)}$ of our concern can be expressed in terms of\ndifferent diagrams:\n\\begin{eqnarray}\nT^{(1)}\n&=&\\sum_{i=1}^{13}(A_{i}+\\bar{A}_{i})+A_{14}\\ .\n\\end{eqnarray}\nHere, the subscripts $i$ denote diagrams in Fig.\\ref{graphs}, and the amplitudes\n$\\bar{A}_{i}$ represent those with quark-antiquark interchange in ${A}_{i}$.\n\n\\subsection{Results of two- and three-point diagrams}\n\nWe categorize the diagrams similar to \\cite{qiao}. The results of\nFigs.\\ref{graphs}.1 and its conjugate one are given below, which\nare split into divergent and finite parts in $\\mathrm{\\overline{MS}}$ scheme:\n\\begin{align}\nA_{1}\\,&=\\, \\frac{N_c}{2}\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}\n\\bigg\\{\\frac{3(H_{T}^a+m H_{p\\varepsilon}^a)c_\\Gamma}{\\epsilon}\n+H_{T}^a\n\\big[6m^2C_0(1)+12C_{00}(1)_{fin}\\nonumber\\\\\n&+(5m^2-2t+3t_a)C_1(1)+(3m^2-t+t_a)C_2(1)+2t_aC_{11}(1)\n+2(m^2-t+t_a)C_{12}(1) \\nonumber\\\\\n&+2m^2C_{22}(1)-2\\big]+\\alpha s\nH_{\\varepsilon}^a\\big[2m^2C_1(1)+(3m^2-t_a)C_2(1)+2t_a\nC_{11}(1)+2m^2C_{22}(1)\\nonumber\\\\\n&+2(t_a+m^2)C_{12}(1) \\big]+m\nH_{p\\varepsilon}^a\\big[3(t_a+m^2)C_0(1)+12C_{00}(1)_{fin}+2(m^2-t+3t_a)C_1(1)\\nonumber\\\\\n&+ (3m^2-2t+5t_a)C_2(1)+2t_aC_{11}(1)\n+2(m^2-t+t_a)C_{12}(1)+2m^2C_{22}(1)-2 \\big]\\nonumber\\\\\n&+2\\alpha s\nmH_{k\\varepsilon}^a\\big[C_1(1)+C_2(1)+C_{11}(1)+2C_{12}(1)+C_{22}(1)\n\\big]\\bigg\\}\\ .\n\\end{align}\nHere $C_i(1)=C_i(t_a,t,m^2,m^2,0,0)$ with $i$ the coefficient index.\nThe definitions of all the loop integrals such as $C_0$,\n$C_{ij}$,$D_0$, $D_{i,j,k}$ are standard and given in the Appendix\nfor reference. The subindex $fin$ means the finite part of the loop\nintegrals. Note, the above result does not contain any infrared\ndivergences, while the ultraviolet divergence remains only in\n$C_{00}(1)$, which can be properly regulated. Numerical evaluation\nof the finite part may be performed by means of LoopTools\n\\cite{looptools} in $\\mathrm{\\overline{MS}}$ scheme. The $C_i$\nfunctions can be reduced into a certain scalar one-loop integrals\nand then expressed as the combination of logarithm and dilogarithm\nfunctions. Nevertheless, such kind of reduction may yield a lengthy\nresult and hard to read. Due to this reason, we do not perform\nfurther reduction on $C_i$ functions, its numerical evaluations are\ncarried by LoopTools directly.\n\nThe result of the conjugate diagram of Figs.\\ref{graphs}.1 can be expressed as\n\\begin{align}\n\\bar{A}_{1}\\,&=\\,\\frac{N_c}{2}\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2}{t(t_b-m^2)}\n\\Gamma_{qq}^{(0),a}\\bigg\\{\\frac{3(H_{T}^a+m\nH_{p\\varepsilon}^a-s H_{\\varepsilon}^a +2(H_{k}^a-m\nH_{m}^a)\\varepsilon\\cdot p+2H_{p}^a\\varepsilon\\cdot\nr)c_\\Gamma}{\\epsilon}\\nonumber\\\\\n&+(H_{T}^a+2H_{p}^a\\varepsilon\\cdot r + 2H_{k}^a\\varepsilon\\cdot\np)\\big[6m^2C_{0}(\\bar{1})+12C_{00}(\\bar{1})_{fin}+2(3m^2-t+t_b)C_{1}\n(\\bar{1})\\nonumber\\\\\n&+(5m^2-2t+3t_b^2)C_{2}(\\bar{1})+2m^2C_{11}(\\bar{1})\n+2(m^2-t+t_b)C_{12}(\\bar{1})+2t_bC_{22}(\\bar{1})-2\\big]\\nonumber\\\\\n&+sH_{\\varepsilon}^a\\big[-6m^2C_{0}(\\bar{1})+(3(\\alpha-3)m^2\n+2t-(1+\\alpha)t_b)C_{1}(\\bar{1})-12C_{00}(\\bar{1})_{fin}\\nonumber\\\\\n&+((2\\alpha-7)m^2+2t-3t_b)\nC_{2}(\\bar{1})+2(\\alpha-2)m^2C_{11}(\\bar{1})\n+2(\\alpha-2)t_bC_{22}(\\bar{1})\\nonumber\\\\\n&+2((\\alpha-2)m^2+t+(\\alpha-2)t_b)C_{12}(\\bar{1})+2\\big]\\nonumber\\\\\n&+2(\\alpha-1)s m (H_{k\\varepsilon}^a+H_{m}^a \\varepsilon\\cdot r)\n\\big[C_{1}(\\bar{1})+C_{2}(\\bar{1})+C_{11}(\\bar{1})\n+2C_{12}(\\bar{1})+C_{22}(\\bar{1})\\big]\\nonumber\\\\\n&+m( H_{p\\varepsilon}^a-2\\varepsilon\\cdot p\nH_{m}^a)\\big[(9m^2-3t_b)C_{0}(\\bar{1})+12C_{00}(\\bar{1})_{fin}+(9m^2-2t-t_b)C_{1}\n(\\bar{1})\\nonumber\\\\\n&+2(4m^2-t)C_{2}(\\bar{1})+2m^2C_{11}(\\bar{1})\n+2(m^2-t+t_b)C_{12}(\\bar{1})+2t_bC_{22}(\\bar{1}) -2\\big] \\bigg\\}\\ .\n\\end{align}\nHere, $C_i(\\bar{1})=C_i(m^2,t,t_b,m^2,0,0)$.\n\nThe calculation procedure of Fig.\\ref{graphs}.4 is similar to that\nof Fig.1, and its result reads:\n\\begin{align}\nA_{4}\\,&=\\,-\\frac{1}{2N_c}\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{4}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}\n\\bigg\\{\\frac{(H_{T}^a+mH_{p\\varepsilon}^a)c_\\Gamma}{2\\epsilon}+\nH_{T}^a\\big[(m^2-t+t_a)C_{0}(4)\\nonumber\\\\\n&+(3m^2-t+t_a)C_{1}(4)+(2m^2-t+2t_a)C_{2}(4)+2C_{00}(4)_{fin}+m^2C_{11}(4)\\nonumber\\\\\n&+(m^2-t+t_a)C_{12}(4) +t_aC_{22}(4)-1\\big]+ \\alpha s\nH_{\\varepsilon}^a\\big[(m^2-t_a)C_{0}(4)-t_aC_{1}(4)\\nonumber\\\\\n& +(m^2-2t_a)2C_{2}(4)-m^2C_{11}(4)-(t_a+m^2)C_{12}(4)-t_aC_{22}(4)\n\\big]\\nonumber\\\\\n&+m H_{p\\varepsilon}^a\\big[(2t_a-t)C_{0}(4)+(2m^2-t+2t_a)C_{1}(4)\n+(m^2-t+3t_a)C_{2}(4)\\nonumber\\\\\n&+2C_{00}(4)_{fin}+m^2C_{11}(4)+(m^2-t+t_a)C_{12}(4)+t_aC_{22}(4)-1\\big]\\nonumber\\\\\n&-\\alpha s m\nH_{k\\varepsilon}^a\\big[C_{1}(4)+C_{2}(4)+C_{11}(4)+2C_{12}(4)+C_{22}(4)\n\\big]\\bigg\\}\\ .\n\\end{align}\nwhere $C_i(4)=C_i(m^2,t,t_a,0,m^2,m^2)$.\n\nThe result of the conjugate diagram of Fig.\\ref{graphs}.4 is:\n\\begin{align}\n\\bar{A}_{4}\\,&=\\,-\\frac{1}{2N_c}\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{4}{t(t_b-m^2)}\\Gamma_{qq}^{(0),a}\n\\bigg\\{\\nonumber\\\\\n&\\frac{(H_{T}^a+mH_{p\\varepsilon}^a-s H_{\\varepsilon}^a\n+2(H_{k}^a-mH_{m}^a)\\varepsilon\\cdot p+2H_{p}^a\\varepsilon\\cdot\nr)c_\\Gamma}{2\\epsilon}\\nonumber\\\\\n&+H_{T}^a\\big[(m^2-t+t_b)C_{0}(\\bar{4})+(3m^2-t+t_b)C_{1}(\\bar{4})\n+(2m^2-t+2t_b)C_{2}(\\bar{4})\\nonumber\\\\\n&+2C_{00}(\\bar{4})_{fin}+m^2C_{11}(\\bar{4})+(m^2-t+t_b)C_{12}(\\bar{4})\n+t_bC_{22}(\\bar{4})-1\\big]\\nonumber\\\\\n&+sH_{\\varepsilon}^a\\big[((\\alpha-2)m^2+t-\\alpha t_b)C_{0}(\\bar{4})\n+(t-3m^2-\\alpha t_b)C_{1}(\\bar{4})\\nonumber\\\\\n&+((\\alpha-3)m^2+t-2\\alpha\nt_b)C_{2}(\\bar{4})-2C_{00}(\\bar{4})_{fin}-\\alpha m^2C_{11}(\\bar{4})\\nonumber\\\\\n& +(t-\\alpha m^2-\\alpha t_b)C_{12}(\\bar{4})-\\alpha\nt_bC_{22}(\\bar{4})+1\\big]\\nonumber\\\\\n&+2(H_{k}^a \\varepsilon\\cdot p+H_{p}^a\\varepsilon\\cdot\nr)\\big[(m^2-t+t_b)C_{0}(\\bar{4})+2C_{00}(\\bar{4})_{fin}\\nonumber\\\\\n&+(3m^2-t+t_b)C_{1}(\\bar{4})+(2m^2-t+2t_b)C_{2}(\\bar{4})\n+m^2C_{11}(\\bar{4})\\nonumber\\\\\n&+(m^2-t+t_b)C_{12}(\\bar{4})+t_bC_{22}(\\bar{4})-1\n\\big]\\nonumber\\\\\n&+m(H_{p\\varepsilon}^a+2H_{m}^a\\varepsilon\\cdot\np)\\big[(2m^2-t)C_{0}(\\bar{4})+(4m^2-t)C_{1}(\\bar{4})+(3m^2-t+t_b)C_{2}(\\bar{4})\\nonumber\\\\\n&+2C_{00}(\\bar{4})_{fin}+m^2C_{11}(\\bar{4})+(m^2-t+t_b)C_{12}(\\bar{4})\n+t_bC_{22}(\\bar{4})-1\\big]\\nonumber\\\\\n&+(1-\\alpha)m(H_{k\\varepsilon}^a+2H_{m}^a\\varepsilon\\cdot\nr)\\big[C_{1}(\\bar{4})+C_{2}(\\bar{4})+C_{11}(\\bar{4})+2C_{12}(\\bar{4})+C_{22}(\\bar{4})\n\\big] \\bigg\\}.\n\\end{align}\nwith $C_i(\\bar{4})=C_i(m^2,t,t_b,0,m^2,m^2)$.\n\nThe Fig.\\ref{graphs}.5 and Fig.\\ref{graphs}.6 diagrams\nbelong to the NLO corrections to the lower vertex\n$\\Gamma_{qq}^{(1),a}$. Their results are\n\\begin{align}\nA_{5}\\,&=\\,-N_c\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}\n(H_{T}^a+mH_{p\\varepsilon}^a)c_\\Gamma\\bigg\\{\\frac{(-t)^{-\\epsilon}}{2\\epsilon}+1\\bigg\\},\n\\end{align}\nand\n\\begin{align}\nA_{6}\\,&=\\,\\frac{1}{2N_c}\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}\n(H_{T}^a+mH_{p\\varepsilon}^a)c_\\Gamma(-t)^{-\\epsilon}\n\\bigg\\{\\frac{2}{\\epsilon^2}+\\frac{3}{\\epsilon}+8\\bigg\\}\\ .\n\\end{align}\nIn the zero mass limit, it is obvious that these two amplitudes\nare just the eqs. $(35)$ and $(36)$ in \\cite{qiao}.\n\nFigs.\\ref{graphs}.7 - \\ref{graphs}.9 contribute to both\nthe upper and lower vertices, and we find\n\\begin{align}\nA_{7+8}\\,&=\\,\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2N_cc_\\Gamma(-t)^{-\\epsilon}}{t(t_a-m^2)}\n\\Gamma_{qq}^{(0),a}(H_{T}^a+mH_{p\\varepsilon}^a)\\bigg(\\frac{5}{3\\epsilon}+\\frac{31}{9}\n\\bigg),\n\\end{align}\nand\n\\begin{align}\nA_{9}\\,&=\\,-n_f\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2c_\\Gamma(-t)^{-\\epsilon}}{t(t_a-m^2)}\n\\Gamma_{qq}^{(0),a}(H_{T}^a+mH_{p\\varepsilon}^a)\\bigg(\\frac{2}{3\\epsilon}\n+\\frac{10}{9}\\bigg)\\ .\n\\end{align}\n\nThe results of conjugate diagrams of Fig.\\ref{graphs}.5 -\n\\ref{graphs}.9 can readily be obtained by substituting\n$(H_{T}^a+mH_{p\\varepsilon}^a)$ with\n$(H_{T}^a+mH_{p\\varepsilon}^a-sH_{\\varepsilon}^a+2(H_{k}^a-mH_{m}^a)\\varepsilon\\cdot\np+2H_{p}^a\\varepsilon\\cdot r)$  and making the replacement\n$t_a\\rightarrow t_b$.\n\nThe result of vertex correction diagram Fig.\\ref{graphs}.10 can be\nexpressed as follows:\n\\begin{align}\nA_{10}\\,&=\\,-\\frac{C_F e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{4}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}\n\\bigg\\{\\frac{-(H_{T}^a+mH_{p\\varepsilon}^a)c_\\Gamma}{2\\epsilon}+\nH_{T}^a\\big[m^2C_{0}(10)+Q^2C_{2}(10)\\nonumber\\\\\n&+(m^2+Q^2+t_a)C_{1}(10)-2C_{00}(10)_{fin}-m^2C_{11}(10)\n+(Q^2+t_a-m^2)C_{12}(10)\\nonumber\\\\\n&+Q^2C_{22}(10)+1\\big]+mH_{p\\varepsilon}^a\\big[m^2C_{0}(10)\n+(Q^2+2t_a)C_{1}(10)+Q^2C_{2}(10)\\nonumber\\\\\n&-2C_{00}(10)_{fin}-m^2C_{11}(10)+(Q^2-m^2+t_a)C_{12}(10)+Q^2C_{22}(10)+1\\big]\\nonumber\\\\\n&+2\\varepsilon\\cdot\nk\\big[(m(2mH_{p}^a+H_{pk}^a)-t_aH_{p}^a)C_{1}(10)\n+m(mH_{p}^a+H_{pk}^a)C_{11}(10)\\nonumber\\\\\n&+H_{p}^a(m^2-t_a)C_{12}(10) \\big]\\bigg\\}.\n\\end{align}\nHere $C_i(10)=C_i(m^2,t_a,-Q^2,m^2,0,m^2)$.\n\nThe NLO amplitude of the conjugate diagram of Fig.\\ref{graphs}.10 is\n\\begin{align}\n\\bar{A}_{10}\\,&=\\,-\\frac{C_F e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{4}{t(t_b-m^2)}\\Gamma_{qq}^{(0),a}\n\\bigg\\{\\nonumber\\\\\n&-\\frac{(H_{T}^a+mH_{p\\varepsilon}^a-sH_{\\varepsilon}^a+2(H_{k}^a-mH_{m}^a)\\varepsilon\\cdot\np+2H_{p}^a\\varepsilon\\cdot r)c_\\Gamma}{2\\epsilon}\\nonumber\\\\\n& +(H_{T}^a-sH_{\\varepsilon}^a+2H_{k}^a\\varepsilon\\cdot p)\\big[\nm^2C_{0}(\\bar{10})+(m^2+Q^2+t_b)C_{1}(\\bar{10})+Q^2C_{2}(\\bar{10})\\nonumber\\\\\n&-2C_{00}(\\bar{10})_{fin}-m^2C_{11}(\\bar{10})+(Q^2-m^2+t_b)C_{12}(\\bar{10})\n+Q^2C_{22}(\\bar{10})+1\\big]\\nonumber\\\\\n&+mH_{p\\varepsilon}^a\\big[m^2C_{0}(\\bar{10})+(2m^2+Q^2)C_{1}(\\bar{10})\n+Q^2C_{2}(\\bar{10})-2C_{00}(\\bar{10})_{fin}-m^2C_{11}(\\bar{10})\\nonumber\\\\\n&+(Q^2-m^2+t_b)C_{12}(\\bar{10})\n+Q^2C_{22}(\\bar{10})+1\\big]+2H_{pk}^a\\big[C_{1}(\\bar{10})\n+C_{11}(\\bar{10})\\big]\\times\\nonumber\\\\\n&(\\varepsilon\\cdot r+\\varepsilon\\cdot\nk)+2mH_{p}^a\\big[((2m^2-t_b)C_{1}(\\bar{10})\n+m^2C_{11}(\\bar{10})+(m^2-t_b)C_{12}(\\bar{10}))\\varepsilon\\cdot k\\nonumber\\\\\n&+(m^2C_{0}(\\bar{10})+(3m^2+Q^2)C_{1}(\\bar{10})+Q^2C_{2}(\\bar{10})\n-2C_{00}(\\bar{10})_{fin}+Q^2C_{12}(\\bar{10})\\nonumber\\\\\n&+Q^2C_{22}(\\bar{10})+1)\\varepsilon\\cdot\nr\\big]+2mH_{m}^a\\big[(-m^2C_{0}(\\bar{10})-(2m^2+Q^2)C_{1}(\\bar{10})\n-Q^2C_{2}(\\bar{10})\\nonumber\\\\\n&+2C_{00}(\\bar{10})_{fin}+m^2C_{11}(\\bar{10})+(m^2-Q^2-t_b)C_{12}(\\bar{10})\n-Q^2C_{22}(\\bar{10})-1)\\varepsilon\\cdot p\\nonumber\\\\\n&-s(C_{1}(\\bar{10})+C_{11}(\\bar{10}))(\\varepsilon\\cdot\nr+\\varepsilon\\cdot k)\\big] \\bigg\\}.\n\\end{align}\nwith $C_i(\\bar{10})=C_i(m^2,t_b,-Q^2,m^2,0,m^2)$.\n\nThe result of the quark self-energy diagram Fig.\\ref{graphs}.11 is\n\\begin{align}\nA_{11}\\,&=\\,\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2C_Fc_\\Gamma}{t(t_a-m^2)^2}\n\\Gamma_{qq}^{(0),a}\\bigg\\{\\frac{(7m^2-t_a)H_T^a+2(t_a\n+2m^2)mH_{p\\varepsilon}^a}{\\epsilon}+\\ln(m^2)[\\nonumber\\\\\n&(t_a-7m^2)H_T^a-2(t_a+2m^2)mH_{p\\varepsilon}^a]+\\frac{4t_a(t_a+m^2)mH_{p\\varepsilon}^a\n-(m^4-10t_am^2+t_a^2)H_T^a}{t_a}\\nonumber\\\\\n&+\\frac{\\ln(1-\\frac{t_a}{m^2})(t_a-m^2)((m^4-6t_am^2+t_a^2)H_T^a\n-2t_a(t_a+m^2)mH_{p\\varepsilon}^a)}{t_a^2}\\bigg\\}.\n\\end{align}\nand the amplitude of the conjugate diagram of Fig.3.11 reads\n\\begin{align}\n\\bar{A}_{11}\\,&=\\,\\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2C_Fc_\\Gamma}{t(t_b-m^2)^2}\n\\Gamma_{qq}^{(0),a}\\bigg\\{\\frac{(7m^2-t_b)(H_T^a-sH_{\\epsilon}^a+2H_k^a\\varepsilon\\cdot\np+2H_p^a\\varepsilon\\cdot r)}{\\epsilon}\\nonumber\\\\\n&+\\frac{2(5m^2-2t_b)m(H_{p\\varepsilon}^a-2H_m^a\\varepsilon\\cdot\np)}{\\epsilon}+\\frac{1}{t_a}\\big[(H_T^a-sH_{\\epsilon}^a+2H_k^a\\varepsilon\\cdot\np+2H_p^a\\varepsilon\\cdot r\\nonumber\\\\\n&)(-t_b(7m^2-t_b)\\ln(m^2)-\\frac{m^2-t_b}{t_b}(m^4-6m^2t_b\n+t_b^2)\\ln(\\frac{m^2-t_b}{m^2})\\nonumber\\\\\n&-m^4+10m^2t_b-t_b^2)+2m(H_{p\\varepsilon}^a-2H_m^a\\varepsilon\\cdot\np)(-t_b(5m^2-2t_b)\\ln(m^2)\\nonumber\\\\\n&-\\frac{m^2-t_b}{t_b}(m^4-5m^2t_b+2t_b^2)\\ln(\\frac{m^2-t_b}{m^2})-m^4+8m^2t_b-3t_b^2)\\big]\\bigg\\}\\ .\n\\end{align}\n\n\\subsection{The Box diagrams}\n\nHere, we present the results of box diagrams. In the calculation of diagrams Fig.\\ref{graphs}.2 and\nFig.\\ref{graphs}.3, the four-point integrals have to be concerned. After taking the high energy limit and eliminating the $s$ suppressed\nterms, in the end the results turn out to be quit simple, that is\n\\begin{align}\nA_{2+3}\\,&=\\,\\frac{e\ne_fN_c}{(4\\pi)^{2-\\epsilon}}\\frac{c_\\Gamma}{t(t_a-m^2)}\n\\Gamma_{qq}^{(0),a}(H_{T}^a+mH_{p\\varepsilon}^a)\\big\\{\\frac{(-t)^{-\\epsilon}}{\\epsilon^2}\n-\\frac{3(m^2)^{-\\epsilon}}{2\\epsilon^2}+\\frac{(m^2)^{-\\epsilon}}{\\epsilon}\n\\big(\\nonumber\\\\\n&\\ln(-\\frac{\\alpha s}{m^2})+\\ln(\\frac{\\alpha\ns}{m^2})+\\ln(\\frac{-t}{m^2})-\\ln(1-\\frac{t_a}{m^2})\\big)\n-\\ln(-\\frac{t}{m^2})\\big[\\ln(-\\frac{\\alpha s}{m^2})+\\ln(\\frac{\\alpha\ns}{m^2})\\big]\\nonumber\\\\\n&-\\ln(1-\\frac{t_a}{m^2})^2-\\frac{3\\pi^2}{4}+C_{0}(1) \\big\\}.\n\\end{align}\nHere, $C_0(1)=C_0(t_a,t,m^2,m^2,0,0)$, as in the $A_1$.\n\nThe results of diagrams conjugating to Fig.\\ref{graphs}.2 and\nFig.\\ref{graphs}.3 can be obtained by taking the following\nreplacements:\n\\\\$t_a\\rightarrow t_b$, $\\alpha\\rightarrow 1-\\alpha$,\n$(H_{T}^a+mH_{p\\varepsilon}^a)\\rightarrow\n(H_{T}^a+mH_{p\\varepsilon}^a-sH_{\\varepsilon}^a+2(H_{k}^a-mH_{m}^a)\\varepsilon\\cdot\np+2H_{p}^a\\varepsilon\\cdot r)$.\n\nNote that there are $\\text{ln} s$ terms in the amplitudes of\nFigs.2.2 and 2.3, and also their conjugate partners.\n\nNext, we give the result of adjacent box as shown in Fig.\\ref{graphs}.12.\n\\begin{align}\nA_{12}\\,&=\\,\\Gamma_{qq}^{(0),a}\\frac{2s}{t}\\bigg(\\frac{e\ne_f}{s}\\bigg)\\bigg(-\\frac{1}{2N_c}\\bigg)\\frac{2}{(4\\pi)^{2-\\epsilon}}\\bigg\\{\n\\frac{c_\\Gamma}{\\epsilon}A_{12}^{-1}+A_{12}^0\\bigg\\}\\ .\n\\end{align}\nin which the divergent part $A_{12}^{-1}$ is\n\\begin{align}\nA_{12}^{-1}\\,&=\\,-(H_{T}^a+mH_{p\\varepsilon}^a)\\frac{M^2x_r\\ln(x_r)}{m^2(1-x_r^2)},\n\\end{align}\nwith \\begin{eqnarray}\nx_r=-\\frac{1-\\sqrt{1-\\frac{4m^2}{2m^2+M^2}}}{1+\\sqrt{1-\\frac{4m^2}{2m^2+M^2}}}\\\n, \\end{eqnarray} and $M^2=(q+r)^2$, defined in the above paragraph.\n\nThe finite term $A_{12}^{0}$ is given in the Appendix. The amplitude\nof the conjugate diagram of Fig.\\ref{graphs}.12 goes as\n\\begin{align}\n\\bar{A}_{12}\\,&=\\,\\Gamma_{qq}^{(0),a}\\frac{2s}{t}\\bigg(\\frac{e\ne_f}{s}\\bigg)\\bigg(-\\frac{1}{2N_c}\\bigg)\\frac{2}{(4\\pi)^{2-\\epsilon}}\\bigg\\{\n\\frac{c_\\Gamma}{\\epsilon}\\bar{A}_{12}^{-1}+\\bar{A}_{12}^0\\bigg\\}\\ ,\n\\end{align}\nwhere the divergent term\n\\begin{align}\n\\bar{A}_{12}^{-1}=-\\frac{M^2x_r\\ln(x_r)}{m^2(1-x_r^2)}\n\\big[H_{T}^a+mH_{p\\varepsilon}^a-sH_{\\varepsilon}^a\n+2(H_{k}^a-mH_{m}^a)\\varepsilon\\cdot p+2H_{p}^a\\varepsilon\\cdot\nr\\big]\\ .\n\\end{align}\nThe finite piece is also given in the Appendix.\n\nThe opposite box diagram Fig.\\ref{graphs}.14 does not contain any\ndivergence, its lengthy analytic expression is presented in the Appendix.\n\n\\subsection{The Pentagon diagram}\nIn this subsection we deal with the pentagon diagram\nFig.\\ref{graphs}.13. The calculation procedure is complicated, and is performed by computer algebra. Due to the fact\nthat the integrals depend upon the large scale $s$,  they can be greatly simplified in the high energy\nlimit.  The results are as follows:\n\\begin{align}\nA_{13}\\,&=\\,\\Gamma_{qq}^{(0),a}\\frac{N_c}{2}\\frac{ee_f}{t(4\\pi)^{2-\\epsilon}}\\bigg\\{\n\\frac{1}{\\epsilon^2}A_{13}^{(-2)}+\\frac{1}{\\epsilon}A_{13}^{(-1)}+\\frac{A_{13}^{(0)}}{\\Delta}\\bigg\\}\\ .\n\\label{pentagon}\n\\end{align}\nHere,\n\\begin{eqnarray}\n\\Delta\\,&=\\,\\alpha(\\alpha-1)m^4-m^2(t+\\alpha(\\alpha-1)(t_a+t_b))+\\alpha(\\alpha-1)(Q^2t+t_at_b)\\ ,\n\\end{eqnarray}\n\\begin{align}\nA_{13}^{(-2)}\\,&=\\,\\bigg\\{-(H_{T}^a+mH_{p\\varepsilon}^a)\n\\big[\\frac{1}{t_a-m^2}+\\frac{1}{t_b-m^2}\\big]\n+\\frac{sH_{\\varepsilon}^a-2(H_{k}^a-mH_m^a)\\varepsilon\\cdot\np-2H_{p}^a\\varepsilon\\cdot r}{t_b-m^2}\\bigg\\}\\ ,\n\\end{align}\nand\n\\begin{align}\nA_{13}^{(-1)}\\,&=\\,2\\bigg\\{(H_{T}^a\n+mH_{p\\varepsilon}^a)\\big[\\ln(m)(\\frac{1}{t_a-m^2}\n+\\frac{1}{t_b-m^2})+\\frac{\\ln(1-\\frac{t_a}{m^2})}{t_a-m^2}\n+\\frac{\\ln(1-\\frac{t_b}{m^2})}{t_b-m^2}\\nonumber\\\\\n&+\\frac{(t_b-t_a)\\ln(\\frac{\\alpha-1}{\\alpha})}{(t_a-m^2)(t_b-m^2)}\\big]\n-\\frac{sH_{\\varepsilon}^a-2(H_{k}^a-mH_m^a)\\varepsilon\\cdot\np-2H_{p}^a\\varepsilon\\cdot\nr}{t_b-m^2}\\ln\\bigg(\\frac{(t_a-m^2)\\alpha}{m(1-\\alpha)}\\bigg)\\bigg\\}\\ .\n\\end{align}\nThe finite piece $A_{13}^{(0)}$ in (\\ref{pentagon}) is listed in the Appendix. Note, we can reproduce the massless result \\cite{qiao} when taking the $m\\rightarrow0$ limit in above expressions.\n\nWe can obtain the amplitude of the conjugate diagram of\nFig.\\ref{graphs}.13 by replacing $s\\rightarrow -s$ in the\n$D_0(1),D_0(2),D_0(3),D_0(4)$ and $D_i(14)\\rightarrow -D_i(14)$.\nWith there replacements one can easily find that the energy\ndependence $\\text{ln}~s$ terms in Fig.\\ref{graphs}.13 cancel the\nterms in its conjugate diagram. Therefore, the energy dependence\nterms merely come from $A_{2+3}+\\bar{A}_{2+3}$.\n\n\\section{Renormalization}\n\nThe results of our concerned process contain both infrared and ultraviolet\ndivergences. The ultraviolet divergences may be renormalized via standard procedure, i.e. canceled by counter terms, in modified minimal subtraction ($\\overline{MS}$) scheme here. The infrared divergences may be canceled out when the soft gluon radiation process $\\gamma^{*}+~reggeon\\longrightarrow Q\\bar{Q}g$ is taken into account.\n\nThe ultraviolet divergences exist only in the self-energy and triangle diagrams, which are\n\\begin{eqnarray} A_1^{UV}&=& \\frac{e e_f}{(4\\pi)^{2-\\epsilon}}\\frac{2(H_{T}^a+m\nH_{p\\varepsilon}^a)}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}\\\n\\frac{3N_cc_\\Gamma}{2\\epsilon_{UV}}, \\end{eqnarray} \\begin{eqnarray} A_4^{UV}&=& \\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2(H_{T}^a+m\nH_{p\\varepsilon}^a)}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}(-\\frac{c_\\Gamma}{2N_c\\epsilon_{UV}})\\\n, \\end{eqnarray} \\begin{eqnarray} A_5^{UV}&=& \\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2(H_{T}^a+m\nH_{p\\varepsilon}^a)}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}\\frac{3N_cc_\\Gamma}{2\\epsilon_{UV}}\\\n, \\end{eqnarray} \\begin{eqnarray} A_6^{UV}&=& \\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2(H_{T}^a+m\nH_{p\\varepsilon}^a)}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}(-\\frac{c_\\Gamma}{2N_c\\epsilon_{UV}})\\\n, \\end{eqnarray} \\begin{eqnarray} A_{7+8+9}^{UV}&=& \\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2(H_{T}^a+m\nH_{p\\varepsilon}^a)}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}\\frac{c_\\Gamma}{\\epsilon_{UV}}\n(\\frac{5}{3}N_c-\\frac{2}{3}n_f) \\ , \\end{eqnarray} \\begin{eqnarray} A_{10}^{UV}&=& \\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2(H_{T}^a+m\nH_{p\\varepsilon}^a)}{t(t_a-m^2)}\\Gamma_{qq}^{(0),a}\\frac{c_\\Gamma\n}{\\epsilon_{UV}}C_F\\ , \\end{eqnarray} and \\begin{eqnarray} A_{11}^{UV}&=& \\frac{e\ne_f}{(4\\pi)^{2-\\epsilon}}\\frac{2}{t(t_a-m^2)^2}\\Gamma_{qq}^{(0),a}\\frac{c_\\Gamma\n}{\\epsilon_{UV}}2C_F((7m^2-t_a)H_{T}^a+2(t_a+2m^2)mH_{p\\varepsilon}^a)\\\n. \\end{eqnarray}\n\n For the ultraviolet divergence discussed above,  when taking the massless limit we can find it is in agreement with the result in \\cite{qiao}. We denote $Z_2, Z_3, Z_m, Z_g$ as the quark-field, gluon-field, mass, and coupling renormalization constants, respectively. Note, in our calculation the renormalization constants $Z_2$ and $Z_m$ are defined in on-shell Scheme, while $Z_3$ and $Z_g$ are given in\n$\\mathrm{\\overline{MS}}$ scheme, which tells:\n\\begin{eqnarray} Z_2\n&=&1-\\frac{\\alpha_s}{4\\pi}C_F\\big[\\frac{1}{\\epsilon_{UV}}+ \\frac{2}{\\epsilon_{UV}}-3\\gamma_E+3\\ln(\\frac{4\\pi\n\\mu^2}{m^2})+4\\big] \\ ,\n\\end{eqnarray}\n\\begin{eqnarray}\nZ_3 &=&\n1+\\frac{\\alpha_s}{4\\pi}(\\frac{5}{3}N_c-\\frac{2}{3}n_f)\\big[\n\\frac{1}{\\epsilon_{UV}}-\\gamma_E+\\ln(4\\pi)\\big]\\ ,\n\\end{eqnarray}\n\\begin{eqnarray}\nZ_g &=&\n1-\\frac{\\alpha_s}{4\\pi}(\\frac{11}{6}N_c-\\frac{1}{3}n_f)\\big[\n\\frac{1}{\\epsilon_{UV}}-\\gamma_E+\\ln(4\\pi)\\big]\\ ,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nZ_m &=&1-3\\frac{\\alpha_s}{4\\pi}C_F\\big[\\frac{1}{\\epsilon_{UV}}-\\gamma_E+\\ln(\\frac{4\\pi\n\\mu^2}{m^2})+\\frac{4}{3}\\big]\\ .\n\\end{eqnarray}\n\nAfter including the counter terms, all above ultraviolet divergences and those divergences from their conjugate diagrams, are canceled out. Hence our results will be ultraviolet finite.\n\n\\section{Numerical Results}\n\nIn the following we show the mass effect numerically in NLO corrections of the  photon impact factor. Since in this work the infrared divergences still exist, the NLO amplitude squared can be expressed as\n\\begin{eqnarray}\n\\mathrm{|M_{NLO}|^2}&=&\\mathrm{M_{NLO}}\\mathrm{M^{*}_{LO}}=(\\mathrm{M_{Born}+M_{Loop}})\\mathrm{M^{*}_{LO}}\\nonumber\\\\\n&=&\\mathrm{|M|^2_{Born}}+\\frac{1}{\\epsilon^2}\\mathrm{|M|^2_{Loop~IR2}}+\\frac{1}{\\epsilon}\\mathrm{|M|^2_{Loop~IR1}}\n +\\mathrm{|M|^2_{Loop~finite}}\\ .\n\\end{eqnarray}\nWe then can evaluate respectively the mass effects for the Born term $\\mathrm{|M|^2_{Born}}$, second-order infrared divergent term\n$\\mathrm{|M|^2_{Loop~IR2}}$, first-order infrared divergent term $\\mathrm{|M|^2_{Loop~IR1}}$, and NLO finite term\n$\\mathrm{|M|^2_{Loop~finite}}$.\n\nIn the numerical evaluation, we take the following inputs:\n\\begin{eqnarray}\nn_f=5,\\ Q=7~\\mathrm{GeV},\\ \\mu=Q^2,\\ \\alpha=0.2,\\\nt_a=t_b=t=50~\\mathrm{GeV^2}.\n\\end{eqnarray}\nIn our calculation, the quark mass $m$ varies from $0~\\mathrm{GeV}$\nto $6~\\mathrm{GeV}$, then $n_f=5$ (in the physical world the charm\nquark mass and bottom quark mass are about 1.4~GeV and 4.7~GeV\nrespectively). In order to demonstrate the results more clearly, we\ndefine the ratio\n$\\mathrm{R(m)}=\\frac{\\mathrm{|M|^2}}{\\mathrm{|M|^2}_{m=0~\\mathrm{GeV}}}$,\nand the ratios of $\\mathrm{|M|^2_{Born}}$,\n$\\mathrm{|M|^2_{Loop~UV}}$, $\\mathrm{|M|^2_{Loop~IR2}}$,\n$\\mathrm{|M|^2_{Loop~IR1}}$ and $\\mathrm{|M|^2_{Loop~finite}}$. The\ninput $s$ are taken as $s=1000~\\mathrm{GeV^2}$ and $\ns=2000~\\mathrm{GeV^2}$, respectively. The curves are shown in\nFig.\\ref{ratio}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.480\\textwidth]{ratio1.eps}%\n\\hspace{5mm}\n\\includegraphics[width=0.480\\textwidth]{ratio2.eps}\\hspace*{\\fill}\n\\caption{\\small the ratios of $\\mathrm{|M|^2_{Born}}$,\n$\\mathrm{|M|^2_{Loop~IR2}}$, $\\mathrm{|M|^2_{Loop~IR1}}$ and\n$\\mathrm{|M|^2_{Loop~finite}}$ versus the quark mass in the process\n$\\gamma^* + q \\to Q\\bar{Q} + q$. }\n\\label{ratio}\n\\vspace{-0mm}\n\\end{figure}\n\n\nFrom Fig.\\ref{ratio} we can see that, for $s=1000~\\mathrm{GeV^2}$\nand $s=2000~\\mathrm{GeV^2}$, the curves change little. As the quark\nmass gets larger, the NLO amplitude squared also gets larger\nquickly, even for the $\\mathrm{|M|^2_{Loop~IR1}}$, the ratio is\nnearly $20$ when the quark mass is $6~\\mathrm{GeV}$. So we may\nconclude that, in the $\\gamma^{*} \\longrightarrow Q\\bar{Q}-Reggeon$\nvertex, the quark mass effects on the photon impact factor are\nsignificant, and may influence the results of high energy\nphoton-photon scattering and heavy quark pair leptoproduction.\n\n\n\n\\newpage\n\\section{Conclusions}\n\nIn this paper, we calculated the process\n$\\gamma^{*}+~q\\longrightarrow Q\\bar{Q}+q$ at NLO with fully virtual corrections in the high energy\nlimit for massive quark pair $Q(\\bar{Q}$, which tells the coupling of the reggeized\ngluon to $\\gamma^{*}\\longrightarrow Q\\bar{Q}$.  This calculation is just the first step of our\nfinal purpose, to obtain the complete NLO corrections to the photon impact\nfactor and checking the BFKL pomeron prediction for the process\n$\\gamma^{*} + \\gamma^{*} \\longrightarrow \\gamma^{*} + \\gamma^{*}$ at\nhigh energies.\n\nIn our calculation all contributions from loop diagrams, i.e., self-energy, triangle,\nbox and pentagon diagrams, are regulated in dimensional\nregularization scheme and the ultraviolet divergences are renormalized by\nadding the corresponding counter terms. In the end, the infrared divergences still\nexist in the result, which would be canceled after taking account of the real corrections.\n\nWe find that for $\\gamma^{*}+~q\\longrightarrow Q\\bar{Q}+q$ process in the\nmassive quark case, the quark mass effects are significant at the next-to-leading order of accuracy, which indicates that\nfor heavy quark diffractive photoproduction, the quark mass is indispensable. Our result might essentially show the quark mass effect, in spite of the absence of real corrections, which are needed in a complete NLO calculation.\nMoreover, for the photon impact factor with\nheavy quarks, the photon is legitimate to be real in order to guarantee the perturbative QCD calculations applicable.\n\n\\vspace{0.7cm} {\\bf Acknowledgments}\n\nThis work was supported by the National Natural Science Foundation\nof China (NSFC) under grants 11905006, 11805042, 11975236 and 11635009.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgments}\n\n\\begin{sloppypar}\nWe thank F.~Ignatov, A.~Nesterenko, and A.E.~Radzhabov for comments and support. \nThe work in this contribution was supported \nby Agence Nationale de la Recherche\t(ANR-19-CE31-001),\nby CERCA program of the Generalitat de Catalunya (2017 SGR 1069), \nby Conacyt 'Paradigmas y Controversias de la Ciencia 2022' (project number 319395), \nby Danmarks Frie Forskningsfond under Grant Number 8021-00122B, \nby Deutsche Forschungsgemeinschaft Collaborative Research Centers CRC~1044, CRC~110, and under Grant Numbers DE 839\/2-1, HA 9289\/1-1, HI 2048\/1-2 (Project No.\\ 399400745), Prisma Cluster for Excellence PRISMA$^+$ EXC2118\/1,\nby DST, Govt.~of India\tINSPIRE Faculty Fellowship (grant number IFA16-PH170),\nby European Research Council under the European Union's Horizon 2020 research and innovation programme under Grant Agreement Number 771971-SIMDAMA, under the Marie Sk\\l{}odowska-Curie grant agreement numbers 860881-HIDDEN, 894103, under grant number\t754510 (EU, H2020-MSCA-COFUND2016), by European Union STRONG 2020 project under Grant Agreement Number 824093, \nby the Excellence Initiative of Aix-Marseille University - A*MIDEX, a French ``Investissements d'Avenir'' program, through grants AMX-18-ACE-005 and AMX-19-IET-008-IPhU, and by the French National Research Agency under contract ANR-20-CE31-0016, \nby FWF (grant numbers I 3845-N27 and W 1252-N27), \nby Fundaci\\'on Marcos Moshinsky\t(C\\'atedra Marcos Moshinsky 2020),\n by the Japan Society for the Promotion of Science under Grant Numbers KAKENHI-16K05317, 20K03926, 20H05625,   \nby the National Research Foundation of South Africa, \nby the Natural Sciences and Engineering Research Council of Canada, \nby Spanish Ministry of Science, Innovation and Universities \t(PID2020-112965GB-I00\/AEI\/10.13039\/501100011033), \nby the Swedish Research Council under Grant Numbers 2016-05996, 2019-03779, \nby the Swiss National Science Foundation under Grant Numbers  PCEFP2\\_181117, PCEFP2\\_194272, PZ00P2\\_193383, 200021\\_200866, 200020\\_175791, \nby the UK Science and Technology Facilities Council (STFC) under Grant Numbers ST\/P000630\/1, ST\/T000988\/1, ST\/S000925\/1,\nby the U.S.\\ Department of Energy, Office of Science, Office of High Energy Physics under Award Numbers DE-SC0007983, DE-SC0010005, DE-SC0010339, DE-SC0011941, DE-SC0012704, DE-SC0013682, DE-SC0015655, by the U.S.\\ Department of Energy, Office of Science, Office of Nuclear Physics under Award Numbers DE-AC02-07CH11359, and\nby the U.S.\\ National Science Foundation under Grant Numbers PHY-1748958\nThis document was prepared by the Muon $g-2$ Theory Initiative using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S.\\ Department of Energy, Office of Science, HEP User Facility.  Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No.\\ DE-AC02-07CH11359.\n\\end{sloppypar}\n\n\\section{Introduction}\n\nThe Run-1 result of the Fermilab $g-2$ experiment~\\cite{Muong-2:2021ojo,Muong-2:2021ovs,Muong-2:2021xzz,Muong-2:2021vma} confirmed the earlier BNL measurement~\\cite{Muong-2:2006rrc}, resulting in a $4.2\\sigma$ tension of the combined experimental value with respect to the recommendation by the White Paper~\\cite{Aoyama:2020ynm} of the Muon $g-2$ Theory Initiative~\\cite{MuonInitiative} (reflecting discussions at a series of workshops~\\cite{FNAL2017,KEK2018,UConn2018,Mainz2018,INT2019}). \nIn Table~\\ref{tab:summary} we reproduce the status of the Standard-Model (SM) prediction as presented therein, as reference point for the prospects of future improvements. With results from subsequent runs of the Fermilab experiment expected soon, poised to reduce the experimental uncertainty by more than another factor of $2$~\\cite{Muong-2:2015xgu}, as well as future $g-2$ experiments at J-PARC~\\cite{Abe:2019thb} and, potentially, PSI~\\cite{Aiba:2021bxe} and Fermilab~\\cite{FutMuon2021}, it is clear that theory needs to be improved concurrently.  \n\n\\begin{table}[t]\n\\begin{centering}\n\\small\n\t\\begin{tabular}{l  r l }\n\t\\toprule\n\t   Contribution  & Value   $\\times 10^{11}$ & References\\\\ \\midrule\n\t   Experiment (E821 + E989)                     &$ 116\\, 592\\, 061(41)$ & Refs.~\\cite{Muong-2:2006rrc,Muong-2:2021ojo}  \\\\\\midrule\n\tHVP LO ($e^+e^-$)  & $ 6931(40) $  &  Refs.~\\cite{Davier:2017zfy,Keshavarzi:2018mgv,Colangelo:2018mtw,Hoferichter:2019gzf,Davier:2019can,Keshavarzi:2019abf}\\\\\n        HVP NLO ($e^+e^-$) & $-98.3(7) $ & Ref.~\\cite{Keshavarzi:2019abf}\\\\\n        HVP NNLO ($e^+e^-$)  & $ 12.4(1) $ & Ref.~\\cite{Kurz:2014wya}\\\\\n        HVP LO (lattice, $udsc$)   & $7116 (184)$ & Refs.~\\cite{Chakraborty:2017tqp,Borsanyi:2017zdw,Blum:2018mom,Giusti:2019xct,Shintani:2019wai,FermilabLattice:2019ugu,Gerardin:2019rua,Aubin:2019usy,Giusti:2019hkz}\\\\\nHLbL (phenomenology)  & $92(19)$ & Refs.~\\cite{Melnikov:2003xd,Masjuan:2017tvw,Colangelo:2017fiz,Hoferichter:2018kwz,Gerardin:2019vio,Bijnens:2019ghy,Colangelo:2019uex,Pauk:2014rta,Danilkin:2016hnh,Jegerlehner:2017gek,Knecht:2018sci,Eichmann:2019bqf,Roig:2019reh}\\\\\nHLbL NLO (phenomenology)  & $2(1)$ & Ref.~\\cite{Colangelo:2014qya}\\\\\nHLbL (lattice, $uds$)  & $79(35)$ & Ref.~\\cite{Blum:2019ugy}\\\\\nHLbL (phenomenology + lattice) & $90(17)$ & Refs.~\\cite{Melnikov:2003xd,Masjuan:2017tvw,Colangelo:2017fiz,Hoferichter:2018kwz,Gerardin:2019vio,Bijnens:2019ghy,Colangelo:2019uex,Pauk:2014rta,Danilkin:2016hnh,Jegerlehner:2017gek,Knecht:2018sci,Eichmann:2019bqf,Roig:2019reh,Blum:2019ugy}\\\\\\midrule\nQED                &  $116\\,584\\,718.931(104)$  & Refs.~\\cite{Aoyama:2012wk,Aoyama:2019ryr}\\\\\nElectroweak      & $153.6(1.0)$   & Refs.~\\cite{Czarnecki:2002nt,Gnendiger:2013pva}\\\\\nHVP ($e^+e^-$, LO + NLO + NNLO)  & $6845(40)$ & Refs.~\\cite{Davier:2017zfy,Keshavarzi:2018mgv,Colangelo:2018mtw,Hoferichter:2019gzf,Davier:2019can,Keshavarzi:2019abf,Kurz:2014wya} \\\\\nHLbL (phenomenology + lattice + NLO)  & $92(18)$ & Refs.~\\cite{Melnikov:2003xd,Masjuan:2017tvw,Colangelo:2017fiz,Hoferichter:2018kwz,Gerardin:2019vio,Bijnens:2019ghy,Colangelo:2019uex,Pauk:2014rta,Danilkin:2016hnh,Jegerlehner:2017gek,Knecht:2018sci,Eichmann:2019bqf,Roig:2019reh,Blum:2019ugy,Colangelo:2014qya}\\\\ \n        Total SM Value   & $ 116\\, 591\\, 810(43) $  & Refs.~\\cite{Aoyama:2012wk,Aoyama:2019ryr,Czarnecki:2002nt,Gnendiger:2013pva,Davier:2017zfy,Keshavarzi:2018mgv,Colangelo:2018mtw,Hoferichter:2019gzf,Davier:2019can,Keshavarzi:2019abf,Kurz:2014wya,Melnikov:2003xd,Masjuan:2017tvw,Colangelo:2017fiz,Hoferichter:2018kwz,Gerardin:2019vio,Bijnens:2019ghy,Colangelo:2019uex,Blum:2019ugy,Colangelo:2014qya}\\\\\n        Difference:    $\\Delta a_\\mu:=\\ensuremath{a_\\mu^\\text{exp}} - \\ensuremath{a_\\mu^\\text{SM}}$  & $ 251(59) $ & \\\\\n        \\bottomrule\n\t\\end{tabular}\n\t\\caption{Summary of the contributions to $\\ensuremath{a_\\mu^\\text{SM}}$, as compiled in Ref.~\\cite{Aoyama:2020ynm}, except for the update of the experimental number to the average of E821 and the first Run of E989. The first block gives the main results for the hadronic contributions as well as the combined result for HLbL scattering from phenomenology and lattice QCD available at the time of Ref.~\\cite{Aoyama:2020ynm}. \n\tThe second block summarizes the quantities entering the final recommendation for the SM contribution, in particular, the total HVP contribution, evaluated from $e^+e^-$ data, and the total HLbL number.     \n\tThe HVP evaluation is mainly based on the experimental Refs.~\\cite{Bai:1999pk,Akhmetshin:2000ca,Akhmetshin:2000wv,Achasov:2000am,Bai:2001ct,Achasov:2002ud,Akhmetshin:2003zn,Aubert:2004kj,Aubert:2005eg,Aubert:2005cb,Aubert:2006jq,Aulchenko:2006na,Achasov:2006vp,Akhmetshin:2006wh,Akhmetshin:2006bx,Akhmetshin:2006sc,Aubert:2007ur,Aubert:2007ef,Aubert:2007uf,Aubert:2007ym,Akhmetshin:2008gz,Ambrosino:2008aa,Ablikim:2009ad,Aubert:2009ad,Ambrosino:2010bv,Lees:2011zi,Lees:2012cr,Lees:2012cj,Babusci:2012rp,Akhmetshin:2013xc,Lees:2013ebn,Lees:2013uta,Lees:2014xsh,Achasov:2014ncd,Aulchenko:2014vkn,Akhmetshin:2015ifg,Ablikim:2015orh,Shemyakin:2015cba,Anashin:2015woa,Achasov:2016bfr,Achasov:2016lbc,TheBaBar:2017aph,CMD-3:2017tgb,TheBaBar:2017vzo,Kozyrev:2017agm,Anastasi:2017eio,Achasov:2017vaq,Xiao:2017dqv,TheBaBar:2018vvb,Anashin:2018vdo,Achasov:2018ujw,Lees:2018dnv,CMD-3:2019ufp}.\n\tIn addition, the HLbL evaluation uses experimental input from Refs.~\\cite{Behrend:1990sr,Gronberg:1997fj,Acciarri:1997yx,Achard:2001uu,Achard:2007hm,Arnaldi:2009aa,Aubert:2009mc,BABAR:2011ad,Berghauser:2011zz,Uehara:2012ag,Babusci:2012ik,Aguar-Bartolome:2013vpw,Ablikim:2015wnx,Masuda:2015yoh,Arnaldi:2016pzu,Adlarson:2016hpp,Adlarson:2016ykr,TheNA62:2016fhr,BaBar:2018zpn,PrimEx-II:2020jwd}.  The lattice QCD calculation of the HLbL contribution builds on crucial methodological advances from Refs.~\\cite{Blum:2014oka,Green:2015mva,Blum:2015gfa,Blum:2016lnc,Asmussen:2016lse,Blum:2017cer,Asmussen:2019act}. Finally, the QED value uses the fine-structure constant obtained from atom-interferometry measurements of the Cs atom~\\cite{Parker:2018vye}, and is affected by the tension with the more recent Rb result~\\cite{Morel:2020dww} only at a level irrelevant for $\\ensuremath{a_\\mu^\\text{SM}}$. Mixed leptonic and hadronic corrections enter at the same order $\\mathcal{O}(\\alpha^4)$ as HVP NNLO and HLbL NLO, but have been estimated as $\\lesssim 1\\times 10^{-11}$~\\cite{Hoferichter:2021wyj}. }\n\\label{tab:summary}\n\\end{centering}\n\\end{table}\n\nThis is particularly pressing given the tension between hadronic vacuum polarization (HVP) extracted from $e^+e^-\\to\\text{hadrons}$ cross section data, upon which the final value from Ref.~\\cite{Aoyama:2020ynm} is based, and the recent lattice calculation by the BMW collaboration~\\cite{Borsanyi:2020mff}. Here the most urgent task is to scrutinize the result of Ref.~\\cite{Borsanyi:2020mff} in detailed comparisons with lattice results of commensurate precision obtained in independent calculations by other lattice collaborations. As discussed in Sec.~\\ref{sec:latticeHVP}, such calculations are forthcoming. If the tensions persist, their phenomenological consequences must also be explored~\\cite{Crivellin:2020zul,Keshavarzi:2020bfy,Malaescu:2020zuc,Colangelo:2020lcg,Ce:2022eix} (see Sec.~\\ref{sec:outlook}).    \nMoreover, also the hadronic light-by-light (HLbL) contribution needs to be further improved to meet the final precision $\\Delta a_\\mu^\\text{E989}=16\\times 10^{-11}$ projected for the Fermilab experiment~\\cite{Muong-2:2015xgu}. \n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{amuHLbLcompare.pdf}\\hfill\n    \\includegraphics[width=0.5\\textwidth]{amuHVPcompare.pdf}\n    \\caption{Left: Comparison of HLbL evaluations, as quoted in Ref.~\\cite{Aoyama:2020ynm}, to earlier estimates~\\cite{Prades:2009tw,Nyffeler:2009tw, Jegerlehner:2009ry,Jegerlehner:2017gek} (orange) and a more recent lattice calculation~\\cite{Chao:2021tvp} (open blue).\n    Right: Comparison of theoretical predictions of $a_\\mu$ with experiment~\\cite{Muong-2:2021ojo,Muong-2:2006rrc} (orange band), adapted from Ref.~\\cite{Aoyama:2020ynm}. Each data point represents a different evaluation of leading-order HVP, to which the remaining SM contributions, as given in Ref.~\\cite{Aoyama:2020ynm}, have been added. Red squares show data-driven results~\\cite{Jegerlehner:2017gek,Benayoun:2019zwh,Davier:2019can,Keshavarzi:2019abf}; filled blue circles indicate lattice-QCD calculations that were taken into account in the WP20 lattice average~\\cite{Borsanyi:2017zdw,Blum:2018mom,Giusti:2019xct,Shintani:2019wai,FermilabLattice:2019ugu,Gerardin:2019rua,Giusti:2019hkz}, while the open ones show results published after the deadline for inclusion in that average~\\cite{Borsanyi:2020mff,Lehner:2020crt}; the purple triangle gives a hybrid of the two~\\cite{Blum:2018mom}.\n        The SM prediction of Ref.~\\cite{Aoyama:2020ynm} is shown as the black square and gray band.}\n    \\label{fig:compare}\n\\end{figure}\n\n A comparison of published results for HVP and HLbL, including those that were published after the March 2020 deadline, is shown in Fig.~\\ref{fig:compare}.  In this contribution, we briefly review the current status from data-driven evaluations and from lattice QCD for both quantities \nand discuss future prospects as well as future plans of the Muon $g-2$ Theory Initiative.       \n\n\\section{Data-driven evaluations of HVP}\n\\label{sec:dataHVP}\n\n\\begin{table}[t]\n\\small\n\t\\centering\n\t\\begin{tabular}{c r r r}\n\t\\toprule\n\t  & Ref.~\\cite{Davier:2019can} & Ref.~\\cite{Keshavarzi:2019abf} & Difference\\\\\\midrule\n\t $\\pi^+\\pi^-$ & $507.85(0.83)(3.23)(0.55)$ & $504.23(1.90)$ & $3.62$\\\\\n\t $\\pi^+\\pi^-\\pi^0$ & $46.21(0.40)(1.10)(0.86)$ & $46.63(94)$ & $-0.42$\\\\\n\t $\\pi^+\\pi^-\\pi^+\\pi^-$ & $13.68(0.03)(0.27)(0.14)$ & $13.99(19)$ & $-0.31$\\\\\n\t $\\pi^+\\pi^-\\pi^0\\pi^0$ & $18.03(0.06)(0.48)(0.26)$ & $18.15(74)$ & $-0.12$\\\\\n\t $K^+K^-$ & $23.08(0.20)(0.33)(0.21)$ & $23.00(22)$ & $0.08$\\\\\n\t $K_SK_L$ & $12.82(0.06)(0.18)(0.15)$ & $13.04(19)$ & $-0.22$\\\\\n\t $\\pi^0\\gamma$ & $4.41(0.06)(0.04)(0.07)$ & $4.58(10)$ & $-0.17$\\\\\\midrule\n\t Sum of the above  & $626.08(0.95)(3.48)(1.47)$ & $623.62(2.27)$ & $2.46$\\\\\\midrule\n\t $[1.8,3.7]\\,\\text{GeV}$ (without $c \\bar c$) & $33.45(71)$ & $34.45(56)$ & $-1.00$\\\\\n\t $J\/\\psi$, $\\psi(2S)$ & $7.76(12)$ & $7.84(19)$ & $-0.08$ \\\\\n\t $[3.7,\\infty)\\,\\text{GeV}$ & $17.15(31)$ & $16.95(19)$ & $0.20$\\\\\\midrule\n\t Total $\\ensuremath{a_\\mu^\\text{HVP, LO}}$ & $694.0(1.0)(3.5)(1.6)(0.1)_\\psi(0.7)_\\textrm{DV+QCD}$ & $692.8(2.4)$ & $1.2$\\\\ \n\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{Comparison of selected exclusive-mode contributions to $\\ensuremath{a_\\mu^\\text{HVP, LO}}$ from Refs.~\\cite{Davier:2019can,Keshavarzi:2019abf}, for the energy range $\\leq 1.8\\,\\text{GeV}$, in units of $10^{-10}$, see Ref.~\\cite{Aoyama:2020ynm} for details.}\n\\label{tab:KNT_DHMZ}\n\\end{table}\n\nThe data-driven evaluation of HVP relies on the master formula from Refs.~\\cite{Bouchiat:1961lbg,Brodsky:1967sr}, a dispersion relation that relates the leading-order HVP contribution $\\ensuremath{a_\\mu^\\text{HVP, LO}}$ to the total cross section for $e^+e^-\\to\\text{hadrons}$.\\footnote{The cross section is defined photon-inclusively, see Ref.~\\cite{Aoyama:2020ynm}, i.e., while $\\ensuremath{a_\\mu^\\text{HVP, LO}}$ is $\\mathcal{O}(\\alpha^2)$, it contains, by definition, one-photon-irreducible\ncontributions of order $\\mathcal{O}(\\alpha^3)$. This convention matches the one used in lattice-QCD calculations.} The main challenges in converting the available data~\\cite{Bai:1999pk,Akhmetshin:2000ca,Akhmetshin:2000wv,Achasov:2000am,Bai:2001ct,Achasov:2002ud,Akhmetshin:2003zn,Aubert:2004kj,Aubert:2005eg,Aubert:2005cb,Aubert:2006jq,Aulchenko:2006na,Achasov:2006vp,Akhmetshin:2006wh,Akhmetshin:2006bx,Akhmetshin:2006sc,Aubert:2007ur,Aubert:2007ef,Aubert:2007uf,Aubert:2007ym,Akhmetshin:2008gz,Ambrosino:2008aa,Ablikim:2009ad,Aubert:2009ad,Ambrosino:2010bv,Lees:2011zi,Lees:2012cr,Lees:2012cj,Babusci:2012rp,Akhmetshin:2013xc,Lees:2013ebn,Lees:2013uta,Lees:2014xsh,Achasov:2014ncd,Aulchenko:2014vkn,Akhmetshin:2015ifg,Ablikim:2015orh,Shemyakin:2015cba,Anashin:2015woa,Achasov:2016bfr,Achasov:2016lbc,TheBaBar:2017aph,CMD-3:2017tgb,TheBaBar:2017vzo,Kozyrev:2017agm,Anastasi:2017eio,Achasov:2017vaq,Xiao:2017dqv,TheBaBar:2018vvb,Anashin:2018vdo,Achasov:2018ujw,Lees:2018dnv,CMD-3:2019ufp} to the corresponding HVP integral include the combination of data sets in the presence of tensions in the data base and the propagation and assessment of the resulting uncertainties. For illustration, the contributions of the main exclusive channels and the inclusive region from the compilations of Refs.~\\cite{Davier:2019can,Keshavarzi:2019abf} are shown in Table~\\ref{tab:KNT_DHMZ}. \n\nIn Ref.~\\cite{Aoyama:2020ynm} a conservative merging procedure was defined \nto obtain a realistic assessment of these underlying uncertainties. The procedure accounts for tensions among the data sets, for differences in methodologies in the combination of experimental inputs, for correlations between systematic errors, and includes constraints from unitarity and analyticity~\\cite{Colangelo:2018mtw,Ananthanarayan:2018nyx,Davier:2019can,Hoferichter:2019gzf}. Further, the next-to-leading-order calculation from Ref.~\\cite{Campanario:2019mjh} suggests that radiative corrections are under control at this level. \n\nRecent developments in the data-driven HVP evaluation that are not yet reflected in \nthe recommendation from Ref.~\\cite{Aoyama:2020ynm} include the crucial $2\\pi$ channel (new data from SND~\\cite{SND:2020nwa} and covariance matrix from BESIII~\\cite{Ablikim:2015orh}) as well as new data for $e^+e^-\\to 3\\pi$~\\cite{BESIII:2019gjz,BABAR:2021cde}, the second-largest channel both in absolute value and error, see Table~\\ref{tab:KNT_DHMZ}.  Moreover, unitarity and analyticity constraints have been analyzed for the $\\pi^0\\gamma$~\\cite{Hoid:2020xjs} and $\\bar K K$ channels~\\cite{Stamen:2022uqh}. However, as of now, none of these developments indicate significant changes compared to the situation described in Ref.~\\cite{Aoyama:2020ynm}. \n\nIn going forward, new data in the critical $2\\pi$ channel at the same level of precision as BaBar~\\cite{Aubert:2009ad,Lees:2012cj} and KLOE~\\cite{Ambrosino:2008aa,Ambrosino:2010bv,Babusci:2012rp,Anastasi:2017eio} are required. \nSuch data are expected in the coming years from BaBar, CMD-3, BESIII, and Belle II, besides new data for other channels as well. To credibly resolve the existing tensions, especially for the $2\\pi$ channel, blind analyses are paramount. Finally, for the success of this program the development of Monte Carlo generators at NNLO accuracy is necessary, see Refs.~\\cite{Abbiendi:2022liz,WorkingGrouponRadiativeCorrections:2010bjp}. \n\nThe precision that can be obtained for data-driven evaluations of HVP strongly depends on whether or not the present tension between the BABAR and KLOE experiments, see Ref.~\\cite{Aoyama:2020ynm}, can be resolved with the upcoming advent of new $2\\pi$ analyses. If the answer to that question is affirmative, a precision of $0.3\\%$ seems feasible by 2025.  \n\n\n\\section{Lattice QCD calculations of HVP}\n\\label{sec:latticeHVP}\n\nHVP can also be computed from first principles in QCD using a non-perturbative lattice regulator. Calculations are performed in Euclidean space, and the HVP contribution is computed by a weighted integration of the correlation functions over Euclidean time. In lattice QCD calculations, the total HVP is obtained from a sum over all quark-flavors and includes connected and disconnected contractions. Almost all gauge-field ensembles generated by the various lattice collaborations to date include light sea quarks in the isospin symmetric limit ($m_u=m_d$). Hence, strong isospin-breaking corrections must be computed alongside the $\\mathcal{O}(\\alpha^3)$ QED corrections that are included in $a_\\mu^{\\rm HVP, LO}$ by definition. \nAbout $90\\%$ of the total HVP is comprised of the light-quark connected contribution, which therefore needs to be computed with subpercent precision. \n\nIn the 2020 White Paper~\\cite{Aoyama:2020ynm} the results of several collaborations published before the March 2020 deadline~\\cite{Chakraborty:2017tqp,Borsanyi:2017zdw,Blum:2018mom,Giusti:2019xct,Shintani:2019wai,FermilabLattice:2019ugu,Gerardin:2019rua,Aubin:2019usy,Giusti:2019hkz} were combined into a lattice HVP average with a total uncertainty of  approximately $2.6\\%$, shown as the blue band in the right panel of Fig.~\\ref{fig:compare}.  In 2021, a first lattice QCD result\nwith sub-percent uncertainty was published by the BMW collaboration \\cite{Borsanyi:2020mff} (BMW20). Taken in isolation, the BMW20 result yields a reduced tension with the experimental average for $a_\\mu$ of approximately\n$1.5\\sigma$, while being, at the same time, $2.1\\sigma$ away from the reference result of Ref.~\\cite{Aoyama:2020ynm} for the data-driven HVP calculation. However, the disagreement with the $R$-ratio approach becomes more pronounced\nin the intermediate Euclidean time window $a_\\mu^{\\rm HVP,~LO,~W}$, introduced\nin RBC\/UKQCD18~\\cite{Blum:2018mom}, where the integration over Euclidean time is restricted to an intermediate time region. For this quantity Ref.~\\cite{Borsanyi:2020mff} finds a\nresult which is $3.7 \\sigma$ above the corresponding data-driven evaluation.  Since the error of BMW20 is dominated by the uncertainty associated with the extrapolation to the continuum limit, it is of crucial importance to obtain results with similar precision from independent calculations employing different discretizations of the QCD action.\n\nThe Euclidean time windows allow for contributions from different Euclidean times to be studied separately. In lattice QCD calculations of the windows  one can disentangle statistical and systematic uncertainties, which affect the various Euclidean time regions differently.  Statistical noise and finite-volume uncertainties are most relevant at large Euclidean times, while discretization errors are typically enhanced at short distances.  The intermediate window, however, can be computed more easily at high precision in lattice QCD, which makes it a particularly attractive target for cross-checks between different lattice QCD calculations.  Indeed, already now multiple results with sub-percent precision are published for the isospin-symmetric light-quark connected contribution to this quantity~\\cite{Blum:2018mom,Aubin:2019usy,Lehner:2020crt,Borsanyi:2020mff}, revealing that the BMW20 result is higher by $2.2\\sigma$ compared to the one of RBC\/UKQCD18~\\cite{Blum:2018mom}, while consistent with the lattice results of Aubin19~\\cite{Aubin:2019usy} and LM20~\\cite{Lehner:2020crt}.\nFurthermore, the Euclidean windows enable a powerful cross check by studying each window quantity (short-distance, intermediate, long-distance) separately in the continuum and infinite volume limits and comparing their sum to the direct evaluation of the total HVP contribution. \n\nAt this point, additional sub-percent calculations of the intermediate Euclidean time window as well as the total HVP are crucial.  Several collaborations (including Aubin {\\it et al.} \\cite{Aubin:2021vej}, ETM \\cite{Giusti:2021dvd}, FNAL\/HPQCD\/MILC \\cite{Lahert:2021xxu,FermilabLattice:2021hzx}, Mainz \\cite{Risch:2021hty}, and RBC\/UKQCD) have on-going efforts for both quantities; however, due to the relative simplicity of $a_\\mu^{\\rm HVP,~LO,~W}$, we may expect additional results for it first.  The next generation of lattice QCD\nresults will also build on recent methodological advances made in the last years.  This includes the use of an exclusive state study for an improved long-distance computation (Mainz~\\cite{Erben:2019nmx}, RBC\/UKQCD~\\cite{Bruno:2019nzm}, and FNAL\/MILC~\\cite{Lahert:2021xxu}).  Most groups plan to include gauge-field ensembles at smaller lattice spacings to test the continuum extrapolations, which is computationally very demanding and requires adequate computational resources. Methods are also being developed to control discretization effects specifically for the short-distance contribution~\\cite{Harris:2021azd}. In addition, new theoretical insights, for example on the quark mass dependence of the light-quark contribution to $a_\\mu^{\\rm HVP}$ \\cite{Colangelo:2021moe}, may improve control over certain systematic effects. Finally, as\nlattice QCD calculations enter the sub-percent-precision era, several collaborations (FNAL\/HPQCD\/MILC and RBC\/UKQCD) have started to implement blind analyses.\n\nThe Euclidean windows can also be evaluated straightforwardly in the data-driven approach.  Once tensions between the results from different lattice collaborations are resolved, detailed comparisons of results for the windows, as well as other sub quantities (see, for example, Ref.~\\cite{Boito:2022rkw}), from lattice QCD and the data-driven approach will yield refined tests of the two approaches to HVP. In addition, assuming that any tensions between the two approaches are understood and resolved, windowed quantities may provide a useful strategy for combining lattice  and data-driven results to yield a better precision on the total $a_\\mu^{\\rm HVP,~LO}$ than each would by itself~\\cite{Blum:2018mom}.\n\nWe expect that by the end of 2022, several new results for the intermediate window quantity and one or more additional sub-percent-precision results for the total HVP will be published.  The workshops organized by the Theory Initiative will continue to provide an open platform to facilitate further cross-checks, to define quality criteria for inclusion in the next iteration of the White Paper, and to develop a method average for lattice HVP based on detailed comparisons of individual contributions and subquantities to take into account any tensions between them and obtain a conservative estimate of the lattice HVP uncertainty.  The next Theory Initiative workshop~\\cite{Higgs2022} will build on first steps towards this goal started at KEK in June 2021~\\cite{KEK2021}.\nBased on preliminary reports by several lattice QCD collaborations and assuming that any tensions between different lattice results are resolved, a lattice HVP average with $\\le 0.5\\%$ errors appears feasible by 2025.\n\n\\section{Data-driven and dispersive approach to HLbL}\n\\label{sec:dataeHLbL}\n\n\\begin{table}\n\\centering\n\\small \n\\begin{tabular}{crrrr}\n\\toprule\n  Contribution & Ref.~\\cite{Prades:2009tw}  & \n  Refs.~\\cite{Nyffeler:2009tw, Jegerlehner:2009ry} & \n  Ref.~\\cite{Jegerlehner:2017gek} & \n  Ref.~\\cite{Aoyama:2020ynm}\\\\\n\\midrule\n $\\pi^0,\\eta,\\eta'$-poles & $ 114(13) $ & $ 99(16) $ & $ 95.45\n (12.40) $ & $ 93.8(4.0) $ \n\\tabularnewline \n\n$\\pi,K$-loops\/boxes & $ -19(19) $ & $ -19(13) $ & $ -20(5) $ & \n $ -16.4(2) $\n\\tabularnewline \n$S$-wave $\\pi\\pi$ rescattering & $ -7(7) $ & $ -7(2) $ & $-5.98(1.20) $ &  \n $ -8(1) $\n\\tabularnewline \n\\midrule\nsubtotal & $88(24)$ & $73(21)$ & $69.5(13.4)$ & $69.4(4.1)$\n\\tabularnewline \n\\midrule\nscalars & $-$ & $-$ & $-$ &    \n  \\multirow{2}{*}{$\\bigg\\}\\qquad -1(3)$} \n \\tabularnewline \n\ntensors & $-$ & $-$ & $ 1.1(1) $ & \n \\tabularnewline \n\naxial vectors & $ 15(10) $ & $ 22(5) $ & $ 7.55(2.71) $ &  \n $ 6(6) $ \n\\tabularnewline \n\n~$u,d,s$-loops \/ short-distance~ & $-$ & $ 21(3) $ & $ 20(4) $\n&   $ 15(10) $   \n\\tabularnewline \\midrule\n$c$-loop & $2.3$ & $-$ & $2.3(2)$\n&   $ 3(1)$   \n\\tabularnewline \n\\midrule\ntotal & $ 105(26) $ & $ 116(39) $ & $ 100.4(28.2) $ &   \n  $ 92(19) $ \n\\tabularnewline \n\\bottomrule\n\\end{tabular}\n\\caption{Comparison of the recommendation from Ref.~\\cite{Aoyama:2020ynm} to two frequently used compilations for HLbL  from 2009 (``Glasgow\n  consensus''~\\cite{Prades:2009tw} and Jegerlehner\/Nyffeler~\\cite{Nyffeler:2009tw, Jegerlehner:2009ry}) and the recent update~\\cite{Jegerlehner:2017gek}; in units of\n  $10^{-11}$.} \n  \\label{tab:compilations}\n\\end{table}\n\nThe organization of the phenomenological estimate of the HLbL contribution from Ref.~\\cite{Aoyama:2020ynm} follows the same guiding principles as the data-driven evaluation of HVP: one considers the lowest-lying singularities of the HLbL tensor in the timelike region and explicitly estimates their contribution in a dispersive approach~\\cite{Colangelo:2014dfa,Colangelo:2014pva,Pauk:2014rfa,Colangelo:2015ama}. That this approach is sensible is shown by the clear hierarchy among the contributions of different intermediate states based on their mass. This strategy needs to be supplemented by short-distance constraints (SDCs), which are relevant for large spacelike momenta and are imposed where applicable. While this procedure becomes significantly more complicated than the analog dispersion relation in the HVP case, it does allow for an, in principle, model-independent evaluation and thus provides a convenient framework for a data-driven approach. Moreover, results from lattice QCD can be used to further constrain the required input.   \n\nThe status according to Ref.~\\cite{Aoyama:2020ynm} is summarized in Table~\\ref{tab:compilations} in comparison to earlier compilations. The first panel shows the dominant contributions from pseudoscalar poles~\\cite{Masjuan:2017tvw,Hoferichter:2018kwz,Gerardin:2019vio,Hoferichter:2018dmo}, boxes, and rescattering corrections~\\cite{Colangelo:2017fiz,Eichmann:2019bqf,Colangelo:2017qdm}, yielding about $75\\%$ of the total with well quantified uncertainties of $\\approx 6\\%$. In particular, for the $\\pi^0$-pole contribution there is agreement among Canterbury approximants~\\cite{Masjuan:2017tvw}, dispersion relations~\\cite{Hoferichter:2018kwz,Hoferichter:2018dmo}, and lattice QCD~\\cite{Gerardin:2019vio}. In this part of the evaluation, ongoing work thus mainly concerns a consolidation of the $\\eta$, $\\eta'$ poles, both using dispersion relations~\\cite{Holz:2015tcg,Holz:2022hwz} and lattice QCD~\\cite{Burri:2021cxr,Gerardin:2021fee} (see Refs.~\\cite{Miramontes:2021exi,Stamen:2022uqh} for recent work on the box contributions).\n\nThe main uncertainty arises from the second panel in Table~\\ref{tab:compilations}, which includes subleading contribution from higher intermediate states (approximated in terms of scalar, tensor, and axial-vector resonances~\\cite{Pauk:2014rta,Danilkin:2016hnh,Jegerlehner:2017gek,Knecht:2018sci,Roig:2019reh}) and the implementation of SDCs~\\cite{Melnikov:2003xd,Bijnens:2019ghy,Colangelo:2019uex,Colangelo:2019lpu}. A clear definition of individual narrow-resonance contributions and a distinction between these and the SDCs is at present affected by ambiguities~\\cite{Colangelo:2017fiz,Danilkin:2021icn,Colangelo:2021nkr}, which yet need to be resolved or better understood. For this reason uncertainties in this panel were added linearly, as the errors in this category are potentially strongly correlated. Ongoing work is aimed at improving estimates of these subleading contributions, including SDCs at higher orders~\\cite{Bijnens:2020xnl,Bijnens:2021jqo}, the implementation of these SDCs~\\cite{Leutgeb:2019gbz,Cappiello:2019hwh,Masjuan:2020jsf,Ludtke:2020moa,Leutgeb:2021mpu,Colangelo:2021nkr}, and the evaluation of narrow resonances~\\cite{Hoferichter:2020lap,Zanke:2021wiq,Danilkin:2021icn,Cappiello:2021vzi}.\n\nCrucial ingredients in this program now concern the two-photon couplings of hadronic states in the $(1\\text{--}2)\\,\\text{GeV}$ region, most prominently of axial-vector resonances: unfortunately only limited experimental information is currently available for their transition form factors~\\cite{Zanke:2021wiq,L3:2007obw,L3:2001cyf}; see also the discussion in Ref.~\\cite{Aoyama:2020ynm}. New experimental results are expected in the future, e.g., from the two-photon program at BESIII~\\cite{BESIII:2020nme}, and further constraints could be obtained from lattice QCD. \n\nIn view of these ongoing developments, and the current error estimate of $\\approx 20\\%$ (based on a linear addition of uncertainties), it seems feasible to obtain a dispersive, data-driven evaluation of the HLbL contribution  \nwith $\\le 10\\%$ total uncertainty by 2025. \n\n\n\\section{Lattice QCD calculations of HLbL}\n\\label{sec:latticeHLbL}\n\nIn Ref.~\\cite{Aoyama:2020ynm} the data-driven evaluation of HLbL scattering was combined with the first complete direct lattice QCD calculation performed by RBC\/UKQCD~\\cite{Blum:2016lnc,Blum:2019ugy} after cross-checks between the RBC\/UKQCD and Mainz group for heavier pion mass were performed.  These cross-checks were facilitated by discussions during the Theory Initiative workshops in the preceding years~\\cite{FNAL2017,KEK2018,UConn2018,Mainz2018,INT2019}.  The RBC\/UKQCD calculation uses a finite-volume regulator for the photon (the QED$_{\\rm L}$ prescription~\\cite{Hayakawa:2008an}) and is based on gauge ensembles at the physical pion mass with chirally symmetric domain-wall fermions.  \n\nAn alternative infinite-volume photon method (QED$_\\infty$) was proposed by the Mainz collaboration~\\cite{Asmussen:2016lse} and refined by both the Mainz~\\cite{Asmussen:2019act,Chao:2020kwq} and RBC\/UKQCD~\\cite{Blum:2017cer} collaborations, culminating in the publication of the second complete direct lattice calculation by the Mainz group~\\cite{Chao:2021tvp} in 2021.  The Mainz calculation uses gauge ensembles with pion mass as low as 200 MeV with Wilson-clover fermions generated by the CLS effort.\nAt this point both lattice results are compatible with each other and with the data-driven result.\n\nThe two groups continue to improve their calculations.  RBC\/UKQCD is focusing on a second result using QED$_\\infty$ as well as improvements targeting a reduction of the statistical noise.  The Mainz group will continue towards adding data at physical pion mass.  In both cases, individual calculations at or below $10\\%$ total uncertainty are feasible by 2025.  It is also expected that over the next years additional lattice collaborations may provide direct calculations of the  HLbL contribution as well.\n\n\\section{Conclusions and Outlook}\n\\label{sec:outlook}\n\nAs outlined in Secs.~\\ref{sec:dataHVP}--\\ref{sec:latticeHLbL}, it is reasonable to expect that by 2025 results for HVP and HLbL from two independent approaches will be available, each at or near the precision required to match the plans of the $g-2$ experiments. If for both HVP and HLbL data-driven and lattice determinations are found to be in good agreement, this will yield a SM prediction for $a_\\mu$ with unprecedented precision, maximizing the discovery potential of the experimental efforts.\n\nIf, on the other hand, significant tensions between data-driven and lattice results are revealed, in particular for HVP, a continued effort will be needed in order to understand where these tensions arise and how sub quantities, such as Euclidean window quantities (see Sec.~\\ref{sec:latticeHVP}), can help clarify the situation. It will also be important to explore in detail the connections between HVP, $e^+e^-$ cross sections and related low-energy parameters, as well as the hadronic corrections to the running of $\\alpha$ and the global electroweak fit~\\cite{Passera:2008jk,Crivellin:2020zul,Keshavarzi:2020bfy,Malaescu:2020zuc,Colangelo:2020lcg,Ce:2022eix}. Further insights into these connections will be provided by another complementary method for HVP, which is expected to become available over the next years at the MUonE experiment~\\cite{CarloniCalame:2015obs,Abbiendi:2016xup,MUonE:LoI,Banerjee:2020tdt}, via a space-like measurement of HVP in muon--electron scattering, with recent work addressing both the experimental realization~\\cite{Ballerini:2019zkk,Abbiendi:2019qtw,Abbiendi:2021xsh,Abbiendi:2022oks} and theory corrections~\\cite{Mastrolia:2017pfy,DiVita:2018nnh,Fael:2018dmz,Alacevich:2018vez,Fael:2019nsf,CarloniCalame:2020yoz,Banerjee:2020rww,Bonciani:2021okt,Budassi:2021twh,Nesterenko:2021byp,Balzani:2021del,Fael:2022rgm,Greynat:2022geu,Budassi:2022kqs}.\nHadronic $\\tau$-decay data can, in principle, also be used to evaluate HVP, which, however, requires a determination of the needed isospin correction~\\cite{Alemany:1997tn}. While phenomenological estimates of this correction are not sufficiently quantified, it may be possible to compute it reliably in lattice QCD~\\cite{Bruno:2018ono}. If precise lattice results for the isospin correction became available, then $\\tau$-decay data could be used to provide interesting cross checks for HVP. We note that precise measurements of $\\tau$-decay spectral functions are expected from Belle II in the coming years.  \n\nTo make optimal use of all these developments, the Muon $g-2$ Theory Initiative continues its work, with two workshops~\\cite{HVP2020,KEK2021} held after the completion of Ref.~\\cite{Aoyama:2020ynm} and the next plenary meeting to be held in September 2022~\\cite{Higgs2022}. At this meeting, concrete plans for White Paper updates will be discussed, with a timeline depending on the availability of new results especially regarding lattice-QCD calculations of HVP. \nA main update is anticipated for 2023 and will include any new available results as well as a method average for lattice HVP and HLbL. Most crucially, the Muon $g-2$ Theory Initiative will continue to facilitate interactions among the different groups and communities involved in the SM prediction of the anomalous magnetic moment of the muon, including experiment, phenomenology, and lattice QCD.   \n\nBeyond 2025, the experimental $g-2$ program will continue at  J-PARC~\\cite{Abe:2019thb}, with a completely independent experimental technique, \nand potentially even further, with the  High-Intensity Muon Beams (HIMB) project at PSI~\\cite{Aiba:2021bxe} and the Muon Campus at Fermilab~\\cite{FutMuon2021}. To fully exploit the final E989 Fermilab result and keep up with these future experiments, it is evident that a sustained effort of further improving the SM prediction for $a_\\mu$ is needed beyond 2025, even if all near-term goals laid out above can be achieved. For the data-driven evaluation of HVP, this requires a sustained program at $e^+e^-$ machines (at Belle II, BES III, and Novosibirsk), together with the calculation of higher-order radiative corrections and the development of MC generators at NNLO accuracy. For HLbL scattering, both improved input data and further theory improvements will be needed.\nFor the lattice HVP and HLbL calculations, systematic and statistical uncertainties can be improved significantly with large-scale access to future leadership-class computing facilities. Further methodological developments, for example, to more efficiently generate gauge field ensembles  with small lattice spacings and large volumes (see, for example, Refs.~\\cite{Albergo:2021bna,Hackett:2021idh,Foreman:2021ljl,Nguyen:2021zgx,Boyda:2022nmh}) as well as improved statistical noise reduction (see, for example, Refs.~\\cite{DallaBrida:2020cik,Detmold:2021ulb,Giusti:2021qhk}) would further enhance the impact of these future computational resources on the precision of the lattice QCD results. \nThe Muon $g-2$ Theory Initiative will continue to coordinate efforts along these lines, to ensure maximal return on the investments made in the experimental $g-2$ program.   \n\n\\input{acknowledgements}\n\n\\bibliographystyle{apsrev4-1_mod}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn this paper we study the ergodic theory of systems with random switchings. Such a system\ncan be described in terms of a finite family of vector fields. We assume that at any given time the\nevolution is driven by one of these vector fields, and at random times the driving vector field changes to another\none from the same family. Systems of this nature arise naturally in applications and we refer to the recent\nmonograph~\\cite{Yin:MR2559912} for motivation and extensive bibliography. \n\n Many long-term asymptotic properties of dynamical systems or random dynamical systems can be\ndescribed in terms of invariant distributions.\nThe existence of invariant measures often can be derived using the Lyapunov function technique that helps to\nestablish recurrence properties or tightness, see, e.g.,\\cite[Sections~3.3--3.4]{Yin:MR2559912}.\n\n\n\n\nThe uniqueness and absolute\ncontinuity of invariant distributions\nare often related to each other and more subtle, especially in the case that we consider in this paper where no\ndiffusion is involved, and the only source of randomness is the random sequence of driving vector fields. Although\nsome claims have been scattered through the literature, no general result is known, see, e.g.,~\\cite[Section\n8.5.2]{Yin:MR2559912}.\n\n\nThe goal of this paper is to close this gap and obtain new general conditions that guarantee uniqueness and absolute\ncontinuity of invariant measures for systems with random switchings. The two conditions that we suggest are\nformulated in terms of Lie algebras associated to the driving vector fields. They are close analogues of the\nclassical H\\\"ormander condition guaranteeing absolute continuity of transition densities of hypoelliptic diffusions.\nIn the diffusion context, this result is usually derived from the variational analysis of diffusion paths known as\nMalliavin calculus, see, e.g.,~\\cite[Chapter VIII]{Bass:MR1483890},\\cite{Bell:MR2250060},\\cite{Nualart:MR2200233}.\n\nIn fact, the central part of this paper is the analysis of transition probabilities of switched systems. Under the\nfirst of our conditions, we prove that all transition probabilities for the system have nontrivial absolutely\ncontinuous components. The second condition is more general, and it allows to prove the existence of absolutely\ncontinuous components not for the transition probabilities themselves, but for their time averages. The extraction of\nthese absolutely continuous components is largely based on classical control theory results that can be found  in\nChapter~$3$ of~\\cite{Jurdjevic:MR1425878}. These control theory results rely on earlier work by\nChow~\\cite{Chow:MR0001880}, Sussmann and Jurdjevic~\\cite{Sussmann-Jurdjevic:MR0338882}, and\nKrener~\\cite{Krener:MR0383206}. \nOur conditions and the\nstructure of our proofs match those of~\\cite{Jurdjevic:MR1425878}, where the nondegeneracy of certain\nmaps is exploited to establish the accessibility property. We use the same nondegeneracy to prove absolute continuity,\nand one can\ninterpret our result as filling the control theory with probabilistic content. In fact, the idea to use the\ngeometric control theory approach to establish regularity of Markov transition kernels along with ergodic\nproperties is not new, see, e.g., \\cite{Agrachev-Kuksin-Sarychev-Shirikyan:MR2329509} where controllability\nof the 2D Navier--Stokes system was used to prove the absolute continuity of finite-dimensional projections of\ntransition kernels. \n\n\n\n\n\n\n \n\nThe article is organized as follows: In Section~\\ref{sec:definitions} we introduce the setting, necessary notation and\nnotions from differential geometry and geometric control theory. We also state the main result on uniqueness and\nabsolute continuity of invariant measures, and two central auxiliary results on regularity of transition probabilities\neach based on one of the H\\\"ormander type assumptions. We prove these regularity results in\nSections~\\ref{sec:AC-component-1-proof} and~\\ref{sec:AC-component-2-proof}. In\nSection~\\ref{sec:AC-of-ergodic-measures} we prove that any ergodic measure has to be absolutely continuous if\nits support contains a point where hypoellipticity holds. Section~\\ref{sec:uniqueness-proof} contains the\nproof of the main result: if the hypoellipticity holds at a point that can be approached from any initial point using\nthe given vector fields as admissible controls, then there exists at most one invariant distribution, and this\ndistribution has to be absolutely continuous. In Section~\\ref{sec:examples}, we apply the main result to a switching system on the $n$-dimenisonal torus and a switching system involving two Lorenz vector fields.     \n\n\n\n{\\bf Acknowledgments:} The idea to study invariant densities for systems with switchings emerged after a discussion\nof ``blinking systems'' with Leonid Bunimovich and Igor Belykh, and we would like to thank them. We are also thankful to\nL.Bunimovich for his comments on the Lorenz system.\nWe are grateful\nto Martin Hairer and especially Jonathan Mattingly for stimulating discussions of other possible approaches to the main\nresults of this paper. The partial support from NSF through a CAREER Award DMS-0742424 is gratefully acknowledged by YB.\n\\section{Definitions, Notation, and Main Results} \\label{sec:definitions}\n\n\\subsection{The dynamics} \\label{sec:dynamics}\n\nWe consider a finite collection $D$ of smooth and forward complete vector fields \non an $n$-dimensional $C^\\infty$-manifold $M$. \n\nWe denote these vector fields by $u_i,i\\in S=\n\\{1,\\ldots,k\\}$.\nEach vector field $u$ in $D$ induces an ordinary differential equation of the\nform \n\\begin{equation*}\n\\dot{x}(t) = u(x(t)).\n\\end{equation*}\nThis differential equation is uniquely solvable if equipped with\nan initial condition\n\\begin{equation*}\nx(0) = \\xi\\in M,\n\\end{equation*}\n and forward completeness means that the solution trajectories are well-defined\nfor all times $t>0$. \n\nWe can define a stochastic process\n$X=(X_t)_{t \\geq 0}$ on $M$ in the following way: Given an initial state $i\\in S$ and an initial\nvalue $\\xi \\in M$, $X_t$ follows the trajectory generated by the vector field $u_i$  and initial\ncondition $\\xi$ for an \nexponentially distributed random time with parameter $\\lambda_i>0$. Then a new state is selected at random from\n$S\\setminus\\{i\\}$, and,\nfor another exponentially distributed random time, $X_t$ follows the new vector field corresponding to that state. \nIterating this construction  we obtain a piecewise smooth\ntrajectory $(X_t)_{t\\ge 0}$ defined for all positive times and driven by one of the vector fields from $D$\nbetween any two switchings.\nWe assume that (i) all the inter-switching times are exponentially distributed and\nindependent conditioned on the sequence of driving vector\nfields, (ii) the parameter $\\lambda_j$ of the exponential time between any two switches depends only on the current\nstate $j$, \n and (iii) the probabilities of\nswitchings between any two states are positive.\n\nIt is convenient to keep track of the driving vector fields at all times. We\ndefine $A_t\\in S$ as the index of the driving vector field at time $t$, also\nreferred to as the regime or state at time $t$. It is a Markov process with continuous time and finitely\nmany states. Its trajectories are right-continuous and piecewise constant. \n\n\nAlthough $X$ alone is not a Markov process, the joint process $(X,A)$ is Markov. We denote elements of the\nassociated Markov family, i.e., the distribution on paths emitted at $(\\xi,i)\\in M \\times S$  and generated by\nthe iterative random procedure above, by $\\mathsf{P}_{\\xi,i}$, and the corresponding transition probability measures by\n$\\mathsf{P}_{\\xi,i}^{t}$,  $t \\geq 0$. The transition probability measures are\ndefined on the product $\\sigma$-algebra $\\mathcal{B}(M) \\otimes \\mathcal{P}(S)$, where $\\mathcal{B}(M)$ is the\nBorel $\\sigma$-algebra on $M$ and $\\mathcal{P}(S)$ is the power set of $S$. We write $\\mathsf{E}_{\\xi,i}$ for expectation with\nrespect to $\\mathsf{P}_{\\xi,i}$.\n\n\n\n\n\nLet us recall that if the initial distribution of the Markov process $(X,A)$ is $\\mu$, then  the distribution of the\nprocess at time $t$ is given by the measure $\\mu \\mathsf{P}^t$ on $M\\times S$ defined by\n\\begin{equation}\n \\mu \\mathsf{P}^t(E\\times\\{j\\}) = \\sum_{i=1}^{k} \\int_{M} \\mathsf{P}_{\\xi,i}^t(E\\times\\{j\\})\\, \\mu(d\\xi\\times\\{i\\}).\n\\label{eq:convolution} \n\\end{equation}\n\n\nA probability measure $\\mu$ on $M \\times S$ is called\ninvariant for $(\\mathsf{P}^t)$ if  \n$\\mu=\\mu \\mathsf{P}^t$ for all $t \\geq 0$. \n\n\nThe main goal of this paper is to give conditions on $D$ that would guarantee absolute\ncontinuity and uniqueness of an invariant measure of the Markov semigroup $(\\mathsf{P}^t)=(\\mathsf{P}^t)_{t\\ge 0}$. The fairly general\nconditions that we suggest are formulated in geometric terms, and we proceed to introduce the\nnecessary\ndefinitions and notation.\n\n\n\\subsection{Auxiliary definitions and notation}\\label{sec:diff_geometry}\n\nLet $V(M)$ denote the set of real smooth vector fields on the manifold $M$, and let $C^{\\infty}(M)$ denote\nthe set of real-valued smooth functions on $M$. As explained above, we assume that $D$ is\ncontained in $V(M)$. Any element of $V(M)$ corresponds uniquely to a derivation on $C^{\\infty}(M)$, that is to\na linear operator $\\delta$ on $C^{\\infty}(M)$ satisfying the Leibniz rule\n\\begin{equation*}\n\\delta(f \\cdot g) = \\delta(f) \\cdot g + f \\cdot \\delta(g).\n\\end{equation*}    \nThe Lie bracket of two vector fields $u$ and $v$ in $V(M)$ is defined as the vector field  \n\\begin{equation*}\n[u,v](f):=u(v(f)) - v(u(f))\n\\end{equation*} \nfor test functions $f$ in $C^{\\infty}(M)$. The set $V(M)$ equipped with the bilinear operator $[.,.]$ becomes\na Lie algebra over the reals. A subset of $V(M)$ is called involutive if it is closed under taking the Lie bracket. An\ninvolutive subspace of $V(M)$ is called a subalgebra of~$V(M)$. \n\n\nThe smallest subalgebra of $V(M)$ that contains $D$ is denoted $\\mathcal{I}(D)$. The\nderived algebra $\\mathcal{I}'(D)$ is the smallest algebra containing \nLie brackets of vector fields in $\\mathcal{I}(D)$. We have $\\mathcal{I}'(D)\\subset \\mathcal{I}(D)$, but\n$\\mathcal{I}'(D)$ might not contain any elements of $D$ and may therefore be strictly contained in $\\mathcal{I}(D)$.\nFurther, we define $\\mathcal{I}_{0}(D)$ as the set of vector fields of the form\n\\begin{equation*}\nv + \\sum_{i=1}^{k}{\\lambda_{i}u_{i}},\n\\end{equation*}\nwhere $v \\in \\mathcal{I}'(D)$, $u_1,\\ldots,u_k \\in D$ and $\\sum_{i=1}^{k}{\\lambda_{i}}=0$. Finally, we set\n\\begin{equation*}\n\\mathcal{I}(D)(\\xi):=\\{u(\\xi):\\ u \\in \\mathcal{I}(D)\\}\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathcal{I}_{0}(D)(\\xi):=\\{u(\\xi):\\ u \\in \\mathcal{I}_{0}(D)\\}\n\\end{equation*}\nfor any $\\xi \\in M$. The sets $\\mathcal{I}(D)(\\xi)$ and $\\mathcal{I}_{0}(D)(\\xi)$ are finite-dimensional\nvector spaces. \n\n\\medskip\n\nOur main results will be based on the following assumptions that can naturally be called hypoellipticity conditions\nin analogy with H\\\"ormander's theory.\nWe say that a point $\\xi\\in M$ satisfies Condition~A if $\\dim \\mathcal{I}_{0}(D)(\\xi)=n$. \nWe say that a point $\\xi\\in M$\nsatisfies Condition~B if  $\\dim \\mathcal{I}(D)(\\xi)=n$. \n\nThe set of points satisfying Condition~A is open and so is the set of\npoints satisfying Condition~B.\n\\medskip\n\n\nFor our absolute continuity results we will need a reference measure on $M$ that will play the role of Lebesgue\nmeasure. As a smooth manifold, $M$ can be endowed with a Riemannian metric. \nThe metric tensor can be used to define measures on coordinate patches of $M$. \nOne can use then a partition of \nunity in a standard way (see, e.g., \\cite[Section 7]{Taylor:MR2245472})\nto construct a Borel measure on $M$ whose\npushforward to $\\mathbb{R}^{n}$ under any chart map is equivalent to Lebesgue measure. We call the measure on $M$\nobtained through this construction Lebesgue measure, denote it by $\\lambda^{M}$, and use\nit as the main reference measure, often omitting ``with respect to Lebesgue measure'' when writing about absolute\ncontinuity. \nThe product of the Lebesgue measure on $M$ and counting measure on $S$ will be called\nthe Lebesgue measure on\n$M\\times S$. We denote the Lebesgue\nmeasure on\n$\\mathbb{R}^{m}$ by~$\\lambda^{m}$. \n\n\\medskip\n\nIt remains to introduce the flows generated by vector fields in $D$ and the concept of reachability.\n\nFor $i\\in S$, we denote the flow function of the vector field $u_i$ by $\\Phi_i$. Due to forward completeness of $u_i$, the\nflow function is uniquely defined for all $t>0$ and $\\eta\\in M$ by\n\\begin{align*}\n \\frac{d}{dt}\\Phi_i(t,\\eta)&=u_i(\\Phi_i(t,\\eta)),\\\\\n \\Phi_i(0,\\eta)&=\\eta.\n\\end{align*}\n\nFor $m\\in\\mathbb{N}$, we will consider vectors \n${\\bf t}=(t_1,\\ldots,t_m)$ of waiting times between subsequent switches and vectors ${\\bf i}=(i_1,\\ldots,i_m)$ of\ndriving states during these waiting intervals. We will restrict ourselves to positive waiting times, but it can also be useful \n(see \\cite{Sussmann-Jurdjevic:MR0338882} and \\cite{Jurdjevic:MR1425878}) to admit flows backwards in\ntime. \n\nWe write $\\mathbb{R}_{+}$ to denote the positive real line $(0;\\infty)$. \n\nFor  ${\\bf t}=(t_1,\\ldots,t_m)\\in \\mathbb{R}_{+}^{m}$ and ${\\bf i}=(i_1,\\ldots,i_m)\\in S^{m}$, we define \n\\begin{equation*}\n\\Phi_{{\\bf i}}({\\bf t},\\xi):=\\Phi_{i_m}(t_m,\\Phi_{i_{m-1}}(t_{m-1},\\ldots\\Phi_{i_1}(t_1,\\xi))\\ldots)\n\\end{equation*}\nas the cumulative flow along the trajectories of $u_{i_1},\\ldots,u_{i_m}$ with starting point $\\xi\\in M$.\n\nThe transition probabilities $\\mathsf{P}_{\\xi,i}^{t}$ can be expressed in terms of cumulative flows. We do not specify\nthese straightforward relations in order to avoid heavy notation.  \n\n\nA point $\\eta\\in M$ is called $D$-reachable from a point $\\xi\\in M$ if there exist a time vector ${\\bf\nt}$ with\npositive components and a vector~${\\bf i}$ of driving states such that\n\\begin{equation*}\n\\eta = \\Phi_{{\\bf i}}({\\bf t},\\xi).\n\\end{equation*}\nIf the components of ${\\bf t}$ sum up to $t$, we say that $\\eta$ is $D$-reachable from $\\xi$ at time~$t$.  \n\nFor $\\xi \\in M$ and $t>0$, let $L_{t}(\\xi)$ denote the set of $D$-reachable points from $\\xi$ at time~$t$, and\nlet $L(\\xi)=\\bigcup_{t>0} L_{t}(\\xi)$ denote the set of $D$-reachable points from $\\xi$.  The points in\nthe closure $\\overline{L(\\xi)}$ can be called $D$-approachable from $\\xi$.\nLet $L=\\bigcap_{\\xi \\in M} \\overline{L(\\xi)}$ denote the set of points that are $D$-approachable from all\nother\npoints.\n\n\n\n \n\n\n\n\\subsection{Main results} \\label{sec:absolute-continuity-invariant}\n\nThe following is the main theorem of this paper.\n   \n\n\\begin{theorem} \\label{thm:uniqueness}\nSuppose Hypoellipticity Condition~B is satisfied at some $\\xi \\in L$.\nIf $(\\mathsf{P}^t)$ has an invariant measure, then it is unique and absolutely continuous with respect\nto the Lebesgue measure on $M\\times S$.\n\\end{theorem}          \n\n\\begin{remark}\\rm\nOf course, Theorem~\\ref{thm:uniqueness} remains true if we replace $L$ by any of its subsets. For example, if\none of the vector fields in $D$ has a minimal global\nattractor, then it is sufficient to check hypoellipticity for some point of the attractor.\n\\end{remark}\n\n\nUniqueness of invariant distributions is tightly connected to the regularity of the Markov semigroup. Various\naspects of regularity in connection with ergodicity have been studied in the literature: the existence of minorizing\nkernels, the strong Feller property, etc. \nThe main task in the proof of Theorem~\\ref{thm:uniqueness} is to establish regularity for\ntransition probabilities under Hypoellipticity Condition~B. However, we begin with a much stronger regularity property that \ncan be established under the stronger Hypoellipticity Condition~A.\n\n\n\n\n\n\\begin{theorem}\\label{thm:AC-component-1}\nIf Condition~A is satisfied at a point $\\xi\\in M$, then for any $i \\in\nS$ and any $t > 0$, the transition kernel $\\mathsf{P}_{\\xi,i}^{t}$ has a nonzero absolutely\ncontinuous component with respect to Lebesgue measure on $M \\times S$. \n\\end{theorem}     \n  \n\nUnder the weaker Condition~B it may happen that none of the transition \nprobability measures $\\mathsf{P}_{\\xi,i}^t$, $t > 0$, has a nonzero absolutely continuous component. For example, let $M$~be\nthe\n$n$-dimensional torus $\\mathbb{T}^n=\\mathbb{R}^n\/\\mathbb{Z}^n$, and let \n$D=\\{u_1,\\ldots,u_n\\}$ be the standard basis in $\\mathbb{R}^{n}$. Fix an arbitrary time $t > 0$. The set of\npoints $D$-reachable from the origin at time $t$ is the image of \n\\begin{equation*}\n\\biggl\\{(s_1,\\ldots,s_n) \\in [0;\\infty)^{n}: \\quad \\sum_{j=1}^{n} s_j =t  \\biggr\\}\n\\end{equation*}\nunder the covering map $\\mathbb{R}^n\\to\\mathbb{T}^n$,\nand has Lebesgue measure zero, so $\\mathsf{P}_{\\xi,i}^t$ is a purely singular measure.\n  \nNevertheless, Condition~B guarantees that time averages of transition probabilities have\nnontrivial absolutely continuous components.  Specifically, we will establish this for\nthe resolvent probability kernel $\\mathsf{Q}_{\\xi,i}$ defined by \n\\begin{equation}\n\\label{eq:resolvent}\n\\mathsf{Q}_{\\xi,i}(E\\times\\{j\\}):= \\int_{\\mathbb{R}_{+}} e^{- t}\\,\\mathsf{P}_{\\xi,i}^t(E\\times\\{j\\}) dt.\n\\end{equation}\nThe resolvent kernels are useful in the study of invariant distributions due to the following straightforward result.\n\\begin{lemma}\\label{lem:Q-invariance}\nIf a measure $\\mu$ is $(\\mathsf{P}^t)$-invariant it is also $(\\mathsf{Q})$-invariant, i.e., \n$\\mu=\\mu \\mathsf{Q}$, where the convolution $\\mu\\mathsf{Q}$ is defined analogously to~\\eqref{eq:convolution}. \n\\end{lemma}\n\n\\begin{theorem}\\label{thm:AC-component-2}\nIf Condition B is satisfied at some point $\\xi\\in M$, then for any $i\\in S$,\nthe measure $\\mathsf{Q}_{\\xi,i}$ defined by~\\eqref{eq:resolvent}\nhas a nonzero absolutely continuous component with respect to Lebesgue measure on $M \\times S$. \n\\end{theorem}\n\n\n\n\n\n\\bigskip\n\n\n\n\nAlthough the convergence of transition probabilities to the invariant measures is out of\nthe scope of the present paper and will be addressed elsewhere, our analysis suggests that under the assumptions of\nTheorem~\\ref{thm:uniqueness} and Condition~A at $\\xi$ this convergence holds true while Condition~B at $\\xi$ implies\nonly Ces\\`aro convergence.\n\n\n\n\nAt the heart of our proofs of Theorems~\\ref{thm:AC-component-1} and~\\ref{thm:AC-component-2} are classical results\nfrom  geometric control theory that can be found in~\\cite{Jurdjevic:MR1425878}. The statements we present are derived\nfrom Theorems~$3.1$,~$3.2$, and~$3.3$ in~\\cite{Jurdjevic:MR1425878}. Analogous results for the special case of analytic\nvector fields on a real analytic manifold are first stated in~\\cite[Theorems~$3.1$ and\n$3.2$]{Sussmann-Jurdjevic:MR0338882}. In their paper, Sussmann and Jurdjevic were able to build on prior\nwork~\\cite{Chow:MR0001880} by Chow who considered symmetric families of analytic vector fields.\nKrener generalized these results to $C^{\\infty}$-vector fields in~\\cite{Krener:MR0383206}.       \n\n\nRecall that a regular point of a function $f:\\mathbb{R}^{m} \\to M$ is a point ${\\bf t} \\in \\mathbb{R}^{m}$ such that the differential $Df({\\bf t})$ has full rank. \nIf $Df({\\bf t})$ has deficient rank, ${\\bf t}$ is called a critical point of~$f$.\n\n\n\\begin{theorem} \\label{thm:Jurdjevic-1} Assume that Condition~A holds at\nsome $\\xi\\in M$.\nThen:\n\\begin{enumerate}\n \\item For any $i,j\\in S$,\nthere are an integer $m>n$\nand  a vector ${\\bf i} \\in S^{m+1}$ with $i_1=i$ and $i_{m+1}=j$ such that for any $t>0$ the mapping $f_\\mathbf{i}:\\mathbb{R}_{+}^{m} \\to M$\ndefined by\n\\begin{equation}\n\\label{eq:flow_for_transition_probability}\nf_\\mathbf{i}(t_1,\\ldots,t_{m})=\\Phi_{{\\bf i}}\\biggl(t_1,\\ldots,t_{m},t-\\sum_{l=1}^{m} t_l, \\xi \\biggr)\n\\end{equation}\nhas a nonempty open set of regular points in the simplex \n\\begin{equation*}\n\\Delta_{t,m}:=\\biggl\\{(t_1,\\ldots,t_m) \\in \\mathbb{R}_{+}^{m}:\\ \\sum_{l=1}^{m} t_l < t \\biggr\\}.\n\\end{equation*}\n\\item The interior of $L(\\xi)$ is nonempty and dense in $L(\\xi)$.\n\\end{enumerate}\n\\end{theorem} \n \n\\begin{theorem} \\label{thm:Jurdjevic-2}\nAssume that Condition~B holds at some $\\xi\\in M$. \nThen:\n\\begin{enumerate}\n \\item For any $i,j\\in S$,\nthere are an integer $m>n$\nand  a vector ${\\bf i} \\in S^{m+1}$ with $i_1=i$ and $i_{m+1}=j$ such that for any $t>0$ the mapping $F_\\mathbf{i}:\\mathbb{R}_{+}^{m+1} \\to M$\ndefined by\n\\begin{equation*}\nF_\\mathbf{i}(t_1,\\ldots,t_{m+1})=\\Phi_{{\\bf i}}(t_1,\\ldots,t_{m+1},\\xi)\n\\end{equation*}\nhas a nonempty open set of regular points in $\\Delta_{t,m+1}$.\n\\item The interior of $L(\\xi)$ is nonempty and dense in $L(\\xi)$.\n\\end{enumerate}\n\\end{theorem}\n\nCondition~A is stronger than Condition~B, so it is not surprising that the conclusion of\nTheorem~\\ref{thm:Jurdjevic-1} implies the conclusion of Theorem~\\ref{thm:Jurdjevic-2}.\n\nThe first statement of Theorem~\\ref{thm:Jurdjevic-2} corresponds to Theorem~$3.1$ in~\\cite{Jurdjevic:MR1425878}, which reads as follows: Under the assumptions of Theorem~\\ref{thm:Jurdjevic-2}, any neighborhood $U$ of $\\xi$ contains points that are normally accessible from $\\xi$ at arbitrarily small times. A point $\\eta$ in $M$ is called normally accessible from $\\xi$ at time $t>0$ if there exist vectors $\\mathbf{i} \\in S^{m+1}$ and $(\\hat{t}_1,\\ldots,\\hat{t}_{m+1}) \\in \\Delta_{t,m+1}$ such that $F_\\mathbf{i}(\\hat{t}_1,\\ldots,\\hat{t}_{m+1}) = \\eta$ and the differential $DF_\\mathbf{i}(\\hat{t}_1,\\ldots,\\hat{t}_{m+1})$ has full rank. It's worth pointing out, though, that in~\\cite{Jurdjevic:MR1425878} only one sequence~$\\mathbf{i}$ resulting in $F$ with a regular point is constructed. But since the flow generated by any vector field is a family of diffeomorphisms, and since the set of points satisying Condition~B is open, one can append any indices in front or at the back of that sequence without destroying the desired properties, and thus recover this part of Theorem~\\ref{thm:Jurdjevic-2} as we state it. \n\nThe fact that the interior of $L(\\xi)$ is nonempty and dense in $L(\\xi)$ follows from Theorem~$3.2.a$ in~\\cite{Jurdjevic:MR1425878}.\n\nTheorem~\\ref{thm:Jurdjevic-1} essentially follows from applying Theorem~$3.1$ (\\cite{Jurdjevic:MR1425878}) to $\\mathbb{R} \\times M$ and\nvector fields ${\\bf 1} \\oplus u_i, i\\in S$, where\n\\begin{equation*}\n({\\bf 1} \\oplus u)(r,\\xi):= (1,u(\\xi)),\\quad (r,\\xi)\\in \\mathbb{R} \\times M,\n\\end{equation*}\nand ${\\bf 1}$ is the unit vector field on $\\mathbb{R}$ corresponding to\nthe derivation $\\partial\/\\partial r$ and identically equal to~$1$ in the natural coordinates on $\\mathbb{R}$. \n\n\n\n\n\\section{Proof of Theorem~\\ref{thm:AC-component-1}} \\label{sec:AC-component-1-proof}   \n\nWe need to prove that for any $t>0$ and $i \\in S$, the measure $\\mathsf{P}^t_{\\xi,i}$ is not singular.\n\n\\medskip\n\nFor any finite sequence ${\\mathbf{i}}$ of indices in $S$ with initial index $i$ (we will call these sequences \nadmissible), let $C_{{\\mathbf{i}}}$ be the event that the driving vector fields up to time $t$ appear in the order determined by ${\\mathbf{i}}$. Since\n$\\mathsf{P}_{\\xi,i}(C_{\\mathbf{i}})>0$ for any admissible $\\mathbf{i}$\nit suffices to find an admissible sequence $\\mathbf{i}$ such that \n$\\mathsf{P}_{\\xi,i}^t(\\cdot| C_{\\mathbf{i}})$ is not singular.\nWe claim that this holds true for the the sequence $\\mathbf{i}$ provided by \nTheorem~\\ref{thm:Jurdjevic-1}. According to\nTheorem~\\ref{thm:Jurdjevic-1}, there is an admissible sequence $\\mathbf{i}=(i_1,i_2,\\ldots,i_{m+1})$ with $i_1 = i$ \nsuch that the function\n$f_\\mathbf{i}$\nhas a regular point in $\\Delta_{t,m}$. Since the set of regular points of a differentiable function is open in its domain, the function $f_\\mathbf{i}$ is regular in a nonempty open set $B \\subset \\Delta_{t,m}$.  \n\nLet $T_1,T_2,\\ldots,T_{m+1}$ be independent and exponentially distributed random variables such that $T_j$ has\nparameter\n$\\lambda_{i_j}$ for $1\\leq j \\leq m+1$.  \n\nOn $C_\\mathbf{i}$ we have $A_t=i_{m+1}$, and the distribution of $X_t$ under $\\mathsf{P}_{\\xi,i}(\\cdot | C_{\\mathbf{i}})$ coincides\nwith the distribution of $f_\\mathbf{i}(T_1,\\ldots,T_m)$ conditioned on the event\n\\begin{equation}\n\\label{eq:event_R}\nR=\\biggl\\{\\sum_{j=1}^{m} T_j < t \\le \\sum_{j=1}^{m+1} T_j \\biggr\\}.\n\\end{equation}\nThe distribution of the random vector $(T_1,\\ldots,T_m)$ conditioned on $R$, \nis equivalent to the uniform distribution on the simplex \n\\begin{equation*}\n\\Delta_{t,m} := \\biggl\\{(t_1,\\ldots,t_m) \\in \\mathbb{R}_{+}^{m}: \\quad \\sum_{j=1}^{m}t_j < t \\biggr\\}.\n\\end{equation*}\nNow the theorem directly follows from the following result:\n\\begin{lemma}\\label{lem:pushforward_full_rank} Let $n,m\\in\\mathbb{N}$, $n\\le m$.\nSuppose that $B$ and $\\Delta$ are nonempty open sets in~$\\mathbb{R}^m$, $B$ is dense in $\\Delta$,\nand $M$ is an $n$-dimensional\nsmooth manifold.\nIf $f:\\Delta\\to M$ is \ndifferentiable on $\\Delta$ and all points in $B$ are regular for $f$, then for any\nabsolutely continuous\nprobability measure $\\mu$ on $\\Delta$ satisfying $\\mu(B)>0$, its\npushforward\n$\\mu f^{-1}$ is not singular with respect to~$\\lambda^M$.\n\\end{lemma}\n         \nWe will prove this lemma only for $M=\\mathbb{R}^n$.  Modifying the proof for the general case using\ncoordinate patches on $M$ amounts only to notational differences. \n\nWe will use the following statement (see, e.g., Proposition~4.4\nin~\\cite{Davydov-et-al:MR1604537}):\n\\begin{lemma}\\label{lem:pushforward_for_nonzero_det}\n Let  $f:B \\to\\mathbb{R}^m$ be a Borel function a.e.-differentiable on an open set $B \\subset\\mathbb{R}^m$\nand satisfying  $\\lambda^m\\{\\mathbf{t}\\in B:\\, \\det Df(\\mathbf{t})=0\\}=0$. If $\\mu\\ll \\lambda^m$, then\n$\\mu f^{-1}\\ll\\lambda^m$, and\n\\[\n \\frac{d(\\mu f^{-1})}{d\\lambda^m}(\\mathbf{s})=\\sum_{\\mathbf{t}\\in B: f(\\mathbf{t})=\\mathbf{s}} |\\det Df(\\mathbf{t})|^{-1}\\frac{d\\mu}{d\\lambda^m}(\\mathbf{t})\n\\]\n\n\\end{lemma}\n\n\n\n\n\\bpf[Proof of Lemma~\\ref{lem:pushforward_full_rank}]We must prove that for any $E \\subset \\mathbb{R}^n$ with $\\lambda^n(E)=0$, \n\\begin{equation}\n\\label{eq:not_singular}\n\\mu\\{\\mathbf{t}\\in\\Delta:\\, f(\\mathbf{t})\\in E\\}<1. \n\\end{equation}\nFor any $\\mathbf{t} \\in B$, we can find some $n$\ncolumns of $Df(\\mathbf{t})$ (without loss of generality, first~$n$ columns) that are linearly\nindependent.\nFor $\\rho:B \\to \\mathbb{R}^n \\times \\mathbb{R}^{m-n}$ defined by\n\\[\n\\rho:\\mathbf{t}=(t_1,\\ldots,t_m) \\mapsto (f(\\mathbf{t}), t_{n+1},\\ldots,t_m),\n\\]\nand any $\\mathbf{t}\\in B$, we have\n\\begin{equation*}\n\\det D\\rho(\\mathbf{t} ) \\neq 0. \n\\end{equation*}\n Since $E \\times \\mathbb{R}^{m-n}$ is a set of measure zero, Lemma~\\ref{lem:pushforward_for_nonzero_det} implies\nthat \n\\begin{equation*}\n0 = \\lambda^{m}\\{\\mathbf{t} \\in B:\\, \\rho(\\mathbf{t}) \\in E \\times \\mathbb{R}^{m-n}\\} = \\lambda^{m}\\{\\mathbf{t} \\in B:\\, f(\\mathbf{t}) \\in E\\}.\n\\end{equation*}\nTherefore, $\\mu\\{\\mathbf{t} \\in B:\\, f(\\mathbf{t}) \\in E\\}=0$, and\n\\[\n \\mu\\{\\mathbf{t}\\in\\Delta:\\, f(\\mathbf{t})\\in E\\}= \\mu\\{\\mathbf{t}\\in \\Delta \\setminus B:\\, f(\\mathbf{t})\\in E\\}\\le 1 -\\mu(B)<1. \n\\]\nso \\eqref{eq:not_singular} is established and the proof is complete.\n{{\\hfill $\\Box$ \\smallskip}}\n \n\n\n\\section{Proof of Theorem~\\ref{thm:AC-component-2}}  \\label{sec:AC-component-2-proof} \n\n\n \n We need to show that $\\mathsf{Q}_{\\xi,i}$ is not a singular measure. The proof is based on\nTheorem~\\ref{thm:Jurdjevic-2}.\n\nFor the $S$-valued process $A$ we denote by $I_t(A)$ the sequence of states visited by $A$ between $0$\nand $t$.\nFor any $m\\in\\mathbb{N}$ and any sequence $\\mathbf{i}\\in S^m$, we can introduce an auxiliary measure $\\mathsf{Q}_{\\xi,i,\\mathbf{i}}$ on $M$ by \n\\[\n\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(B)=\\int_{R_{+}} e^{-t}\\,\\mathsf{P}_{\\xi,i}\\{X_t\\in B\\ \\text{and}\\ I_t(A)=\\mathbf{i}\\}\\,  dt,\\quad B\\in\\mathcal{B}(M).\n\\]\nSince \n\\begin{equation}\n\\label{eq:Qq-via-I_t}\n\\mathsf{Q}_{\\xi,i}(B \\times \\{j\\})=\\sum_{m}\\sum_{\\mathbf{i}=(i,i_2,\\ldots,i_{m-1},j)\\in S^m}\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(B), \n\\end{equation}\nit is sufficient to find $\\mathbf{i}=(i_1,\\ldots,i_m)$ with $i_1=i$ such that $\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(M)>0$ and\n\\[\n\\overline \\mathsf{Q}_{\\xi,i,\\mathbf{i}}(\\cdot)= \\frac{\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(\\cdot)}{\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(M)} \n\\]\n is a nonsingular probability measure. To apply Lemma~\\ref{lem:pushforward_full_rank}, we need to represent\n$\\overline \\mathsf{Q}_{\\xi,i,\\mathbf{i}}$ as the pushforward of a measure, equivalent to Lebesgue measure, under a smooth map with a \nnonempty set of regular points.\n\n\nSince Condition~B holds at $\\xi$, Theorem~\\ref{thm:Jurdjevic-2} yields an integer $m>n$\nand a sequence $\\mathbf{i}=(i_1,i_2,\\ldots,i_{m+1})$ with $i_1=i$, such that the function $F_\\mathbf{i}:\\mathbb{R}_{+}^{m+1} \\to\nM$ defined by\n\\begin{equation*}\nF_\\mathbf{i}(\\mathbf{t})= \\Phi_{\\mathbf{i}}(\\mathbf{t},\\xi)\n\\end{equation*}\nhas a regular point.\nFor this $\\mathbf{i}$ provided by Theorem~\\ref{thm:Jurdjevic-2}, $\\overline \\mathsf{Q}_{\\xi,i,\\mathbf{i}}$ is \nthe distribution of $\\Phi_{{\\bf i}} \\left(T_1,\\ldots,T_m,T-\\sum_{j=1}^{m} T_j, \\xi \\right)$ conditioned on\n\\begin{equation}\nR= \\biggl\\{\\sum_{j=1}^{m} T_j <T \\leq \\sum_{j=1}^{m+1} T_j \\biggr\\},\n\\label{eq:event_R_2}\n\\end{equation}\nwhere $T_1,\\ldots,T_{m+1}$, and $T$ are independent random variables \nexponentially distributed with parameters\n$\\lambda_{i_1},\\ldots,\\lambda_{i_{m+1}}$, and $1$, respectively. \n\n\nSince the joint distribution of $T_1,\\ldots,T_{m+1},T$ is equivalent to Lebesgue measure and since event $R$ has positive probability,\nthe distribution $\\mu$ of $T_1,\\ldots,T_m,T$ conditioned on $R$\ninduces a measure on\n\\[\n\\Delta= \\biggl\\{(t_1,\\ldots,t_m,t) \\in \\mathbb{R}_{+}^{m+1}: \\quad \\sum_{j=1}^{m} t_j < t \\biggr\\}\n\\]\nthat is equivalent to Lebesgue measure. The regularity of $F_\\mathbf{i}$ guaranteed by Theorem~\\ref{thm:Jurdjevic-2} implies\nthat\nthe function $f_\\mathbf{i}:\\Delta\\to M$ defined \nby\n\\begin{equation*}\n f_\\mathbf{i}(t_1,\\ldots,t_m,t)= F_\\mathbf{i}\\biggl(t_1,\\ldots,t_m,t-\\sum_{j=1}^{m} t_j \\biggr)\n\\end{equation*}\nhas a nonempty open set of regular points in~$\\Delta$, and the proof is completed by an application of\nLemma~\\ref{lem:pushforward_full_rank}, since $\\overline \\mathsf{Q}_{\\xi,i,\\mathbf{i}}$ is the pushforward of \n$\\mu$ under $f_\\mathbf{i}$. \n\n\n\n\n\n\\section{Absolute continuity of ergodic invariant measures}\\label{sec:AC-of-ergodic-measures}\n\nAccording to the Ergodic Decomposition Theorem, all invariant measures for a Markov semigroup can be\nrepresented as\nconvex combinations of ergodic ones (see, e.g., \\cite[Theorem~5.7]{Hairer:ergodicity-lectures}). We will use this to derive Theorem~\\ref{thm:uniqueness} from absolute continuity of ergodic invariant distributions.\n\n\nTo define ergodicity, we need to recall the notion of $\\mu$-invariant sets. Let $\\mu$ be an invariant measure for the\nMarkov semigroup $(\\mathsf{P}^t)$. We say that a set $A\\in\\mathcal{B}(M)\n\\otimes \\mathcal{P}(S)$ is $\\mu$-invariant if for every $t\\ge 0$, $\\mathsf{P}^t_{\\xi,i}(A)=1$ for $\\mu$-almost every\n$(\\xi,i)\\in A$. An invariant measure $\\mu$ is called ergodic if for every $\\mu$-invariant set $A$, either  $\\mu(A)=1$ or\n$\\mu(A)=0$.\n\n\nThe following is a basic result on systems with Markov switchings that does not use Conditions A or B.\n\n\n\\begin{theorem}\\label{thm:AC-or-S}\nIf $\\mu$ is $(\\mathsf{P}^t)$-invariant and ergodic then it is either absolutely continuous or singular. \n\\end{theorem}\n\\bpf\nConsider the Lebesgue decomposition $\\mu=\\mu_{ac}+\\mu_{s}$, where $\\mu_{ac}$ is absolutely\ncontinuous and $\\mu_{s}$ is singular with respect to Lebesgue measure.  Let us show that both\n$\\mu_{ac}$ and $\\mu_{s}$ are invariant.\n\nFor any $t>0$, using the\ninvariance of $\\mu$, we can write\n\\begin{align}\n\\mu_{ac}+\\mu_{s} &= \\mu=\\mu \\mathsf{P}^t\n\\label{eq:decomposition_in_ac_and_s}\n=\n\\mu_{ac} \\mathsf{P}^t+\\mu_s\\mathsf{P}^t\n=\\sum_{j=1}^{k} \\nu_j  + \\mu_s\\mathsf{P}^t,\n\\end{align}\nwhere \n\\begin{equation}\n\\label{eq:nu_j}\n\\nu_{j}(\\cdot)= \\int_M \\mathsf{P}^t_{\\xi,j} (\\cdot)\\mu_{ac}(d\\xi\\times\\{j\\}),\\quad j\\in S.\n\\end{equation}\n\n\nWe claim that the measures $\\nu_j$, $j\\in S$,\nare absolutely continuous. To see this we\ncheck that for any sequence $\\mathbf{i}=(i_1,\\ldots,i_{m+1})$ with $i_1=j$, the measure $\\nu_\\mathbf{i}$ defined by\n\\begin{align}\\notag\n \\nu_{\\mathbf{i}}(E)&= \\int_M \\mathsf{P}_{\\xi,j} (X_t\\in E\\, |C_\\mathbf{i})\\mu_{ac}(d\\xi\\times\\{j\\})\\\\\n&=\\int_M \\mathsf{P}\\biggl(\\Phi_{{\\bf i}}\\biggl(T_1,\\ldots,T_m,t-\\sum_{l=1}^{m} T_l,\\xi \\biggr)\\in\nE\\, \\biggr|\\, R\\biggr)\\mu_{ac}(d\\xi\\times\\{j\\})\\label{eq:one_of_continuous_components_disintegrated}\n\\end{align}\n is absolutely continuous (here we use the notation introduced in Section~\\ref{sec:AC-component-1-proof}). Suppose\n$\\lambda^M(E)=0$. For fixed $T_1,\\ldots,T_m,T_{m+1}$,  the map $\\Phi_\\mathbf{i}$ is a diffeomorphism in $\\xi$. Therefore, on\nevent $R$ introduced in~\\eqref{eq:event_R}, we have\n\\[\n \\mu_{ac}\\biggl(\\xi \\times \\{j\\}:\\ \\Phi_{{\\bf i}}\\biggl(T_1,\\ldots,T_m,t-\\sum_{l=1}^{m} T_l,\\xi \\biggr)\\in\nE  \\biggr)=0,\n\\]\nand $\\nu_\\mathbf{i}(E)=0$ follows from disintegrating the right side\nof~\\eqref{eq:one_of_continuous_components_disintegrated} and changing the order of integration.\n\nNow, using~\\eqref{eq:decomposition_in_ac_and_s} and the absolute continuity of $\\nu_j$, $j\\in S$, we can write\n\\begin{equation}\n\\label{eq:decomposition_of_ac_part}\n\\mu_{ac}= \n\\sum_{j=1}^{k} \\nu_j  + (\\mu_s\\mathsf{P}^t)_{ac}\\ .\n\\end{equation}\nSince  $\\mathsf{P}^t_{\\xi,j}(M\\times S)=1$ for all $\\xi$ and $j$, \\eqref{eq:nu_j} implies\n$\\sum_{j=1}^{k} \\nu_j(M\\times S)= \\mu_{ac}(M\\times S)$. Therefore, applying~\\eqref{eq:decomposition_of_ac_part}\nto $M \\times S$, we obtain that the absolutely continuous\ncomponent of the measure $\\mu_s\\mathsf{P}^t$ is zero. In other\nwords, $\\mu_s\\mathsf{P}^t$ is singular, and from~\\eqref{eq:decomposition_in_ac_and_s} and the absolute\ncontinuity of $\\nu_j$, $j\\in S$, we obtain\n\\begin{equation}\n\\label{eq:invariance_of_singular_component}\n\\mu_s = \\mu_s\\mathsf{P}^t.\n\\end{equation}                        \nIn other words, $\\mu_s$ is invariant for $(\\mathsf{P}^t)$. It follows from~\\eqref{eq:decomposition_in_ac_and_s}\nthat $\\mu_{ac}$ is also invariant. Since $\\mu$ is ergodic, it cannot be represented as a sum of two nontrivial\ninvariant measures. This means that either $\\mu=\\mu_{ac}$ or $\\mu=\\mu_s$.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\n\nWe endow the state space $S$ with the discrete topology and recall that a point $(\\xi,i)\\in M\\times S$ is\ncontained in the support of a measure if and\nonly if the measure of every open neighborhood of $(\\xi,i)$ is positive.  \n \n\\begin{theorem}\\label{thm:absolute-continuity-for-ergodic}\nLet $\\mu$ be an ergodic invariant measure for $(\\mathsf{P}^t)$. Assume that the support of $\\mu$ contains a point\n$(\\eta,i)$ such that Condition~B holds at~$\\eta$. \nThen, $\\mu$ is absolutely continuous\nwith\nrespect to Lebesgue measure on $M\\times S$.  \n\\end{theorem}\n\n\n\n\nWe will need several auxiliary statements. \n\n\n\\begin{lemma} \\label{lem:support}\nLet $\\nu$ be a finite Borel measure on $M \\times S$ with support $K$. If $U$ is any open set in $M \\times\nS$ whose intersection with $K$ is nonempty, we have \n\\begin{equation*}\n\\nu(U \\cap K)>0.\n\\end{equation*}\n\\end{lemma} \n \n\\bpf\nAssume that $\\nu(U \\cap K)=0$. The complement of the support of $K$ has measure zero. Therefore\n\\begin{equation*}\n\\nu(U) = \\nu(U \\cap K) + \\nu(U \\cap K^{c})=0.\n\\end{equation*}\nThus, $U^{c}$ is a closed subset of $M \\times S$ whose complement has measure zero. From the\ndefinition of the support, we obtain $K \\subset U^{c}$. But then, $U \\cap K$ must be empty, a contradiction. \n{{\\hfill $\\Box$ \\smallskip}}\n\n\n\n\n\\begin{lemma} \\label{lem:open-starting-points}\nFix a time $t>0$, an open subset $E$ of $M$ and an index $j \\in S$. The set $H$ of points $(\\xi,i) \\in M \\times\nS$ satisfying $\\mathsf{P}^t_{\\xi,i}(E\\times\\{j\\})>0$ is open. \n\\end{lemma}\n\n\\bpf We will use the notation introduced in Section~\\ref{sec:AC-component-1-proof}.\n\nLet us assume that $H$ is not open and there is a sequence $((\\xi_p,i))_{p\\in\\mathbb{N}}$ in $H^c$ \nconverging to a point $(\\xi,i)\\in H$. \n\nSince $(\\xi,i)$ is an element of $H$, there exists an admissible sequence \\\\ $\\mathbf{i}=(i,i_2,...,i_m,j)$ for which\n$\\mathsf{P}_{\\xi,i}(X_t\\in E|C_\\mathbf{i})>0$.\n\nOn the other hand, we have $\\mathsf{P}_{\\xi_p,i}(X_t\\in E|C_\\mathbf{i})=0$ for all $p\\in\\mathbb{N}$. Therefore,\nfor each $p$, the set $B(\\xi_p)$\nhas zero Lebesgue measure, where\n\\[\nB(\\eta)=\\biggl\\{\\mathbf{t}\\in\\Delta_{t,m}:\\, \\Phi_{{\\bf i}}\\biggl(t_1,\\ldots,t_m,t-\\sum_{j=1}^{m} t_j,\\eta \\biggr)\\in E\n\\biggr\\},\\quad \\eta\\in M.\n\\]\n\nIf $\\mathbf{t}\\in B(\\xi_p)^c$ for all $p$,\nthen $\\Phi_{\\mathbf{i}}(\\mathbf{t},\\xi)\\in E^c$ since $\\Phi_{\\mathbf{i}}(\\mathbf{t},\\cdot)$ is a diffeomorphism and $E^c$ is a closed set.\nHence,\nthe Lebesgue measure of $B(\\xi)$ is also zero, so  \n$\\mathsf{P}_{\\xi,i}(X_t\\in E|C_\\mathbf{i})=0$, which contradicts our assumption and finishes the proof.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n  \n\n\\bpf[Proof of Theorem~\\ref{thm:absolute-continuity-for-ergodic}]  According to Theorem~\\ref{thm:AC-or-S} we need to show\nthat $\\mu$ is not singular. If $\\mu$ is singular,\nit is entirely supported on a zero Lebesgue measure set $G\\subset M\\times S$, so\n$\\mu(G^c)=0$. Since $\\mu$ is $(\\mathsf{P}^t)$-invariant, it is also $\\mathsf{Q}$-invariant. Therefore,\n$\\mu(G^c)=\\mu \\mathsf{Q}(G^c)$, and we see that $\\mu(V)=0$ where\n\\[\n V=\\{(\\xi,j)\\in M \\times S:\\ \\mathsf{Q}_{\\xi,j} (G^{c})>0\\}.\n\\]\nLet $U$ be the set of points $\\xi \\in M$ where Condition~B holds.\nDue to Theorem~\\ref{thm:AC-component-2}, $U\\times S \\subset V$, and\nwe conclude that $\\mu(U\\times S)=0$.\n\n\nRecall that $U$ is an open subset of $M$, and $(U\\times S)\\cap \\mathop{\\rm supp} \\mu\\ne\\emptyset$ by assumption.\nLemma~\\ref{lem:support} implies that  $\\mu((U\\times S)\\cap \\mathop{\\rm supp} \\mu)>0$.  The contradiction with $\\mu(U\\times\nS)=0$ completes the proof.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\nOf course, if one replaces Condition~B in Theorem~\\ref{thm:absolute-continuity-for-ergodic} with \nthe stronger Condition~A the resulting statement holds automatically, but one can give a proof that does not involve\nthe resolvent $Q$:\n\n\\begin{theorem}\\label{thm:absolute-continuity-for-ergodic-under-condition-A}\nLet $\\mu$ be an ergodic invariant measure for $(\\mathsf{P}^t)$. Assume that the support of $\\mu$ contains a point\n$(\\eta,i)$ such that Condition~A holds at~$\\eta$.\nThen, $\\mu$ is absolutely continuous\nwith\nrespect to the Lebesgue measure on $M\\times S$.  \n\\end{theorem}\n\n\n\\bpf  According to Theorem~\\ref{thm:AC-or-S} we need to show that $\\mu$ is not singular. If $\\mu$ is singular,\nit is entirely supported on a zero Lebesgue measure set $G\\subset M\\times S$, so\n$\\mu(G^c)=0$. Since $\\mu(G^c)=\\mu \\mathsf{P}^t(G^c)$, we see that $\\mu(V)=0$ where\n\\[\n V=\\{(\\xi,j)\\in M \\times S:\\ \\mathsf{P}^t_{\\xi,j} (G^{c})>0\\}.\n\\]\nLet $U$ be the set of points $\\xi \\in M$ where Condition~A holds.\nDue to Theorem~\\ref{thm:AC-component-1}, $U\\times S \\subset V$, and\n we conclude that $\\mu(U\\times S)=0$.\n\n\nRecall that $U$ is an open subset of $M$, and $(U\\times S)\\cap \\mathop{\\rm supp} \\mu\\ne\\emptyset$ by assumption.\nLemma~\\ref{lem:support} implies that  $\\mu((U\\times S)\\cap \\mathop{\\rm supp} \\mu)>0$.  The contradiction with $\\mu(U\\times\nS)=0$ completes the proof.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{thm:uniqueness}} \\label{sec:uniqueness-proof}\n\nFirst, we establish two properties of the set $E= L\\cap U$, where\n$U$ is the open set of points satisfying Condition B.\n\\begin{lemma} \\label{lem:non-empty-interior}\nThe set $E$ has nonempty interior.\n\\end{lemma}\n\n\\bpf  By assumption,  $\\xi\\in E$ , so $U\\ne\\emptyset$ and $L(\\xi) \\cap U\\ne \\emptyset$ by continuity of the vector fields in $D$. Since $\\xi\\in U$, Theorem~\\ref{thm:Jurdjevic-2} implies that $L(\\xi)$\nhas nonempty interior that is dense in $L(\\xi)$. Therefore, the set\n\\begin{equation*}\nV=L(\\xi)^{\\circ} \\cap U\n\\end{equation*}\nis nonempty and open. Clearly, $V\\subset U$, and it remains to prove that $L(\\xi)^{\\circ}\\subset L$.\nIt is sufficient to show that  $W\\cap L(\\eta)\\ne\\emptyset$ for any nonempty open subset $W$ of $L(\\xi)$ and any $\\eta \\in M$.\n\nLet us take any $\\zeta\\in W$. Since $\\zeta$ is $D$-reachable from $\\xi$, there exist ${\\bf i}$ and~${\\bf t}$ such that \n\\begin{equation*}\n\\zeta = \\Phi_{{\\bf i}}({\\bf t},\\xi).\n\\end{equation*}\nThe mapping $h$ defined by $h(x)=\\Phi_{{\\bf i}}({\\bf t},x), x\\in M,$ is a diffeomorphism on $M$. Therefore, $h^{-1}(W)$ is an open neighborhood of $\\xi$. \nSince $\\xi \\in E\\subset \\overline {L(\\eta)}$, there is a point $\\rho \\in h^{-1}(W)$, $D$-reachable from $\\eta$. Consequently,\n$h(\\rho)\\in W$ is also $D$-reachable from $\\eta$.\n{{\\hfill $\\Box$ \\smallskip}}\n\nAs an immediate corollary of Lemma~\\ref{lem:non-empty-interior}, the set $L$ has nonempty interior.\n\\begin{lemma} \\label{lem:positive-measure}\nSuppose $\\mu$ is an invariant measure for  $(\\mathsf{P}^t)$. If $G$ is a nonempty open subset of~$L$ and $j\\in\nS$, then $\\mu(G\\times\\{j\\})>0$.\n\\end{lemma}      \n  \n\\bpf Let us assume that $\\mu(G\\times\\{j\\})=0$. Since\n$\\mu$ is $(\\mathsf{P}^t)$-invariant, it is also $\\mathsf{Q}$-invariant, and we have\n\\begin{equation*}\n0 = \\mu(G\\times\\{j\\}) = \\sum_{i=1}^{k} \\int_M \\mathsf{Q}_{\\eta,i}(G\\times\\{j\\}) \\mu(d\\eta\\times\\{i\\}).\n\\end{equation*}\nFor all $i\\in S$ and $\\mu(\\cdot \\times\\{i\\})$-almost every $\\eta\\in M$, we thus obtain\n\\begin{equation}\n\\label{eq:Q-0}\n\\mathsf{Q}_{\\eta,i}(G\\times\\{j\\}) = 0.\n\\end{equation}\nLet us choose $\\eta$ such that~\\eqref{eq:Q-0} holds true.\n\n\n\n\n\nBy assumption, we have $G\\subset L\\subset\\overline{L(\\eta)}$. Since $G$ is open,  $G \\cap L(\\eta)\\ne\n\\emptyset$. So there exist  a sequence $\\mathbf{i}=(i,i_2,\\ldots,i_m,j)$ and an interswitching time vector\n$\\mathbf{t}=(t_1,\\ldots,t_m,t_{m+1})$ such that  $\\Phi_{{\\bf i}}({\\bf t}, \\eta)\\in G$. By continuity of $\\Phi_{\\bf i}$ there\nis a neighborhood $W$ of $\\mathbf{t}$ in $\\mathbb{R}^{m+1}_+$\nsuch that $\\Phi_{{\\bf i}}(\\mathbf{s}, \\eta)\\in G$ for all $\\mathbf{s}\\in W$.  Denoting $s=s_1+\\ldots+s_{m+1}$ and using the\nrepresentation of $\\mathsf{P}^s_{\\eta,i}(\\cdot|C_{\\mathbf{i}})$ via exponentially distributed times\nthat we used in the proof of Theorem~\\ref{thm:AC-component-1}, we conclude that   $\\mathsf{P}^s_{\\eta,i}(G \\times \\{j\\})>0$ for \n$s$ sufficiently close to $t=t_1+\\ldots+t_{m+1}$. Therefore, $\\mathsf{Q}_{\\eta,i}(G \\times \\{j\\})>0$ contradicting \\eqref{eq:Q-0}.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\\bigskip\n\n\\bpf[Proof of Theorem~\\ref{thm:uniqueness}]\nAny invariant measure can be written as a convex combination of ergodic invariant measures. Therefore, it suffices to\nshow absolute continuity and uniqueness of an ergodic invariant measure. \n\nLet us begin by deriving absolute continuity. If $\\mu$ is an\nergodic invariant measure that satisfies the assumptions of Theorem~\\ref{thm:uniqueness} then, \ndue to Theorem~\\ref{thm:absolute-continuity-for-ergodic}, it suffices to show that\n$L \\subset \\mathop{\\rm supp} \\mu$. Let $\\xi \\in L$, and let $U$ be a neighborhood of $\\xi$ in $M$, $j \\in S$. By\nLemma~\\ref{lem:positive-measure}, we have $\\mu(U \\times \\{j\\})>0$, hence $\\xi \\in \\mathop{\\rm supp} \\mu$.   \n\nIn order to prove uniqueness of the ergodic invariant measure, let us assume that $\\mu_1$ and $\\mu_2$ are two distinct ergodic invariant probability measures. Birkhoff's ergodic\ntheorem then implies that $\\mu_1$ and $\\mu_2$ are mutually singular. Hence, the set $M \\times S$ can be partitioned into\ntwo disjoint subsets $H_1$ and $H_2$ with\n$\\mu_1(H_2) = \\mu_2(H_1) = 0.$\nThe two sets can be represented as \n\\[H_\\alpha=\\bigcup_{j=1}^{k} M_{\\alpha,j} \\times \\{j\\},\\quad \\alpha=1,2,\\]\nfor some measurable sets $M_{\\alpha,j}$, $j\\in S$, $\\alpha=1,2$. For all $\\alpha$ and $j$,\n\\begin{equation*}\n\\mu_\\alpha( M_{\\alpha,j} \\times \\{j\\})=\\mu_\\alpha( M \\times \\{j\\})>0,\n\\end{equation*}\nsince the left side is a stationary distribution for the Markov chain on $S$ and by our assumptions, transitions\nbetween all states happen with positive probability.\n\nFix a $j$ in $S$. By virtue of Lemma~\\ref{lem:non-empty-interior}, the set $E^{\\circ}$ is nonempty.\nAccording to Lemma~\\ref{lem:positive-measure}, for all $j \\in S$ we have $\\mu_1(E^\\circ\\times\\{j\\})>0$.\n Since $\\mu_1(M_{2,j}\\times\\{j\\}) =0$, we deduce that $\\mu_1(E_1\\times\\{j\\}) > 0$, where $E_1=E^\\circ \\cap\nM_{1,j}$.\n\nThe measure $\\mu_1$ is $(\\mathsf{P}^t)$-invariant, hence it is also $\\mathsf{Q}$-invariant, and we have\n\\begin{equation}\n\\label{eq:invariance-for-M_2_j}\n0=\\mu_1(M_{2,j}\\times\\{j\\}) \\geq \\int_{E_1} \\mathsf{Q}_{\\eta,j}(M_{2,j}\\times \\{j\\}) \\mu_1(d\\eta\\times\\{j\\}).\n\\end{equation}\nSince $\\mu_1(E_1\\times\\{j\\})>0$, it suffices to show that $\\mathsf{Q}_{\\eta,j}(M_{2,j}\\times\\{j\\})>0$  for all $\\eta \\in\nE_1$, to obtain a contradiction with~\\eqref{eq:invariance-for-M_2_j}.\n\nSince $\\eta$ satisfies Condition~B,\nTheorem~\\ref{thm:Jurdjevic-2} guarantees that there exist an integer $m>n$ and a vector ${\\bf i} =\n(j,i_2,\\ldots,i_m,j)$ such\nthat the function $f:\\mathbb{R}^{m+1}_+ \\to M$ defined by\n\\begin{equation*}\n f(\\bf t) =\\Phi_{\\bf i}({\\bf t},\\eta)\n\\label{eq:F_ibf}\n\\end{equation*}\nhas an open set $O$ of regular points such that for all $t>0$,\n\\begin{equation*}\n\\{\\mathbf{t}=(t_1,\\ldots,t_{m+1})\\in O:\\ t_1+\\ldots+t_{m+1}<t\\}\\ne \\emptyset.\n\\end{equation*}\nTherefore the map $F$ defined by\n\\[\n F(t_1,\\ldots,t_{m+1},t)= f\\biggl(t_1,\\ldots,t_m,t-\\sum_{l=1}^{m}t_l\\biggr)\n\\]\non\n\\[\n \\Delta=\\biggl\\{(t_1,\\ldots,t_{m+1},t)\\in\\mathbb{R}_{+}^{m+2}:\\ \\sum_{l=1}^{m}t_l<t<\\sum_{l=1}^{m+1}t_l\\biggr\\}\n\\]\nhas an open set $V\\subset\\Delta$ of regular points such that \n\\begin{equation}\n\\{\\mathbf{t}=(t_1,\\ldots,t_{m+1},t)\\in V:t<s\\}\\ne \\emptyset,\\quad s>0.\n\\label{eq:0_is_limiting_point}\n\\end{equation}\n\n\nUsing the representation of $\\mathsf{Q}$ via~\\eqref{eq:Qq-via-I_t} and the family of exponentially distributed times\n$T_1,\\ldots,T_{m+1},T$, we obtain that it is sufficient to prove that\n\\begin{equation}\n\\mathsf{P}\\left\\{F(T_1,\\ldots,T_{m+1},T) \\in M_{2,j}|\\ R \\right\\} > 0,\n\\label{eq:sufficient_for_contradiction}\n\\end{equation}\nwhere $R$ was introduced in~\\eqref{eq:event_R_2}.\n\n\n Since $E^\\circ$ is an open set containing\n$\\eta$, and $F(V)$ is an open set such that $\\eta\\in \\overline{F(V)}$ (due \nto~\\eqref{eq:0_is_limiting_point} and continuity of $F$ at $0$), we obtain that\n$G= E^{\\circ} \\cap F(V)$ is also a nonempty open set. \n\n\n\nLet us choose a vector ${\\bf r}\\in V$ such that $F({\\bf r}) \\in E^{\\circ}$. Since $\\mathbf{r}$ is a regular point for\n$F$, we see that \nfor an arbitrary choice of local smooth coordinates around $\\mathbf{r}$,\nthere are $n$ independent columns of the matrix $DF(\\mathbf{s})$ for $\\mathbf{s}$\nin a small neighborhood of $\\mathbf{r}$. Without loss of generality we\ncan assume that these are the first $n$ columns. \nThen the map $\\rho:\\mathbb{R}^{m+2}\\to M\\times\\mathbb{R}^{m+2-n}$ defined by\n\\[\n \\rho(s_1,\\ldots,s_{m+1},s)=(F(s_1,\\ldots,s_{m+1},s),s_{n+1},\\ldots ,s_{m+1},s)\n\\]\nhas nonzero Jacobian in that neighborhood. So we can choose an open set $W_V$ containing~$\\mathbf{r}$ so that $\\rho$ is a\ndiffeomorphism between $W_V$ and \n$W_G\\times W_{n-m-2}$, where $W_G\\subset G$ and $W_{m+2-n}\\subset \\mathbb{R}_{+}^{m+2-n}$ are some open sets.\n\n\n\nThe set $W_G$ is an open subset of $L$. It is also not empty since it contains~$F({\\bf r})$.\nLemma~\\ref{lem:positive-measure} implies that $\\mu_2( W_G\\times\\{j\\})>0$.  Since\n$\\mu_2(M_{2,j}^c\\times\\{j\\})=0$, we conclude that\n$\\mu_2(J\\times\\{j\\})>0$ where $J=M_{2,j} \\cap W_G.$ Since $\\mu_2$ is an ergodic measure, it is absolutely continuous, so\n\\begin{equation}\n\\label{eq:J-Lebesgue-positive}\n\\lambda^M(J)>0. \n\\end{equation}\n\n\n\nSince $J\\subset M_{2,j}$, the desired inequality \\eqref{eq:sufficient_for_contradiction} will follow from\n\\begin{equation}\n\\mathsf{P}\\left\\{F(T_1,\\ldots,T_{m+1},T) \\in J|\\ R \\right\\} > 0.\n\\label{eq:sufficient_for_contradiction-2}\n\\end{equation}\n\n\nSince the joint distribution of $T_1,\\ldots,T_m,T_{m+1},T$ is equivalent to the Lebesgue measure on $\\Delta$, \nLemma~\\ref{lem:pushforward_for_nonzero_det} implies that $\\rho(T_1,\\ldots,T_{m+1},T)$ has positive density\nalmost everywhere in $W_G\\times W_{m+2-n}$. Integrating over $W_{m+2-n}$, we see that \\\\ $F(T_1,\\ldots,T_{m+1},T)$ has\npositive density almost everywhere in $W_G$. Now~\\eqref{eq:sufficient_for_contradiction-2} follows from \n\\eqref{eq:J-Lebesgue-positive}.{{\\hfill $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n\\bigskip  \n\n\n\nOf course, Theorem~\\ref{thm:uniqueness} remains true if one replaces Condition~B by the stronger Condition~A.\nHowever, under that condition one can prove this result without referring to the resolvent $\\mathsf{Q}$. Namely, one can\nuse the regularity of transition probabilities established in Theorem~\\ref{thm:AC-component-1} (which is stronger than\nthe regularity established in Theorem~\\ref{thm:AC-component-2}),\nand invoke Theorems~\\ref{thm:Jurdjevic-1} and~\\ref{thm:absolute-continuity-for-ergodic-under-condition-A} instead of\nTheorems~\\ref{thm:Jurdjevic-2} and~\\ref{thm:absolute-continuity-for-ergodic}.\n\n\n\n\n\\section{Examples} \\label{sec:examples}\n\nIn this section, we apply Theorem~\\ref{thm:uniqueness} to two concrete switching systems. In the first example, we have a closer look at the system on the $n$-dimensional torus $\\mathbb{T}^n=\\mathbb{R}^n\/\\mathbb{Z}^n$ that was introduced in Section~\\ref{sec:absolute-continuity-invariant}. In the second example, we switch between two Lorenz vector fields with different parameter sets. For both systems, uniqueness of the invariant measure is derived from Theorem~\\ref{thm:uniqueness}. For the system on $\\mathbb{T}^n$, we point out the invariant measure explicitly.\n\n\\bigskip\n\nLet $M$ be the $n$-dimensional torus $\\mathbb{T}^n$, and let $D=\\{u_1,\\ldots,u_n\\}$ be the standard basis of $\\mathbb{R}^{n}$.\n We assume for simplicity that the parameter $\\lambda$ of the exponential time between any two switches is independent\nof the current state, and that we have a uniform probability of switching between any two states. In\nSection~\\ref{sec:absolute-continuity-invariant} we implicitly argued that Condition~A does not hold for this system: If\nCondition~A was satisfied at some point $\\xi \\in \\mathbb{T}^n$, the transition probability measures $\\mathsf{P}_{\\xi,i}^t$ would not be\nsingular with respect to Lebesgue measure, according to Theorem~\\ref{thm:AC-component-1}. However, as pointed out in\nSection~\\ref{sec:absolute-continuity-invariant}, the measures $\\mathsf{P}_{\\xi,i}^t$ are purely singular. \n\nIt is also instructive to show directly why Condition~A does not hold. As all the vector fields in $D$ are constant, the derived algebra $\\mathcal{I}'(D)$ contains only the zero vector field. Thus, for any $\\xi \\in \\mathbb{T}^n$,\n\\begin{equation*}\n \\mathcal{I}_{0}(D)(\\xi) = \\biggl\\{\\sum_{i=1}^{n} \\lambda_{i} u_{i}:\\ \\sum_{i=1}^{n} \\lambda_{i} = 0\\biggr\\}. \n\\end{equation*}\nDue to the constraint $\\sum_{i=1}^{n} \\lambda_{i} = 0$, the algebra $\\mathcal{I}_{0}(D)(\\xi)$ does not have full\ndimension, so Condition~A is violated at every point in $\\mathbb{T}^n$.\n\nOn the other hand, Condition~B is clearly satisfied at any point $\\xi \\in \\mathbb{T}^n$, as the standard basis of $\\mathbb{R}^n$ applied to $\\xi$ yields a full-dimensional set of vectors in the tangent space. Also note that any point in $\\mathbb{T}^n$ is $D$-reachable from any other point. Therefore, Theorem~\\ref{thm:uniqueness} guarantees that the associated Markov semigroup has a unique invariant measure, provided that such a measure exists. In this elementary example, it is possible to point out the invariant measure explicitly. For Borel sets $E \\subset \\mathbb{T}^n$ and states $i \\in S$, it is given by\n\\begin{equation*}\n \\mu(E \\times \\{i\\}) = \\frac{1}{n} \\cdot \\lambda(E).\n\\end{equation*}\nHere, $\\lambda$ denotes Lebesgue measure on $\\mathbb{T}^n$. \n\n\\bigskip\n\nThe second example provides a situation where (i) the number of vector fields in~$D$ is less than the\ndimension of the manifold $M$, and (ii) each individual vector field in~$D$ gives rise to dynamics with a\nstrange attractor and no absolutely continuous invariant measures, but (iii) the switched system has a unique invariant\nmeasure and it is absolutely continuous.\n\n\nNamely, we consider switching between two Lorenz vector fields with different parameter values. A Lorenz\nvector field is a vector field defined in $\\mathbb{R}^3$, of the form\n\\begin{equation*}\n u(x,y,z) = \\begin{pmatrix}\n             \\sigma \\cdot (y-x) \\\\\n             rx-y-xz \\\\\n             xy-bz\n            \\end{pmatrix},\n\\end{equation*}\n where $\\sigma$, $r$ and $b$ are physical parameters. Let the set $D$ contain exactly two Lorenz vector fields $u_1$ and\n$u_2$ such that $u_1$ has Rayleigh number $r=28$ and $u_2$ has a Rayleigh number different from, but close to, $28$. We\nassume for both vector fields that $\\sigma=10$ and that $b=\\tfrac{8}{3}$, which is the classical parameter choice for\nthe Lorenz system. In~\\cite{Tucker:MR1701385}, Tucker shows that the Lorenz system with parameters $\\sigma=10$, $r=28$\nand $b=\\tfrac{8}{3}$, corresponding to vector field $u_1$, admits a robust strange attractor $\\Lambda$ as well as a\nunique SRB-measure supported on $\\Lambda$. Robustness implies that the dynamical structure of the system remains intact\nunder small parameter changes, so the dynamics induced by $u_2$ share these features if $r$ is sufficiently close to\n$28$. Moreover, the SRB-measure on $\\Lambda$ satisfies a dissipative ergodic theorem, see\ne.g.~\\cite[Section~5.1]{Bunimovich-Sinai:MR755521}. It follows that any point $\\xi\\in\\Lambda$ is\n$\\{u_1\\}$-approachable (and thus $D$-approachable) from every point in a set $S_\\xi\\subset\\mathbb{R}^3$ with zero Lebesgue\nmeasure complement.\n\nAssisted by a computer algebra system, we checked that Condition~A is satisfied for this system at any point in $\\mathbb{R}^3$\nthat does not lie on the $z$-axis. Since the $z$-axis is invariant under the flows of both vector fields, we disregard\nit and set $M$ to be $\\mathbb{R}^3$ without points on the $z$-axis. With this provision, every point on the attractor $\\Lambda$\nis $D$-approachable from any point in $M$: \n\nConsider a point $\\xi \\in \\Lambda$ and a point $\\eta \\in M$. By Theorem~\\ref{thm:Jurdjevic-1}, there is a nonempty open\nset of $D$-reachable points from $\\eta$ (recall that Condition~A holds at any point in $M$). And since this open set has\npositive Lebesgue measure, it contains a point belonging to~$S_\\xi$.\nHence, $\\xi$ is $D$-approachable from $\\eta$. As in the first example, uniqueness and absolute continuity of an invariant measure\nfollow from Theorem~\\ref{thm:uniqueness}.    \n       \n          \n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nPolar codes, introduced by Ar\\i kan in \\cite{Ari08}, are the first provably capacity achieving codes for arbitrary symmetric binary-input discrete memoryless channels (B-DMC) with low encoding and decoding complexity. Polar code construction is based on the following observation: Let \n\\begin{align}\\label{eqn:2by2}\nG_2=\\left[\n\\begin{array}{cc}\n1 & 0 \\\\\n1 & 1\n\\end{array}\n\\right].\n\\end{align}\nConsider applying the transform $G_2^{\\otimes n}$ (where ``$\\phantom{}^{\\otimes n}$'' denotes the $n^{th}$ Kronecker power) to a block of $N = 2^n$ bits and transmitting the output through independent copies of a B-DMC $W$ (see Figure \\ref{fig:transform}). \n As $n$ grows large, the channels seen by individual bits (suitably defined in \\cite{Ari08}) start \\emph{polarizing}: they approach either a noiseless channel or a pure-noise channel, where the fraction of channels becoming noiseless is close to the symmetric mutual information $I(W)$.\n\n\nIt was conjectured in \\cite{Ari08} that polarization is a general phemonenon, and is not restricted to the particular transformation $G_2^{\\otimes n}$. In this note we give a partial affirmation to this conjecture. In particular, we consider transformations of the form $G^{\\otimes n}$ where $G$ is an $\\ell\\times\\ell$ matrix for $\\ell \\geq 3$ and provide necessary and sufficient conditions for such $G$s to polarize symmetric B-DMCs. \n\n\\begin{figure}[ht]\\label{fig:transform}\n\\input{transform.tex}\n\\caption{ }\n\\end{figure}\n\n\\section{Preliminaries}\n\nLet $W: \\{0,1\\} \\to \\mathcal{Y}$ be a B-DMC. Let $I(W) \\in [0,1]$ denote the mutual information between the input and output of $W$ with uniform distribution on the inputs. Also let $Z(W) \\in [0,1]$ denote the Bhattacharyya parameter of $W$, i.e., $Z(W) = \\sum_{y\\in\\mathcal{Y}} \\sqrt{W(y|0)W(y|1)}$. \n\nFix an $\\ell\\geq 3$ and an invertible $\\ell\\times\\ell$ $\\{0,1\\}$ matrix $G$. Consider a random $\\ell$-vector $U_1^\\ell$ that is uniformly distributed over $\\{0,1\\}^\\ell$. Let $X_1^\\ell = U_1^\\ell G$, where the multiplication is performed over GF(2). Also let $Y_1^\\ell$ be the output of $\\ell$ uses of $W$ with the input $X_1^\\ell$. Observe now that the channel between $U_1^\\ell$ and $Y_1^\\ell$ is defined by the transition probabilities \\begin{align*}\nW_\\ell (y_1^\\ell\\mid u_1^\\ell) \\triangleq \\prod_{i=1}^\\ell W(y_i\\mid x_i) = \\prod_{i=1}^\\ell W(y_i\\mid (u_1^\\ell G)_i).\n\\end{align*}\nDefine $W^{(i)}: \\{0,1\\} \\to \\mathcal{Y}^\\ell \\times \\{0,1\\}^{i-1}$ as the channel with input $u_i$, output $(y_1^\\ell,u_1^{i-1})$ and transition probabilities\n\\begin{align*}\nW^{(i)}(y_1^\\ell,u_1^{i-1}\\mid u_i) = \\frac{1}{2^{\\ell-1}} \\sum_{u_{i+1}^\\ell} W_\\ell(y_1^\\ell\\mid u_1^\\ell),\n\\end{align*}\nand let $Z^{(i)}$ denote its Bhattacharyya parameter, i.e., \n\\begin{align*}\nZ^{(i)} = \\sum_{y_1^\\ell,u_1^{i-1}} \\sqrt{W^{(i)}(y_1^\\ell,u_1^{i-1}\\mid 0)W^{(i)}(y_1^\\ell,u_1^{i-1}\\mid 1)}.\n\\end{align*}\nFor $k\\geq 1$, let $W^k: \\{0,1\\}\\to\\mathcal{Y}^k$ denote the B-DMC with transition probabilities \n\\begin{align*}\nW^k(y_1^k\\mid x) = \\prod_{j=1}^k W(y_j\\mid x).\n\\end{align*}\nAlso let $\\tilde{W}^{(i)}: \\{0,1\\} \\to \\mathcal{Y}^\\ell$ denote the B-DMC with transition probabilities\n\\begin{align} \n\\tilde{W}^{(i)}(y_1^\\ell\\mid u_i) = \\frac{1}{2^{\\ell-i}} \\sum_{u_{i+1}^\\ell} W_\\ell(y_1^\\ell\\mid 0_1^{i-1},u_i^\\ell).\n\\end{align} \n\\begin{observation}\\label{obs:equivalent}\nIf $W$ is symmetric, then the channels $W^{(i)}$ and $\\tilde{W}^{(i)}$ are equivalent in the sense that for any fixed $u_1^{i-1}$ there exists a permutation $\\pi_{u_1^{i-1}}: \\mathcal{Y}^\\ell \\to \\mathcal{Y}^\\ell$ such that \n\\[\nW^{(i)}(y_1^\\ell, u_1^{i-1} \\mid u_i) = \\frac{1}{2^{i-1}}\\tilde{W}^{(i)}(\\pi_{u_1^{i-1}}(y_1^\\ell)\\mid u_i).\n\\]\n\\end{observation}\nFinally, let $I^{(i)}$ denote the mutual information between the input and output of channel $W^{(i)}$. Since $G$ is invertible, it is easy to check that \n\\begin{align*}\n\\sum_{i=1}^\\ell I^{(i)} = \\ell I(W). \n\\end{align*}\n\n\n\\section{Polarization}\n\n\nWe will say that $G$ is a \\emph{polarizing} matrix if there exists an $i\\in\\{1,\\dotsc,\\ell\\}$ for which $\\tilde{W}^{(i)}$ is equivalent to $W^k$ for some $k\\geq 2$, in the sense that\n\\begin{align}\\label{eqn:polarizing}\n\\tilde{W}^{(i)}(y_1^\\ell\\mid u_i) = c \\prod_{j\\in A}W(y_j \\mid u_i) \n\\end{align}\nfor some constant $c$ and $A\\subseteq \\{1,\\dots,\\ell\\}$ with $\\vert A \\vert = k$. If $W$ is symmetric, then Observation \\ref{obs:equivalent} implies the equivalence of $W^{(i)}$ and $W^k$ (which we denote by $W^{(i)} \\equiv W^k$) in the sense that \n\\begin{align}\nW^{(i)}(y_1^\\ell,u_1^{i-1} \\mid u_i) = \\frac{c}{2^{i-1}} \\prod_{j\\in A}W((\\pi_{u_1^{i-1}}(y_1^\\ell))_j\\mid u_i).\n\\end{align}\nNote that the equivalence $W^{(i)} \\equiv W^k$ implies $I^{(i)} = I(W^k)$ and $Z^{(i)} = Z(W^k)$.\n\nIt will be shown that channel transformations of the form $G^{\\otimes n}$ polarize symmetric channels if and only if $G$ is polarizing. This statement is made precise in the following theorem:\n\n\\begin{theorem} \\label{thm:main}\nFix a symmetric B-DMC $W$. Let $G^{\\otimes n}$ denote the $n^{th}$ Kronecker power of $G$ and consider the transformation $G^{\\otimes n}: W \\to (W^{(i)}: i=1,\\dotsc, \\ell^n)$. \n\\begin{itemize}\n\\item[i.] If $G$ is polarizing, then for any $\\delta > 0$\n\\begin{align*}\n\\lim_{n \\to \\infty}\\frac{\\#\\left\\{i\\in\\{1,\\dotsc,\\ell^n\\}: I(W^{(i)}) \\in (\\delta, 1-\\delta) \\right\\}}{\\ell^n} = 0.\n\\end{align*}\n\\item[ii.] If $G$ is not polarizing, then \n\\begin{align*}\nI(W^{(i)}) = I(W) \\textrm{ for all } n \\textrm{ and } i\\in\\{1,\\dotsc,\\ell^n\\}.\n\\end{align*}\n\\end{itemize}\n\\end{theorem}\n\nTheorem \\ref{thm:main} is a direct consequence of Lemmas \\ref{lemma:square} and \\ref{lemma:I-converges} below.\n\nNote that any invertible $\\{0,1\\}$ matrix $G$ can be written as a (real) sum $G=P+P'$, where $P$ is a permutation matrix, and $P'$ is a $\\{0,1\\}$ matrix. This fact can be inferred from Hall's Theorem \\cite[Theorem 16.4.]{BoM08}. Therefore, for any such matrix $G$, there exists a column permutation that results in $G_{ii} = 1$ for all $i$. Since the transition probabilities defining $W^{(i)}$ are invariant (up to a permutation of the outputs $y_1^\\ell$) under column permutations on $G$, we only consider matrices with 1s on the diagonal. \n\nThe following lemma gives necessary and sufficient conditions for (\\ref{eqn:polarizing}) to be satisfied:\n\n\\begin{lemma}\\label{lemma:square}\nFor any symmetric B-DMC $W$,\n\\begin{itemize} \n\\item[i.] If $G$ is not upper triangular, then there exists an $i$ for which $W^{(i)} \\equiv W^k$ for some $k\\geq 2$.\n\\item[ii.] If $G$ is upper triangular, then $W^{(i)} \\equiv W$ for all $1\\leq i \\leq \\ell$.\n\\end{itemize} \n\\end{lemma}\n\\begin{proof}\nLet $G^{(\\ell-i)}$ be the $(\\ell-i)\\times (\\ell-i)$ matrix obtained from $G$ by removing its last $i$ rows and columns. Let the number of 1s in the last row of $G$ be $k$. Clearly $W^{(\\ell)} \\equiv W^k$.  If $k\\geq2$ then $G$ is not upper triangular and the first claim of the lemma holds. If $k=1$, then $W^{(\\ell)} \\equiv W$, and $(x_1,\\dotsc,x_{\\ell-1})$ is independent of $u_\\ell$. One can then write\n\\begin{align*}\nW^{(\\ell-i)}(y_1^\\ell,u_1^{\\ell-i-1}\\mid u_{\\ell-i}) &= \\frac{1}{2^{\\ell-1}} \\sum_{u_{\\ell-i+1}^\\ell} W_\\ell(y_1^\\ell \\mid u_1^\\ell) \\\\\n& = \\frac{1}{2^{\\ell-1}} \\sum_{u_{\\ell-i+1}^{\\ell-1},u_\\ell} \\Pr[Y_1^{\\ell-1}= y_1^{\\ell-1}\\mid U_1^{\\ell} = u_1^{\\ell}] \\Pr[Y_\\ell = y_\\ell\\mid Y_1^{\\ell-1} = y_1^{\\ell-1},U_1^{\\ell} = u_1^{\\ell}]\\\\\n&\\stackrel{(a)}{ =} \\frac{1}{2^{\\ell-1}} \\sum_{u_{\\ell-i+1}^{\\ell-1},u_\\ell} W_{\\ell-1}(y_1^{\\ell-1}\\mid u_1^{\\ell-1}) \\Pr[Y_\\ell = y_\\ell\\mid Y_1^{\\ell-1} = y_1^{\\ell-1},U_1^{\\ell} = u_1^{\\ell}]\\\\\n& = \\frac{1}{2^{\\ell-1}} \\sum_{u_{\\ell-i+1}^{\\ell-1}}W_{\\ell-1}(y_1^{\\ell-1}\\mid u_1^{\\ell-1}) \\sum_{u_\\ell} \\Pr[Y_\\ell = y_\\ell\\mid Y_1^{\\ell-1} = y_1^{\\ell-1},U_1^{\\ell} = u_1^{\\ell}]\\\\\n& = \\frac{1}{2^{\\ell-1}} \\big[W(y_\\ell\\mid 0) + W(y_\\ell\\mid 1)\\big]\n\\sum_{u_{\\ell-i+1}^{\\ell-1}} W_{\\ell-1}(y_1^{\\ell-1}\\mid u_1^{\\ell-1})\n\\end{align*}\n\nwhere $(a)$ follows from the fact that $G_{lk} = 0,$ for all $k<\\ell$. Therefore $y_\\ell$ is independent of the inputs to the channels $W^{(\\ell-i)}$ for $i=1,\\dotsc,\\ell-1$. This is equivalent to saying that channels $W^{(1)},\\dotsc,W^{(\\ell-1)}$ are defined by the matrix $G^{(\\ell-1)}$. Applying the same argument to $G^{(\\ell-1)}$ and repeating, we see that if $G$ is upper triangular, then we have $W^{(i)} \\equiv W$ for all $i$. On the other hand, if $G$ is not upper triangular, then there either exists an $i$ for which $G^{(\\ell-i)}$ has at least two 1s in the last row, which in turn implies $W^{(i)} \\equiv W^k$ for some $k\\geq 2$. \\end{proof}\n\n\\begin{remark}\nThe above lemma says that all transformations that are not upper triangular are polarizing. Moreover, upper triangular transformations have no effect on the channel, i.e., each bit sees an independent copy of $W$ after an upper triangular transformation. \n\\end{remark}\n\n\\begin{corollary}\nFor any polarizing transformation $G$, there exists an $i\\in\\{1,\\dotsc,\\ell\\}$ and $k\\geq 2$ for which \n\\begin{align}\nI^{(i)} & = I(W^k) \\label{eqn:prod-channel} \\\\\nZ^{(i)} & = Z(W)^k.\n\\end{align}\n\\end{corollary}\n\\begin{proof}\nThe first claim is trivial. The second claim follows from the fact that the Bhattacharyya parameter of any channel of the form $\\prod_j W_j$ is given by $\\prod_j Z(W_j)$.\n\\end{proof}\n\n\\section{Convergence}\nConsider recursively combining channels $W$ as in \\cite{Ari08}, using a polarizing transformation $G$. Following Ar\\i kan, associate to this construction a tree process $\\{W_n; n\\geq 0\\}$ with \n\\begin{align*}\nW_0 & = W \\\\\nW_{n+1} & = W_{n}^{(B_{n+1})},\n\\end{align*}\nwhere $\\{B_n; n\\geq 1\\}$ is a sequence of i.i.d.\\ random variables defined on a probability space $(\\Omega, \\mathcal{F}, \\mu )$, $B_n$ being uniformly distributed over the set $\\{1,\\dotsc,\\ell\\}$. Define $\\mathcal{F}_0 = \\{\\emptyset, \\Omega\\}$ and $\\mathcal{F}_n = \\sigma (B_1,\\dotsc, B_n)$ for $n \\geq 1$. Define the processes $\\{I_n; n\\geq 0\\} = \\{I(W_n); n\\geq 0\\}$ and $\\{Z_n; n\\geq 0\\} = \\{Z(W_n); n\\geq 0\\}$. \n\n\\begin{observation}\n$\\{(I_n, \\mathcal{F}_n)\\}$ is a bounded martingale and therefore converges a.s.\\ and in $\\mathcal{L}^1$ to a random variable $I_\\infty$.\n\\end{observation}\n\n\\begin{lemma} \\label{lemma:I-converges}\nIf $W$ is symmetric and $G$ is polarizing, then\n\\begin{align*}\nI_\\infty =\n\\begin{cases}\n1 & \\textrm{w.p. } I(W), \\\\\n0 & \\textrm{w.p. } 1 - I(W).\n\\end{cases}\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nBy the convergence in $\\mathcal{L}^1$ of $I_n$ we have $\\mathbb{E} [|I_{n+1} - I_n | ] \\stackrel{n \\to \\infty}{\\longrightarrow}0$. Since $G$ is a polarizing matrix, Lemma \\ref{lemma:square} implies\n\\begin{align*}\nI_{n+1} = I(W_n^k) \\textrm{ with probability at least } \\frac{1}{\\ell},\n\\end{align*}\nfor some $k\\geq 2$. This in turn implies \n\\begin{align}\n\\mathbb{E} [|I_{n+1} - I_n |] \\geq \\frac{1}{\\ell} \\mathbb{E}[I(W_n^k) - I(W_n)] \\to 0. \\label{eqn:I-repetition}\n\\end{align}\nIt is shown in the Appendix that for any symmetric B-DMC $W_n$, if $I(W_n) \\in (\\delta,1-\\delta)$ for some $\\delta > 0$, then there exists an $\\eta (\\delta) > 0$ such that $I(W_n^k) - I(W_n) > \\eta (\\delta)$. We therefore conclude that convergence in (\\ref{eqn:I-repetition}) implies $I_\\infty \\in \\{0,1\\}$ w.p.\\ 1. The claim on the probability distribution of $I_\\infty$ follows from the fact that $\\{I_n\\}$ is a martingale, i.e., $\\mathbb{E} [I_\\infty] = \\mathbb{E} [I_0] = I(W)$.\n\\end{proof}\n\n\\begin{corollary}\nIf $W$ is symmetric and $G$ is polarizing, then $\\{Z_n\\}$ converges a.s.\\ to a random variable $Z_\\infty$ and\n\\begin{align*}\nZ_\\infty =\n\\begin{cases}\n0 & \\textrm{w.p. } I(W), \\\\\n1 & \\textrm{w.p. } 1 - I(W).\n\\end{cases}\n\\end{align*}\n\\end{corollary}\n\\begin{proof}\nThe proof follows from the fact that $I_n \\to I_\\infty$ a.s.\\ and the inequalities \\cite{Ari08} \n\\begin{align*}\nI(Q)^2 + Z(Q)^2 \\leq 1 \\\\\nI(Q) + Z(Q) \\geq 1.\n\\end{align*}\nfor any B-DMC $Q$.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:l-ary}\nGiven a symmetric B-DMC $W$, an $\\ell\\times\\ell$ polarizing matrix $G$, and any $\\beta < 1\/\\ell$,\n\\begin{align*}\n\\lim_{n \\to \\infty} \\Pr[Z_n \\leq 2^{-\\ell^{n\\beta}}] = I(W).\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof}[Proof Idea]\nFor any polarizing matrix it can be shown that $Z_{n+1}\\leq \\ell Z_n$  with probability 1 and that $Z_{n+1} \\leq Z_n^2$  with probability at least $1\/\\ell$.\nThe proof then follows by adapting the proof of \\cite[Theorem 3]{ArT08}.\n\\end{proof}\n\n\\section{Discussion}\nUsing Ar\\i kan's rule for choosing the information bits, polar codes of blocklength $N=\\ell^n$ can be constructed starting with any polarizing $\\ell\\times\\ell$ matrix $G$. The encoding and successive cancellation decoding complexities of such codes are $O(N\\log N)$. Using similar arguments, it is easy to show that polar codes of blocklength $N=\\prod_{i=1}^n \\ell_i$ can be constructed from generator matrices of the form $\\otimes_i G_i$, where each $G_i$ is a polarizing matrix of size $\\ell_i\\times\\ell_i$. The encoding and successive cancellation decoding complexities of these codes are also $O(N\\log N)$.\n\n\\section*{Appendix}\n\nIn this section we prove the following:\n\\begin{lemma}\\label{lem:combinedI}\nLet $W$ be a symmetric B-DMC and let $W^k$ be defined as above. \nIf $I(W) \\in (\\delta, 1-\\delta)$ for some $\\delta > 0$, then there exists an $\\eta (\\delta)>0$ such that $I(W^k) - I(W) > \\eta (\\delta)$.\n\\end{lemma}\nWe will use the following theorem in proving Lemma \\ref{lem:combinedI}:\n\\begin{theorem}[\\cite{SSZ05, HuH05}]\\label{thm:extremes}\nLet $W_1,\\dots,W_k$ be $k$ symmetric B-DMCs with capacities $I_1,\\dots,I_k$\nrespectively. Let $W^{[k]}$ denote the channel with transition probabilities\n\\begin{align*}\nW^{[k]}(y_1^k\\mid x) = \\prod_{i=1}^k W_i(y_i\\mid x).\n\\end{align*}\n\nAlso let $W_{BSC}^{[k]}$ denote the channel with transition probabilities\n\\begin{align*}\nW_{\\BSC{}}^{[k]}(y_1^k\\mid x) =\n\\prod_{i=1}^k W_{\\BSC{(\\epsilon_i)}}(y_i\\mid x),\n\\end{align*} \nwhere $BSC(\\epsilon_i)$ denotes the binary symmetric channel with crossover probability  $\\epsilon_i \\in [0,\\frac{1}{2}]$, $\\epsilon_i \\triangleq h^{-1}(1-I_i)$, where $h$ denotes the binary entropy function. Then, $I(W^{[k]}) \\geq I(W_{\\BSC{}}^{[k]})$.\n\\end{theorem}\n\n\n\\begin{remark}\nConsider the transmission of a single bit $X$ using\n$k$ independent symmetric B-DMCs $W_1,\\dots, W_k$ with capacities\n$I_1,\\dots,I_k$. Theorem \\ref{thm:extremes} states\nthat over the class of all symmetric channels with given mutual informations, \nthe mutual information between the input and the output vector is minimized when each of the\nindividual channels is a BSC. \n\\end{remark}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:combinedI}]\nLet\n$\\epsilon \\in [0,\\frac12]$ be the crossover probability of a BSC with\ncapacity $I(W)$, i.e., $\\epsilon = h^{-1}(1-I(W))$. \nNote that for $k\\geq 2$, \n\\begin{align*}\nI(W^k) \\geq I(W^2).\n\\end{align*}\nBy Theorem \\ref{thm:extremes}, we have $I(W^2) \\geq\nI(W_{BSC(\\epsilon)}^2)$. \nA simple computation shows that \n\\begin{align*}\nI(W_{BSC(\\epsilon)}^2) = 1+h(2\\epsilon\\bar{\\epsilon})- 2 h(\\epsilon).\n\\end{align*}\nWe can then write\n\\begin{align}\nI(W^k) - I(W) & \\geq I(W_{BSC(\\epsilon)}^2) - I(W) \\notag \\\\\n& = I(W_{BSC(\\epsilon)}^2) - I(W_{BSC(\\epsilon)}) \\notag  \\\\\n& = h(2\\epsilon \\bar{\\epsilon})-h(\\epsilon).\n\\end{align}\nNote that $I(W) \\in (\\delta, 1-\\delta)$ implies $\\epsilon \\in (\\phi (\\delta), \\frac 12 - \\phi(\\delta))$ where $\\phi(\\delta)>0$, which in turn implies $h(2\\epsilon \\bar{\\epsilon})-h(\\epsilon) > \\eta(\\delta)$ for some $\\eta(\\delta) > 0$.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nMagnetars are slowly rotating neutron stars with the strongest known magnetic fields in the Universe.\nOutbursts from these highly-magnetized objects are typically manifested by the sudden enhancement of their persistent X-ray emission by up to three orders of magnitude, which is often accentuated by spectral hardening and increased levels of complexity in their X-ray pulse profiles (see \\citealt{CotiZelati2018} for a review).\nThis increase in high-energy activity is sometimes accompanied by the emission of up to hundreds of short-duration, bright hard X-ray bursts.\nFive out of the nearly 30 known magnetars have also been found to display pulsed radio emission during outburst states \\citep{Olausen2014}\\footnote{\\url{https:\/\/www.physics.mcgill.ca\/~pulsar\/magnetar\/main.html}}. \nThese radio-loud episodes are typically preceded by several years of radio quiescence followed by the sudden activation of intense, pulsed radio emission. \nStrong temporal variability in the detected radio flux, pulse-profile shape and polarization fraction are often observed \\citep{Levin2012, Camilo2018, Dai2019, Lower2021a}. Moreover, rotational variations in the form of rapid increases in spin-down and sudden spin-up (glitches) or, less commonly, spin-down events (anti-glitches) are also detected following outbursts \\citep{Archibald2013, Scholz2017, Younes2020, Caleb2022, hu20ApJ}.  These observational properties are likely a result of magnetic stresses from the decaying internal magnetic fields precipitating sudden crustal motion or fracturing, leading to an impulsive release of energy in the form of electromagnetic radiation and occasionally an alteration of the external magnetic field topology \\citep{Duncan1992, Harding1999, Thompson2002, younes22ApJL18301}\n\nWith a spin period of $\\sim 2.1$\\,s, 1E~1547.0$-$5408 (also known as PSR~J1550$-$5418) is the second fastest rotating magnetar found to date.\nFirst discovered as a persistent X-ray source by the \\textit{Einstein Observatory} \\citep{Lamb1981}, it was not until several decades later that it was recognized as a potential magnetar candidate embedded within the putative supernova remnant G327.24$-$0.13, estimated to be $\\sim 4$\\,kpc away \\citep{Gelfand2007}.\nTargeted pulsar-search observations revealed the presence of highly-polarized, radio pulses originating from the source \\citep{Camilo2007}.\nSubsequent timing observations enabled measurements of both the aforementioned spin-period and a spin-down rate of $2.138 \\times 10^{-11}$\\,s\\,s$^{-1}$, thereby confirming its magnetar status (with an equatorial field of $B \\approx 2\\times 10^{14}$\\,G; \\citealt{Camilo2007}) and making it the second radio-loud magnetar to be discovered \\citep{Camilo2007, Camilo2008}.\nThis serendipitous discovery appeared to follow an unnoticed outburst sometime in the mid-2000s \\citep{Camilo2008} (possibly during 2007; see \\citealt{Halpern2008}), and was soon followed by two intense high-energy outbursts in 2008 and 2009 \\citep{Israel2010, Mereghetti2009, Scholz2011, Bernardini2011, vanderHorst12ApJ}.\nDetections of the radio pulses became highly intermittent following the outbursts, but the source stabilized to a persistent radio-loud state after several years (Camilo et al. in prep.).\n\nSince its major outburst of 2009 January, \\src\\ has remained in an unusual high X-ray flux state, bearing a much stronger resemblance to a persistent magnetar as opposed to its initial `transient' classification (\\citealt{CotiZelati2020}; Camilo et al. in prep.).\nNo additional X-ray outbursts above the baseline were observed during this time, barring the detection of two short bursts seen with the the Neil Gehrels \\swift\\ Observatory (thereon Swift) Burst Alert Telescope (BAT) in 2011 and 2013, respectively \\citep{2013GCN.14911....1H}.\nNeither of these two short bursts were followed up by targeted high-cadence monitoring.\n\nIn this work, we present the detection of a new outburst from \\src\\ that began between late-March and early-April 2022.\nAn overview of our radio and high-energy observations are detailed in the following section.\nIn Section~\\ref{sec:analysis} we present our analysis of the timing and radiative properties of \\src\\ during the outburst.\nWe conclude in Section~\\ref{sec:disc} with a discussion on the interplay between changes in the high-energy, radio, and timing properties of \\src, and potential implications for the outburst mechanism.\n\n\\section{Observations} \\label{sec:obs}\n\n\\subsection{\\parkes\\ Observatory}\n\nRegular monitoring of \\src\\ with an approximately 10-day cadence has been undertaken with \\textsl{Murriyang}, the 64-m \\parkes\\ radio telescope, under the P885 project since its discovery as a radio-loud magnetar in 2007 \\citep{Camilo2007}.\nOur observations from 2019 January have been performed using the Ultra-Wideband Low (UWL) receiver system \\citep{Hobbs2020}, where two independent pulsar search-mode data streams are recorded with the pulsar digital filterbank v4 (PDFB4) and GPU-based {\\sc medusa} signal processors.\nIndividual observations lasted between 2--15\\,minutes. \nFor this work, we analyzed the subset of the PDFB4 data spanning from 2021 June to 2022 August and {\\sc medusa} data covering 2021 December to 2022 August.\nData collected by PDFB4 were taken with 1\\,ms time-sampling at a center frequency of 3100\\,MHz with 512 channels covering 1024\\,MHz of bandwidth.\nFull Stokes data collected with {\\sc medusa} were recorded with 128\\,$\\mu$s time resolution with 3328 channels covering the full UWL band from 704--4032\\,MHz, and coherently dedispersed at a dispersion measure (DM) of 830\\,pc\\,cm$^{-3}$.\nBoth data sets were subsequently folded at the rotation period of the magnetar using {\\tt dspsr} \\citep{vanStraten2011}, such that a single pulse period is covered by 1024 phase bins. \nFrequency channels affected by radio-frequency interference (RFI) were excised using the {\\tt paz} and {\\tt pazi} tools in {\\tt psrchive} \\citep{Hotan2004, vanStraten2012}.\nThe folded {\\sc medusa} data were then flux and polarization calibrated following the procedures outlined in \\citet{Lower2020}. \n\n\\subsection{\\nicer}\n\nThe Neutron star Interior Composition ExploreR (\\nicer) is a non-imaging, soft X-ray telescope mounted on the International Space Station, with a $\\approx5\\hbox{$^\\prime$}$ diameter field-of-view (FoV; \\citealt{gendreau16SPIE}). \n\\nicer\\ provides a large collective area peaking at $1900$~cm$^2$ at 1.5~keV and a timing resolution $<300$\\,ns. We initiated a series of \\nicer\\ target of opportunity observations of \\src\\ two days after the 2022 April 7 \\swift-BAT detection of a magnetar-like burst coincident with \\src. The \\nicer\\ monitoring consisted of about 1~ks observations every few days, starting on 2022 April 9. We provide the observations log in Table~\\ref{xObsLog}.\n\n\\begin{table}[t]\n\\centering \n\\caption{List of X-ray observations analyzed in this work}\n\\label{xObsLog}\n\\newcommand\\T{\\rule{0pt}{2.6ex}}\n\\newcommand\\B{\\rule[-1.2ex]{0pt}{0pt}}\n\\begin{center}{\n\\resizebox{0.47\\textwidth}{!}{\n\\begin{tabular}{l c c c}\n\\hline\n\\hline\nTelescope \\T\\B & Observation ID & Date & Total GTI \\\\\n\n          \\T\\B &  & (UTC) & (ks) \\\\\n\\nustar\\ & 30101035002 & 2016-04-23 & 84 \\\\\n\\nustar\\ & 30401008002 & 2019-02-15 & 90 \\\\\n\\swift\/XRT & 00013176022 & 2022-03-05 & 2.6 \\\\\n\\swift\/XRT & 00013176023 & 2022-03-06 & 1.8 \\\\\n\\nicer\\ & 5020300101 & 2022-04-09 & 1.9 \\\\\n\\nicer\\ & 5020300102 & 2022-04-12 & 0.1 \\\\\n\\nicer\\ & 5020300103 & 2022-04-13 & 0.2 \\\\\n\\nicer\\ & 5020300104 & 2022-04-14 & 0.3 \\\\\n\\nicer\\ & 5020300106 & 2022-04-16 & 1.2 \\\\\n\\nicer\\ & 5020300107 & 2022-04-17 & 0.6 \\\\\n\\nicer\\ & 5020300108 & 2022-04-18 & 0.4 \\\\\n\\nicer\\ & 5020300109 & 2022-04-21 & 1.5 \\\\\n\\nustar\\ & 90801309002 & 2022-04-21 & 24 \\\\\n\\nicer\\ & 5020300110 & 2022-04-22 & 1.8 \\\\\n\\nicer\\ & 5020300111 & 2022-04-25 & 0.4 \\\\\n\\swift\/XRT & 00013176025 & 2022-05-07 & 3.0 \\\\\n\\nicer\\ & 5020300113 & 2022-05-08 & 1.6 \\\\\n\\swift\/XRT & 00013176026 & 2022-05-11 & 1.5 \\\\\n\\nicer\\ & 5020300114 & 2022-05-13 & 0.9 \\\\\n\\nicer\\ & 5020300115 & 2022-05-15 & 0.2 \\\\\n\\nicer\\ & 5020300116 & 2022-05-21 & 1.0 \\\\\n\\nicer\\ & 5020300117 & 2022-05-22 & 0.5 \\\\\n\\nicer\\ & 5020300118 & 2022-05-28 & 0.7 \\\\\n\\nicer\\ & 5020300119 & 2022-06-07 & 1.5 \\\\\n\\nicer\\ & 5020300120 & 2022-06-17 & 1.2 \\\\\n\\nicer\\ & 5020300121 & 2022-06-27 & 0.4 \\\\\n\\nicer\\ & 5020300123 & 2022-07-07 & 0.9 \\\\\n\\swift\/XRT & 00013176027 & 2022-07-09 & 4.3 \\\\\n\\nicer\\ & 5020300124 & 2022-07-13 & 0.4 \\\\\n\\nicer\\ & 5020300125 & 2022-07-17 & 2.1 \\\\\n\\nicer\\ & 5020300126 & 2022-07-20 & 1.1 \\\\\n\\nicer\\ & 5020300127 & 2022-07-21 & 0.7 \\\\\n\\nicer\\ & 5020300128 & 2022-07-28 & 1.3 \\\\\n\\nicer\\ & 5020300129 & 2022-07-30 & 0.4 \\\\\n\\nicer\\ & 5020300130 & 2022-08-02 & 0.1 \\\\\n\\nicer\\ & 5020300131 & 2022-08-03 & 0.2 \\\\\n\\nicer\\ & 5020300132 & 2022-08-05 & 0.2 \\\\\n\\nicer\\ & 5020300133 & 2022-08-09 & 0.4 \\\\\n\\nicer\\ & 5020300134 & 2022-08-15 & 1.8 \\\\\n\\nicer\\ & 5020300135 & 2022-08-17 & 2.3 \\\\\n\\hline\n\\hline\n\\end{tabular}}}\n\\end{center}\n\\end{table}\n\n\nWe cleaned and calibrated all \\nicer\\ data utilizing {\\tt NICERDAS} version v008c as part of {\\tt HEASoft} version 6.30.1. We applied the standard filtering criteria as described in the \\nicer\\ Data Analysis Guide\\footnote{\\href{https:\/\/heasarc.gsfc.nasa.gov\/docs\/nicer\/data\\_analysis\/nicer\\_analysis\\_guide.html}{NICER data analysis guide.}}. Moreover, detectors 14 and 34 can show larger than normal background noise, hence, were excluded from all analyses. Finally, we compared the light curve (binned with a 5~second resolution) created in the energy range 1--6~keV (where the source is supposed to dominate the emission) to one created in the 13--15~keV energy range. Almost all the counts in the latter band are due to high energy particle background. We eliminated any simultaneous flaring background intervals that appeared in both light curves. These were mostly present at the start and\/or end of a good time interval (GTI) due to imprecise modelling of the South Atlantic Anomaly. We provide the total GTIs of each \\nicer\\ observation in Table~\\ref{xObsLog}. We utilise {\\tt Xselect} as part of {\\tt HEASoft} version 6.30.1 to extract the source spectra, and estimated the background contribution for each observation utilizing the 3C50 model \\citep{remillard2021:3c50}.\n\n\\subsection{\\nustar}\n\n\\nustar\\ (Nuclear Spectroscopic Telescope ARray; \\citealt{harrison13ApJ:NuSTAR}) is a focusing hard (3--79~keV) X-ray telescope, consisting of two identical focal plane modules, FPMA and FPMB. Each module is formed by an array of $2\\times2$ detectors providing a total of about $12\\hbox{$^\\prime$}\\times12\\hbox{$^\\prime$}$ FoV. Prompted by the increase in the soft X-ray flux, we requested a \\nustar\\ director discretionary time and \\src\\ was observed for 24~ks on 2022 April 21. We utilized {\\tt NuSTARDAS} software version 2.1.2 to clean and calibrate the event files. We employed the task {\\tt nuproducts} to extract source events, light curves, and spectra from a circular region with a 50\\hbox{$^{\\prime\\prime}$}-radius around the source central brightest pixel, which corresponds to 70\\% of the encircled-count fraction of the detectors point spread function. This choice of extraction region is driven by the desire to maximize source counts, while limiting stray-light background, which is present in both \\nustar\\ modules in proximity to the source. Similarly, we extracted the background files from a source-free region on the same detector as the source with the same radius. For comparison purposes, we analysed two archival \\nustar\\ observations taken on 2016 April 23 and 2019 February 15, following the same procedures outlined above. Table~\\ref{xObsLog} summarises the \\nustar\\ observations.\n\n\\subsection{\\swift}\n\nThe X-ray Telescope (XRT) onboard \\swift\\ is an imaging charge-coupled device, sensitive to photons in the energy range of 0.2--10~keV \\citep{burrows05SSRv}. XRT observed \\src\\ on several occasions in 2022 March, May and July, bracketing the time of the outburst. Thus, these observations provide the source soft X-ray spectral state shortly before the outburst onset, and a confirmation of its properties as observed with \\nicer\\ following the outburst. For these observations, XRT was operating in windowed-timing mode, affording a 1D image with a time resolution of 1.7~ms. We utilized {\\tt XRTDAS} version 3.7.0 to reduce and calibrate the data. We extracted source events from each good time interval (GTI) of a given observation separately. We used a circular region with a 20 pixel radius centered on the brightest pixel. We extracted background events from an annulus with inner and outer radii of 80 and 120 pixels centered at the same position as the source. We created the ancillary files using \\texttt{xrtmkarf}, and used the response matrices in CALDB version 20220803. We excluded all GTIs where the source landed within a 3 pixel distance from a bad column or the edge of the CCD due to increase in systematic uncertainties\\footnote{\\url{https:\/\/www.swift.ac.uk\/user\\_objects\/lc\\_docs.php}}. For the remaining good GTIs, we add the source and background spectra, as well as the ancillary and response files using the HEASOFT tool \\texttt{addspec}. We provide the XRT observation IDs, date, and total GTI in Table~\\ref{xObsLog}.\n\nThe Burst Alert Telescope (BAT), also onboard \\swift, is a coded aperture telescope which images a $\\sim 1.5$~sr FoV in the energy range of 15--150~keV. Images are generated by the on-board software in response to count rate increases at timescales from 4 ms to 24 s or, in the absence of rate increases, on minute and longer timescales. \nBursts from sources outside of the FOV can often be detected due to the leakage of photons through the instrument shielding, yielding a count rate increase without a corresponding image peak.  These can be used to confirm the astrophysical nature of count rate increases seen by other instruments and constrain their direction of origin.\n\n\\subsection{\\fermi-GBM}\n\nThe Gamma-ray Burst monitor \\citep[GBM;][]{meegan09ApJ:gbm} is a non-imaging instrument onboard the \\fermi\\ Gamma-ray Space Telescope that is sensitive to photons in the 8--1000\\,keV range. GBM is capable of automatic on-board triggers, with one of the software algorithms tailored to magnetar-like short bursts. Apart from the triggered data, GBM also provides continuous time-tagged events (CTTE), with a time resolution of 2 microseconds. We utilized the CTTE data to search for any untriggered bursts from the direction of \\src\\ from its last radio detection on 2022 March 16 until the confirmed BAT burst on 2022 April 7. This analysis provides an estimate of the start time of the 2022 outburst.\n\n\\section{Analysis and results} \\label{sec:analysis}\n\n\\subsection{Radio disappearance and reactivation}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{Postage_stamps.pdf}\n    \\caption{Examples of the three polarization profile states (A, B and C) of \\src\\ seen at different epochs. Note, the profiles from 2022-03-16 (top-left) and 2022-04-19 (middle-right) correspond to the last pre-outburst and first post-outburst radio detections respectively. Each observation was dedispersed using a dispersion measure of $697$\\,pc\\,cm$^{-3}$ and corrected for a Faraday rotation measure of $-1847$\\,rad\\,m$^{-2}$ prior to averaging in time and frequency.}\n    \\label{fig:postage_stamps}\n\\end{figure*}\n\nUnlike other radio-loud magnetars which fade into radio silence in the months to years after an outburst, \\src\\ has remained radio-loud for well over a decade since the 2009 outburst (Camilo et al., in prep).\nFollowing the 2008 and 2009 outbursts, detections of radio pulses were intermittent \\citep[see e.g.][]{Israel2021}, possibly reflecting the highly dynamic nature of its magnetosphere during that period.\nIntermittent detections continued up to 2013, after which radio pulses were consistently detected during every observation.\n\\src\\ remained in this persistently radio-loud state until our two observations on 2022 March 16 and April 9, where we failed to detect radio pulses from the magnetar.\nRadio pulses were re-detected during the 2022 April 19 observation, $\\sim$12 days after the BAT confirmed trigger, and have persisted up to at least our Parkes-UWL observation on 2022 August 3.\n\nUsing a high $S\/N$ observation taken on 2022 July 9, we measured a structure-optimized dispersion measure (DM) of $697 \\pm 1$\\,pc\\,cm$^{-3}$ with the {\\tt DM\\_phase} package \\citep{Seymour2019}.\nThe updated DM is a significant improvement on the previously reported value of $830 \\pm 50$\\,pc\\,cm$^{-3}$ \\citep{Camilo2007}.\nThis is likely due to differences in the methods applied to compute the DM (profile structure versus S\/N optimisation) in addition to the substantially larger instantaneous bandwidth afforded by the UWL.\nUsing the NE2001 \\citep{Cordes2002} and YMW16 \\citep{Yao2017} electron density models, this updated DM corresponds to estimated distances of 8.3\\,kpc and 5.9\\,kpc, respectively. The latter distance is comparable to the commonly assumed sdistance of $\\sim 4.5$\\,kpc inferred from dust-scattering halos detected after the 2009 outburst \\citep{Tiengo2010}, when considering the large systematic uncertainties that are associated with any of these methods, which is on average 50\\%. Similarly, we obtained an updated rotation measure (RM) of $-1847.6 \\pm 0.5$\\,rad\\,m$^{-2}$ by directly fitting the Stokes $Q$ and $U$ spectra in the 2000--4032 MHz range (to avoid complications from scatter broadening) with {\\tt RMNest} \\citep{Lower2022}.\nOur updated RM is consistent with the previously reported value of $-1860 \\pm 20$\\,rad\\,m$^{-2}$ \\citep{Camilo2008}.\n\nThe shape of the average radio profile detected throughout 2022 January-August was highly variable, and can be broadly categorized into three general archetypes:\n\\begin{itemize}\n    \\item Profile A: Strong leading component with one to three trailing components.\n    \\item Profile B: Complex, multi-component profile, sometimes with a central peak.\n    \\item Profile C: Two dominant profiles joined by weaker sub-components or a flat bridge of emission.\n\\end{itemize}\nExamples of all three profile types are shown in Figure~\\ref{fig:postage_stamps}, all of which were dedispersed and de-Faraday rotated using our updated DM and RM values.\nNote, these categorizations are purely qualitative.\nFor instance, the complex 2022 June 13 profile (bottom-middle panel in Figure~\\ref{fig:postage_stamps}) is largely double peaked, and could also be classified as Profile C.\nTransitions between profile types tend to be sudden and reminiscent of transient profile changes reported in earlier observations of this magnetar (e.g., Fig. 5 of \\citealt{Halpern2008}) and those seen in emission mode-changing\/state-switching pulsars \\citep{Wang2007}.\nThe radio profiles of \\src\\ in Figure~\\ref{fig:postage_stamps} display clear epoch-to-epoch variations in flux density.\nWe computed the frequency-averaged radio flux density as\n\\begin{equation}\n    {\\rm F}_{f} = \\frac{1}{N_{\\rm bin}} \\sum_{i}^{N_{\\rm on}} {\\rm F}_{f,i}\\,,\n\\end{equation}\nwhere $N_{\\rm bin} = 1024$ is the number of phase-bins covering a full rotation of the magnetar, $N_{\\rm on}$ is the number of bins covering the on-pulse region and ${\\rm F}_{f,i}$ is the flux of the $i$-th phase bin.\nUncertainties on the flux density were computed from the root mean square (RMS) of the off-pulse region\n\\begin{equation}\\label{eqn:flux}\n    \\sigma_{{\\rm F},f} = \\frac{\\sqrt{N_{\\rm on}}}{N_{\\rm bin}} \\sqrt{\\sum_{i}^{N_{\\rm off}} {\\rm F}_{f,i}^{2}}\\,,\n\\end{equation}\nwhere $N_{\\rm off}$ is the number of off-pulse phase bins.\nThe measured flux densities from our observations are displayed in the bottom panel of Figure~\\ref{fig:flux}.\nWe used Equation~\\ref{eqn:flux} to set sensitivity limits for our observations on 2022 March 26 and April 9 where we failed to detect radio pulses, obtaining limiting flux densities of $1.3 \\pm 0.3$\\,mJy and $0.9 \\pm 0.2$\\,mJy respectively.\nVisually, there appears to be a shallow increase in the revived radio flux over time.\nHowever, a direct connection between this apparent secular evolution and the outburst cannot be confirmed without a comparison to the long-term flux history of the magnetar.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Flux.pdf}\n    \\caption{Average unabsorbed X-ray (0.5--10\\,keV; top) and radio (0.7--4\\,GHz; bottom) flux of \\src\\ from 2022 January to August. The vertical dashed lines indicate the dates of the \\fermi-GBM (MJD 59672; 2022 April 3) and \\swift-BAT short bursts (MJD 59676; 2022 April 7) respectively, while the blue shaded region corresponds to where pulsed radio emission was undetected. Grey horizontal shading corresponds to the standard deviation about the baseline X-ray flux of $1.7 \\times 10^{-11}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$.}\n    \\label{fig:flux}\n\\end{figure}\n\nAlso indicated in Figure~\\ref{fig:flux} are the corresponding profile types seen during each observation.\nIn terms of relative numbers, we observed the Profile B state in more than half (25\/49) of our observations, while Profile A and C were seen in 17 and 7 observations respectively.\nThe appearance of any one mode appears to be largely stochastic, though there is a clear preference for Profile B after the outburst up until 2022 May 13 (MJD 59712)  where the magnetar was in the Profile A state.\nNo further systematic change in the profile or the mode-changing behavior is apparent after this date.\nIt is possible that this preference may be related to changes in the emission region plasma following the reactivation. \nHowever such profile variations persist among radio-loud magnetars for many months to years after an outburst \\citep{Levin2012, Lower2021a}.\nAdditionally, the swing of the linear polarization position angle (PA) across the profile remains largely flat and unchanged both before and after the outburst.\nIt is consistent with previous observations of \\src\\ in 2007, where rotating vector model fits indicated that the magnetar is a near-aligned rotator \\citep{Camilo2008}.\nOccasional departures from the smooth PA swing are detected in individual epochs, an example of which is shown in the top-right panel of Figure~\\ref{fig:postage_stamps}.\nThe relatively smooth evolution of the PA swing persists at higher observing frequencies where the effects of interstellar scattering are negligible, indicating these features are due to either an unusually smooth transition between two orthogonal polarization modes or the result of birefringent propagation effects in the near-field environment \\citep{Dyks2020}.\nBirefringent propagation effects would also explain the unusually high circular polarization fraction (up to $|V|\/I = 0.54$) as well as changes in handedness between observations.\nA more detailed investigation is left for future work.\n\n\\subsection{Timing}\\label{timSec}\n\nThe variable radio profile of \\src\\ is not conducive to pulsar timing methods that use a single reference template.\nInstead, we obtained pulse times of arrival (ToAs) from the \\parkes\/PDFB4 data by using the {\\tt get\\_toas} tool in {\\tt PRESTO} \\citep{Ransom2011} to fit a single Gaussian template to individual observations via the Fast Fourier Transform based approach of \\citet{Taylor1992}.\nWe added an extra 0.05\\,s uncertainty in quadrature with the formal ToA uncertainties to account for excess epoch-to-epoch scatter in the ToAs that arise from the varying radio profile (often referred to as `jitter').\n\nFor \\nicer, we merged all barycenter-corrected event files and only accepted counts in the 1--5~keV energy range, which maximized the source pulse signal according to the $Z^2$ power \\citep{buccheri83AA}. We then split the data into time intervals that contain $\\approx1000$~counts each, without allowing individual intervals to exceed a 3-day integration. We employed a non-binned maximum likelihood estimation (MLE) method to measure the X-ray pulse ToAs \\citep{livingstone09ApJ,ray11ApJS}. Firstly, we model a high S\/N pulse profile with a Fourier series including only the fundamental harmonic; adding higher order harmonics did not improve the quality of the fit. Secondly, we fit, using MLE, each unbinned data segment to the same model, only allowing for a phase-shift $\\Delta\\phi$. The best-fit phase shift is the value that corresponds to the minimum of the negative log likelihood, $-\\log(\\mathcal{L})_{\\rm min}$. The $1\\sigma$ uncertainty on the phase shift was established by stepping over the full parameter space $\\Delta\\phi \\in [0,2\\pi)$ and noting a change in $-\\log(\\mathcal{L})_{\\rm min}$ by $0.5$ (corresponding to the $\\pm34\\%$ or $1\\sigma$ confidence interval of a $\\chi^2$ distribution with 1~degree of freedom).\n\nWe obtained an initial phase-connected solution by using the Hidden Markov Model (HMM) based glitch detector of \\citet{Melatos2020}\\footnote{\\href{https:\/\/github.com\/ldunn\/glitch_hmm}{https:\/\/github.com\/ldunn\/glitch\\_hmm}} which tracks the spin-frequency ($\\nu$) and spin-down rate ($\\dot{\\nu}$) of the magnetar over time. \nThis enabled robust integer pulse numbering to be applied to each ToA and provided estimates of any changes in $\\nu$ and $\\dot{\\nu}$ at the time of the outburst.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Timing.pdf}\n    \\caption{Top panel shows the pre-fit \\parkes\/\\nicer\\ timing residuals of \\src\\ from 2021 June to 2022 August. Changes in the magnetar spin-frequency and spin-down rate with respect to their nominal values ($0.478836$\\,Hz and $-3.63 \\times 10^{-12}$\\,s$^{-2}$ at MJD 58350) are shown in the middle and bottom panels respectively, where the points correspond to local fits to the ToAs. Purple shading is the $\\nu$ and $\\dot{\\nu}$ posterior distributions that were recovered by the HMM glitch detector at each epoch. Outliers in the local fits to $\\dot{\\nu}$ arise from the limited number of ToAs. Vertical shaded region and dashed lines are the same as in Figure~\\ref{fig:flux}.}\n    \\label{fig:f0f1_fits}\n\\end{figure}\n\nOnce the phase-connected solution was obtained, we then used the {\\tt glitch} plugin to {\\tt tempo2} \\citep{Hobbs2006, Edwards2006} to perform independent local fits to $\\nu$ and $\\dot{\\nu}$.\nThe results of our local fits are shown alongside the HMM tracking in Figure~\\ref{fig:f0f1_fits}.\nBoth sets of spin-frequency\/spin-down rate measurements generally agree with one another and display approximately the same secular evolution.\nA timing anomaly with clear jumps in both $\\nu$ and $\\dot{\\nu}$ occurred in coincidence with the high-energy outburst, with the HMM recovering step-changes of $\\Delta\\nu = (9 \\pm 4) \\times 10^{-7}$\\,Hz and $\\Delta\\dot{\\nu} = (-1.6 \\pm 0.5) \\times 10^{-12}$\\,s$^{-2}$. \nThere is also a clear, secular increase in spin-down rate of the magnetar ($\\dot{\\nu}$ is becoming more negative) over time, where $\\dot{\\nu}$ appears to follow an approximately linear trend over the first 120\\,days post-outburst.\nThis is also reflected by the downward curve in the evolution of $\\nu$ over time.\nIncreasing spin-down rates are also commonly observed in the post-outburst behavior of other magnetars, including recent outbursts in the radio-loud magnetars PSR~J1622$-$4950 \\citep{Camilo2018} and XTE~J1810$-$197 \\citep{Caleb2022}.\nA linear fit to the spin-down evolution gives an apparent $\\ddot{\\nu} = -1.95(5) \\times 10^{-19}$\\,s$^{-3}$.\nSimilar behavior was previously observed in \\src\\ following the 2008 outburst \\citep{Israel2010}.\n\nIn order to better constrain both the longer-term rotational properties of \\src\\ and the outburst-induced changes in its timing, we modelled the anomaly as a pulsar glitch where the corresponding change in rotation phase is given by\n\\begin{eqnarray}\n     \\phi_{\\rm {g}}(t) = \\Delta\\phi_{g} + \\Delta\\nu_{p}(t - t_{\\rm {g}}) + \\frac{1}{2}\\Delta\\dot{\\nu}_{p}(t - t_{\\rm {g}})^{2} \\nonumber \\\\\n     + \\frac{1}{6}\\Delta\\ddot{\\nu}_{p}(t - t_{\\rm {g}})^{3} - \\Delta\\nu_{d}\\tau_{d} e^{-(t - t_{\\rm {g}})\/\\tau_{d}}.\n\\end{eqnarray}\nHere, $t_{g}$ is the glitch epoch, $\\Delta\\phi_{g}$ is a phase-jump to account for residual ambiguities in the rotation phase of the magnetar after the event, $\\Delta\\nu_{p}$ is the permanent step-change in spin-frequency, $\\Delta\\dot{\\nu}_{p}$ the step-change in spin-down rate, and $\\Delta\\nu_{d}$ is the exponentially recovered change in spin-frequency that occurred over a characteristic timescale, $\\tau_{d}$.\nWe also included a glitch-induced second spin-frequency derivative ($\\Delta\\ddot{\\nu}_{p}$) term to account for the post-outburst linear increase in $\\dot{\\nu}$ that we discussed above.\n\nThe gap in our timing between the last \\parkes\\ detection on 2022 March 16 and the first \\nicer\\ observation on April 9 makes it difficult to determine the precise glitch epoch. Since magnetar glitches are often associated with radiative events \\citep{dib14ApJ}, we use the date of the short-burst detected by \\fermi-GBM from the direction of \\src\\ as the nominal glitch epoch (i.e $t_{g} = $\\,MJD 59672; 2022 April 3; see Sec.~\\ref{sec:burSear}). \nNote, the exact choice of glitch epoch between April 1 and April 7 has a strong impact on the recovered $\\Delta\\nu_{p}$, which we discuss further below. \nUsing {\\tt tempo2} and the {\\tt FITWAVES} plugin, we fit the glitch parameters alongside a set of 10 harmonically related sine-waves to account for strong timing noise in the residuals. \nThis returned values of $\\Delta\\nu_{p} = 2.0(1) \\times 10^{-6}$\\,Hz, $\\Delta\\dot{\\nu}_{p} = -1.23(2) \\times 10^{-14}$\\,s$^{-2}$ and $\\Delta\\ddot{\\nu}_{p} = -1.86(3) \\times 10^{-19}$\\,s$^{-3}$.\nA significant $\\Delta\\nu_{d}$ of $-2.1(1) \\times 10^{-6}$\\,Hz was recovered. The negative value of $\\Delta\\nu_{d}$ signifies a resolved rise in the spin-frequency over a period of $\\tau_{d} = 13(1)$\\,days as opposed to a post-glitch decay.\n\nTo verify the existence of this apparent delayed rise in $\\nu$, we sampled the posterior distributions for each parameter using the {\\tt TempoNest} Bayesian inference plugin to {\\tt tempo2} \\citep{Lentati2014}.\nThis also allowed us to simultaneously model the timing noise of the magnetar as a power-law red noise process. The power-spectral density of the red noise model is given by\n\\begin{equation}\n    P(f) = \\frac{A_{\\rm r}^{2}}{12\\pi^{2}} \\Big( \\frac{f}{1\\,{\\rm yr}} \\Big)^{-\\beta_{\\rm r}}\\,{\\rm yr}^{-3},\n\\end{equation}\nwhere $A_{\\rm r}$ is the red noise amplitude (in units of yr$^{3\/2}$) and $\\beta_{\\rm r}$ the spectral index.\nTwo separate fits were performed: one that included a resolved rise in spin frequency, and another where $\\Delta\\nu_{d}$ is fixed to zero for an instantaneous (unresolved) glitch.\nComparing the recovered log-evidence from the resulting fits, we obtain a Bayes' factor of $\\ln\\mathcal{B} = -2.8$ in favor of a resolved rise.\nThe negative value indicates there is no significant evidence for the presence of a resolved versus an instantaneous change in spin-frequency.\nOur results for the un-resolved glitch model are summarized in Table~\\ref{tab:timing}.\n\n\\begin{table}[t!]\n\\centering \n\\caption{Timing and pulsar parameters for \\src\\ from our joint \\parkes\/\\nicer\\ timing. Values in parentheses are the 1-$\\sigma$ uncertainties on the last digit.}\n\\label{tab:timing}\n\\begin{tabular}{ll}\n\\hline\n\\hline\nPulsar parameters &  \\\\\n\\hline\nR. A. (J2000)$^{\\dagger}$                    & 15$^{\\mathrm{h}}$50$^{\\mathrm{m}}$54.12386$^{\\mathrm{s}}$ \\\\\nDecl. (J2000)$^{\\dagger}$                    & $-$54$^{\\circ}$18$'$24.1141$''$ \\\\\n$\\nu$ (Hz)                                   & $0.478836(3)$ \\\\\n$\\dot{\\nu}$ (s$^{-2}$)                       & $-3.63(3) \\times 10^{-12}$ \\\\\n$\\mu_{\\alpha} \\cos\\delta$ (mas)$^{\\dagger}$  & $4.8$ \\\\\n$\\mu_{\\delta}$ (mas)$^{\\dagger}$             & $-7.9$ \\\\\nTiming epoch (MJD TDB)                       & 58350 \\\\\nPosition epoch (MJD TDB)                     & 54795 \\\\\nDM (pc\\,cm$^{-3}$)                           & $697(1)$ \\\\\nRM (rad\\,m$^{-2}$)                           & $-1847.6(5)$ \\\\\nTime span (MJD)                              & 59368--59808 \\\\\nNumber of ToAs                               & 202 \\\\\nSolar system ephemeris                       & DE436 \\\\\n$\\log_{10}(A_{\\rm r})$ (yr$^{3\/2}$)          & $-5.1(2)$ \\\\\n$\\beta_{\\rm r}$                              & $5.7(8)$ \\\\\n\\hline\nGlitch parameters &  \\\\\n\\hline\n$t_{\\rm g}$ (MJD)                   & 59672 \\\\\n$\\Delta\\nu_{p}$ (Hz)                & $2(1) \\times 10^{-7}$ \\\\\n$\\Delta\\dot{\\nu}_{p}$ (s$^{-2}$)    & $-2.4(1) \\times 10^{-12}$ \\\\\n$\\Delta\\ddot{\\nu}_{p}$ (s$^{-3}$)   & $-2.0(1) \\times 10^{-19}$ \\\\\n\\hline\\\\\n\\end{tabular}\n{$^{\\dagger}$}Fixed to values from~\\cite{Deller2012}.\n\\end{table}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{Timing_posteriors.pdf}\n\\caption{One- and two-dimensional posterior distributions for the glitch and red noise in \\src. Darker and lighter shaded regions represent the 68\\% and 95\\% confidence intervals respectively.}\n\\label{fig:glitch_post}\n\\end{center}\n\\end{figure}\n\nWe present the one- and two-dimensional posterior distributions for the glitch and timing noise parameters in Figure~\\ref{fig:glitch_post}. There is a strong correlation between $\\Delta\\dot{\\nu}_{p}$ and $\\Delta\\ddot{\\nu}_{p}$, indicating these two parameters induce a similar signal across the relatively short post-glitch timing of \\src. While $\\Delta\\dot{\\nu}_{p}$ agrees with the value recovered by the HMM at the 68\\% confidence interval, the {\\tt TempoNest} derived value of $\\Delta\\nu_{p}$ is smaller by a factor of $\\sim$4.5. As for the post-outburst rotational-evolution, our recovered value of $\\Delta\\ddot{\\nu}_{p}$ is consistent with that inferred from a simple linear fit to $\\dot{\\nu}$ over time.\n\nThe difference in $\\Delta\\nu_{p}$ values is a direct result of our choice of glitch epoch and the large increase in spin-down rate. This is apparent in {\\tt TempoNest} runs where we assumed earlier and later values of $t_{\\rm g}$, where earlier glitch epochs result in a larger recovered value of $\\Delta\\nu_{p}$ to match the post-outburst timing. Going the other way, setting the glitch epoch to the date of the \\swift-BAT short burst (MJD 59676; 2022 April 7) results in a value of $\\Delta\\nu_{p}$ that is consistent with zero at the 68\\% confidence interval, i.e a pure spin-down glitch. While several glitches in other magnetars have been associated with minimal instantaneous changes in $\\nu$ (e.g. the 2012 outburst of 1E~1048.01$-$5937; \\citealt{Archibald2015}), the lack of ToAs between the radio shutdown of \\src\\ and beginning of our \\nicer\\ monitoring prevents us from ruling out a non-zero $\\Delta\\nu_{p}$.\n\n\\subsection{High-energy outburst}\n\n\\subsubsection{Spectral analysis}\n\nWe employed Xspec version 12.12.1 for our X-ray spectral analysis. We used the \\texttt{tbabs} model to account for the line-of-sight absorption towards the source, along with the \\citet{wilms00ApJ} solar abundances. We binned all spectra to have 5 counts per bin and used the W-statistic (C-stat in Xspec) to estimate the best fit model parameters and their associated uncertainties. For the \\nicer\\ spectra, however, we used the pgstat statistic, which is a more appropriate choice given the modeled nature of the background \\citep{remillard2021:3c50}. Finally, we added a numerical factor factor to account for any cross-calibration uncertainties when fitting spectra from different instruments simultaneously. We find a $\\lesssim2\\%$ and $\\lesssim10\\%$ deviation between the two \\nustar\\ spectra and \\nicer\/\\swift-XRT and \\nustar\\ spectra, respectively, commensurate with the expected values \\citep{madsen15ApJS}.\n\nWe fit the \\nustar\\ and \\nicer\\ observations taken on 2022 April 21 simultaneously, which provides the highest S\/N broad-band spectrum shape of \\src\\ during this most recent outburst. We also fit the pre-outburst 2016 and 2019 simultaneous \\swift-XRT and \\nustar\\ observations \\citep{CotiZelati2020} to better quantify the source spectral variability. We modelled all spectra in the 1--70~keV range using a three-component construction: two absorbed blackbodies ($\\rm BB_{warm}$ and $\\rm BB_{hot}$) plus a power-law (PL), suitably representative of the magnetar broad-band X-ray spectral shape \\citep{Kuiper-2006-ApJ,Goetz-2006-AandA,denHartog-2008a-AandA,denHartog-2008b-AandA,enoto17ApJS}. We linked the absorption hydrogen column density $N_{\\rm H}$ among all the spectra and left the rest of the model parameters free to vary. We establish the goodness of fit by running the Xspec \\texttt{goodness} command to simulate $1000$ spectra drawn from the best-fit model and compare their fit statistic to the observed data. We find that 16\\% of the spectra have a fit statistic smaller than the actual value, implying that the model adequately explains the data. We measure a best-fit $N_{\\rm H} = (3.67 \\pm 0.02) \\times 10^{22}$~cm$^{-2}$, a factor of 1.75 times higher than the value predicted by the $N_{\\rm H}$-DM relation of \\citet{He2013}. We display the source spectrum and best fit model in Figure~\\ref{specFig}, and present the best fit parameters and their $1\\sigma$ uncertainty in Table~\\ref{specParam}.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\hspace{-0.0cm}\n\\includegraphics[angle=0,width=0.48\\textwidth]{spec_1e1547_outburst2022.pdf}\n\\caption{{\\sl Upper panel.} Broad-band $\\rm \\nu F_\\nu$ spectra of \\src\\ derived during the 2022 April outburst (\\nicer+\\nustar, red), 2019 February (XRT+\\nustar, black) and 2016 April (XRT+\\nustar, green). Solid lines represent the best-fit model consisting of two BB and a PL. A flux enhancement is clearly present at soft X-rays coincident with the most recent outburst, in contrast to the stable non-thermal hard X-ray tail. {\\sl Lower panel.} Deviation of the data from the best-fit model in units of $\\sigma$.}\n\\label{specFig}\n\\end{center}\n\\end{figure}\n\n\\begin{table*}[t]\n\\caption{Broad-band X-ray spectral results}\n\\label{specParam}\n\\newcommand\\T{\\rule{0pt}{2.6ex}}\n\\newcommand\\B{\\rule[-1.2ex]{0pt}{0pt}}\n\\begin{center}{\n\\hspace{-2.0cm}\n\\resizebox{1\\textwidth}{!}{\n\\begin{tabular}{l c c c c c c c c}\n\\hline\n\\hline\nDate & $kT_{\\rm warm}$ \\T\\B & $R^2_{\\rm warm}$ & $kT_{\\rm hot}$ & $R^2_{\\rm hot}$ & $\\Gamma$ & $F_{\\rm warm}$\/$F_{\\rm hot}$\/$F_{\\rm PL}$ & $F_{\\rm S}$\/$F_{\\rm H}$ & $L_{\\rm X}$\\\\\n& (keV)       \\T\\B     &    (km$^2$)          &  (keV)            & (km$^2$)          &                     & ($10^{-12}$~erg s$^{-1}$ cm$^{-2}$)                         &   ($10^{-12}$~erg s$^{-1}$ cm$^{-2}$) &  ($10^{34}$~erg s$^{-1}$) \\\\\n\\hline\n2016-03-23 & $0.59\\pm0.03$ \\T\\B & $2.3_{-0.5}^{+0.7}$ & $1.16_{-0.08}^{+0.1}$ & $0.03_{-0.01}^{+0.02}$ & $0.2\\pm0.1$ & $14.1_{-0.8}^{+0.9}$ \/ $2.9_{-0.6}^{+0.7}$ \/ $8.9_{-0.5}^{+0.5}$ & $17.2_{-0.8}^{+0.9}$ \/ $8.7_{-0.4}^{+0.4}$ & $6.3_{-0.2}^{+0.2}$ \\\\\n2019-02-15 & $0.67\\pm0.02$ \\T\\B & $1.3\\pm0.2$       & $1.4\\pm0.2$ & $0.008\\pm0.004$ & $0.0\\pm0.1$ & $13.7_{-0.5}^{+0.5}$ \/ $1.7_{-0.3}^{+0.5}$ \/ $8.7_{-0.6}^{+0.7}$ & $15.5_{-0.5}^{+0.5}$ \/ $8.7_{-0.5}^{+0.5}$ & $5.9_{-0.1}^{+0.2}$ \\\\\n2022-03-21 & $0.64_{-0.05}^{+0.04}$ \\T\\B & $2.0_{-0.3}^{+0.5}$ & $1.0\\pm0.1$ & $0.09_{-0.06}^{+0.14}$ & $0.4\\pm0.2$ & $17_{-3}^{+2}$ \/ $6_{-2}^{+3}$ \/ $9.0_{-1}^{+1}$ & $23_{-1}^{+1}$ \/ $8.7_{-0.8}^{+0.8}$ & $7.7_{-0.3}^{+0.3}$ \\\\\n\\hline\n\\hline\n\\end{tabular}}}\n\\end{center}\n\\begin{list}{}{}\n\\item {\\bf Notes.} BB emitting areas and luminosity are derived assuming a distance of 4.5~kpc \\citep{Tiengo2010}. $F_{\\rm S}$ and $F_{\\rm H}$ are the 0.5-10 keV and 10-70 keV absorption-corrected fluxes, respectively.\n\\end{list}\n\\end{table*}\n\nWe find that the 2022 April 21 observation, 2.5 weeks after the GBM short burst detection (2 weeks after the confirmed BAT detection), exhibits a flux that is about 50\\% larger than the pre-outburst level. This increase is solely due to an increase in the soft 0.5--10~keV X-ray flux (Figure~\\ref{specFig}), and particularly seen in the $\\rm BB_{hot}$ contribution. The flux of this component has increased by a factor 3 and 5 since 2016 and 2019, respectively. On the other hand, the increase in the $\\rm BB_{warm}$ component is a factor 1.2  and 1.3 larger compared to 2016 and 2019, respectively. Finally, the hard X-ray PL flux is consistent between all three observations at the $1\\sigma$ level. This implies that either the hard X-ray flux was not impacted by this most recent outburst, or it followed a steeper temporal decay trend compared to the soft X-ray flux.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width=0.48\\textwidth]{ppCollage_nicer2.pdf}\n\\caption{Time-resolved \\nicer\\ pulse profiles of \\src\\ in days from 2022 April 9. We show the pulsations in the energy range 1.5--3~keV (left column) and 3--6~keV (right column). The red profile corresponds to the \\nicer\\ observation simultaneous to \\nustar\\ taken on 2022 April 21.  All profiles are displayed using 15 phase bins per rotational cycle, with the exception of the red profile, which exhibits 30 phase bins per pulse period (see text for details). Two cycles are shown for clarity.}\n\\label{nicProf}\n\\end{center}\n\\end{figure}\n\nWhile the source exhibited a change in the overall flux, the broad-band spectral curvature in 2022 remained relatively stable compared to the previous epochs. We find no significant variability in the temperature of either of the two BBs, nor in the hardness of the PL component. This points to an increase in the overall area of the hot thermally-emitting region as the origin of the soft X-ray flux increase (Table~\\ref{specParam}). \n\nTo study the long-term high energy evolution of the \\src\\ outburst, we modelled the \\nicer\\ and \\swift-XRT spectra in the 1--6~keV range (beyond 6~keV the data for both instruments was mainly background-dominated). We used a simple absorbed BB model and fixed the hydrogen column density $N_{\\rm H}$ to the best-fit value as derived above. This model sufficed to describe all the spectra except for the \\nicer\\ observation on 2022 April 9 (ID 5020300101, Table~\\ref{xObsLog}) which required a second blackbody component to result in a statistically acceptable fit. The resulting 0.5--10~keV absorption corrected flux evolution is shown in the upper-panel of Figure~\\ref{fig:flux}. At the start of the \\nicer\\ monitoring on 2022 April 9, we find a source flux that is a factor 3.5 larger than the pre-outburst value as established with \\swift-XRT on 2022 March 5 (green circles in Figure~\\ref{fig:flux}, upper panel; see also \\citealt{CotiZelati2020}). This confirmed that \\src\\ was experiencing yet another high energy outburst, its first since its major 2009 outburst. The source flux decayed back to the same pre-outburst persistent flux level on 2022 April 26, i.e. 17 days later, and remained there until the end of our monitoring in 2022 August. We find that the bulk of the flux increase throughout the outburst occurred at energies $>2$~keV, consistent with the result of the \\nicer+\\nustar\\ broad-band modelling.\n\n\\subsubsection{X-ray timing analysis}\n\nGiven the complexity of the glitch timing model  (Section~\\ref{timSec}), we fold the \\nicer\\ barycenter-corrected events with a (strictly) post-glitch variant of the combined radio\/X-ray timing solution in Table~\\ref{tab:timing}.\nTo account for red noise in the timing residuals, we fit for spin-frequency derivatives up to $d^5\\nu\/dt^5$, at which order the residuals are dominated by white noise. Due to the limited exposure of most individual \\nicer\\ observations, we merged several consecutive ones until we accumulated about 6000 counts in the 1.5--6~keV interval. This ensured a proper sampling of the pulse shape in two energy bands, namely 1.5--3~keV and 3--6~keV. During the outburst decay phase (2022 April 9-26), the stretch of each interval that fulfilled our criterion was on average 2 days, and increased to 10~days after the source returned to its pre-outburst flux level. We present these pulse profiles in Figure~\\ref{nicProf}.  Each row displays the pulsations per time interval starting from the date of the initial \\nicer\\ observation on 2022 April 9 (day 0), while the left and right panels show the 1.5--3~keV and 3--6~keV pulse-profiles, respectively; note that the pulse in red is from the \\nicer\\ observation that is simultaneous to the \\nustar\\ one for 2022 April 21.\n\nThe soft 1.5--3~keV pulsations are quasi-sinusoidal throughout our monitoring campaign. On the other hand, the 3--6 keV profile exhibits pronounced variability. For instance, during the outburst decay phase, there is an evident shoulder leading to the pulse peak. Moreover, the peak is slightly broader early in the outburst and displays complex shape, likely formed of two or more peaks. This complexity is confirmed with \\nustar\\ as we show below. The 3--6 keV pulse shape turns quasi-sinusoidal starting on day 17 from the first \\nicer\\ observation, coincident with the return of the flux to the pre-outburst level (Figure~\\ref{fig:flux}). We derive the root-mean-square pulsed fraction (rms PF) for each of the light curves in Figure~\\ref{nicProf} \\citep[see, e.g.,][for details]{woods04ApJ}. For this purpose, we fit each background-subtracted pulse profile to a Fourier series including three harmonics, which was sufficient to provide an acceptable fit to the data. We find pulsed fractions ranging from 19--25\\% and 29--34\\% in the 1.5--3~keV and 3--6~keV energy bands, respectively, with a typical $1\\sigma$ uncertainty of about 3\\%. This implies a lack of significant variability in the pulsed fraction at soft X-rays concurrent with the outburst flux enhancement.\n\nThe \\nustar\\ observation of \\src\\ on 2022 April 21 caught the tail end of the outburst. We display the pulse profiles during this observation in 3 separate energy bands, namely, 3--6~keV, 6--10~keV, and 10--15~keV (Figure~\\ref{Xprof}). We confirm the \\nicer\\ results which show, in the energy range 3--6 keV, a non-symmetric profile with a shoulder leading to a multi-peaked pulse maximum; this fine structure in the peak is reminiscent of that seen during the 2020 outburst of SGR 1830-0645 \\citep{younes22ApJL18301}.  Similar pulse profile morphology is apparent in the 6--10 keV range: the pulse exhibits considerable asymmetry with a shoulder leading to the pulse peak. All these features are clearly absent from the 2016 and 2019 \\nustar\\ monitoring; the pulse profile in the same energy range shows a purely sinusoidal form \\citep[][Figure~5]{CotiZelati2020}. On the other hand, the pulse profile at high energies between 10 and 25~keV exhibits quasi-sinusoidal shape, consistent with the result of the earlier \\nustar\\ observations. Note, however, the lower S\/N in this band compared with that at energies below 10 keV. We do not detect significant pulsations at energies $>25$~keV, despite the detection of the source at the $\\gtrsim10\\sigma$ confidence level.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[angle=0,width=0.48\\textwidth]{ppvsEne_NuS2022.pdf}\n\\caption{The \\src\\ pulse profiles as derived with \\nustar\\ at different energies (top to bottom). See text for more details.}\n\\label{Xprof}\n\\end{center}\n\\end{figure}\n\nFinally, we measure the \\nustar\\ energy-resolved rms PF following the same recipe we utilized for \\nicer. We find an increasing trend at soft energies, from about $30\\%$ and up to $45\\%$ at 10 keV. The PF drops to $30\\%$ in the energy range 10--25~keV. Moreover, the latter profile appears slightly offset from the soft-energy ones (Figure~\\ref{Xprof}), yet a formal measurement is complicated by the difference in shape between the profiles. Lastly, we derive a 3$\\sigma$ upper-limit of $25\\%$ on the pulsed fraction at energies $>25$~keV. These elements are broadly consistent with the 2016 and 2019 \\nustar\\ findings of \\citet{CotiZelati2020}.\n\n\\subsubsection{Short burst searches}\n\\label{sec:burSear}\n\nThe 22-day discrepancy between the time of the first radio non-detection, which occurred on 2022 March 16, and the BAT trigger on 2022 April 7 prompts the question on whether the radio disappearance genuinely preceded the trigger of the high energy outburst. With the absence of any sensitive X-ray observations of \\src\\ during this gap, we relied on burst detection to qualitatively gauge the source activity. Typically, the rate of hard X-ray bursts from an active magnetar is maximal at the start of an outburst, declining thereafter. This behavior is observed during burst storms, e.g., 1E~2259+596 \\citep{gavriil04ApJ2259}, SGR 1935+2154 \\citep{Younes2020}, as well as over the course of weeks following the magnetar outburst onset, e.g., SGR 1830$-$0645 \\citep{younes22ApJ18301}, Swift~J1818.0$-$1607 \\citep{hu20ApJ}. Hence, if the outburst indeed began at the time of the radio non-detection, more bursts should be detected from the direction of \\src\\ by Fermi-GBM during the 20~day interval.\n\nWe searched the CTTE data for untriggered bursts in the energy range 8--100~keV following the procedure detailed in \\citet{Gavriil2004} and \\citet{Younes2020}. In summary, we binned the light curve of each GBM detector separately, at a temporal resolution $t_{\\rm r}$. We search each light curve in consecutive intervals of 100~s, overlapping by 10~s. We note the deviation of the number of counts $x$ in each bin from the long-term average rate in the light curve, assuming a Poisson probability density function $P(x)$. A potential burst is flagged if $P(x)<0.01\/N$, where $N$ is the number of bins per light curve (typically $100\/t_{\\rm r}$). The flagged bins are removed and the procedure is repeated until no more bins are found. We chose $t_{\\rm r}$ to be 8~ms, 32~ms, and 128~ms to sample the full range of magnetar burst durations. We confirm our GBM search results with a Bayesian blocks search technique as detailed in \\citet{lin20ApJ}. We only inspect potential bursts that were found in two or more detectors; single detector enhancements are typically due to particle background rather than an astrophysical event. Moreover, we rejected all bursts when \\src\\ was behind the earth for \\fermi, and those that were registered only in NaI detectors with an angle $>60^\\circ$ from the source since their effective area drops significantly at such large observing angles. The source identification process, however, is complicated by another magnetar, SGR~J1555$-$5402, which is 40\\hbox{$^\\prime$}\\ from \\src\\ and was also active at this time. This separation is not distinguishable by GBM, but is easily resolved in BAT images.\n\nWe find six candidate bursts that may originate from \\src\\ between 2022 March 16 and April 7, including one that corresponds to a GBM trigger on 2022 April 3 (trigger 220403928). \nBAT detected a count rate increase in two cases, in addition to the short-burst detected on 2022~April~7.\nOne of these was identified as the aforementioned SGR~J1555$-$5402, and the other coincidence has a negative value (consistent with zero) at the \\src\\ location and excludes it as the source of the rate increase. \nHence, the apparent lack of any bursting activity associated with \\src\\ over the course of three weeks following the last radio detection suggests that the high energy outburst genuinely commenced later. \nHowever, we cannot establish the precise date of the outburst, and tentatively place it sometime in early April. \nFuture continuous monitoring of X-ray bright radio-loud magnetars is crucial for revealing the dynamic between the radio and X-ray emission at the start of an outburst.\n\n\\section{Discussion} \\label{sec:disc}\n\nPrior to the 2009 outburst, \\src\\ was considered to be a so-called transient magnetar, similar to XTE~J1810$-$197 \\citep{Ibrahim2004}, with a historical quiescent (absorbed, 0.1--10\\,keV) flux of (0.3--2.1)$\\times 10^{-12}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$ \\citep{Gelfand2007}. However, the 2009 outburst was associated with a transition of \\src\\ into a persistent magnetar state where it exhibits a significantly higher steady-state X-ray flux than the pre-2006 quiescent state \\citep{CotiZelati2020}. On account of this increased persistent emission, the 2022 outburst appearing to be both less intense and shorter lived than the 2008 outburst is likely an artifact of the much higher baseline X-ray flux. Indeed the maximum 0.5--10\\,keV flux of $(6.0 \\pm 0.4) \\times 10^{-11}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$ is comparable to the observed peak flux of $\\sim 6 \\times 10^{-11}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$ during the 2008 outburst \\citep{Israel2010}. Had this previous outburst occurred more recently, the X-ray flux of \\src\\ would have fallen below the current baseline of $\\sim 1.7 \\times 10^{-11}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$ after a similarly short 10--20\\,day timescale. Were it not for our triggered \\nicer\\ observations, the enhanced X-ray emission from this outburst may have been missed entirely. A smaller outburst may not have even produced a detectable X-ray enhancement.\n\nMagnetar X-ray outbursts are generally attributed to crustal shifts due to stresses on the surface from internal B-field decay. These shifts could also either modify or enhance toroidal twists in the morphology of external magnetic field lines \\citep[see][and references therein]{turolla15RPPh}. The soft X-ray thermal emission arises due to either internal energy deposition in the outer crust \\citep{beloborodov16ApJ, akgun18MNRAS, lander2019MNRAS}, bombardment of the surface by accelerated particles in the twisted magnetospheric field loops \\citep{Beloborodov2009}, mechanical stresses slowly building in the crust (see \\citealt{Lander2015, Kojima2022} and references therein) or a combination of all three. In this model, a more complex pulse shape is expected during the outburst due to the emergence of new hot spots on the surface \\citep[e.g.,][]{Archibald2017a,younes22ApJL18301}, commensurate with the 3--6 keV pulse profile morphology displayed in Fig.~\\ref{Xprof}. As the outburst abates, the source surface heat map resorts back to its long-term, quasi-sinusoidal configuration.  If surface bombardment is active, the evolving surface temperature profile likely is accompanied by a migration of field line footprints associated with the twisting and untwisting of the magnetospheric field.\n\nThe hard X-ray tail above 10 keV is believed to arise from resonant inverse Compton scattering (RICS) of the soft surface thermal photons by the relativistic electrons accelerated in twisted loops \\citep{baring07ApSS,nobili08MNRAS,wadiasingh2018ApJ}. Figs.~\\ref{specFig} and~\\ref{Xprof} clearly display a lack of any variability in the hard X-ray emission concurrent with the elevated soft thermal emission during this outburst of \\src.  This disconnect between soft and hard X-ray bands is to be expected.  Detailed models of RICS emission generally favor a production of the signal at quasi-equatorial locales of modest altitudes, where scattering kinematics between the electrons and the soft X-ray photons from the surface readily accesses the cyclotron resonance in the cross section \\citep{wadiasingh2018ApJ}.  Thus changes in field line geometry near their footprints (e.g. due to a localized crustal stress) and moreover near the polar regions likely have only a small impact on the RICS emission, as do alterations of the surface hotspot size and temperature.  Evidence for some separation of magnetic co-latitudes for hard X-ray signals and the surface soft X-ray ones may be suggested by the modest 0.1-0.3 phase lag of the peak between the 3--6 keV band and the 10--25 keV one in Fig.~\\ref{Xprof}.  Thus changes in pulsed radio emission emanating from in or near the open field region, might naturally not be accompanied by modifications to the hard X-ray signal.\n\nGenerally, pulsed radio emission from magnetars is closely linked to the outburst mechanism: it switches on following the onset of an outburst before gradually fading into radio silence as the neutron star settles into X-ray quiescence (e.g., \\citealt{Camilo2016, Scholz2017}, but see \\citealt{zhu20ATel14084,younes22arXiv221011518Y}). Unlike other radio-loud magnetars, persistent radio pulses have been detected from \\src\\ since the quasi-stabilization of its X-ray flux to the current high state in $\\sim$2013 (\\citealt{CotiZelati2020}; Camilo et al., in prep.).\nAs shown in Figure~\\ref{fig:flux}, radio pulses were detected up until sometime between 2022 March 16 and 26, and were not seen again until April 19. \nAn initial disappearance and reactivation of the radio emission of \\src\\ was also reported following the 2009 outburst \\citep{Camilo2009ATel, Burgay2009ATel}.\n\nThe 2016 magnetar-like outburst of the high magnetic field strength pulsar PSR~J1119$-$6127 was also associated with a shutdown in the persistent radio emission, which was re-detected around two weeks after the initial X-ray outburst \\citep{Dai2018}. \nMoreover, simultaneous X-ray and radio observations of PSR~J1119$-$6127 revealed that magnetar-like short bursts were associated with a sudden disappearance of pulsed radio emission \\citep{Archibald2017b}.\nThis correlation was interpreted as a burst-induced quenching of the radio emission arising from electron-positron pairs leaking from the fireball trapped above the surface into the particle acceleration gaps, thereby increasing the plasma density within these regions and suppressing the acceleration of charges that precipitate radio emission.\nA similar phenomenon would provide a natural explanation for the disappearance of radio pulses from \\src\\ if it occurred after the onset of the high-energy outburst. \nYet, the dearth of \\fermi-GBM and \\swift-BAT burst candidates up to early April 2022 implies a delayed onset of the high energy outburst with respect to the disappearance of radio pulses.\nAn earlier onset necessitates an unlikely scenario in which all short-bursts emitted over an extended period of time were missed by all large FoV hard X-ray and gamma-ray monitors.\nAlternatively, a slow undetected rise in high-energy activity could explain the observed delay between the radio disappearance and delayed onset of short burst detections. \nSuch behavior was previously seen in the 2002 outburst of 1E~1048.1$-$5937 \\citep{Archibald2020}.\n\nSmall-scale twisting of the magnetosphere from crustal fracturing and motion of the magnetic footprints can lead to a decrease or complete failure of the charged particle currents responsible for producing the pulsed radio emission in addition to variations in the optical depth of the magnetosphere \\citep{Beloborodov2009, Timokhin2010, Hu2022}. \nRegardless of the exact underlying physics, any alteration of the magnetosphere must have been short lived, as we did not detect any substantial change in the average radio pulse profile shape or polarization properties of \\src\\ after re-activation.\nThis is markedly different to the behavior of other radio-loud magnetars, which typically display a broad range of profile variations and a marked increase in radio flux in the immediate aftermath of an outburst (e.g., \\citealt{Scholz2017, Dai2018, Dai2019}). \nFuture observations of similar high-energy outbursts both in \\src\\ and other highly magnetized pulsars will shed further light on the radio-quenching mechanism.\n\nOur joint radio and X-ray observations revealed that the 2022 outburst resulted in a significant spin-up event in the timing of \\src, which bears a strong resemblance to a pulsar glitch.\nThe gap in our timing coverage due to the disappearance of pulsed radio emission, low observing cadence of \\swift\\ and beginning of our triggered \\nicer\\ observations means the exact epoch of the event is unknown, but is constrained to between MJD 59661 to 59672 (March 23 -- April 3, 2022).\nGlitch-like timing anomalies and subsequent rapid changes in $\\dot{\\nu}$ are common feature of the post-outburst timing of magnetars (e.g., \\citealt{dib14ApJ, Archibald2020}), the latter of which are often ascribed to torque variations imparted by twists in the magnetosphere \\citep{Thompson2002}.\nUnlike rotation-powered pulsars, the majority of glitches in magnetars are associated with radiative outbursts that are believed to be caused by either starquakes \\citep{Thompson1995} or magnetic re-connection events \\citep{Lyutikov2002}.\nIn the starquake model, the gradual untwisting of the internal magnetic field induces an enormous strain on the rigid outer crust of the neutron star.\nEventually the tensile strength of the crust is overcome, leading to a sudden plastic deformation or even fracturing of a small region of the crust.\nThis triggers an outward exchange of angular momentum from the superfluid core to the crust via unpinning of vortices at the crust-core boundary \\citep{Ruderman1998}.\nModelling the timing anomaly in \\src\\ as a glitch, we obtained permanent step-changes in $\\nu$ and $\\dot{\\nu}$ of $0.2(1)$\\,$\\mu$Hz and $-2.4(1) \\times 10^{-12}$\\,s$^{-2}$ respectively.\nThe permanent change in spin-frequency is small when compared to most glitches in magnetars (see Figure 2 of \\citealt{Fuentes2017}).\nOn the other hand, the increase in spin-down rate is among the largest recorded in any pulsar \\citep{Lower2021b, Basu2022}.\n\nFollowing the glitch, we also detected a secular increase in the spin-down rate that continued up until at least our last \\nicer\\ observation.\nWe inferred a $\\ddot{\\nu}$ of $-1.95(5) \\times 10^{-19}$\\,s$^{-3}$ from a simple linear fit to the $\\dot{\\nu}$ timeseries and $-2.0(1) \\times 10^{-19}$\\,s$^{-3}$ from our glitch fit in Section~\\ref{timSec}.\nThis corresponds to a net increase in $\\dot{\\nu}$ of $-2.54 \\times10^{-12}$\\,s$^{-2}$ over the 147-days of post-outburst timing.\nA similar linear trend was observed following the 2008 outburst \\citep{Israel2010}, where the resulting $\\ddot{\\nu} = -4.8(9) \\times 10^{-18}$\\,s$^{-3}$ was an order of magnitude larger than we observed after the 2022 outburst. \nSecular increases in spin-down rate are a common feature of outbursts in magnetars and high B-field pulsars and are often attributed to the magnetosphere becoming increasingly twisted up to a maximum value after a few 10's to 100's of days \\citep{Thompson2002, Beloborodov2009}.\nAfter reaching the maximum twist, the field bundle then begins to unwind, causing a turnover in the spin-down evolution as the magnetar relaxes back toward the pre-outburst state.\nA global alteration of the twist can be ruled out in this 2022 event given the lack of hard X-ray evolution, indicating that either a small-scale local twist in bundles of magnetic field lines near the active pole or transient particle winds may be responsible for the secular increase in $\\dot{\\nu}$ \\citep{Harding1999}.\n\nSpin-down rates peak anywhere between $-10^{-13}$ -- $10^{-11}$\\,s$^{-2}$ depending on the strength of the outburst.\nGiven the lack of a turnover in the post-outburst $\\dot{\\nu}$ evolution of \\src, it is likely the maximum twist imparted by the 2022 outburst or peak in the outflowing particle flux has yet to be reached.\nThe recovery back to the initial spin-down state varies between magnetars and from one outburst to the next.\nOccasionally, the initial turnover is followed by an extended period of `noisy' spin-down evolution, including quasi-periodic variations such as those seen in 1E~1048.0$-$5937 \\citep{Archibald2015} and Swift~J1818.0$-$1607 \\citep{Rajwade2022}. \nThese variations are reminiscent of the spin-down state-switching process seen among rotation-powered pulsars \\citep{Lyne2010}, and often not associated with any changes in high-energy emission.\nSuch behavior remains to be seen in the current post-outburst timing of \\src.\n\n\\section{Conclusions}\n\nWe have presented the detection of a new outburst of \\src\\ that began in 2022 April. The outburst was marked by an increased soft X-ray emission and the disappearance of pulsed radio emission, both of which recovered back to their pre-outburst properties $\\sim$3-weeks later. \nThe disappearance of the radio pulses prior to the outburst onset is surprising, and could point to an undetected slow rise in high energy activity. \nSuch behavior could arise from small-scale crustal fracturing, where variations in the magnetospheric currents and optical depth resulted in a quenching of the radio emission mechanism.\nInterestingly, we find no change to the hard non-thermal tail concurrent to the soft X-ray and radio variability. Our joint \\parkes\/\\nicer\\ timing measurements revealed the presence of a glitch-like timing event in the rotation of the magnetar.\nA significant instantaneous increase in spin-down rate was also measured, which continued to increase at a rate of $-2.75 \\times 10^{-19}$\\,s$^{-3}$ over the first 147-days after the outburst.\nSmall-scale shifts in magnetospheric geometry in the lead up to the main quake may have been responsible for the quenching of the pulsed radio emission.\nThese results highlight the importance of conducting high-cadence monitoring of radio-loud magnetars such as \\src\\ at both radio and X-ray frequencies. Future outbursts with greater coverage around the initial trigger time may be able to resolve the radio quenching, delayed rise in spin-frequency, and initial increase X-ray flux.\nSuch observations would provide an even clearer picture of the magnetar outburst mechanism.\nGiven some similarities between this latest event and the 2008 outburst, it is possible that \\src\\ has entered a new state of increased activity.\nIn this case, the 2022 outburst may therefore be a precursor to additional outbursts over the coming years.\n\n\\begin{acknowledgments}\nWe thank George Hobbs and Rahul Sengar for generously giving us some of their Parkes observing time, and a special thanks to James Green and Phil Edwards for granting Parkes director discretionary time. \nWe also thank the anonymous referee for their helpful comments.\nThe Parkes radio telescope (\\textsl{Murriyang}) is part of the Australia Telescope National Facility (\\href{https:\/\/ror.org\/05qajvd42}{https:\/\/ror.org\/05qajvd42}) which is funded by the Australian Government for operation as a National Facility managed by CSIRO. \nWe acknowledge the Wiradjuri people as the traditional owners of the Observatory site. \nRadio data reduction and analysis were performed on the OzSTAR national HPC facility, which is funded by Swinburne University of Technology and the National Collaborative Research Infrastructure Strategy (NCRIS). \nA portion of this work was supported by NASA through the NICER mission and the Astrophysics Explorers Program. \nG.~Y. research is supported by an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center, administered by Oak Ridge Associated Universities under contract with NASA. \nL.~D. is supported by the Australian Research Council Centre of Excellence CE170100004 (OzGrav). M.~G.~B. acknowledges the generous support of the National Science Foundation through grant AST-1813649.  Z.~W. acknowledges support by NASA under award number 80GSFC21M0002. S.~G. acknowledges the support of the National Centre for Space Studies (CNES).\n\\end{acknowledgments}\n\n\\software{{\\tt dspsr} \\citep{vanStraten2011}, {\\tt HEAsoft} (v6.30.1; \\citealt{HEASARC2014}), {\\tt psrchive} \\citep{Hotan2004, vanStraten2012}, {\\tt NuSTARDAS} (v2.1.2), {\\tt DM\\_phase} \\citep{Seymour2019}, {\\tt PRESTO} \\citep{Ransom2011}, {\\tt RMNest} \\citep{Lower2022}, {\\tt tempo2} \\citep{Hobbs2006, Edwards2006}, {\\tt TempoNest} \\citep{Lentati2014}, {\\tt XSPEC} (v12.12.1; \\citealt{Arnaud1996})}\n\n\n\\section{Introduction} \\label{sec:intro}\n\nMagnetars are slowly rotating neutron stars with the strongest known magnetic fields in the Universe.\nOutbursts from these highly-magnetized objects are typically manifested by the sudden enhancement of their persistent X-ray emission by up to three orders of magnitude, which is often accentuated by spectral hardening and increased levels of complexity in their X-ray pulse profiles (see \\citealt{CotiZelati2018} for a review).\nThis increase in high-energy activity is sometimes accompanied by the emission of up to hundreds of short-duration, bright hard X-ray bursts.\nFive out of the nearly 30 known magnetars have also been found to display pulsed radio emission during outburst states \\citep{Olausen2014}\\footnote{\\url{https:\/\/www.physics.mcgill.ca\/~pulsar\/magnetar\/main.html}}. \nThese radio-loud episodes are typically preceded by several years of radio quiescence followed by the sudden activation of intense, pulsed radio emission. \nStrong temporal variability in the detected radio flux, pulse-profile shape and polarization fraction are often observed \\citep{Levin2012, Camilo2018, Dai2019, Lower2021a}. Moreover, rotational variations in the form of rapid increases in spin-down and sudden spin-up (glitches) or, less commonly, spin-down events (anti-glitches) are also detected following outbursts \\citep{Archibald2013, Scholz2017, Younes2020, Caleb2022, hu20ApJ}.  These observational properties are likely a result of magnetic stresses from the decaying internal magnetic fields precipitating sudden crustal motion or fracturing, leading to an impulsive release of energy in the form of electromagnetic radiation and occasionally an alteration of the external magnetic field topology \\citep{Duncan1992, Harding1999, Thompson2002, younes22ApJL18301}\n\nWith a spin period of $\\sim 2.1$\\,s, 1E~1547.0$-$5408 (also known as PSR~J1550$-$5418) is the second fastest rotating magnetar found to date.\nFirst discovered as a persistent X-ray source by the \\textit{Einstein Observatory} \\citep{Lamb1981}, it was not until several decades later that it was recognized as a potential magnetar candidate embedded within the putative supernova remnant G327.24$-$0.13, estimated to be $\\sim 4$\\,kpc away \\citep{Gelfand2007}.\nTargeted pulsar-search observations revealed the presence of highly-polarized, radio pulses originating from the source \\citep{Camilo2007}.\nSubsequent timing observations enabled measurements of both the aforementioned spin-period and a spin-down rate of $2.138 \\times 10^{-11}$\\,s\\,s$^{-1}$, thereby confirming its magnetar status (with an equatorial field of $B \\approx 2\\times 10^{14}$\\,G; \\citealt{Camilo2007}) and making it the second radio-loud magnetar to be discovered \\citep{Camilo2007, Camilo2008}.\nThis serendipitous discovery appeared to follow an unnoticed outburst sometime in the mid-2000s \\citep{Camilo2008} (possibly during 2007; see \\citealt{Halpern2008}), and was soon followed by two intense high-energy outbursts in 2008 and 2009 \\citep{Israel2010, Mereghetti2009, Scholz2011, Bernardini2011, vanderHorst12ApJ}.\nDetections of the radio pulses became highly intermittent following the outbursts, but the source stabilized to a persistent radio-loud state after several years (Camilo et al. in prep.).\n\nSince its major outburst of 2009 January, \\src\\ has remained in an unusual high X-ray flux state, bearing a much stronger resemblance to a persistent magnetar as opposed to its initial `transient' classification (\\citealt{CotiZelati2020}; Camilo et al. in prep.).\nNo additional X-ray outbursts above the baseline were observed during this time, barring the detection of two short bursts seen with the the Neil Gehrels \\swift\\ Observatory (thereon Swift) Burst Alert Telescope (BAT) in 2011 and 2013, respectively \\citep{2013GCN.14911....1H}.\nNeither of these two short bursts were followed up by targeted high-cadence monitoring.\n\nIn this work, we present the detection of a new outburst from \\src\\ that began between late-March and early-April 2022.\nAn overview of our radio and high-energy observations are detailed in the following section.\nIn Section~\\ref{sec:analysis} we present our analysis of the timing and radiative properties of \\src\\ during the outburst.\nWe conclude in Section~\\ref{sec:disc} with a discussion on the interplay between changes in the high-energy, radio, and timing properties of \\src, and potential implications for the outburst mechanism.\n\n\\section{Observations} \\label{sec:obs}\n\n\\subsection{\\parkes\\ Observatory}\n\nRegular monitoring of \\src\\ with an approximately 10-day cadence has been undertaken with \\textsl{Murriyang}, the 64-m \\parkes\\ radio telescope, under the P885 project since its discovery as a radio-loud magnetar in 2007 \\citep{Camilo2007}.\nOur observations from 2019 January have been performed using the Ultra-Wideband Low (UWL) receiver system \\citep{Hobbs2020}, where two independent pulsar search-mode data streams are recorded with the pulsar digital filterbank v4 (PDFB4) and GPU-based {\\sc medusa} signal processors.\nIndividual observations lasted between 2--15\\,minutes. \nFor this work, we analyzed the subset of the PDFB4 data spanning from 2021 June to 2022 August and {\\sc medusa} data covering 2021 December to 2022 August.\nData collected by PDFB4 were taken with 1\\,ms time-sampling at a center frequency of 3100\\,MHz with 512 channels covering 1024\\,MHz of bandwidth.\nFull Stokes data collected with {\\sc medusa} were recorded with 128\\,$\\mu$s time resolution with 3328 channels covering the full UWL band from 704--4032\\,MHz, and coherently dedispersed at a dispersion measure (DM) of 830\\,pc\\,cm$^{-3}$.\nBoth data sets were subsequently folded at the rotation period of the magnetar using {\\tt dspsr} \\citep{vanStraten2011}, such that a single pulse period is covered by 1024 phase bins. \nFrequency channels affected by radio-frequency interference (RFI) were excised using the {\\tt paz} and {\\tt pazi} tools in {\\tt psrchive} \\citep{Hotan2004, vanStraten2012}.\nThe folded {\\sc medusa} data were then flux and polarization calibrated following the procedures outlined in \\citet{Lower2020}. \n\n\\subsection{\\nicer}\n\nThe Neutron star Interior Composition ExploreR (\\nicer) is a non-imaging, soft X-ray telescope mounted on the International Space Station, with a $\\approx5\\hbox{$^\\prime$}$ diameter field-of-view (FoV; \\citealt{gendreau16SPIE}). \n\\nicer\\ provides a large collective area peaking at $1900$~cm$^2$ at 1.5~keV and a timing resolution $<300$\\,ns. We initiated a series of \\nicer\\ target of opportunity observations of \\src\\ two days after the 2022 April 7 \\swift-BAT detection of a magnetar-like burst coincident with \\src. The \\nicer\\ monitoring consisted of about 1~ks observations every few days, starting on 2022 April 9. We provide the observations log in Table~\\ref{xObsLog}.\n\n\\begin{table}[t]\n\\centering \n\\caption{List of X-ray observations analyzed in this work}\n\\label{xObsLog}\n\\newcommand\\T{\\rule{0pt}{2.6ex}}\n\\newcommand\\B{\\rule[-1.2ex]{0pt}{0pt}}\n\\begin{center}{\n\\resizebox{0.47\\textwidth}{!}{\n\\begin{tabular}{l c c c}\n\\hline\n\\hline\nTelescope \\T\\B & Observation ID & Date & Total GTI \\\\\n\n          \\T\\B &  & (UTC) & (ks) \\\\\n\\nustar\\ & 30101035002 & 2016-04-23 & 84 \\\\\n\\nustar\\ & 30401008002 & 2019-02-15 & 90 \\\\\n\\swift\/XRT & 00013176022 & 2022-03-05 & 2.6 \\\\\n\\swift\/XRT & 00013176023 & 2022-03-06 & 1.8 \\\\\n\\nicer\\ & 5020300101 & 2022-04-09 & 1.9 \\\\\n\\nicer\\ & 5020300102 & 2022-04-12 & 0.1 \\\\\n\\nicer\\ & 5020300103 & 2022-04-13 & 0.2 \\\\\n\\nicer\\ & 5020300104 & 2022-04-14 & 0.3 \\\\\n\\nicer\\ & 5020300106 & 2022-04-16 & 1.2 \\\\\n\\nicer\\ & 5020300107 & 2022-04-17 & 0.6 \\\\\n\\nicer\\ & 5020300108 & 2022-04-18 & 0.4 \\\\\n\\nicer\\ & 5020300109 & 2022-04-21 & 1.5 \\\\\n\\nustar\\ & 90801309002 & 2022-04-21 & 24 \\\\\n\\nicer\\ & 5020300110 & 2022-04-22 & 1.8 \\\\\n\\nicer\\ & 5020300111 & 2022-04-25 & 0.4 \\\\\n\\swift\/XRT & 00013176025 & 2022-05-07 & 3.0 \\\\\n\\nicer\\ & 5020300113 & 2022-05-08 & 1.6 \\\\\n\\swift\/XRT & 00013176026 & 2022-05-11 & 1.5 \\\\\n\\nicer\\ & 5020300114 & 2022-05-13 & 0.9 \\\\\n\\nicer\\ & 5020300115 & 2022-05-15 & 0.2 \\\\\n\\nicer\\ & 5020300116 & 2022-05-21 & 1.0 \\\\\n\\nicer\\ & 5020300117 & 2022-05-22 & 0.5 \\\\\n\\nicer\\ & 5020300118 & 2022-05-28 & 0.7 \\\\\n\\nicer\\ & 5020300119 & 2022-06-07 & 1.5 \\\\\n\\nicer\\ & 5020300120 & 2022-06-17 & 1.2 \\\\\n\\nicer\\ & 5020300121 & 2022-06-27 & 0.4 \\\\\n\\nicer\\ & 5020300123 & 2022-07-07 & 0.9 \\\\\n\\swift\/XRT & 00013176027 & 2022-07-09 & 4.3 \\\\\n\\nicer\\ & 5020300124 & 2022-07-13 & 0.4 \\\\\n\\nicer\\ & 5020300125 & 2022-07-17 & 2.1 \\\\\n\\nicer\\ & 5020300126 & 2022-07-20 & 1.1 \\\\\n\\nicer\\ & 5020300127 & 2022-07-21 & 0.7 \\\\\n\\nicer\\ & 5020300128 & 2022-07-28 & 1.3 \\\\\n\\nicer\\ & 5020300129 & 2022-07-30 & 0.4 \\\\\n\\nicer\\ & 5020300130 & 2022-08-02 & 0.1 \\\\\n\\nicer\\ & 5020300131 & 2022-08-03 & 0.2 \\\\\n\\nicer\\ & 5020300132 & 2022-08-05 & 0.2 \\\\\n\\nicer\\ & 5020300133 & 2022-08-09 & 0.4 \\\\\n\\nicer\\ & 5020300134 & 2022-08-15 & 1.8 \\\\\n\\nicer\\ & 5020300135 & 2022-08-17 & 2.3 \\\\\n\\hline\n\\hline\n\\end{tabular}}}\n\\end{center}\n\\end{table}\n\n\nWe cleaned and calibrated all \\nicer\\ data utilizing {\\tt NICERDAS} version v008c as part of {\\tt HEASoft} version 6.30.1. We applied the standard filtering criteria as described in the \\nicer\\ Data Analysis Guide\\footnote{\\href{https:\/\/heasarc.gsfc.nasa.gov\/docs\/nicer\/data\\_analysis\/nicer\\_analysis\\_guide.html}{NICER data analysis guide.}}. Moreover, detectors 14 and 34 can show larger than normal background noise, hence, were excluded from all analyses. Finally, we compared the light curve (binned with a 5~second resolution) created in the energy range 1--6~keV (where the source is supposed to dominate the emission) to one created in the 13--15~keV energy range. Almost all the counts in the latter band are due to high energy particle background. We eliminated any simultaneous flaring background intervals that appeared in both light curves. These were mostly present at the start and\/or end of a good time interval (GTI) due to imprecise modelling of the South Atlantic Anomaly. We provide the total GTIs of each \\nicer\\ observation in Table~\\ref{xObsLog}. We utilise {\\tt Xselect} as part of {\\tt HEASoft} version 6.30.1 to extract the source spectra, and estimated the background contribution for each observation utilizing the 3C50 model \\citep{remillard2021:3c50}.\n\n\\subsection{\\nustar}\n\n\\nustar\\ (Nuclear Spectroscopic Telescope ARray; \\citealt{harrison13ApJ:NuSTAR}) is a focusing hard (3--79~keV) X-ray telescope, consisting of two identical focal plane modules, FPMA and FPMB. Each module is formed by an array of $2\\times2$ detectors providing a total of about $12\\hbox{$^\\prime$}\\times12\\hbox{$^\\prime$}$ FoV. Prompted by the increase in the soft X-ray flux, we requested a \\nustar\\ director discretionary time and \\src\\ was observed for 24~ks on 2022 April 21. We utilized {\\tt NuSTARDAS} software version 2.1.2 to clean and calibrate the event files. We employed the task {\\tt nuproducts} to extract source events, light curves, and spectra from a circular region with a 50\\hbox{$^{\\prime\\prime}$}-radius around the source central brightest pixel, which corresponds to 70\\% of the encircled-count fraction of the detectors point spread function. This choice of extraction region is driven by the desire to maximize source counts, while limiting stray-light background, which is present in both \\nustar\\ modules in proximity to the source. Similarly, we extracted the background files from a source-free region on the same detector as the source with the same radius. For comparison purposes, we analysed two archival \\nustar\\ observations taken on 2016 April 23 and 2019 February 15, following the same procedures outlined above. Table~\\ref{xObsLog} summarises the \\nustar\\ observations.\n\n\\subsection{\\swift}\n\nThe X-ray Telescope (XRT) onboard \\swift\\ is an imaging charge-coupled device, sensitive to photons in the energy range of 0.2--10~keV \\citep{burrows05SSRv}. XRT observed \\src\\ on several occasions in 2022 March, May and July, bracketing the time of the outburst. Thus, these observations provide the source soft X-ray spectral state shortly before the outburst onset, and a confirmation of its properties as observed with \\nicer\\ following the outburst. For these observations, XRT was operating in windowed-timing mode, affording a 1D image with a time resolution of 1.7~ms. We utilized {\\tt XRTDAS} version 3.7.0 to reduce and calibrate the data. We extracted source events from each good time interval (GTI) of a given observation separately. We used a circular region with a 20 pixel radius centered on the brightest pixel. We extracted background events from an annulus with inner and outer radii of 80 and 120 pixels centered at the same position as the source. We created the ancillary files using \\texttt{xrtmkarf}, and used the response matrices in CALDB version 20220803. We excluded all GTIs where the source landed within a 3 pixel distance from a bad column or the edge of the CCD due to increase in systematic uncertainties\\footnote{\\url{https:\/\/www.swift.ac.uk\/user\\_objects\/lc\\_docs.php}}. For the remaining good GTIs, we add the source and background spectra, as well as the ancillary and response files using the HEASOFT tool \\texttt{addspec}. We provide the XRT observation IDs, date, and total GTI in Table~\\ref{xObsLog}.\n\nThe Burst Alert Telescope (BAT), also onboard \\swift, is a coded aperture telescope which images a $\\sim 1.5$~sr FoV in the energy range of 15--150~keV. Images are generated by the on-board software in response to count rate increases at timescales from 4 ms to 24 s or, in the absence of rate increases, on minute and longer timescales. \nBursts from sources outside of the FOV can often be detected due to the leakage of photons through the instrument shielding, yielding a count rate increase without a corresponding image peak.  These can be used to confirm the astrophysical nature of count rate increases seen by other instruments and constrain their direction of origin.\n\n\\subsection{\\fermi-GBM}\n\nThe Gamma-ray Burst monitor \\citep[GBM;][]{meegan09ApJ:gbm} is a non-imaging instrument onboard the \\fermi\\ Gamma-ray Space Telescope that is sensitive to photons in the 8--1000\\,keV range. GBM is capable of automatic on-board triggers, with one of the software algorithms tailored to magnetar-like short bursts. Apart from the triggered data, GBM also provides continuous time-tagged events (CTTE), with a time resolution of 2 microseconds. We utilized the CTTE data to search for any untriggered bursts from the direction of \\src\\ from its last radio detection on 2022 March 16 until the confirmed BAT burst on 2022 April 7. This analysis provides an estimate of the start time of the 2022 outburst.\n\n\\section{Analysis and results} \\label{sec:analysis}\n\n\\subsection{Radio disappearance and reactivation}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{Postage_stamps.pdf}\n    \\caption{Examples of the three polarization profile states (A, B and C) of \\src\\ seen at different epochs. Note, the profiles from 2022-03-16 (top-left) and 2022-04-19 (middle-right) correspond to the last pre-outburst and first post-outburst radio detections respectively. Each observation was dedispersed using a dispersion measure of $697$\\,pc\\,cm$^{-3}$ and corrected for a Faraday rotation measure of $-1847$\\,rad\\,m$^{-2}$ prior to averaging in time and frequency.}\n    \\label{fig:postage_stamps}\n\\end{figure*}\n\nUnlike other radio-loud magnetars which fade into radio silence in the months to years after an outburst, \\src\\ has remained radio-loud for well over a decade since the 2009 outburst (Camilo et al., in prep).\nFollowing the 2008 and 2009 outbursts, detections of radio pulses were intermittent \\citep[see e.g.][]{Israel2021}, possibly reflecting the highly dynamic nature of its magnetosphere during that period.\nIntermittent detections continued up to 2013, after which radio pulses were consistently detected during every observation.\n\\src\\ remained in this persistently radio-loud state until our two observations on 2022 March 16 and April 9, where we failed to detect radio pulses from the magnetar.\nRadio pulses were re-detected during the 2022 April 19 observation, $\\sim$12 days after the BAT confirmed trigger, and have persisted up to at least our Parkes-UWL observation on 2022 August 3.\n\nUsing a high $S\/N$ observation taken on 2022 July 9, we measured a structure-optimized dispersion measure (DM) of $697 \\pm 1$\\,pc\\,cm$^{-3}$ with the {\\tt DM\\_phase} package \\citep{Seymour2019}.\nThe updated DM is a significant improvement on the previously reported value of $830 \\pm 50$\\,pc\\,cm$^{-3}$ \\citep{Camilo2007}.\nThis is likely due to differences in the methods applied to compute the DM (profile structure versus S\/N optimisation) in addition to the substantially larger instantaneous bandwidth afforded by the UWL.\nUsing the NE2001 \\citep{Cordes2002} and YMW16 \\citep{Yao2017} electron density models, this updated DM corresponds to estimated distances of 8.3\\,kpc and 5.9\\,kpc, respectively. The latter distance is comparable to the commonly assumed sdistance of $\\sim 4.5$\\,kpc inferred from dust-scattering halos detected after the 2009 outburst \\citep{Tiengo2010}, when considering the large systematic uncertainties that are associated with any of these methods, which is on average 50\\%. Similarly, we obtained an updated rotation measure (RM) of $-1847.6 \\pm 0.5$\\,rad\\,m$^{-2}$ by directly fitting the Stokes $Q$ and $U$ spectra in the 2000--4032 MHz range (to avoid complications from scatter broadening) with {\\tt RMNest} \\citep{Lower2022}.\nOur updated RM is consistent with the previously reported value of $-1860 \\pm 20$\\,rad\\,m$^{-2}$ \\citep{Camilo2008}.\n\nThe shape of the average radio profile detected throughout 2022 January-August was highly variable, and can be broadly categorized into three general archetypes:\n\\begin{itemize}\n    \\item Profile A: Strong leading component with one to three trailing components.\n    \\item Profile B: Complex, multi-component profile, sometimes with a central peak.\n    \\item Profile C: Two dominant profiles joined by weaker sub-components or a flat bridge of emission.\n\\end{itemize}\nExamples of all three profile types are shown in Figure~\\ref{fig:postage_stamps}, all of which were dedispersed and de-Faraday rotated using our updated DM and RM values.\nNote, these categorizations are purely qualitative.\nFor instance, the complex 2022 June 13 profile (bottom-middle panel in Figure~\\ref{fig:postage_stamps}) is largely double peaked, and could also be classified as Profile C.\nTransitions between profile types tend to be sudden and reminiscent of transient profile changes reported in earlier observations of this magnetar (e.g., Fig. 5 of \\citealt{Halpern2008}) and those seen in emission mode-changing\/state-switching pulsars \\citep{Wang2007}.\nThe radio profiles of \\src\\ in Figure~\\ref{fig:postage_stamps} display clear epoch-to-epoch variations in flux density.\nWe computed the frequency-averaged radio flux density as\n\\begin{equation}\n    {\\rm F}_{f} = \\frac{1}{N_{\\rm bin}} \\sum_{i}^{N_{\\rm on}} {\\rm F}_{f,i}\\,,\n\\end{equation}\nwhere $N_{\\rm bin} = 1024$ is the number of phase-bins covering a full rotation of the magnetar, $N_{\\rm on}$ is the number of bins covering the on-pulse region and ${\\rm F}_{f,i}$ is the flux of the $i$-th phase bin.\nUncertainties on the flux density were computed from the root mean square (RMS) of the off-pulse region\n\\begin{equation}\\label{eqn:flux}\n    \\sigma_{{\\rm F},f} = \\frac{\\sqrt{N_{\\rm on}}}{N_{\\rm bin}} \\sqrt{\\sum_{i}^{N_{\\rm off}} {\\rm F}_{f,i}^{2}}\\,,\n\\end{equation}\nwhere $N_{\\rm off}$ is the number of off-pulse phase bins.\nThe measured flux densities from our observations are displayed in the bottom panel of Figure~\\ref{fig:flux}.\nWe used Equation~\\ref{eqn:flux} to set sensitivity limits for our observations on 2022 March 26 and April 9 where we failed to detect radio pulses, obtaining limiting flux densities of $1.3 \\pm 0.3$\\,mJy and $0.9 \\pm 0.2$\\,mJy respectively.\nVisually, there appears to be a shallow increase in the revived radio flux over time.\nHowever, a direct connection between this apparent secular evolution and the outburst cannot be confirmed without a comparison to the long-term flux history of the magnetar.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Flux.pdf}\n    \\caption{Average unabsorbed X-ray (0.5--10\\,keV; top) and radio (0.7--4\\,GHz; bottom) flux of \\src\\ from 2022 January to August. The vertical dashed lines indicate the dates of the \\fermi-GBM (MJD 59672; 2022 April 3) and \\swift-BAT short bursts (MJD 59676; 2022 April 7) respectively, while the blue shaded region corresponds to where pulsed radio emission was undetected. Grey horizontal shading corresponds to the standard deviation about the baseline X-ray flux of $1.7 \\times 10^{-11}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$.}\n    \\label{fig:flux}\n\\end{figure}\n\nAlso indicated in Figure~\\ref{fig:flux} are the corresponding profile types seen during each observation.\nIn terms of relative numbers, we observed the Profile B state in more than half (25\/49) of our observations, while Profile A and C were seen in 17 and 7 observations respectively.\nThe appearance of any one mode appears to be largely stochastic, though there is a clear preference for Profile B after the outburst up until 2022 May 13 (MJD 59712)  where the magnetar was in the Profile A state.\nNo further systematic change in the profile or the mode-changing behavior is apparent after this date.\nIt is possible that this preference may be related to changes in the emission region plasma following the reactivation. \nHowever such profile variations persist among radio-loud magnetars for many months to years after an outburst \\citep{Levin2012, Lower2021a}.\nAdditionally, the swing of the linear polarization position angle (PA) across the profile remains largely flat and unchanged both before and after the outburst.\nIt is consistent with previous observations of \\src\\ in 2007, where rotating vector model fits indicated that the magnetar is a near-aligned rotator \\citep{Camilo2008}.\nOccasional departures from the smooth PA swing are detected in individual epochs, an example of which is shown in the top-right panel of Figure~\\ref{fig:postage_stamps}.\nThe relatively smooth evolution of the PA swing persists at higher observing frequencies where the effects of interstellar scattering are negligible, indicating these features are due to either an unusually smooth transition between two orthogonal polarization modes or the result of birefringent propagation effects in the near-field environment \\citep{Dyks2020}.\nBirefringent propagation effects would also explain the unusually high circular polarization fraction (up to $|V|\/I = 0.54$) as well as changes in handedness between observations.\nA more detailed investigation is left for future work.\n\n\\subsection{Timing}\\label{timSec}\n\nThe variable radio profile of \\src\\ is not conducive to pulsar timing methods that use a single reference template.\nInstead, we obtained pulse times of arrival (ToAs) from the \\parkes\/PDFB4 data by using the {\\tt get\\_toas} tool in {\\tt PRESTO} \\citep{Ransom2011} to fit a single Gaussian template to individual observations via the Fast Fourier Transform based approach of \\citet{Taylor1992}.\nWe added an extra 0.05\\,s uncertainty in quadrature with the formal ToA uncertainties to account for excess epoch-to-epoch scatter in the ToAs that arise from the varying radio profile (often referred to as `jitter').\n\nFor \\nicer, we merged all barycenter-corrected event files and only accepted counts in the 1--5~keV energy range, which maximized the source pulse signal according to the $Z^2$ power \\citep{buccheri83AA}. We then split the data into time intervals that contain $\\approx1000$~counts each, without allowing individual intervals to exceed a 3-day integration. We employed a non-binned maximum likelihood estimation (MLE) method to measure the X-ray pulse ToAs \\citep{livingstone09ApJ,ray11ApJS}. Firstly, we model a high S\/N pulse profile with a Fourier series including only the fundamental harmonic; adding higher order harmonics did not improve the quality of the fit. Secondly, we fit, using MLE, each unbinned data segment to the same model, only allowing for a phase-shift $\\Delta\\phi$. The best-fit phase shift is the value that corresponds to the minimum of the negative log likelihood, $-\\log(\\mathcal{L})_{\\rm min}$. The $1\\sigma$ uncertainty on the phase shift was established by stepping over the full parameter space $\\Delta\\phi \\in [0,2\\pi)$ and noting a change in $-\\log(\\mathcal{L})_{\\rm min}$ by $0.5$ (corresponding to the $\\pm34\\%$ or $1\\sigma$ confidence interval of a $\\chi^2$ distribution with 1~degree of freedom).\n\nWe obtained an initial phase-connected solution by using the Hidden Markov Model (HMM) based glitch detector of \\citet{Melatos2020}\\footnote{\\href{https:\/\/github.com\/ldunn\/glitch_hmm}{https:\/\/github.com\/ldunn\/glitch\\_hmm}} which tracks the spin-frequency ($\\nu$) and spin-down rate ($\\dot{\\nu}$) of the magnetar over time. \nThis enabled robust integer pulse numbering to be applied to each ToA and provided estimates of any changes in $\\nu$ and $\\dot{\\nu}$ at the time of the outburst.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Timing.pdf}\n    \\caption{Top panel shows the pre-fit \\parkes\/\\nicer\\ timing residuals of \\src\\ from 2021 June to 2022 August. Changes in the magnetar spin-frequency and spin-down rate with respect to their nominal values ($0.478836$\\,Hz and $-3.63 \\times 10^{-12}$\\,s$^{-2}$ at MJD 58350) are shown in the middle and bottom panels respectively, where the points correspond to local fits to the ToAs. Purple shading is the $\\nu$ and $\\dot{\\nu}$ posterior distributions that were recovered by the HMM glitch detector at each epoch. Outliers in the local fits to $\\dot{\\nu}$ arise from the limited number of ToAs. Vertical shaded region and dashed lines are the same as in Figure~\\ref{fig:flux}.}\n    \\label{fig:f0f1_fits}\n\\end{figure}\n\nOnce the phase-connected solution was obtained, we then used the {\\tt glitch} plugin to {\\tt tempo2} \\citep{Hobbs2006, Edwards2006} to perform independent local fits to $\\nu$ and $\\dot{\\nu}$.\nThe results of our local fits are shown alongside the HMM tracking in Figure~\\ref{fig:f0f1_fits}.\nBoth sets of spin-frequency\/spin-down rate measurements generally agree with one another and display approximately the same secular evolution.\nA timing anomaly with clear jumps in both $\\nu$ and $\\dot{\\nu}$ occurred in coincidence with the high-energy outburst, with the HMM recovering step-changes of $\\Delta\\nu = (9 \\pm 4) \\times 10^{-7}$\\,Hz and $\\Delta\\dot{\\nu} = (-1.6 \\pm 0.5) \\times 10^{-12}$\\,s$^{-2}$. \nThere is also a clear, secular increase in spin-down rate of the magnetar ($\\dot{\\nu}$ is becoming more negative) over time, where $\\dot{\\nu}$ appears to follow an approximately linear trend over the first 120\\,days post-outburst.\nThis is also reflected by the downward curve in the evolution of $\\nu$ over time.\nIncreasing spin-down rates are also commonly observed in the post-outburst behavior of other magnetars, including recent outbursts in the radio-loud magnetars PSR~J1622$-$4950 \\citep{Camilo2018} and XTE~J1810$-$197 \\citep{Caleb2022}.\nA linear fit to the spin-down evolution gives an apparent $\\ddot{\\nu} = -1.95(5) \\times 10^{-19}$\\,s$^{-3}$.\nSimilar behavior was previously observed in \\src\\ following the 2008 outburst \\citep{Israel2010}.\n\nIn order to better constrain both the longer-term rotational properties of \\src\\ and the outburst-induced changes in its timing, we modelled the anomaly as a pulsar glitch where the corresponding change in rotation phase is given by\n\\begin{eqnarray}\n     \\phi_{\\rm {g}}(t) = \\Delta\\phi_{g} + \\Delta\\nu_{p}(t - t_{\\rm {g}}) + \\frac{1}{2}\\Delta\\dot{\\nu}_{p}(t - t_{\\rm {g}})^{2} \\nonumber \\\\\n     + \\frac{1}{6}\\Delta\\ddot{\\nu}_{p}(t - t_{\\rm {g}})^{3} - \\Delta\\nu_{d}\\tau_{d} e^{-(t - t_{\\rm {g}})\/\\tau_{d}}.\n\\end{eqnarray}\nHere, $t_{g}$ is the glitch epoch, $\\Delta\\phi_{g}$ is a phase-jump to account for residual ambiguities in the rotation phase of the magnetar after the event, $\\Delta\\nu_{p}$ is the permanent step-change in spin-frequency, $\\Delta\\dot{\\nu}_{p}$ the step-change in spin-down rate, and $\\Delta\\nu_{d}$ is the exponentially recovered change in spin-frequency that occurred over a characteristic timescale, $\\tau_{d}$.\nWe also included a glitch-induced second spin-frequency derivative ($\\Delta\\ddot{\\nu}_{p}$) term to account for the post-outburst linear increase in $\\dot{\\nu}$ that we discussed above.\n\nThe gap in our timing between the last \\parkes\\ detection on 2022 March 16 and the first \\nicer\\ observation on April 9 makes it difficult to determine the precise glitch epoch. Since magnetar glitches are often associated with radiative events \\citep{dib14ApJ}, we use the date of the short-burst detected by \\fermi-GBM from the direction of \\src\\ as the nominal glitch epoch (i.e $t_{g} = $\\,MJD 59672; 2022 April 3; see Sec.~\\ref{sec:burSear}). \nNote, the exact choice of glitch epoch between April 1 and April 7 has a strong impact on the recovered $\\Delta\\nu_{p}$, which we discuss further below. \nUsing {\\tt tempo2} and the {\\tt FITWAVES} plugin, we fit the glitch parameters alongside a set of 10 harmonically related sine-waves to account for strong timing noise in the residuals. \nThis returned values of $\\Delta\\nu_{p} = 2.0(1) \\times 10^{-6}$\\,Hz, $\\Delta\\dot{\\nu}_{p} = -1.23(2) \\times 10^{-14}$\\,s$^{-2}$ and $\\Delta\\ddot{\\nu}_{p} = -1.86(3) \\times 10^{-19}$\\,s$^{-3}$.\nA significant $\\Delta\\nu_{d}$ of $-2.1(1) \\times 10^{-6}$\\,Hz was recovered. The negative value of $\\Delta\\nu_{d}$ signifies a resolved rise in the spin-frequency over a period of $\\tau_{d} = 13(1)$\\,days as opposed to a post-glitch decay.\n\nTo verify the existence of this apparent delayed rise in $\\nu$, we sampled the posterior distributions for each parameter using the {\\tt TempoNest} Bayesian inference plugin to {\\tt tempo2} \\citep{Lentati2014}.\nThis also allowed us to simultaneously model the timing noise of the magnetar as a power-law red noise process. The power-spectral density of the red noise model is given by\n\\begin{equation}\n    P(f) = \\frac{A_{\\rm r}^{2}}{12\\pi^{2}} \\Big( \\frac{f}{1\\,{\\rm yr}} \\Big)^{-\\beta_{\\rm r}}\\,{\\rm yr}^{-3},\n\\end{equation}\nwhere $A_{\\rm r}$ is the red noise amplitude (in units of yr$^{3\/2}$) and $\\beta_{\\rm r}$ the spectral index.\nTwo separate fits were performed: one that included a resolved rise in spin frequency, and another where $\\Delta\\nu_{d}$ is fixed to zero for an instantaneous (unresolved) glitch.\nComparing the recovered log-evidence from the resulting fits, we obtain a Bayes' factor of $\\ln\\mathcal{B} = -2.8$ in favor of a resolved rise.\nThe negative value indicates there is no significant evidence for the presence of a resolved versus an instantaneous change in spin-frequency.\nOur results for the un-resolved glitch model are summarized in Table~\\ref{tab:timing}.\n\n\\begin{table}[t!]\n\\centering \n\\caption{Timing and pulsar parameters for \\src\\ from our joint \\parkes\/\\nicer\\ timing. Values in parentheses are the 1-$\\sigma$ uncertainties on the last digit.}\n\\label{tab:timing}\n\\begin{tabular}{ll}\n\\hline\n\\hline\nPulsar parameters &  \\\\\n\\hline\nR. A. (J2000)$^{\\dagger}$                    & 15$^{\\mathrm{h}}$50$^{\\mathrm{m}}$54.12386$^{\\mathrm{s}}$ \\\\\nDecl. (J2000)$^{\\dagger}$                    & $-$54$^{\\circ}$18$'$24.1141$''$ \\\\\n$\\nu$ (Hz)                                   & $0.478836(3)$ \\\\\n$\\dot{\\nu}$ (s$^{-2}$)                       & $-3.63(3) \\times 10^{-12}$ \\\\\n$\\mu_{\\alpha} \\cos\\delta$ (mas)$^{\\dagger}$  & $4.8$ \\\\\n$\\mu_{\\delta}$ (mas)$^{\\dagger}$             & $-7.9$ \\\\\nTiming epoch (MJD TDB)                       & 58350 \\\\\nPosition epoch (MJD TDB)                     & 54795 \\\\\nDM (pc\\,cm$^{-3}$)                           & $697(1)$ \\\\\nRM (rad\\,m$^{-2}$)                           & $-1847.6(5)$ \\\\\nTime span (MJD)                              & 59368--59808 \\\\\nNumber of ToAs                               & 202 \\\\\nSolar system ephemeris                       & DE436 \\\\\n$\\log_{10}(A_{\\rm r})$ (yr$^{3\/2}$)          & $-5.1(2)$ \\\\\n$\\beta_{\\rm r}$                              & $5.7(8)$ \\\\\n\\hline\nGlitch parameters &  \\\\\n\\hline\n$t_{\\rm g}$ (MJD)                   & 59672 \\\\\n$\\Delta\\nu_{p}$ (Hz)                & $2(1) \\times 10^{-7}$ \\\\\n$\\Delta\\dot{\\nu}_{p}$ (s$^{-2}$)    & $-2.4(1) \\times 10^{-12}$ \\\\\n$\\Delta\\ddot{\\nu}_{p}$ (s$^{-3}$)   & $-2.0(1) \\times 10^{-19}$ \\\\\n\\hline\\\\\n\\end{tabular}\n{$^{\\dagger}$}Fixed to values from~\\cite{Deller2012}.\n\\end{table}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{Timing_posteriors.pdf}\n\\caption{One- and two-dimensional posterior distributions for the glitch and red noise in \\src. Darker and lighter shaded regions represent the 68\\% and 95\\% confidence intervals respectively.}\n\\label{fig:glitch_post}\n\\end{center}\n\\end{figure}\n\nWe present the one- and two-dimensional posterior distributions for the glitch and timing noise parameters in Figure~\\ref{fig:glitch_post}. There is a strong correlation between $\\Delta\\dot{\\nu}_{p}$ and $\\Delta\\ddot{\\nu}_{p}$, indicating these two parameters induce a similar signal across the relatively short post-glitch timing of \\src. While $\\Delta\\dot{\\nu}_{p}$ agrees with the value recovered by the HMM at the 68\\% confidence interval, the {\\tt TempoNest} derived value of $\\Delta\\nu_{p}$ is smaller by a factor of $\\sim$4.5. As for the post-outburst rotational-evolution, our recovered value of $\\Delta\\ddot{\\nu}_{p}$ is consistent with that inferred from a simple linear fit to $\\dot{\\nu}$ over time.\n\nThe difference in $\\Delta\\nu_{p}$ values is a direct result of our choice of glitch epoch and the large increase in spin-down rate. This is apparent in {\\tt TempoNest} runs where we assumed earlier and later values of $t_{\\rm g}$, where earlier glitch epochs result in a larger recovered value of $\\Delta\\nu_{p}$ to match the post-outburst timing. Going the other way, setting the glitch epoch to the date of the \\swift-BAT short burst (MJD 59676; 2022 April 7) results in a value of $\\Delta\\nu_{p}$ that is consistent with zero at the 68\\% confidence interval, i.e a pure spin-down glitch. While several glitches in other magnetars have been associated with minimal instantaneous changes in $\\nu$ (e.g. the 2012 outburst of 1E~1048.01$-$5937; \\citealt{Archibald2015}), the lack of ToAs between the radio shutdown of \\src\\ and beginning of our \\nicer\\ monitoring prevents us from ruling out a non-zero $\\Delta\\nu_{p}$.\n\n\\subsection{High-energy outburst}\n\n\\subsubsection{Spectral analysis}\n\nWe employed Xspec version 12.12.1 for our X-ray spectral analysis. We used the \\texttt{tbabs} model to account for the line-of-sight absorption towards the source, along with the \\citet{wilms00ApJ} solar abundances. We binned all spectra to have 5 counts per bin and used the W-statistic (C-stat in Xspec) to estimate the best fit model parameters and their associated uncertainties. For the \\nicer\\ spectra, however, we used the pgstat statistic, which is a more appropriate choice given the modeled nature of the background \\citep{remillard2021:3c50}. Finally, we added a numerical factor factor to account for any cross-calibration uncertainties when fitting spectra from different instruments simultaneously. We find a $\\lesssim2\\%$ and $\\lesssim10\\%$ deviation between the two \\nustar\\ spectra and \\nicer\/\\swift-XRT and \\nustar\\ spectra, respectively, commensurate with the expected values \\citep{madsen15ApJS}.\n\nWe fit the \\nustar\\ and \\nicer\\ observations taken on 2022 April 21 simultaneously, which provides the highest S\/N broad-band spectrum shape of \\src\\ during this most recent outburst. We also fit the pre-outburst 2016 and 2019 simultaneous \\swift-XRT and \\nustar\\ observations \\citep{CotiZelati2020} to better quantify the source spectral variability. We modelled all spectra in the 1--70~keV range using a three-component construction: two absorbed blackbodies ($\\rm BB_{warm}$ and $\\rm BB_{hot}$) plus a power-law (PL), suitably representative of the magnetar broad-band X-ray spectral shape \\citep{Kuiper-2006-ApJ,Goetz-2006-AandA,denHartog-2008a-AandA,denHartog-2008b-AandA,enoto17ApJS}. We linked the absorption hydrogen column density $N_{\\rm H}$ among all the spectra and left the rest of the model parameters free to vary. We establish the goodness of fit by running the Xspec \\texttt{goodness} command to simulate $1000$ spectra drawn from the best-fit model and compare their fit statistic to the observed data. We find that 16\\% of the spectra have a fit statistic smaller than the actual value, implying that the model adequately explains the data. We measure a best-fit $N_{\\rm H} = (3.67 \\pm 0.02) \\times 10^{22}$~cm$^{-2}$, a factor of 1.75 times higher than the value predicted by the $N_{\\rm H}$-DM relation of \\citet{He2013}. We display the source spectrum and best fit model in Figure~\\ref{specFig}, and present the best fit parameters and their $1\\sigma$ uncertainty in Table~\\ref{specParam}.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\hspace{-0.0cm}\n\\includegraphics[angle=0,width=0.48\\textwidth]{spec_1e1547_outburst2022.pdf}\n\\caption{{\\sl Upper panel.} Broad-band $\\rm \\nu F_\\nu$ spectra of \\src\\ derived during the 2022 April outburst (\\nicer+\\nustar, red), 2019 February (XRT+\\nustar, black) and 2016 April (XRT+\\nustar, green). Solid lines represent the best-fit model consisting of two BB and a PL. A flux enhancement is clearly present at soft X-rays coincident with the most recent outburst, in contrast to the stable non-thermal hard X-ray tail. {\\sl Lower panel.} Deviation of the data from the best-fit model in units of $\\sigma$.}\n\\label{specFig}\n\\end{center}\n\\end{figure}\n\n\\begin{table*}[t]\n\\caption{Broad-band X-ray spectral results}\n\\label{specParam}\n\\newcommand\\T{\\rule{0pt}{2.6ex}}\n\\newcommand\\B{\\rule[-1.2ex]{0pt}{0pt}}\n\\begin{center}{\n\\hspace{-2.0cm}\n\\resizebox{1\\textwidth}{!}{\n\\begin{tabular}{l c c c c c c c c}\n\\hline\n\\hline\nDate & $kT_{\\rm warm}$ \\T\\B & $R^2_{\\rm warm}$ & $kT_{\\rm hot}$ & $R^2_{\\rm hot}$ & $\\Gamma$ & $F_{\\rm warm}$\/$F_{\\rm hot}$\/$F_{\\rm PL}$ & $F_{\\rm S}$\/$F_{\\rm H}$ & $L_{\\rm X}$\\\\\n& (keV)       \\T\\B     &    (km$^2$)          &  (keV)            & (km$^2$)          &                     & ($10^{-12}$~erg s$^{-1}$ cm$^{-2}$)                         &   ($10^{-12}$~erg s$^{-1}$ cm$^{-2}$) &  ($10^{34}$~erg s$^{-1}$) \\\\\n\\hline\n2016-03-23 & $0.59\\pm0.03$ \\T\\B & $2.3_{-0.5}^{+0.7}$ & $1.16_{-0.08}^{+0.1}$ & $0.03_{-0.01}^{+0.02}$ & $0.2\\pm0.1$ & $14.1_{-0.8}^{+0.9}$ \/ $2.9_{-0.6}^{+0.7}$ \/ $8.9_{-0.5}^{+0.5}$ & $17.2_{-0.8}^{+0.9}$ \/ $8.7_{-0.4}^{+0.4}$ & $6.3_{-0.2}^{+0.2}$ \\\\\n2019-02-15 & $0.67\\pm0.02$ \\T\\B & $1.3\\pm0.2$       & $1.4\\pm0.2$ & $0.008\\pm0.004$ & $0.0\\pm0.1$ & $13.7_{-0.5}^{+0.5}$ \/ $1.7_{-0.3}^{+0.5}$ \/ $8.7_{-0.6}^{+0.7}$ & $15.5_{-0.5}^{+0.5}$ \/ $8.7_{-0.5}^{+0.5}$ & $5.9_{-0.1}^{+0.2}$ \\\\\n2022-03-21 & $0.64_{-0.05}^{+0.04}$ \\T\\B & $2.0_{-0.3}^{+0.5}$ & $1.0\\pm0.1$ & $0.09_{-0.06}^{+0.14}$ & $0.4\\pm0.2$ & $17_{-3}^{+2}$ \/ $6_{-2}^{+3}$ \/ $9.0_{-1}^{+1}$ & $23_{-1}^{+1}$ \/ $8.7_{-0.8}^{+0.8}$ & $7.7_{-0.3}^{+0.3}$ \\\\\n\\hline\n\\hline\n\\end{tabular}}}\n\\end{center}\n\\begin{list}{}{}\n\\item {\\bf Notes.} BB emitting areas and luminosity are derived assuming a distance of 4.5~kpc \\citep{Tiengo2010}. $F_{\\rm S}$ and $F_{\\rm H}$ are the 0.5-10 keV and 10-70 keV absorption-corrected fluxes, respectively.\n\\end{list}\n\\end{table*}\n\nWe find that the 2022 April 21 observation, 2.5 weeks after the GBM short burst detection (2 weeks after the confirmed BAT detection), exhibits a flux that is about 50\\% larger than the pre-outburst level. This increase is solely due to an increase in the soft 0.5--10~keV X-ray flux (Figure~\\ref{specFig}), and particularly seen in the $\\rm BB_{hot}$ contribution. The flux of this component has increased by a factor 3 and 5 since 2016 and 2019, respectively. On the other hand, the increase in the $\\rm BB_{warm}$ component is a factor 1.2  and 1.3 larger compared to 2016 and 2019, respectively. Finally, the hard X-ray PL flux is consistent between all three observations at the $1\\sigma$ level. This implies that either the hard X-ray flux was not impacted by this most recent outburst, or it followed a steeper temporal decay trend compared to the soft X-ray flux.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width=0.48\\textwidth]{ppCollage_nicer2.pdf}\n\\caption{Time-resolved \\nicer\\ pulse profiles of \\src\\ in days from 2022 April 9. We show the pulsations in the energy range 1.5--3~keV (left column) and 3--6~keV (right column). The red profile corresponds to the \\nicer\\ observation simultaneous to \\nustar\\ taken on 2022 April 21.  All profiles are displayed using 15 phase bins per rotational cycle, with the exception of the red profile, which exhibits 30 phase bins per pulse period (see text for details). Two cycles are shown for clarity.}\n\\label{nicProf}\n\\end{center}\n\\end{figure}\n\nWhile the source exhibited a change in the overall flux, the broad-band spectral curvature in 2022 remained relatively stable compared to the previous epochs. We find no significant variability in the temperature of either of the two BBs, nor in the hardness of the PL component. This points to an increase in the overall area of the hot thermally-emitting region as the origin of the soft X-ray flux increase (Table~\\ref{specParam}). \n\nTo study the long-term high energy evolution of the \\src\\ outburst, we modelled the \\nicer\\ and \\swift-XRT spectra in the 1--6~keV range (beyond 6~keV the data for both instruments was mainly background-dominated). We used a simple absorbed BB model and fixed the hydrogen column density $N_{\\rm H}$ to the best-fit value as derived above. This model sufficed to describe all the spectra except for the \\nicer\\ observation on 2022 April 9 (ID 5020300101, Table~\\ref{xObsLog}) which required a second blackbody component to result in a statistically acceptable fit. The resulting 0.5--10~keV absorption corrected flux evolution is shown in the upper-panel of Figure~\\ref{fig:flux}. At the start of the \\nicer\\ monitoring on 2022 April 9, we find a source flux that is a factor 3.5 larger than the pre-outburst value as established with \\swift-XRT on 2022 March 5 (green circles in Figure~\\ref{fig:flux}, upper panel; see also \\citealt{CotiZelati2020}). This confirmed that \\src\\ was experiencing yet another high energy outburst, its first since its major 2009 outburst. The source flux decayed back to the same pre-outburst persistent flux level on 2022 April 26, i.e. 17 days later, and remained there until the end of our monitoring in 2022 August. We find that the bulk of the flux increase throughout the outburst occurred at energies $>2$~keV, consistent with the result of the \\nicer+\\nustar\\ broad-band modelling.\n\n\\subsubsection{X-ray timing analysis}\n\nGiven the complexity of the glitch timing model  (Section~\\ref{timSec}), we fold the \\nicer\\ barycenter-corrected events with a (strictly) post-glitch variant of the combined radio\/X-ray timing solution in Table~\\ref{tab:timing}.\nTo account for red noise in the timing residuals, we fit for spin-frequency derivatives up to $d^5\\nu\/dt^5$, at which order the residuals are dominated by white noise. Due to the limited exposure of most individual \\nicer\\ observations, we merged several consecutive ones until we accumulated about 6000 counts in the 1.5--6~keV interval. This ensured a proper sampling of the pulse shape in two energy bands, namely 1.5--3~keV and 3--6~keV. During the outburst decay phase (2022 April 9-26), the stretch of each interval that fulfilled our criterion was on average 2 days, and increased to 10~days after the source returned to its pre-outburst flux level. We present these pulse profiles in Figure~\\ref{nicProf}.  Each row displays the pulsations per time interval starting from the date of the initial \\nicer\\ observation on 2022 April 9 (day 0), while the left and right panels show the 1.5--3~keV and 3--6~keV pulse-profiles, respectively; note that the pulse in red is from the \\nicer\\ observation that is simultaneous to the \\nustar\\ one for 2022 April 21.\n\nThe soft 1.5--3~keV pulsations are quasi-sinusoidal throughout our monitoring campaign. On the other hand, the 3--6 keV profile exhibits pronounced variability. For instance, during the outburst decay phase, there is an evident shoulder leading to the pulse peak. Moreover, the peak is slightly broader early in the outburst and displays complex shape, likely formed of two or more peaks. This complexity is confirmed with \\nustar\\ as we show below. The 3--6 keV pulse shape turns quasi-sinusoidal starting on day 17 from the first \\nicer\\ observation, coincident with the return of the flux to the pre-outburst level (Figure~\\ref{fig:flux}). We derive the root-mean-square pulsed fraction (rms PF) for each of the light curves in Figure~\\ref{nicProf} \\citep[see, e.g.,][for details]{woods04ApJ}. For this purpose, we fit each background-subtracted pulse profile to a Fourier series including three harmonics, which was sufficient to provide an acceptable fit to the data. We find pulsed fractions ranging from 19--25\\% and 29--34\\% in the 1.5--3~keV and 3--6~keV energy bands, respectively, with a typical $1\\sigma$ uncertainty of about 3\\%. This implies a lack of significant variability in the pulsed fraction at soft X-rays concurrent with the outburst flux enhancement.\n\nThe \\nustar\\ observation of \\src\\ on 2022 April 21 caught the tail end of the outburst. We display the pulse profiles during this observation in 3 separate energy bands, namely, 3--6~keV, 6--10~keV, and 10--15~keV (Figure~\\ref{Xprof}). We confirm the \\nicer\\ results which show, in the energy range 3--6 keV, a non-symmetric profile with a shoulder leading to a multi-peaked pulse maximum; this fine structure in the peak is reminiscent of that seen during the 2020 outburst of SGR 1830-0645 \\citep{younes22ApJL18301}.  Similar pulse profile morphology is apparent in the 6--10 keV range: the pulse exhibits considerable asymmetry with a shoulder leading to the pulse peak. All these features are clearly absent from the 2016 and 2019 \\nustar\\ monitoring; the pulse profile in the same energy range shows a purely sinusoidal form \\citep[][Figure~5]{CotiZelati2020}. On the other hand, the pulse profile at high energies between 10 and 25~keV exhibits quasi-sinusoidal shape, consistent with the result of the earlier \\nustar\\ observations. Note, however, the lower S\/N in this band compared with that at energies below 10 keV. We do not detect significant pulsations at energies $>25$~keV, despite the detection of the source at the $\\gtrsim10\\sigma$ confidence level.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[angle=0,width=0.48\\textwidth]{ppvsEne_NuS2022.pdf}\n\\caption{The \\src\\ pulse profiles as derived with \\nustar\\ at different energies (top to bottom). See text for more details.}\n\\label{Xprof}\n\\end{center}\n\\end{figure}\n\nFinally, we measure the \\nustar\\ energy-resolved rms PF following the same recipe we utilized for \\nicer. We find an increasing trend at soft energies, from about $30\\%$ and up to $45\\%$ at 10 keV. The PF drops to $30\\%$ in the energy range 10--25~keV. Moreover, the latter profile appears slightly offset from the soft-energy ones (Figure~\\ref{Xprof}), yet a formal measurement is complicated by the difference in shape between the profiles. Lastly, we derive a 3$\\sigma$ upper-limit of $25\\%$ on the pulsed fraction at energies $>25$~keV. These elements are broadly consistent with the 2016 and 2019 \\nustar\\ findings of \\citet{CotiZelati2020}.\n\n\\subsubsection{Short burst searches}\n\\label{sec:burSear}\n\nThe 22-day discrepancy between the time of the first radio non-detection, which occurred on 2022 March 16, and the BAT trigger on 2022 April 7 prompts the question on whether the radio disappearance genuinely preceded the trigger of the high energy outburst. With the absence of any sensitive X-ray observations of \\src\\ during this gap, we relied on burst detection to qualitatively gauge the source activity. Typically, the rate of hard X-ray bursts from an active magnetar is maximal at the start of an outburst, declining thereafter. This behavior is observed during burst storms, e.g., 1E~2259+596 \\citep{gavriil04ApJ2259}, SGR 1935+2154 \\citep{Younes2020}, as well as over the course of weeks following the magnetar outburst onset, e.g., SGR 1830$-$0645 \\citep{younes22ApJ18301}, Swift~J1818.0$-$1607 \\citep{hu20ApJ}. Hence, if the outburst indeed began at the time of the radio non-detection, more bursts should be detected from the direction of \\src\\ by Fermi-GBM during the 20~day interval.\n\nWe searched the CTTE data for untriggered bursts in the energy range 8--100~keV following the procedure detailed in \\citet{Gavriil2004} and \\citet{Younes2020}. In summary, we binned the light curve of each GBM detector separately, at a temporal resolution $t_{\\rm r}$. We search each light curve in consecutive intervals of 100~s, overlapping by 10~s. We note the deviation of the number of counts $x$ in each bin from the long-term average rate in the light curve, assuming a Poisson probability density function $P(x)$. A potential burst is flagged if $P(x)<0.01\/N$, where $N$ is the number of bins per light curve (typically $100\/t_{\\rm r}$). The flagged bins are removed and the procedure is repeated until no more bins are found. We chose $t_{\\rm r}$ to be 8~ms, 32~ms, and 128~ms to sample the full range of magnetar burst durations. We confirm our GBM search results with a Bayesian blocks search technique as detailed in \\citet{lin20ApJ}. We only inspect potential bursts that were found in two or more detectors; single detector enhancements are typically due to particle background rather than an astrophysical event. Moreover, we rejected all bursts when \\src\\ was behind the earth for \\fermi, and those that were registered only in NaI detectors with an angle $>60^\\circ$ from the source since their effective area drops significantly at such large observing angles. The source identification process, however, is complicated by another magnetar, SGR~J1555$-$5402, which is 40\\hbox{$^\\prime$}\\ from \\src\\ and was also active at this time. This separation is not distinguishable by GBM, but is easily resolved in BAT images.\n\nWe find six candidate bursts that may originate from \\src\\ between 2022 March 16 and April 7, including one that corresponds to a GBM trigger on 2022 April 3 (trigger 220403928). \nBAT detected a count rate increase in two cases, in addition to the short-burst detected on 2022~April~7.\nOne of these was identified as the aforementioned SGR~J1555$-$5402, and the other coincidence has a negative value (consistent with zero) at the \\src\\ location and excludes it as the source of the rate increase. \nHence, the apparent lack of any bursting activity associated with \\src\\ over the course of three weeks following the last radio detection suggests that the high energy outburst genuinely commenced later. \nHowever, we cannot establish the precise date of the outburst, and tentatively place it sometime in early April. \nFuture continuous monitoring of X-ray bright radio-loud magnetars is crucial for revealing the dynamic between the radio and X-ray emission at the start of an outburst.\n\n\\section{Discussion} \\label{sec:disc}\n\nPrior to the 2009 outburst, \\src\\ was considered to be a so-called transient magnetar, similar to XTE~J1810$-$197 \\citep{Ibrahim2004}, with a historical quiescent (absorbed, 0.1--10\\,keV) flux of (0.3--2.1)$\\times 10^{-12}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$ \\citep{Gelfand2007}. However, the 2009 outburst was associated with a transition of \\src\\ into a persistent magnetar state where it exhibits a significantly higher steady-state X-ray flux than the pre-2006 quiescent state \\citep{CotiZelati2020}. On account of this increased persistent emission, the 2022 outburst appearing to be both less intense and shorter lived than the 2008 outburst is likely an artifact of the much higher baseline X-ray flux. Indeed the maximum 0.5--10\\,keV flux of $(6.0 \\pm 0.4) \\times 10^{-11}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$ is comparable to the observed peak flux of $\\sim 6 \\times 10^{-11}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$ during the 2008 outburst \\citep{Israel2010}. Had this previous outburst occurred more recently, the X-ray flux of \\src\\ would have fallen below the current baseline of $\\sim 1.7 \\times 10^{-11}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$ after a similarly short 10--20\\,day timescale. Were it not for our triggered \\nicer\\ observations, the enhanced X-ray emission from this outburst may have been missed entirely. A smaller outburst may not have even produced a detectable X-ray enhancement.\n\nMagnetar X-ray outbursts are generally attributed to crustal shifts due to stresses on the surface from internal B-field decay. These shifts could also either modify or enhance toroidal twists in the morphology of external magnetic field lines \\citep[see][and references therein]{turolla15RPPh}. The soft X-ray thermal emission arises due to either internal energy deposition in the outer crust \\citep{beloborodov16ApJ, akgun18MNRAS, lander2019MNRAS}, bombardment of the surface by accelerated particles in the twisted magnetospheric field loops \\citep{Beloborodov2009}, mechanical stresses slowly building in the crust (see \\citealt{Lander2015, Kojima2022} and references therein) or a combination of all three. In this model, a more complex pulse shape is expected during the outburst due to the emergence of new hot spots on the surface \\citep[e.g.,][]{Archibald2017a,younes22ApJL18301}, commensurate with the 3--6 keV pulse profile morphology displayed in Fig.~\\ref{Xprof}. As the outburst abates, the source surface heat map resorts back to its long-term, quasi-sinusoidal configuration.  If surface bombardment is active, the evolving surface temperature profile likely is accompanied by a migration of field line footprints associated with the twisting and untwisting of the magnetospheric field.\n\nThe hard X-ray tail above 10 keV is believed to arise from resonant inverse Compton scattering (RICS) of the soft surface thermal photons by the relativistic electrons accelerated in twisted loops \\citep{baring07ApSS,nobili08MNRAS,wadiasingh2018ApJ}. Figs.~\\ref{specFig} and~\\ref{Xprof} clearly display a lack of any variability in the hard X-ray emission concurrent with the elevated soft thermal emission during this outburst of \\src.  This disconnect between soft and hard X-ray bands is to be expected.  Detailed models of RICS emission generally favor a production of the signal at quasi-equatorial locales of modest altitudes, where scattering kinematics between the electrons and the soft X-ray photons from the surface readily accesses the cyclotron resonance in the cross section \\citep{wadiasingh2018ApJ}.  Thus changes in field line geometry near their footprints (e.g. due to a localized crustal stress) and moreover near the polar regions likely have only a small impact on the RICS emission, as do alterations of the surface hotspot size and temperature.  Evidence for some separation of magnetic co-latitudes for hard X-ray signals and the surface soft X-ray ones may be suggested by the modest 0.1-0.3 phase lag of the peak between the 3--6 keV band and the 10--25 keV one in Fig.~\\ref{Xprof}.  Thus changes in pulsed radio emission emanating from in or near the open field region, might naturally not be accompanied by modifications to the hard X-ray signal.\n\nGenerally, pulsed radio emission from magnetars is closely linked to the outburst mechanism: it switches on following the onset of an outburst before gradually fading into radio silence as the neutron star settles into X-ray quiescence (e.g., \\citealt{Camilo2016, Scholz2017}, but see \\citealt{zhu20ATel14084,younes22arXiv221011518Y}). Unlike other radio-loud magnetars, persistent radio pulses have been detected from \\src\\ since the quasi-stabilization of its X-ray flux to the current high state in $\\sim$2013 (\\citealt{CotiZelati2020}; Camilo et al., in prep.).\nAs shown in Figure~\\ref{fig:flux}, radio pulses were detected up until sometime between 2022 March 16 and 26, and were not seen again until April 19. \nAn initial disappearance and reactivation of the radio emission of \\src\\ was also reported following the 2009 outburst \\citep{Camilo2009ATel, Burgay2009ATel}.\n\nThe 2016 magnetar-like outburst of the high magnetic field strength pulsar PSR~J1119$-$6127 was also associated with a shutdown in the persistent radio emission, which was re-detected around two weeks after the initial X-ray outburst \\citep{Dai2018}. \nMoreover, simultaneous X-ray and radio observations of PSR~J1119$-$6127 revealed that magnetar-like short bursts were associated with a sudden disappearance of pulsed radio emission \\citep{Archibald2017b}.\nThis correlation was interpreted as a burst-induced quenching of the radio emission arising from electron-positron pairs leaking from the fireball trapped above the surface into the particle acceleration gaps, thereby increasing the plasma density within these regions and suppressing the acceleration of charges that precipitate radio emission.\nA similar phenomenon would provide a natural explanation for the disappearance of radio pulses from \\src\\ if it occurred after the onset of the high-energy outburst. \nYet, the dearth of \\fermi-GBM and \\swift-BAT burst candidates up to early April 2022 implies a delayed onset of the high energy outburst with respect to the disappearance of radio pulses.\nAn earlier onset necessitates an unlikely scenario in which all short-bursts emitted over an extended period of time were missed by all large FoV hard X-ray and gamma-ray monitors.\nAlternatively, a slow undetected rise in high-energy activity could explain the observed delay between the radio disappearance and delayed onset of short burst detections. \nSuch behavior was previously seen in the 2002 outburst of 1E~1048.1$-$5937 \\citep{Archibald2020}.\n\nSmall-scale twisting of the magnetosphere from crustal fracturing and motion of the magnetic footprints can lead to a decrease or complete failure of the charged particle currents responsible for producing the pulsed radio emission in addition to variations in the optical depth of the magnetosphere \\citep{Beloborodov2009, Timokhin2010, Hu2022}. \nRegardless of the exact underlying physics, any alteration of the magnetosphere must have been short lived, as we did not detect any substantial change in the average radio pulse profile shape or polarization properties of \\src\\ after re-activation.\nThis is markedly different to the behavior of other radio-loud magnetars, which typically display a broad range of profile variations and a marked increase in radio flux in the immediate aftermath of an outburst (e.g., \\citealt{Scholz2017, Dai2018, Dai2019}). \nFuture observations of similar high-energy outbursts both in \\src\\ and other highly magnetized pulsars will shed further light on the radio-quenching mechanism.\n\nOur joint radio and X-ray observations revealed that the 2022 outburst resulted in a significant spin-up event in the timing of \\src, which bears a strong resemblance to a pulsar glitch.\nThe gap in our timing coverage due to the disappearance of pulsed radio emission, low observing cadence of \\swift\\ and beginning of our triggered \\nicer\\ observations means the exact epoch of the event is unknown, but is constrained to between MJD 59661 to 59672 (March 23 -- April 3, 2022).\nGlitch-like timing anomalies and subsequent rapid changes in $\\dot{\\nu}$ are common feature of the post-outburst timing of magnetars (e.g., \\citealt{dib14ApJ, Archibald2020}), the latter of which are often ascribed to torque variations imparted by twists in the magnetosphere \\citep{Thompson2002}.\nUnlike rotation-powered pulsars, the majority of glitches in magnetars are associated with radiative outbursts that are believed to be caused by either starquakes \\citep{Thompson1995} or magnetic re-connection events \\citep{Lyutikov2002}.\nIn the starquake model, the gradual untwisting of the internal magnetic field induces an enormous strain on the rigid outer crust of the neutron star.\nEventually the tensile strength of the crust is overcome, leading to a sudden plastic deformation or even fracturing of a small region of the crust.\nThis triggers an outward exchange of angular momentum from the superfluid core to the crust via unpinning of vortices at the crust-core boundary \\citep{Ruderman1998}.\nModelling the timing anomaly in \\src\\ as a glitch, we obtained permanent step-changes in $\\nu$ and $\\dot{\\nu}$ of $0.2(1)$\\,$\\mu$Hz and $-2.4(1) \\times 10^{-12}$\\,s$^{-2}$ respectively.\nThe permanent change in spin-frequency is small when compared to most glitches in magnetars (see Figure 2 of \\citealt{Fuentes2017}).\nOn the other hand, the increase in spin-down rate is among the largest recorded in any pulsar \\citep{Lower2021b, Basu2022}.\n\nFollowing the glitch, we also detected a secular increase in the spin-down rate that continued up until at least our last \\nicer\\ observation.\nWe inferred a $\\ddot{\\nu}$ of $-1.95(5) \\times 10^{-19}$\\,s$^{-3}$ from a simple linear fit to the $\\dot{\\nu}$ timeseries and $-2.0(1) \\times 10^{-19}$\\,s$^{-3}$ from our glitch fit in Section~\\ref{timSec}.\nThis corresponds to a net increase in $\\dot{\\nu}$ of $-2.54 \\times10^{-12}$\\,s$^{-2}$ over the 147-days of post-outburst timing.\nA similar linear trend was observed following the 2008 outburst \\citep{Israel2010}, where the resulting $\\ddot{\\nu} = -4.8(9) \\times 10^{-18}$\\,s$^{-3}$ was an order of magnitude larger than we observed after the 2022 outburst. \nSecular increases in spin-down rate are a common feature of outbursts in magnetars and high B-field pulsars and are often attributed to the magnetosphere becoming increasingly twisted up to a maximum value after a few 10's to 100's of days \\citep{Thompson2002, Beloborodov2009}.\nAfter reaching the maximum twist, the field bundle then begins to unwind, causing a turnover in the spin-down evolution as the magnetar relaxes back toward the pre-outburst state.\nA global alteration of the twist can be ruled out in this 2022 event given the lack of hard X-ray evolution, indicating that either a small-scale local twist in bundles of magnetic field lines near the active pole or transient particle winds may be responsible for the secular increase in $\\dot{\\nu}$ \\citep{Harding1999}.\n\nSpin-down rates peak anywhere between $-10^{-13}$ -- $10^{-11}$\\,s$^{-2}$ depending on the strength of the outburst.\nGiven the lack of a turnover in the post-outburst $\\dot{\\nu}$ evolution of \\src, it is likely the maximum twist imparted by the 2022 outburst or peak in the outflowing particle flux has yet to be reached.\nThe recovery back to the initial spin-down state varies between magnetars and from one outburst to the next.\nOccasionally, the initial turnover is followed by an extended period of `noisy' spin-down evolution, including quasi-periodic variations such as those seen in 1E~1048.0$-$5937 \\citep{Archibald2015} and Swift~J1818.0$-$1607 \\citep{Rajwade2022}. \nThese variations are reminiscent of the spin-down state-switching process seen among rotation-powered pulsars \\citep{Lyne2010}, and often not associated with any changes in high-energy emission.\nSuch behavior remains to be seen in the current post-outburst timing of \\src.\n\n\\section{Conclusions}\n\nWe have presented the detection of a new outburst of \\src\\ that began in 2022 April. The outburst was marked by an increased soft X-ray emission and the disappearance of pulsed radio emission, both of which recovered back to their pre-outburst properties $\\sim$3-weeks later. \nThe disappearance of the radio pulses prior to the outburst onset is surprising, and could point to an undetected slow rise in high energy activity. \nSuch behavior could arise from small-scale crustal fracturing, where variations in the magnetospheric currents and optical depth resulted in a quenching of the radio emission mechanism.\nInterestingly, we find no change to the hard non-thermal tail concurrent to the soft X-ray and radio variability. Our joint \\parkes\/\\nicer\\ timing measurements revealed the presence of a glitch-like timing event in the rotation of the magnetar.\nA significant instantaneous increase in spin-down rate was also measured, which continued to increase at a rate of $-2.75 \\times 10^{-19}$\\,s$^{-3}$ over the first 147-days after the outburst.\nSmall-scale shifts in magnetospheric geometry in the lead up to the main quake may have been responsible for the quenching of the pulsed radio emission.\nThese results highlight the importance of conducting high-cadence monitoring of radio-loud magnetars such as \\src\\ at both radio and X-ray frequencies. Future outbursts with greater coverage around the initial trigger time may be able to resolve the radio quenching, delayed rise in spin-frequency, and initial increase X-ray flux.\nSuch observations would provide an even clearer picture of the magnetar outburst mechanism.\nGiven some similarities between this latest event and the 2008 outburst, it is possible that \\src\\ has entered a new state of increased activity.\nIn this case, the 2022 outburst may therefore be a precursor to additional outbursts over the coming years.\n\n\\begin{acknowledgments}\nWe thank George Hobbs and Rahul Sengar for generously giving us some of their Parkes observing time, and a special thanks to James Green and Phil Edwards for granting Parkes director discretionary time. \nWe also thank the anonymous referee for their helpful comments.\nThe Parkes radio telescope (\\textsl{Murriyang}) is part of the Australia Telescope National Facility (\\href{https:\/\/ror.org\/05qajvd42}{https:\/\/ror.org\/05qajvd42}) which is funded by the Australian Government for operation as a National Facility managed by CSIRO. \nWe acknowledge the Wiradjuri people as the traditional owners of the Observatory site. \nRadio data reduction and analysis were performed on the OzSTAR national HPC facility, which is funded by Swinburne University of Technology and the National Collaborative Research Infrastructure Strategy (NCRIS). \nA portion of this work was supported by NASA through the NICER mission and the Astrophysics Explorers Program. \nG.~Y. research is supported by an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center, administered by Oak Ridge Associated Universities under contract with NASA. \nL.~D. is supported by the Australian Research Council Centre of Excellence CE170100004 (OzGrav). M.~G.~B. acknowledges the generous support of the National Science Foundation through grant AST-1813649.  Z.~W. acknowledges support by NASA under award number 80GSFC21M0002. S.~G. acknowledges the support of the National Centre for Space Studies (CNES).\n\\end{acknowledgments}\n\n\\software{{\\tt dspsr} \\citep{vanStraten2011}, {\\tt HEAsoft} (v6.30.1; \\citealt{HEASARC2014}), {\\tt psrchive} \\citep{Hotan2004, vanStraten2012}, {\\tt NuSTARDAS} (v2.1.2), {\\tt DM\\_phase} \\citep{Seymour2019}, {\\tt PRESTO} \\citep{Ransom2011}, {\\tt RMNest} \\citep{Lower2022}, {\\tt tempo2} \\citep{Hobbs2006, Edwards2006}, {\\tt TempoNest} \\citep{Lentati2014}, {\\tt XSPEC} (v12.12.1; \\citealt{Arnaud1996})}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\n\\noindent The problem of how to unify quantum theory and general relativity into a consistent theoretical framework\nhas baffled physicists for more than eight decades, but has become particularly pressing in recent years.\nEach of the two theories in question is a fundamental theory of physics in its own right - each theory is, in Einstein's terminology,\na \\emph{theory of principle}; see below - and is corroborated by a very impressive body of experimental evidence.\nBut physical reality constitutes an undivided whole and it obviously makes no sense to have \\emph{two} distinct fundamental\ntheories for a single reality.\nThat physicists have been able to get away with this state of affairs for so long is simply due to the fact that the\nintersection of the domains of applicability of the two theories is in some sense small and at any rate is usually thought\nto lie far beyond anything technologically feasible, at present or in the nearby future.\nIndeed, according to the received view, because of gravity's intrinsic weakness as compared to the nuclear and electromagnetic\ninteractions, general relativistic considerations play no role in quantum theory until particle scattering energies reach \nPlanckian dimensions and, conversely, because quantum theory is at first glance effectively restricted to the (sub-)atomic \nregime, it can seemingly play no role in any nontrivial general relativistic effects, as these typically involve distance \nscales many orders of magnitude beyond the atomic scale.\nBut, apart from the questionable nature of these arguments, the lack of a unifying theoretical framework for the entire\nrealm of natural phenomena is arguably a serious problem from the perspective of the foundations of physical theory\\footnote{That is,\nthe lack of such a framework \\emph{is} a serious problem assuming an appropriate form of scientific realism (although that is \nof course the household doctrine of theoretical physicists working on these matters - at least when explaining \ntheir activities to the lay public or when writing up research grants !). It is not necessary however to assume realism to\npoint out the inadequacies of existing theories (see section \\ref{nwthrarguments} for further discussion).}.\\\\\nAs said, both quantum theory and general relativity are theories of principle. This means that they are characterized by \ndefining principles which are supposed to be valid universally, and which should apply, without \nrestriction, to \\emph{all} physical phenomena\\footnote{See \\citeN{Stachel}.} (such theories are to be distinguished from \\emph{constructive theories},\n which are theories which attempt to explain a limited group of natural phenomena by means of some model or set \nof equations, but always so within the context of a specific theory of principle\\footnote{For instance, the\n$\\mbox{SU}(3) \\times \\mbox{SU}(2) \\times \\mbox{U}(1)$ standard model of particle physics and quantum field theory \nmodels more generally are constructive theories within a quantum theoretical context (see subsection \\ref{QTandstmod}),\nwhile the so-called $\\Lambda \\mbox{CDM}$ ``concordance model'' is a constructive theory of cosmology within a general relativistic \ncontext (see subsection \\ref{GRandstmod}). It is sometimes argued that general relativity is (at least partly) a \nconstructive theory in view of the fact that it can be regarded as a classical field theory, but that position will \nnot be adopted here. General relativity describes the properties of space and time in relation to those of ponderable \nmatter and fields and as such it forms the general setting for particular constructive theories of ponderable \nmatter and fields.}).\nBut it is not difficult to see that the two sets of principles associated with these theories are not fully inter-compatible.\nFor instance, according to general relativity, spacetime is a dynamical entity, whereas the standard unitary dynamics \nof states and observables in quantum theory is conditional upon a decomposition of this entity into ``space'' and ``time'',\nwhich are moreover taken as non-dynamical. Similarly, according to \nquantum theory it makes sense to talk of a point mass being in a quantum superposition centred about two different spatial \nlocations, whereas according to general relativity this makes no sense, since such a superposition would (in principle) \nentail a corresponding superposition of two spacetime geometries and there is no natural way to mathematically implement\nsuch a superposition (i.e., to identify the points of two non-identical spacetimes with each other)\\footnote{Cf. \\citeANP{Penrose10} \\citeyear{Penrose10,Penrose9}. See also \\shortciteN{Karolyhazy}, \\citeN{Diosi}.}.\nLogic thus dictates that at least one set of principles cannot be the complete story and it is a matter of simple\nhistorical fact that the ``mainstream'', often implicit, viewpoint in theoretical physics with respect to this issue has \nalways been that the principles of general relativity are somehow to be subjugated to those of quantum theory\n(in a sense, this view is of course also encapsulated in the very phrase ``quantization of gravity'')\\footnote{It is somewhat of an irony\nthat as early as 1916, Einstein himself already argued for a modification of ``the new theory of gravitation'' \n\\emph{because} of quantum theory (of which the general principles had yet to be formulated of course). See \\citeN{Einstein2}.\nFor an explicit quantum-universality claim in this regard, see e.g. \\citeN{Kiefer}.}.\nAs will be seen in the next two sections however, there are ample grounds to question the validity of this viewpoint (at the\nvery least, it is undeniable that it has so far failed to lead to a consistent resolution of the unification problem).\nIf the mainstream perspective is abandoned, the incompatibility of the two sets of principles raises some profound \nconceptual and methodological dilemmas. For instance, on the one hand, it might seem that the most important lesson that can be drawn from the historical \ndevelopment of general relativity is that the strong insistence on a principle of locality, in spite of a number of serious \nearlier obstacles, ultimately payed off\\footnote{The notion of locality will be further explicated and differentiated in subsection \\ref{qnonloc}\nbut the intuitive content of this notion should suffice for present purposes.}. \nOn such an account, it would thus not seem unreasonable to expect that quantum theory will eventually also be seen to be\nfounded upon a more local, ``classical'' basis. On the other hand, it might be\nasked on what precise grounds a physical theory is required to be local. One seemingly obvious answer is that \\emph{if}\nphysical reality is structured that way, human intuition would presumably manage to latch onto such a structure somehow,  \nand so the \\emph{fact} that locality is a very intuitive notion would then (according to this line of reasoning) imply the validity\nof some form of locality as a fundamental principle of nature. Yet, such a position appears dangerously close to the \nKantian notion of the synthetic a priori, whereas the intuitive appeal of a locality principle seems equally well \nexplainable on the basis that the structure of ``macroscopic'' physical reality happens to be local (to a large extent) \nand that human intuition is able to latch onto that structure through sense experience. But there would then be no \nguarantee that such intuition could be validly extended into the ``microscopic'' domain, not directly accessible to \nthe senses. \nIn fact, as recalled in more detail later, it is well established that an underlying local description in the case of quantum \ntheory is \\emph{impossible} for empirical reasons (at least if such a description is interpreted in terms of real spacetime \nphenomena).\\\\\nOn this basis, it is then perhaps tempting to conclude that the entire doctrine of locality - although extremely useful \nin arriving at the successful theories of electromagnetism and relativity - is ultimately untenable and that, because of this,\nthe principles of quantum theory carry more weight than those of general relativity. \nHowever, even if the locality of general relativity and classical electrodynamics ultimately turns out to be ``merely'' an emergent quality \n- as the present evidence strongly suggests - there is an important reason for attaching at least as much weight to the line of \nresearch that culminated in Einstein's general relativity and, consequently, also to the principles of general relativity themselves. \nThis is the circumstance that the development of the theories of relativity - as well as that of classical electromagnetism - \nwere strongly driven by the requirement that physical theories be \\emph{intelligible}, whereas, by contrast, the development\nof quantum theory was strongly dominated by an attitude which was essentially instrumentalist.\\\\ \nThese remarks are further illustrated in subsections \\ref{GRandstmod} and \\ref{QTandstmod} respectively.\nAs to the unification problem, the upshot of the foregoing discussion is this.\nIt is clear that adoption of some set of guiding principles is necessary in order to have a realistic chance of \nformulating a concrete, consistent theory of \\emph{quantum gravity} - i.e., an ``even-handed'' theory of principle\nwhich contains both quantum theory and general relativity in the appropriate limits (and which, in particular, need thus\nnot fully comply with either theory).\nBut given the partial incompatibility of the limiting theories, as well as the plethora of different, often orthogonal\napproaches that have already been attempted in this regard, which principles should be adopted ?\nIt is argued in section \\ref{principles} that if history is to be of any guidance, the best chance to obtain the key \nstructural features of a putative quantum gravity theory is by deducing them, in some form, from the appropriate ``empirical\nprinciples'', i.e., principles which are strongly grounded in observational facts (as opposed to ``theoretical\nprinciples'', which are not so grounded).\nFour candidate empirical principles are subsequently identified in section \\ref{QGprin}.\nIn each case, it is explained - when appropriate - how the principle in question could be implemented mathematically in a \ntheory of quantum gravity, why it is deemed to be of fundamental significance and also why contemporary accounts of it are insufficient.\nIt is important to stress however that the principles listed are not necessarily foreseen to (all) play a role as \\emph{postulates}\nin a future theory of quantum gravity. Rather their roles may be restricted to function ``merely'' as \\emph{guiding principles} (in the\nsame sense, for instance, as the equivalence principle crucially guided the formulation of general relativity, although it is not a\npostulate of that theory).\n\n\n\\section{The Need for a New Theoretical Framework}\\label{nwthrarguments}\n\n\\noindent As noted in the introduction, the standard approach to the quantum gravity problem assumes that general relativity\nsomehow emerges as the ``classical limit'' from a yet-to-be-obtained true \\emph{quantum theory} of gravity of some sorts\n(in the same manner, say, as classical electrodynamics is customarily thought to coincide with the ``classical limit'' of \nquantum electrodynamics).\nConceptually speaking, this standard approach fits well with (some version of) the doctrine of atomist reductionism\nthat underlies modern high-energy physics and according to which it is the supreme task of theoretical physics to\nuncover the ultimate, i.e. smallest, constituents of matter and their governing principles\\footnote{The general view \ntowards the reductionism issue in this work is, first, that there are in fact several such issues, which are\nunfortunately often grouped under the same banner however, and, second, that some notions of reductionism are reasonable,\nor even necessary for science to be possible, while others are not.\nFor instance, there is a clear (and rather uncontroversial) sense in which a science such as physics lies at a lower \nlevel in the descriptional hierarchy than sciences such as chemistry or biology, but this does not mean that the latter \ncan be fully reduced to physics - not even in principle. In fact, there are powerful reasons coming from within science \nto reject such a particular strong form of reductionism. Similar remarks incidentally apply to analogous hierarchical \norderings \\emph{within} physics. See \\citeN{Holman2} for further discussion of these issues.}.\nHowever, apart from the fact that there are a few well-known, general obstructions to atomist reductionism,\nthe standard approach is silent on how to interpret quantum theory, even though this is an issue central to its very \nviability in the first place~!\nIndeed, according to Bohr's Copenhagen interpretation (or some modern version of it), which constituted the prevailing \northodoxy until relatively recently, there is only a quantum-mechanical ``description'' of things and no actual ``quantum world''.\nYet, it is actually very far from clear that such anti-realism can be made conceptually consistent with the type of\nreductionism underlying high-energy physics (i.e., for what reason attempt to formulate a quantum-mechanical\n\\emph{description} of non-real structures supposedly relevant in some way at scales which are at least sixteen orders\nof magnitude beyond currently accessible scales ? for what reason assume that such a description, granted that it can\nbe obtained, would in any meaningful sense be \\emph{final} ?).\nMoreover, even in the unlikely event that the philosophy underlying modern high-energy physics (as just portrayed) can\nbe made conceptually coherent with an anti-realist interpretation of quantum theory, there is the elementary observation\nthat, despite massive efforts for the past thirty years or so, a quantum theory of everything has remained elusive (at best),\nso that it is natural to explore other possible directions.\nOn the other hand, on adopting a realist interpretation of quantum theory of some form, interpretational difficulties\nalso emerge. \nFor instance, a question that immediately emerges when adopting a literal many worlds perspective is : why introduce\nan extravagance of monstrous proportions, with no empirical ramifications whatsoever, just to account for a \nstraightforward discrepancy between theory and observation ?\nIn a similar vein, any ``refined'' interpretation of quantum theory, in which the values of all observables always have \ndeterminate values (such as the de Broglie-Bohm pilot wave theory) necessarily conflicts with relativistic causality\nand it becomes imperative to address the issue of whether initial conditions can always be ``reasonably'' chosen so as \nto avoid standard causal paradoxes (where people kill their own ancestors, for instance).\nAnd of course, a perspective in which the state vector is objectively reduced naturally points to a theory beyond quantum\ntheory (disregarding far-fetched views in which for instance consciousness collapses the state vector).\nThat is, upon such a view it is natural to interpret state reduction and unitary dynamics\nsomehow as different aspects of the same fundamental dynamics.\\\\\nThis is not the place to go into the pros and cons of all the multifarious interpretations and sub-interpretations\nof quantum theory (for some additional comments see subsections \\ref{QTandstmod}, \\ref{qnonloc} and \\ref{qindet}), but the message should be \nclear. Unless it is indicated how to consistently interpret quantum theory, the mainstream view, according to which general \nrelativity is to be somehow seen as the ``classical limit'' of a true microscopic theory, is at best good for heuristic \npurposes\\footnote{In a sense, analogous remarks in fact apply to theories such as classical electrodynamics, given the\nlack of well defined quantum versions of these theories at present.}.\nUpon attempting to pursue a more even-handed approach to the quantum gravity problem, a question which immediately presents\nitself is how to possibly uncover the postulates for a fundamental theory, when the principles of its two limiting theories\nappear so disconnected from each other.\nFor instance, except for the postulated unitary dynamics and projective measurements, the axioms of quantum theory in their \nusual form make no reference to spacetime concepts, which makes it difficult to connect them to the foundations of relativity.\nThe resolution to this dichotomy lies of course in the recognition that within an even-handed approach, the principles \nof \\emph{both} quantum theory and general relativity are likely to be subtly modified, re-appear in a very different form,\nor even be replaced altogether (for instance, it will be argued later that there are good reasons to regard quantum\nnonlocality of EPR type as a key principle of any reformulation of quantum theory relevant to the quantum gravity problem).\nIn fact, there are also motivations coming more or less from within each theory separately that point in this direction.\nFor instance, it is well known that general relativity predicts the existence of spacetime singularities under fairly \ngeneric and physically reasonable conditions\\footnote{Cf. \\citeN{HawPen}.} and that, in a sense, it thus contains its own \nseed of destruction. It is commonly assumed that any such singularities are removed upon including ``quantum gravity effects'' \nin an appropriate manner, but whether this happens through an ordinary quantization of classical structures such as loops\nor strings (as occurs in the standard approaches), or through something more elaborate, it entails a breakdown of \nclassical general relativistic structure.\nQuantum theory, when considered as a theory of principle (i.e., as formulated in terms of its usual postulates involving an\nabstract Hilbert space, linear operators on this space, etc.) is completely consistent, mathematically speaking (there\nare a few subtleties - e.g. domain problems corresponding to unbounded self-adjoint operators - but these can be overcome\nat the price of slight technical complication - e.g. through exponentiating self-adjoint operators to their unitary associates).\nSo, at first sight, it appears that this feature of quantum theory causes it to be slightly better off than general \nrelativity. This way of representing things is somewhat deceptive however.\nFirst, Einstein's equation certainly admits singularity-free solutions and general relativity, when considered as a\ntheory of principle, is certainly also fully consistent mathematically (as it should be, of course).\nIt is only when certain (arguably physically realistic) \\emph{assumptions} are made on the stress-energy content of\nspacetimes representing for instance black holes or the universe at large, that singularities are generically encountered.\nNow, although it is also true that - \\emph{unlike} the case of general relativity - a large number of physically relevant, \nmathematically consistent solutions to quantum theory is known to exist, these solutions are, with very \nfew exceptions, all \\emph{non-relativistic}.\nThey are therefore physically relevant only to the extent that relativistic effects can be safely ignored (something\nwhich can usually be done in atomic physics, for instance, although even here it may be necessary to include\nrelativistic effects, such as fine structure, perturbatively, i.e., ``by hand'').\nIt is usually held that the incorporation of special relativistic structure into quantum theory requires the introduction\nof quantum \\emph{fields}\\footnote{See in particular \\citeN{Weinberg2}.\nThe point is that, although it is possible to write down relativistic wave equations (as famously first found by \nSchr\\\"odinger, Dirac and others and as later derived in a more systematic fashion by Wigner), such equations necessarily\npertain to \\emph{fixed} numbers of particles and are thus unable to describe situations in which particles are annihilated\nor created. Moreover, all these equations suffer from certain paradoxes associated with negative energy solutions (such as\nKlein's paradox and Zitterbewegung) and more importantly, they give rise to discrepancies with experiments (as evidenced\nby e.g. the Lamb shifts of atomic hydrogen).\nThese difficulties were essentially all resolved by re-interpreting the wavefunctions as classical fields, which were then to be \nsubjected to a (second) quantization procedure.\nQuantum field theory streamlines this process by essentially starting from a classical field theory and performing a\nquantization procedure on this theory only once.}.\nThis common wisdom will not be called into question here, but it is noted that for a quantum theory of fields, the issue\nof mathematical consistency changes rather drastically.\nIn fact, at present, no interacting, four-dimensional quantum field theories are known which are (a) mathematically consistent and (b)\ndemonstrably physically relevant (e.g. non-supersymmetric), while there is actually rather\nstrong theoretical evidence that any non-asymptotically free, four-dimensional interacting quantum field theory is mathematically \\emph{in}consistent,\nbecause of the expected presence of a Landau pole\\footnote{That is, because of the probable presence of a \\emph{finite}\nenergy scale for which the coupling constant of the theory diverges (\\citeN{Landau}).\nWhat can be stated with mathematical certainty is that perturbation theory unavoidably breaks down in the ultraviolet\nfor theories that are not asymptotically free (\\citeN{GelLow}, \\citeN{Weinberg3}).\nSince all known renormalizability proofs are perturbative in character, this unfortunately complicates the issue of\ndetermining the ultimate status of the Landau pole.\\label{Landaupole}}.\nSo, any ``realistic'' quantum theory that nontrivially incorporates special relativity most presumably also carries its own seed of\ndestruction.\nAgain, it is commonly assumed that any inconsistencies are removed by including ``quantum gravity effects'' in an \nappropriate manner (for instance, through introduction of a short-distance cutoff associated with a fundamental\nlength scale), implying a breakdown of (special relativistic) quantum structure.\nOf course, quantum theory is also under pressure because of the embarrassing lack of agreement among practising\nphysicists on what the theory is ultimately about and because of the earlier mentioned partial incompatibility of its\nprinciples with those of general relativity. Concerning this latter point, the following observation is also useful.\\\\\nIn recent years some results have been obtained which seem to indicate that, while application of standard quantization\ntechniques to (the weak-field limit of) general relativity results in a theory that is not perturbatively renormalizable\n(and hence - barring any deeper principle constraining the associated infinitely many free parameters to a finite\nnumber - in a theory that fails to have any predictive power), there may nevertheless be reason to believe in the existence\nof a stable ultraviolet fixed point of the renormalization group flow\\footnote{This would mean that there \\emph{is} \nsome deeper principle and that effectively there are only finitely many free parameters. Following Weinberg, the theory \nis then said to be \\emph{asymptotically safe}. For recent reviews of the question of whether quantum gravity meets this\ncriterion, see e.g. \\citeN{NieReu} or \\citeN{Percacci}.\\label{asysafe}}.\nHowever, what is not typically addressed in these treatments is the question what it means for a gravitational field\nto be ``quantized''. In particular, when promoting a metric field, $g_{ab}$, to an operator, metrical properties, such \nas distances and angles, cease to have definite values until they have been ``measured'' (according to the usual Copenhagen \nview, at least), as a result of which the \\emph{causal structure} of spacetime becomes indefinite.\nBut a crucial condition usually postulated for any quantum field theory is that of \\emph{microcausality},\ni.e., the property that any two quantum field variables $\\Phi$, $\\Phi'$ locally (anti)commute\n\\begin{equation}\n[\\Phi (x) , \\Phi' (x') ]_{\\pm} \\; = \\; 0   \\qquad \\quad \\mbox{for} \\quad (x - x')^2 > 0\n\\end{equation}\ni.e., for points $x$, $x'$ at spacelike separation.\nThis local (anti)commutativity of quantum fields guarantees that local \\emph{observables} constructed out of these \nfields \\emph{locally commute} and this last property, together with the linearity of the standard quantum formalism,\nin turn guarantees that spacelike correlations between quantum fields cannot be used for superluminal communication\\footnote{See e.g. \\citeN{Eberhard}.\\label{nosignote}}.\nSo, even though it may be assumed that causality somehow becomes restored in the ``classical limit'', it is far\nfrom clear that a theory in which $g_{ab}$ is quantized does not suffer from features usually regarded as\npathological, such as signalling.\\\\\nOn pursuing a more even-handed approach to the unification of quantum theory and general relativity, it is probably helpful that both\ntheories can be embedded into wider theoretical frameworks that naturally allow for amendments in each\\footnote{See \nrespectively \\citeN{AshSch} and \\citeN{Will2}.}.\nWhat is crucially needed however are guidelines of some sort to separate physically viable amendments from non-viable ones.\nThis is discussed next.\n\n\n\\section{Empirical vs. Theoretical Principles}\\label{principles}\n\n\\noindent Among the multitude of possible principles that could potentially guide the formulation of a theory of\nquantum gravity, a subclass is naturally singled out by the empirical principles, i.e. those principles which are \nstrongly grounded in observational facts. This division is of course not absolute and some remarks are in order.\nFirst, there is the somewhat obvious point that even the most manifest empirical principles always exist within a \ncertain theoretical context (which, according to any reasonable criterion, may still be trivial). \nOr, put slightly differently, observations, as the slogan goes, are always ``theory laden'' - something which is in \nparticular true for modern precision experiments in particle physics and cosmology\\footnote{For a recent discussion of this\npoint in the case of particle physics, see \\citeN{Lyons}. Against this background, one may attempt to assign a measure of \ngenuineness to empirical principles, depending on the amount of non-trivial theoretical context involved.\nFor instance, the equivalence principle would obviously classify as an empirical principle pur sang (by \nany reasonable measure), whereas many entries listed as observational facts in the \\emph{Review of Particle Physics} would \nfall considerably short of approaching this status.\nIt is worth noting in this regard, that the four principles proposed in section \\ref{QGprin} are also genuinely empirical\nin the sense here intended.\\label{moderntests}}.\nFor instance, a well established observational principle in cosmology is that, on the largest astronomical scales,\nthe universe is (spatially) homogeneous and isotropic to a very high degree\\footnote{It is recalled that \na spacetime $(M,g_{ab})$ is said to be (spatially) homogeneous if there exists a \none-parameter family of spacelike hypersurfaces, $\\Sigma_t$, $t \\in \\real{}$, which \nfoliate the spacetime in such a way that for each $t \\in \\real{}$, and for any two points in $\\Sigma_t$, there exists an isometry,\nwhich maps one point into the other point. In addition, $(M, g_{ab})$ is said to\nbe (spatially) isotropic at each point, if there exists a timelike vector field, $u^a$\n(representing the normalized tangent field of a family of observers), such that \nthere exists a subgroup of the isometry group of $(M, g_{ab})$, which for each\n$p \\in M$ acts transitively on the space of vectors orthogonal to $u^a$ at $p$,\nbut which leaves $p$ and $u^a$ fixed.\nIn fact, spatial isotropy at each point implies spatial homogeneity, but not vice versa (cf. \\citeN{Wald1}). \\label{isotropynote}}. \nIt is clear however, that no matter how sophisticated and accurate cosmological tests become, they will be restricted\nto our own galaxy, for any foreseeable time to come. In other words, in extending the factually observed isotropy and \nhomogeneity to an empirical principle for the universe at large, it is assumed that our (galaxy's) position in the \nuniverse is not privileged. But this assumption has of course been generally accepted as increasingly reasonable in \nscience from the time of Copernicus onwards (in fact, nothing short of a scientific revolution would result were this \nassumption proven wrong one day). So, while it is certainly true that observational principles always have to be interpreted \nagainst a certain theoretical background, it appears slightly pedantic (or at very least scientifically counter-productive) to \nstress this too much.\nA second point about empirical principles concerns the fact that if they are taken to be crucial in guiding theory selection\n- as is done here for the case of quantum gravity - some readers may protest that theories are usually\n\\emph{underdetermined} by observational data. That is, there will always be more than one theory consistent with a given set \nof such data and an objective basis is thus lacking, so the argument goes, for choosing one particular theory from a \nnumber of different empirically equivalent rival theories. \nAs is well known however, this argument is only effective if observational data are taken as the \\emph{sole} criterion for \ntheory selection - something which is certainly not intended here\\footnote{Other possible criteria frequently imposed\nin this regard include (amongst others) intelligibility, development potential and (mathematical) naturalness\/aesthetics. For further discussion\nof these issues, see e.g. \\citeN{Cao}.}.\nThis point will be further illustrated in subsection \\ref{GRandstmod} in terms of the equivalence principle and the role\nit played in the development of general relativity.\nA final and related issue concerns potential guiding principles which are not empirical, i.e., the ``theoretical principles''.\nSuch principles are supported by observations only in the most minimal sense, in that they should not lead to conflicts \nwith experiments. For instance, the ``gauge principle'', which is fundamental to the standard model of particle\nphysics belongs to this class. As will be recalled in subsection \\ref{QTandstmod}, all known, nontrivial arguments for \nthis principle are purely theoretical in character (arguments, whether they be empirical or theoretical, are here referred\nto as nontrivial, if they are \\emph{direct} arguments; a posteriori one could always argue that some principle is\nindirectly supported if it occupies a central place in some well-confirmed theory, but such arguments are regarded as trivial).\nAlthough theoretical principles are not necessarily invalid (in fact, as just recalled, they are in general even indispensible\nfor empirical principles to be effective for theory construction), their a priori foundational significance should be viewed\nwith proper caution in any empirical science. In particular, care should be exercised to avoid what might be called\n``epicycle'' principles.\n\n\\subsection{General Relativity and the Standard Model of Cosmology}\\label{GRandstmod}\n\n\\noindent A prime historical example of an observational fact being paramount to the formulation of a new theory\nis provided by the (weak) equivalence principle, i.e., the fact that, ignoring air resistance, (test) bodies in a \ngravitational field fall at the same rate, independent of their composition and internal structure.\nAs is well known, Einstein took this principle as a key hint for obtaining a description of gravitational phenomena \nin accordance with relativistic principles, but in terms of a generalized spacetime geometry \n(rather than to simply try and shoehorn these phenomena into the framework of special relativity).\nIt is instructive to briefly recall the essential steps involved.\nImagine oneself in the year 1905, just following the announcement of special relativity, and suppose one were to attempt\nto describe gravity in terms of basic relativistic ideas.\nWould it be reasonable to indeed pursue development of a \\emph{special} relativistic theory of gravity ?\nThis may seem difficult to answer, given our present-day biasses, but the equivalence principle actually provides a \npowerful hint here.\nThat is, it is exactly as a consequence of this principle that an observer who is located\nwithin an effective vacuum and who is studying the motions of test bodies inside a freely falling laboratory (treated as another \ntest body, thus ignoring possible tidal effects) in the absence of non-gravitational, external forces, would validly \nclassify these motions as \\emph{inertial} (i.e., unaccelerated) motions.\nThus, within any pure special relativistic theory of gravity, the equivalence principle implies that there are \\emph{two}\ndistinct classes of preferred motions in spacetime, the usual inertial motions and the ``free-falling'' motions.\nOf course, the inertial motions of Minkowski spacetime precisely correspond to the geodesics of that spacetime, whereas \nthe free-fall trajectories lack such a clear geometrical interpretation within a special relativistic context. \nBut this suggests that spacetime is perhaps only ``locally Minkowskian'' and that gravitational effects are somehow to be \nattributed to spacetime structure, with the free-fall trajectories corresponding to the geodesics of a more general spacetime geometry. \nIn fact, it is possible to considerably strengthen this intuitive argument and present a powerful case that gravity is \nnecessarily a curved spacetime phenomenon if the so-called \\emph{Einstein Equivalence Principle} (EEP) is satisfied\\footnote{See in \nparticular \\citeANP{Will1} \\citeyear{Will1,Will2}.\nEEP consists of the (weak) equivalence principle together with two additional empirical\nconditions, usually called ``Local Lorentz Invariance'' (LLI) and ``Local Position Invariance'' (LPI), and which essentially\nboil down to a local restatement of the relativity postulate for (test body) systems undergoing free fall. See also \\citeN{Holman2}.\nYet another equivalence principle, the so-called \\emph{Strong Equivalence Principle} (SEP), consists of the extension of EEP\nto include also situations involving gravitational self-interaction effects. There is some evidence that general relativity is singled\nout uniquely by SEP. That is, of all so-called \\emph{metric theories of gravity} - i.e., theories able to incorporate EEP -\nit appears that only general relativity can also incorporate SEP.}.\nSince there is overwhelming experimental evidence that the latter is indeed the case, it is thus possible to draw some\nfar-reaching conclusions about the physical nature of gravity, space and time on an almost purely empirical\nbasis as a result\\footnote{For a relatively recent discussion of the experimental status of LLI and LPI, see \\citeN{Will2}.\nFor other lines of argument casting strong doubt on the possibility of incorporating gravitation within \nthe framework of special relativity, see e.g. \\citeN{Penrose8}, \\shortciteN{MTW}.}.\\\\\nAs is well known of course, in concretely arriving at his field equation, Einstein also made use of other, non-empirical\narguments, most notably the principle of ``general covariance'' and ``Mach's principle''.\nAccording to the most common understanding, the principle of general covariance expresses the form invariance of the\nequations of physics with respect to general coordinate transformations.\nBut as was already clearly realized as early as 1917, \\emph{any} physical theory formulated in terms of tensor fields\nis generally covariant in this sense, and so, as a criterion for theory selection, the principle of general covariance\nin this particular form is rather useless.\nGeneral covariance may also be understood in a more restricted sense however, namely, as imposing the condition of \n``background independence'' (or, equivalently, ``no prior geometry'') and in this form it can (and indeed did in the\ncase of general relativity) act as a sieve for separating viable from non-viable theories of gravity\\footnote{For some further \ndiscussion of these two distinct usages of the principle, see \\citeN{Holman3}.}.\nSimilarly, Mach's principle is based on the intuitive idea that the concept of inertia should somehow \\emph{emerge} \nfrom the various interrelationships between all material bodies present and should thus (in principle) only be meaningful \n\\emph{relative} to the entire matter content of the universe (or, in terms of Wheeler's aphorism : ``mass there governs inertia here'').\nUnfortunately however, it has proven difficult to implement this idea in a mathematically precise way, something \nwhich has generated a vast literature on the meaning and correct implementation of Mach's principle, as well as the question \nof whether general relativity is actually a Machian theory.\nNevertheless, it appears to be generally accepted nowadays that an appropriate form of Mach's principle is in fact contained within general \nrelativity, as exemplified for instance by the experimentally confirmed Lense-Thirring effect (i.e. the ``dragging of inertial frames'') \nproduced by rotating matter and, more generally, by the fact that spacetime geometry - as encapsulated in the metric - is \npartly determined by the matter content of spacetime via Einstein's equation.\\\\\nThe point of the foregoing remarks is to emphasize that, in addition to paying due attention to a key empirical\nprinciple, (i.e., the equivalence principle), the choice to consider some descriptions of physical reality to be \nmore intelligible than others merely on theoretical grounds (for instance, descriptions with a certain ``Machian symmetry'' \nbetween causal agents, as opposed to descriptions that lack such symmetry), was also of crucial importance in fleshing\nout the detailed theoretical structure of general relativity.\nHowever, although of crucial importance, the theoretical arguments were most definitely not \\emph{as} important as the\nempirical principle. First, this is just a \\emph{historical} fact about the actual development of general relativity.\nAs is well known, after formulating his version of the equivalence principle (the ``happiest thought'' of his life),\nEinstein went on to consider its ramifications in connection to spacetime geometry.\nIn particular, according to his famous rotating disc argument\\footnote{\\citeN{Einstein3}.}, a rigid disc at rest with\nrespect to a uniformly rotating coordinate system exhibits non-Euclidean features (i.e., because of Lorentz contraction, \nthe ratio of the circumference of the disc to its diameter appears smaller than $\\pi$ to a non-rotating observer) and the\nequivalence principle then suggests that spacetime itself should have a non-Euclidean character in the presence of a \ngravitational field\\footnote{This latter conclusion was reached shortly afterwards by Einstein and Grossmann \\citeyear{EinGro}, \nwho also generalized the flat metric, $\\eta_{ab}$, of special relativity, to a general curved metric, $g_{ab}$. \nIn fact, the primary role of the equivalence principle in arriving at these conclusions was explicitly acknowledged by these authors :\n\\begin{quote}\nThe theory described here originates from the conviction that the proportionality between the inertial and the gravitational\nmass of a body is an exact law of nature that must be expressed as a foundation principle of theoretical physics. \n\\end{quote}\n(as quoted in \\shortciteN{MTW}). Strictly speaking, this line of argument requires (non-empirically) extending the \nequivalence between uniformly accelerating reference systems in the absence of gravity and stationary reference systems \nlocated inside a uniform gravitational field to the case of arbitrary reference systems and gravitational fields. Nevertheless, it is obvious that the starting point of the whole deduction\nwas firmly grounded in an empirical fact, while it was seen earlier that from a modern vantage point the role of\nnon-empirical considerations can be substantially reduced, in as much as EEP - which is fully empirical - implies that gravitation is necessarily described by a metric theory.}.\nApart from the factual historical course of events, a different (but related) reason for attributing priority to the\nequivalence principle is that it is difficult to see how to even make sense of the other, i.e. theoretical, arguments\nwithout first adducing it as support for curved spacetime geometry (i.e., what could be the meaning of general covariance\nin the sense of ``no prior geometry'' in this context ?)\nFinally, it is a simple truth that, on general grounds, genuine empirical arguments (cf. footnote \\ref{moderntests})\nare more robust than their theoretical counterparts. That is, even though an empirical principle may, strictly speaking, break down at some point -\nas experiments become more refined and accurate - it is more likely to always remain true within its proper context.\nBy contrast, non-empirical arguments, without denying their fruitfulness for theory development (as just pointed out\nand as illustrated by the history of science more generally), are arguably more susceptible to \\emph{fundamental} flaws. \nThat is, although they may also be viewed as always remaining ``true'' as far as the\nempirical structures entailed by them approximately remain true (i.e., in the sense just noted), they may become\ndeeply flawed from a foundational perspective and due caution is therefore warranted in connection to them (as \nthe history of science also shows; one recalls Kant's famous ``synthetic a priori'' statements about e.g. the Euclidean nature\nof space in particular).\\\\\nA second example of an important observational principle connected with general relativity (in particular\ncosmology), is the already mentioned isotropy of the universe at the largest astronomical scales.\nThe strongest support for this principle comes from measurements of temperature fluctuations in the 2.7 K \n\\emph{Cosmic Microwave Background (CMB)}, which, according to Big Bang nucleosynthesis, is the\ndistribution of photons left over from ``decoupling time'', $\\tau \\simeq 380.000$ years, sufficiently cooled down\nto compensate for the universe's overall expansion.\nIndeed, the extraordinary uniformity of the CMB (i.e., to a few parts in $10^5$) implies that the universe must have\nbeen highly isotropic at that time, as the minute temperature variations in the CMB are representative measures\nto a similar degree of accuracy of matter density fluctuations at that epoch\\footnote{See \\shortciteN{Spergel}. \nIn fact, the CMB data give rise to the most accurate agreement between an observed intensity spectrum and a theoretical \nblack-body radiation curve (as given by Planck's famous formula) known in observational science.}.\nThe crucial point is now that isotropy is a mathematically highly restrictive condition, which fixes the cosmological \nmodel to be of \\emph{Friedmann-Lema\\^{i}tre-Robertson-Walker (FLRW) type}. The latter constitutes the so-called standard \nmodel of cosmology and is completely specified in terms of two fundamental equations, involving essentially only four\nexperimental input parameters\\footnote{More precisely, for a FLRW cosmological model, Einstein's equation reduces \nto two independent components, which can be written as \n\\begin{eqnarray}\n1 \\: + \\: \\frac{k}{H^2 a^2} & = & \\Omega_m \\: + \\: \\Omega_{\\Lambda}                \t\t\t\\label{FLRWsol1}  \\\\\nq \t\t\t    & = & \\frac{1}{2} \\Omega_m \\: - \\: \\Omega_{\\Lambda} \\: + \\: 4 \\pi P\/H^2 \t\\label{FLRWsol2}\n\\end{eqnarray}\nwhere geometrized units were used (i.e., $G_{\\mbox{\\scriptsize N}} = c = 1$) and \nwhere $a := a(\\tau)$ denotes an overall scale-factor of the homogeneous three-geometries, $\\Sigma_{\\tau}$ (cf. footnote \\ref{isotropynote}), \n$H := \\dot{a} \/ a$ denotes Hubble's ``constant'' (the dot representing a time derivative), $q := - a \\ddot{a} \/ \\dot{a}^2$ is the so-called deceleration\nparameter, $k$ is a discrete parameter which equals either $-1$, $0$ or $+1$ (according to whether the surfaces of \nhomogeneity are respectively hyperbolic, flat or spherical), $\\Omega_m := 8   \\pi   \\rho \/ 3H^2$ and \n$\\Omega_{\\Lambda} := \\Lambda \/ 3H^2$ are dimensionless parameters that respectively measure the density of matter, $\\rho := \\rho (\\tau)$,\nand the ``energy of the vacuum'', i.e., the cosmological constant, $\\Lambda$, in terms of the critical density,\n$\\rho_c := 3H^2 \/ 8 \\pi$, and $P$, finally, denotes the pressure of the matter distribution (which effectively vanishes \nfor the present ``dust'' dominated epoch).\nMany contemporary authors take the standard model of cosmology to consist of the FLRW context, together with the ideas that\n(a) the very early universe went through an extremely rapid period of inflation and (b) a substantial contribution\nto the overall matter and energy content of the observable universe comes from ``(cold) dark matter'' and ``dark\nenergy''. Given the controversial status of both inflation (as will become clear shortly) and dark matter (in the light\nof the partial successes of MOND-type models), I prefer to reserve the term ``standard model'' to the FLRW context only\nand to refer to the model just described as the ``concordance model'' (the latter terminology being in agreement also with common parlance).}.\nThus, given general relativity, out of all conceivable cosmological models, empirical considerations essentially\nsingle out uniquely a particular constructive theory as the preferred model.\nThere is, however, a major difficulty with this model, if it is assumed that the universe was ``born'' in a completely generic state.\nThat is, at first glance, the extreme uniformity of the CMB appears highly suspect\nbecause of the presence of \\emph{particle horizons} (which exist ``throughout history'' in all physically relevant FLRW models).\nThis means in particular that there were many regions in the early universe which were out of each other's causal influence, \nalthough somehow the photon distributions in all these regions nevertheless ended up at the same temperature to a few parts in $10^5$ !\nNow, the majority of present-day cosmologists seem to regard ``inflation'' as a satisfactory solution to this conceptual\nproblem, but it will be argued in subsection \\ref{CMBuniformity} that inflationary models generate substantial\nconceptual problems of their own and that, moreover, the successful predictions of such models do not necessarily\nconfirm inflation as such\\footnote{Often, the standard cosmological model is said to exhibit a second major\ndifficulty, for which inflation is then also regularly proposed as a solution.\nThis is the remarkable circumstance that, on average, the universe appears very nearly \\emph{(spatially) flat}.\nAs I have argued elsewhere however (\\citeN{Holman4}), it is rather debatable (at the very least) whether this \nobservational fact constitutes a genuine conceptual problem for FLRW cosmologies. See also \\citeN{Rindler}.}.\nIn fact, it will be argued that the observed near isotropy of the large-scale universe should act as a selectional\nguideline for quantum gravity and will be accounted for in terms of the latter through a lawlike initial condition.\nThus, just as the equivalence principle pointed to a conceptual anomaly within special relativistic gravitation theory,\nthe extreme uniformity of the CMB appears to tell us something similarly profound about fundamental physics\n(according to this line of thought, inflationary models would then be quite analogous to initial attempts to shoehorn \ngravitational physics into the framework of special relativity).\n\n\n\\subsection{Quantum Theory and the Standard Model of Particle Physics}\\label{QTandstmod}\n\n\\noindent Just as the equivalence principle does not accord well with special relativistic physics, there were\nseveral observational facts in the late 19th, early 20th century - such as the spectral distribution laws of black-body\nradiation and the interference patterns built up from weak, single-photon sources - that did not really fit into (or were even\nin direct conflict with) the generally accepted theories of mechanics and electrodynamics at the time.\nAnd just as the equivalence principle played a key role in the development of a revolutionary new theory of gravity,\nspacetime and dynamics for the ``macroscopic'' physical realm, these empirical facts were paramount to the development\nof a radical new theory of matter and dynamics for the ``microscopic'' physical domain, viz. quantum theory,\nwhich finally rose on the scene in the late 1920s, as an amalgam of many earlier obtained, partial results by different \ninvestigators. As recollected in the introduction - and in sharp contrast to the situation for general relativity - \nhowever, the final form and physical interpretation of quantum theory was strongly influenced by positivist ideas, \nrather than by a desire to obtain an intelligible understanding of the quantum world at a deeper level.\nIndeed, the very notion of such a world was - and still often is - generally regarded as meaningless and any attempts \nto obtain a further understanding of it were explicitly advised against.\nIn addition and somewhat complementary to this, in the case of quantum theory there simply was no guiding empirical\nprinciple analogous to the equivalence principle.\nFor instance, Planck's quantum hypothesis, even though on extreme secure experimental footing in all relevant cases, was at\nthe time a complete shot in the dark and is moreover not a genuine universal principle (as observables may still have\ncontinuous eigenvalues).\nSimilarly, Heisenberg's uncertainty principle is not really an empirical principle, but rather a feature implied by\nthe quantum formalism that is often invoked (not infrequently in a somewhat cavalier fashion), to theoretically account \nfor phenomena such as virtual quanta.\nThe only empirical principle that played a role in the actual process of formulating quantum theory and which could \nultimately prove \\emph{somewhat} akin to the equivalence principle (even though, according to the views presented\nhere, it will \\emph{not} eventually do so) is the \\emph{principle of linear superposition}.\nIndeed, although at present still at the ``pre-E{\\\"o}tv{\\\"o}s'' stage, if confirmed at a sufficiently macroscopic level\n- and sophisticated experiments are currently under way to attempt to bring light into this matter (see e.g. \\shortciteN{RomeroIsartcs}) -\na strong incentive would be provided to view quantum theory as the ``whole truth about the world'' and to consequently\nadopt an essentially Bohrian, ``extreme statistical interpretation'' - or rather non-interpretation - of it\\footnote{Indeed,\nas has been emphasized by Leggett, any attempts to further elaborate upon the theory's interpretation (be it in terms\nof ``consistent histories'', ``Bohmian trajectories'', or something even more esoteric) would then seem to amount to little more than ``verbal\nwindow dressing'' (cf. pp. 79, 232 in \\citeN{Schlosshauer}).\nFor somewhat similar views, see also the remarks of Mermin on p. 103 of the same reference.}.\\\\\nAccording to the ideas underlying the present work however, a possibility that should seriously be reckoned with is that\nthe outcomes of such experiments will prove to be negative at some point, so that quantum theory is simply not the whole\nstory.\nIn addition to the motivations already given (i.e., the absence of a consistent theory of quantum \ngravity at present, the lack of consensus on how to interpret quantum theory and the partial incompatibility of \nits principles with those of general relativity), a reason for suspecting this to be true comes from high-energy physics.\nThat is to say, quantum theory was initially developed under the assumption of atomism of some form, but has subsequently\nbeen applied to probe the structure of matter at ever deeper levels, something which resulted in the most sophisticated,\nmicroscopic theory of matter and interactions to date, i.e. the $\\mbox{SU}(3) \\times \\mbox{SU}(2) \\times \\mbox{U}(1)$ \nstandard model of particle physics.\nYet, the assumption that a theoretical formalism that was essentially extracted as an ``instrument'' from empirical data\nat the relatively low energies of atomic physics still successfully applies at ever higher energy scales (or even at\nwhatever scale quantum gravity is supposed to make its entry), each time uncovering more sublevels in the process,\nresulted in a theoretical edifice that has become somewhat unwieldy\\footnote{For instance, according to the\nreceived view it is necessary to stabilize the standard model against radiative corrections if its Higgs sector\ncontains relatively light scalar particles (as the results from LHC now in fact seem to have \ndemonstrated) and the most popular mechanism to do this apparently is supersymmetry.\nThis however tends to introduce a great number of extra parameters (typically more than $100$), many of which -\ngiven the recent LHC findings - need to be fine-tuned, thereby giving rise to the very same problem that supersymmetry\nwas intended to resolve in the first place.}.\nAs a constructive quantum theory, the standard model is strongly suggested as the simplest concrete model meeting\n(i) a few basic empirical facts (such as the chiral asymmetry exhibited in particle interactions, the \\emph{global} $\\mbox{SU}(2)$ \nweak isospin and $\\mbox{SU}(3)$ colour symmetries), which partially determine the menu of relevant fields, (ii) the so-called \ngauge principle and (iii) the requirement of (perturbative) renormalizability\\footnote{See e.g. \\citeN{Weinberg}.}.\nNow, in contrast to the principles underlying the standard model of cosmology - which, as noted previously, are empirical and\nactually \\emph{fix} that model - the latter two conditions are of an entirely theoretical nature.\nMoreover, unlike the theoretical principles that played a role in the development of general relativity - which,\nas will be recalled from the previous discussion, are still taken to be valid (in some form) for theory development today \n- the gauge principle has a somewhat dubious status in this regard, whereas it is of course well known (as in fact also follows from earlier\nremarks; cf. footnotes \\ref{Landaupole}, \\ref{asysafe}) that renormalizability is no longer taken to constitute a very useful\nselection principle these days\\footnote{A frequently given argument for fundamental gauge invariance, is that it is\nan implication of including a \\emph{global} symmetry - the presence of which may be dictated on essentially empirical \ngrounds - in the theoretical description in a manner consistent with relativity.\nThere are different versions of this argument - the most advanced being due to Weyl -\nbut none of them is actually conclusive. See e.g. \\citeN{Holman2} for further discussion of these issues.\nIt is of course true that features such as the necessary appearance of gauge potentials in the field equations\nin the non-Abelian case, or the empirical Aharonov-Bohm effect, indicate that the whole notion of gauge is far\nmore subtle than could have been anticipated from an investigation of merely the classical, Abelian case.\nAs of yet however, these features only indicate the need to better understand this notion and they do not establish a\ngauge principle in any form.\nAdditional support for these remarks also comes from the fact that in string theory - presumably still the most popular \napproach to go beyond the standard model of particle physics at present - gauge invariance is no longer taken to constitute\na fundamental principle (cf. \\shortciteN{GSW}).}.\n\n\n\\section{Quantum Gravity as a Principle Theory}\\label{QGprin}\n\n\\noindent If the discussion in section \\ref{principles} is taken to convincingly demonstrate that appropriate \nempirical principles are required for deducing the key structural properties of quantum gravity (assuming history\nto be of guidance in this respect), which principles should consequently be adopted to fulfil this purpose ?\nIn the following subsections, four principles are proposed which (a) are on very firm ground experimentally (in the\none case where this may be less obvious, solid arguments will of course be presented), (b) arguably connect gravity \nto the quantum in a nontrivial fashion and (c) are deemed to be of crucial importance in the search for a \ntheory of quantum gravity.\nMoreover, in each case, it is explained, where relevant, how the principle in question could be implemented \nmathematically in such a theory (section \\ref{epi}), why it is considered to be of fundamental significance and also why contemporary \naccounts of it are insufficient.\nAt the risk of repetitiveness, it is stressed once more that the principles listed here are not necessarily foreseen\nto function as principles of quantum gravity in the true sense of the word (although they will of course have\nto be accommodated in some form in any eventual formulation of the theory).\n\n\\subsection{Quantum Nonlocality}\\label{qnonloc}\n\n\\noindent Probably the most enigmatic feature of the quantum world is its nonlocality.\nIndeed, assuming the (sub)microscopic world of molecules, atoms, quarks, leptons, $\\cdots$, to be as real as the\n``ordinary'' macroscopic world directly accessible to our senses, the violation of Bell-type inequalities - exactly\nin accord with the predictions of quantum theory and as confirmed by countless experimental tests performed over the\npast few decades - has forced theorists to give up on the hope of ever describing this world in a fully local manner\n(there are still a few minor, most presumably soon-to-be-closed loopholes to this conclusion, but with one exception,\nwhich is briefly addressed in the next subsection - these will not be discussed here\\footnote{For a recent discussion\nof some generally acknowledged loopholes and how they could be closed, see e.g. \\citeN{Schlosshauer}.}).\nIn making this statement, it is important however to distinguish between two major notions of the term ``locality''.\nAccording to one notion, locality is synonymous with \\emph{relativistic causality}, i.e., the absence of superluminal\nsignalling (``no-signalling''), which is usually taken to be a fundamental tenet of relativity theories (i.e., in order\nto avoid certain paradoxes already referred to earlier - more on this in subsection \\ref{qindet}).\nA second type of locality corresponds to the concept of supervenience (or, what boils down to the same thing, separability),\ni.e., the feature that the properties of a composite system fully ``supervene'' on the properties of its subsystems.\nIn its usual ``minimalist'' interpretation, quantum theory, by virtue of entanglement, obviously fails to be local in\nthis sense. What is demonstrated by Bell-type experiments is that \\emph{any} more refined interpretation\nof the theory, in which subsystems always have (stochastic) properties in all relevant respects, must be nonlocal\nin one of the two senses just outlined, something which is generally interpreted in terms of some sort of non-separability, \nexactly as entailed by the quantum formalism\\footnote{Cf. footnote \\ref{nosignote}.\nMore precisely, the two types of locality correspond to technical conditions known in the literature as \\emph{parameter} \nand \\emph{outcome independence}, respectively, and the violations of these conditions are often said to be of\na ``controllable'' and ``uncontrollable'' variety, respectively. See e.g. \\citeN{Bub} and references therein.}.\nFurthermore, the nonlocality demonstrated is not to the maximum extent possible, i.e. as allowed by locality in the\nfirst sense. In fact, there is a large class of no-signalling theories which exhibit nonlocal correlations stronger\nthan allowed for by quantum theory, but these so-called ``super-quantum'' theories do not conform to nature\\footnote{The\nstandard way to study such theories is through introduction of the so-called nonlocal PR box - after \\citeN{PopRohr}.\\label{PRbox}}.\nThe nonlocality displayed by Bell-type experiments is indeed in very impressive agreement with the predictions \nof quantum theory and this seems to tell us something profound.\\\\\nEven if nonlocality is an ineliminable part of quantum theory - as the best available evidence indicates it is -\nhow could that have anything to do with quantum gravity ?\nThe link is in fact quite straightforward - although hardly ever made, to my knowledge.\nIndeed, according to general relativity, gravity is directly responsible for spacetime causal structure (even in\nsituations where the ``force'' of gravitation is not directly relevant).\nSo the fact that nature exhibits nonlocal correlations, precisely in accord with quantum theory (i.e., and not with\nany super-quantum theory), suggests a profound connection between gravity and the quantum.\nIn addition, even though not in direct conflict with relativistic causality, nonlocal quantum correlations at least\nbring to the surface an existing \\emph{tension} between quantum theory and relativistic ideas (this is even more\nso for interpretations in which state vector reduction is taken to be a real process).\nIt may be that this tension is again indicative of something profound and that a theory of quantum gravity is\nrequired to ultimately resolve it.\\\\\nThe fact that quantum nonlocality is here taken as a key principle for uncovering the structure of quantum\ngravity does not contradict anything that was said in subsection \\ref{QTandstmod}. \nThat is, even though nonlocality is an \\emph{implication} of quantum theory, rather than a foundational principle, \nthis does not mean that it does not have crucial foundational \\emph{significance}.\nAccording to the above discussion it clearly does and moreover not just for quantum theory\\footnote{It is worth noting \nin this regard that quantum nonlocality has been proposed as a (foundational) principle by other investigators as well.\nSee e.g. \\citeANP{PopRohr}, \\emph{ibid}.}.\n\n\n\\subsection{Irreducible Indeterminacy}\\label{qindet}\n\n\\noindent Since ancient times, thinkers have asked whether Nature is completely governed by deterministic rules, or whether \nit in fact contains processes that are \\emph{intrinsically} random.\nAccording to the classical view -- as expressed most clearly in Laplace's conception of the universe as a giant\nclockwork -- true chance events do not exist and it is a commonplace that this view was decisively overthrown by quantum\ntheory in the early twentieth century.\nGiven the lack of consensus on how to interpret quantum theory however, it is reasonable to ask - especially given\nthe level of technicality of typical modern discussions on quantum foundations - whether such a conclusion might\nperhaps not be premature.\nIn this respect, it is quite remarkable that it is possible to present two rather straightforward, empirical arguments,\nwhich are at first sight completely unrelated and which \\emph{both} point to fundamental randomness.\nHowever, as also emphasized before already, it is important to take note of the fact that observational statements\nare always relative to certain - usually implicit and possibly trivial - theoretical background assumptions.\nNevertheless, it will be argued that in this case the background assumptions are so natural - in fact, arguably to at\nleast the same extent as the Copernican assumption underlying the observed uniformity of the CMB mentioned earlier -\nthat to question them not only seems contrived, but also amounts, in essence, to questioning the very validity of the \nscientific method.\nConsequently, speaking of irreducible indeterminacy as an \\emph{empirical} principle is argued to be fully justified.\nWhat then are the arguments that lead to this conclusion ?\\\\\nFirst, as has only been clearly realized relatively recently, given relativistic causality, violations of Bell type\ninequalities can in fact be utilized to generate true random numbers\\footnote{\\shortciteN{Pironiocs}.}.\nThis is based on the fact that in any deterministic theory nonlocal correlations can in principle be used for signalling\\footnote{\\citeN{Valentini}.}.\nHence, if relativistic causality is to be respected, the outcomes in any Bell-type experiment cannot be predetermined,\nbut are in fact generated at whatever event constitutes an act of measurement.\nNow, it may be surmised, as is indeed often done, that arguments for nonlocality based on Bell inequality violation\nalways involve an assumption somewhere that a human experimenter is free to choose some particular measurement setting.\nIn fact, no such assumption is necessary, as a pair of random number generators may do the job, but the point\nis clear : some presumption of ``free will'' or initial randomness seems to be required for the argument to work\n(this was the caveat mentioned in subsection \\ref{qnonloc}).\\\\\nAlthough this point is not in itself contested here, there are strong grounds to discard the possible invalidity of such\na presumption as a \\emph{serious} loophole in the argument for nonlocality (although it would of course make the argument \nfor indeterminism circular, logically speaking - more on this shortly).\nFirst, even though the notion of ``free will'' is frequently used in a vague sense and has probably been a source of\ndebates since the dawn of humanity, it is an often overlooked fact that in one important sense, this notion is central\nto the whole scientific method\\footnote{See also \\citeN{HawEll}, p. 189, for a clear statement of this point.}.\nSuppose that human experimenters always found that a certain physical magnitude (for instance, the Hubble radius,\nor the level of $\\mbox{CO}_2$ gas in Earth's atmosphere) were increasing.\nNormal scientific practice would then lead them to infer that the physical magnitude in question \\emph{is} in fact\nincreasing. But in a truly deterministic universe, a giant ``conspiracy'' on the part of nature could lead human\nexperimenters to think that there is an increase of the magnitude, when in actual fact there is none.\nNow whether anyone thinks it plausible that our universe is \\emph{actually} like this is perhaps best left as a personal\nmatter, but human scientists surely have to \\emph{behave} at least as if they do not~!\nA related issue is that in standard deterministic physical theories, initial data can be specified arbitrarily.\nIn a ``superdeterministic'' universe in which Bell inequalities are violated however, the initial conditions\n(specified at some arbitrary past, spacelike Cauchy surface) would have to be delicately constrained somehow by\nthe very fact that Bell inequalities are always found to be violated whenever put to the test in the future.\nAs this essentially introduces a teleological element into physical theory, it is quite questionable what\nis actually gained this way, if the objective of the whole enterprise was to retain locality to begin with.\nFinally, concerning the issue of circularity in the argument for randomness, it was recently shown that, although indeed\nsome initial randomness is necessary - so that, strictly speaking, the argument is indeed circular - this initial\nrandomness can be made \\emph{arbitrarily small} (instead of generation of randomness, a more\nappropriate term is therefore randomness \\emph{expansion})\\footnote{\\citeN{ColRen}.}.\\\\\nA second argument for irreducible randomness is not (directly) related to quantum theory and does not appear to be\nvery well known. It is based on the simple observation that in a strictly deterministic physical theory, all of the\n``biological complexity'' - i.e., as exhibited by life in its manifold appearances - clearly present in the universe today\n(or rather, in one very tiny region of the universe), would also have to be present in any prior state of the universe. \nThus, if the universe is represented mathematically as a foliation of spacelike Cauchy surfaces, $\\Sigma_{\\tau}$,\ncomplete knowledge of all physical conditions at instant $\\Sigma_{\\tau_0}$, characterizing the present state of the\nuniverse in all its bewildering complexity, would in principle allow one, in accordance with Laplace, to fully\n``retrodict'' the state of the universe at any (non-singular) prior instant.\nBut this seems highly unlikely, as it would entail that all the complexity observed in the current state would for\ninstance have to be ``conspirationally'' hidden in the state at decoupling time, $\\tau_{\\mbox{\\tiny d}} \\simeq 380.000$ years,\nsince observations of the CMB tell us that this state was essentially featureless.\nMoreover, upon evolving the universe further and further back in time, this complexity would still be present (conspirationally\nhidden or not), at, say, $\\tau_{\\mbox{\\tiny EW}} \\simeq 10^{-12} \\, \\mbox{s.}$, when according to the standard model\nof particle physics, the universe entered its Higgs phase and all particles of ordinary matter acquired their masses.\nBut regardless of whether the Big Bang actually marked a true beginning of things in some definite sense,\n$\\Sigma_{\\tau_{\\mbox{\\tiny EW}}}$, for all physical purposes may be viewed as a true initial data surface (since ``before'' \n$\\tau_{\\mbox{\\tiny EW}}$ essentially did not exist, in view of the absence of any physical time-tracking entities -\non account of all particles being massless).\nThe puzzle is then how to account for an initial state with that much complexity without falling back on some form\nof divine intervention.\\\\\nAlthough it appears to me that the argument for the presence of an irreducible statistical element within the fabric \nof nature just presented is basically correct, its main weakness is of course its qualitative character.\nFor instance, it is clear that the notion of bio-complexity refers to something different from the\nusual notion of complexity of a system in the sense of Kolmogorov (i.e., essentially as the shortest algorithm that \n``generates'' the system when transcribed into a sequence of bits), as systems of maximum Kolmogorov complexity are\ncompletely random - something which is evidently not true for living organisms.\nIn order to capture the kind of complexity encountered in living organisms some sort of trade-off is needed between complete\nrandomness and complete order and it appears the appropriate notion is that of ``effective complexity'', which may roughly \nbe characterized as the length of a ``concise'' description of the system involved\\footnote{\\citeN{GellMann}, \\citeN{GellMannLloyd}.\nHere, care should also be exercised to distinguish between systems with a high measure of such complexity and systems which\nmerely have a large amount of ``logical depth'' (such as e.g. mathematical fractals). Basically, the complexity of systems \nof the latter sort is only apparent (i.e. represents some measure of the time or number of elementary operations\nneeded to generate the system from an ultimately simple system).\nAlthough the issue of how much effective complexity is present in terrestrial biochemistry appears to be unsettled, it\nis clear that there is a huge amount of such complexity present in biology (i.e., a huge amount of additional information\nis still needed to characterize biology, given the -- essentially simple -- laws of physics and chemistry).\nThe overwhelming amount of variety in life on Earth is to a very great extent dependent on historical accidents, i.e.\nchance events that have occurred in the course of four and halve billion years of evolution, which cannot ultimately be \\emph{derived}\nfrom lower-level theories such as chemistry or physics.}.\\\\\nAnother issue is that the foregoing argument appears inconclusive if one refuses to view $\\tau_{\\mbox{\\tiny EW}}$\nas an initial data surface for some reason (for instance, because one does not accept the particle physics explanation\nfor the origin of mass). However, in that case the only way of scientifically resolving the issue of \ncomplex ``initial'' conditions without invoking indeterminism, would be to argue that we basically live\nin an infinitely cyclic universe with eternal recurrences. Such a universe might have had some plausibility in \nNietzsche's time, but surely physical science has moved forward since then.\nSo, while it is true that with both above arguments for irreducible randomness, strictly speaking one cannot avoid a \nsmall amount of circularity, it is necessary to look at the \\emph{complete} picture.\nWhen this is done, it is clear that both arguments complement each other (although, admittedly, the first argument\nis more scientifically rigorous at the present time) and that it moreover seems fully justified to actually speak \nof them as empirical arguments for indeterminism\\footnote{It is of course true that the first argument crucially\ndepends on quantum nonlocality and since the latter was already proposed as an empirical principle previously, by\nitself this argument thus does not \\emph{separately} establish indeterminacy as an empirical fact.\nClearly, this is where the second argument comes in. One advantage of proceeding this way could be that if\nquantum nonlocality and randomness were to both appear in the form of actual \\emph{postulates} in a theory of\nquantum gravity, it may be possible to establish relativistic causality as a \\emph{theorem} in a manner that does \nnot rely on the linearity structure of the standard quantum formalism (cf. footnote \\ref{nosignote}).}.\nIn the context of the present discussion, an element of true chance is clearly suggestive of a more fundamental\nunderlying dynamics that somehow unifies the unitary evolution postulate and the projection postulate of standard\nquantum theory. It is then only natural to link such putative dynamics to a theory of quantum gravity\n(this will become clearer upon discussion of the remaining two empirical principles below).\n\n\\subsection{Thermodynamic Arrow of Time}\\label{TDarrow}\n\n\\noindent One of the most basic facts of physical experience - familiar to anyone who has ever tried to \nwatch a movie backwards - is the apparent directionality of time.\nIn particular, according to the (empirical) second law of thermodynamics, the entropy of an isolated system in \ngeneral increases, but never decreases.\nYet, when it is asked where this ``thermodynamic arrow of time'' comes from, our presently most fundamental\nphysical theories seem unable to provide an answer, as they are (taken to be) completely time-\\emph{symmetric}\\footnote{Indeed, \nthe conventional viewpoint appears to be that, on account of respectively (the unproven, but - within its proper context - \nphysically reasonable conjecture of) strong cosmic censorship and ``post-selection'' of ensembles, general relativity and \nquantum theory (including the projection postulate) are both fully time-symmetric theories.}.\nThe commonly given solution to this paradox traces the observed time-irreversible behaviour to initial conditions. \nThat is, a coarse-graining of phase space in terms of volumes with the same ``macroscopic appearance'' is introduced\nand it is argued that, since the volume of a particular coarse-grained state (i.e., essentially the entropy of that state\nexponentiated) is a measure of the fraction of time a system spends in that state, with overwhelming probability, \nphysical systems tend to evolve into ever larger coarse-graining regions, until they finally enter the largest, \ni.e. maximum entropy, region, in which they then stay forever.\nSo on this account, the second law of thermodynamics is more or less a straightforward consequence of the fact that, \nfor some reason, the very early universe happened to be in an extremely low entropy state - as firm empirical evidence\nactually indicates it was\\footnote{See e.g. \\citeANP{Penrose1} \\citeyear{Penrose1,Penrose9}, \\citeANP{Wald2} \\citeyear{Wald4,Wald2}, \\citeN{Feynman}.}.\\\\\nNow, it should immediately be added that the above arguments are somewhat heuristic and some qualification therefore\nseems to be in order.\nFirst, the extreme uniformity of the CMB mentioned before means that the very early universe was \nessentially in a state of thermal equilibrium and therefore, it would seem, essentially in a state of \n\\emph{maximum} entropy. \nAs is unfortunately not always recognized however, identification of a thermal equilibrium state with a \nstate of maximum entropy is only permissible if gravity (and similar long-range effects) can safely be neglected. \nIn essence, gravitational clumping \\emph{always} increases entropy, so that upon following the standard qualitative\naccount of structure formation and stellar evolution from, say, decoupling time onwards, the universe's entropy\nhas since become (vastly) greater.\nIn fact, this process of entropy build-up is saturated by black hole formation.\nThat is, for a given, sufficiently large distribution of matter, the state of maximum entropy is essentially the\nstate in which the entire matter distribution has collapsed into a black hole.\nThus, upon estimating the amount of matter in the observable universe expected to already have collapsed into\nblack holes and using the Bekenstein-Hawking formula for the entropy of a black hole, it is possible to obtain\na semi-quantitative measure of the ``degree of specialness'', i.e. ultra-low entropy, of the very early universe\\footnote{Taking\nthe estimated total entropy due to black holes, $S_0$, in the current universe as an upper bound for the entropy of the\nvery early universe and taking for the maximum entropy, $S_{\\mbox{\\tiny max}}$, the entropy of a black hole the\nmass of the observable universe, Penrose has obtained an upper bound of $e^{S_0 - S_{\\mbox{\\tiny max}}} \\simeq  10^{10^{- 123}}$  for the fractional phase\nspace volume representing the initial state of the universe.}.\\\\\nA second point is that, when applied to ordinary laboratory systems (not essentially involving gravitational degrees \nof freedom), arguments about evolution in phase space of the type sketched above assume time translation invariance \nand ergodicity and it seems quite clear that these assumptions in general fail to hold within a general relativistic context.\nIt also seems clear however that a theory of quantum gravity is precisely what is needed to properly address these\nissues. In fact, it is probably not reasonable to expect general relativity to have a sensible thermodynamics anyway,\nwhereas features such as the Hawking-Unruh effect and the remarkable similarity between the laws of black hole mechanics\n(which are essentially rigorous theorems in differential geometry) and the (essentially empirical) laws of ordinary\nthermodynamics, nevertheless strongly hint at a deep connection between general relativity, quantum theory and \nthermodynamics, that will be part of a future theory of quantum gravity.\\\\\nAssuming that the puzzles involving gravitational entropy will eventually indeed be resolved by such a theory, the\nquestion that obviously remains about the origins of the thermodynamic arrow of time is : \\emph{why} did the\nearly universe have such an extremely low entropy ?\nHere, two basic types of answers can be sought.\nIn one type of approach it is envisaged that the universe basically started off in an as generic state as possible\nand the idea is to demonstrate that dynamical evolution ``generically'' produces a highly symmetric, ultra-low\nentropy state within a sufficient timeframe.\nHowever, unless a major change in the second law of thermodynamics is contemplated, this would mean that the\ninitial state of the (observable) universe had even lower entropy - and was therefore even more improbable.\nIt therefore does not appear, a priori, that this general type of answer can be successful (in fact, the\nreceived view among contemporary cosmologists appears to be that certain inflationary models successfully address\nthe issue just raised, but this view will be contested in subsection \\ref{CMBuniformity}).\nThe radically different alternative type of answer holds that - for essentially the reason just given - it would\nbe ill-guided to seek a dynamical explanation for the ultra-low entropy origin of the early universe.\nRather, this latter feature should be taken at face value and the most natural inference, arguably, is that\nthe initial entropy, for some reason currently not understood, was \\emph{constrained} to be very low.\nMoreover, since quantum gravity arguably \\emph{does} contain a fundamental arrow of time (so as to e.g. exclude \n``white holes'', i.e., the time-reverses of black holes)\\footnote{Cf. \\citeN{Penrose1}.\nAlthough white holes do not appear to be strictly forbidden, any quantum gravity theory that includes these entities \nmust come to terms with a number of serious difficulties afflicted by them. \nAlso, if information is truly lost in the process of black hole evaporation, CPT and time-reversal invariance would \nnecessarily have to be broken in quantum gravity. See \\citeN{Wald4} for a discussion of these issues.}, it seems quite \nplausible that such a constraint will be provided in the form of a \\emph{lawlike} initial condition in quantum gravity.\n\n\n\\subsection{Uniformity of the Cosmic Microwave Background}\\label{CMBuniformity}\n\n\\noindent According to the cosmological ``concordance model'' (which is generally taken by present-day\ncosmologists as the best description of the available data\\footnote{See for instance \\citeN{Ellis2}.}), the very early universe went through an epoch\nof \\emph{inflation}, during which it expanded by a googol (i.e., $10^{100}$) factor within just an extremely short\ntimeframe ($\\simeq \\, 10^{-30}$ s. seems to be a typical order).\nIt appears that the presence of such an epoch is commonly regarded as a solution to the horizon problem mentioned in\nsubsection \\ref{GRandstmod}, the underlying idea being basically as sketched in the previous subsection,\nnamely that the universe started off in a completely ``generic'' state, but that after many e-folding times of\nexponential expansion, thermal processes had managed to effectively isotropize it.\nAs is well known however, this inflationary explanation of the near FLRW geometry of the observable universe suffers\nfrom a number of problems.\nThe most significant of these is that it is very far from clear whether inflation can actually be said to achieve \nits aims. Highly special initial conditions still appear to be needed to ensure that the universe effectively \nisotropizes at sufficiently late times \\emph{assuming} inflation to take place, while it is moreover far from\nclear that it is actually ``probable'' for the early universe to have undergone an inflationary phase\\footnote{Cf.\n\\citeN{Penrose3}, \\citeN{Ellis1}, \\citeN{Turok}, \\citeN{Wald2}, \\citeN{GibTur}.\nOther difficulties of inflation for instance are (a) the physical nature of the scalar field that is supposed to \ndrive the exponential expansion, (b) the fact that a ``generic'' initial state in a time-symmetric theory\nshould in general include white holes, implying gross violations of the second law of thermodynamics and\n(c) the existence of deep trans-Planckian modes in chaotic inflationary models.}.\\\\\nSomewhat ironically, inflationary models have managed to predict quite successfully minute \\emph{deviations}\nfrom exact uniformity that are well borne out by measurements of the CMB.\nIn particular, inflationary models generically predict that the tiny temperature fluctuations in the CMB are\nGaussian and scale-free, which is indeed what is found in precision measurements by for instance WMAP.\nHowever, alternative cosmological models for the very early universe (without inflation) have been found \nwhich lead to these same predictions\\footnote{Cf. \\citeN{HolWald}, \\citeN{SteTur}.}.\nIt therefore seems highly premature - especially also in view of the aforementioned foundational problems - to conclude\nthat the CMB precision data have actually demonstrated it likely that the early universe went through an inflationary phase.\nOn the other hand, on taking the CMB uniformity at face value, it might be envisaged that the universe was somehow\n``born'' in an (almost) isotropic state.\nSuch a scenario has been argued unlikely on the grounds that any ``reasonable measure'' on the space of all geometries\nassigns zero effective probability to FLRW geometries.\nHowever, this does not appear a very convincing argument.\nFor instance, in view of the highly special properties of four-dimensional manifolds (i.e., $\\mathbb{R}^4$ is the only\nEuclidean space that admits ``non-standard differentiable structures'' and it moreover does so infinitely many times\\footnote{See e.g. \\citeN{DonKro}.}),\nit could equally well be reasoned that it is ``extremely improbable'' that spacetime happens to be four-dimensional\nand that one should therefore attempt to seek a dynamical explanation for this common fact of experience as well.\nJust as with the universe's isotropy, it seems more reasonable to take the specialness of $\\mathbb{R}^4$ at face value\nand to attempt to find out what it can further teach us about the nature of physical reality (see section \\ref{epi}\nfor some additional comments on this issue).\nAt any rate, it would appear that a lawlike initial condition within a theory of quantum gravity, that somehow\nenforces isotropy, would offer a reasonable explanation of both the thermodynamic arrow of time and the \nnear-perfect uniformity of the CMB.\n\n\n\\section{Epilogue}\\label{epi}\n\n\\noindent Do the four principles listed previously naturally point to some mathematical scheme that encompasses them ?\nBefore addressing this it is worth noting that the second principle (indeterminacy) appears to have a somewhat special status\nin the sense that it connects \\emph{directly} to the three remaining principles.\nThe link with quantum nonlocality was already explained. As for the two other principles (which obviously also link\ndirectly with each other), the connection arises as follows.\nThe extreme ``orderliness'' of the very early universe - as implied by both general reasoning based on the second law\nof thermodynamics and the extra-ordinary uniformity of the CMB - means that all (quantum) gravitational degrees of freedom\nwere effectively absent in that epoch (since, in general relativity, gravitational degrees of freedom are represented\nby the Weyl tensor, $C_{abcd}$, this has led to the conjecture that vanishing Weyl curvature must be a basic \\emph{law}\nwithin quantum gravity\\footnote{\\citeN{Penrose2}}).\nSomehow these degrees of freedom became ``activated'' in the course of evolution, in conjunction with a loss of ``order''\n(as entailed by the second law) and with the generation of true complexity (in accordance with the arguments given\nin subsection \\ref{qindet}).\nAlso worth mentioning in this regard is the fact that state vector reduction has recently been invoked in \nan attempt to account for the generation of inhomogeneities in a perfectly smooth gas (which are envisaged to then act\nas seeds for the growth of further structure under the influence of gravity on cosmological time scales\\footnote{\\citeN{Sudarsky}.}).\\\\\nIt is not immediately clear how fundamental time asymmetry - such as for instance implied by a lawlike initial condition enforcing low entropy\n(and near-uniformity) - could be implemented mathematically in quantum gravity, since the usual differential equations of mathematical physics\nare time reversal invariant (although see below)\\footnote{For an interesting recent, mathematically oriented discussion on the distinction\nbetween states and laws and how it might (partially) disappear, see \\citeN{Davies}.}.\nAs for quantum nonlocality however, it relates very naturally to the mathematical ideas of \\emph{spin networks}\nand \\emph{twistor theory}\\footnote{For reviews, see \\citeANP{Penrose4} \\citeyear{Penrose4,Penrose6}.}.\nIn fact, the latter was very much developed with quantum nonlocality acting as a guiding principle.\nOne of the central features of twistor theory is that spacetime is a secondary concept, which should\nsomehow (nonlocally) emerge from the more primitive notion of a twistor space. Basically, ordinary spacetime events are\ntaken to correspond to holomorphic objects (i.e., Riemann spheres) in twistor space, the basic elements of which \nrepresent light rays (labelled also by a helicity parameter) in ordinary spacetime.\nFurthermore and quite remarkably, certain aspects of the twistor formalism seem to relate naturally to some of\nthe other empirical principles listed, although it is yet too early to say whether these relations really exist\nor are merely apparent.\nFor instance, the so-called ``nonlinear graviton'' construction (which relates to a twistorial description of certain\n\\emph{curved}, vacuum spacetimes) seems to hint at a nonlinear modification of the quantum formalism, unifying\n(deterministic) Schr\\\"odinger evolution with (indeterministic) state reduction, while developments to resolve the\nso-called ``googly problem'' (which is the problem of finding the right-handed counterpart of the originally left-handed \nnonlinear graviton) have been tentatively described in terms of a fundamental time-asymmetry present in the formalism.\nIt is finally worth noting that the whole twistor approach is very much based on the fundamental fact of experience\n(in accordance with the views expressed here) that ``physical space'' is three-dimensional and that there is one \ndimension of ``time''.\\enlargethispage*{2cm}\nIn fact, the theory derives its strength from that fact, as it does not really work for any other number of spatial\nand\/or temporal dimensions.\n\n\\newpage\n\n\\bibliographystyle{eigen}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nClusters of galaxies are formed as peaks in the primordial density field collapse and subsequently grow through hierarchical clustering. As probes of the high-mass tail of the spectrum of mass perturbations, these clusters allow us to constrain the evolution of structure in the universe. The basic properties of these clusters are largely determined by initial conditions and the dynamically dominant dissipationless dark matter, although baryonic processes can have a non-trivial impact particularly in the group regime. Models of dark matter driven dissipationless collapse are nonetheless broadly successful in predicting relationships between cluster properties, the most common of which being the self-similar model of clusters \\citep{Kaiser}. In this scenario, groups and clusters are identical when scaled by mass, which is known as strong self-similarity \\citep{Bower:1997}. Under strong self-similarity, the slope of the resulting scaling relations is not expected to evolve with redshift, although there is an expected evolution in the normalisation due to the changing density of the Universe. This model provides us with a clear baseline from which to examine the impact of baryonic physics in cluster formation.\n\nAs matter collapses to form these clusters gas experiences adiabatic compression and shock heating, creating a hot intracluster medium (ICM) with temperatures $T>10^6$K, which emit X-rays as a result of thermal bremsstrahlung as well as metal emission lines. This hot gas cools and settles into hydrostatic equilibrium.  Radiative cooling from dense gas in the cores of all but the most massive systems is expected to create a cooling flow that feeds star formation and accretion onto the central super-massive black hole. However, observed star formation rates (SFRs) are orders of magnitude lower than cooling flow estimates would suggest \\citep{Fabian:2002}, resulting in the cluster cooling flow problem. \n\nFeedback from active galactic nuclei (AGN) is often proposed as the solution to the cooling flow problem \\citep{Bower:2006,Croton:2006,Somerville:2008}, by injecting large amounts of energy into the ICM to keep the gas hot. Several other candidates to solve this problem have been suggested such as magnetic conduction \\citep{NarayanMedvedev:2001, Fabian:2002}, Type~Ia supernova feedback \\citep{Sasaki:2001}, and dynamical heating \\citep{Birnboim:2007}, but the current consensus is that none of these are able to sufficiently counteract cooling.  Meanwhile, bubbles of hot gas seemingly created by AGN jets have been observed with sufficient mechanical inflation work to counteract cooling \\citep{McNamara:2007} lending credence to the idea that AGN jet feedback is responsible for the heating of the intracluster medium (ICM), which quenches massive galaxies and solves the cluster flow problem. \n\nModern models of groups and clusters must therefore include the impact of AGN feedback.  Among hydrodynamic simulations, Illustris \\citep{Vogelsberger:2014, Genel:2014}, Magneticum \\citep{Hirschmann:2014}, Horizon-AGN \\citep{Dubois:2014, Volonteri:2016, Kaviraj:2017}, EAGLE \\citep{Schaye:2015}, MassiveBlack \\citep{Khandai:2015}, Blue Tides \\citep{Feng:2016}, Romulus \\citep{Tremmel:2017}, Illustris-TNG(TNG; \\citep{Springel:2018}, FABLE \\citep{Henden}, and \\mbox{{\\sc Simba}}\\xspace \\citep{Simba} all include black hole growth and the energetic output resulting from black hole accretion. A plausible connection between AGN feedback and quenching of star formation was able to be established by EAGLE, TNG, FABLE, and \\mbox{{\\sc Simba}}\\xspace thanks to their success in broadly reproducing the observed massive red and dead galaxy population.  \n\nDue to the difficult nature of modelling AGN, different simulations have approached the problem in a variety of ways.  EAGLE's implementation employs a single mode of AGN feedback with a fixed efficiency, injecting energy thermally at the location of the black hole (BH) proportional to its accretion rate. This model closely resembles quasar feedback and as such may be effective in breaking self similarity but may lack the effect of jets in opposing cooling and quenching star formation.  TNG conversely employs two forms of feedback, with kinetic black hole-driven winds at low accretion rates, and thermal feedback heating gas surrounding the BH at high accretion rates. FABLE employs a modified version of this, in which quasar mode feedback at high accretion rates is stored in the BH until there is enough energy to raise the temperature of a set number of particles by a given amount. Finally, \\mbox{{\\sc Simba}}\\xspace employs high-velocity jets similar to TNG, but does so in a stably bipolar fashion.  The nature of the torque-limited black hole accretion model used in \\mbox{{\\sc Simba}}\\xspace means that BHs inherently grow along galaxy-black hole scaling relations with no need for self-regulation~\\citep{Angles:2013}, allowing for a more collimated AGN feedback to enact galaxy quenching.\n\nObservations allow us to place constraints on these varying forms of AGN feedback at low redshifts, as we explored in \\citet{Robson:2020}.  X-rays provide an efficient and physically motivated method of detection that allows us to probe the potential well of a cluster through the observation of its X-ray luminosity, $L_X$. Observed properties such as $L_X$ and the X-ray luminosity-weighted temperature $T_X$ allow us to establish proxies for the cluster mass through the use of scaling relations. These scaling relations have been studied in great detail for low redshifts \\citep{Arnaud, Vikhlinin, Pratt, Bohringer}.\nWe are then able to use the halo mass to examine the space density of clusters, and more thoroughly test theoretical models of cluster formation. \n\nMoving to higher redshifts, X-ray observations rapidly become more challenging owing to surface brightness dimming, making it difficult to precisely know the evolution of scaling relations with redshift \\citep{Reichert:2011}. Several studies have found self-similar evolution to redshift $z \\sim 1$ driven primarily by the density of the Universe increasing with redshift \\citep{Vikhlinin:2002,Maughan:2006}. Other works however have found evolution departing from that predicted by self-similarity \\citep{Ettori:2004, Branchesi:2007}.  This lack of consensus may arise from difficulties in managing selection bias in samples of high-redshift clusters drawn from different surveys. Future X-ray telescopes such as {\\it Lynx} and {\\it Athena} are expected to allow us to probe larger samples to higher redshifts.\n\nThus it is timely to make predictions for the expected evolution of groups and clusters within self-consistent models including galaxy formation processes. Such simulations can help us to better understand what constraints on key physical processes such as AGN feedback are possible from present data, as well as improvements to be expected from future facilities.  By further examining how the hot gas content of halos evolves over redshift we can hopefully constrain not only the physical processes driving AGN feedback in these dense environments, but also determine how AGN drive the evolution of intragroup and intracluster gas.  Particularly enabled by simulations, studying the evolution of a set of simulated halos over time may help us better understand these samples and improve our ability to deal with surveys of high-z objects. \n\nIn this paper we examine the redshift evolution of the X-ray scaling relations of halos in the \\mbox{{\\sc Simba}}\\xspace suite of cosmological simulations \\citep{Simba}.  We also track how populations of halos binned by mass evolve along these scaling relations, in order to examine how various galaxy formation processes, particularly AGN feedback, have a differential impact with halo mass.  Finally, we examine these trends in test simulations where we turn off individual feedback modules, in order to isolate the particular physics in \\mbox{{\\sc Simba}}\\xspace driving the evolution.\n\nThis paper is organised as follows. In Section 2 we discuss the \\mbox{{\\sc Simba}}\\xspace suite of simulations and the key tools in reproducing X-ray scaling relations. In Section 3 we discuss the evolution of X-ray scaling relations from $z=3$ to $z=2$. In Section 4 we investigate the evolution of these scaling relations utilising differing levels of feedback within simulations. And finally in Section 5 we discuss the evolution of profiles both within the fiducial run, and within the $50 h^{-1}{\\rm Mpc}$ No-Jet run of \\mbox{{\\sc Simba}}\\xspace to examine the impact of jet feedback. \n\n \n\\section{Simulations and Analysis}\n\n\\subsection{The \\mbox{{\\sc Simba}}\\xspace simulation}\n\nThe \\mbox{{\\sc Simba}}\\xspace simulation \\citep{Simba} is a cosmological hydrodynamic simulation using the \\mbox{{\\sc Gizmo}}\\xspace code \\citep{Hopkins}. The flagship simulation evolves a random cosmological volume of (100$h^-1$ Mpc)$^3$ including $1024^3$ dark matter particles and $1024^3$ gas elements evolved from initial conditions to the present day. \\mbox{{\\sc Simba}}\\xspace adopts a $\\Lambda CDM$ cosmology with $\\Omega_{\\Lambda}  = 0.7,\\Omega_m  = 0.3,\\Omega_b  = 0.048, h = 0.68, \\sigma_8 = 0.82 $, and $n_s = 0.97$ in line with \\citet{Planck_Cosmology}.\n\nAn H$_2$-based model with the H$_2$ fraction computed from the \\citet{KrumholzGnedin2011} sub grid model is used to model star formation from gas in a Monte Carlo fashion. Chemical enrichment is tracked from Type II supernovae (SNe), Type Ia SNe, and Asymptotic Giant Branch (AGB) stars, tracking eleven elements (H,He,C,N,O,Ne,Mg,Si,S,Ca,Fe) during the simulation. Radiative cooling (both primordial and metal-line) and photoionisation is included using {\\sc Grackle-3.1} assuming a \\citet{HaartMadau} ionising background, with on-the-fly self-shielding. Star formation driven galactic winds are modelled as decoupled two-phase winds with a mass loading factor scaled to stellar mass as described by the tracking of individual particles in the FIRE zoom simulations \\citep{2017MNRAS.470.4698A}.\n    \nBlack holes are seeded at $10^4 M_{\\odot}$ in galaxies that exceed stellar masses of $10^{9.5}M_{\\odot}$. There are two modes for the growing of black holes: Torque-limited accretion for gas with $T <10^5$K \\citep{Angles:2017a} in which angular momentum is the primary bottleneck to accretion~\\citep{Hopkins:2011}; and \\citet{Bondi} accretion from higher temperature gas. The galaxy stellar mass threshold is motivated by findings in the FIRE project simulations showing that stellar feedback strongly suppresses black hole growth in low mass galaxies \\citep{Angles:2017c}. \\mbox{{\\sc Simba}}\\xspace also includes AGN feedback, which we detail in the next section owing to its central importance in this work.\n\n\\subsection{AGN feedback in \\mbox{{\\sc Simba}}\\xspace}\n\nWith AGN feedback proposed as a significant mechanism in quenching star formation and contributing the the evolution of groups and clusters, simulations must take care in modelling their impact.  \\citep{Heckman} described two main modes of AGN feedback: Radiative mode at high Eddington ratios ($f_{\\rm Edd} \\equiv \\dot M_{\\rm BH} \/ \\dot M_{\\rm Edd}$); and jet mode at lower $f_{Edd}$. \\mbox{{\\sc Simba}}\\xspace includes these two modes, plus X-ray feedback from and high-energy photon pressure. \n\nThe energy output of radiative mode AGN is dominated by electromagnetic radiation emitted by the accretion disks of central super massive black holes (SMBH). The photon pressure drives multi-phase winds up to speeds of $\\ga 1000\\;{\\rm km}\\,{\\rm s}^{-1}$ or more, entraining surrounding materials and as a result can include molecular \\citep{Sturm_2011} and warm ionised \\citep{Perna:2017} gas.\n\nJet mode AGN on the other hand produce feedback in the form of collimated jets, moving at speeds of order $\\sim 10^4\\;{\\rm km}\\,{\\rm s}^{-1}$ on large scales, and this dominates the energetic output as bulk kinetic energy. This mode is thought to be powered by the advection dominated accretion of hot gas prevalent at low accretion, below $~1-2\\%$ \\citep{BestHeckman:2012} and the spin of SMBHs.\n\nSignificant emission from the BHs in both modes also provide a photon pressure on surrounding material. Cold gas, often found around radiative mode AGN, is able to quickly absorb and subsequently radiate away this X-ray photon pressure. However in systems with low cool gas fractions, the photon pressure is able to drive a spherical outward momentum.\n\nThe way in which these modes interact with their surrounding medium can have a significant impact on the growth and evolution of the galaxies and halos in which they reside. As a result \\mbox{{\\sc Simba}}\\xspace attempts to implement these forms of feedback in a way that mirrors observations as closely as possible. It is not only important to consider how the feedback is implemented, but when, since we are concerned here with redshift evolution.\n\nRadiative and jet feedback in \\mbox{{\\sc Simba}}\\xspace is modelled as purely kinetic and bipolar, continuous outflows, ejected parallel to the axis of angular momentum of the inner disk. Using observations of ionised gas linewidths of X-ray detected AGN \\citep{Perna:2017} the outflow velocity for radiative mode is parametrised in terms of black hole mass $M_{BH}$ as:\n\\begin{equation}\n    v_{\\textrm{w,EL}} = 500 + 500(\\log M_{\\textrm{BH}} - 6)\/3 \\;{\\rm km}\\,{\\rm s}^{-1}  \n\\end{equation}\nThis model is similar to the AGN feedback implemented in \\mbox{{\\sc Gizmo}}\\xspace \\citep{Angles:2017a} however includes a variable outflow velocity.\n\nAs $f_{\\rm Edd}$ drops to $<0.2$ the jet feedback begins to add an additional velocity component, with a  strength depending on the Eddington ratio:\n\\begin{equation}\n    v_{\\textrm{w,jet}} = v_{\\textrm{w,EL}} + 7000 \\log(0.2\/f_{\\rm Edd}) \\;{\\rm km}\\,{\\rm s}^{-1} ,\n\\end{equation}\nwith the velocity increase capped to $7000\\;{\\rm km}\\,{\\rm s}^{-1}$ at $f_{\\rm Edd}\\leq$ 0.02. Additionally jet mode will only be triggered when $M_{BH} > 10^{7.5}M_\\odot$ following observations in which radio jets only occur within galaxies containing black holes with $M_{BH} \\gtrsim 10^8M_{\\odot}$ \\citep{Barisic:2017}. This limit has implications on when jet feedback typically begins in the simulations. As black holes grow in mass with time, we also see a decreasing $f_{\\rm Edd}$ \\citep{Angles:2014}. This combination of $M_{BH}$ beginning to cross the limit of $10^{7.5}$, and a decreasing $f_{\\rm Edd}$ we expect to see jet feedback turning on at $\\approx z\\sim 1.5$ for central galaxies in group-sized halos.\n\nWhile radiative mode winds are ejected at the ISM temperature, gas in jets is raised to the virial temperature of the halo. Jets are typically observed to be made up of synchroton-emitting plasma, eventually thermalising an around $T_{\\rm vir}$ with surrounding hot gas \\citep{Fabian:2012}. To avoid significant radiative losses in the dense gas surrounding the black hole, gas wind elements undergo a short hydrodynamic and radiative cooling decoupling time of $10^{-4}t_H$ where $t_H$ is the Hubble time at launch. \n\nX-ray feedback operates in two forms, depending on the gas it is heating. Non-ISM gas is heated directly according to the heating flux. ISM gas however is low resolution and pressurised such that any added heat would quickly be cooled away. Thus, one half of the energy is applied as a radial outward kick with the other being added as heat. X-ray feedback is only activated at low eddington ratios ($f_{\\rm Edd} < 0.02$) alongside full velocity jets. In an attempt to model the ability of gas rich galaxies to absorb and radiate X-ray energy, galaxies must also have a cold gas fraction of $f_{\\rm gas} < 0.2$. This heating has a minimal effect on the galaxy mass function, however contributes to fully quenching galaxies \\citep{Appleby:2020}.\n\nAlongside the fiducial $100h^{-1}{\\rm Mpc}$ box, we have also run a number of versions excluding various modules of the implemented feedback. Due to the computational requirements of running these simulations, these runs use a $50h^{-1}{\\rm Mpc}$ box with $512^3$ dark matter particles and $512^3$ gas elements. This gives these runs the same resolution as the fiducial run, with $\\frac{1}{8}$th the volume. While this reduces the number of high mass halos, we are still able to examine the impact of the various forms of feedback amongst group-sized halos. The various models run are: a \"No-X\" model (\\mbox{{\\sc Simba}}\\xspace-NoX) with only X-ray feedback turned off and jets are left on; a \"No-Jet\" model (\\mbox{{\\sc Simba}}\\xspace-NoJet) with jet and associated X-ray feedback turned off; a \"No-AGN\" model in which all AGN feedback is turned off; and finally a \"No-Feedback\" model in which all feedback including star formation winds is turned off. The initial conditions for all $50h^{-1}{\\rm Mpc}$ runs are identical. \n\n\n\\subsection{Identifying halos and computing X-ray emission}\n\nHalos are identified via a 3-D friends-of-friends (FoF) finder implemented within \\mbox{{\\sc Gizmo}}\\xspace, using a linking length that is 0.2 times the mean interparticle spacing.  Galaxies are identified in post-processing from within these halos, using a 6-D FoF finder applied to star-forming gas and stars.  The galaxies and halos are identified and catalogued using the {\\sc Caesar} code \\citep{Thompson:2017}.  These catalogs, along with particle snapshots, for all the runs described above are publicly available at {\\tt simba.roe.ac.uk}.\n\nThe python package {\\sc Pygad}\\ \\citep{pygad} is a multi-purpose analysis toolkit for particle-based hydrodynamics simulations, and gives us tools for the general analysis of simulations. Various criteria such as specific particles, defined regions, or particles that fit within a specified parameter range (e.g. temperature or metallicity) can be used to create sub-snapshots with particles contained within individual {\\sc Caesar} halos.\n\nA module for analysing X-ray luminosity is included in {\\sc Pygad}, using XSPEC \\citep{Arnaud} to compute X-ray spectra. XSPEC is fully described in \\citet{Eisenreich_2017}. {\\sc Pygad}\\ calculates the X-ray luminosity of selected gas particles using emission tables in XSPEC along with particle temperatures and metallicities taken from the simulation. In our analysis of \\mbox{{\\sc Simba}}\\xspace's X-ray properties we use the 0.5-2keV X-ray table to be consistent with the work done in \\citet{Robson:2020}. {\\sc Pygad}\\ is publicly available at {\\tt https:\/\/bitbucket.org\/broett\/pygad}.\n\nIn the end, we identify 1379, 1051, 509, and 126 halos with $M_{\\rm 500}>10^{12.3}M_\\odot$ at $z=0,1,2,3$, respectively, in our flagship run.  This constitutes the main sample that we will analyse for this work.\n\n\\section{ Evolution of Scaling Relations}\n\n\nWe begin by examining the evolution of X-ray scaling relations in our selected galaxy groups.  Intragroup gas can be characterised by it's X-ray luminosity $L_X$, X-ray luminosity-weighted temperature $T_X$, the hot gas fraction $f_{\\rm gas}$, and the X-ray luminosity-weighted metallicity $Z_X$.  To understand the physical drivers behind these relations, we first examine the evolution with respect to $M_{500}$, both by looking at the scaling relations at various redshifts, and by tracking individual halos back in time.  In the end, we make predictions for the evolution in the most common observable barometer of hot bound gas, the $L_X-T_X$ relation, and use the evolution in the physical quantities vs. $M_{500}$ to provide a context for the predicted evolution in $L_X-T_X$.\n\n\n\\subsection{$L_X-M_{500}$ evolution}\n\n \\begin{figure}\n     \\centering\n     \\includegraphics[width = 0.5\\textwidth, trim = 50 0 100 0, clip]{EvolvingPlots\/Halo_Mass_Lx_m100n1024Evolution_bh_ratio.pdf}\n     \\caption{$L_X-M_{500}$ scaling relation from $z=3$ to $z=0$ at intervals of 0.5. Colourbar is scaled as $M_{BH}\/M_{500}$. The self-similar scaling normalised to the most massive halos is shown by the dashed line. The slope of the scaling relation shows little evolution. As we move from $z=3$ to $z=0$ the most luminous halos appear to increase in mass, while the lower luminosity halos populate the lower mass end. A significant increase in scatter is also seen in halos with $M_{500} < 10^{13.5} M_{\\odot}$ between $z=1.5$ and $z=1$ when jet feedback starts suggesting jet feedback may have a significant effect on these low mass halos. At a given mass, lower luminosities increasingly coincide with higher $M_{BH}$ from $z=1.5$ to $z=0$}\n     \n     \\label{fig:M500_Lx_Scaling_Evolution}\n \n \\end{figure}\n \n \n\\begin{figure}\n    \\centering\n\n    \\includegraphics[width = 0.5\\textwidth]{ScalingTracks\/Halo_Mass_Lx_m100n1024_ScalingTrack.pdf}\n    \\caption{A plot showing the median evolution of a halo from $z=3$ to $z=0$, binned by final $M_{500}$ with blue, yellow, green, red, and finally purple representing least to most massive mass bins. X marks indicate the positions of $z=2$ and $z=1$ on each track. Light grey dots represent the scaling relation at $z=3$ while black dots represent the scaling relation at $z=0$. Halos with final masses $> 10^{13}M_{\\odot}$ show little change in luminosity over an order of magnitude increase in $M_{500}$. Halos with final masses $< 10^{13}M_{\\odot}$ show an order of magnitude decrease in luminosity. This indicates that the main driver behind significant evolution of $L_X$ mostly impacts the lower mass halos.}\n    \\label{fig:ScalingTrack_M_L}\n\\end{figure}\n\nFigure \\ref{fig:M500_Lx_Scaling_Evolution} shows the $L_X-M_{500}$ scaling relation for all halos with $M_{\\rm halo}>10^{12.3}M_\\odot$ in \\mbox{{\\sc Simba}}\\xspace at $z=3,2.5,2,1.5,1,0.5,0$.  Points colored by the black hole mass fraction  $f_{BH}\\equiv M_{BH}\/M_{500}$, where $M_{BH}$ is the black hole mass of the central galaxy.  The dashed line shows the scaling predicted from the self-similar model, normalized to the most massive halo at that epoch.\n\nA clear $L_X-M_{500}$ is already in place at high redshifts, even though the halos are still relatively small.  As time progresses, the overall scaling relation does not change much, but there is a marked increase in the scatter, moreso at low $M_{500}$.  This starts to become particularly prominent at $z\\la 1.5$.\nAt any given redshift, the most luminous X-ray halos at a given $M_{500}$ follow self-similarity, but there is an increasing number of halos that deviate strongly from it.\n\nA clear trend can be seen in the $f_{BH}$ of lower mass halos. From $z=1.5\\to 0$ we see the drop in $L_X$ of low mass halos correspond to an increase in $f_{BH}$.\n\nThese results can be understood by appealing to the impact of AGN jet feedback in \\mbox{{\\sc Simba}}\\xspace.  As noted earlier, jet feedback turns on at low $f_{\\rm Edd}$.  As shown in \\citet{Angles:2013}, at these mass scales, the Eddington ratio becomes comparable to the jet threshold at $z\\sim 1.5$, albeit with a fair amount of scatter.  Thus we expect that black holes that have managed to grow large by these epochs will be putting out a significant fraction of their AGN feedback in high-energy jets into intragroup gas.  This circumstantially correlates well with driving a decreasing $L_X$ in systems with large $f_{BH}$ at $z\\la 1.5$.  We will show more evidence for this connection with jet feedback when examining the other scaling relations.\n\nFigure \\ref{fig:ScalingTrack_M_L} provide another view on how the $z=3$ population evolves to $z=0$.  Here we examine the median evolutionary tracks of halos binned by mass.  The small points are in two groupings: The high-$L_X$ grouping (red points) is at $z=3$ while the low grouping (black points) shows the individual halos at $z=0$.  The coloured lines show median tracks of halos in various $z=0$ $M_{500}$ bins:  $M_{500}>10^{14.5}M_\\odot$ (purple),  $10^{14}<M_{500}<10^{14.5}M_\\odot$ (red), $10^{13.5}<M_{500}<10^{14}M_\\odot$ (green), $10^{13}<M_{500}<10^{13.5}M_\\odot$ (gold),\n$10^{12.5}<M_{500}<10^{13}M_\\odot$ (blue).  These are constructed by following individual halos back in time via their most massive dark matter progenitor, so long as the progenitor has $M_{500}>10^{12.5}M_\\odot$. \n\nThis shows that halos with $M_{500} > 10^{13.5} M_{\\odot}$ (at $z=0$) show little change in $L_X$ despite an order of magnitude increase in $M_{500}$ from $z=3$ to $z=0$. In contrast, halos with a $z=0$ $M_{500} < 10^{13}$ show over an order of magnitude decrease in $L_X$, with a smaller increase in mass.  The slope of the scaling relation is maintained by low mass halos losing luminosity, along with younger halos forming at the low mass end.  In effect, the $L_X$ scaling relation spreads out in $M_{500}$ over time.\n\nWe checked that including only $z=0$ halos with progenitors all the way back to $z=3$ revealed similar trend in the median tracks.  However, if one only follows the main descendants of halos from $z=3\\to 0$, this reveals that individual scaling tracks almost exclusively retain a constant luminosity as mass increases from $z=3$ to $z=0$. This indicates that the $L_X-M_{500}$ relation evolves via a combination of massive halos gaining hot gas compensated by the lowering density of the cosmos, along with low-mass halos entering into our sample above our mass cut but containing less hot gas. We will later show that this owes primarily to jet feedback removing hot gas from these lower mass halos, as found in \\citet{Robson:2020}.\n\n\\subsection{$T_X-M_{500}$ Evolution}\n\n \\begin{figure}\n     \\centering\n     \\includegraphics[width = 0.5\\textwidth, trim = 50 0 100 0, clip]{EvolvingPlots\/Halo_Mass_Tx_m100n1024Evolution_bh_ratio.pdf}\n     \\caption{$T_X-M_{500}$ scaling relation from $z=3$ to $z=0$ at intervals of 0.5. The colourbar is scaled as $M_{BH}\/M_{500}$. The expected self-similar scaling normalised to the most massive halos is shown by the dashed line. While the slope of the scaling relation at $M_{500} > 10^{13} M_{\\odot}$ shows little evolution, a drop in $T_X$ is introduced at masses below this threshold. This drop in $T_X$ coincides with the occurrence of jet feedback indicating that jets  may be working to lower the temperature of small halos, potentially through the removal of hot gas. From $z=3$ to $z=1.5$ lower $T_X$ corresponds to a lower $M_{BH}$, while from $z=1$ to $z=0$ $M_{BH}$ increases with increasing $M_{500}$.\n     }\n     \n     \\label{fig:M500_Tx_Scaling_Evolution}\n \n \\end{figure}\n \n \n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.5\\textwidth]{ScalingTracks\/Halo_Mass_Tx_m100n1024_ScalingTrack.pdf}\n    \\caption{A plot showing the median evolution of a halo from $z= 3$ to $z= 0$, binned by final $M_{500}$ with blue, yellow, green, red, and finally purple representing least to most massive mass bins. X marks indicate the positions of $z=2$ and $z=1$ on each track. Light grey dots represent the scaling relation at $z=3$ while black dots represent the scaling relation at $z=0$. All but the least massive bin show a clear increase in $T_X$. All but the most massive bin seem to originate from the same area at $z=3$ suggesting there is no clear evolution by population based on $T_X$ alone.}\n    \\label{fig:ScalingTrack_M_T}\n\\end{figure}\n\n\nFigure \\ref{fig:M500_Tx_Scaling_Evolution} shows the $T_X-M_{500}$ scaling relation at $z=3,2.5,2,1.5,1,0.5,0$ with points colour-coded by $f_{BH}=M_{BH}\/M_{500}$, similar to Fig.~\\ref{fig:M500_Lx_Scaling_Evolution}.  The self-similar relation is again shown as the dashed line in each panel.\n\nThe evolution from $z=3\\to 0$ shows a very slight steepening of the slope driven by a decrease of $T_X$ amongst the lowest mass halos. The most notable trend is that, amongst the lower mass halos, we see a large scatter in $T_X$ increase from a span of around 0.5~dex, to $>1$~dex from $z=3\\to 0$, with again the most significant change happening around $z\\sim 1.5$.\n\nLooking at the trend with $f_{BH}$ reveals a muddier situation than with $L_X$.  At high redshifts, there is a clear (though mild) trend that high-$f_{BH}$ halos have high $T_X$.  This makes sense because the integrated AGN feedback energy is higher for high-mass black holes.  Yet, the trend disappears at $z\\la 1$.  This suggest that the switching on of jet feedback is breaking the relation between $T_X$ and $M_{BH}$ amongst smaller halos.  A possible explanation is that jet feedback expels the hottest gas out of low-mass halos altogether, leaving only (relatively) cooler gas. As a result, there is not a simple relationship with $f_{BH}$ for the $T_X-M_{500}$ scaling; in \\mbox{{\\sc Simba}}\\xspace, stronger AGN feedback does not monotonically increase $T_X$.\n\nFigure \\ref{fig:ScalingTrack_M_T} shows median tracks within different halo mass bins, tracked backwards in time from the $z=0$ population, analogous to Fig.~\\ref{fig:ScalingTrack_M_L}.  Again, we show 5 $z=0$ halo mass bins in $\\log\\ M_{500}$ as denoted in the legend, and we plot tick marks along the tracks at integer redshifts.  The tracks start at $z=3$ on the left hand side and evolve rightwards (growing in $M_{500}$) with time.\n\nWe see that the evolution here has some similarities but also some differences relative to the tracks in $L_X-M_{500}$ space.  The highest mass halos show a clear increase in temperature from $z=3\\to 0$, while the least massive halos at $z=3$ show less change in the X-ray weighted gas temperature.  Interestingly, all halos start along the same mildly upwards trend, but while the massive halos continue that trend to $z=0$, \nat $z\\la 1$ the low-mass halos stop evolving in $T_X$, with the lowest mass bin actually dropping in median $T_X$ since $z\\sim 1$.  The net effect is a slight steepening of the $T_X-M_{500}$ slope.\n\nThis differential evolution with halo mass is likely driven by the interplay between jet feedback and structure formation.  At the lower masses, the jets are fast enough to drive substantial material out of halos, heating the surrounding gas but depositing less energy into the halo itself.  At the higher masses, the jets cannot escape the halo as easily, and the kinetic energy is thermalised within the halo.\n\n\n\n\n\\subsection{$f_{\\rm gas}-M_{500}$ evolution}\n\n\\begin{figure}\n     \\centering\n     \\includegraphics[width = 0.5\\textwidth, trim = 50 0 100 0, clip]{EvolvingPlots\/Halo_Mass_Gas_m100n1024Evolution_bh_ratio.pdf}\n     \\caption{$f_{\\rm gas}-M_{500}$ scaling relation from $z=3$ to $z=0$ at intervals of 0.5. The colourbar is scaled as $M_{BH}\/M_{500}$ going from red to blue. A small evolution in scatter of the least massive halos is seen as $M_{500}$ increases up to $z=1.5$. After $z=1.5$ we see a large increase in scatter at the low mass end with a a significant decrease in hot gas mass fraction at these masses. This coincides with the switch on of jet feedback. Halos with $M_{500} < 10^{13.5} M_{\\odot}$ show higher $M_{BH}$ corresponding to lower $f_{\\rm gas}$.}\n     \n     \\label{fig:M500_Gas_Scaling_Evolution}\n\\end{figure}\n\n \n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.5\\textwidth]{ScalingTracks\/Halo_Mass_Gas_m100n1024_ScalingTrack.pdf}\n    \\caption{A plot showing the median evolution of a halo from $z=3$ to $z=0$, binned by final $M_{500}$ with blue, yellow, green, red, and finally purple representing least to most massive mass bins. X marks indicate the positions of $z=2$ and $z=1$ on each track. Light grey dots represent the scaling relation at $z=3$ while black dots represent the scaling relation at $z=0$. Halos with $M_{500} > 10^{13.5} M_{\\odot}$ show a slight increase in $f_{\\rm gas}$ as mass increases by around an order of magnitude. However, halos with $M_{500} < 10^{13.5} M_{\\odot}$ show a clear drop in $f_{\\rm gas}$ suggesting jet feedback is working to remove hot gas from the least massive halos.}\n    \\label{fig:ScalingTrack_M_G}\n\\end{figure}\n\nIn examining $L_X$ and $T_X$ vs. $M_{500}$, we have implicated the halo hot gas fraction $f_{\\rm gas}$ as an important quantity that is directly impacted by jet feedback. \\citet{Robson:2020} showed that without jets on, gas fractions show much larger values in small halos, with not nearly as strong a trend with halo mass.  Thus it is worth examining how the gas fraction evolves with time, to provide insights into the evolution of other more directly observable quantities.\n \nFigure \\ref{fig:M500_Gas_Scaling_Evolution} shows the $f_{\\rm gas}-M_{500}$ scaling relation at $z=3,2.5,2,1.5,1,0.5,0$ with points scaled by the stellar mass fraction $f_{\\rm star}$. We note that the top of each panel corresponds to $f_{\\rm gas}$ reaching the cosmic halo baryon fraction $\\Omega_b\/(\\Omega_m+\\Omega_b)$.  We define hot gas as gas with $T>10^{5.5}$K. \n\nFrom $z=3\\to 1.5$ we see that $f_{\\rm gas}$ has a fairly weak trend with halo mass.  As time goes and particularly at $z\\la 1.5$, the scatter increases dramatically at the low-mass end, with an increasing number of halos with lower $f_{\\rm gas}$ values. This scatter introduced at lower masses results in a steepening of the $f_{\\rm gas}-M_{500}$ scaling relation.  \n\nThe colour scaling shows an inverse relation between $M_{BH}$ and $f_{\\rm gas}$ at $M_{500} < 10^{13.5} M_{\\odot}$, with lower mass black holes corresponding to higher hot gas fractions.  This trend is already beginning at $z=2$, showing the effects of early jet feedback that results in early quenched galaxies, with the effects ramping up most strongly from $z=2\\to 1.5$.  The lower hot gas content in halos with more AGN feedback might seem counterintuitive, since one expects AGN feedback to heat gas.  But the wind speed of the jets is such that it can easily unbind gas at these lowest masses, whereas for the high-mass halos it will be more easily trapped within the halo.  Thus the impact of AGN jets isn't so much to heat the halo gas, as expel it.  This is why we see a correspondingly strong trend with $f_{BH}$ in $L_X$ which is sensitive to gas density, but a weak trend with $T_X$. We will examine gas densities later when we look at the X-ray profiles.\n\nFigure \\ref{fig:ScalingTrack_M_G} tracks the gas fraction over time for these halos.  We see that it is only the less massive halos showing a decrease in $f_{\\rm gas}$ with halos of $M_{500} > 10^{13.5} M_{\\odot}$ demonstrating steady or mildly increasing $f_{\\rm gas}$.  This trend is qualitatively similar to what was seen in tracking $L_X$ (Fig.~\\ref{fig:ScalingTrack_M_L}), highlighting the direct impact of $f_{\\rm gas}$ on $L_X$.\n\n\\subsection{$Z_X-M_{500}$ evolution}\n\n\\begin{figure}\n     \\centering\n     \\includegraphics[width = 0.5\\textwidth, trim = 50 0 100 0, clip]{EvolvingPlots\/Halo_Mass_Zx_m100n1024Evolution.pdf}\n     \\caption{$Z_X-M_{500}$ scaling relation from $z=3$ to $z=0$ at intervals of 0.5. The colourbar is scaled as $f_{star}$ from 0 to 0.1 going from red to blue. A clear increase in metallicity is seen at the low mass end.  }\n     \n     \\label{fig:M500_Zx_Scaling_Evolution}\n \\end{figure}\n \nWhile emission via bremsstrahlung dominates for systems with $T_X \\ga 3 keV$ metal line emission becomes increasingly important towards lower-$T_X$ systems. This includes all but the most massive systems in our sample, and as such an examination of the hot gas metallicity of these systems also provides insight.  Moreover, metals provide a unique tracer of supernova-generated products and their distribution into intragroup gas.\n\nFigure                        \\ref{fig:M500_Zx_Scaling_Evolution} shows the $Z_X-M_{500}$ scaling relation at        $z=3,2.5,2.0,1.5,1.0,0.5,0$ with points scaled by the stellar mass fraction $f_{star}$. $f_{star}$ here is defined as the ratio of stellar mass within $R_{500}$ against $M_{500}$. Through $f_{star}$ we can infer a level of stellar feedback within a cluster, a key driver behind metallicity. Scaling by $M_{BH}\/M_{500}$, as in previous plots, showed no relation motivating us to investigate the effects of different forms of feedback on evolution. \n\nBetween $z=3$ and $z=1.5$ little evolution is seen in the shape of the relation, with an increase in halos above the mass cut we see little change in metallicity at any given mass with the exception of more massive halos become present. However after $z<1.5$ we begin to see a change in metallicities seen among low mass halos. Much like previous scaling relations, we see an increase is seen in the scatter at lower masses. In particular, the lowest mass groups can show high intragroup metallicities, exceeding solar in some cases at $M_{500}<10^{13}M_\\odot$. \n\nNoting the colour-coding, the high intragroup metallicities correlate fairly well with high $f_{star}$ in the central galaxy. This trend is very pronounced at high-$z$, but remains noticeable all the way to $z=0$. At $z>1$, the lowest metallicity systems have very low stellar fractions, which suggests that these systems have been quenched early on~\\citep{Cui:2021}.   This is perhaps not surprising, since the stars are responsible for producing much of the metals in the system, particularly early on.  Moving to lower redshifts, re-accretion of enriched intergalactic gas~\\citep[i.e. ``outside-in enrichment\";][]{Oppenheimer:2012} imparts a metal floor in these high density environments, and dilutes the strong trend arising purely from self-enrichment.  Nonetheless \\mbox{{\\sc Simba}}\\xspace\\ predicts that poor groups with higher stellar masses~\\citep[which also tend to have more star-forming galaxies;][]{Cui:2021} will generally to have higher metallicities.\n\n\n\\subsection{$L_X-T_X$ evolution}\n\n\\begin{figure}\n     \\centering\n     \\includegraphics[width = 0.5\\textwidth, trim = 50 0 100 0, clip]{EvolvingPlots\/Tx_Lx_m100n1024Evolution_bh_ratio.pdf}\n     \\caption{$T_X-L_X$ scaling relation from $z=3$ to $z=0$ at intervals of 0.5. The colourbar is scaled as $M_{BH}\/M_{500}$. The expected self-similar scaling normalise to the most massive halos is shown by the dashed line. A tight scaling is retained amongst the hottest most luminous halos, however a large scatter is introduced at low temperatures and luminosities. The introduction of jet feedback after $z=1.5$ coincides with the introduction of a number of low temperature, low luminosity halos exhibiting a large scatter. }\n     \n     \\label{fig:Lx_Tx_Scaling_Evolution}\n\\end{figure}\n\nThe $L_X-T_X$ relation has long been a key observational diagnostic of hot gas in massive halos. The evolution of this relation will obviously be a combination of the evolution in each quantity versus $M_{500}$ from the previous sections. Here we examine this evolution directly, including examining how groups at different masses evolve within $L_X-T_X$ space.\n\nFigure \\ref{fig:Lx_Tx_Scaling_Evolution} shows the $L_X-T_X$ scaling relation at $z=3,2.5,2,1.5,1,0.5,0$ with points colour-coded by $M_{BH}\/M_{500}$.  We impose a mass cut of $M_{500}>10^{12.3}M_\\odot$ at each redshift.  The dashed line shows the self-similar relation normalized to the highest $T_X$ group.\n\nFrom $z=3\\to 1.5$, the $L_X-T_X$ relation looks broadly very similar. The largest halos with the highest $T_X$ are already on the self-similar relation, and by $z\\sim 1$ there is a fairly tight locus which sets an upper edge following self-similarity.  Over time, there is a growing scatter of points to lower $L_X$.  This becomes particularly prominent at $z\\sim 1-1.5$ in the lower mass halos.  We have similarly seen in $L_X-M_{500}$ and $T_X-M_{500}$ scaling relations that there is a change in evolution for lower-mass groups around this epoch.\n\nThere is a marked trend that the groups with the lowest $L_X$ values have the highest $f_{BH}$ values.  This directly implicates the impact of AGN feedback in growing a population of low-mass groups that deviates from self-similarity.  As we discussed earlier, the effect of the jets is to evacuate the low-mass halos. This lowers the amount of X-ray emitting gas and thus $L_X$, but we showed that it does not have a strong impact on $T_X$ values in low-mass systems because there is the competing effect that the hottest gas is evacuated from these systems altogether.  \\mbox{{\\sc Simba}}\\xspace's prediction of a strong anti-correlation between $L_X$ and $f_{BH}$ at $T_X\\la 1$~keV can be tested with next-generation X-ray facilities.\n\nIn Figure~\\ref{fig:ScalingTrack_L_T} we show the evolution of the $L_X-T_X$ relation in 5 $z=0$  $\\log_{10}M_{500}$ mass bins as indicated in the legend, as before.  In this case, the red points depicting $z=3$ values are not wholly distinct from the $z=0$ black points, since there is not a monotonic increase in $T_X$ as there is in $M_{500}$.\n\nThe scaling tracks offer a very interesting perspective on $L_X-T_X$ evolution.  The overall relation at $z=3$ and $z=0$ are not that different in terms of the slope and amplitude, although at $z=0$ there are many more low-$T_X$ systems with a large scatter in $L_X$.  The apparent lack of evolution in the $L_X-T_X$ relation has been noted in some observations of clusters \\citep{Holden:2002,Borgani:2001}, though not for lower-mass groups since current X-ray data lacks the sensitivity at $z\\gg 0$.  Several other studies have shown this weak evolution at $z>0.3$ however show varying levels of evolution from $z=0.3 \\to z=0$ motivated by the consideration of the effect of cool core clusters \\citep{Vikhlinin:2002, Ettori:2004}. \n\nThe modest evolution in $L_X-T_X$ is seen across all masses, though again there is a different trend for $M_{500}\\ga 10^{14}M_\\odot$ systems that tend to evolve  rightwards with a modest increase in $L_X$, while $M_{500}\\la 10^{13.5}M_\\odot$ evolve downwards in $L_X$ with little change in $T_X$.  The net effect is to leave the overall relation relatively unchanged, but to spread out the groups along the relation.  Hence even though $L_X-M_{500}$ and $T_X-M{500}$ individually show significant change over time, the $L_X-T_X$ relation combines these two coincidentally into a nearly non-evolving relation, albeit with a large increase in scatter at very low $T_X$.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.5\\textwidth]{ScalingTracks\/Tx_Lx_m100n1024_ScalingTrack.pdf}\n    \\caption{A plot showing the median evolution of a halo from $z=3$ to $z=0$, binned by final $M_{500}$ with blue, yellow, green, red, and finally purple representing least to most massive mass bins. X marks indicate the positions of $z=2$ and $z=1$ on each track. Light grey dots represent the scaling relation at $z=3$ while black dots represent the scaling relation at $z=0$. Halos with $M_{500} > 10^{13} M_{\\odot}$ show little evolution in the $L_X-T_X$ space however the least massive halos below this threshold demonstrate a drop of nearly 2 orders of magnitude in $L_X$ while maintaining a consistent $T_X$.}\n    \\label{fig:ScalingTrack_L_T}\n\\end{figure}\n\n\n\n\\section{Effects of Feedback on Scaling Relations}\n\nWe have implicated jet feedback in driving the evolution of the intragroup medium scaling relations particularly at the low-$M_{500}$ end.  We can test this more directly by examining a suite of smaller $50h^{-1}{\\rm Mpc}$ runs having the same resolution, with individual AGN and star-formation modules turned on and off.   This offers us the chance to directly connect the scaling relations to the physics of the implemented feedback processes.\n\nAs previously described, the variant models used for comparison are the full \\mbox{{\\sc Simba}}\\xspace, No-X, No-Jet, No-AGN, and No-Feedback models. No-X turns off only X-ray feedback, No-Jet additionally turns off jet feedback, No-AGN additionally turns off radiative AGN feedback, and No-Feedback additionally turns off star formation winds. Overall, the No-AGN results were very similar to the No-Jet results, showing that the radiative AGN feedback has minimal effect on X-ray scaling relations~\\citep[as likewise found for galaxy properties;][]{Simba}, and as such it has been left out of the following plots for brevity. By investigating the evolution of scaling relations with each feedback variant run we are able to demonstrate each specific model's impact on each of the previously studied scaling relations. \n\n\n\\subsection{Hot Gas Fraction}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.5\\textwidth]{Comparisons\/Halo_Mass_Gas_Z_2x2_small.pdf} \n    \\caption{Evolution from $z=3$ to $z=0$ of the running median of the $f_{\\rm gas}-M_{500}$ scaling relation for \\mbox{{\\sc Simba}}\\xspace, \\mbox{{\\sc Simba}}\\xspace without X-ray feedback, \\mbox{{\\sc Simba}}\\xspace without jet feedback, and \\mbox{{\\sc Simba}}\\xspace with no feedback. All models lacking jet feedback (No-Jet, No-AGN, No-Feedback) show no evolution in $f_{\\rm gas}$ while both models including jet feedback (\\mbox{{\\sc Simba}}\\xspace, No-X) exhibit a very large drop in $f_{\\rm gas}$ in the halos with $M_{500} < 10^{13.5} M_{\\odot}$.}\n    \\label{fig:M_fgas_comp}\n\\end{figure}\n\nSince the hot gas fraction is implicated in driving much of the evolution of the scaling relations, we begin by examining how feedback impacts this quantity.\n\nFigure \\ref{fig:M_fgas_comp} shows the median $f_{\\rm gas}$ as a function of $M_{500}$ at redshifts $z=3\\to 0$ (blue to red) in various colours as indicated in the legend.  The upper left panel shows a $50h^{-1}{\\rm Mpc}$ run with full \\mbox{{\\sc Simba}}\\xspace\\ physics, and the other panels show the No-X run (upper right), No-Jet (lower left), and No-Feedback (lower right).\n\nThere is a dramatic difference in the evolution of $f_{\\rm gas}$ once the jet feedback is turned on, seen by comparing No-Jet in the lower left, and No-X in the upper right.  Turning on star formation feedback and X-ray AGN feedback have in contrast a fairly minor effect.  We clearly see the impact of jet feedback at $M_{500} < 10^{13.5} M_{\\odot}$ at $z\\la 1.5$ while having a limited effect at $M_{500} \\ga 10^{13.5} M_{\\odot}$. Once jets are turned off (No-Jet and No-Feedback runs), there is essentially no evolution in $f_{\\rm gas}$.  This clearly demonstrates that it is jets that are responsible for evacuating the hot gas from halos.  \n\nThe most notable evolution occurs at $z\\sim 1-1.5$, which is consistent with the strong increase in scatter seen in various scaling relations at these lower masses around this epoch.  This suggests that jet feedback drives gas out of halos in a more stochastic manner, thereby creating large scatter in $L_X$.  Since jet feedback becomes strong at $f_{\\rm Edd}$ below a few percent, which tends to occur when the black hole mass grows to above $\\ga 10^8M_\\odot$ \\citet{Thomas:2019}, this then connects the larger black holes with the lower $f_{\\rm gas}$ and hence lower $L_X$.  We can see this more clearly by examining the $L_X$ and $T_X$ relations more directly using our feedback variant runs, which we do next.\n\n\\subsection{$L_X - M_{500}$}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.5\\textwidth]{Comparisons\/Halo_Mass_Lx_Z_2x2_small.pdf}\n    \\caption{Evolution from $z=3$ to $z=0$ of the running median of the $L_X-M_{500}$ scaling relation for \\mbox{{\\sc Simba}}\\xspace, \\mbox{{\\sc Simba}}\\xspace without X-ray feedback, \\mbox{{\\sc Simba}}\\xspace without jet feedback, and \\mbox{{\\sc Simba}}\\xspace with no feedback. Little change is seen between the No-Feedback, No-AGN, and No-Jet runs, with the No-X and \\mbox{{\\sc Simba}}\\xspace runs exhibiting similar larger evolutions. The key difference here being the inclusion of jet feedback driving the evolution of the scaling relation.}\n    \\label{fig:ML_Z_comp}\n\\end{figure}\nFigure \\ref{fig:ML_Z_comp} shows the median $L_X$ as a function of $M_{500}$ at reshifts $z = 3 \\to 0$ (blue to red) in various colours as indicated in the legend. The upper left panel shows a $50h^{-1}{\\rm Mpc}$ run with full \\mbox{{\\sc Simba}}\\xspace physics, and the other panels show the No-X run (upper right), No-Jet (lower left), and No-Feedback (lower right), as in Figure~\\ref{fig:M_fgas_comp}.\n\nIn full-physics \\mbox{{\\sc Simba}}\\xspace we see a steady decrease in the lowest luminosities, with the highest luminosities remaining relatively consistent as masses increase from $10^{13} M_{\\odot}$ to $>10^{14} M_{\\odot}$. This is inline with what we saw in the $100h^{-1}{\\rm Mpc}$ box. We see very little difference in the overall evolution of the No-X run, suggesting that X-ray feedback has minimal effect on the evolution of the X-ray luminosity. \n\nIn contrast, in the No-Jet run we immediately see the impact of jet feedback. Scaling relations at high redshifts match closely with \\mbox{{\\sc Simba}}\\xspace's, but much less evolution in $L_X$ is seen, with minimum values not going below $10^{41}$~erg~s$^{-1}$.  The slope of the relation is mostly invariant.  NoFB shows a similarly low amount of evolution, showing that SF feedback (and radiative AGN feedback) have fairly minimal impact on this scaling relation.  However, there are some non-trivial differences, particularly in the most massive halos that seem to have lower $L_X$ without any feedback.  Hence stellar and radiative AGN feedback seem to work to increase the X-ray luminosities.  The origin is likely from the slight increase in hot gas content from NoFB$\\to$No-Jet as seen in Figure~\\ref{fig:M_fgas_comp}, because the NoFB case locks more of the halo's baryons into stars~\\citep{Appleby:2021}.\n \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{Comparisons\/Halo_Mass_Tx_Z_2x2_small.pdf}\n    \\caption{Evolution from $z=3$ to $z=0$ of the running median of the $T_X-M_{500}$ scaling relation for \\mbox{{\\sc Simba}}\\xspace, \\mbox{{\\sc Simba}}\\xspace without X-ray feedback, \\mbox{{\\sc Simba}}\\xspace without jet feedback, and \\mbox{{\\sc Simba}}\\xspace with no feedback. Both \\mbox{{\\sc Simba}}\\xspace and No-X show similar evolutions while No-Jet, No-AGN, and No-Feedback show much tighter evolutions indicating jet feedbacks effect in lower $T_X$ in low mass halos. No noticeable difference is seen in the scaling relations of the most massive halos.}\n    \\label{fig:MT_Z_comp}\n\\end{figure}\n\\subsection{$T_X - M_{500}$}\nFigure \\ref{fig:MT_Z_comp} shows the median $T_X$ as a function of $M_{500}$ at redshifts $z = 3 \\to 0$ (blue to red) in various colours as indicated in the legend. The panels are ordered among the feedback variants as before. \n\nWe see quantitatively similar behaviour as in the $L_X-M_{500}$ relation, though less dramatically.  Jet feedback has the clearest impact on this scaling relation, particularly on the least massive halos.  This can be seen from the difference between the No-Jet in the lower left, and No-X in the upper right.  The noticeable difference in the evolution between these two runs begins at $z\\sim 1.5$.  The outcome of jets is to, perhaps counter-intuitively, lower the temperature of the hot gas in low-mass halos; we speculated this earlier, but here it is clearly demonstrated.  At high masses, none of the feedback models have a significant effect on $T_X$.  Nonetheless, it is worth noting that stellar feedback impacts the temperature of low mass at early times, producing a steeper scaling relation. Thus $T_X-M_{500}$ evolution owes to a combination of different feedback processes operating at different times.\n\n\\subsection{$L_X-T_X$}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.5\\textwidth]{Comparisons\/Tx_Lx_Z_2x2_small.pdf}\n    \\caption{Evolution from $z=3$ to $z=0$ of the running median of the $T_X-L_X $ scaling relation for \\mbox{{\\sc Simba}}\\xspace, \\mbox{{\\sc Simba}}\\xspace without X-ray feedback, \\mbox{{\\sc Simba}}\\xspace without jet feedback, and \\mbox{{\\sc Simba}}\\xspace with no feedback. Little difference is seen in the NoFB, No-AGN, and No-Jet runs. The No-X run shows an further evolution into the low $T_X-L_X$ region of the plot indicating jets impact on both lower temperatures and luminosities. The \\mbox{{\\sc Simba}}\\xspace run shows even further evolution into the lower temperatures indicating that X-ray feedback may have a small effect in these low temperature halos at late times.}\n    \\label{fig:Tx_Lx_comp}\n\\end{figure}\n\nFigure \\ref{fig:Tx_Lx_comp} shows the median $T_X$ as a function of $M_{500}$ at redshifts $z = 3 \\to 0$ (blue to red) in our various feedback runs.\n\nRemarkably, despite clear evolution in $L_X-M_{500}$ and $T_X-M_{500}$, the $L_X-T_X$ relation seems to show little evolution, and indeed not much sensitivity to feedback.  This occurs because jet feedback lowers both $L_X$ and $T_X$, resulting in halos moving further down the $L_X-T_X$ relation, but not changing its slope or amplitude much.  The net effect is that AGN jet feedback ``stretches out\" the $L_X-T_X$ relation, but doesn't substantially alter it.  In detail, the $L_X-T_X$ relation does get a bit steeper once jets are turned on, because the drop in $L_X$ due to the evacuation of halos is more severe than the lowered $T_X$ owing to the hot gas being unbound from low-mass halos.  Hence AGN jets do break the self-similarity in $L_X-T_X$ at low masses, but the effect is not as dramatic as seen in the individual relations vs. $M_{500}$.\n\nIn \\citet{Robson:2020} we proposed that \\mbox{{\\sc Simba}}\\xspace's scaling relations at $z=0$ were a result of the implemented jet feedback evacuating hot gas in lower mass halos. In Fig \\ref{fig:ML_Z_comp}  and Fig \\ref{fig:MT_Z_comp}, \\mbox{{\\sc Simba}}\\xspace and No-X show a clear deviation in their evolution after $z=1.5$, around the time jet feedback kicks in in the \\mbox{{\\sc Simba}}\\xspace simulations. This lends credence to the idea that jets are responsible for \\mbox{{\\sc Simba}}\\xspace's ability to match X-ray scaling relations, which is mainly driven by the gas fraction evolution.\n\n\\subsection{$Z_X-M_{500}$}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width =0.5\\textwidth]{Comparisons\/Halo_Mass_Zx_Z_2x2_small.pdf}\n    \\caption{Evolution from $z=3$ to $z=0$ of the running median of the $Z_X-M_{500}$ scaling relation for \\mbox{{\\sc Simba}}\\xspace, \\mbox{{\\sc Simba}}\\xspace without X-ray feedback, \\mbox{{\\sc Simba}}\\xspace without jet feedback, and \\mbox{{\\sc Simba}}\\xspace with no feedback. The no feedback run shows a steady increase in $Z_X$ with redshift. With the introduction of stellar feedback in the no jet run we see no evolution from $z=3$ to $z=0$ with stellar winds raising the metallicity to $\\approx 0.3$ before $z=3$. The no xray run shows jets increasing $Z_X$ at late times, especially amongst small halos. Finally the full \\mbox{{\\sc Simba}}\\xspace again shows very little evolution, demonstrating X-ray feedback's effect in full quenching halos in \\mbox{{\\sc Simba}}\\xspace.}\n    \\label{fig:Zx_M500_comp}\n\\end{figure}\n\nFigure~\\ref{fig:Zx_M500_comp} compares the $Z_X-M_{500}$ scaling relations of the various feedback runs, in the same format as the previous plots.\n\nIn contrast to the other quantities, jet feedback appears not to be the main driver of evolution in $Z_X$. In NoFB we see a steady increase in metallicity over time, which disappears in the No-Jet case, which instead has metallicities remaining fairly steady $Z_X\\approx 0.2-0.3$ solar across time (i.e. roughly the $z=0$ value in the No-Feedback run).  We do not show it, but the No-AGN case is similar to No-Jet.  This shows us that stellar feedback is increasing the metallicity of hot gas to observed levels early on and then maintaining them. \n\nNonetheless the jets do have some effect.  In the No-X run, with the introduction of jet feedback we see the X-ray weighted metallicities increase mildly after $z=1.5$ in the most massive halos relative to No-Jet, and slightly later but more noticeably after $z=1$ in the least massive halos.  Hence jets are an important factor for increasing the metallicity in hot intragroup gas at later epochs, presumably by transporting metals out of galaxies in the process of quenching them.\n\nThe inclusion of X-ray feedback in the \\mbox{{\\sc Simba}}\\xspace run then sees these late $Z_X$ brought back down to $\\approx 0.3$ in all but the least massive halos at $z=0$.  This drop relative to No-X is concordant with the drop in the stellar fraction produced in massive galaxies~\\citep{Simba}, thus indicates the effect of X-ray feedback in fully quenching galaxies in \\mbox{{\\sc Simba}}\\xspace.\n\nOverall, jet feedback has the largest impact in altering the scaling relations from the No-Feedback case.  The No-Feedback scalings are, not surprisingly, broadly consistent with self-similar scaling, albeit with deviations at low masses owing to excessive cooling in this model~\\citep{Dave:2002}.  Jets evacuate low mass halos and result in lower temperatures for the gas that remains, giving lower $L_X$ and $T_X$ values.  The metallicity, meanwhile, is primarily deposited by star formation feedback at early epochs, though jet feedback tends to result in higher $Z_X$ in low-mass halos at late epochs.\n\n\n\n\n\n\n\n\n\\section{Profiles}\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width = \\textwidth]{Profiles\/Profiles_m100n1024_s50_135.pdf}\n    \\caption{Median Profiles at $z= 2,1,0$ for electron density, $\\Sigma_{L_X}$, $T_X$, and $S_X$. $z=2$ is represented by a dotted line, $z=1$ by a dashed line, and $z=0$ by a solid line. The median profiles of halos with $M_{500} < 10^{13.5} M_{\\odot}$ are coloured blue while those $> 10^{13.5} M_{\\odot}$ are represented by the red lines. The $\\Sigma_{L_X}$ profiles are scaled to their values at $R_{200}$ to accentuate the evolution in shape however the shift in trends can still be seen in the other plots. $n_e$ shows a profile decreasing with radius at all redshifts, with profiles flattening due to a higher decrease in the core regions as we move from $z=2$ to $z=0$. We also see overall higher densities at all radii within the $M_{500} > 10^{13.5}$ bin at $z=0$, a shift from the similar levels at previous redshift. $\\Sigma_{L_X}$ profiles for $M_{500} > 10^{13.5}$ show a slight flattening with redshift. This evolution matches closely in the outer regions of the $M_{500} < 10^{13.5}$ halos, however these halos exhibit a growing low luminosity core, with the radial extent of the core increasing with redshift to $\\approx 0.1R_{200}$ at $z=0$. $T_X$ profiles show little evolution in shape, however demonstrate a more significant decrease in overall temperature in the lower mass halos, bringing both mass bins in line with each other by $z=0$. $S_X$ profiles flatten with redshift driven by a higher increase in entropy in core regions.   }\n    \\label{fig:Profiles_fiducial}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width = \\textwidth]{Profiles\/Profiles_m50n512_s50nojet_135.pdf}\n    \\caption{Median profils at $z=2,1,0$ for electron density, $\\Sigma_{L_X}$, $T_X$, and $S_X$ in the $50h^{-1}{\\rm Mpc}$ No-Jet \\mbox{{\\sc Simba}}\\xspace run.  $z=2$ is represented by a dotted line, $z=1$ by a dashed line, and z=0 by a solid line. The median profiles of halos with $M_{500} < 10^{13.5} M_{\\odot}$ are coloured blue while those $> 10^{13.5} M_{\\odot}$ are represented by the red lines. The $\\Sigma_{L_X}$ profiles are scaled to their values at $R_{200}$ to accentuate the evolution in shape however the shift in trends can still be seen in the other plots. $n_e$ profiles experience an overall decrease with redshift, maintaining much steeper profiles than the fiducial runs. There is little difference between mass bins. $\\Sigma_{L_X}$ profiles experience very little evolution in shape, with the least massive halos developing a slightly flattened core at $z=0$. $T_X$ profiles experience an overall decrease in both mass bins, however fail to converge at $z=0$.  Little evolution is seen in the shape of the profiles at either mass bin. $S_X$ profiles experience a smaller increase especially at lower masses, with very little evolution seen in the shape of the profiles.}\n    \\label{fig:Profiles_50_s50nojet}\n\\end{figure*}\n\n \nGlobal $L_X-$weighted measures tend to weight central regions more heavily where much of the X-ray emission arises, resulting in the properties used in these scaling relations being more sensitive to the core.  It is interesting to ask where in radius the various feedback processes in \\mbox{{\\sc Simba}}\\xspace impact the observable properties. To garner a more detailed view of where feedback impacts the halo we examine the evolution of these halos' profiles. Thus we examine the evolution of profiles in the fiducial $100h^{-1}{\\rm Mpc}$ \\mbox{{\\sc Simba}}\\xspace run for electron density ($n_e$), X-ray surface brightness $\\Sigma_{L_X}$, $T_X$, and entropy $S_X$, and subsequently compare with the No-Jet run since we have shown that the jets have the largest impact on the hot gas properties.\n\nWe generate projected (2-D) profiles for halos with $10^{12.5}< M_{500} < 10^{13.5} M_{\\odot}$ and $M_{500} > 10^{13.5} M_{\\odot}$, by stacking halos in these mass bins scaling each to their respective $R_{200}$ values. These profiles are plotted for $z=2,1,0$. Profiles at $z=3$ show similar trends as at $z=2$ and so were omitted for clarity. \n\\subsection{ Full \\mbox{{\\sc Simba}}\\xspace Profiles }\n\nFigure~\\ref{fig:Profiles_fiducial} shows the median profiles for electron density ($n_e$) in the top left plot, $L_X$ surface density ($\\Sigma_{L_X}\\equiv L_X\/A$ where $A$ is the surface area of the 2-D radial bin) scaled by its value at $R_{200}$ in the top right, $T_X$ scaled by its value at $R_{200}$ in the bottom left, and Entropy ($S_X$) in the bottom right. Here Entropy is defined as $T_X\/n_e^{2\/3}$ and is calculated by dividing the temperature of each particle by its $n_e^{2\/3}$ and then calculating the $L_X$ weighted average across the radial bin. Profiles are represented by dotted, dashed, and solid lines for $z=2, z=1$, and $z=0$ respectively. Profiles of halos with $M_{500} < 10^{13.5} M_{\\odot}$ are blue and halos with $M_{500} > 10^{13.5} M_{\\odot}$ are red. The shaded areas represent the standard error and are shown on halos with $M_{500} > 10^{13.5} M_{\\odot}$ at $z=0$. We scale $\\Sigma_{L_X}$ and $T_X$ by their values at $R_{200}$ in order to highlight the evolution in the shapes of these profiles; the amplitude evolution for these quantities was quantified in the previous section.\n\nThe top left panel shows the electron density profiles.  The most visible trend in both mass bins is a steady decrease in the amplitude with time.  Less evident but still clear is a slight flattening, indicating a stronger decrease in core density.  At $z=2$ and $z=1$ the profiles for $M_{500} < 10^{13.5} M_{\\odot}$ and $M_{500} > 10^{13.5} M_{\\odot}$ lie very close to each other, consistent with self-similarity.  However, at $z=0$ we see that the two mass bins have diverged in amplitude, with more massive halos having an overall higher electron density.   Notably, this occurs at all radii, and is not limited to the core.  This is consistent with the stronger removal of hot gas due to jet feedback that heats the gas in lower mass halos, but is not expected in a scenario where more efficient cooling in lower mass systems depletes the hot gas in the central region.\n\nMoving to the temperature profiles in the lower left panel of Figure~\\ref{fig:Profiles_fiducial}, we first note that most of the profiles are within a factor of two of being isothermal (note the small range in $y$-axis).  Within this, however, some notable differences in profile shapes are seen.  Even at early epochs, high-mass halos have a steeper profile than low-mass ones.  Both mass bins have steeper profiles with time, and by $z=0$ they exhibit a mildly cooler core at $R\\ll 0.1R_{200}$.  None of these core drops are so dramatic as to identify these systems as ``cool core\" systems, which have $T_X$ profiles dropping well below $T_X(R_{200})$ in the centres.  While some individual profiles do this, the mean profile is not a classic cool core system in either mass bin.  The variation among individual profiles and their central cooling times is an interesting study that we leave for future work; here we simply note that there are mild differences in profile shapes in the two mass bins over time.\n\nThe electron density plus the temperature are the main determinants of the X-ray luminosity.  Figure~\\ref{fig:Profiles_fiducial}, top right, shows $\\Sigma_{L_X}$ profiles scaled to their value at $R_{200}$.  Overall, the profiles in both mass bins becomes slightly shallower with time, showing a pronounced central peak at $z=2$ reflective of the peak in the $n_e$ profile then.  Already at $z=1$, differences emerge in the core, with a slight flattening of the inner profile between $z=2\\to 1$ in the low-mass halos, becoming even more pronounced at $z=0$.  The outer ($R\\ga 0.2R_{200}$) profiles remain fairly consistent at all redshifts, indicating that the drop in X-ray luminosity for low-mass systems at late times (and the associated departures from self-similarity) arises primarily from the core region, likely indicating the effect of jets on the cores of lower mass halos. \n\nFinally we investigate the X-ray weighted entropy profiles in the bottom right plot of Figure~\\ref{fig:Profiles_fiducial}. Here the combined effect of $T_X$\nand $n_e$ can be seen as given by the $S_X \\equiv T_X\/n_e^{2\/3}$ entropy definition. A steady increase in entropy can be seen in both mass bins, with lower mass halos exhibiting more flattening across redshift. While both mass bins show a similar raising of core entropies, lower mass halos do appear to exhibit overall lower entropies at early times. This is consistent with the idea that cooling is more effective in low-mass halos at early epochs, but by $z=0$, the jet feedback has evacuated gas such that it raises the entropy particularly strongly in low-mass systems.  This raised entropy is likely a critical factor to suppress cooling in these systems in order to keep galaxies quenched.\n\n\\subsection{No-Jet Profiles}\n\nWe have implicated jets in impacting the evolution of low-mass profiles in comparison with higher-mass systems.  We can test this more directly by examining the same set of profiles in our No-Jet run.  We note that here we must use the $50h^{-1}{\\rm Mpc}$ run here, but we have checked that the profiles in the $50h^{-1}{\\rm Mpc}$ full \\mbox{{\\sc Simba}}\\xspace run follows the same trends as those in the $100h^{-1}{\\rm Mpc}$ fiducial volume that we presented in the previous section.  However, this does mean that the $M_{500} > 10^{13.5} M_{\\odot}$ bin is notably less populated in the No-Jet case owing to the smaller volume.\n\nFigure \\ref{fig:Profiles_50_s50nojet} shows the profiles in the No-Jet run, \nformatted as in Figure~\\ref{fig:Profiles_fiducial}, displaying profiles at $z=0,1,2$ (solid, dotted, dashed lines) in high and low halo mass bins (red and blue, respectively).\n\nFor the $n_e$ profiles, in the top left plot, we see two interesting effects: Firstly there is significantly less evolution across mass scales, with only an order of magnitude decrease in density, as opposed to the 3 orders of magnitude decrease seen in Figure~\\ref{fig:Profiles_fiducial}. This difference in evolution is most notable in the core regions, with all halos maintain a much higher core density over redshift, even while the electron density at $R_{200}$ has fairly similar evolution with and without jets; the electron density profiles without jets are thus much steeper than with jets.\n\nSecondly the two mass bins evolve in line with each other, with the only difference being a flatter inner core electron density seen in the lower mass halos.  Thus without jets, the profiles are much more in line with self-similarity, with perhaps a mild effect from other feedback processes concentrated at $\\la 0.03R_{200}$ in lower-mass groups.  Comparing with Figure~\\ref{fig:Profiles_fiducial}, this implicates jets in lowering the electron densities across the entire mass range, and doing so across all radii.  \n\nThe $T_X$ profiles in the lower left panel of Figure~\\ref{fig:Profiles_50_s50nojet} also experience a similarly mild evolution. The shapes of the profiles remain fairly constant in both mass bins, with halos maintaining higher temperature cores.  In detail, the $T_X$ profiles are actually steeper for low mass halos vs. high-mass ones, which is opposite to the trend seen in the full-physics \\mbox{{\\sc Simba}}\\xspace case.  By $z=0$, the inner profiles in the low-mass groups shows a significant elevation, possibly due to enhanced energy input from other feedback processes combined with cooling out of the lower-temperature gas to form into stars.  Comparing to Figure~\\ref{fig:Profiles_fiducial}, the No-Jet profiles are clearly much steeper, and the effect of jets here is already evident at $z=2$ with steeper $T_X$ profiles.  This may indicate that even early low level jet feedback works to push hot X-ray emitting gas from cores to the outer regions.  This trend strengthens down to $z=0$, particularly in the low-mass groups.\n\nIn the top right plot of Figure~\\ref{fig:Profiles_50_s50nojet}, the $\\Sigma_{L_X}$ profiles are again notable for showing very little evolution in the shape of the profiles, except for a slightly lower core luminosity seen in the lower mass halos. This directly mimics the trend seen in the electron densities.  Moreover, by $z=0$, the $\\Sigma_{L_X}$ profiles are definitely steeper than in the fiducial run in Figure~\\ref{fig:Profiles_fiducial}.  This most clearly demonstrates that it is AGN jet feedback that is primarily responsible for breaking the self-similar scaling of the X-ray emission.  Other feedback and cooling effects have a much smaller impact.\n\nFinally in the bottom right panel of Figure~\\ref{fig:Profiles_50_s50nojet} we show the entropy profiles.  We see a steady but slow increase in the normalisation of the entropy profiles. The lower mass halos fail to come in line with the most massive halos, and most notably neither bin experiences a rise of core entropy with respect to the outer regions. This is in line with the idea that jets work to remove low entropy gas, especially from the cores of halos.  The low central entropy allows for substantial cooling that keeps galaxies fueled and star-forming even in high-mass halos in the No-Jet run, which does not agree with observations.  This highlights how jets are responsible for quenching star formation in massive systems, via their impact on the hot gas in the cores of massive halos.\n\nOverall we see little to no evolution in the shape of the X-ray profiles in the No-Jet case, along with a smaller evolution in the normalisation of these profiles.  This indicates that jets not only impact the overall properties, but also importantly how these properties are expressed as a function of radius. Interestingly we see jets having an effect in flattening profiles, or creating larger cores (specifically in the low mass $L_X$ profiles).  Finally while jets have a clear impact in the shapes of profiles in both mass bins, larger effects are seen in the normalisations of these profiles in the $M_{500} < 10^{13.5} M_{\\odot}$ bin, highlighting the greater impact of jet feedback on lower mass halos. \n\n\n\n\\section{Summary}\n In this paper we have examined the redshift evolution of the X-ray scaling relations and X-ray profiles of intragroup gas within the \\mbox{{\\sc Simba}}\\xspace suite of cosmological hydrodynamical galaxy formation simulations.  We consider the scaling relations at particular redshifts from $z=3\\to 0$, the median paths of halos in different mass bins through scaling relation space, and the hot gas and X-ray profiles of intragroup gas.  We isolate the physical processes driving this evolution by comparing among feedback variant \\mbox{{\\sc Simba}}\\xspace runs with different feedback models turned on an off, in comparison with the baseline model of self-similar evolution.  We show that: \n \n \\begin{itemize}\n     \\item \\mbox{{\\sc Simba}}\\xspace halos show are consistent with a self-similar slope of their X-ray scaling relations at $z\\ga 1.5$, while at $z\\la 1.5$ there are an increasing number of groups that are substantially deviated from self-similarity.\n     \\item Evolution of the $L_X-M_{500}$ scaling relation sees significant change occurring after $z\\la 1.5$ driven primarily by a decrease in $L_X$ among halos with $M_{500} < 10^{13.5} M_{\\odot}$. This is further corroborated by median tracks of halos showing that the least massive halos show a 2~dex drop in $L_X$ for a 0.3~dex increase in $M_{500}$ while the most massive halos show a slight increase in $L_X$ for an order of magnitude increase in $M_{500}$ from $z=3\\to 0$.\n     \\item Evolution of the $T_X - M_{500}$ appears to be driven by an overall decrease in $T_X$ for halos with $M_{500}<10^{13.5} M_{\\odot}$. Following halo over time indicate that the least massive halos at $z=3$ do not lower in $T_X$ suggesting that halos crossing above $10^{12.3} M_{\\odot}$ at later epochs are responsible for the lowering of $T_X$ in the smaller halos.\n     \\item The hot gas fraction $f_{\\rm gas}$ vs. $M_{500}$ shows dramatic evolution at $z\\la 1.5$ with a large drop and increasing in scatter particularly for $M_{500} < 10^{13.5} M_{\\odot}$ systems. This scatter is highly correlated with the black hole mass fraction $f_{BH}\\equiv M_{BH}\/M_{500}$, such that halos with central galaxies having high $f_{BH}$ have low $f_{\\rm gas}$ and vice-versa.\n     Median tracks show that halos with $M_{500} > 10^{13.5} M_{\\odot}$ have hot gas fractions encompassing the majority of the cosmic baryon fraction, while $M_{500} < 10^{13.5} M_{\\odot}$ halos show an order of magnitude decrease. \n    \n     \\item The X-ray weighted metallicity shows a significant trend with the stellar baryon fraction in halos at early epochs, which gets washed out over time.  Larger groups have a metallicity around one-third solar, while smaller groups generally have slightly higher metallicity but with a very large scatter.\n     \n     \\item The $L_X-T_X$ relation shows an increasing departure from self-similarity with time, with a growing population at $T_X\\la 1$~keV having lower $L_X$ that becomes particularly prominent at $z\\la 1.5$. The halos that are most deviant from self-similarity are the ones with the highest $f_{BH}$, circumstantially implicating black hole feedback as the cause of this deviation.\n     \n     \\item To pinpoint the particular physical processes driving the above evolution, we examine smaller $50h^{-1}{\\rm Mpc}$ \\mbox{{\\sc Simba}}\\xspace runs with specific feedback modules turned off.  Turning off both AGN and star formation feedback results in little evolution in $f_{\\rm gas}$, and modest evolution in $L_X$ and $T_X$ mostly consistent with self-similarity.\n     \n     \\item Introducing stellar (but no AGN) feedback does not change $f_{\\rm gas}$, $L_X$, or $T_X$, but it does result in significantly earlier deposition of metals into the hot gas even by $z=3$.\n     \n     \\item Introducing AGN jet feedback has a dramatic impact on scaling relations.  Jets are clearly implicated in lowering \n     $f_{\\rm gas}$ in $M_{500} < 10^{13.5} M_{\\odot}$ groups, particularly at $z\\la 1.5$.  This reflects in a dramatic reduction in $L_X-M_{500}$ in such groups over time, and an increased reduction in $T_X-M_{500}$ with time. \n     \n     \\item Remarkably, the $L_X-T_X$ evolution undergoes a sort of cancellation, in that the lowering of both $L_X$ and $T_X$ results in groups evolving mostly along the relation.  The net effect is that groups today end up much more spread out along the $L_X-T_X$ relation, but the overall relation is surprisingly similar in slope and amplitude in all the feedback variants.\n     \n     \\item The $Z_X - M_{500}$ evolution is unique in that it is more impacted by stellar feedback than AGN feedback. With no feedback, a steady increase is seen in metallicity. The inclusion of stellar feedback produces hot gas metallicities $>0.1Z_\\odot$ even by $z=3$. The inclusion of jet and X-ray feedback then seems to reintroduce a modest level of upwards evolution in metallicity particularly in low-mass halos, resulting in the final $z=0$ relation that increases mildly to low halo masses.\n     \n     \\item Electron density profiles in \\mbox{{\\sc Simba}}\\xspace drop and flatten with time, at all masses.  The drop in smaller groups from $z\\sim 1\\to 0$ is however significantly greater than in larger systems.  \n     $L_X$ profiles also experience a flattening at all mass ranges, however it is most prominent at $M_{500} < 10^{13.5}M_{\\odot}$, growing a constant surface brightness core ($R\\la 0.1R_{200}$) by $z=0$.  The temperature evolution is not dramatic, but entropy is also flattened, resulting in all groups having median core entropies above 100~keV~cm$^2$. \n     \\item Turning off jet feedback has significant impact on both  the shape and normalisation of these profiles. Electron density, temperature, and luminosity profiles are steeper.  There is much less difference between the two mass bins, with only a very small core appearing at late times in the lower mass groups.  The entropy profile is much steeper, with typical entropies at $\\sim 0.01R_{200}$ below 10~keV~cm$^2$ at all masses. \n    \n\\end{itemize}\n \nThese results overall show that the implementation of AGN jet feedback in \\mbox{{\\sc Simba}}\\xspace has a significant impact on X-ray properties both globally and in terms of their profiles.  It is this feedback that causes strong departures from self-similarity, which tend to become noticeable around $z\\sim 1.5$ as AGN jets turn on in central group galaxies, and that results in intragroup gas that closely matches X-ray observations at $z=0$ as shown in \\citet{Robson:2020}.  The evolution of X-ray profiles provides further constraints on these models, demonstrating the effects of feedback in evacuating low entropy gas and creating X-ray surface brightness cores particularly in lower-mass groups.\n     \nA dearth of observations at these mass scales at $z\\gg 0$, along with an overall lack of consensus caused by selection biases in available results, makes it premature to compare the evolution of \\mbox{{\\sc Simba}}\\xspace's groups to X-ray observations. With future missions such as {\\it Lynx} and {\\it Athena} providing significantly deeper observations that will push toward higher redshifts, it should be possible to further constrain these models providing even greater insight into exactly how AGN feedback impacts the growth of groups and clusters. In future work we plan to examine how these next-generation telescopes will help uncover the physics of gas within groups and clusters, with the successes of \\mbox{{\\sc Simba}}\\xspace suggesting that it provides a suitable framework for interpreting these observations. \n    \n    \n\n\n\\newpage\n\\section*{Acknowledgements}\n\nThe authors thank Weiguang Cui, Nicole Thomas, Sarah Appleby, and Rohit Kondapally for helpful discussions.\nRD acknowledges support from the Wolfson Research Merit Award program of the U.K. Royal Society.\nThis work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility. The equipment was funded by BEIS capital funding via STFC capital grants ST\/P002293\/1, ST\/R002371\/1 and ST\/S002502\/1, Durham University and STFC operations grant ST\/R000832\/1. DiRAC is part of the National e-Infrastructure.\n\n\\section*{Data Availability}\nThe raw simulation data and catalogs for all the runs used in this paper are available at {\\tt https:\/\/simba.roe.ac.uk}.  The X-ray data will be made available on request to the lead author.\n\\label{lastpage}\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\n\nIn the monograph of Gilbert [16], by applying methods of complex analysis, integral representations of solutions of the generalized bi-axially Helmholtz equation\n$$\nH_{\\alpha ,\\beta }^\\lambda  \\left( u \\right) \\equiv u_{xx}  + u_{yy}  + \\frac{{2\\alpha }}\n{x}u_x  + \\frac{{2\\beta }} {y}u_y  - \\lambda ^2 u = 0, \\eqno \\left( H_{\\alpha ,\\beta }^\\lambda \\right)\n$$\nwere constructed via analytic functions. Here $0 < 2\\alpha ,2\\beta  < 1,$ $\\alpha ,\\beta ,\\lambda $ are constants. When $\n\\lambda  = 0 $ this equation is known as the equation of generalized axially symmetric potential theory. This terminology was used for the first time by Weinstein, who first considered fractional dimensional spaces in the potential theory [33, 34]. The special case whith $\\,\\lambda  = 0 $\nhas also been investigated by Erdelyi [5, 6], Gilbert [9-15], Ranger [29], Henrici [21, 22]. There are many works [1-3, 8, 17, 23, 25-28, 30, 32] in which some problems for equation $\\left(H_{\\alpha ,\\beta }^\\lambda\\right)$ were studied. In the paper [18] for equation $\\left(H_{\\alpha ,\\beta }^\\lambda\\right)$ the following fundamental solutions on $ R_2^ +   = \\left\\{ {\\left( {x,y} \\right):x > 0,y > 0} \\right\\}:$ have been constructed\n$$\nq_1 \\left( {x,y;x_0 ,y_0 } \\right) = k_1 \\left( {r^2 } \\right)^{ - \\alpha  - \\beta } A_2^{\\left( 3 \\right)} \\left( {\\alpha  + \\beta ;\\alpha ,\\beta ;2\\alpha ,2\\beta ;\\xi ,\\eta ,\\zeta } \\right),\\eqno(1.1)\n$$\n$$\nq_2 \\left( {x,y;x_0 ,y_0 } \\right) = k_2 \\left( {r^2 } \\right)^{\\alpha  - \\beta  - 1} x^{1 - 2\\alpha } x_0^{1 - 2\\alpha } A_2^{\\left( 3 \\right)} \\left( {1 - \\alpha  + \\beta ;1 - \\alpha ,\\beta ;2 - 2\\alpha ,2\\beta ;\\xi ,\\eta ,\\zeta } \\right),\\eqno(1.2)\n$$\n$$\nq_3 \\left( {x,y;x_0 ,y_0 } \\right) = k_3 \\left( {r^2 } \\right)^{ - \\alpha  + \\beta  - 1} y^{1 - 2\\beta } y_0^{1 - 2\\beta } A_2^{\\left( 3 \\right)} \\left( {1 + \\alpha  - \\beta ;\\alpha ,1 - \\beta ;2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right),\\eqno(1.3)\n$$\n$$\n\\begin{array}{l}\n  q_4 \\left( {x,y;x_0 ,y_0 } \\right) \\hfill \\\\\n   = k_4 \\left( {r^2 } \\right)^{\\alpha  + \\beta  - 2} x^{1 - 2\\alpha } y^{1 - 2\\beta } x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } A_2^{\\left( 3 \\right)} \\left( {2 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right), \\hfill \\\\\n\\end{array}\\eqno(1.4)\n$$\nwhere\n$$\nk_1  = \\frac{{2^{2\\alpha  + 2\\beta } }}\n{{4\\pi }}\\frac{{\\Gamma \\left( \\alpha  \\right)\\Gamma \\left( \\beta  \\right)\\Gamma \\left( {\\alpha  + \\beta } \\right)}}\n{{\\Gamma \\left( {2\\alpha } \\right)\\Gamma \\left( {2\\beta } \\right)}},\\eqno(1.5)\n$$\n$$\nk_2  = \\frac{{2^{2 - 2\\alpha  + 2\\beta } }}\n{{4\\pi }}\\frac{{\\Gamma \\left( {1 - \\alpha } \\right)\\Gamma \\left( \\beta  \\right)\\Gamma \\left( {1 - \\alpha  + \\beta } \\right)}}\n{{\\Gamma \\left( {2 - 2\\alpha } \\right)\\Gamma \\left( {2\\beta } \\right)}},\\eqno(1.6)\n$$\n$$\nk_3  = \\frac{{2^{2 + 2\\alpha  - 2\\beta } }}\n{{4\\pi }}\\frac{{\\Gamma \\left( \\alpha  \\right)\\Gamma \\left( {1 - \\beta } \\right)\\Gamma \\left( {1 + \\alpha  - \\beta } \\right)}}\n{{\\Gamma \\left( {2\\alpha } \\right)\\Gamma \\left( {2 - 2\\beta } \\right)}},\\eqno(1.7)\n$$\n$$\nk_4  = \\frac{{2^{4 - 2\\alpha  - 2\\beta } }}\n{{4\\pi }}\\frac{{\\Gamma \\left( {1 - \\alpha } \\right)\\Gamma \\left( {1 - \\beta } \\right)\\Gamma \\left( {2 - \\alpha  - \\beta } \\right)}}\n{{\\Gamma \\left( {2 - 2\\alpha } \\right)\\Gamma \\left( {2 - 2\\beta } \\right)}},\\eqno(1.8)\n$$\n$$\nr^2=\\left(x-x_0\\right)^2+\\left(y-y_0\\right)^2,\\,\\,r_1^2=\\left(x+x_0\\right)^2+\\left(y-y_0\\right)^2,\\,\\,r_2^2=\\left(x-x_0\\right)^2+\\left(y+y_0\\right)^2,\n$$\n$$\n\\xi  = \\frac{{r^2  - r_1^2 }}\n{{r^2 }},\n\\eta  = \\frac{{r^2  - r_2^2 }}\n{{r^2 }},\n\\zeta  =  - \\frac{{\\lambda ^2 }}\n{4}r^2 ,\\eqno(1.9)\n$$\n$$\nA_2^{\\left( 3 \\right)} \\left( {a;b_1 ,b_2 ;c_1 ,c_2 ;x,y,z} \\right) = \\sum\\limits_{m,n,p = 0}^\\infty  {} \\frac{{\\left( a \\right)_{m + n - p} \\left( {b_1 } \\right)_m \\left( {b_2 } \\right)_n }}\n{{\\left( {c_1 } \\right)_m \\left( {c_2 } \\right)_n m!n!p!}}x^m y^n z^p ,\\eqno(1.10)\n$$\nand $\\left( a \\right)_n  = \\Gamma \\left( {a + n} \\right)\/\\Gamma \\left( a \\right)$ is the Pochhammer symbol.\n\n\\section{Green's formulas.}\n\nWe consider an identity\n$$\nx^{2\\alpha } y^{2\\beta } \\left[ {uH_{\\alpha ,\\beta }^\\lambda  \\left( v \\right) - vH_{\\alpha ,\\beta }^\\lambda  \\left( u \\right)} \\right] = \\frac{\\partial }\n{{\\partial x}}\\left[ {x^{2\\alpha } y^{2\\beta } \\left( {v_x u - vu_x } \\right)} \\right] + \\frac{\\partial }\n{{\\partial y}}\\left[ {x^{2\\alpha } y^{2\\beta } \\left( {v_y u - vu_y } \\right)} \\right].\\eqno(2.1)\n$$\nIntegrating both parts of identity (2.1) over $\\Omega  \\subset R_2^ +$, and using Green's formula we find\n$$\n\\int\\limits_\\Omega  {}x^{2\\alpha } y^{2\\beta } \\left[ {uH_{\\alpha ,\\beta }^\\lambda  \\left( v \\right) - vH_{\\alpha ,\\beta }^\\lambda  \\left( u \\right)} \\right]dxdy = \\int\\limits_S {} x^{2\\alpha } y^{2\\beta } u\\left( {v_x dy - v_y dx} \\right) - x^{2\\alpha } y^{2\\beta } v\\left( {u_x dy - u_y dx} \\right),\\eqno(2.2)\n$$\nwhere $S = \\partial \\Omega$ is the boundary of the domain $\\Omega.$ Formula (2.2) named as Green's formula is deduced under the following assumptions: \\\\\n- the functions $u$ and $v$ are continuous on the closure of the domain $\\Omega$, i.e. on $\\bar \\Omega,$ \\\\\n- the partial derivatives of the first and second orders of $u$ and $v$ are continuous on $\\Omega,$ \\\\\n- the integrals over $\\Omega,$ containing partial derivatives of the first and second orders of $u$ and $v$ have sense. \\\\\nIf  $u$, $v$ are solutions of the equation $\\left(H_{\\alpha ,\\beta }^\\lambda\\right)$, then by formula (2.2) we get\n$$\n\\int\\limits_S {} x^{2\\alpha } y^{2\\beta } \\left( {u\\frac{{\\partial v}}\n{{\\partial n}} - v\\frac{{\\partial u}}\n{{\\partial n}}} \\right)ds = 0,\\eqno(2.3)\n$$\nwhere\n$$\n\\frac{\\partial }\n{{\\partial n}} = \\frac{{dy}}\n{{ds}}\\frac{\\partial }\n{{\\partial x}} - \\frac{{dx}}\n{{ds}}\\frac{\\partial }\n{{\\partial y}}, \\,\\,\\frac{{dy}}\n{{ds}} = \\cos \\left( {n,x} \\right), \\,\\,\\, \\frac{{dx}}\n{{ds}} =  - \\cos \\left( {n,y} \\right),\\eqno(2.4)\n$$\n$n$ is the exterior normal to the curve $S.$ The following identity also takes place:\n$$\n\\int\\limits_\\Omega  {}x^{2\\alpha } y^{2\\beta } \\left[ {u_x^2  + u_y^2  + \\lambda ^2 u^2 } \\right]dxdy = \\int\\limits_S {} x^{2\\alpha } y^{2\\beta } u\\frac{{\\partial u}}\n{{\\partial n}}ds,\\eqno(2.5)\n$$\nwhere $u$ is a solution of the equation $\\left(H_{\\alpha ,\\beta }^\\lambda\\right)$.\n\n\\section{The formulation and the uniqueness of the Dirichlet problem. }\n\nLet $\\Omega  \\subset R_2^ +   = \\left\\{ {\\left( {x,y} \\right):x > 0,y > 0} \\right\\}$ be a domain limited by intervals $I_1  = \\left( {0,a} \\right),a = const > 0$, $I_2  = \\left( {0,b} \\right),b = const > 0$ of the axis $OX$, $OY$ respectively, and a curve $\\Gamma $ with endpoints $A\\left( {a,0} \\right),$ $B\\left( {0,b} \\right).$ The parametrical equation of the curve $\\Gamma $ will be $x = x\\left( s \\right), \\,y = y\\left( s \\right),$ where $s$ is the length of the arc counted from the point $A\\left( {a,0} \\right).$ Concerning the curve $\\Gamma $ we shall assume that: \\\\\n- the functions $x = x\\left( s \\right),y = y\\left( s \\right)$ have continuous derivatives $x'\\left( s \\right),y'\\left( s \\right)$\n on the segment $\\left[ {0,l} \\right],$ not simultaneously equal to zero, where $l$ is length of the curve $\\Gamma$; \\\\\n- the derivatives $x''\\left( s \\right),y''\\left( s \\right)$ satisfy to the H\\\"{o}lder condition on $\\left[ {0,l} \\right]$;  \\\\\n- in neighborhoods of the points $A\\left( {a,0} \\right)$ and $B\\left( {0,b} \\right)$ the conditions\n$$\n\\left| {\\frac{{dx}} {{ds}}} \\right| \\leq Cy^{1 + \\varepsilon } \\left( s \\right), \\,\\,\n\\left| {\\frac{{dy}} {{ds}}} \\right| \\leq Cx^{1 + \\varepsilon } \\left( s \\right), \\,\\, 0 < \\varepsilon  < 1, \\,\\, C = const,\n$$\nare satisfied. \\\\\n\\textbf{Problem $D.$} Find a solution $u$ of equation $\\left(H_{\\alpha ,\\beta }^\\lambda\\right)$ belonging to the class $C\\left( {\\bar \\Omega } \\right) \\cap C^2 \\left( {\\Omega } \\right),$ satisfying the conditions\n$$\n\\left. {u\\left( {x,y} \\right)} \\right|_{y = 0}  = \\tau _1 \\left( x \\right),\\,\\, x \\in \\bar I_1 ,\\eqno(3.1)\n$$\n$$\n\\left. {u\\left( {x,y} \\right)} \\right|_{x = 0}  = \\tau _2 \\left( y \\right), \\,\\, y \\in \\bar I_2 ,\\eqno(3.2)\n$$\n$$\n\\left. {u\\left( {x,y} \\right)} \\right|_\\Gamma   = \\varphi \\left( s \\right),\\,\\,  0 \\leq s \\leq l,\\eqno(3.3)\n$$\nwhere $\\tau _1,\\tau _2,\\varphi $ are given continuous functions and  $\\tau _1 \\left( 0 \\right) = \\tau _2 \\left( 0 \\right),$ $\\tau _1 \\left( a \\right) = \\varphi \\left( 0 \\right),$ $\\tau _2 \\left( b \\right) = \\varphi \\left( l \\right).$\n\n\\textbf{Theorem 1.} If the problem $D$ has a solution in the domain $\\Omega $, then it is unique.\\\\\n\\textbf{Proof.}  Let $\\tau _1 \\left( x \\right) = \\tau _2 \\left( y \\right) = \\varphi \\left( s \\right) = 0$ then by virtue of identity (2.5), we have\n$$\n\\int\\limits_{\\Omega } {}x^{2\\alpha } y^{2\\beta } \\left[ {u_x^2  + u_y^2  + \\lambda ^2 u^2 } \\right]dxdy = 0.\\eqno(3.4)\n$$\nBy (3.4) it follows that $u_x \\left( {x,y} \\right) = u_y \\left( {x,y} \\right) = u\\left( {x,y} \\right) = 0.$ Hence, we have $u\\left( {x,y} \\right) \\equiv 0$ in the domain $\\Omega $. $\\Box$\n\nWe note that the uniqueness of a solution of the problem $D$\nin the domain $\\Omega $ also follows by the extremum principle for elliptic differential equations.\n\n\\section{The existence theorem}\n\nLet $a = b$ and $\\Gamma =:\\{(x,y)\\in R_2^+: x^2  + y^2  = a^2\\}$. We denote this domain by $\\Omega _0$. The function $G_4 \\left( {x,y;x_0 ,y_0 } \\right)$ satisfying the following conditions is called as Green's function of the problem $D:$ \\\\\n- inside the domain $\\Omega _0 ,$ except for the point $\\left( {x_0 ,y_0 } \\right),$ this function is a regular solution of equation $\\left(H_{\\alpha ,\\beta }^\\lambda\\right)$; \\\\\n- it satisfies the boundary condition\n$$\n\\left. {G_4 \\left( {x,y;x_0 ,y_0 } \\right)} \\right|_{\\Gamma  \\cup I_1  \\cup I_2 }  = 0;\\eqno(4.1)\n$$\n- it can be represented in the form\n$$\nG_4 \\left( {x,y;x_0 ,y_0 } \\right) = q_4 \\left( {x,y;x_0 ,y_0 } \\right) - \\left( {R_0^2 } \\right)^{ - \\alpha  - \\beta } q_4 \\left( {x,y;\\bar x_0 ,\\bar y_0 } \\right),\\eqno(4.2)\n$$\nwhere\n$$\nR_0^2  = x_0^2  + y_0^2 , \\,\\, \\bar x_0  = \\frac{{x_0 }} {{R_0^2 }},\\,\\, \\bar y_0  = \\frac{{y_0 }} {{R_0^2 }},\\eqno(4.3)\n$$\n$q_4 \\left( {x,y;x_0 ,y_0 } \\right)$ is a fundamental solution, $q_4 \\left( {x,y;\\bar x_0 ,\\bar y_0 } \\right)$ is a regular solution of equation  $\\left(H_{\\alpha ,\\beta }^\\lambda\\right)$ in the domain $\\Omega _0 .$\n\nLet $\\left( {x_0 ,y_0 } \\right) \\in \\Omega _0.$ We cut out from $\\Omega _0 $ a circle of small radius $\\rho $ with the center at the point $\\left( {x_0 ,y_0 } \\right)$ and the remaining part of $\\Omega _0,$ we denote by $\\Omega _0^\\rho.$ $C_\\rho  $ is a boundary of the cutted out circle. Applying  formula (2.3), we obtain\n$$\n\\begin{array}{l}\n\\displaystyle \\int\\limits_{C_\\rho  } {x^{2\\alpha } y^{2\\beta } u\\frac{{\\partial G_4 \\left( {x,y;x_0 ,y_0 } \\right)}}\n{{\\partial n}}ds}  - \\int\\limits_{C_\\rho  } {x^{2\\alpha } y^{2\\beta } G_4 \\left( {x,y;x_0 ,y_0 } \\right)\\frac{{\\partial u}}\n{{\\partial n}}ds}  \\hfill \\\\\n= \\displaystyle \\int\\limits_0^a {\\left. {x^{2\\alpha } y^{2\\beta } \\tau _1 \\left( x \\right)\\frac{\\partial }\n{{\\partial y}}G_4 \\left( {x,y;x_0 ,y_0 } \\right)} \\right|_{y = 0} dx}  \\hfill \\\\\n+ \\displaystyle \\int\\limits_0^a {x^{2\\alpha } y^{2\\beta } \\tau _2 \\left( y \\right)\\left. {\\frac{\\partial }\n{{\\partial x}}G_4 \\left( {x,y;x_0 ,y_0 } \\right)} \\right|_{x = 0} dy}  - \\int\\limits_\\Gamma  {x^{2\\alpha } y^{2\\beta } \\varphi \\left( s \\right)\\frac{{\\partial G_4 \\left( {x,y;x_0 ,y_0 } \\right)}}\n{{\\partial n}}ds} . \\hfill \\\\\n\\end{array}\\eqno(4.4)\n$$\nUsing the derivation formula\n$$\n\\begin{array}{l}\n\\displaystyle  \\frac{{\\partial ^{i + j + k} }}\n{{\\partial x^i \\partial y^j \\partial z^k }}A_2^{\\left( 3 \\right)} \\left( {a;b_1 ,b_2 ;c_1 ,c_2 ;x,y,z} \\right) \\hfill \\\\\n   = \\displaystyle \\frac{{\\left( a \\right)_{i + j - k} \\left( {b_1 } \\right)_i \\left( {b_2 } \\right)_j }}\n{{\\left( {c_1 } \\right)_i \\left( {c_2 } \\right)_j }}A_2^{\\left( 3 \\right)} \\left( {a + i + j - k;b_1  + i,b_2  + j;c_1  + i,c_2  + j;x,y,z} \\right), \\hfill \\\\\n\\end{array}\\eqno(4.5)\n$$\nand considering the adjacent relation\n$$\n\\begin{array}{l}\n\\displaystyle \\frac{{ab_1 }}\n{{c_1 }}xA_2^{\\left( 3 \\right)} \\left( {1 + a;1 + b_1 ,b_2 ;1 + c_1 ,c_2 ;x,y,z} \\right) + \\frac{{ab_2 }}\n{{c_2 }}yA_2^{\\left( 3 \\right)} \\left( {1 + a;b_1 ,1 + b_2 ;c_1 ,1 + c_2 ;x,y,z} \\right) -  \\hfill \\\\\n- \\displaystyle \\frac{1}\n{{a - 1}}zA_2^{\\left( 3 \\right)} \\left( {a - 1;b_1 ,b_2 ;c_1 ,c_2 ;x,y,z} \\right) \\hfill \\\\\n= \\displaystyle aA_2^{\\left( 3 \\right)} \\left( {1 + a;b_1 ,b_2 ;c_1 ,c_2 ;x,y,z} \\right) - aA_2^{\\left( 3 \\right)} \\left( {a;b_1 ,b_2 ;c_1 ,c_2 ;x,y,z} \\right), \\hfill \\\\\n\\end{array}\\eqno(4.6)\n$$\nwe find that\n$$\n\\begin{array}{l}\nx^{2\\alpha } \\displaystyle \\frac{\\partial }\n{{\\partial x}}q_4 \\left( {x,y;x_0 ,y_0 } \\right) \\hfill \\\\\n= k_4 \\left( {1 - 2\\alpha } \\right)\\left( {r^2 } \\right)^{\\alpha  + \\beta  - 2} x_0^{1 - 2\\alpha } \\left( {yy_0 } \\right)^{1 - 2\\beta } A_2^{\\left( 3 \\right)} \\left( {2 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right) \\hfill \\\\\n- 2k_4 \\left( {2 - \\alpha  - \\beta } \\right)x\\left( {r^2 } \\right)^{\\alpha  + \\beta  - 3} x_0^{2 - 2\\alpha } \\left( {yy_0 } \\right)^{1 - 2\\beta }  A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;2 - \\alpha ,1 - \\beta ;3 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right) \\hfill \\\\\n- 2k_4 \\left( {2 - \\alpha  - \\beta } \\right)x\\left( {r^2 } \\right)^{\\alpha  + \\beta  - 3} x_0^{1 - 2\\alpha } \\left( {yy_0 } \\right)^{1 - 2\\beta } \\left( {x - x_0 } \\right) A_2 \\left( {3 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right) \\hfill \\\\\n\\end{array}\\eqno(4.7)\n$$\nand\n$$\n\\begin{array}{l}\ny^{2\\beta } \\displaystyle \\frac{\\partial }\n{{\\partial y}}q_4 \\left( {x,y;x_0 ,y_0 } \\right) \\hfill \\\\\n= k_4 \\left( {1 - 2\\beta } \\right)\\left( {r^2 } \\right)^{\\alpha  + \\beta  - 2} \\left( {xx_0 } \\right)^{1 - 2\\alpha } y_0^{1 - 2\\beta } A_2^{\\left( 3 \\right)} \\left( {2 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right) \\hfill \\\\\n- 2k_4 \\left( {2 - \\alpha  - \\beta } \\right)y\\left( {r^2 } \\right)^{\\alpha  + \\beta  - 3} \\left( {xx_0 } \\right)^{1 - 2\\alpha } y_0^{2 - 2\\beta } A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;1 - \\alpha ,2 - \\beta ;2 - 2\\alpha ,3 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right) \\hfill \\\\\n- 2k_4 \\left( {2 - \\alpha  - \\beta } \\right)y\\left( {r^2 } \\right)^{\\alpha  + \\beta  - 3} \\left( {xx_0 } \\right)^{1 - 2\\alpha } y_0^{1 - 2\\beta } \\left( {y - y_0 } \\right) \\hfill A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right). \\hfill \\\\\n\\end{array}\\eqno(4.8)\n$$\nIt is easy to prove that the following formulas are true:\n$$\nA_2^{\\left( 3 \\right)} \\left( {a;b_1 ,b_2 ;c_1 ,c_2 ;0,y,z} \\right) = \\sum\\limits_{n,p = 0}^\\infty  {} \\frac{{\\left( a \\right)_{n - p} \\left( {b_2 } \\right)_n }}\n{{\\left( {c_2 } \\right)_n n!p!}}y^n z^p  = H_3 \\left( {a,b_2 ;c_2 ;y,z} \\right),\\eqno(4.9)\n$$\n$$\nA_2 \\left( {a;b_1 ,b_2 ;c_1 ,c_2 ;x,0,z} \\right) = \\sum\\limits_{m,p = 0}^\\infty  {} \\frac{{\\left( a \\right)_{m - p} \\left( {b_1 } \\right)_m }}\n{{\\left( {c_1 } \\right)_m m!p!}}x^m z^p  = H_3 \\left( {a,b_1 ;c_1 ;x,z} \\right),\\eqno(4.10)\n$$\nwhere $H_3 \\left( {a,b;c;x,y} \\right)$ is Kummer's hypergeometric function in two arguments ([7], p. 221, formula (31)). By virtue of equalities (4.7),(4.8),(4.9), (4.10) and taking into account that $\\left. \\xi  \\right|_{x = 0}  = 0,$ $\\left. \\eta  \\right|_{y = 0}  = 0$, we get\n$$\n\\begin{array}{l}\n  y^{2\\beta } \\left. { \\displaystyle  \\frac{\\partial }\n{{\\partial y}}G_4 \\left( {x,y;x_0 ,y_0 } \\right)} \\right|_{y = 0}  = k_4 \\left( {1 - 2\\beta } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } x^{1 - 2\\alpha }  \\hfill \\\\\n   \\times  \\displaystyle \\left\\{ {\\frac{{H_3 \\left( {2 - \\alpha  - \\beta ,1 - \\alpha ;2 - 2\\alpha ;\\rho _1 ,\\rho _1^* } \\right)}}\n{{\\left[ {\\left( {x - x_0 } \\right)^2  + y_0^2 } \\right]^{2 - \\alpha  - \\beta } }} - \\displaystyle \\frac{{H_3 \\left( {2 - \\alpha  - \\beta ,1 - \\alpha ;2 - 2\\alpha ;\\rho _2 ,\\rho _2^* } \\right)}}\n{{\\left[ {\\left( {a - \\displaystyle \\frac{{xx_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x^2 y_0^2 } \\right]^{2 - \\alpha  - \\beta } }}} \\right\\} \\hfill \\\\\n\\end{array}\\eqno(4.11)\n$$\nand\n$$\n\\begin{array}{l}\n  x^{2\\alpha } \\left. {\\displaystyle \\frac{\\partial }\n{{\\partial x}}G_4 \\left( {x,y;x_0 ,y_0 } \\right)} \\right|_{x = 0}  = k_4 \\left( {1 - 2\\alpha } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } y^{1 - 2\\beta }  \\hfill \\\\\n  \\times \\displaystyle \\left\\{ {\\frac{{H_3 \\left( {2 - \\alpha  - \\beta ,1 - \\beta ;2 - 2\\beta ;\\rho _3 ,\\rho _3^* } \\right)}}\n{{\\left[ {x_0^2  + \\left( {y - y_0^2 } \\right)^2 } \\right]^{2 - \\alpha  - \\beta } }} - \\frac{{H_3 \\left( {2 - \\alpha  - \\beta ,1 - \\beta ;2 - 2\\beta ;\\rho _4 ,\\rho _4^* } \\right)}}\n{{\\left[ {\\left( {a - \\displaystyle \\frac{{yy_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x_0^2 y^2 } \\right]^{2 - \\alpha  - \\beta } }}} \\right\\}, \\hfill \\\\\n\\end{array}\\eqno(4.12)\n$$\nwhere\n$$\n\\begin{array}{l}\n \\displaystyle \\rho _1  = \\frac{{ - 4xx_0 }}\n{{\\left( {x - x_0 } \\right)^2  + y_0^2 }},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\rho _1^*  =  - \\frac{{\\lambda ^2 }}\n{4}\\left[ {\\left( {x - x_0 } \\right)^2  + y_0^2 } \\right], \\hfill \\\\\n\\displaystyle  \\rho _2  = \\frac{{ - 4xx_0 }}\n{{\\left( {a - \\displaystyle \\frac{{xx_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x^2 y_0^2 }},\\,\\,\\rho _2^*  =  - \\frac{{a^2 \\lambda ^2 }}\n{{4R_0^2 }}\\left[ {\\left( {a - \\frac{{xx_0 }}\n{a}} \\right)^2  + \\frac{1}\n{{a^2 }}x^2 y_0^2 } \\right], \\hfill \\\\\n\\end{array}\n$$\n\n$$\n\\begin{array}{l}\n \\displaystyle \\rho _3  = \\frac{{ - 4yy_0 }}\n{{x_0^2  + \\left( {y - y_0 } \\right)^2 }},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\rho _3^*  =  - \\frac{{\\lambda ^2 }}\n{4}\\left[ {x_0^2  + \\left( {y - y_0 } \\right)^2 } \\right], \\hfill \\\\\n\\displaystyle  \\rho _4  = \\displaystyle \\frac{{ - 4yy_0 }}\n{{\\left( {a - \\displaystyle \\frac{{yy_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x_0^2 y^2 }},\\,\\,\\,\\rho _4^*  =  - \\displaystyle \\frac{{a^2 \\lambda ^2 }}\n{{4R_0^2 }}\\left[ {\\left( {a - \\displaystyle \\frac{{yy_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x_0^2 y^2 } \\right]. \\hfill \\\\\n\\end{array}\n$$\nNow we shall consider the right-hand side of identity (4.4). Taking into account (4.7) and (4.8), we find that\n$$\n\\begin{array}{l}\n\\displaystyle   \\frac{{\\partial q_4 }}\n{{\\partial n}} =  - k_4 \\left( {2 - \\alpha  - \\beta } \\right)\\left( {r^2 } \\right)^{\\alpha  + \\beta  - 2} \\left( {xx_0 } \\right)^{1 - 2\\alpha } \\left( {yy_0 } \\right)^{1 - 2\\beta }  \\hfill \\\\\n\\times A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right)\\displaystyle \\frac{\\partial }\n{{\\partial n}}\\left[ {\\ln \\,r^2 } \\right] \\hfill \\\\\n+ \\displaystyle k_4 \\left( {r^2 } \\right)^{\\alpha  + \\beta  - 2} x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } x^{ - 2\\alpha } y^{ - 2\\beta } \\left[ {\\left( {1 - 2\\alpha } \\right)y\\frac{{dy}}\n{{ds}} - \\left( {1 - 2\\beta } \\right)x\\frac{{dx}}\n{{ds}}} \\right] \\hfill \\\\\n\\times A_2^{\\left( 3 \\right)} \\left( {2 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right) \\hfill \\\\\n- \\displaystyle 2k_4 \\left( {2 - \\alpha  - \\beta } \\right)\\left( {r^2 } \\right)^{\\alpha  + \\beta  - 3} \\left( {xx_0 } \\right)^{1 - 2\\alpha } \\left( {yy_0 } \\right)^{1 - 2\\beta } A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;2 - \\alpha ,1 - \\beta ;3 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right)\\frac{{dy}}\n{{ds}} \\hfill \\\\\n+ \\displaystyle 2k_4 \\left( {2 - \\alpha  - \\beta } \\right)\\left( {r^2 } \\right)^{\\alpha  + \\beta  - 3} \\left( {xx_0 } \\right)^{1 - 2\\alpha } \\left( {yy_0 } \\right)^{1 - 2\\beta } A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;1 - \\alpha ,2 - \\beta ;2 - 2\\alpha ,3 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right)\\frac{{dx}}\n{{ds}}. \\hfill \\\\\n\\end{array}\\eqno(4.13)\n$$\nFurther we have\n$$\n\\begin{array}{l}\n \\displaystyle  \\int\\limits_{C_\\rho  } {x^{2\\alpha } y^{2\\beta } u\\frac{{\\partial G_4 \\left( {x,y;x_0 ,y_0 } \\right)}}\n{{\\partial n}}ds}  \\hfill \\\\\n   = \\displaystyle \\int\\limits_{C_\\rho  } {x^{2\\alpha } y^{2\\beta } u\\frac{{\\partial q_4 \\left( {x,y;x_0 ,y_0 } \\right)}}\n{{\\partial n}}ds}  - \\left( {R_0^2 } \\right)^{ - \\alpha  - \\beta } \\int\\limits_{C_\\rho  } {x^{2\\alpha } y^{2\\beta } u\\frac{{\\partial q_4 \\left( {x,y;\\bar x_0 ,\\bar y_0 } \\right)}}\n{{\\partial n}}ds}  = J_1  + J_2 . \\hfill \\\\\n\\end{array}\\eqno(4.14)\n$$\nSubstituting (4.13) in (4.14) and passing to the polar coordinates $x\\, = x_0  + \\rho \\cos \\,\\varphi ,\\,\\,y = y_0  + \\rho \\sin \\,\\varphi,$ we have\n$$\n\\begin{array}{l}\n  \\displaystyle   J_1  = 2k_4 \\left( {2 - \\alpha  - \\beta } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } \\int\\limits_0^{2\\pi } {} \\left( {x_0  + \\rho \\cos \\,\\varphi } \\right)\\left( {y_0  + \\rho \\sin \\,\\varphi } \\right)u\\left( {x_0  + \\rho \\cos \\,\\varphi ,y_0  + \\rho \\sin \\,\\varphi } \\right) \\hfill \\\\\n   \\times \\left( {\\rho ^2 } \\right)^{\\alpha  + \\beta  - 2} A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right)d\\varphi  \\hfill \\\\\n   +  \\displaystyle  k_4 x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } \\int\\limits_0^{2\\pi } {} u\\left( {x_0  + \\rho \\cos \\,\\varphi ,y_0  + \\rho \\sin \\,\\varphi } \\right) \\hfill \\\\\n   \\times  \\displaystyle  \\left[ {\\left( {1 - 2\\alpha } \\right)y_0 \\cos \\,\\varphi  + \\left( {1 - 2\\beta } \\right)x_0 \\sin \\varphi  + \\left( {1 - \\alpha  - \\beta } \\right)\\rho \\sin 2\\varphi } \\right] \\hfill \\\\\n  \\times  \\displaystyle  \\left( {\\rho ^2 } \\right)^{\\alpha  + \\beta  - 1} A_2^{\\left( 3 \\right)} \\left( {2 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 -  \\displaystyle  2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right)d\\varphi  \\hfill \\\\\n   -  \\displaystyle  2k_4 \\left( {2 - \\alpha  - \\beta } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } \\int\\limits_0^{2\\pi } {} \\left( {x_0  + \\rho \\cos \\,\\varphi } \\right)\\left( {y_0  + \\rho \\sin \\,\\varphi } \\right)u\\left( {x_0  + \\rho \\cos \\,\\varphi ,y_0  + \\rho \\sin \\,\\varphi } \\right) \\hfill \\\\\n  \\times \\left( {\\rho ^2 } \\right)^{\\alpha  + \\beta  - 2} A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;2 - \\alpha ,1 - \\beta ;3 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right)\\cos \\,\\varphi d\\varphi  \\hfill \\\\\n   -  \\displaystyle  2k_4 \\left( {2 - \\alpha  - \\beta } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } \\int\\limits_0^{2\\pi } {} \\left( {x_0  + \\rho \\cos \\,\\varphi } \\right)\\left( {y_0  + \\rho \\sin \\,\\varphi } \\right)u\\left( {x_0  + \\rho \\cos \\,\\varphi ,y_0  + \\rho \\sin \\,\\varphi } \\right) \\hfill \\\\\n   \\times \\left( {\\rho ^2 } \\right)^{\\alpha  + \\beta  - 2} A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;1 - \\alpha ,2 - \\beta ;2 - 2\\alpha ,3 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right)\\sin \\varphi d\\varphi  \\hfill \\\\\n   = J_{11}  + J_{12}  + J_{13}  + J_{14} . \\hfill \\\\\n\\end{array}\\eqno(4.15)\n$$\nFor evaluation of (4.15) we use the expansion formula ((2.27) in [18], p.678)\n$$\n\\begin{array}{l}\n\\displaystyle  A_2^{\\left( 3 \\right)} \\left( {a;b_1 ,b_2 ;c_1 ,c_2 ;x,y,z} \\right) \\hfill \\\\\n= \\displaystyle \\left( {1 - x} \\right)^{ - b_1 } \\left( {1 - y} \\right)^{ - b_2 } \\sum\\limits_{i,j = 0}^\\infty  {} \\frac{{\\left( a \\right)_{i - j} \\left( {b_1 } \\right)_i \\left( {b_2 } \\right)_i }}\n{{\\left( {c_1 } \\right)_i \\left( {c_2 } \\right)_i i!j!}}\\left( {\\frac{x}\n{{1 - x}}} \\right)^i \\left( {\\frac{y}\n{{1 - y}}} \\right)^i z^j  \\hfill \\\\\n\\times \\displaystyle  F\\left( {c_1  - a + j,b_1  + i;c_1  + i;\\frac{x}\n{{x - 1}}} \\right)F\\left( {c_2  - a + j,b_2  + i;c_2  + i;\\frac{y}\n{{y - 1}}} \\right), \\hfill \\\\\n\\end{array}\\eqno(4.16)\n$$\nwhere $F\\left( {a,b;c;x} \\right)$ is the Gauss hypergeometric function ([7], p. 69, formula (2)). Hence we obtain\n$$\n\\begin{array}{l}\n\\displaystyle A_2^{\\left( 3 \\right)} \\left( {3 - \\alpha  - \\beta ;1 - \\alpha ,1 - \\beta ;2 - 2\\alpha ,2 - 2\\beta ;\\xi ,\\eta ,\\zeta } \\right) \\hfill \\\\\n= \\displaystyle \\left( {\\rho ^2 } \\right)^{2 - \\alpha  - \\beta } \\left( {\\rho ^2  + 4x_0^2  + 4x_0 \\rho \\cos \\,\\varphi } \\right)^{\\alpha  - 1} \\left( {\\rho ^2  + 4y_0^2  + 4y_0 \\rho \\sin \\,\\varphi } \\right)^{\\beta  - 1} P_{11} , \\hfill \\\\\n\\end{array}\\eqno(4.17)\n$$\nwhere\n$$\n\\begin{array}{l}\n\\displaystyle   P_{11}  = \\sum\\limits_{i,j = 0}^\\infty  {} \\frac{{\\left( {3 - \\alpha  - \\beta } \\right)_{i - j} \\left( {1 - \\alpha } \\right)_i \\left( {1 - \\beta } \\right)_i }}\n{{\\left( {2 - 2\\alpha } \\right)_i \\left( {2 - 2\\beta } \\right)_i i!j!}} \\hfill \\\\\n\\displaystyle \\times \\left( {\\frac{{4x_0^2  + 4x_0 \\rho \\cos \\,\\varphi }}\n{{\\rho ^2  + 4x_0^2  + 4x_0 \\rho \\cos \\,\\varphi }}} \\right)^i \\left( {\\frac{{4y_0^2  + 4y_0 \\rho \\sin \\,\\varphi }}\n{{\\rho ^2  + 4y_0^2  + 4y_0 \\rho \\sin \\,\\varphi }}} \\right)^i \\left( { - \\frac{{\\lambda ^2 }}\n{4}\\rho ^2 } \\right)^j  \\hfill \\\\\n \\displaystyle   \\times F\\left( { - \\alpha  + \\beta  - 1 + j,1 - \\alpha  + i;2 - 2\\alpha  + i;\\frac{{4x_0^2  + 4x_0 \\rho \\cos \\,\\varphi }}\n{{\\rho ^2  + 4x_0^2  + 4x_0 \\rho \\cos \\,\\varphi }}} \\right) \\hfill \\\\\n \\times \\displaystyle  F\\left( {\\alpha  - \\beta  - 1 + j,1 - \\beta  + i;2 - 2\\beta  + i;\\frac{{4y_0^2  + 4y_0 \\rho \\sin \\,\\varphi }}\n{{\\rho ^2  + 4y_0^2  + 4y_0 \\rho \\sin \\,\\varphi }}} \\right), \\hfill \\\\\n\\end{array}\\eqno(4.18)\n$$\nUsing equality (46) in ([7], p. 112,)\n$$\nF\\left( {a,b;c;1} \\right) = \\frac{{\\Gamma \\left( c \\right)\\Gamma \\left( {c - a - b} \\right)}}\n{{\\Gamma \\left( {c - a} \\right)\\Gamma \\left( {c - b} \\right)}},\\,\\,\\, c \\ne 0, - 1, - 2,...,\\,\\,Re \\left( {c - a - b} \\right) > 0,\n$$\nit is not complicated to calculate\n$$\n\\mathop {\\lim }\\limits_{\\rho  \\to 0} P_{11}  = \\frac{{\\Gamma \\left( {2 - 2\\alpha } \\right)\\Gamma \\left( {2 - 2\\beta } \\right)}}\n{{\\Gamma \\left( {1 - \\alpha } \\right)\\Gamma \\left( {1 - \\beta } \\right)\\Gamma \\left( {3 - \\alpha  - \\beta } \\right)}}.\\eqno(4.19)\n$$\nBy virtue of (4.17) we calculate $J_{11}$:\n$$\n\\begin{array}{l}\n \\displaystyle J_{11}  = 2k_4 \\left( {2 - \\alpha  - \\beta } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } \\int\\limits_0^{2\\pi } {} \\left( {x_0  + \\rho \\cos \\,\\varphi } \\right)\\left( {y_0  + \\rho \\sin \\,\\varphi } \\right)u\\left( {x_0  + \\rho \\cos \\,\\varphi ,y_0  + \\rho \\sin \\,\\varphi } \\right) \\hfill \\\\\n\\times \\left( {\\rho ^2  + 4x_0^2  + 4x_0 \\rho \\cos \\,\\varphi } \\right)^{\\alpha  - 1} \\left( {\\rho ^2  + 4y_0^2  + 4y_0 \\rho \\sin \\,\\varphi } \\right)^{\\beta  - 1} P_{11} d\\varphi . \\hfill \\\\\n\\end{array}\n$$\nPassing to the limit as $\\rho  \\to 0^+$ and taking into account (1.8), we have\n$$\n\\mathop {\\lim }\\limits_{\\rho  \\to 0} J_{11}  = u\\left( {x_0 ,y_0 } \\right).\\eqno(4.20)\n$$\nSimilarly it can be proved that\n$$\n\\mathop {\\lim }\\limits_{\\rho  \\to 0} J_{12}  = \\mathop {\\lim }\\limits_{\\rho  \\to 0} J_{13}  = \\mathop {\\lim }\\limits_{\\rho  \\to 0} J_{14}  = \\mathop {\\lim }\\limits_{\\rho  \\to 0} J_2  = 0, \\mathop {\\lim }\\limits_{\\rho  \\to 0} \\int\\limits_{C_\\rho  } {x^{2\\alpha } y^{2\\beta } G_4 \\left( {x,y;x_0 ,y_0 } \\right)\\frac{{\\partial u}}\n{{\\partial n}}ds}  = 0.\\eqno(4.21)\n$$\nThus, using on equalities (4.13), (4.14), (4.20) and (4.21), by (4.4) we deduce that\n$$\n\\begin{array}{l}\n\\displaystyle u\\left( {x_0 ,y_0 } \\right) = k_4 \\left( {1 - 2\\beta } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta }  \\hfill \\\\\n\\times  \\displaystyle \\int\\limits_0^a {x\\tau _1 \\left( x \\right)\\left\\{ {\\frac{{H_3 \\left( {2 - \\alpha  - \\beta ,1 - \\alpha ;2 - 2\\alpha ;\\rho _1 ,\\rho _1^* } \\right)}}\n{{\\left[ {\\left( {x - x_0 } \\right)^2  + y_0^2 } \\right]^{2 - \\alpha  - \\beta } }} - \\frac{{H_3 \\left( {2 - \\alpha  - \\beta ,1 - \\alpha ;2 - 2\\alpha ;\\rho _2 ,\\rho _2^* } \\right)}}\n{{\\left[ {\\left( {a - \\displaystyle  \\frac{{xx_0 }}\n{a}} \\right)^2  +  \\displaystyle \\frac{1}\n{{a^2 }}x^2 y_0^2 } \\right]^{2 - \\alpha  - \\beta } }}} \\right\\}dx}  \\hfill \\\\\n+ k_4 \\left( {1 - 2\\alpha } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta }  \\hfill \\\\\n\\times  \\displaystyle \\int\\limits_0^a {y\\tau _2 \\left( y \\right)\\left\\{ {\\frac{{H_3 \\left( {2 - \\alpha  - \\beta ,1 - \\beta ;2 - 2\\beta ;\\rho _3 ,\\rho _3^* } \\right)}}\n{{\\left[ {x_0^2  + \\left( {y - y_0^2 } \\right)^2 } \\right]^{2 - \\alpha  - \\beta } }} - \\frac{{H_3 \\left( {2 - \\alpha  - \\beta ,1 - \\beta ;2 - 2\\beta ;\\rho _4 ,\\rho _4^* } \\right)}}\n{{\\left[ {\\left( {a -  \\displaystyle \\frac{{yy_0 }}\n{a}} \\right)^2  +  \\displaystyle \\frac{1}\n{{a^2 }}x_0^2 y^2 } \\right]^{2 - \\alpha  - \\beta } }}} \\right\\}dy}  \\hfill \\\\\n-  \\displaystyle \\int\\limits_\\Gamma  {x^{2\\alpha } y^{2\\beta } \\varphi \\left( s \\right)\\frac{{\\partial G_4 \\left( {x,y;x_0 ,y_0 } \\right)}}\n{{\\partial n}}ds} . \\hfill \\\\\n\\end{array}\\eqno(4.22)\n$$\nIf we use the formula\n$$\nH_3 \\left( {a,b;c;x,y} \\right) = \\left( {1 - x} \\right)^{ - b} F_{0:1;2}^{1:1;0} \\left[ {\\begin{array}{*{20}c}\n   {c - a:}  \\\\\n   {\\,\\,\\,\\,\\,\\,\\, - :}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {b;}  \\\\\n   {c;}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, - ;}  \\\\\n   {1 - a,c - a;}  \\\\\n \\end{array} \\frac{x}\n{{x - 1}}, - y} \\right],\n$$\nwhich connects Kummer's function with the hypergeometric function of Kampe de Feriet ([4], p. 150, formula (29))\n$$\n\\displaystyle F_{\\displaystyle l:m;n;}^{\\displaystyle p:q;k;} \\left[ {\\begin{array}{*{20}c}\n   {\\left( {a_p } \\right):}  \\\\\n   {\\left( {\\alpha _l } \\right):}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\left( {b_q } \\right);}  \\\\\n   {\\left( {\\beta _m } \\right);}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\left( {c_k } \\right);}  \\\\\n   {\\left( {\\gamma _n } \\right);}  \\\\\n \\end{array} x,y} \\right] = \\displaystyle \\sum\\limits_{ r,s = 0}^\\infty  {} \\frac{{\\prod\\limits_{j = 1}^p {} \\left( {a_j } \\right)_{r + s} \\prod\\limits_{j = 1}^q {} \\left( {b_j } \\right)_r \\prod\\limits_{j = 1}^k {} \\left( {c_j } \\right)_s }}\n{{\\prod\\limits_{j = 1}^l {} \\left( {\\alpha _j } \\right)_{r + s} \\prod\\limits_{j = 1}^m {} \\left( {\\beta _j } \\right)_r \\prod\\limits_{j = 1}^n {} \\left( {\\gamma _j } \\right)_s r!s!}}\\displaystyle x^r y^s ,\n$$\nwe find that solution (4.22) of the problem $D$ may be represented as\n$$\n\\begin{array}{l}\n\\displaystyle u\\left( {x_0 ,y_0 } \\right) = k_4 \\left( {1 - 2\\beta } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } \\int\\limits_0^a {} x\\tau _1 \\left( x \\right)\\left\\{ {\\frac{{F_{0:1;2}^{1:1;0} \\left[ {\\begin{array}{*{20}c}\n   {\\beta  - \\alpha :}  \\\\\n   {\\,\\,\\,\\,\\,\\,\\,\\,\\ - :}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\,\\,\\,1 - \\alpha ;}  \\\\\n   {2 - 2\\alpha ;}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\ - ;}  \\\\\n   {\\alpha  + \\beta  - 1,\\beta  - \\alpha ;}  \\\\\n \\end{array} \\sigma _1 ,\\sigma _1^* } \\right]}}\n{{\\left[ {\\left( {x - x_0 } \\right)^2  + y_0^2 } \\right]^{1 - \\beta } \\left[ {\\left( {x + x_0 } \\right)^2  + y_0^2 } \\right]^{1 - \\alpha } }}} \\right.\\, \\hfill \\\\\n  \\, - \\displaystyle  \\left. {\\frac{{F_{0:1;2}^{1:1;0} \\left[ {\\begin{array}{*{20}c}\n   {\\beta  - \\alpha :}  \\\\\n   {\\,\\,\\,\\,\\,\\,\\,\\,\\ - :}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\,\\,\\,1 - \\alpha ;}  \\\\\n   {2 - 2\\alpha ;}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\ - ;}  \\\\\n   {\\alpha  + \\beta  - 1,\\beta  - \\alpha ;}  \\\\\n \\end{array} \\sigma _2 ,\\sigma _2^* } \\right]}}\n{{\\left[ {\\left( {a - \\displaystyle \\frac{{xx_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x^2 y_0^2 } \\right]^{1 - \\beta } \\left[ {\\left( {a + \\displaystyle \\frac{{xx_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x^2 y_0^2 } \\right]^{1 - \\alpha } }}} \\right\\}dx \\hfill \\\\\n\\end{array}\n$$\n$$\n\\begin{array}{l}\n   + k_4 \\left( {1 - 2\\alpha } \\right)x_0^{1 - 2\\alpha } y_0^{1 - 2\\beta } \\displaystyle \\int\\limits_0^a {} y\\tau _2 \\left( y \\right)\\left\\{ {\\displaystyle \\frac{{F_{0:1;2}^{1:1;0} \\left[ {\\begin{array}{*{20}c}\n   {\\alpha  - \\beta :}  \\\\\n   {\\,\\,\\,\\,\\,\\,\\,\\,\\ - :}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\,\\,\\,1 - \\beta ;}  \\\\\n   {2 - 2\\beta ;}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\ - ;}  \\\\\n   {\\alpha  + \\beta  - 1,\\alpha  - \\beta ;}  \\\\\n \\end{array} \\sigma _3 ,\\sigma _3^* } \\right]}}\n{{\\left[ {x_0^2  + \\left( {y - y_0^2 } \\right)^2 } \\right]^{1 - \\alpha } \\left[ {x_0^2  + \\left( {y + y_0 } \\right)^2 } \\right]^{1 - \\beta } }}} \\right. \\hfill \\\\\n   - \\left. {\\displaystyle \\frac{{F_{0:1;2}^{1:1;0} \\left[ {\\begin{array}{*{20}c}\n   {\\alpha  - \\beta :}  \\\\\n   {\\,\\,\\,\\,\\,\\,\\,\\,\\ - :}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\,\\,\\,1 - \\beta ;}  \\\\\n   {2 - 2\\beta ;}  \\\\\n \\end{array} \\begin{array}{*{20}c}\n   {\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\ - ;}  \\\\\n   {\\alpha  + \\beta  - 1,\\alpha  - \\beta ;}  \\\\\n \\end{array} \\sigma _4 ,\\sigma _4^* } \\right]}}\n{{\\left[ {\\left( \\displaystyle {a - \\displaystyle \\frac{{yy_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x_0^2 y^2 } \\right]^{1 - \\alpha } \\left[ {\\left( \\displaystyle {a + \\displaystyle \\frac{{yy_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x_0^2 y^2 } \\right]^{1 - \\beta } }}} \\right\\}dy \\\\\n- \\displaystyle \\int\\limits_\\Gamma  {x^{2\\alpha } y^{2\\beta } \\varphi \\left( s \\right)\\frac{{\\partial G_4 \\left( {x,y;x_0 ,y_0 } \\right)}}\n{{\\partial n}}ds,}  \\hfill \\\\\n\\end{array}\\eqno(4.23)\n$$\nwhere\n$$\n\\sigma _1  = \\frac{{4xx_0 }}\n{{\\left( {x + x_0 } \\right)^2  + y_0^2 }}, \\,\\,\\,\\,\\,\\,\\ \\sigma _1^*  = \\frac{{\\lambda ^2 }}\n{4}\\left[ {\\left( {x - x_0 } \\right)^2  + y_0^2 } \\right],\n$$\n$$\n\\sigma _2  = \\displaystyle \\frac{{4xx_0 }}\n{{\\left( \\displaystyle {a + \\frac{{xx_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x^2 y_0^2 }},\\,\\,\\,\\,\\ \\sigma _2^*  = \\displaystyle \\frac{{a^2 \\lambda ^2 }}\n{{4R_0^2 }}\\left[ {\\left( \\displaystyle {a - \\frac{{xx_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x^2 y_0^2 } \\right],\n$$\n$$\n\\sigma _3  = \\frac{{4yy_0 }}\n{{x_0^2  + \\left( {y + y_0 } \\right)^2 }}, \\,\\,\\,\\,\\,\\,\\,\\ \\sigma _3^*  = \\frac{{\\lambda ^2 }}\n{4}\\left[ {x_0^2  + \\left( {y - y_0 } \\right)^2 } \\right],\n$$\n$$\n\\sigma _4  = \\displaystyle \\frac{{4yy_0 }}\n{{\\left( \\displaystyle {a + \\frac{{yy_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x_0^2 y^2 }}, \\,\\,\\,\\,\\,\\,\\,\\,\\ \\sigma _4^*  = \\displaystyle \\frac{{a^2 \\lambda ^2 }}\n{{4R_0^2 }}\\left[ {\\left(\\displaystyle  {a - \\frac{{yy_0 }}\n{a}} \\right)^2  + \\displaystyle \\frac{1}\n{{a^2 }}x_0^2 y^2 } \\right].\n$$\n\nNow we can formulate the main result.\n\n\\textbf{Theorem 2.} The problem $D$ has the unique solution defined by formula (4.23).\n\n\tWe note that expansions for the hypergeometric functions of Lauricella $F_A^{\\left( n \\right)} ,\\,\\,F_B^{\\left( n \\right)} ,\\,\\,F_C^{\\left( n \\right)} ,\\,\\,F_D^{\\left( n \\right)}\n$ are found in [19, 20] and applied in [17] for finding fundamental solutions and later for investigating boundary value problems for 3-D singular elliptic equations [24].\n\n\\section{Acknowledgement}. We would like to thank the anonymous referee for valuable suggestions, which made the present paper more readable.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nThe next generation wireless communication system, typically referred to as the fifth generation (5G), is currently under\nintensive research and development. Millimeter wave (mmWave) communication is regarded as one of the enabling components for 5G cellular communication systems, thanks to the large amount of available frequency spectrum \\cite{rappaport2013mwave,andrews2014will,demestichas20135g,Ruisi_1, Ruisi_2,Ke_2}. However, mmWave  transmissions suffer from high propagation loss and blockage, unlike sub-6GHz legacy frequency band. High antenna gain (typically realized with a large-scale antenna structure) is required to overcome the radio propagation loss and achieve the good signal-to-noise ratio (SNR) at mmWave bands. Therefore, mmWave transmissions will be potentially ultra-wideband (UWB) (e.g. with an absolute bandwidth larger than 500 MHz) and involve large-scale antenna systems at both communication ends \\cite{rusek2013scaling,larsson2014massive}. \n\nIt is of importance to understand how the radio signal propagates in the given scenario. A new air interface system  design typically starts with the understanding of the deployment scenario, where channel models should be developed to reflect the physical transmission.\nAccurate knowledge of the delay and angle information of multipath components is essential for many applications in mmWave systems.\n\\added{The extracted multipath components can be further utilized for analysis, clustering and modeling of wireless channels, which is fundamental for system design and performance evaluation \\cite{gustafson2014mm,he2018clustering}. }\nA popular channel sounding setup at mmWave bands is to utilize a nonreal-time channel sounder based on the virtual antenna array concept \\cite{dohler2002vaa,medbo2015vcua,allan2018vcua,Ke_1}. This strategy is simple, flexible and cost-effective, though it is not suitable for dynamic propagation scenarios due to slow mechanical movement of antennas and channel sounding time for each virtual array location. Furthermore, the mutual coupling effects between the array elements are not presented, which is beneficial since it can further simplify the channel estimation. Virtual uniform circular array (UCA) is particularly popular, since it only requires radial mechanical movement of the positioner and presents approximately constant beam patterns over $360^o$ azimuth angle, regardless of the steering angle \\cite{davies1983uca,long1967uca}. \n\n\\replaced{Channel estimation algorithms proposed in the literature for power-angle-delay profile (PADP) estimation were mainly under the far-field and narrowband assumptions. The far-field assumption holds when the distance between the scattering source and the antenna array is larger than the Fraunhofer distance $2D^2\/\\lambda$, with $D$ the antenna array aperture and $\\lambda$ the wavelength. The far field assumption can be violated in short-range mmWave propagation scenarios. The narrowband assumption holds when the condition $D\/\\lambda \\ll f\/B$ is satisfied, where $f$ is the frequency and $B$ is the bandwidth. When the system bandwidth becomes wide (e.g. in the mmWave context) that the narrowband assumption does not hold anymore for a given array aperture and frequency, the propagation delay of each path can be resolved at different delay bins between array elements. The existing algorithms in the literature would fail to work under near-field and wideband scenarios, as briefly summarized below.}{} \n\\begin{itemize}\t\n\t\t\\item[1)] \\added{Beamforming techniques such as classical beamforming (CBF) and Capon beamforming.  As demonstrated in \\cite{fan2016eurasip}, The beam pattern of the CBF method is highly sensitive to near-field conditions (which would introduce a power loss in the target direction) and UWB bandwidth (which would introduce a joint sidelobe in the delay and angle domain and difficult to suppress \\cite{fan2016eurasip}). Capon beamformer degrades as well in near-field and UWB conditions, since the steering vectors are typically calculated at the center frequency and under far-field conditions.}\n\t\t\\item[2)] \\added{Subspace methods. The conventional subspace angle estimation methods are based on the element-space covariance matrix, where the steering vector is the function of center frequency. Thus, it implied the narrowband assumption. UCA is attractive in angle estimation also thanks to its circular symmetry, where beamspace transform of UCA based on the phase mode excitation principle can be applied \\cite{tewfik1992bs,gentile2008fi,chan2002design}. Basically, we can transform the UCA array manifold (in the element space) into uniform linear array (ULA) manifold (in the phase mode space), which enables us to develop computationally efficient and high-resolution subspace algorithms \\cite{zoltowski1992direction,zoltowski1996closed,pesavento2002direction,belloni2003unitary,mathews1994_2d}. However, the aforementioned beam-space multiple signal classification (MUSIC) or estimation of singal parameters by rotational invariance techniques (ESPRIT) algorithms were only investigated under narrowband and far-field assumptions. Subspace methods were also developed for joint angle-delay estimation (JADE). With the JADE strategy, the JADE-MUSIC \\cite{van1997jade_music} and the JADE-ESPRIT \\cite{van1997jade} algorithms exploited both space and time properties by stacking the array element channel impulse responses into a high dimension vector. After performing the eigen-decomposition on the covariance matrix of the constructed vector, JADE can be obtained either with MUSIC or ESPRIT methods. However, both algorithms are unfavorable for the UWB large-scale antenna systems, due to the high complexity introduced by the high dimension of the stacked space-time vector.\t}\n\t\\item[3)] High resolution parametric methods. Maximum likelihood estimator (MLE) is a popular high resolution channel estimation algorithm. However, the MLE is well-known for its high computational complexity due to its joint estimation mechanism, particularly when the channel  parameter dimension is large. An attempt to reduce the MLE computation complexity was proposed in \\cite{ji2017_mle,ji2018channel}. The space-alternating generalized expectation-maximization (SAGE) algorithm is a relatively low-cost expectation-maximization (EM) algorithm, where the multi-dimensional search is replaced with several one-dimensional searches \\cite{bernard1999sage}. However, a prerequisite to utilize the SAGE algorithm is that the likelihood function needs to be independent between different parameter spaces, which might be violated when narrowband and far-field assumptions do not hold, e.g. for the UWB large-scale antenna systems \\cite{bernard1999sage}. A general spherical wave model, i.e. including the distance to the scatterer, can be introduced in the MLE type algorithm implementation, which would, however, further increases the computation complexity. \n\\end{itemize}\n\n\n\\added{There is a strong need for a low-cost and generic channel estimation algorithm for joint angle and delay profile estimation, which works in practical propagation scenarios, regardless of the antenna system scale (i.e. large or small) and frequency bandwidth settings (i.e. narrow or wide). That is, it can be applied for the near-field UWB scenarios without increasing the algorithm computation complexity. However, such algorithm is missing in the literature, to the best knowledge of the authors. In this work, a novel low-cost beamspace UCA beamforming algorithm with a successive cancellation scheme is proposed to jointly detect the angle and delay of the multipath components. The proposed algorithm is insensitive to the system bandwidth and distance to the scatterers, making it suitable for UWB large-scale antenna systems. In this paper, we firstly demonstrated the performance deterioration of CBF of the UCA in the element space when far-field assumption does not hold. To solve the issue, we resort to the beamspace transform of UCA based on the phase mode excitation principle. The beamspace UCA is shown to be robust to phase errors introduced by the spherical wavefront. A novel beamspace UCA algorithm with the successive cancellation of the detected paths is detailed. Both numerical simulations and experimental results at mmWave bands are provided to demonstrate the effectiveness and robustness of the proposed method in channel parameter estimation for UWB large-scale antenna systems.}\n\n\n\n\\section{Problem Statement}\n\\textcolor{black}{Assume a UCA is distributed in the x-y plane and its center is located at the origin of the coordinate system. The UCA consists of $P$ isotropic antenna elements with radius $r$. The angular position of the $p$-th element is $\\varphi_{p}=2\\pi\\cdot(p-1)\/P$, $p\\in[1,P]$. Suppose there are $N$ paths impinging at the UCA, the channel frequency response at the $p$-th UCA element is the superposition of the channel responses of the $N$ paths,\n\\vspace*{-.01in}\n\\begin{equation}\nH_p(f)=\\sum_{n=1}^{N}  \\alpha_n \\exp(-j2\\pi f \\tau_n) \\cdot  a_p(f,\\mathbf{\\Theta}_n),\\label{eq:element frequency response}\n\\end{equation}\nwhere $\\alpha_{n}$ and $\\tau_{n}$ represent the complex amplitude and delay of the $n$-th path, respectively. $\\mathbf{\\Theta}_n = [D_n, \\theta_n, \\phi_n]$ denotes the scatterer location vector of the $n$-th path, where $D_n$ is the distance between the $n$-th scatterer and the UCA center, $\\theta_n$  and $\\phi_n$ the elevation and azimuth angle of the $n$-th path, respectively. $a_p(f,\\mathbf{\\Theta}_n)$ is the transfer function between the $n$-th scatterer and the $p$-th UCA element, which is normalized by the transfer function between the $n$-th scatterer and the UCA center, as\n\\vspace*{-.01in}\n\\begin{equation}\na_p(f,\\mathbf{\\Theta}_n)=\\beta_{p,n}\\exp(-j k \\varPsi_{p,n}).\\label{eq:a_p}\n\\end{equation}\n $\\beta_{p,n} = D_n\/d_{p,n}$ denotes the relative path loss term at the $p$-th element with respect to (w.r.t) the UCA center, where $d_{p,n}$ represents the distance between the $n$-th scatterer and the $p$-th UCA element. Under far-field assumption, we have $\\beta_{p,n}=1$.\n $k = 2 \\pi f \/c$ is the wave number with $c$ denoting the speed of light.\n $\\varPsi_{p,n}$ indicates the relative propagation distance to the $p$-th UCA element w.r.t UCA center, i.e.,\n\\vspace*{-.01in}\n\\begin{equation}\n\\varPsi_{p,n} =d_{p,n} - D_n,\\label{eq:r_dis}\n\\end{equation}\nwhere the distance term $d_{p,n}$ is given by\n\\begin{equation}\nd_{p,n}= \\sqrt{D_n^2 + r^2 - 2rD_n\\sin{\\theta_n}\\cos(\\phi_n-\\varphi_p)}.\\label{eq:dis}\n\\end{equation}\n Following Taylor series expansion, we can approximate $d_{p,n}$ as\n \\begin{equation}\n d_{p,n}= D_n - r\\sin{\\theta_n}\\cos(\\phi_n-\\varphi_p) + \\varDelta_{p,n},\\label{eq:dis_a}\n \\end{equation}\n where $\\varDelta_{p,n}$ is the term introduced by the near-field condition. When the plane-wave assumption holds, we have $\\varDelta_{p,n}=0$.\n By substituting (\\ref{eq:dis_a}) into (\\ref{eq:r_dis}), $\\varPsi_{p,n}$ can be simplified as \n \\begin{equation}\n \\varPsi_{p,n}=  - r\\sin{\\theta_n}\\cos(\\phi_n-\\varphi_p) + \\varDelta_{p,n}.\\label{eq:r_dis_a}\n \\end{equation}}\n \n In this paper, the objective is to  detect path parameters $\\{\\alpha_n, \\tau_n, \\phi_n\\}$ with arbitrary elevation angle $\\theta_n$ present in the near-field condition for $n\\in[1,N]$. The performance of CBF and frequency-invariant beamforming (FIBF) based on UCA for a general 3D multipath scenario \\added{under far-field assumption} has been discussed in \\cite{zhang2017fi}. Below, we limit our discussions on the beamforming properties of CBF and FIBF under near-field conditions, assuming that all impinging paths are confined to the plane of the UCA, i.e. with elevation angle $\\theta_n = \\frac{\\pi}{2}$ for $n\\in [1,N]$ for simplicity. Though the proposed algorithm works for arbitrary 3D propagation scenarios as later discussed in section \\ref{simulation}  and \\ref{measurement}. \n\n\n\n\\subsection{CBF in the UCA element space}  \\label{cb_ff}\n\\textcolor{black}{Using the CBF under plane-wave assumption, the steering weight of the $p$-th UCA element can be written as\n\\begin{equation}\nw_{p}(f,\\phi)=\\frac{1}{P} \\exp[-k r \\cos(\\phi-\\varphi_p)].\\label{eq:w_cb}\n\\end{equation}\nTherefore, the UCA array beam pattern can be obtained by taking the coherent summation of the element responses as\n\\begin{equation}\n\\begin{split}\nB(f,\\phi)& = \\frac{1}{P}\\sum_{p=1}^{P} w_p(f,\\phi) \\cdot H_p(f).\\\\        \n\\end{split}\n\\label{eq:b}\n\\end{equation}\nBy taking (\\ref{eq:element frequency response}) into (\\ref{eq:b}), we have\n\\begin{equation}\n\\begin{split}\nB(f,\\phi)& = \\sum_{n=1}^{N} \\alpha_n \\exp(-j2\\pi f \\tau_n)\\cdot \\sum_{p=1}^{P} w_p(f,\\phi) \\cdot a_p(f,\\mathbf{\\Theta}_n)\/P   \\\\   \n         & = \\sum_{n=1}^{N}  \\alpha_n \\exp(-j2\\pi f \\tau_n) \\cdot v_n(f,\\phi)  \\\\\n         & = \\sum_{n=1}^{N} B_n(f, \\phi),    \n\\end{split}\n\\label{eq:b_1}\n\\end{equation}\nwhere $B_n(f, \\phi)$ represents the beam pattern of the $n$-th path, i.e. the beam pattern of a single path. As seen from (\\ref{eq:b_1}), the linear superposition of the CBF beam patterns of all paths gives the beam pattern of the multiple paths.}\n\\textcolor{black}{$\\arrowvert v_n(f,\\phi)\\arrowvert$ denotes the unit beam pattern term of the $n$-th path with CBF. In ideal case, $\\arrowvert v_n(f,\\phi)\\arrowvert$ mimics a Dirac delta function in $\\phi$ domain, i.e. $\\arrowvert v_n(f,\\phi)\\arrowvert  = \\delta(\\phi-\\phi_n)$. Thus the peak location of $\\arrowvert  v_n(f,\\phi) \\arrowvert $ gives the estimate of $\\phi_n$, $\\alpha_n$ and $\\tau_n$ can be obtained via inverse Fourier transform (IFT) of $B(f,\\phi)$ at $\\phi = \\phi_n$.\nAccording to (\\ref{eq:b_1}), (\\ref{eq:w_cb}) and  (\\ref{eq:a_p}), $v_n(f,\\phi)$ can be given by\n \\begin{equation}\n \\begin{split}\n v_{n}(f,\\phi)=& \\frac{1}{P} \\sum_{p=1}^{P}\\beta_{p,n} \\cdot \\exp(-jk \\varDelta_{p,n}) \\\\\n               &\\cdot \\exp\\{-jkr[\\cos(\\phi-\\varphi_p)-\\cos(\\phi_n-\\varphi_p)]\\}.\\label{eq:v_cb}\n \\end{split}\n \\end{equation}\nUnder the far-field condition (i.e. $\\beta_{p,n}=1$ and $\\varDelta_{p,n} = 0$), we have $\\arrowvert v_{n}(f,\\phi_n)\\arrowvert = 1$ and $ \\arrowvert v_{n}(f,\\phi_{n'})\\arrowvert <1$ with $\\phi_{n'}\\neq \\phi_{n}$, as illustrated by the blue line in Fig.\\ref{bp_cb_dis}. Thus the peak location in the beam pattern gives the angle estimate of the $n$-th path. }\n\n\\added{To investigate whether $\\arrowvert v_{n}(f,\\phi)\\arrowvert$ can still maintain this property under near-field condition, we reduce the distance $D_n$ and its effect on the beam patterns of $v_{n}(f,\\phi)$ is shown in Fig. \\ref{bp_cb_dis}, i.e. the red and black curves, as examples. The results show that the beam patterns are highly susceptible to phase errors introduced by the spherical wavefront in the near-field condition.\nThe near-field condition results in the main beam distortion of the beam pattern, including the power loss (the red curve) and the concave pattern in the target direction (the black curve). The power loss results in underestimating the amplitude $\\alpha_n$ and the concave pattern would result in fake path angle detection.}\n \n\\added{To further investigate the power loss in the target direction under near-field condition, the distance from $3$ m to $70$ m was set and the result is shown by the blue dotted line in Fig. \\ref{cbf_fibf_dis2}. The figure shows that with CBF method, a power loss more than $35$ dB might exist in the target direction for typical indoor scenarios.}\n \nTherefore, besides the joint sidelobe problem introduced by the UWB bandwidth in PADP as shown in \\cite{fan2016eurasip}, the CBF of UCA in the element space would also suffer from significant power loss in target directions or even failure of detecting the true paths, due to phase errors introduced in the near-field condition, which makes it unsuitable for channel parameter estimation of mmWave UWB large-scale antenna systems.  \n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 260bp 695bp 565bp,clip,scale=0.6]{bp_cb_dis_new.pdf}}\n\t\t\\par\\end{centering}\n\t\n\t\\caption{The unit CBF beam pattern $v_{n}(f,\\phi)$ in the UCA element space for different $D_n$ with $\\theta_n = \\pi\/2$, $\\phi_n = \\pi$, $r$ = 0.5 m, $f$ = 29 GHz and $P=720$ are set for the UCA. The calculated Fraunhofer far-field distance $D_{far} = 2(2r)^2\/\\lambda$ is around 193 m. }\n\t\\label{bp_cb_dis}\n\\end{figure}\n\n\\subsection{Beamformer in the UCA phase mode space}\n\n\\textcolor{black}{The frequency response of the UCA element space can be converted to phase mode space as\n\\begin{equation}\n\\begin{split}\n\\underline{H}_m(f) &= \\frac{1}{P}\\sum_{p=1}^{P}\\hat{G}_m(f) \\cdot \\exp(jm\\varphi_p) \\cdot H_{p}(f),\\\\\n\\end{split}\n\\label{eq:hm}\n\\end{equation}\nwhere $\\underline{H}_m(f) $ denotes the $m$-th mode response of the UCA in the phase mode space.\nAs discussed in \\cite{zhang2017fi}, $\\underline{H}_m(f)$ can be approximated to $\\underline{H}_m(f) = \\exp(jm\\phi_n)$ under far-field condition and with all paths confined in the UCA plane, i.e. $\\theta_n = 90^o, n\\in[1, N]$, where the compensation filter was defined as $\\hat{G}_m(f) = 1\/[j^m J_m(kr)]$ with $J_{m}(\\cdot)$ denoting the Bessel function of the first kind with\norder $m$. However, in practical propagation scenario, it is very unlikely that the incident paths are strictly limited to the UCA plane. Therefore, we modified the compensation filter for 3D propagation scenarios as \\cite{zhang2017fi},\n\\begin{equation}\n\\frac{1}{\\hat{G}_m(f)}= 0.5j^{m}[J_{m}(kr)-jJ_{m}^{'}(kr)],\\label{eq:Gm}\n\\end{equation}\n where\n  $(\\cdot)^{'}$ denotes the differential operator.}\n\n\\textcolor{black}{By taking equation (\\ref{eq:element frequency response}) into (\\ref{eq:hm}) and changing the order of the two summations, we have\n\\begin{equation}\n\\begin{split}\n\\underline{H}_m(f) =&  \\sum_{n=1}^{N} \\alpha_n \\exp(-j 2 \\pi f \\tau_n)\\\\\n                  & \\cdot \\bigg\\{\\frac{1}{P}\\sum_{p=1}^{P}\\hat{G}_m(f) \\exp(jm\\varphi_p)\\cdot a_p(f,\\mathbf{\\Theta}_n) \\bigg\\}\\\\\n                  =& \\sum_{n=1}^{N} \\alpha_n \\exp(-j 2 \\pi f \\tau_n) \\cdot \\underline{a}_m(f,\\mathbf{\\Theta}_n),\n\\end{split}\n\\label{eq:hm_1}\n\\end{equation}\nwhere $\\underline{a}_m(f,\\mathbf{\\Theta}_n)$ is given by the $\\{\\cdot\\}$ term in the above equation.\nComparing the above equation with equation (\\ref{eq:element frequency response}), we can see that for the $n$-th path, $\\underline{a}_m(f,\\mathbf{\\Theta}_n)$ is the UCA manifold of the $m$-th mode in the phase mode space and $a_p(f,\\mathbf{\\Theta}_n)$ defined in (\\ref{eq:a_p}) is the UCA manifold of the $p$-th element in the element space.  }\n\n\\textcolor{black}{The beam pattern of the FIBF can be written as\n\\begin{equation}\n\\underline{B}(f,\\phi)= \\frac{1}{2M+1} \\sum_{m=-M}^{M}  \\exp(-jm\\phi) \\cdot \\underline{H}_m(f), \\label{eq:aoa_fi}\n\\end{equation}\nwhere $\\exp(-jm\\phi)\/(2M+1)$ is the steering weight of the $m$-th phase mode.\nSubstituting the array phase mode response $ \\underline{H}_m(f)$ defined in (\\ref{eq:hm_1}) into the above equation and rearranging the summation order, we have\n\\begin{equation}\n\\begin{split}\n\\underline{B}(f,\\phi)&= \\sum_{n=1}^{N}  \\alpha_n \\exp(-j 2 \\pi f \\tau_n) \\cdot \\underline{v}_n(f,\\phi)\\\\\n                     &= \\sum_{n=1}^{N} \\underline{B}_n(f,\\phi), \\label{eq:aoa_fi_1}\n\\end{split}\n\\end{equation}\nwhere $\\underline{B}_n(f,\\phi)$ represents the FIBF beam pattern of the $n$-th path, i.e. the FIBF beam pattern of a single path. The FIBF beam pattern of the multiple paths is obtained by the linear superposition of the FIBF beam patterns of all paths. Similar to the unit beam pattern term $v_n(f,\\phi)$ of CBF,  $\\underline{v}_n(f,\\phi)$ indicates the unit FIBF beam pattern term of the $n$-th path, which is expressed by\n\\begin{equation}\n\\underline{v}(f,\\phi)= \\frac{1}{2M+1} \\sum_{m=-M}^{M} \\underline{a}_m(f,\\mathbf{\\Theta}_n) \\exp(-jm\\phi), \\label{eq:v_fi}\n\\end{equation}\n where $\\underline{a}_m(f,\\mathbf{\\Theta}_n)$ is given in (\\ref{eq:hm_1}).}\n \n \\added{Similar to the discussions about the unit beam pattern term $v_n(f,\\phi)$ of CBF in section \\ref{cb_ff}, we study the property of $\\arrowvert \\underline{v}_n(f,\\phi)\\arrowvert$ for various distance $D_n$. With the same simulation setting as for CBF in Fig. \\ref{bp_cb_dis}, we can plot the unit FIBF beam pattern $\\underline{v}_n(f,\\phi)$ in Fig. \\ref{bp_fi_dis}. The results show that the beam pattern is insensitive to phase error introduced by the spherical wavefront in the near-field conditions, where an approximately constant beam pattern is achieved for different distance $D_n$ within a large dynamic range.}\n \n \\added{We further investigate the power loss in the target direction under near-field conditoins for FIBF and the results are shown by the red dotted curve in Fig. \\ref{cbf_fibf_dis2}. It shows that the power values of the target direction are approximately unchanged with different distances $D_n$. } \n \n \\added{The unit FIBF beam pattern peaks in the target direction and the peak value keeps approximately constant (approximates to 1) for various distances setting, as shown in Fig. \\ref{cbf_fibf_dis2}. It indicates that the path can be accurately detected with FIBF method under either near-field or far-field conditions.}\n\n\\added{As mentioned earlier, the beamformer in the UCA phase mode space with the modified compensation filter works in 3D propagation scenarios when the elevation angle is not restricted to the UCA plane. As detailed in \\cite{zhang2017fi}, the beam patterns are approximately constant with different elevation angles. When the elevation angle gets away from the UCA plane, i.e. with $\\varDelta \\theta_n = |\\theta_n - 90^o|$  getting larger, the main beam peak drops slightly and the sidelobes of the beamforming pattern at $\\phi = \\phi_n \\pm \\pi$ becomes broader. }\n\nThe simulation results indicate that the beamformer in the UCA phase mode space with the modified compensation filter $\\hat{G}_m(f)$ can also be applied in near-field \\added{3D} scenarios, since the beamformer pattern is insensitive to the introduced phase errors. As discussed in \\cite{zhang2017fi}, the joint PADP can be directly obtained with the modified FIBF, which is simple, effective and robust. However, the resulting PADP suffers from high sidelobes,  as shown in Fig. \\ref{bp_fi_dis}. In the next section, a FIBF algorithm with the successive cancellation of the detected paths is detailed, with the objective to eliminate the high sidelobes of the dominant paths.\n\n\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 260bp 695bp 565bp,clip,scale=0.6]{cbf_fibf_dis2.pdf}}\n\t\t\\par\\end{centering}\n\t\n\t\\caption{The main beam peak varies with $D_n$ for CBF and FIBF, where $\\theta_n = \\pi\/2$ and $\\phi_n = \\pi$, \\added{$f=29$ GHz and $P=720$ are set for the UCA}.}\n\t\\label{cbf_fibf_dis2}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 260bp 695bp 565bp,clip,scale=0.6]{bp_fi_dis_new.pdf}}\n\t\t\\par\\end{centering}\n\t\n\t\\caption{The unit FIBF beam pattern $\\underline{v}_{n}(f,\\phi)$ in the UCA phase mode space for variant $D_n$ with $\\theta_n = \\pi\/2$, $\\phi_n = \\pi$, $r$ = 0.5 m, $f$ = 29 GHz and $P=720$. \\added{The calculated Fraunhofer far-field distance $D_{far} = 2(2r)^2\/\\lambda$ is around 193 m.}}\n\t\\label{bp_fi_dis}\n\\end{figure}\n\n\n\n\n\\section{Proposed beamspace UCA with the successive cancellation scheme for near-field scenarios}\n\nAs discussed, the beamspace UCA with the modified compensation filter $\\hat{G}_m(f)$ can achieve constant beam patterns, insensitive to  the distances between the scatterers and the UCA center, which makes it suitable for angle estimation in near-field scenarios. In this section, a novel  FIBF  based on successive cancellation principle to estimate azimuth angle, delay and power of each multipath component is proposed for near-field scenarios, where the paths are detected one by one with the power values in a  descending order. \n\nAs mentioned in paper \\cite{zhang2017fi}, the PADP with modified FIBF can be directly obtained by performing IFT of $\\underline{B}(f,\\phi)$ in (\\ref{eq:aoa_fi_1}) as,\n\\begin{equation}\n\\begin{split}\n\\underline{b}(\\tau,\\phi)&= \\sum_{f=f_1}^{f_L}\\underline{B}(f,\\phi) \\exp(j2\\pi f \\tau)  \\\\          \n&= \\sum_{n=1}^{N}\\sum_{f=f_1}^{f_L}\\underline{v}_n(f,\\phi) \\cdot \\alpha_n \\exp[j2\\pi f (\\tau-\\tau_n)]\\\\\n&=\\sum_{n=1}^{N} \\underline{b}_n(\\tau,\\phi),\n\\end{split}\n\\label{eq:padp}\n\\end{equation}\nwhere $\\underline{b}_n(\\tau,\\phi)$ is the PADP of the $n$-th path.\n\nIn the ideal case, the PADP mimics the Dirac delta function which peaks at the unique angle-delay positions, i.e. $(\\phi_n, \\tau_n)$ with desired power values. However, as discussed, $\\underline{v}_n(f,\\phi)$ presents strong sidelobes at angle around $\\phi_n \\pm \\pi$ for $n  \\in [1,N]$. Therefore, weak desired paths might be buried by undesired sidelobes of strong paths. Below, a novel algorithm following the successive cancellation principle is proposed to tackle this problem.\n\n\\textcolor{black}{The array element response vector $\\mathbf{H}(f)$ is defined as\n\\begin{equation}\n\\mathbf{H}(f) =  [H_1(f);...;H_P(f)],\\label{eq:h_el_v}\n\\end{equation}\nwhere $H_p(f)$ was defined in (\\ref{eq:element frequency response}).}\n\n\\added{\tIn the following algorithm description, we add the superscript numbers to the array element response vector $\\mathbf{H}(f)$ and PADP $\\underline{b}(\\tau,\\phi)$ to indicate that these terms need to update for each iteration. The superscript number $q$ denotes that $q$ path(s) are removed. For example, $\\mathbf{H}^{1}(f)$ denotes the array element response vector with 1 path removed and $\\underline{b}^2(\\tau,\\phi)$ represents the PADP with 2 paths removed. Typically, the original array element response vector and PADP are represented with superscript number $q=0$, i.e. $\\mathbf{H}^{0}(f)$ and $\\underline{b}^0(\\tau,\\phi)$, respectively. }\n\nThe procedure is detailed as below:\n\\begin{itemize}\n\\item[1.] Based on the \\added{current array element response vector $\\mathbf{H}^{0}(f)$}, we apply equations (\\ref{eq:hm}), (\\ref{eq:Gm}), (\\ref{eq:aoa_fi}) and (\\ref{eq:padp}) to obtian the current PADP \\added{$\\underline{b}^0(\\tau,\\phi)$}.\n\\item[2. ] \\added{We find the peak location in the current PADP $\\underline{b}^0(\\tau,\\phi)$, which gives the delay and azimuth angle estimation of path 1, i.e. $\\hat{\\tau}_1$ and $\\hat{\\phi}_1$, respectively. By taking the estimated delay and azimuth angle back to the original PADP $\\underline{b}(\\tau,\\phi)$, we can obtain the amplitude estimation of path 1, i.e. $\\hat{\\alpha}_1 = |\\underline{b}(\\hat{\\tau}_1,\\hat{\\phi}_1)|$. } \n\\item[3.] We remove \\added{path 1} from the array element response vector \\added{$\\mathbf{H}^0(f)$} to obtain the updated array element response $\\mathbf{H}^{1}(f)$ as detailed below.\n\\begin{itemize}\n\t\\item[3.1.] The frequency response vector of the UCA elements corresponding to path 1, can be synthesized based on the detected path parameters under plane-wave assumption as: \n\t\\begin{equation}\n\t\\hat{\\mathbf{H}}(f) =  \\hat{\\mathbf{a}}(f,\\hat{\\phi}_1)\\cdot \\hat{\\alpha}_1  \\exp(-j 2 \\pi f \\hat{\\tau}_1),\\label{eq:h_el_1}\n\t\\end{equation}\n\twhere $\\{\\hat{\\alpha}_1, \\hat{\\phi}_1, \\hat{\\tau}_1\\}$ are parameters estimated in step 2 and $\\hat{\\mathbf{a}}(f,\\hat{\\phi}_1) \\in \\mathbb{C}^{P\\times 1}$ is the array manifold under plane-wave condition for path 1. The $p$-th entry of $ \\hat{\\mathbf{a}}(f,\\hat{\\phi}_1)$ is given by\n\t\\begin{equation}\n\t\\hat{a}_{p}(f,\\hat{\\phi}_1) = \\exp[jkr \\cos(\\hat{\\phi}_1-\\varphi_p) ].\\label{eq:a_s_1}\n\t\\end{equation}\n\t\n\t\\item[3.2.] The synthetic channel impulse response (CIR) vector \\added{$\\hat{\\mathbf{h}}(\\tau)$} corresponding to \\added{path 1} over array elements and the \\added{current CIR vector ${\\mathbf{h}}^0(\\tau)$} can be directly obtained via performing IFT of \\added{$\\hat{\\mathbf{H}}(f)$ and ${\\mathbf{H}}^0(f)$}, respectively.  \n\t\n\t\\item[3.3.] Generate a label vector \\added{$\\mathbf{s}( \\tau)$} with the same size as $\\hat{\\mathbf{h}}(\\tau)$, where the $p$-th entry is obtained as\n\t\n\t\t\t\n   \\begin{equation}\n s_{p}(\\tau) = \\left\\{\n\t\\begin{array}{ll}\n\t0, & \\arrowvert\\hat{h}_{p}(\\tau)\\arrowvert > \\hat{\\alpha}_1 \\cdot 10^{\\frac{-\\eta_t}{20}} \\\\\n\t1, & otherwise \\\\\t\n\t\\end{array} \\right.\n\t\\label{eq:s}\n\t\\end{equation}\n\twhere \\replaced{$\\eta_t$} {} denotes the \\replaced{threshold value}{} in decibels \\added{ and $\\hat{h}_{p}(\\tau)$ the $p$-th entry of synthetic CIR vector $\\hat{\\mathbf{h}}(\\tau)$}. The entries of $\\hat{\\mathbf{h}}(\\tau)$ with dominant power values are labelled to $0$ in the label vector $\\mathbf{s}(\\tau)$.\n\t\n\t\\added{The objective of the label vector $\\mathbf{s}(\\tau)$ is to mark the trajectory of the estimated path in the CIRs over array elements. As explained in step 3.1 and 3.2, we can reconstruct the path trajectory over array elements by using the estimated delay, azimuth angle and amplitude of the path under plane-wave assumption. The label vector would enable us to remove the estimated path from the CIR over array elements in Step 3.4.}\n\t\n\t\\item[3.4.] \\added{Remove path 1 from the current CIR vector $\\mathbf{h}^0(\\tau)$ to obtain the updated CIR vector $\\mathbf{h}^1(\\tau)$ by}\n\t \\begin{equation}\n\t \\mathbf{h}^1(\\tau) = \\mathbf{h}^0(\\tau) \\odot \\mathbf{s}(\\tau), \\label{eq:h_el_11}\n\t \\end{equation}\n\t where $\\odot$ denotes element-wise multiplication.\n\t \n\t \\item[3.5.] By performing Fourier transform (FT) on the updated CIR vector $\\mathbf{h}^1(\\tau)$, we can obtain the updated array element response vector $\\mathbf{H}^{1}(f)$. \n\\end{itemize} \n \\item[4.] Repeat the above steps until the estimated power of the path is not within the preset dynamic range.\n\\end{itemize}\n\n\\added{Note that the superscript numbers of the array element response vector and PADP increase 1 for each iteration. For instance, in the $n$-th iteration, the current array element response vector and PADP are $\\mathbf{H}^{n-1}(f)$ and $\\underline{b}^{n-1}(\\tau,\\phi)$, respectively. After step 3.5, we obtain the path parameters $\\{ \\hat{\\alpha}_n, \\hat{\\phi}_n, \\hat{\\tau}_n\\}$ and the array element response vector is updated to $\\mathbf{H}^{n}(f)$. }\n\n\\added{It is also noted that in step 2, the amplitude estimation of the path is based on the original PADP $\\underline{b}(\\tau,\\phi)$ without superscript, which is kept unchanged for all iterations. Therefore, the amplitude estimations of all the paths are obtained based on the origianl PADP instead of the updated PADP.  The reason is that, when the paths e.g. path $n_1$ and $n_2$ with $n_1>n_2$, have similar or same delays, the trajectories of the two paths are overlapped. As a result, the trajectory of path $n_2$ will be partly removed as we intend to remove path $n_1$ in step 3.4. Thus the amplitude estimation of path $n_2$ based on the updated PADP will be underestimated. This will be further illustrated in the simulation section.}\n\nThe whole procedure of the proposed path estimation algorithm is summarized in Algorithm \\ref{euclid}. \\added{Note that $\\eta$ is the preset dynamic range in decibel.}\n\\begin{algorithm}\n\t\\SetKwInOut{Input}{Input}\n\t\\SetKwInOut{Output}{Output}\n\t\\caption{The proposed algorithm}\\label{euclid}\n\t\\Input{$\\mathbf{H}(f)$}\n\t\\Output{$\\{ \\hat{\\alpha}_n, \\hat{\\phi}_n, \\hat{\\tau}_n\\}, n \\in [1, N]$}\n\t\\BlankLine \n\t$n \\coloneqq 1$\\; \n\t\\replaced{$\\hat{\\alpha}_{n-1} \\coloneqq 1 $ , $\\hat{\\alpha}_{max} \\coloneqq 1 $ }{};  \n\t\t  \n    $\\mathbf{H}^{n-1}(f) \\coloneqq \\mathbf{H}(f)$, $\\mathbf{h}^{n-1}(\\tau) \\coloneqq  IFT(\\mathbf{H}(f))$\\;\n    \t\\While{ $\\hat{\\alpha}_{n-1} > \\added{(\\hat{\\alpha}_{max} \\cdot 10^{-\\eta\/20})} $}{\n  \t\tPerform phase mode space beamforming based on $\\mathbf{H}^{n-1}(f)$ and obtain the PADP $\\underline{b}^{n-1} (\\tau,\\phi)$.  \/\/ Eqs. (\\ref{eq:hm}), (\\ref{eq:Gm}), (\\ref{eq:aoa_fi}) and (\\ref{eq:padp})\\; \n    \tDetect the strongest path of PADP $\\underline{b}^{n-1} (\\tau,\\phi)$ and obtain the path parameters $\\{ \\hat{\\alpha}_n, \\hat{\\phi}_n, \\hat{\\tau}_n\\}$\\; \n    \tRemove the detected path from CIR vector $\\mathbf{h}^{n-1}(\\tau)$ and update the CIR vector to $\\mathbf{h}^{n}(\\tau)$ \\; \n    \t$\\hat{\\alpha}_{max} \\coloneqq  \\hat{\\alpha}_{1}$, $n \\coloneqq  n + 1$. \n    }\t \n\\end{algorithm}\t  \n\nThe basic principle of the low-cost successive cancellation scheme is that the propagation delay value among UCA array elements are insensitive to $D_n$ (i.e. distance between the array center and the $n$-th scatterer location) \\added{and elevation angles}. Therefore, we can remove the detected path based on the azimuth and delay values under the plane-wave assumption. To investigate the effectiveness of the idea, \\added{a single path scenario is considered here.} We can utilize the residual power rate $R_p$ \\added{to evaluate how effective the detected path is removed from the CIR,} which is defined as\n\\begin{equation}\nR_p = \\frac{ \\big\\lVert vec{ \\big\\{\\mathbf{h}^1(\\tau)  \\big\\}} \\big\\lVert ^2}{\\big\\lVert vec{ \\big\\{\\mathbf{h}(\\tau) \\big\\}} \\big\\lVert ^2}\\times 100 \\%,\\label{eq:r_p}\n\\end{equation}\n\\added{where $vec\\{\\cdot\\}$ represents vectorization of a matrix and $\\lVert \\cdot \\lVert $ indicates the Euclidean norm of the vector.}\n\n\\added{The high residual power rate means that the deviation between the trajectory of the synthetic CIR vector $\\hat{\\mathbf{h}}(\\tau)$ and the true CIR vector $\\mathbf{h}(\\tau)$ is large, and the trajectory of the detected path will not be properly removed in the updated CIR vector $\\mathbf{h}^1(\\tau)$. The consequence is that we will estimate the residual trajectory as the fake path if the residual power is within the preset dynamic range.}\n\n\\added{Below, we simulate a single path case to illustrate the residual power rate $R_p$ w.r.t bandwidth, distance $D$ and the elevation angle $\\theta_o$. In the simulation, a single path with a fixed incident azimuth angle $\\phi_0 = 180^o$ impinging at a UCA of radius $r = 0.5$ m is set. The UCA consists of 720 elements with half-wavelength spacing. Besides, we set the center frequency to $29$ GHz, the bandwidth from $400$ MHz to $3$ GHz, distance $D$ from 3 m to 30 m and elevation angle from $90^o$ to $120^o$. }\n\n\\added{The impact of system bandwidth, distance $D$ and elevation angle $\\theta_o$ on the residual power rate is shown in Fig. \\ref{b_dis}.  }\n\\textcolor{black}{\\begin{itemize}\n\t\\item For a given bandwidth and elevation angle $\\theta_o$, $R_p$ decreases as $D$ increases due to the fact that the larger the $D$ is, the closer we approximate the plane-wave condition, resulting in a smaller $R_p $.\n\t\\item For a given distance $D$ and elevation angle $\\theta_o$, $R_p$ increases as bandwidth broadens. The wider the bandwidth, the higher the delay resolution we have. As a result, the larger reconstructed CIR error will be detected as the bandwidth becomes wider, which leads to a larger $R_p$.\n\t\\item For a given bandwidth and distance $D$, $R_p$ increases when elevation angle $\\theta_o$ gets further away from the UCA plane, i.e. with $\\varDelta \\theta =\\arrowvert\\theta_o -90^o \\arrowvert $ becoming larger. This is due to the fact that the synthetic CIR was calculated with $\\hat{\\theta}_o = 90^o$ in (\\ref{eq:h_el_1}) and (\\ref{eq:a_s_1}). The larger the elevation angle away from the UCA plane is, the larger error will be introduced to the reconstructed CIR. The maximum $R_p$ are $0.06\\%$, $0.07\\%$, $0.16\\%$ and $1.1\\%$ when elevation angle $\\theta_o$ are set to $90^o$, $100^o$, $110^o$ and $120^o$, respectively.\n\\end{itemize}\n}\n\n\n\\added{For a practical measurement setting with bandwidth less than\n\t$3$ GHz, the scatterer distance larger than 3 m and the elevation angle $\\arrowvert\\theta_o -90^o \\arrowvert \\leqslant 30 ^o$, the residual power rate is up to $1.1 \\%$, as shown in Fig. \\ref{b_dis}. Therefore, the proposed algorithm works well for practical measurement settings. The proposed cancellation scheme is low-cost, since it only requires the estimated delay and azimuth angle values to remove the detected path. It is effective and robust, as demonstrated in the numerical simulations. }\n \n \n \\added{As a summary, the basic principle of the low-cost successive cancellation scheme is that the propagation delays among the UCA elements are insensitive to the elevation angle of the path and the distance between the UCA center and the scatterer location. Therefore, we can effectively remove the detected path based on azimuth angle and delay of the path in the updated delay profile among elements. Due to the sparsity of mmWave channels, typically only a few iterations are needed to extract all multipath components. As shown, for each iteration, we obtain the delay and azimuth angle estimate based on the power spectra, which essentially is calculated from the one-dimensional beamforming operation in (\\ref{eq:aoa_fi}) and IFT operation in (\\ref{eq:padp}). Therefore, the computational cost is significantly lower, compared to typical high resolution algorithms, where expensive joint search in multiple parameter domains is required.}\n \n \\begin{figure}\n \t\\begin{centering}\n \t\t\\textsf{\\includegraphics[bb=105bp 275bp 477bp 568bp,clip,scale=0.69]{el_D_B.pdf}}\n \t\t\\par\\end{centering}\t\n \t\\caption{\\added{The residual power rate $R_p$ varies with bandwidth, distance $D$ and elevation angle $\\theta_o$, where $f_c = 29$ GHz, $\\phi_o = \\pi$ and $\\tau_o = 0$ ns.}}\n \t\\label{b_dis}\n \\end{figure}\n \n\n\\section{Simulation results} \\label{simulation}\n\\added{In the simulation, we consider a UCA composed of 720 isotropic antennas with radius $0.5$ m and half-wavelength element spacing. The frequency band is from $28$ - $30$ GHz with $750$ frequency points. We simulate a representative yet critical scenario for the channel estimation algorithm, where path 1 and path 3 have the same impinging azimuth angle $\\phi$ at $90^o$. Path 2 has an incident angle of $270^o$, yet it has the same delay as path 3. In addition, the elevation angles are not strictly confined to the UCA plane as detailed in Table \\ref{tab:table1}.}\nThe critical scenario is intentionally set to demonstrate the robustness of the algorithm.\n\n\\replaced{Three beamforming algorithms, i.e. CBF \\cite{fan2016eurasip}, FIBF \\cite{zhang2017fi} and the proposed algorithm, are compared in Fig. \\ref{ad_sim}.}{} As shown in Fig. \\ref{ad_sim} (top), though target paths can be \\added{roughly} detected in the PADP, the CBF algorithm under far-field assumption presents two major drawbacks as explained in Section \\ref{cb_ff}, i.e. susceptible to strong joint side lobes and main lobe distortions. The power loss of the main lobes up to around \\replaced{$13$} {} dB and the concave main lobe (path 1) can be observed in the figure due to the far-field CBF applied in the near-field scenario. The detailed of the concave main lobe can be observed in Fig. \\ref{bp_cb_dis} for $D_n = 3$m. Note that the joint side lobes for the weak path (i.e. path 3) are not shown in the figure due to the limited dynamic range ($40$ dB) set in the simulation. The PADP with the FIBF algorithm is shown in Fig. \\ref{ad_sim} (middle). It shows that the main lobes are not distorted and the peak values are accurate as explained in Fig. \\ref{bp_fi_dis}. However, as discussed, the FIBF suffers from strong side lobes around azimuth angle $\\phi_n \\pm \\pi$ for $n \\in [1,N]$ with a shifted delay. The estimated parameters $\\{\\hat{\\alpha}_n, \\hat{\\phi}_n, \\hat{\\tau}_n\\}, n \\in [1,N]$ with the proposed algorithm are shown in Fig. \\ref{ad_sim} (bottom). The estimated parameters agree well with the targets, with a deviation in path power within $0.3$ dB. The small power deviation is caused by the power variance over the UCA elements in the near-field scenario. \n\\added{Based on the Friis free space propagation equation, the maximal power deviation over the UCA elements for a single path case can be evaluated by the ratio of the maximal and minimal power across the UCA elements, i.e., $\\rho = \\frac{D+r}{D-r}$,  where $r$ denotes the radius of the UCA and $D$ the distance between the scatterer and the UCA center with $D>r$. In the far-field case (i.e. $D\\gg r$), we have $\\rho \\approx 1 $. However, the power over the UCA elements varies in the near-field case. }\n\nBelow, we detail the procedure how the path parameters are estimated with the proposed algorithm.  \n\nTo detect the most dominant path, we can follow the procedure below: \n\t\\begin{itemize}\n\t\t\\item[1.] We can perform the phase mode beamforming based on $\\mathbf{h}^{0}(\\tau)$ (i.e. the raw CIR vector $\\mathbf{h}(\\tau)$), as shown in Fig. \\ref{el_delay} (top), and the obtained PADP $\\underline{b}^0 (\\tau, \\phi)$ is shown in Fig. \\ref{bp_fi} (top).\n\t\t\\item[2.] From the PADP, we can detect the strongest path, i.e. path 1 as shown in Fig. \\ref{bp_fi} (top) with path parameters $\\hat{\\alpha}_1 = 0.2$ dB, $\\hat{\\phi}_1 = 90^o$ and $\\hat{\\tau}_1 = 16.6$ ns.\n\t\t\\item[3.] Then, we remove path 1 from the original CIR vector $\\mathbf{h}^0(\\tau)$ and obtain the updated CIR vector $\\mathbf{h}^1(\\tau)$ as shown in Fig. \\ref{el_delay} \\added{(upper-middle)}.\n\t\t\\item[4.] In the end,  $\\hat{\\alpha}_{max} = \\hat{\\alpha}_1$ is set.\n\t\\end{itemize} \n\\added{We can repeat the above procedure to detect the second and third paths. The detected path parameters of path 2 and path 3 are $ \\{\\hat{\\alpha}_2 = -2.8$ dB,  $\\hat{\\phi}_2 = 270^o,\\hat{\\tau}_2 = 40.1 $ ns $\\}$ and $ \\{\\hat{\\alpha}_3 = -18.1$ dB,  $\\hat{\\phi}_3 = 90^o,\\hat{\\tau}_3 = 40.1$ ns $\\}$, respectively.}\nIn the end, a path with power value within $40$ dB dynamic range can not be found based on $\\mathbf{h}^{3}(\\tau)$ and therefore the channel estimation procedure is complete.\n\n\\added{We can clearly see that the detected path is removed in the updated CIR, e.g. path 1 is removed in the updated CIR vector $\\mathbf{h}^{1}(\\tau)$ as shown in \\ref{el_delay} (upper-middle). Thus the influnce of the path is also eliminated in the updated PADP $\\underline{b}^1 (\\tau, \\phi)$ as shown in \\ref{bp_fi} (upper-middle). In this simulation, path 2 and 3 have the same delays and therefore the trajectories of the two paths are overlapped, as shown in Fig. \\ref{el_delay} (upper-middle). When we remove path 2, the trajectory of path 3 will be partly removed, as illustrated in Fig. \\ref{el_delay} (lower-middle). As discussed in section III, to avoid underestimating the amplitude of path 3 $\\hat{\\alpha}_3$, we estimate the amplitude based on the original PADP $\\underline{b}(\\tau,\\phi)$ as shown in Fig. \\ref{ad_sim} (middle).}\n\n\n\n\\begin{table}\n\t\\caption{Path parameters}\n\t\\label{tab:table1}\n\t\\begin{centering}\n\t\t\\begin{tabular}\t{|c|c|c|c|}\t\t\n\t\t\t\\hline \n\t\t\tPath & $1$  & $2$  & $3$  \\\\ \n\t\t\t\\hline \n\t\t\t$\\alpha$ [dB] & 0 & \\added{-3}   & \\added{-18} \\\\ \n\t\t\t\\hline \n\t\t\t$\\phi$ [deg] &90  &270   &90\\\\ \n\t\t\t\\hline \t\n\t\t\t$\\theta$ [deg] &90  &95  &100 \\\\ \n\t\t\t\\hline\t\n\t\t\t$D$ [m] &4.98   &\\added{12}   &12\\\\ \n\t\t\t\\hline\t\n\t\t\t$\\tau$  [ns] &16.6  &\\added{40.0} &40.0   \\\\ \n\t\t\t\\hline\n\t\t\\end{tabular} \n\t\t\\par\\end{centering}\n\\end{table}\n\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=95bp 225bp 500bp 630bp,clip,scale=0.65]{angle_delay_1_revised.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{The PADPs with CBF (top), FIBF (middle) and the proposed successive FIBF (bottom).}}\n\t\\label{ad_sim}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=80bp 180bp 500bp 665bp,clip,scale=0.58]{element_delay_1_revised_new.pdf}}\n\t\t\\par\\end{centering}\t\n\t\\caption{\\added{The CIR vectors of $\\mathbf{h}^{n-1}(\\tau), n \\in [1,4]$, where the superscript denotes the $(n-1)$ path(s) are removed.}}\n\t\\label{el_delay}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=70bp 185bp 530bp 675bp,clip,scale=0.58]{bf_fi_1_revised_new.pdf}}\n\t\t\\par\\end{centering}\t\n\t\\caption{\\added{The PADPs based on $\\mathbf{h}^{n-1}(\\tau), n \\in [1,4]$, where the superscript denotes the $(n-1)$ path(s) are removed.}}\n\t\\label{bp_fi}\n\\end{figure}\n\n\n\n\n\n\n\\section{Measurement Results}  \\label{measurement}\n\n\\subsection{Introduction}\n\nTo verify how well the proposed algorithm works in practice, we need\nto validate it with practical measurements. The detailed description\nof the measurement campaign was given in \\cite{fan2016eurasip} and only\noutlined here. The measurements were conducted in a typical indoor\nbasement \\added{with the floor dimensions of $7.7$ m $\\times$ $7.9$ m. The basement was empty with few objects including a metallic heater and a metallic ladder leaned against the wall.}\nBoth line-of-sight (LOS) and obstructed LOS (OLOS)\nscenarios were considered. The OLOS scenario was created by placing\na $1.2$ m $\\times$ $1.2 $ m metallic blackboard to block\npaths in LOS directions.\n\n\\added{A wideband biconical antenna with a gain of $6$ dB at 28-30 GHz was used at\n\tthe transmit (Tx) side. The Tx antenna was mounted $0.84$ m above the floor. While an identical biconical antenna was exploited at the receive (Rx) side.\n\tThe Rx antenna was mounted on a turntable with the same height as the Tx antenna. The distance between the Rx antenna and the rotation center was adjusted to $0.5$ m. Then a virtual UCA of radius $r=0.5$ m at Rx side were obtained by rotating the Rx antenna on the turntable. The frequency response of the $p$-th UCA element was measured when the Rx antenna was rotated to the angular position $\\psi_p = 2\\pi\\cdot(p-1)\/P, p\\in [1,P]$ with $P = 720$. \n\tFor each virtual UCA element, the frequency\n\tresponse was measured with a vector network analyzer (VNA) from 28-30\n\tGHz with 750 frequency points. }\n\n\\added{The rotational horn antenna\n\tmeasurements were used as a reference. In the rotational horn antenna measurements, the biconical antenna at the Rx side was replaced by a horn antenna with a gain of $19$ dB at 28-30 GHz. The horn antenna was positioned\tat the rotation center of the turntable (i.e. $r=0$ m) with the same height as the Tx antenna (i.e. $0.84$ m above the floor). The same measurement settings (the same Tx antenna, the same frequency sweep and orientation sweep) adopted in biconical antenna UCA measurements were used in rotational horn antenna measurements for comparison purpose. }\n\nNote that mutual coupling effect is not present with the virtual array measurement, which is desirable\nfor the  channel characterization purpose. The distance between the Tx\nand the center of the Rx array is around 5 m, while the far-field\ndistance at 30 GHz for the UCA is around 200 m. The system bandwidth\nin the measurement is larger than 500 MHz and the UCA array aperture\n(i.e. 1m) is  much larger than the delay resolution multiplied by\nthe speed of light (i.e. 0.15 m). Therefore, for the measurement data,\nboth the far-field assumption and narrowband assumption are violated.\nNote that \\added{the antenna gains of both Tx and Rx antennas are de-embedded in measured CIRs. Further, to focus on the specular and dominant multipath components detection,} a dynamic range of 30 dB is set in the measurement section. \n\n\\subsection{Measured results}\n\nThe measured CIRs over virtual UCA elements (i.e. measured locations) for the LOS scenario are shown\nin Fig. \\ref{LOS_synthetic} (top), where a few specular paths can\nbe clearly detected besides the dominant LOS path. \\added{The mmWave channels are more sparse compared to sub-6GHz channels. In our measurements, the measurement was performed in an empty indoor basement, with no furniture, which also results in sparse channel profiles.} The synthetic CIRs\nover UCA elements  for the LOS scenario are shown in Fig. \\ref{LOS_synthetic}\n(below). The synthetic CIRs over virtual UCA elements are reconstructed based on detected multipath component parameters  $\\{\\hat{\\alpha}_{n},\\hat{\\phi}_{n},\\hat{\\tau}_{n}\\}$\nfor $n\\in[1,N]$ under the plane-wave assumption. The synthetic results agree well with the measured data,\nindicating a consistent estimation result. The \\added{trajectories of the paths} over UCA elements match well with the measured ones, even for weak multipath components. The measured PADP\nwith rotational horn antenna is shown in Fig. \\ref{LOS_horn} (top).\nAs shown in the measured results, paths having the same impinging angle\nyet different delays exist due to the path  bouncing in the LOS direction.\nThe estimated PADP based on the virtual UCA with\nthe proposed algorithm for the LOS scenario is shown in Fig.\n\\ref{LOS_horn} (below), \\added{where in total 10 paths are detected}. An excellent match of the measured PADPs between the rotational\nhorn antenna and virtual UCA in terms of the number of propagation paths,\nazimuth angle, delay and power of each path can be observed. However, the\nmeasured results based on rotational horn antenna suffer from wide\nantenna beam-width, as expected. The proposed algorithm presents consistent\nparameter estimation, with high resolution in the angle and delay\ndomains. \n\\added{ Note that the antenna gains of the horn antenna and biconical antenna are calibrated out in the power spectra. Thus within the same power range, the same number of paths and approximately same path parameters can be observed in the plots. Furthermore, the estimated power of the LOS path (strongest path) is $-76.5 $ dB, which matches well with the calculated path loss according to Friis equation with $D = 5$ m and $f_c = 29$ GHz, i.e. $-75.7$ dB. The deviation might be introduced by the inaccureate data in antenna gains and measurement uncertainties.}\n\nThe measured CIRs and synthetic CIRs over virtual UCA elements for the\nOLOS scenario are shown in Fig. \\ref{OLOS_synthetic}. A good agreement\nbetween the synthetic and measured CIRs can still be observed,\nthough \\added{there exists many weak multipath components in }the measured CIRs. The measured PADPs with rotational horn\nantenna and virtual UCA with the proposed algorithm for the OLOS scenario\nare shown in Fig. \\ref{OLOS_horn}, \\added{where $27$ paths in total are detected within the dynamic range of $30$ dB. Within the same power range, the channel parameters, e.g. the number of paths and path paramters, agree well between the rotational horn and the virtual UCA for\n\tthe more critical OLOS scenario. }\n\n\\added{As observed in the measured CIRs over array elements, channel non-stationarity exists where different channel profiles can be observed by different array elements. This can be caused by several reasons, e.g. the power variation over the UCA elements in the near-field scenarios as explained; the coherent summation of unresolved multipath components due to limited system bandwidth; limited angle of view from the near-field scatterers and the measurement system non-idealities. The channel non-stationarity is not addressed in the proposed algorithm, as seen in the synthetic CIRs over UCA elements. Though channel non-stationarity has been considered in channel modeling works, e.g. \\cite{he2018mobility}, it has not been considered in the channel estimation in the literature so far due to high computation complexity.  }\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 270bp 480bp 567bp,clip,scale=0.58]{los_syn_30db_1.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{Measured CIRs (top) and synthetic CIRs (below) over virtual UCA elements\n\t\tfor the LOS scenario. }}\n\t\\label{LOS_synthetic}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 270bp 480bp 550bp,clip,scale=0.58]{los_horn_30db_1.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{Measured PADPs with rotational horn antenna (top)\n\t\tand virtual UCA with the proposed algorithm (below) for the LOS scenario. }}\n\t\\label{LOS_horn}\n\\end{figure}\n\n\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 270bp 480bp 567bp,clip,scale=0.58]{nlos_syn_30db_1.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{Measured CIRs (top) and synthetic CIRs (below) over virtual UCA elements\n\t\t\tfor the OLOS scenario. } }\n\t\\label{OLOS_synthetic}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 270bp 480bp 550bp,clip,scale=0.58]{nlos_horn_30db_1.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{Measured PADPs with rotational horn antenna (top)\n\t\tand virtual UCA with the proposed algorithm (below) for the OLOS scenario.} }\n\t\\label{OLOS_horn}\n\\end{figure}\n\n\n\n\n\n\n\\section{conclusion}\nAccurate knowledge of the radio propagation parameters is important for system design, applications and performance evaluation of the 5G systems. However, multipath parameter estimation  for UWB large scale antenna systems are challenging,  due to the fact that the well adopted plane-wave  and narrowband assumptions might not hold. In this paper, a novel beamspace UCA algorithm based on phase mode excitation principle is proposed. The proposed beamformer can maintain approximately same beam patterns, independent of distance between the array and scatterer location, and of the system bandwidth, making it suitable for UWB near-field scenarios. The proposed algorithm has low computational cost since it avoids expensive joint estimation in multiple parameter domains. To remove the strong sidelobes of the proposed beamspace beamformer, a novel algorithm based on the successive cancellation principle is proposed. The path cancellation is based on the fact that propagation delays among array elements are insensitive to system bandwidth and near-field effects \\added{for a given elevation angle range, i.e. $\\varDelta \\theta = \\arrowvert \\theta - 90^o\\arrowvert \\leq 30^o$}. The cancellation scheme is effective and robust. For example, the residual power rate of less than 0.2\\% can be achieved for the UCA with radius $r = 0.5$m, system bandwidth 2 GHz and measurement range  $D\\geqslant 3$ m.  To demonstrate the proposed algorithm, a critical scenario is selected in the numerical simulation \\added{with frequency band set the same as in the measurement campaign, i.e. 28-30 GHz}, and the results showed that all the paths can be accurately detected with less than 0.3 dB deviations for the path powers. To validate the algorithm, we applied the proposed algorithm in the practical virtual UCA measurement data in both LOS and OLOS scenarios. The synthetic CIRs obtained based on the detected multipath parameters matched well with the measured CIRs for both measurement scenarios. The detected parameters were further validated against the horn antenna reference measurements. As a summary, both numerical simulations and experimental measurements demonstrated the effectiveness and robustness of the proposed algorithm. The proposed algorithm is a general low-cost channel estimator, since it works in both near-field scenario and UWB system.\n\n\\added{Further, due to the fact that the proposed beamformer pattern is not sensitive\n\tto the elevation angle and scatterer location, the proposed\n\talgorithm would fail to detect the elevation angle and scatter\n\tlocation. A high resolution propagation parameter estimation\n\talgorithm, which is capable of estimating all\n\tpropagation parameters in the 3D near-field conditions is missing\n\tin the literature due to the fatal computation complexity.\n\tOur proposed algorithm, which offers high resolution azimuth\n\tangle, delay and complex amplitude estimation in a low-cost\n\tmanner, can be utilized in the initial stage of the full parameter\n\testimation algorithm, e.g. the maximal likelihood estimator,\n\twhich can significantly reduce the computation complexity\n\tdue to the reduced searching space. This work will be carried out in a future work. }\n \n\n\n\n\n\\bibliographystyle{IEEEtran}\n\\addcontentsline{toc}{section}{\\refname}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supplemental Material}\\setcounter{page}{1}\n\nStarting from the Lindbladian $\\mathcal{L}_\\text{full}$ given in\nEq.~\\eqref{eq:lfull} in the main text, we show how to derive the effective\nmodel describing the slow dynamics. The first step is to define a set of\n(super-)operators, ${O}_q = {O}_+ - {O}_-$ and ${O}_c = ({O}_+ + {O}_-)\/2$\nwith ${O}_+(\\rho)$ = ${O}\\rho$ and ${O}_-(\\rho)=\\rho {O} $ where $\\rho$ is the\ndensity matrix of the system.  Using these definitions for the ladder\noperators, we obtain the commutators $[a_q,a_c^\\dagger] =  [a_c,\na_q^\\dagger]=1 $ (while the rest of the commutators of $a_q, a_c, a_q^\\dag,\na_c^\\dag$ vanish).\n\nAt first, we diagonalize the quadratic part of the Lindbladian\n$\\mathcal{L}_\\text{quad}$  for $s=0$ and without the stabilizing potential in\nthe Hamiltonian $H$ by a symplectic diagonalization to conserve the bosonic\nstructure \\cite{Prosen:10}.  The Lindbladian assumes the form\n\\begin{equation}\n  \\mathcal{L}_\\text{quad} = -\\tfrac12(\\gamma-\\epsilon) v_su_s-\\tfrac12(\\gamma +\\epsilon)v_fu_f\n\\end{equation}\nin terms of ladder operators $v_j, u_j$ such that $[u_j, v_k]=\\delta_{jk}$.\nExplicitly, they are given by\n\\begin{align}\\label{eq:basis}\n  u_s &= x_c +i  \\frac{\\gamma (2\\bar n+1)}{2(\\gamma-\\epsilon)}y_q,&  v_s &= -iy_q, & u_f &=  y_c -i \\frac{\\gamma (2\\bar n+1)}{2(\\gamma+\\epsilon)}x_q,  &v_f &=ix_q,\n\\end{align}\nwhere we introduced the two quadratures $x_{q,c} = (a_{q,c}^\\dagger + a_{q,c})\/\\sqrt{2}$ and\n$y_{q,c} = i(a_{q,c}^\\dagger - a_{q,c})\/\\sqrt{2}$ with $[x_{c},y_{q}] =[ x_{q}, y_{c}]=\ni$. Note, however, that the creation and annihilation operators, $v_j$ and $u_j$, are\nnot the adjoint of each other which is due to the fact that the Lindbladian $\\mathcal{L}_\\text{quad}$ is not Hermitian. \n\nThe slow dynamics of the system is given by the $u_s, v_s$ (or $x_c,y_q$\nrespectively) while $u_f,v_f$ correspond to a mode that decays with the rate\n$\\gamma$ at threshold. The spectrum of the system is given by $\\lambda =\n-\\tfrac12(\\gamma-\\epsilon) n_s -\\tfrac12(\\gamma+\\epsilon) n_f $ with $n_s, n_f \\in \\mathbb{N}_0$. The corresponding right eigenstates are the product states\n$|n_s, n_f\\rangle = |n_s\\rangle |n_f\\rangle$ with $u_j |n_j\\rangle =\n\\sqrt{n_j} |n_j-1\\rangle$ and $v_j |n_j\\rangle = \\sqrt{n_j+1} |n_j+1\\rangle$.\nThe left eigenstates  similarly are characterized by $\\langle n_j| v_j =\n\\sqrt{n_j} \\langle n_j-1|$ and $\\langle n_j| u_j = \\sqrt{n_j+1} \\langle\nn_j+1|$. Note that in the superoperator formalism $|0,0\\rangle$ corresponds to\nthe stationary state $P_s$ and $\\langle 0,0|$ is the trace operation.\n\nNext, we can take the nonlinearity given by the stabilizing potential $V$ into\naccount. It produces the additional term $\\mathcal{L}_V = - i(V_+ - V_-)$ with\n$\\mathcal{L}_\\text{full} = \\mathcal{L}_\\text{quad} + \\mathcal{L}_V + s f\\gamma\na_+ a_-^\\dag $.  The small  parameters $\\alpha$ allows to treat\n$\\mathcal{L}_V$ perturbatively. In particular, we are only interested in the\neffect of $\\mathcal{L}_V$ on the slow mode as the fast mode remains in the\nstationary state $|n_f=0\\rangle$.  Since we are close to threshold, we\nalso set $\\epsilon = \\gamma$ in $\\mathcal{L}_V$.\n\nFor the Josephson oscillator, we can simply project $\\mathcal{L}_V$ onto the subspace with $n_f =0$. Using the expressions \\eqref{eq:basis}, we obtain\n\\begin{equation}\n  \\mathcal{L}_V \\approx \\langle n_f =0 | \\mathcal{L}_V |n_f = 0\\rangle= \\tfrac1{16} \\kappa \\gamma iy_qx_c + \\tfrac1{48}\\kappa\\gamma\\bigl(i y_qx_c^3 + \\tfrac14i y_q^3x_c\\bigr)\\,.\n\\end{equation}\nThe first term corresponds to a small shift of the threshold. This shift,\nwhich vanishes for $\\alpha \\to 0$, can be seen in Fig.~\\ref{fig:fano}. The\nmodel is then given by $y_q \\mapsto p\/x_*$ and $x_c \\mapsto\nx_* q$ with a rescaling factor $x_* \\gg 1$ that is specified in the main\ntext. As discussed in the main text, we have $y_q \\simeq x_*^{-1}$ and $x_c\n\\simeq x_*$ and thus $y_q \\ll x_c$ for $\\alpha \\to 0$. Due to this,  the term\n$i y_qx_c^3$ is the most relevant nonlinearity. We can thus neglect $y_q^3\nx_c$ which allows to identify $m=2$ and $\\alpha = \\kappa\/48$ for the Josephson\noscillator.\n\nThe case of the Duffing potential needs more careful treatment due to the\nrotational symmetry as noted in the main text. In particular, we have $\\langle\nn_f =0 | \\mathcal{L}_V |n_f = 0\\rangle= 0$ due to this\nsymmetry.\nIn order to find the stabilizing effect, we need to include virtual\nexcitations of the fast mode in second-order perturbation theory. We obtain ($\\bar n =0$ for simplicity)\n\\begin{align}\\label{eq:lduff}\n  \\mathcal{L}_V &\\approx \\langle n_f =0 | \\mathcal{L}_V |n_f = 0\\rangle -\\sum_{n_f'> 0}\\frac{\\langle n_f =0 | \\mathcal{L}_V |n_f'\\rangle\\langle n_f'| \\mathcal{L}_V |n_f=0\\rangle}{ \\gamma n_f'} \\\\\n                &=  \\chi^2 \\gamma\\bigl( -\\tfrac{31}{16} iy_qx_c - \\tfrac{13}{4} iy_qx_c^3  + iy_qx_c^5 + \\tfrac{5}{16} y_q^2  - \\tfrac{5}{8} y_q^2x_c^2 + \\tfrac{1}{4} y_q^2x_c^4  + \\tfrac{13}{16} iy_q^3x_c    - \\tfrac{5}{32} y_q^4   + \\tfrac{1}{8} y_q^4x_c^2 + \\tfrac{1}{64} y_q^6  \\bigr).\\nonumber\n\\end{align}\nAs before, the rescaling $y_q \\mapsto p\/x_*$ and $x_c \\mapsto\nx_* q$ leads to $y_q \\ll x_c$ in the limit of weak nonlinearities. Because of this, the most relevant nonlinearity is the term $i \\chi^2 \\gamma y_q x_c^5$. This allows to identify $m=4$ and $\\alpha = \\chi^2$ for the Duffing oscillator. Note that the term $\\propto y_q x_c^3$ has the wrong sign and does not stabilize the mode. Moreover, it is subleading as $y_q x_c^3 \\propto x_*^2$ while $y_q x_c^5 \\propto x_*^4$. While the result \\eqref{eq:lduff} is shown for $\\bar n=0$ for simplicity, the relevant term  $i\\chi^2 \\gamma y_qx_c^5$ is in fact independent of $\\bar n$. \nThe counting field leads to the additional term\n\\begin{equation}\n  s f \\gamma a_+ a_-^\\dag = \\frac{sf\\gamma}2 \\Bigl[ (x_c + \\tfrac{i}2 y_q)^2 + (y_c -\\tfrac{i}2 x_q)^2 \\Bigr]\\,.\n\\end{equation}\nIn the limit $\\alpha \\ll 1$, the dominant term is given by  $\\frac12sf\\gamma \nx_c^2 = \\frac12sf\\gamma  x_*^2 q^2 $ as stated in the main text.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\\noindent\nA Gause type predator-prey model with response\nfunction $f(x)$ is given by\n\\begin{equation}\\label{ppnodelay}\n\\left\\{\n\\begin{aligned}\n&\\dot x(t)=rx(t)\\bigg(1-\\frac{x(t)}{K}\\bigg)-y(t) f(x(t)),\\\\\n&\\dot y(t)=-sy(t)+Y y(t) f(x(t)),\n\\end{aligned}\\right.\n\\end{equation}\nwhere $x(t)$ denotes the density of the prey population and $y(t)$\nthe density of predators. Parameters $r,\\ K,\\ s$, and $Y$ are\npositive constants denoting the intrinsic growth rate and the\ncarrying capacity of the prey, the death rate of the predator in the\nabsence of prey, and the growth yield constant for the conversion of prey\nto viable predator density, respectively.\n\nIf  $f(x)$ is\nof Holling type I form in model (\\ref{ppnodelay}) (i.e. $f(x)=mx$ where\n$m$  is a positive constant denoting the maximal growth rate of\nthe predator), it is well-known (see e.g.\n \\cite{Bazykin,Freedman1980}) that  either\nthe predator population approaches extinction and the prey population\napproaches its carrying capacity, or the predator population and the\nprey population coexist and their density approaches a positive\nequilibrium. Hence, for\nall choices of the parameters, all solutions of this system approach a\nglobally asymptotically stable equilibrium, and so any nontrivial\noscillatory behaviour that arises due to the introduction of delay in\nthe model\ncan be attributed solely to the delay. For this reason, we choose\nHolling type I response functions instead\nof the more realistic Holling type II form, since the Holling type II\nform  results in a model that gives rise to  nontrivial period solutions  without delay (see Rosenzweig \\cite{Rosenzweig1971}).\nOne would also expect that any  exotic dynamics that the model with  Holling\ntype I form admits due to the introduction of delay would  be shared by\nthe model with Holling type II form.   Li et al. \\cite{MR3180725} studied this model with the Holling type II\nresponse function of Monod form and showed that stability switches caused by\nvarying the time delay are accompanied by bounded global Hopf branches, and they proved that when multiple Hopf branches exist, they are nested and the overlap produces coexistence of two or possibly more stable limit cycles.\nHowever, they did not go on to discover the even richer  dynamics that\nour analysis suggests exists in that case.\n\nIncorporating a time delay in (\\ref{ppnodelay}) to model the time between the\ncapture of the prey by the predator and its conversion to viable\npredator biomass, in the case of Holling type I functional response,\n$f(x)=mx, \\ m>0$, we  obtain the following system:\n\\begin{equation}\n\\left\\{\\begin{aligned}\n&\\dot x(t)=rx(t)\\left(1-\\frac{x(t)}{K}\\right)-m y(t)x(t),\\\\\n&\\dot y(t)=-sy(t)+Ye^{-s\\tau}my(t-\\tau)x(t-\\tau). \\label{ppreydelay}\n\\end{aligned}\\right.\n\\end{equation}\nThe term $e^{-s\\tau}y(t-\\tau)$ represents those predators that\n survive the $\\tau \\geq 0$ units of time required to process the\n prey captured at time  $t-\\tau$  in the past. Thus, we have incorporated the delay in the  growth term of the predator equation in a manner that is\nconsistent with  its  decline rate given by the model, as described in Arino,\nWang, and Wolkowicz \\cite{Gail2006Alter}.\n\nWe define\n$\\mathbb{R}_+\\equiv\\{x\\in\\mathbb{R} : \\, x\\geqslant 0\\}$,\nint$\\mathbb{R}_+\\equiv\\{x\\in\\mathbb{R} : \\, x>0 \\}$.   We denote by\n $C([-\\tau,0],  \\mathbb{R}_+)$,  the Banach space of continuous functions   from\n the interval $[-\\tau,0]$ into  $\\mathbb{R}_+$, equipped with the uniform norm.\n We assume initial data for model (\\ref{ppreydelay}) is taken from\n \\begin{equation}\\label{eq:X}\n X= C([-\\tau,0],  \\mathbb{R}_+) \\times  C([-\\tau,0],  \\mathbb{R}_+).\n\\end{equation}\n\nModel (\\ref{ppreydelay}) also has other interpretations.\nGourley and Kuang \\cite{Kuang2004} studied\na stage-structured predator-prey model\nin which they included an equation for the juvenile\npredators and assumed a constant maturation time\ndelay, i.e.,  they assumed that the juvenile predators\ntake a fixed  time to mature.\nUsing the approach developed in Beretta and Kuang\n\\cite{Kuang2002},\nthe authors considered the\npossibility of stability switches, and concluded that\nthere is a range of the parameter modeling the time delay for which\nthere are periodic solutions.\n If the juveniles in their model suffer the\nsame mortality rate as adult predators, their model decouples and\nyields model (\\ref{ppreydelay}).\nForde \\cite{forde2005delay} also considered this model and conjectured\nthat there are periodic orbits whenever the interior equilibrium exists\nand is unstable. He also noted that if the interior equilibrium\nexists and is asymptotically stable without delay, then for small\ndelays it remains globally asymptotically stable. We show that whenever the\ninterior equilibrium exists and is unstable, the system is uniformly\npersistent. Gourley and Kuang\n\\cite{Kuang2004} had already showed that a Hopf bifurcation eventually occurs\nif the delay is increased, destabilizing this equilibrium and  giving\nbirth to a nontrivial periodic solution. Forde  \\cite{forde2005delay}\n left as an open question whether more than one periodic orbit is\n possible and provided a numerical example suggesting chaos is possible\n but does not consider the route to chaos.\n We give numerical evidence that there is a range of parameters for which\n two stable periodic orbits and an unstable periodic orbit all exist and\nwe show a period-doubling route to chaos followed by a\n period-halfing route back to stability of the interior equilibrium.\n\n Cooke, Elderkin, and Huang \\cite{Cooke2006} considered a model\nsimilar to the one in Gourley and Kuang \\cite{Kuang2004}, and\nobtained results concerning Hopf bifurcation of a scaled version.\nThe scaling they used  eliminated the parameter modelling the time\ndelay, the parameter that we focus on and use as a bifurcation\nparameter.  This simplified their analysis,\nsince then, unlike in our case, the components of the coexistence\nequilibrium are independent of the time delay.\n\n\n\n\nIn this manuscript we show that the introduction of\ntime delay cannot only destabilize the  globally asymptotically stable\ncoexistence equilibrium of model (\\ref{ppreydelay}),  it can also be responsible for exotic dynamics for\nintermediate values of the delay as well as the  eventual   disappearance of\nthe coexistence equilibrium with the extinction of the predator\nfor large enough delays.\nAlthough chaotic dynamics has been observed in other models of\npredator-prey interactions, the other models either require at least\nthree trophic levels, or the  response functions  are not as simple and\nso the models admit oscillatory behavior even in the absence of delay,\nor the other models incorporate the delay  in such a way  that   the\npredators   still contribute to\n population growth even if the time required\n to process the prey is longer than the life-span of the predator\n (i.e., the factor $e^{-s\\tau}$ is missing in the $\\dot{y}$ equation),\n or  the delay is used to model different mechanisms (see e.g.,\n\\cite{Ginoux2005,Hastings1991,Morozov2004,JWang2012}).\nThe observation that the resulting strange attractor   resembles the\nstrange attractor for the Mackey-Glass equation  \\cite{Mackey:2009} is also new.\n\nThis paper is organized as follows.  In section~\\ref{basic}, we scale the\nmodel and show that it is well-posed. In section~\\ref{analysis}, we\nconsider the existence and stability of equilibria.\nIf parameters are set so that it is possible for the predator to survive\nwhen there is no delay, it is well-known that the\nequilibrium at which both the prey and the predator survive  is globally asymptotically stable with respect to positive\ninitial conditions   (i.e. solutions\napproach this equilibrium for any choice of positive initial data).\nIn the case of delay,\n  the components of this coexistence equilibrium depend on the\ndelay.\nWe prove that for positive delay,  when this equilibrium exists, both the predator and the prey\npopulations persist uniformly. However, a sufficiently long delay\nresults in   the disappearance of this equilibrium, resulting in  the extinction\nof the predator and convergence to a globally asymptotically stable\nequilibrium with the prey at carrying capacity.\nWe give criteria which when satisfied imply that there are at least two Hopf\nbifurcations that occur before the extinction of the predator, resulting\nin sustained oscillatory behaviour for intermediate values of the delay. Finally, in\nsection~\\ref{sec:example},  by means of\ntime series, time delay embeddings, and orbit (bifurcation) diagrams we show\nthat there are saddle-node bifurcations of limit cycles resulting in\nbistability as well as sequences of period doubling bifurcations leading to\nchaos,\nwith a strange attractor resembling the strange attractor for the\nMackey-Glass equation \\cite{Mackey:2009}.\nWe conclude with a brief discussion.\n\n\n\\section{Scaling and basic properties of solutions}\\label{basic}\nIn order to simplify the analysis, we introduce the following change of\nvariables:\n\\begin{equation}\n\\begin{aligned}\n&\\breve{t}=rt,\\qquad \\breve{x}(\\breve{t})=x(t)\/K,\\qquad\n\\breve{y}(\\breve{t})=m y(t)\/r, \\\\\n&\\breve{\\tau}=r\\tau,\\qquad \\breve{s}=\\frac{s}{r},\\qquad\\qquad\\qquad\n\\breve{Y}=YKm\/r.\\label{rescale}\n\\end{aligned}\n\\end{equation}\nWe drop the $\\breve{} \\ $'s for convenience and study the\nequivalent scaled version of model (\\ref{ppreydelay}):\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\dot x(t)=x(t)(1-x(t))-y(t)x(t),\\\\\n&\\dot y(t)=-sy(t)+Ye^{-s\\tau}y(t-\\tau)x(t-\\tau),\\\\\\label{dimenless}\n& (x(t),y(t))=(\\phi(t),\\psi(t))\\in X, \\, \\mbox{for} \\,\\, t\\in[-\\tau,0],\n\\end{aligned}\\right.\n\\end{equation}\nwhere $X$ was defined in  (\\ref{eq:X}).\n\n\nFirst we address well-posedness of system (\\ref{dimenless}).  For positive delay $\\tau$, the existence and uniqueness of solutions of\nsystem (\\ref{dimenless})\nwas shown in Gourley and Kuang \\cite{Kuang2004}. The following\nproposition, proved in \\ref{app:bounded}, indicates that for positive\ndelay the solutions remain\nnonnegative and provides an  upper bound for each component.\n\n\\begin{proposition}\\label{bounded\n Consider model (\\ref{dimenless}) with initial data in $X$.\n\\begin{enumerate}\n\\item The solutions  exist, are unique, and  remain\n  nonnegative for all $t\\geqslant 0$.\n\\item  $ \\, \\limsup_{t\\rightarrow \\infty}x(t)\\leqslant 1 \\, $ and\n$ \\, \\limsup_{t\\rightarrow \\infty}y(t)\\leqslant\n\\frac{1}{4s}Ye^{-s\\tau}(s+1)^2$.\n  \\item Consider model (\\ref{dimenless}) with initial data in $X^0$ where\n  \\begin{equation}\\label{eq:X0}\n    X^0=\\{ (\\phi(t),\\psi(t)) \\in X : \\, \\phi(0)>0 \\,\\,\n    \\& \\,\\, \\exists \\,  \\theta \\in[-\\tau,0] \\, \\mbox{s.t.} \\, \\,\n  \\phi(\\theta)\\psi(\\theta)>0.\\}\n\\end{equation}\nThen,  $x(t)>0$ for all $t>0$ and there exists $T\\geqslant 0$ such that\n$y(t)>0$ for all $t>T$.\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\\section{Existence and stability of equilibria and uniform persistence} \\label{analysis}\n\n Model\n(\\ref{dimenless}) can have up to three distinct\nequilibria:\n\\begin{eqnarray}\\label{notation1}\n  E_0=(0,0), \\  E_1=(1,0), \\     E_+=(x_+(\\tau),y_+(\\tau))= \\left(\\frac{s}{Y}e^{s\\tau},\n1-\\frac{s}{Y}e^{s\\tau}\\right).\n\\end{eqnarray}\n\nThe components of $E_+$ are nonnegative and $E_+$ is distinct from\n$E_1$, if,\nand only if, $0\\leqslant \\tau < \\tau_c$, where\n\\begin{equation}\\label{notation2}\n\\tau_{c}=\\frac{1}{s}\\ln\\bigg( \\frac{Y}{s} \\bigg).\n\\end{equation}\nThus, when $\\tau_c>0$, i.e., when $Y>s$, the components of $E_+$ are both positive, and   $E_+$ is referred to as the coexistence equilibrium.\n\nWhen there is no delay, i.e. $\\tau=0$ in (\\ref{dimenless}), $E_0$ is\nalways a saddle attracting solutions with $x(0)=0$.\nIf $Y<s$, one of the components of $E_+$ is negative and so it is not\nrelevant, and $E_1$ is globally asymptotically stable with\nrespect to initial conditions satisfying $x(0)>0$ and $y(0)\\geqslant 0$. When $Y=s$,  $E_1$ and $E_+$ coalesce  and are globally\nattracting provided $x(0)>0$. If $Y>s$,  then $E_1$ is  a saddle\nattracting solutions with $x(0)>0$ and $y(0)=0$  and\n$E_+$ sits in int$\\mathbb{R}^2_+$ and is global asymptotically stable\nwith respect to initial conditions in  int$\\mathbb{R}^2_+$.\n\nWhen $\\tau>0$, to determine the local stability of each equilibrium\nsolution, we use the linearization technique for differential equations with\ndiscrete delays (see Hale and Lunel  \\cite{Hale1993}). After\nlinearizing (\\ref{dimenless}) about any one of these equilibria,\n $(x^{\\star},y^{\\star})$, the characteristic equation,\n $P(\\lambda)|_{(x^{\\star},y^{\\star})}=0$,\n is given by,\n\\begin{equation}\n  (\\lambda+s)(\\lambda+y^{\\star}-(1-2x^{\\star}))\n+Ye^{-(s+\\lambda)\\tau}x^{\\star}(1-2x^{\\star})-\\lambda\nYe^{-(s+\\lambda)\\tau}x^{\\star}=0 .\n\\label{chareqn}\n\\end{equation}\n\nWe summarize the results on local and global stability of\nthe equilibrium points and uniform persistence of the populations in the following theorem.  The proof can be found\nin \\ref{app:gasEi}.\n\\begin{theorem\n  \\label{th:gasEi}\nConsider (\\ref{dimenless}).\n\\begin{enumerate}\n  \\item  Equilibrium $E_0$ is always unstable.\n  \\item  Equilibrium $E_1$ is\n    \\begin{enumerate}\n      \\item unstable if\n$0\\leqslant \\tau <\\tau_{c}$, and\n\\item globally asymptotically stable (with respect to $X^0$) if $\\tau>\n  \\tau_{c}$, (i.e. if $\\frac{se^{s\\tau}}{Y}>1$).\n\\end{enumerate}\n  \\item Both components of $E_+$ are positive (i.e., $E_+$ exists), if,\n    and only if,  $0\\leqslant \\tau < \\tau_c$,  (i.e. $\\frac{se^{s\\tau}}{Y}<1$).\n    \\begin{enumerate}\n  \\item When $E_+$ exists and $\\tau=0$, $E_+$ is globally asymptotically\n    stable with respect to int$R^2_+$.\n  \\item When $E_+$ exists, and $\\tau\\geqslant 0$, model (\\ref{dimenless}) is uniformly\n    persistent with respect to initial data in $X^0$, i.e., there exists\n    $\\epsilon>0$ independent of $(\\phi(t),\\psi(t))\\in X^0$ such that \\\\ $\\liminf_{t\\rightarrow\\infty}\n    x(t)>\\epsilon$ and $\\liminf_{t\\rightarrow\\infty}\n    y(t)>\\epsilon$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{theorem}\n\n\n\nThus, for any fixed time delay $\\tau$, if $\\frac{s}{Ye^{-s\\tau}}>1$,  only the prey\npopulation survives and it converges to a steady state. On the other\nhand, if the inequality is reversed,  for appropriate initial data both\nthe prey and the predator populations are uniformly persistent, i.e.\nsurvive indefinitely.\nHowever, we have not yet addressed what form the dynamics takes in the latter\ncase.\n\n\n\\subsection{Local stability  of $E_+$} \\label{HB}\n\nWhen $E_+$ exists, by Theorem~\\ref{th:gasEi} both populations survive\nindefinitely.  To address the possible forms the dynamics can take, we begin\nby investigating the local stability of $E_+$ when it exists, i.e., when\n$0\\leqslant\\tau<\\tau_c$, and hence both components are positive.\nEvaluating the characteristic equation (\\ref{chareqn}) at $E_+$ gives\n\\begin{equation*}\nP(\\lambda)|_{E_+}=\\lambda^2 +\\lambda\ns\\left(1+\\frac{e^{s\\tau}}{Y}\\right)+\\frac{s^2}{Y}e^{s\\tau}+\ne^{-\\lambda\\tau}s\\left(-\\lambda +\\left(1-\\frac{2 s\ne^{s\\tau}}{Y}\\right)\\right)=0.\n\\end{equation*}\nTherefore, $P(\\lambda)|_{E_+}=0$ is of the form\n\\begin{equation}\nP(\\lambda)|_{E_+}=\\lambda^2+p(\\tau)\\lambda+(q\\lambda+c(\\tau))e^{-\\lambda\\tau}+\\alpha(\\tau)=0,\\label{CharE+}\n\\end{equation}\nwhere\n\\begin{equation} \\label{eqn:coeffs}\np(\\tau)=s\\left(1+\\frac{e^{s\\tau}}{Y}\\right),\\ \\ q=-s,\\ \\ c(\\tau)=\ns\\left(1-2\\frac{s e^{s\\tau}}{Y}\\right),\\ \\ \\mbox{and} \\ \\\n\\alpha(\\tau)=\\frac{s^2 e^{s\\tau}}{Y},\n\\end{equation}\nFirst assume that $\\tau=0$. Then (\\ref{CharE+}) reduces to\n\\begin{equation*}\n\\lambda^2 + ( p(0) + q )\\lambda + (\\alpha(0) +c(0) )=0.\n\\end{equation*}\nSince $\\alpha(0)+c(0)=s\\left(1-\\frac{s}{Y}\\right)=s y_+(0)>0$\nand $p(0)+q=\\frac{s e^{s\\tau}}{Y}>0$, by the Routh-Hurwitz\ncriterion  \\cite{Gantmacher1959b},\n all roots of (\\ref{CharE+}) have negative real part.  Therefore, $E_+$ is locally\nasymptotically stable when $\\tau=0$ and hence also for $\\tau>0$ sufficiently\nsmall.\n\nWe  consider the stability of $E_+$ as $\\tau$ varies in the interval\n$0<\\tau<\\tau_c$. Here,  $P(0)|_{E_+}=\\alpha(\\tau)+c(\\tau)=s\\\ny_+(\\tau)>0$ and so   $\\lambda=0$ is not a root of (\\ref{CharE+}).\nTherefore, the only ways that $E_+$ can lose stability is:  (i)\n  when one of the characteristic roots equals zero.  This only occurs\n  when $\\tau=\\tau_c$.  This gives rise to a\n  transcritical bifurcation\nwhere $E_+$ coalesces with $E_1$   and then\ndisappears  as  $\\tau$ increases\nthrough $\\tau_c$; (ii)\nif characteristic roots bifurcate in from infinity; or (iii)  if a pair of\ncomplex  roots with negative real parts and non-zero imaginary parts cross the imaginary axis as\n$\\tau$ increases from $0$, potentially resulting in  Hopf bifurcation.\n In \\ref{app:imaginarycross} we prove that (ii) is impossible to obtain\n the following lemma.\n\n\\begin{lemma}\\label{lem:imaginarycross}\nAs $\\tau$ increases from zero,  the number of\nroots  of (\\ref{CharE+}) with positive\nreal part can change only if a root appears on or crosses the\nimaginary axis as $\\tau$ varies.\n\\end{lemma}\n\n\nIn order to determine when Hopf bifurcations occur, we first\n determine for what values of $\\tau$\npure imaginary roots of (\\ref{CharE+})  exist so that (iii) can occur.\nWe will also be interested in secondary Hopf bifurcations.\n\nSuppose that $\\lambda=i\\omega$ $(\\omega > 0)$ is a root of\n$P(\\lambda)|_{E_+}=0$, where $i=\\sqrt{-1}$. Then\n$$\nP(i\\omega)|_{E_+}=-\\omega^2+ i p(\\tau) \\omega +(i q \\omega +\nc(\\tau)) e^{-i\\tau\\omega}+\\alpha(\\tau)=0.\n$$\nUsing Euler's identity, $e^{i\\theta} = \\cos\\theta + i\\sin\\theta$, and\nequating the real and imaginary parts,   this is equivalent to\n\\begin{eqnarray*}\n       c(\\tau)\\cos(\\tau\\omega) + q \\omega\\sin(\\tau\\omega)=&\\omega^2\n       -\\alpha(\\tau),\n       \\\\\n       c(\\tau)\\sin(\\tau\\omega) - q \\omega\\cos(\\tau\\omega)=&p(\\tau)\\omega.\n       \n\\end{eqnarray*}\nSolving for $\\cos(\\tau\\omega)$ and $\\sin(\\tau\\omega)$ gives\n\\begin{subequations}\\label{chpt1sincos}\n\\begin{align}\n&\\sin(\\tau\\omega)=\\frac{c(\\tau)(p(\\tau)\\omega)+q\\omega(\\omega^2-\\alpha(\\tau))}{c(\\tau)^2+q^2\\omega^2\n, \\label{chpt1sincos:sin}\\\\\n&\\cos(\\tau\\omega)=\\frac{c(\\tau)(\\omega^2-\\alpha(\\tau))+q\\omega(-p(\\tau)\\omega)}{c(\\tau)^2+q^2\\omega^2}.\n\\label{chpt1sincos:cos\n\\end{align}\n\\end{subequations}\nSquaring\nboth sides of the equations in (\\ref{chpt1sincos}), adding, and\nrearranging gives\n \\begin{equation}\n\\omega^4+(p(\\tau)^2-q^2-2\\alpha(\\tau))\\omega^2+\\alpha(\\tau)^2-c(\\tau)^2=0.\\label{omegapower4}\n\\end{equation}\nNoting that \\eqref{omegapower4} is a quadratic function of\n$\\omega^2$, we use  the quadratic formula to obtain\n\\begin{equation*}\n\t\\omega_{\\pm}^2(\\tau)=\\frac{1}{2}\\bigg(q^2-p^2(\\tau)+2\\alpha(\\tau) \\pm\n\\sqrt{(q^2-p^2(\\tau)+2\\alpha(\\tau))^2\n-4\\left(\\alpha^2(\\tau)-c^2(\\tau)\\right)}\\bigg).\\label{omega+-2}\n\\end{equation*}\nSubstituting using \\eqref{eqn:coeffs}, it follows that \n  \\begin{equation}\\label{chpt1omegapm_squared}\n\\omega_{\\pm}^2(\\tau)=\\frac{1}{2}\\bigg(-\\bigg(\\frac{se^{s\\tau}}{Y}\\bigg)^2\n              \\pm\\sqrt{\\bigg(\\frac{se^{s\\tau}}{Y}\\bigg)^4+s^2\\bigg(12\\frac{s^2 e^{2s\\tau}}{Y^2}\n              -16\\frac{se^{s\\tau}}{Y}+4\\bigg)}\\bigg)  \\ .\t\n\\end{equation}\n\nIn order to determine for what values of $\\tau$ there are  positive real\nroots of (\\ref{omegapower4}), and hence candidates for  pure imaginary\nroots, and possibly Hopf bifurcations, we define\n\\begin{equation}\\label{tstardefined}\n\\tau^*=\\frac{1}{s}\\ln\\left(\\frac{Y}{3s}\\right).\n\\end{equation}\nWe will prove that for $\\tau \\geqslant \\tau^*$, there are no postive real\nroots and  for $0 \\leqslant \\tau < \\tau^*$ \n  \\begin{equation}\\label{chpt1omegapm}\n\\omega_{+}(\\tau)=\\sqrt{\\frac{1}{2}\\bigg(-\\bigg(\\frac{se^{s\\tau}}{Y}\\bigg)^2\n              +\\sqrt{\\bigg(\\frac{se^{s\\tau}}{Y}\\bigg)^4+s^2\\bigg(12\\frac{s^2 e^{2s\\tau}}{Y^2}\n              -16\\frac{se^{s\\tau}}{Y}+4\\bigg)}\\bigg)}\n\\end{equation}\nis the only positive real root.\n\n\n\\begin{remark}\n  Note that $\\tau^*>0$, if, and only if, $\\frac{s}{Y}<\\frac{1}{3}$, and\n  then $x_+(\\tau^*)=\\frac{s e^{s\\tau^*}}{Y}=\\frac{1}{3}$.\n\\end{remark}\n\nIn the following theorem, proved in \\ref{app:omega+posicond}, we give\nnecessary conditions on $\\tau$ for  Hopf bifurcations to occur.\n\n\\begin{theorem}\\label{omega+posicond}\n  Consider (\\ref{dimenless}). Assume that both components of $E_+$ are\n  positive.\n \n  \\begin{enumerate}\n\\item   $\\omega_+(\\tau^*)=0$.\nIf $\\tau\\geqslant\\tau^*$, then (\\ref{omegapower4}) has no positive real root.\nTherefore, there can be no pure imaginary roots of (\\ref{CharE+})  in this case. In\nparticular, if $\\frac{s}{Y}\\geqslant \\frac{1}{3}$, then $\\tau^*\\leqslant 0$, and so\nthere can be no Hopf bifurcation of $E_+$ for any $\\tau\\geqslant 0$.\n\\item Assume that $\\frac{s}{Y}<\\frac{1}{3}$, and hence, $\\tau^*>0$.\n      If $\\tau\\in[0,\\tau^*)$, then\n$x_+(\\tau)\\in[\\frac{s}{Y},\\frac{1}{3})$, and  (\\ref{omegapower4}) has\nexactly one positive real\nroot, $\\omega_+(\\tau)$, given by (\\ref{chpt1omegapm}). If\n(\\ref{CharE+}) has pure imaginary roots at $\\tau$, and hence $\\tau$ is a\ncandidate for Hopf bifurcation of $E_+$, then  $\\tau\\in(0,\\tau^*)$ and $\\omega_+(\\tau)$ must\nsatisfy (\\ref{chpt1omegapm}).\n  \\end{enumerate}\n\\end{theorem}\n\n\n\n\\begin{remark} \\label{rem:candidates}\n\t\\begin{enumerate}\n\t\t\\item  Note  that even though $\\omega_+(0)>0$ when\n  $\\frac{s}{Y}<\\frac{1}{3}$, $\\tau=0$ is not a candidate for a Hopf bifurcation,\n  since  in this case all roots of (\\ref{CharE+}) have negative real\n  parts. Also, $\\tau^*$ is not a candidate, since $\\omega_+(\\tau^*)=0$.\n\t\t\\item   Haque \\cite{Haque2012} finds an expression for\n\t\t\t$\\omega_+(\\tau^*)$ for a different scaling of\n\t\t\tthe model.  However, he does not go on as we do\n\t\t\tin what follows, to determine when\n\t\t\tthe equations given in  (\\ref{chpt1sincos})\nare both simultaneously  satisfied for \t$\\omega_+(\\tau^*)$, and to\n\t\t\tshow that the Hopf bifurcations are nested and\n\t\t\thow the number of Hopf bifurcations increases as\n\t\t\tthe death rate of the predator decreases.\n\t\\end{enumerate}\n\\end{remark}\n\nSubstituting   the values of the coefficients given by\n(\\ref{eqn:coeffs}), in the right-hand side of  (\\ref{chpt1sincos}), and recalling\nthat $x_+(\\tau)=se^{s\\tau}\/Y$,  we define\n\\begin{subequations} \\label{h1_h2_defined}\n  \\begin{align}\nh_1(\\omega,\\tau)&=\\frac{\\omega}{s}\\left(\\frac{s+x_+(\\tau)-s\nx_+(\\tau)-2x_+^2(\\tau)-\\omega^2}{(1-2\nx_+(\\tau))^2+\\omega^2}\\right),\\label{h1defined} \\\\\nh_2(\\omega,\\tau)&=\\frac{\\omega^2(1+ s -x_+(\\tau))-(1-2 x_+(\\tau)) s\nx_+(\\tau)}{s\\left((1-2x_+(\\tau))^2+\\omega^2\\right)}.\\label{h2defined}\n\\end{align}\n\\end{subequations}\n\nProperties of the functions $h_1$ and $h_2$ are summarized\nin \\ref{app:h1_h2}.\n\n\\begin{remark}\\label{rem:char_pure_imag}\nBy Theorem~\\ref{omega+posicond} and Lemma~\\ref{lem:h1_pos}, $\\tau$ satisfies\n(\\ref{chpt1sincos}) for $\\omega>0$, (and hence (\\ref{CharE+}) has a pair of pure\nimaginary eigenvalues) if and only if $\\tau\\in(0,\\tau^*)$, and\n\\begin{subequations} \\label{eqn:sin_h1_cos_h2}\n   \\begin{align}\n&\\sin(\\tau\\omega_+(\\tau)) \\, \\, =h_1(\\omega_+(\\tau),\\tau)\n,  \\label{eqn:sin_h1_cos_h2:sin}\n\\\\\n&\\cos(\\tau\\omega_+(\\tau)) \\, \\,  =h_2(\\omega_+(\\tau),\\tau).\n\\label{eqn:sin_h1_cos_h2:cos}\n\\end{align}\n\\end{subequations}\n\\end{remark}\n\n\n\n\n\n\nDefine the function\n\\begin{equation}\\label{eq:theta}\n\\theta:[0,\\tau^*]\\rightarrow [0,\\pi]\n\\end{equation}\n\n$$\\theta(\\tau):=\\arccos(h_2(\\omega_+(\\tau),\\tau)).$$\n By part 3 of Lemma~\\ref{lem:h1_pos}, stated and proved in \\ref{app:h1_h2}, $\\theta(\\tau)$ is a well-defined, continuously\ndifferentiable function.   Replacing  $\\tau\\omega_+(\\tau)$ by\n$\\theta(\\tau)+2 n \\pi$ in the left hand side of\n(\\ref{eqn:sin_h1_cos_h2}),\nwe obtain\n\\begin{subequations} \\label{eqn:sin_h1_cos_h2_theta}\n    \\begin{align}\n&\\sin(\\theta(\\tau)+2 n\\pi) \\, \\, =h_1(\\omega_+(\\tau),\\tau)\n,  \\label{eqn:sin_h1_cos_h2:sin_theta}\n\\\\\n&\\cos(\\theta(\\tau)+2 n\\pi) \\, \\,  =h_2(\\omega_+(\\tau),\\tau).\n\\label{eqn:sin_h1_cos_h2:cos_theta}\n\\end{align}\n\\end{subequations}\n Equation \\eqref{eqn:sin_h1_cos_h2:cos_theta} is satisfied directly by the\n definition of $\\theta(\\tau)$.   Equation\n \\eqref{eqn:sin_h1_cos_h2:sin_theta} is also satisfied,\n from  parts 1 and 2 of Lemma~\\ref{lem:h1_pos}, since $0\\leqslant\n \\theta(\\tau)\\leqslant \\pi .$\n\nBy comparing \\eqref{eqn:sin_h1_cos_h2} and\n\\eqref{eqn:sin_h1_cos_h2_theta}, it follows that solutions of\n\\eqref{eqn:sin_h1_cos_h2} occur at precisely those points\nwhere the curves $\\tau\\omega_+(\\tau)$ and $\\theta(\\tau)+2n\\pi$\nintersect.\n\\begin{remark} \\label{rem:characterization}\n  By Remark~\\ref{rem:char_pure_imag}, (\\ref{CharE+}) has a pair of pure imaginary roots at precisely those points\nwhere the curves  $\\tau\\omega_+(\\tau)$ and $\\theta(\\tau)+2n\\pi$\nintersect  for $\\tau\\in(0,\\tau^*)$,  where $n$ is a\nnonnegative integer.\n\\end{remark}\n\n\n\nFor each integer $n\\geqslant 0$, denote the $j_n$ points of intersection of\nthe  curves $\\tau\\omega_+(\\tau)$ and $\\theta(\\tau)+2n\\pi$\n for $\\tau\\in(0,\\tau^*)$, in increasing order,  by $\\tau_n^j, \\, \\,\n j=1,2\\dots,j_n$,\ni.e., for each $n=0,1,2,\\dots$,\n$$\\tau_n^j\\omega_+(\\tau_n^j)=\\theta(\\tau_n^j)+2n\\pi, \\quad\n\\ j=1,2,\\dots,j_n.$$\n\n\n\\begin{theorem}\\label{chpt3intersections\nConsider system (\\ref{dimenless}).\nThe characteristic equation (\\ref{CharE+}) has a  pair of pure\nimaginary eigenvalues, if and only if, $\\tau=\\tau_n^j\\in(0,\\tau^*)$, a point of\nintersection of the curves $\\tau\\omega_+(\\tau)$ and\n$\\theta(\\tau)+2n\\pi$, for some integer $n\\geqslant0.$\nAt  all  such intersections, the pair of pure imaginary eigenvalues is\n simple\nand no other root of (\\ref{CharE+})\nis an integer multiple of $ i \\omega_+(\\tau_n^j).$\nIf in addition, $\\frac{\\mathrm{d}}{\\mathrm{d}\\tau}(\\tau\\omega_+(\\tau))\n\\Big|_{\\tau=\\tau_n^j} \\neq \\frac{\\mathrm{d}}{\\mathrm{d}\\tau}\\theta(\\tau)\n\\Big|_{\\tau=\\tau_n^j}$,\nthe transversality condition  for Hopf bifurcation,\n$\\frac{\\mathrm{d}}{\\mathrm{d}\\tau}\\mathrm{Re}(\\lambda(\\tau))\n\\Big|_{\\tau=\\tau_{n}^j}\\neq0$,\nholds.\n\\end{theorem}\n\n\\noindent\nThe proof is given in \\ref{app:chpt3intersections}.\n\n\n\n\\begin{remark} \\label{rem:test_transversality}\n   If the slope of the curve  $\\theta(\\tau)+2n\\pi$\n  is less than the slope of   $\\tau\\omega_+(\\tau)$ at an intersection\n  point $\\tau_n^j$, then   a pair of complex  roots of\n  (\\ref{CharE+}) crosses the imaginary axis from left to right as $\\tau$\n  increases through  $\\tau_n^j$.\n  On the other hand, if the slope of the curve  $\\theta(\\tau)+2n\\pi$\n  is greater than the slope of   $\\tau\\omega_+(\\tau)$ at an intersection\n  point $\\tau_n^j$, then   a  pair of complex roots of\n  (\\ref{CharE+}) crosses the imaginary axis from right to left as $\\tau$\n  increases through  $\\tau_n^j$.\n\\end{remark}\n\n\\begin{corollary}\\label{cor:2k+1intersection\nConsider system (\\ref{dimenless}). Assume that $\\tau\\in[0,\\tau^*]$ and\nthat there exists $N\\geqslant 0$ such that\n$(2N+1)\\pi\\leqslant\\max_{\\tau\\in[0,\\tau^*]}\\tau\\omega_+(\\tau)\\leqslant\n2(N+1)\\pi$.\n\\begin{enumerate}\n\\item For $0\\leqslant n\\leqslant N$, $\\theta(\\tau)+2n\\pi$ and\n$\\tau\\omega_+(\\tau)$ have at least two intersections in\n$(0,\\tau^*)$.\n \\item For $n\\geqslant N+1$,  $\\theta(\\tau)+2n\\pi$ and\n$\\tau\\omega_+(\\tau)$  do not intersect in $(0,\\tau^*)$.\n\\item\nIf $\\theta(\\tau)+2n\\pi$ and\n$\\tau\\omega_+(\\tau)$ intersect for any $n\\geqslant 0$, then $\\tau_0^1$ is the\nsmallest  and $\\tau_0^{j_0}$ the largest\nvalue of $\\tau$ for which\n(\\ref{CharE+}) has a pair of pure imaginary eigenvalues.\n\\item\n  The\ncoexistence equilibrium $E_+$ is locally asymptotically stable\nfor $\\tau\\in[0,\\tau_0^1)\\cup(\\tau_0^{j_0},\\tau_c)$.\n\\end{enumerate}\n\\end{corollary}\n\n\n\\noindent\nThe proof is given in \\ref{app:2k+1intersection}.\n\n\n\n \\begin{figure}[bt!]\n  \\begin{center}\n    \\includegraphics[width=5.25cm]{Figures\/theta_omega_s_p02.eps}\n   \n \\includegraphics[width=5.25cm]{Figures\/theta_omega_s_p007.eps}\n   \n\\end{center}\n\\caption[  Intersections of $\\theta(\\tau)+2n\\pi$ and $\\tau\n\\omega(\\tau)$] { Intersections of $\\theta(\\tau)+2n\\pi$\nand $\\tau \\omega_+(\\tau), \\ n=0,1,\\dots, $.\nValues of $\\tau$ at which the characteristic equation has pure imaginary eigenvalues, and hence candidates for critical values\nof $\\tau$ at which there could be Hopf bifurcations.\nIn both graphs, at all such intersections, transversality holds, since\nthe slope of these curves\nat these intersections are different.\nParameters: $ m=1,\\ r=1,\\ K=1, \\ Y=0.6 . $\n{\\scriptsize\\bf (LEFT)}  $s=0.02$.\n For $n=0$ there are two\nintersections (i.e. $j_0=2$), at $\\tau_0^1$ and $\\tau_0^2$, but for $n=1$, and hence\n$n\\geqslant 1$, there are no\nintersections. {\\scriptsize\\bf (RIGHT)}  $s=0.007$.\n There are two\nintersections each (i.e. $j_n=2, \\ n=0,1,2$), at $\\tau_n^1$ and $\\tau_n^2$, for $n=0,1$ and $2$,\nbut for $n=3$, and hence\n$n\\geqslant 3$, there are no\nintersections. In both {\\scriptsize\\bf (LEFT)} and  {\\scriptsize\\bf (RIGHT)},  $E_+$ is\nasymptotically stable for $\\tau\\in[0,\\tau_0^1)\\cup(\\tau_0^2,\\tau_c)$\nand unstable for $\\tau\\in(\\tau_0^1,\\tau_0^2)$.}\n\\label{fig:thetatauomega2}\n\\end{figure}\n\n Our results  differ from those in Gourley and Kuang \\cite{Kuang2004},\nsince  we give explicit formulas   for solutions of \\eqref{omegapower4}\nand define $\\theta(\\tau)$ explicitly. These explicit formulas play an\nimportant role in analysis and make numerical simulations more\nstraightforward.  Although $\\omega_{-}^2(\\tau)$ in\nequation\n\\eqref{chpt1omegapm_squared} is negative in\nthis model and hence its square root is not real, in other models\nequation\n\\eqref{omegapower4} can have two positive real\nsolutions (see \\cite{FanW}, where  both $\\omega_+(\\tau)$ and $\\omega_-(\\tau)$\nare positive). In that case, double Hopf bifurcations are possible.\n\n  In the next section we will demonstrate\n  numerically that in the example shown in\n  Figure~\\ref{fig:thetatauomega2},\n  $E_+$ first loses its stability through a supercritical Hopf\n  bifurcation as $\\tau$  increases through $\\tau_0^1$ and then\n  restabilizes  as a result of a second supercritical Hopf\n  bifurcation as $\\tau$   increases through $\\tau_0^2$.  We will also\n  show that between these two values of $\\tau$ there is a\n  sequence of bifurcations resulting in   interesting\n  dynamics, including a strange attractor.\n\n\n\n\n\n\n\n\\section{An example demonstrating complex dynamics}\\label{sec:example}\nIn this section, unless specified otherwise,  we select the following\nvalues for the parameters in model (\\ref{ppreydelay}):\n\\begin{equation}\n  m=1,\\ r=1,\\ K=1, Y=0.6,\\ s=0.02, \\label{example}\n\\end{equation}\n  and consider $\\tau$ as a bifurcation parameter. Since, with this\n  selection of parameters, the model is already in the form of the scaled\n version of the model (\\ref{dimenless}), it is not necessary to apply the scaling given by (\\ref{rescale}).\nWe first use this example  to  illustrate the analytic\nresults given in section~\\ref{analysis} where we provided necessary and sufficient\nconditions for  a simple  pair of pure imaginary eigenvalues of the\ncharacteristic equation  to occur as $\\tau$ varies. We then provide\nbifurcation diagrams with $\\tau$ as the bifurcation parameter,\nsimulations  including time series and time delay embeddings\nfor various values of $\\tau$, and a return map at a value of $\\tau$\nat which there is a chaotic attractor,\nin order  to illustrate the wide variety of dynamics displayed by the\nmodel,   even in the case when there are only two Hopf\nbifurcations.\nThis includes two supercritical Hopf bifurcations, saddle-node  bifurcations of limit cycles and  sequences of period doublings that\nappear to lead to chaotic dynamics with a strange attractor reminiscent\nof the strange attractor for the well-known Mackey-Glass equation\n\\cite{Mackey:1977,Mackey:2009}\n$$  \\frac{dx}{dt}=\\beta \\frac{x(t-\\tau)}{1+(x(t-\\tau))^n}-\\gamma x(t).$$\n\n\\subsection{Illustration of analytic results}\n\nFor the parameters given by (\\ref{example}), the model has three\nequilibria: $E_0=(0,0)$, $E_1=(1,0)$, that always exist, and\n$E_+=(x_+(\\tau),y_+(\\tau))=( 0.0\\dot{3} e^{0.02\\tau},1-0.0\\dot{3} e^{0.02\\tau})$, given by\n(\\ref{notation1}).\nThe components of $E_+$ are both positive, if, and only if, $\\tau\\in[0,\\tau_c)$,\nwhere $\\tau_c\\approx 170$ (see (\\ref{notation2})), and by part 3(b) of\nTheorem~\\ref{th:gasEi}  the model is uniformly persistent\nfor $\\tau\\in[0,170)$.\n\n   $E_0$ is always a saddle, and hence unstable. $E_1$ is globally\nasymptotically stable for $\\tau>\\tau_c$, and unstable for\n$\\tau\\in[0,\\tau_c)$.  For $\\tau=0$, $E_+$ is asymptotically stable, and\nby Lemma~\\ref{lem:imaginarycross}, can only lose stability by means of a Hopf\nbifurcation.  In order to determine th:e\ncritical values of $\\tau$ at which\n there is a pair of  pure imaginary eigenvalues:\n$\\lambda(\\tau)=\\pm i \\omega(\\tau)$,  we consider the interval\n$(0,115)$, since by Theorem~\\ref{omega+posicond}, the interval on\nwhich $\\omega(\\tau)$ is positive, is bounded above  by $\\tau^*\\approx\n115$ (defined in (\\ref{tstardefined})).\n\nWith the parameters  given in (\\ref{example}),  by \\eqref{chpt1omegapm}  the  function\n\t\\[\t\\omega_+(\\tau)= \n\t \\sqrt{ -.\\dot{5} \\ 10^{-3}(e^{0.02\\tau})^2 + \n\t \\frac{\n\\sqrt{1.235\\  10^{-6}(e^{0.02\\tau})^4 +\n\t.5\\dot{3} \\ 10^{-5}e^{0.04\\tau} - 0.5\\dot{3}e^{0.02\\tau} +\n\t4}}{2} } \\]  \n and recall,  by \\eqref{eq:theta}\n $$\\theta(\\tau)=\\arccos(\\cos(\\tau \\omega_+(\\tau))). $$\nThe intersections of the functions $\\tau\\omega_+(\\tau)$  and\n$\\theta(\\tau)+2n\\pi,$  $n$ a nonnegative integer, in the interval\n$(0,\\tau^*)$,   give  the critical values of\n$\\tau$ for  which there is a pair of pure imaginary eigenvalues.\nThere are only two intersections as can be seen in\nFigure~\\ref{fig:thetatauomega2}~{\\scriptsize\\bf (LEFT)}.\nSince,  $\\pi<\\max_{\\tau\\in[0,\\tau^*]} \\tau\\omega(\\tau)<2\\pi$,\nby Corollary~\\ref{cor:2k+1intersection},\nwe are guaranteed at least\n two  values of $\\tau>0$ at which the characteristic\n equation has a pair of pure imaginary roots.  In fact,\n (see Figure~\\ref{fig:thetatauomega2} ({\\scriptsize (LEFT})),  there are\n precisely two such values, $\\tau_0^1$ and\n $\\tau_0^2$.\nUsing Maple \\cite{maple},  we found that $\\tau^1_0\\approx 1.917$ and\n$\\tau^2_0 \\approx 108.365$. By\nTheorem~\\ref{chpt3intersections},    the slopes of the curves\n$\\theta(\\tau)$ and $\\tau\\omega_+(\\tau)$ are different at these\nintersection points, since these curves cross transversally (see\nFigure~\\ref{fig:thetatauomega2} {\\scriptsize (LEFT)}), and hence the\ntransversality required for Hopf bifurcation holds at each\nroot.  By part 4 of Corollary~\\ref{cor:2k+1intersection} and\nRemark~\\ref{rem:test_transversality}, the stability of $E_+$ changes\nfrom asymptotically stable to unstable as $\\tau$ increases through\n$\\tau_0^1$ and from unstable to asymptotically stable as $\\tau$\nincreases through $\\tau_0^2$, and is unstable for\n$\\tau\\in(\\tau_0^1,\\tau_0^2)$.  But recall, even though $E_+$ is unstable\nhere, by  part 3(b) of Theorem~\\ref{th:gasEi},  the model is still uniformly\npersistent in this interval.\n\n Thus, we have shown that for the parameters chosen, there are exactly\n  two candidates for  Hopf bifurcations (see  \\cite{Smith_delay},\nChapter 6, Theorem~6.1, page 89-90).  That both Hopf bifurcations are\nsupercritical (involving first the birth, and then the disappearance of\norbitally asymptotically stable periodic solutions)\n  will be demonstrated in the next section.  As $\\tau$ increases through\n  $\\tau=1.917$, a family of orbitally asymptotically stable periodic\n  orbits is born.  We will see that these periodic orbits  undergo additional bifurcations\n as $\\tau$ increases, and that they disappear when $\\tau$\n  increases through the critical value $\\tau=108.365$, at the second supercritical  Hopf bifurcation.\n\n By decreasing $s$,  the value of $n$ for which the curves\n\n\n\n   $\\theta(\\tau)+2n\\pi$ and $\\tau\\omega_+(\\tau)$ intersect can increase.\n   See\n  Figure~\\ref{fig:thetatauomega2} {\\scriptsize (RIGHT)} for an example\n  with $s=0.007$ for which the curves $\\theta(\\tau)+2n\\pi$ and\n  $\\tau\\omega_+(\\tau)$ intersect when $n=0,1,2$. In fact, there are $6$\n  points of intersection. We see by\n  Remark~\\ref{rem:test_transversality}, that for this example, $E_+$ is\n  asymptotically stable until $\\tau=\\tau_0^1$, is\n  unstable for $\\tau\\in(\\tau_0^1,\\tau_0^2)$, and finally becomes stable\n  again for $\\tau\\in(\\tau_0^2,\\tau_c)$. Again, even though the model is\n  unstable for $\\tau\\in (\\tau_0^1,\\tau_0^2)$, by part 3(b) of\n  Theorem~\\ref{th:gasEi}, it is uniformly persistent\n  in this interval, since the model is uniformly persistent whenever\n  both components of   $E_+$ are positive, i.e. for $\\tau\\in[0,\\tau_c)$.\n\n\n\\subsection{Saddle-node of limit cycles, period doublings, and chaotic dynamics}\n\\label{numerics}\n\nThe computations and figures in this section\nwere done using Maple \\cite{maple},\nMATLAB  \\cite{MATLAB:2018}, and\nXPPAUT \\cite{Ermentrout2002}\n\n\nIn Figure~\\ref{bif_diag_full}~{\\scriptsize (TOP)}, for each value of\n$\\tau\\in[0,120]$, starting with initial data $x(t)=y(t)=0.1$ for\n$t\\in[-\\tau,0]$, we  integrate long enough for the solution to converge\nto an attractor (e.g., equilibrium, periodic orbit, or strange\nattractor), and then\n plot the local minima and maxima of the $y(t)$ coordinate\n on the attractor.\n Because we are interested in period doubling bifurcations, we then\n eliminate certain local maxima and minima that are due to kinks\n in the\n solutions  (see\n Figure~\\ref{kinks}) rather than actual bifurcations, to obtain the graph in\n Figure~\\ref{bif_diag_full} {\\scriptsize (BOTTOM)}).\n Our solutions have\nkinks for values of $\\tau\\in[55,98]$.  Kinks were also observed in\nthe Mackey-Glass equation  \\cite{Mackey:2009}.\n\n\\begin{figure}[tbhp!]\n  \\begin{center}\n  \n \\includegraphics[width=10cm]{Figures\/orbit_diag_y.eps}\\\\\n \\includegraphics[width=10cm]{Figures\/bif_diag_0_120_steps_p2_bold.eps}\n\n  \\end{center}\n\\caption[Parameter range for\n\tinteresting dynamics]{Orbit diagrams.  Initial\n\tdata was taken to be $x(t)=y(t)=0.1$ for\n\t$t\\in[-\\tau,0]$.  However, we found bistability in the portion\n\tof the diagram beween the vertical dots and varied the initial\n\tdata as explained below.   \n\n\n\n\tExcept for the\n\tportion between the vertical dots, the rest of the diagram was the same \n\tfor all of the initial conditions we tried (not shown).\n{\\scriptsize\\bf (TOP)} All local maxima and minima\nfor the $y(t)$ coordinate of the attractor as $\\tau$ varies,\nincluding kinks.  {\\scriptsize\\bf (BOTTOM)}\nDiagram including local maxes and mins\nfor the $y(t)$ coordinate  as $\\tau$ varies, but with kinks eliminated.\n \tThere are two saddle-node of limit cycle\n\tbifurcations. They occur for\n\t$\\tau$ approximately equal to  76 and 82, where  the curves  in the orbit diagrams stop abruptly and there\n\tappear to be vertical dots. For $\\tau$ between these values, there\n\tis is bistability. Two orbitally asymptotically stable\n\tperiodic orbits (with their maximum and minimum amplitudes\n\tshown) and an unstable periodic orbit with amplitudes between\n\tthem (not shown).  The two stable periodic orbits were found\n\tby producing this part of the orbit diagram varying $\\tau$\n\tforward and then varying  it backwards but startng at the last\n\tpoint of the attractor for the previous value of $\\tau$.}\n \\label{bif_diag_full}\n \\end{figure}\n\n\\begin{figure}[tbhp!]\n  \\begin{center}\n    \\includegraphics[angle=270, width=7cm]{Figures\/ts_kink_70.eps}\n  \\end{center}\n  \\caption[kinks]{The time series for $y(t)$ when $\\tau=70$, depicting\n   kinks. There are two local maxima and two local minima over each period as shown  in  Figure~\\ref{bif_diag_full} {\\scriptsize (TOP)}, but only one local\n  maxima and one local minima in Figure~\\ref{bif_diag_full} {\\scriptsize\n  (BOTTOM)} in which kinks have been removed. }\\label{kinks}\n\\end{figure}\n\n\n\\begin{figure}[tbhp!]\n  \\begin{center}\n\n\t\\includegraphics[width=10cm]{Figures\/orbit_diag_y_zoom.eps}\n \n  \\end{center}\n \\caption{ Zoom-in of orbit diagram shown in  Figure\n\\ref{bif_diag_full} for\n\t$\\tau\\in[75,100]$\n\tincluding kinks.   The vertical dots indicate the boundary of the\n\tregion of bistability, where the  two saddle-node of limit cycle\n\tbifurcations occur.\n}\n\n\t\\label{zoom-in_80_100}\n\\end{figure}\n\n\nFigure \\ref{bif_diag_full} confirms that there are two\nHopf bifurcations at $\\tau\\approx 1.917$ and $\\tau\\approx 108.365$, and\nallows us to conclude that these Hopf bifurcations are both\nsupercritical, since they involve a family of orbitally asymptotically\nstable periodic orbits.\n\nNext we focus on the  more interesting\ndynamics observed for $\\tau\\in[80,100]$ (see  Figure~\\ref{zoom-in_80_100}).\nThere appears to be a discontinuity in the\nbifurcation diagram for $\\tau\\approx 82.225$.  Upon further\ninvestigation we have determined that there is a saddle-node bifurcation\nof limit cycles at this value of $\\tau$ and another saddle-node\nbifurcation of limit cycles for a value of $\\tau$ smaller than\n$\\tau=81$.  For values of $\\tau$ between these two saddle-node bifurcations,\nthere is bistability.  There are two orbitally asymptotically stable\nperiod orbits.  An example of two such orbits is given in\nFigure~\\ref{bistability}, where $\\tau=81$.\n\n\n \\begin{figure}[htbp!]\n\t\\begin{center}\n\t\\includegraphics[angle=270, scale=.25]{Figures\/bistability_81_345_273p7_y_yd.eps}\n\n\\caption{  Time delay embedding  of two orbitally asymptotically stable periodic orbits\ndemonstrating bistability for $\\tau=81$.  The one with larger amplitude (dashed) has initial data\n$x(t)=y(t)=0.1$, for $t \\in [-\\tau,0]$, and period approximately\n$345$.\nThe one with smaller amplitude (solid) has initial data\n$x(t)=y(t)=0.1$, for $t \\in [-\\tau,0)$ and $x(0)=0.3, \\ y(0)=0.83$ and has\nperiod approximately\n$273.7$.}\\label{bistability}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{zoom-in_80_100}  suggests that\n there are sequences of period doubling bifurcations,\n one initiating from the left at  $\\tau\\approx 83$, $86$,  $86.6$,\n $\\dots,$\n and one initiating from the right  at  $\\tau\\approx  98.3$,  $93.2$, $92.2$,\n and $\\tau$ between $92$ and $91.85$.\n To demonstrate these sequences,  time series ($y(t)$ versus\n $t$) and  time delay embeddings\n ($y(t)$ versus $y(t-\\tau)$) at values of $\\tau$ between these\n bifurcations are  shown in Figures~\\ref{PD_left} and \\ref{PD_right}.\n\n \\begin{figure}[tbhp!]\n   \\begin{center}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_82.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_82_340p16.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_85.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_85_564p6.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_86p3.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_86p3_1144p48.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_86p8.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_86p8_2298p26.eps}\n   \\end{center}\n   \\caption[ ]{ {\\scriptsize\\bf (LEFT)} Time series starting from the initial data $x(t)=y(t)=0.1, t\\in[-\\tau,0]$\n   indicating how quickly the orbit gets close to the periodic attractor and\n   {\\scriptsize\\bf (RIGHT)} time delay embeddings of\n   the periodic attractors, demonstrating the sequence of  period doubling bifurcations\n   initiating from the left at\n   $\\tau\\approx 83,\\ 86$, and $86.6$. Values of $\\tau$ selected between these bifurcations:\n    $\\tau=82,\\ 85,\\\n   86.3$, and $86.8$, with periods of the periodic attractor approximately equal to:\n   $340.2,\\ 564.6,\\ 1144.5$, and $2298.3$, respectively, are\n   shown. } \\label{PD_left}\n\\end{figure}\n\n \\begin{figure}[tbhp!]\n   \\begin{center}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_91p95.eps}\n\\includegraphics[angle=270, width=5.25cm]{Figures\/delay_pp_y_yd_91p95_4794p3.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_92.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ytau92_2398p3.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_93.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_y_yd_93_period_1211p2.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_96.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_y_yd_96_period_557p6.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_100.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_y_yd_100_period_295p3.eps}\n   \\end{center}\n   \\caption[ ] { {\\scriptsize\\bf (LEFT)} Time series starting  from the initial data $x(t)=y(t)=0.1, t\\in[-\\tau,0]$\n   indicating how quickly the orbit gets close to the periodic attractor and\n   {\\scriptsize\\bf (RIGHT)} time delay embeddings of\n   the periodic attractors, demonstrating the sequence of  period\n\t halfing bifurcations\n   initiating  from the left for values of $\\tau$ between  $91.5$ and $92$,\n\t and at\n   $\\tau\\approx 92.2, \\  93.2, \\ 98.3$. Graphs shown are for values of $\\tau$  between these bifurcations:\n    $\\tau=91.95,\\ 92,\\\n   93,\\ 96  $, \\ and \\  $100$, \\ with periods approximately equal to:\n   $4794.3, \\  2398.3, \\ 1211.2,\\  557.6, $ and  $295.3$, respectively. } \\label{PD_right}\n\\end{figure}\n\n    Figure~\\ref{zoom-in_80_100} also suggests that between these sequences of period doubling\n    bifurcations there is a window of values of $\\tau$ at which there\n     are  periodic attractors that do not have a\n     period that  results from a bifurcation with period approximately\n     equal to $2^n$  for some integer $n$, as well as chaotic dynamics.\n     An example of the former is illustrated in\n     Figure~\\ref{period_two_times_three}.  For $\\tau=90.7$, the\n     time-series embedding of a periodic orbit with period approximately\n    equal to  1800 time steps involving six loops (2 times 3)  is\n    shown.\n\n     \\begin{figure}\n    \\begin{center}\n\t    \\includegraphics[width=35ex]{Figures\/delay_pp_y_yd_90p7_per1800_not_2n.eps}\n  \\end{center}\n     \\caption [two times three] { Time delay embedding starting at initial\n     data $x(t)=y(t)=0.1, t\\in[\\tau,0]$ for $\\tau=90.7$ showing a\n   periodic attractor with period approximately\n      $1800$, having $6(\\neq2^n)$ loops for some integer $n$.}\n     \\label{period_two_times_three}\n\\end{figure}\n\nThe time series and the time delay embedding of a chaotic attractor for\n$\\tau=90$ is shown in Figure~\\ref{chaotic_attractor}. The projection of\nthis attractor into $(x(t),y(t))$-space is\nalso shown in Figure~\\ref{chaos_xy}. This strange attractor resembles the\nchaotic attractor of the well-known Mackey-Glass equation (\\cite{Mackey:2009}, Figure 2).\nThe return map\nshown in Figure~\\ref{return_map}, for $\\tau=90$, also resembles the return map for the\nMackey-Glass equation (\\cite{Mackey:2009}, Figure 14) in the case of chaotic dynamics.\nSensitivity to initial data is a hallmark of chaotic dynamics.\nFigure~\\ref{sensitivity}{\\scriptsize\\bf (RIGHT)}\ndemonstrates that there is sensitivity to initial data in the case of the solution for\n$\\tau=90$ that\nconverges to the strange attractor,  shown in Figure~\\ref{chaotic_attractor}.\nTo show that this is not just a numerical artifact, in\nFigure~\\ref{sensitivity}{\\scriptsize\\bf (LEFT)} we show that, as\nexpected,  there is no sensitivity for the solution for $\\tau=92$ that\nconverges to the  periodic solution shown in Figure~\\ref{PD_right}.\n\n\\begin{figure}[htbp!]\n   \\begin{center}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_90_240-260000.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_90.eps}\n   \\end{center}\n   \\caption[ ]{ {\\scriptsize\\bf (LEFT)} Time series for $\\tau=90$  starting  from the initial data\n   $x(t)=y(t)=0.1, t\\in[-\\tau,0]$.\n   {\\scriptsize\\bf (RIGHT)} Time delay embedding of\n   the strange attractor for $\\tau=90$.  Only the portion of the orbit from $t=240,000-260,000$\n   is shown.} \\label{chaotic_attractor}\n \\end{figure}\n\n \\begin{figure}[htbp!]\n\t\\begin{center}\n\t\\includegraphics[angle=270, width=6cm]{Figures\/pp_ts_tau_90_x_y.eps}\n\t\\caption[]{\n\tThe strange attractor, for $\\tau=90$,  shown in Figure~\\ref{chaotic_attractor} in\n\t$(x,y)$-space.   Only the\n   portion of the orbit from $t=240,000$ \\,  to \\, $260,000$ is shown.\n   }\\label{chaos_xy}\n\\end{center}\n\\end{figure}\n\n \\begin{figure}[htbp!]\n\t\\begin{center}\n\t\\includegraphics[angle=0, width=7cm,height=5cm]{Figures\/return_map_tau_90.eps}\n\t\\caption[]\n  \n  \n  \n   The return map for $\\tau=90$, computing the minimum value of $y(t)$\n  \n   as a function of\nthe preceding minimum value  of $y(t)$ for $y(t)<0.7$ in both cases, using the data in Figure~\\ref{chaotic_attractor}. }\\label{return_map}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[htbp!]\n\t\\begin{center}\n\t\\includegraphics[angle=0,width=6.75cm]{Figures\/insensitive_tau_92.eps}\n\t\n\t\\includegraphics[angle=0,width=6.75cm]{Figures\/sensitive_tau_90.eps}\n\t\n\n\\caption[]{ Time series {\\scriptsize\\bf (LEFT)} for the solution that converges to a periodic attractor when $\\tau=92$, and {\\scriptsize\\bf (RIGHT)}\nfor the solution that converges to a strange attractor when $\\tau=90$, demonstrating that there is no sensitivity to initial data in the\nformer case, but that there is sensitivity in the latter case. Initial data used for the solid\ncurves:\n$x(t)=y(t)=0.1$ for $t\\in[-\\tau,0]$, and for the dotted curves:  $x(t)=0.11$ and\n$y(t)=0.1$ for $t\\in[\\tau,0]$.} \\label{sensitivity}\n\\end{center}\n\\end{figure}\n\n\nThis example demonstrates that including delay in a simple\npredator-prey model that always has a globally asymptotically stable\nequilibrium point in the absence of delay,\ncannot only destabilize a\nglobally asymptotically stable equilibrium point, but can  even result in\nthe birth of a strange attractor.\n\n\\section{Discussion and Conclusions} \\label{discussion}\n\nWe investigated the effect of the time  required for predators to process\ntheir prey on the possible  dynamics predicted by a mathematical\nmodel of predator-prey interaction.  We incorporated a discrete\ndelay    to model this process in one of the simplest classical\npredator-prey models,    one that only allows convergence to an\nequilibrium  when this delay is ignored.\nWe showed that including the delay results in a model with much\nricher dynamics.  By choosing one of the simplest models when delay is\nignored, one that predicts that no   oscillatory behaviour is possible, the\neffect of the delay on the dynamics is emphasized.\n\n\n\n This model can also be interpreted as a\nmodel of a stage-structured population  with the\ndelay modelling the  maturation time of the juveniles\n(see  Gourley and Kuang \\cite{Kuang2004}).\n\nIn the model we considered,   the prey are assumed to grow logistically in\nthe absence of the predator.   The interaction of the predator and prey\nis described using a linear response function, often referred to as mass\naction or   Holling type I.\nIt is well-known that when delay is ignored this is one of the simplest\npredator-prey models for which all solutions converge\nto a globally asymptotically stable equilibrium point for all choices of the\nparameters.   Therefore, any resulting non-equilibrium dynamics would\nthen be solely attributable to the introduction of the delay in the\ngrowth term of the predator.  We not only found\nnon-trivial periodic solutions, but also bistability, and chaotic dynamics.\nIt is then likely that there is  similar   rich dynamics\nin most predator-prey models with any reasonable response function, when  such a delay is incorporated,  for\nsome selection of the parameters, including the form most used by\necologists, the  Holling type II form.  This form\ngiven mathematically by $f(x)=mx\/(1+bx))$, can be considered a   generalization of the Holling type I form, obtained by simply adding an extra parameter, $b$.\nHowever, we feel that demonstrating  that this wide range of dynamics is\npossible for even for one of the simplest models gives more compelling\nevidence that delay should not be ignored when making policy decisions.\n\nUnderstanding how changes in average\ntemperature   might affect   survivability of endangered populations or\nresult in invasions by undesirable populations is   important.\nSince temperature can affect how quickly predators process the prey\nthat they capture, based on our results there might be important\nimplications   for  populations in the wild.   In most predator\npopulations,   the processing time, $\\tau$, is faster when it is warmer and\nslower when it is colder.  Our results may help us  understand how a change in\naverage temperature might\ninfluence the dynamics of   particular predator-prey systems of interest.\n\n\nOur study\nsuggests that we need to be careful when measuring population sizes\nin the wild and predicting the general health of the population based\nupon whether the population seems to be increasing or decreasing.\nShort term indications  that a population size is changing\n in a system with oscillatory dynamics  may be misleading\nand predicting future population size may be impossible without more\ninformation. It would be necessary to have some idea of the period of the\nintrinsic oscillations  if the population is suspected to\nvary periodically.  If the dynamics are suspected to be chaotic, this  may be\neven  more complicated, due to sensitivity to initial data.\n\n\nIf a predator-prey system has the potential to have  chaotic dynamics,\nbased on our results, is there anything that is predictable?    Can such\nanalyses suggest how to prevent   extinctions or\ninvasions due to a change in average temperature that could result in a\nchange in the processing time of the prey by the predator? We give some observations based on the predictions of our model and the example we considered in Section~\\ref{sec:example}.  However,  more work would need to be done to determine whether  these predictions are consistent for  more realistic\nmodels,  and if so,   long term observations would have to be made by\necologists  to determine if they are relevant for   populations\nin the wild.\n\n\n  First,     at the one extreme,\nif the processing time is too long, the predator population would not\nbe expected to survive,  since it\nis obvious that if the processing time is longer than the life span of\nthe predator,  the predator population has no chance to avoid\nextinction.   If there was\nno such  threshold for extinction predicted by the model, the model should be abandoned.  It is\ntherefore very important to use the term\n$e^{-s\\tau}y(t-\\tau)$ and not just $y(t-\\tau)$ in order to account for the predators that do not survive\nlong enough to affect growth of the population,  in the equation\ndescribing the growth of the predator in\nmodel (\\ref{ppreydelay}). Due to this term,\n in our model there is such a threshold, $\\tau_c$.\n\n From the orbit diagram (see Figure~\\ref{bif_diag_full}),\nfor the example considered in Section~\\ref{sec:example}, we summarize some\nobservations.  The dynamics for both populations are   oscillatory for a wide\nrange of processing times,  and non-oscillatory for only relatively\n(very) short or relatively long processing times  (i.e.  before the first Hopf bifurcation at\n$\\tau\\approx 1.9 $ and after the last one at $\\tau\\approx 108 $).\nFor relatively long processing times, slightly larger than\nthe value of $\\tau$ at the final Hopf\nbifurcation of the coexistence equilibrium,  the population is no longer\noscillatory.  The size of the predator\npopulation decreases relatively slowly as the processing time increases\nfurther.\nAlthough  the size of the predator population gets smaller as the\nprocessing time increases, it does not get  much smaller,\nuntil the processing time gets close to the threshold  for extinction\n$\\tau_c$.\nIf we had shown the diagram extended to the threshold $\\tau_c=170$, one would\nsee that as the processing time gets close to  $\\tau_c$, the size of the\npredator population suddenly decreases\nrelatively quickly to zero.  So our model predicts that for a predator\npopulation with fairly long processing times to\nbegin with,    cooling of the environment could be expected to be\ndetrimental with respect to the survivability of the predator\npopulation. Thus, this effect on the size of the predator population might be minor if the delay is close to its Hopf\nbifurcation value, but  could be  drastic if it  is close to the extinction value.\n\nOur example also suggests that   the\npredator population may not be oscillatory when it becomes\nendangered and hence close to extinction, i.e., for excessively long\nprocessing times. It would  be interesting to\ninvestigate  if this also holds for the  model,\nwith Holling type I response functions replaced by Holling type II.\nIn the   model with Holling type II response functions, if the  carrying capacity of the environment for\nthe prey is\nnot affected by the cooling and it is relatively high, then the model\npredicts that the predator\npopulation could still be   oscillatory, even if  the processing\ntime for the predator is ignored.  However,   cooling might also be expected to\nreduce  the carrying capacity for the prey, moving the parameters to a\nrange where that   model would also predict non-oscillatory dynamics (the paradox of enrichment \\cite{Rosenzweig1971}).  So once again, perhaps alarm\nbells should be sounded   when  there is cooling and the\npredator population has been non-oscillatory and  appears to decline rapidly\nas the average yearly temperature declines.\n\n\nOn the other hand, at the other extreme,  our model predicts a Hopf\nbifurcation at a relatively small value of the delay, resulting in\nthe birth of a family of periodic orbits  with\namplitude  increasing very quickly as the delay\nincreases with\nboth the prey\nand predator populations spending time very close to zero.\nThis remains the case for  small, but\nintermediate values of the delay,  (before the two  saddle node of limit\ncycles bifurcations near $\\tau=80$).\nFor this range of $\\tau$, these populations are therefore   very\nsusceptible to stochastic extinctions.  If the processing time was\noriginally very small (below the critical value for the first Hopf\nbifurcation)  and cooling made it longer, again a stochastic extinction might be\nlikely.  Similarly, if $\\tau$ was close to the first of the\ntwo  saddle node of limit\ncycles bifurcations near $\\tau=80$), warming of the average temperature\ncould result in a stochastic extinction  of one  or both of the populations.\nSince the predator cannot survive without the prey in our model, even if\nit was the  prey population  that experienced the stochastic\nextinction, the predator population would eventually die out as well.\nBetween the smallest value of $\\tau$ at which there is a period doubling\nbifurcation, and the value of $\\tau$ at the largest Hopf bifurcation,\ncooling would probably be advantageous to the predator population size.\n\nIn summary, it seems that whether  cooling is beneficial or detrimental to\n  the size of the predator population depends on where the delay is on\n  the bifurcation diagram, and it is very likely that\n  this is  very difficult to determine. As well,\n  if   values of $\\tau$ were to lie in the chaotic region, then changes\n  in the environment that changed the value of $\\tau$ to obtain regular\n  oscillatory dynamics may or may not be preferable.  As well,\n as can be seen by the various\nattractors  shown in Figures~\\ref{PD_left}-\\ref{chaos_xy}, and the orbit\ndiagrams in Figure~\\ref{zoom-in_80_100},  certain\nproperties of the system in the chaotic region such as\nthe maximum and minimum values of the predator\nsize  were fairly insensitive to the change in the delay.\n This is only a toy model. However, it suggests that ignoring delay\nin a model can result in incorrect predictions. Here, delay could change\nthe dynamics from convergence to \na globally asymptotically stable equilibrium to wild\noscillations.  It is most likely that in nature, it would not be\npossible to distinguish from data, a priori, if the dynamics were chaotic or\nperiodic, but  more importantly it would not necessarily be predictable\nwhat the effect of an increase or a decrease in the delay would be. Hence,\nour analysis indicates that one should be extra cautious if trying to\nmanipulate the delay to achieve a certain result based on model\npredictions.  \n\nFinally it is worth pointing out that\n   the resulting strange attractor in the predator-prey model studied\nhere bares such a close resemblance to the Mackey-Glass attractor, a model\ninvolving  a single delay differential equation\nto model a simple  feedback\nsystem for  respiratory control or hematopoietic diseases. Understanding\nwhether there is a deeper significiance to this relationship may give us\na better understanding of the class of possible strange attractors\nand warrants further investigation.\n\n\n\n\n\\bibliographystyle{siam}\n\n\n\\section*{Funding}\nThe Research of Gail S. K. Wolkowicz was partially supported by\n Natural Sciences and Engineering (NSERC) Discovery Grant \\# 9358 and\n Accelerator supplement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nProbabilistic graphical models for regression problems, where both the model parameters and the observation noise can have complex spatio-temporal correlation structure, are prevalent in neuroscience, neuroimaging and beyond \\citep{shiffrin2008survey}. Example domains where these models are being applied include Gaussian process inference \\citep{rakitsch2013all}, sparse signal recovery and multi-task learning \\citep{candes2006stable,donoho2006compressed,bertrand2019handling,chevalier2020statistical}, array signal processing \\citep{malioutov2005sparse}, fMRI analysis \\citep{cai2020incorporating}, climate science \\citep{beirle2003weekly}, computer hardware monitoring \\citep{ranieri2015near}, and brain-computer interfaces (e.g.~\\citep{lemm2005spatio,NIPS2014_fa83a11a}). The focus in this paper will be on brain source imaging (BSI) \\citep{wu2016bayesian}, i.e. the reconstruction of brain source activity from non-invasive magneto- or electroetoencephalography (M\/EEG) sensors \\citep{baillet2001electromagnetic}. Although a number of generative models that encompass  spatio-temporal correlations have been proposed within the BSI regression problem, inference solutions have either imposed specific simplifications and model constraints \\citep{cai2016bayesian,shvartsman2018matrix,cai2019representational} or have ignored temporal correlations overall \\citep{bertrand2019handling}, and therefore not fully addressed the inherent spatio-temporal problem structure. \n\n\nWe propose a novel and efficient algorithmic framework using Type-II Bayesian regression that explicitly considers the spatio-temporal covariance structure in both model coefficients and observation noise. To this end, we focus on the M\/EEG inverse problem of brain source reconstruction, for which we adopt a multi-task regression approach and formulate the source reconstruction problem as a probabilistic generative model with Gaussian scale mixture priors for sources with spatio-temporal prior covariance matrices that are expressed as a Kronecker product of separable spatial and temporal covariances. Their solutions are constrained to the Riemannian manifold of positive definite (P.D.) temporal covariance matrices. Exploiting the concept of geodesic convexity on the Riemannian manifold of covariance matrices with Kronecker structure, we are then able to derive robust, fast and efficient \\emph{majorization-minimization} (MM) optimization algorithms for model inference with provable convergence guarantees. In addition to deriving update rules for full-structural P.D. temporal covariance matrices, we also show that assuming a Toeplitz structure for the temporal covariance matrix results in computationally efficient update rules within the proposed MM framework. \n\n\n\\section{Spatio-temporal generative model}\n\\label{sec:gen-model}\n\nLet us consider a multi-task linear regression problem, mathematically formulated as\n\\begin{align}\n    {\\bf Y}_{g} = {\\bf L}{\\bf X}_{g} + {\\bf E}_{g} \\quad \\mathrm{for}\\;g=1,\\hdots,G \\;,\\label{eq:linearmodel}\n\\end{align}\nin which a forward matrix, ${\\bf L} \\in \\mathbb{R}^{M\\times N}$, maps a set of coefficients or source components ${\\bf X}_{g} \\in\\mathbb{R}^{N\\times T}$ to measurements ${\\bf Y}_{g} \\in \\mathbb{R}^{M\\times T}$, with independent Gaussian noise ${\\bf E}_{g} \\in\\mathbb{R}^{M\\times T}$. \\ali{$G$ denotes the number of sample blocks in the multi-task problem, while $N$, $M$, and $T$ denote the number of sources or coefficients, the number of sensors or observations, and the number of time instants, respectively.}\nThe problem of estimating $\\{{\\bf X}_{g}\\}_{g=1}^{G}$ given ${\\bf L}$ and $\\{{\\bf Y}_{g}\\}_{g=1}^{G}$ can represent an inverse problem in physics, a multi-task regression problem in machine learning, or a multiple measurement vector (MMV) recovery problem in signal processing \\citep{cotter2005sparse}. \n\n\n\nIn the context of BSI, $\\{{\\bf Y}_{g}\\}_{g=1}^{G}$ refers to M\/EEG sensor measurement data, and $\\{{\\bf X}_{g}\\}_{g=1}^{G}$ refers to brain source activity contributing to the observed sensor data. Here, $G$ can be defined as the number of epochs, trials or experimental tasks. The goal of BSI is to infer the underlying brain activity from the M\/EEG measurements given the lead field matrix ${\\bf L}$. In practice, ${\\bf L}$ can be computed using discretization methods such as the finite element method for a given head geometry and known electrical conductivities using the quasi-static approximation of Maxwell's equations\n\\citep{hamalainen1993magnetoencephalography}. \nAs the number of locations of potential brain sources is dramatically larger than the number of sensors, the M\/EEG inverse problem is highly ill-posed, which can be dealt with by incorporating prior assumptions. Adopting a Bayesian treatment is useful in this capacity because it allows these assumptions to be made explicitly by specifying prior distributions for the model parameters. Inference can be performed either through Maximum-a-Posteriori (MAP) estimation (\\emph{Type-I Bayesian learning}) \\citep{pascual1994low,haufe2008combining,gramfort2012mixed} or, when the model has unknown hyperparameters, through Type-II Maximum-Likelihood (ML) estimation (\\emph{Type-II Bayesian learning}) \\citep{mika2000mathematical,tipping2001sparse,wipf2009unified}. \n\nHere we focus on full spatio-temporal Type-II Bayesian learning, which assumes a family of prior distributions $p({\\bf X}_{g}|{\\bm \\Theta})$ parameterized by a set of hyperparameters $\\bm \\Theta$. We further assume that the spatio-temporal correlation structure of the brain sources can be modeled by a Gaussian $p({\\bf x}_{g}|{\\bm \\Gamma},{\\bf B})\\sim\\mathcal{N}({\\bm 0},{\\bm \\Sigma}_{0})$, where ${\\bf x}_{g}=\\text{vec}({\\bf X}_{g}^{\\top}) \\in \\mathbb{R}^{NT \\times 1}$ and where the covariance matrix ${\\bm \\Sigma}_{0}={\\bm \\Gamma} \\otimes {\\bf B}$ has Kronecker structure, meaning that the temporal and spatial correlations of the sources are modeled independently through  matrices ${\\bm \\Gamma}$ and ${\\bf B}$, where $\\otimes$ stands for the Kronecker product. Note \nthat the decoupling of spatial and temporal covariance is a neurophysiologically plausible assumption as neuronal sources originating from different parts of the brain can be assumed to share a similar autocorrelation spectrum. In this paper, we set ${\\bm \\Gamma}=\\mathrm{diag}({\\bm \\gamma})$, ${\\bm \\gamma} = [\\gamma_{1},\\dots,\\gamma_{N}]^\\top$, and ${\\bf B} \\in \\mathbb{S}^{T}_{++}$, where $\\mathbb{S}^{T}_{++}$ denotes the set of positive definite matrices with size $T \\times T$. Note that using a diagonal matrix ${\\bm \\Gamma}$ amounts to assuming that sources in different locations are independent. This assumption may be relaxed in our future work. On the other hand, modeling a full temporal covariance matrix ${\\bf B}$ amounts to assuming non-stationary dynamics of the sources, which is appropriate in event-related experimental designs. To deal with the more stationary sources (ongoing brain activity), we will also constrain ${\\bf B}$ to have Toeplitz structure later on. Based on these specifications, the prior distribution of the $i$-th brain source is modeled as $p(({\\bf X}_{g})_{i.}|\\gamma_{i},{\\bf B})\\sim\\mathcal{N}(0,\\gamma_{i}{\\bf B}),\\text{for}\\; i=1,\\dots,N$, where $({\\bf X}_{g})_{i.}$ denotes the $i$-th row of source matrix ${\\bf X}_g$. \n\nAnalogous to the definition of the sources, we here also model the noise to have spatio-temporal structure. The matrix ${\\bf E}_{g} \\in \\mathbb{R}^{M\\times T}$ represents $T$ time instances of zero-mean Gaussian noise with full covariance, ${\\bm \\Sigma}_{\\mathrm{\\bf e}}={\\bm \\Lambda} \\otimes {\\bm \\Upsilon}$, where ${\\bf e}_{g}=\\text{vec}({\\bf E}_{g}^{\\top}) \\in \\mathbb{R}^{MT\\times1}\\sim \\mathcal{N}(0,{\\bm \\Sigma}_{\\mathrm{\\bf e}})$, and where ${\\bm \\Lambda}$ and ${\\bm \\Upsilon}$ denote spatial and temporal noise covariance matrices, respectively. Here, we assume that the noise and sources share the same temporal structure, i.e., ${\\bm \\Upsilon}={\\bf B}$. We later investigate violations of this assumption in the simulation section. Analogous to the sources, we consider spatially independent noise characterized by diagonal spatial covariance ${\\bm \\Lambda} =\\mathrm{diag}({\\bm \\lambda})$, ${\\bm \\lambda} = [\\lambda_{1},\\dots,\\lambda_{M}]^\\top$. We consider both the general, \\emph{heteroscedastic}, case in which the noise level may be different for each sensor as well as the \\emph{homoscedastic} case in which the noise level is uniform. \n\nFor later use, we also define augmented versions of the source and noise covariance matrices as well as the lead field. Specifically, we set ${\\bf H}$ and ${\\bm \\Phi}$ so that ${\\bf H}:= [{\\bm \\Gamma}, {\\bf 0}; {\\bf 0}, {\\bm \\Lambda}]$, and ${\\bm \\Phi}:=[{\\bf L},{\\bf I}]$. These definitions allow us to unify source and noise covariance parameters within the single variable ${\\bf H}$, which facilitates the concurrent estimation of both quantities. ${\\bm \\Phi}$ now plays the same role as the lead field ${\\bf L}$, in the sense that it maps ${\\bm \\eta}_g:=[{\\bf x}_g^{\\top},{\\bf e}_g^{\\top}]$ to the measurements; i.e., ${\\bf y}_g = {\\bm \\Phi}{\\bm \\eta}_g$. \\ali{Figure~\\ref{fig:graphprob-geometricmean}-(a) illustrates a probabilistic graphical model summarizing the spatio-temporal generative model of our multi-task linear regression problem.}\n\nThe MMV model Eq.~\\eqref{eq:linearmodel} can be formulated equivalently in \\emph{single measurement vector (SMV)} form by vectorizing the spatio-temporal data matrices and using the Kronecker product as follows: ${\\bf y}_{g}={\\bf D}{\\bf x}_{g}+{\\bf e}_{g}$, where ${\\bf y}_{g}=\\text{vec}\\left({\\bf Y}_{g}^\\top \\right)\\in\\mathbb{R}^{MT\\times1}$, and ${\\bf D}={\\bf L}\\otimes {\\bf I}_{T}$. In a Type-II Bayesian learning framework, the hyper-parameters of the spatio-temporal source model are optimized jointly with the model parameters $\\{\\bm X_g \\}_{g=1}^G$. In our case, these hyper-parameters comprise the unknown source, noise and temporal covariance matrices, i.e., ${\\bm \\Theta} = \\{{\\bm \\Gamma},{\\bm \\Lambda},{\\bf B}\\} $. \n\\section{Proposed method --- full Dugh}\n\\label{sec:full-dugh}\nThe unknown parameters ${\\bm \\Gamma}$, ${\\bm \\Lambda}$, and ${\\bf B}$ can be optimized in an alternating iterative process. Given initial estimates, the posterior distribution of the sources is a Gaussian of the form $p({\\bf x}_{g}|{\\bf y}_{g},{\\bm \\Gamma},{\\bm \\Lambda},{\\bf B}) \\sim \\mathcal{N}(\\bar{\\bf x}_{g},{\\bm \\Sigma}_{\\bf x})$, whose mean and covariance are defined as follows:\n\\begin{align}\n    \\bar{\\bf x}_{g} &={\\bm \\Sigma}_{0}{\\bf D}^{\\top}({\\bm \\Lambda} \\otimes {\\bf B} + {\\bf D}{\\bm \\Sigma}_{0}{\\bf D}^{\\top})^{-1}{\\bf y}_{g}={\\bm \\Sigma}_{0}{\\bf D}^{\\top}\\Tilde{\\bm \\Sigma}_{\\bf y}^{-1}{\\bf y}_{g}\\;,\n    \\label{eq:MeanValue-ST} \\\\\n    {\\bm \\Sigma}_{\\bf x} &= {\\bm \\Sigma}_{0}-{\\bm \\Sigma}_{0}{\\bf D}^{\\top}\\Tilde{\\bm \\Sigma}_{\\bf y}^{-1}{\\bf D}{\\bm \\Sigma}_{0}\n    \\label{eq:CovarianceValue-ST}\\;,\n\\end{align}\nwhere ${\\bm \\Sigma}_{\\bf y}={\\bf L}{\\bm \\Gamma}{\\bf L}^{\\top}+{\\bm \\Lambda}$, and where $\\Tilde{\\bm \\Sigma}_{\\bf y} = {\\bm \\Sigma}_{\\bf y} \\otimes {\\bf B}$ denotes the spatio-temporal statistical model covariance matrix. The estimated posterior parameters $\\bar{\\bf x}_{g}$ and ${\\bm \\Sigma}_{\\bf x}$ are then in turn used to update ${\\bm \\Gamma}$, ${\\bm \\Lambda}$, and $\\bf B$ as the minimizers of the negative log of the marginal likelihood $p({\\bf Y}|{\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$, which is given by:\n\\begin{align}\n    \\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B}) &= T\\log|{{\\bm \\Sigma}_{\\bf y}}|+M\\log|{\\bf B}| +\\frac{1}{G}\\sum_{g=1}^{G}\\text{tr}({{\\bm \\Sigma}_{\\bf y}^{-1}}{\\bf Y}_{g}{\\bf B}^{-1}{\\bf Y}_{g}^{\\top}) \n    \\label{eq:MLKronokCost}\\;,\n\\end{align}\nwhere $|\\cdot|$ denotes the determinant of a matrix. A detailed derivation is provided in Appendix~\\ref{appendix:spatio-temp-cost}. Given the solution of the hyperparameters ${\\bm \\Gamma}$, ${\\bm \\Lambda}$, and $\\bf B$, the posterior source distribution is obtained by plugging these estimates into Eqs.~\\eqref{eq:MeanValue-ST}--\\eqref{eq:CovarianceValue-ST}. This process is repeated until convergence.\n\nThe challenge in this high-dimensional inference problem is to find (locally) optimal solutions to  Eq.~\\eqref{eq:MLKronokCost}, which is a non-convex cost function, in adequate time. Here we propose a novel efficient algorithm, which is able to do so. Our algorithm thus learns the full spatio-temporal correlation structure of sources and noise. \\citet{hashemi2021unification} have previously demonstrated that \n\\emph{majorization-minimization} (MM) \\citep{sun2017majorization}\nis a powerful non-linear optimization framework that can be leveraged to solve similar Bayesian Type-II inference problems. Here we extend this work to our spatio-temporal setting. Building on the idea of majorization-minimization, we construct convex surrogate functions that \\emph{majorize} $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$ in each iteration of the proposed algorithm. Then, we show the minimization equivalence between the constructed majorizing functions and Eq.~\\eqref{eq:MLKronokCost}. These results are presented in theorems~\\ref{Theo:Time-Surrogate} and \\ref{Theo:Space-Surrogate}. Theorems~\\ref{Theo:Time-Sol} and \\ref{Theo:Space-Sol} propose an efficient alternating optimization algorithm for solving $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$, which leads to update rules for the spatial and temporal covariance matrices ${\\bm \\Gamma}$ and ${\\bf B}$ as well as the noise covariance matrix ${\\bm \\Lambda}$. \n\nStarting with the estimation of the temporal covariance based on the current source estimate, we can state the following theorem: \n\\begin{figure}\n\\begin{center}\n\\begin{minipage}[b]{0.49\\textwidth}\n    \\centerline{\\includegraphics[width=7.0cm,keepaspectratio]{figures\/Paper-graphical-probabilistic-models_final.pdf} } \n    \\begin{center}\n        \\centerline{(a)}\n    \\end{center}\n\\end{minipage}\n\\begin{minipage}[b]{0.49\\textwidth}\n    \\centerline{\\includegraphics[width=7.0cm,keepaspectratio]{figures\/PSD-NeurIPS-Temporal_2.pdf}} \n    \\begin{center}\n        \\centerline{(b)}\n    \\end{center}\n\\end{minipage}\n\\end{center}\n\\caption{\\ali{(a) Probabilistic graphical model for the multi-task linear regression problem. (b) Geometric representation of the geodesic path between the pair of matrices \n$\\{{\\bf B}^{k},{\\bf M}_{\\mathrm{time}}^{k}\\}$ on the P.D. manifold and the geometric mean between them, which is used to update ${\\bf B}^{k+1}$.}}\n\\label{fig:graphprob-geometricmean}\n\\vspace{-3mm}\n\\end{figure}\n\n\\begin{theo} \n\\label{Theo:Time-Surrogate}\nOptimizing the non-convex Type-II cost function in Eq.~\\eqref{eq:MLKronokCost},  $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$, with respect to the temporal covariance matrix ${\\bf B}$ is equivalent to optimizing the following convex surrogate function, which \\emph{majorizes} Eq.~\\eqref{eq:MLKronokCost}:  \n\\begin{align}\n    \\mathcal{L}^{\\mathrm{time}}_{\\mathrm{conv}}({\\bm \\Gamma}^{k},{\\bm \\Lambda}^{k},{\\bf B})=\\tr\\left(({\\bf B}^{k})^{-1}{\\bf B}\\right)+\\tr({\\bf M}_{\\mathrm{time}}^{k}{\\bf B}^{-1}),\n    \\label{eq:TimeSurrogateFunction}    \n\\end{align}\nwhere ${\\bf M}_{\\mathrm{time}}^{k}$ is defined as\n\\begin{equation}\n    {\\bf M}_{\\mathrm{time}}^{k}:=\\frac{1}{MG}\\sum_{g=1}^{G}{\\bf Y}_{g}^{\\top}\\left({\\bm \\Sigma}_{\\bf y}^{k}\\right)^{-1}{\\bf Y}_{g}\n    \\label{eq:TempSampleCov}\\;,\n\\end{equation}\nand where ${\\bm \\Gamma}^{k}$, ${\\bm \\Lambda}^{k}$, and ${\\bm \\Sigma}_{\\bf y}^{k}$ denote the source, noise and statistical model covariance matrices at the $k$-th iteration, which are treated as constants when optimizing over ${\\bf B}$. \n\\end{theo}\n\\begin{proof}\nA detailed proof is provided in Appendix \\ref{appendix:Time-Surrogate}.\n\\end{proof}\nThe solution of Eq.~\\eqref{eq:TimeSurrogateFunction} can be obtained by taking advantage of the Riemannian geometry of the search space. The following theorem holds. \n\\begin{theo}\n\\label{Theo:Time-Sol}\nThe cost function $\\mathcal{L}^{\\mathrm{time}}_{\\mathrm{conv}}({\\bm \\Gamma}^{k},{\\bm \\Lambda}^{k},{\\bf B})$ is strictly geodesically convex with respect to the P.D. manifold and its minimum with respect to ${\\bf B}$ can be attained by iterating the following update rule until convergence:\n\\begin{align}\n    {\\bf B}^{k+1} \\leftarrow ({\\bf B}^{k})^{\\nicefrac{1}{2}}\\left(({\\bf B}^{k})^{\\nicefrac{-1}{2}}{\\bf M}_{\\mathrm{time}}^{k}({\\bf B}^{k})^{\\nicefrac{-1}{2}}\\right)^{\\nicefrac{1}{2}}({\\bf B}^{k})^{\\nicefrac{1}{2}}\\;\\label{eq:BGeodesicUpdate}.\n\\end{align}\n\\end{theo}\n\\begin{proof}\n\\ali{A detailed proof is provided in Appendix \\ref{appendix:Time-Sol}.}\n\\end{proof}\n\n\\begin{rem}\n    A geometric interpretation of the update rule \\eqref{eq:BGeodesicUpdate} is that it finds the geometric mean between a spatially whitened version of the empirical temporal covariance matrix, ${\\bf M}_{\\mathrm{time}}^{k}$, and the model temporal covariance matrix from the previous iteration, ${\\bf B}^{k}$. A geometric representation of the geodesic path between the pair of matrices $\\{{\\bf B}^{k},{\\bf M}_{\\mathrm{time}}^{k}\\}$ on the P.D. manifold and the geometric mean between them, representing the update for ${\\bf B}^{k+1}$, is provided in Figure~\\ref{fig:graphprob-geometricmean}-(b).\n\\end{rem}\nWe now derive an update rule for the spatial source and noise covariance matrices using an analogous approach. To this end, we first construct a convex surrogate function that \\emph{majorizes} $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$ in each iteration of the optimization algorithm by considering augmented variables comprising both the source and noise covariances, i.e., ${\\bf H}:= [{\\bm \\Gamma}, \\bf 0; \\bf 0, {\\bm \\Lambda}]$, and ${\\bm \\Phi}:=[{\\bf L},{\\bf I}]$. The following theorem holds. \n\\begin{theo} \n\\label{Theo:Space-Surrogate}\nMinimizing the non-convex Type-II cost function in Eq.~\\eqref{eq:MLKronokCost}, $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$, with respect to the spatial covariance matrix ${\\bf H}$ is equivalent to minimizing the following convex surrogate function, which \\emph{majorizes} Eq.~\\eqref{eq:MLKronokCost}:\n\\begin{align}\n    \\mathcal{L}^{\\mathrm{space}}_{\\mathrm{conv}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B}^{k}) = \\mathcal{L}^{\\mathrm{space}}_{\\mathrm{conv}}({\\bf H},{\\bf B}^{k})=\\tr\\left({\\bm \\Phi}^{\\top}({\\bm \\Sigma}_{\\bf y}^{k})^{-1}{\\bm \\Phi}{\\bf H}\\right)+\\tr\\left({\\bf M}_{\\mathrm{SN}}^{k} {\\bf H}^{-1} \\right) \\;,\n    \\label{eq:SpaceSurrogateFunction}    \n\\end{align}\nwhere ${\\bf M}_{\\mathrm{SN}}^{k}$ is defined as\n\\begin{align}\n        {\\bf M}_{\\mathrm{SN}}^{k} &:= {\\bf H}^{k}{\\bm \\Phi}^{\\top}({\\bm \\Sigma}_{\\bf y}^{k})^{-1}{\\bf M}_{\\mathrm{space}}^{k}({\\bm \\Sigma}_{\\bf y}^{k})^{-1}{\\bm \\Phi} {\\bf H}^{k} \\;, \\mathrm{with} \\nonumber \\\\ \n        {\\bf M}_{\\mathrm{space}}^{k} &:= \\frac{1}{TG}\\sum_{g=1}^{G}{\\bf Y}_{g}({\\bf B}^{k})^{-1}{\\bf Y}_{g}^{\\top} \\;,\n        \\label{eq:SpaceSampleCov}\n\\end{align}\nand where ${\\bf B}^k$ denotes the temporal model covariance matrix estimated in the $k$-th iteration.\n\\end{theo}\n\\begin{proof}\nA detailed proof is provided in Appendix \\ref{appendix:Space-Surrogate}.\n\\end{proof}\n\n\nWhile, in principle, update rules for full-structured spatial source and noise covariances may be conceived, in analogy to the ones presented for the temporal covariance, we here restrict ourselves to the discussion of diagonal spatial covariances. Note, though, that such a choice of prior does not prohibit the reconstruction of parameters with more complex correlation structure. We have ${\\bf H}=\\mathrm{diag}({\\bf h})$, ${\\bf h} = [\\gamma_{1},\\dots,\\gamma_{N}, \\sigma^2_{1},\\dots,\\sigma^2_{M}]^\\top$, and ${\\bm \\Sigma}_{\\bf y}={\\bm \\Phi}{\\bf H}{\\bm \\Phi}^{\\top}$. The update rule for ${\\bf H}$ then takes a simple form, as stated in the following theorem:\n\\begin{theo}\n\\label{Theo:Space-Sol}\nThe cost function $\\mathcal{L}^{\\mathrm{space}}_{\\mathrm{conv}}({\\bf H},{\\bf B}^{k})$ is convex in $\\bf h$, and its minimum with respect to $\\bf h$ can be obtained according to the following closed-form update rule, which concurrently estimates the scalar source variances and heteroscedastic noise variances:\n\\begin{align}\n    {\\bf H}^{k+1}=\\mathrm{diag}({\\bf h}^{k+1}),\\;   h_{i}^{k+1}&\\leftarrow\\sqrt{\\frac{g_{i}^{k}}{z_{i}^{k}}}\\quad \\text{for}\\;i=1,\\hdots,N+M \\;\\label{eq:HDiagonalUpdate}, \\text{where} \\\\ \n    {\\bf g} &:=\\mathrm{diag}({\\bf M}_{\\mathrm{SN}}^{k}) \\label{eq:diagISpace}  \\\\\n    {\\bf z} &:=\\mathrm{diag}({\\bm \\Phi}^{\\top}({\\bm \\Sigma}_{\\bf y}^{k})^{-1}{\\bm \\Phi}) \\label{eq:diagIISpace} \n\\end{align} \n\\end{theo}\n\\begin{proof}\nA detailed proof can be found in Appendix \\ref{appendix:Space-Solution}.\n\\end{proof}\nThe final estimate of the posterior distribution \\eqref{eq:MeanValue-ST} can be obtained by starting from any initial guess for the model parameters, ${\\bf H}^{0}=\\{{\\bm \\Gamma}^{0},{\\bm \\Lambda}^{0}\\}$ and ${\\bf B}^{0}$, and iterating between update rules \\eqref{eq:BGeodesicUpdate}, \\eqref{eq:HDiagonalUpdate}, and \\eqref{eq:MeanValue-ST} until convergence. We call the resulting algorithm \\emph{full Dugh}. Convergence is shown in the following theorem:\n\\begin{theo}\n\\label{theo:MM-Convergence-Guarantees}\nOptimizing the non-convex ML cost function  Eq.~\\eqref{eq:MLKronokCost} with alternating update rules for ${\\bf B}$ and ${\\bf H}=\\{{\\bm \\Gamma},{\\bm \\Lambda}\\}$ in Eq.~\\eqref{eq:BGeodesicUpdate} and Eq.~\\eqref{eq:HDiagonalUpdate} defines an MM algorithm, which is guaranteed to converge to a stationary point. \n\\end{theo}\n\\begin{proof}\nA detailed proof can be found in Appendix~\\ref{appendix:MM-Convergence-Guarantees}.\n\\end{proof}\n\\section{Efficient inference for stationary temporal dynamics --- thin Dugh}\n\\label{sec:thin-Dugh}\nWith full Dugh, we could present a general inference method for data with full temporal covariance structure. This algorithm may be used for non-stationary data, for which the covariance between two time samples depends on the absolute position of the two samples within the studied time window. An example is the reconstruction of brain responses to repeated external stimuli from M\/EEG (event-related electrical potential or magnetic field) data, where data blocks are aligned to the stimulus onset. In other realistic settings, however, stationarity can be assumed in the sense that the covariance between two samples only depends on their relative position (distance) to each other but not to any external trigger. An example is the reconstruction of ongoing brain dynamics in absence of known external triggers. This situation can be adequately modeled using temporal covariance matrices with Toeplitz structure. In the following, we devise an efficient extension of full Dugh for that setting. \n\nTemporal correlation has been incorporated in various brain source imaging models through different priors \\citep[see][and references therein]{pirondini2017computationally,wu2016bayesian}. According to  \\citep{lamus2012spatiotemporal}, the temporal dynamics of neuronal populations can be approximated by a first order auto-regressive (AR(1)) model of the form ${x}_n(t+1)=\\beta {x}_n(t)+\\sqrt{1-\\beta^{2}}\\xi_n(t),~ n=1,\\hdots,N; t=1,\\hdots,T$ with AR coefficient $\\beta \\in(-1,1)$ and innovation noise $\\xi_n(t)$. It can be shown that the temporal correlation matrix corresponding to this AR model has Toeplitz structure, ${\\bf B}_{i,j}=\\beta^{|{i-j}|}$ \\citep{zhang2011sparse}. Consequently,  we now constrain the cost function Eq.~\\eqref{eq:TimeSurrogateFunction} to the set of Toeplitz matrices:\n\\begin{align}\n    {\\bf B}^{k+1} &=\\argmin_{{\\bf B} \\in \\mathcal{B},\\;{\\bf H}= {\\bf H}^{k} } \\tr(({\\bf B}^{k})^{-1}{\\bf B})+\\tr({\\bf M}_{\\mathrm{time}}^{k}{\\bf B}^{-1})\\;,\n    \\label{eq:MinComstrainedTempo}\n\\end{align}\nwhere set $\\mathcal{B}$ denotes the set of real-valued positive-definite Toeplitz matrices of size $T\\times T$. \n\nIn order to be able to derive efficient update rules for this setting, we bring ${\\bf B}$ into a diagonal form. The following proposition holds.\n\\begin{prop} \n\\label{prop:toeplitz}\nLet ${\\bf P} \\in \\mathbb{R}^{L \\times L}$ with $L>T$, be the circulant embedding of matrix ${\\bf B} \\in \\mathcal{B}^L$, where $\\mathcal{B}^L$ is assumed to be the subset of $\\mathcal{B}$ that guarantees that ${\\bf P}$ is a real-valued circulant P.D. matrix.  \nThen the Toeplitz matrix ${\\bf B}$ can be diagonalized as follows:\n\\begin{align}\n    {\\bf B} &= {\\bf Q}{\\bf P}{\\bf Q}^{H} \\quad \\text{with} \\quad \\mathbf{Q} = \n    [{\\bf I}_{M},{\\bf 0}]{\\bf F}_{L}   \\label{eq:FourierDiagonalization},\\;\\text{and}  \\\\\n    [\\mathbf{F}_{L}]_{m,l} &= \\frac{1}{\\sqrt{L}}e^{\\frac{i2\\pi(l-1)}{L}(m-1)} \\label{eq:DFT}\\;,\n\\end{align}\nwhere ${\\bf P}=\\mathrm{diag}({\\bf p}) =\\mathrm{diag}(p_{0},p_{1},\\dots,p_{L-1})$ is a diagonal matrix. The main diagonal coefficients of ${\\bf P}$ are given as the normalized discrete Fourier transform (DFT), represented by the linear operator $\\mathbf{F}_{L}$, of the first row of ${\\bf B}$: ${\\bf p}=\\mathbf{F}_{L}{\\bf B}_{1.}$  \n(Note that a Toeplitz matrix can be represented by its first row or column, ${\\bf B}_{1.}$). \n\\end{prop}\n\\begin{proof}\nThis is a direct implication of the fact that Toeplitz matrices can be embedded into circulant matrices of larger size \\citep{dembo1989embedding} and the result that circulant matrices can be approximately diagonalized using the Fourier transform \\citep{grenander1958toeplitz}. Further details can be found in Appendix~\\ref{appendix:toeplitz}.\n\\end{proof}\nThe solution of Eq.~\\eqref{eq:MinComstrainedTempo} can be obtained by direct application of Proposition~\\ref{prop:toeplitz}. The results is summarized in the following theorem: \n\\begin{theo}\n\\label{Theo:Time-Toeplitz}\nThe cost function Eq.~\\eqref{eq:MinComstrainedTempo} is convex in $\\bf p$, and its minimum with respect to $\\bf p$ can be obtained by iterating the following closed-form update rule until convergence:\n\\begin{align}\n    p_{l}^{k+1} &\\leftarrow\\sqrt{\\frac{\\hat{g}_{l}^{k}}{\\hat{z}_{l}^{k}}}\\;\\text{for}\\;l=1,\\hdots,L \\;\\label{eq:BToeplitzUpdate}, \\text{where} \\\\ \n    \\hat{\\bf g} &:=\\mathrm{diag}({\\bf P}^{k}{\\bf Q}^{H}({\\bf B}^{k})^{-1}{\\bf M}_{\\mathrm{time}}^{k}({\\bf B}^{k})^{-1}{\\bf Q}{\\bf P}^{k}) \\label{eq:diagI}  \\\\\n    \\hat{\\bf z} &:=\\mathrm{diag}({\\bf Q}^{H}({\\bf B}^{k})^{-1}{\\bf Q})\\;. \n    \\label{eq:diagII} \n\\end{align}\n\\end{theo}\n\\begin{proof}\nA detailed proof is provided in Appendix~\\ref{appendix:Time-Toeplitz}.\n\\end{proof}\n\n\n\\subsection{Efficient computation of the posterior}\nGiven estimates of the spatial and temporal covariance matrices, we can efficiently compute the posterior mean by exploiting their intrinsic diagonal structure. \n\\begin{theo}\n\\label{theo:eff-post}\nGiven the diagonalization ${\\bf B}={\\bf Q}{\\bf P}{\\bf Q}^{H}$ of the temporal correlation matrix and the eigenvalue decomposition ${\\bf L}\\bm{\\Gamma}{\\bf L}^{\\top}= {\\bf U}_{\\bf x}{\\bf D}_{\\bf x}{\\bf U}_{\\bf x}^{\\top}$ of ${\\bf L}\\bm{\\Gamma}{\\bf L}^{\\top}$, where ${\\bf D}_{\\bf x}=\\mathrm{diag}(d_1,\\hdots,d_M)$, the posterior mean is efficiently computed as \n\\begin{align}\n    \\bar{\\bf x}_{g} & =\\left(\\bm{\\Gamma}\\otimes{\\bf B}\\right){\\bf D}^{\\top}\\Tilde{\\bm \\Sigma}_{\\bf y}^{-1}{\\bf y}_{g}= \\tr\\left({\\bf Q}{\\bf P} \\left({\\bm \\Pi} \\odot{\\bf Q}^{H}{\\bf Y}_{g}^{\\top}{\\bf U}_{\\bf x}\\right)\\left({\\bf U}_{{\\bf x}}^{\\top}{\\bf L}\\bm{\\Gamma}^{\\top}\\right) \\right) \n    \\label{eq:efficientPosterior}\\;,\n\\end{align}\nwhere $\\odot$ denotes the Hadamard product between the corresponding elements of two matrices of equal size. In addition, ${\\bm \\Pi}$ is defined as follows: $[{\\bm \\Pi}]_{l,m}= \\frac{1}{{\\sigma}_m^2 + p_l d_m}\\;\\text{for}\\; l=1,\\hdots,L\\;\\text{and}\\;\\;m=1,\\hdots,M$.\n\\end{theo}\n\\begin{proof}\nA detailed proof is provided in Appendix \\ref{appendix:eff-post}. \n\\end{proof}\n\nThe resulting algorithm, obtained by iterating between update rules \\eqref{eq:BToeplitzUpdate}, \\eqref{eq:HDiagonalUpdate} and \\eqref{eq:efficientPosterior}, is called \\emph{thin Dugh} (as opposed to \\emph{full Dugh} introduced above).\n\n\\section{Simulations}\n\\label{sec:simulation}\n\\vspace{-3mm}\nWe present two sets of experiments to assess the performance of our proposed methods. In the first experiment, we compare the reconstruction performance of the proposed Dugh algorithm variants to that of Champagne \\citep{wipf2010robust} and two other competitive methods -- eLORETA \\citep{pascual2007discrete} and S-FLEX \\citep{haufe2011large} -- for a range of SNR levels, numbers of time samples, and orders of AR coefficients. In the second experiment, we test the impact of model violations on the temporal covariance estimation.\nAll experiments are performed using \\texttt{Matlab} on a machine with a 2.50 GHz Intel(R) Xeon(R) Platinum 8160 CPU. \\ali{The computational complexity of each method in terms of the average running time in units of seconds for 1000 iterations is as follows: \\emph{Full Dugh}: 67.094s, \\emph{Thin Dugh}: 62.289s, \\emph{Champagne}: 1.533s, \\emph{eLORETA}: 2.653s, and  \\emph{S-FLEX}: 20.963s.}\nThe codes are publicly available at \\href{https:\/\/github.com\/AliHashemi-ai\/Dugh-NeurIPS-2021}{\\textcolor{blue}{https:\/\/github.com\/AliHashemi-ai\/Dugh-NeurIPS-2021}}.\n\n\\vspace{-1mm}\n\\subsection{Pseudo-EEG signal generation and benchmark comparison}\n\\vspace{-1mm}\n\nWe simulate a sparse set of $N_0 =3$ active sources placed at random locations on the cortex. To simulate the electrical neural activity, we sample time series of length $T \\in \\{10,20,50,100\\}$ from a univariate linear autoregressive AR(P) process.\nWe use stable AR systems of order $P \\in \\{1,2,5,7\\}$. \nThe resulting source distribution is then projected to the EEG sensors, denoted by ${{\\bf Y}^{\\mathrm{signal}}}$, using a realistic lead field matrix, ${\\bf L} \\in \\mathbb{R}^{58 \\times 2004}$. We generate ${\\bf L}$ using the New York Head model \\citep{huang2016new} taking into account the realistic anatomy and electrical tissue conductivities of an average human head. Finally, we add Gaussian white noise to the sensor space signal.\nNote that the simulated noise and source time courses do not share a similar temporal structure here -- sources are modeled with a univariate autoregressive AR(P) process while a temporally white Gaussian distribution is used for modeling noise. Thus, we could assess the robustness of our proposed method under violation of the model assumption that the temporal structure of sources and noise is similar.\nThe resulting noise matrix ${\\bf E}=[{\\bf e}(1),\\dots,{\\bf e}(T)]$ is first normalized and then added to the signal matrix ${\\bf Y}^{\\mathrm{signal}}$ as follows: ${\\bf Y}={\\bf Y}^{\\mathrm{signal}}+\\frac{(1-\\alpha)\\left\\Vert {{\\bf Y}^{\\mathrm{signal}}}\\right\\Vert_{F}}{\\alpha \\left\\Vert{\\bf E}\\right\\Vert_{F}}{\\bf E}$, where $\\alpha$ determines the signal-to-noise ratio (SNR) in sensor space. Precisely, SNR is defined as $\\mathrm{SNR} = 20\\mathrm{log}_{10}\\left(\\nicefrac{\\alpha}{1-\\alpha}\\right)$. In this experiment the following values of $\\alpha$ are used: $\\alpha \\in $ \\{0.55, 0.65, 0.7, 0.8\\}, which correspond to the following SNRs: SNR $\\in$ \\{1.7, 5.4, 7.4, 12\\}~(dB). {Interested readers can refer to Appendix~\\ref{appendix-Simulation} and \\citep{haufe2016simulation} for further details on the simulation framework.}\nWe quantify the performance of all algorithms using the \\emph{earth mover's distance} (EMD) \\citep{rubner2000emd,haufe2008combining} and the maximal correlation between the time courses of the simulated and the reconstructed sources (TCE). Each simulation is carried out 100 times using different instances of $\\bf{X}$ and $\\bf E$, and the mean and standard error of the mean (SEM) of each performance measure across repetitions is calculated. \n\n\nIn order to investigate the impact of model violations on the temporal covariance estimation, we generate a random Gaussian source matrix, ${\\bf X} \\in \\mathbb{R}^{2004 \\times 30 \\times G}$ representing the brain activity of $2004$ brain sources at $30$ time instances for different numbers of trials $G \\in 10,20,30,40,50$. In all trials, sources are randomly sampled from a zero-mean normal distribution with spatio-temporal covariance matrix ${\\bm \\Gamma} \\otimes {\\bf B}$, where ${\\bf B} \\in \\mathbb{R}^{30 \\times 30}$ is either a full-structural random PSD matrix or a Toeplitz matrix with $\\beta = 0.8$. Gaussian noise $\\bf E$ sharing the same temporal covariance with the sources is added to the measurements ${\\bf Y}={\\bf L}{\\bf X}$, so that the overall SNR is $0$~dB. We evaluate the accuracy of the temporal covariance reconstruction using Thin and full Dugh.\nThe performance is evaluated using two measures: Pearson correlation between the original and reconstructed temporal covariance matrices, $\\bf B$ and $\\hat{{\\bf B}}$, denoted by $r({\\bf B},\\hat{\\bf B})$, and the normalized mean squared error (NMSE) defined as: $\\text{NMSE} = ||\\hat{{\\bf B}}-{\\bf B}||_{F}^{2} \/ ||{\\bf B}||_{F}^{2}$. The similarity error is defined as: $1- r({\\bf B},\\hat{\\bf B})$. Note that NMSE measures the reconstruction at the true scale of the temporal covariance; while $r({\\bf B},\\hat{\\bf B})$ is scale-invariant and hence only quantifies the overall structural similarity between simulated and estimated noise covariance matrices. \n\n\\begin{figure}\n\\begin{minipage}[b]{0.85\\linewidth}\n  \\centering\n    \\centerline{\\includegraphics[width=8cm]{figures\/legend-final-revised.PNG}}\n\\end{minipage}\n  \\centering\n  \\centerline{\\includegraphics[width=\\textwidth,scale=0.5,keepaspectratio]{figures\/Source-eval-metrics-new-final-revised.pdf}}\n\\caption{Source reconstruction performance (mean $\\pm$ SEM) of the four different brain source imaging schemes, namely Thin Dugh, Champagne, eLORETA, and S-FLEX, for data generated by a realistic lead field matrix. Performance is assessed for different settings including a wide range of SNRs, different numbers of time samples, and different AR model orders. Performance is evaluated in terms of the earth mover's distance (EMD) and time-course correlation error (TCE) between each simulated source and the reconstructed source with highest maximum absolute correlation.}\n\\label{fig:metrics-source}\n\\vspace{-3mm}\n\\end{figure}\n\n\n\n\nIn Figure \\ref{fig:metrics-source}, we show the source reconstruction performance (mean $\\pm$ SEM) of the four different brain source imaging schemes, namely thin Dugh, Champagne, eLORETA and S-FLEX. We notice that Dugh achieves superior performance in terms of EMD metric, whereas it is competitive in terms of TCE. Note that since thin Dugh incorporates the temporal structure of the sources into the inference scheme, its performance with respect to EMD and TCE can be significantly improved by increasing the number of time samples.\nFigure~\\ref{fig:metrics-temporal} demonstrates the estimated temporal covariance matrix obtained from our two proposed spatio-temporal learning schemes, namely thin and full Dugh, indicated by cyan and magenta colors, respectively. The upper panel illustrates the reconstruction results for a setting where the ground-truth temporal covariance matrix has full structure, while the lower panel shows a case with Toeplitz temporal covariance matrix structure. It can be seen that \\emph{full Dugh} can better capture the overall structure of ground truth full-structure temporal covariance as evidenced by lower NMSE and similarity errors compared to \\emph{thin Dugh} that is only able to recover a Toeplitz matrix. As can be expected, the behavior is reversed in the lower-panel of Figure~\\ref{fig:metrics-temporal}, where the ground truth temporal covariance matrix is indeed Toeplitz. Full Dugh, however, still provides reasonable performance in terms of NMSE and similarity error, even though it estimates a full-structure temporal covariance matrix. Finally, the performance of both learning schemes are significantly improved by increasing the number of trials. \n\\begin{figure}\n  \\centering\n  \\centerline{\\includegraphics[width=\\textwidth,trim=0 5cm 0 0,clip]{figures\/NeurIPS-B-reconstrcut-Both.pdf}}\n\\caption{Accuracy of the temporal covariance matrix reconstruction incurred by two different temporal learning approaches, namely thin (cyan) and full (magenta) Dugh, assuming Toeplitz and full temporal covariance structure, respectively. The ground-truth temporal covariance matrix (first column) has full covariance structure in the first row, and Toeplitz structure in the second row. Performance is assessed in terms of the Pearson correlation between the entries of the original and reconstructed temporal covariance matrices, ${\\bf B}$ and $\\hat{{\\bf B}}$, denoted by $r({\\bf B},\\hat{\\bf B})$. Shown is the similarity error $1- r({\\bf B},\\hat{\\bf B})$ (forth column). Further, the normalized mean squared error (NMSE) between ${\\bf B}$ and $\\hat{{\\bf B}}$, defined as $\\text{NMSE} = ||\\hat{{\\bf B}}-{\\bf B}||_{F}^{2} \/||{\\bf B}||_{F}^{2}$ is reported (fifth column).}\n\\label{fig:metrics-temporal}\n\\vspace{-2mm}\n\\end{figure}\n\\vspace{-2mm}\n\\section{Analysis of real MEG and EEG data recordings}\n\\label{sec:real-data-analysis}\n\\vspace{-2mm}\n\\ali{We now demonstrate the performance of the novel algorithms on two MEG datasets (see Figure~\\ref{fig:AEF} and \\ref{fig:VEF}) and one EEG dataset (see Appendix \\ref{appendix:real-data}). All participants provided informed written consent prior to study participation and received monetary compensation for their participation. The studies were approved by the University of California, San Francisco Committee on Human Research. The MEG datasets from two female subjects comprised five trials of visual or auditory stimulus presentation, where the goal was to reconstruct cortical activity reflecting auditory and visual processing. No prior work has shown success in reconstruction in such extreme low SNR data.} \n\n\\ali{Figure~\\ref{fig:AEF} shows the reconstructed sources for auditory evoked fields (AEF) from a representative subject using eLORETA, MCE, thin and full Dugh. In this case, we tested the reconstruction performance of all algorithms with the number of trials limited to 5. As Figure~\\ref{fig:AEF} demonstrates, the reconstructions of thin and full Dugh both show focal sources at the expected locations of the auditory cortex. Limiting the number of trials to as few as 5 does not negatively influence the reconstruction result of Dugh methods, while it severely affects the reconstruction performance of competing methods. For the visual evoked field (VEF) data in Figure~\\ref{fig:VEF}, thin Dugh was able to reconstruct two sources in the visual cortex with their corresponding time-courses demonstrating an early peak at~100 ms and a later peak around~200 ms (similar performance was observed for full Dugh).}\n\\begin{figure}\n  \\centering\n  \\centerline{\\includegraphics[width=\\textwidth,trim=0 17cm 0 0,clip]{figures\/AEF-final.pdf}}\n\\caption{Analysis of auditory evoked fields (AEF) of one representative subject using eLORETA, MCE, thin and full Dugh. As it can be seen, thin and full Dugh can correctly localize bilateral auditory activity to Heschl's gyrus, which is the characteristic location of the primary auditory cortex, with as few as 5 trials. In this challenging setting, all competing methods show inferior performance.}\n\\label{fig:AEF}\n\\vspace{-2mm}\n\\end{figure}\n\\begin{figure}\n  \\centering\n  \\centerline{\\includegraphics[width=\\textwidth]{figures\/Figure4.png}}\n\\caption{Spatial maps and corresponding time-courses reconstructed from five trials of visual stimulus presentation using Dugh and MCE. Distinct activity in visual cortex was reconstructed.}\n\\label{fig:VEF}\n\\vspace{-6mm}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sec:discussion}\nInverse modeling is a challenging problem that can be cast as a multi-task regression problem. Machine learning-based methods have contributed by systematically incorporating prior information into such models (e.g.~\\citep{wipf2009unified,wu2016bayesian,mccann2017convolutional,arridge2019solving,bubba2019learning}). While, so far, explicit models of spatio-temporal dynamics are rare, we contribute in this paper by deriving efficient inference algorithms for regression models with spatio-temporal dynamics in model parameters and noise. Specifically, we employ separable Gaussian distributions using Kronecker products of temporal and spatial covariance matrices. We assume sparsity and independent activations in the spatial domain, while the temporal covariance is modeled to have either full or Toeplitz structure. The proposed Dugh framework is encompassing efficient optimization algorithms for jointly estimating these distributions within a hierarchical Bayesian inference and MM optimization framework. Interestingly, we could theoretically prove convergence for the proposed update rules yielding estimates of the spatial and temporal covariances. In careful simulation studies, we have demonstrated that the inclusion of both spatial and temporal model parameters in the model indeed leads to significantly improved reconstruction performance. Finally, the utility of our algorithms is showcased in challenging real-world applications by reconstructing expected sources from real M\/EEG data based on very few experimental trials.\n\nThe M\/EEG BSI problem has been previously framed as a multi-task regression problem in the context of Type-I learning methods  \\citep{massias2018generalized,bertrand2019handling}. \\citet{bertrand2019handling} proposed a method to extend the group Lasso class of algorithms to the multi-task learning case. Their method shows promising results for large numbers of trials but is not competitive for smaller sample sizes, presumably due to not modeling the temporal structure of the sources. In contrast, Dugh is a Type-II method that learns the full spatio-temporal prior source distribution as part of the model fitting. Note that the assumption of spatially independent noise made here can be readily dropped, as in  \\citep{hashemi2021joint}. A number of alternative approaches have been proposed to estimate the spatio-temporal correlation structure of the \\emph{noise}\n\\citep{huizenga2002spatiotemporal,de2002estimating,bijma2003mathematical,de2004maximum,jun2006spatiotemporal}. These works, however, do not estimate the noise characteristics as part of the source reconstruction problem on the same data but require separate noise recordings. Our proposed algorithm substantially differs in this respect, as it learns the full noise covariance jointly with the brain source distribution. This joint estimation perspective is very much in line with the end-to-end learning philosophy as opposed to a step-wise independent estimation process that can give rise to error accumulation. \n\n\n\n\n \n\n\nWith respect to the limits of our Dugh framework, we would like to note that using the same temporal correlation prior for noise and sources is a potentially restricting assumption in our modeling. It is made here to achieve tractable inference due to the fact that the spatio-temporal statistical model covariance matrix can be formulated in a separable form, i.e., $\\Tilde{\\bm \\Sigma}_{\\bf y} = {\\bm \\Sigma}_{\\bf y} \\otimes {\\bf B}$. Although we have demonstrated empirically that the reconstruction results of our proposed learning schemes are fairly robust against violating this assumption, this constraint may be further relaxed by exploiting eigenvalue decomposition techniques presented in \\citep{rakitsch2013all,wu2018learning}. Another potentially limiting assumption in our model is to assume Gaussian distributions for the sources and noise. Although Gaussian priors are commonly justified, the extension of our framework to heavy-tailed noise distributions, which are more robust to outliers, is a direction of future work. \\ali{Regarding the societal impact of this work, we note that we intend to solve inverse problems that have non-unique solutions, which depend on explicit or implicit assumptions of priors. These have applications in basic, clinical, and translational neuroscience imaging studies. Our use of a hierarchical empirical Bayesian framework allows for explicit specifications of the priors in our model that are learned from data, and a more clear interpretation of the reconstructions with reduced bias. Nevertheless, in our algorithms, we assume sparse spatial priors, and in scenarios where this assumption embodied in the priors is incorrect, the resulting reconstructions will be inaccurate. Users and neuroscientists will need to be cognizant of these issues.} \n\n\n\n\n\\label{sec:conclusion}\nIn conclusion, we could derive novel flexible hierarchical Bayesian algorithms for multi-task regression problems with spatio-temporal covariances. Incorporating prior knowledge, namely, constraining the solutions to Riemannian manifolds, and using ideas of geodesic convexity, circulant embeddings, and majorization-minimization, we derive inference update rules and prove convergence guarantees. The proposed Dugh algorithms show robust and competitive performance both on synthetic and real neural data from M\/EEG recordings and thus contribute to a well-founded solution to complex inverse problems in neuroscience.  \n\n\n\\section*{Broader Impact}\nIn this paper, we focused on sparse multi-task linear regression within the hierarchical Bayesian regression framework and its application in EEG\/MEG brain source imaging. Our algorithm, however, is suitable for a wider range of applications. The same concepts used here for full-structural spatio-temporal covariance learning could be employed in other contexts where hyperparameters like kernel widths in Gaussian process regression \\citep{wu2019dependent} or dictionary elements in the dictionary learning problem \\citep{dikmen2012maximum} need to be inferred from data. Our proposed method, Dugh, may also prove useful for practical scenarios in which model residuals and signals are expected to be correlated, e.g., probabilistic canonical correlation analysis (CCA) \\citep{bach2005probabilistic}, spectral independent component analysis (ICA) \\citep{ablin2020spectral}, direction of arrival (DoA) and channel estimation in massive Multiple Input Multiple Output (MIMO) systems \\citep{prasad2015joint,gerstoft2016multisnapshot,haghighatshoar2017massive}, robust portfolio optimization in finance \\citep{feng2016signal}, covariance matching and estimation \\citep{werner2008estimation,tsiligkaridis2013covariance,zoubir2018robust,benfenati2020proximal,ollila2020shrinking}, graph learning \\citep{kumar2020unified}, thermal field reconstruction \\citep{hashemi2016efficient,flinth2017thermal,flinth2018approximate}, and brain functional imaging \\citep{wei2020bayesian}.\n\n\n\\begin{ack}\nThis result is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 758985).\nAH acknowledges scholarship support from the Machine Learning\/Intelligent Data Analysis research group at Technische Universit{\\\"a}t Berlin. He further wishes to thank the  Charit{\\'e} -- Universit{\\\"a}tsmedizin Berlin, the Berlin Mathematical School (BMS), and the Berlin Mathematics Research Center MATH+ for partial support. CC was supported by the National Natural Science Foundation of China under Grant 62007013. KRM was funded by the German Ministry for Education and Research as BIFOLD -- Berlin Institute for the Foundations of Learning and Data (ref.\\ 01IS18025A and ref.\\ 01IS18037A), and the German Research Foundation (DFG) as Math+: Berlin Mathematics Research Center  (EXC 2046\/1, project-ID: 390685689),\nInstitute of Information \\& Communications Technology Planning \\& Evaluation (IITP) grants funded by the Korea Government (No. 2019-0-00079,  Artificial Intelligence Graduate School Program, Korea University). SSN was funded in part by National Institutes of Health grants (R01DC004855, R01EB022717, R01DC176960, R01DC010145, R01NS100440, R01AG062196, and R01DC013979), University of California MRPI MRP-17\u2013454755, the US Department of Defense grant (W81XWH-13-1-0494).\n\\end{ack}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe gravity duals of gauge theories with running coupling constants have received\nconsiderable attention in the last few\nyears. The original gauge\/gravity duality \\cite{Mal-1}, \\cite{Witt-1} deals exclusively with theories that have no\nrunning of the coupling constants, or with theories that have some running of the coupling constants but eventually\nfall into fixed point surfaces, for example \\cite{klebwitt}. The first kind of dualities that consider the actual\nrunning of the coupling constants leading to, say, confining theories were discussed some time back in\n\\cite{KS}, \\cite{vafa}, \\cite{MN} and its extension to include fundamental flavors in \\cite{ouyang}. The\ntype IIA brane constructions for theories like \\cite{klebwitt} were first discussed in \\cite{dasmukhi}, and for\ntheories with running couplings were discussed in \\cite{ohtatar}. In fact in the fourth reference of \\cite{ohtatar}\nthe precise distinctions between \\cite{KS} and \\cite{vafa} were pointed out in details.\n\nIn recent times we have seen many new advantages of studying theories like \\cite{KS} and \\cite{vafa} that deal with\nrunning couplings. The confining behavior of these theories in the far IR is of course very powerful in extending them\nto more realistic scenarios like high temperature QCD \\cite{FEP, jpsi}. The cascading nature of these theories allow\nthem to remain strongly coupled throughout the RG flow from UV to IR, and therefore {\\it supergravity} duals can\ndescribe the full dynamics of the corresponding gauge theories. For the Klebanov-Strassler (KS) theory \\cite{KS} even the\nfull UV completion, that allow no Landau poles or UV divergences of the Wilson loops, {\\it can} be achieved by\nattaching an UV cap to the KS geometry\\cite{FEP}. An example \nof the full UV completion of the KS geometry both at\nzero and non-zero temperatures has been recently accomplished in \\cite{jpsi}. The UV cap therein is given by an\nasymptotic AdS space that, in the dual gauge theory, will allow for an asymptotic conformal behavior in the UV and\nlinear confinement in the far IR.\n\nFor the model studied by Vafa \\cite{vafa} the full UV completion would be more non-trivial. We expect the UV to be a\nsix-dimensional theory instead of a four-dimensional one. A six-dimensional UV completion that allows no Landau poles\nin the presence of fundamental flavors has not been constructed so far.\nIn fact a proper study of fundamental flavors\n{\\it a-la} \\cite{ouyang} is yet to be done for this case. The F-theory \\cite{vafaF, DM, senF}\nembedding of this model would be crucial\nin analysing the full UV completion. However some aspects of an intemediate UV behavior, for example cascading dynamics,\nhave been discussed in the past \\cite{katzvafa} where the cascade is likened to an infinite sequence of flop transitions.\nThe IR dynamics of the theory where we expect geometric transition to happen is actually the last stage\nin this sequence of transformations\nwhere the flop is immediately followed by a conifold transition. At this point we should\nexpect the wrapped D5-branes to be completely\nreplaced by fluxes (at least in the absence of fundamental flavors) \\cite{katzvafa}.\nWhat happens in the presence of fundamental flavors is rather subtle, and we will not discuss this here anymore. In fact\nwe will only concentrate on the last stage of the transition, namely, the geometric transition in this paper. The\nintermediate cascading dynamics or the UV completion will be discussed elsewhere \\cite{toappear}.\n\nSince the geometric transition leads to a confining theory, the corresponding gauge dynamics is strongly coupled.\nTherefore the physics of this transition can be captured exclusively by supergravity backgrounds. In some\nof our earlier works \\cite{gtpapers} we managed to study this purely using the supergravity backgrounds in the {\\it local}\nlimit, meaning that the sugra background was studied around a specific chosen point in the internal six-dimensional space.\nThe reason for this was the absence of a known globally defined\nsupergravity solution of the wrapped D5-branes on the two-cycle of the\nresolved conifold. The only known global solution i.e \\cite{pandoz} was unfortunately not supersymmetric (see\n\\cite{cvetic, anke} for details) although it satisfied the type IIB EOMs. In this paper, among other things, we will\nbe able to solve this problem and provide a fully supersymmetric globally defined solution for the wrapped D5-branes on a\ncertain resolved conifold. What we will argue soon is that the resolved conifold should have a non-K\\\"ahler metric to\nallow for supersymmetric solutions. This non-K\\\"ahlerity appears exactly from the back-reactions of the wrapped D5-branes.\n\nDespite the absence of supersymetric solutions, in \\cite{gtpapers} we managed to show, at least locally, the\nfull geometric transitons in type II theories. The gravity duals for the IR confining gauge theories on\nthe wrapped D6-branes in type IIA and wrapped\nD5-branes in type IIB were completely captured by non-K\\\"ahler deformations of the resolved and the deformed conifolds\nrespectively. In this paper we will show that globally under some simplifying assumptions\nthis conclusion remains unchanged, but generically\nthese manifolds would become non-geometric (see \\cite{halmagyi} for a recent discussion on this). In the following\nsub-section we will briefly review the state of geometric transition using local supergravity analysis before we proceed\nto compute the full global picture.\n\n\\subsection{Geometric transition and supersymmetric solution}\n\nLet us begin with a bit of historical notes.\nThe original study of open-closed string duality in type II theory\nstarts with D6 branes wrapping a three cycle of a non-compact\ndeformed conifold. Naively one might expect the deformed conifold\nto be a complex K\\\"ahler manifold with a non-zero three cycle.\nHowever as discussed earlier in \\cite{gtpapers} this is not quite correct,\nand the manifold that actually would solve the string equations of\nmotion is a non-K\\\"ahler deformation of the deformed conifold. It\nalso turns out that the manifold has no integrable complex\nstructure, but only has an almost complex structure. This is\nconsistent with the prediction of \\cite{vafaF}.\n\nHowever, as one may recall, in all our earlier papers we managed to study only the\n{\\it local} behavior of the manifolds. This is because the full global picture was\nhard to construct, and any naive procedure always tend to lead to non-supersymmetric\nsolutions. In deriving the local metric, we took a simpler model where all\nthe spheres were replaced by tori with periodic coordinates ($x,\n\\theta_1$) and ($y,\\theta_2$). The coordinate $z$ formed a\nnon-trivial $U(1)$ fibration over the $T^2$ base. The replacement\nof spheres by two tori was directly motivated from the\ncorresponding brane constructions of \\cite{dasmukhi}, where\nnon-compact NS5 branes required the existence of tori instead of\nspheres in the T-dual picture.\n\nLocally\nthe non-K\\\"ahlerity of\nthe underlying metric can be easily seen from its explicit\nform:\n\\bg\\label{iiamet}\n&&ds_{IIA}^2 = g_1~\\left[(dz -\n{b}_{z\\mu}~dx^\\mu) + \\Delta_1~{\\rm cot}~\\hat\\theta_1~(dx -\nb_{x\\theta_i}~d\\theta_i) + \\Delta_2~{\\rm cot}~ \\hat\\theta_2~(dy -\nb_{y\\theta_j}~d\\theta_j)+ ..\\right]^2\\nonumber\\\\\n&& + g_2~ {[} d\\theta_1^2\n+ (dx - b_{x\\theta_i}~d\\theta_i)^2] + g_3~[ d\\theta_2^2 + (dy -\nb_{y\\theta_j}~d\\theta_j)^2{]} +  g_4~{\\rm sin}~\\psi~{[}(dx -\nb_{x\\theta_i}~d\\theta_i)~d \\theta_2 \\nonumber\\\\\n&& ~~~~~~~~~ + (dy -\nb_{y\\theta_j}~d\\theta_j)~d\\theta_1 {]} + ~g_4~{\\rm\ncos}~\\psi~{[}d\\theta_1 ~d\\theta_2 - (dx - b_{x\\theta_i}~d\\theta_i)\n(dy - b_{y\\theta_j}~d\\theta_j)]\n\\nd\nwhere the coefficients $g_i$\nand the coordinates $\\theta_i, \\hat\\theta_i$ etc. are defined in\n\\cite{gtpapers}. The background has\nnon-trivial gauge fields (that form the sources of the wrapped D6\nbranes) and a non-zero string coupling (which could in principle\nbe small).\n\nExistence of such an exact supergravity background helps us to obtain\nthe corresponding mirror type IIB background. One would expect that\nthis can be easily achieved\nusing the mirror rules of \\cite{syz}. It turns out however that\nthe mirror rules of \\cite{syz}, as discussed in \\cite{gtpapers}, do not\nquite suffice.\nA detailed analysis of this is\npresented in \\cite{gtpapers}.\nAs discussed therein, we have to be careful\nabout various subtle issues while doing the mirror transformations:\n\n\\noindent (a) The mirror rules of \\cite{syz} tells us that {\\it any} Calabi-Yau\nmanifold with a mirror admits, at least {\\it locally}, a $T^3$ fibration over a\nthree dimensional base. This seems to fail for the deformed\nconifold as it does not possess enough isometries to represent it as a $T^3$\nfibration.\nOn the other hand, a resolved conifold does have a well defined $T^3$\ntorus over a three-dimensional base, which can be exploited to get the mirror (see also\n\\cite{agavafa}). It also turns out that the $T^3$ torus is a lagrangian\nsubmanifold, so a mirror transformations will not break any\nsupersymmetry.\n\n\\noindent (b) Viewing the mirror transformation naively as three T-dualities\nalong the $T^3$ torus {\\it does~not} give the right mirror metric. There are\nvarious issues here. The rules of \\cite{syz} tell us that the mirror transformation\nwould only work when the three dimensional base is very large. The configuration that\nwe have is exactly opposite of the case \\cite{syz}. Recall that\nour configuration lies at the end of a much larger cascading theory.\nBy UV\/IR correspondences, this means that the\nbase manifold is very small. Furthermore we are at the {\\it tip} of the\ngeometric transition and therefore we have to be in a situation with very\nsmall base (in fact very small fiber too). In \\cite{gtpapers} we showed\nthat we could still apply the rules of \\cite{syz} if we impose a non-trivial large\ncomplex structure on the underlying $T^3$ torus. The complex structure\ncan be integrable or non-integrable. Using an integrable complex structure, we\nshowed in \\cite{gtpapers} that we can come remarkably close to getting\nthe right mirror metric. Our conjecture there was that if we use a\nnon-integrable complex structure we can get the right mirror manifold.\n\nIt seems therefore natural to start with the manifold that exhibits three isometry\ndirections --- the resolved conifold. We can, however, not use the metric for D5\nbranes wrapping the $S^2$ of a resolved conifold as derived in \\cite{pandoz}, because\nit breaks all supersymmetry \\cite{cvetic}.  The metric that we\nproposed in \\cite{gtpapers} (where we kept the harmonic functions undetermined) is very\nclose to the metric of \\cite{pandoz} but differs in some subtle way:\n\n\\noindent (a) The type IIB resolved conifold metric that we proposed in\n\\cite{gtpapers} is a D5 wrapping a two cycle that {\\it preserves} supersymmetry.\nWe will discuss this issue in more detail below.\n\n\\noindent (b) As explained in \\cite{gtpapers}, our IIB manifold\nalso has seven branes (and possibly orientifold planes) along with\nthe type IIB three-form fluxes. The metric constructed in \\cite{pandoz}\ndoesn't have seven branes but allows three-form fluxes.\n\n\\noindent The {\\it local} behavior of the type IIB metric is expressed in terms\nof non-trivial complex structures $\\tau_1$ and $\\tau_2$ as $dz_1 = dx - \\tau_1 d\\theta_1$ and\n$dz_2 = dy - \\tau_2 d\\theta_2$. The local metric then reads\n\\bg\\label{iibmet}\nds^2 = (dz + \\Delta_1 ~ {\\rm cot}~\\theta_1 ~ dx  +\n\\Delta_2 ~{\\rm cot}~\\theta_2 ~dy)^2 + \\vert dz_1 \\vert^2 + \\vert dz_2\\vert^2\n\\nd\nwhere all the warp factors can locally be absorbed in to the coordinate differentials.\nIn this formalism the metric may naively look similar to the one studied in\n\\cite{pandoz} but the global picture is completely different from the\none proposed by \\cite{pandoz}. Our aim in this paper is therefore two-fold: to determine the full global picture (at\nleast without the inclusion of UV caps), and to follow the duality cycle that will lead us to analyse\ngeometric transitions in type II theories.\n\n\\subsection{Organisation of the paper}\n\nThe paper is organised as follows. In section 2\nwe start with geometric preliminaries about the resolved conifold and the\nblown-up conifold.  We then discuss the mathematical construction of a non-K\\\"ahler resolved\nconifold with an $SU(3)$ structure, following \\cite{llt}.\nThe components of the metric of this $SU(3)$ structure\n(which automatically satisfy the\ntorsional equations) are given in {\\bf Appendix 1}, and we use\nthese components to determine the metric of the wrapped D5-branes on the non-K\\\"ahler resolved conifold in\nsection 4.2. Existence of an $SU(3)$ structure will guarantee that the solution we get is supersymmetric, and we\ndiscuss the issue of supersymmetry further in section 4.1.\n\nSections 4.3 to 4.6 are the main sections where we compute the full geometric transitions in the global framework\nusing duality cycle that were used earlier in \\cite{gtpapers}. In the full global picture the fluxes are very involved\ncompared to the local picture. We managed to work out all the fluxes in the type IIA mirror set-up. These flux\ncomponents are given in {\\bf Appendix 2}. It is necessary to track these fluxes because they would eventually determine\nthe non-K\\\"ahler fibration structure in type IIB theory discussed in section 4.6. We discuss the components of the\nmetric in type IIB after geometric transition in {\\bf Appendix 3}.\n\nIn the above discussion we have briefly alluded to the fundamental flavors. In our model they appear from orientifolding\nthe resolved conifold. The orientifolding is subtle, and we give a brief discussion of this in section 3. This\norientifolding will allow us to add seven-branes in type IIB and six-branes in the mirror type IIA picture. The\nseven-branes should be embedded as in \\cite{pandoz} or \\cite{kuperstein}.\n\nWe end with a conclusion and a short discussion on the topics that we will study in \\cite{toappear}.\n\n\n\\section{Mathematical constructions and $SU(3)$ structures}\n\nThe blown-up conifold describes one of the topologies that we will be using.  We will describe it as well as its relation to the resolved conifold which is more familiar\nin string theory.  These ideas are\nwell known in algebraic geometry.  Rather than starting with the standard algebro-geometric constructions, we instead begin with a description more suited\nto the description of an $SU(3)$ structure.  The algebro-geometric description\nwill follow.\n\n\\subsection{The blown-up conifold}\n\\label{blowup}\n\nLet us begin by explaining how the blown-up conifold arises for us.  The\nconifold is a cone over $S^3\\times S^2$ \\cite{candelas}, arising as a quotient\n$(S^3\\times S^3)\/U(1)$, with the $U(1)$ diagonally embedded and on each\nfactor identified with the $U(1)$ of the Hopf fibration $S^3\\to S^2$.\nIf we take the self-product of the Hopf fibration\n$S^3\\times S^3\\to S^2\\times S^2$ and mod out by the diagonal $U(1)$, we are\nleft with a $U(1)$ fibration $S^3\\times S^2\\to S^2\\times S^2$ which restricts\nto the Hopf fibration over either $S^2$ factor.\n\nIn the geometry we will be using which supports the metric (\\ref{susybg}),\nthe internal space is a 6-manifold, the total space of a complex line bundle\n$L$ with base $S^2\\times S^2$ which is described in spherical coordinates\n$(\\theta_i,\\phi_i)$ for $i=1,2$.  The fiber is described by an angular coordinate $\\psi$ describing the nontrivial $U(1)$ bundle\nover $S^2\\times S^2$ just described.  The $U(1)$ bundle is completed to\na complex line bundle by introducing the radial coordinate $r$.\n\nThe topology of either the line bundle or the $U(1)$ bundle is described completely by its Chern class on $S^2\\times S^2$.\nAs noted above, the\n$U(1)$ bundle restricted to either $S^2$ is the Hopf bundle $S^3$, whose Chern class on $S^2$ is well known to have degree $-1$.  So on\n$S^2\\times S^2$, we learn that $L$ has degrees $(-1,-1)$.\n\nWe can now identify $S^2$ with the complex projective line $\\bfP^1$ and switch to the language of algebraic geometry, whereby we see that the internal manifold $X$ is the total space of the line\nbundle ${\\cal O}(-1,-1)$ on $\\bfP^1\\times\\bfP^1$.\n\nThis manifold appears as\nthe blown-up conifold in algebraic geometry.  Rather than refer to known results, we prefer to directly identify $X$ with the blown-up conifold by describing the map from $X$ to the conifold which shrinks $\\bfP^1\\times\\bfP^1$ (identified with the zero section of $L$) to the conifold point.\n\nWe introduce homogeneous coordinates $(u^1,u^2)$ and $(v^1,v^2)$ on the respective\n$\\bfP^1$'s, and identify sections of $L$ with functions on $\\bfP^1\\times\\bfP^1$ which are homogeneous of degree $-1$ with respect to $(u^1,u^2)$ as well as with respect to $(v^1,v^2)$.  Thus a point of $X$ can be described by homogeneous coordinates\n$(u^1,u^2,v^1,v^2,s)$, where $s$ is thought of as a section of $L$.  The homogeneity is\ndescribed by two $\\bfC^*$ actions whose respective weights are\n$(1,1,0,0,-1)$ and $(0,0,1,1,-1)$.\n\nThe map from $X$ to the conifold is realized by the map from $X$ to $\\bfC^4$ given by\n\n\\bg\\label{bdtoconifold}\n(u^1,u^2,v^1,v^2,s)\\mapsto (u^1v^1s,u^2v^2s,u^1v^2s,u^2v^1s).\n\\nd\n\nIf we introduce coordinates $(x_1,\\ldots,x_4)$ on $\\bfC^4$, we see that the image of\n$X$ satisfies the equation\n\\bg\nx_1x_2-x_3x_4=0\n\\nd\nof the conifold.  Identifying $\\bfP^1\\times\\bfP^1$ with the zero section $s=0$, we see\nthat $\\bfP^1\\times\\bfP^1$ is collapsed to the conifold point $(0,0,0,0)$ as claimed.\n\nWhen described by homogeneous coordinates as above, Calabi-Yau manifolds are characterized by the condition that the sum of the weights is zero for any $\\bfC^*$. Since the sum of the weights is one for either $\\bfC^*$, we conclude that the blown-up conifold is not a Calabi-Yau manifold.  We will also check this\ndirectly in the next section by the adjunction formula.\n\n\\subsection{The resolved conifold}\\label{rescon}\n\nWe can relate the blown-up conifold to the more familiar resolved conifold.\nRather than blow down $\\bfP^1\\times\\bfP^1$ to the conifold point as in (\\ref{bdtoconifold}), we can instead partially blow down $\\bfP^1\\times\\bfP^1$ by\nprojecting to one $\\bfP^1$. This gives the usual resolved\nconifold.\n\n\\smallskip\nUsing the coordinates of the blown-up conifold introduced in Section~\\ref{blowup}, the partial blowdown is described by\n\n\\bg\\label{bdtoresolved}\n(u^1,u^2,v^1,v^2,s)\\mapsto\n(u^1,u^2,v^1s,v^2s),\n\\nd\nso that $\\bfP^1\\times\\bfP^1$, identified with $s=0$ as before, is mapped to\n$(u^1,u^2,0,0,0)$, and $\\bfP^1\\times\\bfP^1$ is projected to the first coordinate, as\nclaimed. We let $(z^1,z^2,z^3,z^4)$ be homogeneous coordinates on the image of (\\ref{bdtoresolved}).\n\nOnly one $\\bfC^*$ remains nontrivial on $(z^1,z^2,z^3,z^4)$, with weights $(1,1,-1,-1)$.  The image is the resolved conifold, the total space of ${\\cal O}(-1)\\oplus{\\cal O}(-1)$ on $\\bfP^1$.  The coordinates $(z^1,z^2)$ can be identified\nwith the homogeneous coordinates of $\\bfP^1$, while $z^3$ and $z^4$ are identified with\nsections on the respective copies of ${\\cal O}(-1)$.   The resolved conifold is of course\nCalabi-Yau, which can be seen since the sum of the weights is 0.\n\n\\bigskip\nWe remark that we have taken a circuitous path to get from\nthe blown-up conifold to the more familiar resolved conifold, but we\nhave reached the usual descriptions of the resolved conifold as either\nthe total space of the bundle ${\\cal O}(-1)\\oplus{\\cal O}(-1)$ on $\\bfP^1$\nor as a toric quotient of $\\bfC^4$ by $\\bfC^*$ with weights $(1,1,-1,-1)$.\n\n\\bigskip\nHowever, it is not the Calabi-Yau structure that is relevant in our model, but rather\na non-K\\\"ahler structure.  To construct this non-K\\\"ahler structure, the holomorphic\nhomogeneous coordinates are not particularly useful.\nAt times it will be useful to describe the resolved conifold as a\nsymplectic quotient, or equivalently as the space of vacua of a $U(1)$ gauge theory with four scalar fields with $U(1)$ charges $(1,1,-1,-1)$, whose\nvevs are identified with $(z^1,z^2,z^3,z^4)$.  There is an FI term $u$, which we take to be positive.  If we take $u<0$, we get the flopped version of the resolved conifold.\n\nSo the resolved conifold can be described by\n\\bg\\label{symprescon}\n|z^1|^2+|z^2|^2-|z^3|^2-|z^4|^2=u\n\\nd\nmodulo the $U(1)$ action.\n\nWe can see directly from this description that the resolved conifold is smooth.  We can for\nexample describe the patch in which $z^1\\ne0$ by the complex coordinates $(z^2,z^3,z^4)$\nand solve (\\ref{symprescon}) by\n\\bg\\label{resconcoords}\nz^1=\\sqrt{u-|z^2|^2+|z^3|^2+|z^4|^2}.\n\\nd\nNote that in (\\ref{resconcoords}) we have fixed the gauge by choosing the positive real\nsolution of (\\ref{symprescon}), so that $(z^2,z^3,z^4)$ are in fact local coordinates.\nHowever, they are in no sense to be considered as holomorphic coordinates since\n(\\ref{resconcoords}) is not holomorphic.\n\n\\bigskip\nWe now turn to the holomorphic description.  Start with\nthe conifold singularity $X$ with\nequation\n$$x_1x_2-x_3x_4=0$$\nwhich we denote by $f=0$.\nThe usual resolved conifold $X'$ can be described as the submanifold of\n$\\bfC^4\\times \\bfP^1$ with equations which we informally write as\n$$\\frac{x_1}{x_3}=\\frac{x_4}{x_2}=\\frac{y_2}{y_1},$$ or more formally as\n$$x_1x_2-x_3x_4=0,\\ x_1y_1=x_3y_2,\\ x_4y_1=x_2y_2.$$  In the above,\n$(y_1,y_2)$ are the homogeneous coordinates on $\\bfP^1$.\n\nIf $x=(x_1,x_2,x_3,x_4)\\ne (0,0,0,0),$ then there is a unique solution for\n$y=(y_1,y_2)$, so that $X$ and $X'$ are isomorphic away from the origin.  If\nhowever, $x=0$, then $y$ is unconstrained and we replace the origin by\na $\\bfP^1$ to form $X'$ from $X$.\n\nThis $\\bfP^1$ can be flopped to produce another resolved conifold $X''$.\nThe flop can be realized directly by the equations\n$$\\frac{x_1}{x_4}=\\frac{x_3}{x_2}=\\frac{y_2}{y_1}.$$\n$X'$ and $X''$ are isomorphic in this local model, but in global models\n$X'$ and $X''$ containing a resolved conifold and its flop respectively, the\nCalabi-Yaus $X'$ and $X''$ need not be isomorphic.\n\n\\bigskip\nIn this paper, we use the model where both $\\bfP^1$'s are introduced\nsimultaneously.  This is accomplished by the algebro-geometric construction\nof blowing up the conifold, which complements our description in Section 2.1. We introduce a $\\bfP^3$ with homogeneous\ncoordinates $(y_1,y_2,y_3,y_4)$ and the blowup $\\tilde{X}$ is constructed\nas the submanifold of $\\bfC^4\\times\\bfP^3$ with equation informally expressed as\n$$(x_1,x_2,x_3,x_4)=(y_1,y_2,y_3,y_4).$$  As before, if $x\\ne0$ then there is\na unique solution for $y$ and so $\\tilde{X}$ is isomorphic to $X$ away from\nthe origin.  If $x=0$, there are more solutions, but now there is a\nconstraint $y_1y_2=y_3y_4$.  This is a quadric surface in $\\bfP^3$, isomorphic\nto $\\bfP^1\\times \\bfP^1$ by the isomorphism\n$$\\left((u_1,u_2),(v_1,v_2)\\right)\\mapsto (u_1v_1,u_2v_2,u_1v_2,u_2v_1).$$\n\nNote that the blown-up conifold\n$\\tilde{X}$ is not Calabi-Yau, but we can realize\nthis within string theory by turning on an appropriate flux.\n\n\\smallskip\nWe consider the holomorphic 3-form $\\Omega$ on $X$ given by the usual\nresidue construction\n$$\\Omega=\\frac{dx_2\\wedge dx_3\\wedge dx_4}\n{\\partial f\/\\partial x_1}=\\frac{dx_2\\wedge dx_3\\wedge dx_4}{x_2}$$ and pull it back to a holomorphic\n3-form $\\tilde{\\Omega}$ on $\\tilde{X}$.\nTo see that $\\tilde{X}$ is not Calabi-Yau, it suffices to show that\n$\\tilde\\Omega$ vanishes somewhere on $\\tilde{X}$. It\nsuffices to compute in one coordinate patch, say where $x_1\\ne0$.  In\nthis patch, $(x_1,y_3,y_4)$ are local coordinates.  To see this, we may\nset $y_1=1$, and then\nwe use $y_i=x_i\/x_1$ for $i=2,3,4$ to compute\n$$x_2=x_1y_3y_4,\\ x_3=x_1y_3,\\ x_4=x_1y_4.$$\nIn these coordinates, we have $\\tilde{\\Omega}=x_1dx_1\\wedge dy_3\\wedge dy_4$,\nwhich clearly vanishes on the surface $x_1=0$.\nThus $\\tilde{X}$ is not Calabi-Yau.\n\nFor later use, note that in this coordinate patch, if $x_1=0$ then necessarily\n$x_2=x_3=x_4=0$ as well.  Thus $x_1=0$ is the local equation of the\nexceptional $\\bfP^1\\times\\bfP^1$  of $\\tilde{X}$, which we denote by $E$.\n\nAlternatively we can see the same result by the adjunction formula \\cite{gh}, which\nsays that for any hypersurface $H$ in a complex manifold $M$, we have\n$K_H=(K_M+[H])\\mid_H$.\n\nWe realize $\\tilde{X}$ as a hypersurface in the blowup of $\\bfC^4$ at a point\nand apply the adjunction formula.\n\nLet $E'$ be the $\\bfP^3$ which is the exceptional divisor of the\nblowup $Z$ of $\\bfC^4$ at the origin; then $K_Z=3E'$ \\cite{gh}.  Then the\nadjunction formula yields\n$$K_{\\tilde{X}}=\\left(K_Z+\\tilde{X}\\right)|_{\\tilde{X}}.$$\n\nNow $\\tilde{X}$ is obtained by subtracting off the exceptional divisor from\nthe pullback of $X$ via the blowup.  Since $X$ has a multiplicity~2 singularity\nat the origin, the pullback of $X$ actually contains the exceptional\ndivisor $E'$ with multiplicity~2.\nSince $2E'$ has to be subtracted off to obtain $\\tilde{X}$, we conclude\nthat $\\tilde{X}$  has divisor class $-2E'$.  We\nconclude that\n$$K_{\\tilde{X}}=\\left(3E'-2E'\\right)|_{\\tilde{X}}=E'|_{\\tilde{X}}=E,$$\nso that $K_{\\tilde{X}}$ is nontrivial and $\\tilde{X}$ is not Calabi-Yau.\n\nThis is consistent with the explicit calculation.  Since $x_1=0$ defines\nthe exceptional divisor $E$, the fact that $\\tilde{\\Omega}$ vanished precisely\non $E$ tells us that $K_{\\tilde{X}}=E$.\n\n\\subsection{$A_1$ fibered geometry}\n\nWe start with the blown up $A_1$ geometry fibered over the complex numbers\nwith parameter $x_4\\in\\bfC$.  The equation is just the $A_1$ equation\n\\bg\\label{a1}\nx_1x_2-x_3^2=0\n\\nd\nand the blowup is performed by introducing a $\\bfP^1$ with homogeneous\ncoordinates $(y_1,y_2)$ and imposing the equations\n$$\\frac{x_3}{x_1}=\\frac{x_2}{x_3}=\\frac{y_2}{y_1},$$\nor more formally\n\\bg\\label{resolveda1}\nx_3y_1=x_1y_2,\\qquad x_2y_1=x_3y_2.\n\\nd\nNote that $x_4$ does not appear explicitly, and can be interpreted as a\nparameter for the\nlocation of $\\bfP^1$.  The $\\bfP^1$ corresponding to $x_4=\\phi$ will be written\nas $C_\\phi$.\n\nFor later use, the normal bundle of $C_\\phi$ is $\\cO_{C_\\phi}\\oplus\n\\cO_{C_\\phi}(-2)$.\n\nWe now deform this geometry with deformation parameter $t$:\n\\bg\\label{a1deformed}\nx_1x_2-x_3^2+t^2x_4^{2n}=0\n\\nd\nand the blowup is performed by introducing a $\\bfP^1$ with homogeneous\ncoordinates $(y_1,y_2)$ and imposing the equations\n$$\\frac{x_3-tx_4^n}{x_1}=\\frac{x_2}{x_3+tx_4^n}=\\frac{y_2}{y_1}.$$\n\nThis geometry corresponds to the superpotential $W(\\phi)=t\\phi^{n+1}\/(n+1)$.\nNote that for $t=0$ we have $W(\\phi)\\equiv0$, and the curve $C_\\phi$ is\nholomorphic for all $\\phi$.  For $t\\ne0$, we have $W'(\\phi)=t\\phi^n$, and\n$C_\\phi$ only  persists holomorphically for $\\phi=0$.  To see this, the\nrequirement is\n$$x_3-t\\phi^n=x_1=x_2=x_3+t\\phi^n=0,$$\nwhich implies that $\\phi^n=0$.\n\nThe superpotential can be obtained by integrating the holomorphic\n3-form $\\Omega=dx_2dx_3dx_4\/x_2$ over a three chain $\\Gamma$\nconnecting $C_0$ to $C_\\phi$.  This can be reinterpreted in terms of\nthe relative homology class of $\\Gamma$.  The same relatively\ncohomology group can be realized after blowing up $C_0$ and $C_\\phi$.\n\nInitially putting $t=0$, the blowup of $C_0$ has exceptional divisor isomorphic\nto the Hirzebruch surface $F_2$.  A similar blowup can be performed on\n$C_\\phi$.\n\nThe Hirzebruch surface $F_2$ deforms if $\\phi=0$ but not otherwise.  If\n$n>1$, the deformed surface is still $F_2$.  If $n=1$, then the deformed\nsurface is $\\bfP^1\\times\\bfP^1$.\n\nIf desired, a toric description of the blowup can be given.  The $A_1$\nsurface singularity is a toric variety whose fan has a single\ntwo-dimensional cone with edges spanned by $(1,0)$ and $(1,2)$.  The\nsingularity gets resolved by inserting an extra edge $(1,1)=(1\/2)\n((1,0)+(1,2)$.  This resolved $A_1$ gets fibered over $\\bfC$ in the\nusual way: by adding another coordinate, appending a zero to the\ncoordinates of the vectors spanning the edges, and adding the new\ncoordinate vector.  Hence the fan has edges spanned by\n\n$$(1,0,0), (1,1,0), (1,2,0), (0,0,1).$$\n\nThe curve $C_0$ corresponds to the 2-dimensional cone spanned by $(1,1,0)$\nand $(0,0,1)$, so $C_0$ gets blown up by inserting a new edge spanned by\n$(1,1,1)=(1,1,0)+(0,0,1)$.  In summary, the toric variety has edges\n\n$$(1,0,0), (1,1,0), (1,2,0), (0,0,1), (1,1,1).$$\n\n\n\\subsection{$SU(3)$ structure}\\label{su3}\nWe follow \\cite{llt} which gives a general procedure for constructing string\ncompactifications on toric varieties via a method for producing\n$SU(3)$ structures.  Any $SU(3)$ structure arises as a string compactification \\cite{torsion}.\n\nWe apply the method to the resolved conifold.  The method\nwas designed to apply to compact toric varieties, but since the method has a\nlocal character, it may be applied to the resolved conifold.  We set ourselves\nto that task.\n\nFor the convenience of the reader, we collect some facts about $SU(3)$\nstructures.\n\n\\bigskip\nAn $SU(3)$ structure on a 6-manifold $M$\nis determined by a complex decomposible 3-form $\\Omega$ and a real 2-form $J$ which are\nrelated by\n\\bg\\label{su3conds}\n\\Omega\\wedge J=0, \\qquad \\Omega\\wedge\\overline{\\Omega}=-\\frac{4i}3J\\wedge J\\wedge J.\n\\nd\n\nAt each point $p\\in M$, we can find complex cotangent vectors $dz^1,dz^2,dz^3$ so that $\\Omega=\ndz^1\\wedge dz^2\\wedge dz^3$ at $p$.  The first condition of (\\ref{su3conds}) and the\nreality of $J$ imply that we can ``diagonalize\" $J$, writing it as\n$$J=\\frac{i}2\\left(a_1dz^1\\wedge dz^{\\bar{1}}+a_2dz^2\\wedge dz^{\\bar{2}}+a_3dz^3\\wedge dz^{\\bar{3}}\\right)$$\nfor some real constants $a_i$, while retaining the form of $\\Omega$.  Then the second condition of (\\ref{su3conds})\nimplies that we can rescale the $dz^i$ so that\n\\bg\nJ=\\frac{i}2\\left(dz^1\\wedge dz^{\\bar{1}}+dz^2\\wedge dz^{\\bar{2}}+dz^3\\wedge dz^{\\bar{3}}\\right)\n\\nd\nwhile $\\Omega=dz^1\\wedge dz^2\\wedge dz^3$ still holds.\n\nThere is still the freedom of\nmultiplying the $dz^i$ by phases whose product is 1.\n\nThese coordinates determine a Euclidean metric $g_{i\\bar{j}}=i\/2$ at $p$, which\nis independent of the phase ambiguity.\n\nThis pointwise analysis extends to all of $M$, showing that the data of $\\Omega$ and\n$J$ satisfying (\\ref{su3conds}) completely determines a metric, the metric associated\nwith an $SU(3)$ structure.\n\nThere is an explicit procedure to calculate the metric directly from $\\Omega$ and $J$.\nThe first step is to calculate the (not necessarily integrable) complex structure $I$.  In pointwise Euclidean coordinates at $p$, the complex structure $I$ is the standard one.  But it can be computed intrinsically following \n\\cite{hitchin} as follows\\footnote{For more details on the following analysis, and also to connect to recent conifold \nliterature the readers may refer to \\cite{papat} and references therein.}.\n\nFirst define an unnormalized complex structure by\n\\bg\n\\tilde{I}_j^k=\n\\epsilon^{klmnop}\\left(\\mathrm{Re}\\Omega\\right)_{jlm}\\left(\\mathrm{Re}\\Omega\\right)_{nop},\n\\nd\nwhere $\\epsilon$ is the completely antisymmetric tensor.  It is shown in \\cite{hitchin}\nthat $\\tilde{I}^2$ is a diagonal matrix with negative real entries.  Then\n\\bg\nI=\\frac{\\tilde{I}}{\\sqrt{-\\frac16\\mathrm{Tr}\\tilde{I}^2}}\n\\nd\nis the desired complex structure.\n\nFrom here, the metric is determined by\n\\bg\ng_{ij}=I^k_jJ_{ki}.\n\\nd\n{}From the definitions, $\\Omega$ has type $(3,0)$ and $J$ has type $(1,1)$\nin the complex structure $I$.\nIf $(g,I)$ determines a K\\\"ahler structure, then $J$ is just the usual\nK\\\"ahler form.\n\n\\bigskip\nWe also quickly review the method of \\cite{llt} while applying it to the resolved\nconifold.  A general method is developed for producing $SU(3)$ structures on\nthree-dimensional complex toric varieties, by producing an $\\Omega$ and $J$ satisfying\n(\\ref{su3conds}).  The novel ingredient is to produce a $(1,0)$ form on complex\nEuclidean space satisfying certain conditions, and then then $SU(3)$ structure is\ndetermined by formulae.\n\nWe recall that the resolved conifold has been described as a quotient of $\\bfC^4$ by\nthe $\\bfC^*$ with weights $Q=(1,1,-1,-1)$.\nSo it suffices to produce a $(1,0)$ form\n$K=K_idz^i$ on $\\bfC^4$ satisfying (3.15), (3.16) and the normalization condition\n(3.18) of \\cite{llt}.  We interpret these conditions in concrete terms.\n\nThe condition (3.15) is equivalent to $Q^iz^iK_i=0$.  The condition (3.16)\nsays that $K$ has half the $U(1)$ charge as the holomorphic volume form\n$\\Omega_{\\bfC}=dz^1\\cdots dz^4$ of $\\bfC^4$, which is zero in this case.  Our normalization condition is $\\sum |K_i|^2 =1$, slightly different from\n(3.18) but we will adjust for it later.\n\nAn obvious solution is\n\n\\bg\\label{kdefn}\nK=\\frac{z^3dz^1+z^1dz^3+z^4dz^2+z^2dz^4}{|z|}.\n\\nd\n\nWe now compute the $SU(3)$ structure, following the formulas in \\cite{llt}.  We first construct\nthe standard $SU(3)$ structure on the resolved conifold before modifying it.  The\n$\\bfC^*$ action on $\\bfC^4$ is generated by\n\\bg\nV=z^1\\partial_{z^1}+z^2\\partial_{z^2}-z^3\\partial_{z^3}-z^4\\partial_{z^4}\n\\nd\nand then the standard Calabi-Yau 3-form is\n\\bg\n\\widetilde{\\Omega}&=&~i_V\\Omega_{\\bf C}\\\\\n&=&~z^1dz^2\\wedge dz^3\\wedge dz^4-z^2dz^1\\wedge dz^3\\wedge\ndz^4-z^3dz^1\\wedge dz^2\\wedge dz^4+z^4dz^1\\wedge dz^2\\wedge dz^3.\\nonumber\n\\nd\nThe K\\\"ahler form $\\tilde{J}$ of the resolved conifold arises by modifying the K\\\"ahler form of\n$\\bfC^4$\n\\bg\nJ_{\\bfC}=\\frac{i}2\\sum_{i=1}^4dz^i\\wedge dz^{\\bar{i}}\n\\nd\nby putting\n\\bg\n\\eta=\\bar{z}^1dz^1+\\bar{z}^2dz^2-\\bar{z}^3dz^3-\\bar{z}^4dz^4,\n\\nd\nwith the coefficients coming from the weights, and then putting\n\\bg\n\\tilde{J}=\\frac1{|z^2|}\\left(J_{\\bfC}-\\frac{i}2\\eta\\wedge\\bar{\\eta}\\right).\n\\nd\nNote that $\\eta$ respects the $U(1)$ action but not the $\\bfC^*$ action.  So from this\npoint forward, we have to understand differential forms as $U(1)$-invariant forms,\nsubject to the D-term constraint (\\ref{symprescon}).  In particular, the K\\'ahler\nform implicitly depends on the FI parameter $u$, as it must.\n\n\n\\bigskip\nWe now use $K$ to modify $\\tilde{\\Omega}$ and $\\tilde{J}$, effectively replacing\n$K$ by $\\bar{K}$ throughout.\n\n\\smallskip\nAn auxiliary $SU(2)$ structure is created, characterized by two-forms $\\omega$ and $j$ satisfying\n\\bg\\label{su2cond}\n\\omega\\wedge j=0,\\qquad \\omega\\wedge\\bar{\\omega}=2j\\wedge j,\n\\nd\nwhere\n\\bg\nj=\\tilde{J}-\\frac{i}2K\\wedge\\bar{K}\n\\nd\nand $\\omega$ is given by\n\\bg\\label{su2}\n\\omega_{ij}=-2\\bar{K_l}\\eta^{m\\bar{l}}\\tilde{\\Omega}_{mij},\n\\nd\nwhere $\\eta$ is the Euclidean metric on $\\bfC^4$. The prefactor of 2 here on the right-hand side of (\\ref{su2}) is not present in \\cite{llt} but is required by our\nnormalization condition for $K$.\n\nFrom here, we get a 2-parameter family of $SU(3)$ structures given by\n\\bg\nJ=aj-\\frac{ib^2}2K\\wedge\\bar{K},\\qquad\n\\Omega=ab\\bar{K}\\wedge\\omega.\n\\nd\nUsing (\\ref{su2cond}), it is immediate to see that $J$ and $\\Omega$ satisfy the conditions (\\ref{su3conds}) for an $SU(3)$ structure.\n\n\\smallskip\nThere is an extra phase parameter for $\\Omega$ in \\cite{llt}, but we supress it here\nsince the metric does not depend on this phase. In {\\bf Appendix 1} we write down all the components of the metric.\n\n\\section{T-duality and Orientifold Projection}\n\nThe  blown-up conifold that we discussed above, has a product structure of\n$\\bfP^1\\times\\bfP^1$ and we would like to discuss the T-duality to a IIA model. Our final aim is to see how\norientifold projection effects the T-duality.\nTo do so we first review the case of \\cite{orientifold} for the resolution of a deformed $A_2$ singularity.\nThis involves a\nnatural way to introduce two $\\bfP^1$ cycles.\n\n\\subsection{Brief review of the deformed $A_2$ case}\n\nLet us consider  the singular space $X_0$ realized as:\n\\begin{eqnarray}\n\\label{X0}\nxy=(u-t_0(z))(u-t_1(z))(u-t_2(z))~~, \\end{eqnarray} where $x,y,u,z$ are the\naffine coordinates of $C^4$ and  $t_j(z)$ are polynomials.\n\nThe affine variety (\\ref{X0}) has $A_1$ singularities at\n$x=y=0$ and $z$ one of the double points of the planar\nalgebraic curve: \\begin{eqnarray}\n\\label{Sigma_0}\n\\Sigma_0:~~(u-t_0(z))(u-t_1(z))(u-t_2(z))=0~~.  \\end{eqnarray}\nThe curve has 3 components $C_j$ given by\n$u=t_j(z)$, each a section of the $A_2$ fibration (\\ref{X0}).\n\nThe resolved space ${\\hat X}$ can be described explicitly as\nfollows. Consider two copies of\n$\\bfP^1$ with homogeneous coordinates $[u_1,u_2]$ and $[v_1,v_2]$, respectively, and local\naffine coordinates $\\xi_1:=u_1\/u_2, \\xi_2:=v_1\/v_2$.  Then\n${\\hat X}$ is realized as :\n \\bea\n\\label{hatX}\nu_2 (u-t_0(z))&=&u_1 x\\nn\\\\\nv_1(u-t_1(z))&=&v_2 y\n\\eea\n\nThe IIA construction is obtained by  performing a T-duality with respect to the\nfollowing $U(1)$ action on ${\\hat X}$, which we denote by\n \\begin{eqnarray}\n\\label{rho_global}\n([u_1,u_2], [v_1,v_2],z,u,x,y)\\stackrel{{\\hat \\rho}(\\theta)}{\\longrightarrow}\n([e^{-i\\theta}u_1,u_2],\n[v_1,e^{i\\theta}v_2],z,u,e^{i\\theta}x,e^{-i\\theta} y)~~.  \\end{eqnarray}\nThis projects as follows on the singular space $X_0$:\n\\begin{eqnarray}\n\\label{U1proj}\n(z,~u,~x,~y)\\stackrel{\\rho_0(\\theta)}{\\longrightarrow} (z,~u,~e^{i\\theta}x,~e^{-i\\theta}y)~~.\n\\end{eqnarray}\n\nIn the type IIB set-up, we consider the case\n\\bea\n\\label{tspec}\nt_0(z)&=&t(z)\\nn\\\\\nt_1(z)&=&t(-z)\\\\\nt_2(z)&=&-t(z)-t(-z)\\nn~~.  \\eea\n\nIn this situation, the resolution ${\\hat X}$ admits a $Z_2$ symmetry ${\\hat \\kappa}$ given by:\n\\begin{eqnarray}\n\\label{or_global}\n([u_1,u_2], [v_1,v_2],z,u,x,y)\\stackrel{\\hat \\kappa}{\\longrightarrow}\n([-v_2,v_1], [-u_2,u_1],-z,u,-y,-x) \\end{eqnarray} which acts\nas follows on the affine coordinates $\\xi_j$ of the\ntwo $\\bfP^1$ factors: \\begin{eqnarray} \\xi_1\\longleftrightarrow -1\/\\xi_2~~ \\end{eqnarray} and\nprojects to the following involution $\\kappa_0$ of $X_0$: \\begin{eqnarray}\n\\label{or_projected}\n(z,~x,~y,~u)\\stackrel{\\kappa_0}{\\longrightarrow} (-z,~-y,~-x,~u)~~.  \\end{eqnarray}\n\n\\subsection{The blown-up conifold case}\n\nWe now use  two $\\bfP^1$ cycles but in the blown-up conifold.\nIn this situation we have\n \\begin{eqnarray}\n\\label{rho_globalblowup}\n([u_1,u_2], [v_1,v_2],x_1,x_2,x_3,x_4)\\stackrel{{\\hat \\rho}(\\theta)}{\\longrightarrow}\n([e^{-i\\theta}u_1,u_2],\n[v_1,e^{i\\theta}v_2],x_1,x_2,e^{i\\theta}x_3,e^{-i\\theta} x_4)~~.  \\end{eqnarray}\nThis projects as follows on the singular space $X_0$:\n\\begin{eqnarray}\n\\label{U1projblowup}\n(x_1,~x_2,~x_3,~x_4)\\stackrel{\\rho_0(\\theta)}{\\longrightarrow} (x_1,~x_2,~e^{i\\theta}x_3,~e^{-i\\theta}x_4)~~.\n\\end{eqnarray}\nThe $Z_2$ symmetry ${\\hat \\kappa}$ is given in the homogeneous coordinates\n$(u^1,u^2,v^1,v^2,s)$ by:\n\\bg\n\\label{or_globalblowup}\n(u^1,u^2,v^1,v^2,s)\\stackrel{\\hat\\kappa}{\\longrightarrow}(-v^2,v^1,-u^2,u^1,s)\n\\nd\nwhich projects on $X_0$ as:\n\\bg\n(x_1,x_2,x_3,x_4) \\rightarrow (x_2,x_1,-x_3,-x_4)\n\\nd\nas is seen from (\\ref{bdtoconifold}).\n\nThe action on the homogeneous coordinates $y_i$ of $\\bfP^3$ is then given by\n\\begin{eqnarray}\n(y_1,y_2,y_3,y_4) \\rightarrow (y_2,y_1,-y_3,-y_4)\n\\end{eqnarray}\nand the action on $\\Omega$ is $\\Omega \\rightarrow - \\Omega$.\n\nWhat is the difference between the orientifold of \\cite{orientifold} and the current one? The orientifold of\n\\cite{orientifold} was an O5 orientifold extended on directions orthogonal to the D5 branes wrapped on\n$\\bfP^1$ cycle. After the T-duality, the involution determined an inversion of the radial direction of the\n$S^1$ which implied that the orientifold  became an O6 plane.\n\nIn the current example, the orientifold extends along the direction $s$ of the complex line bundle $L$ but also needs to\nas an inversion on the ${\\bf P^1} \\times {\\bf P^1}$ fiber which means that it wraps a 2-dimensional surface in the\n${\\bf P^1} \\times {\\bf P^1}$ fiber. Therefore we have O7 planes with the action as before.\n\n\n\n\n\\section{Analysis of the global picture and the cycle of geometric transitions}\n\nWith all the mathematical construction at hand, it is time now to discuss the geometrical aspect of the problem i.e\nthe supergravity metric and the fluxes in type II theories. Our starting point would be the\nissue of supersymmetry in the usual resolved conifold background with fluxes and branes in type IIB background. Once\nwe obtain this, it will prepare us for all the subsequent stages of the duality cycle for the\ngeometric transition \\cite{gtpapers}.\n\n\n\\subsection{Analysis of the global picture in type IIB}\n\n{}From our earlier works we know that there are two ways of extending our local configuration of \\cite{gtpapers} to study\nsupersymmetric cases in the full global picture:\n\n\\noindent (a) The full global geometry is a six-dimensional K\\\"ahler manifold with F-theory\nseven-branes distributed in some particular way. These seven-branes contribute to massive\nfundamental flavors in the gauge theory. Orientation of these seven-branes are the generalised version of the\nOuyang \\cite{ouyang}\n(or the Kuperstein \\cite{kuperstein}) embeddings.\n\n\\noindent (b) The full global geometry is non-K\\\"ahler with or without F-theory seven-branes. The seven-branes could\nbe embedded in this picture via Ouyang or the Kuperstein embedding, which in turn would provide fundamental matters\nin the gauge theory. In fact the possibility of such a global completion was already hinted in the second paper of\n\\cite{gtpapers}.\n\nLet us see how from our local picture studied in \\cite{gtpapers} these two possibilities can be realised. In the\nfirst paper of \\cite{gtpapers}, the local metric was argued to be of the following \nform\\footnote{The local metric \\eqref{localmet}\nthat we consider here is that of\na supergravity background studied around a specific chosen point in the internal six-dimensional space. For \nexample we choose a point \n($r_0, \\langle\\theta_i\\rangle, \\langle\\phi_i\\rangle, \\langle\\psi\\rangle$) in \n\\cite{gtpapers} which is away from the $r=0$ conifold point. \n This is because the full global picture was\nhard to construct, and any naive procedure always lead to non-supersymmetric\nsolutions. In deriving the local metric, we took a simpler model where all\nthe spheres were replaced by tori with periodic coordinates ($x,\n\\theta_1$) and ($y,\\theta_2$). The coordinate $z$ formed a\nnon-trivial $U(1)$ fibration over the $T^2$ base. Here ($r, x, y, z, \\theta_1, \\theta_2$) is the coordinate of a point \naway from \n($r_0, \\langle\\phi_1\\rangle, \\langle\\phi_2\\rangle, \\langle\\psi\\rangle, \\langle\\theta_1\\rangle, \n\\langle\\theta_2\\rangle$). \nThe replacement\nof spheres by two tori was directly motivated from the\ncorresponding brane constructions of \\cite{dasmukhi}, where\nnon-compact NS5 branes required the existence of tori instead of\nspheres in the T-dual picture. On the other hand \nthe term {\\it global} means roughly adding back the curvature, warping, etc., replacing tori by\nspheres, so that at the end of the day, we have a supersymmetric\nsolution to the equations of motion. The purpose of this paper \nis to exactly fill in the long-awaited gap, i.e to provide the full global picture of geometric\ntransition. Note also that the only known global solution, i.e \\cite{pandoz}, before our work\nwas unfortunately not supersymmetric (see\n\\cite{cvetic}, \\cite{anke} for details) although it satisfied the type IIB EOMs.}:\n\\bg\\label{localmet}\nds^2 ~=~ && dr^2 + \\Bigg(dz + \\sqrt{\\gamma' \\over \\gamma}~r_0 ~{\\rm\ncot}~\\langle\\theta_1\\rangle~ dx  + \\sqrt{\\gamma' \\over (\\gamma+4a^2)~}~r_0~ {\\rm\ncot}~\\langle\\theta_2\\rangle~dy\\Bigg)^2~ + \\nonumber\\\\\n&& ~~~~~~~~~~~ + \\Bigg[{\\gamma\\sqrt{h} \\over 4}~d\\theta_1^2 + dx^2\\Bigg] +\n\\Bigg[{(\\gamma + a^2)\\sqrt{h} \\over 4}~d\\theta_2^2 + dy^2\\Bigg] + ....\n\\nd\nwhere all the coefficients are measured at a fixed chosen point ($r_0, \\langle\\psi\\rangle, \\langle\\phi_i\\rangle,\n\\langle\\theta_i\\rangle$). For more details see \\cite{gtpapers}. The local $B_{\\rm NS}$ field was taken to be:\n\\bg\\label{bnslocal}\nB_{\\rm NS} = b_{x\\theta_i} dx \\wedge d\\theta_i + b_{y\\theta_i} dy \\wedge d\\theta_i\n\\nd\nwhere $i= 1,2$. The above background is invariant under the orbifold operation:\n\\bg\\label{orbi} {\\cal I}_{xy}: ~~ x ~ \\to ~-x, ~~~~ y ~\\to ~ -y\n\\nd\nand therefore can support D7\/O7 states at the following orientifold points:\n\\bg\\label{opoitla}\n{{\\bf T}^2 \\over {\\cal I}_{xy}~ \\Omega ~(-1)^{F_L}}\n\\nd\nIt is interesting to note that, at the orientifold point, a component like $b_{xy}$ is projected out. However the\norientifold projection may allow components like $b_{xz}, b_{yz}$ which could in principle make our mirror manifold\nnon-geometric. In the local picture advocated in \\cite{gtpapers} we only see components like \\eqref{bnslocal} so\nthe local mirror is non-K\\\"ahler and geometric.\n\nMore interestingly, the orientifolding operation \\eqref{opoitla} allows, along with the wrapped D5-branes,\nthe D7-branes and O7-planes along the internal directions ($r, z, \\theta_1, \\theta_2$) located at the\nfour fixed points of the torus ${\\bf T^2}$ along ($x, y$) directions. Therefore,\nin the local picture, a possible\nsusy preserving Ouyang-type configuration would be D5-branes wrapped on the two-torus ($\\theta_2, \\phi_2$) and the\nseven branes wrapping ($\\theta_1, \\theta_2, \\psi$) and stretched along the radial direction $r$. On the other hand,\nglobally\nin a resolved conifold background the seven-branes are in a configuration that is the {\\it union} of\nbranch 1 and branch 2 (see \\cite{ouyang, sullyj, FEP, jpsi} for details). Recall that\nin branch 1 the seven-branes wrap the ${\\bf P^1}$ parametrised by ($\\theta_2, \\phi_2$) and are\nembedded along ($r, \\psi$) directions at a point on the other ${\\bf P^1}$ parametrised by ($\\theta_1, \\phi_1$);\nwhereas in branch 2 the\nseven-branes are at a point on the ${\\bf P^1}$\nparametrised by ($\\theta_2, \\phi_2$). Thus globally a susy configuration of\nseven-branes is a {\\it two}-dimensional surface in ${\\bf P^1} \\times {\\bf P^1}$\nand stretched along ($r, \\psi$) directions determined\nby the appropriate embedding equation. Therefore in the local limit the two-dimensional susy preserving surface\nin ${\\bf T^2} \\times {\\bf T^2}$ should be the two-cycle parametrised by ($\\theta_1, \\theta_2$) as\nprescribed in \\cite{gtpapers}.\n\nAway from the orientifold point, the local metric takes the following fibration form:\n\\bg\\label{locfib}\nds^2 = && ~h^{-1\/2}ds^2_{0123} +  \\gamma'\\sqrt{h}~ dr^2 +\n(dz + \\Delta_1~{\\rm cot}~\\theta_1~ dx + \\Delta_2~{\\rm cot}~\\theta_2~\ndy)^2 + \\nonumber\\\\\n&& ~~~+ \\left({\\gamma \\sqrt{h} \\over 4} d\\theta_1^2 + dx^2\\right)\n + \\left({(\\gamma + 4a^2)\\sqrt{h} \\over 4} d\\theta_2^2 + dy^2 \\right)\\nonumber\\\\\nH_3 = && d{\\cal J}_1 \\wedge d\\theta_1 \\wedge dx + d{\\cal J}_2 \\wedge\nd\\theta_2 \\wedge dy \\nonumber\\\\\nF_5 = && K(r)~(1 + \\ast) ~dx \\wedge dy \\wedge\ndz \\wedge d\\theta_1\\wedge d\\theta_2\\\\\nF_3 = && c_1~(dz \\wedge d\\theta_2 \\wedge dy - dz \\wedge d\\theta_1 \\wedge dx)\\nonumber\n\\nd\nwith additional axio-dilaton that appear from the seven-brane sources. The above form of orientifold projection only\nallows a non-trivial fibration structure {\\it away} from the orientifold point. However there exist another orientifold\noperation that may be more well suited at the orientifold point. This can be applied locally via:\n\\bg\\label{orbnow}\n{\\cal I}_{x\\theta_1}: ~~~ x ~\\to ~ -x, ~~~~~~~~~ \\theta_1 ~\\to ~ \\pi - \\theta_1\n\\nd\nThe above action gives rise to the following orientifold action:\n\\bg\\label{oriennnw}\n{{\\bf T}^2 \\over {\\cal I}_{x\\theta_1}~ \\Omega ~(-1)^{F_L}}\n\\nd\nthat will keep the wrapped D5 branes only if they\nare {\\it away} from the orientifold point unless of course\nthey exist as bound states with the seven-branes {\\it at} the orientifold point.\nIn addition there is the $B_{\\rm NS}$ field with the following components at the orientifold point:\n\\bg\\label{hecchu}\nB_{\\rm NS} ~=~&& b_{x\\theta_2}~ dx \\wedge d\\theta_2 + b_{y\\theta_1}~ dy \\wedge d\\theta_1 +\nb_{xy}~ dx \\wedge dy + b_{xz}~ dx \\wedge dz + \\nonumber\\\\\n&& b_{rx}~ dr \\wedge dx + b_{r\\theta_1}~ dr \\wedge d\\theta_1 + b_{\\theta_1\\theta_2}~ d\\theta_1 \\wedge d\\theta_2\n+ b_{z\\theta_1}~ dz \\wedge d\\theta_1\n\\nd\nwhich means that at the orientifold point not only is the IIB metric non-trivial, the mirror can also be\nnon-K\\\"ahler and non-geometric. The seven-branes and the orientifold-planes are parallel to the wrapped D5-brane\nbound states\\footnote{These are in fact the dipole-deformed bound states studied in the last paper of\n\\cite{gtpapers}.}.\nIn the following we will argue how susy is preserved in the global set-up when the seven-branes are moved away from the\nwrapped D5 branes. This is the case where the fundamental hypermultiplets are infinitely massive and therefore susy\nremains unbroken at the scale that we want to study.\n\nThe naive global extension of the above configuration along the lines of \\cite{pandoz} will lead to a non-susy\nconfiguration. This is because we have assumed that the global extension of a configuration like \\eqref{locfib} is\nK\\\"ahler in the presence of a $B_{\\rm NS}$ field like \\eqref{bnslocal} {\\it away} from the orientifold point. The\nsimplest global extension that we will study here as the starting point for the IIB geometric transition is a\nnon-K\\\"ahler manifold with D5-branes wrapping two cycles of this manifold. Of course it may be possible to add other\nbranes and fluxes to make the ambient space conformally K\\\"ahler, but we will not do so here.\nWe will use the following set of duality transformations,\nrecently proposed by \\cite{marmal} (see also figure 1), to get\nour type IIB intial configuration.\n\n\\noindent $\\bullet$ Our starting point would be a non-K\\\"ahler type IIB metric with a background dilaton $\\phi$ and\nNS three-form $H_3$ that satisfies the standard relation $H_3 = e^{2\\phi} \\ast d(e^{-2\\phi}J)$\nwith $J$ being the fundamental\n(1,1) form.\n\n\\noindent $\\bullet$ On this background we perform a S-duality that transforms the NS three-form to RR three-form\n$F_3$, and in the process converts the dilaton to $-\\phi$ without changing the metric in the Einstein frame.\n\n\\noindent $\\bullet$ We now make three T-dualities along the spacetime directions $x^{1, 2, 3}$ that takes us to type\nIIA theory. Observe that this is {\\it not} the mirror construction.\n\n\\noindent $\\bullet$ We lift the type IIA configuration to M-theory and perform a boost along the eleventh direction.\nThis boost is crucial in generating D0-brane {\\it gauge} charges in M-theory.\n\n\\noindent $\\bullet$ A dimensional reduction back to IIA theory does exactly what we wanted: it generates the necessary\nnumber of D0-brane charges from the boost, without breaking the underlying supersymmetry of the system.\n\n\\noindent $\\bullet$ Once we have the IIA configuration, we go back to type IIB by performing the three T-dualities\nalong $x^{1, 2, 3}$ directions. From the D0-branes, we get back our three-brane charges namely the five-form. The\nduality cycle also gives us NS three-form $H_3$ as well as the expected RR three-form $F_3$. Therefore the final\nconfiguration is exactly what we required for IR geometric transition: wrapped D5s with necessary sources on a\nnon-K\\\"ahler globally defined ``resolved'' conifold background. Also as expected, the background preserves\nsupersymmetry and therefore should be our starting point. This background should also be compared with the one\ngiven in \\cite{pandoz} that solves the EOM but does not preserve supersymmetry. \nIn figure 1, we illustrate the above dualities.\n\\begin{figure}[htb]\\label{MMdualities}\n       \\begin{center}\n\\includegraphics[height=12cm]{MMdualities.eps}\n        \\caption{This figure illustrates the series of dualities that we used to generate\nthe full supersymmetric background with non-trivial fluxes.}\n        \\end{center}\n        \\end{figure}\nTo start off, we switch on a non-trivial background dilaton $\\phi$ and a NS three-form $H_{\\rm NS}$ on a\nbackground outlined by the following metric:\n\\bg \\label{startmet}\nds^2 = h^{1\/2} e^\\phi d{\\widetilde s}^2_{0123} + h^{-1\/2} e^\\phi ds^2_6\n\\nd\nwhere note that we can choose the dilaton $\\phi$ appropriately so that in the string frame the spacetime metric may not\nhave a warp factor. This will be consistent with the last reference of \\cite{torsion}.\nWe have defined the other variables in the following way:\n\\bg \\label{varde}\n&&h = {e^{-2\\phi} F_0^{-4} \\over e^{-2\\phi} h^{-2}F_0^{-4}{\\rm cosh}^2\\beta - {\\rm sinh}^2 \\beta},\n~~~~~~ d{\\widetilde s}^2_{0123} = F_0 ds^2_{0123}\\\\\n&& ds^2_6 = F_1~ dr^2 + F_2 (d\\psi + {\\rm cos}~\\theta_1 d\\phi_1 + {\\rm cos}~\\theta_2 d\\phi_2)^2  + \\sum_{i = 1}^2 F_{2+i}\n(d\\theta_i^2 + {\\rm sin}^2\\theta_i d\\phi_i^2)\\nonumber\n\\nd\nwith $\\beta$ being an arbitrary constant, $F_i \\equiv F_i(r), i = 1, ..., 4$ \nare functions of the radial coordinate for simplicity and $F_0 = F_0(r, \\theta_1, \\theta_2)$ since this is unconstrained. \nObserve that in the first equation of \\eqref{varde}, $h$ appears on both sides, and once we put in the value of the\ndilaton we can determine the warp factor $h$. For our case the dilaton will take the following form:\n\\bg\\label{dildil}\n\\phi ~ = ~ -{\\rm log}~F_0 - {1\\over 2} {\\rm log}~h\n\\nd\nso that the starting metric in IIB, that preserves spacetime supersymmetry, becomes:\n\\bg\\label{smetru}\nds^2 ~ = ~ ds^2_{0123} ~+~ \\left({1+F_0^2 {\\rm sinh}^2\\beta\\over F_0 {\\rm cosh}^2\\beta}\\right) ds^2_6\n\\nd\nIn\ngeneral we will continue keeping the dilaton $\\phi$ in the metric components to get the general torsion classes for the\nbackground (see the analysis in \\cite{chen2}). \nWe also\nexpect $F_i$ to be functions of all the internal coordinates. We will give an example of this soon when\nwe derive a more precise initial metric. For the time being we will consider \\eqref{startmet} to be our starting point.\nAlso to preserve supersymmetry\\footnote{Or, equivalently, preserving $SU(3)$ structure.}, we expect:\n\\bg\\label{hns}\nH_{\\rm NS} = e^{2\\phi} \\ast d\\left(e^{-2\\phi} J\\right)\n\\nd\nwith the appropriate dilaton. Here\n$J$ is the fundamental form associated with the metric, and we can choose to\nimpose one of the following two conditions on the\nNS three-form:\n\\bg\\label{condhns}\n&&dH_{\\rm NS} ~\\equiv~ d\\ast dJ - d\\ast (d\\phi \\wedge J) ~=~ {\\rm sources}\\nonumber\\\\\n&&dH_{\\rm NS} ~\\equiv~ d\\ast dJ - d\\ast (d\\phi \\wedge J) ~ = ~\\alpha'({\\rm tr} ~R \\wedge R - {\\rm Tr} ~ F \\wedge F)\n\\nd\nThe first condition is what we require here. This will give rise to the IR wrapped D5 branes theory on non-K\\\"ahler\nresolved conifold set-up (after the chain of dualities mentioned above). The latter case will be for the\nheterotic theory. We can use the non-closure of $H_{\\rm NS}$ to study not only\nthe vector bundles $F$ on the heterotic side, but also the possibility of geometric transition in the heterotic theory!\nWe have alluded to this possibility in our earlier papers \\cite{gtpapers} (see also \\cite{israel}).\nWe have completed that side of the\nstory in our follow-up paper \\cite{chen2}.\n\nNow following the chain of dualities mentioned above, we can get the following type IIB \nbackground\\footnote{Due to an unfortunate choice of notation, the RR three-form and the third warp factor have \nthe same notation of $F_3$ (as this is the standard way to represent them!). \nHowever since we use $F_3$ to mostly denote the third warp factor, we hope that there will be \nno confusion.}:\n\\bg\\label{susybg}\n&& F_3 = h~{\\rm cosh}~\\beta~e^{2\\phi} \\ast d\\left(e^{-2\\phi} J\\right), ~~~~~~~\nH_3 = -hF_0^2{\\rm sinh}~\\beta~e^{2\\phi} d\\left(e^{-2\\phi} J\\right)\\nonumber\\\\\n&& F_5 = -{1\\over 4} (1 + \\ast) dA_0 \\wedge dx^0 \\wedge dx^1 \\wedge  dx^2 \\wedge  dx^3, ~~~~~\n\\phi_{\\rm now} = -\\phi\\\\\n&& ds^2 = F_0 ds^2_{0123} + F_1~ dr^2 + F_2 (d\\psi + {\\rm cos}~\\theta_1 d\\phi_1 + {\\rm cos}~\\theta_2 d\\phi_2)^2\n+ \\sum_{i = 1}^2 F_{2+i}\n(d\\theta_i^2 + {\\rm sin}^2\\theta_i d\\phi_i^2)\\nonumber\n\\nd\nwhich is, by construction, supersymmetric and since the RR three-form\n$F_3$ is not closed it represents precisely the IR configuration\nof wrapped D5-branes on warped non-K\\\"ahler resolved conifold. The above set of equations \\eqref{susybg} is one \nof our main results, and as promised in the introduction, provides the fully supersymmetric globally defined \nsolution for the wrapped D5-branes on a non-K\\\"ahler resolved conifold. \nNote that we have left the warp factors $F_i$ undetermined in \\eqref{susybg}. This means that \nthere is a wide range of choices for $F_i$ related to various gauge theory deformations. This is closely related to \na similar class of solutions illustrated in figure 3 of \\cite{chen2}. Thus the procedure will be to identify \ncertain set of $\\{F_i\\}$ related to deformations in ${\\cal N} = 1$ YM theory, and then our duality chain will reproduce \nthe gravity dual of this YM configuration. One may also refer to a recent class of solutions studied in \\cite{pando2} \nwith a given choice of $\\{F_i\\}$. \n\nFinally, the five-form is switched on to satify the equation of\nmotion with\n\\bg\\label{a0def}\nA_0 ~&=~ &{{\\rm cosh}~\\beta~{\\rm sinh}~\\beta (1-e^{2\\phi}h^{-2}F_0^{-4})\n\\over e^{-2\\phi} h^{-2} F_0^{-4}{\\rm cosh}^2\\beta - {\\rm sinh}^2\\beta}\\nonumber\\\\\n&= ~& (F_0^2 -1 ){\\rm tanh}~\\beta \\left[1 + \\left({1-F_0^2\\over F_0^2}\\right){\\rm sech}^2\\beta +  \n\\left({1-F_0^2\\over F_0^2}\\right)^2 {\\rm sech}^4 \\beta\\right]\n\\nd\nThe above equation \\eqref{susybg} \nis our starting metric, whose local forms\nwe studied in details in \\cite{gtpapers}, and therefore should be taken instead of the metric derived in \\cite{pandoz}.\nThe ISD condition for our case gets modified to the following condition on the fluxes:\n\\bg\\label{isdnow}\n\\cosh\\beta~H_3 ~+~ F_0^2~\\sinh\\beta \\ast F_3 ~ = ~ 0\n\\nd\nwhich may be compared to \\cite{marmal}.\nOur derivation\ncould also solve the long standing problem of finding the supersymmetric configuration of wrapped D5-branes on a\nresolved conifold set-up.\n\n\\subsection{More explicit type IIB background before geometric transition}\n\nIn the above section we saw how one could derive the precise intial\nmetric that not only serves as starting point for geometric\ntransition, but is also supersymmetric. One may make this more specific by\nsolving the $SU(3)$ structure condition specified in section 2.4. This is\nworked out in {\\bf Appendix 1}.\nThe metric derived this way has\nmany non-trivial components compared to our initial ansatze\n\\eqref{susybg}. This is not a problem in itself, because we can\nalways do some coordinate transformations to bring the metric to a form that\ndoesn't have components like $g_{r\\mu}$ where $\\mu = \\theta_i,\n\\phi_i, \\psi$. But the metric will have other non-trivial\ncross-terms. It may be possible to make further coordinate\ntransformations to bring the above metric in a form that closely\nresembles \\eqref{susybg}, but we will not pursue this here as this\ndoesn't change the underlying physics. Instead we will continue\nusing the background \\eqref{susybg} and assume that the values of\nthe coefficients $F_i$ are to be fixed using our above metric\nconfiguration\\footnote{See also \\cite{israel} where a non-K\\\"ahler metric\non the resolved conifold is studied. It would be interesting to compare the\nmetric of \\cite{israel} with the metric components given in {\\bf Appendix 1}.}.\nOther possible cross-terms, not considered in\n\\eqref{susybg}, will only make the IIA background more non-trivial\nwithout revealing new physics\\footnote{We have justified this claim in \\cite{chen2} where we explicitly computed the \ntorsion classes for the background \\eqref{startmet}, \\eqref{dildil} and \\eqref{hns} to argue for supersymmetry. See \nsection 3.1 of \\cite{chen2}.}. \nHenceforth our starting point would\nbe \\eqref{susybg} with the assumption that the coefficients are to\nbe derived from the metric discussed in the above subsection.\n\\begin{figure}[htb]\\label{dualitiesGT}\n        \\begin{center}\n\\includegraphics[height=10cm]{dualitiesGT.eps}\n        \\caption{The duality map to generate the full geometric transitions in the supersymmetric global\nset-up of type IIA and type IIB theories.}\n        \\end{center}\n        \\end{figure}\nOnce we know the metric, we can follow up the steps described\nearlier to compute the three-forms. The NS three-form $H_3$ has the\nform: \\begin{eqnarray}\\label{H3}\n\\frac{H_3}{hF_0^2\\sinh\\beta}=&+&\\Big[k^2(2\\phi_{\\theta_1}\\sqrt{F_1F_2}\\cos\\theta_1+\\sqrt{F_1F_2}\\sin\\theta_1+2\\phi_rF_3\\sin\\theta_1-F_{3r}\\sin\\theta_1)\\nonumber\\\\\n&-&2kF_3k_r\\sin\\theta_1-2k\\sqrt{F_1F_2}k_{\\theta_1}\\cos\\theta_1\\Big]dr\\wedge\nd\\theta_1\\wedge d\\phi_1\\nonumber\\\\\n&+&\\Big[k^2(2\\phi_{\\theta_2}\\sqrt{F_1F_2}\\cos\\theta_2+\\sqrt{F_1F_2}\\sin\\theta_2+2\\phi_rF_4\\sin\\theta_2-F_{4r}\\sin\\theta_2)\\nonumber\\\\\n&-&2kF_4k_r\\sin\\theta_2-2k\\sqrt{F_1F_2}k_{\\theta_2}\\cos\\theta_2\\Big]dr\\wedge\nd\\theta_2\\wedge d\\phi_2\\nonumber\\\\\n&-&2k\\sqrt{F_1F_2}\\Big[(k\\phi_{\\theta_1}-k_{\\theta_1})(dr\\wedge\nd\\psi\\wedge d\\theta_1-\\cos\\theta_2dr\\wedge d\\theta_1\\wedge\nd\\phi_2)\\nonumber\\\\\n&+&(k\\phi_{\\theta_2}-k_{\\theta_2} )(dr\\wedge d\\psi\\wedge\nd\\theta_2-\\cos\\theta_1dr\\wedge\nd\\theta_2\\wedge d\\phi_1)\\Big]\\nonumber\\\\\n&-&2kF_3\\sin\\theta_1(k\\phi_{\\theta_2}-k_{\\theta_2})d\\theta_1\\wedge\nd\\theta_2\\wedge\nd\\phi_1\\nonumber\\\\\n&+&2kF_4\\sin\\theta_2(k\\phi_{\\theta_1}-k_{\\theta_1})d\\theta_1\\wedge\nd\\theta_2\\wedge d\\phi_2\n\\end{eqnarray}\nwhere $k^2(r,\\theta_1,\\theta_2)=h^{-1\/2}e^{\\phi}$, and we have\ndefined $\\phi_\\alpha \\equiv \\partial_\\alpha \\phi$ with $\\alpha =\n\\theta_i, r$ as $\\phi \\equiv \\phi (r, \\theta_1, \\theta_2)$ for\nsimplicity. A constant $\\phi$ is not good for us, and also leads to\ncertain issues detailed in \\cite{papad}. Once we have $H_3$, we can\nget $dH_3$ as:\n\\begin{eqnarray}\\label{dH3}\n\\frac{dH_3}{\\sinh\\beta}&=&k\\Big[F_3\\sin\\theta_1(4hk_r\\phi_{\\theta_2}-4hk_{\\theta_2}\\phi_r+2h_{\\theta_2}k_r-2h_rk_{\\theta_2}+2kh_r\\phi_{\\theta_2}-2kh_{\\theta_2}\\phi_r)\\nonumber\\\\\n&&\\sqrt{F_1F_2}\\cos\\theta_1(4hk_{\\theta_1}\\phi_{\\theta_2}-4hk_{\\theta_2}\\phi_{\\theta_1}+2h_{\\theta_2}k_{\\theta_1}-2h_{\\theta_1}k_{\\theta_2}\n+2kh_{\\theta_1}\\phi_{\\theta_2}-2kh_{\\theta_2}\\phi_{\\theta_1})\\nonumber\\\\\n&&-(\\sqrt{F_1F_2}-F_{3r})(2hk\\phi_{\\theta_2}+h_{\\theta_2}k)\\sin\\theta_1\\Big]dr\\wedge\nd\\theta_2\\wedge d\\theta_1\\wedge d\\phi_1\\nonumber\\\\\n&&+k\\Big[F_4\\sin\\theta_2(4hk_r\\phi_{\\theta_1}-4hk_{\\theta_1}\\phi_r+2h_{\\theta_1}k_r-2h_rk_{\\theta_1}+2kh_r\\phi_{\\theta_1}-2kh_{\\theta_1}\\phi_r)\\nonumber\\\\\n&&\\sqrt{F_1F_2}\\cos\\theta_2(4hk_{\\theta_2}\\phi_{\\theta_1}-4hk_{\\theta_1}\\phi_{\\theta_2}+2h_{\\theta_1}k_{\\theta_2}-2h_{\\theta_2}k_{\\theta_1}\n+2kh_{\\theta_2}\\phi_{\\theta_1}-2kh_{\\theta_1}\\phi_{\\theta_2})\\nonumber\\\\\n&&-(\\sqrt{F_1F_2}-F_{4r})(2hk\\phi_{\\theta_1}+h_{\\theta_1}k)\\sin\\theta_2\\Big]dr\\wedge\nd\\theta_1\\wedge d\\theta_2\\wedge\nd\\phi_2\\nonumber\\\\\n&&+2k\\sqrt{F_1F_2}\\Big[h_{\\theta_1}k_{\\theta_2}-h_{\\theta_2}k_{\\theta_1}+kh_{\\theta_2}\\phi_{\\theta_1}-kh_{\\theta_1}\\phi_{\\theta_2}+2hk_{\\theta_2}\\phi_{\\theta_1}-2hk_{\\theta_1}\\phi_{\\theta_2}\\Big]\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~\\times dr\\wedge d\\theta_1\\wedge d\\theta_2\\wedge\nd\\psi\n\\end{eqnarray}\nwith $F_{ir} \\equiv \\partial_r F_i$, $k_{i}=\\partial_i k$ and\n$(hF_0^2)_{i}=\\partial_i (hF_0^2)$. From \\eqref{dH3} it means that\nif we make\n\\begin{eqnarray}\n&&h_{\\theta_1}k_{\\theta_2}-h_{\\theta_2}k_{\\theta_1}+kh_{\\theta_2}\\phi_{\\theta_1}-kh_{\\theta_1}\\phi_{\\theta_2}+2hk_{\\theta_2}\\phi_{\\theta_1}-2hk_{\\theta_1}\\phi_{\\theta_2}=0,\\nonumber\\\\\n&&\\frac{4hk_r\\phi_{\\theta_2}-4hk_{\\theta_2}\\phi_r+2h_{\\theta_2}k_r-2h_rk_{\\theta_2}+2kh_r\\phi_{\\theta_2}-2kh_{\\theta_2}\\phi_r}{2hk\\phi_{\\theta_2}+h_{\\theta_2}k}=\\frac{\\sqrt{F_1F_2}-F_{3r}}{F_3},\\nonumber\\\\\n&&\\frac{4hk_r\\phi_{\\theta_1}-4hk_{\\theta_1}\\phi_r+2h_{\\theta_1}k_r-2h_rk_{\\theta_1}+2kh_r\\phi_{\\theta_1}-2kh_{\\theta_1}\\phi_r}{2hk\\phi_{\\theta_1}+h_{\\theta_1}k}=\\frac{\\sqrt{F_1F_2}-F_{4r}}{F_4}.\\nonumber\n\\end{eqnarray}\nthen $H_3$ will be closed. One however may worry that making $H_3$\nclosed implies too much constraints on the $F_i$'s. For the present\ncase this may still be okay because the initial choice of the\nbackground \\eqref{varde} forms a class of solutions parametrised by\nour choice of $F_i$ and $\\phi$, the dilaton\\footnote{See also \\cite{chen2} where a torsion class analysis reveals \n$k$ to be a function of $r$ only. This means we can choose $F_0$ appropriately such that $h$ and $e^\\phi$ are \nfunctions of ($r, \\theta_i$). One possible choice would be $h = {1\\over k^2 F_0}$ and $e^\\phi = {k\\over \\sqrt{F_0}}$.}. \nA specific choice of the\nbackground with a specified complex structure and K\\\"ahler class is\nexemplified in {\\bf Appendix 1}. For this case we can define a\nclosed three-form with appropriate choice of the dilaton, so that\nour choice remains generic enough. Thus the $B_{NS}$ field can be\ngauge transformed to have only the following components:\n\\begin{eqnarray}\\label{bns}\nb_{r\\psi}&=&\\int-2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_1}-k_{\\theta_1})\\sqrt{F_1F_2}~d\\theta_1,\n\\nonumber\\\\\nb_{r\\phi_1}&=&\\int-2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_2}-k_{\\theta_2})\\sqrt{F_1F_2}\\cos\\theta_1~d\\theta_2,\n\\nonumber\\\\\nb_{r\\phi_2}&=&\\int-2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_1}-k_{\\theta_1})\\sqrt{F_1F_2}\\cos\\theta_2~d\\theta_1,\\nonumber\\\\\nb_{\\theta_1\\phi_1}&=&\\int2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_2}-k_{\\theta_2})\nF_3\\sin\\theta_1~d\\theta_2,\\nonumber\\\\\nb_{\\theta_2\\phi_2}&=&\\int2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_1}-k_{\\theta_1})\nF_4\\sin\\theta_2~d\\theta_1.\n\\end{eqnarray}\nwhere we see that there are three new components of the form $b_{r\\alpha}$ compared to the local case \\cite{gtpapers}.\nThis is expected because we are no longer fixed to $r = r_0$, but have global access. However\nbefore moving ahead we will pause to comment on switching on\nother possible components of the $B_{NS}$ field of the form:\n\\bg\\label{bnsother}\n b_{\\phi_1 \\phi_2}~ d\\phi_1 \\wedge d\\phi_2 ~+~  \\sum_{i = 1}^2 b_{\\phi_i \\psi}~ d\\phi_i \\wedge d\\psi\n\\nd\nSuch choices of $B_{NS}$ fields would make the type IIA background {\\it non-geometric}. So far locally we saw that the\ntype IIA backgrounds remains geometric but does become non-K\\\"ahler. Is there a possibility that the IIA background\nglobally is non-geometric also? We will reflect on this point later, but for the time being we will assume that the\n$B_{NS}$ field is only of the form \\eqref{bns} and doesn't have additional components like \\eqref{bnsother}.\n\nNext comes the RR three-form $F_3$. From our previous set of duality arguments, this is given by:\n\\begin{eqnarray}\\label{F3}\n\\frac{F_3}{h\nF_0^2\\cosh\\beta}=&&2KF_1F_2F_3F_4\\sin\\theta_2\\sin\\theta_1(\\phi_{\\theta_1}\\sin\\theta_1\\cos\\theta_2-\n\\phi_{\\theta_2}\\sin\\theta_2\\cos\\theta_1)dr \\wedge d\\phi_1\\wedge d\\phi_2\\nonumber\\\\\n&&+KF_3^2\\sin^2\\theta_1\\sin\\theta_2(2\\phi_{\\theta_2}\\sqrt{F_1F_2}F_4\\sin\\theta_2-F_2\\sqrt{F_1F_2}\n\\cos\\theta_2\\nonumber\\\\\n&&-2\\phi_rF_2F_4\\cos\\theta_2+F_2F_{4r}\\cos\\theta_2)d\\theta_1\\wedge\nd\\phi_1\\wedge d\\phi_2\\nonumber\\\\\n&&+KF_4^2\\sin^2\\theta_2\\sin\\theta_1\n(-2\\phi_{\\theta_1}\\sqrt{F_1F_2}\\sin\\theta_1+F_2\\sqrt{F_1F_2}\\cos\\theta_1\\nonumber\\\\\n&&+2\\phi_rF_2F_3\\cos\\theta_1-F_2F_{3r}\\cos\\theta_1)d\\theta_2\\wedge d\\phi_1\\wedge d\\phi_2\\nonumber\\\\\n&&-KF_2F_3^2\\sin^2\\theta_1(2\\phi_rF_4\\sin\\theta_2+\\sqrt{F_1F_2}\\sin\\theta_2-F_{4r}\\sin\\theta_2)d\\psi\\wedge\nd\\theta_1\\wedge d\\phi_1\\nonumber\\\\\n&&-KF_2F_4^2\\sin^2\\theta_2(2\\phi_rF_3\\sin\\theta_1+\\sqrt{F_1F_2}\\sin\\theta_1-F_{3r}\\sin\\theta_1)d\\psi\\wedge\nd\\theta_2\\wedge d\\phi_2\\nonumber\\\\\n&&-2\\phi_{\\theta_2}KF_1F_2F_3F_4\\sin\\theta_1\\sin^2\\theta_2dr\\wedge\nd\\psi\\wedge d\\phi_2\\nonumber\\\\\n&&-2\\phi_{\\theta_1}KF_1F_2F_3F_4\\sin\\theta_2\\sin^2\\theta_1dr\\wedge\nd\\psi\\wedge d\\phi_1\n\\end{eqnarray}\nwhere as before $\\phi_\\alpha$ should be understood as derivatives on $\\phi$ i.e $\\partial_\\alpha \\phi$,\nand we have defined $K$ as:\n\\bg\\label{kdef}\nK ={{\\rm cosec}~\\theta_1{\\rm cosec}~\\theta_2\\over \\sqrt{F_1F_2}F_3F_4}\n\\nd\nNote that $dF_3$ is no longer\nclosed, and will be related to delta function sources coming from the wrapped \nD5-branes\\footnote{There is a subtlety \nhere: not every non-closed $F_3$ can be interpreted as a source (see for example the criteria presented in \n\\cite{konkal}). Happily,\nour case does fall into one of the required criteria of \\cite{konkal} as should be clear by writing the \nfluxes in the language of G-structure, or in terms of the torsion classes. A more detailed elaboration of this is \ngiven in \\cite{chen2}.}.\n\nOnce we have the explicit forms for the three-forms, to satisy the type IIB EOMs we will now require RR five-form.\nThis is easy to work out, and is given by:\n\\begin{eqnarray}\\label{F5}\nF_5&=&\\frac{1}{4}\\Big[-A_{0r}dr\\wedge dt\\wedge dx\\wedge dy\\wedge\ndz-A_{0\\theta_1}d\\theta_1\\wedge dt\\wedge dx\\wedge dy\\wedge\ndz\\nonumber\\\\\n&&-A_{0\\theta_2}d\\theta_2\\wedge dt\\wedge dx\\wedge dy\\wedge dz\n-PF_2F_3F_4\\sin^2\\theta_1\\sin^2\\theta_2\\nonumber\\\\\n&&\\times (A_{0r}F_3F_4d\\psi\\wedge d\\theta_1\\wedge d\\theta_2\\wedge\nd\\phi_1\\wedge d\\phi_2+A_{0\\theta_1}F_1F_4dr\\wedge d\\psi\\wedge\nd\\theta_2\\wedge\nd\\phi_1\\wedge d\\phi_2\\nonumber\\\\\n&& +A_{0\\theta_2}F_1F_3dr\\wedge d\\psi\\wedge d\\theta_1\\wedge\nd\\phi_1\\wedge d\\phi_2)\\Big]\n\\end{eqnarray}\nwhere $A_{0\\alpha} \\equiv \\partial_\\alpha A_0$ and $A_0$ is given in \\eqref{a0def}. We have also defined $P$ as:\n\\bg\\label{pdef}\nP={{\\rm cosec}~\\theta_1{\\rm cosec}~\\theta_2\\over \\sqrt{F_1F_2}F_0^2F_3 F_4}\n\\nd\nThus with \\eqref{H3}, \\eqref{F3}, \\eqref{F5}\nand \\eqref{susybg} we have the complete susy background in type IIB before geometric transition. A torsion\nclass analysis with susy constraints has been done in \\cite{chen2} (see eq. (3.15) therein). As long as the\nwarp factors satisfy eq. (3.15) of \\cite{chen2} supersymmetry will be preserved.\nIn the following subsection, we will use the above background to\ncompute the type IIA mirror configuration.\n\n\\subsection{The type IIA mirror configuration}\n\nAs it stands, the metric in \\eqref{susybg} has three obvious\nisometries associated with translation along the three angular\ndirections $\\phi_1, \\phi_2$ and $\\psi$. So there is a natural ${\\bf\nT}^3$ embedded in our configuration, and one might naively think\nthat the mirror would be three T-dualities along ${\\bf T}^3$. Such a\nsimple transformation doesn't work for our case because our\nconfiguration represents the IR limit of a cascading gauge theory\nwhere the base of the three torus is {\\it small}. Mirror\ntransformation {\\it a la} SYZ \\cite{syz} works exactly in the\nopposite limit! So naive T-dualities will not give us the mirror\nmetric, and we need to first make the base, paramerised by\n$\\theta_1, \\theta_2$ and $r$, very large\\footnote{This effectively\nmeans that the distances along the $\\theta_i$ directions have to be\nmade very large, as $r$ is non-compact. See also our earlier works \\cite{gtpapers} where this\nhas been explained in more details.}. The simplest way to do\nthis would be to make the following transformation on the background\n\\eqref{susybg}: \\bg\\label{tranbg}\n&& d\\psi ~ \\to ~ d\\psi ~+~f_1~ {\\rm cos}~\\theta_1~d\\theta_1   ~+~f_2~ {\\rm cos}~\\theta_2~d\\theta_2\\nonumber\\\\\n&& d\\phi_1 ~ \\to ~ d\\phi_1 ~ - ~ f_1~d\\theta_1, ~~~~~~~ d\\phi_2 ~\n\\to ~ d\\phi_2 ~ - ~ f_2~d\\theta_2 \\nd with the assumption that $f_i\n= f_i(\\theta_i)$ so that the transformations \\eqref{tranbg} would be\nintegrable\\footnote{Note also that since $f_i = f_i(\\theta_i)$, the transformations \\eqref{tranbg} on the \none-forms ($d\\psi, d\\phi_i$) are just coordinate transformations of ($\\psi, \\phi_i$). Therefore they don't change the \nEOMs.}.  \nRecall that compatibility with the $SU(3)$ structure will fix $f_i$ in the mirror \\cite{gtpapers}.\nNote also that these transformations are similar in form as\nin the first reference of \\cite{gtpapers} and would change the complex structure of the base accordingly.\n\nUnder these transformations the $B_{\\rm NS}$ field generates extra\ncomponents $b_{r\\theta_1}$, $b_{r\\theta_2}$. It is however interesting to note that\nthey vanish as follows:\n\\begin{eqnarray}\nb_{r\\theta_1}=f_1(b_{r\\psi}\\cos\\theta_1-b_{r\\phi_1})=0,\\;\\;b_{r\\theta_2}=f_2(b_{r\\psi}\\cos\\theta_2-b_{r\\phi_2})=0\n\\end{eqnarray}\nimplying that the $B_{\\rm NS}$ field do not change under the transformation \\eqref{tranbg}. This is similar to the\nlocal case also \\cite{gtpapers}.\n\nOn the other hand the RR three-form {\\it does} change under the coordinate transformation \\eqref{tranbg}. The\nextra components of the three-form are the following:\n\\begin{eqnarray}\\label{extra3}\n&&F_{r\\theta_1\\theta_2}=f_1f_2(F_{r\\phi_1\\phi_2}-\\cos\\theta_1F_{r\\psi\\phi_2}+\\cos\\theta_2F_{r\\psi\\phi_1}),\n~~~F_{r\\psi\\theta_1}=-f_1F_{r\\psi\\phi_1}\\nonumber\\\\\n&&F_{r\\theta_1\\phi_2}=-f_1(F_{r\\phi_1\\phi_2}-\\cos\\theta_1F_{r\\psi\\phi_2}),\\quad\\,\nF_{\\theta_1\\theta_2\\phi_1}=f_2(F_{\\theta_1\\phi_1\\phi_2}-\\cos\\theta_2F_{\\psi\\theta_1\\phi_1}),\n\\nonumber\\\\\n&&F_{r\\theta_2\\phi_1}=f_2(F_{r\\phi_1\\phi_2}+\\cos\\theta_2F_{r\\psi\\phi_1}),\\quad\\,\nF_{\\theta_1\\theta_2\\phi_2}=f_1(F_{\\theta_2\\phi_1\\phi_2}+\\cos\\theta_1F_{\\psi\\theta_2\\phi_2}),\n\\nonumber\\\\\n&&F_{r\\theta_2\\phi_2}=f_2\\cos\\theta_2F_{r\\psi\\phi_2},~~~\nF_{r\\psi\\theta_2}=-f_2F_{r\\psi\\phi_2},~~~\nF_{r\\theta_1\\phi_1}=f_1\\cos\\theta_1F_{r\\psi\\phi_1}\n\\end{eqnarray}\nA physical reason for this change can be easily understood: under the coordinate transformation \\eqref{tranbg} the\nbase parametrised by ($\\theta_1, \\theta_2$) become large. This means that the associated RR three-form\nfield strengths increase\nsimultaneously, which is of course what we see in \\eqref{extra3}. Note that the component\n$F_{r\\theta_1\\theta_2}$ dominates over all other extra components in \\eqref{extra3} because this lies exclusively\non the base parametrised by the coordinates ($r, \\theta_1, \\theta_2$) which is made much bigger than the\n${\\bf T}^3$ fibre parametrised by the coordinates ($\\psi, \\phi_1, \\phi_2$).\n\nOnce the three-form $F_3$ changes, the RR five-form also has to change. Its is easy to show that the\nextra components of the five-form are:\n\\begin{eqnarray}\\label{extra5}\n&&F_{r\\theta_1\\theta_2\\phi_1\\phi_2}=f_1\\cos\\theta_1F_{r\\psi\\theta_2\\phi_1\\phi_2}-f_2\\cos\\theta_2\nF_{r\\psi\\theta_1\\phi_1\\phi_2},\\nonumber\\\\\n&&F_{r\\psi\\theta_1\\theta_2\\phi_2}=f_1F_{r\\psi\\theta_2\\phi_1\\phi_2},\\quad\nF_{r\\psi\\theta_1\\theta_2\\phi_1}=f_2F_{r\\psi\\theta_1\\phi_1\\phi_2}\n\\end{eqnarray}\nsatisfying the background EOMs. All these extra components will give rise to RR four-form in Type IIA\nafter mirror transformation, as we will show soon. But before that lets infer how the metric changes.\nUnder the transformation \\eqref{tranbg} the metric \\eqref{susybg} takes the\nfollowing form:\n\\bg\\label{sysymet} ds^2 = && F_0 ds^2_{0123} + F_1~\ndr^2 + F_2 (d\\psi + {\\rm cos}~\\theta_1 d\\phi_1 + {\\rm cos}~\\theta_2\nd\\phi_2)^2\n+ \\sum_{i = 1}^2 F_{2+i} ~{\\rm sin}^2\\theta_i d\\phi_i^2 \\nonumber\\\\\n&& ~~~~~~~~~~~+ \\sum_{i = 1}^2 \\Big[F_{2+i}\\left(1 + f_i^2~{\\rm sin}^2 \\theta_i\\right) d\\theta_i^2 -2 f_i F_{2+i}~\n{\\rm sin}^2 \\theta_i ~d\\phi_i d\\theta_i\\Big]\n\\nd\nwhich in fact does exactly what we wanted\\footnote{The metric \\eqref{sysymet} also solves the supergravity EOM as \nshould be clear from the discussion presented earlier.}: \nit enlarges the $\\theta_i$-cycles, but doesn't change the $B_{\\rm NS}$\nfield. For SYZ to work properly, we require the base size to be very large, and therefore we will require $f_i$ also to be\nlarge. This conclusion fits well with the local picture that we had in \\cite{gtpapers}. Note that we have also\ngenerated cross terms. These cross terms will be useful soon. The eleven metric components are:\n\\bg\\label{metcom}\n&& j_{rr} = F_1, ~~~~ j_{\\phi_1\\theta_1} = -f_1F_3 {\\rm sin}^2\\theta_1, ~~~~\nj_{\\phi_2\\theta_2} = -f_2 F_4 {\\rm sin}^2\\theta_2 \\nonumber\\\\\n&& j_{\\psi\\psi} = F_2(1-\\epsilon), ~~~~ j_{\\phi_1\\psi} = F_2 {\\rm cos}~\\theta_1, ~~~~\nj_{\\phi_2\\psi} = F_2 {\\rm cos}~\\theta_2\\nonumber\\\\\n&& j_{\\phi_1\\phi_1} = F_2 {\\rm cos}^2 \\theta_1 + F_3 {\\rm sin}^2 \\theta_1, ~~~~\nj_{\\phi_2\\phi_2} = F_2 {\\rm cos}^2 \\theta_2 + F_4 {\\rm sin}^2 \\theta_2\\\\\n&& j_{\\phi_1\\phi_2} = F_2 {\\rm cos}~\\theta_1 \\theta_2, ~~~~\nj_{\\theta_1 \\theta_1} = F_3 (1+f_1^2 {\\rm sin}^2 \\theta_1), ~~~\nj_{\\theta_2 \\theta_2} = F_{4} (1+f_2^2 {\\rm sin}^2\n\\theta_2)\\nonumber \\nd where $\\epsilon$ is a finite but small number\\footnote{Of course $\\epsilon < 1$ to preserve the \nsignature. \nIn fact introducing $\\epsilon$ in\n$j_{\\psi\\psi}$ will help us not only to keep $f_i$ large to satisfy SYZ but also satisfy the susy conditions\nin the mirror dual. This will become clear soon.}.\nLet\nus also define another quantity $\\alpha$ in the following way:\n\\bg\\label{alphadef} \\alpha^{-1} ~\\equiv~ {F_3 F_4 {\\rm\nsin}^2\\theta_1 {\\rm sin}^2\\theta_2 + F_2 F_4 {\\rm cos}^2\\theta_1{\\rm\nsin}^2\\theta_2 + F_2 F_3 {\\rm sin}^2\\theta_1 {\\rm cos}^2\\theta_2}\n\\nd away from the point ($\\theta_1, \\theta_2$) $=0$. Now assuming\nthat $f_1, f_2$ are large, we can perform the mirror\ntransformation along ($\\psi, \\phi_1, \\phi_2$) directions. The mirror metric in type IIA takes the following\nform: \\bg\\label{mirrormet} ds^2_{\\rm mirror} = F_0 ds^2_{0123} +\nds^2_6 \\nd where the six-dimensional internal space is a\nnon-K\\\"ahler deformation of the deformed conifold in the following\nway: \\bg\\label{nkdefco} &&ds^2_6 = F_1 dr^2 + {\\alpha F_2 \\over\n\\Delta_1 \\Delta_2} \\Big[d\\psi-b_{\\psi r}dr + \\Delta_1 {\\rm\ncos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1\nr}dr\\Big)\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+ \\Delta_2\n{\\rm cos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2 r}dr\\Big)\\Big]^2\\nonumber\\\\\n&& ~~~~~~ + \\alpha j_{\\phi_2\\phi_2}\\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)^2 + \\alpha\nj_{\\phi_1\\phi_1}\\Big(d\\phi_2 - b_{\\phi_2\\theta_2}\nd\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\nonumber\\\\\n&& ~~~~~~ -2\\alpha j_{\\phi_1\\phi_2} \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\\\\n&& ~~~~~~ -2\\alpha j_{\\phi_1\\phi_2} \\left({f_1 f_2 \\epsilon\\over\n\\alpha}\\right) d\\theta_1 d\\theta_2 + \\Big(F_3 - \\epsilon ~F_2 f_1^2\n{\\rm cos}^2 \\theta_1\\Big)d\\theta_1^2 + \\Big(F_4 - \\epsilon ~F_2\nf_2^2 {\\rm cos}^2 \\theta_2\\Big)d\\theta_2^2\\nonumber \\nd and we have\ndefined $\\Delta_i$ in the following way: \\bg\\label{deltadef}\n\\Delta_1 ~ = ~ \\alpha F_2 F_4 {\\rm sin}^2 \\theta_2, ~~~ \\Delta_2 ~ =\n~ \\alpha F_2 F_3 {\\rm sin}^2 \\theta_1 \\nd At this stage we can\nextract the consequence of the fact that both $f_1$ and $f_2$ are\nvery large. This fits perfectly well with the mirror metric because\n$f_i^2$ as well as $f_1f_2$ come with the coefficient $\\epsilon$ allowing us to satisfy both SYZ and\nsusy in the mirror.\nThis means that if we impose the following constraint:\n\\bg\\label{consf1f2} f_1 f_2 \\epsilon ~ \\equiv ~ -\\alpha \\nd i.e both\n$f_i$ proportional to $\\epsilon^{-1\/2}$, it will bring the\ncross-terms in the metric to the following suggestive form:\n\\bg\\label{suggest} 2\\alpha j_{\\phi_1\\phi_2} \\Big[d\\theta_1 d\\theta_2\n- \\Big(d\\phi_1 - b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1\nr}dr\\Big)\\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big] \\nd In the limit $b_{\\phi_1\\alpha} =\nb_{\\phi_2\\alpha} =0$ with $\\alpha = r, \\theta_i$, \\eqref{suggest} is in fact a term of the\ndeformed conifold! The above conclusion seems rather encouraging,\nprovided of course \\eqref{consf1f2} is satisfied. In the local\nlimit, similar condition also arose (see the first reference of\n\\cite{gtpapers}) and we argued therein that as long as we can define\n\\bg\\label{fiear} f_i ~\\propto ~ {(-1)^i \\langle\\alpha\\rangle_i \\over\n\\sqrt{\\epsilon}} \\nd where $\\langle\\alpha\\rangle_i$ depend only on\n$\\theta_i$ the constraint \\eqref{consf1f2} is satisfied. Therefore a\ncondition like \\eqref{consf1f2} works perfectly well in the local\ncase. Question is, can we satisfy \\eqref{consf1f2} also for the\nglobal case?\n\nThe answer is now tricky. We demanded that $f_i = f_i(\\theta_i)$, otherwise global coordinate transformation like\n\\eqref{tranbg} {\\it cannot} be defined. This means that $F_i$ appearing in the definition of $\\alpha$ in \\eqref{consf1f2}\nwill have to be highly constrained. Generically this is not possible\\footnote{For the local case $\\alpha$ was\ndefined at $r = r_0$ so this subtlety did not arise and, as we discussed above, we used $\\langle\\alpha\\rangle_i$ to\ndefine $f_i$ so things were perfectly consistent there.} because of the underlying type IIA supersymmetry,\nas constraints on the\ntorsion classes \\cite{torsion} will tend to fix $f_i(\\theta_i)$,\nbut in special case this may happen.\n\nThe special case arises if we allow $F_2$ to depend on the angular coordinates $\\theta_i$\nalso in such a way that the susy constraints on the torsion classes are still satisfied with $F_2$ given by:\n\\bg\\label{f2def}\nF_2(r, \\theta_1, \\theta_2) ~= ~ - {(\\beta_1\\beta_2)^{-1} + F_3 F_4 {\\rm sin}^2\\theta_1 {\\rm sin}^2\\theta_2 \\over\nF_4 {\\rm cos}^2\\theta_1 {\\rm sin}^2\\theta_2 + F_3 {\\rm sin}^2\\theta_1 {\\rm cos}^2\\theta_2}\n\\nd\nwhere $f_i \\equiv {{\\beta}_i\\over \\sqrt{\\epsilon}}$. This tells us that the radial dependence of $F_2$ is fixed by\n$F_3(r)$ and $F_4(r)$, but the angular dependences are pretty much unfixed because $\\beta_i(\\theta_i)$ are arbitrary\nfunctions of $\\theta_i$ respectively\\footnote{Interestingly we can make both $\\beta_i$ and $\\epsilon$ to be\ngeneric functions of the internal coordinates in such a way that $f_i \\equiv {{\\beta}_i\\over \\sqrt{\\epsilon}}$\nremain functions of $\\theta_i$ {\\it only}. The only bound on $\\epsilon$ would be that it never exceeds 1 over any point \nin the internal space.}.\nAll these of course should get fixed once we impose the susy constraints on the\ntorsion classes.\nHowever the above relation \\eqref{f2def} already looks tight, but lets move\non and see how far we can go with these kind of arguments. Our next\nquestion would therefore be: is there a way to fix the angular dependences also?\n\nTo see how to fix the angular dependences, we can go back to the equivalent local limit of\n\\eqref{suggest} where the particular way of writing the\nmetric allows us to make a coordinate rotation to bring the term \\eqref{suggest} into the more familar deformed\nconifold form \\cite{minasian}. This, as we know from \\cite{minasian, gtpapers}, is only possible iff {\\it other}\nterms in the metric remain invariant under the coordinate transformation. If this condition is\nimposed globally, then it would imply the\nfollowing two relations:\n\\bg\\label{2rels}\n&& \\beta_1 ~ = ~ \\pm\\sqrt{F_3 - \\alpha j_{\\phi_2\\phi_2}\\over F_2 {\\rm cos}^2 \\theta_1}\\nonumber\\\\\n&& \\beta_2 ~ = ~ \\mp \\sqrt{F_4 - \\alpha j_{\\phi_1\\phi_1}\\over F_2\n{\\rm cos}^2 \\theta_1} \\nd In the local case, studied in the first\nreference of \\cite{gtpapers}, relations like \\eqref{2rels} are\nconsistent in the sense that \\eqref{consf1f2} is satisfied.\nUnfortunately, this is no longer true for the global case\ngenerically because the above relation along with \\eqref{consf1f2}\nwould lead to inconsistent set of equations, and would probably break susy.\nTherefore in general\nthe mirror metric will take the following form: \\bg\\label{mirmet}\n&&ds^2_6 = F_1 dr^2 + {\\alpha F_2 \\over \\Delta_1 \\Delta_2}\n\\Big[d\\psi-b_{\\psi r}dr + \\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\nonumber\\\\\n&& \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+ \\Delta_2 {\\rm\ncos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]^2\\nonumber\\\\\n&& ~~~~~~ + \\alpha j_{\\phi_2\\phi_2}\\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)^2 + \\alpha\nj_{\\phi_1\\phi_1}\\Big(d\\phi_2 - b_{\\phi_2\\theta_2}\nd\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\nonumber\\\\\n&& ~~~~~~ -2\\alpha j_{\\phi_1\\phi_2} \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2 r}dr\\Big)\\nonumber\\\\ &&\n~~~~~~ -2 j_{\\phi_1\\phi_2} \\beta_1 \\beta_2~ d\\theta_1 d\\theta_2 +\n\\Big(F_3 -  F_2 \\beta_1^2 {\\rm cos}^2 \\theta_1\\Big)d\\theta_1^2 +\n\\Big(F_4 - F_2 \\beta_2^2 {\\rm cos}^2\n\\theta_2\\Big)d\\theta_2^2\\nonumber\\\\ \\nd Only in very special cases,\nwhere \\eqref{2rels} and \\eqref{consf1f2} are both simultaneously\nsatisfied, we expect the mirror to take the following symmetric\nform: \\bg\\label{mirmetspecial} ds^2_6 = && F_1 dr^2 + {\\alpha F_2\n\\over \\Delta_1 \\Delta_2} \\Big[d\\psi -b_{\\psi r}dr+ \\Delta_1 {\\rm\ncos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1\nr}dr\\Big)\n\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\;+ \\Delta_2 {\\rm\ncos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]^2\\nonumber\\\\\n&& + \\alpha j_{\\phi_2\\phi_2}\\Big[d\\theta_1^2 + \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)^2\\Big] \\nonumber\\\\\n&&+ \\alpha j_{\\phi_1\\phi_1}\\Big[d\\theta_2^2 + \\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\Big]\\nonumber\\\\\n&& + 2\\alpha j_{\\phi_1\\phi_2}\\Big[d\\theta_1 d\\theta_2 - \\Big(d\\phi_1\n- b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2 r}dr\\Big)\\Big]\\nonumber\\\\ \\nd\nwhich is strongly reminiscent of the deformed conifold!\nObserve that both the forms of the metrics are finite and well defined.\nThis tells us that our procedure of making the base large before performing\nSYZ \\cite{syz} is logical and correct.\n\nOn the other hand, the cross term that we developed in the metric appears as\nthe $B_{\\rm NS}$ field in type IIA theory. Expectedly, this B-field is large\nand is given by the following form:\n\\begin{eqnarray}\\label{BinIIA}\n\\widetilde{B}&&=~\\alpha\nf_1F_3\\sin^2\\theta_1\\left(F_2\\cos^2\\theta_2+F_4\\sin^2\\theta_2\\right)d\\theta_1\\wedge d\\phi_1\\nonumber\\\\\n&&~+\\alpha\nf_2F_4\\sin^2\\theta_2\\left(F_2\\cos^2\\theta_1+F_3\\sin^2\\theta_1\\right)d\\theta_2\\wedge d\\phi_2\\nonumber\\\\\n&&~+ \\left(1-{\\epsilon \\over \\alpha\nF_2F_4\\sin^2\\theta_1\\sin^2\\theta_2}\\right)\n\\left(f_1\\cos\\theta_1d\\theta_1+f_2\\cos\\theta_2d\\theta_2\\right)\\wedge d\\psi\n\\end{eqnarray}\nIn the limit $\\epsilon \\to 0$, the last two terms are pure gauge. For finite, but small, $\\epsilon < 1$ they cannot be\ngauged away.\nIn the local limit (see the first paper of\n\\cite{gtpapers}) all the $F_i$ were constants, and so $\\widetilde{B}$ became a pure gauge when written in terms\nof $\\langle\\alpha\\rangle_i$. This doesn't seem to be the case globally, unless of course $F_i$'s are of some specific\nforms.\n\nThe wrapped D6 brane two-form charges now come partly from the type IIB three-forms and partly from the five-forms.\nThe three-forms contributions to the IIA two-forms are given by the following\ncomponents:\n\\begin{eqnarray}\\label{2form1}\n&&\\widetilde{F}_{\\psi\\theta_1}=F_{\\phi_1\\phi_2\\theta_1},\\quad\n\\widetilde{F}_{\\psi\\theta_2}=F_{\\phi_1\\phi_2\\theta_2},\\quad\n\\widetilde{F}_{\\psi r}=F_{\\phi_1\\phi_2 r},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_1r}=F_{r\\phi_2\\psi}+\\frac{2j_{\\phi_1\\phi_2}}{j_{\\phi_1\\phi_1}}F_{\\phi_1\nr \\psi}+\\frac{2j_{\\psi\\phi_1}}{j_{\\phi_1\\phi_1}}F_{\nr \\phi_1\\phi_2},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_2 r}=F_{\\phi_1 r\\psi}+2\\alpha\n(j_{\\phi_2\\psi}j_{\\phi_1\\phi_1}-j_{\\phi_1\\phi_2}j_{\\phi_1\\psi})F_{r\\phi_1\\phi_2},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_1\\theta_2}=F_{\\psi\\theta_2\\phi_2}+2\n\\frac{j_{\\phi_1\\psi}}{j_{\\phi_1\\phi_1}}F_{\\phi_1\\phi_2\\theta_2},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_1\\theta_1}=2\\frac{j_{\\phi_1\\phi_2}}{j_{\\phi_1\\phi_1}}F_{\\phi_1\\theta_1\\psi}\n+2\\frac{j_{\\psi\\phi_1}}{j_{\\phi_1\\phi_1}}F_{\\phi_1\\phi_2\\theta_1},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_2\\theta_1}=F_{\\psi\\phi_1\\theta_1}+2\\alpha\n(j_{\\phi_2\\psi}j_{\\phi_1\\phi_1}-j_{\\phi_1\\phi_2}j_{\\phi_1\\psi})F_{\\phi_1\\phi_2\\theta_1},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_2\\theta_2}=2\\alpha\n(j_{\\phi_2\\psi}j_{\\phi_1\\phi_1}-j_{\\phi_1\\phi_2}j_{\\phi_1\\psi})F_{\\phi_1\\phi_2\\theta_2}\n\\end{eqnarray}\nSimilarly, the five-forms contributions to the type IIA two-forms are given in terms of the following components:\n\\begin{eqnarray}\\label{2form2}\n&&\\widetilde{F}_{\\theta_1\\theta_2}=F_{\\psi\\theta_1\\theta_2\\phi_1\\phi_2}+b_{\\theta_2\\phi_2}F_{\\psi\\theta_1\\phi_1}\n+b_{\\theta_1\\phi_1}F_{\\psi\\theta_2\\phi_2}\\nonumber\\\\\n&&\\widetilde{F}_{r\\theta_1}=F_{r\\theta_1\\phi_1\\phi_2\\psi}+b_{\\theta_1\\phi_1}F_{r\\psi\\phi_2}\n+\\left(b_{\\phi_2r}+\\frac{j_{\\phi_1\\phi_2}}{j_{\\phi_1\\phi_1}}b_{r\\phi_1}\\right)F_{\\psi\\theta_1\\phi_1}\n+\\frac{j_{\\psi\\phi_1}}{j_{\\phi_1\\phi_1}}b_{\\theta_1\\phi_1}F_{r\\phi_1\\phi_2}\\nonumber\\\\\n&&~~~~~~~~~~~+\\left(b_{r\\psi}-\\frac{j_{\\psi\\phi_1}}{j_{\\phi_1\\phi_1}}b_{r\\phi_1}\\right)F_{\\theta_1\\phi_1\\phi_2}\n+\\frac{j_{\\phi_1\\phi_2}}{j_{\\phi_1\\phi_1}}b_{\\theta_1\\phi_1}F_{r\\psi\\phi_1}\\nonumber\\\\\n&&\\widetilde{F}_{r\\theta_2}=F_{r\\theta_2\\phi_1\\phi_2\\psi}+b_{r\\phi_1}F_{\\psi\\theta_2\\phi_2}\n+b_{r\\psi}F_{\\theta_2\\phi_1\\phi_2}-b_{\\theta_2\\phi_2}F_{r\\psi\\phi_1}\n\\end{eqnarray}\nAll the above components are finite and give rise to the required D6-branes charges. However since the B-field is\nlarge, to compensate this in the EOMs we need large G-fluxes in type IIA. These fluxes come exactly from the\nextra three- and five-form components \\eqref{extra3} and \\eqref{extra5} respectively.\nThese three- and five-form components give rise to twelve components of the four-form\nfluxes in IIA namely:\n\\bg\\label{twelve}\n&&\\widetilde{F}_{r\\psi\\theta_1\\theta_2},~~~\n\\widetilde{F}_{r\\psi\\theta_1\\phi_1}, ~~~\n\\widetilde{F}_{r\\psi\\theta_1\\phi_2}, ~~~\n\\widetilde{F}_{r\\psi\\theta_2\\phi_1}\\nonumber\\\\\n&&\\widetilde{F}_{r\\psi\\theta_2\\phi_2},~~~\n\\widetilde{F}_{r\\theta_1\\theta_2\\phi_1}, ~~~\n\\widetilde{F}_{r\\theta_1\\theta_2\\phi_2}, ~~~\n\\widetilde{F}_{r\\theta_1\\phi_1\\phi_2}\\nonumber\\\\\n&&\\widetilde{F}_{r\\theta_2\\phi_1\\phi_2}, ~~~\n\\widetilde{F}_{\\psi\\theta_1\\theta_2\\phi_1},~~~\n\\widetilde{F}_{\\psi\\theta_1\\theta_2\\phi_2},~~~\n\\widetilde{F}_{\\theta_1\\theta_2\\phi_1\\phi_2}\n\\nd\nThese components are listed in {\\bf Appendix 2} which the readers may refer to for details. Combined with \\eqref{BinIIA},\nthese fluxes lift to M-theory as G-fluxes with components along the spacetime and the eleventh directions respectively.\nInterestingly, both the metric \\eqref{mirmet} or \\eqref{mirmetspecial} along with the two-form flux components\n\\eqref{2form1} and \\eqref{2form2} lift to a geometrical configuration in M-theory, which we expect to have a\n$G_2$ structure. This is of course expected because both the non-K\\\"ahler deformed conifold as well as the wrapped\nD6-branes tend to become geometrical configurations when the type IIA coupling is made very \nlarge\\footnote{The fact that we get \\eqref{mirmet} instead of \\eqref{mirmetspecial} is nothing too surprising. \nOne {\\it does not} expect to get a K\\\"ahler deformed conifold in type IIA. This was already clear from the \npioneering work of \\cite{vafa}. Here we not only seem to confirm the statement of \\cite{vafa} but also determine \nthe precise form of the IIA metric.}.  \nIn the\nfollowing sub-section we will dwell on this in more details.\n\n\n\\subsection{M theory lift, flop transition and type IIA reduction}\n\nThe lift of our type IIA mirror configuration to M-theory is rather straighforward. The eleven-directional fibration\nis given by gauge fluxes derived from the two-form components \\eqref{2form1} and \\eqref{2form2}. It is easy to\nshow that we need only $A_{\\phi_i}, A_{\\theta_i}$ and $A_r$ components. Using these, the M-theory lift of our IIA\nsymmetric mirror metric \\eqref{mirmetspecial} is:\n\\begin{eqnarray}\\label{mlift}\nds^2_{11}=&&e^{-{2\\phi\\over 3}}\\Bigg\\{F_0ds_{0123}^2+F_1 dr^2 + {\\alpha F_2\n\\over \\Delta_1 \\Delta_2} \\Big[d\\psi -b_{\\psi r}dr\\nonumber\\\\\n&&+ \\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)+ \\Delta_2 {\\rm cos}~\\theta_2\n\\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]^2\\nonumber\\\\\n&& + \\alpha j_{\\phi_2\\phi_2}\\Big[d\\theta_1^2 + \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)^2\\Big] \\nonumber\\\\\n&&+ \\alpha j_{\\phi_1\\phi_1}\\Big[d\\theta_2^2 + \\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\Big]\\nonumber\\\\\n&& + 2\\alpha j_{\\phi_1\\phi_2}\\Big[d\\theta_1 d\\theta_2 - \\Big(d\\phi_1\n- b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]\\Bigg\\}\\nonumber\\\\\n&&+e^{4\\phi\\over 3}\\Big[dx_{11}+A_{\\phi_1}d\\phi_1+A_{\\phi_2}d\\phi_2+A_{\\theta_1}d\\theta_1\n+A_{\\theta_2}d\\theta_2+A_rdr\\Big]^2\n\\end{eqnarray}\nIt is easy to see that the non-symmetric mirror metric \\eqref{mirmet} will also lift to M-theory in an\nidentical way. The local limit of the lift of\n\\eqref{mirmet} is precisely the one discussed in the first paper of\n\\cite{gtpapers} and, as discussed therein, we expect the lift of \\eqref{mirmet} to have a $G_2$ structure to\npreserve supersymmetry. To see this for our case, we have to express the lift of the metric \\eqref{mirmet} in terms of\ncertain one-forms similar to the ones given in \\cite{gtpapers} (see also \\cite{brandhuber}). Following the\nfirst paper of \\cite{gtpapers} we first express the B-fields appearing in the fibration \\eqref{mirmet} in terms\nof periodic angular coordinates $\\lambda_i$ in the following way:\n\\begin{eqnarray}\\label{angles}\n&&\\tan\\lambda_1 ~\\equiv ~b_{\\phi_1\\theta_1},\\quad\n\\tan\\lambda_2~\\equiv~b_{\\phi_2\\theta_2},\\quad \\tan\\lambda_3 ~\\equiv~b_{\\psi\nr}\\nonumber\\\\\n&&\\tan\\lambda_4 ~\\equiv ~b_{\\phi_1 r},\\quad \\;\\,\\tan\\lambda_5 ~\\equiv~ b_{\\phi_2 r}\n\\end{eqnarray}\nUsing these we can define two sets of one-forms. The first set, called $\\sigma_i$ with $i = 1, .., 3$, can be expressed\nin terms of $\\lambda_i$ as:\n\\begin{eqnarray}\\label{lift1}\n\\sigma_1&=&\\sin\\psi_1(d\\phi_1-\\tan\\lambda_4dr)+\\sec\\lambda_1\\cos(\\psi_1+\\lambda_1)d\\theta_1,\\nonumber\\\\\n\\sigma_2&=&\\cos\\psi_1(d\\phi_1-\\tan\\lambda_4dr)-\\sec\\lambda_1\\sin(\\psi_1+\\lambda_1)d\\theta_1,\\nonumber\\\\\n\\sigma_3&=&d\\psi_1-\\frac{1}{2}\\tan\\lambda_3dr+\\Delta_1\\cos\\theta_1(d\\phi_1-\\tan\\lambda_1d\\theta_1-\\tan\\lambda_4dr)\n\\end{eqnarray}\nand the second set can be expressed in terms of $\\lambda_i$ as:\n\\begin{eqnarray}\\label{lift2}\n\\Sigma_1&=&-\\sin\\psi_2(d\\phi_2-\\tan\\lambda_5dr)+\\sec\\lambda_2\\cos(\\psi_2+\\lambda_2)d\\theta_2,\\nonumber\\\\\n\\Sigma_2&=&-\\cos\\psi_2(d\\phi_2-\\tan\\lambda_5dr)-\\sec\\lambda_2\\sin(\\psi_2+\\lambda_2)d\\theta_2,\\nonumber\\\\\n\\Sigma_3&=&d\\psi_2+\\frac{1}{2}\\tan\\lambda_3dr-\\Delta_2\\cos\\theta_2(d\\phi_2-\\tan\\lambda_2d\\theta_2-\\tan\\lambda_5dr)\n\\end{eqnarray}\nAt this stage one may compare these two sets of one-forms to the ones given by eq. (6.2) and eq. (6.3) in the first\npaper of \\cite{gtpapers}. The definitions of $\\psi_1$ and $\\psi_2$ follow exactly as in \\cite{gtpapers}, i.e\n\\bg\\label{psi12}\nd\\psi ~ = ~ d\\psi_1 - d\\psi_2, ~~~~~~~~~ dx_{11} ~ = ~ d\\psi_1 + d\\psi_2\n\\nd\nFurthermore we can perform the following rotation of the coordinates:\n\\begin{equation}\\label{rotc}\n\\begin{pmatrix} {\\cal D}\\phi_2 \\\\ d\\theta_2\\end{pmatrix} ~ \\to ~\n\\begin{pmatrix} \\cos~\\psi_0 & -\\sin~\\psi_0 \\\\ \\sin~\\psi_0 & ~~\\cos~\\psi_0 \\end{pmatrix}\n\\begin{pmatrix} {\\cal D}\\phi_2 \\\\ d\\theta_2\\end{pmatrix}\n\\end{equation}\nwith ${\\cal D}\\phi_2 \\equiv d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2r}dr$ and $\\psi_0$ a constant. If we make\nthis transformation to the symmetric mirror metric of type IIA \\eqref{mirmetspecial}, this will lift to M-theory not as\n\\eqref{mlift}, but to a more {\\it suggestive} configuration:\n\\begin{eqnarray}\\label{mliftnow}\nds^2_{11}=&&e^{-{2\\phi\\over 3}}\\Bigg\\{F_0ds_{0123}^2+F_1 dr^2 + {\\alpha F_2\n\\over \\Delta_1 \\Delta_2} \\Big[d\\psi -b_{\\psi r}dr - b_{\\psi\\theta_2} d\\theta_2\\nonumber\\\\\n&&+ \\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)+ \\Delta_2 {\\rm cos}~\\theta_2 {\\rm cos}~\\psi_0\n\\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]^2\\nonumber\\\\\n&& + \\alpha j_{\\phi_2\\phi_2}\\Big[d\\theta_1^2 + \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)^2\\Big] \\nonumber\\\\\n&&+ \\alpha j_{\\phi_1\\phi_1}\\Big[d\\theta_2^2 + \\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\Big]\\nonumber\\\\\n&& + 2\\alpha j_{\\phi_1\\phi_2}{\\rm cos}~\\psi_0\\Big[d\\theta_1 d\\theta_2 - \\Big(d\\phi_1\n- b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]\\nonumber\\\\\n&& + 2\\alpha j_{\\phi_1\\phi_2}{\\rm sin}~\\psi_0\\Big[\\Big(d\\phi_1- b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big) d\\theta_2\n+ \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2r}dr\\Big)d\\theta_1\\Big]\\Bigg\\}\\nonumber\\\\\n&&+e^{4\\phi\\over 3}\\Big[dx_{11}+{\\widetilde A}_{\\phi_1}d\\phi_1+{\\widetilde A}_{\\phi_2}d\\phi_2+\n{\\widetilde A}_{\\theta_1}d\\theta_1\n+{\\widetilde A}_{\\theta_2}d\\theta_2+{\\widetilde A}_rdr\\Big]^2\n\\end{eqnarray}\nwhere we have introduced a $B$-field fibration using\n$b_{\\psi\\theta_2} \\equiv \\Delta_2 ~{\\rm sin}~\\psi_0 {\\rm cos}~\\theta_2$ to modify the $d\\psi$ fibration structure.\nThe eleven-dimensional fibration structure will also change accordingly because we can always express the\n$A_{\\phi_2} d\\phi_2$ term in $dx_{11}$ of \\eqref{mlift} using ${\\cal D}\\phi_2$. Thus the overall eleven-dimensional\nfibration will retain its form but with shifted $A_\\mu$ fields denoted above by the ${\\widetilde A}_\\mu$ fields.\nIn terms of the fibration components of \\eqref{mlift} one can\nshow that ${\\widetilde A}_{\\phi_1} = A_{\\phi_1}, {\\widetilde A}_{\\theta_1} = A_{\\theta_1}$ and the rest of the\ncomponents can be presented in the following matrix form:\n\\begin{equation}\\label{matricre}\n\\begin{pmatrix}{\\widetilde A}_{\\phi_2}\\\\ {}&{}&{} \\\\ {\\widetilde A}_{\\theta_2}\\\\ {}&{}&{}\\\\\n{\\widetilde A}_{r} \\end{pmatrix} ~ = ~\n\\begin{pmatrix} \\cos\\psi_0 + b_{\\phi_2\\theta_2}\\sin\\psi_0 & \\sin\\psi_0 & 0\\\\ {}&{}&{}\\\\\n-(1+b^2_{\\phi_2\\theta_2})\\sin\\psi_0 & \\cos\\psi_0 - b_{\\phi_2\\theta_2}\\sin\\psi_0 & 0\\\\ {}&{}&{}\\\\\nb_{\\phi_2 r}(1-\n\\cos\\psi_0 - b_{\\phi_2\\theta_2}\\sin\\psi_0) & -b_{\\phi_2 r} \\sin\\psi_0 &1 \\end{pmatrix}\n\\begin{pmatrix}{A}_{\\phi_2}\\\\ {}&{}&{}\\\\ {A}_{\\theta_2}\\\\ {}&{}&{}\\\\ {A}_{r} \\end{pmatrix}\n\\end{equation}\nAdditionally with the above modification, the\nabove metric is surprisingly close to the uplift of a non-K\\\"ahler deformed conifold metric with wrapped D6-branes\nto M-theory provided we can make an additional substitution\\footnote{This is compatible with the\nunderlying $G_2$ structure. An analysis of the $G_2$ torsion classes, along the lines of\nthe first paper in \\cite{gtpapers}, will reveal this. We will discuss this more soon.} in \\eqref{mliftnow}:\n\\bg\\label{psis}\n\\psi_0 ~ ~\\to ~~ \\psi\n\\nd\nMaking such a substitution may lead one to think that the $\\psi$ isometry that we have in \\eqref{mlift} is removed.\nThis is {\\it not} the case locally\nwith the non-K\\\"ahler deformed conifold because the extra B-field component $b_{\\psi\\theta_2}$\nin the $d\\psi$ fibration structure of \\eqref{mliftnow} as well as the vector fields ${\\widetilde A}_\\mu$\nin the $dx_{11}$ fibration structure transform non-trivially under\nshift in $\\psi$ to restore the isometry. One may also do a somewhat similar rotation like \\eqref{rotc} to the\nnon-symmetric type IIA metric \\eqref{mirmet} and bring it in a more suggestive format.\n\nThe rotation \\eqref{rotc} should now be captured by\nthe one-forms \\eqref{lift1} and \\eqref{lift2} appropriately. In fact only the\nsecond set of one-forms \\eqref{lift2} is related to the\nchange \\eqref{rotc}.\nThus the one-forms for\nour purposes will be ($\\sigma_i, \\Sigma_i$) with $\\Sigma_i$ to be viewed as the one got from \\eqref{rotc} directly.\nUsing these one-forms\nwe can rewrite the M-theory metric in two possible ways. The first one is the lift of the non-symmetric type IIA\nmetric \\eqref{mirmet} under the rotation \\eqref{rotc} and transformation \\eqref{psis}:\n\\begin{eqnarray}\\label{liftnow}\nds_7^2 && = g_rdr^2+g_1(\\sigma_3+\\Sigma_3)^2+g_2(\\sigma_3-\\Sigma_3)^2\\nonumber\\\\\n&&+~g_3(\\sin\\psi_1\\sigma_1+\\cos\\psi_1\\sigma_2)^2\n+\\widetilde{g}_3(\\cos\\psi_1\\sigma_1-\\sin\\psi_1\\sigma_2)^2\\nonumber\\\\\n&&+~g_4(\\sin\\psi_2\\Sigma_1+\\cos\\psi_2\\Sigma_2)^2\n+\\widetilde{g}_4(\\cos\\psi_2\\Sigma_1-\\sin\\psi_2\\Sigma_2)^2\\nonumber\\\\\n&&+~g_5(\\sin\\psi_1\\sigma_1+\\cos\\psi_1\\sigma_2)(\\sin\\psi_2\\Sigma_1+\\cos\\psi_2\\Sigma_2)\\nonumber\\\\\n&&-~\\widetilde{g}_5(\\cos\\psi\\sigma_1-\\sin\\psi_1\\sigma_2)(\\cos\\psi_2\\Sigma_1-\\sin\\psi_2\\Sigma_2)\n\\end{eqnarray}\nwhere we have defined the coefficients $g_i, {\\widetilde g}_i$ as:\n\\begin{eqnarray}\\label{ggdef}\n&&g_r=e^{-2\\phi\/3}F_1,\\quad g_1=e^{4\\phi\/3},\\quad\ng_2=e^{-2\\phi\/3}\\frac{\\alpha F_2}{\\Delta_1\\Delta_2},\\quad g_3=\\alpha\nj_{\\phi_2\\phi_2},\n\\nonumber\\\\\n&&\\widetilde{g}_3=F_3-F_2\\beta_1^2\\cos^2\\theta_1,\\quad g_4=\\alpha\nj_{\\phi_1\\phi_1},\\quad\n\\widetilde{g}_4=F_4-F_2\\beta_2^2\\cos^2\\theta_2\\nonumber\\\\\n&&g_5=2\\alpha j_{\\phi_1\\phi_2},\\quad\n\\widetilde{g}_5=2\\beta_1\\beta_2j_{\\phi_1\\phi_2}\n\\end{eqnarray}\nThe second way to rewrite the metric is a little more suggestive of the way to perform the flop operation on the M-theory\nmanifold and has a nice form for the symmetric case \\eqref{mirmetspecial}, again under \\eqref{rotc} and \\eqref{psis}.\nThe local form of this has already appeared in the first reference of \\cite{gtpapers}, and the readers may\nwant to look at that for more details. In fact we can rewrite \\eqref{liftnow} in the following\nway also.\nHere we will simply quote the generic ansatze using parameters $\\alpha_i$ and $\\zeta$:\n\\bg\\label{sugmet}\nds_7^2 = \\alpha_1^2 \\sum_{a=1}^2 (\\sigma_a + \\zeta \\Sigma_a)^2 + \\alpha_2^2 \\sum_{a=1}^2 (\\sigma_a - \\zeta \\Sigma_a)^2\n+ \\alpha_3^2 (\\sigma_3 + \\Sigma_3)^2 + \\alpha_4^2 (\\sigma_3 - \\Sigma_3)^2 + \\alpha_5^2 dr^2\\nonumber\\\\\n\\nd\nThe above is a familiar form by which any $G_2$ structure metric could be expressed. Once we switch off $\\lambda_i$ the\nmanifolds has a $G_2$ holonomy. The coefficients $\\alpha_i$ and $\\zeta$ are not arbitrary. They are fixed by the EOM\nand, for the case \\eqref{mirmetspecial}, they take the following values:\n\\bg\\label{alphazeta}\n&& \\alpha_1 = {1\\over 2} e^{-{\\phi\\over 3}}\\sqrt{2\\alpha\\left(j_{\\phi_2\\phi_2} + {j_{\\phi_1\\phi_2}\\over \\zeta}\\right)},\n~~~~~ \\alpha_2 = {1\\over 2} e^{-{\\phi\\over 3}}\\sqrt{2\\alpha\\left(j_{\\phi_2\\phi_2}\n- {j_{\\phi_1\\phi_2}\\over \\zeta}\\right)}\\nonumber\\\\\n&& \\alpha_3 = e^{2\\phi\\over 3}, ~~~~ \\alpha_4 = e^{-{\\phi\\over 3}} \\sqrt{\\alpha F_2 \\over \\Delta_1 \\Delta_2}, ~~~~\n\\alpha_5 =  e^{-{\\phi\\over 3}} \\sqrt{F_1}, ~~~~ \\zeta = \\sqrt{j_{\\phi_1\\phi_1}\\over j_{\\phi_2\\phi_2}}\n\\nd\nBefore proceeding further let us pause for a while and ask whether the above set of manipulations would preserve\nsupersymmetry. From type IIA point of view we have done the following:\n\n\\vskip.1in\n\n\\noindent $\\bullet$ Shift of the coordinates ($\\psi, \\phi_i$) using variables $f_i(\\theta_i)$. This shifting of the\ncoordinates mixes non-trivially all the three isometry directions as described in \\eqref{tranbg}.\n\n\\noindent $\\bullet$ Shift the metric along $\\psi$ direction by the variable $\\epsilon$, as given in the second line\nof \\eqref{metcom}. This variable doesn't have to be too small in the global limit. As long as it is smaller than 1 \nit'll suffice.\n\n\\noindent $\\bullet$ Make SYZ transformations along the new shifted directions. Thus the three T-dualities are\n{\\it not} made along the three original isometry directions.\n\n\\noindent $\\bullet$ In the new metric of IIA make a further rotation along the ($\\theta_2, \\phi_2$)\ndirections using a $2\\times 2$\nmatrix given as \\eqref{rotc}. The matrix is described using a constant angular variable $\\psi_0$.\n\n\\noindent $\\bullet$ Finally in the transformed metric convert $\\psi_0$ to $\\psi$ as in \\eqref{psis}.\n\n\\vskip.1in\n\n\\noindent However due to steps 2, 4 and 5 above, it is not guaranteed that the metric will preserve supersymmetry.\nFurthermore one might also question whether the SYZ operation itself could preserve supersymmetry. Therefore to verify\nthis we have evaluated all the torsion classes for this background in sec 3.2 of \\cite{chen2}. The supersymmetry\nconstraints are given by eq. (3.21) and Appendix B of \\cite{chen2}. These set of equations along with the constraint\nequation (3.15) of \\cite{chen2}  are enough to guarantee supersymmetry in the type IIA (or the equivalent\nM-theory) background.\n\nOnce the issue of supersymmetry is resolved, we go to the flop operation.\nThe operation of flop on the above metric \\eqref{sugmet} has already been discussed in details in sec. 7 of the\nfirst reference of \\cite{gtpapers}. Using similar techniques for \\eqref{liftnow},\nafter the flop we expect the metric to look like:\n\\begin{eqnarray}\\label{aflop}\nds_7^2=a_1(\\sigma_1^2+\\sigma_2^2)+a_2(\\Sigma_1^2+\\Sigma_2^2)+a_3(\\sigma_3+\\Sigma_3)^2+a_4(\\sigma_3-\\Sigma_3)^2+a_5dr^2\n\\end{eqnarray}\nwith $a_i$, $i=1, ..., 5$ are some coefficients to be determined. Due to the global nature of our metric,\nthe operation of flop can be performed by\na class of transformations parametrized by the values of\n$a$, $b$ etc. in the following way:\n\\begin{eqnarray}\\label{lbaaz}\n&&\\sigma_1\\mapsto a\\sigma_1+b\\Sigma_1,\\quad \\Sigma_1\\mapsto\ne\\sigma_1+f\\Sigma_1,\\nonumber\\\\\n&&\\sigma_2\\mapsto c\\sigma_2+d\\Sigma_2,\\quad \\Sigma_2\\mapsto\ng\\sigma_2+h\\Sigma_2,\\nonumber\\\\\n&&\\sigma_3+\\Sigma_3\\mapsto \\sigma_3-\\Sigma_3, \\quad\n\\sigma_3-\\Sigma_3\\mapsto \\sigma_3+\\Sigma_3\n\\end{eqnarray}\nNow comparing \\eqref{sugmet} and \\eqref{liftnow} one can pretty much fix the coefficients $c, d$ etc. in terms of\n$a, b$ in the following way:\n\\bg\\label{cdef}\n&& c=a\\sqrt{\\frac{k^2G_2+kG_3+G_1}{\\omega^2G_5+\\omega\nG_6+G_4}},~~~~~~~ d=b\\sqrt{\\frac{\\mu^2G_2+\\mu G_3+G_1}{\\tau^2G_5+\\tau\nG_6+G_4}}\\nonumber\\\\\n&& e=ak,~~~~ g=c\\omega,~~~~ f=b\\mu,~~~~ h=d\\tau\n\\nd\nwhere $a, b$ can in turn be fixed by computing the $G_2$-torsion classes and demanding supersymmetry (see also\n\\cite{chen2}).\nThe other coefficients appearing above, namely, $k$,\n$\\omega$, $\\mu$, $\\tau$ satisfy the following equations:\n\\begin{eqnarray}\n&&2G_1+2k\\mu G_2+(k+\\mu)G_3=0, \\quad~ G_7+k\\tau G_8+\\tau G_9+kG_{10}=0,\\nonumber\\\\\n&&2G_4+2\\tau \\omega G_5+(\\tau+\\omega)G_6=0, \\quad G_7+\\omega\\mu\nG_8+\\omega G_9+\\mu G_{10}=0.\n\\end{eqnarray}\nwhose solutions are fixed by the following values of $G_i$ determined from the $G_2$ structure metric \\eqref{liftnow}\nusing ($g_i, \\widetilde{g}_i$) defined earlier in \\eqref{ggdef}:\n\\begin{eqnarray}\n&&G_1=g_3\\sin^2\\psi_1+\\widetilde{g}_3\\cos^2\\psi_1,\\quad\\quad\nG_2=g_4\\sin^2\\psi_2+\\widetilde{g}_4\\cos^2\\psi_2, \\nonumber\\\\\n&&G_3=g_5\\sin\\psi_1\\sin\\psi_2-\\widetilde{g}_5\\cos\\psi_1\\cos\\psi_2,\\nonumber\\\\\n&&G_4=g_3\\cos^2\\psi_1+\\widetilde{g}_3\\sin^2\\psi_1,\\quad\\quad\nG_5=g_4\\cos^2\\psi_2+\\widetilde{g}_4\\sin^2\\psi_2, \\nonumber\\\\\n&&G_6=g_5\\cos\\psi_1\\cos\\psi_2-\\widetilde{g}_5\\sin\\psi_1\\sin\\psi_2,\\nonumber\\\\\n&&G_7=(g_3-\\widetilde{g}_3)\\sin\\psi_1\\cos\\psi_1,\\quad\\;\nG_9=g_5\\sin\\psi_1\\cos\\psi_2+\\widetilde{g}_5\\cos\\psi_1\\sin\\psi_2,\\nonumber\\\\\n&&G_8=(g_4-\\widetilde{g}_4)\\sin\\psi_2\\cos\\psi_2,\\quad\\;\nG_{10}=g_5\\cos\\psi_1\\sin\\psi_2+\\widetilde{g}_5\\sin\\psi_1\\cos\\psi_2.\\nonumber\\\\\n\\end{eqnarray}\nUsing all the above relations, the $a_i$ coefficients in the M-theory metric after flop transition \\eqref{lbaaz} can\nbe determined in terms of ($a, b$). The final form of the metric therefore is given by:\n\\begin{eqnarray}\\label{flopmet}\nds_{11}^2&&= e^{-{2\\phi\\over 3}}F_0 ds_{0123}^2+g_rdr^2+g_1(\\sigma_3-\\Sigma_3)^2+g_2(\\sigma_3+\\Sigma_3)^2\\nonumber\\\\\n&&+a^2(k^2G_2+kG_3+G_1)(\\sigma_1^2+\\sigma_2^2)+b^2(\\mu^2G_2+\\mu\nG_3+G_1)(\\Sigma_1^2+\\Sigma_2^2)\\nonumber\\\\\n\\end{eqnarray}\nWe are now one step away from getting the type IIA metric from the above metric.\nReducing along $x_{11}$ the metric takes the following form in type IIA theory:\n\\begin{eqnarray}\\label{IIAmetric}\nds_{10}^2&& =F_0ds_{0123}^2+F_1dr^2+e^{2\\phi}\\Big[d\\psi-b_{\\psi\n\\mu}dx^\\mu+\\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad +\n\\widetilde{\\Delta}_2 {\\rm cos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2}\nd\\theta_2-b_{\\phi_2 r}dr\\Big)\\Big]^2\\nonumber\\\\\n&&~+e^{2\\phi\\over 3}a^2(k^2G_2+kG_3+G_1)\\Big[d\\theta_1^2+(d\\phi_1^2-b_{\\phi_1\\theta_1}d\\theta_1-b_{\\phi_1\nr}dr)^2\\Big]\\nonumber\\\\\n&&~+e^{2\\phi\\over 3}b^2(\\mu^2G_2+\\mu\nG_3+G_1)\\Big[d\\theta_2^2+(d\\phi_2^2-b_{\\phi_2\\theta_2}d\\theta_2-b_{\\phi_2\nr}dr)^2\\Big]\n\\end{eqnarray}\nwhich has an amazing similarity with the warped resolved conifold! The above metric is completely global and\nsupersymmetric\\footnote{Of course there is a further UV completion that we don't discuss here. The\nUV completion should follow in the same vein as studied recently for the\nKlebanov-Strassler case in \\cite{FEP, jpsi}.}. As before, the full torsion class analysis for this is\ngiven in eq. (3.25) of \\cite{chen2}. Together with equations (3.15), (3.21) and (3.25) of \\cite{chen2} we can\npretty much get most of the warp factor components (plus the other parameters) to demand supersymmetry for all the\nabove backgrounds. Any remaining set of unconstrained parameters would allow us to get a class of gauge\ntheory deformations that span the landscape of solutions in the geometric transition set-up. We will discuss more\non this landscape soon (see also figure 3 of \\cite{chen2}).\n\nTherefore the type IIA background\nshould be viewed as the gravity dual in the IR for the gauge theory on wrapped D6-branes before\ngeometric transition. In this background there are no six-branes. The wrapped D6-branes have {\\it dissolved} in the\ngeometry, and is replaced by the following one-form flux components:\n\\bg\\label{1ffc}\n A=\\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)- \\widetilde{\\Delta}_2 {\\rm\ncos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\n\\nd\nwith $\\widetilde{\\Delta}_2$ is a slight deformation of $\\Delta_2$ appearing from the rotation \\eqref{rotc}\nbefore the flop operation. The type IIA background also supports an effective dilaton, that measures the\nIIA coupling, and is given by:\n\\bg\\label{edil}\n\\phi_{\\rm eff}~ = ~ \\frac{3}{4}\\ln(g_2)\n\\nd\nBefore we end this section, there are a few loose ends that need to be tied up. The first one is related to the M-theory\nG-fluxes. These G-fluxes stem from \\eqref{twelve} and \\eqref{BinIIA} in type IIA,\nand they are in general large\\footnote{As we discussed before $\\epsilon$ in \\eqref{metcom} or \\eqref{consf1f2} is a small\nbut {\\it finite} number less than 1 (otherwise the signature will change), \nthe type IIA flux components \\eqref{BinIIA} and \\eqref{twelve} will be large but finite.\nChoosing a particular \nvalue of $\\epsilon$ and then demanding supersymmetry will consequently fix the resulting fluxes. A range of choices for \n$\\epsilon$ will give a class of backgrounds which are dual to certain continuous deformations parametrised \nby $\\epsilon$ in the gauge theory side. We will discuss more about the class of backgrounds later.}.\nIn the\nlocal picture both \\eqref{BinIIA} as well as \\eqref{twelve} components were all pure gauges, and therefore they\ndid not contribute to the background. Here we expect they would, and therefore we need to see how these fluxes behave\nunder:\n\n\\vskip.1in\n\n\\noindent $\\bullet$ The rotation of coordinates \\eqref{rotc} with shift \\eqref{psis}, and\n\n\\noindent $\\bullet$ The flop transformation \\eqref{lbaaz}.\n\n\\vskip.1in\n\n\\noindent Both these effects can be worked out if we can express our fluxes \\eqref{twelve} and \\eqref{BinIIA}\ncompletely in terms of the one-forms \\eqref{lift1} and \\eqref{lift2}. As we noted before, the rotation\n\\eqref{rotc} and shift \\eqref{psis} is appropriately captured by\nthe one-forms \\eqref{lift2}. Therefore to compensate both the changes, namely rotation\n\\eqref{rotc} (with shift \\eqref{psis})\nand the flop \\eqref{lbaaz}, all we need is to express the M-theory lift of the fluxes in terms of\n\\eqref{lift1} and \\eqref{lift2}. Any {\\it additional} $\\psi$ dependent contributions will come out from\nsusy requirement.\n\nTo achieve all this, we can express the differential coordinates completely in terms of $\\sigma_i$ and $\\Sigma_i$.\nSince there are seven differential coordinates ($dr, d\\theta_1, d\\phi_1, d\\theta_2, d\\phi_2, d\\psi_1, d\\psi_2$) but\nsix one-forms ($\\sigma_i, \\Sigma_i$), we can assume $dr$ goes to itself, and then the rest of the differential\nforms map to the ($\\sigma_i, \\Sigma_i$) in the following way:\n\\begin{eqnarray}\n&&d\\theta_1=\\cos\\psi_1\\sigma_1-\\sin\\psi_1\\sigma_2,\\quad\nd\\theta_2=\\cos\\psi_2\\Sigma_1-\\sin\\psi_2\\Sigma_2,\\nonumber\\\\\n&&d\\phi_1-\\tan\\lambda_4dr=\\sec\\lambda_1\\Big[\\sin(\\psi_1+\\lambda_1)\\sigma_1+\\cos(\\psi_1+\\lambda_1)\n\\sigma_2\\Big],\\nonumber\\\\\n&&d\\phi_2-\\tan\\lambda_5dr=-\\sec\\lambda_2\\Big[\\sin(\\psi_2-\\lambda_2)\\Sigma_1+\\cos(\\psi_2-\\lambda_2)\n\\Sigma_2\\Big],\\nonumber\\\\\n&&d\\psi-\\tan\\lambda_3dr=\\sigma_3-\\Sigma_3-\\Delta_1 {\\rm\ncos}~\\theta_1 \\Big(\\sin\\psi_1\\sigma_1+\\cos\\psi_1\\sigma_2\\Big)\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\;+\n\\Delta_2 {\\rm cos}~\\theta_2\n\\Big(\\sin\\psi_2\\Sigma_1+\\cos\\psi_2\\Sigma_2\\Big)\\nonumber\\\\\n&&dx_{11}=\\sigma_3+\\Sigma_3+\\Delta_1 {\\rm\ncos}~\\theta_1 \\Big(\\sin\\psi_1\\sigma_1+\\cos\\psi_1\\sigma_2\\Big)\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad- \\Delta_2 {\\rm cos}~\\theta_2\n\\Big(\\sin\\psi_2\\Sigma_1+\\cos\\psi_2\\Sigma_2\\Big)\n\\end{eqnarray}\nUnder the rotation, shift and flop the flux components mix in rather\nnon-trivial way.\nWe therefore expect, after the IIA reduction from M-theory, the three-form and four-form flux components of type\nIIA \\eqref{BinIIA} and \\eqref{twelve} respectively before geometric transition go to {\\it new} three- and four-form\nflux components. They can be expressed as:\n\\begin{eqnarray}\\label{BIIA2}\n{B}_{\\rm now}~=~ \\bar{b}_{ij}dx^i\\wedge dx^j, ~~~~~~~~\n{F}_{\\rm now} ~= ~ \\bar{f}_{ijkl}dx^i\\wedge dx^j\\wedge dx^k\\wedge dx^l\n\\end{eqnarray}\nwhere $x^{i,j,k,l}=r, \\theta_i, \\phi_i, \\psi$ and\n$\\bar{b}_{ij}$ and $\\bar{f}_{ijkl}$\nare not only functions of ($r, \\theta_i$) but now also of $\\psi$ because of the shift \\eqref{psis} and rotation\n\\eqref{rotc}.\n\n\n\\subsection{Type IIB after mirror transition}\n\nWith the type IIA picture at hand, we are now at the last chain of the duality transformation that will give us the\nsupergravity dual of the confining gauge theory on the wrapped D5-branes. Our starting points are now:\n\n\\vskip.1in\n\n\\noindent $\\bullet$ The type IIA metric \\eqref{IIAmetric}.\n\n\\noindent $\\bullet$ The remnant of the D6-brane charges, i.e the one-form fluxes \\eqref{1ffc}.\n\n\\noindent $\\bullet$ The type IIA string coupling, or the dilaton \\eqref{edil}, and\n\n\\noindent $\\bullet$ The $B_{\\rm NS}$ and the $F_4$ fluxes \\eqref{BIIA2}\nfrom the remnant of the IIB shift transformations.\n\n\\vskip.1in\n\n\\noindent Before moving ahead, let us make two observations. The first is\nthat now we {\\it do} have components like $\\bar{b}_{\\phi_1\\phi_2}, \\bar{b}_{\\psi\\phi_i}$ in \\eqref{BIIA2}.\nThis would mean that the\nmirror type IIB should become non-geometric! This is what one would have expected generically, and our analysis\ndoes confirm this\\footnote{In fact this tells us that the generic solution spaces we get in type IIB are non-geometric\nmanifolds. For certain choices of parameters (B-fields, and metric components) we can get geometric manifolds like\nKlebanov-Strassler \\cite{KS} or Maldacena-Nunez \\cite{MN}.\nThis is almost like the parameter space of \\cite{butti} but now much bigger,\nand allowing {\\it both} geometric and non-geometric manifolds that cover various\nbranches of the dual gauge theories. One may also note that the susy constraints from the torsion\nclass analysis in equation (3.15), (3.21) and (3.25) of \\cite{chen2} would keep some of the parameters unfixed (one example would be the $\\epsilon$ parameter that we discussed earlier). \nTherefore\nthese parameters could be {\\it varied} to get a class of gauge theory deformations. Typically the duals of\nthese deformations are non-geometric manifolds. This is expressed in figure 3 of \\cite{chen2}.}.\nObserve that locally, as in \\cite{gtpapers}, this aspect of non-geometricity was not visible because\nmost of the extra fluxes were pure gauges. In the global case, the system is rather non-trivial and the dual gravitational\ndescription may become non-geometric. Question now is whether we can look for a special case where we can study the\nsystem as a {\\it geometric} manifold. It turns out at the orientifold point there might be a situation where we can\nswitch off the extra flux components and consider only the standard B-field components. Recall that due to various\nrotations \\eqref{rotc} and shifting \\eqref{tranbg} and \\eqref{psis}\nthe orientifolding is more involved, as all the internal coordinates\nare mixed up in these transformations. However this may not generically remove all the necessary components.\nTherefore to simplify the situation, in the following, we will study the\ntype IIB mirror by first keeping:\n\\bg\\label{nihar} \\bar{b}_{\\phi_1\\phi_2} ~ = ~ \\bar{b}_{\\psi\\phi_i} ~ = ~0\\nd\nso that the mirror could be geometric. Switching on \\eqref{nihar} in the IIA scenario will then make the system\nnon-geometric.\n\n\\noindent Secondly,\nnote that except for the $B_{\\rm NS}$ and the $F_4$ fluxes, rest of the components of the metric or the\none-form fluxes, or even the dilaton are all finite. The $B_{\\rm NS}$ and the $F_4$ fluxes are large, and in the limit\n$\\epsilon$ in \\eqref{metcom} is a small but finite integer, these would also be finite (but large). To proceed further\nlet us define:\n\\bg\\label{defui}\n{\\cal D}\\phi_1 \\equiv d\\phi_1 - b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr,~~~~~~\n{\\cal D}\\phi_2 \\equiv d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2 r}dr\n\\nd\nUsing this, the type IIA metric can be rewritten as:\n\\begin{eqnarray}\\label{IIAmetre}\nds_{10}^2&& =F_0ds_{0,1,2,3}^2+F_1dr^2+e^{2\\phi}\\Big(d\\psi-b_{\\psi\n\\mu}dx^\\mu+\\Delta_1 {\\rm cos}~\\theta_1 ~{\\cal D}\\phi_1 +\n\\widetilde{\\Delta}_2 {\\rm cos}~\\theta_2~ {\\cal D}\\phi_2\\Big)^2\\nonumber\\\\\n&&~+ {\\cal F}_1 \\Big(d\\theta_1^2+ {\\cal D}\\phi_1^2\\Big) +\n{\\cal F}_2 \\Big(d\\theta_2^2+ {\\cal D}\\phi_2^2\\Big)\n\\end{eqnarray}\nIn this form the non-K\\\"ahlerity is obvious in terms of the fibrations ${\\cal D}\\phi_i$ and\nthe resolution parameters of the two two-cycles are determined completely in terms of ${\\cal F}_i$ as:\n\\bg\\label{calfo}\n{\\cal F}_1 = e^{2\\phi\\over 3}a^2(k^2G_2+kG_3+G_1), ~~~~\n{\\cal F}_2 = e^{2\\phi\\over 3}b^2(\\mu^2G_2+\\mu G_3+G_1)\n\\nd\nIt is now clear that to determine the type IIB mirror using SYZ \\cite{syz} we have to make the base bigger as before.\nThe manifold \\eqref{IIAmetre} still retains isometries along ($\\phi_i, \\psi$), so after we enlarge the base we can\nperform SYZ in the usual way\\footnote{This is not as simple as it sounds. As we saw above, due to rotation \\eqref{rotc}\nand shift \\eqref{psis}, the fluxes may develop dependences on $\\psi$ coordinate. A way out of this would be to\nexpress the fluxes in terms of an average $\\langle\\psi\\rangle$ and then perform the SYZ transformations. We can then\ndemand supersymmetry in the\nfinal IIB configuration by analysing the torsion classes, exactly as we did for all the above\ncases. This way all intermediate configurations would be supersymmetric.}.\nSince these details are rather straightforward to work out, we will not redo them again\nnow. To put the type IIB metric in some suggestive format, let us define the following quantities:\n\\begin{eqnarray}\\label{newthings}\n&&\\bar{\\alpha}=\\Big(\\bar{j}_{\\phi_1\\phi_1}\\bar{j}_{\\phi_2\\phi_2}-\\bar{j}_{\\phi_1\\phi_2}+\n\\bar{b}_{\\phi_1\\phi_2}^2\\Big)^{-1}, ~~~~~ \\widetilde{\\cal D}\\psi \\equiv d\\psi+g_{r\\psi}dr \\nonumber\\\\\n&& \\widetilde{\\cal D}\\phi_1 \\equiv \\sqrt{g_{\\phi_1\\phi_1}\\over g_{\\theta_1\\theta_1}}\n\\Big(d\\phi_1+g_{\\phi_1\\theta_1}d\\theta_1+g_{r\\phi_1}dr\\Big), ~~~\n\\widetilde{\\cal D}\\phi_2 \\equiv \\sqrt{g_{\\phi_2\\phi_2}\\over g_{\\theta_2\\theta_2}}\n\\Big(d\\phi_2+g_{\\phi_2\\theta_2}d\\theta_2+g_{r\\phi_2}dr\\Big)\\nonumber\\\\\n\\end{eqnarray}\nwhere\n$\\bar{j}_{\\mu\\nu}$ denote the components of the IIA metric \\eqref{IIAmetre}, and $g_{\\mu\\nu}$ are defined in terms of\n$\\bar{j}_{\\mu\\nu}$ in {\\bf Appendix 3}. Using these definitions, and taking $b_{r\\phi_i} = b_{r\\psi} = 0$,\nthe mirror manifold in type IIB theory\ntakes the following form:\n\\begin{eqnarray}\\label{iibmetfinal}\nds^2 & = & F_0^2ds_{0,1,2,3}^2+g_{rr}dr^2  +g_{\\psi\\psi}\\Big(\\widetilde{\\cal D}\\psi\n+ \\widehat{\\Delta}_1 ~\\widetilde{\\cal D}\\phi_1\n+ \\widehat{\\Delta}_2 ~\\widetilde{\\cal D}\\phi_2\\Big)^2\\\\\n&& +g_{\\theta_1\\theta_1}\\Big(d\\theta_1^2 + \\widetilde{\\cal D}\\phi_1^2\\Big)\n+g_{\\theta_2\\theta_2}\\Big(d\\theta_2^2 + \\widetilde{\\cal D}\\phi_2^2\\Big)\n+g_{\\theta_1\\theta_2}\\Big(d\\theta_1d\\theta_2 + \\widehat{\\Delta}_3\n~\\widetilde{\\cal D}\\phi_1 \\widetilde{\\cal D}\\phi_2\\Big)\\nonumber\n\\nd\nwhich looks surprisingly close to the warped resolved-deformed conifold! Clearly the manifold is non-K\\\"ahler and\n$\\hat{\\Delta}_i$ are defined as:\n\\bg\\label{hatdeldef}\n\\widehat{\\Delta}_1 \\equiv \\sqrt{g_{\\theta_1\\theta_1}g^2_{\\psi\\phi_1}\\over g_{\\phi_1\\phi_1}}, ~~~~~\n\\widehat{\\Delta}_2 \\equiv \\sqrt{g_{\\theta_2\\theta_2}g^2_{\\psi\\phi_2}\\over g_{\\phi_2\\phi_2}}, ~~~~~\n\\widehat{\\Delta}_3 \\equiv \\sqrt{g_{\\theta_1\\theta_1} g_{\\theta_2\\theta_2}g^2_{\\phi_1\\phi_2}\\over\ng_{\\phi_1\\phi_1}g_{\\phi_2\\phi_2}g^2_{\\theta_1\\theta_2}}\n\\nd\nThe type IIB fluxes are rather involved, but they could be worked out exactly as in {\\bf Appendix 2}. We will\nnot do so here, but discuss their implications in our follow-up paper \\cite{toappear}. It is interesting that the\nsolutions that we get in type IIA as well as type IIB for the gravity duals look very close to what has been\nadvocated in the literature so far in the limit where we switch off certain\ncomponents of the $\\bar{b}$-fields as well as $b_{r\\phi_i}, b_{r\\psi}$.\nOnce we keep these components then the metric \\eqref{iibmetfinal} {\\it cannot} be the global description. The global\ndescription will have to be a non-geometric manifold. In the present\npaper we will not discuss the non-geometric aspect anymore, and details on this will be presented in our upcoming\npaper \\cite{toappear}. We end this section by noting that\nthe duality cycle that we\nadvocated here (and also in \\cite{gtpapers} earlier) does\nlead to the correct gauge\/gravity dualities for the confining theories.\n\n\\section{Discussions and conclusion}\n\nIn this paper we have made some progress along two related directions. The first one is to find a supersymmetric\nconfiguration of wrapped D5-branes on a two-cycle of a resolved conifold. We find that the {\\it simplest} way to\nachieve supersymmetry here is to allow for a non-K\\\"ahler metric with an $SU(3)$ structure on the resolved conifold.\nIt is possible that once we add extra fluxes and fundamental seven-branes we can also\nget a K\\\"ahler metric on the resolved conifold that is supersymmetric in the presence of the wrapped D5-branes.\n\nThe second progress that we made here is related to the full cycle of geometric transition. It should now be clear\nthat in the limit where the gauge theories on the wrapped D5-branes\nor the D6-branes were\nstrongly coupled, the geometric transition\neffectively  boiled down to a simple series of mirror transformations \\cite{syz} followed\nin-between by a flop transition to an intermediate M-theory configuration with a $G_2$-structure. The fact that\nsuch a complicated set of gauge\/gravity dualities follow from simple sequences of T-dualities is rather \nremarkable\\footnote{An interesting question to ask at this stage is whether the set of operations (mirror symmetry\nand flop) could be described as a transformation in the space of $SU(3)$ structures, much like the one studied in \n\\cite{minabeta}. The fact that this is indeed the case for our set-up can be inferred from the torsion class\nanalysis before and after geometric transition. In the sequel \\cite{chen2} we have worked out the torsion \nclasses both before and after GT in type IIA theory. These torsion classes could be related to each other by \n$SU(3)$ transformations, and therefore our set of operations can be described as transformations in the space of \n$SU(3)$ structure.}.\n\nIn the global case the mirror transformations are more subtle than the local case.\nFor example the fluxes appearing in the mirror\npictures are very involved (we gave one example in {\\bf Appendix 2}). They are not in general pure gauges, and so\nwe need to follow them through every step of the duality cycle because they would eventally influence the gravity\nduals for the confining gauge theories on the wrapped D6- and D5-branes (at least in the far IR). In this\npaper we have managed to work out these details, and showed that despite the complicated underlining flux-structures, the\ngravity duals for the type IIA and the type IIB configurations in certain cases were\n given by non-K\\\"ahler deformations of the resolved and\nthe deformed conifold respectively. In more generic cases, that allow us to keep all components of the B-fields, the\ngravitational descriptions become non-geometric.\n\nThere are many things that need to be done now. One of the issue is to understand the non-geometric aspects of the\nmirror configurations. We have briefly alluded to this by arguing that there are B-field components (even at the\norientifold points) which could in-principle make the IIA, as well as IIB after geometric transition,\npicture non-geometric (see also the recent discussion in\n\\cite{halmagyi}). Such a non-geometric configuration for example\nin type IIA should lift to a non-geometric configuration in\nM-theory. The question then is: how should we perform the flop operation now? Even if there exist a meaningful\nway to perform the flop, how do we understand the SYZ \\cite{syz} mirror map now? How does the non-geometric\nbehavior reflect in the gauge theory side? How should we compute the exact superpotential for the gauge theory in the\nnon-geometric case? Furthermore if we allow the B-fields components $\\bar{b}_{\\phi_1\\phi_2}$ and\n$\\bar{b}_{\\psi\\phi_i}$ in \\eqref{BIIA2}, how would the metric \\eqref{iibmetfinal} change to reflect the non-geometric\naspect?\n\nClearly these are very interesting questions to resolve, and we have barely scratched the surface of the iceberg. In\nour upcoming paper \\cite{toappear} we plan to address some of these issues. Hopefully these questions\nwill have simple tractable\nsolutions. If not, then it will be even more interesting to reveal the underlying structure.\n\n\\vskip.1in\n\n\\noindent {\\bf Note added:}\nWhile we were writing this draft, \\cite{pandozayas} appeared that has some overlap with sections 4.1 and\n4.2 of this paper. See also the recent set of papers \\cite{lilia} where some new properties of non-K\\\"ahler manifolds\nhave been discussed.\n\n\n\\vskip.1in\n\n\\begin{center}\n{\\bf Acknowledgements}\n\\end{center}\n\n\\noindent We would like to thank N.~Halmagyi, M.~Larfors, D.~L\\\"ust, and D.~Tsimpis for helpful\nconversations. The works of FC, KD and PF were supported in parts by the NSERC grants.\nThe work of SK was supported by NSF grants DMS-02-44412 and\nDMS-05-55678. The work of RT was supported by a PPARC grant.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{intro1}\n\nThe question of how much of a given structure is encoded in its symmetries is one that surfaces in many different ways in many different areas of pure mathematics.  One way in which this question surfaces is reconstruction from automorphism groups of first order structures.  An account of the history of this can be found in \\cite{Pech2Monoid}, by C. Pech and M. Pech.  There are many levels of structure can be placed on a automorphism group, so here is some notation that clarifies what exactly we mean by ``isomorphism''.\n\n\\begin{dfn}\nLet $M$ be an structure and let $\\mathcal{L}_G$ be the language of groups, $\\tau_M$ the topology of pointwise convergence on $\\textnormal{Aut}(M)$ and $\\mathrm{Op}(f,x)$ the group action of $\\textnormal{Aut}(M)$ on $M$.\n$$\\begin{array}{rcl}\n\\textnormal{Aut}(M) \\cong_A \\textnormal{Aut}(N) & \\Leftrightarrow & \\langle \\textnormal{Aut}(M), \\mathcal{L}_G \\rangle \\cong \\langle \\textnormal{Aut}(N), \\mathcal{L}_G \\rangle \\\\\n\\textnormal{Aut}(M) \\cong_T \\textnormal{Aut}(N) & \\Leftrightarrow & \\langle \\textnormal{Aut}(M), \\mathcal{L}_G, \\tau_M \\rangle \\cong \\langle \\textnormal{Aut}(N), \\mathcal{L}_G, \\tau_N \\rangle \\\\\n\\textnormal{Aut}(M) \\cong_P \\textnormal{Aut}(N) & \\Leftrightarrow & \\langle \\textnormal{Aut}(M), M , \\mathcal{L}_G, \\mathrm{Op} \\rangle \\cong \\langle \\textnormal{Aut}(N), N , \\mathcal{L}_G, \\mathrm{Op} \\rangle \\\\\n\\end{array}\n$$\nThe subscript $A$ stands for `abstract', $T$ for `topology' and $P$ for `permutation'.\n\\end{dfn}\n\nOne way in which we can pursue the reconstruction of first order models is the search for `faithful' classes.\n\n\\begin{dfn}\nA class of first order models $K$ is said to be faithful if for all $M,N \\in K$\n$$ \\textnormal{Aut}(M) \\cong_A \\textnormal{Aut}(N) \\Leftarrow M \\cong N$$\n\\end{dfn}\n\nIn this trio of papers, the structures we consider are `cycle-free partial orders' (CFPOs), and our notion of symmetry is the associated automorphism groups in the language of group theory.  CFPOs are a generalisation of trees, or semi-linear orders.  They were introduced in the memoir entitled `The Reconstruction of Trees from Their Automorphism Groups' \\cite{Rubin91}, by M. Rubin, as a family of structures where the methods he used for reconstructing the trees would extend.  Their definition can be found in the `Preliminary Definitions' section.\n\nNotable works concering CFPOs include an extensive study of the transitivity properties of the CFPOs can be found in \\cite{Warren1997}, a memoir of Richard Warren, who defines them using the notion of path.  Warren's study was extended by two papers in 1998, \\cite{Truss1998CFPO} by Truss and \\cite{CreedTrussWarren} by Creed, Truss and Warren, both of which add to cases not fully dealt with by Warren in \\cite{Warren1997}.  Gray and Truss in \\cite{TrussGray09} examine the relationship between ends of a graph and CFPOs, and extract a number of results from this relationship.  This viewpoint is rather illuminating, even if one is not familiar with ends of graphs.\n\nThese papers do not extend Rubin's method for reconstruction to the CFPOs, instead applying different methods from a variety of sources.  The first part shows when we can appeal to his results directly.  The second adapts methods used by Shelah in \\cite{ShelahPermutation, ShelahPermutationErrata} and by Shelah and Truss in \\cite{ShelahTrussQuotients}, while the third uses properties of the wreath products of groups to get a class where the first and second approaches may be used together.\n\nPart I is the draws most strongly from \\cite{Rubin91}, the memoir of Rubin.  In that memoir, Rubin gives an extremely complete reconstruction result for trees.  This part seeks to show when we can appeal to those results directly.  Section \\ref{intro1} contains the preliminary notions required, as well as the definition of CFPOs.  Section \\ref{prelim1} will give the definition of connecting set and path, which will be used extensively throughout this whole work.  Their most immediate use will be to define the class of CFPOs.\n\nSection \\ref{orderconditions} describes three constructions that build from a CFPO with some constraints a tree with the same permutation automorphism group.  These constrains are: possessing a fixed point; not embedding $\\alt{\\omega}$; and not embedding $\\alt{}$.  These constructions allow us to deduce that the theory of a CFPO is dp-minimal, which will be done in Section \\ref{CFPOsMT}.\n\nSection \\ref{gpconditions} shows that if $D_\\infty$, the infinite dihedral group, is a subgroup of an automorphism group of a tree, then it is contained in a supergroup isomorphic to one of a family of groups called `dendromorphic'.  We use this to formulate a condition the abstract automorphism group of a CFPO that shows when it is also the automorphism group of a tree.\n\nThese papers are based on the author's Ph.D. thesis, supervised by Prof. John Truss at The University of Leeds.\n\n\\section{Preliminaries}\\label{prelim1}\n\nThis section contains the definition of CFPOs and the supporting concepts.\n\n\\begin{dfn}[2.3.2 of \\cite{Warren1997}]\\label{dfn:connectingset}\nIf $M$ is a partial order and $a,b \\in M$, then $C$, the n-tuple $\\langle c_1,c_2, \\ldots ,c_n \\rangle$ (for $n \\geq 2$) is said to be a \\textbf{connecting set} from $a$ to $b$ in $M$, written $C \\in C^M \\langle a, b \\rangle$, if the following hold:\n\\begin{enumerate}\n\\item $c_1=a, c_n=b, c_2, \\ldots ,c_{n-1} \\in M^D$\n\\item if $1 \\leq i \\leq n-1$, then $c_i \\not\\parallel c_{i+1}$\n\\item if $1<i<n$, then $c_{i-1} < c_i > c_{i+1}$ or $c_{i-1} > c_i < c_{i+1}$\n\\end{enumerate}\n\\end{dfn}\n\n\\begin{dfn}[2.3.3 of \\cite{Warren1997}]\nLet $M$ be a partial order, $a,b \\in M$, and let $C = \\langle c_1,c_2,\\ldots ,c_n \\rangle$ be a connecting set from $a$ to $b$ in $M$.  Let $\\sigma_k$ (for $ 1< k < n$) be maximal chains in $M^D$ with endpoints $c_k,c_{k+1} \\in \\sigma_k$, such that if $x \\in \\sigma_i \\cap \\sigma_j$ for some $i < j$, then $j = i+1$ and $x = c_{i+1}$.  Then we say that $P = \\bigcup_{0<k<n} \\sigma_k$ is a \\textbf{path} from $a$ to $b$ in $M$.\n\\end{dfn}\n\n\\begin{dfn}\nA partial order $M$ is said to be a \\textbf{cycle-free partial order} (CFPO) if for all $x,y \\in M$ there is at most one path between $x$ and $y$ in $M^D$. \n\\end{dfn}\n\n\\begin{dfn}\nLet $M$ be a CFPO, and let $x,y \\in M$ and $A,B \\subseteq M$.  The unique path between $x,y$ is denoted by $\\path{x,y}$.  We also define:\n$$\n\\begin{array}{r c l}\n\\path{x,B} & := & \\bigcap_{b \\in B} \\path{x,b} \\\\\n\\path{A,y} & := & \\bigcap_{a \\in A} \\path{a,y} \\\\\n\\path{A,B} & := & \\bigcap_{a \\in A} \\path{a,B}\n\\end{array}\n$$\n\\end{dfn}\n\n\\begin{dfn}\nA partial order is said to be \\textbf{connected} if there is a path between any two points, i.e. $\\path{x,y}$ exists for all $x,y \\in M$, and is said to be disconnected otherwise.\n\nLet $M$ be a partial order and let $C \\subseteq M$.  We say that $C$ is a \\textbf{connected component} of $M$ if it is a maximal connected subset of $M$, i.e. for all $x,y \\in M$ if $x \\in C$ and $\\path{x,y}$ exists then $y \\in C$.\n\\end{dfn}\n\nTruss in \\cite{Truss01} shows that the class of cycle-free partial orders is axiomatisable, but not finitely axiomatisable.  Adding colour predicates to the language of partial orders is an inalienable part of Rubin's memoir, as well as this work, so throughout these three papers every partial order discussed may be coloured by infinitely many colour predicates.\n\n\\begin{dfn}\\label{dfn:treelike}\nCFPO $M$ is said to be \\textbf{treelike} if there is a coloured tree $T$ such that\n$$\\textnormal{Aut}(M) \\cong_A \\textnormal{Aut}(T)$$\n\nIf $G \\leq \\textnormal{Aut}(M)$ then the action of $G$ is said to be treelike if there is a tree $T$ such that\n$$G \\cong_A \\textnormal{Aut}(T)$$\n\\end{dfn}\n\n\\begin{prop}\\label{Lemma:NeededInChapter3}\nLet $M$ be a CFPO and let $(a_1, \\ldots, a_n),(b_1, \\ldots, b_n) \\in M$.  Furthermore, we define\n$$\\mathrm{Adj}:= \\lbrace (i,j) \\: : \\: a_i \\leq \\geq a_j \\textnormal{ and } \\forall k \\: a_k \\not\\in ( a_i, a_k) \\rbrace$$\nThen $(a_1, \\ldots a_n)$ and $(b_1, \\ldots, b_n)$ lie in the same orbit if and only if there is an isomorphism of finite structures\n$$\\phi: (a_1, \\ldots a_n) \\rightarrow (b_1, \\ldots, b_n)$$\nsuch that for all $(i,j) \\in \\mathrm{Adj}$, the pair $(a_i,a_j)$ lies in the same 2-orbit as $(b_{\\phi(i)},b_{\\phi(j)})$.\n\\end{prop}\n\\begin{proof}\nThis is a quick consequence of Proposition 4.5 of \\cite{Simon2011}.\n\\end{proof}\n\n\\begin{dfn}\n$\\alt{}$ is the partial order with the domain $ \\lbrace a_i \\; : \\; i \\in \\mathbb{Z} \\rbrace $ ordered by\n\\begin{itemize}\n\\item if $i$ is odd then $ a_{i-1} > a_{i} < a_{i+1} $ \n\\item if $i$ is even then $ a_{i-1} < a_{i} > a_{i+1} $\n\\end{itemize}\n$\\alt{n}$ is defined to be $\\alt{}$ restricted to $\\lbrace a_0, \\ldots a_{n-1} \\rbrace$.  Note that flipping the order does not affect the definition of $\\alt{}$, but does affect $\\alt{n}$.  We will write $\\alt{n}^*$ for the reverse ordering of $\\alt{n}$.\n\n$\\alt{\\omega}$ is defined to be $\\alt{}$ restricted to $\\lbrace a_i \\: : \\: i \\in \\omega \\rbrace$.  Again, the reverse ordering is denoted by $\\alt{\\omega}^*$\n\\end{dfn}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.14]\n\\draw (-25,0) -- (-20,5) -- (-10,-5) -- (0,5) -- (10,-5) -- (20,5) -- (25,0);\n\n\\fill (-20,5) circle (0.5);\n\\fill (-10,-5) circle (0.5);\n\\fill (0,5) circle (0.5);\n\\fill (20,5) circle (0.5);\n\\fill (10,-5) circle (0.5);\n\n\\draw[anchor=west] (-20,5) node {$a_{-2}$};\n\\draw[anchor=west] (-10,-5) node {$a_{-1}$};\n\\draw[anchor=west] (0,5) node {$a_{0}$};\n\\draw[anchor=west] (10,-5) node {$a_{1}$};\n\\draw[anchor=west] (20,5) node {$a_{2}$};\n\n\\draw (-27,0) node {$\\ldots$};\n\\draw (27,0) node {$\\ldots$};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{The Alternating Chain}\n\\end{figure}\n\n\\begin{dfn}\nA CFPO is said to be a $\\cfpo{n}$ if $\\alt{n}$ embeds but $\\alt{n+1}$ does not.  A CFPO is said to be a $\\cfpo{\\omega}$ if $\\alt{\\omega}$ embeds but $\\alt{}$ does not.  A CFPO is said to be a $\\cfpo{\\infty}$ if $\\alt{}$ embeds.\n\\end{dfn}\n\n\\begin{dfn}\\label{dfn:G(t)}\nIf $f \\in \\textnormal{Aut}(M)$ then the \\textbf{support} of $f$ is the following set:\n$$\\textnormal{supp}(f) := \\lbrace x \\in M \\: : \\: f(x) \\not= x \\rbrace$$\nIf $F \\subseteq \\textnormal{Aut}(M)$ and $x \\in M$ then\n$$F(x) := \\lbrace f(x) \\: : \\: f \\in F \\rbrace$$\nand the support of $F$ is the following set:\n$$\\textnormal{supp}(F) := \\bigcup_{f \\in F} \\textnormal{supp}(f)`$$\n\\end{dfn}\n\n\\section{Order Conditions}\\label{orderconditions}\n\nWe start with CFPOs which have points which are fixed by every automorphism (which we call \\textbf{fixed points}).  We will take from the midst of $M$ our fixed point and plant it in the ground, before straightening out the paths of $M$ into branches.\n\nThe colouring of $M$ is largely irrelevant for this work, and so takes a very back-seat role.  Indeed, for the rest of this subsection the term `monochromatic' will mean `monochromatic with respect to $U$', where $U$ is the predicate introduced in the next definition.\n\n\\begin{dfn}\\label{Def:XandY}\nLet $\\langle M, \\leq_M \\rangle$ be a connected CFPO whose automorphism group fixes the point $r$.  We will construct $T(M)$ by specifying a new order on $|M |$.  Let $r$ be the fixed point of $M$, which will become the root of $T(M)$.  The colour of $r \\in M$ is the same in $T(M)$.\n\nWe denote the order on $T$ by $\\leq_T$ and define it as follows:\n\\begin{itemize}\n\\item $r \\leq_{T(M)} s$ for all $s \\in M$\n\\item $s \\leq_{T(M)} t$ if and only if $s \\in \\path{r,t}$\n\\end{itemize}\nWe also add a new unary predicate, which we call $U$.  We define the following sets:\n$$\n\\begin{array}{l c l}\nX_0 & := & \\lbrace t \\in M \\; : \\; r \\leq_M t \\rbrace \\\\\nY_0 & := & \\lbrace t \\in M \\; : \\; t <_M r \\rbrace \\\\\n & \\vdots & \\\\\nX_n & := & \\lbrace t \\in M \\; : \\; y \\leq_M t \\: \\textnormal{for some} \\: y \\in Y_{n-1} \\rbrace \\setminus \\bigcup_{i < n}(X_i \\cup Y_i) \\\\\nY_n & := & \\lbrace t \\in M \\; : \\; t <_M x \\: \\textnormal{for some} \\: x \\in X_{n-1} \\rbrace \\setminus \\bigcup_{i < n}(X_i \\cup Y_i) \\\\\n & \\vdots & \\\\\n\\end{array}\n$$\nWe also define $X := \\bigcup X_i$ and say that $U(t)$ holds whenever $t \\in X$.  Finally\n\n$$\\mathcal{X} := \\lbrace X_i , Y_i \\: : \\: i \\in \\omega \\rbrace$$\n\\end{dfn}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.075]\n\\draw (0,30) -- (15,-30);\n\\draw (0,-30) -- (15,30);\n\\fill[white] (7.5,30) ellipse  (7.5 and 3.75);\n\\fill[white] (7.5,-30) ellipse  (7.5 and  3.75);\n\\draw (7.5,30) ellipse  (7.5 and 3.75);\n\\draw (7.5,-30) ellipse  (7.5 and 3.75);\n\n\\draw (15 ,10) -- (12.5 ,20) -- (25 ,10);\n\\fill[white] (20,10) ellipse (5 and 2.5);\n\\draw (20,10) ellipse (5 and 2.5);\n\n\\draw (15 ,-10) -- (12.5 ,-20) -- (25 ,-10);\n\\fill[white] (20,-10) ellipse (5 and 2.5);\n\\draw (20,-10) ellipse (5 and 2.5);\n\n\\draw (18.75 ,20) -- (18.75 ,15) -- (26.25 ,20);\n\\fill[white] (22.725,20) ellipse (3.725 and 0.75);\n\\draw (22.725,20) ellipse (3.725 and 0.75);\n\n\\fill (7.5,0) circle (1);\n\n\\draw[anchor=east] (7.5,0) node {$r$};\n\n\\draw (7.5,-40) node {$M$};\n\n\\draw[->]  (27.5,0) -- (40,0);\n\n\\draw (45,35) -- (52.5,5) -- (60,35);\n\\fill[white] (52.5,35) ellipse (7.5 and 3.75);\n\\draw (52.5,35) ellipse (7.5 and 3.75);\n\\draw (52.5,25) ellipse (10 and 22.5);\n\\fill (52.5,5) circle (1);\n\n\\draw (45,-35) -- (52.5,-5) -- (60,-35);\n\\fill[white] (52.5,-35) ellipse (7.5 and 2.5);\n\\draw (52.5,-35) ellipse (7.5 and 2.5);\n\\draw (52.5,-25) ellipse (10 and 22.5);\n\n\\draw (72.5,20) -- (70,30) -- (82.5,20);\n\\fill[white] (77.5,20) ellipse (5 and 2.5);\n\\draw (77.5,20) ellipse (5 and 2.5);\n\\draw (76.25,22.5) ellipse (9 and 12);\n\n\\draw (72.5,-20) -- (70,-30) -- (82.5,-20);\n\\fill[white] (77.5,-20) ellipse (5 and 2.5);\n\\draw (77.5,-20) ellipse (5 and 2.5);\n\\draw (76.25,-22.5) ellipse (9 and 12);\n\n\\draw (72.525 ,2.5) -- (72.525 ,-2.5) -- (79.975 ,2.5);\n\\fill[white] (76.25,2.5) ellipse (3.725 and 0.75);\n\\draw (76.25,2.5) ellipse (3.725 and 0.75);\n\\draw (76.25,0) ellipse (6 and 7);\n\n\\draw[anchor=south]  (52.5,47.5) node {$X_0$};\n\\draw[anchor=north]  (52.5,-47.5) node {$Y_0$};\n\\draw[anchor=south] (76.25,34.5) node {$Y_1$};\n\\draw[anchor=north]  (76.25,-34.5) node {$X_1$};\n\\draw[anchor=east]  (70.25,0)  node {$X_2$};\n\n\\draw[->]  (85,0) -- (97.5,0);\n\n\\draw (105,-10) -- (120,-30) -- (115,-10);\n\\fill[white] (110,-10) ellipse (5 and 2.55);\n\\draw (110,-10) ellipse (5 and 2.55);\n\n\\draw[dashed] (125,-10) -- (120,-30) -- (135,-10);\n\\fill[white] (130,-10) ellipse (5 and 2.55);\n\\draw[dashed] (130,-10) ellipse (5 and 2.55);\n\n\\draw[dashed] (107.5,10) -- (110,-10) -- (112.5,10);\n\\fill[white] (110,10) ellipse (2.5 and 1.25);\n\\draw[dashed] (110,10) ellipse (2.5 and 1.25);\n\n\\draw (127.5,10) -- (130,-10) -- (132.5,10);\n\\fill[white] (130,10) ellipse (2.5 and 1.25);\n\\draw (130,10) ellipse (2.5 and 1.25);\n\n\\draw (108,20) -- (110,10) -- (112,20);\n\\fill[white] (110,20) ellipse (2 and 1);\n\\draw (110,20) ellipse (2 and 1);\n\n\\fill (120,-30) circle (1);\n\n\\draw[anchor=south]  (115,-10) node {$X_0$};\n\\draw[anchor=south]  (135,-10) node {$Y_0$};\n\\draw[anchor=west] (112.5,10) node {$Y_1$};\n\\draw[anchor=west]  (132.5,10) node {$X_1$};\n\\draw[anchor=south]  (112,20)  node {$X_2$};\n\n\\draw (120,-40) node {$T$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Turning $M$ with fixed point $r$ into $T(M)$}\n\\end{figure}\n\n\\begin{lemma}\n$\\mathcal{X}$ partitions $|M|$.\n\\end{lemma}\n\\begin{proof}\nBy construction\n$$\n\\begin{array}{r c l}\nX_i \\cap X_j \\not= \\emptyset & \\Rightarrow & i = j \\\\\nY_k \\cap Y_l \\not= \\emptyset & \\Rightarrow & k = l \n\\end{array}\n$$\nso it remains to show that $\\mathcal{X}$ covers $|M|$.  We pick an arbitrary $z \\in |M|$ and consider $\\path{z,r}$, which exists as all CFPOs considered are connected.\n\nLet $z_0 (=z), z_1, \\ldots z_n (=r)$ be the endpoints of $\\path{z,r}$.  We know that $z_n \\in X_0$ as $z_n = r$, and hence $z_{n-1} \\not\\parallel z_n$ implies that $z_{n-1} \\in \\bigcup \\mathcal{X}$.  Similarly $z_{n-2} \\not\\parallel z_{n-1}$ implies that $z_{n-1} \\in \\bigcup \\mathcal{X}$ and so on along $\\path{z,r}$ until we deduce that $z \\in \\bigcup \\mathcal{X}$.\n\\end{proof}\n\nIf we start with a rooted tree, and use the root for our procedure, our construction returns the original structure with an additional predicate which is realised everywhere.  Our eventual goal is to say that the canonical representative of $M$ is the canonical representative of $T(M)$, and to do so we must show that $T(M)$ is a tree with the same automorphism group as $M$.\n\nThis construction has the unfortunate property that we may have to make a choice of fixed point, and the resulting structures depend on this choice.  However, since our claim is that $T(M)$ is a tree, rather than a canonical tree, we may sweep this difficulty under the carpet of Rubin's work.\n\n\\begin{prop}\n$\\langle T(M) , \\leq_{T(M)}, U \\rangle$ is a tree.\n\\end{prop}\n\\begin{proof}\n$M$ is connected so $\\leq_{T(M)}$ is defined everywhere.\n\nIf $s_0, s_1 \\leq_{T(M)} t$ then $\\lbrace s_0, s_1 \\rbrace \\subseteq \\path{t,r}$, and since $M$ is cycle-free this means that either $s_0 \\in \\path{s_1,r}$ or $s_1 \\in \\path{s_0,r}$, showing that $s_0 \\not \\parallel s_1$, and thus all initial sections of $T(M)$ are linearly ordered.  Finally, $r \\in \\path{r,t}$ for all $t$, so every pair from $T$ has a common lower bound, showing that $\\langle T(M), \\leq_{T(M)}, U \\rangle$ is a tree.\n\\end{proof}\n\nOf course, this construction is without merit if it does not preserve the automorphism group.  We work towards that goal with the following lemmas.\n\n\\begin{lemma}\\label{lemma:MinterpretableinT(M)}\n$\\langle M, \\leq_M, r \\rangle$ is interpretable in $\\langle T(M), \\leq_{T(M)}, U \\rangle$.\n\\end{lemma}\n\\begin{proof}\nThe following formulas form an interpretation of $\\langle M, \\leq_M, r \\rangle$ in $\\langle T(M), \\leq_{T(M)}, U \\rangle$:\n\\begin{enumerate}\n\\item $\\phi_{Dom}(x)$, which defines the domain of the interpretation.  We take\n$$x=x$$\n\\item $\\phi_{Eq}(x)$, which defines equivalence classes on the domain of the interpretation.  Again, we take\n$$x=x$$\n\\item A formula $\\phi_{\\leq_M}(x,y)$.  We take the disjunction of the following clauses:\n\\begin{enumerate}\n\\item $(x \\leq _T y \\wedge \\forall z (x \\leq_T z \\leq_T y \\rightarrow U(z) ))$\n\\item $(y \\leq_T x \\wedge \\forall z (y \\leq_T z \\leq_T x \\rightarrow \\neg U(z))) $\n\\item $(U(y) \\wedge \\neg U(x)) \\wedge$\n$$\\exists z \\left( \\begin{array}{l}\n z \\leq_{T(M)} \\lbrace x,y \\rbrace \\wedge \\\\\n\\forall w (z \\leq_{T(M)} w \\leq_{T(M)} y \\rightarrow U(w)) \\wedge \\\\\n\\forall w (z \\leq_{T(M)} w \\leq_{T(M)} x \\rightarrow \\\\\n\\hspace{20pt}\\left( \\begin{array}{c}\n(U(w) \\rightarrow \\forall v (z \\leq_{T(M)} v \\leq w \\rightarrow U(v)))\\wedge \\\\\n(\\neg U(w) \\rightarrow \\forall v (w \\leq_{T(M)} v \\leq x \\rightarrow \\neg U(v)) )\n\\end{array} \\right)\n\\end{array} \n \\right)$$\n\\end{enumerate}\n\n\\item A formula $\\phi_{r}(x)$.  We take\n$$\n\\forall z \\neg (z \\leq x )\n$$\n\\end{enumerate}\nWhile $\\phi_{Dom}$, $\\phi_{Eq}$ and $\\phi_{r}$ are self-explanatory, to show that $\\phi_{\\leq_{M}}$ does what is required of it, we examine it clause by clause.\n\nClause (a) shows that when both $x$ and $y$ lie in the same $X_i$ for some $i$ and $x \\leq_{T(M)} y$ then $x \\leq_M y$.  Clause (b) shows that when both $x$ and $y$ lie in the same $Y_i$ for some $i$ and $y \\leq_{T(M)} x$ then $x \\leq_M y$.  Clause (c) covers when $y \\in X_i$ and $x \\in Y_{i+1} \\cup Y_{i-1}$ for some $i$, one instance of which is depicted in Figure 3.  No clause is required for $y \\in Y_i$ and $x \\not\\in Y_i$, because if $x \\leq_M y$ then $x \\in Y_i$\n\\end{proof}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.13]\n\\draw (10,-10) -- (10,30);\n\\draw (20,30) -- (10,10) -- (30,30);\n\\fill[white] (25,30) ellipse (5 and 2.5);\n\\draw (25,30) ellipse (5 and 2.5);\n\\fill (10,20) circle (0.5);\n\\fill (10,5) circle (0.5);\n\\fill (17.5,20) circle (0.5);\n\\draw[anchor=east] (10,20) node {$y$};\n\\draw[anchor=east] (10,5) node {$z$};\n\\draw[anchor=south] (17.5,20) node {$x$};\n\\draw[anchor=south] (10,30) node {$X_i$};\n\\draw (25,30) node {$Y_{i+1} $};\n\\draw[anchor=north] (20,-12) node {$T(M)$};\n\n\\draw (50,-10) -- (50,30);\n\\draw (60,-10) -- (50,10) -- (70,-10);\n\\fill[white] (65,-10) ellipse (5 and 2.5);\n\\draw (65,-10) ellipse (5 and 2.5);\n\\fill (50,20) circle (0.5);\n\\fill (50,5) circle (0.5);\n\\fill (57.5,0) circle (0.5);\n\\draw[anchor=east] (50,20) node {$y$};\n\\draw[anchor=east] (50,5) node {$z$};\n\\draw[anchor=north] (57.5,0) node {$x$};\n\\draw[anchor=south] (50,30) node {$X_i$};\n\\draw (65,-10) node {$Y_{i+1} $};\n\\draw[anchor=north] (60,-12) node {$M$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Clause (c) of $\\phi_{\\leq_{M}}$ in Lemma \\ref{lemma:MinterpretableinT(M)}}\n\\end{figure}\\label{FigureClause3}\n\n\\begin{lemma}\\label{lemma:T(M)andisomorphism}\nSuppose $M_0$ and $M_1$ are connected CFPOs with fixed points $r_0$ and $r_1$ respectively.  Then $\\langle M_0, \\leq_{M_0}, r_0 \\rangle \\cong \\langle M_1, \\leq_{M_1}, r_1 \\rangle$ if and only if $$\\langle T(M_0), \\leq_{{T(M_0)}}, U_{T(M_0)} \\rangle \\cong \\langle T(M_1), \\leq_{{T(M_1)}}, U_{T(M_1)} \\rangle$$\n\\end{lemma}\n\\begin{proof}\nSince we constructed $\\leq_T$ and $U$ using path-betweenness and $\\leq_M$, both of which are preserved by isomorphism,\n$$\n\\begin{array}{c}\n\\langle M_0, \\leq_{M_0}, r_0 \\rangle \\cong \\langle M_1, \\leq_{M_1}, r_1 \\rangle \\Rightarrow \\\\\n\\langle T(M_0), \\leq_{{T(M_0)}}, U_{T(M_0)} \\rangle \\cong \\langle T(M_1), \\leq_{{T(M_1)}}, U_{T(M_1)} \\rangle\n\\end{array}\n$$\nThe other direction of the isomorphism is a consequence of the fact that in Lemma \\ref{lemma:MinterpretableinT(M)} the domain of the interpretation is $T(M)$ itself.\n\\end{proof}\n\nThis second lemma shows that the construction behaves when we take certain substructures.  We will take from $M$ an extended cone $C$, and show that $T(C)$ is isomorphic to either the corresponding substructure of $T(M)$, or the corresponding substructure with the roles of $U$ and $\\neg U$ reversed.\n\n\\begin{lemma}\\label{lemma:T(M)andsubstructure}\nLet $r$ be a fixed point of $M$ and let $x \\in M$.  We define\n$$N := \\lbrace y \\in M \\: : \\: x \\in \\path{y,r} \\rbrace$$\nIf we add a colour to $N$ which is only realised by $x$ (to ensure that $x$ is a fixed point of $N$ as a structure in its own right), and use $x$ to construct\n$$\\langle T(N), \\leq_{T(N)}, U_{T(N)} \\rangle$$ then if $x \\in X$ (recall Definition \\ref{Def:XandY}) then\n$$\\langle N , \\leq_{T(M)}, U_{T(M)} \\rangle \\cong \\langle T(N), \\leq_{T(N)}, U_{T(N)} \\rangle$$\notherwise $x \\in M \\setminus X$ (recall Definition \\ref{Def:XandY}) implies that\n$$\\langle N , \\leq_{T(M)}, U_{T(M)} \\rangle \\cong \\langle T(N), \\leq_{T(N)}, \\neg U_{T(N)} \\rangle$$\n\\end{lemma}\n\\begin{proof}\nThis is a simple consequence of the fact that $\\path{y,r} = \\path{y,x} \\cup \\path{x,r}$ for all $y \\in N$\n\\end{proof}\n\n\\begin{lemma}\nThe members of $\\mathcal{X}$ are preserved setwise by $\\textnormal{Aut}(M)$.\n\\end{lemma}\n\\begin{proof}\nAll automorphisms fix $r$, so $X_0$, the points greater than $r$, and $Y_0$, the points less than $r$, are fixed setwise.\n\nLet $x_n \\in X_n$ and let $y_{n-1} \\in Y_{n-1}$ with $y_{n-1} \\leq_M x_n $, and assume as an induction hypothesis that for $i < n$ both $X_i$ and $Y_{i}$ are fixed setwise by $\\textnormal{Aut}(M)$.  Let $\\phi \\in \\textnormal{Aut}(M)$ be arbitrarily chosen.  By the induction hypothesis $\\phi(y_{n-1}) \\in Y_{n-1}$, and since $\\phi$ is an automorphism $\\phi(y_{n-1}) \\leq_M \\phi(x_n)$.  If $\\phi(x_n) \\in \\bigcup_{i < n}(X_i \\cup Y_i)$ then $\\phi^{-1}$ violates the induction hypothesis, so $X_n$ is preserved by $\\textnormal{Aut}(M)$.  The argument for $Y_n$ is identical.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:TsetwiseX}\n$\\textnormal{Aut}(T)$ preserves the members of $\\mathcal{X}$ setwise.\n\\end{lemma}\n\\begin{proof}\nLet $x \\in X_n$.  Since $T \\models U(x)$ and $T \\models \\neg U(y)$ for all $y \\in \\bigcup Y_i$, we cannot map $x$ to any member of $\\bigcup Y_i$.  By taking a witness that $x \\in X_n$, and a witness that that witness lies in $Y_{n-1}$ and so on, we obtain a maximal chain $x_1 \\leq_{T(M)} x_2 \\leq_{T(M)} \\ldots x_n (=x)$ such that $U(x_i)$ if and only if $\\neg U(x_{i-1})$ and $\\neg U(x_{i+1})$, with the additional property that for all $x_i \\leq_{T(M)} t \\leq_{T(M)} x_{i+1}$ either $[x_i, t]$ or $[t , x_i]$ is monochromatic.\n\nAny automorphism would have to send this chain to a similar chain below the image of $x$, but the length of this chain is determined by $n$, thus all images of $x$ lie in $X_n$.  A similar argument shows the same for $Y_n$, and so we conclude that $\\textnormal{Aut}(T(M))$ preserves the members of $\\mathcal{X}$ setwise.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:fixedpoint}\n$\\textnormal{Aut}(\\langle M, \\leq_M \\rangle) \\cong_P \\textnormal{Aut}(\\langle T(M), \\leq_{T(M)}, U \\rangle)$\n\\end{theorem}\n\\begin{proof}\nProposition \\ref{Lemma:NeededInChapter3} shows that if all the 1- and 2-orbits of $M$ coincide with the 1- and 2-orbits of $T(M)$ then $\\textnormal{Aut}(T(M)) \\cong_P \\textnormal{Aut}(M)$.  We will start with the 1-orbits, which we will prove by induction on $\\mathcal{X}$.\n\nSince $\\langle X_0, \\leq_M \\rangle$ is a tree \n$$\\langle X_0, \\leq_M \\rangle = \\langle X_0, \\leq_{T(M)} \\rangle$$\nand since $\\langle X_0, \\leq_{T(M)}, U_{T(M)} \\rangle$ is monochromatic,\n$$\\textnormal{Aut}(\\langle X_0, \\leq_M \\rangle) \\cong_P \\textnormal{Aut}(\\langle X_0, \\leq_{T(M)}, U_{T(M)} \\rangle)$$\n\nFrom this we conclude that for all $a,b \\in X_0$, if $a$ and $b$ lie in different orbits of $M$ but the same orbits of $T$ then\n$$\\langle \\lbrace t \\in M \\: : \\: a \\in \\path{t,r} \\rbrace, \\leq_M \\rangle \\not\\cong \\langle \\lbrace t \\in M \\: : \\: b \\in \\path{t,r} \\rbrace, \\leq_M \\rangle$$\nand\n$$\n\\begin{array}{c}\n\\langle \\lbrace t \\in M \\: : \\: a \\leq_{T(M)} t \\rbrace, \\leq_{T(M)}, U_{T(M)} \\rangle \\quad \\quad  \\quad \\quad  \\quad \\\\\n\\cong \\\\\n\\quad \\quad \\quad  \\quad \\quad  \\langle \\lbrace t \\in M \\: : \\: b \\leq_{T(M)} t \\rbrace, \\leq_{T(M)}, U_{T(M)} \\rangle\n\\end{array}\n$$\n\nHowever, this contradicts Lemma \\ref{lemma:T(M)andsubstructure}, so if $a$ and $b$ lie in the same orbit of $T(M)$ then they lie in the same orbit of $M$.  By symmetry, we also conclude that if $a$ and $b$ lie in the same orbit of $M$ then they lie in the same orbit of $T(M)$.  Similarly, if $a,b \\in Y_0$ then $a$ and $b$ lie in the same orbit of $M$ if and only if they lie in the same orbit of $T(M)$.\n\nSo now suppose that for $i < n$ the 1-orbits on $X_i$ and $Y_i$ from $\\textnormal{Aut}(M)$ and $\\textnormal{Aut}(T(M))$ coincide and let $x,y \\in X_n$.  We define, as we did in Lemma \\ref{lemma:TsetwiseX}, $x_1, \\dots x_n$ and $y_1, \\ldots, y_n$, which are linearly ordered by $\\leq_{T(M)}$, are the connecting sets of $\\path{x,r}$ and $\\path{y,r}$ in $\\leq_{M}$.\n\nIf $x_n$ and $y_n$ belong to the same orbit of $M$ then the automorphism that witnesses this also witnesses that $x_{n-1}$ and $y_{n-1}$ lie in the same orbit of $M$, and hence by our induction hypothesis, the same orbit of $T$.  Since there is an automorphism that maps $x_{n-1}$ to $y_{n-1}$,\n$$\\langle \\lbrace z \\in M \\: : \\: x_{n-1} \\in \\path{r,z} \\rbrace, \\leq_M \\rangle \\cong \\langle \\lbrace z \\in M \\: : \\: y_{n-1} \\in \\path{r,z} \\rbrace, \\leq_M \\rangle$$\nand hence (using Lemmas \\ref{lemma:T(M)andisomorphism} and \\ref{lemma:T(M)andsubstructure})\n$$\n\\begin{array}{c}\n\\langle \\lbrace z \\in M \\: : \\: x_{n-1} \\in \\path{r,z} \\rbrace, \\leq_{T(M)}, U_{T(M)} \\rangle \\quad \\quad  \\quad \\quad  \\quad \\\\\n\\cong \\\\\n\\quad \\quad \\quad  \\quad \\quad \\langle \\lbrace z \\in M \\: : \\: y_{n-1} \\in \\path{r,z} \\rbrace, \\leq_{T(M)}, U_{T(M)} \\rangle\n\\end{array}\n$$\nAnd so there is an isomorphism of $T$ that maps $x_n$ to $y_n$.\nThe arguments for $x_n, y_n$ being in the same orbit of $T$, and for $x_n, y_n \\in Y_n$ are, again, extremely similar, and so omitted.\n\nWe now turn out attention to the 2-orbits.  Since $r$ is fixed by both $\\textnormal{Aut}(M)$ and $\\textnormal{Aut}(T)$, the 1-orbits can be thought of as 2-orbits where one of the elements is $r$, and the 2-orbits can be thought of as 3-orbits where $r$ is one of the elements.  This viewpoint is exploited to show the coincidence of the 2-orbits of $\\textnormal{Aut}(M)$ and $\\textnormal{Aut}(T)$.\n\nSuppose $(x_0,x_1)$ and $(y_0,y_1)$ lie in the same orbit of $M$.  We need only consider the case when $x_0 \\in \\path{x_1, r}$ as otherwise we can take $x_2$ to be the intersection of $\\path{x_0,r}$, $\\path{x_0,x_1}$ and $\\path{x_1,r}$, and patch automorphisms together around $x_2$.  Note that $x_2$ would be the meet of $x_0$ and $x_1$ in $T(M)$.\n\nThere is an automorphism of $M$ that maps $x_0$ to $y_0$, and as we have just seen, this means that\n$$\\langle \\lbrace z \\in M \\: : \\: x_{0} \\in \\path{r,z} \\rbrace, \\leq_M \\rangle \\cong \\langle \\lbrace z \\in M \\: : \\: y_{0} \\in \\path{r,z} \\rbrace, \\leq_M \\rangle$$\nSince $(x_0,x_1)$ and $(y_0,y_1)$ lie in the same orbit of $M$, there is an isomorphism from\n$$\\langle \\lbrace z \\in M \\: : \\: x_{0} \\in \\path{r,z} \\rbrace, \\leq_M \\rangle \\; \\mathrm{to} \\; \\langle \\lbrace z \\in M \\: : \\: y_{0} \\in \\path{r,z} \\rbrace, \\leq_M \\rangle$$\nthat maps $x_1$ to $y_1$.  By Lemmas \\ref{lemma:T(M)andisomorphism} and \\ref{lemma:T(M)andsubstructure} this results in an isomorphism from\n$$\\langle \\lbrace z \\in M \\: : \\: x_{0} \\leq_{T(M)} z \\rbrace, \\leq_{T(M)}, U_{T(M)} \\rangle$$\nto\n$$\\langle \\lbrace z \\in M \\: : \\: y_{0} \\leq_{T(M)} z \\rbrace, \\leq_{T(M)}, U_{T(M)} \\rangle$$\nwhich maps $x_1$ to $y_1$.  We call this isomorphism $\\phi$, and we take any automorphism that takes $x_0$ to $y_0$ and call it $\\psi$.  The function\n$$\n\\theta(t) := \\left\\lbrace\n\\begin{array}{c l}\n\\phi(t) & t \\geq_{T(M)} x_0 \\\\\n\\psi(t) & \\mathrm{otherwise}\n\\end{array} \\right.\n$$\nis an automorphism of $T$ which maps $(x_0,x_1)$ to $(y_0,y_1)$, and thus the 2-orbits of $T$ contain the 2-orbits of $M$.\n\n\nOnce again, the argument to show that the 2-orbits of $M$ contain the 2-orbits of $T$ is extremely similar, due to the symmetric nature of Lemmas  \\ref{lemma:T(M)andisomorphism} and \\ref{lemma:T(M)andsubstructure}, and thus we conclude that the 2-orbits of $M$ and $T$ coincide, and so\n$$\\textnormal{Aut}(\\langle M, \\leq_M \\rangle) \\cong_P \\textnormal{Aut}(\\langle T(M), \\leq_{T(M)}, U \\rangle)$$\n\\end{proof}\n\nLots of CFPOs have fixed points, but the CFPOs of the kind discussed in the next lemma reoccur frequently.\n\n\\begin{lemma}\\label{FixedPointPartition} \nLet $M$ be a connected CFPO.  If there are connected $A,B \\subsetneq M$ which are disjoint and fixed setwise by $\\textnormal{Aut}(M)$ then there are $c,d$ which are fixed points of $M$ and $$\\path{A,B}= \\path{c,d}$$\n\\end{lemma}\n\\begin{proof}\nLet $M$ be a connected CFPO, and let $A,B$ be connected proper subsets of $M$ which are disjoint and fixed setwise by $\\textnormal{Aut}(M)$.  We use the notation $$\\path{x,y}^- := \\lbrace z \\in \\path{x,y} \\: : \\: \\exists a,b \\in \\path{x,y} \\; (z=(a\\wedge b) \\vee z=(a \\vee b)) \\rbrace$$\nIn words, $\\path{x,y}^-$ are the local maxima and minima of $\\path{x,y}$.  Just as with $\\path{x,y}$, if $X$ and $Y$ are subsets of $M$ then:\n$$\n\\begin{array}{r c l}\n\\path{x,Y}^- & := & \\bigcap_{y \\in Y } \\path{x,y}^- \\\\\n\\path{X,y}^- & := & \\bigcap_{x \\in X } \\path{x,y}^- \\\\\n\\path{X,Y}^- & := & \\bigcap_{\\substack{ x \\in X \\\\ y \\in Y }} \\path{x,y}^-\n\\end{array}\n$$\nNote that $\\path{x,y}^-$ always has finite cardinality.\n\nWe are going to find a fixed point using (possibly transfinite) induction.  Fix $b \\in B$.\n\n\\begin{description}\n\\item[Base Case] Pick $a_0 \\in A$.  We set $c_0 = a_0$ and let $D_0= \\lbrace x \\in A \\: : \\: c_0 \\in \\path{x,b} \\rbrace$.\n\\item[Successor Step] Suppose we have $a_{\\alpha-1}$, $c_{\\alpha-1}$ and $D_{\\alpha-1}$.\n\nPick $a_\\alpha \\in A \\setminus D_{\\alpha-1}$.  Since $b \\in \\path{c_{\\alpha-1}, b}$ and $b \\in \\path{a_\\alpha, b}$,\n$$\\path{\\lbrace c_{\\alpha-1}, a_\\alpha \\rbrace, b} \\not= \\emptyset$$\n\nLet $$C_\\alpha := \\lbrace x \\in \\path{\\lbrace c_{\\alpha-1}, a_\\alpha \\rbrace, b} \\: : \\: |\\path{\\lbrace c_{\\alpha-1}, a_\\alpha \\rbrace, b}^-| = |\\path{x, b}^-| \\rbrace$$\n$C_\\alpha$ is linearly ordered, and is bounded both above and below by elements of $\\path{c_{\\alpha-1}, b}^- \\cup \\path{a_\\alpha, b}^-$.  Since $M$ is Rubin complete, $C_\\alpha$ has both a maximal and a minimum element.\n\nLet $c_\\alpha \\in C_\\alpha$ be such that $\\path{\\lbrace c_{\\alpha-1}, a_\\alpha \\rbrace, b} = \\path{c_\\alpha, b}$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.075]\n\\draw (0,40) -- (10,30) -- (20,40) -- (40,20);\n\\draw[dashed] (40,20) -- (60,0);\n\\draw (60,0) -- (70,10) -- (80,0);\n\\draw (40, 20) -- (30,10) -- (20,20);\n\n\\fill[white] (0,40) circle (0.5);\n\\fill[white] (20,20) circle (0.5);\n\\fill[white] (80,0) circle (0.5);\n\\draw (0,40) circle (0.5);\n\\draw (20,20) circle (0.5);\n\\draw (80,0) circle (0.5);\n\n\\fill (10,30) circle (0.5);\n\\fill (20,40) circle (0.5);\n\\fill (30,10) circle (0.5);\n\\fill (40,20) circle (0.5);\n\\fill (60,0) circle (0.5);\n\\fill (70,10) circle (0.5);\n\n\\draw[anchor=south] (0,40) node {$a_\\alpha$};\n\\draw[anchor=south] (20,20) node {$c_{\\alpha-1}$};\n\\draw[anchor=west] (40,20) node {$c_\\alpha$};\n\\draw[anchor=north] (80,0) node {$b$};\n\\draw[anchor=south] (51,10) node {$C_\\alpha$};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Finding $c_\\alpha$ in Lemma \\ref{FixedPointPartition}}\n\\end{figure}\n\nSince $A$ is connected, $\\path{c_{\\alpha-1}, a_\\alpha} \\subseteq A$, and since $c_\\alpha \\in \\path{c_{\\alpha-1}, a_\\alpha}$, we have that $c_\\alpha \\in A$.\n\nWe define $D_\\alpha= \\lbrace x \\in A \\: : \\: c_\\alpha \\in \\path{x,b} \\rbrace$.  If $D_\\alpha = A$ then let $c = c_\\alpha$ and stop.\n\n\\item[Limit Step] Let $n_\\lambda = \\min \\lbrace |\\path{c_\\alpha,b}^-| \\: : \\: \\alpha < \\lambda \\rbrace$.\n$$C_\\lambda := \\lbrace x \\in \\path{ c_\\alpha, b} \\: : \\: |\\path{ c_\\alpha, b}^-| = n_\\lambda \\rbrace$$\n$C_\\alpha$ is linearly ordered, and is bounded both above and below by elements of $\\bigcup_{\\alpha < \\lambda} \\path{c_{\\alpha}, b}^-$, so has both a maximal and minimal element.\n\nLet $c_\\lambda \\in C_\\lambda$ be such that $\\path{c_\\lambda, b} \\subseteq \\path{\\lbrace c_{\\alpha}, a_\\alpha \\rbrace, b}$.  We define $D_\\lambda= \\lbrace x \\in A \\: : \\: c_\\lambda \\in \\path{x,b} \\rbrace$.  If $D_\\lambda = A$ then let $c = c_\\lambda$ and stop.\n\\end{description}\nWe have found a $c$ such that $c \\in \\path{A,b}$.  If we repeat this induction, fixing $c$ and choosing $b_\\alpha$ from $B$ then we find a $d$ such that\n$$ \\path{c,d} = \\path{A,B}$$\n\nLet $\\phi \\in \\textnormal{Aut}(M)$.\n$$\\begin{array}{rcl}\n\\path{\\phi(c), \\phi(d)} & = & \\phi (\\path{c,d}) \\\\\n& = & \\phi(\\path{A,B}) \\\\\n& = & \\path{\\phi(A),\\phi(B)} \\\\\n& = & \\path{A,B} \\\\\n& = & \\path{c,d}\n\\end{array}\n$$\nTherefore both $c$ and $d$ are fixed by all automorphisms of $M$. \n\\end{proof}\n\n\\subsection{$\\cfpo{n}$}\\label{Section:CFPOn}\n\n\\begin{lemma}\\label{lemma:CFPO3}\nIf $M$ is a connected $\\cfpo{3}$ then $M$ is treelike.\n\\end{lemma}\n\\begin{proof}\nA $\\cfpo{3}$ can be split into three possibly empty sections, a tree which is above a linear order, which in turn is above a reverse ordering of a tree.  If the tree section is empty the reverse tree cannot be empty, and vice versa.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}[scale= 0.075]\n\\draw (0,60) -- (20,40) -- (40,60);\n\\draw (15,60) -- (10,50);\n\\draw (20,40) -- (27.5,60);\n\n\\draw (20,40) -- (20,20);\n\n\\draw (0,0) -- (20,20) -- (40,0);\n\\draw (15,0) -- (10,10);\n\\draw (25,0) -- (30,10);\n\n\\draw[snake=brace, thick] (50,60)--(50,41);\n\\draw[snake=brace, thick] (50,39)--(50,21);\n\\draw[snake=brace, thick] (50,19)--(50,0);\n\n\\draw (60,50.5) node {A tree};\n\\draw (68,30) node {A linear order};\n\\draw (69,9.5) node {A reversed tree};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{A typical $\\cfpo{3}$}\n\\end{figure}\n\nBy marking the reversed tree with a unary predicate and reversing its order we obtain a tree which has the same automorphism group as the $\\cfpo{3}$.\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}[scale= 0.075]\n\\draw (0,40) -- (20,20) -- (40,40);\n\\draw (15,40) -- (10,30);\n\\draw (20,20) -- (27.5,40);\n\n\\draw (20,20) -- (20,0);\n\n\\draw[dashed] (0,20) -- (20,0) -- (40,20);\n\\draw[dashed] (15,20) -- (10,10);\n\\draw[dashed] (25,20) -- (30,10);\n\\end{tikzpicture}\n\\end{center}\n\\caption{A Tree with the same Automorphism Group}\n\\end{figure}\n\\end{proof}\n\n\\begin{theorem}\\label{thm:cfpo2n+1}\nIf $M$ is a connected $\\cfpo{2n+1}$ then $M$ is treelike.\n\\end{theorem}\n\\begin{proof}\nOur strategy is to find a subset of $M$ which is a $\\cfpo{3}$ and is fixed setwise by $\\textnormal{Aut}(M)$, and add cones to the tree corresponding to this $\\cfpo{3}$ to obtain a tree with the same automorphism group as $M$.\n\nWe consider the $\\phi(a_n)$ and $\\phi(a_n^*)$, the images in $M$ of the midpoints of $\\mathrm{Alt}_{2n+1}$ and $\\mathrm{Alt}_{2n+1}^*$ under all possible embeddings $\\phi$.  Let $C$ be the set of all such $\\phi(a_n)$ and $\\phi(a_n*)$.  This is the candidate for the $\\cfpo{3}$ we require for our strategy, but first we must show that it is indeed a $\\cfpo{3}$, and that it is fixed setwise by $\\textnormal{Aut}(M)$.\n\nSuppose that $C$ contains an antichain ${x_n,y_n}$.  Since $M$ is connected there must be a path between $x_n$ and $y_n$.  We also pick particular copies of either $\\mathrm{Alt}_{2n+1}$ or $\\mathrm{Alt}_{2n+1}^*$ that contain $x_n$ and $y_n$, and label the points using $x_i$ and $y_i$ appropriately.  $X$ is the set $\\lbrace x_i \\rbrace$, while $Y= \\lbrace y_i \\rbrace$.\n\nTo show that the maximum length of a path though $C$ is 3 we consider how the ends of $\\path{x_n,y_n}$ interact with $X$ and $Y$.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.15]\n\\draw (5,10) -- (10,0) -- (15,10);\n\\draw (10,0) -- (15,20);\n\n\\draw (45,0) -- (50,10) -- (55,0);\n\\draw (50,10) -- (55,20);\n\n\\fill (5,10) circle (0.5);\n\\fill (10,0) circle (0.5);\n\\fill (15,10) circle (0.5);\n\\fill (45,0) circle (0.5);\n\\fill (50,10) circle (0.5);\n\\fill (55,0) circle (0.5);\n\n\\draw (1,10) node {$x_{n-1}$};\n\\draw (13,0) node {$x_{n}$};\n\\draw (19,10) node {$x_{n+1}$};\n\\draw (15,22) node {$\\path{x_n,y_n}$};\n\n\\draw (41,0) node {$x_{n-1}$};\n\\draw (48,10) node {$x_{n}$};\n\\draw (59,0) node {$x_{n+1}$};\n\\draw (54,22) node {$\\path{x_n,y_n}$};\n\n\\draw (10,-3) node {Case 1};\n\\draw (50,-3) node {Case 2};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Interactions between $X$ and $Y$}\n\\end{figure}\n\nThe cases where $x_n$ is an upper point of $\\path{x_n,y_n}$ are reverse orderings of Cases 1 and 2, so will not be done explicitly.  Also there is nothing special in our choice of $X$, so these arguments also apply to $Y$.\n\\paragraph*{Case 1}\nIn this case $x_n$ is a lower point of both $X$ and $\\path{x_n,y_n}$.\n\nIf $[x_n,x_{n+1}] \\cap \\path{x_n,y_n} \\not= \\emptyset$ then $[x_n,x_{n-1}] \\cap \\path{x_n,y_n} = \\emptyset$, otherwise $x_{n-1}$ and $x_{n+1}$ would be related.  So the union of at least one of $\\lbrace x_0, \\ldots x_{n-1} \\rbrace$ or $\\lbrace x_{n+1}, \\ldots x_{2n} \\rbrace$ with $ \\path{x_n,y_n}$ is a copy of a finite section of Alt.\n\n\\paragraph*{Case 2}\nIn this case $x_n$ is an upper point of $X$ but a lower point of $\\path{x_n,y_n}$.  As both $x_{n-1}$ and $x_{n+1}$ lie below $x_n$ the two paths $\\path{x_{n-1},y_n}$ and \\linebreak $\\path{x_{n+1},y_n}$ both contain and have the same length as $\\path{x_n,y_n}$.  We also know that $x_{n-2}$ cannot be contained in $\\path{x_{n-1},y_n}$, as this would require $x_{n-2}$ and $x_n$ to be related.  Similarly $x_{n+2}$ cannot be contained in $\\path{x_{n+1},y_n}$.  Thus we see that both $\\lbrace x_0, \\ldots x_{n-2} \\rbrace \\cup \\path{x_{n-1},y_n}$ and $\\lbrace x_{2n}, \\ldots x_{n+2} \\rbrace \\cup \\path{x_{n+1},y_n}$ are copies of a finite section of Alt.\n\nThus in both cases, at least one of $\\lbrace x_0, \\ldots x_{n-1} \\rbrace$ or $\\lbrace x_{n+1}, \\ldots x_{2n} \\rbrace$ with $ \\path{x_n,y_n}$ is a copy of a finite section of Alt.  $M$ is a cycle free partial order so, assuming that the configurations of $X$, $Y$ and $\\path{x_n,y_n}$ result in the shortest possible finite alternating chain,\n$$P := \\lbrace x_0, \\ldots, x_{n-2} \\rbrace \\cup \\path{x_{n-1},y_{n+1}} \\cup \\lbrace y_{n+2}, \\ldots y_{2n} \\rbrace$$\nis a copy of a finite section of Alt.  The length of $P$ is $$2n - 2 + \\mid \\path{x_{n-1},y_{n+1}} \\mid$$\nBy assumption $M$ is a $\\cfpo{2n+1}$, so $P$ has at most $2n+1$ elements, thus $\\mid \\path{x_{n},y_{n}} \\mid \\; \\leq 3$ and $C$ is a $\\cfpo{3}$.\n\nTo see that $C$ is fixed setwise by automorphisms, simply note for any $x \\in C$ and $\\phi \\in \\textnormal{Aut}(M)$, the image of the copy of $\\alt{2n+1}$ that witnesses the fact that $x \\in C$ will witness $\\phi(x) \\in C$.\n\nWe now have the $\\cfpo{3}$ our strategy demands, so now we focus on how we may adjoin cones to it to obtain a tree with the same automorphism group as $M$.\n\nFor each $x \\in C$, we define $B(x):= \\lbrace y \\in M \\: : \\: \\path{x,y} \\cap C = \\lbrace x \\rbrace \\rbrace$.  If we introduce a predicate that fixes $x$ to $B(x)$, then we are able to apply the construction in Definition \\ref{Def:XandY} to $B(x)$ using $x$ as the root to obtain $T(B(x))$.  We also know that if there is an automorphism of $M$ that maps $x_0$ to $x_1$ then $B(x_0) \\cong B(x_1)$.\n\nFor each isomorphism type of $B(x)$, we add a colour predicate $P_x$ to $\\langle C, \\leq \\rangle $ such that $C \\models P_x(y)$ if and only if $B(y) \\cong B(x)$.  We obtain $\\langle C, \\leq_M, \\lbrace P_x \\rbrace \\rangle$, a $\\cfpo{3}$ such that:\n$$\\textnormal{Aut}(\\langle C, \\leq_M, \\lbrace P_x \\rbrace \\rangle) \\cong_P \\lbrace g \\in \\textnormal{Aut}(C) \\, : \\, \\exists h \\in \\textnormal{Aut}(M) \\, h |_C = g \\rbrace$$\nLemma \\ref{lemma:CFPO3} shows that there is a tree, which we call $T(C)$ such that\n$$\\textnormal{Aut}(T(C)) \\cong \\lbrace g \\in \\textnormal{Aut}(C) \\, : \\, \\exists h \\in \\textnormal{Aut}(M) \\, h |_C = g \\rbrace$$\n\nWe define $T$ to be the structure whose domain is\n$$T_C \\cup \\bigcup_{x \\in C} T(B(x))$$\nunder the equivalence relation that identifies the root of $T_{B(x)}$ with the point of $T_C$ that corresponds with $x$.  We give $T$ the transitive closure of the order inherited from $T_C$ and all the $T_{B(x)}$.  This structure is clearly a tree with the automorphism group of $M$.\n\nNote that this method not only gives a tree $T$ such that $\\textnormal{Aut}(M) \\cong_A \\textnormal{Aut}(T)$, but also a tree $T$ such that $\\textnormal{Aut}(T) \\cong_P \\textnormal{Aut}(M)$.\n\\end{proof}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.05]\n\\draw (-10,40) -- (10,20) -- (30,40);\n\\draw (10,20) -- (10,-20);\n\\draw (-10,-40) -- (10,-20) -- (30,-40);\n\n\\draw (-10,30) -- (5,25) -- (-10,10);\n\\fill[white] (-10,20) ellipse (5 and 10);\n\\draw (-10,20) ellipse (5 and 10);\n\n\\draw (30,30) -- (15,25) -- (30,10);\n\\fill[white] (30,20) ellipse (5 and 10);\n\\draw (30,20) ellipse (5 and 10);\n\n\\draw (-10,-30) -- (5,-25) -- (-10,-10);\n\\fill[white] (-10,-20) ellipse (5 and 10);\n\\draw (-10,-20) ellipse (5 and 10);\n\n\\draw (30,-30) -- (15,-25) -- (30,-10);\n\\fill[white] (30,-20) ellipse (5 and 10);\n\\draw (30,-20) ellipse (5 and -10);\n\n\\draw[->] (30,0) -- (49,0);\n\\draw[snake=brace, thick] (50,-80)--(50,80);\n\n\\draw (55,80) -- (75,60) -- (95,80);\n\\draw (75,60) -- (75,40);\n\\draw (55,20) -- (75,40) -- (95,20);\n\n\\draw[->] (90,40) -- (110,40);\n\n\\draw (115,70) -- (135,50) -- (155,70);\n\\draw (135,50) -- (135,30);\n\\draw[dashed] (115,50) -- (135,30) -- (155,50);\n\n\\draw (60,-30) -- (75,-25) -- (60,-10);\n\\fill[white] (60,-20) ellipse (5 and 10);\n\\draw (60,-20) ellipse (5 and 10);\n\n\\draw (95,-30) -- (80,-25) -- (95,-10);\n\\fill[white] (95,-20) ellipse (5 and 10);\n\\draw (95,-20) ellipse (5 and 10);\n\n\\draw (60,-50) -- (75,-55) -- (60,-70);\n\\fill[white] (60,-60) ellipse (5 and 10);\n\\draw (60,-60) ellipse (5 and 10);\n\n\\draw (95,-50) -- (80,-55) -- (95,-70);\n\\fill[white] (95,-60) ellipse (5 and 10);\n\\draw (95,-60) ellipse (5 and 10);\n\n\\draw[->] (90,-40) -- (110,-40);\n\n\\draw[dotted] (110,-15) -- (120,-35) -- (130,-15);\n\\fill[white] (120,-15) ellipse (10 and 5);\n\\draw[dotted]  (120,-15) ellipse (10 and 5);\n\n\\draw[dotted]  (135,-15) -- (145,-35) -- (155,-15);\n\\fill[white] (145,-15) ellipse (10 and 5);\n\\draw[dotted]  (145,-15) ellipse (10 and 5);\n\n\\draw[dotted]  (110,-50) -- (120,-70) -- (130,-50);\n\\fill[white] (120,-50) ellipse (10 and 5);\n\\draw[dotted]  (120,-50) ellipse (10 and 5);\n\n\\draw[dotted]  (135,-50) -- (145,-70) -- (155,-50);\n\\fill[white] (145,-50) ellipse (10 and 5);\n\\draw[dotted]  (145,-50) ellipse (10 and 5);\n\n\\draw[->] (161,0) -- (180,0);\n\\draw[snake=brace, thick] (160,80)--(160,-80);\n\n\\draw[dashed] (175,-30) -- (195,-50) -- (215,-30);\n\\draw (195,-50) -- (195,10);\n\\draw (175,30) -- (195,10) -- (215,30);\n\n\\draw[dotted] (176,-20) -- (185,-40) -- (194,-20);\n\\fill[white] (185,-20) ellipse (9 and 5);\n\\draw[dotted]  (185,-20) ellipse (9 and 5);\n\n\\draw[dotted] (196,-20) -- (205,-40) -- (214,-20);\n\\fill[white] (205,-20) ellipse (9 and 5);\n\\draw[dotted]  (205,-20) ellipse (9 and 5);\n\n\\draw[dotted] (176,40) -- (185,20) -- (194,40);\n\\fill[white] (185,40) ellipse (9 and 5);\n\\draw[dotted]  (185,40) ellipse (9 and 5);\n\n\\draw[dotted] (196,40) -- (205,20) -- (214,40);\n\\fill[white] (205,40) ellipse (9 and 5);\n\\draw[dotted]  (205,40) ellipse (9 and 5);\n\n\\draw (10,-60) node {$M$};\n\\draw (75,80) node {$C$};\n\\draw (135,80) node {$T_C$};\n\\draw (195,-60) node {$T$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Turning a $\\cfpo{2n+1}$ into a Tree}\n\\end{figure}\n\n\\begin{cor}\\label{cor:cfpo2n}\nIf $M$ is a connected $\\cfpo{2n}$ then $M$ is treelike.\n\\end{cor}\n\\begin{proof}\nLet $e \\in M$ be an image of $a_0 \\in \\alt{2n}$ (if $\\alt{2n}$ does not embed into $M$ we may consider $M^*$ instead).  Below every point in $\\mathrm{Or}(e)$ we adjoin a new point, coloured with a new unary predicate.  This new structure is a $\\cfpo{2n+1}$ with the same automorphism group as $M$, so $M$ shares its abstract automorphism group with a tree. \n\\end{proof}\n\nWhile we have found a tree $T$ such that $\\textnormal{Aut}(M) \\cong_A \\textnormal{Aut}(T)$, and thereby proved the corollary, we may do better than that.  We can delete the points we added to $M$ from $T$ without introducing new automorphisms (as we added these points to every point in an orbit of $M$), getting a $T^*$ such that $\\textnormal{Aut}(M) \\cong_P \\textnormal{Aut}(T^*)$.\n\n\\subsection{$\\cfpo{\\omega}$}\n\n\\begin{theorem}\\label{thm:cfpoomega}\nIf $M$ is a connected $\\cfpo{\\omega}$ then $M$ is tree-like.\n\\end{theorem}\n\\begin{proof}\nThis proof works in a similar fashion to the proofs of Theorem \\ref{thm:fixedpoint}, Lemma \\ref{lemma:CFPO3} and Theorem \\ref{thm:cfpo2n+1}; by altering the order on the CFPO we produce a tree, while maintaining the automorphism group.  Let $M$ be a Rubin-complete CFPO.\n\nWe say that $A \\subseteq M$ is a maximal copy of either $\\alt{\\omega}$ or $\\alt{\\omega}^*$ if\n\\begin{itemize}\n\\item $A$ is the image of $\\alt{\\omega}$ (or $\\alt{\\omega}^*$ respectively).\n\\item There is no image of $\\alt{\\omega}$ or $\\alt{\\omega}^*$ that properly contains $A$.\n\\end{itemize}\n\nEvery copy of $\\alt{\\omega}$ is contained in a maximal copy of either $\\alt{\\omega}$ or $\\alt{\\omega}^*$.  To see this, let $\\lbrace A_n \\subseteq M \\: : \\: n \\in \\omega \\rbrace$ be such that each $A_n$ is isomorphic to either $\\alt{\\omega}$ or $\\alt{\\omega}^*$ and if $n < m$ then $A_n \\subsetneq A_m$.  This means that\n$$\\bigcup_{n \\in \\mathbb{N}} (A_n \\setminus A_0) \\cong \\alt{\\omega} \\:\\textnormal{or} \\: \\alt{\\omega}^* $$\nand therefore\n$$A_0 \\cup \\bigcup_{n \\in \\mathbb{N}} (A_n \\setminus A_0) \\cong \\alt{}$$\n\nWe now describe a procedure that transforms $M$ into a tree while preserving its automorphism group.  Again, we add a unary predicate $U$ to remind us when we've changed direction.\n\\begin{enumerate}\n\\item Let $M_0$ be the following set:\n$$\\lbrace x \\in M  :x \\; \\textnormal{is the first element of a maximal copy of either} \\, \\alt{\\omega} \\; \\textrm{or} \\; \\alt{\\omega}^* \\rbrace$$\n\nIf $x \\in M_0$ is witnessed by a maximal copy of $\\alt{\\omega}$ then $x \\in M_0$ cannot be witnessed by a maximal copy of $\\alt{\\omega}^*$.  To see this, let $\\lbrace x, a_1, \\ldots \\rbrace$ be a maximal copy of $\\alt{\\omega}$ and let $\\lbrace x, b_1, \\ldots \\rbrace$ be a maximal copy of $\\alt{\\omega}^*$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.1]\n\\draw (0,10) -- (10,0) -- (20,10) -- (40,-10) -- (50,0) -- (60,-10) ;\n\\fill (   0, 10) circle (0.5);\n\\fill ( 10,   0) circle (0.5);\n\\fill ( 20, 10) circle (0.5);\n\\fill ( 30,   0) circle (0.5);\n\\fill ( 40,-10) circle (0.5);\n\\fill ( 50,   0) circle (0.5);\n\\fill ( 60,-10) circle (0.5);\n\n\\draw[anchor=south] (   0, 10) node {$b_3$};\n\\draw[anchor=north] ( 10,   0) node {$b_2$};\n\\draw[anchor=south] ( 20, 10) node {$b_1$};\n\\draw[anchor=east] ( 30,   0) node {$x$};\n\\draw[anchor=north] ( 40,-10) node {$a_1$};\n\\draw[anchor=south] ( 50,   0) node {$a_2$};\n\\draw[anchor=north] ( 60,-10) node {$a_3$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Witnessing $x \\in M_0$}\n\\end{figure}\n\n$b_1 > a_1$, but $b_2 || a_1$, as $b_2 || x$, so $\\lbrace b_3, b_2, b_1, a_1, \\ldots \\rbrace$ is a copy of $\\alt{\\omega}$, contradicting the assumption that $\\lbrace x, a_1, \\ldots \\rbrace$ was a maximal copy of $\\alt{\\omega}$.\n\nLet $\\sim_C$ be the relation on $M_0$ given by\n$$x \\sim_C y \\Leftrightarrow \\left(\n\\begin{array}{c}\n\\lbrace x, a_1, \\ldots \\rbrace \\:\\textnormal{witnesses}\\: x \\in M_0  \\\\\n\\textnormal{if and only if} \\\\\n\\lbrace y, a_1, \\ldots \\rbrace \\:\\textnormal{witnesses}\\: y \\in M_0.\n\\end{array}\\right)$$\n\nThat $\\sim_C$ is an equivalence relation is readily apparent.  We denote the $\\sim_C$-equivalence classes as $C^0_i$.\n\nLet $x \\in M_0$, and let this be witnessed by $\\lbrace x, a_1, \\ldots \\rbrace$, a copy of $\\alt{\\omega}$.  For every $y \\in [x]_{\\sim_C}$, we know that $y > a_1$, and thus $[x]_{\\sim_C} \\cup a_1$ is a tree.  Similarly, if $x \\in M_0$ is witnessed by a copy of $\\alt{\\omega}^*$ then $[x]_{\\sim_C}$ is a reverse ordered tree.\n\nLet $\\lbrace C^0_i \\rbrace$ be the set of $\\sim_C$-equivalence classes of $M_0$.\n\\item Assume we have defined $M_{n-1}$ and the $C^{n-1}_i$s. We define $M_n$ to be:\n$$\\left\\lbrace x \\in M \\setminus \\bigcup\\limits_{i <n} M_{i} \\; : \\; \n\\begin{array}{c}\nx \\; \\textnormal{is the first element of}\\: A \\:  \\textnormal{which is a maximal}\\\\\n\\textnormal{copy of either} \\, \\alt{\\omega} \\; \\textrm{or} \\; \\alt{\\omega}^*  \\textnormal{in} \\left( M \\setminus \\bigcup\\limits_{i <n} M_{i} \\right)\n\\end{array}\n\\right\\rbrace$$\nAgain, $M_n$ is a disjoint union of trees and reverse ordered trees, which we call $C^{n}_i$.\n\\end{enumerate}\nIf $C^n_i$ is a tree then $T(C^n_i):= \\langle  C^n_i, \\leq, U \\rangle$ where $U$ is realised nowhere, and if $C^n_i$ is a reverse ordered tree then $T(C^n_i):= \\langle (C^n_i)^*, \\leq, U \\rangle$ where $U$ is realised everywhere.\n\nWe define $T_0$ to be the disjoint union of $\\lbrace T(C^0_i) \\rbrace$ with no new relations added to the ordering.  If we have already defined $T_{n-1}$ then\n$$T_n := T_{n-1} \\cup \\bigcup \\lbrace T(C^n_i) \\rbrace$$\nWe add to the order inherited from $T_{n-1}$ and $T(C^n_i)$ pairs of the form $(x,y)$ where\n$$x \\in T(C^n_i) \\: \\textnormal{for some} \\: i$$\nand $y$ is in $T(C^{n-1}_j)$, where $C^{n-1}_j$ is a cone of $x$.  We then take the transitive closure to obtain an ordering.\n\nPut $T(M) := \\bigcup\\limits_{n \\in \\mathbb{N}} T_n$.  Since the $M_n$ partition $M$ and since at each stage we place trees above elements of trees, $T(M)$ is a tree.\n\nIf $\\textnormal{Aut}(M)$ does not preserve the $M_n$ then we would have a map that sends a maximal copy of $\\alt{\\omega}$ or $\\alt{\\omega}^*$ to a non-maximal copy.  $T(M)$ realises $U$ in monochromatic convex subsets.  In the tree obtained by collapsing each of those subsets to a singleton, every maximal chain is isomorphic to $\\omega^*$, so $T(M)$ preserves the $M_n$ set-wise too.\n\nSince each $T(C^n_i)$ is monochromatic, and is order-isomorphic to either $C^n_i$ or $(C^n_i)^*$, if $\\textnormal{Aut}(T(M)) \\not= \\textnormal{Aut}(M)$ then we must either:\n\n\\begin{enumerate}\n\\item be unable send a $T(C^n_i)$ to a $T(C^n_j)$ where we can map $C^n_i$ to a $C^n_j$; or\n\\item be able to send $T(C^n_i)$ to $T(C^n_j)$ where we cannot map $C^n_i$ to $C^n_j$.\n\\end{enumerate}\n\nIf $T(C^n_i) \\cong T(C^n_j)$ but we cannot map one to the other using an automorphism of $T(M)$ then we must eventually attach $T(C^n_i)$ to something different to what we attach $T(C^n_j)$ to, but then $C^n_i$ emanates from a point that is different to the point that $C^n_i$ emanates from, and we cannot map $C^n_i$ to $C^n_i$.\n\nIf we do this argument in reverse we obtain point 2.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.05]\n\\draw (0,30) -- (10,0) -- (20,30);\n\\fill[white] (10,30) ellipse (10 and 5);\n\\draw (10,30) ellipse (10 and 5);\n\n\\draw (15,-15) -- (15,15) -- (35,-15);\n\\fill[white] (25,-15) ellipse (10 and 5);\n\\draw (25,-15) ellipse (10 and 5);\n\n\\draw (10,0) -- (0,-10) -- (-10,10) -- (-15,0);\n\\draw[anchor=east] (-15,0) node {$\\ldots$};\n\n\\draw[->] (35,0) -- (49,0);\n\\draw[snake=brace, thick] (50,-60)--(50,60);\n\n\\draw (55,35) -- (55,55) -- (75,35);\n\\fill[white] (65,35) ellipse (10 and 3);\n\\draw (65,35) ellipse (10 and 3);\n\n\\draw (65,25) node {$\\&$};\n\n\\draw[anchor=east] (56,0) node {$\\ldots$};\n\\draw (56,0) -- (65,10) -- (75,-10) -- (80,0);\n\\draw (70,20) -- (80,0) -- (90,20);\n\\fill[white] (80,20) ellipse (10 and 3);\n\\draw (80,20) ellipse (10 and 3);\n\n\\draw[->] (85,0) -- (95,0);\n\n\\draw (100,20) -- (110,0) -- (120,20);\n\\fill[white] (110,20) ellipse (10 and 3);\n\\draw (110,20) ellipse (10 and 3);\n\n\\draw (110,-5) node {$\\&$};\n\n\\draw[anchor=east] (102,-20) node {$\\ldots$};\n\\draw (102,-20) -- (105,-10) -- (110,-30) -- (112.5,-20); \n\\draw[->] (115,-20) -- (125,-20);\n\\draw[anchor=west] (125,-20) node {$\\ldots$};\n\n\\draw (110, -45) node {$\\vdots$};\n\\draw (165, -45) node {$\\vdots$};\n\n\\draw[->] (145,45) -- (155,45);\n\\draw[dashed] (155,55) -- (165,35) -- (175,55);\n\\fill[white] (165,55) ellipse (10 and 3);\n\\draw[dashed] (165,55) ellipse (10 and 3);\n\n\\draw[->] (145,10) -- (155,10);\n\\draw (155,20) -- (165,0) -- (175,20);\n\\fill[white] (165,20) ellipse (10 and 3);\n\\draw (165,20) ellipse (10 and 3);\n\n\\draw[->] (181,0) -- (190,0);\n\\draw[snake=brace, thick] (180,60)--(180,-60);\n\n\\draw (195,20) -- (205,0) -- (215,20);\n\\fill[white] (205,20) ellipse (10 and 3);\n\\draw (205,20) ellipse (10 and 3);\n\\draw[anchor=north] (205,2) node {$\\vdots$};\n\\draw[dashed] (195,40) -- (205,20) -- (215,40);\n\\fill[white] (205,40) ellipse (10 and 3);\n\\draw[dashed] (205,40) ellipse (10 and 3);\n\n\\draw (7.5,-25) node {$M$};\n\\draw (75,55) node {$C^0_0$};\n\\draw[anchor=south] (110,21) node {$C^1_0$};\n\\draw[anchor=south] (165,56) node {$T(C^0_0)$};\n\\draw[anchor=north] (165,1) node {$T(C^1_0)$};\n\n\\draw (205,-25) node {$T$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Turning a $\\cfpo{\\omega}$ into a Tree}\n\\end{figure}\n\nTherefore every Rubin-complete $\\cfpo{\\omega}$ is treelike.  Let $\\langle M, \\leq_M \\rangle$ be a not necessarily Rubin complete $\\cfpo{\\omega}$, with Rubin completion $\\langle M^R, \\leq_M, I \\rangle$.  There is a tree $T(M)^R$ such that\n$$\\textnormal{Aut}(\\langle M^R, \\leq_M, I \\rangle) \\cong_P \\textnormal{Aut}(\\langle T(M^R), \\leq_T, I, U \\rangle)$$\nWe define $T(M):= \\lbrace x \\in T(M^R) \\: : \\: T(M^R) \\models \\neg I(x) \\rbrace$.  Then\n$$\\textnormal{Aut}(\\langle M, \\leq_M \\rangle) \\cong_P \\textnormal{Aut}(\\langle T(M), \\leq_T, U \\rangle)$$\n\\end{proof}\n\n\\subsection{Disconnected CFPOs}\n\nWhile this section has only proved results about connected CFPOs, they are readily extended to disconnected CFPOs.\n\n\\begin{prop}\\label{prop:FixedPointDisconnected}\nLet $M$ be a possibly disconnected CFPO with connected components $A_i$, where the $i$ are indexed by $I$.  If $A_i$ is treelike for all $i \\in I$ then $M$ is treelike.\n\\end{prop}\n\\begin{proof}\nFor all $i \\in I$, let $\\langle T(A_i), \\leq, U \\rangle$ be the coloured tree such that $\\textnormal{Aut}(\\langle T(A_i), \\leq_i, U_i \\rangle) \\cong_A \\textnormal{Aut}(A_i)$.\n\n$\\mathcal{T} := \\langle \\lbrace r \\rbrace \\cup \\bigcup(T(A_i)), \\leq_T, U_T \\rangle$\nwhere\n\n\\[\n\\begin{array}{rcl}\n\\mathcal{T} \\models (x \\leq_T y) & \\Leftrightarrow & (( \\exists i \\in I \\: (x \\leq_i y)) \\vee (x=r)) \\\\\n\\mathcal{T} \\models U_T(x) & \\Leftrightarrow & \\exists i \\in I \\: U_i(x) \n\\end{array}\n\\]\n\n$\\textnormal{Aut}(M) \\cong_A \\textnormal{Aut}(\\mathcal{T})$, as each of the cones of $r \\in \\mathcal{T}$ share an automorphism group with its corresponding $A_i$, and may only be mapped to one another by an automorphism of $\\mathcal{T}$ if their corresponding $A_i$ are isomorphic.\n\\end{proof}\n\n\\begin{remark}\\label{remark:disconnectedinterpretation}\nIf each of the $T(A_i)$ are obtained using Definition \\ref{Def:XandY}, then we may adapt the interpretation in Lemma \\ref{lemma:MinterpretableinT(M)} by changing $\\phi_{Dom}$ to $x \\not= r$ to obtain an interpretation of $\\langle M, \\leq_M  \\rangle$ in $\\mathcal{T}$.\n\\end{remark}\n\n\\section{CFPOs in Model Theory}\\label{CFPOsMT}\n\nThe theory of trees is known to have certain model theoretic properties.  Parigot showed in 1982 that the theory of trees is NIP, and classified the stable ones \\cite{Parigot1982}, while Simon showed in 2011 that the theory of trees is inp-minimal \\cite{Simon2011}.  The observations that have been made in this section give an easy method for extending these results to the theory of CFPOs.\n\n\\subsection{NIP and Trees}\n\n\\begin{dfn}\nA formula $\\phi(\\bar{x},\\bar{y})$ is said to have the \\textbf{independence property} (for a complete theory $T$) if in every model $M$ of $T$ there is, for each $n<\\omega$, a family of tuples $\\bar{b}_0, \\bar{b}_1, \\ldots \\bar{b}_{n-1}$ such that for every $I \\subseteq \\lbrace 0, 1, \\ldots n-1 \\rbrace$ there is some tuple $\\bar{a} \\in M$ such that\n$$M \\models \\phi(\\bar{a},\\bar{b}_i) \\Leftrightarrow i \\in I$$\n$T$ is said to be \\textbf{NIP} if no formula in $T$ has the independence property.\n\\end{dfn}\n\nNote that if $T$ is interpretable in $S$ then if $\\phi$ has the independence property for $T$ then the interpretation of $\\phi$ has the independence property for $S$.  This means that if $T$ is interpretable in $S$ and $S$ is NIP, then $T$ is NIP.\n\nThe `headline' result of \\cite{Parigot1982} does not mention NIP.\n\n\\begin{theorem}[Parigot, Theorem 2.6 of \\cite{Parigot1982}]\nA type over a tree never has more than $2^{\\aleph_0}$ coheirs.\n\\end{theorem}\n\n`Coheirs' were defined by Poizat, appearing in \\cite{Poizat1981} in 1981, the year before Parigot's paper was published.  If you wish to read the proof of this theorem, but find Poizat's French too daunting, then I recommend the seminar notes of Casanovas \\cite{Casanovas}, which are in English.  I am not aware of any publicly available English translation or account of Parigot's paper.\n\n\\begin{dfn}[Poizat, \\cite{Poizat1981}]\nLet $M,N$ be models such that $M \\prec N$.  Let $p(x) \\subseteq q(x)$ where $q \\in S_1(N)$ and $p \\in S_1(M)$.  We say that $q$ is a \\textit{coheir} of $p$ if $q$ is finitely satisfiable in $M$.\n\\end{dfn}\n\n\\begin{theorem}[Poizat, \\cite{Poizat1981}]\nLet $T$ be a theory.\n\\begin{enumerate}\n\\item If $T$ has the NIP then for all $M$ such that $T \\models M$ and $|M| = \\lambda \\geq |T|$, for all $p \\in  S_1(M)$ there are at most $2^\\lambda$ coheirs of $p$.\n\\item If $T$ has the IP then for every $\\lambda \\geq |T|$ there is an $M$ such that $T \\models M$ and $|M| = \\lambda \\geq |T|$, and there is $p \\in  S_1(M)$ such that $p$ has $2^{2^\\lambda}$ coheirs.\n\\end{enumerate}\n\\end{theorem}\n\nParigot's results do not stop with trees, however.  He extends to `arborescent' structures, defined by Schmerl.\n\n\\begin{dfn}[Schmerl \\cite{SchmerlArborescent}]\nLet $\\mathcal{L} = \\langle R_0, \\ldots, R_{m-1}, U_0, \\ldots ,U_{n-1} \\rangle$ be a finite language where each $R_i$ is a binary predicate and each $U_i$ is a unary predicate.\n\nLet $(x,y) \\equiv (u,v)$ by the following quaternary formula:\n$$x \\not= y \\wedge u \\not= v \\wedge \\bigwedge_{i<m} \\left( (R_i(x,y) \\leftrightarrow R_i(u,v)) \\wedge (R_i(y,x) \\wedge R_i(v,u)) \\right)$$\n\nLet $M$ be an $\\mathcal{L}$-structure.  $M$ is said to be \\textbf{arborescent} if for all finite $B \\subseteq M$, if $|B| \\geq 2$ then there are distinct $a,b \\in B$ such that if $c \\in B \\setminus \\lbrace a,b \\rbrace$ then $(a,c) \\equiv (b,c)$\n\\end{dfn}\n\nFinitely coloured trees are examples of arborescent structures.\n\n\\begin{prop}[Parigot, Corollary 2.8 of \\cite{Parigot1982}]\nAll arborescent structures are NIP.\n\\end{prop}\n\n\\subsection{inp-minimality and Trees}\n\n\\begin{dfn}[Shelah, Definition 7.3 of \\cite{ShelahClassification}]\nAn independence pattern (an inp-pattern) of length $\\kappa$ is a sequence of pairs $(\\phi^\\alpha(x,y), k^\\alpha)_{\\alpha < \\kappa}$ of formulas such that there exists an array $\\langle a_i^\\alpha \\: : \\: \\alpha < \\kappa \\: , i < \\lambda \\rangle$ such that:\n\\begin{itemize}\n\\item Rows are $k^\\alpha$-inconsistent: for each $\\alpha < \\kappa$, the set $\\lbrace \\phi^\\alpha(x, a^\\alpha_i) : i < \\lambda \\rbrace$ is $k^\\alpha$-inconsistent,\n\\item Paths are consistent: for all $\\eta \\in \\lambda^\\kappa$, the set $\\lbrace \\phi^\\alpha(x,a^\\alpha_{\\eta(\\alpha)}) \\: : \\: \\alpha < \\kappa \\rbrace$ is consistent.\n\\end{itemize}\n\\end{dfn}\n\nNote that if $M$ is interpretable in $N$ then any independence pattern in $M$ is also an independence pattern of $N$.\n\n\\begin{dfn}[Goodrick \\cite{Goodrick}]\nA theory is inp-minimal if there is no inp-pattern of length two in a single free variable.\n\\end{dfn}\n\n\\begin{theorem}[Simon, Proposition 4.7 of \\cite{Simon2011}]\nIf $\\langle T, \\leq, C_i \\rangle$ is a coloured tree then $\\mathrm{Th}(\\langle T, \\leq, C_i \\rangle)$ is inp-minimal.\n\\end{theorem}\n\n\\subsection{CFPOs}\n\nHow can we apply these results to CFPOs?\n\nLet $M$ be a CFPO with connected components $A_i$, indexed by $I$.  For each $A_i$, pick an $a_i \\in A_i$ and introduce a new unary predicate $A$ such that\n$$M \\models A(x) \\Leftrightarrow \\exists i \\in I \\: x = a_i$$\nSince we are adding an additional symbol to the language $Th(\\langle M, \\leq_M \\rangle)$ can be interpreted in $Th(\\langle M, \\leq_M, A \\rangle)$ simply by forgetting $A$.\n\n$a_i$ is a fixed point of every $\\langle A_i, \\leq_M, A \\rangle$ so we may invoke Remark \\ref{remark:disconnectedinterpretation} to note that $Th(\\langle M, \\leq_M , A \\rangle$ is interpretable in $ Th(\\mathcal{T})$.\n\nTherefore every CFPO is interpretable in an NIP, inp-minimal theory, and hence is NIP and inp-minimal.\n\nThis shows that if a property that is closed under taking an interpretation is possessed by the theory of coloured trees, then it is possessed by the CFPOs, but the interpretation here is of a special form.  If we are allowed to fix points in a CFPO, we are essentially handling a tree, thus I expect any property of the coloured trees that allows reference to a set of parameters to also be possessed by the CFPOs.\n\n\n\\section{Group Conditions}\\label{gpconditions}\n\n\\begin{dfn}\\label{dfn:dinfinity}\n$D_\\infty$, the \\textbf{infinite dihedral group}, is the group with the following presentation $\\langle \\sigma, \\tau \\, \\vert \\, \\sigma^2 = 1 , \\sigma \\tau \\sigma =  \\tau^{-1} \\rangle$.\n\\end{dfn}\n\nHow $D_\\infty$ occurs as a subgroup of an automorphism group of a CFPO characterises whether it is treelike or not.  We will first examine how $D_\\infty$ can act on trees.\n\n\n\\subsection{Dendromorphic Groups }\n\n\\begin{dfn}\nIf $T$ is a tree that contains points $a$ and $b$ then $$B(a;b) := \\lbrace t \\in T \\, : \\, a< t \\wedge b \\rbrace$$\n$B(a;b)$ is the cone of $a$ that contains $b$.  If $a \\not< b $ then $B(a;b) = \\emptyset$.  If $B$ is a set such that $a \\leq B$ then\n$$B(a,B):= \\bigcup_{b \\in B} B(a;b)$$\n\\end{dfn}\n\n\\begin{dfn}\nGiven an abstract group $G$ and a permutation group $(H,S,\\mu(h,s))$ their wreath product, written as $G \\wr_{S} H$, is the abstract group on domain\n$$H \\times \\lbrace \\eta : S \\rightarrow G \\rbrace$$\nWe use $\\eta(s)$ to denote the function $s \\mapsto \\eta(s)$, and so $\\eta(s_0 s)$ is the function $s \\mapsto \\eta(s_0 s)$.  The group operation of $G \\wr_{S} H$ is given by\n$$(h_0, \\eta_0(x))(h_1, \\eta_1(x))=(h_0 h_1, \\eta_0(\\mu (h_1^{-1},x)) \\eta_1(x))$$\n\nWhen $G=\\textnormal{Aut}(M)$ and $H=\\textnormal{Aut}(N)$ their wreath product $G \\wr H$ is the automorphism group of the structure obtained by replacing every element of $N$ with a copy of $G$.\n\\end{dfn}\n\n\\begin{remark}\n$\\mathbb{Z} \\wr \\mathbb{Z}_2$ is the automorphism group of the structure obtained by replacing the elements of a 2-element antichain by copies of $(\\mathbb{Z},\\leq)$, while $\\mathbb{Z}_2 \\wr \\mathbb{Z}$ is the automorphism group of the structure obtained by replacing the elements of $(\\mathbb{Z}, \\leq)$ with 2-element antichains (the lamplighter group).\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.1]\n\\draw (10,25) -- (10,0);\n\\draw (20,25) -- (20,0);\n\n\\fill (10,20) circle (0.5);\n\\fill (10,15) circle (0.5);\n\\fill (10,10) circle (0.5);\n\\fill (10,5) circle (0.5);\n\n\\fill (20,20) circle (0.5);\n\\fill (20,15) circle (0.5);\n\\fill (20,10) circle (0.5);\n\\fill (20,5) circle (0.5);\n\n\\fill (60,20) circle (0.5);\n\\fill (70,20) circle (0.5);\n\\fill (60,12.5) circle (0.5);\n\\fill (70,12.5) circle (0.5);\n\\fill (60,5) circle (0.5);\n\\fill (70,5) circle (0.5);\n\n\\draw[dashed] (65,25) -- (65,0);\n\n\\draw (65,-5) node {$\\mathbb{Z}_2 \\wr \\mathbb{Z}$};\n\\draw (15,-5) node {$\\mathbb{Z} \\wr \\mathbb{Z}_2$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Mnemonic for the Wreath Product}\n\\end{figure}\n\n\\end{remark}\n\n\\begin{remark}\nIf group $G$ acts on set $X$, with $X_0 \\subseteq X$, then $G_{\\lbrace X_0 \\rbrace}$ is the set-wise stabiliser of $X_0$, while $G_{(X_0)}$ is the point-wise stabiliser.  Similarly for automorphism groups, $\\textnormal{Aut}_{\\lbrace X_0 \\rbrace}(M)$ is the set-wise stabiliser of $X_0$ in $M$, while $\\textnormal{Aut}_{( X_0 )}(M)$ denotes the point-wise stabiliser.  If $X_0 = \\lbrace x \\rbrace$ then these two notions coincide and we use the pithier expression $G_{x}$ or $\\textnormal{Aut}_{X_0}(x)$.\n\\end{remark}\n\n\\begin{dfn}\\label{dfn:regular}\nA tree $T$ is said to be $\\textit{regular}$ if:\n\\begin{enumerate}\n\\item all the maximal chains are isomorphic to each other;\n\\item the maximal chains are isomorphic to a finite linear order or $\\mathbb{N}$;\n\\item the ramification order of any non-maximal element of $T$ is at least 2 but finite; and\n\\item if $|T^{\\leq x}|= |T^{\\leq y}|$ then the ramification order of $x$ equals the ramification order of $y$.\n\\end{enumerate}\nA tree $T$ is said to be $\\textit{fh-regular}$ (finite height) if it is regular and the maximal chains are finite.\n\\end{dfn}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.09]\n\n\\fill (50,0) circle (0.5);\n\n\\fill (25,10) circle (0.5);\n\\fill (75,10) circle (0.5);\n\\draw (25,10) -- (50,0) -- (75,10);\n\n\\fill (10,20) circle (0.5);\n\\fill (20,20) circle (0.5);\n\\draw (20,20) -- (25,10) -- (10,20);\n\\fill (30,20) circle (0.5);\n\\fill (40,20) circle (0.5);\n\\draw (40,20) -- (25,10) -- (30,20);\n\n\\fill (60,20) circle (0.5);\n\\fill (70,20) circle (0.5);\n\\draw (60,20) -- (75,10) -- (70,20);\n\\fill (80,20) circle (0.5);\n\\fill (90,20) circle (0.5);\n\\draw (90,20) -- (75,10) -- (80,20);\n\n\\fill (8,30) circle (0.5);\n\\fill (12,30) circle (0.5);\n\\draw (8,30) -- (10,20) -- (12,30);\n\\fill (18,30) circle (0.5);\n\\fill (22,30) circle (0.5);\n\\draw (18,30) -- (20,20) -- (22,30);\n\\fill (28,30) circle (0.5);\n\\fill (32,30) circle (0.5);\n\\draw (28,30) -- (30,20) -- (32,30);\n\\fill (38,30) circle (0.5);\n\\fill (42,30) circle (0.5);\n\\draw (38,30) -- (40,20) -- (42,30);\n\n\\fill (58,30) circle (0.5);\n\\fill (62,30) circle (0.5);\n\\draw (58,30) -- (60,20) -- (62,30);\n\\fill (68,30) circle (0.5);\n\\fill (72,30) circle (0.5);\n\\draw (68,30) -- (70,20) -- (72,30);\n\\fill (78,30) circle (0.5);\n\\fill (82,30) circle (0.5);\n\\draw (78,30) -- (80,20) -- (82,30);\n\\fill (88,30) circle (0.5);\n\\fill (92,30) circle (0.5);\n\\draw (88,30) -- (90,20) -- (92,30);\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Example of a Regular Tree}\n\\end{figure}\n\n\\begin{remark}\\label{lemma:fh-regular}\nLet $T$ be a finite tree.  $\\textnormal{Aut}(T)$ acts 1-transitively on the maximal elements of $T$ if and only if $T$ is fh-regular.\n\\end{remark}\n\n\\begin{dfn}\nA group $G$ is said to be a dendromorphic group  if it is a Cartesian product of copies at least one of:\n\\begin{enumerate}\n\\item $\\mathbb{Z} \\wr \\mathbb{Z}_2$;\n\\item $\\textnormal{Sym}(\\omega)$;\n\\item $\\textnormal{Sym}(\\omega) \\wr \\mathbb{Z}_2$; and\n\\item the automorphism group of a regular tree;\n\\end{enumerate}\n\\end{dfn}\n\nExamples of the automorphism group of a regular tree include $S_n$, in particular $\\mathbb{Z}_2$,  and $(S_n \\wr \\mathbb{Z}_2)$.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.14]\n\\draw (10,25) -- (10,0);\n\\draw (20,25) -- (20,0);\n\n\\fill (10,20) circle (0.3);\n\\fill (10,15) circle (0.3);\n\\fill (10,10) circle (0.3);\n\\fill (10,5) circle (0.3);\n\n\\fill (20,20) circle (0.3);\n\\fill (20,15) circle (0.3);\n\\fill (20,10) circle (0.3);\n\\fill (20,5) circle (0.3);\n\n\\draw[anchor=north] (10,0) node {$\\vdots$};\n\\draw[anchor=south] (10,25) node {$\\vdots$};\n\\draw[anchor=north] (20,0) node {$\\vdots$};\n\\draw[anchor=south] (20,25) node {$\\vdots$};\n\n\\draw (70,10) -- (60,0) -- (50,10);\n\\draw (40,20) -- (50,10) -- (45,20);\n\\draw (50,20) -- (50,10);\n\n\\draw (60,20) -- (70,10) -- (65,20);\n\\draw (70,20) -- (70,10);\n\n\\fill (60,0) circle (0.3);\n\\fill (70,10) circle (0.3);\n\\fill (50,10) circle (0.3);\n\\fill (40,20) circle (0.3);\n\\fill (45,20) circle (0.3);\n\\fill (50,20) circle (0.3);\n\\fill (60,20) circle (0.3);\n\\fill (65,20) circle (0.3);\n\\fill (70,20) circle (0.3);\n\n\\draw[anchor=west] (51,20) node {$\\ldots$};\n\\draw[anchor=west] (71,20) node {$\\ldots$};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Example of Trees whose automorphism group is a dendromorphic group }\n\\end{figure}\n\n\\begin{dfn}\nLet $M$ be a CFPO, let $x \\in M$ and let $G \\subseteq \\textnormal{Aut}(M)$.\n$$G(x):= \\lbrace y \\in M \\: : \\: \\exists g \\in G \\, g(x)=y \\rbrace$$\n\\end{dfn}\n\n\\begin{theorem}\\label{thm:SuperDinf}\nIf $T$ is a tree and there exists a $G \\leq \\textnormal{Aut}(T)$ such that $G \\cong D_\\infty$ then there exists an $H$ such that $G \\leq H \\leq \\textnormal{Aut}(T)$ and $H$ is a dendromorphic group  .\n\\end{theorem}\n\\begin{proof}\n\nLet $T$ be a tree such that there is $G \\leq \\textnormal{Aut}(T)$ and $G \\cong D_\\infty$.  We use the same presentation of $D_\\infty$ that we gave in Definition \\ref{dfn:dinfinity}, so here $\\sigma$ and $\\tau$ are automorphisms of $T$ that generate $G$ and satisfy the identities $\\sigma^2 = 1$ and $\\sigma \\tau \\sigma =  \\tau^{-1}$.\n\nLet $t \\in T$.  How does $\\sigma$ constrain the structure of $G(t)$?  If $t< \\sigma (t)$ then $\\sigma(t) < \\sigma ^2 (t) =t$, which is a contradiction.  Similarly $\\sigma (t) < t$ also leads to a contradiction, so if $t \\not= \\sigma(t)$ then $t \\parallel \\sigma(t)$.  Since $\\sigma \\tau \\not= \\tau \\sigma$, we know that $ \\textnormal{supp} (\\sigma) \\cap \\textnormal{supp} (\\tau)  \\not= \\emptyset$.\n\nFirst suppose that $t \\in T$ is such that $\\lbrace \\phi|_{G(t)} \\: : \\: \\phi \\in G \\rbrace \\not\\cong D_\\infty$.  This means that there is some $n \\in \\mathbb{Z}$ and $i \\in \\lbrace 0,1 \\rbrace$ such that $\\tau|_{G(t)}^n \\sigma|_{G(t)}^i  = id|_{G(t)}$.\n\\begin{enumerate}\n\\item If $\\sigma|_{G(t)} = id|_{G(t)}$ then the identity $\\sigma \\tau = \\tau^{-1} \\sigma$ becomes $\\tau = \\tau^{-1}$ and we learn that $G(t) = \\lbrace t \\rbrace$ and $\\textnormal{Aut}(G(t))$ is trivial.\n\n\\item If $\\tau|_{G(t)}^n = id|_{G(t)}$ then $G(t)$ is a finite antichain and so $G(t)^+$ is a finite tree whose automorphism group acts transitively on its maximal elements, and by Remark \\ref{lemma:fh-regular} is fh-regular, so $\\textnormal{Aut}(G(t))$ is the automorphism group of the fh-regular tree $G(t)^+$.\n\n\\item If $\\sigma \\tau^n|_{G(t)} = id|_{G(t)}$ then we can deduce that $\\sigma|_{G(t)} = \\tau^n|_{G(t)}$, and thus $ \\tau^{2n}|_{G(t)} = id|_{G(t)}$.\n\\end{enumerate}\n\nNow we suppose $t \\in T$ is such that $\\lbrace \\phi|_{G(t)} \\: : \\: \\phi \\in G \\rbrace \\cong D_\\infty$.\n\nWe now examine the possible action of $\\tau$ on $t$.  Since $\\tau$ has infinite order, $\\lbrace \\tau^n (t) \\: : \\: n \\in \\mathbb{Z} \\rbrace$ and $\\lbrace \\tau^n \\sigma(t) \\: : \\: n \\in \\mathbb{Z} \\rbrace$ are infinite.  We now consider various cases to deduce the structure of $G(t)$.\n\\begin{description}\n\n\\item[Case 1:]$t < \\tau (t)$ or $t > \\tau (t)$\n\nWithout loss of generality we assume that $t < \\tau (t)$.\n\nSince $t< \\tau(t)$ we know that $\\tau^m(t) < \\tau^n(t)$ if and only if $m<n$, where $m,n \\in \\mathbb{Z}$.  Suppose $\\sigma$ fixes one of these $\\tau^m(t)$.  Hence\n$$\\begin{array}{r c l}\n\\sigma \\tau^m (t) & = & \\tau^m(t) \\\\\n\\end{array}$$\nbut in $D_\\infty$ we know that $\\tau^{-m} \\sigma = \\sigma \\tau^{m}$, so \n$$\\begin{array}{r c l}\n\\tau ^{-m} \\sigma(t) & = & \\tau^m (t) \\\\\n\\sigma(t) & = & \\tau^{2m} (t) \\\\\n\\end{array} $$\nwhich means that $\\sigma$ maps $t$ to $\\tau^{2m}(t)$, which in this case is assumed to be greater than $t$, which we have already shown yields a contradiction, and thus $\\sigma$ does not fix any $\\tau^n(t)$.\n\nWe suppose that there is an $n \\in \\mathbb{Z}$ such that $\\tau ^n (t) \\leq t \\wedge \\sigma(t)$.  We know that $\\sigma \\tau^n (t) \\parallel \\tau^n(t)$, which is the situation depicted in Figure 14.\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.16]\n\\draw (0,30)  -- (10,20) -- (20,30);\n\\draw (10,20) -- (10,10) -- (20,0) -- (30,10);\n\n\\fill (0,30) circle (0.3);\n\\fill (10,20) circle (0.3);\n\\fill (20,30) circle (0.3);\n\\fill (10,10) circle (0.3);\n\\fill (20,0) circle (0.3);\n\\fill (30,10) circle (0.3);\n\n\\draw[anchor=west] (1,30) node {$t$};\n\\draw[anchor=west] (21,30) node {$\\sigma(t)$};\n\\draw[anchor=west] (11,20) node {$t \\wedge \\sigma(t)$};\n\\draw[anchor=west] (11,10) node {$\\tau^n(t)$};\n\\draw[anchor=west] (31,10) node {$\\sigma \\tau^n(t)$};\n\\draw[anchor=west] (21,0) node {$\\tau^n(t) \\wedge \\sigma \\tau^n(t)$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Deduced Structure of $G(t)$ if $(\\tau ^n (t) \\leq t \\wedge \\sigma(t))$}\n\\end{figure}\n\nHowever $\\sigma$ maps the pair $(t, \\tau^n(t))$ to $(\\sigma(t), \\sigma \\tau^n (t))$, so $\\tau^n(t) < t$ implies that $\\sigma\\tau^n(t) < \\sigma(t)$, providing a contradiction.\n\nSo there is no $n$ such that $\\tau^n (t) \\leq t \\wedge \\sigma(t)$ and then we are in the situation depicted in Figure 15.\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.1]\n\\draw (0,50)  -- (20,10);\n\\draw (30,10) -- (50,50);\n\n\\fill (25,0) circle (0.5);\n\n\\fill (5,40) circle (0.5);\n\\fill (10,30) circle (0.5);\n\\fill (15,20) circle (0.5);\n\n\\fill (35,20) circle (0.5);\n\\fill (40,30) circle (0.5);\n\\fill (45,40) circle (0.5);\n\n\\draw[anchor=west] (25,0) node {$t \\wedge s(t)$};\n\n\\draw[anchor=west]  (5,40) node {$\\tau(t)$};\n\\draw[anchor=west]  (10,30) node {$t$};\n\\draw[anchor=west] (15,20) node {$\\tau^{-1}(t)$};\n\n\\draw[anchor=west]  (45,40) node {$\\tau^{-1} \\sigma (t)$};\n\\draw[anchor=west]  (40,30) node {$\\sigma (t)$};\n\\draw[anchor=west]  (35,20) node {$\\tau \\sigma (t)$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Deduced Structure of $G(t)$ if $(t \\wedge \\sigma(t) \\leq \\tau^{i}(t))$}\n\\end{figure}\n\nThe automorphism group of this structure is clearly $\\mathbb{Z} \\wr \\mathbb{Z}_2$, and so\n$$\\textnormal{Aut}(G(t)) \\cong \\mathbb{Z} \\wr \\mathbb{Z}_2$$\n\n\\item[Case 2:] $t \\parallel \\tau(t)$ and $\\tau^m(t) \\wedge \\tau^n(t) = \\tau^{m'}(t) \\wedge \\tau^{n'}(t)$ for all $m \\not= n,m' \\not= n'$.  We call denote common ramification point,  $\\tau^m(t) \\wedge \\tau^n(t)$ for $m \\not= n$, by $x$.  In other words, the $\\tau^n(t)$ form an antichain, which ramifies from $x$.\n\nIf $x = \\sigma(x)$ then the whole orbit of $t$ is an infinite (as $G(t)$ is infinite) antichain  above $x$, and thus $\\textnormal{Aut}(T)$ is $\\textnormal{Sym}(\\omega)$.\n\nIf $x \\not= \\sigma(x)$ then the whole orbit of $t$ is two infinite (as both $\\lbrace \\tau^n (t) \\: : \\: n \\in \\mathbb{Z} \\rbrace$ and $\\lbrace \\tau^n \\sigma (t) \\: : \\: n \\in \\mathbb{Z} \\rbrace$ are infinite) antichains , one ramifying from $x$, the other from $\\sigma(x)$.  In this case $\\textnormal{Aut}(T) \\cong \\textnormal{Sym}(\\omega) \\wr \\mathbb{Z}_2$.\n\n\\item[Case 3:] $t \\parallel \\tau(t)$ and $\\tau^m(t) \\wedge \\tau^n(t) \\not= \\tau^{m'}(t) \\wedge \\tau^{n'}(t)$ for some $m,n,m',n'$.\n\nFor $m \\in \\mathbb{N} \\setminus \\lbrace 0 \\rbrace$ let $G_m :=  \\lbrace \\sigma^{i} \\tau^{mn} \\: : \\: i \\in \\lbrace 0 , 1 \\rbrace \\: n \\in \\mathbb{Z}\\rbrace$.  Note that $G_m \\cong D_\\infty$.\n\nFor brevity's sake, $x_n$ will denote $\\tau^{mn}(t) \\wedge \\tau^{m(n+1)}(t)$.  Suppose that $x_i \\not= x_{i+1}$ for all $i$.  Note that $\\tau^{mk} (x_n) = x_{n+k}$ because greatest lower bounds are preserved by automorphisms.  For any $i \\in \\mathbb{Z}$ both $x_i$ and $x_{i+1}$ are below $\\tau^{m(i+1)}(t)$, so $\\lbrace x_i \\: :\\: i \\in \\mathbb{Z} \\rbrace$ is linearly ordered and acted on by $\\tau^m$, showing that $\\tau^m(x_i) < x_i$ or $\\tau^m(x_i) > x_i$.\n\nIf $\\lbrace \\phi|_{G_m(x_0)} \\: : \\: \\phi \\in G_m \\rbrace \\not\\cong D_\\infty$, then $G_m(x_0)$ is an antichain, but we have just established that $\\tau(x_i) < x_i$ or $\\tau(x_i) > x_i$, so $\\lbrace \\phi|_{G_m(x_0)} \\: : \\: \\phi \\in G_m \\rbrace \\cong D_\\infty$, and we may now apply Case 1 to $G_m(x_0)$ and find that $\\textnormal{Aut}(G_m(x_0)) \\cong (\\mathbb{Z} \\wr \\mathbb{Z}_2)$.\n\nSince each $x_i \\not= x_{i+1}$, we can deduce the structure depicted in Figure 16.\n\\begin{figure}[h]\\label{figure:cheapcopout}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.1]\n\\draw (0,50)  -- (20,10);\n\\draw (30,10) -- (50,50);\n\n\\fill (25,0) circle (0.5);\n\n\\fill (5,40) circle (0.5);\n\\fill (10,30) circle (0.5);\n\\fill (15,20) circle (0.5);\n\n\\fill (10,50) circle (0.5);\n\\draw (5,40) -- (10,50);\n\\fill (15,40) circle (0.5);\n\\draw (10,30) -- (15,40);\n\\fill (20,30) circle (0.5);\n\\draw (15,20) -- (20,30);\n\n\\fill (35,20) circle (0.5);\n\\fill (40,30) circle (0.5);\n\\fill (45,40) circle (0.5);\n\n\\fill (30,30) circle (0.5);\n\\draw (30,30) -- (35,20);\n\\fill (35,40) circle (0.5);\n\\draw (35,40) -- (40,30);\n\\fill (40,50) circle (0.5);\n\\draw (40,50) -- (45,40);\n\n\\draw[anchor=west] (25,0) node {$x_0 \\wedge \\sigma (x_0)$};\n\n\\draw[anchor=west]  (5,40) node {$x_1$};\n\\draw[anchor=west]  (10,30) node {$x_0$};\n\\draw[anchor=west] (15,20) node {$x_{-1}$};\n\n\\draw[anchor=west]  (45,40) node {$\\sigma (x_1)$};\n\\draw[anchor=west]  (40,30) node {$\\sigma (x_0)$};\n\\draw[anchor=west]  (35,20) node {$\\sigma (x_{-1})$};\n\n\\draw[anchor=south]  (10,50) node {$\\tau^m(t)$};\n\\draw[anchor=south]  (15,40) node {$t$};\n\n\\draw[anchor=south]  (35,40) node {$\\sigma (t)$};\n\\draw[anchor=south]  (40,50) node {$\\tau^m \\sigma (t)$};\n\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Deduced Structure needed for Case 3}\n\\end{figure}\n\nThus we see that $\\textnormal{Aut} (G_m(t)) \\cong (\\mathbb{Z} \\wr \\mathbb{Z}_2)$.  If we redefine $x_n := \\tau^{mn+k}(t) \\wedge \\tau^{m(n+1) +k}(t)$ and repeat this argument, we see that $\\textnormal{Aut} (G_m(\\tau^k(t))) \\cong (\\mathbb{Z} \\wr \\mathbb{Z}_2)$\n\nLet $m_0$ be the least element of the set \n$$\\lbrace i = \\mathrm{lcm}(n-m,n'-m') \\: : \\: \\tau^m(t) \\wedge \\tau^n(t) \\not= \\tau^{m'}(t) \\wedge \\tau^{n'}(t) \\rbrace$$\nNote that $\\tau^{m_0 n}(t) \\wedge \\tau^{m_0 (n+1)}(t) \\not= \\tau^{m_0 (n+1)}(t) \\wedge \\tau^{m_0 (n+2)}(t)$ for all $n$, so $m_0$ is in fact the least number such that $\\textnormal{Aut} (G_{m_0}(t)) \\cong (\\mathbb{Z} \\wr \\mathbb{Z}_2)$.\n\n$G(t)$ consists of $m_0-1$ copies of $G_{m_0}(t)$, which are preserved by $\\sigma$, and $\\tau$ acts cyclically on them, and indeed their least elements, which we call $L$.  This gives us $\\lbrace \\phi|_{L} \\: : \\: \\phi \\in G \\rbrace \\not\\cong D_\\infty$, and $\\sigma|_L = \\mathrm{id}|_L$, so $L$ is trivial and $\\textnormal{Aut}(G(t)) \\cong (\\mathbb{Z} \\wr \\mathbb{Z}_2)$\n\\end{description}\n\nTherefore for all $t \\in T$ the group $\\textnormal{Aut}(G(t))$ is either trivial or :\n\\begin{enumerate}\n\\item $\\mathbb{Z} \\wr \\mathbb{Z}_2$ (from Cases 1 and 3);\n\\item $\\textnormal{Sym}(\\omega)$ (from Case 2);\n\\item $\\textnormal{Sym}(\\omega) \\wr \\mathbb{Z}_2$ (from Case 2); or\n\\item the automorphism group of an fh-regular tree;\n\\end{enumerate}\neach of which is a dendromorphic group  .\n\nWe pick one $t \\in T$ such that $G(t) \\not= \\lbrace t \\rbrace$, and let $s := \\mathrm{inf} (G(t)^+)$.  The next phase of this proof is to show that the additional automorphisms of $\\textnormal{Aut}(G(t))$ extend to $B(s;G(t))$.  We do this by addressing each of the possibilities in the above list individually.\n\nLet $\\lambda \\in \\textnormal{Aut}(G(t)) \\setminus G$.  We wish to extend $\\lambda$ to $B(s;G(t))$ and show that the group of the extensions of elements of $\\textnormal{Aut}(G(t))$ is a dendromorphic group  .\n\n\\begin{enumerate}\n \\item Suppose $\\textnormal{Aut}(G(t)) \\cong ( \\mathbb{Z} \\wr  \\mathbb{Z}_2)$.  Then $\\lambda$ is characterised by where it maps $t$ and $\\sigma(t)$.  Let's suppose that $\\lambda(t) = \\tau^n(t)$ and $\\lambda(\\sigma(t)) = \\tau^m \\sigma(t)$.  Then we define $\\bar{\\lambda}$ to be the following:\n$$\n\\bar{\\lambda} :x \\mapsto \\left\\lbrace\n\\begin{array}{l c}\n\\tau^n(x) & x \\in B(s;t) \\\\\n\\tau^m(x) & x \\in B(s;\\sigma(t))\n\\end{array}\n\\right.\n$$\nIf $\\lambda(t) = \\tau^n \\sigma(t)$ and $\\lambda(\\sigma(t)) = \\tau^m (t)$ then\n$$\n\\bar{\\lambda} :x \\mapsto \\left\\lbrace\n\\begin{array}{l c}\n\\tau^m \\sigma(x) & x \\in B(s;t) \\\\\n\\tau^n \\sigma(x) & x \\in B(s;\\sigma(t))\n\\end{array}\n\\right.\n$$\nThus we may extend $\\lambda$ to a unique element of $\\textnormal{Aut}((B(s;G(t)))$, so\n$$\\textnormal{Aut}((B(s;G(t))) \\cong ( \\mathbb{Z} \\wr  \\mathbb{Z}_2)$$\n\n\\item Suppose $\\textnormal{Aut}(G(t)) \\cong \\textnormal{Sym}(\\omega)$.  If there is some $b \\in G(t)$ such that $\\sigma(b) = b$ and $\\sigma|_{B(s;b)} \\not= \\mathrm{id}|_{B(s;b)}$ then there are two possible extensions of $\\lambda$.  If $x \\in B(s,a)$ and $\\tau^n(a) = \\lambda(a) = \\tau^m \\sigma (a)$ then\n$$\n\\begin{array}{l}\n\\bar{\\lambda}_0 :x \\mapsto \\tau^n (x) \\\\\n\\bar{\\lambda}_1 :x \\mapsto \\tau^m \\sigma (x)\n\\end{array}\n$$\nSince each $\\lambda$ may be extended to two elements of $\\textnormal{Aut}((B(s;G(t)))$, we know that\n$$\\textnormal{Aut}((B(s;G(t))) \\cong ( \\mathbb{Z}_2 \\times  \\textnormal{Sym}(\\omega))$$\nOtherwise if $x \\in B(s; a)$ and $\\lambda(a) = \\tau^n \\sigma^i (t)$ then\n $$\n\\bar{\\lambda} :x \\mapsto \\tau^n \\sigma^i(x) \\\\\n$$\nand we uniquely extend $\\lambda$, showing\n$$\\textnormal{Aut}((B(s;G(t))) \\cong \\textnormal{Sym}(\\omega)$$\n\n \\item Suppose $\\textnormal{Aut}(G(t)) \\cong (\\textnormal{Sym}(\\omega) \\wr \\mathbb{Z}_2)$.  If $x \\in B(s; a)$ and $\\lambda(a) = \\tau^n \\sigma^i (t)$ then\n $$\n\\bar{\\lambda} :x \\mapsto \\tau^n \\sigma^i(x) \\\\\n$$\nso we can uniquely extend $\\lambda$, showing\n$$\\textnormal{Aut}((B(s;G(t))) \\cong (\\textnormal{Sym}(\\omega) \\wr \\mathbb{Z}_2)$$\n\n \\item Suppose $G(t)^+$ is an fh-regular tree, and suppose that there is an $x \\in B(s;G(t))$ such that $\\lbrace \\phi|_{G(x)} \\: : \\: \\phi \\in G \\rbrace \\cong D_\\infty$.  Clearly $G$ preserves $B(s;G(t))$, so\n $$G(x) \\subseteq  B(s;G(t))$$\n \n Suppose that $x \\in B(s;t)$.  Then $\\tau^n \\sigma^i (x) \\in B(s;\\tau^n \\sigma^i (t))$ for all $n \\in \\mathbb{Z}$ and $i \\in \\lbrace 0,1 \\rbrace$, therefore for all $y \\in G(t)$\n $$G(x) \\cap B(s;y) \\not= \\emptyset$$\n  \n Rather than look at $\\lambda \\in \\textnormal{Aut}(G(t))$, we instead extend every $\\mu \\in \\textnormal{Aut}(G(x))$ to obtain a dendromorphic supergroup of $G$ in $B(s,G(t))$.\n \n Now we suppose that there is no $x \\in B(s;G(t))$ such that $\\lbrace \\phi|_{G(x)} \\: : \\: \\phi \\in G \\rbrace \\cong D_\\infty$.  We will define by induction a family of sets that we will call $X_k$ which will help us extend $\\lambda$.\n \n Let $X_0$ be the maximal subset of $B(s,G(t))$ such that for all $\\phi, \\psi \\in G$\n $$\\phi|_{G(t)} = \\psi|_{G(t)} \\Rightarrow \\phi|_{X_0} = \\psi|_{X_0}  $$\n Let $x \\in B(s;y)$ and let $\\phi \\in G$ be such that $\\lambda(y)=\\phi(y)$.\n   $$\n\\bar{\\lambda} :x \\mapsto \\phi(x) \\\\\n$$\nSince all the possible $\\phi$ agree, this map is a well-defined, unique extension of $\\lambda$, so $\\textnormal{Aut}(X_0) \\cong \\textnormal{Aut}(G(t)^+)$.  If $X_0 = B(s;G(t))$ then we have extended $\\lambda$ to $B(s;G(t))$ and we are done.\n\nSuppose that we have defined $X_{k-1}$, but $X_{k-1} \\not= B(s;G(t))$.  Let $x_k \\in B(s,G(t)) \\setminus X_{k-1}$.  Let $X_k$ be the maximal subset of $B(s,G(x_1))$ such that for all $\\phi, \\psi \\in G$\n $$\\phi|_{G(t)} = \\psi|_{G(t)} \\Rightarrow \\phi|_{X_1} = \\psi|_{X_1} $$\n Again, $\\textnormal{Aut}(X_k) \\cong \\textnormal{Aut}(G(x_k)^+)$ and if $X_k = B(s;G(t))$ then we have extended $\\lambda \\in \\textnormal{Aut}(X_k)$ to $B(s;G(t))$ and we are done.\n \n If $X_k \\not= B(s;G(t))$ then we define $X:= \\bigcup\\limits_{k \\in \\mathbb{N}} X_k$.   We know how to extend $\\lambda$ to $X$, so if we can show that:\n \\begin{enumerate}\n \\item $X = B(s;G(t))$; and\n \\item there is a regular tree $F$ such that $\\textnormal{Aut}(X) = \\textnormal{Aut}(F)$;\n \\end{enumerate}\n then we will have shown that $\\textnormal{Aut}(B(s,G(t)) \\cong \\textnormal{Aut}(F)$.\n \n \\begin{enumerate}\n \\item For all $k$, the orbit $|G(x_{k})| > |G(x_{k-1})|$, as there are $\\phi, \\psi \\in G$ such that $\\phi(x_{k-1}) = \\psi(x_{k-1})$ but $\\phi(x_{k}) \\not= \\psi(x_{k})$, so the set $\\lbrace |G(x_{k})| \\: : \\: k \\in \\mathbb{N} \\rbrace$ is unbounded.\n \n  If $y \\in B(s;G(t)) \\setminus X$ then for all $k$\n  $$\\tau^{|G(x_{k})|}(y) \\not= y$$\n  so $G$ acts as $D_\\infty$ on $G(y)$, and we have already seen how to extend $\\lambda$ to $B(s;G(t))$ in this case, so we may assume now that $X = B(s;G(t))$.\n  \\item Since $X_k$ extends $X_{k-1}$ and since $s$ is the root of both $G(x_{k-1})^+$ and $G(x_{k})^+$, we know that $G(x_k)^+$ is an extension of $G(x_{k-1})^+$.  Therefore we consider the tree $F:=\\bigcup\\limits_{k \\in \\mathbb{N}} G(x_k)^+$.\n  \n  Let $(s, y_1 \\ldots )$ and $(s, z_1, \\ldots )$ denote maximal chains of $F$.  Since each $G(x_k)^+$ is an fh-regular tree, given any two maximal chains of $F$ there is a partial automorphism from the initial $k$ elements of the first to the initial $k$ elements of the second.  The union of all these partial automorphisms will be an automorphism of $F$, and thus $\\textnormal{Aut}(F)$ acts transitively on every maximal chain, which is Condition 1 of Definition \\ref{dfn:regular}.\n  \n  The initial section of every maximal chain of $F$ finite, so every maximal chain is isomorphic to $\\mathbb{N}$, Condition 2 of Definition \\ref{dfn:regular}.\n  \n  If $y \\in F$ then $y \\in G(x_k)^+$ for some $k$, so the ramification order of any non-maximal element of $F$ is at least 2 but finite, showing that $F$ satisfies Condition 3 of Definition \\ref{dfn:regular}.\n  \n  Finally, if $|F^{\\leq y}|= |F^{\\leq z}|$ then there is a $k$ such that $y, z \\in G(x_k)^+$ and $|(G(x_k)^+)^{\\leq y}|= |(G(x_k)^+)^{\\leq z}|$, so the fact that $G(x_k)^+$ is fh-regular implies that $F$ satisfies Condition 3 of Definition \\ref{dfn:regular}, and is regular.\n \\end{enumerate}\n \n Therefore there is a regular tree $F$ such that $\\textnormal{Aut}(B(s,G(t)) \\cong \\textnormal{Aut}(F)$.\n\\end{enumerate}\n\nFor any $t \\in T$ let $s_t$ be the root of $G(t)^+$.  Consider the set\n$$ \\mathcal{B}:= \\lbrace B(s_t;G(t)) \\: : \\: |G(t)| \\not= 1 \\rbrace \\cup \\lbrace \\lbrace t \\rbrace \\: : \\: |G(t)| \\not= 1 \\rbrace $$\nLet $H$ be the group of all automorphisms of $T$ that fix every $B \\in \\mathcal{B}$ setwise.\n$$ H = \\prod_{B \\in \\mathcal{B}} \\textnormal{Aut}(B) $$\nSince the Cartesian product of dendromorphic groups is dendromorphic, $H$ is also dendromorphic.  We have already seen that $G$ fixes every $B \\in \\mathcal{B}$ setwise, so $G \\leq H$.\n\\end{proof}\n\nIf you are familiar with automorphism groups as topological groups, you may have realised that in the proof of Theorem \\ref{thm:SuperDinf} we are essentially calculating the closure of the copy of $D_\\infty$.  In Theorem \\ref{thm:dinfinityinCFPOs} we will see that a CFPO is not treelike if and only if its automorphism group contains a closed copy of $D_\\infty$.\n\nWhile describing this situation using the language of topological groups might have been more elegant, I prefer this approach as it makes it clear that these properties are recognisable from the abstract group.\n\n\\subsection{$D_\\infty$ in CFPOs}\n\n\\begin{cor} \\label{prop:ALTnotTree}\n$\\textnormal{Aut}(\\alt{}) \\not\\cong \\textnormal{Aut}(T)$ for all trees $T$.\n\\end{cor}\n\\begin{proof}\n$\\textnormal{Aut}(\\alt{}) \\cong D_\\infty$, so if $\\textnormal{Aut}(T) \\cong \\textnormal{Aut}(\\alt{})$ then the whole automorphism group is a copy of $D_\\infty$, and so cannot be contained in a dendromorphic group  .\n\\end{proof}\n\nSo we've established that $D_\\infty$ can occur as a subgroup of the automorphism group of a CFPO in a different way than it can as a subgroup of the automorphism group of a tree.  The rest of this subsection is devoted to finding out how copies of $D_\\infty$ that aren't contained in a dendromorphic group  can act on a CFPO.\n\n\\begin{dfn}\\label{dfn:connectionclosure}\nLet $M$ be a CFPO.  If $X \\subseteq M$ then $X^{cc}$, the \\textbf{connection closure} of $X$, is the following set\n$$\\bigcup_{x,y \\in X} \\path{x,y}$$\nIn particular, if $G \\leq \\textnormal{Aut}(M)$ and $x \\in M$ then this combines with the notation of Definition \\ref{dfn:G(t)} to give:\n$$G(x)^{cc}:= \\bigcup\\limits_{g,h \\in G} \\path{g(x),h(x)}$$\n\\end{dfn}\n\n\\begin{theorem}\\label{thm:dinfinityinCFPOs}\nLet $M$ be a Rubin complete CFPO and let $G \\leq \\textnormal{Aut}(M)$.  If $G \\cong D_\\infty$ then either $G$ is contained in a dendromorphic group  or $G$ acts on a copy of Alt in $M$, but not both.\n\\end{theorem}\n\\begin{proof}\nIf $M$ is a $\\cfpo{n}$ for some $n \\in \\mathbb{N}$ or a $\\cfpo{\\omega}$ then by Theorem \\ref{thm:cfpo2n+1}, Corollary \\ref{cor:cfpo2n} and Theorem \\ref{cor:cfpo2n} there is a tree $T$ such that $\\textnormal{Aut}(M) \\cong \\textnormal{Aut}(T)$.  Thus Theorem \\ref{thm:SuperDinf} shows that $G$ is contained in a dendromorphic group  and $G$ cannot act on a copy of Alt, as $M$ does not contain a copy of Alt.  We now suppose that $M$ is a connected $\\cfpo{\\infty}$.\n\nIf $G$ fixes $a \\in M$ then $G \\leq \\textnormal{Aut}_{a}(M)$.  By adding a colour predicate to $M$ that only $a$ realises, we find a CFPO with a fixed point whose automorphism group is $\\textnormal{Aut}_{a}(M)$.  Since this CFPO has a fixed point it is treelike (Theorem \\ref{thm:fixedpoint}), and Theorem \\ref{thm:SuperDinf} shows that there is a dendromorphic group  $X$ which is contained in $\\textnormal{Aut}_{a}(M)$ and contains $G$.  Therefore if $M \\setminus \\textnormal{supp}(G) \\not= \\emptyset$ then $G$ is contained in a dendromorphic group {.}\n\nNow suppose that $G$ has no fixed point and that $G(m)^{cc}$ is not a $\\cfpo{\\infty}$ for any $m \\in M$.\n\nWe can view the connected components of $M \\setminus G(m)^{cc}$ as extended cones of elements of $G(m)^{cc}$.  For all $a \\in G(m)^{cc}$\n$$C(a) := \\lbrace x \\in M \\setminus G(m)^{cc} \\: : \\: a \\in \\path{x, G(m)^{cc}} \\rbrace$$\ni.e. $C(a)$ is the union of all the extended cones of $M \\setminus G(m)^{cc}$ that ramify from $a$.  If $\\phi \\in \\textnormal{Aut}(\\langle G(m)^{cc}, \\leq_M \\rangle)$ does not extend to an automorphism of $M$ then $\\phi$ must map $a$ to $b$ but $C(a) \\not\\cong C(b)$.\n\nIf for all CFPOs $C$ such that $\\exists a \\in G(m)^{cc} \\: C \\cong C(a)$ we introduce a colour predicate $P_C$ to $\\langle G(m)^{cc}, \\leq_M \\rangle$ such that\n$$\\langle G(m)^{cc}, \\leq_M \\rangle \\models P_C(a) \\: \\Leftrightarrow \\: C(a) \\cong C$$\nEvery automorphism of $\\langle G(m)^{cc}, \\leq_m, P_C \\rangle$ is a restriction of an automorphism of $M$.\n\nEach $G(m)^{cc}$ is $G$-invariant, as otherwise we would be able to map a path inside $G(m)^{cc}$ to one outside by an element of $G$, but this map must take the endpoints of this path with it, and these endpoints are elements of $G(m)^{cc}$.\n\nWe choose one $m \\in M$.  Since $G(m)^{cc}$ is not a $\\cfpo{\\infty}$, it is treelike.  All of the extended cones that are contained in $M \\setminus G(m)^{cc}$ are treelike if we fix the point in $G(m)^{cc}$ that they emanate from, so by replacing $G(m)^{cc}$ and the extended cones, we may find a tree $T$ such that $G \\leq \\textnormal{Aut}(T) \\leq \\textnormal{Aut}(M)$, and so $G$ is contained in a dendromorphic group {.}\n\nSo now suppose that $a \\in M$ is such that $G(a)^{cc}$ is a $\\cfpo{\\infty}$.  From such an $a$ we define\n$$b := \\path{a,\\path{\\tau^{-1}(a),\\tau(a)}}$$\n(because $\\path{\\tau^{-1}(b),b} \\cap \\path{b,\\tau(b)} = \\lbrace b \\rbrace$) and consider $G(b)^-$, the set of maximal and minimal points of $G(b)^{cc}$.  If $\\tau^{\\mathbb{Z}}(b) = G(b)$ then $G(b)^-$ is a copy of Alt on which $G$ acts.\n\nIf $\\sigma(b) \\not\\in \\tau^{\\mathbb{Z}}(b)$ then we consider $G$'s action on\n$$\\path{\\tau^{\\mathbb{Z}}(b),\\tau^{\\mathbb{Z}}(\\sigma(b))}$$\nwhich, if non-empty, will be fixed pointwise by $\\tau$, and on which $\\sigma$ will have a fixed point, contradicting the assumption that $G$ has no fixed points.\n\\begin{figure}[h!]\\label{FigureNextSoon}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.13]\n\\draw (0,30) -- (10,20) -- (20,30) -- (30,20) -- (40,30);\n\\draw (5,5) -- (15,-5) -- (25,5) -- (35,-5) -- (45,5);\n\\draw (25,25) -- (24,12) -- (23,14) -- (22,12) -- (21,14) -- (20,0);\n\n\\draw (0,32) node {$\\tau^{-1}(b)$};\n\\draw (20,32) node {$b$};\n\\draw (40,32) node {$\\tau(b)$};\n\n\\draw (5,7) node {$\\tau^{-1}(\\sigma(b))$};\n\\draw (27,7) node {$\\sigma(b)$};\n\\draw (45,7) node {$\\tau(\\sigma(b))$};\n\n\\draw[anchor=west] (25,15) node {$\\path{\\tau^{\\mathbb{Z}}(b),\\tau^{\\mathbb{Z}}(\\sigma(b))}$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{$\\path{\\tau^{\\mathbb{Z}}(b),\\tau^{\\mathbb{Z}}(\\sigma(b))}$}\n\\end{figure}\n\nIf $\\path{\\tau^{\\mathbb{Z}}(b),\\tau^{\\mathbb{Z}}(\\sigma(b))}$ is empty then we are in the situation depicted in Figure 17.\n\\begin{figure}[hb]\\label{FigureNextSoon2}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.14]\n\\draw (-20,5) -- (-10,-5) -- (0,5) -- (10,-5) -- (20,5) -- (30,-5);\n\\draw (-8,5) -- (-10,2) -- (-11,3) -- (-12,2) -- (-13,3) -- (-15,0);\n\\draw (12,5) -- (10,2) -- (9,3) -- (8,2) -- (7,3) -- (5,0);\n\\draw (32,5) -- (30,2) -- (29,3) -- (28,2) -- (27,3) -- (25,0);\n\n\\draw (-20,7) node {$\\tau^{-1}(b)$};\n\\draw (0,7) node {$b$};\n\\draw (20,7) node {$\\tau(b)$};\n\n\\draw (-8,7) node {$c_{-1}$};\n\\draw (12,7) node {$c_0$};\n\\draw (32,7) node {$c_1$};\n\n\\draw (-18,0) node {$d_{-1}$};\n\\draw (3,0) node {$d_0$};\n\\draw (23,0) node {$d_1$};\n\\end{tikzpicture}\n\\end{center}\n\n\\caption{$\\path{\\tau^{i}(b),c_i}$}\n\n\\end{figure}\n\nIn Figure 18 $c_0$ is $\\sigma(b)$ and $c_{k}:=\\tau^k(c_0)$, which forces $\\sigma \\tau^k (b)$ to be $c_j$ for some $j$ (whose relationship with $k$ will be deduced shortly).  Note that $\\sigma$ and $\\tau$ satisfy the identity $$\\sigma \\tau = \\tau^{-1} \\sigma$$ which implies the following equations:\n\n\n$$\\begin{array}{r c l}\nc_i & = &  \\sigma \\tau^j(b) \\\\\n & = & \\tau^{-j} \\sigma (b) \\\\\n \\tau^j (c_i) & = & c_0 \\\\\n c_{i+j} & = & c_0\n\\end{array}$$\n\nso $c_i = \\sigma(\\tau^{-i}(b))$.  Let the $d_i$ be the points fixed by $\\sigma$ on $\\path{\\tau^{i}(b),c_i}$ respectively.  Then $\\bigcup \\path{d_i,d_j}$ is a copy of Alt which is acted on as desired.\n\nLet $\\mathcal{A}$ be the family of copies of Alt in $M$.  We now show that if $\\textnormal{Act}(A,\\textnormal{Aut}(M))$ (the action of $\\textnormal{Aut}(M)$ on $A$) is isomorphic to $D_\\infty$ for some $A \\in \\mathcal{A}$ then $\\textnormal{Act}(A,\\textnormal{Aut}(M))$ cannot be contained in a dendromorphic group , thus showing the exclusivity of the theorem.\n\nIf for some $A \\in \\mathcal{A}$ the action of $\\textnormal{Aut}(M)$ is $D_\\infty$ then $\\textnormal{Act}(A,\\textnormal{Aut}(M)) \\cong D_\\infty$ and there is no dendromorphic group  contained in $\\textnormal{Aut}_{\\lbrace A \\rbrace}(M)$ that contains $\\textnormal{Act}(A,\\textnormal{Aut}(M))$.  Therefore if $\\textnormal{Act}(A,\\textnormal{Aut}(M))$ is contained in a dendromorphic group  $X$, then\n$$X \\not\\leq \\textnormal{Aut}_{\\lbrace A \\rbrace}(M)$$\nIn particular this implies that if $g \\in X \\setminus \\textnormal{Act}(A,\\textnormal{Aut}(M))$ then $g(A) \\not= A$.\n\nLet $A^U$ be the set of upper points of $A$, enumerated by $\\lbrace \\ldots, a_{-2}, a_0, a_2, \\ldots \\rbrace$.\n\nSince $(A^U,\\textnormal{Act}(A,\\textnormal{Aut}(M)))$ is 1-transitive so is $(X(A^U),X)$, and so\n\n$$(X(A^U),X) \\cong (X(A^U), X_0)$$\n\nwhere $X_0$ is one of the factors of $X$ (i.e. $\\textnormal{Sym}(\\omega)$, $\\mathbb{Z} \\wr \\mathbb{Z}_2$ or $\\prod S_n$).  This $X_0$ cannot be $\\textnormal{Sym}(\\omega)$ as then it would be possible to map the triple $(a_{-2}, a_0, a_2)$ to $(a_{-2}, a_2, a_0)$, but any map that does this has to change the length of $\\path{ a_{-2},a_2 }$, and so cannot be an isomorphism.  This same argument prevents $X_0 \\cong \\prod S_n$.\n\nLet $\\sigma$ be the infinite order generator of $\\textnormal{Act}(A,\\textnormal{Aut}(M))$ and $\\tau$ be the finite order generator.  Suppose $X_0 \\cong \\mathbb{Z} \\wr \\mathbb{Z}_2$, generated by $\\alpha$, $\\beta$ and $\\gamma$, where $\\alpha$ and $\\beta$ have infinite order and $\\gamma$ has finite order.  Since $\\textnormal{Act}(A,\\textnormal{Aut}(M))$ contains an element of finite order, both $\\textnormal{supp}(\\alpha)$ and $\\textnormal{supp}(\\beta)$ must have a non-empty intersection with $A^U$.\n\nSince $\\alpha$, $\\beta$ and $\\gamma$ generate $X$ and either preserve or switch $\\textnormal{supp}(\\alpha)$ and $\\textnormal{supp}(\\beta)$, every member of $\\textnormal{Act}(A,\\textnormal{Aut}(M))$ either preserves or switches $\\textnormal{supp}(\\alpha)$ and $\\textnormal{supp}(\\beta)$.  So both $\\textnormal{supp}(\\alpha) \\cap A^U$ and $\\textnormal{supp}(\\beta) \\cap A^U$ cannot both be singletons, as only the identity will preserve $\\textnormal{supp}(\\alpha) \\cap A^U$ and $\\textnormal{supp}(\\beta) \\cap A^U$ and no member of $\\textnormal{Act}(A,\\textnormal{Act}(M))$ will swap them.  Since $\\textnormal{supp}(\\alpha) \\cap A^U$ is not a singleton, the action on it determines the action on the whole of $A^U$, and so $\\alpha$ and $\\beta$ cannot act independently.  So $X_0$ cannot be isomorphic to $\\mathbb{Z} \\wr \\mathbb{Z}_2$.\n\\end{proof}\n\n\\begin{cor}\nLet $M$ be a CFPO.  If there is an $A \\subseteq M$ and a $G \\leq \\textnormal{Aut}(M)$ such that:\n\\begin{enumerate}\n\\item $A$ is a copy of Alt;\n\\item $G \\cong D_\\infty$; and\n\\item $G$ acts on $A$.\n\\end{enumerate}\nthen $M$ is not treelike.\n\\end{cor}\n\\begin{proof}\nIf $M$ is treelike then Theorem \\ref{thm:SuperDinf} shows that $G$ is contained in a dendromorphic group, but Theorem \\ref{thm:dinfinityinCFPOs} shows that this is impossible.\n\\end{proof}\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWith the rapid growth of e-commerce, recommender systems have become an obligatory tool for interconnecting customers with relevant items. Early schemes suffer from cold-start problems caused by data insufficiency. To tackle this problem, auxiliary information of data such as social relations \\cite{fu2021dual}, the trustworthiness of reviewers \\cite{banerjee2017whose}, item images \\cite{yu2018aesthetic} are exploited. Especially, textual or linguistic information such as reviews are commonly available, and many text-aided recommendation algorithms have been introduced \\cite{zheng2017joint,chen2018neural,chen2019co,dong2020asymmetrical,wang2021leveraging,shi2021wg4rec,xiong2021counterfactual,shuai2022review}.\nFor example, some studies infer the preferences of users by applying natural language processing (NLP) techniques such as topic modeling \\cite{mcauley2013hidden,bao2014topicmf}. Recently, some apply convolutional neural network (CNN) on textual reviews \\cite{zheng2017joint}, others \\cite{chen2018neural,dong2020asymmetrical} further utilize attention mechanism to deal with important reviews. Many text analysis modules are considerable, but here, we employ a simple text analysis module \\cite{zheng2017joint} to focus on a cross-domain recommendation scenario.\n\nAlong with the text-based recommender systems, numerous cross-domain recommendation \\cite{hu2018conet,yuan2019darec,fu2019deeply} and transfer learning \\cite{yuan2020parameter,yang2021autoft,hande2021domain} approaches have been introduced. They leverage the information learned from source domains to improve the quality of recommendation in a target domain. Some of them suggest a fine-tuning network, but they may suffer from catastrophic forgetting \\cite{chen2019catastrophic}. Context mappings \\cite{yuan2019darec,li2020ddtcdr,liu2021collaborative,guo2021gcn} are another branch of transfer learning, which map shareable information from a source to the target domain. However, these approaches require the same contexts or overlapped users, which may restrict their applicability severely \\cite{kang2019semi}. In a real-world dataset, this kind of information might be scarce, and thus, their contributions might be insignificant, which still requires further investigation. \n\nTo solve such constraints, we focus on another branch of methods that are independent of specific users or contexts \\cite{cai2019learning,zhao2020catn,krishnan2020transfer}. However, it is still challenging to achieve knowledge transfer between heterogeneous domains (e.g., totally different users or items), while debilitating noises.\nFor one solution, domain adaptation (DA) \\cite{ben2010theory,ganin2016domain,li2020maximum,li2021learning} minimizes source risk as well as H-divergence, capturing domain-indiscriminative information. Nonetheless, these methods are highly dependent on the consistency (maximum mean discrepancy) between domains. Though disentangled representation learning algorithms \\cite{gretton2005measuring,bousmalis2016domain,li2019learning,peng2019domain} concurrently extract domain-specific, and domain-invariant knowledge to distill pertinent knowledge from multiple seemingly counterproductive domains, they also suffer from measuring theoretical bound and tractability of a log-partition function \\cite{gabrie2019entropy,mcallester2020formal}.\n\nIn this paper, we suggest a novel method of disentangled representation learning that utilizes textual reviews common to most e-commerce systems. Here, we focus on debilitating noises of another domain and claim that this can be achieved by reinforcing the discrimination power of a domain discriminator. Specifically,\nusing the connections between mutual information (MI) estimation and DA \\cite{kang2019contrastive,saito2019semi,park2020joint,zhao2021domain}, we integrate the role of the MI estimator with a single domain discriminator. Thus, the domain-shareable information is further utilized as an input for the domain discriminator, which can reinforce the discrimination power. In terms of adversarial training, a well-trained discriminator guides three types of FEs to secure robustness and improve the recommendation quality for both domains.\nWe perform extensive experiments on the publicly available dataset to compare our model under single and cross-domain recommendation scenarios. The quantitative and qualitative analysis demonstrates the superiority of our suggestions. The contributions can be summarized as follows.\n\\begin{itemize}\n\\item We propose $SER$ that adaptively disentangles features depending on the characteristics of the source and target domains, where the novel optimization strategy leads to a promising solution for a recommendation. Consequently, our approach is also applicable to heterogeneous scenarios, in case the source and target domain have less similar characteristics (e.g., non-overlapping users or contexts).\n\\item Our approach is comprehensive that is closely integrated with text-based feature extraction. Unlike previous cross-domain recommendation schemes that require duplicate entities from heterogeneous domains, we focus on retrieving review information that is common for most domains. \n\\item We perform extensive experiments to answer the important research questions described in Section \\ref{experiments}. The results indicate the superiority and robustness of our proposed method.\n\\end{itemize}\n\n\\section{Related Work}\nIn this section, we describe an overview of existing recommender systems; text-aided single-domain recommendations, extending to a cross-domain scenario, and applying domain adaptation to achieve disentangled representation learning, which can enhance the knowledge transfer between domains.\n\n\\subsection{Text-aided Recommender System}\nTextual information is the most popular side information and many text-based methods \\cite{zheng2017joint,chen2019dynamic,chen2019co,dong2020asymmetrical} have been proposed recently. Previous techniques simply integrate DNN-based feature extraction with MF for a rating prediction, while \\cite{zheng2017joint} utilizes two parallel CNNs, and \\cite{li2017neural} adopts Gated Recurrent Units for review analysis. Attention mechanisms are widely used also to pinpoint useful words and reviews \\cite{seo2017interpretable,chen2018neural,tay2018multi,dong2020asymmetrical}. \nEven though prior works show the usefulness of textual information, the limitation of review information due to the limited size of training data, and the irrelevance of reviews toward target items have been raised also \\cite{sachdeva2020useful,zeng2021zero}. To tackle this problem, we now introduce recent algorithms for cross-domain recommender systems which adopt supplementary domains.\n\n\\subsection{Cross-Domain Recommendation (CDR)}\nCDR utilizes information from source domains to alleviate the cold-start problem in the target domain.\nEspecially, most of them assume embedding-based or rating pattern-based transfer scenario \\cite{zhu2021cross}.\nIn detail, early studies \\cite{elkahky2015multi,man2017cross} adopt a feature mapping technique that requires overlapped users. For example, RC-DFM \\cite{fu2019deeply} applies Stacked Denoising Autoencoder (SDAE) to each domain, where the learned knowledge of the same set of users is transferred from a source to the target domain. More recently, to overcome the requirements of overlapped users, there was an attempt \\cite{wang2018cross,zhao2020catn} that employs similar users for feature mapping. However, they implicate the limitation of debilitating noises from a source domain. In this paper, we adopt review texts that are common to most e-commerce systems without requiring overlapping users or contexts. Further, we suggest a novel disentangled representation learning for a knowledge transfer between domains which can reduce noises from a source domain. We now introduce some traditional\nmethods for disentangled representation learning below.\n\n\\subsection{Disentangled Representation Learning}\nRecently, many efforts have been dedicated to capturing domain-shareable information. Especially, with the powerful mechanism of adversarial training, DA has been adopted for various fields; question answering tasks \\cite{ramakrishnan2018overcoming}, and recommendation scenarios \\cite{yuan2019darec,bonab2021cross}. Some adopt textual reviews under no user or context overlap for a cross-domain recommendation \\cite{yu2020semi}. However, these approaches only focus on domain-shareable knowledge, while ignoring domain-specific ones. MMT \\cite{krishnan2020transfer} further captures domain-specific knowledge, but the disentanglement between domain-common features is not applied. Though DADA \\cite{peng2019domain} introduced domain-agnostic learning, the domain discriminator is only utilized for the extraction of a domain-common feature. Since the domain-specific knowledge is solely guided using MI minimization \\cite{belghazi2018mutual}, they have shown to implicate some defects \\cite{mcallester2020formal}. In this paper, we clear the aforementioned limitations, adopting a framework for the simultaneous extraction of domain-specific and domain-common knowledge by integrating MI estimator with domain discriminator. Consequently, our model enhances the quality of disentangled representation learning without requiring overlapping users or contexts for a recommendation.\n\n\n\\begin{figure*}[ht]\n  \\includegraphics[width=\\textwidth]{assets\/model.png}\n  \\caption{The overall framework of SER for a cross-domain recommendation scenario}\n  \\label{model}\n\\end{figure*}\n\n\\begin{table}[htbp]\\caption{Notations}\n\\vspace{-1.5em}\n\\begin{center}\n     \\begin{tabular}{@{}l|l@{}}\n\\toprule\n\\multicolumn{1}{c|}{\\textbf{Symbol}} & \\textbf{Explanation} \\\\\n\\toprule\n\\multicolumn{1}{c|}{$D^s,D^t$} & Set of source and target domain dataset\\\\\n\\multicolumn{1}{c|}{$u,i,y_{u,i},r_{u,i}$} & Set of user, item, rating, and individual review\\\\\n\\multicolumn{1}{c|}{$R_u,R_i$} & Set of aggregated reviews for user and item \\\\\n\\multicolumn{1}{c|}{$p_u, q_i$} & Rating embedding of user and item  \\\\\n\\multicolumn{1}{c|}{$O$} & Extracted feature of $R_u,R_i$ \\\\\n\\multicolumn{1}{c|}{$I$} & Extracted feature of $r_{u,i}$ \\\\\n\\multicolumn{1}{c|}{$\\widehat{d}$} & Predicted domain label of $O$ \\\\\n\\multicolumn{1}{c|}{$\\widehat{y}_E,\\widehat{y}_I$} & Predicted label of $E$ and $I$ \\\\\n\\multicolumn{1}{c|}{MI} & Mutual information \\\\\n\\multicolumn{1}{c|}{$N_s,N_t$} & Mini-batch training samples of $s$ and $t$ \\\\\n\\multicolumn{1}{c|}{$\\phi$} & Word embedding function \\\\\n\\multicolumn{1}{c|}{$\\oplus$} & Concatenation operator \\\\\n\\multicolumn{1}{c|}{$\\mathcal{L}$} & Loss function \\\\\n\\bottomrule\n    \\end{tabular}\n\\end{center}\n    \\label{notations}\n\\end{table}\n\n\n\\section{Problem Formulation}\nWe formulate the cross-domain recommendation task as follows. Assume two datasets, $D^s$ and $D^t$, be the data of the source and target domains. Each of them consists of tuples, $(u, i, y_{u, i}, r_{u,i})$ which represents an individual review $r_{u,i}$ provided by a user $u$ for an item $i$ with rating $y_{u,i}$. \nOur goal is to improve the recommendation quality in a target domain with the aid of a source domain. To achieve this, we suggest a domain-aware knowledge transfer module with encoding networks. The overall notations can be seen in Table \\ref{notations}.\n\n\n\\section{Our Approach}\n\nFigure \\ref{model} describes our model with the following key components: \n\n\\begin{itemize}\n    \\item \\textbf{Review embedding layer} \n    Vectorize the aggregated and individual review using pre-trained word embedding function, which can be word2vec\\footnote{https:\/\/code.google.com\/archive\/p\/word2vec} \\cite{mikolov2013distributed} or GloVe\\footnote{https:\/\/nlp.stanford.edu\/projects\/glove} \\cite{pennington2014glove}.\n    \\item \\textbf{Domain-aware feature extraction} Integrated with domain discriminator, three feature extractors (FEs) capture domain-aware knowledge from vectorized aggregated reviews. Especially, the common FE acts as a knowledge transfer module and utilizes both source and target domain data.\n    \\item \\textbf{Encoding network} The encoding networks generate a single vector from vectorized aggregated reviews by aligning them with the latent of an individual review. \n\\end{itemize}\nWe now illustrate the details of each module precisely.\n\n\\subsection{\\textbf{Review Embedding Layer}}\nGiven the input data (\\textit{u, i, $y_{u,i}$, $r_{u,i}$}), we first ensemble the reviews of user and item. For example, given user $u$, all reviews written by her are treated as a single document $R_u$. The item reviews $R_i$ can be retrieved similarly. However, for each training sample, we exclude an individual review $r_{u,i}$ that $u$ has written after purchasing $i$ since it cannot be used during the inference phase. Here, we do not consider a sequence (or time) of their purchasing histories.\n\nWe utilize first $n$ words in $R$ and apply pre-trained word embedding function \\cite{mikolov2013distributed,pennington2014glove} for vectorization. The words are mapped to $c$-dimensional vectors, and column-wisely concatenated to form document embedding $V=\\phi(w_1) \\oplus \\phi(w_2) \\oplus ...  \\oplus \\phi(w_n)$, where $\\phi$ and $\\oplus$ is a embedding and concatenation operation. \n\n\\subsection{\\textbf{Domain-aware Feature Extraction}}\nAs described in Figure \\ref{model}, we utilize three types of feature extractors (FEs) to separate the domain-specific and domain-common knowledge. In this paper, each FE adopts the widely used text analysis method \\cite{zheng2017joint}. More specifically, using the document embedding $V$, three convolutional $FEs$ produce outputs, followed by a non-linear activation function (e.g., ReLU) with a row-wise max-pooling layer. \nFinally, by concatenating the outputs of stacked convolution layers, we can derive a unified embedding $O$ from user and item reviews (please refer to \\cite{zheng2017joint} for text convolutions).\nSimilarly, we can retrieve the embedding of an individual review $I$ using three FEs.\n\nTo summarize, the domain-specific FEs (source and target) generate $O^s_{spe},I^s_{spe}$ and $O^t_{spe},I^t_{spe}$ for each domain. Also, the domain-common FE extracts $O^s_{com},I^s_{com}$, and $O^t_{com},I^t_{com}$ using inputs from source and target domain, respectively. Here, the superscript $s,t$ acts as an identifier of two domains.\nThough we assume three FEs, they may contain similar information without any constraints. \nTo achieve a better disentanglement, we suggest an optimization strategy from two perspectives: (1) for the extraction of domain-common knowledge, we employ domain adaptation to serve common FE as an intermediate agent for knowledge transfer. (2) for the disentanglement of domain-specific features, we suggest a novel constraint for adversarial training.\n\n\n\n\n\n\\subsubsection{\\textbf{Extracting domain-common features}}\nFor a knowledge transfer between domains, we focus on domain adaptation (DA) \\cite{ganin2016domain} that has proven to be effective in case a source domain has a richer label than the target \\cite{ben2010theory}.\nDA employs domain discriminator with Gradient Reversal Layer (GRL) to reduce the cross-domain discrepancy. The discriminator gives additional penalties (or loss) to common FE for capturing domain-discriminative information, which has proven to be effective for debilitating noises. Here, we adopt two layers of a fully-connected neural network as domain discriminator $F_{disc}$, which takes the output of common FE as below:\n\n\\begin{equation} \n\\label{domain_label}\n\\widehat{d}_{com}^s=F_{disc}(O_{com}^s),\\, \\,\\widehat{d}_{com}^t=F_{disc}(O_{com}^t).\n\\end{equation}\nHere, $\\widehat{d}_{com}$ stands for predicted domain probability of common feature $O_{com}$, where the error can be calculated through binary cross-entropy loss between true label $d_{com}$ as below:\n\\begin{equation}\n\\begin{gathered}\n\\mathcal{L}_{com}^s= -{1 \\over N_s}\\sum_{s=1}^{N_s} log(1-\\widehat{d}^{s}_{com}),\\,\\,\\mathcal{L}_{com}^t = -{1 \\over N_t}\\sum_{t=1}^{N_t} log(\\widehat{d}^{t}_{com}).\n\\end{gathered}\n\\label{d_inv_loss}\n\\end{equation}\nThe true label $d_{com}$ is a binary value, \\{0, 1\\} for source and target domain. Here, $N$ is the mini-batch training samples from two domains. GRL is positioned between common FE and domain discriminator (please refer to Figure \\ref{model}), multiplies a negative constant during back-propagation. Consequently, the common FE is reinforced to capture domain-indiscriminative knowledge to fool discriminator.\nNonetheless, DA itself implicates some limitations, which can be sensitive to domain divergence, and prohibitive applicability \\cite{chen2019transferability}.  \n\n\\begin{figure}\n  \\centering\n  \\vspace{-2mm}\n  \\includegraphics[width=0.47\\textwidth]{assets\/domain_adaptation.png}\n  \\caption{(a) Conventional domain disentanglement \\cite{peng2019domain} using mutual information (MI) estimator $T$, and (b) ours integrating MI estimator with domain discriminator}\n  \\label{domain adaptation}\n  \\vspace{-3mm}\n\\end{figure}\n\n\\subsubsection{\\textbf{Disentangling domain-specific features}}\nTo deal with the aforementioned problems, many approaches endeavored for the parallel extraction of domain-specific knowledge. One of the promising solutions is disentangled representation learning, which can secure pairwise independence between generated factors \\cite{gretton2005measuring,li2019learning,nema2021disentangling}. \nFor example, DSN \\cite{bousmalis2016domain} optimizes the source, common, and target FEs by securing orthogonality between domain-specific and domain-common features. More recently, estimating mutual information (MI) between extracted features has been proposed \\cite{peng2019domain,cheng2020improving}. Specifically, IDEL \\cite{cheng2020improving} suggests a novel MI bound to retrieve disentangled representation of texts. DADA \\cite{peng2019domain} captures domain-specific knowledge using MI estimator (see Figure \\ref{domain adaptation}-a). In terms of independence maximization, DADA aims to minimize MI between domain-specific $O_{spe}$ and domain-common $O_{com}$ features as follows: \n\\begin{equation}\n\\begin{gathered}\n\\mathcal{L}^s_{MI}= {1 \\over N_s} \\sum_{s=1}^{N_s} T(O^{s}_{spe},O^{s}_{com}) - log ({1 \\over N_s} \\sum_{s=1}^{N_s} e^{T(O^{s}_{spe},O'^{s}_{com})}), \\\\\n\\mathcal{L}^t_{MI}= {1 \\over N_t} \\sum_{t=1}^{N_t} T(O^{t}_{spe},O^{t}_{com}) - log ({1 \\over N_t} \\sum_{t=1}^{N_t} e^{T(O^{t}_{spe},O'^{t}_{com})}).\n\\end{gathered}\n\\label{mi_loss}\n\\end{equation}\nSince MI cannot be directly calculated, they adopt approximation theory MINE \\cite{belghazi2018mutual} with a neural network $T$: \n\\begin{equation}\n\\begin{gathered}\n\\label{mine}\nT(x, z) = F_3(\\sigma(F_1(x) + F_2(z))).\n\\end{gathered}\n\\end{equation}\nWith two inputs $x$ and $z$, a fully-connected layer $F$ with non-linear activation $\\sigma$ estimates MI. Back into Equation \\ref{mi_loss}, as an input of $T$, ($O_{spe},O_{com}$) pairs are sampled from joint distribution, while the same number of ($O_{spe},O'_{com}$) pairs are from marginals (please refer to \\cite{peng2019domain} for more details). \nAs described in Figure \\ref{domain adaptation}-a, domain-specific FEs are trained to minimize $\\mathcal{L}^s_{MI},\\mathcal{L}^t_{MI}$ (red arrows), while GRL guides a common FE (blue arrows) through Equation \\ref{d_inv_loss}.\nNonetheless, MINE implicates the following limitations: a statistical drawback of measuring bounds \\cite{mcallester2020formal}, the estimators become intractable even for a known distribution as the dimension increases \\cite{gabrie2019entropy} and is highly dependent on the number of symbols and the size of dataset \\cite{panzeri2007correcting}.  \n\nUsing the observations \\cite{kang2019contrastive,saito2019semi,park2020joint,zhao2021domain}, minimizing $\\mathcal{H}$-divergence is closely related to MI maximization. We integrate the MI estimator with a domain discriminator from the following perspectives: (1) maximizing domain loss (equal to MI maximization) for domain-common features, while (2) minimizing domain loss for domain-specific outputs. To achieve this, we exclude GRL between domain-specific FEs and domain discriminator (please refer to Figure \\ref{domain adaptation}-b), to guide these FEs to generate domain-discriminative knowledge.\nThe advantages of this suggestion are as follows. Firstly, domain-specific features ($O^s_{spe}, O^t_{spe}$) are additionally utilized as an input of a domain discriminator, where they reinforce the discriminating power of a domain discriminator. Then, a discriminator well distinguishes the origin of the input vectors, leading three types of FEs better extract domain-aware knowledge. Secondly, our model can disentangle inter-domain outputs ($O^s_{spe} \\leftrightarrow O^t_{spe}$), while the MI-based methods only separate intra-domain ones ($O^s_{spe} \\leftrightarrow O^s_{com}$ or $O^t_{spe} \\leftrightarrow O^t_{com}$).\nLastly, the large values of MI can mislead the parameters, while the ground truth of our model lies between 0 and 1. \nUsing Equation \\ref{domain_label} and \\ref{d_inv_loss}, we can define domain-specific losses as:\n\\begin{equation} \n\\begin{gathered}\n\\widehat{d}^s_{spe}=F_{disc}(O^s_{spe}), \\, \\,\\widehat{d}^t_{spe}=F_{disc}(O^t_{spe}), \\\\\n\\mathcal{L}_{spe}^s= -{1 \\over N_s}\\sum_{s=1}^{N_s} log(1-\\widehat{d}^s_{spe}),\\,\\,\\mathcal{L}_{spe}^t = -{1 \\over N_t}\\sum_{t=1}^{N_t} log(\\widehat{d}^t_{spe}).\n\\end{gathered}\n\\label{d_spe_loss}\n\\end{equation}\n\nFinally, integrating Equation \\ref{d_inv_loss} and \\ref{d_spe_loss}, the overall domain loss function can be abbreviated as below:\n\\begin{equation}\n\\begin{gathered}\n\\mathcal{L}_{dom} = a(\\mathcal{L}_{com}^s +\n\\mathcal{L}_{spe}^s) + (1-a)(\\mathcal{L}_{com}^t +  \\mathcal{L}_{spe}^t).\n\\end{gathered}\n\\label{da_loss}\n\\end{equation}\n$a={N_s \\over N_s+N_t}$ controls the weight of source and target losses based on the size of two domains \\cite{ben2010theory}. For training, the domain discriminator is updated using Equation \\ref{da_loss}. Then, GRL is applied for domain-common losses $-a\\mathcal{L}^s_{com},-(1-a)\\mathcal{L}^t_{com}$ (blue arrows in Figure \\ref{domain adaptation}-b) to update common FE, while domain-specific losses $a\\mathcal{L}^s_{spe},(1-a)\\mathcal{L}^t_{spe}$ (red arrows) without GRL update domain-specific FEs. \n\n\\subsubsection{Theoretical analysis}\nReferring to the estimated bounds of semi-supervised DA \\cite{li2021learning}, we can infer the theoretical bounds of DADA \\cite{peng2019domain} and our SER. \nSince both of them captures domain-common knowledge through DA, the difference lies in domain disentanglement strategy and their performance. \nWe first discuss the classification errors between true domain labels and predicted ones.\nDADA adopts dual representations of KL-divergence, which has proven \\cite{belghazi2018mutual} that a neural network $T_{\\theta}$ with parameter $\\theta$ satisfies the conditions below as the number of samples $n$ goes to infinity:\n\\begin{equation}\n\\begin{gathered}\n|\\widehat{I(X;Z)}_n-I_{\\theta}(X,Z)| \\leq \\epsilon\n\\end{gathered}\n\\end{equation}\nSimilarly, our SER adopts stochastic gradient descent with binary domain labels, which has proven to \\cite{gower2019sgd} converge under the arbitrary sampling scenario.\nBased on these results, both approaches well predict the true labels within $\\epsilon$. However, it is notable that the value of MI ranges from 0 to infinity. Referring to Equation \\ref{mi_loss}, a large domain-specific loss might impede the stable convergence of three FEs. Comparatively, we adopt binary cross-entropy loss of Equation \\ref{da_loss}, and thus, the loss rarely diverges even for completely domain-indistinguishable features ($\\widehat{d}^s_{spe}\\approx\\widehat{d}^t_{spe}\\approx0.5$). In Section \\ref{dd_analysis}, we scrutinize two methods that use Equation \\ref{mi_loss} and \\ref{d_spe_loss}.\n\n\n\n\\subsection{\\textbf{Encoding Network and Regressor}} \n\\label{sec_re}\nIn this section, except for Equation \\ref{enc_loss} and \\ref{mse_loss}, we exclude domain identifier $d$ to simplify notation.\n\n\\textbf{Encoding network} This layer employs a transfer network \\cite{wu2020learning} (MLP) to align the latent of aggregated reviews with an individual review, which is proposed by TransNet \\cite{catherine2017transnets}. Using this strategy, FEs are trained to mimic the latent of an individual review that a user has written after purchasing a specific item (please refer to \\cite{catherine2017transnets} for more details).\nHere, we extend this mechanism from a single to cross-domain recommendation scenario, while most text-aided schemes \\cite{zheng2017joint,chen2018neural,chen2019co,dong2020asymmetrical,wang2021leveraging} underestimate this kind of information.  \nSince we have obtained the latents of aggregated review $O_{spe},O_{com}$, the encoded representation can be derived as below:\n\\begin{equation}\n\\label{E_out}\nO  = F_{enc}(O_{spe}+O_{com})\n\\end{equation}\n, where $F_{enc}$ is an encoding network with two layers.\nSimilarly, the representation of an individual review can be retrieved as follows:\n\\begin{equation}\n\\label{I_out}\nI = I_{spe} + I_{com}.\n\\end{equation}\n\nTo align the two vectors $O\\leftrightarrow I$ of Equation \\ref{E_out} and \\ref{I_out}, for each domain $d \\in \\{s,t\\}$, we adopt Euclidean distance as a loss function: \n\\begin{equation}\n\\label{enc_loss}\n\\mathcal{L}^d_{enc} = {1 \\over N_d}\\sum_{d=1}^{N_d}||O^d-I^d||_2^2.\n\\end{equation}\n\n\\textbf{Regressor} Finally, we can utilize two representations $O,I$ to predict a recommendation score through regressor $F_{reg}$ (MLP). Here, we further apply a widely used latent factor model \\cite{tan2016rating}, where $p_u$ and $q_i$ stand for the embedding of user and item:\n\\begin{equation}\n\\label{regress_out}\n\\begin{split}\n\\widehat{y}_I=F_{reg}(I)+p_u\\cdot q_i^T, \\,\\,\\widehat{y}_O=F_{reg}(O)+p_u\\cdot q_i^T.\n\\end{split}\n\\end{equation}\nWe utilize the prediction of an individual review $\\widehat{y}_I$, which can guide the regressor precisely. The regression loss is defined in Equation \\ref{mse_loss}, which is the Mean Squared Error (MSE) between two predicted scores $\\widehat{y}_I$, $\\widehat{y}_O$ and true label $y$ in domain $d$ as below:\n\\begin{equation}\n\\label{mse_loss}\n\\begin{split}\n\\mathcal{L}^d_{reg} = {1 \\over 2N_d}\\sum_{d = 1}^{N_d} \\left((\\widehat{y}^d_I-y^d)^2 + (\\widehat{y}^d_O-y^d)^2\\right).\n\\end{split}\n\\end{equation}\n\n\\subsection{Optimization and Inference}\n\\label{sec_opt}\n\\textbf{Optimization} We jointly optimize SER by minimizing the weighted sum of three losses defined in Equation \\ref{da_loss}, \\ref{enc_loss}, \\ref{mse_loss}:\n\\begin{equation}\n\\label{total_loss}\n\\min_{\\theta}\\mathcal{L} = \\alpha \\mathcal{L}_{dom} + \\beta (\\mathcal{L}^s_{enc}+\\mathcal{L}^t_{enc}) + \\gamma (\\mathcal{L}^s_{reg} + \\mathcal{L}^t_{reg}) + \\delta ||\\theta||.\n\\end{equation}\n\nIn Figure \\ref{model}, we describe how the gradients are propagated with three basic shapes (the horizontal stripe denotes a source loss). The hyper-parameters $\\alpha$, $\\beta$, and $\\gamma$ balance the weight of three losses. We set $\\alpha=0.1$, $\\beta=0.05$, and $\\gamma=1$ through a grid search. $\\theta$ denotes parameters of our model, where $\\delta$ is a regularization term. We adopt early stopping under 300 iterations with Adam optimizer and learning ratio as $lr=1e^{-4}$. \n\n\n\\textbf{Inference} During inference phase (lower side of Figure \\ref{model}), SER recommends an item using aggregated reviews of user $R^t_u$ and item $R^t_i$.\nThen, the common and target FEs are applied for $R^t_u$ and $R^t_i$, retrieving their features $O^t_{com},O^t_{spe}$, respectively. Finally, a target encoder generates meaningful feature $O^t$, followed by a regressor.\n\n\\subsection{Computational Complexity Analysis}\nIn this section, we analyze the computational cost of our model. Firstly, let us assume parameters of vanilla text analysis model DeepCoNN \\cite{zheng2017joint} as $A+B$, which adopts a single FE ($A$) and regressor ($B$). The running time can be approximated as $\\mathcal{O}((A+B) \\cdot N_t \\cdot e)$, where $N_t$ is the size of training samples and $e$ is a training epoch.\n\nOur model further utilizes a domain discriminator with encoding networks. With the slight abuse of notation, let us assume the number of additional parameters as $C$. Also, we utilize the source domains that contain $N_s$ samples, the computational complexity can be abbreviated as $\\mathcal{O}((3A+2B+C) \\cdot (N_s+N_t) \\cdot e)$. To summarize, we can infer that a cost linearly depends on the input size of the source domain $N_s$, which can be quite efficient.\n\n\n\\section{Experiments} \\label{experiments}\nWe aim to answer the following research questions:\n\\begin{itemize}\n    \\item \\textbf{RQ1:} Can SER enhance the recommendation quality compared to the state-of-the-art approaches? \n    \\item \\textbf{RQ2:} Does SER effectively disentangles features for a cross-domain recommendation?\n    \\item \\textbf{RQ3:} How much does the domain-aware feature extractor and encoding network contribute to the overall quality? \n    \\item \\textbf{RQ4:} Is a domain-common feature necessary for CDR? If so, how does the knowledge is shared across domains?\n\\end{itemize}\n\n\n\n\\subsection{Dataset and Experimental Setup}\n\\textbf{Dataset description} We systematically conduct investigations with publicly available dataset \\textit{Amazon}\\footnote{http:\/\/jmcauley.ucsd.edu\/data\/amazon\/} and \\textit{Yelp}\\footnote{https:\/\/www.yelp.com\/dataset}. The target domain includes the following four categories of Amazon 5-core: \\textit{Office Products, Instant Video, Automotive, Patio Lawn and Garden}. We designate the source domain with more interactions \\cite{ben2010theory}. Three categories from \\textit{Amazon}: \\textit{Baby, Kindle Store (KS), Toys and Games (TG)} and one another from \\textit{Yelp}. Here, \\textit{Yelp} data is used to show the effect of excluding duplicate users. The statistical details of the dataset are summarized in Table \\ref{dataset}.\nThough some studies employ a source domain of seemingly relevant categories, it is natural to ask: \\textit{how can we choose the most relevant domain? if they are truly relevant, how does the knowledge transferred across domains?} Here, we focus on the second question rather than figuring out relevant domains. Multi-source adaptation is also considerable, but here, we leave it for future work. \n\n\\begin{table}\n\\caption{Statistical details of the dataset}\n\\label{dataset}\n\\centering\n\\begin{adjustbox}{width=0.48\\textwidth}\n\\begin{tabular}{lllll}\n\\multicolumn{1}{l}{}    & \\multicolumn{1}{l}{}       &         &         &                \\\\ \n\\Xhline{2\\arrayrulewidth}\n\\multicolumn{1}{l}{}        & Dataset                       & \\# users & \\# items & \\# reviews  \\\\ \n\\Xhline{2\\arrayrulewidth}\n\\multirow{4}{*}{Source} \n                        & Baby                       & 19,445  & 7,050   & 160,792        \\\\\n                        & Kindle Store (KS)               & 68,223  & 61,934  & 982,619        \\\\\n                        & Toys and Games (TG)             & 19,412  & 11,924  & 167,597        \\\\\n                        & Yelp             & 1.9 M  & 0.2 M  & 8.1 M        \\\\\n\\hline\n\\multirow{4}{*}{Target} & Office Products            & 4,905   & 2,420   & 53,258         \\\\\n                        & Instant Video              & 5,130   & 1,685   & 37,126         \\\\\n                        & Automotive                 & 2,928   & 1,835   & 20,473         \\\\\n                        & Patio Lawn and Garden      & 1,686   & 962     & 13,272         \\\\\n\\Xhline{2\\arrayrulewidth}\n\\end{tabular}\n\\end{adjustbox}\n\\end{table}\n\n\\begin{table*}[!htbp]\n\\centering\n\\caption{ MSE ($\\downarrow$) on four target domain  dataset. Bold and underline indicate $1^{st}$ and $2^{nd}$ best.}\n\\label{performance_mse}\n\\small\n\\begin{tabularx}{\\textwidth}{@{}c|cccc|cccc|cccc|cccc@{}}\n\\toprule\nTarget domain &  \\multicolumn{4}{c|}{\\underline{Office Product}} & \\multicolumn{4}{c|}{\\underline{Instant Video}} & \\multicolumn{4}{c|}{\\underline{Automotive}} & \\multicolumn{4}{c}{\\underline{Patio Lawn and Garden}} \\\\\n\nSource domain & Baby & KS & TG & Yelp & Baby & KS & TG & Yelp & Baby & KS & TG & Yelp & Baby & KS & TG & Yelp \\\\\n\\midrule\nPMF &  \\multicolumn{4}{c|}{1.085} & \\multicolumn{4}{c|}{1.129} & \\multicolumn{4}{c|}{1.162} & \\multicolumn{4}{c}{1.177} \\\\\nNeuMF &  \\multicolumn{4}{c|}{0.974} & \\multicolumn{4}{c|}{1.014} & \\multicolumn{4}{c|}{1.087} & \\multicolumn{4}{c}{1.143} \\\\\nDeepCoNN &  \\multicolumn{4}{c|}{0.902} & \\multicolumn{4}{c|}{0.949} & \\multicolumn{4}{c|}{0.979} & \\multicolumn{4}{c}{1.128} \\\\\nNARRE &  \\multicolumn{4}{c|}{0.863} & \\multicolumn{4}{c|}{0.914} & \\multicolumn{4}{c|}{0.887} & \\multicolumn{4}{c}{1.108} \\\\\nAHN &  \\multicolumn{4}{c|}{0.859} & \\multicolumn{4}{c|}{0.892} & \\multicolumn{4}{c|}{0.863} & \\multicolumn{4}{c}{\\underline{1.094}} \\\\\n\\Xhline{2\\arrayrulewidth}\nDANN & 0.966 & 0.939 & 0.943 & 1.118 & 0.986 & 0.946 & 0.987 & 1.147 & 0.946 & 0.881 & 0.945 & 1.183 & 1.129 & 1.189 & 1.199 & 1.395\\\\\nDAREC & 0.989 & 0.988 & 0.972 & 0.994 & 1.060 & 1.045 & 1.043 & 1.073 & 1.001 & 0.997 & 0.993 & 1.004 & 1.123 & 1.151 & 1.131 & 1.150\\\\\nDDTCDR & 0.954 & 0.947 & 0.926 & 0.965 & 0.974 & 0.981 & 0.967 & 0.988 & 0.961 & 0.959 & 0.954 & 0.969 & 1.109 & 1.111 & 1.105 & 1.133\\\\ \nRC-DFM & 0.834 & 0.839 & 0.828 & \\underline{0.841} & 0.878 & \\underline{0.855} & \\underline{0.868} & 0.872 & \\underline{0.792} & \\underline{0.800} & \\underline{0.794} & \\underline{0.802} & \\underline{1.094} & 1.096 & 1.109 & 1.112\\\\\nCATN & 0.875 & 0.872 & 0.873 & 0.876 & 0.915 & 0.906 & 0.892 & 0.919 & 0.824 & 0.831 & 0.826 & 0.837 & 1.141 & 1.144 & 1.129 & 1.149\\\\\nMMT & \\underline{0.815} & \\underline{0.820} & \\underline{0.822} & 0.856 & \\underline{0.862} & 0.855 & 0.878 & \\underline{0.871} & 0.818 & \\textbf{0.798} & 0.800 & 0.833 & 1.116 & 1.099 & \\underline{1.094} & 1.117\\\\\n\\Xhline{2\\arrayrulewidth}\n\\textbf{SER} & \\textbf{0.789} & \\textbf{0.815} & \\textbf{0.810} & \\textbf{0.806} & \\textbf{0.852} & \\textbf{0.833} & \\textbf{0.855} & \\textbf{0.847} & \\textbf{0.785} & \\textbf{0.798} & \\textbf{0.769} & \\textbf{0.784} & \\textbf{1.028} & \\textbf{1.029} & \\textbf{1.039} & \\textbf{1.033}\\\\\n\\bottomrule\n\\end{tabularx}\n\\end{table*}\n\n\\begin{table*}[!htbp]\n\\centering\n\\small\n\\caption{ nDCG@5 ($\\uparrow$) on four target domain dataset. Bold and underline indicate $1^{st}$ and $2^{nd}$ best. }\n\\label{performance_ndcg}\n\\begin{tabularx}{\\textwidth}{@{}c|cccc|cccc|cccc|cccc@{}}\n\\toprule\nTarget domain &  \\multicolumn{4}{c|}{\\underline{Office Product}} & \\multicolumn{4}{c|}{\\underline{Instant Video}} & \\multicolumn{4}{c|}{\\underline{Automotive}} & \\multicolumn{4}{c}{\\underline{Patio Lawn and Garden}} \\\\\nSource domain & Baby & KS & TG & Yelp & Baby & KS & TG & Yelp & Baby & KS & TG & Yelp & Baby & KS & TG & Yelp \\\\\n\\midrule\nPMF &  \\multicolumn{4}{c|}{0.737} & \\multicolumn{4}{c|}{0.759} & \\multicolumn{4}{c|}{0.764} & \\multicolumn{4}{c}{0.770} \\\\\nNeuMF &  \\multicolumn{4}{c|}{0.756} & \\multicolumn{4}{c|}{0.788} & \\multicolumn{4}{c|}{0.781} & \\multicolumn{4}{c}{0.776} \\\\\nDeepCoNN &  \\multicolumn{4}{c|}{0.856} & \\multicolumn{4}{c|}{0.840} & \\multicolumn{4}{c|}{0.816} & \\multicolumn{4}{c}{0.842} \\\\\nNARRE &  \\multicolumn{4}{c|}{0.861} & \\multicolumn{4}{c|}{0.872} & \\multicolumn{4}{c|}{0.851} & \\multicolumn{4}{c}{0.844} \\\\\nAHN &  \\multicolumn{4}{c|}{\\underline{0.874}} & \\multicolumn{4}{c|}{0.879} & \\multicolumn{4}{c|}{0.862} & \\multicolumn{4}{c}{\\underline{0.878}} \\\\\n\\Xhline{2\\arrayrulewidth}\nDANN & 0.843 & 0.847 & 0.840  & 0.829  & 0.851 & 0.849  & 0.846  & 0.835  & 0.844 & 0.858  & 0.836  & 0.831  & 0.830 & 0.823  & 0.818  & 0.812 \\\\\nDAREC & 0.859 & 0.842  & 0.835  & 0.827  & 0.844 & 0.848  & 0.841  & 0.823  & 0.865 & 0.872  & 0.872  & 0.860  & 0.854 & 0.852  & 0.861  & 0.835 \\\\\nDDTCDR & 0.854 & 0.853 & 0.860 & 0.847 & 0.852 & 0.858 & 0.849 & 0.840 & 0.877 & 0.874 & 0.881 & 0.865 & 0.846 & 0.851 & 0.849 & 0.839\\\\\nRC-DFM & 0.875  & 0.871  & \\underline{0.880}  & 0.869  & \\underline{0.890}  & 0.884  & \\underline{0.881}  & 0.879  & 0.884  & 0.895  & \\underline{0.899}  & \\underline{0.902}  & \\underline{0.878}  & 0.873  & 0.871  & \\underline{0.879} \\\\\nCATN & 0.869  & 0.865  & 0.871  & 0.842  &  0.873 & 0.857  & 0.860  & 0.873  & 0.866  & 0.863  & 0.872  & 0.875  & 0.864  & 0.861  & 0.858  & 0.854 \\\\\nMMT & \\underline{0.881} & \\underline{0.874} & 0.870 & \\underline{0.883}  & 0.888 & \\underline{0.885} & 0.876 & \\underline{0.883} & \\underline{0.886} & \\underline{0.896} & 0.892 & 0.877 & 0.867 & 0.869 & 0.871 & 0.871\\\\\n\\Xhline{2\\arrayrulewidth}\n\\textbf{SER} & \\textbf{0.891} & \\textbf{0.885} & \\textbf{0.888} & \\textbf{0.889} & \\textbf{0.896} & \\textbf{0.892} & \\textbf{0.889} & \\textbf{0.895} & \\textbf{0.892} & \\textbf{0.901} & \\textbf{0.908} & \\textbf{0.913} & \\textbf{0.889} & \\textbf{0.882} & \\textbf{0.885} & \\textbf{0.883} \\\\\n\\bottomrule\n\\end{tabularx}\n\\end{table*}\n\n\\textbf{Environmental settings and baselines} Following previous studies, we divide the target domain dataset into three parts: 80 percent for training, 10 percent for validation, and another 10 percent for testing. We repeatedly consume a target domain w.r.t. the size of a source domain for one iteration. Early stopping is applied based on the validation score for 300 iterations. The word embedding is set to 100 with 100 convolution filters of size $\\mathbb{R}^{5 \\times 100}$. \nNow, we introduce state-of-the-art single and cross-domain methods below. \n\n\\textit{Single-Domain Approaches:}\n\\begin{itemize}\n    \\item \\textbf{PMF} \\cite{mnih2008probabilistic} is a classical probabilistic matrix factorization method, which only utilizes rating information. \n    \\item \\textbf{NeuMF} \\cite{he2017neural} combines deep neural networks with a probabilistic model, and use rating information only.\n    \\item \\textbf{DeepCoNN} \\cite{zheng2017joint} leverages review texts for rating prediction. They jointly encode the latent of user and item, respectively. \n    \\item \\textbf{NARRE} \\cite{chen2018neural} maintains the overall architecture of \\textit{DeepCoNN}, while employing attention mechanism. They firstly measure the usefulness of each review using attention scores.\n    \\item \\textbf{AHN} \\cite{dong2020asymmetrical} proposes a hierarchical attention mechanism: from a sentence to review-level attention for a better representation learning. It achieves state-of-the-art performance for a review-based single-domain recommendation.\n\\end{itemize}\n\n\\textit{Cross-Domain Approaches:}\n\\begin{itemize}\n    \\item \\textbf{DANN} \\cite{ganin2016domain} proposes the seminal domain adversarial technique that extracts domain-common features from two different domains. Here, review texts are embedded as 5,000-dimensional feature vectors.\n    \\item \\textbf{DAREC} \\cite{yuan2019darec} assume the same set of users between two domains. For each domain, the high-dimensional rating vectors are mapped to low-dimensional feature vectors using AutoEncoder, followed by a domain discriminator. They extract shareable rating patterns between two domains.\n    \\item \\textbf{DDTCDR} \\cite{li2020ddtcdr} assumes that if two users have similar preferences in a source domain, it should be preserved in a target domain through an orthogonal mapping function.  \n    \\item \\textbf{RC-DFM} \\cite{fu2019deeply} suggest fusing review texts with rating information. With SDAE, they adequately preserve the latent features with rich semantic information. Under our experimental setting, we fine-tune the text convolution layer. \n    \\item \\textbf{CATN} \\cite{zhao2020catn} extracts multiple aspects from reviews. For a knowledge transfer between domains, they assume an aspect correlation matrix with an attention mechanism. \n    \\item \\textbf{MMT} \\cite{krishnan2020transfer} suggest component-wise transfer mechanism, while also preserving domain-specific modules. Here, we assume the text convolution layer as a knowledge transfer module and fine-tune the parameters in a target domain.\n\\end{itemize}\n\n\n\n\\subsection{Results and Discussion (RQ1)}\nFor quantitative analysis, we assume two metrics: Mean Squared Error (MSE) for a rating prediction and normalized Discounted Cumulative Gain (nDCG$@5$) for ranking-based recommendations.\n\n\\textbf{Utilizing additional domain generally enhances recommendation quality} Table \\ref{performance_mse}, \\ref{performance_ndcg} shows the performance of our model and previous methods based on MSE and nDCG@5 score. For single-domain methodologies, it is not surprising that rating-based PMF and NeuMF performed worse than other review-based methods, which indicates the usefulness of textual information. Among review-based methods, NARRE and AHN outperform DeepCoNN with an attention mechanism. Nonetheless, it is noticeable that textual-based CDR (RC-DFM, CATN, MMT) generally outperforms single-domain methods under our experimental scenario.\n\n\\textbf{The quality of CDR can be degraded w.r.t. domain discrepancy} \nTo test a domain discrepancy, we adopt DANN and MMT, where they transfer knowledge independent of user overlapping. For DANN, the result varies significantly w.r.t. the selection of source domain. Rather, MMT secures stability through domain-specific information (rating).\nAn important thing is that a CDR does not always achieve the best performance.\nIn the case of \\textit{Patio Lawn and Garden}, AHN shows the best result among all baselines. It indicates that utilizing additional domains without debilitating noises can degrade the recommendation quality. Comparatively, SER manages to achieve stable performance, where the domain disentanglement alleviates noises of the auxiliary domain. \n\n\\textbf{Instead of overlapping users, review-based knowledge transfer further improves the recommendation quality} \nFor example, RC-DFM integrates textual data with rating information, which significantly outperforms DAREC and DDTCDR. \nHowever, the problem is that all these methods require duplicate users for a knowledge transfer. This can be quite restrictive for the selection of a source domain, and some argued \\cite{kang2019semi} that user-based knowledge transfer has a limited impact. Even excluding duplicate users (selecting $Yelp$), their performance varies quite insignificant (please refer to \\cite{fu2019deeply} for more details) compared to the other methods (CATN, MMT, SER). Comparatively, MMT and SER (review-based transfer) improve their score even by selecting \\textit{Yelp} as a source, which fairly is different from \\textit{Amazon} dataset.\nThough CATN utilizes auxiliary reviews of another user, does not show outstanding performance in our experiments. Further, MMT shows lower performance than SER because of the limitation of transfer learning and the absence of review disentanglement.\nUsing the above results, we show that SER surpasses the SOTA methods without duplicate users or contexts.\n\n\n\\subsection{\\textbf{Analysis of Domain Discriminator (RQ2)}} \\label{dd_analysis}\nIn this section, we compare state-of-the-art the domain disentanglement strategy \\cite{peng2019domain} with our model, while numerous optimization methods can be considered \\cite{gretton2005measuring,li2019learning,bousmalis2016domain}. We assume two different models; (1) $SER_{MI}$ minimizes mutual information \\cite{peng2019domain}, while other modules remain the same as $SER$, and (2) our original $SER_{SA}$. To implement $SER_{MI}$, we utilize the publicly available source \\textit{code}\\footnote{https:\/\/github.com\/VisionLearningGroup\/DAL}. Specifically, we train them to minimize the absolute value of estimated MI (please refer to Equation \\ref{mi_loss}).\n\n\n\\begin{figure}\n  \\vspace{-2mm}\n\\includegraphics[width=0.49\\textwidth]{assets\/domain_performance_plot.png}\n  \\caption{To show the influence of domain loss function, we describe nDCG@5 ($\\uparrow$) of two SER variants; SER with MI minimization $SER_{MI}$, and our proposed model $SER_{SA}$}\n  \\label{da.per}\n\\end{figure}\n\n\\begin{figure}\n  \\hspace{-3mm}\n \\includegraphics[width=0.48\\textwidth]{assets\/domain_loss_plot.png}\n  \\caption{For two SER variants; $SER_{MI}$ and $SER_{SA}$, we plot their domain losses \\textit{w.r.t.} training epoch}\n  \\label{da.loss}\n\\end{figure}\n\n\\textbf{Analysis of the recommendation quality \\textit{w.r.t.} domain loss} To scrutinize how the domain loss affects the overall performance, we measure the nDCG@5 of two variants using four target domain data, which is illustrated in Figure \\ref{da.per}.\nHere, we notice that our $SER_{SA}$ slightly outperforms $SER_{MI}$ for all CDR scenarios. We insist that these results are closely related to domain losses. In Figure \\ref{da.loss}, using $Baby$ as a source with two target domain datasets (\\textit{Office Products, Automotive}), we plot the variation of domain losses for $SER_{MI}$ and $SER_{SA}$ \\textit{w.r.t.} training epoch. Specifically, (1) for domain-common loss, both of them adopts adversarial training of Equation \\ref{d_inv_loss}, $\\mathcal{L}_{com}^{MI}=\\mathcal{L}_{com}^{SA}$=$\\mathcal{L}_{com}^s+\\mathcal{L}_{com}^t$. (2) instead, their domain-specific loss ($\\mathcal{L}_{spe}^{MI}$, $\\mathcal{L}_{spe}^{SA}$) differs. $SER_{MI}$ adopts Equation \\ref{mi_loss}, while $SER_{SA}$ utilizes Equation \\ref{d_spe_loss}.\nReferring Figure \\ref{da.loss}, for $SER_{MI}$, $\\mathcal{L}_{com}^{MI}$ (red line) rarely changes, while $\\mathcal{L}_{spe}^{MI}$ (green) converges so fast. Comparatively, $\\mathcal{L}_{com}^{SA}$ and $\\mathcal{L}_{spe}^{SA}$ slowly converges, consistently pushing three FEs for domain disentanglement.\nThis result indicates that integrating the MI estimator with a domain discriminator not only secures robustness but also achieves better performance.\n\n\\begin{figure}\n  \\hspace{-2mm}\n  \\includegraphics[width=0.49\\textwidth]{assets\/da_wda.png}\n  \\caption{Visualization of disentangled representations for two SER variants; $SER_{MI}$ and $SER_{SA}$. From the source and target domains, we randomly sample user and item pairs, and plotted their reviews using t-SNE}\n  \\label{da.viz}\n  \\vspace{-3mm}\n\\end{figure}\n\n\\textbf{Visualization of disentangled representations}\nIn Figure \\ref{da.viz}, we visualize the disentangled representations of two \\textit{SER} variants. For an experiment, we pre-trained our model using \\textit{Baby} and \\textit{Patio Lawn and Garden} as source and target domain data.\nHere, we randomly select one thousand user-item pairs, and extract four types of features $O^s_{spe},O^s_{com},O^t_{com},O^t_{spe}$ from their aggregated reviews. The visualization is conducted using t-distributed Stochastic Neighbor Embedding (t-SNE), which is called a nonlinear dimensionality reduction method for high dimensional representations. In detail, t-SNE has shown to reflect the similarity between data points, where we set the perplexity as 30.\n\nThrough Figure \\ref{da.viz}-a and b, we can see that both methods well capture domain-common knowledge ($O^s_{com},O^t_{com}$ are overlapped each other). Further, two methods seem to be well disentangle domain-specific features ($O^s_{spe},O^t_{spe}$) from domain-common outputs.\nHowever, for a disentanglement between domain-specific features ($O^s_{spe}$ $\\leftrightarrow$ $O^t_{spe}$), $SER_{MI}$ has shown to implicate some defects. Firstly, in Figure \\ref{da.viz}-a, $O^s_{spe}$ (cluster of green points) are positioned between $O^t_{spe}$ (cluster of blue points). As mentioned above, t-SNE preserves a point-wise distance after dimensionality reduction, but the clusters derived from $SER_{MI}$ fall apart from each other. Secondly, some points appear out of their original cluster (marked with a circle). We argue that the objective function of $SER_{MI}$ only minimizes MI of each domain independently while disregarding discrimination between domain-specific features. \nComparatively, in Figure \\ref{da.viz}-b, we notice that $SER_{SA}$ well captures domain-specific knowledge, where the outputs are included in a dense region without outliers.\n\n\\subsection{Ablation Study (RQ3)}\nTo test the significance of domain-aware feature extractor and encoding network, we assume three variants: (1) \\textit{SER w\/o DD}: excluding domain discriminator, (2) \\textit{SER w\/o EN}: excluding encoding network, and (3) our original \\textit{SER}. Like previous studies, we conduct experiments under four target domain data.\nIn Figure \\ref{Fig.module}, for each domain, we plot the min, median, and max value of MSE scores under 5 iterations. We do not assume a model excluding both components, which is identical to the vanilla text analysis model \\cite{zheng2017joint}.\n\n\\begin{figure}\n\\hspace{-3mm}\n    \\includegraphics[width=0.49\\textwidth]{assets\/ser_dd_en.png}\n  \\caption{Ablation study (RQ3) of three SER variants. We adopt MSE ($\\downarrow$) score for a comparison}\n  \\label{Fig.module}\n\\end{figure}\n\n\\textbf{Domain-aware feature extractor is fundamental for debilitating noises} We discuss \\textit{SER w\/o DD} (blue lines), removing domain discriminator from SER. In Figure \\ref{Fig.module}, we can see that the performance of \\textit{SER w\/o DD} shows higher variance compared to \\textit{SER w\/o EN} (green lines), suggesting the domain discrepancy is critical for the overall performance in a target domain (the column-wise width denotes the variance of the results). For example, in Figure \\ref{Fig.module}-b (\\textit{Instant Video}), we notice that the adoption of \\textit{Yelp} as a source domain improves the performance, whereas utilizing \\textit{Toys and Games} can degrade the recommendation quality.\n\n\\textbf{Encoding network contributes to the recommendation quality significantly} Compared to \\textit{SER w\/o DD}, \\textit{SER w\/o EN} shows relatively stable results, which can debilitate noises from a source domain. Though the domain-aware feature extractor effectively controls the mismatches between two different domains, we notice that excluding the encoding layer is critical for the overall performance. Specifically, for all datasets, it is notable that \\textit{SER w\/o DD} slightly outperforms \\textit{SER w\/o EN}.\n\n\\subsection{Case Study (RQ4)}\nIn this section, we scrutinize the interpretability of SER through a real-world dataset $Amazon$. In Figure \\ref{case_study}, using \\textit{Automotive} as a target data, we describe the nDCG@5 of three methods; SER w\/o domain-specific features $O_{spe}$, SER w\/o domain-common features $O_{com}$, and our original $SER$. As can be seen, excluding a single type of information downgrades the performance. Here, we notice that domain-specific information contributes slightly more to the overall quality since $SER$ $w\/o$ $O_{com}$ outperforms $SER$ $w\/o$ $O_{spe}$ $1.2\\%$ on average. We now scrutinize this phenomenon below.\n\nIn Figure \\ref{eval_ex}, we pre-trained a model using $Baby$ for $Automotive$. Then, we apply common and target FEs for each review, retrieving domain-common $O_{com}$, and domain-specific $O_{spe}$ outputs. We \\textit{italicized} the most similar words (\\textit{e.g., cosine similarity}) of domain-common feature $O_{com}$, and colorized most similar words of domain-specific ones $O_{spe}$. As can be seen, the italicized words further contain semantic meaning (\\textit{quality, price, and money}), while the colorized one contains the user's own preference (\\textit{cable and battery}). Throughout these results, we can infer that SER well disentangles domain knowledge since two features are complimentary, while none of them are negligible for the improvement of overall quality.\n\n\\begin{figure}\n  \\includegraphics[width=0.48\\textwidth]{assets\/case_plot.png}\n  \\caption{Case study (RQ4) of three SER variants. We adopt nDCG@5 ($\\uparrow$) score for a comparison} \n  \\label{case_study}\n\\end{figure}\n\n\\begin{figure}\n  \\includegraphics[width=0.48\\textwidth]{assets\/evaluation_example.png}\n  \\caption{Explainability analysis for user and item reviews}\n  \\label{eval_ex}\n  \\vspace{-3mm}\n\\end{figure}\n\n\n\\section{Conclusion}\nIn this paper, we propose a novel method of utilizing review texts in multiple domains without requiring overlapping users or contexts. Our optimization strategies for domain disentanglement well achieve the knowledge transfer between domains, while also securing robustness. Further, we successfully extend a transfer network from a single to multiple domains, which can infer the latent of an individual review regarding domain-aware knowledge. The extensive experiments and ablation studies demonstrate the superiority and advantages of our method for the CDR scenario.\n\n\\subsubsection*{\\normalfont{\\textbf{Acknowledgments}}}\nThis work was supported by the National Research Foundation of Korea (NRF) (No. 2016R1A5A1012966, No. 2020R1A2C110168713),  Institute of Information \\& communications Technology Planning \\& Evaluation (IITP) (No. 2021-0-02068 Artificial Intelligence Innovation Hub, No. RS-2022-00156287 Innovative Human Resource Development for Local Intellectualization support program) grant funded by the Korea government (MSIT).\n\n\n\\bibliographystyle{ACM-Reference-Format}\n\\balance\n\n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\verb|acmsmall|}: The default journal template style.\n\\item {\\verb|acmlarge|}: Used by JOCCH and TAP.\n\\item {\\verb|acmtog|}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\verb|acmconf|}: The default proceedings template style.\n\\item{\\verb|sigchi|}: Used for SIGCHI conference articles.\n\\item{\\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\verb|sigplan|}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\verb|anonymous,review|}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\verb|\\acmSubmissionID| command to print the\n  submission's unique ID on each page of the work.\n\\item{\\verb|authorversion|}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\verb|screen|}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. Multiple authors may share one affiliation. Authors'\nnames should not be abbreviated; use full first names wherever\npossible. Include authors' e-mail addresses whenever possible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\verb|acmsmall|}: The default journal template style.\n\\item {\\verb|acmlarge|}: Used by JOCCH and TAP.\n\\item {\\verb|acmtog|}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\verb|acmconf|}: The default proceedings template style.\n\\item{\\verb|sigchi|}: Used for SIGCHI conference articles.\n\\item{\\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\verb|sigplan|}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\verb|anonymous,review|}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\verb|\\acmSubmissionID| command to print the\n  submission's unique ID on each page of the work.\n\\item{\\verb|authorversion|}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\verb|screen|}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. Multiple authors may share one affiliation. Authors'\nnames should not be abbreviated; use full first names wherever\npossible. Include authors' e-mail addresses whenever possible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\verb|acmsmall|}: The default journal template style.\n\\item {\\verb|acmlarge|}: Used by JOCCH and TAP.\n\\item {\\verb|acmtog|}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\verb|acmconf|}: The default proceedings template style.\n\\item{\\verb|sigchi|}: Used for SIGCHI conference articles.\n\\item{\\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\verb|sigplan|}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\verb|anonymous,review|}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\verb|\\acmSubmissionID| command to print the\n  submission's unique ID on each page of the work.\n\\item{\\verb|authorversion|}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\verb|screen|}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. Multiple authors may share one affiliation. Authors'\nnames should not be abbreviated; use full first names wherever\npossible. Include authors' e-mail addresses whenever possible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces).  {\\bfseries Figure descriptions\n  should not repeat the figure caption \u2013 their purpose is to capture\n  important information that is not already provided in the caption or\n  the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph.  For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\nThe rapid evolution of the threat landscape in recent years has caused many organisations to reassess the risks associated with their digital infrastructure \\cite{eling2021cyber}. For example, during the 2020-2021 financial year, almost 500 ransomware cybercrimes were reported in Australia, an increase of 15\\% compared to the previous year \\cite {ACSC-CT-report-2020-21}. As a result, many organisations have started integrating Cyber Threat Intelligence (CTI) into traditional Cyber Security Risks Management (CSRM) frameworks to try and achieve a more proactive security posture \\cite{kure2019cyber}. CTI can be defined as the collection and synthesis of evidence-based knowledge about actors, threats, and vulnerabilities of harmful events occurring in cyberspace. Subsequently, the contents of CTI are highly variable in nature and can range from Indicators of Compromise (IoC) to sophisticated information detailing an attacker's Techniques, Tactics and Procedures (TTPs) \\cite{wagner2019cyber}.\n\nGiven that integration of CTI data into traditional CSRM frameworks can be utilised to provide organisations with a more proactive defense posture, inter organisational CTI sharing holds additional benefits \\cite{kure2019cyber}. In the case where one organisation (e.g., a CTI producer) has been attacked, sharing CTI has the potential to allow other organisations (e.g., a CTI consumer) to proactively implement mitigation strategies \\cite{wagner2019cyber}. Note, we refer to CTI producers as the entities who share CTI, while CTI consumers are the entities that use shared CTI. As a result, in recent years several \\textit{Open-source\/Vendor-created} CTI sharing platforms have been developed.\n\n\n\n\n\n        \n        \n        \n        \n         \n         \n         \n         \n         \n         \n        \n        \n\n        \n        \n        \n        \n         \n         \n         \n         \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{Motivation and Problem Statement}\nIn many cases, sharing CTI data in an uncontrolled way can result in CTI producers sustaining reputational damage.\nSubsequently, for cases where CTI contains sensitive data, trust-based data sharing has been highlighted as a possible solution \\cite{wagner2019cyber}. Trust-based CTI sharing ensures that potentially sensitive CTI data is shared with consumers with which pre-established or developed trust exists. Moreover, trust issues are significant when sharing takes place in large-scale industrial systems, e.g., Industry 4.0~\\cite{suhail2022towards}\n\nWhile trust-based sharing has the potential to reduce sharing risks, CTI producers require more fine-grained control over what data they share. For instance, in \\cite{wagner2019cyber}, it is highlighted that organisations are likely to differ in their perceptions of what data is considered sensitive.\nAs a result, CTI sharing platforms need to provide organisations with the flexibility to share CTI in a way that caters to this variation. Organisations are also likely to differ in their perception of how trust-based sharing should be facilitated. For instance, an organisation A may determine their trust in another organisation B based on B's attribute or using identity-based policies specific to B. Subsequently, constraining organisations to share in a fixed way, either through  share or deny or policies, or group-based policies that share or deny CTI with specific groups, does not provide the granularity required for versatile CTI sharing. We propose a framework that allows CTI producers to share sensitive CTI with consumers in a \\textit{differential} way. We refer to differential as the ability of CTI producers to control the amount of information exchanged with CTI consumers at a fine-grained level (more discussion in Section~\\ref{differential}). \n\nFurthermore, CTI consumers should have the ability to verify requested information to ensure that the supplied intelligence has not be tampered with or altered by the CTI producer. Subsequently, an efficient verification mechanism which simultaneously preserves the privacy in data sharing is required.\nThus, we argue that CTI sharing must be trusted, performed in a verifiable way, and without compromising the privacy of sensitive CTI data. \n\n\n\n\n\nSeveral blockchain-based CTI sharing proposals have been proposed \\cite{allouche2021trade} in recent years. Blockchain has a number of salient properties that can be leveraged to facilitate CTI sharing. For example, among others, blockchain's consensus, auditability, and immutability properties allow organisations to interact with one another in a distributed way without the need of a trusted third party. Another advantage of using blockchain to facilitate CTI sharing is verifiability. Considering that the information stored on the blockchain is immutable, CTI consumers are able to validate the integrity of received data and, therefore, can be sure it has not been tampered with or changed~\\cite{dai2019blockchain}.\n\n\n\n\n\n\n\\subsection{Contributions}\n\\label{contributions}\nExisting proposals only allow CTI producers to share intelligence in fixed way using a platform defined trust-based policy scheme. While these approaches do provide verifiable and trust-based sharing, constraining CTI producers to share in a fixed way, e.g. binary or group-based sharing, has the potential to limit the effectiveness of sharing between organisations. Subsequently, in our work we define a blockchain-based framework which provides CTI producers with the ability to share CTI \\textit{differentially} using  policies\/metrics which they define. Moreover, we further demonstrate how CTI producers can share in this way without compromising the verifiability of the intelligence from the CTI consumers perspective. To the best of our knowledge, our work is the first to use blockchain for trusted, verifiable, and differential CTI sharing. The major contributions of this paper are: \n\n\n\n\\begin{itemize}\n   \n    \n   \n    \n    \\item We propose the concept of \\textit{differential sharing} within the CTI sharing context.\n    \n    \\item We detail how \\textit{differential sharing} can provide more granuality compared to existing solutions by allowing data segmentation and \\textit{pluggable} trust-based policy\/metric schemes, without compromising verifiability through the use of two integrity hash schemes.  \n    \n   \n    \n   \n    \n   \n    \n   \n    \n    \\item We discuss a detailed design for the proposed CTI sharing framework, its system functionality, and communications among the various components.\n    \n    \\item We demonstrate and evaluate a detailed proof of concept implementation using Ethereum private blockchain. The experimental results indicate that the proposed framework is feasible and creates insignificant overheads compared to a baseline sharing scenario\n    \n    \\item We present a privacy and trust analysis of the proposed framework.  \n    \n\\end{itemize}\n\nNext, in Section~\\ref{related-work}, we discuss the related work. In Section~\\ref{proposed-architecture}, we introduce the proposed CTI sharing framework. Section~\\ref{sytem-design} describes and evaluates the proof of concept implementation of the proposed framework and present the key results. Moreover, this section also analysis the privacy and trust of the proposed framework. Finally, Section~\\ref{conclusion} concludes the paper and highlights the future work. \n\n\n\n\n\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Available blockchain-based CTI sharing proposals and their comparison with our work. \n    }\n    \\label{tab:realted_work}\n    \\scalebox{0.765}{\n    \\begin{tabular}{ccccccc}\n        \\toprule\n            {Ref.} & {Differential} & {Sensitive} & {Producer-Defined} & {Verifiable} & {Blockchain}  \\\\\n            {} & {Sharing} & {Data Grouping} & {Trust Scheme} & {Sharing} & {Platform} \\\\\n        \\toprule\n        \n        \\cite{shi2022threat} & \\xmark &  \\xmark & \\xmark & \\checkmark & CITA \\\\ [0.5ex]\n        \n        \\cite{badsha2020blocynfo} & \\xmark & \\xmark & \\xmark & \\checkmark & Ethereum \\\\ [0.5ex]\n         \n         \\cite{allouche2021trade} & \\xmark &  \\xmark & \\xmark & \\checkmark & Ethereum \\\\ [0.5ex]\n         \n         \\cite{preuveneers2020distributed} & \\xmark & \\xmark & \\xmark & \\checkmark & Hyperledger   \\\\ [0.5ex]\n         \n         \\cite{nguyen2022blockchain} & \\xmark & \\xmark & \\xmark & \\checkmark & Hyperledger \\\\ [0.5ex]\n         \n         \\cite{moubarak2021dissemination} & \\xmark & \\xmark & \\xmark & \\checkmark & Hyperledger \\\\ [0.5ex]\n         \n         \\cite{homan2019new} & \\xmark & \\xmark & \\xmark & \\checkmark & Hyperledger \\\\ [0.5ex]\n         \n        \\textbf{[Our Work]} & \\checkmark & \\checkmark & \\checkmark & \\checkmark & Ethereum \\\\ \n        \n        \\bottomrule\n        \n    \\end{tabular}\n    }\n\\end{table}\n\n\\section{Related Work}\n\\label{related-work}\nIn Table~\\ref{tab:realted_work}, we provide a list of the available related works on blockchain-based CTI sharing and their comparison with our work. For example, \nThe authors in \\cite{shi2022threat} developed a blockchain-based CTI sharing platform using the CITA model. This model provides CTI producers with the ability to store sensitive data off-chain. When sharing this CTI data with authorised CTI consumers, the authors propose using point-to-point communication to ensure fine-grained access control over which CTI consumers receive the sensitive data. To ensure verifiability, a hash of the sensitive data is stored on-chain such that CTI consumers can validate the integrity of data received off-chain. While~\\cite{shi2022threat} provides  trust-based verifiable sharing, CTI producers are constrained to share CTI data with consumers in a coarse-grained and binary way.\n\n`BloCyNfo-Share' is a blockchain-based CTI sharing framework proposed in \\cite{badsha2020blocynfo}. It utilises proxy re-encryption and attribute-based encryption to achieve fine-grained access control. When sharing, CTI producers provide an encrypted copy of the data to a cloud server. When a CTI consumer requests access to this CTI, the CTI producer generates a re-encryption key and adds it to a smart contract. This re-encryption key is subsequently used by the cloud server to generate a new cipher text. The resulting cipher text is subsequently sent to the CTI consumer, who can decrypt it given they satisfy the policy specified by the CTI producer.\nSimilar to \\cite{shi2022threat}, in this framework, CTI producers are limited to making binary sharing decisions. Thus, unlike our approach, proposals \\cite{badsha2020blocynfo} and  \\cite{shi2022threat} do not provide differential sharing.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale=0.55]{share-type.pdf}\n    \\caption{Illustration of our differential CTI sharing compared to traditional binary and group-based CTI sharing approaches.}\n    \\label{fig:turst-group-df}\n\\end{figure*}\n\nIn \\cite{allouche2021trade}, the blockchain-based CTI sharing platform TRusted Anonymous Data Exchange (TRADE) is proposed. To facilitate the exchange of CTI data, the TRADE platform proposes the use of off-chain data storage, policy-based access control, on-chain policy validation and off-chain data exchange using Trusted Automated Exchange of Intelligence Information (TAXII)~\\cite{jordan_varner_2021}. When a CTI producer wants to share,\nthey add an integrity hash and access policy (e.g., eXtensible Access Control Markup Language (XACML) policy) to the blockchain. \nHowever, TRADE again limits CTI producers to binary sharing decisions. \n\nIn contrast to \\cite{badsha2020blocynfo}, proposal \\cite{preuveneers2020distributed} utilises the existing CTI sharing platform MISP \\cite{wagner2016misp} to facilitate the exchange of CTI between organisations. To ensure that the privacy of sensitive data is maintained, a \\textit{Trustworthy API for Threat Intelligence Sharing} (TATIS) Reverse Proxy is used. This proxy uses ciphertext attribute-based encryption to restrict the access of sensitive data to trusted organisations. To provide verifiability, CTI producers store a hash of the original data on-chain. However, like the above proposals, \\cite{preuveneers2020distributed} also limits producers to binary sharing decisions.\n\nIn a few other proposals, e.g., \\cite{nguyen2022blockchain},  \\cite{moubarak2021dissemination}, and \\cite{homan2019new}, a group-based approach for sensitive CTI sharing is utilised. Instead of limiting CTI producers to binary sharing decisions, a group-based approach allows CTI producers and consumers to exchange sensitive CTI as part of a trusted group. Moreover, these proposals utilise the Traffic Light Protocol (TLP). Instead of limiting CTI producers to share in a binary way, more or less sensitive data can be shared using a layered approach. In this case, layered refers to a grouping of organisations into trust groups (e.g., red, amber, green, and white), in which CTI of varying sensitivity can be shared.\nHowever, the major limitation of a group-based approach is that it assumes organisations trust each other in a pre-defined and fixed way. While these approaches extend sharing beyond binary decisions, we argue they still lack the granularity required for more effective CTI sharing. Subsequently, our proposal allows CTI producers to share differentially, and therefore supports dynamic and time-varying trust relationships. \n\n\nIn summary, the existing blockchain-based CTI sharing proposals limit a CTI producer's ability to share sensitive CTI in a coarse-grained manner, either based on binary share\/deny policies or as part of a pre-defined trust group. In most cases, the existing solutions provide trust-based policies\/metrics that limit the dissemination of sensitive data to trusted organisations, but they fail to deliver more granular sharing. Given that a more fine-grained sharing mechanism is required to facilitate effective CTI sharing, we propose a framework with the concept of differential sharing discussed in the next section.\n\n\n\\section{Proposed Framework}\n\\label{proposed-architecture}\n\n\n\nIn this section, we present the proposed blockchain-based CTI sharing framework. First, we discuss the concept of differential CTI sharing in detail. Then we present the various components of the architecture. We also discuss the system functionality and communication among these components. \n\n\\subsection{Differential Sharing}\n\\label{differential}\nIn Section~\\ref{introduction}, we highlight that organisations are likely to differ in their perceptions of data sensitivity and what trust-based policy\/metrics should be used to determine a CTI consumers access. We propose the concept of \\textit{differential} CTI sharing, which aims to address this challenge.  Figure~\\ref{fig:turst-group-df} compares differential CTI sharing with binary and trusted group-based CTI sharing. From this comparison, we highlight that differential sharing can tailor the CTI data to be shared with each CTI consumer. As a result, the proposed \\textit{differential} sharing approach does not restrict sharing to only highly trusted CTI consumers, and therefore, allows other CTI consumers to benefit from CTI data they otherwise would not have received.\n\nGiven that CTI data contains many attributes, e.g., \\textit{chained objects} in Structured Threat Information Expression (STIX), their individual sensitivity can be evaluated by the CTI producer. Furthermore, as these attributes are likely to vary in their sensitivity, attributes of similar sensitivities can be grouped together. For example, \\textit{chain-objects} of type vulnerability often contain specific software versions that could be valuable information to potential attacks, while \\textit{chain-objects} of type indicator often only contain IoC. As a result, based on the information contained within specific STIX \\textit{chain-objects}, Fig.~\\ref{fig:stix-cti} demonstrates how CTI attributes can be broken down into a variable number of sensitive data groups. \n\n\n\\begin{figure} [ht]\n\\includegraphics[scale=0.69]{tbdg.pdf}\n\\caption{Sensitive data groups based on CTI attributes.}\n\\label{fig:stix-cti}\n\\end{figure}\n\nOnce the CTI has been segmented using the above method, a subset of these sensitive data groups can be given to CTI consumers based on a single or set of trust-based policies\/metrics (e.g., XACML policy). By allowing CTI producers to segment CTI data into groups based on a sensitivity assessment, a variable proportion of these groups can be shared with each CTI consumer. To ensure verifiability of the received data, we propose the use of two group-based hashing approaches: single hash and multi-hash. Given that CTI consumers receive a distinct number of the sensitive data groups, a set of integrity hashes constructed using either of these two approaches is required. In Fig.~\\ref{fig:intgrity-hashes} we illustrate how these group-based hashes can be constructed using a single and multi hash approach.\n\n\n\\begin{figure} [t]\n\\centering\n\\includegraphics[scale=0.82]{ih.pdf}\n\\caption{Proposed single hash (a) and multi hash (b) approaches. Note, dotted boxes represent integrity hashes. ($N_i$ = One-time random nonce).}\n\\label{fig:intgrity-hashes}\n\\end{figure}\n\nSharing CTI using the proposed \\textit{differential} approach gives CTI producers the ability to share with more granularity compared to traditional solutions, e.g., binary and group-based. We further note that greater granularity is achieved in two ways: (i) CTI producers are given the ability to define what subset of the CTI data is shared with an individual or group of CTI consumers, and (ii) CTI producers are also able to change how they segment CTI data and the policies\/metrics they use to evaluate trust each time they share.\n\n\n\n\n    \n    \n    \n    \n    \n    \n    \n\n\n\n\n\\subsection{Architectural Components}\n\\label{components}\nIn Fig.~\\ref{fig:architecture}, we illustrate the proposed blockchain-based CTI sharing framework. It contains the following components: \n\n\n\n\n\\begin{itemize}\n    \\item \\textbf{Organisation:} An entity that participates in CTI sharing. Organisations can assume the role of a producer and\/or consumer. Organisations can participate in both roles over time. However, we assume that an organisation performs only one role at a time for a specific interaction.\n    \n    \\item \\textbf{Request Manager:} A component within an organisation responsible for handling and submitting access requests of CTI data. The request manager role is primarily broken into two tasks, \\textit{submitting} an access request and \\textit{processing} an access request. They are as follows:  \n    \\begin{enumerate}\n        \\item Submitting an access request occurs when new CTI is made available. Given that another organisation has added a CTI, the request manager submits an access request to the blockchain. \n        \n        \\item When an organisation adds a new CTI, the organisation's request manager is responsible for processing the access requests added by other organisations.\n    \\end{enumerate}\n    \n   \n    \n   \n    \n   \n    \n    \\item \\textbf{Data Manager:} A component inside of an organisation responsible for storing sensitive CTI data and their corresponding access policies. This component is also responsible for indicating to other organisations that new CTI is available. Subsequently, it adds content on and off-chain when an organisation acts as a CTI producer\n    \n    \\item \\textbf{Access Manager:} A component inside an organisation responsible for determining what portion of requested CTI should be shared with a CTI consumer. Using the access policies stored by the data manager and credentials (e.g., identity-proof) provided by the CTI consumer, the access manager determines what portion of the CTI data should be given to a particular CTI consumer.\n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n    \n   \n    \n   \n    \n   \n    \n    \n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.755]{whole.pdf}\n\\caption{Proposed blockchain-based CTI sharing framework.}\n\\label{fig:architecture}\n\\end{figure}\n\n  \n    \\item \\textbf{Blockchain:} A distributed digital ledger of information that is shared, duplicated and distributed across the entire network of computer systems. It can be seen as a shared database that stores a series of digital records using cryptographic operations (called hash).\n    Ethereum blockchain is used for our proof-of-concept implementation. \n    \n    \\item \\textbf{Smart Contract:} \n    Self-executable programs stored on a blockchain. We introduce a \\textit{data storage} smart contract in our framework. It is composed of three functions: (i) \\textit{share} function provided by the smart contract is used by a CTI producer to indicate the availability of CTI, (ii) \\textit{request} function provided by the smart contract is used by a CTI consumer to request access to sensitive CTI data, and (iii) \\textit{response} function provided by the smart contract is used by a CTI producer to respond to a request submitted to the \\textit{request} function.\n    \n   \n    \n   \n    \n    \\item \\textbf{Interplanetary File System (IPFS):} A peer-to-peer hypermedia protocol that is designed to distribute the storage of data across a network of computers. IPFS provides highly available and consistent data storage of large files that otherwise would be expensive to store using a traditional blockchain network (e.g., Ethereum). IPFS is utilised to store off-chain data associated with the sharing process \\cite{benet2014ipfs}.\n   \n    \n   \n    \n   \n    \n   \n\\end{itemize}\n\n\n\n\n\n\n\n\\subsection{System Functionality}\n\\label{system-functionality}\n\nThis section provides an overview of the proposed CTI sharing framework's functionality. The framework follows five main phases to share and consume CTI. They are:\n\n\n\\underline{{\\textbf{\\textit{Share CTI:}}}} The objective of this phase is for the CTI producer to create and share all of the data\nrequired to facilitate the exchange of CTI. Given that an organisation has produced CTI data, a sensitivity analysis is conducted by the CTI producer. This analysis identifies sensitive CTI attributes and segments them into a number of sensitivity groups (cf. Fig.~\\ref{fig:stix-cti}). \n\n\n\n\n\n\nOnce the sensitive data groups have been defined, a set of \\textit{integrity hashes} are created. To construct these hashes, the single or multi hash approach (cf. Fig.~\\ref{fig:intgrity-hashes}) can be used by the CTI producer. For both of the proposed approaches, the sensitive data groups and a set of one-time random nonce's are used to construct the integrity hashes. In this case, the set of one-time random nonce's are used to protect the privacy of sensitive data (more discussion in Section~\\ref{privacy-trust}). To generate these integrity hashes, a single or set of data groups are hashed using these nonce's in sequence. \n\n\n\n\nAs discussed in Section~\\ref{related-work}, several trust-based metric\/policy schemes have been presented by existing works. Subsequently, our framework allows CTI producers to plug-in a scheme dynamically. For example, an organisation might assign an individual XACML policy for each of the sensitive data groups.\n\n\n\n\nOnce the above process has been completed, the sensitive data groups and access policy are stored securely off-chain by the CTI producer. Moreover the non-sensitive CTI data group, set of integrity hashes and access policy are added to IPFS, while its associated content identifier (CID) (IPFS File Pointer) and CTI metadata are added to the \\textit{data storage} smart contract.\n\n\n\n\\underline{{\\textbf{\\textit{Request CTI:}}}} During this phase, a CTI consumer requests access to a particular CTI of interest. An access request is submitted via a \\textit{data storage} smart contract. This smart contract is the same one used during the \\textit{Share CTI} phase. The access request indicates which CTI data the CTI consumer wishes to access and contains the credentials (e.g., identity-proof) required to meet the associated access policy.\n\nTo ensure that these credentials are shared in a privacy-preserving way, they are encrypted using the CTI producer's public key before being added to IPFS. By encrypting their credentials, a CTI consumer can ensure that only the intended CTI producer can access these them. Once the access request has been submitted, the CTI producer receives the request, accesses the encrypted credentials from IPFS using the provided IPFS CID, and decrypts them using their private key. \n\n\\underline{{\\textbf{\\textit{Policy Evaluation:}}}} During this phase, the CTI producer processes the access request submitted by the CTI consumer. To determine the sensitive data groups that the CTI consumer should have access to, the set of credentials provided during the \\textit{Request CTI} phase and the access policy associated with the requested CTI is used. Once the provided credentials are validated, the CTI producer evaluates the access policy to determine which subset of the sensitive data groups the requesting CTI consumer should receive.\n\n\n\n\\underline{{\\textbf{\\textit{Response CTI:}}}} During this phase, the CTI producer provides the CTI consumer with the sensitive data groups they are able to access. Note, in the \\textit{Policy Evaluation} phase above, the CTI producer determines which subset of the sensitive data groups a CTI consumer is able to access. To ensure that the CTI consumer can validate the integrity of the provided sensitive data groups, the corresponding one-time random nonce values are also included in the response. To ensure that the response data is exchanged in a privacy-preserving way, it is encrypted using the CTI consumer's public key. Once encrypted, the response data is added to IPFS and the associated IPFS CID is added to the \\textit{data storage} smart contract by the CTI producer. The CTI consumer can then access the encrypted response data using the provided IPFS CID. To decrypt the response data, the CTI consumers uses their private key.\n\n\n\n\n\\underline{{\\textbf{\\textit{Validate CTI:}}}} During this phase, the CTI consumer validates the integrity of the received sensitive data. As part of the \\textit{Share CTI} phase, the CTI producer added a set of integrity hashes to IPFS. Subsequently, the CTI consumer can access these integrity hashes and use them to validate the sensitive CTI data received during the \\textit{Response CTI} phase.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=.48]{id.pdf}\n\\caption{Communication between various components of the proposed CTI sharing framework (on-chain interactionss).}\n\\label{fig:interactions}\n\\end{figure}\n\n\\subsection{Communications}\n\\label{communication}\n\n\nIn Fig.~\\ref{fig:interactions}, we illustrate the communication flow within our CTI framework. Note that for simplicity, we contemplate the interactions of various components with the blockchain. Therefore, only on-chain interactions are shown, and interactions with IPFS are excluded. The steps are as follows:\n\n\\textbf{Step 0:} CTI producer completes the initial setup steps. Firstly, the CTI data is segmented into a variable number of sensitivity data groups (cf. Fig.\\ref{fig:stix-cti}). The sensitive data groups and a set of one-time random nonces are used to construct a set of integrity hashes using either of the proposed methods (cf. Fig.\\ref{fig:intgrity-hashes}). The non sensitive data group, CTI producer-defined access policy, and integrity hashes are added to IPFS. The access policy and sensitive data groups are stored off-chain by the CTI producer's \\textit{access manager}. Lastly, the IPFS CID and CTI metadata (e.g., threat type) are added to the \\textit{data storage} smart contract using the \\textit{share} function.\n\n\n\\textbf{Step 1a:} The CTI consumer's \\textit{request manager} adds an access request to the \\textit{data storage} smart contract using the \\textit{request} function. As discussed in Section~\\ref{system-functionality}, the request contains a IPFS CID that is associated with an encrypted set of the CTI consumers' credentials. \n\n\n\\textbf{Step 1b:} The CTI producer's \\textit{request manager} receives the access request. The \\textit{request manager} decrypts the credentials associated with the IPFS CID provided by the CTI consumer using their private key.  \n\n\n\\textbf{Step 2:} The CTI producer's \\textit{request manager} forwards the decrypted credentials and requested CTI to the \\textit{access manager}.\n\n\\textbf{Step 3:} The CTI producers \\textit{access manager} validates the received credentials (e.g., certificate signature is checked). Using the access policy associated with the requested CTI, and the verified credentials, the \\textit{access manager} assess the subset of sensitive data that should be provided to the CTI consumer.\n\n\n\n\\textbf{Step 4:} The CTI producers \\textit{access manager} returns the accessible sensitive data groups and their associated one-time random nonce to the CTI producers \\textit{request manager}. \n\n\n\n\\textbf{Step 5a:} The CTI producer's \\textit{request manager} responds to the access request submitted during \\textit{Step 1a} using the \\textit{data storage} smart contract's \\textit{response} function. As discussed in Section~\\ref{system-functionality}, this response contains an IPFS CID that is associated with the encrypted set of the accessible sensitive data groups and one-time random nonce's.\n\n\n\\textbf{Step 5b:} The CTI consumer's \\textit{request manager} receives the response. The \\textit{request manager} decrypts the accessible sensitive data groups and one-time random nonce's associated with the IPFS CID using their private key.   \n\n\n\\textbf{Step 6:} The CTI consumer's \\textit{request manager} forwards the received sensitive data, one-time random nonce's and original IPFS CID referencing the original CTI data, to the CTI consumers \\textit{data manager}.\n\n\n\\textbf{Step 7:} Using the original IPFS CID, the CTI consumers \\textit{data manager} gets the set of integrity hashes shared by the CTI producer during \\textit{Step 0}. \n\n\\textbf{Step 8:} Using the received sensitive data groups and one-time random nonce's, the CTI consumer's \\textit{data manager} computes the integrity hashes. This hash(s) is then compared to the set of integrity hashes received during \\textit{Step 7}.\n\n\n   \n    \n   \n    \n   \n    \n   \n    \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{System Design and Evaluation}\n\\label{sytem-design}\nThis section discusses the\nimplementation of the proposed CTI sharing framework. We evaluate the developed proof-of-concept prototype using several key on- and off-chain metrics, e.g. \\textit{gas cost} and \\textit{computational performance}. In addition, we also provide a privacy and trust analysis. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.42]{test-data-both.pdf}\n    \\caption{Sample data integrity hash (a) \\textit{generation}, and (b) \\textit{validation} time (ms) using the single (Single) and multi (Multi) hash approaches.}\n   \n    \\label{fig:sample_both}\n\\end{figure}\n\n\n\n\\subsection{Implementation}\n\\label{implementation}\nWe implemented a proof-of-concept of our proposed framework using Ethereum blockchain, in which we deploy our \\textit{data storage} smart contract written in Solidity v0.8.11. We select the Ethereum platform as it can be deployed as a public or private blockchain, allowing higher flexibility when evaluating our framework. We wrote a Python script using \\texttt{web3.py} library \\cite{web3-lib} to evaluate the blockchain implementation.\n\nIn addition, we developed the off-chain components of our framework in the Python programming language (e.g., request, access, and data managers). Specifically, we used \\texttt{hashlib} library \\cite{hash-lib} to implement the proposed integrity hashing mechanisms (cf. Fig.~\\ref{fig:intgrity-hashes}), where we use the SHA-256 hashing function and a public CTI dataset available on GitHub~\\cite{STIX-GITHUB} to evaluate its performance.\nIn addition, we also generate a number of synthetic data files larger and smaller than the CTI sample data. For both the sample and synthetic files, the data is segmented into various groups. We deployed a resource-constrained Amazon Elastic Compute Cloud instance (i.e., \\texttt{t2.micro}) to benchmark the off-chain operations, as CTI sharing may also occur in settings with computational limitations (e.g., in an IoT setting). As such, we consider these constrained environments as a \\textit{worst case} scenario when measuring the effect of additional hashing operations.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Results}\n\\label{results}\n\nTo evaluate the on-chain components of the developed \\textit{data storage} smart contract, we evaluated the transaction cost associated with the \\textit{share}, \\textit{request} and \\textit{response} functions under two different market conditions, \\textit{current} (1 gas = 20 gwei, 1 Eth = \\$1,097.2) and all time \\textit{high} (1 gas = 979 gwei, 1 Eth = \\$4891.70). In this context, gas cost represents the execution cost of the smart contact that is payed for in Eth (i.e., a gas fee that is required for an Ethereum blockchain network user to conduct a transaction on the network) based on the cost per gas unit, measured in gwei (note, each gwei is equal to $10^{-9}$ Eth).\n\nIn Table~\\ref{tab:cost}, the gas cost associated with the \\textit{data storage} smart contract functions are calculated using the two above-defined market conditions. These results show that under \\textit{current} market conditions, the transaction cost associated with sharing and requesting CTI is US\\$0.96 and US\\$2.57 respectively. As a result, the additional overhead created by our proposed differential sharing mechanism is US\\$2.57. Given that this increased cost is incurred by CTI consumer who in-exchange receive potentially valuable threat information, we argue this additional cost is unlikely to prevent participating.\n\nFurthermore, Table~\\ref{tab:cost} \\textit{High (US\\$)}, also highlights that transaction costs increase drastically (21762\\%) in the case where the market price for both gas and Eth rise significantly. While extreme, this result highlights the potential effect fluctuations in transaction costs may have on the effectiveness of the proposed framework. Subsequently, we argue that a private Ethereum network should be considered in a real world context, as this would allow participating organisations to better moderate these costs.    \n\nAdditionally, we also evaluated the off-chain computational overheads associated with the proposed integrity hashing schemes (cf. Fig.~\\ref{fig:intgrity-hashes}) using the sample and synthetic data. For each of these data types, the execution time associated with the generation and validation of an integrity hash set using both of the proposed hashing schemes was measured. In addition, the execution time associated with hashing the sample and synthetic data was also measured as a baseline. Given that the generation and validation execution times are highly dependent on the number of additional hashing operations, the baseline case indicates the additional overheads created by the two proposed hashing schemes relative to the size of the input data. As discussed in Section \\ref{implementation}, these operations were performed on an Amazon Elastic Compute Cloud \\texttt{t2.micro} instance, with the average of 1000 iterations recorded.\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[scale=0.38]{var-data-gen.pdf}\n    \\caption{Synthetic data integrity hash \\textit{generation} time using the single (Single) and multi (Multi) hash approaches.}\n   \n    \\label{fig:datasize_gen}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[scale=0.38]{var-data-val.pdf}\n    \\caption{Synthetic data integrity hash \\textit{validation} time using the single (Single) and multi (Multi) hash approaches.}\n    \\label{fig:datasize_val}\n\\end{figure}\n\nIn Table~\\ref{tab:compute_results}, a summary of the key results presented in Figs.~\\ref{fig:sample_both}-\\ref{fig:datasize_val} is provided. These results show that both the single hash and multi hash approaches result in an increased execution time compared to the baseline. However, we argue that these additional overheads are outweighed by the broader access to CTI and the greater granularity achieved by the proposed \\textit{differential} approach. Moreover, given that these results were obtained using a resource-constrained test bed, they indicate that the proposed \\textit{differential} sharing mechanism can operate efficiently in resource-constrained scenarios, e.g., with IoT.\n\nWhen the performance of these two approaches (i.e, single and multi hash) using the sample data are compared (cf. Fig.~\\ref{fig:sample_both}), two trends can be observed. In terms of generating the integrity hashes, the multi hash approach can be observed to have an increased execution time when compared to the single hash approach. This result is  due to the redundancy associated with hashing the same data exponentially more times as the number of data groups get larger for the multi hash case (cf. Fig.~\\ref{fig:intgrity-hashes}). In contrast, the opposite relationship can be observed if the validation times of these two approaches are compared. As the number of data groups increases, it can be observed that the single hash approach increases in execution time compared to the multi hash approach. This result is due to the redundancy associated with the number of comparisons required to validate a set of groups being equal to number of groups for the single hash case (cf. Fig.~\\ref{fig:intgrity-hashes}). \n\nFor the synthetic data, similar trends were also observed. In Figs.~\\ref{fig:datasize_gen} and \\ref{fig:datasize_val}, the single hash generation and multi hash validation efficiency differences are mostly consistent. However, we observed that the multi hash validation efficiency reduced proportional to the size of synthetic data. More specifically, after a data size of 150(KB), we observed that the single hash approach was either comparable or more efficient in terms of validation performance. \n\nBased on both the sample and synthetic data results, we argue that the multi-hash approach is more efficient. Given that CTI data is shared with \\textit{C} number of CTI consumers, integrity hash validation can be considered a \\textit{C}-time operation, and therefore, additional validation overheads are exacerbated in a real-world setting where \\textit{C} is denoted as a large number. In contrast, integrity hash generation is a one-time operation completed by the CTI producer each time they share. However, we note that in circumstances where CTI data is significantly larger than the sample data, the single hash approach may become more efficient, considering the validation time results for the 150-500KB synthetic data in Fig.~\\ref{fig:datasize_val}. \n\nUsing the discussed results, we make three major findings: (i) the proposed differential sharing mechanism does not significantly increase transaction costs, (ii) market conditions could affect the effectiveness of CTI sharing, and (iii) the proposed integrity hashing mechanisms (cf. Fig.~\\ref{fig:intgrity-hashes}) do not create large computational overheads, with the multi hash approach being more efficient for CTI data size less than 150KB. \n\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{table}[]\n    \\centering\n    \\caption{Gas cost and price in US\\$ for each \\textit{data storage} smart contract function.}\n    \\label{tab:cost}\n    \\scalebox{1}{\n    \\begin{tabular}{cccc}\n        \\toprule \n        Function & Gas Cost & Current (US\\$) & High (US\\$)  \\\\\n        \\toprule \n        Share & 43,897 & \\$0.96 & \\$210.22 \\\\ [0.5ex]\n        Request & 66,628 & \\$1.46 & \\$319.07 \\\\ [0.5ex]\n        Response & 50,625 & \\$1.11 & \\$242.44 \\\\ [0.5ex]\n        \\textbf{Total} & \\textbf{161,150} & \\textbf{\\$3.53} & \\textbf{\\$771.73} \\\\\n        \n        \\bottomrule\n    \n    \\end{tabular}\n    }\n\\end{table}\n\n\\begin{table}[]\n    \\centering\n    \\caption{Summary of computational times associated with Figs.~\\ref{fig:sample_both}-\\ref{fig:datasize_val}. (Gen = Generation Time, Val = Validation Time, SH = Single Hash, MH = Multi Hash).}\n    \\scalebox{0.94}{\n    \\begin{tabular}{cccccccc}\n        \\toprule\n   \n        Data & Size & Baseline & Number of & \\multicolumn{2}{c}{Gen (ms)} & \\multicolumn{2}{c}{Val (ms)} \\\\\n        {Type} & {(KB)} & {(ms)} & {Group} & SH & MH & SH & MH \\\\\n        \\toprule     \n        \\multirow{3}{*}{Sample} & \n            \\multirow{3}{*}{94} & \n                \\multirow{3}{*}{0.24} &\n                10 & 0.38 & 1.69 & 0.27 & 0.26 \\\\ [0.5ex]\n                & & & 20 & 0.51 & 3.30 & 0.30 & 0.27  \\\\ [0.5ex]\n                & & & 50 & 0.89 & 8.10 & 0.36 & 0.26  \\\\\n                \n        \\midrule\n        \n        \\multirow{9}{*}{Synthetic} & \n                \n            \\multirow{3}{*}{50} & \n                \\multirow{3}{*}{0.14} &\n                5 & 0.22 & 0.54 & 0.16 & 0.16   \\\\ [0.5ex]\n                & & & 20 & 0.41 & 1.9 & 0.2 & 0.17  \\\\ [0.5ex]\n                & & & 50 & 0.78 & 4.6 & 0.26 & 0.17  \\\\ [0.5ex]\n              \\cline{2-8} \n            & \\multirow{3}{*}{200} & \n                \\multirow{3}{*}{0.55} &\n                5 & 0.67 & 2.1 & 0.61 & 0.83  \\\\ [0.5ex]\n                & & & 20 & 0.88 & 7.2 & 0.66 & 0.83  \\\\ [0.5ex]\n                & & & 50 & 1.2 & 17 & 0.72 & 0.83  \\\\  [0.5ex]\n                  \\cline{2-8} \n            & \\multirow{3}{*}{500} & \n                \\multirow{3}{*}{1.4} &\n                5 & 1.6 & 5.1 & 2.5 & 1.9  \\\\ [0.5ex]\n                & & & 20 & 1.8 & 18 & 1.5 & 1.9  \\\\ [0.5ex]\n                & & & 50 & 2.2 & 45 & 1.6 & 2.1  \\\\\n               \n        \\bottomrule\n    \\end{tabular}}\n    \\label{tab:compute_results}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Privacy and Trust Analysis}\n\\label{privacy-trust}\nNow we examine a number of privacy and trust-based considerations with our proposed framework. From a CTI producer's perspective, it is important that sensitive data is shared in a privacy-preserving way. Subsequently, the proposed differential approach seeks to provide a more granular approach that maintains privacy at rest and in transit. To ensure that privacy is maintained at rest, sensitive data groups are stored securely off-chain by CTI producers, and therefore cannot be accessed by untrusted entities. Moreover, when sensitive data groups are exchanged with trusted CTI consumers, encryption is used to ensure that \\textit{man-in-the-middle} and \\textit{eavesdropping} attacks (e.g., sniffing or snooping) are not possible. Finally, the use of one-time random nonce's as part of the integrity hash generation and validation process, ensures that the privacy of sensitive data common across multiple sharing instances or which contains low entropy (e.g., software versions), are protected from dictionary and brute force attacks. \n\n\nFurther, two major trust-based factors must also be considered. Firstly, CTI producers must be provided with a dynamic approach to trust management that ensures that sensitive data is only shared with trusted entities. Subsequently, our framework allows CTI producers to define a trust-based policy\/metric scheme that best suits their requirements. Finally, the verifiability of CTI data from the perspective of CTI consumers must also be considered. For say, if a CTI consumer receives a subset of the CTI data, it could be possible for a malicious CTI producer to modify this data depending on the identity of the requesting organisation (e.g., a competitor). In such a case, to protect CTI consumers from this attack, our proposed integrity hashing schemes ensure that CTI consumers can validate that the received subset of the CTI data is contained within the original set of CTI data shared by the CTI producer. \n\n\n\n\n\n\n\\section{Conclusion}\n\\label{conclusion}\n\nWe have proposed a blockchain-based CTI sharing framework that allows for the trusted, verifiable, and differential exchange of CTI between producers and consumers. We refer to differential as the ability of a CTI producer to control the amount of information exchanged with a CTI consumer. The framework thus allows CTI producers to segment CTI data into sensitive data groups, which can be variably shared with CTI consumers in a differential manner.\nOur results showed that the proposed framework could facilitate the exchange of CTI between producers and consumers in a differential way without compromising trust or variability. An interesting direction for future work is to conduct a more comprehensive study to examine the impact of incentives on CTI sharing. \n\n\n\\section*{Acknowledgment}\nThe authors acknowledge the support of the Commonwealth of Australia and Cybersecurity Research Centre Limited.\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Introduction}\n\\label{introduction}\n\nThe rapid evolution of the threat landscape in recent years has caused many organisations to reassess the risks associated with their digital infrastructure \\cite{eling2021cyber}. For example, during the 2020-2021 financial year, almost 500 ransomware cybercrimes were reported in Australia, an increase of 15\\% compared to the previous year \\cite {ACSC-CT-report-2020-21}. As a result, many organisations have started integrating Cyber Threat Intelligence (CTI) into traditional Cyber Security Risks Management (CSRM) frameworks to try and achieve a more proactive security posture \\cite{kure2019cyber}. CTI can be defined as the collection and synthesis of evidence-based knowledge about actors, threats, and vulnerabilities of harmful events occurring in cyberspace. Subsequently, the contents of CTI are highly variable in nature and can range from Indicators of Compromise (IoC) to sophisticated information detailing an attacker's Techniques, Tactics and Procedures (TTPs) \\cite{wagner2019cyber}.\n\nGiven that integration of CTI data into traditional CSRM frameworks can be utilised to provide organisations with a more proactive defense posture, inter organisational CTI sharing holds additional benefits \\cite{kure2019cyber}. In the case where one organisation (e.g., a CTI producer) has been attacked, sharing CTI has the potential to allow other organisations (e.g., a CTI consumer) to proactively implement mitigation strategies \\cite{wagner2019cyber}. Note, we refer to CTI producers as the entities who share CTI, while CTI consumers are the entities that use shared CTI. As a result, in recent years several \\textit{Open-source\/Vendor-created} CTI sharing platforms have been developed.\n\n\n\n\n\n        \n        \n        \n        \n         \n         \n         \n         \n         \n         \n        \n        \n\n        \n        \n        \n        \n         \n         \n         \n         \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{Motivation and Problem Statement}\nIn many cases, sharing CTI data in an uncontrolled way can result in CTI producers sustaining reputational damage.\nSubsequently, for cases where CTI contains sensitive data, trust-based data sharing has been highlighted as a possible solution \\cite{wagner2019cyber}. Trust-based CTI sharing ensures that potentially sensitive CTI data is shared with consumers with which pre-established or developed trust exists. Moreover, trust issues are significant when sharing takes place in large-scale industrial systems, e.g., Industry 4.0~\\cite{suhail2022towards}\n\nWhile trust-based sharing has the potential to reduce sharing risks, CTI producers require more fine-grained control over what data they share. For instance, in \\cite{wagner2019cyber}, it is highlighted that organisations are likely to differ in their perceptions of what data is considered sensitive.\nAs a result, CTI sharing platforms need to provide organisations with the flexibility to share CTI in a way that caters to this variation. Organisations are also likely to differ in their perception of how trust-based sharing should be facilitated. For instance, an organisation A may determine their trust in another organisation B based on B's attribute or using identity-based policies specific to B. Subsequently, constraining organisations to share in a fixed way, either through  share or deny or policies, or group-based policies that share or deny CTI with specific groups, does not provide the granularity required for versatile CTI sharing. We propose a framework that allows CTI producers to share sensitive CTI with consumers in a \\textit{differential} way. We refer to differential as the ability of CTI producers to control the amount of information exchanged with CTI consumers at a fine-grained level (more discussion in Section~\\ref{differential}). \n\nFurthermore, CTI consumers should have the ability to verify requested information to ensure that the supplied intelligence has not be tampered with or altered by the CTI producer. Subsequently, an efficient verification mechanism which simultaneously preserves the privacy in data sharing is required.\nThus, we argue that CTI sharing must be trusted, performed in a verifiable way, and without compromising the privacy of sensitive CTI data. \n\n\n\n\n\nSeveral blockchain-based CTI sharing proposals have been proposed \\cite{allouche2021trade} in recent years. Blockchain has a number of salient properties that can be leveraged to facilitate CTI sharing. For example, among others, blockchain's consensus, auditability, and immutability properties allow organisations to interact with one another in a distributed way without the need of a trusted third party. Another advantage of using blockchain to facilitate CTI sharing is verifiability. Considering that the information stored on the blockchain is immutable, CTI consumers are able to validate the integrity of received data and, therefore, can be sure it has not been tampered with or changed~\\cite{dai2019blockchain}.\n\n\n\n\n\n\n\\subsection{Contributions}\n\\label{contributions}\nExisting proposals only allow CTI producers to share intelligence in fixed way using a platform defined trust-based policy scheme. While these approaches do provide verifiable and trust-based sharing, constraining CTI producers to share in a fixed way, e.g. binary or group-based sharing, has the potential to limit the effectiveness of sharing between organisations. Subsequently, in our work we define a blockchain-based framework which provides CTI producers with the ability to share CTI \\textit{differentially} using  policies\/metrics which they define. Moreover, we further demonstrate how CTI producers can share in this way without compromising the verifiability of the intelligence from the CTI consumers perspective. To the best of our knowledge, our work is the first to use blockchain for trusted, verifiable, and differential CTI sharing. The major contributions of this paper are: \n\n\n\n\\begin{itemize}\n   \n    \n   \n    \n    \\item We propose the concept of \\textit{differential sharing} within the CTI sharing context.\n    \n    \\item We detail how \\textit{differential sharing} can provide more granuality compared to existing solutions by allowing data segmentation and \\textit{pluggable} trust-based policy\/metric schemes, without compromising verifiability through the use of two integrity hash schemes.  \n    \n   \n    \n   \n    \n   \n    \n   \n    \n    \\item We discuss a detailed design for the proposed CTI sharing framework, its system functionality, and communications among the various components.\n    \n    \\item We demonstrate and evaluate a detailed proof of concept implementation using Ethereum private blockchain. The experimental results indicate that the proposed framework is feasible and creates insignificant overheads compared to a baseline sharing scenario\n    \n    \\item We present a privacy and trust analysis of the proposed framework.  \n    \n\\end{itemize}\n\nNext, in Section~\\ref{related-work}, we discuss the related work. In Section~\\ref{proposed-architecture}, we introduce the proposed CTI sharing framework. Section~\\ref{sytem-design} describes and evaluates the proof of concept implementation of the proposed framework and present the key results. Moreover, this section also analysis the privacy and trust of the proposed framework. Finally, Section~\\ref{conclusion} concludes the paper and highlights the future work. \n\n\n\n\n\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Available blockchain-based CTI sharing proposals and their comparison with our work. \n    }\n    \\label{tab:realted_work}\n    \\scalebox{0.765}{\n    \\begin{tabular}{ccccccc}\n        \\toprule\n            {Ref.} & {Differential} & {Sensitive} & {Producer-Defined} & {Verifiable} & {Blockchain}  \\\\\n            {} & {Sharing} & {Data Grouping} & {Trust Scheme} & {Sharing} & {Platform} \\\\\n        \\toprule\n        \n        \\cite{shi2022threat} & \\xmark &  \\xmark & \\xmark & \\checkmark & CITA \\\\ [0.5ex]\n        \n        \\cite{badsha2020blocynfo} & \\xmark & \\xmark & \\xmark & \\checkmark & Ethereum \\\\ [0.5ex]\n         \n         \\cite{allouche2021trade} & \\xmark &  \\xmark & \\xmark & \\checkmark & Ethereum \\\\ [0.5ex]\n         \n         \\cite{preuveneers2020distributed} & \\xmark & \\xmark & \\xmark & \\checkmark & Hyperledger   \\\\ [0.5ex]\n         \n         \\cite{nguyen2022blockchain} & \\xmark & \\xmark & \\xmark & \\checkmark & Hyperledger \\\\ [0.5ex]\n         \n         \\cite{moubarak2021dissemination} & \\xmark & \\xmark & \\xmark & \\checkmark & Hyperledger \\\\ [0.5ex]\n         \n         \\cite{homan2019new} & \\xmark & \\xmark & \\xmark & \\checkmark & Hyperledger \\\\ [0.5ex]\n         \n        \\textbf{[Our Work]} & \\checkmark & \\checkmark & \\checkmark & \\checkmark & Ethereum \\\\ \n        \n        \\bottomrule\n        \n    \\end{tabular}\n    }\n\\end{table}\n\n\\section{Related Work}\n\\label{related-work}\nIn Table~\\ref{tab:realted_work}, we provide a list of the available related works on blockchain-based CTI sharing and their comparison with our work. For example, \nThe authors in \\cite{shi2022threat} developed a blockchain-based CTI sharing platform using the CITA model. This model provides CTI producers with the ability to store sensitive data off-chain. When sharing this CTI data with authorised CTI consumers, the authors propose using point-to-point communication to ensure fine-grained access control over which CTI consumers receive the sensitive data. To ensure verifiability, a hash of the sensitive data is stored on-chain such that CTI consumers can validate the integrity of data received off-chain. While~\\cite{shi2022threat} provides  trust-based verifiable sharing, CTI producers are constrained to share CTI data with consumers in a coarse-grained and binary way.\n\n`BloCyNfo-Share' is a blockchain-based CTI sharing framework proposed in \\cite{badsha2020blocynfo}. It utilises proxy re-encryption and attribute-based encryption to achieve fine-grained access control. When sharing, CTI producers provide an encrypted copy of the data to a cloud server. When a CTI consumer requests access to this CTI, the CTI producer generates a re-encryption key and adds it to a smart contract. This re-encryption key is subsequently used by the cloud server to generate a new cipher text. The resulting cipher text is subsequently sent to the CTI consumer, who can decrypt it given they satisfy the policy specified by the CTI producer.\nSimilar to \\cite{shi2022threat}, in this framework, CTI producers are limited to making binary sharing decisions. Thus, unlike our approach, proposals \\cite{badsha2020blocynfo} and  \\cite{shi2022threat} do not provide differential sharing.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale=0.55]{share-type.pdf}\n    \\caption{Illustration of our differential CTI sharing compared to traditional binary and group-based CTI sharing approaches.}\n    \\label{fig:turst-group-df}\n\\end{figure*}\n\nIn \\cite{allouche2021trade}, the blockchain-based CTI sharing platform TRusted Anonymous Data Exchange (TRADE) is proposed. To facilitate the exchange of CTI data, the TRADE platform proposes the use of off-chain data storage, policy-based access control, on-chain policy validation and off-chain data exchange using Trusted Automated Exchange of Intelligence Information (TAXII)~\\cite{jordan_varner_2021}. When a CTI producer wants to share,\nthey add an integrity hash and access policy (e.g., eXtensible Access Control Markup Language (XACML) policy) to the blockchain. \nHowever, TRADE again limits CTI producers to binary sharing decisions. \n\nIn contrast to \\cite{badsha2020blocynfo}, proposal \\cite{preuveneers2020distributed} utilises the existing CTI sharing platform MISP \\cite{wagner2016misp} to facilitate the exchange of CTI between organisations. To ensure that the privacy of sensitive data is maintained, a \\textit{Trustworthy API for Threat Intelligence Sharing} (TATIS) Reverse Proxy is used. This proxy uses ciphertext attribute-based encryption to restrict the access of sensitive data to trusted organisations. To provide verifiability, CTI producers store a hash of the original data on-chain. However, like the above proposals, \\cite{preuveneers2020distributed} also limits producers to binary sharing decisions.\n\nIn a few other proposals, e.g., \\cite{nguyen2022blockchain},  \\cite{moubarak2021dissemination}, and \\cite{homan2019new}, a group-based approach for sensitive CTI sharing is utilised. Instead of limiting CTI producers to binary sharing decisions, a group-based approach allows CTI producers and consumers to exchange sensitive CTI as part of a trusted group. Moreover, these proposals utilise the Traffic Light Protocol (TLP). Instead of limiting CTI producers to share in a binary way, more or less sensitive data can be shared using a layered approach. In this case, layered refers to a grouping of organisations into trust groups (e.g., red, amber, green, and white), in which CTI of varying sensitivity can be shared.\nHowever, the major limitation of a group-based approach is that it assumes organisations trust each other in a pre-defined and fixed way. While these approaches extend sharing beyond binary decisions, we argue they still lack the granularity required for more effective CTI sharing. Subsequently, our proposal allows CTI producers to share differentially, and therefore supports dynamic and time-varying trust relationships. \n\n\nIn summary, the existing blockchain-based CTI sharing proposals limit a CTI producer's ability to share sensitive CTI in a coarse-grained manner, either based on binary share\/deny policies or as part of a pre-defined trust group. In most cases, the existing solutions provide trust-based policies\/metrics that limit the dissemination of sensitive data to trusted organisations, but they fail to deliver more granular sharing. Given that a more fine-grained sharing mechanism is required to facilitate effective CTI sharing, we propose a framework with the concept of differential sharing discussed in the next section.\n\n\n\\section{Proposed Framework}\n\\label{proposed-architecture}\n\n\n\nIn this section, we present the proposed blockchain-based CTI sharing framework. First, we discuss the concept of differential CTI sharing in detail. Then we present the various components of the architecture. We also discuss the system functionality and communication among these components. \n\n\\subsection{Differential Sharing}\n\\label{differential}\nIn Section~\\ref{introduction}, we highlight that organisations are likely to differ in their perceptions of data sensitivity and what trust-based policy\/metrics should be used to determine a CTI consumers access. We propose the concept of \\textit{differential} CTI sharing, which aims to address this challenge.  Figure~\\ref{fig:turst-group-df} compares differential CTI sharing with binary and trusted group-based CTI sharing. From this comparison, we highlight that differential sharing can tailor the CTI data to be shared with each CTI consumer. As a result, the proposed \\textit{differential} sharing approach does not restrict sharing to only highly trusted CTI consumers, and therefore, allows other CTI consumers to benefit from CTI data they otherwise would not have received.\n\nGiven that CTI data contains many attributes, e.g., \\textit{chained objects} in Structured Threat Information Expression (STIX), their individual sensitivity can be evaluated by the CTI producer. Furthermore, as these attributes are likely to vary in their sensitivity, attributes of similar sensitivities can be grouped together. For example, \\textit{chain-objects} of type vulnerability often contain specific software versions that could be valuable information to potential attacks, while \\textit{chain-objects} of type indicator often only contain IoC. As a result, based on the information contained within specific STIX \\textit{chain-objects}, Fig.~\\ref{fig:stix-cti} demonstrates how CTI attributes can be broken down into a variable number of sensitive data groups. \n\n\n\\begin{figure} [ht]\n\\includegraphics[scale=0.69]{tbdg.pdf}\n\\caption{Sensitive data groups based on CTI attributes.}\n\\label{fig:stix-cti}\n\\end{figure}\n\nOnce the CTI has been segmented using the above method, a subset of these sensitive data groups can be given to CTI consumers based on a single or set of trust-based policies\/metrics (e.g., XACML policy). By allowing CTI producers to segment CTI data into groups based on a sensitivity assessment, a variable proportion of these groups can be shared with each CTI consumer. To ensure verifiability of the received data, we propose the use of two group-based hashing approaches: single hash and multi-hash. Given that CTI consumers receive a distinct number of the sensitive data groups, a set of integrity hashes constructed using either of these two approaches is required. In Fig.~\\ref{fig:intgrity-hashes} we illustrate how these group-based hashes can be constructed using a single and multi hash approach.\n\n\n\\begin{figure} [t]\n\\centering\n\\includegraphics[scale=0.82]{ih.pdf}\n\\caption{Proposed single hash (a) and multi hash (b) approaches. Note, dotted boxes represent integrity hashes. ($N_i$ = One-time random nonce).}\n\\label{fig:intgrity-hashes}\n\\end{figure}\n\nSharing CTI using the proposed \\textit{differential} approach gives CTI producers the ability to share with more granularity compared to traditional solutions, e.g., binary and group-based. We further note that greater granularity is achieved in two ways: (i) CTI producers are given the ability to define what subset of the CTI data is shared with an individual or group of CTI consumers, and (ii) CTI producers are also able to change how they segment CTI data and the policies\/metrics they use to evaluate trust each time they share.\n\n\n\n\n    \n    \n    \n    \n    \n    \n    \n\n\n\n\n\\subsection{Architectural Components}\n\\label{components}\nIn Fig.~\\ref{fig:architecture}, we illustrate the proposed blockchain-based CTI sharing framework. It contains the following components: \n\n\n\n\n\\begin{itemize}\n    \\item \\textbf{Organisation:} An entity that participates in CTI sharing. Organisations can assume the role of a producer and\/or consumer. Organisations can participate in both roles over time. However, we assume that an organisation performs only one role at a time for a specific interaction.\n    \n    \\item \\textbf{Request Manager:} A component within an organisation responsible for handling and submitting access requests of CTI data. The request manager role is primarily broken into two tasks, \\textit{submitting} an access request and \\textit{processing} an access request. They are as follows:  \n    \\begin{enumerate}\n        \\item Submitting an access request occurs when new CTI is made available. Given that another organisation has added a CTI, the request manager submits an access request to the blockchain. \n        \n        \\item When an organisation adds a new CTI, the organisation's request manager is responsible for processing the access requests added by other organisations.\n    \\end{enumerate}\n    \n   \n    \n   \n    \n   \n    \n    \\item \\textbf{Data Manager:} A component inside of an organisation responsible for storing sensitive CTI data and their corresponding access policies. This component is also responsible for indicating to other organisations that new CTI is available. Subsequently, it adds content on and off-chain when an organisation acts as a CTI producer\n    \n    \\item \\textbf{Access Manager:} A component inside an organisation responsible for determining what portion of requested CTI should be shared with a CTI consumer. Using the access policies stored by the data manager and credentials (e.g., identity-proof) provided by the CTI consumer, the access manager determines what portion of the CTI data should be given to a particular CTI consumer.\n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n    \n   \n    \n   \n    \n   \n    \n    \n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.755]{whole.pdf}\n\\caption{Proposed blockchain-based CTI sharing framework.}\n\\label{fig:architecture}\n\\end{figure}\n\n  \n    \\item \\textbf{Blockchain:} A distributed digital ledger of information that is shared, duplicated and distributed across the entire network of computer systems. It can be seen as a shared database that stores a series of digital records using cryptographic operations (called hash).\n    Ethereum blockchain is used for our proof-of-concept implementation. \n    \n    \\item \\textbf{Smart Contract:} \n    Self-executable programs stored on a blockchain. We introduce a \\textit{data storage} smart contract in our framework. It is composed of three functions: (i) \\textit{share} function provided by the smart contract is used by a CTI producer to indicate the availability of CTI, (ii) \\textit{request} function provided by the smart contract is used by a CTI consumer to request access to sensitive CTI data, and (iii) \\textit{response} function provided by the smart contract is used by a CTI producer to respond to a request submitted to the \\textit{request} function.\n    \n   \n    \n   \n    \n    \\item \\textbf{Interplanetary File System (IPFS):} A peer-to-peer hypermedia protocol that is designed to distribute the storage of data across a network of computers. IPFS provides highly available and consistent data storage of large files that otherwise would be expensive to store using a traditional blockchain network (e.g., Ethereum). IPFS is utilised to store off-chain data associated with the sharing process \\cite{benet2014ipfs}.\n   \n    \n   \n    \n   \n    \n   \n\\end{itemize}\n\n\n\n\n\n\n\n\\subsection{System Functionality}\n\\label{system-functionality}\n\nThis section provides an overview of the proposed CTI sharing framework's functionality. The framework follows five main phases to share and consume CTI. They are:\n\n\n\\underline{{\\textbf{\\textit{Share CTI:}}}} The objective of this phase is for the CTI producer to create and share all of the data\nrequired to facilitate the exchange of CTI. Given that an organisation has produced CTI data, a sensitivity analysis is conducted by the CTI producer. This analysis identifies sensitive CTI attributes and segments them into a number of sensitivity groups (cf. Fig.~\\ref{fig:stix-cti}). \n\n\n\n\n\n\nOnce the sensitive data groups have been defined, a set of \\textit{integrity hashes} are created. To construct these hashes, the single or multi hash approach (cf. Fig.~\\ref{fig:intgrity-hashes}) can be used by the CTI producer. For both of the proposed approaches, the sensitive data groups and a set of one-time random nonce's are used to construct the integrity hashes. In this case, the set of one-time random nonce's are used to protect the privacy of sensitive data (more discussion in Section~\\ref{privacy-trust}). To generate these integrity hashes, a single or set of data groups are hashed using these nonce's in sequence. \n\n\n\n\nAs discussed in Section~\\ref{related-work}, several trust-based metric\/policy schemes have been presented by existing works. Subsequently, our framework allows CTI producers to plug-in a scheme dynamically. For example, an organisation might assign an individual XACML policy for each of the sensitive data groups.\n\n\n\n\nOnce the above process has been completed, the sensitive data groups and access policy are stored securely off-chain by the CTI producer. Moreover the non-sensitive CTI data group, set of integrity hashes and access policy are added to IPFS, while its associated content identifier (CID) (IPFS File Pointer) and CTI metadata are added to the \\textit{data storage} smart contract.\n\n\n\n\\underline{{\\textbf{\\textit{Request CTI:}}}} During this phase, a CTI consumer requests access to a particular CTI of interest. An access request is submitted via a \\textit{data storage} smart contract. This smart contract is the same one used during the \\textit{Share CTI} phase. The access request indicates which CTI data the CTI consumer wishes to access and contains the credentials (e.g., identity-proof) required to meet the associated access policy.\n\nTo ensure that these credentials are shared in a privacy-preserving way, they are encrypted using the CTI producer's public key before being added to IPFS. By encrypting their credentials, a CTI consumer can ensure that only the intended CTI producer can access these them. Once the access request has been submitted, the CTI producer receives the request, accesses the encrypted credentials from IPFS using the provided IPFS CID, and decrypts them using their private key. \n\n\\underline{{\\textbf{\\textit{Policy Evaluation:}}}} During this phase, the CTI producer processes the access request submitted by the CTI consumer. To determine the sensitive data groups that the CTI consumer should have access to, the set of credentials provided during the \\textit{Request CTI} phase and the access policy associated with the requested CTI is used. Once the provided credentials are validated, the CTI producer evaluates the access policy to determine which subset of the sensitive data groups the requesting CTI consumer should receive.\n\n\n\n\\underline{{\\textbf{\\textit{Response CTI:}}}} During this phase, the CTI producer provides the CTI consumer with the sensitive data groups they are able to access. Note, in the \\textit{Policy Evaluation} phase above, the CTI producer determines which subset of the sensitive data groups a CTI consumer is able to access. To ensure that the CTI consumer can validate the integrity of the provided sensitive data groups, the corresponding one-time random nonce values are also included in the response. To ensure that the response data is exchanged in a privacy-preserving way, it is encrypted using the CTI consumer's public key. Once encrypted, the response data is added to IPFS and the associated IPFS CID is added to the \\textit{data storage} smart contract by the CTI producer. The CTI consumer can then access the encrypted response data using the provided IPFS CID. To decrypt the response data, the CTI consumers uses their private key.\n\n\n\n\n\\underline{{\\textbf{\\textit{Validate CTI:}}}} During this phase, the CTI consumer validates the integrity of the received sensitive data. As part of the \\textit{Share CTI} phase, the CTI producer added a set of integrity hashes to IPFS. Subsequently, the CTI consumer can access these integrity hashes and use them to validate the sensitive CTI data received during the \\textit{Response CTI} phase.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=.48]{id.pdf}\n\\caption{Communication between various components of the proposed CTI sharing framework (on-chain interactionss).}\n\\label{fig:interactions}\n\\end{figure}\n\n\\subsection{Communications}\n\\label{communication}\n\n\nIn Fig.~\\ref{fig:interactions}, we illustrate the communication flow within our CTI framework. Note that for simplicity, we contemplate the interactions of various components with the blockchain. Therefore, only on-chain interactions are shown, and interactions with IPFS are excluded. The steps are as follows:\n\n\\textbf{Step 0:} CTI producer completes the initial setup steps. Firstly, the CTI data is segmented into a variable number of sensitivity data groups (cf. Fig.\\ref{fig:stix-cti}). The sensitive data groups and a set of one-time random nonces are used to construct a set of integrity hashes using either of the proposed methods (cf. Fig.\\ref{fig:intgrity-hashes}). The non sensitive data group, CTI producer-defined access policy, and integrity hashes are added to IPFS. The access policy and sensitive data groups are stored off-chain by the CTI producer's \\textit{access manager}. Lastly, the IPFS CID and CTI metadata (e.g., threat type) are added to the \\textit{data storage} smart contract using the \\textit{share} function.\n\n\n\\textbf{Step 1a:} The CTI consumer's \\textit{request manager} adds an access request to the \\textit{data storage} smart contract using the \\textit{request} function. As discussed in Section~\\ref{system-functionality}, the request contains a IPFS CID that is associated with an encrypted set of the CTI consumers' credentials. \n\n\n\\textbf{Step 1b:} The CTI producer's \\textit{request manager} receives the access request. The \\textit{request manager} decrypts the credentials associated with the IPFS CID provided by the CTI consumer using their private key.  \n\n\n\\textbf{Step 2:} The CTI producer's \\textit{request manager} forwards the decrypted credentials and requested CTI to the \\textit{access manager}.\n\n\\textbf{Step 3:} The CTI producers \\textit{access manager} validates the received credentials (e.g., certificate signature is checked). Using the access policy associated with the requested CTI, and the verified credentials, the \\textit{access manager} assess the subset of sensitive data that should be provided to the CTI consumer.\n\n\n\n\\textbf{Step 4:} The CTI producers \\textit{access manager} returns the accessible sensitive data groups and their associated one-time random nonce to the CTI producers \\textit{request manager}. \n\n\n\n\\textbf{Step 5a:} The CTI producer's \\textit{request manager} responds to the access request submitted during \\textit{Step 1a} using the \\textit{data storage} smart contract's \\textit{response} function. As discussed in Section~\\ref{system-functionality}, this response contains an IPFS CID that is associated with the encrypted set of the accessible sensitive data groups and one-time random nonce's.\n\n\n\\textbf{Step 5b:} The CTI consumer's \\textit{request manager} receives the response. The \\textit{request manager} decrypts the accessible sensitive data groups and one-time random nonce's associated with the IPFS CID using their private key.   \n\n\n\\textbf{Step 6:} The CTI consumer's \\textit{request manager} forwards the received sensitive data, one-time random nonce's and original IPFS CID referencing the original CTI data, to the CTI consumers \\textit{data manager}.\n\n\n\\textbf{Step 7:} Using the original IPFS CID, the CTI consumers \\textit{data manager} gets the set of integrity hashes shared by the CTI producer during \\textit{Step 0}. \n\n\\textbf{Step 8:} Using the received sensitive data groups and one-time random nonce's, the CTI consumer's \\textit{data manager} computes the integrity hashes. This hash(s) is then compared to the set of integrity hashes received during \\textit{Step 7}.\n\n\n   \n    \n   \n    \n   \n    \n   \n    \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{System Design and Evaluation}\n\\label{sytem-design}\nThis section discusses the\nimplementation of the proposed CTI sharing framework. We evaluate the developed proof-of-concept prototype using several key on- and off-chain metrics, e.g. \\textit{gas cost} and \\textit{computational performance}. In addition, we also provide a privacy and trust analysis. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.42]{test-data-both.pdf}\n    \\caption{Sample data integrity hash (a) \\textit{generation}, and (b) \\textit{validation} time (ms) using the single (Single) and multi (Multi) hash approaches.}\n   \n    \\label{fig:sample_both}\n\\end{figure}\n\n\n\n\\subsection{Implementation}\n\\label{implementation}\nWe implemented a proof-of-concept of our proposed framework using Ethereum blockchain, in which we deploy our \\textit{data storage} smart contract written in Solidity v0.8.11. We select the Ethereum platform as it can be deployed as a public or private blockchain, allowing higher flexibility when evaluating our framework. We wrote a Python script using \\texttt{web3.py} library \\cite{web3-lib} to evaluate the blockchain implementation.\n\nIn addition, we developed the off-chain components of our framework in the Python programming language (e.g., request, access, and data managers). Specifically, we used \\texttt{hashlib} library \\cite{hash-lib} to implement the proposed integrity hashing mechanisms (cf. Fig.~\\ref{fig:intgrity-hashes}), where we use the SHA-256 hashing function and a public CTI dataset available on GitHub~\\cite{STIX-GITHUB} to evaluate its performance.\nIn addition, we also generate a number of synthetic data files larger and smaller than the CTI sample data. For both the sample and synthetic files, the data is segmented into various groups. We deployed a resource-constrained Amazon Elastic Compute Cloud instance (i.e., \\texttt{t2.micro}) to benchmark the off-chain operations, as CTI sharing may also occur in settings with computational limitations (e.g., in an IoT setting). As such, we consider these constrained environments as a \\textit{worst case} scenario when measuring the effect of additional hashing operations.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Results}\n\\label{results}\n\nTo evaluate the on-chain components of the developed \\textit{data storage} smart contract, we evaluated the transaction cost associated with the \\textit{share}, \\textit{request} and \\textit{response} functions under two different market conditions, \\textit{current} (1 gas = 20 gwei, 1 Eth = \\$1,097.2) and all time \\textit{high} (1 gas = 979 gwei, 1 Eth = \\$4891.70). In this context, gas cost represents the execution cost of the smart contact that is payed for in Eth (i.e., a gas fee that is required for an Ethereum blockchain network user to conduct a transaction on the network) based on the cost per gas unit, measured in gwei (note, each gwei is equal to $10^{-9}$ Eth).\n\nIn Table~\\ref{tab:cost}, the gas cost associated with the \\textit{data storage} smart contract functions are calculated using the two above-defined market conditions. These results show that under \\textit{current} market conditions, the transaction cost associated with sharing and requesting CTI is US\\$0.96 and US\\$2.57 respectively. As a result, the additional overhead created by our proposed differential sharing mechanism is US\\$2.57. Given that this increased cost is incurred by CTI consumer who in-exchange receive potentially valuable threat information, we argue this additional cost is unlikely to prevent participating.\n\nFurthermore, Table~\\ref{tab:cost} \\textit{High (US\\$)}, also highlights that transaction costs increase drastically (21762\\%) in the case where the market price for both gas and Eth rise significantly. While extreme, this result highlights the potential effect fluctuations in transaction costs may have on the effectiveness of the proposed framework. Subsequently, we argue that a private Ethereum network should be considered in a real world context, as this would allow participating organisations to better moderate these costs.    \n\nAdditionally, we also evaluated the off-chain computational overheads associated with the proposed integrity hashing schemes (cf. Fig.~\\ref{fig:intgrity-hashes}) using the sample and synthetic data. For each of these data types, the execution time associated with the generation and validation of an integrity hash set using both of the proposed hashing schemes was measured. In addition, the execution time associated with hashing the sample and synthetic data was also measured as a baseline. Given that the generation and validation execution times are highly dependent on the number of additional hashing operations, the baseline case indicates the additional overheads created by the two proposed hashing schemes relative to the size of the input data. As discussed in Section \\ref{implementation}, these operations were performed on an Amazon Elastic Compute Cloud \\texttt{t2.micro} instance, with the average of 1000 iterations recorded.\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[scale=0.38]{var-data-gen.pdf}\n    \\caption{Synthetic data integrity hash \\textit{generation} time using the single (Single) and multi (Multi) hash approaches.}\n   \n    \\label{fig:datasize_gen}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[scale=0.38]{var-data-val.pdf}\n    \\caption{Synthetic data integrity hash \\textit{validation} time using the single (Single) and multi (Multi) hash approaches.}\n    \\label{fig:datasize_val}\n\\end{figure}\n\nIn Table~\\ref{tab:compute_results}, a summary of the key results presented in Figs.~\\ref{fig:sample_both}-\\ref{fig:datasize_val} is provided. These results show that both the single hash and multi hash approaches result in an increased execution time compared to the baseline. However, we argue that these additional overheads are outweighed by the broader access to CTI and the greater granularity achieved by the proposed \\textit{differential} approach. Moreover, given that these results were obtained using a resource-constrained test bed, they indicate that the proposed \\textit{differential} sharing mechanism can operate efficiently in resource-constrained scenarios, e.g., with IoT.\n\nWhen the performance of these two approaches (i.e, single and multi hash) using the sample data are compared (cf. Fig.~\\ref{fig:sample_both}), two trends can be observed. In terms of generating the integrity hashes, the multi hash approach can be observed to have an increased execution time when compared to the single hash approach. This result is  due to the redundancy associated with hashing the same data exponentially more times as the number of data groups get larger for the multi hash case (cf. Fig.~\\ref{fig:intgrity-hashes}). In contrast, the opposite relationship can be observed if the validation times of these two approaches are compared. As the number of data groups increases, it can be observed that the single hash approach increases in execution time compared to the multi hash approach. This result is due to the redundancy associated with the number of comparisons required to validate a set of groups being equal to number of groups for the single hash case (cf. Fig.~\\ref{fig:intgrity-hashes}). \n\nFor the synthetic data, similar trends were also observed. In Figs.~\\ref{fig:datasize_gen} and \\ref{fig:datasize_val}, the single hash generation and multi hash validation efficiency differences are mostly consistent. However, we observed that the multi hash validation efficiency reduced proportional to the size of synthetic data. More specifically, after a data size of 150(KB), we observed that the single hash approach was either comparable or more efficient in terms of validation performance. \n\nBased on both the sample and synthetic data results, we argue that the multi-hash approach is more efficient. Given that CTI data is shared with \\textit{C} number of CTI consumers, integrity hash validation can be considered a \\textit{C}-time operation, and therefore, additional validation overheads are exacerbated in a real-world setting where \\textit{C} is denoted as a large number. In contrast, integrity hash generation is a one-time operation completed by the CTI producer each time they share. However, we note that in circumstances where CTI data is significantly larger than the sample data, the single hash approach may become more efficient, considering the validation time results for the 150-500KB synthetic data in Fig.~\\ref{fig:datasize_val}. \n\nUsing the discussed results, we make three major findings: (i) the proposed differential sharing mechanism does not significantly increase transaction costs, (ii) market conditions could affect the effectiveness of CTI sharing, and (iii) the proposed integrity hashing mechanisms (cf. Fig.~\\ref{fig:intgrity-hashes}) do not create large computational overheads, with the multi hash approach being more efficient for CTI data size less than 150KB. \n\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{table}[]\n    \\centering\n    \\caption{Gas cost and price in US\\$ for each \\textit{data storage} smart contract function.}\n    \\label{tab:cost}\n    \\scalebox{1}{\n    \\begin{tabular}{cccc}\n        \\toprule \n        Function & Gas Cost & Current (US\\$) & High (US\\$)  \\\\\n        \\toprule \n        Share & 43,897 & \\$0.96 & \\$210.22 \\\\ [0.5ex]\n        Request & 66,628 & \\$1.46 & \\$319.07 \\\\ [0.5ex]\n        Response & 50,625 & \\$1.11 & \\$242.44 \\\\ [0.5ex]\n        \\textbf{Total} & \\textbf{161,150} & \\textbf{\\$3.53} & \\textbf{\\$771.73} \\\\\n        \n        \\bottomrule\n    \n    \\end{tabular}\n    }\n\\end{table}\n\n\\begin{table}[]\n    \\centering\n    \\caption{Summary of computational times associated with Figs.~\\ref{fig:sample_both}-\\ref{fig:datasize_val}. (Gen = Generation Time, Val = Validation Time, SH = Single Hash, MH = Multi Hash).}\n    \\scalebox{0.94}{\n    \\begin{tabular}{cccccccc}\n        \\toprule\n   \n        Data & Size & Baseline & Number of & \\multicolumn{2}{c}{Gen (ms)} & \\multicolumn{2}{c}{Val (ms)} \\\\\n        {Type} & {(KB)} & {(ms)} & {Group} & SH & MH & SH & MH \\\\\n        \\toprule     \n        \\multirow{3}{*}{Sample} & \n            \\multirow{3}{*}{94} & \n                \\multirow{3}{*}{0.24} &\n                10 & 0.38 & 1.69 & 0.27 & 0.26 \\\\ [0.5ex]\n                & & & 20 & 0.51 & 3.30 & 0.30 & 0.27  \\\\ [0.5ex]\n                & & & 50 & 0.89 & 8.10 & 0.36 & 0.26  \\\\\n                \n        \\midrule\n        \n        \\multirow{9}{*}{Synthetic} & \n                \n            \\multirow{3}{*}{50} & \n                \\multirow{3}{*}{0.14} &\n                5 & 0.22 & 0.54 & 0.16 & 0.16   \\\\ [0.5ex]\n                & & & 20 & 0.41 & 1.9 & 0.2 & 0.17  \\\\ [0.5ex]\n                & & & 50 & 0.78 & 4.6 & 0.26 & 0.17  \\\\ [0.5ex]\n              \\cline{2-8} \n            & \\multirow{3}{*}{200} & \n                \\multirow{3}{*}{0.55} &\n                5 & 0.67 & 2.1 & 0.61 & 0.83  \\\\ [0.5ex]\n                & & & 20 & 0.88 & 7.2 & 0.66 & 0.83  \\\\ [0.5ex]\n                & & & 50 & 1.2 & 17 & 0.72 & 0.83  \\\\  [0.5ex]\n                  \\cline{2-8} \n            & \\multirow{3}{*}{500} & \n                \\multirow{3}{*}{1.4} &\n                5 & 1.6 & 5.1 & 2.5 & 1.9  \\\\ [0.5ex]\n                & & & 20 & 1.8 & 18 & 1.5 & 1.9  \\\\ [0.5ex]\n                & & & 50 & 2.2 & 45 & 1.6 & 2.1  \\\\\n               \n        \\bottomrule\n    \\end{tabular}}\n    \\label{tab:compute_results}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Privacy and Trust Analysis}\n\\label{privacy-trust}\nNow we examine a number of privacy and trust-based considerations with our proposed framework. From a CTI producer's perspective, it is important that sensitive data is shared in a privacy-preserving way. Subsequently, the proposed differential approach seeks to provide a more granular approach that maintains privacy at rest and in transit. To ensure that privacy is maintained at rest, sensitive data groups are stored securely off-chain by CTI producers, and therefore cannot be accessed by untrusted entities. Moreover, when sensitive data groups are exchanged with trusted CTI consumers, encryption is used to ensure that \\textit{man-in-the-middle} and \\textit{eavesdropping} attacks (e.g., sniffing or snooping) are not possible. Finally, the use of one-time random nonce's as part of the integrity hash generation and validation process, ensures that the privacy of sensitive data common across multiple sharing instances or which contains low entropy (e.g., software versions), are protected from dictionary and brute force attacks. \n\n\nFurther, two major trust-based factors must also be considered. Firstly, CTI producers must be provided with a dynamic approach to trust management that ensures that sensitive data is only shared with trusted entities. Subsequently, our framework allows CTI producers to define a trust-based policy\/metric scheme that best suits their requirements. Finally, the verifiability of CTI data from the perspective of CTI consumers must also be considered. For say, if a CTI consumer receives a subset of the CTI data, it could be possible for a malicious CTI producer to modify this data depending on the identity of the requesting organisation (e.g., a competitor). In such a case, to protect CTI consumers from this attack, our proposed integrity hashing schemes ensure that CTI consumers can validate that the received subset of the CTI data is contained within the original set of CTI data shared by the CTI producer. \n\n\n\n\n\n\n\\section{Conclusion}\n\\label{conclusion}\n\nWe have proposed a blockchain-based CTI sharing framework that allows for the trusted, verifiable, and differential exchange of CTI between producers and consumers. We refer to differential as the ability of a CTI producer to control the amount of information exchanged with a CTI consumer. The framework thus allows CTI producers to segment CTI data into sensitive data groups, which can be variably shared with CTI consumers in a differential manner.\nOur results showed that the proposed framework could facilitate the exchange of CTI between producers and consumers in a differential way without compromising trust or variability. An interesting direction for future work is to conduct a more comprehensive study to examine the impact of incentives on CTI sharing. \n\n\n\\section*{Acknowledgment}\nThe authors acknowledge the support of the Commonwealth of Australia and Cybersecurity Research Centre Limited.\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\\input{sections\/introduction}\n\n\\section{Related Work}\n\\label{sec:related}\n\\input{sections\/related}\n\n\\section{Mathematical background}\n\\label{sec:mathback}\n\\input{sections\/background}\n\n\\section{Method}\n\\label{sec:method}\n\\input{sections\/method}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\input{sections\/experiments}\n\n\\section{Conclusions and Future Work}\n\\label{sec:conclusion}\n\\input{sections\/conclusions}\n\n\n\\subsubsection{Acknowledgments.} We thank Bharat Bhatnagar, Verica Lazova, Ilya Petrov and Garvita Tiwari for their help and feedback. This  work  is  partly  funded  by  the DFG - 409792180 (Emmy Noether Programme, project:  Real Virtual Humans), German Federal Ministry of Education and Research (BMBF): T\u00fcbingen AI Center, FKZ: 01IS18039A and ERC Consolidator Grant 4DRepLy (770784),\nthe EU Horizon 2020 project RICAIP (grant agreeement No.857306), and the European Regional Development Fund under project IMPACT (No. CZ.02.1.01\/0.0\/ 0.0\/15\\_003\/0000468). Gerard Pons-Moll is a member of the Machine Learning Cluster of Excellence, EXC number 2064\/1 - Project number 390727645.\n\\clearpage\n\n\\input{supplementary_arxiv}\n\n\\clearpage\n\n\\bibliographystyle{splncs04}\n\n\n\n\\subsubsection{Limitations and future work.}\nTo the best of our knowledge, our approach is the first to tackle the problem of human-object interaction from egocentric data while positioning both in a scene. \nIt thus represents a first and important step towards novel immersive applications, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, by enhancing AR\/VR experiences via interaction, and thus towards the recently often-mentioned Metaverse~\\cite{Aria}. \nYet, many challenges still need to be overcome: \ncurrently, our approach focuses on a single object movement which is segmented semi-automatically. Ultimately, the goal should be tracking every dynamic motion happening in the scene, without the need to guide the segmentation.\nUsing learned human motion priors, it would be also interesting to capture interactions using a reduced set of body-worn IMUs~\\cite{yi2021transpose,SIP,DIP:SIGGRAPHAsia:2018}. \nFurther, automatically anticipating the time and location of contacts, as well as introducing learned human-object interaction priors to handle more \ncomplicated activities and avoid interpenetration are important directions for future work. \nTo promote future research in this new exciting direction, we will release our datasets and code.\n\\subsubsection{Ethical implications.} As with any approach that analyzes human behavior, there is potential for misuse. \nThis includes, but is not limited to, surveillance, building profiles of individual users based on their interactions (and selling that data, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, for advertisement purposes), and identifying individuals based on characteristic interactions and movements. \nOur approach is based on ego-centric data, which, in combination with on-device processing, offers a user some level of control over what data is shared with other parties. \nFurther, one could report generic rather than individual body poses to make it harder to prevent identifying individuals based on their movement. \nStill, it is not possible to prevent user surveillance or data aggregation as a post-processing step in later stages of an applications. \nWe believe that these issues would best be addressed by law makers.\n\n\n\\subsubsection{Comparisons.}\nTo prove the usefulness of being able to adapt to dynamic environment, we compared our method to the baseline human-only localization method from Sec.~\\ref{subsec:meth_bodyloc}, which is essentially an improved version of HPS~\\cite{guzov2021human} featuring the revised localization pipeline and minor adaptations to improve stability. We refer to it as \\textbf{HPS~(Improved)}.\nTo show the advantage of using camera and IMU information jointly, we designed the version of the algorithm which does not use body motion to infer the object position and only relies on the visual object localization, which is a combination on the improved HPS algorithm and a visual object localization. As it is not possible to visually localize the object during the interaction, the algorithm linearly interpolates the object motion between starting and ending positions. We refer to this method as~\\textbf{HPS (Improved) + Visual object localization}.\nNote that the body localization part (Sec.~\\ref{subsec:meth_bodyloc}) still uses information from both the camera and the IMUs, but the human-object interaction refinement step~\\ref{subsec:meth_joint} is not applied. \nThe results are presented in Fig.~\\ref{fig:with_wo_tracking} and supplementary video.\nWe observe that, compared to the HPS baseline, \n``HPS (Improved) + Visual object localization'' method improves performance, however, as it is only possible to reliably localize the object before and after the interaction, the interpolation between those positions does not reflect the real motion (see supp. video).\nA lack of pose correction also leads to inaccurate contact position. HOPS benefits from both visual and motion clues resulting in the most accurate representation of a dynamic scene.\n\n\n\\subsection{Human body localization} \n\\label{subsec:meth_bodyloc}\n\\subsubsection{IMU-based pose estimation.}\nCurrent IMU-based pose estimation solutions, such as XSens~\\cite{xsensawinda} used in our setup, allow us to get a continuous human pose estimate using a wearable and small IMU sensors. Xsens system consists of 17 IMUs attached to the body with velcro-straps. Using proprietary algorithm based on the Kalman filter and kinematic body, the system provides human pose $\\tilde{\\pose}^I_t$ and translation estimates $\\tilde{\\trans}^I_t$ in its own coordinate frame. While these estimates can be considered accurate, the system has two major flaws: 1) motions are not registered within the 3D scene and 2) these estimates accumulate significant drift over time, especially in the global body position and orientation components.\n\\newline\n\\textbf{Head-mounted camera localization.} \nFor every camera frame, we get the initial head pose estimates using  hierarchical localization algorithm~\\cite{sarlin2019coarse}, which operates on RGB images. Given the dataset of images with known position in the pre-scanned scene, the algorithm establishes correspondences between those images and frames coming from the camera and finds the position of the the head camera $\\tilde{\\mat{C}}_t$ by minimizing the reprojection loss.\n\\newline\n\\textbf{Combining camera and IMU.} \nInspired by HPS~\\cite{guzov2021human}, we use a data from the head-mounted camera to register the body within the scene and correct the IMU drift. \nOur body localization module uses these head camera locations $\\tilde{\\mat{C}}_t$ and initial IMU estimates $(\\tilde{\\pose}^I_t, \\tilde{\\trans}^I_t)$ to derive the body position within the scene and correct body and camera positions using the bending energy minimization introduced in Sec.~\\ref{subsec:meth_control}. As a result, we get improved camera locations $\\mat{C}_t$ and human pose parameters $(\\pose^I_t, \\trans^I_t)$ registered within the scene. Please see the supplementary for more detailed description of the algorithm. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/method\/bending.pdf}\n        \\caption{\\textbf{Trajectory and pose parameters fitting with bending energy.} A) bending trajectories with $F_{tr}$ using different rigidness coefficients $\\lambda$, purple marks the original trajectory, green marks the result; B) Visualization of the SMPL model after fitting parameters with $F_{\\pose}$ to minimize the distance between the hand and the control point. Orange dots denote control points.}\n        \\label{fig:meth_physics_simulation}\n\\end{figure} \n\n\\subsection{Object localization}\n\\label{subsec:meth_objloc}\n\\subsubsection{Robust visual object localization for anchor keyframes.}\n\\label{subsec:meth_objvisloc}\n\nIn contrast to localizing the human head camera which is in permanent motion, the objects we consider are static and can only move during the interaction with the human. \nWe use this observation and restrict object visual localization to detecting a few anchor keyframes, which will serve as control points during interaction reasoning here and in  Sec.~\\ref{subsec:meth_joint}.\n\nAfter localizing the camera $\\mat{C}_t \\in \\sethree$ for every frame $t \\in T^C,\\, T^C \\subset \\real$, we compute the object locations $\\mat{O}^C_t \\in \\sethree$ relative to the camera. Here, we use the same camera localization method from Sec.~\\ref{subsec:meth_bodyloc}, but use only keypoints inside the object mask (both for the head camera frame, and the database images).\nThe position and orientation of the object $\\mat{O}_t$ in world coordinates is obtained as: $\\mat{O}_t=\\mat{C}_t(\\mat{O}^C_t)^{-1}$.\nSince objects only move when the human interacts with them, we leverage temporal information to accurately locate them. We split object positions in $K+1$ groups, where $K$ is the expected number of dynamic interaction intervals. Each group contains object positions detected before interaction interval $K+1$ and after $K$. To filter the noise, for each group, we run the clustering algorithm DBSCAN~\\cite{dbscan} on the translation part of every location in the group and leave only the largest cluster. We compute the mean position per group, and sort them temporally, which gives us $K+1$ reliable positions $\\{\\mat{O}_t\\}_{t=1}^{K+1}$.\n\n\\subsubsection{Tracking objects from human motion, even when they are not visible.}\n\\label{subsec:meth_imuloc}\n\nThe set of reliable object localization estimates $\\{\\mat{O}_t\\}_{t=1}^{K+1}$ from the previous paragraph is limited -- the object should be visible and contain distinct features on the surface. As we are dealing with real-world interactions, we cannot guarantee that these conditions are met during the interaction.\nThe key idea here is to leverage the human motion and the physical constraints of object motion to address an otherwise impossible task: tracking objects that are not visible. Every object type has its own physical behaviour and restrictions -- \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot some of the objects can be carried everywhere, but some of them are fixed or cannot be lifted and can only be dragged. Therefore the solution is class-specific: here, we describe our approach for objects moved by hands along the floor plane (chairs, tables, \\emph{etc}\\onedot} \\def\\vs{\\emph{vs}\\onedot), or fixed to a vertical hinge (doors), but the method can be extended to the other object classes with different physical restrictions (drawers, sliding doors, \\emph{etc}\\onedot} \\def\\vs{\\emph{vs}\\onedot).  \n\n\\paragraph{Human trajectory.} For each hand marked to be in contact at the start of the interval (input to our method), we find the closest point to the hand on the object $\\vec{c}^i, i\\in \\{L,R\\}$ (left, right) and mark it as a contact. Then we compute the offset we need to apply to the human root joint such that hands $\\vec{h}^i$ are in contact by $r_\\mathrm{offset}=\\frac{1}{2}\\sum_{i\\in\\{L,R\\}}(\\vec{c}^i-\\vec{h}^i)$ in case of two hands or simply $r_\\mathrm{offset}=(\\vec{c}^i-\\vec{h}^i)$ in case of one hand. \nThe desired SMPL root location $r_{\\mathrm{root}}$ at the start of interaction is therefore $\\vec{p}_1=r_{\\mathrm{root}}+r_\\mathrm{offset}$. Hence, we deform the full trajectory using the method described in Sec.~\\ref{subsec:meth_control} with a single control point $\\hat{\\trans}^I = F_{tr}(\\trans^I, \\{\\vec{p}_1\\})$.\n\\paragraph{Tracking objects without seeing them.}\nFor \\emph{objects dragged on the floor} with two hands, \nwe set the object translation $\\Delta\\vec{t}^O=\\Delta\\vec{h}^R$ to the translation of the right hand $\\Delta\\vec{h}^R$ (the choice of left or right is mathematically equivalent). The change of rotation $\\Delta\\mat{R}^O$ is determined by the angle change between left and right hand joints. If the object is dragged with one hand, then $\\Delta\\mat{t}^O=\\Delta\\vec{h}^{X}$, where $X=L$ or $R$ depending on the hand in contact. The rotation cannot be determined from 1 contact point and remains unchanged, however it will be interpolated if visual localization is available (Sec.~\\ref{subsec:meth_joint}).\nFor the \\emph{objects attached to vertical hinge} \nwe restrict inference to a planar rotation $\\mat{R}(\\alpha)$ along the hinge with angle $\\alpha$. This is done by computing the angle between previous and current hand to hinge joint vectors $\\Delta \\alpha = \\widehat{(\\vec{v}_i, \\vec{v}_{i-1})}, \\vec{v}_i=\\vec{h}^X_i-\\vec{j}$, where $\\vec{j}$ is hinge joint position on the floor, and $X=L$ or $R$ depending on the hand in contact.\n\n\\subsection{Human-object interaction refinement: Inferring human and object motion coherently with object localization}\n\\label{subsec:meth_joint}\nWith the previously explained human-object reasoning method, we can track objects, but the human motion might drift during the interaction (since motion relies on IMU). Here, we can leverage the object localized at the end of the interaction (note that only one good visual localization frame is needed to localize the object after interaction) to self-correct both the human and object trajectories. \n\nWithout loss of generality, we explain our method for a single interaction interval. Let $\\mat{O}_s$ and $\\mat{O}_e$ be the object visual localizations at the start and end of the interaction, obtained using the method in Sec.~\\ref{subsec:meth_objvisloc}.\nGiven the starting contact positions $\\vec{c}_s$ from Sec.~\\ref{subsec:meth_imuloc} (to simplify notation we use $\\vec{c}_s$ to denote the contact location of the left, right hand or both depending on the situation), we determine the contact points at the end of the interaction as $\\vec{c}_e = {{\\mat{O}}_e}{{\\mat{O}}_s}^{-1}\\vec{c}_s$.\n\nNext, we deform the hand trajectories $\\vec{h} = \\{c^\\mathrm{body}(\\pose_t,\\trans_t)\\}_{t=s}^e$ to fit start and end control points $\\{\\mat{c}_s, \\mat{c}_e\\}$ using again the bending energy $\\hat{\\vec{h}} = F_{tr}(\\vec{h}, \\{\\vec{c}_s, \\mat{c}_e\\})$ (Fig.~\\ref{fig:overview},~D). If the object has a hinge, hand trajectories $\\hat{\\vec{h}}$ are additionally projected to the subspace of possible trajectories for this object.\nThe resulting pose trajectory is then obtained by deforming the pose estimates to fit the newly found hand trajectories as\n $\\pose, \\trans = F_{\\pose}(\\pose^I, \\hat{\\trans}^I, \\hat{\\vec{h}})$.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/exp\/exp_comparison.pdf}\n\t\t\\caption{\\textbf{Effect of different components of HOPS.} Here, the different interactions are shown. While other methods are either losing contact with the object, or does not track the object at all, HOPS ensures that the object and the body are both correctly positioned and interacting in a realistic way.}\n\t\t\\label{fig:with_wo_tracking}\n\\end{figure} \n\n\n\n\\subsection{Capturing setup}\nWe use a combination of Apple iPhone 12 with a head mount to capture first-person view and Xsens Awinda~\\cite{xsensawinda} IMU system to capture body motions.\nThe iPhone 12 camera is capturing at a resolution of $1920 \\times 1440$, $30$ FPS. While the camera features automatic stabilization, we turn it off explicitly to get a better idea of the head motion. We use OpenCV~\\cite{opencv_library} to get the camera intrinsics.  \nXsens Awinda consists of 17 IMU sensors attached to the body with velcro-straps and captures the body motion at $60$ FPS. The camera and IMU systems record their data independently, for that reason the data need to be temporally and spatially syncronized (\\emph{cf}\\onedot} \\def\\Cf{\\emph{Cf}\\onedot supp. Sec.~\\ref{subsec:meth_imucam_calib}) afterwards.\n\nXsens Awinda outputs the data in form of skeleton joint angles which is not directly compatible to SMPL~\\cite{smpl2015loper} pose parameters format. To convert between these formats we developed the following retargeting algorithm: we export the motion from Xsens internal format (MVN) into Autodesk FBX format and use Autodesk FBX Python SDK~\\cite{autodesk_fbx} to read and extract rotation for each joint in form of quaternions. We convert those quaternions to axis-angle representation used in SMPL. We map joint rotations from FBX skeleton to SMPL skeleton according to manually designed mapping. As a result, we can produce SMPL pose parameter vector by concatenating the mapped axis-angle joint rotations in the right order. While this method works well in most cases, we notice some pose artifacts during specific motions, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot the stomach is sometimes bent too much forward while the person leans forward to grab an object (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot supp. Fig.~\\ref{fig:exp_gt_vs_reproduction}). This artifact appears in all of the methods we explored (HPS (Improved), HPS (Improved) + Visual object localization, HOPS) because the error happens on the data preprocessing stage and does not depend on the chosen method. One possible way to improve the quality of retargeting is to switch to fiting the SMPL model to chosen Xsens skeleton joints instead of copying joint rotations directly~--~we aim to explore this in the future work. \n\n\\subsection{Algorithms implementation and performance}\nWe implement our algorithms on Python using Pytorch~\\cite{NEURIPS2019_9015} library for the visual localization pipeline and the bending energy optimization algorithm. For optimization of trajectories and pose parameters with bending energy (eq.~\\refg{2} and~\\refg{5} in the paper), we use Adam~\\cite{kingma2014adam} optimizer with number of iterations, learning rate and rigidness coefficient $\\lambda$ acting as hyperparameters. \n\nPerformance-wise, the most computationally expensive part of our algorithm is the visual localization pipeline, requiring $8$ seconds per frame to run on NVIDIA Q8000 GPU. Other parts of the algorithm does not add significantly to this time estimation. We use such computationally expensive pipeline to achieve the best localization results as the performance of visual localization network itself is not the aim of our study. We expect that this pipeline can be replaced by less demanding alternative, but this can lead to a localization quality drop.\n\\subsection{Human body localization based on a head-mounted camera and IMUs}\n\\label{subsec:meth_visloc}\nTo effectively reason about object contact and fine-grained interactions, we need a good initialization of the human body pose first. For that, we need to register the IMU pose estimation within the scene coordinate system.\nTo achieve so, we use a 5-step initialization method: \n    \\textbf{Step 1: Initial camera estimates.} To align IMU coordinate frames with the world and camera coordinate frames, we compute initial camera localization estimates at each frame $\\tilde{\\mat{C}}_t$. The localization method is similar to the one used in HPS~\\cite{guzov2021human} with several modifications. \n    For each camera frame time $t$, we detect keypoints~\\cite{detone2018superpoint} and match them with keypoints with known 3D position obtained using the prescanned scene. This is done with the SuperGlue~\\cite{sarlin2020superglue} matching network, which outputs matches $\\{m^i_t\\}_{i=1}^M $ along with their confidence scores $\\{s^i_t\\}_{i=1}^M$. To find the camera location, we minimize the reprojection loss of the matching 3D points w.r.t\\onedot} \\def\\dof{d.o.f\\onedot the camera pose.\n    We save the average matching confidence $s_t=\\frac{1}{M} \\sum_{i=1}^M s^i_t$ to reason about the localization precision in our pipeline. \n    \n    \\textbf{Step 2: Finding the mapping between IMU, world and camera coordinate frames.}\n    IMU poses $\\tilde{\\pose}^I_t$ and translation estimates $\\tilde\\trans^I_t$ obtained from the IMU mocap system~\\cite{xsensawinda} need to be transformed to the world coordinate frame (\\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, the pre-scanned 3D scene). Specifically, we find transformation matrices from the global IMU coordinate frame ``$Z$\" to scene ``$W$\") $\\mat{X}^{WZ} \\in \\sethree$ and local (camera ``$C$\" to IMU sensor ``$I$\") $\\mat{X}^{IC} \\in \\sethree$ coordinate frames -- see supp. Sec.~\\ref{subsec:meth_imucam_calib} for details. With this we can obtain the absolute head camera pose estimated by the head IMU, as $\\mat{H}^W_t=\\mat{X}^{WZ} \\mat{H}^Z_t \\mat{X}^{IC}$, where $\\mat{H}^Z_t \\in \\sethree$ are the raw head IMU positions. \n    \n    \\textbf{Step 3: Improving the camera estimates.} We rerun the camera localization with the search space for 2D-3D matches restricted to a vicinity of the current local scene in the human field of view. To this end, we filter out matches based on the IMU head pose estimates $\\mat{H}^W_t$. We render a depth map of the camera view as seen from the head IMU; with the depth map, we filter out all keypoints in the database images that are not visible in this virtual view. This drastically restricts the search space from the full building database to only the current view. \n     Localizing the camera with only filtered keypoints results in camera estimates $\\mat{C}_t$ (and their corresponding scores $s_t$), which are significantly more accurate than the ones in Step 1. \n    Hence, we also recompute $\\mat{X}^{WZ}$ and $\\mat{X}^{IC}$ with $\\mat{C}_t$ to increase the IMU-to-scene mapping precision.\n    \n    \\textbf{Step 4: Correcting IMU drift with visual localizaton.} Pose estimation inferred from the IMUs can drift over time. We solve this problem by minimally bending the trajectory of the head from the IMU pose estimation to comply with the head-mounted camera localizations. For that, we consider only reliable camera estimates $\\mat{C}_t$ with confidence $s_t\\geq0.5$, which are used as a set of control points in the bending energy minimization framework described in Sec.~\\refg{3} in the paper:\n    \\begin{equation}\n    \\widehat{\\trajectory}^{H} = F_{tr}(\\trajectory^{H}, \\{\\mat{C}_t|s_t \\geq 0.5\\}) \\enspace ,\n    \\end{equation}\n    where $\\trajectory^{H} = \\{\\trajectory^{H}_t\\}$ is the translation part of the head position $\\mat{H}^W_t$. \n    The corrected head trajectory and initial IMU pose estimates $\\tilde{\\pose}^I_t, \\tilde\\trans^I_t$ are used in the next step to produce the human body pose and translation parameters $\\pose^I_t,\\gamma^I_t$.\n    \n    \\textbf{Step 5: SMPL parameters adaptation to the new IMU trajectories.}\n    After correcting the head IMU trajectory with camera localization, we adapt SMPL parameters $\\pose^I_t$, $\\trans^I_t$ to fit the new trajectory. We initialize $\\pose^I_t$, $\\trans^I_t$ with $\\tilde{\\pose}^I_t, \\tilde\\trans^I_t$. We map the parameters from the global IMU coordinate frame ``$Z$'' to the scene coordinate frame ``$W$'' with $\\mat{X}^{WZ}$. Translation part is mapped as $\\trans^I_t = \\mat{X}^{WZ}\\tilde\\trans^I_t$ and the body is rotated according to the rotation part of $\\mat{X}^{WZ}$ by modifying the first 3 values of $\\pose^I_t$ (which correspond to the global body rotation).\n    \n    Next, we take the original trajectory $\\trajectory^{H}$ and the refined trajectory $\\hat{\\trajectory}^{H}$ and compute difference in rotation and translation. Our goal is to apply the same rotation and translation difference to the SMPL parameters $\\pose^I_t$, $\\trans^I_t$. To do so, we treat trajectories as a discrete set of points $\\trajectory^{H} = \\{\\trajectory^{H}_i\\}$. As a result, for each SMPL pose and translation parameter set $\\pose^I_i$, $\\trans^I_i$ there is a corresponding trajectory point $\\trajectory^{H}_i$ (same for $\\hat{\\trajectory}^{H}_i$). We  project trajectories to the XY plane, zeroing out Z axis component (our experiments show that Z axis coordinate component usually does not require correction). For each SMPL pose index $i$ we compute the trajectory pose and rotation angle change: $\\Delta \\trans^I_i = \\hat{\\trajectory}^{H}_i - \\trajectory^{H}_i$, $\\Delta \\alpha_i = (\\widehat{\\hat{v}_i, v_i}), \\hat{v}_i = \\hat{\\trajectory}^{H}_i - \\hat{\\trajectory}^{H}_{i-1}, v_i = \\trajectory^{H}_i - \\trajectory^{H}_{i-1}$. The translation parameters are updated as $\\trans^I_i=\\trans^I_i+\\Delta \\trans_i$. The first 3 pose parameters $\\pose^I_i$ corresponding to the root SMPL joint are modified so that the root joint rotates by $\\Delta \\alpha_i$ around the Z axis. The resulting human pose parameters $(\\pose^I_i,\\trans^I_i)$ are drift-free and registered within the scene.\n\n\\subsection{IMU-Camera coordinate frames calibration.}\n\\label{subsec:meth_imucam_calib}\nAs stated in supp. Sec.~\\ref{subsec:meth_visloc}, Step 2, to use IMU pose estimations we need to find the mapping between the IMU and camera local and global coordinate frames. Using the notation introduced in supp. Sec.~\\ref{subsec:meth_visloc}, we first find transformation matrices from the global IMU coordinate frame ``$Z$\" to scene ``$W$\" $\\mat{X}^{WZ} \\in \\sethree$.\nFor that, we rigidly align a sequence of head IMU positions in 3D to the initial noisy camera positions (without considering the orientations). In our data, the Z axis is aligned to the gravity vector for both IMU frame ``$Z$'' and the scene frame ``$W$'', so we project IMU and camera positions to the XY plane and only perform the alignment in the XY plane.\nTo align IMU positions to noisy camera observations, we use RANSAC~\\cite{ransac} algorithm, which features noise detection in the data. To synchronize camera and IMU timings we repeat RANSAC fitting with several time offsets, performing a grid search; the time synchronization is corrected manually if needed. \n\nNext, we obtain transformation matrix between the local (camera ``$C$\" to IMU sensor ``$I$\") $\\mat{X}^{IC} \\in \\sethree$ coordinate frames. For this, we select $K=5$ camera positions $\\{\\tilde{\\mat{C}}_k\\}_{k=1}^{K}$ (transformations from local to global camera coordinate frame). We select the camera positions by measuring the distance between all the camera translations and the IMU head translations $\\{\\mat{X}^{WZ}\\vec{p}^{H}_t\\}$ mapped to the scene coordinate frame, where $\\vec{p}^{H}_t$ is the translation part of the raw head IMU position $\\mat{H}^Z_t$. We select $K$ camera positions with the smallest distance. We also select the corresponding head IMU positions $\\{\\mat{H}^Z_k\\}_{k=1}^{K}$. For each $k=1..K$, we compute the local camera to IMU transformation $\\hat{\\mat{X}}^{IC}_k$: $\\hat{\\mat{X}}^{IC}_k = {\\mat{H}^Z_k}^{-1}(\\mat{X}^{WZ})^{-1}\\tilde{\\mat{C}}_k$. To obtain the target $\\mat{X}^{IC}$, we average all $\\hat{\\mat{X}}^{IC}_k$ with the following algorithm: As $\\hat{\\mat{X}}^{IC}_k \\in \\sethree \\forall k=1..K$, each of them can be subdivided to rotation matrix $\\hat{\\mat{R}}^{IC}_k \\in SO(3)$ and translation vector $\\hat{\\vec{t}}^{IC}_k \\in \\real^3$. For translation, we simply find an arithmetic average $\\vec{t}^{IC} = \\frac{1}{K}\\sum_{k=1}^{K}\\hat{\\vec{t}}^{IC}_k$, for rotation $\\mat{R}^{IC}$ we use chordal $L_2$ mean~\\cite{hartley2013rotation}. Averaged rotation and translation $(\\mat{R}^{IC}, \\vec{t}^{IC})$ form the resulting transformation $\\mat{X}^{IC}$.\n\n\\textbf{SMPL parameters optimization details.}\nTo ensure smooth parameters change while optimizing SMPL parameters to fit the trajectory during human-object interaction refinement (Sec.~\\refg{4.3} in the paper), we additionaly allow $w$ parameters before and after the interaction interval to change ($w$ is a hyperparameter), therefore avoiding abrupt transition. Additionaly, to stabilize motion we restrict the parameters change only to particular joints, namely hands, arms, shoulders and pelvis joints.\n\\section{Method details}\n\\label{supp_sec:method}\n\\input{supp_sections\/method}\n\n\\section{Implementation details}\n\\label{supp_sec:imp}\n\\input{supp_sections\/implementation}\n\n\\section{Additional experiments and experimental setup}\n\\label{supp_sec:exp}\n\\input{supp_sections\/experiments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:Intro}\nThe reconstruction of 3D shape and appearance is among the classical challenges in computer vision.\nWhile we have witnessed significant progress on this task with suitably designed neural representations, the resulting reconstructions of shape and appearance are typically limited by the resolution of the input images where high-quality models invariably require high-resolution input images.\nAt the same time, the concept of super-resolution modeling has been studied intensively in classical variational optimization approaches.\nThe aim of this work is to bring both of these developments together and introduce super resolution modeling into neural differentiable volume rendering approaches in order to allow high-resolution reconstructions of 3D shape and reflectance even from low-resolution input images -- see Figure~\\ref{fig:teaser} for high quality geometry and super-resolution image reconstruction on two real-world datasets.\n\\input{figures\/01teaser}\nMore specifically, we consider a setting where we capture images of an opaque non-metallic object from various viewpoints. The so called \\textit{photometric} images are illuminated only from a white point light source co-located with the camera. We want to separately recover geometry and different components of an isotropic BRDF.\nNote that due to the nature of the setup, we can only reconstruct the slice of the BRDF for which illumination direction is equal to viewing direction.\nThe images are assumed to be calibrated, i.e. camera ex- and intrinsics are given or precomputed using e.g. COLMAP~\\cite{schonberger2016pixelwise}, which largely follows the works of~\\cite{mildenhall2021nerf,zhang2021physg,zhang2022iron}.\n\\\\\nIn detail, our contributions are as follows: \n\\begin{itemize}\n    \\item Given a set of photometric images, we put forward an end-to-end inverse rendering approach for jointly estimating high quality shape of arbitrary topology and its corresponding reflectance properties.\n    \\item Within the image formation model, we explicitly parameterize the degradation process induced by the camera's sensor via modeling its point spread function, allowing us to estimate super-resolved shape and material properties.%\n    \\item In numerous experiments, we validate that the proposed method gives rise to state-of-the-art reconstructions of shape and reflectance.\n    In particular, the reconstructed objects are significantly more detailed and sharper than the individual input images. \n\\end{itemize}\n\n\\section{Related Work}\n\\label{sec:Related_work}\nIn the following we recall neural inverse rendering and view synthesis as well as super-resolution approaches for 3D reconstruction.\n\n\\subsection{Neural Inverse Rendering and View Synthesis}\n\n\n\n\n\nNeural approaches for inverse rendering~\\cite{kaya2021uncalibrated, kellnhofer2021neural,mueller2022instant,sengupta2019neural,taniai2018neural,tewari2022advances,tewari2020state,zhang2022modeling} and novel view synthesis~\\cite{gao2021dynamic,hedman2021baking,li2021neural,liu2021neural,mildenhall2021nerf,peng2021neural,riegler2021stable,tancik2022block,wizadwongsa2021nex} have gained a lot of attention over the recent years.\nTheir expressivity within a lightweight architecture such as a multilayer perceptron (MLP) form a great basis for these complex tasks~\\cite{liu2020neural,martin2021nerf,mildenhall2021nerf, pumarola2021d,wang2022nerf,yu2021pixelnerf,zhang2020nerf++}.\nHowever, those approaches can only recover geometry of reduced quality due to their underlying volume rendering\nbased on the scene's density.\nIn contrast, surface rendering approaches~\\cite{niemeyer2020differentiable,yariv2020multiview} perform better when estimating geometry, but require mask supervision and can still get stuck at unsatisfactory local minima with severe geometry artefacts, which limits their usage to relatively simple objects.\nTo get the best out of both worlds, \\cite{oechsle2021unisurf, wang2021neus, yariv2021volume} express the underlying shape implicitly via occupancy fields~\\cite{oechsle2021unisurf} or SDFs~\\cite{wang2021neus, yariv2021volume}.\n\\cite{yariv2021volume} even provide a sampling procedure of the volume integral to theoretically upper bound the opacity error, which is a missing desired property in~\\cite{oechsle2021unisurf, wang2021neus}.\nWhile~\\cite{oechsle2021unisurf, wang2021neus, yariv2021volume} allow to reliably reconstruct accurate geometry of complex scenes without mask supervision, these approaches lack the important ability to estimate other scene properties such as reflectance.\n\nOftentimes it is desired to recover the object's BRDF properties along with the geometry, as this allows for relighting the scene under novel illumination.\nWhile some works focus on relighting based on static, unknown illumination of the scene~\\cite{boss2021nerd,srinivasan2021nerv,zhang2021nerfactor, zhang2021physg}, others enforce a change of illumination~\\cite{bi2020deep,luan2021unified,nam2018practical,zhang2022iron} by resorting to %\nphotometric images. %\nIn particular, this leads to a well-posed optimization problem of the underlying scene properties~\\cite{brahimi2020springer}.\n\nDensity based volume rendering approaches such as~\\cite{bi2020deep,srinivasan2021nerv,zhang2021nerfactor,boss2021nerd} also estimate material parameters, but they inherit the same limitations as~\\cite{mildenhall2021nerf}: \ninaccurate geometry and the fact that the object's material parameters are not properly defined on the surface, but everywhere in the volume.\nThis severely limits the editing capability, and a volume rendering step has to be used during inference, making it unusable in common graphics pipelines.\n\\cite{luan2021unified,nam2018practical} are mesh-based classical inverse rendering approaches and require masks and a good initialization of the geometry, causing their convergence to be fragile, as mesh-based optimization is inherently non-differentiable at depth discontinuities and difficult to handle if topological changes or self-intersections arise.\nMore recently, \\cite{zhang2022iron} proposed an inverse rendering approach which eliminates the disadvantages of mesh-based approaches to some extent due to the use of a neural SDF representation and an edge-aware surface renderer.\nHowever, the weak convergence properties of surface rendering made them resort to a two-step approach, where the first step consists of a volume rendering step~\\cite{wang2021neus} which is used to initialize the second step based on surface rendering.\n\nCompared to prior works using photometric images, our approach is based solely on volume rendering using SDFs, causing reliable geometry and reflectance reconstruction \nwithout resorting to other rendering techniques such as surface rendering, thus avoiding a multi-step pipeline. Additionally, thanks to our novel problem formulation our method is still applicable for standard graphics pipelines, as a mesh and each surface point's reflectance property can be easily recovered.\nNext we discuss the state-of-the-art in super-resolution for 3D reconstruction and how we leverage that to further improve our methodology.\n\n\n\\subsection{Super-Resolution for 3D Reconstruction}\n\n\nThe problem of super-resolution (SR) has been extensively studied in the past~\\cite{anwar2020deep,nasrollahi2014super,park2003super,tian2011survey,van2006image,wang2020deep,yang2019deep,yue2016image}. Different problem statements of SR lead to different approaches, e.g. the case of single image SR~\\cite{glasner2009super,huang2015single,niu2020single}, video SR~\\cite{chan2021basicvsr,chan2022basicvsr++,isobe2022look}, or depth SR~\\cite{riegler2016atgv,haefner2019photometric,sang2020inferring,voynov2019perceptual,yang2007spatial}. Given that we are interested in a 3D reconstruction of the scene from a set of photometric images, we do not perform SR in 2D image space, but in 3D scene space~\\cite{goldlucke2014super,maier2015super,wang2022nerf}.\n\\cite{wang2022nerf} are able to synthesize images of higher resolution than the individual input images by resorting to supersampling, i.e. a low-resolution pixel is sampled at each super-resolution pixel's center, allowing for a denser sampled radiance field.\n\\cite{goldlucke2014super,maier2015super} are classical approaches optimizing SR geometry and textures.\nWhile~\\cite{maier2015super} integrate low-resolution depth and color from and RGB-D sensor into SR keyframes and fuse these keyframes into a texture map,~\\cite{goldlucke2014super} describe their SR process using a convolution with a Gaussian kernel.\nThis is a well motivated image formation model of a camera sensor element and straightforward to carry out as they work in a discrete pixel grid.\nHowever, this not easily applicable in a continuous case, i.e. when using neural networks to implicitly express shape and reflectance. \nWhile~\\cite{goldlucke2014super,maier2015super,wang2022nerf} share the benefit of increasing the input resolution of the individual input images to result in a sharper, high detailed output, they all lack the possibility of representing the scene's intrinsic properties, i.e. shape and material.\nEither the full scene is represented in a neural network~\\cite{wang2022nerf} or the estimated textures consist of lighting cues baked in to the reflectance properties, making faithful relighting impossible~\\cite{goldlucke2014super,maier2015super}.\n\nContrary to the existing SR works mentioned here, we mathematically formulate the camera's image formation process in the continuous case leading to a principled sampling heuristic applicable for neural approaches which allows us to invert the camera's image formation model resulting in reconstructions beyond the input in terms of resolution and quality.\nAdditionally, our model can recover super-resolved geometry and BRDF parameters, free from baked in lighting cues, allowing faithful photorealistic reconstructions with full control over the scene's properties.\n\\section{Preliminaries: VolSDF}\n\n\\label{sec:Preliminaries}\nVolSDF~\\cite{yariv2021volume} leverages volume rendering similarly to NeRF~\\cite{mildenhall2021nerf}, however aims at overcoming certain limitations of NeRF by decoupling geometric representation and appearance.\nTo this end, VolSDF models the scene geometry within a volume $\\Omega\\subset\\mathbb{R}^3$ by means of a density~$\\sigma:\\Omega\\to\\mathbb{R}_{\\geq0}$, which is related to its signed distance function (SDF)\n$d:\\Omega\\to\\mathbb{R}$ by the transformation\n\\begin{equation}\n\\label{eq:preliminaries:density}\n\\sigma(\\mathbf{x}) = \\alpha\\Psi_\\beta(-d(\\mathbf{x})).\n\\end{equation}\nHere, $\\Psi_\\beta$ is the Cumulative Distribution Function of the Laplace distribution with zero mean and scale $\\beta$, and both~$\\alpha, \\beta>0$ are learnable parameters.\n\\input{figures\/03volvssurf}\nGiven this parameterization of the density in terms of the underlying SDF, we can set up the volume rendering equation to obtain irradiance~$\\mathcal{L}_p$ at a pixel~$p\\in\\mathbb{R}^2$ within the image of a camera located at~$\\mathbf{c}\\in\\mathbb{R}^3$. Let~$\\mathbf{v}\\in\\S^2$ be the viewing direction from~$\\mathbf{c}$ through~$p$, then \n\\begin{equation}\n  \\mathcal{L}_p = \\int_{0}^\\infty w(t) L(\\mathbf{x}(t),\\mathbf{n}(t),\\mathbf{v})\\dx{t}\n  \\label{eq:volume_integral},\n\\end{equation}\nwhere we integrate along the ray~$\\mathbf{x}(t) = \\mathbf{c} + t \\mathbf{v}, t \\in \\mathbb{R}$.\nThe weights~$w(t)$ form a probability distribution along the ray~\\cite{yariv2021volume},\nand are given by %\n\\begin{equation}\n  w(t)=\\sigma(\\mathbf{x}(t))\\,\\exp\\left( -\\int_0^t\\sigma(\\mathbf{x}(s))\\dx{s} \\right ).\n\\label{eq:weights}\n\\end{equation}\nFinally, $L:\\Omega\\times\\S^2\\times\\S^2\\to\\mathbb{R}^3$ is the radiance field, which depends on location,\nnormal and viewing direction.\nSince positive values of~$d$ inside the surface are assumed, the normal vector is obtained as %\n$\\mathbf{n}=\\nabla d\/\\norm{\\nabla d}$. %\nIn practice, the integral~\\eqref{eq:volume_integral} is being approximated using the well-known quadrature rule at a discrete set of samples\n$t_1 < t_2 < \\dots < t_{m}$\nfor each pixel,\n\\begin{equation}\n  \\mathcal{L}_p \\approx \\sum^{m-1}_{i=1} (t_{i+1}-t_i)w(t_i)L(\\mathbf{x}(t_i),\\mathbf{n}(t_i),\\mathbf{v}).\n  \\label{eq:discrete_volume_integral}\n\\end{equation}\nNote that the integral in~\\eqref{eq:weights} to compute~$w(t_i)$ is accumulated in a similar way while iteratively computing the sum~\\eqref{eq:discrete_volume_integral}.\nVolSDF represents the scene using two separate MLPs, one is used to describe the SDF, $d_{\\varphi}$ \\textit{and} a global geometry feature map $z_\\varphi:\\Omega\\to\\mathbb{R}^{256}$, while a second MLP is used to describe the radiance $L_{\\psi}$, both with their corresponding learnable network parameters $\\varphi,\\psi$.\nAdditionally to its stable convergence compared to surface rendering approaches~\\cite{niemeyer2020differentiable,yariv2020multiview}, cp. Figure~\\ref{fig:volvssurf}, VolSDF satisfies a the theoretical guarantee to upper bound the opacity error compared to similar approaches~\\cite{oechsle2021unisurf, wang2021neus}.\nSpecifically, after convergence the rendered image showing the geometry of the SDF using a volume renderer is almost indistinguishable from the same rendered image based on a surface renderer using e.g. a sphere tracing algorithm~\\cite{hart1996sphere}.\nThis observation was shown in~\\cite{oechsle2021unisurf} and has the consequence that the object can be rendered using standard surface renderer frameworks~\\cite{li2018differentiable,nimier2019mitsuba,zhang2020path}, as well as an appearance defined \\textit{on} the object's surface, despite being learned as a volumetric quantity.\nNevertheless, VolSDF is unable to recover the reflectance properties, since both the material and lighting are baked into the radiance network.\nHence, the reconstructed 3D model can only be rendered with the same material under the same static illumination.\nIn the next section, we show how we extend VolSDF to enable joint estimation of shape and material, which allows material editing and view synthesis under novel lighting conditions using a traditional graphics pipeline for surface rendering.\n\n\\section{Method}\n\\label{sec:Method}\nWe will first show how to decouple appearance into reflectance and lighting, which allows to estimate a high quality material in addition to the shape.\nFollowing this, we will extend this into a novel framework for super-resolved shape and BRDF estimation to allow rendering of novel views with more detail than the individual input images.\n\n\n\\subsection{Radiance field with explicit BRDF model}\n\nWe express the radiance field~$L(\\mathbf{x},\\mathbf{n},\\mathbf{v})$ in~\\eqref{eq:volume_integral} in terms of the BRDF and lighting using the rendering equation~\\cite{kajiya1986rendering}\n\\begin{equation}\n  L(\\mathbf{x},\\mathbf{n},\\mathbf{v}) = \\int_{H_\\mathbf{n}} L_i(\\mathbf{x},{\\bm\\omega}) f_\\text{r}(\\mathbf{x}, \\mathbf{n}, \\mathbf{v}, {\\bm\\omega}) ( {\\bm\\omega} \\cdot \\mathbf{n}) \\dx{\\bm\\omega}\n  \\label{eq:rendering_equation}\n\\end{equation}\nwhere $H_\\mathbf{n} \\subset \\S^2$ is the half-sphere in direction~$\\mathbf{n}$, $L_i(\\mathbf{x},{\\bm\\omega})$ denotes the radiance incoming at~$\\mathbf{x}$ from direction~${\\bm\\omega}$,\nand $f_\\text{r}$ the spatially varying BRDF (SVBRDF). %\nSince we assume an achromatic point light source co-located with the camera center $\\mathbf{c}$, \\eqref{eq:rendering_equation} simplifies to\n\\begin{equation}\n  L(\\mathbf{x},\\mathbf{n},\\mathbf{v}) = \\dfrac{L_0}{\\norm{x - c}^2} f_\\text{r}(\\mathbf{x},\\mathbf{n},\\mathbf{v},\\mathbf{v}) (\\mathbf{v} \\cdot \\mathbf{n}),\n  \\label{eq:point_light_model}\n\\end{equation}\nwhere $L_0$ corresponds to the scalar light intensity.\nWe use use the simplified Disney BRDF \\cite{karis2013real}, as this provides a compact model expressive enough to represent a wide variety of materials, which was successfully used in several prior works~\\cite{luan2021unified,zhang2021physg}.\nHere, the SVBRDF is parametrized with a diffuse RGB albedo~$\\rho:\\Omega\\to\\mathbb{R}^3_{\\geq0}$,\na roughness $\\alpha_\\text{r}:\\Omega\\to\\mathbb{R}_{+}$ and a specular albedo $\\alpha_\\text{s}:\\Omega\\to[0,1]$.\n\nWe implement these three components using two MLPs. The first MLP~$\\rho(\\mathbf{x};\\brdfparam_1)$\nis used to compute the diffuse component of the BRDF at a point $\\mathbf{x}$,\nthe second MLP $\\alpha(\\mathbf{x};\\brdfparam_2) = \\left(\\alpha_\\text{s}(\\mathbf{x};\\brdfparam_2),\\alpha_\\text{r}(\\mathbf{x};\\brdfparam_2)\\right)$\ncomputes the respective specular components.\nThe combined network parameters for BRDF are denoted with~$\\gamma=(\\brdfparam_1,\\brdfparam_2)$.\nAs mentioned earlier, we model the geometry using a third MLP~$d(\\mathbf{x};\\theta)$\nfor the SDF with its own network parameters $\\theta$. \nNote that we do not incorporate a global geometric feature map into our framework.\nThe main motivation behind such a map is for the radiance field $L$ to account for indirect lighting and self-shadows.\nWe follow the spirit of multiple works~\\cite{bi2020deep,haefner2021recovering,luan2021unified,nam2018practical,zhang2022iron} showing that satisfactory results can be achieved without modeling indirect lighting explicitly, furthermore we can successfully treat these effects as outliers thanks to the robust $L^1$-norm in~\\eqref{eq:data_term}.\nOn the other hand, self-shadows are not present in our captures anyway, due to our co-located camera-light setup.\n\nIn order perform inverse rendering, we train our neural networks from the available input images.\nFor comparing the images to the rendered image in the loss function, we model the physical camera capturing process in the next section and show how this leads to a super-resolved reconstruction of the scene.\n\\subsection{Super-resolution image formation model}\nAs common in many volume rendering based approaches~\\cite{mildenhall2021nerf,yariv2021volume,wang2021neus},\nit is assumed that the image brightness at a given pixel corresponds exactly to the accumulated radiance of the volume rendering~\\eqref{eq:volume_integral},\n\\begin{equation}\n  I_p = \\mathcal{L}_p(\\theta,\\gamma).\n  \\label{eq:image_formation}\n\\end{equation}\nAs described in \\cite{wang2022nerf}, even if a framework can render novel views at any resolution during inference, the performance will significantly decrease when the inference resolution becomes larger than the one of the input images.\nIndeed, during training, the networks are sampled exactly at the pixel locations of the training images, which means that there is no training data for points of the surface whose projections do not coincide with these locations.\nThanks to the interpolation property of neural networks, we can usually still compute a reasonable value for the points which were not seen during training, in particular since they are typically in between points which have been trained.\nHowever, the sampling rate of the input images band-limits the components of the reconstructed SDF and BRDF, and higher frequency details are not magically generated if one only renders at higher resolution.\n\nIn order to restore higher frequency content, we can exploit that the camera capturing process does not only sample the exact centers of the pixels.\nInstead, a camera performs an integration over a subset of incoming rays which can be modeled by the so-called point spread function (PSF).\nIn essence, it describes blur during the capturing process, for example caused by integration across a sensor element, diffraction, lens aberrations, or objects being not perfectly in focus~\\cite{delbracio2013two}.\nThis intrinsic blur can actually be beneficial to avoid aliasing on the captured low resolution images, as it reduces high-frequency content in the captured scene, but leads to unavoidable loss of detail.\n\nTaking this physical process into account, we follow ~\\cite{goldlucke2014super,baker2002limits} and\nconsider the effect of the PSF in our image formation model.\nThus, generalizing~\\eqref{eq:image_formation}, we convolve the accumulated radiance with the~$\\PSF$ kernel\nin order to obtain image irradiance,\n\\begin{equation}\n  I_p = (\\mathcal{L}(\\theta,\\gamma)\\ast \\PSF)(p) = \\int_{\\mathbb{R}^2} \\mathcal{L}_{p-q}(\\theta,\\gamma) \\PSF(q)\\dx q.\n  \\label{eq:psf_image_model}\n\\end{equation}\nFor the special case that the~$\\PSF$ is a Dirac delta distribution, one arrives at the original model~\\eqref{eq:image_formation}, so this is indeed a generalization.\nIn practice, we assume that the PSF is a Gaussian distribution, as~\\cite{peter2001image} have shown the validity of such an approximation, and~\\cite{goldlucke2014super} successfully used it to achieve texture super-resolution.\nIn our experiments, we choose half the size of a pixel in the low-resolution input images as the\nstandard deviation.\nFor computational efficiency, we approximate the convolution in~\\eqref{eq:psf_image_model} by Monte Carlo integration\n\\begin{equation}\n  I_p \\approx \\dfrac{1}{\\sum^{N_s}_{k=1} \\PSF(q_k)} \\sum^{N_s}_{k=1}\\mathcal{L}_{p-q_k}(\\theta,\\gamma) \\PSF(q_k),\n  \\label{eq:psf_discrete_image_model}\n\\end{equation}\nwhere $(q_k)_{k=1\\hdots N_s}$ are uniform samples within a square with a radius of\nthree standard deviations of the PSF. Note that the formula is slightly different from\nthe usual form to make sure we always obtain a weighted average of the radiance samples.\nIn the limit, it also converges to the value of the integral.\nSee Figure~\\ref{fig:srexample} for a comparison of the sampling process based on a Dirac kernel and a Gaussian kernel.\n\\input{figures\/04srexample}\n\n\n\n\n\\subsection{Final training objective}\nOur final objective consists of three terms.\nThe first term $E_\\text{RGB}$ is the data term, ensuring that the rendered images fit the input images,\n\\begin{equation}\n  E_\\text{RGB}(\\theta,\\gamma) = \\sum_p \\norm{I_p - \\(\\mathcal{L}(\\theta,\\gamma)\\ast \\PSF\\)(p)}_1.\n  \\label{eq:data_term}\n\\end{equation}\nNote that we use an $L^1$-norm to improve robustness against outliers.\nThe second term is the Eikonal term $E_\\text{eik}$, which encourages $d(\\mathbf{x};\\theta)$ to approximate\nan SDF~\\cite{gropp2020implicit},\n\\begin{equation}\n  E_\\text{eik}(\\theta) = \\sum_\\mathbf{x} (\\norm{\\nabla_\\mathbf{x} d(\\mathbf{x};\\theta)} - 1 )^2.\n  \\label{eq:eikonal_term}\n\\end{equation}\nFinally, we largely follow~\\cite{wang2021neus} and introduce an \\textit{optional} mask loss $E_\\text{mask}(\\theta)$, allowing to impose silhouette consistency, \n\\begin{equation}\n  E_\\text{mask}(\\theta) = \\sum_p \\text{BCE}(M_p,W_p(\\theta)),\n  \\label{eq:mask_term}\n\\end{equation}\nwhere $M_p$ is the given binary mask value at the pixel $p$,\n$W_p(\\theta) = \\sum_{i=1}^{m-1} w(t_i;\\theta)$\nis the sum of the weights at the sampling locations~$t_i$ used in~\\eqref{eq:discrete_volume_integral}, and BCE is the binary cross entropy loss~\\cite{wang2021neus}.\nWe would like to emphasize that the only aim of the mask loss is to reduce computation time as it allows us to use more samples per pixel inside the mask, and one single sample per pixel outside.\nAdditionally, if the mask loss is used, we also truncate the volume rendering integral~\\eqref{eq:volume_integral} to the unit sphere, as the background is already modeled by the mask loss, further reducing the computational requirements.\nIf the mask loss is not used, we use an inverse sphere parameterization for the background~\\cite{zhang2020nerf++}.\nBoth use cases are directly inspired from~\\cite{wang2021neus}.\n\nThe final loss becomes the sum of the three terms with additional weighting parameters $\\lambda_1, \\lambda_2>0$,\n\\begin{equation}\n  E(\\theta,\\gamma) = E_\\text{RGB}(\\theta,\\gamma) + \\lambda_1 E_\\text{eik}(\\theta) + \\lambda_2 E_\\text{mask}(\\theta).\n  \\label{eq:loss_function}\n\\end{equation}\nThis loss function can in principle be used to estimate a super-resolved geometry and BRDF with one single training pass.\nHowever, since the full super-resolution model is computationally expensive, we first run an initialization pass\nwith a Dirac kernel for the PSF and without the mask loss.\nIndeed, in Section~\\ref{sec:Results} we show that this super-resolution free approach\nalready retrieves state of the art results even without mask supervision.\nAfter that, we have accurate silhouettes from projecting the estimated geometry into the input images,\nand can significantly decrease computation time by optimizing sampling as described above.\nFurthermore, convergence of the full super-resolution model is accelerated since we already start from a reasonable initialization of the scene.\n\n\n\\section{Results}\n\\label{sec:Results}\nWe evaluate our framework SupeRVol on both synthetic and real world data.\nTo this end, we create a synthetic dataset which consists of a combination of two geometries and two materials, designated in the following as \\textit{dog1}{}, \\textit{dog2}{}, \\textit{girl1}{} and \\textit{girl2}{}. Each of them is rendered from 60 different viewpoints for training, and 30 other viewpoints for testing. Our real-world dataset consist of two scans: \\textit{bird}{}, which we captured ourselves, and \\textit{pony}{} from \\cite{bi2020deep}. From these, we consider around 40 images for training and 20 images for testing.\n\\paragraph{Evaluation}\n\\input{tables\/table_HR_training}\nWe consider an ablation study that we name \"noSR\", which consists of our framework SupeRVol but with a Dirac kernel instead of a Gaussian kernel, and we evaluate both noSR and SupeRVol together with IRON\\cite{zhang2022iron}. IRON has demonstrated state of the art performance for inverse rendering with photometric images, significantly outperforming prior work~\\cite{bi2020deep,luan2021unified} for which no open-source code is available online.\nFurther, we consider two complementary scenarios.\nIn {\\em high resolution training}, we use the original (high resolution) images of the scans for both training and testing, while in {\\em low resolution training}, we down-sample the resolution of the training images of the scans by a factor of 16 (four in each direction) to mimic the effect of a low-resolution image capture, and then test on the original test images.\n\nThe first scenario is typically considered in inverse rendering works, and allows to assess the quality of our differentiable volume renderer with explicit BRDF model. Since both training and testing resolutions are the same in this case, we only compare noSR against IRON. For the second scenario, we include SupeRVol in the comparison, allowing to measure the impact of modeling the PSF correctly.\n\n\n\\paragraph{High resolution training}\n\\input{figures\/06HR_training_concave}\nAs can be seen in Table~\\ref{tab:1}, noSR is quantitatively superior to IRON~\\cite{zhang2022iron} for both the geometric and rendering quality. This is further confirmed qualitatively in Figure~\\ref{fig:HR_rendering}, where noSR demonstrates more faithful novel-view synthesis and extraction of the diffuse albedo.\nMoreover, as shown in Figure~\\ref{fig:HR_concave_parts}, noSR not only yields much more detailed shape reconstruction in convex parts of the objects, but is also more reliable in highly concave parts where IRON~\\cite{zhang2022iron} as a surface rendering based approach can fail badly.\nThis further exhibits the advantage of using volume rendering instead of surface rendering for a better convergence, as previously noted in~\\cite{yariv2021volume,wang2021neus} and shown in Figure~\\ref{fig:volvssurf}. Our evaluation reveals a better inverse rendering performance when using our differentiable volume renderer with explicit BRDF model on high resolution images even without properly modeling the PSF.\n\n\\paragraph{Low resolution training}\n\\input{tables\/table_LR_training}\n\\input{figures\/05HR_training_rendering}\nTable~\\ref{tab:2} and Figure~\\ref{fig:LR_rendering} demonstrate both quantitatively and qualitatively the effectiveness of correct~PSF modeling, where our framework SupeRVol outperforms both IRON~\\cite{zhang2022iron} and noSR by a large margin.\nIts results show some details that are barely visible in the individual low resolution images and not properly reconsutrcted by the baselines. Furthermore, as shown in Figure~\\ref{fig:LR_detailed_geometry}, the refinement induced by a correct PSF is not limited to an increased quality of the reflectance properties, but also has a significant impact on the geometry, which also exhibits some details that are difficult to see at low resolution level, and at most partially recovered by the baseline. \n\n\n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nWe propose to enhance suitable neural representations of shape and material parameters with a physical model of the camera that includes the blurring and down-sampling due to the point spread function.  This leads to a neural approach for recovering shape, albedo and material properties at a resolution that is superior to that of baseline methods and even superior to that of the input images.  We carefully motivate the choice of representation including the use of volume rendering over surface rendering and propose an efficient stochastic sampling scheme for modeling the effect of the point spread function. In qualitative and quantitative studies we demonstrate that the proposed approach outperforms the state-of-the-art and offers a method for high resolution 3D modeling of shape and appearance even from low-resolution input imagery. \n\n\n\\clearpage\n\\input{figures\/07LR_training_rendering}\n\\input{figures\/08LR_training_details_geometry}\n\n\\section{Network Details}\n\n\\subsection{Architecture}\nAs mentioned in the main paper, we use three multilayer perceptrons (MLPs).\nOne describes the geometry via an SDF, $d_\\theta$, one describes the BRDF's diffuse albedo, $\\rho_{\\brdfparam_1}$, and one is used for the specular parameters of the material, $\\alpha_{\\brdfparam_2}$.\nThe MLP of $d_\\theta$ consists of $5$ layers of width $512$, with a skip connection at the $4$-th layer.\nThe MLPs of $\\rho_{\\brdfparam_1}$ and $\\alpha_{\\brdfparam_2}$ consist of $4$ layers of width $512$, and $3$ layers of width $256$, respectively.\\\\\nIn order to compensate the spectral bias of MLPs \\cite{mildenhall2021nerf}, the input is encoded by positional encoding using $6$ frequencies for both $d_\\theta$ and $\\alpha_{\\brdfparam_2}$, and $12$ frequencies for $\\rho_{\\brdfparam_1}$. \n\n\\subsection{Parameters and Cost Function}\nSimilarly to \\cite{zhang2022iron,yariv2020multiview}, we assume that the scene of interest lies within the unit sphere, which can be achieved by normalizing the camera positions appropriately.\nTo approximate the Volume rendering integral (2) using (4), we use $m=98$ samples which are also used to approximate (3), all with the sampling strategy of~\\cite{yariv2021volume}.\\\\\nIn the following, we distinguish between the ablation study noSR of the main paper %\nand SupeRVol.\\\\\nFor SupeRVol, we set the objective's function trade-off parameters $\\lambda_1 = \\lambda_2=0.1$. Furthermore, in order to approximate the convolution with a Gaussian PSF (8), we use $N_s=25$ in (9), and\nthe terms of the objective function (10) and (11) consist of a batch size of $100$ (inside the silhouette) and $1000$, respectively.\nFor the mask term (12) of the objective function, we use the same batch as (10), and add around $500$ additional rays outside the silhouette whose rays still intersect with the unit sphere.\\\\\nConcerning the noSR parameters, we set the objective's function trade-off parameters $\\lambda_1 = 0.1$, $\\lambda_2=0$, i.e. we \\textit{turn off} mask supervision, and the terms of the objective function (10) and (11) consist of a batch size of $2000$ and $1000$, respectively.\\\\\nNote, that we always normalize each objective function's summand with its corresponding batch size.\n\n\\subsection{Training}\nWe train our networks using the Adam optimizer\\cite{kingma2014adam} with a learning rate initialized with $5\\text{e}-4$ and decayed exponentially during training to $5\\text{e}-5$, except for the MLP $\\alpha_{\\brdfparam_2}$ whose learning rate is constantly equal to $1\\text{e}-5$. The remaining parameters are kept to Pytorch's default.\\\\\nWe train for $2000$ epochs, which lasts about $2$ days for noSR, and less than $3$ days for SupeRVol using a single NVIDIA P6000 GPU with $24$GB memory and $60$ input images.\nFor SupeRVol, %\nwe fix the geometry after the end of the training, and refine the BRDF's parameters using a larger batch size of $500$ -- all within the object's silhouette.\n\n\\section{Data Acquisition}\nIn this section we describe how we generated the data sets used in this paper\n\\subsection{Synthetic Data}\nThe synthetic data sets \\textit{dog1}, \\textit{dog2}, \\textit{girl1}, \\textit{girl2}{} were generated using Blender \\cite{blender} and Matlab\\cite{MATLAB:2020}, where Blender\\cite{blender} is used to render depth, normal and BRDF parameter maps for each viewpoint, and Matlab\\cite{MATLAB:2020} is used to render images %\nusing equation (6) and (7) of the main paper. \\\\\nExemplary low-resolution rendered images of size $320 \\times 240$, as well as the object's GT high-resolution shape, diffuse albedo, roughness and specular albedo of size $1280\\times960$ are shown in Figures~\\ref{fig:dogA_data}--\\ref{fig:girlB_data}, where the low-resolution images are obtained by blurring and downsampling high-resolution images by a factor 16 (four in each direction).\n\\input{figures\/supp\/dogA_data}\n\\input{figures\/supp\/dogB_data}\n\\input{figures\/supp\/girlA_data}\n\\input{figures\/supp\/girlB_data}\n\n\\subsection{Real-World Data}\nThe real-world data of \\textit{pony}{} was shared by the authors of~\\cite{bi2020deep}, and the real-world data of \\textit{bird}{} was created by ourselves.\nWe use a Samsung Galaxy Note 8 smartphone and the application \"CameraProfessional\"\\footnote{\\url{https:\/\/play.google.com\/store\/apps\/details?id=com.azheng.camera.professional}, accessed 18-th Nov. 2022, 5.21PM} to generate RAW images as well as the smartphone's images in parallel.\nWe use the RAW images for our algorithm, and we pre-processed those using Matlab \\cite{MATLAB:2020} by following \\cite{sumner2014processing}, where we avoid non-linear operations such as gamma correction. %\nExemplary low-resolution images of both real-world datasets are shown in Figure \\ref{fig:real_data}, where \\textit{pony}{} (top) has a resolution of size $270 \\times 480$ and \\textit{bird}{} (bottom) has a resolution of size $504 \\times 378$.\nThese $16\\times$ ($4$ in each direction) downsampled low-resolution images are obtained similarly to synthetic data.\n\\input{figures\/supp\/real_data}\n\n\\section{Novel Renderings}\nTo validate that our approach results in the scene's parameters which can be used to alter the material and visualize it under novel illumination with standard software (Blender\\cite{blender}), we show novel renderings in Fig.~\\ref{fig:material_novel}.\n\\input{figures\/supp\/material_novel}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nSelf-healing is one of the most fascinating properties of diffraction-free optical fields\\,\\cite{Durnin1987}. These fields have the ability to reconstruct if they are partially disturbed by an obstruction placed in their propagation path. Diffraction-free beams have found applications in fields such as imaging~\\cite{Fahrbach2010,Planchon2011,Fahrbach2012}, optical trapping~\\cite{McGloin2005,McGloin2003,ARLT2001,Volke2002}, laser material processing~\\cite{Dubey2008}, amongst many others. Arguably, the most  well-known propagation invariant (self-healing) fields are Bessel modes of light, first introduced in 1987 by J. Durin\\,\\cite{Durnin1987,Durnin1987b}. However, the self-healing property is not limited to so called non-diffracting beams, but also appears in helico-conical\\,\\cite{Hermosa2013}, caustic, or self-similar fields, namely Airy\\,\\cite{Broky2008}, Pearcey\\,\\cite{Ring2012}, Laguerre-Gaussian\\,\\cite{Bouchal2002,Mendoza2015} and even standard Gaussian beams\\,\\cite{Aiello2017}. Furthermore, within the last years, it has been shown that self-healing can also be observed at the quantum level, for example, McLaren {\\it et al.} demonstrated experimentally the self-reconstruction of quantum entanglement\\,\\cite{McLaren2014}. Importantly, self-healing is not only an attribute of scalar fields but it can also apply to beams with spatially variant polarization \\cite{Milione2015d,Wu2014,Li2017}. \\\\\nBessel beams also appear as complex vector light fields, where polarisation and spatial shape can be coupled in a non-separable way\\,\\cite{Zhan2009,Rosales2017,Otte2018}. This property has fueled a wide variety of applications, from industrial processes, such as, drilling or cutting\\,\\cite{Meier2007,Niziev1999,Dubey2008}, to optical trapping\\,\\cite{Donato2012,Zhao2005,Zhan2004,Roxworthy2010,Kozawa2010,Huang2012}, high resolution microscopy\\,\\cite{Torok2004}, quantum and classical communication\\,\\cite{Zhan2002,Ndagano2017,Ndagano2018}, amongst many others. Controversially, such non-separable states of classical light are sometimes referred to as classically or non-quantum entangled \\,\\cite{simon2010nonquantum}. This stems from the fact that the quintessential property of quantum entanglement is non-separability, which is not limited to quantum systems. Indeed, the equivalence has been shown to be more than just a mathematical construct \\cite{Ndagano2017}.  While such classical non-separable fields do not exhibit non-locality, they manifest all other properties of local entangled states.\\\\\nHere, we demonstrate that the decay in such local entanglement after an obstruction can be counteracted if the carried field is Bessel. We create higher-order vector Bessel beams that are non-separable in orbital angular momentum (the azimuthal component of the spatial mode) and polarisation, and show that self-healing also comprises the non-separability of the beams. This is at a first glance surprising since self-healing is traditionally attributed to the radial component of the spatial mode, which in our field is entirely separable. In order to demonstrate the far-reaching concept of self-healing, we unambiguously quantify the degree of non-separability in different scenarios ranging from fully to partially reconstructed fields by performing a state tomography on the classical field \\cite{McLaren2015,Ndagano2016}. We show both theoretically and experimentally that even though the non-separability reduces after the obstruction, it recovers again upon propagation, proportionally to the level of self-reconstruction. Further, we confirm our findings by a Bell-like inequality measurement\\,\\cite{bell1964js} in its most commonly used version for optics, namely, the Clauser-Horne-Shimony-Holt (CHSH) inequality \\,\\cite{clauser1969}, confirming that the non-separability of vector Bessel beams also features self-healing properties. Although our tests are exerted on purely classical fields, the results are expected to be identical for the local entanglement of internal degrees of freedom of a single photon, and may be beneficial where such entanglement preservation is needed, e.g., transporting single photons through nano-apertures for plasmonic interactions.\n\n\n\n\\begin{figure*}[ht]\n\\centering\n\\def\\svgwidth{0.9\\linewidth}\\sffamily\n\\input{Fig1_Concept1.eps_tex}  \n\\caption{(a) Formation of Bessel beams by off-axis interfering plane waves. If obstacles are included (radius $R$) a shadow region is formed. Scalar BG modes (intensity profile at $z_0$ in (c)) are realized by applying the binary Bessel function (b1) in combination with a blazed grating and Gaussian aperture (b2). (d) Concept of the realization and analysis of non-separable vBG modes (SLM: spatial light modulator; L$_{(\\mathcal{F})}$: (Fourier) lens with focal distance $f$; A: aperture; $\\lambda\/n$, $n=\\lbrace2,\\,4\\rbrace$: wave plates; q: $q$-plate; CCD: camera). (e) Propagation behavior of azimuthal vBG mode (e1) without obstruction and (e2) obstructed by on-axis absorbing object with $R=200\\,\\mbox{\\textmu m}$, indicated by white circle.\\label{fig:Concept}} \n\\end{figure*}\n\n\\section{Vector Bessel modes} \\label{sec:VectorBesselModes}\n\\subsection{Bessel-Gaussian beams}\n\\noindent Over finite distances, a valid approximation of Bessel beams is given by the so called Bessel-Gaussian (BG) modes\\,\\cite{Gori1987}. Besides other properties, these BG fields have the same ability of self-reconstruction in amplitude and phase\\,\\cite{McGloin2003,Litvin2009}. In polar coordinates $(r,\\varphi, z)$, BG modes are defined as\n\\begin{align}\nE_{\\ell}^{\\text{BG}}(r,\\varphi, z) = &\\sqrt{\\frac{2}{\\pi}}J_{\\ell}\\left(\\frac{z_R k_r r}{z_R-\\text{i}z}\\right)\\exp\\left(\\text{i}\\ell\\varphi-\\text{i}k_z z\\right) \\nonumber \\\\\n&\\cdot \\exp\\left(\\frac{\\text{i}k_r^2z w_0-2kr^2}{4(z_R-\\text{i}z)}\\right),\n\\label{eq:BGmodes}\n\\end{align}\nwhereby $\\ell$ represents the azimuthal index (topological charge), and $k_r$ and $k_z$ are the radial and longitudinal wave numbers, respectively. Further, $J_{\\ell}(\\cdot)$ defines the Bessel function, whereas the Gaussian information is encoded in the last factor with the initial beam waist $w_0$ of the Gaussian profile and the Rayleigh range $z_R = \\pi w_0^2\/\\lambda$, $\\lambda$ being the wavelength. The finite propagation distance of BG modes (``non-diffracting length\") is limited by $z_{\\text{max}}$. This distance describes the length of a rhombus-shaped region created by the superposition of plane waves with wave vectors lying on a cone described by the angle $\\alpha=k_r\/k$ (wave number $k=2\\pi\/\\lambda$)\\,\\cite{Gori1987}, as indicated in Fig. \\ref{fig:Concept}(a). The center of the rhombus-shaped region is positioned at $z_0$. For small $\\alpha$, i.e., $\\sin \\alpha \\approx \\alpha$, the non-diffracting distance is given by $z_{\\text{max}} =2\\pi w_0\/\\lambda k_r$ \\cite{McGloin2005}. If an obstruction is included within the non-diffracting distance, a shadow region is formed of length $z_{\\text{min}} \\approx R\/\\alpha \\approx \\frac{2\\pi R}{k_r \\lambda}$ \\cite{Bouchal1998} (Fig.~\\ref{fig:Concept}(a)). Here, $R$ describes the radius of the obstruction. After this distance $z_{\\text{min}}$, the beam starts to recover due to the plane waves passing the obstruction \\cite{McGloin2003,Litvin2009}. A fully reconstructed BG beam will be observed at $2z_{\\text{min}}$, as visualised in Fig.~\\ref{fig:Concept}(a).\n\n\\subsection{Realization of obstructed BG modes}\n\\noindent An established tool for the realization of complex beams are spatial light modulators (SLMs). These modulators allow for an on-demand dynamic modulation of structured beams by computer generated holograms \\cite{SPIEbook}. For the formation of BG modes, we choose a binary Bessel function as phase-only hologram, defined by the transmission function\n\\begin{equation}\nT(r,\\varphi) = \\text{sign}\\lbrace J_{\\ell}(k_r r)\\rbrace \\exp (\\text{i}\\ell\\varphi), \\label{eq:BinaryBessel}\n\\end{equation}\nwith the sign function $\\text{sign}\\lbrace\\cdot \\rbrace$ \\cite{Turunen1988,Cottrell2007}. This approach has the advantage of generating a BG beam immediately after the SLM. An example of this function is shown in Fig. \\ref{fig:Concept}(b1). Note that for encoding this hologram we use a blazed grating (see Fig.~\\ref{fig:Concept}(b2)), so that the desired beam is generated in the first diffraction order of the grating\\,\\cite{Davis1999}.\\\\\nHere, we set $k_r = 18\\,\\mbox{rad\\,mm}^{-1}$ and $\\ell =0$ for the fundamental Bessel mode. Furthermore, we multiply the hologram by a Gaussian aperture function for the realization of a Gaussian envelope with $w_0 = 0.89\\,\\mbox{mm}$ (see Eq. (\\ref{eq:BGmodes}), Fig. \\ref{fig:Concept}(b2)). These settings result in a BG beam with $z_{\\text{max}} = 49.16\\,\\mbox{cm}$ for a wavelength of $\\lambda = 633\\,\\mbox{nm}$, whose intensity profile in the $z_0$-plane is depicted in Fig.~\\ref{fig:Concept}(c).\\\\\nBeyond the generation of BG modes, the SLM can also be used for the realization of obstructions within the $z_0$-plane (see Fig. \\ref{fig:Concept}(d1)): Absorbing obstacles are created by including a circular central cut in the hologram, such that within this area no blazed grating is applied. This means, the respective information of the BG mode is deleted in the first diffraction order. Furthermore, phase obstructions can be realized by adding the chosen phase object to the hologram. Hence, this artificial generation of obstructions facilitates the realisation of any chosen kind of obstacle of defined radius $R$ in the $z_0$-plane. Moreover, the relation $z_{\\text{min}} \\approx \\frac{2\\pi R}{k_r \\lambda}$ shows that a decrease in the radius R of the circular obstruction at $z_0$ results in a decrease in the length $z_{\\text{min}}$ of the shadow region, which is equivalent to moving the detection plane in $\\pm z$-direction (see Fig. \\ref{fig:Concept}(d2)), in order to analyse the evolution from a partially to a fully reconstructed beam.\n\n\\subsection{Self-healing vector Bessel modes}\n\\noindent In order to investigate the relation between the self-healing of propagation invariant beams and the non-separability of light modes, we apply vector Bessel beams, or, more precisely, vector Bessel-Gaussian (vBG) modes. These modes are classically entangled in their spatial and polarisation degrees of freedom (DoF) as explained in the next section. As illustrated in Fig. \\ref{fig:Concept}(d1), these beams are generated by a suitable combination of an SLM (SLM$_1$), half wave plates ($\\frac{\\lambda}{2}$) and a $q$-plate (q), a device capable of correlating the polarisation and spatial DoFs\\,\\cite{Marrucci2006}. First, we create the fundamental scalar BG mode (linearly polarised in the horizontal direction) by encoding the binary Bessel hologram (Fig. \\ref{fig:Concept}(b2)) on SLM$_1$. Within a $4f$-system we filter the first diffraction order with an aperture (A). If we now position, for example, a half wave plate, whose fast axis is oriented in a $45^{\\circ}$ ($\\pi\/4$) angle with respect to the incoming horizontal polarisation, in combination with a $q$-plate ($q=1\/2$) in the beam path, an azimuthally polarised vBG mode is created in the image plane of SLM$_1$ (see Fig. \\ref{fig:Concept}(d1)). The desired mode is generated from the $q$-plate by coupling the polarisation DoF with the orbital angular momentum (OAM) via a geometric phase control, imprinting an OAM charge of $\\pm 2q$ per circular polarisation basis to the passing beam\\,\\cite{Marrucci2006}. Note that wave and $q$-plate(s) do not need to be placed within the non-diffracting distance as we work in the paraxial regime ($\\sin \\alpha \\approx \\alpha$). The plates could even be located within the Fourier plane of SLM$_1$\\,\\cite{Milione2015d}. Moreover, we are of course not limited to azimuthally polarised vBG modes. Depending on the chosen number and orientation of wave plates, different polarisation structures are accessible\\,\\cite{Cardano2012} as depicted in Fig. \\ref{fig:Polarization}. Here, we demonstrate the intensity distribution of different vBG beams in the $z_0$-plane analysed by a polariser (orientation indicated by white arrows).\\\\ \n\n\\begin{figure}[h]\n\\centering\n\\def\\svgwidth{1.0\\linewidth}\\sffamily\n\\input{Polarization2.eps_tex}  \n\\caption{Examples of non-separable vBG modes with respective polarisation analysis. The normalised intensity distribution in the $z_0$-plane for different orientations of a polariser are shown. The respective orientation is indicated by white arrows in (a). The according polarisation distribution is highlighted by black arrows. \\label{fig:Polarization}} \n\\end{figure}\nIn Fig. \\ref{fig:Concept}(e1) we present the experimentally measured propagation invariant properties of these modes by the example of the azimuthal vBG beam (cf. Fig. \\ref{fig:Polarization}(a)). The transverse intensity profile is shown for different positions $z\\in [z_0,\\,z_{\\text{max}}]$. Consider that the outer rings of the vBG mode disappear with increasing $z$ due to the rhombus shape of the non-diffracting region.\\\\\nAs explained above, SLM$_1$ enables the inclusion of an obstruction within the holographically created scalar BG field. Following this, we are also able to apply the SLM for imparting an obstruction within the vBG mode: As SLM$_{1}$ is imaged by a $4f$-system to the $z_0$-plane of the vBG beam, an obstacle created by SLM$_1$ is also imaged to the $z_0$-plane of formed vector mode (cf. Fig.~\\ref{fig:Concept}(d1)). As an example, we investigated the propagation properties of the azimuthal vBG mode if obstructed by an absorbing object with $R=200\\,\\mbox{\\textmu m}$ created by SLM$_1$. Here, we included an additional horizontally oriented polariser to analyse the polarisation properties simultaneously. Results are shown in Fig. \\ref{fig:Concept}(e2). For the programmed obstacle we calculate a self-healing distance of $z_{\\text{min}} = 11.02\\,\\mbox{cm}$. The shown intensity distributions reveal a self-reconstruction of the beam including its polarisation properties after approximately $2z_{\\text{min}}$, as expected.\n\n\n\\section{Non-separability of self-healing beams} \n\\noindent\nWe define the non-separability of a self-healing vBG beam in the framework of quantum mechanics. This can be easily understood through the formal definition of the non-separability or entanglement of quantum systems; two systems A and B are separable if they can be written as a factorisable product of the two subsystems, i.e, $\\ket{\\Psi_{AB}}=\\ket{A}\\otimes\\ket{B}$, conversely, the systems can be entangled if ($\\ket{\\Psi_{AB}} \\neq \\ket{A}\\otimes\\ket{B}$). Here, the two subsystems are analogously replaced with the internal DoF of the photons in the vBG beam. Importantly, a non-separable vector mode has maximally entangled polarisation and spatial components.\\\\\nTo quantify the non-separability between the polarisation and spatial components of the field, we employ simple measures borrowed from quantum mechanics  \\cite{McLaren2015,Ndagano2016}. Consider an arbitrary (vBG) field with each photon described by the following state  \n\\begin{equation}\n\\ket{\\Psi}_{k_r \\ell} = \\cos(\\theta)\\ket{u_{k_r,  \\ell}} \\ket{R} + \\sin(\\theta)\\ket{u_{k_r,  -\\ell}} \\ket{L}, \\label{eq: Field}\n\\end{equation}\n\\noindent where $\\ket{R}$ and $\\ket{L}$ are the canonical right and left circular polarisation states spanning the qubit Hilbert space $\\mathcal{H}_2$. The infinite dimensional state vectors  $\\ket{u_{k_r, \\pm\\ell}} \\in \\mathcal{H_\\infty}$ represent the self-healing transverse eigenstates, namely scalar Bessel or BG modes of light (cf. Eq. (\\ref{eq:BGmodes})), characterised by the continuous radial wave number $k_r$ and the topological charge $\\ell$. The parameter $\\theta$ determines whether $\\ket{\\Psi}_{k_r \\ell}$  is purely vector (non-separable; $\\theta=(2n+1)\\pi\/4,\\, n\\in \\mathbf{Z})$, scalar (separable; $\\theta=n\\pi\/2,\\,\\, n\\in \\mathbf{Z})$, or some intermediate state.\n\nThe ``vectorness\" of a given classical state can be determined via a vector quality analysis which is equivalent to measuring the concurrence of a quantum state $C$ \\cite{Wootters2001}. Mathematically, the respective vector quality factor (VQF) has the form\n\\begin{equation}\n\\text{VQF}=\\text{Re}(C)=\\sqrt{1-s^2}=|\\sin(2\\theta)|. \\label{eq:VQFpure}\n\\end{equation}\n\\noindent Here, $s$ is the Bloch vector defined as $s=\\sum_{i}^{3}\\langle \\sigma_i\\rangle$ with $\\sigma_i$, $i=\\lbrace 1,2,3\\rbrace$, being the traceless Pauli operators spanning the so-called higher-order Poincar\\'{e}  sphere (HOPS, see Fig. \\ref{fig:HOPS})\\,\\cite{Millione2011}. The VQF takes values in the interval $[0,1]$, with 0 corresponding to a separable scalar mode and  1 representing a maximally non-separable vector mode. We employ the VQF measure as a figure of merit for determining the non-separability of partially obstructed and consequently self-healing vBG modes.\n\n\\subsection{Vector quality factor of obstructed beams}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\def\\svgwidth{1.0\\linewidth}\\sffamily\n\t\\input{HOPs2.eps_tex}\n\n\t\\caption{Higher-order Poincar\\'e spheres of self-healing BG beams spanned by the Stokes parameters (a) $S^{+1}_{1,2,3}$ for $\\mathcal{H}_{2,\\sigma,\\ell=1}$ and (b) $S^{-1}_{1,2,3}$ for $\\mathcal{H}_{2,\\sigma,\\ell=-1}$. The poles represent separable BG states with circular polarisation and $\\ell=\\pm1$, while non-separable vBG states are found on the equator (i.e. the plane intersecting the $S^{\\pm1}_{1,2}$ plane).}\t\n\t\\label{fig:HOPS}\n\\end{figure}\n\\noindent\nFrom wave optics, it is well-known that an obstructed beam can be modified in both phase and amplitude due to the interaction between the beam and the outer edges of an obstruction, resulting in diffraction. At the single photon level, this can be understood as modal coupling\\,\\cite{Sorelli2018}. That is, if some field with a transverse profile given by $\\ket{ u_{k_{r_1}, \\ell_1} }$ interacts with an obstruction, the state evolves following the mapping \n\\begin{equation}\n\\ket{ u_{k_{r_1},\\ell_1 }} \\rightarrow \\int\\sum_{\\ell} \\alpha_{\\ell}(k_r) \\ket{ u_{{k_r},\\ell}} \\text{d}k_r, \\label{eq. Diff}\n\\end{equation}\nwhere the input mode spreads over all eigenmodes $\\ket{u_{{k_r},\\ell}}$. The coefficients $|\\alpha_{\\ell}(k_r)|^2$ represent the probability of the state $\\ket{ u_{k_{r_1  },\\ell_1}}$ scattering into the eigenstates $\\ket{u_{{k_r},\\ell}}$ with the property that $\\int\\sum_{\\ell}|\\alpha_{\\ell}(k_r)|^2\\text{d}k_r=1$. Applying the mapping of Eq. (\\ref{eq. Diff}) to the scattering of the self-healing vBG mode presented in Eq. (\\ref{eq: Field}) and post selecting a particular $k_r$ and $\\ell$ values yields the state \n\\begin{eqnarray}\n\\ket{\\Phi_{k_r, \\ell}}=& a \\ket{ u_{{k_r},\\ell}}\\ket{R} + b \\ket{ u_{{k_r},-\\ell}}\\ket{R}\\nonumber\\\\ \n&+ c \\ket{ u_{{k_r},\\ell}}\\ket{L} + d \\ket{u_{k_r,\\ell}}\\ket{L}, \\label{post}\n\\end{eqnarray}\nwhere $|a|^2 +|b|^2+|c|^2+|d|^2=1$. Consequently, $\\ket{\\Phi_{k_r, \\ell}}$ restricts the measurement of the photons to the four-dimensional Hilbert space  $\\mathcal{H}_4=\\text{span}\\big(\\{\\ket{R}, \\ket{L}\\}\\otimes\\{\\ket{ u_{{k_r},\\ell}}, \\ket{ u_{{k_r},-\\ell}}\\}\\big)$ which can be written as the direct sum\n\\begin{equation}\n\t\\mathcal{H}_4= \\mathcal{H}_{2,\\sigma,\\ell} \\oplus \\mathcal{H}_{2,\\sigma,-\\ell}.\\label{eq:Hilbert}\n\\end{equation}\n\\noindent The subspaces $\\mathcal{H}_{2,\\sigma,\\pm\\ell}=\\text{span}(\\ket{u_{k_r, \\pm \\ell}}\\ket{R}, \\ket{u_{k_r,\\mp\\ell}}\\ket{L})$ are topological unit spheres for spin-orbit coupled beams belonging to the family of HOPS\\,\\cite{Millione2011}. \\\\\n\\noindent Invoking the equivalence between VQF and concurrence enables us to exploit the definition of concurrence with $C=\\sqrt{1-\\text{Tr}(\\rho_{\\psi}^2)}$ for a given density matrix $\\rho_\\psi=\\ket{\\psi}\\bra{\\psi}$. In quantum mechanics, if the state of a two qubit system written in the logical computation basis is given by the pure state\n\\begin{equation}\n \\ket{\\Psi} = c_1\\ket{0}\\ket{0}+c_2\\ket{0}\\ket{1}+c_3\\ket{1}\\ket{0}+c_4\\ket{1}\\ket{1},\n\\end{equation}\n\\noindent satisfying the normalization condition $\\sum_{j=1}^4 |c_{j}|^2=1$, the concurrence is $C=2|c_1c_4-c_3c_2|$. By replacing the two qubit logical basis with the $\\mathcal{H}_4$ basis from the two HOPSs, we can equivalently write the VQF as \n\\begin{equation}\n\\text{VQF}=2|ad-cb|, \\label{VQF_VM}\n\\end{equation}\n\\noindent following Eq. (\\ref{post}).\\\\\nAs an illustrative example, consider the azimuthally polarised vBG mode given by \n\\begin{equation}\n\\ket{\\psi}_{k_r,1} =\\frac{1}{\\sqrt{2}}\\big( \n \\ket{u_{k_r, 1}} \\ket{R} - \\ket{u_{k_r,  -1}} \\ket{L}\\big). \\label{eq:Fieldradial}\n\\end{equation}\n\n\\noindent Upon diffracting off the edges of an obstruction, one expects the mode coupling profiled in Eq. (\\ref{post}) to occur (by restricting $|\\ell|=1$). However, since the radial profile of BG modes enables the transverse structure to self-heal, $cb\\rightarrow0$ and $ad\\rightarrow-\\frac{1}{2}$, and therefore $\\text{VQF}\\rightarrow 1$ giving rise to the self-healing of the non-separability. This behavior is predicated to occur best after twice of the minimum self-healing distance $z_{\\text{min}}$.\\\\\n\n\\noindent To experimentally quantify the characteristics of vectorness, i.e. classical entanglement, in relation to the self-healing properties of vGB modes, we apply a configuration as indicated in Fig. \\ref{fig:Concept}(d2). By this configuration, consisting of a quarter wave plate ($\\frac{\\lambda}{n}$, $n=4$), a polarisation sensitive SLM (SLM$_2$), a Fourier lens (L$_{\\mathcal{F}}$) and a CCD camera, we determine the expectation values of the Pauli operators $\\langle\\sigma_{i}\\rangle$, $i = \\lbrace 1,\\,2,\\,3\\rbrace$ by 12 on-axis intensity measurements or six identical measurements for two different basis states\\,\\cite{McLaren2015,Ndagano2016}. These values are used to determine the VQF according to Eq. (\\ref{eq:VQFpure}). \\\\\nIf circular polarisation is chosen as basis ($\\ket{R},\\,\\ket{L}$), the projection measurements represent two OAM modes of topological charge $\\ell$ and $-\\ell$, namely $\\ket{u_{k_r,\\pm\\ell}} = E_{\\pm\\ell}^{\\text{BG}}$, as well as four superposition states $\\ket{u_{k_r,\\ell}} + \\exp(\\text{i}\\gamma) \\ket{u_{k_r,-\\ell}}$ with $\\gamma = \\lbrace 0, \\pi \\ell \/2, \\pi \\ell, 3\\pi\\ell\/2 \\rbrace$. As we use a $q$-plate with $q=1\/2$, our measurements are performed for $\\ell = 1$. Following Tab. \\ref{tab:Tomography}, we calculate expectation values $\\langle\\sigma_i \\rangle$ from\n\\begin{align*}\n&\\langle\\sigma_1\\rangle = I_{13}+I_{23}-(I_{15}+I_{25}),\\\\\n&\\langle\\sigma_2\\rangle = I_{14}+I_{24}-(I_{16}+I_{26}),\\\\\n&\\langle\\sigma_3\\rangle = I_{11}+I_{21}-(I_{12}+I_{22}).\n\\end{align*}\nOn-axis intensity values $I_{uv}$, $u = \\lbrace1,2\\rbrace$ , $v = \\lbrace1,2,...,6\\rbrace$, are normalised by $I_{11}+I_{12}+I_{21}+I_{22}$. The respective polarisation projections are performed by inserting a quarter wave plate, set to $\\pm 45^{\\circ}$, in combination with the polarisation selective SLM$_2$. Further, SLM$_2$ is responsible for the OAM projections. For this purpose, we encode the OAM as well as superposition states as phase-only holograms according to the binary Bessel function in Eq.~(\\ref{eq:BinaryBessel}). Finally, the on-axis intensity is measured in the focal plane of a Fourier lens by means of a CCD camera.\\\\\nCrucially, the decoding SLM$_2$ is placed at a adequately chosen distance $\\Delta z = 23\\,\\mbox{cm}$ from the $z_0$-plane so that we are able to access different levels of self-healing without the need to move the detection system. That is, by changing the radius $R$ of the digitally created obstruction, the vBG mode can fully self-heal in front of ($2z_{\\text{min}}< \\Delta z$) or behind ($2z_{\\text{min}}> \\Delta z$) this SLM$_2$, as indicated in Fig.~\\ref{fig:Concept}(d2) by the black shadow region or the green and yellow dashed lines, respectively. \n  \\begin{table}[h\n \\caption{Normalised intensity measurements $I_{uv}$ for the determination of expectation values $\\langle \\sigma_i \\rangle$. \\label{tab:Tomography}}\n \\begin{ruledtabular}\n \\begin{tabular}{c|c c c c c c c}\nBasis states & $l=1$ & $-1$ & $\\gamma =0$ & $\\pi\/2$ & $\\pi$ & $3\\pi\/2$& \\\\ \\hline \\hline\nLeft circular $\\vert L\\rangle$& $I_{11}$ & $I_{12}$ & $I_{13}$ & $I_{14}$ & $I_{15}$ & $I_{16}$ &\\\\ \nRight circular $\\vert R\\rangle$& $I_{21}$ & $I_{22}$ & $I_{23}$ & $I_{24}$ & $I_{25}$ & $I_{26}$& \\\\ \n \\end{tabular}\n \\end{ruledtabular}\n \\end{table}\n \n\n\\begin{figure*}[tb]\n\\centering\n\\def\\svgwidth{1.0\\linewidth}\\sffamily\n\\input{stateTomography.eps_tex}  \n\\caption{Vector quality analysis of self-healing vBG mode (azimuthally polarised) without obstacle (a), with absorbing obstacles of radius (b) $R = 150\\,\\mbox{\\textmu m}$, (c) $R=200\\,\\mbox{\\textmu m}$, (e) $R = 500\\,\\mbox{\\textmu m}$, and (f) $R=600\\,\\mbox{\\textmu m}$, as well as (d) phase obstacle (homogeneous phase shift of $\\pi$) of radius $R=200\\,\\mbox{\\textmu m}$. Results for beams which (do not) fully self-heal before the analysis are marked (blue) red with respective VQF within the measured normalized matrices of on-axis intensity values arranged according to Table.~\\ref{tab:Tomography}.  \\label{fig:StateTomography}} \n\\end{figure*}\n\n\\subsection{Experimental quantification of vectorness}\n\\noindent First, we prove that pure undisturbed vBG modes show the maximum degree of non-separability. This is exemplified by the analysis of an azimuthally polarised vBG beam. The experimentally measured on-axis intensity values are visualised in Fig. \\ref{fig:StateTomography}(a), arranged according to Tab. \\ref{tab:Tomography}. The measurements result in a VQF of $0.99$, verifying our expectation. In a next step, we digitally impart different obstacles and determine the respective VQF. On the one hand, we chose obstacles allowing the beam to self-reconstruct within $\\Delta z =23\\,\\mbox{cm}$, namely absorbing obstructions with $R=150\\,\\mbox{\\textmu m}$ ($2z_{\\text{min}} = 2\\cdot 8.27\\,\\mbox{cm} = 16.54\\,\\mbox{cm}$) or $R=200\\,\\mbox{\\textmu m}$ ($2z_{\\text{min}} = 2\\cdot 11.03\\,\\mbox{cm} = 22.06\\,\\mbox{cm}$) and a phase obstacle with $R=200\\,\\mbox{\\textmu m}$ creating a homogeneous phase shift of $\\pi$ (Fig. \\ref{fig:StateTomography}(b)-(d)). On the other hand, we program absorbing obstructions with $R = 500\\,\\mbox{\\textmu m}$ ($2z_{\\text{min}} = 2\\cdot 27.57\\,\\mbox{cm} = 55.14\\,\\mbox{cm}$) and $R = 600\\,\\mbox{\\textmu m}$ ($2z_{\\text{min}} = 2\\cdot 33.03\\,\\mbox{cm} = 66.06\\,\\mbox{cm} $), for which $2z_{\\text{min}}>\\Delta z$ and even $z_{\\text{min}}>\\Delta z$ (Fig. \\ref{fig:StateTomography}(e), (f)). The respective measured VQFs are shown within each subfigure. \\\\\nObviously, the degree of non-separability, i.e. the VQF, decreases with increasing absorbing obstacle size $R$ (Fig. \\ref{fig:StateTomography}). For self-healed beams with absorbing obstacles (b)-(c), the VQF differs only minimally from the non-obstructed case (a). Note that a phase obstruction of the same radius (d) results in a slightly larger deviation. In contrast to absorbing obstacles, causing a loss of information, phase obstructions do not cut but vary information. Since in the case of absorbing obstacles the cut information is also included within the passing plane waves, the loss can be compensated within $2z_{\\text{min}}$. In the phase obstruction case, the varied information represent additional information, i.e. noise, within the non-diffracting beam, which is not eliminated when being decoded by the SLM. Hence, the beam stays disturbed and, as a consequence, the VQF decreases.\\\\\nFurther, if the beam cannot fully reconstruct before being decoded by SLM$_2$, thus, if we analyse the degree of entanglement within the self-healing distance (Fig. \\ref{fig:StateTomography}(e), (f)), the decrease in VQF is relatively large in comparison to the self-healed versions in Fig. \\ref{fig:StateTomography}(b), (c). However, note that in both cases, the self-reconstructed and non-reconstructed beams, the VQF is $\\geq 0.88$. Consequently, the beams are closer to being vector or non-separable (VQF $=1$) than scalar or separable (VQF $=0$). This is due to the fact that there is always undisturbed information reaching the SLM$_2$, and only little scattering into other modes, i.e., little modal coupling. We thus may conclude that the beam is always non-separable, but due to noise caused by the obstacle the measurement shows deviations from pure non-separability. As the noise is annihilated with propagation distance (for absorbing obstacles), the quantitative value for non-separability, namely VQF, recovers. This effect is similar to what we call ``self-healing\" in the case of amplitude, phase and polarisation of vBG modes: Obstacles add noise to the information on these degrees of freedom. Upon propagation, these perturbations vanish and the pure vBG mode information is left so that the beam and its properties seem to self-reconstruct.\\\\\nConsequently, we conclude that the lower the level of self-healing, i.e. the larger the obstacle, the smaller the VQF. This means, not only amplitude, phase and polarisation properties of the vBG mode reconstruct with distance behind an obstruction, but also the degree of non-separability. \\\\\n\n\\begin{figure*}[tb]\n\\centering\n\\def\\svgwidth{1.0\\linewidth}\\sffamily\n\\input{BellAnalysis.eps_tex}  \n\\caption{Bell-type curves for azimuthal vBG mode with four different orientations $2\\theta_A$  of the half wave plate to determine the Bell parameter $|S|$. The investigations were performed undisturbed (a) as well as for differently sized obstacles (radii $R=\\lbrace 150,\\,200,\\,500,\\,600\\rbrace\\,\\mbox{\\textmu m}$ in (b)-(f)) and a $\\pi$-phase obstacle ($R=200\\,\\mbox{\\textmu m}$) (d). Shown on-axis intensity measurements $I'(\\theta_A, \\theta_B)$ for (a)-(f) are normalised according to the maximum intensity measured without obstacle (a). Dashed curves represent $\\cos^2$-fits used to determine $S$. \\label{fig:BellAnalysis}} \n\\end{figure*}\n\n\n\\subsection{CHSH Bell-like inequality violation}\n\\noindent To confirm our results with respect to the VQF, we performed an additional investigation of the degree of entanglement or non-separability using the Bell parameter\\,\\cite{McLaren2015}. More specifically, we perform a Clauser-Horne-Shimony-Holt (CHSH) inequality measurement\\,\\cite{clauser1969}, the most commonly used Bell-like inequality for optical systems, to demonstrate the degree of entanglement between polarisation and spatial DoFs. Instead of measuring a single DoF, e.g. polarisation or OAM, non-locally, we analyse two DoF locally on the same classical light field\\,\\cite{McLaren2015}. For this purpose, we placed a half wave plate ($\\frac{\\lambda}{n}$, $n=2$) in front of SLM$_2$ (see Fig. \\ref{fig:Concept}(d2)) and measured the on-axis intensity $I(\\theta_A,\\theta_B)$ for different angles $2\\theta_A = \\lbrace 0,\\, \\pi\/8,\\,\\pi\/4,\\,3\\pi\/4\\rbrace$ of the half wave plate. Here, $\\theta_B \\in [0,\\,\\pi]$ represents the rotation angle of the hologram encoded on SLM$_2$ by $\\ket{u_{k_r,\\ell}} + \\exp(\\text{i}2\\theta_B) \\ket{u_{k_r,-\\ell}}$ ($\\ell = 1$, $k_r = 18\\,\\mbox{rad\\,mm}^{-1}$). \\\\\nWe define the CHSH-Bell parameter $S$ as\n\\begin{equation}\nS = E(\\theta_A,\\theta_B)-E(\\theta_A,\\theta_B')+E(\\theta_A',\\theta_B)+E(\\theta_A',\\theta_B'), \\label{eq:BellParameter}\n\\end{equation}\nwith $E(\\theta_A,\\theta_B)$ being calculated from measured on-axis intensity according to\n\\begin{align}\n& E(\\theta_A,\\theta_B) = \\frac{A(\\theta_A,\\theta_B)-B(\\theta_A,\\theta_B)}{A(\\theta_A,\\theta_B)+B(\\theta_A,\\theta_B)}.\\\\\n& A(\\theta_A, \\theta_B) = I(\\theta_A,\\theta_B)+I\\left(\\theta_A+\\frac{\\pi}{2},\\theta_B+\\frac{\\pi}{2}\\right),\\nonumber\\\\\n& B(\\theta_A, \\theta_B) = I\\left(\\theta_A+\\frac{\\pi}{2},\\theta_B\\right)+I\\left(\\theta_A,\\theta_B+\\frac{\\pi}{2}\\right).\\nonumber\n\\label{eq:BellParameterE}\n\\end{align}\nThe values of $S$ ranges from $|S|\\leq 2$ for separable states, up to $|S|= 2\\sqrt{2}$ for entangled or non-separable states. Our experimental results with according Bell parameters $|S|$ are presented in Fig. \\ref{fig:BellAnalysis}. As before, we performed our investigation without obstacle (a) as well as with different absorbing obstacles of radii $R = \\lbrace 150,\\, 200,\\, 500,\\, 600\\rbrace\\,\\mbox{\\textmu m}$ (b)-(f), or (d) phase-obstructing. Note that the intensity values are normalised with respect to the maximum value measured in the unobstructed case ($I'(\\theta_A, \\theta_B)$). Obviously, the ratio of measurable maximum intensity ($I'_{\\text{max}}(R)$) decreases dramatically depending on the size of the obstruction, as it can be seen on the $I'(\\theta_A, \\theta_B)$-axis (vertical) of Fig. \\ref{fig:BellAnalysis}. However, the Bell parameter does not change significantly if we measure in the fully reconstructed regime ($2z_{\\text{min}}<\\Delta z$ for (b)-(d)) compared to the undisturbed vBG mode (a). In accordance with our VQF analysis, $|S|$ reveals bigger changes if the beam is not fully self-healed when it is analysed (e), (f). In total, even if the intensity lowers to some percentage (f) of the original maximum value (a), all measurements validate a violation of the Bell inequality, matching our non-separability analysis results based on vectorness. \n\n\\subsection{The relation of self-healing and non-separability}\n \\begin{table}[h]\n \\caption{Quantification of non-separability properties of self-reconstructing vBG modes as function of the self-healing level, given by the obstacle radius $R$.  \\label{tab:Summarize}}\n \\begin{ruledtabular}\n \\begin{tabular}{c|c c c c c c c}\n$R$ in \\textmu m & $0$ & $150$ & $200$ & $200$, $\\pi$-obst. & $500$ & $600$& \\\\ \\hline \\hline\nVQF & $0.99$ & $0.98$ & $0.97$ & \\textcolor{gray}{$0.95$} & $0.94$ & $0.88$ &\\\\ \n$|S|$ & $2.81$ & $2.81$ & $2.79$ & \\textcolor{gray}{$2.79$} & $2.74$ & $2.75$& \\\\ \n$I'_{\\text{max}}$ & $1$ & $0.93$ & $0.82$ & \\textcolor{gray}{$0.53$} & $0.09$ & $0.03$ & \\\\\n \\end{tabular}\n \\end{ruledtabular}\n \\end{table}\n \n\\noindent In Table \\ref{tab:Summarize} we summarize our results emphasizing the dependence of the VQF, the maximum intensity $I'_{\\text{max}}$ as well as Bell parameter $|S|$ on the size $R$ of included obstructions, i.e. on the self-healing level. The maximum intensity $I'_{\\text{max}}(R)$ reveals an approximately Gaussian decrease with increasing obstacle size which reflects the Gaussian envelope of investigated vBG mode. Simultaneously to the intensity, the VQF as well as $|S|$ decrease as demonstrated in previous sections. However, only small changes are observed as only minor noise is disturbing vBG modes if obstructions are included. In short, both, vector quality as well as Bell analysis reveal a similar behavior of non-separability with respect to changes in the obstruction size, demonstrating the self-healing of the degree of entanglement within obstructed vBG beams. \n\n\n\\section{Conclusion and Discussion}\n\n\\noindent It is well-known that the phase and amplitude of vector Bessel beams self-heal in the presence of an obstacle that partially blocks its path. However, there are no reports about the effect of this on the coupling between phase and polarisation, known as classical entanglement. In this work, we presented for the first time to our knowledge experimental evidence that even though the coupling between these two degrees of freedom decreases after passing an obstruction, it eventually restores itself to its maximum value. For this purpose, we dynamically realized vector Bessel Gaussian (vBG) modes with digital obstructions by combining holography-based generation of structured light with a q-plate. We quantified the degree of non-separability between the spatial shape and polarisation using two different means: the vector quality factor (VQF) and a classical version of the CHSH Bell-like inequality. By a specific design of our detection system combined with digital variation of the obstruction size, different levels of self-healing were accessible, which enabled the relation between degree of non-separability and self-healing level to be analysed.\nThe measured VQF values showed that the degree of classical entanglement increases as function of decreasing object size. Analogously, the measured CHSH $S$ parameter values show a similar dependence on the objects size or self-healing level of the vBG beam, showing in all cases a clear violation of the CHSH inequality. This behavior can be interpreted as a self-healing in the degree of non-separability as function of the distance from an obstacle since a decrease in obstacle size is comparable to placing the detector further from the object.\\\\\nThe complex fields used in this study were separable in radial profile, defined by the $k_r$ vector of the conical waves, but non-separable in the azimuthal profile and polarisation, the latter being defined by the OAM of each polarisation component. Intriguingly, it is the radial profile that leads to the self-healing of the non-separability, despite itself being separable. Thus the local entanglement in angular momentum (spin and OAM) can be made more resilient to decay from obstructions by engineering the unused degree of freedom in a judicious manner.\n  \n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nTopological phases of matter are classified into two categories\\cite{Wen11,Wen21}: symmetry-protected topological\\,(SPT) phase and topologically ordered\\,(TO) phase. The SPT phase is distinguished by short-ranged entanglement, whereas the TO phase is distinguished by long-range entanglement. Furthermore, the SPT phase protects boundary gapless states, and it cannot be adiabatically connected to a trivial product state under perturbations preserving a certain symmetry. In contrast, in the TO phase, the global pattern of entanglement causes topological ground state degeneracy, which is robust to local perturbation regardless of symmetry. Furthermore, the topological degeneracy generates a universal subleading term in the entanglement entropy, which is known as the topological entanglement entropy (TEE)\\cite{Kitaev11, Levin11}. This entanglement entropy has been mainly used to detect the topological order\\cite{Bal}. Topological order frequently results in topological excitations with fractional quantum numbers. Entanglement entropy may also show signs of topologically ordered insulators\\cite{Haldane191}.\n\n\n\n\nRecently, it was revealed that undoped interacting disordered  graphene\\cite{Nov,Zhang,Neto} zigzag ribbons\\cite{Fujita} are a new TO Mott--Anderson insulator displaying $e^-\/2$ fractional charges,\nspin-charge separation, and two degenerate ground states\\cite{Jeong11,eeyang11}. The disorder is a singular perturbation that couples electrons on opposing zigzag edges, resulting in instantons. This effect converts zigzag ribbons from a STP to a TO phase and generates $e^-\/2$  fractional charges on the opposite zigzag edges. These fractional charges are protected\\cite{Girvin117} by an exponentially decaying soft gap\\cite{Efros,Mac1} $\\Delta_s$, as shown in Fig. \\ref{degGap}. Furthermore, numerical work\\cite{Kim11} showed that an interacting disordered zigzag nanoribbon has a finite TEE.\n\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Figure1.eps}\n\\caption{\\normalfont (\\textbf{a}) Schematic band structure of a disorder-free interacting zigzag graphene nanoribbon at half-filling with a hard Mott gap $\\Delta\\sim 0.1t$, where the hopping parameter $t\\sim 3$\\;eV. Soliton zigzag edge states are near $k=\\pm \\pi\/a_0$, and their charge is $e^-$ (the ribbon period is $a_0$). (\\textbf{b}) Schematic density of states (DOS) of a half-filled disordered zigzag ribbon (dashed line). It decays exponentially with an energy scale of $\\Delta_s$ (a soft gap), which decreases with increasing disorder strength\\cite{Jeong11}. The van Hove singularities of the DOS at $\\epsilon=\\pm \\Delta\/2$, originating from   the band structure displayed in a, are reduced due to the formation of the gap states. These gap states are spin--split, and many of them are soliton states with fractional charge $e^-\/2$. Note that the DOS has particle-hole symmetry after disorder averaging. The soft gap $\\Delta_s$ protects the fractional charges from quantum fluctuations.}\n\\label{degGap}\n\\end{center}\n\\end{figure}\n\n\n\n\nRecent advancements in fabrication methods have enabled the production of atomically precise graphene nanoribbons\\cite{Cai2,Kolmer}.  But it is unclear how to unequivocally measure the presence of fractional charges. We believe that doped zigzag nanoribbons are ideal for observing exotic anyons with fractional charges. The properties of doped disordered zigzag ribbons, on the other hand, are largely unknown.\nA doped ribbon is not expected to be a topologically ordered insulator because there is no hard gap (the density of states (DOS) at the Fermi is non-zero but small). However,\n  the system is still an insulator with localized edge states near the Fermi energy, displaying doubly degenerate ground states. In the dilute limit the added fractional charges will still be well defined. Let us explain this, following  Ref.\\cite{Girvin117}. These fractional charges are analogous to quasiparticles of the fractional quantum Hall effect's $1\/m$ Laughlin state ($m$ is an odd integer). In such a system's low doped regime, the added electrons divide into fractional charges.  \\textcolor{black}{Recent experimental works provide evidence for these anyons\\cite{Nakamura01,Barto1}.}   Suppose one adds $\\delta N$ electrons to such a state. In the dilute limit, each of these electrons fractionalizes into $m$ quasiparticles that are well separated from each other (the charge of a quasiparticle is $e^-\/m$). The total energy of the new system is thus $E_-=E_m+\\delta N m \\Delta_-$, where $E_m$ is the ground state energy and $\\Delta_{-}$ is the quasiparticle excitation energy. \\textcolor{black}{Despite that the quasiparticles form quasi-degenerate states, the} excitation gap $\\Delta$ and localization of quasiparticles protect fractional charges against quantum fluctuations~\\textcolor{black}{\\cite{eeyang11,Girvin117}}.\n \n \nThe role of fractional charges in low doped disordered systems is one of the fundamental questions in doped disordered zigzag ribbons. What exactly is the ground state? This concerns the applicability of mean field approaches to such a system: quantum fluctuations\\cite{Girvin117} not included in the Hartree--Fock (HF) approximation may be significant because gap states are no longer empty.\nFurthermore, in contrast to the uniform spin density of undoped ribbons, the ground state of a doped disorder-free ribbon exhibits an edge spin density wave. It is unknown how localization and charge quantization affect the nature of the ground state.\nWe use the density matrix renormalization group \\,(DMRG) approach in the matrix product states (MPS) representation to investigate the ground state of a doped ribbon and the importance of quantum fluctuations beyond the HF approach. The MPS representation is a powerful tool for solving eigenvalue problems of quantum many-body systems\\cite{White1992,Schollwock2011}.\n\n\n\n\nRibbons are in a new disordered anyon phase, according to our investigation of the low doping regime\\cite{Lei13,Wilczek03}.\nWe discover that a low doped disordered zigzag ribbon contains a large number of anyons with a fractional charge (but as doping concentration increases they disappear).\nThey cause numerous magnetic domain walls and localized magnetic moments residing on the zigzag edges.\nAlso, objects that display spin-charge separation proliferate in this phase.\nAs a result, the ground state is drastically reorganized, with highly distorted edge charge and spin modulations, as well as non-local correlations between the left and right zigzag edges.\nWe will define this new phase as a disordered anyon phase because its electron and spin densities are highly inhomogeneous.  Furthermore, we make the following new experimentally testable predictions.\n(1) The disordered anyon phase has an unusual shape of tunneling density of states (TDOS), depending on the number of extra electrons \\textcolor{black}{(for experimental measurement of a soft gap in the TDOS, see, for example, Refs. \\cite{Ashoori01,JPEisen})}.\nThe TDOS has one sharp peak at the midgap energy and two other peaks, one on each side of the sharp peak at the midgap energy, at the low doping limit. (2) However, the midgap peak disappears as the doping concentration increases. The detection of these peaks will provide strong evidence for the presence of $e^-\/2$ fractional charges. Furthermore, our findings indicate that doped zigzag ribbons may have unusual transport, magnetic, and inter-edge tunneling properties. \\textcolor{black}{Theoretical calculations\\cite{Pisa1} of disorder-free zigzag ribbons show \\textcolor{black}{that} antiferromagnetism is favored over ferromagnetism  for ribbon widths $< 100$\\AA. In the presence of disorder, the new disordered anyon phase is expected for these width values.} \n\\section*{Results}\n\\subsection*{Model}\nTo model the graphene zigzag nanoribbons, we apply the Hubbard model with the nearest neighbor hopping and a diagonal disorder $V_i$.\n\\begin{eqnarray}\n H=-t\\sum_{\\langle ij \\rangle,\\sigma} c^{\\dag}_{i,\\sigma}c_{j,\\sigma} +\\sum_{i,\\sigma} V_i c_{i,\\sigma}^{\\dag}c_{i,\\sigma}\n +U\\sum_i n_{i,\\uparrow} n_{i,\\downarrow},\n\\label{Hubbard}\n\\end{eqnarray}\nwhere $i=(x,y)$ denotes the site indices (see Fig. \\ref{dmrg-res}a), $c^{\\dagger}_{i,\\sigma}$\/$c_{i,\\sigma}$ are the creation\/destruction operators at site $i$, $t$ is the nearest neighbor hopping parameter and  $U$ is the on-site repulsion. The ratio of the numbers of impurities and carbon atoms is given by $n_{imp}=N_I\/N_s$. The values of the disorder strength $V_i$ at $N_I$ impurity sites  are  uniformly distributed in the interval \n$ [-\\Gamma,\\Gamma]$ (the sum in the second term of $H$ is only over impurity sites). The dimensionless coupling constant of the problem is the ratio of the disorder strength and on-site repulsion $g= \\frac{\\Gamma\\sqrt{n_{imp}}}{U}$. The doping concentration is defined as $\\delta N\/N_s$, where $\\delta N$ is the total number of added electrons. The mean field version of this Hamiltonian for a doped ribbon is given  in method, see Eq. (\\ref{MFhspin}).\nThe HF results of undoped zigzag ribbons show that fractionalization occurs  independent of the disorder potential range, density, and strength. Note that disorder is a singular perturbation\\cite{Jeong11,eeyang11}.\n\n\n\n\n\n\n\nIn graphene systems mean field approximations are widely used because they give accurate results\\cite{Stau}.   \\textcolor{black}{ However, there are several nearly degenerate HF ground states in graphene nanoribbons that can be generated using different HF initial states,and one does not know which of these states is close to the true ground state because quantum fluctuations are missing in the HF approximation. In this paper, we conducted the DMRG to determine which HF initial state generates the  HF state that is close to the true ground state. We will concentrate on two types of HF initial states in this section. The first, labeled AF, is generated from an antiferromagnetic initial state, while the second, labeled PM, is generated from a paramagnetic initial state with a small spin-splitting.}\nThe DMRG found that the \\textcolor{black}{undoped} ground state at clean limit exhibits the N\\'eel magnetic ordering, where spins at two zigzag edges align antiparallel to each other \\textcolor{black}{(these results agree with those results obtained using the AF initial state)}. Nonetheless, the addition of enough extra electrons results in an edge spin density wave. The corresponding DMRG results, presented in Supplementary material,    \\textcolor{black}{ agree with those results obtained using  the PM initial state, see Fig. \\ref{dmrg-res}b.}\nThe results were then tested at half-filling for an undoped disordered interacting zigzag nanoribbon. The PM initial state produces a state with fractional charges. Site spin values computed from this ground state agree well qualitatively with those of the DMRG approach, as shown in \\textcolor{black}{Figs.~\\ref{dmrg-res}c,d.} We will show below\nthat DMRG results for doped interacting disordered zigzag ribbons also support the results obtained from  the PM initial state. \\textcolor{black}{The methods of both DMRG and HF approximation are explained in detail in Supplementary material.}\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Figure2.eps}\n\\caption{\\normalfont (\\textbf{a}) Vertical and horizontal lines of carbon atoms are numbered. All lengths and widths in this paper are measured in the number of these lines. (\\textbf{b})  Site spin values $ s_{iz}$ of a disorder-free  doped zigzag ribbon. This state is generated from  $N_e=N_s+20$ ($\\delta N\/N_s=0.017$), $L_x=301$, $L_y=4$, and $U=t$. (\\textbf{c}) DMRG result of the ground state site spins $s_{iz}$ at half-filling for $U=t$, $n_{imp}=1$, and $\\Gamma=0.5t$ ($g=0.5$). We discover that other spin components $s_{ix}$ and $s_{iy}$ are very small. (\\textbf{d}) HF site spin values  at half-filling are shown.  Here, $U=t$, $n_{imp}=0.1$, and $\\Gamma=0.5t$ ($g=0.16$ is smaller compared with the value used in (\\textbf{c})).}\n\\label{dmrg-res}\n\\end{center}\n\\end{figure}\n\n\\subsection*{New Anyon Phase and TDOS of Low Doping Region}\nUsing the PM initial state, we investigated the shape of the TDOS as a function of doping concentration \\textcolor{black}{(all the HF results below are generated by using this HF initial state).}  As shown in Fig. \\ref{SolPhas}, adding a few extra electrons to the half-filled ribbon results in a sharp peak near the midgap energy $E=0$ inside an exponentially decaying small soft gap. The peak's physical origin is as follows: A tunneling electron enters into a soliton state and divides into two fractionally charged quasiparticles\n\\begin{equation}\ne^-\\rightarrow e^-\/2+e^-\/2.\n\\end{equation}\n(Ref.\\cite{Girvin} gives a good account of this process). A soliton state is described by a non-local wave function, as shown in the upper left inset of Fig. \\ref{SolPhas}.\nThe width of the central peak is $\\sim 0.02\\Delta\/2$.\n\\textcolor{black}{In the low doping limit, when an entering electron has a non-zero energy $E\\neq 0$, it has a significant chance not to split into $e^-\/2$ charges because fractionalization is only approximate at non-zero energies\\cite{eeyang11}. The lower left inset of Fig. \\ref{SolPhas} shows the highly non-linear dependence of the peak value at E=0 on doping concentration $\\delta N\/N_s$. The zero energy peak in the DOS disappears for $\\delta N\/N_s>0.005$ (the shape of this DOS will be shown below). Such non-linear behavior is unusual and provides compelling evidence for fractional charges. The shape of the TDOS at the low doping limit differs significantly from that of the half-filled undoped state} (there are also two side peaks, one on each side of the sharp peak at the midgap energy. These peaks are not shown in Fig. \\ref{SolPhas} because their energies  lie outside the energy range $|E|>0.05\\Delta\/2$). Edge site occupation numbers and site spins are displayed in Fig. \\ref{SolPhas3}. These findings show modulated ferromagnetic edges.\n\n\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Figure3.eps}\n\\caption{\\normalfont The HF results for $N_e=N_s+3$ ($\\delta N\/N_s=0.0037$), $\\Gamma=0.01 t $, $L_x=101$, $L_y=8$, $n_{imp}=0.1$, and $U=t$ $(g=0.0032)$. DOS of a slightly doped ribbon away from half-filling is shown \\textcolor{black}{in the weak disorder regime}. Sharp peak is present inside the soft gap at the midgap energy $E=0$  (the magnitude of this peak is rather {\\it small} in comparison to the peaks at $E=\\pm\\Delta\/2$ shown in Fig. \\ref{degGap}). Since there are excess electrons, the Fermi energy $E_F\/(\\Delta\/2)=0.14 $ is {\\it above} the mid gap energy. \\textcolor{black}{The DOS of $L=300$ in a larger  energy interval $E<\\Delta$  is shown in the lower right inset.}\nA charge fractionalized HF eigenstate is shown in the upper left inset. Note that energy is measured in units of $\\Delta\/2$. The number of disorder realization is $N_D\\sim 400$. A tunneling electron is fractionalized in the upper right inset. The DOS is determined by measuring the differential I-V.  \\textcolor{black}{Lower left  inset displays  the dependence of the midgap peak on doping concentration for $L=100$.}}\n\\label{SolPhas}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Figure4.eps}\n\\caption{\\normalfont The HF results for $N_e=N_s+3$ ($\\delta N\/N_s=0.0037$), $\\Gamma=0.01 t $, $L_x=101$, $L_y=8$, $n_{imp}=0.1$, and $U=t$ $(g=0.0032)$. Their disorder-free values are represented by dashed lines. (\\textbf{a}) A disorder realization of zigzag edge site occupation numbers $n_{i\\sigma}$ for a doped ribbon. (\\textbf{b}) Total site occupation numbers $n_i$ are shown. Some charges are transferred between the zigzag edges on the left and right. (\\textbf{c}) Site spins $s_{iz}$ are plotted. Their disorder-free values are represented by dotted lines.}\n\\label{SolPhas3}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\pagebreak\n\n Many of the added electrons have $q_A$ values $\\sim 1\/2$,  which can be seen by comparing Figs. \\ref{SolPhas1}a,b (the quantity $q_A$ gives the total probability to find an electron with energy $E$ on A-type carbon atoms. Charge fractionalization occurs when $q_A=1\/2$). As a result, the extra electrons enter soliton states with well-defined fractional charges. Thus, our results for low doped ribbons indicate that doping does not destroy anyons. The average energy cost to create\\cite{Girvin117} an $e^-\/2$ fractional charge from the undoped ground state with an exponentially small gap is $\\Delta_s\/2$, which corresponds to the midgap energy $E=0$ in the excitation spectrum. This effect is similar to the formation of polyacetylene soliton midgap states\\cite{yang1}. An undoped zigzag ribbon, unlike the chiral edges of Laughlin fractional quantum Hall states, lacks significant gapless edge excitations.\n\n\n\n\n\n\n\n Following  Ref.\\cite{Wilczek03}  let us argue that an $e^-\/2$ fractional charge of a disordered ribbon is an anyon.\nConsider two single-particle HF mixed chiral states that display $e^-\/2$ fractional charges, as shown in Fig. \\ref{semi}. If we exchange these two electrons, the total many-body wave function of $N$ electrons acquires a statistical phase of $e^{i\\pi}=-1$. This exchange is also equivalent to exchanging two $e^-\/2$ charges on the left zigzag edge and two others on the right zigzag edge. Thus, we expect that each of these exchanges generates the statistical phase of $e^{i\\pi\/2}$ to yield the final phase of $e^{i\\pi}=-1$. An anyon with the statistical phase $e^{i\\pi\/2}$ is called \\textcolor{black}{a semion\\cite{Kalme,Canri}.}\nThe presence of semions is consistent with the presence of anyons in TO phases.\nIt is also consistent with \nthe shape of the entanglement spectrum.\n It is believed that the ground state entanglement spectrum of a TO phase resembles the corresponding edge spectrum of the system\\cite{Haldane191,Van,fid}.\n The shape of the entanglement spectrum is computed and found to be similar to the DOS of the edge states (see Supplementary material).  The entanglement spectrum of undoped interacting disordered zigzag ribbons differs from that of zigzag ribbons in the disorder-free SPT phase\\cite{Kim11}.\n\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Figure5.eps}\n\\caption{\\normalfont $q_A$ values of the HF eigenstates of a ribbon are plotted for (\\textbf{a}) $N_e=N_s$  and (\\textbf{b}) $N_e=N_s+3$ ($\\delta N\/N_s=0.0037$). A gap state electron with $q_A=1\/2$ is fractionalized. Here $L_x=101$, $L_y=8$, $N_D=400$, $n_{imp}=0.1$, $U=t$, $\\Gamma=0.01t$, and $N_e=N_s+3$ $(g=0.0032)$. In case of (\\textbf{b}), the spectrum does not have particle-hole symmetry. }\n\\label{SolPhas1}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{Figure6.eps}\n\\caption{\\normalfont  The HF results for Two HF mixed chiral states are shown. Exchanges of two $e^-\/2$ charges on the left zigzag edge and two others on the right zigzag edge are displayed.}\n\\label{semi}\n\\end{center}\n\\end{figure}\n\nAs more electrons are added, the sharp peak at the midgap energy  {\\it disappears} in the DOS, but the two side peaks near $E\\sim \\pm 0.05\\Delta\/2 $ {\\it persist}, as shown in Fig. \\ref{SolPhas0}.\nSimultaneously, the edge occupation number profile becomes highly nonuniform, as shown in Fig. \\ref{SolPhas2}a.  These edge occupation numbers and site spin profiles appear to be quite different from those with fewer electrons, see Figs.~\\ref{SolPhas3}a-c.\nThe nature of the disordered ground state of the doped system is as follows. To begin, it is important to note that the ground state of doped disorder-free zigzag ribbons differs from that of undoped ribbons, which have ferromagnetic edges that are antiferromagnetically coupled.  In sufficiently doped disorder-free ribbons both the DMRG (see Supplementary material) and HF display {\\it spin density} type periodic modulations on the zigzag edges, see Fig. \\ref{dmrg-res}b (the opposite edges are still antiferromagnetically coupled). The ground state changes again in the presence of disorder, and the periodic spin density is destroyed (a disorder potential is a {\\it singular} perturbation\\cite{eeyang11}). As the vertical dashed lines in Fig. \\ref{SolPhas2}a show many HF values of $s_{iz}$ of each zigzag edge change\nsign at sites where $n_i$ abruptly changes (see Fig. \\ref{SolPhas2}b).\nSimilar behavior is also observed in the DMRG calculation, as shown in Fig. \\ref{SolPhas2}c. \nMoreover, sites $i$, where the values of $n_{i\\sigma}$ and $s_{iz}$ abruptly change, have almost identical values for the $x$-coordinate on the left and right zigzag edges. This effect is a consequence of the {\\it nonlocal} correlation between the left and right zigzag edges. This correlation between opposite zigzag edges strongly suggests that the formation of non-local soliton states is responsible for the drastic reorganization of the ground state, as well as the zigzag edge modifications.\nFurthermore, we find {\\it local magnetic moments} with non-zero values of $s_{iz}$ that is extended over several sites.\nThese objects\n proliferate in comparison to the case of undoped ribbons.  There are also objects extended over several sites with rather small values of\n$s_i=\\frac{1}{2}(n_{i\\uparrow}-n_{i\\downarrow})\\approx 0$, see Fig. \\ref{SolPhas2}b. In such an object, {\\it spin-charge separation} would take place. The following procedure is used to create these objects. An $e^-\/2$ fractional charge moves along the zigzag edges from left to right, while another fractional charge with the opposite spin moves in the opposite direction (see Ref.\\cite{eeyang11} for a detailed explanation).\n\n\n\nThe resulting ground state displays a highly distorted edge spin density.\nThis phase is characterized by localized edge magnetic moments, spin-charge separation, and correlation between the left and right zigzag edges, a {\\it disordered anyon phase} of zigzag nanoribbons.  \\textcolor{black}{Charge fractionalization is not exact in this phase since some of the nearly zero energy states are not fractionalized, see the left inset in Fig. \\ref{SolPhas0}. This is in contrast to the results of slightly doped and undoped disordered ribbons, where the fractional charge of zero energy  states is well-defined (see Fig. \\ref{SolPhas} of the current manuscript and Fig.9 in Ref.\\cite{eeyang11}, respectively).}\nWhen doping concentration is increased further ($\\delta N\/N_s\\sim 0.04$), the distorted edge spin density wave   and   charge fractionalization almost disappear. We also discover that the HF gap states are no longer localized along  the ribbon direction.   \\textcolor{black}{ These findings imply that as doping concentration increases from zero, a topological phase transition with a significant crossover region occurs.}\n\n\n\n\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Figure7.eps}\n\\caption{\\normalfont  The HF results for $N_e=N_s+20$ ($\\delta N\/N_s=0.0083$), $\\Gamma=0.06 t$, $L_x=301$, $L_y=8$, $n_{imp}=0.1$, and $U=t$ ($g=0.019$). Two side peaks on the DOS. In the limit of large ribbon length or, equivalently, in the limit of zero doping, the profile of these two peaks becomes symmetric. Since there are excess electrons, the Fermi energy $E_F\/(\\Delta\/2)=0.46$. The number of disorder realization is $N_D\\sim 200$.}\n\\label{SolPhas0}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{Figure8.eps}\n\\caption{\\normalfont The HF results for $N_e=N_s+20$ ($\\delta N\/N_s=0.0083$), $\\Gamma=0.06 t$, $L_x=301$, $L_y=8$, $n_{imp}=0.1$, and $U=t$ ($g=0.019$):   (\\textbf{a}) Vertical lines indicate sites where $n_{i}$ or $s_{iz}$ abruptly change. On the zigzag edges, there are numerous localized magnetic moments. (\\textbf{b}) Site occupation numbers $n_{i\\sigma}$ of a disordered ribbon. The arrows point to locations where spin-charge separation occurs. (\\textbf{c}) DMRG results for $\\delta n_i \\equiv n_i - n_i^{\\rm clean}$ plotted as a function of $x$ and $y$. Here, $n_i^{\\rm clean}$ is the site occupation number for $\\Gamma=0$. Site spins $s_{iz}$ are also shown. The parameters are as follows:\n$N_e=N_s+12$ ($\\delta N\/N_s=0.025$), $\\Gamma=t$, $L_x=120$, $L_y=4$, $n_{imp}=0.2$, and $U=t$. Note that the length of this ribbon is considerably shorter than the one used in (\\textbf{a}) and (\\textbf{b}). These are the results for a more strongly disordered ribbon with $g=0.45$, and the overall magnetization is significantly reduced.  }\n\\label{SolPhas2}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\\newpage\n\\section*{Discussion}\nQuantum fluctuations beyond the HF approximation do not mitigate charge fractionalization in a ribbon at low doping concentration, according to our findings. Despite the presence of disorder, as doping increases, the edge magnetic ordering weakens and charge fractionalization disappears. Furthermore, we discovered that the low doped state is a new disordered anyon phase with highly distorted edge charge and spin modulations, as well as localized magnetic moments with non-local correlations between the left and right zigzag edges. Anyons play a key role in the formation of this new phase.\n\n As a result of spin-charge separation, our findings suggest that doped zigzag nanoribbons may exhibit new magnetic and low temperature transport properties: the conductivity may display a usual behavior while the spin susceptibility may be rather small, as was observed in polyacetylene\\cite{Chung15}.\nFurthermore, we demonstrated that the TDOS profile is significantly affected by doping concentration. The measurement of the differential  I-V curve may reveal this effect and may provide a strong test for the presence of $e^-\/2$ fractional charges. \\textcolor{black}{Ribbons with width less than $100$~\\AA\\,  are\nwell-suited for the observation of these fractional charges as the  antiferromagnetic phase  is more stable than  the ferromagnetic phase\\cite{Pisa1}.}\n\n\n\nThe following additional investigations may be interesting to pursue. A worthwhile but challenging task is to compute the anyon statistical phase using a microscopic approach. Recently such an adiabatic DMRG simulation, utilizing the quantized Hall response,  was successfully conducted\\cite{Zhu} for the non-Abelian Moore-Read state on a Haldane honeycomb lattice model.\nA similar DMRG calculation in a Mott-Anderson insulator of disordered zigzag nanoribbon with Abelian quasiparticles, where electron localization is critical, is not clear.\nAnother method is to compute the statistical Berry phase of Abelian quasiparticles using a trial wave function\\cite{Arovas}. But a good trial wave function is not yet available for disordered zigzag ribbons.\n\n\\textcolor{black}{In the limit of small doping and in weak disorder regime, ribbons  with a sharp midgap peak in the DOS have a universal value of the  TEE\\cite{Kim11}. When doping is high enough, the midgap peak disappears, as does the exact fractionalization of zero energy states. When this occurs, we can expect non-universal TEE values. It may be worthwhile }to  probe the topological phase transition as  doping concentration increases. The following issues must be addressed: \\textcolor{black}{Does} the TEE of doped disordered ribbons decay to zero as a function of doping concentration, and does the transition exhibit non-universal dependence on physical parameters\\cite{Kim11}?  This investigation  may require very high cpu resources to compute accurately small values of the TEE\\cite{Kim11}. This type of calculation could provide more information about the phase transition from modulated ferromagnetic edges at zero doping to distorted spin-wave edges at finite doping\n\nThe following experiments would also be fascinating. Investigation of tunneling between zigzag edges, as seen in fractional quantum Hall bar systems \\cite {Kang}, may be fruitful.\n Scanning tunneling microscopy can reveal the presence of fractional charges by measuring the electron density on the zigzag edges\\cite{Andrei}. Finally, it would be interesting to look into the new disordered anyon phase in other antiferromagnetic zigzag nanoribbon systems, e.g., silicene and boron nitride nanoribbons \\cite{Yao,Bar}. Chiral gauge theory can be used to describe $e^-\/2$ fractional charges\\cite{ChiralTH}.  It would be fascinating to look into the new anyon phase using random chiral gauge fields.\n\\section*{Methods}\n\\subsection*{HF Approximation}\nTo model graphene zigzag nanoribbons, the mean field Hubbard model is commonly used \\begin{eqnarray}\nH_{MF}=-t\\sum_{\\langle ij\\rangle,\\sigma} c^{\\dag}_{i,\\sigma}c_{j,\\sigma} +\\sum_{i,\\sigma} V_ic_{i,\\sigma}^{\\dag}c_{i,\\sigma}\n+U\\sum_i[ n_{i,\\uparrow}\\langle n_{i,\\downarrow}\\rangle +n_{i,\\downarrow}\\langle n_{i,\\uparrow}\\rangle-\\langle n_{i,\\downarrow}\\rangle\\langle n_{i,\\uparrow}\\rangle]\n+ \\sum_i [s_{ix} \\langle h_{ix}\\rangle+s_{iy} \\langle h_{iy}\\rangle]\n\\label{MFhspin}\n\\end{eqnarray}\nwhere $ \\langle h_{ix}\\rangle=-2U\\langle s_{ix}\\rangle$ and $ \\langle h_{iy}\\rangle=-2U\\langle s_{iy}\\rangle$ are the self-consistent ``magnetic fields.\"\nThe last term of Eq. (\\ref{MFhspin}) describes spin flips and is present but only separately from half-filling. This term mixes spin-up and spin-down. Note that the band structure no longer has particle-hole symmetry when away from half-filling.\\\\\n\n\\subsection*{DMRG}\nWe apply the DMRG\\cite{White1992, Rommer,Schollwock2011} to obtain the ground state of the model Eq.~(\\ref{Hubbard}) in the MPS representation.\nFurthermore, we illustrate the geometry of the MPS for the graphene zigzag nanoribbon of the size $(L_x \\times L_y)$ (see Supplementary material).\nFor a quasi-one-dimensional system, the complexity of the DMRG scales exponentially in the width of the system\\,($L_y$), whereas it scales polynomially in the length\\,($L_x$) of the system. Therefore, our MPS setup allows us to consider the graphene strip with long zigzag edges, and we focus on the system with $(L_x,L_y) = (120,4)$, which is far beyond the reach of exact diagonalization in the present calculations. The precision of the DMRG can be controlled by the number of basis states kept or the maximum bond dimension of the MPS ($\\chi_{\\rm max}$), and we use up to $\\chi_{\\rm max} = 1600$ to achieve the typical error of the total energy lower than $10^{-6}$ (For a short introduction to MPS, see Supplementary material). To fix a gauge redundancy of MPS specially, i.e., the canonical form, the DMRG optimizes each tensor considering the global information of the wave function, which makes the algorithm extremely stable and reliable\\cite{Schollwock2011}. Nonetheless, the DMRG can become trapped in a local minimum, particularly for models with a quasi-one-dimensional lattice. To avoid local minima, we apply the noise perturbation\\cite{White05} with the two-site algorithm\\cite{Schollwock2011} at each optimization step.\nMoreover, we exploit the $U(1)$ symmetry of the model such that the DMRG preserves the total number of electrons, e.g., $\\sum_{i,\\sigma} c^\\dag_{i,\\sigma} c_{i,\\sigma} = N_s\/2$, thereby improving greatly its convergence speed and accuracy\\cite{Schollwock2011}.\n\n\\section*{Data availability}\nOn reasonable request, the corresponding author will provide all relevant data in this paper.\n\n\\section*{Code availability}\nOn reasonable request, the corresponding author will provide all numerical codes in this paper.\n\n\n\\section{DMRG.}\n \n\n\n\\section{Matrix Product States}\n\nA large class of quantum many-body wavefunctions, $\\Psi$, can be efficiently factorized into a product of tensors as follows:\n\\begin{align}\n\t|\\psi\\rangle &= \\sum_{ \\{ s_i \\} } \\Psi_{\\cdots s_{i-1} s_i s_{i+1} \\cdots} \\,\\, \\big| \\cdots s_{i-1} s_i s_{i+1} \\cdots \\big\\rangle \\nonumber\\\\\n\t&= \\sum_{ \\{ s_i \\} } {\\rm tTr}[ \\cdots A_{i-1}^{s_{i-1}} A_i^{s_i} A_{i+1}^{s_{i+1}} \\cdots ] \\big|\\cdots s_{i-1} s_i s_{i+1} \\cdots \\big\\rangle,\n\\end{align}\nwhere $A_{i}^{s_i}$ stands for a tensor at site $i$, $\\cdots A_{i-1}^{s_{i-1}} A_i^{s_i} A_{i+1}^{s_{i+1}} \\cdots$ indicates the product of tensors forming a network, depending on how tensors are connected, and ${\\rm tTr[\\cdots]}$ stands for the tensor trace or contraction of all connected indices in the network. Here the index $s_i$ denotes a local state at site $i$, e.g., $|s_i\\rangle = |\\uparrow\\rangle, |\\downarrow\\rangle$ for a spin-half fermion system or $|s_i\\rangle = |0\\rangle, |\\uparrow\\rangle, |\\downarrow\\rangle, |\\uparrow\\downarrow\\rangle$ for a spin-full fermion system. Particularly, the  so-called MPS, which is a chain-like product of rank-3 tensors, may represent quantum states for one-dimensional and quasi-one-dimensional systems accurately. Specifically, the many-body wave function in the MPS representation is written as follows:\n\n\\begin{align}\n\t\\Psi_{\\cdots s_{i-1} s_i s_{i+1} \\cdots}\t\n \t= \\sum_{\\{l_i\\}, \\{r_i\\}} \\cdots \\,\\, \\delta_{r_{i-2} l_{i-1}} \\,\\, [A_{i-1}^{s_{i-1}}]_{l_{i-1}, r_{i-1}} \\,\\, \\delta_{r_{i-1} l_i} \\,\\, [A_i^{s_i}]_{l_i, r_i} \\,\\, \\delta_{r_i l_{i+1}} \\,\\, [A_{i+1}^{s_{i+1}}]_{l_{i+1}, r_{i+1}} \\,\\,\\delta_{r_{i+1} l_{i+2}} \\,\\, \\cdots,\n\\end{align}\n\n\nwhere $[A_i^{s_i}]_{l_i, r_i}$ is a rank-3 tensor with two virtual indices $l_i$ and $r_i$\\,(say left and right, respectively), which are traced out, $s_i$ is the physical index, and $\\delta_{ij}$ stands for the Kronecker delta or the identity matrix. Note that the Kronecker delta contracts the right and left indices of tensor at $i$ and $i+1$, respectively. Hence, the element of the wavefunction for a given set of $\\{s_i\\}$ is identical to the product of {\\it matrices} $\\{A_i^{s_i}\\}$, i.e., $\\Psi_{\\cdots s_{i-1} s_i s_{i+1} } = \\cdots A_{i-1}^{s_{i-1}} A_i^{s_{i}} A_{i+1}^{s_{i+1}} \\cdots$. It is also convenient to introduce a graphical representation for the tensor and its network. A tensor is depicted by an object with open legs denoting its indices. For instance, the tensor $A_i^{s_i}$ can be illustrated as follows:\n\\begin{align}\n\t\\includegraphics[width=0.2\\textwidth]{mps_tensor.eps},\n\\end{align}\nwhere the vertical open leg $s_i$ denotes the physics index and the horizontal ones $l_i$ and $r_i$ stand for the virtual indices. Furthermore, a contraction of two indices, particularly one left and one right horizontal indices, occurs by connecting the legs\\,(or indices) as shown below:\n\\begin{align}\n\t\\includegraphics[width=0.4\\textwidth]{contraction.eps}.\n\\end{align}\nThus, the total wavefunction in the MPS representation is illustrated as follows:\n\\begin{align}\n\t\\includegraphics[width=0.48\\textwidth]{mps.eps}.\n\\end{align}\nHere, the size of the matrix $A_i^{s_i}$, which is referred to as the bond dimension $\\chi$, determines the expressibility of the MPS. In other words, the accuracy of the MPS can be systematically enhanced by increasing the value of $\\chi$. We define the MPS on the zigzag nanoribbon in a so-called snake pattern, as illustrated in Fig.~\\ref{fig:mps_geometry}. This definition allows us to consider the graphene strip with long zigzag edges.\n\n\\begin{figure}[htpb]\n\\begin{center}\n\t\\includegraphics[width=0.4\\textwidth]{mps_geometry.eps}\n\\caption{Schematic figure of the snake pattern of the matrix product state defined on the zigzag nanoribbon lattice.}\n\\label{fig:mps_geometry}\n\\end{center}\n\\end{figure}\n\n\n\nSimilarly, a large class of Hamiltonians can be represented exactly as a product of matrices or as the matrix product operator\\,(MPO) as follows:\n\\begin{align}\n\t\\hat{H} &= \\sum_{\\{s_i\\},\\{s_i'\\}} H_{\\cdots s_{i-1} s_i s_{i+1} \\cdots}^{\\cdots s_{i-1}' s_i' s_{i+1}' \\cdots} \\,\\,\\big|\\cdots s_{i-1}' s_i' s_{i+1}' \\cdots \\big \\rangle \\big\\langle \\cdots s_{i-1} s_i s_{i+1} \\cdots \\big| \\nonumber \\\\\n\t& = \\sum_{\\{s_i\\},\\{s_i'\\}} {\\rm tTr}[ \\cdots W_{i\\!-\\!1,s_{i\\!-\\!1}}^{s_{i\\!-\\!1}'} W_{i,s_{i}}^{s_{i}'} W_{i\\!+\\!1,s_{i\\!+\\!1}}^{s_{i\\!+\\!1}'} \\cdots ] \\big|\\cdots s_i' \\cdots \\big \\rangle \\big\\langle \\cdots s_i \\cdots \\big|,\n\\end{align}\nwhere $W_{i,s_i}^{s_i'}$ is a rank-4 tensor with its graphical representation given as\n\\begin{align}\n\t\\includegraphics[width=0.2\\textwidth]{mpo_tensor.eps}.\n\\end{align}\nThen, the Hamiltonian is represented in the graphical representation as follows:\n\\begin{align}\n\t\\includegraphics[width=0.48\\textwidth]{mpo.eps}.\n\\end{align}\nFor example, the Hamiltonian of the one-dimensional Anderson--Hubbard model is produced with the tensor $W_{i,s}^{s'}$ as follows:\n\n\\begin{align}\n\t[W_{i,s}^{s'}]_{lr} = \t\n\t\\begin{bmatrix}\n\t\t(I_i)_{ss'}\t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(c_{i\\uparrow}^\\dagger)_{ss'} \t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(c_{i\\downarrow}^\\dagger)_{ss'} \t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(c_{i\\uparrow})_{ss'} \t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(c_{i\\downarrow})_{ss'} \t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(U n_{i\\uparrow} n_{i\\downarrow} + v_i (n_{i\\uparrow} + n_{i\\downarrow}))_{ss'} & t (c_{i\\uparrow})_{ss'} & t (c_{i\\downarrow})_{ss'} & -t (c_{i\\uparrow}^\\dagger)_{ss'} & -t (c_{i\\downarrow}^\\dagger)_{ss'} & (I_i)_{ss'}\n\t\\end{bmatrix}_{lr},\n\\end{align}\n\nwhere $I_i$ is the trivial operator acting on site $i$, $c_i^{(\\dagger)}$ is the annihilation\\,(creation) operator, $v_i$ is the Anderson random potential, and $U$ is the Hubbard interaction. One can easily generalize the above $W-$tensor and MPO into the proper quantities for the model of the quasi-one-dimensional honeycomb lattice.\n\nThe total energy of a given wavefunction is given as\n\\begin{align}\n\tE = \\frac{\\langle \\Psi | H | \\Psi \\rangle}{\\langle \\Psi | \\Psi \\rangle}\n\t= \\frac{\\sum_{\\{s_i\\},\\{s_i'\\}} (\\Psi_{\\cdots s_i' \\cdots})^* H_{\\cdots s_i \\cdots}^{\\cdots s_i' \\cdots} (\\Psi_{\\cdots s_i \\cdots})}\n\t{\\sum_{\\{s_i\\}} (\\Psi_{\\cdots s_i \\cdots})^* (\\Psi_{\\cdots s_i \\cdots})},\n\\end{align}\nand this total energy is graphically recast as\n\\begin{align}\n\t\\includegraphics[width=0.4\\textwidth]{energy.eps}.\n\\end{align}\nThus, to measure the total energy $E$, one should contract two different tensor networks in the denominator and numerator in the above equation. The contraction of tensor networks can be efficiently conducted by choosing the order of the contraction properly~\\cite{Schollwock2011}. With the above expression, one can apply a variational principle to optimize tensors $\\{A_i\\}$ by minimizing the energy, e.g.,\n\t$\\partial E\/\\partial A_i^* = 0$,\nThereby leading to an eigenvalue problem to update the tensor $A_i$\\,(see Ref.~\\cite{Schollwock2011} for more details).\n\n\n\n\n\\subsection{Density Matrix Renormalization Group: Clean limit}\n\n\n\\begin {figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{idmrg_scaling.eps}\n\\caption{The bond dimension scaling of entanglement entropy (a), energy density (b),  edge magnetization ($\\hbar=1$) (c),  and  correlation length (d).  Here  $U\/t=1$ is used.   The energy  unit is $t$ and the length unit is $\\sqrt{3}a$, where  $a$ is the carbon-carbon distance.}\n\\label{fig:idmrg_scaling}\n\\end{center}\n\\end{figure}\n\n\nAs discussed in the previous section, the bond dimension of the MPS determines the amount of the MPS. The number of MPS should be carefully chosen to appropriately represent the ground state. Specifically, the MPS is optimal for representing quantum states that satisfy the area law of the entanglement, i.e., its entanglement entropy scales as the size of the boundary between two subsystems. By contrast, the entanglement entropy of a gapless ground state diverges logarithmically with the subsystem size. Thus, we first check whether the ground state of our Hamiltonian is gapped or gapless. Then, we determine how large the bond dimension should be to appropriately represent the ground state. To this end, we perform the infinite-size variant of the DMRG ~\\cite{Schollwock2011} and see how the energy density, total magnetization, entanglement entropy scale and correlation length as a function of the maximum bond dimension $\\chi_{\\rm max}$. The result is presented in Fig.~\\ref{fig:idmrg_scaling}. It is certain that the entanglement entropy does not diverge but converges to a finite value as $\\chi_{\\rm max} \\rightarrow \\infty$\\,[see red dashed line in Fig.~\\ref{fig:idmrg_scaling}(a)], which implies that the ground state is gapped or satisfies the area law of the entanglement. Also, based on the scaling result of the energy density and magnetization,\\, as shown in Figs.~\\ref{fig:idmrg_scaling}(a) and \\ref{fig:idmrg_scaling}(b), we conclude that keeping $1600$ states, i.e., $\\chi_{\\rm max}=1600$, is enough to capture the essential physics of the ground state. Thus, we fix $\\chi_{\\rm max}$ to 1600 in all calculations. Although we did not present the results here, we however directly confirmed that keeping 1600 states provides convergent results in the finite system regardless of disorder.\n\nAs a benchmark result, we present in Fig.~\\ref{fig:dmrg_clean} the DMRG results of magnetization profile in the clean system. The system is at half-filling\\,($\\delta N = 0$) with size $(L_x, L_y) = (30,8)$, as shown in Fig.~\\ref{fig:dmrg_clean}(a). Expectedly, the ground state shows the N\\'eel order where spins at each zigzag edge align antiparallel to each other. In Fig.~\\ref{fig:dmrg_clean}(b), the system is slightly away from the half-filling, i.e., $\\delta N = 12$ with size $(L_x, \\, L_y) = (120,4)$. The doping introduces a spin-density wave. Hence, the magnetization oscillates along the ribbon direction.\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{dmrg_Mz_clean.eps}\n\\caption{Magnetization profile at (a) $(L_x,L_y, \\delta N) = (30, 8, 0)$ and (b) $(L_x, L_y, \\delta N) = (120, 4, 12)$ in the clean limit with $U\/t=1$.}\n\\label{fig:dmrg_clean}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{HF Approximation and Topological Entanglement Spectrum.}%\n\n\n\n\n\n\n\n\n\n\nThe ground state entanglement spectrum~\\cite{Haldane191} is computed at half-filling using the HF approach. The  spectrum differs from that of disorder-free SPT zigzag ribbons: degenerate eigenvalues of the SPT phase are split and distributed similarly to that of the DOS of the edge states of a disordered TO zigzag ribbon.\nThe entanglement spectrum may be obtained from the reduced density matrix.\nTo compute the bulk entanglement spectrum, we choose the region~\\cite{Bal} $A$ separate from the zigzag edges. We use the HF approximation; thus, the relevant reduced density matrix for a region $A$ can be written as~\\cite{Peschel119}\n\\begin{eqnarray}\n\\rho_A=K e^{ -\\tilde{h}}.\n\\end{eqnarray}\n\n\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{W72G03ESfit.eps}\n\\caption{Consider a rectangular region with length $l_x=76$ and width $l_y=36$. The rectangular region is inside a ribbon with length $L_x=150$ and width $L_y=72$. Distribution of eigenvalues $\\tilde{\\epsilon}_k $ of the reduced density matrix of this region is plotted (there are also positive values of $\\tilde{\\epsilon}_k $, but the distribution is identical). The distribution follows the exponential curve $B[e^{(\\tilde{\\epsilon}-\\tilde{\\epsilon_0})^2\/\\delta^2}-1]$ (black solid line). The parameters are $\\Gamma=0.3 t$, $n_{imp}=0.1$, and $U=0.5 t$ $(g=0.19)$. The number of disorder realization is $N_D \\sim 50$.}\n\\label{calg0g3}\n\\end{center}\n\\end{figure}\n\n\n\nWhen the operator $\\tilde{h}$ is diagonalized, we get the following Hamiltonian matrix\n\\begin{eqnarray}\n\\tilde{h}_{ij}=\\sum_{k}\\psi^*_k(i)\\psi_k(j)\\tilde{\\epsilon}_k,\n\\end{eqnarray}\nwhere $\\tilde{\\epsilon}_k$ and $\\psi_k(j)$ are eigenvalues and eigenstates of the ``Hamiltonian\" $\\tilde{h}$, respectively.\nNote that this particular density matrix describes a Fermi gas at temperature $k_B T=1$.\nThe reduced density matrix of either spin-up or spin-down electrons is equal to\n\\begin{eqnarray}\n\\rho_{ij}= Tr(\\rho c^{\\dagger}_ic_j)=\\sum_k \\psi^*_k(i)\\psi_k(j)\\frac{1}{ e^{ \\tilde{\\epsilon}_k }+1 },\n\\end{eqnarray}\nwhere $c_i=\\sum_k \\psi_k(i)a_k$, and $a_k$ is the electron destruction operator corresponding to the eigenstate $\\psi_k(i)$. The distribution of the eigenvalues $\\tilde{\\epsilon}_k $ of $\\tilde{h}$ is called the entanglement spectrum. (Note that $\\tilde{\\epsilon}_k $ are not the eigenenergies of the Hartree--Fock Hamiltonian.) The eigenvalues of a density matrix are given as\n\\begin{eqnarray}\n\\lambda_k=\\frac{1}{ e^{ \\tilde{\\epsilon}_k }+1 }.\n\\end{eqnarray}\nThe values $\\tilde{\\epsilon}_k \\approx 0$, corresponding to $\\lambda_k\\approx 1\/2$, dominate the entanglement~\\cite{KunY}.\nThe entanglement spectrum of the SPT phase of a disorder-free zigzag nanoribbon exhibits numerous nearly degenerate eigenvalues, thereby reflecting the presence of nearly degenerate edge states. Fig.~\\ref{calg0g3} shows the HF entanglement spectrum of a disordered interacting graphene zigzag ribbon at half-filling. This entanglement spectrum is different from that of the disorder-free SPT phase of zigzag ribbons.\nIn the presence of disorder, the degenerate eigenvalues of the entanglement spectrum are split and exponentially distributed in a way similar to that of the DOS of edge states in TO zigzag ribbons, as shown in Fig. 1(b) in main article. \nIn contrast, the DOS of disorder-free SPT phase of zigzag ribbons has a van Hove singularity, see  Fig. 1 in main article.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nLet $k$ be a field of characteristic different from $2$ or $3$. By an elliptic curve $E$ over $k$ we mean a smooth curve of genus one together with a specified base point $0$. These curves are described by equations   \n$$y^2z = x^3 + Axz^2 + Bz^3,$$ with $A,B \\in k^\\times. $ The Brauer group of $E$, $\\Br(E)$, is an important invariant of the elliptic curve and has applications in both arithmetic and algebraic geometry. In a famous construction, Artin and Mumford used the Brauer group of the function field of $\\mathbb{P}^2$ to give a counterexample to the L\\\"uroth problem; they constructed a complex unirational $3$-fold which is not rational \\cite{artinmumford}. Furthermore, Manin described an obstruction to the Hasse principle for varieties that lies in the Brauer group \\cite{Manin, Skorobogatov:Beyond-the-Manin-Obstruction}. There has been an ongoing effort to gain a better understanding of the Brauer group. \\\\\n\nThe goal of this paper is an explicit description of $\\Br(E)$. Note that the Brauer group of $E$ is naturally isomorphic to the unramified Brauer group of $k(E)$, the function field of $E$ \\cite{colliot-thelene-sansuc:therationalityproblem,Gabber:ANoteOnTheUnramifiedBrauerGroupAndPurity}. Furthermore, $\\Br(E)$ is a torsion abelian group and therefore it will suffice to study its prime power torsion subgroups. Fix an integer $d\\geq 2$, coprime to the characteristic of $k$, and suppose that $k$ contains a primitive $d$-th root of unity $\\rho$. Some well-understood elements in the $d$-torsion of $\\Br k(E)$ are symbol (or cyclic) algebras, a generalization of quaternion algebras. These are $k(E)$-algebras with two generators $x$ and $y$ subject to the relations\n$x^d= a, y^d = b,$ and $xy=\\rho yx$ for some $a,b \\in k(E)^\\times$.  \nThe Merkurjev-Suslin theorem \\cite{merkurjev-suslin} implies that every element in the $d$-torsion of the Brauer group of $E$, $\\tor[d][\\Br(E)]$, can be described as a tensor product of these symbol algebras over $k(E)$. We aim to describe a complete set of generators and relations of $\\tor[d][\\Br(E)]$ in terms of these products. \\\\\n\nThe initial theory for the techniques used in the case $d=2$ was developed in \\cite{GuletskiiV.Yanchevskii1998} and \\cite{Pumplun1998}. This case was investigated in \\cite{Guletskii1997} and \\cite{chernousov-guletskii:2-torsion}. In the present paper, we give generators and relations of $\\tor[d][\\Br(E)]$ in the following cases:\n\\begin{itemize} \n\t\\item Any $d$ coprime to the characteristic of $k$, assuming that $k$ contains a primitive $d$-th root of unity and that the $d$-torsion of $E(\\kbar)$ is $k$-rational (see \\cref{Prop:main result in split case}).\n\t\\item $d=q$, an odd prime coprime to the characteristic of $k$, assuming only that $k$ contains a primitive $q$-th root of unity (see \\cref{sec:Algorithm}). \n\\end{itemize} \n The abstract methods we use were developed in \\cite[Chapter 4]{skorobogatov}. We describe elements in the $d$-torsion of the Brauer group as a cup-product of certain torsors over $E$ under the $d$-torsion of $E(\\kbar)$. There is an ongoing effort to calculate \\etale cup-products explicitly (see for example \\cite{Poonen-Rains:SelCupProducts} and \\cite{Aldrovandi-Ramachandra:Cup-products-the-Heisenberg-group-and-codimension-two-algebraic-cycles}). In this paper, we overcome this difficulty by calculating the cup product at the generic point and using results from Galois cohomology. We will now discuss our methods in some detail. \\\\\n\nConsider the Hochschild-Serre spectral sequence $\\H^i\\left( k, \\H^j \\left(\\overline{E}, \\mathbb{G}_m \\right) \\right) \\Rightarrow \\H^{i+j}\\left(E, \\mathbb{G}_m \\right).$\nThe $d$-torsion of its sequence of low degree terms is\n\\begin{align}\\label{Eq:exact sequence on tor[d]Br(E)} \n\\xymatrix{0 \\ar[r]& \\tor[d][\\Br(k)] \\ar[r]^i & \\tor[d][\\Br(E)] \\ar[r]^(.4)r & \\tor[d][\\H^1\\left(k, E\\left(\\kbar\\right)\\right)]  \\ar[r] & 0 },\n\\end{align}\nwhere the first map sends the class of a central simple algebra $A$ to the class of $A \\otimes k(E)$ under the canonical map $E \\rightarrow \\Spec k(E)$. For more details on this sequence, see also \\cite{Faddeev:SimpleAlgebrasOverAFieldOfAglebraicFunctionsOfOneVariable} and \\cite{Lichtenbaum1969}. We will discuss the second map in more detail in \\cref{Mkrational}.\nDenote the zero-section of $E$ by $s: \\Spec(k) \\ra E$. Sequence (\\ref{Eq:exact sequence on tor[d]Br(E)}) is split on the left by the induced map $s^*: \\Br(E) \\ra \\Br(k)$.\nThus the $d$-torsion of the Brauer group decomposes as $$\\tor[d][\\Br(E)] = \\tor[d][\\Br(E)] \\oplus \\tor[d][\\H^1\\left(k, E\\left(\\kbar\\right)\\right)].$$ Hence, in order to calculate $\\tor[d][\\Br(E)]$ it is sufficient to describe a split to $r$ in sequence (\\ref{Eq:exact sequence on tor[d]Br(E)}). To that end, consider the Kummer sequence \n\\begin{align*}\n\\xymatrix{0 \\ar[r]& M \\ar[r]& E(\\kbar) \\ar[r]^{[d]}& E(\\kbar) \\ar[r] & 0 },\n\\end{align*} \nwhere $[d]$ denotes multiplication by $d$ on the elliptic curve and $M$ is the full $d$-torsion of $E$. This induces a short exact sequence on group cohomohology\n\\begin{align}\\label{eq:Kummer induces on cohomology} \n\\xymatrix{ 0 \\ar[r] & E(k)\/[d]E(k) \\ar[r]^\\delta & \\H^1(k,M) \\ar[r]^(0.45){\\lambda} & \\tor[d][\\H^1\\left(k,E(\\kbar)\\right)] \\ar[r]& 0 }.\n\\end{align} \nWe want to relate the sequences (\\ref{Eq:exact sequence on tor[d]Br(E)}) and (\\ref{eq:Kummer induces on cohomology}). \nDefine $\\epsilon: \\H^1(k,M) \\ra \\tor[d][\\Br(E)]$ as the following composition \n\\begin{equation}\\label{Eq:epsilon}\n\\mbox{\\large{$\\epsilon$ :}}\\left\\{ \n\\begin{gathered}\n\\xymatrix{\n\t\\H^1(k,M) \\ar[r]^(.35)\\sim & \\H^1_{\\text{\\etale}}\\left(\\Spec(k),M\\right) \\ar[r]^(.6){p^*}& \\H^1_{\\text{\\etale}}(E,M)}\\\\\n\\xymatrix{\\ar[r]^(.2){- \\cup [\\mathcal{T}]}& \\H^2_{\\text{\\etale}}(E,M \\otimes M) \\ar[r]^(.55)\\e& \\H^2_{\\text{\\etale}}\\left(E, \\mu_d \\right)  \\ar[r]& \\tor[d][\\Br(E)] },\n\\end{gathered}\\right.\n\\end{equation} \nwhere \n\\begin{itemize} \n\t\\item $p: E \\ra \\Spec k$ is the structure map,\n\t\\item $\\mathcal{T}$ is the torsor given by multiplication by $d$ on $E$, and \n\t\\item $e$ is the map induced by the Weil-pairing. \n\\end{itemize} In \\cite[Chapter 4]{skorobogatov}, the author proves, using abstract methods, that $\\epsilon$ induces a split of squence (\\ref{Eq:exact sequence on tor[d]Br(E)}). We reprove this result using explicit methods and compute $\\epsilon$ in terms of group cohomology. We use this explicit description to give generators and a complete set of relations of $\\tor[d][\\Br(E)]$ under the assumption that $M$ is $k$-rational. Here is our first result. \n\\begin{thm}\\label{Prop:main result in split case} \n\tSuppose that the $d$-torsion $M$ of $E$ is $k$-rational and fix two generators $P$ and $Q$ of $M$. Denote by $0$ the identity of the group law on $E$. Let $t_P,t_Q \\in k(E)$ with divisors  $\\divisor(t_P) = d(P) - d(0)$ and $\\divisor(t_Q) = d(Q)- d(0)$ so that $t_P\\circ [d], t_Q \\circ [d] \\in (k(E)^\\times)^d$. Then the $d$-torsion of $\\Br(E)$ decomposes as \n\t$$\\tor[d][\\Br(E)] =\\tor[d][\\Br(k)] \\oplus I$$ and every element in $I$ can be represented as a tensor product \n\t$$\\left( a, t_P \\right)_{d,k(E)} \\otimes \\left( b, t_Q \\right)_{d,k(E)}$$ with $a,b \\in k^\\times$. Such a tensor product is trivial if and only if it is similar to one of the following \n\t\\begin{itemize} \n\t\t\\item $\t\\left( t_Q(P), t_P \\right)_{d,k(E)} \\otimes \\left( \\frac{t_P(P+Q)}{t_P(Q)}, t_Q \\right)_{d,k(E)}$, \n\t\t\\item $\t\\left( \\frac{ t_Q(P+Q)}{t_Q(P)}, t_P \\right)_{d,k(E)} \\otimes \\left( t_P(Q), t_Q  \\right)_{d,k(E)}$, or \n\t\t\\item $ \\left( t_Q (R), t_P(R) \\right)_{d,k(E)} $ for some $R\\in E(k), R \\neq 0,P,Q$.\\end{itemize}  \n\\end{thm}\nIn addition to recovering the generators of $\\tor[d][\\Br(E)]$ as calculated in \\cite[Remark 6.3]{Chernousov2016}, we give a full set of relations of the group. For the proof of \\cref{Prop:main result in split case} see \\cref{Mkrational}.\n\\\\\n\nThe main result of this paper is the algorithm given in \\cref{sec:Algorithm}. Let $q$ be an odd prime coprime to the characteristic of $k$. The algorithm can be used to determine a complete set of generators and relations of the $q$-torsion of $\\Br(E)$ over an arbitrary field $k$ containing a primitive $q$-th root of unity. This algorithm is proved in \\cref{Mnotkrational}. The main idea of the proof is the following: Let $L$ be the smallest Galois extension of $k$ so that the $q$-torsion $M$ of $E$ is $L$-rational. Use \\cref{Prop:main result in split case} to describe the Brauer group of $E \\times_{\\Spec k} \\Spec L$. Note that the order of $L$ over $k$ divides the order of $SL_2\\left( \\mathbb{F}_q \\right)$, which is $q (q-1) (q+1)$. If $q$ does not divide the order of $L$ over $k$, we use the fact that restriction followed by corestriction is an isomorphism to determine generators and relations of $\\Br(E)$. If the order of $L$ over $k$ equals $q$, we apply the inflation restriction exact sequence to our problem. Finally, if $q$ divides the order of $\\Br(E)$, we combine the two previous cases. \\\\\n\nA direct consequence of the algorithm is the following corollary on the symbol length. \n\n\\begin{cor}Let $q$ be an odd prime coprime to the characteristic of $k$ and assume that $k$ contains a primitive $q$-th root of unity. For any elliptic curve $E$ over $k$, the $q$-torsion of the Brauer group decomposes as $\\tor[q][\\Br(E)] = \\tor[q][\\Br(k)] \\oplus I$ and every element in $I$ can be written as a tensor product of at most $n_q = 2(q-1)(q+1)$ symbol algebras over $k(E)$. This means that the symbol length in $\\tor[q][\\Br(E)]\/\\tor[q][\\Br(k)]$ is at most $n_q$.\n\\end{cor} \n\nFinally, we calculate multiple examples in \\cref{ch:Examples}. For further examples, see also \\cite{dissertation}. \\\\\n\n\\noindent\\textbf{Notation.} \nThroughout this paper, $k$ denotes a base field of characteristic different from $2$ and $3$ and $E$ is an elliptic curve over $k$. We denote the addition on $E$ by $\\oplus$ and the point at infinity by $0$. Let $d\\geq 2$ be an integer coprime to the characteristic of $k$ so that $k$ contains a primitive $d$-th root of unity. $M$ denotes the $d$-torsion of $E$, and $[d]$ the multiplication by $d$-map on $E$. $P$ and $Q$ will always be generators of $M$ and $\\e(.,.)$ the Weil-pairing on $E$ with $\\e(P,Q) = \\rho$. Fix an isomorphism $[.]_\\rho: \\mu_{d} \\rightarrow \\Z\/d\\Z$ with $\\left[\\rho^i\\right]_\\rho = i $. Furthermore, identify $\\Z\/d\\Z$ with the subset $\\{0, \\ldots, d-1\\}$ of the integers and denote the image of $\\rho^i$ under the composition by $\\left[\\rho^i\\right]_\\rho^\\Z= i \\in \\Z$. When we assume that $d$ is an odd prime $q$, we write $d=q$. \\\\\n\nFor a field $F$, denote an algebraic closure by $\\overline{F}$ and its absolute Galois group by $G_F$. Furthermore, $H^i(F,A) = H^i(G_F,A)$ is the $i$-th (profinite) group cohomology group of $G_F$ with coefficients in a $G_F$-module $A$. $\\res, \\Cor,$ and $\\inflation$ denote the restriction, the corestriction, and the inflation map, respectively. \\\\\n\n\\noindent\\textbf{Acknowledgements.}\nI would like to thank my advisor Rajesh Kulkarni, for his expertise and support throughout the process of writing this paper. I also thank Igor Rapinchuk for helpful conversations and Alexei N. Skorobogatov for a useful correspondence. I thank David Saltman and Adam Chapman for comments on an earlier draft of this article. \n\n\\section{The Algorithm}\\label{sec:Algorithm}\nLet $k$ be a field of characteristic different from $2$ or $3$ and let $q$ be an odd prime. Assume that $q$ is coprime to the characteristic of $k$ and that $k$ contains a primitive $q$-th root of unity. Let $E$ be an elliptic curve over $k$. Denote by $M$ the $q$ torsion of $E(\\kbar)$. The Brauer group of $E$ decomposes as \n$$\\tor[q][\\Br(E)] = \\tor[q][\\Br(k)] \\oplus I$$ \nand generators $\\mathcal{G}$ and relations $\\mathcal{R}$ of $I$ can be calculated using the following algorithm. \n\\begin{enumerate}\n\t\\item Determine the kernel of the natural Galois representation \n\t$$\\Psi: G_k \\rightarrow GL_2\\left(\\mathbb{F}_q\\right).$$\n\tSince $k$ contains a primitive $q$-th root of unity, the image of $\\Psi$ lies in $SL_2(\\mathbb{F}_q)$ (for more details see \\cref{Mnotkrational}). Denote by $L$ the fixed field of this kernel. \n\t\\item \n\t\\begin{enumerate} \n\t\t\\item If $q$ divides the order of $L\/k$, fix some intermediate field $L'$ so that $L\/L'$ is a Galois extension of degree $q$. Let $P$ and $Q$ be elements in $M$ so that $\\Gal(L\/L')$ is generated by $\\overline{\\sigma}$ with $\\sigma(P) = P$ and $\\sigma(Q) = P\\oplus Q$. Set \n\t\t$$\\mathcal{G}_{L'} = \\left\\{\\left( l^q, n_Q \\right)_{L'(E)}, (a,t_P)_{L'(E)}:\n\t\ta \\in L'^\\times \\right\\},$$\n\t\twhere $t_P \\in L'(E)$ with $\\divisor(t_P)= q(P)-q(0)$ and $n_Q \\in L'(E)$ with $\\divisor(n_Q) = \\sum_{i=0}^{q-1} \\sigma^i(Q) = \\sum_{i=0}^{q-1} (iP+Q).$ Furthermore, let $t_Q \\in L(E)$ with $\\divisor(t_Q) = q(Q) - q(0)$ and $n_Q^q = N_{L(E)\/k(E)}(t_Q)$. \n\t\t\\item Else fix generators $P$ and $Q$ of $M$. Set $$\\mathcal{G}_{L'} = \\left\\{ \\left( a,t_P \\right)_{L(E)}, \\left( b,t_Q\\right)_{L(E)}: a,b \\in L^\\times  \\right\\},$$\n\t\twhere $t_P, t_Q \\in L(E)$ with $\\divisor(t_P) = q(P) - q(0)$ and $\\divisor(t_Q) = q(Q) -q(0)$. \n\t\\end{enumerate}\n\t\\item \n\tSet \n\t\\begin{align*} \n\t\\mathcal{R}_L = &\n\t\\left\\{\n\t\\left( t_Q(P), t_P \\right)_{L(E)}  \\otimes \\left( \\frac{t_P(P\\oplus Q)}{t_P(Q)} , t_Q \\right)_{L(E)},\n\t\\left( \\frac{ t_Q(P\\oplus Q)}{t_Q(P)}, t_P \\right)_{L(E)} \\otimes \\left( t_P(Q), t_Q  \\right)_{L(E)} \\right\\} \\\\\n\t&\\cup \\left\\{\n\t \\left( t_Q (R), t_P \\right)_{L(E)} \\otimes \\left( t_P(R), t_Q \\right)_{L(E)} : R \\in E(L), R \\neq 0,P,Q\\right\\}.\n\t\\end{align*} \n\t\t\\begin{enumerate} \n\t\t\\item If $q$ divides the order of $L\/k$ let \n\t\t$$\\mathcal{R}_{L',\\res} = \\left\\{ (a,t_P)_{L'(E)} : \\res(a,t_P)_{L'(E)} \\in \\mathcal{R}_L \\right\\}.$$\n\t\tFurther, if the quotient $\\frac{E(k) \\cap [q]E(L)}{[q]E(k)}$ is not trivial let \n\t\t$$\\mathcal{R}_{L',\\inflation} = \\left\\{ \\left( l^q,n_Q \\right)_{L'(E)} \\right\\}.$$ Otherwise, let $\\mathcal{R}_{L',\\inflation} = \\emptyset$. Set $\\mathcal{R}_{L'} = \\mathcal{R}_{L', \\res} \\cup \\mathcal{R}_{L',\\inflation}.$\n\t\t\\item Else let $L=L'$ and $\\mathcal{R}_{L'} = \\mathcal{R}_L$.\n\t\\end{enumerate}\n\t\\item Set \n\t$$\\mathcal{G} = \\left\\{ \\Cor(A): A \\in \\mathcal{G}_{L'} \\right\\}$$\n\tand \n\t$$\\mathcal{R} = \\left\\{ \\Cor(A): A \\in \\mathcal{R}_{L'}\\right\\}.$$\n\\end{enumerate}\n\nNote that there are additional relations that may arise from the fact that the corestriction map is not injective. These relations require a more careful treatment. See for example \\cref{subsec:Ex degree L\/k coprime to 3}.\n\n\n\\begin{rem} \n\tWe assume that the characteristic of $k$ is different form $2$ or $3$ for simplicity of the presentation of the elliptic curves and its torsion subgroups. The general results still hold in characteristic equal to $2$ or $3$. \n\\end{rem}  \n\n\n\\section{Background}\\label{ch:background}\nLet $E$ be an elliptic curve defined over a field $k$ of characteristic different from $2$ and $3$ described by the affine equation\n$$y^2 = x^3 + Ax + B$$\nwith $A,B \\in k$. Denote the point addition on $E(\\kbar)$ by $\\oplus$, the point subtraction by $\\ominus$, and the point at infinity by $0$. Furthermore, for any natural number we denote by $[d]$ the multiplication by $d$ map on $E(\\kbar)$. \\\\\n\nWe now proceed to describe the correspondence between torsors and elements in the first cohomology group. For more details we refer the reader to \\cite{skorobogatov}. Let $A$ be an algebraic group defined over a field $F$. An $F$-torsor under $A$ is a non-empty $F$-variety $T$ equipped with a right-action of $A$ so that $T(\\overline{F})$ is a principal homogeneous space under $A(\\overline{F})$. There is a bijection  \n$$ H^1(F,A) \\leftrightarrow \\left\\{ \\begin{gathered} \\text{$F$-torsors under $A$}\\\\\\text{ up to isomorphism }\\end{gathered} \\right\\}$$\nthat is explicitly given as follows. Let $T$ be an $F$-torsor under $A$. Choose an $\\overline{F}$-point $x_0$ of $T$. By the definition of $F$-torsor, for any $\\sigma \\in G_F$, there exists a unique $a_\\sigma \\in A(\\overline{F})$ so that \n$\\sigma(x_0) = x_0. a_\\sigma$. The map $\\sigma \\mapsto a_\\sigma$ defines the cocycle in $H^1(F,A)$ corresponding to $T$. \nNow consider the more general case, where $X$ is an abelian variety over a field $k$. An $X$-torsor under an $X$-group scheme $\\mathcal{A}$ is a scheme $\\mathcal{T}$ over $X$ together with an $\\mathcal{A}$-action compatible with the projection to $X$ that is \\'etale-locally trivial. As before, there is a one-to-one correspondence between $X$-torsors under $\\mathcal{A}$ and elements of the \\etale cohomology $H^1_{\\text{\\et}}(X,\\mathcal{A})$.  \nA $d$-covering of an abelian variety $X$ is a pair $(\\mathcal{T},\\psi)$, where $\\mathcal{T}$ is a $k$-torsor under $X$ and $\\psi: \\mathcal{T} \\ra X$ is a morphism such that $\\psi(x.t) = d x + \\psi(t)$ for any $t \\in \\mathcal{T}(\\kbar), x \\in X(\\kbar).$ By \\cite[Proposition 3.3.4 (a)]{skorobogatov}, any $d$-covering is an $X$-torsor under the $d$-torsion $\\tor[d][X]$.\\\\\n\nIn the following, we describe the  correspondence between the Brauer group and the second cohomology group. Let $F$ be a field and let $d\\geq 2$ be an integer. We say that two central simple algebras $A$ and $B$ are Morita equivalent if there exist some natural numbers $n$ and $m$ so that $A \\otimes M_n(F)$ and $B \\otimes M_m(F)$ are isomorphic as $F$-algebras. Elements in the Brauer group are given by equivalence classes of central simple algebras modulo Morita equivalence and the group structure is given by the tensor product. There is a group isomorphism between $\\Br(F)$ and $H^2(F, \\overline{F}^\\times)$ that can be described as follows. Let $K\/F$ be a finite Galois extension with Galois group $G$ and let $f$ be a cocycle representing an element in $H^2(G,K^\\times)$. Consider the $F$-vector space \n$A = F \\left<x_g : g \\in G \\right>$ with multiplication $\\lambda x_g = x_g g(a)$ and $x_g x_h = f(g,h) x_{gh}$. This turns $A$ into a finite dimensional central simple algebra over $F$. \n\nFrom now on suppose that the field $F$ contains a primitive $d$-th root of unity $\\rho$. Fix an isomorphism $[.]_\\rho: \\mu_{d} \\rightarrow \\Z\/d\\Z$ with $\\left[\\rho^i\\right]_\\rho = i $. Furthermore, identify $\\Z\/d\\Z$ with the subset $\\{0, \\ldots, d-1\\}$ of the integers and denote the image of $\\rho^i$ under the composition by $\\left[\\rho^i\\right]_\\rho^\\Z= i \\in \\Z$. For $a,b \\in F^\\times$, the symbol algebra \n\t\\begin{equation}\\label{defn:symbolalgebra} \n\t\t(a,b)_{d,F}= (a,b)_{d,F,\\rho} := F \\left< x,y : x^d = a , y^d = b, xy = \\rho yx \\right>\n\t\t\\end{equation} \n\tis a central simple algebra over $F$. The element in $H^2(F,\\overline{F}^\\times)$ corresponding to $(a,b)_{d,F,\\rho}$ can be represented by the cocycle\n\t\\begin{equation}\\label{eqn:cocyclesymbolalgebra}  \n\t(\\gamma, \\tau) \\mapsto \\begin{cases} a & \\text{if }\\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[ \\frac{\\tau\\left( \\sqrt[d]{b} \\right)}{\\sqrt[d]{b}}\\right]_\\rho^\\Z \\geq d \\\\ 1 & \\text{else} \\end{cases}.\n\t\\end{equation} \n\tFor more details we refer the reader to \\cite[Chapter 7 \\S 29]{Reiner:Maximal-orders}. The following cocycle representing the symbol algebra $(a,b)_{d,F}$ will prove more useful for our purposes.\n\n\\begin{prop}\\label{prop:sumbolalgebra-cocycle} \n\tLet $M$ be the $d$-torsion of an elliptic curve $E$ over $k$ with generators $P$ and $Q$. Assume that the Weil-pairing satisfies $\\e(P,Q)=\\rho$. Let $F$ be a field containing a primitive $d$-th root of unity $\\rho$. The symbol algebra $(a,b)_{d,F}$ can be represented by the cocycle \n\t$$(\\gamma,\\tau) \\mapsto \\e \\left(\\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\left( \\sqrt[d]{a} \\right)} P, \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b} \\right)} Q \\right)^{-1}$$\n\tfor every pair $a,b \\in F^\\times$. \n\tWe will often consider the case $F = k(E)$. \n\\end{prop}\n\n\\begin{proof}\n\tConsider the map $$g: \\gamma \\rightarrow {\\sqrt[d]{a}}^{ \\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b}\\right)}\\right]_\\rho^\\Z }.$$\n\tThe differential of $g$ is \n\t\\begin{align*}\n\tdg(\\gamma,\\tau) &= \\gamma \\left( {\\sqrt[d]{a}}^{ \\left[ \\frac{\\tau\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b}\\right)}\\right]_\\rho^\\Z } \\right) \n\t{\\sqrt[d]{a}}^{ \\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b}\\right)}\\right]_\\rho^\\Z }\n\t{\\sqrt[d]{a}}^{ - \\left[ \\frac{\\gamma\\tau\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b}\\right)}\\right]_\\rho^\\Z }\\\\\n\n\n\n\n\t&= \\begin{cases} \n\ta \\e \\left(\\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\left( \\sqrt[d]{a} \\right)} P, \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b} \\right)} Q \\right) &\\text{if } \\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[ \\frac{\\tau\\left( \\sqrt[d]{b} \\right)}{\\sqrt[d]{b}}\\right]_\\rho^\\Z \\geq d \\\\\n\t\\e \\left(\\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\left( \\sqrt[d]{a} \\right)} P, \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b} \\right)} Q \\right)& \\text{else} \t\t\t\t\t\n\t\\end{cases} \n\t\\end{align*}\n\tSubtracting this trivial cocycle from the cocycle in \\cref{eqn:cocyclesymbolalgebra} gives the desired result.\n\\end{proof}\n\n\\section{$M$ is $k$-rational}\\label{Mkrational}\n\nLet $k$ be a field of characteristic different from $2$ or $3$. Let $d\\geq 2$ be an integer coprime to the characteristic of $k$. Assume additionally that $k$ contains a primitive $d$-th root of unity $\\rho$. Fix an isomorphism $[.]_\\rho: \\mu_{d} \\rightarrow \\Z\/d\\Z$ with $\\left[\\rho^i\\right] = i $. Furthermore, for $\\rho^i \\in \\mu_d$, let $\\left[\\rho^i \\right]_\\rho^\\Z = i \\in \\left\\{ 0, \\ldots, d-1 \\right\\}\\subset \\Z$. Let $E$ be an elliptic curve over $k$ and denote its $d$-torsion by $M$. Assume throughout this section that $M$ is $k$-rational. Fix two generators $P$ and $Q$ of $M$. Denote by $\\e(.,.)$ the Weil pairing and assume that $\\e(P,Q) = \\rho$. Let $t_P, t_Q \\in \\kbar(E)$ with $\\divisor(t_P) = d(P) - d(0)$ and $\\divisor(t_Q) = d(Q) - d(0)$. We may assume that $t_P, t_Q \\in k(E)$ since their divisors are invariant under the Galois action of $G_k$. Let $\\mathcal{T}$ be the torsor given by multiplication by $d$ on $E$.\n\n\\begin{prop}\\label{Prop:torsor mult by 3 over kbar}\n\tThe pull-back  $\\eta^*\\left( \\mathcal{T}\\right)$ along the generic point $\\eta: \\Spec k(E) \\ra E$ corresponds to the element in $H^1\\left(k(E),M\\right)$ given by the cocycle\n\t$$\\gamma \\mapsto \\left[ \\frac{\\gamma\\left(\\alpha_Q\\right)}{\\alpha_Q} \\right]_\\rho P - \\left[\\frac{\\gamma\\left(\\alpha_P\\right)}{\\alpha_P} \\right]_\\rho Q,$$\n\twhere $\\alpha_P, \\alpha_Q \\in \\overline{k(E)}$ with $\\alpha_P^d = t_P$ and $\\alpha_Q^d = t_Q$.   \n\\end{prop}\n\n\\begin{proof}\n\tFor the correspondence between torsors and elements in the first cohomology group see \\cref{ch:background}. Let $P' \\in E(\\kbar)$ so that $[d]P' = P$. Then there is some $g_P \\in \\kbar(E)$ with $$\\divisor\\left(g_P \\right)= [d]^* (P) - [d]^* (0) = \\sum_{R \\in M} \\left((P' \\oplus R) - (R) \\right).$$ Note that we may choose $g_P \\in k(E)$ since the divisor is invariant under the action of the absolute Galois group of $k$. Now $\\divisor\\left(g_P^d\\right) = \\divisor\\left( [d]^* t_P\\right)$ and thus we may assume that $g_P^d = [d]^*t_P$. Similarly we find $g_Q \\in k(E)$ with $g_Q^d = [d]^*t_Q$. \\\\\n\tNow consider the pullback of the torsor $\\mathcal{T}$ along the generic point $\\eta: \\Spec k(E) \\rightarrow \\Spec k$. Fix a $\\overline{k(E)}$-point $x_0$ of this pullback, i.e. a map of algebras so that $x_0 \\circ [d]^* = \\iota$, where $\\iota: k(E) \\rightarrow \\overline{k(E)}$ is the inclusion. This means, that the following diagramms commute\n\t$$\n\t\\xymatrix{\n\t\t& \\Spec k(E) \\ar[r]^-\\eta \\ar[d]^{[d]} & E \\ar[d]^{[d]} \\\\\n\t\t\\Spec \\overline{k(E)} \\ar[ru] \\ar[r] & \\Spec k(E) \\ar[r]^-\\eta & E}\n\t\\qquad \t\n\t\\xymatrix{ & k(E) \\ar[ld]_{x_0} \\\\\n\t\t\\overline{k(E)}& k(E)  \\ar[u]_{[d]^*} \\ar[l]^\\iota}$$\n\tAfter possibly renaming $\\alpha_P$ and $\\alpha_Q$, we may assume that $x_0\\left(g_P\\right) = \\alpha_P$ and $x_0\\left(g_Q\\right) = \\alpha_Q$. \n\tThere is a group isomorphism\n\t$$\tM \\rightarrow \\Gal\\left(k(E)\/[d]^*k(E) \\right): R\\mapsto \\tau_R^*,$$\n\twhere $\\tau_R: E \\rightarrow E$ is the translation by $R$-map; $\\tau_R: E \\rightarrow E: S \\mapsto S \\oplus R$. By the definition of the Weil-pairing $\\e(R,P) = \\frac{g_P (X\\oplus S)}{g_P(X)} = \\frac{ \\tau_S^*(X)}{g_P(X)},$ for any $R \\in M$, $X \\in E(\\kbar)$ any point so that $g_P(X)$ and $g_P(X \\oplus S)$ are defined. The analogous result holds for $g_Q$ as well. Finally, we calculate   \n\t\\begin{align*} \n\tx_0 \\circ \\tau^*_{\\left[\\frac{\\gamma(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\gamma(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q} \\left( g_P\\right)\n\t&= x_0 \\left( \\e \\left( \\left[\\frac{\\gamma(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\gamma(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q, P \\right) g_P \\right)\\\\\n\t&= x_0 \\left( \\e\\left(-\\left[\\frac{\\gamma(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q, P\\right) g_P\\right)\\\\\n\t&= x_0 \\left({\\frac{\\gamma(\\alpha_P)}{\\alpha_P}}g_P \\right)\\\\\n\t&= \\gamma(\\alpha_P)\n\t\\end{align*}\n\tand similarly\n\t\\begin{align*} \n\tx_0 \\circ \\tau^*_{\\left[\\frac{\\gamma(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\gamma(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q} \\left( g_Q \\right)\n\t&= \\gamma(\\alpha_Q).\t\n\t\\end{align*}\n\tThe statement follows since $k(E)\/[d]^*k(E)$ is generated by $g_P$ and $g_Q$.\n\\end{proof}\n\nRecall that $\\epsilon$ is the composition \n\\begin{equation*}\n\\mbox{\\large{$\\epsilon$ :}} \\qquad \n\\begin{gathered}\n\\xymatrix{\n\tH^1(k,M) \\ar[r]^-\\sim & H^1_{\\text{\\et}}\\left(\\Spec(k),M\\right) \\ar[r]^-{p^*}& H^1_{\\text{\\et}}(E,M)}\\\\\n\\xymatrix{ \\ar[r]^-{- \\cup [\\mathcal{T}]}& H^2_{\\text{\\et}}(E,M \\otimes M) \\ar[r]^-\\e& H^2_{\\text{\\et}}\\left(E, \\mu_d\\right)  \\ar[r]& \\tor[d][\\Br(E)]}.\n\\end{gathered}\n\\end{equation*}\n\nIn \\cite[Theorem 4.1.1 and the following example]{skorobogatov}, the author proves abstractly that $\\epsilon$ induces a split to \\cref{Eq:exact sequence on tor[d]Br(E)} using general properties of torsors and the cup-product. We will now determine $\\epsilon$ explicitly and prove directly that the map induces the desired split. \n\n\\begin{prop}\\label{Prop:epsilonk}On the level of cocycles $\\epsilon$ coincides with the map that assigns to a 1-cocycle $f: G_k \\rightarrow M$ the 2-cocycle \n\t\\begin{equation*}\n\t\\begin{gathered}  \n\t\\epsilon(f): G_{k(E)} \\times G_{k(E)} \\ra \\overline{k(E)}^\\times\\\\\n\t(\\gamma,\\tau) \\mapsto \\e\\left( f(\\gamma), \\gamma \\left( \\left[\\frac{\\tau(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\tau(\\alpha_P)}{\\alpha_P}\\right]_\\rho  Q \\right)  \\right).\n\t\\end{gathered} \n\t\\end{equation*} \n\\end{prop}\n\n\\begin{proof}The cup-product commutes with $\\eta^*$ by \\cite[Chapter 2, 8.2]{Bredon:Sheaf-Theory}. Consider the following diagram with commutative squares\n\t$$\n\t\\xymatrix{\n\t\tH^1(k,M) \\ar[r]^{p^*} & H^1_{\\text{\\et}}(E,M) \\ar[d]^{\\eta^*} \\ar[rr]^-{-\\cup [\\mathcal{T}]} && H^2_{\\text{\\et}}(E,M \\otimes M) \\ar[r]^-\\e_-\\sim \\ar[d]^{\\eta^*}\t& H^2_{\\text{\\et}}(E,\\mu_d)\\ar[d] \\\\ \n\t\t& H^1_{\\text{\\et}}(k(E),M) \\ar[rr]^-{-\\cup [\\eta^*\\mathcal{T}]} && H^2_{\\text{\\et}}(k(E), M \\otimes M ) \\ar[r]^-\\e_-\\sim & \\tor[d][\\Br k(E)]\n\t}$$\n\tThe statement follows from \\cref{prop:torsormult3} and by the definition of the cup-product in group cohomology \\cite[Chapter VIII, Section 3]{serrelocal}. Recall that by \\cite[Theorem 5.11]{colliot-thelene-sansuc:therationalityproblem} the map on the right is given by the injection that identifies $\\Br(E)$ with the unramified Brauer group of $k(E)$. \n\\end{proof}\n\nWe are now ready to calculate a set of generators of $\\tor[d][\\Br(E)]$. Since we assume that $M$ is $k$-rational, Kummer theory implies that there is an isomorphism \n\\begin{align}\\label{Eq:phi definition}  \\phi: k^\\times\/(k^\\times)^d \\times k^\\times\/(k^\\times)^d \\rightarrow H^1(k,M): (a,b) \\mapsto c_{a,b},\\end{align} \nwhere $c_{a,b}$ can be represented by the cocycle\n$$G_k \\rightarrow M: \\gamma \\mapsto \\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right) }{\\sqrt[d]{a}} \\right]_\\rho P \\oplus \\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right) }{\\sqrt[d]{b}} \\right]_\\rho Q.$$ \n\n\\begin{prop}\\label{Prop:epsilon-split-case}The cocycle\n\t$\\epsilon \\circ \\phi (a,b)$ corresponds to the Brauer class of $$\\left(a,t_P\\right)_{d,k(E)} \\otimes \\left( b, t_Q \\right)_{d,k(E)}$$ for any \n\t$(a,b) \\in k^\\times\/(k^\\times)^d \\times k^\\times\/(k^\\times)^d$. \n\\end{prop}\n\n\\begin{proof}Observe that  $\t\\epsilon_k\\circ \\phi(a,b)$ can be represented by the cocycle that takes $(\\gamma,\\tau) \\in G_{k(E)} \\times G_{k(E)}$ to \n\t\\begin{align*}\n\t&\\e\\left( \\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right) }{\\sqrt[d]{a}} \\right]_\\rho P \\oplus \\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right) }{\\sqrt[d]{b}} \\right]_\\rho Q, \\left[\\frac{\\tau(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\tau(\\alpha_P)}{\\alpha_P}\\right]_\\rho  Q  \\right)\\\\\n\t=& \\e\\left( \\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right) }{\\sqrt[d]{a}} \\right]_\\rho P, \\left[\\frac{\\tau(\\alpha_P)}{\\alpha_P}\\right]_\\rho  Q  \\right)^{-1} \\e\\left(\\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right) }{\\sqrt[d]{b}} \\right]_\\rho Q, \\left[\\frac{\\tau(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P \\right).\n\t\\end{align*}\n\tThe statement follows from \\cref{prop:sumbolalgebra-cocycle}.\n\\end{proof}\nRecall that we need to prove that $\\epsilon$ induces a split to the sequence (\\ref{Eq:exact sequence on tor[d]Br(E)}) on the right, i.e. we need to show that $r \\circ \\epsilon = \\lambda$ and $\\epsilon(\\ker (\\lambda))= 0$. We first describe $r$ explicitly. For more details see also \\cite{Faddeev:SimpleAlgebrasOverAFieldOfAglebraicFunctionsOfOneVariable} or \\cite{Lichtenbaum1969}. Let \n$\\alpha \\in \\Br(E) \\subset \\Br k(E)$. Using Tsen's theorem we view $\\alpha$ as an element in $\\H^2\\left( G_k, \\kbar(E)^\\times \\right) \\cong \\Br k(E)$. \nConsider the exact sequence \n$$0 \\rightarrow \\kbar(E)^\\times \\rightarrow \\Prin (\\overline{E}) \\rightarrow \\Div(\\overline{E}) \\rightarrow 0,$$\nwhere $\\Prin(\\overline{E})$ denotes the set of pricipal divisors on $\\overline{E}$ and $\\Div(\\overline{E})$ the set of divisors on $\\overline{E}$. \nThe sequence induced on group cohomology is \n$$\\H^2(G_k,\\kbar(E)^\\times) \\rightarrow \\H^2(G_k, \\Prin(\\overline{E})) \\rightarrow \\H^2(G_k, \\Div(\\overline{E})),$$\nwhere the first map takes $\\alpha$ to some $\\alpha'$ in the kernel of the second map. Now consider the degree sequence \n$$0 \\rightarrow \\Div^0(\\overline{E}) \\rightarrow \\Div(\\overline{E}) \\rightarrow \\Z \\rightarrow 0, $$\nwhere $\\Div^0(\\overline{E})$ is the group of degree zero divisors. Since $\\H^1(G_k, \\Z) =0$,  \nthe map $$\\H^2(G_k, \\Div^0(\\overline{E})) \\rightarrow \\H^2(G_k, \\Div(\\overline{E}))$$ is injective. Finally, the exact sequence\n$$0\\rightarrow \\Prin(\\overline{E}) \\rightarrow \\Div^0(\\overline{E}) \\rightarrow E(\\kbar) \\rightarrow 0$$\ninduces an exact sequence\n$$1 \\rightarrow \\H^1(G_k,E(\\kbar)) \\rightarrow \\H^2(G_k, \\Prin(\\overline{E})) \\rightarrow \\H^2(G_k, \\Div^0(\\overline{E})).$$\nThe element $\\alpha'$ is in the kernel of the second map, and therefore there exists a unique $\\alpha'' \\in \\H^1(G_k, E(\\kbar))$ with image $\\alpha'$. Set $r(\\alpha) = \\alpha''$. \n\n\\begin{prop}\\label{prop:kappcircepsilon=lambda_split}\n\t$r \\circ \\epsilon = \\lambda$. \n\\end{prop} \n\n\\begin{proof}We will only prove that $r \\circ \\epsilon\\circ \\phi(a,1) = \\lambda\\circ \\phi(a,1).$ The other cases are similar. We showed previously that $\\epsilon \\circ \\phi(a,1) = (a,t_P)_{d,k(E)}$, which corresponds to the cocycle \n\t$$(\\gamma,\\tau) \\mapsto \\begin{cases} 1 & \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z < d \\\\\n\tt_P  & \\text{else}\\end{cases}$$\n\tin $H^2(G_k, \\kbar(E)^\\times)$. This gives an element in $H^2(G_k, \\Prin(E))$ via \n\t$$(\\gamma,\\tau) \\mapsto \\begin{cases} 1 & \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z < d \\\\\n\td(P) - d(0)  & \\text{else} \\end{cases}.$$\n\tOn the other hand for any $\\gamma \\in G_k$\n\t$$\\lambda(\\phi(a,1)) (\\gamma) = \\phi(a,1)(\\gamma) =  \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z P.$$\n\tNow we follow the proof of the snake lemma to calculate the image of $\\lambda(f)$ under the connecting homomorphism \n\t$$H^1(k, E(\\kbar)) \\rightarrow H^2(k, \\Prin(E))$$\n\tinduced by the sequence \n\t$$0 \\ra \\kbar(E)^\\times \\ra \\Prin(\\overline{E}) \\ra \\Div (\\overline{E}) \\ra 0. $$\n\tFirst lift it to \n\t$$\\gamma \\mapsto \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right) \\in \\Div^0(\\overline{E}).$$\n\tNow use the boundary map to get \n\t\\begin{align*} \n\t(\\gamma,\\tau) \\mapsto &\\gamma \\left( \\left[ \\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right)\\right) -  \\left[ \\frac{\\gamma\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right) +  \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right)\\\\\n\t&= \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right) - \\left[ \\frac{\\gamma\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right) + \\left[ \\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}} \\right]_\\rho^\\Z \\left(  (P) - (0) \\right)\\\\\n\t&=  \\begin{cases} d(P) - d(0) & \\text{if } \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\geq d \\\\\n\t1 & \\text{else} \\end{cases},\n\t\\end{align*}\n\twhich coincides with what we previously calculated. The statement follows.  \n\\end{proof} \n\n\\begin{prop}\\label{prop:epsilonkerlambda=0} \n\t$\\epsilon\\left( \\ker(\\lambda) \\right) = 0.$ \n\\end{prop}\n\\begin{proof} \n\tRecall that $\\ker(\\lambda) = \\text{Im}(\\delta)$ and let $R \\in E(k)$. By the previous proposition $r\\circ\\epsilon\\circ \\delta (R) = \\lambda \\circ \\delta(R)$ is trivial. Thus the algebra $\\epsilon\\circ \\delta (R) $ is in the image of the $\\Br(k) \\rightarrow \\Br(E)$. It remains to show that the specialization of $\\epsilon \\circ \\delta(R)$ at $0$ is trivial. The cup-product commutes with specialization at a closed point \\cite[Chapter 2, 8.2]{Bredon:Sheaf-Theory}, i.e. $\\left( [\\mathcal{T}'] \\cup [\\mathcal{T}]\\right)_S = \\left( [\\mathcal{T}']\\right)_S \\cup \\left( [\\mathcal{T}]\\right)_S$ for every $S \\in E(k)$ and every $[\\mathcal{T}']\\in H^1_{\\text{\\et}} (E, M)$. By definition of $\\epsilon$ $$\\left( \\epsilon\\circ \\delta (R) \\right)_S =  \\left( \\epsilon\\circ \\delta (R) \\right)_S =   \\delta(R) \\cup \\mathcal{T}_S \\in \\Br(k)$$ for any $S \\in E(k)$. \n\tIn particular, $\\left( \\epsilon\\circ \\delta (R) \\right)_0 = \\delta(R) \\cup \\mathcal{T}_0$. The specialization of $\\mathcal{T}$ at $0$ admits a point (the point $0$) and is therefore the trivial torsor. We deduce that  $\\left( \\epsilon\\circ \\delta (R) \\right)_0 $ is trivial and thus so is $\\epsilon\\circ \\delta (R)$. \n\\end{proof} \n\nWe conclude that under the above assumptions, the $d$-torsion of $\\Br(E)$ decomposes as \n\t$$\\tor[d][\\Br(E)] =\\tor[d][\\Br(k)] \\oplus I$$ and every element in $I$ can be represented as a tensor product \n\t$$\\left( a, t_P \\right)_{d,k(E)} \\otimes \\left( b, t_Q \\right)_{d,k(E)}$$ with $a,b \\in k^\\times$.\n\tWe now proceed to determine the relations in $I$.  \nRecall that that an element in $I$ as before is trivial if and only if it is in the image of the composition \n$$\\xymatrix{\n\tE(k)\/[d]E(k) \\ar[r]^-\\delta & H^1(k,M) \\ar[r]^-\\epsilon & \\tor[d][\\Br(k)]\n}.$$\nWe first need to describe the image of $\\phi^{-1} \\circ \\delta$ explicitly, where $\\phi$ is the isomorphism from \\cref{Eq:phi definition}.\n\\begin{prop}Let $R \\in E(k)\/[d]E(k)$ and let $t_P, t_Q \\in k(E)$ with $\\divisor(t_P) = d(P) - d(0)$, $\\divisor (t_Q) = d(Q) -d(0)$. Assume that  $t_P\\circ [d], t_Q \\circ [d] \\in (k(E)^\\times)^d$. Then \n\t$$\\phi^{-1} \\circ \\delta (R) = \\begin{cases} \n\t(1,1) & R = 0\\\\\n\t\\left( t_Q(P), \\frac{t_P(P\\oplus Q)}{t_P(Q)} \\right) & R = P \\\\\n\t\\left( \\frac{ t_Q(P\\oplus Q)}{t_Q(P)}, t_P(Q)  \\right) & R= Q \\\\ \n\t\\left( t_Q (R), t_P(R) \\right) & \\text{else} \\end{cases} $$\n\\end{prop}\t\n\nThe proof of this proposition is inspired by a computation of the Kummer pairing in \\cite[Ch. X, Theorem 1.1]{silverman}.  \n\\begin{proof} \n\tLet $R \\in E(k)\/dE(k)$ and suppose that $R \\neq 0,P,Q$. Fix some $S \\in E(\\kbar)$ with $[d]S = R$. Let $t_P, t_Q$ as above and fix $g_P, g_Q \\in \\kbar(E)$ with $g_P^d = t_P \\circ [d]$ and $g_Q^d = t_Q \\circ [d]$. Since the divisors of $g_P$ and $g_Q$ are $G_k$-invariant, we may choose $g_P, g_Q \\in k(E)$.\n\tBy the definition of $\\phi$ we see that $\\phi(f)=(a,b)$ for some cocycle $f$ representing a class in $ H^1(k,M)$ means exactly that \n\t$\\e\\left(f(\\gamma),P\\right) = \\frac{\\gamma\\left( \\sqrt[d]{b}\\right) }{\\sqrt[d]{b}} \\text{ and } \\e\\left(f(\\gamma),Q\\right) = \\frac{\\gamma\\left( \\sqrt[d]{a}\\right) }{\\sqrt[d]{a}}.$\n\tThe Weil pairing satisfies \n\t$$\\e(\\gamma(S) \\ominus S,P ) = \\frac{ g_P(\\gamma(S) \\ominus S \\oplus S)}{g_P(S)}=  \\frac{ g_P(\\gamma(S))}{g_P(S)} =\\frac{\\gamma( g_P(S))}{g_P(S)}.$$\n\tAdditionally by definition of $g_P$, we see that $g_P(S)^d = t_P \\circ [d] (S) = t_P(R)$. A similar result holds for $Q$ as well. Therefore $\\phi^{-1}\\circ \\delta(R) = \n\t\\left( t_Q (R), t_P(R) \\right)$. The other results follow by linearity of the Weil pairing. \n\\end{proof} \n\nSummarizing these results we conclude \\cref{Prop:main result in split case}. \n\n\\section{$M$ is not $k$-rational}\\label{Mnotkrational}\n\nNow let $k$ be any field and assume that $d=q$ is an odd prime coprime to the characteristic of $k$. Assume that $k$ contains a primitive $q$-th root of unity $\\rho$. Let $E$ be an elliptic curve over $k$ and denote its $q$-torsion by $M$. We do not assume that $M$ is $k$-rational throughout this section. Consider the Galois representation \n$$\\Psi: G_k \\rightarrow GL_2(\\mathbb{F}_q)$$ given by the action of $G_k$ on $M$ and denote the fixed field of its kernel by $L$. Since $k$ contains a primitive $q$-th root of unity $\\rho$, the image of $\\Psi$ lies in $SL_2(\\mathbb{F}_q)$. For $\\sigma \\in G_k$ with $\\Psi(\\sigma) = \\begin{pmatrix} a&b\\\\c&d \\end{pmatrix}$, we have that \n$$\\rho = \\sigma(\\rho) = \\sigma \\e (P,Q) = \\e \\left(\\sigma(P), \\sigma(Q) \\right) = \\e \\left( aP + cQ, bP + dQ \\right) = \\e(P,Q)^{ad-bc} = \\rho^{ad-bc}.$$\nTherefore $ad-bc=1$ and $\\Psi(\\sigma) \\in SL_2(\\mathbb{F}_q)$. \n\nConsider the tower of field extensions \n$$\\xymatrix{\n\t& \\overline{k(E)}\\ar@{-}[d]\\\\\n\t& L(E) \\ar@{-}[ld]_M \\ar@{-}[rd]^{\\Gal(L\/k)}\\\\\n\t[q]^*L(E)\\ar@{-}[rd]^{\\Gal(L\/k)} & & k(E) \\ar@{-}[ld]\\\\\n\t& [q]^* k(E)\n}$$\nFix a set $\\tilde{G}_{L\/k} \\subset G_{k(E)}$ of coset representatives of $G_{k(E)} \/ G_{L(E)} \\cong \\Gal(L\/k)$. Note that $\\tilde{G}_{L\/k}$ is also a set of coset representatives of \n$G_{[q]^*k(E)} \/ G_{[q]^*L(E)} \\cong \\Gal(L\/k)$ and every $\\tilde{\\sigma} \\in \\tilde{G}_{L\/k}$ fixes $k(E)$. Let $\\gamma \\in G_{[q]^* k(E)}$, then $\\gamma$ decomposes as $\\gamma= \\gamma' \\tilde{\\sigma}$ for some $\\gamma' \\in G_{[q]^*L(E)}$ and some $\\tilde{\\sigma} \\in \\tilde{G}_{L\/k}$. \\\\\n\nLet $\\mathcal{T}$ be the torsor given by multiplication by $q$ on $E$. Fix a set of coset representatives $\\tilde{G}_{L\/k}$ as before. We now describe the cocycle corresponding to the pullback of $\\mathcal{T}$ along the generic point using the correspondence in  \\cref{ch:background}.\nLet $x_1$ be a $\\overline{k(E)}$ point. We may assume, that $x_0 = \\iota_1 \\circ x_0$ for some $x_0$ as in the following commutative diagram. \n$$\n\\xymatrix{\n\tk(E) \\ar[r]^{\\iota_1}\\ar@\/^3pc\/[drr]^{x_1} & L(E) \\ar[rd]^{x_0} \\\\ \n\tk(E) \\ar[u]^{[q]^*} \\ar[r]^{\\iota_1}  &L(E) \\ar[u]^{[q]^*} \\ar[r]^\\iota & \\overline{k(E)} }.\n$$\nAny $\\gamma = \\gamma' \\tilde{\\sigma} \\in G_{k(E)}$ with $\\gamma' \\in G_{L(E)}, \\tilde{\\sigma} \\in \\tilde{G}_{L\/k}$ acts on $x_1$ by \n$$\\gamma.x_1 = \\gamma' \\circ \\tilde{\\sigma} x_0 \\circ \\iota_1 =  \\gamma'. x_1.$$\nFinally, $\\gamma'.x_1$ can be computed as in \\cref{Prop:torsor mult by 3 over kbar}.\nSummarizing this we conclude the following proposition. \n\n\\begin{prop}\\label{prop:torsormult3}\n\tThe pull-back of $\\mathcal{T}$ to the generic point corresponds to the element in $H^1(k(E),M)$ given by the cocycle \n\t$$G_{k(E)} \\ra M: \\gamma \\mapsto \\left[\\frac{\\gamma'(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[ \\frac{\\gamma'(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q,$$\n\twhere $\\gamma = \\gamma' \\tilde{\\sigma}$ as above, for some $\\tilde{\\sigma} \\in \\tilde{G}_{L\/k}$ and $\\gamma' \\in G_{L(E)}$, $\\alpha_P, \\alpha_Q\\in \\overline{k(E)}$ so that $\\alpha_P^q = t_P$ and $\\alpha_Q^q = t_Q$.  \n\\end{prop}\n\nAs in the previous section, we describe the map $\\epsilon$ explicitly. \n\n\\begin{prop}On the level of cocycles $\\epsilon$ coincides with the map that assigns to a 1-cocycle $f: G_k \\rightarrow M$ the 2-cocycle \n\t\\begin{equation*}\n\t\\begin{gathered}  \n\t\\epsilon(f): G_{k(E)} \\times G_{k(E)} \\ra \\overline{k(E)}^\\times\\\\\n\t(\\gamma,\\tau) \\mapsto \\e\\left( f(\\gamma), \\gamma \\left( \\left[\\frac{\\tau'(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\tau'(\\alpha_P)}{\\alpha_P}\\right]_\\rho  Q \\right)  \\right),\n\t\\end{gathered} \n\t\\end{equation*} \n\tfor $\\gamma,\\tau \\in G_{k(E)}, \\tau = \\tau' \\tilde{\\sigma}$ for some $\\sigma \\in \\tilde{G}_{L\/k}$ and $\\tau' \\in G_{\\kbar(E)}$. \n\\end{prop}\n\n\\subsection{$[L:k]$ is coprime to $q$ }\\label{sec:Lkcoprimetoq}\n\nSuppose throughout this section that $q$ does not divide the order $[L:k]$. \nLet $\\mathcal{T}$ be the torsor given by multiplication by $d$ on $E$. Consider the pull-back $\\eta_L^*(\\mathcal{T})$ of $\\mathcal{T}$ to $L(E)$ \nand the pull-back $\\eta_k^*(\\mathcal{T})$ of $\\mathcal{T}$ to $k(E)$. By \\cref{Prop:torsor mult by 3 over kbar} and \\cref{prop:torsormult3} we see immediately that $\\res\\left( \\eta^*_k (\\mathcal{T}) \\right) = \\eta^*_L (\\mathcal{T})$. By \\cite[Ch. 1, Proposition 1.5.3 (iii) and (iv)]{Neukirch-Schmidt-Wingberg-Cohomology-of-number-fields} and the construction of $\\epsilon$ the following diagram commutes\n$$\\xymatrix{\n\tH^1(k,M) \\ar[r]^\\res \\ar[d]^{\\epsilon_k} & H^1(L,M) \\ar[r]^\\Cor \\ar[d]^{\\epsilon_L} & H^1(k,M) \\ar[d]^{\\epsilon_k}\\\\\n\t\\tor[q][\\Br(E)] \\ar[r]^\\res & \\tor[q][\\Br(E \\otimes L)] \\ar[r]^\\Cor & \\tor[q][\\Br(E)]\n}.$$\nTherefore, the corestriction map $$\\Cor: \\tor[q][\\Br(E \\otimes L)] \\rightarrow \\tor[q][\\Br(E)]$$ is surjective and every element in $I$ can be written as $\\Cor(A)$ with $A \\in \\tor[q][\\Br(E)]$. We summarize this observation in the following theorem. \n\\begin{thm}\n\tLet $t_P,t_Q \\in L(E)$ with divisors  $\\divisor(t_P) = q(Q) - q(0)$ and $\\divisor(t_Q) = q(Q)- q(0)$. Then the $q$-torsion of $\\Br(E)$ decomposes as \n\t$$\\tor[q][\\Br(E)] =\\tor[q][\\Br(k)] \\oplus I$$ and every element in $I$ can be represented as a tensor product \n\t$$\\Cor\\left( a, t_P \\right)_{q,L(E)} \\otimes \\Cor\\left( b, t_Q \\right)_{q,L(E)}$$ with $a,b \\in L^\\times$. \n\\end{thm}\n\n\\begin{rem}\\label{rem:image of restriction} The corestriction map is in general not injective. To get a smaller set of generators remark that by \\cite[Ch, 1, Corollary 1.5.7]{Neukirch-Schmidt-Wingberg-Cohomology-of-number-fields} the image of the restriction map $H^1(k,M) \\ra H^1(L,M)$ coincides with the image of the norm map $$N_{L\/k}: H^1(L,M) \\ra H^1(L,M).$$ Now by Kummer theory $H^1(L,M) \\cong L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ via the isomorphism $\\phi$. Let $g \\in G_k$ and $(a,b) \\in L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$. Suppose that $g^{-1}(P) = c_1 P \\oplus c_2 Q$ and $g^{-1}(Q) = c_3P \\oplus c_4Q$. The action of $g$ compatible with $\\phi$ is  \n\t\\begin{align*}\n\tg.(a,b) =& \\phi^{-1} g. \\phi(a,b) \\\\\n\t=& \\left( \\left(  g^{-1}(a) \\right)^{c_1} \\left( g^{-1}(b) \\right)^{c_3}, \\left(  g^{-1}(a) \\right)^{c_2} \\left( g^{-1}(b) \\right)^{c_4}\\right) .\n\t\\end{align*}\n\tNow the image of the restriction followed by $\\phi$ coincides with the image of the norm on $L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ under the above action. \n\\end{rem} \nTo calculate the relations consider the following commutative diagram \n$$\\xymatrix{\n\tE(k)\/[q]E(k) \\ar[r]^{\\res} \\ar[d]^{\\delta_k}& E(L)\/[q]E(L) \\ar[r]^{\\Cor} \\ar[d]^{\\delta_L}& E(k)\/[q]E(k) \\ar[d]^{\\delta_k}\\\\\n\tH^1(k,M) \\ar[r]^{\\res} & H^1(L,M) \\ar[r]^{\\Cor} & H^1(k,M) \n},$$where the horizontal compositions coincide with multiplication by $[L:k]$, which is an isomorphism. Thus, the image of $\\delta_k$ is also given by the image of the composition $\\delta_k \\circ \\Cor = \\Cor \\circ \\delta_L$. Using the description of the image of $\\delta_L$ in the previous section we deduce the following result. \n\n\\begin{prop} \n\tSuppose that $[L:k]$ is not divisible by $q$. Fix two generators $P$ and $Q$ of $M$ and and let $t_P, t_Q \\in L(E)$ with $\\divisor(t_P) = q(P) - q(0)$ and $\\divisor(t_Q) = q(Q) - q(0)$. Assume additionally that $t_P \\circ [q], t_Q \\circ[q] \\in \\left( L(E)^\\times \\right)^d.$ An element \n\t$$\\Cor(a,t_P)_{L(E)} \\otimes \\Cor(b,t_Q)_{L(E)}$$in $I$ is trivial if it is similar to one of the following \n\t\\begin{itemize}\n\t\t\\item $\\Cor\\left( t_Q(P), t_P \\right)_{k(E)}  \\otimes \\Cor\\left( \\frac{t_P(P\\oplus Q)}{t_P(Q)} , t_Q \\right)_{k(E)}$,\n\t\t\\item $ \\Cor\\left( \\frac{ t_Q(P \\oplus Q)}{t_Q(P)}, t_P \\right)_{k(E)} \\otimes \\Cor\\left( t_P(Q), t_Q  \\right)_{k(E)},$ or  \n\t\t\\item $\\Cor\\left( t_Q (R), t_P \\right)_{k(E)} \\otimes \\Cor\\left( t_P(R), t_Q \\right)_{k(E)}$ for some $R \\in E(k)\\setminus\\{0,P,Q\\}$. \n\t\\end{itemize}\n\\end{prop} \n\nThe following observation will be useful to calculate these corestrictions explicitly. Consider the following commutative diagram\n\\begin{equation}\\label{eqn:relationscorrescorres} \n\\xymatrix{\n\tE(k) \\ar[r] \\ar[d]^{\\delta_k}& E(L) \\ar[r] \\ar[d]^{\\delta_L} & E(k) \\ar[r] \\ar[d]^{\\delta_k}& E(L) \\ar[r] \\ar[d]^{\\delta_L} & E(k) \\ar[d]^{\\delta_k}\\\\\n\tH^1(k,M) \\ar[r]^\\res \\ar[d]^{\\epsilon_k}& H^1(L,M) \\ar[r]^\\Cor \\ar[d]^{\\epsilon_L}& H^1(k,M)\\textbf{} \\ar[r]^\\res\\ar[d]^{\\epsilon_k}  & H^1(L,M) \\ar[r]^\\Cor \\ar[d]^{\\epsilon_L}& H^1(k,M) \\ar[d]^{\\epsilon_k}\\\\\n\t\\tor[q][\\Br E] \\ar[r] & \\tor[q][\\Br E_L] \\ar[r] &  \\tor[q][\\Br E] \\ar[r] & \\tor[q][\\Br E_L] \\ar[r] &  \\tor[q][\\Br E]\n},\\end{equation} \nwhere the rows compose to multiplication by $[L:k]^2$, which is an isomorphism. Furthermore, the composition $\\res \\circ \\Cor$ coincides with the norm map \\cite[Ch. 1, Corollary 1.5.7]{Neukirch-Schmidt-Wingberg-Cohomology-of-number-fields}. Therefore, an element in $I$ is trivial if it lies in the image of the composition $\\Cor\\circ \\epsilon_L \\circ N_{L\/k} \\circ \\delta_L \\circ \\res$. In \\cref{subsec:Ex degree L\/k coprime to 3}, we see how this observation can be applied to the calculation of the relations in $I$. \n\n\\subsection{$[L:k]$ equals $q$}\\label{sec:Generators-div-q} \n\nSuppose for this section that $L$ is of degree $q$ over $k$. After renaming $P$ and $Q$ we may assume without loss of generality that there is some $\\sigma \\in G_k$ such that $\\sigma(Q) = P\\oplus Q$ and $\\overline{\\sigma}$ generates $G_k\/G_L$.\nFix a coset representative $\\tilde{\\sigma}$ of $\\sigma$ in $G_{k(E)}$.\nAdditionally denote a primitive element for the extension $L\/k$ by $l$.\nConsider the diagram\n$$\\xymatrix{ 0 \\ar[r]& H^1\\left(G_k\/G_L, M\\right) \\ar[r]^-{\\inflation}& H^1(k,M)\\ar[rr]^-\\res\\ar[d]^{\\epsilon_k} && H^1(L,M)^{G_k\/G_L}\\ar[d]^{\\epsilon_L}\\\\\n\t& & \\tor[q][\\Br(E)]\\ar[rr]^-\\res && \\tor[q][\\Br(E \\times_k \\Spec L)] },$$\nwhere the first row is the inflation-restriction exact sequence. The diagram commutes by construction of $\\epsilon$ and since the restriction map and the cup-product commute \\cite[Ch. 1 Proposition 1.5.3 (iii)]{Neukirch-Schmidt-Wingberg-Cohomology-of-number-fields}. We will first describe the image of the inflation map and explore the restriction afterwards. We will apply the following technical lemma throughout. \n\n\\begin{lem}\\label{lem:technical}\n\t$\\sum_{i=0}^{q-1} \\sigma^i(R) = 0$ for every $R \\in M$.\n\\end{lem} \n\n\\begin{proof} \n\tLet $R= mP\\oplus nQ \\in M$. We calculate directly that\n\t$$\n\t\\sum_{i=0}^{q-1} \\sigma^i(mP \\oplus nQ) =  \\sum_{i=0}^{q-1} \\left( mP \\oplus inP\\oplus nQ \\right) = mqP \\oplus \\frac{q (q-1)}2 n P \\oplus n q Q = 0.\n\t$$  \n\\end{proof}\n\n\\begin{lem} The group\n\t$H^1\\left( G_k\/G_L, M \\right)$ is cyclic of rank $q$ with generator the class of the cocycle $f_L$ defined by $f_L(\\overline{\\sigma}) = Q$. \n\\end{lem} \n\n\\begin{proof} \n\t\\Cref{lem:technical} implies that  $f_L(\\overline{\\sigma}^q) = \\sum_{i=0}^{q-1} \\sigma^i f_L(\\overline{\\sigma}) = 0$ and thus $f_L$ defines a cocycle. Since $G_k\/G_L$ is cyclic with generator $\\overline{\\sigma}$, every element $f$ in $H^1(G_k,G_L,M)$ is determined by $f(\\overline{\\sigma})$. Furthermore, if $f(\\overline{\\sigma}) = mP\\oplus nQ$, then \n\t\\begin{align*}\n\tf(\\overline{\\sigma}) - \\overline{\\sigma}(mP) = mP \\oplus nQ \\ominus mP =  nQ = f_L^n (\\overline{\\sigma}).\n\t\\end{align*} \n\tThe statement follows. \n\\end{proof} \nLet $\\alpha_Q \\in \\overline{k(E)}$ with $\\alpha_Q^q = t_Q$ and consider $n_Q = \\prod_{i=0}^{q-1} \\tilde{\\sigma}^i \\left(\\alpha_Q\\right)$. Fix $\\gamma \\in G_{L(E)}$. By our previous calculations and with $x_0$ and $g_Q$ as in the proof of \\cref{prop:torsormult3} we deduce that there is some $R \\in M$ such that $\\gamma(\\alpha_Q) = R.x_0(g_Q)$. Then \nby \\cref{lem:technical} \n\\begin{align}\n\\gamma \\left( \\prod_{i=0}^{q-1} \\tilde{\\sigma}^i \\left(\\alpha_Q\\right) \\right)  = \\left( \\sum_{i=0}^{p-1} \\sigma^i(R) \\right). x_0(g_Q) = \\alpha_Q.\n\\end{align} Now $\\prod_{i=0}^{q-1} \\tilde{\\sigma}^i \\left(\\alpha_Q \\right) $ is obviously fixed by $\\tilde{\\sigma}$ and therefore $\\prod_{i=0}^{q-1} \\tilde{\\sigma}^i \\left(\\alpha_Q\\right) \\in k(E).$ Thus $n_Q$ is defined to be the element in $k(E)$ with $n_Q^q = N_{L(E)\/k(E)} (t_Q)$. Note additionally that $\\divisor n_Q = \\sum_{i=0}^{q-1} \\left(\\sigma^i(Q) - (0)\\right)$.\n\n\\begin{prop}\\label{prop:ImageofInflation}\n\t$\\epsilon_k\\left(\\inflation\\left(f_L\\right)\\right)$ is the inverse of the Brauer class of the symbol algebra\n\t$\\left(l^q, n_Q\\right)_{q,k(E)},$ where $\\alpha_Q \\in \\overline{k(E)}$ with $\\alpha_Q^q = t_Q$.\n\\end{prop} \n\n\\begin{proof} Let $\\gamma, \\tau \\in G_{k(E)}$ and denote $\\gamma = \\gamma' \\tilde{\\sigma}^i, \\tau = \\tau' \\tilde{\\sigma}^j$ with $\\gamma',\\tau' \\in G_{L(E)}$. Then by definition of $\\epsilon$ we see that  \n\t\\begin{align*} \n\t\\epsilon_k \\left( \\inflation\\left( f_L \\right) \\right) (\\gamma,\\tau) \n\t&=\\e \\left( \\frac{(i-1) i }{2} P \\oplus iQ , \\sigma^i \\left( \\left[\\frac{ \\tau'\\left( \\alpha_Q \\right)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{ \\tau' \\left( \\alpha_P \\right)}{\\alpha_P}\\right]_\\rho Q \\right) \\right)  \\\\\n\n\n\n\t&= \\left( \\frac{ \\tau' \\left( \\alpha_P \\right)}{\\alpha_P} \\right)^{- \\frac{(i-1)i}{2}} \\left(\\frac{  \\tau'(\\alpha_Q)}{\\alpha_Q} \\right)^{-i} \\left( \\frac{ \\tau' \\left( \\alpha_P \\right)}{\\alpha_P} \\right)^{i^2}\\\\\n\t&= \\left( \\frac{ \\tau' \\left( \\alpha_P \\right)}{\\alpha_P} \\right)^{ \\frac{(i+1)i}{2}} \\left(\\frac{  \\tau'(\\alpha_Q)}{\\alpha_Q} \\right)^{-i}\\end{align*} \n\tNow consider the map \n\t\\begin{equation}\n\tg: G_{k(E)} \\rightarrow \\overline{k(E)}^\\times : \\gamma \\mapsto \\gamma' \\left( \\prod_{n=0}^{i-1} \\tilde{\\sigma}^n(\\alpha_Q) \\right),\n\t\\end{equation}\n\twhere $\\gamma= \\gamma' \\tilde{\\sigma}^i$ for some $\\gamma' \\in G_{L(E)}$. The differential of $g$ is \n\t$$dg(\\gamma,\\tau) = \n\t\\begin{cases} \n\t\\frac{ \\prod_{n=0}^{i-1} \\tilde{\\sigma}^n(\\alpha_Q) }{ \\tilde{\\sigma}^i \\tau' \\tilde{\\sigma}^{-i} \\left( \\prod_{n=0}^{i-1} \\tilde{\\sigma}^n(\\alpha_Q) \\right)} & \\text{ if } i + j < q\\\\\n\t\\left(  \\prod_{n=0}^{q-1} \\tilde{\\sigma}^n(\\alpha_Q)\\right) \\left( \\frac{ \\alpha_P}{\\tau'(\\alpha_P)} \\right)^{-\\frac{(i+1)i}{2}}  \\left(  \\frac{ \\alpha_Q}{\\tau'(\\alpha_Q)}\\right)^i & \\text{ else}\n\t\\end{cases}. $$\n\tThe statement follows by subtracting this trivial cocycle and applying \\cref{prop:sumbolalgebra-cocycle}.\n\\end{proof} \n\n\\begin{prop} \n\t$\\epsilon$ induces a split to sequence (\\ref{Eq:exact sequence on tor[d]Br(E)}).\n\\end{prop} \n\n\\begin{proof}\n\tIt suffices to show that $r \\circ \\epsilon \\circ \\inflation (f_L) = \\lambda \\circ \\inflation (f_L)$. \n\tBy the previous lemma, $\\epsilon(\\inflation(f_L))$ gives\n\t$$(\\gamma'\\sigma^i,\\tau'\\sigma^j) \\mapsto \\begin{cases} 1 & \\\\ i+j < q \\\\\n\t\\sum_{i=0}^{q-1} \\left(\\sigma(Q) - (0)\\right) & i+j \\geq q \\end{cases}$$\n\tin $H^2(G_k, \\Prin(E))$, where $\\gamma',\\tau' \\in G_k$. On the other hand, $r(\\inflation(f_L))$ can be presented by the cocycle \n\t$$\\gamma' {\\sigma}^i \\mapsto \\sum_{m=0}^i {\\sigma}^m(Q) - 0.$$\n\tThis lifts to the map\n\t$$\\gamma' {\\sigma}^i \\mapsto \\sum_{m=0}^i \\left( \\sigma^m(Q)\\right)  - (0) \\in \\Div^0(\\overline{E}),$$\n\tand a direct computation of the boundary map gives \n\t\\begin{align*} \n\t(\\gamma' {\\sigma}^i,\\tau'{\\sigma}^j) \\mapsto &\n\t\\begin{cases}  \\sigma^i \\left( \\sum_{m=0}^j \\left( \\sigma^m(Q)\\right)  - (0)\\right) - \\left( \\sum_{m=0}^{i+j} \\left( \\sigma^m(Q)\\right)  - (0)\\right) \\\\\n\t+ \\left( \\sum_{m=0}^i \\left( \\sigma^m(Q)\\right)  - (0)\\right) & i+ j < q \\\\\n\t\\sigma^i \\left( \\sum_{m=0}^j \\left( \\sigma^m(Q)\\right)  - (0)\\right) - \\left( \\sum_{m=0}^{i+j-q} \\left( \\sigma^m(Q)\\right)  - (0)\\right) \\\\\n\t+ \\left( \\sum_{m=0}^i \\left( \\sigma^m(Q)\\right)  - (0)\\right) & i+ j \\geq q \\end{cases} \\\\\n\t&=  \\begin{cases} 1 & i+ j < q \\\\\n\t\\left( \\sum_{m=0}^{q-1} \\left( \\sigma^m(Q)\\right)  - (0)\\right) & i+ j \\geq q \\end{cases},\n\t\\end{align*}\n\twhich coincides with the previous calculation.\n\\end{proof}\n\nWe now calculate the image of the composition $\\phi^{-1} \\circ \\res$ with $\\phi$ as in \\cref{Eq:phi definition}. By \\cite[Chapter VII, Section 5]{serrelocal} the action of $G_k$ on $L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ compatible with $\\phi$ is given by \n$$\\sigma.(a,b) = \\phi^{-1} \\sigma. \\phi(a,b) = \\left( \\frac{\\sigma^{-1}(a)}{ \\sigma^{-1}(b)} , \\sigma^{-1}(b) \\right).$$  \n\n\\begin{lem}\\label{lem:fixedSet}\n\tUnder the isomorphism $\\phi^{-1}$ the fixed set $H^1(L,M)^{G_k\/G_L}$ corresponds to \n\t$$\\left\\{ \\left( a, \\frac{a}{\\sigma(a)} \\right) : \\sigma(a)^2 \\equiv \\sigma^2(a) a \\mod (L^\\times)^q \\right\\}.$$\n\\end{lem} \n\n\\begin{proof} \n\tLet $(a,b) \\in L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ be fixed by the above action. Then \n\t$a \\equiv \\frac{\\sigma^{-1}(a)}{ \\sigma^{-1}(b)}$, which implies that $b \\equiv \\frac{a}{\\sigma(a)}$. Now $b \\equiv \\sigma^{-1}(b)$ and thus \n\t$ \\frac{a}{\\sigma(a)} \\equiv \\frac{ \\sigma^{-1}(a)}{a}$ which implies that $a^2 \\equiv \\sigma(a) \\sigma^{-1}(a)$ or equivalently $\\sigma(a)^2 \\equiv \\sigma^2(a) a$. \n\\end{proof} \n\n\\begin{lem}\\label{lem:ImageRes1} \n\t$f \\in H^1\\left( L,M\\right)^{G_k\/G_L}$ is in the image of the restriction map if and only if \n\t$f(\\gamma^q) = 0$ for any $\\gamma \\in G_k$. \n\\end{lem} \n\n\\begin{proof}\n\tLet $\\gamma \\in G_k$ and suppose that $f$ is in the image of the restriction map with preimage $g$. Then we can write $\\gamma = \\gamma' \\sigma^i$ for some $\\gamma' \\in G_L$. We calculate directly using \\cref{lem:technical} that\n\t$$f(\\gamma^q) = g(\\gamma^q) = \\sum_{i=0}^{q-1} \\gamma^q g(\\gamma) = \\sum_{i=0}^{q-1} \\sigma^{iq} g(\\gamma) = \\sum_{i=0}^{q-1} \\sigma^i  g(\\gamma) = 0.$$\n\tFor the converse assume that $f$ satisfies the condition that $f(\\gamma^q) = 0$ for any $\\gamma \\in G_k$. In particular $f(\\sigma^q) = 0$. Define $g \\in H^1(k,M)$ by setting $g(\\gamma) = f(\\gamma')$, where $\\gamma = \\gamma' \\sigma^i$ for $\\gamma' \\in G_L$. This is well-defined as for any $\\gamma,\\tau\\in G_k$ with $\\gamma = \\gamma' \\sigma^i, \\tau = \\tau' \\sigma^j$ and $\\gamma',\\tau' \\in G_L$ we have that \n\t$\n\tg\\left( \\gamma\\tau\\right) = g\\left( \\gamma' \\left( \\sigma^i \\tau' \\sigma^{-i}\\right)  \\sigma^{i+j}\\right)  \n\t= g\\left( \\gamma'\\right) g\\left( \\sigma^i \\tau' \\sigma^{-i}\\right)\n\t= g\\left( \\gamma' \\right) \\sigma^i g(\\tau') \n\t= g\\left( \\gamma \\right) \\gamma g(\\tau).\n\t$\\end{proof} \n\nWe will now prove a technical lemma that will be useful to determine the image of the restriction. \n\n\\begin{lem}\\label{lem:technical-lemma-Galois-theory}\n\tLet $k$ be a field of characteristic prime to $q$ containing a primitive $q$-th root of unity. Let $k \\subset L \\subset F$ be a tower of field extensions so that each extension is Galois of degree $q$. Let $a \\in L^\\times$ such that $F = L\\left( \\sqrt[q]{a}\\right)$. Fix a representative $\\sigma \\in G_k$ that generates $\\Gal(L\/k)$. Suppose that for every $\\gamma \\in G_k$ we have that $\\gamma^q \\left( \\sqrt[q]{\\sigma^i(a)}\\right)= \\sqrt[q]{\\sigma^i(a)}$ for $0 \\leq i < q$. Then there exists some $b \\in k^\\times$ such that $a \\equiv b \\mod (L^\\times)^q$. \t\n\\end{lem} \n\n\\begin{proof}Assume that $a \\not\\in k^\\times$. Fix $\\sigma \\in G_k$ such that $\\overline{\\sigma}$ generates $\\Gal(L\/k)$. Denote the Galois closure of the extension $F$ over $k$ by $\\tilde{L}$. By Galois theory \n\t$\\tilde{L} = L\\left( \\sqrt[q]{a}, \\sqrt[q]{\\sigma(a)}, \\ldots, \\sqrt[q]{\\sigma^{q-1}(a)} \\right)$ and $\\Gal(\\tilde{L}\/L)$ is isomorphic to $\\left( \\Z\/q\\Z \\right)^{r}$ for $r=1$, or $r=q$. \n\tWe prove the lemma by contradiction. Assume that $r=q$. Let $\\tau \\in \\Gal(\\tilde{L}\/L)$ be the element with $\\tau \\left( \\sqrt[q]{a} \\right) = \\rho \\sqrt[q]{a}$ and $\\tau \\left( \\sigma^i \\sqrt[q]{a} \\right) = \\sigma^i \\sqrt[q]{a}$ for $1 \\leq i <q$. \n\tNow $\\left( \\sigma\\tau \\right) \\left( \\sqrt[q]{a} \\right)= \\left( \\rho \\sigma^q\\right) \\left( \\sqrt[q]{a} \\right) =\\rho \\left(   \\sqrt[q]{a}\\right)$, which is a contradiction to our assumptions. We conclude that $F\/k$ is Galois of degree $q^2$. We want to show that $\\Gal(F\/k) = \\Z\/q\\Z \\times \\Z\/q\\Z$.\t\n\tNext suppose that $\\Gal(F\/k) = \\Z\/q^2 \\Z$ and denote the generator by $\\tau$. Then $\\tau^q$ fixes $L$, as $\\tau^q(\\sqrt[q]{a}) =\\sqrt[q]{a}$ implies that $\\tau^q(a) = a$ and $L= k(a)$. Hence $\\tau \\in \\Gal(F\/L)$, which implies that $\\tau$ is of order $q$, a contradiction. We conclude that $\\Gal(F\/k) \\cong \\Z\/q\\Z \\times \\Z\/q\\Z$. Consider the fixed field $F^{\\left< \\sigma \\right>}$, which is a degree $q$ extension of $k$. By Kummer theory, there exists an element $b \\in k^\\times$ so that  $F^{\\left< \\sigma \\right>} = k\\left( \\sqrt[q]{b}\\right)$. Finally, $F = L \\left( \\sqrt[q]{b}\\right) = L \\left( \\sqrt[q]{a}\\right)$ and thus by Kummer theory $a \\equiv b \\mod (L^\\times)^q$. \n\\end{proof} \n\n\\begin{prop}\\label{prop:ImageOfRestrictionmap}\n\tThe image of the restriction map corresponds to the set $$k^\\times\/\\left((L^\\times)^q\\cap k^\\times\\right) \\times \\{1\\} \\subset L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$$ under the isomorphism $\\phi^{-1}$. \n\\end{prop} \n\n\\begin{proof} \n\tLet $(a,b) \\in L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ be in the image of the restriction map. There exists some $f \\in H^1(k,M)$ so that $\\phi^{-1}\\circ \\res(f) = (a,b)$. Then $(a,b)$ is necessarily in the preimage $\\phi^{-1} \\left( H^1(L,M)^{G_k\/G_L} \\right) $ and by \\cref{lem:fixedSet} we see that $(a,b) \\equiv \\left( a, \\frac{a}{\\sigma(a)} \\right)$ so that $\\sigma(a)^2 \\equiv \\sigma^2(a) a \\mod (L^\\times)^q$. It remains to show that we can choose $a \\in k^\\times$. \\\\\n\tBy definition of $\\phi$ and by \\cref{lem:ImageRes1} we get that $\\gamma^q\\left(\\sqrt[q]{a}\\right) = \\sqrt[q]{a}$ and $\\gamma^q\\left( \\sqrt[q]{\\frac{a}{\\sigma(a)}}\\right) = \\sqrt[q]{\\frac{a}{\\sigma(a)}}$ for any $\\gamma \\in G_k$ and for any choice of root. Using the condition that $\\sigma(a)^2 \\equiv \\sigma^2(a) a$, we deduce that $\\gamma^p \\left( \\sqrt[q]{\\sigma^i}{a} \\right) = \\sqrt[q]{\\sigma^i}{a}$ for any $i$. The statement follows from \\cref{lem:technical-lemma-Galois-theory}.\t\n\n\n\n\\end{proof} \n\nThe following theorem summarizes the results of this section. \n\n\\begin{thm}\\label{thm:Br(E) three case} \n\tSuppose that $[G_k:G_L]$ is of order $q$ and assume that there is some $\\sigma \\in G_k$ with $\\sigma(Q) = P\\oplus Q$ such that $\\overline{\\sigma}$ generates $G_k\/G_L$.  Additionally denote a primitive element for the extension $L\/k$ by $l$.  \n\t$$\\tor[q][\\Br(E)] = \\tor[q][\\Br(k)] \\oplus I$$\n\tand $I$ has generators $\\left\\{\t\\left( l^q, n_Q \\right)_{k(E)}, (a,t_P)_{k(E)}:\n\ta \\in k^\\times\n\t\\right\\},$ where $n_Q \\in k(E)$ with $n_Q^q = N_{L(E)\/k(E)}(t_Q)$.\n\\end{thm} \nFor the relations consider the commutative diagram with exact rows and columns \n$$\\xymatrix{\n\t& & 0 \\ar[d] & 0 \\ar[d] \\\\\n\t0 \\ar[r] & \\frac{E(k) \\cap [q]E(L)}{[q]E(k)} \\ar[r]^{\\inflation} \\ar[d]^{\\delta_{L\/k}} & E(k)\/[q]E(k) \\ar[r]^{\\res}\\ar[d]^{\\delta_k} & E(L)\/[q]E(L) \\ar[d]^{\\delta_L}   \\\\\n\t0 \\ar[r]& H^1(\\Gal(L\/k),M) \\ar[r]^{\\inflation} & H^1(k,M) \\ar[r]^{\\res} & H^1(L,M) \n},$$\nwhere $\\delta_{L\/k}$ is the map induced by $\\delta_k$. It is immediate that $\\delta_{L\/k}$ is injective. Recall that $H^1\\left( \\Gal(L\/k),M\\right)$ is cyclic of order $q$ with generator $f_L$. Furthermore, we saw that $\\epsilon_L\\left( \\inflation(f_L) \\right) = \\left( l^q, n_Q \\right)_{k(E)}$ (\\cref{prop:ImageofInflation}). We deduce the following results. \n\n\\begin{enumerate} \n\t\\item The Brauer class of $\\left( l^q, n_Q \\right)_{k(E)} $ is trivial, that is it is in the image of the map $\\Br(k) \\rightarrow \\Br(E)$, if and only if the quotient \n\t$\\frac{E(k) \\cap [q]E(L)}{[q]E(k)} $ is non-trivial. \n\t\\item \tThe Brauer class of $\\left( a, t_P \\right)_{k(E)}$ is trivial if and only if there is some $R \\in E(k)\/[q]E(k)$ so that $\\phi^{-1}(a,1) = \\delta_L (R)$. \n\\end{enumerate}  \n\n\n\\subsection{$q$ divides the degree $[L:k]$}\\label{sec:Generators:qdividesLk} \n\nSuppose for this section that $q$ divides $[L:k]$. Let $k \\subset L' \\subset L$ be an intermediate field so that $L\/L'$ is a Galois extension of degree $q$ and $q$ does not divide the degree $L'\/k$. After renaming $P$ and $Q$ we may assume that there is some generator $\\overline{\\sigma}$ of $\\Gal(L\/L')$ so that $\\sigma(P) = P$ and $\\sigma(Q) = P\\oplus Q$. Furthermore, let $l \\in L$ with $l^q \\in L'$ and $L = L'(l)$.\nFix $t_P, t_Q \\in L(E)$ with $\\divisor(t_P) = q(P) - q(0)$ and $\\divisor(t_Q) = q(Q) - q(0)$. Assume additionally that $t_P \\circ [q], t_Q \\circ[q] \\in \\left( L(E)^\\times \\right)^d.$  Furthermore, let $\\alpha_Q\\in \\overline{k(E)}$ so that $\\alpha_Q^q = t_Q$. \\\\\n\nAlthough the field extension $L'\/k$ might not be Galois, restriction followed by corestriction coincides with multiplication by $[L':k]$, which is an isomorphism. Using the previous section we deduce the following result. \n\n\\begin{prop} \n\tUnder the above assumptions, the Brauer group decomposes as\n\t$$\\tor[q][\\Br(E)] =  \\tor[q][\\Br(k)] \\oplus I$$\n\tand every element in $I$ can be expressed using the generators \n\t$$\\left\\{\t\\Cor\\left( l^q, n_Q \\right)_{L'\n\t\t(E)}, \\Cor(a,t_P)_{L'(E)}:\n\ta \\in L'^\\times\n\t\\right\\}.$$\n\\end{prop}\n\n\nSuppose that $q$ divides $[L:k]$ and use the notation used in \\cref{sec:Generators:qdividesLk}. Recall that every element in $I$ can be written as $\\Cor(A)$ for some $A \\in \\tor[q][\\Br(E\\times \\Spec(L'))]$. Such an element is trivial if and only if it is similar to $\\epsilon_{L'} \\circ \\delta_{L'}$. \nRemark that some corestrictions of element in $\\tor[q][\\Br(E_L)]$ may coincide and we do not account for this in our description. \n\n\\section{Examples}\\label{ch:Examples} \n\nIn this section, we calculate the $q$-torsion of the Brauer group for some elliptic curves $E$, where $q$ is an odd prime. For computational reasons, we only consider the case $q=3$. As before, we will consider various cases depending on the extension $L$, that is the smallest Galois extension of $k$, so that $M$ is $L$-rational. \n\n\\subsection{$M$ is $k$-rational over a number field}\\label{sec:Ex:split}\nLet $k = \\Q(\\omega)\\subset \\mathbb{C}$, where $\\omega$ is a primitive third root of unity. In \\cite{Paladino2010} the author describes a family of elliptic curves such that $M$ is $\\mathbb{Q}(\\omega)$-rational, for example $E$ given by the affine equation $y^2 = x^3 + 16$. In this case, the three torsion of $E$ is generated by \n$P= (0,4)$ and $Q= (-4, 8 \\omega + 4)=(-4,4\\sqrt{3}i)$. Furthermore, the tangent lines at $P$ and $Q$, respectively, are given by $t_P = y-4$ and $$t_Q= y - \\frac{6}{2 \\omega + 1 } (x+4) - 8 \\omega - 4 = y - 4 \\sqrt{3}i x - 20 \\sqrt{3} i.$$ \nBy the previous discussion $\\tor[3][\\Br(E)] = \\tor[3][\\Br(k)] \\oplus I$ and every element in $I$ can be written as a tensor product \n\\begin{equation}\\label{eq:Ex1}\n(a, y-4)_{3,k(E)} \\otimes\\left(b,  y - 4 \\sqrt{3}i x - 20 \\sqrt{3} i\\right)_{3,k(E)}\n\\end{equation} for some $a,b \\in k^\\times.$\nWe calculate with magma, that $E(k) = M$ and therefore also $E(k)\/3E(k) = M$. Using the algorithm we see that a tensor product as in \\cref{eq:Ex1} is trivial if and only if it is similar to an element in the subgroup generated by \n$$\\left( 4- 20 \\sqrt{3} i , t_P \\right)_{3,k(E)}$$\nand \n$$\\left( \\frac6{19} - \\frac{8}{19} \\sqrt{3} i, t_P\\right)_{3,k(E)} \\otimes \\left( 4\\sqrt{3} i -4,  t_Q \\right)_{3,k(E)}.$$\n\n\n\\subsection{Degree $L\/k$ coprime to $q$ for $k$ a number field}\\label{subsec:Ex degree L\/k coprime to 3} \n\nLet $k = \\mathbb{Q}(\\omega)$ and $E$ the elliptic curve given by the affine equation \n$$y^2 = x^3 + B,$$\nwhere $B \\equiv 2 \\mod (\\mathbb{Q}^\\times )^3$ and $B \\not\\equiv 1,-3 \\mod (\\mathbb{Q}^\\times)^2$. By \\cite[Theorem 3.2 and Corollary 3.3]{Bandini2012} we see that $L = k(\\sqrt{B})$. Let $\\sigma$ be given by $\\sigma(\\sqrt{B}) = - \\sqrt{B}$. The three torsion of $E$ has generators $P$ and $Q$ with $P = (0,\\sqrt{B})$ and $Q =  \\left(\\sqrt[3]{-4B},\\sqrt{-3B}\\right).$ Then $\\sigma(P) = 2P$ and $\\sigma(Q) = 2Q$. We need to calculate \n$$\\Cor_{L(E)\/k(E)}\\left((a,t_P)_{3,L(E)} \\otimes (b,t_Q)_{3,L(E)}\\right).$$ \tRecall that by \\cref{rem:image of restriction} it will be enough to consider $(a,b)$ a norm in $L^\\times\/(L^\\times)^3 \\times L^\\times\/(L^\\times)^3$. For $(a,b) \\in L^\\times\/(L^\\times)^3 \\times L^\\times\/(L^\\times)^3$ we have $N_{L\/k}(a,b) = \\left(\\frac{a}{\\sigma(a)}, \\frac{b}{\\sigma(b)}\\right)$, or equivalently we may assume that $N_{L\/k}(a) = 1$ and $N_{L\/k}(b) = 1$ and $a,b \\in L^\\times \\setminus k^\\times$. Note that $$t_P = y - \\sqrt{B}$$ and $$t_Q = y - \\frac{3 \\sqrt[3]{-4B}}{2\\sqrt{-3B}} x - 3 \\sqrt{-3B}.$$ Furthermore, $\\sqrt{-3} \\in k$ as $\\omega \\in k$ and $\\sqrt[3]{16B^2} \\in k^\\times$ since $B \\equiv 2 \\mod (\\mathbb{Q}^\\times )^3$. \\\\\n\nLet $a = a_1\\sqrt{B} + a_2$ with $a_1 \\neq 0$ and $N_{L\/k}(a) = 1$. We use the algorithm given in \\cite[Section 3]{Rosset1983} to calculate the corestriction explicitly as\n\\begin{align*} \n\\Cor(a,t_P)_{3,L(E)} &= \\left( a_2 - a_1y, a_1^2\\right)_{3,k(E)}.\n\\end{align*}\nFinally, let $b= b_1 \\sqrt{B} + b_2$ with $b_1 \\neq 0$ and $N_{L\/k}(b) = 1$. Then\n\\begin{align*} \n\\Cor(b, t_Q)_{3,L(E)} &= \\left(  \\frac{ y b_1}{ \\sqrt{3} i\\left(x+1\\right) } + b_2, 1 - b_2^2 - \\frac{b_1^2 (x^3 + B)}{3(x+1)^2} \\right)_{3,k(E)}.\n\\end{align*} \nOverall, the three torsion of the Brauer group decomposes as\n\t$$\\tor[3][\\Br(E)] = \\tor[3][\\Br(k)]\\oplus I$$\n\tand every element in $I$ can be written as a tensor product \n\t$$\\left( a_2 - a_1y, a_1^2\\right)_{3,k(E)} \\otimes  \\left(  \\frac{ y b_1}{ \\sqrt{3} i\\left(x+1\\right) } + b_2, 1 - b_2^2 - \\frac{b_1^2 (x^3 + B)}{3(x+1)^2} \\right)_{3,k(E)}$$\n\tfor some $a_1, a_2, b_1, b_2 \\in k^\\times$ with $a_1, b_1 \\neq 0$, and $a_2^2 - B a_1^2 = b_2^2 - B b_1^2 = 1$.  \\\\\n\nTo calculate the relations we need to specify $B$. Consider the case $B= -1024$. Using magma we calculate that $E(k) = 0$. Thus there are no additional relations. Note that some elements might still become trivial due to the fact that the corestriction map is not surjective. \n\n\\subsection{Degree $L\/k= q$ for $k$ a number field}\\label{sec:Ex:degreeq}\nLet $k = \\Q(\\omega)$, $\\omega = -\\frac{1}{2} + \\frac{i\\sqrt{3}}{2}$ and let $E$ be the elliptic curve given by the affine equation \n$$y^2 = x^3+4.$$\nGenerators of the three torsion are given by $P= (0, 2)$ and $Q=(-2\\sqrt[3]{2} , 2i\\sqrt{3})$. Let $l = \\sqrt[3]{2}$. In our previous notation $L= k\\left(\\sqrt[3]{2}\\right)$ and the Galois group $\\Gal(L\/k)$ is generated by $\\overline{\\sigma}$ with $\\sigma(Q) = P+Q = (-\\omega 2 \\sqrt[3]{2}, 2i \\sqrt{3})$. It can be seen that \n\\begin{align*}\nt_P & = y - 2 \\\\\nt_Q & = y + i \\sqrt{3} \\sqrt[3]{4}x + 2 i \\sqrt{3}\\\\\nn_Q &= y - 2 \\sqrt{3}i )^3. \n\\end{align*}\nTherefore the $3$-torsion of $\\Br(E)$ is $$\\tor[3][\\Br(E)] = \\tor[3][\\Br(k)] \\oplus I$$ and every element in $I$ can be written as a tensor product of the symbol algebras \n\t$$\\left( 2, y - 2 \\sqrt{3}i \\right)_{3,k(E)} \\text{ and } (a,y-2)_{3,k(E)}$$ for some $a \\in k^\\times$. \nWe calculate that $E(k) = \\left<P\\right>$ and $E(L) \\cong \\Z\/6\\Z \\times \\Z\/6\\Z$. Therefore \n$E(k)\/3E(k) = \\left< P\\right>$ and $E(L)\/3E(L) = M$ and the quotient $\\frac{E(k) \\cap [3] E(L)}{[3]E(k)}$ is trivial. Therefore the symbol algebra \n$\\left(  2, y - 2 \\sqrt{3}i \\right)_{k(E)}$ is not trivial. Finally, $\\epsilon \\circ \\delta \\circ \\res (P) = \\left( 2 + i \\sqrt{3} , y-2 \\right)_{L(E)}$ and therefore a symbol algebra $(a, y-2)_{k(E)}$ is trivial if and only if it is similar to one of the following \n$$\\left\\{ \n(1,1)_{3,k(E)}, \\left( 2+ i \\sqrt{3}, y-2 \\right)_{3,k(E)}, \\left( -8 + 8 i \\sqrt{3}, y-2 \\right)_{3,k(E)}\n\\right\\}.\n$$\n\n\\subsection{Over a local field}\\label{sec:local} \n\nDenote by $\\Q_7$ the $7$-adic numbers. Note that since three divides $7-1$, the field $\\Q_7$ contains a primitive third root of unity $\\omega$. Let $E$ be the elliptic curve \n$$E: y^2 = x^3 + 16$$ over $k$. Consider the reduction $\\tilde{E}$ of $E$ modulo $7$. Then $\\tilde{E}$ is a non-singular curve and using magma we see that \n$$\\tilde{E}\\left(\\mathbb{F}_7\\right) = \\Z\/3\\Z \\oplus \\Z\/3\\Z = \\left\\{ \n0, (0,3), (0,4), (3,1), (3,6), (5,1),(5,6), (6,1), (6,6)\n\\right\\}. $$\nDenote by $\\hat{E}$ the formal group associated to $E$ and consider the group $\\hat{E}\\left(7\\Z_7\\right)$. By \\cite[IV Theorem 6.4]{silverman} there is an isomorphism \n$\\hat{E} \\left( 7 \\Z_7 \\right) \\rightarrow \\hat{\\mathbb{G}}_a \\left( 7 \\Z_7 \\right)$, where ${\\mathbb{G}_a}$ denotes the additive group. By \\cite[IV.3 and VII.2]{silverman} there is an exact sequence \n$$\\xymatrix{ 0 \\ar[r]&\\hat{\\mathbb{G}}_a\\left(7\\Z_7\\right) \\ar[r]& E(\\Q_7) \\ar[r]&  \\tilde{E}\\left(\\mathbb{F}_7\\right) \\ar[r]& 0}.$$\nFurthermore, by \\cite[VII.3 Proposition 3.1]{silverman} the reduction map $\\tor[3][E(\\Q_7)] \\rightarrow \\tilde{E}\\left(\\mathbb{F}_7\\right)$ is injective. Thus $E$ has $k$-rational $3$-torsion. Since $3$ is a unit in $\\Z_7$, we further deduce that $E(\\Q_7)\/[3]E(\\Q_7) = \\tilde{E}(\\mathbb{F})\/[3]\\tilde{E}(\\mathbb{F}) =M.$ Finally, \n$$\\Q_7^\\times \/ \\left( \\Q_7^\\times \\right)^3 \\cong \\left( \\Z_7^\\times \\times \\left(7\\Z_7\\right) \\right)\/ \\left( \\Z_7^\\times \\times \\left(7\\Z_7\\right) \\right)^3 \\cong \\Z\/3\\Z \\times \\Z\/3\\Z.$$\nTherefore, $H^1(k,M) \\cong \\left( \\Q_7^\\times \/ \\left( \\Q_7^\\times \\right)^3  \\right)^2 \\cong \\left( \\Z\/3\\Z \\right)^4$. \nBy the algorithm and using \\cite[Corollaire 2.3]{GroupedeBrauerIII}, the three torsion of the Brauer group decomposes as follows \n$$\\tor[3][\\Br(E)] \\cong \\tor[3][\\Br\\left(\\Q_7\\right)] \\oplus \\left( \\Z\/3\\Z  \\right)^2 = \\tor[3][\\left( \\Q\/\\Z \\right)] \\oplus \\left( \\Z\/3\\Z \\right)^2 \n= \\left( \\Z\/3\\Z \\right)^3.\n$$ \n\n\\begin{rem}\n\tThe above computations also show that $\\tor[3][\\Br\\left( \\tilde{E}\\right)] = \\tor[3][\\Br\\left( \\mathbb{F}_7 \\right)] = 0$. \t\n\\end{rem} \n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAs one of the nearest powerful radio-loud active galactic nuclei (AGN), Cygnus A provides an excellent laboratory to study the environment and activity of powerful AGN. The luminosity of the AGN in Cygnus A (which is predominantly expressed in the infrared) is high enough to classify it as quasar \\citep[e.g.,][]{Djorgovski91}. While not seen in total intensity, broad H$\\alpha$ is seen in polarized light \\citep{Ogle97}, lending support for the existence of an obscured broad-line region (BLR). The polarized broad lines were detected within the ionization cone seen by \\citet{Jackson96}, giving them a possible scattering origin in the NLR. \n\nThe host galaxy of Cygnus A is a cD elliptical near the center of a cluster which appears to be undergoing a merger with cluster of similar size \\citep{Ledlow05}. \\citet{Tadhunter03} obtained Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph (STIS) spectra of the nuclear region. Stepping the slit across the nucleus, a velocity gradient indicative of rotation around the radio axis was observed. Modeling the velocity as due to the potential of a supermassive black hole (SMBH) and stellar mass distribution (measured from a $1.6\\mu$m NICMOS image) gives a SMBH mass of $2.5\\pm0.7\\times10^{9}$ $M_{\\odot}$. This measurement is consistent with black-hole-mass--host-galaxy relations \\citep[e.g.,][]{Magorrian98,Ferrarese00,Gebhardt00,Gultekin09}.\n\nThe large scale radio morphology shows prominent hotspots, lobes, and a radio core. Both a jet and counterjet are visible in Very Large Array (VLA) and Very Long Baseline Array (VLBA) observations \\citep[e.g.,][]{Sorathia96}. Due to the edge-brightened morphology it is classified as an FR II source \\citep[Fanaroff-Riley Type II;][]{Fanaroff74}.\n\nX-ray observations of Cygnus A are consistent with the presence of a hidden quasar. Chandra ACIS observations from $0.7$ to $9$ keV by \\citet{Young02} show the hard X-ray flux to be peaked at a location consistent with that of the radio core and unresolved (less than $0.\\arcsec 4$ in size, determined by comparison with a model PSF). Higher energy INTEGRAL observations show emission between $20$ and $100$ keV \\citep{Beckmann06}, although there may be some contamination from intracluster gas.  This hard X-ray emission is likely due to accretion disk emission Comptonized in the AGN corona, although the UV\/optical emission is obscured.\n\nIn addition to the AGN activity, HST imaging has also revealed star formation in the central region of Cygnus A, which began $< 1$ Gyr ago \\citep{Jackson98}. It is located in a $4$ kpc ring around the nucleus, oriented orthogonal to the radio axis.\n\nBased on adaptive optics observations showing a secondary point source near the nucleus, \\citet{Canalizo03} suggest that Cygnus A may be in the late stages of a merger event. This merger event and related accretion may be related to the current epoch of nuclear activity. \n\nA near-infrared spectrum presented by \\citet{Bellamy04} showed complicated emission line properties suggesting an infalling molecular cloud, consistent with the \\citet{Canalizo03} picture of a minor merger.  The H$_2$ lines were seen in several components, both redshifted and blueshifted relative to the systemic velocity, interpreted as emission from a rotating torus. The observed near-infrared line ratios are consistent with excitation by X-rays (from the AGN), while likely ruling out shocks as a possible excitation method.\n\n\\begin{figure}\n\\includegraphics[angle=270,width=0.5\\textwidth]{fig1.eps}\n\\caption{Spitzer IRS spectrum for Cygnus A. Extracted in a $20\\arcsec$ circular aperture from spectral mapping data.}\n\\label{fig:irs}\n\\end{figure}\n\nFor a more detailed summary of Cygnus A, including properties of the larger host galaxy and environment, see the review by \\citet{Carilli96}.\n\nWhile the bulk of the infrared emission is likely related to the presence of an AGN, the star formation can contribute to the infrared emission as well. Additionally, the bolometric luminosity of the AGN is uncertain. Estimates have been made using template spectral energy distributions (SEDs) and X-ray observations. However, a significant portion of the bolometric luminosity comes out in the infrared, via dust reprocessing of the UV\/optical continuum. An accurate determination of the bolometric luminosity then requires an understanding of the contributions to the observed infrared luminosity. \n\nIn this paper, we present modeling of the SED of Cygnus A. To accomplish this we combined new infrared measurements obtained by the Spitzer Space Telescope's Infrared Spectrograph \\citep[IRS;][]{Werner04} with existing Spitzer observations with the Imaging Array Camera \\citep[IRAC;][]{Fazio04} and the Multiband Imaging Photometer for SIRTF \\citep[MIPS;][]{Rieke04}, and measurements of the radio core from the literature to construct a radio through infrared ($\\sim2-10^{5}$ GHz; $4-10^{5}$ $\\mu$m) SED of the inner regions of Cygnus A. The resulting SED was subsequently modeled using components intended to replicate the physical processes likely to produce the observed emission. Using the results of the fitting we have been able to decompose the infrared emission and determine the bolometric luminosity of the AGN in Cygnus A. This in turn also provides an estimate of the star formation rate.\n\nThe paper is organized as follows: in Section \\ref{sec:data}, we discuss the new Spitzer IRS observations, new reductions of Spitzer IRAC and MIPS data, as well as the data from the literature. In Section \\ref{sec:modeling}, we describe the components used to model the SED, for which the results are described in Section \\ref{sec:results}. We conclude with a general discussion in Section \\ref{sec:discussion} and a summary in Section \\ref{sec:summary}.\n\n\\section{Data}\n\\label{sec:data}\n\nNew observations were combined with data compiled from multiple archives and new reductions in order to obtain a SED covering almost five orders of magnitude in frequency. Our analysis is focused on the properties of the continuum emission in this system. Below we discuss the reduction and analysis of the new Spitzer IRS observations and the re-reduction of archival IRAC and MIPS observations\n\n\\subsection{Spitzer IRS Observations}\n\nCygnus A was observed (PI: Baum) in ``mapping mode'' using the low resolution mode of IRS on board the Spitzer Space Telescope. Both the short- and long-wavelength slits were stepped across the source in increments of half the slit width. After pipeline calibration at the Spitzer Science Center, the observations were combined into a spectral data cube using the \\emph{Cube Builder for IRS Spectra Maps} \\citep[CUBISM;][]{Smith07b}. From this data cube, a spectrum was extracted in a $20\\arcsec$ aperture (Figure \\ref{fig:irs}).\n\n\\subsubsection{Removal of Mid-infrared Emission Lines}\n\nThe IRS spectrum was fit using PAHFIT \\citep{Smith07a} to measure and remove contributions from narrow emission lines. Integrated line fluxes and widths are provided in Table \\ref{table:mirlines}. While a study of the emission line properties is beyond the scope of this paper, we note the detection of multiple high ionization lines such as [O IV], [Ne V] and [Ne VI]. These are all consistent with the presence of an AGN \\citep[][and references therein]{Genzel98,Armus07}. Of particular note are [Ne V] and [Ne VI] which require the presence of ionizing photons of at least 97.1 and 125.8 eV respectively. All the measured emission lines in Table \\ref{table:mirlines} were subtracted from the IRS spectrum before fitting the SED.\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Mid-infrared Emission Line Properties}\n\\tablehead{\\colhead{Line} & \\colhead{Flux} & \\colhead{FWHM}\\\\\n\\colhead{} & \\colhead{\\emph{($\\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$)}} & \\colhead{\\emph{($\\mu$m)}}}\n\\startdata\n[Ar II] 6.9\t& $1.93\\pm0.08$\t& 0.11 \\\\\n{}[Ne VI] 7.6\t& $1.37\\pm0.09$\t& 0.11 \\\\\n{}[Ar III] 8.9\t& $1.11\\pm0.04$ & 0.11 \\\\\n{}[S IV\t10.5\t& $1.55\\pm0.03$ & 0.09 \\\\\n{}[Ne II] 12.8\t& $2.67\\pm0.03$ & 0.11 \\\\\n{}[Ne V] 14.3\t& $1.51\\pm0.04$ & 0.10 \\\\\n{}[Ne III] 15.6\t& $4.13\\pm0.04$ & 0.15 \\\\\n{}[S III] 18.7\t& $2.42\\pm0.06$ & 0.13 \\\\\n{}[Ne V] 24\t& $2.51\\pm0.06$ & 0.37 \\\\\n{}[O IV] 25.9\t& $4.58\\pm0.08$ & 0.32 \\\\\n{}[S III] 33\t& $3.33\\pm0.08$ & 0.31 \\\\\n\\hline\nH$_2$ S(7)\t& $0.944\\pm0.130$ & 0.06 \\\\\nH$_2$ S(5)\t& $0.853\\pm0.112$ & 0.06 \\\\\nH$_2$ S(3)\t& $0.286\\pm0.030$ & 0.09 \\\\\nH$_2$ S(2)\t& $0.333\\pm0.025$ & 0.11 \\\\\nH$_2$ S(0)\t& $0.128\\pm0.038$ & 0.31\n\\enddata\n\\tablecomments{Only detections $>3\\sigma$ are listed. Errors quoted from PAHFIT output.}\n\\label{table:mirlines}\n\\end{deluxetable}\n\n\\subsubsection{Dust Features}\n\nIn addition to fitting the emission lines, we have also fit the mid-infrared dust features using PAHFIT. Table \\ref{table:dustfeatures} shows the integrated flux and profile FWHM based on the PAHFIT output. In contrast with the emission lines, the dust features were not removed as they are expressed in the starburst models.\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Mid-infrared Dust Features}\n\\tablehead{\\colhead{$\\lambda$} & \\colhead{Flux} & \\colhead{FWHM}\\\\\n\\colhead{($\\mu$m)} & \\colhead{\\emph{($\\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$)}} & \\colhead{\\emph{($\\mu$m)}}}\n\\startdata\n$6.2$   &       $1.76\\pm0.23$ &       $0.19$ \\\\\n$6.7$   &       $5.99\\pm0.49$ &       $0.47$ \\\\\n$7.4$   &       $22.3\\pm0.8$ &       $0.94$ \\\\\n$7.8$   &       $2.26\\pm0.32$ &       $0.42$ \\\\\n$8.3$   &       $5.50\\pm0.23$ &       $0.42$ \\\\\n$8.6$   &       $4.42\\pm0.20$ &       $0.34$ \\\\\n$11.3$  &       $0.685\\pm0.101$ &       $0.36$ \\\\\n$12.0$  &       $2.71\\pm0.12$ &       $0.54$ \\\\\n$12.6$  &       $3.82\\pm0.21$ &       $0.53$ \\\\\n$13.5$  &       $2.49\\pm0.12$ &       $0.54$ \\\\\n$14.2$  &       $1.28\\pm0.11$ &       $0.36$ \\\\\n$17.0$  &       $2.52\\pm0.27$ &       $1.11$ \\\\\n$17.4$  &       $0.397\\pm0.065$ &       $0.21$ \\\\\n$17.9$  &       $0.802\\pm0.099$ &       $0.29$ \\\\\n$18.9$  &       $2.19\\pm0.14$ &       $0.36$ \\\\\n$33.1$  &       $3.54\\pm0.51$ &       $1.66$ \n\\enddata\n\\tablecomments{Only detections $>3\\sigma$ are listed. Errors quoted from PAHFIT output.}\n\\label{table:dustfeatures}\n\\end{deluxetable}\n\n\\begin{deluxetable}{lcccc}\n\\tablecaption{Additional Infrared Flux Densities from Spitzer}\n\\tablehead{\\colhead{Instrument\/Channel} & \\colhead{$\\lambda$ ($\\mu m$)} & \\colhead{F$_{\\nu}$ (Jy)} & \\colhead{$\\sigma$ (Jy)} & \\colhead{Aperture ($\\arcsec$)}}\n\\startdata\nIRAC-2\t& $4.5$     & $0.010$ & $0.003$ & $12.2$ \\\\\nIRAC-4\t& $8$       & $0.054$ & $0.013$ & $12.2$ \\\\\nMIPS-70\t& $70$      & $2.20$ & $0.11$ & $30$ \\\\\nMIPS-160\t& $160$     & $0.668$ & $0.033$ & $48$\n\\enddata\n\\label{table:spitzer}\n\\end{deluxetable}\n\n\\citet{Spoon07} developed a diagnostic diagram using the $6.2~\\mu$m polycyclic aromatic hydrocarbon (PAH) and the strength of the $9.7~\\mu$m silicate feature (S$_{sil}$), which is seen in absorption in Cygnus A. These two spectral features can be used in tandem to classify the relative dominance of PAH emission, continuum emission, and silicate absorption. We follow their method of spline fitting to measure the depth of the silicate feature, finding S$_{sil}\\approx-0.8$, where S$_{sil}$ is the negative of the apparent optical depth. The EQW of the  $6.2~\\mu$m PAH is $0.0552~\\mu$m, placing Cygnus A on the border of region 1A and 2A in their diagnostic diagram, corresponding to objects dominated by continuum emission in the mid-infrared. This is consistent with the presence of a strong AGN.\n\n\\subsection{Archival Spitzer IRAC+MIPS Data}\n\nThe nucleus and hotspots were imaged using IRAC at $4.5$ and $8.0~\\mu$m \\citep[see ][for a presentation of the data and analysis of hotspot properties]{Stawarz07}. As fluxes for the core were not presented, the archival data were retrieved, re-reduced and calibrated according to the IRAC instrument manual. The images showed strong emission at the location of both radio hotspots and the radio core in both channels. The emission was unresolved in the core component, the measured flux densities for this component are given in Table \\ref{table:spitzer}. Extraction apertures of $12\\arcsec.2$ were used for both channels. \n\n\\citet{Shi05} presented observation of Cygnus A using the MIPS instrument on Spitzer at 24, 70, and 160 $\\mu$m. Their 24 $\\mu$m flux is consistent with our IRS observations. The slope between their 70 and 160 $\\mu$m points is steeper than that of the Rayleigh--Jeans tail. The 70 and 160 $\\mu$m data were re-reduced from the BCD products in the Spitzer Science Center archive. Our measured flux densities are given in table \\ref{table:spitzer}. The core of Cygnus A was unresolved in both the $70$ and $160~\\mu$m bands. Apertures of $30\\arcsec$ and $48\\arcsec$ were used at 70 and 160 $\\mu$m, respectively. Our 160 $\\mu$m measurement has larger error bars, but is broadly consistent with their value.\n\n\\subsection{Published Data}\n\nCygnus A's strong synchrotron emission at radio frequencies suggests that synchrotron may contribute to the infrared as well. To anchor the synchrotron spectrum we supplemented the Spitzer observations with radio measurements from the literature. Core fluxes were obtained from \\citet{Eales89,Salter89,Alexander84,Wright84}. Submillimeter nuclear fluxes were also taken from \\citet{Robson98}. Based on these archival data the unresolved radio core\\footnote{Here ``radio core'' refers to the core seen by the VLA with on kpc-scale resolution. This encompasses flux from the VLBI scale core and jet.} is flat spectrum ($\\alpha=0.18$, $F_{\\nu}\\propto \\nu^{-\\alpha}$), until $\\sim1$ THz, where thermal emission from dust begins to dominate the SED. The compiled radio through infrared SED is shown in Figure \\ref{fig:SED} and the flux densities taken from the literature are provided in Table \\ref{table:literature}. The resolution of the observations is also provided.\n\nThere is some uncertainty associated with the highest frequency sub-mm observations. The $450~\\mu$m observation suggests that the synchrotron break may be occurring in this spectral regime. It is unclear if this is a genuine break or if the measurements are affected by variability. Additional observations at these frequencies may be able to clarify this issue. \n\nA consideration of the apertures is important when assembling data across several orders of magnitude in frequency. The resolution of the data used to assembled the SED in Figure \\ref{fig:SED} varies by over a factor of 10, but in all cases the core component is unresolved. At lower frequencies the emission is due solely to synchrotron emission from the flat spectrum radio core. There is no evidence to suggest that the larger scale steep spectrum jet will contribute emission in the infrared. Accordingly we use the unresolved VLA-scale core to anchor the synchrotron emission in the nucleus, so the difference in apertures should not affect the construction of this SED for the nucleus.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.5\\textwidth]{fig2.eps}\n\\caption{Radio through mid-infrared SED with new Spitzer observations and other data from the literature. See the text for details and references.}\n\\label{fig:SED}\n\\end{figure}\n\n\\begin{deluxetable}{lcccc}\n\\tablecaption{Flux densities from the literature}\n\\tablehead{\\colhead{$\\lambda$ ($\\mu m$)} & \\colhead{F$_{\\nu}$ (Jy)} & \\colhead{$\\sigma$ (Jy)} & \\colhead{Resolution ($\\arcsec$)} & \\colhead{Ref}}\n\\startdata\n450 & 0.34 & 0.06 & $8$ & 1\\\\\n800 & 0.56 & 0.08 & $13$ & 2 \\\\\n850 & 0.53 & 0.05 & $14$ & 1\\\\\n1100 & 0.58 & 0.06 & $19$ & 2 \\\\\n1300 & 0.59 & 0.07 & $11$ & 3 \\\\\n$3.3\\times10^3$ (89 GHz) & 0.70 & 0.07 & $2$ & 4\\\\\n$19.5\\times10^3$ (15.4 GHz)& 1.22 & 0.20 & Not provided in ref & 5 \\\\\n$60.6\\times10^3$ (5 GHz)& 0.97 & 0.20 & $2.0\\times3.1$ & 5\\\\\n$109\\times10^3$ (2.7 GHz) & 1.5 & 0.4 & $3.7\\times5.8$ & 5\n\\enddata\n\\label{table:literature}\n\\tablerefs{1 - \\citet{Robson98}, 2 - \\citet{Eales89}, 3 - \\citet{Salter89}, 4 - \\citet{Wright84}, 5 - \\citet{Alexander84}.}\n\\end{deluxetable}\n\n\\section{Modeling}\n\\label{sec:modeling}\n\nWe aim to reproduce the major features in the SED: the powerlaw emission at radio wavelengths, the strong thermal emission at infrared wavelengths, and the overall character of the silicate absorption. Extrapolating the powerlaw from the radio to the infrared requires a modification of the powerlaw spectrum to avoid exceeding the observed mid-infrared flux. The infrared emission is thermal in nature, coming from dust at a variety of temperatures ranging from the cold ISM ($T\\sim20$ K) through hot dust near the AGN, up to the sublimation temperature ($T\\sim1500$ K). The power source for this dust heating is a combination of star formation and AGN activity, with an uncertain balance between the two.\n\nAs noted in the introduction previous studies of Cygnus A have found evidence for simultaneous ongoing AGN activity and star formation, both of which can contribute to the infrared emission. We model the continuum emission from $\\sim2-10^5$ GHz ($3\\times10^{6}-5$ $\\mu$m; see Figure \\ref{fig:SED}) using components to represent the AGN torus model, a starburst, and synchrotron emission. The selection of models has $15$ free parameters, and the particular choices are discussed in the following sub-sections.\n\n\\subsection{AGN Torus Model}\n\nAccording to the unified scheme for radio loud AGN, the SMBH and accretion disk can be hidden from view along some lines of sight by an obscuring torus. Along these obscured lines of sight the UV\/optical radiation is absorbed and re-radiated in the mid- and far-infrared. As noted above, Cygnus A shows evidence for a hidden BLR through observations of H$\\alpha$ in polarized light. This suggests that Cygnus A harbors a ``hidden'' accretion disk and a BLR which is obscured along our line of sight.\n\n\\citet{Nenkova08} have constructed a model for an obscuring AGN torus, where the obscuration is due to the presence of multiple clouds along the line of sight to the AGN. We select this set of models because clumpy models seem to provide better fits to Sil features than smooth dust distributions \\citep[e.g.][]{Baum10}. In order to reproduce the observed ratio of Type I\/II AGNs, the obscuring clouds collectively populate a rough toroidal structure with some opening angle. \n\nThe CLUMPY torus model is specified by multiple geometrical parameters. The outer radius of the torus ($R_o$) is $Y$ times the inner radius ($R_d$), where the inner radius is determined from a dust sublimation temperature of $T=1500$ K ($R_d=0.4\\times L_{45}^{0.5}$ pc. $L_{45}$ is the bolometric luminosity of the AGN in units of $10^{45}$ erg s$^{-1}$). The geometry of the model is shown in Figure \\ref{fig:clumpygeom}. The clumps have a Gaussian angular distribution,  with $\\sigma$ parameterizing the width of the angular distribution from the mid-plane. The radial distribution is a power law with index $q$: $r^{-q}$. The inclination of the torus symmetry axis to the line of sight is $i$ and the average number of clouds along a given line of sight is $N$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.5\\textwidth]{fig3.eps}\n\\caption{Geometry of the CLUMPY torus model. Figure from \\citet{Nenkova08}, used with permission. See the text for an explanation of the labels.}\n\\label{fig:clumpygeom}\n\\end{figure}\n \nIn addition to the geometrical parameters, the models also vary the optical depth of each clump ($\\tau_V$), using the \\citet{Ossenkopf92} dust composition and \\citet{Mathis77} grain size distribution. The overall scaling of the model flux is $F_{AGN}$, the bolometric flux of the AGN's accretion disk component (treating the synchrotron emission separately). \n\nThe clumpy nature of the obscuration implies that there is a finite probability of a direct line of sight to the BLR, even at high inclinations. However, as Cygnus A shows no evidence for directly observed broad lines, we have only fit models where the central regions are fully obscured along our line of sight.\n\n\\subsection{Starburst}\n\nStar formation is represented using the \\citet{Siebenmorgen07} models which assume spherical symmetry and an ISM with dust properties characteristic of the Milky Way. Emission is broken down into two components: an old stellar population uniformly distributed through the volume and hot, luminous O and B stars embedded in dusty hot spots. The density of OB stars is centrally peaked although the model output is the emission integrated over the entire starburst.\n\nThe free parameters for the model are: starburst radius $r$, total luminosity $L_{SB}$, ratio of luminosity in O and B stars to total luminosity $f_{OB}$, visual extinction from the center to the edge of the nucleus $A_V$, and dust density in hotspots around O and B stars $n$. \n\nOwing to the spherical symmetry of this starburst model and the known ring morphology of the star formation in Cygnus A, parameters such as $A_V$ and the radius $r$ do not have straight forward interpretations here. The assumption of optically thick star formation does not have strong evidence (either for or against), given the absence of high resolution far infrared observations.\n\n\\subsection{Synchrotron}\n\\label{sec:sync-model}\n\nThe strong radio emission in Cygnus A justifies the final model component. VLA core fluxes are consistent with powerlaw emission to the submillimeter where thermal emission begins to dominate. Extrapolating the powerlaw to higher frequencies suggests that the synchrotron spectrum must either be modified or subjected to attenuation. The AGN is known to sit behind a dust lane with significant extinction \\citep[$A_V=50\\pm30$;][]{Djorgovski91}. Though some attenuation of the synchrotron flux is expected at higher frequencies, this alone is not sufficient to explain why the synchrotron powerlaw does not continue through the mid-infrared. With extinction alone, the flux densities at 30 and $10~\\mu$ m would be similar for the observed spectral index of $\\alpha=0.18$. For $A_V=50$, the expected flux from the extrapolated synchrotron emission would by itself exceed the measured infrared flux at 10$\\mu m$. We conclude that there must be a break in the population of emitting electrons.\n\nOne possible explanation is a simple cutoff in the population of relativistic electrons at some energy (Case I). This would manifest itself as a cutoff in the synchrotron spectrum:\n\n\\begin{equation}\nF_{\\nu} \\propto \\nu^{-\\alpha_1} e^{-\\frac{\\nu}{\\nu_c}}e^{-\\tau_r(\\nu)}\n\\label{eq:synchrotron}\n\\end{equation}\n\nwhere $\\nu_c$ is the frequency corresponding to the cutoff in the energy distribution of the particles, $\\alpha_1$ is the spectral index in the optically thin regime, and $\\tau_r(\\nu)$ is the dust screen between the synchrotron emitting region and the observer. $\\tau_r(\\nu)$ follows \\citet{Draine84}.\n\nIn Cygnus A the spectral index $\\alpha_1$ is measured from VLA core radio fluxes, and the amplitude of the powerlaw fixed from the same observations. The only free parameters are the cutoff frequency $\\nu_c$ and the optical depth $\\tau_{r}$. These parameters art partly degenerate in that in trial runs we experienced a situation where $\\tau_r$ and $\\nu_c$ would both increase, effectively offsetting each other. To combat this we limited $\\tau_r$ such that $A_V$ does not exceed 500 towards the radio source.\n\nAn alternate model for the synchrotron emission at higher frequencies is a broken powerlaw behind a dust screen (Case II). Aging of the population of relativistic electrons results in a broken powerlaw whose spectral index increases for frequencies higher than a break frequency \\citep{Kardashev62}. The functional form adopted is:\n\n\\begin{equation}\nF_{\\nu}  \\propto e^{-\\tau_r(\\nu)} \\times\n  \\begin{cases} \n    \\nu^{-\\alpha_1}  & \\text{if } \\nu < \\nu_{break} \\\\ \n    \\nu^{-\\alpha_2}  & \\text{if } \\nu \\geq \\nu_{break}\n  \\end{cases}\n\\label{eq:brokenpw}\n\\end{equation}\n\nFor a simple aging of the electron population, the post-break spectral index in Cygnus A would be $\\alpha_2=1.24$. As with Case I, we fit a dust screen in front of the synchrotron emission ($\\tau_r$).\n\n\\subsection{Stellar Contribution to the Mid-infrared}\n\nStarlight can potentially contribute to the mid-infrared flux, especially in a large aperture. Using the flux of the stellar component from \\citet{Jackson98} and their brightness profile, we determined the contribution due to starlight in a $20\\arcsec$ aperture. The relative fluxes were consistent with an elliptical galaxy template from \\citet{Silva98}. Expected flux densities at 2.2, 5, and 10 $\\mu$m were computed using the same template spectrum. After subtracting a nuclear point source, the K-band flux density in starlight is consistent with the flux in a $20\\arcsec$ aperture as measured using Two Micron All Sky Survey (2MASS) data products \\citep{Skrutskie06}. At 5 and 10 $\\mu$m, the starlight contributes roughly 14\\% and 2\\% respectively, of the flux measured by IRS. Therefore, we do not include any contribution to the mid-infrared flux from an old stellar population in our modeling.\n\n\\subsection{Dust in the NLR}\n\nWarm dust in the NLR directly illuminated by the AGN can also contribute to the mid- and far-infrared emission \\citep[e.g.,][]{Groves06}. \\citet{Ramos09} find that a significant contribution to the mid-infrared flux can come from sources other than the AGN torus, particularly additional hot dust. In Cygnus A \\citet{Radomski02} find an extended component to the mid-infrared emission which is consistent with $T\\sim150$ K dust. Some of this extended emission is co-spatial with sites of possible star formation. Dust of this temperature is well reproduced with our choice of starburst models (with the implicit assumption that this dust is heated by star formation). \n\nHigher temperature dust is also present in the inner regions of Cygnus A. However the resolved mid-infrared images of the nuclear regions by \\citet{Radomski02} are unable to distinguish between dust in the ``torus'' and dust in the NLR heated by the AGN. Without strong observational motivation for an additional dust component we opt not to include one. As will be shown in the next section we are able to reproduce the observed emission without this additional component.\n\n\\subsection{Prior Constraints on Model Parameters}\n\nPrior to fitting the models, the available parameter space was constrained using results from previous studies of Cygnus A.  The opening angle and covering fraction of the CLUMPY torus are determined from the $\\sigma$ and $N$ parameters \\citep[see Equations (3) and (4) in ][]{Mor09}. We define the half opening angle of the torus as the angle at which the escape probability of a photon drops below $e^{-1}$:\n\n\\begin{equation}\n\\theta_{half}=90-\\sigma\\sqrt{\\ln{N_0}}\n\\label{eq:halfopening}\n\\end{equation}\n\n\\citet{Tadhunter99} measured the opening angle of the ionization cone in Cygnus A using near infrared HST data, finding $\\theta_{half}=(58\\pm4)^{\\circ}$. We use this value as the torus opening angle and limit the parameter range of $\\sigma$ and $N$ according to Equation \\ref{eq:halfopening}.\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=0.5\\textwidth]{fig4a.eps}\\includegraphics[angle=270,width=0.5\\textwidth]{fig4b.eps}\n\\caption{LEFT: Case I fit to the Cygnus A SED. The shaded region shows all acceptable model fits within the 95.4\\% confidence interval. The lines are the components for the best-fit (lowest $\\chi^2$). RIGHT: Same as left, but for Case II fit to the Cygnus A SED. Insets are a zoom of the region around the $9.7~\\mu$m Sil absorption.}\n\\label{fig:SEDfit}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=270,width=0.8\\textwidth]{fig5.eps}\\\\\n\\end{center}\n\\caption{Histogram of bolometric AGN luminosity from best fit torus parameters (left) and bolometric starburst luminosity (right) for Cases I (blue, solid line) and II (red, dashed line).}\n\\label{fig:lumhist}\n\\end{figure*}\n\nThe inclination range of the torus $i$ was limited by very long baseline interferometry (VLBI) observations and modeling of the inner pc-scale jet \\citep[$50^{\\circ}\\leq i \\leq 85^{\\circ}$,][]{Sorathia96}. Additionally, the available VLA core radio fluxes were used to fix the synchrotron spectrum at radio frequencies ($\\alpha_1$, and the amplitude). \n\n\n\\section{Results}\n\\label{sec:results}\n\nTo fit the observed SED, the flux from each model was summed in each wavelength bin and compared to the observed flux within a $20\\arcsec$ aperture. Optimization was performed using Levernberg--Marquardt least-squares minimization. \n\n\nSome degeneracy between parameters was seen in the model results. Comparison model runs with fewer prior constraints on the parameters resulted in similar $\\chi^2$ values to those quoted below for regions of parameter space which are unlikely to be physically reasonable matches to Cygnus A (e.g., CLUMPY torus fits with covering fractions of unity and $\\theta_{half}=0^{\\circ}$). Prior constraints from other observations are thus important in eliminating regions of parameter space which may be statistically reasonable but physically unrealistic.\n\nAfter the models were fit, confidence intervals were determined separately for Case I and Case II through bootstrapping with replacement \\citep{Efron81}. These were used to determine the ``acceptable'' range of parameter values.\n\nThe results of the fits are shown in Figure \\ref{fig:SEDfit}. The range of parameter values for fits within at $95.4\\%$ confidence interval are shown in Figures \\ref{fig:lumhist}-\\ref{fig:synchist}. Section \\ref{ssec:sync-exp} discusses the Case I fits (exponential cutoff in the relativistic electron population) and Section \\ref{ssec:sync-break} discusses the Case II fits (broken powerlaw for the synchrotron emission).\n\n\\subsection{Case I: Synchrotron Exponential Cutoff}\n\\label{ssec:sync-exp}\n\nThe best fit combination of parameters for Case I has $\\chi^2$\/DOF = $1.35$. The range of acceptable matches for the 1641 model combinations is shown in Figure \\ref{fig:SEDfit} (left). Histograms of the best fit parameters are given in Figures \\ref{fig:lumhist} (left) and \\ref{fig:agnhist}, where $f$ is the fraction of models within the 95.4\\% confidence interval in a given bin. Parameters for the best fit model are marked with a small blue horizontal bar.\n\n\\subsubsection{AGN\/Torus Properties and Contribution}\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=\\textwidth]{fig6.eps}\n\\caption{Histograms of torus parameters for acceptable fits within the 95.4\\% confidence interval for Cases I (blue, solid line) and II (red, dashed line). The small horizontal bars denote the parameter value for the best-fit model. From left to right, top to bottom: viewing angle of the torus ($90$ meaning the torus is viewed edge-on), N - average number of clouds along an equatorial line of sight, A$_V$ - for an individual cloud, $\\sigma$ - angular width of the distribution of torus clouds, Y - ratio of inner to outer radii, q - power law index of the radial distribution of torus clouds, covering fraction of the torus (equivalent to the escape probability of optical\/UV photons, assuming optically thick clouds), integrated A$_V$ along an equatorial line of sight, integrated A$_V$ along our line of sight (calculated using N, $\\sigma$, the cloud A$_V$, and the inclination). Parameters in the first two rows are discrete and the histograms represent the relative number of models with those specific values. Parameters in the bottom row are derived from the parameters in the top two rows as well as the luminosity of the AGN component (shown in Figure \\ref{fig:lumhist}), and are continuous.}\n\\label{fig:agnhist}\n\\end{figure*}\n\nThe fits favor a bolometric accretion disk luminosity of $log(L_{AGN}\/L_{\\odot})\\sim 11.8-12.0$ (median of $11.82$ with an interquartile spread of $0.09$). The fits clearly favor an extended torus ($Y=200$) with low opacity clouds ($A_V\\sim10-30$). For the median bolometric luminosity, the inner radius of the torus is $R_d=0.6$ pc, giving a corresponding outer torus edge of $\\sim125$ pc (for Y$=200$). \n\nThe range of values for $i$ was limited based on previous work in the radio regime. The fits prefer an inclination for the torus on the high end of the range, $i\\approx80^{\\circ}$. $N$ has a bimodal distribution, with $25$\\% of fits having $N\\leq6$ clouds along an equatorial line of sight and $\\sim60$\\% having $N\\geq20$ clouds. Most of the fits within the 95.4\\% confidence interval have a flat radial distribution of clouds ($q=0$). \n\nThe opacity of individual clouds is anti-correlated with the radial extent of the torus; small torus sizes favor high $A_V$ values. High values of $N$ (i.e., many clouds along a given line of sight) are favored for small torus sizes, however for large torus sizes, both small and large $N$ values are acceptable.\n\nWe calculate the total extinction through the torus along both an equatorial line of sight (Figure \\ref{fig:agnhist} middle, bottom row), and along our line of sight to the torus (Figure \\ref{fig:agnhist} right, bottom row). The integrated equatorial $A_V$ spans a range between $100$ and $250$. When viewing angle is taken into consideration, the range narrows, with most fits showing the line of sight $A_V$ between $80$ and $120$. However the $A_V$ is poorly constrained beyond having a well-defined lower bound.\n\nThe torus covering fraction was computed using the method of \\citet[their Equations (3) and (4)]{Mor09}. By design, the derived covering fractions for the torus are between $50\\%$ and $70\\%$. \n\n\\subsubsection{Starburst}\n\nThe median value of the starburst luminosity for Case I is log$(L_{SB}\/L_{\\odot})\\sim10.8$, with a tail up to $\\sim11.6$ containing approximately $40\\%$ of the fits (Figure \\ref{fig:lumhist}). This covers a range of star formation rates from $10$ to $70$ M$_{\\odot}$ yr$^{-1}$, as determined by the $L_{IR}$ calibration from \\citet{Kennicutt98a} \\citep[for a review of SFR estimates, see][]{Kennicutt98b}.\n\n\\begin{equation}\n\\frac{SFR}{(M_{\\odot}~yr^{-1})} = 4.5\\times10^{-44}~\\frac{L_{FIR}}{erg~s^{-1}}=1.72\\times10^{-10}~\\frac{L_{FIR}}{L_{\\odot}}\n\\label{eq:sfr}\n\\end{equation}\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=\\textwidth]{fig7.eps}\n\\caption{Histograms of starburst parameters for acceptable fits within the 95.4\\% confidence interval for Cases I (blue, solid line) and II (red, dashed line). The small horizontal bars denote the parameter value for the best-fit model. From left to right, top to bottom: the fraction of luminosity in O and B stars, integrated extinction through the starburst, size of the starburst, dust density in hotspots around O and B stars. f$_{OB}$ and the size are discrete and the histograms represent the relative frequency of models with those specific values. The A$_V$ and $n$ parameters are more finely sampled than the histogram represents; the plots show the relative fraction of models with parameters in the noted ranges.}\n\\label{fig:sbhist}\n\\end{figure*}\n\nRoughly half the fits within this confidence interval have $40$\\% of the starburst luminosity in the form of OB stars. The distribution of size parameters is relatively flat, although even the largest sizes would be unresolved by our Spitzer observations. The distribution of dust density $n$ peaks around $10^3$ cm$^{-3}$. The extinction through this starburst is relatively unconstrained.\n\nThe strengths of dust features in starburst models falling within the 95.4\\% confidence interval were also measured using PAHFIT to compare with the intensities of observed dust feature. Figure \\ref{fig:dust} shows histograms of the predicted intensities from our models divided by the measured intensity from the IRS spectrum. Only dust features which are detected in the IRS spectrum have been plotted. In general the agreement is good with most models matching the measured line intensities. However for some dust features a significant number of the models within the confidence interval predict emission in excess of what is observed. While a detailed comparison of the relative strengths of dust features is beyond the scope of the paper, we suggest the models are able to generally reproduce the dust features seen in the spectrum. The dust features which show the greatest discrepancy between observed fluxes and those predicted from the modeling ($6.2$, $7.8$, and $11.3~\\mu$m) comprise 3 of the 5 lowest signal-to-noise dust feature fits in the spectrum. The discrepancy may then be due to the difficulty of fitting low equivalent width dust features.\n\nPAHFIT also attempted to fit eight other dust features in the IRS spectrum. The upper limits for three ($5.7$, $14.0$, and $15.9~\\mu$m) are consistent with expectations from the starburst models. One dust feature ($16.4~\\mu$m) has a measured limit below that expected from the best fitting starburst model, and so is discrepant. The other four features ($7.6$, $10,7$, $11.2$, and $12.7~\\mu$m) also have upper limits from PAHFIT which are lower than the expected value from the starburst models. However, these are coincident with or near other spectral features (e.g., narrow emission lines or the Sil absorption). Thus an accurate measurement of these dust features in the IRS spectrum would rely more heavily on fitting the wings, which could prove difficult in a continuum dominated source such as Cygnus A. \n\n\\subsubsection{Synchrotron Radiation}\n\nThe synchrotron emission amplitude was fixed using the non-thermal emission from the radio core, assuming it to be a point source at all frequencies observed. The synchrotron model, therefore, has only two parameters: cutoff frequency ($\\nu_c$) and extinction due to dust ($\\tau_r$).\n\nThe distribution of cutoff frequencies is somewhat broad, covering the range of $\\nu\\approx10-60$ THz ($5-30~\\mu$m) (Figure \\ref{fig:synchist}). The fits show a range of acceptable extinction for the dust screen, peaking in the range of $A_V\\approx60-80$, slightly lower than the predicted line of sight $A_V$ from the torus models. The integrated luminosity of the ``core'' synchrotron flux is log$(L_{sync}\/L_{\\odot})\\sim11.2$. \n\nThis break in the synchrotron spectrum is consistent other powerful FR II radio sources, where an extrapolation of the radio synchrotron emission significantly exceeds the observed optical flux \\citep[e.g.,][]{Schwartz00,Sambruna04,Mehta09}. In these cases, the similarity of the radio and X-ray spectral indices suggests that the X-ray may be produced by inverse Compton (IC) scattering.\n\nWith the modeled synchrotron spectrum, a magnetic field strength, and the assumption that each electron emits at a single frequency, the energy distribution of relativistic electrons can be determined:\n\n\\begin{equation}\n\\gamma(\\nu)= \\sqrt{\\frac{4\\pi m_e c \\nu}{3eB}}\n\\label{eq:e-gamma}\n\\end{equation}\n\nwhere all constants and values are in the cgs system of units.\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=\\textwidth]{fig8.eps}\n\\caption{Histograms of log(Predicted Flux \/ Measured Flux) for dust features detected in the IRS spectrum of Cygnus A. The predicted values were measured from the \\citet{Siebenmorgen07} models using PAHFIT. The solid blue line denotes Case I while the red dashed line denotes Case II. (see Table \\ref{table:dustfeatures} for the measured flux values).}\n\\label{fig:dust}\n\\end{figure*}\n\n\\clearpage\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=\\textwidth]{fig9.eps}\n\\caption{Histograms of synchrotron parameters for acceptable fits within the 95.4\\% confidence interval for Cases I and II. For Case I $\\nu_{crit}\\equiv\\nu_{cutoff}$ and for Case II $\\nu_{crit}\\equiv\\nu_{break}$. $A_V=1.08\\tau_r (\\nu= \\text{V-Band} )$.}\n\\label{fig:synchist}\n\\end{figure*}\n\nThe magnetic field strength has been estimated from VLBI observations to be roughly $17$ mG ($100$ mG) in the jet (core) \\citep{Roland88,Kellermann81}. Using $\\nu\\approx30$ THz and Equation \\ref{eq:e-gamma} we find $\\gamma\\sim2\\times10^4$ ($8\\times10^3$) for the jet (core).\n\nIn addition to radiating energy via synchrotron emission, relativistic electrons can also lose energy via IC scattering. The ratio of the energy lost through synchrotron to the energy lost to IC is simply given by the ratio of the magnetic and radiation energy densities, with the cosmic microwave background (CMB) setting a minimum value for the radiation energy density. Using the magnetic field strength assumed above ($17$ mG), and $T_{CMB}=2.73$ K, synchrotron losses are dominant by a factor $>10^8$. A larger magnetic field would further enhance this ratio, so IC losses from scattering of CMB photons have a negligible effect on the population of relativistic particles.  \n\n\\subsection{Case II: Synchrotron Broken Power Law}\n\\label{ssec:sync-break}\n\nThe range of best fit models for Case II is shown in Figure \\ref{fig:SEDfit} (right). The best fit model had $\\chi^2$\/DOF = $1.03$. Figure \\ref{fig:SEDfit} (right) shows the range of the 1580 model fits within the 95.4\\% confidence interval. Histograms of the best fit parameters are given in Figures \\ref{fig:lumhist} (left) and \\ref{fig:agnhist}. A red horizontal bar marks the parameter for the parameters with the lowest $\\chi^2$ value.\n\n\\subsubsection{AGN\/Torus}\n\nIn Case II fits the bolometric accretion disk luminosity is roughly log$(L_{AGN}\/L_{\\odot})\\sim 11.8$ (median $11.88$ with an interquartile range of $0.08$). The distribution of luminosities is more strongly peaked than for Case I, with over 80\\% of fits occupying the log$(L\/L_{\\odot})=11.8-12.0$ bin. A large torus ($Y=200$) is exclusively preferred, corresponding to $R_{out}\\approx135$ pc (using the median value for the luminosity). Again, by design, the torus covering fraction is between 0.5 and 0.7. An inclination of $i=80^{\\circ}$ is exclusively preferred.\n\nThe distribution of the average number of clumps along a line of sight $N$ is again bimodal with either large ($<14$) or small ($<8$) numbers of clouds preferred. The number of clouds is anti-correlated with the extinction through individual clouds. The spread in equatorial $A_V$ is somewhat large ($120-260$), but the line of sight $A_V$ is more constrained ($100-160$ for 80\\% of fits). \n\n\\subsubsection{Starburst}\n\nThe starburst component in Case II fits show qualitatively similar behavior to Case I. The typical luminosity is comparable, but shows a smaller tail up to log$(L_{SB}\/L_{\\odot})=11.6$ ($\\sim30\\%$ of fits). Other parameters are similar to those in Case I fits, with the exception of the size which appears to have a very slight preference for a larger size, suggesting an overall cooler dust temperature for a given luminosity. This may indicate a degeneracy between the starburst and synchrotron models. The critical frequency for the synchrotron spectrum (see the next section) is at lower frequency for Case II, resulting in a smaller contribution to the far-infrared flux. The starburst model compensates with a larger overall size to provide additional cool dust emission (for a given starburst luminosity).  The dust feature strengths expressed in the models generally compare favorably with Case I.\n\n\\subsubsection{Synchrotron Properties and Contribution}\n\nAn alternate mechanism for limiting the influence of synchrotron emission at shorter wavelengths is for the spectrum to break at some frequency (see Section \\ref{sec:sync-model}). Case II fits used a fixed pre-break spectral index of $\\alpha_1=0.18$ and a post-break spectral index of $\\alpha_2=1.24$, consistent with aging of the relativistic population \\citep[without injection of additional particles;][]{Kardashev62}. As the flux density decreases in a slower fashion when compared to an exponential cutoff, the powerlaw must break at lower frequencies to ensure that the observed mid-infrared flux is not exceeded. \n\nThe break frequency is roughly 5 THz ($60~\\mu$m). The unobscured synchrotron luminosity is found to be log$(L_{sync}\/L_{\\odot})\\sim11.1$. Following the same arguments as Case I, the electrons emitting at the break frequency have $\\gamma\\sim7\\times10^3$ ($3\\times10^{3}$) for jet (core) magnetic field values. Again, this is consistent with the observed properties of jets in other FR II radio sources.\n\nThe extinction of the dust screen in front of the radio source is lower than Case I fits, with $A_V<60$, still within range of the extinction to the central source estimated by \\citet{Djorgovski91}, but lower than the computed line of sight $A_V$ from the torus. The larger discrepancy compared with Case I is due to the enhanced short wavelength emission of the broken power law at shorter wavelengths (when compared to the Case I exponential cutoff). The equatorial torus $A_V$ increases to provide an overall cooler temperature, and the modeled dust screen in front of the synchrotron component remains small (in order to contribute sufficient flux at shorter wavelengths).\n\n\\section{Discussion of General SED Results}\n\\label{sec:discussion}\n\n\\subsection{Luminosity and Kinetic Power in Cygnus A}\n\nOur best estimate of the bolometric AGN luminosity in Cygnus A (log$(L\/L_{\\odot})\\sim12$, including the synchrotron component and X-ray emission) is above the \\citet{Whysong04} estimate using Keck mid-infrared observations and a PG-quasar spectrum ($3.9\\times10^{11}$ $L_{\\odot}$), but is somewhat below the estimates of \\citet{Tadhunter03} who find $L_{bol}=12-55\\times10^{11}$ $L_{\\odot}$ (although consistent with the low end of their range). The bolometric AGN luminosity is the best constrained parameter, and is insensitive to the synchrotron model adopted.\n\nThe kinetic power in the expansion of the radio lobes in Cygnus A can be inferred from X-ray observations of the cluster environment \\citep{Wilson06}. From this analysis of Chandra observations Cygnus A has a kinetic power of $L_{kin} = 1.2 \\times 10^{46}$ erg s$^{-1}$ (log$(L_{kin}\/L_{\\odot})=12.5$), a factor of three larger than the bolometric AGN luminosity inferred from the modeling. Given the uncertainties in the determinations of both the kinetic and bolometric power, it is unclear if the kinetic power dominates significantly over the power emitted as radiation.\n\n\\subsection{Implications of the Modeling}\n\nThe probable torus sizes from the SED modeling give outer radii of $R_o=130$ pc ($\\approx0\\arcsec.2$ at the distance of Cygnus A). Generally, the torus parameters provide reasonable estimates of the properties of the obscuring structure. This predicted outer radius is significantly larger than what is generally assumed to be reasonable for the obscuring torus. A torus with the angular diameter of the $\\approx0.\\arcsec2$ is well within range of multiple mid and near-infrared instruments. At large radii, the distinction between torus clouds and narrow line region clouds may be somewhat arbitrary. The observations by \\citet{Canalizo03} show extended emission on scales including and larger than our fit torus sizes. The degree of contamination by emission lines is unclear, and it would be interesting to attempt to obtain a line-free continuum image to ascertain the true size and shape of the continuum emitting region.\n\nThe torus size parameters are consistent with those found from modeling of the $9.7~\\mu$m Sil feature by \\citet{Imanishi00}. Their radiative transfer modeling suggested some torus properties similar to those presented here, namely a small inner radius ($<10$ pc) and an inner-to-outer radius of $80-500$. In contrast they find a steeper radial dependence for the dust distribution $q\\sim2-2.5$, possibly due to the use of a smooth dust distribution.\n\nThe preferred parameters for the obscuring torus; low $q$, high $A_V$, and large $Y$ suggest the model is being driven toward a cooler spectrum. This could influence the starburst component, leading to a lower inferred star formation rate. In order to combat this it would be beneficial to place tighter constraints on the far-infrared SED to enable better modeling of the star formation in Cygnus A.\n\nA comparison of our IRS data with mid-infrared observations by \\citet{Radomski02} is consistent with the suggestion of our model that at $10$ and $18~\\mu$m, the emission is dominated by the torus+synchrotron component. \\citet{Tadhunter99} measure a K-band nuclear point source with $F_{\\nu}=(49\\pm10)~\\mu$Jy. This flux limit is broadly consistent with some model fits for the torus+synchrotron component.\n\nFor the estimated bolometric luminosity $10^{12}$ $L_{\\odot}$, and a black hole mass of $2.5\\times10^9$ $M_{\\odot}$, the Eddington Ratio ($L\/L_{edd}$) is $\\sim1.3\\times10^{-2}$. This is similar to, but slightly lower than previous estimates from \\citet{Tadhunter03}, likely due to the fact that our model attributes some of the IR luminosity to the heating of dust from star formation. This starburst luminosity is relatively well constrained, with consistent values across the bulk of fits. The AGN (torus plus jet) contributes $\\sim90\\%$ of the infrared luminosity and star formation produces the remaining $10\\%$.\n\n\\subsection{Future Work and Observations}\n\nThe results of our modeling can be tested and improved with the help of future observations in various wavelength regimes.\n\nAround 1 THz ($\\sim300\\mu$m) the synchrotron and starburst model components are of similar flux density, with the synchrotron contribution decreasing and the thermal contribution from star formation beginning to dominate. Unfortunately, data at this location are limited in resolution, sensitivity, and wavelength coverage. Disentangling the emission from the starburst and emission from the AGN at this frequency will be possible with the Atacama Large Millimeter Array \\citep[ALMA; e.g.,][]{Carilli05}, particularly with the availability of Band 10 and full science capabilities.\n\nThe far-infrared suffers from poor sampling of the SED. In this regime the Herschel Space Observatory \\citep{Pilbratt10} provides an opportunity to improve on the understanding of the continuum emission. SPIRE and PACS cover wavelength ranges of interest: the region of the possible jet-break and the long wavelength side of the infrared bump. Observations here can provide improved constraints for the synchrotron and starburst models. \\citet{Gonzales10} demonstrate decomposition of far-infrared emission for Mrk 231, with the Herschel SPIRE observations providing important constraints. Similar observations of Cygnus A will provide important data on this sparsely observed portion of the SED. PACS observations would contribute measurements of far-infrared fine structure lines which could be used in concert with the mid-infrared lines to provide an alternate method of determining the relative contribution of star formation and AGN activity to the infrared luminosity \\citep[e.g.,][for Mrk 231]{Fischer10}.\n\nAlthough the synchrotron spectrum breaks in our fits, it may still dominate the flux between $5$ and $10~\\mu$m. This result can be tested by future mid-infrared polarimetry or variability studies. Synchrotron emission from a compact source such as the radio core or jet knots should be subject to flaring. If the emission in this wavelength range is instead of a thermal origin (e.g., hot dust in the host galaxy), the $10~\\mu$m flux will remain relatively stable.\n\n\\section{Summary}\n\\label{sec:summary}\n\nUsing a combination of a new mid-infrared spectrum from the Spitzer Space Telescope and radio data from the literature, the nuclear emission in Cygnus A has been modeled as a combination of powerlaw emission from a synchrotron jet, reprocessed AGN emission from a dusty torus, and emission from a dusty circumnuclear starburst.\n\nThe data are well fit by a combination of these three models, and all three are necessary to reproduce the observed emission. Statistically acceptable fits were found for both an exponential cutoff in the population of relativistic electrons (Case I) as well as emission from an aging electron population (Case II), however we are unable to distinguish between the two cases. For Case I we find the cutoff frequency to be between $10$ and $50$ THz ($5-30~\\mu$m) while for Case II fits the predicted break frequency is $5$ THz ($60~\\mu$m). Degeneracy between the starburst and synchrotron components makes a more precise determination of the break or cutoff frequency difficult. Better observations on the long-wavelength side of the thermal bump will provide tighter constraints on the models.\n\nFrom this modeling, we find the following:\n\\begin{itemize}\n  \\item{The bolometric luminosity of the AGN in Cygnus A is $\\sim10^{12}$ $L_{\\odot}$}\n  \\item{The mid-infrared emission is consistent with emission from a clumpy obscuring torus with an outer size of $\\sim130$ pc.}\n  \\item{The far-infrared emission is consistent with being dominated by star formation which is occurring at a rate between $10$ and $70$ $M_{\\odot}$ yr$^{-1}$.}\n\\end{itemize}\n\nIn Cygnus A, the infrared emission is a combination of AGN and starburst heated dust, with the AGN contributing $\\sim90\\%$ of the luminosity. \n\n\n\\acknowledgements\n\nThe authors thank the anonymous referee who's comments have improved the quality of this paper. \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. Support for this work was provided by NASA through an award issued by JPL\/Caltech. Computing resources were provided in part by the Research Computing group of the Rochester Institute of Technology. This research has made use of the NASA\/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made use of NASA's Astrophysics Data System. This work is supported in part by the Radcliffe Institute for Advanced Study at Harvard University.\n\nG.P. thanks D. Whelan, D. M. Whittle, and A. Evans for helpful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOur understanding of galaxy evolution has immensely progressed over the last decades thanks to deep photometric observations mainly at ultraviolet, optical and near-infrared wavelengths enabled by the {\\it Hubble Space Telescope} (HST). The {\\it Spitzer} and {\\it Herschel} space telescopes have complemented such datasets at mid- and far-infrared wavelengths, allowing us to characterize galaxy populations out to high redshifts in unprecedented detail.\nThese observations suggest that the Universe has undergone a peak of star formation activity at $1<z<3$ (e.g.~\\citealt{hopkins2006,bouwens2007,cucciati2012}). Mapping this epoch of intense galaxy assembly is thus crucial to understand galaxy evolution.\nThe new WFC3 camera on board HST has revolutionized the study of the galaxy population at this epoch. In particular, the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey (CANDELS; \\citealt{grogin2011,koekemoer2011}) HST Multi-Cycle Treasury Survey is providing high spatial resolution images of the emission in the most extensively studied extragalactic fields (GOODS, COSMOS, EGS and UDS) for which a wealth of ancillary photometric data from the ultraviolet to the infrared have been compiled over the years. At the same time, the 3D-HST WFC3-G141 grism survey has been obtaining spatially resolved rest-frame optical spectra of $\\sim 7000$ galaxies ($H < 23$) at $z>1$ in these fields, providing redshift estimates with an accuracy of $\\sigma(z)=0.0034(1+z)$, i.e. an order of magnitude better than that of photometric redshift based on broad-band observations \\citep{brammer2012}. Crucially, these new spectroscopic data also enable detailed studies of the ionized gas and dust geometry (via rest-frame optical emission lines) and stellar population properties (via stellar absorption features) of unprecedentedly large galaxy samples at $z>1$ (e.g.~\\citealt{brammer2012,fumagalli2012,fumagalli2013,schmidt2013,whitaker2013,price2014}).\n\nA main caveat in current statistical studies of the galaxy population at $z\\ga1$ is that the way in which the physical properties of galaxies are generally derived from rich multi-wavelength data-sets does not reflect recent advances in the sophisticated modelling of galaxy spectral energy distributions (SEDs). For example, spectral analyses often rely on oversimplified colour diagnostics and\/or limited modelling of the stellar spectral continuum using simple star formation histories (SFHs), such as exponentially declining  $\\tau$-models.\\footnote{SFHs in which the star formation rate depends on time $t$ as $\\propto\\exp(-t\/\\tau)$ are often referred to as `$\\tau$-models'.} As shown for example by \\cite{maraston2010}, such models do not account well for the observed colours of $z\\approx 2$ galaxies from the GOODS-South sample, which these authors reproduce by appealing to exponentially rising SFHs \\cite[see also][]{pforr2012}.  \\cite{simha2014} propose a 4-parameter model (delayed exponential + linear ramp at some transition time) to best represent the SFHs of galaxies from smoothed particle hydrodynamics simulations. Other studies have already shown that more sophisticated SFH parametrizations provide better agreement with the data (e.g.~\\citealt{lee2010,pacifici2013,behroozi2013,lee2014}). The inclusion of nebular emission is also important to interpret observed spectral energy distributions of galaxies. While most studies including nebular emission rely on empirically calibrated emission-line template spectra (e.g.~\\citealt{fioc1997,anders2003,schaerer2009,schaerer2010}), more elaborate prescriptions have been proposed, based on combination of stellar population synthesis and photoionization codes (e.g., \\citealt{charlot2001,groves2008}).\n\n\\cite{pacifici2012} provide the sophisticated modelling framework required to interpret multi-wavelength photometric and spectroscopic galaxy observations in a physically and statistically consistent way. This approach overcomes several main limitations of galaxy spectral modelling mentioned above. Specifically, \\cite{pacifici2012} build a comprehensive library of model galaxy SEDs, which can be used to derive statistical constraints on physical parameters. This combines: (i) physically motivated star formation and chemical enrichment histories from cosmological simulations; (ii) state-of-the-art stellar population synthesis and nebular emission modelling computed consistently using the photoionization code {\\small CLOUDY} \\citep{ferland1996} and (iii) a sophisticated treatment of dust attenuation, which includes uncertainties in the spatial distribution of dust and in galaxy orientation \\citep{chevallard2013}. One of the main features of the \\cite{pacifici2012} approach is that it allows one to interpret simultaneously the stellar and nebular emission from galaxies at any spectral resolution.\n\nIn this paper, we use the wealth of multi-wavelength data provided by the 3D-HST survey to investigate, in a systematic way, how different SED modelling approaches (regarding star formation and chemical enrichment histories, dust attenuation and nebular emission lines) and the availability of spectroscopic (in addition to photometric) information affect the constraints derived on the physical parameters of high-redshift galaxies. We use the \\cite{pacifici2012} models to reproduce simultaneously the observed $0.35$ to $3.6\\mu$m photometry and emission line strengths of a sample of 1048 galaxies at $0.7<z<2.8$. We investigate the advantages provided by these sophisticated models when deriving galaxy physical parameters, with respect to standard approaches relying on widely used, oversimplified parametrizations of star formation histories (continuous, exponentially declining $\\tau$-models), fixed assumptions about metallicity and dust properties and neglecting nebular emission (e.g.~\\citealt{noll2009,kriek2009,forster2009,marchesini2009,pozzetti2010}). We also quantify the importance of accounting for nebular emission in the interpretation of broad-band galaxy observations by comparing the results obtained using the original \\cite{pacifici2012} models with those obtained using the same sophisticated prescriptions for the star formation and chemical enrichment histories and attenuation by dust, but not including nebular emission lines.\nTo further illustrate the importance of using appropriate spectral models to derive physical properties of galaxies and, more generally, to constrain galaxy evolution, we also explore how the different SED-fitting methods considered in this paper affect the interpretation of the correlation between SFR and stellar mass of galaxies (commonly referred to as the `star-formation main sequence' - \\citealt{noeske2007}) and the scatter about this relation.\n\nOur paper is organized as follows. In Section~\\ref{sec:data}, we describe the sample of 1048 galaxies in GOODS-South extracted from the 3D-HST catalogue, for which multi-wavelength photometry and HST\/WFC3 grism spectra are available. In Section~\\ref{sec:modelintro}, we introduce three model libraries of galaxy SEDs with different levels of sophistication to interpret these observations. We also assess how well each of these libraries can account for the data. Then, in Section~\\ref{sec:results}, we quantify the differences in the constraints derived on the stellar masses, SFRs and attenuation by dust for the galaxies in the sample when adopting the different spectral libraries. We investigate in Section~\\ref{sec:ms} how these differences affect the measured correlation between SFR and stellar mass of galaxies. Finally, in Section~\\ref{sec:summary}, we summarize our results. Throughout this paper we express magnitudes in the AB system and we adopt the \\cite{chabrier2003} stellar IMF and the following cosmology: $\\Omega_\\Lambda=0.7$, $\\Omega_m=0.3$, $H_0=70$ km s$^{-1}$ Mpc$^{-1}$.\n\n\\section{Data}\n\\label{sec:data}\n\nWe select a sample of galaxies from the 3D-HST Treasury Survey\\footnote{http:\/\/3dhst.research.yale.edu\/} \\citep[PI: van Dokkum]{brammer2012}, which provides a multi-wavelength photometric catalogue \\citep{skelton2014} and low-resolution WFC3 grism observations. In this Section, we present the photometric catalogue, spectroscopic observations and describe the sample selection.\n\n\\subsection{Photometry}\n\\label{sec:phot}\n\nWe use version 4.1 of the 3D-HST Survey photometric catalogue for the GOODS-South field covering an area of 171 arcmin$^2$ \\citep{skelton2014}. The catalogue contains broad- and narrow-band photometry in 38 filters sampling ultraviolet (UV) to mid-infrared (MIR) wavelengths, from both ground- and space-based observatories, for a total of 50,507 objects detected in the HST\/WFC3-F160W band. To make the present investigation relevant to a wide range of studies of high-redshift galaxies, we focus on the following commonly used nine broad bands: $U$, ACS-F435W, ACS-F606W, ACS-F775W, ACS-F850lp, WFC3-F125W, WFC3-F140W, WFC3-F160W and {\\em Spitzer}\/IRAC 3.6$\\mu$m. These bands allow us to reasonably sample the rest-frame UV, optical and near-infrared emission of galaxies at all redshifts between 0.7 and 2.8, optimally probing the emission by stellar populations of different ages and the effects of dust attenuation. We exclude {\\em Spitzer}\/IRAC data at wavelengths longer than $3.6\\mu$m because the spectral modelling (see Section~\\ref{sec:modelintro}) does not include dust emission.\\footnote{Adding dust emission increases the complexity of the analysis and is beyond the scope of the current paper; we leave a detailed analysis combining the \\citealt{pacifici2012} model with the models by \\citealt{dacunha2008} to a future study.}\n\nThe photometry in all bands is extracted consistently across all wavelengths as described in \\cite{skelton2014}. To summarize, the 3D-HST photometric catalogue is based on WFC3 selection using a combination of the three WFC3 bands (F125W, F140W and F160W) for detection. In order to measure consistent colours across different HST bands, each of the HST images is convolved with the same point spread function (PSF) and aperture photometry is performed on the PSF-matched images. For the lower resolution ground-based optical and {\\em Spitzer}\/IRAC bands, the photometry is extracted using the method described in \\cite{labbe2006}, \\cite{wuyts2008} and \\cite{whitaker2011}, which uses the HST images as a high-resolution prior to correct for the larger PSF and confusion caused by neighboring sources.\n\n\\subsection{Grism spectroscopy}\n\\label{sec:spec}\n\nThe 248 orbits dedicated to the 3D-HST Survey are distributed among 124 individual pointings, each observed for two orbits. 34 of these pointings cover the GOODS-South field. For all 50,507 galaxies in this field, WFC3-G141 grism spectra are available, while a spectroscopic redshift ($z_\\mathrm{grism}$) has been estimated for a subset of 2699 galaxies to $H<23$ \\citep[here the $H$ band is HST\/WFC3-F160W;][]{brammer2012}.\nThe WFC3 grism spectra cover observer-range wavelengths between 1.1 and 1.6 $\\mu$m. The spectral scale is $\\approx 23$ {\\AA}\/pixel and the effective resolution depends on the spatial extent of the galaxy (see \\citealt{brammer2012}, Sec. 2.1 and 4.2). For example, the observed full width at half maximum (FWHM) of the {\\hbox{H$\\alpha$}} line (which in these spectra is blended with $\\hbox{[N\\,{\\sc ii}]}\\lambda 6548$ and $\\hbox{[N\\,{\\sc ii}]}\\lambda 6584$) can span a range as wide as 75--200 \\AA. The emission-line sensitivity is $\\sim 5 \\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ at $5\\sigma$. \nGrism redshifts are derived from a combination of the 1D spectra and broad- and medium-band photometry from the catalogue described above (see \\citealt{brammer2012} and references therein for details). The precision of grism redshifts is $\\sigma(z)=0.0034(1+z)$, i.e. an order of magnitude better than what is typically achieved with high-quality broad-band photometry alone.\n\nTo characterize the emission features in the observed spectra, we measure rest-frame equivalent widths (EWs) of the most prominent lines. Because of the low resolution, some emission lines are blended together, thus we extract the EWs of: ${\\hbox{[O\\,{\\sc ii}]}}\\lambda\\lambda3726,\\,3729$; {\\hbox{H$\\beta$}}; ${\\hbox{[O\\,{\\sc iii}]}}\\lambda\\lambda4959,\\,5007$; $\\hbox{[N\\,{\\sc ii}]}\\lambda6548+\\hbox{H$\\alpha$}+\\hbox{[N\\,{\\sc ii}]}\\lambda6584$; and ${\\hbox{[S\\,{\\sc ii}]}}\\lambda\\lambda6717,\\,6731$. In the following, we refer to these (blends of) lines simply as {\\hbox{[O\\,{\\sc ii}]}}, {\\hbox{H$\\beta$}}, {\\hbox{[O\\,{\\sc iii}]}}, {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}}. To measure the continuum underneath the emission lines, we first smooth the spectra over 17 pixels (i.e., about 200 \\AA\\ in rest-frame at $z=1$). Then, we mask $\\approx350${\\AA}-wide regions centred on the 5 (groups of) emission lines, interpolate the spectra in the masked regions and finally smooth again over 17 pixels to remove possibly persisting features. Using the resulting continuum, we can calculate the emission-line EWs of {\\hbox{[O\\,{\\sc ii}]}}, {\\hbox{H$\\beta$}}, {\\hbox{[O\\,{\\sc iii}]}}, {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} as defined above. The uncertainties on the EWs are measured through the propagation of the errors on the flux.\\footnote{Since the continuum is obtained by smoothing the flux, we scale the error on the continuum with the fraction between the standard deviations of the flux and the continuum in a wavelength window free of line contamination.} The signal-to-noise ratio (S\/N) of {\\hbox{H$\\alpha$}} and {\\hbox{[O\\,{\\sc iii}]}} can reach up to $\\sim30$, while that of the other lines ({\\hbox{[O\\,{\\sc ii}]}}, {\\hbox{H$\\beta$}} and {\\hbox{[S\\,{\\sc ii}]}}) rarely exceeds $\\sim10$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{.\/figures\/fig_01.png}\n\\caption{Redshift distributions of the 3D-HST photometric sample (gray shaded histogram; 1048 galaxies) and emission-line sub-sample (black open histogram; 364 galaxies) are shown in the top panel. Coloured boxes indicate the redshift windows in which selected emission lines can be detected in the WFC3-G141 grism spectra. In the bottom panel, we show the observer-frame grism spectrum of one of the highest S\/N objects in the sample, detected at $z=1.316$. The black solid line and the grey shaded region represent the observed flux and the flux error, respectively. Dashed lines indicate the position of the four detected (blends of) emission lines.}\n\\label{fig:zz}\n\\end{center}\n\\end{figure}\n\n\\subsection{Sample selection}\n\\label{sec:sample}\n\n\\begin{table}\n\\begin{center}\n\\caption{Typical magnitudes and signal-to-noise ratios of the 3D-HST photometric sample in the nine bands considered in this paper.}\n\\label{tab:phot}\n\\begin{tabular}{l r r r}\n\\hline\n\\hline\nFilter & Effective $\\lambda$\/{\\AA} & Median magnitude & Median S\/N \\\\\n\\hline\n$U$ & 3446 & 25.1 & 52\\\\\nACS-F435W & 4302 & 24.9 & 23\\\\\nACS-F606W & 5925 & 24.3 & 52\\\\\nACS-F775W & 7705 & 23.6 & 62\\\\\nACS-F850lp & 9059 & 23.2 & 72\\\\\nWFC3-F125W & 12,516 & 22.5 & 91\\\\\nWFC3-F140W & 13,971 & 22.4 & 61\\\\\nWFC3-F160W & 15,392 & 22.2 & 105\\\\\nIRAC 3.6$\\mu$m & 35,573 & 21.5 & 106\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe first select a \\textit{photometric} sample by extracting from the 3D-HST v4.1 photometric catalogue all galaxies with a measured grism redshift in the range $0.7<z_\\mathrm{grism}<2.8$ and use\\_flag $=1$ (see Section 3.8 in \\citealt{skelton2014}). This sample includes 1140 galaxies down to $H=23$. From this sample, we select an \\textit{emission-line} sub-sample including only galaxies with at least one emission line detected in the WFC3-G141 grism spectrum with equivalent-width S\/N greater than 5. We visually inspect all selected grism spectra to check the accuracy in redshift and to make sure that contiguous emission lines ({\\hbox{H$\\beta$}} with {\\hbox{[O\\,{\\sc iii}]}} and {\\hbox{H$\\alpha$}} with {\\hbox{[S\\,{\\sc ii}]}}) are well separated. In the end, we retain 392 emission-line galaxies. We then reject all sources detected by \\textit{Chandra} (\\citealt{xue2011}, $\\log(L_\\mathrm{X})>42$) as potential AGNs, which leaves us with 1048 galaxies in the photometric sample and 364 in the emission-line sample. In Table~\\ref{tab:phot}, we summarize the median magnitudes and S\/N in the nine filters of interest to us for the photometric sample. We note that the typical S\/N on the photometry is very high for most bands. We discuss the influence of such small uncertainties on the derivation of statistical constraints on physical parameters in Section~\\ref{sec:fit} and \\ref{sec:results}.\n\nThe redshift distributions of the photometric and emission-line samples are shown in Fig.~\\ref{fig:zz} (top panel) as shaded and open histograms, respectively. The peak at $z\\approx0.7$ is associated to an over-density in the field (\\citealt{salimbeni2009}). Most of the galaxies in the emission-line sample lie at $z\\sim1$, which corresponds to where {\\hbox{H$\\alpha$}} falls in the G141 spectral range. The most `data-rich' window is the redshift range $1.2 \\lesssim z \\lesssim 1.5$, where four emission lines can be detected simultaneously; 17 per cent of the galaxies in the emission-line sample are selected from this redshift window. As an example, we show, in the bottom panel of Fig.~\\ref{fig:zz}, the observer-frame grism spectrum of one of the highest S\/N objects in the sample, detected at $z=1.316$. This galaxy shows two emission lines with S\/N~$>20$ ({\\hbox{H$\\alpha$}} and {\\hbox{[O\\,{\\sc iii}]}}), one with S\/N~$\\approx10$ ({\\hbox{H$\\beta$}}) and one with S\/N~$\\approx5$ ({\\hbox{[S\\,{\\sc ii}]}}).\n\n\\section{Comparison between different spectral modelling approaches}\n\\label{sec:modelintro}\n\nTo derive galaxy physical parameters (such as stellar mass, SFR and optical depth of the dust) from the multi-wavelength observations described above, we must appeal to spectral modelling techniques. A main focus of the present paper is to establish the appropriateness of different spectral modelling approaches to interpret photometric and spectroscopic observations of distant galaxies. Specifically, we consider three modelling approaches relying on different assumptions about several main ingredients of spectral interpretation techniques: the explored (prior) ranges of star formation and chemical enrichment histories; attenuation by dust; and nebular emission. In the next paragraphs, we describe these competing approaches in order of increasing complexity. We also quantify the extent to which the different models can account for the data of Section~\\ref{sec:data} by comparing prior libraries of predicted colours with these multi-wavelength data with a bayesian approach. We refer to Appendix A for a specific comparison between physical parameters extracted using our most sophisticated spectral library (P12, see below) and a tool widely used in the 3D-HST collaboration \\citep[FAST]{kriek2009}.\n\n\\subsection{Different model spectral libraries}\n\\label{sec:model}\n\n\\subsubsection{The `classical' spectral library (CLSC)}\n\\label{sec:basic}\n\nWe first assemble a `classical' model spectral library, corresponding to standard simplistic descriptions of the stellar and interstellar content of galaxies, which are the most widely used to derive galaxy physical properties from fits of ultraviolet to near-infrared SEDs (e.g.~\\citealt{kriek2009,pozzetti2010}). Galaxy SFHs in this spectral library are parametrized as exponentially declining functions of the form $\\psi(t)\\propto \\exp(\\gamma \\, t)$. Here $\\gamma$ is the inverse star formation timescale, drawn randomly in the range $0<\\gamma\/\\mathrm{Gyr}^{-1}<3$,\\footnote{We note that allowing for larger values of $\\gamma$ would increase the number of low-specific-SFR galaxies in the prior at fixed evolutionary stage. We have checked that, on the one hand, this would bias the SFR estimates low \\citep{wuyts2011,price2014}. On the other hand, very young ages (and thus low masses) would be required in order to match the photometry of the galaxies in the sample. In Appendix A, we show the physical parameters extracted using the code FAST \\citep{kriek2009}, in which very short $e$-folding star-formation timescales are allowed.} and $t$ is the lookback time. In this CLSC spectral library, all stars in a given galaxy are assumed to have the same metallicity, drawn randomly from the logarithmic range $-1.6<\\log(Z\/Z_{\\odot})<0.4$ (we adopt the solar metallicity $Z_{\\odot}=0.017$). By analogy with \\cite{pacifici2012}, we generate a sample of 4 million galaxy SFHs, selecting randomly the redshift of observation in the range $0.6<z<3.0$ (appropriate for the 3D-HST sample described in Section~\\ref{sec:data}) and the evolutionary stage between 0.5 Gyr and the age of the Universe at the redshift of observation (see Sections 2.1 and 3.1.2 of \\citealt{pacifici2012}).\\footnote{In \\cite{pacifici2012}, the evolutionary stage is drawn randomly from a linear range in redshift. Here, we choose to draw the evolutionary stage randomly from a linear range in lookback time to avoid having in the spectral library too many galaxies which started forming stars less than 1 Gyr before the redshift of observation.} We combine this set of exponentially declining star formation histories with the latest version of the the \\cite{bruzual2003} stellar population synthesis models, which allows one to compute the spectral evolution of stellar populations for given SFH, metallicity and IMF (see Appendix B for a comparison with the original version of the \\citealt{bruzual2003} stellar models). We model interstellar dust attenuation using a two-component model, where the $V$-band attenuation optical depth seen by stars older than 10Myr in the diffuse interstellar medium (ISM) is a fraction ($\\mu$) of the optical depth $\\hbox{$\\hat{\\tau}_{V}$}$ seen by younger stars still embedded in their birth clouds \\citep{charlot2000}. We adopt a fixed slope of the attenuation curve in both the birth clouds and the diffuse ISM ($\\hbox{$\\hat{\\tau}_\\lambda$} \\propto \\lambda^n$, with $n=-0.7$). In this spectral library, we draw randomly the total attenuation optical depth $\\hbox{$\\hat{\\tau}_{V}$}$ in the range between 0 and 4 and fix $\\mu=0.3$. The overall (i.e. galaxy-wide) attenuation law produced by the Charlot \\& Fall (2000; see their fig. 5) model depends on the relative contributions by young and old stars to the emission of a galaxy and for young starbursts, can resemble the \\cite{calzetti2000} attenuation law (as used in several studies, e.g.~\\citealt{kriek2009,forster2009,marchesini2009}).\n\n\\subsubsection{The `P12 without emission lines' spectral library (P12nEL)}\n\\label{sec:p12}\n\nThe second model spectral library we consider is based directly on the approach of \\cite{pacifici2012}. It includes physically motivated star formation and chemical enrichment histories derived from a post-treatment of the Millennium cosmological simulation \\citep{springel2005} using the semi-analytic models of \\cite{delucia2007}. This approach allows us to explore a wide range of (non-parametric) SFHs, including declining, rising, roughly constant, bursty and smooth evolutionary shapes, in addition to realistic chemical enrichment histories. As in the case of the CLSC spectral library, we first select randomly the redshift of observation and the evolutionary stage of each model galaxy. Then, following \\cite{pacifici2012}, to further widen the range of physical properties probed by this library, and we re-sample the `current' (i.e. averaged over a period of 10Myr before a galaxy is looked at) star formation and chemical properties of each model galaxy by redrawing the specific SFR (the SFR divided by the stellar mass; $\\psi_S = $~SFR~$\/M_{\\ast}$) in the range $-2<\\log(\\psi_S\/\\mathrm{Gyr}^{-1})<+2$ and the gas-phase oxygen abundance in the range $7<\\hbox{$12+\\log\\textrm{(O\/H)}$}<9.4$. For each star formation and chemical enrichment history in the original library we draw 10 different realizations of these current properties and assemble in this way a library of 4 million star formation and chemical enrichment histories. We then generate a library of 4 million model spectra by combining this set of star formation and chemical enrichment histories with the same stellar population synthesis models as used to build the CLSC spectral library in Section~\\ref{sec:basic} above. We include dust attenuation using a more sophisticated implementation of the \\cite{charlot2000} two-component dust model than in the CLSC spectral library. In this prescription, the slope of the attenuation curve is fixed in the birth clouds ($n=-1.3$), but is drawn randomly in the range $-1.1<n<-0.4$ in the diffuse ISM to reflect uncertainties about the spatial distribution of dust and the orientation of a galaxy (see \\citealt{pacifici2012,chevallard2013}). As in the case of the CLSC spectral library, we take $\\hbox{$\\hat{\\tau}_{V}$}$ to be randomly distributed between 0 and 4. In summary, the P12nEL spectral library differs from the CLSC spectral library of Section~\\ref{sec:basic} in that it includes far more realistic star formation and chemical enrichment histories and a more sophisticated prescription for attenuation by dust.\n\n\\subsubsection{The `P12 with emission lines' spectral library (P12)}\n\\label{sec:p12el}\n\nFinally, the third model spectral library we consider is the original one of \\cite{pacifici2012}. This is identical to the P12nEL library described in the previous section, except that it also includes a component of nebular emission. \\cite{pacifici2012} used the standard photoionization code {\\small CLOUDY} \\citep{ferland1996} to compute the emission lines consistently with the emission by stellar populations of different ages (neglecting the contribution by stars older than 10 Myr, which produce negligible ionizing radiation). The nebular emission in a model spectrum is parametrized in terms of the gas-phase metallicity, $Z$, the zero-age ionization parameter, $U_0$ and the dust-to-metal (mass) ratio, $\\xi_d$ (which characterizes the depletion of metals onto dust grains) of the photoionized gas (see section 2.2.2 of \\citealt{pacifici2012} for more details). These parameters are randomly drawn while re-sampling of the current chemical properties of a galaxy (Section~\\ref{sec:p12}). We build in this way a P12 library of 4 million model galaxy spectra.\n\n\\subsection{Confronting spectral models with the observations}\n\\label{sec:mockobs}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/figures\/fig_02.png}\n\\caption{Optical-NIR colour-colour diagrams comparing the two 3D-HST samples with the three model libraries. The photometric sample and the emission-line sub-sample (including only galaxies with at least one emission line detected with equivalent-width S\/N~$>5$) are represented by gray and black symbols, respectively; open circles mark objects for which the error in at least one of the two colours is larger than 0.2 magnitudes. Contours show the colour-colour space covered by the three spectral libraries: CLSC (left-hand column, orange), built using exponentially declining SFHs, fixed metallicity, simple dust attenuation and no nebular emission (Section~\\ref{sec:basic}); P12nEL (middle column, green), built using physically motivated star formation and chemical enrichment histories and a sophisticated treatment for dust attenuation (Section~\\ref{sec:p12}); P12 (right-hand column, blue), same as previous, including a component of nebular emission (Section~\\ref{sec:p12el}). In each panel, the three contours mark 50, 16 and 2 per cent of the maximum density. While the CLSC spectral library leaves few observed galaxies with no model counterpart, the P12 spectral library allows us to cover reasonably well the entire observed colour-colour space.}\n\\label{fig:col3lib}\n\\end{center}\n\\end{figure*}\n\nHence, we have built three model spectral libraries relying on different prescriptions to describe galaxy star formation and chemical enrichment histories, attenuation by dust and nebular emission. Each spectral library contains 4 million galaxy SEDs covering the rest-frame wavelength range $912\\,\\mathrm{\\AA}<\\lambda<5\\,\\mu \\mathrm{m}$.\\footnote{We do not include here reradiation by dust grains, which is expected to dominate the emission at wavelengths greater than $\\sim 5\\,\\mu$m (e.g., \\citealt{dacunha2008}).} To compare these model SEDs with the observations of galaxies in the 3D-HST photometric and emission-line samples, we convolve the observer-frame model SEDs with the response functions of the filters presented in Section~\\ref{sec:phot}. For P12, we also compute the EW of the emission lines in each model spectrum in the same way as we compute the observed EWs from the WFC3 grism spectra. In brief, we first reproduce the average resolution of the observations ($100$ {\\AA} FWHM and 22.5{\\AA} bin width) and then calculate the EWs of the {\\hbox{[O\\,{\\sc ii}]}}, {\\hbox{H$\\beta$}}, {\\hbox{[O\\,{\\sc iii}]}}, {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} emission lines (as defined in Section~\\ref{sec:spec}).\n\nIn Fig.~\\ref{fig:col3lib}, we compare the observations of both the photometric and emission-line 3D-HST samples described in Section~\\ref{sec:sample}, in two ACS and two WFC3 bands, with the predictions of the three model spectral libraries.  We plot the ACS (F435W $-$ F775W) vs WFC3 (F125W $-$ F160W) observer-frame colours in different redshift ranges (rows) for the three different spectral libraries (columns). Contours show the colour-colour space covered by the different spectral libraries. The photometric sample (grey symbols) and the emission-line sample (black symbols) are shown on top. Since the emission-line sample can be analysed only with the P12 spectral library, we plot it only in the right hand-side column. This figure shows that the CLSC spectral library leaves few observed galaxies with no model counterpart. Thus, SED fits for these galaxies will be biased towards the models that lie the closest to the observations, at the very edge of the spectral library. The P12nEL spectral library, which does not include nebular emission, can cover reasonably well the bulk of the observations at all redshifts. This shows the importance of accounting for more realistic ranges of star formation (and chemical enrichment) histories and dust properties than included in the widely used CLSC spectral library. Few observed galaxies fall outside the contours of the P12nEL model spectral library, presumably because of the contamination of the WFC3-F160W flux by strong {\\hbox{H$\\alpha$}} emission. In fact, the P12 spectral library, which includes contamination of broad-band fluxes by nebular emission, spans a much larger range of the colour-colour space than the CLSC and P12nEL libraries in Fig.~\\ref{fig:col3lib}. This spectral library allows us to cover reasonably well the entire observed colour-colour space, with the exception of some fairly red galaxies with large photometric errors (empty circles).\n\nWe note that, the emission-line 3D-HST sample in Fig.~\\ref{fig:col3lib} (right-hand panel), although including only galaxies with at least one well-detected emission line, is not biased towards the strongest starburst galaxies (characterized by blue ACS colours), but populates a similar colour-colour space as the photometric sample. This is because the selection in equivalent-width S\/N also gathers those massive galaxies with good continuum S\/N and relatively faint emission lines. The global properties of the photometric and emission-line 3D-HST samples in Fig.~\\ref{fig:col3lib} are thus quite similar.\n\n\\subsection{Spectral fits}\n\\label{sec:fit}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/figures\/fig_03.png}\n\\vspace{-0.3cm}\n\\caption{An example of spectral fit of a galaxy at redshift $z=1.261$ using the three model spectral libraries. (a) Observed broad-band magnitudes (black crosses) and best-fitting observer-frame model spectra in full resolution computed with the CLSC (orange), P12nEL (green) and full P12 (blue) spectral libraries. (b, c, d) Residuals between the observed magnitudes and the magnitudes of the best-fitting models. The error bars are roughly the size of the symbols or smaller. Probability density functions (PDFs) of (e) stellar mass, (f) SFR, (g) gas-phase oxygen abundance and (h) dust attenuation optical depth derived with the three libraries using the same colour code as above. (i, j, k) SFHs of the best-fitting models (black solid lines), likelihood-weighted average SFHs (black dashed lines) and associated confidence ranges (likelihood-weighted standard deviation, shades) from the three fits. The residuals show that all three fits are reasonably good, but the PDFs reveal large differences in the extracted parameters. The PDFs derived using the CLSC spectral library in (g) and (h) look unrealistically narrow and hit the edge of the prior (see text for details). The original P12 library yields narrower PDFs than the P12nEL library. This illustrates how accounting for the contamination of broadband fluxes by emission lines can help constrain the parameters better (see Section~\\ref{sec:resphot}). The SFHs estimated using the P12nEL (j) and P12 (k) spectral libraries rise as a function of time, in contrast to those derived using the CLSC spectral library (i).}\n\\label{fig:fitp}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{.\/figures\/fig_04.png}\n\\vspace{-0.25cm}\n\\caption{Distribution of fit residuals between the observed flux $F_\\mathrm{obs}$ and best-fitting model flux $F_\\mathrm{best-fitting}$, in units of the observational error $\\sigma_\\mathrm{obs}$, for all 1048 galaxies in the 3D-HST photometric sample. Each panel refers to a different photometric band (indicated in the top-right corner). The different histograms refer to the CLSC (yellow shaded histogram), the P12nEL (green hatched histogram) and the original P12 (blue solid histogram) model spectral libraries. We note that the residuals extend to large values because the quote photometric uncertainties on the observed fluxes are very small, of the order of 2 percent of the total flux (as discussed in Section~\\ref{sec:sample}). The three different spectral libraries provide reasonable fits to the data, although the best results (i.e. narrowest histograms most centred on zero) are obtained with the P12 library.}\n\\label{fig:resid}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{.\/figures\/fig_05a.png}\n\\includegraphics[width=0.47\\textwidth]{.\/figures\/fig_05b.png}\n\\vspace{-0.25cm}\n\\caption{Best-fitting models and parameter PDFs for an example galaxy in the 3D-HST emission-line sample fitting the photometry alone (\\textit{left-hand panel}) and fitting the photometry and the observed emission-line EWs simultaneously (\\textit{right-hand panel}) with the P12 spectral library. (a, h) Observed SED (black crosses) and best-fitting models (blue solid lines). (b, i) Observed grism spectrum (black solid lines) and best-fitting model spectra at similar resolution (blue solid lines). PDFs of stellar mass (c, j), SFR (d, k), gas-phase oxygen abundance (e, l) and optical depth of the dust (f, m). 50th (solid line), 16th and 84th (dashed lines) percentiles of the PDFs are marked on top. (g, n) SFHs of the best-fitting models (black solid lines), likelihood-weighted average SFHs (black dashed lines) and associated confidence ranges (likelihood-weighted standard deviation, blue shades) in the two cases. The photometric fits look reasonable in both top panels, but the emission-line fluxes predicted when fitting the photometry alone do not match the observed ones. When including the emission-line EWs in the fit, the best-fitting model spectrum reproduces them satisfactorily and the constraints on the physical parameters are significantly tighter.}\n\\label{fig:fit}\n\\end{center}\n\\end{figure*}\n\nWe can quantify further the appropriateness of the three spectral modelling approaches investigated here to interpret photometric and spectroscopic observations of distant galaxies by comparing the constraints derived in each case on physical parameters, such as stellar mass, SFR and dust attenuation optical depth. To do so, we use the bayesian approach described in \\cite{pacifici2012} (see also \\citealt{kauffmann2003a,gallazzi2005,dacunha2008}). In brief, we compare the observational constraints (photometry$+$spectroscopy or photometry alone) on each 3D-HST galaxy to the same observable quantities predicted for each model in a given spectral library, such that $z_{\\mathrm{model}}=z_{\\mathrm{grism}} \\pm 0.02$ (in practice, this implies that each observed galaxy is compared to about 65,000 models). We then use the likelihood of each model to build probability density functions (PDFs) of selected physical parameters for that galaxy: stellar mass ($M_{\\ast}$); SFR ($\\psi$); gas-phase oxygen abundance [$\\hbox{$12+\\log\\textrm{(O\/H)}$}$]; and attenuation optical depth of the dust ($\\hbox{$\\hat{\\tau}_{V}$}$).\n\nFor galaxies in the 3D-HST photometric sample, we fit the fluxes in all nine broad bands from 0.35 to 3.6$\\mu$m.\\footnote{Since the formal photometric errors can be extremely small (Section~\\ref{sec:phot}), we adopt a minimum photometric error of 5 per cent to enlarge the number of models contributing to each fit.} In Fig.~\\ref{fig:fitp}, we show an example of spectral fit of a galaxy at redshift $z=1.261$ using the three model spectral libraries described in Section~\\ref{sec:model}. In Fig.~\\ref{fig:fitp}a, we plot the observed magnitudes (black crosses) and the best-fitting model spectra from the CLSC (orange), P12nEL (green) and P12 (blue) spectral libraries. The residuals in Fig.~\\ref{fig:fitp}bcd show that all three fits are reasonably good, but the PDFs plotted in Fig.~\\ref{fig:fitp}efgh reveal large differences in the extracted physical parameters. Since all fits are performed in the same way, these discrepancies arise purely from differences in the model spectral libraries. Not only the median values, but also the widths of the PDFs are different in the three fits. In particular, the fit obtained using the CLSC spectral library shows narrow histograms of the gas-phase oxygen abundance (g) and the dust attenuation optical depth (h). If a narrow PDF can in some occasions reflect a good fit, this must be interpreted in the context of the number of fitted data points, the errors in the data, the distribution of the priors, the number of models in the spectral library and the number of models which actually contribute to the fit. In Fig.~\\ref{fig:fitp}, differences in the derived PDFs of $M_{\\ast}$ (e), $\\psi$ (f), $\\hbox{$12+\\log\\textrm{(O\/H)}$}$ (g) and $\\hbox{$\\hat{\\tau}_{V}$}$ (h) using different libraries can arise only from differences in the prior distributions and the number of models contributing to each fit. In particular, the narrowness of the PDFs obtained using the CLSC library arises from the fact that only few models in this library are as blue as the observed galaxy (with both ACS and WFC3 colours close to zero in Fig.~\\ref{fig:col3lib}), driving the fit to the most actively star-forming, most metal-poor and least attenuated models. The observed colours are better sampled by the other libraries, the original P12 library yielding narrower PDFs than the P12nEL library. This illustrates how accounting for the contamination of broadband fluxes by emission lines can help constrain these parameters better, as we discuss in Section~\\ref{sec:resphot}. This point was emphasized by \\cite{pacifici2012}, whose models were specifically designed to allow one to maximize the constraining power of photometric observations in the absence of direct spectroscopic information on emission lines. In Fig.~\\ref{fig:fitp}ijk, we show the SFHs of the best-fitting models (black solid lines), the likelihood-weighted average SFHs (black dashed lines) and associated confidence ranges (likelihood-weighted standard deviation, shades) derived with the three spectral libraries. We note that the SFHs derived using the P12nEL (j) and P12 (k) spectral libraries rise as a function of time, in contrast to those derived using the CLSC spectral library.\n\nTo quantify in a more global way the ability of the different spectral libraries to account for the photometric properties of 3D-HST galaxies, we plot in Fig.~\\ref{fig:resid} the distribution of the residuals between best-fitting model and observed fluxes, in each photometric band, for all 1048 galaxies in the sample. In each panel, the different histograms refer to the CLSC (yellow shaded histogram), the P12nEL (green hatched histogram) and the original P12 (blue solid histogram) model spectral libraries. Fig.~\\ref{fig:resid} shows that, overall, the three different spectral libraries provide reasonable fits to the data, although as expected, the best results (i.e. narrowest histograms most centred on zero) are obtained with the P12 library. The CLSC spectral library shows slight systematic offsets in the $U$ band (where the best-fitting flux overestimates the observed one) and in the F606W and F775W bands (where the best-fitting flux underestimates the observed one). These discrepancies imply that the corresponding models do not reproduce the rest-frame ultraviolet-optical slopes of observed galaxies as well as those in the P12 spectral library, which we attribute to the oversimplified dust attenuation prescription and lack of stochasticity in the star formation histories in the CLSC spectral library. The most extended residual distributions in Fig.~\\ref{fig:resid} pertain to the F125W band, in which the best-fitting fluxes in both the CLSC and P12nEL spectral libraries can severely underestimate the observed fluxes. This is because both models fail to reproduce the contamination of the F125W band flux by the {\\hbox{H$\\alpha$}} emission line. This happens at $z\\lesssim1.2$, thus for more than half the galaxies in the photometric sample. Remarkably, when emission lines are included in the model spectra, as is the case for the P12 spectral library, the fit residuals improve significantly. We further discuss and quantify the contamination of observed broadband fluxes by emission lines in Section~\\ref{sec:resspec}.\n\nIn the case of the 3D-HST emission-line sample, we can add to the constraints on the nine broad-band fluxes those on up to four emission-line EWs in the rest-frame optical wavelength range. Fig.~\\ref{fig:fit} shows an example of fit (performed with the P12 spectral library) of such a galaxy at $z=1.781$ with available measurements of the {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} emission lines. The left-hand panels show the results obtained when fitting the photometry alone and the right-hand panels those obtained when including also the constraints on the {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} emission lines. In Fig.~\\ref{fig:fit}ah, we show the observed SED (black crosses) together with the best-fitting models (blue solid line) and in Fig.~\\ref{fig:fit}bi, the observed grism spectrum (black solid line) together with the best-fitting model spectra at similar resolution (blue solid line). The photometric fits performed with the P12 spectral library look reasonable in both top panels, but the emission-line fluxes predicted when fitting the photometry alone do not match the observed ones (b). When including the emission-line EWs in the fit, the best-fitting model spectrum reproduces them satisfactorily (i). In Fig.~\\ref{fig:fit}, we show the constraints derived in both cases on the same physical parameters as in Fig.~\\ref{fig:fitp}, i.e. stellar mass (c, j), SFR (d, k), gas-phase oxygen abundance (e, l) and attenuation optical depth of the dust (f, m). Except for the stellar mass (which is usually well constrained by multi-band photometry alone), the constraints obtained when including information on the {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} emission lines (k, l, m) are significantly tighter than those derived from photometry alone (d, e, f). In particular for the SFR, the uncertainty decreases from 0.7 to 0.35 dex. This is expected, since the nebular emission lines arise from the obscured {\\mbox{H\\,{\\sc ii}}} regions ionized by young massive stars; thus measuring these lines allows one not only to probe current, massive star formation, but also the metallicity and dust content in those regions \\citep{charlot2001,pacifici2012}. In Fig.~\\ref{fig:fit}gn, we show the SFHs of the best-fitting models (black solid lines), the likelihood-weighted average SFHs (black dashed lines) and associated confidence ranges (likelihood-weighted standard deviation, blue shades) in the two cases. Including the emission-line equivalent widths in the fit improves the constraints on the \\textit{current} (last 10 Myr) values of physical parameters, but generally does not affect the constraints on the full SFH.\n\nIn the next Section, we apply the above fitting procedure to all galaxies in the photometric and emission-line 3D-HST samples and inter-compare the results obtained with the different spectral libraries in terms of the accuracy and uncertainty of statistical constraints on physical parameters (stellar mass, SFR and attenuation optical depth of the dust) in a more global way.\n\n\\section{Statistical constraints on galaxy physical parameters}\n\\label{sec:results}\n\nThe accuracy and uncertainty of the constraints on galaxy physical parameters derived from statistical fits of observations depend both on the type of observations considered (photometry, spectroscopy) and on the model spectral library used to interpret these. In this Section, we first explore the effects of using the three different model spectral libraries of Section~\\ref{sec:model}, which describe with different levels of sophistication the stellar and interstellar components of galaxies, to interpret the same set of observations (photometric 3D-HST sample). Then, we quantify the improvement introduced by the ability with the P12 spectral library to fit a combination of photometric and spectroscopic data (available for the 3D-HST emission-line sample) compared to fitting broad-band photometry alone. \n\n\\subsection{Fits to the photometry: classical vs realistic models}\n\\label{sec:resphot}\n\n\\begin{figure*}\n\\begin{center}\n\\vspace{-0.9cm}\n\\includegraphics[width=0.95\\textwidth]{.\/figures\/fig_06.png}\n\\caption{Comparison between constraints of physical parameters derived for all 1048 galaxies in the 3D-HST photometric sample using the CLSC (top panels, in orange) and P12nEL (bottom panels, in green) model spectral libraries to those obtained using the more sophisticated P12 library. In each box, we show the comparison between the medians of the PDFs. The shades, from dark to light, mark 75, 50, 30, 10 and 1 per cent of the maximum density. The black contours mark 50 per cent of the maximum density. In the top part of each panel, we show also the average uncertainty in three bins. Each column represents a different physical parameter: stellar mass, SFR and optical depth of the dust, from left to right. The use of simple exponentially declining SFHs (CLSC spectral library) can cause strong biases on both the stellar mass and the SFR. The uncertainties on physical parameters estimated using this library may appear to be artificially narrow, hiding the fact that the models span a too narrow prior range. Not including the emission lines in the broad-band fluxes (P12nEL spectral library) does not strongly affect the estimates of stellar mass, but can induce a slight overestimation of the SFR.}\n\\label{fig:paramp}\n\\end{center}\n\\end{figure*}\n\n\\begin{table*}\n\\begin{center}\n\\caption{16, 50 and 84 percentiles of the distributions of the differences between best estimates and uncertainties of stellar mass, SFR and optical depth of the dust when comparing constraints with different libraries as shown in Fig.~\\ref{fig:paramp}: CLSC vs P12 and P12nEL vs P12.}\n\\label{tab:diff}\n\\begin{tabular}{l c c c c c c | c c c c c c}\n\\hline\n\\hline\n& \\multicolumn{6}{c}{CLSC $-$ P12} & \\multicolumn{6}{c}{P12nEL $-$ P12} \\\\\n& \\multicolumn{3}{c}{bias} & \\multicolumn{3}{c}{$\\Delta$ uncertainty} & \\multicolumn{3}{c}{bias} & \\multicolumn{3}{c}{$\\Delta$ uncertainty} \\\\\n& 16th & 50th & 84th &  16th & 50th & 84th & 16th & 50th & 84th & 16th & 50th & 84th  \\\\\n\\hline\n$\\log(M_{\\ast}\/M_{\\sun})$ \t\t\t\t&{0.27}\t& {0.08}\t&{ -0.03}\t&{0.02}\t&{-0.02}\t& {-0.07}\t& 0.06\t& 0.00\t& -0.13\t& 0.08\t& 0.01\t& -0.02 \\\\\n$\\log[\\psi\/(M_{\\sun} \\mathrm{yr}^{-1})]$ \t&{ -0.14}\t& {-0.63}\t& {-2.23}\t&{0.03}\t&{ -0.16}\t& {-0.61}\t& {0.62}\t&{0.12}\t& -0.15\t&{0.35}\t& {0.06}\t& {-0.09} \\\\\n$\\hat{\\tau}_V$ \t\t\t\t\t\t&{ 0.28}\t& {-0.18}\t& {-1.62}\t&{0.26}\t& {-0.03}\t& {-0.41}\t& {0.42}\t& 0.07\t& {-0.12}\t& {0.17}\t& 0.01\t& {-0.16} \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nIn Fig.~\\ref{fig:paramp}, we compare the constraints on stellar mass, SFR and optical depth of the dust derived for all 1048 galaxies in the 3D-HST photometric sample using the CLSC (top panels, in orange) and P12nEL (bottom panels, in green) model spectral libraries to those obtained using the more sophisticated P12 library. In all cases, the constraints are derived from fits of the 9-band photometry at observer-frame wavelengths between 0.35 and 3.6 $\\mu$m, as described in Section~\\ref{sec:fit}.\n\n\\paragraph*{Stellar mass.} The left-hand panels of Fig.~\\ref{fig:paramp} show the differences in the constraints derived on galaxy stellar mass. For each set of spectral libraries, each panel shows the comparison between the median likelihood estimates of stellar mass (colour-coded according to the number of galaxies falling into each bin of the diagram). The results show that stellar-mass estimates derived using the CLSC spectral library are systematically $\\sim0.08$ dex greater than those derived using the P12 library. This is because in the CLSC spectral library, simple exponentially declining SFHs tend to produce larger mass-to-light ratios and smaller specific SFRs than SFHs including late bursts of star formation. Such a trend can cause a bias in the mass estimate towards large values (although this depends on the allowed age range of the models; see below). Fig.~\\ref{fig:paramp} also does not reveal any systematic bias in the stellar masses obtained using the P12nEL rather than the original P12 spectral library, only a large scatter. This suggests that stellar-mass estimates are not strongly affected by the contamination of broad-band fluxes by emission lines, at least for masses $M_{\\ast}>10^9~\\hbox{$M_\\odot$}$.\n\n\\paragraph*{SFR.} The middle panels of Fig.~\\ref{fig:paramp} show the analog to the left-hand panels for the SFR. In this case, the constraints derived using the CLSC spectral library produce SFR estimates significantly smaller (by $\\sim 0.63$ dex) than those derived from the more sophisticated P12 library, again because the SFH prior favours low specific-SFR values. When comparing the constraints derived from the P12nEL and P12 spectral libraries, we see that neglecting nebular emission causes a slight bias towards larger SFRs ($\\sim 0.12$ dex). This is likely because the total (stellar$+$nebular) observed emission is interpreted as a larger amount of young stars when emission lines are not included in the broad band fluxes predicted by the models. The uncertainties are also slightly larger when the emission lines are not included in the models.\n\n\\paragraph*{Dust optical depth.} The right-hand panels of Fig.~\\ref{fig:paramp} pertain to the attenuation optical depth of the dust. In the CLSC spectral library, dust attenuation is computed using a two-component model with a fixed slope of the attenuation curve ($n=-0.7$) both in stellar birth clouds and in the diffuse ISM (as recalled in Section~\\ref{sec:basic}, the galaxy-wide attenuation curve then depends on the SFH). The P12nEL and P12 spectral libraries rely on the same two-component model, but with a steeper slope of the attenuation curve in stellar birth clouds ($n=-1.3$) and a random slope of the attenuation curve in the diffuse ISM ($n$ between $-0.4$ and $-1.1$). On the one hand, this difference would cause a bias in the estimate of the optical depth of the dust, giving a larger value when applying a shallow attenuation curve. On the other hand, the large mass-to-light ratios and small SFRs favoured by the CLSC spectral library tend to produce redder intrinsic SEDs and favour low dust attenuation. These two effects conspire to produce a large scatter ($\\pm$ 1 dex), and an offset ($\\sim -$0.18 dex) between the results derived using the CLSC and P12 spectral libraries in the upper right panel of Fig.~\\ref{fig:paramp}. The lower-right panel further shows that neglecting nebular emission in the P12 spectral library introduces a slight bias upward ($\\sim$ 0.07) in the derived attenuation optical depth with a large scatter, showing the strong degeneracy of this parameter with age, metallicity and SFR. We note that, for some dusty galaxies ($\\hbox{$\\hat{\\tau}_{V}$}>2.5$), the measurements of $\\hbox{$\\hat{\\tau}_{V}$}$ derived with the P12nEL and P12 spectral libraries are in good agreement. This is because nebular emission lines are strongly attenuated at high dust optical depths and therefore have a negligible influence on the broad-band fluxes. As a consequence, in the lower-right panel of Fig.~\\ref{fig:paramp}, the density along the identity relation is high when $\\hbox{$\\hat{\\tau}_{V}$}$ is large.\n\n\\paragraph*{} We conclude from the above analysis that the widely used, CLSC spectral library is not appropriate to extract reliable stellar masses and SFRs from photometric observations of distant galaxies such as those in the 3D-HST survey, for two main reasons: firstly, because the limited parameter range probed by this library cannot account for all observations (as discussed in Sections~\\ref{sec:mockobs} and \\ref{sec:fit}); and secondly, because this discrepancy implies important systematic biases in derived galaxy physical parameters, even if the resulting PDFs can appear artificially narrow (see Section~\\ref{sec:fit}). We note that the results presented here for the CLSC spectral library are consistent with the claim by \\citet[see also \\citealt{pforr2012}]{maraston2010} that models with exponentially declining SFHs overestimate the stellar mass and underestimate the SFR if galaxies are assumed to be at least $\\sim1\\,$Gyr old (see their section~3.2). The constraints on stellar mass, SFR and optical depth of the dust derived from 9-band photometry using the P12nEL and P12 spectral libraries are roughly similar, except for a slight bias in SFR toward large values when nebular emission is neglected. For reference, we list in Table~\\ref{tab:diff} the 16th, 50th and 84th percentiles of the distributions of differences between the median-likelihood estimates of stellar mass, SFR and optical depth of the dust (and associated uncertainties) derived using the CLSC and P12nEL spectral libraries with respect to those derived using the more sophisticated P12 library.\n\n\\subsection{Including emission-line constraints in the spectral fits}\n\\label{sec:resspec}\n\n\\begin{figure*}\n\\begin{center}\n\\vspace{-0.9cm}\n\\includegraphics[width=0.95\\textwidth]{.\/figures\/fig_07.png}\n\\vspace{-0.25cm}\n\\caption{Comparison between the constraints derived on stellar mass, SFR and optical depth of the dust for all 364 galaxies in the 3D-HST emission-line sample when fitting the photometry alone and when fitting both the photometry and emission-line EWs, using the P12 model spectral library. The format is the same as in Fig.~\\ref{fig:paramp}. Including the emission-line EWs in the fit does not affect strongly the best estimates, but improves considerably the uncertainties on all physical parameters.}\n\\label{fig:params}\n\\end{center}\n\\end{figure*}\n\n\\begin{table*}\n\\begin{center}\n\\caption{16, 50 and 84 percentiles of the distributions of the differences between best estimates and uncertainties of the stellar mass, SFR and attenuation optical depth of the dust as shown in Fig.~\\ref{fig:params}, comparing simultaneous fits to the photometry and emission-line EWs vs. fits to the photometry alone (using the P12 model spectral library in both cases).}\n\\label{tab:diffew}\n\\begin{tabular}{l c c c c c c}\n\\hline\n\\hline\n&  \\multicolumn{6}{c}{P12 (phot) $-$ P12 (phot $+$ EW)}\\\\\n&  \\multicolumn{3}{c}{bias} & \\multicolumn{3}{c}{$\\Delta$ uncertainty}\\\\\n&16th & 50th & 84th & 16th & 50th & 84th  \\\\\n\\hline\n$\\log(M_{\\ast}\/M_{\\sun})$ \t\t\t\t& 0.09\t& 0.01\t& -0.07\t& 0.07\t& 0.03\t& -0.01\\\\\n$\\log[\\psi\/(M_{\\sun} \\mathrm{yr}^{-1})]$ \t& 0.50\t& {0.13}\t& {-0.21}\t& {0.48}\t& {0.14}\t& 0.02\\\\\n$\\hat{\\tau}_V$ \t\t\t\t\t\t& {0.45}\t& {0.03}\t& {-0.23}\t& {0.35}\t& {0.06}\t& {-0.09}\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.475\\textwidth]{.\/figures\/fig_08.png}\n\\vspace{-0.5cm}\n\\caption{(a): comparison between the rest-frame emission EW of {\\hbox{H$\\alpha$}}  measured from grism spectroscopy and the EW predicted from fits of the 9-band photometry using the P12 model spectral library, for the 314 galaxies with \\hbox{H$\\alpha$}\\ measurements in the 3D-HST emission-line sample. For reference, the black solid line indicates the identity relation and the black dashed lines mark deviations by a factor of 2 ($\\pm0.3\\,$dex) between the measured and predicted EWs. (b): distribution of the difference between predicted and measured \\hbox{H$\\alpha$}\\ rest-frame EW. (c) and (d):  same as (a) and (b), but for the EW of \\hbox{[O\\,{\\sc iii}]}\\ for the 99 galaxies with \\hbox{[O\\,{\\sc iii}]}\\ measurements in the 3D-HST emission-line sample.}\n\\label{fig:ewmvsbfit}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.475\\textwidth]{.\/figures\/fig_09a.png}\n\\includegraphics[width=0.475\\textwidth]{.\/figures\/fig_09b.png}\n\\vspace{-0.5cm}\n\\caption{Contribution by nebular emission to the WFC3-F140W magnitude of the best-fitting (P12) model for the galaxies in the 3D-HST emission-line sample as a function of the observer-frame line EWs directly measured in the grism spectra (left-hand panel) and as a function of stellar mass (right-hand panel). Crosses represent galaxies with measured {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} (235 galaxies at $0.8<z<1.4$), while circles represent galaxies with measured {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} (47 galaxies at $1.5<z<2.2$). The points are colour-coded according to star formation activity from low (red) to high (black) specific SFR. In the left-hand panel, the grey lines show the same relation considering the best-fitting model EWs instead of the observed ones. In the right-hand panel, grey symbols represent median and 16-to-84 percentiles of the contamination in bins ($\\approx 0.5$ dex wide) of stellar mass for {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} (squares) and for {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} (filled circles). The contamination of the emission lines in the broad-band WFC3-F140W filter increases from $\\approx 0.01$ to $\\approx 0.5\\,$mag as the stellar mass decreases.}\n\\label{fig:ewband}\n\\end{center}\n\\end{figure*}\n\nThe 3D-HST emission-line sample includes only those galaxies from the photometric sample with well-detected (S\/N~$>5$) emission lines, which allows us to test the effect of adding emission-line information when deriving constraints on physical parameters. In Fig.~\\ref{fig:params}, we compare (in a similar way to Fig.~\\ref{fig:paramp}) the constraints derived on stellar mass, SFR and optical depth of the dust for all 364 galaxies in this sample when fitting the photometry alone and when fitting both the photometry and emission-line EWs, using the P12 model spectral library. For the stellar mass (left-hand panel), there is no significant difference between the constraints derived in both ways and the uncertainties remain similar. The SFR (middle panel) appears to be slightly overestimated when the constraints on the emission-line EWs are not included in the fit (by $\\sim0.13$~dex). This is because emission-line EWs help break the degeneracy between star formation activity (i.e. specific SFR) and attenuation by dust (right-hand panel in Fig.~\\ref{fig:params}) at fixed SED shape. The fact that the SFR is biased high when fitting only the photometry can be considered as a consequence of the selection criteria of the emission line sample, because, for an emission line to be well detected in a low-resolution 3D-HST spectrum, attenuation by dust must be low. In the absence of emission-line information, $\\hbox{$\\hat{\\tau}_{V}$}$ is less well constrained and the corresponding PDF broadens to larger values (implying larger SFR at fixed observed colours). The introduction of line-EW constraints breaks the SFR-dust degeneracy in these lightly obscured galaxies therefore leads on average to slightly lower SFR estimates. Adding constraints on emission-line EWs reduces the uncertainties in both SFR and $\\hbox{$\\hat{\\tau}_{V}$}$ by $\\sim0.14$\\,dex and 0.06, respectively (Table~\\ref{tab:diffew}).\n\nIt is important to note that, for galaxies in the 3D-HST emission-line sample, the emission-line strengths predicted by the P12 model providing the best fit to 9-band photometry alone are in fair agreement with direct EW measurements from grism spectroscopy. This is illustrated by Fig.~\\ref{fig:ewmvsbfit}, in which we compare the rest-frame EWs of {\\hbox{H$\\alpha$}} (Fig.~\\ref{fig:ewmvsbfit}a) and {\\hbox{[O\\,{\\sc iii}]}} (Fig.~\\ref{fig:ewmvsbfit}c) emission lines as measured from the grism spectra with the predictions obtained when fitting the photometry alone. The distributions of the differences between predicted and measured EWs are plotted in Figs~\\ref{fig:ewmvsbfit}b and \\ref{fig:ewmvsbfit}d for {\\hbox{H$\\alpha$}} and {\\hbox{[O\\,{\\sc iii}]}}, respectively. Fig.~\\ref{fig:ewmvsbfit} shows that, for 52 per cent (20 per cent) of the galaxies with spectroscopic {\\hbox{H$\\alpha$}} ({\\hbox{[O\\,{\\sc iii}]}}) emission-line measurements, the EWs inferred from fits of 9-band photometry agree to within a factor of 2 with those measurements. Considering only galaxies with EW({\\hbox{H$\\alpha$}}) larger than 100\\,{\\AA} makes this fraction rise to 73 per cent. This is not surprising, as the P12 model spectral library accounts for contamination of broad-band fluxes by nebular emission, which increases and hence is more easily identifiable in the most actively star-forming galaxies. We note that, despite this good agreement, 9-band photometric fits tend to systematically overestimate EW(\\hbox{H$\\alpha$}) in Fig.~\\ref{fig:ewmvsbfit}a relative to direct measurements. This is again a consequence of the SFR-dust degeneracy described in the previous paragraph. There is no systematic bias in the predicted EW(\\hbox{[O\\,{\\sc iii}]}) but a larger scatter, because the strength of this line is strongly affected by the metallicity of the gas and is not easily constrained by purely photometric fits.\n\nTo further quantify how emission lines contaminate observed broad-band fluxes, we record, for each galaxy in the 3D-HST emission-line sample, the contribution by nebular emission to the WFC3-F140W magnitude of the best-fitting P12 model (as derived when including the constraints from both 9-band photometry and EW measurements). This is shown in Fig.~\\ref{fig:ewband} (left-hand panel) against the line EWs directly measured in the grism spectra. For galaxies at redshifts $0.8 < z < 1.4$, both {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} fall in the WFC3-F140W filter. The combined observer-frame EW of these lines is plotted in Fig.~\\ref{fig:ewband} (crosses). In the same way, we plot the combined observer-frame EW of {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} for galaxies in the range $1.5 < z < 2.2$ (empty circles). Each galaxy is colour-coded according to star formation activity, from low (red) to high (black) specific SFR. As expected, the contamination of broad-band fluxes by emission lines increases with the level of star formation activity. The bulk of the sample shows observed emission-line EWs between 200 and 700~\\AA, which corresponds roughly to a contamination of $0.1\\,$magnitudes in the F140W broad-band magnitude. We also show for reference the relation obtained when considering the EWs of the best-fitting model instead of the observed ones (grey lines). In the right-hand panel of Fig.~\\ref{fig:ewband}, we plot the emission-line contamination of the WFC3-F140W broad-band magnitude as a function of stellar mass. For galaxies at $0.8 < z < 1.4$, where {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} are both sampled in the band, the contamination decreases from $\\approx 0.1$ to $\\approx 0.02\\,$magnitudes as the stellar mass increases from $\\approx 10^{9.5}$ to $\\approx 10^{11}$  M$_{\\odot}$.  At higher redshift, where {\\hbox{[O\\,{\\sc iii}]}} and {\\hbox{H$\\beta$}} are sampled in the band, the contamination is slightly larger because the SFR is on average larger at higher than at lower redshifts \\citep{noeske2007}.\n\n\\section{Implications for the scatter in the star-formation main sequence}\n\\label{sec:ms}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/figures\/fig_10.png}\n\\vspace{-0.5cm}\n\\caption{Constraints on stellar mass and SFR (star-formation main sequence) derived in Section~\\ref{sec:resphot} from fits of the 9-band 3D-HST photometry using the CLSC (left-hand panels), P12nEL (middle panels) and P12 (right-hand panels) model spectral libraries, in different redshift ranges (rows). In each panel, we show the median and 16-to-84 percentile range of the SFR in bins of stellar mass (coloured squares) and linear least-squares fit to all points (black solid line; red solid line in the right-hand panels). For each redshift bin, we report in the 2 left-most panels the fit obtained in the right-hand panel using the P12 spectral library (red dashed line). The use of the CLSC spectral library, which relies on oversimplified prescriptions of SFHs and dust attenuation, cause the main sequence to lie, at all redshifts, significantly below the relation obtained using the more realistic P12 spectral library. The relative zero-point of the main sequence derived using the P12nEL spectral library is shifted towards higher SFRs compared to that obtained using the original P12 spectral library. The scatter about the derived star formation main sequence is reduced including the emission lines as contaminants of the broad-band fluxes. In the right-hand panel, we also report for comparisons the fits to the main-sequence by \\protect\\citet[black long-dashed line]{karim2011} and \\protect\\citet[black dot-dashed line]{whitaker2012} in the mass ranges allowed by their samples.}\n\\label{fig:msall}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/figures\/fig_11.png}\n\\vspace{-0.5cm}\n\\caption{The main sequence derived from the 3D-HST emission-line sample when fitting the photometry alone (left-hand panels; analog of the right-hand panels of Fig.~\\ref{fig:msall}) and when conbining the photometry with the emission-line EWs detected in the grism spectra (right-hand panels). The format is the same as in Fig.~\\ref{fig:msall}. Estimates are extracted using the P12 spectral library. For this sample, the estimates of SFR are in general less uncertain than for the photometric sample given the strength of the emission lines. This contributes to reducing the scatter. Including the information from the emission-line EWs further reduces the scatter at all redshifts.}\n\\label{fig:ms}\n\\end{center}\n\\end{figure*}\n\nWe have shown in the previous sections that interpreting the same 3D-HST observations of distant galaxies using model spectral libraries of different levels of sophistication can lead to different constraints on galaxy stellar masses and SFRs. We also showed that accounting for nebular emission is important to interpret (even purely photometric) observations of distant star-forming galaxies. Here we explore how the biases introduced by the use of oversimplified model spectral libraries might affect the interpretation of the correlation between stellar mass and SFR observed for star-forming galaxies, commonly referred to as the star-formation main sequence. It is now established that this `main sequence' is in place both in the local Universe and at high redshift (up to at least $z\\approx3$; \\citealt{brinchmann2004,noeske2007,elbaz2007,daddi2007,karim2011,reddy2012,whitaker2012}), but the normalization, slope and scatter of the relation and the dependence of these on redshift, are still under debate. The normalization of the main sequence appears to increase with time (from redshift 0 to roughly 3) in such a way that, at fixed stellar mass, galaxies at higher redshift form stars at higher rates. The slope also appears to change with redshift, as shown by \\cite{karim2011} and \\cite{whitaker2012}. The intrinsic scatter should contain important information about the physical processes that drive the formation of galaxies \\citep{dutton2010}, but it is always contaminated by observational uncertainties and thus is hard to quantify \\citep{guo2013}.\n\nIn this Section, we assess how the use of different model spectral libraries and the availability or not of emission-line measurements can affect the determinations of the shape and scatter of the main sequence at $0.7<z<1.8.$\\footnote{Because of the lack of statistics, we limit the analysis of the star-formation main sequence to $z=1.8$. We note also that the samples we use reach a magnitude of $H=23$ (WFC3-F160W) and thus are complete only to a stellar mass of $\\approx10^{10}\\hbox{$M_\\odot$}$ at $z=1$ ($\\approx10^{10.7}\\hbox{$M_\\odot$}$ at $z=2$). Such completeness limits do not allow us to draw reliable conclusions about the absolute slope or the zero point of the main sequence. It is thus beyond the scope of this paper to assess the evolution of the main sequence in redshift.} We use the estimates of stellar mass and SFR derived in Section~\\ref{sec:results} to draw the star-formation main sequence in four redshift bins for both the 3D-HST photometric and the emission-line samples.\\footnote{We include only galaxies with specific SFR larger than 0.01 Gyr$^{-1}$.} Fig.~\\ref{fig:msall} shows the results obtained when adopting the constraints on stellar mass and SFR derived in Section~\\ref{sec:resphot} from fits of the 9-band 3D-HST photometry using the CLSC (left-hand panels), P12nEL (middle panels) and P12 (right-hand panels) model spectral libraries. In each panel, on top of the single points pertaining to individual galaxies, we plot the median and 16-to-84 percentile range of the SFR in bins of stellar mass (coloured squares) and a linear least-squares fit to all points (black solid line in the 2 left-most panels; red solid line in the right-hand panel). For the purpose of comparison, for each redshift bin, we report in the 2 left-most panels the fit obtained in the right-hand panel using the P12 spectral library (as a red dashed line). Fig.~\\ref{fig:msall} confirms our finding in Section~\\ref{sec:resphot} above that the use of the CLSC spectral library, which relies on oversimplified prescriptions of SFHs and dust attenuation, can severely bias estimates of galaxy stellar masses and SFRs from photometric data. Such biases cause the main sequence to lie, at all redshifts, significantly below the relation obtained using the more realistic P12 spectral library. The primary cause for this can be traced to the prior distribution of the specific SFR, which in the CLSC spectral library is relatively narrow with a peak around $\\log(\\psi_{S}\/\\mathrm{Gyr}^{-1})\\sim -0.75$.\n\nThe middle panels of Fig.~\\ref{fig:msall} show the results obtained when including more sophisticated treatments of star formation and chemical enrichment histories and attenuation by dust, but without accounting for nebular emission (P12nEL spectral library). This produces a stronger correlation between stellar mass and SFR than obtained with the CLSC library, but with a very large scatter spanning almost the entire range of the prior distribution of the specific SFR ($-2<\\log(\\psi_{S}\/\\mathrm{Gyr}^{-1})<2$). This is caused by the strong degeneracy between SFR and dust attenuation, which affects the uncertainties in SFR estimates at all redshifts (Section~\\ref{sec:resphot}). We also find that the relative zero-point of the main sequence derived using the P12nEL spectral library is shifted towards higher SFRs compared to that obtained using the original P12 spectral library. This is in agreement with the result of Fig.~\\ref{fig:paramp} that the SFR is slightly overestimated when the models do not account for the contamination of broadband fluxes by emission lines.\n\nAppealing to the original P12 library to include the contamination of broadband fluxes by nebular emission reduces the scatter about the derived star formation main sequence ($\\approx0.7$~dex), especially at those redshifts where the {\\hbox{H$\\alpha$}} line falls in one of the observer-frame near-infrared bands ($0.7<z<1.5$); at $z>1.5$, the scatter remains very large ($\\approx1.2$~dex) because the parameters are not as tightly constrained.\n\nThe star-formation main sequence has been studied in the past using different approaches tailored to specific redshift ranges and datasets. For comparison, we plot in the right-hand panels of Fig.~\\ref{fig:msall} the main-sequence fits derived by \\citet[black long-dashed line]{karim2011} and \\citet [black dot-dashed line]{whitaker2012} over a redshift range similar to that sampled by the 3D-HST survey. In both studies, stellar masses are derived using a classical approach (exponentially declining $\\tau$-models, possibly allowing for very young ages to widen the specific-SFR and mass-to-light-ratio priors), while SFRs are derived in different ways: \\cite{karim2011} estimate galaxy SFRs from stacked 1.4 GHz data using the prescription by \\cite{bell2003}; and \\cite{whitaker2012} from ultraviolet and infrared data using the prescription by \\cite{kennicutt1998}. The main sequences obtained in these two studies are both in good agreement with that derived from 3D-HST data using the P12 library in Fig.~\\ref{fig:msall}. An advantage of the approach presented here is the ability to derive, for each galaxy, simultaneous constraints on the stellar mass and SFR from a library of star formation and chemical enrichment histories, which allows one to also constrain other physical parameters (age, dust attenuation, metallicity).\n\nWe can now turn to the 3D-HST emission-line sample and assess whether the addition of spectroscopic emission-line measurements can improve determinations of the star formation main sequence with the P12 model spectral library. Fig.~\\ref{fig:ms} shows the main sequences derived for this sample in the same redshift bins as for the photometric sample in Fig.~\\ref{fig:msall}, both from fits of the photometry alone (left-hand panels; this is the analog of the right-hand panels of Fig.~\\ref{fig:msall}, but for the sub-sample of 3D-HST galaxies with emission-line measurements) and from combined fits of the photometry and emission-line EWs (right-hand panels). Even when fitting the photometry alone, the scatter in Fig.~\\ref{fig:ms} is, to some extent, reduced ($\\approx 0.5$~dex) compared to that in the right-hand panels of Fig.~\\ref{fig:msall}. This is because the average uncertainty in the SFR derived from 9-band photometry  in the emission-line sample (which includes only galaxies with well-detected emission lines) is slightly smaller than that for the entire photometric sample and so is the scatter in the inferred relation between stellar mass and SFR.  When we add the constraints on the emission-line EWs, the uncertainty on the SFR is further reduced (Section~\\ref{sec:resspec}) and the main sequence becomes tighter at all redshifts (with a scatter $\\approx 0.4$ dex; right-hand panels in Fig.~\\ref{fig:ms}). The slight bias towards lower SFRs at large stellar masses when including the emission-line EWs in the fits is associated to the selection of the sample as discussed in Section~\\ref{sec:resspec}.\n\n\\section{Summary and conclusions}\n\\label{sec:summary}\n\nInterpreting ultraviolet-to-infrared observations of distant galaxies in terms of constraints on physical parameters -- such as stellar mass, SFR and attenuation by dust -- requires spectral synthesis modelling. In this paper, we have investigated how increasing the level of sophistication of standard simplifying assumptions of such models can improve estimates of galaxy physical parameters. To achieve this, we have compiled a sample of 1048 galaxies at $0.7<z<2.8$ with accurate photometry at rest-frame ultraviolet to near-infrared wavelengths from the 3D-HST survey, 364 of which have strong enough optical-line emission to be measured with good S\/N from grism spectroscopy. Such spectroscopic information is important to allow us to quantify, from a pure observational point of view, the contamination of broadband photometric fluxes by emission lines. We have compared the SEDs of the galaxies in this sample with those from different model spectral libraries (of different levels of sophistication) to derive bayesian estimates of stellar mass, star formation rate and optical depth of the dust. We have found the following.\n\\begin{enumerate}\n\\item Classical spectral libraries, in which galaxy SFHs are assumed to be exponentially declining functions of time ($\\tau$-models), cannot reproduce the observed SEDs of all galaxies in the sample, because such libraries do not account for the colours of galaxies with old populations undergoing a new starburst. \\cite{maraston2010} and \\cite{pforr2012} introduced exponentially {\\em rising} SFHs to overcome this limitation, but such parametrization cannot reproduce the potential rise and fall of galaxy SFHs. The more realistic SFHs implemented in the P12 library \\citep{pacifici2012}, which rely on the versatile post-treatment of cosmological simulations, allow us to adequately reproduce the observed colours of all galaxies in the 3D-HST sample.\n\\item As a result, stellar masses derived using classical spectral libraries (generally associated with an oversimplified prescription for dust attenuation and the neglect of nebular emission) tend to be systematically overestimated (by a median $\\sim 0.1$ dex), and SFRs systematically underestimated (by a median $\\sim 0.6\\,$dex), relative to the values derived adopting the more realistic P12 spectral library (which includes a sophisticated and versatile prescription for dust attenuation as well as a careful account for nebular emission). Moreover, the uncertainties derived using classical spectral libraries can be misleadingly small because of the inappropriate sampling of the physical parameter space (Section~\\ref{sec:fit}).\n\\item Neglecting the contamination of broadband fluxes by emission lines, while using the same SFHs and dust prescription as in the P12 spectral library, would lead to systematic SFR overestimates (by a median $\\sim 0.1\\,$dex), leaving stellar-mass estimates nearly unaffected. We note that galaxies in the 3D-HST sample have masses $M \\ga10^9\\hbox{$M_\\odot$}$. In lower-mass galaxies with high specific SFR, we expect SFR and stellar-mass estimates obtained when neglecting nebular emission to be more severely biased than found here \\citep[e.g.][]{atek2011,stark2013}.\n\\item The simultaneous fit of photometric broad-band fluxes and emission-line EWs, as enabled by the \\cite{pacifici2012} approach, helps break a fundamental degeneracy between SFR and dust attenuation, which reduces considerably the uncertainties in these derived parameters (by $\\sim0.1$--0.5\\,dex; the uncertainty in stellar-mass estimates dropping by only $\\sim0.05\\,$dex).\n\\end{enumerate}\n\nThe results obtained in this paper reveal the importance of choosing appropriate spectral models to interpret deep galaxy observations. In particular, the biases introduced by the use of classical spectral libraries to derive estimates of SFR, stellar mass and attenuation by dust in distant galaxies can significantly affect the interpretation of standard diagnostics diagrams of galaxy evolution. These include, for example, the galaxy stellar-mass function \\citep[see also][]{marchesini2009,muzzin2013}; the main sequence of star-forming galaxies and its intrinsic scatter; and the relations between stellar mass, SFR and other galaxy physical parameters, such as stellar age and metallicity \\citep[e.g.][]{mannucci2010}. In this context, the spectral library developed by \\cite{pacifici2012} offers the possibility to interpret these and other fundamental diagnostics on the basis of more realistic, and at the same time more versatile models. This is all the more valuable in that the approach can be straightforwardly tailored to the analysis of any combination of photometric and spectroscopic observations of galaxies at any redshift.\n\n\n\\section*{acknowledgements}\n\nWe thank the Referee for the very useful report that helped improving the Paper. This work was funded in part by the Marie Curie Initial Training Network ELIXIR of the European Commission under contract PITN-GA-2008-214227 and in part by the KASI-Yonsei Joint Research Program for the Frontiers of Astronomy and Space Science funded by the Korea Astronomy and Space Science Institute. E.d.C. acknowledges funding through the ERC grant `Cosmic Dawn'. S.C. acknowledges support from the European Research Council via an Advanced Grant under grant agreement no. 321323-NEOGAL. S.K.Y was supported by the National Research Foundation of Korea (Doyak 2014003730).\n\n\n\\defAJ{AJ}\n\\defARA\\&A{ARA\\&A}\n\\defApJ{ApJ}\n\\defApJ{ApJ}\n\\defApJS{ApJS}\n\\defAp\\&SS{Ap\\&SS}\n\\defA\\&A{A\\&A}\n\\defA\\&A~Rev.{A\\&A~Rev.}\n\\defA\\&AS{A\\&AS}\n\\defMNRAS{MNRAS}\n\\defPASP{PASP}\n\\defPASJ{PASJ}\n\\defQJRAS{QJRAS}\n\\defNature{Nature}\n\n\\defAstrophys.~Lett.{Astrophys.~Lett.}\n\\defAAS{AAS}\n\\let\\astap=A\\&A\n\\let\\apjlett=ApJ\n\\let\\apjsupp=ApJS\n\\let\\applopt=\\ao\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nAdiabatic quantum computing~\\cite{farhiQuantum00} and quantum annealing~\\cite{kadowakiQuantum98}\ncan be used to prepare ground states and thermal states of quantum\nsystems, a central desideratum of quantum computation. The quality\nof ground state preparation and Gibbs sampling is central to the performance\nof quantum algorithms for optimization as well as machine learning\nproblems~\\cite{Amin:2016}. The performance of these computational\nparadigms is quantified by the adiabatic theorem in its different\nforms, whether for closed or open quantum systems. The theorem, or\nfamily of theorems, places a bound on the adiabatic error: the distance\nbetween the state that we set out to prepare (usually the solution\nto a computational problem) and the state the system actually ends up in through\nthe dynamical evolution. In general, if a spectral gap condition is\nsatisfied, the adiabatic error is bounded by $C\/t_{f}$ for some constant\n$C$, where $t_{f}$ is the total evolution time~\\cite{katoAdiabatic50,jansenBounds07,joye_general_2007}.\nThe boundary cancellation theorem (BCT) shows that an improvement\nis possible if the schedule from the initial to the final generator\n(Hamiltonian or Liouvillian) has certain properties. In the closed\nsystem case the adiabatic error is bounded by $C_{k}\/t_{f}^{k+1}$\nfor some constant $C_{k}$ independent of $t_f$, if the time-dependent Hamiltonian $H(t)$\nhas vanishing derivatives up to order $k$ at the \\emph{beginning\nand end} of the evolution~\\cite{garridoDegree62}, a bound that\ncan be improved to exponentially small in $t_{f}$ under additional\nassumptions~\\cite{nenciu_linear_1993,Hagedorn:2002kx,lidarAdiabatic09,wiebeImproved12,Ge:2015wo}.\nThis theorem has recently been extended to the open system setting\nfor a general class of time-dependent Liouville operators $\\mathcal{L}(t)$,\nwhere it can be used to prepare steady states of Liouvillians instead\nof ground states of Hamiltonians. The theorem can be succinctly stated\nas follows: For a time-dependent Liouvillian $\\mathcal{L}(t)$ with\na unique steady state $\\sigma(t)$ separated by a gap at each time\n$t$, the adiabatic error is likewise bounded by $C_{k}\/t_{f}^{k+1}$,\nbut it is sufficient for the derivatives to vanish up to order $k$\n\\emph{only at the end time} $t_{f}$~\\cite{camposvenutiError18}.\nBoundary cancellation (BC) plays a significant role in the theoretical\nanalysis of error suppression in Hamiltonian quantum computing~\\cite{albashDecoherence15,Lidar:2019ab}.\n\nIn this work we set out to demonstrate the BCT predictions in experiments\nusing the D-Wave 2000Q (DW2KQ) quantum annealer. The DW2KQ implements the transverse field Ising model $H(t) = A(t)H_X +B(t)H_Z$, where $H_X$ and $H_Z$ are transverse and longitudinal (Ising) Hamiltonians, respectively~\\cite{johnsonQuantum11,DW2KQ}.\nIt allows\nlimited programmability of the control schedules $A(t)$ and $B(t)$\nof the system Hamiltonian, which we exploit to implement boundary cancellation\nprotocols. \n\nTo guide our intuition we model the behavior of the D-Wave\nquantum annealer with the adiabatic master equation (AME) derived in \\cite{albashQuantum12}.\nThis is a time-dependent Davies-like master equation~\\cite{davies_markovian_1974} which has been successfully used to interpret several D-Wave experiments, e.g.,~\\cite{albashConsistency15,albashReexamination15} (though not always~\\cite{bando2021breakdown}).\nWhen combining the AME for dephasing with boundary cancellation, we encounter a problem. Namely, as we explain in detail below, the Liouvillian gap vanishes at $t=t_f$, which prevents us  from directly applying the BCT in the form given in Ref.~\\cite{camposvenutiError18}. \n\nTo circumvent this problem, in this work we generalize the BCT and identify the conditions on the Liouvillian gap under which BC does or does not remain effective. We find that the generalized BCT\nplays an important role in the D-Wave implementation.\n\n\nThere is another \nsignificant consideration regarding\nthe implementation of BC on D-Wave:\nthe phenomenon of freezing. In essence, freezing refers to\na significant increase in all relaxation timescales well before the\nend of the anneal ($t<t_{f}$), in such a manner that the system is\nfrozen in a state that does not correspond to the steady state of\n$\\mathcal{L}(t_{f})$~\\cite{albashQuantum12}. However, the frozen state is not truly static but quasi-static~\\cite{Amin:2015qf}, i.e., relaxation is not fully switched off but instead proceeds on a timescale that may be much longer than practically realizable anneal times $t_f$. Such glassy-like behavior can take place even in very simple systems that do not have a glassy landscape. We give an explanation of freezing in Sec.~\\ref{sec:freezing} below, \nbased on the AME. Essentially, it arises due to the fact that the system Hamiltonian commutes with the system-bath\ncoupling when $A(s)=0$, and the fact that the schedule $A(t)$  has nearly vanished long before $t=t_{f}$\n(see Fig.~\\ref{fig:dw2kq_schedule}). This is true in particular \nof currently available commercial quantum annealers manufactured by D-Wave~\\cite{boothby2020nextgeneration}, which can be effectively\ndescribed by longitudinal field coupling to a bosonic bath, which\ncommutes with the system Hamiltonian at sufficiently low transverse\nfields. The state of the system is insensitive\nto the details of the schedules past the freezing point, so that BC becomes ineffective. This freezing mechanism plays an adverse role in demonstrating the predictions of the BCT, since the experimentally accessible anneal times are significantly shorter than the relaxation time required to reach the true steady state of $\\mathcal{L}(t_f)$.\n\nIn order to overcome both issues, i.e., the gaplessness of the Liouvillian at\n$t=t_{f}$ and freezing at $t\\approx0.5t_{f}$, we\ndesign the programmable DW2KQ schedule so that it flattens (and thus implements BC) at a point $t_{BC}$ before freezing, followed by a ramp to the final values of $B(t_{f})$\nand $A(t_{f})$, upon which the system is measured in computational basis (eigenbasis of $H_Z$).  \n\nWe apply the protocol to two 8-qubit gadgets: 1) a ``ferromagnetic chain gadget''\n(FM-gadget), and 2) a ``tunneling gadget'' (T-gadget) with a larger\ntunneling barrier between the ground state and the first excited state\naround the minimum gap~\\cite{AlbashDemonstration18}. We estimate\nthe scaling of the adiabatic error for the FM-gadget  and compare\nthe experimental results of the boundary cancellation protocol (BCP)\nwith those predicted by open system simulations using the AME. Based on these simulations,\nwe discuss the conditions under which the BCT-predicted scaling may\nbe observed. We also apply the protocol to an error-corrected version of\nthe FM-gadget using quantum annealing correction\n(QAC)~\\cite{pudenzErrorcorrected14}. The QAC encoding mitigates\nthe effects of Hamiltonian programming errors~\\cite{youngAdiabatic13,pearsonAnalog19}\nand effectively lowers the temperature of the environment~\\cite{Matsuura:2016aa,Vinci:2017ab}; we find that QAC improves the performance of the BCP.\n\nThe structure of this paper is as follows. In Sec.~\\ref{sec:Theory} we first review the BCT, and the formulate a new version which accounts for the possibility of the Liouvillian gap closing at the same point as where the boundary cancellation conditions are enforced, which is relevant for our experiments. The rest of the section is devoted to a discussion of freezing, the effect of a ramp after the BC point, and anomalous heating, all of which are phenomena affecting the performance of the BCP due to their presence in our experiments. In Sec.~\\ref{sec:methods} we describe our methods: the BCP we used in detail, the two $8$-qubit gadgets used in our experiments, and encoding with quantum annealing correction. Our results are presented in Sec.~\\ref{sec:results}; we start with a standard linear control schedule and then present the results for the experimental BCP implementation. We include simulation results for a higher precision version of the BCP for reference, and then report on experiments from the QAC-encoded version of one of our gadgets. Finally, we compare the performance of the BCP to that of the pausing protocol. We conclude with a discussion and outlook in Sec.~\\ref{sec:discussion}. The appendix includes background on the adiabatic master equation, proofs of various Propositions, supporting numerical results,  additional information needed to reproduce our experiments and analysis, and experimental results from two other D-Wave processors.\n\nFor convenience, we include a glossary of the acronyms, terminology, and key symbols we used in Table~\\ref{tab:gloss}.\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n$\\eta$ & Adiabatic error scaling exponent & Sec.~\\ref{sec:everything} \\\\\n$\\rho$ & Solution of the master equation & Eq.~\\eqref{eq:MEode}\\\\\n$\\sigma$ & Gibbs state at end of BCP & Sec.~\\ref{sec:ramp} \\\\\n$s_{\\text{BC}}$ & Termination point of BCP & Sec.~\\ref{sec:bcp}\\\\\n$s_0,t_0$ & Dynamical freezing point & Eq.~\\eqref{eq:freezing_point}\\\\\nAME & Adiabatic master equation & Sec.~\\ref{sec:Theory-A}, \\cite{albashQuantum12}\\\\\nBCP & Boundary cancellation protocol & Sec.~\\ref{sec:bcp} \\\\\nBCT  & Boundary cancellation theorem & Sec.~\\ref{sec:Theory}, \\cite{camposvenutiError18}\\\\\n$D_{\\text{GS}}$ & adiabatic error & Eq.~\\eqref{eq:D_GS} \\\\\nDW2KQ & D-Wave 2000Q annealer & \\cite{dwave-manual,DW-manual}\\\\\nFM-gadget & Ferromagnetic chain gadget & Sec.~\\ref{sec:8QG} \\\\\nNTS & Number of tries-to-solution & Eq.~\\eqref{eq:NTS} \\\\\n$P_{\\text{GS}}$ & Measured ground state prob. & Eq.~\\eqref{eq:19a} \\\\\n$P_{\\text{GS}}^*$ & Gibbs ground state prob. & Eq.~\\eqref{eq:19b} \\\\\nPR & Pause-ramp protocol & Sec.~\\ref{sec:tries} \\\\\nQAC & Quantum annealing correction & Sec.~\\ref{sec:QAC}, \\cite{pudenzErrorcorrected14}\\\\\nT-gadget & Tunneling gadget & Sec.~\\ref{sec:8QG}, \\cite{AlbashDemonstration18}\n\\end{tabular}\n\\end{center}\n\\label{tab:gloss}\n\\caption{Glossary of key symbols and acronyms used in this work in alphabetical order, along with the sections, equations, or references in which they are defined.}\n\\end{table}%\n\n\n\\section{Theory}\n\\label{sec:Theory}\n\n\\subsection{Review of the BCT results of Ref.~\\cite{camposvenutiError18}}\n\\label{sec:Theory-A}\n\nWe consider a finite-dimensional system whose density matrix $\\rho_{t_{f}}$\nsatisfies the master equation $d\\rho_{t_{f}}\/dt=\\mathcal{L}_{t_{f}}(t)\\rho_{t_{f}}(t)$,\nwhere $\\mathcal{L}_{t_{f}}(t)$ is a time-dependent Liouvillian superoperator.\nIf $\\mathcal{L}_{t_{f}}(t)$ depends on the time $t$ only through the ``anneal parameter'' $\\tau:=t\/t_{f}$,\nwhere $t_{f}$ is the total evolution time, then defining $\\rho(\\tau):=\\rho_{t_{f}}(t)$\nand $\\mathcal{L}(\\tau):=\\mathcal{L}_{t_{f}}(t)$, the master equation\ntakes the form \n\n\\begin{equation}\n\\frac{d\\rho}{d\\tau}=t_{f}\\mathcal{L}(\\tau)\\rho(\\tau)\\ .\n\\label{eq:MEode}\n\\end{equation}\nThe main result of Ref.~\\cite{camposvenutiError18} is summarized\nin the following proposition:\n\n\\begin{prop}\n\\label{prop:BCT} \nAssume that $\\mathcal{L}(\\tau)$ is such that $\\left\\Vert \\mathcal{L}(\\tau)\\right\\Vert _{1,1}$\nis summable in $\\tau\\in[0,1]$,\\footnote{The superoperator norm we use is the induced $1,1$ norm: \\unexpanded{$\\left\\Vert \\mathcal{A}\\right\\Vert_{1,1} :=\\sup_{\\left\\Vert y\\right\\Vert _{1}=1}\\left\\Vert \\mathcal{A}y\\right\\Vert _{1}$}\nfor a superoperator $\\mathcal{A}$, where the trace-norm is defined\nas \\unexpanded{$\\|A\\|_{1}:=\\mathrm{Tr}\\sqrt{A^{\\dagger}A}$}, for any operator\n$A$, i.e., the sum of $A$'s singular values. A function is summable if its integral is finite on every compact subset of its domain.} differentiable to order $k+2$ in\na neighborhood of $\\tau=1$, and generates a trace-preserving and\nhermiticity-preserving contractive semigroup, i.e., $\\left\\Vert e^{r\\mathcal{L}(\\tau)}\\right\\Vert _{1,1}\\le1$\nfor all $r\\ge0$, $\\tau\\in\\left[0,1\\right]$. Suppose $\\mathcal{L}(\\tau)$ has a unique steady state $\\sigma(\\tau)$ with eigenvalue zero\nseparated from the rest of the spectrum by a nonzero gap $\\forall \\tau\\in[0,1]$. Let $\\rho(\\tau)$\nbe the solution of Eq.~\\eqref{eq:MEode} with initial condition $\\rho(0)=\\sigma(0)$.\nIf $\\mathcal{L}(\\tau)$ has vanishing derivatives at $\\tau=1$ to\norder $k$, i.e., $\\left.\\partial_{\\tau}^{(j)}\\mathcal{L}\\right|_{\\tau=1}=0$\n$\\forall j=1,2,\\ldots,k$, then there is a constant $C_{k}$ independent\nof $t_{f}$ such that \n\\begin{equation}\n\\left\\Vert \\rho(1)-\\sigma(1)\\right\\Vert _{1}\\leq\\frac{C_{k}}{t_{f}^{k+1}}\\ .\n\\label{eq:BCT}\n\\end{equation}\n\\end{prop}\nWe note that the smoothness assumptions of\nProp.~\\ref{prop:BCT} are more relaxed than\nin Ref.~\\cite{camposvenutiError18}.  \n\nFrom here on we focus in particular on the AME, which we briefly review in App.~\\ref{app:AME}. \nIt was shown in \\cite{camposvenutiError18} that, under the action\nof the AME Liouvillian, it is possible to enforce the BCT as required in\nProp.~\\ref{prop:BCT} by controlling just the time-dependent system\nHamiltonian $H_S$:\n\n\\begin{prop}\n\\label{prop:AME_gap}\nAssume the system evolves according to the time\ndependent Liouvillian in Eq.~\\eqref{eq:AME}. Assume further that\n$\\left.\\partial_{t}^{(j)}H_{S}(t)\\right|_{t=t_{f}}=0$, for $j=1,2,\\ldots,k$.\nThen $\\left.\\partial_{t}^{(j)}\\mathcal{L}\\right|_{t=t_{f}}=0$ $\\forall j=1,2,\\ldots,k$. \n\\end{prop}\n\nThis means that rather than having to directly check that the entire Liouvillian\nsatisfies the conditions of vanishing derivatives at the end, it suffices\nto check that the system Hamiltonian $H_{S}(t)$ satisfies the\nsame conditions. However, the Prop.~\\ref{prop:BCT} requirement that the Liouvillian has a finite\ngap above its zero eigenvalue throughout the evolution window\nmust be checked independently. For the finite dimensional systems that we consider here, this is equivalent\nto the requirement that the steady state is unique throughout the\nevolution, or alternatively, that there are no ``level crossings''\nof $\\mathcal{L}(\\tau)$'s zero eigenvalue throughout the evolution.\\footnote{In principle one must also check that \\unexpanded{$\\left\\Vert \\mathcal{L}(\\tau)\\right\\Vert _{1,1}\\in L^{1}\\left(\\left[0,1\\right]\\right)$},\nbut this follows automatically if the control functions satisfy\n\\unexpanded{$A\\left(\\tau\\right),B\\left(\\tau\\right)\\in L^{1}\\left(\\left[0,1\\right]\\right)$},\nwhich is a natural assumption.} Since the system Hamiltonian is naturally the controllable component\nof the total Hamiltonian or Liouvillian, this is also a physically\nsensible controllability condition. Moreover it was also shown in\n\\cite{camposvenutiError18} that the conclusions of Prop.~\\ref{prop:BCT}\nare more robust than the assumptions of Prop.~\\ref{prop:AME_gap}\nmay seem to imply. Indeed, the scaling with $t_{f}$ of Eq.~\\eqref{eq:BCT} was verified numerically for a more general Liouvillian\nof time-dependent Redfield form, which does not satisfy Prop.~\\ref{prop:AME_gap},\ni.e., for which vanishing of the derivatives of the system Hamiltonian\ndoes not imply vanishing of the derivatives of the Liouvillian. \n\nWe next discuss a modified form of the BCT, which is tailored to the situation we encounter in our experiments using the D-Wave annealer.\n\n\n\\subsection{A modified BCT}\n\\label{sec:Theory-B}\n\n\\begin{figure}\n\\includegraphics[width=.95\\columnwidth]{figs\/dw2kq6_sched}\n\\caption{Native annealing schedule of the D-Wave 2000Q low noise processor.}\n\\label{fig:dw2kq_schedule}\n\\end{figure}\n\nWe now consider the quantum annealing problem defined by a system\nHamiltonian of the form \n\\begin{equation}\nH_{S}(s)=\\frac{A(s)}{2}H_{X}+\\gamma\\frac{B(s)}{2}H_{Z}\\,,\\label{eq:QA_Hamiltonian}\n\\end{equation}\nwhere \n\\begin{equation}\nH_{X}=-\\sum_{i}\\sigma_{i}^{x},\\qquad H_{Z}=\\sum_{i}h_{i}\\sigma_{i}^{z}+\\sum_{i<j}J_{ij}\\sigma_{i}^{z}\\sigma_{j}^{z}\\ .\n\\label{eq:HXHZ}\n\\end{equation}\nand $\\gamma\\in(0,1]$ is a scaling factor that controls the energy\nscale of the computational problem. We refer to $A(s)$ and $B(s)$\nas the \\emph{physical schedules} of the anneal. They are fixed properties\nof the quantum annealing hardware (see Fig.~\\ref{fig:dw2kq_schedule}), but $s$ can be made to depend\non $\\tau=t\/t_{f}$, in the adjustable way that we describe below for the DW2KQ. We call the function $s(\\tau):[0,1]\\mapsto[0,1]$\nthe \\emph{control schedule}. \n\nWe model the anneal according to the AME \nwith the Hamiltonian in Eq.~\\eqref{eq:QA_Hamiltonian}, where the\nsystem is independently or collectively coupled to the bath through the longitudinal field of each qubit (see App.~\\ref{app:AME} for details).\n\nUnfortunately, we cannot directly apply Prop.~\\ref{prop:BCT} \nto annealers with $A(t_{f})=0$ (such as the DW2KQ) because of the following result (proven in App.~\\ref{app:Proof-of-degeneracy}):\n\n\\begin{prop}\n\\label{prop:degeneracy}\nAssume that $\\mathcal{L}(\\tau)$ is the AME Liouvillian [Eq.~\\eqref{eq:AME}]\nwith the Hamiltonian in Eq.~\\eqref{eq:QA_Hamiltonian} and \nlongitudinal system-bath coupling (i.e., that commutes with $H_Z$).\nThen, at the end of the anneal when $A(t_{f})=0$, the zero eigenvalue\nof $\\mathcal{L}(t_{f})$ is at least $d$-fold degenerate, where $d$ is the system's Hilbert space dimension.\n\\end{prop}\n\nSince presumably, when $A(t)\\neq0$ for $t<t_{f}$ the Lindbladian\n$\\mathcal{L}(t)$ has a unique steady state $\\sigma(t)$ which corresponds\nto its zero eigenvalue, Prop.~\\ref{prop:degeneracy} means that the Liouvillian\ngap closes at $t_{f}$ and we cannot use Prop.~\\ref{prop:BCT}.\n\nWe now generalize the BC result Prop.~\\ref{prop:BCT} to the situation where one enforces the BC conditions at the same point as where the Liouvillian gap closes, e.g., at $t=t_f$.\n\n\\begin{prop}\n\\label{prop:BC_gapless_end} \nAssume that $\\left\\Vert \\mathcal{L}(\\tau)\\right\\Vert _{1,1}$\nis summable in $\\tau\\in[0,1]$, differentiable to order $k+2$ in\na neighborhood of $\\tau=1$, and generates a trace-preserving and\nhermiticity-preserving contractive semigroup, i.e., $\\left\\Vert e^{r\\mathcal{L}(\\tau)}\\right\\Vert _{1,1}\\le1$\nfor all $r\\ge0$, $\\tau\\in\\left[0,1\\right]$. Suppose $\\mathcal{L}(\\tau)$\nhas a unique steady state $\\sigma(\\tau)$ at each time $\\tau\\in[0,1]$\nexcept at the point $\\tau_{0}=1$ where the Liouvillian gap closes as\n$\\Delta\\simeq v(\\tau_0-1)^{\\alpha}$ for\na linear schedule (i.e., in the absence of a BC schedule). Let $\\rho(\\tau)$ be the solution of Eq.~\\eqref{eq:MEode}\nwith initial condition $\\rho(0)=\\sigma(0)$ and $\\sigma(1^{-}):=\\lim_{r\\to1^{-}}\\sigma(r)$.\nAssume we enforce boundary cancellation at the end, i.e., $\\mathcal{L}(\\tau)$\nhas vanishing derivatives at $\\tau=1$ to order $k$: $\\left.\\partial_{\\tau}^{(j)}\\mathcal{L}\\right|_{\\tau=1}=0$\n$\\forall j=1,2,\\ldots,k$ [note that here $\\mathcal{L}(s(\\tau))$ is written for\nsimplicity as $\\mathcal{L}(\\tau)$]. Then boundary cancellation is only mildly effective\nin that the adiabatic error satisfies\n\\bes\n\\begin{align}\n\\left\\Vert \\rho(1)-\\sigma(1^{-})\\right\\Vert _{1} & \\leq\\frac{C}{t_{f}^{\\eta}}\\ , \\label{eq:adia_error_scaling}\\\\\n\\mathrm{with\\,\\,} & \\eta=\\frac{k+1}{k\\alpha+\\alpha+1} <\\frac{1}{\\alpha}\n\\label{eq:eta_gapless_end} .\n\\end{align}\n\\ees\n\\end{prop}\nThe proof is given in App.~\\ref{app:Proof-of-Proposition_end}, which also discusses the relaxed smoothness assumptions of Prop.~\\ref{prop:BCT} relative to Ref.~\\cite{camposvenutiError18}.\nThe fact that the gap closes for the Lindbladian modeling the D-Wave annealers was already\nnoted in Ref.~\\cite{venutiAdiabaticity16}, and indeed Eq.~\\eqref{eq:eta_gapless_end} with $k=0$ was \nderived there.\n\nFor completeness we comment also on the case where the Liouvillian gap closes in the middle of the evolution and one enforces boundary cancellation at the end. \nWe have performed numerical simulations under these conditions (see App. \\ref{app:gap_mid}), all agreeing with a scaling of the adiabatic error as in Eq.~\\eqref{eq:adia_error_scaling}, but with an exponent given by $\\eta=k+1$, as in Eq.~\\eqref{eq:BCT}. Analytical results will be presented elsewhere. \n\nSummarizing, if the gap closes at the end of the evolution, implementing BC is mildly advantageous in that the best one can do is to change the exponent $\\eta$ from $1\/(\\alpha+1)$ to $1\/\\alpha$ for an infinitely flat schedule ($k\\to\\infty$). If, instead, the gap closes in the middle of the evolution, we infer from our numerical results that is has no effect on the scaling of the adiabatic error and implementing BC results in a favorable exponent $\\eta=k+1$, just as in the case where the Liouvillian gap does not close at all. \n\n\n\\subsection{Freezing}\n\\label{sec:freezing}\nAs the Hamiltonian starts to commute with the system-bath coupling operators near the end of the anneal,\nthe dynamics of the system will begin to slow down. This gives another significant consideration \nregarding the successful implementation of BC schedules, \nwhich is that the system system state appears to stop evolving at a point $t_0 < t_f$, a phenonemon of quantum annealing called freezing.\nTo explain freezing, we write the master equation in the instantaneous energy eigenbasis $\\{|n\\rangle\\}$, $n=0,1,\\ldots,d-1$\n(we omit the time dependence for clarity), i.e., the basis which diagonalizes the Hamiltonian $H_{S}(t)$. Consider the region of the anneal where $A\\left(t\\right)$ is small but non-zero, and also $B(t) > 0$. It is then natural to assume that the spectrum of\n$H_{S}(t)$ is non-degenerate. The density\nmatrix in the energy basis is $\\rho(t)=\\sum_{mn}\\rho_{mn}|m\\rangle\\!\\langle n|$.\nThe diagonal elements of $\\rho$ in this basis evolve according to\nthe Pauli master equation~\\cite{Pauli-master-equation} (for a modern derivation see, e.g., Ref.~\\cite{Lidar:2019aa}). In particular, the ground state probability\n$\\rho_{00}=\\langle0|\\rho(t)|0\\rangle$ evolves according to \n\\bes\n\\label{eq:freezing}\n\\begin{align}\n\\label{eq:freezing-a}\n\\dot{\\rho}_{00} & =\\sum_{n}\\left(\\rho_{nn}W_{0n}-\\rho_{00}W_{n0}\\right) \\\\\n & =\\sum_{n>0}W_{0n}\\left(\\rho_{nn}-\\rho_{00}e^{-\\beta\\left(E_{n}-E_{0}\\right)}\\right)\\ ,\n \\label{eq:freezing-b}\n\\end{align}\n\\ees\nwhere the transition rate matrix $W$ has the following matrix elements:\\footnote{The matrix $W$ satisfies detailed balance, i.e., $W_{nm}e^{-\\beta E_{m}}=W_{mn}e^{-\\beta E_{n}}$,\nwhich follows from an analogous equation for $\\gamma(\\omega)$, which\nin turns follows from the Kubo-Martin-Schwinger (KMS) conditions on the bath correlation function.} \n\\begin{equation}\nW_{mn}=\\sum_{i,j}\\gamma_{ij}(E_{n}-E_{m})\\langle m|\\sigma_{j}^{z}|n\\rangle\\langle n|\\sigma_{i}^{z}|m\\rangle.\n\\label{eq:matrix_M}\n\\end{equation}\n\nOn the basis of Eqs.~\\eqref{eq:freezing} and~\\eqref{eq:matrix_M},\nfreezing is seen to be a consequence of the following argument. Since\nwe are in the adiabatic regime, only the lowest levels are populated,\ni.e., $\\rho_{nn}\\simeq0$ for $n$ greater than some $n_{A}$. Eq.~\\eqref{eq:freezing}\nis then replaced by \n\\bes\n\\label{eq:freezing-2}\n\\begin{align}\n\\label{eq:freezing-2a}\n\\dot{\\rho}_{00}&\\simeq\\sum_{n=1}^{n_{A}}W_{0n}\\left(\\rho_{nn}-\\rho_{00}e^{-\\beta\\left(E_{n}-E_{0}\\right)}\\right) \\\\\n&\\qquad -\\rho_{00}\\sum_{n=n_{A}+1}^{d}W_{0n}e^{-\\beta\\left(E_{n}-E_{0}\\right)}\\ ,\n\\label{eq:freezing-2b}\n\\end{align}\n\\ees\nwhere $d$ is the system's Hilbert space dimension. When $A(t)=0$, the Hamiltonian is diagonal in the computational basis. Let the Hamming distance between the ground state\n$|0\\rangle$ and the excited states $\\{|n\\rangle\\}_{n=1}^{n_{A}}$ when $A(t)=0$\nbe at least $q$ (note that $q\\ge 1$). Using perturbation theory\naround $A(t)=0$ it can be shown that \n$\\langle0|\\sigma_{j}^{z}|n\\rangle=O\\left(A^{q}\\right)$;\nsee App.~\\ref{app:Proof-of-freezing}. Hence, at the time $t_0$ at which  $A$ is sufficiently small (but non-zero)\none has $W_{0n}= O(A^{2q}) \\simeq0$ for $n=1,2,\\ldots,n_{A}$, and we can neglect the sum in  line~\\eqref{eq:freezing-2a}; we determine $t_0$ below.\nAs for the term in line~\\eqref{eq:freezing-2b}, the transition rates between higher excited states\nare typically smaller, and moreover, the terms are exponentially suppressed\nbecause of the larger gaps (i.e., for sufficiently small temperatures $E_{n}-E_{0}\\gg 1\/\\beta$, given that $n\\ge n_A+1 \\ge 2$). As a\nconsequence, Eq.~\\eqref{eq:freezing-2} becomes $\\dot{\\rho}_{00}\\approx0$,\ni.e., the ground state population is effectively frozen for $t\\ge t_0$. \n\nThe location of the freezing point for the $n$th excited state is determined by the\npoint where \nthe relaxation time for the $n$th excited state, given by $\\tau_{\\text{rel}}^{(n)} = W_{0n}^{-1}$, becomes longer than the anneal time. As long as just one of these relaxation times, with $n\\in\\{1,\\ldots,n_{A}\\}$ is too long, the system will not thermalize. For the system to freeze, i.e., $\\dot{\\rho}_{00}\\approx0$, we need all transitions to cease. Hence, we define the freezing point as the solution $t_0$ of $\\min_{n\\in\\{1,\\ldots,n_{A}\\}}\\tau_{\\text{rel}}^{(n)} =t_f$, or:\\footnote{A shortcut to deriving Eq.~\\eqref{eq:freezing_point} is to interpret $1\/\\tau_{\\text{rel}}^{(n)} = \\gamma\\left(E_{n}-E_{0}\\right)\\left|\\langle 0|V|n\\rangle\\right|^{2} = W_{0n}$ as a statement of Fermi's golden rule for the transition rate, with $V = \\sum_{j}\\sigma_{j}^{z}$ playing the role of the perturbation, and $\\gamma$ the density of states.}\n\\begin{equation}\nt_0 := \\{\\min t\\in[0,t_f] \\text{ s.t. }\\max_{n\\in\\{1,\\ldots,n_{A}\\}} W_{0n}(t)=1\/t_{f}\\}\\ .\n\\label{eq:freezing_point}\n\\end{equation}\nIn terms of the control schedule $s(\\tau)$, we have $s_0 := s(\\tau_0)$, where $\\tau_0 = t_0\/t_f$. In practice, to determine the freezing point $s_0$ we implement a closely related procedure described in App.~\\ref{app:AME}.\n\nNote that, as a consequence of the perturbative argument, (for sufficiently\nsmall $A$) the rates $W_{0n}$ are decreasing functions of $A$.\nSo if $A(t)$ is decreasing in $t$, $W_{0n}$ can be considered\nzero for $t\\ge t^{\\ast}$. A similar argument works when substituting $0\\leftrightarrow m$,\nand one finds that the population in the $m$th excited state, $\\rho_{mm}$, is frozen,\nalbeit with possibly different freezing points than given by Eq.~\\eqref{eq:freezing_point}. \n\nAnother consequence of this argument is that the phenomenon of freezing\nshould be more pronounced (i.e., occur for smaller $t_0$ and resulting in smaller $\\dot{\\rho}_{00}$ for $t>t_0$) for those problems where the ground state\nis separated by a large Hamming distance from the excited states, i.e., a larger tunneling barrier such as the T-gadget compared to the FM-gadget, discussed in Sec.~\\ref{sec:8QG} below.\nThese problems are the ones which are harder to simulate with standard classical\nsimulated annealing with single spin flip moves~\\cite{kirkpatrick_optimization_1983} (cluster flip moves~\\cite{PhysRevLett.115.077201} would not necessarily be similarly affected). \n\nFinally, note that a \\emph{simulation} of a master equation to which the considerations above apply, such as the AME (see App.~\\ref{app:AME}), is also expected to exhibit freezing.\n\n\n\\subsection{Boundary cancellation with a ramp at the end}\n\\label{sec:ramp}\n\nSince the ground state population does not change past the freezing\npoint, no change in the schedule would be effective if performed\nafter freezing. In view of these considerations we perform BC before\nfreezing sets in, which --- for the standard control schedule $s(\\tau)=\\tau$  (see Fig.~\\ref{fig:dw2kq_schedule}) --- happens for $s\\approx 0.55$,\ndepending on the problem. Our strategy will be the following. First,\nevolve from $t=0$ to $t_{f}$ with a BC schedule.\nTo avoid freezing, we ensure that $A(t_{f})\\neq0$ at the end of the BC schedule. \nThus, $t_f$ does \\emph{not} correspond to the usual total anneal time for which $A(t_f)=0$. Right after the BC schedule ends, we perform\na linear ramp\\footnote{The term ``quench'' is used in the D-Wave documentation instead of ramp~\\cite{dwave-manual}.} of duration $t_r = 1\\,\\mu$s until the schedules reach their final values (in particular $A(t_{f}+t_r)=0$), after which the system is measured in the computational\nbasis. The state after the entire evolution can be written as \n$\\mathcal{E}_{\\mathrm{ramp}}\\mathcal{E}_{\\mathrm{BC}}\\rho(0)$,\nwhere $\\mathcal{E}_{\\mathrm{BC}}$ ($\\mathcal{E}_{\\mathrm{ramp}}$) is the\nevolution through the BC schedule (ramp). However, random local fields\nand coupler perturbations [integrated control errors (ICE)] result\nin a Hamiltonian that does not behave as intended in the ideal case~\\cite{Albash:2019ab},\nand these errors have been well documented in the D-Wave processors~\\cite{dwave-manual}.\nThe effect of such random perturbations can be controlled with error\nsuppression and correction \\cite{youngAdiabatic13,pudenzErrorcorrected14,pearsonAnalog19},\nwhich will be employed below for the ferromagnetic chain gadget. Due\nto this ICE effect, the measured state is better represented by \n\\begin{equation}\n\\rho_{\\mathrm{final}}:=\\mathsf{E}_{J}\\left[\\mathcal{E}_{\\mathrm{ramp}}\\mathcal{E}_{\\text{BC}}\\rho(0)\\right]\\ ,\n\\label{eq:rho_f}\n\\end{equation}\nwhere we denoted by $\\mathsf{E}_{J}\\left[\\bullet\\right]$ the\naverage over the noise on the random couplings $J_{ij}$ and fields\n$h_{i}$. \n\n\nLet us now define $\\rho(t_{f}):=\\mathcal{E}_{\\text{BC}}\\rho(0)$, while\n$\\sigma(t_{f})$ is the Gibbs state at the end of the BC schedule [recall that $\\rho(0)= \\sigma(0)$],\ncorresponding to the Hamiltonian in Eq.~\\eqref{eq:QA_Hamiltonian}. The\nadiabatic theorem in its various forms, including with boundary cancellation,\nprovides an upper bound on the pre-ramp distance $\\|\\delta\\|_1$, where $\\delta:=\\rho(t_{f})-\\sigma(t_{f})$. Defining \n\\begin{equation}\n\\sigma_{\\mathrm{final}}:=\\mathsf{E}_{J}\\left[\\mathcal{E}_{\\mathrm{ramp}}\\sigma(t_{f})\\right]\\ ,\n\\label{eq:sigma_f}\n\\end{equation}\n$\\delta$ can be related to $\\rho_{\\mathrm{final}}$ and $\\sigma_{\\mathrm{final}}$\nvia the following bound:\n\\bes\n\\label{eq:18}\n\\begin{align}\n\\left\\Vert \\rho_{\\mathrm{final}}-\\sigma_{\\mathrm{final}}\\right\\Vert _{1} & =\\left\\Vert \\mathsf{E}_{J}[\\mathcal{E}_{\\mathrm{ramp}}\\delta]\\right\\Vert _{1}\\label{eq:bound}\\\\\n & \\le\\mathsf{E}_{J}\\left[\\left\\Vert \\mathcal{E}_{\\mathrm{ramp}}\\delta\\right\\Vert _{1}\\right]\\label{eq:Jensen}\\\\\n & \\le\\mathsf{E}_{J}\\left[\\left\\Vert \\delta\\right\\Vert _{1}\\right]\\ .\n \\label{eq:CPTP}\n\\end{align}\n\\ees\nHere Eq.~\\eqref{eq:Jensen} follows from from Jensen's inequality~\\cite{Jensen:1906up} and the fact that every norm is convex (implying $\\left\\Vert \\mathsf{E}_{J}\\left[x\\right]\\right\\Vert _{1}\\le\\mathsf{E}_{J}\\left[\\left\\Vert x\\right\\Vert _{1}\\right]$),\nwhile Eq.~\\eqref{eq:CPTP} follows because $\\mathcal{E_{\\mathrm{ramp}}}$\nis a CPTP map (implying $\\left\\Vert \\mathcal{E}_{\\mathrm{ramp}}\\delta\\right\\Vert _{1}\\le\\left\\Vert \\delta\\right\\Vert _{1}$).\nNote that $\\rho_{\\mathrm{final}}$ is the empirically measured state,\nwhile $\\sigma_{\\mathrm{final}}$ differs from the state that we\nsought to prepare $\\sigma(t_{f})$, by the presence of the extra operations\n$\\mathsf{E}_{J}\\left[\\mathcal{E}_{\\mathrm{ramp}}\\bullet\\right]$.\n\nThe bound~\\eqref{eq:18} implies a similar bound for the ground state probabilities.\nLet us define \n\\bes\n\\label{eq:19}\n\\begin{align}\n\\label{eq:19a}\nP_{\\text{GS}} &:=\\mathrm{Tr}\\left[|\\text{GS}\\rangle\\!\\langle \\text{GS}|\\rho_{\\mathrm{final}}\\right]\\\\\n\\label{eq:19b}\nP_{\\text{GS}}^{*} &:=\\mathrm{Tr}\\left[|\\text{GS}\\rangle\\!\\langle \\text{GS}|\\sigma_{\\mathrm{final}}\\right]\\ ,\n\\end{align}\n\\ees\nwhere $|\\text{GS}\\rangle$ is the ground state of the Hamiltonian at the\nend of the anneal, i.e., the ground state of $H_{Z}$. \nSince\\footnote{Here \\unexpanded{$\\left\\Vert y\\right\\Vert _{\\infty}$} is the operator\nnorm of $y$, i.e., its maximum singular value, which is $1$ for an orthogonal\nprojection.} \n\\begin{equation}\n\\left|\\mathrm{Tr}\\left[|\\text{GS}\\rangle\\!\\langle \\text{GS}|x\\right]\\right|\\le\\left\\Vert |\\text{GS}\\rangle\\!\\langle \\text{GS}|\\right\\Vert _{\\infty}\\left\\Vert x\\right\\Vert _{1}=\\left\\Vert x\\right\\Vert _{1}\\ ,\\end{equation}\n it follows that the \\emph{adiabatic error} defined as\n\\begin{equation}\nD_{\\text{GS}}:=\\left|P_{\\text{GS}}-P_{\\text{GS}}^{*}\\right|\\ ,\n\\label{eq:D_GS}\n\\end{equation}\nsatisfies\n \\begin{equation}\nD_{\\text{GS}} \\le \\left\\Vert \\rho_{\\mathrm{final}}-\\sigma_{\\mathrm{final}}\\right\\Vert _{1}\\ .\n \\label{eq:16}\n \\end{equation}\nThe adiabatic error quantifies the difference between the ground state overlaps of the experimentally measured state ($\\rho_{\\mathrm{final}}$) and the Gibbs state  ($\\sigma_{\\mathrm{final}}$).\n\n\\subsection{An adiabatic error bound that combines everything}\n\\label{sec:everything}\n\nCombining Eqs.~\\eqref{eq:18} and~\\eqref{eq:16} with Props.~\\ref{prop:BCT} and~\\ref{prop:BC_gapless_end} and our numerical evidence,\nwe obtain \n\\begin{align}\n\\label{eq:21}\nD_{\\text{GS}} \\le \\left\\Vert \\rho_{\\mathrm{final}}-\\sigma_{\\mathrm{final}}\\right\\Vert _{1} &\\le \\frac{C}{t_{f}^{\\eta}}\\ ,\n\\end{align}\nwhere $C$ is now the noise averaged constant ($\\mathsf{E}_{J}\\left[\\bullet\\right]$) and $\\eta$ depends\non the physical assumptions. Namely, $\\eta=k+1$ if the Liouvillian\neither has a non-zero spectral gap throughout the anneal or if BC is enforced after the Liouvillian gap has already closed somewhere along the anneal, while $\\eta=(k+1)\/(k\\alpha+\\alpha+1)$\nif the Liouvillian gap closes at the same point at which BC is enforced (Prop.~\\ref{prop:BC_gapless_end}). \n\nOf course we may reformulate Eq.~\\eqref{eq:21} as a bound on $P_{\\text{GS}}$:\n\\begin{equation}\nP_{\\text{GS}}^* - \\frac{C}{t_{f}^{\\eta}} \\leq P_{\\text{GS}} \\leq P_{\\text{GS}}^* + \\frac{C}{t_{f}^{\\eta}}\\ .\n\\label{eq:PGS-bound}\n\\end{equation}\n\n\n\\subsection{Anomalous heating}\n\\label{sec:anom-heat}\n\nOur discussion so far has assumed that the effective temperature of the system remains constant as a function of both the anneal parameter $\\tau = t\/t_f$ and the total anneal time $t_f$. However, there is evidence to suggest that the latter is in fact not the case in the D-Wave devices, i.e., the temperature is $t_f$-dependent. The reason is an unintentional but omnipresent high-energy photon flux that enters the D-Wave chip from higher temperature stages through cryogenic filtering, which accumulates over long anneal times and manifests as an effectively higher on-chip temperature~\\cite{anomalous-heating}. This anomalous heating phenomenon will hinder our ability to test the BCP, since it means that in fact $P_{\\text{GS}}^*$ is a function of $t_f$, which complicates testing the BCT prediction as summarized by Eq.~\\eqref{eq:PGS-bound}. Indeed, for a Gibbs state $\\sigma_{\\text{final}} = e^{-\\beta H_S}\/Z$ (with $\\beta$, $H_S$, and the partition function $Z$ all evaluated at $t=t_f$), expanding $H_S$ in its eigenbasis as $H_S =\\sum_{i=0} E_i \\ketb{i}{i}$ we readily find\n\\begin{equation}\nP_{\\text{GS}}^* = \\langle \\text{GS}|\\sigma_{\\mathrm{final}}|\\text{GS}\\rangle = \\frac{1}{1+\\sum_{i=1}e^{-\\beta(t_f)\\Delta_{i}}} \\ ,\n\\end{equation}\nwhere $\\ket{0}$ is the ground state and $\\Delta_{i}:= E_i-E_0$.\n\nAssuming for simplicity that $\\beta(t_f) = \\beta_0 + a\/t_f$, i.e., a temperature that depends linearly on $t_f$ with rate $a>0$, and that $1 \\ll \\beta(t_f)\\Delta_1 \\ll \\beta(t_f)\\Delta_i$ for $i\\ge 2$, we can write this as \n\\begin{equation}\nP_{\\text{GS}}^* \\approx 1 - e^{-\\beta(0)\\Delta_1}e^{-a\\Delta_1\/t_f} \\ ,\n\\end{equation}\ni.e., the algebraic scaling with $t_f$ of Eq.~\\eqref{eq:PGS-bound} becomes obscured by an exponential scaling due to $P_{\\text{GS}}^*$.\n\nHowever, in reality we do not know the functional form of $\\beta(t_f)$, and the assumption $1 \\ll \\beta(t_f)\\Delta_1 \\ll \\beta(t_f)\\Delta_i$ may not hold. Therefore, in the analysis of our experimental results in Sec.~\\ref{sec:results} below, we instead use an ansatz of the form \n\\begin{equation}\nP_{\\text{GS}} = \\bar{P}_{\\text{GS}}^* + \\frac{C'}{t_{f}^{\\eta'}}\\ ,\n\\label{eq:PGS-fit}\n\\end{equation}\nwhere $\\bar{P}_{\\text{GS}}^*$ is a free fitting parameter representing an averaged value of the true (unknown) $P_{\\text{GS}}^*(t_f)$,  $C'$ becomes another fitting parameter which already accounts for noise averaging as explained in Sec.~\\ref{sec:everything}, and \n$\\eta'$ plays the role of the effective scaling exponent, i.e., our proxy for $\\eta$ in Eq.~\\eqref{eq:PGS-bound}. \n\n\n\n\n\n\\section{Methods}\n\\label{sec:methods}\n\n\\subsection{Boundary Cancellation Protocol (BCP)}\n\\label{sec:bcp}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.95\\columnwidth]{figs\/dw2kq6_param_b2}\n\\includegraphics[width=0.95\\columnwidth]{figs\/dw2kq6_sched_b2} \n\\caption{An example of the D-Wave boundary cancellation protocol with the $\\beta_{2}$\nschedule ramped at $s_{\\text{BC}}=0.50$, with the constructed parametric\nschedule (top) and the corresponding physical schedules (bottom).\nFor the schedule shown here we chose an anneal time of $t_{f}=100\\,\\mu$s. The ramp duration is $t_r=1\\mu$s. Contrast with the native schedule of the DW-LN processor shown in Fig.~\\ref{fig:dw2kq_schedule}. The most significant impact is on the $B(s)$ schedule, which is not natively flat, in contrast to the $A(s)$ schedule, which natively approaches $0$ in a very flat manner.}\n\\label{fig:sched}\n\\end{figure}\n\n\\begin{figure}\n\\centering \\subfigure[\\ ]{\\includegraphics[height=1.84in]{figs\/fm_gadget}}\n\\hspace{0.1in} \\subfigure[\\ ]{\\includegraphics[height=1.8in]{figs\/t_gadget}}\n\\caption{Illustration of the FM-gadget (a) and T-gadget (b) embedded into the\nChimera architecture as represented by an Ising Hamiltonian $H_Z$ [Eq.~\\eqref{eq:HXHZ}]. Ferromagnetic\n(blue) and anti-ferromagnetic (dashed purple) couplings all have the\nsame strength of $J_{ij}=-1$ and $J_{ij}=1$, respectively. Local fields $h_i$ are indicated by the\narrows and their value beside them. The ordering of the qubits in\nthe Chimera architecture is also enumerated in (a).}\n\\label{fig:fm_chain}\n\\end{figure}\n\n\\begin{figure*}\n\\centering \\subfigure[\\ FM-gadget]{\\includegraphics[width=0.46\\textwidth]{figs\/dw2kq6_fer_a_ct0_gaps}\\label{fig:fm_gaps-a}}\n\\hfill{}\\subfigure[\\ T-gadget]{\\includegraphics[width=0.46\\textwidth]{figs\/dw2kq6_t_half_gaps}\\label{fig:fm_gaps-b}}\n\\caption{Spectral gaps of (a) the FM-gadget and (b) the T-gadget (with $\\gamma=0.5$),\nas the schedule is varied according to typical physical schedules\n$A(s)$ and $B(s)$ of the D-Wave processor (App.~\\ref{app:betasched}),\nshown in units of $\\mathrm{ns}^{-1}$ ($\\hbar=1$). The notation $\\Delta_{n,n+1}$\ndenotes the energy gap between the $n$th and $n+1$th Hamiltonian\neigenstates, with $n=0$ being the ground state. The $13.5\\,\\mathrm{mK}$\nenergy scale is also shown, which is the reported dilution fridge temperature of the\nDW-LN processor. The minimum gap location is $s^{*}=0.43$ for the\nFM-gadget and $s^{*}=0.44$ for the T-gadget. The minimum gap region\nis marked by the light blue shading, while the estimated start of\nfrozen dynamics at $s_0$ is marked by pink shading,\nfound by solving Eq.~\\eqref{eq:freezing_point} (see App.~\\ref{app:AME} for details).\nNote that for both gadgets, the first, second, and third excited states\nare all initially degenerate in the subspace of a single $\\ket{-}$\nexcitation of the transverse-field ground state. The nonzero excited\nstate gaps $\\Delta_{12}$ and $\\Delta_{23}$ of the FM gadget ($\\Delta_{12}$ only for the T gadget) at $s=1$ are the result\nof symmetry breaking due to a cross-talk term we included in the\nHamiltonian, of strength $\\chi = 0.02$ (see App.~\\ref{app:xtalk}). The avoided level\ncrossing of the T-gadget is slightly narrower than that of the FM-gadget.\n}\n\\label{fig:fm_gaps}\n\\end{figure*}\n\n\nWe performed most of our experiments with the D-Wave 2000Q low noise (DW-LN) processor\naccessed through D-Wave Leap. We also performed additional experiments\nwith the D-Wave 2000Q processor at the NASA Quantum Artificial Intelligence\nLaboratory (DW-NA) as well as the D-Wave Advantage (DWA) processor\nthrough D-Wave Leap. \nIn the processor specifications, the hardware-determined schedules\n$A(s)$ and $B(s)$ are parametrized by the user-defined control schedule\n$s(\\tau)$. In the standard case the control is linear, i.e., $s(\\tau)=\\tau$\nbut in general $s(\\tau)$ can be programmed in a piecewise linear\nmanner as a function of time. The processor permits a maximum of $12$\npoints to specify the piecewise linear function $s(\\tau)$. We take\nadvantage of this enhanced capacity to approximate a BC schedule.\n The allowed range of programmable anneal times is $t_{f}\\in[1,2000]\\,\\mu$s.\n\nEven though $A(s)$ and $B(s)$ themselves need not satisfy the vanishing\nderivative requirement of the BCP, it follows from the chain rule that\n$A(s(t))$ and $B(s(t))$ do, as long as the control schedule $s(t)$\nsatisfies this requirement and $A(s)$ and $B(s)$ are differentiable\nto the same order as $s(t)$. To be concrete, we take \n\\begin{align}\ns(\\tau)=\\beta_{k}(\\tau)\\ ,\\qquad\\tau=t\/t_{f}\\ ,\\label{eq:s=00003Dbeta}\n\\end{align}\nwhere \n\\bes\n\\begin{align}\n\\beta_{k}(\\tau) &:=\\frac{\\mathrm{B}_{\\tau}(1,k+1)}{\\mathrm{B}_{1}(1,k+1)} \\\\\n &= 1-(1-\\tau)^{k+1}  \\label{eq:beta_kt}\n\\end{align}\n\\ees\nis the regularized incomplete beta function of order $k$~\\cite{RPL:10,camposvenutiError18},\nwith\n\\begin{equation}\n\\mathrm{B}_{x}(a,b):=\\int_{0}^{x}y^{a-1}(1-y)^{b-1}dy\n\\label{eq:ibf}\n\\end{equation}\nbeing the incomplete beta function. As is apparent from Eq.~\\eqref{eq:beta_kt}, \nthe function $\\beta_{k}(\\tau)$\nhas exactly $k$ vanishing $\\tau$-derivatives at $\\tau=1$,\nas required by the BCT. We refer to this class of control schedules\nsimply as \\emph{beta schedules}. \n\nAs we discussed above, in contrast to the theoretical setup of Ref.~\\cite{camposvenutiError18},\ndue to freezing there is no practical advantage to flattening the\nschedule as $s$ approaches $1$. The effectiveness of the BCP is\nmost apparent when the flattening of the schedule occurs after the\navoided level crossing of the anneal (at $s=s^*$), but before the dynamics freeze (at $s=s_0$).\nThus, rather than using the standard schedules $\\{A(s),B(s)\\}$ with\nlinear $s(\\tau)$ that freeze out around $s\\simeq0.5$, the beta schedule\nneeds to be adjusted so that it terminates at an appropriate point $s_{\\text{BC}}$ corresponding to $t=t_f$ and satisfying\n$0< s^* < s_{\\text{BC}}<s_0\\le 1$, rather than always at $s=1$, when the open system\ndynamics are already frozen out. The anneal is then ramped to the final values $A(t_f+t_r)=0$ and $B(t_f+t_r)$. The precise construction of the\ncontrol schedule is given in App.~\\ref{app:betasched}. \n\nNote that the piecewise linear approximation of Eq.~\\eqref{eq:s=00003Dbeta}\ncannot exactly satisfy the requirement of vanishing derivatives at\nthe end. However, Ref.~\\cite[Prop.~3]{camposvenutiError18}\nshows that (at least for the case $k=1$) if one tries to enforce\na vanishing derivative but achieves only approximate vanishing, the\noverall adiabatic error is diminished nonetheless. A reasonable expectation\nis that the scaling will be milder than the theoretical ideal of $1\/t_{f}^{k+1}$ for Liouvillians with a spectral gap (Prop.~\\ref{prop:BCT}), i.e., will\nbe of the form $1\/t_{f}^{\\eta}$ with $\\eta<k+1$ for schedules that\nonly approximate the desired $\\beta_{k}$ schedule. An example of\nthis schedule construction is shown in Fig.~\\ref{fig:sched}. \n\n\n\\subsection{8 Qubit Gadgets}\n\\label{sec:8QG}\n\nWe briefly describe the two gadgets utilized in our experiments. The\nfirst gadget we use is an 8-qubit ferromagnetic (FM) chain, illustrated\nin Fig.~\\ref{fig:fm_chain}. This gadget breaks the degeneracy present\nin both the ground subspace and first excited subspace of the same chain without local fields, creating a large Hamming distance between its ground\nstate and first exited state without introducing additional two-body interactions.\nIt is also simple to tile and embed its QAC encoding (see below) on the Chimera\ngraph.\n\nThe second gadget we utilize is the one introduced in Ref.~\\cite{AlbashDemonstration18}.\nThis gadget was thoroughly analyzed there and established to pose\na small-gap tunneling barrier between the ground state and the first\nexcited state, so we refer to it as the ``T-gadget''. For finer\ncontrol of the energy gap, the T-gadget is scaled to $\\gamma=0.5$\n[recall Eq.~\\eqref{eq:QA_Hamiltonian}]. The ground state of\nboth gadgets is the all-$0$ state, denoted by $|\\mathbf{0}\\rangle$.\nBoth gadgets also share the all-$1$\nstate $|\\mathbf{1}\\rangle$ as their first excited state.\n\n\nThe spectral gaps of the gadgets are shown in Fig.~\\ref{fig:fm_gaps}.\nThe most crucial distinction between them is that the FM-gadget has\na minimum gap $\\Delta_{01}(s^*)$ to the first excited state that is larger than\nthe dilution fridge operating temperature, while the opposite holds for the T-gadget.\nFigure~\\ref{fig:fm_gaps} suggests that implementing the BCP with the termination point $s_{\\text{BC}}$ somewhere in the range\nof $[0.45,0.50]$ will be the most effective for both gadgets, since $s^*\\approx 0.43$ and $s_0\\approx 0.5$. Before\nthis range, the BCP spends more time slowing down before the minimum\ngap, while after this range it goes through the minimum gap too quickly\nand only slows down going into the freezing phase.\n\n\n\\subsection{Encoding with Quantum Annealing Correction}\n\n\\label{sec:QAC}\n\nWe supplement the BCP with quantum annealing correction (QAC) \\cite{pudenzErrorcorrected14}\nand compare the scaling improvement versus QAC with a simple linear\nanneal. The QAC encoding of the computational Hamiltonian in \\eqref{eq:QA_Hamiltonian}\nhas the form \n\\begin{equation}\nH_{\\text{QAC}}=\\overline{H}_{Z}+\\lambda H_{P}\\label{eq:QAC}\n\\end{equation}\nwhere $\\overline{H}_{Z}$ is a $k$-qubit repetition of the problem\nHamiltonian $H_{Z}$ wherein each $\\sigma_{i}^{z}$ and $\\sigma_{i}^{z}\\sigma_{j}^{z}$\nterm is replaced by its encoded counterpart $\\overline{\\sigma_{i}^{z}}=\\sum_{\\ell=1}^{m}\\sigma_{i_{\\ell}}^{z}$\nand $\\overline{\\sigma_{i}^{z}\\sigma_{j}^{z}}=\\sum_{\\ell=1}^{m}\\sigma_{i_{\\ell}}^{z}\\sigma_{j_{\\ell}}^{z}$,\nrespectively, and the penalty Hamiltonian imposes an energy cost by\nferromagnetic coupling to an ancilla penalty-qubit, \n\\begin{equation}\nH_{P}=-\\sum_{i}(\\sigma_{i_{1}}^{z}+\\ldots+\\sigma_{i_{m}}^{z})\\sigma_{i_{p}}^{z}.\n\\end{equation}\nHere the index $i$ is over the logical qubits, $i_{1}\\ldots i_{m}$\nare the physical qubits corresponding to the logical qubit $i$, and\n$i_{p}$ is the ancilla for qubit $i$. A majority vote over the results\nof the measurement on the computational basis of the (odd) $m$ physical\nqubits decides the spin value of logical qubit $i$. This strategy\nof combining an energy penalty against errors anticommuting with $H_{P}$\n(the sum of the stabilizer generators of the bit-flip code)~\\cite{jordan2006error,Marvian:2014nr,Bookatz:2014uq,Jiang:2015kx,Marvian-Lidar:16,Marvian:2017aa,Lidar:2019ab},\nalong with a bit-flip repetition code, has been demonstrated numerous\ntimes to be successful at enhancing the performance of quantum annealing,\ndespite not being able to protect against pure dephasing errors (which\ncommute with $H_{P}$)~\\cite{pudenzQuantum15,Mishra:2015,MNAL:15,Vinci:2015jt,Matsuura:2016aa,Pearson:2019aa}.\nWe only consider $m=3$ QAC encoding, which is naturally embedded\nonto the Chimera graph of all the DW2KQ processors. As the encoded FM-gadget\ncan be naturally embedded with no additional minor embedding, we restrict\nour attention to it, and do not consider the T-gadget under QAC, which\nwould require minor embedding. For the same reason we do not consider\nthe more powerful nested QAC strategy~\\cite{vinci2015nested,Vinci:2017ab,Matsuura:2018,Li:2020aa},\nwhich is left for future work. We estimate the optimal penalty strength\n$\\lambda$ for QAC by finding the value that \nmaximizes\nthe ground state probability of a simple linear anneal. \n\n\\begin{figure*}\n\\centering \\subfigure[\\ %\nFM-gadget -- $P_{\\text{GS}}$]%\n{\\includegraphics[width=0.87\\columnwidth]{figs\/dw2kq6_fm_nl_pgs}\\label{fig:dwsim_lin-a}}\n\\centering \\subfigure[\\ %\nFM-gadget -- $D_{\\text{GS}}$]%\n{\\includegraphics[width=0.87\\columnwidth]{figs\/dw2kq6_fm_lin_dgs}\\label{fig:dwsim_lin-b}}\n\\centering \\subfigure[\\ %\nT-gadget -- $P_{\\text{GS}}$]%\n{\\includegraphics[width=0.87\\columnwidth]{figs\/dw2kq6_t_nl_pgs}\\label{fig:dwsim_lin-c}}\n\\centering \\subfigure[\\ %\nT-gadget -- $D_{\\text{GS}}$]%\n{\\includegraphics[width=0.87\\columnwidth]{figs\/dw2kq6_t_lin_dgs}\\label{fig:dwsim_lin-d}}\n\\caption{%\n    DW-LN and AME results for $P_{\\text{GS}}$ with a simple linear control schedule (without a ramp) for the FM-gadget [(a) and (b)] and the T-gadget [(c) and (d)]. \n    A non-linear fit to \\eqref{eq:PGS-fit} is applied on the data and yields estimates for $\\eta'$ as described in App.~\\ref{app:data}.\n    The initial constrained nonlinear fit to  \\eqref{eq:PGS-fit} yields values for $\\eta'$ and $\\bar{P}_{\\text{GS}}^*$ that may have high variance.  \n    A linear refit on $D_{\\text{GS}}$ is done assuming $P_{\\text{GS}}^* = \\bar{P}_{\\text{GS}}^*$ to refine $\\eta'$ with a smaller error.\n   \n   \n    The values of $\\eta'$ and their associated uncertainties we find in this manner are reported in the legends. \n    Error bars on the measured ground state probabilities represent $95\\%$ confidence intervals.\n    AME simulation parameters for both gadgets: $\\eta_0 g^{2}=5.0\\times10^{-4}$,\n$T=13.5$mK (see App.~\\ref{app:AME}) and crosstalk strength $\\chi=0.02$ (see App.~\\ref{app:xtalk}).\n}\n\\label{fig:dwsim_lin}\n\\end{figure*}\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nIn this section we report on our results for $P_\\text{GS}$, arising from both runs on the DW-LN and numerical simulations. As explained in Sec.~\\ref{sec:anom-heat}, we use Eq.~\\eqref{eq:PGS-fit} to fit our empirical results. \nOur data collection procedure is described in App.~\\ref{app:data}.\n\n\\begin{figure*}\n\\centering \\subfigure[\\ $s_{\\text{BC}}=0.45$]{\\includegraphics[width=0.329\\textwidth]{figs\/dw2kq6_fm_sq45}\\label{fig:6a}}\n\\subfigure[\\ $s_{\\text{BC}}=0.47$]{\\includegraphics[width=0.329\\textwidth]{figs\/dw2kq6_fm_sq47}\\label{fig:6b}}\n\\subfigure[\\ $s_{\\text{BC}}=0.50$]{\\includegraphics[width=0.329\\textwidth]{figs\/dw2kq6_fm_sq50}\\label{fig:6c}}\n\\subfigure[\\ $s_{\\text{BC}}=0.45$]{\\includegraphics[width=0.329\\textwidth]{figs\/dw2kq6_qac_sq45}\\label{fig:6d}}\n\\subfigure[\\ $s_{\\text{BC}}=0.50$]{\\includegraphics[width=0.329\\textwidth]{figs\/dw2kq6_qac_sq50}\\label{fig:6e}}\n\\caption{DW-LN adiabatic distance scaling fits for the FM-gadget with BCP at\n(a) $s_{\\text{BC}}=0.45$, (b) $s_{\\text{BC}}=0.47$, and (c) $s_{\\text{BC}}=0.50$ past the\nminimum gap location of $s^{*}\\approx0.43$ but before freeze-out at $s_0\\approx0.51$.\n(d) and\n(e): Adiabatic distance scaling of QAC encoded FM-gadget with BCP\nat $s_{\\text{BC}}=0.45$ and $s_{\\text{BC}}=0.50$ (see Sec.~\\ref{sec:QAC-results}).\nSymbols represent the experimentally measured data points, solid lines are fits to\n$1\/t_{f}^{\\eta'}$. The $\\eta'$ values reported in the legends an well as other parameters are obtained from nonlinear fits to the measured data as explained in App.~\\ref{app:data}. Error bars represent $95\\%$ confidence intervals.\n}\n\\label{fig:fm_sq_scal}\n\\end{figure*}\n\n\\subsection{Linear control schedule ($k=0$)}\n\\label{sec:lcs}\n\nWe first consider the FM and T-gadgets under the native $A(s)$ and $B(s)$ schedules\nwith linear control $s(\\tau)=\\tau$ and no ramp ($s_{\\text{BC}}=1$), i.e., a standard quantum annealing protocol. This means that the anneal is stopped deep inside the frozen regime, i.e., freezing is expected to have a noticeable effect. \n\nIn Fig.~\\ref{fig:dwsim_lin} we plot, for both the FM-gadget and the T-gadget, the measured ground state probability $P_\\text{GS}$\nfor the DW-LN, along with the same quantity obtained through AME simulations. \nWe found $P_{\\text{GS}}^*$ for DW-LN by a non-linear fit to the function $C'\/t_f^{\\eta'} + \\bar{P}_\\text{GS}^*$, where $C'$ and $\\eta'$ are free parameters (left column of Fig.~\\ref{fig:dwsim_lin}). In the AME case only the last four data points were used in the fit (since otherwise $t_f$ is too small, as evidenced by the curvature observed for small $t_f$ in the right column), while in the DW-LN case we dropped the first data point for both gadgets for the same reason. In the T-gadget case we also dropped the last two data points, as they are likely corrupted by anomalous heating (see Sec.~\\ref{sec:anom-heat}).\n    This leads to the values $P_{\\text{GS}}^*=0.97$ and $P_{\\text{GS}}^*=0.78$ for the FM-gadget [Fig.~\\ref{fig:dwsim_lin-a}] and T-gadget [Fig.~\\ref{fig:dwsim_lin-c}], respectively.\n    For the AME, we took $P_{\\text{GS}}^*$ to be $1$, as this is consistent with the BCT for a large gap and the nonlinear fits did not find a smaller value. \n    We then refined the value of $\\eta'$ by a linear fit to $\\log_{10} (P_{\\text{GS}}-P_{\\text{GS}}^*)$ as a function of $\\log_{10} t_f$, since we expect $P_{\\text{GS}}-P_{\\text{GS}}^*$ to scale as $C\/t_f^\\eta$ by Eq.~\\eqref{eq:PGS-bound}. Using Eqs.~\\eqref{eq:21} and~\\eqref{eq:PGS-bound}, we interpret $\\log_{10} (P_{\\text{GS}}-P_{\\text{GS}}^*)$ as a proxy for $\\log_{10}D_{\\text{GS}}$, plotted in Fig.~\\ref{fig:dwsim_lin-b} and Fig.~\\ref{fig:dwsim_lin-d} for the FM-gadget and T-gadget respectively. \n    \nSeveral observations are in order from Fig.~\\ref{fig:dwsim_lin}. \n\\begin{itemize}\n\\item The $P_{\\text{GS}}$ values at the largest $t_f$ are larger for the FM-gadget than for the T-gadget, in both the AME and DW-LN cases. This is consistent with the larger FM-gadget gap (see Fig.~\\ref{fig:fm_gaps}) and the larger tunneling barrier of the T-gadget (quantified by the ground state being separated by a larger Hamming distance from the excited states), which causes an earlier onset of freezing and hence ``locks in'' the smaller ground state population of the T-gadget.\n\\item The FM gadget AME simulation has a larger $\\eta'$ ($0.43$) than the T-gadget ($0.21$). This signifies a faster convergence to the adiabatic limit of $P_{\\text{GS}}=1$.  This too is consistent with the larger FM-gadget gap and the larger tunneling barrier of the T-gadget. However, the opposite holds for the DW-LN $\\eta'$ values ($0.19$ \\textit{vs} $0.37$, for the FM and T-gadgets, respectively).\n\\item The AME data for the T-gadget is in closer agreement with the power-law bound of the adiabatic theorem than the FM-gadget. This is evidenced by the good agreement with the linear fit in Fig.~\\ref{fig:dwsim_lin-d}, but the persistence of some curvature at large $t_f$ in Fig.~\\ref{fig:dwsim_lin-b}. We may speculate that the FM-gadget will transition to a power-law behavior at even larger $t_f$ than we were able to simulate, though one should keep in mind that in any case, the power-law bound of the adiabatic theorem is not tight.\n\\item For the DW-LN at large $t_f$, we observe an appreciable reduction in $P_{\\text{GS}}$ [Fig.~\\ref{fig:dwsim_lin-c}], and a concomitant increase in $D_{\\text{GS}}$ [Fig.~\\ref{fig:dwsim_lin-d}] in the T-gadget case, and a small positive curvature in the FM-gadget case. This suggests that an anomalous excitation (heating) mechanism is at work in the D-Wave case for $t_f \\gtrsim 300\\mu$s. The source of this anomaly was discussed in Sec.~\\ref{sec:anom-heat}.\nThe anomalous heating phenomenon hinders our ability to test the BCP, since it means that the effective temperature is itself $t_f$-dependent and increasing. As explained above, this is why we treated $P_{\\text{GS}}^*$ as a fitting parameter.\n\\item The DW-LN FM-gadget scaling [Fig.~\\ref{fig:dwsim_lin-b}] is much closer to the expected $D_{\\text{GS}} \\sim 1\/t_f^\\eta$ than the T-gadget [Fig.~\\ref{fig:dwsim_lin-d}], which is obscured by anomalous heating. It is unclear why this phenomenon should more severely impact the T-gadget. \n\\end{itemize}\n\nWe report the values of $\\eta'$ extracted from our fits in Fig.~\\ref{fig:dwsim_lin}.\nTo understand what value of $\\eta'$ to expect, note that according to our model of the D-Wave annealer and in view of Proposition\n\\ref{prop:degeneracy}, the\nLiouvillian gap $\\Delta(\\tau)$ must close at the end of the anneal. Recall that $\\Delta\\simeq v(\\tau-\\tau_{0})^{\\alpha}$ with $\\tau_0=1$ (or $t=t_f$) being the end of the anneal. \nWe do not know the associated value of the exponent $\\alpha$; \nhowever, given that $\\mathcal{L}$ is a normal operator, if the Liouvillian depends smoothly on the schedule in a small interval around $t_{f}$, \na standard result \\cite{kato_perturbation_1995} implies that $\\alpha$ must\nbe an integer. On the other hand, $\\alpha=1$ is incompatible\nwith the fact that the lowest eigenvalue of $\\mathcal{L}$ is zero,\nand that the gap must close smoothly. \nThe value observed in the single qubit case is $\\alpha=2$~\\cite{venutiAdiabaticity16}. According to Prop.~\\ref{prop:BC_gapless_end} and more specifically Eq.~\\eqref{eq:eta_gapless_end} with $k=0$ (due to the linear schedule), this would correspond to\nan exponent $\\eta=1\/3$, which is also the scaling observed in the\nsingle qubit case~\\cite{venutiAdiabaticity16}. \n\nHowever, as seen in Fig.~\\ref{fig:dwsim_lin-d}, the AME T-gadget case has $\\eta' = 0.21(1)$, which -- when equated with $1\/(\\alpha+1)$ according to Eq.~\\eqref{eq:eta_gapless_end} with $k=0$ -- yields $\\alpha=4$.\\footnote{One might be tempted to associate the flatness of the native $A(s)$ schedule (see Fig.~\\ref{fig:dw2kq_schedule}) with the enforcement of a BC-type schedule with a $k\\gg 0$ value, which would in turn correspond to $\\eta=1\/\\alpha$ according to Eq.~\\eqref{eq:eta_gapless_end}. However, the native $B(s)$ schedule is not flat at all, and it dominates the end of the anneal.} This larger value of $\\alpha$ (and corresponding smaller value of $\\eta'$) than was found in the single qubit case is not unexpected given the very different spectrum associated with the T-gadget.\n\nAs remarked above, in contrast to the T-gadget, the AME value of $\\eta'$ for the FM-gadget is at least $0.43$, and is likely to be slightly higher since the largest $t_f$ value for which we were able to run AME simulations appears to be too small for convergence. From Eq.~\\eqref{eq:21} we expect $D_{\\text{GS}}$ to scale with an exponent $\\eta' \\ge \\eta$, where for $k=0$ we have $1\/(\\alpha+1) \\leq \\eta \\leq 1\/\\alpha$ [Eq.~\\eqref{eq:eta_gapless_end}]. Thus $\\eta'\\approx 0.43$ is consistent with these expectations for any integer $\\alpha \\ge 2$.\n\n\nThe DW-LN scaling is consistent with the expectation $\\eta' \\ge \\eta$ for the T-gadget, for which $\\eta'=0.37$. However, anomalous heating makes it difficult to be confident in the extracted $\\eta'$ value for this gadget. For the FM-gadget we find a good fit to $D_{\\text{GS}} \\sim 1\/t_f^{\\eta'}$ with $\\eta'=0.19$, which could be explained by $\\alpha=4$, as in the AME T-gadget case. However, the small positive curvature in the DW-LN FM-gadget data at large $t_f$ suggests that the true scaling may be obscured by anomalous heating here as well.\n\nSince anomalous heating plays such a significant role in the T-gadget behavior at large $t_f$, we abandoned it for our experiments involving BCP schedules with $k\\ge1$. We focus for the rest of this work on the FM-gadget.\n\n\n\n\\subsection{BCP schedules with $k\\ge1$}\n\nWe now turn to actual boundary cancellation schedules. \nIn light of our discussion in Sec.~\\ref{sec:Theory}, a successful BCP must occur before the closing of the Liouvillian gap and freeze-out, while at the same time the Hamiltonian gap remains large enough to promote relaxation to the ground state.\nWe thus apply the BCP at a point $s^* < s_{\\text{BC}} < s_0$, such that $A(s_{\\text{BC}})\\neq0$. \nThe BC schedule is followed by a ramp of  $1\\,\\mu\\mathrm{s}$ to the\nfinal values of $A$ and $B$. \n\nFollowing the same nonlinear fit procedure as in Sec.~\\ref{sec:lcs}, we plot the DW-LN $P_{\\text{GS}}$ results for the FM-gadget in Fig.~\\ref{fig:fm_sq_scal}. More specifically, Fig.~\\ref{fig:6a}-\\ref{fig:6c} shows the results for $s_{\\text{BC}}\\in\\{0.45,0.47,0.50\\}$, while \nFig.~\\ref{fig:6d}-\\ref{fig:6e} shows analogous plots after\napplying QAC, which we discuss in Sec.~\\ref{sec:QAC-results}. \n\nThe first observation that emerges from Fig.~\\ref{fig:fm_sq_scal} is that the $\\beta_0$ schedule results in a significantly larger (i.e., better) scaling exponent than the $k=0$ schedule of Sec.~\\ref{sec:lcs}, for which we found $\\eta'=0.19$. The difference between the two is that for the case shown in Fig.~\\ref{fig:fm_sq_scal} the schedule includes a ramp at $s=s_{\\text{BC}}$. As the ramp point grows from $0.45$ to $0.47$ to $0.50$, the value of $\\eta'$ correspondingly declines from $0.47$ to $0.23$ to $0.21$. This confirms that stopping the BC schedule before freeze-out (at $s_0\\approx 0.52$) has a significant positive impact on the performance of the BCP.\n\nIf the Liouvillian is gapped at the endpoint of the BC schedule, the theoretical expectation for the scaling of the adiabatic error is $\\eta=k+1$ for the $\\beta_{k}$ schedule (recall Sec.~\\ref{sec:everything}). Instead, Figure~\\ref{fig:fm_sq_scal} shows that the measured exponents $\\eta'$ exhibit a strong dependence on the value of $s_{\\text{BC}}$. The most favorable case is $s_{\\text{BC}}=0.45$ [Fig.~\\ref{fig:6a}], for which the exponents $\\eta'$ are $\\{0.47\\ (k=0),\\ 0.98\\ (k=1),\\ 1.1\\ (k=2)\\}$, i.e., roughly half that value for $k=0,1$. The $k=2$ exponent is statistically indistinguishable from that for $k=1$. It is likely that this is due to the fact that the relatively small number of schedule interpolation points does not allow us to approximate the ideal $\\beta_2$ schedule sufficiently well (see Sec.~\\ref{sec:highprecBCP} below).\nOther sources of error, such as ICE and anomalous heating, may also play a role.\n\nWhile this level of agreement with the BCT for the gapped Liouvillian case is far from impressive, the dependence of $\\eta'$ on $k$ is less consistent with the BCT prediction\nfor when the Liouvillian\ngap closes at $s_{\\text{BC}}$; in this case Prop.~\\ref{prop:BC_gapless_end} for $\\alpha=1$\nyields $\\eta=\\{0.5,\\,0.66,\\,0.75\\}$ for $k=0,1,2$ with an asymptotic\nvalue of $1$ for a flat schedule ($k\\to\\infty$), while using $\\alpha=2$\nyields $\\eta=\\{0.33,\\,0.4,\\,0.42\\}$ for $k=0,1,2$ and an asymptotic\nvalue of $0.5$. It is thus reasonable to conclude that the predictions of \nProp.~\\ref{prop:BC_gapless_end} are less consistent with the data than those of Prop.~\\ref{prop:BCT}, and in this sense we can interpret Fig.~\\ref{fig:fm_sq_scal}\nas evidence that the Liouvillian describing \nthe experiment\ndoes not become gapless at the end of the BC schedule.\n\nThe improvement in the scaling of the adiabatic error provided by\nthe BC schedule is most prominent for $s_{\\text{BC}}=0.45$. As we increase\n$s_{\\text{BC}}$ to $0.47$ and $0.5$ the improvement gradually disappears. We interpret this\nas being due to the fact that the freezing point $s_0\\approx 0.52$ is being approached, and so any change\nmade to the schedule can be expected to be effective only on much longer timescales\n$t_{f}$. \n\nWe note that in contrast to the scaling behavior, which favors the smallest $s_{\\text{BC}}$ value, the asymptotic value of $P_{\\text{GS}}$ is highest for the largest $s_{\\text{BC}}$ value. This is consistent with the the larger gap at $s_{\\text{BC}}=0.50$ [see Fig.~\\ref{fig:fm_gaps-a}]. This effect is in fact more significant than the improvement due to the BCP. \n\nFor additional experiments performed using the the NASA Quantum Artificial Intelligence\nLaboratory as well as the D-Wave Advantage processor,\nsee App.~\\ref{app:altdw}. These results are in qualitative agreement with those shown in Fig.~\\ref{fig:fm_sq_scal}, and do not modify our conclusions in any significant way.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=0.9\\columnwidth]{figs\/dw2kq6_fm_ame_hp_sq50}\n\\caption{Adiabatic error of AME simulations of the D-Wave programmed BCP and\nhigh-precision variants (denoted HP in the legend), for the FM-gadget. Here $s_{\\text{BC}}=0.50$, using the schedule of the DW-LN processor\nand a cross-talk parameter of $0.02$ (see App.~\\ref{app:xtalk}). The adiabatic error is evaluated relative to the Gibbs state ground state probability $\\bar{P}_{\\text{GS}}^{*}=0.9690$ at $s=0.50$.\n}\n\\label{fig:fm_ame_hp}\n\\end{figure}\n\n\\subsection{High precision BCP}\n\\label{sec:highprecBCP}\n\nTo test the effect of the relatively low precision of the implementation of the BCP schedules (using only $12$ interpolation points), we simulated\nthe AME for an additional set of higher-precision constructions of the\nBC schedule, with $8$ logarithmic partitions up to $s_{\\text{BC}}$, rather\nthan the $4$ allotted for the D-Wave schedule (see the construction\nof App.~\\ref{app:betasched}). The scaling of the adiabatic error for the FM-gadget\nis shown in Fig.~\\ref{fig:fm_ame_hp}. We see that the high precision\nschedule brings the scaling of the adiabatic error significantly closer\nto the value predicted by the Prop.~\\ref{prop:BCT} version of the BCT within anneal times that are currently accessible to D-Wave devices: we find $\\eta'= 1.66,2.69$ for the higher precision schedules, compared to the theoretically expected values of $k+1 = 2,3$, respectively. The lower precision schedules result in $\\eta'= 1.49,1.74$, respectively.\n\nWe also simulated the AME with a precision of $100$ evenly spaced points in the control\nschedule, to approximate reaching the limit of the analytically exact beta schedule.\nThe results are indiscernible from the high-precision piecewise\nschedule shown in Fig.~\\ref{fig:fm_ame_hp}, indicating that the latter already implements the beta schedule\nat the smallest precision that is meaningful within the simulated\nannealing times.\n\nWe conclude that the relatively poor agreement with the Prop.~\\ref{prop:BCT} BCT predictions we observed in Figure~\\ref{fig:fm_sq_scal}(a)-(c), is likely to be improved in future higher precision implementations of the beta schedule.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{figs\/dw2kq6_qac_scale_sqn}\n\\caption{Adiabatic error for a linear anneal of the QAC-encoded FM-gadget with\nno ramp. Different curves correspond to different values of the penalty\nstrengths, ranging from no penalty (NP; $\\lambda=0$) to $\\lambda=0.09$\n(see symbol legend). The largest slope $\\eta'=0.69\\pm0.03$ is obtained\nfor $\\lambda=0.07$.  \nWe directly assumed a linear model with $\\bar{P}_{\\text{GS}}^*=1$ for comparability between the different penalties; performing a non-linear fit first does not change the slopes for $\\lambda > 0.04$.\n}\n\\label{fig:fm_qac_np}\n\\end{figure}\n\n\n\n\\subsection{QAC-Encoded FM Gadget}\n\\label{sec:QAC-results}\n\nThe scaling of the adiabatic error of the QAC-encoded FM-gadget for\nvarious penalty strengths $\\lambda$ [recall Eq.~\\eqref{eq:QAC}]\nis shown in Fig.~\\ref{fig:fm_qac_np}, for the case of the standard\nlinear control schedule with no ramp. We observe that\nfrom the set of values tried, the optimal penalty strength is $\\lambda=0.07$.\nUsing this optimal penalty value, the ground state probabilities of\nBCP with QAC are shown in Fig.~\\ref{fig:6d}-\\ref{fig:6e}. The\nQAC-protected $P_{\\text{GS}}$ results for both $s_{\\text{BC}}$ values shown are significantly higher than their unprotected ($\\lambda=0$, NP) counterparts in Fig.~\\ref{fig:6a}-\\ref{fig:6c}, consistent\nwith many prior results on the improved performance offered by QAC.\n\nHowever, we are primarily interested here in QAC's effect on the BCP.\nIn this regard, at $s_{\\text{BC}}=0.45$, there is a notable distinction\nbetween different $k$ values, so that QAC amplifies the effect of the BC schedule. \n\nAt $s_{\\text{BC}}=0.50$, QAC-protected BCP still improves over the linear anneal\n($k=0$) by $P_{\\text{GS}}$ value, but the improvement is less distinct\nfor different $k$ values. Moreover, $\\eta'$ is smaller (at most $0.61$) than that for the optimal-$\\lambda$\nlinear QAC anneal in Fig.~\\ref{fig:fm_qac_np} (for which $\\eta'=0.69$),\nsignaling that freezing has already mostly occurred before $s=0.50$,\nand that the optimal $\\lambda$ value is $s_{\\text{BC}}$-dependent.\n\n\nOur results demonstrate that the combination of BCP\nand QAC as error suppression methods is more powerful in terms of\nthe scaling of the adiabatic distance than either method alone: while\n$k=2$ BCP has $\\eta'=1.1$ [Fig.~\\ref{fig:6a}] and optimal-$\\lambda$ linear\n($k=0$) QAC has $\\eta'=0.69$ (Fig.~\\ref{fig:fm_qac_np}), their combination yields $\\eta'=1.3$ [Fig.~\\ref{fig:6d}] and the largest $P_{GS}$ of any other schedule for $t_f=12\\,\\mathrm{\\mu s}$.\n\n\n\\begin{figure*}\n\\includegraphics[width=1\\textwidth]{figs\/dw2kq6_t_nts_hi_sq} \n\\caption{Number of tries to solution for the T-gadget, comparing the pausing\nprotocols (empty symbols) and $\\beta_{0},\\beta_{1}$ BCP (all with\na ramp at $s_{\\text{BC}}$), along with the linear anneal without a ramp (denoted\n``Lin.''; the same data for all panels), for reference. Results shown are for DW-LN, as a function\nof the total anneal time of each protocol. Different panels have different\nvalues of $s_{\\text{BC}}$, from $0.45$ to $0.55$. The pausing protocol\nis shown for different initial anneal times $t_{0}$ of $1$,$5$,\nand $20\\,\\mathrm{\\mu s}$ (empty symbols), followed by a pause of\nvarying length $t_{p}$ and a $1\\,\\mathrm{\\mu s}$ long ramp (so that\nthe paused portion of the schedule is comparable to BCP). For BCP\n(filled symbols), $t_{a}=t_{f}+1\\mathrm{\\mu s}$. For pausing, $t_{a}=t_{0}+t_{p}+1\\mathrm{\\mu s}$.\nBoth schedules are plotted against $t_{a}$ rather than their natural\nparameters ($t_{f}$ and $t_{p}$ respectively) for direct comparison.\n\\label{fig:t_pau_nts_hi}}\n\\end{figure*}\n\n\n\n\\subsection{Comparison of the BCP to the Pause-Ramp protocol}\n\\label{sec:tries}\n\nA different protocol that attempts to exploit slowing down the anneal is the pausing protocol, which interrupts an ordinary linear anneal by a single pause \\cite{marshallPower19,chenWhy20,izquierdo2020ferromagnetically,albash2020comparing,izquierdoAdvantage22}. \nPausing directly uses thermal relaxation at a single point in the anneal \nto try to increase the ground state probability. As in the BCP, this point should also be after \nan avoided level crossing, but before the open system dynamics freeze~\\cite{chenWhy20}. \n\nWe compare the BCP to a variant of the usual pausing schedule~\\cite{marshallPower19} referred to here as the pause-ramp (PR) schedule. While the pausing protocol is typically constructed by interrupting a linear anneal with a pause (but no ramp), PR is constructed by making the first linear segment last for $t_0$, pausing for $t_p$, and ramping to the end. As with the BCP, we use a ramp that is $1\\,\\mathrm{\\mu s}$ long in all cases. PR is better suited for comparison with BCP than the usual pausing schedule since they differ only in their equilibrium state preparation and not in their behavior during the frozen phase.\n\nAs our metric for comparison we use not the ground state probability but rather the number of tries-to-solution (NTS) at a fixed anneal time (with 90\\% confidence):\n\\begin{equation}\n\\mathrm{NTS}_{90\\%}(P_{\\text{GS}}) =\n\\frac{\\log(1-0.90)}{\\log(1-P_{\\text{GS}} )}\\,\n\\label{eq:NTS}\n\\end{equation}\nThis metric is simply another way to study the ground state probability, interpreted as the expected number of tries needed to find the ground state at least once with 90\\% confidence, at the given anneal time. We use this rather than the standard time-to-solution (TTS) metric, since the latter requires identifying the optimal anneal time~\\cite{speedup,AlbashDemonstration18,Hen:2015rt}. However, neither the FM-gadget nor the T-gadget exhibited an optimal anneal time in our experiments (not shown). \n\n\nFigure~\\ref{fig:t_pau_nts_hi} shows the NTS results for the T-gadget\nat various values of $s_{\\text{BC}}$. It compares the standard schedule (no BCP or pausing), the BCP protocol at $k=0$ and $k=1$, and the PR schedule with different initial anneal and pause times. Starting at $s_{\\text{BC}}=0.45$ (top left)\nthe gap is small and thermalization is fast. BCP exhibits two distinct\noptimal anneal times (minima) for $k=0$ and $k=1$, both of which\nresult in slightly smaller NTS than pausing. Also, the $k=1$ case\nis optimal at smaller $t_{a}$ than $k=0$, which can be interpreted\nas as advantage of the higher order protocol. This trend persists\nfor most of the $s_{\\text{BC}}$ values shown. Right at $s_{\\text{BC}}=0.46$, pausing\nachieves its optimal NTS curve, with a slight advantage over BCP,\nand the BCP of each order $k$ is very distinct. However, the advantage\nof pausing is only present at $s_{\\text{BC}}=0.46$, and disappears already\nat $s_{\\text{BC}}=0.47$, showing that pausing performance is highly sensitive\nto the pause point. Since the BCP slowdown is smooth, the influence\nof the optimal thermalization at $s_{\\text{BC}}=0.46$ remains present, although\nthe distinction between BCP orders diminishes as $s_{\\text{BC}}$ grows.\nWell before $s_{\\text{BC}}$ approaches the freeze-out point ($s_0\\approx0.51$), \npausing becomes detrimental for the T-gadget due to excitations (witnessed by the increase in the NTS metric), while BCP mitigates the excitations until the anneal time becomes too long to avoid them.\n\nThese results paint a picture of the two protocols as follows. \nThe relaxation induced by the BCP depends on\nthe properties of the Liouvillian over a \\emph{neighborhood} $(s_{\\text{BC}}-\\epsilon,s_{\\text{BC}}]$.\nIf this neighborhood overlaps with the neighborhood of an avoided\nlevel crossing, then the BCP has an opportunity for an advantage,\nas the speed at which the crossing is traversed is minimized and detrimental\nexcitations to the excited state are reduced. In contrast, PR schedules\nmust try to discontinuously stop the anneal as close as possible to\nthe crossing, and the time required by a paused schedule could be increased\ndue to the longer pause time needed to recover the ground state population,\nunless the precise optimal pause point is quickly found. \n\nThis suggests an important practical point: optimization\nof the BCP can be simpler than that of PR schedules, by the need to optimize just one discrete parameter\n$k$ and one continuous parameter $s_{\\text{BC}}$. Moreover, $s_{\\text{BC}}$ need not be exactly\ntuned, since even an approximate value can allow the BCP to\nclosely approach its optimal performance. On the other hand, pausing requires\n$s_{\\text{BC}}$ to be optimized very precisely, as well as the anneal and pause\ntimes $t_{0}$ and $t_{p}$.\n\n\n\n\\section{Discussion and Outlook}\n\\label{sec:discussion}\n\nWe have extended the boundary cancellation theorem for open systems\nto the case where the Liouvillian gap vanishes at the end\nof the anneal, and derived the asymptotic scaling of the adiabatic\nerror with the anneal time $t_{f}$. Armed with the corresponding theoretical\nexpectation of the scaling of the adiabatic error for the gapped and gapless\ncases,\nwe set out\nto test the scaling predictions and to improve the success probability of quantum\nannealing hardware by implementing boundary cancelling schedules.  The\nspecific functional form of these schedules induces a smooth slowdown\nin accordance with the boundary cancellation theorem. We experimentally tested boundary\ncancellation protocols for open systems and \nevaluated their performance and error-suppression characteristics on specifically designed\n$8$-qubit gadgets embedded on the DW-LN annealer. \n\nWhile a quantitative agreement with the theoretical predictions was not observed, we did demonstrate that as long as the protocol terminates before the onset of freezing, it can increase the ground state population in the examples studied here beyond what is achievable with simple linear anneals, and it does so with shorter anneal times. These results are in qualitative agreement with the theoretical scaling predictions of the BCT.\n\nIn conjunction with quantum annealing correction (QAC), the boundary\ncancellation protocol is also capable of improved adiabatic error\nscaling over what would be achieved with either method alone. While\nthis does not immediately translate to a ground state solution speedup\nwithin the annealing problems studied here, we have shown that BCP-QAC\nis a novel error suppression strategy successfully combining two complimentary\nmethods: the suppression of environmentally induced logical errors\nand the promotion of relaxation via boundary cancellation.\n\nIn contrasting the BCP with the pause-ramp protocol, we found that the BCP is significantly less sensitive to the location of the ramp point, and achieves better performance except at the exact ramp point where the pause-ramp protocol is optimal.\n\nWith the small system size of $8$ qubits used in this work, it was\npossible to collect a large number of annealing samples as well as\nvalidate the protocol behavior against the energy spectrum and open\nsystem simulations. Future work will assess the protocol for larger\nsystem sizes and the impact it has on the scaling of time-to-solution\nas a function of problem size. The largest expected improvement in the protocol's performance, based on our simulations, will arise from an increase in the number and resolution of interpolation points along the annealing schedule, which will allow a more faithful experimental implementation of the ideally smooth annealing schedules demanded by the theoretical protocol.\n\n\\acknowledgments This research is based upon work (partially) supported\nby the Office of the Director of National Intelligence (ODNI), Intelligence\nAdvanced Research Projects Activity (IARPA) and the Defense Advanced\nResearch Projects Agency (DARPA), via the U.S. Army Research Office\ncontract W911NF-17-C-0050. The views and conclusions contained herein\nare those of the authors and should not be interpreted as necessarily\nrepresenting the official policies or endorsements, either expressed\nor implied, of the ODNI, IARPA, DARPA, ARO, or the U.S. Government.\nThe U.S. Government is authorized to reproduce and distribute reprints\nfor Governmental purposes notwithstanding any copyright annotation\nthereon. The authors acknowledge the Center for Advanced Research\nComputing (CARC) at the University of Southern California for providing\ncomputing resources that have contributed to the research results\nreported within this publication. URL: \\url{https:\/\/carc.usc.edu}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Section Title}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\IEEEPARstart{T}{his} is how to write IEEE first section with 2 line initial drop letter \\cite{WilliamMellette2018}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\n\n\n\n\n\n\n\n\n\\IEEEPARstart{B}{y} 2021, annual global data centre network (DCN) traffic will reach $20.6\\times 10^{21}$ bytes, 90\\% of which will be intra-DCN \\cite{Cisco2018}. Additionally, the proportion of requests being serviced by central processing units (CPUs) is expected to decrease from 75\\% today to 50\\% in 2025 as specialised bandwidth-hungry hardware is installed to enable new machine learning applications \\cite{McKinsey2019}. Furthermore, the increasingly common approach of clustering compute resources for large-scale data processing is requiring more network-intensive server-server communication \\cite{Andreades2019}. These trends are exerting a growing strain on internal DCNs, in which many of the interconnects are electronic switches. Electronic switches have limited scalability, limited bandwidth, high latency and high power consumption \\cite{Zervas2019}, \\cite{Wang2018}. As such, switching is presenting a problematic bottleneck for DCN performance, and current network architectures are unfit to meet next-generation DCN requirements.   \n\nOptical switches offer the potential to alleviate many of these network performance issues. With an optical circuit switch (OCS) implementation, there is no packet inspection, buffering, or optical-electrical-optical (OEO) conversion overhead, therefore latency times are significantly lower \\cite{Liu2015}. They also have much higher bandwidth, allowing more servers to be connected to the same switch without increasing oversubscription-related buffering, thus improving scalability. Furthermore, the lack of OEO conversion, the transparency to signal modulation format, and the lower heat generation reduces the number of expensive transceiver components needed, the hardware changes required when new transmission protocols are adopted, and the overall network power consumption respectively. The latter is particularly important since networking can account for $>$50\\% of the \\$20 bn annual DCN power costs, with $CO_2$ emissions equal in volume to the entire aviation industry \\cite{Abts2010}. In addition, optical switches have a more compact physical design than their electronic counterparts, allowing for a smaller footprint in DCs.\n\nThe difficulty of implementing all-optical DCN switching derives from the bursty nature of most DCN traffic and the lack of an all-optical memory alternative. Since no all-optical memory or processor architectures exist, current DCN packet-switched protocols cannot be implemented with an exclusively optical network architecture based on all-optical switches since header information must be processed and payload information stored on a per-hop basis. An alternative to packet switching is circuit switching, which is possible with an all-optical architecture. However, current state-of-the-art commercial optical switches have slow (100s $\\mu$s) switching times. Such long switching times are not compatible with the small data packets that dominate DCN traffic $(90\\% < 576 \\text{ bytes})$ \\cite{Zervas2019} since the switching time would be comparable or greater in size than the forwarding time making for an inefficient network.\n\nFor optical circuit switching (OCS) to be compatible with current DCN demands, it must be possible to switch circuits at the packet timescale \\cite{Zervas2019}, \\cite{Balanici2019}. This requires minimal switching overhead when switching for epochs of the order of 10s-100s of ns.\n\nA promising candidate for realising such a high-speed switch is the semiconductor optical amplifier (SOA). SOAs can be used for either space switching or wavelength switching due to their high and relatively flat optical gain bandwidth. Further benefits of SOAs over other potential optical switching technologies such as MEMS or holograms include fast inherent switching times (theoretically limited only by their $\\approx 100$ ps carrier recombination lifetimes \\cite{Huang2003}), high extinction\/optical contrast ratio, and relatively compact design, making them ideal for low latency-, scalability-, and footprint-constrained DCN applications \\cite{Assadihaghi2010}.\n\nThe sub-ns off-on time of SOAs allows for an SOA-based optical switch architecture that avoids the issues presented by the lack of all-optical memory\/processor alternatives discussed above.  This SOA-based OCS solution is generally more simple and better performing than others suggested by the literature such as optical loop memory \\cite{Srivastava2009}, optical burst switching (OBS) \\cite{Qiao2004}, \\cite{Praveen2005}, \\cite{Kiran2007} and hybrid optical packet switching (OPS) \\cite{Benjamin2017}, \\cite{Wang2018}. However, SOAs have an intrinsic optical overshoot and oscillatory response to electronic drive currents due to exciton density variations and spontaneous emission in the gain region \\cite{Paradisi2019}. As demonstrated in this paper, the overshoot and oscillatory optical output result in the key advantage of SOA switching (rapid switching times) being negated, preventing sub-ns switching.\n\nA previous attempt to optimise SOA output applied a `pre-impulse step injection current' (PISIC) driving signal to the SOA \\cite{Gallep2002}. This PISIC signal pre-excited carriers in the SOA's gain region, increasing the charge carrier density and the initial rate of stimulated emission to reduce the 10\\% to 90\\% rise time from 2 ns to 500 ps. However, this technique only considered rise time when evaluating SOA off-on switching times. A more accurate off-on time is given by the settling time, which is the time taken for the signal to settle within $\\pm 5\\%$ of the `on' steady state. Before settling, bits experience variable signal to noise ratio, which impacts the bit error rate (BER) and makes the signal unusable until settled, therefore the switch is effectively `off' during this period.\n\nA later paper looked at applying a `multi-impulse step injection current' (MISIC) driving signal to remedy the SOA oscillatory and overshoot behaviour \\cite{Figueiredo2015}. As well as a pre-impulse, the MISIC signal included a series of subsequent impulses to balance the oscillations, reducing the rise time to 115 ps and the overshoot by 50\\%. However, the method for generating an appropriate pulse format was trial-and-error. Since each SOA has slightly different properties and parasitic elements, the same MISIC format cannot be applied to different SOAs, therefore a different format must be generated through this inefficient manual process for each SOA, of which there will be thousands in a real DC. As such, MISIC is not scalable.  Critically, the MISIC technique did not consider the settling time, therefore the effective off-on switching time was still several ns.\n\nMore recent work expands on the driving signal modification shown in \\cite{Figueiredo2015}. \\cite{reviewer_reference_1} applies the MISIC signal detailed in \\cite{Figueiredo2015}, but in addition applies a Wiener filter, where the filter is determined by the steady state value of the SOA response and the mean squared error (MSE) between the output and the filter is minimised by means of finding optimal weight-coefficients of the filter. The work accomplishes a roughly 60\\% reduction in guard time, with the goal of reducing guard time as much as possible such that the BER of the output does not exceed a particular level. While the objective of the work (reduce guard time with respect to BER guarantees) is different to that of this work (minimise the settling time of the SOA output), and thus direct comparison is difficult, it is interesting to acknowledge this analogous approach of MSE \\& weight optimisation to optimising the output of an SOA.\n\nSimilarly, \\cite{reviewer_reference_2} explores the optimisation of an SOA by means of both modification of the driving signal and optimisation of the SOA's microwave mounting. A best case of 33\\% reduction in guard time is accomplished with an improved microwave mounting architecture and a step driving signal, where various MISIC and PISIC driving signals were also tested. This work demonstrates that significant improvements in guard time can be derived exclusively from improvements being made to the microwave mounting of the SOA - something that is not dealt with in this paper - and that the improvement of the SOA's output by optimisation of the driving signal does not preclude the simultaneous improvement by optimisation of the microwave mounting. The results therefore are complementary to those presented in this work, which improves the SOA output purely by means of driving signal optimisation. It is speculated here that the optimisation of the SOA's driving signal by the methods presented in this work combined with the optimisation of its microwave mounting could achieve greater improvements in its output than seen in either \\cite{reviewer_reference_2} or this work.\n\nThe previous solutions discussed so far have had a design flow of first manually coming up with a heuristic for a simplified model of an SOA, followed by meticulous testing and tuning of the heuristic until good real-world performance is achieved. If some aspect of the problem is changed such as the SOA type used or the desired shape of the output signal, this process must be repeated. \n\nThis paper presents a novel and scalable approach to optimising the SOA driving signal in an automated fashion with artificial intelligence (AI) techniques, namely `Particle Swarm Optimisation' (PSO), `Ant Colony Optimisation' (ACO) and `Genetic Algorithms' (GA) \\cite{Mata2018}. These algorithms were chosen on the basis that they had previously been applied to proportional-integral-derivative (PID) tuning in control theory \\cite{7873803}. Moreover, AI techniques propose the benefit of not requiring prior knowledge of the SOA and therefore provide a means of developing an optimisation method that is generalisable to any SOA-based switch. All algorithms were shown to reduce the settling and rise times to the $O$(100 ps) scale. The algorithms' hyperparameters were tuned in an SOA equivalent circuit (EC) simulation environment and their efficacy was demonstrated in an experimental setup. AI performance was compared to that of step, PISIC and MISIC driving signals as well as the popular raised cosine and PID control approaches to optimising oscillating and overshooting systems, all of which the AI algorithms outperformed. Of the AI algorithms, PSO was found to have both the best performance and generalisability due to the additional hyperparameters and search space restrictions that were required for GA and ACO. All code and plotted data are freely available at \\cite{parsonson_shabka_chlupka_2020} and \\cite{parsonson_shabka_chlupka_goh_zervas_2020} respectively.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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{Simulation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nSOAs are typically modelled using simple rate equations. However, as shown in \\cite{Ghafouri-Shiraz2004}, the electrical parasitics of an SOA and its surrounding packaging degrade optical signals by broadening the output optical pulse width, reducing the peak optical power (thereby reducing optical contrast), and causing a slight time delay in the emitted optical pulse. Additionally, they alter the relaxation frequency of the SOA output oscillations. As such, modelling the electrical parasitics was crucial to building a simulation environment in which to optimise switching. As described in \\cite{Ghafouri-Shiraz2004}, \\cite{Figueiredo2011}, and \\cite{Tucker1984}, assuming a small circuit model, microwave ECs can be used to more accurately simulate semiconductor diodes by accounting for these electrical parasitics. Therefore, ECs were the chosen approach to SOA modelling for this paper. \n\nSince at low voltages ($<0.8 V$) the current ($I$) - voltage ($V$) relationship can be described by (\\ref{eq:IVRelationship}) ($q$~=~charge, $K_b$~=~Boltzmann constant, $T$~=~temperature), the ideality factor $\\eta$ and the saturation current $I_s$ could be calculated as 1.59 and $3.48 \\times 10^{-11}$ A respectively using the semi-logarithmic $I$-$V$ curve of the SOA in Fig.~\\ref{fig:soaIVCharacteristics}. Using $\\eta$, $I_s$ and the internal and external SOA constants taken from the literature for a typical silicon laser diode \\cite{Ghafouri-Shiraz2004}, \\cite{Figueiredo2011}, the SOA was modelled below the current threshold $I_{TR}$ (2-50 mA) and above $I_{TR}$ (70-110 mA).\n\n\n\\begingroup\n\\begin{equation} \\label{eq:IVRelationship}\nln\\big(I\\big) = ln\\big(I_s\\big) + \\bigg(\\frac{1}{\\eta}\\bigg) \\bigg(\\frac{q V}{K_b T}\\bigg)\n\\end{equation}\n\\endgroup\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.23]{Figures\/parso1.png}\n\\caption{Semi-logarithmic I-V plot for the SOA used to calculate $\\eta$ and $I_s$.}\n\\label{fig:soaIVCharacteristics}\n\\end{figure}\n\n\nThe SOA in the experimental setup had the optimum trade-off between gain and signal noise at a bias current of 70 mA, therefore the simulated SOA was biased at this current. Using Matlab's Simulink tool, a transfer function (TF) for the SOA EC was obtained and simplified as shown in (\\ref{eq:transferFunction}) with the constants defined in Table~\\ref{tab:transfer_func_table}. This allowed for custom drive signals to be generated, sent to the biased SOA EC and an optical output measured. \n\n\n\\begin{equation} \\label{eq:transferFunction}\n\\begin{split}\nTF                &= \\frac{2.01\\times 10^{85}} {\\sum_{i=0}^9a_is^i }\n\\end{split}\n\\end{equation}\n\n\n\\begin{table}[]\n    \\caption{Constants used in EC transfer function.}\n    \\label{tab:transfer_func_table}\n    \\centering\n    \\tabcolsep 0.1in\n    \\begin{tabular}{|c|c|c|c|}\n             \\hline\n        $a_9$ & 1.65 & $a_4$ &  $1.37\\times 10^{52}$\\\\\n         \\hline\n        $a_8$ &  $4.56\\times 10^{10}$ & $a_3$ & $2.82\\times 10^{62}$\\\\\n         \\hline\n        $a_7$ &  $3.05\\times 10^{21}$ & $a_2$ & $9.20\\times 10^{71}$\\\\\n         \\hline\n        $a_6$ & $4.76\\times 10^{31}$ &  $a_1$ & $1.69\\times 10^{81}$\\\\\n         \\hline\n        $a_5$ & $1.70\\times 10^{42}$ & $a_{0}$ & $2.40\\times 10^{90}$ \\\\\n         \\hline\n\n    \\end{tabular}\n\n\\end{table}\n\nIn the experimental setup (described later), an arbitrary waveform generator (AWG) with 12 GSPS sampling frequency was used allowing for signal bit windows of 83.3 ps, therefore for 20 ns time periods each signal had 240 points. As such, the optimisation algorithms were searching for a solution in a 240-dimensional search space. Additionally, the oscilloscope had 8-bit resolution, therefore each dimension in the solution could take one of 256 values. The EC simulation environment enabled different driving signals to be rapidly tested.\n\nThe constants used for the EC model were taken from the literature (\\cite{Ghafouri-Shiraz2004}, \\cite{Figueiredo2011}, and \\cite{Tucker1984}). The difficulty with SOA modelling, and subsequently also SOA switching, is that there are many variables whose values are difficult to experimentally measure, and which vary significantly even for same-specification SOAs due to parasitics introduced during manufacturing and packaging. Re-measuring these constants for a new SOA would be cumbersome, difficult, and unfruitful since broad assumptions would still need to be made. Furthermore, scaling this bespoke-modelling to 1,000s of SOAs in a single DC would be unrealistic. As such, analytical solutions to SOA switching are not beneficial. Additionally, different driving circuit setups with different amplifiers, bias tees, cabling etc. influence the shape of the driving signal that arrives at the SOA, thereby (if the methods described before this paper are used) requiring more manual tuning every time the equipment surrounding the SOA is changed. This highlights the need for the partially `model-free' AI approaches proposed in this paper, which neither make or require any assumptions about the SOA or the surrounding driving circuit they are optimising, resulting in their optimised driving signals being superior both in terms of performance and scalability relative to traditional analytical and\/or manual methods. Here, we borrow the term `model-free' from the field of reinforcement learning, meaning an algorithm that does not initially know anything about the environment in which it must perform its optimisation \\cite{10.5555\/3312046}.\n\nFig.~\\ref{fig:freq_response} compares the frequency response of the theoretical TF with the experimental SOA. The TF had a -3dB bandwidth of 0.5 GHz (around 700 ps rise time) compared to the experimental SOA's 0.6 GHz (around 550 ps rise time). These values were similar to one-another and consistent with both the theoretical and experimental optical responses. The differences between the responses were due to the use of EC parameters from the literature which do not exactly match those of our SOA.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.4]{Figures\/parso2.png}\n\n\\caption{Frequency responses of the theoretical transfer function (TF) and the experimental SOA (Exp).}\n\n\\label{fig:freq_response}\n\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{table}[]\n    \\caption{Factor(s) used on the EC transfer function coefficients to simulate different SOAs (factor = 1 unless stated otherwise).}\n    \\label{tab:diff_transfer_func_table_of_factors}\n    \\centering\n    \\tabcolsep=0.11cm\n    \\begin{tabular}{|c|c|c|c|c|}\n         \\hline\n        \\textbf{TF Component:} & Numerator & $a_0$ & $a_1$ & $a_2$    \\\\\n         \\hline\n         \n        \\textbf{Factor(s)}: & 1.0, 1.2, 1.4 & 0.8 & 0.7, 0.8, 1.2 & 1.05, 1.1, 1.2 \\\\\n         \\hline\n         \n    \\end{tabular}\n\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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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{Conclusion}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nSimulation and experimental results of SOA off-on switching were presented for various driving signal formats. The paper outlined a novel approach to SOA driving signal generation with AI algorithms which made no assumptions about the SOA and therefore were general, required no historic data collection and could be scaled to any SOA-based switch, opening up the possibility of rapid all-optical switching in real data centres. Experimental settling times (and therefore effective off-on times) of 547 ps were achieved using PSO, offering an order of magnitude performance improvement with respect to settling time over our implementation of the PISIC and MISIC techniques from the literature. Additionally, the standard PID control and raised cosine techniques from control theory were shown to be inadequate for the problem of ultra-fast SOA switching. Although ACO and GA demonstrated slightly faster rise times than PSO, PSO had a faster settling time and also a significantly lower 1.8\\% cost spread, giving greater reliability that any given PSO run had found the optimum solution. Furthermore, due to the fewer restrictions placed on the search space and the lower number of fine-tuned hyperparameters compared to ACO and GA, PSO was found to be more easy to generalise to unseen SOAs. Future work expanding on the presented methods could examine the robustness of the method with respect to hardware limitations\/irregularities (e.g. temperature\/bias current variations, bit resolution of driving voltage values or the sampling frequency of the AWG). Additionally, future work could extend the method to a scenario consisting of multiple, possibly cascaded, SOAs.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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{Experimental Setup}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe experimental setup is shown in Fig.~\\ref{fig:experimentalSetup}. An INPhenix-IPSAD1513C-5113 SOA with a 3dB bandwidth of 69 nm, a small signal gain of 20.8 dB, a 0-140 mA bias current range, a saturation output power of 10 dBm, a response frequency of 0.6 GHz, and a noise figure of 7.0 dB was used. An SHF 100 BP RF amplifier was selected by calculating the amplified MSE relative to the direct signal for different amplifiers, enabling a full dynamic range peak-to-peak voltage of 7V. A $50\\Omega$ resistor was placed before the SOA, allowing for the maximum allowed dynamic current range of 140 mA to be applied across the SOA.\n\nThe 70 mA optimum SOA bias current was found by measuring how MSE, optical signal-to-noise ratio (OSNR), rise time, overshoot, and optical gain varied with current. A 70 mA bias using a -2.5 dBm SOA input laser power produced the lowest rise time and MSE. The SOA was therefore driven between 0 and 140 mA centred at 70 mA. The other equipment used included a Lightwave 7900b lasing system, an Agilent 8156A optical attenuator, an LDX-3200 Series bias current source, a Tektronix 7122B AWG with 12 GSPS sampling frequency, an Anritsu M59740A optical spectrum analyser (OSA), and an Agilent 86100C oscilloscope (OSC) with an embedded photodiode. The RF signal going into the SOA had a rise time of 180 ps, therefore this was the best possible rise time (and settling time) that the SOA could have achieved. Throughout the experiments, a wavelength of 1,545 nm was used.\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.35]{Figures\/parso16.png}\n\\caption{Diagram of the SOA experimental setup used.}\n\\label{fig:experimentalSetup}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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{Experimental Results}\n\n\n\n\n\n\n\n\n\n\n\n\\begin{table*}[h]\n\\begin{center}\n    \\tabcolsep 0.1in\n    \\caption{\n        Comparison of SOA Optimisation Techniques. (Best in bold).}\n        \\begin{itemize}\n            \\item \\scriptsize $^{a}$ Though exact value not reported in \\cite{reviewer_reference_2}, it is referred to as being `below 500 ps'.\n            \\item \\scriptsize $^{b}$ Comparison of the ASM mounting against the commercial STF mounting.\n            \\item \\scriptsize $^{c}$ Exact value not reported in \\cite{reviewer_reference_2} so percentage improvement is (approximately) inferred from a graph presented in \\cite{reviewer_reference_2}. Comparison made at bias current value corresponding to the best case performance of the best performing ASM mount + drive combination and is compared against the STF mount + drive at the same bias and for the same drive (step was best performing in the reported metrics).\n            \\item \\scriptsize $^{d}$ Comparison is made between the best and worst cases presented in \\cite{reviewer_reference_1}.\n            \\item \\scriptsize $^{e}$ Several variants of the `MISIC' format were tested in \\cite{Figueiredo2015} and the best is used here for comparison.\n            \\item \\scriptsize $^{f}$ Comparison made with respect to the performance of the STEP driving signal presented in \\cite{Figueiredo2015}.\n        \\end{itemize}\n    \n    \\begin{tabular}{|c|c|c|c|c|c|}\n        \\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{Method \\\\ (Technique)}}\n        & \\textbf{Reference}\n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering \\textbf{Rise Time, ps \\\\ (Reduction, \\%)}}\n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering \\textbf{Settling Time, ps \\\\ (Reduction, \\%)}}\n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering \\textbf{Overshoot, \\% \\\\ (Reduction, \\%)}}\n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering \\textbf{Guard Time, ps \\\\ (Reduction, \\%)}}\n        \\\\\\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{PSO \\\\ (Signal Optimisation)}}\n        & This work \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 454 ps \\\\ (35\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering \\textbf{547 ps \\\\ (85\\%)}} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 5\\% \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        \\\\\\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{ACO \\\\ (Signal Optimisation)}}\n        & This work \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 413 ps \\\\ (41\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 560 ps \\\\ (85\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 4.8\\% \\\\ -}\n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        \\\\\\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{GA \\\\ (Signal Optimisation)}}\n        & This work \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering \\textbf{340 ps \\\\ (51\\%)}} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 825 ps \\\\ (78\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 10.3\\% \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        \\\\\\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{PISIC \\\\ (Signal Optimisation)}}\n        & This work \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 502 ps \\\\ (28\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 4350 ps \\\\ (-17\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 40.5\\% \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        \\\\\\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{MISIC1 \\\\ (Signal Optimisation)}}\n        & This work \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 502 ps \\\\ (28\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 4020 ps \\\\ (-8\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering undershot \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        \\\\\\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{Raised Cosine \\\\ (Signal Optimisation)}}\n        & This work \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 921 ps \\\\ (-32\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 4690 ps \\\\ (-26\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering undershot \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        \\\\\\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{PID Control \\\\ (Signal Optimisation)}}\n        & This work \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 501 ps \\\\ (28\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 4020 ps \\\\ (-8\\%)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 2.3\\% \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        \\\\\\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{ASM Mounting + STEP Drive \\\\ (Microwave Mounting Optimisation)}} \n        & \\cite{reviewer_reference_2} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering \\textbf{$\\approx$ 5\\% $^{[c]}$ \\\\ ($\\approx$ 75\\% $^{[b,c]}$)}}\n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering $\\approx$ 500 ps $^{[a]}$ \\\\ ($\\approx$ 33\\% $^{[b,c]}$)} \n        \\\\\\hline\n        \n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{STEP Drive + Wiener Filtering \\\\ (Signal Optimisation + Filtering)}} \n        & \\cite{reviewer_reference_1} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -}\n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering \\textbf{286 ps} \\\\ \\textbf{(60\\% $^{[d]}$)}} \n        \\\\\\hline\n        \n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{PISIC Drive \\\\ (Signal Optimisation)}} \n        & \\cite{Figueiredo2015} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 115 ps \\\\ (34\\% $^{[f]}$)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 25\\% \\\\ (-56\\% $^{[f]}$)}\n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        \\\\\\hline\n        \\multicolumn{1}{|p{3cm}|}{\\centering \\textbf{MISIC-6 Drive $^{[e]}$ \\\\ (Signal Optimisation)}} \n        & \\cite{Figueiredo2015} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 115 ps \\\\ (34\\% $^{[f]}$)} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering 12.5\\% \\\\ (22\\% $^{[f]}$)}\n        & \\multicolumn{1}{|p{2.2cm}|}{\\centering - \\\\ -} \n        \\\\\\hline\n    \\end{tabular}\n    \\label{tab:review_comparison_table}\n    \\end{center}\n\\end{table*}\n\n\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.26]{Figures\/parso17.png}\n\\caption{Experimental SOA responses to the step, PISIC, MISIC1, raised cosine and PID driving signals.}\n\\label{fig:expSOAResponses}\n\\end{figure}\n\nIn this section the experimental results for the SOA responses to step, PISIC, MISIC, raised cosine, PID and AI driving signals have been compared. The objective was to reduce the off-on switching time and power oscillations (measured by the settling time and overshoot metrics).\n\nA step driving signal was the simplest format used to drive the SOA. Fig.~\\ref{fig:expSOAResponses} (which  has  been  normalised  with  respect  to  the steady  state  value  as done in \\cite{Figueiredo2015} for easy comparison) shows the SOA optical response to a step driving signal, resulting in a rise time, settling time and overshoot of 697 ps, 3.72 ns and 0.0\\% (since it undershot the steady state) respectively. \n\nThe PISIC format proposed by \\cite{Gallep2002} was applied to the SOA with 2.95V step + 4.05V impulse, and the response is shown in Fig.~\\ref{fig:expSOAResponses} with a rise time, settling time and overshoot of 502 ps, 4.35 ns and 40.5\\% respectively. The form of the PISIC pulse used was optimised for the SOA in use, where different step-impulse voltage combinations (as done in [18]) were tested, as well as varying widths of the pre-impulse section of the PISIC signal as a percentage of the total signal length centered at the percentage used in [18]. It was found that a 500ps pulse width gave the best results.\n\nThe MISIC 1-6 bit-sequences proposed by \\cite{Figueiredo2015} were applied with 2.95V step + 4.05V impulse, where the same step-impulse voltage combinations were tested as for PISIC. The format with the best performance was MISIC1, whose response is shown in Fig.~\\ref{fig:expSOAResponses} with a rise time, settling time and overshoot of 502 ps, 4.02 ns and 0.0\\% (undershot) respectively. \n\nA popular approach to optimising oscillating systems in control theory is the raised cosine approach, whereby the rising step for a signal of period $T$ is adapted to a rising cosine defined by the frequency-domain piecewise function in (\\ref{eq:raisedCosine}). As $\\beta$ increases ($0 \\leq \\beta \\leq 1$), the rate of signal rise decreases. The best performing raised cosine was $\\beta = 0.5$, whose response is shown in Fig.~\\ref{fig:expSOAResponses} and whose rise time, settling time and overshoot were 921 ps, 4.69 ns and 0.0\\% (undershot) respectively. \n\n\\begingroup\n\\small\n\\begin{equation} \\label{eq:raisedCosine}\nH(f) = \\begin{cases}\n    1, & \\text{if $f \\leq \\frac{1-\\beta}{2T}$} \\\\\n    \\frac{1}{2} \\left[ 1 + cos \\left( \\frac{\\pi T}{\\beta} \\left[f - \\frac{1-\\beta}{2T} \\right] \\right) \\right], & \\text{if $ \\frac{1-\\beta}{2T} < f \\leq \\frac{1+\\beta}{2T} $} \\\\\n    0, & \\text{otherwise}\n    \\end{cases}\n\\end{equation}\n\\endgroup\n\nAnother popular approach in control theory is the PID controller. The optical response of the PID control signal is shown in Fig.~\\ref{fig:expSOAResponses}, with a rise time, settling time and overshoot of 501 ps, 4.02 ns and 2.3\\% respectively. In order to quickly obtain values for the 3 PID parameters, $K_c, K_i \\text{ and } K_d$, a First Order Plus Dead Time (FOPDT) model was applied to the SOA, where the key parameters for this model ($K_p, \\tau_p \\text{ and } \\theta_p$) can be measured directly from the step response of the device. The PID tuning parameter, $\\tau_c$, which is inversely proportional to the magnitude of the response to offset, was tested with values between that of an `aggressive' tuning regime $(\\tau_c \\approx 0.1)$ and a `conservative' one $(\\tau_c \\approx 10.0)$. The results shown in Fig.~\\ref{fig:expSOAResponses} are with $\\tau_c = 5.0$ which was found to be the best performing value.\n\n\nThe PSO algorithm used in the simulation environment was applied to the real SOA. The SP and the PSO response are shown in Fig.~\\ref{fig:experimentalResults}, with a rise time, settling time and overshoot of 454 ps, 547 ps and 5.0\\% respectively.\n\n\n\n\\begin{figure*}[!t]\n\\centering\n\n    \\begin{tabular}{ccc}\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso18.png} \n        \\put(15, 68){(a)}\n    \\end{overpic}\n    &\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso19.png}\n        \\put(15, 68){(b)}\n    \\end{overpic}\n    &\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso20.png}  \n        \\put(15, 68){(c)}\n    \\end{overpic}\n      \n      \n    \\end{tabular}\n\n\\caption{Experimental results showing the optimised SOA optical outputs for (a) PSO, (b) ACO, and (c) GA.}\n\\label{fig:experimentalResults}\n\\end{figure*}\n\n\n\\begin{figure*}[!t]\n\\centering\n\n    \\begin{tabular}{ccc}\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso21.png} \n        \\put(15, 68){(a)}\n    \\end{overpic}\n    &\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso22.png}\n        \\put(15, 68){(b)}\n    \\end{overpic}\n    &\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso23.png}  \n        \\put(15, 68){(c)}\n    \\end{overpic}\n      \n      \n    \\end{tabular}\n\n\\caption{Experimental results showing the optimised SOA electrical driving signal inputs for (a) PSO, (b) ACO, and (c) GA.}\n\\label{fig:ai_optimised_ops}\n\\end{figure*}\n\n\n\n\n\n\n\n\n      \n      \n\n\n\nAn ACO run with 200 ants accomplished a rise time, settling time and overshoot of 413 ps, 560 ps and  4.8\\% respectively, performing similarly well to the PSO algorithm. The ACO result is shown in Fig.~\\ref{fig:experimentalResults}\n\nSimilarly, the GA result shown in Fig.~\\ref{fig:experimentalResults} had a rise time, settling time, and overshoot of 340 ps, 825 ps, and 10.3\\% respectively. The rise times of the AI algorithms were an order of magnitude improvement on the step's, and the settling times (and therefore the effective off-on switching time) were several factors faster than the previous MISIC1 optimum from the literature, bringing SOA switching times truly down to the hundred ps scale. A scatter plot comparing these data is shown in Fig.~\\ref{fig:Standard_3D_Scatter}. \n\nBy comparison, PSO had the lowest settling time and therefore the lowest overall switch time. We hypothesise that this was due to the fact that PSO, being less memory-hungry than ACO and having superior convergence properties compared to GA as a result of having fewer hyperparameters to fine-tune and a smaller search space with the PISIC shell, was able to be given a better search space-hyperparameter tuning trade-off, and therefore was able to find a more optimum driving signal. This larger search space also enabled PSO to explore a wider variety of drive signal solutions without needing a large number of hyperparameters tuned (which adds complexity), allowing PSO to generalise to a more diverse set of SOAs than either ACO or GA were able to. Therefore, although in theory all AI algorithms used were powerful and generalisable, due to the number of hyperparameters and search space restrictions that were required in practice, PSO had both the best performance and generalisability, although GA came close to matching PSO.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.26]{Figures\/parso24.png}\n\\caption{Scatter plot comparing the experimental rise times, settling times and overshoots of all the driving signals tested. The outlined target region highlights the performance required for truly sub-nanosecond optical switching.}\n\\label{fig:Standard_3D_Scatter}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\nTable \\ref{tab:review_comparison_table} shows results (both absolute, and relative improvement for cross-comparison) of the rise time, settling time, overshoot and guard time for all methods implemented in this work, as well as a variety from the literature. The rows associated with \\cite{Figueiredo2015} are the results for the optimised PISIC and MISIC-6 signals defined and implemented in this work. This is distinguished from the other two columns with `PISIC' and `MISIC' methods referred to as coming from this work, which are a re-implementation of the methods described in \\cite{Figueiredo2015} but applied to and optimised for a different experimental setup.\n\n\n\nFinally, Fig.~\\ref{fig:ai_optimised_ops} shows the electrical drive signals found by each algorithm. Whilst we stress that the main focus of this paper is the \\textit{method} rather than the \\textit{specific drive signal}, the drive signal is important for real-World implementation and general understanding of the search space restrictions used. As Fig.~\\ref{fig:ai_optimised_ops} shows, the derived driving signals are noisy despite a smooth resultant optical output. This is likely because the AWG (arbitrary waveform generator using an 8-bit digital to analogue converter) drive signal frequency was  6 GHz offering 12 GSa\/s whereas the SOA used had a -3dB  frequency response of 0.6 GHz, therefore we over-sampled the drive signal by approximately ~10x. In a real DCN scenario, to implement our algorithms' driving signals in practice, we would likely use an FPGA or ASIC with an embedded on-chip DAC  for multilevel signal generation, and there are already existing FPGAs (a.k.a. RF System on Chip (RFSoC)) that  support multiple DACs at 6 GSa\/s. Therefore in practice the search space would be lower (fewer dimensions\/number of points to optimise) than assumed in this paper, and we would expect this to improve the AI convergence characteristics. Further experiments using fewer points in the drive signal\/a slower AWG are necessary to see what the true effects are on the AI algorithms. This is beyond the scope of this paper, and we intend to further investigate it in our future work.\n\nWithin the context of a DCN implementation of the presented methods, some considerations were made with respect to the effect that the algorithms have on the signal to noise ratio (SNR). Namely, it should be considered if the oscillations caused by the algorithms (all of which are of the order of 5\\%) have a negative effect on the SNR of the ON period of the output, particularly in comparison to the output of a step driving signal, where the ON period considered is defined as starting when the signal enters the $\\pm 5\\%$ (with respect to the steady state) region for a 20 ns pulse length. Following from the model of amplifier noise given in \\cite{agrawal} and accounting for Shot noise, intrinsic amplifier noise (the noise figure of the SOA) and the additional noise due to the fluctuations in the output, we consider the penalty on the noise figure (as defined in \\cite{agrawal}) due to the deviations of the output from its steady state value throughout the duration of its ON period. Assuming (based on intrinsic and Shot noise contributions) a base noise figure (i.e. if the driving method caused no deviations at all) of 7.1dB, the measured noise figure penalties for ACO, PSO, GA and step were 1.05 dB, 0.65 dB, 1.12 dB and 0.53 dB with SNR values of 28.52 dB, 28.90 dB, 28.54 dB and 29.06 dB respectively, showing that the additional noise figure penalty due to the AI methods ranges between  0.08 dB (PSO) and 0.59 dB (GA) compared to a step in the case of the best performing algorithm (PSO).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n      \n      \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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{Optimisation Algorithms}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nAll AI algorithms had the goal of minimising the MSE between the actual SOA output and an ideal SOA step output with 0 rise time, settling time and overshoot. The closer the driving signal's corresponding output `process variable' (PV) was to achieving this ideal `set point' (SP), the lower its MSE (defined in (\\ref{eq:meanSquaredError})).\n\n\\begingroup\n\\begin{equation} \\label{eq:meanSquaredError}\nMSE = \\frac{1}{m} \\sum_{g=0}^{m} \\left( PV - SP \\right)^2\n\\end{equation}\n\\endgroup\n\n\n\\subsection{Particle Swarm Optimisation (PSO)}\n\n\\subsubsection{Implementation}\n\nAn overview of PSO is given in \\cite{Kiranyaz2014}, \\cite{Iqbal2015a}. PSO is a population-based AI metaheuristic for optimising continuous nonlinear functions. First proposed in 1995 by \\cite{Kennedy1995}, it combines swarm theory by observing natural phenomena such as bird flocks and fish schools with evolutionary programming. In this paper, PSO is adapted to be applicable to SOA drive signal optimisation. \n\nTo apply PSO to SOA optimisation, $n$ particles (driving signals) were initialised at random positions in a hyper-dimensional search space with $m=240$ dimensions (number of points in the signal). Since experimental results showed spurious overshoots after the rising edge and therefore an increase in the settling time, the PSO search space was bounded by a PISIC-shaped `shell' beyond which the particle dimensions could not assume values. An added benefit of the shell was a reduction in the complexity of the problem and therefore also the convergence time. The shell area was a PISIC signal with a leading edge whose width was defined as some fraction of the `on' period of the signal. At each generation, in order to evaluate a given particle position, the MSE in (\\ref{eq:meanSquaredError}) was used to calculate the fitness (which was to be minimised). As discussed in \\cite{Clerc1999}, the particle inertia weights ($w$) and personal and social cognitive acceleration constants ($c_1$ and $c_2$) can be dynamically adapted as the PSO population evolves. This was done using the update rules in (\\ref{eq:updateW}), (\\ref{eq:updateM}), and (\\ref{eq:updateC}) \\cite{Clerc1999} at the start of each generation, where $p_{best_{j}}$ was the historic personal best position of particle $j$, $x_j$ was the position (amplitude taken) of particle $j$, $w(0)$ was the initial inertia weight constant ($0~\\leq~w(0)~<~1$), $w(n_t)$ was the final inertia weight constant ($w(0) > w(n_t)$), $m_j(t)$ was the relative fitness improvement of particle $j$ at time $t$, and $c_{max}$ and $c_{min}$ were the maximum and minimum values for the acceleration constants. So long as these values satisfied (\\ref{eq:conditionForConvergence}), PSO was guaranteed to converge on some driving signal \\cite{VanDenBergh2001}. Using dynamic PSO significantly improved the algorithm's performance (see the `Hyperparameter Tuning' section below).\n\n\n\\begingroup\n\\begin{equation} \\label{eq:updateW}\nw_j(t+1) = w(0) + \\left[ \\left( w(n_t) - w(0) \\right) \\cdot \\left( \\frac{e^{m_j(t)} -1} {e^{m_j(t)} + 1} \\right) \\right]\n\\end{equation}\n\\endgroup\n\n\\begingroup\n\\begin{equation} \\label{eq:updateM}\nm_j(t) = \\frac{p_{best_j}(t) - x_j(t)}{p_{best_j}(t) + x_j(t)}\n\\end{equation}\n\\endgroup\n\n\\begingroup\n\\begin{equation} \\label{eq:updateC}\nc_{1,2}(t) = \\frac{c_{min} + c_{max}}{2} + \\frac{c_{max} - c_{min}}{2} + \\frac{e^{-m_j(t)}-1}{e^{-m_j(t)} + 1}\n\\end{equation}\n\\endgroup\n\n\\begingroup\n\\begin{equation} \\label{eq:conditionForConvergence}\n0 \\leq \\frac{1}{2} \\left( c_1 + c_2 \\right) - 1 < w < 1\n\\end{equation}\n\\endgroup\n\n\n\n\n\n\nThis PSO process could be repeated until the particles converged on a position with the best fitness (i.e. the optimum SOA driving signal). To help with convergence time and performance, some additional constants were defined:\n\n\\begin{itemize}\n    \\item $iter_{max}$ = Maximum number of iterations that PSO could evolve through before termination. Higher gives more time for convergence but longer total optimisation time.\n    \n    \\item $max\\_v\\_f$ = Factor controlling the maximum velocity a particle could move with at each iteration. Higher can improve convergence time but, if too high, particles may oscillate around the optimum and never converge.\n    \n    \\item $on\\_s\\_f$ and $off\\_s\\_f$ = `On' and `off' suppression factors used to set the minimum and maximum driving signal amplitudes the particle positions could take when the step signal was `on' and `off' respectively. Lower will restrict the particle search space to make the problem tractable for the algorithm, but too low will impact the generalisability of the algorithm to any SOA.\n    \n    \\item $shell\\_w\\_f$ = Factor by which to multiply the `on' time of the signal to get the width of the leading edge of the PISIC shell. Higher (wider) value will give the algorithm more freedom to rise over a longer period at the leading edge of the signal and improve generalisability, but will also increase the size of the search space and impact convergence.\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\\subsubsection{Hyperparameter Tuning}\n\nThe simulation environment enabled the PSO hyperparameters to be rapidly tuned by plotting the PSO learning curve (MSE vs. number of iterations). Since the same PSO algorithm ran multiple times may converge on different minima, each PSO version with its unique hyperparameters was ran 10 times and the 10 corresponding learning curves plotted on the same graph to get a `cost spread' (i.e. how much the converged solution's MSE varied between PSO runs). A lower cost spread gave greater reliability that PSO had converged on the best solution that it could find rather than getting stuck in a local minimum. \n\nTo begin with, it was found that using dynamic PSO whereby $w$, $c_1$ and $c_2$ were adapted at the beginning of each generation led to multiple advantages. Firstly, the solution found by 10 dynamic particles had the same MSE as that found by 2,560 static particles, reducing the computation time by a factor of 256. Secondly, the final driving signal found by adaptive PSO was significantly less noisy since it was less prone to local minima. Thirdly, the final MSE found was 63\\% lower. Fourthly, although the relative cost spread of dynamic PSO was 72\\% compared to 50\\% due to the lower MSE, the absolute cost spread was just $8.7\\times 10^{-13}$ compared to $140\\times 10^{-13}$. Pursuing with dynamic PSO, it was found that placing a `PISIC shell' on the search space (with $shell\\_w\\_f = 0.1$) beyond which the particles could not travel led to an absolute cost spread of $6.9\\times 10^{-13}$ and a further 14\\% reduction in the final cost (despite initial costs being higher due to the fact that PISIC signals lead to greater overshoot and subsequently also greater oscillations). Finally, it was also found that initialising one of the $n$ particle positions as a step driving signal improved the convergence time by a factor of two. Using dynamic PSO, a PISIC shell and an embedded step, the following hyperparameter values were found to give the best spread, final cost and convergence time: $iter_{max} = 150$, $n = 160$, $max\\_v\\_f = 0.05$, $w(0) = 0.9$, $w(n_t) = 0.5$, $c_{min} = 0.1$, $c_{max} = 2.5$, $on\\_s\\_f = 2.0$, and $off\\_s\\_f = 0.2$. This final tuning resulted in a cost spread of just 1.8\\%. The evolution of this PSO tuning process is summarised in Fig.~\\ref{fig:simulatedPsoOutputs}, where the learning curves for the above sets of hyperparameters have been plotted in red, orange, blue and green respectively. The final PSO SOA output, shown in Fig.~\\ref{fig:simulatedPsoOutputs}, had a rise time, settling time and overshoot of 669 ps, 669 ps and 3.7\\% respectively. Fig.~\\ref{fig:simulatedPsoOutputs} also shows the optical response to a step driving signal, showing a rise time, settling time and overshoot of 669 ps, 4.85 ns and 31.1\\% respectively. Thus, the simulations indicated that the settling time (and therefore the effective off-on switching time) could be reduced by a factor of 7.2 and the overshoot by a factor of 8.4 compared to a step. Although rise time remained unimproved, the experimental results section shows that, for a real SOA with optical drift, PSO improves all three parameters.\n\n\n\n\\begin{figure*}[!t]\n\\centering\n\n    \\begin{tabular}{ccc}\n    \n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso3.png}\n        \\put(15, 68){(a)}\n    \\end{overpic}\n    & \n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso4.png}\n        \\put(15, 68){(b)}\n    \\end{overpic}\n    &\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso5.png}\n        \\put(15, 68){(c)}\n    \\end{overpic}\n  \\\\\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso6.png} \n        \\put(15, 68){(d)}\n    \\end{overpic}\n    &\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso7.png}\n        \\put(15, 68){(e)}\n    \\end{overpic}\n    &\n    \\begin{overpic}[width=0.3\\textwidth]{Figures\/parso8.png}  \n        \\put(15, 68){(f)}\n    \\end{overpic}\n        \n    \\end{tabular}\n\n\\caption{Simulated SOA optical response to (a) PSO, (b) ACO, and (c) GA driving signals relative to a standard step input. For reference, the target SPs used have also been plotted. Learning curves showing how both the cost spread and the optimum solution improved as the (d) PSO, (e) ACO, and (f) GA algorithms were tuned, showing 10 learning curves for each set of hyperparameters. The curves for the optimum hyperparameters have been plotted in green. For PSO in (d), some additional information has been plotted: i) No dynamic PSO, PISIC shell, or embedded step (red), ii) no PISIC shell or embedded step (blue), iii) no embedded step (orange), and iv) the final PSO algorithm (green, also plotted on separate graph (inserted)). For GA, the i) default DEAP (red) and ii) optimised (green) hyperparameter learning curves have been plotted. For ACO, the blue curve is for a run with a larger pheromone exponent (0.5) value than the optimum, and the red is for a larger dynamic range on the signal search space ($\\pm50 \\%$).}\n\n\\label{fig:simulatedPsoOutputs}\n\n\\end{figure*}\n\n\n\n\n\n    \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{Ant Colony Optimisation (ACO)}\n\n\n\\subsubsection{Implementation}\n\nACO is primarily a path-finding evolutionary algorithm modelled on observations of how ant colonies find food sources in nature. As such, it optimises to find an optimum path along nodes in a graph, $G = \\{g_{i}\\}$ by means of probabilistic exploration, and colony exploitation across generations of ants. A more comprehensive explanation of ACO can be found in \\cite{aco_book}. Of several ACO variants, the `Ant Colony System' algorithm was used in this work.\n\nSince ACO is typically applied to routing problems, considerations must be made as to how to represent parameter selection as such a problem. A system with $N$ parameters each having $M$ possible values can be modelled as a graph with $|N|$ clusters of $|M|$ nodes, where each node maps to a possible value of a particular parameter. A path can then be found that visits one node in each cluster, defining a set of parameter values after each cluster has been visited once and only once.\n\nFor example, consider an $N=3$ parameter $(a, b, c)$ system where each parameter can take 1 of $M=2$ possible values which are selected in the order $a\\rightarrow b\\rightarrow c$. A $(N \\times N) \\times (M \\times M)$ matrix representing the probability of choosing a value for one parameter, given a previous value choice for another, can be written as in (\\ref{eq:aco_matrix}) where $\\alpha^{xy}_{ij}$ is the probability of choosing value $j$ for parameter $y$, given that value $i$ for parameter $x$ was just chosen. Zeroing the matrix entries appropriately ensures that parameter values are selected in order.\n\n\\begin{equation} \\label{eq:aco_matrix}\n    \\begin{pmatrix}\n        0 & 0 & \\alpha^{ab}_{00} & \\alpha^{ab}_{01} & 0 & 0 \\\\\n        0 & 0 & \\alpha^{ab}_{10} & \\alpha^{ab}_{11} & 0 & 0 \\\\\n        0 & 0 & 0 & 0 & \\alpha^{bc}_{00} & \\alpha^{bc}_{01} \\\\\n        0 & 0 & 0 & 0 & \\alpha^{bc}_{10} & \\alpha^{bc}_{11} \\\\\n        0 & 0 & 0 & 0 & 0 & 0 \\\\\n        0 & 0 & 0 & 0 & 0 & 0 \\\\\n    \\end{pmatrix}\n\\end{equation}\n\n\n\\subsubsection{Hyperparameter Tuning}\n\nThe important hyperparameters with respect to ACO (specifically the Ant Colony System algorithm used here) are the pheromone exponent (where higher values encourage more exploitation of previously found paths), the evaporation exponent (where higher values discourage exploitation of previously found paths) and the probability of an ant travelling along a randomly selected path. Additionally, the number of ants and generations must be selected.\n\nParameters were tuned by means of running optimisation routines with one hyperparameter varying across a range of values and the rest kept constant. For each MSE value, the learning curve from 10 different runs were plotted against each other. Just as with PSO, parameter values were selected to prioritise the minimisation of cost spread to ensure that the optimisation technique could give consistent results when used on different occasions. Firstly, it was found that beyond 200 ants, the cost spread did not improve significantly. Similarly, regardless of the spread, the ACO routine was typically converging after between 60 and 75 generations, so a generation cap of 100 was imposed since this was sufficient to guarantee convergence. The values for the other parameters were the pheromone constant $\\alpha = 0.25$, the evaporation constant $\\rho = 0.5$ and the exploration probability $p = 0.1$. It was also found that minimising the search space by reducing the dynamic range of the signal to $\\pm25\\%$ centred at $50\\%$ of the maximum shortened convergence time without degradation of the final signal, which had the advantage of making matrices memory sizes manageable. No further hyperparameters, such as the PISIC shell applied with the PSO method, were utilised, which is more desirable since fewer hyperparameters simplify the tuning process.\n\nAs seen in Fig.~\\ref{fig:simulatedPsoOutputs}, the spread of the ACO routine was reduced from 23\\% to 14.9\\% through tuning, but was still less consistent than the 1.8\\% spread of the PSO algorithm. Fig.~\\ref{fig:simulatedPsoOutputs} shows the convergence of the Ant Colony System algorithm for various hyperparameter combinations (described in the figure's caption). While the spread in the early iterations of the routine is explained by the embedding of a square signal in the PSO routine described above (since it is very unlikely to randomly initialise a signal better than a square and the ACO does not use any sort of initial signal embedding), the spread in the later stages is thought to be due to some practical limitations of the ACO optimisation method. For $N$ parameters with $M$ values each, the ACO routine requires 2 $(N^2 \\times M^2)$ matrices (point-wise multiplied to make a third). A 100 point signal with 100 possible values per point gives a matrix with $100,000,000$ elements. Implemented with the popular NumPy Python library, a minimum of 8 bytes per floating point means such a matrix is on the order of gigabytes. Given the relatively low power PC used in the experiment, restrictions on the state space had to be imposed due to memory limitations. This meant that rather than optimising each point on the signal (240) with the maximum resolution allowed by the AWG (8 bit = 256 points), only 180 points (those in the HIGH state of the initial driving step signal) were optimised with a resolution of 50 points. This means that the state space viewed by the ACO routine was more strongly discretised than that viewed by a method (such as PSO) with lower memory requirements, limiting how optimum the generated signal can be and how well ACO could generalise to other SOAs. Nevertheless, as will be seen, the ACO still produced driving signals that improved upon previous methods. The final ACO tuning output, shown in Fig. \\ref{fig:simulatedPsoOutputs}, had a rise time, settling time and overshoot of 753 ps, 1.58 ns and 9.1\\% respectively.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Genetic Algorithm (GA)}\n\n\\subsubsection{Implementation}\nGAs are a group of nature-inspired population-based metaheuristics. The term `Genetic Algorithm' relates to the model proposed by John Holland in 1975 \\cite{Holland1975}. A detailed explanation of GAs can be found in \\cite{Whiteley1994}.\n\nThe DEAP Python library \\cite{fortin2012deap} was used to implement the canonical GA. Each optimisation started with an initial population of 100 individuals with random positions in order to span as much of the search space as possible. Each individual was represented by an array of 240 points with values within the 7V range, therefore representing a driving signal.\n\nDuring the evolutionary process, the mutation stage was performed by applying Gaussian noise to some points of each individual. Any individuals with points which went beyond the supported 7V range were discarded.\n\n\n\n\\subsubsection{Hyperparameter Tuning}\n\nAs described in \\cite{Whiteley1994}, there are three parts to the evolutionary process: selection, crossover, and mutation. Each of these can be implemented in a few different ways (e.g. Proportionate, Ranking or Tournament for selection \\cite{sivaraj2011review}), and each of these implementations use different hyperparameters (e.g. $tournsize$ for Tournament Selection; or $\\mu$, $\\sigma$, and $indpb$ for Gaussian Mutation). This results in an overall high number of hyperparameters which might significantly impact the probability of the GA getting stuck in a local minimum as well as the speed of convergence. The high number of hyperparameters also meant that there were more values to fine-tune, which made tuning both more complex and time consuming, thereby reducing its generalisability. Since the high number of hyperparameters already impacted generalisability, we refrained from restricting the search space (as done with ACO and with the PSO PISIC shell) to try to still allow for as much generalisability as possible, but this would have the knock-on effect of poorer convergence and a lesser settled signal. However, as demonstrated in Fig. \\ref{fig:diff_tf_pvs_pso}, GA was still able to generalise fairly well to 10 different SOAs.\n\nThe DEAP library documentation came with a set of suggested default hyperparameter values. These were varied using grid search over 61 optimisations. A limit on the number of generations was set to 500, which was found to be sufficient for convergence.\n\nMutation was implemented using Gaussian Mutation, which has a probability $indpb$ (mutation rate) of changing each of an individual's points by applying normally distributed noise of mean $\\mu$ and standard deviation $\\sigma$. Using a negative $\\mu$ led to a solution with lower values, while a positive $\\mu$ did the opposite - each leading to a lower overall performance, so $\\mu$ was set to 0. Decreasing $indpb$ or $\\sigma$ slowed down the process as it reduced the overall mutation speed, but increasing either one too much led to the GA getting stuck at local minima. By performing grid search on the hyperparameters, the optimal values were found to be 0.06 and 0.15 respectively. A population size of 60 led to the fastest initial convergence speed (per number of fitness function evaluations), however, the higher number of 100 individuals in a population led to a better overall solution after many generations. Additionally, both $cxpb$ (the probability of mating two individuals), and $mutpb$ (the probability of mutating an individual), were increased significantly from 0.6 to 0.9 and from 0.05 to 0.3 respectively. Increasing $tournsize$ (which controls the number of randomly selected individuals from which to choose the best one for the next generation \\cite{miller1995genetic}) above 4 did not have an impact on the convergence, whereas using the values of 2 and 3 significantly slowed down the process. Most hyperparameters did not change by much from the DEAP library's default values since the initial values were almost optimal and changing them led to a slower convergence.\n\nFig.~\\ref{fig:simulatedPsoOutputs} shows the 10 learning curves for the default hyper parameters (red) and the optimised parameters (green), where the cost spread was reduced from 58.6\\% to 10.8\\%. Fig.~\\ref{fig:simulatedPsoOutputs} also shows the simulated SOA output of the tuned GA algorithm with a rise time, settling time and overshoot of 799 ps, 2.55 ns, and 9.0\\% respectively.\n\nThe hyperparameters of the AI algorithms can be used to address the general problem of `SOA optimisation'. This is because the hyperparameters are only for restricting the search space to reduce the size of the problem, and restricting how much the algorithm can change its solution between iterations; they are specific to the general SOA optimisation problem, but \\textit{not} to a specific SOA. The EC simulation environment provided a useful test bed in which to tune the algorithm hyperparameters and allow optimisation of any SOA (even though drive signal solutions derived from simulations are not directly transferable to experiment).\n\n\\begin{figure}[!t]\n\\centering\n\n    \\begin{tabular}{cc}\n    \n    \\begin{overpic}[scale=0.17]{Figures\/parso9.png}\n        \\put(87, 69){(a)}\n    \\end{overpic}\n    \n    \\\\\n    \n    \\begin{overpic}[scale=0.17]{Figures\/parso10.png}\n        \\put(87, 71){(b)}\n    \\end{overpic}\n    \n    \\begin{overpic}[scale=0.17]{Figures\/parso11.png}\n        \\put(87, 69){(c)}\n    \\end{overpic}\n    \n    \\\\\n    \\begin{overpic}[scale=0.17]{Figures\/parso12.png}\n        \\put(87, 71){(d)}\n    \\end{overpic}\n    \n    \\begin{overpic}[scale=0.17]{Figures\/parso13.png}\n        \\put(87, 69){(e)}\n    \\end{overpic}\n    \n    \\\\\n    \n    \\begin{overpic}[scale=0.17]{Figures\/parso14.png}\n        \\put(87, 69){(f)}\n    \\end{overpic}\n    \n    \\begin{overpic}[scale=0.17]{Figures\/parso15.png}\n        \\put(87, 69){(g)}\n    \\end{overpic}\n    \n    \n    \n        \n    \\end{tabular}\n\n\\caption{Simulated SOA optical responses of 10 different SOAs (each with a different transfer function) to (a) step, (c) PSO, (e) ACO, and (g) GA, and the corresponding driving signals for (b) PSO, (d) ACO, and (f) GA. All AI optimisations were done with the same hyperparameters and a common target SP.}\n\n\\label{fig:diff_tf_pvs_pso}\n\n\\end{figure}\n\n\nTo test the above claim that these algorithms can in theory be generalised to any SOA, we generate 10 different transfer functions each modelling a different SOA. These were generated by multiplying the coefficients in Table~\\ref{tab:transfer_func_table} by various factors (summarised in Table~\\ref{tab:diff_transfer_func_table_of_factors} so as to be reproducible), thereby simulating SOAs with different characteristics. The optical outputs of these different SOAs in response to the same step driving signal are shown in Fig.~\\ref{fig:diff_tf_pvs_pso}. Using the PSO and GA algorithms with the \\textit{same hyperparameters}, all 10 of these SOAs are able to be optimised with no changes to the algorithms, as shown in Fig.~\\ref{fig:diff_tf_pvs_pso} (where the AI electrical drive signals have been included for reference). Due to search space restrictions, ACO could not generalise. For all 10 SOAs, a common target set point was chosen. The set point was defined as a perfect 0 overshoot, rise time and settling time step response based on the steady states of the initial step response of one of the simulated SOA's. However, the target can be arbitrarily defined by the user if a different optical response is required, demonstrating the flexibility of the AI algorithms to optimise optical outputs with respect to specific problem requirements. Relative to this target set point, the performances are summarised in Table~\\ref{tab:diff_transfer_func_comparison_table}. Signals that did not settle have been marked as `-' and excluded from performance summary metrics. PSO had the greatest generalisability to optimising the settling times of different SOAs. Researchers in our field should therefore be able to black box our PSO AI approach and optimise their SOAs even though they will have different equivalent circuit components from the specific device(s) optimised in this paper.\n\n\n\\begin{table}[]\n\\footnotesize\n    \\caption{Performance summary for the techniques applied to the 10 different simulated SOAs, given in the format min | max | mean | standard deviation (best in bold).}\n    \\begin{itemize}\n        \\item \\scriptsize{Signals marked `-' never settled.}\n    \\end{itemize}\n    \\label{tab:diff_transfer_func_comparison_table}\n    \\centering\n    \\tabcolsep=0.11cm\n    \\begin{tabular}{|c|c|c|c|}\n         \\hline\n        {\\textbf{Technique}} & \\pbox{20cm}{\\textbf{Rise Time (ps)}} & {\\textbf{Settling Time (ns)}} & {\\textbf{Overshoot (\\%)}} \\\\\n         \\hline\n         \n         Step & \\scriptsize{\\textbf{502}, \\textbf{753}, 653, 86.4} & \\scriptsize{3.1, -, 5.8, 3.0} & \\scriptsize{16.5, 70.4, 39.2, 14.1} \\\\\n         \\hline\n         \n         PSO & \\scriptsize{669, 837, 703, \\textbf{58.5}} & \\scriptsize{\\textbf{0.67}, \\textbf{1.3}, \\textbf{0.87}, \\textbf{0.20}} & \\scriptsize{\\textbf{2.51}, \\textbf{6.01}, \\textbf{4.46}, \\textbf{1.22}} \\\\\n         \\hline\n         \n         ACO & \\scriptsize{\\textbf{502}, \\textbf{753}, \\textbf{644}, 79.4} & \\scriptsize{1.6, -, 2.6, 0.82} & \\scriptsize{11.1, 70.4, 32.6, 17.0} \\\\\n         \\hline\n         \n         \n         \n         GA & \\scriptsize{760, 930, 793, \\textbf{58.5}} & \\scriptsize{1.0, 1.5, 1.3, 1.5} & \\scriptsize{4.31, 9.36, 7.04, 1.54} \\\\\n         \\hline\n         \n         \n         \n         \n         \n    \\end{tabular}\n\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\n\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{Previous Work}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nA previous attempt to optimise SOA output applied a `pre-impulse step injection current' (PISIC) driving signal to the SOA \\cite{Gallep2002}. This PISIC signal pre-excited carriers in the SOA's gain region, increasing the charge carrier density and the initial rate of stimulated emission to reduce the 10\\% to 90\\% rise time from 2 ns to 500 ps. However, this technique only considered rise time when evaluating SOA off-on switching times. A more accurate off-on time is given by the settling time, which is the time taken for the signal to settle within $\\pm 5\\%$ of the `on' steady state. Before settling, bits experience variable signal to noise ratio, which impacts the bit error rate (BER) makes the signal unusable until settled, therefore the switch is effectively `off' during this period.\n\nA later paper looked at applying a `multi-impulse step injection current' (MISIC) driving signal to remedy the SOA oscillatory and overshoot behaviour \\cite{Figueiredo2015}. As well as a pre-impulse, the MISIC signal included a series of subsequent impulses to balance the oscillations, reducing the rise time to 115 ps and the overshoot by 50\\%. However, the method for generating an appropriate pulse format was trial-and-error. Since each SOA has slightly different properties and parasitic elements, the same MISIC format cannot be applied to different SOAs, therefore a different format must be generated through this inefficient manual process for each SOA, of which there will be thousands in a real DC. As such, MISIC is not scalable.  Critically, the MISIC technique did not consider the settling time, therefore the effective off-on switching time was still several ns.\n\nThe previous solutions discussed so far have had a design flow of first manually coming up with a heuristic for a simplified model of an SOA, followed by meticulous testing and tuning of the heuristic until good real-world performance is achieved. If some aspect of the problem is changed such as the SOA type used or the desired shape of the output signal, this process must be repeated. This paper presents a novel and scalable approach to optimising the SOA driving signal in an automated fashion with artificial intelligence (AI), requiring no prior knowledge of the SOA and being able to be generalised to any device without laborious manual configuration. Three algorithms were chosen on the basis that they had previously been applied to proportional-integral-derivative (PID) tuning in control theory \\cite{7873803}: Particle swarm optimisation (PSO), ant colony optimisation (ACO), and a genetic algorithm (GA). All algorithms were shown to reduce the settling and rise times to the $O$(100ps) scale. The algorithms' hyperparameters were tuned in an SOA equivalent circuit (EC) simulation environment and their efficacy was demonstrated in an experimental setup. AI performance was compared to that of step, PISIC and MISIC driving signals as well as the popular raised cosine and PID control approaches to optimising oscillating and overshooting systems, all of which the AI algorithms outperformed.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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\nFor a finite subset $A\\subset \\mathbb N$, let $E_A$ denote the set of all $x\\in(0,1)$ such that the digits\n$a_1(x), a_2(x),\\ldots$ in the continued fraction expansion \n$$\nx = \n[a_1(x), a_2(x), a_3(x), \\ldots ]\n= \n\\frac{1}{a_1(x) + \\frac{1}{a_2(x) + \n\\frac{1}{a_3(x) + \\cdots}\n}}\n$$\nall belong to $A$.\nSets of the form $E_A$ are said to be of \\emph{bounded type} (see e.g.~\\cite{kontorovich, shallit});\nin particular they are Cantor sets, and study of their Hausdorff dimension has attracted significant attention.\n\nOf particular interest have been the sets $E_n=E_{\\{1,\\ldots,n\\}}$,\nwith $E_2=E_{\\{1,2\\}}$ the most studied of these, serving as a test case for various general methods\nof approximating Hausdorff dimension. \nJarnik \\cite{jarnik} showed that $\\text{dim}(E_2)>1\/4$,\nwhile Good \\cite{good} improved this to $0.5306  < \\text{dim}(E_2)<0.5320$,\nBumby \\cite{bumby2}\nshowed that $0.5312 < \\text{dim}(E_2) < 0.5314$,\nHensley \\cite{hensley1989} showed that\n$0.53128049$ $< \\text{dim}(E_2) < 0.53128051$,\nwhile\nFalk \\& Nussbaum \\cite{falknussbaum}\\footnote{This preprint has been split into the two articles\n\\cite{falknussbaum1} and \\cite{falknussbaum2}, with \\cite{falknussbaum1} containing the approximation to\n$\\text{dim}(E_2)$.} \nrigorously justified the first 8 decimal digits of $\\text{dim}(E_2)$,\nproving that\n$0.531280505981423 \\le \\text{dim}(E_2) \\le 0.531280506343388\\,.$\nA common element in the methods \\cite{bumby2, falknussbaum, hensley1989} is \nthe study of a transfer operator, while for the higher accuracy estimates \n\\cite{falknussbaum, hensley1989} there is some element of\ncomputer-assistance involved in the proof.\n\nIn \\cite{jp} we outlined a different approach to approximating the Hausdorff dimension of bounded type sets,\nagain using a transfer operator, but exploiting the real analyticity of the maps defining continued fractions to\nconsider the determinant $\\Delta$ of the operator, and its\n approximation in terms of periodic points\\footnote{The periodic points are precisely those numbers in $(0,1)$\n with periodic continued fraction expansion, drawn from digits in $A$.\n The reliance on periodic points renders the method \\emph{canonical}, inasmuch as it does not involve any arbitrary choice of coordinates or partition of the space.}\n  of an underlying dynamical system.\nWhile some highly accurate \\emph{empirical} estimates of Hausdorff dimension were given, for example a 25 decimal digit approximation to $\\text{dim}(E_2)$, these were not rigorously justified. Moreover, although the \nalgorithm was proved to generate a sequence of approximations $s_n$ to the Hausdorff dimension\n(depending on points of period up to $n$),\nwith convergence rate faster than any exponential, \nthe derived error bounds were sufficiently conservative (see Remark \\ref{conservative} below) that it was unclear\nwhether  they could be combined with \nthe computed approximations to yield any effective \\emph{rigorous} estimate.\n\nIn the current paper we investigate the possibility of sharpening the approach of \\cite{jp} \nso as to obtain rigorous computer-assisted estimates on $\\text{dim}(E_A)$, with particular focus on $E_2$.\nThere are several ingredients in this sharpening.\nThe first step is to locate a disc $D$ in the complex plane with the property that\nthe images of $D$ under the mappings $T_n(z)=1\/(z+n)$, $n\\in A$, are contained in $D$.\nIt then turns out to be preferable to consider the transfer operator as acting on a \\emph{Hilbert} space of analytic functions on $D$, rather than the Banach space of \\cite{jp};\nthis facilitates an estimate on the Taylor coefficients of $\\Delta$ in terms of the \\emph{approximation numbers}\n(or \\emph{singular values}) of the operator, which is significantly better than those bounds derived from Banach space methods. \nThe specific Hilbert space used is Hardy space, consisting of those analytic functions on the disc which extend as $L^2$\nfunctions on the bounding circle.\nThe contraction of $D$ by the mappings $T_n(z)=1\/(n+z)$, $n\\in A$,\nprompts the introduction of the \\emph{contraction ratio}, which captures\nthe strength of this contraction, and leads to estimates on the convergence of\nthe approximations to the Hausdorff dimension.\nThe $n^{th}$ Taylor series coefficient of $\\Delta$ can be expressed in terms of periodic points of period up to $n$, and for sufficiently small $n$ these can be evaluated\nexactly,  to arbitrary precision.\nFor larger $n$, we show it is advantageous to\nobtain two distinct types of upper bound on the Taylor coefficients: we refer to these\nas the \\emph{Euler bound} and the \\emph{computed Taylor bound}.\nThe Euler bound is used for all sufficiently large $n$, while the\ncomputed Taylor bound is used for a finite intermediate range of $n$\ncorresponding to those Taylor coefficients which are deemed to be computationally inaccessible,\nbut \nwhere the Euler bound is insufficiently sharp.\nIntrinsic to the definition of the computed Taylor bounds is the sequence of\n\\emph{computed approximation bounds}, which we introduce as \ncomputationally accessible upper bounds on\nthe approximation numbers \nof the transfer operator.\n\nAs an example of the effectiveness of the resulting method we rigorously justify the \nfirst 100 decimal digits\\footnote{The choice of 100 decimal digits in the present article is motivated by a number of factors.\nOn the one hand 100 is considered a particularly round number, and an order of magnitude larger than  the number of decimal digits obtained (even non-rigorously) for the dimension of $E_2$ in previous works.\nOn the other hand, readily available computer resources (namely, a program written in Mathematica running on a modestly equipped laptop) performed the necessary calculations, in particular the high accuracy evaluation of points of period up to 25, in a reasonable timeframe (approximately one day), and it turns out that this choice of maximum period is sufficient to rigorously justify 100 decimal digits.}\nof the Hausdorff dimension of $E_2$, \nthereby improving on the rigorous estimates in \\cite{bumby2, falknussbaum, good, hensley1989, jarnik}.\nSpecifically, we prove (see Theorem \\ref{e2theorem}) that \n\\begin{align*}\n\\text{dim}(E_2) =\n0.&53128050627720514162446864736847178549305910901839 \\\\\n& 87798883978039275295356438313459181095701811852398\\ldots \\,,\n\\end{align*}\n using the \n  periodic points of period up to 25.\n\n\n\n\\section{Preliminaries}\n\nIn this section we collect a number of results (see also \\cite{jp}) which underpin our algorithm for approximating Hausdorff dimension.\n\n\\subsection{Continued fractions}\n\nLet $E_A$ denote the set of all $x\\in(0,1)$ such that the digits\n$a_1(x), a_2(x),\\ldots$ in the continued fraction expansion \n$$\nx = \n[a_1(x), a_2(x), a_3(x), \\ldots ]\n= \n\\frac{1}{a_1(x) + \\frac{1}{a_2(x) + \n\\frac{1}{a_3(x) + \\cdots}\n}}\n$$\nall belong to $A$.\nFor any $i\\in\\mathbb N$ we define the map $T_i$ by\n$$\nT_i(x)=\\frac{1}{i+x}\\,,\n$$\nand for a given $A\\subset \\mathbb N$, the collection $\\{T_i:i\\in A\\}$ is referred to as the corresponding\n\\emph{iterated function system}.\nIts \\emph{limit set}, consisting of limit points of sequences\n$T_{i_1}\\circ \\cdots\\circ T_{i_n}(0)$, where each $i_j\\in A$, is precisely the set $E_A$.\n\nEvery set $E_A$ is invariant under the \\emph{Gauss map} $T$, defined by\n$$\nT(x) = \\frac{1}{x} \\pmod 1\\,.\n$$\n\n\\subsection{Hausdorff dimension}\n\nFor a set $E \\subset \\mathbb R$,\ndefine\n$$\nH_\\varepsilon^\\delta(E) = \\inf\n\\left\\{\\sum_i\\text{diam}(U_i)^\\delta : \\text{$\\mathcal{U} = \\{U_i\\}$ is an open cover of $E$ such that each $\\text{diam}(U_i) \\leq \\varepsilon$}\\right\\}\\,,\n$$\nand set\n$H^\\delta(E) =\\lim_{\\varepsilon \\to 0} H_\\varepsilon^\\delta(E)$.\nThe \\emph{Hausdorff dimension} $\\text{dim}(E)$\nis then defined as\n$$\n\\text{dim}(E) = \\inf\\{\\delta \\hbox{ : } H^\\delta(E) = 0\\} \\,.\n$$\n\n\\subsection{Pressure formula}\n\nFor a continuous function $f: E_A \\to \\mathbb R$,\nits \\emph{pressure} $P(f)$ \nis given by\n$$\nP(f) = \\lim_{n\\to +\\infty}\\frac{1}{n}\n\\log \\left(\\sum_{T^nx=x \\atop x\\in E_A}\ne^{f(x) + f(Tx) + \\ldots + f(T^{n-1}x)}\n\\right)\\,,\n$$\nand if\n$f = -s\\log|T'|$ then we have the following implicit characterisation of the Hausdorff\ndimension of $E_A$ (see \\cite{bedford, bowen, falconer, mauldinurbanski}):\n\n\\begin{lemma}\\label{pressurelemma}\nThe function $s\\mapsto P(-s\\log |T'|)$ is strictly decreasing,\nwith a unique zero at $s=\\text{dim}(E_A)$.\n\\end{lemma}\n\n\\subsection{Transfer operators}\n\nFor a given $A\\subset \\mathbb N$, and $s\\in\\mathbb R$, the \\emph{transfer operator} $\\mathcal L_{A,s}$, defined by\n$$\n\\mathcal L_{A,s} f (z) = \\sum_{i\\in A} \\frac{1}{(z+i)^{2s}}\\, f\\left(\\frac{1}{z+i}\\right) \\,,\n$$\npreserves various natural function spaces, for example the Banach space of Lipschitz functions on $[0,1]$.\nOn this space it has a simple positive eigenvalue $e^{P(-s\\log|T'|)}$, which is the unique eigenvalue whose modulus equals \nits spectral radius,\nthus by Lemma \\ref{pressurelemma} \nthe Hausdorff dimension of $E_A$ is the unique value $s\\in\\mathbb R$ such that $\\mathcal L_{A,s}$ has spectral radius equal to 1.\n\n\\subsection{Determinant}\nThe \\emph{determinant} for $\\mathcal L_{A,s}$ is the entire function defined for $z$ of sufficiently small modulus\\footnote{The power series $\\sum_{n=1}^\\infty \\frac{z^n}{n} \\text{tr}(\\mathcal L_{A,s}^n)$, and hence the expression (\\ref{determinantformula}), is convergent for $|z|< e^{-P(-s\\log|T'|)}$.}\nby\n\\begin{equation}\\label{determinantformula}\n\\Delta(z,s) = \\exp -\\sum_{n=1}^\\infty \\frac{z^n}{n} \\text{tr}(\\mathcal L_{A,s}^n) \\,,\n\\end{equation}\nand for other $z\\in\\mathbb{C}$ by analytic continuation;\nhere the trace $\\text{tr}(\\mathcal L_{A,s}^n)$ is given (see \\cite{jp, ruelle}) by \n\\begin{equation}\\label{traceformula}\n\\text{tr}(\\mathcal L_{A,s}^n) = \\sum_{\\underline i\\in A^n}  \\frac{|T_{\\underline i}'(z_{\\underline i})|^s}{1-T_{\\underline i}'(z_{\\underline i})} \n = \\sum_{\\underline i\\in A^n}  \\frac{\\prod_{j=0}^{n-1}T^j(z_{\\underline i})^{2s}}{1-(-1)^n\\prod_{j=0}^{n-1}T^j(z_{\\underline i})^2 } \n\\,,\n\\end{equation}\nwhere the point $z_{\\underline i}$, which has period $n$ under $T$, is the unique fixed point of the \n$n$-fold composition $T_{\\underline i} = T_{i_1}\\circ T_{i_2}\\circ\\cdots\\circ T_{i_n}$.\n\n\nWhen acting on a suitable space of holomorphic functions, the eigenvalues of $\\mathcal L_{A,s}$ are precisely the reciprocals\nof the zeros of its determinant.\nIn particular, the zero of minimum modulus for $\\Delta(s,\\cdot)$ is $e^{-P(-s\\log|T'|)}$, so the Hausdorff dimension\nof $E_A$ is characterised as the value of $s$ such that 1 is the zero of minimum modulus of $\\Delta(s,\\cdot)$.\n\nIn fact we shall later show that, when $\\mathcal L_{A,s}$ acts on such a space of holomorphic functions, its \napproximation numbers decay at an exponential rate (see Corollary \\ref{ks}), so that $\\mathcal L_{A,s}$ belongs to an exponential class (cf.~\\cite{bandtlow, bj}) and is in particular a trace class operator, from which the existence and above\nproperties of trace and determinant follow (see \\cite{simon}).\n\nAs outlined in \\cite{jp}, this suggests the possibility of expressing $\\Delta(z,s)$ as a power series\n$$\n\\Delta(z,s) = 1 + \\sum_{n=1}^\\infty \\delta_n(s) z^n\\,,\n$$\nthen \ndefining $\\frak D$ by\n$$\n\\frak D(s) := \\Delta(1,s) =1 + \\sum_{n=1}^\\infty \\delta_n(s) \\,.\n$$\nThe function $\\frak D$ is an entire function of $s$ (see \\cite{jp}),\nand solutions $s$ of the equation\n\\begin{equation}\\label{propereqn}\n0 = 1+\\sum_{n=1}^\\infty \\delta_n(s) = \\frak D(s)\n\\end{equation}\nhave the property that the value 1 is an eigenvalue for $\\mathcal L_{A,s}$;\nin particular,\nthe unique zero of $\\frak D$ in the interval $(0,1)$ is precisely $\\text{dim}(E_A)$,\nbeing the unique value of $s$ for which 1 is the eigenvalue of maximum modulus for $\\mathcal L_{A,s}$.\n\nAs a result of the trace formula (\\ref{traceformula}), the coefficients $\\delta_n(s)$ are computable\\footnote{By this we mean that for a given $s$, the $\\delta_n(s)$ are computable exactly, to arbitrary precision.}\nin terms of the periodic points of $T|_{E_A}$ of period no greater than $n$, so for some suitable\n$N\\in\\mathbb N$, chosen so that $\\delta_1(s),\\ldots, \\delta_N(s)$ can be computed to a given precision in reasonable time,\nwe can define $\\frak D_N$ by\n\\begin{equation}\\label{DNdefn}\n\\frak D_N(s) := 1+\\sum_{n=1}^N \\delta_n(s)\\,.\n\\end{equation}\nA solution to the equation\n\\begin{equation}\\label{truncatedequation}\n\\frak D_N(s) = 0 \n\\end{equation}\nwill be an approximate solution to (\\ref{propereqn}), where the quality of this approximation will be related to the \nsmallness of the discarded tail\n\\begin{equation}\\label{discardedtail}\n\\sum_{n=N+1}^\\infty \\delta_n(s)\\,.\n\\end{equation}\nIn particular, any rigorous estimate of the closeness of a given approximate solution $s_N$ of (\\ref{truncatedequation}) to the true Hausdorff dimension $\\text{dim}(E_A)$ will require a rigorous upper bound\non the modulus of the tail (\\ref{discardedtail}).\n\n\\begin{remark}\\label{conservative}\nIn \\cite{jp} we considered the set $E_2=E_{\\{1,2\\}}$ and, although the empirical estimates of its\nHausdorff dimension appeared convincing,\nthe estimate on the tail (\\ref{discardedtail}) was not sharp enough to permit any effective rigorous bound.\nEssentially\\footnote{In \\cite{jp} we actually worked with $\\det(I-z\\mathcal L_{A,s}^2)$ rather than\n$\\det(I-z\\mathcal L_{A,s})$, though the methods there lead to very similar bounds for both determinants.}, the bound in \\cite{jp} was\n$\n|\\delta_n(s)| \\le \\varepsilon_n := C K^n n^{n\/2} \\theta^{n(n+1)}\n$\nwhere\n$\nC= \\gamma \\prod_{r=1}^\\infty(1-\\gamma^r)^{-1} \\approx 122979405533\n$,\n$\nK= \\frac{45}{16\\pi} \\approx 0.895247\n$,\nand\n$\n\\theta = \\left( \\frac{8}{9}\\right)^{1\/4} \\approx 0.970984\n$.\nAlthough the bounding sequence $\\varepsilon_n$ tends to zero, and does so at super-exponential rate $O(\\theta^{n^2})$, \nthe considerable inertia in this convergence \n(e.g.~the sequence increases for $1\\le n\\le 39$ to the value $\\varepsilon_{39}\\approx 1.31235 \\times 10^{22}$,\nand remains larger than $1$ until\n$n=85$)\nrenders the bound ineffective in practice,\nin view of the exponentially increasing computation time required to calculate the $\\delta_n(s)$\n(as seen in this article, we can feasibly compute several million periodic points, but performing calculations involving more\n than $2^{85}$ points is out of the question).\n\\end{remark}\n\n\n\\begin{remark}\nThe specific rigorous approximation of dimension is performed in this article only for the set $E_2$\n(see \\S \\ref{E2section}), corresponding to the iterated function system consisting of the maps $T_1(x)=1\/(x+1)$\nand $T_2(x)=1\/(x+2)$.\nIn principle, however, it can be performed for arbitrary iterated function systems consisting of real analytic maps $T_1,\\ldots, T_l$ satisfying the open set condition (i.e.~there exists a non-empty open set $U$ such that\n$T_i(U)\\cap T_j(U)=\\emptyset$ for $i\\neq j$, and $T_i(U)\\subset U$ for all $i$).\nIn this setting the accuracy of our \nHausdorff dimension estimate depends principally on the contractivity of the maps $T_i$ and the number $l$ of such maps, with stronger contraction and a smaller value of $l$ corresponding to increased accuracy.\nStronger contraction (as reflected by smallness of the \\emph{contraction ratio} defined in \\S \\ref{contractionratiossubsection}) is associated with more rapid decay of the Taylor coefficients of the determinant\n$\\Delta(z,s)$, implying greater accuracy of the polynomial truncations, while for $l>2$ the time required to locate the points of period up to $n$ increases by a factor of roughly $(l\/2)^n$ relative to the case $l=2$\n(note that for \\emph{infinite} iterated function systems, i.e.~$l=\\infty$, our method is rarely applicable, since it is usually impossible to locate all period-$n$ points for a given $n$, though here non-rigorous approximations may be obtained by suitable approximation).\nIf the $T_i$ are not M\\\"obius maps then for practical purposes there is some minor decrease in the efficiency of our method: the compositions $T_{\\underline i}$ are more highly nonlinear than in the M\\\"obius case, so evaluation of their fixed points typically takes slightly longer.\n\\end{remark}\n\n\n\\begin{remark}\nWork of Cusick \\cite{cusick1,cusick2}\non continuants with bounded digits characterised the Hausdorff dimension of $E_n=E_{\\{1,\\ldots,n\\}}$ in terms of the abscissa of convergence of a certain Dirichlet series,\nand Bumby \\cite{bumby1, bumby2} showed that $0.5312 < \\text{dim}(E_2) < 0.5314$.\nHensley \\cite{hensley1989} obtained the bound\n$0.53128049$ $< \\text{dim}(E_2) < 0.53128051$ using a recursive procedure,\nand in \\cite[Thm.~3]{hensley1996} introduced a general approach for\napproximating the Hausdorff dimension of $E_A$, obtaining in particular the empirical estimate\n$\\text{dim}(E_2)=\n0.5312805062772051416\\ldots$\n\\end{remark}\n\n\n\n\n\n\\section{Hilbert Hardy space, approximation numbers, approximation bounds}\n\nIn this section we introduce the Hilbert space upon which the transfer operator acts, then make the connection between\napproximation numbers for the operator and Taylor\ncoefficients of its determinant, leading to so-called Euler bounds on these Taylor coefficients.\n\n\\subsection{Hardy space}\n\nLet $D\\subset \\mathbb{C}$ be an open disc of radius $r$, centred at $c$.\nThe \\emph{Hilbert Hardy space}\n$H^2(D)$ consists of those functions $f$\nwhich are holomorphic on $D$ and such that\n$\\sup_{\\varrho<r}\n\\int_0^{1} |f(c+\\varrho e^{2\\pi it})|^2\\, dt<\\infty$.\nThe inner product on $H^2(D)$ is defined by\n$(f,g)=\n\\int_0^{1} f(c+r e^{2\\pi it}) \\overline{g(c+r e^{2\\pi it})}\\, dt$,\nwhich is well-defined since any element of $H^2(D)$\nextends as an $L^2$ function of the boundary $\\partial D$.\nThe norm of $f\\in H^2(D)$ will be simply written as $\\|f\\|=(f,f)^{1\/2}$.\n\nAn alternative characterisation of $H^2(D)$ \n(see e.g.~\\cite{shapiro})\nis as the set of functions $f$ which are holomorphic on $D$\nand such that if $m_k(z)=r^{-k}(z-c)^k$ for $k\\ge0$, then\n$$\nf = \\sum_{k=0}^\\infty \\hat f(k)\\, m_k\n$$\nwhere the sequence $(\\hat f(k))_{k=0}^\\infty$ is square summable.\nThe norm $\\|f\\|$ can then be expressed as\n$$\n\\|f\\|^2 = \\sum_{k=0}^\\infty |\\hat f(k)|^2\\,.\n$$\n\n\\subsection{Approximation numbers}\n\nGiven a compact operator $L:H\\to H$\non a Hilbert space $H$, its\n$i^{th}$ \\emph{approximation number} $s_i(L)$\nis defined as\n$$\ns_i(L)=\\inf\\{ \\| L-K \\| : \\text{rank}(K) \\leq i-1 \\} \\,,\n$$\nso that in particular $s_1(L)=\\| L \\|$.\n\nThe following result exploits our Hilbert space setting,\nand represents an improvement over analogous Banach space estimates\nin \\cite{jp} (where e.g.~a multiplicative factor $n^{n\/2}$ reduces the quality of the bound on $|\\delta_n(s)|$).\n\n\\begin{lemma}\\label{gohberglemma}\nIf $\\mathcal L_{A,s}:H^2(D)\\to H^2(D)$, then the $n^{th}$ Taylor coefficient $\\delta_n(s)$ of its determinant can be bounded by\n\\begin{equation}\\label{Taylorapproximationbound}\n|\\delta_n(s)|\n\\leq\n \\sum_{i_1 < \\ldots < i_n}\n\\prod_{j=1}^n  s_{i_j}(\\mathcal L_{A,s})\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nIf $\\{\\lambda_n(s)\\}$ is the eigenvalue sequence for $\\mathcal L_{A,s}$, \n ordered by\ndecreasing modulus and counting algebraic multiplicities,\nthen\n(see e.g.~\\cite[Lem.~3.3]{simon}) we have\n$$\\delta_n(s)\n= \\sum_{i_1 < \\ldots < i_n}\n\\prod_{j=1}^n\n\\lambda_{i_j}(s)\n\\,,$$\nand\n\\begin{equation*}\n\\left|\\sum_{i_1 < \\ldots < i_n}\n\\prod_{j=1}^n\n\\lambda_{i_j}(s)\n\\right|\n\\leq\n \\sum_{i_1 < \\ldots < i_n}\n \\prod_{j=1}^n  s_{i_j}(\\mathcal L_{A,s})\n\\end{equation*}\nby \\cite[Cor.~VI.2.6]{ggk}, so\nthe result follows.\n\\end{proof}\n\n\nIn view of the link between Hausdorff dimension\nerror estimates and the tail (\\ref{discardedtail}),\ntogether with the bounding of terms in this tail by sums of products of approximation numbers provided by Lemma \\ref{gohberglemma},\nit will be important to establish upper bounds on the \nTaylor coefficients $\\delta_n(s)$ for those $n$ where it is not computationally feasible to evaluate \nexactly\nvia periodic points.\nWe shall derive two distinct types of such upper bound, \nwhich we refer to as \\emph{Euler bounds} and \\emph{computed Taylor bounds}.\nThere is an Euler bound on $\\delta_n(s)$ for each $n$, given as a simple closed form; \nthis bound will be used for all sufficiently large values of $n$,\nthough for low values of $n$ may be too conservative for our purposes.\nThe finitely many\ncomputed Taylor bounds\nwill be on the Taylor coefficients $\\delta_{P+1}(s),\\ldots, \\delta_Q(s)$ where $P$ is the largest integer\nfor which we locate all period-$P$ points, and $Q$ is chosen so that the Euler bounds on $|\\delta_n(s)|$ are\nsufficiently sharp when $n>Q$.\n In view of Lemma \\ref{gohberglemma}, the computed Taylor bounds will be derived by first bounding the finitely\n many approximation numbers $s_1(\\mathcal L_{A,s}),\\ldots, s_N(\\mathcal L_{A,s})$, for some $N\\in\\mathbb N$, by explicitly computable quantities that we call \\emph{computed approximation bounds}.\n The computations required to derive the computed approximation bounds are not onerous, the main task being the \nevaluation of numerical integrals defining certain $H^2$ norms\n (of the transfer operator images of a chosen orthonormal basis).\n\n\nWe shall approximate $\\mathcal L_{A,s}$\nby first projecting $H^2(D)$ onto the space of polynomials up to a given degree.\nLet $\\mathcal L_{A,s}:H^2(D)\\to H^2(D)$ be a transfer operator,\nwhere $D\\subset \\mathbb C$ is an open disc of radius $\\varrho$ centred at $c$, and \n$\\{m_k\\}_{k=0}^\\infty$ is the corresponding orthonormal basis of monomials, given by\n\\begin{equation}\\label{monomial}\nm_k(z)=\\varrho^{-k}(z-c)^k\\,.\n\\end{equation}\n\n\\subsection{Approximation bounds}\n\n\\begin{defn}\\label{alphadefn}\nFor $n\\ge1$, define the $n^{th}$ \\emph{approximation bound} $\\alpha_n(s)$ to be\n\\begin{equation}\\label{alphaexpression}\n\\alpha_n(s) = \\left( \\sum_{k=n-1}^\\infty  \\|\\mathcal L_{A,s} (m_k)\\|^2 \\right)^{1\/2} \\,.\n\\end{equation}\n\\end{defn}\n\n\n\\begin{prop}\\label{alphaapproximationbound}\nFor each $n\\ge1$,\n\\begin{equation}\\label{snalphanbound}\ns_n(\\mathcal L_{A,s}) \\le \\alpha_n(s)\\,.\n\\end{equation}\n\\end{prop} \n\\begin{proof}\nFor $f\\in H^2(D)$ we can write\n$$\nf = \\sum_{k=0}^\\infty \\hat f(k)\\, m_k\n$$\nwhere the sequence $(\\hat f(k))_{k=0}^\\infty$ is square summable.\nDefine the rank-$(n-1)$ projection $\\Pi_n:H^2(D)\\to H^2(D)$ by\n$$\n\\Pi_n(f) = \\sum_{k=0}^{n-2} \\hat f(k)\\, m_k \\,,\n$$\nwhere in particular $\\Pi_1\\equiv 0$.\n\nThe transfer operator $\\mathcal L_{A,s}$ is approximated by the rank-$(n-1)$ operators $$\\mathcal L_{A,s}^{(n)} := \\mathcal L_{A,s} \\Pi_n \\,,$$\nand  $\\| \\mathcal L_{A,s}- \\mathcal L_{A,s}^{(n)} \\|$ can be estimated\nusing the Cauchy-Schwarz inequality  as follows:\n\\begin{multline*}\n\\| (\\mathcal L_{A,s} - \\mathcal L_{A,s}^{(n)})f \\|\n= \\| \\sum_{k=n-1}^\\infty \\hat f(k)\\, \\mathcal L_{A,s}(m_k) \\| \n \\le \\sum_{k=n-1}^\\infty | \\hat f(k)| \\| \\mathcal L_{A,s}(m_k) \\| \\cr\n \\le \\left( \\sum_{k=n-1}^\\infty   \\| \\mathcal L_{A,s}(m_k) \\|^2  \\right)^{1\/2}  \\left( \\sum_{k=n-1}^\\infty |\\hat f(k)|^2\\right)^{1\/2}  \\cr\n \\le \\left( \\sum_{k=n-1}^\\infty   \\| \\mathcal L_{A,s}(m_k) \\|^2  \\right)^{1\/2} \\|f\\| \\,,\\cr\n\\end{multline*}\nand therefore \n$\\| \\mathcal L_{A,s}- \\mathcal L_{A,s}^{(n)} \\| \\le \\left( \\sum_{k=n-1}^\\infty   \\| \\mathcal L_{A,s}(m_k) \\|^2  \\right)^{1\/2} = \\alpha_n(s)$.\nSince $\\mathcal L_{A,s}^{(n)}$ has rank $n-1$,\nit follows that $s_n(\\mathcal L_{A,s}) \\le \\alpha_n(s)$, as required.\n\\end{proof}\n\n\\subsection{Contraction ratios}\\label{contractionratiossubsection}\n\nLet $C_i:H^2(D)\\to H^2(D)$ be the \\emph{composition operator}\n$$\nC_i f = f\\circ T_i\\,.\n$$\n\nThe estimate arising in the following lemma motivates our definition below (see Definition \\ref{hardydefinition})\nof the \\emph{contraction ratio} associated to a disc $D$ and subset $A\\subset \\mathbb N$.\n\n\\begin{lemma}\\label{H2contractionratio}\nLet $D$ and $D'$ be concentric discs, with radii $\\varrho$ and $\\varrho'$ respectively.\nIf, for $i\\in A$, the image $T_i(D)$ is contained in $D'$, then for all $k\\ge0$,\n\\begin{equation}\n\\| C_i(m_k)\\| \\le \\left( \\frac{\\varrho'}{\\varrho}\\right)^k\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet $c$ denote the common centre of the discs $D, D'$.\nIf $z\\in D$ then\n$$|C_i(m_k)(z)| = \\varrho^{-k} |T_i(z)-c|^k < \\varrho^{-k} (\\varrho')^k = (\\varrho' \/\\varrho)^k\\,,$$\n so $\\|C_i(m_k)\\|\\le  (\\varrho'\/\\varrho)^k$, as required.\n\\end{proof}\n\nFor each $i\\in A$, $s\\in\\mathbb R$, if the open disc $D$ is such that $-i\\notin D$ then\ndefine the \\emph{weight function} $w_{i,s}:D\\to\\mathbb{C}$ by\n$$\nw_{i,s}(z) = \\left(\\frac{1}{z+i}\\right)^{2s}\\,, \n$$\nand the \\emph{multiplication operator} $W_{i,s}:H^2(D)\\to H^2(D)$ by\n$$\nW_{i,s} f = w_{i,s} f\\,.\n$$\nWe may write\n$$\n\\mathcal L_{A,s}  = \\sum_{i\\in A} W_{i,s} C_i\\,,\n$$\nso that\n$$\n\\| \\mathcal L_{A,s}(m_k) \\| \\le \n\\sum_{i\\in A} \\| W_{i,s} C_i(m_k) \\|\n\\le\n\\sum_{i\\in A} \\| w_{i,s}\\|_\\infty  \\|C_i(m_k) \\|\n \\,,\n$$\nand if $\\varrho_i'$ is such that $T_i(D)$ is contained in the concentric disc $D_i'$ of radius $\\varrho_i'$ then\nLemma \\ref{H2contractionratio} implies that\n\\begin{equation}\\label{rhoprimei}\n\\| \\mathcal L_{A,s}(m_k) \\| \\le \\sum_{i\\in A} \\| w_{i,s}\\|_\\infty (\\varrho_i'\/\\varrho)^k\\,.\n\\end{equation}\n\nFor our purposes it will be more convenient to work with a slightly simpler (and less sharp) version of\n(\\ref{rhoprimei}). This prompts the following definition:\n\n\\begin{defn}\\label{hardydefinition}\nLet $A\\subset\\mathbb N$ be finite, and $D\\subset \\mathbb{C}$ an open disc of radius $\\varrho$ such that\n$\\cup_{i\\in A} T_i(D)\\subset D$. Let $D'$ be the smallest disc, concentric with $D$, such that\n$\\cup_{i\\in A} T_i(D)\\subset D'$, and let\n$\\varrho'$ denote the radius of $D'$.\nThe corresponding \\emph{contraction ratio} $h=h_{A,D}$ is defined to be\n\\begin{equation}\\label{hardyratiodefn}\nh = h_{A,D} = \\frac{\\varrho'}{\\varrho}\\,. \n\\end{equation}\n\\end{defn}\n\n\n\\begin{lemma}\\label{hardyfirst}\nLet $A\\subset \\mathbb N$ be finite, and $D$ an admissible disc, with contraction ratio $h=h_{A,D}$.\nFor all $k\\ge0$,\n\\begin{equation}\\label{normdecay}\n\\| \\mathcal L_{A,s}(m_k) \\| \\le  h^k  \\sum_{i\\in A} \\| w_{i,s}\\|_\\infty \\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nIf $D'$ is as in Definition \\ref{hardydefinition} then $\\varrho'=\\max_{i\\in A} \\varrho_i'$ \nin the notation of (\\ref{rhoprimei}), and the result follows from (\\ref{rhoprimei}).\n\\end{proof}\n\n\\begin{cor}\\label{ks}\nLet $A\\subset \\mathbb N$ be finite, and $D$ an admissible disc, with contraction ratio $h=h_{A,D}$.\nFor all $n\\ge1$,\n\\begin{equation}\\label{salphabound}\ns_n(\\mathcal L_{A,s}) \\le \\alpha_n(s) \\le  K_s  h^n\n\\end{equation}\nwhere\n\\begin{equation}\\label{ksdefn}\nK_s =\n\\frac{  \\sum_{i\\in A} \\| w_{i,s}\\|_\\infty }{ h  \\sqrt{1 - h^2}}  \\,.\n\\end{equation}\n\\end{cor}\n\\begin{proof}\nNow\n$$\n\\alpha_n(s) = \\left( \\sum_{k=n-1}^\\infty  \\|\\mathcal L_{s} (m_k)\\|^2 \\right)^{1\/2} \n$$\nfrom Definition \\ref{alphadefn} and Proposition \\ref{alphaapproximationbound},\nso Lemma \\ref{hardyfirst} gives\n$$\n\\alpha_n(s) \\le   \\left( \\sum_{k=n-1}^\\infty h^{2k} \\right)^{1\/2} \\sum_{i\\in A} \\| w_{i,s}\\|_\\infty\n= \\frac{ h^{n-1}}{\\sqrt{1 - h^2}}  \\sum_{i\\in A} \\| w_{i,s}\\|_\\infty \\,,\n$$\nand the result follows.\n\\end{proof}\n\n\\subsection{Euler bounds}\n\nWe can now derive the \\emph{Euler bound} on the $n^{th}$ Taylor coefficient of the determinant:\n\n\\begin{prop}\\label{eulerprop}\nLet $A\\subset \\mathbb N$ be finite, and $D$ an admissible disc, with contraction ratio $h=h_{A,D}$.\nIf the transfer operator $\\mathcal L_{A,s}$ has determinant $\\det(I-z\\mathcal L_{A,s})=1+\\sum_{n=1}^\\infty \\delta_n(s)z^n$,\n then for all $n\\ge1$,\n \\begin{equation}\\label{eulerequation}\n|\\delta_n(s)| \\le \\frac{K_s^n h^{n(n+1)\/2}}{ \\prod_{i=1}^n (1 - h^i)}\\,.\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nBy Lemma \\ref{gohberglemma},\n\\begin{equation*}\n|\\delta_n(s)|\n\\leq\n \\sum_{i_1 < \\ldots < i_n}\n\\prod_{j=1}^n\ns_{i_j}(\\mathcal L_{A,s})\n \\,,\n\\end{equation*}\nso Corollary \\ref{ks} gives\n\\begin{equation*}\n|\\delta_n(s)|\n\\leq\nK_s^n\n \\sum_{i_1 < \\ldots < i_n} h^{i_1+\\ldots+i_n} \\,,\n\\end{equation*}\nand the result follows by repeated geometric summation\n(as first noted by Euler \\cite[Ch.~16]{euler}).\n\\end{proof}\n\nHenceforth we use the notation\n\\begin{equation}\\label{eulerformula}\nE_n(r) := \\frac{r^{n(n+1)\/2}}{\\prod_{i=1}^n (1-r^i)} = \\sum_{i_1< \\ldots < i_n} r^{i_1+\\ldots + i_n}\\,,\n\\end{equation}\nso that (\\ref{eulerequation}) can be written as\n\\begin{equation}\\label{eulerequation2}\n|\\delta_n(s)| \\le K_s^n E_n(h)\\,,\n\\end{equation}\nand we define the righthand side of (\\ref{eulerequation2}) \n(or equivalently of (\\ref{eulerequation}))\nto be the \\emph{Euler bound} on the $n^{th}$ Taylor coefficient of the determinant.\n\n\\section{Computed approximation bounds}\\label{cabsection}\n\nFor all $n\\ge1$, the $n^{th}$ approximation bound \n$$\\alpha_n(s) = \\left( \\sum_{k=n-1}^\\infty  \\|\\mathcal L_{A,s} (m_k)\\|^2 \\right)^{1\/2}$$\nis, as noted in Proposition \\ref{alphaapproximationbound}, an upper bound on the $n^{th}$ approximation number \n$s_n(\\mathcal L_{A,s})$. \n\nEach $m_k$ is just a normalised monomial (\\ref{monomial}), and the operator $\\mathcal L_{A,s}$ is available in closed form,\nso that \n$$\\mathcal L_{A,s}(m_k)(z) = \\sum_{i\\in A} \\frac{(T_i(z)-c)^k}{\\varrho^{k} (z+i)^{2s}}\\,,$$\nand we may use numerical integration \nto compute\\footnote{Numerical integration capability\nis available in computer packages such as Mathematica,\nand these norms can be computed to arbitrary precision; although higher precision requires greater computing time,\nthese computations are relatively quick \n(e.g.~for the computations in \\S \\ref{E2section} these integrals were computed with 150 digit accuracy).}\n each Hardy norm $\\|\\mathcal L_{A,s} (m_k)\\|$ as\n\\begin{equation}\\label{hardynormexplicit}\n\\|\\mathcal L_{A,s} (m_k)\\|^2 = \\int_0^1 \\left| \\sum_{i\\in A}  \\frac{(T_i(\\gamma(t))-c)^k}{\\varrho^{k} (\\gamma(t)+i)^{2s}}\\right|^2 \\, dt\\,,\n\\end{equation}\nwhere $\\gamma(t)=c+\\varrho e^{2\\pi it}$.\n\n\nEvaluation of $\\alpha_n(s)$ involves the tail sum\n$\\sum_{k=n-1}^\\infty  \\|\\mathcal L_{A,s} (m_k)\\|^2$,\nand in practice we can bound this by the sum of an exactly computed long finite sum\n$\\sum_{k=n-1}^N  \\|\\mathcal L_{A,s} (m_k)\\|^2$, for some $N \\gg n$,\nand a rigorous upper bound on \n$\\sum_{k=N+1}^\\infty  \\|\\mathcal L_{A,s} (m_k)\\|^2$\nusing (\\ref{normdecay}).\nMore precisely, we have the following definition: \n\n\\begin{defn}\nGiven $n,N\\in\\mathbb N$, with $n\\le N$, define the\n\\emph{lower and upper computed approximation bounds}, $\\alpha_{n,N,-}(s)$ and $\\alpha_{n,N,+}(s)$, respectively, by\n\\begin{equation}\\label{alphanN}\n\\alpha_{n,N,-}(s) = \\left( \\sum_{k=n-1}^N \\| \\mathcal L_{A,s}(m_k)\\|^2 \\right)^{1\/2}\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{alphanNplus}\n\\alpha_{n,N,+}(s) \n=\n\\left( \\alpha_{n,N,-}(s)^2\n+ \\left(\\sum_{i\\in A} \\|w_{i,s}\\|_\\infty\\right)^2 \\frac{h^{2(N+1)}}{1-h^2}\\right)^{1\/2}\n\\,.\n\\end{equation}\n\\end{defn}\n\nEvidently the lower computed approximation bound $\\alpha_{n,N,-}(s)$ is a lower bound for $\\alpha_n(s)$, \nin view of the positivity of the summands in (\\ref{alphaexpression}) and (\\ref{alphanN}), while\nLemma \\ref{upperlowerbounds} below establishes that the upper computed approximation bound\n$\\alpha_{n,N,+}(s)$ is an upper bound for $\\alpha_n(s)$.\nMoreover, both $\\alpha_{n,N,+}(s)$ and $\\alpha_{n,N,-}(s)$ are readily computable: they are given by finite sums and,\nas already noted, the summands $\\| \\mathcal L_{A,s}(m_k)\\|^2$ are computable to arbitrary precision.\n\n\\begin{lemma}\\label{upperlowerbounds}\nLet $s\\in\\mathbb R$. For all $n,N\\in \\mathbb N$, with $n\\le N$,\n\\begin{equation}\\label{alphaupperlower}\n\\alpha_{n,N,-}(s) \\le \\alpha_n(s) \\le \\alpha_{n,N,+}(s)\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe inequality $\\alpha_{n,N,-}(s) \\le \\alpha_n(s)$ is \nimmediate from the definitions.\nTo prove that $\\alpha_n(s) \\le \\alpha_{n,N,+}(s)$ note that\n$$\\alpha_n(s)^2 = \\sum_{k=n-1}^N \\|\\mathcal L_{A,s}(m_k)\\|^2 +  \\sum_{k=N+1}^\\infty \\|\\mathcal L_{A,s}(m_k)\\|^2\\,,$$\nwhich together with (\\ref{normdecay}) gives\n$$\\alpha_n(s)^2 \\le  \\sum_{k=n-1}^N \\|\\mathcal L_{A,s}(m_k)\\|^2 + \\left( \\sum_{i\\in A} \\| w_{i,s}\\|_\\infty \\right)^2 \\frac{h^{2(N+1)}}{1-h^2}\\,,\n$$\nand the result follows.\n\\end{proof}\n\n\\begin{remark}\nThe upper bound $\\alpha_{n,N,+}(s)$ will be used in the sequel, as a tool\nin providing rigorous estimates on Hausdorff dimension.\nIn practice $N$ will be chosen so that the values\n$\\alpha_{n,N,-}(s)$ and $\\alpha_{n,N,+}(s)$ are close enough together that \nthe inequality \n(\\ref{alphaupperlower})\ndetermines $\\alpha_n(s)$\nwith precision far higher than that of the desired Hausdorff dimension estimate;\nin particular, $N$ will be such that the difference \n$\\alpha_{n,N,+}(s)-\\alpha_{n,N,-}(s) = O(h^N)$ is\nextremely small relative to the size of $\\alpha_{n}(s)$.\n\\end{remark}\n\nCombining (\\ref{salphabound}) with (\\ref{alphaupperlower}) immediately gives the exponential bound\n\\begin{equation}\n\\label{salphaboundN}\n\\alpha_{n,N,-}(s) \\le K_s  h^n\\quad\\text{for all }n \\le N\\,,\n\\end{equation}\nthough the analogous bound for $\\alpha_{n,N,+}(s)$ (which will be more useful to us in the sequel)\nrequires some extra care:\n\n\\begin{lemma}\\label{nNboundlemma}\nLet $s\\in\\mathbb R$. For all $n,N\\in \\mathbb N$, with $n\\le N$,\n\\begin{equation}\\label{nNbound}\n\\alpha_{n,N,+}(s) \\le K_s(1 + h^{2(N+2-n)})^{1\/2} h^n\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nCombining (\\ref{salphaboundN}) with (\\ref{alphanNplus})\ngives \n$$\n\\alpha_{n,N,+}(s) \n\\le\n\\left( (K_sh^n)^2\n+ \\left(\\sum_{i\\in A} \\|w_{i,s}\\|_\\infty\\right)^2 \\frac{h^{2(N+1)}}{1-h^2}\\right)^{1\/2}\\,,\n$$\nbut (\\ref{ksdefn}) gives\n$$ \\frac{ \\left(\\sum_{i\\in A} \\|w_{i,s}\\|_\\infty\\right)^2}{1-h^2} = K_s^2h^2\\,,$$\nso\n$$\n\\alpha_{n,N,+}(s) \n\\le\n\\left( (K_sh^n)^2\n+ K_s^2 h^{2(N+2)} \\right)^{1\/2} \\,,\n$$\nand the result follows.\n\\end{proof}\n\nThe utility of (\\ref{nNbound}) stems from the fact that in practice $N-n$ will be large, and that \n for sufficiently small values of $n$ the following\nmore direct analogue of (\\ref{salphaboundN}) can be used:\n\n\\begin{cor}\\label{jqnscor}\nLet $s\\in\\mathbb R$. \nSuppose $N,Q\\in\\mathbb N$, with $Q \\le N$.\nIf \n\\begin{equation}\\label{jqns}\nJ = J_{Q,N,s} := K_s\\left( 1+h^{2(N+2-Q)}\\right)^{1\/2}\n\\end{equation}\nthen\n\\begin{equation}\\label{jqnshbound}\n\\alpha_{n,N,+}(s) \\le J h^n\\quad\\text{for all }1\\le n\\le Q\\,.\n\\end{equation}\n\\end{cor}\n\\begin{proof}\nImmediate from Lemma \\ref{nNboundlemma}.\n\\end{proof}\n\n\\begin{remark}\nIn practice $Q$ will be of some modest size, dictated by the computational resources at our disposal;\nspecifically, it will be chosen slightly larger than the largest $P\\in \\mathbb N$ for which it is feasible to compute all periodic points of period\n$\\le P$ (e.g.~in \\S \\ref{E2section}, when estimating the dimension of the set $E_2=E_{\\{1,2\\}}$,\nwe explicitly compute all periodic points up to period $P=25$, and in the proof of Theorem \\ref{e2theorem} we choose $Q=28$).\nThe value $N$ will be chosen to be significantly larger than $Q$ (e.g.~in \nthe proof of Theorem \\ref{e2theorem}  we choose $N=600$).\nSince $N+2-Q$ is large, $h^{N+2-Q}$ will be extremely small, and \n $J=J_{Q,N,s}$ will be extremely close to $K_s$; ideally this closeness ensures that\n the two constants $J_{Q,N,s}$ and $K_s$ \n\n are indistinguishable to the chosen level of working precision\n (e.g.~in the proof of Theorem \\ref{e2theorem}, $N+2-Q=574$ and $h\\approx 0.511284$,\n so $h^{N+2-Q}\\approx 5.9 \\times 10^{-168}$, whereas computations are performed to 150 decimal digit precision).\n\\end{remark}\n\n\n\\section{Computed Taylor bounds}\\label{taylorsection}\n\nIn order to use the computed approximation bounds to provide a rigorous upper bound on the \nTaylor coefficients of the determinant $\\det(I-z\\mathcal L_{A,s})$,\nwe now fix a further natural number $M$, satisfying $M\\le N$.\nFor any such $M$, it is convenient to define\nthe sequence $(\\alpha_{n,N,+}^{M}(s) )_{n=1}^\\infty$\nto be the one whose $n^{th}$ term equals $\\alpha_{n,N,+}(s)$ until $n=M$,\nand whose subsequent terms are given\nby the exponential upper bound on $s_n(\\mathcal L_{A,s})$ and $\\alpha_n(s)$ (cf.~(\\ref{salphabound})):\n\\begin{equation}\\label{alph}\n\\alpha_{n,N,+}^{M}(s) :=\n\\begin{cases}\n\\alpha_{n,N,+}(s) & \\text{for }1\\le n\\le M\\,, \\cr\nK_s h^n & \\text{for }n>M\\,.\n\\end{cases}\n\\end{equation}\n\nThis allows us to make the following definition:\n\n\\begin{defn}\nLet $s\\in\\mathbb R$. For $n,M,N\\in\\mathbb N$ with $n\\le M\\le N$,\nthe \\emph{Taylor bound} $\\beta_{n,N,+}^{M}(s)$ is defined by\n\\begin{equation}\\label{firstbeta}\n\\beta_{n,N,+}^{M}(s) := \\sum_{i_1<\\ldots< i_n} \\prod_{j=1}^n \\alpha_{i_j,N,+}^{M}(s) \\,,\n\\end{equation}\nwhere the sum is over those $\\underline i=(i_1,\\ldots,i_n)\\in\\mathbb N^n$\nwhich satisfy $i_1< i_2 < \\ldots< i_n$.\n\\end{defn}\n\nAs the name suggests, the Taylor bound $\\beta_{n,N,+}^{M}(s)$ bounds the $n^{th}$\nTaylor coefficient of the determinant $\\det(I-z\\mathcal L_{A,s})=1+\\sum_{n=1}^\\infty \\delta_n(s)z^n$:\n\n\\begin{lemma}\\label{betacbound}\nLet $s\\in\\mathbb R$.\nFor $n, M, N\\in \\mathbb N$ with $n\\le M\\le N$,\n\\begin{equation}\\label{cnalphatilde}\n|\\delta_n(s)|\n\\leq\n \\beta_{n,N,+}^{M}(s)\n \\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nCombining\n(\\ref{salphabound}), (\\ref{alphaupperlower}) and (\\ref{alph})\ngives\n\\begin{equation*}\\label{snalphanNplus}\ns_n(\\mathcal L_{A,s}) \\le \\alpha_{n,N,+}^{M}(s)\\quad\\text{for all }1\\le n\\le M \\le N\\,,\n\\end{equation*}\nand combining this with Lemma \\ref{gohberglemma} gives (\\ref{cnalphatilde}).\n\\end{proof}\n\n\nNote that $ \\beta_{n,N,+}^{M}(s)$ is precisely the $n^{th}$ power series coefficient\nfor the infinite product $\\prod_{i=1}^\\infty (1+ \\alpha_{i,N,+}^{M}(s) z)$, and\nthat the sum in (\\ref{firstbeta}) is an infinite one; thus we will seek a computationally accessible approximation to \n$ \\beta_{n,N,+}^{M}(s)$.\nWe expect that $ \\beta_{n,N,+}^{M}(s)$ is well approximated by the $n^{th}$ power series coefficient\nfor the \\emph{finite} product $\\prod_{i=1}^M (1+ \\alpha_{i,N,+}^{M}(s) z) = \\prod_{i=1}^M (1+ \\alpha_{i,N,+}(s) z)$,\nnamely the value $\\beta_{n,N,+}^{M,-}(s)$ \ndefined as follows:\n\n\\begin{defn}\nLet $s\\in\\mathbb R$. For $n,M,N\\in\\mathbb N$ with $n\\le M\\le N$,\nthe \\emph{lower computed Taylor bound} $\\beta_{n,N,+}^{M,-}(s)$ is defined as\n\\begin{equation}\\label{betanNplusMminusalternative}\n\\beta_{n,N,+}^{M,-}(s)\n:=\n  \\sum_{i_1<\\ldots< i_n\\le M} \\prod_{j=1}^n \\alpha_{i_j,N,+}(s)\\,.\n \\end{equation}\n\\end{defn}\n\n\\begin{remark}\n\\item[\\, (i)]\nThe fact that $\\beta_{n,N,+}^{M,-}(s)$ is defined in terms of upper computed approximation bounds \n$\\alpha_{i_j,N,+}(s)$,\ntogether with the finiteness of the sum (and product) in (\\ref{betanNplusMminusalternative}), \nensures that $\\beta_{n,N,+}^{M,-}(s)$ can be computed (to arbitrary precision).\n\\item[\\, (ii)]\nClearly, an equivalent definition of \n$\\beta_{n,N,+}^{M,-}(s)$ is\n\\begin{equation}\\label{betanNplusMminus}\n\\beta_{n,N,+}^{M,-}(s)\n=\n \\sum_{i_1<\\ldots< i_n \\le M} \\prod_{j=1}^n \\alpha_{i_j,N,+}^{M}(s)\\,.\n\\end{equation}\n\\end{remark}\n\n\n\n\n\n\nThe lower computed Taylor bound\n$\\beta_{n,N,+}^{M,-}(s)$ is obviously smaller than\nthe Taylor bound\n $\\beta_{n,N,+}^{M}(s)$, though in view of (\\ref{cnalphatilde})\n we require an \\emph{upper computed Taylor bound} (introduced in Definition \\ref{uppercomputedtaylorbound} below)\n that is larger than $\\beta_{n,N,+}^{M}(s)$.\nThe following result estimates\nthe difference \n$\\beta_{n,N,+}^{M}(s) - \\beta_{n,N,+}^{M,-}(s)$, and subsequently\n(see Definition \\ref{uppercomputedtaylorbound})\n provides the inspiration for the definition of the\nupper computed Taylor bound:\n\n\\begin{lemma}\\label{betadifference}\nLet $s\\in\\mathbb R$.\nGiven $Q,M,N\\in\\mathbb N$ with $Q\\le M\\le N$,\n and $J=J_{Q,N,s}$ defined by (\\ref{jqns}),\n\\begin{equation}\\label{betadifferencerequiredbound}\n\\beta_{n,N,+}^{M}(s) - \\beta_{n,N,+}^{M,-}(s)\n\\le \\sum_{l=0}^{n-1} J^{n-l} \\beta_{l,N,+}^{M,-}(s)\\, h^{M(n-l)} E_{n-l}(h)\\quad\\text{for all }1\\le n\\le Q \\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet $n$ be such that $1\\le n\\le Q$.\nThe set\n$\n\\mathcal I_n := \\{ \\underline i = (i_1,\\ldots,i_n) \\in \\mathbb N^n: i_1 <\\ldots < i_n\\} \n$\ncan be partitioned as \n$\\mathcal I_n = \\bigcup_{l=0}^n \\mathcal I_n^{(l)}$,\nwhere the $\\mathcal I_n^{(l)}$ are defined by\n\\begin{equation*}\n\\mathcal I_n^{(l)} =\n\\begin{cases}\n\\{ \\underline i = (i_1,\\ldots,i_n)\\in\\mathcal I_n:  M < i_{1}\\} & \\text{if }l=0 \\,, \\cr\n\\{ \\underline i = (i_1,\\ldots,i_n)\\in\\mathcal I_n: i_l \\le M < i_{l+1}\\} & \\text{if } 1\\le l \\le n-1\\,, \\cr\n\\{ \\underline i = (i_1,\\ldots,i_n)\\in\\mathcal I_n: i_n \\le  M \\} & \\text{if }l=n\\,.\n\\end{cases}\n\\end{equation*}\nDefine\n$$\n\\beta_{n,N,+}^{M, (l)}(s) \n:= \\sum_{\\underline i \\in \\mathcal I_n^{(l)}} \n\\prod_{j=1}^n \\alpha_{i_j,N,+}^{M}(s) \n\\quad\\text{for each }0\\le l\\le n\\,,\n$$\nso that in particular\n\\begin{equation}\\label{inparticularnminus}\n\\beta_{n,N,+}^{M, (n)}(s) = \\beta_{n,N,+}^{M, -}(s) \\,.\n\\end{equation}\n\nWith this notation, and since $\\mathcal I_n = \\bigcup_{l=0}^n \\mathcal I_n^{(l)}$,\n we can express $\\beta_{n,N,+}^{M}(s)$ as\n\\begin{equation}\\label{canexpress}\n\\beta_{n,N,+}^{M}(s) \n= \\sum_{\\underline i \\in \\mathcal I_n} \\prod_{j=1}^n \\alpha_{i_j,N,+}^{M}(s)   \n= \\sum_{l=0}^{n} \\beta_{n,N,+}^{M, (l)}(s)  \\,.\n\\end{equation}\nCombining (\\ref{inparticularnminus}) and (\\ref{canexpress}) gives\n\\begin{equation}\\label{epsilonsum}\n\\beta_{n,N,+}^{M}(s) - \\beta_{n,N,+}^{M,-}(s)\n= \\sum_{l=0}^{n-1} \\beta_{n,N,+}^{M, (l)}(s) \\,.\n\\end{equation}\n\nIn order to bound each $\\beta_{n,N,+}^{M, (l)}(s)$ in (\\ref{epsilonsum}) we\nuse the fact that $\\alpha_{i,N,+}^{M}(s) \\le J h^i$ for all $1\\le i\\le Q$ (see Corollary \\ref{jqnscor}) to obtain\n\\begin{equation}\\label{manipulation}\n\\beta_{n,N,+}^{M, (l)}(s) \n= \\sum_{\\underline i \\in \\mathcal I_n^{(l)}} \\prod_{j=1}^n \\alpha_{i_j,N,+}^{M}(s)\n\\le J^{n-l} \\sum_{\\underline i \\in \\mathcal I_n^{(l)}}  h^{i_{l+1} +\\ldots + i_n} \\prod_{j=1}^l \\alpha_{i_j,N,+}^{M}(s) \\,,\n\\end{equation}\nand introducing $\\underline \\iota = (\\iota_1,\\ldots,\\iota_{n-l})\\in \\mathcal I_{n-l}$ with $i_{l+k} = \\iota_k +M$ for $1\\le k\\le n-l$, we can re-express the righthand side of (\\ref{manipulation}) to obtain\n\\begin{equation*}\n\\beta_{n,N,+}^{M, (l)}(s)\n\\le J^{n-l} \\left( \\sum_{\\underline i \\in \\mathcal I_l^{(l)}}  \\prod_{j=1}^l \\alpha_{i_j,N,+}^{M}(s) \\right)\n\\left(  \\sum_{\\underline \\iota\\in \\mathcal I_{n-l}} h^{(n-l)M} h^{\\iota_1+\\ldots+\\iota_{n-l} }  \\right) \\,,\n\\end{equation*}\nand therefore\n\\begin{equation}\\label{betanlM}\n\\beta_{n,N,+}^{M, (l)}(s) \n\\le J^{n-l} \\beta_{l,N,+}^{M,-}(s)\\,  h^{M(n-l)} \\ E_{n-l}(h)\\,.\n\\end{equation}\n\nNow combining (\\ref{epsilonsum}) and (\\ref{betanlM}) gives the required bound (\\ref{betadifferencerequiredbound}).\n\\end{proof}\n\n\\begin{remark}\nIn practice the $l=n-1$ term on the righthand side of\n(\\ref{betadifferencerequiredbound}) tends to be the dominant one, as $M$ is chosen large enough so that\n$h^M$ is extremely small.\n\\end{remark}\n \n\\begin{defn}\\label{uppercomputedtaylorbound} \nLet $s\\in\\mathbb R$. For $n, Q,M,N\\in\\mathbb N$ with $n\\le Q\\le M\\le N$,\ndefine the \\emph{upper computed Taylor bound} $ \\beta_{n,N,+}^{M,+}(s)$ by\n$$\n \\beta_{n,N,+}^{M,+}(s)\n :=\n \\beta_{n,N,+}^{M,-}(s) +  \\sum_{l=0}^{n-1} J_{Q,N,s}^{n-l}\\, \\beta_{l,N,+}^{M,-}(s)\\, h^{M(n-l)} E_{n-l}(h) \\,.\n $$\n \\end{defn}\n \n From\n Lemma \\ref{betadifference} it then follows that the upper computed Taylor bound\n $\\beta_{n,N,+}^{M,+}(s)$\n  is indeed larger than the Taylor bound\n $\\beta_{n,N,+}^{M}(s)$:\n \n \n \\begin{cor}\\label{betadifferencecor}\nLet $s\\in\\mathbb R$.\nIf $Q,M,N\\in\\mathbb N$ with $Q\\le M\\le N$,\nthen \n$$\n\\beta_{n,N,+}^{M}(s) \\le \\beta_{n,N,+}^{M,+}(s)\\quad\n\\text{for all }1\\le n\\le Q \\,.\n$$\n\\end{cor}\n\\begin{proof}\nImmediate from Lemma \\ref{betadifference} and Definition \\ref{uppercomputedtaylorbound}.\n\\end{proof}\n\n Finally, we deduce that the $n^{th}$ Taylor coefficient $\\delta_n(s)$ of the determinant $\\det(I-z\\mathcal L_{A,s})$\n can be bounded in modulus by the upper computed Taylor bound\n $\\beta_{n,N,+}^{M,+}(s)$ (a quantity we can compute to arbitrary precision):\n\n\\begin{prop}\\label{taylorboundprop}\nLet $s\\in\\mathbb R$.\nIf $Q,M,N\\in\\mathbb N$ with $Q\\le M\\le N$,\nthen \n$$\n|\\delta_n(s)| \\le \\beta_{n,N,+}^{M,+}(s)\\quad\\text{for all } 1\\le n\\le Q\\,.\n$$\n\\end{prop}\n\\begin{proof}\n\nLemma \\ref{betacbound} gives\n$ |\\delta_n(s)| \\le \\beta_{n,N,+}^{M}(s)$, and\nCorollary \\ref{betadifferencecor} gives\n$\\beta_{n,N,+}^{M}(s) \\le \\beta_{n,N,+}^{M,+}(s)$,\nso the result follows.\n\\end{proof}\n\n\\begin{remark}\nIn \\S \\ref{E2section}, \nfor the computations in the proof of Theorem \\ref{e2theorem},\nwe choose $N=600$, $M=400$, and $Q=28$, using\nProposition \\ref{taylorboundprop} to obtain the upper bound on $|\\delta_n(s)|$ for $P+1=26\\le n\\le 28$,\nhaving explicitly evaluated $\\delta_n(s)$ for $1\\le n\\le 25$ using periodic points of period up to $P= 25$.\n\\end{remark}\n\n\n\n\n\n\n\n\\section{The Hausdorff dimension of $E_2$}\\label{E2section}\n\nHere we consider the set $E_2$, corresponding to the choice $A=\\{1,2\\}$.\nWe shall suppress the set $A$ from our notation, writing $\\mathcal L_s$ instead of $\\mathcal L_{A,s}$.\n\nThe approximation $s_N$ to $\\text{dim}(E_2)$, based on periodic points of period up to $N$,\nis the zero (in the interval $(0,1)$) of the function $\\frak D_N$ defined by (\\ref{DNdefn});\nthese approximations are tabulated in Table 1 for $18\\le n\\le 25$. \nWe note that the 24th and 25th approximations to $\\text{dim}(E_2)$ share\nthe first 129 decimal digits\n\\begin{align*}\n0.&5312805062772051416244686473684717854930591090183987798883978039 \\\\\n& 27529535643831345918109570181185239880428057243075187633422389339 \\,\n\\end{align*}\nthough the rate of convergence gives confidence that the first 139 digits\n\\begin{align*}\n0.&531280506277205141624468647368471785493059109018398779888397803927529 \\\\\n& 5356438313459181095701811852398804280572430751876334223893394808223090 \\,\n\\end{align*}\nof $s_{25}$ are in fact correct digits of $\\text{dim}(E_2)$.\n\nIt turns out that we can \\emph{rigorously} \njustify around three quarters of these decimal digits,\nproving that the first 100 digits are correct.\nIn fact we prove slightly more than that, \nby setting $s^-$ to be the value\n\\begin{align*}\ns^- =\n0.&531280506277205141624468647368471785493059109018398 \\\\\n& 77988839780392752953564383134591810957018118523987\\,,\n\\end{align*}\nand setting $s^+ = s^- + 2\/10^{101}$ to be the value\n\\begin{align*}\ns^+ =\n0.&531280506277205141624468647368471785493059109018398 \\\\\n& 77988839780392752953564383134591810957018118523989\\,.\n\\end{align*}\n\n\\begin{table}\n\\begin{equation*}\n\\begin{tabular}{|r|r|}\n\\hline\n$n$  & $s_n$ \\qquad\\qquad\\qquad\\qquad\\quad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\\\\n\\hline \n$18$ & 0.531280506277205141624468647368471785493059109018398779888397803927529535645 \\cr\n$$ & 596972005085668529391352118806494054592120629038239974478243258576620540205 \\cr\n$19$ & 0.531280506277205141624468647368471785493059109018398779888397803927529535643 \\cr\n$$ & 831345931151408384198942403518425963034455124305471103063941900681921725781 \\cr\n$20$ & 0.531280506277205141624468647368471785493059109018398779888397803927529535643 \\cr\n$$ & 831345918109570144457186603287266737112934351614056377793361034907544181115 \\cr\n$21$ & 0.531280506277205141624468647368471785493059109018398779888397803927529535643 \\cr\n$$ & 831345918109570181185239840988322512589524907498366765561230541095944497891 \\cr\n$22$ & 0.531280506277205141624468647368471785493059109018398779888397803927529535643\\cr\n$$ & 831345918109570181185239880428057259226147992212780800516214656456345194120 \\cr\n$23$ & 0.531280506277205141624468647368471785493059109018398779888397803927529535643 \\cr\n$$ &    831345918109570181185239880428057243075187635944921448427780108909724612227 \\cr\n$24$ & 0.531280506277205141624468647368471785493059109018398779888397803927529535643 \\cr\n$$ & 831345918109570181185239880428057243075187633422389339330546198723829886067 \\cr\n$25$ & 0.531280506277205141624468647368471785493059109018398779888397803927529535643 \\cr\n$$ & 831345918109570181185239880428057243075187633422389339480822309014454563836 \\cr\n\\hline\n\\end{tabular}\n\\end{equation*}\n\\caption{Approximations $s_n\\approx \\text{dim}(E_2)$; each $s_n$ is a zero of a truncation $\\frak D_n$ (formed using only periodic points of period $\\le n$) of the function $\\frak D$}\n\\end{table}\n\n\nWe then claim:\n\n\\begin{theorem}\\label{e2theorem}\nThe Hausdorff dimension of $E_2$ lies in the interval $(s^-,s^+)$.\n\\end{theorem}\n\\begin{proof}\nWe will show that $\\frak D(s^-)$ and $\\frak D(s^+)$ take opposite signs, \nand deduce that $\\dim(E_A)$, as the zero of $\\frak D$, lies between $s^-$ and $s^+$.\n\n\nLet $D\\subset\\mathbb{C}$ be the open disc centred at $c$, of radius $\\varrho$,\nwhere $c$ is the largest real root of the polynomial\n$$\n128c^7 + 768c^6 + 1296c^5  - 192c^4 - 1764c^3 - 108c^2 + 819c -216   \\,,\n$$\nso that\n$$c\\approx\n0.758687144013554292899790137015621955739402945444266741967051997691009 \\,,\n$$\nand\n\\begin{equation}\\label{crho}\n\\varrho=\n\\frac{-c+\\sqrt{-6c+5c^2+12c^3+4c^4}}{2c}\\,,\n\\end{equation}\nso that\n$$\n\\varrho\\approx\n0.957589818521375342814351002388265920293251603461349541441037951859499 \\,.\n$$\nThe relation (\\ref{crho}) ensures that $T_1(c-\\varrho)$ and $T_2(c+\\varrho)$ are equidistant from $c$, and this common distance is denoted by $\\varrho' = T_1(c-\\varrho) - c = c - T_2(c+\\varrho)$, so that\n\\begin{equation*}\\label{rhoprime}\n\\varrho' \\approx \n0.48960063348666271539624547964205669003751747416510762619582637319401 \\,.\n\\end{equation*}\nThe specific choice of $c$ is to ensure that the \ncontraction ratio $h = \\varrho'\/\\varrho$ is minimised, taking the value\n\\begin{equation*}\nh = \\frac{\\varrho'}{\\varrho} \\approx\n0.51128429314616176482942956363790038479511374855036304746799036536341 \\,.\n\\end{equation*}\n\n\n\\begin{figure}[!h]\n\\begin{center}\n \\includegraphics[]{e2_optimal_contraction_ratio_disc.pdf}\n\\caption{Inner disc $D'$ (dashed) contains images $T_1(D)$, $T_2(D)$ of the outer disc $D$, in the rigorous bound on the dimension of $E_2$}\n\\end{center}\n\\end{figure}\n\n\n\n\nHaving computed the points of period up to $P= 25$ we can \nform the functions\n$s\\mapsto \\delta_n(s)$ for $1\\le n\\le 25$,\nand evaluate these at $s=s^-$\n(cf.~Table 2)  to give\n\\begin{equation}\\label{sminus22}\n\\frak D_{25}(s^-) = 1+\\sum_{n=1}^{25} \\delta_n(s^-) =\n(-1.584605810787991617286291643870\\ldots)\n\\times 10^{-101}\n< 0\\,,\n\\end{equation}\nand at $s=s^+$ to give\n\\begin{equation}\\label{splus22}\n\\frak D_{25}(s^+) =\n1+\\sum_{n=1}^{25} \\delta_n(s^+) =\n(1.454514082498475271478438451769\\ldots)\n \\times 10^{-101} >0\\,.\n\\end{equation}\n\nWe now aim to show that the  approximation $\\frak D_{25}$ is close enough to \n$\\frak D$ for (\\ref{sminus22}) and (\\ref{splus22}) to imply, respectively, the negativity of $\\frak D(s^-)$\nand the positivity of $\\frak D(s^+)$.\nIn other words, we seek to bound the tail $\\sum_{n=26}^\\infty \\delta_n(s)$, and this will\nbe achieved by bounding the individual\nTaylor coefficients $\\delta_n(s)$, for $n\\ge 26=P+1$.\nIt will turn out that for $n\\ge29$ the cruder Euler bound on $\\delta_n(s)$ is sufficient,\nwhile for $26\\le n\\le 28$ we will use the Taylor bounds described in \\S \\ref{taylorsection}.\nMore precisely, for $P+1=26\\le n\\le 28=Q$ we will use the upper computed Taylor bound\\footnote{As will be noted shortly,\nthe upper computed Taylor bound we use agrees with the corresponding Taylor bound to over 200 decimal digits,\nso in particular the two quantities are indistinguishable at the 150 digit precision level of these computations.}\n$\\beta_{n,N,+}^{M,+}(s)$ for suitable $M,N\\in\\mathbb N$.\n\nHenceforth let $Q=28$, $M=400$, $N=600$ (so that in particular $Q\\le M\\le N$, as was assumed throughout \\S \\ref{taylorsection})\nand consider the case $s=s^-$.\n\nWe first evaluate\\footnote{As described in \\S \\ref{cabsection}, \n(\\ref{hardynormexplicit}) can be readily evaluated to\narbitrary precision using numerical integration; for this particular computation the precision level used was 150 decimal places.}  \nthe $H^2(D)$ norms of the monomial images $\\mathcal L_{s} (m_k)$ for $0\\le k\\le N=600$.\nThese norms are decreasing in $k$;\nTable 3 contains the first few evaluations, for $0\\le k\\le 10$,\nwhile for $k= 600$ we have\n$$\n\\| \\mathcal L_{s}(m_{600}) \\|= \n(2.297607298251023508986187604945746\\ldots) \\times 10^{-176} \\,.\n$$\n\nUsing these norms $\\|\\mathcal L_s(m_k)\\|$  we then evaluate, for $1\\le n\\le M=400$,  the\nupper computed approximation bounds\n$\\alpha_{n,N,+}(s) = \\alpha_{n,600,+}(s)$ defined (cf.~(\\ref{alphanNplus})) by\\footnote{Note that \n$h\\approx 0.511284$ and $N=600$,\nso $\\frac{h^{2(N+1)}}{1-h^2} \\le 8.8 \\times 10^{-351}$.\nMoreover (\\ref{weightboundsrange3}) gives\n$\\sum_{i=1}^2 \\|w_{i,s}\\|_\\infty \\le 1.81$, thus\n$ (\\sum_{i=1}^2 \\|w_{i,s}\\|_\\infty)^2 \\frac{h^{2(N+1)}}{1-h^2} \\le 2.9 \\times 10^{-350}$.\nCombining these bounds with the values taken by $\\alpha_{n,N,+}(s) $, \nit follows that for $1\\le n\\le 400$, \nthe approximation bound $\\alpha_n(s) = ( \\sum_{k=n-1}^\\infty  \\|\\mathcal L_{s} (m_k)\\|^2 )^{1\/2}$ agrees with both computed approximation bounds\n$\\alpha_{n,N,-}(s)$ and $\\alpha_{n,N,+}(s)$ to at least 200 decimal places, a level well beyond the\ndesired precision used in these calculations.} \n$$\n\\alpha_{n,N,+}(s) \n= \\left( \\sum_{k=n-1}^N \\| \\mathcal L_{s}(m_k)\\|^2 \n+ \\left(\\sum_{i=1}^2 \\|w_{i,s}\\|_\\infty\\right)^2 \\frac{h^{2(N+1)}}{1-h^2}\\right)^{1\/2} \\,.\n$$\nThese bounds are decreasing in $n$; Table 4 contains the first few evaluations, for $1\\le n\\le 10$,\nwhile for $n=400$ we have\n$$\n\\alpha_{400,600,+}(s) = \n(3.806826780744825698066314723072781\\ldots) \\times 10^{-147} \\,.\n$$\n\nThe upper computed approximation bounds $\\alpha_{n,600,+}(s)$ are then used to form the \nupper computed Taylor bounds\\footnote{The difference\n$ \\beta_{n,N,+}^{M,+}(s) -  \\beta_{n,N,+}^{M,-}(s)\n=\n\\sum_{l=0}^{n-1} J_{Q,N,s}^{n-l}\\, \\beta_{l,N,+}^{M,-}(s)\\, h^{M(n-l)} E_{n-l}(h)$ is smaller than\n$1.86 \\times 10^{-210}$ for $26\\le n\\le 28=Q$, so in fact the upper and lower computed Taylor bounds,\nand the Taylor bound $\\beta_{n,N,+}^M(s)$, agree to well beyond the 150 decimal place precision used in these computations.}\n $\\beta_{n,N,+}^{M,+}(s)\n=  \\beta_{n,N,+}^{M,-}(s) +  \\sum_{l=0}^{n-1} J_{Q,N,s}^{n-l}\\, \\beta_{l,N,+}^{M,-}(s)\\, h^{M(n-l)} E_{n-l}(h)$,\nwhere\n$$\n\\beta_{n,N,+}^{M,-}(s) = \\beta_{n,600,+}^{400,-}(s) =  \\sum_{i_1<\\ldots< i_n\\le 400} \\prod_{j=1}^n \\alpha_{i_j,600,+}(s)\\,,\n$$\nwhich for $26\\le n\\le 28=Q$ are\\footnote{See also Table 6\nfor computations of $\\beta_{n,N,+}^{M,+}(s)$ for $1\\le n\\le 28=Q$.}\n$$\n\\beta_{26,N,+}^{M,+}(s)\n=\n(7.0935010683530957339350457686786431427508\\ldots) \\times 10^{-103},\n$$\n$$\n\\beta_{27,N,+}^{M,+}(s)\n=\n(7.0379118021870691622913562125699156503586\\ldots) \\times 10^{-111},\n$$\n$$\n\\beta_{28,N,+}^{M,+}(s)\n= \n(3.5360715444914082167026977943200738452867\\ldots) \\times 10^{-119},\n$$\nso in particular Proposition \\ref{taylorboundprop} gives\n\\begin{equation}\\label{intermediateE2}\n\\sum_{n=26}^{28} |\\delta_n(s)|\n\\le\n\\sum_{n=26}^{28} \\beta_{n,N,+}^{M,+}(s)\n< 7.1\n\\times 10^{-103} .\n\\end{equation}\n\nIt remains to derive the Euler bounds on the Taylor coefficients $\\delta_n(s)$ for $n\\ge 29$.\nFor $s>0$, the functions $w_{1,s}(z) = 1\/(z+1)^{2s}$ and $w_{2,s}(z)=1\/(z+2)^{2s}$ have maximum modulus on $D$ when $z=c-\\varrho$, so \n\\begin{equation}\\label{weightbound}\n\\|w_{1,s}\\|_\\infty = 1\/(1+c-\\varrho)^{2s}\\quad \\text{and} \\quad \\|w_{2,s}\\|_\\infty = 1\/(2+c-\\varrho)^{2s}\\,.\n\\end{equation}\n\n\n\n\n\n\n\n\nA computation using (\\ref{weightbound}) gives\n\\begin{equation}\\label{weightboundsrange1}\n\\|w_{1,s}\\|_\\infty \n\\le \n1.2657276413750668025007241047661655434034644495987711959332997\n\\end{equation}\nand\n\\begin{equation}\\label{weightboundsrange2}\n\\|w_{2,s}\\|_\\infty \n\\le  \n0.5351507690357290789991731014616306223833750046974228167583536\n\\,,\n\\end{equation}\nthus\n\\begin{equation}\\label{weightboundsrange3}\n\\|w_{1,s}\\|_\\infty + \\|w_{2,s}\\|_\\infty \n\\le \n1.8008784104107958814998972062277961657868394542961940127\n\\,,\n\\end{equation}\nand therefore $K_s= (\\|w_{1,s}\\|_\\infty + \\|w_{2,s}\\|_{\\infty})\/( h \\sqrt{1-h^2})$ \nis bounded by\n\\begin{equation}\nK_s \n\\le  \n4.098460062897625162727128104751085223751087056801141844 \\,.\n\\end{equation}\n\nNow $|\\delta_n(s)| \\le K_s^n E_n(h)$, and we readily compute (see also Table 5) that\n$$\nK_s^{29} E_{29}(h) \n< 3.991837779947559\n\\times 10^{-109}\\,,\n$$\n$$\nK_s^{30} E_{30}(h) \n< 2.976234382308237\n\\times 10^{-117}\\,,\n$$\nand we easily bound\n\\begin{equation}\\label{easybound2}\n\\left| \\sum_{n=29}^\\infty \\delta_n(s) \\right| \\le\n\\sum_{n=29}^\\infty K_s^n E_n(h) <  4 \\times 10^{-109}\\,.\n\\end{equation}\n\nCombining (\\ref{easybound2})\nwith (\\ref{intermediateE2}) gives, for $s = s^-$,\n\\begin{equation}\\label{easyboundcombined}\n\\left| \\sum_{n=26}^\\infty \\delta_n(s) \\right| \n< 7.2 \\times 10^{-103}\\,.\n\\end{equation}\n\nCombining (\\ref{easyboundcombined}) with\n(\\ref{sminus22}) \nthen gives\n\\begin{equation}\n\\frak D(s^-) = 1+\\sum_{n=1}^\\infty \\delta_n(s^-) < 0 \\,.\n\\end{equation}\n\nIt remains to show that $\\frak D(s^+)$ is positive.\nIn view of (\\ref{splus22}), \nfor this it is sufficient to show that $\\left| \\sum_{n=26}^\\infty \\delta_n(s) \\right| < 10^{-101}$\nfor $s=s^+$.\nIn fact the stronger inequality (\\ref{easyboundcombined}) (which we have proved for $s=s^-$) can also be established\nfor $s=s^+$, using the same general method as for $s=s^-$,\nsince the intermediate computed values for the\nnorms $\\|\\mathcal L_s(m_k)\\|$, computed approximation bounds $\\alpha_{n,N,+}(s)$, computed Taylor bounds\n$\\beta_{n,N,+}^{M,+}(s)$,\nand Euler bounds $K_s^nE_n(h)$,\nare sufficiently close to those for $s=s^- = s^+ - 2\/10^{101}$. \nCombining (\\ref{splus22}) with inequality (\\ref{easyboundcombined}) for $s=s^+$ gives\nthe required positivity\n\\begin{equation}\n\\frak D(s^+) = 1+\\sum_{n=1}^\\infty \\delta_n(s^+) > 0\\,.\n\\end{equation}\n\nThe map $s\\mapsto \\frak D(s)$ is continuous and increasing, so the fact that $\\frak D(s^-) < 0 < \\frak D(s^+)$\nimplies that its unique zero (which is equal to the dimension) is contained in $(s^-,s^+)$.\n\\end{proof}\n\n\n\\begin{remark}\nIf, as in Theorem \\ref{e2theorem}, our aim is to rigorously justify 100 decimal places of the computed approximation $s_P$\nto the Hausdorff dimension, then roughly speaking $P$ should be chosen so that the modulus of the\ntail $\\sum_{n=P+1}^\\infty \\delta_n(s)$ can be shown to be somewhat smaller than $10^{-100}$ for $s\\approx s_P$. Since $|\\delta_n(s)|$ is bounded above by the upper computed Taylor bound $\\beta_{n,N,+}^{M,+}(s)$, the fact that\n$\\beta_{26,N,+}^{M,+}(s) < 7.1 \\times 10^{-103}$ (see Table 6) for suitably large $M, N$, together with the rapid decay \n(as a function of $n$) of these bounds, suggests that we may choose $P=25$, i.e.~that it suffices to explicitly locate the periodic points of period $\\le 25$.\n\nThe choice of the value $Q$ is relatively unimportant, as the upper computed Taylor bounds are only slightly more time consuming to compute than the (instantaneously computable) Euler bounds; in the proof of Theorem  \\ref{e2theorem} we chose $Q$ such that the Euler bounds $K_s^nE_n(h)$ were substantially smaller than $10^{-100}$ for $n>Q$ (our choice $Q=28$ has this property, as does any larger $Q$, and indeed the choice $Q=27$ may also be feasible, cf.~Table 5).\n\nThe values $M$ and $N$ \nare chosen large enough to ensure that the bound (\\ref{Taylorapproximationbound}) on $|\\delta_n(s)|$ is rendered essentially as sharp as possible using our method (see Proposition \\ref{alphaapproximationbound}) of bounding approximation numbers by approximation bounds; equally, the values $M$ and $N$ are of course chosen small enough to allow the $\\beta_{n,N,+}^{M,+}(s)$ to be evaluated in reasonable time.\n\\end{remark}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\Acp{afm} are promising materials for spintronic devices. Among the advantages over \\acp{fm} are the lack of stray fields, the very low susceptibility to magnetic fields, the abundance of materials and much faster spin dynamics \\cite{jungwirth_antiferromagnetic_2016,zelezny_spin_2018,baltz_antiferromagnetic_2018}. However, the antiferromagnetic order parameter in \\acp{afm} is difficult to read and to control because of a lack of macroscopic magnetization, a fact which is strongly related to some of their advantages. A major step in the field of antiferromagnetic spintronics \\cite{jungwirth_antiferromagnetic_2016,zelezny_spin_2018,baltz_antiferromagnetic_2018}\nwas the discovery of electrically induced \\ac{nsot} \\cite{zelezny_relativistic_2014,wadley_electrical_2016,olejnik_antiferromagnetic_2017,bodnar_writing_2018,olejnik_terahertz_2018} in specific antiferromagnetic materials. \nThese torques are a result of a special magnetic structure, where, for the magnetic state, global inversion symmetry is broken but one sublattice forms the inversion partner of the other, in combination with the inverse spin-galvanic or (Rashba--)Edelstein effect \\cite{zelezny_relativistic_2014}, which is the generation of a nonequilibrium spin polarization by electrical currents.\nCurrently, \\ce{CuMnAs} and \\ce{Mn2Au} are the two known materials that provide antiferromagnetic order at room temperature and possess the specific crystal structure required for \\ac{nsot}. The latter is the more promising material as its critical temperature is extremely high---higher than the peritectic temperature of about \\SI{950}{\\kelvin}, where the material decomposes \\cite{barthem_revealing_2013}---and it is easier to handle due to the lack of toxic components.\n\nDespite the fact that several studies clearly demonstrate that it is possible to switch the order parameter of \\ce{Mn2Au} via the application of an electrical current by $90^{\\circ}$ \\cite{zelezny_relativistic_2014,roy_robust_2016,meinert_electrical_2018,bodnar_writing_2018,salemi_orbitally_2019}, the switching mechanism---whether deterministic or thermally activated, coherent or via domain wall motion---remains concealed. The employed models and simulations so far rest on phenomenological descriptions \\cite{roy_robust_2016} and macrospin approximations \\cite{meinert_electrical_2018}. A microscopic and quantitative model of the switching process is missing. \n\nHere, we combine \\emph{ab initio} calculations with atomistic spin dynamics simulations to develop and employ a multi-scale model of the current induced switching in \\ce{Mn2Au}.\nThe three ingredients for this multi-scale model are \\emph{ab initio} calculations of the exchange interactions and anisotropies (section II), first-principles calculations of the current induced magnetic moments (section III), and atomistic spin model simulations (section IV), that include the results from the first-principles calculations and investigate the switching mechanism and its dynamics. We show that the switching is fast, on a time scale of some tens of picoseconds, but not purely deterministic, requiring some degree of thermal activation to overcome the anisotropy energy barrier during the switching process.\n\n\n\\section{Derivation of the spin model from ab initio calculations}\nWe employ the fully relativistic \\ac{skkr} method  \\cite{zabloudil_electron_2005} to determine the electronic structure and magnetic interactions of \\ce{Mn2Au}. \\ce{Mn2Au} crystallizes in the \\ce{MoSi2} structure with the lattice constants $a_{\\mathrm{2d}}=\\SI{3.328}{\\angstrom}$ and $c=\\SI{8.539}{\\angstrom}$ \\cite{wells_structure_1970,shick_spin-orbit_2010,barthem_revealing_2013}.\nThe \\ce{MoSi2}-type lattice geometry is depicted in Fig.~\\ref{fig:exchange}. The potentials were treated within the \\ac{asa} with an angular momentum cutoff of $\\ell_\\text{max}=2$ to describe the electron scattering. For energy integrations we used 15 energy points on a semicircular contour on the upper complex semiplane, and up to 7260 \n$k$-points in the irreducible wedge of the Brillouin zone near the Fermi energy for the calculation of spin model parameters. \n\n\nWe perform self-consistent calculations for the layered \\ac{afm} state shown in Fig.~\\ref{fig:exchange}, which has been identified as the magnetic ground state by neutron diffraction experiments \\cite{barthem_revealing_2013},\nbut also for the \\ac{fm} state.\nWe find the layered \\ac{afm} state lower in energy than the \\ac{fm} state by 25.8 mRy\/atom, which compares fairly well to the value reported in Ref.~\\cite{khmelevskyi_layered_2008} (21.5 mRy\/atom).\nAlso in agreement with Ref.\\ \\cite{khmelevskyi_layered_2008} we obtain a larger magnetic moment for the \\ce{Mn} atoms in the layered \\ac{afm} state ($\\SI{3.74}{\\mu_{\\mathrm{B}}}$) than in the \\ac{fm} state ($\\SI{3.70}{\\mu_{\\mathrm{B}}}$).\nFor the description of the switching process we consider the following spin model:\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\mathcal{H} =&- \\frac{1}{2}\\sum_{i\\neq j}J_{ij}\\vec{S}_{i}\\cdot\\vec{S}_{j}\n\t\t-\\sum_{i} d_{z}S_{i,z}^2\\\\\n\t\t&-\\sum_{i} d_{zz}S_{i,z}^4\n\t\t-\\sum_{i} d_{xy}S_{i,x}^2S_{i,y}^2 \\ ,\n\t\\end{aligned}\n        \\label{eq:spin_model}\n\\end{equation}\nwhere the isotropic exchange interactions $J_{ij}$ are obtained from \nthe \\ac{rtm} \\cite{udvardi_first-principles_2003},\nwhile the anisotropy parameters $d_{z}$, $d_{zz}$ and $d_{xy}$ are derived from band energy calculations in the spirit of the magnetic force theorem \\cite{weinberger_magnetic_2009}. \n\nThe isotropic exchange interactions calculated from the layered \\ac{afm} state as reference are plotted in Fig.~\\ref{fig:exchange} as a function of the interatomic distance. We can identify three dominant Heisenberg couplings: antiferromagnetic ones for the two nearest neighbors, $J_{1}=\\SI{-43.84}{\\milli\\electronvolt}$ and $J_{2}=\\SI{-81.79}{\\milli\\electronvolt}$, but a ferromagnetic one for the third nearest neighbor, $J_{3}=\\SI{39.28}{\\milli\\electronvolt}$. \nThese values show good qualitative agreement with those calculated in Ref.~\\cite{khmelevskyi_layered_2008} also in terms of the KKR-ASA method, but using a cutoff of $\\ell_{\\rm max}=3$ for the partial waves,\n$J_{1}=\\SI{-68.30}{\\milli\\electronvolt}$,\n$J_{2}=\\SI{-91.70}{\\milli\\electronvolt}$ and\n$J_{3}=\\SI{19.86}{\\milli\\electronvolt}$.\nSince the interactions $J_1$ and $J_2$ act between sublattices (layers), while $J_3$ is the leading interaction within a sublattice (cf.\\ Fig.~\\ref{fig:exchange}), these couplings clearly favor the layered \\ac{afm} state as the ground state of the system.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{{exchange_and_structure}.pdf}\n\t\\caption{Left: Isotropic exchange interactions as a function of distance calculated by using the \\acs*{rtm} and the \\acs*{sce} methods. Right: Crystal structure of \\ce{Mn2Au}. The two \\ce{Mn} sublattices are illustrated by red and blue spheres. The ground state orientations of the magnetic moments are indicated by arrows. The first three nearest neighbor magnetic exchange interactions $J_i$ are also visualized in the figure. \\label{fig:exchange}}\n\\end{figure}\n\nIt turns out that taking into account only the first three nearest neighbor interactions is not sufficient for a precise determination of the inter- and intra-sublattice interactions. In our simulations we, hence,  consider interactions up to a distance of $2.7\\ a_{\\mathrm{2d}}$, resulting in an inter-sublattice exchange interaction of $J_{\\mathrm{inter}}=\\SI{-371.13}{\\milli\\electronvolt}$ and an intra-sublattice exchange interaction of\n$J_{\\mathrm{intra}}=\\SI{182.36}{\\milli\\electronvolt}$. \nConsidering exchange interactions only in the first three shells yields $J_{\\mathrm{inter}}=4J_1+J_2=\\SI{-257.15}{\\milli\\electronvolt}$ and $J_{\\mathrm{intra}}=4J_3=\\SI{157.12}{\\milli\\electronvolt}$, being thus 30 and 14 \\% smaller in magnitude than the ones calculated with a spatial cutoff of $2.7\\ a_{\\mathrm{2d}}$.\n\nExperimental values for the effective inter-sublattice exchange coupling, $J_{\\mathrm{eff}}=-J_{\\mathrm{inter}}\/4$ \\cite{sapozhnik_experimental_2018},  were previously provided based on susceptibility measurements for \\ce{Mn2Au} powder \\cite{barthem_revealing_2013} and thin films \\cite{sapozhnik_experimental_2018}, $J_{\\mathrm{eff}}=\\SI{75}{\\milli\\electronvolt}$ and $J_{\\mathrm{eff}}=\\SI{22 \\pm 5}{\\milli\\electronvolt}$, respectively. The corresponding values from our calculations, $J_{\\mathrm{eff}}=\\SI{92.8}{\\milli\\electronvolt}$, and the one derived from the exchange interactions in Ref.~\\onlinecite{khmelevskyi_layered_2008}, $J_{\\mathrm{eff}}=\\SI{90}{\\milli\\electronvolt}$, compare remarkably well and are also in good agreement with the experimental result for the powder sample \\cite{barthem_revealing_2013}.    \n\nFrom our spin dynamics simulations we obtain a N\\'{e}el temperature of $\\SI{1680\\pm3}{\\kelvin}$, which is in good agreement with the value of \\SI{1610\\pm10}{\\kelvin} calculated in Ref.~\\cite{khmelevskyi_layered_2008} via Monte-Carlo simulations using nine nearest neighbor shells (the numerical values of which, however, were not provided beyond the first three shells).\nNote that due to a peritectic temperature of \\SI{950}{\\kelvin}, the N\\'eel temperature can only be extrapolated from experiments, yielding values in the range of \\SIrange{1300}{1600}{\\kelvin} \\cite{barthem_revealing_2013}.\n\nIn order to support the validity of our spin model description relying on the assumption of rigid magnetic moments that are stable against magnetic disorder, we also perform calculations using the \\ac{rdlm} theory \\cite{gyorffy_dlm_1985,staunton_rdlm_2006}. This approach assumes a fully spin disordered reference state, and also enables the extraction of spin model parameters by means of the so-called \\ac{sce} \\cite{drautz_spin-cluster_2004,szunyogh_atomistic_2011}, which maps the adiabatic magnetic energy surface onto a spin model.\n\nThe resulting isotropic Heisenberg couplings are also displayed in Fig.~\\ref{fig:exchange}. There is a remarkable similarity between the two spin model parameter sets, despite their quantitative differences especially for the first and third neighbor shells. \nObviously, the interactions obtained from the \\ac{sce}-\\ac{rdlm} calculation are also consistent with the layered \\ac{afm} structure as ground state and we obtain a N\\'{e}el temperature of $\\SI{1786\\pm3}{\\kelvin}$, which is in good agreement with the \\ac{rtm}. \n\nConceptually, the \\ac{rtm} gives a good approximation near the ground state, whereas the \\ac{sce} corresponds to a high-temperature phase. The fact that the two sets of parameters agree well despite this fundamental difference between the two methods can be explained by the rigidity of the \\ce{Mn} local spin moments.\nIn order to support this point we compare the \\ac{dos} for the two magnetic states in Fig.~\\ref{fig:DOS}. As also noted in Ref.~\\cite{khmelevskyi_layered_2008}, the narrow bandwidth of the \\ce{Mn} $\\mathrm{d}$-bands and the formation of a pseudogap around the Fermi level are visible in the \\ac{afm} state. The expected smearing of the \\ac{dos} in the \\ac{dlm} state due to spin disorder is clearly seen in the bottom panel of  Fig.~\\ref{fig:DOS}, but the large exchange splitting between the two spin channels prevails. \nThis shows up also in the spin moment of \\ce{Mn} calculated in the \\ac{dlm} state of $\\SI{3.71}{\\mu_{\\mathrm{B}}}$ being practically the same as in the layered AFM state.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{{DOS_total_comparison_eV}.pdf}\n\t\\caption{Density of states per atom from the electronic structure calculations in the \\ac{afm} (top panel) and \\ac{dlm} (bottom panel) states. The DOS for only one \\ce{Mn} sublattice is shown. Positive values correspond to spin up states, negative ones to spin down states. \\label{fig:DOS}}\n\\end{figure}\n\nAs for the anisotropies in Eq.~\\eqref{eq:spin_model}, we calculate a second order anisotropy of $d_{z}=\\SI{-0.62}{\\milli\\electronvolt}$, and fourth order anisotropies $d_{zz}=\\SI{-0.024}{\\milli\\electronvolt}$ and $d_{xy}=\\SI{0.058}{\\milli\\electronvolt}$. These values compare fairly well to those that can be derived from the anisotropy constants reported in Ref.~\\cite{shick_spin-orbit_2010},  \n$d_{z}=\\SI{-1.19}{\\milli\\electronvolt}$, $d_{zz}=\\SI{-0.015}{\\milli\\electronvolt}$, and $d_{xy}=\\SI{0.04}{\\milli\\electronvolt}$, in particular considering that the latter ones were calculated in terms of a full-potential density functional method  contrary to the \\ac{asa} we used in our calculations. This result is also in agreement with experimental reports of an upper bound for the in-plane anisotropy $d_{xy}$ of $\\SI{0.068}{\\milli\\electronvolt}$ \\cite{sapozhnik_direct_2018}.\nThus, in agreement with Refs.~\\cite{shick_spin-orbit_2010,barthem_easy_2016,sapozhnik_direct_2018} we find the magnetic easy axis along the $\\langle110\\rangle$ direction as illustrated in Fig.~\\ref{fig:exchange}. However, in our results the anisotropy responsible for the confinement in the basal plane is only about half in magnitude as compared to Ref.~\\cite{shick_spin-orbit_2010}. Note though that the out-of-plane anisotropy plays only a minor role in the switching process discussed in our work. \n\nFor our atomistic spin dynamics simulations we combine these anisotropies with the \\ac{rtm} exchange parameters since both are calculated from the same converged potential in contrast to the \\ac{sce} exchange parameters.\n\n\\section{First-principles calculations of the induced moments}\nThe inverse spin-galvanic or Rashba--Edelstein effect leads to electrically induced magnetic moments. These induced spin and orbital polarizations can be computed using the Kubo linear-response formalism.\nSpecifically, the locally induced polarizations can be expressed as\n\\begin{equation}\n    \\delta \\vec{S} = \\vec{\\chi}^S \\vec{E}, ~{\\textrm{and}} ~~ \\delta \\vec{L} = \\vec{\\chi}^L \\vec{E},\n\\end{equation}\nwith $\\boldsymbol{\\chi}^{S}$ and $\\boldsymbol{\\chi}^{L}$ \nthe spin and orbital Rashba--Edelstein  susceptibility tensors, respectively,\nand $\\vec{E}$ the applied electric field.\nThe magneto-electric susceptibility tensors can be obtained by evaluating the response to a perturbing electric field, $\\hat{V} = -e \\hat{\\vec{r}} \\cdot \\vec{E}$ where $e$ is the electron charge.\n\nEmploying DFT-based single-electron states, the susceptibility tensors are given by \\cite{salemi2021}\n\\begin{align}\n\\label{eq:LinearResponse}\n\\chi^{S,L}_{ij} &= -\\frac{ie}{m_e} \\int_{\\Omega} \\frac{d\\vec{k}}{\\Omega}\n\\sum_{n\\neq m} \\frac{f_{n\\vec{k}} - f_{m\\vec{k}} }{\\hbar \\omega_{nm\\vec{k}}}~\n\\frac{A^{(S,L)i}_{mn\\vec{k}} ~ p^j_{nm\\vec{k}} }{-\\omega_{nm\\vec{k}} + i\\tau_{\\text{inter}}^{-1}} \\nonumber \\\\\n& ~~~ -\\frac{ie}{m_e} \\int_{\\Omega} \\frac{d\\vec{k}}{\\Omega}\n\\sum_{n} \\frac{\\partial f_{n\\vec{k}} }{\\partial \\epsilon}~\n\\frac{A^{(S,L)i}_{nn\\vec{k}} ~ p^j_{nn\\vec{k}} }{i\\tau_{\\text{intra}}^{-1}} \\, .\n\\end{align}\nHere,  $ \\hbar \\omega_{nm\\vec{k}} = \\epsilon_{n\\vec{k}} - \\epsilon_{m\\vec{k}}$, with $\\epsilon_{n\\vec{k}}$ the unperturbed relativistic Kohn--Sham single electron energies, $\\Omega$ is the Brillouin zone volume, $p^j_{nm\\mathbf{k}}$ is the matrix element of the $j^\\text{th}$ component of the momentum-operator, and\n$f_{n\\vec{k}}$ is the occupation of Kohn--Sham state $|n\\vec{k}\\rangle$. $\\vec{A}^{(S,L)}_{mn\\vec{k}}$ stands for a matrix element of the spin or orbital angular momentum operator, i.e., \n$\\vec{A}^{S}_{mn\\vec{k}} = \\hat{\\vec{S}}_{mn\\vec{k}}$  for $\\vec{\\chi}^S$ and $\\vec{A}^{L}_{mn\\vec{k}} = \\hat{\\vec{L}}_{mn\\vec{k}}$ for $\\vec{\\chi}^L$.\nThe parameters $\\tau_{\\text{inter}}$ and $\\tau_{\\text{intra}}$ are the electronic lifetimes for inter- and intraband scattering processes, respectively. These parameters capture the decay of an electron state $|n\\vec{k}\\rangle$ due to electron-electron scattering and interactions with external baths, e.g., phonons and defect scattering. In this work, we use an effective decay $\\tau = \\tau_{\\text{inter}} = \\tau_{\\text{intra}} = \\SI{50}{\\femto\\second}$.\n\nTo compute the current induced spin and orbital polarizations on the individual atoms in \\ce{Mn2Au} we employ the relativistic DFT package WIEN2k \\cite{Blaha2018}, which gives the Kohn--Sham energies $\\epsilon_{n\\vec{k}}$ and wave functions $|n\\vec{k}\\rangle$ that are then used in Eq.\\ (\\ref{eq:LinearResponse}). We calculate the induced magnetic moments \nfor different orientations of the electrical field with respect to the magnetic easy axes, as reversible switching was reported for both the \\mbox{[110]} and \\mbox{[100]} directions \\cite{bodnar_writing_2018}. Furthermore, we evaluate both the induced spin and orbital polarizations. \nThe local magnetic moments induced by the electric field are finally given as\n$\\vec{\\mu}= \\vec{\\mu}_S + \\vec{\\mu}_L = (2\\vec{\\chi}^S + \\vec{\\chi}^L)\\vec{E}$.\n  \n\n\\begin{figure*}[ht]\n\t\\begin{minipage}[H]{.48\\linewidth}\n\t\t\\includegraphics[width=0.9\\linewidth]{{orbital_angledep}.pdf}\n\t\\end{minipage}%\n\t\\begin{minipage}[H]{.03\\linewidth}\n\t\t\\hspace{\\linewidth}\n\t\\end{minipage}%\n\t\\begin{minipage}[H]{.48\\linewidth}\n\t\t\\includegraphics[width=0.9\\linewidth]{{spin_angledep}.pdf}\n\t\\end{minipage}\n\t\\caption{Calculated induced orbital ($\\mu_L$) and spin ($\\mu_S$) moments on the two \\ce{Mn} sublattices as a function of the electric field direction for a field of $E=\\SI{1e7}{\\volt\\per\\meter}$ and local magnetic moments oriented along the \\mbox{[110]} direction.\n\ta) In-plane direction of the induced orbital moments on the two \\ce{Mn} sublattices (black vs.\\ red). The arrows in the center depict the local magnetic moments. b) Cartesian components of the induced orbital moments on the two \\ce{Mn} sublattices (solid vs.\\ dashed) as a function of the in-plane angle of the electric field with respect to the \\mbox{[100]} axis. c) and d) same as in a) and b) for the induced spin moments. \\label{fig:induced_moments}}\n\\end{figure*}\n\nThe calculated induced orbital and spin magnetic moments on the two \\ce{Mn} sublattices are presented in Fig.\\ \\ref{fig:induced_moments} as a function of the electric field direction. The orbital moments $\\mu_L$ are always induced perpendicular to the electric field direction and are antisymmetric (staggered) for the \\ce{Mn} atoms of two sublattices. The spin moments $\\mu_S$, on the other hand, are not necessarily perpendicular to the electric field direction, but their in-plane components are staggered as well. Additionally, the spin moments display a homogeneous out-of-plane component, i.e., a non-N\\'eel-type contribution. \n\nInterestingly, in all configurations the induced orbital moments are more than one order of magnitude larger than the induced spin moments, yet the former were not included in previous studies \\cite{zelezny_relativistic_2014,zelezny_spin-orbit_2017}.\nTo summarize, there are always quite large staggered orbital moments induced on the \\ce{Mn} sublattices and small induced spin moments with nonstaggered as well as staggered components that can be parallel on antiparallel to the orbital moments depending on the direction of the electric field, see also \\cite{salemi_orbitally_2019}.\n\n\\section{Atomistic spin dynamics simulations}\n\nTo include our first-principles calculations in a spin dynamics simulation we extend the semi-classical Heisenberg Hamiltonian [Eq.\\ (\\ref{eq:spin_model})] by contributions from induced spin and orbital moments,\n\\begin{equation}\n\\begin{aligned}\n\t\\mathcal{H} =&\n\t- \\frac{1}{2}\\sum_{i\\neq j}J_{ij}(\\vec{S}_{i}+\\vec{s}_{i})\\cdot(\\vec{S}_{j}+\\vec{s}_{j})\\\\\n\t& - \\sum_{i}J_{}^{\\mathrm{sd}}\\vec{S}_{i}\\cdot\\vec{s}_{i}\n\t+ \\sum_{i} \\xi_{}\\vec{S}_{i}\\cdot\\vec{l}_{i}\\\\\n\t&-\\sum_{i} d_{z}S_{i,z}^2 -\\sum_{i} d_{xy}S_{i,x}^2S_{i,y}^2 \\ ,\n\\end{aligned}\n\\end{equation}\nwhere $\\vec{S}_{i} =\\vec{\\mu}^{\\mathrm{d}}_{S,i}\/{\\mu}^{\\mathrm{d}}_{S}$ is the local magnetic moment of the $\\mathrm{d}$-electrons, $\\vec{s}_{i} =\\vec{\\mu}^{\\mathrm{s}}_{S,i}\/{\\mu}^{\\mathrm{d}}_{S}$ the induced magnetic moment from the conduction $\\mathrm{s}$-electrons and $\\vec{l}_{i} =\\vec{\\mu}_{L,i}\/{\\mu}^{\\mathrm{d}}_{S}$ the induced orbital magnetic moment. \nAll magnetic moments are normalized with respect to the local magnetic moment.\nThus, the Hamiltonian consists of five different contributions: the inter-atomic exchange with exchange constant $J_{ij}$, an additional intra-atomic $\\mathrm{sd}$-exchange with exchange constant $J^{\\mathrm{sd}}_{}$, a \\ac{soc} term with strength $\\xi_{}$, as well as second and fourth order anisotropy terms constituting the tetragonal anisotropy. \n\nAs our classical spin model employs quantum mechanical and statistical averages of the spin and orbital moments, we also use a classical description of the \\ac{soc}  replacing the spin and orbital momentum operators by their averages. Note that this effective model for the \\ac{soc} was used by Bruno \\cite{bruno_physical_1993} in order to provide a simple physical interpretation of magnetic anisotropy.   \nIn this model only the spin moments couple via the inter-atomic exchange interaction, which is in agreement with the conclusions from \\cite{sapozhnik_experimental_2018}.\n\nAll the contributions from the induced moments can also be represented by a simple Zeeman-like term with a sublattice-specific effective field that represents the staggered field, which was used in previous phenomenological descriptions,\n\\begin{align}\n\t{\\mu}^{\\mathrm{d}}_{S}\\vec{B}_{i}^{\\mathrm{ind}} =  \\sum_{j}J_{ij}\\vec{s}_{j} + J_{}^{\\mathrm{sd}}\\vec{s}_{i} - \\xi_{}\\vec{l}_{i} \\ .\n\\end{align}\nFor the intra-atomic exchange we estimate from the shift in the up and down s-states $J^{\\mathrm{sd}}=\\SI{50}{\\milli\\electronvolt}$. The \\ac{soc} strength is calculated from the energy difference between the $\\mathrm{d}_{3\/2}$ and $\\mathrm{d}_{5\/2}$ resonances yielding $\\xi=\\SI{46}{\\milli\\electronvolt}$. \nTogether with the exchange interactions derived in Sec.\\ II and the induced moment calculated for an electrical field of $\\SI{1e7}{\\volt\\per\\meter}$, this yields staggered fields of about $\\SI{76}{\\milli\\tesla}$.\nHere, the contribution from the induced orbital moments dominates \\cite{salemi_orbitally_2019}. It is about a factor of five larger than the contribution from the inter-atomic exchange and more than one order of magnitude larger than that of the intra-atomic exchange.\nThis explains also why the staggered fields calculated here are much larger than those estimated and predicted before \\cite{roy_robust_2016,zelezny_spin-orbit_2017,meinert_electrical_2018} as the orbital contribution was previously not taken into account.\n\nThe time evolution of the localized \\ce{Mn} moments stemming from the $\\mathrm{d}$-electrons is described by the stochastic \\ac{llg} equation\n\\begin{align}\n\\dot{\\vec{S}}_{i}=-\\frac{\\gamma}{\\left(1+\\alpha^2\\right){\\mu}^{\\mathrm{d}}_{S}} \\vec{S}_{i}\\times \\Big[\\vec{H}_{i}+\\alpha \\vec{S}_{i}\\times\\vec{H}_{i}\\Big] \\ ,\n\\end{align}\nwhere $\\gamma=\\SI{1.76e11}{\\per\\second\\tesla}$ is the gyromagnetic ratio and $\\alpha$ a dimensionless damping constant. Temperature is included via Langevin dynamics by adding a random thermal noise $\\vec{\\zeta}_{i}$ to the effective field $\\vec{H}_{i}=-\\frac{\\partial\\mathcal{H}}{\\partial\\vec{S}_i}+\\vec{\\zeta}_{i}$ \\cite{kronmuller_classical_2007}. The field from the induced moments $\\vec{B}_{i}^{\\mathrm{ind}}$ is part of this effective field.\n\nThe damping constant is a free parameter as there are no experimental values for it in the literature.\nFor comparison with \\cite{roy_robust_2016} we use a plausible value of $\\alpha=0.01$. Similar, for the electrical field a rectangular pulse with pulse length of \\SI{20}{\\pico\\second} was simulated to compare the results with those from a phenomenological model \\cite{roy_robust_2016}.\nSince the samples in experiments are mostly of granular type \\cite{meinert_electrical_2018}, we simulate a system of $\\SI{20.3}{\\nano\\meter}\\times\\SI{20.3}{\\nano\\meter}\\times\\SI{20.5}{\\nano\\meter}$ size with open boundary conditions, resembling one grain of a typical sample.\n\nIn our simulations we consider electrical fields along \\mbox{[110]}, i.e.\\ parallel to the local magnetic moments, and along \\mbox{[100]}, since reversible switching was reported for both directions \\cite{bodnar_writing_2018}.\nFor both field configurations our model does not switch at $T=0$ for  $E=\\SI{1e7}{\\volt\\per\\meter}$ corresponding to currents of about \\SIrange{1e10}{1e11}{\\ampere\\per\\meter\\squared}, which are used in experiments \\cite{bodnar_writing_2018,bodnar_imaging_2019}. Instead, we need a field strength of at least  $E=\\SI{1.9e7}{\\volt\\per\\meter}$ for the field along the \\mbox{[110]} direction, where torques on the local magnetic moments are maximal. For the \\mbox{[100]} direction an even larger field of $\\SI{3.1e7}{\\volt\\per\\meter}$ is required for switching at zero temperature. However, once  the system switches, it switches within a few picoseconds, see Fig.~\\ref{fig:T0_comparison}. This is even faster than predicted in the phenomenological model in Ref.~\\cite{roy_robust_2016}, probably because of the inclusion of the orbital induced moments and the exchange interactions beyond the first three nearest neighbors.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{{T0_comparison}.pdf}\n\t\\caption{Time evolution of the magnetic order parameter during $90^{\\circ}$ switching at $T=\\SI{0}{\\kelvin}$. The electric field (applied in the shaded area) is \\SI{1.9e7}{\\volt\\per\\meter} in \\mbox{[110]} direction (top) and \\SI{3.1e7}{\\volt\\per\\meter} in \\mbox{[100]} direction (bottom).\\label{fig:T0_comparison}}\n\\end{figure}\n\nAs was already pointed out in Ref.~\\cite{roy_robust_2016}, the reason for this rapid switching is the so-called exchange enhancement, which is characteristic for antiferromagnetic dynamics \\cite{dannegger2021}. The staggered fields do not only rotate the magnetic moments via the damping term in the \\ac{llg} but also induce a canting between the sublattices via the much stronger precession term. \nThis leads to a very small magnetization resulting in huge torques due to the inter-sublattice exchange field, which tries to realign the sublattices. Here, the damping term is responsible for the realignment. The precession term, on the other hand, rotates the magnetic moments towards the direction of the staggered field. The out-of-plane component of the order parameter remains zero during the process (see Fig.\\ \\ref{fig:switching_path}). Hence, the inter-sublattice exchange field governs the switching process and, in contrast to the switching in \\acp{fm}, lower damping allows for faster switching  \\cite{rozsa_reduced_2019}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{switching_path.pdf}\n\t\\caption{Switching path of the two sublattice magnetization vectors. The antiparallel \\ce{Mn} moments switch over 90$^{\\circ}$ from the initial \\mbox{[110]} configuration (semi-transparent) to the final \\mbox{[-110]} configuration (opaque). During the switching process the sublattices are canted slightly resulting in huge torques from the inter-sublattice exchange field enhancing the switching process significantly. This exchange enhancement is characteristic for antiferromagnetic dynamics \\cite{dannegger2021}. The out-of-plane component is here scaled by a factor of 100!\n\t\\label{fig:switching_path}}\n\\end{figure}\n\nThe electric fields considered so far are much larger than those applied in experiments, but  temperature plays an additional major role. A finite temperature does not only lower the energy barrier, here the fourth order in-plane magnetic anisotropy, but thermal fluctuations can also support  probabilistic switching.\nFig.~\\ref{fig:T_comparison} shows the time evolution of the order parameter at elevated temperatures as well as the switching probability as a function of temperature for electrical fields of $E=\\SI{1e7}{\\volt\\per\\meter}$. For the \\mbox{[110]} direction the system does not switch at temperatures below \\SI{250}{\\kelvin}, between \\SI{250}{\\kelvin} and \\SI{350}{\\kelvin} the process is probabilistic and above \\SI{350}{\\kelvin} deterministic.\nIn the deterministic regime the energy barrier is so low that the system switches in a few picoseconds, similar to simulations with larger electric fields. In the probabilistic regime, however, it can take several attempts to cross the energy barrier due to thermal agitation. Of course, here the switching probability also depends on the pulse length of the external electric field as longer time scales allow for more stochastic attempts to cross the barrier.\nFor the electric field along the \\mbox{[100]} direction the probabilistic regime lies between \\SI{400}{\\kelvin} and \\SI{550}{\\kelvin}, above which the switching is deterministic.\n\nReversible switching for pulse currents along the \\mbox{[100]} direction was also observed in experiments \\cite{bodnar_writing_2018}. In the same paper, also significant heating resulting in temperatures up to \\SI{300}{\\celsius} was reported and thermal activation was considered to play an important role in the process. A key role of thermal activation was also reported by \\textcite{meinert_electrical_2018}.\nOf course, for thermal switching of nanoparticles the system size is crucial as well, especially for antiferromagnets as their thermal stability is much lower than that of ferromagnets \\cite{rozsa_reduced_2019}. Here, the system size was chosen such to avoid  a purely superparamagnetic switching which would lead to a forth and back switching.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{{T_comparison}.pdf}\n\t\\caption{100 trajectories of the order parameter during the electric field pulse of $E=\\SI{1e7}{\\volt\\per\\meter}$ (shaded area). Top: at $T=\\SI{300}{\\kelvin}$ for an electric field in \\mbox{[110]} direction. Bottom: at $T=\\SI{500}{\\kelvin}$ for an electric field in \\mbox{[100]} direction. The inset shows the switching probability as a function of temperature.\\label{fig:T_comparison}}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\nModeling the current induced switching process in \\ce{Mn2Au} with all its different contributing terms in a quantitative manner is a challenging task. Here, we have presented the first multi-scale model combining first-principles calculations of exchange and anisotropy constants, as well as electrically induced spin and orbital moments in an extended atomistic spin model.\nWe predict much higher effective staggered fields due to the formerly neglected contributions from induced orbital moments.\nWithin the framework of atomistic spin dynamics simulations, we have shown that these fields---combined with inter-sublattice exchange interactions---result in switching processes on the time scale of few picoseconds. However, this switching requires significantly higher electrical fields than in experiments or, alternatively, elevated temperatures. This applies for both considered electrical field directions, \\mbox{[110]} and \\mbox{[100]}, which is in agreement with experimental findings \\cite{bodnar_writing_2018}.\nHence, in agreement with previous experimental studies \\cite{bodnar_writing_2018,meinert_electrical_2018} we find that thermal activation plays a key role in the current induced switching process and we have consequently distinguished temperature regimes for probabilistic and deterministic switching. \n\n\n\\begin{acknowledgments}\nThe authors gratefully acknowledge valuable discussions with Karel Carva. L.S.\\ and P.M.O.\\ acknowledge funding from the Swedish Research Council (VR) and the European Union's Horizon2020 Research and Innovation Programme under FET-OPEN Grant agreement No.\\ 863155 (s-Nebula), and acknowledge computer resources provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC Center for High Performance Computing and the Uppsala Multidisciplinary Center for Advanced Computational Science (UPPMAX). The work of A.D., E.S.\\ and L.Sz.\\ was supported by National Research, Development, and Innovation Office under projects\nNo.\\ PD134579 and No.\\ K131938. The work in Konstanz was supported by the {Deutsche Forschungsgemeinschaft} via the {Sonderforschungsbereich 1432}.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nStars form mainly due to gravitational contraction of a molecular cloud.\nMolecular cloud is like a vast laboratory for space chemistry. From the verge \nof collapse to star formation, a molecular cloud goes through different stages. \nChemical changes during this evolution acts as \na hint to understand physical and chemical processes associated with the source. \nFor example, SiO is a good shock tracer, high-velocity CO line wings are a good tracer \nof outflows; CH$_3$OH can trace other kinds of environments, including outflows\n\\citep{taquet2015} and even cold gas in pre-stellar cores \\citep{vastel2014}, C$^{17}$O helps to \nknow disk properties in some cases, etc. After the collapse of a molecular cloud, \na young stellar object forms at the central region. This central region is embedded in a thick rotating \nenvelope which further forms a disk like region. Some of the matter is often ejected in the form of bipolar outflow\ncarrying significant amount of angular momentum.\n\\citet{lada87} classified an evolutionary sequence of young stellar objects depending on the spectral energy \ndistribution, e.g., class I, class II, class III. Younger protostars are often called class 0 object \\citep{andre93}.\nHH212 is thought to be a Class 0 object of low-mass proto stellar system, located in Orion at 400pc \\citep{koun2017}.\n\nThe inner region of the low-mass star-forming cores may be enriched with complex molecules, e.g.,\nmethyl formate (HCOOCH$_3$), ethyl cyanide ($\\mathrm{C_2H_5CN}$), dimethyl ether \n($\\mathrm{CH_3OCH_3}$), methyl alcohol ($\\mathrm{CH_3OH}$) and formaldehyde (H$_2$CO) \\citep{biss2008, bott2004, cazaux2003}.\nTo differentiate between the `hot core' region that are present in high mass star \nforming region, a term `hot corino' \\citep{ceccar2007} is often used in the context of a low mass \nstar forming region. Molecules can form both in the gas phase and on the grain surface --\nduring the evolution of molecular cloud in gas phase, \nmolecules can deplete on grain surface and start new chemistry there. \nGrain surfaces help formation of complex molecules by hydrogenation; \nthere are also routes by which complex molecules can\nalso form in the gas phase too. Complex molecules like HCOOCH$_3$,\nCH$_3$OCH$_3$ \\citep{balu2015}, NH$_2$CHO \\citep{barone2015}\ncan effectively form in gas-phase following sublimation of key\nsimpler precursors, e.g., CH$_3$OH, NH$_2$, HDCO from grain mantles.\nComplex molecules which form on grain mantles can populate the gas-phase \nby desorption from grain mantels. Due to thermal\ndesorption in high-temperature region e.g., hot core, hot corinos,\ncomplex molecules or their precursors come out of the grain surfaces in gas phase.\nFor this reason, high deuteration of complex species in grain surfaces\nmay be reflected to gas phase (e.g., \\citealp{cc2014, das2015a, das2015b}). Deuterated water\n has been reported by \\citet{code2016} in the hot-corino of HH212. Very recently \n\\citet{lee2017a} reported D$_2$CO and singly deuterated methanol (CH$_2$DOH),\nfirst deuterated complex molecule to be observed in central\nregion of HH212.\n\nDue to limited spatial resolution of observational facilities in the past, only a few hot-corinos have\nbeen reported till now, on $<$100 AU scale, e.g., IRAS 16293-2422, NGC 1333 IRAS2A, IRAS4A etc.\n(e.g., \\citealp{imai16,jorgen2012} and references therein).\nRecently, \\citet{code2016} has suggested a `hot-corino' region in HH212\nsystem. The HH212 region is an ideal system to investigate different processes (e.g.,\ninfall, rotation, hot corino, bipolar outflow) related to the formation of a low mass protostar. \nThe HH212 source has been observed in the past using Sub-millimeter array \n(SMA)\\citep{lee2006}, the IRAM Plateau de Bure (PdB) interferometer \\citep{code2007} and Atacama Large Millimeter Array (ALMA)\n\\citep{lee2014, code2014}. ALMA, due to its high spatial resolution and sensitivity has revealed much more \ndetail than any other previous observations. The HH212 system has bipolar jets as observed by SiO, SO, and SO$_2$ \nemission lines \\citep{code2014,podio2015}. It has a central hot-corino surrounded by a flattened, \ninfalling and rotating envelope as observed by \nC$^{17}$O and HCO$^+$ \\citep{lee2006,lee2014}. Also, from HCO$^+$ and C$^{17}$O emission line \nobservations, \\citet{lee2014}, \\citet{code2014} suggested a compact disk ($\\sim 90$AU) rotating around the \nsource of $\\cong 0.2-0.3$M$\\odot$.  Later observation \\citep{lee2017b} resolved the disk and\nsuggested the disk size to be 40 AU.\n\nIn this paper, we use ALMA archival dataset 2011.0.00647.S \nand report emission of deuterated formaldehyde line from central hot-corino region. We compare the HDCO emission \nwith methanol (CH$_3$OH), C$^{34}$S and C$^{17}$O and try to explain chemistry around the central region of the source.\nThe line search is performed using splatalouge (www.splatalouge.net) and molecular data has been taken from\nthe CDMS \\citep{muller01,muller05} and JPL molecular database \\citep{pic1998}.\n\n\\section {Observations}\nThe HH212 protostar system was observed with ALMA (Band -7) using 24 12m antennas on 2012 December 1 \n(Early Science Cycle 0 phase\n, \\citealt{code2014}). In this observation, the shortest and the longest baselines were respectively 20m and 360m. We report \nHDCO (Table~\\ref{tab:lines}) and include the CH$_3$OH, C$^{34}$S and C$^{17}$O lines \nto compare with deuterated formaldehyde emission. The datacubes have a spectral resolution \n488 KHz ($\\sim 0.43$ km s$^{-1}$), a typical beam FWHM of $0.''65 \\times 0.''47$ at position angle (PA) $\\sim 49^{\\circ}$.\nThe observed spectral windows were 333.7-337.4 GHz and 345.6-349.3 GHz, the typical rms level was 3-4 mJy beam$^{-1}$\nin 0.43 km s$^{-1}$ channels.\nThe data were calibrated with the CASA package, with\nquasars J0538-440 and J0607-085 as the passband calibrators, quasar\nJ0607-085 as the gain calibrator, and Callisto and Ganymede as the\nflux calibrators. We generated spectral cubes by subtracting the continuum emission in visibility data.\nWe used Briggs\nweighting with robustness parameter 0.5 for CLEANing the image.\nPositions are given with respect to peak continuum of the MM1 protostar located at \n$\\alpha(J2000)= 05^h43^m51^s.41,\n \\delta(J2000)=-01^{\\circ}02'53''.17$ \\citep{lee2014}.\n\n\\section{Results and Discussions}\n\n\n\n\n\nALMA band 7 is found to be rich in numerous spectral transitions in a region \ntowards MM1 protostar. Here, we discuss \nthree lines, C$^{17}$O, deuterated formaldehyde line (HDCO 5(1,4)-4(1,3)) and methanol (CH$_3$OH(v0 7(1,7)- 6(1,6)).\nThe details of the lines are given in Table~\\ref{tab:lines}. The C$^{17}$O (3-2)  line transition was used by \\citet{code2014}\nto describe the rotating envelope around the central region. Here, we consider C$^{17}$O (3-2) \nemission as optically thin and have calculated \nits column density around different regions of the source. We compare C$^{17}$O column density with HDCO and CH$_3$OH\ncolumn density to study the chemistry around the MM1 protostar.\nFrom Figure~\\ref{fig:hdco_overlap}, HDCO emission is seen in central region of the protostar system and \npartially resolved (synthesized beam size $0.66'' \\times 0.47'', {\\rm PA}\\quad 49.6^{\\circ}$ and image size $0.70''\n\\times 0.49'', {\\rm PA}\\quad 57.2^{\\circ}$).\nFor the first time, we are reporting deuterated formaldehyde emission around the MM1 protostar position.\nRecent work by \\citet{leurini2016} \nhas described the kinematics of methanol in HH212. Moreover, from a recent observation (ALMA, 2015)\n\\citet{lee2017a} found that methanol is from a rotating disk environment. Here, we compare formaldehyde emission \nwith the most intense methanol line (E$_u$=79 K). Both the line profiles of formaldehyde and methanol\nhave peaks at $\\sim$ 1.9 km s$^{-1}$, i.e., close to the systematic velocity $\\sim$1.7 km s$^{-1}$ \\citep{lee2014}.\nThough \\citet{code2014} suggested a systematic velocity of $\\sim$ 1.3 km s$^{-1}$,\nhere we consider the systemic velocity to be $\\sim$ 1.7 km s$^{-1}$. \n\n\\begin{table}\n\t\\centering\n\t\\caption{List of unblended transitions detected towards HH212-MM1 and line properties.}\n\t\\label{tab:lines}\n\t\\begin{tabular}{lccc c}\n\t\t\\hline\n\t\tSpecies & Transition& Frequency& $E_u$& S$\\mu^2$\\\\\n\t\t        &        & (GHz)       &(K)    & D$^2$  \\\\\n\t\t\\hline\n\t\tCH$_3$OH &v0 7(1,7)- 6(1,6) & 335.58202 &78.97& 5.55\\\\ \n\t\tHDCO & 5(1,4)-4(1,3)& 335.09678 &56.25&26.05422\\\\\n\t\tC$^{17}$O & J=3-2 & 337.06112&  32.35  & 0.01411\\\\\n\t\tC$^{34}$S & J=7-6 & 337.396459 &   50.23 & 25.57\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\begin{figure}\n\n\\includegraphics[width=\\columnwidth]{figure1.eps}\n\n\\caption{HH212 system, observed by ALMA -Band 7 \\citep{code2014}. Here, integrated emissions (moment 0 map) of three\n     lines, SiO, HDCO and C$^{17}$O are overlaid on source continuum (gray scale). \n     Magenta contours show the SiO(8-7) bipolar jet; \n     first contour and steps are 10$\\sigma$\n     ($\\sigma\\sim$ 5 mJy\/beam km\/s)\n     Blue contour is of HDCO; contours are from 7$\\sigma$ and steps are 8$\\sigma$(15 mJy\/beam km\/s).\n     Red contour is of C$^{17}$O, contours are from 5$\\sigma$ in steps of 5$\\sigma$ (1.5 mJy\/beam km\/s)}\n     \n\\label{fig:hdco_overlap}\n\\end{figure}\n\\subsection{HDCO emission}\nThe HDCO emission is almost symmetric around the systematic velocity ($\\sim 1.6-2.0$ km s$^{-1}$), so it can be assumed that \nit originates near the central hot region. In this region, the dust temperature is \nhigh enough, and molecules on the grain surface (i.e., on the dust) easily desorb to gas phase. \nTemperature of hot-corinos may \nvary from few 10's K to few 100's K \\citep{parise2009, code2016}. Here, we have observed only one HDCO transition, \nso the temperature can not be derived directly. \\citet{code2016} used the same dataset and listed five acetaldehyde \n(CH$_3$CHO) transitions; from optically thin LTE analysis of CH$_3$CHO they suggested a temperature of $87\\pm47$K.\n\\citet{leurini2016} reported a rotational temperature 295 K for methanol emissions, while \n\\citet{lee2017a} finds an excitation temperature of 165$\\pm$85 K for deuterated methanol transitions.\nConsidering all these results and roughly the temperature variation along the radius \\citep{lee2014}\nwe have considered five excitation temperatures (20, 40, 90, 160, 300 K) for our calculations.\nWe consider some low excitation temperatures e.g., 20 \\&\n40 K for regions away from hot-corino and high-temperatures for a hot region.\n Here we assume the HDCO emission region to be optically thin and local thermodynamic equilibrium (LTE) condition\nis satisfied.\nThe details of column density calculation  and the variation of it, is described in Section 3.3.\n\nIn Figure~\\ref{fig:hdco_overlap}, we see that HDCO emission is concentrated in a circular\nregion around the central protostar position, i.e., the peak of continuum emission. The \nemission is elongated along the jet direction. \nAs the emission is only partially resolved, we can not infer conclusively whether the elongation \nis real or because of the effect of synthesized beam size. Figure 2 shows that near the systematic\nvelocity, the HDCO emission is most extended; though the extended emission feature is weak ($\\sim 3\\sigma$), \nit is absent in higher velocity channels from the systematic velocity. \nThis extended emission feature is similar to the `X' shaped outflow cavity as traces by C$^{34}$S emission.\nTo draw a further conclusion, in Figure~\\ref{fig:channel_map} we plot three velocity channels near\nthe systematic velocity\nfor HDCO emission and compare it with C$^{34}$S channel maps. The C$^{34}$S emission is \ntracing a dense gas component. `X' shaped outflow is closely related to the bipolar jet or outflow\nnear the systematic velocity \\citep{code2014}. From \nFigure ~\\ref{fig:channel_map}, it can be seen that though the emissions of HDCO away from the central source\nis weak, it is significantly similar to the C$^{34}$S emission near the systematic velocity. \nTo find whether HDCO emission shows any rotation, we plot centroid emission positions of various\nvelocity channels of C$^{34}$S and HDCO line in Figure ~\\ref{fig:uv1}.  \\citet{code2014} finds an\nevident rotation around jet for C$^{34}$S  emission in the southern lobe. From \nFigure ~\\ref{fig:uv1} we can see that the there is definite signature of rotation for \nC$^{34}$S as the blue shifted and red-shifted emission are situated away from the \ncentral peak position and the jet axis. \nAt low velocities ($<1.0$), it is clearly showing rotation (in southern lobe) as emission\ncentroids of blue-shifted and red-shifted emission situated roughly symmetrically away from\nthe jet-axis and below the disk plane. At a high velocity, the rotation feature is not \nclear but becomes more collimated      \ntowards the jet axis. The HDCO emission centroids also show a  signature of rotation which is\nclear from Figure~\\ref{fig:uv1}. Looking to the southern portion \nof emission there seems to be some similarity with \nC$^{34}$S features. As the intensity of emission away from the source is very faint for HDCO \nand the shift of different velocity channels is less than the beam size, this inference \nmay not be conclusive. We can speculate that as the emission centroid (red-shifted southern\nportion in Fig. 4) of HDCO is away from the disk plane and shifts along the jet axis. So, it \nis not associated with disk rotation; at smaller scales, it may be associated with disk wind \nor small-scale outflow or cavity rotation but with current spatial resolution of the ALMA data we cannot confirm it.\n\\citet{leurini2016} finds that methanol (CH$_3$OH) could trace the base of the low-velocity of the small \nscale outflow and another higher resolution observation (0.04$''$, \\citealp{lee2017a}) suggests that\nit is from a warm environment near the disk surface. Hence, to discuss the origin HDCO of emission, we \ncompare HDCO emission with CH$_3$OH and C$^{17}$O emission in the next section. \n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{figure2.eps}\n \\caption{\n HDCO colour map and contour map overplotted for the 1.63 km s$^{-1}$ channel which is close to the systematic velocity\n 1.7 km s$^{-1}$. The contours are from 3$\\sigma$ ($\\sigma \\sim$7 mJy) with a step of 2$\\sigma$.\n    The HDCO emission map at systematic velocity is quite similar to 'X' shaped outflow.}\n\n    \\label{fig:hdco_sysve}\n\\end{figure}\n\n\\begin{figure*}\n \\includegraphics[width=8cm, angle=270]{HDCO_C34S.eps}\n \\caption{Comparison of channel map of C$^{34}$S and HDCO. Value of velocities are near the systematic velocity \n $\\sim$ 1.7 Km s$^{-1}$. First contour is at 3$\\sigma$ and steps are 2$\\sigma$ for \n both HDCO and C$^{34}$S. $\\sigma$ is $\\sim$ 6mJy\/beam and 3.5 mJy\/beam for HDCO and C$^{34}$S\n contours, respectively.}\n \\label{fig:channel_map}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\includegraphics[width=18cm]{Figure_4.eps}\n\t\\vskip -0.1cm\n    \\caption{Distribution of the centroid positions of various velocity channels of \n C$^{34}$S and HDCO line. Velocities are colour-coded according to the \n color-bar shown in\nthe figure and value of velocities are subtracted from systematic velocity (1.7 Km s$^{-1}$).\nThe direction of jet (PA 22) and disk are (PA 112) are shown by lines.  }\n    \\label{fig:uv1}\n\\end{figure*}\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{Figure_5.eps}\n    \\caption{\n    Distribution of the centroid positions of various velocity channels of \n C$^{17}$O, CH$_3$OH and HDCO line. Velocities are colour-coded according to the \n color-bar shown in\nthe Figure and value of velocities are subtracted from systematic velocity (1.7 Km s$^{-1}$).\nThe direction of jet (PA 22) and disk are (PA 112) are shown by lines.\n   }\n    \\label{fig:uv2}\n\\end{figure}\n\n\\subsection{CH$_3$OH, and C$^{17}$O emission}\nMethyl alcohol emission of v0 7(1,7)- 6(1,6) transition\nis also observed around the MM1 protostar position.\nFrom the spectral signature (Figure~\\ref{fig:spectra}) it is seen that CH$_3$OH emission is from\na smaller region than HDCO around the protostar position.\nThe CH$_3$OH line was identified using CDMS\/JPL molecular database, having E$_u$=79K. The kinematics of methanol \nemission is already described by \\citet{leurini2016}. \n\nMethanol emission is from a smaller region  ($0.64'' \\times 0.45'' PA \\quad 49.6^{\\circ}$) \nthan that from HDCO emission region. \nIn Figure~\\ref{fig:uv2} we compare HDCO emission centroid for different velocity channel\nwith C$^{17}$O and CH$_3$OH. C$^{17}$O\nemission traces disk rotation at high velocity and rotating outflow cavity at low-velocity \\citep{code2014}. Methanol \nemission shows rotation as the same sense of C$^{17}$O and traces small-scale outflow.\nThough the spatial resolution ($0.6 ''$) is not enough to ascertain the fact,\nwe have compared HDCO emission with these emissions. At small velocities, HDCO emission shows \nsome rotation near the disk plane. This rotation may trace small scale outflow or \na rotating environment \\citep{lee2017a} near the disk plane. At high velocities, the red-shifted centroid shifts towards\njet axis and away from disk plane in southern portion of the outflow region. This is very different than the \nmethanol velocity centroids at high velocities.  Hence at small velocity it may trace some rotation near\nthe disk which may be because of small-scale outflow or some rotation environment \\citep{lee2017a}.\nAt systematic velocities HDCO emission traces `X' shaped outflow cavity similar to the C$^{34}$S emission; it \nmay trace small scale outflow too but due to limited spatial resolution we are not sure about this.\n \n\\subsection{Column density calculation and abundances}\nWe calculate the column densities of C$^{17}$O and HDCO, assuming the lines are optically thin.\nAs we do not know the rotational temperature of HDCO transition, we consider a range \nof temperature (20 K-300 K) as the excitation temperature of the molecular transition.\nNow, \n\\begin{equation}\nN_u\/g_u=\\frac{N_{tot}}{Q_{T_{rot}}} e^{\\frac{-E_u}{T_{rot}}}\n           =\\frac{8\\pi\\nu^2 k\\int T_b dv}{hc^3Ag_u}\n           =\\frac{3k\\int T_b dv}{8\\pi^3\\mu^2\\nu S} \n\\end{equation}\nwhere, g$_u$ is the statistical weight of the upper level u, N$_{tot}$ is the total \ncolumn density of the molecule,$ Q_{T_{rot}}$is the rotational partition function, $T_{rot}$ is rotational\ntemperature, $E_u$ is the upper energy level, $k$ is the Boltzman constant, $\\nu$ is the frequency \nof the line transition, $A$ is the Einstein co-efficient of the transition. $\\int T dv$ is the \nintegrated line intensity. Calculated column density is a beam average column density, so we consider the\nbeam dilution factor, given by,\n\\begin {equation}\n\\eta_{BD}=\\frac{\\theta_s^2}{\\theta_s^2+\\theta_B^2}\n\\end {equation}\nwhere, $\\theta_S$ is the source size and $\\theta_B$ is the beam size. \nAs the source is marginally resolved in HDCO emission, we are unsure about the source size.\n\\citet{code2016} considered  source size $\\sim 0.3''$, which is about the size of dusty disk \\citep{lee2017b}.\n\\citet{leurini2016} considered source size  $\\sim 0.2''$. Here, we consider source size $\\sim 0.2''$.\nIn that case, the calculated column density would be multiplied by $\\simeq 10$ for central \nregion only (region `M' in Fig. 6), if we consider beam dilution.\nThe spectral profile for different region is shown in Figure~\\ref{fig:spectra}.\nCalculated average column densities without beam dilution correction \nover different regions (Figure~\\ref{fig:region}) are enlisted in Table~\\ref{tab:colden}.\nConsequences of beam dilution effect are discussed section 3.5.\n\\begin{table*}\n\t\\centering\n\t\\caption{Column density of HDCO around MM1 protostar for different regions depicted in Fig~\\ref{fig:region}}\n        \\label{tab:colden}\n\t\\begin{tabular}{|l| c| c| c| c |c|c|}\n\t\t\\hline\n\t\tRegions & \\multicolumn{5}{c|}{Column densities for various T$_{rot}$ in unit of $10^{14}$ (cm$^{-2}$)}&Error \\\\\n\t\t        &  300 K   & 160 K &  90 K  & 40 K    & 20 K & \\\\\n\t\t\\hline        \n\t\tM       & 12.3& 5.6 & 3.1  &   2.0  &  ..          &22.8\\%\\\\\n\t\tL2      & ..   & 1.0  & 0.5   & 0.36    & 0.52     & 25.6\\%\\\\\n\t\tL3      &  ..  & 1.3 & 0.74  &   0.48  & 0.69     &23.6\\%   \\\\\n\t\tD2      &  . . & 0.86 & 0.48  & 0.31    & 0.44      &25.2\\%  \\\\\n\t\tD3      &  ..  & 1.3  &  0.74 & 0.48    & 0.69     & 25.4\\% \\\\\n\t\\hline\n\t\\end{tabular}\n\\end{table*}\nIn the Table ~\\ref{tab:colden} we have shown the column \ndensity for different regions along the jet and perpendicular to the jet axis. \nThe central hot-corino is expected to be hot, so we have excluded the lowest temperature (20 K)\nin our range of temperature to calculate column density in the central region. Similarly, in regions away from the central\nsource we have not considered the 300 K temperature for the calculation. Considering the H$_2$ column density \n$\\sim 10^{24}$cm$^{-2}$ (see next section), X$_{HDCO} \\sim 10^{-10}$. \nThe errors for column density \nare calculated for noise and statistical  (Gaussian) fitting. If we consider \n a calibration error of 20\\% then the same uncertainty will be added to column density calculation in \n addition to the statistical error; the resultant error shown in Table~\\ref{tab:colden}.\nDue to low signal to noise ratio we have not estimated the column densities in outermost \nregions e.g. L1, D1 in Figure~\\ref{fig:region}\n\n\\subsection{C$^{17}$O emission and disk mass}\n\nIn earlier Section it was mentioned that C$^{17}$O traces a Keplerian disk at high velocities. Assuming\nC$^{17}$O emission to be optically thin, we can calculate disk mass from the beam-averaged C$^{17}$O\ncolumn density and converting it to H$_2$ volume density. For conversion of C$^{17}$O column density to H$_2$,\nwe use $X_{CO}\/X_{C^{17}O}$=1792  \\citep{wilson1994} and $X_{CO}=N_{CO}\/N_{H_2}\n \\sim 10^{-4}$.\n In region `M' (Figure~\\ref{fig:region} ) C$^{17}$O column density for T$_{rot}$ = 90-300 K  is \n 1.3--3.3$\\times 10^{16}$ cm$^{-2}$, so N$_{H_2}\\sim  2.3-5.9\\times 10^{23}$ cm$^{-2}$, as a beam \n averaged column density. As the disk is not resolved and has been seen edge on, considering beam \n dilution H$_2$ volume density becomes ${\\rm 8\\times N_{H_2}\/D_{disk}} $ where the disk size is $\\sim 90$AU \n \\citep{code2014}. Using the above information H$_2$ volume density becomes 1.3--3.4$\\times 10^{9}$cm$^{-3}$.\n Considering H$_2$ volume density $\\sim 10^{9}$cm$^{-3}$, we derive the disk mass to be 0.016 M$_\\odot$.\n It is close to the value 0.014  M$_\\odot$, estimated by \\citet{lee2014}.\n Here, we have considered the formula, $M_D \\sim 1.4 \\times m_{H_2} \\times \\pi r^2 \\times 2H$, where `r' \nis the disk radius and `H' is the disk height from mid-plane; here the factor, 1.4 accounts for \n the mass in the form of helium. We consider H$\\sim 40$AU and the disk radius to be 90 AU \\citep{lee2014}. The  \n disk mass can also be calculated from continuum emission. We have considered that the continuum from HH212 disk is\n optically thin and isothermal. Though the disk mass is not constant as suggested by \\citet{lee2014}, we have \n calculated it by assuming a dust temperature to be $\\sim$ 90 K.\n Following  earlier work \\citep{tobin2012}, we assume the spectral index\n $\\beta$=1 \\citep{kwon2009}, and the dust opacity $\\kappa_0$=0.035 g cm$^{-2}$ at 850 $\\mu$m \\citep{andrew2005}.\n We use the following formula \\citep{tobin2012} to calculate the disk mass\n $$ M_{dust}=\\frac{D^2F_\\lambda}{\\kappa_0 (\\frac{\\lambda}{850\\mu m})^\n {-\\beta}B_\\lambda(T_{dust}).}$$\n With {T$_{dust}$=90 K}, the disk mass is 0.0159 M$_\\odot$, which is close to 0.016 M$_\\odot$, \n the disk mass calculated based on hydrogen mass.\n\n\\subsection{Discussion}\n\\subsubsection{chemistry}\nFormaldehyde can be formed in both the gas and the grain phase. On grain surfaces, \nformaldehyde forms through sequential reactions of H or D atom with CO:\\\\\nCO $\\rightarrow$ HCO $\\rightarrow$ H$_2$CO and \nCO $\\rightarrow$ DCO $\\rightarrow$ HDCO\/D$_2$CO  \\citep{cazaux2011, taquet2012}. The grain phase reactions\noccur mainly in cold temperature condition ($<$50K). Gas phase formation of formaldehyde and its deuterated form is\nalso possible through reaction involving CH$_2$D$^+$. More specifically, at relatively high temperatures, T$\\sim$ \n100K or higher, this reaction is relevant to the central hot-corino region of HH212\n\\citep{wootten87, oberg2012}. Reaction involving CH$_2$D$^+$ is not active in cold region due to \nits high exothermicity ($\\Delta$E of 654 K, see \\citealt{roueff13}) but in high temperature region this reaction \nmay take part and favours deuterium fractionation in gas-phase:\n\nCH$_3^+$ + HD$\\rightarrow$ CH$_2$D$^+$ + H$_2$ + $\\Delta$E.\n\n\\citet{fontani14} first time disentangled the emission of deuterated formaldehyde \nform on grain surfaces from its gas phase production. In our observation \nwe have not seen any H$_2$CO line but still we can infer about the production region and possible production\nroute of deuterated formaldehyde (HDCO). HDCO emission is from two regions, one is from central hot-corino\nregion another is from outflow cavities. The impact of bipolar jet on the cavities may release HDCO from grain\nsurfaces due to sputtering or sublimation from grain mantle (e.g., see \\citealt{code2012}). As the regions in \nthe outflow cavities have low-temperature, hence we can expect the production of HDCO is mainly from grain \nsurfaces. If we consider the column density of formaldehyde in central region assuming a temperature 160 K and \n40 K in one of the outflow region (south-west lobe, centered at $\\alpha(J2000)= 05^h43^m51^s.37.4,\n \\delta(J2000)=-01^{\\circ}02'54''.69$; see Fig~\\ref{fig:hdco_sysve}),\nthen the column densities are respectively 5.6$\\times 10^{14}$\nand 0.18$\\times 10^{14}$ cm$^{-2}$. In the outflow region this is a factor $\\sim 31$ less than the central \nregion column density of HDCO.\nFrom C$^{17}$O emission we can get a rough idea about the density difference in these two regions. In the same \nregion (south-west lobe) we have calculated the C$^{17}$O column density, and it is a factor of\n10 less than the centre region (`M'). Hence, if HDCO comes from only grain surfaces then their column densities should be \ndecreased similarly by the density factor. Considering the density differences, the column density in the \ncentral region is a factor of three ($\\sim 31\/10$) higher.\nHence, indirectly it implies that there is active gas phase HDCO production in central hot-corino region\nalong side the grain-phase desorbed HDCO. Recently, another doubly deuterated formaldehyde (D$_2$CO) has been \nreported by \\citet{lee2017a} for the same source for a beam size of $\\sim 0.04''$. In this work, HDCO line emission \nis marginally resolved, if we consider emission region $0.2''$ then considering the beam dilution,  a factor of 10 \n will be  multiplied by central region column density. In that case, \nthe column density of HDCO and D$_2$CO would be comparable ($\\sim 10^{15}$). \nIf we consider smaller regions of the emission region such as the D$_2$CO beam,  then the column density \nwill be higher for HDCO. Hence, considering the source size $0.2''-0.04''$, the D$_2$CO\/HDCO ratio becomes\n$1-0.04$. We guess the emission region for HDCO will not be as small as that for D$_2$CO, in that case the deuteration of formaldehyde \nin quite higher than methanol for the same source. Methanol deuteration (D\/H) for the HH212 is 2.4$\\pm 0.4 \\times 10^{-2}$\nas reported by \\citet{bian2017}. On the other hand  \\citet{lee2017a} suggested D\/H\n$\\sim 0.27$ for methanol in the disk environment. If we consider D\/H ratio 0.27 to be true\nthen methanol is still produced effectively on the grain surface in the disk environment. \\citet{bian2017} used\n $\\rm{^{13}CH_3OH}$ and LTE approximation to calculate methanol column density. If we consider D\/H ratio\nof 2.4$\\pm 0.4 \\times 10^{-2}$ to be more likely then probably due to high temperature deuterated \nmethanol production is not efficient in the hot-corino region. We speculate that deuterated formaldehyde can still be\nproduced in this region through gas phase reaction network unlike the methanol production. \n\\subsubsection{kinematics\/morphology}\n In Figure~\\ref{fig:region} we have \ndefined different circular regions ($0.5''$ diameter) comparable to the beam size and the observed spectra\ntowards these regions are shown in Figure~\\ref{fig:spectra}. We can see from \nFigure~\\ref{fig:uv1},~\\ref{fig:uv2} there may be emission from rotating environment or small-scale outflow near the disc \nbut it is very uncertain. At high velocity the emission is certainly shifted above the disk along the jet\nwhich signifies that the emission is affected by outflow or disk-wind at the base of the jet.\nThe column density (Table~\\ref{tab:colden}) \non both sides (L2, L3) of the central circular region of $0.5''$ diameter is less than that at the central region. \nWe can see a jump in column density for HDCO. This is due to the fact that \nat the disk ($\\sim 90$AU) and envelope interface there is a sharp rise of temperature\nand density due to accretion shock. Also, from Figure~\\ref{fig:spectra}\nwe see that the spectral signature of methanol emission is absent from the outermost regions (L1, L4, D1, D4)\nthough a weak signature of HDCO emission is present. From this also we can say that HDCO emission is more extended than \nmethanol. \\citet{lee2017a} at a scale of $0.04''$ found that difference of emission regions of the complex\nmolecules is partly due to different A-coefficients. In that observation of HH212, it was found that molecules\nwith comparatively low A-coefficients are seen to be more extended than the that with high A-\ncoefficients. Here, A-coefficient of methanol transition is lower than that of HDCO, but the region seems \nto be more compact than that of HDCO. Higher A-value may correspond to higher critical \ndensity. \\citet{guzman2011} described H$_2$CO critical densities $\\sim 10^6$; \ndue to high density ($\\sim 10^8-10^9$ cm$^{-3}$) in the central region it is expected that LTE condition \nis maintained for the reported molecular transitions here.\nHence, we can speculate that it is due to local physico-chemical condition and high line \nstrength, comparatively lower E$_u$ of HDCO which may be responsible for\nthe emission difference. Another difference we can see for methanol and HDCO emission in central region (`M') is this:\nHDCO has a red-shifted peak which is absent in CH$_3$OH emission. This peak may be from the line contamination of other \nmolecular transition, but we have not found any such line from other molecules. Alternatively, this may be \nrelated to the high velocity outflow from the base of the jet. \n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{region_figure.eps}\n    \\caption{Different region along and perpendicular to the jet axis. The region circle's diameter is 0.5 arcsec\n    similar to the beam width. The gray scale image is the continuum image of the central source, namely, HH212. \nThe contours are for SiO jet emission along a PA 22$^o$.}\n\\label{fig:region}\n    \\end{figure}\n\n\\begin{figure*}\n\t\\includegraphics[width=12cm]{plot2.eps}\n    \\caption{Spectral profile in different regions as shown in Figure~\\ref{fig:region}.}\n    \\label{fig:spectra}\n\\end{figure*}\n\n\\section{Conclusions}\nIn this letter we described the emission of deuterated formaldehyde (HDCO) from the hot inner region \nof HH212. This emission is limited mainly to the inner $\\sim 200$ AU. The kinematics of \nHDCO is quite similar to that of the C$^{34}$S emission \\citep{leurini2016} near systematic velocity.\nHDCO traces large scale outflow cavity near systematic velocity; \nit may trace small-scale outflow or disk wind\nbut due to limitation of spatial resolution in this observation we are uncertain about this.\nOn the other hand, both methanol and HDCO emission peak values are $\\sim 1.9$ km s$^{-1}$ \nand the spectral profile is symmetric in low to medium velocity range ($<2.4 {\\rm km s^{-1}}$). \nThe asymmetry at high velocity for HDCO may be associated with the outflow near the disk plane.\nThe HDCO rotation may be associated with the disk wind or rotating environment \\citep{lee2017a};\nor to the rotating cavity wall, similar to C$^{34}$S. Due to limited resolution of the observation\nwe cannot conclude about the rotation with certainly. The emission \nis assumed to be optically thin. Here we observe only one transition of HDCO.  Thus we cannot \ndetermine the excitation temperature. We have considered a range of possible temperature depending on earlier \nstudies and typical hot-corino temperature assumed in the literature. \nThe column density of HDCO is $\\sim 10^{14} {\\rm cm^{-2}}$. Though \nwe have not observed any H$_2$CO transition but comparing results of D$_2$CO given in \\citet{lee2017a} \nwe speculate that the deuterium fractionation of formaldehyde is relatively higher than methanol in the central region.\nWe guess that the gas phase formation of deuterated formaldehyde is active in the central hot region in the low-mass \nprotostar HH212.\n\\section*{Acknowledgements}\nWe acknowledge the anonymous referee for the constructive comments.\nWe also thank Dr. Nirupam Roy (IISc, India) for his helpful suggestions.\nDS is thankful to Department of Space, Govt. of India (PRL) for support in continuing research and  also \nwant to thank Young Visitor Programme (KASI) for financial help to work in Korea Astronomy\nSpace science Institute for a short time. AD want to acknowledge ISRO Respond (Grant no. ISRO\/RES\/2\/402\/16-17).\nC. -F.L. acknowledges grants from the Ministry of Science and Technology of Taiwan (MoST 104-2119-M\n-001-015-MY3) and Academia Sinica (Career Development Award).\nThis paper makes use of the following ALMA data: ADS\/JAO.ALMA\\#2011.0.00647.S.\nALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan),\ntogether with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea),\nin cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI\/NRAO and NAOJ.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\nSpeyer \\cite{Speyer:2016:VOATOKW} recently published a very short and elegant proof\nfor Kasteleyn's Theorem \\cite{Kasteleyn:1967:GTACP}. This proof involves some higher topology arguments; and\nthe purpose of this note is to replace them by elementary graph--theoretical\narguments.\nWe shall, however, present only a sketch of a proof, since the technical details\nof a fully rigorous proof would likely conceal the beauty and simplicity of the basic idea.\n\n\n\\section{Basic definitions and notation}\n\nFor reader's convenience and to fix our notation, we shall briefly\nrecall some basic definitions.\n\n\\subsubsection*{Graph}\n\nA finite graph $G$ consists\n\\begin{itemize}\n\\item of a finite set of\n\\EM{vertices} $V\\of G$\n\\item and a finite (multi--)set of \\EM{edges} $E\\of G$,\n\\end{itemize}\nwhere each $e\\in E\\of G$\n\\EM{joins} two different vertices $v_1, v_2\\in V\\of G$ (we also say that $e$ is \\EM{incident}\nwith $v_1$ and $v_2$), which we denote by $e=\\setof{v_1,v_2}$.\nWe shall not consider \\EM{loops}, which are\nedges ``joining a single vertex'' ($e=\\setof{v}$): If such loop arises during the constructions\ndescribed below, we shall remove it immediately.\n\n\\subsubsection*{Graph embedding}\n\nA finite graph $G$ can always be \\EM{drawn} in the plane ${\\mathbb R}^2$, such that\n\\begin{itemize} \n\\item the vertices correspond bijectively to \\EM{points} in the plane (we shall call them \\EM{vertex--points}),\n\\item and the edges correspond bijectively to smooth \\EM{curves} $\\brk{0,1}\\to{\\mathbb R}^2$ in the plane (we shall call them \\EM{edge--curves})\n\twhich \\EM{connect} the respective vertex--point, i.e.,\n\tsome edge $e=\\setof{v_1,v_2}$ corresponds to an edge--curve\n\t\\begin{itemize}\n\t\\item which starts in the vertex--point corresponding to $v_1$\n\tand ends in the vertex--point corresponding to $v_2$ (or vice versa: the orientation of the curve is irrelevant in our context),\n\t\\item but \\EM{does not} ``intersect itself'' (i.e., does not pass twice through the same point;\n\tin particular, edge--curves do not ``return'' to their starting point or end point),\n\t\\item and does not pass through any other vertex--point.\n\t\\end{itemize}\n\\end{itemize}\nSuch drawing is called an \\EM{embedding} of the graph $G$ in the plane.\n\nBy definition, the edge--curves of edges incident to the same vertex $v$ have\nthe vertex--point corresponding to $v$ in common,\nbut there might be other such intersections in \\EM{nonvertex--points}.\n\nFor our purpose, we additionally assume that:\n\\begin{enumerate}\n   \n   \n\n\\item Any two curves intersecting in some nonvertex--point $p$\n\tdo actually \\EM{cross} each other (i.e., they are \\EM{not tangent} in $p$),\n\twhence we call such point $p$ a \\EM{crossing} of the two curves.\n\\item There is only a finite number of such \\EM{crossings}.\n\\end{enumerate}\nWe call an embedding which fulfils these additional assumptions a \\EM{proper embedding}.\n\nSee \\figref{fig:simple-graph} for an\nillustration of these concepts: In all graphical representations of proper embeddings in this\nnote, we shall indicate\n\\begin{itemize}\n\\item vertex--points by \\EM{black dots},\n\\item curves corresponding to edges by \\EM{gray lines},\n\\item and crossings by \\EM{small white circles}.\n\\end{itemize}\n\\begin{figure}\n\\label{fig:simple-graph}\n\\caption{Example of a proper embedding of some graph $G$ in the plane.}\n{\\small\nLet $V\\of G = \\setof{v_1,v_2,\\dots, v_6}$ and $E\\of G = \\setof{\n\\setof{v_1,v_2},\n\\setof{v_2,v_3}^2,\n\\setof{v_3,v_4},\n\\setof{v_5,v_6}}$, where the notation $\\setof{v_2,v_3}^2$ indicates that there are\n\\EM{two} edges joining vertices $v_2$ and $v_3$. The picture shows a proper embedding of $G$\nwith a single crossing $p$: The points corresponding to the vertices are indicated by black\ndots, the curves corresponding to edges are indicated by gray lines, and the crossing is indicated by a small\nwhite circle.\n}\n\\begin{center}\n\\input graphics\/simple-graph.tex\n\\end{center}\n\\end{figure}\n\n\\subsubsection*{Simplified notation}\nIf we are given some graph $G$ and an embedding $\\eta$ of $G$, let us denote\n\\begin{itemize}\n\\item by $\\eta\\of v$ the vertex--point representing vertex $v\\in V\\of G$,\n\\item by $\\eta\\of e$ the edge--curve representing edge $e\\in E\\of G$.\n\\end{itemize}\nBut we shall abuse this notation in the following by making no distinction\n\\begin{itemize}\n\\item between $v\\in V\\of G$ and $\\eta\\of v$, i.e., instead of saying\n\t``the vertex--point in the plane representing $v$'' we simply shall say ``$v$'',\n\\item and between $e\\in E\\of G$ and $\\eta\\of e$, i.e., instead of saying\n\t``the edge--curve in the plane representing $e$'' we simply shall say ``$e$''.\n\\end{itemize}\nThis slight imprecision should cause no confusion in our context.\n\n\\subsubsection*{Planar graph}\n\nA proper embedding without any intersection of edge--curves in non--vertex points is called a\n\\EM{planar embedding}.\nA graph which admits a planar embedding is called a \\EM{planar graph}.\n\nNot all graphs are planar, but we may \\EM{view} every proper embedding of some\ngraph $G$ as planar embedding of a planar graph $G^\\prime$ by \\EM{reinterpreting} all\ncrossings as vertex--points: $G=G^\\prime$ (in the sense of graph isomorphisms) if the\nembedding has no crossings at all. (In \\figref{fig:simple-graph}, this reinterpretation would amount\nto replacing the single crossing $p$ by a new vertex--point $v_7$.)\n\n\\subsubsection*{Perfect matching}\n\nA \\EM{perfect matching} $M\\subseteq E\\of G$ of $G$ is a subset of edges\nsuch that every vertex of $G$ is incident with precisely one edge in $M$.\n\n\\subsubsection*{Edge weight}\n\nWe assume that we are given some (nowhere--zero) \\EM{weight function} (called \\EM{edge weight}) $\\omega$\n$$\\omega:E\\of G\\to R\\setminus\\setof{0}$$\non the multiset of edges of $G$, where $R$ is some (nontrivial) commutative\nring (in most cases, $R$ is ${\\mathbb Z}$ or some polynomial ring; the constant edge weight \n$\\omega\\equiv 1$ is used for enumeration purposes).\n\n\n\\subsubsection*{Generating function of perfect matchings}\n\nThe weight of some perfect matching $M\\subseteq E\\of G$ is the product of the weights of the edges in $M$:\n$$\n\\omega\\of M:=\\prod_{e\\in M} \\omega\\of e.\n$$\n\nThe \\EM{generating function} of  perfect matchings of some graph $G$ with edge weight $\\omega$ is\ndefined as\n$$\nm\\of{G,\\omega}:=\\sum_{M}\\omega\\of M,\n$$\nwhere $M$ ranges over all perfect matchings of $G$.\n\n\\subsubsection*{Sign of perfect matchings}\n\\def\\operatorname{img}{\\operatorname{img}}\nAssuming a fixed proper embedding $\\eta$ of graph $G$, the \\EM{sign} of some perfect matching\n$M$ of $G$ in the embedding $\\eta$ is defined as\n$$\n\\sgn\\of{M,\\eta}:=\\pas{-1}^{C\\of{M,\\eta}},\n$$\nwhere $C\\of{M,\\eta}$ is the \\EM{number of all crossings of all edges} in $M$,\ni.e., denoting by $\\operatorname{img}\\of{e}$ the image of the (edge--curve corresponding to) edge $e$:\n\\begin{equation}\n\\label{eq:crossing}\nC\\of{M,\\eta}:=\\sum_{\\setof{e_1,e_2}\\subseteq M} \\absof{\\operatorname{img}\\of{e_1}\\cap\\operatorname{img}\\of{e_2}}.\n\\end{equation}\n\nIt is possible to modify some proper embedding $\\eta$ to another embedding $\\eta^\\prime$ so that\nthe signs of all perfect matchings are the same for $\\eta$ and $\\eta^\\prime$: See \\figref{fig:crossing}\nfor a simple example of such \\EM{sign--preserving} modification.\n\n\\subsubsection*{Signed generating function of perfect matchings}\n\nThe \\EM{signed generating function} of perfect matchings\n$s\\of{G,\\omega,\\eta}$ (which depends not only on the graph $G$, but also on the proper embedding $\\eta$)\nis defined as\n$$\ns\\of{G,\\omega,\\eta}:=\\sum_{M}\\sgn\\of{M,\\eta}\\cdot\\omega\\of M,\n$$\nwhere $M$ ranges over all perfect matchings of $G$.\n\nClearly, $m\\of{G,\\omega}=s\\of{G,\\omega,\\eta}\\equiv 0$ if\n$\\absof{V\\of G}$ is {odd}.\n\n\n\\begin{figure}\n\\caption{Untangling of a multiple crossing.}\n\\label{fig:crossing}\n{\\small\nThe left picture shows \\EM{four} edge--curves passing through the \\EM{same} nonvertex--point: \nObserve that this situation can be ``modified locally'' (in the sense that the embedding is left\nunchanged outside some small neighbourhood of the crossing) to a situation like the one\nshown in the right picture, and  that such ``local modification''  does not change the sign\nof any perfect matching. For instance, if we assume that the edges\nshown here belong to some perfect\nmatching $M$, both ``local situations'' contribute a factor $\\pas{-1}$ to the sign of $M$\nfor \\EM{every two--element subset} of the four curves (i.e., $\\pas{-1}^{\\binom{4}2}=\\pas{-1}^6=1$).\n}\n\\begin{center}\n\\input graphics\/star.tex\n\\end{center}\n\\end{figure}\n\n\n\\subsubsection*{Stembridge's embedding and the Pfaffian}\nFor every graph $G$ with vertex set $V\\of G=\\setof{v_1,v_2,\\dots,v_n}$\nthere is a specific proper embedding which we shall call \\EM{Stembridge's embedding}\n(see \\cite{Stembridge:1990:NPPAPP}):\nEvery edge $e=\\setof{v_i,v_j}$, $i\\neq j$, is represented by the half--circle in the upper half--plane\nwith center $\\pas{\\frac{i+j}2,0}$ and radius $\\frac{\\absof{i-j}}2$,\nand every vertex $v_i$ is represented by the point $\\pas{i,0}$.\n(See the left picture in \\figref{fig:pfaffian} for an example of Stembridge's embedding.)\n\nDenote this specific embedding by $\\overline{\\eta}$: The \\EM{Pfaffian} of a graph $G$ is defined\nas the signed generating function for the embedding $\\overline{\\eta}$, i.e.:\n$$\n\\operatorname{Pf}\\of{G,\\omega}:= s\\of{G,\\omega,\\overline{\\eta}}.\n$$\n\n\\begin{figure}\n\\caption{Stembridge's embedding and the Pfaffian.}\n\\label{fig:pfaffian}\n{\\small\nThe left picture shows Stembridge's embedding $\\overline{\\eta}$ of a graph $G$.\n\nThe right picture shows a specific perfect matching $M$ of $G$ for which the number $C\\of{M,\\overline{\\eta}}$\nof crossings according to \\eqref{eq:crossing} equals $3$, whence the sign\n$\\sgn\\of{M,\\overline{\\eta}}$ of this matching is $\\pas{-1}^3=\\pas{-1}$. \n}\n\\begin{center}\n\\input graphics\/pfaffians.tex\n\\end{center}\n\\end{figure}\n\n\\subsubsection*{Sign--modifications of edge weights and Kasteleyn's Theorem}\nLet $G$ be some simple graph with edge weight $\\omega$: Another edge weight $\\omega^\\prime$\nis called a \\EM{sign--modification} of $\\omega$ if for all $e\\in E\\of G$ there holds\n$$\n\\omega^\\prime\\of e = \\omega\\of e \\text{ or }\\omega^\\prime\\of e = -\\omega\\of e.\n$$\n\nThen we may formulate Kasteleyn's Theorem \\cite{Kasteleyn:1967:GTACP} as follows:\n\n\\begin{thm}[Kasteleyn's Theorem]\n\\label{thm:kasteleyn}\nLet $G$ be a \\EM{planar} finite simple graph\nwith edge weight $\\omega$. Let $\\eta$ be an arbitrary proper\nembedding of $G$. Then there exists a sign--modification $\\omega^\\prime$\nof $\\omega$ such that\n$$\nm\\of{G,\\omega} =\n\\operatorname{Pf}\\of{G,\\omega^\\prime}.\n$$\n\\end{thm}\n\nNote that for any \\EM{planar} embedding $\\eta$ of  $G$ we have $m\\of{G,\\omega}=s\\of{G,\\omega,\\eta}$.\n\nWe will rephrase Speyer's elegant argument \\cite{Speyer:2016:VOATOKW} to\ngive a sketch of proof for the following slight generalization:\n\\begin{thm}\n\\label{thm:kasteleyn2}\nLet $G$ be a finite simple graph\nwith edge weight $\\omega$. Let $\\eta$ be a\nproper embedding of $G$.\nThen for \\EM{every} proper embedding $\\pas{G,\\eta^\\prime}$ \nthere is a sign--modification $\\omega^\\prime$ of $\\omega$ such that\n$$\ns\\of{G,\\omega,\\eta}\\equiv s\\of{G,\\omega^\\prime,\\eta^\\prime}.\n$$\n\\end{thm}\n\n\\section{Speyer's argument, in simple pictures}\nThe simple idea for the proof of Theorem~\\ref{thm:kasteleyn2} is to \\EM{successively transform} the\nembedding $\\eta$\n\\begin{itemize}\n\\item by \\EM{local modifications of the embedding}\n\\item and \\EM{corresponding modifications of the edge weight} (if necessary)\n\\end{itemize}\nsuch that the signed generating function of perfect matchings stays \\EM{unchanged},\nuntil the embedding $\\eta^\\prime$ is obtained.\n\nNote that smooth bijections ${\\mathbb R}^2\\to{\\mathbb R}^2$ will transform the embedding without changing the (numbers\nof) crossings, thus leaving the signed generating function of perfect matchings unchanged.\nMore informally: If we think of the embedding\nas a ``web of infinitely thin and ductile strings'' (corresponding to the edges) glued together\nat their endpoints (corresponding to the vertices), then we can imagine that we may ``deform and\ndrag around'' these strings, and such transformation will not change the signed generating function of\nperfect matchings if we do not \\EM{remove} crossings or \\EM{introduce} new ones.\n\nMoreover, there are modifications which leave the sign of every perfect matching unchanged,\nbut cannot be achieved by a continuous mapping ${\\mathbb R}^2\\to{\\mathbb R}^2$: One example is the ``untangling of\na multiple crossing''\nillustrated in \\figref{fig:crossing}.\n\nIn the following, we shall consider ``local modifications of proper embeddings''. For simplicity, we assume that\nevery crossing\nis the intersection of \\EM{precisely two} edge--curves\n(which we can achieve by the ``untangling of\na multiple crossing'' illustrated in \\figref{fig:crossing}; without changing the signed generating function\nwe are interested in).\n\n\\subsubsection*{Crossing edges incident to the the same vertex}\nIf two edges $e_1$, $e_2$ have a vertex $v$ in common, then they can never both belong to\nthe \\EM{same} perfect matching, hence crossings of $e_1$, $e_2$ are\nirrelevant for the sign of any perfect matching. If the curve--segments between\n$v$ and some crossing $p$ do not contain any other crossing, then $p$\ncan simply be removed (or introduced)\nby the modification illustrated in the following picture:\n\\begin{center}\n\\input graphics\/remove-intersection-neighbours\n\\end{center}\nThis modification has no effect on the sign of any perfect matching. If we modify the situation\n``from left to right'', we shall call this \\EM{straightening out a single crossing} of $e_1$ with $e_2$.\n\n\\subsubsection*{Self--crossing edges}\nWe ruled out self--intersections of edge--curves for proper embeddings, but they might arise by one\nof the modifications described below. The following picture makes clear that we can ``change the\nsituation locally'' and remove (or introduce) such self--crossing.\n\\begin{center}\n\\input graphics\/remove-selfintersection\n\\end{center}\nThis modification has no effect on the sign of any perfect matching, since only crossings of \\EM{different}\nedges contribute to the sign.\n If we modify the situation\n``from left to right'', we shall call this \\EM{straightening out a single self--crossing} of $e$.\n\n\\subsubsection*{Dragging a segment of an edge--curve over another segment}\nAssume that segments of edges\n$e$, $e^\\prime$ can be ``dragged over one another''\nsuch that precisely two crossings arise (or vanish), as the following picture illustrates:\n\\begin{center}\n\\input graphics\/edge_over_edge\n\\end{center}\nThis modification has  no effect on the sign of \\EM{any} perfect matching.\n If we modify the situation\n``from left to right'', we shall call this \\EM{straightening out a double crossing} of $e$ with $e^\\prime$.\n\n\\subsubsection*{Transition of a vertex through an edge}\nAssume that a segment of\nsome edge $e$ can be ``dragged over some vertex $v$''\nsuch that precisely one crossing arises (or vanishes) for every edge incident with $v$,\nas the following picture illustrates:\n\\begin{center}\n\\input graphics\/edge_over_vertex\n\\end{center}\nSince every perfect matching $M$ must contain \\EM{precisely one} edge incident to $v$, the sign of $M$\nis \\EM{reversed} by this modification if and only if $e\\in M$: But we can easily offset this by replacing\n$\\omega\\of e$ by $-\\omega\\of e$.\n\n\n\\section{Sketch of proof by example}\n\\label{sec:example}\n\nA rigorous proof of Theorem~\\ref{thm:kasteleyn2} \nwould involve tedious technical details: We shall avoid them by \nonly illustrating the \\EM{idea of proof}, i.e, the ``successive modification of embeddings'', \nin an example.\n\nThe example we shall consider is the \\EM{complete bipartite graph} $K_{3,3}$: Three\ndifferent proper embeddings of this graph are shown in \\figref{fig:proper}.\n\nObserve that the embedding shown in the middle of \\figref{fig:proper} can easily\nbe transformed to the embedding\nshown in the right by the ``untangling of a multiple crossing''\nillustrated in \\figref{fig:crossing}.\n\n\\figref{fig:example} illustrates the process of successively modifying the embedding $\\eta$\nshown in the left picture  of \\figref{fig:proper} until the embedding $\\eta^\\prime$ shown in the middle\npicture of \\figref{fig:proper} is obtained.\nWe shall explain the single steps in the following, where we denote the edge connecting\nvertices $v_i$ and $v_j$ by $e_{i,j}$.\n\n\\begin{figure}\n\\caption{Three different embeddings of the same graph $G$.}\n\\label{fig:proper}\n{\\small\nThe graph $G$ we are considering here is the \\EM{complete bipartite graph} $K_{3,3}$:\nThe fact that $K_{3,3}$ is \\EM{not planar} corresponds to the fact that we are\nunable to present an embedding without crossings.\n}\n\\begin{center}\n\\input graphics\/proper.tex\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Example: Succesive modification of embeddings.}\n\\label{fig:example}\n{\\small\nConsider the embeddings of the graph $G$ depicted in \\protect{\\figref{fig:proper}}:\nWe want to transform the embedding $\\eta$ shown in the left picture to the embedding $\\eta^\\prime$\nshown in the middle picture of \\protect{\\figref{fig:proper}}.\n}\n\\begin{center}\n\\input graphics\/example.tex\n\\end{center}\n\\end{figure}\n\n\nThe left upper picture in \\figref{fig:example} shows the result of a smooth deformation of the plane,\nwhere vertices $v_2$, $v_4$ and $v_6$ already have arrived at their desired positions\n(according to the hexagonal configuration in $\\eta^\\prime$). Now we want to drag $v_5$ to its \ndesired position along the path indicated by the dashed arrow. This involves a transition\nof $v_5$ through edge $e_{3,6}$, whose weight must therefore change its sign.\n\nThe upper picture in the middle shows the result of this operation: Note that we may now\nstraighten out the double crossing of $e_{4,5}$ with $e_{3,6}$ and the single\ncrossing of $e_{5,6}$ with  $e_{3,6}$.\n\nNow we want to drag $v_1$ to its \ndesired position along the path indicated by the dashed arrow. This involves a transition\nof $v_1$ through edge $e_{2,3}$, whose weight must therefore change its sign. Note that after performing\nthis operation, we may straighten out the double crossing of\n$e_{1,4}$ with $e_{2,3}$.\n\nThe right upper picture shows the result of these operations: Observe that we may now straighten\nout the double crossing of $e_{2,3}$ with $e_{1,6}$ and the single crossing of $e_{1,2}$ with $e_{2,3}$.\n\nNow we want to drag $v_3$ to its \ndesired position along the path indicated by the dashed arrow: This can be done without introducing\nor removing any crossing.\n\nThe left lower picture shows the result of these operations. By dragging $e_{1,6}$ over vertex\n$v_4$ and straightening out the single crossing with $e_{1,4}$ thus introduced, we arrive at the lower\npicture in the middle: This involves a transition\nof $v_4$ through $e_{1,6}$, whose weight must therefore change its sign. \n\nNow dragging $e_{3,6}$ over vertices $v_4$ \\EM{and} $v_5$ (and changing the sign of $\\omega\\of{e_{3,6}}$\n\\EM{twice}, accordingly), and straightening out\n\\begin{itemize}\n\\item the two single crossings of $e_{3,6}$ with $e_{5,6}$ and $e_{3,4}$\n\\item and the double crossing of $e_{3,6}$ with $e_{4,5}$\n\\end{itemize}\nintroduced by this modification gives the right lower picture. \n\nRepeating the last step by dragging $e_{2,5}$ over vertices $v_3$ and $v_4$, we arrive at the right proper drawing\nin {\\figref{fig:proper}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection {section}{1}{\\z@}{-3.5ex plus -1ex minus \n-.2ex}{2.3ex plus .2ex}{\\normalsize\\bf}}\n\\makeatother\n\n\\makeatletter\n\\def\\subsection{\\@startsection {subsection}{1}{\\z@}{-3.5ex plus -1ex minus \n-.2ex}{2.3ex plus .2ex}{\\normalsize\\bf}}\n\\makeatother\n\n\n\n\\begin{document}\n\n\n\\title{\\textbf{\\large On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations}}\n\\author{\\textsc{\\normalsize Marjorie G.\\ Hahn, Kei Kobayashi, Jelena Ryvkina and Sabir Umarov}\\thanks{Department of Mathematics, \n                        Tufts University, 503 Boston Avenue, Medford, MA 02155, USA; marjorie.hahn@tufts.edu, kei.kobayashi@tufts.edu, jelena.ryvkina@tufts.edu, sabir.umarov@tufts.edu}\n        \\vspace{1mm} \\\\ \\textit{\\small Tufts University}}\n\\date{ }\n\\maketitle                                                       \n\\vspace{-6mm}\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\\begin{abstract}\nThis paper establishes Fokker-Planck-Kolmogorov type equations for time-changed\nGaussian processes.    \nExamples include those equations for a time-changed fractional Brownian motion with time-dependent Hurst parameter and for a time-changed Ornstein-Uhlenbeck process.  The time-change process considered is the inverse of either a stable subordinator or a mixture of independent stable subordinators.\\footnote[0]{\\textit{AMS 2010 subject classifications:} Primary 60G15, 35Q84; secondary 60G22. \n     \\textit{Keywords:} time-change, inverse subordinator, Gaussian process, Fokker-Planck equation, Kolmogorov equation, fractional Brownian motion, time-dependent Hurst parameter, Volterra process.} \n\\end{abstract}\n\n\\section{Introduction} \n\\label{Sec1}\n\n\nA one-dimensional stochastic process $X=(X_t)_{t\\ge 0}$ is called a \\textit{Gaussian process} if the random vector $(X_{t_1},\\ldots,X_{t_m})$ has a multivariate Gaussian distribution for all finite sequences $0\\le t_1<\\cdots<t_m<\\infty$.\nThe joint distributions are characterized by the mean function $\\bbE[X_t]$ and the covariance function $R_X(s,t)=\\textrm{Cov}(X_s,X_t)$.  \nThe class of Gaussian processes contains some of the most important stochastic processes in both theoretical and applied probability, including Brownian motion, fractional Brownian motion \\cite{book,Cheridito03,turbulence,solar}, and Volterra processes \\cite{AlosNualart01,Decr-volterra}.\n\nIn this paper, Fokker-Planck-Kolmogorov type \nequations (FPKEs) for time-changed Gaussian processes are derived.   These FPKEs are partial differential equations (PDEs) satisfied by the transition probabilities of those processes. \nMain features of these equations involve 1){\\nolinebreak} fractional order derivatives in time and 2) a new class of operators acting on the time variable; see \\eqref{FP-volterra3-2}--\\eqref{FP-volterra3.5}.    \nBy definition, the \\textit{fractional order derivative $D_{\\ast}^{\\beta}$ of order $\\beta\\in (0,1)$ in the sense of Caputo-Djrbashian} is given by\n\\begin{equation}\n\\label{Caputo}\n  {D}_{\\ast}^{\\beta} g(t)={D}_{\\ast,t}^{\\beta} g(t) =\n      \\frac{1}{\\Gamma(1-\\beta)}\n      \\int_0^{t} \\frac{g^{\\prime}(\\tau) }{(t-\\tau)^{\\beta}}\\hspace{1pt} d \\tau, \n\\end{equation}\nwith $\\Gamma(\\cdot)$ being Euler's Gamma function. \nBy convention, set $D^1_{\\ast,t}=d\/dt$.  \nIntroducing the fractional integration operator\n\\[\nJ^{\\alpha}g(t)=J^{\\alpha}_t g(t)=\\frac{1}{\\Gamma(\\alpha)}\\int_0^{t}\n(t-\\tau)^{\\alpha-1}g(\\tau)\\hspace{1pt}d\\tau, ~~ \\alpha >0,\n\\]\none can represent $D_{\\ast,t}^{\\beta}$ in the form\n$D_{\\ast,t}^{\\beta}=J^{1-\\beta}_t\\circ (d\/dt)$ (see \\cite{GM97} for details). \n\n\n\nFor the last few decades, time-fractional order FPKEs have appeared as an essential tool for the study of dynamics of various complex processes arising in anomalous diffusion in physics\n\\cite{MetzlerKlafter00,Zaslavsky}, finance \\cite{GMSR,Janczura}, hydrology\n\\cite{BWM} and cell biology \\cite{Saxton}.\nUsing several different methods, many authors derive FPKEs associated with time-changed stochastic processes.\nFor example, in \\cite{MMM-TAMS}, FPKEs for time-changed continuous Markov processes are obtained via the theory of semigroups.   \nPaper \\cite{HKU} identifies a wide class of stochastic differential equations whose associated FPKEs are represented by time-fractional distributed order pseudo-differential equations.  \nThe driving processes for these stochastic differential equations are time-changed L\\'evy processes.   \nTwo different approaches are taken, one based on the semigroup technique and the other on the time-changed It\\^o formula in \\cite{Kobayashi}.   \nPaper \\cite{HKU-2} provides FPKEs associated with a time-changed fractional Brownian motion from a functional-analytic viewpoint.  \nContinuous time random walk-based approaches to derivations of time-fractional order FPKEs are illustrated in \\cite{GM1,MMM-ctrw,UG,US}.   \nIn the current paper, we follow the method presented in \\cite{HKU-2}.\n\n\nConsider a time-change process $E^\\beta=(E^\\beta_t)_{t\\ge 0}$ given by the \\textit{inverse}, or the \\textit{first hitting time process}, of a $\\beta$-stable subordinator $W^\\beta=(W^\\beta_t)_{t\\ge 0}$ with $\\beta\\in(0,1)$.  The relationship between the two processes is expressed as \n$E^\\beta_t=\\inf\\{s>0\\hspace{1pt};\\hspace{1pt} W^\\beta_s>t \\}$.\nTo make precise the problem being pursued in this paper, first recall the following results.  For proofs of these results as well as analysis on the associated classes of stochastic differential equations, \nsee e.g.\\ \\cite{GM1,HKU,HKU-2,MMM-ctrw}.\nThroughout the paper, all processes are assumed to start at 0.\n\\begin{enumerate}\n   \\item[(a)] If $E^\\beta$ is independent of an $n$-dimensional Brownian motion $B$, then $B$ and the time-changed Brownian motion $(B_{E^\\beta_t})$ have respective transition probabilities $p(t,x)$ and $q(t,x)$ satisfying the PDEs\n\\begin{align}\\label{Brownian FPK}\n   \\partial_t \\hspace{1pt}p(t,x)=\\frac 12 \\varDelta \\hspace{1pt}p(t,x) \\ \\ \\ \\textrm{and} \\ \\ \\  D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)=\\frac 12 \\varDelta \\hspace{1pt}q(t,x),  \n\\end{align}\nwhere $\\partial_t=\\frac{\\partial}{\\partial t}$ and $\\varDelta=\\sum_{j=1}^n \\partial_{x^j}^2=\\sum_{j=1}^n \\bigl(\\frac{\\partial}{\\partial {x^j}}\\bigr)^2$, with the vector $x\\in\\bbR^n$ denoted as $x=(x^1,\\ldots,x^n)$.\n   \\item[(b)]  Let $L$ be a L\\'evy process whose characteristic function is given by $\\bbE[e^{i(\\xi,L_t)}]=e^{t \\psi(\\xi)}$ with symbol $\\psi$ (see \\cite{Applebaum,Sato}).     \nIf $E^\\beta$ is independent of $L$, then $L$ and the time-changed L\\'evy process $(L_{E^\\beta_t})$ have respective transition probabilities $p(t,x)$ and $q(t,x)$ satisfying the PDEs \n\\begin{align}\\label{Levy FPK}\n   \\partial_t \\hspace{1pt}p(t,x)=\\mathcal{L}^\\ast  p(t,x) \\ \\ \\ \\textrm{and} \\ \\ \\ \n   D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)=\\mathcal{L}^\\ast q(t,x), \n\\end{align}\nwhere $\\mathcal{L}^\\ast$ is the conjugate of the pseudo-differential operator with symbol $\\psi$.   \n   \\item[(c)] If $E^\\beta$ is independent of an $n$-dimensional fractional Brownian motion $B^H$ with Hurst parameter $H\\in(0,1)$ (definition is given in Example \\hyperlink{Application-1}{1}), then $B^H$ and the time-changed fractional Brownian motion $(B^H_{E^\\beta_t})$ have respective transition probabilities $p(t,x)$ and $q(t,x)$ satisfying the PDEs\n\\begin{align}\\label{FBM FPK}\n   \\partial_t \\hspace{1pt}p(t,x)=H t^{2H-1}\\varDelta \\hspace{1pt}p(t,x) \\ \\ \\ \\textrm{and} \\ \\ \\ \n   D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)=H G^\\beta_{2H-1,t}\\varDelta \\hspace{1pt}q(t,x), \n\\end{align}\nwhere $G^\\beta_{\\gamma,t}$ with $\\gamma\\in(-1,1)$ is the operator acting on $t$ given by\n\\begin{equation} \\label{presentation}\nG^\\beta_{\\gamma,t} \\hspace{1pt}g(t)= \\beta \\Gamma(\\gamma + 1) J^{1-\\beta}_t\n\\mathcal{L}^{-1}_{s \\to t}\\biggl[\\frac{1}{2\\pi i} \\int_{C-i\n\\infty}^{C+i\\infty}\n\\frac{\\tilde{g}(z)}{(s^{\\beta}-z^{\\beta})^{\\gamma+1}}\\hspace{1pt}dz \\biggr](t),\n\\end{equation}\nwith $0 < C <s$ and $z^{\\beta}=e^{\\beta \\hspace{1pt} \\textrm{Ln} (z)},$\n$\\textrm{Ln}(z)$ being the principal value of the complex logarithmic function $\\ln(z)$ with cut along the negative real axis.\nHere $\\tilde{g}(s)=\\mathcal{L}_{t\\to s}[g(t)](s)=\\mathcal{L}_{t}[g(t)](s)$ and $\\mathcal{L}_{s\\to t}^{-1}[f(s)](t)$ denote the Laplace transform and the inverse Laplace transform, respectively. \n\\end{enumerate}  \n\n Obviously (a) provides a special case of both (b) and (c).  Note that in \\eqref{Brownian FPK} and \\eqref{Levy FPK} the second PDE  \nis obtained upon replacing the first order time derivative $\\partial_t$\nin the first PDE\nby the fractional order derivative $D^\\beta_{\\ast,t}$ and the right-hand side remains unchanged.   On the other hand, as \\eqref{FBM FPK} shows, this is not the case for a time-changed fractional Brownian motion; the right hand side of the time-fractional order FPKE has a different form than that of the FPKE for the untime-changed fractional Brownian motion.\nNamely, the operator $G^\\beta_{2H-1,t}$ instead of $t^{2H-1}$ appears, which is ascribed to dependence between increments over non-overlapping intervals of the fractional Brownian motion $B^H$.   \nWhen $H=1\/2$ so that the fractional Brownian motion coincides with a usual Brownian motion, the FPKEs in \\eqref{FBM FPK} match with those in \\eqref{Brownian FPK}.   \nThe operators $\\{G^\\beta_{\\gamma,t}; \\gamma\\in(-1,1)\\}$ are known to satisfy the semigroup property (Proposition 3.6 in \\cite{HKU-2}).  \n\n\nFollowing the functional-analytic technique presented in \\cite{HKU-2} to obtain the time-fractional order FPKE in \\eqref{FBM FPK}, we will establish in Theorem \\ref{theorem_FP-volterra} the FPKE for the time-changed Gaussian process $(X_{E^\\beta_t})$ under the assumption that the time-change process $E^\\beta$ is independent of the Gaussian process $X$.\nMoreover, generalization to time-changes which are the inverses of mixtures of independent stable subordinators will be considered in Theorem \\ref{theorem_FP-volterra_general}. \nIn Section \\ref{Sec4}, applications of these results yield FPKEs for time-changed mixed fractional Brownian motions as well as those for time-changed Volterra processes.  \nFractional Brownian motions with time-dependent Hurst parameter $H=H(t)\\in(1\/2,1)$ are among the Volterra processes considered; see Example \\hyperlink{Application-3}{3}.\nTwo equivalent forms of FPKEs for a time-changed Ornstein-Uhlenbeck process are compared in Example \\hyperlink{Application-5}{5}.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{Sec2}\n\n\nLet $E^\\beta$ be the inverse of a stable subordinator $W^\\beta$ starting at $0$ with stability index $\\beta \\in(0,1)$.  The process $W^\\beta$ is a L\\'evy process with Laplace transform $\\mathbb{E}[e^{-sW^\\beta_t}]=e^{-ts^{\\beta}}$ and self-similarity: $(W^\\beta_{ct})_{t\\ge 0}=(c^{1\/ \\beta} W^\\beta_t)_{t\\ge 0}$ in the sense of finite dimensional distributions for all $c>0$ (see \\cite{Applebaum,Sato}).   \nSince $W^\\beta$ is strictly increasing, its inverse $E^\\beta$ is\ncontinuous and nondecreasing, but no longer a L\\'evy process (see \\cite{MMM-infinitemean}).  \nThe density $f_{E_t^\\beta}$ of $E_t^\\beta$ can be expressed using the density $f_{W_\\tau^\\beta}$ of $W_\\tau^\\beta$ as \n\\begin{align*} \nf_{E_t^\\beta}(\\tau) = \\partial_\\tau \\mathbb{P}(E^\\beta_t\\le \\tau)\n=\\partial_\\tau \\bigl\\{1-\\mathbb{P}(W^\\beta_\\tau < t)\\bigr\\}\n=-\\partial_\\tau \\bigl\\{[J^1_t f_{W_\\tau^\\beta}](t) \\bigr\\},\n\\end{align*}\nfrom which it follows that\n\\begin{align}\\label{laplace_Ebeta}\n   \\mathcal{L}_{t \\to s}[f_{E_t^\\beta}(\\tau)](s)= -\\partial_\\tau \\biggl[\\frac{\\widetilde{f_{W_\\tau^\\beta}}(s)}{s}\\biggr]\n=s^{\\beta-1} e^{-\\tau s^\\beta}, ~ s>0, ~ \\tau \\ge 0.\n\\end{align}\nThe function $f_{E_t^\\beta}(\\tau)$ is $C^{\\infty}$ with respect to the two variables $t$ and $\\tau$.\n\n\n\nThe notion of time-change can be extended to the more general case where\nthe time-change process is given by the inverse of an arbitrary\nmixture of independent stable subordinators. \nLet $\\rho_\\mu(s)=\\int_0^1\ns^{\\beta}d \\mu (\\beta),$ where \n$\\mu$ is a finite measure with $\\textrm{supp}\\, \\mu \\subset (0,1).$\nLet $W^{\\mu}$ be a nonnegative stochastic\nprocess satisfying $\\mathbb{E}[ e^{-sW^{\\mu}_t}]=e^{-t \\rho_\\mu(s)}$ and let\n$E^{\\mu}_t=\\inf \\{\\tau>0\\hspace{1pt};\\hspace{1pt} W^{\\mu}_{\\tau} > t\\}.$\nClearly, $W^\\mu=W^{\\beta_0}$ if $\\mu=\\delta_{\\beta_0}$, the Dirac measure on $(0,1)$ concentrated on a single point $\\beta_0$.  \nThe process\n$W^{\\mu}$ represents a weighted mixture of independent stable subordinators.    \nSimilar to the identity \\eqref{laplace_Ebeta}, the density $f_{E_t^\\mu}$ of $E_t^\\mu$ has Laplace transform\n\\begin{align}\\label{laplace_Emu}\n   \\mathcal{L}_{t \\to s}[f_{E_t^\\mu}(\\tau)](s)\n=\\frac{\\rho_\\mu(s)}{s} e^{-\\tau \\rho_\\mu(s)}, ~ s>0, ~ \\tau \\ge 0.\n\\end{align}\nFor further properties of $f_{E_t^\\beta}(\\tau)$ and $f_{E_t^\\mu}(\\tau)$, see \\cite{HKU-2,MMM-ctrw}.\n\n\n\nThe above time-change process $E^\\mu$ is connected with the \\textit{fractional derivative $D^\\mu_{\\ast}$ with distributed orders} given by \n\\begin{align}\\label{distributed-derivative}\n   D^\\mu_{\\ast}\\hspace{1pt}g(t)=D^\\mu_{\\ast,t}\\hspace{1pt}g(t)=\\int_0^1 D^\\beta_\\ast g(t)\\hspace{1pt} d\\mu(\\beta).   \n\\end{align}\nNamely, if $E^\\beta$ is replaced by $E^\\mu$ in items (a) and (b) of the list of known results in Section \\ref{Sec1}, then the FPKEs for the time-changed Brownian motion $(B_{E^\\mu_t})$ and the time-changed L\\'evy process $(L_{E^\\mu_t})$ are respectively given by (see \\cite{HKU})\n\\[\n   D^\\mu_{\\ast,t} \\hspace{1pt}q(t,x)=\\frac 12 \\varDelta\\hspace{1pt} q(t,x)\n   \\ \\ \\ \\textrm{and} \\ \\ \\  \n   D^\\mu_{\\ast,t} \\hspace{1pt}q(t,x)=\\mL^\\ast q(t,x).  \n\\] \nFor investigations into PDEs with fractional derivatives with distributed orders, see \\cite{Kochubey,MMM-ultraslow,UG05}.\n\n\n\nThe covariance function of a given zero-mean Gaussian process is symmetric and positive semi-definite; conversely, every symmetric, positive semi-definite function on $[0,\\infty)\\times[0,\\infty)$ is the covariance function of some zero-mean Gaussian process (see e.g., Theorem 8.2 of \\cite{Janson}).  Examples of such functions include $R_X(s,t)=s \\wedge t$ for Brownian motion and $R_X(s,t)=\\sigma_0^2+s\\cdot t$ which is obtained from linear regression\n (see \\cite{RW}).  \nThe sum and the product of two covariance functions for Gaussian processes are again covariance functions for some Gaussian processes.  For more examples of covariance functions, consult e.g.\\ \\cite{RW}.  \n\n\n\nAn $n$-dimensional Gaussian process $X=(X^1,\\ldots,X^n)$ is a process whose components $X^j$ are independent one-dimensional Gaussian processes with possibly distinct covariance functions $R_{X^j}(s,t)$.\nThe variance functions $R_{X^j}(t)=R_{X^j}(t,t)$ will play\nan important role in establishing FPKEs for Gaussian and time-changed Gaussian processes.  \nFor differentiable variance functions $R_{X^j}(t)$, let\n\\begin{align}\\label{FP-volterra0.5}\n   A=A_X=\\frac 12\\sum_{j=1}^n R_{X^j}'(t)\\hspace{1pt}\\partial_{x^j}^2.\n\\end{align}\nClearly $A=\\frac{1}{2}\\hspace{1pt}\\varDelta$ if $X$ is an $n$-dimensional Brownian motion.\n\n\n\n\n\n\\begin{proposition} \\label{proposition_FP-volterra}\nLet $X=(X^1,\\ldots,X^n)$ be an $n$-dimensional zero-mean Gaussian process with covariance functions $R_{X^j}(s,t)$, $j=1,\\ldots,n$.  Suppose the variance functions $R_{X^j}(t)=R_{X^j}(t,t)$ are differentiable on $(0,\\infty)$.  \nThen the transition probabilities $p(t,x)$ of $X$ satisfy the PDE\n\\begin{align} \\label{FP-volterra1}\n   \\partial_t \\hspace{1pt}p(t,x)= A\\hspace{1pt} p(t,x), \\ t>0, \\ x\\in\\bbR^n,\n\\end{align}\nwhere $A$ is the operator in \\eqref{FP-volterra0.5}.\n\\end{proposition}\n\n\\begin{proof}\nSince the components $X^j$ of $X$ are independent zero-mean Gaussian processes, it follows that\n\\begin{align}\np(t,x)=\\prod_{j=1}^n\\bigl(2\\pi R_{X^j}(t)\\bigr)^{-1\/2}\\times \\exp\\biggl\\{-\\sum_{j=1}^n\\frac{(x^j)^2}{2R_{X^j}(t)}\\biggr\\}.\n\\end{align}\nDirect computation of partial derivatives of $p(t,x)$\nyields the equality in \\eqref{FP-volterra1}.\n\\end{proof}\n\n\n\\begin{remark}\n\\begin{em}\na)  In Proposition \\ref{proposition_FP-volterra}, if the components $X^j$ are independent Gaussian processes with a \\textit{common} variance function $R_{X}(t)$ which is differentiable, then \n\\eqref{FP-volterra1} reduces to the following form which agrees with the classical FPKE:\n\\begin{align} \n   \\partial_t \\hspace{1pt}p(t,x)=\\frac 12 R_X'(t) \\varDelta \\hspace{1pt}p(t,x).\n\\end{align}\n\nb) If $X=(X^1,\\ldots,X^n)$ is an $n$-dimensional Gaussian process with mean functions $m_{X^j}(t)$ and covariance functions $R_{X^j}(s,t)$, and if both $m_{X^j}(t)$ and $R_{X^j}(t)=R_{X^j}(t,t)$ are differentiable,  \nthen the associated FPKE contains an additional term:\n\\begin{align} \n   \\partial_t \\hspace{1pt}p(t,x)=A \\hspace{1pt}p(t,x)\n   + B \\hspace{1pt}p(t,x) \\ \\ \\ \\ \\textrm{where} \\ \\ \\ \\ \n    B=-\\sum_{j=1}^n m_{X^j}'(t)\\hspace{1pt} \\partial_{x^j}.\n\\end{align}\nSuch Gaussian processes include e.g.\\ the process defined by the sum of a Brownian motion and a deterministic differentiable function.\n\nc) The initial distribution of the Gaussian process $X$ in Proposition \\ref{proposition_FP-volterra} is not given; however, it needs to be specified in order to guarantee uniqueness of the solution to PDE \\eqref{FP-volterra1}.  The same argument applies to the PDEs to be established in Theorems \\ref{theorem_FP-volterra} and \\ref{theorem_FP-volterra_general} as well.\n\\end{em}\n\\end{remark}\n\n\n\n\n\n\\section{FPKEs for time-changed Gaussian processes}\n\\label{Sec3}\n\n\nTheorems \\ref{theorem_FP-volterra} and \\ref{theorem_FP-volterra_general} formulate the FPKE for a time-changed Gaussian process under the assumption that the time-change process is independent of the Gaussian process.  \nThe case where those processes are dependent is not discussed in this paper.  \nAs in Proposition \\ref{proposition_FP-volterra}, the variance function plays a key role here.  \n\n\n\\begin{theorem} \\label{theorem_FP-volterra} \nLet $X=(X^1,\\ldots,X^n)$ be an $n$-dimensional \nzero-mean Gaussian process with covariance functions $R_{X^j}(s,t)$, $j=1,\\ldots,n$,\nand let $E^\\beta$ be the inverse of a stable subordinator $W^\\beta$ of index $\\beta\\in(0,1)$, independent of $X$.   \n Suppose the variance functions $R_{X^j}(t)=R_{X^j}(t,t)$ are differentiable on $(0,\\infty)$ and Laplace transformable.  \nThen the transition probabilities $q(t,x)$ of the time-changed Gaussian process $(X_{E^\\beta_t})$  satisfy the equivalent PDEs\n\\begin{align} \\label{FP-volterra3-2}\n   {D}^\\beta_{\\ast,t} \\hspace{1pt}q(t,x)&=\n\\sum_{j=1}^n J^{1-\\beta}_t \\Lambda^\\beta_{X^j,t}\\hspace{1pt} \\partial_{x^j}^2 \\hspace{1pt}q(t,x), \\ t>0, \\ x\\in\\bbR^n,\\\\\n\\label{FP-volterra3-1}\n   \\partial_t \\hspace{1pt}q(t,x)&= \n\\sum_{j=1}^n  \\Lambda^\\beta_{X^j,t}\\hspace{1pt} \\partial_{x^j}^2 \\hspace{1pt}q(t,x), \\ t>0, \\ x\\in\\bbR^n,\n\\end{align}\nwhere $\\Lambda^\\beta_{X^j,t}$, $j=1,\\ldots,n$, are the operators acting on $t$ given by\n\\begin{align} \\label{FP-volterra3.5}\n  \\Lambda^\\beta_{X^j,t}\\hspace{1pt} g(t)=\n \\frac\\beta 2\\hspace{1pt} \\mL^{-1}_{s\\to t} \n   \\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}}\n    (s^\\beta-z^\\beta)\\widetilde{R_{X^j}}(s^\\beta-z^\\beta) \n   \\hspace{1pt}\\tilde{g}(z)\\hspace{1pt} dz\\biggr] (t),\n\\end{align}\nwith $z^\\beta=e^{\\beta \\hspace{1pt}\\textrm{\\rm Ln}(z)}$, $\\textrm{\\rm Ln}(z)$ being the principal value of the complex logarithmic function $\\ln(z)$ \nwith cut along the negative real axis,\nand \n$\\bm{\\mathcal{C}}$ being a curve in the complex plane obtained via the transformation $\\zeta=z^\\beta$ which leaves all the singularities of $\\widetilde{R_{X^j}}$ on one side.\n\\end{theorem}\n\n\n\\begin{proof}\nLet $p(t,x)$ denote the transition probabilities of the Gaussian process $X$.   \nFor each $x\\in\\bbR^n$, it follows from the independence assumption between $E^\\beta$ and $X$ that \n\\begin{equation} \\label{relation}\n   q(t,x)=\\int_0^{\\infty} f_{E_t^\\beta}(\\tau)\\hspace{1pt}p(\\tau,x)\\;d\\tau, \\ t>0.\n\\end{equation}\nRelationship \\eqref{relation} and equality \\eqref{laplace_Ebeta} together yield \n\\begin{align} \\label{FP-volterra4}\n   \\tilde{q}(s,x)=s^{\\beta-1}\\tilde{p}(s^\\beta,x), \\ s>0.\n\\end{align}\nSince $R_{X^j}(t)$ is Laplace transformable, \n$\\widetilde{R_{X^j}}(s)$ exists for all $s>a$, for some constant $a\\ge 0$.   \nTaking Laplace transforms on both sides of \\eqref{FP-volterra1}, \n\\begin{align}  \\label{FP-volterra4.5}\n   s \\,\\tilde{p}(s,x)-p(0,x)\n   &=\\frac 12\\sum_{j=1}^n\\mL_{t\\to s}[  R_{X^j}'(t) \\partial_{x^j}^2 \\hspace{1pt} p(t,x)](s)\\\\\n   &=\\frac 12 \\sum_{j=1}^n \\bigl(\\mL_{t}[R_{X^j}'(t)]\\ast \\mL_{t}[\\partial_{x^j}^2 \\hspace{1pt} p(t,x)] \\bigr)(s)\\notag \\\\\n   &=\\frac 12 \\sum_{j=1}^n  \\biggl[ \\frac 1{2\\pi i} \\int_{c-i\\infty}^{c+i\\infty} \\widetilde{R_{X^j}'}(s-\\zeta) \\partial_{x^j}^2 \\hspace{1pt} \\tilde{p}(\\zeta,x) \\hspace{1pt}d\\zeta\\biggr]\\notag\\\\\n   &=\\frac \\beta 2 \\sum_{j=1}^n\\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}} \\widetilde{R_{X^j}'}(s-z^\\beta) \\partial_{x^j}^2  \\hspace{1pt}  \\tilde{p}(z^\\beta,x)  z^{\\beta-1}\\hspace{1pt}dz\\biggr], \\ s>a, \\notag\n\\end{align}\nwhere $\\ast$ denotes the convolution of Laplace images and \nthe function \n\\begin{align} \\label{FP-volterra4.6}\n   \\widetilde{R_{X^j}'}(s)=s\\hspace{1pt}\\widetilde{R_{X^j}}(s), \\ s>a,\n\\end{align}\nexists by assumption.    \nEquation \\eqref{FP-volterra4.6} is valid since $R_{X^j}(0)=0$ due to the initial condition $X^j(0)=0$.\nSince $E^\\beta_0=0$ with probability one, it follows that $p(0,x)=q(0,x)$.  Replacing $s$ by $s^\\beta$ and using the identity \\eqref{FP-volterra4} yields\n\\begin{align}\\label{FP-volterra5}\n   s \\,\\tilde{q}(s,x)-q(0,x)\n   =\\frac \\beta 2 \\sum_{j=1}^n \\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}}   \\widetilde{R_{X^j}'}(s^\\beta-z^\\beta) \\partial_{x^j}^2  \\hspace{1pt}\\tilde{q}(z,x) \\hspace{1pt}dz\\biggr], \\ s>a^{1\/\\beta}.\n\\end{align}\nSince the left hand side equals $\\mL_{t\\to s}[\\partial_t \\hspace{1pt}q(t,x)](s)$, PDE \\eqref{FP-volterra3-1} follows upon\nsubstituting \\eqref{FP-volterra4.6}\nand taking the inverse Laplace transform on both sides.\nMoreover, applying the fractional integral operator $J^{1-\\beta}_t$ to both sides of \\eqref{FP-volterra3-1} yields \\eqref{FP-volterra3-2}.\n\\end{proof}\n\nThe next theorem extends the previous theorem to time-changes which are the inverses of mixtures of independent stable subordinators.  \n\n\n\\begin{theorem} \\label{theorem_FP-volterra_general} \nLet $X=(X^1,\\ldots,X^n)$ be an $n$-dimensional \nzero-mean Gaussian process with covariance functions $R_{X^j}(s,t)$, $j=1,\\ldots,n$, and let\n$E^\\mu$ be the inverse of a mixture $W^\\mu$ of independent stable subordinators, independent of $X$.  \nSuppose the variance functions $R_{X^j}(t)=R_{X^j}(t,t)$ are differentiable on $(0,\\infty)$ and Laplace transformable.  \nThen the transition probabilities $q(t,x)$ of the time-changed Gaussian process $(X_{E^\\mu_t})$ satisfy the PDEs\n\\begin{align} \\label{FP-volterra13-2}\n   D^\\mu_{\\ast,t} \\hspace{1pt}q(t,x)= \n\\sum_{j=1}^n \n\\int_0^1 J^{1-\\beta}_t \\Lambda^\\mu_{X^j,t}\\hspace{1pt} \\partial_{x^j}^2 \\hspace{1pt} q(t,x) \\hspace{1pt} d\\mu(\\beta), \\ t>0, \\ x\\in\\bbR^n,\n\\end{align}\nand\n\\begin{align} \\label{FP-volterra13-1}\n   \\partial_t \\hspace{1pt}q(t,x)=\n\\sum_{j=1}^n  \\Lambda^\\mu_{X^j,t}\\hspace{1pt} \\partial_{x^j}^2  \\hspace{1pt}q(t,x), \\ t>0, \\ x\\in\\bbR^n,\n\\end{align}\nwhere $D^\\mu_{\\ast,t}$ is the operator in \\eqref{distributed-derivative} and $\\Lambda^\\mu_{X^j,t}$, $j=1,\\ldots,n$, are the operators acting on $t$ given by\n\\begin{align}\n   \\Lambda^\\mu_{X^j,t}\\hspace{1pt} g(t)\n   =\\frac {1}{2} \\hspace{1pt} \\mL^{-1}_{s\\to t}  \n   \\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}}\n   \\bigl(\\rho_\\mu(s)-\\rho_\\mu(z)\\bigr) \\widetilde{R_{X^j}}\\bigl(\\rho_\\mu(s)-\\rho_\\mu(z)\\bigr) \n   \\hspace{1pt}m_\\mu(z) \\hspace{1pt}\\tilde{g}(z)\\hspace{1pt}dz\\biggr] (t),\n\\end{align}\nwith $\\rho_\\mu(z)=\\int_0^1 e^{\\beta \\hspace{1pt} \\emph{Ln} (z)}\\hspace{1pt}d\\mu(\\beta)$, $m_{\\mu}(z)=\\frac{1}{\\rho_\\mu(z)}\\int_0^1 \\beta z^{\\beta}\\hspace{1pt} d \\mu(\\beta)$, and \n$\\bm{\\mathcal{C}}$ being a curve in the complex plane obtained via the transformation $\\zeta=\\rho_\\mu(z)$ which leaves all the singularities of $\\widetilde{R_{X^j}}$ on one side.\n\\end{theorem}\n\n\n\n\\begin{proof}\nWe only sketch the proof since it is similar to the proof of Theorem\n\\ref{theorem_FP-volterra}.  \nLet $p(t,x)$ denote the transition probabilities of the Gaussian process $X$.   For each $x\\in\\bbR^n$, it follows from relationship \\eqref{relation} with $f_{E^\\beta_t}$ replaced by $f_{E^\\mu_t}$ together with equality \\eqref{laplace_Emu} that\n\\begin{align} \\label{FP-volterra14}\n   \\tilde{q}(s,x)=\\frac{\\rho_\\mu(s)}{s}\\hspace{1pt}\\tilde{p}\\bigl(\\rho_\\mu(s),x\\bigr), \\ s>0.\n\\end{align}\nTaking Laplace transforms on both sides of \\eqref{FP-volterra1} leads to the second to last equality in \\eqref{FP-volterra4.5}.  Letting $\\zeta=\\rho_{\\mu}(z)$ yields\n\\begin{align*} \n   s \\,\\tilde{p}(s,x)-p(0,x)=\\frac 1 2 \\sum_{j=1}^n \\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}} \\widetilde{R_{X^j}'}\\bigl(s-\\rho_\\mu(z)\\bigr)  \\partial_{x^j}^2 \\hspace{1pt} \\tilde{p}\\bigl(\\rho_\\mu(z),x\\bigr)  \\frac {\\rho_\\mu(z)}{z}\\hspace{1pt} m_\\mu(z) \\hspace{1pt}dz\\biggr],\n\\end{align*}\nwhich is valid for all $s$ for which $\\widetilde{R_{X^j}}(s)$ exists.\nReplacing $s$ by $\\rho_\\mu(s)$ and using the identity \\eqref{FP-volterra14} yields an equation similar to \\eqref{FP-volterra5}.  \nPDE \\eqref{FP-volterra13-1} is obtained upon taking the inverse Laplace transform on both sides.  Finally, applying the fractional integral operator $J^{1-\\beta}_t$ and integrating with respect to $\\mu$ on both sides of \\eqref{FP-volterra13-1} yields \\eqref{FP-volterra13-2}.\n\\end{proof}\n\n\n\\begin{remark}\n\\begin{em}\na) If $\\mu=\\delta_{\\beta_0}$ with $\\beta_0\\in(0,1)$, then $\\Lambda^\\mu_{X^j,t} \\hspace{1pt}g(t)=\\Lambda^{\\beta_0}_{X^j,t}\\hspace{1pt}g(t)$ and the FPKEs in \\eqref{FP-volterra13-2} and \\eqref{FP-volterra13-1}\nrespectively reduce to the FPKEs in \\eqref{FP-volterra3-2} and \\eqref{FP-volterra3-1} with $\\beta=\\beta_0$, as expected. \n\nb) In Theorem \\ref{theorem_FP-volterra_general}, if the components $X^j$ are independent Gaussian processes with a \\textit{common} variance function $R_{X}(t)$ which is differentiable and Laplace transformable, then the FPKEs in \\eqref{FP-volterra13-2} and \\eqref{FP-volterra13-1} respectively reduce to the following simple forms:\n\\begin{align} \n D^\\mu_{\\ast,t} \\hspace{1pt}q(t,x) &=  \\hspace{1pt}  \\int_0^1 J^{1-\\beta}_t \\Lambda^\\mu_{X,t} \\hspace{1pt}\\varDelta \\hspace{1pt} q(t,x)\\hspace{1pt} d\\mu(\\beta);\\\\\n  \\partial_t \\hspace{1pt}q(t,x)&= \\Lambda^\\mu_{X,t} \\varDelta \\hspace{1pt}  q(t,x).\n\\end{align}\n\nc) As Example \\hyperlink{Application-1}{1} in Section \\ref{Sec4} shows, Theorem \\ref{theorem_FP-volterra} extends the time-fractional order FPKE in \\eqref{FBM FPK} for a time-changed fractional Brownian motion to that for a general time-changed Gaussian process, revealing the role of the variance function in describing the dynamics of the process.\n\\end{em}\n\\end{remark}\n\n\n\\section{Applications} \\label{Sec4}\n\n\n\nThis section is devoted to applications of the results established in this paper concerning FPKEs for Gaussian and time-changed Gaussian processes.  For simplicity of discussion, we will consider the time-changed process $E^\\beta$ given by the inverse of a stable subordinator $W^\\beta$ of index $\\beta\\in(0,1)$, rather than the more general time-change process $E^\\mu$.\n\n\\vspace{3.2mm}\n\n\\noindent\n\\textsl{Example \\hypertarget{Application-1}{1}. Fractional Brownian motion.}\nOne of the most important Gaussian processes in applied probability is a fractional Brownian motion $B^H$ with Hurst parameter $H\\in(0,1)$.  \nA one-dimensional \\textit{fractional Brownian motion} is a zero-mean Gaussian process with covariance function\n\\begin{equation}\\label{FBM-cov}\nR_{B^H}(s,t)=\\bbE[B_s^H B_t^H]=\\frac{1}{2} (s^{2H}+t^{2H}- |s-t|^{2H}).\n\\end{equation}\nIf $H=1\/2,$ then \n$B^H$ becomes a usual Brownian motion.  \nAn $n$-dimensional fractional Brownian motion is\nan $n$-dimensional process whose components are independent fractional Brownian motions with a common Hurst parameter.\n\nStochastic processes driven by a fractional Brownian motion $B^H$ are of increasing interest for both theorists and applied researchers due\nto their wide application in fields such as mathematical finance\n\\cite{Cheridito03}, solar activities \\cite{solar} and\nturbulence \\cite{turbulence}.\nThe process $B^H,$ like the usual Brownian motion,\nhas nowhere differentiable paths and stationary increments; however, it does not have independent increments.  \n$B^H$ has the integral representation \n$B_t^H=\\int_0^t K_H(t,s)\\hspace{1pt}d B_s,$ where $B$ is a Brownian motion and $K_H(t,s)$ is a deterministic kernel.  \n$B^H$ is not a semimartingale unless $H=1\/2$, so the usual It\\^o's stochastic calculus is not valid. \nFor details of the above properties, see \\cite{book,Nualart}.   \n\n\n\nLet $B^H$ be an $n$-dimensional fractional Brownian motion and let $E^\\beta$ be the inverse of a stable subordinator of index $\\beta\\in(0,1)$, independent of $B^H$.  Then the components of $B^H$ share the common variance function $R_{B^H}(t)=t^{2H}$ and its Laplace transform $\\widetilde{R_{B^H}'}(s)=2H\\hspace{1pt} \\Gamma(2H)\/s^{2H}$.  \nHence, Proposition \\ref{proposition_FP-volterra} and Theorem \\ref{theorem_FP-volterra} immediately recover both the FPKEs in \\eqref{FBM FPK} for the fractional Brownian motion $B^H$ and the time-changed fractional Brownian motion $(B^H_{E^\\beta_t})$.  \nIn this case,\n\\begin{align} \\label{correspondence-operators}\nJ^{1-\\beta}_t \\Lambda^\\beta_{B^H,t}=H G^\\beta_{2H-1,t},\n\\end{align}\nwhere $G^\\beta_{\\gamma,t}$ is the operator given in \\eqref{presentation}.   \nNote that the curve $\\bm{\\mathcal{C}}$ appearing in the expression of the operator $\\Lambda^\\beta_{B^H,t}$ in \\eqref{FP-volterra3.5} can be replaced by a vertical line $\\{C+i r \\hspace{1pt}; \\hspace{1pt}r\\in \\bbR\\}$ with $0<C<s$ since the integrand has a singularity only at $z=s$. \n\n\n\n\n\\vspace{3.2mm}\n\n\\noindent\n\\textsl{Example \\hypertarget{Application-2}{2}.  Mixed fractional Brownian motion.}  \nLet $X$ be an $n$-dimensional process defined by a finite linear combination of independent zero-mean Gaussian processes $X_{1},\\ldots, X_{m}$: $X_t=\\sum_{\\ell=1}^m a_\\ell X_{\\ell,t}$ with $a_1,\\ldots,a_m\\in\\bbR$. \nFor simplicity, assume that for each $\\ell=1,\\ldots,m$, the components of the vector $X_\\ell=(X_\\ell^1,\\ldots,X_\\ell^n)$ share a common variance function $R_{X_\\ell}(t)$.  \nThen the process $X$ is again a Gaussian process whose components have the same variance function $R_X(t)=\\sum_{\\ell=1}^m a_\\ell^2 R_{X_\\ell}(t)$. \nTherefore, it follows from Proposition \\ref{proposition_FP-volterra} and Theorem \\ref{theorem_FP-volterra} that the FPKEs for $X$ and $(X_{E^\\beta_t})$, under the independence assumption between $E^\\beta$ and $X$, are respectively given by \n\\begin{align}  \\label{mixed FPK}\n   \\partial_t \\hspace{1pt}p(t,x)=\\phi(t) \\hspace{1pt}\\varDelta \\hspace{1pt}p(t,x) \\ \\ \\ \\textrm{and} \\ \\ \\   \n   D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)=\\Phi^\\beta_t \\hspace{1pt}\\varDelta \\hspace{1pt} q(t,x),\n\\end{align}\nwhere $\\phi(t)=\\frac 12 \\sum_{\\ell=1}^m a_\\ell^2 R_{X_{\\ell}}' (t)$ and $\\Phi^\\beta_t=\\sum_{\\ell=1}^m a_\\ell^2 J^{1-\\beta}_t \\Lambda^\\beta_{X_{\\ell},t}$.  Notice that $\\phi(t)$ simply denotes the multiplication by a function of $t$ whereas $\\Phi^\\beta_t$ is an operator acting on $t$.  \nThis generalizes the correspondence between the function $t^{2H-1}$ and the operator $G^\\beta_{2H-1,t}$ observed in the FPKEs in \\eqref{FBM FPK} for the fractional Brownian motion and the time-changed fractional Brownian motion.  \n\n\nA \\textit{mixed fractional Brownian motion} is a finite linear combination of independent fractional Brownian motions (see \\cite{MRR,Thale} for its properties). \nIt was introduced in \\cite{Cheridito01} to discuss the price of a European call option on an asset driven by the process.  The process $X$ considered in that paper is of the form $X_t=B_t+a \\hspace{1pt}B_t^H$, where $a\\in\\bbR$, $B$ is a Brownian motion, and $B^H$ is a fractional Brownian motion with Hurst parameter $H\\in(0,1)$.   In this situation, the FPKEs in \\eqref{mixed FPK}, with the help of \\eqref{correspondence-operators}, yield\n\\begin{align}\n   \\partial_t \\hspace{1pt}p(t,x)&=\\frac 12\\hspace{1pt}\\varDelta \\hspace{1pt}p(t,x)+a^2 H t^{2H-1}\\hspace{1pt}\\varDelta \\hspace{1pt}p(t,x); \\\\ \n   D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)&=\\frac 12 \\hspace{1pt}\\varDelta\\hspace{1pt}q(t,x)+a^2 H  G^\\beta_{2H-1,t} \\hspace{1pt}\\varDelta \\hspace{1pt} q (t,x). \n\\end{align}\n\n\n\n\n\n\n\\vspace{3.2mm}\n\n\\noindent\n\\textsl{Example \\hypertarget{Application-3}{3}. Fractional Brownian motion with variable Hurst parameter.}  \n\\textit{Volterra processes} form an important subclass of Gaussian processes.  \nThey are continuous zero-mean Gaussian processes $V=(V_t)$ defined on a given finite interval $[0,T]$ with integral representations of the form $V_t=\\int_0^t K(t,s)\\hspace{1pt} dB_s$ for some deterministic kernel $K(t,s)$ and Brownian motion $B$ (see \\cite{AlosNualart01,Decr-volterra} for details).   Fractional Brownian motions are clearly an example of a Volterra process.  In particular, if $B^H$ is a fractional Brownian motion with Hurst parameter $H\\in(1\/2,1)$, then $B^H=\\int_0^t K_H(t,s)\\hspace{1pt} dB_s$ with the kernel\n\\begin{align} \\label{K_H-1}\n   K_H(t,s)=c_H s^{1\/2-H}\\int_s^t (r-s)^{H-3\/2}\\hspace{2pt}r^{H-1\/2} \\hspace{1pt}dr, \\ \\  t>s, \n\\end{align}\nwhere the positive constant $c_H$ is chosen so that the integral\n$\\int_0^{t\\wedge s} K_H(t,r)K_H(s,r) \\hspace{1pt}dr$ coincides with \n$R_{B^H}(s,t)$ in \\eqref{FBM-cov}.\nIncrements of $B^H$ exhibit long range dependence.\n\n\nA particular interesting Volterra process\nis the fractional Brownian motion with time-dependent Hurst parameter $H(t)$ suggested in Theorem 9 of \\cite{Decr-volterra}.\nNamely, suppose $H(t): [0,T]\\rightarrow (1\/2,1)$ is a deterministic function satisfying the following conditions:\n\\begin{align}\\label{condition_star}\n    \\inf_{t\\in[0,T]} H(t)>\\frac 12 \\ \\hspace{3pt} \\textrm{and} \\ \\hspace{3pt}  \n    H(t)\\in \\mathcal{S}_{1\/2+\\alpha,2} \\ \\hspace{3pt}\\textrm{for some} \\ \\hspace{3pt}  \n     \\alpha\\in\\biggl(0,\\hspace{2pt}\\inf_{t\\in[0,T]}H(t)-\\frac 12\\biggr), \n\\end{align}\nwhere $\\mathcal{S}_{\\eta,2}$ is the Sobolev-Slobodetzki space given by the closure of the space $C^1[0,T]$ with respect to the semi-norm \n\\begin{align}\n   \\|  f \\|^2= \\int_0^T\\hspace{-1.2mm} \\int_0^T \\frac{|f(t)-f(s)|^2}{|t-s|^{1+2\\eta}}\\hspace{1pt} dt \\hspace{1pt}ds.  \n\\end{align}\nThen representation \\eqref{K_H-1} with $H$ replaced by $H(t)$ induces a covariance function $R_{V}(s,t)=\\int_0^{t\\wedge s} K_{H(t)}(t,r) \\hspace{1pt}K_{H(s)}(s,r)\\hspace{1pt}dr$ for some Volterra process $V$ on $[0,T]$.   \nThe variance function is given by $R_{V}(t)=t^{2H(t)}$ and is necessarily continuous due to the Sobolev embedding theorem, which says $\\mathcal{S}_{\\eta,2}\\subset C[0,T]$ for all $\\eta>1\/2$ (see e.g.\\ \\cite{Holmander3}).\nTherefore, $H(t)$ is also continuous.   \n\n\nLet $H(t): [0,\\infty) \\rightarrow (1\/2,1)$ be a differentiable function\nwhose restriction to any finite interval $[0,T]$ satisfies the conditions in  \\eqref{condition_star}.   \nFor each $T>0$, let $K_{V^T}(s,t)$ be the kernel inducing the covariance function $R_{V^T}(s,t)$ of the associated Volterra process $V^T$ defined on $[0,T]$ as above.  \nThe definition of $K_{V^T}(s,t)$ is consistent; i.e.\\ $K_{V^{T_1}}(s,t)=K_{V^{T_2}}(s,t)$ for any $0\\le s,\\hspace{1pt} t\\le T_1\\le T_2<\\infty$.  \nHence, so is that of $R_{V^T}(s,t)$, which implies that the function $R_X(s,t)$ given by $R_X(s,t)=R_{V^T}(s,t)$ whenever $0\\le s,\\hspace{1pt} t\\le T<\\infty$ is \na well-defined covariance function of a Gaussian process $X$ on $[0,\\infty)$ whose restriction to each interval $[0,T]$ coincides with $V^T$.\nThe process $X$ represents a fractional Brownian motion with variable Hurst parameter.  \nThe variance function $R_X(t)=t^{2H(t)}$ is differentiable on $(0,\\infty)$ by assumption and Laplace transformable due to the estimate $R_X(t)\\le t^2$.   Therefore, Proposition \\ref{proposition_FP-volterra} and Theorem \\ref{theorem_FP-volterra} can be applied to yield the FPKEs for $X$ and the time-changed process $(X_{E^\\beta_t})$ under the independence assumption between $E^\\beta$ and $X$.\n\n\n\n\n\n\n\\vspace{3.2mm}\n\n\\noindent\n\\textsl{Example \\hypertarget{Application-4}{4}. Fractional Brownian motion with piecewise constant Hurst parameter.}  \nThe fractional Brownian motion discussed in Example \\hyperlink{Application-3}{3} has a continuously varying Hurst parameter $H(t): [0,\\infty) \\rightarrow (1\/2,1)$.  Here we consider a piecewise constant Hurst parameter $H(t): [0,\\infty) \\rightarrow (0,1)$ which is described as\n\\begin{align}\\label{H(t)}\nH(t)=\\sum_{k=0}^N H_k \\textbf{\\textit{I}}_{[T_k,T_{k+1})}(t),\n\\end{align} \nwhere $\\{H_k\\}_{k=0}^N$ are constants in $(0,1)$, $\\{T_k\\}_{k=0}^N$ are fixed times such that $0=T_0<T_1<\\cdots<T_N<T_{N+1}=\\infty$, and $\\textbf{\\textit{I}}_{[T_k,T_{k+1})}$ denotes the indicator function over the interval $[T_k,T_{k+1})$. \n\nFor each $k=0,\\ldots, N$, let $B^{H_k}$ be an $n$-dimensional fractional Brownian motion with Hurst parameter $H_k$.  Let $X$ be the process defined by \n\\begin{align}\nX_t=\\sum_{j=0}^{k-1} (B^{H_j}_{T_{j+1}}-B^{H_j}_{T_j})+(B^{H_k}_t-B^{H_k}_{T_k}) \\ \\ \\ \\textrm{whenever} \\ \\ \\  t\\in[T_k,T_{k+1}).\n\\end{align}  \nThen $X$ is a continuous process representing a fractional Brownian motion which involves finitely many changes of mode of Hurst parameter (described in \\eqref{H(t)}).   \n\nThe transition probabilities of the process $X$ are constructed as follows.  \nFor each $k=0,\\ldots,N$, let $\\theta_k(t)=H_k \\hspace{1pt}t^{2H_k -1}$ for $t\\in[T_k,T_{k+1})$.  \nLet $\\{p_k(t,x)\\}_{k=0}^N$ be a sequence of the unique solutions to the following initial value problems, each defined on $[T_k,T_{k+1})\\times\\bbR^n$:\n\\begin{align}\n\\partial_t \\hspace{1pt} p_0(t,x)&=\\theta_0(t) \\hspace{1pt} \\varDelta \\hspace{1pt} p_0(t,x), \\ t\\in(0,T_1), \\ x\\in\\bbR^n,\\\\ \np_0(0,x)&=\\delta_0(x), \\ x\\in\\bbR^n,\n\\end{align}\nwhere $\\delta_0(x)$ is the Dirac delta function with mass on $0$, and for $k=1,\\ldots, N$,\n\\begin{align}\n\\partial_t \\hspace{1pt} p_k(t,x)&=\\theta_k(t) \\hspace{1pt} \\varDelta \\hspace{1pt} p_k(t,x), \\ t\\in(T_k,T_{k+1}), \\ x\\in\\bbR^n,\\\\\n p_k(T_k,x)&=p_{k-1}(T_k-0,x), \\ x\\in\\bbR^n. \n\\end{align}\nDefine functions $\\theta(t)$ and $p(t,x)$ respectively by $\\theta(t)=\\theta_k(t)$ and  $p(t,x)=p_k(t,x)$ whenever $t\\in[T_k,T_{k+1})$.  \nThen the transition probabilities of $X$ are given by $p(t,x)$ and satisfy\n\\begin{align}\n \\partial_t \\hspace{1pt} p(t,x)&=\\theta(t) \\hspace{1pt} \\varDelta \\hspace{1pt} p(t,x), \\ t\\in{\\textstyle \\bigcup_{k=0}^N}(T_k,T_{k+1}), \\ x\\in\\bbR^n,\\\\ \np(0,x)&=\\delta_0(x), \\ x\\in\\bbR^n.\\\\\np(T_k,x)&=p(T_k-0,x), \\ x\\in\\bbR^n, \\ k=1,\\ldots,N. \n\\end{align}\nDiscussion on existence and uniqueness of the solution to this type of \ninitial value problems\nis found in \\cite{US09}.  \nTo ensure transition probabilities which are continuous in time, the initial value problem associated with the time-changed process $(X_{E^\\beta_t})$ is given by\n\\begin{align}\n D^\\beta_{\\ast,t} \\hspace{1pt} q(t,x)&=\\Theta^\\beta_t \\hspace{1pt} \\varDelta \\hspace{1pt} q(t,x), \\ t\\in(0,\\infty), \\ x\\in\\bbR^n,\\\\ \nq(0,x)&=\\delta_0(x), \\ x\\in\\bbR^n,\\\\\nq(T_k,x)&=q(T_k-0,x), \\ x\\in\\bbR^n, \\ k=1,\\ldots,N,\n\\end{align}\nwhere $\\Theta^\\beta_{t}$ is the operator acting on $t$ defined by $\\Theta^\\beta_{t}=\\sum_{k=0}^N H_k \\hspace{1pt}G^\\beta_{2H_k-1,t}\\hspace{1pt} \\textbf{\\textit{I}}_{[T_k,T_{k+1})}(t)$.  \n\n\n\\begin{remark}\n\\begin{em}\nCombining ideas in Examples \\hyperlink{Application-3}{3} and \\hyperlink{Application-4}{4}, it is possible to construct a fractional Brownian motion having variable Hurst parameter $H(t)\\in(1\/2,1)$ with finitely many changes of mode and to establish the associated FPKEs.\n\\end{em}\n\\end{remark}\n\n\n\n\n\n\\vspace{1mm}\n\n\\noindent\n\\textsl{Example \\hypertarget{Application-5}{5}. Ornstein-Uhlenbeck process.}\nConsider the one-dimensional Ornstein-Uhlenbeck process $Y$ given by \n\\begin{align} \\label{example_OU-1}\nY_t=y_0\\hspace{1pt}e^{-\\alpha t}+\\sigma \\int_0^t e^{-\\alpha(t-s)}\\hspace{1pt}dB_s, \\ t\\ge 0,\n\\end{align}\nwhere $\\alpha\\ge 0$, $\\sigma>0$, $y_0\\in\\bbR$ are constants and $B$ is a standard Brownian motion.  \nIf $\\alpha=0$, then $Y_t=y_0+\\sigma B_t$, a Brownian motion multiplied by $\\sigma$ starting at $y_0$.\nSuppose $\\alpha>0$.  The process $Y$ defined by \\eqref{example_OU-1} is the unique strong solution to the inhomogeneous linear SDE\n   \\begin{align}\n          dY_t=-\\alpha Y_tdt +\\sigma dB_t \\ \\ \\textrm{with} \\ \\ Y_0=y_0,  \\label{SDE81-1}\n     \\end{align}\nwhich is associated with the SDE \n   \\begin{align}\n          d\\bar{Y}_t=-\\alpha \\bar{Y}_tdE^\\beta_t +\\sigma dB_{E^\\beta_t} \\ \\ \\textrm{with} \\ \\ \\bar{Y}_0=y_0,  \\label{SDE81-2}\n     \\end{align}\nvia the dual relationships $\\bar{Y}_t=Y_{E^\\beta_t}$ and $Y_t=\\bar{Y}_{W^\\beta_t}$; \nsee \\cite{Kobayashi} for details.\n\nConsider the zero-mean process $X$ defined by\n\\begin{align}\n   X_t=Y_t-y_0\\hspace{1pt}e^{-\\alpha t}=\\sigma \\int_0^t e^{-\\alpha(t-s)}\\hspace{1pt}dB_s.\n\\end{align} \n$X$ is a Gaussian process since each random variable $X_t$ is a linear transformation of the It\\^o stochastic integral of the deterministic integrand $e^{\\alpha s}$.\nDirect calculation yields \n$R_X(t)=\\frac{\\sigma^2}{2\\alpha}(1-e^{-2\\alpha t})$ and\n$\\widetilde{R'_{X}}(s)=\\frac{\\sigma^2}{s+2\\alpha}$.  \nTherefore, due to Proposition \\ref{proposition_FP-volterra} and Theorem \\ref{theorem_FP-volterra}, the initial value problems associated with $X$ and $(X_{E^\\beta_t})$, where $E^\\beta$ is independent of $X$, are respectively given by \n\\begin{align}\\label{example_OU-2}\n   \\partial_t \\hspace{1pt}p(t,x)&=\\frac{\\sigma^2}{2}\\hspace{1pt} e^{-2\\alpha t}\\hspace{1pt}\\partial_x^2 \\hspace{1pt}p(t,x), \\ p(0,x)=\\delta_{0}(x);\\\\\n\\label{example_OU-3}   D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)&=\\frac{\\sigma^2 \\beta}{2} \\hspace{1pt}J^{1-\\beta}_t\n\\mathcal{L}^{-1}_{s \\to t}\\biggl[\\frac{1}{2\\pi i} \\int_{\\bm{\\mathcal{C}}}\n\\frac{\\partial_x^2\\hspace{1pt}\\tilde{q}(z,x)}{s^\\beta-z^\\beta+2\\alpha}\\; dz \\biggr](t), \\ q(0,x)=\\delta_{0}(x).\n\\end{align}\nThe unique representation of the solution to the initial value problem \\eqref{example_OU-2} is obtained via the usual technique using the Fourier transform.  Moreover, expression \\eqref{relation} guarantees uniqueness of the solution to \\eqref{example_OU-3} as well.\n\n\n\nNotice that the two processes $X$ and $(X_{E^\\beta_t})$ are unique strong solutions to SDEs \\eqref{SDE81-1} and \\eqref{SDE81-2} with $y_0=0$, respectively.  Therefore, it is also possible to apply Theorem 4.1 of \\cite{HKU} to obtain the following forms of initial value problems which are understood in the sense of generalized functions: \n\\begin{align}\\label{example_OU-4}\n   \\partial_t \\hspace{1pt}p(t,x)&=\\alpha \\hspace{1pt}\\partial_x \\bigl\\{x p(t,x) \\bigr\\} + \\frac{\\sigma^2}{2}\\hspace{1pt} \\partial_x^2 \\hspace{1pt}p(t,x), \\ p(0,x)=\\delta_{0}(x);\\\\\n\\label{example_OU-5}   D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)&=\\alpha \\hspace{1pt}\\partial_x \\bigl\\{x q(t,x) \\bigr\\} + \\frac{\\sigma^2}{2}\\hspace{1pt} \\partial_x^2 \\hspace{1pt}q(t,x), \\ q(0,x)=\\delta_{0}(x).\n\\end{align}\nActually these FPKEs hold in the strong sense as well.  \nFor uniqueness of solutions to \\eqref{example_OU-4} and \\eqref{example_OU-5}, see e.g.\\ \\cite{Friedman} and Corollary 3.2 of \\cite{HKU}.\n\n\nThe above discussion yields the following two sets of equivalent initial value problems: \\eqref{example_OU-2} and \\eqref{example_OU-4}, and \\eqref{example_OU-3} and \\eqref{example_OU-5}.\nAt first glance, PDE \\eqref{example_OU-2} might seem simpler or computationally more tractable than PDE \\eqref{example_OU-4}; however, PDE \\eqref{example_OU-3} which is associated with the time-changed process has a more complicated form than PDE \\eqref{example_OU-5}.    \nA significant difference between PDEs \\eqref{example_OU-2} and \\eqref{example_OU-4} is the fact that the right-hand side of \\eqref{example_OU-4} can be expressed as $A^\\ast p(t,x)$ with the \\textit{spatial} operator $A^\\ast=\\alpha \\hspace{1pt}\\partial_x x + \\frac{\\sigma^2}{2}\\hspace{1pt} \\partial_x^2$ whereas the right-hand side of \\eqref{example_OU-2} involves both the spatial operator $\\partial_x^2$ and the \\textit{time-dependent} multiplication operator by $e^{-2\\alpha t}$.  This observation suggests: 1) establishing FPKEs for time-changed processes via several different forms of FPKEs for the corresponding untime-changed processes, and 2) choosing appropriate forms for handling specific problems.  \n\n\n\n\n\\begin{remark}\n\\begin{em}\nExample \\hyperlink{Application-5}{5} treated a Gaussian process given by the It\\^o integral of the deterministic integrand $e^{-\\alpha(t-s)}$.  \nMalliavin calculus, which is valid for an arbitrary Gaussian integrator, can be regarded as an extension of It\\^o integration (see \\cite{Janson,Malliavin,Nualart}). It is known that Malliavin-type stochastic integrals of deterministic integrands are again Gaussian processes.  Therefore, if the variance function of such a stochastic integral satisfies the technical conditions specified in Theorem \\ref{theorem_FP-volterra}, then the FPKE for the time-changed stochastic integral is explicitly given by \n\\eqref{FP-volterra3-2}, or equivalently, \\eqref{FP-volterra3-1}.  \n\\end{em}\n\\end{remark}\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the most important observable predictions for star formation theory is the resulting spectrum of masses, known as the initial mass function (IMF). The extreme ends of this distribution are of particular interest for constraining star formation physics. At the lowest masses, brown dwarfs (BDs) are thought to be approximately as numerous in the Milky Way as hydrogen-burning stars \\citep{Cha03}. However, what determines the IMF, and whether it varies with environment, remain the topic of debate \\citep[see][for a review]{Kru14}.  \n\n\nBDs are usually defined to be objects less massive than required to burn ordinary hydrogen ($\\lesssim 0.08\\, M_\\odot$) and greater than the deuterium-burning limit ($\\gtrsim 13\\, M_\\mathrm{J}$). If the IMF in this regime is set by the local thermal Jeans mass, then the lower mass limit for fragmentation is set by the requirement that the centre of the clump can cool \\citep[i.e. the opacity limit --][]{Low76}. In this case, as the metallicity increases, the cooling becomes more efficient and the minimum mass for BDs decreases \\citep{Bat05}. However, dynamical processes and competitive accretion might further influence the distribution of stellar masses \\citep[e.g.][]{Bon97, Kle98,Bon08, Bat12}. Dynamical interactions can eject stars from their formation environment, shutting off accretion and stunting growth before the hydrogen mass burning limit is reached \\citep{Rei01, Bat02b}. Alternatively, the photoevaporation of the gas in an accreting envelope due to irradiation by neighbouring OB stars can have a similar effect \\citep{Hes96, Whi04}.\n\nThe low-mass IMF appears to have a similar shape across numerous local star forming regions \\citep{Andersen08}. However, a number of regions also appear to exhibit deviations from a `universal' IMF. \\citet{Scholz13} and \\citet{Luhman16} find that the cluster NGC 1333 has a greater fraction of sub-stellar objects than IC348, which is interpreted as evidence of enhanced low mass star formation in dense environments. For the most massive and dense local star forming region, the Orion Nebula cluster (ONC), contradictory results have been inferred by different authors. \\citet{DaRio12} find a deficiency of BDs, in direct contrast to a previously inferred enhancement \\citep{Muench02}. Such results might be reconciled by a {bimodal} distribution, as found by \\citet{Drass16}. In this case, the enhancement in occurrence rates at the lowest BD masses may be the result of a distinct formation mechanism, possibly within a protoplanetary disc. \n\nThe distinction between BDs and planets, while motivated by a physical threshold, may not cleanly delineate formation mechanisms \\citep{Chabrier14}. In particular, planets with masses greater than $\\sim 4\\, M_\\mathrm{J}$ may form by gravitational instability rather than core accretion \\citep{Schlaufman18}. Interestingly, the masses of `planets' thought to be responsible for the gaps observed in the dust emission in protoplanetary discs with ALMA \\citep[e.g.][]{HLTau_15, Muller18} can be several Jupiter masses, approaching or exceeding the canonical minimum BD mass \\citep[e.g.][]{Haffert19, Christaens19}. The questions of BD and massive planet occurrence rates may therefore be related. \n\nGlobular clusters are massive populations of stars that remain bound against galactic tides despite their old ages. Having low metallicities and high densities, they represent the present day remnant of formation in a completely different environment to local star forming regions. As such, if the formation of stars or planets is in any way dependent on environment, one should expect to see differences in globular cluster populations with respect to those in the field. \\citet[][hereafter \\citetalias{Gil00}]{Gil00} presented a HST survey of the globular cluster 47 Tuc in search of transiting giant planets. The null result put upper-limits on the frequency of hot Jupiters (with orbits $\\lesssim 10$~days), suggesting a frequency at least an order of magnitude lower than the solar neighbourhood average \\citep[$\\sim 1$~percent --][]{Wright12}. \n\n\nThe potential significance of the dearth of low mass companions of stars in 47 Tuc compounds when one considers the potential for tidal capture in such a dense stellar environment. Tidal capture for close binary formation in globular clusters was suggested by \\citet{Fab75} as a means to explain their high observed X-ray luminosity-to-mass ratios. {This two-body capture mechanism has since been incorporated into Monte Carlo simulations of globular clusters alongside three-body gravitational interactions between `point masses' \\citep{Stodolkiewicz85, Stodolkiewicz86}.} \\citet[][hereafter \\citetalias{Bon03}]{Bon03} pointed out that the same principle also applies to the capture of BDs. This means that if the fraction of BDs that formed in globular clusters is similar to the galactic field, then a significant number of close BD binaries should also exist. Thus, the absence of detected transits in 47 Tuc not only suggests a reduced occurrence rate of hot Jupiters, but also a dearth of BDs. \n\n{In this mini-series of two papers, we explore the dearth of short period sub-stellar companions in 47 Tuc in terms of the expected rates of both BDs (this paper) and hot Jupiters (a second paper). To this end we apply the \\textsc{Mocca} Monte Carlo code \\citep{Hyp13, Giersz13} to accurately compute the theoretical capture and scattering rates over the lifetime of 47 Tuc. In this, the first of the two papers, we focus on the possibility of tidal BD capture in 47 Tuc while a second paper w ill deal with the migration of massive planets to short period orbits. We first discuss the theory of tidal capture in Section~\\ref{sec:tidal_capt_calcs}. We then introduce our Monte Carlo model for 47 Tuc in Section~\\ref{sec:Num_Method}. In Section~\\ref{sec:results} we present the resultant BD capture rates over the lifetime of 47 Tuc, compare to the observational constraints and discuss future observations and generalisation to other globular clusters. Conclusions are summarised in Section~\\ref{sec:conclusions}.  }\n\n\n\n\n\n\n\\section{Tidal capture theory}\n\\label{sec:tidal_capt_calcs}\n\\subsection{Tidal capture cross section}\n\\label{sec:tid_cross}\nThe tidal capture condition for a star and BD pair is discussed in detail by \\citetalias{Bon03}; we briefly review the relevant equations here. During the close passage between stars, the orbital energy can be dissipated by non-radial oscillations within the stellar interiors \\citep{Rob68}. It follows that if the passage is sufficiently close, the tidal dissipation can result in capture and the formation of a tight binary. \\citet{Fab75} applied this mechanism to explain low mass X-ray binaries found in globular clusters. We consider a primary star of radius $R_*$, mass $m_*$. Then for a periastron distance $a_\\mathrm{p}$ during an encounter with a much smaller (point) mass $m$, the condition for capture can be approximated \\citep{Fab75}:\n\\begin{equation}\n\\label{eq:capt_rad}\n \\frac{a_\\mathrm{p}}{R_*} < \\frac{a_\\mathrm{capt}}{R_*} = \\left[ \\frac{Gm_*}{R_* v_\\infty^2} q (1+q) \\right]^{1\/6}\n\\end{equation} where $q=m\/m_*$, for secondary mass $m$, and $v_\\infty$ is the relative velocities of the two stars at infinite separation, $a_\\mathrm{capt}$ is the capture radius. The point mass approximation is justified since tides can only be excited in a much smaller secondary if the impact parameter is within the collisional cross section. Equation~\\ref{eq:capt_rad} remains valid for low mass main-sequence primaries \\citep[$n=3\/2$ polytropes,][]{Lee86}. \n\nThe capture radius must also exceed the sum of the radii of the two interacting bodies, otherwise the objects would collide (i.e. periastron distance $a_\\mathrm{p}> R_*+R_\\mathrm{bd}$, where $R_\\mathrm{bd}$ is the BD radius). We will hereafter assume that all BDs have $R_\\mathrm{bd}=0.1 \\, R_\\odot$. For a star and BD pair the capture cross section is:\n\\begin{equation}\n\\label{eq:sigma_capt}\n    \\sigma_\\mathrm{capt}=\\begin{cases}\n    \\sigma_\\mathrm{capt}' - \\sigma_\\mathrm{coll}\\, & \\sigma_\\mathrm{capt}' >\\sigma_\\mathrm{coll}\\\\\n    0   \\, &\\rm{otherwise}\n    \\end{cases},\n\\end{equation} where $\\sigma_\\mathrm{coll}$ is the collisional cross section (including gravitational focusing) and \n\\begin{equation}\n\\label{eq:sigmacapt_dash}\n\\sigma_\\mathrm{capt}' = \\pi a_\\mathrm{capt}^2 \\left[ 1 + \\frac{2Gm_*(1+q)}{v_\\infty^2 a_\\mathrm{capt}}\\right]\n\\end{equation} is the capture cross section if collisions are ignored. \n\nWithin the \\textsc{Mocca} framework, we implement the tidal capture scenario in the same way as stellar collisions \\citep{Freitag02}. In brief, this involves looping over all stars within a local subset \\citep[`zone' -- see][]{Gie98}, and finding a corresponding BD pair at random. {We compute the local number density $n_\\mathrm{bd}$ of BDs in the same way as stars in the \\textsc{Mocca} framework. In brief, this involves finding a number of the closest objects in cluster radius ($r$) space and normalising by the minimum spherical shell volume that encloses them.} The probability of capture between the pair is:\n\\begin{equation}\n    P_{\\mathrm{capt}} = n_\\mathrm{bd} \\sigma_\\mathrm{capt} v_\\infty \\Delta t, \n\\end{equation} for time-step $\\Delta t$. In this way, due to the normalisation by the BD density, it is only necessary to loop over all stars and not both stars and BDs. In the case of capture, a star-BD binary is produced with a circular orbit (assuming a  short circularisation time-scale) and semi-major axis $a_\\mathrm{bd} = 2 a_\\mathrm{p}$ \\citep{Mardling96}. {To determine $a_\\mathrm{p}$ for a given encounter, we first draw the encounter cross section uniformly between $\\sigma_\\mathrm{coll}$ and $\\sigma'_\\mathrm{capt}$, then assign the corresponding $a_\\mathrm{p}$. Apart from for tidal capture scenarios, BDs are treated as main sequence stars in \\textsc{Mocca}, with associated collision probabilities and dynamical interactions.}\n\n\n\\subsection{Capture rates}\n\\label{sec:theory_rates}\n\n{As discussed in Section~\\ref{sec:post-process}, it is useful to not only compute the cross section and capture probability for BD-star pairs in the Monte Carlo simulation but also post-process the encounter probability for individual stars. This allows us to compute the scaling of the capture probability over the physical parameter space, where the Monte Carlo may only yield a small number of encounters that result in large uncertainties. Therefore we compute the encounter rates as a function of stellar and environmental parameters in this section. We make an analytic estimate assuming no collisions to give an intuition as to how the capture rate scales (Section~\\ref{sec:analytic_approx}). We then compute the capture rates with star-BD collisions included, demonstrating how these collisions reduce the capture rates (Section~\\ref{sec:full_calc}).  } \n\n\\begin{figure}\n    \\centering\n   \\subfloat[\\label{subfig:sigvar}Encounter cross sections with relative speed]{\\includegraphics[width= 0.8\\columnwidth]{sigcolcapt.pdf}}\\\\\\vspace{-10pt}\n   \\subfloat[\\label{subfig:dgcdv}Differential capture rate]{\\includegraphics[width= 0.8\\columnwidth]{dgcaptdv.pdf}}\\\\ \\vspace{-10pt}\n        \\subfloat[\\label{subfig:Gamma_capt}Overall per star capture rates ]{\\includegraphics[width= 0.8\\columnwidth]{gamma_capt.pdf}}\\\\\\vspace{-3pt}\n    \\caption{Computations of the key quantities in determining the tidal capture rates of BDs in a given stellar environment. Figure~\\ref{subfig:sigvar} shows the effective cross sections of collision (dotted lines) and capture (solid lines) for varying relative speeds at infinite separation. Figure~\\ref{subfig:dgcdv} shows the corresponding differential capture rates (equation~\\ref{eq:Gamma_capt}) for different velocity dispersions. Figure~\\ref{subfig:Gamma_capt} shows the integrals of this differential across all velocities (solid lines) compared to the approximation in equation~\\ref{eq:approx_captrate} (dotted lines). In all cases, the lines are coloured by the assumed stellar properties. Results in Figures~\\ref{subfig:dgcdv} and~\\ref{subfig:Gamma_capt} are shown for BD number density $n_\\mathrm{bd}\/10^6$~pc$^{-3} = n_{\\mathrm{bd},6} = 1$ and scale linearly with this value. }\n    \\label{fig:Gammacapt_sigv}\n\\end{figure}\n\n\\subsubsection{Estimated encounter rate by \\citetalias{Bon03}}\n\n {Given a local velocity dispersion and BD density, the instantaneous tidal capture rate for an individual star can be estimated. This rate $\\Gamma_\\mathrm{capt}$ is the rate at which neighbours pass within the effective cross section for capture, equation~\\ref{eq:sigma_capt}. In general, if all objects have mass $m$, velocity dispersion $\\sigma_v$ and density $n$ then the encounter rate within a given radius $a_\\mathrm{enc}$ can be written \\citep{Binney08}:}\n \\begin{equation}\n     \\Gamma_\\mathrm{enc} = 16\\sqrt{\\pi} \\cdot n \\sigma_v a_\\mathrm{enc}^2 \\cdot \\left( 1+\\frac{G m}{2\\sigma_v^2 a_\\mathrm{enc}}\\right).\n \\end{equation}{In the gravitationally focused regime, the second term in brackets dominates and $\\Gamma_\\mathrm{enc} \\propto a_\\mathrm{enc}\/\\sigma_v$. If $a_\\mathrm{enc} = a_\\mathrm{capt}- (R_*+R_{\\mathrm{bd}})$ is not strongly dependent on $v_\\infty$, then for BD density $n_\\mathrm{bd}$ the BD capture rate for a star of mass $m_*$ is:\n\\begin{multline}\n\\label{eq:Gammacapt_Bonnell}\n    \\Gamma_\\mathrm{capt}^\\mathrm{BCB+03} \\approx 1.4 \\times 10^{-4} \\, \\left( \\frac{n_\\mathrm{bd}}{10^6\\, \\rm{pc}^{-3}}\\right) \\times \\\\ \n    \\times \\left( \\frac{\\sigma_v}{10\\, \\rm{km\\, s}^{-1}}\\right)^{-1} \\frac{a_\\mathrm{enc}}{1\\, R_\\odot} \\frac{m_*}{1\\, M_\\odot} \\,\\rm{Myr},\n\\end{multline}where \\citetalias[][]{Bon03} apply this expression with fixed $v_\\infty = \\sigma_v = 10$~km\/s. Adopting equation~\\ref{eq:Gammacapt_Bonnell} and assuming $v_\\infty = \\sigma_v$ to obtain $a_\\mathrm{enc}$ is accurate when the $\\sigma_v$ is sufficiently small  -- i.e. when $\\sigma_\\mathrm{capt}' \\gg \\sigma_\\mathrm{coll}$ for $v_\\infty \\approx \\sigma_v$. However, both $\\sigma'_\\mathrm{capt}$ and $\\sigma_\\mathrm{coll}$ are dependent on the relative velocities of the pairs, stellar mass and radius in different ways, such that understanding the scaling of $\\sigma_\\mathrm{capt}$ on these properties is non-trivial when $\\sigma_\\mathrm{capt}' \\approx \\sigma_\\mathrm{coll}$ for $v_\\infty \\approx \\sigma_v$. It is therefore unclear what kind of encounters dominate the overall capture rate in general and we will find that $\\sigma_v$ significantly exceeds $10$~km\/s in our dynamical model for 47 Tuc. For this reason, we must generally integrate over the full differential capture rate, including the capture cross section. }\n\n\n\\subsubsection{Analytic approximation without collisions}\n\\label{sec:analytic_approx}\nThe differential capture rate as a function of the velocity at infinity $v_\\infty$ is:\n\\begin{equation}\n\\label{eq:Gamma_capt}\n   \\mathrm{d} \\Gamma_\\mathrm{capt} = \\sigma_\\mathrm{capt}(v_\\infty; q, R_*) \\,n_{\\rm{bd}} \\,  v_\\infty g(v_\\infty; \\sigma_v) \\,\\rm{d}v_\\infty  \n\\end{equation}where\n\\begin{equation}\n    g(v_\\infty; \\sigma_v) = \\frac{v_\\infty^2}{2\\sqrt{\\pi} \\sigma_v^3} \\exp \\left(\\frac{-v_\\infty^2}{4\\sigma_v^2} \\right)\n\\end{equation} is the Maxwell-Boltzmann distribution, or the relative asymptotic speed $v_\\infty$ distribution for dispersion $\\sigma_v$. In the limit $\\sigma_\\mathrm{capt}\\gg \\sigma_\\mathrm{coll}$ (small $v_\\infty$, large $m_*$) for the dominant capture scenarios, equation~\\ref{eq:Gamma_capt} can be integrated over all velocities to yield the analytic upper-limit to the capture rate:\n\\begin{multline}\n\\label{eq:approx_captrate}\n    \\Gamma_\\mathrm{capt} < 3.7 \\times 10^{-7}  \\left[\\left(\\frac{R_*}{1\\,R_\\odot}\\right)^5 \\frac{m_*}{1\\,M_\\odot}  \\frac{\\sigma_v}{10\\,\\rm{km \\, s}^{-1} }q (1+q) \\right]^{1\/3}\\times \\\\\n    \\times \\left(1+ \\phi_\\mathrm{grav} \\right) \\cdot \\left( \\frac{n_{\\rm{bd}}}{10^6\\, \\rm{pc}^{-3}}\\right) \\, \\rm{Myr}^{-1},\n\\end{multline} where \n\\begin{align}\n\\nonumber\n    \\phi_{\\rm{grav}} &\\approx 423\\, q^{-1\/6}(1+q)^{5\/6} \\quad\\times \\\\ &\\qquad\\times \\left(\\frac{m_*}{1\\,M_\\odot}\\right)^{5\/6} \\left(\\frac{R_*}{1\\,R_\\odot}\\right)^{-5\/6} \\left(\\frac{\\sigma_v}{10\\, \\rm{km\\,s}^{-1}} \\right)^{-5\/3}\n\\end{align} is the gravitational focusing factor. While these expressions are approximate, they highlight three things:\n\\begin{enumerate}\n    \\item The relevant encounters are in practice always gravitationally focused ($\\phi_\\mathrm{grav}\\gg1$), independently of the type of star under consideration, where we assume $R_*\\propto m_*^\\beta$ with $\\beta\\sim 1$. The velocity dispersion required for $\\phi_\\mathrm{grav}\\sim 1$ is $\\sigma_v \\sim 100$~km~s$^{-1}$. \n    \\item The capture rate scales steeply with the mass (radius) of the star: $\\Gamma_\\mathrm{capt} \\propto q^{1\/6} R_*^{5\/6} m_*^{7\/6} \\propto m_*^{1 +5\\beta\/6}$. This steep scaling highlights the importance of the mass function in assessing the total number of encounters. In particular, for mass function $\\xi$ we have $\\Gamma_\\mathrm{capt}\\cdot \\xi \\,\\mathrm{d}m_* \\propto m_*^{-\\gamma}\\,\\mathrm{d}m_*$ for $\\gamma \\lesssim 1$, such that the integral diverges for large $m_*$. The overall capture rate is therefore initially dependent on the choice of maximum stellar mass $m_\\mathrm{max}$.\n    \\item The capture time-scale ($\\propto 1\/\\Gamma_\\mathrm{capt}$) scales super-linearly with the velocity dispersion ($\\Gamma_\\mathrm{capt} \\propto \\sigma_v^{-4\/3}$). The temporal and spatial evolution of the local velocity dispersion within a globular cluster is therefore an important factor in determining capture frequency.\n\\end{enumerate}\n\n\n\\subsubsection{Full calculation with collisions}\n\\label{sec:full_calc}\n\nThe stellar mass and velocity dispersion become even  more important  when one numerically integrates the full expression for equation~\\ref{eq:Gamma_capt}. {We show the relevant quantities in computing the capture rate in Figure~\\ref{fig:Gammacapt_sigv}. The comparison of the cross sections for capture and collisions are in Figure~\\ref{subfig:sigvar}. The collision cross section exceeds the cross section for capture above some relative speed $v_\\infty$, which increases with stellar mass. The effect of this on the differential encounter rate for given velocity dispersion $\\sigma_v$ is shown in Figure~\\ref{subfig:dgcdv}, the form of which is non-trivially dependent on the stellar properties. }\n\nThe analytic approximation in the limit $\\sigma_\\mathrm{coll} \\ll \\sigma_\\mathrm{capt}$ (equation~\\ref{eq:approx_captrate}) is compared with this numerical integration in Figure~\\ref{subfig:Gamma_capt} as a function of the velocity dispersion. As $\\sigma_v$ becomes large, from equation~\\ref{eq:capt_rad} we have that $a_\\mathrm{capt}\/R_*$ becomes small for all possible encounters, such that neglecting collisions is not possible. The result is a (stellar mass dependent) steep decline in the capture rate with $\\sigma_v$, much steeper than the analytic approximation of $\\Gamma_\\mathrm{capt}\\propto \\sigma_v^{-4\/3}$. Since the approximation is only valid for smaller $\\sigma_v$ than is typical for globular clusters, we will hereafter always adopt the full numerical integration when estimating capture rates. \n\nThe deviation from our approximation particularly affects the lower mass stars, for which gravitational focusing becomes weaker at a lower velocity dispersion relative to higher mass stars. The decrease in capture efficiency for low mass stars is precipitous for $m_*\\lesssim 0.5\\,M_\\odot$, such that the stellar mass function must be considered in order to compute global capture rates. We further consider how the evolving mass function alters the overall capture rates in Appendix~\\ref{app:massfunc}, where we justify adopting $m_* = 0.7\\,M_\\odot$ to approximate the global capture rate over the lifetime of 47 Tuc. \n\n\n\\section{Dynamical modelling}\n\n\\label{sec:Num_Method}\n\n\n\\subsection{Monte Carlo simulations}\n\nSimulating the dynamical evolution of globular clusters directly using $N$-body calculations over their $\\sim 10$~Gyr evolution is not computationally practicable. For this reason, the \\textsc{Mocca} code \\citep{Gie98, Gie01} has been developed as a {Monte Carlo} approach to statistically computing the evolution of massive, dense stellar clusters by solving the Fokker--Planck equation \\citep[see also][]{Sto82}. This approach allows a fast and accurate calculation of the dynamical evolution of the stellar population in a globular cluster over its lifetime. The added bonus of using this code is that it has already been applied to model the evolution of 47 Tuc \\citep[][hereafter \\citetalias{Gie11}]{Gie11}. We are therefore able to adopt the parameters obtained in this previous modelling effort. Where appropriate, we make similar comparisons to observational constraints as \\citetalias{Gie11}. \\textsc{Mocca} has since been updated to incorporate the \\textsc{Fewbody} code \\citep{Fregeau04, Fregeau12} into an improved prescription for interactions between multiple systems, as described by \\citet{Hyp13}. Stellar evolution modules by \\citet{Hurley00, Hurley02} are used to compute single star and binary evolution. We do not include tidal forces between a star and a companion in this evolution.  \n\n\\subsection{Initial conditions for 47 Tuc}\n\n\\label{sec:ICs}\n\\begin{figure*}\n \\subfloat[\\label{subfig:surfmag}Surface brightness]{\\includegraphics[width= 0.47\\textwidth]{surfbright.pdf}}\n \\subfloat[\\label{subfig:los_vdisp}Line of sight velocity dispersion]{\\includegraphics[width= 0.47\\textwidth]{los_vdisp.pdf}}\n       \n     \\caption{Observational constraints on the dynamical properties of 47 Tuc compared to the results of our \\textsc{Mocca} (Monte Carlo) model with the parameters listed in Table~\\ref{table:dyn_mods}. Figure~\\ref{subfig:surfmag} shows the visible surface brightness profile of the Monte Carlo model (red circles) compared to the observed profile found by \\citet[][black crosses]{Tra95}. The line-of-sight velocity dispersion is compared in Figure~\\ref{subfig:los_vdisp}, where the Monte Carlo results are again red circles. The relative line of sight velocities measured by \\citet{Gebhardt95} are shown by faint points and the inferred dispersion shown as a black line. The sampling uncertainties are indicated by dashed lines.  }\n     \\label{fig:47Tuc_obs}\n\\end{figure*}\n\n\\begin{table}\n\\centering \n \\begin{tabular}{c c c } \n \\hline\n Parameter & Property & Value     \\\\ [0.5ex] \n \\hline\n$N_*$ & Number of stars &  $2 \\times 10^6$ \\\\\n$N_\\mathrm{bd}$& Number of  BDs  &  $2\\times 10^6$ \\\\\n$N_\\mathrm{bin}$ & Number of binaries &  $4.4\\times10^4$ \\\\ \n$N_\\mathrm{pl}$ & Number of planets &  $2 \\times10^4$ \\\\ \n$W_0$ & Central concentration & $7.5$ \\\\\n$m_\\mathrm{br}\/M_\\odot$ & Break mass & 0.8 \\\\\n$m_\\mathrm{max}\/M_\\odot$ &  Max. mass  & 50\\\\\n$\\alpha_1$ &  IMF slope $m<m_\\mathrm{br}$ & 0.4 \\\\\n$\\alpha_2$ &  IMF slope $m>m_\\mathrm{br}$ & 2.8\\\\\n$T_\\mathrm{age}\/$Gyr & Age & $12$ \\\\\n$Z\/Z_\\odot$ & Metallicity (dex) & -0.6 \\\\\n$e_0$ & Planet eccentricity &  0.9 \\\\ \n$a_0\/$au & Planet semi-major axis &  $5$\\\\ \n \\hline\n\\end{tabular}\n\\caption{Initial condition parameters used for the Monte Carlo globular cluster model discussed in the text. Where appropriate, choices are made to match the model of \\citetalias{Gie11} for 47 Tuc. BDs and planets are the same except that planets are initially companions to stars.}\n\\label{table:dyn_mods}\n\\end{table}\n\nThe initial conditions that we adopt are motivated by the findings of \\citetalias{Gie11}, who reproduced the key observable properties of 47 Tuc. The main parameters are summarised in Table~\\ref{table:dyn_mods}. The binary population are drawn from a log-uniform distribution between $1{-}100$~au. {The initial conditions include a number of choices that are somewhat artificial (such as the low maximum stellar mass, small binary fraction and no mass fallback for black hole formation). These choices were invoked to reproduce the unusual surface brightness and velocity dispersion profiles. A low binary fraction is also convenient in our context, because we compute tidal capture rates only between BDs and single stars. More recent models incorporating a higher binary fraction and maximum stellar mass can reproduce the frequency of special objects (e.g. black holes binaries and pulsars) and central surface brightness, but do not presently reproduce the observed surface brightness profile (A. Askar -- private communication). The most important property for computing the BD capture rate is the stellar density profile. We therefore retain the parameters of \\citetalias{Gie11} with which we find good agreement with the observed density and velocity dispersion profiles (see Section~\\ref{sec:47Tuc_obs_comp}). Provisional checks using the unpublished alternative models yield BD capture rates similar to those we obtain with our fiducial model. However, we emphasise initial conditions that can reproduce the properties of globular clusters are degenerate. Although models that yield a similar density profile probably yield similar capture rates (see discussion of caveats in Section~\\ref{sec:caveats}), an extensive parameter study investigating these choices is outside of the scope of this work.} \n\n\n\nWe additionally include a BD population, with equal numbers as the stars. The masses of the BDs have initial masses $m_\\mathrm{bd} = 0.079\\, M_\\odot$, to ensure that their masses are lower than that of the least massive stars. In the case that an object has a mass that exceeds $0.08 \\, M_\\odot$ (for example, via a collision\/merger), then the object is no longer defined as a BD. The initial spatial distribution of BDs is assumed to be the same as the stellar population (i.e. no primordial mass segregation).\n\nFinally, we add a population of `migrating planets' around 1 percent of the initial stellar population. {The orbital evolution of this population will be considered in Paper II, but are not relevant in this work.} We adopt the same mass as the BDs for simplicity. While this results in a greater mass ratio $q$ than for planets, the scattering cross section for binaries is only weakly dependent on the mass ratio, particularly for $q\\lesssim 0.1$ \\citep{Fregeau04}. We initialise all of the planet orbits to have eccentricity $e=0.9$ and semi-major axes $a = 5$~au, reflecting a Jupiter analogue with high eccentricity. Planets are paired with stars drawn from the same IMF as single stars. {Because the companion (planet) population is a small fraction of the stellar population, as well as low mass and with small semi-major axis compared to the binaries, this population has a negligible influence on the overall evolution of the \\textsc{Mocca} simulation (confirmed by performing runs without them).  We do not include these systems in the tidal capture Monte Carlo routine, although this too has a negligible effect on the overall capture rate due to the low companion fraction. They are treated as binaries within the \\textsc{Mocca} framework, undergoing single-binary and binary-binary interactions integrated with \\textsc{Fewbody}. We will only refer to this population again in Paper II. }\n\n\\subsection{Comparison with observed properties of 47 Tuc}\n\\label{sec:47Tuc_obs_comp}\nWe wish to ensure that the model approximately reproduces the key physical properties of 47 Tuc at its present age. These are the stellar density and velocity dispersion profile. \\citetalias{Gie11} fitted their models to the surface brightness profile as measured by \\citet{Tra95} and the line-of-sight velocity dispersion inferred from the measurements of \\citet{Gebhardt95}. For direct comparison, we perform the same comparisons for our model. \n\nThe $V$-band surface magnitude profile after integrating the model for 12 Gyr is shown in Figure~\\ref{subfig:surfmag}, adopting a distance of $4$~kpc. The profile is calculated by averaging over the $V$-band luminosity contribution of concentric shells of stars for distance $d$ from the cluster centre:\n\\begin{equation}\n\\label{eq:Vband_surf}\n   \\Sigma_V(d) =\\sum_{r_i>d}\\frac{L_V}{2\\pi r_i^2} \\frac{r_i}{\\sqrt{r_i^2-d^2}}.\n   \\end{equation}\n   We then convert this to a surface magnitude by the expression:\n\\begin{equation}\n    \\mu_V = V_\\odot -2.5\\log \\Sigma_{V}' + A_V,\n\\end{equation}where $V_\\odot = 4.80$ is the solar $V$-band magnitude and $\\Sigma_{V}' $ is $\\Sigma_{V} $ in units of solar $V$-band luminosity per square arcsecond. {We assume small visual extinction $A_V$. Commonly, the \\citet{Harris96} value of $E(B-V) = 0.04$ is used \\citep[although smaller value $E(B-V)=0.024$ may also be adopted --][]{Crawford75}, to give $A_V \\approx 0.12$ for relative visibility $R_V = 3.1$.} Despite the updated version of \\textsc{Mocca} and the inclusion of BDs compared to \\citetalias{Gie11}, we find reasonable agreement between the model and the observed profile obtained by \\citet{Tra95} similarly to \\citetalias{Gie11}. \n\nThe line-of-sight velocity dispersion profile can be computed in an analogous fashion to the surface brightness and is shown in Figure~\\ref{subfig:los_vdisp}. The dispersion at projected separation $d$ from the centre is the contribution of the projected contributions of the radial and tangential velocities ($v_r$ and $v_{\\rm{t}}$ respectively):\n\\begin{equation}\n    \\sigma_{v,\\rm{los}}^2(d)=   \\frac{1}{n_{d}}\\sum_{r_i>d} \\frac{d}{r_i\\sqrt{r_i^2-d^2}}\\left[v_r^2\\frac{r_i^2-d^2}{r_i^2} + \\frac {v_{\\rm{t}}^2} 2 \\frac{d^2}{r_i^2} \\right] , \n\\end{equation} where \n\\begin{equation}\n    n_d=  \\sum_{r_i>d} \\frac{d}{r_i\\sqrt{r_i^2-d^2}}.\n\\end{equation}The corresponding observed dispersion can be extracted from the measured line-of-sight velocities by computing the dispersion relative to the mean velocity, binned by separation from the cluster centre. The comparison in Figure~\\ref{subfig:los_vdisp} shows that the two dispersions are similar across the separations with observational constraints. Given that both the surface density and velocity dispersion profiles are comparable to the observational constraints, we adopt this model without further (computationally expensive) parameter study. \n\n\n\\subsection{Stellar density and velocity evolution}\n\nThe main quantities of interest for computing the rate of close encounters in a stellar cluster are the local number density and velocity dispersion. We post-process the output of our Monte Carlo integration to track the density and velocity dispersion evolution over the 12 Gyr lifetime of 47 Tuc. We divide the stars by cluster radii into $30$ log-uniformly spaced bins between $10^{-1.5}$ and $10^1$~pc, then normalise the number in each shell by the volume to yield the density. For velocity, we perform a similar binning but then compute the dispersion $\\sigma_v$ in the one dimensional velocities: $\\sqrt{ v_r^2+v_\\mathrm{t}^2}$, where $v_r$ is the velocity in the radial direction, and $v_\\mathrm{t}$ is the tangential component (combined azimuthal and polar). \n\n\\subsubsection{Density evolution}\n\nWe show the density profile evolution in Figure~\\ref{fig:n_evol}, separated into stars (dashed) and BDs (dotted). The initial core stellar density distribution is $n_*\\sim 10^6$~pc$^{-3}$ within $\\sim 0.5$~pc, and stellar densities $\\gtrsim 10^5$~pc$^{-3}$ are retained for several Gyr. However, this is not true for the BD population. During relaxation, dynamical interaction leads to energy equipartition, resulting in lower masses being pushed to regions of a shallower gravitational potential; this is known as mass segregation \\citep{Binney08}. By this process highest density regions at the cluster centre are quickly vacated of BDs to yield densities $n_\\mathrm{bd}\\lesssim 10^5$~pc$^{-3}$ within $\\sim 1$~Gyr. This will have significant consequences on the efficiency of tidal capture over the lifetime of 47 Tuc  (see Section~\\ref{sec:bd_capt}). \n\n\n\\begin{figure} \n    \\centering\n    \n    \\includegraphics[width=\\columnwidth]{density_evol.pdf}\n    \\caption{Density profile evolution for stars (dashed) and BDs (dotted) in our \\textsc{Mocca} model for the dynamical evolution of 47 Tuc. The lines are shown every $500$~Myr, coloured by the time in the simulation. Both stars and BDs initially have the same density profile.}\n    \\label{fig:n_evol}\n\\end{figure}\n\n\\subsubsection{Velocity dispersion evolution}\n\\label{sec:sigmav_evol}\n\n{The evolution of the velocity dispersion profile is shown in Figure~\\ref{fig:sigmav_evol}. In this case the velocity dispersion across the majority of the cluster decreases as it relaxes. The very inner region retains a high velocity dispersion, which is due to the fact that the majority of trajectories passing through this region are at the pericentre of an eccentric orbit. The physical (three dimensional) dispersion is a factor $\\sqrt{3}$ greater than the one dimensional dispersion. However, the line-of-sight dispersions inferred from radial velocity measurements and our reconstructed `observation' of the model ($\\sim 10{-}12$~km~s$^{-1}$ in the central regions; Section~\\ref{sec:ICs}) are lower again by a factor of order unity. This is due to projection effects. The stellar density profile means that the majority of stars are a few parsec from the centre in three dimensions. When one infers the dispersion of radial velocities at projected (two dimensional) separations smaller than this, the measurements are mostly made for outer, lower velocity stars that fall along the line-of-sight. Thus the apparent dispersion is an underestimate of the physical one dimensional dispersion in the central regions. }\n\n\nAs a result of these considerations, the inner physical three dimensional dispersion is much larger in the highest density regions than the fiducial $10$~km~s$^{-1}$ assumed by \\citetalias{Bon03}. In Section~\\ref{sec:theory_rates} we show that the capture rate decreases much steeper than linearly with $\\sigma_v$. These two findings demonstrate why the approach we take is necessary in computing accurate capture rates over the lifetime of a globular cluster. A velocity dispersion of $\\sigma_v= 30$~km~s$^{-1}$ can result in order of magnitude decreases in the capture rate compared to $\\sigma_v=10$~km~s$^{-1}$, depending on the stellar mass. The full, local velocity dispersion evolution is therefore a necessary ingredient in the calculations we perform. \n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{sigmav_evol.pdf}\n    \\caption{Velocity dispersion profile evolution in the \\textsc{Mocca} model for the dynamical evolution of 47 Tuc. Lines are coloured by time in the simulation, each separated by $500$~Myr.}\n    \\label{fig:sigmav_evol}\n\\end{figure}\n\n\n\n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\subsection{Overall capture rate}\n\\label{sec:bd_capt}\n\\label{sec:overall_capt}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{captrate_evol.pdf}\n    \\caption{Top: The instantaneous rate $\\Gamma_\\mathrm{capt}$ of tidal BD capture for a star of mass $m_*=0.7\\, M_\\odot$ with the mass-radius relation given by equation~\\ref{eq:approx_mr} at a given cluster radius within 47 Tuc. Contours are computed by integrating over equation~\\ref{eq:Gamma_capt} given the BD number density and velocity dispersion shown in Figures~\\ref{fig:n_evol} and~\\ref{fig:sigmav_evol} respectively. {Bottom: The same countours but now weighted by $4\\pi r^2  n_*$, giving the estimated rate of overall tidal capture per unit radius. This is an approximation for the integrand in equation~\\ref{eq:totcaptrate}. }}\n    \\label{fig:Gammacapt_evol}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{tidal_capts.pdf}\n    \\caption{Cumulative number of tidal BD captures through the lifetime of 47 Tuc. The black line shows the result obtained directly from the Monte Carlo calculation. The red line is an estimate using the local stellar\/BD density and velocity dispersion with equation~\\ref{eq:Gamma_capt}, assuming $m_*=0.7\\,M_\\odot$ for all stars (see discussion in Appendix~\\ref{app:massfunc}). The poor agreement after $\\sim 1$~Gyr is due to {stellar} mass segregation (see text for details).}\n    \\label{fig:MCvest_capts}\n\\end{figure}\n\nThe total number of BD captures in the Monte Carlo simulation of 47 Tuc is 377. In order to interpret this directly computed capture rate, we take the density and velocity dispersion profiles from our models to approximate the global capture rates by post-processing {the density and velocity dispersion profiles to obtain the encounter rates according to equation~\\ref{eq:Gamma_capt}}. {To perform this calculation, we need to adopt an expression that encapsulates the stellar mass averaged encounter rate:\n\\begin{equation}\n     \\hat{\\Gamma}_\\mathrm{capt} =  \\int_{m_{\\mathrm{min}}}^{m_\\mathrm{max}}  \\, \\mathrm{d} m_* \\, \\xi(m_*)  \\Gamma_\\mathrm{capt} (m_*),\n\\end{equation}which strictly requires calculating the temporal and spatial evolution of the mass function $\\xi$. However, we show in Appendix~\\ref{app:massfunc} that if the mass function remains constant then after a short time-scale ($\\sim 100$~Myr) the removal of the most massive stars results in $ \\hat{\\Gamma}_\\mathrm{capt} \\approx \\Gamma_\\mathrm{capt} (0.7\\,M_\\odot)$, which remains true over the majority of the lifetime of 47 Tuc. This approximates the global temporal evolution of the mass function, but not the spatial variation. We will see that the assumption of a spatially homogeneous mass function is an important omission.}\n\nIntegrating over equation~\\ref{eq:Gamma_capt} yields a local capture rate over the dynamical history of 47 Tuc. The results of this calculation are shown in the top panel of Figure~\\ref{fig:Gammacapt_evol}. This in turn can be integrated to give a global capture rate:\n\\begin{equation}\n\\label{eq:totcaptrate}\n\\dot{N}_\\mathrm{capt}= \\int_0^\\infty\\,\\mathrm{d}r  \\,  4\\pi r^2 \\,  n_*(r)  \\hat{\\Gamma}_\\mathrm{capt} (r),\n\\end{equation}where we adopt the stellar mass averaged capture rate $ \\hat{\\Gamma}_\\mathrm{capt}  \\approx \\Gamma_\\mathrm{capt} (0.7\\, M_\\odot)$ (see Appendix~\\ref{app:massfunc}). \n\nThe integrand of equation~\\ref{eq:totcaptrate} is shown in the bottom panel of Figure~\\ref{fig:Gammacapt_evol}. The evolution of the capture rate reflects the fact that in the early stages BDs occupy the central, high density regions, but are quickly evacuated to the outer regions as the cluster becomes mass segregated. During the segregation, the capture rate profile quickly transitions from being centrally concentrated to relatively flat with radius. In general, the capture rates are slightly lower than the estimates by \\citetalias{Bon03} due to the considerations discussed in Section~\\ref{sec:theory_rates}; principally the strong dependence of capture efficiency on the stellar mass and local velocity dispersion. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{mmed_evol.pdf}\n    \\caption{Median stellar mass binned by radius from the cluster centre over the lifetime of 47 Tuc in our Monte Carlo model. Lines are coloured by time in the simulation, each separated by 500~Myr.}\n    \\label{fig:mmed_profile}\n\\end{figure}\n\nIn Figure~\\ref{fig:MCvest_capts} we compare the result of the estimated encounter rate computed using equation~\\ref{eq:totcaptrate} {(assuming a constant stellar mass function over space and time)} with the number of captures obtained directly from the Monte Carlo simulation. The rate of capture is initially well reproduced by our estimate, but after approximately $1$~Gyr we overestimate the global rate of captures. This highlights a problem with approximating the encounter rates: we have not accounted for the segregation of \\textit{stellar} mass throughout the cluster (only the BDs). Figure~\\ref{fig:mmed_profile} shows the median stellar mass in radial bins over the dynamical evolution of our Monte Carlo model. The model is not initially segregated, which is why we have good agreement between the Monte Carlo and the post-processed approximation. However, a gradient in stellar masses quickly emerges as mass segregation operates. The result is that lower mass stars preferentially occupy the same regions as the BDs. Capture for low mass stars is inefficient due to the steep decline in the cross section with decreasing stellar mass when the local velocity dispersion is large (Section~\\ref{sec:theory_rates}). Hence the capture rates are further suppressed by the separation of the BDs and high mass stars by which they can be efficiently captured. \n\n\\subsection{Semi-analytic encounter rate calculations}\n\\label{sec:post-process}\n\n{The Monte Carlo approach we describe in Section~\\ref{sec:tid_cross} has the benefit that we can generate a realistic number of capture or scattering events across a complex parameter space. However, there are benefits to complementing the Monte Carlo with semi-analytical estimates. In particular, this allows us to more easily understand the scaling of the results with the stellar parameters and cluster properties. In the Monte Carlo calculation, the number of events may be low in some regions of parameter space (e.g. stellar mass and position), which results in large uncertainties in recovering the probability from sampling. To construct probability functions for encounter rates, a better approach is to take a subset of stars at the end of the simulation and track their encounter rate throughout their lifetime. This has the added benefit that we can use the analytic expressions to scale our results based on assumed physical properties.}\n\nTo recover the encounter rate evolution we must integrate over orbits which are much shorter than is possible to temporally resolve with the output time-step. At each snapshot we therefore recover the orbital solution by first fitting an approximate analytic double power-law density profile:\n\\begin{equation}\n\\label{eq:rho_pot}\n    \\rho_* = \\frac{M_\\mathrm{s}}{4\\pi a_\\mathrm{s}^3} (r\/a_\\mathrm{s})^{-\\alpha}(1-r\/a_\\mathrm{s})^{\\alpha-\\beta}\n\\end{equation} to the stellar mass density, where $M_\\mathrm{s}$, $a_\\mathrm{s}$, $\\alpha$ and $\\beta$ are fitting constants. We then construct the corresponding spherically symmetric potential using the \\texttt{TwoPowerSphericalPotential} class of \\textsc{Galpy}\\footnote{\\url{http:\/\/github.com\/jobovy\/galpy}} \\citep{Bovy15}. {Although orbits in the potential described by equation~\\ref{eq:rho_pot} are not closed, since the density and velocity dispersion profiles are spherically symmetric we are only interested in the radial oscillations of the star with respect to the respect the centre of mass of the cluster. We therefore define the period $P_\\mathrm{orb}$ for the star to make a single epicycle. Then the} orbitally averaged {capture} rate at the specified time-step is:\n\\begin{equation}\n\\label{eq:orbavg}\n    \\langle \\Gamma_\\mathrm{capt} \\rangle (\\Theta(t_\\mathrm{step}))= \\frac{1}{P_\\mathrm{orb}} \\int_{t_\\mathrm{step}}^{t_\\mathrm{step}+P_\\mathrm{orb}} \\,\\mathrm{d}t \\, \\Gamma_\\mathrm{capt}(\\Theta(t)),\n\\end{equation}where $t$ is the time coordinate, $t_\\mathrm{step}$ is the time of the snapshot and all other pertinent parameters are enclosed in $\\Theta$. In practice, if the time-step between updating orbital solution $\\Delta t<P_\\mathrm{orb}$, then $P_\\mathrm{orb}$ is replaced with $\\Delta t$ in equation~\\ref{eq:orbavg}. {The capture probability in equation~\\ref{eq:orbavg} is computed by integrating over equation~\\ref{eq:Gamma_capt} with the BD density and velocity dispersion interpolated over radius space at time $t_\\mathrm{step}$.} In this way, we can estimate the probability of {capture} for a given star $i$ within a certain time $T_\\mathrm{age}$:\n\\begin{equation}\n\\label{eq:Penc}\n    P_{\\mathrm{capt},i} =1- \\exp\\left[ - \\int_0^{T_\\mathrm{age}}\\, \\mathrm{d}t \\,   \\langle \\Gamma_\\mathrm{capt} \\rangle ( \\Theta_i(t))\\right], \n\\end{equation}where $\\Theta_i$ are now the star-specific and time-dependent parameters that determine the {capture} rate. {In practice, we apply equation~\\ref{eq:Penc} as a sum over discrete time-steps} to understand the variation in {capture} rate with the properties of the star, particularly the stellar mass. \n\n\n\\subsection{Brown dwarf capture and stellar mass}\n\n{In this work, we have demonstrated the importance of stellar mass for the rates of tidal BD capture. The sample of \\citetalias{Gil00} comprised $34,091$ stars in the $V$-band magnitude range $17.1 < V < 21.1$. Based on the models of \\citet{Bergbusch92} for [Fe\/H]$=-0.78$ at $12$~Gyr, the brightest stars included therefore have mass $m_*\\approx 0.88\\, M_\\odot$ (approximately the turn-off from main sequence for 47 Tuc) and the lowest masses are $m_* \\approx 0.52\\,M_\\odot$. For comparison, we must therefore consider the capture probabilities in this stellar mass range. }\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{mfunc_capt.pdf}\n    \\caption{{Cumulative distribution function of the masses of stars that capture a BD over the course of the Monte Carlo simulation of 47 Tuc. The vertical red line is placed at $m_* =0.88\\,M_\\odot$, for which the visual magnitude in 47 Tuc would be $V\\approx 17.1$ according to the models of \\citet{Bergbusch92}. This is approximately the mass of the turn off from the main sequence, which are the brightest stars included in the sample of \\citetalias{Gil00}. The vertical blue line is at $m_*=0.52\\, M_\\odot$, which is approximately the cut-off at lower stellar masses adopted by \\citetalias[][visual magnitude $V= 21.1$]{Gil00}.}}\n    \\label{fig:mfunc_capt}\n\\end{figure}\n\n\n{In Figure~\\ref{fig:mfunc_capt} we show the distribution of stellar masses for stars that capture BDs in the Monte Carlo simulation. We find that many of these captures are for stars of high mass that reach the end of their main sequence lifetime by the present day. Indeed, in the Monte Carlo simulation we find that $210\/377$ ($56$ percent) of the captured BDs subsequently undergo a merger with their host due to stellar evolution. The caveat for this finding is that the stellar evolution code employed in \\textsc{Mocca} is not necessarily adapted to deal with close BD companions. Whether a closely orbiting BD may survive the end of the main sequence, for example, is therefore unclear. However, we do not explore this concern in this work since \\citetalias{Gil00} only included main sequence stars in their survey.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{capture_integrals.pdf}\n    \\caption{Orbitally averaged BD capture rates for a subset of five stars over the lifetime of 47 Tuc in our Monte Carlo model, {where each star is shown by a different colour line.} The orbital solutions are updated every 500~Myr {and the resultant average capture rate at each update time is denoted by a square marker.}  }\n    \\label{fig:capt_ints}\n\\end{figure}\n\nTo illustrate how the capture rates vary as a function of stellar mass and final position in the cluster, we compute the orbitally averaged capture rates as outlined in Section~\\ref{sec:post-process}. Despite the poor estimate when averaging over the full mass function in Section~\\ref{sec:overall_capt} due to mass segregation, we can still estimate per star capture rates. We draw a subset of 1000 stars from the final snapshot, chosen semi-randomly to cover a range of stellar masses and radii within the cluster. We then compute the orbitally averaged capture rate $\\langle \\Gamma_\\mathrm{capt} \\rangle$ as defined by equation~\\ref{eq:orbavg}, with the orbital solutions and stellar mass updated every $500$~Myr. A subset of five examples are illustrated in Figure~\\ref{fig:capt_ints}. We can then integrate these encounter rates over the star lifetimes to give the capture probability (equation~\\ref{eq:Penc}). \n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.63\\textwidth]{capture_probs.pdf}\n    \\caption{Probability that individual stars have captured a BD over their lifetimes in our dynamical model, as a function of stellar mass. Points are coloured by their radius within the cluster at the end of the simulation. The solid red line is the best power-law fit, while the shaded red region is the mass range of the \\citetalias{Gil00} sample. {The corresponding expected range of values of $P_\\mathrm{capt}$ for the range of stellar masses surveyed by \\citetalias{Gil00} according to the power-law fit (assuming equal numbers of BDs and stars) is shaded in blue. The upper limit on $P_\\mathrm{capt}$ inferred by non-detection in that sample is represented by the dashed black line ($P_\\mathrm{capt}= 4\\times 10^{-4}$).} }\n    \\label{fig:capture_mfunc}\n\\end{figure*}\n\n\nThe resulting capture probabilities are shown in Figure~\\ref{fig:capture_mfunc}. We find that the probability of capture is a relatively weak function of {final} location, with large scatter. This is the combined influence of the mixing of the stellar population over time and the competing influence of higher velocity dispersion and higher density on the capture rates in the central region (see Section~\\ref{sec:theory_rates}).\nHowever, the capture efficiency remains a strong function of stellar mass. {Since masses are spatially segregated, this in turn influences overall spatial dependence of the capture rate such that our assertion that capture probability is not strongly dependent on position is somewhat dependent on sampling in mass-position space. Nonetheless, in an observational context the mass and radial position can be (approximately) measured, such that neither need be marginalized out. }\n\nThe exact scaling cannot be computed analytically because it depends non-trivially on the local velocity dispersion evolution and the degree of mass segregation. We therefore fit a function:\n\\begin{equation}\n\\label{eq:Pcapt_mst}\n    P_\\mathrm{capt}(m_*) = A \\cdot \\left( \\frac{m_*}{1\\,M_\\odot} \\right)^{\\gamma}\n\\end{equation} to the capture probabilities for main sequence stars using \\texttt{optimize.curve\\_fit} in the \\textsc{Scipy} software library \\citep{Virtanen20}. We obtain $A=1.1\\times 10^{-3}$ and $\\gamma=2.7$, with the resulting evaluation of equation~\\ref{eq:Pcapt_mst} shown as a red line in Figure~\\ref{fig:capture_mfunc}. \n\n\\subsection{Comparisons to observational constraints}\n\n\\subsubsection{47 Tuc}\n\n{The absence of hot Jupiter detection by \\citetalias{Gil00} puts an upper limit on both the fraction of hot Jupiters and tight BD binaries. A close companion with radius $R_\\mathrm{comp}$ and separation $a_\\mathrm{comp}$ can only be detected by transit if the line-of-sight inclination $i$ satisfies:\n\\begin{equation}\n    \\sin i < \\frac{R_\\mathrm{comp}+R_*}{a_\\mathrm{comp}}.\n\\end{equation} \\citetalias{Gil00} adopt $R_\\mathrm{comp} = 1.3 R_\\mathrm{J}$ and a period of $3.5$~days, corresponding to $a_\\mathrm{comp} \\approx 9 \\, R_\\odot$ for a star of mass $m_* \\approx 0.7\\, M_\\odot$. These assumptions lead to the conclusion that $10$ percent of companions should have favourable alignment. This detection efficiency is reduced by a factor $\\sim 2$ when aggregating over the stellar radii and time series noise in the sample. However, as noted by \\citetalias{Bon03}, the typical tidal capture separations ($a_\\mathrm{comp}\\approx 4 R_*$) are smaller than the separations assumed by \\citetalias{Gil00}. Given this orbit, both the geometric probability and the signal-to-noise are enhanced with respect to the hot Jupiter estimates \\citepalias[see Figure 4 of][]{Gil00}. \\citetalias{Bon03} estimate a $\\sim 20$ percent detection probability for a tidally captured BD. The non-detection of any transit therefore implies an upper limit on the occurrence rate of short period BD binaries of $\\lesssim 4\\times 10^{-4}$. This limit is represented by the horizontal dashed black line in Figure~\\ref{fig:capture_mfunc} and can be compared to the grey region which is the range of $P_\\mathrm{capt}$ from equation~\\ref{eq:Pcapt_mst} over the range of masses in the \\citetalias{Gil00} sample. The theoretical capture probabilities are approximately coincident with the empirical upper limit such that non-detection remains (marginally) consistent with the present empirical constraints.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{sigma_nd.pdf}\n    \\caption{The probability of not detecting any BD in a 47 Tuc-like globular cluster given an initial BD-to-star ratio $f_\\mathrm{bd}\\equiv N_\\mathrm{bd}\/N_*$. We have assumed that the total detection efficiency of a transit is $\\epsilon_\\mathrm{det}=0.2$. We show the present sample size from the \\citetalias{Gil00} study in black and an order of magnitude larger sample size in blue. The red lines represent the $2\\sigma$ (dashed) and $3\\sigma$ (solid) significance for non-detection. The stellar masses are chosen between $m_* = 0.52\\,M_\\odot$ and $m_*=0.88\\, M_\\odot$ to bracket the mass range in the study by \\citetalias{Gil00}. }\n    \\label{fig:sigma_nd}\n\\end{figure}\n\n{More generally, we can write the probability $P_\\mathrm{nd}$ of no detection given the number of survey stars $N_\\mathrm{surv}$, the capture probability $P_\\mathrm{capt}$ and the effective detection efficiency $\\epsilon_\\mathrm{det}$:\n\\begin{equation}\n    P_\\mathrm{nd} = { N_\\mathrm{surv}\\choose 0 } (1-\\epsilon_\\mathrm{det} P_\\mathrm{capt})^{N_\\mathrm{surv}}. \n\\end{equation}Adopting $\\epsilon_\\mathrm{det}=0.2$ and scaling $P_\\mathrm{capt}$ in equation~\\ref{eq:Penc} by the ratio of BDs to stars $f_\\mathrm{bd} \\equiv N_\\mathrm{bd}\/N_*$ yields the results shown in Figure~\\ref{fig:sigma_nd}. Adopting $m_*=0.7\\, M_\\odot$ as a representative star in the sample of \\citetalias{Gil00}, we see that $f_\\mathrm{bd} \\lesssim 1$ to $2\\, \\sigma$ significance with the current sample of size $N_\\mathrm{surv}=3.4\\times 10^4$. Increasing the available sample size would (linearly) decrease the maximum $f_\\mathrm{bd}$ that is consistent with non-detection. {Future samples should focus on the most massive stars because these offer the best discrimination of the initial BD-to-stellar ratio.  }}\n\n\\subsubsection{Other local globular clusters}\n\n\\begin{figure}\n    \\centering\n \\includegraphics[width= \\columnwidth]{GC_Pcapt.pdf}\n    \\caption{Local globular cluster properties and the estimated capture BD probability for a star of mass $m_*=0.7\\, M_\\odot$ if the BD-to-star ratio is $f_\\mathrm{bd}=1$. Markers are coloured by the one-dimensional half-mass velocity dispersion and {sized proportionally to the total mass.} 47 Tuc (NGC 104) is highlighted in red and all values are summarised in Table~\\ref{table:GC_scaled}. }\n    \\label{fig:GC_plots}\n\\end{figure}\n\n{To further motivate and guide future observations, we now estimate the equivalent BD capture probabilities in 47 Tuc with respect to other globular clusters. To do this, we adopt parameters from the catalogue of \\citet{Hilker20}, who fit N-body simulations to observational data to yield physical property estimates for a large sample, which have been made publicly available.\\footnote{\\url{https:\/\/people.smp.uq.edu.au\/HolgerBaumgardt\/globular\/}} As an order of magnitude estimate, we can then write:\n\\begin{equation}\n    \\label{eq:GC_scaled}\n    P_\\mathrm{capt}^\\mathrm{GC} \\approx P_\\mathrm{capt}^\\mathrm{47 Tuc} \\frac{n_{*}^{\\mathrm{GC}}}{n_{*}^\\mathrm{47 Tuc}} \\left(\\frac{\\sigma_{v}^{\\mathrm{GC}}}{\\sigma_{v}^\\mathrm{47 Tuc}} \\right)^{-4\/3}\n\\end{equation}where the super-script `GC' denotes an arbitrary globular cluster value and the `47 Tuc' super-scripts are the values for 47 Tuc. We have assumed the power-law $-4\/3$ in the velocity scaling, which is the low velocity limit and somewhat shallower than the true scaling. However, without full modelling it is not clear what absolute velocity dispersion for the dominant capture encounters should be adopted. For stars of mass $m_*\\approx 0.7\\,M_\\odot$ and $\\sigma_v \\lesssim 50$~km~s$^{-1}$ the scaling we adopt is reasonable to make estimates in lieu of full dynamical models.  Note that the exact definitions of both the density $n_*$ and velocity dispersion $\\sigma_v$ only matter insofar as we can compare across all clusters, including our reference cluster 47 Tuc. We adopt the half-mass quantities in the database of \\citet{Hilker20}, wherein the one dimensional velocity dispersions are quoted.} \n\n{The relevant physical parameters for a subset of 17 globular clusters within 7~kpc and with masses $>10^5\\,M_\\odot$ are summarised in Table~\\ref{table:GC_scaled} and represented in Figure~\\ref{fig:GC_plots}. {We have listed the approximate number of stars $N_{*, 0.52-0.88}$ in the mass range surveyed by \\citetalias{Gil00} by integrating the mass function we assume for the 47 Tuc model, truncated above $0.88 \\, M_\\odot$ in the relevant range.} A number of clusters have significantly higher densities than 47 Tuc, in particular M 28, M 62, Ter1, Ter 5 and Ter 9, which also have considerably higher BD capture probabilities for similar $f_\\mathrm{bd}$. {However, these clusters are also relatively close to the galactic centre with low galactic latitude. Their high densities, large distances and possible extinction make these targets challenging for future transit surveys. A number of clusters (e.g. NGC 6656, NGC 6752 and $\\omega$ Cen) are relatively nearby and have similar $P_\\mathrm{capt}$ to 47 Tuc, thus representing promising targets for future transit surveys. However, overall, 47 Tuc remains possibly the best target given it has a large number of high mass stars that were not surveyed by \\citetalias{Gil00} and has the highest estimated $P_\\mathrm{capt}$ of the globular clusters that are not in the galactic centre.  }}\n\n\\begin{table*}\n\\centering \n \\begin{tabular}{c c c c c c c c c} \n \\hline\nCluster & $d$ [kpc] & $l$ [$^\\circ$] & $b$ [$^\\circ$] & $\\log \\, N_{*,0.52-0.88}$ & $\\log n_{*,\\mathrm{hm}}$ [pc$^{-3}$] & $\\sigma_{v,\\mathrm{hm}}$ [km\/s] &  $\\log [P_\\mathrm{capt}(0.7 \\, M_\\odot)\/f_\\mathrm{bd}]$  \\\\\n\\hline\n\\rowcolor{Gray}\nNGC 104 (47 Tuc) &  $4.52 $& $305.89$ &  $-44.89 $& $5.73$ & $2.35$ & $3.65 $ & $-3.38$ \\\\\nNGC 3201 &  $4.74 $& $277.23$ &  $8.64 $& $4.98$ & $1.49$ & $2.83 $ & $-4.09$ \\\\\n\\rowcolor{Gray}\nNGC 4372 &  $5.71 $& $300.99$ &  $-9.88 $& $5.07$ & $1.28$ & $2.75 $ & $-4.28$ \\\\\nNGC 4833 &  $6.48 $& $303.60$ &  $-8.02 $& $5.09$ & $2.06$ & $3.26 $ & $-3.60$ \\\\\n\\rowcolor{Gray}\nNGC 5139 ($\\omega$ Cen) &  $5.43 $& $309.10$ &  $14.97 $& $6.34$ & $2.30$ & $3.84 $ & $-3.46$ \\\\\nNGC 6218 &  $5.11 $& $15.72$ &  $26.31 $& $4.80$ & $1.99$ & $3.11 $ & $-3.64$ \\\\\n\\rowcolor{Gray}\nNGC 6254 (M 10) &  $5.07 $& $15.14$ &  $23.08 $& $5.09$ & $2.04$ & $3.24 $ & $-3.62$ \\\\\nNGC 6266 (M 62) &  $6.41 $& $353.57$ &  $7.32 $& $5.56$ & $3.41$ & $4.29 $ & $-2.41$ \\\\\n\\rowcolor{Gray}\nNGC 6304 &  $6.15 $& $355.83$ &  $5.38 $& $4.87$ & $2.00$ & $3.13 $ & $-3.64$ \\\\\nTer 1 &  $5.67 $& $357.56$ &  $0.99 $& $4.95$ & $2.97$ & $3.80 $ & $-2.78$ \\\\\n\\rowcolor{Gray}\nTer 5 &  $6.62 $& $3.84$ &  $1.69 $& $5.75$ & $3.04$ & $4.11 $ & $-2.76$ \\\\\nTer 9 &  $5.77 $& $3.60$ &  $-1.99 $& $4.85$ & $3.02$ & $3.80 $ & $-2.73$ \\\\\n\\rowcolor{Gray}\nNGC 6553 &  $5.33 $& $5.25$ &  $-3.02 $& $5.23$ & $2.26$ & $3.43 $ & $-3.43$ \\\\\nNGC 6626 (M 28) &  $5.37 $& $7.80$ &  $-5.58 $& $5.25$ & $3.19$ & $4.05 $ & $-2.60$ \\\\\n\\rowcolor{Gray}\nNGC 6656 (M 22) &  $3.30 $& $9.89$ &  $-7.55 $& $5.45$ & $2.29$ & $3.52 $ & $-3.42$ \\\\\nNGC 6752 &  $4.12 $& $336.49$ &  $-25.63 $& $5.22$ & $2.06$ & $3.28 $ & $-3.60$ \\\\\n\\rowcolor{Gray}\nNGC 6809 (M 55) &  $5.35 $& $8.79$ &  $-23.27 $& $5.06$ & $1.54$ & $2.91 $ & $-4.06$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Local globular cluster parameters from the N-body models of \\citet{Hilker20}, selected to be closer than $7$~kpc and more massive than $10^5\\, M_\\odot$. {The number of stars $N_{*, 0.52-0.88}$ is estimated by dividing the total mass by $0.5\\,M_\\odot$ then multiplying by $0.297$, the approximate fraction of stars with masses $0.52 - 0.88\\,M_\\odot$.} The stellar number density, $n_*$, and (one dimensional) velocity dispersion, $\\sigma_v$ are taken inside the half-mass radius. The last column is obtained by scaling the results for 47 Tuc using equation~\\ref{eq:GC_scaled}. }\n\\label{table:GC_scaled}\n\\end{table*}\n\n\n\n\\subsection{Caveats for capture rates}\n\\label{sec:caveats}\n\n{We have explored BD capture rates in detail and suggested that sufficiently large transit surveys can put upper limits on BD formation rates. However, it is possible that the present day short period companion rates are influenced by other physical mechanisms. Factors that may alter the rates of short period BD companions include (although not necessarily limited to): }\n\\begin{itemize}\n    \\item \\textit{Primordial mass segregation}: We have demonstrated that once a cluster become mass segregated, BD tidal capture becomes inefficient. If a population is primordially segregated, this would similarly reduce the capture efficiency.\n    \\item \\textit{Time-scale for circularisation:} In Paper II we explore the time-scale on which a migrating planet may undergo a dynamical perturbation while circularising. Following \\citetalias{Bon03}, we have assumed that this time-scale is short for a tidally captured BD \\citep{Mardling96}. However, if this is not the case then perturbations after the initial tidal encounter may curtail tidal circularisation and therefore prevent the formation of the {tight BD-star binary that can be detected through transit}. \n    \\item \\textit{Tidal inspiral of BDs:} {Evidence for the correlation of hot Jupiter occurrence with cold stellar kinematics may originate from the inspiral of close companions onto the central star on Gyr timescales \\citep[][]{Hamer19}. If close sub-stellar companions do inspiral on these time-scales, then a similar process {may operate on} tidally captured BDs. However, hot Jupiters appear to be retained in the dense cluster M67 \\citep[][]{Brucalassi16} which has an age of $\\sim 4.5$~Gyr, such that this would require a relatively narrow range of inspiral timescales \\citep[see also discussion in Section 4.3 of][]{Winter21}.}\n    \\item \\textit{Evacuation of BDs:} Apart from mass segregation due to two-body relaxation, low mass stars and BDs can be further evacuated from the central regions of the globular cluster by alternative heating mechanisms. For example, black hole subsystems may induce dynamically heating and eject low mass objects such as BDs to the cluster halo \\citep{Breen13,Giersz19}. However, this process occurs on a time-scale longer than the half-mass relaxation time-scale ($\\sim 3$~Gyr for 47 Tuc). The time-scale on which the majority of BD captures occurs in our models is $\\lesssim 2$~Gyr, while segregation on longer time-scales may not strongly influence capture rates. Similarly, heating due to tidal shocks during to passages through the galactic plane may operate time-scales comparable to two-body relaxation \\citep{Gnedin99}. If these mechanisms significantly reduce the stellar density after $\\sim 12$~Gyr, this would also suggest a moderately higher initial stellar density required to reproduce the present day density profile. These considerations may therefore increase the initial capture rate and subsequently reduce it due to enhanced mass segregation. We do not explore these possibilities quantitatively in this work. \n\\end{itemize}\n\n{An absence of close sub-stellar companions would therefore suggest either that BD formation is suppressed or that one of the above processes (or unconsidered alternative) is operating. In any case, non-detection in an increased sample of stars would require explanation and future survey campaigns are therefore merited.}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this work, we have explored the apparent absence of close-in sub-stellar companions in the globular cluster 47 Tuc from a theoretical perspective. We applied a Monte Carlo model using the \\textsc{Mocca} code \\citep{Hyp13, Giersz13} for the dynamical evolution of the globular cluster. Using this model, we compute the rates of tidal BD capture over its lifetime.\n\nOur results indicate lower capture efficiency than previous estimates \\citep{Bon03}. The reasons for this are subtle, but fundamentally originate from the rapid decrease of the tidal capture cross section with decreasing stellar mass. This is particularly true for environments with velocity dispersions as high as globular clusters. Once mass segregation operates, BDs and low mass stars are preferentially found in the same spatial location. Therefore the global tidal capture efficiency drops precipitously, such that the current constraints {cannot rule out that the frequency of BDs in the IMF is as high in 47 Tuc as in the galactic field.}\n\nThese considerations also lead to a steep scaling of the capture probability with stellar mass. For initial number of BDs $N_\\mathrm{bd}$ and stars $N_*$, those stars that have not reached the end of their main sequence have a lifetime capture probability:\n\\begin{equation}\n    P_\\mathrm{capt} = 1.1 \\times 10^{-3} \\frac{N_\\mathrm{bd}}{N_*}\\cdot \\left(\\frac{m_*}{1\\, M_\\odot} \\right)^{2.7}.\n\\end{equation}The large exponent means that any constraints on the initial BD ratio are strongly dependent on the mass function of stars that are surveyed for close companions. For the typical masses of the stars surveyed by \\citet[][$0.52 \\,M_\\odot \\lesssim m_* \\lesssim 0.88\\, M_\\odot$]{Gil00} and equal numbers of BDs and stars, this yields capture probabilities that are comparable to the upper limit constraint on close sub-stellar companions ($P_\\mathrm{capt}\\lesssim 4\\cdot 10^{-4}$). \n\n\n\nFinally we conclude that, while the current constraints on the frequency of close sub-stellar companions {cannot rule out that the incidence of BDs in 47 Tuc is as high as it is in the field,} stronger constraints can be obtained by surveying a larger number of relatively high mass stars. {Such an exercise may also be achieved aggregating across several globular clusters.} We therefore estimate the capture rates in local globular clusters for a similar mass range of stars to those surveyed in 47 Tuc. The estimated capture rates are summarised in Table~\\ref{table:GC_scaled}. {We suggest that 47 Tuc remains among the most promising targets for follow up, with a convenient location and a large number of relatively high mass stars that have not yet been monitored for short period sub-stellar companions. A number of other globular clusters, such as $\\omega$ Cen, may also represent feasible targets for transit surveys. Our results offer motivation and interpretation for future transit surveys of globular clusters.}\n\n\n\\section*{Acknowledgements}\n\nWe thank the anonymous referee for their careful reading that helped clarify the manuscript and Abbas Askar for his helpful comments on dynamical models for 47 Tucanae. AJW acknowledges funding from an Alexander von Humboldt Stiftung Postdoctoral Research Fellowship. GR acknowledges support from the Netherlands Organisation for Scientific Research (NWO, program number 016.Veni.192.233) and from an STFC Ernest Rutherford Fellowship (grant number ST\/T003855\/1). This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 681601) and been supported by the DISCSIM project, grant agreement 341137 funded by the ERC under ERC-2013-ADG. \n\n\\section*{Data availability}\n\nAll data in this article is available from the corresponding author upon reasonable request.\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we study the structure of parabolic induction of admissible representations of reductive groups defined over a non-archimedean local field $F$ of characteristic 0. Of course, the problem goes back to the era of Harish-Chandra and marches on in the century of Langlands for its deep connection with number theory, many great mathematicians have devoted their efforts on this topic (cf. \\cite{knappstein,knapp1976classification,langlands,jacquet,howe1976any,bernstein1977induced,speh1980reducibility,kazhdan1987proof,casselman1995introduction,shahidi1990proof,moeglin2002construction,moeglin2003points}).\n\nLet $G$ be a connected reductive group defined over $F$, and $P=MN$ be a parabolic subgroup of $G$ with $M$ its Levi subgroup and $N$ nilpotent radical of $P$. For an irreducible admissible representation $(\\sigma, V_\\sigma)$ of $M$, a fundamental question originating from Harish-Chandra's theory is to give a reasonable criterion of the reducibility of the parabolic induction $I^G_P(\\sigma)$:\n\\[I^G_P(\\sigma)=Ind^G_P(\\sigma):=\\{f:G\\rightarrow V_\\sigma \\mbox{ smooth }|f(mng)=\\delta_P(m)^{\\frac{1}{2}}\\sigma(m)f(g), \\forall m\\in M,n\\in N,g\\in G \\}. \\]\nIf $G$ is a finite group, such a question is answered perfectly by Mackey's theory. Indeed, analogous Mackey theory does exist for our $G$ which is the so-called Bernstein--Zelevinsky geometrical lemma (cf. \\cite{bernstein1977induced,casselman1995introduction,waldspurger2003formule}), and analogous simple criterion does exist for tempered inductions $I^G_P(\\sigma)$, i.e. $\\sigma$ is a discrete series representation. Such a criterion is the so-called Knapp--Stein $R$-group theory (cf. \\cite{knappstein,knapp1975singular,silberger1978knapp,luo2017R}). But there is an essential obstruction for general $\\sigma$: the restriction functor, i.e. Jacquet functor does not preserve unitarity as opposed to the finite group case. If $\\sigma$ is a supercuspidal representation, i.e. does not arise from proper parabolic inductions, a simple criterion has been discovered recently which originates from Muller's criterion for principal series, i.e. $\\sigma$ is a character (cf. \\cite{muller1979integrales,kato1982irreducible,luo2018muller}). For general $\\sigma$ and general $G$, there is no essential progress (to my best knowledge). Inspired by our Muller type irreducibility criterion for $\\sigma$ supercuspidal, we give a first structural answer for a large class of $\\sigma$. Denote by $\\rho$ the supercuspidal support of $\\sigma$ on $P_0=M_0N_0\\subset P=MN$, i.e. $\\sigma\\in JH(I^M_{M\\cap P_0}(\\rho))$ the set of Jordan--H\\\"{o}lder constituents of $I^M_{M\\cap P_0}(\\rho)$, then our universal hierarchical irreducibility criterion is as follows: please see the context for the detail,\n\\begin{thm}(cf. Theorem \\ref{prod}) Assume the reducibility conditions given by Muller type irreducibility criterion for $I^G_{P_0}(\\rho)$ lie in $M$, then\n\t$I^G_P(\\tau)$ is always irreducible for any $\\tau\\in JH(I^M_{M\\cap P_0}(\\rho))$.\n\\end{thm} \nSuch a criterion exists for tempered inductions which is a key ingredient of proving Howe's finiteness conjecture (cf. \\cite{clozel1989orbital,luo2017howe}). Indeed, Clozel also proposed a beautiful conjecture as a key ingredient in his first attempt to prove Howe's finiteness conjecture (see \\cite{clozel1985conjecture}). As an application of the above theorem, we serve you a simple intuitive proof of such a beautiful conjecture under the assumption that it holds for co-rank one cases.\n\nLet us take a brief look at the main idea of the proof of Theorem \\ref{prod} for regular $\\rho$, i.e. no non-trivial Weyl element fixes it. Our argument is quite classical. It relies heavily on the Bernstein--Zelevinsky geometrical lemma and the exactness of Jacquet functor (cf. \\cite{bernstein1977induced,casselman1995introduction,silberger2015introduction,waldspurger2003formule}): the constituents of the generalized principal series $I^G_{P_0}(\\rho)$ are parametrized by the partition of the set of the relative Weyl group $W_{M_0}:=N_G(M_0)\/M_0$. If our reducibility conditions of $I^G_{P_0}(\\rho)$ given by our Muller type irreducibility criterion lie in some standard Levi subgroup $M$, then it gives rise to a partition of $W_{M_0}^M:=N_M(M_0)\/M_0\\subset W_{M_0}$. Then to show the irreducibility of $I^G_P(\\sigma)$ for any $\\sigma\\in JH(I^M_{M\\cap P_0}(\\rho))$, the novelty of our argument is to find a good set $S$ of representatives of the quotient $W_{M_0}^M\\backslash W_{M_0}$, such that    \n\\[I^G_P(\\rho^{w_1})\\simeq I^G_P(\\rho^{w_1w}) \\]\nfor any $w_1\\in W^M_{M_0}$ and $w\\in S$.\n\nIt seems that our argument is quite restrictive. But it also seems that this may be the only universal argument we could come up with nowadays which has been practiced intelligently by Bernstein--Zelevinsky, Silberger, Jantzen, Moeglin, the Tadi{\\'c} school etc (please refer to \\cite{bernstein1977induced,jantzen1997supports,moeglin2002construction,tadic1998regular,lapid2017some} for a glimpse). Inspired by a conjectural Muller type irreducibility criterion for general parabolic inductions (see \\cite[Conjecture 4.4]{luo2018muller}), a detailed study of the structure of Jacquet modules of a general parabolic induction may give us hope to prove the following conjectural universal irreducibility structure (Please see the context for the details):\n\\begin{conj}(see Conjecture \\ref{conj})\n\tGiven a parabolic induction $I^G_{P=MN}(\\sigma)$ with $\\sigma$ an irreducible admissible representation of $M$, if the ``reducibility conditions'' of $I^G_P(\\sigma)$ lie in some standard Levi subgroup $L$ of a parabolic subgroup $Q=LV\\subset P=MN$, then $I^G_Q(\\tau)$ is always irreducible for any $\\tau\\in JH(I^L_{L\\cap P}(\\sigma))$.\n\\end{conj}\n\\begin{rem}\n\tAt first glance, Conjecture \\ref{conj} may look meaningless and ridiculous. But some supportive examples of a conjectural Muller type irreducibility criterion for general parabolic induction, i.e. \\cite[Conjecture 4.4]{luo2018muller} could guide us to a right direction. Those examples are parabolic induction representations inducing from essentially discrete series and standard modules which many mathematicians have investigated since the era of Harish-Chandra.\n\\end{rem}\nInspired by Goldberg and Jantzen's product formulas for quasi-split classical groups (cf. \\cite{goldberg1994reducibility,jantzen1997supports,jantzen2005duality}), another direction we could cook up with the help of our new notion of $R$-group is a Goldberg--Jantzen type product formula as follows: (Please see the context for the details)\n\\begin{conj}(see Conjecture \\ref{gjconj})\n\tLet $\\nu\\in \\mathfrak{a}_M^*$ and $\\sigma$ be a unitary supercuspidal representation of $M$. For the generalized principal series representation $I^G_P(\\nu,\\sigma):=Ind^G_P(\\sigma\\otimes\\nu)$ of $G$, there is a one-one correspondence:\n\t\\[JH(I^G_P(\\nu,\\sigma))\\leftrightarrows \\prod_i JH(I^{M_i}_{M_i\\cap P}(\\nu,\\sigma)). \\]\n\\end{conj}\nHere $\\Phi_\\sigma:=\\{\\alpha\\in\\Phi_M:~w_\\alpha.\\sigma=\\sigma \\}$ is a root system and decomposes into irreducible pieces, i.e. $\\Phi_\\sigma=\\sqcup_i\\Phi_{\\sigma,i}$. Each irreducible root system $\\Phi_{\\sigma,i}$ gives rise to a Levi subgroup $M_i\\supset M$ which is defined by\n\\[M_i:=C_G((\\bigcap\\limits_{\\alpha\\in\\Phi_{\\sigma,i}}Ker(\\alpha) )^0). \\]\nIn the meantime we assume that $R_\\sigma$ associated to $\\sigma$ decomposes into a product of small pieces in the same pattern as does $\\Phi_\\sigma$, i.e.\n\\[R_\\sigma=\\prod_i R_{\\sigma,i} \\]\nwith $R_{\\sigma,i}$ a subgroup of the relative Weyl group $W^{M_i}_M=N_{M_i}(M)\/M$ of $M$ in $M_i$.\n\nAs a by-product of Theorem \\ref{prod}, we could reduce Clozel's finiteness conjecture of special exponents to co-rank one cases. There is no harm to assume that $G$ is of compact center. For a discrete series representation $\\sigma$ of $G$, we denote its associated supercuspidal support to be $\\rho$ which is a supercuspidal representation of some Levi subgroup $M$ of $G$. Denote by $\\omega_\\rho$ the unramified part of the central character of $\\rho$, i.e. $\\omega_\\rho\\in \\mathfrak{a}_{M,\\mathbb{C}}^\\star:=Hom_F(M,\\mathbb{G}_m)\\otimes \\mathbb{C}$. Such a character is called a \\underline{special exponent}. Clozel conjectured that the set of special exponents is finite (see \\cite{clozel1985conjecture,clozel1989orbital}).\n\n\\begin{thm}(cf. Theorem \\ref{clozel})\n\tThe set of special exponents is finite provided it is true for co-rank one cases.\n\\end{thm}\nIndeed, the generic co-rank one case is a result of the profound Langlands--Shahidi theory, and the general co-rank one case follows from two conjectures of Shahidi (cf. \\cite[Conjectures 9.2 \\& 9.4]{shahidi1990proof}). Those two conjectures for classical groups in some sense are by-products of Arthur's standard model argument in his monumental book \\cite{arthur2013endoscopic}. The main idea of the proof of Clozel's finiteness conjecture is as follows:\n\\begin{enumerate}[(i)]\n\t\\item A full induced representation $I^G_P(\\sigma)$ can never be discrete series.\n\t\\item An invertible matrix has only one solution.\n\\end{enumerate}\nTo illustrate the simple ideas, let us take a look at the real parts of special exponents, if $I^G_P(\\rho)$ contains a discrete series subquotient, then the reducibility conditions must generate the whole vector space $\\mathfrak{a}^\\star_{M,\\mathbb{C}}$. Otherwise, there exists a proper parabolic subgroup $Q=LV\\supset P$ such that the reducibility conditions lie in $L$, then our universal hierarchical irreducibility criterion implies that $I^G_Q(\\tau)$ is always irreducible for any $\\tau\\in JH(I^L_{L\\cap P}(\\rho))$, whence any constituent of $I^G_P(\\rho)$ can never be discrete series. Contradiction. Thus $\\mathfrak{a}^\\star_{M,\\mathbb{C}}$ is generated by the reducibility conditions. Indeed, we recently learned that such a claim is also a corollary of an old result of Harish-Chandra (cf. \\cite[Theorem 5.4.5.7]{silberger2015introduction}, \\cite[Theorem 3.9.1]{silberger1981discrete} or \\cite[Corollary 8.7]{heiermann2004decomposition}). Under the assumption that the set of special exponents is finite for co-rank one cases, those reducibility conditions form a finite set of hyperplanes in $\\mathfrak{a}^\\star_{M,\\mathbb{C}}$, therefore the finiteness for high rank cases follows easily from the fact that there exists only one solution for an invertible matrix. At last, we would like to mention that Clozel proposed an analogous strong global conjecture of finiteness of poles of Eisenstein series constructed from cusp forms which would simplify Arthur's trace formula machine significantly. We hope that the simple intuitive proof of Clozel's local finiteness conjecture could shed some light on his global conjecture in our future work.\n\nWe end the introduction by recalling briefly the structure of the paper. In Section 2, we prepare some necessary notation. In Section 3, we state and prove the universal hierarchical irreducibility criterion for Harish-Chandra parabolic inductions. In the end, as its first application, a new understanding of the generic irreducibility property of parabolic inductions is provided. In Section 4, with the aide of the universal hierarchical irreducibility criterion in the previous section, a simple intuitive proof of Clozel's finiteness conjecture is served under the assumption that it holds for the co-rank one case.    \n\n\\section{Preliminaries}\nLet $G$ be a connected reductive group defined over a non-archimedean local field $F$ of characteristic 0. Denote by $|-|_F$ the absolute value, by $\\mathfrak{w}$ the uniformizer and by $q$ the cardinality of the residue field of $F$. Fix a minimal parabolic subgroup $B=TU$ of $G$ with $T$ a minimal Levi subgroup and $U$ a maximal unipotent subgroup of $G$, and let $P=MN\\supset B=TU$ be a standard parabolic subgroup of $G$ with $M$ the Levi subgroup and $N$ the unipotent radical.\n\n\\subsection{Structure theory}Let $X(M)_F$ be the group of $F$-rational characters of $M$, and set \n\\[\\mathfrak{a}_M=Hom(X(M)_F,\\mathbb{R}),\\qquad\\mathfrak{a}^\\star_{M,\\mathbb{C}}=\\mathfrak{a}^\\star_M\\otimes_\\mathbb{R} \\mathbb{C}, \\]\nwhere\n\\[\\mathfrak{a}^\\star_M=X(M)_F\\otimes_\\mathbb{Z}\\mathbb{R} \\]\ndenotes the dual of $\\mathfrak{a}_M$. Recall that the Harish-Chandra homomorphism $H_P:M\\longrightarrow\\mathfrak{a}_M$ is defined by\n\\[q^{\\left< \\chi,H_P(m)\\right>}=|\\chi(m)|_F \\] \nfor all $\\chi\\in X(M)_F$.\n\nNext, let $\\Phi$ be the root system of $G$ with respect to $T$, and $\\Delta$ be the set of simple roots determined by $U$. For $\\alpha\\in \\Phi$, we denote by $\\alpha^\\vee$ the associated coroot, and by $w_\\alpha$ the associated reflection in the Weyl group $W=W^G$ of $T$ in $G$ with\n\\[W:=N_G(T)\/T=\\left<w_\\alpha:~\\alpha\\in\\Phi\\right>. \\]\nDenote by $w_0^G$ the longest Weyl element in $W$, and similarly by $w_0^M$ the longest Weyl element in the Weyl group $W^M:=N_M(T)\/T$ of a Levi subgroup $M$. \n\nLikewise, we denote by $\\Phi_M$ (resp. $\\Phi_M^L$) the reduced relative root system of $M$ in $G$ (resp. the Levi subgroup $M\\subset L$), by $\\Delta_M$ the set of relative simple roots determined by $N$ and by $W_M:=N_G(M)\/M$ (resp. $W_M^L$) the relative Weyl group of $M$ in $G$ (resp. $L$). In general, a relative reflection $\\omega_\\alpha:=w_0^{M_\\alpha}w_0^M$ with respect to a relative root $\\alpha$ does not preserve our Levi subgroup $M$. Denote by $\\Phi^0_M$ (resp. $\\Phi^{L,0}_M$) the set of those relative roots which contribute reflections in $W_M$ (resp. $W_M^L$). It is easy to see that $W_M$ preserves $\\Phi_M$, and further $\\Phi_M^0$ as well, as $\\omega_{w.\\alpha}=w\\omega_\\alpha w^{-1}$. Note that $W_M$ (resp. $W_M^{L,0}$) in general is larger than $W_M^0$ (resp. $W_M^{L,0}$) the one generated by those relative reflections in $G$ (resp. $L$). Denote by $\\Phi_M(P)$ the set of reduced roots of $M$ in $P$.\n\nRecall that the canonical pairing $$\\left<-,-\\right>:~\\mathfrak{a}^\\star_M\\times \\mathfrak{a}_M\\longrightarrow\\mathbb{R}$$ suggests that each  $\\alpha\\in \\Phi_M$ will enjoy a one parameter subgroup $H_{\\alpha^\\vee}(F^\\times)$ of $M$ satisfying: for $x\\in F^\\times$ and $\\beta\\in \\mathfrak{a}^\\star_M$,\n\\[\\beta(H_{\\alpha^\\vee}(x))=x^{\\left<\\beta,\\alpha^\\vee\\right>}. \\]\n\n\\subsection{Parabolic induction and Jacquet module}For $P=MN$ a parabolic subgroup of $G$ and an admissible representation $(\\sigma,V_\\sigma)~ (resp.~(\\pi,V_\\pi))$ of $M~(resp.~G)$, we have the following normalized parabolic induction of $P$ to $G$ which is a representation of $G$\n\\[I_P^G(\\sigma)=Ind_P^G(\\sigma):=\\{\\mbox{smooth }f:G\\rightarrow V_\\sigma|~f(nmg)=\\delta_P(m)^{1\/2}\\sigma(m)f(g), \\forall n\\in N, m\\in M~and~g\\in G\\} \\]\nwith $\\delta_P$ stands for the modulus character of $P$, i.e., denote by $\\mathfrak{n}$ the Lie algebra of $N$,\n\\[\\delta_P(nm)=|det~Ad_\\mathfrak{n}(m)|_F, \\]\nand the normalized Jacquet module $J_M(\\pi)$ with respect to $P$ which is a representation of $M$\n\\[\\pi_N:=V_\\pi\/\\left<\\pi(n)e-e:~n\\in N,e\\in V_\\pi\\right>. \\] \nGiven an irreducible admissible representation $\\sigma$ of $M$ and $\\nu\\in \\mathfrak{a}^\\star_{M}$, let $I(\\nu,\\sigma)$ be the representation of $G$ induced from $\\sigma$ and $\\nu$ as follows:\n\\[I(\\nu,\\sigma)=Ind_P^G(\\sigma\\otimes q^{\\left<\\nu,H_P(-)\\right>}) .\\]\nWe denote by $JH(I_P^G(\\sigma))$ the set of Jordan--H\\\"{o}lder constituents of the parabolic induction $I^G_P(\\sigma)$, and define the action of $w\\in W_M$ on representations $\\sigma$ of $M$ to be $w.\\sigma=\\sigma\\circ Ad(w)^{-1}$ and $\\sigma^w=\\sigma\\circ Ad(w)$.\n\n\\subsection{$R$-group}\nIn \\cite{muller1979integrales}, for a principal series $I(\\lambda)$ of $G$, she defines a subgroup $W_\\lambda^1$ of the Weyl group $W$ governing the reducibility of the ``unitary'' part of principal series on the Levi level, which is indeed the Knapp--Stein $R$-group as follows (cf. \\cite{winarsky1978reducibility,keys1982decomposition}), \n\\begin{align*}\n\\Phi_{\\lambda}^0&:=\\{\\alpha\\in \\Phi:~\\lambda_\\alpha=Id \\},\\\\\nW_{\\lambda}^0&:=\\left<w_\\alpha:~\\alpha\\in \\Phi_\\lambda^0 \\right>,\\\\\nW^1_\\lambda&:=\\{w\\in W_\\lambda:~w.(\\Phi_\\lambda^0)^+>0 \\},\\\\\nW_\\lambda&:=\\{w\\in W:~w.\\lambda=\\lambda \\}.\n\\end{align*} \nIn view of \\cite[Lemma I.1.8]{waldspurger2003formule}, one has\n\\[W_\\lambda=W_\\lambda^0\\rtimes W_\\lambda^1. \\]\nFollowing the Knapp--Stein R-group theory (cf. \\cite{silberger2015introduction}), we denote by $R_\\lambda$ the subgroup $W_\\lambda^1$.\n\nLikewise, for generalized principal series $I^G_P(\\sigma)$ (cf. \\cite{luo2018muller}),\n\\begin{align*}\n\\Phi_{\\sigma}^0&:=\\{\\alpha\\in \\Phi_M^0:~w_\\alpha.\\sigma=\\sigma \\},\\\\\nW_{\\sigma}^0&:=\\left<w_\\alpha:~\\alpha\\in \\Phi_{\\sigma}^0 \\right>,\\\\\nW^1_{\\sigma}&:=\\{w\\in W_{\\sigma}:~w.(\\Phi_{\\sigma}^0)^+>0 \\},\\\\\nW_{\\sigma}&:=\\{w\\in W_M:~w.\\sigma=\\sigma \\}.\n\\end{align*}\nVia \\cite[Lemma I.1.8]{waldspurger2003formule}, we have\n\\[W_{\\sigma}=W_{\\sigma}^0\\rtimes W_{\\sigma}^1, \\]\nand we denote $R_{\\sigma}$ to be $W_{\\sigma}^1$ following tradition, but it is not the exact Knapp--Stein $R$-group in the sense of Silberger.\n\n\\subsection{Special exponent} There is no harm to assume that $G$ is of compact center. For a discrete series representation $\\pi$ of $G$, we denote its associated supercuspidal support to be $\\sigma$ which is a supercuspidal representation of some Levi subgroup $M$ of $G$. Denote by $\\omega_\\sigma$ the unramified part of the central character of $\\sigma$, i.e. $\\omega_\\sigma\\in \\mathfrak{a}_{M,\\mathbb{C}}^\\star$. Such a character is called a \\underline{special exponent}.\n\n\\section{Universal Hierarchical Structure of Reducibility}\n\\subsection{A product formula}\nIn this subsection, we prove a key observation of the decomposition of parabolic induction which opens a gate to understand some of the classical results\/conjectures, for example the generic irreducibility property of parabolic induction and Clozel's finiteness conjecture of special exponents.\n\nRecall that $G$ is a connected reductive group defined over $F$ with the set of simple roots $\\Delta$, $P=MN$ is a standard parabolic subgroup of $G$ associated to $\\Theta_M\\subset \\Delta$ and $\\sigma$ is a supercuspidal representation of $M$ (not necessary unitary), one forms a parabolic induction $I_P^G(\\sigma)$. Then our ``product formula'' is designed to ask the following question\n\\[\\mbox{When does the reducibility of $I^G_P(\\sigma)$ only happen on the Levi-level?\\tag*{$(\\star)$}} \\]\ni.e.\n\\[\\mbox{What is a reasonble condition for the irreducibility of $I^G_{Q=LV}(\\tau)$ for all $\\tau\\in JH(I_{L\\cap P}^L(\\sigma))$?} \\]\nThe answer traces back to a beautiful theorem of I. Muller \\cite{muller1979integrales,kato1982irreducible} which provides a natural criterion of the irreducibility of principal series, and its generalized version for generalized principal series \\cite{luo2018muller} using the Knapp--Stein $R$-group and co-rank one reducibility. As the irreducibility is governed by the Knapp--Stein $R$-group and the co-rank one reducibility, a natural candidate for ($\\star$) is to assume that those governing conditions occur only on the Levi-level. To be more precise, let $Q=LV$ be a standard parabolic subgroup associated to $\\Theta_L$ with $\\Theta_M\\subset\\Theta_L\\subset \\Delta$, then our working assumption for $(\\star)$ is as follows.\n\n{\\bf \\underline{Working Hypothesis}}:\n\\leavevmode\n\\begin{enumerate}[(i)]\n\t\\item (\\underline{Rank-one reducibility}) The co-rank one reducibility only occurs within $L$, i.e.\n\t\\[I_M^{M_\\alpha}(\\sigma) \\mbox{ is reducible only for some }\\alpha\\in \\Phi_M^{L},~i.e.~\\alpha\\in \\Phi_M^{L,0}\\mbox{ (cf. \\cite[Theorem 7.1.4]{casselman1995introduction})}. \\] \n\t\\item (\\underline{$R$-group}) The $R$-group $R_\\sigma$ associated to $\\sigma$ is a subgroup of $W^L_M$, i.e.\n\t\\[R_\\sigma\\subset W^L_M. \\]\n\\end{enumerate}\nUnder those hypothesis, via the Jacquet module machine, the confirmation of $(\\star)$ results from the associativity property of intertwining operators and the following observation\/fact (cf. \\cite{bernstein1977induced,casselman1995introduction,moeglin_waldspurger_1995,waldspurger2003formule}).\n\\begin{enumerate}[(i)]\n\t\\item (Bernstein--Zelevinsky geometrical lemma)\n\t\\[J_M( I_P^G(\\sigma))=\\sum\\limits_{w\\in W_M}\\sigma^w. \\]\n\t\\item (Bruhat--Tits decomposition)\n\t\\[W_M=W_M(L)W^L_M, \\tag{$\\star\\star$}\\]\n\twhere $W_M(L)$ is defined as follows:\n\t\\[W_M(L):=\\{w\\in W_M:~w.(\\Phi_M^{L,0})^+>0 \\}\/W_M^{L,1}, \\]\n\there $W_M^{L,1}$ is defined as in \\cite[Lemma 2.1]{luo2018muller}), i.e.\n\t\\[W_M^{L,1}:=\\{w\\in W_M^L:w.(\\Phi_M^{L,0})^+>0 \\}. \\]\n\\end{enumerate}\nTo be precise,\n\\begin{thm}(Universal hierarchical irreducibility criterion)\\label{prod}\n\tKeep the notions as above. Under the above {\\bf Working Hypothesis}, we have the following cardinality equality\n\t\\[\\#JH(I^G_P(\\sigma))=\\#JH(I_{L\\cap P}^L(\\sigma)). \\]\n\tIndeed, this is equivalent to saying that \n\t\\[I^G_Q(\\tau)\\mbox{ is always irreducible for any }\\tau \\in JH(I^L_{L\\cap P}(\\sigma)). \\]\n\\end{thm}\nBefore turning to the proof, let us first prove the Bruhat-Tits decomposition, i.e. $(\\star\\star)$, which is a generalization of \\cite[Lemma 1.1.2]{casselman1995introduction} for $M=T$, in what follows.\n\\begin{proof}[Proof of $(\\star\\star)$]\n\tFirst note that (cf. \\cite[Lemma 2.1]{luo2018muller}) \n\t\\[W_M:=W_M^0\\rtimes W_M^1 \\]\n\tand \n\t\\[W_M^L:=W_M^{L,0}\\rtimes W_M^{L,1}, \\]\n\twith\n\t\\[W_M^{1}:=\\{w\\in W_M:w.(\\Phi_M^{0})^+>0 \\}=\\{w\\in W_M:w.(\\Phi_M^{0})^+=(\\Phi_M^{0})^+ \\}. \\]\n\tSo \n\t\\[W_M\/W_M^L=\\{w\\in W_M:w.(\\Phi_M^{L,0})^+>0 \\}\/W_M^{L,1}. \\]\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{prod}]\n\tNote that the decomposition of $I^G_P(\\sigma)$ is a partition of $W_M$. Recall that $R_\\sigma$ is in general not the exact R-group in the sense of Knapp--Stein, as it is defined by\n\t\\[W_\\sigma=W_\\sigma^0\\rtimes R_\\sigma, \\]\n\twhere\n\t\\[W_\\sigma:=\\{w\\in W_M:~w.\\sigma=\\sigma \\},\\]\n\tand \n\t\\[W_\\sigma^0:=\\left<w_\\alpha:~w_\\alpha.\\sigma=\\sigma,~\\alpha\\in\\Phi_M  \\right>. \\]\n\tBut reducibility coming from $W_\\sigma^0$ has been taken care of by the co-rank one reducibility condition which only occurs within $L$ by the assumption. On the other hand, the Knapp-- Stein $R$-group theory helps us control the multiplicity issue.\n\t\n\tTherefore it reduces to show that the non-zero intertwining operator $A(w,\\sigma)$ associated to $w\\in W_M(L)$ is an isomorphism, i.e.\n\t\\[A(w,\\sigma):~I_P^G(\\sigma)\\stackrel{\\sim}{\\longrightarrow} I_P^G(\\sigma^w). \\]\n\tRecall that $A(w,\\sigma)$ is defined as follows:\n\t\\[J_{P|P^w}(\\sigma^w)\\circ \\lambda(w):~I^G_P(\\sigma)\\longrightarrow I^G_{P^w}(\\sigma^w)\\longrightarrow I^G_P(\\sigma^w). \\]\n\tThus the above isomorphism claim follows from the associativity property of intertwining operators (cf.\\cite[IV.3 (4)]{waldspurger2003formule} or \\cite[Lemma 3.5]{luo2018R}), i.e.\n\t\\[J_{P|P^w}(\\sigma^w)J_{P^w|P}(\\sigma)=\\prod j_\\alpha(\\sigma)J_{P|P}(\\sigma), \\]\n\twhere $\\alpha$ runs over\n\t\\[\\Phi_M(P)\\bigcap \\Phi_M(\\overline{P^w}) \\]\n\twith $\\overline{P^w}$ the opposite parabolic subgroup of $P^w$, and $\\Phi_M(P)$ (resp. $\\Phi_M(\\overline{P^w})$) is the set of restricted roots of $M$ in $P$ (resp. $\\overline{P^w}$).\n\t\n\tNote that for $\\alpha\\in \\Phi_M(P)-\\Phi_M^0$, the associated co-rank one induction is always irreducible and $j_\\alpha(\\sigma)\\neq 0,\\infty$ (cf. \\cite[Corollary 1.8]{silberger1980special}). \n\t\n\tNote also that for $\\alpha\\in \\Phi_M^{0}-\\Phi_M^{L,0}$, the associated co-rank one induction is always irreducible by our Working Hypothesis, which implies that either $j_\\alpha(\\sigma)\\neq 0,\\infty$ for non-unitary induction (cf. \\cite[Corollary 1.8]{silberger1980special}), or $j_\\alpha(\\sigma)$ has a pole of order 2 (cf. \\cite[Proposition 2]{savin2007}). For the latter case, one can take the residue to get an isomorphism. \n\t\n\tTherefore one only needs to consider those $j_\\alpha(\\sigma)$ with $\\alpha\\in \\Phi_M^{L,0}$. \n\t\n\tFor those $\\alpha\\in \\Phi_M^{L,0}$, we have \n\t\\[w.(\\Phi_M^{L,0})^+>0, \\]\n\tso\n\t\\[\\Phi_M(P)\\bigcap \\Phi_M(\\overline{P^w})\\bigcap \\Phi_M^{L,0}=\\emptyset.  \\]\n\tThus $A(w,\\sigma)$ is an isomorphism.\n\\end{proof}\nIt seems that our proof of Theorem \\ref{prod} is quite restrictive. But it also seems that this may be the only universal argument we could come up with nowadays which has since been practiced intelligently by Bernstein--Zelevinsky, Silberger, Jantzen, Moeglin, the Tadi{\\'c} school etc (please refer to \\cite{bernstein1977induced,jantzen1997supports,moeglin2002construction,tadic1998regular,lapid2017some} for a glimpse). Inspired by a conjectural Muller type irreducibility criterion for general parabolic inductions (see \\cite[Conjecture 4.4]{luo2018muller}), a detailed study of the structure of Jacquet modules of a general parabolic induction may give us hope to prove the following conjectural universal irreducibility structure:\n\\begin{conj}\\label{conj}\n\tGiven a parabolic induction $I^G_{P=MN}(\\sigma)$ with $\\sigma$ an irreducible admissible representation of $M$, if the reducibility conditions of $I^G_P(\\sigma)$ lie in some standard Levi subgroup $L$ of a parabolic subgroup $Q=LV\\subset P=MN$, then $I^G_Q(\\tau)$ is always irreducible for any $\\tau\\in JH(I^L_{L\\cap P}(\\sigma))$.\n\\end{conj}\n\\begin{rem}\n\tAt first glance, Conjecture \\ref{conj} may look meaningless and ridiculous. But some supportive examples of a conjectural Muller type irreducibility criterion for general parabolic induction, i.e. \\cite[Conjecture 4.4]{luo2018muller} could guide us to a right direction. Those examples are parabolic induction representations inducing from essentially discrete series and standard modules which many mathematicians have investigated since the era of Harish-Chandra.\n\\end{rem}\nDenote by $\\Theta_Q$ the associated subset of $\\Delta$ which determines the parabolic subgroup $Q=LV\\supset P=MN$ of $G$. Explicitly, we decompose $\\Theta_L=\\Theta_1\\sqcup\\cdots \\sqcup \\Theta_t $ into irreducible pieces, and accordingly $\\Theta_M=\\Theta_1^M\\sqcup \\cdots \\sqcup \\Theta_t^M$. Assume that $R_\\sigma$ decomposes into $R_\\sigma=R_1\\times \\cdots \\times R_t$ with respect to the decomposition of $\\Theta_L$, and a similar decomposition pattern holds for the co-rank one reducibility, i.e. co-rank one reducibility only occurs within $P_{\\Theta_i}=M_{\\Theta_i}N_{\\Theta_i}$ for $1\\leq i\\leq t$. Then we have  \n\\begin{cor}[Product formula]\\label{prod1}\n\t\\[\\#(JH(I^G_P(\\sigma)))=\\prod_{i=1}^{t}\\#(JH(I_{M_{\\Theta_i^M}}^{M_{\\Theta_i}}(\\sigma))).\\]\n\\end{cor}\n\\begin{rem}\n\tVery recently, we learned that Jantzen has a beautiful product formula for split $Sp_{2n}$, $SO_{2n+1}$ and $O_{2n}$ which in some sense originates from Goldberg's product formula for tempered inductions (cf. \\cite{jantzen1997supports,jantzen2005duality,goldberg1994reducibility}). Note that our new $R$-group is always trivial for those groups in Goldberg and Jantzen's works. Inspired by their beautiful theorems, we would like to investigate what kind of general product formula we could prove under our new notion of $R$-group in our future work.\n\\end{rem}\nAs an instance, one version of Goldberg--Jantzen type product formula is as follows. Consider the tempered generalized principal series $I^G_P(\\sigma)$ with $\\sigma$ unitary supercuspidal representation of $M$, we know that $\\Phi_\\sigma:=\\{\\alpha\\in\\Phi_M:~w_\\alpha.\\sigma=\\sigma \\}$ is a root system, may be reducible. Decomposing $\\Phi_\\sigma$ into irreducible pieces, i.e. $\\Phi_\\sigma=\\sqcup_i\\Phi_{\\sigma,i}$. Each irreducible root system $\\Phi_{\\sigma,i}$ gives rise to a Levi subgroup $M_i\\supset M$ which is defined by\n\\[M_i:=C_G((\\bigcap\\limits_{\\alpha\\in\\Phi_{\\sigma,i}}Ker(\\alpha) )^0). \\]\nAssume that our new $R$-group $R_\\sigma$ associated to $\\sigma$ decomposes into a product of small pieces in the same pattern as does $\\Phi_\\sigma$, i.e.\n\\[R_\\sigma=\\prod_i R_{\\sigma,i} \\]\nwith $R_{\\sigma,i}$ a subgroup of the relative Weyl group $W^{M_i}_M=N_{M_i}(M)\/M$ of $M$ in $M_i$, then we have\n\\begin{conj}(Goldberg--Jantzen type product formula)\\label{gjconj}\n\tKeep the notions as above. For $\\nu\\in \\mathfrak{a}_M^*$, there is a one-one correspondence:\n\t\\[JH(I^G_P(\\nu,\\sigma))\\leftrightarrows \\prod_i JH(I^{M_i}_{M_i\\cap P}(\\nu,\\sigma)). \\]\n\\end{conj} \n\n\\subsection{Generic irreducibility property of parabolic induction}\nIn this subsection, we provide a new simple proof of the generic irreducibility property of parabolic inductions which plays an essential role in Harish-Chandra's Plancherel formula \\cite[IV.3]{waldspurger2003formule}.\n\nGiven an irreducible smooth representation $\\sigma$ of the Levi subgroup $M$ of a parabolic subgroup $P=MN$ in reductive group $G$, we form a family of normalized parabolic induction representations $I(\\nu,\\sigma)=Ind^G_P(\\sigma\\otimes \\nu)$ of $G$, where $\\nu$ varies in $\\mathfrak{a}_{M,\\mathbb{C}}^\\star$. The generic irreducibility property says that \n\\begin{thm}(Generic Irreducibility Theorem cf. \\cite{sauvageot})\n\tKeep the notions as above, we have\n\t\\[Irred_\\sigma:=\\{\\nu\\in \\mathfrak{a}_{M,\\mathbb{C}}^\\star:~I(\\nu,\\sigma) \\mbox{ is irreducible} \\}\\mbox{ is a non-trivial Zariski open subset.} \\]\n\\end{thm}\n\\begin{proof}\n\tIn view of the Langlands classification theorem, it reduces to the case where $\\pi$ is supercuspidal. Based on Theorem \\ref{prod} (or Muller type irreducibility criterion \\cite{luo2018muller}), the reducibility conditions are controlled by co-rank one reducibility and $R$-group. As there exists a unique reducibility point for the co-rank one case (cf. \\cite[Lemma 1.2 \\& 1.3]{silberger1980special}), thus it reduces to consider the $R$-group condition. Note that the $R$-group is a subgroup of $W_{\\sigma_\\nu}:=\\{w\\in W_M:~w.(\\sigma\\otimes\\nu)=\\sigma\\otimes \\nu \\}$, whence the non-trivial set $Irred_\\sigma$ is Zariski open.  \n\\end{proof}  \n\\section{Clozel's Finiteness Conjecture Of Special Exponents}\nIn this section, let us start with quoting Clozel's remark on Clozel's finiteness conjecture of special exponents in his second paper on Howe's finiteness conjecture \\cite[P. 3]{clozel1989orbital} as follows:\n\n\\textit{``We would like to finish the introduction with the remark that the stronger conjecture still retains some interest, although we do not know any obvious application. Here the analogy with the theory of automorphic forms is interesting. In the automorphic case, the analogue of the finiteness assumption about exponents would be the fact that the poles of Eisenstein series constructed from cusp forms on a given parabolic subgroup lie in a fixed, finite set independent of the inducing cusp form. This is a very strong conjecture, unknown even for $GL(n)$, although it would result from the conjectural description of the residues stated by Jacquet in [11]. If this conjecture was true, that would trivially imply that the operator defined by a smooth, $K$-finite function on the adelic group acting on the discrete spectrum is trace-class.''}\n\nAs an application of Theorem \\ref{prod}, under some conditions, we prove Clozel's finiteness conjecture of special exponents proposed in \\cite{clozel1985conjecture} which plays an essential role in Clozel's firt attempt to proving Howe's finiteness conjecture. Note that Clozel's finiteness conjecture may be checked directly for classical groups from Moeglin--Tadic's work on the classification of discrete series (cf. \\cite{moeglin2002construction,moeglin2007classification}). As the conjecture is much of a quantitative result, it should be proved with little forces, instead of resorting to such a big stick. Indeed, our proof is quite natural and may shed some lights on the global analogy. In what follows, we first recall some notions.\n\nThere is no harm to assume that $G$ is of compact center. Recall that for a discrete series representation $\\pi$ of $G$, we write its associated supercuspidal support as $\\sigma$ which is a supercuspidal representation of some Levi subgroup $M$ of $G$. Denote by $\\omega_\\sigma$ the unramified part of the central character of $\\sigma$, i.e. $\\omega_\\sigma\\in \\mathfrak{a}_{M,\\mathbb{C}}^\\star$. Such a character is called a \\underline{special exponent}.\n\\begin{thm}(Clozel's finiteness conjecture)\\label{clozel}\n\tThe set of special exponents is finite provided it holds for co-rank one cases.\n\\end{thm}\n\nBefore turning to the proof, let us first talk about the main idea.\n\nUnder the {\\bf Induction Assumption}, i.e.\n\\[\\mbox{Clozel's finiteness conjecture holds for the co-rank one case.}\\]\nOur proof of Clozel's finiteness conjecture for the general case rests on the following two novel observations:\n\\begin{enumerate}[(i)]\n\t\\item Theorem \\ref{prod}, or ``Product Formula''.\n\t\\item Irreducible induced representation can never be a discrete series.\n\\end{enumerate}\nRoughly speaking, with the help of Muller type theorem of generalized principal series, one knows that the decomposition of $I_P^G(\\sigma)$ is governed by the co-rank one reducibility and the $R$-group $R_\\sigma$. On the other hand, one knows that a full induced representation can not be a discrete series. In view of Theorem \\ref{prod}, thus in order to ensure $\\omega_\\sigma$ is a special exponent, those roots associated to the co-rank one reducibility and $R_\\sigma$ must generate the whole space $\\mathfrak{a}_{M,\\mathbb{C}}^\\star$. Then the conjecture follows from an easy fact of linear algebra, i.e. an invertible matrix has only one solution.\n\nTo be more precise, let $P_\\Theta=M_\\Theta N$ be a standard parabolic subroup of $G$ with $\\Theta\\subset \\Delta$, and let $\\sigma$ be a supercuspidal representation of $M_\\Theta$. Decomposing $\\Theta$ into irreducible pieces\n\\[\\Theta=\\Theta_1\\sqcup\\cdots\\sqcup\\Theta_n \\]\nAs $W_{M_\\Theta}$ acts on $M_\\Theta$, then it preserves the decomposition up to sign, so does $R_\\sigma$, i.e. preserving\n\\[\\pm\\Theta=\\pm\\Theta_1\\sqcup\\cdots\\sqcup\\pm\\Theta_n.\\] \nUnder the action of $R_\\sigma$ on $\\pm\\Theta$, we have a new decomposition of $\\pm\\Theta$ into irreducible pairs, i.e.\n\\[R_\\sigma\\hookrightarrow S_{\\pm n}, \\]\nwhere $S_{\\pm n}$ is the ``pseudo''-permutation group, i.e.\n\\[S_{\\pm n}:=\\{\\left( a_{i_1}\\cdots  a_{i_k}\\right):~ a_{i_j}\\in \\{\\pm 1,\\cdots,\\pm n\\} \\}\/\\pm. \\]\nHere $\\pm $-equivalence means that\n\\[\\left(a_{i_1}\\cdots a_{i_k}\\right)=\\left((- a_{i_1})\\cdots (-a_{i_k})\\right) .\\]\nFor each simple permutation $s=\\left(a_{i_1}\\cdots  a_{i_k}\\right)$, we define the associated roots as, up to scalar,\t\n\\[\\Phi_s:=\\{e_{i_j}-e_{i_l}:~1\\leq j<l\\leq k \\}, \\]\nwhere $e_{i_j}$ is the component character on $\\Theta_{i_j}$.\n\n\n\\begin{proof}\n\tIt suffices to prove the finiteness of special exponents for only one parabolic subgroup, like $P_\\Theta=M_\\Theta N$ with $\\sigma$ supercuspidal representations of $M_\\Theta$ and $\\Theta\\subset \\Delta$.\n\t\n\tConsidering the set $\\Phi_\\sigma$ of roots associated to the co-rank one reducibility and the $R$-group $R_\\sigma$, if \n\t\\[Span_\\mathbb{C}\\Phi_\\sigma\\neq \\mathfrak{a}_{M,\\mathbb{C}}^\\star, \\]\n\tthen one knows that, up to associated forms, \n\t\\[Span_\\mathbb{C}\\Phi_\\sigma= \\mathfrak{a}_{L,\\mathbb{C}}^\\star, \\]\n\tfor some parabolic subgroup $Q=LV$ with Levi $L\\supset M$. Then by Theorem \\ref{prod}, we know that \n\t\\[I^G_Q(\\tau)\\mbox{ is irreducible for all }\\tau\\in JH(I_{L\\cap P}^L(\\sigma)), \\]\n\twhich can not be discrete series. Therefore\n\t\\[Span_\\mathbb{C}\\Phi_\\sigma= \\mathfrak{a}_{M,\\mathbb{C}}^\\star.\\tag{SP} \\]\n\tWhich in turn says that the set of real parts of $\\{\\omega_\\sigma\\}_\\sigma$ is finite as there are only finitely many rank-one reducibility hyperplanes in $\\mathfrak{a}_M^\\star$ by the {\\bf Induction Assumption}. Indeed, we recently learned that such a claim (SP) is also a corollary of an old result of Harish-Chandra (cf. \\cite[Theorem 5.4.5.7]{silberger2015introduction}, \\cite[Theorem 3.9.1]{silberger1981discrete} or \\cite[Corollary 8.7]{heiermann2004decomposition}).\n\t\n\tAs for the set of imaginary parts of $\\{\\omega_\\sigma\\}_\\sigma$, the finiteness follows from the facts that\n\t\\begin{enumerate}[(i)]\n\t\t\\item There are finitely many $R_\\sigma$, and $R_\\sigma$ is finite. This says that there are only finitely many linearly independent subsets.\n\t\t\\item Each element in $R_\\sigma$ is of finite order. This says that there are only finitely many solutions for each linearly independent subset.\n\t\\end{enumerate}\nCombining the finiteness of the real parts and imaginary parts, Clozel's finiteness conjecture holds under the {\\bf Induction Assumption}.\t\n\\end{proof} \n\\begin{cor}\n\tLet $\\tilde{G}$ be a finite central covering group of $G$, then Clozel's finiteness conjecture holds for $\\tilde{G}$ under the {\\bf Induction Assumption}. \n\\end{cor}\n\\begin{proof}\n\tThis follows from the Induction Assumption that there are only finitely many rank-one reducibility hyperplanes in $\\mathfrak{a}_{M}^\\star$ and the R-group theory in \\cite{luo2017R}.\n\\end{proof}\n\\begin{rem}\n\tThe Induction Assumption is a byproduct of the profound Langlands--Shahidi theory for generic $\\sigma$s (cf. \\cite{shahidi1990proof}). But for non-generic $\\sigma$s, it follows from two conjectures of Shahidi (see \\cite[Conjectures 9.2 and 9.4]{shahidi1990proof}). Indeed, for classical groups, those two conjectures are by-products of applying Arthur's (twisted) stable trace formula machine in his monumental book \\cite{arthur2013endoscopic}. Moreover an explicit description of co-rank one reducibility points for classical groups is given in \\cite{moeglin2003points,moeglin2002classification,moeglin2002construction,moeglin2007classification}).\n\\end{rem}\n\\begin{rem}\n\tClozel's finiteness conjecture is also known for low rank groups of which their unitary duals are completely known (cf. \\cite{hanzer2006unitary,hanzer2010unitary,konno2001induced,luo2018unitary,luo2018unitary2,matic2010new,muic1998unitary,sally1993induced,schoemann2014unitary}).\n\\end{rem}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\nWe would like to thank K.S.~Babu, G.R.~Farrar, S.F.~King and H.-P.~Nilles\nfor conversations. The work of CK is supported \nby Department of Energy contract DE-FG02-90ER40542 and through \nthe generosity of Helen and Martin Chooljian. The work of SP is supported\nby the Polish Committee for\nScientific Research, grant 2P03B0412 (1997-8)\nand by the US-Poland Maria Sklodowska-Curie Fund MEN\/DOE-96-264.\nThe work of NP is\nsupported by National Science Foundation grant PHY-94-23002.\nWe would all like to thank\nthe Aspen Center for Physics, where this work was initiated, for its\nhospitality.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe fundamental role of conformal group $SO(4,2)$ for Minkowski space-time was\nfirst stressed by Dirac, \\cite{Dirac}. Its covering group $G =\nSU(2,2)$ describes conformal properties of spinning particles, see\n\\cite{Todorov}, where one can found a systematic introduction to\nthe subject. The group $G$ and its orbits are fundamental for the\ntwistor theory, \\cite{Penrose}. It is also of essential importance\nfor the ADS-CFT theory\\cite{Malda}.\n\nAll unitary irreducible representations of the group $G$ were\nclassified by \\cite{Knapp} and \\cite{Kihlberg}. More general case of $SU(m,n)$ is\ntreated in \\cite{Klimyk}. The discrete series $SU(2,2)$\nrepresentations were used by \\cite{Mack} and \\cite{Ruhl} for the\ninvestigation of conformal properties of fields on Minkowski\nspace. The highest\/lowest weight of the discrete series of representations has been studied by\n\\cite{Mack} and \\cite{Dobrev}.\n\nThe importance of the deformations theory for quantum systems was\nfirst stressed by \\cite{Bayen}. The deformation method was\ngeneralized to linear Poisson structures (related to Lie algebras)\nin \\cite{Rieffel} and to general Poisson structures\nin\\cite{Konts}. The relation between both approaches was described\nin \\cite{BenAmar}.\n\nThe star-product formula represents an approach going beyond\ndeformation theory. A general star-product approach, based on\ncoherent states on co-adjoint orbits \\cite{Perelomov}, was\nproposed in \\cite{GP}, for $SU(2)$ case the star-product formula\nwas found in \\cite{P}, see also \\cite{Samann} for $SU(n)$ orbits\n(the deformed algebra can be represented as a matrix algebra). A\ngeneral formula for compact Lie groups was derived in\n\\cite{Aleks}.\n\nFor non-compact Lie groups the situation is more complicated. The\ncase $SU(1,1)$ was briefly sketched in \\cite{GP}. Similar approach\nwas applied in \\cite{Jakim} to a particular $SU(2,2)$ orbit - the\ncomplex Minkowski space. The corresponding deformed noncommutative\nalgebra was represented in terms of 4 bosonic oscillator pairs.\n\nA noncommutative field theory may be defined provided the\nnoncommutative algebra of functions, with some additional\nstructures, is specified on a configuration space(time). For\n$SU(2)$ case this was done in \\cite{GP1} and followed by various\nother papers. For the noncommutative Heisenberg group, the\nformulation of a noncommutative quantum field theory on a Moyal\nspace, was given in \\cite{DFR}. Much more work has been done for\nmodels defined over the Euclidean deformed space-time. This\nculminated in \\cite{GW} and \\cite{Riv}, where a renormalizable\nnontrivial 4D model was found and studied. For a recent review see\n\\cite{Rivasseau}.\n\nThe noncommutative space-time model proposed  by H. S. Snyder and\nC. N. Yang (see \\cite{Snyder}), based on non-compact groups\n$SO(4,1)$ and $SO(5,1)$ has not been developed much further,\nmainly due to the success of renormalization theory approach to\nquantum field theory. In our opinion it could be a right time to\nreturn back to those old ideas.\n\nThe paper is organized as follows. In Section 1 we describe the\nLie group $G = SU(2,2)$ and its Lie algebra $\\textbf{g} =\nsu(2,2)$. The relevant mathematical background can be found, e.g.,\nin \\cite{Kirillov} and \\cite{Molchanov}. In Section 3 we present a\nsimple oscillator realization (in terms of 8 bosonic oscillator\npairs) of most degenerate discrete series of representations which\ngeneralizes more common (Schwinger-Jordan) oscillator realizations\nused in the case of compact groups. In Section 4 we construct the\nsystem of coherent states for the representation in question and we give\na corresponding star-product formula for the algebra of functions\non a Bergman domain $D$. Finally, in Section 5 we construct\na quantum field theory model on the quantized Bergman domain $D$.\n\n\n\n\\section{The group $\\boldsymbol{SU(2,2)}$ and its Lie algebra}\n\\subsection{The definition of $\\boldsymbol{SU(2,2)}$}\nThe group $G\\,=\\,SU(2,2)$ is the subgroup of $SL(4,\\bf{C})$\nmatrices satisfying\n\\begin{equation}\\label{1.1} g\\ =\\\n\\left(\\begin{array}{cc}a&b\\\\c&d\\end{array}\\right)\\, \\in\\, G\\\n\\Rightarrow\\ g^\\dagger\\,\\Gamma\\,g\\ =\\ \\Gamma\\,,\\ \\Gamma\\ =\\\n\\left(\\begin{array}{cc}E&0\\\\0&-E\\end{array}\\right)\\,.\\end{equation}\nHere all the entries $a$, $b$, $c$, $d$ are $2\\times2$- matrices\nand the symbols $0$ and $E$ denote the $2\\times2$ zero matrix and\nthe unit matrix, respectively. Inserting this into (\\ref{1.1}) one\nobtains two sets of equivalent constraints\n\\begin{equation}\\label{1.1a} \\begin{array}{c} a^\\dagger\\,a\\ =\\\nE\\,+\\,c^\\dagger\\,c\\,,\\ \\ c^\\dagger\\,d\\ =\\ E\\,+\\,b^\\dagger\\,b\\,,\\ \\\na^\\dagger\\,b\\ =\\ c^\\dagger\\, d\\,,\\end{array}\\end{equation}\nor,\n\\begin{equation}\\label{1.1b} \\begin{array}{c} a\\,a^\\dagger\\ =\\\nE\\,+\\,b\\,b^\\dagger\\,,\\ \\ d\\,d^\\dagger\\ =\\, E\\,+\\,c\\,c^\\dagger\\,,\\\n\\ a\\,c^\\dagger\\ =\\ b\\,d^\\dagger .\\end{array}\\end{equation}\n\\subsection{Maximal compact subgroup and Bergman domain }\nThe maximal compact subgroup of $SU(2,2)$ is $K=S(U(2)\\times U(2))$ which\nconsists of the matrices\n\\begin{equation}\n k\\ =\\ \\begin{pmatrix}\nk_1 & 0 \\\\\n0 & k_2 \\\\\n\\end{pmatrix},\\  k_{1,2}\\in U(2),\\ \\ \\  \\det(k_1)\\,\\det(k_2)=1.\n\\end{equation}\nThe corresponding Bergman domain $D$ is a kind of Type $1$ Cartan domain\nwhich defined as the group coset space:\n\\begin{equation}\n D\\,=\\,G\/K.\n\\end{equation}\nIt can be represented as the set of all complex $2\\times 2$ matrix\n\\begin{equation}\nZ=\\begin{pmatrix}\nz_{11} & z_{12} \\\\\nz_{21} & z_{22} \\\\\n\\end{pmatrix}\n\\end{equation}\nwith $Z^\\dagger Z<E$. The group action on $D$ is given by:\n\\begin{equation}\\label{action}\n Z'=gZ=(aZ+b)(cZ+d)^{-1}\n\\end{equation}\n\nThe Bergman domain $D$ is a pseudo-convex domain where we could define\nHilbert space $L^2(D, d\\mu_N)$ (see (\\ref{measure})) of holomorphic functions with reproducing kernel\n$K(Z,W)$, where $Z, W\\in D$. This reproducing kernel is also called\nBergman kernel and it is well known that\n\\begin{equation}\\label{n4}\n K(Z, W)=\\det(E-ZW^{\\dagger})^{-N}.\n\\end{equation}\nThe Bergman domain $D$ is an $8$ dimensional rank $2$ Hermitian\nsymmetric space. It is also a K\\\" ahler manifold of $4$ complex\ndimensions, with the metric given by\n\\begin{equation}\ng_{i\\bar j}=\\frac{\\partial^2}{\\partial_{z_i}\n\\partial_{\\bar{z_j}}}\\log( K(Z,\\bar Z))\n\\end{equation}\nThe topology of Bergman domain $D$ is nontrivial. It not simply connected but has genus $4$. One way to\ncalculate the genus is by studying the corresponding complex\nJordan triple. The interested reader could find more details in\n(\\cite{Upmeier}).\n\n\\subsection{The Lie algebra $\\textbf{g}$ and the Haar\nmeasure}\\label{root} Let $\\textbf{g}\\,=\\,su(2,2)$ be the Lie\nalgebra of G, so it is real and semisimple. It is formed by\nmatrices satisfying\n\\begin{equation}\\label{1.1b0} X^\\dagger\\,\\Gamma\\ +\\ \\Gamma\\,X\\ =\\ 0\\end{equation}\n\nConsider the Cartan decomposition of $\\textbf{g} $(see\n\\cite{Molchanov}, \\cite{PZh}, \\cite{Bob}):\n\\begin{equation}\\label{phys} \\textbf{g}=\\textbf{k}+\\textbf{p},\\end{equation}\nwhere $\\textbf{k}$ is the subset of all anti-hermitian matrices in\n$\\textbf{g}$\n\\begin{equation}\\textbf{k}=\\big\\{\\begin{pmatrix}\nA & 0 \\\\\n0 & D \\\\\n\\end{pmatrix}: A^\\dagger=-A,\\,D^\\dagger=-D,\\,\n \\mbox{tr}(A+D)=0,\\,\nA,D \\in M_2(C)\\big\\}.\\end{equation}\nThe set $\\textbf{k}$ is the Lie algebra of the maximal compact\nsubgroup $K$ in $G$. The subset $\\textbf{p}$  of all hermitian\nmatrices in $\\textbf{g}$\n\\begin{equation}\\label{B} \\textbf{p} =\n\\big\\{\\begin{pmatrix}\n0 & B \\\\\nB^\\dagger & 0 \\\\\n\\end{pmatrix}: B \\in M_2(C)\\big\\}\n\\end{equation}\nis just a linear space and not a Lie algebra.\n\nLet $\\textbf{a}$ be a maximal Abelian subalgebra in $\\textbf{p}$ .\nWe may choose for $\\textbf{a}$ the set of all matrices of the form\n\\begin{equation}\nH_\\Lambda=\\begin{pmatrix}\n0 & \\Lambda \\\\\n\\Lambda & 0 \\\\\n\\end{pmatrix}\n\\end{equation}\nwhere $0$ means the $2\\times 2$ matrix with entries zeros and\n$\\Lambda = \\mbox{diag}(\\lambda_1,\\lambda_2)$ is diagonal $2\\times\n2$ with $\\lambda_1$, $\\lambda_2$ real.\nThe corresponding subgroup consists of all matrices of the type:\n\\begin{equation}\\label{boost}\n\\delta_\\Lambda =\\begin{pmatrix}\nC & S \\\\\nS & C \\\\\n\\end{pmatrix},\\ \\ \\\n\\begin{array}{c}C=\\mbox{diag}(\\mbox{ch}\\lambda_1,\\mbox{ch}\n\\lambda_2),\\\\\nS=\\mbox{diag}(\\mbox{sh}\\lambda_1,\\mbox{sh}\\lambda_2).\\end{array}\n\\end{equation}\nDefine the dual space $\\textbf{a}^\\star$ spanned by the elements\n$\\alpha_i$ satisfying $\\alpha_i(H_\\Lambda)=\\lambda_i$.\n\nThen the roots of $(\\textbf{g},\\textbf{a})$ are given by\n\\begin{equation}\n\\pm 2\\alpha_1,\\  \\pm 2\\alpha_2,\\  \\pm (\\alpha_1-\\alpha_2)\n\\end{equation}\nwith multiplicities $m_{2\\alpha_1}=m_{2\\alpha_2}=1$ and\n$m_{\\alpha_1\\pm\\alpha_2}=2$.\n\nOn the root system we choose that the positive Weyl chamber  given\nby $C^{+}=\\{\\lambda_1, \\lambda_2\\}$ with $\\lambda_1>\\lambda_2>0$.\nThen the positive roots are $2\\alpha_1$, $2\\alpha_2$ and\n$(\\alpha_1\\pm \\alpha_2)$. We use $\\Sigma$ donate the set of all\nroots and $\\Sigma^+$ the set of positive roots.\n\n\nDefine\n\\begin{equation}\n\\rho=\\frac{1}{2}\\sum_{\\alpha\\in\\Sigma^+}m_\\alpha \\alpha\n\\end{equation}\nSo we have\n\\begin{equation}\n\\rho=\\alpha_1+\\alpha_2+(\\alpha_1+\\alpha_2)+\n(\\alpha_1-\\alpha_2)=3\\alpha_1+\\alpha_2\n\\end{equation}\n\nLet $\\textbf{a}_c$ be the complex extension of $\\textbf{a}$. Follow the same procedure as shown above we could define the complex roots\n$\\alpha_c\\in\\textbf{a}_c^*$ as\n\\begin{equation}\\label{harish}\n\\alpha_c^i(\\textbf{a}_c)=\\tau_i,\\ \\  i=1, 2.\n\\end{equation}\nHere $\\tau_i$ are complex numbers. The formula of $\\rho$ and $\\tau_i$ will be used\nfor constructing the eigenfunction of the radial part of the\ninvariant Laplacian. See section \\ref{field}.\n\nNow we have some physical interpretation of the Lie algebra: the\nfirst summand in formula (\\ref{phys}) represents compact\ngenerators of {\\it rotations}, whereas the second one represents\nnon-compact generators {\\it boosts}. Since the Lie algebras\n$su(2,2)$ and $so(4,2)$ are isomorphic we shall label rotations as\n$X_{05}$ and $x_{ab}$, $a,b\\,=\\,1,2,3,4$, and boosts as $X_{0a}$\nand $X_{a5}$ considering them as generators $X_{AB}\\,=\\,-X_{BA}$,\n$A,B\\,=\\,0,1,\\,\\dots,\\,5$, satisfying $so(4,2)$ commutation\nrelations\n\\begin{equation}\\label{1.1b1} [X_{AB},X_{CD}]\\ =\\ \\eta^{AC}\\,X_{BD}\\,-\\,\n\\eta^{BC}\\,X_{AD}\\,+\\, \\eta^{BD}\\,X_{AC}\\,-\\, \\eta^{AD}\\,X_{BC}\\,,\\end{equation}\nwith the metric tensor $\\eta^{AB}\\,=\\,\n\\mbox{diag}(+1,-1,-1,-1,-1,+1)$. Explicitly, the compact Lie\nalgebra $\\textbf{k}$ is spanned by 7 anti-hermitian matrices and\nthe basis of $\\textbf{p}$ is formed by 8 hermitian matrices given\nbelow:\n\\[ S_{05} = \\frac{i}{2}\\left(\\begin{array}{cc}1&0\\\\\n0&-1\\end{array}\\right),\\ S_{j4}\\ =\\ \\frac{i}{2}\\,\\left(\n\\begin{array}{cc}\\sigma_j&0\\\\0&-\\sigma_j\\end{array}\\right),\\\nS_{ij} = \\frac{i}{2}\\varepsilon_{ijk}\\left(\n\\begin{array}{cc}\\sigma_k&0\\\\0&\\sigma_k\\end{array}\\right),\\]\n$$ S_{k5} = \\frac{i}{2}\\left(\\begin{array}{cc}0&\\sigma_k\\\\\n-\\sigma_k&0\\end{array}\\right),\\ S_{0k} = \\frac{1}{2}\n\\left(\\begin{array}{cc}0&\\sigma_k\\\\\\sigma_k&0\n\\end{array}\\right),$$\n\\begin{equation}\\label{1.7} S_{45} = \\frac{1}{2}\\left(\\begin{array}{cc}0&1\n\\\\1&0\\end{array}\\right),\\ S_{04} = \\frac{1}{2}\\,\\left(\n\\begin{array}{cc}0&i\\\\-i&0\\end{array}\\right),\\end{equation}\nwhere $i,j,k\\,=\\,1,2,3$ and $\\sigma_1,\\,\\sigma_2,\\,\\sigma_3$ are\nusual Pauli matrices. \\vskip0.5cm\n\n\nThe principal Cartan subalgebra\n\\begin{equation}\\label{1.1b3} \\textbf{h}\\ =\\ \\textbf{a}\\,\\oplus\\,\\textbf{u}\\end{equation}\nof $G$ is spanned by three commuting generators: two noncompact\n$X_{45}$ and $X_{03}$ span $\\textbf{a}$, and the additional\ncompact one $X_{12}$ spans $\\textbf{u}$. The corresponding\nsubgroups we shall denote as $H$, $A$ and $U$:  $H\\,=\\,A\\times U$.\n\n\nAny element of $G$ possesses a unique Cartan decompositions\n\\begin{equation}\\label{1.1b4} g\\ =\\ k\\,\\delta\\,\\tilde{q}\\ =\\ \\tilde{k}\\,\\delta\\,q\\,,\\end{equation}\nwhere $\\delta$ is some pure positive non-compact element of $H$ with\npositive $\\lambda_1$ and $\\lambda_2$ given by formula (\\ref{boost}).\nFurther, $k\\,=\\,\\tilde{k}\\,u$, and $q\\,=\\,u\\,\\tilde{q}$ are\nelements of $K$, and $u$ is the element of a compact subgroup $U$\nin $H$.\n\nThe Haar measure $dg$ on $G\\,=\\,SU(2,2)$ is, in the parametrization\n(\\ref{1.1b4}), given as\n\\begin{equation}\\label{1.4b} dg\\ \\equiv\\ dg(\\lambda,k,q)\\ =\\ \\rho(\\lambda_1,\n\\lambda_2)\\,d\\lambda_1\\,d\\lambda_2\\,d\\tilde{k}\\,d\\tilde{q}\\,du\\,,\\end{equation}\nwhere $d\\tilde{k}$ and $d\\tilde{q}$ denotes the normalized\nmeasures on $K\/U$, and $du$ is the usual measure on $U$ (see\n\\cite{Ruhl}). The explicit form of $\\rho(\\lambda_1, \\lambda_2)$ is\nconstructed from the positive roots (see Section \\ref{root}):\n\\begin{equation}\n\\rho(\\Lambda)=\\prod_{\\alpha\\in\\Sigma^+}|\\sinh\\alpha(T)|^{m_\\alpha}\n\\end{equation}\nwhere $m_\\alpha$ is the multiplicity of the positive roots. So we\nhave:\n\\begin{equation} \\rho(\\lambda_1,\\lambda_2)\\ =\\\n\\mbox{sh}^2\\,(\\lambda_1+\\lambda_2)\n\\,\\mbox{sh}^2\\,(\\lambda_1-\\lambda_2)\\,\\mbox{sh}\\,(2\\lambda_1)\\,\n\\mbox{sh}\\,(2\\lambda_2)\\,.\n\\end{equation}\n\\vskip1cm\n\n\\section{Discrete series representation of $\\boldsymbol{SU(2,2)}$}\n\nThe group $G\\,=\\,SU(2,2)$ possesses principal, supplementary and\ndiscrete series of unitary irreducible representations, see e.g.,\n\\cite{Kihlberg}, \\cite{Klimyk}. The discrete series of\nrepresentations is given by:\n\\begin{equation}\\label{rep}\n T_g f(Z)=[\\det(C Z+ D)]^{-N}f(Z'), \\ N=4, 5, 6\\cdots\n\\end{equation}\nwhere $Z'$ is given by formula (\\ref{action}) and \n$f(Z)\\in L^2(D,d\\mu_N)$ with the measure\n\\begin{equation}\\label{measure} d\\mu_N(\\bar{Z}, Z)\\ =\\ \nc_N\\,d\\bar{Z}\\,dZ\\,\\mbox{det}\n\\,(E-Z^\\dagger Z)^{N-4}. \\end{equation}\nThe normalization constant $c_N\\,=\\,\\pi^{-4}(N-1)(N-2)^2 \n(N-3)$ guarantees that the function $f_0(Z)\\,\\equiv 1$ \nhas a unit norm, see \\cite{Ruhl}.\n\nWe introduce a $4\\times 2$ matrix $\\hat{Z} = (\\hat{z}_{a\\alpha})$,\n$a\\,=\\,1,\\,\\dots,\\,4$, $\\alpha\\,=\\,1,2$, of bosonic oscillators\nacting in Fock space and satisfying commutation relations\n$$ [\\hat{z}_{a\\alpha},\\hat{z}^\\dagger_{b\\beta}]\\ =\\ -\\Gamma_{ab}\\,\n\\delta_{\\alpha\\beta}\\,, $$\n\\begin{equation}\\label{2.1}  [\\hat{z}_{a\\alpha},\\hat{z}_{b\\beta}]\\ =\\\n[\\hat{z}^\\dagger_{a\\alpha},\\hat{z}^\\dagger_{b\\beta}]\\ =\\ 0\\,,\\end{equation}\nwhere $\\Gamma$ is a $4\\times 4$ matrix defined in (\\ref{1.1}). It can\nbe easily seen that for all  $g\\,\\in\\,SU(2,2)$ these commutation\nrelations are invariant under transformations:\n\\begin{equation}\\label{2.1a} \\hat{Z}\\ \\mapsto\\ g\\,\\hat{Z},\\ \\ \\ \\hat{Z}^\\dagger\\\n\\mapsto\\ \\hat{Z}^\\dagger\\,g^\\dagger .\\end{equation}\nSince, $\\Gamma\\,=\\,\\mbox{diag}(+1,+1,-1,-1)$ the upper two rows in\n$\\hat{Z}$ corresponds to creation operators whereas the lower ones\nto annihilation operators:\n\\begin{equation}\\label{2.2}  \\hat{Z}\\ =\\ \\left(\\begin{array}{c}\\hat{a}^\\dagger\\\\\n\\hat{b}\\end{array}\\right):\\ \\   [\\hat{a}_{\\alpha\\beta},\n\\hat{a}^\\dagger_{\\gamma\\delta}]\\ =\\  [\\hat{b}_{\\alpha\\beta},\n\\hat{b}^\\dagger_{\\gamma\\delta}]\\ =\\ \\delta_{\\alpha\\beta}\\,\n\\delta_{\\gamma\\delta},\\ \\  \\alpha,\\beta,\\gamma,\\delta\\,=\\,1,2. \\end{equation}\nand all other commutation relations among oscillator operators vanish.\nThe Fock space  ${\\cal F}$ in question is generated from a normalized\nvacuum state $|0\\rangle$, satisfying  $\\hat{a}_{\\alpha\\beta}\\,\n|0\\rangle\\,=\\,\\hat{b}_{\\alpha\\beta}\\,|0\\rangle\\,=\\,0$, by repeated\nactions of creation operators:\n\\begin{equation}\\label{2.3} |m_{\\alpha\\beta},\\,n_{\\alpha\\beta}\\rangle\\ =\\\n\\prod_{\\alpha\\beta}\\,\\frac{(\\hat{a}^\\dagger_{\\alpha\\beta}\n)^{m_{\\alpha\\beta}}\\,(\\hat{b}^\\dagger_{\\alpha\\beta}\n)^{n_{\\alpha\\beta}}}{\\sqrt{m_{\\alpha\\beta}!\\,n_{\\alpha\\beta}!}}\n\\,|0\\rangle\\,.\\end{equation}\nWe shall use the terminology that the state $|m_{\\alpha\\beta},\n\\,n_{\\alpha\\beta}\\rangle$ contains $m=\\sum m_{\\alpha\\beta}$\nparticles $a$ and $n=\\sum n_{\\alpha\\beta}$ particles\n$b$.\\vskip0.5cm\n\nThe Lie algebra $su(2,2)=so(4,2)$ acting in Fock space can be\nrealized in terms of oscillators. To any $4\\times 4$ matrix\n$X\\,=\\,(X_{ab})$ we assign the operator\n\\begin{equation}\\label{2.3a} \\hat{X}\\,=\\,-\\mbox{tr}(\\hat{Z}^\\dagger\\Gamma\nX\\hat{Z})\\ =\\ -\\hat{z}^\\dagger_{a\\alpha}\\,\n\\Gamma_{ab}\\,X_{bc}\\,\\hat{z}_{c\\alpha} \\,,\\end{equation}\nwith $\\hat{Z}^\\dagger$ and $\\hat{Z}$ given in (\\ref{2.1}) and\n(\\ref{2.2}) in terms of oscillators. Using commutation relations\nfor annihilation and creation operators the commutator of\noperators $\\hat{X}\\,=\\,-\\mbox{tr}(\\hat{Z}^\\dagger\\Gamma X\\hat{Z})$\nand $\\hat{Y}\\,=-\\,\\mbox{tr}(\\hat{Z}^\\dagger\\Gamma Y\\hat{Z})$ can\nbe easily calculated:\n\\begin{equation}\\label{2.4} [\\hat{X},\\hat{Y}]\\ =\\ [\\mbox{tr}\n(\\hat{Z}^\\dagger\\Gamma X\\hat{Z}),\\mbox{tr}(\\hat{Z}^\\dagger\\Gamma\nY\\hat{Z})]\\ =\\ -\\mbox{tr}[\\hat{ Z}^\\dagger\\,\\Gamma\\,(XY-YX)\\,\n\\hat{Z}]\\,.\\end{equation}\nIt can be easily seen that the {\\it anti-hermitian operators}\n\\begin{equation}\\label{2.3b} \\hat{X}_{AB}\\,=\\,-\\mbox{tr}(\\hat{Z}^\\dagger\\Gamma\nX_{AB}\\hat{Z}),\\ \\ A,B=0,1, \\dots,5,\\end{equation}\nwith $X_{AB}$ given in (\\ref{1.7}), satisfy in Fock space the\n$su(2,2)\\cong so(4,2)$ commutation relations (\\ref{1.1b1}).  The\nassignment\n\\begin{equation}\\label{2.4a}  g\\,=\\,e^{\\xi^{AB}X_{AB}}\\,\\in\\,SU(2,2)\n\\Rightarrow\\ \\hat{T}(g)\\,=\\,e^{\\xi^{AB}\\hat{X}_{AB}} \\end{equation}\nthen defines a {\\it unitary $SU(2,2)$ representation} in Fock\nspace.\n\nThe adjoint action of $\\hat{T}(g)$ on operators reproduces\n(\\ref{2.1a}). In terms of $a$ and $b$-oscillators in block-matrix\nnotation this can be rewritten as\n\\begin{equation}\\label{2.4b}  g=\\left(\\begin{array}{cc}a&b\\\\c&d\\end{array}\n\\right):\\ \\begin{array}{cc} \\hat{T}(g)\\,\\hat{a}^\\dagger\\,\n\\hat{T}^\\dagger(g)\\,=\\,a\\,\\hat{a}^\\dagger+b\\,\\hat{b}, &\n\\hat{T}(g)\\,\\hat{a}\\hat{T}^\\dagger(g)\\,=\\,\n\\hat{a}\\,a^\\dagger+\\hat{b}^\\dagger\\,b^\\dagger,\\\\\n\\hat{T}(g)\\,\\hat{b} \\,\\hat{T}^\\dagger(g)\\,=\\,\nd\\,\\hat{b}+c\\,\\hat{a}^\\dagger, & \\hat{T}(g)\\,\\hat{b}^\\dagger\\,\n\\hat{T}^\\dagger(g)\\,=\\, \\hat{b}^\\dagger\\,d^\\dagger+\n\\hat{a}\\,c^\\dagger.\\end{array}\\end{equation}\nSince any $g\\,\\in\\,SU(2,2)$ possesses Cartan decomposition\n(\\ref{1.1b4}) we shall discuss separately rotations and special\nboosts given in (9). For rotations we obtain a mixing of\nannihilation and creation operators of a same type:\n\\begin{equation}\\label{k} k=\\left(\\begin{array}{cc} k'&0\\\\0&k^{\\prime\\prime}\n\\end{array}\\right):\\ \\begin{array}{cc} \\hat{T}(k)\\,\\hat{a}^\\dagger\n\\,\\hat{T}^\\dagger(k)\\,=\\,k'\\,\\hat{a}^\\dagger, & \\hat{T}(k)\\,\n\\hat{a}\\hat{T}^\\dagger(g)\\,=\\,\\hat{a}\\,k^{\\prime\\dagger},\\\\\n\\hat{T}(g)\\,\\hat{b}\\,\\hat{T}^\\dagger(k)\\,=\\,k^{\\prime\\prime}\\,\\hat{b},\n&\\hat{T}(k)\\,\\hat{b}^\\dagger\\,\\hat{T}^\\dagger(k)\\,=\\,\\hat{b}^\\dagger\n\\,k^{\\prime\\prime\\dagger}.\\end{array} \\end{equation}\nHowever, for special boosts from $\\textbf{a}$ we obtain Bogolyubov\ntransformations:\n\\begin{equation} \\delta =\\left(\\begin{array}{cc}C&S\\\\S&C\\end{array} \\right):\\\n\\begin{array}{cc} \\hat{T}(\\delta)\\,\\hat{a}^\\dagger\\,\\hat{T}^\\dagger(g)\n\\,=\\,C\\,\\hat{a}^\\dagger+S\\,\\hat{b}, & \\hat{T}(\\delta)\\,\\hat{a}\n\\hat{T}^\\dagger(\\delta)\\,=\\,\\hat{a}\\,C+\\hat{b}\\,S,\\\\\n\\hat{T}(\\delta)\\,\\hat{b}\\,\\hat{T}^\\dagger(\\delta )\\,=\\,C\\,\n\\hat{b}+S\\,\\hat{a}^\\dagger, &\\hat{T}(\\delta )\\,\\hat{b}^\\dagger\\,\n\\hat{T}^\\dagger(\\delta )\\,=\\,\\hat{b}^\\dagger\\,C+\\hat{a}\\,S,\n\\end{array}\\end{equation}\nwith $C$ and $S$ determined in (\\ref{boost}).\n\nUsing the explicit form of matrices $X_{AB}$, following from\n(\\ref{1.1b0}), the action of generators can be described in terms\ncreation and annihilation of $a$- and $b$-particles:\n\n(i) The action of {\\it rotation} generators results in a replacement\nof some $a$-particle by an other $a$-particle and by replacement of\n$b$-particle by other $(ab)$-particle.\n\n(ii)  The action of {\\it boost} generators results in creation of a\npair $(ab)$ of particles or in a destruction of  $ab$  pair.\n\nIn this context it is useful to consider lowering and rising\noperators labeled by arbitrary $2\\times 2$ complex\\ matrix\n$B\\,=\\,(B_{\\beta\\gamma})$ entering (\\ref{B}), that annihilate and\ncreate $ab$ pairs:\n\\begin{equation}\\label{2.5} \\hat{T}^B_-\\ =\\\n\\hat{a}_{\\alpha\\beta}\\,B_{\\beta\\gamma}\\,\\hat{b}_{\\gamma\\alpha},\\ \\\n\\ \\hat{T}^B_+\\ =\\ (\\hat{T}^B_+)^\\dagger\\ =\\\n\\hat{a}^\\dagger_{\\alpha\\beta}\\,B^*_{\\beta\\gamma}\\,\n\\hat{b}^\\dagger_{\\gamma\\alpha}.\\end{equation}\nMore specifically, we can consider $B = E_{\\beta \\gamma}$ (the\nmatrix with $1$ in the intersection of $\\beta$-th row and\n$\\gamma$-th column and $0$'s otherwise). The corresponding\nlowering and rising operators are:\n\\begin{equation}\\label{2.5a} \\hat{T}^{\\beta \\gamma}_-\\ =\\\n\\hat{a}_{\\alpha\\beta}\\,\\hat{b}_{\\gamma\\alpha},\\ \\ \\ \\hat{T}^{\\beta\n\\gamma}_+\\ =\\ (\\hat{T}^{\\beta \\gamma}_+)^\\dagger\\ =\\\n\\hat{a}^\\dagger_{\\alpha\\beta}\\,\\hat{b}^\\dagger_{\\gamma\\alpha}.\\end{equation}\nIn (\\ref{2.5}) and (\\ref{2.5a}) the summation over $\\alpha$ is\nunderstood. Any boost can be uniquely expressed as complex\ncombinations of operators $T^{\\beta \\gamma}_-$ and $T^{\\beta\n\\gamma}_+$. It follows from (\\ref{2.5}) that the operator\n\\begin{equation}\\label{2.6}   \\hat{N}\\equiv\n\\frac{1}{2}(\\hat{N}_{\\hat{b}}-\\hat{N}_{\\hat{b}})\\ =\\\n\\frac{1}{2}(\\hat{b}^\\dagger_{\\alpha\\beta}\\hat{b}_{\\alpha\\beta}\n\\,-\\,\\hat{a}^\\dagger_{\\alpha\\beta}\\hat{a}_{\\alpha\\beta})\\ =\\\n\\frac{1}{2}Tr(\\hat{Z}^\\dagger\\Gamma\\hat{Z})\\,-\\,2 \\end{equation}\ncommutes with all generators $\\hat{X}_{AB}$, $A,B\\,=\\,0,1,\\,\n\\dots,\\,5$. \\vskip0.5cm\n\nBelow, we shall restrict ourselves to most degenerate discrete\nseries representations which are specified by the eigenvalue\nof the operator $ \\hat{N}$  in the representation subspace.\nWe start to construct the representation space ${\\cal F}_N$\nfrom a distinguished normalized state containing lowest number\nof particles:\n\\begin{eqnarray}\\label{2.7} |x_0\\rangle\\ &=&\\\n\\frac{\\mbox{det}(\\hat{b}^\\dagger)^{N-1}}{(N-1)!\\,\\sqrt{N}}\\ |0\\rangle,\\ N=1, 2, \\cdots\n\\nonumber\n\\\\\n &=&\\ \\frac{1}{\\sqrt{N}}\\sum_{n=0}^{N-1}\n(-1)^n\\,|\\,0,0,0,0\\,;\\,N-1-n,n,n,N-1-n\\rangle\\,.\\end{eqnarray}\nHere, $\\hat{b}^\\dagger\\,=\\,(\\hat{b}^\\dagger_{\\alpha\\beta})$ is\n$2\\times 2$ matrix of $b$-particle creation operators, and $N$ is\na natural number that specifies the representation: $\n\\hat{N}\\,|x_0\\rangle\\,=\\,(N-1)\\,|x_0\\rangle$. All other states in the\nrepresentation space are obtained by the action of rising\noperators given in (\\ref{2.5}): such states contain besides $(2N-2)$\n$b$-particles a finite number of $ab$ pairs. \\vskip0.5cm\n\nThe maximal compact subgroup $K\\,=\\,S(U(2)\\times U(2))$ is the\nstability group of the state $|x_0\\rangle$. The group action for\n$k\\,=\\,\\mbox{diag}(k',k^{\\prime\\prime})$ reduces just to the phase\ntransformation  (see (\\ref{k})):\n\\begin{equation}\\label{2.7a} \\hat{T}(k)\\,|x_0\\rangle\\ =\\\n\\mbox{det}(k^{\\prime\\prime\\dagger})^{N-1}\\,\\mbox{det}(k')\\,\n|x_0\\rangle\\ =\\\n\\mbox{det}(k^{\\prime\\prime})^{-N}\\,|x_0\\rangle\\,.\\end{equation}\nThe first factor comes from $\\hat{b}^\\dagger\\,\\mapsto\\,\n\\hat{b}^\\dagger\\,k^{\\prime\\prime\\dagger}$ (see (\\ref{k})), whereas\nthe second factor comes from\n$\\hat{T}(k)\\,|0\\rangle\\,=\\,\\mbox{det}(k')\\,|0\\rangle$ (due to the\nanti-normal ordering of the compact generators containing\n$\\hat{a}$ and $\\hat{a}^\\dagger$).\n\nLet us calculate  the mean values of the operator $\\hat{T}(g)$\nin the state $|x_0\\rangle$: $\\omega_0(g) = \\langle x_0|\\,\n\\hat{T}(g)\\,|x_0\\rangle$.  Using Cartan decomposition $g = k\\,\n\\delta\\,q$  and the action(\\ref{2.7}) of rotations we obtain:\n$$ \\omega_0(g)\\ =\\ \\langle x_0|\\,\\hat{T}(g)\\,|x_0\\rangle\\ =\\\n\\langle x_0|\\,\\hat{T}(k)\\,\\hat{T}(\\delta)\\,\\hat{T}(q)\\,|x_0\n\\rangle $$\n\\begin{equation}\\label{2.7b} =\\ \\mbox{det}(k^{\\prime\\prime})^{-N}\\,\n\\mbox{det}(q^{\\prime\\prime})^{-N}\\, \\langle\nx_0|\\,\\hat{T}(\\delta)\\,|x_0\\rangle .\\end{equation}\nThus it is enough to calculate the mean value for the special\nboost:\n\\begin{equation}\\label{2.8}  \\delta\\,=\\,\\left(\\begin{array}{cc}C&S\\\\\nS&C\\end{array}\\right) = \\left(\\begin{array}{cc}E&0\\\\\nT&E\\end{array}\\right)\\left(\\begin{array}{cc}C&0\\\\\n0&C^{-1}\\end{array}\\right)\\left(\\begin{array}{cc}E&T\\\\\n0&E\\end{array}\\right)\\,=\\,t_+\\,t_0\\,t_-.\\end{equation}\n Here $C=\\mbox{diag}(\\mbox{ch}\\,\\lambda_1,\\mbox{ch}\\,\\lambda_2)$,\n $S=\\mbox{diag}(\\mbox{sh}\\,\\lambda_1,\\mbox{sh}\\,\\lambda_2)$ and\n $T=\\mbox{diag}(\\mbox{th}\\,\\lambda_1,\\mbox{th}\\,\\lambda_2)$.\n In the representation in question, the matrices $t_+$ and $t_-$\n are exponents of rising and lowering operators respectively:\n$$ \\hat{T}(t_+)\\,=\\, e^{\\mbox{tr}(\\hat{b}^\\dagger\\,T\\,\n\\hat{a}^\\dagger)},\\ \\ \\hat{T}(t_-)\\,=\\, e^{-\\mbox{tr}(a\\,T\\,b)}.\n$$\nSince $ \\hat{T}(t_-)$  contains  $\\hat{a}$ its action does not\naffect  $|x_0\\rangle$, and similarly $\\hat{T}(t_+)$  containing\n$\\hat{a}^\\dagger$ does not affect $\\langle x_0|$. The only\nnon-trivial action comes from\n\\begin{equation}\\label{2.9} \\hat{T}(t_0)\\,=\\, \ne^{-\\,\\mbox{tr}(\\hat{a}\\,\\Lambda\\,\\hat{a}^\\dagger)\\,-\\,\n\\mbox{tr}(\\hat{b}^\\dagger\\,\n\\Lambda\\,\\hat{b})},\\ \\ \\ \n\\Lambda\\,=\\,\\ln C\\,.\\end{equation}\nConsequently,\n\\begin{equation}\\label{2.9a} \\hat{T}(t_0)|x_0\\rangle\n\\ =\\ \\mbox{det}\\,(C)^{-N}\\,|x_0\\rangle\\,. \\end{equation}\nThe last equality follows from the identity\n$$  e^{-\\mbox{tr}(\\hat{b}^\\dagger\\,\\Lambda\\,b)}\\,\n\\mbox{det}\\,(\\hat{b}^\\dagger)^N\\ =\\ e^{-N\\,\\mbox{tr}(\\Lambda)}\\,\n\\mbox{det}(\\hat{b}^\\dagger)^N\\, e^{-\\mbox{tr}(\\hat{b}^\\dagger\\,\n\\Lambda\\,b)} $$\n(which can be proven, e.g. by induction in $N$). From equations\n(\\ref{2.7a}) and  (\\ref{2.9}) we obtain remarkably simple\nresults:\n\\begin{equation}\\label{2.10} \\omega_0(g)\\ =\\ \\langle\nx_0|\\,\\hat{T}(g)\\,|x_0\\rangle\\ =\\\n\\mbox{det}(k^{\\prime\\prime})^{-N}\\,\\mbox{det}\n(C)^{-N}\\,\\mbox{det}(q^{\\prime\\prime})^{-N}\n =\\ \\det(d)^{-N}.\\end{equation}\nHere $d\\,=\\,k^{\\prime\\prime}\\,C\\,q^{\\prime\\prime}$ is the\nlower-right block of matrix $g$ (see the Cartan decomposition in\n(\\ref{2.7b})). We recovered the results valid in the holomorphic\nrepresentation (\\ref{rep}). \n\\section{The star product}\n\nStarting from the normalized state $|x_0\\rangle\\,\\in \\,{\\cal F}_N$\nwe shall construct the Perelomov's system of coherent states for\nthe representation in question. We choose a set of boosts of the\nform:\n\\begin{equation}\\label{2.11} g_x =  k\\,\\delta\\,k^\\dagger\\ =\\\n\\left(\\begin{array}{cc}k'\\,C\\,k^{\\prime\\dagger}&k'\\,\nS\\,k^{\\prime\\prime\\dagger}\\\\ k^{\\prime\\prime}\\,S\\,\nk'&k^{\\prime\\prime}\\,C\\,k^{\\prime\\prime\\dagger}\\end{array}\\right)\n\\equiv \\left(\\begin{array}{cc}C' &\\tilde{S}\\\\ \\tilde{S}^\\dagger\n& C^{\\prime\\prime}\\end{array}\\right) \\in G\/K,\\end{equation}\nwhere $k\\equiv\\tilde{k}= \\mbox{diag}(k',k^{\\prime\\prime})$ is an\nelement of $K$ with eliminated Cartan $U(1)$ factor,\n$C=\\mbox{diag}(\\mbox{ch\\,}\\lambda_1,\\mbox{ch\\,}\\lambda_2)$ and\n$S=\\mbox{diag}(\\mbox{sh\\,}\\lambda_1,\\mbox{sh\\,}\\lambda_2)$. Thus $ x\n= x(k,\\delta)$ depends on 8 parameters: 6 of them, hidden in $k$,\nare compact, and there are 2 non-compact ones $\\lambda_1$ and\n$\\lambda_2$ representing special boosts. To any boost $g_x$, given\nin (\\ref{2.11}), we assign coherent coherent state (see,\n\\cite{Perelomov})\n\\begin{equation}\\label{2.12} |x\\rangle\\ =\\  \\hat{T}(g_x)\\,|x_0\\rangle\\ =\\\n\\hat{T}(k\\,\\delta\\,k^\\dagger)\\,|x_0\\rangle,\\ \\ x = x(k,\\delta).\n\\end{equation}\n\n{\\it Note}: Any $ x = x(k,\\delta),$  can be uniquely assigned to\nthe $2\\times 2$ complex matrix $z =\nk'\\,T\\,k^{\\prime\\prime\\dagger}$,\n$T=\\mbox{diag}(\\mbox{th\\,}\\lambda_1,\\mbox{th\\,} \\lambda_2)$,\nforming the bounded Bergman domain $D\\cong G\/K$.\n\nLet us consider operators in the representation space of the form\n\\begin{equation}\\label{2.13} \\hat{F}\\ =\\ \\int_G dg\\,\\tilde{F}(g)\\,\n\\hat{T}(g),\\end{equation}\nwhere $\\tilde{F}(g)$ is a distribution on a group $G$ with\ncompact support $\\mbox{supp}\\,\\tilde{F}$. To any such operator\nwe assign function on $G\/K$ by the prescription\n\\begin{equation}\\label{2.14} F(x)\\ =\\ \\langle x|\\,\\hat{F}\\,|x\\rangle\\ =\\\n\\int_G dg\\,\\tilde{F}(g)\\,\\omega(g,x),\\end{equation}\nwhere\n\\begin{equation}\\label{2.14a} \\omega(g,x)\\ \\equiv\\ \\langle x|\\,\\hat{T}(g)\n\\,|x\\rangle\\ =\\  \\omega_0(g^{-1}_x\\,g\\,g_x).\\end{equation}\nThis equation combined with (\\ref{2.10}) offers an explicit\nform of $\\omega(g,x)$ and is well suited for calculations.\n\nThe star-product of two functions $ F(x)= \\langle x|\\,\\hat{F}\n\\,|x\\rangle$ and $ G(x)= \\langle x|\\,\\hat{G}\\,|x\\rangle$ was\ndefined in \\cite{GP}:\n$$ (F\\star G)(x)\\ =\\ \\langle x|\\,\\hat{F}\\hat{G}\\,|x\\rangle\\\n=\\ \\int_{G\\times G} dg_1 dg_2\\,\\tilde{F}(g_1)\\,\\tilde{G}(g_2)\\,\n\\omega(g_1g_2,x)$$\n\\begin{equation}\\label{2.15} =\\ \\int_G dg\\,(\\tilde{F}\\circ\\tilde{G})(g)\\,\n\\omega(g,x).\\end{equation}\nIn (\\ref{2.15}) the symbol $\\tilde{F}\\circ\\tilde{G}$ denotes the\nconvolution in the group algebra $\\tilde{\\cal A}_G$ of compact\ndistributions:\n\\begin{equation}\\label{2.16} (\\tilde{F}\\circ\\tilde{G})(g)\\ =\\ \\int_G dh\\,\n\\tilde{F}(gh^{-1})\\,\\tilde{G})(h).\\end{equation}\nObviously, the mapping $\\tilde{F}\\,\\mapsto\\,F$ given in\n(\\ref{2.14}) is a homomorphism of the group algebra $\\tilde{\\cal\nA}_G$ into the star-algebra ${\\cal A}^\\star_G$ of functions\n(\\ref{2.14}) on $D=G\/K$. The quantized Bergman domain $D_\\star$ we\nidentify with the noncommutative algebra of functions ${\\cal\nA}^\\star_G$ on $D$. \\vskip0.5cm\n\n{\\it Note}: We point out that as in the case of usual\ndistributions, the convolution product may exist even for\ndistribution with non-compact support provided there are satisfied\nspecific restrictions at infinity. \\vskip0.5cm\n\nIt can be easily seen that\n$$ \\mbox{supp}\\,(\\tilde{F}\\circ\\tilde{G})\\,\\subset\\,(\\mbox{supp}\n\\,\\tilde{F})\\,(\\mbox{supp}\\,\\tilde{G})\\,\n\\equiv\\,\\{ g=g_1 g_2\\,|\\,g_1\\in\\mbox{supp}\\,\\tilde{F},\\,g_2\\in\n\\mbox{supp}\\,\\tilde{G}\\}. $$\nConsequently, for a non-compact group there are two classes\nof group algebras:\n\n(i) The first one is generated by distributions with a general\ncompact support and the corresponding group algebra is simply\nthe full algebra $\\tilde{\\cal A}_G$ defined in (\\ref{2.15}).\n\n(ii) The second one is formed by distributions $\\tilde{F}$ with\n$\\mbox{supp}\\,\\tilde{F}$ subset of a subgroup $H\\subset K$, form a\nsub-algebra $\\tilde{\\cal A}_H$ of  the group algebra $\\tilde{\\cal\nA}_G$. \\vskip0.5cm\n\nIn the second class there are are two interesting extremal cases:\n\n(a) $\\tilde{\\cal A}_{\\{e\\}}$ corresponding to the trivial subgroup\n$H = \\{e\\}$ in $G = SU(2,2)$ ($\\tilde{\\cal A}_{\\{e\\}}$ is\nisomorphic to the enveloping algebra ${\\cal U}(su(2,2))$, see\ne.g., \\cite{Kirillov} or \\cite{Molchanov}).\n\n(b) $\\tilde{\\cal A}_K$ corresponding to the maximal compact\nsubgroup $K$ in $G$. \\vskip0.5cm\n\nThe deformation quantization on Lie group co-orbits in terms of\nthe Lie group convolution algebra  $\\tilde{\\cal A}_{\\{e\\}}$  was\nintroduced by \\cite{Rieffel}. Here we follow the related coherent\nstate construction of the star-star  on Lie group co-orbits\nproposed in \\cite{GP}.\n\nAny distribution  $\\tilde{F}$ can be given as a linear combination\nof finite derivatives of the group $\\delta$-function, i.e., as a\nlinear combination of distributions\n\\begin{equation}\\label{2.17}  \\tilde{F}_{A_1B_1\\dots A_nB_n}(g)\\ =\\\n({\\cal X}_{A_1B_1}\\,\\dots\\, {\\cal X}_{A_nB_n}\\delta)(g),\\end{equation}\nwhere ${\\cal X}_{AB}$ is the left-invariant vector field\nrepresenting the generator $X_{AB}$. The explicit form of ${\\cal X}_{AB}$ in terms of the coordinates of the Bergman domain has been given by \\cite{Esteve} and \\cite{barut}.\n\nInserting this into (\\ref{2.14}) we obtain the corresponding function from\n${\\cal A}^\\star_{\\{e\\}}$\n\\begin{equation}\\label{2.18}  F_{A_1B_1\\dots A_nB_n}(x)\\ =\\ (-1)^n\n({\\cal X}_{A_nB_n}\\,\\dots\\, {\\cal X}_{A_1B_1}\\omega)(g,x)|_{g=e}.\\end{equation}\nHere we used the fact that the operators ${\\cal X}_{AB}$ are\nanti-hermitian differential operators with respect to the group\nmeasure $dg$. From (\\ref{2.14}) it follows directly that\n$$ (F_{A_1B_1\\dots A_nB_n}\\star F_{C_1D_1\\dots C_mD_m})(x) $$\n\\begin{equation}\\label{2.19}   =\\ (-1)^{n+m}({\\cal X}_{A_nB_n}\\,\\dots\\,\n{\\cal X}_{A_1B_1}\\,{\\cal X}_{C_mD_m}\\,\\dots\\,\n{\\cal X}_{C_1D_1}\\omega)(g,x)|_{g=e}.\\end{equation}\nEquations (\\ref{2.18}) and (\\ref{2.19}) describe explicitly the\nhomomorphism ${\\cal U}(su(2,2))\\to{\\cal A}^\\star_{\\{e\\}}$.\n\\vskip0.5cm\n\nUsing exponential parametrization of the group element $g =\ne^{\\xi^{AB}X_{AB}}$ formula for the symmetrized function\n(\\ref{2.18}) takes simple form:\n$$ F_{\\{A_1B_1\\dots A_nB_n\\}}(x)\\ =\\ (-1)^n\n(\\partial_{\\xi_{A_1B_1}}\\,\\dots\\, \\partial_{\\xi_{A_nB_n}}\n\\omega)( e^{\\xi^{AB}X_{AB}},x)|_{\\xi=0} $$\n\\begin{equation}\\label{2.20} =\\ (-1)^n\\,\\langle x|\\,\\hat{X}_{\\{A_1B_1}\\,\\dots\\,\n\\hat{X}_{A_nB_n\\}}\\\\|x\\rangle, \\end{equation}\nwhere $\\{\\,\\dots\\,\\}$ means symmetrization of double indexes and\n$\\xi=0$ means the evaluation at $\\xi_{AB}=0$ for\n$A,B=0,1,\\dots,5$. Symmetrized form a basis of the algebra in\nquestion and symmetrized elements from the center of algebra\ncorrespond to Casimir operators. In the series of representation\nin question all Casimir operators are given in terms of a single\noperator $\\hat{N}$ given in (\\ref{2.6}) which is represented by a\nconstant function $N(x) = \\langle x|\\hat{N}|x\\rangle = N$.\n\\vskip0.5cm\n\n{\\it Example 1.: The function $\\omega(g,x)$}. Let us calculate the\nfunction $\\omega(g,x) = \\omega_0(g_x^{-1}\\,g\\,g_x)$ explicitly.\nTaking $g$ and $g_x$ as in (\\ref{1.1}) and (\\ref{2.11}) we have to\ncalculate the product of three matrices:\n $$ g_x^{-1}\\,g\\,g_x\\ =\\ \\left(\\begin{array}{cc}C' & -\\tilde{S}\\\\\n-\\tilde{S}^\\dagger & C^{\\prime\\prime}\\end{array}\\right) \n\\left(\\begin{array}{cc}a&b\\\\ c&d\\end{array}\\right) \n\\left(\\begin{array}{cc}C' &\\tilde{S}\\\\\n\\tilde{S}^\\dagger & C^{\\prime\\prime}\\end{array}\\right)\\ \\equiv\\\n\\left(\\begin{array}{cc}a_x&b_x\\\\ c_x&d_x\\end{array}\\right),$$\nwhere $d_x\\,=\\,C^{\\prime\\prime}\\,d\\,C^{\\prime\\prime}\n\\,+\\,C^{\\prime\\prime}\\,c\\,\\tilde{S}^\\dagger\\,-\\,\n\\tilde{S}\\,c\\,C^{\\prime\\prime}\\,-\\,\\tilde{S}^\\dagger\\,a\\,\n\\tilde{S}$. Using equation (\\ref{2.10}) for $\\omega_0(g)$ we obtain\n\\begin{eqnarray}\\label{2.20b} &&\\omega(g,x)=\\det(d_x)^{-N}\\nonumber\\\\\n&=&\\det(E-z_x^\\dagger z_x)^{N}\\det(d\\,+\\,cz^\\dagger_x\\,-\\,z_xb\\,-\\,z_xaz^\\dagger_x)^{-N}\n\\end{eqnarray}\nHere $z_x = C'^{-1}\\tilde{S} = k'Tk^{\\prime\\prime\\dagger}$ is\n$2\\times 2$ complex matrix from the Bergman domain. In this \nform the expression is convenient for calculations.\\vskip0.5cm\n\n{\\it Example 2.: Fock space realization of the co-adjoint orbit}:\nOur aim is to calculate the coordinates\n\\begin{equation}\\label{2.21} \\xi_{AB}(x) = \\frac{1}{N}\\,\\langle\nx|\\hat{X}_{AB}|x\\rangle = \\frac{1}{N}\\,\\langle x_0|\\hat{T}^\\dagger\n(g_x)\\,\\hat{X}_{AB}\\hat{T}(g_x)|x_0\\rangle, \\end{equation}\nfor $A,B\\,=\\,0,1,\\,\\dots,\\,5$. Taking into account (\\ref{2.4b}) we\nsee that $\\xi_{AB}(x) = D^{CD}_{AB}(g_x)\\,\\xi_{AB}(x_0)$, where\n$(D^{CD}_{AB}(g))= Ad^*_g$ is the matrix corresponding to the\ngroup action in co-adjoint algebra. Therefore it is sufficient to\nevaluate the coordinates at $x_0$: $\\xi_{AB}(x_0) =\n\\frac{1}{N}\\,\\langle x_0|\\hat{X}_{AB}|x_0\\rangle$. A simple\ncalculation gives: $\\xi_{45}(x_0) = 1$ with all other\n$\\xi_{AB}(x_0) = 0$. We see that $\\xi_{AB}(x)$ just forms the\nco-adjoint orbit generated from $\\xi_{AB}(x_0)$. \\vskip0.5cm\n\n{\\it Example 3. The star-product of coordinates $\\xi_{AB}(x)$}: We\nhave\n$$ (\\xi_{AB}\\star\\xi_{CD})(x)\\ =\\ \\frac{1}{2N^2}\\,\\langle\nx|\\{\\hat{X}_{AB},\\hat{X}_{CD}\\}|x\\rangle\\ +\\ \\frac{1}{2N^2}\\,\n\\langle x|[\\hat{X}_{AB},\\hat{X}_{CD}]|x\\rangle ,$$\nwhere $\\{\\,\\dots\\,\\}$ denotes anti-commutator and $[\\,\\dots\\,]$ is\ncommutator. Therefore, the second term is\n$$ \\frac{1}{2N^2}\\,\\langle x|[\\hat{X}_{AB},\\hat{X}_{CD}]|x\n\\rangle\\ =\\ \\frac{1}{2N}\\,f^{EF}_{AB,CD}\\,\\xi_{EF}(x), $$\nwhere we used the definition of $\\hat{X}_{AB}$ and the short-hand\nnotation for the commutator (\\ref{1.1b1}): $[X_{AB},X_{CD}] =\nf^{EF}_{AB,CD} X_{EF}$. The first term is proportional to the\nsymmetrized function $F_{\\{AB,CD\\}}$ and we can use (\\ref{2.20}):\n$$ \\frac{1}{2N^2}\\,\\langle x|\\{\\hat{X}_{AB},\\hat{X}_{CD}\\}\n|x\\rangle\\ =\\ (1 + A_N)\\,\\xi_{AB}(x)\\,\\xi_{CD}(x)\\,+\\,B_N\\,\n\\delta_{AB,CD}, $$\nwhere we have a usual point-wise product of functions in the first\nterm and $\\delta_{AB,CD} = (1\/2)(\\delta_{AC}\\delta_{BD}\\,\n-\\,\\delta_{AD}\\delta_{BC})$ in the second one. The coefficients\n$A_N$ and $B_N$ are of order $1\/N$. Last two equations give\n\\begin{equation}\\label{2.22} (\\xi_{AB}\\star\\xi_{CD})(x)\\ =\\ (1+ A_N)\\,\n\\xi_{AB}(x)\\,\\xi_{CD}(x) \\,+\\,\\frac{1}{2N}\\,f^{EF}_{AB,CD}\n\\,\\xi_{EF}(x)\\,+\\,B_N\\, \\delta_{AB,CD}. \\end{equation}\nWe see that the parameter of the non-commutativity is $\\lambda_N =\n1\/N$. For $N\\,\\to\\,\\infty$ we recover the commutative product.\n\n\\begin{theorem}\n The star product (\\ref{2.15}, \\ref{2.22}) is associative and\n invariant under the transformation of $SU(2,2)$ group.\n\\end{theorem}\nThe proof of this theorem follows directly the definition of the\nstar product.\n\n\n\n\\section{Quantum field on a Bergman domain\n$\\boldsymbol{D}$}\\label{field}\n\\subsection{The invariant Laplacian on $\\boldsymbol{D}$}\nThe invariant Laplacian $\\Delta_N$ is defined by:\n\\begin{equation}\n \\Delta_N \\hat {T_g}=\\hat{ T_g}\\Delta_N\n\\end{equation}\nwhere $\\hat{T_g}$ is the representation operator given by\n(\\ref{2.4a}). We have (see \\cite{ Hua}, \\cite{PZh}):\n\\begin{eqnarray}\n&&\\Delta_N=tr[(E-ZZ^{\\dagger})\\bar{\\partial}_Z\\cdot\n(E-Z^{\\dagger}Z)\\cdot \\partial'_Z]\\\\\n&+&\\det(E-Z^{\\dagger}\nZ)^{-N}tr[(E-ZZ^{\\dagger})\\bar{\\partial}_Z\\cdot\n(E-Z^{\\dagger}Z)\\cdot\n\\partial'_Z(\\det(E-Z^{\\dagger}Z)^N)]\\nonumber\n\\end{eqnarray}\nwhere $\\partial_Z = (\\partial_{z_{ij}})$ is the $2\\times 2$ matrix\nof differential operators in the variables $Z=(z_{ij})\\in D$,\n$\\bar{\\partial}_Z$ and $\\partial'_Z$ denote, respectively the\ncomplex conjugate and the transpose of the matrix operator\n$\\partial_Z$. It is understood that the operators\n$\\bar{\\partial}_Z$ and $\\partial'_Z$ do not differentiate the\nmatrices $E-ZZ^{\\dagger}$ and $E-Z^{\\dagger}Z$. \n\nThe Laplacian\n$\\Delta_N$ is self-adjoint on $L^2(D,d\\mu_N)$ with respect to the measure\ngiven by (\\ref{measure}).\n\nIn what follows we consider only the radial part of the invariant\nLaplacian $\\Delta_N$, as this part that contains information about\nmost interesting physical quantities, e.g., the energy levels. The\nradial part of the invariant Laplacian could be constructed from\nthe roots system introduced in section \\ref{root}, (see\n(\\cite{PZh}, \\cite{Bob}) for more details). The radial part of the\ninvariant Laplacian reads:\n\\begin{equation}\n\\Delta^r_N=\\omega^{-1}(\\sum_{i=1}^2\\frac{1}{4}L_i\\,-\\,\n\\frac{N}{2}\\mbox{th}\\,\\lambda_i\\,\\partial_{\\lambda_i})\\omega\n\\end{equation}\nwhere\n\\begin{equation}\n\\omega=2(\\mbox{ch}\\,2\\lambda_1\\,-\\,\\mbox{ch}\\,2\\lambda_2)\n\\end{equation}\nand\n\\begin{equation}\nL_i=\\frac{\\partial^2}{\\partial{\\lambda_i}^2}+2\\mbox{cth}\n\\,2\\lambda_i\\,\\partial_{\\lambda_i}\n\\end{equation}\n\nLet $\\Phi(N,\\tau_1, \\tau_2)$ be the eigenfunction of $\\Delta^r_N$,\nwe have (see\\cite{PZh}):\n\\begin{equation}\n\\Delta^r_N\\Phi(N, \\tau_1,\\tau_2)=-\\frac{1}{4}[2(N-1)^2\n+\\tau_1^2+\\tau_2^2]\\Phi(N,\\tau_1,\\tau_2)\n\\end{equation}\nwhere $\\tau_1$ and $\\tau_2$ are given by (\\ref{harish}).\n\nThe operator $-\\Delta_N$ is positive: it has the continuous\nspectrum\n\\begin{equation}\n\\big[-\\frac{1}{2}(N-1)^2,+\\infty\\big)\n\\end{equation}\nfor arbitrary $\\tau_i$, and the discrete finite spectrum\n\\begin{equation}\n(N-1)(l_1+l_2)\\,+\\,l_1^2+l_2^2\n\\end{equation}\nfor $\\tau_j$, $\\ j=1,2$, imaginary:\n\\begin{equation} \\tau_j=-i(N-1-2l_j),\\\nl_j=0, 1,\\cdots,\\big[\\frac{N-1}{2}\\big].\n    \\end{equation}\nHere $\\big[\\frac{N-1}{2}\\big]$ means the integer part of\n$\\frac{N-1}{2}$. The discrete spectrum consists of $\\frac{1}{2}\\\nk(k-1)$ points, where $k=\\big[\\frac{N-1}{2}\\big]$.\n\n\\subsection{A quantum field theory model}\nWe shall present a construction of a real scalar (Euclidean) field\ntheory on a quantized Bergman domain. The action of the model in\nquestion reads:\n\\begin{equation}\\label{model}\nS[\\Phi]\\ =\\ \\int d\\mu_N(\\xi)\\ [-\\frac{1}{2}\\Phi\\star\\Delta_N\\Phi+\n\\frac{1}{2}m^2 \\Phi\\star\\Phi+V_\\star(\\Phi)]\\,,\n\\end{equation}\nwhere $\\Phi=\\Phi(\\xi)$ is a real scalar field depending on the\nnoncommutative coordinates $\\xi$ and $d\\mu_N(\\xi)$ is the measure\n(\\ref{measure}) expressed in terms of $\\xi$.\nWe suppose that $V$ is a polynomial of $\\Phi$ bounded from below.\nThen we expand the scalar field $\\Phi$ in terms of the\neigenfunction of the invariant Laplacian:\n\\begin{equation}\n\\Phi=\\int d\\tau_1 d\\tau_2\\,C_N(\\tau_1,\\tau_2)\\Phi(N,\n\\tau_1,\\tau_2)+\\sum_{l_1,l_2} C_{N,l_1,l_2}\\Phi(N, l_1,l_2),\n\\end{equation}\nwhere we integrated over the continuous part of spectrum and\nsummed up over the discrete part of the spectrum. The coefficients\n$C_N(\\tau_1, \\tau_2)$ and $C_{N, l_1, l_2}$ are arbitrary real\nnumbers.\n\nThe quantum mean value of some polynomial field functional\n$F[\\Phi]$ is defined as the functional integral over fields\n$\\Phi$:\n\\begin{equation}\n \\langle F[\\Phi]\\rangle\\ =\\ \\frac{\\int D\\Phi\\,e^{-S[\\Phi]}\\,\n F[\\Phi]}{\\int D\\Phi\\,e^{-S[\\Phi]}},\n\\end{equation}\nwhere $D\\Phi=D_x d\\Phi(x)\\cong\\prod_{\\tau_j} dC_N(\\tau_j)$ and\n$S[\\Phi]$ denotes the corresponding action (\\ref{model}).\n\nFor the free field propagator we recover the quantum field theory\nresults on a commutative Bergman domain $D$\n\\begin{equation}\n<\\Phi, \\Phi>=\\frac{1}{m^2+\\frac{1}{4}[2(N-1)^2+\\tau_1^2+\\tau^2_2]}\n\\end{equation}\nwhich is valid for arbitrary $\\tau_j$:\n\\begin{itemize}\n \\item For the discrete part of the spectrum where\n $\\tau_j=-i(N-1-2l_j)$, the quantum field theory is finite, it\n possesses a cutoff at the maximal energy level $N$.\n\\item For the continuous part of the spectrum where the parameters\n$\\tau_j$ are arbitrary, the theory is divergent. But it could be\nmade finite after proper renormalization. This point will be\nstudied in more detail in future publications.\n\\item When $N=\\rm{finite}$, we have $\\frac{1}{N}$ corrections for\nthe vertices coming from the lowest order of the star product. The\ndivergent behavior is similar to the semiclassical case.\n\\end{itemize}\n\n\\section{Concluding remarks}\nIn this paper we introduced an oscillator realization of the\ndiscrete series of $SU(2,2)$ representations. We performed a\ndeformation quantization over the corresponding coset space\n$D=SU(2,2)\/S(U(2)\\times U(2))$. We presented an explicit\nexpression of the star-product over $D$. Using this star product\nwe constructed a QFT model over this noncommutative Bergman domain\n$D_\\star$. This method can be applied to other $SU(m,n)$ type I\nCartan domains, see \\cite{PGW}, where $SU(2,1)$ is discussed in\ndetail. Such results are of interest for both physics and\nmathematics. From the physical point of view, $SU(2,2)$ is the\nmaximal symmetry group of the (compactified) Minkowski space. It\nis also of interest for the ADS-CFT correspondence, as $SU(2,2)$\nis the double cover of $SO(4,2)$ conformal group. As the $SU(2,2)$\nmodule, the Bergman domain $D$ is a K\\\" ahler manifold which is\nimportant in supersymmetric Quantum field theory and string\ntheory. In addition the Bergman domain $D$ has nontrivial Shilov\nboundary, and the quantization of $D$ could help us to understand\nthis boundary problem in the framework of noncommutative geometry.\nThese aspects are under study and will be discussed in a\nforthcoming paper. \\newpag\n\n{\\bf Acknowledgements} \\vskip0.5cm The author ZT Wang is very\ngrateful to the Physics department of University of Vienna and the\nErwin-Schr\\\" odinger Institute  for hospitality and financial\nsupport. The work of P Pre\\v snajder was supported by the project\nVEGA 1\/100809 of the Slovak Ministry of Education .\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Bipartite and Multipartite\nEntanglement of Gaussian States} \\markboth{G. Adesso and F.\nIlluminati}{Bipartite and Multipartite Entanglement of Gaussian\nStates}\n\n\\author{Gerardo Adesso and Fabrizio Illuminati}\n\n\\address{Dipartimento di Fisica ``E. R. Caianiello'',\nUniversit\\`a di Salerno; CNR-Coherentia, Gruppo di Salerno; and INFN\nSezione di Napoli-Gruppo Collegato di Salerno, Via S. Allende,\n84081 Baronissi (SA), Italy\\\\\nE-mail: gerardo@sa.infn.it, illuminati@sa.infn.it}\n\n\\begin{abstract}\nIn this chapter we review the characterization of entanglement in\nGaussian states of continuous variable systems. For two-mode\nGaussian states, we discuss how their bipartite entanglement can be\naccurately quantified in terms of the global and local amounts of\nmixedness, and efficiently estimated by direct measurements of the\nassociated purities. For multimode Gaussian states endowed with\nlocal symmetry with respect to a given bipartition, we show how the\nmultimode block entanglement can be completely and reversibly\nlocalized onto a single pair of modes by local, unitary operations.\nWe then analyze the distribution of entanglement among multiple\nparties  in multimode Gaussian states. We introduce the\ncontinuous-variable tangle to quantify entanglement sharing in\nGaussian states and we prove that it satisfies the\nCoffman-Kundu-Wootters monogamy inequality. Nevertheless, we show\nthat pure, symmetric three--mode Gaussian states, at variance with\ntheir discrete-variable counterparts, allow a promiscuous sharing of\nquantum correlations, exhibiting both maximum tripartite residual\nentanglement and maximum couplewise entanglement between any pair of\nmodes. Finally, we investigate the connection between multipartite\nentanglement and the optimal fidelity in a continuous-variable\nquantum teleportation network. We show how the fidelity can be\nmaximized in terms of the best preparation of the shared entangled\nresources and, viceversa, that this optimal fidelity provides a\nclearcut operational interpretation of several measures of bipartite\nand multipartite entanglement, including the entanglement of\nformation, the localizable entanglement, and the continuous-variable\ntangle.\n\\end{abstract}\n\n\\section{Introduction}\n\nOne of the main challenges in fundamental quantum theory as well as in quantum\ninformation and computation sciences lies in the characterization and quantification\nof bipartite entanglement for mixed states, and in the definition and interpretation\nof multipartite entanglement both for pure states and in the presence of mixedness.\nWhile important insights have been gained on these issues in the context of\nqubit systems, a less satisfactory understanding has been achieved until\nrecent times on higher-dimensional systems, as the structure of entangled states\nin Hilbert spaces of high dimensionality exhibits a formidable degree of complexity.\nHowever, and quite remarkably, in infinite-dimensional Hilbert spaces of\ncontinuous-variable systems, ongoing and coordinated efforts by different\nresearch groups have led to important progresses in the understanding of the\nentanglement properties of a restricted class of states, the so-called Gaussian\nstates. These states, besides being of great importance both from a fundamental\npoint of view and in practical applications, share peculiar features that make\ntheir structural properties amenable to accurate and detailed theoretical analysis.\nIt is the aim of this chapter to review some of the most recent results on the\ncharacterization and quantification of bipartite and multipartite entanglement\nin Gaussian states of continuous variable systems, their relationships with standard\nmeasures of purity and mixedness, and their operational interpretations in practical\napplications such as quantum communication, information transfer, and quantum\nteleportation.\n\n\\section{Gaussian States of Continuous Variable Systems}\n\nWe consider a continuous variable (CV) system consisting of $N$\ncanonical bosonic modes, associated to an infinite-dimensional\nHilbert space ${\\cal H}$ and described by the vector $\\hat{X}=\\{\\hat\nx_1,\\hat p_1,\\ldots,\\hat x_N,\\hat p_N\\}$ of the field quadrature\n(``position'' and ``momentum'') operators. The quadrature phase\noperators are connected to the annihilation $\\hat a_i$ and creation\n$\\hat a^{\\dag}_i$ operators of each mode, by the relations $\\hat\nx_{i}=(\\hat a_{i}+\\hat a^{\\dag}_{i})$ and $\\hat p_{i}=(\\hat\na_{i}-\\hat a^{\\dag}_{i})\/i$. The canonical commutation relations for\nthe $\\hat X_i$'s can be expressed in matrix form: $[\\hat X_{i},\\hat\nX_j]=2i\\Omega_{ij}$, with the symplectic form\n$\\Omega=\\oplus_{i=1}^{n}\\omega$ and $\\omega=\\delta_{ij-1}-\n\\delta_{ij+1},\\, i,j=1,2$.\n\nQuantum states of paramount importance in CV systems are the\nso-called Gaussian states, {\\em i.e.}~states with Gaussian\ncharacteristic functions and quasi--probability\ndistributions\\cite{cvbook}. The interest in this special class of\nstates (important examples include vacua, coherent, squeezed,\nthermal, and squeezed-thermal states of the electromagnetic field)\nstems from the feasibility to produce and control them with linear\noptical elements, and from the increasing number of efficient\nproposals and successful experimental implementations of CV quantum\ninformation and communication processes involving multimode Gaussian\nstates (see Ref.~\\refcite{review} for recent reviews). By\ndefinition, a Gaussian state is completely characterized by first\nand second moments of the canonical operators. When addressing\nphysical properties invariant under local unitary transformations,\nsuch as mixedness and entanglement, one can neglect first moments\nand completely characterize Gaussian states by the $2N\\times 2N$\nreal covariance matrix (CM) $\\gr{\\sigma}$, whose entries are\n$\\sigma_{ij}=1\/2\\langle\\{\\hat{X}_i,\\hat{X}_j\\}\\rangle\n-\\langle\\hat{X}_i\\rangle\\langle\\hat{X}_j\\rangle$. Throughout this\nchapter, $\\gr{\\sigma}$ will be used indifferently to indicate the CM\nof a Gaussian state or the state itself. A real, symmetric matrix\n$\\gr{\\sigma}$ must fulfill the Robertson-Schr\\\"odinger uncertainty\nrelation\\cite{simon87}\n\\begin{equation}\\label{bonafide}\n\\gr{\\sigma}+i\\Omega \\geq 0\\,,\n\\end{equation}\nto be a {\\em bona fide} CM of a physical state. Symplectic\noperations ({\\em i.e.}~belonging to the group $Sp_{(2N,\\mathbb{R})}= \\{S\\in\nSL(2N,\\mathbb{R})\\,:\\,S^T\\Omega S=\\Omega\\}$) acting by congruence on CMs in\nphase space, amount to unitary operations on density matrices in\nHilbert space. In phase space, any $N$-mode Gaussian state can be\ntransformed by symplectic operations in its Williamson diagonal\nform\\cite{williamson36}\n $\\gr\\nu$, such that $\\gr{\\sigma}= S^T \\gr{\\nu} S$,\nwith $\\gr{\\nu}=\\,{\\rm diag}\\,\\{\\nu_1,\\nu_1,\\ldots\\nu_N,\\nu_N\\}$. The\nset $\\Sigma=\\{\\nu_i\\}$ of the positive-defined eigenvalues of\n$|i\\Omega\\gr{\\sigma}|$ constitutes the symplectic spectrum of\n$\\gr{\\sigma}$ and its elements, the so-called symplectic\neigenvalues, must fulfill the conditions $\\nu_i\\ge 1$, following\nfrom \\eq{bonafide} and ensuring positivity of the density matrix\nassociated to $\\gr{\\sigma}$. We remark that the full saturation of\nthe uncertainty principle can only be achieved by pure $N$-mode\nGaussian states, for which $\\nu_i=1\\,\\,\\forall i=1,\\ldots, N$.\nInstead, those mixed states such that $\\nu_{i\\le k}=1$ and\n$\\nu_{i>k}>1$, with $1\\le k\\le N$, partially saturate the\nuncertainty principle, with partial saturation becoming weaker with\ndecreasing $k$. The symplectic eigenvalues $\\nu_i$ are determined by\n$N$ symplectic invariants associated to the characteristic\npolynomial of the matrix $|i\\Omega\\gr{\\sigma}|$. Global\ninvariants include the determinant $\\,{\\rm\nDet}\\,\\gr{\\sigma}=\\prod_{i}\\nu_i^2$ and the quantity\n $\\Delta=\\sum_i \\nu_i^2$, which is the sum of the\ndeterminants of all the $2\\times 2$ submatrices of $\\gr{\\sigma}$\nrelated to each mode\\cite{serafozzi}.\n\nThe degree of information about the preparation of a quantum state\n$\\varrho$ can be characterized by its \\emph{purity} $\\mu\\equiv\\,{\\rm\nTr}\\,\\varrho^2$, ranging from $0$ (completely mixed states) to $1$\n(pure states). For a Gaussian state with CM $\\gr{\\sigma}$ one\nhas\\cite{paris} \\begin{equation} \\label{muu} \\mu=1\/\\sqrt{\\,{\\rm\nDet}\\,\\gr{\\sigma}}\\,.\\end{equation} As for the entanglement, we recall that\npositivity of the CM's partial transpose (PPT)\\cite{ppt} is a\nnecessary and sufficient condition of separability for $(M+N)$-mode\nbisymmetric Gaussian states (see Sec. 4) with respect to the $M|N$\nbipartition of the modes\\cite{unitarily}, as well as for\n$(M+N)$-mode Gaussian states with fully degenerate symplectic\nspectrum\\cite{botero}. In the special, but important case $M=1$, PPT\nis a necessary and sufficient condition for separability of all\nGaussian states\\cite{simon00,wernerwolf}. For a general Gaussian\nstate of any $M|N$ bipartition, the PPT criterion is replaced by\nanother necessary and sufficient condition stating that a CM $\\gr{\\sigma}$\ncorresponds to a separable state if and only if there exists a pair\nof CMs $\\gr{\\sigma}_{A}$ and $\\gr{\\sigma}_{B}$, relative to the subsystems $A$ and\n$B$ respectively, such that the following inequality\nholds\\cite{wernerwolf}: $\\gr{\\sigma} \\geq \\gr{\\sigma}_{A} \\oplus \\gr{\\sigma}_{B}$. This\ncriterion is not very useful in practice. Alternatively, one can\nintroduce an operational criterion based on a nonlinear map, that is\nindependent of (and strictly stronger than) the PPT\ncondition\\cite{giedkemappa}.\n\nIn phase space, partial transposition amounts to a mirror reflection\nof one quadrature in the reduced CM of one of the parties. If\n$\\{\\tilde{\\nu}_i\\}$ is the symplectic spectrum of the partially\ntransposed CM $\\tilde{\\gr{\\sigma}}$, then a $(1+N)$-mode (or\nbisymmetric $(M+N)$-mode) Gaussian state with CM $\\gr{\\sigma}$ is\nseparable if and only if $\\tilde{\\nu}_i\\ge 1$ $\\forall\\, i$. A\nproper measure of CV entanglement is the \\emph{logarithmic\nnegativity}\\cite{vidwer} $E_{{\\cal N}}\\equiv \\log\\|\\tilde{\\varrho}\\|_{1}$,\nwhere $\\| \\cdot \\|_1$ denotes the trace norm, which constitutes an\nupper bound to the {\\em distillable entanglement} of the state\n$\\varrho$. It can be computed in terms of the symplectic spectrum\n$\\tilde{\\nu}_i$ of $\\tilde{\\gr{\\sigma}}$:\n\\begin{equation}\\label{ensy}\nE_{{\\cal N}}= \\max\\left\\{0,\\,-{\\sum}_{i :\n\\tilde{\\nu}_i<1}\\log\\tilde{\\nu}_i\\right\\}\\,.\n\\end{equation}\n$E_{{\\cal N}}$ quantifies the extent to which the PPT condition\n$\\tilde{\\nu}_i\\ge 1$ is violated.\n\n\n\\section{Two--Mode Gaussian States: Entanglement and Mixedness}\\label{secduemodi}\nTwo--mode Gaussian states represent the prototypical quantum states\nof CV systems, and constitute an ideal test-ground for the\ntheoretical and experimental investigation of CV\nentanglement\\cite{francesi}. Their CM can be written is the\nfollowing block form\n\\begin{equation}\n\\boldsymbol{\\sigma}\\equiv\\left(\\begin{array}{cc}\n\\boldsymbol{\\alpha}&\\boldsymbol{\\gamma}\\\\\n\\boldsymbol{\\gamma}^{T}&\\boldsymbol{\\beta}\n\\end{array}\\right)\\, , \\label{espre}\n\\end{equation}\nwhere the three $2\\times 2$ matrices $\\boldsymbol{\\alpha}$,\n$\\boldsymbol{\\beta}$, $\\boldsymbol{\\gamma}$ are, respectively, the\nCMs of the two reduced modes and the correlation matrix between\nthem. It is well known\\cite{simon00} that for any two--mode CM\n$\\boldsymbol{\\sigma}$ there exists a local symplectic operation\n$S_{l}=S_{1}\\oplus S_{2}$ which takes $\\boldsymbol{\\sigma}$ to its\nstandard form $\\boldsymbol{\\sigma}_{sf}$, characterized by\n\\begin{equation}\n\\gr\\alpha={\\rm diag}\\{a,\\,a\\},\\quad\\gr\\beta={\\rm\ndiag}\\{b,\\,b\\},\\quad \\gr\\gamma={\\rm diag}\\{c_+,\\,c_-\\}\\,.\n\\label{stform}\n\\end{equation}\nStates whose standard form fulfills $a=b$ are said to be\nsymmetric. Any pure state is symmetric and fulfills\n$c_{+}=-c_{-}=\\sqrt{a^2-1}$.  The uncertainty principle\nIneq.~(\\ref{bonafide}) can be recast as a constraint on the\n$Sp_{(4,{\\mathbb R})}$ invariants ${\\rm Det}\\gr{\\sigma}$ and\n$\\Delta(\\gr{\\sigma})={\\rm Det}\\boldsymbol{\\alpha}+\\,{\\rm\nDet}\\boldsymbol{\\beta}+2 \\,{\\rm Det}\\boldsymbol{\\gamma}$, yielding\n$\\Delta(\\gr{\\sigma})\\le1+\\,{\\rm Det}\\boldsymbol{\\sigma}$.\nThe standard form covariances $a$, $b$, $c_{+}$, and $c_{-}$ can\nbe determined in terms of the two local symplectic invariants\n\\begin{equation}\\label{mu12}\n\\mu_1 = (\\det\\gr\\alpha)^{-1\/2} = 1\/a\\,,\\quad \\mu_2 =\n(\\det\\gr\\beta)^{-1\/2} = 1\/b\\,,\n\\end{equation}\nwhich are the marginal purities of the reduced single--mode\nstates, and of the two global symplectic invariants\n\\begin{equation}\\label{globinv}\n\\mu = (\\det\\gr\\sigma)^{-1\/2} =\n[(ab-c_{+}^2)(ab-c_{-}^2)]^{-1\/2}\\,,\\quad \\Delta =\na^2+b^2+2c_+c_-\\,,\n\\end{equation}\nwhere $\\mu$ is the global purity of the state.\nEqs.~(\\ref{mu12}-\\ref{globinv}) can be inverted to provide the\nfollowing physical parametrization of two--mode states in terms of\nthe four independent parameters $\\mu_1,\\,\\mu_2,\\,\\mu$, and\n$\\Delta$\\cite{extremal}:\n\\begin{equation}\na  \\,\\,=\\,\\, \\frac{1}{\\mu_1}\\,, \\quad b = \\frac{1}{\\mu_2}\\,, \\quad\nc_{\\pm}\\,\\,=\\,\\,\\frac{\\sqrt{\\mu_1 \\mu_2}}4 \\, \\big( \\epsilon_- \\pm\n\\epsilon_+ \\big)\\,, \\label{gabc}\n\\end{equation}\nwith  $\\epsilon_\\mp \\equiv \\sqrt{\\left[ \\Delta - (\\mu_1 \\mp\n\\mu_2)^2\/(\\mu_1^2 \\mu_2^2)\\right]^2-4\/\\mu^2}$. The uncertainty\nprinciple and the existence of the radicals appearing in \\eq{gabc}\nimpose the following constraints on the four invariants in order to\ndescribe a physical state\n\\begin{eqnarray}\n\\mu_1 \\mu_2 &\\le& \\mu \\,\\,\\le\\,\\,\n\\frac{\\mu_1 \\mu_2}{\\mu_1 \\mu_2 + \\abs{\\mu_1-\\mu_2}}\\,, \\label{consmu} \\\\\n\\frac{2}{\\mu} + \\frac{(\\mu_1 - \\mu_2)^2}{\\mu_1^2 \\mu_2^2}\n&\\le& \\Delta  \\,\\,\\le\\,\\,  1+\\frac{1}{\\mu^2}  \\, . \\label{deltabnd}\n\\end{eqnarray}\nThe physical meaning of these constraints, and the role of the\nextremal states ({\\em i.e.~}states whose invariants saturate the\nupper or lower bounds of Eqs.~(\\ref{consmu}-\\ref{deltabnd})) in\nrelation to the entanglement, will be investigated soon.\n\nIn terms of symplectic invariants, partial transposition\ncorresponds to flipping the sign of ${\\rm Det}\\,\\gr{\\gamma}$, so\nthat $\\Delta$ turns into $\\tilde{\\Delta}=\\Delta-4\\,{\\rm\nDet}\\,\\gr{\\gamma} = -\\Delta + 2\/\\mu_1^2 + 2\/\\mu_2^2$. The\nsymplectic eigenvalues of the CM $\\gr{\\sigma}$ and of its partial\ntranspose $\\tilde{\\gr{\\sigma}}$ are promptly determined in terms\nof symplectic invariants\n\\begin{equation}\n2\\nu_{\\mp}^2 = \\Delta\\mp\\sqrt{\\Delta^2\n-4\/\\mu^2}\\,, \\quad\n2\\tilde{\\nu}_{\\mp}^2 = \\tilde{\\Delta}\\mp\\sqrt{\\tilde{\\Delta}^2\n-4\/\\mu^2}\\,, \\label{n1}\n\\end{equation}\nwhere in our naming convention $\\nu_- \\le \\nu_+$ in general, and\nsimilarly for the $\\tilde\\nu_\\mp$. The PPT criterion yields a\nstate $\\gr{\\sigma}$ separable if and only if\n$\\tilde{\\nu}_{-}\\ge 1$.\nSince $\\tilde\\nu_+ > 1$ for all two--mode Gaussian states, the\nquantity $\\tilde\\nu_-$ also completely quantifies the entanglement,\nin fact the logarithmic negativity \\eq{ensy} is a monotonically\ndecreasing and convex function of $\\tilde\\nu_-$,\n$E_{{\\cal N}}=\\max\\{0,-\\log\\,\\tilde{\\nu}_{-}\\}$. In the special instance\nof symmetric Gaussian states, the \\emph{entanglement of\nformation}\\cite{entfor} $E_F$ is also computable\\cite{giedke03} but,\nbeing again a decreasing function of $\\tilde\\nu_-$, it provides the\nsame characterization of entanglement and is thus fully equivalent\nto $E_{\\cal N}$ in this subcase.\n\nA first natural question that arises is whether there can exist\ntwo-mode Gaussian states of finite maximal entanglement at a given\namount of mixedness of the global  state. These states would be the\nanalog of the maximally entangled mixed states (MEMS) that are known\nto exist for two-qubit systems\\cite{hishizaka00}. Unfortunately, it\nis easy to show that a similar question in the CV scenario is\nmeaningless. Indeed, for any fixed, finite global purity $\\mu$ there\nexist infinitely many Gaussian states which are infinitely\nentangled. However, we can ask whether there exist maximally\nentangled states at fixed global {\\it and} local purities. While\nthis question does not yet have a satisfactory answer for two-qubit\nsystems, in the CV scenario it turns out to be quite interesting and\nnontrivial. In this respect, a crucial observation is that, at fixed\n$\\mu$, $\\mu_1$ and $\\mu_2$, the lowest symplectic eigenvalue\n$\\tilde\\nu_-$ of the partially transposed CM is a monotonically\nincreasing function of the global invariant $\\Delta$. Due to the\nexistence of exact {\\em a priori} lower and upper bounds on $\\Delta$\nat fixed purities (see Ineq.~\\ref{deltabnd}), this entails the\nexistence of both maximally {\\it and} minimally entangled Gaussian\nstates. These classes of extremal states have been introduced in\nRef.~\\refcite{prl}, and completely characterized (providing also\nschemes for their experimental production) in\nRef.~\\refcite{extremal}, where the relationship between entanglement\nand information has been extended considering generalized entropic\nmeasures to quantify the degrees of mixedness. In particular, there\nexist maximally and minimally entangled states also at fixed global\nand local generalized Tsallis $p$-entropies\\cite{extremal}. In this\nshort review chapter, we will discuss only the case in which the\npurities (or, equivalently, the linear entropies) are used to\nmeasure the degree of mixedness of a quantum state. In this\ninstance, the Gaussian maximally entangled mixed states (GMEMS) are\ntwo--mode squeezed thermal states, characterized by a fully\ndegenerate symplectic spectrum; on the other hand, the Gaussian\nleast entangled mixed states (GLEMS) are states of partial minimum\nuncertainty ({\\em i.e.} with the lowest symplectic eigenvalue of\ntheir CM being equal to 1). Studying the separability  of the\nextremal states (via the PPT criterion), it is possible to classify\nthe entanglement properties of all two--mode Gaussian states in the\nmanifold spanned by the purities:\n\\begin{equation} \\label{entsum}\n\\begin{array}{ll}\n\\mu_1 \\mu_2 \\; \\le \\; \\mu \\; \\le \\; \\frac{\\mu_1 \\mu_2}{\\mu_1\n+ \\mu_2 - \\mu_1 \\mu_2}, & \\Rightarrow\\ \\hbox{separable;} \\\\\n\\frac{\\mu_1 \\mu_2}{\\mu_1 + \\mu_2 - \\mu_1 \\mu_2} < \\mu \\le\n\\frac{\\mu_1 \\mu_2}{\\sqrt{\\mu_1^2 + \\mu_2^2 - \\mu_1^2 \\mu_2^2}},\n& \\Rightarrow\\ \\hbox{coexistence;} \\\\\n\\frac{\\mu_1 \\mu_2}{\\sqrt{\\mu_1^2 + \\mu_2^2 - \\mu_1^2 \\mu_2^2}} <\n\\mu \\le \\frac{\\mu_1 \\mu_2}{\\mu_1 \\mu_2 + \\abs{\\mu_1-\\mu_2}},\n& \\Rightarrow\\ \\hbox{entangled.} \\\\\n\\end{array}%\n\\end{equation}\n\n\nIn particular, apart from a narrow ``coexistence'' region where both\nseparable and entangled Gaussian states can be found, the\nseparability of two--mode states at given values of the purities is\ncompletely characterized. For purities that saturate the upper bound\nin Ineq.~(\\ref{consmu}), GMEMS and GLEMS coincide and we have a\nunique class of states whose entanglement depends {\\it only} on the\nmarginal purities $\\mu_{1,2}$. They are Gaussian maximally entangled\nstates for fixed marginals (GMEMMS). The maximal entanglement of a\nGaussian state decreases rapidly with increasing difference of\nmarginal purities, in analogy with finite-dimensional\nsystems\\cite{adesso}. For symmetric states $(\\mu_1=\\mu_2)$ the upper\nbound of Ineq.~(\\ref{consmu}) reduces to the trivial bound $\\mu \\le\n1$ and GMEMMS reduce to pure two--mode states.\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=7.5cm]{Table3Dlow.eps}\n\\caption{ Classification of the entanglement for two--mode Gaussian\nstates in the space of marginal purities $\\mu_{1,2}$ and normalized\nglobal purity $\\mu\/\\mu_1\\mu_2$. All physical states lie between the\nhorizontal plane of product states $\\mu=\\mu_1\\mu_2$, and the upper\nlimiting surface representing GMEMMS. Separable states (dark grey\narea) and entangled states are well distinguished except for a\nnarrow coexistence region (depicted in black). In the entangled\nregion the average logarithmic negativity (see text) grows from\nwhite to medium grey. The expressions of  the boundaries between all\nthese regions are collected in \\eq{entsum}.} \\label{fig2D}\n\\end{figure}\nKnowledge of the global and marginal purities thus accurately\ncharacterizes the entanglement of two-mode Gaussian states,\nproviding strong sufficient conditions and exact, analytical lower\nand upper bounds. As we will now show, marginal and global purities\nallow as well an accurate quantification of the entanglement.\nOutside the region of separability, GMEMS attain maximum logarithmic\nnegativity $E_{{{\\cal N}}\\max}$  while, in the region of nonvanishing\nentanglement (see Eq.~(\\ref{entsum})), GLEMS acquire minimum\nlogarithmic negativity $E_{{{\\cal N}}\\min}$. Knowledge of the global\npurity, of the two local purities, and of the global invariant\n$\\Delta$ ({\\it i.e.}, knowledge of the full covariance matrix) would\nallow for an exact quantification of the entanglement. However, we\nwill now show that an estimate based only on the knowledge of the\nexperimentally measurable global and marginal purities turns out to\nbe quite accurate. We can in fact quantify the entanglement of\nGaussian states with given global and marginal purities by the {\\em\naverage logarithmic negativity} $\\bar{E}_{{\\cal N}} \\equiv\n(E_{{{\\cal N}}\\max}+E_{{{\\cal N}}\\min})\/2$ We can then also define the relative\nerror $\\delta \\bar{E}_{{\\cal N}}$ on $\\bar{E}_{{\\cal N}}$ as $\\delta\n\\bar{E}_{{\\cal N}} (\\mu_{1,2},\\mu) \\equiv (E_{{\\cal N}\\max}\n-E_{{\\cal N}\\min})\/(E_{{\\cal N}\\max} +E_{{\\cal N}\\min})$. It is easy to see that this\nerror decreases {\\em exponentially} both with increasing global\npurity and decreasing marginal purities, {\\it i.e.} with increasing\nentanglement, falling for instance below  $5\\%$ for symmetric states\n($\\mu_1=\\mu_2\\equiv\\mu_i$) and $\\mu > \\mu_i$.\n\\begin{figure}[t!]\n\\centering\n\\begin{minipage}[t]{6.2cm}\n\\centering\n\\includegraphics[width=6.2cm]{GS3Dclow.eps}\n\\end{minipage} \\hspace*{.3cm}\n\\begin{minipage}[b]{4.7cm}\n\\centering\n\\caption{Maximal and minimal logarithmic\nnegativities as functions of the global and marginal purities of\nsymmetric two-mode Gaussian states. The darker (lighter) surface represents GMEMS\n(GLEMS). In this space, a generic two--mode mixed symmetric state\nis represented by a dot lying inside the narrow gap between the\ntwo extremal surfaces. \\newline  \\newline}\n\\label{fig3D}\n\\end{minipage}\n\\end{figure}\nThe reliable quantification of quantum correlations in genuinely\nentangled two-mode Gaussian states is thus always assured by\nthe experimental determination of the purities, except at most for a\nsmall set of states with very weak entanglement (states with $E_{{\\cal N}}\n\\lesssim 1$). Moreover, the accuracy is even greater in the general\nnon-symmetric case $\\mu_1 \\neq \\mu_2$, because the maximal\nachievable entanglement decreases in such an instance. In\nFig.~\\ref{fig3D}, the surfaces of extremal logarithmic negativities\nare plotted versus $\\mu_i$ and $\\mu$ for symmetric states. In the\ncase $\\mu=1$ the upper and lower bounds coincide, since\nfor pure states the entanglement is completely quantified by the\nmarginal purity. For mixed states this is not the case, but, as the\nplot shows, knowledge of the global and marginal purities strictly\nbounds the entanglement both from above and from below. This\nanalysis shows that the average logarithmic negativity\n$\\bar{E}_{{\\cal N}}$ is a reliable estimate of the logarithmic negativity\n$E_{{\\cal N}}$, improving as the entanglement increases. We remark that the\npurities may be directly measured experimentally, without the need\nfor a full tomographic reconstruction of the whole CM, by exploiting\nquantum networks techniques\\cite{network} or single--photon detections\nwithout homodyning\\cite{cerf}.\n\nFinally, it is worth remarking that most of the results presented\nhere (including the sufficient conditions for entanglement based on\nknowledge of the purities), being derived for CMs using the\nsymplectic formalism in phase space, retain their validity for\ngeneric non Gaussian states of CV systems. For instance, any\ntwo-mode state with a CM equal to that of an entangled two-mode\nGaussian state is entangled as well\\cite{vanlok}. Our methods may\nthus serve to detect entanglement for a broader class of states in\ninfinite-dimensional Hilbert spaces. The analysis briefly reviewed\nin this paragraph on the relationships between entanglement and\nmixedness, can be generalized to multimode Gaussian states endowed\nwith special symmetry under mode permutations, as we will show in\nthe next section.\n\n\\section{Multimode Gaussian States: Unitarily Localizable\nEntanglement}\\label{secuni}\n\nWe will now consider Gaussian states of CV systems with an arbitrary\nnumber of modes, and briefly discuss the simplest instances\nin which the techniques introduced for two--mode Gaussian states\ncan be generalized and turn out to be useful for the quantification\nand the scaling analysis of CV multimode entanglement.\nWe introduce the notion of {\\em bisymmetric} states, defined as those\n$(M+N)$-mode Gaussian states, of a generic bipartition $M|N$, that\nare invariant under local mode permutations\non the $M$-mode and $N$-mode subsystems. The CM $\\gr{\\sigma}$ of a\n$(M+N)$-mode bisymmetric Gaussian state results from a correlated\ncombination of the fully symmetric blocks $\\gr{\\sigma}_{\\gr{\\alpha}^M}$ and\n$\\gr{\\sigma}_{\\gr{\\beta}^N}$: \\begin{equation} \\gr{\\sigma} = \\left(\\begin{array}{cc}\n\\gr{\\sigma}_{\\gr{\\alpha}^M} & \\gr\\Gamma\\\\\n\\gr\\Gamma^{\\sf T} & \\gr{\\sigma}_{\\gr{\\beta}^N}\n\\end{array}\\right) \\; , \\label{fulsim}\n\\end{equation} where $\\gr{\\sigma}_{\\gr{\\alpha}^M}$ ($\\gr{\\sigma}_{\\gr{\\beta}^N}$) describes a $M$-mode\n($N$-mode) reduced Gaussian state completely invariant under mode\npermutations, and $\\gr\\Gamma$ is a $2M\\times 2N$ real matrix formed\nby identical $2\\times 2$ blocks $\\gr\\gamma$. Clearly, $\\gr\\Gamma$ is\nresponsible for the correlations existing between the $M$-mode and\nthe $N$-mode parties. The identity of the submatrices $\\gr\\gamma$ is\na consequence of the local invariance under mode exchange, internal\nto the $M$-mode and $N$-mode parties. A first observation is that\nthe symplectic spectrum of the CM $\\gr{\\sigma}$ \\eq{fulsim} of a\nbisymmetric $(M+N)$-mode Gaussian state includes two degenerate\neigenvalues, with multiplicities $M-1$ and $N-1$. Such eigenvalues\ncoincide, respectively, with the degenerate eigenvalue\n$\\nu_{\\alpha}^{-}$ of the reduced CM $\\gr{\\sigma}_{\\alpha^M}$ and the\ndegenerate eigenvalue $\\nu_{\\beta}^{-}$ of the reduced CM\n$\\gr{\\sigma}_{\\beta^N}$, with the same respective multiplicities. Equipped\nwith this result, one can prove\\cite{unitarily} that $\\gr{\\sigma}$ can be\nbrought, by means of a local unitary operation, with respect to the\n$M|N$ bipartition, to a tensor product of single-mode uncorrelated\nstates and of a two-mode Gaussian state with CM $\\gr{\\sigma}^{eq}$. Here we\ngive an intuitive sketch of the proof (the detailed proof is given\nin Ref.~\\refcite{unitarily}). Let us focus on the $N$-mode block\n$\\gr{\\sigma}_{\\beta^{N}}$. The matrices $i\\gr{\\Omega}\\gr{\\sigma}_{\\beta^N}$ and\n$i\\gr{\\Omega}\\gr{\\sigma}$ possess a set of $N-1$ simultaneous eigenvectors,\ncorresponding to the same (degenerate) eigenvalue. This fact\nsuggests that the phase-space modes corresponding to such\neigenvectors are the same for $\\gr{\\sigma}$ and for\n$\\gr{\\sigma}_{\\beta^{N}}$. Then, bringing by means of a local\nsymplectic operation the CM $\\gr{\\sigma}_{\\beta^{N}}$ in Williamson\nform, any $(2N-2)\\times(2N-2)$ submatrix of $\\gr{\\sigma}$ will be\ndiagonalized because the normal modes are common to the global and\nlocal CMs. In other words, no correlations between the $M$-mode\nparty with reduced CM $\\gr{\\sigma}_{\\alpha^M}$ and such modes will be left:\nall the correlations between the $M$-mode and $N$-mode parties will\nbe concentrated in the two conjugate quadratures of a single mode of\nthe $N$-mode block. Going through the same argument for the $M$-mode\nblock with CM $\\gr{\\sigma}_{\\alpha^M}$ will prove the proposition and show\nthat the whole entanglement between the two multimode blocks can\nalways be concentrated in only two modes, one for each of the two\nmultimode parties.\n\nWhile, as mentioned, the detailed proof of this result can be found\nin Ref.~\\refcite{unitarily} (extending the findings obtained in\nRef.~\\refcite{adescaling} for the case $M=1$), here we will focus on\nits relevant physical consequences, the main one being that the\nbipartite $M\\times N$ entanglement of bisymmetric $(M + N)$-mode\nGaussian states is {\\em unitarily localizable}, {\\em i.e.}, through\nlocal unitary operations, it can be fully concentrated on a single\npair of modes, one belonging to party (block) $M$, the other\nbelonging to party (block) $N$. The notion of ``unitarily\nlocalizable entanglement'' is different from that of ``localizable\nentanglement'' originally introduced by Verstraete, Popp, and Cirac\nfor spin systems\\cite{localiz}. There, it was defined as the maximal\nentanglement concentrable on two chosen spins through local {\\em\nmeasurements} on all the other spins. Here, the local operations\nthat concentrate all the multimode entanglement on two modes are\n{\\em unitary} and involve the two chosen modes as well, as parts of\nthe respective blocks. Furthermore, the unitarily localizable\nentanglement (when computable) is always stronger than the\nlocalizable entanglement. In fact, if we consider a generic\nbisymmetric multimode state of a $M|N$ bipartition, with each of the\ntwo target modes owned respectively by one of the two parties\n(blocks), then the ensemble of optimal local measurements on the\nremaining (``assisting'') $M+N-2$ modes belongs to the set of local\noperations and classical communication (LOCC) with respect to the\nconsidered bipartition. By definition the entanglement cannot\nincrease under LOCC, which implies that the localized entanglement\n(\\`a la Verstraete, Popp, and Cirac) is always less or equal than\nthe original $M\\times N$ block entanglement. On the contrary, {\\it\nall} of the same $M\\times N$ original bipartite entanglement can be\nunitarily localized onto the two target modes, resulting in a\nreversible, of maximal efficiency, multimode\/two-mode entanglement\nswitch. This fact can have a remarkable impact in the context of\nquantum repeaters\\cite{briegel} for communications with continuous\nvariables. The consequences of the unitary localizability are\nmanifold. In particular, as already previously mentioned, one can\nprove that the PPT (positivity of the partial transpose) criterion\nis a necessary and sufficient condition for the separability of $(M\n+ N)$-mode bisymmetric Gaussian states\\cite{unitarily}. Therefore,\nthe multimode block entanglement of bisymmetric (generally mixed)\nGaussian states with CM $\\gr{\\sigma}$, being equal to the bipartite\nentanglement of the equivalent two-mode localized state with CM\n$\\gr{\\sigma}^{eq}$, can be determined and quantified by the logarithmic\nnegativity in the general instance and, for all multimode states\nwhose two--mode equivalent Gaussian state is symmetric, the\nentanglement of formation between the $M$-mode party and the\n$N$-mode party can be computed exactly as well.\n\nFor the sake of illustration, let us consider fully symmetric\n$2N$-mode Gaussian states described by a $2N \\times 2N$ CM\n$\\gr{\\sigma}_{\\beta^{2N}}$. These states are trivially bisymmetric under\nany bipartition of the modes, so that their block entanglement is\nalways localizable by means of local symplectic operations. This\nclass of states includes the pure, CV GHZ--type states (discussed in\nRefs.\\cite{vloock03,adescaling}) that, in the limit of infinite\nsqueezing, reduce to the simultaneous eigenstates of the relative\npositions and the total momentum and coincide with the proper\nGreenberger-Horne-Zeilinger\\cite{ghzs} (GHZ) states of CV\nsystems\\cite{vloock03}. The standard form CM $\\gr{\\sigma}^{p}_{\\beta^{2N}}$\nof this particular class of pure, symmetric multimode Gaussian\nstates depends only on the local mixedness parameter $b\\equiv\n1\/\\mu_\\beta$, which is the inverse of the purity of any single-mode\nreduced block, and it is proportional to the single-mode squeezing.\nExploiting our previous analysis, we can compute the entanglement\nbetween a block of $K$ modes and the remaining $2N-K$ modes for pure\nstates (in this case the block entanglement is simply the Von\nNeumann entropy of any of the reduced blocks) and, remarkably, for\nmixed states as well.\n\\begin{figure}[t!]\n\\centering\n\\begin{minipage}[t]{6.7cm}\n\\centering\n\\includegraphics[width=6.7cm]{fiscalb.eps}\n\\end{minipage} \\hspace*{.3cm}\n\\begin{minipage}[b]{4.2cm}\n\\centering\n\\caption{Hierarchy of block entanglements of fully symmetric\n$2N$-mode Gaussian states of $K \\times (2N-K)$ bipartitions ($N=10$)\nas a function of the single-mode mixedness $b$,  for pure states\n(solid lines) and for mixed states obtained from $(2N+4)$-mode pure\nstates by tracing out $4$ modes (dashed lines). \\newline \\newline} \\label{fiscalb}\n\\end{minipage}\n\\end{figure}\n\nWe can in fact consider a generic $2N$-mode fully symmetric mixed\nstate with CM $\\gr{\\sigma}_{\\beta^{2N}}^{p\\backslash Q}$, obtained from a\npure fully symmetric $(2N+Q)$-mode state by tracing out $Q$ modes.\nFor any $Q$ and any dimension $N$ of the block ($K \\leq N$), and for\nany nonzero squeezing ({\\em i.e.~}for any $b>1$) one has that the\nstate exhibits genuine multipartite entanglement, as first remarked\nin Ref.~\\refcite{vloock03} for pure states: each $K$-mode party is\nentangled with the remaining $(2N-K)$-mode block. Furthermore, the\ngenuine multipartite nature of the entanglement can be precisely\nquantified by observing that the logarithmic negativity between the\n$K$-mode and the remaining $(2N-K)$-mode block is an increasing\nfunction of the integer $K \\le N$, as shown in\nFig.~\\ref{fiscalb}.The optimal splitting of the modes, which yields\nthe maximal, unitarily localizable entanglement, corresponds to\n$K=N\/2$ if $N$ is even, and $K=(N-1)\/2$ if $N$ is odd. The multimode\nentanglement of mixed states remains finite also in the limit of\ninfinite squeezing, while the multimode entanglement of pure states\ndiverges with respect to any bipartition, as shown in\nFig.~\\ref{fiscalb}. For a fixed amount of local mixedness, the\nscaling structure of the multimode entanglement with the number of\nmodes can be analyzed as well, giving rise to an interesting\nresult\\cite{unitarily}. Let us consider, again for the sake of\nillustration, the class of fully symmetric $2N$-mode Gaussian\nstates, but now at fixed single-mode purity. It is immediate to see\nthat the entanglement between any two modes decreases with $N$,\nwhile the $N|N$ entanglement increases (and diverges for pure states\nas $N \\rightarrow \\infty$): the quantum correlations become\ndistributed among all the modes. This is a clear signature of\ngenuine multipartite entanglement and suggests a detailed analysis\nof its sharing properties, that will be discussed in the next\nsection. The scaling structure of multimode entanglement also\nelucidates the power of the unitary localizability as a strategy for\nentanglement purification, with its efficiency improving with\nincreasing number of modes. Finally, let us remark that the local\nsymplectic operations needed for the unitary localization can be\nimplemented by only using passive\\cite{wolfito} and active linear\noptical elements such as beam splitters, phase shifters and\nsqueezers, and that the original multimode entanglement can be\nestimated by the knowledge of the global and local purities of the\nequivalent, localized two--mode state (see\nRefs.~\\cite{adescaling,unitarily} for a thorough discussion), along\nthe lines presented in Section~\\ref{secduemodi} above.\n\n\\section{Entanglement Sharing of Gaussian States}\n\nHere we address the problem of entanglement sharing among multiple\nparties, investigating the structure of multipartite\nentanglement\\cite{contangle,sharing}. Our aim is to analyze the\ndistribution of entanglement between different (partitions of) modes\nin CV systems. In Ref.~\\refcite{CKW} Coffman, Kundu and Wootters\n(CKW) proved for a three-qubit system ABC, and conjectured for $N$\nqubits (this conjecture has now been proven by Osborne and\nVerstraete\\cite{Osborne}), that the entanglement between, say, qubit\nA and the remaining two--qubits partition (BC) is never smaller than\nthe sum of the A$|$B and A$|$C bipartite entanglements in the\nreduced states. This statement quantifies the so-called {\\em\nmonogamy} of quantum entanglement\\cite{monogam}, in opposition to\nthe classical correlations which can be freely shared. One would\nexpect a similar inequality to hold for three--mode Gaussian states,\nnamely\n\\begin{equation}\\label{CKWine}\nE^{i|(jk)}- E^{i|j} - E^{i|k} \\ge 0\\,,\n\\end{equation}\nwhere $E$ is a proper measure of CV entanglement and the indices\n$\\{i,j,k\\}$ label the three modes. However, an immediate computation\non symmetric states shows that Ineq.~(\\ref{CKWine}) can be violated\nfor small values of the single-mode mixedness $b$  using either the\nlogarithmic negativity $E_{\\cal N}$ or the entanglement of formation $E_F$\nto quantify the bipartite entanglement. This is not a\nparadox\\cite{sharing}; rather, it implies that none of these two\nmeasures is the proper candidate for approaching the task of\nquantifying entanglement sharing in CV systems. This situation is\nreminiscent of the case of qubit systems, for which the CKW\ninequality holds using the tangle $\\tau$ \\cite{CKW}, but fails if\none chooses equivalent measures of bipartite entanglement such as\nthe concurrence\\cite{Wootters} ({\\em i.e.}~the square root of the\ntangle) or the entanglement of formation itself. Related problems on\ninequivalent entanglement measures for the ordering of Gaussian\nstates are discussed in Ref.~\\refcite{ordering}.\n\nWe then wish to define a new measure of CV entanglement able to\ncapture the entanglement distribution trade-off via the monogamy\ninequality (\\ref{CKWine}). A rigorous treatment of this problem is\npresented in Ref.~\\refcite{contangle}. Here we briefly review the\ndefinition and main properties of the desired measure that\nquantifies entanglement sharing in CV systems. Because it can be\nregarded as the continuous-variable analogue of the tangle, we will\nname it, in short, the {\\it contangle}.\n\nFor a pure state $\\ket{\\psi}$ of a $(1+N)$-mode CV system,\nwe can formally define the contangle as\n\\begin{equation}\\label{etaupure}\nE_\\tau (\\psi) \\equiv \\log^2 \\| \\tilde \\varrho \\|_1\\,,\\quad \\varrho =\n\\ketbra\\psi\\psi\\,.\n\\end{equation}\n$E_\\tau (\\psi)$ is a proper measure of bipartite entanglement, being\na convex, increasing function of the logarithmic negativity\n$E_{\\cal N}$, which is equivalent to the entropy of entanglement in\nall pure states. For pure Gaussian states\n$\\ket\\psi$ with CM $\\gr{\\sigma}^p$, one has $E_\\tau (\\gr{\\sigma}^p) = \\log^2 (1\/\\mu_1 -\n\\sqrt{1\/\\mu_1^2-1})$, where $\\mu_1 = 1\/\\sqrt{\\det\\gr{\\sigma}_1}$ is the\nlocal purity of the reduced state of mode $1$, described by a CM\n$\\gr{\\sigma}_1$ (considering $1 \\times N$ bipartitions).\nDefinition (\\ref{etaupure}) is extended to generic mixed\nstates $\\varrho$ of $(N+1)$-mode CV systems through the convex-roof\nformalism, namely:\n\\begin{equation}\\label{etaumix}\nE_\\tau(\\varrho) \\equiv \\inf_{\\{p_i,\\psi_i\\}} \\sum_i p_i\nE_\\tau(\\psi_i)\\,,\n\\end{equation}\nwhere the infimum is taken over the decompositions of $\\varrho$ in\nterms of pure states $\\{\\ket{\\psi_i}\\}$. For infinite-dimensional\nHilbert spaces the index $i$ is continuous, the sum in \\eq{etaumix}\nis replaced by an integral, and the probabilities $\\{p_i\\}$ by a\ndistribution $\\pi(\\psi)$. All multimode mixed Gaussian states $\\gr{\\sigma}$\nadmit a decomposition in terms of an ensemble of pure Gaussian\nstates. The infimum of the average contangle, taken over all pure\nGaussian decompositions only, defines the {\\em Gaussian contangle}\n$G_{\\tau}$, which is an upper bound to the true contangle $E_\\tau$,\nand an entanglement monotone under Gaussian local operations and\nclassical communications (GLOCC)\\cite{ordering,geof}. The Gaussian\ncontangle, similarly to the Gaussian entanglement of\nformation\\cite{geof}, acquires the simple form $G_\\tau (\\gr{\\sigma}) \\equiv\n\\inf_{\\gr{\\sigma}^p \\le \\gr{\\sigma}} E_\\tau(\\gr{\\sigma}^p)$, where the infimum runs over\nall pure Gaussian states  with CM $\\gr{\\sigma}^p \\le \\gr{\\sigma}$.\n\nEquipped with these properties and definitions, one can prove\nseveral results\\cite{contangle}. In particular, the general\n(multimode) monogamy inequality $E^{i_m|(i_1\\ldots i_{m-1}\ni_{m+1}\\ldots i_{N})} - \\sum_{l \\ne m} E^{i_m|i_l} \\ge 0$  is\nsatisfied by all pure three-mode and all pure {\\em symmetric}\n$N$-mode Gaussian states, using either $E_\\tau$ or $G_\\tau$ to\nquantify bipartite entanglement, and by all the corresponding mixed\nstates using $G_\\tau$. Furthermore, there is numerical evidence\nsupporting the conjecture that the general CKW inequality should\nhold for all {\\it nonsymmetric} $N$-mode Gaussian states as\nwell.\\footnote{The conjectured monogamy inequality for all (pure or\nmixed) $N$-mode Gaussian states has been indeed proven by\nconsidering a slightly different version of the continuous-variable\ntangle, defined in terms of the (convex-roof extended) squared\nnegativity instead of the squared logarithmic negativity [T.\nHiroshima, G. Adesso and F. Illuminati, Phys. Rev. Lett. {\\bf 98},\n050503 (2007)].} The sharing constraint (\\ref{CKWine}) leads to the\ndefinition of the {\\em residual contangle} as a tripartite\nentanglement quantifier. For nonsymmetric three-mode Gaussian states\nthe residual contangle is partition-dependent. In this respect, a\nproper quantification of tripartite entanglement is provided by the\n{\\em minimum} residual contangle\n\\begin{equation}\\label{etaumin}\nE_\\tau^{i|j|k}\\equiv\\min_{(i,j,k)} \\left[\nE_\\tau^{i|(jk)}-E_\\tau^{i|j}-E_\\tau^{i|k}\\right]\\,,\n\\end{equation}\nwhere $(i,j,k)$ denotes all the permutations of the three mode\nindexes. This definition ensures that $E_\\tau^{i|j|k}$ is invariant\nunder mode permutations and is thus a genuine three-way property of\nany three-mode Gaussian state. We can adopt an analogous definition\nfor the minimum residual Gaussian contangle $G_\\tau^{i|j|k}$. One\nfinds that the latter is a proper measure of genuine tripartite CV\nentanglement, because it is an entanglement\nmonotone under tripartite GLOCC for pure three-mode Gaussian\nstates\\cite{contangle}.\n\n\nLet us now analyze the sharing structure of multipartite CV\nentanglement, by taking the residual contangle as a measure of\ntripartite entanglement. We pose the problem of identifying\nthe three--mode analogues of the two inequivalent classes of fully\ninseparable three--qubit states, the GHZ state\\cite{ghzs}\n$\\ket{\\psi_{\\rm GHZ}} = (1\/\\sqrt{2}) \\left[\\ket{000} +\n\\ket{111}\\right]$, and the $W$ state\\cite{wstates} $\\ket{\\psi_{W}} =\n(1\/\\sqrt{3}) \\left[\\ket{001} + \\ket{010} + \\ket{100}\\right]$. These\nstates are both pure and fully symmetric, but the GHZ state\npossesses maximal three-party tangle with no two-party quantum\ncorrelations, while the $W$ state contains the maximal two-party\nentanglement between any pair of qubits and its tripartite residual\ntangle is consequently zero.\n\nSurprisingly enough, in symmetric three--mode Gaussian states, if\none aims at maximizing (at given single--mode squeezing $b$) either\nthe two--mode contangle $E_\\tau^{i|l}$ in any reduced state ({\\it\ni.e.}~aiming at the CV $W$-like state), or the genuine tripartite\ncontangle ({\\it i.e.}~aiming at the CV GHZ-like state), one finds\nthe same, unique family of pure symmetric three--mode squeezed\nstates. These states, previously named ``GHZ-type''\nstates\\cite{vloock03}, have been introduced for generic $N$--mode CV\nsystems in the previous Section, where their multimode entanglement\nscaling has been studied\\cite{adescaling,unitarily}. The peculiar\nnature of entanglement sharing in this class of states, now baptized\nCV GHZ\/$W$ states, is further confirmed by the following\nobservation. If one requires maximization of the $1 \\times 2$\nbipartite contangle $E_\\tau^{i|(jk)}$ under the constraint of\nseparability of all two--mode reductions, one finds a class of\nsymmetric mixed states whose tripartite residual contangle is\nstrictly smaller than the one of the GHZ\/$W$ states, at fixed local\nsqueezing\\cite{3modi}. Therefore, in symmetric three--mode Gaussian\nstates, when there is no two--mode entanglement, the three-party one\nis not enhanced, but frustrated.\n\nThese results, unveiling a major difference between\ndiscrete-variable and CV systems, establish the {\\em promiscuous}\nstructure of entanglement sharing in symmetric Gaussian states.\nBeing associated with degrees of freedom with continuous spectra,\nstates of CV systems need not saturate the CKW inequality to achieve\nmaximum couplewise correlations. In fact, without violating the\nmonogamy inequality (\\ref{CKWine}), pure symmetric three--mode\nGaussian states are maximally three-way entangled and, at the same\ntime, maximally robust against the loss of one of the modes due, for\ninstance, to decoherence, as demonstrated in full detail in\nRef.~\\refcite{3modi}. This fact may promote these states,\nexperimentally realizable with the current technology\\cite{3mexp},\nas candidates for reliable CV quantum communication. Exploiting a\nthree--mode CV GHZ\/$W$ state as a quantum channel can ensure for\ninstance a tripartite quantum information protocol like a\nteleportation network or quantum secret sharing; or a standard,\nhighly entangled two--mode channel, after a unitary (reversible)\nlocalization has been performed through a single beam splitter; or,\nas well, a two--party quantum protocol with better-than-classical\nefficiency, even if one of the modes is lost due to decoherence. We\nwill next focus on a relevant applicative setting of CV multipartite\nentanglement, in which various of its properties discussed so far\nwill come in a natural relation.\n\n\\section{Exploiting Multipartite Entanglement: Optimal Fidelity of Continuous Variable Teleportation}\n\nIn this section we analyze an interesting application of\nmultipartite CV entanglement: a quantum teleportation-network\nprotocol, involving $N$ users who share a genuine $N$-partite\nentangled Gaussian resource, completely symmetric under permutations\nof the modes. In the standard multiuser protocol, proposed by Van\nLoock and Braunstein\\cite{bra00}, two parties are randomly chosen as\nsender (Alice) and receiver (Bob), but, in order to accomplish\nteleportation of an unknown coherent state, Bob needs the results of\n$N-2$ momentum detections performed by the other cooperating\nparties. A nonclassical teleportation fidelity (i.e. ${\\cal F} >\n{\\cal F}^{cl}$) between {\\em any} pair of parties is sufficient for\nthe presence of genuine $N$-partite entanglement in the shared\nresource, while in general the converse is false (see {\\it\ne.g.}~Fig.~1 of Ref.~\\refcite{bra00}). The {\\em fidelity}, which\nquantifies the success of a teleportation experiment, is defined as\n${\\cal F} \\equiv \\bra{\\psi^{in}} \\varrho^{out}\\ket{\\psi^{in}}$,\nwhere ``in'' and ``out'' denote the input and the output state.\n${\\cal F}$ reaches unity only for a perfect state transfer,\n$\\varrho^{out} = \\ket{\\psi^{in}}\\!\\bra{\\psi^{in}}$, while without\nentanglement in the resource, by purely classical communication, an\naverage fidelity of ${\\cal F}_{cl}=1\/2$ is the best that can be\nachieved if the alphabet of input states includes all coherent\nstates with even weight\\cite{bfkjmo}. This teleportation network has\nbeen recently  demonstrated experimentally\\cite{naturusawa} by\nexploiting three-mode squeezed Gaussian states\\cite{3mexp}, yielding\na best fidelity of ${\\cal F} = 0.64 \\pm 0.02$, an index of genuine\ntripartite entanglement. Our aim is to determine the optimal\nmulti-user teleportation fidelity, and to extract from it a\nquantitative information on the multipartite entanglement in the\nshared resources. By ``optimal'' here we mean maximization of the\nfidelity over all local single-mode unitary operations, at fixed\namounts of noise and entanglement in the shared resource. We\nconsider realistically mixed $N$-mode Gaussian resources, obtained\nby combining a mixed momentum-squeezed state (with squeezing\nparameter $r_1$) and $N-1$ mixed position-squeezed states (with\nsqueezing parameter $r_2 \\neq r_1$ and in principle a different\nnoise factor) into an $N$-splitter\\cite{bra00} (a sequence of $N-1$\nsuitably tuned beam splitters). The resulting state is a completely\nsymmetric mixed Gaussian state of a $N$-mode CV system. For a given\nthermal noise in the individual modes (comprising the unavoidable\nexperimental imperfections), all the states with equal average\nsqueezing $\\bar r \\equiv (r_1+r_2)\/2$ are equivalent up to local\nsingle--mode unitary operations and possess, by definition, the same\namount of multipartite entanglement with respect to any partition.\nThe teleportation efficiency, instead, depends separately on the\ndifferent single--mode squeezings. We have then the freedom of\nunbalancing the local squeezings $r_1$ and $r_2$ without changing\nthe total entanglement in the resource, in order to single out the\noptimal form of the resource state, which enables a teleportation\nnetwork with maximal fidelity. This analysis is straightforward (see\nRef.~\\refcite{telepoppate} for details), but it yields several\nsurprising side results. In particular, one finds that the optimal\nform of the shared $N$-mode symmetric Gaussian states, for $N>2$, is\nneither unbiased in the $x_i$ and $p_i$ quadratures (like the states\ndiscussed in Ref.~\\refcite{bowen} for $N=3$), nor constructed by $N$\nequal squeezers ($r_1=r_2= \\bar r$). This latter case, which has\nbeen implemented experimentally\\cite{naturusawa} for $N=3$, is\nclearly not optimal, yielding fidelities lower than $1\/2$ for\n$N\\ge30$ and $\\bar r$ falling in a certain interval\\cite{bra00}.\nAccording to the authors of Ref.~\\refcite{bra00}, the explanation of\nthis paradoxical behavior should lie in the fact that their\nteleportation scheme might not be optimal. However, a closer\nanalysis shows that the problem does not lie in the choice of the\nprotocol, but rather in the choice of the resource states. If the\nshared $N$-mode squeezed states are prepared, by local unitary\noperations, in the optimal form (described in detail in\nRef.~\\refcite{telepoppate}), the teleportation fidelity ${\\cal\nF}^{opt}$ is guaranteed to be nonclassical (see Fig.~\\ref{finopt})\nas soon as $\\bar r>0$ for any $N$, in which case the considered\nclass of pure states is genuinely multiparty entangled, as we have\nshown in the previous sections. In fact, one can\nshow\\cite{telepoppate} that this nonclassical optimal fidelity is\n{\\em necessary and sufficient} for the presence of multipartite\nentanglement in any multimode symmetric Gaussian state used as a\nshared resource for CV teleportation.\n\\begin{figure}[t!]\n\\centering\n\\begin{minipage}[t]{6.7cm}\n\\centering\n\\includegraphics[width=6.7cm]{telefig.eps}\n \\end{minipage} \\hspace*{.3cm}\n\\begin{minipage}[b]{4.2cm}\n\\centering  \\caption{ Plot of the optimal fidelity for teleporting\nan arbitrary coherent state from any sender to any receiver chosen\nfrom $N$ ($N=2, \\ldots, 50$) parties, exploiting $N$-party\nentangled, pure symmetric Gaussian states as resources. A\nnonclassical fidelity ${\\cal F}^{opt}_N > 0.5$ is always assured for\nany $N$, if the shared entangled resources are prepared in their\noptimal form.\n\\newline}\n\\label{finopt}\n\\end{minipage}\n\\end{figure}\nThese findings yield quite naturally a direct operative way to\nquantify multipartite entanglement in $N$-mode (mixed) symmetric\nGaussian states, in terms of the so-called {\\em Entanglement of\nTeleportation}\\cite{telepoppate}, defined as the normalized optimal\nfidelity\n\\begin{equation}\\label{et}\nE_T \\equiv \\max\\left\\{0, \\big({\\cal F}_N^{opt}-{\\cal F}_{cl}\\big)\/\n\\big(1-{\\cal F}_{cl}\\big)\\right\\}\\,,\n\\end{equation}\ngoing from 0 (separable states) to 1 (CV GHZ\/$W$ state). Moreover,\none finds that the optimal shared entanglement that allows for the\nmaximal fidelity is {\\it exactly} the CV counterpart of the\nlocalizable entanglement, originally introduced for spin systems by\nVerstraete, Popp, and Cirac\\cite{localiz}. The CV localizable\nentanglement (not to be confused with the unitarily localizable\nentanglement introduced in Section \\ref{secuni}) thus acquires a\nsuggestive operational meaning in terms of teleportation processes.\nIn fact, the localizable entanglement of formation (computed by\nfinding the optimal set of local measurements\n--- unitary transformations and nonunitary momentum detections ---\nperformed on the assisting modes to concentrate the highest possible\nentanglement onto Alice and Bob pair of modes) is a monotonically\nincreasing function of $E_T$: $E_F^{loc} = f[(1-E_T)\/(1+E_T)]$, with\n$f(x) \\equiv \\frac{(1+x)^2}{4x} \\log{\\frac{(1+x)^2}{4x}} -\n\\frac{(1-x)^2}{4x} \\log{\\frac{(1-x)^2}{4x}}$. For $N=2$ (standard\ntwo-user teleportation\\cite{brakim}) the state is already localized\nand $E_F^{loc} = E_F$, so that $E_T$ is equivalent to the\nentanglement of formation $E_F$ of two-mode Gaussian states.\nRemarkably, for $N=3$, {\\it i.e.~} for three-mode pure Gaussian\nresource states, the residual contangle $E_\\tau^{i|j|k}$ introduced\nin Section 5 (see \\eq{etaumin}) turns out to be itself a\nmonotonically increasing function of $E_T$:\n\\begin{equation}\nE_\\tau^{i|j|k} = \\log^2{\\frac{2\\sqrt2\nE_T-(E_T+1)\\sqrt{E_T^2+1}}{(E_T-1)\\sqrt{E_T(E_T+4)+1}}}-\\frac12\n\\log^2\\!{\\frac{E_T^2+1}{E_T(E_T+4)+1}}\\,.\n\\end{equation}\nThe quantity $E_T$ thus represents another {\\em\nequivalent} quantification of genuine tripartite CV\nentanglement and provides the latter with an\noperational interpretation associated to the success of a\nthree-party teleportation network. This suggests a possible\nexperimental test of the promiscuous sharing of CV entanglement,\nconsisting in the successful (with nonclassical optimal fidelity)\nimplementation of both a three-user teleportation network exploiting\npure symmetric Gaussian resources, and of two-user standard\nteleportation exploiting any reduced two-mode channel obtained\ndiscarding a mode from the original resource.\n\nBesides their theoretical aspects, the results\nreviewed in this section are of direct practical\ninterest, as they answer the experimental need for the best\npreparation recipe of an entangled squeezed resource, in order to\nimplement quantum teleportation and in general CV communication\nschemes with the highest possible efficiency.\n\n\\section{Conclusions and Outlook}\n\nWe have reviewed some recent results on the entanglement of Gaussian\nstates of CV systems. For two-mode Gaussian states we have shown how\nbipartite entanglement can be qualified and quantified via the\nglobal and local degrees of purity. Suitable generalizations of the\ntechniques introduced for two-mode Gaussian states allow to analyze\nvarious aspects of entanglement in multimode CV systems, and we have\ndiscussed recent findings on the scaling, localization, and sharing\nproperties of multipartite entanglement in symmetric, bisymmetric,\nand generic multimode Gaussian states. Finally, we have shown that\nmany of these properties acquire a clear and simple operational\nmeaning in the context of CV quantum communication and teleportation\nnetworks. Generalizations and extensions of these results appear at\nhand, and we may expect further progress along these lines in the\nnear future, both for Gaussian and non Gaussian states.\n\nA good portion of the material reported in this chapter originates\nfrom joint work with our friend and colleague Alessio Serafini, whom\nwe warmly thank for the joy of collaborating together. It is as well\na pleasure to acknowledge stimulating exchanges over the last two\nyears with Nicolas Cerf, Ignacio Cirac, Jens Eisert, Jarom\\'ir\nFiur\\'a\\u sek, Ole Kr\\\"uger, Gerd Leuchs, Klaus M{\\o}lmer, Tobias Osborne, Matteo\nParis, Eugene Polzik, Gustavo Rigolin, Peter van Loock, Frank\nVerstraete, David Vitali, Reinhard Werner, Michael Wolf,\nand Bill Wootters.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSuperstring theory is currently the most promising candidate for\na theory of everything. Yet, it is not clear what superstring theory is.\nMoreover, even the perturbation theory of the standard (RNS)\nformulation of string theory is not yet completely established beyond some\nloop level, due to complications related to supermoduli\nspaces~\\cite{D'Hoker:2002gw,Grushevsky:2008zm,DuninBarkowski:2009ej}.\nA possible way to define superstring theory that might also resolve the\nproblems with supermoduli spaces, is as a superstring field theory, i.e.,\nas a field theory of strings (see~\\cite{Fuchs:2008cc} for a recent review).\nIt is natural to expect from a reliable formulation of superstring field\ntheory that it respects the underlying symmetries\nof string theory, i.e., it should be covariant and, moreover,\nuniversal~\\cite{Sen:1999xm}.\nThere are several variants of superstring field theories of this kind, e.g.,\nthe heterotic~\\cite{Berkovits:2004xh} and\nopen~\\cite{Arefeva:1989cp,Preitschopf:1989fc,Berkovits:1995ab,Kroyter:2009rn}\ntheories. However, it was never checked if any of those is really well\ndefined at the quantum level. It is our intention to address this question\nusing lattice techniques.\n\nSome of the formulations of superstring field theory cannot be expected, at\nthis stage, to be consistent at the quantum level, since, e.g., they don't\ninclude a consistent Ramond sector. The fermions of the Ramond sector, as\nwell as the notion of ``picture''~\\cite{Friedan:1985ge} introduce, in any\ncase, some complications for the theory.\nHence, it might be advisable to start with a simpler model,\nsuch as that of the bosonic string. The bosonic closed string field\ntheory~\\cite{Zwiebach:1992ie} is much more complicated than the open\none~\\cite{Witten:1986cc}. Hence, we concentrate on the later.\n\nIt is known that the bosonic string theory is consistent in flat space only\nin 26 dimensions. Simulating any theory on a 26 dimensional lattice is almost\na hopeless task. Moreover, the theory has tachyons, both in the open and in\nthe closed string spectrum. It is by now understood that the open string\ntachyon is related to a condensation of an unstable\nD-brane~\\cite{Sen:1999xm,Sen:1999mh,Sen:1999nx,Schnabl:2005gv}. However,\nno analogous understanding exists regarding the fate of the closed string\ntachyon~\\cite{Yang:2005rw,Yang:2005rx,Moeller:2006cv}.\nBoth problems can be avoided by considering ``non-critical'' string theory.\nThe non-critical theory lives at lower dimensions and for $d\\leq 2$ the\ntachyon is absent. On the other hand, a new complication is introduced,\nnamely the theory includes a linear dilaton, which breaks Poincar\\'e\ninvariance. Moreover, all the coupling constants are proportional to the\ndilaton vacuum expectation value, which runs to infinity in one direction. These issues pose\na challenge to a lattice simulation.\n\nA simple consideration of world-sheet gauge symmetry and degrees of freedom\nreveals that two of the $d$ dimensions in which the string lives are\nunphysical. Indeed, the two-dimensional string theory is already not a\nstring theory, but a field theory, in the sense that only one physical field\nout of the infinitely many ones, remains. This field is the ``tachyon'',\nwhich is now a massless field. Going on to ``lower dimensions'' is possible.\nThe dimension is replaced by the central charge of the conformal field\ntheory. This is natural, since $d$ flat dimensions correspond to $c=d$,\nwhere $c$ is the central charge. The two-dimensional theory includes a single\nflat scalar field, i.e., a $c=1$ system, coupled to a single linear dilaton\ndirection. Theories with $0\\leq c<1$ exist and are well studied. They go\nunder the name of ``minimal models''~\\cite{Belavin:1984vu,9304011}. For\nsimplicity we are starting this study with the simplest model of all,\nthe $c=0$ theory that includes only the one dimensional linear dilaton\ndirection with $c=26$ and the canonical ghost system with $c=-26$.\n\nWhile minimal models have only a finite number of degrees of freedom,\nthe string field theory that describe them still contains an infinite\nnumber of fields. Almost all of the degrees of freedom of the theory should\nthen be removed by the very large gauge symmetry present. This fact raises\ntwo further problem that one should address. First, we have to truncate the\ninfinite number of fields to a finite number while taking this number to\ninfinity eventually. Second, we have to address the existence of the gauge\nsymmetry. The first issue of ``level truncation'' was much employed in the\nstring field theory\nliterature~\\cite{Kostelecky:1990nt,Kostelecky:1988ta,Gaiotto:2002wy},\nalbeit only at the classical level. It is\nnot even a-priori clear that it would be a consistent regularization at\nthe quantum level. Here, we apply an ``experimental approach'' towards this\nquestion. The issue of gauge symmetry arises only at the next level.\nSince in this report we only concentrate on preliminary results from the\nlowest level, we ignore this issue for now. \n\n\\section{Methods}\n\nThe ``level'' of level truncation is, up to an additive constant, the\neigenvalue of the ``Hamiltonian'' $L_0$ (the zeroth Virasoro generator).\nAs such, the level includes two contributions,\nthat of the field itself, which is different for any of the infinitely many\nfields of which the string field is composed and a momentum contribution for\neach possible mode of the field. The former contribution is denoted $l_0$ and\nthe total level is given by,\n\\begin{equation}\nl=l_0+\\al' p^2\\,,\n\\end{equation}\nwhere $p$ is the momentum and $\\al'$ is a dimensional constant setting the\nstring scale. We also assumed that the fields were properly redefined,\ne.g., instead of the canonical ``tachyon field'' $T(x)$ we consider\n$\\tau(x)=e^{-\\frac{Vx}{2}}T(x)$. This is the only field with $l_0=0$.\n\nThe $l_0=0$ level action is\n\\begin{equation}\nS=-\\frac{1}{2}\\int d x\\, \\big(m_0^2\\tau^2+(\\nabla \\tau)^2\\big)\n -\\frac{g_o K^{3\\big(1-\\frac{\\al' V^2}{4}\\big)}}{3}\\int d x\\,\n e^{-\\frac{V \\cdot x}{2}}\\tilde \\tau (x)^3 \\,,\n\\end{equation}\nwhere $K=\\frac{3\\sqrt{3}}{4}$, $g_o$ is the open string coupling constant,\n$V$ is the dilaton gradient\n$V=-\\sqrt{\\frac{25}{6\\al'}}$ and $m_0^2$ is the mass squared of the\n``tachyon field'', $m_0^2=\\frac{V^2}{4}-\\frac{1}{\\al'}=\\frac{1}{24\\al'}$.\nThe second term depends on a non-local variant of $\\tau$, namely\n$\\tilde \\tau(x)=K^{\\al' \\nabla^2} \\tau(x)$.\nThis action is both non-local and space-dependent. Furthermore, we\ncannot use periodic boundary conditions, since this would unphysically\nglue together the strong-coupling region at large $x$ with the\nweak-coupling region at small $x$. Instead we choose Dirichlet\nboundary conditions and expand $\\tau(x)$ on an interval $x_{min}< x <\nx_{max}=x_{min}+L$ in sine waves:\n\\begin{equation}\n\\tau(x)=\\sqrt{\\frac{2}{L}}\\sum_{n=1}^N\n  \\tau_n \\sin\\Big(\\frac{\\pi n x}{L}\\Big)\\,.\n\\end{equation}\nHere the level of each mode is given by $l(\\tau_n)=\\al' (\\frac{\\pi\n  n}{L})^2$. We choose $N$ so that all modes have $l < 1$ since we are\nworking at zero $l_0$ level. We also set $g_o=1$, which amounts to a shift\nin $x$.\n\nIn terms of the $\\tau_n$, the action is\n\\begin{equation}\nS=-\\frac{1}{2}\\sum_{n=1}^N \\Big(\\frac{1}{24\\al'}\n  +\\big(\\frac{\\pi n}{L}\\big)^2\\Big) \\tau_n^2\n-\\frac{g_o K^{3\\big(1-\\frac{\\al' V^2}{4}\\big)}}{3} \\sum_{n_{1,2,3}=1}^N\n   K^{-\\al'\\big(\\frac{\\pi}{L}\\big)^2(n_1^2+n_2^2+n_3^2)}\n     \\tau_{n_1}\\tau_{n_2}\\tau_{n_3}f_{n_1,n_2,n_3}\\,,\n\\label{tau_n action}\n\\end{equation}\nwhere\n\\begin{equation}\nf_{n_1,n_2,n_3} = \\Big(\\frac{2}{L}\\Big)^\\frac{3}{2}\\int_{x_{min}}^{x_{max}} dx\n  \\,e^{-\\frac{V x}{2}}\\sin\\Big(\\frac{\\pi n_1 x}{L}\\Big)\n    \\sin\\Big(\\frac{\\pi n_2 x}{L}\\Big)\\sin\\Big(\\frac{\\pi n_3\n      x}{L}\\Big).\n\\label{f_n1n2n3}\n\\end{equation}\nThis is the action we want to consider. Note that the weight of a\nconfiguration in the path integral is $e^S$ rather than $e^{-S}$ due\nto the way we Wick-rotated.\n\nWe see an immediate problem: the action~(\\ref{tau_n action}) has a\ncubic instability. To proceed, we consider the integral $\\int d\n\\tau_n$ over each mode as a complex integral, and deform the\nintegration contour to be a straight line at an angle $\\gamma$ to the\nreal axis.\nIf we choose $\\gamma=\\pi\/6$, the cubic part of the action becomes pure\nimaginary and so the action is no longer unstable. This is similar to\nthe contour deformation used to consider an analytic continuation of\nChern-Simons theory~\\cite{Witten:2010cx}.\n\nHowever, taking the $\\tau_n$ to be complex introduces another problem;\nthe action also becomes complex and so cannot be interpreted as a\nweight for a Markov chain. Instead we simulate in the phase-quenched\nensemble and reweight. That is, we split $e^{S}$ into an amplitude and\na phase:\n\\begin{equation}\ne^{S}=|e^{S}| e^{i\\theta},\n\\end{equation}\nand calculate the expectation value of an observable $\\mathcal{O}$\nusing the identity\n\\begin{eqnarray}\n\\langle \\mathcal{O} \\rangle & = & \\frac{\\int \\mathcal{O}|e^{S}|e^{i \\theta}}{\\int\n  |e^{S}|e^{i \\theta}} \\\\\n& = & \\frac{\\langle \\mathcal{O}e^{i \\theta} \\rangle_\\mathrm{PQ}}{\\langle e^{i\n    \\theta} \\rangle_\\mathrm{PQ}},\n\\end{eqnarray}\nwhere the label $\\mathrm{PQ}$ means the expectation value is evaluated in\n  the phase-quenched ensemble, i.e. with the weight $|e^{S}|$. This\n  is a real, positive weight, so can be used in a Monte Carlo simulation.\n\nWe generate configurations in the phase-quenched ensemble using a\nMetropolis algorithm. The observables we measure are the action $S$\nand the fields $\\tau_n$. We estimate errors with the jackknife method.\n\n\n\\section{Results}\nIn principle, we expect that the theory will be unstable for all\nvalues of the parameters, since there is always a cubic term in the\naction. However, due to the factor $e^{-\\frac{V x}{2}}$ in\n$f_{n_1,n_2,n_3}$, this term can be exponentially small, in which case\nthe theory will be stable for all practical purposes. We have\nperformed scans in parameter space to search for the onset of instability.\n\nThe instability can be seen by looking at the imaginary parts of\n$\\langle S \\rangle$ and $\\langle \\tau_n \\rangle$, which will be zero\nfor a stable set of parameters and will become non-zero as the\ninstability increases. In practice we find that the errors are smaller\nfor the $\\langle \\tau_n \\rangle$ than for the action, so we will\nconcentrate on the former from now on; however the behaviour of the\naction is very similar.\n\nWe find that the onset of the instability is rather rapid. We show an\nexample in Fig.~\\ref{complx_fig}. Here we have $N=6$, which is the\nmaximum allowed for $L=20$. In each case we observe that the $\\langle\n\\tau_n \\rangle$ oscillate, but at $x_{min}=-20.5$ they have negligible\nimaginary parts, whereas\nby $x_{min}=-19.5$ the imaginary parts are as large as the real parts.\nHence the instability appears roughly when $x_{max}=0$, that is when\nwe start to include the region $x>0$ where the cubic terms become\nlarge. The behaviour for other values of $L$ is very similar, with the\ninstability first appearing around $x_{max}=0$ in each case. \n\n\\begin{figure}\n  \\centering\n  \\psfrag{Re<tau>}{$\\mathrm{Re}\\langle \\tau_n \\rangle$}\n  \\psfrag{Im<tau>}{$\\mathrm{Im}\\langle \\tau_n \\rangle$}\n  \\epsfig{file=figs\/tau_6b.eps,scale=0.92,clip}  \n  \\caption{$\\langle \\tau_n \\rangle$ in the complex plane for\n    $x_{min}=-20.5$ (red),\n    $-20$ (green), and $-19.5$ (blue). All for $L=20$, $N=6$, $\\al'=1$,\n    $V=-\\sqrt{\\frac{25}{6\\al'}}$. In each case $\\langle \\tau_1\n    \\rangle$ is the point furthest to the left.}\n  \\label{complx_fig}\n\\end{figure}\n\nSince the action is space-dependent, the meaning of the\ninfinite-volume limit is unclear. It is straightforward to decrease $x_{min}$,\nsince the cubic term becomes exponentially small at small\n$x$. However, it is not clear if the limit $x_{max} \\rightarrow\n\\infty$ is well-defined, since the cubic term continues to get\nstronger in this direction. Indeed, we find that the imaginary parts\nof the $\\tau_n$ continue to increase rapidly when we increase $x_{max}$.\n\n\n\\subsection{Continuum limit}\nIn momentum space, the continuum limit is approached by increasing the\nnumber of modes~$N$. We can only increase $N$ up to a\nmaximum value of $L\/\\pi\\sqrt{\\al'}$ since we require $l<1$. In this\nrange we find that the instability becomes stronger as $N$ is\nincreased, presumably because the number of unstable cubic terms\nincreases rapidly with $N$. We show an example in\nFig.~\\ref{contlimit_fig},\nwhere the maximum level increases from 0.05 (where $N=1$)\nto 0.95 ($N=6$).\nTo approach closer to the continuum limit,\nwe will have to include more fields at higher\n$l_0$-levels. Calculations at level-1 are in progress.\n\n\\begin{figure}\n  \\centering\n  \\psfrag{l}{$l$}\n  \\psfrag{Im<tau1>}{$\\mathrm{Im}\\langle \\tau_1 \\rangle$}\n  \\epsfig{file=figs\/contlimit.eps,scale=0.46,angle=-90,clip}  \n  \\caption{$Im \\langle \\tau_1 \\rangle$ as a function of level $l$, for\n    $x_{min}=-20$, $L=20$, $\\al'=1$,\n    $V=-\\sqrt{\\frac{25}{6\\al'}}$.}\n  \\label{contlimit_fig}\n\\end{figure}\n\n\n\\subsection{Dilaton}\nFinally, we have considered what happens when we vary the dilaton\n$V$. In the full string field theory we require\n$V=-\\sqrt{\\frac{25}{6\\al'}}$. We might expect that varying $V$ away\nfrom this value would increase the instability. However, this is not\nthe case, at least for the level-0 theory. Increasing $V$ above\n$-\\sqrt{\\frac{24}{6\\al'}}$ changes the sign of $m_0^2$, making the\ntheory tachyonic and hence more unstable; on the other hand decreasing\n$V$ makes $m_0^2$ larger and the cubic terms smaller and hence\ndecreases the instability. Our results when we vary $V$ are \nconsistent with this picture. However, it should be noted that\ndecreasing $V$ does not remove the instability entirely, and it will\nalways become strong at some sufficiently large value of $x$.\n\n\n\\section{Conclusions}\nWe have implemented a Monte Carlo simulation of the 1-d linear\ndilation truncated to zero $l_0$-level. We observe non-trivial\nquantum effects: the classical solution to the equations of motion is\n$\\tau_n=0$, but we observe non-zero $\\tau_n$. We also find that as\nexpected, the theory is unstable at large $x$, as shown by the large\nimaginary parts the expectation values of the field develop.\n\nThere are several possible explanations for our result. Firstly, it\nmay be that the instability is a real feature of the full,\nnon-truncated theory. This should not be the case for the theory at hand.\nAnother possibility is that the instability is just an artifact of the\nlevel-truncation. In this scenario the higher-level fields, which we\nhave not included, would stabilise the theory. Alternatively, it might\nalso be the case that level-truncation is not a consistent regularization\nof the quantum theory.\nFinally it is also possible that the instability represents some\nfundamental problem with open string field theory as a method for\nquantising string theory. This could be attributed, e.g., to the lack of\ncontrol over closed string degrees of freedom or to the somewhat\nsingular nature of the star product.\n\nCalculations including level-1 fields are underway. At this level it\nis not obvious how to deal with the gauge and ghosts degrees of\nfreedom, and there are several possible choices. It will be\ninteresting to see how these compare. A practical issue is that\nGrassmann-odd fields will appear and will have to be dealt with.\n\nLooking further ahead, it would be interesting to increase the number\nof dimensions. Ultimately the target would be to work in ten dimensions\nand to include fermionic degrees of freedom, with the aim of reaching a full\nquantum, non-perturbative definition of superstring theory.\n\n\n\\section*{Acknowledgements}\nWe would like to thank Y.~Oz, L.~Rastelli and B.~Zwiebach for discussions.\nThe research of M.~K is supported by a Marie Curie OIF. The views presented\nare those of the authors and do not necessarily reflect those of the European\nCommunity.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\text{{\\hspace{0.48em}\\raisebox{0.8ex}{${\\scriptscriptstyle\\sim}$}}}{ \\text{{\\hspace{0.48em}\\raisebox{0.8ex}{${\\scriptscriptstyle\\sim}$}}}\n                    \\hspace{-0.65em}{\\rightarrow}\\hspace{0.3em}}\n\\newcommand{\\hookrightarrow}{\\hookrightarrow}\n\\newcommand{\\twoheadrightarrow}{\\twoheadrightarrow}\n\\newcommand{\\longrightarrow}{\\longrightarrow}\n\\renewcommand{\\iff}{\\Longleftrightarrow}\n\n\\newcommand{\\rightlabel}[1]{\\stackrel{#1}{\\longrightarrow}}\n\\newcommand{\\leftlabel}[1]{\\stackrel{#1}{\\longleftarrow}}\n\\newcommand{\\rightlabel{\\sim}}{\\rightlabel{\\sim}}\n\\newcommand{\\xxrightarrow}[1]{\\xrightarrow{\\raisebox{-0.1ex}{\\ensuremath{{\\scriptscriptstyle #1}}}}}\n\n\n\\newcommand{\\sqmat}[4]{{\\big(\\genfrac{.}{.}{0pt}{1}{#1}{#3} \\,\n                        \\genfrac{.}{.}{0pt}{1}{#2}{#4}\\big) }}\n\\newcommand{\\colmat}[2]{{\\big(\\genfrac{.}{.}{0pt}{1}{#1}{#2}\\big) }}\n\n\n\\renewcommand{\\phi}{\\varphi}\n\\renewcommand{\\epsilon}{\\varepsilon}\n\n\n\n\n\n\\newcommand{\\bib}[6]{{\\bibitem{#2} #3: {\\emph{#4},} #5#6.}}\n\n\\newcommand{\\bibno}[1]{}\n\n\\section*{Introduction}\n\\addtocontents{toc}{\\protect{\\setcounter{tocdepth}{1}}}  \n\n\\noindent\nThroughout, $X$ is a smooth, projective genus $g$ curve over an algebraically closed field $k$.\nLet $\\Pic \\mathrel{\\mathop:}= \\Pic^0(X)$ be the Picard variety of $X$ and ${\\mathcal P}$ the Poincar\\'e line bundle on $X\\times \\Pic$.\n\nFor a vector bundle $E\\in\\Coh(X)$, its \\emph{Picard complex} is the Fourier--Mukai (or integral) transform\n$\\shat{E} \\mathrel{\\mathop:}= \\mathsf{FM}_{\\mathcal P}(E)$, an object of $\\Db(\\Pic)$.\nWe denote its two cohomology sheaves by $\\shat{E}^0$ and $\\shat{E}^1$ and call these the \\emph{Picard sheaves} of $E$.\nOur goal is to show that $\\shat{E}$ is (semi)stable on $\\Pic$ for general, (semi)stable\nbundles $E$ on $X$ for certain slopes. In fact, we prove this by showing that $\\shat{E}$ is semistable when\nrestricted to curves $i\\colon X \\hookrightarrow \\Pic$.\nWe have the following result; see Corollaries \\ref{cor:res1}, \\ref{cor:STAB1}, \\ref{cor:res2}, and \\ref{cor:STAB2}:\n\n\\begin{introtheorem} \\label{thm:intro1}\nIf $E$ is a stable bundle on $X$ of slope $\\mu(E) > 2g-1$, then the Picard sheaf $\\shat{E}^0$ is stable on $\\Pic$.\nDually, if $E$ is stable of slope $\\mu(E) < -1$, then the Picard sheaf $\\shat{E}^1$ is stable.\nThe analogous statements hold for semistability, using the non-strict inequalities.\n\\end{introtheorem}\n\nUsing our concept of orthogonality, we obtain results for Picard sheaves for\ngeneric semistable bundles of slope $\\mu\\in[g-2,g]$ unless $\\mu=g-1$; see \\hyref{Proposition}{prop:STAB3} and \\hyref{Corollary}{cor:STAB4}.\n\n\\begin{introtheorem}\nFor $\\mu \\in (g-1,g]$, there exists a semistable bundle $E$ on $X$ of slope $\\mu$ such that its Picard sheaf $\\shat{E}^0$ is semistable. Dually, for $\\mu \\in [g-2,g-1)$, there exists a semistable bundle $E$ on $X$ of slope $\\mu$ such that its Picard sheaf $\\shat{E}^1$ is semistable.\n\\end{introtheorem}\n\nIn order to show \\hyref{Theorem}{thm:intro1}, we generalise Clifford's theorem about estimating global sections, from divisors to not necessarily semistable vector bundles. If $E = L_1\\oplus\\cdots\\oplus L_r$ is a direct sum of line bundles with all $\\deg(L_i) \\in [0,2g-2]$, then $h^0(L_i) - 1 \\leq \\deg(L_i)\/2$ by the classical Clifford theorem. This sums up to $h^0(E) - r \\leq \\deg(E)\/2$. Therefore, the best generalisation one can hope for is the following result; see \\hyref{Proposition}{Clifford}, where we  also give precise information about the equality case.\n\n\\begin{introproposition}\nLet $E$ be a vector bundle of rank $r$ and degree $d$ on the smooth projective curve $X$ of genus $g$.\nIf $\\mu_{\\max}(E) \\leq 2g-2 $ and $\\mu_{\\min}(E) \\geq 0$, then we have   the estimate\n\\[ h^0(E)-r \\leq \\frac{d}{2} \\, .\\]\n\\end{introproposition}\n\nThe special case of semistable vector bundles of slope $\\mu \\in [0,2g-2]$ was already proved in \\cite[Theorem 2.1]{BGN}. If the slope of a semistable bundle $E$ is not in this interval, then either $H^0(E)=0$ or $H^1(E)=0$, and the dimension of the remaining cohomology group is computed by the Riemann--Roch theorem.\n\n\n\\subsection*{Known results}\n\n\\noindent\nClassically, Picard sheaves are pushforwards of Poincar\\'e bundles along the projection $X\\times\\Pic^d(X) \\to \\Pic^d(X)$, see \\cite{Mattuck, Schwarzenberger, Mukai}. Later on, this notion was extended to pushforwards of universal bundles onto the moduli space.\n\nKempf \\cite{Kempf} has shown that the Picard bundle on $\\Pic^d(X)$ is stable for $d=2g-1$, the smallest degree where the Picard complex $\\shat{{\\mathcal O}}_X = (\\shat{{\\mathcal O}}_X)^0$ is a vector bundle. Here and later on, stability on the Picard variety is meant with respect to the polarisation by the theta divisor.\nIn \\cite{EL}, Ein and Lazarsfeld proved the stability of Picard bundles on $\\Pic^d(X)$ for $d\\geq 2g$. They do this by restricting the Picard bundles to the canonical curves $X\\subset\\Pic^d(X)$ and $(-X)\\subset\\Pic^d(X)$.\n\nLi \\cite{Li} considers the moduli space $\\mathscr{M}_{r,d}$ of stable bundles on $X$ of rank $r$ and degree $d$ with $d > 2r(g-1)$ and $(d,r)=1$. If $\\mathscr{U}$ denotes the universal bundle on $X\\times\\mathscr{M}_{r,d}$, then Li shows that the Picard bundle $\\mathsf{pr}_{2*}\\mathscr{U}$ on $\\mathscr{M}_{r,d}$ is stable if $d>2gr$.\n\nIn \\cite{BBGN}, the authors consider the same question for the moduli space of stable bundles of rank $r$ and fixed determinant $L$. Then the associated Picard bundle on the moduli space is stable (with respect to the unique polarisation) if $\\deg(L) > 2r(g-1)$ and $(r,\\deg(L))=1$.\n\nHere, we go back to the case of Jacobians, but now we consider the Poincar\\'e bundle twisted by a vector bundle pulled back from the curve. In other words, we study preservation of (semi)stability for the Fourier--Mukai transform along the Poincar\\'e bundle, taking bundles on the curve to sheaves on its Picard variety.\n\n\\subsubsection*{Acknowledgments}\nThis work has been supported by SFB\/TR 45\n``Periods, moduli spaces and arithmetic of algebraic varieties''.\n\n\\subsubsection*{Conventions}\n$X$ is always a smooth, projective curve of genus $g$ over a fixed algebraically closed field.\nThe \\emph{slope} of a vector bundle is denoted $\\mu(E) = \\deg(E)\/\\mathrm{rk}(E)$.\nNote $\\mu(E\\otimes F) = \\mu(E) + \\mu(F)$.\nWe repeatedly use the Riemann--Roch formula, always in the form $\\chi(E) = \\mathrm{rk}(E)(\\mu(E) + 1-g)$.\nWe write $h^0(E) = \\dim H^0(E)$ and $h^1(E) = \\dim H^1(E)$.\nWe denote projections by $\\mathsf{pr}_X,\\mathsf{pr}_Y\\colon X\\times Y\\to X$ or by $\\mathsf{pr}_1,\\mathsf{pr}_2\\colon X\\times X\\to X$.\nGiven sheaves $E$ and $F$ on $X$, then as usual we write $E\\boxtimes F = \\mathsf{pr}_1^*E\\otimes\\mathsf{pr}_2^*F$.\nSometimes, we follow standard usage and pack two statements into one, using (semi)stability and $\\mathrel{(\\leq)}$.\n\n\\section{Orthogonality and stability}\n\n\\subsection{Definition of orthogonality and first properties}\n\n\\noindent\nWe first recall a classical result of Faltings \\cite[Theorem~1.2]{Faltings}, which expresses semistability of a vector bundle on a curve as an orthogonality condition on $X$:\n\n\\begin{theorem}[Faltings 1993] \\label{thm:Faltings}\nA vector bundle $E$ on $X$ is semistable if and only if there exists a vector bundle $0\\ne F$ such that $H^*(E\\otimes F)=0$.\n\\end{theorem}\n\n\\begin{proof}\nFor the convenience of reader, we prove the easy implication of this statement. Assume $H^*(E\\otimes F)=0$, and let $0\\ne G\\subset E$ be a destabilising subsheaf, i.e.\\ $\\mu(G) > \\mu(E)$.\n\nThen $G\\otimes F \\subset E\\otimes F$ with $\\mu(G\\otimes F) > \\mu(E\\otimes F)$.\nSince $H^*(E\\otimes F)=0$, Riemann--Roch gives $\\chi(G\\otimes F) > \\chi(E\\otimes F) = 0$. Hence $h^0(G\\otimes F) > 0$, which contradicts\n $H^0(G\\otimes F) \\subseteq H^0(E\\otimes F) = 0$.\n\\end{proof}\n\nHere, we introduce two other orthogonality notions, on $X\\times X$, which work well with Picard sheaves:\n\n\\begin{definition}\nTwo coherent sheaves $E$ and $F$ on $X$ are called \\emph{orthogonal with respect to $-\\Delta$} if the sheaf\n $\\mathsf{pr}_1^*E \\otimes {\\mathcal O}_{X\\times X}(-\\Delta) \\otimes \\mathsf{pr}_2^* F$ on $X\\times X$ has vanishing cohomology. Analogously, $E$ and $F$ are \\emph{orthogonal with respect to $\\Delta$} if the cohomology groups of $\\mathsf{pr}_1^*E \\otimes {\\mathcal O}_{X\\times X}(\\Delta) \\otimes \\mathsf{pr}_2^* F$ vanish. In short, we write\n\\begin{align*}\n   E  \\orthstack{-\\Delta}  F &\\iff H^*(E \\boxtimes F(-\\Delta)) = 0 , \\\\\n   E  \\orthstack{\\Delta}  F  &\\iff H^*(E \\boxtimes F(\\Delta))  = 0 .\n\\end{align*}\n\\end{definition}\n\nOrthogonality has the following simple numerical description:\n\n\\begin{lemma} \\label{lem:numORTH}\nLet $E$ and $F$ be coherent sheaves on $X$.\n\\begin{enumerate}[label=(\\roman*)]\n  \\item If $H^*(E\\otimes F)=0$, then $\\mu(F) = -\\mu(E)+g-1$. \\\\\n  \\item If $E \\orthstack{-\\Delta}  F$, then $\\mu(F) = g + \\displaystyle\\frac{g}{\\mu(E)-g}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(i) follows readily from Riemann--Roch.\n\n(ii) The cohomology of the exact sequence $0 \\to E\\boxtimes F(-\\Delta) \\to E\\boxtimes F \\to E\\boxtimes F|_\\Delta \\to 0$, using\n $E\\boxtimes F|_\\Delta \\cong E\\otimes F$ and $H^*(E\\boxtimes F(-\\Delta))=0$ from $E \\orthstack{-\\Delta}  F$ gives:\n $H^*(E\\otimes F) \\cong H^*(E\\boxtimes F) \\cong H^*(E) \\otimes H^*(F)$, the latter isomorphism from the K\\\"unneth formula. Hence\n $\\chi(E\\otimes F) = \\chi(E)\\chi(F)$, or\n $(\\mu(E)+\\mu(F)+1-g) = (\\mu(E)+1-g)(\\mu(F)+1-g)$ by Riemann--Roch.\nManipulating this equation yields the claimed formula.\n\\end{proof}\n\nWe collect the following statements for referability; the proofs are immediate:\n\n\\begin{lemma} \\label{lem:sym}\n For vector bundles $E$ and $F$ on $X$, we have the three equivalences:\n\\begin{flalign*}\n  &\\qquad\\text{(i)}    & E  \\orthstack{-\\Delta}  F &\\iff F  \\orthstack{-\\Delta}  E, & \\\\[0.5ex]\n  &\\qquad\\text{(ii)}   & E  \\orthstack{\\Delta}  F  &\\iff F  \\orthstack{\\Delta}  E,  & \\\\[0.5ex]\n  &\\qquad\\text{(iii)}  & E  \\orthstack{-\\Delta}  F &\\iff (\\omega_X \\otimes E ^\\vee )  \\orthstack{\\Delta}  (\\omega_X \\otimes F ^\\vee)\n                        \\qquad \\text{(Serre duality). } &\n\\end{flalign*}\n\\end{lemma}\n\n\\begin{definition}\nWe define two functors $\\mathsf{F}_{\\!\\!+}, \\mathsf{F}_{\\!\\!-} \\colon \\Coh(X) \\to \\Db(X)$ by\n\\begin{align*}\n  \\mathsf{F}_{\\!\\!+}(E) &= \\mathbf{R}\\mathsf{pr}_{2*}({\\mathcal O}_{X \\times X}(\\Delta) \\otimes \\mathsf{pr}_1^*E) , \\\\\n  \\mathsf{F}_{\\!\\!-}(E) &= \\mathbf{R}\\mathsf{pr}_{2*}({\\mathcal O}_{X \\times X}(-\\Delta) \\otimes \\mathsf{pr}_1^*E) .\n\\intertext{We denote the cohomology sheaves by}\n  \\mathsf{F}_{\\!\\!+}^i &= R^i\\mathsf{pr}_{2*}({\\mathcal O}_{X \\times X}(\\Delta) \\otimes \\mathsf{pr}_1^*E)\n\\end{align*}\nand similarly for $\\mathsf{F}_{\\!\\!-}^i(E)$. Since the fibres are 1-dimensional and objects in $\\Db(X)$ decompose into their cohomology sheaves, we have\n  $\\mathsf{F}_{\\!\\!+}(E) = \\mathsf{F}_{\\!\\!+}^0(E) \\oplus \\mathsf{F}_{\\!\\!+}^1(E)[-1]$, and\n  $\\mathsf{F}_{\\!\\!-}(E) = \\mathsf{F}_{\\!\\!-}^0(E) \\oplus \\mathsf{F}_{\\!\\!-}^1(E)[-1]$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:ortho1}\nFor vector bundles $E$ and $F$ on $X$, we have the two equivalences\n\\[\\begin{array}{lrcll}\n       (i) &  E  \\orthstack{-\\Delta}  F  & \\iff & H^*(\\mathsf{F}_{\\!\\!-}^0(E) \\otimes F) = 0 \\text{, and } H^*(\\mathsf{F}_{\\!\\!-}^1(E) \\otimes F) = 0\n    & \\text{ and} \\\\[0.5ex]\n      (ii) &  E  \\orthstack{\\Delta}  F   & \\iff & H^*(\\mathsf{F}_{\\!\\!+}^0(E) \\otimes F) = 0 \\text{, and } H^*(\\mathsf{F}_{\\!\\!+}^1(E) \\otimes F) = 0 .\n\\end{array}\\]\n\\end{lemma}\n\n\\begin{proof}\nWe only show (i), as the proof of (ii) works analogously.\n\nLet\n $G=\\mathsf{pr}_1^*E \\otimes {\\mathcal O}_{X \\times X}(-\\Delta) \\otimes \\mathsf{pr}_2^* F$.\nWe compute the cohomology of $G$\nusing the Leray spectral sequence for $\\mathsf{pr}_2$. For dimension reasons, the spectral sequence degenerates, thus\n\\[  H^0(G)= H^0(\\mathsf{pr}_{2*}G) , \\qquad  H^2(G)= H^1(R^1\\mathsf{pr}_{2*}G) \\,, \\]\nand a short exact sequence\n\\[ 0 \\to H^1(\\mathsf{pr}_{2*}G) \\to H^1(G) \\to  H^0(R^1\\mathsf{pr}_{2*}G) \\to 0 \\,.\\]\nTherefore, we get $R^i\\mathsf{pr}_{2*}(G) = \\mathsf{F}_{\\!\\!-}^i(E) \\otimes F$, using the projection formula, together with the definitions of $G$ and $\\mathsf{F}_{\\!\\!-}^i(E)$. This proves both implications of the assertion, noting that $E  \\orthstack{-\\Delta}  F$ is tantamount to $H^*(G)=0$.\n\\end{proof}\n\n\\begin{proposition} \\label{prop:delta-orthogonality}\nFor a coherent sheaf $E$ on $X$, the following conditions are equivalent:\n\\begin{enumerate}[itemindent=1em,labelsep=1em]\n    \\item There exists a coherent $F \\ne 0$ such that $E  \\orthstack{-\\Delta}  F$.\n    \\item  $\\mathsf{F}_{\\!\\!-}^0(E)$ and $\\mathsf{F}_{\\!\\!-}^1(E)$ are semistable sheaves of the same slope.\n\\end{enumerate}\nSimilarly there exists such an equivalence for orthogonality with respect  to $+\\Delta$.\n\\begin{enumerate}[itemindent=1em,labelsep=1em]\n    \\item[(1')] There exists a coherent $F \\ne 0$ such that $E  \\orthstack{\\Delta}  F$.\n    \\item[(2')]  $\\mathsf{F}_{\\!\\!+}^0(E)$ and $\\mathsf{F}_{\\!\\!+}^1(E)$ are semistable sheaves of the same slope.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} We start with $(1) \\implies (2)$. Assume $E  \\orthstack{-\\Delta}  F$ for some $F \\ne 0$.\nFrom \\hyref{Lemma}{lem:ortho1} we conclude\n $H^*(\\mathsf{F}_{\\!\\!-}^i(E) \\otimes  F) = 0$ for $i \\in \\{0,1\\}$.\nBy the easy direction of \\hyref{Theorem}{thm:Faltings}, $\\mathsf{F}_{\\!\\!-}^i(E)$ is semistable.\nMoreover, we get $\\mu(\\mathsf{F}_{\\!\\!-}^i(E)) = -\\mu(F) + g - 1$ from \\hyref{Lemma}{lem:numORTH}(i)\n\n$(2) \\implies (1)$. With $\\mathsf{F}_{\\!\\!-}^0(E)$ and $\\mathsf{F}_{\\!\\!-}^1(E)$ semistable of the same slope, their direct sum $\\mathsf{F}_{\\!\\!-}^0(E) \\oplus \\mathsf{F}_{\\!\\!-}^1(E)$ is semistable as well. Thus by \\hyref{Faltings' Theorem}{thm:Faltings}, there exists a sheaf $F \\ne 0$  such that\n $H^*((\\mathsf{F}_{\\!\\!-}^0(E) \\oplus \\mathsf{F}_{\\!\\!-}^1(E) )\\otimes F)=0$.\nBy \\hyref{Lemma}{lem:ortho1} we are done.\n\\end{proof}\n\n\\begin{remark}\nWhenever we have an orthogonal pair $E  \\orthstack{-\\Delta}  F$ with non-zero sheaves $E$ and $F$, then we conclude that the six sheaves\n $E$, $\\mathsf{F}_{\\!\\!-}^0(E)$, $\\mathsf{F}_{\\!\\!-}^1(E)$, $F$, $\\mathsf{F}_{\\!\\!-}^0(F)$, and $\\mathsf{F}_{\\!\\!-}^1(F)$\nare semistable.\n\nHowever, in most situations we consider here, one of the two sheaves $\\mathsf{F}_{\\!\\!-}^0(E)$ or $\\mathsf{F}_{\\!\\!-}^1(E)$ will be zero.\nAnyway, they cannot be both zero, as the following argument shows:\nSet $r=\\mathrm{rk}(E)$, $R=\\mathrm{rk}(\\mathsf{F}_{\\!\\!-}^0(E))-\\mathrm{rk}(\\mathsf{F}_{\\!\\!-}^1(E))$, $d=\\deg(E)$, and $D=\\deg(\\mathsf{F}_{\\!\\!-}^0(E))-\\deg(\\mathsf{F}_{\\!\\!-}^1(E))$.\nA short Riemann--Roch computation along the lines of \\hyref{Lemma}{lem:numORTH}(ii) gives\n\\[\n    \\left( \\begin{array}{c} R\\\\D\\end{array} \\right) =\n    \\left( \\begin{array}{rr} -g & 1\\\\0 & -1\\end{array} \\right)\n    \\left( \\begin{array}{c} r\\\\d\\end{array} \\right)  .\n\\]\nThus, we can deduce the pair $(r,d)$  from $(R,D)$ unless $g=0$.\n\\end{remark}\n\n\\subsection{Picard sheaves, and embedding $X$ into the Picard scheme}\nWe denote by $\\Pic = \\Pic^0(X)$ the Picard scheme of the smooth curve $X$ and by ${\\mathcal P}$ the Poincar\\'e bundle on $X\\times\\Pic$. Fixing a point $P_0 \\in X(k)$, we normalise the Poincar\\'e bundle by the additional assumption that\n ${\\mathcal P}|_{\\{P_0\\} \\times \\Pic} \\cong {\\mathcal O}_{\\Pic}$.\nThe projections from $X\\times\\Pic$ will be denoted\n\\[ \\xymatrix{ X & X \\times \\Pic \\ar[l]_(0.57){\\mathsf{pr}_X} \\ar[r]^-{\\mathsf{pr}_P} & \\Pic} . \\]\nFor a coherent sheaf $E$ on $X$, we define its \\emph{Picard complex} to be the object\n\\[ \\shat{E} \\mathrel{\\mathop:}= \\mathsf{FM}_{\\mathcal P}(E) = \\mathbf{R}\\mathsf{pr}_{P*}({\\mathcal P} \\otimes \\mathsf{pr}_X^* E) \\]\nin the derived category $\\Db(\\Pic)$.\nSince ${\\mathcal P}$ is $\\mathsf{pr}_1$-flat, and $\\mathsf{pr}_2$ is of dimension 1, we have only two\ncohomology sheaves of our complex, and we call these the \\emph{Picard sheaves} of $E$:\n\\begin{align*}\n \\shat{E}^0 &\\mathrel{\\mathop:}= \\mathsf{FM}^0_{\\mathcal P}(E) =    \\mathsf{pr}_{P*}({\\mathcal P} \\otimes \\mathsf{pr}_X^* E)  \\, , \\quad \\mbox{ and } \\\\\n \\shat{E}^1 &\\mathrel{\\mathop:}= \\mathsf{FM}^1_{\\mathcal P}(E) = R^1\\mathsf{pr}_{P*}({\\mathcal P} \\otimes \\mathsf{pr}_X^* E) .\n\\end{align*}\nWe are interested mainly in the case when one of $\\shat{E}^0$ or $\\shat{E}^1$ is zero.\nTo study their semistability, we will restrict them to the curves $(X)_M$ and\n$(-X)_N$, to be defined next.\n\nFor any line bundle $M$ of degree 1, we have an embedding of\n $\\iota_M\\colon X \\to \\Pic$ given by $P \\mapsto M(-P)$. The image of $\\iota_M$ is denoted by\n$(-X)_M$. In the same way, any line bundle $N$ of degree $-1$ defines an embedding\n $\\iota_N \\colon X \\to \\Pic$ by $\\iota_N(P)=N(P)$ with image $(X)_N$.\nThe next proposition gives the restriction of\n$\\shat{E}^i$ to the curves $(X)_N$ and $(-X)_M$.\n\n\\begin{proposition} \\label{prop:FM-restriction}\nLet $E$ be a coherent sheaf on $X$. \\\\\nFor arbitrary line bundles $N\\in\\Pic^{-1}(X)$ and $M\\in\\Pic^1(X)$, there are isomorphisms\n\\begin{align*}\n \\mathsf{FM}_{\\mathcal P}(E) \\otimes^\\mathbf{L} {\\mathcal O}_{(X)_N}  &\\cong \\mathsf{F}_{\\!\\!+}(E \\otimes N) \\otimes {\\mathcal O}_X(-P_0) , \\\\\n \\mathsf{FM}_{\\mathcal P}(E) \\otimes^\\mathbf{L} {\\mathcal O}_{(-X)_M} &\\cong \\mathsf{F}_{\\!\\!-}(E \\otimes M) \\otimes {\\mathcal O}_X(P_0)\n\\end{align*}\nin the derived category of $(X)_N \\cong X \\cong (-X)_M$.\nMoreover, for general line bundles $N\\in\\Pic^{-1}(X)$ and $M\\in\\Pic^1(X)$, there are isomorphisms of sheaves\n\\[ \\begin{array}{l@{~}c@{~}l @{\\qquad} l@{~}c@{~}l}\n      \\shat{E}^0|_{(X)_N}   &\\cong& \\mathsf{F}_{\\!\\!+}^0(E \\otimes N) \\otimes {\\mathcal O}_X(-P_0) , &\n      \\shat{E}^1|_{(X)_N}   &\\cong& \\mathsf{F}_{\\!\\!+}^1(E \\otimes N) \\otimes {\\mathcal O}_X(-P_0) \\\\[0.5ex]\n      \\shat{E}^0|_{(-X)_M}  &\\cong& \\mathsf{F}_{\\!\\!-}^0(E \\otimes M) \\otimes {\\mathcal O}_X(P_0) , &\n      \\shat{E}^1|_{(-X)_M}  &\\cong& \\mathsf{F}_{\\!\\!-}^1(E \\otimes M) \\otimes {\\mathcal O}_X(P_0) .\n  \\end{array}\\]\n\\end{proposition}\n\n\\begin{proof}\nFor the formula for $\\mathsf{FM}_{\\mathcal P}(E) \\otimes^\\mathbf{L} {\\mathcal O}_{(X)_N}$, we compute\n\\begin{align*}\n         \\mathsf{FM}_{\\mathcal P}(E) \\otimes^\\mathbf{L} {\\mathcal O}_{(X)_N} \n  &=     \\mathbf{R}\\mathsf{pr}_{P*}( \\mathsf{pr}_X^*E  \\otimes {\\mathcal P}) \\otimes^\\mathbf{L} {\\mathcal O}_{(X)_N}      & \\text{(definition of $\\mathsf{FM}_{\\mathcal P}$)} \\\\\n  &\\cong \\mathbf{R}\\mathsf{pr}_{P*}( \\mathsf{pr}_X^*E  \\otimes {\\mathcal P} \\otimes \\mathsf{pr}_P^* {\\mathcal O}_{(X)_N})  & \\text{(projection formula)} \\\\\n  &\\cong \\mathbf{R}\\mathsf{pr}_{P*}( \\mathsf{pr}_X^*E  \\otimes {\\mathcal P}|_{ X \\times (X)_N})            & \\text{(${\\mathcal P}$ locally free)} \\\\\n  &\\cong \\mathbf{R}\\mathsf{pr}_{2*}( \\mathsf{pr}_1^*(E \\otimes N) \\otimes {\\mathcal O}_{X \\times X}(\\Delta) \\otimes \\mathsf{pr}_2^*{\\mathcal O}_X(-P_0))\n             \\hspace{-1em}                                             & \\text{($\\star$)} \\\\\n  &\\cong \\mathbf{R}\\mathsf{pr}_{2*}( \\mathsf{pr}_1^*(E \\otimes N) \\otimes {\\mathcal O}_{X \\times X}(\\Delta) ) \\otimes {\\mathcal O}_X(-P_0)\n                                                                       & \\text{(projection formula)} \\\\ \n  &=     \\mathsf{F}_{\\!\\!+}(E \\otimes N) \\otimes {\\mathcal O}_X(-P_0)                          & \\text{(definition of $\\mathsf{F}_{\\!\\!+}$)} \n\\end{align*}\nwhere in $\\text{($\\star$)}$, we identify $(X)_N$ with $X$, so the universal family ${\\mathcal P}$ restricted to  $X \\times (X)_N$ is the line bundle\n $\\mathsf{pr}_1^*N \\otimes {\\mathcal O}_{X \\times X}(\\Delta) \\otimes \\mathsf{pr}_2^*{\\mathcal O}_X(-P_0)$ on $X \\times X$.\n\nChoosing a nice resolution $0 \\to E_1 \\to E_0 \\to E \\to 0$ where the $E_i$ are vector bundles with $\\mu_{\\max }(E_i) < 0$, we see that $\\mathsf{FM}_{\\mathcal P}(E)$ can be represented by the complex of vector bundles $\\shat{E}_1^1 \\to \\shat{E}_0^1$ on $\\Pic$. For a general $N$, the curve $(X)_N$ does not contain any of the associated components of the cohomologies of that complex. Thus tensoring with ${\\mathcal O}_{(X)_N}$ is exact. Similarly for $M$.\n\\end{proof}\n\n\\subsection{Orthogonality proofs of semistability of Picard sheaves}\n\n\\begin{proposition} \\label{prop:generic-Picard-semistability}\nFor a coherent sheaf $E$ on $X$, the following implications hold\n\\[ (1) \\iff (2) \\implies (3) \\impliedby (2') \\iff (1') \\]\namong these five conditions:\n\\begin{enumerate}\n\\item[(1)] $(E \\otimes M)  \\orthstack{-\\Delta}  F$ for some line bundle $M$ of degree 1,\n           and a sheaf $F \\ne 0$.\n\\item[(2)] For a general line bundle $M$ of degree 1, the restrictions of the\n           Picard sheaves $\\shat{E}^0$ and $\\shat{E}^1$ to the curve $(-X)_M$ are\n           both semistable of the same slope.\n\\item[(3)] The sheaves $\\shat{E}^0$ and $\\shat{E}^1$ are $\\mu$-semistable with\n           respect to the theta divisor on $\\Pic$.\n\\item[(1')] $(E \\otimes N)  \\orthstack{\\Delta}  F$ for some line bundle $N$ of degree $-1$,\n            and a sheaf $F \\ne 0$.\n\\item[(2')] For a general line bundle $N$ of degree $-1$, the restrictions of\n            the Picard sheaves $\\shat{E}^0$ and $\\shat{E}^1$ to the curve $(X)_N$\n            are both semistable of the same slope.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThe proof of the equivalence $(1) \\iff (2)$ follows from Propositions\n\\ref{prop:delta-orthogonality} and \\ref{prop:FM-restriction}, having in mind that twisting with the line bundle ${\\mathcal O}_X(P_0)$ does not affect semistability. The implication $(2) \\implies (3)$ is standard: given a sheaf $F$ on a variety $Y$ such that $F|_C$ is semistable, where $C\\subset Y$ is a curve cut out by divisors in the linear system $|H|$ of the polarisation $H$, then $F$ is $\\mu_H$-semistable on $Y$ --- a destabilising subsheaf $F'\\subset F$ would induce a destabilising subsheaf $F'|_C \\subset F|_C$.\n\\end{proof}\n\n\\begin{remark}\nIndeed, the implications $(2) \\implies (3) \\Longleftarrow (2')$ were also\nused in \\cite{EL} as a main tool. Using orthogonality we can not detect\nwhether a semistable sheaf is stable.\n\\end{remark}\n\n\\subsection{A first example}\n\n\\begin{lemma} \\label{lem:orth-pair}\nLet $E$ be a semistable coherent sheaf on $X$ of degree 0.\nThen there exists a coherent sheaf $F \\ne 0$ such that $E  \\orthstack{-\\Delta}  F$.\n\\end{lemma}\n\n\\begin{proof}\nBy \\hyref{Faltings' Theorem}{thm:Faltings}, there exists a vector bundle $F$ with $H^*(E \\otimes F)=0$. Since $F$ can be taken in an open subset in the moduli space of rank $R$ and degree $R(g-1)$ vector bundles, we may furthermore assume that  $H^*(F)=0$.\nTensoring the short exact ideal sheaf sequence of $\\Delta$ on $X \\times X$ with $E \\boxtimes F=\\mathsf{pr}_1^*E \\otimes \\mathsf{pr}_2^*F$, we obtain\n\\[ 0 \\to E \\boxtimes F(-\\Delta) \\to E \\boxtimes F \\to E\\boxtimes F|_\\Delta \\to 0 .\\]\nWe have $H^*(E \\boxtimes F) = H^*(E) \\otimes H^*(F)=0$ by the K\\\"unneth formula, and $H^*(E\\boxtimes F|_\\Delta) = H^*(E \\otimes F) =0$ by our assumption. Hence $H^*(E \\boxtimes F(-\\Delta))=0$, i.e.\\ $E \\orthstack{-\\Delta}  F$.\n\\end{proof}\n\n\\begin{corollary} \\label{cor:res1}\nLet $E$ be a semistable vector bundle on $X$ with $\\mu(E)=-1$.\nThen $\\shat{E}^0 = 0$ and $\\shat{E}^1$ is a vector bundle of rank $g \\cdot \\mathrm{rk}(E)$ which is also semistable. \nMoreover, the restriction of $\\shat{E}^1$ to any curve $(-X)_M$ is semistable.\n\\end{corollary}\n\n\\begin{proof}\nSince $E$ is semistable of negative degree, we have $\\shat{E}^0 =0$, and it follows that $\\shat{E}^1$ is a vector bundle of the given rank.\nNow for any line bundle $M$ of degree 1, the tensor product $E \\otimes M$ is semistable of degree 0.\nThus by \\hyref{Lemma}{lem:orth-pair}, there exists a sheaf $F\\neq0$ such that $(E \\otimes M)  \\orthstack{-\\Delta}  F$, and \\hyref{Proposition}{prop:generic-Picard-semistability} shows the semistability of $\\shat{E}^1$.\n\nFinally, $\\shat{E}^1$ is semistable when restricted to any curve $(-X)_M$ because the base change argument from the proof of \\hyref{Proposition}{prop:FM-restriction} simplifies, with $\\mathsf{FM}_{\\mathcal P}(E) = \\shat{E}^1[-1]$ a shifted vector bundle.\n\\end{proof}\n\n\\subsection{A more subtle example}\n\n\\begin{lemma}\\label{orth-pair2}\nAssume that the genus $g \\ne 1$.\nLet $E$ be a vector bundle on $X$ of slope $\\mu(E)=g-1$. Then the following three conditions are equivalent:\n\\begin{enumerate}[itemindent=1em,labelsep=1em]\n\\item $H^*(E)=0$.\n\\item $E  \\orthstack{-\\Delta}  {\\mathcal O}_X$.\n\\item There exists a coherent sheaf $F \\ne 0$ such that $E  \\orthstack{-\\Delta}  F$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n$(1) \\implies (2)$:\nFrom $H^*(E)=0$ we get $H^*(E\\boxtimes{\\mathcal O}_X) = H^*(E) \\otimes H^*({\\mathcal O}_X) = 0$ and $H^*(E\\otimes{\\mathcal O}_X|_\\Delta) = H^*(E) = 0$.\nTherefore, the long exact cohomology sequence of $0\\to E\\boxtimes{\\mathcal O}_X(-\\Delta) \\to E\\boxtimes{\\mathcal O}_X \\to E\\otimes{\\mathcal O}_X|_\\Delta \\to 0$ gives $E\\boxtimes{\\mathcal O}_X(-\\Delta)) = 0$, i.e.\\ $E \\orthstack{-\\Delta}  {\\mathcal O}_X$.\n\n$(2) \\implies (3)$: This implication is obvious.\n\n$(3) \\implies (1)$:\nIf we have such an orthogonality, then $E$ and $F$ are semistable. Note that $\\deg(F)=0$ by \\hyref{Lemma}{lem:numORTH}(ii).\nMoreover, $E$ is also orthogonal to the direct sums $F^{\\oplus N}$ for all $N>0$, which are semistable sheaves of rank $N \\cdot \\mathrm{rk}(F)$ and degree 0.\nWe may take $N$ big enough such that the theta divisor $\\Theta_E$ in the moduli space of semistable degree 0 bundles on $X$ of rank $N \\cdot \\mathrm{rk}(F)$ is effective. By semicontinuity, being orthogonal to $E$ is an open condition. Thus, we may assume there exists an $F'$ outside $\\Theta_E$, i.e.\\ $E  \\orthstack{-\\Delta}  F'$ and $H^*(E \\otimes F') = 0$.\nAgain we consider the long exact cohomology sequence of $0\\to E\\boxtimes F'(-\\Delta) \\to E\\boxtimes F' \\to E\\otimes F'|_\\Delta \\to 0$; here it yields $H^*(E) \\otimes H^*(F') =0$.\nWe have $\\chi(F') \\ne 0$ from Riemann--Roch, if $g\\ne 1$. So $H^*(F') \\ne 0$, hence $H^*(E)=0$.\n\\end{proof}\n\n\\begin{corollary}\nLet $E$ be a semistable sheaf of slope $g-2$.\nIf there exists a line bundle $M$ of degree 1 with $H^*(E \\otimes M)=0$, then $\\shat{E}^0=0$, and $\\shat{E}^1$ is semistable of rank $\\mathrm{rk}(E)$.\n\\end{corollary}\n\n\\begin{remark}\nRaynaud proved in \\cite{Raynaud} the existence of stable sheaves $E$ having integral slope with the following property: $H^*(E \\otimes M) \\ne 0$ for all line bundles $M$. These base points of the theta divisor form a proper closed subset of the moduli space. Thus, we can only hope that the Picard sheaves $\\shat{E}$ are semistable for general semistable sheaves $E$.\n\\end{remark}\n\n\\section{A classical proof for the stability of $\\shat{E}$}\n\n\\noindent\nWe take $E$ to be a globally generated vector bundle on $X$. Thus, we have\n\\[ 0 \\to \\mathsf{F}_{\\!\\!-}^0( E) \\to H^0(E) \\otimes {\\mathcal O}_X \\to E \\to 0     \\qquad  \\text{ and } \\qquad\n   \\mathsf{F}_{\\!\\!-}^1(E) \\cong H^1(E) \\otimes {\\mathcal O}_X . \\]\nNote that a semistable bundle of slope $>2g-1$ is globally generated.\n\nFor a subsheaf $E' \\subset E$ we obtain an injective map $H^0(E') \\hookrightarrow H^0(E)$.\nFrom both morphisms we eventually obtain a subsheaf $\\mathsf{F}_{\\!\\!-}^0(E') \\subset \\mathsf{F}_{\\!\\!-}^0(E)$.\nThe next result tells us, that these subsheaves are enough to test\n(semi)stability.\n\n\\begin{lemma}\\label{lem:good1}\nThe sheaf $\\mathsf{F}_{\\!\\!-}^0(E)$ is (semi)stable if for all globally generated subsheaves $E' \\subset E$ we have the\ninequality $\\mu(\\mathsf{F}_{\\!\\!-}^0(E')) \\mathrel{(\\leq)} \\mu(\\mathsf{F}_{\\!\\!-}^0(E) )$.\n\\end{lemma}\n\n\\begin{proof}\nWe have the short exact sequence\n $0 \\to \\mathsf{F}_{\\!\\!-}^0(E) \\to H^0(E)\\otimes{\\mathcal O}_X \\to E \\to 0$,\nas $E$ is globally generated.\nLet $U\\subset\\mathsf{F}_{\\!\\!-}^0(E)$ be a subbundle. The inclusion $U\\hookrightarrow H^0(E)\\otimes{\\mathcal O}_X$ induces a surjection\n $H^0(E)^\\vee\\otimes{\\mathcal O}_X \\twoheadrightarrow H^0(U^\\vee)\\otimes{\\mathcal O}_X$. Denote by $V^\\vee\\subseteq H^0(U^\\vee)$ the image of the induced map on global sections, $H^0(E)^\\vee\\to H^0(U^\\vee)$. The commutative diagram\n\\[ \\xymatrix{\n   H^0(E)^\\vee \\otimes {\\mathcal O}_X \\ar@{->>}[d] \\ar@{->>}[rrd] \\\\\n       V^\\vee \\otimes {\\mathcal O}_X  \\ar@{^{(}->}[r] & H^0(U^\\vee) \\otimes {\\mathcal O}_X \\ar[r] & U^\\vee\n} \\]\nshows that $V^\\vee\\otimes{\\mathcal O}_X\\to U^\\vee$ is surjective. We get inclusions\n $U\\hookrightarrow V\\otimes{\\mathcal O}_X\\hookrightarrow H^0(E)\\otimes{\\mathcal O}_X$, which combine into a commutative diagram of short exact sequences\n\\[ \\xymatrix{\n   0 \\ar[r] & U  \\ar[d] \\ar[r] & V\\otimes{\\mathcal O}_X \\ar[d] \\ar[r] & E' \\ar[r] \\ar[d] & 0 \\\\\n   0 \\ar[r] & \\mathsf{F}_{\\!\\!-}^0(E) \\ar[r] & H^0(E)\\otimes{\\mathcal O}_X   \\ar[r]  & E  \\ar[r]        & 0\n} \\]\nwhere all vertical arrows are injective. Denote $v=\\dim(V)$ and $r' = \\mathrm{rk}(E')$. Then\n\\[ \\mu(U) = -\\frac{\\deg(E')}{v-r'} \\leq -\\frac{\\deg(E')}{h^0(E')-r'} = \\mu(\\mathsf{F}_{\\!\\!-}^0(E')) , \\]\nwhich shows that testing the (semi)stability condition only on subsheaves of the form $\\mathsf{F}_{\\!\\!-}^0(E')$ suffices to deduce it for arbitrary subsheaves.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:general1}\nIf $E$ is a (semi)stable vector bundle of slope $\\mu(E) > 2g$, then $\\mathsf{F}_{\\!\\!-}^0(E)$ is (semi)stable.\n\\end{corollary}\n\n\\begin{proof}\nSince $\\mu \\mathrel{\\mathop:}= \\mu(E) > 2g-2$, we have $h^1(E)=0$ and $h^0(E)=\\deg(E)+(1-g)\\mathrm{rk}(E)$.\nFrom the short exact sequence\n $0 \\to \\mathsf{F}_{\\!\\!-}^0(E) \\to H^0(E) \\otimes {\\mathcal O}_X \\to E \\to 0$,\nwe deduce\n\\[ \\mu(\\mathsf{F}_{\\!\\!-}^0(E)) = \\frac{-\\deg(E)}{h^0(E)-\\mathrm{rk}(E)} = \\frac{-\\deg(E)}{\\chi(E)-\\mathrm{rk}(E)}\n                 = \\frac{-\\deg(E)}{\\deg(E)-g\\cdot\\mathrm{rk}(E)} = \\frac{-\\mu}{\\mu-g} . \\]\nAssume that $E$ is stable.\nLet $E' \\subsetneq E$ be a globally generated proper subsheaf of $E$.\nIn \\hyref{Corollary}{cor:CT1} (\\hyref{Section}{sec:clifford} is independent of the rest of the article), we draw the following consequence from the generalised Clifford theorem:\n $h^0(E')-\\mathrm{rk}(E') < \\frac{\\mu -g}{\\mu} \\deg(E')$.\nThus\n\\[    \\mu(\\mathsf{F}_{\\!\\!-}^0(E')) = \\frac{-\\deg(E')}{ h^0(E')-\\mathrm{rk}(E')}\n   <  \\frac{-\\mu}{\\mu-g } = \\mu(\\mathsf{F}_{\\!\\!-}^0(E)) \\, . \\]\nBy \\hyref{Lemma}{lem:good1}, to show the stability of $\\mathsf{F}_{\\!\\!-}^0(E)$ it suffices to check this inequality for the subsheaves of type $\\mathsf{F}_{\\!\\!-}^0(E')$.\nThe claim about semistability follows from this by the Jordan--H\\\"older filtration of $E$.\n\\end{proof}\n\n\\begin{corollary} \\label{cor:STAB1}\nLet $E$ be a (semi)stable vector bundle of slope $\\mu(E) > 2g-1$.\nThen the restriction of $\\shat{E}^0$ to any curve $(-X)_M$ is (semi)stable.\nIn particular, $\\shat{E}^0$ is (semi)stable.\n\\end{corollary}\n\n\\begin{proof} \nWe have $\\shat{E}^0|_{(-X)_M} \\cong \\mathsf{F}_{\\!\\!-}^0(E\\otimes M) \\otimes {\\mathcal O}_X(P_0)$ from \\hyref{Proposition}{prop:FM-restriction}. This holds for all curves $(-X)_M$ because $E$ semistable of slope $>2g-1$ implies that $\\shat{E}^0$ is locally free.\nMoreover, $E\\otimes M$ is a (semi)stable bundle of slope $\\mu(E\\otimes M) > 2g$, so \\hyref{Corollary}{cor:general1} applies and yields the (semi)stability of $\\mathsf{F}_{\\!\\!-}^0(E\\otimes M)$ and hence of $\\shat{E}^0|_{(-X)_M}$.\n\\end{proof}\n\n\n\\section{Application of orthogonality}\n\n\\noindent\nThere are two ways how to apply the orthogonality condition $E  \\orthstack{-\\Delta}  F$, in order to deduce the semistability of the Picard bundle $\\shat{F}$ from the semistability of another Picard sheaf $\\shat{F}$. Either we use the symmetry of orthogonality, i.e.\\ \\hyref{Lemma}{lem:sym}(i) and (ii), or  we employ Serre duality, i.e.\\ \\hyref{Lemma}{lem:sym}(iii).\nWe start with the latter method.\n\n\\begin{corollary} \\label{cor:res2}\nLet $E$ be a semistable vector bundle on $X$ with $\\mu(E)=2g-1$.\nThen $\\shat{E}^1 = 0$ and $\\shat{E}^0$ is a vector bundle of rank $g \\cdot \\mathrm{rk}(E)$ which is also semistable. \nMoreover, the restriction of $\\shat{E}^0$ to any curve $(X)_N$ is semistable.\n\\end{corollary}\n\n\\begin{proof}\nThe vanishing of $\\shat{E}^1$ follows from cohomology and base change, and for the same reason, $\\shat{E}^0$ is a vector bundle of the given rank.\nAs $E$ is semistable, so is its dual $E ^\\vee$. Thus, $E^\\vee \\otimes \\omega_X$ is semistable of degree $-1$. By \\hyref{Lemma}{lem:orth-pair}, there exists a sheaf $F$ such that $(E^\\vee \\otimes \\omega_X \\otimes M)  \\orthstack{-\\Delta}  F$\nfor any line bundle $M$ of degree 1. By Serre duality,  \\hyref{Lemma}{lem:sym}(iii), this implies\n $(E \\otimes M^\\vee)  \\orthstack{\\Delta}  (F^\\vee \\otimes \\omega_X)$.\nNow we proceed as in the proof of \\hyref{Corollary}{cor:res1}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:general1a}\nApplying \\hyref{Corollary}{cor:res2} to a degree $2g-1$ line bundle $L$, we obtain the semistability of the Picard bundle $P_{2g-1} = \\shat{L}^0$. Thus, the above result is a generalisation of Kempf's result \\cite{Kempf}.\nIn fact, Kempf shows the stability of $P_{2g-1}$. The stability follows along the lines of Corollaries~\\ref{cor:general1} and \\ref{cor:CT1}.\nIndeed, if $X$ is not hyperelliptic, and $L \\otimes M\\otimes \\omega_X^\\vee$ is not effective, then the restriction of $\\shat{L}^0$ to $(-X)_M$ is stable.\n\\end{remark}\n\n\\begin{lemma}\\label{lem:general2}\nIf $E$ is a stable vector bundle on $X$ with $\\mu(E)<-2$, then $\\mathsf{F}_{\\!\\!+}^1(E)$ is stable.\n\\end{lemma}\n\n\\begin{proof}\nTensoring the short exact sequence\n $0 \\to {\\mathcal O}_{X \\times X} \\to {\\mathcal O}_{X\\times X}(\\Delta) \\to {\\mathcal O}_\\Delta(\\Delta) \\to 0$\non $X \\times X$ with $E\\boxtimes{\\mathcal O}_X$, and applying $\\mathsf{pr}_{2*}$ yields the following short exact sequence on $X$:\n\\[ 0 \\to E \\otimes \\omega^\\vee \\to H^1(E) \\otimes  {\\mathcal O}_X \\to \\mathsf{F}_{\\!\\!+}^1(E) \\to 0 ,\\]\nbecause $\\mathsf{F}_{\\!\\!-}^0(E)=0$ from stability of $E$ with $\\mu(E)<-2$.\nDualising this sequence yields\n\\[ 0 \\to \\left( \\mathsf{F}_{\\!\\!+}^1(E) \\right)^\\vee \\to H^1(E) ^\\vee \\otimes  {\\mathcal O}_X \\to E ^\\vee \\otimes \\omega_X \\to 0 . \\]\nThus, by classical Serre duality, $\\mathsf{F}_{\\!\\!+}^1(E)$ is the dual of $\\mathsf{F}_{\\!\\!-}^0(E')$ for\n$E'=E^\\vee \\otimes \\omega_X$. However $E'$ is also stable, of slope\n $\\mu(E') = 2g-2 -\\mu(E) > 2g$.\nSo by \\hyref{Corollary}{cor:general1} the sheaf $\\mathsf{F}_{\\!\\!-}^0(E')$ is stable.\nThis proves the lemma.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:STAB2}\nIf the vector bundle $E$ on $X$ is (semi)stable of slope $\\mu(E)< -1$, then\nthe Picard sheaf $\\shat{E}^1$ is (semi)stable when restricted to any curve $(X)_N$.\nIn particular, $\\shat{E}^1$ is (semi)stable.\n\\end{corollary}\n\n\\begin{proof}\nFor stability, this follows from \\hyref{Proposition}{prop:FM-restriction} and \\hyref{Lemma}{lem:general2}.\n\nThe claim for semistable $E$ then follows using the Jordan--H\\\"older filtration of $E$.\n\\end{proof}\n\nNext, we give examples for how to apply the symmetry property of orthogonality.\nAs usual, for a rational number $x$ we denote by $\\lceil x\\rceil$ the round up of $x$.\nFor any $r,h\\in{\\mathbb{N}}$, we introduce the number used by Popa in \\cite[Theorem 5.3]{Popa} for an effective version of \\hyref{Faltings' Theorem}{thm:Faltings}:\n\\[ P(r,h) \\mathrel{\\mathop:}= 2h \\left\\lceil\\frac{h^2r^2+1}{8h}\\right\\rceil \\,. \\]\n\n\\begin{theorem}[Popa 2001] \\label{thm:Popa}\nFor any semistable vector bundle $G$ of rank $r \\cdot h$ and degree $d \\cdot h$ with\ncoprime integers $r$ and $d$, there exists for any $k \\geq P(r,h)$ a vector\nbundle $F$ of rank $r \\cdot k$ such that $H^*(G \\otimes F)=0$.\n\\end{theorem}\n\n\n\\begin{lemma}\\label{lem:general3}\nLet $\\mu=\\frac{d}{r} \\in {\\mathbb{Q}}$ with $\\mu\\in (g,g+1]$, let $k \\geq P(r,g)$ and $R = k \\cdot r$. \nThen there exists a vector bundle $F$ on $X$ of slope $\\mu$ and rank $R$ such that $\\mathsf{F}_{\\!\\!-}^0(F) \\oplus \\mathsf{F}_{\\!\\!-}^1(F)$ is semistable.\n\\end{lemma}\n\n\\begin{proof}\nWe begin with the involution ${\\mathbb{Q}}\\to{\\mathbb{Q}}, \\mu \\mapsto \\mu^- \\mathrel{\\mathop:}= g+\\smash{\\frac{g}{\\mu-g}}$. It is decreasing on ${\\mathbb{Q}}_{>g}$.\nBy \\hyref{Lemma}{lem:numORTH}, if $E  \\orthstack{-\\Delta}  F$, then $\\mu(E)=\\mu(F)^-$.\n\nNow let $\\mu = \\frac{d}{r} \\in (g,g+1]$, then $\\mu^- \\in [2g, \\infty)$.\nLet $E$ be a stable vector bundle on $X$ of rank $d-rg$ and degree $gd-g^2r+rg$, i.e.\\\n$\\mu(E)=\\mu^-$. By \\hyref{Corollary}{cor:general1} or \\hyref{Remark}{rem:general1a}, we have that\n$\\mathsf{F}_{\\!\\!-}^0(E)$ is semistable. Since $\\mu(E) >2g-1$, we also conclude $\\mathsf{F}_{\\!\\!-}^1(E)=0$.\nSo we can use the Riemann--Roch formula to compute $\\mathrm{rk}(\\mathsf{F}_{\\!\\!-}^0(E))=gr$, and\n$\\deg(\\mathsf{F}_{\\!\\!-}^0(E))=gr-gd-g^2r$. Popa's result Theorem \\ref{thm:Popa} implies that for\nany $R=k \\cdot r$ with $k$ as above there exists a vector bundle $F$ of rank $R$ such\nthat $H^*(\\mathsf{F}_{\\!\\!-}^0(E) \\otimes F)=0$.\n\nBy \\hyref{Lemma}{lem:ortho1}, this yields $E  \\orthstack{-\\Delta}  F$.\nSymmetry, i.e.\\ \\hyref{Lemma}{lem:sym}(i), then gives $F  \\orthstack{-\\Delta}   E$.\nAnd so, again by \\hyref{Lemma}{lem:ortho1}, it follows that\n $H^*((\\mathsf{F}_{\\!\\!-}^0(F) \\oplus \\mathsf{F}_{\\!\\!-}^1(F)) \\otimes E)=0$.\nThis implies the semistability of the direct sum $\\mathsf{F}_{\\!\\!-}^0(F) \\oplus \\mathsf{F}_{\\!\\!-}^1(F)$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:STAB3}\nLet $\\mu=\\frac{d}{r} \\in {\\mathbb{Q}}$ with $\\mu\\in(g-1,g]$, let $k \\geq P(r,g)$ and $R=k \\cdot r$.\nThen there exists a vector bundle $F$ on $X$ of slope $\\mu$ and rank $R$ such that $\\shat{F}^0$ is semistable. Indeed, the restriction of $\\shat{F}^0$ to the general curve $(-X)_M$ is semistable.\n\\end{proposition}\n\n\\begin{proof}\nIt is enough to show the existence of some $F$ such that $\\shat{F}^0$ restricted to $(-X)_M$ is semistable. Let $\\mu$ and $R$ be as in the\nproposition. Let $F'$ be a vector bundle of slope $\\mu+1$ and rank $R$ such that $\\mathsf{F}_{\\!\\!-}^0(F) \\oplus \\mathsf{F}_{\\!\\!-}^1(F)$ is semistable, which exists by \\hyref{Lemma}{lem:general3}. Then $\\mathsf{F}_{\\!\\!-}^0(F')  \\otimes {\\mathcal O}_X(P_0)$ is also semistable.\n\nWe set $F=F' \\otimes M^\\vee$ for some line bundle $M$ of degree 1. By \\hyref{Proposition}{prop:FM-restriction}, the restriction of $\\shat{F}^0$ to $(-X)_M$ is the semistable sheaf $\\mathsf{F}_{\\!\\!-}^0(F') \\otimes {\\mathcal O}_X(P_0)$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:STAB4}\nLet $\\mu=\\frac{d}{r} \\in {\\mathbb{Q}}$ with $\\mu \\in [g-2,g-1)$, let $k \\geq P(r,g)$ and $R=k \\cdot r$. \nThen there exists a vector bundle $F$ on $X$ of slope $\\mu$ and rank $R$ such that $\\shat{F}^1$ is semistable. Indeed, the restriction of $\\shat{F}^1$ to the general curve $(X)_N$ is semistable.\n\\end{corollary}\n\n\\begin{proof}\nThis follows from \\hyref{Proposition}{prop:STAB3}, using Serre duality as in \\hyref{Lemma}{lem:sym}(iii).\n\\end{proof}\n\n\n\\section{Clifford's theorem for vector bundles on a curve} \\label{sec:clifford}\n\\noindent\nLet us remind the reader that $\\mu_{\\max }(E)$ denotes the maximum of all\nslopes of subbundles of $E$, and $\\mu_{\\min}(E)$ denotes the minimal slope of\nall quotient bundles of $E$.\n\n\\begin{proposition} \\label{Clifford}\nLet $E$ be a vector bundle of rank $r$ and degree $d$ on the smooth projective curve $X$ of genus $g$.\nIf $\\mu_{\\max}(E) \\leq 2g-2 $ and $\\mu_{\\min}(E) \\geq 0$, then we have   the estimate\n\\[ h^0(E)-r \\leq \\frac{d}{2} \\, .\\]\nMoreover, if $\\mu_{\\max}(E) < 2g-2 $ and $\\mu_{\\min}(E) > 0$ and $h^0(E)-r = \\frac{d}{2}$, then $X$ is hyperelliptic, and the determinant line bundle $\\det(E)$ is a multiple of the hyperelliptic line bundle $M$, and $E$ possesses a filtration with graded object ${\\rm gr}(E)= \\bigoplus_{i=1}^r M^{ \\otimes a_i}$ with $0<a_i<g-1$.\n\\end{proposition}\n\n\\begin{proof}\nWe first prove the inequality $h^0(E)-r \\leq \\frac{d}{2}$ by induction on $r$.\n\nIf $r=1$, then $E={\\mathcal O}(D)$ is a line bundle associated to a divisor $D$. In this case $d=\\mu_{\\max}({\\mathcal O}(D))=\\mu_{\\min}({\\mathcal O}(D))$. If $D$ is effective and special, then the claim is precisely the well known theorem of Clifford, see for example \\cite[Theorem IV.5.4]{Hartshorne}. If $D$ is not effective, then $h^0({\\mathcal O}(D))=0$ and the statement is trivial. If $D$ is non-special, then by Riemann--Roch\n $h^0({\\mathcal O}(D))-1 = \\chi({\\mathcal O}(D))-1 = d-g < \\frac{d}{2}$,\nthe inequality following from $d = \\mu_{\\max} \\leq 2g-2$.\n\nNow suppose that $E$ is of rank $r\\geq2$, and the inequality holds for all vector bundles of rank smaller than $r$ which meet the slope conditions. We consider two cases:\n\\\\\nCase 1: $E$ is not semistable. Take the subsheaf $E_1$ of $E$ of slope $\\mu_{\\max}(E)$  and of maximal possible rank.\nThis $E_1$ is the first sheaf appearing in the Harder--Narasimhan filtration of $E$. We obtain a short exact sequence\n\\[ 0 \\to E_1 \\to E \\to E_2 \\to 0 \\, .\\]\nWe have\n $\\mu_{\\max}(E_1)=\\mu_{\\min}(E_1)=\\mu_{\\max}(E)$, $\\mu_{\\max}(E) > \\mu_{\\max}(E_2)$, and $\\mu_{\\min}(E)=\\mu_{\\min}(E_2)$.\nIn particular, we see that the induction hypothesis applies to the vector bundles $E_1$ and $E_2$. Hence\n $h^0(E_i) -\\mathrm{rk}(E_i) \\leq \\frac{1}{2}\\deg(E_i)$\nfor $i\\in \\{ 1,2\\}$. Taking global sections of the above short exact sequence, we conclude\n $h^0(E) \\leq h^0(E_1)+h^0(E_2)$.\nSo we get\n\\[ \\begin{array}{rclcl}\n        h^0(E)-r & \\leq &  h^0(E_1)+h^0(E_2)-r\n& = &   \\left( h^0(E_1) -\\mathrm{rk}(E_1) \\right) + (h^0(E_2) -\\mathrm{rk}(E_2)) \\\\\n&\\leq&  \\frac{1}{2} \\deg(E_1) + \\frac{1}{2} \\deg(E_2) &=& \\frac{1}{2} \\deg(E) .\n\\end{array} \\]\nCase 2: $E$ is semistable.\nAgain, we distinguish two cases, by inspecting the slope of $E$.\n\\\\\nCase 2.1: $\\mu(E) \\leq g-1$.\nWe may assume $h^0(E)>0$. Let $E_1$ be a line subbundle of $E$ of maximal possible degree $d_1$. From $h^0(E)>0$ we conclude that $d_1 \\geq 0$. We obtain a short exact sequence\n\\[ 0 \\to E_1 \\to E \\to E_2 \\to 0 \\, .\\]\nSince any quotient of $E_2$ is also a quotient of $E$ we conclude $\\mu_{\\min}(E_2) \\geq \\mu_{\\min}(E) \\geq 0$. We want to show that $\\mu_{\\max}(E_2) \\leq 2g-2$.\nAssume the contrary. Then we have a subsheaf $E_3 \\subset E_2$ of rank $r_3$ and slope $\\mu(E_3) > 2g-2$. The kernel $K$ of the composition of surjections\n\\[ E \\to E_2 \\to E_2\/E_3 \\]\nis of rank $r_3+1$ and of slope\n $\\mu(K) = \\frac{d_1+\\mu(E_3)\\cdot r_3}{r_3+1}\n      \\geq \\frac{\\mu(E_3)\\cdot r_3}{r_3+1}\n         = \\frac{\\mu(E_3)}{1+1\/r_3}\n         > \\frac{2g-2    }{1+1\/r_3}\n      \\geq g-1$.\nThis contradicts the semistability of $E$. Thus, for both sheaves $E_1$ and $E_2$ the induction hypothesis applies, and we can proceed like in Case 1.\n\\\\\nCase 2.2: $\\mu(E_1) > g-1$.\nThe Serre dual bundle $E'=E^\\vee \\otimes \\omega_X$ has slope\n $\\mu(E') = 2g-2 -\\mu(E) < g-1$.\nTherefore, as we have seen in Case 2.1\n\\[ h^0(E') \\leq \\frac{\\deg(E')}{2} = \\frac{\\mathrm{rk}(E)(2g-2)-\\deg(E)}{2} \\, .\\]\nBy Serre duality $h^0(E') = h^1(E)$. So when adding the Riemann--Roch formula\n $h^0(E)-h^1(E) = \\deg(E)+\\mathrm{rk}(E)(1-g)$\nto the above inequality, we obtain the stated inequality.\n\nThe statement for the case $h^0(E)-r = \\frac{d}{2}$ follows along the same lines. Indeed, we must have this equality for $E_1$ and $E_2$ and can proceed by induction since\n $\\det(E) \\cong \\det(E_1) \\otimes \\det(E_2)$.\nThe passage from $E$ to the Serre dual $E'=E^\\vee \\otimes \\omega_X$ sends a vector bundle $E$ with $\\det(E)=M^{\\otimes a}$ to a bundle with \n $\\det(E')=M^{\\otimes(r(g-1)-a)}$\nwhere $M$ denotes the hyperelliptic line bundle.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:CT1}\nLet $E$ be a stable vector bundle of slope $\\mu = \\mu(E) > 2g$.\nFor any globally generated subsheaf $E' \\subsetneq E$ which is not a trivial bundle,\nwe have the strict inequality\n\\[  h^0(E')-\\mathrm{rk}(E') < \\Big( 1 - \\frac{g}{\\mu}\\Big) \\deg(E') . \\]\n\\end{corollary}\n\n\\begin{proof}\nAs $E' \\subset E$ is globally generated, we have $\\mu_{\\min}(E') \\geq 0$.\nFor one sheaf $E_1'$ in the Harder--Narasimhan filtration of $E'$, we have $\\mu_{\\min}(E_1')>2g-2$, and $\\mu_{\\max}(E'\/E_1') \\leq 2g-2$.\nWe set $E_2'=E'\/E_1'$. Now $E_1'$ is semistable with $\\mu_{\\min}(E_1')>2g-2$, hence $h^1(E_1')=0$. So\n\\[ h^0(E_1')-\\mathrm{rk}(E_1') = \\chi(E_1') - \\mathrm{rk}(E_1')\n                         = \\deg(E_1') - g\\cdot\\mathrm{rk}(E_1')\n                         = \\Big( 1 - \\frac{g}{\\mu(E_1')} \\Big) \\deg(E_1') \\]\nby Riemann--Roch. Since $\\mu(E_1') < \\mu$, and the function $x \\mapsto 1-\\frac{g}{x}$ is strictly increasing for $x>0$, we deduce the inequality\n\\[ h^0(E_1')-\\mathrm{rk}(E_1') < \\Big( 1- \\frac{g}{\\mu} \\Big) \\deg(E_1') . \\]\nThe sheaf $E_2'$ satisfies the assumptions of \\hyref{Proposition}{Clifford}. Moreover, $E'_2$ is itself a globally generated sheaf, and we have\n $\\mu_{\\min}(E_2') \\geq \\mu_{\\min}(E') > 0$,\nthe latter inequality from the assumption that $E'$ is not a trivial bundle. Hence $\\deg(E_2')>0$ and so we have \n\\[ h^0(E_2')-\\mathrm{rk}(E_2') \\leq \\frac{1}{2} \\deg(E_2') < \\Big( 1 - \\frac{g}{\\mu} \\Big) \\deg(E_2') . \\]\nThis last inequality holds, because $\\mu > 2g$ implies $\\frac{1}{2} < 1-g\/\\mu$. Adding the two inequalities for $h^0(E_i')-\\mathrm{rk}(E_i')$ for $i =1,2$, we obtain the statement of the corollary.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nSocial media platorms have been seeing growing usage and adoption over time. This growth has been further accelerated with the lockdowns in the pandemic where people had no choice but to interact and express themselves online. This has led to increase in new users as well as daily active users. India has seen tremendous adoption in social media usage with the Jio revolution when Reliance Jio made internet charges affordable. Reliance Jio allowed democratizing internet which has benefitted social media platforms operating in India. With the rise of intenet access to Indian people, social media platforms are seeing rise in Indic native content. It is in interest of social media platforms to adapt their systems to understand the regional content to better serve the users and attract more user base. One such company which has been focusing on serving the regional content since its foundation is Sharechat. Sharechat is an Indian company which is one of the leading social networking space in catering to regional content. Their short video app is called Moj which went viral in the lockdown and has been since seeing tremendous growth in its user base and content. \n\nIncreasing social media usage has made it important to keep the platform safe from abusive content to enable individual safety in the online world. There is more abusive content in the online world when compared to the offline world because Internet enables anonymity. This can be used to share abusive content without being held responsible. Text analysis of English social media content is a well explored domain \\cite{sentiment-svm}, \\cite{sentiment}. Multilingual NLP is a relative new subfield and has been picking up pace due to globalization \\cite{xlm-r-twitter}. Most of the multilingual transfomer models were trained on Wikipedia dataset of Common Crawl dataset \\cite{mbert}, \\cite{xlm-r}. This is different from  training on a social media text dataset as social media text content have vast amount of acceptable misspellings. On top of this, non-English social media content have their own unique set of challenges comprising of code-mixing, transliteration, mixing of scriptures in the same sentences. Code-Mixing is the usage of different languages in the same sentence or utterance. This mostly used in informal conversation and social media. This phenonmenon exits in multilingual societies where the speaker is fluent in more than one language \\cite{bertologicomix}. Transliteration is a way of writing a native language in the scripture of a different language. The pronunciation of the transliterated langugae, by following the rules of the scripture used, is similar to the one written using the native scripture. Most common form of transliteration is to use the Latin scripture to express a non-English sentence. Transliteration aggravates the misspellings and variants of the same word. This is because not all syllables and sounds in the native language would be available in the Latin language so the writer has to use non-standard ways to approximate the syllable. \n\nPrior works have performed text analysis on non-English languages and have attempted to tackle a subset of the above challenges associated with non-English social media text \\cite{2020-semeval}. \\cite{sentiment-arabic}, \\cite{sentiment-czech}, and \\cite{sentiment-telegu} have extended text analysis to Arabic, Czech, and Telugu languages. In our work, we identify the common challenges associated with non-English social media text and propose an approach to tackle all of these challenges in multilingual setting. Our contributions include a spell-correction algorithm that does not rely on a manually defined dictionary but instead infers its own from the training data. This allows our spell-correction algorithm to learn new words by retraining on the any new corpus. This is in contrast to the traditional dictionary based approach where the words need to be added manually. Our approach is scalable in the age of neologism which is prevalant in context of social media \\cite{neologism}.\n\nThe paper is organized as follows. Sec. \\ref{sec:dataset} describes the Moj Dataset used in this work. Sec \\ref{sec:problem-statement} defines the problem statement and challenges associated with it. Sec. \\ref{sec:proposed-method} explains our approach in detail. Sec. \\ref{sec:exp-setup} provides the experimental setup. Sec. \\ref{sec:result} lists the result of our approach and in  \\ref{sec:ablation} we perform ablation studies on our approach. Sec. \\ref{sec:future-work} discusses the future work and we conclude our work in Sec. \\ref{sec:conclusion}\n\n\\section{Dataset}\n\\label{sec:dataset}\nThe dataset provided by Moj consists of multilingual comments in $10+$ low-resource Indic languages. The comments in the dataset have been human annotated with a label indicating whether the comment is abusive or not. The dataset also has \\textit{meta-data} like the \\textit{language} of the comment and the number of \\textit{likes} and \\textit{reports} on each comment and the corresponding post on which the comment was made. The label distribution across different language is show in Table \\ref{tab:label-dist} \n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Label distribution across languages in Moj Dataset}\n\t\\label{tab:label-dist}\n\t\n\t\\bgroup\n\t\\def\\arraystretch{1.2\n\t\\begin{tabular}{ccc}\n\t\t\\hline\n\t\t\\textbf{Language} & \\textbf{Non-Abusive} & \\textbf{Abusive}\\\\\n\t\t\\hline\n\t\tAssamese & 1496 & 1284\\\\\n\t\tBengali & 11428 & 11407\\\\\n\t\tBhojpuri & 2917 & 2887\\\\\n\t\tGujarati & 4426 & 4402\\\\\n\t\tHaryanvi & 4395 & 4417\\\\\n\t\tHindi & 153433 & 153747\\\\\n\t\tKannada & 6954 & 6989\\\\\n\t\tMalayalam & 31749 & 9216\\\\\n\t\tMarathi & 44677 & 27367\\\\\n\t\tOdia & 5475 & 5499\\\\\n\t\tRajasthani & 2183 & 2185\\\\\n\t\tTamil & 34792 & 34705\\\\\n\t\tTelugu & 48461 & 48551\\\\\n\t\t\n\t\t\\hline\n\t\\end{tabular}\n\t\\egroup\n\\end{table}\n\n\n\\section{Problem Statement}\n\\label{sec:problem-statement}\nThe main objective of the paper is to develop a multi-lingual approach to identify abusive comments for social media platforms. Upon exploring the Moj dataset, we identify the following challenges associated with multi-lingual non-English social media content:\n\\begin{itemize}\n\t\\item \\textbf{Code-Mixing}: Social media text is not written in just one language in multi-lingual socities.\n\t\\item \\textbf{Transliteration}: People often use Latin scripture to write sentences in their native language. Pronunciation of both original native sentence and transliterated sentence is almost similar.\n\t\\item \\textbf{Mixing of Scriptures}: Some people write sentences which contain words from different scriptures.\n\t\\item \\textbf{Misspelling and variations} in social media text.\n\\end{itemize}\nFig. \\ref{fig:challenge} shows different representation of the same Hindi sentence. Our approach attempts to tackle such challenges in multi-lingual setting.\n\n\\begin{figure}[!ht]\n\t\\includegraphics[width=0.43\\textwidth]{image\/challenge-fig.png}\n\t\\caption{\\label{fig:challenge} Example of challenges described in Sec. \\ref{sec:problem-statement}.}\n\\end{figure}\n\n\n\\section{Proposed Method}\n\\label{sec:proposed-method}\n\\subsection{Data Cleaning}\n\\label{ssec:data-cleaning}\nThe dataset includes multiple comments from the same post as well. We replace the post level meta-data of each comment, i.e., post like count and post report count, with the maximum value of respective meta-data feature among all the comments corresponding to the same post.\n\nWe pre-process the comments in the Moj dataset with the following steps:\n\\begin{itemize}\n\t\\item Unicode normalization using \\verb|unicodedata| python package\n\t\\item Remove special characters, e.g, $@$, $\"$, $\\$$, $\\#$\n\t\\item Remove Emoji using \\verb|demoji|\\footnote{https:\/\/pypi.org\/project\/demoji\/} python package\n\t\\item Replace characters which occur more than twice consecutively with the same character, e.g., hellooo \\textrightarrow hello\n\t\\item Transliterate the comments using \\verb|indic-trans| python package \\cite{indic-trans} to the native laguage scripture to tackle the challenge of transliteration mentioned in Sec. \\ref{sec:problem-statement}. The effect of this pre-processing step is discussed in Sec. \\ref{sec:ablation}.\n\\end{itemize}\n\n\\begin{figure*}[!ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{image\/graph.png}\n\t\\caption{\\label{fig:spell-correct} Overview of the training phase of Spell Correction using Graph Clustering: \n\t\t(a) Graph is constructed using words in the training corpus. Two words have an edge in the graph if they satisfy the nearness criteria specified in Sec. \\ref{sssec:train-spell-correct}.\n\t\t(b) Maximal clique is found in this graph. The circled subgraph in the figure is the maximal clique in this graph. This subgraph is considered to contain words that are misspellings of one another and are representing the same concept.\n\t\t(c) This subgraph\/cluster is removed from the graph and is processed to obtain its \\textit{Parent Word} and \\textit{Anchor Words} as explained in Sec. \\ref{sssec:train-spell-correct}. The whole process is again repeated using the remaining smaller graph untill all the nodes have been processed. Please refer to Sec.\n\t\t\\ref{ssec:spell-correct} for more details.}\n\\end{figure*}\n\n\\subsection{Spell Correction using Graph Clustering}\n\\label{ssec:spell-correct}\nOur approach builds a graph of words using the training corpus of Moj dataset. Sec. \\ref{sssec:train-spell-correct} shows how graph clustering is used to identify the set of related words and the correct spelling of those words. Sec. \\ref{sssec:test-spell-correct} shows the strategy used to identify the correct spelling. This approach works even for the wrong words that were not seen in the training as long as the correct word was seen in training. Sec. \\ref{sec:ablation} shows the utility of our approach.\n\\subsubsection{Training: Correct Spelling Identification}\n\\label{sssec:train-spell-correct}\nWe intend to spell-correct the abusive words and ignore words which occur predominantly in non-abusive comments. This is done to prevent non-abusive words from being auto-corrected to abusive words or vice-versa. Since this is a classification problem, presence of abusive word is more important than presence of non-abusive word. This is because any combination of abusive and non-abusive word will be abusive in nature. \n\nConsider words that have occured minimum $5$ times in the abusive comments in the training corpus. Ignore words that have length less than $6$ and have number of consonants less than $4$.  Construct a graph where nodes are these words. Connect words with an edge if they have the same first $k$ letters and the Levenshtein Distance between them is above a threshold score $t$. In this work, $k$ was $3$ and score $t$ was $85$ based on hyper-parameter tuning on a validation set. Levenshtein Distance was computed using \\verb|rapidfuzz|\\footnote{https:\/\/pypi.org\/project\/rapidfuzz\/}. Algorithm \\ref{alg:Graph} is a psuedo-code of the graph construction.\n\nFind the maximal clique from the above constructed graph of words. A maximal clique is the largest subgraph where nodes are connected to one another by an edge. This maximal clique represents a subgraph of words which our algorithm thinks are variants or misspellings of one another and represent the same thing, e.g, \\verb|bewakoof|, \\verb|bewakoofi|, \\verb|bewakouf| where \\verb|bewakoof| means idiot in Hindi. The word in this cluster with highest occurring frequency in the training corpus is chosen as the correct spelling for this cluster. We refer to this word as the \\textit{Parent Word} of the cluster. Next, we find the \\textit{Anchor Words} for this cluster. \\textit{Anchor Words} are those words which will represent this cluster during Testing phase explained in Sec. \\ref{sssec:test-spell-correct}. Top five frequent words of this cluster are chosen as the candidates for the \\textit{Anchor Words}. Among them, those words that have frequency above one-fourth of the the \\textit{Parent Word} is included in \\textit{Anchor Words}.\n\nThe above maximal clique is removed from the graph once that cluster is processed and its \\textit{Parent Words} and \\textit{Anchor Words} have been computed. This process of finding a cluster using maximal clique, processing the cluster, and removing it is repeated on the remaining smaller graph obtained from the previous rounds of clustering until all the nodes in the original graph have been processed. Algorithm \\ref{alg:train} shows the psuedo-code for training phase.\n\n\\subsubsection{Testing: Auto-correction during inference}\n\\label{sssec:test-spell-correct}\nBy the end of training phase, we have a set of clusters with \\textit{Parent Word} and its \\textit{Anchor Words} that will be used during inference to spell-correct the words in train and test corpus. Given a word $w$ during inference, we find out the clusters whose \\textit{Parent Word}  has the same first $k$ letters as that of word $w$. The association $score_{wi}$ of a cluster $C_{i}$ obtained from the previous filtering step is given as:\n\\begin{equation}\n\\label{eq:y}\nscore_{wi} = \\max_{a \\in A(C_{i})} \\{Levenshtein Distance(w, a)\\}\n\\end{equation}\nwhere $A(C_{i})$ is the set of \\textit{Anchor Words} of cluster $C_{i}$. The cluster $C_{j}$ that has the highest association $score_{wj}$ with the word $w$ is chosen as the appropriate cluster $C^{*}$ if the score $score_{wj}$ is above a threshold $t$. The word $w$ is replaced with the \\textit{Parent Word} of the cluster $C^{*}$ if the word $w$ was able to find its appropriate cluster $C^{*}$, e.g., \\verb|bewakoufi| will be corrected to \\verb|bewakoof|. We spell-correct all the comments in both training and testing data before using them for classification. Algorithm \\ref{alg:test} shows the psuedo-code for the testing phase.\n\n\\subsection{Multilingual Models}\n\\label{ssec:model}\nGiven that our task involves low-resource Indic languages, we use multiple pre-trained multilingual transformer models and finetune them on the comments of Moj dataset. The following models were used in our work:\n\\begin{itemize}\n\t\\item \\textbf{mBERT} \\cite{mbert}: is a multilingual variant of BERT \\cite{bert} trained on multilingual Wikipedia data from $104$ languages.\n\t\\item \\textbf{XLM-R} \\cite{xlm-r}: is trained on cross-lingual masked language modeling task on $100$ languages.\n\t\\item \\textbf{XLM-T} \\cite{xlm-r-twitter}: is XLM-R trained on Twitter comments\n\t\\item \\textbf{Multilingual DistilBERT} \\cite{distilbert}: is a distilled version of mBERT\n\t\\item \\textbf{MURIL} \\cite{muril}: is a variant of BERT trained on monolingual, translated and transliterated data from $17$ Indic languages\n\t\\item \\textbf{Indic-BERT} \\cite{indic-bert}: is a variant of ALBERT \\cite{albert} trained on IndicCorp dataset \\cite{indic-bert}.\n\t\\item \\textbf{Canine} \\cite{canine}: is a tokenizer and vocabulary free transformer model which operates directly on sequences of characters. Canine-S and Canine-C are two variants trained on different loss functions defined in \\cite{canine}.\n\\end{itemize}\n\nWe also trained a logistic regression model on an embedding created by using Naive-Bayes over TF-IDF feautures \\cite{nb-svm}\\footnote{https:\/\/www.kaggle.com\/jhoward\/nb-svm-strong-linear-baseline}.\n\n\n\\subsection{Meta-Data}\nThe meta-data associated with the comments, i.e, likes and report count of the comments as well as the post on which the comment was made, was leveraged to further improve the predictions. We trained a Random Forest classifier over this meta-data only to provide the predictions. These meta-data specific predictions are later combined with the predictions of the comment-specific models as discussed later in Sec. \\ref{sec:result}.\n\n\\begin{algorithm}[!ht]\n\t\\caption{Graph Construction}\n\t\\label{alg:Graph}\n\t\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\t\\Input{~Words $W$ present in abusive sentences; Frequency count $F$ containing number of times a word has occurred in the corpus; Length of words $L$; Number of consonants in words $C$; Function $T(w)$ gives the first three letters of input word $w$; Levenshtein Distance function $score(a, b)$}\n\t\\Output{~Word Graph $G(V, E)$ where $V$ is the nodes and $E$ is the edges in the graph $G$}\n\t\\BlankLine\n\t$V \\leftarrow \\phi$ \\\\\n\t\\For{$i=1...|W|$}\n\t{\n\t\t\\If{$F_{i}\\geq 5$}\n\t\t{\n\t\t\t\\If{$L_{i} \\geq 6$}\n\t\t\t{\n\t\t\t\t\\If{$C_{i} \\geq 4$}\n\t\t\t\t{\n\t\t\t\t\t\\If{$V = \\phi$}\n\t\t\t\t\t{\n\t\t\t\t\t\t$V_{1} \\leftarrow W_{i}$\n\t\t\t\t\t}\n\t\t\t\t\t\\Else\n\t\t\t\t\t{\n\t\t\t\t\t\t\\For{$j=1...|V|$}\n\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\\If{$T(W_{i}) = T(V_{j})$}\n\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\\If{$score(W_{i}, V_{j}) \\ge 85$}\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t$k \\leftarrow |V| + 1$\\\\\n\t\t\t\t\t\t\t\t\t$V_{k} \\leftarrow W_{i}$\\\\\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t$E_{j} \\leftarrow E_{j} + V_{k}$\\\\\n\t\t\t\t\t\t\t\t\t$E_{k} \\leftarrow E_{k} + V_{j}$\\\\\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\t}\n\\end{algorithm}\n\n\\begin{algorithm}[!ht]\n\t\n\t\\caption{\\textsc{Training: Correct Spelling Identification} algorithm}\n\t\\label{alg:train}\n\t\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\\SetKwInOut{Intermediate}{Intermediate}\n\t\\Input{~Graph $G(V,E)$; Frequency count $F$ containing number of times a word has occurred in the corpus}\n\t\\Intermediate{~$C$ is the maximal cluster obtained in a particular step; $p$ is the parent element for a particular cluster; $a$ is the anchor elements for a particular cluster}\n\t\\Output{~Parent $P$ containing list of parent member for the clusters that will be obtained; Anchor $A$ containing list of anchor members for the clusters that will be obtained}\n\t\\BlankLine\n\t$G' \\leftarrow G$\\\\\n\t$A \\leftarrow \\phi$\\\\\n\t$P \\leftarrow \\phi$\\\\\n\t\n\t\\While{$G' \\ne \\phi$}\n\t{\n\t\t$C(V^{c}, E^{c}) = MaximalClique(G')$\\\\\n\t\t$j = argmax(\\{F_{i} | i \\in \\{1...|V^{c}|\\} \\})$\\\\\n\t\t$p \\leftarrow V^{c}_{j}$\\\\\n\t\t$p_{freq} = F_{j}$\\\\\n\t\t$a \\leftarrow \\phi$\\\\\n\t\t\\For{$k \\in \\{1...|V^{c}|\\}\\}$}\n\t\t{\n\t\t\t\\If{$F_{k} > \\frac{p_{freq}}{4}$}\n\t\t\t{\n\t\t\t\t$a \\leftarrow a + V^{c}_{k}$\\\\\n\t\t\t}\n\t\t$G' \\leftarrow G' - C$\\\\\n\t\t$P \\leftarrow P + p$\\\\\n\t\t$A \\leftarrow A + a$\\\\\n\t\t}\n\t}\n\t\n\\end{algorithm}\n\n\\begin{algorithm}[!ht]\n\t\n\t\\caption{\\textsc{Testing: Auto-Correction duing Inference} algorithm}\n\t\\label{alg:test}\n\t\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\t\\Input{~$w_{T}$ is the test word that needs to be corrected; Parent $P$ containing list of parent member for the clusters obtained during training; Anchor $A$ containing list of anchor members for the clusters obtained during training; $score(a, b)$ is the Levenshtein distance between words $a$ and $c$; Function $T(w)$ gives the first three letters of input word $w$}\n\t\\Output{~$w^{*}$ is the corrected word for $w_{T}$}\n\t\\BlankLine\n\t$w^{*} = w_{T}$\\\\\n\t$score^{*} = 0$\\\\\n\t\\For{$i \\in \\{1...|P|\\}$}\n\t{\n\t\t\\If{$T(w_{T}) = T(P_{i})$}\n\t\t{\n\t\t\t\\For{$j \\in {1...|A_{i}|}$}\n\t\t\t{\n\t\t\t\t$s = score(w_{T}, A_{i,j})$\\\\\n\t\t\t\t\\If{$s \\ge 85$}\n\t\t\t\t{\t\n\t\t\t\t\t\\If{$s \\ge score^{*}$}\n\t\t\t\t\t{\n\t\t\t\t\t\t$s = score^{*}$\\\\\n\t\t\t\t\t\t$w^{*} = P_{i}$ \\\\\n\t\t\t\t\t}\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\t}\n\t\n\\end{algorithm}\n\n\n\\section{Experimental Setup}\n\\label{sec:exp-setup}\nAll transformer models except for Canine are finetuned using Adam optimizer \\cite{adam} with learning rate of $1e$-$5$. Canine models were finetuned using AdamW optimizer \\cite{adamw} using $2e$-$5$ learning rate. All the models were trained for $3$ epochs with binary-cross entropy as the loss function. Threshold of $0.5$ was used to get the output class from output probabilites. Huggingface library \\cite{huggingface} was used for training all the transformer models.\n\n\\section{Result}\n\\label{sec:result}\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Comparison of Models on Moj Dataset}\n\t\\label{tab:model-result}\n\t\n\t\\bgroup\n\t\\def\\arraystretch{1.2\n\t\\begin{tabular}{ccc}\n\t\t\\hline\n\t\t\\textbf{Model} & \\textbf{Val. F1} & \\textbf{Test F1}\\\\\n\t\t\\hline\n\t\tMURIL & \\textbf{0.8675} & - \\\\\n\t\tIndic-BERT & 0.8475 & - \\\\\n\t\tXLM-R & 0.8623 & - \\\\\n\t\tXLM-T & 0.8633 & - \\\\\n\t\tMultilingual DistilBERT & 0.8553 & - \\\\\n\t\tmBERT & 0.8551 & - \\\\\n\t\tCanine-C & 0.8646 & - \\\\\n\t\tCanine-S & 0.8622 & - \\\\\n\t\tLogistic Regression + NB + TF-IDF & 0.8559  & - \\\\\n\t\tRandom Forest (meta-data) & 0.7109 & - \\\\\n\t\tEnsemble & - & \\textbf{0.8933}\\\\ \n\t\t\n\t\t\\hline\n\t\\end{tabular}\n\t\\egroup\n\\end{table}\n\nWe create a stratified validation set from $10\\%$ of the training set. Table \\ref{tab:model-result} shows the comparison of all the models discussed in Sec. \\ref{ssec:model}. Logistic Regression + Naive Bayed on TF-IDF provides a good baseline even though it is a very simple model. MURIL scores the highest F1 score on the validation set. XLM-T performed slightly better than XLM-R as XLM-T is trained on multilingual Twitter comment which is closer to the Moj Dataset than the Wikipedia dataset on which XLM-R was trained on. The meta-data contains signals essential for this task as the Random Forest obtains an F1 score of $0.71$ using meta-data alone.  Our final model is an ensemble of all the models from all the $3$ epochs in Sec. \\ref{ssec:model}. A logistic regression model was trained on the validation set to combine the predictions from all models for ensembling.\n\n\\section{Ablation Analysis}\n\\label{sec:ablation}\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Effect of Data Cleaning, Native Transliteration, and Spell correction using Logistic Regression}\n\t\\label{tab:ablation}\n\t\n\t\\bgroup\n\t\\def\\arraystretch{1.2\n\t\\begin{tabular}{cc}\n\t\t\\hline\n\t\t\\textbf{Model} & \\textbf{Val. F1}\\\\\n\t\t\\hline\n\t\tRaw Dataset & 0.8493 \\\\\n\t\tCleaned Dataset & 0.8508 \\\\\n\t\tCleaned + Native Transliteration Dataset & 0.8547 \\\\\n\t\tCleaned + Native Transliteration + Spell-Corrected Dataset & \\textbf{0.8567} \\\\\n\t\tCleaned + English Transliteration Dataset & 0.8496 \\\\ \n\t\t\\hline\n\t\\end{tabular}\n\t\\egroup\n\\end{table}\n\n\\subsection{Data Cleaning}\nLogistic regression was used to measure the effect of our pre-processing steps as its simple to train, test and iterate. Table. \\ref{tab:ablation} measures the validation F1 score on raw data and the various pre-processing steps. It shows that using the data cleaning as specified in Sec. \\ref{ssec:data-cleaning} improves the F1 score. \n\n\\subsection{Transliteration}\n\\cite{how-multilingual-is-mbert} shows that mBERT is unable to perform good on transliterated dataset as it was not trained on such data. Since only MURIL was originally trained on transliterated data, other transformer models would face domain mismatch when finetuned on Moj Dataset. To tackle transliteration and mixture of scriptures in same sentences as discussed on Sec. \\ref{sec:problem-statement}, we experimented with transliterating the comments before feeding it to our classifiers. Table. \\ref{tab:ablation} shows that transliterating all the comments to the native scripture shows the largest improvement among all the steps. We also experimented with transliterating everything to English so that there is knowledge sharing between languages that have common words but have different scriptures. This would increase the number of training data for lower resource language. However, this led to a decrease in F1 score. Transliterating eveything to native might be working better than transliterating to English because there could be same sounding words that could mean different things in different language, e.g, \\verb|kutta| means \"dog\" in Hindi whereas it means \"new\" in Telugu. Transliterating to English also leads to loss of syllables which are not present in English language but are present in the native language.\n\n\\subsection{Spell Correction}\nSpelling correction allows the number of training samples per concept to increase as different variants of words are mapped to the same underlying concept, e.g, \\verb|bakwas| and \\verb|bakvas| where \\verb|bakwas| means \"useless\" in Hindi. Increased number of training samples per concept allows the model to learn better representation for that particular concept. Table. \\ref{tab:ablation} indicates that our clustering approach in Sec. \\ref{ssec:spell-correct} boosts the F1 score.\n\n\\section{Future Work}\n\\label{sec:future-work}\nTable. \\ref{tab:label-dist} shows that there is class imbalance for Malayalam language. Dravidian-CodeMix-FIRE \\cite{dravidiancodemix2021-overview} can be used to augment Moj Dataset to overcome this. Adversarial training \\cite{miyato2016adversarial}, \\cite{pk-nutcracker}, \\cite{pk-paw} can be used to increase robustness of the system. Emojis could be replaced with words instead of completely removing them as done in Sec. \\ref{ssec:data-cleaning}. Incorporating homophones in the graph construction in Sec. \\ref{ssec:spell-correct} might bring more gains. \n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, we propose an approach to identify abusive comments from a multilingual dataset. We leverage pre-trained multilingual transformer models along with classical machine learning models for abusive identification. We identify the inherent challenges associated with this task such as code-mixing, transliteration, mixing of scriptures, and misspellings. Our approach attempts to tackle all of these challenges to build a robust system. We propose a spell correction algorithm using graph clustering that  does not need manual creation of a dictionary for each language. It can also correct unseen incorrect words during inference if the correct word was observed in training. Results shows that our approach is well-suited for non-English social media text analysis.\n\n\\bibliographystyle{.\/IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nSuppose we have $n$ keys, $n$ access probabilities for the keys, and $n+1$\naccess probabilities for the gaps between the keys.\nThe problem to construct a binary search tree for these $n$ keys\nthat minimizes\nthe expected access time is known as the\n\\emph{optimal binary search tree problem}.\nKnuth presented in \\cite{knu71} a well-known dynamic programming algorithm\nthat solves this problem in $O(n^2)$ time.\n\nApart from the original problem,\nthe construction of optimal binary search trees whose heights are restricted \nhas been considered in the literature.\nBy the height restriction the maximum number of comparisons during a search can\nbe bounded. Thus, an optimal height restricted binary search tree\nperforms well in both the worst and the average case.\nItai \\cite{itai76} and Wessner \\cite{wess76} independently discovered \nconstruction algorithms for height restricted binary search trees.\nTheir algorithms have running time $O(L n^2)$, where $L$ is the maximal permitted\nheight.\n\nLet $h_{\\min}(n) = \\lceil \\log_2 (n+1)\\rceil$\nbe the \\emph{minimal height of a binary search tree for $n$ keys}.\nIn this paper, we show that for any fixed $\\Delta$ an optimal\nbinary search tree with height $h \\leq h_{\\min}(n) + \\Delta$\ncan be constructed in time $O(n^2)$.\nThis improves the results from Itai and Wessner \\cite{itai76,wess76}.\nBecause $L \\geq \\lceil \\log_2 (n+1) \\rceil$, the algorithms of Itai and Wessner\nhave running time $O(n^2 \\log n)$ if we use them to construct optimal search\ntrees with height $h \\leq h_{\\min}(n) + \\Delta$.\n\nGagie \\cite{gag03, gag05} presents a $O(n)$ time algorithm\nfor the restructuring of optimal binary search trees.\nHis algorithm restructures an existing optimal binary search\nin such a way that the resulting tree\nhas nearly optimal height and cost.\nIn contrast to Gagie's algorithm our algorithm always selects\nthe best binary search tree from the set of all trees with restricted height.\n\nOther interesting facts about optimal binary search trees can be found in the\narticle of Nagaraj \\cite{nag97}. This article gives a\ncomprehensive survey about optimal binary search trees.\n\nAll algorithms for the construction of optimal binary search trees, whether height\nrestricted or not, are based on dynamic programming.\nThey all use step by step construction of larger trees from smaller subtrees.\nInstead of step by step construction from smaller subtrees we use a\ndecision model where the keys are placed by a sequential decision process in such a way\ninto the tree, that the costs become optimal. This approach is adopted from\nthe construction algorithm for optimal B-trees \\cite{bec94}.\n\nThe rest of the paper is structured in the following way: in\nSection 2 a formal description of the problem is given.\nIn Section 3 we present our approach:\nthe decision model is explained and\nthe attached dynamic program is formulated.\nSection 4 states the solution algorithm\nand gives the complexity results.\nSection 5 summarizes the results.\n\n\\section{The Problem}\n\nNow we give the problem formulation.\nWe have $n$ keys $k_1 < k_2 \\ldots < k_n$\nand $2n+1$ probabilities\n$\\alpha_0 , \\beta_1 , \\alpha_1 , \\beta_2 , \\ldots , \\beta_n , \\alpha_n $.\n\n$\\beta_i$ are the \\emph{key weights} and $\\alpha_j$ are the\n\\emph{gap weights}.\n$\\beta_i$ is the probability that key $k_i$ is requested, and\n$\\alpha_j$ is the\nprobability, that a search is made for a key $d$ with\n$k_j < d < k_{j+1}$.\nWe assume that we have artificial keys $k_0 = - \\infty$ and $k_{n+1} = \\infty$.\n\nLet $b_i$ be the level resp.\\ the depth of the $i$-th internal node where key\n$k_i$ is stored, and\nlet $a_j$ be the level of the external node for the gap between $k_j$ and $k_{j+1}$.\nThe root is on level $0$. \nFor a binary search tree $T$ we define the\n\\emph{weighted path length} $\\textnormal{wpl}(T)$ by\n\\[\n\\textnormal{wpl}(T) := \\sum_{i=1}^n \\beta_i (b_i+1) + \\sum_{j=0}^n \\alpha_j a_j\n\\]\nThe weighted path length is the expected number\nof node visits resp.\\ comparisons in a search.\n\nThe height $h(T)$ of a tree $T$ is defined as the level of the deepest external node.\nThe \\emph{minimal height} $h_{\\min}(n)$ of a binary search tree for $n$ keys\nis then given by\n\\[\n   h_{\\min}(n) = \\lceil \\log_2(n+1) \\rceil\n\\]\n\nWe want to construct search trees whose heights are nearly minimal.\nLet $\\Delta \\geq 0$ be some fixed value.\nThe problem is to find a binary search tree $T$\nthat minimizes the weighted path length\n$\\textnormal{wpl}(T)$ subject to the constraint\n$h(T) \\leq h_{\\min}(n) + \\Delta$. \nSuch a tree is denoted as an\n\\emph{optimal binary search tree with near minimal height}.\n\n\\section{Dynamic Programming Model}\n\nWe model the process of constructing an optimal binary search tree\nwith near minimal height\nas a decision problem with $n$ stages. For every key $k_i$\nwe have to decide, on which level this key should be placed.\nWhether placing on some level is feasible,\ndepends on the former decisions for the keys $k_1$ to $k_{i-1}$,\nwhich define a certain state in the decision process.\nThen placing the key $k_i$ on any level results in an increasing\nweighted path length and a new state. The amount of increasing\nas well as the new state depend on our decision.\n\nUsing this approach, the\noptimal tree is the result of a sequence of optimal\ndecisions starting in a unique initial state. This leads to\na dynamic program $DP$ of the form $DP = (S_\\nu ,A_\\nu ,D_\\nu ,\nT_\\nu , c_\\nu , C_{n+1})$, where $n$ is the number of the\nstages of $DP$, $S_\\nu$ is the \\emph{state set} of stage\n$\\nu , 1 \\leq \\nu \\leq n+1$, and $A_\\nu$ is the \\emph{decision set}\nof stage $\\nu , 1 \\leq \\nu \\leq n$. The sets\n$D_\\nu \\subseteq S_\\nu \\times A_\\nu$\ndefine the \\emph{feasible decisions} for the states of stage $\\nu$.\nIt holds: $(s,a) \\in D_\\nu$, if and only if $a$\nis feasible in state $s$ on stage $\\nu$.\nThe set $D_\\nu (s) := \\{ a \\in A_\\nu | (s,a) \\in D_\\nu \\}$\ncontains all feasible decisions for state $s$ on stage $\\nu$.\n$T_\\nu : D_\\nu \\rightarrow S_{\\nu +1}$ is\nthe \\emph{transition function}.\nMaking decision $a$ in state $s$ at stage $\\nu$ results in state\n$T_\\nu(s,a)$ at stage $\\nu + 1$.\n$c_\\nu : D_\\nu \\rightarrow \\mbox{I}\\!\\mbox{R}$ is the \\emph{cost function} of\nstage $\\nu$. $c_\\nu(s,a)$ gives the costs that arise if we\ndecide to make decision $a$ in state $s$ on stage $\\nu$.\n$C_{n+1} : S_{n+1} \\rightarrow \\mbox{I}\\!\\mbox{R}$ is the\n\\emph{terminal cost} function.\n$C_{n+1}(s)$ gives the costs that arise if our final state\nis $s$.\n\n\\begin{figure}[htb]\n\\postscript{states.eps}{0.8}\n\\caption{Tree states in the construction process}\n\\label{figstates}\n\\end{figure}\n\nNow we have to define the components of the dynamic program\nin such a way that the decision process models the\nconstruction of a binary search tree with restricted height.\nFirst we give the definition of the states.\nFor motivation take a look at Figure~\\ref{figstates}.\nSuppose we have $h_{\\max} := h_{\\min}(n) + \\Delta = 3$, \nthat means we can place the keys on levels from $0$ to $2$.\n\nFor a correct placing\nof a key in the partial tree only the\nrightmost path fragments from the actual root to the node that contains the largest key\nis relevant. Due\nto this fact we can represent a state $s \\in S_\\nu$\nby a binary vector with $h_{\\max}$ components.\nWe number the vector components from $0$ to $h_{\\max} -1$.\nVector component $s_i$ is related to level $i$.\n\\[\n    s = \\svec{s_0}{\\vdots}{s_{h_{\\max}-1}} \\textnormal{ with } s_i \\in \\{0,1\\}.\n\\]\nEach vector component $s_i$ determines, whether the level $i$ \nin the rightmost path is occupied.\nMore formally, vector component $s_i$ is $1$ if and only if the largest\nkey on level $i$ is greater than any key on the levels from $0$ to $i-1$.\nFor instance the state $s$ resulting from tree (a)\nin Figure~\\ref{figstates} is represented\nby\n\\[\n   s = \\svec{1}{1}{0}\n\\]\nand the state $s'$ resulting from tree (c) by \n\\[\n   s' = \\svec{1}{0}{1}\n\\]\nObserve, that different trees may have the same associated states.\nFor instance the trees (a) and (b) of Figure \\ref{figstates} are both\nrepresented by the same state.\n\nThe set $S_\\nu$ is defined to be the set of all vectors\nthat are possible after the assignment of $\\nu - 1$ keys.\nThe initial state set $S_1$ consists of a single state:\n\\[\n   S_1 := \\left\\{ \\svec{0}{\\vdots}{0} \\right\\}\n\\]\n\nA decision is characterized by the level on which a key is placed.\nSo we define $A = A_\\nu = \\{0 ,\\ldots, h_{\\max}-1 \\}$.\nMaking decision $a$ means that the corresponding key is placed on level $a$.\nFor instance, the tree (a) in Figure~\\ref{figstates} is constructed by the\ndecision sequence $DS = (1,0,2,1)$.\n\n\\begin{figure}[hbt]\n\\postscript{infeasible.eps}{0.8}\n\\caption{Feasible and infeasible decision}\n\\label{infeasible}\n\\end{figure}\n\nLet $s = (s_0 ,\\ldots, s_{h_{\\max}-1})$ be a state.\nA feasible decision $a$ for state $s$\nhas to fulfill the following conditions:\n\\begin{itemize}\n\\item[(i)]\n   We can place keys only on unoccupied levels:\n   \\[\n       s_a = 0\n   \\]\n\\item[(ii)]\n   If a key is placed above some path fragment, this path fragment\n   has to be the deepest path\n   fragment and the key has to be placed directly above this path fragment:\n   \\[\n      \\not\\exists i,j : a < i < j \\textnormal{ and } s_i = 0 \\textnormal{ and } s_j = 1\n   \\]\n\\end{itemize}\nCondition (i) is obvious. Figure~\\ref{infeasible} demonstrates condition (ii).\nThe next key $k_4$ has to be placed on level $2$, because $k_3$ becomes the left son\nof $k_4$. If we place $k_4$ on level $1$, the left son would not be on the next\ndeeper level.\n\nSo we can define\n\\[\n   D_\\nu := \\{ (s,a) | s \\in S_\\nu, a\\;\\textnormal{fulfills (i) and (ii)} \\}\n\\]\nObserve that the feasible decisions of a state $s$ are independent\nof the stage $\\nu$. So we define\n\\[\n    D(s) := \\{ a \\in A | a\\;\\mbox{fulfills (i) to (ii)} \\}\n\\]\nas the \\emph{set of feasible decisions for state $s$}.\nFor every binary search tree (with near minimal height) there exists a unique\nfeasible decision sequence that constructs the tree. As an\nexample see the decision sequence to construct tree (a) of Figure~1 (see\nabove). Using this definition each feasible decision sequence\nleads to trees that are valid binary search trees with the exception of\nthe rightmost path.\nTrees with invalid rightmost path on stage $n+1$\nare filtered by the terminal cost function $C_{n+1}$ (see below).\n\n\\begin{figure}[htb]\n\\postscript{transition.eps}{0.8}\n\\caption{Example for a transition}\n\\label{transition}\n\\end{figure}\n\nMaking a decision $a$ has two effects. First, the\nlevel $a$ of the rightmost path becomes occupied and\nsecond, the levels from $a+1$ to $h_{\\max}-1$ become unoccupied.\nSo the definition for the transition function is:\n\\[\n   T(s,a) := T_\\nu(s,a) = \\left(\\begin{array}{c} \n      s_0 \\\\ \n      \\vdots \\\\\n      s_{a-1} \\\\\n      1 \\\\\n      0 \\\\\n      \\vdots\\\\\n      0 \\end{array}\\right)\n\\]\nFigure~\\ref{transition} shows an example for a single transition. \nThe following state and decision sequence shows the transitions from the\ninitial state to the right tree of Figure~\\ref{transition}.\n\\[\n\\svec{0}{0}{0} \\stackrel{a=1}{\\longrightarrow}\n\\svec{0}{1}{0} \\stackrel{a=0}{\\longrightarrow}\n\\svec{1}{0}{0} \\stackrel{a=2}{\\longrightarrow}\n\\svec{1}{0}{1} \\stackrel{a=1}{\\longrightarrow}\n\\svec{1}{1}{0}\n\\]\n\nIf we have a state $s \\in S_\\nu$, we can deduce from $s$ the preceding decision, \ni.e.\\ the decision on stage $\\nu -1$ that induced $s$.\nTake a look at the transition function $T(s,a)$:\nthe largest $i$ with $s_i = 1$ defines this preceding decision.\n\\[\n   \\textnormal{precdec}(s) := \\left\\{\n   \\begin{array}{ll}\n   0 & \\textnormal{ if } s_0 = \\cdots = s_{h_{\\max}-1} = 0 \\\\\n   \\max \\{ 0\\leq i \\leq h_{\\max}-1 | s_i = 1 \\} & \\textnormal{ otherwise}\n   \\end{array}\n   \\right.\n\\]\n\nOur cost function $c_\\nu(s,a)$\nhas to consider two aspects: the level of key $k_\\nu$ and\nthe level of the gap $(k_{\\nu-1},k_\\nu)$. The first is simple: the\nlevel of key $k_\\nu$ is determined by the decision $a$.\nWith the following Lemma, we are able the determine the level of the gap $(k_{\\nu-1},k_\\nu)$.\n\\begin{lemma}\n\\label{gaplevel}\nLet $\\textnormal{klevel}(k_\\nu)$ denote the level of key $k_\\nu$ and\nlet $\\textnormal{glevel}(k_{\\nu-1},k_\\nu)$\ndenote the level of the gap $(k_{\\nu-1},k_\\nu)$. Then we have\n\\[\n   \\textnormal{glevel}(k_{\\nu-1},k_\\nu)= 1 + \\max\\{\\textnormal{klevel}(k_{\\nu-1}),\n                                                   \\textnormal{klevel}(k_\\nu)\\}\n\\]\n\\end{lemma}\n\n\\begin{proof}\nAdjacent keys cannot be on the same level. So we have either\n$\\textnormal{klevel}(k_{\\nu-1}) < \\textnormal{klevel}(k_\\nu)$\nor\n$\\textnormal{klevel}(k_{\\nu-1}) > \\textnormal{klevel}(k_\\nu)$.\n\nIn the case of $\\textnormal{klevel}(k_{\\nu-1}) < \\textnormal{klevel}(k_\\nu)$,\nthe key $k_\\nu$ is in the right subtree of key $k_{\\nu-1}$ and the\ngap $(k_{\\nu-1},k_\\nu)$ is the left son of the node that contains $k_\\nu$.\nIn the other case\nthe key $k_{\\nu-1}$ is in the left subtree of key $k_\\nu$ and the\ngap $(k_{\\nu-1},k_\\nu)$ is the right son of the node that contains $k_{\\nu-1}$.\nIn both cases the equation of Lemma~\\ref{gaplevel} is valid.\n\\end{proof}\n\nThe cost functions $c_\\nu(s,a)$ are defined by:\n\\[\n   c_\\nu(s,a) := \n                (1+\\max\\{\\textnormal{precdec}(s),a\\})\\cdot\\alpha_{\\nu -1} + \n                (a+1)\\cdot\\beta_\\nu\n\\]\nThis definition utilizes Lemma~\\ref{gaplevel}: $\\textnormal{klevel}(k_{\\nu-1})$\nis equivalent to $\\textnormal{precdec}(s)$ and $\\textnormal{klevel}(k_\\nu)$ to\nthe decision $a$.\n\nThe terminal costs $C_{n+1}$ model whether our final\nstate fulfills the tree conditions.\nIn particular, we have to check whether the right most path contains\nunoccupied levels above occupied levels.\nFor instance, tree (c) of Figure~\\ref{figstates} is not a valid search tree\nbecause level~$1$ is not occupied but level~$2$ is.\nWe have:\n\\[\n   C_{n+1}(s) = \\left\\{\n   \\begin{array}{ll}\n      (1 + \\textnormal{precdec}(s))\\cdot\\alpha_n & \\textnormal{ if } \n         s_0 = 1 \\textnormal{ and } \n\t \\not\\exists i < j: s_i = 0 \\wedge s_j = 1 \\\\\n      \\infty & \\textnormal{ otherwise}\n   \\end{array}\n   \\right.\n\\]\nTo check whether there exists an unoccupied level we use an adaption of\ncondition (ii) of the feasible decision set $D(s)$. If the root level\nis occupied and there exists no unoccupied level\nabove an occupied level\nthe terminal costs consist of the access probability $\\alpha_n$\nof the last gap  multiplied\nby the level of key $k_n$ plus $1$.\n\nNow the definition of the dynamic program $DP$ is\ncomplete. Using this definition the optimization problem is\n\\[\nF :=\n\\sum_{\\nu = 1}^n c_\\nu (s_\\nu , a_\\nu ) + C_{n+1} (s_{n+1})\n\\rightarrow \\min\n\\]\nsubject to:\n\\[\n\\begin{array}{ll}\n&   s_1 = (0 \\cdots 0) \\\\\n&   a_\\nu \\in D (s_\\nu ), 1 \\leq \\nu \\leq n \\\\\n&   s_{\\nu + 1} = T(s_\\nu , a_\\nu ), 1 \\leq \\nu \\leq n\n\\end{array}\n\\]\nThe value $F$ of the objective function yields the\nminimum weighted path length and the tree is given by\nthe optimal sequence $(a_1,\\ldots,a_n)$ of feasible decisions.\n\n\\section{Algorithm and Complexity}\n\nFor the solution of this optimization problem we use a common\ndynamic programming algorithm, cf.\\ \\cite{nm02}.\n\n\\begin{algorithm}\n\\label{dpalgo}\\mbox{}\\\\\n\\begin{tabular}{rl}\n(0) & \/* Initialization *\/ \\\\\n(1) & {\\bf forall} $s \\in S_{n+1}$ \\\\\n(2) & \\hspace*{0.5cm} $V_\\nu(s) \\leftarrow C_{n+1}(s)$ \\\\\n(3) & \/* Backward Computation *\/ \\\\\n(4) & {\\bf for} $\\nu \\leftarrow n$ {\\bf downto} $1$ {\\bf do} \\\\\n(5) & \\hspace*{0.5cm} {\\bf forall} $s \\in S_\\nu$ {\\bf do} \\\\\n(6) & \\hspace*{1.0cm} $V_\\nu(s) \\leftarrow \\infty$ \\\\\n(7) & \\hspace*{1.0cm} $\\pi_\\nu(s) \\leftarrow \\textnormal{undefined}$ \\\\\n(8) & \\hspace*{1.0cm} {\\bf forall} $a \\in D(s)$ {\\bf do} \\\\\n(9) & \\hspace*{1.5cm} {\\bf if} $c_\\nu(s,a) + V_{\\nu+1}(T(s,a)) < V_\\nu(s)$ {\\bf then} \\\\\n(10) & \\hspace*{2.0cm} $V_\\nu(s) \\leftarrow c_\\nu(s,a) + V_{\\nu+1}(T(s,a))$ \\\\\n(11) & \\hspace*{2.0cm} $\\pi_\\nu(s) \\leftarrow a$ \\\\\n(12) & \/* Forward Computation *\/ \\\\\n(13) & $s \\leftarrow (0 \\cdots 0) $ \\\\\n(14) & $F \\leftarrow  V_1(s)$ \\\\\n(15) & {\\bf for} $\\nu \\leftarrow 1$ {\\bf to} $n$ {\\bf do} \\\\\n(16) & \\hspace*{0.5cm} $a_\\nu \\leftarrow \\pi_\\nu(s)$ \\\\\n(17) & \\hspace*{0.5cm} $s \\leftarrow T(s, a_\\nu)$ \\\\\n\\end{tabular}\n\\end{algorithm}\n\n$V_\\nu(s)$ is the \\emph{value function} which represents the minimal costs to reach a terminal state\nfrom state $s$ on stage $\\nu$. In line (1) and (2) we initialize the value function with\nthe terminal costs.\n$\\pi_\\nu(s)$ represents the optimal decision for state $s$ on stage $\\nu$. \nThe value function $V_\\nu(s)$ and the optimal decision\n$\\pi_\\nu(s)$ is\ndetermined by the \\emph{Bellman equation}\n\\[\n    V_\\nu(s) = \\min_{a \\in D(s)} \\{c_\\nu(s,a) + V_{\\nu+1}(T(s,a)) \\}\n\\]\nwhich is solved for all states on\nall stages in lines (4) to (11).\n\nAfter the backward computation terminates, the $\\pi_\\nu$ define an\n\\emph{optimal policy}. To get the\noptimal decision sequence\nwe apply the $\\pi_\\nu$ in a forward computation (line (13) to (17))\nbeginning with our initial state. As a result\nthe $a_\\nu$ represent the decision sequence to build an optimal tree and the value of $F$ is the weighted\npath length of the optimal tree.\n\nWith the decision sequence\n$DS = (a_1 ,\\ldots, a_n)$ that defines the optimal binary search tree\nwe are able to build the corresponding tree in linear time,\nas for each key $k_\\nu$ the level where $k_\\nu$ has to be placed\nis given by the decision $a_\\nu$.\n\n\\begin{example}\nSuppose we have keys $k_1,\\ldots,k_4$ with access probabilities $\\beta_1 = \\frac{3}{16},\n\\beta_2 = \\frac{1}{16}, \\beta_3 = \\frac{1}{2}, \\beta_4 = \\frac{1}{4}$ and\n$\\alpha_0 = \\cdots = \\alpha_4 = 0$. Let $\\Delta=0$, that means we have\nto construct a tree of height $\\lceil \\log_2(5)\\rceil = 3$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\include{exproblem}\n\\end{center}\n\\caption{State space for the example problem}\n\\label{ssexample}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\postscript{opttree.eps}{0.3}\n\\caption{Optimal binary search tree for the example problem}\n\\label{ssexampletree}\n\\end{figure}\n\n\nFigure~\\ref{ssexample} shows the search graph for this problem.\nThe number adjacent to an arc represents the cost $c_\\nu(s,a)$ of the corresponding transition.\nThe terminal costs $C_5(s)$ are shown below the states of state set $S_5$ and the value function $V_\\nu(s)$\nis shown\nright beside the states for the state sets $S_1$ to $S_4$. Observe, that the the value function\nof state $(1,1 1) \\in S_4$ yields $\\infty$ because of an empty decision set.\n\nThe best decision sequence $DS=(1,2,0,1)$ is given by the bold arcs. Its overall cost is\n$\\frac{25}{16}$, that means the corresponding\noptimal binary search tree has a weighted path\nlength of $\\frac{25}{16}$. Figure~\\ref{ssexampletree}\nshows the corresponding tree.\n\\end{example}\n\n\n\nOur complexity results are based on bounds for the cardinality of the state sets $S_\\nu$ and the\ndecision sets $D_\\nu$.\n\n\\begin{theorem}\nFor all state sets $S_\\nu\\,(\\nu=1,\\ldots,n+1)$ we have:\n\\[\n|S_\\nu | \\leq 2^{\\Delta+1} (n+1)\n\\]\n\\end{theorem}\n\n\\begin{proof}\nLet $h_{\\max}(n) := h_{\\min}(n)+\\Delta$ and\n$S:=\\{0,1\\}^{h_{\\max}(n)}$. With these definitions we get\n\\[\n   |S_\\nu | \\leq |S| = 2^{h_{\\max}(n)} = 2^{h_{\\min}(n)+\\Delta}\n\\]\nUsing $h_{\\min}(n) = \\lceil \\log_2(n+1) \\rceil$ we get\n\\begin{eqnarray*}\n   |S| & \\leq & 2^{\\lceil \\log_2(n+1) \\rceil + \\Delta} \\\\\n       & \\leq    & 2^{\\Delta+1}\\cdot 2^{\\log_2(n+1)} \\\\\n       & =       & 2^{\\Delta+1}\\cdot(n+1)\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{corollary}\nFor any fixed $\\Delta$ the cardinality of the state sets $S_\\nu$\nis bounded by $O(n)$.\n\\end{corollary}\n\n\\begin{theorem}\nFor all feasible decision sets $D_\\nu\\,(\\nu=1,\\ldots,n+1)$ we have:\n\\[\n   |D_\\nu| \\leq 2^{\\Delta+2} (n+1) \n\\]\n\\end{theorem}\n\n\\begin{proof}\nLet $h_{\\max}(n) := h_{\\min}(n)+\\Delta$,\n$S:=\\{0,1\\}^{h_{\\max}(n)}$ and $D := \\{(s,a)| s \\in S, a \\textnormal{ is feasible for } s\\}$.\nWith these definitions we get $|D_{\\nu}| \\leq |D|$ for all $\\nu=1,\\ldots,n$.\n\nHow many feasible decisions exists for a state $s \\in S$? Take a look at condition (ii) in the\ndefinition of $D(s)$ (see Section~3).\nIf $s_{h_{\\max}-1} = 1$ there is at most one feasible decision $a$, which is\ndetermined by the highest index $a$ with $s_a = 0$. That means, that half of all the states in $S$ have only\none feasible decision. States with $s_{h_{\\max}-1} = 0$ and $s_{h_{\\max}-2} = 1$, which comprise a quarter\nof all states in $S$, have at most two decisions. Generalized, $\\frac{1}{2^k}|S|$ states \nof all the states in $S$ have $k$ feasible\ndecisions. We get:\n\\begin{eqnarray*}\n   |D_{\\nu}| & \\leq & |D| \\\\\n             & \\leq & 1\\cdot\\frac{1}{2}|S| + 2\\cdot\\frac{1}{4}|S| +\n\t              3\\cdot\\frac{1}{8}|S| + \\cdots \\\\\n\t     & \\leq    & \\sum_{k=0}^{\\infty} \\frac{k}{2^k} \\cdot |S| \\\\\n             & =       & \\left(\\sum_{k=0}^{\\infty} \\frac{k+1}{2^k} -\n\t                 \\sum_{k=0}^{\\infty} \\frac{1}{2^k}\\right) \\cdot |S| \\\\\n\t     & =       & \\left(\\frac{1}{(1-\\frac{1}{2})^2} - \n\t                       \\frac{1}{1-\\frac{1}{2}}\\right) \\cdot |S| \\\\\n\t     & =       & 2\\cdot |S| \\\\\n\t     & \\leq    & 2^{\\Delta+2} (n+1)\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{corollary}\nFor any fixed $\\Delta$ \nAlgorithm~\\ref{dpalgo} constructs an optimal binary search with\nheight $h \\leq h_{\\min}(n)+\\Delta$ in time $O(n^2)$.\n\\end{corollary}\n\n\\begin{proof}\nWe have to iterate over the $n$ stages from $n$ down to $1$.\nIn doing so, the cardinality of \neach state set $S_\\nu$ and each feasible decision set $D_\\nu$\nis bounded by $O(n)$ for fixed $\\Delta$. All operations can be executed in\nconstant time. It follows, that the overall running time is $O(n^2)$.\n\\end{proof}\n\n\\section{Summary}\n\nWe have presented a quadratic time algorithm to compute optimal binary search trees with near\nminimal height, i.e.\\ with height $h \\leq h_{\\min}(n) + \\Delta$ and fixed $\\Delta$.\nThe algorithm was adopted  \nfrom the construction algorithm for optimal B-tress. The construction process was modeled by a\ndecision oriented dynamic program: In the model we have to decide key by key, on which level the\nkey should be placed. The tree conditions are represented by additional constraints and a\nterminal cost function.\n\nIt seems to be easy to apply this approach to other kinds of trees.\nBy applying the construction algorithm of \\cite{bec94},\nit should be possible to construct\noptimal B-trees with near minimal height and fixed order in quadratic time, too.\nThe construction of unrestricted optimal B-trees needs time $O(n^{2+\\frac{\\log 2}{\\log k+1}})$.\nA generalization of the binary tree model to multiway trees of a fixed order\nshould also lead to a quadratic time algorithm in constrast to the cubic time\nalgorithms for the unrestricted case \\cite{got81, vkw80}. This means for both cases, that\noptimal trees with near minimal height can be constructed faster than unrestricted trees. \nIf we consider that optimal trees have typically a low height, the approach of height restriction may lead to\nfast construction algorithms, which generate optimal trees with high probability.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe \\textsc{Fortran} file \\T{ARWV\\_v1.7.f} contains an updated and modified version of the algorithmic regularization code originally written by Seppo Mikkola. It is an open source program intended to integrate a few-body problem ($\\T{N} \\lesssim 500$) with high precision via the use of `chained variables' for the coordinates and velocities of the stellar bodies.  The set of equations of motions is that corresponding to the pair gravitational interaction (see \\cite{RCDbook} for reference). In addition to its high computational accuracy, this program includes a number of interesting features among which Post-Newtonian (PN) terms in the computation of the acceleration (see \\cite{Memmesheimer2004,Mikkola2008}). \n\nThis program has been extended by other people. For example, an inclusion of the treatment of external regular potential and dynamical friction is due to M. Arca-Sedda \\cite{ASCD19}.\n\nLater, P. Chassonnery and R. Capuzzo-Dolcetta, in particular, have added a relativistic recoil velocity to the merger routine, and modified and improved the treatment of external, analytic, potentials and dynamical friction. \n\nThe zipped package containing all the necessary files is downloadable\nhere:\n\\url{https:\/\/sites.google.com\/uniroma1.it\/}\\\\\n\\url{astrogroup\/hpc-html}\n\n\nEveryone is free to use it, upon proper citation of Mikkola's original papers and of this user manual.\n\n\\medskip\n\nThe \\textsc{Fortran} file should be compiled with the command:\n\\begin{verbatim}\n\t> gfortran -o arwv.exe ARWV_v1.7.f\n\\end{verbatim}\n\nTo work, \\T{arwv.exe} needs three files:\n\\begin{itemize}[leftmargin=3em]\n\t\\item \\T{ARCCOM2e2.CH}, which lists the \\textit{common} variables and may not be modified except in a special case, as mentioned below;\n\t\\item \\T{PARAMETERS.TXT}, a set of working parameters;\n\t\\item the data file \\T{INPUT.TXT}, which contains the initial conditions (mass, position and velocity) for the set of \\T{N} bodies.\n\\end{itemize} \n\n\nAt a chosen frequency, the code produces an output file called \\T{100000n.dat}, where $n$ is an integer incrementing from $0$ (thus, you will have \\T{1000000.dat}, \\T{1000001.dat}, etc.). These files are formatted as the original input file, with the mass, coordinates and velocity components of all the stellar bodies belonging to the system. The integration time corresponding to each data snapshot is written at the end of the file.\n\n\n\n\n\n\\begin{table*}[ht]\n\\begin{center}\n\\begin{tabular}{ccp{9cm}}\n\tParameter & Type and range & Meaning \\\\ \\hline\n\t\\T{N}         & $1 < \\T{N} \\leqslant \\T{NMX}$ & Number\n of stellar bodies in the system \\\\\n\t\\T{Nbh}       & $0 \\leqslant \\T{Nbh} \\leqslant \\T{N}$ & Number of black-holes in the system \\\\\n\t\\T{TypeOfUnits} & $0\/1$ & Indicates if the input data are given in \\textit{autonomous} ($0$) or astrophysical units ($1$) \\\\\n\t\\T{TMAX}      & real $> 0.0$ & Time over which the integration is conducted \\\\\n\t\\T{DeltaT}  & real $> 0.0$ & Output interval ($\\T{DeltaT} = 0.2$ recommanded) \\\\\n\t\\T{cmet}      & real(3) $\\geqslant 0.0$ & Indicates the regularization method to use, must be different from $(0,0,0)$ ($\\T{cmet} = (1,0,0)$ or $(1,10^{-16},0)$ recommanded). \\\\\n\t\\T{Clight}    & real $\\geqslant 0.0$ & Value of the speed of light in the autonomous units (relevant only if \\T{TypeOfUnits} is set to $0$) \\\\\n\t\\T{Ixc}       & $0\/1\/2$ & Indicates if exact output times are required ($\\T{Ixc} = 1$ recommanded) \\\\\n    \\T{ComputeOrbitalElements}  & yes\/no & If yes, the code compute the keplerian orbital elements of the trajectory of each body with respect to the first body \\\\\n\t\\T{tolerance} & real $\\geqslant 10^{-14}$ & Tolerance in the accuracy of the integration ($\\T{tolerance} = 10^{-12}$ recommanded) \\\\\n\t\\T{soft}      & real $\\geqslant 0.0$ & Softening of the potential ($\\T{soft} = 0$ recommanded) \\\\\n\t\\T{DehnenYes}, \\T{DehnenDF} & yes\/no ; yes\/no & Activate Dehnen external potential and dynamical friction \\\\\n\t$\\gamma_{\\subD}$, $\\T{r}_{\\subt{D}}$, $\\T{M}_{\\subt{D}}$ & reals $\\geqslant 0.0$ & Slope, scale radius and total mass of the Dehnen profile (ignored for \\T{DehnenYes} = no) \\\\\n\t\\T{PlummerYes}, \\T{PlummerDF} & yes\/no ; yes\/no & Activate Plummer external potential and dynamical friction \\\\\n\t$\\T{r}_{\\subt{P}}$, $\\T{M}_{\\subt{P}}$ & reals $\\geqslant 0.0$ & Scale radius and total mass of the Plummer profile (ignored for \\T{PlummerYes} = no) \\\\\n\t\\T{VirialYes}, \\T{Q} & yes\/no ; real $> 0.0$ & If yes the system is rescaled to the indicated virial ratio \\T{Q} (rescaling to $\\T{Q} = 1$ recommanded)\n\\end{tabular}\n\\end{center}\n\\caption{Overview of the user-defined parameters required by the code, as they are organized in the file \\T{PARAMETERS.TXT}. For details on the available choices see the corresponding section.}\n\\label{Table_param}\n\\end{table*}\n\n\n\\section*{Input data}\n\nThe file \\T{INPUT.TXT} is an ASCII file which must be provided by the user. It is composed of \\T{N} rows, each giving the mass $m_i$, the cartesian coordinates $x_i,y_i,z_i$, and the cartesian velocity components $v_x^i, v_y^i, v_z^i$, of every body ($i=1, 2, \\dots, N$) in the system. The order in which the data are given is arbitrary, except that the first \\T{Nbh} rows refer to \\textit{black-holes}. \n\nAt the beginning of the code execution, the coordinates and velocities of the bodies are rescaled so that the center of mass of the system is in the origin of the coordinate frame and has a zero velocity. It is worth noticing that the external gravitational field is always centered in the center of mass of the \\T{N}-body system. \\\\\n\nThe input data units can be chosen by the user. One possibility is by setting in the last line of \\T{PARAMETERS.TXT} (see Table \\ref{Table_param}) the parameter \\T{TypeOfUnits} to $0$, so that the units of the entries in the file \\T{INPUT.TXT} are ``free'' as long as the universal constant of gravitation, $G$, is set to $1$. We call these units \\textit{autonomous}. The major constraint is that the parameters \\T{TMAX}, \\T{DeltaT}, \\T{Clight} and, if relevant, $\\T{r}_{\\subt{D}}$, $\\T{M}_{\\subt{D}}$, $\\T{r}_{\\subt{P}}$ and $\\T{M}_{\\subt{P}}$ (see Table \\ref{Table_param}) must be given in the same user-defined units.\n\nIf the user chooses \\textit{astrophysical} units, the parameter \\T{TypeOfUnits} must be set to $1$, and the various input data and physical parameters have to be given in solar masses (\\mbox{$\\mbox{M}_{\\odot}$}), parsecs (pc) and km\/s. The value of \\T{Clight} is consequently $\\sim 299,792$ and any value written for it inside \\T{PARAMETERS.TXT} is not considered (the user must still write a dummy argument, to preserve the file format). To avoid both too large and too small numbers (with associated round-up error issues), the code will automatically converts all input data and parameters to the standard \\T{N}-body units (for more details, see \\cite{HeggieHut} p.$7$), defined as follow~:\n\\begin{itemize}\n\t\\item the length unit is the initial size of the stellar system, i.e. $U_{\\subt{l}} = \\max\\limits_{1\\leqslant i \\leqslant \\T{N}}\\sqrt{x_i^2 + y_i^2 + z_i^2}$;\n\t\\item the mass unit is the total mass (\\T{N} bodies + diffuse matter) inside the sphere of radius $U_{\\subt{l}}$, that is \n\t\\begin{equation*}\n\t\tU_{\\subt{m}} = \\sum\\limits_{i=1}^N m_i +\n\t\t\\int_{V} \\rho(\\mathbf{r})\\,d^3\\mathbf{r},\n\t\\end{equation*}\n\twhere $\\rho(\\mathbf{r})$ is the density distribution generating, via solution of the Poisson's equation, the external potential;\n\t\\item to ensure $G = 1. U_{\\subt{l}}^3 U_{\\subt{m}}^{-1} U_{\\subt{t}}^{-2}$, the time scale must be $U_{\\subt{t}} = \\sqrt{U_{\\subt{l}}^3\/(G U_{\\subt{m}})}$;\n\t\\item accordingly, the unit of velocity is $\\displaystyle U_{\\subt{v}} = \\frac{U_{\\subt{l}} }{ U_{\\subt{t}} } = \\sqrt{\\frac{ G U_{\\subt{m}} }{ U_{\\subt{l}} }}$.\n\\end{itemize} \n\nNote that \\T{N}-body units are a peculiar case of autonomous units.\n\t\nThe output data will \\textit{always} be provided in the same units than the input data.\n\n\n\n\n\n\\section*{Number and type of gravitating bodies}\n\n\\subsection*{Number of stellar bodies}\n\nThe number \\T{N} of stellar bodies must be $\\geqslant 2$. It is also capped at $\\T{NMX} = 400$ (see file ARCCOM2e2.CH).\n\nAlthough this is not recommended, the upper limit \\T{NMX} can be changed. To do so, one has first to modify the variable \\T{NMX} to the wanted value in the file \\T{ARCCOM2e2.ch} and the routine \\T{NotAtAllNeeded} of the main file. \n\nThen, the variable with same name \\T{NMX} present in the routines \\T{ARC}, \\T{Iterate2ExactTime} and \\T{DIFSYAB} of the main code must be modified accordingly : if the common variable \\T{NMX} is (for example) multiplied by $2$, then the user must also multiply the homonimous variable by $2$ in the three routines indicated above.\n\n\n\n\n\\subsection*{Number of black-holes}\n\nThe code discriminates the \\T{N} stellar bodies into two populations~: the ``classic'' stars and the ``black-holes'', these latter being the \\T{Nbh} first bodies entered into the file \\T{INPUT.TXT}. The motion of a classic star is governed by the standard Newtonian equations, while for black-holes a PN approximation up to $2.5$ order is adopted (see \\cite{Memmesheimer2004} for details of the equations and \\cite{Mikkola2008} for their implementation).\n\nThe number \\T{Nbh} of black-holes may range from $0$ (no PN effect at all) to \\T{N} (all stellar bodies are black-holes). However, \\T{Nbh} greater than \\T{N} will be accepted by the code and interpreted as $\\T{Nbh} = \\T{N}$.\n\nTo compute the PN terms, the code needs the value of the speed of light in the units used for the simulation. If the input data are entered in physical units ($\\T{TypeOfUnits} = 1$), then this value is computed simultaneously at the conversion into \\T{N}-body units, with:\n$$\\T{Clight} = \\frac{2.99792\\times 10^5 \\mbox{ km\/s}}{U_{\\subt{v}}}.$$\n\n\nIf input data are given in autonomous units ($\\T{TypeOfUnits} = 0$), the user must indicate in the file \\T{PARAMETERS.TXT} the correct value of \\T{Clight} in these units. It has to be noted that setting $\\T{Clight} = 0$ will switch off all PN terms and effects (including the merger process, see related section), even if $\\T{Nbh} \\neq 0$. Doing things this way can prove a little tricky for the user~: it is more straight-forward to set $\\T{Nbh} = 0$. Note, also, that setting \\T{Clight} at a very large value makes the role of PN terms negligible at a useless comptutational cost. \n\n\nWhen removing the PN part by setting $\\T{Nbh} = 0$, the integration proceeds much faster. On the other hand, the simulation as a whole can get utterly slow if there is any close two-body encounter, because the code cannot proceed to a merger; then, to ensure sufficient accuracy in the modelization of the binary motion, the time-steps reduce enormously and the simulation process get stuck.\n\n\n\n\n\\section*{Time-related parameters}\n\n\\subsection*{Total time of integration}\n\nThe input variable \\T{TMAX} represents the total time over which the evolution of the system will be simulated. If $\\T{TypeOfUnits} = 0$, its value is assumed to be in the autonomous time unit of the system. If $\\T{TypeOfUnits} = 1$, it is considered to be given in million years ($10^6$ yrs). \n\n\n\n\\subsection*{Output interval} \n\nWhile computing the evolution of the system, the code uses an adaptative time-step chosen so as to ensure enough accuracy in the treatment of tight binaries (see the section about time-transformation). This time-step can be very small, so that writing down the state of the system at each step would both slow down the computation and produce an enormous amount of output files which could overflow the memory storage.\n\nTo avoid this, the user may define the output time interval, hereafter called \\T{DeltaT}. If $\\T{TypeOfUnits} = 0$, its value is assumed to be in the autonomous time unit of the system. If $\\T{TypeOfUnits} = 1$, it is considered to be given in million years, like \\T{TMAX}. It is worth noticing that \\T{DeltaT} is also used for initializing the computation of the adaptative time-step. \n\n\nThe parameter \\T{Ixc} allows the user to select the way the output is triggered. Indeed, since the optimal integration time-step is computed independently by the code, the actual integration time $t$ is \\textit{not} a priori a multiple of \\T{DeltaT}.\n\nLet $t_0$ be the integration time at which the last output file has been produced, and $t_{\\subt{next}} = t_0 + \\T{DeltaT}$ the time for the, possible, next output.\nIf $\\T{Ixc} = 0$, then a new output file is written as soon as the integration has gone far enough -- that is when $t>t_{\\subt{next}}$. This is the most straightforward method but, depending on the system, the actual global time interval can be much longer than the one the user asked for.\n\n\nIf $\\T{Ixc} = 2$ on the other hand, the code will respect the exact output times required by integrating to $t = t_{\\subt{next}}$. This can be slow if many outputs are required, that is if $\\T{DeltaT} \\ll \\T{TMAX}$. \n\nA compromise between these two methods is provided by the choice $\\T{Ixc} = 1$. In such case the code tries to keep the time interval close to the wanted \\T{DeltaT} by approximating the exact output time $t_{\\subt{next}}$ with the Stumpff-Weiss method \\cite{stu67}. \nThis option is often the fastest and most suitable one. \\\\\n\n\nIt must be noted that the setting $\\T{Ixc} = 0$, and to a lesser extent $\\T{Ixc} = 1$, works well only when \\T{DeltaT} is reasonably small. If \\T{DeltaT} is large, one may prefer the use of the slower $\\T{Ixc}=2$ choice. However, the notion of what ``small'' is depends on the specific case.\n\n\n\n\n\\subsection*{Time-transformation or regularization}\n\nIn few-body systems, close approaches happen frequently. Consequently, to ensure accuracy in numerical simulations, one must {\\it regularize} the strong interaction by a proper computation of coordinates and velocities (which is done by the leapfrog algorithm). Several methods exist for that purpose, which can efficiently be used, not only for the purely Newtonian few-body problem but also for other problems with external perturbations, including velocity dependent ones (in our case, external gravitational field with dynamical friction, this latter depending on the object speed).\n\\par\\noindent\nThe \\T{ARWV} code proposes a combination of three of them~: \n\\begin{itemize}\n\t\\item the {\\it logarithmic-Hamiltonian} (logH) method, proposed by Mikkola and Tanikawa (see \\cite{Mikkola99a} and \\cite{Mikkola99b}) as well as Preto and Tremaine \\cite{PretoTremaine}, gives accurate trajectories for the two-body problem and also satisfactory results for perturbed two-body encounters; \n\t\\item the Time-Transformed-Leapfrog (Mikkola and Aarseth \\cite{Mikkola2002}), which turned out to be close to the logH-method, but, in some respects, more general;\n\t\\item the auxiliary velocity algorithm, presented by Hellstr\\\"{o}m and Mikkola in \\cite{Mikkola2010}, which is not a real regularization method, but rather  a trick which allows to apply the ``normal'' leapfrog algorithm to a velocity-dependent acceleration.\n\\end{itemize}\n\n\nThe parameter \\T{cmet} is a three-dimensional vector that determines the way these methods will be combined. $\\T{cmet} = (1,0,0)$ is the logarithmic-Hamiltonian, $(0,1,0)$ is the Time-Transformed-Leapfrog and $(0,0,1)$ corresponds to no time-transformation.\n\nSetting $\\T{cmet} = (1,0,0)$ is usually the best choice, but a small value for \\T{cmet}(2) may be advisable in case of star-star close encounter. If the system contains big and small bodies, one may use $\\T{cmet} = (1,10^{-16},0)$ instead of $(1,0,0)$ : this causes smaller stepsizes when two very small bodies come close to each other. Please note that, in this case, the computational speed depends heavily on the formation of binaries !\n\n \n\n\n\n\n\n\\section*{Merging and recoil velocity}\n\nThe code includes a \\textit{merging} procedure intended for the fusion of a black-hole binary or for the swallowing of a classic star by a black-hole (but not for collisions between two stars; this means that collisions are handled out only if at least one of the two objects is in the list of the \\T{Nbh} black-holes).\n\nThe classic recipe for the merging of two bodies is that the remnant of the merger will be located at the center of mass of the previous pair (considered as the location where the collision actually happens, though the code stops the integration of the separate trajectories just before that point).\n\nFor the velocity of the remnant (or ``recoil'' velocity) however, there is no general consensus. For a purely Newtonian simulation, possible alternatives are giving to the remnant a null velocity or the velocity of the center of mass of the two progenitors. In the original version of \\T{ARWV}, the latter choice has been done. \\\\\n\n\nThe code was later modified by Roberto Capuzzo-Dolcetta and Pauline Chassonnery to follow the prescription given by Lousto and Healy in \\cite{LoustoHealy}. \n\nLet $m_{\\sub{1}}$ and $m_{\\sub{2}}$ be the masses of two merging black-holes. We use the convention $m_{\\sub{1}} \\leqslant m_{\\sub{2}}$ and introduce the auxiliary variables:\n\\begin{align*} \n\t&m = m_{\\sub{1}} + m_{\\sub{2}}, \\hspace{1em} \\eta = \\frac{m_{\\sub{1}} m_{\\sub{2}}}{m^2}, \\hspace{1em} 0< q = \\frac{m_{\\sub{1}}}{m_{\\sub{2}}} \\leqslant 1, \\\\\n\t&\\delta_m = \\frac{m_{\\sub{1}} - m_{\\sub{2}}}{m} \\leqslant 0.\n\\end{align*}\n\n\nEach stellar body is assumed to be auto-rotating, with a dimensionless spin $\\bm{\\alpha}_i$ such that $|\\bm{\\alpha}_i| \\leqslant 1$. Two spin-related variables are~: \n\\begin{align*}\n\t\\bm{S} &= \\frac{ \\bm{\\alpha}_{\\sub{1}} m_1^2 + \\bm{\\alpha}_{\\sub{2}} m_2^2 }{m^2} = \\frac{ \\bm{\\alpha}_{\\sub{2}} + q^2 \\bm{\\alpha}_{\\sub{1}}}{(1 + q)^2}, \\\\\n\t\\bm{\\Delta} &= \\frac{ \\bm{\\alpha}_{\\sub{2}} m_{\\sub{2}} - \\bm{\\alpha}_{\\sub{1}} m_{\\sub{1}} }{m} = \\frac{ \\bm{\\alpha}_{\\sub{2}} - q \\bm{\\alpha}_{\\sub{1}}}{1 + q}.\n\\end{align*}\n$ $\n\nThe dimensionless spin of each of the \\T{N} bodies can be generated randomly at the beginning of the simulation, or read from a separated file. The latter is best for reproductibility purpose, but need the user to provide a file named \\T{SPIN.TXT}, formatted as \\T{N} rows containing each the three coordinates of the spin of one body. The user must also uncomment, in the main routine of the code, the part from line 157 to 182 included, i.e. the section \"SET INITIAL SPINS\". \\\\\n\n\nThe recoil velocity is finally given by:\n$$\\textbf{v}_{\\subt{recoil}} = v_{\\subt{m}} \\textbf{e}_{\\sub{1}} + v_{\\sub{\\perp}} \\left( \\cos \\xi\\textbf{e}_{\\sub{1}} + \\sin \\xi\\textbf{e}_{\\sub{2}} \\right),$$\nwhere $\\textbf{e}_{\\sub{1}}$ is the unit vector pointing from $m_{\\sub{1}}$ to $m_{\\sub{2}}$ and $\\textbf{e}_{\\sub{2}}$ an orthogonal unit vector in the two-body orbital plane, such that the basis formed by $\\textbf{e}_{\\sub{1}}$, $\\textbf{e}_{\\sub{2}}$ and the orbital angular momentum of the pair is direct.\n\nThe values of $\\xi$, $v_{\\subt{m}}$ and $v_{\\sub{\\perp}}$ are given by the following fitting formulas to numerical data:\n\\begin{align*}\n\t&\\xi = a + b S_{||} + c \\delta_m \\Delta_{||}, \\\\\n\t&v_{\\subt{m}} = \\eta^2 \\delta_m \\left(A + B \\delta_m^2 + C \\delta_m^4 \\right), \\\\\n\t&v_{\\sub{\\perp}} = H \\eta^2 \\left( \\Delta_{||} + H_{\\subt{2a}} S_{||} \\delta_m + H_{\\subt{2b}} \\Delta_{||} S_{||} \\right. \\\\\n\t &\\hspace{2em} + H_{\\subt{3a}} \\Delta_{||}^2 \\delta_m + H_{\\subt{3b}} S_{||}^2 \\delta_m + H_{\\subt{3c}} \\Delta_{||} S_{||}^2 \\\\\n\t &\\hspace{2em} + H_{\\subt{3d}} \\Delta_{||}^3 + H_{\\subt{3e}} \\Delta_{||} \\delta_m^2 + H_{\\subt{4a}} \\Delta_{||}^2 S_{||} \\delta_m \\\\\n\t &\\hspace{2em} + H_{\\subt{4b}} S_{||}^3 \\delta_m + H_{\\subt{4c}} S_{||} \\delta_m^3 + H_{\\subt{4d}} \\Delta_{||} S_{||} \\delta_m^2 \\\\\n\t &\\hspace{2em} \\left. + H_{\\subt{4e}} \\Delta_{||} S_{||}^3 + H_{\\subt{4f}} \\Delta_{||}^3 S_{||} \\right).\n\\end{align*}\nwhere the subscript $||$ denotes the projection of vectors along the direction of the orbital angular momentum of the pair. The numerical values of the different constants above are given in \\cite{LoustoHealy}. Note that $A$, $B$, $C$ and $H$ are given in km\/s and must be converted into the units used for the simulation. This is done automatically by the code, no matter the value of \\T{TypeOfUnits} (unless $\\T{Nbh} = 0$ and\/or $\\T{Clight} = 0$, in which cases PN terms are switched off and mergers are not permitted).\n\n\n\n\\section*{External gravitational fields}\n\nThis code allows the user to add to the simulated stellar system an external regular and spherical distribution of matter, accounting for its gravitation field and dynamical friction over the \\T{N} bodies.\n\nThe code allows two possible density profiles for the external field : a {\\it Dehnen} profile \\cite{Dehnen93} and\/or a {\\it Plummer} profile \\cite{Plummer}. The user may activate either of these by setting to ``yes'' the corresponding option \\T{DehnenYes} and\/or \\T{PlummerYes} (the two options can be activated at once, to get a superposition of the two density profiles).\n\nThe matter distribution is defined by its total mass (parameter $\\T{M}_{\\subt{D}}$ or $\\T{M}_{\\subt{P}}$) and its scale radius (parameter $\\T{r}_{\\subt{D}}$ or $\\T{r}_{\\subt{P}}$). These two values may be given either in $\\mbox{$\\mbox{M}_{\\odot}$}$ and parsec (option $\\T{TypeOfUnits} = 1$) or in the same autonomous units than the other input data (option $\\T{TypeOfUnits} = 0$). For the Dehnen profile, the user also needs to define a central slope (parameter $\\gamma_{\\subD}$), whose value must be $0\\leqslant \\gamma_{\\subD}< 5\/2$. \\\\\n\nThe spherical matter distribution acts over the \\T{N} objects by mean of its gravitational field and dynamical friction braking.  \n\nThe dynamical friction (df) effect is taken into account via a deceleration term in the classical Chandrasekhar's form in local approximation \\cite{Chandrasekhar}: its magnitude depends upon a proper average of the velocity dispersion of the particles composing the regular matter distribution. For a Plummer profile, the average velocity dispersion is :\n$$\\sigma^2_{\\subt{P}} = 3 \\pi G M_{\\subt{P}}\/(32 r_{\\subt{P}}).$$\n\n\nFor a Dehnen profile, the formula for the gravitational energy converges only if $\\gamma_{\\subD} < 5\/2$, to give the following expression :\n$$\\sigma^2_{\\subt{D}} = \\frac{G M_{\\subt{D}}}{2 r_{\\subt{D}}} \\frac{1}{5-2\\gamma_{\\subD}}.$$\n\n\nThen, for the superposition of a Dehnen and a Plummer profile, the global mass-averaged velocity dispersion is:\n$$\\sigma^2 = \\frac{M_{\\subt{D}} \\sigma^2_{\\subt{D}} + M_{\\subt{P}} \\sigma^2_{\\subt{P}}}{M_{\\subt{D}} + M_{\\subt{P}}}.$$\n\n\n\n\n\n\n\\section*{Initial virial ratio}\n\nA physical system is said to be in virial equilibrium if the second derivative $\\ddot{I}$ of its polar moment of inertia is null, at least on average. \n\nLet $K$ denote the kinetic energy of a system and $\\Omega$ its potential energy. We can show that $\\ddot{I}\/2 = 2K +\\Omega$ (see \\cite{HeggieHut}, chapter $9$ for proof).\nCalling ``virial ratio'' the quantity $Q = 2K\/|\\Omega|$, the condition $\\langle\\ddot{I}\\rangle = 0$ is equivalent to $\\langle Q \\rangle = 1$. \\\\\n\n\nWhen studying systems embedded in a regular mass distribution, the gravitational attraction exerted by the regular field is, generally, non negligible and deserves to be included in the computation of the total potential energy $\\Omega$ of the system.\n\nLet $Q_{\\sub{0}}$ denote the virial ratio corresponding to the input data given by the user (file \\T{INPUT.TXT}). If the parameter \\T{VirialYes} (in the file \\T{PARAMETERS.TXT}) is set to ``yes'', then the code  rescales the system's velocities so that the virial ratio gets the value \\T{Q} assigned by the user in the \\T{PARAMETERS.TXT} file. This is, simply, done by multiplying the velocities of all the stellar bodies by $\\sqrt{\\T{Q}\/Q_{\\sub{0}}}$. \\\\\n\n\n\n\n\n\n\n\n\n\\section*{Keplerian orbital elements}\n\nAn interesting case of \\T{N}-body system are systems where small bodies (stars as well as black-holes) are moving around a massive rotating black-hole. Most of the small bodies can then be considered as orbiting around the central one in a binary relationship (like the planets around the Sun). In this case, the user may be more interested in the characteristics of these binary-orbits (semi-major axis, eccentricity, period...) than by the raw coordinates and velocities of each body.\n\nThe user may thus place the data of the massive black-hole in the first line of the file \\T{INPUT.TXT} and activate the binary motion analysis by setting to ``yes'' the option \\T{ComputeOrbitalElements} in the parameters file.\n\n\n\n\nIn PN theory, the characteristic variables of a binary motion, or orbital elements, depend on the spin of the two members of the binary. If we assume a large unbalance between the masses, the spin of the small body can be neglected and the code only require the spin of the massive black-hole. \n\nThis (dimensionless) spin $\\bm{\\alpha}_{\\sub{1}}$ has already been defined, since it is necessary to compute the recoil velocity (see related section). We recall here that this spin can be defined via a file \\T{SPIN.TXT} provided by the user, or randomly generated by the code at its initialization. It must have a norm less than $1$ and, unless the spin of the other bodies which are constant, it will evolve along the computation.\n\n\n\nAt each output time, the code will compute the value of the orbital elements for all pairs $(m_{\\sub{1}},m_{\\sub{j}})_{1<j\\leqslant N}$ and save them in separate files numbered from $71$ to $76$~:\n\\begin{itemize}\n\t\\item file \\T{71.dat} contains the semi-major axis $a$ of each pair at each output time,\n\t\\item file \\T{72.dat} contains the eccentricity, $e$, of each pair, \n\t\\item file \\T{73.dat} contains the inclination $i$ (or vertical tilt of the orbital plane with respect to the reference plane,\n\n\t\\item file \\T{74.dat} contains the longitude of the ascending node $\\Omega$ (or orientation of the orbital plane with respect to the reference frame's vernal point),\n\t\\item file \\T{75.dat} contains the argument of periapsis $\\omega$ (or orientation of the ellipse in the orbital plane),\n\t\\item file \\T{76.dat} contains the time, the current spin $\\bm{\\alpha}_{\\sub{1}}(t)$ of the first body, and the absolute variation of the norm of this spin (i.e. $\\alpha_{\\sub{1}}(t) - \\alpha_{\\sub{1}}(0)$).\n\\end{itemize}\n\n\n\\section*{Miscellanous}\n\n\\subsection*{Tolerance}\nThe parameter \\T{tolerance} sets the integration accuracy in the routine \\T{DIFSYAB}. A small value  is suggested, but not less than $10^{-14}$ (if \\T{tolerance} is smaller, the code sets it at $10^{-14}$). Of course, the code is faster when requiring low accuracy, but it must be remembered that \\T{ARWV} is designed to conduct highly accurate simulations.\n\n\\subsection*{Softening}\nThe parameter \\T{soft} allows the user to {\\it smooth} the straight $1\/r$ Newtonian potential with one of the form $1\/\\sqrt{r^2 + \\T{soft}^2}$. To have a purely Newtonian interaction this parameter must be set to $0$.\n\n\\section*{Acknowledgements}\nWe thank M. Arca-Sedda for providing us a version of the archain code with some modifications he made, that constituted a starting version form our further changes and additions which eventually lead to the version described in this user manual and that can be downloaded.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nRecently, with the advent of large pre-trained language models~\\cite{devlin2018bert,brown2020language,smith2022using}, dense retrieval \\cite{DBLP:conf\/emnlp\/KarpukhinOMLWEC20,DBLP:conf\/iclr\/XiongXLTLBAO21,DBLP:journals\/corr\/abs-2210-11773} has become a popular topic for information retrieval, which aims to identify relevant contents from a large collection of passages.\nDespite the great success of dense retrieval on benchmarks like MS MACRO~\\cite{DBLP:conf\/nips\/NguyenRSGTMD16}, the inference efficiency is still left as a problem.\nTherefore, small and efficient models are welcomed in the practical scenarios of dense retrieval.\n\nConsidering the trade-off between efficiency and effectiveness, researchers resort to knowledge distillation (KD) \\cite{DBLP:journals\/corr\/HintonVD15,sanh2019distilbert,jiao2019tinybert}, which aims to transfer knowledge from a strong and large teacher model to a small yet efficient student model.\nPopular distillation methods involve response-based methods and feature-based methods.\nResponse-based methods \\cite{DBLP:journals\/corr\/HintonVD15, kim2018paraphrasing, mirzadeh2020improved} train the student model to predict the output of the teacher model, which is simple yet effective in different tasks. \nHowever, compared with feature-based methods, response-based methods ignore the intermediate signals of teachers, which are proven to be important in deep and thin neural network~\\cite{DBLP:journals\/corr\/RomeroBKCGB14} and lead to lower upper limits than feature-based methods.\n\nFeature-based methods mainly use the representations or attention maps of the intermediate layers as features, directly matching the features of the teacher and the student \\cite{DBLP:journals\/corr\/RomeroBKCGB14,zagoruyko2016paying,DBLP:conf\/iclr\/ZagoruykoK17,passalis2018learning,xu2020feature,DBLP:journals\/corr\/abs-2112-04195,haidar2021rail}.\nHowever, these methods suffer from two major weaknesses:\n(1) In our preliminary study, feature-based methods do not work in dense retrieval, whose performances are even lower than response-based methods.\n(2) Matching the features requires the teacher and the student to share the same tokenizer-vocabulary set or internal output size, which would somehow constrain the flexibility choices of the teacher model.\n\nTo fill the gap, we propose a liberal feature-based distillation method (LEAD) for dense retrieval.\nSpecifically, we use the internal \\texttt{[CLS]} embedding vectors to calculate the similarity distribution of the passages as features in the mini-batch.\nThen, we align the features between the teacher and the student, to accomplish the feature matching without tokenizer of architecture constraints.\nWe also randomly select layers from the model to participate in the feature matching, adaptively assigning weights to the selected layers based on their informativeness.\nBesides, we jointly train the teacher model and student model to let them learn from each other.\nThe reason why we call our method \\textbf{liberal} comes from two folds:\n(1) LEAD supports various teacher architectures like dual encoder, ColBERT, cross encoder, and so on, only assuming that the teacher has a layer number no less than the student.\n(2) LEAD uses the label distributions to calculate distillation loss, which is free of the constraints on vocabularies, tokenizers, or model architectures.\n\nTo evaluate the effectiveness of LEAD, we conduct experiments on several public benchmarks, such as MS MARCO \\cite{DBLP:conf\/nips\/NguyenRSGTMD16} and TREC Deep Learning Track \\cite{DBLP:journals\/corr\/abs-2003-07820,DBLP:journals\/corr\/abs-2102-07662}, where LEAD achieves better performances than traditional response-based and feature-based methods.\nFurthermore, we also find that LEAD has two main advantages:\n(1) LEAD is extendable to the broad sense ``layers'', which include Transformers layers and linear layers that can be appended after the models depending on the downstream tasks.\n(2) LEAD is portable for the existing popular distillation pipeline for dense retrieval \\cite{DBLP:conf\/sigir\/ZengZV22,lin2022prod}, which means it not only achieves superior performance but can also achieve further improvements with other effective pipelines.\n\n\n\\section{Related Work}\n\n\n\n\\subsection{Dense Retrieval}\n\nDense retrieval methods \\cite{DBLP:conf\/sigir\/ZhanM0G0M21, lin2021batch, lin2022prod, lu2022ernie} have the potential of identifying semantically relevant but text-independent query-passage pairs.\nThe query and passage representations in dense retrieval models are usually obtained using dual encoders \\cite{DBLP:conf\/emnlp\/KarpukhinOMLWEC20}.\nColBERT~\\cite{DBLP:conf\/sigir\/KhattabZ20} pushes performance higher by introducing the late interaction while requiring higher computation and storage costs.\nIn spite of the model structure, recent work also focuses on utilizing training strategies to obtain better results, ranging from data-centric studies~\\cite{DBLP:conf\/acl\/RenLQLZSWWW21, sciavolino2021simple} and negative sampling~\\cite{gao2021your, DBLP:conf\/iclr\/XiongXLTLBAO21, DBLP:journals\/corr\/abs-2210-11773} to distillation~\\cite{DBLP:journals\/corr\/abs-2010-02666, DBLP:conf\/sigir\/ZengZV22, DBLP:journals\/corr\/abs-2205-09153}.\n\nOur work is in line with the distillation methods, where LEAD can be a better alternative to the widely-used response-based methods.\n\n\n\\subsection{Knowledge Distillation}\n\nKnowledge distillation~\\cite{DBLP:journals\/corr\/HintonVD15} aims to transfer knowledge from an effective teacher model to an efficient student model, which can be divided into two categories: response-based methods and feature-based methods.\nThe main idea of response-based methods~\\cite{lin2022prod, DBLP:conf\/sigir\/ZengZV22} is to let student model mimic the final predictions of the teacher model~\\cite{gou2021knowledge}, which is simple yet effective in different tasks.\nHowever, these methods fail to take into account the intermediate signals of the teacher model, which turns out to be very important in very deep neural networks~\\cite{romero2014fitnets}.\n\nFeature-based methods are thought to be the extension of response-based methods and are especially useful in thinner and deeper networks~\\cite{gou2021knowledge}, which focus on aligning the internal representations or attention scores.\n\\citet{DBLP:journals\/corr\/RomeroBKCGB14,xu2020feature,wang2020exclusivity,sun2019patient,haidar2021rail} directly match the internal representations of the teacher and the student, which is not suitable for cross encoders.\nBesides, \\citet{zagoruyko2016paying, passban2021alp, huang2017like,li2021virt} directly enclose the attention maps of the teacher and the student, which are constrained on used vocabularies and tokenizers.\n\nUnlike distillation work on multiple teachers \\cite{DBLP:journals\/corr\/abs-2205-09153,lin2022prod}, LEAD focuses on single teacher, taking advantage from both response-based and feature-based methods, enclosing the label distributions of intermediate layers, thus leaving no constraints on the vocabularies, tokenizers, or model architectures.\n\n\\section{Method}\n\nWe first introduce the task description of dense retrieval and then offer a unified view of retrieval models.\nBased on the unified view, we elaborate the liberal feature-based distillation method.\n\n\n\\subsection{Task Description}\n\nThe target of dense retrieval is to retrieve passages based on learned distributed representations of queries and passages.\nFormally, given a query set ${\\mathbb{Q}}=\\{ {\\bm{q}}_1, {\\bm{q}}_2, \\dots, {\\bm{q}}_{n} \\}$ containing $n$ queries and a passage set ${\\mathbb{P}}=\\{ {\\bm{p}}_1, {\\bm{p}}_2, \\dots, {\\bm{p}}_{m} \\}$ containing $m$ passages, dense retrieval aims to find the most relevant passages in corpus ${\\mathbb{P}}$ for each query ${\\bm{q}}_i$.\n\n\n\\subsection{Unified View of Retrieval Models}\n\\label{sec:view}\n\nIn the research line of dense retrieval, there are several popular models consisting of stacked neural networks, such as dual encoder \\cite{DBLP:conf\/emnlp\/KarpukhinOMLWEC20}, ColBERT \\cite{DBLP:conf\/sigir\/KhattabZ20}, and cross encoder \\cite{DBLP:conf\/naacl\/QuDLLRZDWW21}.\nAlthough these models have diverse architectures, they shall have common characteristics.\nTherefore, we offer a unified view of these models in this section.\n\nConsider dividing the above-mentioned models into two parallel aligned modules, $E_1$ and $E_2$, which are two piles of neural layers (including Transformers and linear layers) with the same layer numbers.\nGiven a model with $N$ layers, let's use $E_{1}^{i}$ and $E_{2}^{i}$ ($1 \\leq i \\leq N$) to denote the encoding of the first $i$ layers of $E_1$ and $E_2$, respectively.\nThe two major differences between those models are (1) the instantiations of $E_1$ and $E_2$ and (2) the similarity calculation function $f(\\cdot)$.\nTo better elaborate the unified view, we show the following examples.\n\n\\textbf{Dual Encoder (DE)} \\cite{DBLP:conf\/emnlp\/KarpukhinOMLWEC20} is the most widely used dense retrieval architecture, which encodes queries and passages into dense vectors separately, calculating the relevance score through the inner product.\nFor DE, $E_1$ is the query encoder and the $E_2$ is the passage encoder.\nBoth of them are Transformer encoders.\nThe similarity calculation function $f_{\\textrm{DE}}(\\cdot)$ is defined as:\n\\begin{equation}\n\\footnotesize\nf_{\\textrm{DE}}({\\bm{q}}, {\\bm{p}}) = E_{Q}({\\bm{q}})^{T} \\cdot E_{P}({\\bm{p}})\n\\label{equ:descore}\n\\end{equation}\n\n\\textbf{ColBERT (CB)} \\cite{DBLP:conf\/sigir\/KhattabZ20} can be viewed as a more expressive dual-encoder, which delays the interaction between query and passage after encoding.\nThe instantiation of $E_1$ and $E_2$ is the same as DE.\nBut the similarity calculation function $f_{\\textrm{CB}}(\\cdot)$ is defined as:\n\\begin{equation}\n\\footnotesize\nf_{\\textrm{CB}}({\\bm{q}}, {\\bm{p}}) = \\sum_{1 \\leq x \\leq X} \\mathop{\\textrm{max}}\\limits_{1 \\leq y \\leq Y} E_{Q}({\\bm{q}})_x \\cdot E_{P}({\\bm{p}})_y\n\\label{equ:colscore}\n\\end{equation}\nwhere $X$ and $Y$ denote the length of the query and passage token sequence, respectively.\nPlease note that, following \\citet{DBLP:journals\/corr\/abs-2010-02666}, we remove the punctuation filter and the last linear layer of the encoders to focus on distillation.\n\n\\textbf{Cross Encoder (CE)} \\cite{DBLP:conf\/naacl\/QuDLLRZDWW21} has strong abilities to capture the fine-grained relationships between queries and passages within the Transformer encoding.\nMuch different from DE and CB, for CE, $E_1$ is the query-passage pair encoder $E_{\\textrm{CE}}$ and $E_2$ is the projection layer ${\\bm{w}}$ after the Transformer encoder, which is used in a shared manner.\nThe similarity calculation function $f_{\\textrm{CE}}(\\cdot)$ is defined as:\n\\begin{equation}\n\\footnotesize\nf_{\\textrm{CE}}({\\bm{q}}, {\\bm{p}}) = {\\bm{w}}^T \\cdot E_{\\textrm{CE}}([{\\bm{q}};{\\bm{p}}])\n\\label{equ:descore}\n\\end{equation}\nwhere $[;]$ is the concatenation operation.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[page=1,width=0.8\\textwidth]{figures\/framework7.pdf}\n\\caption{\nThe unified framework of LEAD, which supports various teacher architectures.\nAssume the teacher and student model has $N$ and $M$ layers ($N \\geq M$), respectively.\nThe \\textit{Module1} and \\textit{Module2} correspond to the $E_1$ and $E_2$ of the teacher model, respectively.\n${\\bm{t}}_i$ and ${\\bm{s}}_i$ is the similarity distribution of the passages in the current batch, which is calculated with the $i$-th layer for the teacher and student model using $f(*)$ and $g(*)$, respectively.\n$a_1, a_2, ..., a_K$ is the current random selection of teacher layers for layer-wise distillation.\n$b_1, b_2, ..., b_K$ is the current random selection of student layers for layer-wise distillation.\nThe selected layers preserve the original partial order $1 \\leq a_1 < a_2 < ... < a_K \\leq N$ and $1 \\leq b_1 < b_2 < ... < b_K \\leq M$.}\n\\label{fig:framework}\n\\end{figure*}\n\n\n\\subsection{Distilling with Liberal Layer Feature}\n\nPrevious response-based distillation methods \\cite{DBLP:conf\/naacl\/QuDLLRZDWW21,DBLP:journals\/corr\/abs-2010-11386,luan2021sparse} mainly distill knowledge from the output logits of the entire teacher model to the student model.\nHowever, considering that different model layers try to capture the relationships between queries and passages at different granularity, we believe distilling knowledge from intermediate layers may be good for the student model.\nThus, based on the unified view, we propose a liberal feature-based distillation method (LEAD) based on layer-wise alignment, which is shown in \\cref{fig:framework}.\n\n\n\\paragraph{Layer Feature Definition.}\n\nIn general, the backbone of our model is a pile of Transformer encoders, which abstracts the input information from low-level structure to high-level semantics.\nFor each layer, we calculate the score distribution of the input query-passage pairs in a minibatch, which is defined as the \\textbf{layer feature}:\n\\begin{equation}\n\\footnotesize\n{\\bm{t}}_{i} = \\mathop{\\forall}_{{\\bm{p}} \\in {\\mathbb{P}}^+ + {\\mathbb{P}}^-} \\frac{\\exp(f({\\bm{q}}, {\\bm{p}}))}{\\sum_{{\\bm{p}}' \\in {\\mathbb{P}}^+ + {\\mathbb{P}}^-}\\exp(f({\\bm{q}}, {\\bm{p}}'))}\n\\label{equ:studentscorepd}\n\\end{equation}\n\\begin{equation}\n\\footnotesize\n{\\bm{s}}_{i} = \\mathop{\\forall}_{{\\bm{p}} \\in {\\mathbb{P}}^+ + {\\mathbb{P}}^-} \\frac{\\exp(g({\\bm{q}}, {\\bm{p}}))}{\\sum_{{\\bm{p}}' \\in {\\mathbb{P}}^+ + {\\mathbb{P}}^-}\\exp(g({\\bm{q}}, {\\bm{p}}'))}\n\\label{equ:studentscorepd}\n\\end{equation}\nwhere ${\\mathbb{P}}^+$ and ${\\mathbb{P}}^-$ is the relevant and negative passage pool of ${\\bm{q}}$, respectively;\n$f(*),g(*)$ is the similarity score function of the student model and teacher model, respectively;\n$i$ denotes the $i$-th layer.\n\n\n\\paragraph{Layer Selection.}\n\nIn each layer-wise alignment, we need to find a teacher layer for each student layer.\nWithout loss of the generality, we randomly select $K$ layers ($N \\geq M \\geq K$) from teacher model $A = \\{a_1, a_2,..., a_K\\} $ and $K$ layers from student model $B = \\{b_1, b_2,..., b_K\\}$, respectively, keeping their original partial orders.\nThen, we distill knowledge from the $i$-th selected teacher layer $a_i$ to the $i$-th selected student layer $b_i$ by forcing the layer features ${\\bm{t}}_{a_i}$ and ${\\bm{s}}_{b_i}$ to be close:\n\\begin{equation}\n\\footnotesize\n\\mathcal{L}_{\\textrm{lyr}}^{i}=D_{\\mathrm{KL}}({\\bm{t}}_{a_i}\/\\tau, {\\bm{s}}_{b_i}\/\\tau)\n\\label{equ:ctdsoftloss}\n\\end{equation}\nwhere $\\tau$ is the temperature.\n\n\n\\paragraph{Layer Re-weighting.}\n\nAlthough different teacher layers contain different information, some layers may contain noisy information which may be harmful to the model performance.\nThus, we propose a layer re-weighting technique by assigning different weights to involved teacher layers.\nThe weights are calculated based on the teacher layer features and the ground truth label distribution:\n\\begin{equation}\n\\footnotesize\nw_i = \\frac{\\exp(-D_{\\mathrm{KL}}({\\bm{y}}, {\\bm{t}}_{a_i},) \/ \\tau)}{\\sum_{{\\bm{j}} = 1}^K \\exp(-D_{\\mathrm{KL}}({\\bm{y}}, {\\bm{t}}_{a_j}) \/ \\tau)}\n\\end{equation}\nwhere ${\\bm{y}}$ is the ground truth one-hot label.\n\nAt last, the layer distillation loss is the weighted sum of the loss items of all the selected layers.\n\\begin{equation}\n\\footnotesize\n\\mathcal{L}_{\\textrm{lyr}} = \\sum_{i=1}^K w_i \\times \\mathcal{L}_{\\textrm{lyr}}^{i}\n\\label{equ:layer_sum}\n\\end{equation}\n\n\n\\subsection{Joint Training of Teacher and Student}\n\nPrevious distillation methods for dense retrieval \\cite{DBLP:journals\/corr\/abs-2010-02666, lin2020distilling, lin2022prod, DBLP:conf\/sigir\/ZengZV22} usually tune the student with a fixed teacher, while the supervision signals come from the differences in the output label distributions.\nWe think that the pre-learned teacher models that are converged on the training data could further benefit from distillation.\nTherefore, we jointly train the teacher model and the student model, allowing them to learn from each other.\n\nFirst, we adopt the bi-directional response-based loss as:\n\\begin{equation}\n\\footnotesize\n\\mathcal{L}_{\\textrm{rep}}=D_{\\mathrm{KL}}({\\bm{t}}_{N}\/\\tau, {\\bm{s}}_{M}\/\\tau) + D_{\\mathrm{KL}}({\\bm{s}}_{M}\/\\tau, {\\bm{t}}_{N}\/\\tau)\n\\label{equ:ctdsoftloss}\n\\end{equation}\nSecond, we calculate the hard loss items by the output distributions and ground truth as:\n\\begin{equation}\n\\footnotesize\n\\mathcal{L}_{\\textrm{tch}} = -\\sum_{{\\bm{p}}^+ \\in {\\mathbb{P}}^+} \\log\\frac{\\exp(f({\\bm{q}}, {\\bm{p}}^+))}{\\sum_{{\\bm{p}} \\in {\\mathbb{P}}^+ + {\\mathbb{P}}^-} \\exp(f({\\bm{q}}, {\\bm{p}}))}\n\\label{equ:dtdhardloss}\n\\end{equation}\n\\begin{equation}\n\\footnotesize\n\\mathcal{L}_{\\textrm{stu}} = -\\sum_{{\\bm{p}}^+ \\in {\\mathbb{P}}^+} \\log\\frac{\\exp(g({\\bm{q}}, {\\bm{p}}^+))}{\\sum_{{\\bm{p}} \\in {\\mathbb{P}}^+ + {\\mathbb{P}}^-} \\exp(g({\\bm{q}}, {\\bm{p}}))}\n\\label{equ:dtdhardloss}\n\\end{equation}\nAs last, the total loss is the combination of the layer distillation loss, the response-based loss and the hard loss:\n\\begin{equation}\n\\footnotesize\n\\mathcal{L}_{\\textrm{total}} = \\mathcal{L}_{\\textrm{lyr}} + \\mathcal{L}_{\\textrm{rep}} + \\mathcal{L}_{\\textrm{tch}} + \\mathcal{L}_{\\textrm{stu}}\n\\label{equ:layer_sum}\n\\end{equation}\n\n\n\\subsection{Discussion}\n\n\n\\paragraph{Extendibility.}\n\nSince LEAD is based on the unified view of retrieval models presented in \\cref{sec:view}, in which the models are abstracted as piles of neural layers, we think LEAD is extendable to different types of layers.\nFor example, LEAD also supports variants of DE and CB models, such as appending linear layers to the naive output of DE and CB, reducing the output dimensionalities, which supports building a smaller vector-based index for passages.\n\n\n\\paragraph{Portability.}\n\nLEAD is a good alternative to the traditional response-based distillation method, which can also be applied in the state-of-the-art distillation pipelines for dense retrieval, such as curriculum learning \\cite{DBLP:conf\/sigir\/ZengZV22} and learning with multiple teachers \\cite{lin2022prod}.\n\n\\section{Experiments}\n\n\\subsection{Experimental Setting}\n\n\\paragraph{Datasets.}\nExperiments are conducted on several popular retrieval datasets: MS MACRO Passage Ranking (\\textsc{MS-Pas}) \\cite{DBLP:conf\/nips\/NguyenRSGTMD16}, TREC 2019 DL Track (\\textsc{TREC-Pas-19} and \\textsc{TREC-Doc-19}) \\cite{DBLP:journals\/corr\/abs-2003-07820}, MS MARCO Document Ranking (\\textsc{MS-Doc}) \\cite{DBLP:conf\/nips\/NguyenRSGTMD16} and TREC 2020 DL Track (\\textsc{TREC-Pas-20} and \\textsc{TREC-Doc-20}) \\cite{DBLP:journals\/corr\/abs-2102-07662}.\nThe statistics are shown in \\cref{sec:data}.\n\n\n\\paragraph{Metrics.}\nFollowing previous works~\\cite{DBLP:conf\/sigir\/ZhanM0G0M21,lin2022prod}, for \\textsc{MS-Pas},  we report  \\textbf{MRR@10}, \\textbf{MAP@1k} and \\textbf{R@1k} on the dev set.\nFor \\textsc{TREC-Pas-19} and \\textsc{TREC-Pas-20}, we report \\textbf{nDCG@10} and \\textbf{MAP@1k}. \nFor \\textsc{MS-Doc}, we report \\textbf{MRR@10} and \\textbf{R@100} on the dev set.\nFor \\textsc{TREC-Doc-19} and \\textsc{TREC-Doc-20}, we report \\textbf{nDCG@10} and \\textbf{R@100}.\n\n\n\\subsection{Baselines}\n\nTo verify the effectiveness of our methods, we compare LEAD with three groups of methods.\n\nThe first group contains sparse retrieval methods and popular dense retrieval methods without distillation, including \n\\textbf{BM25} \\cite{DBLP:conf\/sigir\/Yang0L17},\n\\textbf{DeepCT} \\cite{DBLP:conf\/sigir\/DaiC19},\n\\textbf{docT5query} \\cite{DBLP:journals\/corr\/abs-1904-08375},\n\\textbf{ANCE} \\cite{DBLP:conf\/iclr\/XiongXLTLBAO21}, \\textbf{ME-BERT} \\cite{luan2021sparse} and\n\\textbf{ADORE} \\cite{DBLP:conf\/sigir\/ZhanM0G0M21}.\n\nThe second group consists of recent dense retrieval methods enhanced with distillation, including \n\\textbf{Margin-MSE} \\cite{DBLP:journals\/corr\/abs-2010-02666},\n\\textbf{TCT-ColBERT} \\cite{DBLP:journals\/corr\/abs-2010-11386}\nand\n\\textbf{RocketQA v1} \\cite{DBLP:conf\/naacl\/QuDLLRZDWW21}.\n\nThe third group contains different knowledge distillation strategies like response-based method (\\textbf{RD}) \\cite{DBLP:conf\/naacl\/QuDLLRZDWW21} and feature-based method (\\textbf{FD}) \\cite{DBLP:journals\/corr\/abs-2112-04195}.\nFurthermore, we use \\textbf{12-layer DE}, \\textbf{12-layer CB} and \\textbf{12-layer CE} as the teacher model and conduct knowledge distillation with a 6-layer DE student.\n\n\n\\subsection{Implementation Details}\n\n\\paragraph{Model Initialization.}\nWe use \\textsc{luyu\/co-condenser-marco} from huggingface\\footnote{\\url{https:\/\/huggingface.co\/}} to initialize the 12-layer teacher models and \\textsc{distilbert-base-uncased} for the 6-layer dual encoder student model.\n\n\n\\paragraph{Training Process.}\nFirst, we use random or BM25 negatives to train a 12-layer DE and retrieve top-100 hard negatives.\nThen, we train the teacher model and student model with the mined hard negatives as warming up.\nFinally, we load the trained checkpoints for the teacher model and student model, conducting the knowledge distillation.\nThe experiments are conducted on 8 NVIDIA V100 GPUs.\nDetailed hyper-parameters for reproducing our experiments are listed in \\cref{sec:hyps}.\n\n\n\n\\paragraph{Metric Calculation.}\nFor DEs, we follow the previous work \\cite{DBLP:conf\/iclr\/XiongXLTLBAO21,DBLP:conf\/naacl\/QuDLLRZDWW21} and compute the metrics using the \\texttt{IndexFlatIP} of faiss\\footnote{\\url{https:\/\/faiss.ai\/}}.\nWhile for CB and CE teacher models, we follow \\citet{DBLP:journals\/corr\/abs-2010-02666,lin2020distilling,lin2022prod} and rerank the top-1000 results and the top-100 results retrieved by the 12-layer DE teacher model for \\textsc{MS-Pas} and \\textsc{MS-Doc}, respectively.\n\n\n\\subsection{Main Results}\n\n\\paragraph{Comparing with existing methods.}\n\nWe compare LEAD with other state-of-the-art text retrieval methods on \\textsc{MS-Pas}, \\textsc{TREC-Pas-19}, \\textsc{MS-Doc} and \\textsc{TREC-Doc-19}.\nWe use ``LEAD'' to denote using 12-layer CB as teacher model.\nThe results are shown in \\cref{tab:main_results_pas1} and \\cref{tab:main_results_doc1}.\n\nFrom the result, we can find that: (1) LEAD achieves the best performance on almost all the metrics on both \\textsc{MS-Pas} and \\textsc{MS-Doc}.\n(2) Among the baselines, knowledge distillation methods include: TCT-ColBERT \\cite{DBLP:journals\/corr\/abs-2010-11386} which uses CB as teacher model, RocketQA v1 \\cite{DBLP:conf\/naacl\/QuDLLRZDWW21} which uses CE as teacher model, and Margin-MSE \\cite{DBLP:journals\/corr\/abs-2010-02666} which uses the ensemble of multi CE as teachers.\nThe best setting of LEAD which uses CB as teacher model performs better than all of them.\n\n\n\n\n\\paragraph{Comparing with different KD strategies.}\n\nTo further compare with other knowledge distillation methods in a fair setting, we conduct experiments using different kinds of teacher models, such as DE, CB and CE.\nThe results on \\textsc{MS-Pas}, \\textsc{TREC-Pas-19}, \\textsc{TREC-Pas-20}, \\textsc{MS-Doc}, \\textsc{TREC-Doc-19} and \\textsc{TREC-Doc-20} are illustrated in \\cref{tab:main_results2}. \n\nAs we can see from the table, it is obvious that we can get the following conclusions:\n(1) Compared to the original 6-layer DE student, LEAD can further achieve improvements with various teacher models, especially for MRR@10, MAP and nDCG@10.\n\n(2) When training with the same teacher model, generally speaking, RD is better than FD in all the comparisons, indicating that FD is not easy to work on distillation for dense retrieval.\nFurthermore, LEAD outperforms other KD strategies, such as FD and RD, on all the datasets and metrics, showing the effectiveness of our method.\n\n(3) Among the explored teacher models, 12-layer CB is the best for boosting a 6-layer DE student when using LEAD.\nWe guess the reason may be two folds.\nFirst, CB and DE share a similar architecture while the late interaction brings CB better performances, which further leads to better distillation results.\nSecond, although the CE teacher outperforms the CB teacher, training with CB in RD and LEAD can benefit from the cross-batch negatives \\cite{DBLP:conf\/naacl\/QuDLLRZDWW21}, which is limited in CE.\nTherefore, we use 12-layer CB as the teacher model for LEAD in the rest experiments.\n\n\n\\subsection{Effect on Layer Selection}\n\\label{sec:layer_selection}\n\nWe explore the effects of two major factors in the layer selection of LEAD.\n\n\\input{tables\/layer_selection1}\n\\paragraph{Study on Layer Number $K$.}\nWe tune the distillation layer number $K$ in the range of $\\{1, 2, 3, 4, 5, 6\\}$ on \\textsc{MS-Pas} and \\textsc{MS-Doc}, and observe the trends of student performances, which is shown in \\cref{tab:layer_selection1}.\nWe can observe that the model performance gradually improves with the increase of $K$.\nHowever, the model performance reaches the peak at $K=5$ on \\textsc{MS-Pas} and $K=6$ on \\textsc{MS-Doc}, respectively.\nThis is because, with more distillation layers, the student model can absorb more knowledge from the teacher model, which benefits model performance, but eventually the best layer number highly depends on the dataset characteristic.\nTherefore, we use $K=5$ for \\textsc{MS-Pas} and $K=6$ for \\textsc{MS-Doc} in later experiments.\n\n\n\\input{tables\/layer_selection2}\n\\paragraph{Study on Selection Strategies.}\nIn LEAD, we randomly select intermediate layers from teacher model and student model for knowledge distillation. We further investigate the performance of different layer selection strategies including Random, Last and Skip in \\citet{sun2019patient,haidar2021rail}. Random refers to randomly selecting intermediate layers for knowledge distillation. Last refers to distilling information from the last layers. And Skip refers to distilling information across every $k$ layers.\nMore precisely, on \\textsc{MS-Pas}, the layer choice for student model is \\{2, 3, 4, 5, 6\\}.\nWe set $k=2$ and the layer choice for teacher model is \\{8, 9, 10, 11, 12\\} in Last and \\{1, 3, 5, 7, 9\\} in Skip.\nIt is worth mentioning that the layer selection choice keeps dynamically changing during training in Random while remaining fixed in Last and Skip. The result of the selection strategies on \\textsc{MS-Pas} is shown in \\cref{tab:layer_selection2}. \n\nFrom the result, we can find that our layer selection strategy Random is better than Last and Skip. Presumably, it might be due to the fact that Last and Skip only distill information from part of intermediate layers, which may lose information from unchosen layers. However, by distilling information from dynamically changing intermediate layers, Random captures more diverse distributions of richer semantics from low-level to high-level.\n\n\n\\subsection{Ablation Study}\n\\input{tables\/ablation}\n\nTo show the effectiveness of each component, we conduct an ablation study on LEAD.\nSince the layer selection has been discussed in \\cref{sec:layer_selection}, we focus on the joint training and the layer re-weighting, whose results are shown in \\cref{tab:ablation}.\n\nFirst, to completely forbid the joint training, we directly remove the teacher's hard loss item $\\mathcal{L}_{\\textrm{tch}}$ and $D_{\\mathrm{KL}}({\\bm{s}}_{M}\/\\tau, {\\bm{t}}_{N}\/\\tau)$ in $\\mathcal{L}_{\\textrm{rep}}$ from the total loss, freezing the teacher model during training.\nIt is clear that after removing the joint training from LEAD, the student's performances on nearly all the metrics decreased.\nThe reason may be that the information of the teacher model and the student model can complement each other.\n\nSecond, we replace the weight $w_i$ with a constant $1\/K$ in \\cref{equ:layer_sum} to show the effects of the layer re-weighting.\nAs we can see, removing the layer re-weighting causes a larger performance drop in \\textsc{MS-Pas} compared with the joint training, but still outperforms response-based methods. \nWe think the reason may be that by assigning higher weights to more informative layers, the noises from uninformative layers can be largely reduced.\n\n\n\\subsection{Further Study}\n\n\\input{tables\/extendability.tex}\n\\paragraph{Extendibility.}\nTo verify the extendibility of LEAD, we append a linear layer 768 $\\rightarrow$ 768 after the student model and teacher model, and observe the performance of RD, FD and LEAD on \\textsc{MS-Pas}, which is shown in \\cref{tab:extend}.\n\nWe can find that after appending the linear layer, the performance of all distillation methods drops. However, our method LEAD+linear still achieves the best performance, which demonstrates the extendibility of our method.\n\n\n\\input{tables\/plug_in.tex}\n\\paragraph{Portability.}\nTo verify the portability of LEAD, we follow PROD~\\cite{lin2022prod} and choose three teacher models for continual distillation, which include DE, CB and CE.\nPROD* refers to using RD in the distillation process, while PROD*+LEAD refers to using LEAD in the distillation process.\nThe results on \\textsc{MS-Pas} are shown in \\cref{tab:plug_in}.\n\nFrom the result, we can find that:\n(1) The performance of student model gradually improves in each step for both PROD* and PROD*+LEAD. However, the performance gain of PROD*+LEAD in each step is better than that of PROD*, which leads to better performance of the final student model. The reason may be that compared with RD, LEAD can fully utilize the information from the intermediate layers of the teacher model.\n(2) Compared with LEAD which solely uses CB as teacher model, the result of PROD*+LEAD after continual distillation gets little improvement. This may be due to the fact that the multi-teachers used in the continual distillation share similar abilities.\nOverall, the experiment results prove the feasibility of plugging LEAD into the existing distillation paradigm and further improving the model performance.\n\n\n\\input{tables\/teacher.tex}\n\\paragraph{Joint training benefits teacher model.}\nWe propose to jointly train the teacher and student with the intuition that they can complement each other.\nTo verify it, we evaluate the performance of the teacher model before and after LEAD on \\textsc{MS-Pas} and \\textsc{MS-Doc}, which is shown in \\cref{tab:teacher}.\n\nWe can find that the performance of teacher model can be greatly improved after LEAD on both datasets, which verifies that teacher model can learn from student model. Besides, as shown in \\cref{tab:ablation}, when we remove joint training from LEAD, the model performance degrades severely. However, the performance of LEAD without joint training is still better than that of RD and FD.\n\n\\section{Conclusion}\nIn this paper, we propose a novel knowledge distillation method LEAD for dense retrieval.\nTo be specific, LEAD aligns the distribution of the student model and teacher model and pays more attention to the most informative layers using the layer re-weighting technique. LEAD is extendable to more linear layers and is easy to plug into the existing distillation pipelines.\nExtensive experiments on six widely-used benchmarks show that LEAD can significantly improve the performance of the student model after distillation compared with response-based methods and feature-based methods.\nWe further carried out some hyper-parameter analysis and ablation experiments to prove the effectiveness of LEAD, which also inspires future research direction on the distillation of dense retrievers.\n\\section*{Acknowledgement}\n\n\n\n\n\n\\section{Hyper-parameters}\n\\label{sec:hyps}\n\n\\input{tables\/hyper}\n\nThe detailed hyper-parameters are shown in \\cref{tab:hyper}.\n\n\\section{Dataset Statictics}\n\\label{sec:data}\n\n\\input{tables\/dataset_statistics}\n\nThe statistics are shown in \\cref{tab:dataset}.\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nVery-high-energy cosmic rays and gamma rays induce extensive air showers (EAS) when entering the atmosphere. Cherenkov and fluorescence light emitted by secondary charged particles is used as a proxy for studying the primary particles that initiate the particle cascades.\n\nDesign, calibration and data analysis of cosmic-ray and gamma-ray observatories strongly rely on Monte Carlo simulations of both the air shower and detector response. \\texttt{CORSIKA} program \\citep{corsika} is widely used for carrying out the first step of the simulation, whereas the second step depends on the detection technique. For example, in the case of imaging atmospheric Cherenkov telescopes (IACT), the program \\texttt{sim\\_telarray} \\citep{Bernlohr2008} is commonly used. These detailed simulations are currently very demanding computationally.\n\nWe present a fast python package called \\texttt{ShowerModel} to compute the light emission in air showers and its detection by an array of telescopes \\citep{ShowerModel}. This tool can speed up calculations that do not require a full simulation or that may serve to complement complex Monte Carlo studies and data analyses (e.g., as a cross-check). It can also be used for educational purposes. A similar approach was presented previously in \\citet{Vuillaume2017}.\n\nFunctionalities and examples of results obtained with \\texttt{ShowerModel} are shown. Several possible remarkable applications of this software are also briefly discussed.\n\n\\section{ShowerModel}\nThe package comprised several functions and classes to model air-shower development and detection (see Figure \\ref{block_diagram}). Both gamma and proton-like 1D air showers can be generated using analytical Greisen or Gaisser-Hillas formulas \\citep{Greisen1956, GaisserHillas1977}. Simulation-generated longitudinal profiles can also be input. Different flat atmospheric models are available.\n\n\\articlefigure[width=.8\\textwidth]{P4-176_f1.eps}{block_diagram}{Class dependency of \\texttt{ShowerModel}.}\n\nThe code uses detailed parameterizations of both fluorescence and Cherenkov light emission and angular distribution \\citep{Morcuende2019, Nerling2006}. The time-varying light intensity reaching a telescope is readily computed from geometry. Rayleigh scattering losses are also included.\n\nTelescope objects are highly configurable (e.g., quantum efficiency, pointing direction, field of view, detection area) to model fluorescence telescopes, imaging air Cherenkov telescopes, wide-angle Cherenkov detectors or hybrid designs. Camera images are produced using the Nishimura-Kamata-Greisen (NKG) function \\citep{Nishimura1958, Greisen1956} to describe the lateral distribution of electrons in the air shower.\n\n\n\\section{Functionalities}\n\n\\texttt{ShowerModel} permits the calculation of:\n\n\\begin{itemize}\n    \\item Projection of shower tracks in local coordinates \\texttt{alt}\/\\texttt{az} as well as the telescope field of view coordinates \\texttt{theta}\/\\texttt{phi} (Figure \\ref{fluorescence_telescope}, left).\n    \n    \\articlefigure[width=.8\\textwidth]{P4-176_f2.eps}{fluorescence_telescope}{\\emph{Left:} Shower track projections of an air shower observed by a fluorescence telescope.  \\emph{Right:} Signal produced from the two light components.}\n    \n    \\item Longitudinal shower profiles: energy deposit and light production.\n    \n    \\item Cherenkov and fluorescence photon densities on ground (Figure \\ref{2d_distr}).\n    \n    \\articlefigure[width=0.8\\textwidth]{P4-176_f3.eps}{2d_distr}{Cherenkov and fluorescence photon densities at ground level.}\n    \n    \\item Time evolution of signal in a telescope (Figure \\ref{fluorescence_telescope}, right).\n    \n    \\item Shower events detected by an array of telescopes (Figure \\ref{obs_event}).\n    \n    \\articlefigure[width=.4\\textwidth]{P4-176_f4.eps}{obs_event}{Air shower observed by an observatory of 6 telescopes. Green dots mark the telescope positions and their relative signal sizes.}\n    \n    \\item Camera images in customized telescopes (Figure \\ref{camera_images}).\n\n    \\articlefigure[width=.8\\textwidth]{P4-176_f5.eps}{camera_images}{Camera images of an air shower observed by an array of six IACTs.}\n    \n\\end{itemize}\n\n\n\\section{Possible remarkable use cases}\n\nThe proposed tool can serve as a complement of the analysis pipeline of cosmic-ray observatories. For instance, it might be used to fast evaluate if changes in the observation conditions (e.g., atmospheric parameters and night sky background) make necessary new simulations or instrument response functions. In addition, it may be useful to explore new observatory configurations or detection techniques that exploit both Cherenkov and fluorescence signals (see \\citet{ICRC2015} and \\citet{Thesis_Sailer2020}).\n\n\\acknowledgements We gratefully acknowledge support from Spanish MINECO (contract FPA2017-82729-C6-3-R) and the European Commission (E.U. Grant Agreement 653477). D.~Morcuende acknowledges a predoctoral grant UCM-Harvard University (CT17\/17-CT18\/17) from Universidad Complutense de Madrid.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\n\\subsection{Cohen-Lenstra-Martinet and Malle} \\label{CLMM}\n\nIn~\\cite{CL}, Cohen and Lenstra presented their famous heuristic principle concerning the distribution of ideal class groups of quadratic number fields.\n\\begin{clheur}[Cohen and Lenstra, 1984] \\label{clheur}\nFor any odd prime $\\ell$, a finite abelian $\\ell$-group should appear as the $\\ell$-Sylow subgroup of the ideal class group of an imaginary quadratic extension of $\\ensuremath{{\\mathbb{Q}}}$ with frequency inversely proportional to the order of its automorphism group.\n\\end{clheur}\n\\noindent With a bit more notation, we can reframe this heuristic.\nLet $\\mathcal{G}$ be the poset of isomorphism classes of finite abelian $\\ell$-groups and for any number field $K$, let $\\cl{(K)}$ denote the ideal class group of $K$.\nFor any group $G$, let $G\\left[\\ell^\\infty\\right]$ denote its $\\ell$-Sylow subgroup.\nNow, since $\\sum_{A\\in\\mathcal{G}}{1\/\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}=\\prod_{i=1}^{\\infty}{(1-\\ell^{-i})^{-1}}$ (a fact first proved by Hall in~\\cite{Hall}), the map from $\\mathcal{G}\\to\\ensuremath{{\\mathbb{R}}}$ given by $A\\mapsto\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}^{-1}\\prod_{i=1}^{\\infty}{\\lp1-\\ell^{-i}\\ensuremath{\\right)}}$ defines a discrete probability distribution on $\\mathcal{G}$.\n\\hyperref[clheur]{Heuristic~\\ref*{clheur}} is the claim that the statistics of this distribution match the statistics of $\\ell$-Sylow subgroups of imaginary quadratic extensions (when ordered by fundamental discriminant).\nIn other words, \\hyperref[clheur]{Heuristic~\\ref*{clheur}} is equivalent to the following assertion: for any $A\\in\\mathcal{G}$,\n\\[\n\\lim_{X\\to\\infty}\n{\\frac{\\ensuremath{\\left\\vert}\\lb0\\leq D\\leq X\n\\,\\,\\Big\\vert\\,\\,\\substack{-D\\text{ a fundamental discriminant}\\\\\n\\cl{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Q}}}\\ensuremath{\\left(}\\sqrt{-D}\\ensuremath{\\right)}\\rp}\\left[\\ell^\\infty\\right]\\simeq A}\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}}\n{\\ensuremath{\\left\\vert}\\lb0\\leq D\\leq X\n\\mid\\substack{-D\\text{ a fundamental discriminant}}\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}}}\n=\\frac{1}{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}\\prod_{i=1}^{\\infty}{\\lp1-\\ell^{-i}\\ensuremath{\\right)}}.\n\\]\n(We remark that this assertion remains unproven; in fact, the above limit is not even known to exist.)\nThis heuristic explains many previously observed tendencies of class groups of imaginary quadratic fields, such as: their orders should be divisible by three with probability\n\\[\n1-\\prod_{i=1}^{\\infty}{\\lp1-3^{-i}\\ensuremath{\\right)}}\n=\\frac{1}{3}+\\frac{1}{9}+\\cdots\n\\approx.44.\n\\]\n\nIn 1990, Cohen and Martinet~\\cite{CM} extended their heuristics to include relative class groups of finite extensions of arbitrary number fields, placing different distributions on $\\mathcal{G}$ depending on properties of the family of extensions they study.\nOnce again, they proved that these distributions imply many numerical observations, thereby obtaining a new family of conjectures.\n(Recall that relative ideal class groups are defined as follows: if $K\/K_0$ is an extension of number fields, the relative class group $\\cl{\\ensuremath{\\left(} K\/K_0\\ensuremath{\\right)}}$ is the kernel of the norm map $\\text{N}_{K\/K_0}:\\cl{(K)}\\to\\cl{\\ensuremath{\\left(} K_0\\ensuremath{\\right)}}$.)\n\nRecently, however, Malle~\\cite{Mal0} presented new computational data that called into question some of the Cohen-Lenstra-Martinet conjectures.\nFor example, he studied the 3-parts of the relative class groups of quadratic extensions of $\\ensuremath{{\\mathbb{Q}}}\\ensuremath{\\left(}\\sqrt{-3}\\ensuremath{\\right)}$, which has third roots of unity.\nCohen-Lenstra-Martinet predicted that the class numbers of such extensions should be coprime to 3 with probability\n\\[\n\\prod_{i=2}^\\infty{\\lp1-3^{-i}\\ensuremath{\\right)}}\\approx.840.\n\\]\nHowever, when Malle computed the class numbers of the first million of these extensions with discriminant at least $10^{20}$, he discovered that the proportion of them with class number coprime to 3 was about .852.\nHe conjectured that the proportion of all such class groups that have class number coprime to 3 should be exactly\n\\[\n\\frac{4}{3}\\prod_{i=1}^\\infty{\\lp1+3^{-i}\\ensuremath{\\right)}^{-1}}\\approx.852,\n\\]\nwhich is in much better agreement with his data.\nTwo years later, in~\\cite{Mal}, Malle presented more computational evidence calling into question more Cohen-Lenstra-Martinet conjectures, once again when there are $\\ell$th roots of unity in the base field.\nIn this paper, he also presented a new family of distributions on $\\mathcal{G}$ to describe relative class groups when the base field of the extension has $\\ell$th roots of unity but not $\\ell^2$th roots of unity (see Conjecture 2.1 in~\\cite{Mal}).\nThese distributions on $\\mathcal{G}$ imply rank statistics that seem to be a much better fit for his new data.\nA special case of his conjecture is the following:\n\n\\begin{mallec}[Malle, 2010] \\label{mallec}\nSuppose that $A\\in\\mathcal{G}$ and that $A$ has $\\ell$-rank $r$.\nLet $K_0$ be a number field with $\\ell$th but not $\\ell^2$th roots of unity.\nLet $\\mathcal{S}$ be the set of quadratic extensions $K\/K_0$ with a fixed signature (with fixed relative unit rank $u$).\nThen\n\\[\n\\lim_{X\\to\\infty}\n{\\frac{\\ensuremath{\\left\\vert}\\ensuremath{\\left\\{} K\\in\\mathcal{S}\n\\,\\,\\big\\vert\\,\\,\\substack{\\ensuremath{\\left\\vert}\\Disc{K}\\ensuremath{\\right\\vert}\\leq X,\\\\\n\\cl{\\ensuremath{\\left(} K\/K_0\\ensuremath{\\right)}}\\left[\\ell^{\\infty}\\right]\\simeq A}\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}}\n{\\ensuremath{\\left\\vert}\\ensuremath{\\left\\{} K\\in\\mathcal{S}\n\\mid\\substack{\\ensuremath{\\left\\vert}\\Disc{K}\\ensuremath{\\right\\vert}\\leq X}\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}}}\n=\\frac{\\prod_{i=u+1}^{u+r}{\\ensuremath{\\left(}\\ell^i-1\\ensuremath{\\right)}}}\n{\\ell^{r(u+1)}\\ensuremath{\\left\\vert} A\\ensuremath{\\right\\vert}^u\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}\n\\cdot\\prod_{i=u+1}^{\\infty}{\\lp1+\\ell^{-i}\\ensuremath{\\right)}^{-1}}.\n\\]\n\\end{mallec}\n\nIn this paper, we study a random matrix model of ideal class groups of function fields when the base field has $\\ell$th roots of unity (i.e., the function field analog of the situation Malle studies in \\hyperref[mallec]{Conjecture~\\ref*{mallec}} ).\nWe compute the distributions on $\\mathcal{G}$ given by this matrix model in two cases (see \\hyperref[final]{Theorem~\\ref*{final}}):\nin the case when base field has $\\ell$th roots of unity but not $\\ell^2$th roots of unity and in the case when the base field with $\\ell^2$th roots of unity but not $\\ell^3$th roots of unity.\nIn the former case, our distribution matches the distribution proposed by Malle.\nMoreover, we compute all the moments of the distribution given by this matrix model in the general case when the base case has $\\ell^\\xi$th but not $\\ell^{\\xi+1}$th roots of unity for any $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ (see~\\hyperref[moments]{Corollary~\\ref*{moments}}).\n\n\nThe work in this paper is based on my 2012 PhD dissertation~\\cite{GD}.\nThe matrix distributions were computed independently in the 2014 PhD dissertation of M. Adam~\\cite{AdamD} as well as in his 2014 paper~\\cite{AdamP}.\nThey are also used in a recent paper of Adam and Malle~\\cite{AM}.\n\n\n\\subsection{The function field case} \\label{FFC}\n\nComplementing the work described in \\hyperref[CLMM]{Section~\\ref{CLMM}}, investigators have been studying analogous phenomena in function fields defined over finite fields.\nIn 1989, Friedman and Washington (in~\\cite{FW}) addressed the case of quadratic extensions of the field $\\ensuremath{{\\mathbb{F}}}_{p^n}(t)$ for a prime $p\\neq2$ and $n\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$.\nMore precisely, if $f(t)\\in\\ensuremath{{\\mathbb{F}}}_{p^n}[t]$ is monic with distinct roots of degree $2g+1$, let $C_f$ be the hyperelliptic curve (defined over $\\ensuremath{{\\mathbb{F}}}_{p^n}$) of genus $g$ given by $y^2=f(t)$.\nNote that the curve $C_f$ has exactly one point at infinity, just as imaginary quadratic extensions of $\\ensuremath{{\\mathbb{Q}}}$ have exactly one place at infinity.\nThus, $\\Pic^0_{\\ensuremath{{\\mathbb{F}}}_{p^n}}{\\ensuremath{\\left(} C_f\\ensuremath{\\right)}}$ is isomorphic to the ideal class group of the field extension $\\mathbb{F}_{p^n}(t)[\\sqrt{f(t)}]\/\\mathbb{F}_{p^n}(t)$.\n\nTo study these groups, Friedman and Washington introduced a new heuristic principle, one that comes from the geometry of hyperelliptic curves over finite fields.\nSpecifically, for $f(t)\\in\\ensuremath{{\\mathbb{F}}}_{p^n}[t]$ monic with distinct roots of degree $2g+1$, let $T_{\\ell}\\ensuremath{\\left(} C_f\\ensuremath{\\right)}$ be the $\\ell$-adic Tate module of $C_f$, which is a free $2g$-dimensional $\\ensuremath{{\\mathbb{Z}}}_\\ell$-module.\nIn addition, let $\\Frob_{p^n}$ be the $p^n$-power Frobenius map acting on $T_\\ell\\ensuremath{\\left(} C_f\\ensuremath{\\right)}$.\nThinking of $\\Frob_{p^n}$ as a matrix over $\\ensuremath{{\\mathbb{Z}}}_\\ell$, it is well-known that $\\coker{\\ensuremath{\\left(}\\Id-\\Frob_{p^n}\\ensuremath{\\right)}}$ is isomorphic to the $\\ell$-Sylow subgroup of $\\Pic^0_{\\ensuremath{{\\mathbb{F}}}_{p^n}}{\\ensuremath{\\left(} C_f\\ensuremath{\\right)}}$ (see the appendix of~\\cite{FW} for a proof of this fact).\nFriedman and Washington conjectured that the statistics of $\\ell$-Sylow subgroups of ideal class groups of quadratic extensions of $\\ensuremath{{\\mathbb{F}}}_{p^n}(t)$ match the statistics of $\\ell$-adic matrices.\nSpecifically, if we let\n\\[\nF(g,p^n,\\ell,A)\n:=\\frac{\\ensuremath{\\left\\vert}\\ensuremath{\\left\\{} f(t)\\in\\ensuremath{{\\mathbb{F}}}_{p^n}[t]\n\\,\\Big\\vert\\,\\substack{f\\text{ monic with distinct roots}\\\\\n\\deg{f}=2g+1,\n\\,\\Pic^0_{\\ensuremath{{\\mathbb{F}}}_{p^n}}{\\ensuremath{\\left(} C_f\\ensuremath{\\right)}}[\\ell^{\\infty}]\\simeq A}\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}}\n{\\ensuremath{\\left\\vert}\\ensuremath{\\left\\{} f(t)\\in\\ensuremath{{\\mathbb{F}}}_{p^n}[t]\n\\,\\Big\\vert\\,\\substack{f\\text{ monic with distinct roots}\\\\\n\\deg{f}=2g+1}\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}},\n\\]\nthen they proposed the following:\n\\begin{FWheur}[Friedman and Washington, 1989] \\label{FWheur}\nIf $A\\in\\mathcal{G}$, then\n\\[\n\\lim_{g\\to\\infty}\n{F(g,p^n,\\ell,A)}\n=\\lim_{g\\to\\infty}\n{\\alpha_{2g}\\ensuremath{\\left(}\\ensuremath{\\left\\{}\\phi\\in\\Mat_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}}\n\\mid\\coker{(\\Id-\\phi)}\\simeq A\\ensuremath{\\right\\}}\\ensuremath{\\right)}},\n\\]\nwhere $\\alpha_{2g}$ is the normalized Haar measure on $\\Mat_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}}$.\n\\end{FWheur}\n\\noindent (See Sections~\\ref{notdef} and~\\ref{THM} for more details on Haar measures.)\nA few years later, Katz and Sarnak~\\cite{KS} vastly extended the philosophy of considering the action of Frobenius as a random matrix, especially when the size of the base field is large.\nFriedman and Washington show that the limit on the right hand side of~\\hyperref[FWheur]{Heuristic~\\ref*{FWheur}} exists, and that it defines exactly the same distribution on $\\mathcal{G}$ as Cohen-Lenstra's original heuristic for imaginary quadratic extensions of \\ensuremath{{\\mathbb{Q}}}.\nHowever, just as the work of Malle calls into question the appropriateness of certain Cohen-Lenstra-Martinet distributions, it also calls into question the appropriateness of Friedman-Washington's proposed distribution.\nIndeed, note the Friedman-Washington heuristic does not depend at all on the presence of $\\ell$th roots of unity in the base field $\\ensuremath{{\\mathbb{F}}}_{p^n}(t)$, while Malle's work suggests that distributions modeling $\\ell$-Sylow subgroups of class groups ought to change in the presence presence of $\\ell$th roots of unity.\nThus, the new data of Malle suggests that \\hyperref[FWheur]{Heuristic~\\ref*{FWheur}} might be flawed when $\\ensuremath{{\\mathbb{F}}}_{p^n}(t)$ has $\\ell$th roots of unity.\n\n\n\n\n\n\n\nA possible explanation for this flaw is that $\\Frob_{p^n}$ is a symplectic similitude with respect to the Weil pairing on $T_\\ell(C_f)$.\nIndeed, it scales the Weil pairing by $p^n$, so when considered as a matrix, $\\Frob_{p^n}\\in\\GSp_{2g}^{(p^n)}{(\\ensuremath{{\\mathbb{Z}}}_\\ell)}$.\n(See \\hyperref[notdef]{Section~\\ref*{notdef}} for more details on this notation.)\nSince the presence of $\\ell$th roots of unity in $\\ensuremath{{\\mathbb{F}}}_{p^n}(t)$ depends on the congruence class of $p^n\\pmod{\\ell}$, the set of symplectic similitudes that scale the Weil pairing by $p^n$ does indeed change change when $\\ensuremath{{\\mathbb{F}}}_{p^n}(t)$ has $\\ell$th roots of unity.\nThese facts led Friedman and Washington (and Achter~\\cite{A2}) to suggest\n\\begin{Aheur} \\label{Aheur}\nIf $A\\in\\mathcal{G}$, then\n\\[\n\\lim_{g\\to\\infty}\n{F(g,p^n,\\ell,A)}\n=\\lim_{g\\to\\infty}\n{\\mu_{2g}^{(p^n)}\\ensuremath{\\left(}\\ensuremath{\\left\\{}\\phi\\in\\GSp^{(p^n)}_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}}\n\\mid\\coker{(\\Id-\\phi)}\\simeq A\\ensuremath{\\right\\}}\\ensuremath{\\right)}},\n\\]\nwhere $\\mu_{2g}^{(p^n)}$ is the unique normalized multiplicative Haar measure on $\\Sp_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}}$ translated to $\\GSp_{2g}^{(p^n)}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}}$.\n\\end{Aheur}\n\\noindent(Again, see Sections~\\ref{notdef} and~\\ref{THM} for more details on Haar measures.)\nFriedman and Washington hoped that this new heuristic would turn out to describe the same distribution as \\hyperref[FWheur]{Heuristic~\\ref*{FWheur}}, but Achter~\\cite{A1} proved that\n\\begin{align*}\n\\lim_{g\\to\\infty}\n&{\\mu_{2g}^{(p^n)}\\ensuremath{\\left(}\\ensuremath{\\left\\{}\\phi\\in\\GSp^{(1)}_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}}\n\\mid\\coker{(\\Id-\\phi)}\\simeq\\{0\\}\\ensuremath{\\right\\}}\\ensuremath{\\right)}}\\\\\n&\\hspace{110px}\\neq\\lim_{g\\to\\infty}\n{\\alpha_{2g}\\ensuremath{\\left(}\\ensuremath{\\left\\{}\\phi\\in\\Mat_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}}\n\\mid\\coker{(\\Id-\\phi)}\\simeq\\{0\\}\\ensuremath{\\right\\}}\\ensuremath{\\right)}},\n\\end{align*}\nrevealing that this was not the case, providing an early indication of the importance of the  presence of $\\ell$th roots of unity in the base field.\nIn~\\cite{A2}, Achter uses work of Katz-Sarnak~\\cite{KS} to prove a revised version of~\\hyperref[FWheur]{Heuristic~\\ref*{FWheur}}:\n\\begin{Athm}[Achter, 2008] \\label{Athm}\nIf $A\\in\\mathcal{G}$, then\n\\[\n\\lim_{p^n\\to\\infty}\n{\\ensuremath{\\left\\vert} F(g,p^n,\\ell,A)\n-\\mu_{2g}^{(p^n)}\\ensuremath{\\left(}\\ensuremath{\\left\\{}\\phi\\in\\GSp^{(p^n)}_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}}\n\\mid\\coker{(\\Id-\\phi)}\\simeq A\\ensuremath{\\right\\}}\\ensuremath{\\right)}\\ensuremath{\\right\\vert}}\n=0.\n\\]\n\\end{Athm}\n\\noindent We remark that this limit in \\hyperref[Athm]{Theorem~\\ref*{Athm}} leaves $g$ fixed while letting $p^n$ increase, whereas the limit in \\hyperref[Aheur]{Heuristic~\\ref*{Aheur}} does the opposite.\n\nThe work of Ellenberg, Venkatesh, and Westerland~\\cite{EVW} uses the topology of Hurwitz spaces to study \\hyperref[Aheur]{Heuristic~\\ref*{Aheur}}.\nOne consequence of their work is that\n\\[\n\\lim_{g\\to\\infty}\n{\\lim_{\\substack{p^n\\to\\infty\\\\\np^n\\nequiv1\\imod{\\ell}}}\n{F(g,p^n,\\ell,A)}}\n=\\frac{1}{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}\\prod_{i=1}^{\\infty}{\\lp1-\\ell^{-i}\\ensuremath{\\right)}}.\n\\]\nSince $p^n\\equiv1\\pmod{\\ell}$ exactly when $\\ensuremath{{\\mathbb{F}}}_{p^n}(t)$ has $\\ell$th roots of unity, this result only addresses the case when the base field does not have $\\ell$th roots of unity (and only when $p^n\\to\\infty$).\nThe remaining case is when $p^n\\equiv1\\pmod{\\ell}$; that is, the case where there are $\\ell$th roots of unity in the base field.\n\\hyperref[mallec]{Conjecture~\\ref*{mallec}} suggests that a different distribution is needed to describe this case.\nIn fact, \\hyperref[finalc]{Corollary~\\ref*{finalc}} gives such a distribution.\nUsing Achter's result, \\hyperref[Athm]{Theorem~\\ref*{Athm}}, \\hyperref[finalc]{Corollary~\\ref*{finalc}} implies the following theorem:\n\n\\begin{prettyfinal} \\label{prettyfinal}\nIf $A$ is a finite abelian $\\ell$-group with $\\ell$-rank $r$ and $\\ell^2$-rank $s$, then\n\\begin{align*}\n\\lim_{g\\to\\infty}&{\n\\lim_{\\substack{p^n\\to\\infty\\\\\np\\nequiv1\\imod{\\ell^\\xi},\\\\\np\\nequiv1\\imod{\\ell^{\\xi+1}}}}\n{F(g,p^n,\\ell,A)}}\\\\\n&\\phantom{======}=\\begin{cases}\n\\ell^{\\frac{r(r-1)}{2}}\\cdot\\ensuremath{\\left(}\\ell^{-1};\\ell^{-1}\\ensuremath{\\right)}_r\n\\cdot\\frac{\\prod_{i=1}^{\\infty}{(1+\\ell^{-i})^{-1}}}{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}^{-1}},\n&\\text{if }\\xi=1\\\\\n\\ell^{\\frac{r(r-1)}{2}+\\frac{s(s-1)}{2}}\n\\cdot\\ensuremath{\\left(}\\ell^{-1};\\ell^{-1}\\ensuremath{\\right)}_s\n\\cdot\\ensuremath{\\left(}\\ell^{-1};\\ell^{-2}\\ensuremath{\\right)}_{\\lceil \\frac{r-s}{2}\\rceil}\n\\cdot\\frac{\\prod_{i=1}^{\\infty}{(1+\\ell^{-i})^{-1}}}{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}^{-1}},\n&\\text{if }\\xi=2,\n\\end{cases}\n\\end{align*}\nwhere $\\ensuremath{\\left(}\\ell^{-1};\\ell^{-j}\\ensuremath{\\right)}_k$ is the $\\ell^{-j}$-Pochhammer symbol, defined for any $j\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ and $k\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$ (see \\hyperref[Not6]{Notation~\\ref*{Not6}}).\n\\end{prettyfinal}\n\n\\noindent \\hyperref[prettyfinal]{Theorem~\\ref*{prettyfinal}} extends \\hyperref[mallec]{Conjecture~\\ref*{mallec}} by including the case where $\\ensuremath{{\\mathbb{F}}}_{p^n}(t)$ has $\\ell^2$th roots of unity but not $\\ell^3$th roots of unity.\nWe remark that since imaginary hyperelliptic curves have only one place at infinity, the function field version of \\hyperref[mallec]{Conjecture~\\ref*{mallec}} should set $u=0$; making this substitution in \\hyperref[mallec]{Conjecture~\\ref*{mallec}} yields the $\\xi=1$ case of \\hyperref[prettyfinal]{Theorem~\\ref*{prettyfinal}}.\n\n\n\\section{Prelinimaries}\n\n\\subsection{Notation and definitions} \\label{notdef}\n\nAs above, let $\\ell$ be an odd prime and let $\\mathcal{G}$ be the poset of isomorphism classes of finite abelian $\\ell$-groups, with the relation $\\left[A\\right]\\leq\\left[B\\right]$ if and only if there exists an injection $A\\hookrightarrow B$.\n(For notational simplicity, we will conflate finite abelian $\\ell$-groups and the equivalence classes containing them.)\nFor any $A\\in\\mathcal{G}$, we denote the exponent of $A$ by $\\exp{A}$.\nIf $i\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, let\n\\[\n\\rank_{\\ell^i}{A}:=\\dim_{\\ensuremath{{\\mathbb{F}}}_\\ell}{\\ensuremath{\\left(}\\ell^{i-1}A\/\\ell^iA\\ensuremath{\\right)}}.\n\\]\nWe will abbreviate $\\rank_\\ell{A}$ by $\\rank{A}$.\nIf $r_1,\\ldots,r_{i-1}\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ and $r_i\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$, let $\\mathcal{G}(r_1,\\ldots,r_i)$ be the following subposet of $\\mathcal{G}$:\n\\[\n\\mathcal{G}(r_1,\\ldots,r_i)\n:=\\ensuremath{\\left\\{} A\\in\\mathcal{G}\\mid\\rank_{\\ell^j}{A}=r_j\\text{ for all }j\\in\\lb1,\\ldots,i\\ensuremath{\\right\\}}\\rb.\n\\]\nNext, for any $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, set $R_\\rho=\\ensuremath{{\\mathbb{Z}}}_\\ell\/\\ell^\\rho\\ensuremath{{\\mathbb{Z}}}_\\ell\\simeq\\ensuremath{{\\mathbb{Z}}}\/\\ell^\\rho\\ensuremath{{\\mathbb{Z}}}$.\nFor any $g,\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, let $\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}_{2g,\\rho}$ be a fixed choice of symplectic form on $(R_\\rho)^{2g}$; that is, $\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}_{2g,a}:(R_\\rho)^{2g}\\times(R_\\rho)^{2g}\\to(R_\\rho)$ is $R_\\rho$-bilinear, alternating, and nondegenerate.\nBy nondegenerate, we mean that the matrix associated to $\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}_{2g,\\rho}$ is invertible; see Theorem~III.2 of~\\cite{McD} for more details on symplectic spaces.\nSimilarly, let $\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}_{2g}$ be a choice of symplectic form on $\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}^{2g}$.\nFor any ring $R$ and any $g\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, if $R^{2g}$ has a symplectic form $\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}$, then the \\emph{symplectic group of $R$} is\n\\[\n\\Sp_{2g}{(R)}\n\\simeq\\Sp{\\ensuremath{\\left(} R^{2g},\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}\\ensuremath{\\right)}}\\\\\n=\\ensuremath{\\left\\{}\\phi\\in\\GL{\\ensuremath{\\left(} R^{2g}\\ensuremath{\\right)}}\n\\mid\\ensuremath{\\left\\langle} \\phi(x),\\phi(y)\\ensuremath{\\right\\rangle}=\\ensuremath{\\left\\langle} x,y\\ensuremath{\\right\\rangle}\\text{ for all }x,y\\in R^{2g}\\ensuremath{\\right\\}}.\n\\]\nNote that a different choice of symplectic form on $R^{2g}$ yields an isotropic space, so the choice is immaterial (see page 188 of~\\cite{McD} for more details).\nSimilarly, the \\emph{group of symplectic similitudes of $R$} is\n\\[\n\\GSp_{2g}{(R)}\n\\simeq\\GSp{\\ensuremath{\\left(} R^{2g},\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}\\ensuremath{\\right)}}\n=\\ensuremath{\\left\\{}\\phi\\in\\GL{\\ensuremath{\\left(} R^{2g}\\ensuremath{\\right)}}\n\\,\\Big\\vert\\,\\substack{\\text{there exists }m(\\phi)\\in R^{\\times}\\text{ such that}\\\\\n\\ensuremath{\\left\\langle}\\phi(x),\\phi(y)\\ensuremath{\\right\\rangle}=m(\\phi)\\cdot\\ensuremath{\\left\\langle} x,y\\ensuremath{\\right\\rangle}\n\\text{ for all }x,y\\in R^{2g}}\\ensuremath{\\right\\}}.\n\\]\nFor concreteness, we will always assume that the rings $(R_\\rho)^{2g}$ and $(\\ensuremath{{\\mathbb{Z}}}_\\ell)^{2g}$ are equipped with the forms $\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}_{2g,\\rho}$ and $\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}_{2g}$ fixed above.\nThe map $m:\\GSp_{2g}{\\ensuremath{\\left(} R\\ensuremath{\\right)}}\\to R^{\\times}:\\phi\\mapsto m(\\phi)$ given above is a homomorphism called the \\emph{multiplier map}, and the element $m(\\phi)\\in R^{\\times}$ is called the \\emph{multiplier} of~$\\phi$.\nFor any $g\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, let $\\mu_{2g}$ be the unique normalized Haar measure on $\\Sp_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$, noting that this measure is invariant under both left and right multiplication since $\\Sp_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$ is a unimodular group.\nFinally, for any $g\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ and any unit $x$ in a ring $R$, let $\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} R\\ensuremath{\\right)}}=m^{-1}(x)$.\n\nNow, for any $x\\in\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}^{\\times}$ and $\\phi\\in\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} \\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$ we define a measure $\\mu_{2g}^{(x)}$ on $\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} \\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$ as follows: for any $\\mu_{2g}$-measurable subset $S\\subseteq\\Sp_{2g}{\\ensuremath{\\left(} \\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$, define\n\\[\n\\mu_{2g}^{(x)}\\ensuremath{\\left(} S\\phi\\ensuremath{\\right)}:=\\mu_{2g}\\ensuremath{\\left(} S\\ensuremath{\\right)}.\n\\]\nWe remark that this measure is independent of the choice $\\phi\\in\\GSp_{2g}^{(x)}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$.\nIndeed, given some other $\\psi\\in\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} \\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$ there exists a unique $\\phi_\\psi\\in\\Sp_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$ such that $\\phi_\\psi\\phi=\\psi$; ie, $S\\psi=S\\phi_\\psi\\phi$.\nSince $\\mu_{2g}$ is translation invariant, we know that\n\\[\n\\mu_{2g}\\ensuremath{\\left(} S\\ensuremath{\\right)}=\\mu_{2g}\\ensuremath{\\left(} S\\phi_\\psi\\ensuremath{\\right)},\n\\]\nas desired.\nMoreover, since $\\mu_{2g}$ is translation invariant (by $\\Sp_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$) and normalized, so is $\\mu_{2g}^{(x)}$.\nSimilarly, for any $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, let $\\nu_{2g,\\rho}$ be the unique normalized Haar measure on $\\Sp_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}$, and for any $x\\in R_\\rho^{\\times}$, define $\\nu_{2g,\\rho}^{(x)}$ on $\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}$ as above.\nFor any $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, $x\\in R_\\rho^{\\times}$, and $S\\subseteq\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}$, we know $\\nu_{2g,\\rho}^{(x)}(S)=\\ensuremath{\\left\\vert} S\\ensuremath{\\right\\vert}\\cdot\\ensuremath{\\left\\vert}\\Sp_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}^{-1}$, since $\\Sp_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}$ is a finite group.\nTo ease notation, for any $A\\in\\mathcal{G}$, $g\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, and $x\\in\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}^{\\times}$, we set\n\\[\n\\mu_{2g}^{(x)}\\ensuremath{\\left(} A\\ensuremath{\\right)}\n:=\\mu_{2g}^{(x)}\\ensuremath{\\left(}\\ensuremath{\\left\\{}\\phi\\in\\GSp_{2g}^{(x)}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}\n\\,\\,\\big\\vert\\,\\,\\coker{\\ensuremath{\\left(}\\Id-\\phi\\ensuremath{\\right)}}\\simeq A\\ensuremath{\\right\\}}\\ensuremath{\\right)}.\n\\]\nFurthermore, if $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ and $x\\in R_\\rho^{\\times}$, set\n\\[\n\\nu_{2g,\\rho}^{(x)}\\ensuremath{\\left(} A\\ensuremath{\\right)}\n:=\\nu_{2g,\\rho}^{(x)}\\ensuremath{\\left(}\\ensuremath{\\left\\{}\\gamma\\in\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}\n\\,\\,\\Big\\vert\\,\\,\\coker{\\ensuremath{\\left(}\\Id-\\gamma\\ensuremath{\\right)}}\\simeq A}\\ensuremath{\\right\\}}\\ensuremath{\\right)}.\n\\]\n\n\\subsection{The Haar measures} \\label{THM}\n\nThe measures defined in \\hyperref[notdef]{Section~\\ref*{notdef}} have an important relationship, given in the following lemma.\n\n\\begin{haar} \\label{haar}\nSuppose $A\\in\\mathcal{G}$, $g\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, $x\\in\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_\\ell\\ensuremath{\\right)}^{\\times}$, and $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$.\nLet $\\overline{\\phantom{\\vert}\\cdot\\phantom{\\vert}}:\\ensuremath{{\\mathbb{Z}}}_\\ell\\to R_\\rho$ denote reduction mod $\\ell^\\rho$.\nIf $\\ell^\\rho>\\exp{A}$, then\n\\[\n\\mu_{2g}^{(x)}\\ensuremath{\\left(} A\\ensuremath{\\right)}=\\nu_{2g,\\rho}^{(\\overline{x})}\\ensuremath{\\left(} A\\ensuremath{\\right)}.\n\\]\n\\end{haar}\n\\begin{proof}\nChoose any $\\phi\\in\\GSp_{2g}^{(x)}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$.\nThen for any measurable $S\\subseteq\\GSp_{2g}^{(x)}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$, we know that $\\mu_{2g}^{(x)}\\ensuremath{\\left(} S\\ensuremath{\\right)}=\\mu_{2g}^{(x)}\\ensuremath{\\left(} S\\phi^{-1}\\phi\\ensuremath{\\right)}=\\mu_{2g}\\ensuremath{\\left(} S\\phi^{-1}\\ensuremath{\\right)}$ by the definition of $\\mu_{2g}^{(x)}$.\nSince $\\mu_{2g}$ is invariant under translation, every coset of the kernel of the reduction map $\\overline{\\phantom{\\vert}\\cdot\\phantom{\\vert}}:\\Sp_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}\\to R_\\rho$ has the same measure; namely,\n\\[\n\\left[\\Sp_{2g}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}:\\ker{\\ensuremath{\\left(}\\overline{\\phantom{i}\\cdot\\phantom{i}}\\ensuremath{\\right)}}\\right]^{-1}=\\ensuremath{\\left\\vert}\\Sp_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}^{-1}.\n\\]\nMoreover, note that if $\\psi\\in\\GSp_{2g}^{(x)}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}}$, then $m\\ensuremath{\\left(}\\overline{\\psi}\\ensuremath{\\right)}=\\overline{m(\\psi)}$ and $\\coker{\\ensuremath{\\left(}\\Id-\\psi\\ensuremath{\\right)}}\\simeq A$ if and only if $\\coker{\\ensuremath{\\left(}\\Id-\\overline{\\psi}\\ensuremath{\\right)}}\\simeq A$, since $\\ell^\\rho>\\exp{A}$.\nThe result follows.\n\\end{proof}\n\n\\begin{Not} \\label{Not}\nSuppose that $g\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$.\nFor $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ satisfying $\\rho\\geq\\xi$, we define an important subgroup of $\\GSp_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}$:\n\\[\n\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\n:=\\ensuremath{\\left\\{}\\gamma\\in\\GSp_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\n\\mid m(\\gamma)\\in\\ell^\\xi R_\\rho+1\\ensuremath{\\right\\}}.\n\\]\nNote that $\\GSp^{\\ensuremath{\\left\\langle}\\rho\\ensuremath{\\right\\rangle}}_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}=\\Sp_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}$ and $\\GSp^{\\la0\\ensuremath{\\right\\rangle}}_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}=\\GSp_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}$.\nFor any $A\\in\\mathcal{G}$, we adopt the suggestive notation:\n\\[\nN_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}\\ensuremath{\\left(} A\\ensuremath{\\right)}\n:=\\ensuremath{\\left\\vert}\\ensuremath{\\left\\{}\\gamma\\in\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\n\\mid\\coker{(\\Id-\\gamma)}\\simeq A\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}\n\\]\nand, if $\\rho>\\xi$,\n\\[\n\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}\\ensuremath{\\left(} A\\ensuremath{\\right)}\n:=\\frac{N_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}\\ensuremath{\\left(} A\\ensuremath{\\right)}\n-N_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}\\ensuremath{\\left(} A\\ensuremath{\\right)}}\n{\\ensuremath{\\left\\vert}\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}\n-\\ensuremath{\\left\\vert}\\GSp^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}}.\n\\]\n\\end{Not}\n\n\\begin{goa} \\label{goa}\nWe can now state the matrix-theoretic analog of the situation about which Malle made Conjecture~2.1.\nFollowing \\hyperref[Aheur]{Conjecture~\\ref*{Aheur}}, for $A\\in\\mathcal{G}$, $x\\in\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}_{\\ell}\\ensuremath{\\right)}^{\\times}$ and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, with $x\\equiv1\\pmod{\\ell^\\xi}$ but $x\\nequiv1\\pmod{\\ell^{\\xi+1}}$, we must evaluate\n\\[\n\\mu_x(A):=\\lim_{g\\to\\infty}\n{\\mu_{2g}^{(x)}\\ensuremath{\\left(} A\\ensuremath{\\right)}}.\n\\]\nIf we let $\\overline{\\phantom{\\vert}\\cdot\\phantom{\\vert}}:\\ensuremath{{\\mathbb{Z}}}_\\ell\\to R_\\rho$ denote reduction mod $\\ell^\\rho$, then we know by \\hyperref[haar]{Lemma~\\ref*{haar}} that this amounts to calculating\n\\[\n\\lim_{g\\to\\infty}\n{\\nu_{2g,\\rho}^{(\\overline{x})}\\ensuremath{\\left(} A\\ensuremath{\\right)}}\n\\]\nfor any $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ satisfying both $\\ell^\\rho>\\exp{A}$ and $\\rho>\\xi$.\nIn \\hyperref[subcos]{Note~\\ref*{subcos}} we will see that for all such $\\rho$:\n\\[\n\\nu_{2g,\\rho}^{(\\overline{x})}\\ensuremath{\\left(} A\\ensuremath{\\right)}=\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}\\ensuremath{\\left(} A\\ensuremath{\\right)},\n\\]\nso we will turn our attention to computing\n\\[\n\\lim_{g\\to\\infty}\n{\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}\\ensuremath{\\left(} A\\ensuremath{\\right)}},\n\\]\nwhich we compute explicitly for $\\xi=1,2$ in \\hyperref[finalc]{Corollary~\\ref*{finalc}}.\nUsing Achter's result, \\hyperref[Athm]{Theorem~\\ref*{Athm}}, we then obtain \\hyperref[prettyfinal]{Theorem~\\ref*{prettyfinal}} as a corollary.\n\\end{goa}\n\n\\begin{momentsn} \\label{momentsn}\nSuppose that $x\\in\\ensuremath{{\\mathbb{Z}}}_\\ell$.\nIn addition to explicitly computing the distribution $\\mu_x:\\mathcal{G}\\to\\ensuremath{{\\mathbb{R}}}$ if $x\\equiv1\\pmod{\\ell^\\xi}$ but $x\\nequiv1\\pmod{\\ell^{\\xi+1}}$ for $\\xi=1,2$, we also compute the moments of this distribution for any $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$.\nSpecifically, in \\hyperref[moments]{Corollary~\\ref*{moments}} we find that if $A\\in\\mathcal{G}$, then\n\\[\n\\sum_{B\\in\\mathcal{G}}{\\ensuremath{\\left\\vert}\\Surj{(B,A)}\\ensuremath{\\right\\vert}\\mu_x(B)}\n=\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} A\/\\ell^\\xi\\ensuremath{\\right)}\\ensuremath{\\right\\vert}.\n\\]\n(See \\hyperref[Not22]{Notation~\\ref*{Not22}} for the definition of $\\Lambda:\\mathcal{G}\\to\\ensuremath{{\\mathbb{Z}}}^{>0}$.)\n\\end{momentsn}\n\n\\section{The symplectic action} \\label{TSA}\n\n\\subsection{Basic properties} \\label{bp}\n\n\\begin{Not2} \\label{Not2}\nFor any $A,B\\in\\mathcal{G}$, let $\\Inj{\\ensuremath{\\left(} A,B\\ensuremath{\\right)}}$ and $\\Surj{\\ensuremath{\\left(} A,B\\ensuremath{\\right)}}$ be the set of injective homomorphisms and surjective homomorphisms from $A$ to $B$.\n\\end{Not2}\n\nIn what follows, we will consider either injections or surjections (as well as either kernels or cokernels) depending on which is more convenient at the time.\nThe next two lemmas justify this shifting point of view.\n\n\\begin{dual} \\label{dual}\nSuppose that $A\\in\\mathcal{G}$, $g,\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$.\nIf $\\rho\\geq\\xi$, then $\\GSp^{\\ensuremath{\\left\\langle} \\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}$ acts on $\\Inj{(A,(R_\\rho)^{2g})}$ and $\\Surj{((R_\\rho)^{2g},A)}$ by post- and pre-composition, respectively.\nThe number of orbits of these actions are the same.\n\\end{dual}\n\\begin{proof}\nIf $\\ell^\\rho<\\exp{A}$, the result is trivial, so suppose $\\ell^\\rho\\geq\\exp{A}$.\nIn this case, we can think of $A$ as an $R_\\rho$-module.\nMoreover, we know that $R_\\rho$ is an injective $R_\\rho$-module by Baer's criterion, so the functor\n\\[\n\\ensuremath{\\left(}\\cdot\\ensuremath{\\right)}^\\vee\n:=\\Hom(\\,\\cdot\\,,R_\\rho):R_\\rho\\mathtt{-mod}\\to R_\\rho\\mathtt{-mod}\n\\]\nis exact.\nThus, for any $\\gamma\\in\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}$:\n\\begin{gather*}\nf,h\\in\\Inj{(A,(R_\\rho)^{2g})}\n\\text{ with }\\gamma\\circ f=h\\\\\n\\text{ if and only if }\\\\\nf^{\\vee},h^{\\vee}\\in\\Surj{(((R_\\rho)^{2g})^\\vee,A^\\vee)}\n\\text{ with }f^\\vee\\circ\\gamma^{\\vee}=h^{\\vee}.\n\\end{gather*}\nAfter choosing $R_\\rho$-bases for $(R_\\rho)^{2g}$ and $A$, it is easy to see that $((R_\\rho)^{2g})^\\vee\\simeq(R_\\rho)^{2g}$, $A^\\vee\\simeq A$, and $\\gamma^\\vee=\\gamma^\\top\\in\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}$, giving the result.\n\\end{proof}\n\nThe number orbits of the action described above turn out to be very important, so we bestow a name upon them:\n\n\\begin{Not21} \\label{Not21}\nSuppose that $A\\in\\mathcal{G}$, $g,\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$.\nIf $\\rho\\geq\\xi$, let $o^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}$ be the number of orbits of $\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}$ acting on $\\Inj{\\ensuremath{\\left(} A,(R_\\rho)^{2g}\\ensuremath{\\right)}}$ or $\\Surj{\\ensuremath{\\left(} (R_\\rho)^{2g},A\\ensuremath{\\right)}}$.\n\\end{Not21}\n\n\\begin{dual2} \\label{dual2}\nFor $A,g,\\rho,\\xi$ as above:\n\\[\nN_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}\\ensuremath{\\left(} A\\ensuremath{\\right)}\n=\\ensuremath{\\left\\vert}\\ensuremath{\\left\\{}\\gamma\\in\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\n\\mid\\ker{(\\Id-\\gamma)}\\simeq A\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}.\n\\]\n\\end{dual2}\n\\begin{proof}\nAs in \\hyperref[dual]{Lemma~\\ref*{dual}}, this follows from the exactness of $\\ensuremath{\\left(}\\cdot\\ensuremath{\\right)}^\\vee$.\nNote that for any $\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{(R_\\rho)}$:\n\\[\n\\ensuremath{\\left(}\\coker{\\ensuremath{\\left(}\\Id-\\gamma\\ensuremath{\\right)}}\\ensuremath{\\right)}^{\\vee}\n=\\ker{\\ensuremath{\\left(}\\lp\\Id-\\gamma\\ensuremath{\\right)}^{\\vee}\\ensuremath{\\right)}}\n=\\ker{\\ensuremath{\\left(}\\Id-\\gamma^{\\top}\\ensuremath{\\right)}},\n\\]\ngiving the result.\n\\end{proof}\n\nIn \\hyperref[goa]{Goal~\\ref*{goa}}, we turned our attention from the measures of cosets of the symplectic group to subgroups of the group of symplectic similitudes.\nThe following note justifies this turn.\n\n\\begin{subcos} \\label{subcos}\nSuppose that $A\\in\\mathcal{G}$, $g\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, $x\\in\\ensuremath{{\\mathbb{Z}}}_{\\ell}$ and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, with $x\\equiv1\\pmod{\\ell^\\xi}$ but $x\\nequiv1\\pmod{\\ell^{\\xi+1}}$.\nIf $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ satisfies $\\rho>\\xi$, then\n\\[\n\\nu_{2g,\\rho}^{(\\overline{x})}(A)=\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A)\n\\]\n\\end{subcos}\n\\begin{proof}\nThis amounts to showing that if $x,y\\in R_\\rho$ such that $x\\equiv y\\equiv1\\pmod{\\ell^\\xi}$ but neither $x$ nor $y$ is equivalent to $1\\pmod{\\ell^{\\xi+1}}$, then\n\\[\n\\nu_{2g,\\rho}^{\\ensuremath{\\left(}\\overline{x}\\ensuremath{\\right)}}(A)=\\nu_{2g,\\rho}^{\\ensuremath{\\left(}\\overline{y}\\ensuremath{\\right)}}(A).\n\\]\nBy our assumptions on $x$ and $y$, there exists some $m_0$ such that that $\\ell\\nmid m_0$ and $\\overline{x}^{\\hspace{1px}m_0}=\\overline{y}$.\nChoose some $m$ in the arithmetic progression $\\ensuremath{\\left\\{} m_0+\\ell^{\\rho-\\xi}j\\ensuremath{\\right\\}}_{j=0}^\\infty$ such that\n\\[\n\\gcd{\\ensuremath{\\left(} m,\\ensuremath{\\left\\vert}\\GSp_{2g}{(R_\\rho)}\\ensuremath{\\right\\vert}\\ensuremath{\\right)}}=1,\n\\]\nand choose $k$ such that $mk\\equiv1\\pmod{\\ensuremath{\\left\\vert}\\GSp_{2g}{(R_\\rho)}\\ensuremath{\\right\\vert}}$.\nNow, the map\n\\begin{align*}\n\\ensuremath{\\left(}\\cdot\\ensuremath{\\right)}^m:\\GSp_{2g}^{\\ensuremath{\\left(}\\overline{x}\\ensuremath{\\right)}}{(R_\\rho)}&\\to\\GSp_{2g}^{\\ensuremath{\\left(}\\overline{y}\\ensuremath{\\right)}}{(R_\\rho)}\\\\\n\\gamma&\\mapsto\\gamma^m\n\\end{align*}\nis bijective with inverse $\\ensuremath{\\left(}\\cdot\\ensuremath{\\right)}^k$.\nMoreover, for any $z\\in(R_\\rho)^{2g}$ and any $\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left(}\\overline{x}\\ensuremath{\\right)}}{(R_\\rho)}$, it is clear that $\\gamma z=z$ if and only if $\\gamma^mz=z$.\nThus, we obtain\n\\[\n\\ensuremath{\\left\\vert}\\ensuremath{\\left\\{}\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left(}\\overline{x}\\ensuremath{\\right)}}{(R_\\rho)}\n\\mid\\ker{\\ensuremath{\\left(}\\Id-\\gamma\\ensuremath{\\right)}}\\simeq A\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}\n=\\ensuremath{\\left\\vert}\\ensuremath{\\left\\{}\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left(}\\overline{y}\\ensuremath{\\right)}}{(R_\\rho)}\n\\mid\\ker{\\ensuremath{\\left(}\\Id-\\gamma\\ensuremath{\\right)}}\\simeq A\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert},\n\\]\nand we conclude by \\hyperref[dual2]{Lemma~\\ref*{dual2}}.\n\\end{proof}\n\n\\subsection{Orbit counting} \\label{oc}\n\n\\begin{Not22} \\label{Not22}\nFor any $A\\in\\mathcal{G}$, let $\\Lambda\\ensuremath{\\left(} A\\ensuremath{\\right)}$ be the set of alternating bilinear forms on $A$ thought of as a $\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}\/\\exp{(A)}\\ensuremath{\\right)}$-module.\n\\end{Not22}\n\n\\begin{forms} \\label{forms}\nSuppose that $A=\\ensuremath{{\\mathbb{Z}}}\/\\ell^{\\alpha_1}\\oplus\\cdots\\oplus\\ensuremath{{\\mathbb{Z}}}\/\\ell^{\\alpha_r}$ with $\\alpha_1\\geq\\cdots\\geq\\alpha_r>0$.\nThen\n\\[\n\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} A\\ensuremath{\\right)}\\ensuremath{\\right\\vert}=\\ell^{\\sum_{i=2}^r{(i-1)\\alpha_i}}\n\\]\n\\end{forms}\n\\begin{proof}\nLet $\\ensuremath{\\left\\{}\\mathbf{e}_i\\ensuremath{\\right\\}}_{i=1}^r$ be an $\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}\/\\exp{(A)}\\ensuremath{\\right)}$-basis for $A$ such that $\\mathbf{e}_i$ has order $\\ell^{\\alpha_i}$ for all $i\\in\\lb1,\\ldots,r\\ensuremath{\\right\\}}$.\nEvery alternating bilinear form $\\ensuremath{\\left\\langle}\\cdot,\\cdot\\ensuremath{\\right\\rangle}$ on $A$ corresponds to an antisymmetric matrix $\\ensuremath{\\left(}\\ensuremath{\\left\\langle}\\mathbf{e}_i,\\mathbf{e}_j\\ensuremath{\\right\\rangle}\\ensuremath{\\right)}\\in\\Mat_{r\\times r}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}\/\\exp{(A)}\\ensuremath{\\right)}}$.\nMoreover, any antisymmetric matrix $\\ensuremath{\\left(} a_{ij}\\ensuremath{\\right)}\\in\\Mat_{r\\times r}{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}\/\\exp{(A)}\\ensuremath{\\right)}}$ corresponds to an alternating bilinear form on $A$, as long it has 0's along its main diagonal and $\\ell^{\\alpha_j}a_{ij}=0$ whenever $i<j$ (the $i>j$ case follows from the the fact that $\\ensuremath{\\left(} a_{ij}\\ensuremath{\\right)}$ is antisymmetric).\nThere are $\\ell^{\\alpha_j}$ such elements of $\\ensuremath{{\\mathbb{Z}}}\/\\exp{(A)}$, so the result follows.\n\\end{proof}\n\n\\begin{orb} \\label{orb}\nSuppose that $r\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$, $A\\in\\mathcal{G}(r)$, $g,\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$.\nIf $\\ell^\\rho\\geq\\exp{A}$, $\\rho\\geq\\xi$, and $2g\\geq r$, then\n\\[\no^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\n\\leq\\ell^{-(\\rho-\\xi)}\\ensuremath{\\left\\vert}\\Lambda(A)\\ensuremath{\\right\\vert}+\n(\\ell-1)\\sum_{i=0}^{\\rho-\\xi-1}{\\ell^{-(i+1)}}{\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} A\/\\ell^{\\xi+i}\\ensuremath{\\right)}\\ensuremath{\\right\\vert}}.\n\\]\nFurthermore, when $g\\geq r$, the upper bound above is an equality.\n(In particular, $o^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}$ is independent of $g$ for large enough $g$.)\n\\end{orb}\nAs pointed out in \\hyperref[goa]{Goal~\\ref*{goa}}, we need only calculate\n\\[\n\\lim_{g\\to\\infty}{\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A)}.\n\\]\nDespite this fact, the inequality for small $g$ in \\hyperref[orb]{Lemma~\\ref*{orb}} does indeed turn out to be useful.\nThis is due to the fact that $\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A)$ can be expressed as a sum of orbit data for finite abelian groups of rank up to $2g$. \n(See \\hyperref[infi]{Corollary~\\ref*{infi}}.)\n\n\\begin{proof}[Proof of the Lemma]\nThe result is obviously true when $r=0$, so suppose that $r>0$.\nTheorem~2.14 of~\\cite{Mic} shows that the set of orbit representatives of $\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}=\\Sp_{2g}{(R_\\rho)}$ acting on $\\Surj{\\ensuremath{\\left(} (R_\\rho)^{2g},A\\ensuremath{\\right)}}$ injects into $\\Lambda(A)$.\nIn fact, when $g\\geq r$, this injection is a bijection.\nFurthermore, this injection is equivariant with respect to the natural actions of $(R_\\rho)^{\\times}=\\GSp_{2g}{(R_\\rho)}\/\\Sp_{2g}{(R_\\rho)}$ on these two sets.\nThus, computing the number of orbits of $\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}$ acting on $\\Surj{\\ensuremath{\\left(}(R_\\rho)^{2g},A\\ensuremath{\\right)}}$ is a straightforward application of Burnside's counting theorem.\nIndeed, suppose that $A=\\ensuremath{{\\mathbb{Z}}}\/\\ell^{\\alpha_1}\\oplus\\cdots\\oplus\\ensuremath{{\\mathbb{Z}}}\/\\ell^{\\alpha_r}$ with $\\alpha_1\\geq\\cdots\\geq\\alpha_r>0$, then use \\hyperref[forms]{Note~\\ref*{forms}} to note that\n\\begin{align*}\no^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\n&\\leq\\frac{1}{\\ensuremath{\\left\\vert}\\ell^\\xi R_\\rho+1\\ensuremath{\\right\\vert}}\n\\cdot\\sum_{\\upsilon\\in\\ell^\\xi R_\\rho+1}{\\ensuremath{\\left\\vert}\\Fix{(\\upsilon)}\\ensuremath{\\right\\vert}}\\\\\n&=\\frac{1}{\\ensuremath{\\left\\vert}\\ell^\\xi R_\\rho+1\\ensuremath{\\right\\vert}}\\ensuremath{\\left(}\\lp\\sum_{i=0}^{\\rho-\\xi-1}{\n\\sum_{\\upsilon\\in(\\ell^{\\xi+i}R_\\rho+1)\\setminus(\\ell^{\\xi+i+1}R_\\rho+1)}\n{\\ensuremath{\\left\\vert}\\Fix{(\\upsilon)}\\ensuremath{\\right\\vert}}}\\ensuremath{\\right)}+\n\\sum_{\\upsilon\\in\\ell^\\rho R_\\rho+1=\\{1\\}}\n{\\ensuremath{\\left\\vert}\\Fix{(\\upsilon)}\\ensuremath{\\right\\vert}}\\ensuremath{\\right)}\\\\\n&=\\frac{1}{\\ell^{\\rho-\\xi}}\n\\ensuremath{\\left(}\\sum_{i=0}^{\\rho-\\xi-1}\n{(\\ell^{\\rho-\\xi-i}-\\ell^{\\rho-\\xi-i-1})\\ensuremath{\\left(}\n\\ell^{\\sum_{j=2}^{r}\n{(j-1)\\min{\\ensuremath{\\left\\{}\\xi+i,\\alpha_j\\ensuremath{\\right\\}}}}}\\ensuremath{\\right)}}\n+\\ell^{\\alpha_2+2\\alpha_3+\\cdots+(r-1)\\alpha_r}\\ensuremath{\\right)}\\\\\n&=\\frac{1}{\\ell^{\\rho-\\xi}}\n\\ensuremath{\\left\\vert}\\Lambda(A)\\ensuremath{\\right\\vert}\n+(\\ell-1)\\sum_{i=0}^{\\rho-\\xi-1}\n{\\ell^{-(i+1)}}{\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} A\/\\ell^{\\xi+i}\\ensuremath{\\right)}\\ensuremath{\\right\\vert}},\n\\end{align*}\nwith equality when $g\\geq r$.\n\\end{proof}\n\n\\begin{Not3} \\label{Not3}\nSuppose that $A\\in\\mathcal{G}$, $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$.\nIf $\\ell^\\rho\\geq\\exp{A}$ and $\\rho\\geq\\xi$, use \\hyperref[orb]{Lemma~\\ref*{orb}} to define $o^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_\\rho:=o^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}$ for any $g\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ such that $g\\geq\\rank{A}$.\n\\end{Not3}\n\nWe now mention an identity which will be useful later.\n(See \\hyperref[moments]{Corollary~\\ref*{moments}} and \\hyperref[nutime]{Note~\\ref*{nutime}}.)\n\n\\begin{subt} \\label{subt}\nSuppose $A\\in\\mathcal{G}$ and $\\rho,\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$.\nIf $\\ell^\\rho\\geq\\exp{A}$ and $\\rho>\\xi$, then by \\hyperref[orb]{Lemma~\\ref*{orb}} and \\hyperref[forms]{Note~\\ref*{forms}}, we see that\n\\[\n\\ell o^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_\\rho-o^{A,\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_\\rho\n=(\\ell-1)\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} A\/\\ell^\\xi\\ensuremath{\\right)}\\ensuremath{\\right\\vert}.\n\\]\n\\end{subt}\n\n\nBelow is a simple observation, which has the important consequence \\hyperref[moments]{Corollary~\\ref*{moments}}.\nThis corollary gives the moments of the probability distributions $\\mu_x:\\mathcal{G}\\to\\ensuremath{{\\mathbb{R}}}$ for any $x\\in\\ensuremath{{\\mathbb{Z}}}_\\ell$, as promised in \\hyperref[THM]{Section~\\ref*{THM}}.\n\n\\begin{bij} \\label{bij}\nSuppose that $A\\in\\mathcal{G}$, $g,\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$.\nFurthermore, suppose $\\rho\\geq\\xi$, let $\\gamma\\in\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}$, and consider $\\Inj{\\ensuremath{\\left(} A,\\ker{(\\Id-\\gamma)}\\ensuremath{\\right)}}\\subseteq\\Inj{(A,(R_\\rho)^{2g})}$.\nThere is a 1-1 correspondence between $\\Inj{\\ensuremath{\\left(} A,\\ker{(\\Id-\\gamma)}\\ensuremath{\\right)}}$ and $\\Fix{(\\gamma)}$.\nDually, $\\Surj{\\ensuremath{\\left(}\\coker{(\\Id-\\gamma)},A\\ensuremath{\\right)}}\\leftrightarrow\\Fix{(\\gamma)}$.\n\\end{bij}\n\\begin{proof}\nSuppose that $f\\in\\Inj{(A,(R_\\rho)^{2g})}$.\nNote that $f\\in\\Inj{\\ensuremath{\\left(} A,\\ker{(\\Id-\\gamma)}\\ensuremath{\\right)}}$ if and only if $(\\Id-\\gamma)f=0$ if and only if $f=\\gamma f$.\nThe dual proof is similar.\n\\end{proof}\n\n\\begin{moments} \\label{moments}\nLet $x\\in\\ensuremath{{\\mathbb{Z}}}_\\ell$ and suppose that $x\\equiv1\\pmod{\\ell^\\xi}$ but $x\\nequiv1\\pmod{\\ell^{\\xi+1}}$ for some $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$.\nIf $A\\in\\mathcal{G}$, then\n\\[\n\\sum_{B\\in\\mathcal{G}}{\\ensuremath{\\left\\vert}\\Surj{(B,A)}\\ensuremath{\\right\\vert}\\mu_x(B)}\n=\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} A\/\\ell^\\xi\\ensuremath{\\right)}\\ensuremath{\\right\\vert}.\n\\]\n\\end{moments}\n\\begin{proof}\nChoose any $g,\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ such that $g\\geq\\rank{A}$, $\\ell^\\rho\\geq\\exp{A}$, and $\\rho>\\xi$.\nTo begin with, note that\n\\begin{align*}\n\\sum_{B\\in\\mathcal{G}}\n{\\ensuremath{\\left\\vert}\\Surj{\\ensuremath{\\left(} B,A\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}\\nu_{2g,\\rho}^{(x)}(B)}\n&=\\ensuremath{\\left\\vert}\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}^{-1}\n\\cdot\\sum_{B\\in\\mathcal{G}}\n{\\ensuremath{\\left\\vert}\\Surj{\\ensuremath{\\left(} B,A\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}\n\\cdot\\ensuremath{\\left\\vert}\\ensuremath{\\left\\{}\\gamma\\in\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\n\\mid\\coker{(\\Id-\\gamma)}\\simeq B\\ensuremath{\\right\\}}\\ensuremath{\\right\\vert}}\\\\\n&=\\ensuremath{\\left\\vert}\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}^{-1}\n\\cdot\\sum_{\\gamma\\in\\GSp_{2g}^{(x)}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}}\n{\\ensuremath{\\left\\vert}\\Surj{\\ensuremath{\\left(}\\coker{(\\Id-\\gamma),A}\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}}.\n\\end{align*}\nNow, thanks to \\hyperref[subcos]{Lemma~\\ref*{subcos}}, we can turn our attention to the following quantity:\n\\[\n\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\n\\setminus\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}^{-1}\n\\cdot\\sum_{\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\n\\setminus\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}}\n{\\ensuremath{\\left\\vert}\\Surj{\\ensuremath{\\left(}\\coker{\\ensuremath{\\left(}\\Id-\\gamma\\ensuremath{\\right)}},A\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}}.\n\\]\nUsing the fact that $\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{(R_\\rho)}\\ensuremath{\\right\\vert}=\\ell\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{(R_\\rho)}\\ensuremath{\\right\\vert}$ and applying \\hyperref[bij]{Lemma~\\ref*{bij}} to $\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{(R_\\rho)}$ acting on $\\Surj{((R_\\rho)^{2g},A)}$, then using Burnside's counting theorem and \\hyperref[Not3]{Notation~\\ref*{Not3}}, we see that\n\\begin{align*}\n&\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\n\\setminus\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}^{-1}\n\\cdot\\sum_{\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\n\\setminus\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}}\n{\\ensuremath{\\left\\vert}\\Surj{\\ensuremath{\\left(}\\coker{\\ensuremath{\\left(}\\Id-\\gamma\\ensuremath{\\right)}},A\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}}\\\\\n&\\qquad=\\frac{\\ell}\n{(\\ell-1)\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}}\n\\ensuremath{\\left(}\\sum_{\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}}\n{\\ensuremath{\\left\\vert}\\Fix{\\gamma}\\ensuremath{\\right\\vert}}\n-\\sum_{\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}}\n{\\ensuremath{\\left\\vert}\\Fix{\\gamma}\\ensuremath{\\right\\vert}}\\ensuremath{\\right)}\\\\\n&\\qquad=\\frac{\\ell}{(\\ell-1)}\n\\ensuremath{\\left(}\\sum_{\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}}\n{\\frac{\\ensuremath{\\left\\vert}\\Fix{\\gamma}\\ensuremath{\\right\\vert}}{\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}}}\n-\\sum_{\\gamma\\in\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}}\n{\\frac{\\ensuremath{\\left\\vert}\\Fix{\\gamma}\\ensuremath{\\right\\vert}}{\\ell\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{\\ensuremath{\\left(} R_\\rho\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}}}\\ensuremath{\\right)}\\\\\n&\\qquad=\\frac{\\ell}{(\\ell-1)}\n\\ensuremath{\\left(} o^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}-\\frac{1}{\\ell}o^{A,\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_{2g,\\rho}\\ensuremath{\\right)}\\\\\n&\\qquad=\\frac{1}{(\\ell-1)}\n\\ensuremath{\\left(}\\ell o^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_\\rho-o^{A,\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_\\rho\\ensuremath{\\right)},\n\\end{align*}\nso we can conclude by applying \\hyperref[subt]{Note~\\ref*{subt}} and \\hyperref[haar]{Lemma~\\ref*{haar}}.\n\\end{proof}\n\n\\section{A weighted M\\\"{o}bius function} \\label{AWMF}\n\n\\subsection{First observations}\n\nLet $\\mathcal{P}$ be a locally finite poset.\nThe \\emph{M\\\"{o}bius function} on $\\mathcal{P}$, denoted by $\\mu_\\mathcal{P}$, is defined by the following criteria: for any $x,z\\in\\mathcal{P}$,\n\\begin{align*}\n\\hspace{73px}&\\mu_\\mathcal{P}\\ensuremath{\\left(} x,z\\ensuremath{\\right)}&&\\hspace{-83px}=0&&\n\\hspace{-73px}\\text{ if }x\\nleq z,\\\\\n\\hspace{73px}&\\mu_\\mathcal{P}\\ensuremath{\\left(} x,z\\ensuremath{\\right)}&&\\hspace{-83px}=1&&\n\\hspace{-73px}\\text{ if }x=z,\\\\\n\\hspace{73px}\\sum_{x\\leq y\\leq z}\n&{\\mu_\\mathcal{P}\\ensuremath{\\left(} x,y\\ensuremath{\\right)}}&&\\hspace{-83px}=0&&\n\\hspace{-73px}\\text{ if }x<z.\n\\end{align*}\nA classic reference for M\\\"{o}bius functions is~\\cite{Rota}.\nIn this section, we need to study a variant of the M\\\"{o}bius function on the poset of subgroups of a finite group (ordered by inclusion). \nFor a history of the work on the M\\\"{o}bius function on this poset, see~\\cite{HIO}.\nNow, for any finite group $G$, let $\\mathcal{P}_G$ be the poset of subgroups of $G$ ordered by inclusion.\nFor $A\\in\\mathcal{G}$, we study an amalgam of the M\\\"{o}bius functions on $\\mathcal{P}_A$ and $\\mathcal{G}$, which we define below.\n\n\\begin{Not4} \\label{Not4}\nFor any $A,B\\in\\mathcal{G}$, let $\\sub{(A,B)}$ be the number of subgroups of $B$ that are isomorphic to $A$.\nIf $A\\in\\mathcal{G}$, an \\emph{$A$-chain} is a finite (possibly empty) linearly ordered subset of $\\ensuremath{\\left\\{} B\\in\\mathcal{G}\\mid B>A\\ensuremath{\\right\\}}$.\nNow, given an $A$-chain $\\mathfrak{C}=\\ensuremath{\\left\\{} A_j\\ensuremath{\\right\\}}_{j=1}^i$, with $A_j<A_{j+1}$ for all $j\\in\\lb1,\\ldots,i-1\\ensuremath{\\right\\}}$, define\n\\[\n\\sub{\\ensuremath{\\left(}\\mathfrak{C}\\ensuremath{\\right)}}\n:=(-1)^i\\sub{\\ensuremath{\\left(} A,A_1\\ensuremath{\\right)}}\\prod_{j=1}^{i-1}{\\sub{\\ensuremath{\\left(} A_j,A_{j+1}\\ensuremath{\\right)}}}.\n\\]\n(We set $\\sub{\\ensuremath{\\left(}\\mathfrak{C}\\ensuremath{\\right)}}=1$ if $\\mathfrak{C}$ is empty.)\nFinally, for any $A,B\\in\\mathcal{G}$, let\n\\[\nS(A,B)\n=\\begin{cases}\n\\hspace{18px}0&\\text{if }A\\nleq B,\\\\\n\\hspace{18px}1&\\text{if }A=B,\\\\\n\\displaystyle{\n\\sum_{\\substack{A\\text{-chains }\\mathfrak{C},\\\\\n\\max{\\mathfrak{C}}=B}}\n{\\sub{\\ensuremath{\\left(}\\mathfrak{C}\\ensuremath{\\right)}}}}&\\text{if } A<B.\n\\end{cases}\n\\]\n\\end{Not4}\n\n\\begin{amalgam} \\label{amalgam}\nThough $S$ is defined on the poset $\\mathcal{G}$, it is closely related to the classical work on the M\\\"{o}bius function on the subgroup lattice of a fixed group.\nIndeed, by applying Lemma~2.2 of~\\cite{HIO}, we see that if $A,B\\in\\mathcal{G}$, then\n\\[\nS(A,B)\n=\\sum_{\\substack{C\\leq B\n\\\\ensuremath{{\\mathbb{C}}}\\simeq A}}\n{\\mu_B(C,B)}.\n\\]\n\\end{amalgam}\n\nGiven $x\\in\\ensuremath{{\\mathbb{Z}}}_\\ell$, we can use the function $S$ defined in \\hyperref[Not4]{Notation~\\ref*{Not4}} to begin our analysis of the measure $\\mu_x$, following the outline in \\hyperref[goa]{Goal~\\ref*{goa}}.\n\n\\begin{burn} \\label{burn}\nSuppose $A\\in\\mathcal{G}$, $g,\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$, with $\\rho\\geq\\xi$ and $\\ell^\\rho\\geq\\exp{A}$.\nThen\n\\[\no^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\\ensuremath{\\left\\vert}\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}\\ensuremath{\\right\\vert}\n=\\sum_{\\substack{B\\in\\mathcal{G}\\\\B\\leq(R_\\rho)^{2g}}}\n{N^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}(B)\\ensuremath{\\left\\vert}\\Inj{(A,B)}\\ensuremath{\\right\\vert}}.\n\\]\n\\end{burn}\n\\begin{proof}\nApplying \\hyperref[bij]{Lemma~\\ref*{bij}} and Burnside's counting theorem, we see that\n\\[\no^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\\ensuremath{\\left\\vert}\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}\\ensuremath{\\right\\vert}\n=\\sum_{\\gamma\\in\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}}{\\ensuremath{\\left\\vert}\\Fix{(\\gamma)}\\ensuremath{\\right\\vert}}\n=\\sum_{\\gamma\\in\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}}\n{\\ensuremath{\\left\\vert}\\Inj{\\ensuremath{\\left(} A,\\ker{(\\Id-\\gamma)}\\ensuremath{\\right)}}\\ensuremath{\\right\\vert}}\n=\\sum_{\\substack{B\\in\\mathcal{G}\\\\B\\leq(R_\\rho)^{2g}}}{N^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}(B)\\ensuremath{\\left\\vert}\\Inj{(A,B)}\\ensuremath{\\right\\vert}},\n\\]\nwhere the last step follows from \\hyperref[dual2]{Lemma~\\ref*{dual2}}.\n\\end{proof}\n\nFor $A,g,\\rho,\\xi$ as above, \\hyperref[burn]{Lemma~\\ref*{burn}} gives us an ``upper triangular\" system of equations, which we will solve for $N^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}(A)$.\n(The quotes indicate that the system is indexed by the poset $\\mathcal{P}_{(r_\\rho)^{2g}}$.)\n\\hyperref[deter]{Proposition~\\ref{deter}} is the first step along this path.\n\n\\begin{deter} \\label{deter}\nSuppose $A,g,\\rho,\\xi$ are as above.\nThen\n\\[\n\\frac{N^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}(A)}{\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{(R_\\rho)}\\ensuremath{\\right\\vert}}\n=\\sum_{\\substack{B\\in\\mathcal{G}\\\\B\\leq(R_\\rho)^{2g}}}\n{o^{B,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\\cdot \\frac{S(A,B)}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}}.\n\\]\n\\end{deter}\n\\begin{proof}\nWe use strong induction on $\\ensuremath{\\left\\vert}(R_\\rho)^{2g}\\ensuremath{\\right\\vert}\/\\ensuremath{\\left\\vert} A\\ensuremath{\\right\\vert}$.\nIn light of \\hyperref[burn]{Lemma~\\ref*{burn}}, the base case $A=(R_\\rho)^{2g}$ is trivial.\nNow suppose the result is true for all $B\\in\\mathcal{G}$ with $B\\leq(R_\\rho)^{2g}$ and $\\ensuremath{\\left\\vert}(R_\\rho)^{2g}\\ensuremath{\\right\\vert}\/\\ensuremath{\\left\\vert} B\\ensuremath{\\right\\vert}<\\ensuremath{\\left\\vert}(R_\\rho)^{2g}\\ensuremath{\\right\\vert}\/\\ensuremath{\\left\\vert} A\\ensuremath{\\right\\vert}$.\nUsing \\hyperref[burn]{Lemma~\\ref*{burn}}, we see that\n\\begin{align*}\n\\frac{N^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}(A)}{\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{(R_\\rho)}\\ensuremath{\\right\\vert}}\n&=\\frac{1}{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}\\cdot\\ensuremath{\\left(} o^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\n-\\frac{1}{\\ensuremath{\\left\\vert}\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}\\ensuremath{\\right\\vert}}\n\\cdot\\sum_{\\substack{B\\in\\mathcal{G}\\\\B\\leq(R_\\rho)^{2g}\\\\B\\neq A}}\n{N^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}(B)\\ensuremath{\\left\\vert}\\Inj{(A,B)}\\ensuremath{\\right\\vert}}\\ensuremath{\\right)}\\\\\n&=\\frac{o^{A,\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}}{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}\n-\\sum_{\\substack{B\\in\\mathcal{G}\\\\B\\leq(R_\\rho)^{2g}\\\\B\\neq A}}\n{\\frac{N^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}(B)}{\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{(R_\\rho)}\\ensuremath{\\right\\vert}}\\cdot\\sub{(A,B)}},\n\\end{align*}\nso the result follows by the induction hypothesis.\n\\end{proof}\n\n\\subsection{Vanishing of the M\\\"{o}bius function}\n\nBefore proceeding, we need a bit more notation, and a result from~\\cite{GMob}.\n\n\\begin{Not23} \\label{Not23}\nFor any $A\\in\\mathcal{G}$ and any $i\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$, let\n\\[\nA_{\\oplus i}\n:=A\\oplus\\overbrace{\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}\/\\ell\\ensuremath{\\right)}\\oplus\\cdots\\oplus\\ensuremath{\\left(}\\ensuremath{{\\mathbb{Z}}}\/\\ell\\ensuremath{\\right)}}^{i\\text{ times}}.\n\\]\n\\end{Not23}\n\nIn 1934, Hall~\\cite{Hall2} proved that if $G$ is an $\\ell$-group of order $\\ell^n$, then $\\mu_G(1,G)=0$ unless $G$ is elementary abelian, in which case $\\mu_G(1,G)=(-1)^n\\ell^{\\binom{n}{2}}$.\nThe paper~\\cite{GMob} proves an analogous property of the function $S$:\n\n\\begin{finalm}[\\cite{GMob}] \\label{finalm}\nIf $A,B\\in\\mathcal{G}$, then $S(A,B)=0$ unless there exists an injection $\\iota:A\\hookrightarrow B$ with $\\coker{(\\iota)}$ elementary abelian.\n\\end{finalm}\n\\noindent \\hyperref[finalm]{Theorem~\\ref*{finalm}} has the following corollary:\n\n\\begin{infi} \\label{infi}\nSuppose $A,g,\\rho,\\xi$ are as above, and let $r=\\rank{A}$.\nIf in addition we know $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ and $\\rho$ satisfies $\\rho>\\xi$ and $\\ell^\\rho>\\exp{A}$, then\n\\[\n\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A)\n=\\sum_{B\\in\\mathcal{G}(r)}\n{S(A,B)\\cdot\\sum_{i=0}^{2g-r}\n{\\frac{\\ell o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\n-o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_{2g,\\rho}}\n{\\ell-1}\n\\cdot \\frac{S(B,B_{\\oplus i})}{\\ensuremath{\\left\\vert}\\Aut{B_{\\oplus i}}\\ensuremath{\\right\\vert}}}}.\n\\]\n\\end{infi}\n\\begin{proof}\nUsing the fact that $\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{(R_\\rho)}\\ensuremath{\\right\\vert}=\\ell\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{(R_\\rho)}\\ensuremath{\\right\\vert}$ along with \\hyperref[deter]{Proposition~\\ref*{deter}} and \\hyperref[finalm]{Theorem~\\ref*{finalm}}, we see that\n\\begin{align*}\n\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A)\n&=\\frac{N_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A)\n-N_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}(A)}\n{\\ensuremath{\\left\\vert}\\GSp^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}\\ensuremath{\\right\\vert}\n-\\ensuremath{\\left\\vert}\\GSp^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_{2g}{(R_\\rho)}\\ensuremath{\\right\\vert}}\\\\\n&=\\frac{\\ell N_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A)}\n{(\\ell-1)\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}{(R_\\rho)}\\ensuremath{\\right\\vert}}\n-\\frac{N_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}(A)}\n{(\\ell-1)\\ensuremath{\\left\\vert}\\GSp_{2g}^{\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}{(R_\\rho)}\\ensuremath{\\right\\vert}}\\\\\n&=\\sum_{B\\in\\mathcal{G}(r)}\n{S(A,B)\\cdot\\sum_{i=0}^{2g-r}\n{\\frac{\\ell o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\n-o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_{2g,\\rho}}\n{\\ell-1}\n\\cdot\\frac{S(B,B_{\\oplus i})}{\\ensuremath{\\left\\vert}\\Aut{B_{\\oplus i}}\\ensuremath{\\right\\vert}}}}.\n\\end{align*}\n\\end{proof}\n\n\\begin{nutime} \\label{nutime}\nSuppose $A,g,\\rho,\\xi,r$ are as in \\hyperref[infi]{Corollary~\\ref*{infi}}, and suppose that $g\\geq r$.\nThen for any $B\\in\\mathcal{G}(r)$ and $i\\in\\lb0\\ldots,g-r\\ensuremath{\\right\\}}$, we know by \\hyperref[subt]{Note~\\ref*{subt}} and \\hyperref[forms]{Lemma~\\ref*{forms}} that\n\\[\n\\frac{\\ell o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\n-o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_{2g,\\rho}}\n{\\ell-1}\n=\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} B_{\\oplus i}\/\\ell^\\xi\\ensuremath{\\right)}\\ensuremath{\\right\\vert}\n=\\ell^{ir+\\frac{i(i-1)}{2}}\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} B\/\\ell^\\xi B\\ensuremath{\\right)}\\ensuremath{\\right\\vert},\n\\]\nand for any $i\\in\\ensuremath{\\left\\{} g-r+1\\ldots,2g-r\\ensuremath{\\right\\}}$, we can use the proof of \\hyperref[orb]{Lemma~\\ref*{orb}} to note that\n\\[\n\\frac{\\ell o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\n-o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_{2g,\\rho}}\n{\\ell-1}\n\\leq\\ell o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\n\\leq\\ell\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} B_{\\oplus i}\\ensuremath{\\right)}\\ensuremath{\\right\\vert}\n=\\ell^{ir+\\frac{i(i-1)}{2}+1}\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} B\\ensuremath{\\right)}\\ensuremath{\\right\\vert}.\n\\]\nThus, if\n\\[\n\\sum_{i=0}^\\infty\n{\\ell^{ir+\\frac{i(i-1)}{2}}\\frac{S(B,B_{\\oplus i})}{\\ensuremath{\\left\\vert}\\Aut{B_{\\oplus i}}\\ensuremath{\\right\\vert}}}\n\\]\nconverges absolutely (and it does, see \\hyperref[product]{Lemma~\\ref*{product}}, \\hyperref[converge]{Lemma~\\ref*{converge}}, and \\hyperref[final]{Theorem~\\ref*{final}}), then so does \n\\[\n\\sum_{i=0}^\\infty\n{\\frac{\\ell o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}_{2g,\\rho}\n-o^{B_{\\oplus i},\\ensuremath{\\left\\langle}\\xi+1\\ensuremath{\\right\\rangle}}_{2g,\\rho}}\n{\\ell-1}\n\\cdot \\frac{S(B,B_{\\oplus i})}{\\ensuremath{\\left\\vert}\\Aut{B_{\\oplus i}}\\ensuremath{\\right\\vert}}},\n\\]\nand\n\\[\n\\lim_{g\\to\\infty}\n{\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A)}\n=\\sum_{B\\in\\mathcal{G}(r)}\n{S(A,B)\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} B\/\\ell^\\xi B\\ensuremath{\\right)}\\ensuremath{\\right\\vert}\\cdot\\sum_{i=0}^\\infty\n{\\ell^{ir+\\frac{i(i-1)}{2}}\\frac{S(B,B_{\\oplus i})}{\\ensuremath{\\left\\vert}\\Aut{B_{\\oplus i}}\\ensuremath{\\right\\vert}}}}.\n\\]\nAnalyzing the inner series is the subject of the next section.\n(Note that the above limit does not depend on $\\rho$, once $\\rho$ is large enough; this is consistent with \\hyperref[haar]{Lemma~\\ref*{haar}}.)\n\\end{nutime}\n\n\\section{$q$-series and convergence} \\label{QSAC}\n\n\\subsection{$q$-series} \\label{QS}\n\nBefore continuing, we make a small foray into some $q$-series notations and calculations.\n\n\\begin{Not6} \\label{Not6}\nFor $z,q\\in\\ensuremath{{\\mathbb{C}}}$ with $\\ensuremath{\\left\\vert} q\\ensuremath{\\right\\vert}<1$ and $i\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$, let\n\\[\n\\ensuremath{\\left(} z;q\\ensuremath{\\right)}_i:=\\prod_{j=0}^{i-1}{\\lp1-q^jz\\ensuremath{\\right)}}.\n\\]\nTo ease notation, set $(q)_i:=(q;q)_i$.\nRecall the definition of the $q$-binomial coefficients: for any $k,m\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$, let\n\\[\n\\binom{k}{m}_q:=\\frac{(q)_k}{(q)_m(q)_{k-m}},\n\\]\nwith $\\binom{k}{m}_q:=0$ if $k<m$.\n\nFor $i\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$, let $r_i=-1\/\\ensuremath{\\left(}\\ell^{\\frac{i(i+1)}{2}}(\\ell^{-1})_i\\ensuremath{\\right)}$.\nWe define the next object in terms of any finite set of nonnegative integers $S$ and any $i\\in\\ensuremath{{\\mathbb{Z}}}$ satisfying $i>\\max{S}$.\nIf $S\\cup\\{0\\}=\\{s_0,\\ldots,s_j\\}$, where $0=s_0<s_1<\\cdots<s_{j+1}:=i$, define $r^i_S=\\prod_{i=0}^{j}{r_{s_{i+1}-s_i}}$.\n\nFinally, let $t_0=1$, let $t_1=r^1_{\\varnothing}$, and for $i>1$, let\n\\[\nt_i=\\sum_{S\\subseteq\\{1,\\ldots i-1\\}}{r^i_S}.\n\\]\n\\end{Not6}\n\n\\begin{product} \\label{product}\n\\[\n\\sum_{i=0}^{\\infty}{t_i}=\\prod_{i=1}^{\\infty}{\\lp1+\\ell^{-i}\\ensuremath{\\right)}^{-1}}.\n\\]\n\\end{product}\n\\begin{proof}\nLet $R=r_1+r_2+\\cdots$ and, to get into the spirit of a $q$-series calculation, let $q=\\ell^{-1}$.\nUsing a product formula of Euler (see~\\cite{And}, p~19), we note that\n\\[\nR\n=-\\sum_{i=1}^{\\infty}{\\frac{q^{\\frac{i(i+1)}{2}}}{(1-q^i)\\cdots(1-q)}}\n=-\\sum_{i=1}^{\\infty}{\\frac{q^iq^{\\frac{i(i-1)}{2}}}{(1-q^i)\\cdots(1-q)}}\n=1-\\prod_{i=1}^{\\infty}{(1+q^i)}.\n\\]\nNow, by the definition of $t_i$ (and by using \\hyperref[converge]{Lemma~\\ref*{converge}} to rearrange the terms of the sum), we know\n\\[\n\\sum_{i=0}^{\\infty}{t_i}\n=1+R+R^2+R^3+\\cdots\n=\\frac{1}{1-R}\n=\\prod_{i=1}^{\\infty}{\\lp1+\\ell^{-i}\\ensuremath{\\right)}^{-1}},\n\\]\nas desired.\n\\end{proof}\n\nNext, we justify the reordering of the summands in \\hyperref[product]{Lemma~\\ref*{product}}:\n\n\\begin{converge} \\label{converge}\nFor any finite set of nonnegative integers $S$ and $i\\in\\ensuremath{{\\mathbb{Z}}}$ satisfying $i>\\max{S}$, let $\\rho^i_S:=\\ensuremath{\\left\\vert} r^i_S\\ensuremath{\\right\\vert}$.\nNext, let $\\tau_0=1$, let $\\tau_1=\\rho^1_{\\varnothing}$, and for any $i>1$, let\n\\[\n\\tau_i:=\\sum_{S\\subseteq\\{1,\\ldots i-1\\}}{\\rho^i_S}.\n\\]\nThen $\\sum_{i=0}^{\\infty}{\\tau_i}$ converges.\n\\end{converge}\n\\begin{proof}\nFor fun, we will give two proofs: a simple proof that holds for any $\\ell>3$, and a more complicated one that holds for $\\ell\\geq3$.\nNote that the sum clearly diverges for $\\ell=2$ since it includes infinitely many 1's.\n\nFor the simple proof, note that for any finite set $S$ of nonnegative integers and any $i>\\max{S}$, we know $\\rho^i_S\\leq(\\ell-1)^{-i}$.\nIt follows that for any $i\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$, we have that $\\tau_i\\leq2^{i-1}(\\ell-1)^{-i}$, so $\\sum_{i=0}^{\\infty}{\\tau_i}$ converges for $\\ell>3$.\n\nOf course, the above argument fails for $\\ell=3$.\nIn this case, for a finite set $S$ of nonnegative integers and an $i>\\max{S}$, we must use a (slightly) better bound than $\\rho^i_S\\leq(\\ell-1)^{-i}$.\nLet $\\lambda=(\\ell-1)^{-1}$.\nSince $(\\ell^m-1)^{-1}\\leq(\\ell-1)^{-m}$ for any $m\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$, if we let $S\\cup\\{0\\}=\\{s_0,\\ldots,s_j\\}$, where $0=s_0<s_1<\\cdots<s_{j+1}:=i$, then\n\\begin{equation} \\label{trianglebound}\n\\rho^i_S\n=\\prod_{k=0}^{j}{\\ensuremath{\\left\\vert} r_{s_{k+1}-s_k}\\ensuremath{\\right\\vert}}\n\\leq\\prod_{k=0}^{j}{\\lambda^{\\frac{1}{2}(s_{k+1}-s_k)(s_{k+1}-s_k+1)}}.\n\\end{equation}\nLet $T_i$ be the number of compositions of $i$ by triangular numbers.\nThen by rearranging the terms of $\\sum_{i=0}^{\\infty}{\\tau_i}$ to order them by the exponent of $\\lambda$ appearing in the bound~\\hyperref[trianglebound]{(\\ref*{trianglebound})}, we see that if $\\sum_{i=1}^{\\infty}{T_i\\lambda^i}$ converges, then so does $\\sum_{i=0}^{\\infty}{\\tau_i}$.\nSince the generating function for the number of compositions of positive triangular numbers is\n\\begin{equation} \\label{gftricomps}\n\\sum_{i=0}^{\\infty}{T_ix^i}=\\frac{1}{1-\\ensuremath{\\left(}\\sum_{j=1}^{\\infty}{x^{\\frac{1}{2}j(j+1)}}\\ensuremath{\\right)}},\n\\end{equation}\nwe need only show that the radius of convergence of~\\hyperref[gftricomps]{(\\ref*{gftricomps})} is at least $\\lambda$.\nSince $\\ell\\geq3$, we know that $\\lambda\\leq\\frac{1}{2}$, and\n\\[\n1>\\frac{1}{2}+\\ensuremath{\\left(}\\frac{1}{2}\\ensuremath{\\right)}^3+\\ensuremath{\\left(}\\frac{1}{2}\\ensuremath{\\right)}^6+\\ensuremath{\\left(}\\frac{1}{2}\\ensuremath{\\right)}^{10}+\\cdots,\n\\]\nso the lemma is true.\n\\end{proof}\n\nWe can now finish proving the result mentioned in \\hyperref[nutime]{Note~\\ref*{nutime}}.\n\n\\begin{final} \\label{final}\nSuppose $A\\in\\mathcal{G}$, $\\rho,\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$, and let $r=\\rank{A}$.\nIf $\\rho>\\xi$ and $\\ell^\\rho>\\exp{A}$, then\n\\[\n\\lim_{g\\to\\infty}\n{\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A)}\n=\\prod_{i=1}^{\\infty}{\\lp1+\\ell^{-i}\\ensuremath{\\right)}^{-1}}\\cdot\\sum_{B\\in\\mathcal{G}(r)}\n{\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} B\/\\ell^\\xi B\\ensuremath{\\right)}\\ensuremath{\\right\\vert}\\cdot\\frac{S(A,B)}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}}.\n\\]\n\\end{final}\n\\begin{proof}\nLet $B\\in\\mathcal{G}(r,s)$, let $S$ be a finite set of nonnegative integers, and let $i$ a positive integer with $i>\\max{S}$.\nSuppose $S\\cup\\lb0\\ensuremath{\\right\\}}=\\ensuremath{\\left\\{} s_0,\\ldots,s_j\\ensuremath{\\right\\}}$, where $0=s_0<\\cdots<s_{j+1}:=i$.\nNow, we know by~\\cite{GAb} that for any $k,m\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$ with $k\\leq m$\n\\[\n\\sub{\\ensuremath{\\left(} B_{\\oplus k},B_{\\oplus m}\\ensuremath{\\right)}}\n=\\frac{\\ell^{(r+k)(m-k)}\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{r-s+m}}\n{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{r-s+i}\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{m-k}}\n\\]\nand\n\\[\n\\ensuremath{\\left\\vert}\\Aut{B_{\\oplus i}}\\ensuremath{\\right\\vert}\n=\\frac{\\ell^{2ir+i^2}\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{r-s+i}}{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{r-s}}\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert},\n\\]\nso\n\\begin{align*}\n(-1)^{j+1}\\cdot\\frac{\\ell^{ir+\\frac{i(i-1)}{2}}}{\\ensuremath{\\left\\vert}\\Aut{B_{\\oplus i}}\\ensuremath{\\right\\vert}}\n\\cdot\\prod_{k=0}^j\n{\\sub{\\ensuremath{\\left(} B_{\\oplus s_k},B_{\\oplus s_{k+1}}\\ensuremath{\\right)}}}\n&=(-1)^{j+1}\n\\cdot\\frac{\\ell^{-ir-\\frac{i(i+1)}{2}}}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}\n\\cdot\\frac{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{r-s}}{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{r-s+i}}\n\\cdot\\prod_{k=0}^j\n{\\frac{\\ell^{(r+s_k)(s_{k+1}-s_k)}\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{r-s+s_{k+1}}}\n{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{r-s+s_k}\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{s_{k+1}-s_k}}}\\\\\n&=(-1)^{j+1}\n\\cdot\\frac{\\ell^{-ir-\\frac{i(i+1)}{2}}}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}\n\\cdot\\prod_{k=0}^j\n{\\frac{\\ell^{(r+s_k)(s_{k+1}-s_k)}}\n{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{s_{k+1}-s_k}}}\\\\\n&=(-1)^{j+1}\n\\cdot\\frac{1}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}\n\\cdot\\prod_{k=0}^j\n{\\frac{\\ell^{-\\frac{1}{2}\\ensuremath{\\left(} s_{k+1}-s_k\\ensuremath{\\right)}\\ensuremath{\\left(} s_{k+1}-s_k+1\\ensuremath{\\right)}}}\n{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_{s_{k+1}-s_k}}}.\n\\end{align*}\nBut by \\hyperref[product]{Lemma~\\ref*{product}}, this means that \n\\[\n\\sum_{i=0}^\\infty\n{\\ell^{ir+\\frac{i(i-1)}{2}}\\frac{S(B,B_{\\oplus i})}{\\ensuremath{\\left\\vert}\\Aut{B_{\\oplus i}}\\ensuremath{\\right\\vert}}}\n=\\frac{1}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}\\cdot\\sum_{i=0}^\\infty{t_i}\n=\\frac{1}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}\\cdot\\prod_{i=1}^{\\infty}{\\lp1+\\ell^{-i}\\ensuremath{\\right)}^{-1}},\n\\]\nso we conclude by \\hyperref[nutime]{Note~\\ref{nutime}}.\n\\end{proof}\n\n\\subsection{The main results}\n\nTo conclude we mention two corollaries of \\hyperref[final]{Theorem~\\ref*{final}}, one trivial and one nontrivial.\n\n\\begin{trivial} \\label{trivial}\n\nIf $x\\in\\ensuremath{{\\mathbb{Z}}}_\\ell$ satisfies $x\\equiv1\\pmod{\\ell}$, then\n\\[\n\\lim_{g\\to\\infty}\n{\\mu_{2g}^{(x)}(\\lb0\\ensuremath{\\right\\}})}\n=\\prod_{i=1}^{\\infty}{\\lp1+\\ell^{-i}\\ensuremath{\\right)}^{-1}}.\n\\]\n\\end{trivial}\n\nThe nontrivial corollary relies heavily on calculations from~\\cite{GAb}.\n\n\\begin{finalc} \\label{finalc}\nSuppose $r,s\\in\\ensuremath{{\\mathbb{Z}}}^{\\geq0}$ with $r\\geq s$.\nFurthermore, suppose that $x\\in\\ensuremath{{\\mathbb{Z}}}_\\ell$ and $\\xi\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ with $x\\equiv1\\pmod{\\ell^\\xi}$ but $x\\nequiv1\\pmod{\\ell^{\\xi+1}}$.\nIf $A\\in\\mathcal{G}(r,s)$, then\n\\[\n\\lim_{g\\to\\infty}\n{\\mu_{2g}^{(x)}(A)}\n=\\begin{cases}\n\\ell^{\\frac{r(r-1)}{2}}\\cdot\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_r\n\\cdot\\frac{\\prod_{i=1}^{\\infty}{(1+\\ell^{-i})^{-1}}}{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}},\n&\\text{if }\\xi=1\\\\\n\\ell^{\\frac{r(r-1)}{2}+\\frac{s(s-1)}{2}}\n\\cdot\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_s\\ensuremath{\\left(}\\ell^{-1};\\ell^{-2}\\ensuremath{\\right)}_{\\lceil \\frac{r-s}{2}\\rceil}\n\\cdot\\frac{\\prod_{i=1}^{\\infty}{(1+\\ell^{-i})^{-1}}}{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}},\n&\\text{if }\\xi=2.\n\\end{cases}\n\\]\n\\end{finalc}\n\\begin{proof}\nChoose any $\\rho\\in\\ensuremath{{\\mathbb{Z}}}^{>0}$ with $\\rho>\\xi$ and $\\ell^\\rho>\\exp{A}$.\nThen by \\hyperref[haar]{Lemma~\\ref*{haar}} we know\n\\[\n\\mu_{2g}^{(x)}(A)=\\nu_{2g,\\rho}^{\\ensuremath{\\left\\langle}\\xi\\ensuremath{\\right\\rangle}}(A).\n\\]\nNow, we know from~\\cite{GAb} that\n\\[\n\\sum_{B\\in\\mathcal{G}(r)}\n{\\frac{S(A,B)}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}}\n=\\frac{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_r}{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}},\n\\]\nand for any $i\\in\\ensuremath{\\left\\{} s,\\ldots,r\\ensuremath{\\right\\}}$:\n\\[\n\\sum_{B\\in\\mathcal{G}(r,i)}\n{\\frac{S(A,B)}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}}\n=(-1)^{i-s}\\cdot\\ell^{\\frac{s(s+1)}{2}-\\frac{i(i+1)}{2}}\\cdot\n\\binom{r-s}{r-i}_{\\ell^{-1}}\n\\cdot\\frac{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_s}\n{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}.\n\\]\nThe $\\xi=1$ case follows from \\hyperref[forms]{Note~\\ref*{forms}}.\nFor $\\xi=2$, use \\hyperref[forms]{Note~\\ref*{forms}} again to see that\n\\begin{align*}\n\\sum_{B\\in\\mathcal{G}(r)}\n{\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} B\/\\ell^2B\\ensuremath{\\right)}\\ensuremath{\\right\\vert}\n\\cdot\\frac{S(A,B)}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}}\n&=\\sum_{i=s}^r\n{\\sum_{B\\in\\mathcal{G}(r,i)}\n{\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} B\/\\ell^2B\\ensuremath{\\right)}\\ensuremath{\\right\\vert}\n\\cdot\\frac{S(A,B)}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}}}\\\\\n&=\\sum_{i=s}^r\n{(-1)^{i-s}\\cdot\\ell^{\\frac{r(r-1)}{2}+\\frac{s(s+1)}{2}-i}\n\\cdot\\binom{r-s}{r-i}_{\\ell^{-1}}\n\\cdot\\frac{\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_s}\n{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}}\\\\\n&=\\frac{\\ell^{\\frac{r(r-1)}{2}+\\frac{s(s+1)}{2}}\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_s}\n{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}\n\\cdot\\sum_{i=s}^r\n{(-1)^{i-s}\\cdot\\binom{r-s}{r-i}_{\\ell^{-1}}\\cdot\\ell^{-i}}.\n\\end{align*}\nLetting $k=r-s$ and $q=1\/\\ell$, we apply formula (1.10) from~\\cite{Kup}, which is a corollary of formula (1.12), to obtain\n\\begin{align*}\n\\sum_{B\\in\\mathcal{G}(r)}\n{\\ensuremath{\\left\\vert}\\Lambda\\ensuremath{\\left(} B\/\\ell^2B\\ensuremath{\\right)}\\ensuremath{\\right\\vert}\\cdot\\frac{S(A,B)}{\\ensuremath{\\left\\vert}\\Aut{B}\\ensuremath{\\right\\vert}}}\n&=\\frac{\\ell^{\\frac{r(r-1)}{2}+\\frac{s(s-1)}{2}}\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_s}\n{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}\n\\cdot\\sum_{i=0}^k{(-1)^i\\binom{k}{i}_qq^i}\\\\\n&=\\frac{\\ell^{\\frac{r(r-1)}{2}+\\frac{s(s-1)}{2}}\\ensuremath{\\left(}\\ell^{-1}\\ensuremath{\\right)}_s}\n{\\ensuremath{\\left\\vert}\\Aut{A}\\ensuremath{\\right\\vert}}\n\\cdot\\ensuremath{\\left(} q;q^2\\ensuremath{\\right)}_{\\lceil \\frac{k}{2}\\rceil},\n\\end{align*}\nas desired.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec_intro}\n\\vspace{-.2cm}\n\nSince we first developed language, humans have always told stories. Fashioning a good story is an act of creativity and developing algorithms to replicate this has been a long running challenge. Adding pictures as input \ncan provide information for guiding story construction by offering visual illustrations of the storyline.\nIn the related task of image captioning, most methods try to generate descriptions only for individual images or for short videos depicting a single activity.  Very recently, datasets have been introduced that extend this task to longer temporal sequences such as movies or photo albums~\\cite{rohrbach2016movie,pan2016jointly,lu2013story,huang2016visual}. \n\nThe type of data we consider in this paper provides input illustrations for story generation in the form of photo albums, sampled over a few minutes to a few days of time. For this type of data, generating textual descriptions involves telling a temporally consistent story about the depicted visual information, where stories must be coherent and take into account the temporal context of the images. Applications of this include constructing visual and textual summaries of albums, or even enabling search through personal photo collections to find photos of life events. \n\nPrevious visual storytelling works can be classified into two types, vision-based and language-based, where image or language stories are constructed respectively.\nAmong the vision-based approaches, unsupervised learning is commonly applied:\ne.g.,~\\cite{sigurdsson2016learning} learns the latent temporal dynamics given a large amount of albums, and~\\cite{kim2014reconstructing} formulate the photo selection as a sparse time-varying directed graph. \nHowever, these visual summaries tend to be difficult to evaluate and selected  photos may not agree with human selections. For language-based approaches, a sequence of natural language sentences are generated to describe a set of photos.\nTo drive this work ~\\cite{park2015expressing} collected a dataset mined from Blog Posts. However, this kind of data often contains contextual information or loosely related language. A more direct dataset was recently released~\\cite{huang2016visual}, where multi-sentence stories are collected describing photo albums via Amazon Mechanical Turk. \n\n\nIn this paper, we make use of the Visual Storytelling Dataset~\\cite{huang2016visual}.\nWhile the authors provide a seq2seq baseline, they only deal with the task of generating stories given 5-representative (summary) photos hand-selected by people from an album.\nInstead, we focus on the more challenging and realistic problem of end-to-end generation of stories from entire albums. This requires us to either generate a story from all of the album's photos or to learn selection mechanisms to identify representative photos and then generate stories from those summary photos. We evaluate each type of approach. \n\nUltimately, we propose a model of hierarchically-attentive recurrent neural nets, consisting of three RNN stages. The first RNN encodes the whole album context and each photo's content, the second RNN provides weights for photo selection, and the third RNN takes the weighted representation and decodes to the resulting sentences. Note that during training, we are only given the full input albums and the output stories, and our model needs to learn the summary photo selections latently.\n\n\nWe show that our model achieves better performance over baselines under both automatic  metrics and human evaluations. As a side product, we show that the latent photo selection also reasonably mimics human selections. Additionally, we propose an album retrieval task that can reliably pick the correct photo album given a sequence of sentences, and find that our model also outperforms the baselines on this task.\n\n\n\n\n\n\n\n\n\n\\section{Related work}\n\\vspace{-.2cm}\n\nRecent years have witnessed an explosion of interest in vision and language tasks, reviewed below. \n\n\n\\noindent{\\bf Visual Captioning:}\nMost recent approaches to image captioning~\\cite{vinyals2015show, xu2015show} have used CNN-LSTM structures to generate descriptions. For captioning video or movie content~\\cite{venugopalan2015sequence, pan2016jointly}, sequence-to-sequence models are widely applied, where the first sequence encodes video frames and the second sequence decodes the description. Attention techniques~\\cite{xu2015show, yu2016video, yao2015describing} are commonly incorporated for both tasks to localize salient temporal or spatial information.\n\n\\noindent{\\bf Video Summarization:}\nSimilar to documentation summarization~\\cite{rush2015neural, cheng2016neural, mei2016selective, woodsend2010automatic} which extracts key sentences and words, video summarization selects key frames or shots.\nWhile some approaches use unsupervised learning~\\cite{lu2013story, khosla2013large} or intuitive criteria to pick salient frames, recent models learn from human-created summaries~\\cite{gygli2015video, zhang2016video, zhang2016summary, gong2014diverse}. \nRecently, to better exploit semantics,~\\cite{choi2017textually} proposed textually customized summaries. \n\n\\noindent{\\bf Visual Storytelling:}\nVisual storytelling tries to tell a coherent visual or textual story about an image set. \nPrevious works include storyline graph modeling~\\cite{kim2014reconstructing}, unsupervised mining~\\cite{sigurdsson2016learning}, blog-photo alignment~\\cite{kim2015joint}, \nand language re-telling~\\cite{huang2016visual, park2015expressing}.\nWhile~\\cite{park2015expressing} collects data by mining Blog Posts,~\\cite{huang2016visual} collects stories using Mechanical Turk, providing more directly relevant stories.\n\n\n\n\\section{Model}\n\\vspace{-.2cm}\n\nOur model (Fig.~\\ref{fig:model}) is composed of three modules: Album Encoder, Photo Selector, and Story Generator, jointly learned during training.\n\n\\subsection{Album Encoder}\\label{sec:album_encoder}\n\\vspace{-.1cm}\n\nGiven an album $A=\\{a_1, a_2, ..., a_n\\}$, composed of a set of photos, we use a bi-directional RNN to encode the local album context for each photo.\nWe first extract the 2048-dimensional visual representation $f_i\\in R^k$ for each photo using ResNet101~\\cite{he2016deep}, then\na bi-directional RNN is applied to encode the full album.\nFollowing~\\cite{huang2016visual}, we choose a Gated Recurrent Unit (GRU) as the RNN unit to encode the photo sequence.\nThe sequence output at each time step encodes the local album context for each photo (from both directions).\nFused with the visual representation followed by ReLU, our final photo representation is (top module in Fig.~\\ref{fig:model}):\n\\vspace{-.2cm}\n\\begin{equation}\n\\begin{split}\\nonumber\nf_i &= \\mbox{ResNet}(a_i) \\\\ \n\\vec{h}_i &= \\vec{\\mbox{GRU}}_{album}(f_i, \\vec{h}_{i-1}) \\\\\n\\cev{h}_i &= \\cev{\\mbox{GRU}}_{album}(f_i, \\cev{h}_{i+1}) \\\\\nv_i &= \\mbox{ReLU}([\\vec{h}_i, \\cev{h}_i ] + f_i).\n\\end{split}\n\\end{equation}\n\\vspace{-.3cm}\n\n\\vspace{-.4cm}\n\\subsection{Photo Selector}\\label{sec:photo_selector}\n\\vspace{-.1cm}\nThe Photo Selector (illustrated in the middle yellow part of Fig.~\\ref{fig:model}) identifies representative photos to summarize an album's content. As discussed, we do not assume that we are given the ground-truth album summaries during training, instead regarding selection as a latent variable in the end-to-end learning.\nInspired by Pointer Networks~\\cite{vinyals2015pointer}, we use another GRU-RNN to perform this task \\footnote{While the pointer network requires grounding labels, we regard the labels as latent variables}.\n\n\nGiven the album representation $V^{n\\times k}$, the photo selector outputs probabilities $p_t\\in R^n$ (likelihood of selection as $t$-th summary image) for all photos using soft attention. \n\\vspace{-.1cm}\n\\begin{equation}\\nonumber\n\\begin{split}\n&\\bar{h}_t = \\mbox{GRU}_{select}(p_{t-1}, \\bar{h}_{t-1}), \\\\\n&p(y_{a_i}(t)=1) = \\sigma(\\mbox{MLP}([\\bar{h}_t, v_i])),\n\\end{split}\n\\end{equation}\nAt each summarization step, $t$, the GRU takes the previous $p_{t-1}$ and previous hidden state as input, and outputs the next hidden state $\\bar{h}_t$.\n$\\bar{h}_t$ is fused with each photo representation $v_i$ to compute the $i^{th}$ photo's attention $p_t^i = p(y_{a_i}(t)=1)$.\nAt test time, we simply pick the photo with the highest probability to be the summary photo at step $t$.\n\n\\subsection{Story Generator}\n\\vspace{-.1cm}\n\n\nTo generate an album's story, given the album representation matrix $V$ and photo summary probabilities $p_t$ from the first two modules, we compute the visual summary representation $g_t \\in R^k$ (for the $t$-th summary step).\nThis is a weighted sum of the album representations, i.e., $g_t = p_t^T V$.  Each of these 5 $g_t$ embeddings (for $t = 1$ to $5$) is then used to decode 1 of the 5 story sentences respectively, as shown in the blue part of Fig.~\\ref{fig:model}.\n\nGiven a story $S = \\{s_t\\}$, where $s_t$ is $t$-th summary sentence. \nFollowing~\\newcite{donahue2015long}, the $l$-th word probability of the $t$-th sentence is:\n\\vspace{-.1cm}\n\\begin{equation}\n\\begin{split}\nw_{t,l-1} &= W_e s_{t,l-1}, \\\\\n\\tilde{h}_{t, l} &= \\mbox{GRU}_{story}(w_{t, l-1}, g_t, \\tilde{h}_{t, l-1}),\\\\\np(s_{t,l}) &= \\mbox{softmax}(\\mbox{MLP}(\\tilde{h}_{t, l})),\n\\end{split}\n\\end{equation}\n\\vspace{-.1cm}\nwhere $W_e$ is the word embedding.\nThe GRU takes the joint input of visual summarization $g_t$, the previous word embedding $w_{t,l}$, and the previous hidden state, then outputs the next hidden state.\nThe generation loss is then the sum of the negative log likelihoods of the correct words: $L_{gen}(S) = -\\sum_{t=1}^T\\sum_{l=1}^{L_t} \\log p_{t,l}(s_{t,l})$. \n\\vspace{.1cm}\n\n\nTo further exploit the notion of temporal coherence in a story, we add an order-preserving constraint  to order the sequence of sentences within a story (related to the story-sorting idea in~\\newcite{agrawal2016sort}).\nFor each story $S$ we randomly shuffle its 5 sentences to generate negative story instances $S'$.\nWe then apply a max-margin ranking loss to encourage correctly-ordered stories: $L_{rank}(S, S') = \\max(0, m-\\log p(S')+\\log p(S))$.\nThe final loss is then a combination of the generation and ranking losses:  \n\\begin{equation}\nL = L_{gen}(S) + \\lambda L_{rank}(S, S').\\label{eqn:loss}\n\\end{equation}\n\n\n\\section{Experiments}\n\nWe use the Visual Storytelling Dataset~\\cite{huang2016visual}, consisting of 10,000 albums with 200,000 photos.\nEach album contains 10-50 photos taken within a 48-hour span with two annotations: 1) 2 album summarizations, each with 5 selected representative photos, and 2) 5 stories describing the selected photos. \n\n\\subsection{Story Generation}\n\nThis task is to generate a 5-sentence story describing an album.\nWe compare our model with two sequence-to-sequence baselines: 1) an encoder-decoder model (enc-dec), where the sequence of album photos is encoded and the last hidden state is fed into the decoder for story generation,\n2) an encoder-attention-decoder model~\\cite{xu2015show} (enc-attn-dec) with weights computed using a soft-attention mechanism.\nAt each decoding time step, a weighted sum of hidden states from the encoder is decoded. For fair comparison, we use the same album representation (Sec.~\\ref{sec:album_encoder}) for the baselines.\n\n\\begin{table}[t]\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{l|c c c c}\n\\multicolumn{5}{c}{beam size=3}\\\\\n\\hline\n& Bleu3 & Rouge & Meteor & CIDEr \\\\\n\\hline\nenc-dec & 19.58 & 29.23 & 33.02 & 4.65 \\\\\nenc-attn-dec & 19.73 & 28.94 & 32.98 & 4.96 \\\\\nh-attn & 20.53 & 29.82 & 33.81 & 6.84 \\\\\nh-attn-rank & \\bf{20.78} & \\bf{29.82} & \\bf{33.94} & \\bf{7.38} \\\\\n\\hline\nh-(gd)attn-rank & 21.02 & 29.53 & 34.12 & 7.51\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-.4cm}\n\\caption{Story generation evaluation.}\n\\vspace{-.1cm}\n\\label{table:generation}\n\\end{table}\n\n\n\n\\begin{table}[t]\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{l | l}\n\\hline\nenc-dec (29.50\\%) & h-attn-rank (70.50\\%)\\\\\n\\hline\nenc-attn-dec (30.75\\%) & h-attn-rank (69.25\\%) \\\\\n\\hline\n\\hline\nh-attn-rank (30.50\\%) & gd-truth (69.50\\%) \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-.4cm}\n\\caption{Human evaluation showing how often people prefer one model over the other.}\n\\vspace{-.2cm}\n\\label{table:human}\n\\end{table}\n\n\\begin{table}[t]\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{l | c c }\n\\hline\n& precision & recall \\\\\n\\hline\nDPP & 43.75\\% & 27.41\\% \\\\\n\\hline\nenc-attn-dec & 38.53\\% & 24.25\\% \\\\\n\\hline\nh-attn \t\t & 42.85\\% & 27.10\\% \\\\\n\\hline\nh-attn-rank  & \\bf{45.51}\\% & \\bf{28.77}\\% \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-.4cm}\n\\caption{Album summarization evaluation.}\n\\vspace{-.1cm}\n\\label{table:summarization}\n\\end{table}\n\n\\begin{table}[t]\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{l | c c c c}\n\\hline\n& R@1 & R@5 & R@10 & MedR \\\\\n\\hline\nenc-dec      & 10.70\\% & 29.30\\% & 41.40\\% & 14.5 \\\\\n\\hline\nenc-attn-dec & 11.60\\% & 33.00\\% & 45.50\\% & 11.0 \\\\\n\\hline  \nh-attn       & 18.30\\% & \\bf{44.50}\\% & \\bf{57.60}\\% & \\bf{6.0} \\\\\n\\hline\nh-attn-rank  & \\bf{18.40}\\% & 43.30\\% & 55.50\\% & 7.0 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-.4cm}\n\\caption{1000 album retrieval evaluation.}\n\\vspace{-.1cm}\n\\label{table:retrieval}\n\\end{table}\n\n\n\nWe test two variants of our model trained with and without ranking regularization by controlling $\\lambda$ in our loss function, denoted as h-attn (without ranking), and h-attn-rank (with ranking).\nEvaluations of each model are shown in Table~\\ref{table:generation}.\nThe h-attn outperforms both baselines, and h-attn-rank achieves the best performance for all metrics.\nNote, we use beam-search with beam size=3 during generation for a reasonable performance-speed trade-off (we observe similar improvement trends with beam size = 1).\\footnote{We also compute the $p$-value of Meteor on 100K samples via the bootstrap test~\\cite{efron1994introduction}, as Meteor has better agreement with human judgments than Bleu\/Rouge~\\cite{huang2016visual}.\nOur h-attn-rank model has strong statistical significance ($p=0.01$) over the enc-dec and enc-attn-dec models (and is similar to the h-attn model).}\nTo test performance under optimal image selection, we use one of the two ground-truth human-selected 5-photo-sets as an oracle to hard-code the photo selection, denoted as h-(gd)attn-rank.\nThis achieves only a slightly higher Meteor compared to our end-to-end model.\n\nAdditionally, we also run human evaluations in a forced-choice task where people choose between stories generated by different methods.\nFor this evaluation, we select 400 albums, each evaluated by 3 Turkers. Results are shown in Table~\\ref{table:human}. \nExperiments find significant preference for our model over both baselines. As a simple Turing test, we also compare our results with human written stories (last row of Table~\\ref{table:human}), indicating room for improvement of methods.\n\n\n\\subsection{Album Summarization}\nWe evaluate the precision and recall of our generated summaries (output by the photo selector) compared to human selections (the combined set of both human-selected 5-photo stories). \nFor comparison, we evaluate enc-attn-dec on the same task by aggregating predicted attention and selecting the 5 photos with highest accumulated attention.\nAdditionally, we also run DPP-based video summarization~\\cite{kulesza2012determinantal} using the same album features.\nOur models have higher performance compared to baselines as shown in Table~\\ref{table:summarization} (though DPP also achieves strong results, indicating that there is still room to improve the pointer network).\n\n\\vspace{-0.2cm}\n\\subsection{Output Example Analysis}\nFig.~\\ref{fig:generation_a} and Fig.~\\ref{fig:generation_b} shows several output examples of the joint album summarization and storytelling generation.\nWe compare our full model h-attn-rank with the baseline enc-attn-dec, as both models are able to do the album summarization and story generation tasks jointly.\nIn Fig.~\\ref{fig:generation_a} and Fig.~\\ref{fig:generation_b}, we use blue dashed box and red box to indicate the album summarization by the two models respectively.\nAs reference, we also show the ground-truth album summaries by randomly selecting 1 out of 2 human album summaries, which are highlighted with green box.\nBelow each album are their generated stories.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.98\\textwidth]{figures\/all_example_with_gd.jpg}\n\\caption{Examples of album summarization and storytelling by enc-attn-dec (blue), h-attn-rank (red), and ground-truth (green). We randomly select 1 out of 2 human album summaries as ground-truth here.}\n\\label{fig:generation_a}\n\\end{figure*}\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.98\\textwidth]{figures\/all_example_with_gd2.jpg}\n\\caption{More examples of album summarization and storytelling by enc-attn-dec (blue), h-attn-rank (red), and ground-truth (green). We randomly select 1 out of 2 human album summaries as ground-truth here.}\n\\label{fig:generation_b}\n\\end{figure*}\n\n\n\\subsection{Album Retrieval}\nGiven a human-written story, we introduce a task to retrieve the album described by that story. \nWe randomly select 1000 albums and one ground-truth story from each for evaluation. \nUsing the generation loss, we compute the likelihood of each album $A_m$ given the query story $S$ and retrieve the album with the highest generation likelihood, $A = \\mbox{argmax}_{A_m} p(S|A_m)$.\nWe use Recall@k and Median Rank for evaluation.\nAs shown in Table~\\ref{table:retrieval}), we find that our models outperform the baselines, but the ranking term in Eqn.\\ref{eqn:loss} does not improve performance significantly.\n\n\\section{Conclusion}\n\\vspace{-0.3cm}\nOur proposed hierarchically-attentive RNN based models for end-to-end visual storytelling can jointly summarize and generate relevant stories from full input photo albums effectively. Automatic and human evaluations show that our method outperforms strong sequence-to-sequence baselines on selection, generation, and retrieval tasks.\n\n\n\\section*{Acknowledgments}\n\\vspace{-0.4cm}\nWe thank the anonymous reviewers for their helpful comments. This research is supported by NSF Awards \\#1633295, 1444234, 1445409, 1562098.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\n    The \\emph{Tur\\'an number} of a graph $H$, denoted $\\ex(n, H)$, is the maximum possible number of edges in an $n$-vertex graph that does not contain a copy of $H$. In this paper we study a rainbow variant of Tur\\'an numbers, introduced by Keevash, Mubayi, Sudakov and Verstra\\\"ete \\cite{keevash2007rainbow}. A \\emph{proper edge-colouring} of a graph is an assignment of colours to its edges so that edges that share a vertex have distinct colours. A \\emph{rainbow} subgraph of an edge-coloured graph is a subgraph whose edges have distinct colours.\n    The \\emph{rainbow Tur\\'an number} of a graph $H$, denoted $\\ex^*(n, H)$, is the maximum possible number of edges in a properly edge-coloured graph on $n$ vertices with no rainbow copy of $H$. One can define $\\ex(n, \\mathcal{H})$ and $\\ex^*(n, \\mathcal{H})$ analogously for a family of graphs $\\mathcal{H}$.\n    \n    It was shown in \\cite{keevash2007rainbow} that\n    $\\ex^*(n, H) = (1 + o(1))\\ex(n, H)$ for non-bipartite $H$. Perhaps unsurprisingly, little is known about rainbow Tur\\'an numbers of bipartite graphs. The authors of \\cite{keevash2007rainbow} raised two problems concerning rainbow Tur\\'an numbers of even cycles, one concerning an even cycle of fixed length $2k$ and the other concerning the family $\\mathcal{C}$ of all cycles. \n    For all $k\\geq 2$,  they showed that $\\ex^*(n, C_{2k}) = \\Omega(n^{1 + 1\/k})$ and conjectured that\n    $\\ex^*(n, C_{2k}) = \\Theta(n^{1 + 1\/k})$. The authors of \\cite{keevash2007rainbow} verified the conjecture\n    for $k \\in \\{2, 3\\}$. Following further progress on the conjecture by Das, Lee and Sudakov \\cite{das2013rainbow},  Janzer \\cite{janzer2020rainbow} recently resolved the conjecture. \n    \n    Regarding the rainbow Tur\\'an number of the family $\\mathcal{C}$ of all cycles,\n    Keevash, Mubayi, Sudakov and Verstra\\\"ete \\cite{keevash2007rainbow} showed that $\\ex^*(n, \\mathcal{C}) = \\Omega(n \\log n)$, by considering a naturally defined proper edge-colouring of the hypercube $Q_k$, where $k=\\lfloor \\log n \\rfloor$. They also showed that $\\ex^*(n, \\mathcal{C}) = O(n^{4\/3})$ and\n    asked if $\\ex^*(n, \\mathcal{C}) = O(n^{1+o(1)})$ and furthermore, if $\\ex^*(n, \\mathcal{C}) = O(n \\log n)$.\n    Das, Lee and Sudakov \\cite{das2013rainbow} answered the first question affirmatively, by showing that $\\ex^*(n, \\mathcal{C})  \\le ne^{(\\log n)^{\\frac{1}{2}+o(1)}}$. Recently, Janzer \\cite{jiang2021rainbow} improved this bound by establishing that $\\ex^*(n, \\mathcal{C}) = O\\big(n (\\log n)^4\\big)$, which is tight up to a polylogarithmic factor. Very recently, Jiang, Methuku and Yepremyan \\cite{jiang2021rainbow} proved the following generalisation of Das, Lee and Sudakov \\cite{das2013rainbow} on $\\ex^*(n,\\mathcal{C})$.\n   \n\n    \\begin{theorem}[Jiang, Methuku, Yepremyan~\\cite{jiang2021rainbow}]\n    \\label{thm:JMY}\n        For every integer $m \\geq 2$ there exists a constant $c>0$ such that for every integer $n\\geq m$ the following holds. If $G$ is a properly edge-coloured graph  on $n$ vertices with at least $ne^{c\\sqrt{\\log n}}$ edges, then $G$ contains a rainbow subdivision of $K_m$, where each edge is subdivided at most $1300 \\log^2 n$ times.\n    \\end{theorem}\n    The method used in \\cite{jiang2021rainbow} utilises robust expanders in the coloured setting together with a density increment argument, inspired in part by the method introduced by Sudakov and Tomon~\\cite{sudakov2020extremal}.\n    \n    In this paper, we lower the $e^{O(\\sqrt{\\log n})}$ error term in Theorem~\\ref{thm:JMY} to a polylogarithmic term, which in conjunction with the above-mentioned $\\Omega(n\\log n)$ lower bound on $\\ex^*(n,\\mathcal{C})$ determines the rainbow Tur\\'an number of the family of $K_m$-subdivisions up to a polylogarithmic factor.\n    \n    \n    \\begin{theorem}\\label{thm:main}\n        Fix an integer $m \\geq 2$ and let $n$ be sufficiently large.\n        Suppose that $G$ is a properly edge-coloured graph on $n$ vertices with at least $n (\\log n)^{60}$ edges. Then $G$ contains a rainbow subdivision of $K_m$, where each edge is subdivided at most $(\\log n)^{6}$ times.\n    \\end{theorem}\n\n    Our proof exploits the connection between mixing time of random walks and edge expansion.\n    This connection is used in conjunction with counting lemmas developed by Janzer in \\cite{janzer2020rainbow} regarding homomorphisms of cycles in graphs.\n    We also prove a strengthening of \\Cref{thm:main}, regarding `rooted' rainbow subdivisions of $K_m$ in expanders (see \\Cref{thm:main-rooted}). For this stronger version, in addition to the ingredients used for proving Theorem~\\ref{thm:main}, we use the framework of \\cite{jiang2021rainbow} and an additional idea used by Letzter in  \\cite{letzter2021tight} \n    (see Lemma~\\ref{lem:reachable-robust-new}).\n   \n    The framework that we develop in this paper is quite general. In addition to using it to prove Theorem~\\ref{thm:main}, we also apply\n    it to obtain a generalisation of the following result of Janzer \\cite{janzer2020rainbow} concerning the Tur\\'an number of  blow-ups of cycles. For an integer $r\\geq 1$ and a graph $F$, the \\emph{$r$-blow-up} of $F$, denoted $F[r]$, is the graph obtained by replacing each vertex of $F$ with an independent set of size $r$ and each edge of $F$ by a $K_{r,r}$. Let $\\mathcal{C}_0[r] \\coloneqq \\{C_{2k}[r] : k \\ge 2\\}$. Answering a question of Jiang and Newman~\\cite{jiang2017small}, Janzer~\\cite{janzer2020rainbow} proved the following.\n    \n    \\begin{theorem}[Janzer~\\cite{janzer2020rainbow}] \\label{thm:oliver-blowup}\n       Let $r \\ge 1$ be a fixed integer. Then $\\ex(n, \\mathcal{C}_0[r]) = O\\big(n^{2-1\/r} (\\log n)^{7\/r}\\big)$.\n    \\end{theorem}\n    This bound is tight up to a polylogarithmic factor as random graphs show that $\\ex(n, \\mathcal{C}_0[r])=\\Omega(n^{2-1\/r})$.   We generalise Theorem~\\ref{thm:oliver-blowup} by proving the following. \n  \n    \\begin{theorem} \\label{thm:main-blowups}\n        Let $r, m \\ge 1$ be fixed integers and let $n$ be sufficiently large. Suppose that $G$ is a graph on $n$ vertices with at least $n^{2-\\frac{1}{r}} (\\log n)^{\\frac{60}{r}}$ edges. Then $G$ contains an $r$-blow-up of a subdivision of $K_m$, where each edge is subdivided at most $(\\log n)^{6}$ times.\n    \\end{theorem}  \n\n    This bound is also tight up to a polylogarithmic factor as shown  by random graphs (see Proposition~\\ref{prop:lowerboundblowup}).\n    \n    The proof of \\Cref{thm:main-blowups} follows the proof of our first main result, \\Cref{thm:main}, with the additional use of a `balanced supersaturation' result, due to Morris and Saxton \\cite{morris2016number}.   Such a result gives us a collection of $K_{r,r}$'s in a sufficiently dense graph such that no copy of $K_{1, r}$ is contained in too many copies of $K_{r,r}$ (the result in \\cite{morris2016number} is more general but this condition suffices for our purposes).\n    This sort of result is usually used in conjunction with the container method in order to upper bound the number of $H$-free graphs. So it is quite interesting that we use this `balanced supersaturation' result for $K_{r,r}$'s in a new setting.\n    \n    The rest of the paper is organised as follows. In  \\Cref{sec:overview},\n    we give a short overview of our proofs, focusing on the proof of \\Cref{thm:main} and a strengthening of it (\\Cref{thm:main-rooted}).\n    In \\Cref{sec:prelims}, we mention various preliminary results, regarding the existence of expanders which are close to being regular and properties of expanders. In \\Cref{sec:non-rainbow-walks}, we state three lemmas due to Janzer \\cite{janzer2020rainbow} and some consequences of these lemmas. \\Cref{sec:random-walks} contains the main new ideas of the paper, exploiting a connection between the mixing time of a random walk and expansion properties in a graph. In \\Cref{sec:main-proof},\n    we prove \\Cref{thm:main} and a strengthening of it regarding rooted subdivisions in almost regular expanders. \n    In \\Cref{sec:blowups}, we prove \\Cref{thm:main-blowups} about blow-ups of subdivisions. We complete the paper with concluding remarks in \\Cref{sec:conc}.\n    \n    Throughout the paper, for convenience, we drop floor and ceiling signs for large numbers, and logarithms are in base $2$.\n\n\\section{Overview of the proofs} \\label{sec:overview}\n    Our main idea is to use the connection between the mixing time of random walks, the notion of `conductance' (see \\Cref{def:conductance}) and our notion of expansion.     \n    It is a well-known and very useful fact that `large' conductance implies `small' mixing time (see, e.g., Lov\\'asz \\cite{lovasz1993random}). We show  that expanders which are close to being regular have large conductance, and thus conclude that long enough paths are close to being uniformly distributed in such expanders.     \n    We also use two counting lemmas of Janzer from~\\cite{janzer2020rainbow}. Below we describe these lemmas and the main ideas in more detail.\n    \n\n    \n    In a properly edge-coloured graph, say that a closed walk is \\emph{degenerate} if it is either not rainbow or visits a vertex more than once. \n    The first lemma from \\cite{janzer2020rainbow} implies that in a properly edge-coloured graph which is close to being regular, the number of degenerate closed $2k$-walks is significantly smaller than the number of closed $2k$-walks, provided that $k$ is sufficiently large.\n    \n    Given two vertices $x$ and $y$, a closed $2k$-walk $W$ is said to \\emph{be hosted} by $x$ and $y$ if it starts at $x$ and reaches $y$ after $k$ steps. We call a pair of vertices $(x, y)$ \\emph{good} if the number of degenerate closed $2k$-walks hosted by $x$ and $y$ is significantly smaller than the number of closed $2k$-walks hosted by $x$ and $y$. The second lemma from \\cite{janzer2020rainbow} that we use shows that if a pair $(x, y)$ is good then there are many short pairwise colour-disjoint and internally vertex-disjoint $k$-paths from $x$ to $y$.\n\n    Using results about random walks on graphs, which relate mixing time to expansion,  we show that in an expander $G$ on $n$ vertices which is close to being regular, for $k$ suitably large (at least polylogarithmic in $n$), the numbers of closed $2k$-walks hosted by any two pairs of vertices are within a suitable polylogarithmic factor (in $n$) of each other. This, combined with the fact that the number of degenerate closed $2k$-walks is small compared to the total number of closed $2k$-walks (due to the first lemma above), implies that almost all pairs of vertices are good. Thus, using Tur\\'an's theorem, we find a copy of $K_m$ in the graph formed by good pairs.\n    This, together with the fact that there are many short colour-disjoint and internally vertex-disjoint rainbow paths between any good pair of vertices (due to the second lemma above) allows us to greedily build the desired rainbow-subdivision of $K_m$.\n    \n    \n    We also prove a stronger version of Theorem~\\ref{thm:main} (see \\Cref{thm:main-blowups}) asserting that in an expander $G$ which is close to being regular and whose average degree is large enough, for any set $S$ of $m$ vertices, there exists a rainbow $K_m$-subdivision with the vertices of $S$ being the branching vertices. The main step in this proof shows that for any two vertices $x$ and $y$ in $G$ there is a short rainbow $x,y$-path avoiding a prescribed small set $C$ of vertices and colours. By iterating this over all pairs of vertices in $S$, we can build the desired rainbow $K_m$-subdivision.\n    \n    To show that there is a short rainbow $x,y$-path in $G$, we first apply tools due to Jiang, Methuku and Yepremyan \\cite{jiang2021rainbow} and Letzter \\cite{letzter2021tight} to show that there is a set of vertices $U$ of size $\\Omega(n)$ such that for each $v\\in U$ there is such a short rainbow $x,v$-path $P(v)$ and a short rainbow $y,v$-path $Q(v)$, both of which avoid $C$, such that no colour is used on too many of these paths $P(v)$ and $Q(v)$. It easily follows that for almost all pairs $(u, v)$ with $u, v\\in U$, the paths $P(u)$ and $Q(v)$ are colour-disjoint.\n    This, combined with the fact that most pairs in $U$ are good (in the sense mentioned earlier), implies that there exists at least one good pair $(u,v)$ for which $P(u)$ and $Q(v)$ are colour-disjoint. This allows us to find a suitable short rainbow $u,v$-path $L$ such that $P(u)\\cup L\\cup Q(v)$ is a rainbow $x,y$-walk which contains the desired rainbow $x,y$-path.\n    \n    To prove  \\Cref{thm:main-blowups} about $r$-blow-ups of subdivisions of cliques, we use similar arguments as the ones for proving Theorem~\\ref{thm:main}, albeit tailored to the setup of $r$-blow-ups.    Recall that we are given a graph $G$ on $n$ vertices with at least $n^{2 - \\frac{1}{r}} (\\log n)^{\\frac{60}{r}}$ edges.  A supersaturation result due to Erd\\H{o}s and Simonovits \\cite{erdos1983supersaturated} implies that $G$ has many copies of $K_{r,r}$. \n    One natural approach is to take a large collection $\\mathcal{F}$ of copies of $K_{r,r}$ in $G$ and\n    consider an auxiliary graph $\\mathcal{H}$ whose vertices are $r$-sets of vertices in $G$, and whose edges correspond to copies of $K_{r,r}$ in $\\mathcal{F}$. To find a blow-up of a $K_m$-subdivision in $G$, it would suffice to find a `clean' subdivision of $K_m$ in $\\mathcal{H}$, denoted $\\mathcal{K}$, where the $r$-sets in $G$ corresponding to the vertices of $\\mathcal{K}$ are pairwise disjoint.\n    However, in order for our framework to be applicable,  we need a crucial additional property of $\\mathcal{F}$ that any $r$-set of vertices $A$ in $G$ and vertex $u \\notin A$ do not lie in too many $K_{r,r}$-copies in this collection $\\mathcal{F}$. Fortunately, the existence of such a collection $\\mathcal{F}$ is guaranteed by a `balanced supersaturation' result due to Morris and Saxton \\cite{morris2016number}.\n   \n\n\n\n\\section{Preliminaries} \\label{sec:prelims}\n\n    Let $G$ be a graph. We denote by $d(G)$ the average degree of $G$. For a subset $S \\subseteq V(G)$, let $e(S) = e(G[S])$, and for subsets $S, T \\subseteq V(G)$, let $e(S, T) = e(G[S, T])$.  We will use the notions of $d$-minimality and expanders, defined below, following \\cite{jiang2021rainbow}.\n     \n    \\begin{definition}\n        A graph $G$ is said to be $d$-\\emph{minimal} if $d(G) \\geq d$ but $d(H) < d$ for every proper subgraph $H\\subseteq G$.\n    \\end{definition}\n\n    It is easy to see that every graph $G$ contains a $d(G)$-minimal subgraph.\n    The following observation was used in \\cite{jiang2021rainbow}. For completeness, we include its short proof.\n        \n    \\begin{observation} \\label{obs:d-expand}\n        If $G$ is $d$-minimal, then every subset $S\\subseteq V(G)$ satisfies $e(S)+e(S,S^c)\\geq \\frac{d |S|}{2}$.\n        In particular, $\\delta(G) \\ge \\frac{d}{2}$.\n    \\end{observation}\n    \n    \\begin{proof} \n        Suppose otherwise. Then $e(S^c) \\geq \\frac{d|V|}{2} - \\frac{d|S|}{2} \\geq \\frac{d(|V| - |S|)}{2}$, contradicting $d$-minimality.\n    \\end{proof}\n    \n    \\begin{definition} \\label{def:expander-new}\n        Given $d \\ge 1$, $\\eta \\in (0, 1)$ and $\\varepsilon\\in (0,\\frac{1}{2}]$, an $n$-vertex graph $G$ is called a \\emph{$(d, \\eta, \\varepsilon)$-expander} if $G$ is $d$-minimal, and for every subset $S \\subseteq V(G)$ of size at most $(1 - \\varepsilon)n$, we have $d(S) \\le (1 - \\eta)d$. \n    \\end{definition}\n\n    Note that, by definition, for $0<\\varepsilon'\\leq\\varepsilon\\leq\\frac{1}{2}$ and $0 < \\eta \\le \\eta' < 1$, every $(d,\\eta',\\varepsilon')$-expander is  also a $(d,\\eta,\\varepsilon)$-expander. Also, if $G$ is a $(d, \\eta, \\varepsilon)$-expander then it is a $(d(G), \\eta, \\varepsilon)$-expander.\n    It will be useful to note the following `edge-expansion' property of $(d, \\eta, \\varepsilon)$-expanders.\n    \n    \\begin{observation} \\label{obs:edge-expansion-new}\n        Let $n, d \\ge 1$, let $\\eta\\in (0, 1)$ and let $\\varepsilon\\in (0,\\frac{1}{2}]$.\n        Suppose that $G$ is a $(d, \\eta, \\varepsilon)$-expander on $n$ vertices. Then every $S \\subseteq V(G)$ with $|S| \\le (1 - \\varepsilon)n$ satisfies $e(S, S^c) \\ge \\frac{\\eta d}{2}|S|$.\n    \\end{observation}\n    \n    \\begin{proof}\n        Let $S \\subseteq V(G)$ satisfy $|S| \\le (1 - \\varepsilon)n$.\n        Since $G$ is $d$-minimal and by \\Cref{obs:d-expand}, we have $e(S) + e(S, S^c) \\ge \\frac{d|S|}{2}$. Since $G$ is a $(d, \\eta, \\varepsilon)$-expander, by definition, we also have $e(S) = \\frac{d(S)}{2}|S| \\le \\frac{(1 - \\eta)d}{2}|S|$. It follows that $e(S, S^c) \\ge \\frac{\\eta d}{2}|S|$, as claimed.\n    \\end{proof}\n    \n    \\begin{lemma}[Lemma 2.5 from \\cite{jiang2021rainbow}]\n    \\label{lem:existence-expanders-new}\n        Let $n, d \\ge 1$, let $\\varepsilon \\in (0,\\frac{1}{2}]$ and let $\\eta = \\frac{\\varepsilon}{2\\log n}$.\n        Suppose that $G$ is a graph on $n$ vertices with average degree $d$. Then $G$ contains a $(d', \\eta, \\varepsilon)$-expander, with $d' \\ge \\frac{d}{2}$. \n    \\end{lemma}\n    \n    The following lemma from \\cite{jiang2021rainbow} asserts that in a properly edge-coloured expander, one can reach almost every vertex by a short rainbow path starting at a specified vertex.\n    \n    \\begin{lemma}[Lemma 2.7 from \\cite{jiang2021rainbow}] \\label{lem:reachable-new}\n        Let $n, \\ell, d, M \\ge 1$, let $\\eta \\in (0,1)$, and let $\\varepsilon \\in (0, \\frac{1}{2}]$. Suppose that $\\ell = \\frac{4 \\log n}{\\eta}$ and $d \\ge \\frac{4\\ell + 8M}{\\eta}$.\n        Let $G$ be a properly edge-coloured $(d, \\eta, \\varepsilon)$-expander on $n$ vertices, let $x \\in V(G)$ and let $F$ be a set of vertices and colours of size at most $M$. Then at least $(1 - \\varepsilon)n$ vertices can be reached from $x$ by a rainbow path of length at most $\\ell+1$ that avoids the vertices and colours in $F$.\n    \\end{lemma}\n    \n    We will need a stronger version of the previous lemma, where we require that no colour is used too many times in the short rainbow paths. A similar idea was used in \\cite{letzter2021tight} (see Lemma 5) in the context of tight paths.\n    \n    \\begin{lemma} \\label{lem:reachable-robust-new}\n        Let $n, \\ell, d, q, M \\ge 1$, let $\\eta \\in (0,1)$ and let $\\varepsilon \\in (0, \\frac{1}{2}]$.\n        Suppose that $\\ell = \\frac{4\\log n}{\\eta}$ and $d \\ge \\frac{20q\\ell + 8M}{\\eta}$.\n        Let $G$ be a properly edge-coloured $(d, \\eta, \\varepsilon)$-expander on $n$ vertices, let $x \\in V(G)$, and let $F$ be a set of colours and vertices of size at most $M$. Then there is a set $U \\subseteq V(G)$ of size at least $(1 - \\varepsilon)n$, and a collection  $\\mathcal P = \\{ P(u) : u \\in U\\}$ where for each $u \\in U$, $P(u)$ is a rainbow path from $x$ to $u$ of length at most $\\ell + 1$ that avoids the vertices and colours in $F$ and no colour appears in more than $\\frac{n}{q}$ of the paths in $\\mathcal P$.\n    \\end{lemma}\n    \n    \\begin{proof}\n        Let $U$ be a largest set satisfying that for every $u \\in U$ there is a rainbow path $P(u)$ from $x$ to $u$ of length at most $\\ell + 1$ that avoids $F$, such that no colour appears in more than $\\frac{n}{q}$ of the paths $P(u)$. Say that a colour is \\emph{bad} if it appears on exactly $\\frac{n}{q}$ of the paths $P(u)$ with $u \\in U$, and let $C_{\\bad}$ be the set of bad colours. Since each path $P(u)$ has length at most $\\ell + 1$, we have \n        \\begin{equation*}\n            |C_{\\bad}| \\le \\frac{n(\\ell + 1)}{n\/q} \\le 2q\\ell.\n        \\end{equation*}\n        \n       Since $d\\ge \\frac{20q\\ell + 8M}{\\eta}\\geq \\frac{4\\ell+8(M+|C_{\\bad})|}{\\eta}$, by \\Cref{lem:reachable-new} with $F\\cup C_{\\bad}$ playing the role of $F$, there is a set $U'$ with $|U'| \\ge (1 - \\varepsilon)n$ such that for every $v \\in U'$, there is a rainbow path $P'(v)$ from $x$ to $v$ of length at most $\\ell + 1$ that avoids the colours and vertices in $F \\cup C_{\\bad}$. If $|U| < (1 - \\varepsilon)n$, then there is a vertex $v \\in U' \\setminus U$. The set $U \\cup \\{v\\}$ (along with the paths $P(u)$ for $u \\in U$ and $P'(v)$) contradicts the maximality of $U$. It follows that $|U| \\ge (1 - \\varepsilon)n$, as required. \n    \\end{proof}\n    \n    We would like to work with expanders that are close to regular. For this, we use the following lemma, which is a slight adaptation of Lemma 3.2 of \\cite{montgomery2021C4}.\n    \\begin{lemma}\\label{lem:regularization}\n        Let $n \\ge 2$ and $d \\ge 36 \\log n$.\n        Let $G$ be a bipartite graph on $n$ vertices with minimum degree at least $d$. Then there exists a subgraph $H$ of $G$ with average degree at least $\\frac{d}{12 \\log{n}}$ and maximum degree at most $d$.\n    \\end{lemma}\n        \n    \\begin{proof}\n        Let $\\{A,B\\}$ denote a bipartition of $G$ with $|A|\\geq |B|$. Let $G'$ be obtained from $G$ by keeping exactly $d$ edges incident to each vertex in $A$. Then for each $v\\in A$ we have $d_{G'}(v)=d$, and hence $e(G')=d|A|$.\n        \n        Let $m=\\lceil \\log n\\rceil$. For each $i\\in [m]$, let $B_i=\\{v\\in B: 2^{i-1}\\leq d_{G'}(v)<2^i\\}$. Denoting the set of isolated vertices in $B$ by $B_0$, we have $B \\setminus B_0 = \\cup_{i\\in [m]} B_i$.\n        By the pigeonhole principle, there exists an $i\\in [m]$ for which $e(G'[A,B_i])\\geq \\frac{e(G')}{m} \\geq \\frac{d|A|}{2\\log n}$. \n        Fix such $i$ and let $t = 2^{i-1}$.\n        Then, by definition, each $v\\in B_i$ has degree between $t$ and $2t$ in $G'[A,B_i]$.\n        If $2t\\leq d$, then $G'[A,B_i]$ has maximum degree at most $d$ and\n        average degree at least $\\frac{2 e(G'[A,B_i])}{|A|+|B_i|}\\geq \\frac{d}{2\\log n}$.\n        So the lemma holds. Hence, we may assume that $2t > d$.\n        \n        Set $p = \\frac{d}{4t}$; then $0 < p < \\frac{1}{2}$. Now let $A'\\subseteq A$ be chosen by including each vertex in $A$ independently with probability $p$. \n        For convenience, write $G_i=G'[A,B_i]$ and $G_i'=G'[A',B_i]$. Then\n        \\begin{equation} \\label{A'B'}\n            \\mathbb{E}[e(G_i')] =p \\cdot e(G_i)\\geq \\frac{pd|A|}{2\\log n}.\n        \\end{equation}\n        Now, let $B'=\\{v\\in B_i: d_{G'_i}(v)\\leq d\\}$. \n\n        For each $v\\in B_i$, the degree $d_{G'_i}(v)$ is binomially distributed with expectation $p \\cdot d_{G_i}(v)$. Since $t\\leq d_{G_i}(v)\\leq 2t$, we have\n        $\\frac{d}{4}=pt\\leq \\mathbb{E}[d_{G'_i}(v)]\\leq 2pt=\\frac{d}{2}$. Therefore, using Chernoff's bound (see, e.g., Appendix A of \\cite{alonspencer2016}), we have\n        \\begin{align*}\n           \\mathbb{E}[|B_i\\setminus B'|]&=\\sum_{v\\in B_i} \\mathbb{P}(v\\in B_i\\setminus B')= \\sum_{v\\in B_i} \\mathbb{P}\\!\\left[d_{G'_i}(v)\\geq d\\right]\n            \\,\\leq\\, \\sum_{v\\in B_i} \\mathbb{P}\\!\\left[d_{G'_i}(v)\\geq 2 \\mathbb{E}[d_{G'_i}(v)]\\right] \\\\\n            &\\,\\leq\\, \\sum_{v\\in B_i} 2\\cdot \\rm exp\\!\\left(-\\frac{\\mathbb{E}[d_{G'_i}(v)]}{3}\\right) \n            \\leq\\ \\sum_{v\\in B_i}2\\cdot \\rm exp\\!\\left(-\\frac{d}{12}\\right)\n            \\leq\\ n \\cdot 2e^{-3\\log n}<\\frac{1}{n}.\n        \\end{align*}\n        This together with the fact that for any $A'$ and the corresponding $B'$, $e(G'[A',B_i\\setminus B']) \\leq n|B_i\\setminus B'|$, implies\n        \\begin{equation} \\label{A'notB'}\n        \\mathbb{E}[e(G'[A',B_i\\setminus B'])]\\leq n \\cdot \\mathbb{E}[|B_i\\setminus B'|]  \\le 1.\n        \\end{equation}\n        \n        By \\eqref{A'B'} and \\eqref{A'notB'},\n        \\[\\mathbb{E}[e(G'[A',B'])\\geq \\frac{pd|A|}{2\\log n}-1.\\]\n        Note that  $\\mathbb{E}[|A'|]=p|A|$. Also, as $t|B_i|\\leq e(G'[A, B_i]) \\leq d|A|$, we have\n        $|B'|\\leq |B_i|\\leq \\frac{d}{t}|A|=4p|A|$. Hence $|A'|+|B'|\\leq 5p|A|$.\n        Let $d_0=\\frac{d}{12 \\log n}$.\n        We have\n        \\begin{equation*}\n            \\mathbb{E}\\!\\left[e(G'[A', B'])-d_0(|A'|+|B'|)\\right]\n            \\geq \\frac{pd|A|}{2\\log n} - 1  - \\frac{5dp|A|}{12 \\log n} \n            \\geq \\frac{pd|A|}{12 \\log n} - 1\n            \\ge 0,\n        \\end{equation*}\n        where the last inequality used $pd|A| = \\frac{d^2 |A|}{4t} \\ge 18^2(\\log n)^2 \\ge 12 \\log n$ which holds since $t \\le |A|$ and $d \\ge 36 \\log n$.\n        Thus, there is a choice of $A'$ for which $e(G'[A', B'])-d_0(|A'|+|B'|)\\geq 0$. Taking $H = G'[A', B']$, we have $d(H)\\geq d_0= \\frac{d}{12 \\log n}$ and $\\Delta(H)\\leq d$, as desired.\n    \\end{proof}\n    \n    Our final preliminary result combines \\Cref{lem:existence-expanders-new,lem:regularization} to show that every relatively dense graph contains an expander which is close to regular.\n    \n    \n    \\begin{lemma} \\label{lem:bounded-max-deg-expander}\n        Let $n, d \\ge 1$ and let $\\varepsilon \\in (0,\\frac{1}{2}]$, and suppose that $d \\ge 10^7 (\\log n)^3$.\n        Suppose that $G$ is a bipartite graph on $n$ vertices with average degree at least $d$. Then $G$ has a subgraph $H$ with the following properties.\n        \\begin{enumerate} [ref = \\rm\\arabic*]\n            \\item \\label{itm:property-1}\n                $H$ is a $(d', \\eta, \\varepsilon)$-expander on $n'$ vertices, where $d' \\ge \\frac{d}{2500(\\log n)^2}$ and $\\eta \\ge \\frac{\\varepsilon}{100(\\log n')^2}$,\n            \\item \\label{itm:property-2}\n                $H$ has maximum degree at most $2500 (\\log n')^2 d'$.\n        \\end{enumerate}\n    \\end{lemma}\n    \n    \\begin{proof}\n        Let $G_0 = G$. We run the following process, generating graphs $G_i$ for $i \\ge 0$. For each graph $G_i$, we write $d_i = d(G_i)$ and $n_i = |V(G_i)|$.\n        \\begin{enumerate} [label = \\rm(\\alph*)]\n            \\item \\label{itm:step-a}\n                Let $H_i$ be a subgraph of $G_i$ with average degree at least $\\frac{d_i}{24 \\log n_i}$ and maximum degree at most $d_i$. Such a subgraph $H_i$ exists by \\Cref{lem:regularization}, using the fact that every graph with average degree $d$ contains a subgraph with minimum degree at least $\\frac{d}{2}$. To apply the lemma we need to verify that $d_i \\ge 72 \\log n_i$, which we shall do below.\n            \\item \\label{itm:step-b}\n                Write $n_i' = |V(H_i)|$.\n                Let $G_{i+1}$ be a subgraph of $H_i$ which is a $(d_{i+1}, \\eta_{i+1}, \\varepsilon)$-expander, where $d(G_{i+1}) = d_{i+1} \\ge \\frac{d(H_i)}{2} \\ge \\frac{d_i}{48 \\log n_i}$ and $\\eta_{i+1} = \\frac{\\varepsilon}{2 \\log n_i'}$. Such a subgraph exists by \\Cref{lem:existence-expanders-new}, using the observation that if $G$ is a $(d, \\eta, \\varepsilon)$-expander then it is a $(d(G), \\eta, \\varepsilon)$-expander.\n        \\end{enumerate}\n        \n        \\begin{claim}\n            If $48 \\log n_i < \\sqrt{48 \\log n_{i-1}}$ for $i \\in [t]$ then the graphs $G_1, \\ldots, G_t$ can be defined as above and are non-empty.\n        \\end{claim}\n        \n        \\begin{proof}\n            Notice that to prove the statement, it suffices to show that for $t$ as described, $d_i \\ge 72 \\log n_i$ for $i \\in [t-1]$.\n            \n            We prove this by induction.\n            It is easy to check that the statement is true for $t = 0$ (note that the condition on $t$ holds vacuously here). Indeed, we just need to check that $d_0 \\ge 72 \\log n_0$, which is the case as $d_0 = d$, $n_0 = n$ and $d \\ge 10^7(\\log n)^3$.\n            \n            Now suppose that $48 \\log n_i < \\sqrt{48 \\log n_{i-1}}$ for $i \\in [t]$ and that the inductive statement holds for $t' \\le t$. This means that the process runs as described for all $i \\in [t]$, and it remains to show that the $(t+1)$-st step can be performed, namely that $d_{t} \\ge 72 \\log n_{t}$.\n            By the assumption on $t$, the following holds for every $i \\in [t]$.\n            \\begin{equation*}\n                48 \\log n_i \n                \\le (48 \\log n_{i-1})^{1\/2}\n                \\le \\ldots \\le (48 \\log n_0)^{2^{-i}}\n                = (48 \\log n)^{2^{-i}}.\n            \\end{equation*}\n            Since $d_i \\ge \\frac{d_{i-1}}{48 \\log n_{i-1}}$ for $i \\in [t]$ (see \\ref{itm:step-b}), it follows that\n            \\begin{equation} \\label{eqn:d}\n                d_t \\ge\n                \\frac{d_{t-1}}{48 \\log n_{t-1}} \n                \\ge \\ldots \n                \\ge \\frac{d_0}{48 \\log n_{t-1} \\cdot \\ldots \\cdot 48 \\log n_{0}} \n                \\ge \\frac{d}{(48 \\log n)^{2^0 + \\ldots + 2^{-(t-1)}}}\n                \\ge \\frac{d}{(48 \\log n)^2}.\n            \\end{equation}\n            Using $d \\ge 10^7 \\log n$ and $n \\ge n_t$, we find that $d_t \\ge 10^3 \\log n > 72 \\log n_t$, as required.\n        \\end{proof}\n        \n        Let $\\ell$ be minimum such that $48 \\log n_{\\ell} \\ge \\sqrt{48 \\log n_{\\ell - 1}}$. We claim that such $\\ell$ exists. If not, then by the previous claim the process can be run forever and $n_i \\ge 1$ for all $i \\ge 0$, implying that $(48 \\log n_i)_{i \\ge 0}$ is decreasing, and thus $(n_i)_{i \\ge 0}$ is an infinite decreasing sequence of positive integers, a contradiction. \n        \n        We will show that $G_{\\ell}$ satisfies the requirements of \\Cref{lem:bounded-max-deg-expander}.\n        Indeed, $G_{\\ell}$ is a $(d_{\\ell}, \\eta_{\\ell}, \\varepsilon)$-expander. \n        Using the proof of the above claim, inequality \\eqref{eqn:d} holds for $t = \\ell$, showing $d_{\\ell} \\ge \\frac{d}{2500(\\log n)^2}$.\n        By choice of $G_{\\ell}$, we have $\\eta_{\\ell} = \\frac{\\varepsilon}{2\\log n_{\\ell-1}} \\ge \\frac{\\varepsilon}{96(\\log n_{\\ell})^2} \\ge \\frac{\\varepsilon}{100(\\log n_{\\ell})^2}$ (using $48 \\log n_{\\ell} \\ge \\sqrt{48 \\log n_{\\ell-1}}$). It follows that property \\ref{itm:property-1} of the lemma holds.\n        To see property \\ref{itm:property-2}, note that $G_{\\ell}$ has maximum degree at most $d_{\\ell - 1}$ and $d_{\\ell} \\ge \\frac{d_{\\ell-1}}{48 \\log n_{\\ell-1}} \\ge \\frac{d_{\\ell-1}}{(48 \\log n_{\\ell})^2} \\ge \\frac{d_{\\ell-1}}{2500 (\\log n_{\\ell})^2}$ (again using $48 \\log n_{\\ell} \\ge \\sqrt{48 \\log n_{\\ell-1}}$).\n    \\end{proof} \n    \n    \n    \n\\section{Counting rainbow cycles} \\label{sec:non-rainbow-walks}\n    \n    In a graph $G$, denote by $\\Hom_{x, y}(P_k)$ the family of walks of length $k$ from $x$ to $y$. Similarly, let $\\Hom_{x, y}(C_{2k})$ be the family of closed walks of length $2k$ that start at $x$ and reach $y$ after $k$ steps. \n    Write $\\hom_{x,y}(P_k)=|\\Hom_{x,y}(P_k)|$ and $\\hom_{x,y}(C_{2k})=|\\Hom_{x,y}(C_{2k})|$.\n    \n    The following relation between $\\hom_{x, y}(P_k)$ and $\\hom_{x, y}(C_{2k})$ is very useful.\n    \\begin{equation} \\label{eqn:hom-cycle-paths}\n        \\hom_{x, y}(C_{2k}) = (\\hom_{x, y}(P_k))^2. \n    \\end{equation}\n    We also define $\\hom(C_{2k})$ to be the total number of homomorphic copies of $C_{2k}$, namely \n    \\begin{equation*}\n        \\hom(C_{2k}) = \\sum_{\\substack{x, y \\in V(G)}}\n        \\hom_{x,y}(C_{2k}).\n    \\end{equation*}\n    Similarly, we define $\\hom(P_k) = \\sum_{x, y \\in V(G)}\\hom_{x,y}(P_k)$.\n    \n    We will make use of the following two lemmas from a recent paper of Janzer \\cite{janzer2020rainbow}. \n    \n    \\begin{lemma}[Lemma 2.1 from \\cite{janzer2020rainbow}] \\label{lem:janzer-edges}\n        Let $k \\ge 2$ be an integer and let $G = (V, E)$ be a graph on $n$ vertices. Let $\\sim$ be a symmetric binary relation on $E$ such that for every $uv \\in E$ and $w \\in V$, there are at most $t$ neighbours $z$ of $w$ for which $uv \\sim zw$. Then the number of homomorphic $2k$-cycles $(x_1, \\ldots, x_{2k})$ in $G$ such that $x_i x_{i+1} \\sim x_j x_{j+1}$ for some $i \\neq j$ is at most\n        \\begin{equation*}\n            32k^{3\/2}t^{1\/2}\\Delta(G)^{1\/2}n^{\\frac{1}{2k}}\\hom(C_{2k})^{1 - \\frac{1}{2k}}.\n        \\end{equation*}\n    \\end{lemma}\n    \n    \\begin{lemma}[Lemma 2.2 from \\cite{janzer2020rainbow}] \\label{lem:janzer-vertices}\n        Let $k \\ge 2$ be an integer and let $G = (V, E)$ be a graph on $n$ vertices. Let $\\sim$ be a symmetric binary relation on $V$ such that for every $u, v \\in V$, there are at most $t$ neighbours $w$ of $v$ for which $u \\sim w$. Then the number of homomorphic $2k$-cycles $(x_1, \\ldots, x_{2k})$ in $G$ such that $x_i \\sim x_j$ for some $i \\neq j$ is at most\n        \\begin{equation*}\n            32k^{3\/2}t^{1\/2}\\Delta(G)^{1\/2}n^{\\frac{1}{2k}}\\hom(C_{2k})^{1 - \\frac{1}{2k}}.\n        \\end{equation*}\n    \\end{lemma}\n    \n    In a properly edge-coloured graph $G$, let $\\Hom^*_{x, y}(C_{2k})$ be the family of all the closed walks in $\\Hom_{x,y}(C_{2k})$ that do not form a rainbow copy of $C_{2k}$. \n    Write $\\hom^*_{x,y}(C_{2k})=|\\Hom^*_{x,y}(C_{2k})|$.\n    Let \n    \\begin{equation*}\n        \\Hom^*(C_{2k}) \n        = \\bigcup_{x,y\\in V(G)} \\Hom^*_{x,y}(C_{2k}),\n    \\end{equation*} \n    and write $\\hom^*(C_{2k})=|\\Hom^*(C_{2k})|$. \n\n    \\begin{lemma} \\label{lem:non-rainbow-hom}\n        Let $n, d, \\mu, k, S \\ge 1$ and let $\\eta \\in (0, 1)$. Suppose that $d \\ge 2^{14} k^3 S^2 \\mu n^{1\/k}$.\n        Let $G$ be a properly edge-coloured graph with minimum degree at least $\\frac{d}{2}$ and maximum degree at most $\\mu d$. \n        Then $\\hom^*(C_{2k}) \\le \\frac{1}{S} \\hom(C_{2k})$.\n    \\end{lemma}\n    \n    \\begin{proof}\n        We first give a lower bound on $\\hom(C_{2k})$. For this, note that $\\hom(P_k) \\ge n \\cdot (\\frac{d}{2})^k$. Hence,\n        \\begin{equation}\n        \\label{eq:lowerboundhomC2k}\n            \\hom(C_{2k}) \n           \n            = \\sum_{x, y \\in V(G)} \\big(\\hom_{x, y}(P_k)\\big)^2\n            \\ge \\frac{1}{n^2} \\Big(\\sum_{x, y \\in V(G)} \\hom_{x, y} (P_k) \\Big)^2\n           \n            = \\bigg(\\frac{\\hom(P_k)}{n}\\bigg)^2\n            \\ge \\left(\\frac{d}{2}\\right)^{2k},\n        \\end{equation}\n        where the first inequality follows by convexity.\n        \n        Let $\\sim_e$ be the binary relation defined on $E(G)$ where $e \\sim_e f$ if and only if $e$ and $f$ have the same colour.  Because $G$ is properly edge-coloured, for every edge $uv$ and vertex $w$, there is at most one neighbour $z$ of $w$ for which $uv \\sim_e zw$.\n        Let $\\sim_v$ be the binary relation defined on $V(G)$ where $u \\sim_v v$ if and only if $u = v$. Then, trivially, for every $u, v \\in V$ there is at most one neighbour $w$ of $v$ for which $u \\sim_v w$.\n        Apply \\Cref{lem:janzer-edges,lem:janzer-vertices} with $\\sim$ (so $t$ can be taken to be $1$ in both lemmas), to obtain the desired upper bound on $\\hom^*(C_{2k})$, as follows. \n        \\begin{align*}\n            \\hom^*(C_{2k}) \n            & \\le 64 k^{3\/2} (\\mu d)^{1\/2} n^{\\frac{1}{2k}} \\cdot \\hom(C_{2k})^{1 - \\frac{1}{2k}} \\\\\n            & \\overset{\\eqref{eq:lowerboundhomC2k}}{\\le} \n            \\frac{128 k^{3\/2} (\\mu d)^{1\/2} n^{\\frac{1}{2k}}}{d}  \\cdot \\hom(C_{2k}) \\\\\n            & = \\bigg(\\frac{2^{14} k^3 \\mu n^{\\frac{1}{k}}}{d}\\bigg)^{1\/2} \\cdot \\hom(C_{2k})\n            \\le \\frac{1}{S} \\cdot \\hom(C_{2k}). \n        \\end{align*}\n        Here we used the inequality $\\hom(C_{2k}) \\ge (\\frac{d}{2})^{2k}$ proved in \\eqref{eq:lowerboundhomC2k}, and the lower bound on $d$ assumed in the lemma.\n    \\end{proof}\n    \n\n    \n    It would be useful to be able to find many pairwise colour-disjoint and vertex-disjoint paths between a given pair $(x, y)$ of vertices. To that end, we use Lemma~\\ref{lem:theta} which will immediately follow from Lemma~\\ref{lem:theta-general} stated below, whose proof uses the arguments of Theorem 3.7 in \\cite{janzer2020rainbow}. Lemma~\\ref{lem:theta-general} will also be used in \\Cref{sec:blowups}.\n    For a graph $G$, and vertices $x, y \\in V(G)$, and walks $P, Q \\in \\Hom(P_k)$, let $PQ$ denote the closed walk in $\\Hom_{x, x}(C_{2k})$ obtained by concatenating $P$ and the reverse of $Q$. \n\n   \n       \n   \n    \\begin{lemma} \\label{lem:theta-general}\n        Let $k, s \\ge 1$ be integers. Let $G$ be a graph and let $x, y \\in V(G)$. Let $\\mathcal B$ be a subfamily of $\\Hom_{x,y}(C_{2k})$ satisfying $|\\mathcal B| < \\frac{1}{s^2}\\hom_{x, y}(C_{2k})$. Then there exist walks $P_1,\\dots, P_s \\in \\Hom_{x, y}(P_k)$ such that $P_iP_j\\notin \\mathcal B$ for every distinct $i, j \\in [s]$.\n    \\end{lemma}\n    \n    \\begin{proof}\n        Let $P_1, \\ldots, P_s$ be walks of length $k$ from $x$ to $y$ in $G$, each chosen uniformly at random among all such walks, independently. Note that for any distinct $i, j \\in [s]$, the walk $P_iP_j$ is a uniformly chosen random element in $\\Hom_{x, y}(C_{2k})$. Thus, the probability that $P_iP_j\\in \\mathcal B$ is \n        $\\frac{|\\mathcal B|}{\\hom_{x,y}(C_{2k})} < \\frac{1}{s^2}$. A union bound implies that, with positive probability, $P_i P_j \\notin \\mathcal{B}$ for every distinct $i, j \\in [s]$.\n    \\end{proof}\n    \n    \\begin{lemma} \\label{lem:theta}\n        Let $k, s \\ge 1$ be integers. Let $G$ be a properly edge-coloured graph and let $x, y \\in V(G)$. Suppose that $\\hom^*_{x, y}(C_{2k}) < \\frac{1}{s^2}\\hom_{x, y}(C_{2k})$. Then there are $s$ pairwise colour-disjoint  and internally vertex-disjoint rainbow paths of length $k$ from $x$ to $y$.   \n    \\end{lemma}\n    \n    \\begin{proof}\n        By our assumption, $|\\Hom^*_{x,y}(C_{2k})|<\\frac{1}{s^2}\\hom_{x, y}(C_{2k})$. By Lemma~\\ref{lem:theta-general}, there exist walks $P_1, \\dots, P_s \\in \\Hom_{x, y}(P_k)$ satisfying $P_i P_j \\notin \\Hom^*_{x, y}(C_{2k})$ for every distinct $i, j \\in [s]$ \n        In other words, $P_i P_j$ is a rainbow copy of $C_{2k}$ for every distinct $i, j \\in [s]$. Now, $P_1, \\dots, P_s$ are pairwise colour-disjoint internally vertex-disjoint paths of length $k$ from $x$ to $y$, as desired. \n    \\end{proof}\n    \n\\section{Counting walks in expanders} \\label{sec:random-walks}\n\n    In this section we exploit the connection between the mixing time of a random walk on a graph $G$ and expansion properties of $G$. A lot of the notation and results that we use can be found in~\\cite{lovasz1993random}.\n    \n    Suppose $G=(V,E)$ is a connected graph where $V = [n]$. \n    Consider a random walk on $V(G)$, where we start at some vertex $v_0$ and at the $i$-th step we move from $v_i$ to one of its neighbours, denoted by $v_{i+1}$, where\n    each neighbour of $v_i$ is chosen as $v_{i+1}$ with probability $\\frac{1}{d(v_i)}$. The sequence of vertices $(v_i)_{i \\ge 0}$ defines a Markov chain. \n    Let $M$ be the $n \\times n$ matrix of transition probabilities of the Markov chain, namely $M_{v, u}$ is the probability of stepping from $v$ to $u$; so $M_{v, u}= \\frac{1}{d(v)}$ if $vu \\in E(G)$, and $M_{v, u} = 0$ otherwise. Denote by $D$ the $n \\times n$ diagonal matrix with $D_{v, v} = \\frac{1}{d(v)}$ for $v \\in [n]$, and let $A$ be the adjacency matrix of $G$. Then $M=D A$. \n    So the probability that a random walk starting at vertex $v$ reaches $u$ in $t$ steps is $(M^t)_{v, u}$.\n\n    \\begin{definition}\n    \\label{def19}\n        Let $G$ be a graph on the vertex set $[n]$. \n        Let $D = D(G)$ denote the diagonal $n \\times n$ matrix where  $D_{v,v}=\\frac{1}{d(v)}$ for each $v\\in [n]$. Let $A = A(G)$ be the adjacency matrix of $G$, let $M(G) = D A$ and let $N(G) =D^{1\/2}AD^{1\/2}$. Note that the matrix $N(G)$ is symmetric, so it has $n$ real eigenvalues. Let $\\lambda_1(N)\\geq \\lambda_2(N)\\geq \\dots \\geq \\lambda_n(N)$ denote the eigenvalues of $N:=N(G)$. \n    \\end{definition}\n\n  \n    \\begin{lemma}\\label{lem:markovchain}\n        Let $G$ be a bipartite graph, with a bipartition $\\{X, Y\\}$, on the vertex set $[n]$ with $m$ edges and no isolated vertices. \n        Let $M=D(G)A(G)$ and $N = N(G)$.\n        Then for every $v,u \\in V(G)$ and integer $k \\ge 1$, we have\n        \\begin{equation*}\n            \\left|(M^k)_{v, u} - \\frac{d(u)}{2m}\\left(1 + (-1)^{k + \\mathbbm{1}(v \\in X) + \\mathbbm{1}(u \\in X)}\\right) \\right|\n            \\leq \\sqrt{\\frac{d(u)}{d(v)}} \\cdot \\big(\\lambda_2(N)\\big)^k.\n        \\end{equation*}\n    \\end{lemma}\n    \n    Note that Lemma~\\ref{lem:markovchain} says that when $k$ is even and both $v,u$ are in the same part or when $k$ is odd and $v,u$ are in different parts then   $ \\left|(M^k)_{v, u} - \\frac{d(u)}{m} \\right|\n            \\leq \\sqrt{\\frac{d(u)}{d(v)}} \\cdot \\big(\\lambda_2(N)\\big)^k.$ Note that when $k$ is even and $v$ and $u$ are in different parts or when $k$ is odd and $v$ and $u$ are in the same part then $(M^k)_{v, u}=0$. \n    \\begin{proof}\n        For any vector $w = (w_1, \\ldots, w_n)^T$, let $\\overline{w}$ be the vector $(w_1', \\ldots, w_n')^T$ where $w_i' = w_i$ when $i \\in X$ and $w_i' = -w_i$ when $i \\in Y$. It is easy to check that $N \\overline{w} = -\\overline{Nw}$. Hence, if $w$ is an eigenvector of $N$ with eigenvalue $\\lambda$, then $\\overline{w}$ is an eigenvector of $N$ with eigenvalue $-\\lambda$. It follows that $\\lambda_i = -\\lambda_{n+1-i}$ for $i \\in [n]$. In particular, $|\\lambda_i| \\le \\lambda_2$ for every $i \\in \\{2, \\ldots, n-1\\}$.\n        \n        One can check that $w_1$, defined as follows, is a unit eigenvector of $N$ with eigenvalue $1$. \n        \\begin{equation*}\n            w_1 = \\frac{1}{\\sqrt{2m}}\\left(\\sqrt{d(1)}, \\sqrt{d(2)}, \\dots, \\sqrt{d(n)}\\right)^T.\n        \\end{equation*}\n        By the Frobenius--Perron theorem, since the entries of $N$ are non-negative and the entries of $w_1$ are positive, we have $\\lambda_1 = 1$. As explained above, it follows that $\\overline{w_1}$ is a unit eigenvector of $N$ with eigenvalue $\\lambda_n = -1$. Write $w_n = \\overline{w_1}$, and let $w_i$ be a unit eigenvector of $N$ with eigenvalue $\\lambda_i$ for $i \\in \\{2, \\ldots, n-1\\}$, such that $w_1, \\ldots, w_n$ are orthogonal to each other. Note that $w_i$ is an eigenvector of $N^k$ with eigenvalue $(\\lambda_i)^k$, for each $i \\in [n]$.\n        Since $N^k$ is a symmetric matrix and $\\{w_1, \\ldots, w_n\\}$ is an orthonormal eigenbasis for $N^k$, we may write $N^k$ in the spectral form as follows (using $\\lambda_1 = 1$ and $\\lambda_n = -1$).\n        \\begin{equation*}\n            N^k\n            = \\sum_{i=1}^n{(\\lambda_i)^k w_i (w_i)^T} \n            = w_1(w_1)^T + (-1)^k \\,\\overline{w_1}(\\overline{w_1})^T + \\sum_{i=2}^{n-1}{(\\lambda_i)^k w_i (w_i)^T}.\n        \\end{equation*}\n We also have $D^{1\/2}ND^{-1\/2} = DA = M$.\n        Therefore,\n        \\begin{align*}\n            M^k \n            &= D^{1\/2}N^k D^{-1\/2} \\\\\n            &= D^{1\/2} w_1 (w_1)^T  D^{-1\/2} + (-1)^kD^{1\/2} \\,\\overline{w_1} (\\overline{w_1})^T  D^{-1\/2} + \\sum_{i=2}^{n-1}{(\\lambda_i)^k D^{1\/2} w_i (w_i)^T D^{-1\/2} }\n        \\end{align*}\n        Let $Q=  D^{1\/2} w_1 (w_1)^T  D^{-1\/2} + (-1)^kD^{1\/2} \\,\\overline{w_1} (\\overline{w_1})^T  D^{-1\/2}$. Then\n              \\begin{align*}\n            M^k \n           = Q + \\sum_{i=2}^{n-1}{(\\lambda_i)^k D^{1\/2} w_i (w_i)^T D^{-1\/2} }.\n        \\end{align*}  \n        Hence\n        \\begin{equation}\n        \\label{eq:P't}\n        (M^k)_{v, u} \n            = Q_{v, u} + \\sum_{i=2}^{n-1}{(\\lambda_i)^k  w_{i, v} w_{i, u} \\sqrt{\\frac{d(u)}{d(v)}}}.\n        \\end{equation}\n        \n        Let $W$ be the matrix whose rows are $w_1, \\ldots, w_n$. Then $W W^T = I$, implying $W^TW = I$. For each $v \\in [n]$, since $(W^TW)_{v,v}=1$, we have $\\sum_{i = 1}^n |w_{i, v}|^2 = 1$, so $\\sum_{i = 2}^{n-1} |w_{i, v}|^2 \\le 1$.\n        By the Cauchy-Schwarz inequality,  $\\sum_{i = 2}^{n-1} | w_{i, v}w_{i, u}| \\le \n        \\sqrt{\\sum_{i=2}^{n-1} |w_{i,v}|^2} \\sqrt{\\sum_{i=2}^{n-1}  |w_{i,u}|^2}\\leq 1$. \n        Since $|\\lambda_i| \\le \\lambda_2$ for every $i \\in \\{2, \\ldots, n-1\\}$, the inequality in \\eqref{eq:P't} implies\n        \\begin{equation*}\n            \\left| (M^k)_{v, u} - Q_{v, u} \\right| \n            \\le \\sum_{i = 2}^{n-1} |\\lambda_i|^k |w_{i, u} w_{i, v}| \\cdot \\sqrt{\\frac{d(u)}{d(v)}}\n            \\le (\\lambda_2)^k \\sqrt{\\frac{d(u)}{d(v)}}.\n          \n        \\end{equation*}\n        Finally, a straightforward calculation shows that  \n        $Q_{v, u} = \\frac{d(u)}{2m}\\left(1 + (-1)^{k + \\mathbbm{1}(v \\in X) + \\mathbbm{1}(u \\in X)}\\right)$ for all $v,u \\in [n]$, as desired.\n    \\end{proof}\n  \n    \\begin{definition} \\label{def:conductance}\n        For a graph $G$ with $m$ edges, let $\\pi(v)= \\frac{d(v)}{2m}$, and for any $S \\subseteq V(G)$,  let $\\pi(S) \\coloneqq \\sum_{s\\in S}{\\pi(s)}$; observe that $\\pi(S) \\le 1$ for every $S \\subseteq V(G)$. Define the \\emph{conductance} of a set $S$, denoted $\\Phi(S)$, by\n        \\begin{equation*}\n            \\Phi(S) \\coloneqq \\frac{e(S,S^c)}{2m \\cdot \\pi(S)\\pi(S^c)}, \n        \\end{equation*}\n        and let the \\emph{conductance} of a graph $G$, denoted $\\Phi_G$, be defined by\n        \\begin{equation*}\n            \\qquad \\Phi_G \\coloneqq \\min_{S \\subseteq V(G)} \\Phi(S).\n        \\end{equation*}\n    \\end{definition}\n   \n    \\begin{theorem}[Theorem 5.3 in \\cite{lovasz1993random}] \\label{thm:upperboundoneigenvalue} \n        Let $G$ be a graph and let $\\lambda_2 = \\lambda_2(N(G))$. Then $\\lambda_2 \\le 1 - \\frac{\\Phi_G^2}{8}$.\n    \\end{theorem}\n    \n    In light of \\Cref{thm:upperboundoneigenvalue} and \\Cref{lem:mixing}, it will be useful to have a lower bound on $\\Phi_G$ for a $(d, \\eta, \\varepsilon)$-expander $G$. This is easy to achieve, as can be seen in the following lemma.\n    \n    \\begin{lemma} \\label{lem:large-phi} \n        Let $d\\geq 1$, $\\eta\\in (0,1)$, $\\varepsilon\\in (0,\\frac{1}{2}]$.\n        Let $G$ be a $(d, \\eta, \\varepsilon)$-expander on $n$ vertices. Then $\\Phi_G \\geq \\frac{\\eta}{3}$.\n    \\end{lemma}\n    \n    \\begin{proof}\n        Let $S\\subseteq V(G)$. Since $\\Phi(S)=\\Phi(S^c)$, we may assume that\n        $|S|\\leq \\frac{n}{2}\\leq (1-\\varepsilon) n$. Observation~\\ref{obs:edge-expansion-new} thus implies  \n        $e(S,S^c)\\geq \\frac{1}{2}\\eta d|S|$.\n        Let $\\gamma$ be such that $e(S,S^c)=\\gamma d|S|$, so $\\gamma\\geq \\frac{\\eta}{2}$. By definition of a $(d,\\eta,\\varepsilon)$-expander, $G$ is also $d$-minimal.\n        Hence $e(G[S])\\leq \\frac{1}{2}d|S|$.\n        Hence, $\\sum_{v\\in S} d(v)=2e(G[S])+e(S,S^c)\\leq d|S|+\\gamma d|S|$. \n        Also, observe that $\\pi(S^c)\\leq 1$. Hence\n   \n        \\begin{align*}\n            \\Phi(S)& =\\frac{e(S,S^c)}{2e(G)\\pi(S)\\pi(S^c)} \n            \\geq \\frac{e(S,S^c)}{\\sum_{v\\in S} d(v)}\n            \\geq \\frac{\\gamma d|S| }{d |S|+\\gamma d|S|}\\geq \\frac{\\gamma}{1+\\gamma} \\geq \\frac{\\eta}{\\eta+2}\\geq \\frac{\\eta}{3}.\n        \\end{align*}\n        The above inequality thus implies $\\Phi_G \\ge \\frac{\\eta}{3}$.\n    \\end{proof}\n    \n    Recall that given a graph $G$ and two vertices $x,y$, the quantity $\\hom_{x,y}(P_k)$ denotes the number of walks of length $k$ from $x$ to $y$. The following lemma and its immediate corollary will allow us to compare the values of $\\hom_{x, y}(P_k)$ for different pairs of vertices $(x, y)$ in $G$. \n    \n    \\begin{lemma} \\label{lem:mixing} \n        Let $d\\geq 1$, $\\eta\\in (0,1)$, $\\varepsilon\\in (0,\\frac{1}{2}]$.\n        Let $G$ be a bipartite $(d, \\eta, \\varepsilon)$-expander on $n$ vertices with bipartition $\\{X, Y\\}$. The following holds for any two vertices $x,y\\in V(G)$\n        and every integer $k\\geq 1$.\n        \\begin{equation*}\n            \\left|\\frac{\\hom_{x,y}(P_k)}{\\sum_{z\\in V(G)}{\\hom_{x,z}(P_k)}} - \\frac{d(y)}{2e(G)}\\left(1 + (-1)^{k + \\mathbbm{1}(x \\in X) + \\mathbbm{1}(y \\in X)}\\right)\\right| \n            \\leq \\sqrt{\\frac{d(y)}{d(x)}}\\left(1-\\frac{\\eta^2}{72}\\right)^k.\n        \\end{equation*}\n    \\end{lemma}\n    \n    \\begin{proof}\n        Let $M=M(G)$, $N = N(G)$ and let $\\lambda_2$ be the second largest eigenvalue of $N$. Notice that $\\frac{\\hom_{x,y}(P_k)}{\\sum_{z\\in V(G)}{\\hom_{x,z}(P_k)}}$ is the probability that a random walk starting at $x$ reaches $y$ in $k$ steps. This, along with our discussion before Definition~\\ref{def19}, yields $(M^k)_{x,y}=\\frac{\\hom_{x,y}(P_k)}{\\sum_{z\\in V(G)}{\\hom_{x,z}(P_k)}}$.\n        By Lemma~\\ref{lem:markovchain}, \n        \\begin{equation}\n        \\label{eqlem18}\n             \\left|\\frac{\\hom_{x,y}(P_k)}{\\sum_{z\\in V(G)}{\\hom_{x,z}(P_k)}}-\\frac{d(y)}{2e(G)}\\left(1 + (-1)^{k + \\mathbbm{1}(x \\in X) + \\mathbbm{1}(y \\in X)}\\right)\\right| \n            \\leq \\sqrt{\\frac{d(y)}{d(x)}} \\cdot (\\lambda_2)^k.\n        \\end{equation}\n        \\Cref{thm:upperboundoneigenvalue} gives that $\\lambda_2 \\le 1 - \\frac{\\Phi_G^2}{8}$, and by \\Cref{lem:large-phi} we have $\\Phi_G \\ge \\frac{\\eta}{3}$. It follows that $\\lambda_2 \\leq 1-\\frac{\\eta^2}{72}$. \n        Combining this inequality with \\eqref{eqlem18}, the lemma follows.\n    \\end{proof}\n    \n      \\begin{corollary} \\label{cor:mixing} \n        Let $d\\geq 1$, $\\eta\\in (0,1)$, $\\varepsilon\\in (0,\\frac{1}{2}]$. Let $G$ be a bipartite $(d, \\eta, \\varepsilon)$-expander on $n$ vertices with a bipartition $\\{X, Y\\}$. The following holds for any two vertices $x,y\\in X$\n        and even integer $k\\geq 2$.\n        \\begin{equation*}\n            \\left|\\frac{\\hom_{x,y}(P_k)}{\\sum_{z\\in V(G)}{\\hom_{x,z}(P_k)}} - \\frac{d(y)}{e(G)}\\right| \n            \\leq \\sqrt{n}\\left(1-\\frac{\\eta^2}{72}\\right)^k.\n        \\end{equation*}\n    \\end{corollary}\n    \n    \n    The next lemma contains the main takeaway from our discussion about random walks in almost regular bipartite expanders. It tells us that for relatively large $k$, the values of $\\hom_{x, y}(C_{2k})$ do not differ by much over the range of pairs $(x, y)$ where $x$ and $y$ are in the largest part of the bipartition.\n    \n    \\begin{lemma} \\label{lem:rhomin-rhomax}\n        Let $n, d, \\mu, k \\ge 1$ and $\\eta \\in (0, 1)$, $\\varepsilon\\in (0,\\frac{1}{2}]$.\n        Suppose that $k$ is an even integer satisfying $k \\ge \\frac{2^9 \\log n}{\\eta^2}$.\n        Let $G$ be a bipartite $(d, \\eta, \\varepsilon)$-expander with maximum degree at most $\\mu d$; denote the bipartition of $G$ by $\\{X, Y\\}$ and suppose that $|X| \\ge \\frac{n}{2}$. \n        Let \n        \\begin{align*}\n            \\rho_{\\min} \n            & = \\min \\{ \\hom_{x, y}(C_{2k}) \\,:\\, x, y \\in X \\} \\\\\n            \\rho_{\\max}\n            & = \\max \\{ \\hom_{x, y}(C_{2k}) \\,:\\, x, y \\in X \\}.\n        \\end{align*}\n        Then $\\rho_{\\max} \\le 2^{12} \\mu^4 \\cdot \\rho_{\\min}$.\n    \\end{lemma}\n    \n    \\begin{proof}\n        By \\Cref{cor:mixing}, for every $x, y \\in X$ we have \n        \\begin{align*}\n            \\left| \\frac{\\hom_{x, y}(P_k)}{\\sum_{z \\in X}\\hom_{x, z}(P_k)} - \\frac{d(y)}{e(G)} \\right| \n            & \\le \\sqrt{n}\\left(1 - \\frac{\\eta^2}{72}\\right)^k \\\\\n            & \\le \\sqrt{n} \\cdot \\rm exp\\left(-\\frac{k\\eta^2}{72}\\right)\n            \\le \\sqrt{n} \\cdot \\rm exp(-4 \\log n) \n            \\le \\frac{1}{n^3}\n            \\le \\frac{d(y)}{2e(G)}.\n        \\end{align*}\n        It follows that\n        \\begin{equation*}\n            \\frac{d(y)}{2e(G)}\n            \\le \\frac{\\hom_{x, y}(P_k)}{\\sum_{z \\in X}\\hom_{x, z}(P_k)}\n            \\le \\frac{2d(y)}{e(G)}.\n        \\end{equation*}\n        Hence, every $x, y, z \\in X$ satisfy $\\frac{\\hom_{x, y}(P_k)}{\\hom_{x, z}(P_k)} \\le \\frac{4d(y)}{d(z)} \\leq 8 \\mu$ since the minimum degree of $G$ is at least $\\frac{d}{2}$ (by Observation~\\ref{obs:d-expand}) and the maximum degree is at most $\\mu d$. \n        \n        Observe that $\\hom_{x, y}(P_k) = \\hom_{y, x}(P_k)$ for $x, y \\in X$. It follows that, for every $x, y, z, w \\in X$,\n        \\begin{equation*}\n            \\frac{\\hom_{x, y}(P_k)}{\\hom_{z, w}(P_k)}\n            = \\frac{\\hom_{x, y}(P_k)}{\\hom_{x, z}(P_k)} \\cdot \\frac{\\hom_{z, x}(P_k)}{\\hom_{z, w}(P_k)} \n            \\le 64 \\mu^2.\n        \\end{equation*}\n        Let $x, y, z, w \\in X$ satisfy $\\rho_{\\max} = \\hom_{x, y}(C_{2k})$ and $\\rho_{\\min} = \\hom_{z, w}(C_{2k})$.\n        Then \n        \\begin{equation*}\n            \\frac{\\rho_{\\max}}{\\rho_{\\min}}\n            = \\frac{\\hom_{x, y}(C_{2k})}{\\hom_{z, w}(C_{2k})}\n            = \\left(\\frac{\\hom_{x, y}(P_k)}{\\hom_{z, w}(P_k)}\\right)^2\n            \\le 2^{12} \\mu^4,\n        \\end{equation*}\n        proving the lemma.\n    \\end{proof}\n    \n    Recall that, by \\Cref{lem:theta}, pairs of vertices $(x, y)$ for which $\\hom^*_{x, y}(C_{2k})$ is considerably smaller than $\\hom_{x, y}(C_{2k})$ can be used to build many colour-disjoint rainbow paths. \n    The following lemma shows that for large enough $k$ and $d$, almost all pairs of vertices in one of the parts of an almost regular bipartite expander satisfy this property.\n\n    \\begin{lemma} \\label{lem:few-bad-pairs}\n        Let $n, d, \\mu, k, s, p\\ge 1$ and $\\eta \\in (0, 1)$, $\\varepsilon\\in (0,\\frac{1}{2}]$.\n        Suppose that $k$ is even and satisfies $k \\ge \\frac{2^9  \\log n}{\\eta^2}$ and that $d \\ge 2^{40} k^3 \\mu^9 s^4 p^2 n^{1\/k}$.\n        Let $G$ be a bipartite $(d, \\eta, \\varepsilon)$-expander on $n$ vertices with maximum degree at most $\\mu d$; denote the bipartition of $G$ by $\\{X, Y\\}$ and suppose that $|X| \\ge \\frac{n}{2}$.\n        Then for all but at most $\\frac{n^2}{p}$ pairs $(x, y)$ with $x,y \\in X$ the following holds.   \n        \\begin{equation*}\n            \\hom^*_{x, y}(C_{2k}) \\le \\frac{1}{s^2} \\hom_{x, y}(C_{2k}).\n        \\end{equation*}\n    \\end{lemma}\n    \n    \\begin{proof}\n        Let $S = 2^{13} \\mu^4 s^2 p$, and let $\\rho_{\\min}$ and $\\rho_{\\max}$ be defined as in the statement of \\Cref{lem:rhomin-rhomax}. Then, by the same lemma we have $\\rho_{\\max} \\le 2^{12} \\mu^4 \\rho_{\\min}$.\n        Let $A$ be the collection of (ordered) pairs $(x, y)$ with $x, y \\in X$ that satisfy  $\\hom^*_{x, y}(C_{2k}) \\ge \\frac{1}{s^2}\\hom_{x, y}(C_{2k})$.\n        Then\n        \\begin{align*}\n            \\hom^*(C_{2k}) \n            \\ge \\sum_{(x, y) \\in A} \\hom^*_{x, y}(C_{2k}) \n            \\ge \\frac{1}{s^2} \\sum_{(x, y) \\in A} \\hom_{x, y}(C_{2k})\n            \\ge \\frac{|A| \\cdot \\rho_{\\min}}{s^2}.\n        \\end{align*}\n      \n        Note that $\\sum_{x, y \\in X} \\hom_{x,y}(C_{2k}) = \\sum_{x,y \\in Y} \\hom_{x,y}(C_{2k})$, and so $\\hom(C_{2k}) = 2\\sum_{x, y \\in X} \\hom_{x,y}(C_{2k})$.\n        Hence, by \\Cref{lem:non-rainbow-hom} (which applies since $d \\ge 2^{14} k^3 S^2 \\mu n^{1\/k}$),\n        \\begin{equation*}\n            \\hom^*(C_{2k}) \n            \\le \\frac{1}{S} \\cdot \\hom(C_{2k})\n            = \\frac{2}{S} \\sum_{x, y \\in X} \\hom_{x, y}(C_{2k}) \n            \\le \\frac{2n^2 \\cdot \\rho_{\\max}}{S}.\n        \\end{equation*}\n        Combining the lower and upper bounds on $\\hom^*(C_{2k})$, we obtain the required inequality, as follows.\n        \\begin{equation*}\n            |A|\n            \\le \\frac{2n^2 \\cdot s^2 \\cdot \\rho_{\\max}}{S \\cdot \\rho_{\\min}}\n            \\le \\frac{n^2 \\cdot 2^{13} \\mu^4 s^2}{S} \n            = \\frac{n^2}{p}.\n            \\qedhere\n        \\end{equation*}\n    \\end{proof}\n   \n\n\\section{Rainbow paths and subdivisions in expanders} \\label{sec:main-proof}    \n\n    We now prove our first main result about rainbow subdivisions.\n    \n    \\begin{proof}[Proof of \\Cref{thm:main}.]  \n        \\begin{enumerate}\n            \\item \n                $H$ is a $(d, \\eta, \\varepsilon)$-expander on $n'$ vertices, where $d \\ge \\frac{d(G')}{2500(\\log n)^2} \\ge (\\log n)^{56}$, $\\varepsilon=\\frac{1}{2}$ and $\\eta = \\frac{\\varepsilon}{100(\\log n')^2} = \\frac{1}{200(\\log n')^2}$,\n            \\item\n                $H$ has maximum degree at most $\\mu d$, where $\\mu = 2500(\\log n')^2$.\n        \\end{enumerate}\n        \n        Denote the bipartition of $H$ by $\\{X, Y\\}$, and suppose that $|X| \\ge \\frac{n'}{2}$. \n        Let $k$ be the smallest even integer which is at least $\\frac{2^9 \\log n'}{\\eta^2}$, so $k = \\Theta\\!\\left((\\log n')^{5}\\right)$. \n        Let $s=2\\binom{m}{2} k$ and $p = 8m$. \n        Let $A$ be the set of pairs $(x, y)$ with $x, y \\in X$ that satisfy $\\hom^*_{x, y}(C_{2k}) > \\frac{1}{s^2}\\hom_{x, y}(C_{2k})$.\n        Note that $2^{40} k^3 \\mu^9 s^4 p^2 (n')^{1\/k} = \\Theta\\!\\left((\\log n')^{53}\\right) < d$ and so, by \\Cref{lem:few-bad-pairs}, we have $|A| \\le \\frac{(n')^2}{p} =\\frac{(n')^2}{8m}< \\frac{1}{m-1}\\binom{|X|}{2}$,\n        noting that $n'$ is sufficiently large.\n            \n        Hence by Tur\\'an's theorem, there is a subset $Z \\subseteq X$ of size $m$, such that $(x, y) \\notin A$ for every distinct $x, y \\in Z$. \n        By \\Cref{lem:theta}, for any two vertices $x$ and $y$ in $Z$, there exist $s$ many pairwise colour-disjoint and internally vertex-disjoint rainbow paths of length $k$ from $x$ to $y$. By the choice of $s$, one can find greedily $\\binom{m}{2}$ many paths of length $k$, which are pairwise colour-disjoint and internally  vertex-disjoint and are internally vertex-disjoint from $Z$, each of which connecting a different pair of vertices in $Z$. This gives us the desired subdivision of $K_m$.\n    \\end{proof}\n\n    Given a set of $m$ vertices in a graph $G$, a \\emph{$K_m$-subdivision rooted at $Z$} is a subgraph consisting of $\\binom{m}{2}$ paths, each joining a different pair of distinct vertices in $Z$, whose interiors are pairwise vertex-disjoint and disjoint from $Z$. By slightly adapting the proof of\n    \\Cref{thm:main}, one can show that any bipartite $(d, \\eta, \\varepsilon)$-expander $G$ (with suitable parameters) contains a rainbow $K_m$-subdivision rooted at $Z$ for almost all the $m$-sets $Z$ in $V(G)$ (not just in $X$). \n     By using some additional tools, we next show that in a bipartite $(d,\\eta,\\varepsilon)$-expander with suitable parameters, in fact, one can find  a rainbow $K_m$-subdivision rooted at $Z$ for every $m$-set $Z$ in $V(G)$.\n\n    \\begin{theorem} \\label{thm:main-rooted} \n        Let $n, L, d, \\mu, m \\geq 2$, $\\eta\\in (0,1)$ and $\\varepsilon\\in (0,\\frac{1}{8}]$, and suppose that $L = \\frac{2^{10} \\log n}{\\eta^2}$ and $d \\ge \\frac{2^{135} m^8 \\mu^{9} (\\log n)^7}{\\eta^{14}}$. Let $G$ be a bipartite $(d, \\eta, \\varepsilon)$-expander with maximum degree at most $\\mu d$, and let $Z$ be a set of $m$ vertices in $G$. Then there is  a rainbow $K_m$-subdivision, rooted at $Z$, where every edge is subdivided at most $L$ times.\n    \\end{theorem}\n    \n    \\begin{proof}\n        Let $\\ell = \\frac{4\\log n}{\\eta}$ and let $k$ be the smallest even integer satisfying $k \\ge \\frac{2^9 \\log n}{\\eta^2}$. One can check that $L \\ge 2(\\ell + 1) + k$.\n       \n        \n        \\begin{claim} \\label{claim:rainbow-connect}\n        Let $M$ be any set of colours and vertices such that $|M|\\leq 2\\binom{m}{2}(L+1) $. Let $x,y$ be any two vertices in $G$. There exists a rainbow $x,y$-path of length at most $L$ in $G$ that avoids $M$.    \n        \\end{claim}\n        \\begin{proof}[Proof of Claim~\\ref{claim:rainbow-connect}]\n            \n\n        Let $q = 256\\ell$. \n        By \\Cref{lem:reachable-robust-new} (using $d \\ge \\frac{20q\\ell + 8[2\\binom{m}{2}(L+1)]}{\\eta}$), there exists a subset $U_{x} \\subseteq V(G)$ of size at least $(1-\\varepsilon)n$ and a collection of paths $\\mathcal P = \\{P(u) : u \\in U_x \\}$, where for each $u \\in U_x$ the path $P(u)$ is a rainbow path from $x$ to $u$ of length at most $\\ell+1$ that avoids $M$, and no colour appears in more than $\\frac{n}{q}$ of the paths in $\\mathcal P$. Similarly, there exists a subset $U_{y} \\subseteq V(G)$ of size at least $(1-\\varepsilon)n$ and a collection of paths $\\mathcal Q = \\{Q(u) : u \\in U_y\\}$ where for each $u \\in U_y$ the path $Q(u)$ is a rainbow path from $y$ to $u$ of length at most $\\ell+1$ that avoids $M$, and no colour appears in  more than $\\frac{n}{q}$ of the paths in $\\mathcal Q$. Write $U = U_x \\cap U_y$; then $|U| \\ge (1 - 2\\varepsilon)n \\ge \\frac{3n}{4}$.\n        \n        We call an ordered pair $(u,v)$ with $u, v\\in U$ \\emph{colour-bad} if there is a colour that appears on both paths $P(u)$ and $Q(v)$. We next show that the number of colour-bad pairs in $U$ is small compared to all pairs. Indeed, let $H$ be the auxiliary graph on the vertex set $U$ where $uv$ is an edge whenever at least one of $(u, v)$ and $(v, u)$ is colour-bad.\n        Note that $d_H(u)\\leq \\frac{2(\\ell+1)n}{q}$, for every $u \\in U$, since $P(u)$ has length at most $\\ell+1$ and any colour on $P(u)$ can appear in the collection $\\mathcal{Q}$ at most $\\frac{n}{q}$ times (and similarly with the roles of $P$ and $Q$ reversed). Thus, $e(H) \\leq \\frac{(\\ell+1)n}{q}|U| \\le \\frac{2\\ell n}{q}|U| \\leq \\frac{n^2}{128}$.\n        \n        Let $s = |M| + 2(\\ell + 1) + 1$; then $s \\le 2\\binom{m}{2}(L+1) + 2\\ell + 3 \\le 2 m^2 L$.\n        Denote the bipartition of $G$ by $\\{X, Y\\}$ and suppose that $|X| \\ge \\frac{n}{2}$.\n        Call a pair $(u,v)$, with $u, v \\in X$, \\emph{$s$-bad} if $\\hom^*_{u, v}(C_{2k}) > \\frac{1}{s^2} \\hom_{u, v}(C_{2k})$. \n        Applying \\Cref{lem:few-bad-pairs} with $p = 64$ and verifying the condition on $d$ (namely, that $d \\ge 2^{40} k^3 \\mu^9 s^4 p^2 n^{1\/k}$), at most $\\frac{n^2}{64}$ ordered pairs $(u, v)$, with $u, v \\in X$, are $s$-bad. \n        \n        We claim that there is a pair $(u, v)$, with $u, v \\in U \\cap X$, which is neither colour-bad nor $s$-bad. Indeed, the total number of ordered pairs $(u, v)$ with $u, v \\in U \\cap X$ where $(u, v)$ is either colour-bad or $s$-bad is at most $\\frac{n^2}{128}+\\frac{n^2}{64}\\leq \\frac{n^2}{32}$. Since $|U| \\ge \\frac{3n}{4}$ and $|X| \\ge \\frac{n}{2}$, we have $|X \\cap U| \\ge \\frac{n}{4}$, so the number of ordered pairs $(u, v)$ with $u, v \\in U \\cap X$ is certainly more than $\\frac{n^2}{32}$. Hence, there is a pair $(u, v)$ which is neither colour-bad nor $s$-bad, as claimed. By Lemma~\\ref{lem:theta} there are $s$ many pairwise colour-disjoint and internally vertex-disjoint rainbow paths of length $k$ from $u$ to $v$. By the choice of $s$, there is at least one such path $T(uv)$ which shares no colours with the paths $P(u)$ and $Q(v)$ and also avoids $M$. Since $P(u)$ and $Q(v)$ are also colour-disjoint, it follows that $P(u)T(uv)Q(v)$   is a rainbow $x,y$-walk avoiding $M$ which contains a rainbow $x,y$-path, as desired.\n        \\end{proof}\n                \n        Let $(x_1, y_1), \\ldots, (x_{\\binom{m}{2}}, y_{\\binom{m}{2}})$ be an arbitrary ordering of the unordered pairs $(x, y)$ where $x, y \\in Z$ and $x \\neq y$.\n        We iteratively build paths $P_i$ for $i \\in [\\binom{m}{2}]$, as follows. Let $P_1$ be any rainbow $x_1,y_1$-path of length at most $L$, which exists\n        by Claim~\\ref{claim:rainbow-connect}. In general, suppose $P_1,\\dots, P_i$ have been defined, where $i<\\binom{m}{2}$. We let $M_i$ denote the set of vertices  and colours used in $\\cup_{j=1}^i P_j$ and let $P_{i+1}$ be a rainbow $x_{i+1},y_{i+1}$-path of length at most $L$ that avoids $M_i\\setminus\\{x_{i+1},y_{i+1}\\}$.\n        Since $|M_i|\\leq 2\\binom{m}{2}(L+1)$, by Claim~\\ref{claim:rainbow-connect} such a path $P_{i+1}$ exists. Hence, we are able to find $P_1,\\dots, P_{\\binom{m}{2}}$\n        as described above. Now, $\\bigcup_{i=1}^{\\binom{m}{2}} P_i$ forms a rainbow $K_m$-subdivision rooted at $Z$ in which each edge is subdivided\n        at most $L$ times.\n    \\end{proof}\n\n\\section{Blow-ups} \\label{sec:blowups}\n\n    In this section we use the ideas employed in the proof of our main result about rainbow subdivisions to prove \\Cref{thm:main-blowups} about blow-ups of subdivisions.\n\n    We will represent a copy of $K_{r, r}$ in a graph $G$ by a pair of $r$-sets $(A, B)$ such that all $A$-$B$ edges are present in $G$. We shall use the following `balanced supersaturation' result, due to Morris and Saxton \\cite{morris2016number}.\n\n    \\begin{theorem}[Special case of Theorem 7.4 in \\cite{morris2016number}] \\label{thm:morris-saxton}\n        For every $r \\ge 2$ there exist $c_1, c_2 > 0$ such that the following holds for every integer $n \\ge 1$ and sufficiently large $\\alpha$.\n        Suppose that $G$ is a graph on $n$ vertices with at least $\\alpha n^{2 - 1\/r}$ edges. Then there is a collection $\\mathcal{H}$ of copies of $K_{r, r}$ in $G$ satisfying\n        \\begin{enumerate}\n            \\item \n                $|\\mathcal{H}| \\ge c_1 \\alpha ^{r^2} n^r$,\n            \\item\n                For every $r$-set $A$ and vertex $u \\notin A$, there are at most $c_2 \\alpha^{r(r-1)}$ sets $B$ of size $r-1$ such that $(A, B \\cup \\{u\\}) \\in \\mathcal{H}$.\n        \\end{enumerate} \n    \\end{theorem}\n \n    \n    \\begin{proof}[Proof of \\Cref{thm:main-blowups}]\n \n        Let $G$ be a graph on $n$ vertices with at least $n^{2 - \\frac{1}{r}} (\\log n)^{\\frac{60}{r}}$ edges, and suppose that $n$ is large. We will show that $G$ contains an $r$-blow-up of a subdivision of $K_m$.\n        Let $\\alpha = (\\log n)^{\\frac{60}{r}}$, and let $c_1, c_2$ be as in \\Cref{thm:morris-saxton}. Observe that since $n$ is large, $\\alpha$ is large enough for \\Cref{thm:morris-saxton} to be applicable. \n        \n        Consider the auxiliary graph $\\mathcal{H}$ whose vertices are $r$-sets of vertices in $G$, and where $AB$ is an edge if and only if $(A, B)$ is a copy of $K_{r, r}$ in $G$. By \\Cref{thm:morris-saxton}, there is a subgraph $\\mathcal{H}'$ of $\\mathcal{H}$ with $d(\\mathcal{H}') \\ge c_1 \\alpha^{r^2} = c_1  (\\log n)^{60 r}$, such that for every $r$-set $A$ and every vertex $u \\notin A$ in $G$, there are at most $c_2 \\alpha^{r(r-1)} = c_2 (\\log n)^{60(r-1)}$ neighbours of $A$ in $\\mathcal{H}'$ that contain the vertex $u$. \n        \n        Write $t = rc_2 (\\log n)^{60(r-1)}$.\n        Consider a bipartite subgraph $\\mathcal{H}''$ of $\\mathcal{H}'$ with average degree at least $\\frac{c_1}{2}(\\log n)^{60r}$, and note that $|V(\\mathcal{H}')| \\le n^r$.\n        By \\Cref{lem:bounded-max-deg-expander} (using that $\\frac{c_1}{2} (\\log n)^{60r} \\ge 10^7 (\\log n)^3$), there is a bipartite subgraph $\\mathcal{F}$ of $\\mathcal{H}''$ satisfying the following properties.\n     \n        \\begin{enumerate}\n            \\item \n                $\\mathcal{F}$ is a $(d, \\eta, \\frac{1}{2})$-expander on $N$ vertices, where $d \\ge \\frac{c_1 (\\log n)^{60 r} }{5000(r\\log n)^2} \\ge t (\\log n)^{55}$ (using that $n$ is large) and $\\eta = \\frac{1}{200(\\log N)^2}$, \n            \\item\n                $\\mathcal{F}$ has maximum degree at most $\\mu d$, where $\\mu = 2500 (\\log N)^2$. \n        \\end{enumerate}\n        \n        Let $k$ be the smallest even integer such that $k\\geq  \\frac{2^9 \\log N}{\\eta^2}$. Then $k = \\Theta\\!\\left((\\log N)^{5}\\right)$.\n        \n        For each $u\\in V(\\mathcal{F})$, let $R_u$ denote the $r$-set in $G$ that $u$ corresponds to. Let $\\sim$ be a relation defined on $V(\\mathcal{F})$, where $u \\sim v$ whenever $R_u\\cap R_v\\neq\\emptyset$.\n        For vertices $x, y$ in $\\mathcal{F}$, let $\\Hom^{**}_{x, y}(C_{2k})$ be the family of all the closed walks $(x_1, \\ldots, x_{2k})$ in  $\\mathcal{F}$ where $x_1 = x$, $x_{k+1} = y$, and $x_i \\sim x_j$ for some distinct $i, j \\in [2k]$. \n        Let $\\Hom^{**}(C_{2k}) = \\cup_{x, y \\in V(G) }\\Hom^{**}_{x,y}(C_{2k}).$ Let $\\hom^{**}(C_{2k}) = |\\Hom^{**}(C_{2k})|$ and let $\\hom^{**}_{x,y}(C_{2k}) = |\\hom^{**}_{x,y}(C_{2k})|$. \n       \n        \n\n\n       \n\n        Let $S = \\left(\\frac{d}{2^{10} k^3t \\mu N^{1\/k}}\\right)^{1\/2}$. Since $\\frac{d}{t} \\ge (\\log n)^{55}$, $N^{1\/k} \\le N^{\\frac{1}{\\log N}} = 2$ and $\\log N = O(\\log n)$, we have \n        \\begin{equation*}\n            S             = \\Omega\\left( (\\log N)^{19}\\right).\n        \\end{equation*}\n        \\begin{claim}\n        \\label{homstarrtohom}\n            $\\hom^{**}(C_{2k}) \\le \\frac{1}{S}\\hom(C_{2k})$.\n        \\end{claim}\n        \n        \\begin{proof}[Proof of Claim~\\ref{homstarrtohom}]\n           Since $\\mathcal{F}$ is a $(d,\\eta,\\frac{1}{2})$-expander, $\\delta(F)\\geq \\frac{d}{2}$.\n           As in the proof of \\Cref{lem:non-rainbow-hom}, we have $\\hom(C_{2k}) \\ge \\left(\\frac{d}{2}\\right)^{2k}$. \n           Since $\\mathcal{F}\\subseteq \\mathcal{H}'$, by our assumption on $\\mathcal{H}'$, for every two vertices $u$ and $v$ in $\\mathcal{F}$ there are at most $rc_2 (\\log n)^{60(r-1)} = t$ neighbours $w$ of $v$ for which $u \\sim w$. By \\Cref{lem:janzer-vertices}, we get \n            \\begin{align*}\n                \\hom^{**}(C_{2k}) \n                & \\le 32 k^{3\/2} t^{1\/2} (\\mu d)^{1\/2} N^{\\frac{1}{2k}} \\cdot \\hom(C_{2k})^{1 - \\frac{1}{2k}} \\\\\n                & \\le \\left(\\frac{2^{10} k^3 t \\mu N^{\\frac{1}{k}}} {d} \\right)^{1\/2} \\cdot \\hom(C_{2k}) \\\\\n                &= \\frac{\\hom(C_{2k})}{S},\n            \\end{align*} \n            as desired.\n        \\end{proof}\n         \n        Let $\\rho_{\\min}$ and $\\rho_{\\max}$ be defined as in \\Cref{lem:rhomin-rhomax} with respect to the graph $\\mathcal{F}$. Then, by the same lemma, $\\frac{\\rho_{\\max}}{\\rho_{\\min}} \\le 2^{12} \\mu^4 = O\\!\\left( (\\log N)^8\\right)$.\n        Let $s = \\binom{m}{2} r k$; then $s = \\Theta\\!\\left((\\log N)^{5}\\right)$.\n      \n        Denote the bipartition of $\\mathcal{F}$ by $\\{X, Y\\}$, and suppose that $|X| \\ge \\frac{N}{2}$. Let \n        \\[A=\\left \\{(x,y) : \\, x,y\\in X \\text{ and } \\hom^{**}_{x, y}(C_{2k}) \\ge \\frac{1}{s^2}\\hom_{x, y}(C_{2k}) \\right\\}.\\]\n    \n        Then \n        \\begin{equation*}\n            \\hom^{**}(C_{2k}) \n            \\ge \\sum_{(x, y) \\in A} \\hom^{**}_{x, y}(C_{2k})\n            \\ge \\frac{1}{s^2} \\sum_{(x, y) \\in A} \\hom_{x, y}(C_{2k}) \\ge \\frac{|A| \\rho_{\\min}}{s^2}.\n      \\end{equation*}\n        By Claim~\\ref{homstarrtohom},\n        \\begin{equation*}\n            \\hom^{**}(C_{2k}) \n            \\le \\frac{\\hom(C_{2k})}{S} \n            = \\frac{2}{S}\\sum_{x, y \\in X} \\hom_{x, y}(C_{2k}) \n            \\le \\frac{2 N^2 \\rho_{\\max}}{S},\n        \\end{equation*}\n        using $\\sum_{x, y \\in X} \\hom_{x, y}(C_{2k}) = \\sum_{x, y \\in Y}\\hom_{x, y}(C_{2k})$.\n        Combining the two bounds, and using $\\frac{\\rho_{\\max}}{\\rho_{\\min}} = O\\!\\left((\\log N)^8\\right)$, $s = O\\!\\left( (\\log N)^{5}\\right)$ and $S = \\Omega\\!\\left( (\\log N)^{19}\\right)$, we get\n        \\begin{align*}\n            \\frac{|A|}{N^2}\n            \\le \\frac{2 s^2 \\cdot \\rho_{\\max}}{S \\cdot \\rho_{\\min}}\n            = O \\left( \\frac{(\\log N)^{18}}{(\\log N)^{19}}\\right)\n            = O\\left( \\frac{1}{\\log N}\\right).\n        \\end{align*}\n        As $N$ is large, $m$ is a constant and $|X| \\ge \\frac{N}{2}$, it follows that $|A| \\le \\frac{1}{m-1}\\binom{|X|}{2}$.\n        By Tur\\'an's theorem, there is a set $Z \\subseteq X$ of size $m$, such that for every distinct $x, y \\in Z$ we have $(x, y) \\notin A$.\n        \n        \\begin{claim} \\label{blowup-theta}\n            For every pair of distinct vertices $x,y$ in $Z$, there exist $x,y$-paths $P_1,\\dots, P_s$ of length $k$ in $\\mathcal{F}$ such that\n            for any distinct $u,v$ in $V(\\cup_{i=1}^s P_i)\\setminus \\{x,y\\}$ we have $R_u\\cap R_v=\\emptyset$.\n        \\end{claim}\n        \\begin{proof}[Proof of Claim~\\ref{blowup-theta}] \n            By choice of $Z$ and definition of $A$, we have $\\hom^{**}_{x, y}(C_{2k}) < \\frac{1}{s^2}\\hom_{x, y}(C_{2k})$.\n            By Lemma~\\ref{lem:theta-general}, there exist $P_1, \\ldots, P_s \\in \\Hom_{x, y}(P_k)$ such that $P_i P_j \\notin \\Hom^{**}_{x, y}(C_{2k})$ for every distinct $i, j \\in [s]$. The paths $P_1, \\ldots, P_s$ satisfy the requirements, using the definitions of $\\Hom^{**}_{x, y}(C_{2k})$ and the relation $\\sim$.\n        \\end{proof}\n        \n        Let $L$ be a maximal collection of pairs of distinct vertices $u, v \\in Z$ with the property that there are paths $P_{u,v}$ of length $k$ in $\\mathcal{F}$ such that the $r$-sets in $G$ corresponding to the vertices on $W\\coloneqq \\cup_{(u,v)\\in L} V(P_{u,v})$ are pairwise disjoint. Suppose $|L|<\\binom{m}{2}$. Then there exists a pair of distinct vertices $x,y\\in Z$ such that $(x,y)\\not \\in L$. Also, $|W|<\\binom{m}{2}k$ and hence\n        $|\\cup_{w\\in W} R_w|<\\binom{m}{2}kr=s$. By Claim~\\ref{blowup-theta}, there exists a $P_{x,y}$ of length $k$ from $x$ to $y$ in $\\mathcal{F}$\n        such that $\\cup_{v\\in V(P_{x,y})} R_v$ is disjoint from $\\cup_{w\\in W} R_w$. Hence, the pair $(x,y)$ can be added to $L$, contradicting the maximality of $L$. Therefore, $L=\\binom{Z}{2}$, and so $\\bigcup_{(u,v)\\in L} P_{u,v}$ is a $K_m$-subdivision in $\\mathcal{F}$ \n        in which each edge is subdivided at most $k$ times. Since the $r$-sets in $G$ corresponding to the vertices of $L$ are pairwise disjoint, this yields an $r$-blow-up of a subdivision of $K_m$ in $\\mathcal F$.\n    \\end{proof}\n\n\n\\section{Conclusion} \\label{sec:conc} \nIn this paper we showed that there is a constant $c\\leq 60$ such that any  $n$-vertex properly edge-coloured graph $G$ \nwith at least $n(\\log n)^c$ edges contains a rainbow subdivision of $K_m$. \nOn the other hand, an immediate lower bound is given by the best known lower bound from \\cite{keevash2007rainbow} on $\\ex^*(n,\\mathcal{C})$, which is $\\Omega(n\\log{n})$. This shows that our bound is tight up to a polylogarithmic factor.\nWe pose the following question.\n    \n    \\begin{question} \\label{question:correct-bound}\n        Fix $m\\geq 2$. What is the smallest $c$ such that for\n        all sufficiently large $n$ the following holds: if $G$ is a properly edge-coloured graph on $n$ vertices with at least $\\Omega(n(\\log{n})^c)$ edges, then it contains a rainbow subdivision of $K_m$? \n        In particular, is $c=1$?\n    \\end{question}\n\nFor the clarity of presentation, we did not optimise our arguments to obtain\nthe best possible value of $c$. However,\nto answer Question~\\ref{question:correct-bound}, new ideas will be needed.\nNote that even the correct order of magnitude of $\\ex^*(n,\\mathcal{C})$ is still unknown.   \n    Recall that Janzer~\\cite{janzer2020rainbow} proved $\\ex^*(n,\\mathcal{C})=O(n(\\log n)^4)$, using the counting lemmas on\n    closed walks.  We can give an alternative proof of $\\ex^{*}(n,\\mathcal{C})=O(n(\\log{n})^5)$ by using the basic\n    expansion property of an expander (Lemma~\\ref{lem:reachable-robust-new}) and a digraph idea used by Letzter~\\cite{letzter2021tight} regarding the Tur\\'an number of the family of tight cycles.\n    \n \n   \n    \n    In this paper, we also showed that any $n$-vertex graph with at least $n^{2-\\frac{1}{r}} (\\log n)^{\\frac{60}{r}}$ edges contains an $r$-blow-up of a subdivision of $K_m$. This bound is tight up to $(\\log n)^{\\frac{60}{r}}$ factor,  due to the following proposition,\n    which was mentioned in~\\cite{janzer2020rainbow} as a remark. Since no proof was given there, we include one here for completeness. Let $\\mathcal{C}[r] \\coloneqq \\{C_k[r] : k \\ge 1\\}$.\n    \n    \\begin{proposition}\\label{prop:lowerboundblowup}\n        Let $r \\ge 1$ be a fixed integer. Then $\\ex(n, \\mathcal{C}[r]) = \\Omega\\big(n^{2-1\/r})$.\n    \\end{proposition}\n    \\begin{proof}\n    \n    For each $i,$ let $F_i$ be the $r$-blow-up of a cycle of length $i$, let $v_i = v(F_i)$ and $e_i=e(F_i)$. Then $e_i = r v_i$ since $F_i$ is $2r$-regular. \n    Let $\\mathcal{F} \\coloneqq \\{F_i : i \\le n\/r\\}$. Consider the random graph $G \\coloneqq G(n, p)$ with $p = n^{-1\/r}$. \n    For each $i$, let $X_i$ be the number of copies of $F_i$ in $G$, and let $X \\coloneqq \\sum_i X_i$. \n    Then \n    $\\mathbb{E}[X_i] \\le n^{v_i}p^{e_i} = n^{v_i}(n^{-1\/{r}})^{rv_i} = 1.$\n    By linearity of expectation, $\\mathbb{E}[X]=\\sum_i \n    \\mathbb{E}[X_i] \\le n$.\n    On the other hand,\n    $\\mathbb{E}[e(G)] \\ge  \\Omega(n^{2-1\/r}).$\n    Thus, \n    $\\mathbb{E}[e(G) - X] \\ge  \\Omega(n^{2-1\/r}).$\n    \n    So there exists a graph $G$ such that $e(G) - X \\ge \\Omega(n^{2-1\/r}).$ Now remove an edge from every copy of a member of $\\mathcal{F}$ in $G$, to obtain an $\\mathcal{F}$-free graph with  $\\Omega(n^{2-1\/r})$ edges, as desired.\n    \\end{proof}\n    \n    We pose the following question, which strengthens Question 6.2 in \\cite{janzer2020rainbow}.\n    \\begin{question} \n        Fix $m \\ge 2$. Is it true that if $G$ is an $n$-vertex graph with at least $\\Omega(n^{2-\\frac{1}{r}})$ edges then it contains an $r$-blow-up of a subdivision of $K_m$?\n    \\end{question}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgments}\nWe thank Xian Teng and Muheng Yan for their help in data annotation. We would like to acknowledge the support from NSF \\#2027713, ARC DP180101985, and AFOSR projects 19IOA078, 19RT0797, 20IOA064, and 20IOA065. Any opinions, findings, and conclusions or recommendations expressed in this material do not necessarily reflect the views of the funding sources.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nSeveral recent studies have documented the ideological asymmetries between the left-wing and right-wing activism~\\cite{brady2019ideological,schradie2019revolution,freelon2020false,waller2021quantifying}. Some highlight the dominance of conservative voices on social media~\\cite{brady2019ideological}; others portray the widespread symbolic support for progressive social movements~\\cite{jackson2020hashtagactivism}. The term ``conservative advantage'' is coined to describe the strategic dissemination of right-wing users to spread their messages~\\cite{schradie2019revolution}. However, most of the existing research bases on the analysis of a single platform or a single political topic. Relatively little is known about how different ideological groups garner attention across platforms, and whether the group advantage of gaining visibility remains across topics and over time. To answer these questions, this work designs several sets of cross-platform measurements on the collective attention dynamics of two different ideological groups across three controversial political topics.\n\nOnline platforms, such as Twitter, YouTube, Reddit, and Facebook, are social-technological artifacts that segregate online attention into silos defined by the underlying software and hardware systems. Video views on YouTube are known to be driven by discussions outside the platform~\\cite{rizoiu2017expecting}, and to be part of users' broader information diet~\\cite{hosseinmardi2021examining}. What is not known, however, is how groups of related content comparatively evolve across different social platforms. Collective attention on political content have been studied on one topic, such as the Occupy Movement~\\cite{thorson2013youtube}, Gun Control\/Rights~\\cite{zhang2019whose}, and Black Lives Matters~\\cite{de2016social,stewart2017drawing}. Yet, cross-cutting studies that compare different movements are rare. With data from three long-running controversial topics, this work seeks to provide measures across YouTube and Twitter and paint a nuanced picture about the temporal patterns of attention from left to right.\n\n\\input{images\/intro_teaser}\n\nWe choose three topics: {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and \\texttt{Black Lives Matter} ({\\tt BLM}\\xspace). We rely on video hyperlinks to connect the content from YouTube to Twitter. A motivating example is given in~\\Cref{fig:intro_teaser}. We plot the time series of daily view count for the collected {\\tt BLM}\\xspace videos from YouTube (top panel) and daily volume of tweets mentioning these {\\tt BLM}\\xspace videos from Twitter (bottom panel). Both time series are further disaggregated by video uploaders' political leanings. Visually, the view count dynamics of both left- and right-leaning videos are relatively stable in year 2017, except a sharp spike caused by the ``Unite the Right rally''\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/Unite_the_Right_rally}} event in Charlottesville, USA. On the bottom panel, the tweet count dynamics of right-leaning videos have many spikes, which can be attributed to the upload of new videos from far-right YouTube political commentators. The measures on YouTube and Twitter present a contrasting story here: if we focus on the two weeks period after the rally, left-leaning videos attracted more attention on YouTube (measured by views, left: 27.2M, right: 13.9M) while right-leaning videos had higher exposure on Twitter (measured by tweets, left: 37.5K, right: 52.3K). This example demonstrates the need for cross-platform analysis -- findings on one platform may not generalize to another.\n\nWe design a set of metrics from publicly available data on YouTube and Twitter, which include total views, video watch engagement, tweet reactions, the evolution of attention over time, and early adopter networks among tweets and Twitter users. On YouTube, we find that left-leaning videos accumulate more views, are more engaging, and have higher viral potential than right-leaning videos. In contrast, right-leaning videos have higher numbers of total tweets and retweets on Twitter. Statistics on the unfolding speed for views and tweets show that the attention on left-leaning videos attenuates faster, while that on right-leaning videos persists for longer. Note that these observations are not generalized unanimously across topics, e.g., for some metrics, we observe significant differences for {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace, but not for {\\tt BLM}\\xspace. These findings expand current wisdom on ideological asymmetries in two ways: the first is exposing the novel facet that left-leaning content attracts more attention in a shorter period of time; the second is the need of contrasting temporal attention statistics between platforms, such as right-leaning tweet cascades tend to start earlier and YouTube views on right-leaning content sustain longer. In sum, our observations paint a richer picture of attention patterns across the political spectrum, provide a basis for further studying political framing and group behavior, and supply fundamental metrics for understanding influences that transcend platforms.\n\nThe main contributions of this work include:\n\n\\begin{itemize}[leftmargin=*]\n  \\item a data curation procedure linking content on YouTube and Twitter for longitudinal topic monitoring.\\footnote{Our datasets and analysis code are publicly available at \\url{https:\/\/github.com\/picsolab\/Measuring-Online-Information-Campaigns}}\n  \\item several sets of cross-platform metrics that support statistical comparisons for different ideological groups, encompassing the volume and quality of attention, networks of tweets and users, as well as relative temporal evolution.\n  \\item adding the temporal and cross-platform dimensions to recent observations on ideological asymmetries. We find that polarized content engages users in distinct ways -- more views, more engagement, and faster reactions for videos on the left; comparing to more tweets, more sustained attention for videos on the right.\n\\end{itemize}\n\n\n\\section{Measures for Cross-Platform Attention}\n\\label{sec:metric}\n\nThis section designs several sets of metrics for the cross-platform data, in order to compare content across different political ideologies, and examine whether the differences are consistent across topics, across platforms, and over time.\n\n\\subsection{Aggregate Attention on YouTube and Twitter}\n\\label{ssec:aggregate_metrics}\n\nWe present four metrics for the total video attention on YouTube.\n\n\\header{\\em Total view count} sums up a video's view count time series until day 120.\n\n\\header{\\em Relative engagement} is a metric proposed in~\\cite{wu2018beyond} for quantifying the average video watching behavior. Specifically, for each video, we first compute {\\em average watch percentage}, defined as the total watch time divided by total number of views (both at 120 days) and then normalized by the video length (in seconds). The relative engagement score is the percentile ranking of average watch percentage among videos of similar lengths. It is a normalized score between 0 and 1. A higher score means more engaging, e.g., a score of 0.8 suggests that this video is on average watched for longer time than 80\\% videos of similar length. Note that relative engagement is shown to be stable over time, hence there is no need to examine the temporal variations of watch time, as it would strongly correlate with view counts. In this work, relative engagement is computed based on a publicly available collection of 5.3M YouTube videos~\\cite{wu2018beyond}, with details described in Section D of \\cite{appendix}.\n\n\\header{\\em Fraction of likes} measures the video reaction -- provided by YouTube as the total counts of likes and dislikes and collected via the thumb-up and thumb-down icon on the video page. A relatively lower fraction of likes indicates a more diverse audience reaction to the video content. Note that the majority of videos receive a lot more likes than dislikes.\\footnote{YouTube announced that the dislike count will no longer be available to the public on Nov 10, 2021. \\url{https:\/\/blog.youtube\/news-and-events\/update-to-youtube\/}}\n\n\\header{\\em Viral potential} is a positive number, representing the {\\em expected} number of views that a YouTube video will obtain if mentioned by an {\\em average} tweet on Twitter~\\cite{rizoiu2017online}. More specifically, it is the area under the impulse response function of an integral equation known as Hawkes Intensity Process (HIP)~\\cite{rizoiu2017expecting}, which is learned for each video by using the first 120 days of tweeting and viewing history. We choose this quantity rather than simply dividing the number of views by the number of tweets, because the model takes into account views that are yet to unfold due to its sustained circulation via sharing and tweeting. A self-contained summary about HIP and viral potential computation is given in Section E of \\cite{appendix}.\n\nOn Twitter, tweets can be categorized into four types: original tweets, retweets, quotes, and replies. This leads to five counting metrics: the total number of {\\bf\\em tweets, original tweets, retweets, quoted tweets}, and {\\bf\\em replies}. \n\n\\subsection{Views and Tweets over Time}\n\\label{ssec:temporal_metrics}\n\n\\header{\\em Viewing half-life} is computed as the number of days to achieve half of its total views at day 120.\n\n\\header{\\em Tweeting half-life} is computed as the number of days to achieve half of its total tweets at day 120.\n\n\\header{\\em Tweeting lifetime} is time gap between the first and the last tweets. We do not measure lifetime on viewing because the view count of a video rarely becomes zero even towards the end of the measurement period, but tweets tend to exhaust much sooner.\n\n\\header{\\em Tweeting inter-arrival time} is the average time difference between every two consecutive tweets about each video.\n\n\\header{\\em Accumulation of views and tweets.} \nIn addition to the summary metrics above, we also compare the attention accumulation on the left- and right-leaning content on a daily basis. On each day $t$, we compute the fraction of the total views that each video has achieved. This leads to two sets of samples $\\{v^{(L)}_i\\}_{i=1}^n$ and $\\{v^{(R)}_j\\}_{j=1}^m$, where $n$ is the number of left-leaning videos and $m$ is the number of right-leaning videos. We then compute the normalized Mann-Whitney U (MWU) statistic~\\cite{mann1947test}, \n$$\\bar U_t = \\frac{1}{nm}\\sum_i\\sum_j \\{ I[v_i^{(L)} > v^{(R)}_j] + 0.5\\,I[v_i^{(L)} = v^{(R)}_j] \\}$$ \n\nHere $I[\\cdot]$ is the indicator function that takes value 1 when the argument is true, 0 otherwise. The U statistic intuitively corresponds to the fraction of sample pairs $(v^{(L)}_i, v^{(R)}_j)$ where the sample from left-leaning distribution is larger, accounting for ties. If the distributions of $v^{(L)}$ and $v^{(R)}$ are indistinguishable, then $\\bar U$ would be around 0.5. We compute the statistic $\\bar U_t$ on tweets in the same fashion, and both statistics are computed for each day. These two series of statistics allows us to quantify the differences between left- and right-leaning content, and compare the trends on the accumulation of views and tweets over time.\n\n\\subsection{Videos' Tweet Cascades}\n\\label{ssec:cascade_metrics}\n\n\\header{\\textit{Cascade size.}}\nWe define that a {\\em cascade} consists of a root tweet and all of its retweets, replies, and quotes. It is well-known that the vast majority of cascades in online diffusion networks are very small and only a very small fraction of cascades would become very big~\\cite{goel2012structure}. Based on the number of tweets in a cascade, we divide the cascades into isolated (only root tweet), small (2-4 tweets), and large ($\\geq 5$ tweets) groups. For videos of each leaning on each topic, we compute the fractions of isolated\/small\/large cascades and the fraction of tweets in each cascade group. These metrics quantify the structure of online diffusion and allow us to compare behavior on controversial political topics with what was known about tweeted videos in general.\n\n\\header{\\textit{Cascade start time}} is the percentage of accumulated views of the video when the root tweet of the cascade is posted. It measures how much view attention is accumulated on YouTube before the infusion on Twitter starts. We choose to describe cascade timing relative to the accumulation of view, rather than in absolute number of days since upload, because (a) such relative time more directly correlates the amount of cascades with respect to the views they can potentially drive (rather than through another variable, days); and (b) the percentage of views provides more granularity, since many videos have all views and tweets unfold within a few days after upload.\n\n\\subsection{Networks among Early Adopters on Twitter}\n\\label{ssec:network_metrics}\n\nFor each video, we obtain its follower network among the early adopters. If there exists a following relationship between a pair of users, a directed edge is established. This results in one network for each shared video. We compute a set of metrics per video, and then compare their distributions on each topic for left- and right-leaning videos. We describe two key metrics here, and discuss four additional metrics in Section H of~\\cite{appendix}.\n\n\\header{\\textit{Gini coefficient of indegree centrality.}} \nWe calculate the indegree centrality for each node in the network. To have a video-level metric, we use the Gini coefficient, which ranges from 0 to 1 and measures the distribution inequality. Specifically, the Gini coefficient of indegree centrality quantifies the degree of inequality of the indegree distribution. A higher value indicates that a few early adopters are followed more by other early adopters, and a lower value indicates that the indegree distribution is more equal. \n\n\\header{\\textit{Gini coefficient of closeness centrality}} captures the dispersion in inverse of average shortest path length from one early adopter to all other early adopters of a given video. Higher coefficient implies that a few early adopters can reach the rest of the early adopters within a few hops.\n\n\\section{Conclusion and Discussion}\n\\label{sec:conclusion}\n\nThis work presents a quantitative study that links collective attention towards online videos across YouTube and Twitter over three political topics: {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace. For each topic, we curated a cross-platform datatset that contained hundreds of videos and hundreds of thousands of tweets spanning 16 months. The extracted videos all have a non-trivial amount of views and tweets. The key contributions include several sets of video-centric metrics for comparing attention consumption patterns between left-leaning and right-leaning videos across two platforms. We find that left-leaning videos are more viewed and more engaging, while right-leaning videos are more tweeted and have longer attention spans. We also found that the follower networks of early adopters on left-leaning videos are of higher centrality, whereas tweet cascades for right-leaning videos start earlier in the attention lifecycle. This study enriches the current understanding of ideological asymmetries by adding a set of temporal and cross-platform analyses.\n\n\\header{Limitations.} \nExtensive discussions about social data biases are presented in~\\cite{olteanu2019social}. The biases can be introduced due to the choice of social platforms, data (un)availability, sampling methods, etc. Here we discuss three limitations in our data collection process.\n\nA recent study found that Twitter filtered streaming API subsamples high-volume data streams that consist of more than 1\\% of all tweets~\\cite{wu2020variation}. The authors proposed a method of using Twitter rate limit messages to quantify the data loss. Based on this method, we find our 16-month Twitter stream has a sampling rate of 79.4 \\% -- we collected 1,802,230,572 out of 2,270,223,254 estimated total tweets. Under a Bernoulli process assumption, the chance of collecting a video tweeted more than once in our tweet stream is 95.8\\%. Since most missing videos are tweeted sporadically, the sampling loss from Twitter APIs is small, which minimally affects the measures on tweeting activities, including attention volumes, timing, and cascade sizes. Confidence intervals for simple measures such as volume can be derived~\\cite{wu2020variation}.\n\nIn this paper, we present various measurements focused on YouTube videos, which are the main entities that link the two platforms. YouTube viewers are unknowable (via publicly available data) and Twitter users are hard to track consistently over time. Therefore, we track videos which attract views and tweets. All the presented metrics are video-centric and we do not assume that the viewers or tweeters of the videos represent specific groups of users. We believe that each set of videos ({\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace) represent YouTube videos that are relevant to the topic, curated by keyword queries and semi-manual coding. The number of videos belonging to each topic is not large but we attempted to include all relevant videos shared on Twitter which have non-trivial activities. Thus our results intend to explain attention gathering behavior of topic-relevant videos. Nevertheless, it is unclear that our observations about ideological asymmetries can generalize to videos with less attention and\/or videos about other topics. We leave this generalization validation as future work.\n\nOne data integrity limitation is the time gap between tweet collection in 2017--2018 and early adopters' follower networks collected in early 2020.\nOur data collection is limited by the Twitter search API quota, which restrains collecting tweets and Twitter user followers simultaneously. The tweeted videos stream has on average 3.7M tweets in each day. Collecting the follower network for all these tweets far exceeds the capacity of Twitter API, focusing on the early-adopter network is a practical trade-off between still having informative results and making data collection feasible. A related issue comes from unavailable YouTube videos and Twitter users since content publicly available in 2017 may be deleted, banned, or protected in 2020. We found that between 17\\% to 19\\% candidate videos become unavailable in our dataset.\n\n\\header{Practical implications and future work.}\nWe believe this work adds a new dimension to the understanding of online political behavior and discourse -- cross platform links. Further examination in this direction could bear theoretical and empirical fruits. The measurements presented in this work are mostly quantitative. One direction of future work is to complement qualitative analysis. For example, to gain deeper insight into our observations about how the user attention to left-leaning YouTube videos was driven by a group of elite early adopters, one can examine typical diffusion networks from both left- and right-leaning groups and study the diffusion process of the video spreading. One could also examine the framing of left- and right-leaning content in both video descriptions and tweets about them. For example, \\citet{linDynamicsTwitterUsers2020} used mixed-methods approaches to identify the primary framing and rhetorics in online conversations related to gun control, which can be expanded to enrich the quantitative analyses, such as investigating the linguistic features of YouTube descriptions and tweet cascades, and their relationships to the changes in collective attitudes. Finally, understanding the collective attention across multiple social platforms is important for content producers, who could devise better strategies in promoting their content in another domain.\n\n\\section*{Ethical Statement}\nAll data that we obtained was publicly available at the time of data collection. We discarded deleted, protected, and private content at the time of analysis. In our released dataset, we anonymized user identities. Therefore, the analyses reported in this work do not compromise any user privacy.\n\n\\input{tables\/table-measures}\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\subsubsection{Online behavior of political groups.} \nMeasurement studies have quantified different aspects of users, contents, and their interactions under political polarization on social media. \\citet{conover2011political} presented one of the first profiling studies of polarized political groups on Twitter. \nThere have also been evidences that liberal and conservative groups attract online attention in different manners. \\citet{abisheva2014watches} focused on a set of influential Twitter users who promoted YouTube videos, and they found that conservatives tweeted more diverse topics than liberals and that conservatives shared new videos faster. \\citet{bakshy2015exposure} quantified the extent to which Facebook users were exposed to politically opposing contents, and they found that conservatives tended to seek out more cross-partisan content. \n\\citet{linDynamicsTwitterUsers2020} distinguished online behavioral signals, such as linguistic and narrative characteristics, of two ideology groups in response to mass shooting events.\n\\citet{garimella2018political} defined several consumption and production metrics and profiled key user behavior patterns. \\citet{ottoni2018analyzing} showed that conservatives used more specific language to discuss political topics and showed more negative emotions in the language. On YouTube, a recent study from \\citet{wu2021cross} found that left-leaning videos attracted more comments from conservatives than right-leaning videos from liberals. However, all of these works are conducted platform-wide, and are not specialized into particular topics or movements. \n\nOnline activism, also known as online social movement has been actively studied as a form of digital political campaigns.\nFor example, \\citet{de2016social} presented one of the first studies on the {\\tt BLM}\\xspace movement, measuring geographical differences in participation, and relationships to offline protests. \\citet{stewart2017drawing} constructed a shared audience network of users who talked about {\\tt BLM}\\xspace on Twitter and found the existence of superclusters among liberals and conservatives. \\citet{zhang2015modeling} performed policy decision prediction based on tweet texts analysis on \\texttt{same-sex marriage}. In a follow-up work, \\citet{zhang2016gender} discussed gender disparity by linking tweet texts to the state-level {\\tt Abortion}\\xspace policy events. \\citet{ertugrul2019activism} examined the relation between offline protest events and their social and geographical contexts. \\citet{freelon2020false} explained different tactics of liberals and conservatives when approaching audience on social media and articulated the asymmetries of measured behavior between conservatives and liberals. However, all of these works focus on a single controversial topic. In contrast, we examine three political topics and assess consistency of findings across the topics.\n\n\\subsubsection{Cross-platform measurement studies.} One early attempt in linking Twitter and YouTube data is from \\citet{abisheva2014watches}, in which the authors found that the features of early adopters on Twitter were predictive for the video view counts on YouTube. \\citet{rizoiu2017expecting} proposed the Hawkes Intensity Process that linked the time series of tweets and views, which led to a metric called {\\it viral potential} for measuring the expected number of views that a video would obtain if mentioned by an average tweet~\\cite{rizoiu2017online}. \\citet{zannettou2017web} measured the sharing of alternative and mainstream news articles on three different platforms -- Twitter, Reddit, and 4chan. This seminal study characterized the role of fringe communities in spreading news. \\citet{hosseinmardi2021examining} used browsing histories to infer video watch behavior, and quantified the consumption and driver of extreme content with respect to users' information diet. However, all of these works cover a breadth of content sources and categories, but none is focused around social movements or consistent political topics. \n\nThis presented work bridges the gap of cross-platform measurement studies on multiple social movements, aiming to provide richer understanding that balances the ideological asymmetry between the left and the right. \n\n\n\n\n\n\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{Identifying Topical Videos and Tweets}\n\\label{sec:app-data}\n\nWe curated a keyword list for each of the three controversial topics -- {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace. Keywords for {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace came from another recent work \\cite{guo2020inflating} and were expanded using contextual expansion with sample tweets. Keywords for {\\tt BLM}\\xspace were manually curated, separating into pro- and anti-{\\tt BLM}\\xspace keywords respectively. We expanded the latter with white-supremacy related keywords. \\Cref{table:app-keyword} lists all keywords.\n\nWe consider a video is potentially relevant if (a) it contains at least one keyword in the video title or description; or (b) it is mentioned in a tweet that contains the keywords in the tweet text. We obtained 815 candidate videos for {\\tt Abortion}\\xspace, 974 candidate videos for {\\tt Gun Control}\\xspace, and 8,571 candidate videos for {\\tt BLM}\\xspace.\n\nFor {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace, we manually annotated all candidate videos. All videos were randomly divided into five buckets (with 163 or 195 videos in each) and each bucket was labeled by an annotator. The five authors of this paper, along with two postgraduate students who had extensive research experience in computational social science, consisted of the annotator team. Annotators were instructed to watch a video for at least five minutes before making a decision of whether or not this video is relevant to the target topic. To measure the inter-rater reliability, we randomly selected a bucket, and had two more annotators labeling the same bucket. We computed the Fleiss' Kappa. {\\tt Abortion}\\xspace had Fleiss' Kappa of 0.830 and {\\tt Gun Control}\\xspace had that of 0.826, suggesting a high agreement among annotators.\n\nFor {\\tt BLM}\\xspace, we used a semi-automatic approach because the number of candidate videos (8,571) is much larger. Based on our observations on {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace, we designed the following protocol:\n\n\\begin{itemize}[leftmargin=*]\n  \\item For videos that contained any topic-relevant keywords in the video titles or descriptions,\n\n  \\begin{itemize}[leftmargin=*]\n    \\item if the videos were also mentioned by tweets with topic-relevant keywords, 89.0\\%\/86.7\\% of the videos were labeled \\textit{relevant} (in {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace respectively, same below). These videos consisted of 16.1\\%\/12.5\\% of all candidate videos. We labeled such videos as relevant, there were 418 in {\\tt BLM}\\xspace.\n    \\item if the videos were not mentioned by tweets with topic-relevant keywords, 77.8\\%\/89.4\\% of the videos were labeled \\textit{irrelevant}. These videos consisted of 6.7\\%\/11.9\\% of all candidate videos. We labeled such videos as irrelevant.\n  \\end{itemize}\n\n  \\item For videos that did not contain any topic-relevant keywords in the titles nor descriptions, but had been mentioned by some topic-relevant tweets,\n\n  \\begin{itemize}[leftmargin=*]\n    \\item if the videos were tagged as ``Music'', or did not have English captions, or did not contain any topic-relevant keywords in the captions, 95.2\\%\/80.6\\% of the videos were labeled \\textit{irrelevant}. We labeled such videos as irrelevant.\n    \\item we used the manual annotation steps for {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace to label the remaining 749 videos, out of which 359 videos were labeled relevant. The Fleiss' Kappa was 0.711.\n  \\end{itemize}\n\\end{itemize}\n\nIn total, we identified 179 {\\tt Abortion}\\xspace videos, 268 {\\tt Gun Control}\\xspace videos, and 777 {\\tt BLM}\\xspace videos. We extracted all tweets and Twitter users that mentioned any of these topic-relevant videos from our 16 months data stream.\n\n\\input{tables\/table-topical-keywords}\n\n\\clearpage\n\n\\section{Estimating and Validating the Leanings of Twitter Users}\n\\label{sec:app-estimate-twitter}\n\nWe estimated the political leanings of early adopters on Twitter in four steps with increasing coverage.\n\n\\header{1) Curating seed hashtags.}\nTwitter users are known to use political hashtags in their user profiles as a marker of identity and a tool of network gatekeeping, such as marking participation in an ongoing activism, facilitating the establishment of social network ties around specific topics, or expressing community identities~\\cite{kulshrestha2017quantifying}. Based on this insight, we started by identifying users who clearly signaled political ideologies in their Twitter profile descriptions. We curated two sets of ideologically descriptive hashtags that were typically used by \\rev{liberal} and \\rev{conservative} users. We described a hashtag as ideologically descriptive if it either supported or criticized political leaders, their statements, and their campaigns (e.g., \\textit{\\#trump}, \\textit{\\#america1st}, \\textit{\\#maga}). We called these hashtags as political seed hashtags. The whole list can be found in \\Cref{table:seed_hashtag}. There are 25 left-leaning hashtags used by 260, 552, 817 users, and 35 right-leaning hashtags used by 1023, 2065, 3798 users in {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace, respectively.\n\n\\input{tables\/table-seed-hashtags}\n\n\\header{2) Expanding seed hashtags.}\nTo obtain new political hashtags, we followed an expansion protocol based on co-occurrence with the political seed hashtags in the user profile descriptions. Firstly, we required that a candidate hashtag must co-occur with the seed hashtags in at least 0.1\\% of all users' profile descriptions. Secondly, the candidate hashtag should be relevant to only one ideology (either left or right). To achieve this, we computed the Shannon entropy based on the co-occurrence of the candidate hashtag with the seed left- and right-leaning hashtags. We selected the hashtags with entropy less than or equal to 0.1. \\Cref{table:expanded_hashtag} presents the expanded political hashtags. The number of users using left- and right-leaning hashtags are 264 and 1250 for {\\tt Abortion}\\xspace, 677 and 2375 for {\\tt Gun Control}\\xspace, 970 and 4412 for {\\tt BLM}\\xspace, respectively. We noticed that the number of \\rev{conservative} users are about four times that of \\rev{liberal} users across the three topics.\n\n\\input{tables\/table-expanded-hashtags}\n\n\\header{3) Identifying seed users.}\nAfter curating and expanding the left- and right-leaning hashtags (jointly called {\\em political hashtags}), we assigned the political ideologies to the Twitter users. For users whose profile descriptions included at least one political hashtag, we counted the numbers of left- and right-leaning hashtags in their profile descriptions. The average numbers of political hashtags per such user were 1.96, 1.97, and 2.04 for {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace and {\\tt BLM}\\xspace, respectively. If the ratio of one leaning was greater than or equal to 0.9, we assigned the corresponding ideology to the user. We called the users {\\em seed users}, since their labels were later used to estimate the ideologies of other users. The numbers of liberal and conservative seed users are given in~\\Cref{table:seed_users}. \n\n\\input{tables\/table-seed-users}\n\n\\header{4) Propagating labels to other users.}\nWe aimed to infer the ideologies of users who had no explicit political hashtags in their profile descriptions. We first created a user-to-user network where the edge weight between two users $(u, v)$ represented their {\\em shared audience}, calculated using the Jaccard similarity of their follower lists:\n\n\\input{equations\/eq-jaccard}\n\nThe shared audience network is shown to be an effective method for splitting users with different political behaviors~\\cite{stewart2017drawing}. When estimating the user ideologies, our assumption is that users with the same ideology share similar audience. The shared audience network was a dense network since we created edges between any two users regardless of whether they had a direct follower-following relation. The initial networks consisted of approximately $28M$, $119M$, and $394M$ edges for {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace, respectively. The corresponding network densities were 18.1\\%, 15.2\\%, and 15.7\\%. To reduce the network complexity, we used a disparity filtering algorithm~\\cite{serrano2009extracting} to extract the network backbone. The disparity filtering algorithm can identify important edges in a network. It has a hyperparameter $\\alpha$ to control the degree of edge importance. We used $\\alpha = 0.05$ as suggested in the initial work~\\cite{serrano2009extracting}. After disparity filtering, the resulting networks contained approximately $1.9M$, $7.8M$, and $25.5M$ edges for {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace, respectively. The densities of these networks ranged between 1\\% and 1.2\\%.\n\nNext, we applied a label propagation algorithm~\\cite{zhou2004learning} on the filtered network to infer the political leanings of the unknown users from those of the seed users. Label propagation is a semi-supervised algorithm that builds on two intuitions: (a) the seed labels should not change too much after propagation; (b) the inferred labels should not be too different with their neighbours. A teleport factor $\\beta$ controls the trade-off between these two intuitions. We used $\\beta = 0.85$ as suggested in the prior work~\\cite{yan2020mimicprop}. The label propagation algorithm returned a liberal score and a conservative score for each user in the network, which we then re-normalized into a score between 0 and 1, where 0 indicated strong liberal and 1 indicated strong conservative. Re-normalization was done by computing the ratio of conservative score to the summation of liberal and conservative scores.\n\nWe stratified one-tenth of seed users in each fold and hided their labels. Next, we compared the inferred political leanings to actual leanings of these users. Note that we employed whole network during the label propagation validation. \\Cref{table:seed_users} shows the evaluation results for the label propagation.\n\n\\clearpage\n\n\\section{Estimating and Validating the Leanings of YouTube Videos}\n\\label{sec:app-estimate-youtube}\n\nOnce we obtained the ideology probabilities of early adopters on Twitter, we computed the leaning score for each YouTube video by averaging its promoters' ideology scores. To assign discrete leaning labels (i.e. \\textit{Left}, \\textit{Right} and \\textit{Center}) to the videos, we leveraged the Recfluence dataset~\\cite{ledwich2020algorithmic} as an external source to annotate YouTube videos' leanings. This dataset includes 816 YouTube channels where each channel has 10k+ subscribers and more than 30\\% of its content is relevant to US politics, or cultural news, or cultural commentary. It consists of channels of YouTubers in addition to those of traditional news outlets. In this dataset, each channel is assigned to \\textit{Left (L)}, \\textit{Center (C)} or \\textit{Right (R)} leaning. The annotations were conducted based on two well known media bias resources\\footnote{Media Bias\/Fact Check: \\url{https:\/\/mediabiasfactcheck.com\/}; Ad Fontes Media: \\url{https:\/\/www.adfontesmedia.com\/}} as well as manual labeling.\n\nThe videos whose channels' leanings are available in the Recfluence dataset are assigned to the same leaning. We were able to label 109 videos (L: 38, R: 66, C:5) from 38 channels for {\\tt Abortion}\\xspace, 153 videos (L: 52, R: 88, C: 13) from 62 channels for {\\tt Gun Control}\\xspace, and 521 videos (L: 194, R: 285, C:42) from 120 channels for {\\tt BLM}\\xspace. To label the videos whose channels are not available in the Recfluence dataset, we identified two thresholds for each topic -- $thr_{(L, C)}$ and $thr_{(C, R)}$ where $thr_{(L, C)} < thr_{(C, R)}$. For a given video $\\nu$, we assigned a discrete leaning label, $leaning\\_label(\\nu)$, based on its leaning score, $leaning\\_score(\\nu)$, as follows:\n\n\\[\nleaning\\_label(\\nu) = \n\\begin{cases}\n  L & \\text{if $leaning\\_score(\\nu) < thr_{(L, C)}$} \\\\\n  R & \\text{if $leaning\\_score(\\nu) > thr_{(C, R)}$} \\\\\n  C & \\text{otherwise.}\n\\end{cases}\n\\]\n\nWe used videos with labels from Recfluence to find optimum thresholds $thr_{(L, C)}$ and $thr_{(C, R)}$. We obtained a leaning score distribution for each group (i.e., L, R and C). We identified outliers for each group using inter-quartile rule and filtered them out. The leaning score distributions by groups for each topic are given in \\Cref{fig:recfluence_thr_plots}. Next, we found the thresholds $thr_{(L, C)}$ and $thr_{(C, R)}$ using posterior probabilities as follows:\n\\begin{eqnarray}\n     P(L|leaning\\_score(\\nu)) = P(C|leaning\\_score(\\nu)),&\\text{where}~ leaning\\_score(\\nu) = thr_{(L, C)} \\\\\n    P(C|leaning\\_score(\\nu)) = P(R|leaning\\_score(\\nu)),&\\text{where}~ leaning\\_score(\\nu) = thr_{(C, R)}   \n\\end{eqnarray}\n\n\\input{images\/app_recfluence_thr_dist}\n\nThe resulting threshold pairs ($thr_{(L, C)}$, $thr_{(C, R)}$) are $(0.538, 0.749)$ for {\\tt Abortion}\\xspace, $(0.478, 0.716)$ for {\\tt Gun Control}\\xspace and $(0.524, 0.695)$ for {\\tt BLM}\\xspace, respectively. Based on these thresholds, we inferred the leaning labels of 70 videos (L: 20, R: 45, C: 5) from 58 channels for {\\tt Abortion}\\xspace, 115 videos (L: 29, R: 66, C: 20) from 98 channels for {\\tt Gun Control}\\xspace, 256 videos (L: 103, R: 111, C: 42) from 197 channels for {\\tt BLM}\\xspace. \n\nTo validate the video leaning estimation, we conducted a manual leaning annotation for videos in {\\tt Gun Control}\\xspace. We used stratified sampling where we sampled 5 videos from each 10 percentile bin of their leaning scores. The leaning of these 50 videos were annotated independently by three authors considering three categories (\\textit{left-leaning}, \\textit{right-leaning}, \\textit{unknown}). The Fleiss' Kappa is 0.691, suggesting a moderate agreement among annotators. We used the majority vote to assign a video label. Table \\ref{table:gun_stance_manual_annotation} summarizes the annotation results. \n\nWe observed that the only decile bin with both manually annotated left- and right-leaning video is the 30th-40th percentile of leaning scores $(0.465 - 0.665)$. Suggesting that the annotator had high agreement with the estimated leaning scores, and that the threshold should be in this interval -- which is indeed the case for {\\tt Gun Control}\\xspace with $thr_{(L, C)}$ lies within its left boundary, and $thr_{(C, R)}$ only 0.05 away from its right boundary. In addition, most of the videos which were assigned \\textit{unknown} by the raters are the footage of shooting of Philando Castile and mass shooting events including Las Vegas shooting and TX massacre. These videos do not include any stance on their contents. However, we observed that the leaning scores of these videos align well with the leaning of channels. One example is ``Comments on TX church shooting'' where the video leaning score belongs to the second bin $(80th - 90th)$ percentile. The channel owner is a far-right YouTuber. The video leaning score matches with the channel owner's leaning, our approach is thus successful for solving such ambiguities.\n\n\\Cref{table:gun_stance_manual_annotation} shows manual leaning annotation results of 50 Gun Control videos. Compared to the leaning score range in the second column (which is estimated), the first three rows of 14 videos (which are labeled as left-leaning) are consistently labeled (the result of manual labeling and estimated leanings agree) as well as the last five rows of 19 videos (which are labeled as right-leaning). The computed thresholds for Gun Control videos, as presented in~\\Cref{fig:recfluence_thr_plots}, are $0.478$ and $0.716$.\n\n\\input{tables\/table-gun-stance-annotation}\n\n\\clearpage\n\n\\section{Computing Relative Engagement of a Video}\n\\label{sec:app_relengagement}\n\nFrom the collected videos we extracted topic ({\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace and {\\tt BLM}\\xspace) relevant videos for our topical analysis, but all collected videos were used to compute relative engagement scores.\nWe filter out videos that receive less than 100 views within their first 30 days after upload, which is the same filter used by~\\citet{brodersen2012youtube}.\nWe summarize how relative engagement score is computed. For a given video, first we compute two aggregate metrics:\n\\begin{itemize}\n    \\item average watch time ($\\bar{\\omega}_t$): the total watch time $x_{w}[1 : t]$ divided by the total view count $x_v[1 : t]$ up to day $t$\n    \\[\\bar{\\omega}_t = \\frac{\\Sigma^t_{i=1}x_{w}[i]}{\\Sigma^t_{i=1}x_{v}[i]}\\]\n    \\item average watch percentage ($\\bar{\\mu}_t$): the average watch time $\\bar{\\omega}_t$ normalized by video duration D\n    \\[\\bar{\\mu}_t = \\frac{\\bar{\\omega}_t}{D}\\]\n\\end{itemize}\n\nThen {\\it Engagement map} of tweeted videos is constructed as follows. Two maps are created; the first where x-axis\nshows video duration D, and the y-axis shows average\nwatch time over the first 120 days ($\\bar{\\omega}_{120}$) (\\Cref{fig:appendix_relative_engagement}(a)) and the second with the same axis with average watch percentage over the first 120 days ($\\bar{\\mu}_{120}$) (\\Cref{fig:appendix_relative_engagement}(b)) as y-axis. All videos in the tweeted videos dataset are projected onto both maps. The x-axis is split into 1,000 equally wide bins in log scale. These 2 maps are logically identical but \\Cref{fig:appendix_relative_engagement}(b) is easier to read as y-axis is bounded between [0,1]. This second map is denoted as {\\it Engagement map}.\n\nBased on this {\\it Engagement map},  relative engagement $\\bar{\\eta}_t\\in [0,1]$ defined\nas the rank percentile of video in its duration bin.  This is an average engagement measure in the first t days. \\Cref{fig:appendix_relative_engagement}(b) illustrates the relation between video duration D, watch percentage $\\bar{\\mu}_{120}$ and relative engagement $\\bar{\\eta}_{120}$ for three example videos. $V_1$ is a left-leaning video in {\\tt Abortion}\\xspace topic with duration of 136 seconds. $V_2$ is a right-leaning video in {\\tt Abortion}\\xspace topic with duration of 141 seconds. $V_1$ and $V_2$ have similar lengths but different watch percentages, $\\bar{\\mu}_{120}(V_1) = 0.83$ and $\\bar{\\mu}_{120}(V_2) = 0.59$. The difference becomes more apparent with relative engagement scores, $\\bar{\\eta}_{120}(V_1) = 0.94$ and $\\bar{\\eta}_{120}(V_2) = 0.27$ as shown in \\Cref{fig:appendix_relative_engagement}(b).\n$V_3$ is also a right-leaning video in {\\tt Abortion}\\xspace which is significantly long than the others. Although $V_3$ is not as much watched as the previous two, $\\bar{\\mu}_{120}(V_3) = 0.40$, relative engagement of $V_3$ is much higher than $V_2$, $\\bar{\\eta}_{120}(V_3) = 0.86$ as longer videos tend to have lower average watch percentages. \n\n\\input{images\/app_rel_engagement}\n\n\\clearpage\n\n\\section{Computing Viral Potential of a Video via the Hawkes Intensity Process}\n\\label{sec:app_hip}\n\n\\header{Viral potential}\nestimates the expected number of views a video will obtain if mentioned by an  {\\em average} tweet ~\\cite{rizoiu2017online}.\nMore specifically, it is the area under the impulse response function of an integral equation known as Hawkes Intensity Process (HIP)~\\cite{rizoiu2017expecting}, which are learned for each video by using the first 120 days of tweeting and viewing history. The HIP model is designed to describe the phenomena of tweets driving or attracting video views including factors such as network effect (such as views beget views), system memory (such as interests in news videos wane within a week, but interests on music videos sustain for several years), and video quality (including factors such as sensitivity to tweet mentions and inherent attractiveness). \nWe estimate the number of views per tweet via the HIP model rather than simply dividing the number of views by the number of tweets, because the model takes into account future views that are expected to unfold after the 120-day cut-off. \nThe same reason motivates the name viral {\\em potential} rather than viral score.\nOne limitation of HIP and viral potential is the notion of an  ``average'' tweet being implemented by marginalizing over power-law distributed number of followers. This means\nnetwork size variations across different topics are not taken into account.\nA self-contained summary of HIP and viral potential computation is included below. The viral potential is a positive number, since views cannot be negative (i.e., no video loses views). However, it can take values less than one, corresponding to the effect of tweets being dampened (one view per several tweets) rather than amplified, as often seen in videos linked by spam tweets. \n\nHawkes Intensity Process (HIP)~\\cite{rizoiu2017expecting} extends the well-known Hawkes (self-exciting) process~\\cite{hawkes1971spectra} to describe the volume of activities within a fixed time interval (e.g. daily).\nThis is done by taking expectations over the stochastic event history.\nSpecifically, this model describes a self-exciting phenomenon that is commonly observed in online social network~\\cite{rizoiu2017online}.\nIt models the target quantity $x[t]$ as a self-consistent equation into three parts: the unobserved external influence, the effects of external promotions, and the influence from historical events.\nFormally, it can be written as\n\n\\input{equations\/eq-hip}\n\nThe first two terms represent unobserved external influences.\n$\\gamma$ and $\\eta$ model the strengths of an initial impulse and a constant background rate, respectively.\nIn the middle component, $\\alpha$ is the sensitivity to external promotion, $s[t]$ is the volume of promotion, and $\\alpha s(t)$ is the instantaneous response to promotion.\nIn the last component, $\\theta$ is the exponent of a power-law memory kernel $(\\tau+c)^{-(1+\\theta)}$.\n$c$ is a nuisance parameter for keeping the kernel bounded, and $C$ accounts for the latent content quality.\nOverall, this last component models the impact over its own event history $x[\\tau]$ for $\\tau{=}1:t{-}1$.\n\nIn our case, $x[t]$ is the time series of daily views $x_v[t]$ on YouTube.\n$s[t]$ is the daily number of tweets on Twitter.\nThe parameter set $\\{\\gamma, \\eta, \\alpha, C, c, \\theta\\}$ is estimated from the first 90-day interval of each video using the constrained L-BFGS algorithm in \\texttt{SciPy} Python package.\\footnote{\\url{https:\/\/github.com\/andrei-rizoiu\/hip-popularity}}\n\n\\clearpage\n\n\\section{View and Tweets over time: additional plots}\n\\label{sec:app-overtime_additional}\n\n\\Cref{fig:app_gun_ccdf} and \\Cref{fig:app_blm_ccdf} visualize the attention accumulation for {\\tt Gun Control}\\xspace and {\\tt BLM}\\xspace topic.\n\n\\begin{figure*}[!htb]\n  \\centering\n \\includegraphics[width=0.66\\linewidth]{image-files\/app_gun_ccdf.pdf}\n  \\caption{Comparing view and tweet accumulations of left- and right-leaning videos in {\\tt Gun Control}\\xspace. (a) and (b) show CCDF of views and tweets accumulated for Day 1 and Day 30,  respectively. As seen in (c), the differences between left- and right-leaning videos are larger than the differences of {\\tt Abortion}\\xspace videos, but the differences between accumulation of views and tweets are smaller than {\\tt Abortion}\\xspace videos.}\n  \\label{fig:app_gun_ccdf}\n\\end{figure*}\n\n\\begin{figure*}[!htb]\n  \\centering\n \\includegraphics[width=0.66\\linewidth]{image-files\/app_blm_ccdf.pdf}\n  \\caption{Comparing view and tweet accumulations of left- and right-leaning videos in {\\tt BLM}\\xspace. (a) and (b) show CCDF of views and tweets accumulated for Day 1 and Day 30,  respectively. (c) shows that the differences between left- and right-leaning videos are the largest among all topics.}\n  \\label{fig:app_blm_ccdf}\n\\end{figure*}\n\n\\clearpage\n\n\\section{Tweet cascades: additional plots}\n\\label{sec:app-cascade_additional}\n\n\\Cref{fig:app_cascade_additional} visualizes attention distribution of the isolated, small, and large cascades for {\\tt Gun Control}\\xspace and {\\tt BLM}\\xspace topic.\n\n\\begin{figure}[!htb]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\linewidth}\n    \\includegraphics[width=\\linewidth]{image-files\/app_gun_cascade_l.pdf}\n    \\caption{Tweet cascades in {\\tt Gun Control}\\xspace left-leaning videos.}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\linewidth}\n      \\includegraphics[width=\\linewidth]{image-files\/app_gun_cascade_r.pdf}\n      \\caption{Tweet cascades in {\\tt Gun Control}\\xspace right-leaning videos.}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\linewidth}\n    \\includegraphics[width=\\linewidth]{image-files\/app_blm_cascade_l.pdf}\n    \\caption{Tweet cascades in {\\tt BLM}\\xspace left-leaning videos.}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\linewidth}\n      \\includegraphics[width=\\linewidth]{image-files\/app_blm_cascade_r.pdf}\n      \\caption{Tweet cascades in {\\tt BLM}\\xspace right-leaning videos.}\n  \\end{subfigure}\n  \\caption{Cascade start times measured at view accumulation process. \\Cref{table:measures} reports that retweet cascades for left-leaning videos start significantly later in view accumulation process for all topics.}\n  \\label{fig:app_cascade_additional}\n\\end{figure}\n\n\\clearpage\n\n\\section{Additional metrics on early adopter networks}\n\\label{sec:app_centralized_left_right}\n\nWe measured additional metrics on early-adopter networks, the comparative findings are summarized in \\Cref{table:measures}.\n\n\\header{\\textit{Network density}} is the fraction of existing edges among all possible pair-wise edges. It quantifies the connectivity of the follower network. A higher value indicates the early adopters are more cohesively followed among themselves.\n\n\\header{\\textit{Maximum indegree}} is the maximum indegree value in the network. It provides information about the most influential user in the network. \n\n\\header{\\textit{Global efficiency}} is computed as the number of closed triplets over the total number of triplets. It measures the average efficiency over all pairs of distinct users. Higher global efficiency reflects higher capacity to diffuse information over the network.\n\n\\header{\\textit{Gini coefficient of betweenness centrality}} measures the dispersion in fraction of all shortest paths in the network that pass through a given user. Higher coefficient implies that a few early adopters are on the shortest path for most node-pairs, but the rest of the early adopters are not.\n\n\n\\section{Intersecting relative engagement with view and tweet count}\n\\label{sec:app_intersect}\n\n\\input{images\/app_intersecting}\n\nWe link the measure of relative engagement of YouTube videos to the number of tweets, the number of followers, and the number of views. The number of tweets and views are accumulated for the first 120 days after upload.\nRather than looking at the overall correlation, which is dominated by large variations, we choose to group videos according to relative engagement. This allows us to interpret the change in Twitter user base, tweeting behavior, and YouTube attention across videos of different engagement levels.\n\nWe compute relative engagement score for each video and rank the videos in descending order of relative engagement score. Then, the videos are divided into 5 groups, each including the top 20\\% videos, top 20\\%\\textasciitilde{}40\\% videos, top 40\\%\\textasciitilde{}60\\% videos, top 60\\%\\textasciitilde{}80\\% videos and top 80\\%\\textasciitilde{}100\\% videos. \nFor each group, left and right-leaning videos are shown as boxplots of \\textit{Tweet120}, \\textit{Follower count}, and \\textit{View120} (\\Cref{fig:app_intersecting}). \n\n\nAs seen earlier, left-leaning videos have less number of tweets on average which can be seen with {\\it Tweet120}. Especially, in the case of {\\tt Abortion}\\xspace, the middle buckets (20\\%, 40\\% and 60\\%) show that left-leaning videos have significantly less tweets but have more views on average. The same pattern holds for {\\tt Gun Control}\\xspace. This is consistent with the previous finding that left-leaning videos are more effectively promoted (higher viral potential) with fewer number of tweets.\n\nWe can observe that higher numbers of followers does not imply higher numbers of tweets on average as witnessed by most of middle buckets. However, we note that the top three outliers in left-leaning videos in 40\\% bucket of {\\it Follower count} boxplot for {\\tt Gun Control}\\xspace corresponds to the top three outliers of left-leaning videos in the same bucket of {\\it Tweet120}.\n\nIn terms of {\\it View120}, the average trends support findings in \\Cref{ssec:aggregate_obs} in that left-leaning videos in {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace have higher views whereas left-leaning videos in {\\tt BLM}\\xspace have lower views. Although view count is considered as an important measure of video popularity in general, no noticeable correlation is shown between relative engagement and {\\it View120}. This is expected as engagement metric and popularity metric capture very different perspective of online attention~\\cite{wu2019estimating}.\n\n\\section{Curating Tweeted Video Datasets}\n\\label{sec:data}\n\nWe constructed three new cross-platform datasets by tracking videos on YouTube and posts on Twitter over three controversial topics: {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace. Those topics have been studied extensively by social and political scientists~\\cite{zhang2016gender,de2016social,stewart2017drawing,garimella2018political}. In this section, we first describe the data collection strategy. We then introduce our methods for estimating the political leanings of Twitter users and YouTube videos. \\Cref{table:datasets} summarizes the overall statistics of the three topical datasets.\n\n\\input{tables\/table-datasets.tex}\n\n\\subsection{Finding YouTube Videos and Twitter Posts of Controversial Topics}\n\\label{ssec:data_find}\n\nWe are interested in topical YouTube videos and the discussions about them on Twitter. Following the approach used in~\\cite{wu2018beyond}, we collected public tweets that mentioned any YouTube URLs via the Twitter filtered streaming API. Our raw Twitter stream spanned 16 months (2017-01-01 to 2018-04-30), and contained more than 1.8 billion tweets. To subsample videos that attracted a reasonable amount of attention before the end of the observation period, we required that the videos must be published in 2017, receive at least 100 tweets and at least 100 views within the first 120 days after upload. We make this filtering choice because (a) analyzing videos with little attention (${<}1$ view per day) is bound to generate noises when comparing different groups; (b) characterizing the timing and structure of a video's tweeting cascades requires a non-trivial number of tweets. This yielded 328,557 videos, which were mentioned in 242M tweets by 29.9M users. For each video, we collected its metadata, daily time series of view count and watch time, and all tweets mentioning it.\n\nTo identify topic-relevant videos and tweets, we first curated three separate keyword lists for three controversial topics -- {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace. We consider a video is potentially relevant if (a) it contains at least one keyword in the video title or description; or (b) it is mentioned in a tweet that contains at least one keyword in the tweet text. Next, we used a mix of manual and semi-automated approaches to annotate the potentially relevant videos. Section A of \\cite{appendix} details the topical keywords, their curation, and our video annotation protocol. In total, we obtained 179 {\\tt Abortion}\\xspace, 268 {\\tt Gun Control}\\xspace, and 777 {\\tt BLM}\\xspace videos, which were mentioned in 970K+ tweets.\n\n\\input{images\/data_example}\n\n\\Cref{fig:data_overtime} shows the daily view count series and tweet cascades for an example video. A notable contrast between the two sources is that YouTube attention data is only available in daily aggregates, i.e., without individual user logs, while tweets have precise timestamp, user information, and relations between tweets and users. \\Cref{sec:metric} is dedicated to designing measures for such cross-platform multi-relational temporal data. We bootstrapped the political information about Twitter users and YouTube videos. In particular, we gathered more information about videos' early adopters, defined as the first 20\\% users who tweeted about each video. We collected the follower lists for all early adopters. Network measures of early adopters are found to indicate future popularity~\\cite{romero2013interplay}. The threshold (first 20\\%) is chosen to balance the need for data and the burden of collecting network information within practical API limits. After filtering out banned and protected users, we extracted 132K early adopter tweets posted by tens of thousands of users across the three topics (see \\Cref{table:datasets}).\n\n\\subsection{Estimating Political Leanings of Twitter Users and YouTube Videos}\n\\label{ssec:data_estimate}\n\nWe classified the political leanings of early adopters on Twitter into liberal, neutral, and conservative. Meanwhile, we classified the video leanings into left, center, and right. Twitter users and YouTube videos are related in that left-leaning contents (e.g., {\\tt Gun Control}\\xspace videos) are generally shared by liberal users while right-leaning contents (e.g., {\\tt Gun Rights} videos) are shared by conservative users.\n\nWe estimated Twitter users' political leanings by first identifying a group of seed users who included political hashtags in their profile descriptions, and then by using a label propagation algorithm~\\cite{zhou2004learning} to propagate the labels of seed users to other users based on the shared follower network. This is a common approach for classifying user leaning on Twitter~\\cite{stewart2017drawing} and the {\\it follow} relation is found to be the most important in predicting user ideology~\\cite{xiao2020timme}. We performed 10-fold cross-validation to evaluate the classification performance. We observed very high scores in precision, recall, and F-score across all three topics ($>95\\%$ in all metrics). Section B of~\\cite{appendix} describes our classification and evaluation methods for Twitter users in more detail.\n\nWe estimated YouTube videos' political leanings by first averaging the leaning scores of videos' early adopters on Twitter. We then used an external YouTube media bias dataset~\\cite{ledwich2020algorithmic} to label the video leanings and identified optimal classification thresholds. We were able to find 58\/10\/111 left, center, and right-leaning videos for {\\tt Abortion}\\xspace (analogously, 81\/33\/154 for {\\tt Gun Control}\\xspace and 297\/84\/396 for {\\tt BLM}\\xspace). To validate our estimation, we performed one round of manual annotation for videos in {\\tt Gun Control}\\xspace. We used stratified sampling to sample 50 videos based on the video leaning scores. These videos were annotated independently by three authors. The Fleiss' Kappa was 0.69, suggesting a moderate inter-rater agreement. Section C of~\\cite{appendix} details our classification and evaluation methods for YouTube videos.\n\n\\section{Observations on Cross-Platform Attention}\n\\label{sec:measure}\n\nWe report the results on all metrics described in \\Cref{sec:metric}, in mirroring subsections to aid navigation. Many results in this section are presented as violin plots. The outlines are kernel density estimates for the left-leaning (blue) and right-leaning (red) videos, respectively. The center dashed line is the median, whereas the two outer lines denote the inter-quartile range. To compare the distributions of each metric for the left- and right-leaning videos, we adopt the one-sided Mann\u2013Whitney U test. We summarize our results in~\\Cref{table:measures} at the end of this paper.\n\n\\subsection{Aggregate Attention on YouTube and Twitter}\n\\label{ssec:aggregate_obs}\n\n\\header{Total view count.}\n\\Cref{fig:obs_aggregate}(a) shows the distribution of video views at day 120 after upload. Using the view count at the same day removes the effects of video age, so that the videos published for longer time are not taking an unfair advantage. In {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace, the median, as well as 25$^{th}$ and 75$^{th}$ percentile of views of left-leaning videos are higher than that of right-leaning videos. The median views for left-leaning videos are 107,346 for {\\tt Abortion}\\xspace and 153,482 for {\\tt Gun Control}\\xspace, versus 62,780 and 103,373 for right-leaning ones. The differences in view distribution are statistically significant ($p < 0.01$, \\Cref{table:measures} row 1). For {\\tt BLM}\\xspace, right-leaning videos have higher median and 75$^{th}$ of views, but the effect is not significant.\n\n\\input{images\/measure_aggregate}\n\n\\header{Relative engagement.}\nFrom \\Cref{fig:obs_aggregate}(b), we can see that videos in all three topics are highly engaging, with mean relative engagement at 0.834 for {\\tt Abortion}\\xspace, 0.824 for {\\tt Gun Control}\\xspace, and 0.831 for {\\tt BLM}\\xspace. This is because our data processing procedure requires videos to have at least 100 tweets and 100 views, which tends to select videos with significant amount of interests. Left-leaning videos are significantly more engaging than their right-leaning counterparts across all three topics ($p < 0.05$, \\Cref{table:measures} row 2).\n\n\\header{Fraction of likes.}\n\\Cref{fig:obs_aggregate}(c) presents the proportion of likes in videos' reactions. Left-leaning videos across all topics have significantly smaller fraction of likes than right-leaning videos ($p < 0.001$, \\Cref{table:measures} row 3). This may be explained by the observation that there are far more cross-partisan talks on left-leaning videos~\\cite{wu2021cross}.\n\n\\header{Viral potential.}\n\\Cref{fig:obs_aggregate}(d) shows the distributions of viral potential. We find that the left-leaning videos have significantly higher viral scores than the right-leaning videos across all three topics ($p < 0.05$, \\Cref{table:measures} row 4), meaning that given the same amount of tweets exposing the video on Twitter, an average left-leaning video can effectively attracts more views than an average right-leaning video. The difference is most notable in {\\tt Abortion}\\xspace: a typical left-leaning video receives 224 views from an average tweet, whereas a typical right-leaning video receives only 63 views.\n\n\\header{Tweet counts.}\n\\Cref{fig:obs_aggregate}(e) and (f) show the distributions of total tweets and retweets. Contrasting to the observation that left-leaning videos are more viewed, here we find that right-leaning videos are significantly more tweeted, especially with more retweets and more replies ($p < 0.001$, \\Cref{table:measures} row 5-7) in {\\tt Abortion}\\xspace and {\\tt BLM}\\xspace. On the other hand, we do not observe a significant difference in original tweets and quotes, except for {\\tt BLM}\\xspace where right-leaning videos have prevailing volume across all tweet types.\n\nTo examine the robustness of presented results in this section, we bootstrapped videos for each topic and for each ideological group. Specifically, for each group, we created $1,000$ bootstrapped sets of videos that are of the same size as the original group (shown in~\\Cref{table:datasets}). Next, we computed the mean of proposed metrics (shown in~\\Cref{fig:obs_aggregate}) for each bootstrapped set. Lastly, we used the {\\it independent t-test} to check the statistical significance between left- and right-leaning groups. The results of the {\\it t-tests} support all reported relations in~\\Cref{table:measures} row 1-6 with $p < 0.001$.\n\n\\subsection{Views and Tweets over Time}\n\\label{ssec:temporal_obs}\nWe measure how quickly left- and right-leaning videos attract views and tweets. We find that left-leaning videos are reacted on YouTube and Twitter quicker across all topics.\n\n\\header{Viewing half-life} and \\header{Tweeting half-life.}\nWe notice that there are significant differences in the attention consumption patterns: right-leaning videos have more prolonged attention spans on YouTube across all topics ($p < 0.01$, \\Cref{table:measures} row 10). Right-leaning videos also have longer attention spans on Twitter for {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace ($p < 0.05$, \\Cref{table:measures} row 11). For example, \\Cref{fig:obs_temporal}(a) shows that right-leaning videos for {\\tt Abortion}\\xspace have the longest attention span -- taking 9 days for 75\\% videos to achieve viewing  half-life, while left-leaning videos only take 3 days. Comparing \\Cref{fig:obs_temporal}(a) to \\Cref{fig:obs_temporal}(b), we find that attention spans on Twitter are shorter than that on YouTube. In {\\tt Abortion}\\xspace, left-leaning videos take 2 days for 75\\% of videos to reach tweeting half-life (vs. 3 days for views) and right-leaning videos take 5 days for 75\\% of videos to reach tweeting half-life (vs. 9 days for views).\n\n\\input{images\/measure_temporal}\n\n\\header{Tweeting lifetime} and \\header{Tweeting inter-arrival time.} \n\\Cref{fig:obs_temporal} (c) and (d) shows the distributions of tweeting lifetime and inter-arrival time. The results are mixed across topics. For {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace, the MWU test results show that left-leaning videos have significantly less scores in both metrics than right-leaning videos ($p < 0.05$, \\Cref{table:measures} row 12-13). But for {\\tt BLM}\\xspace, both metrics show similar distributions between left- and right-leaning videos.\n\nThese results suggest that left-leaning videos have shorter circulation duration and are mentioned more quickly (except {\\tt BLM}\\xspace). The most notable difference is in {\\tt Abortion}\\xspace where the median of tweeting lifetime and inter-arrival time for right-leaning videos are more than three times of those for left-leaning videos. For instance, the median tweeting lifetime is 1811.6 minutes for left-leaning videos, while the median is 8453.8 minutes for right-leaning videos.\n\n\\header{Accumulation of views and tweets.}\nWe examine how much views and tweets are accumulated each day. \\Cref{fig:obs_accumulation}(a) and (b) show the Complementary Cumulative Distribution Function (CCDF) of views and tweets percentages accumulated for the first day (video published date) and $30^{th}$ day for {\\tt Abortion}\\xspace videos. We observe that left-leaning videos tend to achieve views\/tweets faster than right-leaning videos, which is consistent with \\Cref{fig:obs_temporal}. For example, after day 1, 56.9\\% left-leaning videos have achieved viewing half-life, but only 26.1\\% right-leaning videos achieved the same. For tweet accumulation, after 1 day, the gap of tweeting half-life between left- and right- leaning videos is 11.4\\% (46.6\\% and 35.1\\%, respectively). By day 30, only 3 left-leaning videos have not accumulated 80\\% of views.\n\n\\Cref{fig:obs_accumulation}(c) compares the normalized MWU statistic values of left- and right-leaning videos in views $\\bar U_v$ and tweets $\\bar U_t$ on each of the 120 days since upload. It shows that the difference between left- and right-leaning videos is larger in the beginning and decreases towards 0.5 over time. At the $120^{th}$ day (as the observation period ends), both will be 0.5 by definition, we thus truncate the plot at day 30. We also observe that the differences in views is larger than that of tweets across left- and right- leaning videos initially, whereas the discrepancy between views and tweets narrows as videos get older. This is because more videos have already fully achieved all the views and tweets. \n\n\\input{images\/measure_accumulation}\n\n\\subsection{Videos' Tweet Cascades}\n\\label{ssec:cascade_obs}\n\n\\header{Cascade size.} \n\\citet{goel2012structure} found that one-node-cascades (isolated cascades) account for $96\\%$ of all cascades in their {\\it Twitter Videos} dataset. \\Cref{fig:obs_cascades}(a) shows that the proportions of isolated cascades in our datasets are lower, measured at $91.5\\%$, $91.2\\%$, and $91.9\\%$ respectively for {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace. Notwithstanding a confounder that the dynamics on Twitter has changed significantly since \\cite{goel2012structure}, this may still suggest that tweets on controversial topics are less isolated than tweets of videos about any topics. \\Cref{fig:obs_cascades}(b) shows the volume of tweets involved in each cascade group. It is interesting to find that most tweets belong to either isolated cascades or large cascades. Aggregated over left and right-leaning videos in {\\tt Abortion}\\xspace, {\\tt Gun Control}\\xspace, and {\\tt BLM}\\xspace, 47\\%, 43.4\\%, and 47.3\\% of tweets are isolated, whereas 44.7\\%, 48.4\\%, and 44.3\\% are in large cascades of size 5 and above, dominated by a handful of cascades over 1,000 tweets.\n\n\\input{images\/measure_cascades}\n\n\\header{Cascade start time.} \nWe compare when tweet cascades start in the process of view accumulation, grouped by different cascade sizes. \\Cref{fig:obs_cascades}(c) and (d) show the distribution of cascade start times of left- and right-leaning videos in {\\tt Abortion}\\xspace relative to percentages of views. For right-leaning videos, there is a peak of isolated cascades started at the end of the videos' viewing lifetime. We also observe that for left-leaning videos, the peaks for isolated, small and large cascades are concentrated near $70\\%$ of view accumulation while peaks for right-leaning videos are more distributed over different stages of view accumulation. Moreover, in all topics, more right-leaning tweet cascades start before viewing half-life regardless of cascades size. For example, 41\\% of right-leaning isolated cascades started before viewing half-life while 25\\% of left-leaning isolated cascades started before viewing half-life in {\\tt Abortion}\\xspace. This is consistent with the observation that views of right-leaning videos unfold much slower (See \\Cref{fig:obs_temporal}(a) and \\Cref{fig:obs_accumulation}(a)), allowing tweet cascades to start at earlier stages of view accumulation process. The difference in cascade start time is significant between left- and right-leaning videos in all topics for isolated and small cascades ($p < 0.001$, \\Cref{table:measures} row 14-15) and is significant for {\\tt Abortion}\\xspace and {\\tt Gun Control}\\xspace for large cascades ($p < 0.001$, \\Cref{table:measures} row 16).\n\n\\subsection{Networks among Early Adopters on Twitter}\n\\label{ssec:network_obs}\n\n\\input{images\/measure_network}\n\n\\Cref{fig:obs_network}(c) and (d) show the distributions of the Gini coefficient of indegree centrality and the Gini coefficient of closeness centrality. Gini coef. of indegree centrality of left-leaning videos' networks is significantly larger in {\\tt Gun Control}\\xspace and {\\tt BLM}\\xspace ($p < 0.001$, \\Cref{table:measures} row 17). For Gini coefficient of closeness centrality, the MWU test results indicate that left-leaning videos' networks have significantly greater Gini index than those of right-leaning videos' networks across all topics ($p < 0.05$, \\Cref{table:measures} row 18). \n \nThis suggests that the networks of early adopters for left-leaning videos have more users serving as hubs, i.e., who are followed by more early adopters and have shorter path to other early adopters. This also suggests that in the networks of early adopters for right-leaning videos, users are more equally facilitating dissemination of political information which is consistent with the findings shown in~\\cite{conover2012partisan}. As an example of this, we present the follower networks of early adopters of one left-leaning video and one right-leaning video in {\\tt Abortion}\\xspace in \\Cref{fig:obs_network}(a,b). To have a fair comparison we sample two videos having similar network size (57 and 59 for left and right-leaning videos, respectively). The left-leaning video has Gini coef. of indegree centrality: 0.918, Gini coef. of closeness centrality: 0.90. The right-leaning video has Gini coef. of indegree centrality: 0.748, Gini coef. of closeness centrality: 0.536. It can be observed that the sharing of this left-leaning video relies more on central users who are followed by more early adopters and have shorter path to others. On the other hand, in the follower network of the early adopters of this right-leaning video, indegree and closeness centrality distributions are more equal.\n\nApart from the reported metrics, we have also performed preliminary examination on the correlation and trends between two and more metrics. An example on linking relative engagement to the view and tweet counts is presented in Section I of~\\cite{appendix}. We have not seen consistent and salient patterns that are not already captured by individual measures. \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\textbf{Introduction}}\nIn this work, focusing on the decumulation phase, we fix the annuitization time and investigate the optimal investment-consumption strategies prior to annuitization in a Brownian market model with time dependent mortality rate. We follow the framework developed in \\cite{kn:Ger3} in which a target for the consumptions during the decumulation phase and a target for the terminal accumulated wealth is considered. Moreover, motivated from \\cite{kn:G}, we consider a minimum guarantee for the final annuity. \n\nAssuming a fixed rate of consumption throughout the whole period of time before annuitization, which is usually a long period, is far from the optimality. On the other hand, it is quite reasonable to consider a minimum consumption rate for a retiree to cover the essential expenses. Therefore, we consider the rate of consumption as a control variable which varies between the two limits, $C_1$ and $C_2$, in which $C_1 > C_2$.  We will see from the simulation results that considering a variable consumption rate yields much higher final annuities. To compare the optimal portfolios obtained from different scenarios of the admissible consumptions, we take into account the present market value of the future cash flows before and after the anniuty purchase. \n\nGerrard et al. \\cite{kn:Ger1} study the portfolio optimization problem post-retirement when the loss function is defined totally by the wealth process and the annuitization time and  consumption rate are fixed.  In \\cite{kn:Ger2}, they violate the fixed consumption rate assumption. In a similar framework, Di Giacinto et al. \\cite{kn:G} explore the optimal investment strategy when a minimum guarantee for the final wealth is assumed and the consumption rate is fixed. When the running cost is neglected, they obtain the closed form of solution. \nIn this paper, in a same framework, we develop a numerical algorithm based on the policy iteration method to find approximations of the optimal investment-consumption strategies when the consumption rate is variable and a running cost for the loss function based on the consumptions is considered.  The  policy iteration method is a well studied method in finding approximations of  solutions of optimal control problems, see \\cite{kn:FL}, \\cite{kn:SR}, in which the value function and the optimal policies are deriving iteratively to converge to the correct solution of the corresponding HJB equation. \n   \nThe portfolio optimization problem post-retirement has been studied by many scholars by considering different utility or loss functions, different control variables and different wealth dynamics. Furthermore,  different constraints are considered on the control variables and on the wealth dynamics. We mention here a few related works. Milevsky and Young \\cite{kn:MY} study extensively the optimal annuitization and investment-consumption problem for time dependent mortality function in the\\emph{ all or nothing} market and also the more general \\emph{anything anytime}s market, where gradual annuitization strategies are allowed. Milevsky et al. \\cite{kn:MMY} derive the optimal investment and annuitization strategies for a retiree whose objective is to minimize the ruin probability when the consumption rate is fixed. Stabile \\cite{kn:S} studies the optimal investment-consumption problem and investigates the optimal time for purchasing the annuity subject to a constant force of mortality by considering different utility functions defined on the consumption and the final annuity. Blake et al. \\cite{kn:BCD} compare the immediate purchasing the annuity at the retirement age with distribution programs involving differing exposures to equities during retirement. Albrecht and Maurer \\cite{kn:AM} compare the immediate annuitization and the income drawdown option and determine the probability of running out of money before the uncertain date of death. Gerrard et al. \\cite{kn:Ger3}, by considering  targets for the consumptions during the decumulation phase and a target for the final wealth, investigate the optimal annuitization time together with the optimal consumption-investment strategies.  \n\nThe article is organized as follows. In the next section, we identify the market model, the loss function and also the set of admissible strategies. In the third section, to apply the dynamic programming principle, we specify the value function. Furthermore, by considering a safety level for the wealth process we specify the set of admissible strategies  and write the HJB equation obtained via the dynamic programming principle corresponding to the optimal control problem. In section 4, we express the numerical algorithm to find approximations of solution of the HJB equation. Finally in section 5, we present the simulation results for the final annuity, optimal investment-consumption strategies and optimal wealth process. \n\n\n\\section{The Market Model}\nWe consider a Brownian market model consists of a risky and a risk-less asset with the dynamics: \n\\begin{align}\n&dS_t=S_t(\\mu dt+\\sigma dB_t), \\nonumber\\\\\n&dA_t=rA_tdt,\\nonumber\n\\end{align}\nwhere $B(\\cdot)$ is a Brownian motion on the filtered probability space $(\\Omega, \\mathcal{F}, \\mathbb{F}, \\mathbf{P})$ and $r$ is the fixed interest rate. So, the risky asset is a geometric Brownian motion with constant volatility $\\sigma$ and expected return $\\mu=r+\\sigma \\beta$, in which $\\beta $ is its   Sharpe ratio.\n\nAt any time $t$ and when the fund value is $x$, let $y(t,x)$ and $1-y(t,x)$ denote the proportions of the fund's portfolio that are invested in the risky and in the riskless asset, respectively. Moreover,  denote the consumption rate at the point $(t, x)$ by $c(t,x)$. Therefore, we have the following dynamics of the wealth process (or the fund value) \n\\begin{align}\ndX_t=\\{[y(\\mu-&r)+r]X_t-c \\} dt+\\sigma y X_t dB_t,\\label{wealtheq}\\\\\n&X(0)=x_0. \\nonumber\n\\end{align}\n\nLet the decumulation phase be denoted by the time interval $[0,T]$. We consider the functions of two variables $y:[0,T]\\times[0,\\infty )\\rightarrow \\mathbb R$ and $c:[0,T]\\times[0,\\infty )\\rightarrow [C_2, C_1]$ as control variables that must be determined by solving an optimal control problem. We let the variable $y$ to be greater than one, $y>1$, or to be negative, $y<0$,  which correspond to borrowing from the money market and  short-selling of the risky asset, respectively. However, our numerical results justify that the short-selling does not appear as the optimal investment strategy. \nWe restrict the consumption rate to the interval $[C_2, C_1]$. Actually $C_1$ is the desired rate of consumption and usually is set to be equal to the annuity that is purchasable by the accumulated wealth at the retirement and $C_2$ corresponds to the minimum required consumption during the decumulation phase.  \n\nOur market model can be extended in several directions. Among many other models, Boulier  et al. \\cite{kn:BHT} consider a model with stochastic interest rate. Han and Hung \\cite{kn:HH} equip the market model with the inflation rate which has important consequences on the optimal investment strategy. Gao \\cite{kn:Ga}, in a stochastic interest rate framework, considers a market model that consists of three assets, a risky asset, a risk-less asset and a bond. Hainaut and Deelstra investigate the optimal time for annuity purchase in a market model with jump-diffusion dynamics when the economic utility function is replaced by the  expected present value operator. Considering the jump-diffusion dynamics, which is much more justified by the empirical data, is the aim of our ongoing work. \n\nWe consider a retiree with age 60 who is going to postpone the annuitization until the age 75, which means $T=15$.  The main purpose of a retiree in postponing the annuity purchase is to reach the desired annuity. Let $F$ be the target for the terminal accumulated wealth, $X(T)$, by which the retiree can purchase the desired annuity at the age 75. In other words, if $a_{75}$ is the actuarial value of the unitary lifetime annuity at the age 75, then $\\frac{ F}{a_{75}}$ would be the desired annuity. Moreover, during the decumulation phase, part of the  retiree's concern is  on the consumption and he or she would like to have the maximum consumption rate $C_1$.  \nTherefore, we write the loss function by two terms. One term, as the running cost, is formulated by  the consumption and is considered as the distance from the desired rate, $C_1$. Moreover, this term is weighted  by the impact factor $\\kappa$. The second term is based on the final annuity and is written as the distance from the specified target $\\frac{ F}{a_{75}}$.\n\nSo, supposing the mortality rate after the retirement, $\\mu(t), t\\geq 60 $, to be independent of the asset dynamics, we write the loss function as\n  \n\\begin{align}\n\\kappa \\int^T_0  e^{-\\int^t_0 (\\rho+\\mu(s))ds } (C_1-c(t))^2dt+ e^{-\\int^T_0(\\rho+\\mu(s))ds }\\left(\\frac{ F-X(T)}{a_{75}}\\right)^2, \\label{lossfunc}\n\\end{align}\n in which the constant factor $\\rho$ is the subjective discount factor. \n\nFor any $0\\leq t\\leq T$, consider the filter $\\mathbb{F}^t:=(\\mathcal{F}^t_s)_{s\\in [t,T]}$ where $\\mathcal{F}^t_s$ is the $\\sigma$-algebra generated by the random variables $(B(u)-B(t))_{u\\in [t,s]}$. \nThen at any time $t\\geq 0$, we choose the admissible strategies $\\pi_1(\\cdot)\\in \\mathcal{L}^2(\\Omega \\times [t,T]; \\mathbb{R})$ and $ \\pi_2(\\cdot)\\in \\mathcal{L}^2(\\Omega \\times [t,T]; [C_2, C_1])$, which are $\\mathbb{F}^t$-progressively measurable. It should be noted that if the variables $y$ and $c$ in Eq. (\\ref{wealtheq}) be replaced by $\\pi_1(\\cdot)$ and $\\pi_2(\\cdot)$, respectively, the equation will have a unique strong solution, see \\cite[Section 5.6.C]{kn:KS}. We denote this solution by $X(\\cdot ; t, x, \\pi_1(\\cdot), \\pi_2(\\cdot)).$ \n\nThe main feature of our framework is designing a minimum guarantee for the final annuity or equivalently for the terminal wealth amount. So, if we denote $S$ as the safety level of the terminal wealth, then the set of admissible investment and consumption strategies reduces to \n\\begin{align}\n\t\\tilde{\\Pi}_{ad}(t):=\\{&\\pi_1(\\cdot)\\in \\mathcal{L}^2(\\Omega \\times [t,T]; \\mathbb{R}), \\pi_2(\\cdot)\\in \\mathcal{L}^2(\\Omega \\times [t,T]; [C_1, C_2])\\nonumber\\\\\n\t & | \\pi_1(\\cdot), \\pi_2(\\cdot)\\; are \\;\\mathbb{F}^t- prog. meas., X(T; t,x,\\pi_1(\\cdot), \\pi_2(\\cdot)) \\geq S \\; a.s.\\}\\label{adm}\n\\end{align}\n\\section{\\textbf{The HJB Equation}}\n\nConsidering the loss function (\\ref{lossfunc}), we define the objective functional $\\tilde{J}$, for any $(t,x)\\in [0,T]\\times \\mathbb R^+$,  on the set of admissible strategies $ \\tilde{\\Pi}_{ad}(t)$, as\n\n\\begin{align}\n\\tilde{J}(t,x;\\pi_1(\\cdot), \\pi_2(\\cdot)):=\\mathbb{E}^x &[\\kappa \\int^T_t e^{-\\int^s_0 (\\rho+\\mu(r))dr } (C_1-\\pi_2(s))^2ds\\nonumber\\\\\n&+ e^{-\\int^T_0(\\rho+\\mu(s))ds }\\left(\\frac{F-X(T)}{a_{75}}\\right)^2 ],\\label{oldfunc} \n\\end{align}\nwhere $X(s), t\\leq s\\leq T$ is the wealth process obtained by employing the investment and consumption strategies $\\pi_1(\\cdot), \\pi_2(\\cdot)\\in \\tilde{\\Pi}_{ad}(t)$ and $\\mathbb{E}^x$ denotes the expectation subject to $X(t)=x$. \nOur goal is to find the admissible strategies that minimize the above functional. Therefore, to solve this stochastic optimal control problem via the dynamic programming, we define the value function \n\\begin{align}\n\\tilde{V}(t,x):=inf_{\\pi_1(\\cdot), \\pi_2(\\cdot)\\in \\tilde{\\Pi}_{ad}(t)} \\tilde{J}(t,x;\\pi_1(\\cdot), \\pi_2(\\cdot)).\\label{value.f}\n\\end{align}\n\nThe minimum guarantee constraint at the terminal time imposes consequently a constraint on the wealth process before the annuitization. Actually, the following curve is as a  barrier for the wealth process, \n\\begin{align}\nS(t)=\\frac{C_2}{r}-(\\frac{C_2}{r}-S)e^{-r(T-t)}, \\qquad 0\\leq t\\leq T. \\label{safe}\n\\end{align}\nBecause, whenever the wealth process hits this curve, investing any portion of the wealth in the risky asset or consuming at any rate higher $C_2$, yields  a positive probability for that final wealth to be less than $S$ at the terminal time $T$. Therefore, thereafter, the only admissible investment and consumption policies will be the null strategy and consuming at the  minimum rate, $\\pi_1(s)=0$ and $\\pi_2(s)=C_2$, $ s\\geq t$, respectively, which keeps the wealth process on this curve until the terminal time $T$.\n\nMoreover, at any time $t\\in [0,T]$, the wealth amount \n\\begin{align}\nF(t)=\\frac{C_1}{r}+(F-\\frac{C_1}{r})e^{-r(T-t)}, \\label{targfun}\n\\end{align}\nguarantees reaching the desired target $F$ at the terminal time T, by investing the whole portfolio during the time interval $[t,T]$ in the risk-less asset and consuming at the maximum rate $C_1$. This strategy keeps the wealth process on the curve $\\{F(t), 0\\leq t\\leq T \\}$ and yields zero cost. Therefore, it is the optimal strategy. \n\nThese observations indicate that the curves $\\{F(t), 0\\leq t\\leq T\\}$ and $\\{S(t), 0\\leq t\\leq T\\}$ are as the attractors of the wealth process. In the other words, if the initial value lies in $[S(0), F(0)]$ then the wealth process will remain in a domain bounded at the top and at the bottom by the curves $\\{F(t), 0\\leq t\\leq T\\}$ and $\\{S(t), 0\\leq t\\leq T\\}$, respectively. \n\nDue to the definition of the value function $\\tilde{V}$, (\\ref{value.f}), the Bellman principle indicates the following HJB equation, see \\cite[Chapter 11]{kn:O},\n\\begin{align}\n\\inf_{y, c\\in \\mathbb R}\\{ \\frac{\\partial \\tilde{V}}{\\partial t}&+ \\tilde{\\mathcal{A}}\\tilde{V}(t,x)+ \\kappa e^{-\\int^t_0(\\rho+\\mu(s))ds } (C_1-c(t))^2 \\}=0, \\label{main1}\n\\end{align}\nwhere $\\tilde{\\mathcal{A}}$ is the generator of the diffusion process (\\ref{wealtheq}), \n$$\\tilde{\\mathcal{A}} =\\{(y[\\mu-r]+r)x-c\\} \\frac{\\partial }{\\partial x}+ \\frac{1}{2} \\sigma^2 y^2 x^2 \\frac{\\partial^2 }{\\partial x^2}.$$ \nAdditionally, the definition of $\\tilde{V}$ indicates the following boundary conditions\n\\begin{align}\n\\tilde{V}(T,x)= &e^{-\\int^T_0(\\rho+\\mu(s))ds }\\left(\\frac{F-x}{a_{75}}\\right)^2, \\qquad \\qquad \\qquad  \\quad x\\in [S,F],\\nonumber\\\\\n\\tilde{V}(t,F(t))=&0,\\qquad \\qquad  \\qquad\\qquad \\qquad \\qquad \\qquad\\qquad \\quad t\\in [0,T],\\label{mainboundary}\\\\\n\\tilde{V}(t,S(t))=&e^{-\\int^T_0(\\rho+\\mu(s))ds }\\left(\\frac{ F-S}{a_{75}}\\right)^2\\nonumber\\\\\n&+\\kappa (C_1-C_2)^2\\left(\\int^T_te^{-\\int^s_0(\\rho+\\mu(r))dr } ds\\right), \\quad  t\\in [0,T].\\nonumber\n\\end{align}\n\nThe above equation has been defined on an irregular domain, in which the top and the bottom borders are curved.  Since we are going to apply the finite difference method as part of our algorithm, we apply a change of variable that converts the domain to a rectangle.\nTo this end, we define the diffeomorphism $\\mathcal{L}: \\mathcal{C} \\rightarrow \\mathcal{C}^{\\prime}$, where $\\mathcal{C}:=\\{(t,x) | t\\in[0,T], S(t)\\leq x\\leq F(t) \\}$, $\\mathcal{C}^{\\prime}:=\\{(t,z) | t\\in[0,T], S\\leq z\\leq F \\}$  and \n\n\\begin{align}\n(t,x)\\rightarrow (t,z)&=\\mathcal{L}(t,x)=(t,\\mathcal{L}_1(t,x))\\nonumber\\\\\n&:=\\left(t,xe^{r(T-t)}+\\left[C_1+(C_2-C_1)\\frac{F(t)-x}{F(t)-S(t)} \\right]\\frac{1-e^{r(T-t)}}{r}\\right).\\nonumber\n\\end{align}\nNotice that \n\\begin{align}\n\\mathcal{L}_1(t,S(t))=S, \\qquad \\mathcal{L}_1(t,F(t))=F. \\label{borders}\n\\end{align}\nNow, we define the process $Z(\\cdot)$ as \n$$Z(t):=\\mathcal{L}_1(t,X(t))=X(t)G(t)+H(t), \\quad t\\in [0,T], $$\nin which \n$$G(t)=e^{r(T-t)}+ \\frac{C_1-C_2}{F(t)-S(t)} \\frac{1-e^{r(T-t)}}{r},$$\n$$H(t)=\\left(C_1+ \\frac{F(t)(C_2-C_1)}{F(t)-S(t)}\\right) \\frac{1-e^{r(T-t)}}{r}.$$ \nApplying the It\\^{o} formula, the process $Z_t$ satisfies the dynamics \n\\begin{align}\ndZ_t=&X_tdG_t+G_tdX_t+dH_t  \\nonumber\\\\\n=&X_t\\left[\\left(\\frac{C_1-C_2}{F(t)-S(t)}-r\\right)e^{r(T-t)}+\\frac{(C_1-C_2)(r(F_S)-C_1+C_2) }{(F(t)-S(t))^2}\\frac{1-e^{-r(T-t)}}{r}\\right]dt\\nonumber\\\\\n&+G(t)\\left\\{[(y(\\mu-r)+r)X_t-c ]dt+\\sigma yX_tdW_t \\right \\}\\nonumber\\\\\n&+\\left[C_1+\\frac{F(t)(C_2-C_1)}{F(t)-S(t)}\\right]e^{r(T-t)}dt\\nonumber\\\\\n&+(C_2-C_1)\\frac{e^{-r(T-t)-1}}{r}\\frac{(F(t)-S(t))(rF-C_1)-F(t)[r(F-S)-C_1+C_2]}{(F(t)-S(t))^2}dt.\\label{zproc}\n\\end{align}\nMoreover, by a few manipulations we get\n$$X_t=\\frac{r(F(t)-S(t))e^{-r(T-t)}Z_t-[F(t)C_2-S(t)C_1](e^{-r(T-t)}-1) }{r(F(t)-S(t))+(C_1-C_2)(e^{-r(T-t)}-1)}. $$\n\nWe  define  for any $\\pi_1(\\cdot), \\pi_2(\\cdot) \\in \\Pi_{ad}(t)$ the process \n$$Z(s;t,z,\\pi_1(s), \\pi_2(s)):=\\mathcal{L}_1(s,X(s;t,x,\\pi_1(s), \\pi_2(s))), \\qquad t\\leq s\\leq T,$$ \nin which $Z(t)=z=\\mathcal{L}_1(t,x)$. Then, because of the relations (\\ref{borders}), the set of admissible strategies turns to \n\\begin{align}\n\\Pi_{ad}(t)=\\{&\\pi_1(\\cdot)\\in \\mathcal{L}^2(\\Omega \\times [t,T]; \\mathbb{R}), \\pi_2(\\cdot)\\in \\mathcal{L}^2(\\Omega \\times [t,T]; [C_2, C_1]) \\nonumber\\\\ \n&| \\pi_1(\\cdot), \\pi_2(\\cdot)\\; are \\; \\mathbb{F}^t- prog. meas. , S\\leq Z(s)\\leq F, s\\in [t,T] \\}.\\nonumber\n\\end{align}\nAnalogous to (\\ref{oldfunc}), we have for any $(t,z)\\in \\mathcal{C}^{\\prime}$ the following functional defined on $\\Pi_{ad}(t)$ whose one term is represented by the process $Z(\\cdot;t,z,\\pi_1(\\cdot), \\pi_2(\\cdot))$, \n\\begin{align} \nJ(t,z,\\pi_1(\\cdot), \\pi_2(\\cdot)):=\\mathbb{E}^x [&\\kappa\\int^T_t e^{-\\int^s_0(\\rho+\\mu(r))dr } (C_1-\\pi_2(s))^2ds\\nonumber\\\\\n&+e^{-\\int^T_0(\\rho+\\mu(s))ds } \\left(\\frac{F-Z(T)}{a_{75}}\\right)^2 ]. \\nonumber\n\\end{align}\nNow, defining the value function $V$ as \n$$V(t,z):=inf_{\\pi_1(\\cdot), \\pi_2(\\cdot)\\in\\Pi_{ad}(t)} J(t,z;\\pi_1(\\cdot), \\pi_2(\\cdot)), \\quad (t,z)\\in \\mathcal{C}^{\\prime},$$\nwe get the following HJB equation on the domain $\\mathcal{C}^{\\prime}$ \n\\begin{align}\n\\inf_{y, c\\in\\mathbb R}\\left\\{ \\frac{\\partial V}{\\partial t}+ \\mathcal{A}V(t,z) +\\kappa e^{-\\int^t_0(\\rho+\\mu(s))ds } (C_1-c)^2\\right \\}=0, \\label{maineq}\n\\end{align}\nwhere $\\mathcal{A}$, the generator of the diffusion process $Z(\\cdot)$, is written as \n\\begin{align}\n\\mathcal{A}=&\\{K(t,z) G^{\\prime}(t)+G(t) [(y(\\mu-r)+r)z-c ]+H^{\\prime}(t) \\}  \\frac{\\partial }{\\partial z}\\nonumber\\\\\n&+\\frac{1}{2}G^2(t)\\sigma^2y^2z^2 \\frac{\\partial^2 }{\\partial z^2},\\nonumber\n\\end{align}\nin which \n$$K(t,z)=\\frac{r(F(t)-S(t))e^{-r(T-t)}z-[F(t)C_2-S(t)C_1](e^{-r(T-t)}-1) }{r(F(t)-S(t))+(C_1-C_2)(e^{-r(T-t)}-1)}.$$  \n\nSince during the decumulation phase the loss function just depends on the consumptions, we get  the following boundary conditions that are similar to (\\ref{mainboundary}), \n\\begin{align}\n V(T,z)=& e^{-\\int^T_0(\\rho+\\mu(s))ds}\\left(\\frac{F-z}{a_{75}}\\right)^2,\\qquad    \\qquad \\quad z\\in [S,F],\\nonumber\\\\\nV(t,F)=&0, \\qquad \\qquad\\qquad \\qquad \\qquad \\qquad \\qquad \\quad\\quad t\\in [0,T],\\label{boundary}\\\\\nV(t,S)=&e^{-\\int^T_0(\\rho+\\mu(s))ds }\\left(\\frac{F-S}{a_{75}}\\right)^2\\nonumber\\\\\n&+\\kappa (C_1-C_2)^2(\\int^T_te^{-\\int^s_0(\\rho+\\mu(r))dr } ds), \\quad  t\\in [0,T].\\nonumber\n\\end{align}\n\nThe coefficients of Eq. (\\ref{maineq}) are Lipschitz continuous w. r. t. the state variables $(t,z)$ and the control variables $y$ and $c$. Therefore, they are bounded on the domain. Moreover, the control variable $c$ is chosen from a compact interval. Although, we do not assume a bound for the variable $y$, but practically it is located inside a bounded interval, too.    Furthermore, the boundaries of the domain are the attractors of the wealth process. From these observations, we can conclude that the above equation has the non-degeneracy property. Now, \\cite[Theorem 2.1] {kn:BS} indicates the strong comparison property of Eq. (\\ref{maineq}) which implies the existence of viscosity solution for this equation. \n\n\n\\section{\\textbf{Numerical Algorithm}}\n\nWe apply the policy iteration method in which the value function $V$ and the control variables $y, c$ are improved iteratively until they converge to the correct solution of the HJB Eq. (\\ref{maineq}). \n For the fixed control variable functions, we  apply a fully implicit  backward in time finite difference scheme to obtain the value function. \n\nIn discretizing the domain $\\mathcal{C}^{\\prime}= [0, T] \\times [S, F]$, the  time horizon $[0, T]=[0, 15]$ is divided to 780 subintervals with the length $\\Delta t=\\frac{1}{52}$, which corresponds to the length of one week in a year. Moreover, the interval $[S, F]$ is divided to $N+1$ subintervals with $N$ interior nodes, in which the length of each subinterval is $\\Delta z$.  Starting from the last column $t=T-\\Delta t$ inside the domain $\\mathcal{C}^{\\prime}$, we apply to each column the following algorithm.   \\\\ \n\n\\textbf{ I)} \\emph{Initial value:} For the starting point, we can take the investment function $y=\\frac{1}{2}$ and the consumption rate as $c=\\frac{C_1+C_2}{2}$.\n\n\\textbf{II)} \\emph{Policy evaluation:} For the given investment and consumption functions $y, c$, apply a fully implicit finite difference scheme    to solve the PDE (\\ref{maineq}) for the value function $V$ with the corresponding boundary conditions (\\ref{boundary}).  \n\n\\textbf{III)} \\emph{Policy improvement:} For the fixed $t$, at any node $(t,z_j), 2\\leq j\\leq N(i)+1$, which correspond to all nodes on the fixed column $t=t_i$ that are inside the domain $\\mathcal{C}^{\\prime}$,  solve the static optimization problem (\\ref{maineq}) to find  new optimal investment and consumption functions $y^{new}$ and $c^{new}$, respectively.  \n\n\\textbf{IV)} \\emph{End of iteration:} Applying the functions $y, c$ and $V$ obtained from the steps (II) and (III), whenever, at any node $(t,z_j), 2\\leq j\\leq N+1$, the absolute value of the left hand side of Eq. (\\ref{maineq}) is not less than $10^{-6}$ or one of the following convergence criterions does not satisfy, return to the step (II)   \n$$max_{j} \\abs{c^{new}(t,z_j)-c^{old}(t,z_j)}\\leq \\{ max_{j} \\abs{c^{new}(t,z_j)} \\}\\times 10^{-6}, $$\n$$max_{j} \\abs{y^{new}(t,z_j)-y^{old}(t,z_j)}\\leq \\{ max_{j} \\abs{y^{new}(t,z_j)} \\}\\times 10^{-6}, $$\n$$max_{j} \\abs{V^{new}(t,z_j)-V^{old}(t,z_j)}\\leq \\{ max_{j} \\abs{V^{new}(t,z_j)} \\}\\times 10^{-6}. $$\n\n\\textbf{V)} \\emph{Iteration policy in the preceding column:} If $t > 0$ go to the time step $t-\\Delta t$ and return to the step (II). \\\\\n\nTo apply the finite difference method in step (II), we consider the forward scheme for the time and the space first derivatives and the central difference scheme for the space second derivative. So, denoting  $V(i,j)=V(t_i,z_j)$, the value function at the node $(t_i,z_j), \\; 1\\leq i\\leq 781, 1\\leq j\\leq N+2$, the discretization of the PDE (\\ref{maineq}) at the node $(t_i,z_j)$ that is  inside the domain $\\mathcal{C}^{\\prime}$  is given by\n\n\\begin{align}\n\\frac{V(i+1,j)-V(i,j)}{\\Delta t}&+\\kappa e^{-\\int^{t_i}_0(\\rho+\\mu(s))ds } (C_1-c(t_i,z_j))^2\\nonumber\\\\\n&+\\left(\\frac{\\alpha(i,j)}{\\Delta z}+\\frac{\\beta(i,j)}{(\\Delta z)^2}\\right) V(i,j+1)+\\frac{\\beta(i,j)}{(\\Delta z)^2}V(i,j-1)\\nonumber\\\\\n&-\\left(\\frac{\\alpha(i,j)}{\\Delta z}+2\\frac{\\beta(i,j)}{(\\Delta z)^2}\\right)V(i,j)=0, \\label{discrete}\n\\end{align}\nin which \n\n$$\\alpha(i,j)=K(t_i,z_j) G^{\\prime} (t_i)+G(t_i) [(y(\\mu-r)+r)z_j-c ]+H^{\\prime} (t_i) ,$$ \n$$\\beta(i,j)=\\frac{1}{2}G^2(t_i)\\sigma^2y^2z^2_j.$$\nThen, applying the fully implicit scheme for the nodes on a fixed  column, $(t_i,z_j), 2\\leq j\\leq N+1$,  we get the linear equation $Aw=b$, in which $w=(V(t_i,z_2), \\cdots ,V(t_i, z_{N+1}))$ is the unknown vector and $A=[a_{ij}]_{N\\times N}$ and $b=[b_j]_{1\\times N}$ are a tridiagonal matrix and a positive vector, respectively, with the following entries \n$$a_{j,j}=\\frac{1}{\\Delta t}+\\frac{\\alpha(i,j+1)}{\\Delta z}+\\frac{2\\beta(i,j+1)}{(\\Delta z)^2},\\quad 1\\leq j\\leq N,$$\n$$a_{j,j+1}=-\\frac{\\alpha(i,j+1)}{\\Delta z}-\\frac{\\beta(i,j+1)}{(\\Delta z)^2}, \\quad 1\\leq j\\leq N-1, $$\n$$a_{j,j-1}=-\\frac{\\beta(i,j+1)}{(\\Delta z)^2}, \\qquad \\qquad \\qquad 2\\leq j\\leq N,$$ \n$$b_j=\\frac{V(i+1,j+1)}{\\Delta t}+\\kappa e^{-\\int^{t_i}_0(\\rho+\\mu(s))ds } (C_1-c(t_i,z_{j+1}))^2, \\quad 2\\leq j\\leq N,$$\n$$b_1=\\frac{V(i+1,2)}{\\Delta t}+\\kappa e^{-\\int^{t_i}_0(\\rho+\\mu(s))ds } (C_1-c(t_i,z_2))^2+\\frac{ \\beta(i,2)}{(\\Delta z)^2}V(t_i,S).$$\n\nIt is clear from the definitions of the functions  $\\alpha$ and $\\beta$ that the discrete representation (\\ref{discrete}) has the positive coefficient property, see \\cite{kn:FL} for the definition of this property, which consequently indicates that the matrix $A$ is an M-matrix.\nThe positive coefficient property and the boundary condition (\\ref{boundary}) easily yield the stability, consistency and the monotonicity of our numerical scheme, see \\cite{kn:FL}.  Therefore, from  \\cite{kn:B} and \\cite{kn:BS}, the convergence of our numerical approximations to the viscosity solution of Eq. (\\ref{maineq}) is guaranteed. Here we show just the stability property. \n\n\\begin{prop}\nThe discretization (\\ref{discrete}) satisfies the $l_{\\infty}$- stability property, \n\\begin{align}\n\\norm{V(t_i,\\cdot)}_{\\infty} \\leq \\left(\\frac{F-S}{a_{75}}\\right)^2+ \\kappa (C_1-C_2)^2, \\qquad  1\\leq i\\leq 780.\\label{stability}\n\\end{align}\n\n\\begin{proof}\nWe have for every $1\\leq i\\leq 780$ and $2\\leq j\\leq N+1$\n\n\\begin{align}\nV(i,j)=&V(i+1,j)+\\Delta t \\alpha(i,j)\\frac{V(i,j+1)-V(i,j)}{\\Delta z}\\nonumber\\\\\n&+\\Delta t \\beta(i,j) \\frac{V(i,j+1)-2V(i,j)+V(i,j-1)}{(\\Delta z)^2}\\nonumber\\\\\n&+\\Delta t\\kappa e^{-\\int^{t_i}_0(\\rho+\\mu(s))ds } (C_1-c(t_i,z_j))^2.\\nonumber\n\\end{align}\nSo, we have \n\n\\begin{align}\n\\abs{V(i,j)}&\\left(1+\\frac{\\Delta t}{\\Delta z}\\alpha(i,j)+2\\frac{\\Delta t}{(\\Delta z)^2}\\beta(i,j)\\right ) \\leq \\nonumber\\\\\n&\\norm{V(i+1,\\cdot)}_{\\infty}+\\norm{V(i,\\cdot)}_{\\infty} \\left (\\frac{\\Delta t}{\\Delta z}\\alpha(i,j)+2\\frac{\\Delta t}{(\\Delta z)^2}\\beta(i,j)\\right ) \\nonumber\\\\\n&+\\Delta t\\kappa e^{-\\int^{t_i}_0(\\rho+\\mu(s))ds }(C_1-C_2)^2.\\nonumber\n\\end{align}\nIf $V(i,j_1)=max_{1\\leq j\\leq N+2}V(i,j)=\\norm{V(i,\\cdot)}_{\\infty}$, then we can write \n\\begin{align}\n\\norm{V(i,\\cdot)}_{\\infty}&\\left (1+\\frac{\\Delta t}{\\Delta z}\\alpha(i,j_1)+2\\frac{\\Delta t}{(\\Delta z)^2}\\beta(i,j_1)\\right ) \\leq \\nonumber\\\\\n&\\norm{V(i+1,\\cdot)}_{\\infty}+\\norm{V(i,\\cdot)}_{\\infty} \\left (\\frac{\\Delta t}{\\Delta z}\\alpha(i,j_1)+2\\frac{\\Delta t}{(\\Delta z)^2}\\beta(i,j_1)\\right ) \\nonumber\\\\\n&+\\Delta t \\kappa e^{-\\int^{t_i}_0(\\rho+\\mu(s))ds }(C_1-C_2)^2,\\nonumber\n\\end{align}\nwhich means \n$$\\norm{V(i,\\cdot)}_{\\infty}\\leq  \\norm{V(i+1,\\cdot)}_{\\infty}+\\Delta t\\kappa e^{-\\int^{t_i}_0(\\rho+\\mu(s))ds }(C_1-C_2)^2.$$\nThen, from the terminal condition of the value function we obtain the bound (\\ref{stability}). \n\\end{proof}\n\n\\end{prop}\n\n\n\n\n\n\\section{\\textbf{Simulation Results }}\nFor comparison purposes, we consider the same market parameters as in \\cite{kn:Ger1} and \\cite{kn:G}. So, we consider the interest rate $r=0.03$ and the expected return and the volatility of the risky asset $\\mu=0.08$ and $\\sigma=0.15$, respectively, which implies a Sharpe ratio equal to $\\beta=0.33$. Furthermore, we consider a retiree with age 60 and with the initial wealth $x_0=100$ and set the decumulation period equal to $T=15$ years. The maximum consumption rate is set to be $C_1=6.5155$, which is assumed to be equal to the payments of a lifetime annuity purchasable by a retiree at age 60 with the wealth $x_0=100$, regarding the mortality rate given in this section. \n\nWe consider four different rates for the minimum consumption, $C_2=C_1$, $C_2=\\frac{3}{4}C_1$, $C_2=\\frac{2}{3}C_1$ and $C_2=\\frac{1}{2}C_1$, which correspond to different minimum cost of living of a retiree.   Moreover, we consider the target level $F=1.75C_1 a_{75}$ and the safety level $S=0.5 C_1a_{75}$ for the wealth process, which in the literature correspond  to the medium level of risk aversion, see  \\cite{kn:Ger1} and \\cite{kn:G}. Actually, in this level, the final annuity that the retiree will get is at most 1.75 times $C_1$ and at least half of $C_1$. \n\nBy employing the dynamics (\\ref{wealtheq}) and the optimal investment and consumption functions obtained from the previous section, we simulate the optimal wealth process. To this end, we apply a same 5000 generated stream of pseudo random numbers for four different ranges of consumptions. \nUsing the simulated optimal wealth processes when the impact factor of the running cost, $\\kappa$, is equal to $0.5$, we present the histograms of the final annuities, Figs. \\ref{fig1}, \\ref{fig2}, \\ref{fig4} and \\ref{fig5} and  reveal some percentiles  of the optimal wealth amounts, Figs. \\ref{fig3}, \\ref{fig6}, \\ref{fig7} and \\ref{fig10}, and the optimal investment strategies, Figs. \\ref{fig8}, \\ref{fig9}, \\ref{fig11} and \\ref{fig12}, during the decumulation period. \n\nThe graphs show that although the optimal strategies are similar regarding different scenarios of admissible consumption interval, we get higher final annuities when this interval is more restricted. The remarkable indication of the simulation results is the difference between the fixed consumption case and three other scenarios. Actually, the results show that by considering a variable consumption rate, although restricted,  we can get much more valuable   final annuity and  higher percentiles of  wealth amounts. \n\n \\begin{figure}\n  \\begin{multicols}{2}\n   \n    \\hspace{-2.5cm}\n    \\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Final Annuity Amount}}\n        \\includegraphics[width=0.9\\textwidth]{FA05K05.png}\n        \\caption{ \\footnotesize{$C_2=\\frac{1}{2}C_1$}}\n        \\label{fig1}\n    \\end{minipage}\n\n\\vspace{1cm}\n \\hspace{-2.5cm} \n\\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{FInal Annuity Amount}} \n        \\includegraphics[width=0.9\\textwidth]{FA075K05.png}\n        \\caption{ \\footnotesize{$C_2=\\frac{3}{4}C_1$}}\n        \\label{fig2}\n    \\end{minipage}\n\n\n\\vspace{1cm}\n  \\hspace{-2.5cm}\n    \\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Optimal Wealth Amounts}}\n        \\includegraphics[width=0.9\\textwidth]{OWA05K05.png}\n        \\caption{ \\footnotesize{$C_2=\\frac{1}{2}C_1$}}\n        \\label{fig3}\n    \\end{minipage}\n\n    \\begin{minipage}{0.8\\textwidth}\n            \\centering\n         \\title{\\footnotesize{Final Annuity Amount}}\n        \\includegraphics[width=0.9\\textwidth]{FA23K05.png}\n        \\caption{ \\footnotesize{$C_2=\\frac{2}{3}C_1$}}\n        \\label{fig4}\n    \\end{minipage}    \n\n\\vspace{1cm}\n    \\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Final Annuity Amount}}\n        \\includegraphics[width=0.9\\textwidth]{FAfixK05.png}\n        \\caption{ \\footnotesize{$C_2=C_1$}}\n        \\label{fig5}\n    \\end{minipage}\n\n\\vspace{1cm}\n    \\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Optimal Wealth Amount}}\n        \\includegraphics[width=0.9\\textwidth]{OWA23K05.png}\n        \\caption{ \\footnotesize{$C_2=\\frac{2}{3}C_1$}}\n        \\label{fig6}\n    \\end{minipage}\n\n \\end{multicols}\n\\end{figure}\n\n\n\n \\begin{figure}\n \\begin{multicols}{2}\n\n\\vspace{-4cm}\n \\hspace{-2.5cm}\n    \\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Optimal Wealth Amount}}\n        \\includegraphics[width=0.9\\textwidth]{OWA075K05.png}\n        \\caption{ \\footnotesize{$C_2=\\frac{3}{4}C_1$}}\n        \\label{fig7}\n    \\end{minipage}\n\n\\vspace{1cm}\n \\hspace{-2.5cm}\n\\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Optimal Investment Strategy}}\n        \\includegraphics[width=0.9\\textwidth]{OP05K05.png}\n        \\caption{ \\footnotesize{$C_2=\\frac{1}{2}C_1$}}\n        \\label{fig8}\n    \\end{minipage}\n\n\\vspace{1cm}\n \\hspace{-2.5cm}\n \\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Optimal Investment Strategy}} \n        \\includegraphics[width=0.9\\textwidth]{OP075K05.png}\n        \\caption{ \\footnotesize{$C_2=\\frac{3}{4}C_1$}}\n        \\label{fig9}\n    \\end{minipage}\n   \n\n    \\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Optimal Wealth Amount}}\n        \\includegraphics[width=0.9\\textwidth]{OWAfixK05.png}\n        \\caption{ \\footnotesize{$C_2=C_1$}}\n        \\label{fig10}\n    \\end{minipage}\n\n\\vspace{1cm}\n    \\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Optimal Investment Strategy}} \n        \\includegraphics[width=0.9\\textwidth]{OP23K05.png}\n        \\caption{ \\footnotesize{$C_2=\\frac{2}{3}C_1$}}\n        \\label{fig11}\n    \\end{minipage}\n\n  \\vspace{1cm}\n     \\begin{minipage}{0.8\\textwidth}\n        \\centering\n         \\title{\\footnotesize{Optimal Investment Strategy}}\n        \\includegraphics[width=0.9\\textwidth]{OPfixK05.png}\n        \\caption{ \\footnotesize{$C_2=C_1$}}\n        \\label{fig12}\n    \\end{minipage}\n    \n \\end{multicols}\n\\end{figure}\n\n\\\n\nTo compare our results corresponding to different scenarios, we consider the market present value of the cash flows that a retiree would have before and after the annuitization. Actually, the cash flow before the annuitization consists of withdrawals from the fund, that in \\cite{kn:Ger1} and  \\cite{kn:G}  are supposed to be constant  and in this paper vary between the two limits $C_2, C_1$. Furthermore, we consider the accumulated wealth at the time of death of an individual if this occurs before the annuitization time $T$. The cash flow after the annuitization consists of the constant payments of the lifetime annuity that has been purchased by the accumulated wealth at the terminal time $T$.  \n\nFor the mortality rate, we consider the Gompertz-Makeham distribution, which for an individual of age $t$ is written as  \n$$\\mu(t)=A+BC^t.$$\nThe parameters $A, B$ and $C$ are those defined by the Belgian regulator for the pricing of life annuities purchased by males, as in \\cite{kn:HD}. So, we assume $A=0.00055845$, $B=0.000025670$ and $C=1.1011$.\n\nTherefore, considering $X(t)$ and $c(t)$ as the fund value and the consumption rate at time $t$, respectively, the market present value of the future cash flows is written as \n\\begin{align}\nP.V.=&\\int^{\\tau_d\\wedge15}_0e^{-\\rho t }c(t) dt+e^{-\\rho\\tau_d}X_{\\tau_d} 1_{\\{\\tau_d <15\\}}\\nonumber \\\\\n& +\\int^{\\tau_d}_{ \\tau_d\\wedge 15}  \\frac{X(15)}{a_{75}}e^{-\\rho t }dt.\\nonumber\n\\end{align}\nin which $\\tau_d$ is the time of death.  Then, given that the time of death is independent of the filtration of financial returns, we get\n\\begin{align}\nP.V.=&\\int^{15}_0e^{-\\int^t_0(\\rho+\\mu(60+s))ds }\\left(c(t)+\\mu(60+t)X(t)\\right) dt\\nonumber\\\\\n&+\\int^{T_m-60}_{15} \\frac{X(15)}{a_{75}} e^{-\\int^t_0(\\rho+\\mu(60+s))ds }dt, \\nonumber\n\\end{align}\nin which $T_m=100$ is considered  as the  maximum lifespan and  $\\mu(t)$ is the mortality rate of an individual of age $t$. \n\\begin{table} \n\\caption{\\footnotesize{Distribution of final annuities, consumptions and cash flows}} \n\\begin{tabular}{c  c  c  c  c  c}\n\\hline \n&\\footnotesize{ $C_2 =\\frac{1}{2}C_1$}\n&\\footnotesize{$C_2=\\frac{2}{3}C_1$} \n&\\footnotesize{ $C_2=\\frac{3}{4}C_1$}\n&\\footnotesize{ $C_2=C_1$}\\\\\n&$\\kappa=0.25$  &$\\kappa=0.25$  &$\\kappa=0.25$   \\\\\n&$\\kappa= 0.5$  &$\\kappa=0.5$  &$\\kappa=0.5$    \\\\\n&$\\kappa=0.75$  &$\\kappa=0.75$ &$\\kappa=0.75$    \\\\\n&$\\kappa= 1$  &$\\kappa=1$  &$\\kappa=1$     \\\\\n\\hline\nmean FA\\footnotemark                        &9.59                     &9.43                          &9.28                        &5.69 \\\\\n                                                             &9.08                     &8.96                         & 8.85                                      \\\\ \n                                                             &8.77                      &8.67                        &8.57                           \\\\\n                                                              &8.54                     &8.45                       & 8.37                                \\\\\n\\hline\nsd. FA                                                  &2.14                       &2.24                             &2.33                      &2.77   \\\\\n                                                            &2.43                       & 2.50                           &2.55                                       \\\\\n                                                            &2.58                       &2.63                            &2.67                                      \\\\\n                                                            &2.68                       & 2.71                           &2.74                                   \\\\\n\\hline\nmean PV\\footnotemark                               &105.21                                 &104.15                              &103.25          &96.17   \\\\\n                                                                   &105.34                                &104.59                              & 103.92                           \\\\\n                                                                   &105.06                                 &104.46                              &103.91                           \\\\\n                                                                   &104.71                               & 104.22                                 & 103.76                            \\\\\n\n\\hline\nsd. PV                                                           &13.95                              &13.77                              &13.49            &12.17   \\\\\n                                                                    &14.11                                &14.14                             & 14.08                             \\\\\n                                                                    &14.12                                &14.19                              &14.21                           \\\\\n                                                                     &14.16                               &14.18                                & 14.21                         \\\\\n\\hline\n\\footnotesize{Prob(FA $>  C_1$) (\\%)}                                    &89.62                          &88.10                      &86.22       &37.44  \\\\\n                                                                                                &83.72                          &82.42                      & 80.98                       \\\\\n                                                                                                &80.10                           &79.08                     &77.92                   \\\\\n                                                                                                &77.42                           & 76.54                    & 75.76                  \\\\\n\\hline\n\\footnotesize{Prob(FA =$\\frac{1}{2}C_1$)} (\\%)                         &1.62                &1.94                         &2.16                 &6.94     \\\\\n                                                                                                   &1.66                &2.60                         & 2.86                              \\\\\n                                                                                                   &1.44                 &1.94                         &3.24                            \\\\\n                                                                                                   &0.94               &2.18                          &3.26                                 \\\\ \n\\hline \nmean consumption                                                                     &5.7752              & 5.7466                    &5.7341                 &6.5155             \\\\\n                                                                                                   &5.9869              &5.9698                     &5.9595                                              \\\\\n                                                                                                   &6.0873               &6.0756                    &6.0668                                            \\\\\n                                                                                                   &6.1470               &6.1398                     &6.1330                                    \\\\\n\\hline                                                                                                                     \n\n$^1$FA=Final Annuity\\\\\n $^2$ PV=Present Value\n\\end{tabular}\n\\label{tab2}\n\\end{table}\nTable \\ref{tab2} gives more information concerning the distribution, the mean and standard deviation, of the final annuity and the market present value of the cash flows for different ranges of consumption and different impact factors of the running costs, $\\kappa=0.25, 0.5, 1$. Furthermore, it reports the mean of the consumptions before the annuitization and the probability that a member get a final annuity that is higher than the annuity purchasable at the retirement, $C_1$. \n\nAs it is expected, when the minimum consumption rate, $C_2$, decreases or the admissible  consumption interval becomes wider, we get more valuable final annuity and higher cash flows. Furthermore, in this case, the probability of getting a final annuity higher than $C_1$ increases.  \n\nThe interesting point of this table is the big difference between the outcomes of the fixed consumption rate case, $C_2=C_1$, and the three other cases. Actually,  when we assume a fixed consumption rate, we get much less valuable final annuity and, moreover, the present value of the cash flows is remarkably less than the corresponding outcomes of the variable consumption rate. However,  it is clear that, since in the fixed consumption  case the cash flow before the annuitization, and of course before the time of death, is constant, the deviation of cash flows is less than the other cases. \n\nFor a fixed admissible consumption interval, when we compare the results corresponding to different impact factors of the running cost, we observe finer final annuity (higher mean with lower standard deviation) for lower impact factors. This is quite reasonable, since when the impact factor $\\kappa$ is small, more weight is devoted to the second term of the loss function which is based on the final annuity. The surprising result for a fixed admissible consumption interval is that the maximum present value of cash flows is attained when $\\kappa=0.5$. This can be interpreted by considering the two terms of loss function. Actually, for a smaller impact factor $\\kappa$,  more weight of the loss function is devoted to the final annuity and therefore higher cash flow after the annuitization is expected. On the other side, for a greater  impact factor $\\kappa$, more weight is devoted to the consumptions  and therefore higher cash flow before the annuitization time is expected. \n\nThe last rows of the table report the mean of consumptions before the annuitization. It is clear that, on every fixed admissible consumption interval, the greater impact factor of running cost, the more weight to the first term of loss function and therefore the more consumptions. But, it is surprising that when the minimum consumption rate, $C_2$,  decreases, the mean of consumptions slightly decreases. This can be interpreted by the  previous rows in the table that when the admissible consumption interval becomes wider the probability of hitting the lower border slightly increases. \n\n\\section{\\textbf{Conclusions }}\nStudying the optimal investment-consumption problem post-retirement with a minimum guarantee for final annuity for a member of a defined contribution plan yields an HJB equation on a bounded domain. Applying the policy iteration method in solving this equation, the value function and the optimal strategies are improved iteratively. To find the approximation of the value function in one step of our algorithm, we apply the backward in time implicit scheme of finite difference method. Our scheme has the positive coefficient property which guarantees the convergence to the viscosity solution. \n\nHowever, the simulation results show that by assuming variable, although very restricted, consumption rate, we can get much finer final annuity. Moreover, considering the cash flows before and after the annuitization, the variable consumption rate yields higher present value of cash flows. \n\nIn the next work, we are going to consider the jump-diffusion dynamics which is more justified by the empirical data. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe identification of motion as a manifestation of biological life\ndates back to the earliest records of science itself. The Greek\nphysician Erasistratos of Ceos studied biological motion on the length\nscale of muscles already in the 3rd century BC. He imagined muscles to\nfunction in the way of a piston contracting and relaxing from\npneumatic origin.  It was not until the invention of the microscope in\nthe 17th century by van Leeuwenhoek that this theory could be\ndevalidated with Swammerdams observation that muscles contract at\nconstant volume.\n\nConcerning biological motion on a microscopic scale, scientists\nfavored concepts of ``living forces'' for many centuries until this was\nfinally ruled out by the observations of the Scottish botanist Robert\nBrown in 1828 who found all kind of matter to undergo erratic motion\nin suspensions.  A satisfactory explanation was provided by Einstein\nin 1905 by the interaction with thermally fluctuating molecules in the\nsurroundings. However, the molecular details remained unknown in the\nfog of low microscope resolution. Modern experimental techniques\n\\cite{metha} have lately revealed the causes of sub-cellular motion\nand transport.\n\nToday we know that every use of our muscles is the collective effort\nof a class of proteins called myosin that ``walk'' on actin filaments.\nGenerally spoken, we refer to all proteins that convert the chemical\nenergy of ATP (adenosine-triphospate) in a hydrolysis reaction into\nmechanical work as molecular motors. These motors are highly\nspecialized in their tasks and occur in a large variety: ribosomes\nmove along mRNA strands while translating the codons into proteins,\ndynein is responsible for cilia motion and axonal transport, and\nkinesins play a key role in cytoskeletal traffic and spindle formation\n(for an overview see \\cite{howard} and references therein).\n\nWhile the exact details of the molecular structure and function of\nmotor proteins~\\cite{schliwa} remain a topic of ongoing research, on a\ndifferent level attention was drawn to phenomena that arise out of the\ncollective interaction of many motors. Early research along this line\nwas motivated by mRNA translation that is managed by ribosomes.\nRibosomes are bound to the mRNA strand with one subunit and step\nforward codon by codon. The codon information is translated into\ncorresponding amino acids that are taken up from the cytoplasm and\nassembled into proteins. To increase the protein synthesis many\nribosomes can be bound to the same mRNA strand simultaneously. This\nfact might induce collective properties as was first realized by\nMacDonald \\cite{macdonald} who set up a theoretical model for the\ntranslation of highly expressed mRNA. The importance of effects caused\nby the concerted action of many motors can be deduced from a very\nsimple example that has yet drastic consequences: the slow down of\nribosomes due to steric hindrance caused by another ribosome in front\n-- comparable to an intracellular traffic jam that might significantly\nslow down protein synthesis.\n\n\nA theoretical approach to collective phenomena in intracellular\ntraffic will try to simplify the processes of molecular motion\ndown to a single step rate rather than focus on the chemical or \nmechanical details\non the molecular level of motor steps. Then it becomes\npossible to model and analyze the behavior of several motors with the\ntools of many-body and statistical physics. We will start this review\nwith a short introduction on this single step model in Sec.\n\\ref{s_model} before we introduce the total asymmetric exclusion\nprocess (TASEP) as a theoretical model for intracellular transport.\nSec. \\ref{s_phase} describes the stationary states and density\ndistributions and their phase diagram as a function of boundary\nconditions. After a review on several recent extensions in Sec.\n\\ref{s_extensions}, we will focus on the competition of TASEP and bulk\ndynamics in Sec. \\ref{s_pff}. Before concluding, Sec.\n\\ref{s_outlook} contains further recent developments.\n\n\n\\section{Model and Methods} \\label{s_model}\n\nIn the quest for a theoretical model for the motion of molecular\nmotors the first and simplest choice may be the use of a Poisson\nprocess. The ``Poisson stepper'' is assumed to be an extensionless\nobject advancing stochastically in discrete steps along a one-dimensional periodic\nlattice. \nThe process is uni-directional as the position of the stepper\ncan be described by $ x(t)=a \\ n(t)$ with the discrete step size $a$ and\nthe random variable $n(t)$ being a sequence of growing integers. Step\nevents occur stochastically with a rate $r$ constant in both space and\ntime. Consequently, the average time between two steps is then given\nby the dwell time $\\tau = 1\/r$ and the probability to find the\n``Poisson stepper'' at a position\n$n$ after time $t$ by the Poisson distribution \\cite{vankampen}.\n\nAfter we have defined a model for the translocation of a single motor,\nwe proceed with our original task which aims at the understanding of\ncollective properties of many motors. Of course, more elaborate\nmodels have been established \\cite{ps04,jp97} that account for several rate\nlimiting steps -- examples are the ATP supply or the availability of\namino acids for ribosomal mRNA translation. However, the very basic\n``Poisson stepper'' is chosen for reasons of simplicity and in order to\nprevent unnecessary molecular details from masking collective effects.\nStill, the validity and limitations of this simplification have to be\nkept in mind.\n\n\nBeing supplied with the dynamics of a single motor, a stage for the\nconcerted action of many can now be set up.  This was first done in a\npioneering work by MacDonald \\cite{macdonald} and is now widely know\nas the total asymmetric exclusion process (TASEP). It consists of an\none-dimensional lattice (Fig. \\ref{tasepmodel}) with N sites labeled\nby $i={1,\\cdots,N}$ and with a spacing of $a=L\/N$, where $L$ is the total\nlength of the lattice. For convenience, $L$ is often set to $1$ and\nthe lattice spacing then referred to as $\\varepsilon=1\/N$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{.\/tasepmodel.eps}\n\\caption{Schematic model of TASEP (particles are injected with rate $\\alpha$, move exclusively to\nthe right being subject to hard-core exclusion, and are removed with rate $\\beta$)}\n\\label{tasepmodel}\n\\end{figure}\n\nParticles have an extension of the size of the lattice spacing and are\nsubjected to hard core exclusion due to steric hindrance. Therefore\nthe occupation number $n_i$ of site $i$ can only take the values 0 or\n1.  Particles on the lattice attempt jumps to their right neighboring\nsite with a rate $r$, which will be set to unity in the following.\nHereby, a reference time scale is set. The effective frequency of\njumps can be much smaller than $r$ when attempted jumps are rejected\ndue to an already occupied target site. The attempted jump rate to the\nleft is zero, since we deal with a total asymmetric exclusion process,\nin contrast to the asymmetric exclusion process or the symmetric\nexclusion process, where the jump rate to the left is non-zero or even\nequal to the jump rate to the right.\n\n\nUnless one uses periodic boundary conditions, specific dynamic rules\nhave to be defined at the boundaries, which play a crucial role in the\nsolution of the process. Among different other conditions (reflective,\nopen with a blockage) the most common type are open boundaries, which\nwe will use as well: at the left boundary ($i=1$) particles attempt to\nattach with a rate $\\alpha$, while they detach at the right boundary\n($i=N$) with rate $\\beta$. This is equivalent to two additional sites\n$i=0$ and $i=N+1$ at the boundaries, which are connected to the system\nby the bulk dynamics described above, and are constantly set to the\ndensity $\\alpha$ and $1-\\beta$ respectively.\n\n\nIn spite of its simplicity, TASEP shows a wide range of interesting\nproperties. Since it was propelled into the scope of statistical\nphysicists,  \nit has become a paradigm for non-equilibrium physics. In contrast to\nequilibrium systems it lacks detailed balance but evolves into a\nsteady state where a non-vanishing current is maintained between\nboundaries. Upon varying these boundary conditions, TASEP was found to\nexhibit phase transitions which -- following general theorems\n\\cite{wagner} -- are not even allowed for one-dimensional equilibrium\nsystems in the absence of long-range interactions. However, the\nanalysis of non-equilibrium systems is considerably complicated by the\nlack of universal concepts like the Boltzmann-Gibbs ensemble theory.\nFeasible methods exist nevertheless and will be explained in the next\nsection.\n\n\\section{Density and Current in Stationary States} \\label{s_phase}\n\nIn analyzing exclusion processes research can focus on a multitude of\ndifferent properties. The probably most obvious to address is the\ndensity and current distribution in the stationary state. This is \nmotivated by two reasons. On the one\nhand, one intuitively attributes a strong importance to density\ninformation with respect to the biological background as e.g. the\nribosome density is connected to the rate of protein synthesis. On the\nother hand, promising experimental techniques can measure motor\ndensities and may allow for validation of theoretical models. Of\ncourse, quite extensive research has also been attracted to a\nmultitude of different properties like correlation functions\n\\cite{de93,schuetz93}, relaxation properties \\cite{ds00} or\nsuper-diffusive spreading of fluctuations \\cite{bks85} which will not\nbe the topic of this review. We will focus on analytical methods\n(supported by numerical simulations) that are designed to investigate\nspatial density distributions in the stationary state of the system.\nTo this end we will introduce some basic tools that have proven useful in\nthe exploration of TASEP properties. These are based on mean-field\napproximations and reproduce many results that can also be derived exactly.\nWe are well aware that this approach neglects correlations as included\nin the exact solutions that have been achieved for the TASEP density\nprofile by applying either recursion relations \\cite{ddm92} or a\nquantum Hamilton formalism with Bethe ansatz \\cite{schuetz01}.\n\n\\subsection{Quantum Mechanics and Statistic Properties}\n\nAs an introduction we will outline some general statistical properties.\nAt any given moment, the system can be found in a certain\nconfiguration $\\mu$ made up of the occupation numbers at each lattice\nsite. The next occurring stochastic event (i.e. the jump of one\nparticle to a neighboring site) will therefore change the system to\nanother configuration $\\mu'$. The transition probability $p_{\\mu \\to\n  \\mu'}$ is independent of the way the system had reached the initial\nconfiguration. Since there is no memory of the system's history, but\nany transition probability solely depends on the preceding state,\nTASEP is a Markov process. In order to describe the system's\nevolution, we can then use a master equation for the probability to\nfind the system in a certain state.\n\\begin{equation}\n\\frac{dP(\\mu)}{dt}=\\sum_{\\mu' \\neq \\mu} \n\\left[\\omega_{\\mu' \\to \\mu}P_{\\mu'}(t)-\\omega_{\\mu \\to \\mu'}P_{\\mu}(t)\\right] \\;,\n\\end{equation}\nwhere the $\\omega_{\\mu \\to \\mu'}$ are the transition rates from one \nconfiguration $\\mu$ to another $\\mu'$. \n\nHow can we now translate this general property into a description of\nTASEP? To this end we will use a convenient notation, which applies\nmethods from the quantum mechanics toolbox in order to formulate the\nmaster equation in terms of operators. It was introduced as ``quantum\nHamiltonian formalism'' and allows for exact solutions\n\\cite{schuetz01}.  We introduce operators $\\hat n_i(t)$, which act as\noccupation number operators, measuring the presence ($n_i=1$) or\nabsence ($n_i=0$) of a particle at site $i$. This results in the\nHeisenberg equation (for an introduction see e.g. \\cite{schuetz01})\n\\begin{equation}\n\\frac{d}{dt}\\hat n_i(t)= \\hat n_{i-1}(t)[1-\\hat n_i(t)] - \n                         \\hat n_i(t)[1-\\hat n_{i+1}(t)] \\;,\n\\label{heisenberg}\n\\end{equation}\nwhere the first term on the right hand side constitutes the jump of a\nparticle from the left neighboring site to site $i$ (and thus a\nparticle gain) and the second term a jump from site $i$ to the adjacent\nlattice site on the right (a particle loss). Note the intrinsic\nexclusion constraint in both terms that prevents jump events if the\ndestination site is occupied, i.e. the expression in brackets equals\nzero.  If one expresses these gains and losses in current, it becomes\nconvenient to use the current operator\n\\begin{equation}\n\\hat j_i(t)=\\hat n_i(t)[1-\\hat n_{i+1}(t)] \\;.\n\\end{equation}\nThis allows to rewrite (\\ref{heisenberg}) as a discretized form of a\ncontinuity equation with the discrete divergence $\\nabla \\hat\nj_i(t)=\\hat j_i(t) - \\hat j_{i-1}(t)$:\n\\begin{equation}\n\\frac{d}{dt}\\hat n_i + \\nabla \\hat j_i(t)=0 \\;.\n\\label{continuity}\n\\end{equation}\nSimilar equations for the boundaries are readily derived in the same\nway.  Since we are interested in the average density on a certain\nlattice site, we need to compute the time (or ensemble) average of the\noperators. Equation (\\ref{heisenberg}) gives an equation of motion for\nthe operator and allows to solve for the time evolution of $\\langle\n\\hat n_i(t) \\rangle$.  In executing the ensemble average of\n(\\ref{heisenberg}) two-point correlation functions like $\\langle \\hat\nn_{i-1}(t)(1-\\hat n_i(t)) \\rangle$ appear on the right hand side. These\ncorrelation functions again are connected to higher order correlations\nvia their equations of motion. The resulting infinite series of\ncorrelation functions can be solved exactly for special cases only.\nGenerally, one is required to use mean-field approaches.\n\n\\subsection{Mean-Field Solution} \\label{s_mf}\nThe rather blurry term ``mean-field theory'' is based on the concept of\nusing time or space averages (e.g. by neglecting temporal or spatial\nfluctuations) and has found a wide range of applications in\nstatistical physics (see \\cite{goldenfeld}). In this chapter, we\nwill explain its implementation for the TASEP and show a possible solution.\nTo point out the use of mean-field theory in TASEP, we look again at\nthe average of the operator $\\hat n_i(t)$. We are interested in the\nstationary state and therefore averaging signifies either a time or an\nensemble average, since the system is ergodic. Then the average\nreturns the density at the considered site $i$ as $ \\varrho_i =\n\\langle \\hat n_i \\rangle $.  Performing the average over\n(\\ref{heisenberg}), leaves us with the difficulty of the infinite\nseries of correlation functions mentioned earlier. The mean field\napproximation consists now in neglecting any correlations by setting\ne.g. $\\langle \\hat n_i \\hat n_j \\rangle = \\langle \\hat n_i \\rangle\n\\langle \\hat n_j \\rangle $ (see \\cite{mahan}). In our case, we obtain\nfor the current\n\\begin{equation}\n\\langle \\hat n_{i}(t)(1-\\hat n_{i+1}(t)) \\rangle = \n\\langle \\hat n_{i}(t) \\rangle (1-\\langle\\hat n_{i+1}(t)\\rangle) \\;.\n\\end{equation}\nThis allows then, to rewrite (\\ref{heisenberg}) in the stationary\nstate ($d \\varrho_i(t)\/dt=0$) as\n\\begin{equation}\n0=\\varrho_{i-1}(1-\\varrho_i)-\\varrho_i(1-\\varrho_{i+1}) \\;.\n\\label{e_steady}\n\\end{equation}\nObviously, (\\ref{e_steady}) could easily be solved numerically, since\nit forms a system of N difference equations.  However, it is possible\nto reduce the set of equations to arrive at an explicit solution by\nmaking two assumptions. First, note that the stationary state\ncondition $d \\varrho_i(t)\/dt=0$ implies a conservation of current\nthroughout the bulk, as can be seen from (\\ref{e_steady}). Ergo, we\njust have to solve one equation out of the set, to determine the\nstationary, homogeneous current (and density).  For that purpose, we\nuse the continuum approximation, which turns the spatial lattice\nvariable quasi-continuous.  This is achieved for a large number $N$ of\nlattice sites on a lattice of normalized length $L=1$.  The\ndistribution of sites then approaches a continuum as $\\varepsilon=L\/N\n\\ll 1$ and $x=i\/N$ is rescaled to the interval $0 \\leq x \\leq 1$.\nThereby an expansion in powers of $\\varepsilon$ is allowed.\n\\begin{equation}\n\\varrho(x \\pm \\varepsilon)=\\varrho(x) \\pm \\varepsilon \\partial_x\n\\varrho(x) \n+ \\frac{1}{2}\\varepsilon^2 \\partial_x^2 \\varrho(x) + O(\\varepsilon^3)\n\\label{e_power}\n\\end{equation}\nUsing this expansion in (\\ref{e_steady}) and the corresponding\nequations for the boundaries, results in the following first-order\ndifferential equation if we neglect all terms with higher orders in\n$\\varepsilon$:\n\\begin{equation}\n(2 \\varrho - 1) \\partial_x \\varrho  = 0 \\;. \n\\label{e_ode}\n\\end{equation}\nThe corresponding boundary conditions are $\\varrho(0)=\\alpha$ and\n$\\varrho(1)=1-\\beta$. Because we have a first order differential equation\nthat needs to satisfy two boundary conditions, we are evidently\nconcerned with an over-determined boundary value problem. There are\nthree solutions to (\\ref{e_ode}): $\\varrho_\\mathrm{bulk}(x) = 1\/2$\ndoes not satisfy either boundary condition (except for the special\ncase $\\alpha=\\beta=1\/2$), while $\\varrho(x)=C$ can satisfy either the\nleft or the right boundary condition, resulting in\n$\\varrho_\\mathrm{\\alpha}(x)=\\alpha$ and\n$\\varrho_\\mathrm{\\beta}(x)=1-\\beta$, respectively.\n\n\\subsection{Phase Diagram and Domain Wall Theory}\n\nTo obtain a general solution satisfying the boundary conditions, both\nsolutions need to be matched.  Consider the case $\\alpha,\\beta < 1\/2$.\nTo meet the boundary condition at both sides, the global density\nfunction $\\varrho(x)$ has to be $\\varrho_\\mathrm{\\alpha}$\n($\\varrho_\\mathrm{\\beta}$) in an environment close to the left (right)\nboundary. Since both $\\varrho_\\mathrm{\\alpha}$ and\n$\\varrho_\\mathrm{\\beta}$ are uniform, the two solutions do not\nintersect. At this point, we have to go beyond mean-field theory and\nassume that at any given time both solutions are valid in\nnon-overlapping areas of the system. Where those areas border, they\nare connected by a sharp domain wall (DW) at position $x_\\mathrm{w}$\n(see Fig.~\\ref{f_matching})\n\\begin{equation} \\label{e_dw}\n\\varrho(x)= \n\\cases{ \\varrho_\\mathrm{\\alpha} & for $0 \\leq x \\leq x_\\mathrm{w} \\;,$ \\cr\n                \\varrho_\\mathrm{\\beta} & for $x_\\mathrm{w} \\leq x \\leq 1 \\;.$}\n\\end{equation}\n\\begin{figure}\n\\centering\n\\includegraphics[height=4cm]{.\/matching.eps}\n\\hspace{1 cm}%\n\\includegraphics[height=4cm]{.\/tasep_phase.eps}\n\\caption{({\\bf left}) Schematic density distribution for \n  $\\alpha=.3,\\beta=.2$: in this situation the particle current\n  $j_\\mathrm{\\alpha}$ exceeds the current $j_\\mathrm{\\beta}$, thus\n  carrying more particles to the domain wall which is then shifted to\n  the left. ({\\bf right}) Phase diagram of TASEP in $\\alpha,\\beta$\n  phase space shows a low density (LD), high density (HD) and maximal\n  current (MC) phase}\n\\label{f_matching}\n\\end{figure}\nFrom the dynamics of this domain wall \\cite{kolomeisky-etal:98} we can deduce\nimportant information about the system. The key point of domain wall\ntheory is the identification of particle currents as the cause of\ndomain wall motion. If for example the current\n$j_\\mathrm{\\alpha}=\\varrho_\\mathrm{\\alpha}(1-\\varrho_\\mathrm{\\alpha})$\nof the left density solution $\\varrho_\\mathrm{\\alpha}$ is higher than\nthe current in the right part of the system (corresponding to\n$\\alpha>\\beta$), particles are transported faster to the DW from the\nleft end then they can head on to the right. Thus, the domain wall is\nshifted to the left. The system is filled up, until finally the whole\nbulk density has taken the value of $\\varrho_\\mathrm{\\beta}$ except\nfor a small boundary layer \\footnote{The boundary layer is necessary\n  in order to fulfill the both boundary conditions. Its extend is\n  finite for small systems, but will vanish in the thermodynamic limit\n  $N \\to \\infty$.}  at the system's left boundary. The opposite\nhappens, if $j_\\mathrm{\\beta}>j_\\mathrm{\\alpha}$, which is the case\nfor $\\beta<\\alpha$. Hence we can empirically state that the boundary\nwith the smaller rate acts as a bottleneck and imposes its density\ndistribution on the system.  To quantify this behavior, one can use a\ntraveling wave solution \\cite{kolomeisky-etal:98} of the form\n$\\varrho(x-Vt)$ to obtain the domain wall velocity as\n\\begin{equation}\nV = \\frac{j_\\mathrm{\\beta}-j_\\mathrm{\\alpha}}\n    {\\varrho_\\mathrm{\\beta}-\\varrho_\\mathrm{\\alpha}}=\\beta - \\alpha \\;.\n\\end{equation}\nTo arrive at a phase diagram in $\\alpha,\\beta$-space we will analyze\nthe bulk density.  A positive DW velocity is obtained for\n$\\alpha<\\beta$ and pushes the DW to the right side of the system\nresulting in a bulk density smaller $1\/2$ called low-density phase\n(LD). On the contrary, $\\beta<\\alpha$ leads to a high-density phase\n(HD), as illustrated in the phase diagram (Fig.~\\ref{f_matching}). LD\nand HD phase are connected by a first-order transition. Note, that\nthis phase diagram is equally obtained by the\nabove mentioned exact methods.\n\nThe transition line $\\alpha=\\beta<1\/2$ in the phase diagram requires\nspecial treatment. The velocity of the domain wall yields zero here,\nbut Monte Carlo simulations show that the DW actually will make random\nsteps to either side.  This behavior is caused by the stochasticity of\nthe input and output events that are described as rate processes.\nHence, the DW makes random steps to either side with equal\nprobability, being nothing else than the famous random walk in a\ndomain with reflecting boundaries.  An average over a sufficiently\nlong time will therefore result in a homogeneous probability density\nover space. The stationary density profile will just be a linear slope\nconnecting the two boundary conditions:\n\\begin{equation}\n\\varrho(x)=\\alpha+(1-\\beta-\\alpha)x \\;.\n\\end{equation}\nNote that mean-field theory neglects fluctuations and hence fails to\npredict this density distribution, but gives the result of a\nstationary domain wall. However, coming back to the picture of the\ndomain wall as a random walker, the linear density distribution can\nreadily be made plausible. Interpreting the random walk as free\ndiffusion of the domain wall and keeping in mind that the density\nconstraints $\\varrho(0)=\\alpha$ and $\\varrho(1)=1-\\beta$ hold at all\ntimes, the linear density profile is easily derived as solution of\nthe one dimensional diffusion equation with boundary conditions.\n\nFinally, consider an increase of e.g. $\\alpha$ to values that exceed\n$1\/2$, while $\\beta>1\/2$ (crossing the boundary from the upper left to\nthe upper right quadrant in the phase diagram).  In this situation\nparticles are removed sufficiently fast from the system and the supply\nof particles on the left side acts as a bottleneck limiting density\nand current. For $\\alpha<1\/2$ any increase in $\\alpha$ results in an\nincreased current (think of a highway, where an additional car will\nresult in a higher overall traffic). But above a certain value\n$\\alpha_\\mathrm{C}=1\/2$ any increase in particle input will not\nincrease the current further \\footnote{As\n  $j=\\varrho(1-\\varrho)=\\varrho-\\varrho^2$ has a maximum at\n  $\\varrho=1\/2$. This density-current relation is a convenient tool to\n  characterize traffic processes and its plot is often referred to as\n  fundamental diagram.} but will cause the current to\ndiminish again (as an additional car will further slow down traffic\nduring rush hour).  As a result the bulk will keep its current maximum\nat $\\varrho=1\/2$ and a boundary layer will form at the left side of\nthe system to match the boundary condition. This is of course nothing\nelse than the bulk solution $\\varrho_\\mathrm{bulk}$ with two boundary\nlayers and was baptized maximal current phase (MC). It is reached via\nsecond-order transitions for values $\\alpha>1\/2$ and $\\beta>1\/2$. A\nmore rigorous treatment of this behavior can be gained by computing\nthe collective velocity of the particles \\cite{kolomeisky-etal:98}.\n\n\\section{Biologically Motivated Generalizations of TASEP} \n\\label{s_extensions}\n\nThe adoption of lattice gas models for biological systems was followed\nby a variety of efforts to fit TASEP to different realistic\nenvironments. For that purpose some of the simplifying assumptions\nTASEP is based on had to be questioned. For example, experimental\nobservations have shown that ribosomes typically cover an area on the\nmRNA that exceeds the lattice spacing by a multiple. To account for\nthis situation the particles in TASEP have to extend over several\nlattice sites. However it was found that this does not change the\nphase diagram quantitatively \\cite{szl03}. Another direction of\nresearch went back to the original field of MacDonald to elucidate the\nimportance of initiation and prolongation of ribosomes \\cite{heinrich}\nor formation of mRNA loops to facilitate the back transport of\nribosomes from the termination site \\cite{chou03}. \n\n\nWhile particle interactions in TASEP is limited to hard-core\npotentials, even a small increase in the interaction radius -- as it\ncould be caused by charged molecules -- leads to qualitative changes in\nthe phase diagram \\cite{ps99}.  It was shown that lattice gases with\nshort-range repulsive interaction exhibit a density-current relation\nwith two local maxima in contrast to simple TASEP that leads to one\nmaximum. This behavior results in a qualitative change of the phase\ndiagram that is enriched by four more regions one being a\nminimal-current phase.\n\nFurther work was dedicated to the scenario of the interaction of\ndifferent species of molecular motors that move in opposite\ndirections. This can either happen on the same filament when two\ndifferent particles are able to surpass each other with a jump rate\nthat differs from the jump rate of either particle to a free site\n\\cite{efgm95} or on two adjacent one-dimensional filaments\n\\cite{pp01}. In both cases, spontaneous symmetry breaking was\nobserved.\n\nIntracellular transport along cytoskeletal filaments has also served\nas a source of inspiration for driven lattice gas models. While in the TASEP\nmodel motors can only bind and unbind on the left and the right\nboundary respectively, cytoskeletal motors are known to detach from the track\nto the cytoplasm \\cite{howard} where they perform Brownian motion and \nsubsequently reattach to the track. The interplay between diffusion  \nin the cytoplasm and directed motion along the filament was studied \\cite{lkn01} \nboth in open and closed compartments, focussing on anomalous drift and diffusion \nbehavior, and on maximal current and traffic jams as a function of the \nmotor density.\n\nIn \\cite{pff03} it has been realized that the on-off kinetics may not only \ngive rise to quantitative changes in the transport efficiency but also to \na novel class of driven lattice gas models. It was shown that the interplay \nbetween bulk on-off kinetics and driven transport results in a stationary \nphase exhibiting phase separation. This was achieved by an appropriate \nscaling of the on-off rates that ensures that particles travel a finite \nfraction on the lattice even in the limit of large systems. Then, particles \nspend enough time to ``feel'' the their mutual interaction and, eventually, \nproduce collective effects. In the following section, we will review the \nresults of these studies \\cite{pff03,pff04}.\n\n\n\\section{Phase Coexistence} \\label{s_pff}\n\nThe essential features of cytoskeletal transport are the possibility of \nbulk attachment and detachment and a finite residence time on the lattice. \nThe latter can be understood as a effect of thermal fluctuations that may overcome the\nbinding energy of the motors that is only of the order of several\n$k_\\mathrm{B} T$. Hence, attachment and detachment is a stochastic\nprocess whose dynamic rules have to be defined.  Parmeggiani {\\it et\n  al.} \\cite{pff03} chose to use Langmuir kinetics (LK) known\nas adsorption-desorption kinetics of particles on a one- or\ntwo-dimensional lattice coupled to a bulk reservoir \\cite{vilfan}.\nParticles can adsorb at empty sites and desorb from occupied sites and\nmicroscopic reversibility demands that the kinetic rates obey detailed\nbalance leading to an evolution towards an equilibrium steady state\ndescribable by standard concepts of equilibrium statistical mechanics.\nIn this sense the choice of LK is especially tempting as we are now\nfaced with the competition of two representatives of both equilibrium\nand non-equilibrium systems.  The system -- in the following referred\nto as TASEP\/LK -- is defined as follows: the well-known TASEP is\nextended with the possibility of particles to attach to the filament\nwith rate $\\omega_\\mathrm{A}$ and to detach from an occupied lattice\nsite to the reservoir with rate $\\omega_\\mathrm{D}$.  According to the\ntype of ensemble (canonical, grand canonical) the reservoir is either\nfinite or infinite.  Here, the reservoir is assumed to be infinite and\nhomogeneous throughout space and time. The density on a lattice\nreached in the equilibrium state of LK is only dependent on the ratio\n$K=\\omega_\\mathrm{D}\/\\omega_\\mathrm{A}$ and is completely uncorrelated\nin both space and time for neglection of any particle interaction\nexcept hard-core. This is justified by the assumption that the\ndiffusion in the cytoplasm is fast enough to flatten any deviations of\nthe homogeneous reservoir density.  The resulting density profile on\nthe lattice is homogeneous and given as Langmuir isotherm\n$\\varrho_\\mathrm{L}=K\/(K+1)$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{.\/pffmodel.eps}\n\\caption{Schematic model of TASEP\/LK: \n  the TASEP is extended by possible particle attachment and detachment\n  in the bulk with rate $\\omega_\\mathrm{A}, \\omega_\\mathrm{D}$}\n\\label{f_pff}\n\\end{figure}\nIf we now consider the combination of TASEP and LK into the model\ndisplayed in Fig.~\\ref{f_pff}, attention has to be paid to the\ndifferent statistical nature of both processes. TASEP evolves into a\nnon-equilibrium state carrying a finite current. Since particles are\nconserved in the bulk, the system is very sensitive to the boundary\nconditions, whereas LK as a equilibrium process is expected to be\nrobust to any boundary effects especially for large systems. Combining\nboth processes would thus lead to a trivial domination of LK as the\nbulk rates $\\omega_\\mathrm{A}$ and $\\omega_\\mathrm{D}$ that apply to a\nlarge number of bulk sites become predominant over the rates $\\alpha$\nand $\\beta$ that only act on the two boundary sites.  To observe any\ninteresting behavior (i.e. real interplay) between the two dynamics,\none needs competition. A prerequisite for the two processes to compete\nare comparable jump rates.  To ensure that the rates are of the same\norder independently of the system size, an appropriate scaling is\nneeded. To this end a $N$-independent global detachment rate\n$\\Omega_\\mathrm{D}$ is introduced, while the local rate per site\nscales as\n\\begin{equation}\n\\omega_\\mathrm{D}=\\frac{\\Omega_\\mathrm{D}}{N} \\;.\n\\label{e_scaling}\n\\end{equation}\nFor the attachment one proceeds similarly.  What does this scaling\nsignify physically? For an explanation it is instructive, to have a\nlook at the time scales involved: a particle on the lattice will\nperform a certain move on an average time scale, which is the inverse\nof that moves rate. Therefore, a particle spends an average time $\\tau\n\\approx 1\/\\omega_\\mathrm{D}$ on the lattice before it detaches.\nBearing in mind that the TASEP jump rate is set to unity, a particles\nwill jump to its adjacent site after a typical time of one unit time\nstep. Therefore the particle will travel a number $N_\\mathrm{T} =\n1\/\\omega_\\mathrm{D}$ of sites before leaving the lattice. Compared to\nthe lattice length this corresponds to a fraction of\n$n_\\mathrm{T}=N_\\mathrm{T}\/N=1\/(N \\omega_\\mathrm{D})$. In order to\nkeep this fraction finite in the thermodynamic limit,\n$\\omega_\\mathrm{D}$ needs to scale as defined in (\\ref{e_scaling}).\nOnly if the fraction $n_\\mathrm{T}$ is finite, a given particle can\nexperience interaction with other particles and give rise to\ncollective phenomena \\cite{pff04}. \n\n\\subsection{Mean Field Solution of TASEP\/LK}\n\nTo obtain density and current distributions of the TASEP\/LK, we use\nagain a mean-field approach and proceed along the lines of Sec.\n\\ref{s_mf}.\n\nFirst of all, we need to account for the Langmuir kinetics. This is\ndone by adding the following terms to the Heisenberg equation\n(\\ref{heisenberg}) of the simple TASEP to obtain\n\\begin{equation}\n\\frac{d}{dt}\\hat n_i(t) = \n\\hat n_{i-1}(t)(1-\\hat n_{i}(t))-\\hat n_{i}(t)(1-\\hat n_{i+1}(t))\n-\\omega_\\mathrm{D} \\hat n_i(t)+\\omega_\\mathrm{A} (1-\\hat n_i(t)) \\;.\n\\end{equation}\nwhere the first added term captures the detachment events and the\nlater the attachment.  Neglecting correlations as done before gives\nfor the stationary state\n\\begin{equation}\n0 = \\varrho_{i-1}(1-\\varrho_i)-\\varrho_i(1-\\varrho_{i+1})-\n    \\omega_\\mathrm{D} \\varrho_i + \\omega_\\mathrm{A} (1-\\varrho_i) \\;.\n\\label{e_steady_pff}\n\\end{equation}\nThe boundary conditions are not altered by LK. Using again the power\nseries expansion (\\ref{e_power}) and keeping in mind the scaling of\nthe on and off rates $\\omega$ as in (\\ref{e_scaling}), we obtain the\nfollowing ODE:\n\\begin{equation}\n\\frac{\\varepsilon}{2}\\partial_x^2 \\varrho + \\partial_x \\varrho (2\n\\varrho - 1) - \n\\omega_\\mathrm{D} \\varrho + \\omega_\\mathrm{A} (1-\\varrho)=0 \\;.\n\\label{e_ode_pff}\n\\end{equation}\nThe ratio $K=\\omega_\\mathrm{A}\/\\omega_\\mathrm{D}$ between the\nattachment and detachment rates will prove an important parameter in\nthe analysis of this differential equation. Since the case $K \\neq 1$\nis considerably complicated in its mathematical analysis we refer the\nreader to reference \\cite{pff04} and restrain our discussion to the\ncase $K=1$.  In the thermodynamic limit ($\\varepsilon \\to 0$) and with\n$\\omega_\\mathrm{D}=\\omega_\\mathrm{A}=\\Omega$ the ODE(\\ref{e_ode_pff})\nsimplifies to first order:\n\\begin{equation}\n(\\partial_x \\varrho - \\Omega)(2 \\varrho - 1)=0 \\;.\n\\label{e_ode_pff3}\n\\end{equation}\nObviously, there are two solutions to this general ODE problem. The\nhomogeneous density $\\varrho_\\mathrm{L}=1\/2$ given by the Langmuir\nisotherm and the linear slope $\\varrho(x)=\\Omega x + C$. The value of\nC is determined by the boundary conditions and leads to the two\nsolutions $\\varrho_\\mathrm{\\alpha}(x)=\\alpha + \\Omega x$ and\n$\\varrho_\\mathrm{\\beta}(x)=1-\\beta-\\Omega + \\Omega x$.  The complete\ndensity profile $\\varrho(x)$ is the combination of one or several of\nthe three densities above.  Depending on how they are matched, we\ndistinguish several phases as explained in the following.\n\n\\subsection{Phase Diagram and Density Distributions}\n\nThe only area in the phase diagram that does not change compared to\nsimple TASEP is the upper right quadrant. This is not surprising since\nthe maximal current phase is a bulk controlled regime anyway.\nTherefore the additional bulk dynamics with the Langmuir isotherm at\n$\\varrho_\\mathrm{L}=1\/2$ do not result in any changes in the density\ndistribution. In this case, non-equilibrium and equilibrium dynamics\ndo not compete but cooperate. \n\nAs in TASEP different solutions can be matched in various ways, the\nsimplest being the connection by a domain wall between the left\nsolution $\\varrho_\\mathrm{\\alpha}$ and the right solution\n$\\varrho_\\mathrm{\\beta}$.  Depending on the current distribution, two\npossibilities have to be distinguished. As both solutions are\nnon-homogeneous, the corresponding currents $j_\\mathrm{\\alpha}$ and\n$j_\\mathrm{\\beta}$ will be strictly monotonic (Fig. \\ref{f_pff_phase}\n(left)).  If the currents equal each other inside the system at a\nposition $x_\\mathrm{w}$, the DW is localized at this position, as a\ndisplacement to either side would result in a current inequality that\ndrives the DW back to $x_\\mathrm{w}$ (see Fig. \\ref{f_pff_phase}\n(left)). Ergo, the TASEP\/LK exhibits multi-phase existence of low and\nhigh density regions (LD-HD phase) in the stationary state on all\ntime-scales, opposed to TASEP where this behavior is only observed for\nshort observation times. Recently, this DW localization has been\nobserved experimentally \\cite{nosc}.\n\nIf the matching of left and right current is not possible inside the\nsystem, the known LD and HD phases are found. This is the case for one\nboundary condition being considerably larger than the other and is\nevidently depending quantitatively on the slope of the density\nsolutions. This slope is determined by the ratio of the TASEP step\nrate and the bulk interchange rate $\\Omega$. For large $\\Omega$ any\ndensity imposed by the boundary relaxes fast against the Langmuir\nisotherm of $\\varrho_\\mathrm{L}=1\/2$ resulting a steep slope of the\ndensity profile.\n\nThis fact allows for the existence of two other phases with\nmulti-regime coexistence. We could imagine a scenario in which the\nboundary imposed density solutions decay fast enough towards the\nisotherm to enable a three-regime coexistence of low density, maximal\ncurrent and high density (LD-MC-HD phase). Furthermore, a combination\nof a MC phase with a boundary layer on one side and a LD or HD region\nof finite extend at the other boundary can be imagined. Not all these\nphases will be realized for every value of $\\Omega$. Instead, the\nphase topology of two-dimensional cuts through the\n$\\alpha,\\beta,\\Omega$-phase space changes. An example is shown in Fig.\n\\ref{f_pff_phase} (right).\n\\begin{figure}\n\\centering\n\\includegraphics[height=7cm]{.\/pff_dens.eps}\n\\hspace{1 cm}%\n\\includegraphics[height=7cm]{.\/pff_phase.eps}\n\\caption{({\\bf left}) The DW connects the two densities $\\rho_\\alpha$ \n  and $\\rho_\\beta$ ({\\it both dashed}) and is localized at the point\n  where the correspondent currents $j_\\alpha$ and $j_\\beta$ match.\n  Note the finite extend of the DW (localization length) that is only\n  produced for Monte-Carlo simulations ({\\it solid wiggly line}) and\n  is not captured by mean-field results ({\\it solid line}).  ({\\bf\n    right}) Topological changes in the phase diagram of TASEP\/LK for\n  (a) $\\Omega=0.3$, (b) $\\Omega=0.5$, (c) $\\Omega=1$, from\n  \\cite{pff04}}\n\\label{f_pff_phase}\n\\end{figure}\n\n\n\\subsection{Domain Wall Theory}\n\\label{s_dw}\n\nAfter we have derived an analytical solution for the density profile\nbased on the mean-field differential equation (\\ref{e_ode_pff}), we\nnow have a closer look on the domain wall and its stochastic\nproperties.\n\n\nAs mentioned before, the density profile exhibits a discontinuity at\n$x_\\mathrm{w}$ that is actually a finite size continuous transition\nbetween the high and low density for systems of finite size.  Only\nupon increasing the system size $N \\to \\infty$ a sharp transition\nbetween the left and the right solution occurs. However, this is only\ndue to the fact that the lattice spacing decreases to $\\varepsilon \\to\n0^+$. So compared to the lattice length $L$, usually normalized to\n$1$, the domain wall is discontinuous, while on the length scale of\nlattice sites it will still have a finite extend. This extend is an\nintrinsic statistical feature and is usually referred to as\nlocalization length.\n\nThe domain wall in its random walk behavior can be either subjected to\nequal rates (unbiased) as in TASEP or the rates of movement to the\nleft\/right can be different (biased). In general, the rates will not\nonly be different, but also depend on the space variable.\nTo begin with, we will show a way how to derive these rates by taking\ninto account fluctuations of particle number\n\\cite{kolomeisky-etal:98,santen-appert:02}. Consider a situation where\nall events that can change the particle number ($\\alpha,\\beta,\\Omega$)\nhave a typical time scale that is considerably larger than that of\njump processes on the lattice. In this case, the time between any\nentry and exit events is so long, that the system has enough time to\n``rearrange'' (to reach a temporary steady state) in between . Then it\nis possible to identify jump rates $\\omega_\\mathrm{l}(x)$\n($\\omega_\\mathrm{r}(x)$) for DW movement to the left (right) with the\noverall rate for entry and exit of any particle at any site.\nSpecifically, if a particle enters the system, the DW is shifted to\nthe left by a distance of $\\approx\n\\varepsilon\/[\\varrho_\\mathrm{\\beta}(x_\\mathrm{w})-\\varrho_\\mathrm{\\alpha}(x_\\mathrm{w})]$.\nTherefore the rate for the DW to move one lattice site to the left is\n$\\omega_\\mathrm{l} =\n\\omega_\\mathrm{entry}\/[\\varrho_\\mathrm{\\beta}(x_\\mathrm{w})-\\varrho_\\mathrm{\\alpha}(x_\\mathrm{w})]$.\n\nIf the density distribution $\\varrho(x)$ is known analytically, it is\npossible to calculate $\\omega_\\mathrm{entry}$, the overall rate of\nparticle entrance, as the sum over all possible entrance events\n\\begin{equation} \\label{e_partfluct}\n\\omega_\\mathrm{entry}=\\alpha(1-\\alpha)+\\int_0^1 dx \\ \\omega_\\mathrm{A} (1-\\varrho(x)) \\;.\n\\end{equation}\nThe first term captures entrance events from the left boundary\nreservoir and the integral accounts for the Langmuir kinetics. The\nfirst multiplicand in both terms is the attempted rate of a jump,\nwhereas the difference in brackets states the probability of the\ndestination site to be vacant. Along the same lines, the exit rate is\ncomputed.  As we know the analytical density distribution as\n$\\varrho(x)=\\alpha+\\Omega x + \\Delta \\Theta(x-x_\\mathrm{w})$ with the\nHeavyside function $\\Theta$ and the DW height~\\footnote{The term \n  $-1 \\Omega$ accounts for the diminished height that is caused \n  by the Langmuir kinetics on the whole length $1$ of the system.} \n$\\Delta=1-\\alpha-\\beta-\\Omega$, we can execute the integrals\nto obtain\n\\begin{equation}\n\\omega_\\mathrm{entry}=\\alpha(1-\\alpha)+\\Omega(\\beta+\\frac{\\Omega}{2}) + x \\Omega \\Delta \\;.\n\\end{equation}\nKnowing these rates, one can complete the description of the domain\nwall as a random walker by calculating the position dependent jump\nrates to the left and right $\\omega_\\mathrm{l}(x)$ and\n$\\omega_\\mathrm{r}(x)$. The two rates constitute an effective\npotential displayed in Fig.~\\ref{f_potential}.\n\\begin{figure}\n  \\centering \\includegraphics[height=3.5cm]{.\/potential.eps}\n\\hspace{1 cm}%\n\\includegraphics[height=3.5cm]{.\/roger4.eps}\n\\caption{({\\bf left}) The DW will be localized at the point where the\n  jump rate to the left $\\omega_\\mathrm{l}$ and right\n  $\\omega_\\mathrm{r}$ intersect. This can be interpreted as an\n  potential $|\\Delta \\omega|$. ({\\bf right}) Variance of DW\n  probability distribution over $\\Omega$ for $\\alpha=\\beta=0.01$ and\n  $\\alpha=\\beta=0.1$ in a system of $N=500$, predictions according to\n  \\cite{evans-juhasz-santen:03} ({\\it solid}) and based on particle\n  number fluctuations ({\\it dashed}) compared to MC simulations ({\\it\n    dots})}\n\\label{f_potential}\n\\end{figure}\nThe DW will always be driven to that point $x_\\mathrm{S}$ in the\nsystem where\n$\\omega_\\mathrm{exit}(x_\\mathrm{S})=\\omega_\\mathrm{entry}(x_\\mathrm{S})$.\nThis position is the center of the DW probability density in a\nstochastic picture and the stationary DW position in a mean-field\npicture. It is in agreement with mean field results and yields\n\\begin{equation}\nx_\\mathrm{S}=\\frac{\\Omega-\\alpha+\\beta}{2\\Omega} \\;.\n\\end{equation}\nThe other quantity of interest which - in contrast to the DW position\n- cannot be determined by mean-field calculations is the localization\nlength. In order to compute this quantity, fluctuations have to be\ntaken into account as we will show in the following.  If we use the\nnotation $p(x)$ for the probability that the domain wall is at\nposition $x$, then the condition for a stationary DW reads in the\ncontinuum limit\n\\begin{equation}\n\\omega_\\mathrm{r}(x)p(x)=\\omega_\\mathrm{l}(x+\\varepsilon)p(x+\\varepsilon) \\;.\n\\label{e_stationary_dw}\n\\end{equation}\nIntroducing now $y(x)=\\omega_\\mathrm{l}(x)p(x)$ and approximate\n$y'(x)=|y(x+\\varepsilon)-y(x)|\/\\varepsilon$ we obtain:\n\\begin{equation}\ny'(x)+ N y(x) (1-\\frac{\\omega_\\mathrm{r}(x)}{\\omega_\\mathrm{l}(x)})=0 \\;.\n\\end{equation}\nThe solution then is given by\n\\begin{equation}\np(x)=\\frac{\\tilde p(x)}{Z} = \n\\frac{1}{Z \\omega_\\mathrm{l}(x)} exp[-N \\int_{x_0}^x dx' (1-\\frac{\\omega_\\mathrm{r}(x)}{\\omega_\\mathrm{l}(x)})] \\;.\n\\label{e_p_dw}\n\\end{equation}\nwhere $Z$ accounts for normalization. In general, $Z$ is not available\nexplicitly, but it has been shown \\cite{evans-juhasz-santen:03} that\nthe unnormalized probability function can be approximated by a\nGaussian\n\\begin{equation}\n\\tilde p(x) \\propto e^{-C(x-x_\\mathrm{S})^2} \\;,\n\\label{e_gaussian}\n\\end{equation}\nwhere $C$ is given by the second order derivative of the exponent in\n(\\ref{e_p_dw}) as\n\\begin{equation}\nC=\\frac{1}{2} \\frac{d^2}{dx^2}[N \\int_{x_0}^x dx' \\left(1-\\frac{\\omega_\\mathrm{r}(x)}{\\omega_\\mathrm{l}(x)}\\right)]=\n\\frac{N(\\omega_\\mathrm{l}-\\omega_\\mathrm{r})'(x_\\mathrm{S})}{2 \\omega_\\mathrm{l}(x_\\mathrm{S})} \\;.\n\\label{e_C}\n\\end{equation}\nHence, the variance $\\sigma=\\sqrt{1\/(2 C)}$ of the domain wall can be\neasily be obtained provided that the jump rates $\\omega_\\mathrm{l}(x)$\nand $\\omega_\\mathrm{r}(x)$ are available. Evans {\\it et al.}\n\\cite{evans-juhasz-santen:03} have assumed those rates to be\n$\\omega_\\mathrm{l,r}(x)=\nj_\\mathrm{\\alpha,\\beta}\/(\\varrho_\\mathrm{\\beta}-\\varrho{\\alpha})$.\nKouyos has shown \\cite{roger} that using the rates (\\ref{e_partfluct})\nderived from the fluctuations of particle number, one arrives at more\naccurate results compared to Monte Carlo simulations (see Fig.\n\\ref{f_potential}). In this case $C$ evaluates to\n\\begin{equation}\nC=\\frac{2 N \\Omega \\Delta}{\\alpha(1-\\alpha)+\\beta(1-\\beta)+\\Omega} \\;. \n\\end{equation}\nAs the width of the DW is given by $\\sigma=1\/\\sqrt{2C}$, the\nlocalization of the DW scales with $N^{-1\/2}$.\n\n\\section{Conclusions and Outlook} \n\\label{s_outlook}\n\nMuch in the same way as MacDonald's pioneering paper \\cite{macdonald} on\nmRNA translation, recent work on kinesin motors walking along\nmicrotubules has spurred progress in nonequilibrium transport\nphenomena. The ubiquitous exchange of material between the cytoplasm\nand the molecular track, which originally was thought to only lead to\nquantitative modifications of the dynamics \\cite{lkn01}, has recently\nbeen identified \\cite{pff03} as the source for qualitatively new phenomena such as\nphase separation. This introduced a completely new class of\nnon-equilibrium transport models.\n\nThese lattice gas models are characterized by a scaling of the on-off \nrates with system size which enables competition of driven motion \nand equilibrium Langmuir kinetics. Hereby, a finite residence time on the \nlattice ensures cooperative effects to establish multi-phase coexistence, \nlocalized shocks and an enriched phase behavior compared to prior TASEP \nresults.\n\nThere are now various routes along which one could proceed. The first\none is to add more realistic features of the molecular motors, such as\nthe fact that they are dimers \\cite{pierobon} or\nthat there is more than one chemo-mechanical state \\cite{nosc}. Such\ninvestigations are crucial for a quantitative understanding of\t\nintracellular traffic in various ways.  One might ask how robust the\nfeatures of minimal models are with respect to the addition of more\nmolecular details. In the case of dimers the answer is far from\nobvious since the non-equilibrium dynamics of dimer adsorption shows\nrich dynamic behavior with anomalously slow relaxation towards the\nequilibrium state \\cite{vilfan}. How this combines with the\ndriven transport along the molecular track was recently analyzed\nthoroughly \\cite{pierobon}.  While correlation effects due to the extended\nnature of dimers invalidate a simple mean-field picture it was found\nthat an extended mean-field scheme can be developed which\nquantitatively describes the stationary phases. Surprisingly, the\ntopology of the phase diagram and the nature of the phases is similar\nto the minimal model with monomers. The physical origin of this\nrobustness can be traced back to the form of the current-density\nrelation which exhibits only a single maximum.\n\nThe second line of research generalizing the minimal model \\cite{pff03} asks\nfor the effect of interactions, more than one molecular traffic\nlane, ``road blocks'' such as microtubule associated proteins and\nvarious other kinds of ``disorder'',\nbi-directional traffic, coupling of driven and diffusive transport \nand the like on the stationary density\nprofiles and the dynamics. In almost every instance it is found that\nthis leads to an even richer behavior with new phenomena emerging.\n\nFor TASEP it is known that isolated defects (slow sites) depending on\ntheir strength may either give rise to local density perturbations for\nlow particle densities or yield macroscopic effects for densities\nclose to the carrying capacity \\cite{tb98}.\nThe interplay between coupling to the motor reservoir in the\ncytoplasm and the fluctuations in the capacity limit due to disorder\nalong the track gives rise to a number of interesting collective\neffects \\cite{kouyos}.\n\nCoupling two lanes by allowing particle exchange at a constant rate\nalong the molecular track also results in novel phenomena. Similar to\nequilibrium phase transitions described by field theories with two\ncoupled order parameters higher order critical points may emerge\n\\cite{reichenbach}.\n\nCoupling diffusive and driven transport, the origin of new phenomena\nis due to a competition of different processes of comparable time\nscales. The qualitative failure of mean-field theory in some of these \nsystems \\cite{hinsch} comes as quite a surprise, since mean-field has proven to \npredict phase diagrams for large systems with an astonishing accuracy \nin the lattice gas models mentioned above.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nCompressed sensing has recently been intensively studied as a novel technique that aims to  undersample  sparse signals with low complexity \\cite{Candes05}. The nonadaptive undersmapling process can be simply formulated as a system of linear equations $y=Ax$, where $x\\in R^n$ is a $k$-sparse signal with at most $k$ nonzero elements, $A \\in R^{m\\times n}$ is a sensing matrix with $m\\ll n$, and $y\\in R^{m}$ is the linear observation. Given $y$ and $A$, the sparse signal $x$ can be recovered by solving the following $\\ell_1$-based minimization problem\n\\begin{equation}\n\\min ||\\hat{x}||_1 ~~ s.~t.~~  y=A\\hat{x}\n\\end{equation}\nif   the sensing matrix $A$ can satisfy  restricted isometry property (RIP) or null space property (NSP) \\cite{Tillmann12}. The solution to formula (1) can be perfectly derived with  linear programming (LP) \\cite{Boyd04}. Furthermore, for simpler computation,  a number of greedy solution algorithms are successively proposed with guaranteed performance \\cite{Pati93} \\cite{Daiwei09}.  Currently the explicit construction of sensing matrices well satisfying RIP or NSP is still a challenging task, since two conditions above are hard to be calculated or evaluated in practice. It is known that the random matrices with elements drawn from some general distributions, such as standard Gaussian distribution and Bernoulli distribution, can provide guaranteed sensing performance \\cite{Candes05} \\cite{Candes06near}. But they are computationally difficult to implement in practice. Then a few deterministic sensing matrices  are developed based on coding theory \\cite{Amini11} \\cite{DeVore07} \\cite {Calderbank10} \\cite{Liuxj13}. Specifically, the codes constructed over finite fields are collected as the columns of sensing matrices because they tend to suffer relatively small mutual correlations due to  relatively large distance\/difference between each other.  However, this method is not perfect. In practice, there is no  explicit way to collect an ensemble of codes with as large average distances as possible. In addition, the codes based on primitive polynomials usually cannot constitute   arbitrary sizes of sensing matrices.\n\n\nIt is known that the sensing performance is inversely proportional to the maximum correlation $\\mu$ between columns \\cite{Donoho11}, i.e. the sparsity $k$ is upper bounded with\n\\begin{equation}\nk<1\/2(\\mu^{-1}+1).\n\\end{equation}\nHence it is desirable if one could minimize the maximum correlations. This paper aims to approach this goal by constructing a family of deterministic matrices which is required to  have at most \\emph{one} common nonzero position between two arbitrary columns. Clearly the matrices defined above  take correlation values between \\emph{normalized} columns inversely proportional to the column degrees, namely the number of nonzero elements in each column. In this sense, the major task of this paper is to propose an explicit method to maximize the column degrees of the matrices defined above.\n\n By exploring the connection between bipartite graph and binary matrix, the construction of the desired matrices can be transformed into the construction of bipartite graphs with as many edges as possible but without cycles of length 4.  As a combinatorial optimization problem, in practice the desired bipartite graph is hard to be constructed. A greedy algorithm  initially proposed for LDPC codes and termed progressive edge-growth (PEG) \\cite{Huxiaoyu05}, seems suitable for the construction of the desired bipartite graph, since it aims to  build a bipartite graph edge-by-edge without introducing short cycles at each step.  However, this method  is  imperfect on both performance and  complexity. Specifically, it has to evaluate all possible distributions of matrix column degrees for building the underlying best bipartite graph. Apparently it is a NP-hard problem.  Further, the bipartite graph constructed with PEG cannot  guarantee to have no  cycles of length 4. Thus one has to calculate the cycles of each generated bipartite graph to exclude the case of cycles of length 4.\n\nTo overcome these problems, this paper proposes a novel bipartite graph-based construction (BGC) method specially designed to construct the desired sensing matrices described above. As  will be detailed latter, the proposed BGC method outperforms PEG algorithm on both performance and complexity.  In practice, the matrices constructed with BGC method  present comparable and even better phase transition curves over Gaussian random matrices.\n\n\n\nThe rest of this paper is organized as follows. In the next Section, after introducing the  basic connection between binary matrix and bipartite graph, we describe and analyze the proposed BGC method. In Section \\uppercase\\expandafter{\\romannumeral3}, the matrices constructed with BGC are evaluated in terms of  column degrees and phase transitions. Finally,  the paper is concluded in Section \\uppercase\\expandafter{\\romannumeral4}.\n\n\n\\section{Main Results}\nIn this section, we first associate binary matrix with bipartite graph, and then illustrate and analyze the proposed BGC method.\n\\subsection{Preliminaries on bipartite graph}\n  \\begin{figure}[ftp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[height=60mm,width=0.19\\textwidth]{bipartite_graph_2}&\\includegraphics[height=60mm,width=0.21\\textwidth]{sub_graph}\\\\\n(a)&(b)\n\\end{tabular}\n\\caption{ (a) An example of binary matrix being associated with bipartite\ngraph. The bipartite graph has variable nodes and check nodes\ndenoted by circles and squares, respectively. (b)\nThe form of subgraph expanded from a variable node. }\n\\end{figure}\n The bipartite graph consists of two disjoint sets of vertices and a few edges having both endpoints  from above two sets. Two sets of vertices  are typically called variable nodes and check nodes.   A  \\emph{path} is defined by a sequence of vertices with two arbitrary adjacent vertices connected with an edge. A closed path starting and ending at the same vertex is called a \\emph{cycle}. The length of a cycle is measured with the number of edges included in the cycle. Obviously the cycle length of a bipartite graph should be an even number not less than 4. The  length of the shortest cycle is called the \\emph{girth} of bipartite graph.\n\n As the example shown in Figure 1(a), a sensing matrix can be  associated with a  bipartite graph  by relating columns and rows  to variable nodes and check nodes, respectively. The edges of bipartite graph are determined by the nonzero positions of sensing matrix. Hence the desired bipartite graph should have as many edges as possible. Note that  the bipartite graph will hold  cycles of length 4 through two variable nodes, if  the associated sensing matrix share two same nonzero positions between some columns. So to derive the desired sensing matrices with at most one same nonzero positions between columns, the desired bipartite graph also has to render a girth larger than 4.\n\n From each variable  node, as shown in Figure 1(b), a subgraph possibly with multiple floors can be generated by recursively traversing all reachable nodes through edges. Interestingly, for a given subgraph, if a check node outside the subgraph is added and connected to the root variable node, no cycles will be generated. Otherwise, a cycle of length 4 will occur if  a check node in  floor-2 is selected; likewise, a cycle of length $2i$ will appear if  a check node in  floor-$i$ is introduced.   It is thus  inevitable that the bipartite graph will have more and more shorter cycles as the edge number increases. Hence the desired bipartite graph with  as many edges as possible should have all cycles of length 6.\n\n\n\n\n\n\n\\subsection{Bipartite graph based construction (BGC)  method}\n\n As stated before, the desired bipartite graph should have all cycles of length 6. The proposed BGC method is thus developed to generate such kind of graphs with as many edges as possible. Similarly to PEG algorithm, BGC method proceeds in the way of expanding an underlying subgraph edge-by-edge. But their principles of selecting each edge are different. In each step of edge selection, BGC method prefers introducing a cycle of length 6 rather than no cycles while PEG algorithm attempts to avoid introducing cycles or short cycles. More importantly, unlike PEG, BGC does not need to enumerate all possible distributions of column degrees to seek the underlying best sensing matrix.\n\n The BGC method can be implemented with multiple iterations. In each iteration,  each variable node is allowed to connect at most one check node to its current  subgraph. Specifically, if the subgraph can reach floor-3 with a set of check nodes, then a check node of this floor is randomly selected to generate a cycle of length 6; otherwise,  if  current subgraph does not include all check nodes,  a check node outside of current subgraph is randomly chosen to avoid the cycle of length 4. The  procedure above is repeated  until no variable nodes have check nodes to update. Then the desired bipartite graph with as many edges as possible is derived with the following two special features:\n  \\begin{itemize}\n  \\item all existing cycles have length of 6, and  any further added edge would lead to  shorter cycles of length 4;\n   \\item any variable node can spread to be a subgraph with  two and only two floors  containing all check nodes.\n  \\end{itemize}\nTwo features above implies that the BGC method indeed proposes the maximum column degrees that it can greedily approach. For clarity, the proposed BGC method is sketched in  Algorithm 1.\n\nNote that to obtain relatively uniform  column degrees, the variable nodes are updated in a random sequence in each iteration. Likewise, the candidate check nodes are also randomly selected for relatively uniform row degrees. In fact, as in \\cite{Huxiaoyu05}, the row degrees can be further evened out by randomly selecting the candidate check nodes with minimum number of adjacent variable nodes. However, it is not applied in the proposed method, since in practice  it  does not lead to noticeable advances on maximizing the number of edges.\n\n\\begin{algorithm}[htp]\n\\caption{ BGC  method}\n\\label{alg:Framwork}\n\\textbf{Initializations}:  Let $\\mathcal{C}$ and $\\mathcal{V}$  refer to the set of $m$ check nodes and  the set of $n$ variable nodes, respectively. $\\mathcal{I} $ is defined as the set of variable nodes still to be updated in current bipartite graph, which is initialized as $\\mathcal{I}=\\mathcal{V}$. $\\mathcal{I}_i$  indicates the $i$-th element of $\\mathcal{I} $. Searching for new edges will stop if $\\mathcal{I} $ turns to be empty.  Three subsets $\\mathcal{C}_i$ with subscript $1\\leq i\\leq 3$ are further defined to contain the check nodes appearing on the $i$-th floor of  subgraph. If the subgraph has no check nodes on  floor-$i$, the corresponding $\\mathcal{C}_i=\\varnothing$. The selected edges are collected in the set $\\mathcal{E}$ which is initialized as $\\varnothing$.\n\\begin{algorithmic}[1]\n\\FOR {$i$=1 to $m$}\n\\IF{$\\mathcal{I}=\\varnothing $ }\n\\STATE break; \/\/ terminate the program and output the edge set $\\mathcal{E}$;\n\\ENDIF\n\\FOR{$j$=1 to $|\\mathcal{I}|$}\n\\STATE Try to expand a subgraph from variable node $\\mathcal{I}_j$ to  floor-3 with current edge set $\\mathcal{E}$; and the the check nodes on the $k$-th floor are collected in the empty-initialized set $\\mathcal{C}_k$, $1\\leq k\\leq 3$;\n\\IF{$\\mathcal{C}_3=\\varnothing$ }\n\\IF{$\\mathcal{C}\\backslash\\{\\mathcal{C}_1\\bigcup\\mathcal{C}_2\\}\\neq \\varnothing$}\n\\STATE Introduce a new edge $(\\mathcal{I}_j, c)$ to the edge set $\\mathcal{E}$ by $\\mathcal{E}=\\mathcal{E}\\bigcup (\\mathcal{I}_j, c)$, where $c$ is a check node randomly selected from the set $\\mathcal{C}\\backslash\\{\\mathcal{C}_1\\bigcup\\mathcal{C}_2\\}$;\n\\ELSE\n\\STATE Exclude the variable node  $\\mathcal{I}_j$ from  $\\mathcal{I} $, namely $\\mathcal{I}=\\mathcal{I}\\backslash\\mathcal{I}_j$;\n\\ENDIF\n\\ENDIF\n\\IF{$\\mathcal{C}_3\\neq\\varnothing$}\n\\STATE Introduce a new edge $(\\mathcal{I}_j, c)$ to the edge set $\\mathcal{E}$ by $\\mathcal{E}=\\mathcal{E}\\bigcup (\\mathcal{I}_j, c)$, where $c$ is a check node randomly selected from the set $\\mathcal{C}_3$;\n\\ENDIF\n\\ENDFOR\n\\ENDFOR\n\\end{algorithmic}\n\n\\end{algorithm}\n At the end of this subsection, three crucial problems concerning the practical implementation of BGC method are discussed as follows.\n\n\\subsubsection{Performance and complexity} the proposed BGC method outperforms PEG algorithm on both performance and complexity. Precisely, in performance, BGC method can provide a distribution of larger average column degrees as shown in the following Table \\uppercase\\expandafter{\\romannumeral1}.  As for complexity, BGC shows two obvious advantages. First,  it only needs to be given the size of target matrix, while besides matrix size, PEG can proceed only with a given distribution of column degrees. In other words, theoretically PEG has to enumerate all possible distributions of column degrees to find the underlying best matrix. Second,  PEG algorithm cannot avoid  the case of girth equal to 4, and so additional computation has to be introduced to evaluate the girth of each matrix constructed with PEG; in contrast, this case does not happen in BGC method. Secondly, during the spreading of each subgraph, BGC needs to spread  at most 3 floors, while PEG has to expand  as far as possible.\n\\vspace{3pt}\n\\subsubsection{Bounds for column degrees} \\emph{i) upper bound}: the maximum edge number of bipartite graph with girth larger than 4 has been early studied as a combinatorial problem  in \\cite{Naor2005} and related references therein. However, empirically these asymptotic results  are far away from the real  values that BGC or PEG can achieve. Here we provide a more accurate method to tackle this problem.  Considering an ideal case where the generated matrix is regular, i.e. both column degrees and row degrees are uniform, its uniform column degree $d$ can be approximately estimated with\n    \\begin{equation}\n    \\mathop{\\text{argmin}}\\limits_d ~\\{|d+d(dn\/m-1)(d-1)-m|\\},\n    \\end{equation}\n     since we can  regard $\n    d+d(dn\/m-1)(d-1)= m\n    $ from the fact that each subgraph finally spread two floors in which all check nodes are included. Note that theoretically it cannot be ensured that the regular  matrix described above really exists. For instance, the solution $d$ to formula (3) is usually not an integer provided a pair of $(m,n)$.    But compared to  pervious impractical  theoretical estimation \\cite{Naor2005}, it is  much closer to the real values as shown in Table \\uppercase\\expandafter{\\romannumeral1}. \\emph{ ii) lower bound}: it is necessary to point out that the matrices constructed with BGC probably have some columns of degrees equal to 1, when the column size $n$ is very small or the compression rates $n\/m$ is considerably large. Obviously for overcomplete sensing matrices, it is necessary to render most column  degrees larger than 1  to avoid producing same columns. In practice, as  shown in the simulations, this condition can be well satisfied as $n$ or $m\/n$ increases. \\vspace{3pt}\n\\subsubsection {Binary and Ternary matrices} with given bipartite graph, in practice we can construct two types of matrices: deterministic binary matrix and random ternary matrix by taking the nonzero values as  1 and  random binary values $\\pm 1$, respectively. Both of them share the same distribution on the magnitudes of column correlation. But the same performance cannot be assured in practice. Recall that, empirically, the sensing performance is  sensitive to the sign of sparse signals \\cite{Donoho10112009}.  It is thus reasonable to conjecture that  the \\emph{unsigned} binary matrix and the \\emph{signed} ternary matrix also probably differ in performance.  This conjecture is confirmed with the following simulations.\n\n\n\n\n\n\\section{Simulations}\n\n\\begin{table*}[htp]\n\\caption{For sensing matrices with diverse sizes, the \\emph{average} column degrees  provided by BGC are compared with the maximum\\emph{ uniform} column degrees achieved by PEG. The theoretical estimation derived with formula (3), here denoted as 'Th', is illustrated as well.}\n\\centering\n\\renewcommand\\arraystretch{1.0}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\n\\hline\n\\multicolumn{3}{|c|}{m\/n}&0.1 &0.2&0.3 &0.4&0.5 &0.6& 0.7&0.8 &0.9&1\\\\\n\\hline\\hline\n\n \\multirow{9}{*}{n}&\\multirow{3}{*}{\\rotatebox{90}{100}}& BGC& \\textbf{1.45}& \\textbf{ 2.37}& \\textbf{3.2}&\\textbf{3.99}&\\textbf{4.68}& \\textbf{5.27}& \\textbf{5.93}& \\textbf{6.55}& \\textbf{7.21}&\\textbf{7.69}\\\\\n&& PEG&  1&   2&3&  3& 4& 5& 5&6& 7& 7\\\\\n&&Th&1.44&2.00&2.50&2.96&3.39&3.79&4.18&4.56&4.90&5.25\\\\\n\n\\cline{2-13}\n&\\multirow{3}{*}{\\rotatebox{90}{500}}& BGC& \\textbf{2.68}&  \\textbf{4.45}& \\textbf{5.99}&\\textbf{7.40}& \\textbf{8.67}& \\textbf{9.86}& \\textbf{10.99}& \\textbf{12.08}& \\textbf{13.07}&\\textbf{14.21}\\\\\n&& PEG&  2&   4&5&  7& 8& 9& 10& 11& 12& 13\\\\\n&&Th&2.12&3.12&3.98&4.76&5.48&6.16&6.80&7.41&8.00&8.57\\\\\n\n\n\n\n\\cline{2-13}\n\n & \\multirow{3}{*}{\\rotatebox{90}{1000}}& BGC& \\textbf{3.48}&  \\textbf{5.82}&7\\textbf{.80} &\\textbf{9.59}&\\textbf{11.26} &\\textbf{12.78}& \\textbf{14.36}& \\textbf{15.75}&\\textbf{17.06} &\\textbf{18.45}\\\\\n&& PEG&  3&   5&7 & 9& 11 &12& 14 &15& 16& 17\\\\\n&&Th&2.55&3.83&4.91&5.88&6.78&7.63&8.43&9.20&9.93 &10.64\\\\\n\n\\hline\n\n\\end{tabular}\n\\end{table*}\n\n\\begin{figure}[t]\n\\centering\n\\begin{tabular}{c}\n\\includegraphics[width=0.42\\textwidth]{input_pm1_slight}\\\\ (a) \\\\ \\includegraphics[width=0.42\\textwidth]{input_1_slight}\\\\ (b)\n\n\\end{tabular}\n\\caption{The phase transitions of binary matrices and ternary matrices constructed by BGC, as well as Gaussian random matrix. The sparse signals have nonzero elements  being $\\pm1$ equiprobably in (a), and being $ 1$ in (b).}\n\\end{figure}\nThis section first compares the maximum column degrees achieved by  BGC and  PEG, and then evaluates  the phase transitions  of  matrices constructed with BGC.\n\\subsection{Maximum column degrees}\n\nThe maximum column degrees achieved by BGC and PEG are compared in Table \\uppercase\\expandafter{\\romannumeral1}. Simultaneously, the theoretical estimation derived with formula (3) is also illustrated.  Apparently BGC obtains larger average column degrees  over PEG. The requirement of avoiding  column degrees equal to 1 is also well satisfied as $n $ or $m\/n$ increases. Note that PEG is only used to seek the desired matrices of uniform column degrees, because  we cannot enumerate all possible distributions of column degrees.  In practice PEG tends to provide larger average column degrees if the column degrees are distributed non-uniformly. For instance, for the case of $n=500$ and $m\/n=0.1$ in Table \\uppercase\\expandafter{\\romannumeral1}, PEG in fact can achieve an average column degree 2.5 by having  half of the columns with degree 2, and  the rest columns with degree 3. Even so,  PEG can hardly  achieve the performance level of BGC, since we do not know the better  distribution of column degrees  derived with BGC.  Moreover, it can be observed that the practical matrices constructed with PEG or BGC  outperform the theoretical estimation determined with  formula (3). This estimation error can be partially explained by the fact that, the theoretical estimation is derived under the ideal assumption that both row degrees and column degrees are uniform, while the condition above is not  followed by PEG or BGC. In particular,   BGC has no requirement for uniform column or row degrees, and PEG here is exploited to merely produce uniform column degrees.\n\n\n\n\\subsection{Phase transitions}\n\n\n\nThe phase transitions \\cite{Donoho10} are illustrated in Figure 2. Besides the binary matrices and ternary matrices constructed with BGC, Gaussian random matrix are evaluated as well. These matrices are of column size $n=300$. The curves are depicted by the maximum nonzero number $k $ allowed by  the correct LP decoding rates larger than $99\\%$.  The decoding rates are measured with $||\\hat{x}-x||_2\/||x||_2$.  Note that  $\\delta=m\/n$ ranges discretely  from 0.1 to 1 in nine equal steps. Each point is derived from 1000 simulations.  In each simulation, the ternary matrix has nonzero elements randomly taking $\\pm1$ signs. Similarly to \\cite{Donoho10112009}, here we test both \\emph{signed} and \\emph{unsigned} sparse signals with nonzero elements i.i.d drawn from the sets  $\\{ \\pm1\\}$ and $\\{ 1\\}$, respectively.\n\nDue to the low resolution of Figure 2, it is necessary to point out that the as $\\delta$ increases, the performance order is Ternary matrix>Gaussian matrix>Binary matrix in Figure 2(a), and Binary matrix>Ternary matrix>Gaussian matrix in Figure 2(b). In terms of the average performance, obviously the deterministic binary matrix is more attractive in practice.\n\n\n\n\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{Conclusion}\n\nThis paper has proposed a bipartite graph based construction method, called BGC method to deterministically construct binary or ternary sensing matrices with  column correlations as small as possible. This method is developed by equivalently regarding the desired matrix as a bipartite graph with as many edges as possible but without short cycles of length 4. The matrices constructed with BGC show comparable and even better performance over Gaussian random matrices. Note that, in practice the PEG algorithm initially proposed for the construction of LDPC codes can also be applied to construct such kind of graphs. But this method can hardly seek the underlying best distribution of column degrees unless we could enumerate all possible distributions. The proposed BGC method not only successfully works out this problem but significantly reduces computation complexity. In the future, it can be easily adapted to generate more hardware-friendly  matrices of quasi-cyclic structure.\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Some spaces of mod 2 modular forms of level \\bm{$\\Gamma_{0}(5)$}}\n\\label{section1}\n\nNicolas and Serre \\cite{3}, \\cite{4} have proved various results about the action of the Hecke algebra on the space of mod~2 modular forms of level 1. In \\cite{1} we gave a variant of their results in level $\\Gamma_{0}(3)$.  Here we find close analogues in level $\\Gamma_{0}(5)$.\n\nWe first summarize results from \\cite{3}, \\cite{4} just as we did at the start of \\cite{1}.  There are commuting formal Hecke operations $T_{p}:Z\/2[[x]]\\rightarrow Z\/2[[x]]$, one for each odd prime $p$. Here $T_{p}(\\sum c_{n}x^{n}) = \\sum c_{pn}x^{n} + \\sum c_{n}x^{pn}$. Let $F$ in $Z\/2[[x]]$ be $\\sum_{n \\mathit{odd},\\ n>0}x^{n^{2}}$. Using modular forms of level 1, Nicolas and Serre show that the $T_{p}$ stabilize the space spanned by $F, F^{3}, F^{5}, F^{7}, \\ldots $ and that the associated (completed) Hecke algebra is a power series ring in $T_{3}$ and $T_{5}$. Indeed they make the space into a faithful $Z\/2[[X,Y]]$-module with $X$ and $Y$ acting by $T_{3}$ and $T_{5}$, and show that each $T_{p}$ is multiplication by an element of the maximal ideal $(X,Y)$.\n\nIn \\cite{1} we took $D$ in $Z\/2[[x]]$ to be $\\sum_{(n,6)=1,\\ n>0}x^{n^{2}}$, so that $D=F(x)+F(x^{9})$. $W1$ had a basis consisting of the $D^{k}$ with $k\\equiv 1\\mod{6}$ and $W5$ a basis consisting of the $D^{k}$, $k\\equiv 5\\mod{6}$; $W$ was $W1\\oplus W5$. We showed that the $T_{p}$ with $p\\equiv 1\\mod{6}$ stabilize $W1$ and $W5$, while when $p\\equiv 5\\mod{6}$, $T_{p}(W1)\\subset W5$ and $T_{p}(W5)\\subset W1$.  We further showed that $W1$ has a basis $m_{i,j}$ ``adapted to $T_{7}$ and $T_{13}$'' with $m_{0,0}=D$. We deduced that the (completed shallow) Hecke algebra attached to $W$ is a power series ring in $T_{7}$ and $T_{13}$ with an element of square $0$ adjoined. Though $W$ is a space of mod~2 modular forms of level $\\Gamma_{0}(9)$, it identifies as Hecke-module with a certain subquotient of the space of odd mod~2 modular forms of level $\\Gamma_{0}(3)$.\n\nIn the present work we change notation. Now $D=\\sum_{(n,10)=1,\\ n>0}x^{n^{2}}$, so that $D=F(x)+F(x^{25})$. Let $G=F(x^{5})$. For $k>0$ with $(k,10)=1$ define $D_{k}$ as follows: $D_{1}=D$, $D_{3}=D^{8}\/G$, $D_{7}=D^{2}G$ and $D_{9}=D^{4}G$, while $D_{k+10}=G^{2}D_{k}$. Let $W$ be spanned by the $D_{k}$. (As $D_{k}=x^{k}+\\cdots $, these are linearly independent over $Z\/2$.) Then $W=W_{a}\\oplus W_{b}$ where the $D_{k}$, $k\\equiv 1, 3, 7, 9\\mod{20}$, are a basis of $W_{a}$, and the $D_{k}$, $k\\equiv 11, 13, 17, 19\\mod{20}$ are a basis of $W_{b}$. We will establish the following analogues to the results of \\cite{1}.\n\n\\begin{enumerate}\n\\item[(1)]\nThe $T_{p}$, $p\\ne 5$, stabilize $W$. If $p\\equiv 1, 3, 7, 9\\mod{20}$, $T_{p}$ stabilizes $W_{a}$ and $W_{b}$, while if $p\\equiv 11, 13, 17, 19\\mod{20}$, $T_{p}(W_{a})\\subset W_{b}$ and $T_{p}(W_{b})\\subset W_{a}$.\n\\item[(2)]\nThough $W$ is a space of mod~2 modular forms of level $\\Gamma_{0}(25)$, it identifies as Hecke-module with a certain subquotient of the space of odd mod~2 modular forms of level $\\Gamma_{0}(5)$.\n\\item[(3)]\n$W_{a}$ admits a basis $m_{i,j}$ adapted to $T_{3}$ and $T_{7}$ in the sense of Nicolas and Serre, with $m_{0,0}=D$. \n\\item[(4)]\nThe (completed shallow) Hecke algebra attached to $W$ is a power series ring in $T_{3}$ and $T_{7}$ with an element of square $0$ adjoined.\n\\end{enumerate}\n\nThe proofs of (1) and (2) occupy sections \\ref{section1} and \\ref{section2}. Some ingredients are the mod~2 level~5 modular equation $(F+G)^{6}=FG$ of Theorem \\ref{theorem1.12} and the relation $D^{15}+G^{4}D^{3}+G^{3}= 0$ of Lemma \\ref{lemma2.4}.\n\nThe proofs of (3) and (4) resemble those of corresponding results in \\cite{1}. We identify $W_{a}$ with a  subspace $V^{\\prime}$ of the polynomial ring $Z\/2[w]$, making $D_{k}$ correspond to $w^{k}$, and show that under this identification $T_{3}(D_{k})$ corresponds to a certain $P_{k}$ defined in \\cite{2}, with $P_{k+80}=w^{80}P_{k}+w^{20}P_{k+20}$. Lemma 5.5 of \\cite{2}, which deals with this recursion, gives insight into the action of $T_{3}$. Suppose in particular that $q$ is a power of 2, and let $W_{a}(q)$ be spanned by the $D_{k}$ in $W_{a}$ with $k<40q^{2}$. Using this Lemma 5.5 we find that the kernel of $T_{3}:W_{a}(q)\\rightarrow W_{a}(q)$ has dimension at most $2q$. Ideal theory in $Z[\\sqrt{-10}]$, developed in section \\ref{section4}, then shows that this kernel is a space $DI(q)$ of theta-series attached to binary quadratic forms. A study of the action of $T_{7}$ on $DI(q)$, together with formalism from \\cite{1}, leads, in section \\ref{section5}, to a proof of (3). And the further study in section \\ref{section6} of $T_{11}:W_{a}\\rightarrow W_{b}$ and $W_{b}\\rightarrow W_{a}$ gives a proof of (4).\n\nWe begin our proofs by introducing elements $P$,$E_{4}$, and $B$ of $\\ensuremath{\\mathbb{C}} [[x]]$. These are the expansions at infinity of classical modular forms of level $\\Gamma_{0}(5)$.\n\n\\begin{definition}\n\\label{def1.1}\n$P=1+\\cdots$, $E_{4}=1+\\cdots$ and $B=x+\\cdots$ are the expansions at infinity of:\n\\vspace{-2ex}\n\\begin{enumerate}\n\\item[(1)] The normalized weight~2 Eisenstein series for $\\Gamma_{0}(5)$.\n\\item[(2)] The normalized weight~4 Eisenstein series of level~1.\n\\item[(3)] The normalized weight~4 Eisenstein series, vanishing at infinity, for $\\Gamma_{0}(5)$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark*}{Remark}\nOne usually views expansions at infinity of modular forms as elements of $\\ensuremath{\\mathbb{C}} [[q]]$ where $q=\\me^{2\\pi iz}$. But as in \\cite{1} we'll use the letter $x$ rather than the letter $q$.\n\\end{remark*}\n\n\\begin{definition}\n\\label{def1.2}\n$r$ in $Z\/2[[x]]$ is $\\sum_{n>0}(x^{n^{2}}+x^{2n^{2}}+x^{5n^{2}}+x^{10n^{2}})$.\n\\end{definition}\n\nClassical formul\\ae\\ for the coefficients of Eisenstein series show that $P$, $E_{4}$ and $B$ are in $Z[[x]]$ with mod~2 reductions $1$, $1$ and $r$.\n\n\\begin{definition}\n\\label{def1.3}\n$C$ in $\\ensuremath{\\mathbb{C}}[[x]]$ is the expansion at infinity of the normalized weight~4 cusp form $(\\eta(z)\\eta(5z))^{4}$ for  $\\Gamma_{0}(5)$.\n\\end{definition}\n\nNow the expansion of $\\eta(z)$ at infinity is $x^{1\/24}$(an element $1-x-x^{2}+\\cdots$ of $Z[[x]]$). We deduce:\n\n\\begin{lemma}\n\\label{lemma1.4}\n$C$ is in $Z[[x]]$ and is $x-4x^{2}+\\cdots$.\n\\end{lemma}\n\nWe now show that the mod~2 reduction $\\bar{C}$ of $C$ is $r^{2}+r$.\n\n\\begin{lemma}\n\\label{lemma1.5}\nLet $n$ be a positive integer. The number of $(a,b)$ in $Z\\times Z$ with $a^{2}+5b^{2}=6n$ and $a\\equiv b\\mod{3}$ is $2$~mod~$4$ if $n$ is either a square or $5$(square), and $0$~mod~$4$ otherwise.\n\\end{lemma}\n\n\\begin{proof}\nLet $S(n)$ be the set of such $(a,b)$. Then $T:(a,b)\\rightarrow (-a,-b)$ and $U:(a,b)\\rightarrow \\left(\\frac{2a-5b}{3},\\frac{-a-2b}{3}\\right)$ are commuting involutions of $S(a)$. $T$ has no fixed points, while the fixed points of $U\\ (\\mbox{resp.}\\ TU)$ are of the form $(5k,-k)\\ (\\mbox{resp.}\\ (k,k))$. In the first case, $5k^{2}=n$ while in the second $k^{2}=n$. So we have an action of $Z\/2\\times Z\/2$ on $S(n)$ in which all orbits are of size 4 with the following exceptions. When $n=5k^{2}$ there is a size 2 orbit $\\pm (5k,-k)$. Where $n=k^{2}$ there is a size 2 orbit $\\pm (k,k)$. The result follows.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma1.6}\nLet $n$ be an integer. The number of $(a,b)$ in $Z\\times Z$ with $a^{2}+5b^{2}=6n$ and $a\\equiv b\\equiv 1\\mod{6}$ is odd if $n$ is an odd square or $5\\cdot$(an odd square), and even otherwise.\n\\end{lemma}\n\n\\begin{proof}\nWe may assume $n\\ge 0$. Since $a^{2}+5b^{2}$ is $2$~mod~$4$, there are no such pairs when $n$ is even. For fixed odd $n>0$ the pairs of Lemma \\ref{lemma1.5} come in three types according as $a\\equiv b\\equiv 1\\mod{6}$, $a\\equiv b\\equiv 5\\mod{6}$, or $a\\equiv b\\equiv 3\\mod{6}$.  There are as many pairs of the first type as there are of the second. So in view of Lemma \\ref{lemma1.5} it suffices to show that the number of pairs of the third type is a multiple of 4.  But these pairs come in sets of 4, $(\\pm a, \\pm b)$.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma1.7}\nThe mod~2 reduction $\\bar{C}$ of $C$ is $r^{2}+r$.\n\\end{lemma}\n\n\\begin{proof}\nUsing the familiar expansion of $\\eta(z)$ at infinity we find that $\\bar{C}=\\left(\\sum_{a\\equiv 1\\mod{6}}x^{a^{2}\/6}\\right)\\left(\\sum_{b\\equiv 1\\mod{6}}x^{5b^{2}\/6}\\right)$. The coefficient of $x^{n}$ in this element of $Z\/2[[x]]$ is the mod~2 reduction of the number of $(a,b)$ in $Z\\times Z$ with $a^{2}+5b^{2}=6n$ and $a\\equiv b\\equiv 1\\mod{6}$. So by Lemma \\ref{lemma1.6}, $\\bar{C}=\\sum_{n\\ \\mathit{odd},\\ n>0}(x^{n^{2}}+x^{5n^{2}})$. This is precisely $r^{2}+r$.\n\\qed\n\\end{proof}\n\n\\begin{definition}\n\\label{def1.8}\nIf $k\\ge 0$ and even, $M_{k}$ consists of those $f$ in $Z\/2[[x]]$ for which there is a weight $k$ modular form of level $\\Gamma_{0}(5)$ whose expansion at infinity lies in $Z[[x]]$ and reduces to $f$.\n\\end{definition}\n\nUsing multiplication by $E_{4}$ we see that $M_{0}\\subset M_{4}\\subset M_{8}\\subset \\ldots$\n\n\\begin{definition}\n\\label{def1.9}\n$M=\\cup M_{4m}$ is ``the space of mod~2 modular forms of level $\\Gamma_{0}(5)$.'' $\\ensuremath{M(\\mathit{odd})}$ consists of the odd elements of $M$, i.e.\\ those elements lying in $x\\cdot Z\/2[[x^{2}]]$.\n\\end{definition}\n\nNote that $M$ is a subring of $Z\/2[[x]]$. Since the reductions of $E_{4}$ and $B$ ane $1$ and $r$, $M\\supset Z\/2[r]$\/ Since $r^{2}+r=\\sum_{n\\ \\mathit{odd},\\ n>0}(x^{n^{2}}+x^{5n^{2}})$ is odd, each $r^{2k}(r^{2}+r)$ is in $\\ensuremath{M(\\mathit{odd})}$.\n\n\\begin{theorem}\n\\label{theorem1.10}\nFix $m\\ge 0$ and suppose $0\\le i \\le 2m$. Then there is a weight $4m$ modular form $u_{i}$ of level $\\Gamma_{0}(5)$ whose expansion at infinity has the following properties. It is $x^{i}+\\ldots$, lies in $Z[[x]]$, and reduces to $r^{i}$.\n\\end{theorem}\n\n\\begin{proof}\nIt suffices to prove this when $m=1$, so the weight is $4$. For $i=0$ we take $P^{2}$ whose expansion reduces to $1$, while for $i=1$ we take $B$ whose expansion reduces to $r$. Now $E_{4}=1+240x+2160x^{2}+\\cdots$, $P^{2}=(1 +6x+18x^{2}+\\cdots)^{2}$ and $B=x+9x^{2}+\\cdots$ are expansions at infinity of weight $4$ forms of level $\\Gamma_{0}(5)$, and $E_{4}-P^{2}-228B=36x^{2}+\\cdots$. Also $B-C=13x^{2}+\\cdots$, so a $Z$-linear combination of these last two expansions such as $4(\\mathit{first})-11(\\mathit{second})$ is $x^{2}+\\cdots$. And the mod~2 reduction of this linear combination is $\\bar{B}-\\bar{C}$ which is $r^{2}$ by Lemma \\ref{lemma1.7}.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem1.11}\nThe $r^{i}$, $0\\le i \\le 2m$, are a basis of $M_{4m}$ over $Z\/2$. It follows that $M=Z\/2[r]$. Furthermore the $r^{2k}(r^{2}+r)$ are a basis of $\\ensuremath{M(\\mathit{odd})}$ over $Z\/2$.\n\\end{theorem}\n\n\\begin{proof}\nTheorem \\ref{theorem1.10} shows that the $r^{i}$, $0\\le i \\le 2m$, lie in $M_{4m}$. Now the $u_{i}$, $0\\le i \\le 2m$, of Theorem \\ref{theorem1.10} are linearly independent over $\\ensuremath{\\mathbb{C}}$. Classical dimension formul\\ae\\ then show that over $\\ensuremath{\\mathbb{C}}$ they span the space of weight $4m$ modular forms of level $\\Gamma_{0}(5)$. Suppose $u$ lies in this space and that the expansion of $u$ at infinity lies in $Z[[x]]$. Writing $u$ as a $\\ensuremath{\\mathbb{C}}$-linear combination of the $u_{i}$ and examining the expansions we see that $u$ is a $Z$-linear combination of the $u_{i}$. So the reduction of $u$ is a $Z\/2$-linear combination of $r^{i}$, $0\\le i \\le 2m$, giving the first two results. It follows also that the $r^{2i}$ with $0\\le i \\le m$ together with the $r^{2i}(r^{2}+r)$ with $0\\le i \\le m-1$ are a $Z\/2$-basis of $M_{4m}$. We conclude that the $r^{2i}(r^{2}+r)$ with $0\\le i \\le m-1$ are a basis for the subspace of $M_{4m}$ consisting of odd power series.\n\\qed\n\\end{proof}\n\nRecall now that $F$ in $Z\/2[[x]]$ is $\\sum_{n\\ \\mathit{odd},\\ n>0} x^{n^{2}}$, while $G=F(x^{5})$. We have seen that $r^{2}+r=F+G$.\n\n\\begin{theorem}\n\\label{theorem1.12}\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] $F$ and $G$ lie in $M_{12}$.\n\\item[(2)] $G=r^{5}(r+1)$, $F=r(r+1)^{5}$, and $(F+G)^{6}=FG$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nLet $\\Delta$ be Ramanujan's weight~12 level~1 cusp form. Then $F$ and $G$ are the reductions of the expansions at infinity of $\\Delta(z)$ and $\\Delta(5z)$, and so lie in $M_{12}$. Theorem \\ref{theorem1.11} shows that $G$ is a $Z\/2$-linear combination of $r^{2}+r$, $r^{4}+r^{3}$ and $r^{6}+r^{5}$. As $G=x^{5}+\\cdots$ it can only be $r^{6}+r^{5}$. Then $F=r^{6}+r^{5}+r^{2}+r=r(r+1)^{5}$. The final result is immediate.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem1.13}\n$\\ensuremath{M(\\mathit{odd})}$ is spanned by the $F^{i}G^{j}$ with $i+j$ odd.\n\\end{theorem}\n\n\\begin{proof}\n$r^{2}+r=F+G$, while $r^{4}+r^{3}=(r^{2}+r)^{3}+r^{6}+r^{5}=(F+G)^{3}+G$. Furthermore, $(r^{4}+r^{2})(r^{2k+2}+r^{2k+1})+r^{2k+4}+r^{2k+3}=r^{2k+6}+r^{2k+5}$. Since $r^{4}+r^{2}=F^{2}+G^{2}$, an induction on $k$ shows that each $r^{2k+2}+r^{2k+1}$ is a sum of $F^{i}G^{j}$ with $i+j$ odd.\n\\qed\n\\end{proof}\n\nSuppose now that $p$ is prime, $p\\ne 2$ or $5$. Then we have commuting formal Hecke operators $T_{p}:Z\/2[[x]]\\rightarrow Z\/2[[x]]$ with $T_{p}$ taking $\\sum c_{n}x^{n}$ to $\\sum c_{pn}x^{n}+\\sum c_{n}x^{pn}$.\n\n\\begin{theorem}\n\\label{theorem1.14}\nThe $T_{p}$ stabilize $M$ and $\\ensuremath{M(\\mathit{odd})}$. In fact they stabilize the space spanned by the $r^{i}$ with $0\\le i \\le 2m$, as well as the space spanned by the $r^{2i}(r^{2}+r)$, $0\\le i \\le m-1$.\n\\end{theorem}\n\n\\begin{proof}\nWe may assume $m>0$. Consider the $Z$-submodule of $Z[[x]]$ consisting of those elements of $Z[[x]]$ that are expansions at infinity of weight $4m$ modular form of level $\\Gamma_{0}(5)$. This module is stabilized by each classical Hecke operator $T_{p}:\\sum c_{n}x^{n}\\rightarrow \\sum c_{pn}x^{n}+p^{4m-1}\\sum c_{n}x^{pn}$, $p\\ne 5$. Reducing mod~2 and using Theorem \\ref{theorem1.11} we get the result.\n\\qed\n\\end{proof}\n\n\\begin{remark*}{Remark}\n\\label{remark1.15}\nMultiplication by $G^{2}$ stabilizes $\\ensuremath{M(\\mathit{odd})}$. Since $(F+G)^{6}=FG$, Theorem \\ref{theorem1.13} shows that $\\ensuremath{M(\\mathit{odd})}$, viewed as $Z\/2[G^{2}]$-module, is spanned by $F$, $F^{3}$, $F^{5}$, $G$, $F^{2}G$ and $F^{4}G$. As $F$ has degree 6 over $Z\/2(G)$, a basis of $\\ensuremath{M(\\mathit{odd})}$ as $Z\/2[G^{2}]$-module is $\\{G, F, F^{2}G, F^{3}, F^{4}G, F^{5}\\}$.\n\\end{remark*}\n\n\\begin{definition}\n\\label{def1.16}\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] $N2\\subset \\ensuremath{M(\\mathit{odd})}$ has $Z\/2[G^{2}]$-basis $\\{G, F, F^{2}G, F^{3}, F^{4}G\\}$.\n\\item[(2)] $N1\\subset N2$ has $Z\/2[G^{2}]$-basis $\\{G\\}$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\n\\label{def1.17}\n$J_{1}$, $J_{3}$, $J_{5}$, $J_{7}$ and $J_{9}$ are $F$, $F^{8}\\!\/G$, $G$, $F^{2}G$ and $F^{4}G$.\n\\end{definition}\n\n$J_{1}$, $J_{5}$, $J_{7}$ and $J_{9}$ are evidently in $N2$. Since $(F+G)^{8}=FG(F+G)^{2}$, $F^{8}=G^{8}+FG(F+G)^{2}$, and $J_{3}=G^{7}+F(F+G)^{2}$ is also in $N2$. Now $G$, $F$, $F^{2}G$, $F^{3}$ and $F^{4}G$ evidently generate the same $Z\/2[G^{2}]$-module as do $J_{5}$, $J_{1}$, $J_{7}$, $J_{3}$ and $J_{9}$. Consequently:\n\n\\begin{theorem}\n\\label{theorem1.18}\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] $J_{1}$, $J_{3}$, $J_{5}$, $J_{7}$ and $J_{9}$ are a $Z\/2[G^{2}]$-basis of $N2$, while $J_{1}$, $J_{3}$, $J_{7}$, $J_{9}$ are a $Z\/2[G^{2}]$-basis of $N2\/N1$.\n\\item[(2)] Define $J_{k}$, $k$ odd, $k>0$, by taking $J_{k+10}$ to be $G^{2}J_{k}$. Then the $J_{k}$ with $(k,10)=1$ are a $Z\/2$-basis of $N2\/N1$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark*}{Remarks}\n\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] The space spanned by the $F^{k}$. $k$ odd and $>0$, is stabilized by the $T_{p}$ with $p\\ne 2$. It follows that $N1$ is stabilized by the $T_{p}$ with $p\\ne 2$ or $5$.\n\\item[(2)] One may describe $N2$ more elegantly as follows. $Z\/2(F,G)$ is a degree 6 field extension of $Z\/2(G)$, and we have a trace map, $Z\/2(F,G)\\rightarrow Z\/2(G)$. Using the identity $(F+G)^{6}=FG$ we find that $F^{i}$, $0\\le i\\le 4$, have trace $0$, while $F^{5}$ has trace $G$. So $N2$ consists of those elements of $\\ensuremath{M(\\mathit{odd})}$ of trace $0$. We'll see in the next section that the $T_{p}$, $p\\ne 2$ or $5$ stabilize not only $\\ensuremath{M(\\mathit{odd})}$ and $N1$, but also $N2$.\n\\end{enumerate}\n\\end{remark*}\n\n\\section{The spaces \\bm{$W$}, \\bm{$W_{a}$} and \\bm{$W_{b}$}. A decomposition of \\bm{$N2\/N1$}}\n\\label{section2}\n\n\\begin{definition}\n\\label{def2.1}\n$H=F(x^{25})=G(x^{5})$.\n\\end{definition}\n\nAs we noted in section \\ref{section1}, $D=\\sum_{(n,10)=1,\\ n>0}x^{n^{2}}$ is $F+H$.\n\n\\begin{definition}\n\\label{def2.2}\n$pr: Z\/2[[x]]\\rightarrow Z\/2[[x]]$ takes $\\sum c_{n}  x^{n}$ to $\\sum_{(n,5)=1}  c_{n} x^{n}$.\n\\end{definition}\n\nSince $G$ lies in $Z\/2[[x^{5}]]$, $pr$ is $Z\/2[G]$-linear and $pr(N1)=0$. The effect of $pr$ on the $Z\/2[G^{2}]$-basis $\\{J_{1}, J_{3}, J_{7}, J_{9}\\}$ of $N2\/N1$ is easily described: \n\n\\begin{lemma}\n\\label{lemma2.3}\n$pr$ takes $J_{1}, J_{3}, J_{7}, J_{9}$ to $D, D^{8}\\!\/G, D^{2}G, D^{4}G$.\n\\end{lemma}\n\n\\begin{proof}\n$pr(J_{3}) = pr(F^{8}\\!\/G) = pr((H^{8}+D^{8})\\!\/G) = pr(D^{8}\\!\/G)$. Since all exponents appearing in $D$ are prime to 5, the same holds for all exponents appearing in $D^{8}\\!\/G$ and $pr(D^{8}\\!\/G)=D^{8}\\!\/G$. The other results have similar proofs.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma2.4}\n$D^{15}+G^{4}D^{3}+G^{3}=0$.\n\\end{lemma}\n\n\\begin{proof}\n$(F+G)^{6}= FG$. Replacing $x$ by $x^{5}$ we find that $(G+H)^{6}= GH$. So $(D+F+G)^{6}= G(D+F)$. Adding $FG$ to both sides, expanding in powers of $D$ and dividing by $D$ we find that $D^{5}+(F+G)^{2}D^{3}+ (F+G)^{4}D=G$. So if we set $A=(F+G)^{2}$, then:\n\n\\begin{proofresult}\n$DA^{2}+D^{3}A + (D^{5}+G)=0$.\n\\end{proofresult}\n\nOn the other hand, $A^{3}=(F+G)^{6}=FG$. So $A^{6}=(A+G^{2})G^{2}$, and:\n\n\\begin{proofresult}\n$A^{6}+G^{2}A + G^{4}=0$.\n\\end{proofresult}\n\nWe can now eliminate $A$ from (1) and (2) to get our result. Explicitly, when we multiply (1) by $DA^{4}+D^{3}A^{3}+GA^{2}+D^{7}A$, (2) by $D^{2}$, and add, all terms involving $A^{6}$, $A^{5}$, $A^{4}$ or $A^{3}$ drop out, and we find that $(D^{10}+D^{5}G+G^{2})(A^{2}+D^{2}A) + D^{2}G^{4}=0$. Let $B=A^{2}+D^{2}A$. Then $(D^{10}+D^{5}G+G^{2})B=D^{2}G^{4}$. Also, (1) tells us that $DB=D^{5}+G$. So $(D^{10}+D^{5}G+G^{2})(D^{5}+G)=D^{3}G^{4}$, as desired.\n\\qed\n\\end{proof}\n\nMultiplying by $D^{5}\\!\/G^{4}$ gives:\n\n\\begin{corollary}\n\\label{corollary2.5}\n$(D^{5}\\!\/G)^{4}+D^{5}\\!\/G=D^{8}$.\n\\end{corollary}\n\n\\begin{definition}\n\\label{def2.6}\nWhen $(i,10)=1$, $D_{i}=pr(J_{i})$.\n\\end{definition}\n\nAs we've seen, $D_{1}$, $D_{3}$, $D_{7}$ and $D_{9}$ are $D$, $D^{8}\\!\/G$, $D^{2}G$ and $D^{4}G$. And since $pr$ is $Z\/2[G^{2}]$-linear, $D_{i+10}=G^{2}D_{i}$.\n\n\\begin{theorem}\n\\label{theorem2.7}\nThe $D_{i}$ are linearly independent over $Z\/2$. So $pr$ maps $N2\/N1$ bijectively to the space $W$ spanned by the $D_{i}$. (Recall that the $J_{k}$ with $(k,10)=1$ are a $Z\/2$-basis of $N2\/N1$.)\n\\end{theorem}\n\n\\begin{proof}\n$G$ is transcendental over $Z\/2$. It follows from Lemma \\ref{lemma2.4} that $D$ has degree 15 over $Z\/2(G)$, so that $D$, $D^{8}$, $D^{2}$ and $D^{4}$ are linearly independent over this field. Consequently, $D_{1}=D$, $D_{3}=D^{8}\\!\/G$, $D_{7}=D^{2}G$ and $D_{9}=D^{4}G$ are linearly independent over $Z\/2[G^{2}]$, giving the result.\n\\qed\n\\end{proof}\n\n\\begin{remark*}{Remark}\n$W$ does not consist of mod~2 modular forms of level~5. In fact the elements of $W$ ``are of level 25.''\n\\end{remark*}\n\nWe shall see that the $T_{p}$ ($p\\ne 2$ or $5$ as usual) stabilize both $W$ and $N2$. To this end we use a real Dirichlet character $\\chi$, of modulus 20, to write $W$ as a direct sum of $Z\/2[G^{4}]$-submodules, $W_{a}$ and $W_{b}$. Then we show that the $T_{p}$ with $\\chi(p)=1$ stabilize $W_{a}$ and $W_{b}$, while those with  $\\chi(p)=-1$ take $W_{a}$ to $W_{b}$ and $W_{b}$ to $W_{a}$.\n\n\\begin{definition}\n\\label{def2.8}\n$\\chi$ is the mod~20 Dirichlet character taking $1,3,7,9$ to $1$, and taking $11,13,17,19$ to $-1$.\n\\end{definition}\n\n\\begin{definition}\n\\label{def2.9}\n$W_{a}\\ (\\mbox{resp.}\\ W_{b})$ is the subspace of $W$ spanned by the $D_{i}$ with $\\chi(i)=1\\ (\\mbox{resp.}\\ -1)$.\n\\end{definition}\n\nEvidently $W_{a}$ and $W_{b}$ are $Z\/2[G^{4}]$-submodules of $W$ with bases $\\{D_{1},D_{3},D_{7},D_{9}\\}$ and $\\{D_{11},D_{13},D_{17},D_{19}\\}$. And $W=W_{a}\\oplus W_{b}$.\n\n\\begin{lemma}\n\\label{lemma2.10}\nIf $x^{n}$ appears in $D_{k}$, $n$ is $k$ or $9k\\bmod{40}$.  In particular when $x^{n}$ appears in $D_{k}$, $\\chi(n)=\\chi(k)$.\n\\end{lemma}\n\n\\begin{proof}\nSince $D_{k+10}=G^{2}D_{k}$ and all exponents appearing in $G^{2}$ are $10\\bmod{40}$, it suffices to prove the lemma for $k$ in$\\{1,3,7,9\\}$. Suppose for example $k=9$, so that $D_{k}=D^{4}G$. If $(i,10)=1$, $i^{2}\\equiv 1$ or $9\\bmod{40}$. So all exponents appearing in $D$ are $1$ or $9\\bmod{40}$, and all exponents in $D^{4}G$ are $4+5=9$ or $36+5=41\\bmod{40}$. The other cases are handled similarly.\n\\qed\n\\end{proof}\n\n\\begin{definition}\n\\label{def2.11}\n\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[] $p_{a}:Z\/2[[x]]\\rightarrow Z\/2[[x]]$ takes $\\sum c_{n}x^{n}$ to $\\sum_{\\chi(n)=1} c_{n}x^{n}$.\n\\item[] $p_{b}:Z\/2[[x]]\\rightarrow Z\/2[[x]]$ takes $\\sum c_{n}x^{n}$ to $\\sum_{\\chi(n)=-1} c_{n}x^{n}$.\n\\end{enumerate}\n\\end{definition}\n\n$p_{a}$ and $p_{b}$ are evidently $Z\/2[G^{4}]$-linear. Furthermore:\n\n\\begin{lemma}\n\\label{lemma2.12}\n$p_{a}(G^{2}f)$ and $p_{b}(G^{2}f)$ are $G^{2}p_{b}(f)$ and $G^{2}p_{a}(f)$.\n\\end{lemma}\n\n\\begin{proof}\nWe may assume that all the exponents appearing in $f$ are congruent to some fixed $k\\bmod{20}$. Then the exponents in $G^{2}f$ are congruent to $k+10$, and we use the fact that $\\chi(k+10)=-\\chi(k)$.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma2.13}\n\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[] If $\\chi(k)=1$, $p_{a}(J_{k})=D_{k}$ and $p_{b}(J_{k})=0$.\n\\item[] If $\\chi(k)=-1$, $p_{a}(J_{k})=0$ and $p_{b}(J_{k})=D_{k}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSince $pr(J_{k})=D_{k}$, $p_{a}(J_{k})=p_{a}(D_{k})$ while $p_{b}(J_{k})=p_{b}(D_{k})$.  Lemma \\ref{lemma2.10} then gives the result.\n\\qed\n\\end{proof}\n\nAs $Z\/2[G^{4}]$-module, $N2\/N1$ has basis $\\{J_{1},J_{3},J_{7},J_{9},J_{11},J_{13},J_{17},J_{19}\\}$. Evidently $N2\/N1=N2a\\oplus N2b$, where $N2a$ and $N2b$ are the $Z\/2[G^{4}]$-submodules with bases $\\{J_{1},J_{3},J_{7},J_{9}\\}$ and $\\{J_{11},J_{13},J_{17},J_{19}\\}$. The $J_{k}$ with $\\chi(k)=1$ are a $Z\/2$-basis of $N2a$; those with $\\chi(k)=-1$ are a $Z\/2$-basis of $N2b$. Lemma \\ref{lemma2.13} now shows that $p_{a}$ maps $N2\/N1$ onto $W_{a}$ with kernel $N2b$, while $p_{b}$ maps $N2\/N1$ onto $W_{b}$ with kernel $N2a$.\n\nSuppose now that $f$ is in $\\ensuremath{M(\\mathit{odd})}\/N1$. We'll show that if $p_{b}(f)=0$, then $f$ is in $N2a$, while if $p_{a}(f)=0$, $f$ is in $N2b$.  So in either case, $f$ is in $N2\/N1$. This is key to showing that the $T_{p}$ stabilize $N2$.\n\n\\begin{lemma}\n\\label{lemma2.14}\n$p_{a}(F(F+G)^{4})=p_{a}(r^{8}G)$ and $p_{b}(F(F+G)^{4})=p_{b}(r^{8}G)$.\n\\end{lemma}\n\n\\begin{proof}\n$F(F+G)^{4}=r(r+1)^{5}(r^{2}+r)^{4}=(r+1)^{8}(r^{6}+r^{5})=r^{8}G+G$. Now apply $p_{a}$ and $p_{b}$.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma2.15}\nLet $S=p_{a}(r^{8}G)$, $T=p_{b}(r^{8}G)$. Then $T=D^{5}+G$, $S=D^{10}\\!\/G + G$.\n\\end{lemma}\n\n\\begin{proof}\nSince $r+r^{2}=F+G$, $r^{8}G=G((F+G)^{8}+(F+G)^{16}+(F+G)^{32}+\\cdots )$. So $pr(r^{8}G)=G(D^{8}+D^{16}+D^{32}+\\cdots)$. Now all exponents $n$ appearing in any of $GD^{8}$, $GD^{32}$, $GD^{128}$, \\ldots are $13$ or $37\\bmod{40}$, while those in any of $GD^{16}$, $GD^{64}$, \\ldots are $21$ or $69\\bmod{40}$. Applying $p_{b}$ to our identity we find that $T=G(D^{8}+D^{32}+D^{128}+\\cdots)$. Then $(T\/G)^{4}+(T\/G)=D^{8}$. So by Corollary \\ref{corollary2.5}, $T\/G$ and $D^{5}\\!\/G$ differ by a constant, and comparing expansions we see that the constant is $1$. So $T=D^{5}+G$. Finally $S=G(D^{16}+D^{64}+D^{256}+\\cdots )=T^{2}\\!\/G$.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma2.16}\n\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] \\parbox{2.5in}{$p_{a}(F(F+G)^{4})=D^{10}\\!\/G + G$,} $p_{b}(F(F+G)^{4})=D^{5}+G$. \n\\item[(2)] \\parbox{2.5in}{$p_{a}(FG^{2}(F+G)^{4})=D^{5}G^{2}+G^{3}$,} $p_{b}(FG^{2}(F+G)^{4})=D^{10}G+G^{3}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLemmas \\ref{lemma2.14} and \\ref{lemma2.15} give (1), and Lemma \\ref{lemma2.12} yields (2).\n\\qed\n\\end{proof}\n\nWe saw in the remark following the proof of Theorem \\ref{theorem1.14} that a basis of $\\ensuremath{M(\\mathit{odd})}$ as $Z\/2[G^{2}]$-module is  $\\{G, F, F^{2}G, F^{3}, F^{4}G, F^{5}\\}$. The last five of those elements then give a  basis of $\\ensuremath{M(\\mathit{odd})}\/N1$. It follows that another $Z\/2[G^{2}]$-basis of\n$\\ensuremath{M(\\mathit{odd})}\/N1$ is $\\{J_{1},J_{3},J_{7},J_{9},F(F+G)^{4}\\}$. Then a $Z\/2[G^{4}]$-basis of $\\ensuremath{M(\\mathit{odd})}\/N1$ is  $\\{J_{1},J_{3},J_{7},J_{9},J_{11},J_{13},J_{17},J_{19},F(F+G)^{4},FG^{2}(F+G)^{4}\\}$.\n\n\n\\begin{theorem}\n\\label{theorem2.17}\nThe kernels of $p_{b}:\\ensuremath{M(\\mathit{odd})}\/N1\\rightarrow Z\/2[[x]]$ and $p_{a}:\\ensuremath{M(\\mathit{odd})}\/N1\\rightarrow Z\/2[[x]]$ are $N2a$ and $N2b$ where $N2a$ and $N2b$ have $Z\/2[G^{4}]$-module bases  $\\{J_{1},J_{3},J_{7},J_{9}\\}$ and $\\{J_{11},J_{13},J_{17},J_{19}\\}$.\n\\end{theorem}\n\n\\begin{proof}\nConsider the 10 element $Z\/2[G^{4}]$-module basis of $\\ensuremath{M(\\mathit{odd})}\/N1$ given in the sentence preceding Theorem \\ref{theorem2.17}. $p_{b}$ annihilates $J_{1}$, $J_{3}$, $J_{7}$ and $J_{9}$ and sends the last 6 basis elements to $D_{11}=DG^{2}$, $D_{13}=D^{8}G$, $D_{17}=D^{2}G^{3}$, $D_{19}=D^{4}G^{3}$, $D^{5}+G$ and $D^{10}G+G^{3}$; see Lemma \\ref{lemma2.16}. As we've seen, $D$ has degree 15 over $Z\/2(G)$, and so $D$, $D^{8}$, $D^{2}$, $D^{4}$, $D^{5}+G$ and $D^{10}+G^{2}$ are linearly independent over this field. So no non-trivial $Z\/2[G^{4}]$-linear combination of $D_{11}$, $D_{13}$, $D_{17}$, $D_{19}$, $D^{5}+G$ and $D^{10}+G^{3}$ is zero, and the result for $p_{b}$ follows. The result for $p_{a}$ is proved similarly.\n\\qed\n\\end{proof}\n\n\\begin{corollary}\n\\label{corollary2.18}\n\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] If $\\chi(p)=1$, $T_{p}$ stabilizes $N2a$ and $N2b$.\n\\item[(2)] If $\\chi(p)=-1$, $T_{p}$ maps $N2a$ to $N2b$ and $N2b$ to $N2a$.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nFor example suppose $f$ is in $N2b$ with $\\chi(p)=-1$. Then the exponents, $n$, appearing in (a representative of) $f$ have $\\chi(n)=-1$ or $0$. Since $\\chi$ is multiplicative, the exponents appearing in $T_{p}$ applied to this representative have $\\chi(n)=1$ or $0$. In other words, $T_{p}(f)$ is an element of $\\ensuremath{M(\\mathit{odd})}\/N1$ in the kernel of $p_{b}$. By Theorem \\ref{theorem2.17}, $T_{p}(f)$ is in $N2a$. The other cases are treated similarly.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem2.19}\nIf $p\\ne 2$ or $5$, $T_{p}$ stabilizes $N2$ and $W$.\n\\end{theorem}\n\n\\begin{proof}\nIf $h$ is in $N2$, the image of $h$ in $N2\/N1$ is the sum of an element of $N2a$ and an element of $N2b$. By Corollary \\ref{corollary2.18}, the same holds for the image of $T_{p}(h)$ in $\\ensuremath{M(\\mathit{odd})}\/N1$. So this image is in $N2\/N1$ and $h$ is in $N2$. Suppose $f$ is in $W$. Then $f=pr(h)$ with $h$ in $N2$. Then $T_{p}(h)$ is in $N2$, and $T_{p}(f)=pr(T_{p}(h))$ is in $W$.\n\\qed\n\\end{proof}\n\n\\begin{corollary}\n\\label{corollary2.20}\nSuppose $p\\ne 2$ or $5$, and $(n,10)=1$. Then $T_{p}(D_{n})$ is a sum of distinct $D_{k}$. In such a decomposition each $k$ is either $pn$ or $9pn\\bmod{40}$, and in particular $\\chi(k)=\\chi(p)\\chi(n)$.\n\\end{corollary}\n\n\\begin{proof}\nSince $D_{n}$ is in $W$, so is $T_{p}(D_{n})$, giving the first result. Also the exponents appearing in $T_{p}(D_{n})$ are all congruent to $pn$ or $9pn\\bmod{40}$ (by Lemma \\ref{lemma2.10} and the definition of $T_{p}$) while those appearing in a $D_{k}$ are $k$ or $9k\\bmod{40}$. It follows easily that those $D_{k}$ appearing in the sum for which $k$ is neither congruent to $pn$ nor to $9pn\\bmod{40}$ sum to $0$. So there are no such $k$.\n\\qed\n\\end{proof}\n\n\\begin{remark*}{Remark}\nSince $N2\/N1=N2a\\oplus N2b$, $pr\\hspace{-.4em}:\\hspace{-.3em}N2\/N1 \\rightarrow Z\/2[[x]]$ is 1--1 with image $W_{a}\\oplus W_{b}=W$. So $pr$ gives an identification of the subquotient $N2\/N1$ of $\\ensuremath{M(\\mathit{odd})}$ with $W$. Now the $T_{p}$, $p\\ne 2$ or $5$, stabilize $N2\/N1$ and $W$, and the identification preserves the action of $T_{p}$. Corollary \\ref{corollary2.20} and this identification are the quoted results (1) and (2) at the beginning of section \\ref{section1}.\n\\end{remark*}\n\nWe show next that when $p\\equiv 3$ or $7\\mod{10}$, each $k$ in Corollary \\ref{corollary2.20} is $<n$. (This is also true when $p\\equiv 1$ or $9$, but this will only be proved in the final section.)\n\n\\begin{definition}\n\\label{def2.21}\n$v_{2}=r^{2}+r$, $v_{4}=r^{4}+r^{3}$, $v_{6}=r^{6}+r^{5}$, $v_{10}=r^{10}+r^{9}+r^{8}+r^{7}$, $v_{12}=r^{12}+r^{11}+r^{10}+r^{9}$.\n\\end{definition}\n\nNote that $v_{2}=F+G$, $v_{4}=(F+G)^{3}+G$, $v_{6}=G$, $v_{10}=(F+G)^{2}G$ and $v_{12}=(F+G)^{4}G+(F+G)G^{2}$. So $v_{2}$, $v_{4}$, $v_{6}$, $v_{10}$ and $v_{12}$ generate the same $Z\/2[G^{2}]$-submodule of $\\ensuremath{M(\\mathit{odd})}$ as do $G$, $F$, $F^{2}G$, $F^{3}$ and $F^{4}G$; that is to say they generate $N2$. Since $G^{2}=r^{12}+r^{10}$, $G^{2s}v_{j}$ is an element of $N2$ whose degree in $r$ is $12s+j$.\n\n\\begin{lemma}\n\\label{lemma2.22}\n$T_{p}(D_{10m+3})$ and $T_{p}(D_{10m+7})$ are sums of $D_{k}$ with $k\\le 10m+7$. \n\\end{lemma}\n\n\\begin{proof}\n$J_{7}=F^{2}G\\equiv v_{10}\\bmod{N1}$. And $J_{3}=F^{8}\\!\/G=(F+G)^{8}\\!\/G+G^{7}=F(F+G)^{2}+G^{7}$. So mod $N1$, $J_{3}+J_{7}\\equiv (F+G)^{3}\\equiv v_{4}$. It follows that $J_{10m+3}=G^{2m}J_{3}$ and $J_{10m+7}=G^{2m}J_{7}$ are each congruent mod $N1$ to polynomials in $r$ of degree $\\le 12m+10$. By Theorem \\ref{theorem1.14} the same is true of $T_{p}(J_{10m+3})$ and $T_{p}(J_{10m+7})$. (Note also that $J_{1}=F\\equiv v_{2}\\bmod{N1}$ while $J_{9}+J_{11}=F^{4}G+FG^{2}\\equiv v_{12}$.) Now take an $h$ of degree $\\le 12m+10$ in $r$ which is congruent mod $N1$ to $T_{p}(J_{10m+3})$ (or to $T_{p}(J_{10m+7})$). Since $J_{10m+3}$ and $J_{10m+7}$ are in $N2$, so is $h$. Write $h$ as a sum of distinct $G^{2s}v_{i}$ with each $i$ in $\\{2,4,6,10,12\\}$. The degree in $r$ of $G^{2s}v_{i}$ is $12s+i$. These degrees are distinct, and since the degree of $h$ is $\\le 12m+10$, the $s$ appearing in those $G^{2s}v_{i}$ with $i=12$ are all $<m$, while the remaining $s$ are all $\\le m$. Since $v_{2}$, $v_{4}$, $v_{6}$, $v_{10}$ and $v_{12}$ are congruent mod $N1$ to $J_{1}$, $J_{3} +J_{7}$, $0$, $J_{7}$ and $J_{9}+J_{11}$, and each of $10m+1$, $10m+7$, $10m+7$, $10(m-1)+11$ is $\\le 10m+7$, each $G^{2s}v_{i}$ appearing in the sum for $h$ is, mod $N1$, a sum of $J_{k}$ with $k\\le 10m+7$. Applying $pr$ to the identity $h=\\sum G^{2s}v_{i}$, noting that $pr$ preserves the action of $T_{p}$ and that $pr(J_{k})$ is either $D_{k}$ or $0$, we get the result.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma2.23}\n$T_{p}(D_{10m+9})$ and $T_{p}(D_{10m+11})$ are sums of $D_{k}$ with $k\\le 10m+11$. \n\\end{lemma}\n\n\\begin{proof}\n$J_{11}=G^{2}J_{1}\\equiv G^{2}v_{2}\\bmod{N1}$, while $J_{9}+J_{11}\\equiv v_{12}$. It follows that $J_{10m+9}$ and $J_{10m+11}$ are congruent mod $N1$ to polynomials in $r$ of degree $\\le 12m+14$. By Theorem \\ref{theorem1.14}, the same is true of $T_{p}(J_{10m+9})$ and $T_{p}(J_{10m+11})$. Now take an $h$ of degree $\\le 12m+14$ in $r$ which mod $N1$ is $T_{p}(J_{10m+9})$ (or $T_{p}(J_{10m+11})$). $h$ is in $N2$ and we write it as a sum of distinct $G^{2s}v_{i}$, $i$ in $\\{2,4,6,10,12\\}$. Arguing as in the proof of Lemma \\ref{lemma2.22} we find that the $s$ appearing in the $G^{2s}v_{i}$ with $i=2$ are $\\le m+1$, while the remaining $s$ are $\\le m$, and we continue as in the proof of Lemma \\ref{lemma2.22}.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem2.24}\nSuppose $p\\equiv 3$ or $7\\mod{10}$. When we write $T_{p}(D_{n})$ as a sum of distinct $D_{k}$, each $k<n$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $n=10m+3$ or $10m+7$. By Lemma \\ref{lemma2.22}, $T_{p}(D_{n})$ is a sum of distinct $D_{k}$, $k\\le 10m+7$. By Corollary \\ref{corollary2.20}, each $k\\equiv pn$ or $9pn\\mod{10}$ and so is $1$ or $9$ mod~10. So no $k$ can be $10m+3$ or $10m+7$. If  $n=10m+9$ or $10m+11$, $T_{p}(D_{n})$ is, by Lemma \\ref{lemma2.23}, a sum of $D_{k}$, $k\\le 10m+11$. Then each $k\\equiv pn$ or $9pn\\mod{10}$ and so is $3$ or $7$ mod~10. So no $k$ is $10m+9$ or $10m+11$. Finally, $T_{p}(D_{1})=T_{p}(D)=0$.\n\\qed\n\\end{proof}\n\nWe shall write down a linear recursion satisfied by the $T_{3}(D_{n})$, $n\\equiv 1,3,7,9\\mod{20}$. This recursion, together with some initial condition results, proved with the help of Theorem \\ref{theorem2.24} when $p=3$, allow us to relate $T_{3}(D_{n})$ to a polynomial $P_{n}$ appearing in section 5 of \\cite{2}.\n\n\\begin{lemma}\n\\label{lemma2.25}\nFor $u$ in $Z\/2[[x]]$, $T_{3}(G^{16}u)=G^{16}T_{3}(u)+G^{4}T_{3}(G^{4}u)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $u$ be the $2$-variable polynomial $A^{4}+B^{4}+AB$. We have the ``level 3 modular equation for $F$,'' $U(F(x^{3}),F(x))=0$. Replacing $x$ by $x^{5}$ shows that $U(G(x^{3}),G(x))=0$. Now proceed as in the proof of Lemma 2.19 of \\cite{1} (though now we have 4 imbeddings $\\varphi_{i}$, the first taking $f(x)$ to $f(x^{3})$, and the others taking $f(x)$ to $f(\\lambda x^{1\/3})$ where $\\lambda$ runs over the cube roots of $1$ in an algebraic closure of $Z\/2$).\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem2.26}\nThe $T_{3}(D_{n})$. $n\\equiv 1,3,7,9\\mod{20}$ satisfy the recursion $T_{3}(D_{n+80})=G^{16}T_{3}(D_{n})+G^{4}T_{3}(D_{n+20})$. And $T_{3}$ takes \n\\vspace{-1ex}\n\\[\nD_{1},D_{3},D_{7},D_{9},D_{21},D_{23},D_{27},D_{29},D_{41},D_{43},D_{47},D_{49},D_{61},D_{63},D_{67},D_{69}\n\\]\n\\vspace{-4ex}\nto:\n\\vspace{-4ex}\n\\begin{eqnarray*}\n0,D_{1},0,D_{3},D_{7},D_{21},D_{9},D_{23},0,D_{41},D_{21},D_{43}+D_{27},\\\\\nD_{47}+D_{23},D_{61}+D_{29}+D_{21},D_{49}+D_{41},D_{63}+D_{47}+D_{23}.\n\\end{eqnarray*}\n\\vspace{-4ex}\n\\end{theorem}\n\n\\begin{proofsketch}\nTaking $u=D_{n}$ in Lemma \\ref{lemma2.25}, we ge the recursion. I'll calculate $T_{3}(D_{47})$ explicitly, the other 15 initial values being derived in a similar way. By Corollary \\ref{corollary2.20} and Theorem \\ref{theorem2.24}, $T_{3}(D_{47})$ is a $Z\/2$-linear combination of $D_{21}$ and $D_{29}$. Now $D_{47}=G^{8}(D^{2}G)=(x^{40}+x^{360}+\\cdots)(x^{2}+x^{18}+x^{98}+\\cdots)(x^{5}+x^{45}+\\cdots)$. So the coefficients of $x^{7}$, $x^{63}$ and $x^{87}$ in $D_{47}$ are $0$, $1$ and $1$. It follows that the coefficients of $x^{21}$ and $x^{29}$ in $T_{3}(D_{47})$ are each $1$. But $D_{21}=G^{4}D=x^{21}+x^{29}+\\cdots$ while $D_{29}=x^{29}+\\cdots$. It follows that $T_{3}(D_{47})=D_{21}$. \n\\qed\n\\end{proofsketch}\n\nNow let $w$ be an indeterminate over $Z\/2$.\n\n\\begin{definition}\n\\label{def2.27}\n$V^{\\prime}\\subset Z\/2[w]$ is the space spanned by the $w^{k}$ with $k\\equiv 1,3,7,9\\mod{20}$.\n\\end{definition}\n\nThere is a $Z\/2$-linear identification of $W_{a}$ with $V^{\\prime}$ taking $D_{k}$ to $w^{k}$. Then $T_{3}:W_{a}\\rightarrow W_{a}$ goes over to a map $V^{\\prime}\\rightarrow V^{\\prime}$ which we'll still call $T_{3}$. Now in section 5 of \\cite{2} (see Theorem 5.1, the paragraphs preceding it, and Remark 5.2) we defined certain $P_{k}$, $k\\equiv 1,3,7,9\\mod{20}$ in $V^{\\prime}$.\n\n\\begin{theorem\n\\label{theorem2.28}\n$T_{3}:V^{\\prime}\\rightarrow V^{\\prime}$ takes $w^{k}$ to the $P_{k}$ of \\cite{2} section 5.\n\\end{theorem}\n\n\\begin{proof}\nLet $A_{k}=T_{3}(w^{k})$. The recursion of Theorem \\ref{theorem2.26} tells us that $A_{k+80}=w^{80}A_{k}+w^{20}A_{k+20}$, while Theorem 5.1 of \\cite{2} tells us that $P_{k+80}=w^{80}P_{k}+w^{20}P_{k+20}$. Putting the initial values in Theorem \\ref{theorem2.26} together with Theorem 5.1 of \\cite{2} we see that $P_{k}=A_{k}$ whenever $k<80$, $k\\equiv 1,3,7,9\\mod{20}$. The recursions then show that $P_{k}=A_{k}$ whenever $k\\equiv 1,3,7,9\\mod{20}$.\n\\qed\n\\end{proof}\n\n\\section{Coding a basis of \\bm{$W_{a}$}. The effect of \\bm{$T_{3}$} on the code}\n\\label{section3}\n\nDefine a total ordering on $N\\times N$ as follows.  $(c,d)$ ``precedes'' (or ``is earlier than'') $(a,b)$ if $c+d<a+b$ or $c+d=a+b$ and $d<b$.\n\nConsider the space $W_{a}$ with its basis $\\{D_{k}, k\\equiv 1,3,7,9\\mod{20}\\}$. In this section we ``code the basis'' by identifying it with $N\\times N$; $(c,d)^{*}$ will denote the particular $D_{k}$ corresponding to $(c,d)$. Under this identification, the total ordering on $N\\times N$ given in the last paragraph goes over to a total ordering on our basis, and we use the ``precedence'' language of the last paragraph and show:\n\n\\begin{enumerate}\n\\item[(1)] $T_{3}(a,b)^{*}$ is a sum of $(c,d)^{*}$ with $c+d<a+b$.\n\\item[(2)] If $a>0$, $T_{3}(a,b)^{*}=(a-1,b)^{*}+$ a sum of earlier $D_{j}$.\n\\end{enumerate}\n\nThe hard work in establishing (1) and (2) has already been done in Lemma 5.5 of \\cite{2} and Theorem \\ref{theorem2.28} of the last section, so most of this section is a summary of results from \\cite{2}.\n\n\\begin{definition}\n\\label{def3.1}\n$g:N\\rightarrow N$ is the function with $g(2n)=4g(n)$ and $g(2n+1)=g(2n)+1$.\n\\end{definition}\n\n\\begin{definition}\n\\label{def3.2}\n$V\\subset Z\/2[t]$ is spanned by the $t^{k}$, $k$ odd. If $(a,b)$ is in $N\\times N$, $[a,b]$ in $V$ is $t^{1+2g(a)+4g(b)}$.\n\\end{definition}\n\n$(a,b)\\rightarrow [a,b]$ sets up a 1--1 correspondence between $N\\times N$ and the monomial basis of $V$. The total ordering of $N\\times N$ goes over to a total ordering of the basis. Once again we use the language of ``precedence.''  We now pass from $V$ to $V^{\\prime}$. \n\n\\begin{definition}\n\\label{def3.3}\nIf $V^{\\prime}$ is as in Definition \\ref{def2.27}, $\\varphi:V\\rightarrow V^{\\prime}$ is the $Z\/2$-linear bijection with:\n\n\\vspace{-3ex}\n\\begin{eqnarray*}\n&&\\varphi(t^{16m+1}) = w^{40m+1},\\quad \\varphi(t^{16m+3}) = w^{40m+3}\\\\\n&&\\varphi(t^{16m+5}) = w^{40m+7},\\quad \\varphi(t^{16m+5}) = w^{40m+21}\\\\\n&&\\varphi(t^{16m+9}) = w^{40m+9},\\quad \\varphi(t^{16m+11}) = w^{40m+27}\\\\\n&&\\varphi(t^{16m+13}) = w^{40m+23},\\quad \\varphi(t^{16m+15}) = w^{40m+29}\n\\end{eqnarray*}\n\\end{definition}\n\n\\begin{definition}\n\\label{def3.4}\n$\\langle a,b\\rangle$ in $V^{\\prime}$ is $\\varphi[a,b]$. $(a,b)^{*}$ in $W_{a}$ is the image of $\\langle a,b\\rangle$ under the identification of $V^{\\prime}$ with $W_{a}$ taking $w^{k}$ to $D_{k}$.\n\\end{definition}\n\nNote that $\\varphi:V\\rightarrow V^{\\prime}$ identifies the monomial basis (in $t$) of $V$ with the monomial basis (in $w$) of $V^{\\prime}$. So Definition \\ref{def3.4} results in a coding of the basis $\\{D_{k}\\}$ of $W_{a}$.\n\n\\begin{theorem}\n\\label{theorem3.5}\n\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] $T_{3}(a,b)^{*}$ is a sum of $(c,d)^{*}$ with $c+d<a+b$.\n\\item[(2)] If $a>0$, $T_{3}(a,b)^{*}=(a-1,b)^{*}+$ a sum of earlier $D_{j}$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nWe saw in the last section that $T_{3}$ stabilizes $W_{a}$. In view of the identification of $W_{a}$ with $V^{\\prime}$ it suffices to show that $T_{3}\\langle a,b \\rangle$ is a sum of $\\langle c,d\\rangle$ with $c+d<a+b$, and that if $a>0$, $T_{3}\\langle a,b \\rangle =\\langle a-1,b \\rangle +$ a sum of earlier monomials. Suppose then that $\\langle a,b \\rangle =w^{k}$. By Theorem \\ref{theorem2.28}, $T_{3}\\langle a,b \\rangle$ is the $P_{k}$ of \\cite{2}, Theorem 5.1.  So our result is precisely Lemma 5.5 of \\cite{2}.\n\\qed\n\\end{proof}\n\nWe will need one further property of our code.\n\n\\begin{theorem}\n\\label{theorem3.6}\nIf $D_{i}=(c,d)^{*}$ precedes $D_{j}=(0,b)^{*}$, then $i<j$.\n\\end{theorem}\n\n\\begin{proof}\nIn view of our identification of $W_{a}$ with $V^{\\prime}$, it's enough to show that if $w^{i}=\\langle c,d \\rangle$ precedes $w^{j}=\\langle 0,b\\rangle$, then $i<j$.\n\nNow there is the following corresponding result for $V$. If $t^{i}=[c,d]$ precedes $t^{j}=[0,b]$ then $i<j$. For $c+d\\le b$, so by Lemma 4.1 of \\cite{1}, $4g(c)+4g(d)\\le 4g(b)$. Then $i=1+2g(c)+4g(d)\\le 1+4g(b)=j$, and since $i\\ne j$, $i<j$.\n\nWe'll deduce Theorem 3.6 from this result for $V$. Suppose first that $b$ is even, and let $m=g(b\/2)$. Then $1+4g(b)=16m+1$, and $[0,b]=t^{16m+1}$. Write $[c,d]$ as $t^{16m^{\\prime}+r}$ with $r$ in $\\{1,3,5,7,9,11,13,15\\}$. Since $\\langle c,d \\rangle$ precedes $\\langle 0,b \\rangle$, $[c,d]$ precedes $[0,b]$. By the result of the last paragraph, $16m^{\\prime}+r < 16m+1$ and so $m^{\\prime}<m$. Now $\\langle 0,b \\rangle = w^{40m+1}$ so that $j=40m+1$. Similarly, $i\\le 40m^{\\prime}+29\\le 40m-11$, and this is $<j$. The argument when $b$ is odd is similar. Let $m=g((b-1)\/2)$, so that $g(b)=4m+1$, and $[0,b]=t^{16m+5}$. Let $m^{\\prime}$ and $r$ be as in the case of even $b$. Arguing as in the case of even $b$ we find that $16m^{\\prime}+r<16m+5$. If $m^{\\prime}<m$ we proceed as in the case of even $b$. If $m^{\\prime}=m$ then $r$ must be $1$ or $3$. So $\\langle c,d \\rangle$ is either $w^{40m+1}$ or $w^{40m+3}$, while $\\langle 0,b \\rangle=\\varphi(t^{16m+5})=w^{40m+7}$, and once again $i<j$.\n\\qed\n\\end{proof}\n\n\\section{Type \\bm{$a$} ideals of \\bm{$Z[\\sqrt{-10}]$} and Gauss-classes}\n\\label{section4}\n\nThis section is the counterpart to section 3 of \\cite{1}. Fix a power, $q$, of 2. We shall (essentially) use binary quadratic forms of discriminant $-640q^{2}$ and their associated theta-series to construct a subspace $DI(q)$ of $W_{a}$ of dimension $2q$, stable under the $T_{p}$ with $\\chi(p)=1$, and annihilated by $T_{3}$. and we'll give a simple description of the action of $T_{7}$ on $DI(q)$ involving Gaussian composition of forms. We will not use all primitive positive forms of discriminant $-640q^{2}$; the Dirichlet character $\\chi$ of Definition \\ref{def2.8} may be thought of as a genus character, and we'll only consider forms on which this character takes the value $1$. We'll see that the $SL_{2}(Z)$-classes of such forms make up a cyclic group of order $4q$, and that the class of a form representing $7$ is a generator.\n\nAs in section 3 of \\cite{1}, we'll avoid the explicit language of binary forms. Instead we'll consider ideals $I$ in $Z[\\sqrt{-10}]$ for which $\\chi(\\mathrm{norm}(I))= 1$. We'll say that such an ideal is ``of type $a$.''  We fix a power $q$ of $2$, and introduce an equivalence relation, depending on $q$, on the type $a$ ideals. We will call our equivalence relation ``Gauss-equivalence,'' and the equivalence classes under it ``Gauss-classes.''\n\nThe class number of $Z[\\sqrt{-10}]$ is $2$, with an ideal of norm $7$ representing the non-principal class. We begin by defining Gauss-equivalence for principal ideals $(b+c\\sqrt{-10})$ of type $a$.\n\nSince the norm of $(b+c\\sqrt{-10})$ is $b^{2}+10c^{2}$, $(b+c\\sqrt{-10})$ is of type $a$ precisely when $(b,10)=1$ and $c$ is even. Note also that the generator of a principal ideal is defined up to multiplication by $\\pm 1$.\n\n\\begin{definition}\n\\label{def4.1}\nPrincipal type $a$ ideals $(\\alpha)$ and $(\\beta)$ are equivalent if there is an integer $N$ with $(N,10)=1$ such that $N\\alpha\\equiv\\beta\\bmod{4q}$ in the ring $Z[\\sqrt{-10}]$.\n\\end{definition}\n\nEvidently this does not depend on the choices of generators for the ideals, and is an equivalence relation. Also ideal multiplication makes the set of equivalence classes into a semigroup. Since $(\\alpha)(\\bar{\\alpha})=(\\mathrm{norm}(\\alpha))$ which is equivalent to $(1)$, the semigroup is a group.\n\n\\begin{lemma}\n\\label{lemma4.2}\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] Any principal type $a$ ideal $(\\alpha)$ is equivalent to $(1+2d\\sqrt{-10})$ for some $d$.\n\\item[(2)] $(1+2c\\sqrt{-10})$ and $(1+2d\\sqrt{-10})$ are equivalent if and only if $c\\equiv d\\bmod{2q}$. So there are $2q$ equivalence classes.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWrite $\\alpha$ as $(b+2c\\sqrt{-10})$. Then $(b,10)=1$ and we can choose $N$ with $(N,10)=1$ so that $Nb\\equiv 1\\mod{4q}$. Then $(Na)$ is of type $a$, and $N\\alpha\\equiv 1+2Nc\\sqrt{-10}\\bmod{4q}$, proving (1). Turning to (2), if $c\\equiv d\\bmod{2q}$ we may take $N=1$. Conversely if $N(1+2c\\sqrt{-10})\\equiv 1+2d\\sqrt{-10}\\bmod{4q}$, then $N\\equiv 1\\mod{4q}$. So mod $4q$, $2d\\equiv 2Nc\\equiv 2c$.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem4.3}\nThe order $2q$ group of classes of principal type $a$ ideals is cyclic, generated by the class of any $(c+2d\\sqrt{-10})$ with $(c,10)=1$ and $d$ odd.\n\\end{theorem}\n\\pagebreak\n\n\\begin{proof}\nOne argues as in the last few sentences of the proof of Theorem 3.2 of \\cite{1}.\n\\qed\n\\end{proof}\n\nNow fix a type $a$ non-principal ideal $L$; for example $P=(7,2-\\sqrt{-10})$. If $I$ is another such ideal, $IL$ is principal of type $a$.\n\n\\begin{definition}\n\\label{def4.4}\nSuppose $I$ and $J$ are type $a$ non-principal. $I$ and $J$ are Gauss equivalent if $IL$ and $JL$ are equivalent in the sense of Definition \\ref{def4.1}.\n\\end{definition}\n\nThe definition appears to depend on the choice of $L$. But if one replaces $L$ by $\\gamma L$ where $\\chi(\\mathrm{norm}(\\gamma))=1$, one finds that the notion of Gauss-equivalence is unchanged. It follows at once that the dependence on $L$ is illusory. One sees further that the Gauss-classes of all type $a$ ideals form a group of order $4q$ under ideal multiplication and that the inverse of $I$ is $\\bar{I}$.\n\n\\begin{theorem}\n\\label{theorem4.5}\nThe group of Gauss-classes of type $a$ ideals is cyclic of order $4q$; any class consisting of non-principal ideals is a generator.\n\\end{theorem}\n\n\\begin{proof}\nLet $P=(7,2-\\sqrt{-10})$. $P$ has norm $7$, and is of type $a$. Also, $P^{2}=(3+2\\sqrt{-10})$. By Theorem \\ref{theorem4.3}, $P^{2}$ has order $2q$. So $P$ has order $4q$, and the result follows.\n\\qed\n\\end{proof}\n\n\\begin{definition}\n\\label{def4.6}\nIf $R$ is a Gauss-class, $\\theta(R)$ in $Z[[x]]$ is $\\sum x^{\\mathrm{norm}(I)}$, where $I$ runs over the ideals in $R$.\n\\end{definition}\n\n\\begin{definition}\n\\label{def4.7}\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] $e$ is the Gauss-class of (1).\n\\item[(2)] $\\mathit{AMB}$ is the Gauss-class of order 2; that is to say the Gauss-class of $(1+2q\\sqrt{-10})$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{lemma}\n\\label{lemma4.8}\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] The mod~2 reduction of $\\theta(e)$ is $D$.\n\\item[(2)] $\\theta(\\mathit{AMB})$ is in $2Z[[x]]$, and the mod~2 reduction of $\\frac{1}{2}\\theta(\\mathit{AMB})$ is $D_{40q^{2}+1}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n$I$ is in $e$ if and only if $\\bar{I}$ is in $e$; also they each make the same contribution to $\\theta(e)$. So in calculating $\\theta(e)\\bmod{2}$ we only need consider $I$ in $e$ with $\\bar{I}=I$. These are just the $(N)$ with $N$ in $Z$, $(N,10)=1$ and $N>0$, and we get (1). $\\mathit{AMB}$ consists of principal ideals. These are of the form $(b+2cq\\sqrt{-10})$ with $(b,10)=1$, $b>0$ and $c$ odd. Since $(b+2cq\\sqrt{-10})$ and $(b-2cq\\sqrt{-10})$ make the same contribution to $\\theta(\\mathit{AMB})$, $\\theta(\\mathit{AMB})$ is in $2Z[[x]]$. Also, $\\frac{1}{2}\\theta(\\mathit{AMB})$ is $\\sum x^{b^{2}+40q^{2}c^{2}}$, the sum running over all positive $b$ and $c$ with $(b,10)=1$ and $c$ odd. Reducing mod~2 we get $DG^{8q^{2}}=D_{40q^{2}+1}$.\n\\qed\n\\end{proof}\n\nWe next define a Hecke operator $T_{p}:Z[[x]]\\rightarrow Z[[x]]$ for each $p$ with $\\chi(p)=1$.\n\n\\begin{definition}\n\\label{def4.9}\nIf $\\chi(p)=1$, $T_{p}$ is the map $\\sum c_{n}x^{n}\\rightarrow \\sum c_{pn}x^{n}+\\left(\\frac{-10}{p}\\right)\\sum c_{n}x^{pn}$.\n\\end{definition}\n\nNote that the mod~2 reduction of $T_{p}$ is our usual mod~2 Hecke operator $T_{p}:Z\/2[[x]]\\rightarrow Z\/2[[x]]$.\n\n\\begin{remark*}{Remark}\nThe motivation for Definition \\ref{def4.9} is the following. It can be shown that each $\\theta(R)$ is the expansion at infinity of a modular form of weight 1 for some $\\Gamma_{1}(N)$, with character $n\\rightarrow \\left(\\frac{-10}{n}\\right)$. And Definition \\ref{def4.9} is the standard definition of the Hecke action on such expansions. (But in what follows we won't use the connection of the $\\theta(R)$ with weight 1 modular forms.)\n\\end{remark*}\n\n\\begin{theorem}\n\\label{theorem4.10}\nSuppose $\\chi(p)=1$ and $p$ is inert in $Z[\\sqrt{-10}]$. Then $T_{p}$ annihilates each $\\theta(R)$ in $Z[[x]]$.\n\\end{theorem}\n\n\\begin{proof}\n$I\\rightarrow pI$ sets up a 1--1 correspondence between ideals of norm $n$ in $R$ and ideals of norm $p^{2}n$ in $R$. Also, if $(n,p)=1$ there are no ideals of norm $pn$ in $R$. Since $\\left(\\frac{-10}{n}\\right)=-1$, the result follows directly from Definition \\ref{def4.9}.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem4.11}\nSuppose $\\chi(p)=1$ and $p$ splits in $Z[\\sqrt{-10}]$, so that $(p)=P\\cdot\\bar{P}$ with $P\\ne \\bar{P}$. Then if $R$ is a Gauss-class, $T_{p}(\\theta(R))=\\theta(PR)+\\theta(\\bar{P}R)$. (Since $\\chi(p)=1$, $P$ and $\\bar{P}$ are type $a$.)\n\\end{theorem}\n\n\\begin{proof}\nThe argument follows that in the proof of Theorem 3.6 of \\cite{1}, the essential points being the multiplicativity of the norm and unique factorization at the ideal level in $Z[\\sqrt{-10}]$.\n\\qed\n\\end{proof}\n\n\\begin{definition}\n\\label{def4.12}\n$\\alpha(R)$ is the mod~2 reduction of $\\theta(R)$. $DI(q)\\subset Z\/2[[x]]$ is the space spanned by the $\\alpha(R)$, as $R$ runs over the $4q$ Gauss-classes of type $a$ ideals.\n\\end{definition}\n\nNow let $C$ be a Gauss-class containing one of the ideals of norm 7. By Theorem \\ref{theorem4.5}, $C$ generates the group of Gauss-classes of type $a$ ideals.\n\n\\begin{theorem}\n\\label{theorem4.13}\nLet $\\alpha_{i}=\\alpha(C^{i})$. Then $\\alpha_{0}=D$, $\\alpha_{2q}=0$, and the $\\alpha_{i}$, $0\\le i <2q$ span $DI(q)$.\n\\end{theorem}\n\n\\begin{proof}\nSince $C$ is a generator, $\\alpha_{0}, \\ldots, \\alpha_{4q-1}$ span $DI(q)$. By Lemma \\ref{lemma4.8}, $\\alpha_{0}=D$ and $\\alpha_{2q}=0$. Finally $I\\rightarrow\\bar{I}$ sets up a 1--1 norm preserving correspondence between the ideals in $C^{i}$ and the ideals in $C^{4q-i}$, so $\\alpha_{i}=\\alpha_{4q-i}$.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem4.14}\n\\hspace{2em}\\\\\n\\vspace{-4ex}\n\\begin{enumerate}\n\\item[(1)] $T_{7}(\\alpha_{i})=\\alpha_{i-1}+\\alpha_{i+1}$ if $0<i<2q$. So $T_{7}$ stabilizes $DI(q)$.\n\\item[(2)] $T_{7}(D_{40q^{2}+1})=\\alpha_{2q-1}$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nBy Theorem \\ref{theorem4.11}, $T_{7}(\\theta_{i})=\\theta_{i-1}+\\theta_{i+1}$; reducing mod~2 we get (1). Also $T_{7}(\\theta(\\mathit{AMB}))=\\theta_{2q-1}+\\theta_{2q+1}=2\\theta_{2q-1}$. Dividing by 2, reducing mod~2 and using Lemma \\ref{lemma4.8} we get (2).\n\\qed\n\\end{proof}\n\n\\begin{definition}\n\\label{def4.15}\n$U_{n}$ is the element of $Z\/2[t]$ with $U_{n}(t+t^{-1})=t^{n}+t^{-n}$. Note that $U_{0}=0$, $U_{1}(Y)=Y$, and that $U_{n+2}(Y)=YU_{n+1}(Y)+U_{n}(Y)$. Furthermore $U_{2n}=U_{n}^{2}$, and it follows that $U_{q}(Y)=Y^{q}$. Finally $Y$ divides each $U_{n}(Y)$.\n\\end{definition}\n\n\\begin{lemma}\n\\label{lemma4.16}\nLet $Y$ be the operator $T_{7}:Z\/2[[x]]\\rightarrow Z\/2[[x]]$. Then for $0\\le i \\le 2q$, $\\alpha_{2q-i}=U_{i}(Y)\\cdot (D_{40q^{2}+1})$. \n\\end{lemma}\n\n\\begin{proof}\nIf $i\\le 2q-2$, Theorem \\ref{theorem4.14} and the recursion in Definition \\ref{def4.15} give:\n\\begin{enumerate}\n\\item[(1)] $\\alpha_{2q-i-2}=Y(\\alpha_{2q-i-1})+\\alpha_{2q-i}$.\n\\item[(2)] $U_{i+2}(Y)=Y\\cdot U_{i+1}(Y)+U_{i}(Y)$.\n\\end{enumerate}\nSo, by induction on $i$, it suffices to show that $\\alpha_{2q}$ and $\\alpha_{2q-1}$ are $U_{0}(Y)(D_{40q^{2}+1})$ and $U_{1}(Y)(D_{40q^{2}+1})$.  But Lemma \\ref{lemma4.8} and Theorem \\ref{theorem4.14} show that $\\alpha_{2q}=0$ and $\\alpha_{2q-1}=Y\\cdot D_{40q^{2}+1}$. \n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem4.17}\nAs $Z\/2[Y]$-module, $DI(q)$ is cyclic with generator $\\alpha_{2q-1}=Y\\cdot D_{40q^{2}+1}$.\n\\end{theorem}\n\n\\begin{proof}\n$\\alpha_{2q-i}=U_{i}(Y)\\cdot D_{40q^{2}+1}$ for $1\\le i\\le 2q$. Since $Y$ divides each $U_{i}(Y)$ we're done.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem4.18}\n$Y^{2q-1}(\\alpha_{2q-1})=D$, while $Y^{2q}(\\alpha_{2q-1})=0$. Also, $DI(q)$ has dimension $2q$ over $Z\/2$ and is isomorphic as $Z\/2[Y]$-module with $Z\/2[Y]\/Y^{2q}$.\n\\end{theorem}\n\n\\begin{proof}\n$Y^{2q-1}(\\alpha_{2q-1})= Y^{2q}(D_{40q^{2}+1})=U_{2q}(Y)\\cdot D_{40q^{2}+1}$. By Lemma \\ref{lemma4.16} this is $\\alpha_{0}=D$. So $Y^{2q}(\\alpha_{2q-1})= T_{7}(D)=0$. It follows that the annihilator of $\\alpha_{2q-1}$ in $Z\/2[Y]$ is the ideal $(Y^{2q})$. Theorem \\ref{theorem4.17} then gives the final assertions.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem4.19}\nIf $p$ is as in Theorem \\ref{theorem4.10} then $T_{p}$ annihilates $DI(q)$. In particular, $X=T_{3}$ annihilates $DI(q)$.\n\\end{theorem}\n\n\\begin{proof}\nThis is immediate from Theorem \\ref{theorem4.10}\n\\qed\n\\end{proof}\n\n\\section{The action of \\bm{$T_{3}$} and \\bm{$T_{7}$} on \\bm{$W_{a}$}}\n\\label{section5}\n\nThis section is the counterpart to section 4 of \\cite{1}. In that paper we defined certain subspaces $W1$ and $W5$ of $Z\/2[[x]]$; see section \\ref{section1} of the present paper for a summary.  In section 4 of \\cite{1} we made $W5$ into a $Z\/2[X,Y]$-module with $X$ and $Y$ acting by $T_{7}$ and $T_{13}$, and we showed the existence of an ``adapted basis'' $m_{i,j}$ where $D=\\sum_{n>0,\\ (n,6)=1}x^{n^{2}}$ and $m_{0,0}=D^{5}$. Here we'll derive a similar result with $W5$, $T_{7}$ and $T_{13}$ replaced by $W_{a}$, $T_{3}$ and $T_{7}$.\n\nSince $\\chi(3)=\\chi(7)=1$, the commuting operators $T_{3}$ and $T_{7}:Z\/2[[x]]\\rightarrow Z\/2[[x]]$ stabilize $W_{a}$; see Corollary \\ref{corollary2.20}. So one can make $W_{a}$ into a $Z\/2[X,Y]$-module with $X$ and $Y$ acting by  $T_{3}$ and $T_{7}$. We shall filter $W_{a}$ by finite-dimensional $Z\/2[X,Y]$-stable subspaces, $W_{a}(q)$, $q$ running over the powers of $2$.\n\nIn the paragraph following Definition \\ref{def2.27} we constructed an isomorphism between $W_{a}$ and a certain subspace $V^{\\prime}$ of $Z\/2[w]$. Using this identification we may speak of the ``$w$-degree'' of an element of $W_{a}$. Theorem \\ref{theorem2.24} shows that the maps $X$ and $Y:W_{a}\\rightarrow W_{a}$ lower the $w$-degree.\n\n\\begin{definition}\n\\label{def5.1}\n$W_{a}(q)$ is the subspace of $W_{a}$, of $Z\/2$-dimension $8q^{2}$, consisting of elements of $w$-degree $<40q^{2}$. (Since $X$ and $Y$ lower the $w$-degree they stabilize $W_{a}(q)$.)\n\\end{definition}\n\n\\begin{theorem}\n\\label{theorem5.2}\nThe space $DI(q)$ of Definition \\ref{def4.12}, of $Z\/2$-dimension $2q$, is a $Z\/2[X,Y]$-submodule of $W_{a}(q)$ annihilated by $X$.\n\\end{theorem}\n\n\\begin{proof}\nBy Theorems \\ref{theorem4.17}, \\ref{theorem4.18} and \\ref{theorem4.19}, $DI(q)$ has $Z\/2$-dimension $2q$, is annihilated by $X$ and as $Z\/2[Y]$-module is cyclic generated by $\\alpha_{2q-1}=Y\\cdot D_{40q^{2}+1}$. So it suffices to show that $\\alpha_{2q-1}$ is in $W_{a}(q)$. But $Y$ lowers the $w$-degree.\n\\qed\n\\end{proof}\n\nWe'll use the results of section \\ref{section3} to show that the kernel of $X:W_{a}(q)\\rightarrow W_{a}(q)$ is precisely $DI(q)$. Note that $g(2q)=4q^{2}$. So in the language of section \\ref{section3}, $\\langle 0, 2q\\rangle=w^{40q^{2}+1}$, and so the $w$-degree of $(0,2q)^{*}$ is $40q^{2}+1$.\n\n\\begin{lemma}\n\\label{lemma5.3}\nSuppose $f\\ne 0$ is in $W_{a}(q)$, with $Xf = 0$. Write $f$ as $(a,b)^{*}+$ a sum of earlier $D_{i}$, as in section \\ref{section3}. Then $a=0$, and $0\\le b< 2q$.\n\\end{lemma}\n\n\\begin{proof}\nIf $a>0$, then by Theorem \\ref{theorem3.5}, $Xf =(a-1,b)^{*}+$ a sum of earlier $D_{i}$, and so cannot be $0$. So $f=(0,b)^{*}+$ a sum of earlier $D_{i}$. By Theorem \\ref{theorem3.6}, the $w$-degree of $f$ is the $w$-degree of $(0,b)^{*}$. If $b\\ge 2q$, then this last $w$-degree is $\\ge$ the $w$-degree, $40q^{2}+1$, of $(0,2q)^{*}$. So $f$ cannot be in $W_{a}(q)$.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem5.4}\nThe kernel of $X:W_{a}(q)\\rightarrow W_{a}(q)$ is $DI(q)$.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma \\ref{lemma5.3} this kernel has dimension $\\le 2q$, and we use Theorem \\ref{theorem5.2}.\n\\qed\n\\end{proof}\n\n\\begin{corollary}\n\\label{corollary5.5}\n$DI(1)\\subset DI(2)\\subset DI(4)\\subset \\ldots$, and the kernel of $X:W_{a}\\rightarrow W_{a}$ is the union, $DI$ of the $DI(q)$.\n\\end{corollary}\n\n\\begin{theorem}\n\\label{theorem5.6}\nThe only elements of $W_{a}$ annihilated by $X$ and $Y$ are $0$ and $D$.\n\\end{theorem}\n\n\\begin{proof}\nIf $(X,Y)f=0$, $f$ is in $DI$ by Corollary \\ref{corollary5.5}, and so is in some $DI(q)$. But $DI(q)$, as $Z\/2[Y]$-module, is isomorphic to $Z\/2[Y]\/(Y^{2q})$. It follows that the kernel of $Y:DI(q)\\rightarrow DI(q)$ has dimension $1$ over $Z\/2$. \n\\qed\n\\end{proof}\n\n\\begin{definition}\n\\label{def5.7}\n$S_{m}$ is the subspace of $W_{a}$. of dimension $m(m+1)\/2$, spanned over $Z\/2$ by the $(a,b)^{*}$ with $a+b<m$.\n\\end{definition}\n\nNote that $S_{0}=(0)$ while $S_{1}$ is spanned by $(0,0)^{*}=D$. So $X\\cdot S_{1}=Y\\cdot S_{1} = S_{0}$.\n\n\\begin{lemma}\n\\label{lemma5.8}\n$X:W_{a}\\rightarrow W_{a}$ is onto. In fact $X$ maps $S_{m+1}$ onto $S_{m}$.\n\\end{lemma}\n\n\\begin{proof}\nBy Theorem \\ref{theorem3.5}, $X\\cdot S_{m+1}\\subset S_{m}$, so it suffices to show that the kernel of $X: S_{m+1}\\rightarrow S_{m}$ has dimension at most $m+1$. Suppose $f\\ne 0$ is in this kernel. The proof of Lemma \\ref{lemma5.3} shows that $f=(0,b)^{*}+$ a sum of earlier $D_{i}$, and that the $w$-degree of $f$ is the $w$-degree of $(0,b)^{*}$. But Theorem \\ref{theorem3.6} tells us that every element of $S_{m+1}$ has $w$-degree $\\le$ the $w$-degree of $(0,m)^{*}$. So $0\\le b \\le m$, and the result follows.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem5.9}\n$Y\\cdot S_{m+1}\\subset S_{m}$.\n\\end{theorem}\n\n\\begin{proof}\nWe argue by induction on $m$, $m=0$ being clear. Suppose $f$ is in $S_{m+1}$ with $m>0$. Then $Xf$ is in $S_{m}$, so by the induction hypothesis, $X(Yf)=Y(Xf)$ is in $S_{m-1}$. By Lemma \\ref{lemma5.8}, there is an $h$ in $S_{m}$ with $Xh=XYf$. Then $h+Yf$, being in the kernel of $X$, is $(0,b)^{*}+$ a sum of earlier $D_{i}$ for some $b$. Since $h+Yf$ is in $S_{m+1}$, $b\\le m$, and it will suffice to show that $b\\ne m$. Suppose on the contrary that $h+Yf=(0,m)^{*}+$ a sum of earlier $D_{i}$. Then since $f$ is in $S_{m+1}$ and $Y$ lowers the $w$-degree, the $w$-degree of the left hand side is $<$ the $w$-degree of $(0,m)^{*}$. But that of the right hand side equals the $w$-degree of $(0,m)^{*}$, giving a contradiction.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma5.10}\nFor each $m$ there is an element of $DI$ of the form $(0,m)^{*}+$ a sum of earlier $D_{i}$.\n\\end{lemma}\n\n\\begin{proof}\nFix $q>m$. Each $f\\ne 0$ in $DI(q)$ can be written as $(0,b)^{*}+$ a sum of earlier $D_{i}$ for some $b$ with $0\\le b <2q$. Since there are only $2q$ choices for $b$, and $DI(q)$ has dimension $2q$, the result follows.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma5.11}\n$DI\\cap S_{m}$ has dimension $m$. Furthermore, $Y$ maps $DI\\cap S_{m+1}$ onto $DI\\cap S_{m}$.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lemma5.10}, $DI\\cap S_{m+1} \\ne DI\\cap S_{m}$. Now $Y$ maps $DI\\cap S_{m+1}$ into $DI\\cap S_{m}$ by Theorem \\ref{theorem5.9}; Theorem \\ref{theorem5.6} shows that the kernel of this map is contained in $\\{0,D\\}$. So the map is onto, and the dimensions of $DI\\cap S_{m+1}$ and $DI\\cap S_{m}$ differ by $1$.\n\\qed\n\\end{proof}\n\nThe machinery is now in place to establish the results analogous to those of section 4 of \\cite{1}. The arguments are exactly the same as those that were used to derive Theorems 4.12, 4.14, 4.15, 4.16, 4.17 and Corollary 4.13 of \\cite{1}.\n\n\\begin{theorem}\n\\label{theorem5.12}\nLet $f$ and $h$ be in $S_{m}$ with $Yf=Xh$. Then there is an $e$ in $S_{m+1}$ with $Xe=f$ and $Ye=h$.\n\\end{theorem}\n\n\n\\begin{corollary}\n\\label{corollary5.13}\nThere are $m_{a,b}$ in $S_{a+b+1}$ such that:\n\\vspace{-2ex}\n\\begin{enumerate}\n\\item[(1)] $m_{0,0}=D$.\n\\item[(2)] $X\\cdot m_{a,b}= m_{a-1,b}$ or $0$ according as $a>0$ or $a=0$.\n\\item[(3)] $Y\\cdot m_{a,b}= m_{a,b-1}$ or $0$ according as $b>0$ or $b=0$.\n\\item[(4)] The $m_{a,b}$ are a $Z\/2$-basis of $W_{a}$.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{theorem}\n\\label{theorem5.14}\nMake $W_{a}$ into a $Z\/2[[X,Y]]$-module with $X$ and $Y$ acting by $T_{3}$ and $T_{7}$. (This is possible since $T_{3}$ and $T_{7}$ lower the $w$-degree.) Then the action of $Z\/2[[X,Y]]$ on $W_{a}$ is faithful.\n\\end{theorem}\n\n\\begin{theorem}\n\\label{theorem5.15}\nIf $\\chi(p)=1$, $T_{p}:W_{a}\\rightarrow W_{a}$ is multiplication by some $u$ in the ideal $(X,Y)$ of $Z\/2[[X,Y]]$. In other words, $T_{p}$ in its action on $W_{a}$ is a power series with $0$ constant term in $T_{3}$ and $T_{7}$.\n\\end{theorem}\n\nWe next prove an analogue to Theorem 4.18 of \\cite{1}.  Since $\\chi(11)=-1$, $T_{11}(W_{a})\\subset W_{b}$, $T_{11}(W_{b})\\subset W_{a}$, and $T_{11}^{2}$ stabilizes $W_{a}$. We now decompose $W_{a}$ into a direct sum of 4 summands. The first is spanned by the $D_{k}$ with $k\\equiv 1$ or $9\\mod{40}$. For the second, $k\\equiv 3$ or $27$, for the third, $k\\equiv 7$ or $23$, and for the fourth, $k\\equiv 21$ or $29$. By Lemma \\ref{lemma2.10}, the exponents appearing in elements of the first summand are $1$ or $9\\bmod{40}$, and corresponding results hold for the other summands.\n\n\\begin{theorem}\n\\label{theorem5.16}\n$T_{11}^{2}:W_{a}\\rightarrow W_{a}$ is multiplication by $\\lambda^{2}$ for some $\\lambda$ in the ideal $(X,Y)$ of $Z\/2[[X,Y]]$. \n\\end{theorem}\n\n\\begin{proof}\nAs in the proofs of Theorem 4.16 and 4.17 of \\cite{1} we use the $Z\/2[[X,Y]]$-linearity of our map to show that it is multiplication by some $u$ in $(X,Y)$ and we write $u$ as $a+bX+cY+dXY$ with $a,b,c,d$ in $Z\/2[[X^{2},Y^{2}]]$. Let $a=\\lambda^{2}$. We'll show that for each $h$ in $W_{a}$, $T_{11}^{2}(h)=\\lambda^{2}h$. We may assume that $h$ is in one of the 4 subspaces of the above direct sum decomposition. Suppose for example that it is in the second. Then $T_{11}^{2}(h)$ and $ah$ are sums of $D_{k}$, $k\\equiv 3$ or $27\\mod{40}$, $(bX)h$ is a sum of $D_{k}$, $k\\equiv 1$ or $9\\mod{40}$, $(cY)h$ is a sum of $D_{k}$, $k\\equiv 21$ or $29\\mod{40}$ and $(dXY)h$ is a sum of $D_{k}$, $k\\equiv 7$ or $23\\mod{40}$.  Since $T_{11}^{2}(h) = uh$ is the sum of $\\lambda^{2}h$, $(bX)h$, $(cY)h$ and $(dXY)h$, and the decomposition is direct, the result follows.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma5.17}\nWrite the $\\lambda$ of Theorem \\ref{theorem5.16} as $cX+dY+\\cdots$, with $c$ and $d$ in $\\{0,1\\}$. Then $c=d=1$.\n\\end{lemma}\n\\pagebreak\n\n\\begin{proof}\nOn $W_{a}$, $T_{11}^{2}=\\lambda^{2}=cX^{2}+dY^{2}+\\cdots$. Applying both sides to $D_{9}$ we find that $0=cD_{1}+dD_{1}$, while applying both sides to $D_{41}$ shows that $D_{1}=dD_{1}$.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem5.18}\n$T_{11}^{2}$ maps $S_{m+2}$ onto $S_{m}$ for all $m$. So $T_{11}^{2}:W_{a}\\rightarrow W_{a}$ is onto.\n\\end{theorem}\n\n\\begin{proof}\nUsing the explicit description of the action of $T_{3}$ and $T_{7}$ on $W_{a}$ provided by Corollary \\ref{corollary5.13} we see that $X^{2}+Y^{2}$ maps $S_{m+2}$ onto $S_{m}$. Since $T_{11}^{2}=X^{2}+Y^{2}+\\cdots$, $T_{11}^{2}$ and $X^{2}+Y^{2}$ induce the same map $S_{m+3}\/S_{m+2}\\rightarrow S_{m+1}\/S_{m}$. So this map is onto for all $m$, and an induction gives the result.\n\\qed\n\\end{proof}\n\n\\section{\\bm{$T_{11}:W_{b}\\rightarrow W_{a}$} is bijective}\n\\label{section6}\nSince $\\chi(11)=-1$, $T_{11}$ maps $W_{a}$ to $W_{b}$ and $W_{b}$ to $W_{a}$. In this section we show that $T_{11}(D_{k})$ is a sum of $D_{i}$, $i<k$, and that $T_{11}:W_{b}\\rightarrow W_{a}$ is bijective. Let $C_{k}$, $(k,10)=1$, be $T_{11}(D_{k})$.\n\n\\begin{lemma}\n\\label{lemma6.1}\nIf $k<120$ then:\n\\vspace{-2ex}\n\\begin{enumerate}\n\\item[(a)] When $k\\equiv 1,3,7$ or $9\\mod{20}$, $C_{k}$ is a sum of $D_{i}$, $i<k$.\n\\item[(b)] When $k\\equiv 11\\ (\\mbox{resp.}\\ 19,13,17) \\bmod{20}$, $C_{k}=D_{j}+$ a sum of $D_{i}$, $i<j$, where $j=k-10\\ (\\mbox{resp.}\\ k-10,k-6,k-14)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proofsketch}\nUsing Corollary \\ref{corollary2.20} and Lemmas \\ref{lemma2.22} and \\ref{lemma2.23} we see that it suffices to prove (a) when $k\\equiv 9\\mod{20}$, i.e.\\ when $k$ is in $\\{9,29,49,69,89,109\\}$. In fact the corresponding $C_{k}$ are $0$, $0$, $D_{19}+D_{11}$, $D_{39}+D_{31}$, $D_{11}$ and $D_{39}$. To calculate $C_{109}$, for example, we use Corollary \\ref{corollary2.20} and Lemma \\ref{lemma2.23} to see that it is a sum of $D_{i}$ with each $i$ in $\\{31,39,71,79,111\\}$. Examining the expansions of $D_{109}$ and the $D_{i}$ and arguing as in the proof of Theorem \\ref{theorem2.26} we find that $C_{109}=D_{39}$. (b) follows from the more precise results:\n\\begin{enumerate}\n\\addtolength{\\itemindent}{.2em}\n\\item[(b1)] $C_{11}$, $C_{31}$, $C_{51}$, $C_{71}$, $C_{91}$ and $C_{111}$ are $D_{1}$, $D_{21}$, $D_{41}+D_{9}$, $D_{61}+D_{29}$, $D_{81}+D_{41}+D_{9}$, $D_{101}+D_{61}+D_{21}$.\n\n\\item[(b2)] $C_{19}$, $C_{39}$, $C_{59}$, $C_{79}$, $C_{99}$ and $C_{119}$ are $D_{9}$, $D_{29}$, $D_{49}+D_{9}$, $D_{69}+D_{29}$, $D_{89}+D_{49}+D_{9}$, $D_{109}+D_{69}+D_{29}+D_{21}$.\n\n\\item[(b3)] $C_{13}$, $C_{33}$, $C_{53}$, $C_{73}$, $C_{93}$ and $C_{113}$ are $D_{7}$, $D_{27}+D_{3}$, $D_{47}$, $D_{67}+D_{43}+D_{27}$, $D_{87}+D_{47}$, $D_{107}+D_{83}+D_{67}+D_{43}+D_{27}$.\n\n\\item[(b4)] $C_{17}$, $C_{37}$, $C_{57}$, $C_{77}$, $C_{97}$ and $C_{117}$ are $D_{3}$, $D_{23}+D_{7}$, $D_{43}$, $D_{63}+D_{47}+D_{23}+D_{7}$, $D_{83}+D_{43}$, $D_{103}+D_{87}+D_{63}+D_{47}+D_{23}$.\n\\end{enumerate}\n\\vspace{-4ex}\n\\hspace{2em}\\\\\nTo calculate $C_{93}$, for example, we use Corollary \\ref{corollary2.20} and Lemma \\ref{lemma2.22} to see that it is a sum of $D_{i}$ where each $i$ is in $\\{7,23,47,63,87\\}$. Examining the expansions of $D_{93}$ and the $D_{i}$ and arguing as in the proof of Theorem \\ref{theorem2.26} we find that $C_{93}=D_{87}+D_{47}$.\n\\qed\n\\end{proofsketch}\n\n\\begin{lemma}\n\\label{lemma6.2}\nFor $u$ in $Z\/2[[x]]$, $T_{11}(uG^{24})$ is the sum of $G^{2i}T_{11}(uG^{2j})$ where $(i,j)$ runs over the 9 pairs $(12,0)$, $(8,4)$, $(4,8)$, $(6,2)$, $(2,6)$, $(9,1)$, $(1,9)$, $(3,3)$, $(1,1)$.\n\\end{lemma}\n\n\\begin{proof}\nWe argue as in the proof of Lemma \\ref{lemma2.25}. Let $U$ be the 2 variable polynomial $(A+B)^{12}+A^{6}B^{2}+A^{2}B^{6}+A^{9}B+AB^{9}+A^{3}B^{3}+AB$. Then $U(F(x^{11}), F(x))=0$; this is the level 11 modular equation for $F$. Replacing $x$ by $x^{5}$ we find that $U(G(x^{11}), G(x))=0$. Now let $L$ be an algebraic closure of $Z\/2$. We have 12 imbeddings $\\varphi_{k}:Z\/2[[x]]\\rightarrow L[[x^{1\/11}]]$, the first of which takes $f$ to $f(x^{11})$, while each of the others takes $f$ to $f(\\lambda x^{1\/11})$ for some $\\lambda$ in $L$ with $\\lambda^{11}=1$. Replacing $x$ by $\\lambda x^{1\/11}$ in the identity $U(G(x^{11}), G(x))=0$ and using the symmetry of $U$ we find that each $U(\\varphi_{k}(G),G)$ is $0$. Squaring and expanding we find that $\\varphi_{k}(G^{24})$ is the sum of the $G^{2i}\\varphi_{k}(G^{2j})$ where $(i,j)$ runs over the 9 pairs above. Now the definition of $T_{11}$ shows that if $f$ is in $Z\/2[[x]]$, $T_{11}(f)$ is the sum of the $\\varphi_{k}(f)$. Multiplying the $k$\\textsuperscript{th} of our identities by $\\varphi_{k}(u)$ and summing we get the result.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma6.3}\nThe conclusions (a) and (b) of Lemma \\ref{lemma6.1} hold for all $k$ with $(k,10)=1$.\n\\end{lemma}\n\n\\begin{proof}\nWe argue by induction on $k$. For $k<120$, Lemma \\ref{lemma6.1} applies. Suppose $k=n+120$ with $n>0$. By Lemma \\ref{lemma6.2}, $C_{n+120}$ is a sum of 9 terms each corresponding to one of the 9 pairs $(i,j)$ of the lemma. These terms are $G^{24}C_{n}$, $G^{16}C_{n+40}$, $G^{8}C_{n+80}$, $G^{12}C_{n+20}$, $G^{4}C_{n+60}$, $G^{18}C_{n+10}$, $G^{2}C_{n+90}$, $G^{6}C_{n+30}$ and $G^{2}C_{n+10}$. The induction hypothesis shows that the term corresponding to $(i,j)$ is a sum of $D_{m}$ with $m<(n+10j)+10i$. Since $i+j\\le 12$, each $m<n+120$, and in particular (a) holds for $k$. We turn to (b). For each of the last 6 terms in our sum, $i+j\\le 10$. So the corresponding term is a sum of $D_{m}$ with $m<n+100$. Now consider the first 3 terms. Suppose for example that $n\\equiv 17\\mod{20}$. By the induction hypothesis, each of $G^{24}C_{n}$, $G^{16}C_{n+40}$ and $G^{8}C_{n+80}$ is $D_{n+106}+$ a sum of $D_{m}$ with $m<n+106$. It follows that $C_{n+120}$ itself is $D_{n+106}+$ a sum of $D_{m}$ with $m<n+106$. The proof of (b) when $n\\equiv 11,19$ or $13\\mod{20}$ is identical.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem6.4}\nIf $(k,10)=1$, $T_{11}(D_{k})$ is a sum of $D_{i}$ with $i<k$. Furthermore, $T_{11}:W_{b}\\rightarrow W_{a}$ is bijective.\n\\end{theorem}\n\n\\begin{proof}\nLemma \\ref{lemma6.3} gives the first result. Also the $D_{k}$ where $k>0$ and $\\equiv 11,19,13$ or $17\\mod{20}$ form a basis of $W_{b}$, while the corresponding $D_{k-10}$, $D_{k-10}$, $D_{k-6}$ and $D_{k-14}$ form a basis of $W_{a}$. So Lemma \\ref{lemma6.3} gives the second result as well.\n\\qed\n\\end{proof}\n\n\\begin{corollary}\n\\label{corollary6.5}\n$T_{11}:W_{a}\\rightarrow W_{b}$ is onto.\n\\end{corollary}\n\n\\begin{proof}\nBy Theorem \\ref{theorem5.18}, $T_{11}$ maps the subspace $T_{11}(W_{a})$ of $W_{b}$ onto $W_{a}$, and we use the last theorem to see that $T_{11}(W_{a})$ is all of $W_{b}$.\n\\qed\n\\end{proof}\n\n\\begin{corollary}\n\\label{corollary6.6}\nLet $\\lambda$ be as in Theorem \\ref{theorem5.16}. Then in its action on $W=W_{a}\\oplus W_{b}$, $T_{11}^{2}$ is multiplication by $\\lambda^{2}$. (This follows from Corollary \\ref{corollary6.5} and Theorem \\ref{theorem5.16}.)\n\\end{corollary}\n\n\\section{The algebra \\bm{$\\ensuremath{\\mathcal{O}}$} acting on \\bm{$W_{a}$} }\n\\label{section7}\n\nTake $\\lambda$ in $(X,Y)$ as in Theorem \\ref{theorem5.16}, and let $U:W\\rightarrow W$ be the map $h\\rightarrow \\lambda(X,Y)h+T_{11}(h)$. Since $U$ is $Z\/2[[X,Y]]$-linear, and $U^{2}$ annihilates $W$ (by Corollary \\ref{corollary6.6}), $W$ has the structure of $Z\/2[[X,Y]][\\varepsilon]$ module with $\\varepsilon^{2}=0$, $\\varepsilon$ acting by $U$, and $X$ and $Y$ by $T_{3}$ and $T_{7}$. Let $\\ensuremath{\\mathcal{O}}$ be the local ring  $Z\/2[[X,Y]][\\varepsilon]$.\n\n\\begin{lemma}\n\\label{lemma7.1}\nThe only element of $W_{b}$ annihilated by $\\varepsilon$ is $0$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $\\varepsilon h_{b} =0$, with $h_{b}$ in $W_{b}$. Then $T_{11}(h_{b})=\\lambda(T_{3},T_{7})h_{b}$. Since the first of these is in $W_{a}$ and the second in $W_{b}$, $T_{11}(h_{b})=0$. By Theorem \\ref{theorem6.4}, $h_{b}=0$.\n\\qed\n\\end{proof}\n\nIn Corollary \\ref{corollary5.13} we constructed a $Z\/2$-basis $m_{i,j}$ of $W_{a}$ ``adapted to $X=T_{3}$ and $Y=T_{7}$.''  Since $T_{11}:W_{b}\\rightarrow W_{a}$ is bijective, this basis pulls back under $T_{11}$ to a basis $n_{i,j}$ of $W_{b}$ ``adapted to $T_{3}$ and $T_{7}$.'' \n\n\\begin{theorem}\n\\label{theorem7.2}\\hspace{2em}\\\\\n\\vspace{-2ex}\n\\begin{enumerate}\n\\item[(1)] The $n_{i,j}$ and the $\\varepsilon n_{i,j}$ form a $Z\/2$-basis of $W$.\n\\item[(2)] $\\ensuremath{\\mathcal{O}}$ acts faithfully on $W$.\n\\item[(3)] Each $T_{p}$, $p\\ne 2$ or $5$, acts on $W$ by multiplication by some $r+t\\varepsilon$ with $r$ and $t$ in $Z\/2[[X,Y]]$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nThe arguments are just like those giving Theorems 5.2, 5.3 and 5.4 of \\cite{1}, using the basis $n_{i,j}$ of $W_{b}$.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem7.3}\\hspace{2em}\\\\\n\\vspace{-2ex}\n\\begin{enumerate}\n\\item[(1)] If $\\chi(p)=1$, $T_{p}:W\\rightarrow W$ is multiplication by some $t$ in the maximal ideal $(X,Y)$ of $Z\/2[[X,Y]]$.\n\\item[(2)] If $\\chi(p)=-1$, $T_{p}:W\\rightarrow W$ is the composition of $T_{11}$ with multiplication by some $t$ in the maximal ideal $(X,Y)$ of $Z\/2[[X,Y]]$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nSuppose $\\chi(p)=1$. By Theorem \\ref{theorem5.15}, $T_{p}:W_{a}\\rightarrow W_{a}$ is multiplication by some $t$ in the maximal ideal $(X,Y)$. Since $T_{11}:W_{a}\\rightarrow W_{b}$ is onto, $T_{p}:W_{b}\\rightarrow W_{b}$ is multiplication by the same $t$, and we get (1). Now $T_{11}$ is multiplication by $\\lambda+\\varepsilon$. Suppose $\\chi(p)=-1$. By Theorem \\ref{theorem7.2}, $T_{p}$ is multiplication by some $r+t\\varepsilon$ with $r$ and $t$ in $Z\/2[[X,Y]]$.  Then $T_{p}+tT_{11}$ is multiplication by $r+\\lambda t$. Since $T_{p}+tT_{11}$ and multiplication by $r+\\lambda t$ map $W_{b}$ into $W_{a}$ and $W_{b}$ respectively, $T_{p}+tT_{11}=0$, giving (2).\n\\qed\n\\end{proof}\n\n\\begin{corollary}\n\\label{corollary7.4}\nIf $p\\ne 2$ or $5$ and $(k,10)=1$, then $T_{p}(D_{k})$ is a sum of $D_{i}$ with $i<k$.\n\\end{corollary}\n\n\\begin{proof}\nWe have seen that this holds when $p=3$, $7$ or $11$; Theorem \\ref{theorem7.3} then gives the general result.\n\\qed\n\\end{proof}\n\nTheorems \\ref{theorem7.2} (2) and \\ref{theorem7.3} tell us that when we complete the Hecke algebra generated by the $T_{p}$ acting on $W$ with respect to the maximal ideal generated by the $T_{p}$, then the completed Hecke algebra we get is just the (non-reduced) local ring $\\ensuremath{\\mathcal{O}}$. This is completely analogous to the results of \\cite{1}; see Theorems 5.3 and 5.5 of that paper.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}  \n\nImages collected with ground based telescopes show temporal variation in image quality, primarily due to fluctuations in the index of refraction of the air along the light path.  These index of refraction fluctuations are caused by differences in air temperature arising from turbulent mixing. These turbulent motions happen on smaller scales than the gross motions of the winds. Still it is known that the wind has a part in the strength of these fluctuations.  Hufnagel \\cite{hufnagel1974} derived a model for the vertical profile of strength of turbulence $C_n^2(z)$ which depended on a wind speed profile centered at a particular altitude. Furthermore, it has been observed that the index of refraction fluctuations are enhanced where there are shears in winds \\cite{beland1993}, so that direction as well as speed can be a factor, although this is difficult to quantify \\cite{dewan1993}.\n\nAside from the effects of wind on turbulence strength, the movement of turbulent regions past an observer induces temporal variations in seeing. If wind speeds are sufficiently high that turbulence does not evolve significantly over a measurement period, we apply Taylor's hypothesis and treat the temperature fluctuations as quantities that are advected by the wind \\cite{taylor1938}. \n\nThe following wind model is often used in the modeling of adaptive optics and other imaging systems (\\textit{e.g.} \\cite{parenti1994, andrews1998,tyson1998}),\n\n\\begin{equation}\n\\label{greenwood_fit}\n   v(z) = 8 + 30 \\exp\\left(-\\left[\\frac{(z\\sin\\theta -9400)}{4800}\\right]^2\\right),\n\\end{equation}\n\n\\noindent where $v(z)$ is the wind speed in m\/s, $z$ is the height in m, and $\\theta$ is the angle from the zenith. Bufton \\cite{bufton1973} is often cited as the source for this model (\\textit{e.g.} \\cite{parenti1994, andrews1998,tyson1998}), but that paper does not present an explicit wind model. \n\nThe basic form of the model was presented by Greenwood \\cite{greenwood1977}.  Readers often over look the statement in the body of the original paper that the height $z=0$ corresponds to a mean sea level altitude of 3048 m.  This is the height of the Maui Space Surveillance System on Haleakala, where the analysis was being used.  So the height of the tropopause term, 9400 m, should really be 12448 m when the model is used for sites other than Haleakala.   This caveat has been overlooked in a number of references, leading to misleading results.\n\nThe model from Greenwood \\cite{greenwood1977} was derived by averaging radiosonde data collected from balloons launched from Lihue on the island of Kauai from 1950-1970 and from Hilo on the island of Hawaii from 1950-1974.  This was more fully detailed in a technical report, which did not explicitly generalize the equation, but the authors clearly understood what the various terms signified  \\cite{greenwood1975}.  \n\n\nThe wind model is generalized as,\n\n\\begin{equation}\n\\label{gaussian_fit}\nv(z)=v_G + v_T\\exp\\left(-\\left[\\frac{z\\cos\\zeta-h_T}{L_T}\\right]^2\\right)\n\\end{equation}\n\n\n\\noindent where $v$ is the wind speed, $z$ is the altitude,  $v_G$ is the wind speed at the ground or low altitude, $v_T$ is the wind speed at tropopause, $h_T$ is the altitude of the tropopause, $L_T$ is the thickness of the tropopause layer and $\\zeta$ is the zenith angle of the observation \\cite{hardy1998}.\n\n\nThere are several uses for a wind profile.  It can be used in conjunction with a turbulence model to predict the performance of instrumentation.  Most commonly this is applied to the servo bandwidth of closed loop systems  \\cite{greenwood1977}.  The minimum bandwidth of the system (to limit servo lag) is related to the Greenwood frequency. Because the Greenwood frequency depends on a product of $C_n^2$ and a power of the wind speed $v^{5\/3}$, it is important to get the wind profile and turbulence profiles as a function of altitude.  A wind profile can also be used to simulate turbulence in conjunction with phase screens.   \n\n \nCurrently, there is an increased emphasis on modeling effects of turbulence on telescopes with diameters of 20-50 meters. Significantly more spatial variation of wavefront optical path difference is captured with such large telescopes in a short exposure than on a smaller telescope. Some of that variation is due to the presence of multiple layers of turbulence, moving at different speeds and in different directions. The patterns may not simply translate across the telescope aperture as implied by the Taylor hypothesis, but are more likely to``boil\" with a complicated wind profile. This can have an impact on the design and performance of an adaptive optics system.   \n\nIn order to better model atmospheric turbulence, we used archival radiosonde data to compute the wind profiles for a number of astronomical observatories.  We also show that the wind profile has a strong seasonal variation. We also extend the wind profiles, by including wind direction. Unfortunately, the wind directions do not always lend themselves to an analytic expression, so we only show them in graphs.\n\nUsing the same radiosonde data, we examined the variation of the gradient Richardson number \\cite{kundu1990}), which is an indicator of the stability of parcels of air. Since it depends upon strength of wind shear, as opposed to wind speed, it also serves to inform the modeler that wind speed alone is not sufficient. As will be seen in our plots, the wind speed can be modeled with a few simple terms, but the form of the Richardson number plots indicates the true nature of atmospheric turbulence: layered, with sharp boundaries (although these are muted by the resolution of our data).\n\n\\section{Data Analysis}\\label{data_analysis}\n\nWorldwide there are over 900 sites that launch radiosondes on a routine basis. Radiosondes are small instrument packages mounted on weather balloons.  They record atmospheric pressure, temperature, potential temperature, mixing ratio, dew point, and relative humidity as a function of height. Data are taken at about 6 second intervals, but databases may only have the data recorded at 60 second intervals. Readings at certain specific standard barometric pressures are made regardless of when they occur.  Since the balloons rise at about 5 m\/s, the best vertical resolution would be approximately\n30 meters, while the usual recorded resolution would be 300 meters and frequently the recorded data are at irregular intervals.  Wind speed and wind direction are obtained from either radar observation or a navigation system such as LOng RAnge Navigation (LORAN) and newer systems may use the Global Positioning System. Almost all of these sites launch balloons at 0 UT and 12 UT. The information is used as entries to weather prediction and simulation programs. \n\n\n\nWe downloaded atmospheric sounding data for 1973 January till 2006 September for a number of stations from the University of Wyoming's Weather Web (http:\/\/weather.uwyo.edu\/upperair\/sounding.html).  The stations were chosen based upon their proximity to an astronomical observatory.   Not all the sites had data for the entire time period; the dates for the available data are also listed in the table. In addition there are gaps of varying length in each data set, which are most likely due to equipment problems.  These data drop outs are not a significant problem due to the large number of data points that are available.  The names of stations and their details along with the nearby observatories are listed in Table \\ref{station_table}. The name of the station is given, along with its latitude, longitude and altitude, the dates when data were recorded and the names of nearby observatories. In parenthesis are the distance in km between the observatory and the radiosonde launch site and the altitude of the observatory in km. \n\n\n\n\n\\begin{table}[htb]\n{\\bf \\caption{\\label{station_table}Details of the radiosonde launch sites used}}\n\\begin{center}\n\\begin{tabular}{lcccclcc}\n\\hline\nStation     & Lat.         & Lon.        & Alt.& Dates & Nearby & Dist. & Alt.\\\\\n            &   ($^\\circ$) &  ($^\\circ$) &  (m) &  & Observatories& (km) & (km)\\\\\n\\hline\nAntofagasta & -23.43  & -70.43    &    135    & 1973-2006      & Paranal & 133  &  2.3  \\\\\n  &   &   &         &     &                                     Armazones &132 &  3.0 \\\\\nFlagstaff     &  35.23  &  -111.82  &   2192    & 1995-2006      & Lowell &14  &  2.2 \\\\\nHilo           &  19.71  &   -155.06  &   11     & 1973-2006      & Haleakala &168 &   3.1 \\\\\n  &   &   &         &     &                                       Mauna Kea &46  &  4.4\\\\\nOakland        &  37.72  &   -122.20  &   3      & 1973-2006      & Lick &65  &  1.3 \\\\\nSan Diego     &  32.84  &   -117.11  &   128    & 1990-2006      & Palomar &61  &  1.7 \\\\\nTenerife    &  28.47  &   -16.38   &   105    & 2002-2006      & R. Muchachos &150 &  2.3  \\\\\n &   &   &         &     &                                         Teide &23  &  2.4 \\\\\nTucson         &  32.11  &   -110.93  &   779    & 1973-2006      & Kitt Peak &64  &  2.1 \\\\\n &   &   &         &     &                                         Mt. Graham &117 &  3.1\\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFor each sounding the pressure, temperature, dew point, relative humidity, wind speed and direction are measured as a function of height.   The heights are irregularly measured and not repeated, so the data were interpolated onto a fixed height grid with an increment of 400 m and the mean was computed for various time periods.  The height grid is measured from mean sea level, rather than the local surface.  The different soundings have data measured to varying maximum altitudes, depending on when the balloons burst.  We did not analyze any data from altitudes higher than 30 km, since there were too few data points for a meaningful analysis. \n\nFig. \\ref{speed_variations} shows the yearly variations that occur for January and July in Hilo, Hawaii.   Plotted over this are the mean wind speeds for the respective month for the entire 33 year period of data. This shows the variation in monthly mean wind speeds.  Some of this difference is caused by large scale weather patterns and some is the natural chaotic behavior of weather. The overall shape of the wind profiles is highly consistent over the 33 years of data.   \n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=4.5cm]{roberts_fig1a.eps}\n\\includegraphics[height=4.5cm]{roberts_fig1b.eps}\n\\caption{\\label{speed_variations}   The solid line in the figure on the left is the model fit to the mean monthly wind speed for Hilo, Hawaii in the month of January. The dotted lines are the mean wind speed for Hilo, Hawaii in each January from 1973-2006.  The figure on the right is the same, but for July.  These figures illustrate the variation in the monthly mean wind speed.}\n\\end{figure}\n \n\n\nIn addition to the sites listed in Table \\ref{station_table}, we also downloaded the wind profiles for Lihue HI.  As mentioned above this was used in Greenwood's original model.  We computed the correlation coefficient between the Hilo and Lihue data sets for each month of the mean wind profiles.  The correlation was highest in February (0.999) and lowest in August (0.981).  The average correlation was 0.992 with a standard deviation of 0.006.  This correlation was high enough that we felt it was valid to only use the Hilo data.  This does not say that Hilo and Lihue constantly experience the exact same wind conditions, but that on average the wind profiles are very similar. This also shows that it is valid to use the same wind model for the entire state of Hawaii including the observatories on Haleakala and Mauna Kea. \n\nSurface winds are highly dependent on local geography, such as being in the lee of a mountain, and are also time dependent due to on-shore and off-shore breezes for the island sites and katabatic flows for sites near mountains.  They are also dependent on the solar heating of the ground.    This is less of a problem for astronomical uses of the wind profiles since mountain top observatories are usually situated above these surface effects. In addition, the wind profiles do not capture the effect that the local orography will have on the wind profile.\n\nWe created the mean wind speed and direction for each of the 12 months for each of the sites listed in Table \\ref{station_table}.  We used a non-linear least squares fit algorithm to fit the data to\n\n\\begin{equation}\n\\label{gaussian_fit_eqn}\n   v(z) = A_0 + A_1 \\exp\\left(-\\left[\\frac{(z-A_2)}{A_3}\\right]^2\\right).\n\\end{equation}\n\n\\noindent Tables \\ref{hilo_coeffs}-\\ref{anto_coeffs} list the monthly coefficients and their associated error bars for all sites listed in Table \\ref{station_table}.  The error bars are computed by the non-linear least squares fit.  The errors bars increase as the winds vary from the pure Gaussian model. \nWe fit the Gaussian to the wind speed data from the ground to a height of 16 km, beyond this the stratospheric winds start to affect the fit.  The effect is marginal and often below the error bars, except for the summer data sets.\n\n The Gaussian wind model of \\cite{greenwood1977} (Eq. \\ref{greenwood_fit}) only fit the troposphere winds and ignored the stratospheric winds.  It is possible to fit both of those winds if a Gaussian is summed with a second order polynomial.  This has limited utility to astronomical imaging, as the turbulence above 20km is relatively weak, but high altitude airships are being designed to operate between 20 and 30 km \\cite{fesen2006} and require  a knowledge of the wind speeds and directions in this region. \n\n\n\n\\begin{table}[htbp]\n{\\bf \\caption{\\label{hilo_coeffs}Hawaii Wind Speed Model Coefficients}}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\nMonth& A0 & A1 & A2 & A3 \\\\\n\\hline\nJan. &  4.1 $\\pm$  0.4 &    25.8 $\\pm$  0.5 &    12007 $\\pm$   69 &    4047 $\\pm$ 115\\\\\nFeb. &  4.2 $\\pm$  0.4 &    27.6 $\\pm$  0.4 &    11952 $\\pm$   61 &    3957 $\\pm$  99\\\\\nMar. &  5.0 $\\pm$  0.3 &    29.6 $\\pm$  0.4 &    12335 $\\pm$   55 &    3405 $\\pm$  79\\\\\nApr. &  5.0 $\\pm$  0.3 &    29.4 $\\pm$  0.4 &    12776 $\\pm$   51 &    3001 $\\pm$  68\\\\\nMay. &  4.8 $\\pm$  0.4 &    22.9 $\\pm$  0.5 &    12909 $\\pm$   88 &    3161 $\\pm$ 117\\\\\nJun. &  4.8 $\\pm$  0.3 &    17.0 $\\pm$  0.5 &    12637 $\\pm$   95 &    2877 $\\pm$ 127\\\\\nJul. &  5.0 $\\pm$  0.3 &    14.0 $\\pm$  0.5 &    12406 $\\pm$  102 &    2605 $\\pm$ 133\\\\\nAug. &  4.6 $\\pm$  0.3 &    12.8 $\\pm$  0.5 &    12527 $\\pm$   99 &    2575 $\\pm$ 128\\\\\nSep. &  4.2 $\\pm$  0.3 &    13.6 $\\pm$  0.4 &    12263 $\\pm$   93 &    2868 $\\pm$ 126\\\\\nOct. &  4.5 $\\pm$  0.3 &    17.2 $\\pm$  0.4 &    12222 $\\pm$   75 &    2845 $\\pm$ 101\\\\\nNov. &  5.1 $\\pm$  0.3 &    17.4 $\\pm$  0.5 &    12313 $\\pm$   92 &    3214 $\\pm$ 129\\\\\nDec. &  5.0 $\\pm$  0.4 &    21.1 $\\pm$  0.5 &    12199 $\\pm$   84 &    3617 $\\pm$ 126\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\\begin{table}[htbp]\n{\\bf \\caption{\\label{oakland_coeffs}Oakland Wind Speed Model Coefficients}}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\nMonth& A0 & A1 & A2 & A3\\\\\n\\hline\nJan. & -1.4 $\\pm$  1.8 &    30.5 $\\pm$  1.7 &    10730 $\\pm$   81 &    8329 $\\pm$ 318\\\\\nFeb. & -0.7 $\\pm$  1.6 &    30.0 $\\pm$  1.5 &    10695 $\\pm$   77 &    8047 $\\pm$ 283\\\\\nMar. & -0.9 $\\pm$  1.4 &    29.5 $\\pm$  1.3 &    10548 $\\pm$   66 &    7878 $\\pm$ 245\\\\\nApr. &  0.6 $\\pm$  1.0 &    28.3 $\\pm$  1.0 &    10472 $\\pm$   59 &    7195 $\\pm$ 188\\\\\nMay. &  2.8 $\\pm$  0.7 &    22.8 $\\pm$  0.7 &    10649 $\\pm$   70 &    6367 $\\pm$ 170\\\\\nJun. &  3.1 $\\pm$  0.7 &    19.7 $\\pm$  0.7 &    11164 $\\pm$   98 &    6430 $\\pm$ 215\\\\\nJul. &  4.0 $\\pm$  0.6 &    16.5 $\\pm$  0.6 &    11581 $\\pm$  122 &    5540 $\\pm$ 210\\\\\nAug. &  4.4 $\\pm$  0.5 &    17.3 $\\pm$  0.6 &    11602 $\\pm$  113 &    5187 $\\pm$ 184\\\\\nSep. &  3.2 $\\pm$  0.6 &    18.8 $\\pm$  0.6 &    11400 $\\pm$   95 &    6240 $\\pm$ 192\\\\\nOct. &  1.3 $\\pm$  0.8 &    22.3 $\\pm$  0.8 &    11073 $\\pm$   79 &    7297 $\\pm$ 217\\\\\nNov. & -2.0 $\\pm$  1.6 &    32.8 $\\pm$  1.5 &    10878 $\\pm$   72 &    8251 $\\pm$ 263\\\\\nDec. & -1.8 $\\pm$  2.0 &    31.1 $\\pm$  1.8 &    10611 $\\pm$   81 &    8358 $\\pm$ 333\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\begin{table}[htbp]\n{\\bf \\caption{\\label{sandiego_coeffs}San Diego  Wind Speed Model Coefficients}}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\nMonth& A0 & A1 & A2 & A3\\\\\n\\hline\n Jan. & -2.9 $\\pm$  1.4 &    33.9 $\\pm$  1.3 &    11470 $\\pm$   76 &    6104 $\\pm$ 249\\\\\n Feb. &  1.4 $\\pm$  1.1 &    32.4 $\\pm$  1.0 &    11400 $\\pm$   81 &    5081 $\\pm$ 199\\\\\n Mar. & -1.3 $\\pm$  1.2 &    32.7 $\\pm$  1.1 &    11701 $\\pm$   82 &    5793 $\\pm$ 226\\\\\n Apr. & -2.4 $\\pm$  1.4 &    32.7 $\\pm$  1.3 &    11656 $\\pm$   88 &    6062 $\\pm$ 266\\\\\n May. &  0.2 $\\pm$  1.2 &    25.6 $\\pm$  1.1 &    11577 $\\pm$  113 &    5471 $\\pm$ 295\\\\\n Jun. &  3.6 $\\pm$  0.9 &    17.6 $\\pm$  0.9 &    12081 $\\pm$  201 &    4546 $\\pm$ 364\\\\\n Jul. &  3.7 $\\pm$  0.6 &    10.6 $\\pm$  0.6 &    11483 $\\pm$  180 &    3909 $\\pm$ 317\\\\\n Aug. &  4.4 $\\pm$  0.4 &    10.5 $\\pm$  0.5 &    11821 $\\pm$  161 &    3318 $\\pm$ 239\\\\\n Sep. &  4.4 $\\pm$  0.6 &    18.8 $\\pm$  0.6 &    11897 $\\pm$  121 &    3745 $\\pm$ 193\\\\\n Oct. &  2.2 $\\pm$  0.8 &    22.7 $\\pm$  0.7 &    11805 $\\pm$  111 &    4843 $\\pm$ 230\\\\\n Nov. & -1.6 $\\pm$  1.3 &    30.6 $\\pm$  1.2 &    11511 $\\pm$   88 &    5776 $\\pm$ 256\\\\\n Dec. & -1.3 $\\pm$  1.2 &    32.2 $\\pm$  1.1 &    11354 $\\pm$   76 &    5639 $\\pm$ 223\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\begin{table}[htbp]\n{\\bf \\caption{\\label{tucson_coeffs}Tucson Wind Speed Model Coefficients}}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\nMonth & A0 & A1 & A2 & A3\\\\\n\\hline\nJan. &   -1.1 $\\pm$  0.7 &    34.1 $\\pm$  0.6 &    11467 $\\pm$   34 &    5549 $\\pm$ 112\\\\\nFeb. &    1.6 $\\pm$  0.6 &    34.1 $\\pm$  0.6 &    11620 $\\pm$   40 &    4993 $\\pm$ 104\\\\\nMar. &    1.4 $\\pm$  0.7 &    34.0 $\\pm$  0.6 &    11779 $\\pm$   46 &    5163 $\\pm$ 120\\\\\nApr. &    2.6 $\\pm$  0.8 &    29.1 $\\pm$  0.7 &    11804 $\\pm$   63 &    5028 $\\pm$ 155\\\\\nMay. &    4.6 $\\pm$  0.7 &    21.2 $\\pm$  0.6 &    11729 $\\pm$   85 &    4465 $\\pm$ 181\\\\\nJun. &    6.5 $\\pm$  0.4 &    14.7 $\\pm$  0.5 &    12107 $\\pm$  109 &    3358 $\\pm$ 167\\\\\nJul. &    5.4 $\\pm$  0.2 &     6.4 $\\pm$  0.3 &    11852 $\\pm$  128 &    2933 $\\pm$ 184\\\\\nAug. &    5.0 $\\pm$  0.2 &     7.8 $\\pm$  0.3 &    11955 $\\pm$  101 &    2779 $\\pm$ 141\\\\\nSep. &    5.7 $\\pm$  0.3 &    16.3 $\\pm$  0.4 &    12076 $\\pm$   80 &    3282 $\\pm$ 120\\\\\nOct. &    4.8 $\\pm$  0.5 &    22.4 $\\pm$  0.5 &    11889 $\\pm$   76 &    4158 $\\pm$ 144\\\\\nNov. &    0.5 $\\pm$  0.8 &    30.1 $\\pm$  0.8 &    11453 $\\pm$   50 &    5297 $\\pm$ 152\\\\\nDec. &    0.0 $\\pm$  0.7 &    32.3 $\\pm$  0.7 &    11462 $\\pm$   41 &    5295 $\\pm$ 123\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}[htbp]\n{\\bf \\caption{\\label{flagstaff_coeffs}Flagstaff Wind Speed Model Coefficients}}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\nMonth& A0 & A1 & A2 & A3\\\\\n\\hline\n Jan. & -8.1 $\\pm$  4.0 &    36.8 $\\pm$  3.9 &    11486 $\\pm$   63 &    6939 $\\pm$ 528\\\\\n Feb. &  1.3 $\\pm$  1.6 &    29.2 $\\pm$  1.5 &    11404 $\\pm$   55 &    5210 $\\pm$ 244\\\\\n Mar. & -0.6 $\\pm$  0.9 &    27.0 $\\pm$  0.9 &    11487 $\\pm$   29 &    5702 $\\pm$ 155\\\\\n Apr. &  2.9 $\\pm$  0.8 &    24.6 $\\pm$  0.8 &    11280 $\\pm$   34 &    5106 $\\pm$ 151\\\\\n May. &  6.3 $\\pm$  0.7 &    17.7 $\\pm$  0.7 &    11569 $\\pm$   68 &    4266 $\\pm$ 192\\\\\n Jun. &  8.7 $\\pm$  0.3 &    11.8 $\\pm$  0.4 &    12084 $\\pm$   80 &    2865 $\\pm$ 123\\\\\n Jul. &  6.0 $\\pm$  0.2 &     7.7 $\\pm$  0.2 &    11847 $\\pm$   68 &    2606 $\\pm$ 101\\\\\n Aug. &  5.7 $\\pm$  0.2 &    10.3 $\\pm$  0.3 &    11967 $\\pm$   61 &    2583 $\\pm$  88\\\\\n Sep. &  8.0 $\\pm$  0.4 &    16.2 $\\pm$  0.5 &    12097 $\\pm$   82 &    3133 $\\pm$ 137\\\\\n Oct. &  4.5 $\\pm$  0.8 &    19.8 $\\pm$  0.7 &    11669 $\\pm$   59 &    4587 $\\pm$ 183\\\\\n Nov. & -0.2 $\\pm$  1.7 &    29.0 $\\pm$  1.6 &    11443 $\\pm$   54 &    5484 $\\pm$ 266\\\\\n Dec. & -2.9 $\\pm$  2.8 &    31.7 $\\pm$  2.7 &    11246 $\\pm$   61 &    5969 $\\pm$ 398\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}[htbp]\n{\\bf \\caption{\\label{tenerife_coeffs}Tenerife Wind Speed Model Coefficients}}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\nMonth& A0 & A1 & A2 & A3\\\\\n\\hline\nJan. &  2.5 $\\pm$  0.9 &    20.3 $\\pm$  0.9 &    12371 $\\pm$  159 &    7964 $\\pm$ 341\\\\\nFeb. &  4.0 $\\pm$  0.6 &    29.7 $\\pm$  0.6 &    12102 $\\pm$   82 &    6478 $\\pm$ 149\\\\\nMar. &  1.7 $\\pm$  1.1 &    27.5 $\\pm$  1.0 &    12847 $\\pm$  173 &    8041 $\\pm$ 325\\\\\nApr. &  3.0 $\\pm$  0.9 &    30.6 $\\pm$  0.8 &    12332 $\\pm$  120 &    6784 $\\pm$ 216\\\\\nMay. &  4.6 $\\pm$  0.5 &    24.9 $\\pm$  0.5 &    12294 $\\pm$   92 &    5646 $\\pm$ 143\\\\\nJun. &  4.9 $\\pm$  0.8 &    14.9 $\\pm$  0.8 &    12711 $\\pm$  282 &    5899 $\\pm$ 420\\\\\nJul. &  6.7 $\\pm$  0.4 &     8.1 $\\pm$  0.6 &    12536 $\\pm$  232 &    3760 $\\pm$ 304\\\\\nAug. &  6.3 $\\pm$  0.4 &     8.3 $\\pm$  0.7 &    12137 $\\pm$  226 &    3630 $\\pm$ 296\\\\\nSep. &  6.0 $\\pm$  0.4 &    11.5 $\\pm$  0.6 &    12137 $\\pm$  172 &    4341 $\\pm$ 241\\\\\nOct. & -0.2 $\\pm$  2.4 &    22.9 $\\pm$  2.2 &    11545 $\\pm$  218 &    8465 $\\pm$ 661\\\\\nNov. &  4.5 $\\pm$  0.7 &    16.2 $\\pm$  0.7 &    12313 $\\pm$  191 &    6579 $\\pm$ 336\\\\\nDec. &  5.6 $\\pm$  0.7 &    20.9 $\\pm$  0.7 &    12074 $\\pm$  139 &    6033 $\\pm$ 238\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[htbp]\n{\\bf \\caption{\\label{anto_coeffs}Antofagasta Wind Speed Model Coefficients}}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\nMonth& A0 & A1 & A2 & A3\\\\\n\\hline\n Jan. &  3.8 $\\pm$  0.3 &    13.4 $\\pm$  0.3 &    12169 $\\pm$   92 &    3357 $\\pm$ 134\\\\\n Feb. &  3.5 $\\pm$  0.3 &    14.1 $\\pm$  0.3 &    12090 $\\pm$   86 &    3598 $\\pm$ 131\\\\\n Mar. &  4.0 $\\pm$  0.3 &    17.3 $\\pm$  0.4 &    12186 $\\pm$   91 &    3540 $\\pm$ 135\\\\\n Apr. &  4.0 $\\pm$  0.4 &    25.3 $\\pm$  0.5 &    12318 $\\pm$   75 &    3727 $\\pm$ 112\\\\\n May. &  2.7 $\\pm$  0.4 &    29.7 $\\pm$  0.4 &    12419 $\\pm$   68 &    4367 $\\pm$ 111\\\\\n Jun. &  3.0 $\\pm$  0.5 &    30.5 $\\pm$  0.5 &    12179 $\\pm$   66 &    4298 $\\pm$ 110\\\\\n Jul. &  3.1 $\\pm$  0.6 &    32.5 $\\pm$  0.6 &    11797 $\\pm$   65 &    4148 $\\pm$ 114\\\\\n Aug. &  2.7 $\\pm$  0.5 &    31.1 $\\pm$  0.5 &    11671 $\\pm$   53 &    4274 $\\pm$  98\\\\\n Sep. &  3.4 $\\pm$  0.5 &    32.1 $\\pm$  0.5 &    11785 $\\pm$   59 &    3948 $\\pm$  98\\\\\n Oct. &  3.7 $\\pm$  0.4 &    29.1 $\\pm$  0.4 &    12052 $\\pm$   55 &    3785 $\\pm$  86\\\\\n Nov. &  4.0 $\\pm$  0.3 &    26.0 $\\pm$  0.4 &    12192 $\\pm$   60 &    3394 $\\pm$  88\\\\\n Dec. &  4.0 $\\pm$  0.3 &    21.1 $\\pm$  0.4 &    12240 $\\pm$   65 &    3131 $\\pm$  91\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\nThe results are shown in Figs. \\ref{hilo_wind}--\\ref{anto_wind} respectively.    Each figure is composed of 12 graphs corresponding to the 12 months of the year.  The first letter of the corresponding month is in the upper right corner.  Each graph has the the wind speed plotted with a solid line and the fitted wind speed computed from the coefficients in Tables \\ref{hilo_coeffs}--\\ref{anto_coeffs} is plotted as a dashed line. The wind direction is plotted as a solid red line.  The wind speed is marked along the bottom of the graph, while the wind direction is marked on the top of the graph. The wind directions often show sharp spikes at the edges of jets. Those are an artificial result of the averaging of directions and can be ignored. \n\n\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig2.eps}\n\\caption{\\label{hilo_wind}  The wind speed and direction for Hilo, Hawaii.  Each of the graphs in this figure has the the measured wind speed as a function of height plotted with a solid black line and the wind speed computed from the model coefficients in Table \\ref{hilo_coeffs} is shown as a black dashed line. The wind direction is plotted as a solid red line.  The wind speed is marked along the bottom of the graph, and the wind direction is marked on the top of the graph.  The first letter of the corresponding month is in the upper right corner. }\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig3.eps} \n\\caption{ \\label{oakland_wind} The wind speed and direction for Oakland, California.  The layout of the figure is the same as in Fig. \\ref{hilo_wind}, except the fitted wind model comes from Table \\ref{oakland_coeffs}. }\n\\end{figure}\n \n\n\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig4.eps}\n\\caption{\\label{sandiego_wind} The wind speed and direction for San Diego, California.  The layout of the figure is the same as in Fig. \\ref{hilo_wind}, except the fitted wind model comes from Table \\ref{sandiego_coeffs}.}\n\\end{figure}\n \n\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig5.eps} \n\\caption{\\label{tucson_wind}  The wind speed and direction for Tucson, Arizona. The \nlayout of the figure is the same as in Fig. \\ref{hilo_wind}, except the fitted wind model comes from Table \\ref{tucson_coeffs}.  }\n\\end{figure}\n \n\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig6.eps}\n\\caption{\\label{flagstaff_wind}  The wind speed and direction for Flagstaff, Arizona. \nThe layout of the figure is the same as in Fig. \\ref{hilo_wind}, except the fitted wind model comes from Table \\ref{flagstaff_coeffs}. }\n\\end{figure}\n\n \n\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig7.eps} \n\\caption{\\label{tenerife_wind}  The wind speed and direction for Tenerife, Canary Islands, Spain, which is useful for the observatories located in the Canary Islands. The layout of the figure is the same as in Fig. \\ref{hilo_wind}, except the fitted wind model comes from Table \\ref{tenerife_coeffs}.     }\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig8.eps} \n\\caption{\\label{anto_wind}  The wind speed and direction for Antofagasta, Chile. The layout of the figure is the same as in Fig. \\ref{hilo_wind}, except the fitted wind model comes from Table \\ref{anto_coeffs}. }\n\\end{figure}\n \n\n\n\nFor both hemispheres, the tropospheric winds increase in the winter months and decrease in the summer months. Stratospheric winds increase in the summer months, the opposite of the tropospheric winds.  During the summer months, Tenerife has a low level ($\\approx5$ km) jet form. This is most obvious in July and August, but first appears in June and persists to October.   In the Greenwood model, $A_0$ is the low altitude wind speed and that is true for our model for Hawaii.  It is not always true for other sites, as these values are sometimes negative. The reason for this is that the wind profiles for those sites has a slightly different shape than the Hawaiian wind profile.  This can be seen in Figs.  \\ref{oakland_wind} and\\ref{flagstaff_wind}.  \n\nGreenwood \\cite{greenwood1977} only provides a model for the wind speed, though \\cite{greenwood1975} does show the annual mean wind direction for Hilo and Lihue.  As shown by Figs. \\ref{hilo_wind}-\\ref{anto_wind} the wind direction is not an easily modeled function.  It can most easily be modeled as a square wave.  Since stability of flows does depend upon the amount of shear present, an analysis that computed the wind shear at the upper and lower boundaries would tend to overestimate the instability.    Looking at Figs. \\ref{hilo_wind}--\\ref{anto_wind}, the wind direction does not change instantaneously, but it does change very quickly, often changing direction by $180^\\circ$ in a km.  Some regions of slowly changing direction, most commonly above 25 km,  are not very well modeled by a square wave. A square wave model will suffice for modeling where phase screens at only a few altitudes are used to model the distribution of turbulence. \n\nNote that the reversal of wind direction at altitudes above 20 km is a well known feature of the zonal wind flow. It is due to a higher altitude jet whose center is north  of the equator in winter, and south of the equator in summer.  There are also northern and southern jets at higher altitudes which move in the same direction as the jets in the lower stratosphere. These are seasonal and impact the wind speed and directions in our data \\cite{newell1972}.\n\n\n\n\\section{Richardson Number}\\label{richardson}\n\n\nThe gradient Richardson number is a dimensionless ratio, $R_i$, related to the buoyant production or consumption of turbulence divided by the shear production of turbulence \\cite{kundu1990}. It is defined as \n\n\\eq\n\\label{eq_richardson}\nR_i = \\frac{g\\frac{\\partial\\ln\\Theta}{\\partial~z}}{\\left (\\frac{\\partial u}{\\partial z}\\right)^2 + \\left(\\frac{\\partial v}{\\partial z}\\right)^2},\n\\en\n\nwhere $g$ is the acceleration of gravity, $\\Theta$ is the potential temperature, $z$ is the altitude, $u$ and $v$ are the horizontal components of the wind vector.  Normally a gradient Richardson number with a value less than 0.25 is considered to be turbulent and a value greater than 0.25 to be non-turbulent.  It is used to indicate dynamic stability and the formation of turbulence.\n\nIn order to see at which altitudes turbulence forms, we used data from the radiosondes to compute the monthly binned Richardson number for each site in Table \\ref{station_table} using Eq. \\ref{eq_richardson} at each of the fixed grid points used in \\S \\ref{data_analysis}.  The grid heights points are every 400m and some of the finer atmospheric layers may not be captured with our analysis.  The derivatives were computed using a three-point Lagrangian interpolation routine.  \n\nThe computed Richardson numbers were then binned into two bins: $R_i < 0.25$, and  $R_i \\geq 0.25$. \nFor a given site, at each altitude the numbers in each bin were normalized by the total number of data points at that altitude.  This gives the Richardson number as a function of altitude.  These are shown in Figs. \\ref{hilo_rich}-\\ref{anto_rich}. As expected, the Richardson numbers above 20 km are very high and have little impact on astronomical seeing; as a result we only plotted the values below 20 km. It is important to note that these values are for radiosondes launched from sites at lower altitudes than astronomical observatories.  The highest Richardson number values occur at these low altitudes and are not a concern to most astronomical observatories.  \n\n\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig9.eps} \n\\caption{\\label{hilo_rich} \nThe binned Richardson number for Hilo, Hawaii The figure shows the fraction of Richardson numbers falling into two bins.  Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \\ge 0.25$).}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig10.eps} \n\\caption{\\label{oakland_rich}   The binned Richardson number for Oakland, California.  Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \\ge 0.25$).}\n\\end{figure}\n \n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig11.eps} \n\\caption{\\label{sandiego_rich}  The binned Richardson number for San Diego, California.   Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \\ge 0.25$). }\n\\end{figure}\n \n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig12.eps} \n\\caption{ \\label{tucson_rich}   The binned Richardson number for Tucson, Arizona.  Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \\ge 0.25$). }\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig13.eps} \n\\caption{ \\label{flagstaff_rich}   The binned Richardson number for Flagstaff, Arizona.  Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \\ge 0.25$). }\n\\end{figure}\n\n\\\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig14.eps} \n\\caption{\\label{tenerife_rich}   The binned Richardson number for Tenerife, Canary Islands, Spain.  Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \\ge 0.25$). }\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[height=7.5cm]{roberts_fig15.eps} \n\\caption{\\label{anto_rich}  The binned Richardson number for Antofagasta, Chile.  Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \\ge 0.25$).}\n\\end{figure}\n \n\n\nExamining the individual months reveals that the figures are not as smooth as the wind speed figures, but instead have many fine layers.  The layers are significant, even if they do not seem to explicitly show turbulence (ie their value of $R_i$ is > 0.25). Because the $R_i$ calculation has a height increment of 400m, and because the precise positions and altitudes of jets vary over periods of a day or less, the effective values of the velocity derivatives in Equation \\ref{eq_richardson} are lower in our results than they might be at any given time. This averaging leads to relatively large calculated $R_i$ values. It also means that our data will not show many instances of $R_i < 0.25$.  But where is does occur it is significant. Moreover, any drop in the plotted $R_i$ number indicates a persistent region of instability,  and it is likely that some turbulence is present there for extended periods,\n\nSince low gradient Richardson numbers indicate the presence of atmospheric turbulence, they can be an indicator of seeing. In Fig. \\ref{hilo_rich}, the winter months for Hawaii have the highest proportion of Richardson numbers lower than 0.25 (yellow values), and the summer months have the lowest. In other words the winter months have more turbulent conditions than the summer months. Studies of seeing from Mauna Kea show that this seasonal behavior of seeing correlates with this very well \\cite{bely1987,seigar2002, skidmore2009}.  There is relatively little seasonal variation in for Antofagasta, Chile (Fig. \\ref{anto_rich}) and this correlates well with seeing measurements carried at nearby Cerro Armazones \\cite{skidmore2009} which show no clear evidence for any seasonal dependency.  The trend in Fig. \\ref{tenerife_rich} agrees with seeing measurements for  La Palma, Canary Islands, showing that seeing is best May-September \\cite{munoz1998, wilson1999}.  Flagstaff, Arizona (Fig. \\ref{flagstaff_rich} has more turbulent values than Oakland, California, which is agreement with site comparison done between Anderson Mesa (outside of Flagstaff) and Lick Observatory (near Oakland) \\cite{walters1997}.  This shows that Richardson numbers computed from radiosonde data can be used to differentiate between astronomical sites at a macro level which can be of use for narrowing site selection for new telescopes.  It is important to remember that the Richardson numbers computed from radiosonde data do not include local effects such as the turbulent surface boundary layer that can have a profound impact on seeing.  Small differences in altitudes and location can alter the observed seeing significantly \\cite{walters1997}.  \n\nWhile the Richardson number does not indicate the strength of turbulence, knowing that there are persistent turbulent layers may be useful to site planners.  For example, knowing roughly where and when one might expect upper layers of turbulence to be present could be useful for measurements, such as laser communications, which are strongly influenced by scintillation. On the other hand, we have seen that some layers closer to the ground have $R_i$<0.25. These must also be persistent to show up over the averaging. Knowing that these layers exist is useful for large telescopes\nthat plan to compensate for near ground effects.\n\n\n\\section{Other Wind Models}\\label{theothers}\n\nThere have been a few other studies of wind speed for different sites, though they have been done for only a limited number of sites and with different methodology.  Sarazin \\& Tokovinin \\cite{sarazin2002} suggested that the wind speed at the 200 mbar pressure isobar (\\vtwo)~is a good indication of atmospheric turbulence at a given site (\\vtwo~is well monitored for meteorology).    The advantage of this is that there are models of \\vtwo~for the entire planet, while there are only a fixed number of radiosondes launched each day.  Several groups have used this idea to characterize and compare several astronomical sites \\cite{chueca2004, carrasco2005, garcia-lorenzo2005} including several sites which are included in this study.    During the year, the altitude at which \\vtwo~occurs varies, this makes it difficult to use the values in Eq. \\ref{greenwood_fit}. It is useful for site comparisons, but it supplements rather than replaces the knowledge of the vertical wind profile. \n\nGarc\\'{i}a-Lorenzo et al. \\cite{garcia-lorenzo2005} studied \\vtwo~for Paranal and La Silla in Chile, San Pedro M\\'{a}rtir in Mexico, Mauna Kea in Hawaii and La Palma in the Canary Islands.  The last two observatories are also characterized in this paper. We show a seasonal variation in $A_1$ in Tables \\ref{hilo_coeffs} and \\ref{tenerife_coeffs} which has the highest tropospheric wind speed in the spring and the lowest tropospheric winds in the summer. Garc\\'{i}a-Lorenzo et al. showed that the \\vtwo wind speeds at Mauna Kea and La Palma have this same seasonal behavior.   \n\n\\section{Conclusions}  \n\nWe have presented upper-level wind models computed from archival radiosonde data suitable for the major astronomical observatories in the United States, Chile and the Canary Islands.    We find that the commonly used Greenwood model  is not suitable for sites other than Hawaii. In addition it produced a single wind profile for the entire year, when it is more correct to use different wind models over the course of the year. We find that, as may be expected from geophysical fluid dynamics, that sites with similar latitudes share similar wind profiles. In addition we have computed Richardson number profiles from the same datasets. Those results indicate the presence of turbulence at different altitudes with seasonal variations that seem to agree with variations in seeing.  These models and results will be of use to modelers of atmospheric turbulence and instrument developers.\n\n\n\\section*{Acknowledgments}\n\nThis research was funded by the Air Force Office of Scientific Research and by the Air Force Research Laboratory's Directed Energy Directorate. A portion of the research in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Radiosonde data were obtained from the Wyoming Weather Web, maintained by the Department of Atmospheric Science of the University of Wyoming.  \n\n \n\n\\end{document} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe deep reinforcement learning (RL) setting in the control of continuous systems still requires large amounts of data to obtain optimal policies and construct general and scalable reinforcement learning agents \\citep{mach_1}. In order to improve data and sampling efficiency, agents store their experiences in a buffer, experience replay memory \\citep{experience_replay}, and reuse them multiple times to perform gradient steps on their policies and value functions approximated by deep neural networks. Methods that employ experience replay \\citep{experience_replay} are \\textit{off-policy} as the samples contained in the buffer which are used to learn a \\textit{target} policy are generated by a group of different policies than the one being optimized \\citep{sutton_book}. Although the past experiences of the agents may be beneficial for learning in later stages, \\textit{deadly triad} can occur in off-policy deep RL, a phenomenon that refers to the instability of an RL algorithm when it employs off-policy learning, function approximation, and bootstrapping simultaneously \\citep{sutton_book}. Due to the combination of the mentioned entities, a sudden divergence may arise even if the policy is very close to an optimal one \\citep{sutton_book}. To mitigate the effects of the deadly triad, a common approach in off-policy learning is to disregard or completely discard transitions whose underlying distributions are uncorrelated to the policy distribution to be optimized, also known as \\textit{off-policy correction} \\citep{dogan}. This is done through the relative divergence between the action probabilities generated by the current and other behavioral policies that collected the experiences, which is performed through \\textit{importance sampling} or \\textit{off-policy policy gradients}.\n\nImportance sampling methods weigh the gradients of the approximate value functions according to this relative divergence using off-policy transitions sampled from the experience replay buffer \\citep{experience_replay}. Importance sampling can consider temporally correlated transitions through the trajectories, e.g., \\citep{dayan_watkins_q_learning, degris, munos_safe, harutyunyan, impala} or through a single transition, e.g., \\citep{trust_region_estimators}. Although the widely known importance sampling techniques are shown to be successful in off-policy learning, there are several issues introduced by these approaches \\citep{trust_region_estimators}. As the trajectory-based approaches conservatively treat samples, there may be unnecessary trajectory terminations, a large amount of variance or biased action probability estimates \\citep{munos_safe}. For single-transition methods, inaccurate importance weights are one of the broadly observed issues \\citep{trust_region_estimators}.\n\nPrior to the employment of function approximators in reinforcement learning, many off-policy techniques, specifically Monte-Carlo, did not have an option to evaluate the off-policy policies in terms of the action probabilities \\citep{harutyunyan}. However, with the utilization of value function approximators, temporal difference \\citep{temporal_difference} methods provide a continuous link between one-step and Monte-Carlo approaches through \\textit{eligibility traces} \\citep{learning_from_delayed_rewards, mach_3}. As the value functions in temporal difference methods assess actions under the cumulative reward objective, they offer a direct correction to the discrepancy between the target and behavioral policies \\citep{harutyunyan}. Off-policy correction in terms of action probabilities via eligibility traces \\citep{learning_from_delayed_rewards,mach_3} is a well-studied artifact for discrete action spaces \\citep{impala,harutyunyan, munos_safe, precup_2001, degris, trust_region_estimators}. Even though actor-critic algorithms are on-policy by construction due to the Bellman optimization \\citep{bellman}, \\citep{trust_region_estimators}, similar work for the continuous action spaces is very few, e.g., \\citep{ris, geoff_pac, ace, ipg}. Furthermore, for large action spaces, the gradients can vanish due to the product of importance weights \\citep{dimension_wise_is}. As an environment's observation and action spaces start to increase and more challenging tasks are introduced, the vanishing gradient problem due to importance sampling can be detrimental \\citep{mach_2}. Additionally, if the continuous policy is deterministic, then off-policy correction in terms of the action probability estimates through eligibility traces \\citep{learning_from_delayed_rewards,mach_3} is not possible by construction \\citep{sutton_book}.\n\nMotivated by the mentioned problems in the current importance sampling approaches and limited generalization of off-policy correction to continuous action domains, we introduce Actor-Critic Off-Policy Correction (AC-Off-POC), an alternative one-step off-policy correction method for actor-critic algorithms to overcome the effects of deadly triad when learning from off-policy samples. The main contributions of this study can be summarized as follows:\n\\begin{itemize}\n    \\item We derive a parameter-free importance weighting rule that can operate using randomly sampled transitions on both types of continuous policies, deterministic or stochastic. Through this rule, we correct the off-policy error to achieve efficient off-policy learning in continuous control and make the standard on-policy policy gradients closer to their on-policy nature.\n    \\item By operating using a randomly sampled batch of transitions and in continuous action spaces, our algorithm prevents the high-variance build-up, vanishing gradients, and unnecessary trajectory terminations observed in traditional importance sampling methods. Therefore, the presented method can achieve a \\textit{safe} off-policy learning by constructing a contraction mapping around the discount factor with a unique fixed point. \n    \\item Our method can readily be adapted to off-policy actor-critic algorithms and experience replay \\citep{experience_replay} sampling methods that are currently available and will be introduced in the future, which we validate through remarks and empirical results.\n    \\item We make remarks to indicate the generalizability and behavior of our approach. An extensive set of empirical studies support our claims in the provided remarks and demonstrate that, in the majority of the tasks, our algorithm improves the performance of the baseline algorithms and outperforms the competing off-policy correction methods in challenging OpenAI Gym \\citep{gym} continuous control tasks. Moreover, we conduct additional experiments to give insights into the introduced off-policy correction scheme and perform a sensitivity analysis based on the components that may affect off-policy learning. \n    \\item To ensure reproducibility, our source code is made publicly available at the GitHub repository\\footnote{\\label{our_repo}\\url{https:\/\/github.com\/baturaysaglam\/AC-Off-POC.git}}. \n\\end{itemize}\n\n\\section{Related Work}\nOff-policy RL considers learning from experiences generated by the current or past versions of the agent's policy. Learning from the past experiences of the policy can cause action-values to be erroneously estimated and may degrade the performance in environments that vastly require learning from on-policy samples, i.e., data collected by the current policy of interest. Importance sampling or off-policy correction can prevent such a discrepancy in off-policy learning. This can be done through a known or estimated probability distribution of the policy \\citep{harutyunyan, munos_safe, trust_region_estimators, impala}. Initial studies in off-policy correction and importance sampling have been done by \\cite{precup_2000, harutyunyan, learning_from_delayed_rewards}. Before the deep function approximators, basic approaches in importance sampling through eligibility traces have been studied by \\cite{precup_2000, harutyunyan, mach_3}. \\cite{learning_from_delayed_rewards} proposed an off-policy correction method to learn from delayed rewards. Before employing deep neural networks, \\cite{weighted_is} introduced a weighted importance sampling with linear function approximators. \\cite{degris} introduced a multi-step formulation for the expected return in terms of the importance sampling. Monte-Carlo is employed to compute the value estimates' importance weights when the action probabilities cannot be computed due to large state and action spaces. The high variance due to the product of importance coefficients motivated further variance reduction techniques such as the work of \\cite{precup_2000}, which reduces the number of steps before the bootstrapping. \n\nWith the introduction of deep function approximators in RL, \\cite{munos_safe} utilized full returns as in the on-policy case in their RETRACE algorithm, yet \\cite{trust_region_estimators} shows that it introduces a zero-mean random variable at each step, yielding an increased variance in both on- and off-policy settings. The variance build-up in importance sampling is overcome by the V-trace algorithm \\citep{impala} by trading off the variance for a biased return estimate. Differently from RETRACE \\citep{munos_safe}, V-trace \\citep{impala} fully recovers the Monte-Carlo return under on-policy learning. \\cite{trust_region_estimators} presented LASER and show to eliminate the bias in the V-trace \\citep{impala} algorithm without introducing an unbounded variance. Finally, \\cite{mach_4} replaces the behavioral policy's action probabilities by their maximum likelihood estimate under the observed data. They show that this general technique reduces variance due to sampling error in Monte Carlo style estimators. These methods were mainly introduced for discrete action spaces and have improved performance in classic Atari games and control problems.\n\nIn continuous action domains, importance sampling through the probability distributions of the policies is impossible by constructing the deterministic policies \\citep{sutton_book}. However, \\cite{policy_optim_with_is} proposed to derive a high-confidence bound for importance weight estimates in on-policy learning. They define a surrogate objective function that optimizes the policy offline when a new batch of trajectories is collected. Detrimental estimation bias due to the clipped importance weights to avoid the large variances is addressed by \\cite{dimension_wise_is} and overcome by an adaptive structure that separately clips the importance weight of each action dimension for on-policy algorithms. In addition, \\cite{dogan} previously introduced an experience replay \\citep{experience_replay} sampling algorithm that prioritizes relatively on-policy samples in training off-policy actor-critic algorithms. Their approach, Batch Prioritizing Experience Replay via KL-divergence (KLPER) \\citep{dogan}, exhibits substantial results in environments that vastly require on-policy experiences to be solved.\n\nFor off-policy learning with stochastic actors, the instability in the policy gradients due to the discrepancy between the target and behavioral policy distributions is addressed by \\cite{ris, ace} and \\cite{ipg}. \\cite{ris} alleviate such a discrepancy by employing a smooth variant of the relative importance sampling in their algorithm, Relative Off-Policy Actor-Critic (RIS-Off-PAC), which has a parameter $\\beta \\in [0, 1]$ that controls the smoothness. Additionally, when the behavior policy does not always aim to learn and adopt the optimal policy for the task in off-policy learning, the policy gradient theorem becomes insufficient \\citep{ace}. \\cite{ace} introduced a novel off-policy policy gradient through an emphatic weighting scheme in their algorithm, Actor-Critic with Emphatic Weights (ACE). However, the excursion objective used in \\citep{ace} can be misleading about the performance of the target policy. Generalized Off-Policy Actor-Critic (Geoff-PAC) \\citep{geoff_pac} solves this problem by containing a new objective for off-policy learning to compute the policy gradient of the counterfactual objective and use an emphatic approach to get an unbiased sample from this policy gradient. Moreover, Interpolated Policy Gradient (IPG) \\citep{ipg} merges on- and off-policy updates for actor-critic methods. They show that off-policy updates with a value function estimator can be interpolated with on-policy policy gradient updates while the performance can still satisfy the bounds. Overall, these methods are well-known off-policy correction techniques in the control of continuous systems and have been shown to exhibit remarkable results. Thus, we employ them as competing methods in our experiments.\n\n\\section{Background and Notation}\nIn reinforcement learning, an \\textit{agent} interacts with its \\textit{environment} and collect \\textit{rewards}. At each discrete time step $t$, the agent observes its state $s$ and chooses an action $a$; it receives a reward $r$ and observes a new state $s'$. In fully observable environments, the RL problem is represented by a Markov Decision Process (MDP), a tuple $(\\mathcal{S}, \\mathcal{A}, P, \\gamma)$ where $\\mathcal{S} \\in \\mathbb{R}^{m}$ and $\\mathcal{A} \\in \\mathbb{R}^{n}$ denotes the state and action spaces, respectively, $P$ is the transition dynamics such that $s', r \\sim P(s, a)$ and $\\gamma$ is the constant discount factor. The goal of an agent is to maximize the \\textit{value} defined as the expected cumulative discounted returns $\\mathbb{E}[R_{t}] = \\sum_{i = 0}^{\\infty}\\gamma^{i}r_{t + i}$. \n\nThe agent selects actions depending on its observation of states $s \\in \\mathcal{S}$ and with respect to a policy $\\pi: \\mathcal{S} \\rightarrow \\mathcal{A}$. Policy of an agent $\\pi(\\cdot)$ is stochastic if it maps states to action probabilities, $a \\sim \\pi(\\cdot | s)$, or deterministic if it maps states to unique actions, $a = \\pi(s)$. The policy of an agent follows a distribution $\\eta^{\\pi}(s)$ over the states $s \\in \\mathcal{S}$ visited by the policy. There exists an action-value function $Q^{\\pi}(s, a) = \\mathbb{E}_{s_{i} \\sim \\pi, a_{i} \\sim \\pi}[R_{t}|s, a], $ corresponding to each policy $\\pi$, which represents the expected return while following the policy after taking action $a$ in state $s$. For a given policy $\\pi$, the action-value function is computed through the Bellman operator $\\mathcal{T}^{\\pi}$ \\citep{bellman}:\n\\begin{equation}\n    \\mathcal{T}^{\\pi}Q^{\\pi}(s, a) = \\mathbb{E}_{s'}[r + \\gamma Q^{\\pi}(s^{\\prime}, a^{\\prime})]; \\quad a^{\\prime} \\sim \\pi(s^{\\prime}).\n\\end{equation}\nThe Bellman operator \\citep{bellman} is a contraction for $\\gamma \\in [0, 1)$ with unique fixed point $Q^{\\pi}(s, a)$, i.e., the resulting error is bounded by $\\gamma$ \\citep{bertsekas}. The optimal action-value function, $Q^{*}(s, a) = \\mathrm{max}_{a}Q^{\\pi}(s, a)$, is obtained through the greedy actions of the corresponding policy, i.e., actions that yield the highest action-value.\n\nIn deep reinforcement learning, the action-value functions are modeled by differentiable function approximators $Q_{\\theta}(s, a)$, parameterized by $\\theta$, also known as Q-networks. The deep action-value function is obtained by temporal difference (TD) learning \\citep{sutton_1988}, an update rule based on the Bellman equation \\citep{bellman} which represents a fundamental relationship used to learn the action-value function by bootstrapping from the estimate of the current state-action pair $(s, a)$ to the subsequent state-action pair $(s^{\\prime}, a^{\\prime})$, i.e., the Deep Q-Networks (DQN) algorithm \\citep{dqn}:\n\\begin{equation}\n    Q_{\\theta}(s, a) \\leftarrow r + \\gamma Q_{\\theta^{\\prime}}(s^{\\prime}, a^{\\prime}),\n\\end{equation}\nwhere $\\theta'$ is the target network parameters of the Q-network. The usage of target networks is a practical approach in deep reinforcement learning to provide a fixed objective to Q-networks and ensure stability in the updates. The target networks are usually updated by a small proportion $\\tau$ at each step, i.e., $\\theta' \\leftarrow \\tau\\theta + (1 - \\tau)\\theta'$, called soft-update, or periodically to exactly match the behavioral networks called hard-update. Q-networks are optimized by minimizing:\n\\begin{gather}\n    J(\\theta) =  \\delta^{2}\\label{eq:ddpg_critic_loss}; \\\\\n    y \\coloneqq r + \\gamma Q_{\\theta'}(s', a'), \\\\\n    \\delta \\coloneqq y - Q_{\\theta}(s, a),\n\\end{gather}\nwhere $y$ is the fixed objective for the Q-network and $\\delta$ is the resulting TD-error.\n\nIn continuous action spaces, maximum of the action-value function over possible actions, i.e., $\\mathrm{max}_{a}Q^{\\pi}(s, a)$, is intractable. Therefore, a separate network that selects actions on the observed states is employed, called the actor network. The deep reinforcement learning algorithms that utilize actor networks are called actor-critic methods, and they enable to control of continuous systems through the employed actor network. In actor-critic algorithms, actor networks $\\pi_{\\phi}$ parameterized by $\\phi$, are optimized by one-step gradient ascent over the policy gradient $\\nabla_{\\phi}J(\\phi)$, where the loss for the actor networks are the Q-values of the selected actions estimated by the Q-network:\n\\begin{equation}\n    J(\\phi) = Q_{\\theta}(s, \\pi_{\\phi}(s)).\n\\end{equation}\nFor deterministic policies, the policy gradient is found through maximizing the Q-value of the given state-action pair, also known as the Deterministic Policy Gradient algorithm (DPG) \\citep{dpg}. DPG \\citep{dpg} computes the gradient of the deterministic policy as:\n\\begin{equation}\n\\label{eq:det_pg}\n    \\nabla_{\\phi}J_{det}(\\phi) = \\nabla_{a}Q_{\\theta}(s, a)|_{a = \\pi_{\\phi}(s)}\\nabla_{\\phi}\\pi_{\\phi}(s).\n\\end{equation}\nIn stochastic policy gradient, the log-derivative trick is used to compute the gradient over the expected cumulative reward:\n\\begin{equation}\n\\label{eq:sto_pg}\n    \\nabla_{\\phi}J_{sto}(\\phi) = \\nabla_{\\phi}\\mathrm{log}\\pi_{\\phi}(a|s)Q_{\\theta}(s, a).\n\\end{equation}\n\nIn off-policy deep reinforcement learning, the \\textit{behavioral} policy of agents collects experiences and stores them into the experience replay buffer \\citep{experience_replay} to use in updating their deep policies and Q-networks. As the agents progressively learn the environment and improve their policies, the distribution of the current policy may not match the distribution under the previously collected transitions, such as experiences collected in the initial time steps. In this case, those experiences are \\textit{off-policy samples}, i.e., experiences gathered by other policies, and the ones collected by the current agent's policy are \\textit{on-policy samples}. At every update step, the agent samples a batch of transitions through a sampling algorithm that may contain on- and off-policy samples:\n\\begin{equation}\n\\label{eq:transition_tuple_definition}\n    (\\boldsymbol{S}^{|\\mathcal{B}| \\times m}, \\boldsymbol{A}^{|\\mathcal{B}| \\times n}, \\boldsymbol{R}^{|\\mathcal{B}| \\times 1}, \\boldsymbol{S'}^{|\\mathcal{B}| \\times m}) \\sim \\mathcal{B},\n\\end{equation}\nwhere $|\\mathcal{B}|$ is the number of sampled transitions, and $\\boldsymbol{S}^{|\\mathcal{B}| \\times m}$, $\\boldsymbol{A}^{|\\mathcal{B}| \\times n}$, $\\boldsymbol{R}^{|\\mathcal{B}| \\times 1}$, $\\boldsymbol{S'}^{|\\mathcal{B}| \\times m}$ are the state, action, reward and next state matrices or vectors in $\\mathcal{B}$, respectively, i.e.: \n\\begin{align}\n    &\\boldsymbol{S}^{|\\mathcal{B}| \\times m} = [\\boldsymbol{s}_{1}^{m}, \\boldsymbol{s}_{2}^{m}, \\dots, \\boldsymbol{s}_{|\\mathcal{B}|}^{m}]^{\\mathrm{T}}, \\\\\n    &\\boldsymbol{A}^{|\\mathcal{B}| \\times n} = [\\boldsymbol{a}_{1}^{n}, \\boldsymbol{a}_{2}^{n}, \\dots, \\boldsymbol{a}_{|\\mathcal{B}|}^{n}]^{\\mathrm{T}}, \\\\\n    &\\boldsymbol{R}^{|\\mathcal{B}| \\times 1} = [r_{1}, r_{2}, \\dots, r_{|\\mathcal{B}|}]^{\\mathrm{T}}, \\\\\n    &\\boldsymbol{S'}^{|\\mathcal{B}| \\times m} = [\\boldsymbol{s'}_{1}^{m}, \\boldsymbol{s'}_{2}^{m}, \\dots, \\boldsymbol{s'}_{|\\mathcal{B}|}^{m}]^{\\mathrm{T}}.\n\\end{align}\nNote that we use batch or mini-batch interchangeably throughout the paper. Through batch learning, we consider that the agent optimizes its actor and critic networks using the sampled batch of transitions that do not require to be necessarily temporally correlated. Following Equation (\\ref{eq:transition_tuple_definition}), policy gradients denoted by Equation (\\ref{eq:det_pg}) and Equation (\\ref{eq:sto_pg}), and critic loss denoted by Equation (\\ref{eq:ddpg_critic_loss}) can be adapted to mini-batch setting:\n\\begin{gather}\n    \\boldsymbol{y}^{|\\mathcal{B}| \\times 1} = \\boldsymbol{R}^{|\\mathcal{B}| \\times 1} + \\gamma Q_{\\theta'}(\\boldsymbol{S'}^{|\\mathcal{B}| \\times m}, \\pi_{\\phi}(\\boldsymbol{S'}^{|\\mathcal{B}| \\times m})),\\\\\n    \\boldsymbol{\\delta}^{|\\mathcal{B}| \\times 1} = \\boldsymbol{y}^{|\\mathcal{B}| \\times 1} - Q_{\\theta}(\\boldsymbol{S}^{|\\mathcal{B}| \\times m}, \\boldsymbol{A}^{|\\mathcal{B}| \\times n});\\label{eq:td_error_batch} \\\\\n    J(\\theta) = \\frac{\\Vert \\boldsymbol{\\delta}^{|\\mathcal{B}| \\times 1} \\Vert^{2}_{2}}{|\\mathcal{B}|},\\label{eq:critic_loss_batch} \\\\\n    \\nabla_{\\phi}J_{det}(\\phi) = \\nabla_{\\phi}\\frac{1}{|\\mathcal{B}|}\\sum_{|\\mathcal{B}|}Q_{\\theta}(\\boldsymbol{S}^{|\\mathcal{B}| \\times m}, \\pi_{\\phi}(\\boldsymbol{S}^{|\\mathcal{B}| \\times m}))\\label{eq:det_pg_batch}, \\\\\n    \\nabla_{\\phi} J_{sto}(\\phi) = \\nabla_{\\phi}\\frac{1}{|\\mathcal{B}|}\\sum_{\\mathcal{B}}\\mathrm{log}\\pi_{\\phi}(\\boldsymbol{A}^{|\\mathcal{B}| \\times n}|\\boldsymbol{S}^{|\\mathcal{B}| \\times m})\\circ Q_{\\theta}(\\boldsymbol{S}^{|\\mathcal{B}| \\times m}, \\boldsymbol{A}^{|\\mathcal{B}| \\times n})\\label{eq:sto_pg_batch},\n\\end{gather}\nwhere sum operators compute the loss over the row vectors of $\\mathcal{B}$, $\\circ$ is the Hadamard product, and $\\Vert \\cdot \\Vert_{2}$ is the $\\mathrm{L}_{2}$ norm.\n\n\\section{Off-Policy Correction for Actor-Critic Algorithms}\nOur goal is to achieve faster and more efficient learning by disregarding transitions that the distribution under which does not match the distribution of the current policy. To enable safe and efficient off-policy learning, we first aim to mitigate the effects of off-policy error induced by the data collected by divergent behavioral policies. In the following subsections, we introduce AC-Off-POC with two variants that measure the similarity between deterministic or stochastic policies. We show that the effect of each off-policy sample in the optimization can be corrected through the Jensen-Shannon divergence without action probability estimates under a basic multivariate Gaussian assumption on the actions. AC-Off-POC projects the similarity measure within the interval $[0, 1]$, where 1 indicates identical policies and 0 indicates uncorrelated ones, to weigh the gradients of the actor and critic networks. We then introduce a general off-policy actor-critic framework with AC-Off-POC for continuous control and support our claims with theoretical derivations. We show that AC-Off-POC provides a safe off-policy correction through the mentioned bounded $\\gamma$-contraction mapping. We make remarks throughout the study to emphasize certain points and findings and discuss the empirical results further. \n\n\\subsection{Deterministic Policies}\n\\label{sec:deterministic_AC-Off-POC}\nTo measure the mismatch between the distributions under the off-policy experiences and current policy, we assume without loss of generality that at every time step $t$, the multi-dimensional continuous actions are samples of a multivariate Gaussian distribution which is not known during the training. As the agent is progressively learning, each action selected by a different behavioral policy may be a sample of a different multivariate Gaussian distribution. However, each dimension in an action vector may be correlated to the rest of the dimensions. This is realistic since each dimension in action vectors often has effects on other dimensions \\citep{mujoco, box2d}. For instance, consider a continuous control task where a robot lies on the ground and tries to stand up, and the $n$-dimensional action space represents the angles of the robot's joints. Intuitively, an angle of a joint may restrict or affect the motion of other joints. This implies the correlation across the action dimensions. A multivariate probability distribution can represent such an action vector, specifically, multivariate Gaussian distribution, for which dimensions of actions are correlated. In particular, the mean vector can represent the deterministic action chosen by the policy, and the covariance matrix can represent the noise introduced by the exploration and approximation through deep neural networks. \n\nHowever, an agent's policy may have multiple modes in some environments, e.g., jumping over an obstacle or passing it by crouching in a 2D view. In such cases, the distributions under the actions selected by each policy mode will be another multivariate Gaussian. Therefore, transitions collected by each policy mode can be treated as if a different behavioral policy generated them. Hence, scenarios consisting of different policy modes still yield the made assumption.\n\\begin{assumption}\nIn off-policy learning, multivariate Gaussian distribution can represent the deterministic policies, having single or multiple modes. The mean vector represents the deterministic action chosen by the policy, and the covariance matrix represents the noise introduced by the exploration and approximation through deep neural networks.\n\\end{assumption}\n\nOff-policy correction requires importance sampling that measures the mismatch between the underlying distributions of the off-policy samples and the current policy to be optimized. Given the sampled batch of transitions denoted by Equation (\\ref{eq:transition_tuple_definition}), and using Equation (\\ref{eq:det_pg_batch}), the deterministic policy gradient weighted by importance sampling can be written as:\n\\begin{equation}\n\\label{eq:det_actor_loss}\n    \\nabla_{\\phi}\\Tilde{J}_{det}(\\phi) = \\nabla_{\\phi}\\frac{\\lambda}{|\\mathcal{B}|}\\sum_{|\\mathcal{B}|}Q_{\\theta}(\\boldsymbol{S}^{|\\mathcal{B}| \\times m}, \\pi_{\\phi}(\\boldsymbol{S}^{|\\mathcal{B}| \\times m})),\n\\end{equation}\nwhere $\\lambda$ is the importance weight, $\\boldsymbol{\\delta}^{|\\mathcal{B}| \\times 1}$ is the temporal difference error vector, i.e., Equation (\\ref{eq:td_error_batch}); $\\phi$, $\\phi'$, $\\theta$ and $\\theta'$ are the parameters of the actor, target actor, critic and target critic networks, respectively. The weight $\\lambda$ measures the overall discrepancy of the sampled batch of transitions. If the majority of behavioral policies within the sampled batch are distant from the current policy, the gradient step on the policy and Q-network parameters will be small since $\\lambda$ and loss are computed over the sampled batch in this case. Then, using Equation (\\ref{eq:critic_loss_batch}) and Equation (\\ref{eq:det_actor_loss}) in a similar fashion, the loss for the temporal difference learning \\citep{sutton_1988} becomes:\n\\begin{equation}\n\\label{eq:det_critic_loss}\n        \\Tilde{J}(\\theta) = \\lambda\\frac{\\Vert \\boldsymbol{\\delta}^{|\\mathcal{B}| \\times 1}\\Vert^{2}_{2}}{|\\mathcal{B}|}.\n\\end{equation}\n\nWe now show how to derive importance weights for the off-policy experiences. The agent samples a batch of off-policy transitions corresponding to different behavioral policies in each gradient step. We know that given the states $\\boldsymbol{S}^{|\\mathcal{B}| \\times m}$, each action in the experience replay buffer \\citep{experience_replay} corresponds to a multivariate Gaussian distribution $\\mathcal{N}(\\boldsymbol{\\mu}^{1 \\times n}, \\boldsymbol{\\Sigma}^{n \\times n})$ with mean vector $\\boldsymbol{\\mu}^{1 \\times n}$ and covariance matrix $\\boldsymbol{\\Sigma}^{n \\times n}$. To measure the similarity between the current policy and the policies that executed the off-policy transitions from the sampled batch $\\mathcal{B}$, we first forward pass the states from $\\mathcal{B}$ through the actor-network corresponding to the current policy:\n\\begin{equation}\n\\label{eq:det_current_actions}\n    \\boldsymbol{\\hat{A}}^{|\\mathcal{B}| \\times n} = \\pi_{\\phi}(\\boldsymbol{S}^{|\\mathcal{B}| \\times m}).\n\\end{equation}\nWe now have the batch of current policy's action decisions on the states from the off-policy transitions $\\boldsymbol{\\hat{A}}^{|\\mathcal{B}| \\times n}$, and the batch of past policies' decisions $\\boldsymbol{A}^{|\\mathcal{B}| \\times n}$ from $\\mathcal{B}$. Let $\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}$ be the batch of numerical differences in action decisions, that is:\n\\begin{equation}\n\\label{eq:action_diff_batch_def}\n    \\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n} \\coloneqq \\boldsymbol{A}^{|\\mathcal{B}| \\times n} - \\boldsymbol{\\hat{A}}^{|\\mathcal{B}| \\times n}.\n\\end{equation}\nObserve that $\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}$ indicates the numerical deviation between the action decisions of the current policy and previous behavioral policies of the agent that generated the sampled off-policy transitions. To construct a multivariate Gaussian distribution from the action difference batch $\\mathcal{N}(\\boldsymbol{\\dot{\\mu}}^{1 \\times n}, \\boldsymbol{\\dot{\\Sigma}}^{n \\times n})$, let:\n\\begin{gather}\n    \\boldsymbol{\\dot{\\mu}}^{1 \\times n} = \\frac{1}{|\\mathcal{B}|}\\smashoperator{\\sum_{i = 1}^{|\\mathcal{B}|}}\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}_{i},\\label{eq:multi_var_gaussian_mean} \\\\\n    \\boldsymbol{\\dot{\\Sigma}}^{n \\times n} = \\frac{1}{|\\mathcal{B}| - 1}\\smashoperator{\\sum_{i = 1}^{|\\mathcal{B}|}}(\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}_{i} - \\boldsymbol{\\dot{\\mu}}^{1 \\times n})^{\\mathrm{T}}(\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}_{i} - \\boldsymbol{\\dot{\\mu}}^{1 \\times n}),\\label{eq:multi_var_gauss_cov_mat} \n\\end{gather}\nwhere $\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}_{i}$ represents the $i^{\\mathrm{th}}$ row vector in $\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}$ which corresponds to the $i^{\\mathrm{th}}$ transition. Then, define the dissimilarity measure as:\n\\begin{equation}\n\\label{eq:complete_dissimilarity_measurement}\n    \\rho = \\mathrm{JSD}(\\mathcal{N}(\\boldsymbol{\\dot{\\mu}}^{1 \\times n}, \\boldsymbol{\\dot{\\Sigma}}^{n \\times n}) \\| \\mathcal{N}(\\boldsymbol{0}^{1 \\times n}, \\sigma\\boldsymbol{I}^{n \\times n})),\n\\end{equation}\nwhere $\\sigma$ is the standard deviation of the exploration noise, and $\\boldsymbol{I}^{n \\times n}$ is the identity matrix. In addition, JSD denotes the Jensen-Shannon divergence, the symmetrized and smoothed version of the Kullback\u2013Leibler (KL) divergence defined by:\n\\begin{gather}\n    \\mathrm{JSD}(P \\| Q) = \\frac{1}{2}\\mathrm{KL}(P \\| M) + \\frac{1}{2}\\mathrm{KL}(Q \\| M); \\\\\n    M = \\frac{1}{2}(P + Q),\n\\end{gather}\nwhere $\\mathrm{KL}$ denotes the KL-divergence, $P$ and $Q$ can be any probability distribution. We choose JSD for symmetric similarity measurement as the similarity of two policy distributions should not be assumed to be directed. Although KL-divergence is well-known for penalizing a distribution that is completely different from the distribution in interest, two policies in the same environment cannot be completely distinct~\\citep{sutton_book}. Hence, we do not need to penalize two distinct distributions heavily.\n\\begin{remark}\n\\label{rem:why_jsd}\nJensen-Shannon divergence provides a symmetric similarity measurement between two probability distributions by utilizing Kullback-Leibler divergence. Moreover, KL-divergence heavily penalizes distributions that are entirely different from each other. Therefore, the smoothed version of KL-divergence, JS-divergence, should be used to measure the similarity between two behavioral policies throughout the training, as two policies in the same environment cannot be completely distinct.\n\\end{remark}\n\nRather than directly comparing the action difference with a zero multivariate Gaussian in Equation (\\ref{eq:complete_dissimilarity_measurement}), we compute JSD using a reference matrix of zero-mean Gaussian with a diagonal covariance matrix having the standard deviation of the exploration noise in the diagonal entries. Otherwise, policies closer to the current policy that executed a transition with exploration noise may be rejected as the action decisions numerically deviate from the actual policy due to the additive exploration noise such as additive Gaussian noise \\citep{williams_1992} or Ornstein-Uhlenbeck noise process \\citep{ou_noise} utilized by the Deep Deterministic Policy Gradient algorithm \\citep{ddpg}. Thus, we couple the standard deviation of the exploration noise with the reference matrix to form a parameter-free algorithm. \n\nNaturally, if all transitions in the sampled batch match the distribution under the current policy, we have $\\rho = 0$ and $\\rho \\in (0, \\infty)$. To project the similarity measure into the interval $[0, 1]$ for bounded weights, a non-linear transformation can be applied:\n\\begin{equation}\n\\label{eq:complete_similarity_measurement}\n    \\lambda = e^{-\\rho}.\n\\end{equation}\nNotice that for two identical policies we have $\\lambda = 1$, and for completely distinct policies we have $\\lambda = 0$, making Equation (\\ref{eq:complete_similarity_measurement}) a similarity measure between two policies.\n\nIntuitively, the deterministic variant of AC-Off-POC first computes the numerical difference between the actions chosen by the target and behavioral policies, then compares the difference with zero. This approach is equivalent to an implicit comparison of the distributions under the current policy and the policies that executed the off-policy transitions. However, one concern with the deterministic variant of AC-Off-POC may be that the minority of the transitions within the sampled batch may be executed by policies very similar to the current agent's policy. Since we take the average of the action batch in computing the importance weight, i.e., Equation (\\ref{eq:multi_var_gaussian_mean}), those transitions may be weighted by a fixed weight close to 0, which results in loss of information. We address this issue and conclude our off-policy correction scheme for deterministic policies in Remark \\ref{rem:det_policies_conc}, and provide an intuitive analysis on the impact of mini-batch size in Observation \\ref{rem:det_batch_size_analysis}.\n\\begin{remark}\n\\label{rem:det_policies_conc}\n    The deterministic variant of AC-Off-POC measures the overall discrepancy between the sampled batch of transitions and the underlying distribution of the current policy. If the distribution under the majority of the transitions in a sampled batch diverges from the current agent's policy, the corresponding importance weights will be small, or vice versa. Nevertheless, it should be expected that the policies corresponding to the majority of the transitions in a sampled batch must be close to the current policy since the networks in actor-critic methods are already optimized through mini-batch learning.\n\\end{remark}\n\\begin{observation}\n\\label{rem:det_batch_size_analysis}\n    Let $|\\mathcal{B}|$ and $|\\mathcal{R}|$ are the fixed mini-batch and experience replay buffer \\citep{experience_replay} sizes used in the training. If $|\\mathcal{B}| \\rightarrow |\\mathcal{R}|$, then the mean indices of the sampled transitions will be 0.5 in the expectation, and the deterministic AC-Off-POC weights become slightly less than 0.5 due to the non-linear transformation. If $|\\mathcal{B}| \\rightarrow 0$, then the AC-Off-POC weights cannot be accurately estimated. Therefore, very large or small mini-batch sizes may result in poor performance.  \n\\end{observation}\n\n\\subsection{Stochastic Policies}\n\\label{sec:stochastic_AC-Off-POC}\nThe extension of AC-Off-POC to stochastic policies is trivial as probability distributions represent the policies. Although the known probability distributions can be directly used in the traditional importance sampling methods through eligibility traces \\citep{learning_from_delayed_rewards, mach_3}, they require sufficiently long trajectories executed by each policy, such as in \\citep{impala, trust_region_estimators, munos_safe, harutyunyan}. Instead, stochastic AC-Off-POC can apply an off-policy correction on randomly selected off-policy transitions to control continuous systems. \n\nIn this variant of AC-Off-POC, we do not need assumptions on the type of distribution the actions are sampled. Therefore, we can store the parameters of the policy distributions in the replay buffer \\citep{experience_replay}. For example, if the stochastic policy of an agent is represented by a Beta distribution, $\\mathrm{B}(\\alpha, \\beta)$, we store the parameters $\\alpha$ and $\\beta$ in the experience replay buffer \\citep{experience_replay} and sample these parameters along with states, actions, and rewards at every update step.\n\nAt each gradient step, the agent samples a batch of off-policy transitions along with the parameters that define the policy distribution:\n\\begin{equation}\n\\label{eq:transition_tuple_definition_stochastic}\n    (\\boldsymbol{S}^{|\\mathcal{B}| \\times m}, \\boldsymbol{A}^{|\\mathcal{B}| \\times n}, \\boldsymbol{R}^{|\\mathcal{B}| \\times 1}, \\boldsymbol{S'}^{|\\mathcal{B}| \\times m}, \\alpha_{1}, \\dots, \\alpha_{|\\mathcal{B}|}) \\sim \\mathcal{B},\n\\end{equation}\nwhere $\\alpha_{1}, \\dots, \\alpha_{|\\mathcal{B}|}$ represent the distribution parameters for each behavioral policy that executed the transitions indexed by $1, \\dots, \\mathcal{|\\mathcal{B}|}$, respectively. Each behavioral policy $\\eta_{i}$ that executed the $i^\\mathrm{th}$ transition can be represented with the corresponding distribution parameters, $\\eta_{i}(\\cdot; \\alpha_{i})$. Instead of comparing the difference batch with a reference Gaussian, we can directly measure the similarity between the distribution under the current policy $\\eta^{\\pi}(\\cdot)$ and $\\eta_{i}(\\cdot; \\alpha_{i})$:\n\\begin{gather}\n    \\rho_{i} = \\mathrm{JSD}(\\eta^{\\pi}(\\cdot)\\|\\eta_{i}(\\cdot; \\alpha_{i})),\\label{eq:complete_dissimilarity_measurement_stochastic}\\\\\n    \\lambda_{i} = e^{-\\rho_{i}}.\\label{eq:complete_similarity_measurement_stochastic}\n\\end{gather}\nThen, we construct an importance weight vector containing the weights for the sampled batch of transitions:\n\\begin{equation}\n    \\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1} \\coloneqq [\\lambda_{1} \\dots \\lambda_{|\\mathcal{B}|}]^{\\mathrm{T}}.\n\\end{equation}\nSimilar to Equation (\\ref{eq:det_actor_loss}) and Equation (\\ref{eq:det_critic_loss}), we can derive the weighted policy gradient and critic loss for stochastic actor-critic as:\n\\begin{gather}\n    \\boldsymbol{\\nabla}_{\\phi}\\Tilde{J}_{sto}(\\phi) = \\frac{1}{\\Vert \\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1 }\\Vert_{1}}\\smashoperator{\\sum_{i = 1}^{|\\mathcal{B}|}}\\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1}_{i}\\mathrm{log}\\pi_{\\phi}(\\boldsymbol{A}^{|\\mathcal{B}| \\times n}|\\boldsymbol{S}^{|\\mathcal{B}| \\times m})\\circ Q_{\\theta}(\\boldsymbol{S}^{|\\mathcal{B}| \\times m}, \\boldsymbol{A}^{|\\mathcal{B}| \\times n}),\\label{eq:sto_actor_loss} \\\\\n    \\Tilde{J}(\\theta) = \\frac{\\Vert \\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1} \\circ \\boldsymbol{\\delta}^{|\\mathcal{B}| \\times 1} \\Vert^{2}_{2}}{\\Vert \\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1 }\\Vert_{1}},\\label{eq:sto_critic_loss}\n\\end{gather}\nwhere $\\Vert \\cdot \\Vert_{1}$ represents the $\\mathrm{L}_{1}$ norm.\n\nIn the stochastic variant, each off-policy transition has a unique weight as it can be computed using a known mean vector and covariance matrix for each policy corresponding to that transition. Thus, we compute the loss for actor and critic parameters using a weighted sum of the off-policy transitions. However, for deterministic policies, all off-policy transitions in $\\mathcal{B}$ are weighted by the same importance weight $\\lambda$. Therefore, the deterministic variant weighs the sampled batch through the overall discrepancy instead of a weighted sum. As a result, the concern with the loss of information for the deterministic variant is overcome in the stochastic policy case. Nonetheless, the loss of information with deterministic policies can still be neglected as the sampled batch should be similar to the current policy on average \\citep{off_policy}. We adapt Remark \\ref{rem:det_policies_conc} to the stochastic variant of AC-Off-POC in the following.\n\\begin{remark}\n\\label{rem:sto_policies_conc}\n    Due to the transition-wise similarity measurement in the stochastic variant of AC-Off-POC, the off-policy correction is not affected by the mini-batch size. However, the impact of the batch size can still be observable on the underlying off-policy actor-critic algorithm, which may affect the performance of stochastic AC-Off-POC application.\n\\end{remark}\n\nWe refer to the resulting parameter-free algorithm as \\textit{Actor-Critic Off-Policy Correction} (AC-Off-POC) and provide pseudocode in Algorithm \\ref{alg:js_importance_weighting}. In the next section, we describe a generic off-policy actor-critic framework with AC-Off-POC, explain the generalizability of our method, and make a theoretical analysis of the contraction mapping produced by the off-policy correction coefficients.\n\n\\begin{algorithm}[!hbt]\n    \\caption{Actor-Critic Off-Policy Correction (AC-Off-POC)}\n    \\begin{algorithmic}[1]\n        \\STATE \\textbf{Input:} $\\pi_{\\phi}, \\mathcal{B}$\n        \\STATE \\textbf{Output:} $\\lambda$ $\\vee$ $\\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1}$\n        \\IF{$\\pi_{\\phi}$ is deterministic}\n            \\STATE Compute the current policy's action decisions on the sampled batch of off-policy states: \\\\\n            $\\boldsymbol{\\hat{A}}^{|\\mathcal{B}| \\times n} = \\pi_{\\phi}(\\boldsymbol{S}^{|\\mathcal{B}| \\times m})$.\n            \\STATE Compute the numerical differences between the current agent's actions and off-policy actions: \\\\ $\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n} \\coloneqq \\boldsymbol{A}^{|\\mathcal{B}| \\times n} - \\boldsymbol{\\hat{A}}^{|\\mathcal{B}| \\times n}.$\n            \\STATE Construct the multivariate Gaussian distribution, $\\mathcal{N}(\\boldsymbol{\\dot{\\mu}}^{1 \\times n}, \\boldsymbol{\\dot{\\Sigma}}^{n \\times n})$ where: \\\\\n            $\\boldsymbol{\\dot{\\mu}}^{1 \\times n} = \\frac{1}{|\\mathcal{B}|}\\smashoperator{\\sum_{i = 1}^{|\\mathcal{B}|}}\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}_{i},$ \\\\\n            $\\boldsymbol{\\dot{\\Sigma}}^{n \\times n} = \\frac{1}{|\\mathcal{B}| - 1}\\smashoperator{\\sum_{i = 1}^{|\\mathcal{B}|}}(\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}_{i} - \\boldsymbol{\\dot{\\mu}}^{1 \\times n})^{\\mathrm{T}}(\\boldsymbol{\\dot{A}}^{|\\mathcal{B}| \\times n}_{i} - \\boldsymbol{\\dot{\\mu}}^{1 \\times n}).$\n            \\STATE Compute the similarity $\\lambda$: $\\rho = \\mathrm{JSD}(\\mathcal{N}(\\boldsymbol{\\dot{\\mu}}^{1 \\times n}, \\boldsymbol{\\dot{\\Sigma}}^{n \\times n}) \\| \\mathcal{N}(\\boldsymbol{0}^{1 \\times n}, \\sigma\\boldsymbol{I}^{n \\times n})) \\Rightarrow \\lambda = e^{-\\rho}.$\n        \\ELSE\n            \\STATE Obtain the batch containing the policy distribution parameters: \\\\\n            $(\\boldsymbol{S}^{|\\mathcal{B}| \\times m}, \\boldsymbol{A}^{|\\mathcal{B}| \\times n}, \\boldsymbol{R}^{|\\mathcal{B}| \\times 1}, \\boldsymbol{S'}^{|\\mathcal{B}| \\times m}, \\alpha_{1}, \\dots, \\alpha_{|\\mathcal{B}|}) \\sim \\mathcal{B}.$\n            \\STATE Compute the similarities $\\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1} = [\\lambda_{1} \\dots \\lambda_{|\\mathcal{B}|}]^{\\mathrm{T}}$: \n            $\\rho_{i} = \\mathrm{JSD}(\\eta^{\\pi}(\\cdot)\\|\\eta_{i}(\\cdot; \\alpha_{i})) \\Rightarrow \\lambda_{i} = e^{-\\rho_{i}}.$\n        \\ENDIF\n        \\RETURN $\\lambda$ $\\vee$ $\\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1}$\n    \\end{algorithmic}\n    \\label{alg:js_importance_weighting}\n\\end{algorithm}\n\n\\subsection{Off-Policy Actor-Critic with AC-Off-POC}\nAs discussed, trajectory-based importance sampling methods can introduce high-variance build-up or vanishing gradients. Specifically, when each importance weight in a trajectory is larger than 1, the multiplication of the weights causes a detrimental accumulated variance that can diminish the accuracy of the trajectory weight. In contrast, when the weights are closer to 0, the trajectory weight becomes nearly 0, which yields the gradients to vanish; therefore, no updates are to be performed. However, these issues cannot be observed in AC-Off-POC since our method does not consider temporally correlated samples. Instead, single transitions are weighted by coefficients constrained in the interval of $[0, 1]$, i.e., weights of the off-policy transitions in a single update do not affect the weights in the next update. Thus, continuous control with AC-Off-POC offers safe and efficient off-policy learning. We summarize through the following remark how AC-Off-POC can overcome the mentioned issues observed in the importance sampling methods and why the introduced approach is preferable to providing off-policy corrections compared to standard importance sampling.\n\\begin{remark}\n\\label{rem:problems_in_current_is_methods}\n    As a single coefficient weighs each transition, vanishing or exploding gradient problem due to the product of weights in the trajectory-based importance sampling methods does not occur in AC-Off-POC. This is also valid for the unnecessary trajectory termination issue since AC-Off-POC only considers transition-wise or temporally uncorrelated batch-wise similarities. \n\\end{remark}\n\nOne concern with the introduced approach is that the mean difference could be zero even when the behavior policies are quite different from the current policy. Suppose the action range is $[-10, 10]$, and the current policy outputs action 0, half of past actions are -10, and the other half are 10. The fitted normal distribution will have a mean zero even though the policies are quite different. In this particular example, the variance in the fitted normal distribution would mitigate this issue, i.e., the empirical variance would be greater than the exploration noise added to the current policy, and hence, the similarity weights will be small.\n\nAnother concern with the off-policy correction is that some environments may broadly require off-policy samples to be solved, e.g., transitions of a random policy can also be used to improve a suboptimal policy. In such cases, an off-policy correction that filters out the on-policy samples may degrade the performance \\citep{sutton_book}. This is addressed by Equation (\\ref{eq:complete_similarity_measurement}) and Remark \\ref{rem:why_jsd}. Therefore, the AC-Off-POC weights should be near zero only when the distributions under two policies are nearly distinct, allowing beneficial off-policy samples to be included in the training. How AC-Off-POC performs this is explained in Remark \\ref{rem:off_policy_environments}.\n\\begin{remark}\n\\label{rem:off_policy_environments}\n    In environments that vastly require off-policy transitions to be solved, AC-Off-POC can still include transitions that diverge from the current agent's policy to an extent due to the non-linear transformation $e^{-\\rho}$ and Jensen-Shannon divergence used in the similarity measurements, which does not heavily penalize two different distributions unless they are nearly distinct.\n\\end{remark}\n\nFurthermore, as AC-Off-POC operates as a learning rate scheduler to adjust the contribution of each gradient with respect to off-policy samples, it can be employed in learning from a fixed dataset such as batch constrained RL algorithms and imitation learning. By allowing off-policy transitions correlated to the distribution under the current policy, AC-Off-POC can inhibit the extrapolation error, a phenomenon caused by the mismatch between the distributions corresponding to the off-policy data collected by a different agent and the latest agent's policy~\\citep{off_policy}. Thus, it can be applied to various off-policy reinforcement learning applications. However, batch constraint RL and imitation learning are not in the scope of this work as they require additional data generation modules. \n\nWe introduce a general off-policy actor-critic framework with AC-Off-POC in Algorithm \\ref{alg:actor_critic_with_AC-Off-POC} and perform an intuitive complexity analysis in Remark \\ref{rem:complexity_analysis}. Furthermore, we emphasize the ``safeness\" of our off-policy correction method in Corollary \\ref{cor:safe_off_policy}, where we show that the presented method defines a contraction mapping around the produced weights, i.e., the Q-value error interval shrinks as there exists a large similarity between behavioral policies corresponding to the sampled off-policy transitions and the policy of interest.\n\n\\begin{algorithm}[!t]\n    \\caption{A General Framework for Off-Policy Actor-Critic with AC-Off-POC}\n    \\begin{algorithmic}[1]\n        \\STATE Initialize the agent with actor $\\pi_{\\phi}$ and critic $Q_{\\theta}$ networks with parameters $\\phi, \\theta$, respectively. \n        \\STATE Initialize target and additional behavioral networks if required.\n        \\STATE Initialize the experience replay buffer $\\mathcal{R}$.\n        \\FOR{each exploration time step}\n            \\STATE Obtain transition tuple $\\tau$.\n            \\STATE Store transition tuple $\\tau$ in $\\mathcal{R}$.\n        \\ENDFOR\n        \\FOR{each training iteration}\n            \\STATE Sample a batch of transitions $\\mathcal{B}$ from $\\mathcal{R}$ by a sampling algorithm.\n            \\STATE Obtain the importance weight(s): $\\lambda$ $\\vee$ $\\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1}$ = \\textbf{AC-Off-POC($\\pi_{\\phi}$, $\\mathcal{B}$)}. \n            \\STATE Update actor and critic networks with an off-policy continuous actor-critic algorithm using policy gradients and losses weighted by $\\lambda$ $\\vee$ $\\boldsymbol{\\lambda}^{|\\mathcal{B}| \\times 1}$, e.g., Equation (\\ref{eq:det_actor_loss}), Equation (\\ref{eq:det_critic_loss}), Equation (\\ref{eq:sto_actor_loss}) and Equation (\\ref{eq:sto_critic_loss}).\n            \\STATE Update target networks if required.\n        \\ENDFOR\n    \\end{algorithmic}\n    \\label{alg:actor_critic_with_AC-Off-POC}\n\\end{algorithm}\n\n\\begin{remark}\n\\label{rem:complexity_analysis}\nIn the worst case, i.e., AC-Off-POC consists of a single forward pass through the actor network and several arithmetic matrix operations for deterministic policies. It is known that a single pass through a feedforward neural network has a time complexity of $\\mathcal{O}(n^{4})$. This value dominates the rest of the operations, which have at most quadratic runtime of $\\mathcal{O}(n^{2})$. Considering a simple deep actor-critic algorithm that involves a single actor and critic networks, the backpropagation operations will run in $\\mathcal{O}(n^{5})$. Therefore, even under the most straightforward deep actor-critic algorithm, the complexity of AC-Off-POC will be dominated by the backpropagation through the actor and critic networks.\n\\end{remark}\n\n\\begin{definition}\n\\label{def:exp_op_definition}\n    The general expectation operator for one-step importance sampling in return-based off-policy algorithms is defined in the bootstrapped form by:\n    \\begin{equation}\n        \\label{eq:def_eq}\n            \\mathcal{H}Q(s, a) \\coloneqq Q(s, a) + \\mathbb{E}_{\\eta}[\\sum_{t \\geq 0}\\gamma^{t}\\lambda_{t}\\left(r_{t} + \\gamma \\mathbb{E}_{\\pi}Q(s_{t + 1}, \\cdot) - Q(s_{t}, a_{t})\\right)],\n        \\end{equation}\n    for transition tuple $(s_{t}, a_{t}, r_{t}, s_{t + 1})$ and some non-negative one-step importance sampling coefficient $\\lambda_{t}$ at time step $t$, and any behavioral policy $\\eta$, where we write $\\mathbb{E}_{\\pi}Q(s, \\cdot) \\coloneqq \\sum_{a}\\pi(a | s)Q(s, a)$. \n\\end{definition}\n\n\\begin{lemma}\n\\label{lem:difference}\n    The difference between $\\mathcal{H}Q$ and its fixed point $Q^{\\pi}$ is expressed by:\n        \\begin{equation}\n            \\mathcal{H}Q(s, a) - Q^{\\pi}(s, a) = \\mathbb{E}_{\\eta}[\\sum_{t \\geq 1}\\gamma^{t}(\\mathbb{E}_{\\pi}[(Q - Q^{\\pi})(s_{t}, \\cdot)] - \\lambda_{t}(Q - Q^{\\pi})(s_{t}, a_{t}))]\n        \\end{equation}\n\\end{lemma}\n\\begin{proof}\n    The proof reduces the proof of Lemma 1 by \\cite{munos_safe} to one-step importance sampling. First, rewrite Equation (\\ref{eq:def_eq}) in terms of the next state-action pair:\n   \\begin{equation}\n       \\mathcal{H}Q(s, a) = \\sum_{t \\geq 0}\\gamma^{t}\\mathbb{E}_{\\eta}\\left[\\lambda_{t}(r_{t} + \\gamma[\\mathbb{E}_{\\pi}Q(s_{t + 1}, \\cdot) - \\lambda_{t + 1}Q(s_{t + 1}, a_{t + 1}))]\\right],\n   \\end{equation}\n   Since $Q^{\\pi}$ is the fixed point of $\\mathcal{H}$, we have:\n   \\begin{equation}\n           Q^{\\pi}(s, a) = \\mathcal{H}Q^{\\pi}(s, a)\n           = \\sum_{t \\geq 0}\\gamma^{t}\\mathbb{E}_{\\eta}\\left[\\lambda_{t}(r_{t} + \\gamma[\\mathbb{E}_{\\pi}Q^{\\pi}(s_{t + 1}, \\cdot) - \\lambda_{t + 1}Q^{\\pi}(s_{t + 1}, a_{t + 1}))]\\right].\n   \\end{equation}\n   Letting $\\Delta Q \\coloneqq Q - Q^{\\pi}$, we infer that:\n   \\begin{equation}\n           \\mathcal{H}Q(s, a) - Q^{\\pi}(s, a) = \\sum_{t \\geq 0}\\gamma^{t}\\mathbb{E}_{\\eta}\\left[\\lambda_{t}(\\gamma[\\mathbb{E}_{\\pi}\\Delta Q(s_{t + 1}, \\cdot) - \\lambda_{t + 1}\\Delta Q(s_{t + 1}, a_{t + 1})]\\right].\n   \\end{equation}\n   Decrementing a single step, the latter equation is equal to the following due to the bootstrapping in the temporal difference learning \\citep{sutton_1988}:\n   \\begin{equation}\n        \\mathcal{H}Q(s, a) - Q^{\\pi}(s, a) = \\sum_{t \\geq 1}\\gamma^{t}\\mathbb{E}_{\\eta}\\left[\\mathbb{E}_{\\pi}[\\Delta Q(s_{t}, \\cdot) - \\lambda_{t}\\Delta Q(s_{t}, a_{t})]\\right].\n    \\end{equation}\n    Taking the $\\sum_{t \\geq 1}\\gamma^{t}$ term into the expectation as $\\gamma$ is fixed and expanding $\\Delta Q$, we have:\n    \\begin{equation}\n        \\mathcal{H}Q(s, a) - Q^{\\pi}(s, a) = \\mathbb{E}_{\\eta}[\\sum_{t \\geq 1}\\gamma^{t}(\\mathbb{E}_{\\pi}[(Q - Q^{\\pi})(s_{t}, \\cdot)] - \\lambda_{t}(Q - Q^{\\pi})(s_{t}, a_{t}))].\n    \\end{equation}\n\\end{proof}\n\n\\begin{theorem}\n\\label{thm:safe_off_policy}\n    The operator $\\mathcal{H}$ defined by Definition \\ref{def:exp_op_definition} has a unique fixed point $Q^{\\pi}$. Moreover, if for each action selected by the policy $a_{t} \\in \\mathcal{A}$ and sampled batch of transitions $\\mathcal{B}_{t}$ at time step $t$, we have $\\lambda_{t} = \\lambda(a_{t}, \\mathcal{B}_{t}) \\in [0, e^{-\\rho}]$. Then for any Q-function $Q$, we have $\\mathcal{H}$, which is a $\\gamma$-contraction mapping around $Q^{\\pi}$:\n    \\begin{equation}\n        \\| \\mathcal{H}Q - Q^{\\pi} \\| \\leq \\gamma \\|Q - Q^{\\pi}\\|,\n    \\end{equation}\n    under the current policy $\\pi$.\n\\end{theorem}\n\\begin{proof}\n    Follows from the adaptation of proof of Theorem 1 by \\cite{munos_safe} to one-step importance sampling. It is trivial to observe from Definition \\ref{def:exp_op_definition} that $Q^{\\pi}$ is the fixed point of the operator $\\mathcal{H}$ since:\n    \\begin{equation}\n        \\mathbb{E}_{s_{t + 1} \\sim P(\\cdot|s_{t}, a_{t})}[r_{t} + \\gamma\\mathbb{E}_{\\pi}Q^{\\pi}(s_{t + 1}, \\cdot) - Q^{\\pi}(s_{t}, a_{t})] = (\\mathcal{H}^{\\pi}Q^{\\pi} - Q^{\\pi})(s_{t}, a_{t}) = 0,\n    \\end{equation}\n    as $Q^{\\pi}$ is the fixed point of $\\mathcal{H}^{\\pi}$. Let $\\Delta Q \\coloneqq Q - Q^{\\pi}$ and from Lemma \\ref{lem:difference}, we have:\n    \\begin{align}\n        \\mathcal{H}Q(s, a) - Q^{\\pi}(s, a) &= \\sum_{t \\geq 1}\\gamma^{t}\\underset{\\substack{s_{1:t} \\\\ a_{1:t}}}{\\mathbb{E}}\\left[\\mathbb{E}_{\\pi}[\\Delta Q(s_{t}, \\cdot) - \\lambda_{t}\\Delta Q(s_{t}, a_{t})]\\right], \\\\\n        &= \\sum_{t \\geq 1}\\gamma^{t}\\underset{\\substack{s_{1:t} \\\\ a_{1:t - 1}}}{\\mathbb{E}}\\left[\\mathbb{E}_{\\pi}[\\Delta Q(s_{t}, \\cdot) - \\mathbb{E}_{a_{t}}[\\lambda_{t}(a_{t}, \\mathcal{B}_{t})\\Delta Q(s_{t}, a_{t}) | \\mathcal{B}_{t}]]\\right], \\\\\n        &= \\sum_{t \\geq 1}\\gamma^{t}\\underset{\\substack{s_{1:t} \\\\ a_{1:t - 1}}}{\\mathbb{E}}\\left[\\sum_{b}(\\pi(b|s_{t}) - \\eta(b|s_{t})\\lambda_{t}(b, \\mathcal{B}_{t}))\\Delta Q(s_{t}, b)\\right].\n    \\end{align}\n    Since $\\lambda \\in [0, 1]$, we have $\\pi(a|s_{t}) - \\eta(a|s_{t})\\lambda_{t}(a, \\mathcal{B}_{t}) \\geq 0$ \\citep{munos_safe}. This yields:\n    \\begin{equation}\n        \\mathcal{H}Q(s, a) - Q^{\\pi}(s, a) = \\sum_{y, b}w_{y, b}\\Delta Q(y, b),\n    \\end{equation}\n    which is a linear combination of $\\Delta Q(y, b)$ weighted by non-negative coefficients $\\lambda$:\n    \\begin{equation}\n        w_{y, b} \\coloneqq \\sum_{t \\geq 1}\\gamma^{t}\\underset{\\substack{s_{1:t} \\\\ a_{1:t-1}}}{\\mathbb{E}}[\\pi(b|s_{t}) - \\eta(b|s_{t})\\lambda_{t}(b, \\mathcal{B}_{t})\\mathbb{I}\\{s_{t} = y\\}],\n    \\end{equation}\n    where $\\mathbb{I}(\\cdot)$ is the indicator function. The sum of those coefficients over $y$ and $b$ is:\n    \\begin{align}\n    \\sum_{y, b}w_{y, b} &= \\sum_{t \\geq 1}\\gamma^{t}\\underset{\\substack{s_{1:t} \\\\ a_{1:t - 1}}}{\\mathbb{E}}[\\sum_{b}(\\pi(b|s_{t}) - \\eta(b|s_{t})\\lambda_{t}(b, \\mathcal{B}_{t}))], \\\\\n    &= \\sum_{t \\geq 1}\\gamma^{t}\\underset{\\substack{s_{1:t} \\\\ a_{1:t - 1}}}{\\mathbb{E}}[\\mathbb{E}_{a_{t}}[1 - \\lambda_{t}(a_{t}, \\mathcal{B}_{t})|\\mathcal{B}_{t}]] = \\sum_{t \\geq 1}\\gamma^{t}\\underset{\\substack{s_{1:t} \\\\ a_{1:t}}}{\\mathbb{E}}[1 - \\lambda_{t}], \\\\\n    &= \\mathbb{E}_{\\eta}[\\sum_{t \\geq 1}\\gamma^{t} - \\sum_{t \\geq 1}\\gamma^{t}\\lambda_{t}]\n    \\end{align}\n     Clearly, we have $\\sum_{y, b}w_{y, b} \\leq \\gamma$ for any $t \\geq 1$ and $\\lambda_{t} \\in [0, 1]$. Therefore, $\\mathcal{H}Q(s, a) - Q^{\\pi}(s, a)$ is a sub-convex combination of $\\Delta Q(y, b)$ weighted by non-negative coefficients $\\omega_{y, b}$ which sum to at most $\\gamma$. Hence, $\\mathcal{H}$ is a $\\gamma$-contraction mapping around $Q^{\\pi}$.\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:safe_off_policy}\n    In the proof of Theorem \\ref{thm:safe_off_policy}, notice that the term $\\mathbb{E}_{\\eta}[\\sum_{t \\geq 1}\\gamma^{t}\\lambda_{t}]$ depends on $(s, a)$ since $\\lambda_{t}$ already depends on the sampled batch $\\mathcal{B}$. Let:\n    \\begin{equation}\n        \\xi(s, a) \\coloneqq \\sum_{y, b}w_{y, b}.\n    \\end{equation}\n    Then, we can show that:\n    \\begin{equation}\n        |\\mathcal{H}Q(s, a) - Q^{\\pi}(s, a)| \\leq \\xi(s, a)\\|Q - Q^{\\pi}\\|. \n    \\end{equation}\n    Thus, $\\xi(s, a) \\in [0, \\gamma]$ is a contraction coefficient based on $(s, a)$ where $\\xi(s, a) = \\gamma$ if $\\lambda = 0$, i.e., when there is no similarity, and close to zero when the behavioral policies corresponding to the sampled batch $\\mathcal{B}$ match the current policy.\n\\end{corollary}\n\n\\section{Experiments}\nWe conduct experiments to evaluate the effectiveness of our off-policy correction approach. In Section \\ref{sec:evaluation}, we present the simulation results for the comparative evaluation with different importance sampling methods and off-policy policy gradient techniques. We conduct experiments under different batch and replay memory sizes, experience replay sampling methods, and divergence measures in Section \\ref{sec:sensitivity_analysis} for the sensitivity analysis of the introduced off-policy correction scheme. In addition, weights produced by AC-Off-POC are visualized and discussed in Section \\ref{sec:weight_analysis}. We use the MuJoCo \\citep{mujoco} and Box2D \\citep{box2d} benchmarks interfaced by OpenAI Gym \\citep{gym}. For reproducibility, we do not modify the environment dynamics and reward functions. The computing infrastructure for reproducibility is summarized in our repository\\footref{our_repo}.\n\n\\subsection{Experimental Setup and Implementation}\n\\label{sec:exp_setup_and_implementation}\nWe apply our method to three baseline off-policy actor-critic methods: Deep Deterministic Policy Gradient (DDPG) \\citep{ddpg}, Soft Actor-Critic \\citep{sac} and Twin Delayed DDPG (TD3) \\citep{td3}. For comparative evaluation, we consider each baseline with and without AC-Off-POC, the continuous importance sampling method, RIS-Off-PAC \\citep{ris}, and off-policy policy gradient methods, ACE \\citep{ace}, Geoff-PAC \\citep{geoff_pac} and IPG \\citep{ipg}. In the sensitivity analysis, we use the experience replay \\citep{experience_replay} sampling methods of Combined Experience Replay (CER) \\citep{cer}, Experience Replay Optimization (ERO) \\citep{ero} and Prioritized Experience Replay (PER) \\citep{per}.\n\nWe use the stochastic variant of the SAC algorithm \\citep{sac} to show the performance improvement when stochastic AC-Off-POC is applied. Environment-specific and shared parameters of SAC \\citep{sac} follows the tuned hyper-parameters from OpenAI Baselines3 Zoo\\footnote{\\label{footnote:zoo}\\url{https:\/\/github.com\/DLR-RM\/rl-baselines3-zoo}} \\citep{zoo}. As the actions are sampled from the environment's action space in the initial exploration steps, policy distribution parameters are not accessible for stochastic AC-Off-POC. Rather, we start storing these parameters after a multiple of the number of exploration time steps and remove the exploration transitions from the experience replay buffer \\citep{experience_replay}. Our implementation of DDPG \\citep{ddpg} also closely matches the hyper-parameter setting outlined in the original paper. Different from the proposed setting, we add 1000 exploration time steps with frozen parameters at the beginning of each training and increase the batch size from 64 to 256 for the agents to encounter more off-policy samples in each update. Moreover, we replace Ornstein\u2013Uhlenbeck exploration noise with a zero-mean Gaussian with a standard deviation of 0.1, shown by \\cite{td3} to be effective for exploration and also to couple with the reference Gaussian distribution. To implement TD3 \\citep{td3}, we use the author's GitHub repository\\footnote{\\url{https:\/\/github.com\/sfujim\/TD3}} that contains the fine-tuned version of the TD3 algorithm \\citep{td3}. Exact implementations of the baseline algorithms can be accessed from our code\\footref{our_repo}.\n\nMoreover, the policy gradient methods ACE \\citep{ace}, Geoff-PAC \\citep{geoff_pac} and IPG \\citep{ipg} are applied to the baselines by replacing the policy gradient computation. The importance sampling method RIS-Off-PAC \\citep{ris} is directly applied on top of the baseline algorithms. We follow the original papers to implement ACE \\citep{ace}, IPG \\citep{ipg}, and RIS-Off-PAC \\citep{ris}. Our implementation of Geoff-PAC \\citep{geoff_pac} is based on the code from authors' GitHub repository\\footnote{\\url{https:\/\/github.com\/ShangtongZhang\/DeepRL}}. We follow the exact structure given in the original papers to implement CER \\citep{cer} and ERO \\citep{ero}. Additionally, the repository\\footnote{\\url{https:\/\/github.com\/sfujim\/LAP-PAL}} is used to implement PER \\citep{per}.\n\nThe considered competing algorithms introduce hyper-parameters that either control the bias-variance trade-off or on-policyness of the updates. We use the tuned hyper-parameters provided in the original articles. In particular, we use the bias-variance trade-off value of $v = 1$ for IPG \\citep{ipg} as it is defined for actor-critic methods with corrected off-policy samples. As stated by \\cite{geoff_pac}, ACE \\citep{ace} is not sensitive for the bias parameter $\\lambda_{1}$ on OpenAI Gym \\citep{gym} benchmarks and $\\lambda_{1} = 0$ can produce sufficiently good results. Hence, we set $\\lambda_{1} = 0$ for ACE \\citep{ace}. For Geoff-PAC \\citep{geoff_pac}, however, we employ the tuned hyper-parameters of $\\lambda_{1} = 0.7$, $\\lambda_{2} = 0.6$, and $\\hat{\\gamma} = 0.2$ since they are reported to result in good empirical performance. Finally for RIS-Off-PAC \\citep{ris}, we use $0.2$ for the smoothing control parameter $\\beta$. \nFor all simulations, each algorithm is run for 1 million time steps with evaluations occurring every 1000 time steps, where an evaluation of an agent records the average reward over ten episodes in a distinct evaluation environment without exploration noise and updates. We report the average evaluation return of ten random seeds for initializing networks, simulators, and dependencies. Uniform sampling and an experience replay buffer \\citep{experience_replay} of size 1 million are used in all experiments. \n\n\\subsection{Comparative Evaluation}\n\\label{sec:evaluation}\n\\begin{sidewaystable*}[!hbpt]\n    \\centering\n    \\scalebox{0.75}{\n    \\begin{tabular}[t]{lcccccccc}\n    \\toprule%\n    \\textbf{Method}  & \\textbf{Ant} & \\textbf{BipedalWalker} & \\textbf{HalfCheetah} & \\textbf{Hopper} & \\textbf{Humanoid} & \\textbf{LunarLanderContinuous} & \\textbf{Swimmer} & \\textbf{Walker2d}  \\\\        \n    \\midrule\n    DDPG & 816.99 $\\pm$ 165.29 & -139.73 $\\pm$ 36.10 & 3798.16 $\\pm$ 1586.74 & 2137.20 $\\pm$ 537.64 & 206.80 $\\pm$ 105.63 & 26.69 $\\pm$ 73.32 & 19.71 $\\pm$ 5.90 & 1270.86 $\\pm$ 332.23 \\\\\n    DDPG + AC-Off-POC & \\textbf{1052.28 $\\pm$ 239.11} & \\textbf{-104.79 $\\pm$ 38.39} & \\textbf{5012.32 $\\pm$ 2155.64} & \\textbf{2335.00 $\\pm$ 765.7}2 & \\textbf{1115.81 $\\pm$ 641.76} & 173.02 $\\pm$ 85.59 & \\textbf{28.95 $\\pm$ 7.88} & \\textbf{2854.85 $\\pm$ 555.45} \\\\\n    DDPG + ACE & 829.96 $\\pm$ 133.42 & -122.38 $\\pm$ 33.63 & 4043.05 $\\pm$ 1672.13 & 1742.74 $\\pm$ 488.32 & 204.14 $\\pm$ 100.09 & 161.61 $\\pm$ 58.27 & 20.82 $\\pm$ 12.27 & 1742.84 $\\pm$ 414.89 \\\\\n    DDPG + Geoff-PAC & 784.67 $\\pm$ 133.77 & -125.71 $\\pm$ 36.54 & 4660.93 $\\pm$ 1951.33 & 1767.06 $\\pm$ 676.08 & 255.75 $\\pm$ 115.61 & \\textbf{210.53 $\\pm$ 70.60} & 27.80 $\\pm$ 5.45 & 2030.25 $\\pm$ 625.53 \\\\\n    DDPG + IPG & 833.51 $\\pm$ 88.32 & -121.91 $\\pm$ 39.11 & 4248.09 $\\pm$ 1796.15 & 1905.45 $\\pm$ 731.71 & 208.46 $\\pm$ 84.78 & 207.11 $\\pm$ 69.60 & 24.55 $\\pm$ 5.47 & 1601.75 $\\pm$ 516.69 \\\\\n    DDPG + RIS-Off-PAC & 837.10 $\\pm$ 107.24 & -107.97 $\\pm$ 31.83 & 4171.61 $\\pm$ 1753.42 & 1819.95 $\\pm$ 450.00 & 198.50 $\\pm$ 81.86 & 196.59 $\\pm$ 66.94 & 21.93 $\\pm$ 6.86 & 1644.48 $\\pm$ 462.71 \\\\ \\\\\n    \n    SAC & 4864.24 $\\pm$ 296.94 & 285.37 $\\pm$ 50.74 & 9871.42 $\\pm$ 273.98 & 3351.43 $\\pm$ 317.07 & 2689.22 $\\pm$ 1499.29 & 271.37 $\\pm$ 9.82 & 53.11 $\\pm$ 4.68 & 4103.54 $\\pm$ 805.03 \\\\\n    SAC + AC-Off-POC & \\textbf{5983.03 $\\pm$ 341.04} & \\textbf{343.96 $\\pm$ 6.93} & \\textbf{11320.82 $\\pm$ 262.23} & \\textbf{3846.05 $\\pm$ 777.89} & \\textbf{5098.33 $\\pm$ 529.73} & \\textbf{284.83 $\\pm$ 7.09} & \\textbf{130.29 $\\pm$ 19.46} & \\textbf{5366.81 $\\pm$ 819.00} \\\\\n    SAC + ACE & 5019.17 $\\pm$ 549.69 & 307.18 $\\pm$ 17.98 & 9375.53 $\\pm$ 410.10 & 3367.48 $\\pm$ 313.79 & 4545.63 $\\pm$ 415.48 & 277.18 $\\pm$ 7.91 & 60.89 $\\pm$ 5.46 & 3239.37 $\\pm$ 1193.46 \\\\\n    SAC + Geoff-PAC & 5281.92 $\\pm$ 471.66 & 297.24 $\\pm$ 29.89 & 10604.40 $\\pm$ 173.26 & 2636.44 $\\pm$ 791.47 & 4897.81 $\\pm$ 508.99 & 279.75 $\\pm$ 9.46 & 96.24 $\\pm$ 14.80 & 4367.23 $\\pm$ 1217.94 \\\\\n    SAC + IPG & 5173.16 $\\pm$ 274.65 & 318.80 $\\pm$ 9.41 & 9993.52 $\\pm$ 399.77 & 2906.38 $\\pm$ 609.73 & 4768.96 $\\pm$ 457.70 & 276.29 $\\pm$ 11.56 & 84.20 $\\pm$ 16.11 & 3456.34 $\\pm$ 1486.13 \\\\\n    SAC + RIS-Off-PAC & 4613.55 $\\pm$ 430.51 & 313.67 $\\pm$ 12.80 & 10267.73 $\\pm$ 325.33 & 3036.12 $\\pm$ 534.42 & 4646.68 $\\pm$ 459.60 & 276.06 $\\pm$ 11.19 & 61.90 $\\pm$ 9.43 & 4032.21 $\\pm$ 976.43 \\\\ \\\\\n    \n    TD3 & 4846.66 $\\pm$ 418.23 & 271.77 $\\pm$ 79.60 & 10050.95 $\\pm$ 168.03 & 3174.56 $\\pm$ 401.05 & 4929.35 $\\pm$ 331.30 & 277.36 $\\pm$ 5.42 & 52.18 $\\pm$ 4.45 & 4107.08 $\\pm$ 349.81 \\\\\n    TD3 + AC-Off-POC & \\textbf{5472.19 $\\pm$ 276.49} & \\textbf{322.38 $\\pm$ 6.92} & \\textbf{11735.35 $\\pm$ 196.97} & \\textbf{3658.85 $\\pm$ 354.52} & \\textbf{5297.80 $\\pm$ 308.16} & \\textbf{308.05 $\\pm$ 15.24} & \\textbf{114.02 $\\pm$ 21.51} & \\textbf{5219.89 $\\pm$ 583.36} \\\\\n    TD3 + ACE & 5064.34 $\\pm$ 376.07 & 306.96 $\\pm$ 15.10 & 10543.87 $\\pm$ 148.98 & 3215.57 $\\pm$ 379.31 & 5115.29 $\\pm$ 219.01 & 279.49 $\\pm$ 5.27 & 57.18 $\\pm$ 5.30 & 4179.44 $\\pm$ 267.09 \\\\\n    TD3 + Geoff-PAC & 5261.21 $\\pm$ 416.70 & 319.52 $\\pm$ 15.19 & 11467.90 $\\pm$ 148.70 & 3041.78 $\\pm$ 516.34 & 4805.35 $\\pm$ 397.55 & 268.35 $\\pm$ 6.36 & 80.36 $\\pm$ 16.91 & 4795.22 $\\pm$ 232.31 \\\\\n    TD3 + IPG & 5332.47 $\\pm$ 367.49 & 295.09 $\\pm$ 44.95 & 11064.32 $\\pm$ 143.90 & 3524.52 $\\pm$ 247.26 & 4908.21 $\\pm$ 482.69 & 280.46 $\\pm$ 5.32 & 59.94 $\\pm$ 8.01 & 4751.56 $\\pm$ 153.17 \\\\\n    TD3 + RIS-Off-PAC & 5045.43 $\\pm$ 488.06 & 307.22 $\\pm$ 16.90 & 10811.51 $\\pm$ 166.86 & 3532.53 $\\pm$ 128.64 & 5015.17 $\\pm$ 340.12 & 278.38 $\\pm$ 6.68 & 56.18 $\\pm$ 5.13 & 4421.31 $\\pm$ 194.74 \\\\\n    \\bottomrule\n    \\end{tabular}\n    }\n    \\caption{Average return of last 10 evaluations over 10 trials\n    of 1 million time steps. $\\pm$ captures a 95\\% confidence interval over the trials. Bold values represent the maximum under each baseline algorithm and environment.}\n    \\label{table:eval}\n\\end{sidewaystable*}\n\nEvaluation results for the DDPG \\citep{ddpg}, SAC \\citep{sac} and TD3 \\citep{td3} algorithms are reported in Figure \\ref{fig:eval_results_ddpg}, \\ref{fig:eval_results_sac}, and \\ref{fig:eval_results_td3}, respectively. As learning curves are intertwined and hard to follow for some of the environments, we also report the average of the last ten evaluation returns, i.e., where the algorithms converge, in Table \\ref{table:eval}. Note that for some of the tasks, the baseline actor-critic algorithms have worse performance than reported in the original articles. This is due to the stochasticity of the simulators and used random seeds. Nevertheless, the performance difference between competing methods would be consistent if we used different sets of random seeds, regardless of where the baselines converge. Therefore, our comparative evaluations are fair per the standard deep RL benchmarking \\citep{deep_rl_that_matters}. \n\n\\begin{figure*}[!b]\n\\centering\n    \\begin{align*}\n        &\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_blue, solid, line width=2.3pt](0,0) -- (5mm,0);}}} DDPG}  &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_orange, solid, line width=2.3pt](0,0) -- (5mm,0);}}} DDPG + AC-Off-POC} &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_green, solid, line width=2.3pt](0,0) -- (5mm,0);}}} DDPG + ACE} \\\\ &\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_purple, solid, line width=2.3pt](0,0) -- (5mm,0);}}} DDPG + Geoff-PAC} &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_red, solid, line width=2.3pt](0,0) -- (5mm,0);}}} DDPG + IPG} &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_pink, solid, line width=2.3pt](0,0) -- (5mm,0);}}} DDPG + RIS-Off-PAC}\n    \\end{align*}\n\t\\subfigure{\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/DDPG_Ant-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/DDPG_BipedalWalker-v3.jpg} \n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/DDPG_HalfCheetah-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/DDPG_Hopper-v2.jpg} \n\t}\n\t\\\\\n\t\\subfigure{\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/DDPG_Humanoid-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/DDPG_LunarLanderContinuous-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/DDPG_Swimmer-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/DDPG_Walker2d-v2.jpg} \n\t}\n\t\\caption{Evaluation curves for the set of OpenAI Gym continuous control tasks for 1 million training time steps over 10 random seeds under the DDPG algorithm. The shaded region represents a 95\\% confidence interval over the trials. A sliding window of window size 10 smoothes the curves for visual clarity.}\n\t\\label{fig:eval_results_ddpg}\n\\end{figure*}\n\nWe observe that both variants of AC-Off-POC significantly improve the performance of the baseline methods and either match or outperform the competing off-policy correction techniques in all of the domains and baselines tested. This shows that our off-policy correction method is scalable to many off-policy actor-critic algorithms, as previously discussed. The performance improvement obtained by AC-Off-POC is mainly more observable when the baselines are stuck at the local optima in challenging tasks, e.g., Ant, BipedalWalker, Humanoid, and Swimmer \\citep{deep_rl_that_matters}. The adversity of these environments usually comes from the dimensionality, e.g., large state-action spaces or state-action intervals, or internal simulation dynamics \\citep{deep_rl_that_matters}. Thus, the fatal effects of off-policy samples are further amplified in these tasks, and the resulting improvement offered by an effective off-policy correction technique becomes more prominent.\n\nFurthermore, we observe more substantial improvement by AC-Off-POC in the BipedalWalker, Hopper, Swimmer, and Walker2d environments. As shown by \\cite{deep_rl_that_matters}, on-policy algorithms, i.e., which learn from the data that the policy of interest collected, outperform the off-policy methods by a considerable margin in these tasks. Hence, we deduce that these environments may vastly require on-policy samples to be solved for a stable and better learning process. Hence, corrected off-policy transitions maximize the improvement compared to other environments where off-policy algorithms are often superior to the on-policy algorithms, e.g., Ant, HalfCheetah, and Humanoid \\citep{deep_rl_that_matters}. Nonetheless, we still observe improved performance in these domains. We believe that this improvement is caused by only eliminating very divergent off-policy samples collected throughout the learning process since the employed JSD measure assigns very low scores only to transitions that correspond to very distinct behavioral policies. Therefore, although transitions corresponding to different behavioral policies are not similar, they are still included in the Q-learning and policy optimization. Consequently, the latter two empirical findings validate Remarks \\ref{rem:why_jsd} and \\ref{rem:off_policy_environments}.\n\nAlthough the performance enhancements achieved under the SAC \\citep{sac} and TD3 \\citep{td3} algorithms remain practically the same, the overall performance improvement for stochastic policies is slightly more notable than the deterministic policies when DDPG \\citep{ddpg} is also considered. Nevertheless, the performance improvement on deterministic policies becomes prominent when mini-batch learning is applied as AC-Off-POC completely disregards transitions in the sampled batch. These simulation results verify Remark \\ref{rem:det_policies_conc} and \\ref{rem:sto_policies_conc}, that is, more improvement for stochastic policies is expected, yet if the distribution of the sampled transitions is similar to the distribution under the current policy on average, off-policy deterministic actor-critic methods can be further improved. These empirical results suggest that our method effectively boosts the learning of off-policy methods and makes the agents converge to higher evaluation returns in a shorter time by disregarding highly divergent off-policy samples.\n\n\\begin{figure*}[!t]\n\\centering\n    \\begin{align*}\n        &\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_blue, solid, line width=2.3pt](0,0) -- (5mm,0);}}} SAC}  &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_orange, solid, line width=2.3pt](0,0) -- (5mm,0);}}} SAC + AC-Off-POC} &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_green, solid, line width=2.3pt](0,0) -- (5mm,0);}}} SAC + ACE} \\\\ &\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_purple, solid, line width=2.3pt](0,0) -- (5mm,0);}}} SAC + Geoff-PAC} &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_red, solid, line width=2.3pt](0,0) -- (5mm,0);}}} SAC + IPG} &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_pink, solid, line width=2.3pt](0,0) -- (5mm,0);}}} SAC + RIS-Off-PAC}\n    \\end{align*}\n\t\\subfigure{\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/SAC_Ant-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/SAC_BipedalWalker-v3.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/SAC_HalfCheetah-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/SAC_Hopper-v2.jpg} \n\t} \n\t\\\\\n\t\\subfigure{\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/SAC_Humanoid-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/SAC_LunarLanderContinuous-v2.jpg} \n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/SAC_Swimmer-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/SAC_Walker2d-v2.jpg} \n\t} \n\t\\caption{Evaluation curves for the set of OpenAI Gym continuous control tasks for 1 million training time steps over 10 random seeds under the SAC algorithm. The shaded region represents a 95\\% confidence interval over the trials. A sliding window of window size 10 smoothes the curves for visual clarity.}\n\t\\label{fig:eval_results_sac}\n\\end{figure*}\n\n\\begin{figure*}[!hbt]\n\\centering\n    \\begin{align*}\n        &\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_blue, solid, line width=2.3pt](0,0) -- (5mm,0);}}} TD3}  &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_orange, solid, line width=2.3pt](0,0) -- (5mm,0);}}} TD3 + AC-Off-POC} &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_green, solid, line width=2.3pt](0,0) -- (5mm,0);}}} TD3 + ACE} \\\\ &\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_purple, solid, line width=2.3pt](0,0) -- (5mm,0);}}} TD3 + Geoff-PAC}  &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_red, solid, line width=2.3pt](0,0) -- (5mm,0);}}} TD3 + IPG} &&\\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_pink, solid, line width=2.3pt](0,0) -- (5mm,0);}}} TD3 + RIS-Off-PAC}\n    \\end{align*}\n\t\\subfigure{\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/TD3_Ant-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/TD3_BipedalWalker-v3.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/TD3_HalfCheetah-v2.jpg}  \n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/TD3_Hopper-v2.jpg}\n\t}\n\t\\\\\n\t\\subfigure{\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/TD3_Humanoid-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/TD3_LunarLanderContinuous-v2.jpg} \n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/TD3_Swimmer-v2.jpg}\n\t\t\\includegraphics[width=1.55in, keepaspectratio]{Figures\/TD3_Walker2d-v2.jpg} \n\t}\n\t\\caption{Evaluation curves for the set of OpenAI Gym continuous control tasks for 1 million training time steps over 10 random seeds under the TD3 algorithm. The shaded region represents a 95\\% confidence interval over the trials. A sliding window of window size 10 smoothes the curves for visual clarity.}\n\t\\label{fig:eval_results_td3}\n\\end{figure*}\n\nCompeting methods exhibit notable stability in unstable environments such as Ant, Hopper, and Walker2d. However, the performance improvement is still not promising as they directly disregard off-policy samples through policy gradients or importance sampling. In detail, we observe that Geoff-PAC \\citep{geoff_pac} and IPG \\citep{ipg} perform best with similar performances among the competing algorithms, while RIS-Off-PAC \\citep{ris} obtains the worst performance and ACE \\citep{ace} remains in the middle. As discussed, RIS-Off-PAC \\citep{ris} uses the action-value generated from the behavioral policy in the reward function to train the algorithm rather than from the target policy. However, the behavioral policy that generates the action-value is usually suboptimal throughout the learning process, which yields underperformed baselines. Moreover, the empirical performance of ACE \\citep{ace} is limited to simple domains such as simple Markov chains or cart-pole balancing, as it was previously proposed for linear function approximation \\citep{geoff_pac}. Its generalization to deep neural network function approximation, Geoff-PAC \\citep{geoff_pac}, obtains an improved baseline performance, as expected. Although the emphatic approach in Geoff-PAC \\citep{geoff_pac} can get an unbiased sample from the policy gradient, we believe its estimation of the policy gradient is still erroneous and insufficient for optimal performance. Finally, as stated by \\cite{ipg}, IPG yields an on-policy deterministic actor-critic method when $v = 1$ is used. Our previous discussion suggests that a complete on-policy policy gradient can completely disregard the off-policy samples. However, off-policy samples may sometimes be beneficial depending on the task. Therefore, we do not observe the maximal performance by IPG \\citep{ipg} but AC-Off-POC. \n\nIn summary, off-policy samples usually degrade the performance due to the underlying distribution that substantially diverges from the current agent's policy. Nonetheless, off-policy methods may still require off-policy samples to learn the environment \\citep{deep_rl_that_matters}. AC-Off-POC solves this issue by employing the Jensen-Shannon divergence, which obtains a smooth similarity measurement prior to the non-linear transformation. Its symmetric similarity measurement prevents the off-policy transitions from being heavily penalized and allows them to contribute to the learning progress even with a small proportion. Considering the increased computational complexity of the introduced hyper-parameters in the competing approaches and the superior performance of parameter-free AC-Off-PAC, we believe that our method provides significant gains over the prior approaches.\n\n\\subsection{Sensitivity Analysis}\n\\label{sec:sensitivity_analysis}\nAs our technique is an off-policy method, batch and replay memory sizes and different experience replay sampling methods can considerably impact the learning. Therefore, we perform experiments on both variants of AC-Off-POC under different replay buffer and batch sizes and experience replay \\citep{experience_replay} sampling algorithms. We evaluate the baseline and AC-Off-POC with mini-batch sizes of 32, 256, 1024, and 2048, and replay memory \\citep{experience_replay} of sizes 1 million (1M) and 100,000 (100K). We also investigate the performance improvement by AC-Off-POC under the sampling algorithms of CER \\citep{cer}, ERO \\citep{ero}, and PER \\citep{per}. The same experimental setting given in Section \\ref{sec:exp_setup_and_implementation} is used. Unless otherwise stated, a mini-batch size of 256, replay memory of size 1 million transitions, and uniform sampling are used in all experiments. \n\n\\subsubsection{Impact of the Batch Size}\n\\begin{table*}[!hbt]\n    \\centering\n    \\begin{tabular}[t]{lcccc}\n    \\toprule\n    \\textbf{Setting}  & \\textbf{HalfCheetah} & \\textbf{Hopper} & \\textbf{Walker2d} \\\\        \n    \\midrule\n        SAC (32) & 9092.21 $\\pm$ 312.47 & 3304.77 $\\pm$ 348.16 & 3707.69 $\\pm$ 844.41 \\\\\n        SAC (256) & 9871.42 $\\pm$ 273.98 & 3351.43 $\\pm$ 317.07 & 4103.54 $\\pm$ 805.03 \\\\\n        SAC (1024)& 8766.95 $\\pm$ 290.90 & 2700.17 $\\pm$ 342.32 & 3848.68 $\\pm$ 927.21 \\\\\n        SAC (2048)& 8477.68 $\\pm$ 322.43 & 2930.11 $\\pm$ 324.41 & 3222.36 $\\pm$ 930.57 \\\\\n        \n        SAC + AC-Off-POC (32) & 11022.58 $\\pm$ 276.83 & 3470.82 $\\pm$ 777.71 & 5135.93 $\\pm$ 860.30 \\\\\n        SAC + AC-Off-POC (256)& \\textbf{11320.82 $\\pm$ 262.23} & \\textbf{3846.05 $\\pm$ 777.89} & \\textbf{5366.81 $\\pm$ 819.00} \\\\\n        SAC + AC-Off-POC (1024)& 10459.58 $\\pm$ 284.13 & 3269.79 $\\pm$ 833.98 & 5242.90 $\\pm$ 823.80 \\\\\n        SAC + AC-Off-POC (2048) & 9215.60 $\\pm$ 303.78 & 3558.28 $\\pm$ 893.85 & 4300.63 $\\pm$ 918.97 \\\\ \\\\ \n        \n        TD3 (32)& 8955.77 $\\pm$ 181.85 & 2961.18 $\\pm$ 434.44 & 3966.54 $\\pm$ 377.57 \\\\\n        TD3 (256)& 10050.95 $\\pm$ 168.03 & 3174.56 $\\pm$ 401.05 &  4107.08 $\\pm$ 349.81 \\\\\n        TD3 (1024)& 9963.48 $\\pm$ 185.11 & 2847.66 $\\pm$ 408.16 & 3881.20 $\\pm$ 351.81 \\\\\n        TD3 (2048)& 8712.29 $\\pm$ 194.69 & 2630.64 $\\pm$ 455.97 & 4033.33 $\\pm$ 351.51 \\\\\n        TD3 + AC-Off-POC (32)& 9104.90 $\\pm$ 249.96 & 3112.20 $\\pm$ 411.33 & 4951.56 $\\pm$ 698.94 \\\\\n        TD3 + AC-Off-POC (256)& \\textbf{11735.35 $\\pm$ 196.97} & \\textbf{ 3658.85 $\\pm$ 354.52} & \\textbf{5219.89 $\\pm$ 583.36} \\\\\n        TD3 + AC-Off-POC (1024)& 8791.17 $\\pm$ 252.00 & 3126.42 $\\pm$ 426.60 & 5040.96 $\\pm$ 607.94 \\\\\n        TD3 + AC-Off-POC (2048)& 6982.50 $\\pm$ 248.80 & 2176.53 $\\pm$ 469.73 & 4713.68 $\\pm$ 730.17 \\\\\n    \\bottomrule\n    \\end{tabular}\n     \\caption{Average return over the last 10 evaluations over 10 trials\n    of 1 million time steps, comparing the impact of mini-batch sizes $\\{32, 256, 1024, 2048\\}$. $\\pm$ captures a 95\\% confidence interval over the trials. Bold values represent the maximum under each baseline algorithm and environment.}\n    \\label{table:ablation_bs}\n\\end{table*}\nTable \\ref{table:ablation_bs} presents the batch size sensitivity analysis over ten random seeds. First, the batch size of 256 gives the best performance for all methods in terms of both mean and confidence as they are already tuned, as outlined in the original papers of the baselines \\citep{sac,td3}. For stochastic AC-Off-POC used in the SAC algorithm \\citep{sac}, we observe that the relative performance improvement by AC-Off-POC practically remains constant over different batch sizes. This is due to the transition-wise similarity measurement offered by the stochastic variant. Hence, when applied to stochastic policies, AC-Off-POC is not affected by the mini-batch learning and mini-batch size. Moreover, the mini-batch size considerably affects the batch-wise similarity measurement in the deterministic variant for TD3 \\citep{td3}. For all environments, a larger batch size degrades the performance of AC-Off-POC and underperforms the baseline. Similarly, a performance drop exists for smaller batch sizes caused by the inaccurate estimation of JSD of two Gaussians with an insufficient number of samples. Overall, our sensitivity analysis on the batch size confirms Observation \\ref{rem:det_batch_size_analysis} and Remark \\ref{rem:sto_policies_conc}.\n\n\\subsubsection{Impact of the Replay Memory Size}\n\\begin{table*}[!b]\n    \\centering\n    \\begin{tabular}[t]{lcccc}\n        \\toprule\n        \\textbf{Setting}  & \\textbf{HalfCheetah} & \\textbf{Hopper} & \\textbf{Walker2d} \\\\      \n        \\midrule\n            SAC (100K)& 8596.65 $\\pm$ 326.68 & 3446.68 $\\pm$ 368.41 & 4117.61 $\\pm$ 937.39 \\\\\n            SAC (1M)& 9871.42 $\\pm$ 273.98 & 3351.43 $\\pm$ 317.07 & 4103.54 $\\pm$ 805.03 \\\\\n            SAC + AC-Off-POC (100K)& 8290.28 $\\pm$ 309.43 & \\textbf{3959.37 $\\pm$ 744.74} & \\textbf{5471.21 $\\pm$ 788.24} \\\\\n            SAC + AC-Off-POC (1M)& \\textbf{11320.82 $\\pm$ 262.23} & 3846.05 $\\pm$ 777.89 & 5366.81 $\\pm$ 819.00 \\\\ \\\\\n            \n            TD3 (100K)& 9343.67 $\\pm$ 188.52 & 3200.30 $\\pm$ 403.49 & 4183.48 $\\pm$ 350.12 \\\\\n            TD3 (1M)& 10050.95 $\\pm$ 168.03 & 3174.56 $\\pm$ 401.05 &  4107.08 $\\pm$ 349.81 \\\\\n            TD3 + AC-Off-POC (100K)& 9068.78 $\\pm$ 255.80 & \\textbf{3790.57 $\\pm$ 494.83} & \\textbf{5302.83 $\\pm$ 670.45} \\\\\n            TD3 + AC-Off-POC (1M)& \\textbf{11735.35 $\\pm$ 196.97} & 3658.85 $\\pm$ 354.52 & 5219.89 $\\pm$ 583.36 \\\\\n        \\bottomrule\n    \\end{tabular}\n     \\caption{Average return over the last 10 evaluations over 10 trials\n    of 1 million time steps, comparing the impact of replay buffer sizes $\\{100000, 1000000\\}$. $\\pm$ captures a 95\\% confidence interval over the trials. Bold values represent the maximum under each baseline algorithm and environment.}\n    \\label{table:ablation_rb}\n\\end{table*}\nImpact of the replay memory sizes is reported in Table \\ref{table:ablation_rb}. When the size of the experience replay memory \\citep{experience_replay} changes, we observe a significant performance variation in terms of both mean rewards and confidences. In the HalfCheetah environment, a smaller replay memory \\citep{experience_replay} significantly degrades the performance under all methods. As discussed, this is caused by the environment dynamics of HalfCheetah, which usually requires off-policy samples to be efficiently solved. In all experiments, we consider First-In, First-Out (FIFO) buffer. Thus, the smaller replay memory \\citep{experience_replay} contains more on-policy samples than the buffer of size 1 million transitions. For Hopper and Walker2d, smaller memory yields better performance which supports the evaluation results found in Section \\ref{sec:evaluation}, i.e., off-policy correction in these environments yields a further performance improvement compared to HalfCheetah as the policy is trained in a more on-policy fashion. When we decrease the size of the experience replay buffer \\citep{experience_replay}, baseline SAC \\citep{sac} produces better results in HalfCheetah as AC-Off-POC further filters out the off-policy transitions that may be contained in the limited replay buffer and thus spoil the off-policy learning, which is outlined in Remark \\ref{rem:off_policy_environments}. Nonetheless, stochastic AC-Off-POC can still correct for off-policy samples and improve the performance of SAC \\citep{sac} in the rest of the environments. Furthermore, similar results can be found in the deterministic variant, that is, AC-Off-POC with a smaller buffer underperforms the baseline in HalfCheetah while substantially improving the deterministic baseline algorithm in Hopper and Walker2d.\n\n\\subsubsection{Impact of Different Sampling Algorithms}\n\\begin{table*}[!b]\n    \\centering\n    \\begin{tabular}[t]{lcccc}\n        \\toprule\n        \\textbf{Setting}  & \\textbf{HalfCheetah} & \\textbf{Hopper} & \\textbf{Walker2d} \\\\      \n        \\midrule\n            SAC (uniform) & 9871.42 $\\pm$ 273.98 & 3351.43 $\\pm$ 317.07 & 4103.54 $\\pm$ 805.03 \\\\\n            SAC (CER) & 10539.72 $\\pm$ 285.12 & 3442.93 $\\pm$ 311.60 & 4105.39 $\\pm$ 712.35 \\\\\n            SAC (ERO) & 9497.97 $\\pm$ 284.17 & 3393.43 $\\pm$ 315.16 & 4033.84 $\\pm$ 843.72 \\\\\n            SAC (PER) & 8953.30 $\\pm$ 296.70 & 2891.34 $\\pm$ 317.16 & 3844.24 $\\pm$ 832.49 \\\\\n            SAC + AC-Off-POC (uniform) & \\textbf{11320.82 $\\pm$ 262.23} & 3846.05 $\\pm$ 777.89 & 5366.81 $\\pm$ 819.00 \\\\\n            SAC + AC-Off-POC (CER) & 9642.60 $\\pm$ 209.10 & \\textbf{3905.79 $\\pm$ 760.39} & \\textbf{5449.23 $\\pm$ 802.84} \\\\ \n            SAC + AC-Off-POC (ERO) & 11006.71 $\\pm$ 272.90 & 3865.37 $\\pm$ 765.77 & 5409.00 $\\pm$ 827.40 \\\\ \n            SAC + AC-Off-POC (PER) & 10388.39 $\\pm$ 297.94 & 3493.84 $\\pm$ 840.43 & 4402.69 $\\pm$ 963.22 \\\\ \\\\\n            \n            TD3 (uniform) & 10050.95 $\\pm$ 168.03 & 3174.56 $\\pm$ 401.05 &  4107.08 $\\pm$ 349.81 \\\\\n            TD3 (CER) & 10628.20 $\\pm$ 173.55 & 3190.81 $\\pm$ 400.69 & 4151.20 $\\pm$ 342.95 \\\\\n            TD3 (ERO) & 9591.56 $\\pm$ 168.77 & 3187.00 $\\pm$ 403.67 & 4013.53 $\\pm$ 361.32 \\\\\n            TD3 (PER) & 9578.80 $\\pm$ 172.61 & 2637.58 $\\pm$ 407.95 & 3447.57 $\\pm$ 350.60 \\\\\n            TD3 + AC-Off-POC (uniform) & \\textbf{11735.35 $\\pm$ 196.97} & 3658.85 $\\pm$ 354.52 & 5219.89 $\\pm$ 583.36 \\\\\n            TD3 + AC-Off-POC (CER) & 9851.44 $\\pm$ 177.24 & \\textbf{3692.22 $\\pm$ 335.98} & \n            \\textbf{5367.62 $\\pm$ 486.86} \\\\\n            TD3 + AC-Off-POC (ERO) & 11345.45 $\\pm$ 205.72 & 3664.95 $\\pm$ 352.06 & 5039.16 $\\pm$ 589.69 \\\\\n            TD3 + AC-Off-POC (PER) & 10648.33 $\\pm$ 231.66 & 3467.80 $\\pm$ 415.79 & 4357.27 $\\pm$ 622.60 \\\\\n            \\bottomrule\n    \\end{tabular}\n     \\caption{Average return over the last 10 evaluations over 10 trials\n    of 1 million time steps, comparing the impact of uniform, CER, ERO and PER sampling methods. $\\pm$ captures a 95\\% confidence interval over the trials. Bold values represent the maximum under each baseline algorithm and environment.}\n    \\label{table:ablation_sm}\n\\end{table*}\nTable \\ref{table:ablation_sm} shows the resulting performances under the considered sampling algorithms. We first observe that CER \\citep{cer} is superior among all the sampling methods. This is due to the most recent collected transition, which is included in each update. Hence, all updates always occur with at least a single on-policy sample. A performance improvement can still be obtained when AC-Off-POC is applied to the baselines while sampling is performed through CER \\citep{cer}. However, AC-Off-POC slightly underperforms the baselines in the HalfCheetah environment under CER \\citep{cer} because of the discussed off-policy sample requirement. ERO \\citep{ero} performs poorly when applied to the baseline algorithms due to the internal structure of the algorithm, which is not in the scope of this work. Nevertheless, AC-Off-POC still improves ERO \\citep{ero} when applied to the baseline algorithms. In addition, PER \\citep{per} usually underperforms when applied to an actor-critic method. This is an expected result since it was initially proposed for discrete action spaces where a single network, i.e., Q-network, is used to select and evaluate action. Transitions with high prediction errors for Q-network are prioritized. However, in actor-critic methods, the actor network cannot be effectively trained through Bellman optimization \\citep{bellman} where the prediction error for the critic is large \\citep{sutton_book}. Although ERO \\citep{ero} and PER \\citep{per} exhibit a poor performance due to the discussed drawbacks, AC-Off-POC can still attain higher rewards. Therefore, we infer that AC-Off-POC can be readily adapted to any experience replay \\citep{experience_replay} sampling algorithm.\n\n\\subsubsection{Impact of the Divergence Measure}\nAs previously discussed in Remark \\ref{rem:why_jsd}, we employ Jensen-Shannon divergence in our approach, a symmetric measure of the KL-divergence, to measure the distance between two multivariate Gaussians. This is due to the asymmetric characteristics of the JSD, that is, it does not penalize distributions that are very different as firmly as KL-divergence, which allows a further performance improvement in environments that usually require off-policy samples to be solved, e.g., HalfCheetah, as discussed.\n\nWe test the resulting performance of AC-Off-POC when JSD is replaced with KL-divergence. These results are provided in Table \\ref{table:ablation_kl}. We obtain higher results with JSD in all of the environments. However, for the HalfCheetah environment, KL-divergence significantly degrades the performance. As it severely penalizes transitions under which the policy distribution differs from the policy of interest, most off-policy transitions are not included in the gradient computation due to the corresponding small weights. As discussed, the HalfCheetah environment usually requires those off-policy transitions, and since the agent does not effectively use them, the performance drops. Overall, these empirical studies also verifies Remark \\ref{rem:why_jsd}.\n\n\\begin{table*}[!t]\n    \\centering\n    \\begin{tabular}[t]{lcccc}\n        \\toprule\n        \\textbf{Setting}  & \\textbf{HalfCheetah} & \\textbf{Hopper} & \\textbf{Walker2d} \\\\      \n        \\midrule\n            SAC + AC-Off-POC (KL) & 10747.20 $\\pm$ 219.16 & 3567.88 $\\pm$ 373.95 & 4980.37 $\\pm$ 620.16 \\\\\n            SAC + AC-Off-POC (JSD) & \\textbf{11320.82 $\\pm$ 262.23} & \\textbf{3846.05 $\\pm$ 777.89} & \\textbf{5366.81 $\\pm$ 819.00} \\\\ \\\\\n            \n            TD3 + AC-Off-POC (KL) & 10112.36 $\\pm$ 200.21 & 3449.99 $\\pm$ 362.96 & 4915.94 $\\pm$ 612.65 \\\\\n            TD3 + AC-Off-POC (JSD) & \\textbf{11735.35 $\\pm$ 196.97} & \\textbf{3658.85 $\\pm$ 354.52} & \n            \\textbf{5219.89 $\\pm$ 583.36} \\\\\n            \\bottomrule\n    \\end{tabular}\n     \\caption{Average return over the last 10 evaluations over 10 trials\n    of 1 million time steps, comparing the divergence measures of Jensen-Shannon divergence and KL-divergence. $\\pm$ captures a 95\\% confidence interval over the trials. Bold values represent the maximum under each baseline algorithm and environment.}\n    \\label{table:ablation_kl}\n\\end{table*}\n\n\n\\subsection{Weight Analysis}\n\\label{sec:weight_analysis}\nWe highlight that the deterministic variant of AC-Off-POC does not employ temporally correlated trajectories to compute the importance weights as in the state-of-the-art importance sampling methods \\citep{degris, munos_safe, harutyunyan, dayan_watkins_q_learning, impala}, and instead, compute the weights through only randomly sampled off-policy transitions. We perform experiments to show the accuracy of the weights produced by deterministic AC-Off-POC through the experiences of an actor trained under the TD3 algorithm  \\citep{td3} on the OpenAI Gym tasks HalfCheetah and Hopper \\citep{gym}, over 1 million time steps. Throughout the training, we store the transitions executed by the agent in the temporal order. Then, we start by sampling the transitions with a window size of 256 centered at each transition. We compute the importance weight for each sampled batch with the deterministic variant of AC-Off-POC. Figure \\ref{fig:deterministic_weight_analysis} depicts the normalized values for time steps and importance weights produced by AC-Off-POC. Note that these weights are derived according to the expert agent's policy. From the first transition executed by a random policy to the last transition executed by the expert agent, the sampled batches become on-policy with respect to the trained agent's policy. We observe that as the center of the sliding window approaches the last transition, the AC-Off-POC weights approach the value 1. Hence, the weights increases as the on-policyness of the sampled batch increases, which we observe from the linear path taken by the AC-Off-POC weights. Overall, we infer that the deterministic variant of the AC-Off-POC produces accurate estimates, although it does not employ temporally correlated trajectories or any action probability estimate. Furthermore, some of the AC-Off-POC weights are above the time step line; that is, off-policy transitions are valued more than they should be, which shows that AC-Off-POC can mix on- and off-policy data.  \n\n\\begin{figure*}[!b]\n    \\centering\n    \\begin{equation*}\n        \\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_orange, solid, line width=2.3pt](0,0) -- (5mm,0);}}} Time step of the sliding window center} \\qquad \\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_purple, solid, line width=2.3pt](0,0) -- (5mm,0);}}} Deterministic AC-Off-POC Weights}\n    \\end{equation*}\n    \t\\subfigure{\n\t\t\\includegraphics[width=2.3in, keepaspectratio]{Figures\/TD3_HalfCheetah-v2_weight_analysis.jpg}\n\t\t\\includegraphics[width=2.3in, keepaspectratio]{Figures\/TD3_Hopper-v2_weight_analysis.jpg}\n\t}\n\t\\caption{Temporally ordered batches of transitions versus importance weights produced by deterministic AC-Off-POC under the TD3 algorithm for the set of OpenAI Gym continuous control tasks over 1 million training time steps. The shaded region represents a 95\\% confidence interval of the normalized values over the trials.}\n\t\\label{fig:deterministic_weight_analysis}\n\\end{figure*}\n\n\\begin{figure*}[!hbt]\n    \\centering    \n    \\begin{equation*}\n        \\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_orange, solid, line width=2.3pt](0,0) -- (5mm,0);}}} Time step of the transition} \\qquad\n        \\text{{\\raisebox{2pt}{\\tikz{\\draw[custom_purple, solid, line width=2.3pt](0,0) -- (5mm,0);}}} Stochastic AC-Off-POC Weights}\n    \\end{equation*}\n\t\\subfigure{\n\t\t\\includegraphics[width=2.3in, keepaspectratio]{Figures\/SAC_HalfCheetah-v2_weight_analysis.jpg}\n\t\t\\includegraphics[width=2.3in, keepaspectratio]{Figures\/SAC_Hopper-v2_weight_analysis.jpg}\n\t}\n\t\\caption{Temporally ordered transitions versus importance weights produced by stochastic AC-Off-POC under the SAC algorithm for the set of OpenAI Gym continuous control tasks over 1 million training time steps. The shaded region represents a 95\\% confidence interval of the normalized values over the trials.}\n\t\\label{fig:stochastic_weight_analysis}\n\\end{figure*}\nWe conduct the same set of experiments to verify the accuracy of the stochastic variant of AC-Off-POC. As the stochastic variant assigns unique weights to each off-policy sample, we directly depict the weights assigned to each transition contained in the expert agent's experience replay buffer \\citep{experience_replay}. In Figure \\ref{fig:stochastic_weight_analysis}, we observe that the stochastic variant also produces accurate importance weights to schedule the contribution of each gradient with respect to off-policy samples. Moreover, the confidence of the weights is stronger, and weights are more accurate due to the direct use of the policy distributions in computing the dissimilarity between the target and behavioral policies. In conclusion, as discussed in Remark \\ref{rem:problems_in_current_is_methods}, AC-Off-POC can produce accurate importance weights by preventing vanishing or exploding gradients and the employment of trajectories due to the transition-wise or temporally uncorrelated batch-wise similarities. \n\n\\section{Conclusion}\nIn this paper, we discuss the theme of off-policy correction, which mitigates the potential effects of off-policy samples whose underlying distribution is divergent from the current policy of interest. As the action probabilities cannot be known and estimated, off-policy correction through eligibility traces for deterministic policies is not available in continuous control. To make the actor-critic methods closer to their on-policy nature, we introduce a novel off-policy correction approach with two variants for stochastic and deterministic policies, AC-Off-POC, which achieves an efficient one-step off-policy correction, enables a safe off-policy learning and improves the data-efficiency by reweighting the contribution of off-policy samples to the gradients of the value functions. We support our claims with solid theoretical analysis that the introduced off-policy correction schemes obtain a bounded contraction mapping which enables a safe one-step importance sampling.\n\nAn extensive set of empirical studies shows that AC-Off-POC improves the baseline state-of-the-art and outperforms the competing off-policy correction methods by a considerable margin. By overcoming the perturbation induced by the off-policy samples contained in the experience replay buffer \\citep{experience_replay}, our method can attain faster convergence and optimal policies by disregarding the transitions executed by behavioral policies that highly deviate from the current policy in terms of the numerical action decisions. Our method also resolves the issues regarding the computational cost by not introducing any hyper-parameters or networks. Furthermore, we support our remarks with experimental results throughout the paper and show that AC-Off-POC can readily be applied to any off-policy actor-critic method through a generic approach. \n\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nInterval valued fuzzy logic \\cite{baczynski2013advances,LI20114754,CORNELIS2006622,BSFZEEE2007,BSLuka2013,SANTIAGO20191} and interval valued fuzzy sets  \\cite{bustince2008survey,bustince2010interval,dubois2005interval,DASILVA2016169} are  widely developed fields  of investigation. Intervals provide a way to express imprecision. In this case they are used to express the imprecision of a fuzzy degree. For example, the interval $[0.3,0.4]$ expresses an imprecision with respect to a fuzzy degree which belongs to this range of values. This enables the application of such interval uncertainty in many fields, like: decision making \\cite{nguyen2016new,mathew2019interval,bentkowska2015decision,lanbaran2020evaluation}, control \\cite{zeng2010interval,hassanzadeh2014interval,kavikumar2019non,bounouara2020stable}, image processing \\cite{barrenechea2011construction,bigand2010fuzzy} and others.\n\n\\medskip \n\nIn the usual  fuzzy setting, many important applications and theories are based on the usual order on $[0,\\!1]$ as well as the standard notion of monotonicity of functions. Some of them, like the theory of aggregations \\cite{beliakov2007aggregation,grabisch2009aggregation,Agreg2021}, mixture functions \\cite{pereira2003a,vspirkova2019mixture,farias2016,farias2016b}  and decision making \\cite{blanco2017fuzzy,li1995fuzzy,albahri2021novel} rest on intrinsic properties of this usual order. For example, the linearity is an essential requirement for decision making, whereas monotonicity is necessary for aggregation functions. In order to include some functions which clearly aggregate data but are not monotonic, like \\textit{mode}, Wilkin and Beliakov  \\cite{Wilkin2015} and Bustince et al. in \\cite{BustinceDirMon2015} both in 2015, generalized the notion of monotonicity to weak and directional monotonicity respectively. Then the notion of aggregation was relaxed to pre-aggregation  \\cite{bustince2016pre,dimuro2016new}  to include those functions.  All of this contributions were developed to the usual fuzzy setting in $[0,\\!1]$.\n\n\\medskip \n\nThe interval fuzzy setting has been also deeply developed.  Interval fuzzy connectives  like: $ t $-norms, $ t $-conorms, overlaps, implications, etc. \\cite{reiser2010analyzing,bedregal2009approach,bedregal2013interval,bedregal2006best,deschrijver2008representation,dimuro2011interval,bedregal2017generalized,paiva2021lattice} as well as applications like decision making have been developed over time. Since most of these notions are connected with monotonicity, and some  with the order-linearity, the same attitude must be taken here. In terms of linearity Bustince et al. \\cite{santana2020admissible,bentkowska2015decision} introduced the notion of admissible orders for intervals, which are refinements of the Kulisch-Miranker (non-linear) partial order (see below). This was a very important contribution in applications like decision making. On the other hand, the development of interval pre-aggregation functions theory requires the introduction of interval versions of weak and directional monotonicities. This work has been developed by  Sesma-Sara et al. \\cite{MikelIVPreAgg2019}, but in terms of Kulisch-Miranker order. They do not provide a development of  such relaxed monotonicities for admissible orders.\n\n\\medskip \n\nBesides, in 2021 and 2022  Santiago et al.  proposed the notion of $g$-weak increasing function \\cite{SantiAbsHomo2021}  and $\\mathcal{F}$-increasing function in \\cite{FHomogSant2022}. These notions expand weak-monotonicity on $[0,\\!1]$ by involving more general functions.\n\n\\medskip \n\nWhereas Sesma-Sara et al. \\cite{MikelIVPreAgg2019} develop the notions of weak\/directional monotonicity and pre-aggregation functions in terms of the \\emph{Kulisch-Miranker order} and Santiago et al. \\cite{SantiAbsHomo2021} develop $ G $-monotonicity for $ [0,\\!1] $, we develop these subjects for intervals endowed with admissible orders. In this way the set of intervals becomes a chain and the theories\/applications of these types of monotonicities (like the theory of pre-aggregations and its application in decision making and other fields) can be developed in this setting. We also show: (i) that notions like \\emph{minimum} and \\emph{maximum} between intervals become non-increasing when they are ``carried'' to the admissible orders setting; (ii) how some implications behave in this context; (iii) some relationships between abstract homogeneity and the monotonicities introduced. \n\n\\medskip \n\nThe remainder of this paper is structured as follows. In section \\ref{sec:Prelim} we recall some basic concepts. In section \\ref{sec:Monot} we provide notions of monotonicity for IV-functions. In section \\ref{sec:mon&prop} we  supply some properties related to the concepts of monotonicity. In section \\ref{sec:preaggreg}, we  define a pre-aggregation IV-function considering on $\\mathbb{I}([0,\\!1])$ admissible orders and provide some examples and results. In section \\ref{sec:mon&absthomo}, we investigate the relationships between abstract homogeneity and the monotonicities introduced. Finally, in section \\ref{sec:Final} we provided our final considerations. \n\n\n\\section{\\label{sec:Prelim} Preliminaries}\n\nThroughout this paper the following notation will be used: $\\mathbb{I}(\\mathbb{R})$  denotes the set of all closed intervals with real numbers, uppercase letters denote its elements, $\\mathbb{I}(\\mathbb{R}_{-})$ denotes the subset  consisting of all closed intervals with negative real numbers, $\\mathbb{I}(\\mathbb{R}_{+})$ denotes the subset of closed intervals with positive real numbers. The set $\\mathbb{I}([0,\\!1])$  contains all closed subintervals of $[0,\\!1]$. Further, we use uppercase letters $X = [\\underline{X},\\overline{X}]$ to denote any interval variable $X$ and use bold fonts to denote degenerated intervals \\footnote{An interval $X$ is said to be a \\emph{degenerated} iff $\\underline{X} = \\overline{X}$.}: e.g. $\\boldsymbol{0.8} = [0.8,0.8]$; $\\vec{0}$ and $\\vec{1}$ to denote the vectors $(0, \\ldots, 0)$ and $(1, \\ldots, 1)$, respectively; $\\vec{\\boldsymbol{0}}$ and $\\vec{\\boldsymbol{1}}$, in bold, to denote the vectors $(\\boldsymbol{0}, \\ldots, \\boldsymbol{0})$ and $(\\boldsymbol{1}, \\ldots, \\boldsymbol{1})$, respectively, where $\\boldsymbol{0} = [0,0]$ and $\\boldsymbol{1} = [1,1]$. The prefix ``\\emph{IV}'' will mean ``\\emph{interval-valued}'' in general.\n\n\\medskip\n\n\\begin{definition}[Operations over intervals]\n\t\\label{def_intoperations}\n\tLet $X,Y\\in \\mathbb{I}(\\mathbb{R})$ and $\\alpha\\in\\mathbb{R}$.\n\t\\begin{enumerate}[leftmargin=*,label={\\rm($\\mathbf{OI}$-\\arabic*)}]\n\t\t\n\t\t\\item The \\emph{sum of intervals} $X$ and $Y$, denoted by $X + Y$, is defined by $X + Y = [\\underline{X} + \\underline{Y}, \\overline{X} + \\overline{Y}]$.\n\t\t\n\t\t\\item The \\emph{opposite of an interval} $X$, denoted by\\  $-X$, is defined by\\  $-X = [-\\overline{X}, - \\underline{X}]$.\n\t\t\n\t\t\\item The \\emph{difference of intervals} $X$ and $Y$, denoted by $X-Y$, is  $X+(-Y)$, i.e. $X - Y = [\\underline{X} - \\overline{Y},  \\overline{X} - \\underline{Y}]$. \\footnote{Note that $X-X = \\boldsymbol{0}$ iff $X$ is a degenated interval.}\n\t\t\n\t\t\\item The \\emph{product of intervals} $X,Y\\in\\mathbb{I}(\\mathbb{R}_{+})$, denoted by $X \\cdot Y$, is defined by $X \\cdot Y = [\\underline{X} \\cdot \\underline{Y}, \\overline{X} \\cdot \\overline{Y}]$.\n\t\t\n\t\t\\item The \\emph{$\\alpha$-multiple of an interval} $X$, denoted by $\\alpha\\cdot X$, is defined by $\\alpha\\cdot X=[\\alpha \\cdot \\underline{X}, \\alpha \\cdot \\overline{X}]$ if $\\alpha \\geq 0$ and  $\\alpha\\cdot X=[\\alpha \\cdot \\overline{X}, \\alpha \\cdot \\underline{X}]$ if $\\alpha <0$.\n\t\t\n\t\t\\item For $K\\in\\mathbb{I}(\\mathbb{R}_{+})$, the \\emph{power $K$ of} $X$, denoted by $X^{K}$, is defined by:\n\t\t\n\t\t$$X^{K} = \n\t\t\\begin{cases}\n\t\t\t[\\underline{X}^{\\overline{K}}, \\overline{X}^{\\underline{K}}] & if\\  \\overline{X} \\leq 1\\\\\n\t\t\t[\\underline{X}^{\\overline{K}}, \\overline{X}^{\\overline{K}}] & if\\  \\underline{X} < 1 \\leq\\overline{X}\\\\\n\t\t\t[\\underline{X}^{\\underline{K}}, \\overline{X}^{\\underline{K}}] & if\\  \\underline{X} > 1 \\\\\n\t\t\\end{cases}. $$\n\t\\end{enumerate}\n\\end{definition}\n\n\\subsection{Admissible Orders}\n\n\\vspace{0,1cm}\n\nGiven a non-empty set $L$, \\textit{a partial order} $\\preceq$ on this set is a binary relation on $L$ which is:\n\n\\begin{enumerate}[leftmargin=*,label={\\rm($\\mathbf{PO}$-\\arabic*)}]\n\t\\item Reflexive: $x \\preceq x, \\forall x \\in L$;\n\t\\item Anti-symmetric: if $x \\preceq y$ and $y \\preceq x$, then $x=y$;\n\t\\item Transitive: if $x \\preceq y$ and $y \\preceq z$, then $x \\preceq z$.\n\\end{enumerate}\n\nWe use $x \\preceq y$ and $y \\succeq x$ interchangeably. Whenever $x \\preceq y$ and $x \\not= y$, we write $x \\prec y$. If $x \\not\\preceq y$ and $y \\not\\preceq x$ then $x$ and $y$ are called \\textit{incomparable} and we write $x \\| y$. A set $L$ with a partial order $\\preceq$ is called a \\textit{partially ordered set} (\\textit{poset}) and is denoted by $(L,\\preceq)$. If in a poset $(L,\\preceq)$ any two elements of $L$ are comparable then $\\preceq$ is called a \\textit{linear order} and $L$ is said to be a \\textit{chain} \\cite{Davey_2002, Bustince2013}. \n\n\\medskip\n\nNext, we provide some important examples of partial orders. \n\n\\begin{example}\n\t\\ \n\t\\begin{enumerate}[leftmargin=*,label=(\\emph{\\alph*})]\n\t\t%\n\t\t\\item The usual order on $[0,\\!1]$ is a linear order;\n\t\t%\n\t\t\\item Given the partial orders $\\langle A_{i}, \\leq_{i} \\rangle$ for $i = 1, \\ldots, n$ and the cartesian product $\\displaystyle\\prod_{i=1}^{n} A_i$, the relation $(x_1, \\ldots, x_n) \\leq (y_1, \\ldots,y_n)$ iff $ x_i \\leq_{i} y_i$ for all $ i = 1, \\ldots ,n $, is a partial order on $\\displaystyle\\prod_{i=1}^{n} A_i$ called product order \\cite{Neggers1998, Ghorpade2010} or coordinatewise order \\cite{Davey_2002, Harzheim2006}, or even componentwise order \\cite{Guerraggio1994, DANG2018}. \n\t\t%\n\t\t\\item  The relation $X \\ {\\preceq}_{KM} \\ Y$ iff $ \\left( \\underline{X} \\leq \\underline{Y} \\ \\ \\text{and} \\ \\  \\overline{X} \\leq \\overline{Y} \\right) $ is a partial order on $ \\mathbb{I}(\\mathbb{R}) $ called Kulisch-Miranker order \\cite{Kulisch1981, Takahashi2012, Flores2013, Younus2015}.\n\t\\end{enumerate}\n\\end{example}\n\n\\begin{definition} \n\t[\\cite{Bustince2013}]\n\t\\label{def_admorder}\n\tA partial order $\\preceq$ on $\\mathbb{I}([0,\\!1])$ is called an admissible order if\n\t\n\t\\begin{enumerate}[leftmargin=*,label={\\rm($\\mathbf{AO}$-\\arabic*)}]\n\t\t\n\t\t\\item $\\preceq$ is a linear order on $\\mathbb{I}([0,\\!1])$.\n\t\t\n\t\t\\item $\\preceq$ refines $\\leq_{KM}$, i.e., given $X, Y \\in \\mathbb{I}([0,\\!1])$, if $X {\\preceq}_{KM} Y$ then $X \\preceq Y$.    \n\t\\end{enumerate}  \n\t\n\tIn the same sense we will use the notion of an admissible order on $\\mathbb{I}(\\mathbb{R}_{+})$ and $\\mathbb{I}(\\mathbb{R})$.\n\\end{definition}\n\n\\begin{example} \n\t\\label{example_lexorders} \n\tMotivated by the lexicographical order of points in $\\mathbb{R}^2$, Bustince et al. \\cite{Bustince2013} defined the following admissible orders on $\\mathbb{I}([0,\\!1])$:\n\t\n\t\\begin{enumerate}[leftmargin=*,label=(\\alph*)]\n\t\t\n\t\t\\item Lexicographical-1 order: $X \\preceq_{L1} Y$ iff ($\\underline{X} < \\underline{Y}$) or ($\\underline{X} = \\underline{Y}$ and $\\overline{X} \\leq \\overline{Y})$;\n\t\t%\n\t\t\n\t\t\\item Lexicographical-2 order: $X \\preceq_{L2} Y$ iff ($\\overline{X} < \\overline{Y}$) or ($\\overline{X} = \\overline{Y}$ and $\\underline{X} \\leq \\underline{Y})$. \n\t\\end{enumerate}\n\\end{example}\n\n\\begin{example} \n\t\\label{example_xyorder} \n\tIn \\cite{XuYager2006} Xu and Yager introduced the XuYa order on $\\mathbb{I}([0,\\!1])$, defined by:\n\t\\begin{center}\n\t\t$X \\sqsubseteq Y$ iff ($\\underline{X} + \\overline{X} < \\underline{Y} + \\overline{Y}$) or ($\\underline{X} + \\overline{X} = \\underline{Y} + \\overline{Y}$ and $\\overline{X} - \\underline{X} \\leq \\overline{Y} - \\underline{Y}$), \n\t\\end{center}\n\twhich is an admissible order. \n\\end{example}\n\nIn what follows we fix the following notation: the symbol ``$\\preceq$'' will always mean an admissible order and any set of real intervals $\\mathbb{I} \\subseteq \\mathbb{I}(\\mathbb{R})$ (e.g $\\mathbb{I}([0,\\!1])$) is assumed to be equipped with $\\preceq$.\n\n\\medskip\n\nThe pair $(\\mathbb{I}([0,\\!1]),\\preceq)$ is a lattice with $\\boldsymbol{0}=[0,0]$ as the smallest element and $\\boldsymbol{1}=[1,1]$ as the greatest element \\cite{Bustince2013}.\n\n\\medskip\n\nThis paper requires the notion of stardard monotonicity of an IV-function $F : \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$:\n\n\\begin{definition} \\label{def_intmonotonicity}\n\tAn IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is said to be \\emph{increasing}, if it is increasing with respect to the product order on $\\mathbb{I}([0,\\!1])^n$, i.e.  \n\t\\begin{center}\n\t\t$Y_i  \\succeq \tX_i, \\ for \\ i = 1, \\ldots, n \\Rightarrow F(Y_1, \\ldots, Y_n) \\succeq F(X_1, \\ldots, X_n)$.\n\t\\end{center}\t\n\tDually, we define a  \\emph{decreasing} IV-function.\n\\end{definition}\n\n\\begin{example} \n\t\\label{example_intmonotonicity}\n\t\\\t\n\t\\begin{enumerate}[leftmargin=*,label=(\\alph*)]\n\t\t\n\t\t\\item For any positive integer $n$, the IV-function minimum, $\\min: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$, defined by $\\min(X_1, \\ldots, X_n) = X_i$ if $X_i \\preceq X_j$ for all $j = 1, \\ldots, n$, is increasing. Dually we define the IV-function \\emph{maximum} which is also increasing.\n\t\t\n\t\t\\item The IV-function $N : \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$, given by $N(X) = \\boldsymbol{1} - X$, is   decreasing. \n\t\\end{enumerate}\n\\end{example}\n\n\\medskip \n\nIn the next example, we show that notions of \\emph{minimum} and \\emph{maximum}  between intervals become non-increasing when they are ``carried'' to the context of admissible orders.\n\n\\medskip \n\n\\begin{example}\n\t\\label{example_FminFmax}\n\tThe IV-functions $F_{min},F_{max}: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$, given by \\begin{center}\n\t\t$F_{min}(X,Y) = \\left[\\min(\\underline{X}, \\underline{Y}), \\min(\\overline{X}, \\overline{Y}) \\right]$\n\t\t\\ and \\ $F_{max}(X,Y) = \\left[\\max(\\underline{X}, \\underline{Y}), \\max(\\overline{X}, \\overline{Y}) \\right]$,\n\t\\end{center} \n\tare not increasing. Indeed, consider $Y,Z \\in \\mathbb{I}([0,\\!1])$ non-degenerated intervals such that $ \\underline{Z} > \\underline{Y} $ and $ \\overline{Z} < \\overline{Y} $. There are two possibilities: $Y \\prec Z$ or $Z \\prec Y$. If $Y \\prec Z$, take $X \\in \\mathbb{I}([0,\\!1])$ non-degenerated such that $Y \\subset X$. Since $\\underline{X} \\leq \\underline{Y} \\leq \\underline{Z} < \\overline{Z} < \\overline{Y} \\leq \\overline{X}$, it follows that   \n\t\\begin{center}\n\t\t$F_{min}(X, Z) = [\\underline{X}, \\overline{Z}] \\ {\\prec}_{KM} \\  [\\underline{X}, \\overline{Y}] = F_{min}(X, Y)$.\n\t\\end{center} \n\tSuppose now $Z \\prec Y$. Taking $X \\in \\mathbb{I}([0,\\!1])$ non-degenerated such that $X \\subset Z$, we have $\\underline{Y} < \\underline{Z} \\leq \\underline{X} < \\overline{X} \\leq \\overline{Z} < \\overline{Y}$. Thus,   \n\t\\begin{center}\n\t\t$F_{min}(X, Y) = [\\underline{Y}, \\overline{X}] \\ {\\prec}_{KM} \\  [\\underline{Z}, \\overline{X}] = F_{min}(X, Z)$.\n\t\\end{center} \n\tThe fact that $F_{\\max}$ is not increasing either can be proved analogously.\n\\end{example}\n\n\\medskip \n\nA function $F : \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ can be increasing\/decreasing with respect just one of its arguments.\n\n\\begin{definition} \n\t\\label{def_intmonoteacharg}\n\tGiven a positive integer $n$ and fixed $i \\in \\{1, \\ldots, n\\}$, an IV-function $F : \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is said to be \\emph{increasing in the $i$-th component} if $ F(X_1, \\ldots, X_n) \\preceq F(Y_1, \\ldots, Y_n) $ whenever $ X_i  \\preceq Y_i$ and $X_j = Y_j \\ for \\ j \\in \\{1, \\ldots, n\\} \\setminus \\{i\\} $. Dually, we define an IV-function \\emph{decreasing in the $i$-th component}.\n\\end{definition}\n\n\\begin{example} \n\t\\label{example_intmonoteacharg}\n\tGiven a positive integer $n$ and fixed $i \\in \\{1, \\ldots, n\\}$,\n\t\n\t\\begin{enumerate}[leftmargin=*,label=(\\alph*)]\n\t\t\\item the IV-function $F : \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$, given by $F(X_1, \\ldots, X_i, \\ldots,  X_n) = \\min(2X_i, \\boldsymbol{1})$, is increasing in the $i$-th component.\n\t\t\n\t\t\\item the IV-function $G : \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$, given by $G(X_1, \\ldots, X_i, \\ldots,  X_n) = \\boldsymbol{1} - X_i$, is decreasing in the $i$-th component.\n\t\\end{enumerate}\n\\end{example}\n\n\\begin{definition} [Order preserving IV-function]\n\t\\label{def:ordpreservfunct}\n\tLet $\\mathbb{I},\\mathbb{J}\\subseteq\\mathbb{I}(\\mathbb{R})$. Let $\\preceq_{\\mathbb{I}}$ and $\\preceq_{\\mathbb{J}}$ admissible orders on $\\mathbb{I}$ and $\\mathbb{J}$, respectively. An IV-function $\\varphi : \\mathbb{I}\\to\\mathbb{J}$ is said to be \\emph{order preserving} iff\n\t\\begin{equation*}\n\t\tX\\,\\preceq_{\\mathbb{I}}\\, Y \\quad \\Rightarrow\\quad \\varPhi(X)\\,\\preceq_{\\mathbb{J}}\\,\\varPhi(Y),\\quad X,Y \\in\\mathbb{I}.\n\t\\end{equation*}\n\\end{definition}\n\n\\medskip \n\nThroughout this paper, the order preserving function $\\varPhi: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$ which preserves the same order $\\preceq$ imposed on $\\mathbb{I}([0,\\!1])$ will be exclusively used.\n\n\n\n\\section{\\label{sec:Monot} Monotonicities of IV-Functions}\n\nSome part of this paper is focused on aggregation function applications. An aggregation function can be -- in a simplified way -- characterized as a multi-variable function that preserves bottom and top elements, and any its input increase does not cause any output decrease. Thus, its definition requires some monotonicity notion.\n\n\\medskip\n\nIn what follows, it will be applied monotonicity with respect to an admissible order in the frame of IV-functions as well as its weakened\/relaxed versions. \n\n\\medskip\n\nUnder certain circumstances there is not necessary to demand monotonicity of an IV-function over its whole domain. In these cases it suffices that the function is increasing\/decreasing on some suitable subdomain only. Referring to probability theory terminology, the corresponding notion can be called \\emph{conditional monotonicity}. In the following definition, IV-functions on $\\mathbb{I}([0,\\!1])^n$ which are \\emph{conditionally increasing} and \\emph{conditionally decreasing}, respectively, \\emph{given by a transformation} will be introduced. This indicates that the subset of a domain on which the function is monotonic is selected due to some suitable IV-function.\n\n\\begin{definition}[Conditional monotonicity given a transformation]\n\t\\label{def_condmonotonicity}\n\tLet $\\Theta: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$ be an IV-function. An IV-function $F\\!:\\,\\mathbb{I}^n\\to\\mathbb{I}$ is said to be \\emph{conditionally increasing given the transformation $\\Theta$} iff\n\t\\begin{equation*}\n\t\t\\Theta(X_{i}) \\succeq X_{i} \\ \\text{for all} \\  i=1,\\ldots,n \\ \\Rightarrow\\,\n\t\tF\\big(\\Theta(X_1),\\ldots,\\Theta(X_n)\\big) \\succeq F(X_1, \\ldots,X_n), \n\t\\end{equation*}\n\tand $F\\!:\\,\\mathbb{I}^n\\to\\mathbb{I}$ is said to be \\emph{conditionally decreasing given the transformation $\\Theta$} iff\n\t\\begin{equation*}\n\t\t\\Theta(X_{i}) \\succeq X_{i} \\ \\text{for all} \\  i=1,\\ldots,n \\ \\Rightarrow\\,\n\t\tF\\big(\\Theta(X_1),\\ldots,\\Theta(X_n)\\big) \\preceq F(X_1, \\ldots,X_n).\n\t\\end{equation*}\n\\end{definition}\n\n\n\\medskip\n\nReferring to \\emph{directional monotonicity}, introduced in~\\cite{Sesma2020}, in the context of admissible order and conditional monotonicity we have the following: \n\n\\begin{definition}[Conditional monotonicity given a vector of degenerated intervals]\n\t\\label{def:condmonotgvdi}\n\tGiven $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n}) \\in \\mathbb{I}(\\mathbb{R})^n \\setminus \\{\\vec{\\boldsymbol{0}}\\}$ a vector of degenerated intervals, an IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is said to be \\emph{conditionally increasing(resp. decreasing) given a vector of degenerated intervals} iff it is conditionally increasing (resp. decreasing) given the transformation $\\Theta: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$ defined as\n\t\\begin{equation*}\n\t\t\\Theta(X_{i})=X_i + k \\cdot \\boldsymbol{V_i}\n\t\\end{equation*}\n\tfor all $0 < k \\in \\mathbb{R}$ and all $X_i \\in\\mathbb{I}([0,\\!1])$, $i=1,\\ldots,n$, such that $X_i + k \\cdot \\boldsymbol{V_i} \\in\\mathbb{I}([0,\\!1])$. In the case that $F$ is simultaneously conditionally increasing and conditionally decreasing given a vector of degenerated intervals, $F$ is said to be a \\emph{conditionally constant given a vector of degenerated intervals}  IV-function.\n\\end{definition}\n\n\\medskip \n\nImplications $I: [0,\\!1]^2 \\to [0,\\!1]$ (cf. \\cite{Bustince2020} and \\cite{Lucca2015}) are examples of IV-functions conditionally increasing given a vector of degenerated intervals. In what follows we recall the definition of IV implications with respect to admissible orders provided by Zapata et al. in \\cite{Zapataetal2017}.\n\n\\medskip \n\n\\begin{definition} [IV-Implication]\n\t\\label{def_IVFuzzyImplication}\n\tAn IV-implication is an IV-function $I: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ satisfying the following properties:  \n\t\n\t\\begin{enumerate}[leftmargin=*,label={\\rm($\\mathbf{I}$-\\arabic*)}]\n\t\t\n\t\t\\item $ I $ is decreasing in the first component and increasing in the second component;\n\t\t\n\t\t\\item $I(\\boldsymbol{0}, \\boldsymbol{0}) = I(\\boldsymbol{1}, \\boldsymbol{1}) =  \\boldsymbol{1}$;\n\t\t\n\t\t\\item $I(\\boldsymbol{1}, \\boldsymbol{0}) =  \\boldsymbol{0}$. \n\t\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\n\t\\label{prop_directmonotImp}\n\tEvery IV-implication $I: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ is a conditionally increasing given the vector $(\\boldsymbol{-1}, \\boldsymbol{1})$ IV-function.\n\\end{proposition}\n\n\\begin{proof}\n\tIndeed, if $I: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ is an IV-implication, then for every  $0 < k \\in \\mathbb{R}$ and $(X, Y) \\in \\mathbb{I}([0,\\!1])^2$ such that $(X + k \\cdot (\\boldsymbol{-1}), Y + k \\cdot \\boldsymbol{1}) \\in \\mathbb{I}([0,\\!1])^2$ we have: \n\t\\begin{center}\n\t\t$I(X + k \\cdot (\\boldsymbol{-1}), Y + k \\cdot \\boldsymbol{1}) = I(X - [k,k], Y + [k,k]) \\succeq I(X, Y + [k,k]) \\succeq I(X, Y)$,\n\t\\end{center}\n\tsince $ I $ is an IV-function  decreasing  in the first component and increasing in the second component.\n\\end{proof}\n\nBesides being conditionally increasing given the vector $(\\boldsymbol{-1}, \\boldsymbol{1})$ IV-functions, some IV-implications are conditionally increasing\/decreasing\/constant given a vector of degenerated intervals with respect to specific vectors. Below we provide some examples.\n\n\\begin{example} \n\t\\label{example_LukaImpweakfunction}\n\tThe IV-\\L{ukasiewicz}~implication, $I_{\\text{\\L}}: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$, given by $I_{\\text{\\L} }(X,Y) = \\min(\\boldsymbol{1}, \\boldsymbol{1}-X+Y)$, is conditionally constant given the vector  $\\vec{\\boldsymbol{1}}$.\n\t\\begin{flushleft}\n\t\tReally, for all  $0 < k \\in \\mathbb{R}$ and $(X, Y) \\in \\mathbb{I}([0,\\!1])^2$ such that $(X + k \\cdot  \\boldsymbol{1}, Y + k \\cdot  \\boldsymbol{1}) \\in \\mathbb{I}([0,\\!1])^2$, we have:\n\t\t\\\\ \\ \\\\\n\t\t$I_{\\text{\\L}}(X + k \\cdot \\boldsymbol{1}, Y + k \\cdot \\boldsymbol{1}) = \\min(\\boldsymbol{1}, \\boldsymbol{1}-(X + k \\cdot  \\boldsymbol{1})+(Y + k \\cdot  \\boldsymbol{1}))= \\min(\\boldsymbol{1}, \\boldsymbol{1}-X+Y)= I_{\\text{\\L}}(X,Y)$.\n\t\\end{flushleft}\n\\end{example}\n\n\\medskip\n\nThe following is an example of an IV-function that is conditionally decreasing given the vector $(\\boldsymbol{1},\\!\\boldsymbol{0})$ and conditionally increasing given the vector $(\\boldsymbol{0},\\!\\boldsymbol{1})$ but is not conditionally monotonic given the vector $(\\boldsymbol{1},\\!\\boldsymbol{1})$.\n\n\\medskip \n\n\\begin{example} \n\t\\label{example_FodorImplic}\t\n\tLet $I_{FD}: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$, the IV-Fodor implication, given by $I_{FD }(X,Y) = \\left\\{\n\t\\begin{array}{ccccccc}\n\t\t\\ \\ \\ \\ \\ \\ \\ \\ \\boldsymbol{1}, \\ \\ \\ \\ \\ \\ \\ \\ \\ if \\ X \\preceq Y\\\\\n\t\t\\max(\\boldsymbol{1}-X, Y), \\ if \\ X \\succ Y\\\\\n\t\\end{array}\n\t\\right.$. Given $0 < k \\in \\mathbb{R}$ and $(X, Y) \\in \\mathbb{I}([0,\\!1])^2$ such that $(X + k \\cdot \\boldsymbol{1}, Y + k \\cdot \\boldsymbol{1}) \\in \\mathbb{I}([0,\\!1])^2$, there are two cases: (i) $X \\succ Y$ and (ii) $X \\preceq Y$. In the first case, we have $X + k \\cdot \\boldsymbol{1} \\succ X \\succ Y$. Thus, \n\t\\begin{center}\n\t\t$I_{FD}(X + k \\cdot \\boldsymbol{1}, Y) = \\max\\left( \\boldsymbol{1} - (X + k \\cdot \\boldsymbol{1}), Y \\right) \\prec \\max (\\boldsymbol{1} - X, Y) = I_{FD}(X,Y)$.\n\t\\end{center}\n\tIn the second case, there are two subcases: (ii.a) $X \\prec X + k \\cdot \\boldsymbol{1} \\prec Y$ and (ii.b) $X \\preceq Y \\prec X + k \\cdot \\boldsymbol{1}$. If $X \\prec X + k \\cdot \\boldsymbol{1} \\prec Y$, then $I_{FD}(X + k \\cdot \\boldsymbol{1}, Y) = \\boldsymbol{1} = I_{FD}(X,Y)$; and if $X \\preceq Y \\prec X + k \\cdot \\boldsymbol{1}$, so \n\t\\begin{center}\n\t\t$I_{FD}(X + k \\cdot \\boldsymbol{1}, Y) = \\max\\left(\\boldsymbol{1} - (X + k \\cdot \\boldsymbol{1}), Y \\right) \\prec \\max(\\boldsymbol{1} - X, Y) = I_{FD}(X, Y)$.\n\t\\end{center} \n\tIn any case, $I_{FD}(X + k \\cdot \\boldsymbol{1}, Y) \\preceq I_{FD}(X, Y)$, for every $0 < k \\in \\mathbb{R}$ and $(X, Y) \\in \\mathbb{I}([0,\\!1])^2$ such that $(X + k \\cdot \\boldsymbol{1}, Y + k \\cdot \\boldsymbol{1}) \\in \\mathbb{I}([0,\\!1])^2$. Therefore, $ I_{FD} $ is conditionally decreasing given the vector $(\\boldsymbol{1},\\!\\boldsymbol{0})$. \n\t\n\t\\medskip \n\tFurther, $I_{FD}$ is conditionally increasing given the vector $(\\boldsymbol{0}, \\boldsymbol{1})$. Indeed, again let us consider two cases: (i) $X \\succ Y$ and (ii) $X \\preceq Y$. In the first case, we have two subcases: (i.1) $X \\succ Y + k \\cdot \\boldsymbol{1} \\succ Y$ and (i.2) $Y + k \\cdot \\boldsymbol{1} \\succ X \\succ  Y$. If $X \\succ Y + k \\cdot \\boldsymbol{1} \\succ Y$, then \n\t\\begin{center}\n\t\t$I_{FD}(X,Y+ k \\cdot \\boldsymbol{1}) = \\max(\\boldsymbol{1} - X, Y + k \\cdot \\boldsymbol{1}) \\succ \\max(\\boldsymbol{1} - X, Y) = I_{FD}(X, Y)$;\n\t\\end{center} \n\tand if $Y + k \\cdot \\boldsymbol{1} \\succ X \\succ Y$, so \n\t\\begin{center}\n\t\t$I_{FD}(X,Y + k \\cdot \\boldsymbol{1}) = \\boldsymbol{1} \\succeq \\max(\\boldsymbol{1} - X, Y) = I_{FD}(X, Y)$.\n\t\\end{center}\n\tIn the second case, we have $X \\preceq Y \\prec Y + k \\cdot \\boldsymbol{1}$. Then, $I_{FD}(X, Y + k \\cdot \\boldsymbol{1}) = \\boldsymbol{1} = I_{FD}(X, Y)$. In any case, $I_{FD}(X, Y + k \\cdot \\boldsymbol{1}) \\succeq I_{FD}(X,Y)$.\n\t\n\t\\medskip \n\t\n\tHowever, taking $X = [0.2, 0.3]$, $Y = [0.1, 0.2]$ and $k = 0.3$, we have:\n\t\n\t\\begin{center}\n\t\t$I_{FD}(X + k \\cdot \\boldsymbol{1}, Y + k \\cdot \\boldsymbol{1}) = I_{FD}([0.5,0.6], [0.4, 0.5]) = [0.4, 0.5] \\prec [0.7, 0.8] = I_{FD}(X,Y)$;\n\t\\end{center}  \n\tand taking $k = 0.7$, we have:\n\t\n\t\\begin{center}\n\t\t$I_{FD}(X + k \\cdot \\boldsymbol{1}, Y + k \\cdot \\boldsymbol{1}) = I_{FD}([0.9,1], [0.8, 0.9]) = [0.8, 0.9] \\succ [0.7, 0.8] = I_{FD}(X, Y)$.\n\t\\end{center}  \n\tThus, $I_{FD}$ is neither conditionally increasing given the vector $(\\boldsymbol{1}, \\boldsymbol{1})$-increasing nor conditionally decreasing given the vector $(\\boldsymbol{1}, \\boldsymbol{1})$. \n\\end{example}\n\n\\medskip \n\n\nIn what follows we provide examples of IV-functions that are conditionally increasing\/decreasing given a vector of degenerated intervals with respect to a family of vectors $\\overrightarrow{V} \\in \\mathbb{I}(\\mathbb{R})^2$.\n\n\\medskip\n\n\\begin{example} \n\t\\label{ex:multsentdirincrf}\t\n\tThe IV-function $F: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$,  given by \\begin{center}\n\t\t$F(X, Y) = \\left\\{\n\t\t\\begin{array}{ccccccc}\n\t\t\tX(\\boldsymbol{1} - X), \\ if \\ Y \\preceq \\boldsymbol{0.8}\\\\\n\t\t\t\\ \\ \\ \\ \\ \\boldsymbol{1}, \\ \\ \\ \\ \\ otherwise\\\\\n\t\t\\end{array}\n\t\t\\right.$,\n\t\\end{center} \n\tis  conditionally increasing given any vector $\\overrightarrow{V} = (\\boldsymbol{0}, \\boldsymbol{A})$, where $\\boldsymbol{0} \\prec \\boldsymbol{A} \\in \\mathbb{I}(\\mathbb{R})$ is a degenerated interval. Indeed, for every $0 < k \\in \\mathbb{R}$ and $(X, Y) \\in \\mathbb{I}([0,\\!1])^2$ such that $(X, Y + k \\cdot \\boldsymbol{A}) \\in \\mathbb{I}([0,\\!1])^2$, there are two possibilities: (i) $Y + k  \\cdot \\boldsymbol{A} \\preceq \\boldsymbol{0.8}$ and (ii) $Y + k \\cdot \\boldsymbol{A} \\succ \\boldsymbol{0.8}$. In the first case, since $Y \\preceq Y + C$, we also have $Y \\preceq \\boldsymbol{0.8}$. Thus, $F(X, Y + k \\cdot \\boldsymbol{A}) = X \\cdot (\\boldsymbol{1} - X) = F(X, Y)$; in the second one, there are two subcases to consider: $Y \\preceq \\boldsymbol{0.8}$ and $Y \\succ \\boldsymbol{0.8}$. If $Y \\preceq \\boldsymbol{0.8}$, then $F(X, Y + k \\cdot \\boldsymbol{A}) = \\boldsymbol{1} \\succeq X \\cdot (\\boldsymbol{1} - X) = F(X,Y)$. Otherwise, we have: $F(X, Y + k \\cdot \\boldsymbol{A}) = \\boldsymbol{1} = F(X,Y)$. \\end{example}\n\n\\begin{example} \n\t\\label{example_v1v2monotonousfunction}\t\n\tLet $F: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ be the IV-function given by $F(X,Y) = X(\\boldsymbol{1}-Y)$ and  $0 < k \\in \\mathbb{R}$. Consider $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\boldsymbol{0})$, where $\\boldsymbol{0} \\prec \\boldsymbol{V_1} \\in \\mathbb{I}(\\mathbb{R})$ is a degenerated interval. For every $(X,Y) \\in \\mathbb{I}([0,\\!1])^2$ such that $(X + k \\cdot \\boldsymbol{V_1} , Y) \\in \\mathbb{I}([0,\\!1])^2$, we have: \n\t\n\t\\begin{center}\n\t\t$F(X + k \\cdot \\boldsymbol{V_1},Y) = (X + k \\cdot \\boldsymbol{V_1}) \\cdot (\\boldsymbol{1} - Y) \\succ X \\cdot (\\boldsymbol{1} - Y) = F(X,Y)$.\n\t\\end{center}\n\t%\n\tAnalogously, taking $\\overrightarrow{V} = (\\boldsymbol{0}, \\boldsymbol{V_2})$, where $\\boldsymbol{0} \\prec \\boldsymbol{V_2} \\in \\mathbb{I}(\\mathbb{R})$ is a degenerated interval, for every $(X,Y) \\in \\mathbb{I}([0,\\!1])^2$ such that $(X,Y + k \\cdot \\boldsymbol{V_2} ) \\in \\mathbb{I}([0,\\!1])^2$, we have: \n\t\n\t\\begin{center}\n\t\t$F(X,Y + k \\cdot \\boldsymbol{V_2}) = X \\cdot [\\boldsymbol{1} - (Y + k \\cdot \\boldsymbol{V_2})] \\prec X \\cdot (\\boldsymbol{1} - Y) = F(X,Y)$.\n\t\\end{center}\n\tThus, $F$ is a conditionally increasing given any vector of degenerated intervals   $(\\boldsymbol{V_1}, \\boldsymbol{0})$ and a conditionally decreasing given any vector of degenerated intervals  $(\\boldsymbol{0}, \\boldsymbol{V_2})$ IV-function. \n\\end{example}\n\n\\medskip \n\n\nIn~\\cite{Sesma2020}, there was studied the monotonicity determined by a constant vector of non-degenerated intervals. The corresponding functions were called \\emph{weak increasing} and \\emph{weak decreasing}. In the context of admissible order and conditional monotonicity we have the following:\n\n\\medskip\n\n\\begin{definition}[Conditional monotonicity given a constant vector]\n\t\\label{def:condmonotgcv}\n\tAn IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is said to be \\emph{conditionally increasing(resp. decreasing) given a constant vector} iff it is conditionally increasing (resp. decreasing) given the transformation $\\Theta: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$ defined as\n\t\\begin{equation*}\n\t\t\\Theta(X_{i})=X_i + C\n\t\\end{equation*}\n\tfor all $X_i \\in\\mathbb{I}([0,\\!1])$, $i=1,\\ldots,n$ and for all $C \\in \\mathbb{I}([0,\\!1])\\!\\setminus\\!\\{\\boldsymbol{0}\\}$ such that $X_i + C \\in\\mathbb{I}([0,\\!1])$. \n\\end{definition}\n\n\\begin{remark}\n\t\\label{remark_intweakdirmonotonicity} Let $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ be a conditionally increasing IV-function given a constant vector of degenerated intervals $C=[k,k]$, $k\\in\\mathbb{R}$, $k>0$. Then, for all $C,X_i\\in\\mathbb{I}([0,\\!1])$, $i=1,\\ldots,n$, such that $X_i + C\\in\\mathbb{I}([0,\\!1])$,\n\t\\begin{align*}\n\t\tF(X_1 + k\\cdot\\boldsymbol{1}, \\ldots, X_n + k\\cdot\\boldsymbol{1}) \n\t\t&=F(X_1 + [k,k], \\ldots, X_n + [k,k])\\\\\n\t\t&= F(X_1 + C, \\ldots, X_n + C)\\\\\n\t\t&\\succeq F(X_1, \\ldots, X_n).\n\t\\end{align*}\n\t\n\tSo, conditional increasingness given a constant vector implies conditional increasingness given the vector $\\vec{\\boldsymbol{1}}$ (similarly, decreasing version).\n\\end{remark}\n\n\\medskip \n\n\\begin{remark}\n\t\\label{rem_intweakmonotonicity}\n\tEvery increasing (resp. decreasing) IV-function is a conditional increasing given a constant vector (resp. conditional decreasing given a constant vector) IV-function since taking $X_i + C \\succeq X_i$ for $i = 1, \\ldots,n$, we have: $F(X_1+C, \\ldots, X_n+C) \\succeq F(X_1, \\ldots, X_n)$ (resp. $F(X_1+C, \\ldots, X_n+C) \\preceq F(X_1, \\ldots, X_n)$) for all $X_1, \\ldots, X_n, C \\in \\mathbb{I}([0,\\!1])$ such that   $(X_1 + C, \\ldots, X_n + C) \\in \\mathbb{I}([0,\\!1])^n$.\n\\end{remark}\n\n\\medskip \n\nHowever, not every conditionally increasing given a constant vector IV-function is an increasing function (as we can see in the next example, cf. Remark 6.12 from Sesma-Sara et al. \\cite{Sesma2020}) and not every conditionally decreasing given a constant vector IV-function is a decreasing IV-function.\n\n\\vspace{0,1cm}\n\n\\begin{example} \n\t\\label{example_weakfunction}\t\n\tLet $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ the IV-function defined by $F(X_1, \\ldots, X_n) = [\\overline{X_1} - \\underline{X_1}, \\overline{X_1} - \\underline{X_1}]$. If $X_i, X_i + C \\in \\mathbb{I}([0,\\!1])$ for $i = 1, \\ldots, n$, we have: \n\t\\begin{align*}\n\t\tF(X_1 + C, \\ldots, X_n + C) &= [\\overline{X_1 + C } - \\underline{X_1+ C}, \\overline{X_1 + C} - \\underline{X_1+ C}]\\\\\n\t\t& = [(\\overline{X_1} - \\underline{X_1}) + (\\overline{C} - \\underline{C}), (\\overline{X_1} - \\underline{X_1}) + (\\overline{C} - \\underline{C})]\\\\ \n\t\t&= F(X_1, \\ldots, X_n) + F(C, \\ldots, C)\\\\ \n\t\t&\\succeq F(X_1, \\ldots, X_n)\n\t\\end{align*}\t\n\tThus, $F$ is an IV-function conditionally increasing given a constant vector. But $F$ is not  increasing: $[0.1,0.3] \\prec [0.2,0.3]$, however $F([0.1,0.3], \\boldsymbol{0}, \\ldots, \\boldsymbol{0}) = [0.2,0.2] \\succ [0.1,0.1] =  F([0.2,0.3], \\boldsymbol{0}, \\ldots, \\boldsymbol{0})$.\n\\end{example}\n\n\\medskip \n\nNext, we present an IV-function that is neither conditionally increasing given a constant vector nor conditionally decreasing given a constant vector.\n\n\\medskip \n\n\\begin{example} \n\t\\label{example_notweakincrfunction}\t\n\tLet $F: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ be the IV-function  defined by $F(X,Y) = \\left\\{\n\t\\begin{array}{ccccccc}\n\t\tX(\\boldsymbol{1} - X), \\ if \\ Y \\preceq \\boldsymbol{0.8}\\\\\n\t\t\\ \\ \\ \\ \\ \\boldsymbol{1}, \\ \\ \\ \\ \\ otherwise\\\\\n\t\\end{array}\n\t\\right.$ (see Example \\ref{ex:multsentdirincrf}). Taking $X = [0.5,0.7]$,  $C = [0.2,0.3]$ and any $Y \\in \\mathbb{I}([0,\\!1])$ such that $Y, Y+C \\preceq \\boldsymbol{0.8}$, we have: \n\t\n\t\\begin{center}\n\t\t$F(X, Y) = [0.5,0.7] \\cdot [0.3,0.5] = [0.15,0.35] \\succ [0,0.3] =  [0.7,1] \\cdot [0,0.3] = F(X + C, Y + C)$. \n\t\\end{center}\n\t%\n\tOn the other hand, taking $X = [0.1,0.2]$,  $C = [0.3,0.4]$ and any $Y \\in \\mathbb{I}([0,\\!1])$ such that $Y \\preceq \\boldsymbol{0.8}$ and  $Y+C \\succ \\boldsymbol{0.8}$, we have: \n\t%\n\t\\begin{center}\n\t\t$F(X,Y) = [0.1,0.2] \\cdot [0.8,0.9] = [0.08,0.18] \\prec \\boldsymbol{1} = F(X + C, Y + C)$. \n\t\\end{center}\n\t%\n\tSo, $ F $ is not conditionally increasing given a constant vector, nor conditionally decreasing given a constant vector.\n\\end{example}\n\n\nIn \\cite{SantiAbsHomo2021}, Santiago et al. introduced the notion of $ G $-weak monotonicity for $ [0,\\!1] $. It can be adapted for our context in the following way:\n\n\\begin{definition}[Conditional monotonicity with respect to $ G $ and a parameter]\n\t\\label{def:condmonotglambda}\n\tLet $ \\Lambda \\in \\mathbb{I}([0,\\!1]) $ and $G: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ an IV-function such that $G(X,Y) \\succeq Y $ for all $X,Y \\in \\mathbb{I}([0,\\!1])$ \\footnote{In \\cite{SantiAbsHomo2021}, the authors have not attempted to impose the condition $ G(x,y)\\geq y $ for all $x,y \\in [0,\\!1]$.}. An IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is said to be \\emph{conditionally increasing(resp. decreasing) with respect to $ G $ and $ \\Lambda $} iff it is conditionally increasing (resp. decreasing) given the transformation $\\Theta: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$ defined as\n\t\\begin{equation*}\n\t\t\\Theta(X_{i})= G(\\Lambda,X_i)\n\t\\end{equation*}\n\tfor all $X_i \\in\\mathbb{I}([0,\\!1])$, $i=1,\\ldots,n$. \n\\end{definition}\n\n\\begin{example}\n\t\\label{example_Fweakmonotonocity} Given $\\Lambda\\in\\mathbb{I}([0,\\!1])$,  if $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is an increasing IV-function, then it is conditionally increasing with respect to $ S_{\\text{\\L}} $ and $\\Lambda$, where $S_{\\text{\\L}}$ is the IV-\\L{ukasiewicz} \\ $t$-conorm: the IV-function $ S_{\\text{\\L}} : \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ defined by $S_{\\text{\\L}}(X,Y) = \\min(X+Y,\\boldsymbol{1})$. \t\n\\end{example}\n\n\\medskip\n\nThe function in the following example is the interval-valued version of the function presented in Example 3 of \\cite{Katarzyna2017}.\n\n\\begin{example} \n\t\\label{example_gdecreasingfunction}\n\tGiven $\\Lambda\\in\\mathbb{I}([0,\\!1])$ and fixed $\\Theta \\in \\mathbb{I}([0,\\!1])$, the IV- function $F : \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$, given by \n\t\\begin{center}\n\t\t$F(X_1, \\ldots, X_n) = (\\boldsymbol{1} - \\Theta) \\cdot \\max\\left(X_1, \\ldots, X_n \\right)+ \\Theta \\cdot \\min\\left(X_1, \\ldots, X_n \\right)$,\n\t\\end{center} is conditionally decreasing with respect to $ P $ and $\\Lambda$, where $P$ is the IV-function product $P : \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$, given by $ P(X,Y) = X \\cdot Y $ for all $ X,Y \\in \\mathbb{I}([0,\\!1])$. Indeed, since $\\Lambda \\cdot X_i \\preceq X_i$ for $i = 1, \\ldots, n$, we have:\n\t\\begin{align*}\n\t\tF(P(\\Lambda,X_1), \\ldots, P(\\Lambda,X_n)) &= (\\boldsymbol{1} - \\Theta) \\cdot \\max\\left(\\Lambda \\cdot X_1, \\ldots, \\Lambda \\cdot x_n \\right)+ \\Theta \\cdot  \\min\\left(\\Lambda \\cdot X_1, \\ldots, \\Lambda \\cdot X_n \\right)\\\\\n\t\t& \\preceq (\\boldsymbol{1} - \\Theta) \\cdot \\max\\left(X_1, \\ldots, X_n \\right)+ \\Theta \\cdot \\min\\left(X_1, \\ldots, X_n \\right)\\\\\n\t\t& = F(X_1, \\ldots, X_n)\n\t\\end{align*}\n\tfor all $\\Lambda \\in \\mathbb{I}([0,\\!1])$ and $(X_1, \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$.\n\\end{example}\n\n\n\\section{\\label{sec:mon&prop} Properties}\n\nIn what follows we proceed by showing properties for the previous  monotonicities.\n\n\\begin{definition}\n\t\\label{def_shiftinvweakmonot}\n\tAn IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is  shift-invariant if \n\t\\begin{center}\n\t\t$F(X_1 + \\Lambda, \\ldots, X_n + \\Lambda) = F(X_1, \\ldots, X_n) + \\Lambda$ \n\t\\end{center}\n\twhenever $(X_1 + \\Lambda, \\ldots, X_n + \\Lambda) \\in \\mathbb{I}([0,\\!1])^n$ and $F(X_1, \\ldots, X_n) + \\Lambda \\in \\mathbb{I}([0,\\!1])$.\n\\end{definition}\n\n\\begin{proposition}\n\t\\label{prop_shiftinvweakmonot}\n\tIf an IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is conditionally decreasing given a constant vector and $\\lambda + F(X_1, \\ldots, X_n) \\leq F(\\Lambda + X_1, \\ldots, \\Lambda+ X_n)$, for all $\\Lambda, X_1, \\ldots, X_n \\in \\mathbb{I}([0,\\!1])$, then $F$ is shift-invariant.\n\\end{proposition}\n\n\\begin{proof}\n\tBy hypotheses, for all $\\Lambda, X_1, \\ldots, X_n \\in \\mathbb{I}([0,\\!1])$, we have:\n\t\n\t\\begin{center}\n\t\t$F(X_1 + \\Lambda, \\ldots, X_n + \\Lambda) \\leq F(X_1, \\ldots, X_n) \\leq F(X_1, \\ldots, X_n) + \\Lambda \\leq F(X_1 + \\Lambda, \\ldots, X_n + \\Lambda)$.\n\t\\end{center}\n\tThus, $F(X_1 + \\Lambda, \\ldots, X_n + \\Lambda) =  F(X_1, \\ldots, X_n) + \\Lambda$.\n\\end{proof}\n\n\\begin{remark}\n\tDue to Remark \\ref{rem_intweakmonotonicity}, Proposition \\ref{prop_shiftinvweakmonot} also holds if $F$ is a decreasing function.  \n\\end{remark}\n\n\\medskip \n\nThe next result was proved in \\cite{Katarzyna2017} for vectors $\\vec{v} = (c, \\ldots, c) \\in {\\mathbb{R}}^n$ with $0 < c \\in \\mathbb{R}$ and functions $F: [0,\\!1]^n \\to [0,\\!1]$.   \n\n\\medskip \n\n\\begin{proposition}\n\t\\label{prop_intrmonotonicity}\n\tLet $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ be an IV-function  and  $\\overrightarrow{V} = (\\boldsymbol{C}, \\ldots, \\boldsymbol{C}) \\in \\mathbb{I}(\\mathbb{R})^n$ such that $\\boldsymbol{C} = [a,a]$ with $0 < a \\in \\mathbb{R}$. Then, $F$ is a  conditionally increasing (resp. decreasing) given the  vector $ \\overrightarrow{V} $ IV-function iff it is a  conditionally increasing (resp. decreasing) given the vector $ \\overrightarrow{\\boldsymbol{1}} $ IV-function.\n\\end{proposition}\n\n\\begin{proof}\n\tSuppose $F$ is a conditionally increasing given the  vector $ \\overrightarrow{V} = (\\boldsymbol{C}, \\ldots, \\boldsymbol{C}) $ IV-function. Then, for all  $0 < k \\in \\mathbb{R}$ and $(X_1,  \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot \\boldsymbol{1}, \\ldots, X_n + k \\cdot \\boldsymbol{1}) \\in \\mathbb{I}([0,\\!1])^n$, we have:\n\t\\begin{align*}\n\t\tF(X_1 + k \\cdot \\boldsymbol{1}, \\ldots, X_n + k \\cdot  \\boldsymbol{1}) &= F \\left(X_1 + \\dfrac{k}{a} \\cdot a \\cdot \\boldsymbol{1}, \\ldots, X_n + \\dfrac{k}{a} \\cdot a \\cdot \\boldsymbol{1} \\right)\\\\ \n\t\t&= F \\left( X_1 + \\dfrac{k}{a} \\cdot \\boldsymbol{C}, \\ldots, X_n + \\dfrac{k}{a} \\cdot \\boldsymbol{C} \\right)\\\\\n\t\t&\\succeq F(X_1, \\ldots, X_n).\n\t\\end{align*}\n\tThus, $F$ is a conditionally increasing given the vector $\\vec{\\boldsymbol{1}}$ IV-function. Reciprocally, if $F$ is a conditionally increasing given the vector $\\vec{\\boldsymbol{1}}$ IV-function, for all  $0 < k \\in \\mathbb{R}$ and $(X_1,  \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot \\boldsymbol{C}, \\ldots, X_n + k \\cdot \\boldsymbol{C}) \\in \\mathbb{I}([0,\\!1])^n$, we have:\n\t\\begin{center}\n\t\t$F(X_1 + k \\cdot \\boldsymbol{C}, \\ldots, X_n + k \\cdot \\boldsymbol{C}) = F \\left( X_1 + k \\cdot a \\cdot \\boldsymbol{1}, \\ldots, X_n + k \\cdot a \\cdot \\boldsymbol{1} \\right) \\succeq F \\left( X_1, \\ldots, X_n \\right)$,\n\t\\end{center}\n\tsince $0 < k \\cdot a \\in \\mathbb{R}$. Therefore, $F$ is a conditionally increasing given the  vector $ \\overrightarrow{V} = (\\boldsymbol{C}, \\ldots, \\boldsymbol{C}) $ IV-function. Analogously we have that $F$ is a  conditionally decreasing given the  vector $ \\overrightarrow{V} = (\\boldsymbol{C}, \\ldots, \\boldsymbol{C}) $ IV-function iff it is a  conditionally decreasing given the vector $ \\overrightarrow{\\boldsymbol{1}} $ IV-function.\n\\end{proof}\n\n\\begin{corollary}\n\t\\label{cor_intweakrmonotonicity}\n\tLet $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ be an IV function and $\\overrightarrow{V} = (\\boldsymbol{C}, \\ldots, \\boldsymbol{C}) \\in \\mathbb{I}(\\mathbb{R})^n$ such that $\\boldsymbol{C} = [a,a]$ with $0 < a \\in \\mathbb{R}$. Then, $F$ is a conditionally increasing given a constant vector (resp. conditionally decreasing given a constant vector) IV-function iff it is a conditionally increasing (resp. decreasing) given the  vector $ \\overrightarrow{V} $ IV-function.  \n\\end{corollary}\n\n\\begin{proof}\n\tStraight from Remark \\ref{remark_intweakdirmonotonicity} and Proposition \\ref{prop_intrmonotonicity}.  \n\\end{proof}\n\n\\medskip\n\nThe following proposition was proved in \\cite{MikelIVPreAgg2019} for vectors $\\overrightarrow{V} = (a_1,b_1, \\ldots, a_n,b_n) \\in (\\mathbb{R}^2)^n$ such that $(a_i,b_i) \\not= (0,0)$ for some $i \\in \\{1, \\ldots, n\\}$ considering on $ \\mathbb{I}([0,\\!1]) $ the \\textit{Kulisch-Miranker order}. \n\n\\begin{proposition}\n\t\\label{prop_intrmonotonicity2}\n\tGiven $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n}) \\in \\mathbb{I}(\\mathbb{R})^n \\setminus \\{\\vec{\\boldsymbol{0}}\\}$ such that each $\\boldsymbol{V_i}$ is a degenerated interval for $i = 1, \\ldots, n$, an IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is a  conditionally increasing given the  vector $ \\overrightarrow{V} $ IV-function iff it is a conditionally decreasing given the  vector $\\overrightarrow{(\\text{-}V)}$ IV-function.\n\\end{proposition}\n\n\\begin{proof}\n\tSuppose $F$ is conditionally increasing given the  vector $ \\overrightarrow{V} $. Then, for all $0 < k \\in \\mathbb{R}$ and $(X_1,  \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot (-\\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (-\\boldsymbol{V_n})) \\in \\mathbb{I}([0,\\!1])^n$, we have:  \n\t\\begin{align*}\n\t\tF(X_1 + k \\cdot (-\\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (-\\boldsymbol{V_n})) &\\preceq F([X_1 + k \\cdot (-\\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (-\\boldsymbol{V_n})] + (k \\cdot \\boldsymbol{V_1}, \\ldots, k \\cdot \\boldsymbol{V_n}))\\\\\n\t\t&= F(X_1, \\ldots, X_n)\n\t\\end{align*}\n\tsince each $\\boldsymbol{V_i}$ is a degenerated interval. Thus, $F$ is a conditionally decreasing given the  vector $\\overrightarrow{(\\text{-}V)}$ IV-function. Analogously, we can prove that if $ F $ is  conditionally decreasing given the  vector $\\overrightarrow{(\\text{-}V)}$  then it is conditionally increasing given the  vector $ \\overrightarrow{V} $. \n\\end{proof}\n\n\\medskip \n\n\\begin{corollary}\n\t\\label{cor_intrmonotonicity2}\n\tEvery IV-implication $I: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ is a conditionally decreasing given the  vector $(\\boldsymbol{1}, \\boldsymbol{-1})$ IV-function.\n\\end{corollary}\n\n\\begin{proof}\n\tIt follows from Propositions \\ref{prop_directmonotImp} and \\ref{prop_intrmonotonicity2}.\n\\end{proof}\n\n\\medskip\n\nWe can obtain directional conditionally increasing\/decreasing given a vector of degenerated intervals functions from increasing\/decreasing and conditionally constant given a vector of degenerated intervals functions as shown in the next two results (cf. \\cite{MikelIVPreAgg2019}). \n\n\\medskip \n\n\\begin{proposition}\n\t\\label{prop_intrmonotonicity3}\n\tIf $F_1, \\ldots, F_m: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ are  directional conditionally increasing(resp. decreasing) given a vector of degenerated intervals IV-functions and $F: \\mathbb{I}([0,\\!1])^m \\to \\mathbb{I}([0,\\!1])$ is an increasing (resp. decreasing) IV-function, then $G = F(F_1, \\ldots, F_m)$, such that  $F(F_1, \\ldots, F_m)(x_1,\\ldots,x_n)=F(F_1(x_1,\\ldots,x_n), \\ldots, F_m(x_1,\\ldots,x_n))$ is a  directional conditionally increasing(resp. decreasing) given a vector of degenerated intervals IV-function.\n\\end{proposition}\n\n\\begin{proof}\n\tLet $F_1, \\ldots, F_m: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ directional conditionally increasing given the vector  $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n})$ IV-functions. For all $0 < k \\in \\mathbb{R}$ and $(X_1, \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot \\boldsymbol{V_1}, \\ldots, X_n + k \\cdot \\boldsymbol{V_n}) \\in \\mathbb{I}([0,\\!1])^n$ we have:  \n\t\n\t\\begin{center}\n\t\t$F_i(X_1 + k \\cdot \\boldsymbol{V_1}, \\ldots, X_n + k \\cdot \\boldsymbol{V_n}) \\succeq F_i(X_1, \\ldots, X_n)$, \n\t\\end{center}\n\tfor $i = 1, \\ldots, m$, which implies:\n\t\\begin{align*}\n\t\t&G(X_1 + k \\cdot \\boldsymbol{V_1},\\ldots, X_n + k \\cdot \\boldsymbol{V_n})\\\\ \n\t\t&= F\\left(  F_1(X_1 + k \\cdot \\boldsymbol{V_1},\\ldots, X_n + k \\cdot \\boldsymbol{V_n}), \\ldots, F_m(X_1 + k \\cdot \\boldsymbol{V_1},\\ldots, X_n + k \\cdot \\boldsymbol{V_n}) \\right)\\\\\n\t\t& \\succeq F\\left( F_1(X_1, \\ldots, X_n), \\ldots, F_m(X_1, \\ldots, X_n) \\right)\\\\\n\t\t&= G(X_1, \\ldots, X_n)\n\t\\end{align*}\n\tsince $F$ is increasing. Therefore, $G$ is directional conditionally increasing given the vector  $\\overrightarrow{V}$. Similarly, if $F_1, \\ldots, F_m$ are directional conditionally decreasing given the vector $\\overrightarrow{V}$ and $F: \\mathbb{I}([0,\\!1])^m \\to \\mathbb{I}([0,\\!1])$ is   decreasing, then $G = F(F_1, \\ldots, F_m)$ is directional conditionally decreasing given the vector $\\overrightarrow{V}$.\n\\end{proof}\n\n\\begin{corollary}\n\t\\label{cor_intrmonotonicity3}\n\tIf $F: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$ is an  increasing (resp. decreasing) IV-function and $\\Phi: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is a  directional conditionally increasing (resp. decreasing) given a vector of degenerated intervals IV-function, then $G = F \\circ \\Phi$ is a directional conditionally increasing (resp. decreasing) given a vector of degenerated intervals IV-function.\n\\end{corollary}\n\\begin{proof}\n\tStraigh from Proposition \\ref{prop_intrmonotonicity3} taking $F_1 = \\ldots = F_m = \\Phi$.\n\\end{proof}\n\n\\medskip\n\nA similar version for functions $F: [0,\\!1] \\to [0,\\!1]$ of the following result appears in \\cite{BustinceDirMon2015} as Corollary 3. \n\n\\begin{proposition}\n\t\\label{prop_intrmonotonicity6}\n\tAn IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ and its dual, the function $F^{d}: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ given by $F^{d}(X_1, \\ldots, X_n) = \\boldsymbol{1} - F(\\boldsymbol{1} - X_1, \\ldots, \\boldsymbol{1} - X_n)$, are directional conditionally increasing (resp. decreasing) given a vector of degenerated intervals with respect to the same set of vectors.\n\\end{proposition}\n\n\\begin{proof}\n\tLet $F$ be a directional conditionally increasing given the vector $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n})$ IV-function. Then, by Proposition \\ref{prop_intrmonotonicity2}, it is directional conditionally decreasing given the vector  $\\overrightarrow{\\text{-}V}$. Thus, for all $0 < k \\in \\mathbb{R}$ and $(X_1, \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot \\boldsymbol{V_1}, \\ldots, X_n + k \\cdot \\boldsymbol{V_n}) \\in \\mathbb{I}([0,\\!1])^n$ we have:  \n\n\t\\begin{center}\n\t\t$F(X_1 + k \\cdot (-\\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (-\\boldsymbol{V_n})) \\preceq F((X_1, \\ldots, X_n)$,\n\t\\end{center}\n\twhich gives us \n\t\\begin{center}\n\t\t$\\boldsymbol{1} - F(X_1 + k \\cdot (-\\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (-\\boldsymbol{V_n})) \\succeq \\boldsymbol{1} - F((X_1, \\ldots, X_n)$.\n\t\\end{center} \n\tSo,\n\t\\begin{align*}\n\t\tF^{d}(X_1 + k \\cdot \\boldsymbol{V_1}, \\ldots, X_n + k \\cdot \\boldsymbol{V_n}) &= \\boldsymbol{1} - F(\\boldsymbol{1} - (X_1 + k \\cdot \\boldsymbol{V_1}), \\ldots, \\boldsymbol{1} - (X_n + k \\cdot \\boldsymbol{V_n}))\\\\\n\t\t&= \\boldsymbol{1} - F((\\boldsymbol{1} - X_1) + k \\cdot (-\\boldsymbol{V_1}), \\ldots, (\\boldsymbol{1} - X_n) + k \\cdot (-\\boldsymbol{V_n}))\\\\\n\t\t&\\succeq \\boldsymbol{1} - F(\\boldsymbol{1} - X_1, \\ldots, \\boldsymbol{1} - X_n)\\\\\n\t\t&= F^{d}(X_1, \\ldots, X_n).\n\t\\end{align*}\n\t%\n\tTherefore, $F^{d}$ is directional conditionally increasing given the vector $\\overrightarrow{V}$. On the other hand, if $F$ is a directional conditionally decreasing given the vector $\\overrightarrow{V}$ IV-function then, it is directional conditionally increasing given the vector $\\overrightarrow{\\text{-}V}$. Thus, for all $0 < k \\in \\mathbb{R}$ and $(X_1, \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot \\boldsymbol{V_1}, \\ldots, X_n + k \\cdot \\boldsymbol{V_n}) \\in \\mathbb{I}([0,\\!1])^n$ we have:  \n\n\t\\begin{center}\n\t\t$F(X_1 + k \\cdot (-\\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (-\\boldsymbol{V_n})) \\succeq F((X_1, \\ldots, X_n)$,\n\t\\end{center}\n\twhich gives us \n\t\\begin{center}\n\t\t$\\boldsymbol{1} - F(X_1 + k \\cdot (-\\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (-\\boldsymbol{V_n})) \\preceq \\boldsymbol{1} - F((X_1, \\ldots, X_n)$.\n\t\\end{center} \n\tSo,\n\t\\begin{align*}\n\t\tF^{d}(X_1 + k \\cdot \\boldsymbol{V_1}, \\ldots, X_n + k \\cdot \\boldsymbol{V_n}) &= \\boldsymbol{1} - F((\\boldsymbol{1} - X_1) + k \\cdot (-\\boldsymbol{V_1}), \\ldots, (\\boldsymbol{1} - X_n) + k \\cdot (-\\boldsymbol{V_n}))\\\\\n\t\t&\\preceq \\boldsymbol{1} - F(\\boldsymbol{1} - X_1, \\ldots, \\boldsymbol{1} - X_n)\\\\\n\t\t&= F^{d}(X_1 + k \\cdot \\boldsymbol{V_1}, \\ldots, X_n + k \\cdot \\boldsymbol{V_n}).\n\t\\end{align*}\n\tTherefore, $F^{d}$ is directional conditionally decreasing given the vector $\\overrightarrow{V}$. The proof of the reciprocal is analogous. \n\\end{proof}\n\n\\medskip\n\n\\begin{lemma}\n\t\\label{lemma_intrmonotonicity}\n\tLet $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ a function, $\\alpha \\in \\mathbb{R} \\setminus \\{0\\}$ and $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n}) \\in \\mathbb{I}(\\mathbb{R})^n \\setminus \\vec{\\boldsymbol{0}}$ such that each $\\boldsymbol{V_i}$ is a degenerated interval for $i = 1, \\ldots, n$. The following statements holds.\n\t\n\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\n\t\t\\item If $\\alpha > 0$ and $F$ is directional conditionally increasing(resp. decreasing) given the vector $\\overrightarrow{V}$, then it is directional conditionally increasing (resp. decreasing) given the vector $\\overrightarrow{\\alpha V}$.\n\t\t\n\t\t\\item If $\\alpha < 0$ and $F$ is is directional conditionally increasing(resp. decreasing) given the vector $\\overrightarrow{V}$, then it is directional conditionally decreasing (resp. increasing) given the vector $\\overrightarrow{\\alpha V}$.\n\t\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\n\t\t\\item Suppose $0 < \\alpha \\in \\mathbb{R}$ and $F$ a directional conditionally increasing given the vector $\\overrightarrow{V}$ IV-function. Then, for all $0 < k \\in \\mathbb{R}$ and $(X_1,  \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot (\\alpha \\cdot \\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (\\alpha \\cdot \\boldsymbol{V_n})) \\in \\mathbb{I}([0,\\!1])^n$, we have:     \n\t\n\t\t\\begin{align*}\n\t\t\t&F(X_1 + k \\cdot (\\alpha \\cdot \\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (\\alpha \\cdot \\boldsymbol{V_n}))\\\\ \n\t\t\t&= F(X_1 + (k\\alpha) \\cdot \\boldsymbol{V_1}, \\ldots, X_n + (k \\alpha) \\cdot \\boldsymbol{V_n})\\\\\n\t\t\t&\\succeq F(X_1, \\ldots, X_n),\n\t\t\\end{align*}\n\t\tsince $k \\alpha >0$. Thus, $F$ is directional conditionally increasing given the vector $\\overrightarrow{\\alpha V}$. Similarly, if $F$ is directional conditionally decreasing given the vector $\\overrightarrow{V}$ IV-function, then it is directional conditionally decreasing given the vector $\\overrightarrow{\\alpha V}$.  \n\t\t\n\t\t\\item Let $0 > \\alpha \\in \\mathbb{R}$ and $F$ a directional conditionally increasing given the vector $\\overrightarrow{\\alpha V}$ IV-function. From Proposition \\ref{prop_intrmonotonicity2}, $F$ is directional conditionally decreasing given the vector $\\overrightarrow{-V}$. Then, for all $0 < k \\in \\mathbb{R}$ and $(X_1,  \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot (\\alpha \\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (\\alpha \\boldsymbol{V_n})) \\in \\mathbb{I}([0,\\!1])^n$, we have:     \n\t\n\t\t\\begin{align*}\n\t\t\t& F(X_1 + k \\cdot (\\alpha\\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (\\alpha \\boldsymbol{V_n}))\\\\\n\t\t\t&= F(X_1 + k \\cdot (-\\alpha) \\cdot (-\\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (-\\alpha)\\cdot (-\\boldsymbol{V_n}))\\\\\n\t\t\t&\\preceq F(X_1, \\ldots, X_n),\n\t\t\\end{align*}\n\t\tsince $k \\cdot (-\\alpha) >0$. Therefore, $F$ is directional conditionally decreasing given the vector $\\overrightarrow{\\alpha V}$. Analogously, if $F$ is directional conditionally decreasing given the vector $\\overrightarrow{V}$, then it is directional conditionally increasing given the vector $\\overrightarrow{\\alpha V}$.\n\t\\end{enumerate}\n\\end{proof}\n\nIn the following theorem we provide sufficient conditions for an IV-function that is directional conditionally increasing given a pair of vectors of degenerated intervals to be directional conditionally increasing given a linear combination of these vectors. A similar result, but considering on $ \\mathbb{I}([0,\\!1]) $ the \\textit{Kulisch-Miranker order}, can be found in \\cite{MikelIVPreAgg2019}. \n\n\\medskip\n\n\\begin{theorem}\n\t\\label{theo_intrmonotonicity4}\n\tLet $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ be a given function, $\\alpha, \\beta \\in \\mathbb{R}$ and $\\overrightarrow{U} = (\\boldsymbol{U_1}, \\ldots, \\boldsymbol{U_n})$ and $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n})$ be vectors in  $\\mathbb{I}(\\mathbb{R})^n \\setminus \\vec{\\boldsymbol{0}}$ where $\\boldsymbol{U_i}$ and $\\boldsymbol{V_i}$ are  degenerated intervals, for $i = 1, \\ldots, n$,  and such that for every $0 < k \\in \\mathbb{R}$, either \n\t\\begin{center}\n\t\t$(X_1 + k \\alpha \\cdot  \\boldsymbol{U_1}, \\ldots, X_n + k\\alpha \\cdot \\boldsymbol{U_n}) \\in \\mathbb{I}([0,\\!1])^n$\n\t\\end{center} \n\tor \n\t\\begin{center}\n\t\t$(X_1 + k\\beta \\cdot  \\boldsymbol{V_1}, \\ldots, X_n + k\\beta \\cdot \\boldsymbol{V_n}) \\in \\mathbb{I}([0,\\!1])^n$,\n\t\\end{center} \n\twhenever $(X_1 + k \\cdot (\\alpha \\boldsymbol{U_1} + \\beta \\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (\\alpha \\boldsymbol{U_n} + \\beta \\boldsymbol{V_n})) \\in \\mathbb{I}([0,\\!1])^n$. The following statements holds.\n\t\n\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\n\t\t\\item If $\\alpha \\ge 0$ and $\\beta > 0$ and $F$ is directional conditionally increasing (resp. decreasing) given the vector $\\overrightarrow{U}$ and directional conditionally increasing (resp. decreasing) given the vector $\\overrightarrow{V}$, then it is directional conditionally increasing (resp. decreasing) given the vector $(\\alpha \\overrightarrow{U} + \\beta \\overrightarrow{V})$.\n\t\t\n\t\t\\item If $\\alpha > 0$ and $\\beta \\leq 0$ and $F$ is directional conditionally increasing (resp. decreasing) given the vector $\\overrightarrow{U}$ and directional conditionally decreasing (resp. increasing) given the vector $\\overrightarrow{V}$, then it is directional conditionally increasing (resp. decreasing) given the vector $(\\alpha \\overrightarrow{U} + \\beta \\overrightarrow{V})$.\n\t\t\n\t\t\\item If $\\alpha < 0$ and $\\beta \\geq 0$ and $F$ is directional conditionally decreasing (resp. increasing) given the vector $\\overrightarrow{U}$ and directional conditionally increasing (resp. decreasing) given the vector $\\overrightarrow{V}$, then it is directional conditionally decreasing (resp. increasing) given the vector $(\\alpha \\overrightarrow{U} + \\beta \\overrightarrow{V})$.\t\t\n\t\t\n\t\t\\item If $\\alpha \\leq 0$ and $\\beta < 0$ and $F$ is directional conditionally increasing (resp. decreasing) given the vector $\\overrightarrow{U}$ and directional conditionally increasing (resp. decreasing) given the vector $\\overrightarrow{V}$, then it is directional conditionally decreasing (resp. increasing) given the vector $(\\alpha \\overrightarrow{U} + \\beta \\overrightarrow{V})$.\t\t\n\t\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\tLet $0 < k \\in \\mathbb{R}$ and $(X_1,  \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot (\\alpha U_1 + \\beta V_1), \\ldots, X_n + k \\cdot (\\alpha U_n +  \\beta V_n)) \\in \\mathbb{I}([0,\\!1])^n$. Suppose  $(X_1 + k\\alpha \\cdot  U_1, \\ldots, X_n + k\\alpha \\cdot U_n) \\in \\mathbb{I}([0,\\!1])^n$ whenever $(X_1 + k \\cdot (\\alpha U_1 + \\beta V_1), \\ldots, X_n + k \\cdot (\\alpha U_n + \\beta V_n)) \\in \\mathbb{I}([0,\\!1])^n$. \n\t\n\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\\item Consider $F$ directional conditionally increasing given the vectors $\\overrightarrow{U}$ and $\\overrightarrow{V}$ and $\\alpha,\\beta \\in \\mathbb{R}$ such that $\\alpha \\geq 0$ and $\\beta > 0$. If $\\alpha = 0$, then from Lemma \\ref{lemma_intrmonotonicity}, $F$ is directional conditionally increasing given the vector $(\\alpha \\overrightarrow{U} + \\beta \\overrightarrow{V})$. If $\\alpha > 0$ then, since $k\\alpha, k \\beta > 0$, we have:\n\t\n\t\t\\begin{align*}\n\t\t\t&F(X_1 + k \\cdot (\\alpha \\boldsymbol{U_1} + \\beta \\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (\\alpha \\boldsymbol{U_n} + \\beta \\boldsymbol{V_n}))\\\\\n\t\t\t&= F((X_1 + k \\alpha \\cdot \\boldsymbol{U_1}) + k\\beta \\cdot \\boldsymbol{V_1}, \\ldots, (X_n + k \\alpha \\cdot \\boldsymbol{U_n}) + k\\beta \\cdot \\boldsymbol{V_n})\\\\\n\t\t\t&\\succeq F(X_1 + k \\alpha \\cdot \\boldsymbol{U_1}, \\ldots, X_n + k \\alpha \\cdot \\boldsymbol{U_n})\\\\\n\t\t\t&\\succeq F(X_1, \\ldots, X_n),\n\t\t\\end{align*}    \n\t\twhich shows that $F$ is directional conditionally increasing given the vector $(\\alpha\\overrightarrow{U} + \\beta \\overrightarrow{V})$. On the other hand, if $F$ is directional conditionally decreasing given the vectors $\\overrightarrow{U}$ and $\\overrightarrow{V}$, then $F$ is directional conditionally decreasing given the vector $(\\alpha \\overrightarrow{U} + \\beta \\overrightarrow{V})$ for any real numbers $\\alpha \\geq 0$ and $\\beta > 0$.\n\t\t\n\t\n\t\t\n\t\t\\item Consider $F$ directional conditionally increasing given the vector $\\overrightarrow{U}$ and directional conditionally decreasing given the vector $\\overrightarrow{V}$ and $\\alpha,\\beta \\in \\mathbb{R}$ such that $\\alpha > 0$ and $\\beta \\leq 0$. By Proposition \\ref{prop_intrmonotonicity2}, $F$ is directional conditionally increasing given the vector $\\overrightarrow{-V}$. If $\\beta = 0$ then, from Lemma \\ref{lemma_intrmonotonicity}, $F$ is directional conditionally increasing given the vector $(\\alpha \\overrightarrow{U} + \\beta \\overrightarrow{V})$. If $\\beta < 0$ then, since $k\\alpha, k (-\\beta) > 0$, we have:\n\t\n\t\t\\begin{align*}\n\t\t\t&F(X_1 + k \\cdot (\\alpha \\boldsymbol{U_1} + \\beta \\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (\\alpha \\boldsymbol{U_n} + \\beta \\boldsymbol{V_n}))\\\\\n\t\t\t&= F((X_1 + k \\alpha \\cdot \\boldsymbol{U_1}) + k(-\\beta) \\cdot (-\\boldsymbol{V_1}), \\ldots, (X_n + k \\alpha \\cdot \\boldsymbol{U_n}) + k(-\\beta) \\cdot (-\\boldsymbol{V_n}))\\\\\n\t\t\t&\\succeq F(X_1 + k \\alpha \\cdot \\boldsymbol{U_1}, \\ldots, X_n + k \\alpha \\cdot \\boldsymbol{U_n})\\\\\n\t\t\t&\\succeq F(X_1, \\ldots, X_n).\n\t\t\\end{align*}    \n\t\tThus $F$ is directional conditionally increasing given the vector $(\\alpha \\overrightarrow{U} + \\beta \\overrightarrow{V})$. On the other hand, if $F$ is directional conditionally decreasing given the vector $\\overrightarrow{U}$ and directional conditionally increasing given the vector $\\overrightarrow{V}$, then $F$ is directional conditionally decreasing given the vector $(\\alpha \\overrightarrow{U} + \\beta \\overrightarrow{V})$ for any real numbers $\\alpha \\geq 0$ and $\\beta > 0$. \n\t\t\n\t\n\t\\end{enumerate}\n\tThe proofs of (iii) and (iv) follow analogously and all of these proofs are similar if $(X_1 + k\\beta \\cdot  \\boldsymbol{V_1}, \\ldots, X_n + k\\beta \\cdot \\boldsymbol{V_n}) \\in \\mathbb{I}([0,\\!1])^n$ whenever $(X_1 + k \\cdot (\\alpha \\boldsymbol{U_1} + \\beta \\boldsymbol{V_1}), \\ldots, X_n + k \\cdot (\\alpha \\boldsymbol{U_n} + \\beta \\boldsymbol{V_n})) \\in \\mathbb{I}([0,\\!1])^n$.\n\\end{proof}\n\n\\medskip \n\nNext, we characterize the standard monotonicity of functions $F: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$ in terms of directional conditionally monotonicity given a vector of degenerated intervals. For this, we use the canonical basis of the vector space $\\mathbb{I}(\\mathbb{R})^n$.\n\n\\medskip \n\n\\begin{theorem} \n\t\\label{theo_canonicalbasisofI(R)^n}\t\n\tA function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is increasing (resp. decreasing) iff it is directional conditionally increasing (resp. decreasing) given the vector  $\\overrightarrow{E_i}$  for each vector of the set $\\{\\overrightarrow{E_1}, \\ldots, \\overrightarrow{E_n}\\}$ where $\\overrightarrow{E_i} = (\\boldsymbol{\\epsilon_1}, \\ldots, \\boldsymbol{\\epsilon_n})$ with $\\boldsymbol{\\epsilon_i} = \\boldsymbol{1}$ and $\\boldsymbol{\\epsilon_j} = \\boldsymbol{0}$ for every $j \\not= i$:\n\t\\begin{align*}\n\t\t\\overrightarrow{E_1} &= (\\boldsymbol{1}, \\boldsymbol{0}, \\ldots, \\boldsymbol{0});\\\\\n\t\t& \\ \\vdots\\\\\n\t\t\\overrightarrow{E_n} &= (\\boldsymbol{0}, \\boldsymbol{0}, \\ldots, \\boldsymbol{1}).\n\t\\end{align*} \n\\end{theorem}\n\n\\begin{proof}\n\tSuppose $F$ increasing. Then, for all  $0 < k \\in \\mathbb{R}$ and $(X_1,  \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])$ such that $X_i + k \\cdot \\boldsymbol{1} \\in \\mathbb{I}([0,\\!1])$ for $i = 1, \\ldots, n$, since $X_i + k \\cdot \\boldsymbol{1} \\succeq X_i$ for $i = 1, \\ldots, n$, we have:\n\t\\begin{align*}\n\t\tF(X_1 + k \\cdot \\boldsymbol{1}, \\ldots, X_n) &\\succeq F(X_1, \\ldots, X_n);\\\\ \n\t\t& \\ \\vdots\\\\\n\t\tF(X_1, \\ldots, X_n+ k \\cdot \\boldsymbol{1})& \\succeq F(X_1, \\ldots, X_n).\n\t\\end{align*}\t\n\tSo, $F$ is directional conditionally increasing given the vector  $\\overrightarrow{E_i}$ for every $\\overrightarrow{E_i}$ with $i = 1, \\ldots, n$. Reciprocally, suppose $F$ is directional conditionally increasing given the vector $\\overrightarrow{E_i}$ for $i = 1, \\ldots,n$. If $(X_1, \\ldots, X_n), (Y_1, \\ldots, Y_n) \\in \\mathbb{I}([0,\\!1])$ and $Y_i  \\succeq X_i, \\ for \\ i = 1, \\ldots, n$, we have: \n\n\t\\begin{align*}\n\t\tF(Y_1, \\ldots, Y_n) &= F(Y_1\\overrightarrow{E_1}+ \\ldots + Y_n\\overrightarrow{E_n})\\\\ \n\t\t&\\succeq F((Y_1, \\ldots, \\boldsymbol{0}) + \\ldots + (\\boldsymbol{0}, \\ldots, Y_n))\\\\\n\t\t&\\succeq F((X_1, \\ldots, \\boldsymbol{0}) + \\ldots + (\\boldsymbol{0}, \\ldots, X_n))\\\\\n\t\t&= F(X_1, \\ldots, X_n).\n\t\\end{align*}\t\n\tTherefore, $F$ is increasing iff it is directional conditionally increasing given the vector  $\\overrightarrow{E_i}$ for every $\\overrightarrow{E_i}$ with $i = 1, \\ldots, n$. Analogously, $F$ is decreasing iff it is directional conditionally decreasing given the vector $\\overrightarrow{E_i}$ for every $\\overrightarrow{E_i}$ with $i = 1, \\ldots, n$. \n\\end{proof}\n\n\n\\section{\\label{sec:preaggreg} Pre-Aggregation IV-Functions}\n\nNow we recall the concept of an aggregation IV-function and, based in notion of a pre-aggregation IV-function given by Mikel et al. in \\cite{MikelIVPreAgg2019}, define a pre-aggregation IV-function considering on $\\mathbb{I}([0,\\!1])$ admissible orders. Further, we provide some examples and results.    \n\n\\begin{definition} [IV Aggregation Function \\cite{BedregalGenOWA2016}] \n\t\\label{def_intaggregfunction}\n\tAn aggregation IV-function is a funtion $A : \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ such that  \n\t\\begin{enumerate}[leftmargin=*,label={\\rm($\\mathbf{A}$-\\arabic*)}]\n\t\t\\item $A$ is increasing;\n\t\t\\item $A(\\boldsymbol{0}, \\ldots,\\boldsymbol{0}) = \\boldsymbol{0}$ and $A(\\boldsymbol{1}, \\ldots,\\boldsymbol{1}) = \\boldsymbol{1}$.\\\\ \n\t\\end{enumerate}\n\\end{definition}\n\n\n\\begin{example}\n\t\\label{example_aggregfunctions1}\n\tThe IV-arithmetic mean, $M: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$, given by\n\t\\begin{center}\n\t\t$M(X_1, \\ldots ,X_n) = \\left[ \\frac{1}{n} \\sum_{i=1}^{n} \\underline{X_i} \\ , \\  \\frac{1}{n} \\sum_{i=1}^{n} \\overline{X_i}\\right]$, \n\t\\end{center} \n\tand the IV-geometric mean, $G: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ given by\n\t\\begin{center}\n\t\t$G(X_1, \\ldots ,X_n) = \\left[ \\left( \\prod_{i=1}^{n} \\underline{X_i} \\right)^{1\/n} , \\left( \\prod_{i=1}^{n} \\overline{X_i} \\right)^{1\/n} \\right]$, \n\t\\end{center} \n\tare examples of aggregations.\n\\end{example}\n\nAn important class of IV aggregation functions are $t$-norms and $t$-conorms, whose definition we state below.\n\n\\begin{definition} [IV-$t$-norm]\n\t\\label{def_inttnorm}\n\tA function $\\mathbb{T}: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ is said to be an IV-t-norm if it is associative, comutative, increasing and has identity element $\\boldsymbol{1}$.\n\\end{definition}\n\n\\begin{definition} [IV-$t$-conorm]\n\t\\label{def_inttconorm}\n\tA function $\\mathbb{S}: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ is said to be an IV-$t$-conorm if it is associative, comutative, increasing and has identity element $\\boldsymbol{0}$.\n\\end{definition}\n\n\\begin{remark} \n\t\\label{rem_inttnorm&conorm}\n\tAn IV-t-norm $\\mathbb{T}: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$, an IV-t-conorm $\\mathbb{S}: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ and $X \\in \\mathbb{I}([0,\\!1])$, we have: \n\t\\begin{center}\n\t\t$\\boldsymbol{0} \\preceq \\mathbb{T}(X, \\boldsymbol{0}) \\preceq \\mathbb{T}(\\boldsymbol{1}, \\boldsymbol{0}) = \\boldsymbol{0}$;\\\\\n\t\t$\\boldsymbol{1} = \\mathbb{S}(\\boldsymbol{0}, \\boldsymbol{1}) \\preceq \\mathbb{S}(X, \\boldsymbol{1}) \\preceq \\boldsymbol{1}$.\n\t\\end{center}\n\tTherefore, $\\mathbb{T}(X, \\boldsymbol{0}) = \\boldsymbol{0}$ and $\\mathbb{S}(X, \\boldsymbol{1}) = \\boldsymbol{1}$ for all $X \\in \\mathbb{I}([0,\\!1])$.\n\\end{remark}\n\n\\begin{example}\n\t\\label{example_intuninorm1}\n\tThe minimum is an IV-$t$-norm and the maximum is an IV-$t$-conorm.\n\t\n\t\\begin{flushleft}\n\t\tIn fact,  \n\t\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\t\n\t\t\t\\item associativity holds for both functions: for all $X, Y, Z \\in \\mathbb{I}([0,\\!1])$, we have:\n\t\t\t\\begin{center}\n\t\t\t\t$\\min( \\min(X, Y), Z) = \\min(X, Y, Z) = \\min( X, \\min(Y, Z))$,\n\t\t\t\t\\\\ \\ \\\\\n\t\t\t\t$\\max( \\max(X, Y), Z) = \\max(X, Y, Z) = \\max( X, \\max(Y, Z))$;\t\n\t\t\t\\end{center}\n\t\t\t\n\t\t\t\\item commutativity holds for both functions: $\\min(X, Y) = \\min(Y, X)$ and $\\max(X, Y) = \\max(Y, X)$, for all $X, Y \\in \\mathbb{I}([0,\\!1])$;\n\t\t\t\n\t\t\t\\item both functions are increasing: given $X_1, X_2, Y_1, Y_2 \\in \\mathbb{I}([0,\\!1])$, with $X_1 \\preceq X_2$ and $Y_1 \\preceq Y_2$, we have the results shown in Table 1.\n\t\t\t\n\t\t\t\\begin{table}[htbp!]\n\t\t\t\t\\label{table_maxminincreasing}\n\t\t\t\t\\caption{}\n\t\t\t\t\\centering\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\t\\hline\n\t\t\t\t\tCase & $\\min(X_1, Y_1)$ & $\\min(X_2, Y_2)$ & $\\max(X_1, Y_1)$ & $\\max(X_2, Y_2)$ \\\\ \\hline\n\t\t\t\t\n\t\t\t\t\t$X_1 \\preceq X_2 \\preceq Y_1 \\preceq Y_2$ & $X_1$ & $X_2$ & $Y_1$ & $Y_2$ \\\\ \\hline\n\t\t\t\t\t$X_1 \\preceq Y_1 \\preceq X_2 \\preceq Y_2$ & $X_1$ & $X_2$ & $Y_1$ & $Y_2$ \\\\ \\hline\n\t\t\t\t\t$X_1 \\preceq Y_1 \\preceq Y_2 \\preceq X_2$ & $X_1$ & $Y_2$ & $Y_1$ & $X_2$ \\\\ \\hline\n\t\t\t\t\t$Y_1 \\preceq Y_2 \\preceq X_1 \\preceq X_2$ & $Y_1$ & $Y_2$ & $X_1$ & $X_2$ \\\\ \\hline\n\t\t\t\t\t$Y_1 \\preceq X_1 \\preceq Y_2 \\preceq X_2$ & $Y_1$ & $Y_2$ & $X_1$ & $X_2$ \\\\ \\hline\n\t\t\t\t\t$Y_1 \\preceq X_1 \\preceq X_2 \\preceq Y_2$ & $Y_1$ & $X_2$ & $X_1$ & $Y_2$ \\\\ \\hline\n\t\t\t\t\\end{tabular} \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\end{table}\n\t\t\t\n\t\t\tIn any case, $\\min(X_1, Y_1) \\preceq \\min(X_2, Y_2)$ and $\\max(X_1, Y_1) \\preceq \\max(X_2, Y_2)$;  \n\t\t\t\n\t\t\t\\item finally, for every $X \\in \\mathbb{I}([0,\\!1])$, we have: $\\min(X, \\boldsymbol{1}) = \\min(\\boldsymbol{1} , X) =  X$ and $\\max(X, \\boldsymbol{0}) = \\max( \\boldsymbol{0}, X) = X$.   \n\t\t\\end{enumerate}\n\t\\end{flushleft}\n\\end{example}\n\n\\medskip \n\nThe exchange in Definition \\ref{def_intaggregfunction} of standard monotonicity for directional conditionally monotonicity given a vector of degenerated intervals allows us to provide the following concept.\n\n\\begin{definition} [IV-Pre-Aggregation Function] \n\t\\label{def_intpreaggregfunction}\n\tAn IV-pre-aggregation is a funtion $PA : \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ such that  \n\t\\begin{enumerate}[leftmargin=*,label={\\rm($\\mathbf{PA}$-\\arabic*)}]\n\t\t\\item $PA$ is directional conditionally increasing given a vector of degenerated intervals $\\overrightarrow{V}= (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n}) \\in \\mathbb{I}(\\mathbb{R}_{+})^n \\setminus \\{\\vec{\\boldsymbol{0}}\\}$;\n\t\t\\item $PA(\\boldsymbol{0}, \\ldots,\\boldsymbol{0}) = \\boldsymbol{0}$ and $PA(\\boldsymbol{1}, \\ldots,\\boldsymbol{1}) = \\boldsymbol{1}$.\n\t\\end{enumerate}\n\tIf for every $\\overrightarrow{X} = (X_1, \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ it holds $PA(\\overrightarrow{X}) = X_i$, for some $i \\in \\{1, \\ldots, n\\}$, $PA$ is called an internal pre-aggregation function.\n\\end{definition}\n\n\\begin{remark}\n\t\\label{rem_IVAggreg&Preag}\n\tAny IV-aggregation function is an IV-pre-aggregation function. Indeed, if $A : \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is an aggregation function, then \n\t\n\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\\item $A$ is a directional conditionally increasing given the  vector $\\vec{\\boldsymbol{1}}$ IV-function, since $X_i + k \\cdot \\boldsymbol{1} \\succeq X_i$ for $i = 1, \\ldots, n$ implies\n\t\t\\begin{center}\n\t\t\t$A(X_1 + k \\cdot \\boldsymbol{1}, \\ldots, X_n + k \\cdot \\boldsymbol{1}) \\succeq A(X_1, \\ldots, X_n)$,\n\t\t\\end{center}    \n\t\tfor all  $0 < k \\in \\mathbb{R}$ and $(X_1,  \\ldots, X_n) \\in \\mathbb{I}([0,\\!1])^n$ such that $(X_1 + k \\cdot \\boldsymbol{1}, \\ldots, X_n + k \\cdot \\boldsymbol{1}) \\in \\mathbb{I}([0,\\!1])^n$;\n\t\t\n\t\t\\item $A(\\boldsymbol{0}, \\ldots,\\boldsymbol{0}) = \\boldsymbol{0}$ and $A(\\boldsymbol{1}, \\ldots,\\boldsymbol{1}) = \\boldsymbol{1}$.\n\t\\end{enumerate}\t\n\\end{remark}\n\n\\begin{example}\n\t\\label{example_IVAggreg&Preag1} \n\tLet $mode: \\mathbb{I}([0,\\!1])^4 \\to {\\cal P}(\\mathbb{I}([0,\\!1]))$ the function that gives back the value(s) that appear(s) most times in the considered 4-tuple and $\\max: {\\cal P}(\\mathbb{I}([0,\\!1])) \\to  \\mathbb{I}([0,\\!1])$ the function that gives back the maximum value of the set. The composition function $ \\max \\circ \\ \\! mode: \\mathbb{I}([0,\\!1])^4 \\to \\mathbb{I}([0,\\!1])$ is not an aggregation, but it is a pre-aggregation function. Indeed, consider $A,B, C \\in \\mathbb{I}([0,\\!1])$ such that $C \\succ B \\succ A$. Then, \n\t\\begin{align*}\n\t\t(\\max \\circ \\ \\! mode)(A, A, B, B) &= \\max\\{A,B\\} = B\\\\ \n\t\t&\\succ A = \\max\\{A\\}\\\\ \n\t\t&= (\\max \\circ \\ \\! mode)(A, A, B, C). \n\t\\end{align*}\n\tThus, $\\max \\circ \\ \\! mode$ is not increasing. However, for all  $0 < k \\in \\mathbb{R}$ and $(X_1, X_2, X_3, X_4) \\in \\mathbb{I}([0,\\!1])^4$ such that $(X_1 + k \\cdot \\boldsymbol{1}, X_2 + k \\cdot \\boldsymbol{1}, X_3 + k \\cdot \\boldsymbol{1}, X_4 + k \\cdot \\boldsymbol{1}) \\in \\mathbb{I}([0,\\!1])^4$, we have:    \n\t\\begin{align*}\n\t\t(\\max \\circ \\ \\! mode)(X_1 + k \\cdot \\boldsymbol{1}, X_2 + k \\cdot \\boldsymbol{1}, X_3 + k \\cdot \\boldsymbol{1}, X_4 + k \\cdot \\boldsymbol{1}) &= mode(X_1, X_2, X_3, X_4) + k \\cdot \\boldsymbol{1}\\\\\n\t\t& \\succ (\\max \\circ \\ \\! mode)(X_1, X_2, X_3, X_4),\n\t\\end{align*}\n\twhich shows that $\\max \\circ \\ \\! mode$ is directional conditionally increasing given the  vector  $\\vec{\\boldsymbol{1}}$. Moreover, $(\\max \\circ \\ \\! mode)(\\boldsymbol{0}, \\boldsymbol{0}, \\boldsymbol{0}, \\boldsymbol{0}) = \\boldsymbol{0}$ and $(\\max \\circ \\ \\! mode)(\\boldsymbol{1}, \\boldsymbol{1}, \\boldsymbol{1}, \\boldsymbol{1}) = \\boldsymbol{1}$. Therefore, $\\max \\circ \\ \\! mode$ is a pre-aggregation function.\n\\end{example}\n\n\\begin{example} \n\t\\label{example_IVAggreg&Preag2}\t\n\tThe IV-function $F: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$,  given by \\begin{center}\n\t\t$F(X, Y) = \\left\\{\n\t\t\\begin{array}{ccccccc}\n\t\t\tX(\\boldsymbol{1} - X), \\ if \\ Y \\preceq \\boldsymbol{0.8}\\\\\n\t\t\t\\ \\ \\ \\ \\ \\boldsymbol{1}, \\ \\ \\ \\ \\ otherwise\\\\\n\t\t\\end{array}\n\t\t\\right.$ (see Example \\ref{ex:multsentdirincrf}),\n\t\\end{center}\n\tis a pre-aggregation function. Indeed, as seen earlier, $F$ is directional conditionally increasing given any vector $\\overrightarrow{V} = (\\boldsymbol{0}, A)$, where $\\boldsymbol{0} \\prec A \\in \\mathbb{I}(\\mathbb{R})$ is a degenerated interval. Further, $F(\\boldsymbol{0}, \\boldsymbol{0}) = \\boldsymbol{0} \\cdot \\boldsymbol{1} = \\boldsymbol{0}$ and $F(\\boldsymbol{1}, \\boldsymbol{1}) = \\boldsymbol{1}$.\n\\end{example}\n\n\\begin{proposition}\t\n\t\\label{prop_IVAggreg&Preag1}\n\tAn IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ such that $F(\\boldsymbol{0}, \\ldots,\\boldsymbol{0}) = \\boldsymbol{0}$ and $F(\\boldsymbol{1}, \\ldots,\\boldsymbol{1}) = \\boldsymbol{1}$ is a pre-aggregation function if one of the following statements holds. \n\t\n\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\\item $F$ is a conditionally increasing given a constant vector function;\n\t\t\n\t\t\\item $F$ is a directional conditionally increasing given some vector $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n}) \\in \\mathbb{I}(\\mathbb{R}_{-})^n \\setminus \\{\\vec{\\boldsymbol{0}}\\}$ such that each $\\boldsymbol{V_i}$ is a degenerated interval for $i = 1, \\ldots, n$.\n\t\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\t\n\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\n\t\t\\item Straight from Remark \\ref{remark_intweakdirmonotonicity} and Definition \\ref{def_intpreaggregfunction}.    \n\t\t\n\t\t\\item By Proposition \\ref{prop_intrmonotonicity2}, $F$ is $-\\overrightarrow{V}$-increasing, where $-\\overrightarrow{V} = (-\\boldsymbol{V_1}, \\ldots, -\\boldsymbol{V_n}) \\in \\mathbb{I}(\\mathbb{R}_{+})^n \\setminus \\{\\vec{\\boldsymbol{0}}\\}$ and each $-\\boldsymbol{V_i}$ is a degenerated interval for $i = 1, \\ldots, n$. Thus, from Definition \\ref{def_intpreaggregfunction}, $F$ is a pre-aggregation function.\n\t\\end{enumerate}\n\\end{proof}\n\n\\begin{proposition}\n\t\\label{prop_intrmonotonicity3cor}\n\tIf $F_1, \\ldots, F_k: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ are pre-aggregation functions and $A: \\mathbb{I}([0,\\!1])^k \\to \\mathbb{I}([0,\\!1])$ is an aggregation function, then $PA = A(F_1, \\ldots, F_k)$ is a pre-aggregation function.\n\\end{proposition}\n\n\\begin{proof}\n\tLet $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n}) \\in \\mathbb{I}(\\mathbb{R}_{+})^n \\setminus \\{\\vec{\\boldsymbol{0}}\\}$ such that each $\\boldsymbol{V_i}$ is a degenerated interval for $i = 1, \\ldots, n$ the vector for which $F_1, \\ldots, F_k$ are directional conditionally increasing given the  vector $\\overrightarrow{V}$ functions. Since $A$ is increasing, by Proposition \\ref{prop_intrmonotonicity3}, $PA$ is directional conditionally increasing given the vector $\\overrightarrow{V}$. Moreover, \n\t\\begin{align*}\n\t\tPA(\\boldsymbol{0}, \\ldots, \\boldsymbol{0}) &= A(F_1(\\boldsymbol{0}, \\ldots, \\boldsymbol{0}), \\ldots, F_k(\\boldsymbol{0}, \\ldots, \\boldsymbol{0})) = A(\\boldsymbol{0}, \\ldots, \\boldsymbol{0}) = \\boldsymbol{0};\\\\\n\t\tPA(\\boldsymbol{1}, \\ldots, \\boldsymbol{1}) &= A(F_1(\\boldsymbol{1}, \\ldots, \\boldsymbol{1}), \\ldots, F_k(\\boldsymbol{1}, \\ldots, \\boldsymbol{1})) = A(\\boldsymbol{1}, \\ldots, \\boldsymbol{1}) = \\boldsymbol{1}.\n\t\\end{align*}\n\tTherefore, $PA$ is a pre-aggregation function.\n\\end{proof}\n\n\\begin{proposition}\n\t\\label{prop_intrmonotonicity4cor}\n\t$F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is a pre-aggregation function iff its dual, the function $F^{d}: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ given by $F^{d}(X_1, \\ldots, X_n) = \\boldsymbol{1} - F(\\boldsymbol{1} - X_1, \\ldots, \\boldsymbol{1} - X_n)$, is a pre-aggregation function.\n\\end{proposition}\n\n\\begin{proof}\n\tFirst, suppose $F$ a pre-aggregation function and let $\\overrightarrow{V} = (\\boldsymbol{V_1}, \\ldots, \\boldsymbol{V_n}) \\in \\mathbb{I}(\\mathbb{R}_{+})^n \\setminus \\{\\vec{\\boldsymbol{0}}\\}$ such that each $\\boldsymbol{V_i}$ is a degenerated interval for $i = 1, \\ldots, n$ the vector for which $F$ is directional conditionally increasing given the vector $\\overrightarrow{V}$. By Proposition \\ref{prop_intrmonotonicity6}, $F^{d}$ is a $\\overrightarrow{V}$-increasing function. Furthermore,\n\t\\begin{align*}\n\t\tF^{d}(\\boldsymbol{0}, \\ldots, \\boldsymbol{0}) &= \\boldsymbol{1} - F(\\boldsymbol{1}, \\ldots, \\boldsymbol{1}) = \\boldsymbol{1} - \\boldsymbol{1} = \\boldsymbol{0};\\\\\n\t\tF^{d}(\\boldsymbol{1}, \\ldots, \\boldsymbol{1}) &= \\boldsymbol{1} - F(\\boldsymbol{0}, \\ldots, \\boldsymbol{0}) = \\boldsymbol{1} - \\boldsymbol{0} = \\boldsymbol{1}.\n\t\\end{align*}\n\tThus, $F^d$ is a pre-aggregation function. The proof of the reciprocal follows analogously.\n\\end{proof}\n\n\\section{\\label{sec:mon&absthomo}Monotonicities and Abstract Homogeneous  IV-Functions}\n\nLet $G: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ an IV-function and $\\varPhi: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$ an automorphism. An IV function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is said to be \\emph{abstract homogeneous with respect to $G$ and $\\varPhi$}, or just \\emph{$(G, \\varPhi)$-homogeneous}, if for all $\\Lambda, X_1, \\ldots, X_n \\in \\mathbb{I}([0,\\!1])$ we have\n\\begin{center}\n\t$F(G(\\Lambda,X_1), \\ldots,G(\\Lambda,X_n)) = G(\\varPhi(\\Lambda,F(X_1, \\ldots, X_n))$.\n\\end{center}\nIf $\\varPhi$ is the IV-identity function, $F$ is called  $G$-homogeneous instead of $(G, \\varPhi)$-homogeneous.\n\n\\medskip  \n\n\\begin{example}\n\t\\label{example_homarithmeticmean}\n\t\\\n\t\\begin{enumerate}[label=(\\alph*)]\n\t\t\\item The IV-arithmetic mean (see Example \\ref{example_aggregfunctions1}) is abstract homogeneous with respect to $M_2: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$, given by $M_2(X,Y) = \\left[ \\frac{1}{2}(\\underline{X}+\\underline{Y}) , \\frac{1}{2} (\\overline{X}+\\overline{Y})\\right]$.\n\t\t\n\t\t\\item $F_{min}$, $F_{max}$, $\\min$ and $\\max$ are $P$-homogeneous. \n\t\\end{enumerate}\n\\end{example}\n\n\\begin{definition}[Idempotent IV-function]\t\n\t\\label{def_idempotency}\n\tAn IV-function $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is said to be idempotent, if \n\t\\begin{center}\n\t\t$F(X, \\ldots, X) = X$\n\t\\end{center}  \n\tfor every $X \\in \\mathbb{I}([0,\\!1])$.\n\\end{definition}\n\n\\begin{example}\n\t\\label{example_idempotency}\n\tExamples of idempotent functions are the IV-functions: the minimum and the maximum (Example \\ref{example_intmonotonicity}); $F_{min}$ and $ F_{max} $ (Example \\ref{example_FminFmax}); and the arithmetic mean (Example \\ref{example_aggregfunctions1}).\n\\end{example}\n\n\\begin{proposition}[\\cite{MonteiroAbstHomogIV2022}]\n\t\\label{prop_homidempfunction}\n\tLet $G: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ be an IV-function. If $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is a  $G$-homogeneous IV-function and there exists $A \\in \\mathbb{I}([0,\\!1])$ such that $F(A, \\ldots, A) = A$ and $G_{A}: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$, given by $G_{A}(X) = G(X,A)$,  is a bijection, then $F$ is idempotent.\n\\end{proposition}\n\n\\begin{proposition}\n\t\\label{prop_Gweakincfunction2}\n\tLet $\\Lambda \\in \\mathbb{I}([0,\\!1])$ and  $\\mathbb{S}: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ an IV-$t$-conorm. If $F: \\mathbb{I}([0,\\!1])^n \\to \\mathbb{I}([0,\\!1])$ is $\\mathbb{S}$-homogeneous, then it is conditionally increasing with respect to $ \\mathbb{S} $ and $\\Lambda$.\n\\end{proposition}\n\\begin{proof}\n\tSince $\\mathbb{S}$ is increasing, for all $X,Y \\in \\mathbb{I}([0,\\!1])$, we have\n\t\\begin{center}\n\t\t$\\mathbb{S}(X,Y) \\succeq \\mathbb{S}(\\boldsymbol{0},Y) = Y$.\n\t\\end{center}\n\tThus, since $F$ is $\\mathbb{S}$-homogeneous, it follows that   \n\t\\begin{center}\n\t\t$F(\\mathbb{S}(\\Lambda,X_1), \\ldots, \\mathbb{S}(\\Lambda,X_n)) = \\mathbb{S}(\\Lambda, F(X_1, \\ldots, X_n)) \\succeq F(X_1, \\ldots, X_n)$,\n\t\\end{center}  \n\tfor all $\\Lambda, X_1, \\ldots, X_n \\in \\mathbb{I}([0,\\!1])$.\n\\end{proof}\n\n\\medskip \n\nIn \\cite{MonteiroAbstHomogIV2022}, Monteiro et al., considering admissible orders  on $\\mathbb{I}([0,\\!1])$, showed that the only idempotent IV-$ t $-norm is the IV-minimum and the only idempotent IV-$ t $-conorm is the maximum: \n\n\\medskip \n\n\\begin{proposition}  \n\t\\label{prop_uninormsidempotent}\t\n\tThe following holds:\n\t\n\t\\begin{description}\n\t\t\\item[(i)] $\\mathbb{T}$ is an IV idempotent t-norm iff $\\mathbb{T} = \\min$; \n\t\t\\item[(ii)] $\\mathbb{S}$ is an IV idempotent t-conorm iff $\\mathbb{S} = \\max$.\n\t\\end{description}\n\\end{proposition}\n\n\\begin{corollary} \n\t\\label{cor_uninormprodhomog}\t\n\tThe following holds:\n\t\n\t\\begin{description}\n\t\t\\item[(i)] $\\mathbb{T}$ is an IV-$t$-norm $P$-homogeneous iff $\\mathbb{T} = \\min$; \n\t\t\\item[(ii)] $\\mathbb{S}$ is an IV idempotent t-conorm $P$-homogeneous iff $\\mathbb{S} = \\max$.\n\t\\end{description}\n\\end{corollary}\n\n\\begin{proof}\n\tFrom Example \\ref{example_homarithmeticmean}, $\\min$ and $\\max$ are $P$-homogeneous functions. Now, suppose $\\mathbb{T}$ an IV-$t$-norm and $\\mathbb{S}$ an IV-$t$-conorm, both $P$-homogeneous. By Remark \\ref{rem_inttnorm&conorm}, we have: \n\t\\begin{center}\n\t\t$\\mathbb{T}(\\boldsymbol{1},\\boldsymbol{1}) = \\boldsymbol{1}$ and $\\mathbb{S}(\\boldsymbol{1},\\boldsymbol{1}) = \\boldsymbol{1}$.\n\t\\end{center}\n\tFurther, the function $P_{\\boldsymbol{1}}: \\mathbb{I}([0,\\!1]) \\to \\mathbb{I}([0,\\!1])$, given by $P_{\\boldsymbol{1}}(X) = P(X,\\boldsymbol{1})$, is the identity IV-function. Then, by Proposition  \\ref{prop_homidempfunction}, $\\mathbb{T}$ and $\\mathbb{S}$ are idempotents, which implies, from Proposition   \\ref{prop_uninormsidempotent}, that $\\mathbb{T} = \\min$ and $\\mathbb{S} = \\max$.  \n\\end{proof}\n\n\\medskip \n\nFrom Example \\ref{example_homarithmeticmean}, we know that the IV-arithmetic mean, $M(X,Y) = \\left[ \\frac{1}{2}(\\underline{X}+\\underline{Y}) , \\frac{1}{2} (\\overline{X}+\\overline{Y})\\right]$, is abstract homogeneous with respect to itself. In this case, we say that $M$ is self-homogeneous. \n\n\\begin{definition} [IV Abstract Self -Homogeneous Function]\n\t\\label{def_intselfhomog}\n\tA function $F: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ is called an IV  self-homogeneous function if it is abstract homogeneous with respect to itself, i.e.\n\t\\begin{equation}\n\t\t\\label{eq_intselfhomog}\n\t\tF( F(\\Lambda,X), F(\\Lambda, Y)) = F(\\Lambda, F(X,Y))\n\t\\end{equation}\n\tfor all $\\Lambda,X,Y \\in \\mathbb{I}([0,\\!1])$.\n\\end{definition}\n\nThe projections $P_1(X,Y) = X$ and $P_2(X,Y) = Y$ are examples of self-homogeneous functions.\n\n\\begin{proposition}\n\t\\label{prop_intselfhomog3}\n\tIf $F: \\mathbb{I}([0,\\!1])^2 \\to \\mathbb{I}([0,\\!1])$ is self-homogeneous, then\n\t\\begin{equation}\n\t\t\\label{eq_intselfhomog2}\n\t\tF( F(X,X), F(X,Y)) = F(X, F(X,Y))\n\t\\end{equation}\n\tfor all $X,Y \\in \\mathbb{I}([0,\\!1])$. In particular, if $F$ is injective, so it is idempotent. \n\\end{proposition}\n\n\\begin{proof}\n\tTaking $\\Lambda = X$ in \\eqref{eq_intselfhomog}, we obtain \\eqref{eq_intselfhomog2}. Further, if $F$ is injective, we will have $F(X,X) = X$ for every $X \\in \\mathbb{I}([0,\\!1])$.    \n\\end{proof}\n\n\n\\section{\\label{sec:Final}Final Considerations}\n\nIn this paper we have developed the notions of conditional monotonicity given a transformation, conditional monotonicity given a vector of degenerated intervals, \nconditional monotonicity with respect to $ G $ and a parameter $\\Lambda\\in \\mathbb{I}([0,\\!1])$. The definition of conditional monotonicity given a vector of degenerated intervals  enabled $ \\mathbb{I}([0,\\!1]) $ to be endowed with pre-aggregation IV-functions in this context. Further investigations on the classes of monotonicities introduced deserve attention. The generalization of conditional monotonicity with respect to $ G $ and a parameter to $ \\mathcal{F} $-monotonicity and its respective applications is a step forward.\n\n\\section*{Acknowledgments}\n\\addcontentsline{toc}{section}{Acknowledgments}\nThis work has been supported by: the National Council\nfor Scientific and Technological Development (CNPq-Brazil) within the Project 312899\/2021-1 and 403167\/2022-1; the Spanish research projects  PID2019-108392GBI00(3031138640\/AEI\/10.13039\/501100011033) and APVV-18-0052 and by the Slovak Research and Development Agency under the contract No. APVV-20-0069.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\n\nInterest rate markets are a large and important part of global financial markets.\nThe figures published by the Bank for International Settlements (BIS) show that\ninterest rate derivatives represent more than 60\\% of the over-the-counter\nmarkets over the years, in terms of notional amount; cf. Table \\ref{BIS}. Hence,\nit is important to have models that can adequately describe the dynamics and\nmechanics of interest rates.\n\nThere is a notable difference between interest rate markets and stock or foreign\nexchange (FX) markets. While in the latter there is a single underlying to be\nmodeled, the stock price or the FX rate, in interest rate markets there is a\nwhole \\emph{family} of underlyings to be modeled, indexed by the time of maturity.\nThis poses unique challenges for researchers in mathematical finance and has led\nto some fascinating developments.\n\nThe initial approaches to interest rate modeling considered short rates or\ninstantaneous forward rates as modeling objects, and then deduced from them\ntradable rates. More recently, \\textit{effective rates}, i.e. tradable market\nrates such as the LIBOR or swap rate, were modeled directly. Models for\neffective rates consider only a discrete set of maturity dates, the so-called\n\\textit{tenor structure}, which consists of the dates when these rates are\nfixed. A review of the different approaches to modeling interest rates is beyond\nthe scope of the present article. There are many excellent books available,\nfocusing on the theoretical and practical aspects of interest rate theory. We\nrefer the reader e.g. to \\citeN{Bjoerk04}, \\citeN{MusielaRutkowski97},\n\\citeN{Filipovic09}, or \\citeN{BrigoMercurio06}.\n\nThe aim of this article is to review the construction and basic properties of\nmodels for LIBOR\\xspace rates. We consider the following popular approaches: LIBOR market models\\xspace,\nforward price models\\xspace and Markov-functional models\\xspace, as well as the recently developed class of affine LIBOR models\\xspace. In section\n\\ref{axioms} we will present and discuss some basic requirements that models\nfor LIBOR rates should satisfy. These are briefly: \\textit{positivity} of LIBOR\\xspace\nrates, \\textit{arbitrage freeness} and \\textit{analytical tractability}.\n\nThere are two natural starting points for modeling LIBOR rates: the rate itself\nand the forward price. Although they differ only by an additive and a\nmultiplicative constant, cf. \\eqref{basic}, the model dynamics are noticeably\ndifferent, depending on whether the model is based on the LIBOR or the forward\nprice. In addition, the consequences from the point of view of econometrics are\nalso significant.\n\nModeling LIBOR rates directly, leads to positive rates and arbitrage-free\ndynamics, but the model is not analytically tractable. On the other hand, models\nfor the forward price are analytically tractable, but LIBOR\\xspace rates can become\nnegative. The only models that can respect all properties simultaneously are\nMarkov-functional models\\xspace and affine LIBOR models\\xspace.\n\nThe article is organized as follows: in section \\ref{markets} we introduce some\nbasic notation for interest rates and in section \\ref{axioms} we describe the\nbasic requirements for LIBOR\\xspace models. In section \\ref{lmm} we review the\nconstruction of LIBOR market models\\xspace, describe its shortcomings and discuss some approximation\nmethods developed to overcome them. In section \\ref{fpm} we review forward price models\\xspace and in\nsection \\ref{mfm} we discuss Markov-functional models\\xspace. Finally, in section \\ref{alm} we present\naffine LIBOR models\\xspace and in section \\ref{extend} we outline the extensions of LIBOR models to\nthe multi-currency and default risk settings.\n\n\\begin{table}\n \\begin{center}\n  {\\renewcommand{\\arraystretch}{1.05}\n  \\begin{tabular}{lrrrr}\n                       & Dec 2006  & Dec 2007 & Jun 2008 & Dec 2008\\\\\n\\hline\n  Foreign exchange     &   40,271  &  56,238  &  62,983  &  49,753\\\\\n  Interest rate        &  291,581  & 393,138  & 458,304  & 418,678\\\\\n  Equity-linked        &    7,488  &   8,469  &  10,177  &   6,494\\\\\n  Commodity            &    7,115  &   8,455  &  13,229  &   4,427\\\\\n  Credit default swaps &   28,650  &  57,894  &  57,325  &  41,868\\\\\n  Unallocated          &   43,026  &  71,146  &  81,708  &  70,742\\\\\n\\hline\n  Total                &  418,131  & 595,341  & 683,726 & 591,963\\\\\n   \\end{tabular}}~\\\\[1ex]\n  \\caption{Amounts outstanding of over-the-counter (OTC) derivatives\n           by risk category and instrument (in billions of US dollars).\n           Source: BIS Quarterly Review, September 2009.}\n  \\label{BIS}\n  \\end{center}\n\\end{table}\n\n\n\n\\section{Interest rate markets -- notation}\n\\label{markets}\n\nLet us consider a discrete tenor structure $0=T_0<T_1<\\dots<T_N$, with constant\ntenor length $\\delta=\\tk[k+1]-T_k$. The following notation is introduced for\nconvenience; $K:=\\{1,\\dots,N-1\\}$ and $\\overline{K}:=\\{1,\\dots,N\\}$. Let us denote\nby $B(t,T)$ the time-$t$ price of a \\textit{zero coupon bond} maturing at time\n$T$; by $L(t,T)$ the time-$t$ \\textit{forward LIBOR\\xspace rate} settled at time $T$\nand received at time $T+\\delta$; and by $F(t,T,U)$ the time-$t$ \\textit{forward\nprice} associated to the dates $T$ and $U$. These fundamental quantities are\nrelated by the following basic equation:\n\\begin{align}\\label{basic}\n  1+\\delta L(t,T) = \\frac{B(t,T)}{B(t,T+\\delta)} = F(t,T,T+\\delta).\n\\end{align}\n\nThroughout this work, $\\mathscr{B}=(\\Omega,\\ensuremath{\\mathcal{F}},\\mathbf{F},\\ensuremath{\\mathrm{I\\kern-.2em P}})$ denotes a complete\nstochastic basis, where $\\ensuremath{\\mathcal{F}}=\\ensuremath{\\mathcal{F}}_T$, $\\mathbf{F}=(\\ensuremath{\\mathcal{F}}_t)_{t\\in[0,T]}$, and\n$T_N\\le T<\\infty$. We denote by $\\mathcal{M}(\\ensuremath{\\mathrm{I\\kern-.2em P}})$ the class of martingales on\n$\\mathscr{B}$ with respect to the measure $\\ensuremath{\\mathrm{I\\kern-.2em P}}$.\n\nWe associate to each date $T_k$ in the tenor structure a \\emph{forward martingale\nmeasure}, denoted by $\\pt[k]$, $k\\in\\overline{K}$. By the definition of forward measures,\ncf. \\citeN[Def. 14.1.1]{MusielaRutkowski97}, the bond price with maturity $T_k$\nis the numeraire for the forward measure $\\pt[k]$. Thus, we have that forward\nmeasures are related to each other via\n\\begin{align}\\label{Pk-to-next}\n\\frac{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k]}}{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k+1]}} \\Big|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\frac{F(t,\\tk[k],\\tk[k+1])}{F(0,\\tk[k],\\tk[k+1])}\n  = \\frac{B(0,\\tk[k+1])}{B(0,\\tk[k])}\n    \\times \\frac{B(t,\\tk[k])}{B(t,\\tk[k+1])},\n\\end{align}\nwhile they are related to the terminal forward measure via\n\\begin{align}\\label{Pk-to-final}\n\\frac{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k]}}{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[N]}} \\Big|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\frac{F(t,\\tk[k],\\tk[N])}{F(0,\\tk[k],\\tk[N])}\n  = \\frac{B(0,\\tk[N])}{B(0,\\tk[k])}\n    \\times \\frac{B(t,\\tk[k])}{B(t,\\tk[N])}.\n\\end{align}\nAll forward measures are assumed to be equivalent to the measure $\\ensuremath{\\mathrm{I\\kern-.2em P}}$.\n\n\n\n\\section{Axioms for LIBOR\\xspace models}\n\\label{axioms}\n\nIn this section, we present and discuss certain requirements that a model for\nLIBOR\\xspace rates should satisfy. These requirements are motivated both by the economic\nand financial aspects of LIBOR\\xspace rates, as well as by the practical demands\nfor implementing and using a model in practice. The aim here is to unify the\nline of thought in \\shortciteN{HuntKennedyPelsser00} and in \n\\shortciteANP{KellerResselPapapantoleonTeichmann09}\n\\citeyear{KellerResselPapapantoleonTeichmann09}.\n\nA model for LIBOR\\xspace rates should satisfy the following \\emph{requirements}:\n\\begin{itemize}\n\\item[(\\textbf{A1})] LIBOR\\xspace rates should be \\emph{non-negative}:\n            $L(t,T)\\ge0$ for all $t\\in[0,T]$.\n\\item[(\\textbf{A2})] The model should be \\emph{arbitrage-free}:\n            $L(\\cdot,T)\\in\\mathcal{M}(\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T+\\delta})$.\n\\item[(\\textbf{A3})] The model should be \\emph{analytically tractable},\n            easy to implement and quick to calibrate to market data.\n\\item[(\\textbf{A4})] The model should provide a \\emph{good calibration} to\n            market data of liquid derivatives, i.e. caps and swaptions.\n\\end{itemize}\n\nRequirements (A1) and (A2) are logical conditions originating from economics and\nmathematical finance. Below, we briefly elaborate on (A3) and (A4); they stem\nfrom practical demands and are more difficult to quantify precisely. In order to\nclarify their difference, we point out that e.g. the Black--Scholes model\nobviously satisfies (A3) but not (A4).\n\nRequirement (A3) means that we can price liquid derivatives, e.g. caps and\nswaptions in ``closed form'' in the model, so that the model can be calibrated\nto market data in a fast and easy manner. Ideally, of course we would like to\nbe able to price as many derivatives as possible in closed form. Here, ``closed\nform'' is understood in a broad sense meaning e.g. Fourier transform methods;\nreally closed form solutions \\'a la Black--Scholes are typically hard to achieve.\n\\shortciteN{HuntKennedyPelsser00} say that the model is analytically tractable\nif it is driven by a \\emph{low-dimensional} Markov process. In \\shortciteN{KellerResselPapapantoleonTeichmann09}, as well as\nin the present article, we say that a model is analytically tractable if the\nstructure of the driving process is \\emph{preserved} under the different forward\nmeasures.\n\nFinally, requirement (A4) means that the model is able to describe the\nobserved data accurately, without overfitting them. We will not examine this\nrequirement further in this article. On an intuitive level, since the models we\nwill describe in the sequel are driven by general Markov processes or general\nsemimartingales, we can always find a driving process that provides a good\ncalibration to market data. However, an empirical analysis should be performed\nin order to identify such a driving process (cf. e.g. \\shortciteNP{JarrowLiZhao07}\nand \\citeNP[Ch. III]{Skovmand08}).\n\n\n\n\\section{LIBOR market models}\n\\label{lmm}\n\nLIBOR market models\\xspace were introduced in the seminal papers of\nMiltersen et al. \\citeyear{MiltersenSandmannSondermann97},\n\\shortciteN{BraceGatarekMusiela97}\nand \\citeN{Jamshidian97}. In this framework, LIBOR\\xspace rates are modeled as the exponential\nof a Brownian motion under their corresponding forward measure, hence they are\nlog-normally distributed. This is the so-called \\emph{log-normal LIBOR market model}.\nCaplets are then priced by Black's formula (cf. \\citeNP{Black76}), which is in\naccordance with standard market practice. Later, LIBOR market models\\xspace were extended to\naccommodate more general driving processes such as L\\'{e}vy\\xspace processes, stochastic volatility\nprocesses and general semimartingales, in order to describe more accurately the market\ndata; cf. \\citeN{Jamshidian99}, Glasserman and Kou \\citeyear{GlassermanKou03},\n\\citeN{EberleinOezkan05}, \\citeANP{AndersenBrothertonRatcliffe05}\n\\citeyear{AndersenBrothertonRatcliffe05}, \\citeN{BelomestnySchoenmakers06} and\n\\shortciteN{BelomestnyMathewSchoenmakers06}, to mention just a fraction of the existing\nliterature.\n\nConsider an initial tenor structure of non-negative LIBOR\\xspace rates $L(0,T_k)$, $k\\in\\overline{K}$,\nand let $\\lambda(\\cdot,T_k):[0,T_k]\\to\\ensuremath{\\mathbb{R}}$ denote the volatility of the forward LIBOR\\xspace\nrate $L(\\cdot,T_k)$, $k\\in K$; the volatilities are assumed deterministic, for\nsimplicity. Let $H$ denote a semimartingale on $(\\Omega,\\ensuremath{\\mathcal{F}},\\mathbf{F},\\pt)$ with triplet\nof semimartingale characteristics $\\mathbb{T}(H|\\pt)=(B,C,\\nu)$ and $H_0=0$ a.s.;\n$H$ satisfies certain integrability assumptions which are suppressed for brevity\n(e.g. finite exponential moments, absolutely continuous characteristics). The\nprocess $H$ is driving the dynamics of LIBOR\\xspace rates and is chosen to have a\n\\emph{tractable} structure under $\\pt$ (e.g. $H$ is a L\\'{e}vy\\xspace or an affine process).\n\nIn LIBOR market models\\xspace, forward LIBOR\\xspace rates are modeled as follows:\n\\begin{align}\\label{LMM-1}\nL(t,T_k) &= L(0,T_k)\n \\exp\\left( \\int_0^t \\beta(s,T_k)\\ud s + \\int_0^t \\lambda(s,T_k)\\ensuremath{\\mathrm{d}} H_s \\right),\n\\end{align}\nwhere $\\beta(\\cdot,T_k)$ is the drift term that makes\n$L(\\cdot,T_k)\\in\\mathcal{M}(\\pt[k+1])$, for all $k\\in K$. Therefore, the model\nclearly satisfies requirements (A1) and (A2).\n\nNow, using Theorem 2.18 in \\citeN{KallsenShiryaev02}, we have that\n\\begin{align}\\label{LMM-2}\n\\beta(s,T_k)\n &= -\\lambda(s,T_k)b_s^{\\tk[k+1]}\n    -\\frac12\\lambda^2(s,T_k)c_s \\nonumber\\\\\n &\\quad\n    -\\int_{\\ensuremath{\\mathbb{R}}}\\big( \\mathrm{e}^{\\lambda(s,T_k)x}\n                     - 1 - \\lambda(s,T_k)x \\big) F_s^{\\tk[k+1]}(\\ud x),\n\\end{align}\nsuch that indeed $L(\\cdot,T_k)\\in\\mathcal{M}(\\pt[k+1])$. Here\n$(b_s^{\\tk[k+1]}, c_s, F_s^{\\tk[k+1]})$ denote the differential characteristics\nof $H$ under $\\pt[k+1]$.\nTherefore, in order to completely understand the dynamics of the model we have\nto calculate the characteristics $(b_s^{\\tk[k+1]}, c_s, F_s^{\\tk[k+1]})$.\n\nThese characteristics follow readily from Girsanov's theorem for semimartingales\n(cf. \\citeNP[III.3.24]{JacodShiryaev03}) once we have the density between the\nmeasure changes at hand. It is convenient to express this density as a stochastic\nexponential. Keeping \\eqref{Pk-to-final} in mind, and denoting \\eqref{LMM-1} as\nfollows\n\\begin{align}\\label{LMM-3}\n\\ensuremath{\\mathrm{d}} L(t,T_k) = L(t-,T_k)\\ensuremath{\\mathrm{d}} \\widetilde{H}_t^k,\n\\end{align}\ni.e. $\\widetilde{H}^k$ is the exponential transform of the exponent in\n\\eqref{LMM-1}, we get from \\eqref{basic} that\n\\begin{align}\\label{LIBOR-3}\n\\ensuremath{\\mathrm{d}} F(t,T_k,T_{k+1})\n &= \\delta\\, \\ensuremath{\\mathrm{d}} L(t,T_k) \\nonumber\\\\\n &= \\delta L(t-,T_k)\\ensuremath{\\mathrm{d}} \\widetilde{H}_t^k \\nonumber\\\\\n &= F(t-,T_k,T_{k+1}) \\frac{\\delta L(t-,T_k)}{1+\\delta L(t-,T_k)}\n    \\ensuremath{\\mathrm{d}} \\widetilde{H}_t^k \\nonumber\\\\\n \\Rightarrow\nF(t,T_k,\\tk[k+1])\n &= F(0,T_k,\\tk[k+1])\\,\n    \\mathcal{E}\\!\\left( \\int_0^\\cdot\\frac{\\delta L(s-,T_k)}{1+\\delta L(s-,T_k)}\n                                  \\ensuremath{\\mathrm{d}} \\widetilde{H}_s^k \\right)_{\\!t}.\n\\end{align}\nTherefore, the density between the measure changes takes the form\n\\begin{align}\\label{LMM-4}\n\\frac{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k+1]}}{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[N]}} \\Big|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\frac{B(0,\\tk[N])}{B(0,\\tk[k+1])}\n    \\times \\prod_{l=k+1}^{N-1}\n     \\mathcal{E}\\!\\left( \\int_0^\\cdot\n       \\frac{\\delta L(s-,T_l)}{1+\\delta L(s-,T_l)}\\ensuremath{\\mathrm{d}} \\widetilde{H}_s^l \\right)_{\\!t}.\n\\end{align}\n\nThis calculation reveals the problem of LIBOR market models\\xspace: the density process between the\nmeasure changes -- and thus the characteristics of $H$ under the forward measures\n-- does not depend \\emph{only} on the dynamics of $\\widetilde{H}^k$, or\nequivalently on the dynamics of $\\int\\lambda(s,T_k)\\ensuremath{\\mathrm{d}} H_s$, as is the case in e.g.\nHJM models. It also crucially depends on \\emph{all subsequent} LIBOR\\xspace rates, as the\nproduct and the terms $\\frac{\\delta L(\\cdot,T_l)}{1+\\delta L(\\cdot,T_l)}$ in\n\\eqref{LMM-4} clearly indicate. This means, in particular, that the structure of\nthe model is \\emph{not preserved} under the different forward measures; e.g. if\n$H$ is a L\\'{e}vy\\xspace or an affine process under the terminal measure $\\pt$, then $H$ is\nneither a L\\'{e}vy\\xspace nor an affine process under any other forward measure -- not even\na time-inhomogeneous version of those. Therefore, LIBOR market models\\xspace do not satisfy\nrequirement (A3).\n\nThe semimartingale $H$, that drives the dynamics of LIBOR rates, has the following\ncanonical decomposition under the terminal martingale measure $\\pt$\n\\begin{align}\\label{LMM-4.5}\n H_t = B_t + \\int_0^t \\sqrt{c_s}\\ensuremath{\\mathrm{d}} W_s + \\int_0^t\\int_\\ensuremath{\\mathbb{R}} x(\\mu^H-\\nu)\\ensuremath{(\\ud s, \\ud x)},\n\\end{align}\n(cf. \\citeNP[II.2.38]{JacodShiryaev03} and \\citeNP[Theorem 3.4.2]{KaratzasShreve91})\nwhere $W$ denotes the $\\pt$-Brownian motion and $\\mu^H$ denotes the random measure\nof the jumps of $H$. The $\\pt$-compensator of $\\mu^H$ is $\\nu$ and\n$C=\\int_0^\\cdot c_s\\ud s$. Straightforward calculations using the density in\n\\eqref{LMM-4} \\linebreak (cf. e.g. \\citeNP{Kluge05} or \\citeNP{PapapantoleonSiopacha09})\nyield that the $\\pt[k+1]$-Brownian motion $W^{T_{k+1}}$ is related to the\n$\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_N}$-Brownian motion via\n\\begin{align}\\label{LMM-5}\nW_t^{T_{k+1}}\n & = W_t - \\int_0^t \\left(\\sum_{l=k+1}^{N-1}\n            \\frac{\\delta L(t-,T_l)}\n                 {1+\\delta L(t-,T_l)}\\lambda(t,T_l)\\right)\\sqrt{c_s} \\ensuremath{\\mathrm{d}} s,\n\\end{align}\nwhile the $\\pt[k+1]$-compensator of $\\mu^H$, $\\nu^{T_{k+1}}$, is related to the\n$\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_N}$-compensator of $\\mu^H$ via\n\\begin{align}\\label{LMM-6}\n\\nu^{T_{k+1}}\\ensuremath{(\\ud s, \\ud x)}\n &= \\left(\\prod_{l=k+1}^{N-1}\\gamma(s,x,T_l)\\right)\\nu\\ensuremath{(\\ud s, \\ud x)},\n\\end{align}\nwhere\n\\begin{align}\\label{LMM-7}\n\\gamma(s,x,T_l,)\n = \\frac{\\delta L(s-,T_l)}{1+\\delta L(s-,T_l)}\\Big(\\mathrm{e}^{\\lambda(s,T_l)x}-1\\Big) +1.\n\\end{align}\nIn addition, the drift term of the LIBOR\\xspace rate $L(\\cdot,T_k)$ relative to the\n$\\pt$ differential characteristics of $H$, i.e. $(b,c,F)$, is\n\\begin{align}\\label{LMM-8}\n\\widehat{\\beta}(s,\\ts[k])\n & = -\\frac12 \\lambda^2(s,T_k) c_s\n     - c_s \\lambda(s,T_k)\n        \\sum_{l=k+1}^{N-1}\\frac{\\delta L(s-,T_l)}{1+\\delta L(s-,T_l)}\\lambda(s,T_l)\n  \\nonumber\\\\ &\\,\\,\\,\\,\n     - \\int_{\\ensuremath{\\mathbb{R}}}\\left(\\Big(\\mathrm{e}^{\\lambda(s,T_k) x}-1\\Big)\n          \\prod_{l=k+1}^{N-1}\\gamma(s,x,T_l) - \\lambda(s,T_k) x\\right)F_s(\\ensuremath{\\mathrm{d}} x).\n\\end{align}\n\n\nThe consequences of the intractability of LIBOR market models\\xspace are the following. When the\ndriving process is a \\emph{continuous} semimartingale, then\n\\begin{itemize}\n\\item caplets can be priced in closed form;\n\\item swaptions and other multi-LIBOR products cannot be priced in closed form;\n\\item Monte-Carlo simulations are particularly time consuming, since one is\n      dealing with coupled high dimensional stochastic processes.\n\\end{itemize}\nWhen the driving process is a \\emph{general} semimartingale, then\n\\begin{itemize}\n\\item even caplets cannot be priced in closed form, let alone swaptions\n      or other multi-LIBOR\\xspace derivatives;\n\\item the Monte-Carlo simulations are equally time consuming.\n\\end{itemize}\n\nSeveral approximation methods have been developed in the literature in order to\novercome these problems. We briefly review three of the proposed methods below;\nfor other methods and empirical comparison we refer the interested reader to\nthe review paper by \\citeN{JoshiStacey08}.\n\n\n\\subsection{``Frozen drift'' approximation}\nThe first and easiest solution to the problem is the so-called ``frozen drift''\napproximation, where one replaces the random terms in \\eqref{LMM-8} or\n\\eqref{LMM-4} by their deterministic initial values, i.e.\n\\begin{align}\\label{LMM-FD1}\n\\frac{\\delta L(s-,T_l)}{1+\\delta L(s-,T_l)}\n \\approx \\frac{\\delta L(0,T_l)}{1+\\delta L(0,T_l)}.\n\\end{align}\nThis approximation was first proposed by \\shortciteN{BraceGatarekMusiela97}, and\nhas been used by many authors ever since. Under this approximation, the\nmeasure change becomes a structure preserving one -- observe that the density in\n\\eqref{LMM-4} depends now only on the driving process and the volatility\nstructure -- and the resulting \\emph{approximate} LIBOR market model\\xspace is analytically tractable;\ne.g. caplets and swaptions can now be priced in closed form even in models\ndriven by semimartingales with jumps.\n\nHowever, this method yields poor empirical results. Comparing the prices of\neither liquid options, or long-dated options, using the frozen drift approximation\nto the prices obtained by the simulation of the actual dynamics for the LIBOR rates,\nwe can observe notable differences both in terms of prices and in terms of\nimplied volatilities. See Figure \\ref{diffs} for an example. We refer to\n\\shortciteN{KurbanmuradovSabelfeldSchoenmakers}, \\citeN{SiopachaTeichmann07},\nand \\citeN{PapapantoleonSiopacha09} for further numerical illustrations.\n\n\\begin{figure}\n \\centering\n  \\includegraphics[width=177pt]{IV-SDE-FD}\n  \\includegraphics[width=177pt]{IV-SDE-ST}\n \\caption{Difference in implied volatilities between the actual LIBOR and the\n          frozen drift prices (left), and between the actual LIBOR and the\n          Taylor approximation prices (right). Source: Papapantoleon\n          and Siopacha (2009).}\n \\label{diffs}\n\\end{figure}\n\n\n\\subsection{Log-normal approximations}\nThe following approximation schemes for the log-normal LIBOR market model\\xspace were developed by\nKurbanmuradov et al.\n\\citeyear{KurbanmuradovSabelfeldSchoenmakers}. Consider the log-normal LIBOR market model\\xspace\ndriven by a one-dimensional Brownian motion, for simplicity. The dynamics of\nLIBOR rates (expressed under the terminal measure) take the form\n\\begin{align}\\label{LMM-LG-1}\n\\ensuremath{\\mathrm{d}} L(t,T_k)\n = L(t,T_k)\\big( \\lambda_t(T_k)\\ensuremath{\\mathrm{d}} W_t + \\beta_t(T_k)\\ud t \\big),\n\\end{align}\nwhere the drift term equals\n\\begin{align}\\label{LMM-LG-1+1\/2}\n\\beta_t(T_k)\n &= -\\lambda_t(T_k) \\sum_{l=k+1}^{N-1}\n     \\frac{\\delta L(t-,T_l)}{1+\\delta L(t-,T_l)}\\lambda_t(T_l),\n\\end{align}\ncf. \\eqref{LMM-8}; w.l.o.g. we can set $c\\equiv1$. A very crude log-normal\napproximation is to ``neglect'' the non-Gaussian terms in the SDE, i.e. to set\n$\\beta_t(\\ts[k])\\equiv0$. Of course, this approximation does not yield\nsatisfactory results -- in principle, results are even worse than the frozen\ndrift approximation.\n\nOne can develop more refined log-normal approximations as follows: let\n$f(x)=\\frac{\\delta x}{1+\\delta x}$, and define the process $Z$, which equals the\nterms that need to be approximated; i.e.\n\\begin{align}\\label{LMM-LG-2}\nZ(t,T_k)\n = \\frac{\\delta L(t,T_k) }{1+\\delta L(t,T_k)}\n =f\\big( L(t,T_k) \\big).\n\\end{align}\nApplying It\\^o's formula, we derive the SDE that $Z(\\cdot,T_k)$ satisfies\n\\begin{align}\\label{LMM-LG-3}\n\\ensuremath{\\mathrm{d}} Z(t,T_k)\n &= f'(L(t,T_k))L(t,T_k)\\lambda_t(T_k)\\ensuremath{\\mathrm{d}} W_t \\nonumber\\\\\n &\\,\\,+ \\big\\{f'(L(t,T_k))L(t,T_k)\\beta_t(T_k)\n  + \\frac12 f''(L(t,T_k))L^2(t,T_k)\\lambda^2_t(T_k)\\big\\}\\ud t \\nonumber\\\\\n &=: A_k(t,Z)\\ud t + B_k(t,Z)\\ensuremath{\\mathrm{d}} W_t,\n\\end{align}\nwith the initial condition $Z(0,T_k)=f(L(0,T_k))$. Note that the coefficients\n$A_k$ and $B_k$ can be calculated explicitly, by solving \\eqref{LMM-LG-2} for\n$L$ and substituting into \\eqref{LMM-LG-3}. Moreover, $Z$ in $A_k$ and $B_k$ denotes\nthe dependence on the whole vector $Z=[Z(\\cdot,T_1),\\dots,Z(\\cdot,T_N)]$. The\nfirst and second \\emph{Picard iterations} for the solution of this SDE are\n\\begin{align}\\label{LMM-LG-4}\nZ^0(t,T_k)\n &= Z(0,T_k)\n  = \\frac{\\delta L(0,T_k) }{1+\\delta L(0,T_k)},\n\\end{align}\nand\n\\begin{align}\\label{LMM-LG-5}\nZ^1(t,T_k)\n &= Z(0,T_k) + \\int_0^t A_k(s,Z^0)\\ud s + \\int_0^t B_k(s,Z^0)\\ensuremath{\\mathrm{d}} W_s.\n\\end{align}\nNote that $Z^0$ is constant, while $Z^1(t,T_k)$ has a Gaussian distribution\nsince the coefficients $A_k(\\cdot,Z^0)$ and $B_k(\\cdot,Z^0)$ are deterministic.\n\nNow, replacing the random terms $Z(\\cdot,T_k)$ in $\\beta(T_k)$ with the Picard iterates\n$Z^0(\\cdot,T_k)$ and $Z^1(\\cdot,T_k)$ leads to two different \\emph{log-normal\napproximations} to the dynamics of LIBOR\\xspace rates. Obviously, the approximation by\n$Z^0$ is nothing else than the frozen drift approximation. The approximation by\n$Z^1$ is again log-normal, cf. \\eqref{LMM-LG-1+1\/2} and \\eqref{LMM-LG-2}, and\nyields very good empirical results. This latter approximation has the advantage\nthat the law of the approximate LIBOR\\xspace rate is known at any time $t$, hence the\ntime-consuming Monte Carlo simulations can be avoided. For the empirical and\nnumerical analysis of these approximations we refer to\n\\shortciteN{KurbanmuradovSabelfeldSchoenmakers} and \\citeN[Ch. 6]{Schoenmakers05}.\n\n\n\\subsection{Strong Taylor approximation}\nAnother approximation method has been recently developed by\n\\citeN{SiopachaTeichmann07} and Papapantoleon and Siopacha\n\\citeyear{PapapantoleonSiopacha09}. The main idea\nbehind this method is to replace the random terms in the drift \\eqref{LMM-8} by\ntheir first-order strong Taylor approximations. The Taylor approximation is developed\nby a perturbation of the SDE for the LIBOR\\xspace rates and a subsequent Taylor expansion.\n\nLet us denote the log-LIBOR\\xspace rates by $G(\\cdot,T_k):=\\log L(\\cdot,T_k)$. Then, they\nsatisfy the linear SDE (under the terminal measure)\n\\begin{align}\\label{LMM-ST-1}\n\\ensuremath{\\mathrm{d}} G(t,T_k) = \\widehat{\\beta}(t,T_k)\\ud t + \\lambda(t,T_k)\\ensuremath{\\mathrm{d}} H_t,\n\\end{align}\nwith initial condition $G(0,T_k)=\\log L(0,T_k)$; cf. \\eqref{LMM-1} and \\eqref{LMM-8}.\nWe perturb this SDE by a real parameter $\\epsilon$, i.e.\n\\begin{align}\\label{LMM-ST-2}\n\\ensuremath{\\mathrm{d}} G^\\epsilon(t,T_k) = \\epsilon\\big(\\widehat{\\beta}^\\epsilon(t,T_k)\\ud t + \\lambda(t,T_k)\\ensuremath{\\mathrm{d}} H_t\\big),\n\\end{align}\nwhere $G^\\epsilon(0,T_k)=G(0,T_k)$ for all $\\epsilon$. The superscript $\\epsilon$ in the\ndrift term $\\widehat{\\beta}^\\epsilon(\\cdot,T_k)$ is a reminder that this term is\nalso perturbed by $\\epsilon$, since it contains all subsequent LIBOR\\xspace rates; see\n\\eqref{LMM-8} again. Now, the \\emph{first order strong Taylor approximation}\nof $G^\\epsilon$, denoted by $\\mathbf{T}G^\\epsilon$, is\n\\begin{align}\\label{LMM-ST-3}\n\\mathbf{T}G^\\epsilon(t,T_k)\n = G(0,T_k)\n  + \\epsilon \\frac{\\partial}{\\partial\\epsilon}\\big|_{\\epsilon=0} G^\\epsilon(t,T_k).\n\\end{align}\nWe denote the ``first variation'' process\n$\\frac{\\partial}{\\partial\\epsilon}\\big|_{\\epsilon=0} G^\\epsilon(\\cdot,T_k)$ by $Y(\\cdot,T_k)$,\nand then we can deduce, after some calculations, that it has the decomposition\n\\begin{align}\\label{LMM-ST-4}\nY(t,T_k)\n &= \\int_0^t \\widehat{\\beta}^0(s,T_k)\\ud s + \\int_0^t \\lambda(s,T_k)\\ensuremath{\\mathrm{d}} H_s,\n\\end{align}\nwhere $\\widehat{\\beta}^0(s,T_k):=\\widehat{\\beta}^\\epsilon(s,T_k)|_{\\epsilon=0}$. Hence,\nthis is a \\emph{deterministic} drift term, obtained by replacing the random\nterms in \\eqref{LMM-8} by their deterministic initial values. In particular, we\ncan easily deduce from \\eqref{LMM-ST-4} that, for example, if $H$ is a L\\'{e}vy\\xspace\nprocess then $Y(\\cdot,T_k)$ is a \\emph{time-inhomogeneous} L\\'{e}vy\\xspace process.\n\nConcluding, we have developed the following approximation scheme:\n\\begin{align}\\label{LMM-ST-5}\n\\log L(t,T_k) \\approx \\log L(0,T_k) + Y(t,T_k),\n\\end{align}\nwhere $Y(\\cdot,T_k)$ has the decomposition \\eqref{LMM-ST-4}; compare with\n\\eqref{LMM-1}.\n\nThe advantage of this method is threefold: (a) it is universal, and can be applied\nto LIBOR\\xspace models with stochastic volatility and\/or jumps, (b) it is faster and easier\nto simulate than the actual SDE for the LIBOR\\xspace rates, and (c) the empirical performance\nis very satisfactory; cf. Figure \\ref{diffs} and the aforementioned articles for\nfurther numerical illustrations. The drawback is that it is based on Monte Carlo\nsimulations, hence computational times can become long.\n\n\n\n\\section{Forward price models}\n\\label{fpm}\n\nForward price models were developed by \\citeN{EberleinOezkan05}, and further\ninvestigated by \\citeN{Kluge05}; see also \\citeN{EberleinKluge06}.\\!\nWe consider a setting similar to LIBOR market models\\xspace, i.e. an\ninitial tenor structure of non-negative LIBOR\\xspace rates, $\\lambda(\\cdot,T_k)$\ndenotes the volatility of the forward LIBOR\\xspace rate $L(\\cdot,T_k)$, and $H$ denotes\na semimartingale on $(\\Omega,\\ensuremath{\\mathcal{F}},\\mathbf{F},\\pt)$ with triplet of characteristics\n$(B,C,\\nu)$; again some assumptions on $H$ are suppressed. The process $H$ is\ndriving the dynamics of LIBOR\\xspace rates and has a \\emph{tractable} structure under\n$\\pt$ (e.g. $H$ is a L\\'{e}vy\\xspace or an affine process).\n\nInstead of modeling the forward LIBOR\\xspace rate directly, one now models the forward\nprice in a similar fashion; that is\n\\begin{align}\\label{fp-model-1}\n1\\!+\\!\\delta L(t,T_k)\n &= (1\\!+\\!\\delta L(0,T_k))\n    \\exp\\left( \\int_0^t \\beta(s,T_k)\\ud s + \\int_0^t\\lambda(s,T_k)\\ensuremath{\\mathrm{d}} H_s \\right),\n\\end{align}\nwhere again the drift term is such that $L(\\cdot,T_k)\\in\\mathcal{M}(\\pt[k+1])$,\nfor all $k\\in K$; i.e. $\\beta(\\cdot,T_k)$ has similar form to \\eqref{LMM-2}.\nTherefore the model obviously satisfies requirement (A2).\n\nNow, from \\eqref{Pk-to-final} and \\eqref{fp-model-1}, we get that the density\nbetween the forward measures is\n\\begin{align}\\label{fp-model-2}\n\\frac{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k+1]}}{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[N]}} \\Big|_{\\ensuremath{\\mathcal{F}}_t}\n &= \\frac{B(0,\\tk[N])}{B(0,\\tk[k+1])}\n    \\times \\prod_{l=k+1}^{N-1}\\big(1+\\delta L(t,T_l)\\big)\\\\\n &= \\frac{B(0,\\tk[N])}{B(0,\\tk[k+1])}\n    \\times \\exp\\left( \\int_0^t\\!\\sum_{l=k+1}^{N-1} \\beta(s,T_l)\\ud s +\n     \\int_0^t\\!\\sum_{l=k+1}^{N-1} \\lambda(s,T_l)\\ensuremath{\\mathrm{d}} H_s \\right). \\nonumber\n\\end{align}\nObserve that this density only depends on the driving process $H$ and the volatility\nstructures, hence we can deduce that the measure changes between forward\nmeasures are \\emph{Esscher transformations}; cf. \\citeANP{KallsenShiryaev02} \n\\linebreak\\citeyear{KallsenShiryaev02} for\nthe Esscher transform. Analogously to eqs.\n\\eqref{LMM-4.5}--\\eqref{LMM-6}, we have now that the $\\pt[k+1]$-Brownian motion\nis related to the $\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_N}$-Brownian motion via\n\\begin{align}\\label{fp-model-3}\nW_t^{T_{k+1}}\n & = W_t\n   - \\int_0^t \\left(\\sum_{l=k+1}^{N-1}\\lambda(t,T_l)\\right)\\sqrt{c_s} \\ensuremath{\\mathrm{d}} s,\n\\end{align}\nwhile the $\\pt[k+1]$-compensator of $\\mu^H$, is related to the\n$\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_N}$-compensator of $\\mu^H$ via\n\\begin{align}\\label{fp-model-4}\n\\nu^{T_{k+1}}\\ensuremath{(\\ud s, \\ud x)}\n &= \\exp\\left(x\\sum_{l=k+1}^{N-1}\\lambda(s,T_l)\\right)\\nu\\ensuremath{(\\ud s, \\ud x)}.\n\\end{align}\nThus the structure of the driving process is preserved. For example, if $H$\nis a L\\'{e}vy\\xspace or an affine process under $\\pt$, then it becomes a\n\\emph{time-inhomogeneous} L\\'{e}vy\\xspace or affine process respectively under any forward\nmeasure $\\pt[k+1]$. This implies that requirement (A3) is satisfied, so that\ncaplets and swaptions can be priced in closed form. In this class of models, we\nhave the additional benefit that we can even price some exotic path-dependent\noptions easily using Fourier transform methods; see \\citeN{KlugePapapantoleon06}\nfor an example.\n\nThe main shortcoming of forward price models\\xspace is that \\emph{negative} LIBOR\\xspace rates can occur,\nsimilarly to HJM models, since\n\\begin{align*}\n1+\\delta L(t,T_k)>0 \\,\\,\\nRightarrow\\,\\, L(t,T_k)>0\n\\qquad\\text{for all }\\, t\\in[0,T_k].\n\\end{align*}\nTherefore, this model can violate requirement (A1).\n\n\n\n\\section{Markov-functional models}\n\\label{mfm}\n\nMarkov-functional models\\xspace were introduced in the seminal paper of Hunt, Kennedy, and Pelsser\n\\citeyear{HuntKennedyPelsser00}. In\ncontrast to the other approaches described in this review, the aim of Markov-functional models\\xspace is\nnot to model some fundamental quantity, e.g. LIBOR\\xspace or swap rates, directly. Instead,\nMarkov-functional models\\xspace are constructed by inferring the model dynamics, as well as their functional\nforms, through matching the model prices to the market prices of certain liquid\nderivatives. That is, they are \\emph{implied interest rate models}, and should be\nthought of in a fashion similar to local volatility models and implied trees in\nequity markets.\n\nThe main idea behind Markov-functional models\\xspace is that bond prices and the numeraire are, at any point\nin time, a function of a \\emph{low-dimensional} Markov process under some martingale\nmeasure. The functional form for the bond prices is selected such that the model\naccurately calibrates to the relevant market prices, while the freedom to choose the\nMarkov process makes the model realistic and tractable. Moreover, the functional form\nfor the numeraire can be used to reproduce the marginal laws of swap rates or other\nrelevant instruments for the calibration.\n\nMore specifically, let $(M,\\mathbf{M})$ denote a numeraire pair, and consider a\n(time-inhomogeneous) Markov process \\prozess[X] under the measure $\\mathbf{M}$.\nIn the framework of Markov-functional models, one assumes that bond prices have\nthe functional form\n\\begin{align}\\label{MFM-1}\nB(t,S) = \\mathrm{B}(t,S;X_t),\n \\quad 0\\le t\\le \\partial_S\\le S,\n\\end{align}\nwhere $\\partial_S$ denotes some ``boundary curve''. In applications, the boundary\ncurve typically  has the form\n\\begin{align}\\label{MFM-2}\n\\partial_S\n = \\left\\{\n     \\begin{array}{ll}\n       S,   & S\\le T_{*}, \\\\\n       T_*, & S>T_{*},\n     \\end{array}\n   \\right.\n\\end{align}\nwhere $T_*$ is a common time of maturity. One further assumes that the numeraire\n$M$ is also a function of the driving Markov process $X$, i.e.\n\\begin{align}\\label{MFM-3}\nM_t = \\mathrm{M}(t;X_t),\n \\quad 0\\le t\\le T.\n\\end{align}\nTherefore, in order to specify a Markov-functional model\\xspace, the following quantities are required:\n\\begin{itemize\n\\item[(P1)] the law of $X$ under the measure $\\mathbf{M}$;\n\\item[(P2)] the functional form $\\mathrm{B}(\\partial_S,S;\\cdot)$ for $S\\in[0,T]$;\n\\item[(P3)] the functional form $\\mathrm{M}(t;\\cdot)$ for $0\\le t\\le T$.\n\\end{itemize}\n\nIn applications, the Markov process is specified first and is typically a diffusion\nprocess with time-dependent volatility. Then, the functional forms for the bond\nprices and the numeraire are implied by calibrating the model to market prices of\nliquid options. The choice of the calibrating instruments depends on the exotic\nderivative that should be priced or hedged with the model. If the exotic depends\non LIBOR rates, e.g. the flexible cap, then the model is calibrated to digital caplets,\nwhich leads to the \\emph{Markov-functional LIBOR model}. If the exotic depends on\nswap rates, e.g. the Bermudan swaption, then the model is calibrated to digital swaptions,\nwhich leads to the \\emph{Markov-functional swap rate model}. Let us point out that the\nfunctional forms are typically not known in closed form, and should be computed\nnumerically. These models typically satisfy requirements (A1), (A2) and (A3).\nFor further details and concrete applications we refer the reader to the books\nby \\citeN{HuntKennedy04} and \\citeN{Fries07}, and the references therein.\n\n\\begin{remark}\nLet us point out that forward price models\\xspace and affine LIBOR models\\xspace, that will be introduced in section \\ref{alm},\nbelong to the class of Markov-functional models\\xspace, while LIBOR market models\\xspace do not. In LIBOR market models\\xspace the LIBOR\\xspace rates are\nfunctions of a \\textit{high-dimensional} Markov process. \n\\end{remark}\n\n\n\\subsection{Markov-functional LIBOR model}\nIn order to gain a better understanding of the construction of Markov-functional models\\xspace, we will\nbriefly describe a Markov-functional model\\xspace calibrated to LIBOR\\xspace rates. This model is called the\n\\textit{Markov-functional LIBOR model}.\n\nThe set of relevant market rates are LIBOR\\xspace rates $L(\\cdot,T_k), k\\in K$. We will\nconsider the numeraire pair $(M,\\mathbf{M})=(B(\\cdot,T_N),\\pt)$.\n\nIn order to be consistent with Black's formula for caplets, we assume that\n$L(\\cdot,T_{N-1})$ is a log-normal martingale under \\pt, i.e.\n\\begin{align}\\label{MFL-1}\n\\ensuremath{\\mathrm{d}} L(t,T_{N-1})\n= \\sigma(t,T_{N-1}) L(t,T_{N-1}) \\ensuremath{\\mathrm{d}} W_t,\n\\end{align}\nwhere $W$ denotes a standard Brownian motion under $\\pt$ and $\\sigma(\\cdot,T_{N-1})$\nis a deterministic, time-dependent volatility function. Hence, we have that\n\\begin{align}\\label{MFL-2}\nL(t,T_{N-1})\n= L(0,T_{N-1}) \\exp\\Big( -\\frac12\\Sigma_t + X_t\\Big),\n\\end{align}\nwhere $\\Sigma=\\int_0^\\cdot \\sigma^2(s,T_{N-1})\\ud s$, and $X$ is a deterministic\ntime-change of the Brownian motion, that satisfies\n\\begin{align}\\label{MFL-3}\n\\ensuremath{\\mathrm{d}} X_t = \\sigma(t,T_{N-1})\\ensuremath{\\mathrm{d}} W_t.\n\\end{align}\nWe will use $X$ as the driving process of the model, which specifies (P1).\n\nRegarding (P2), the boundary curve is exactly of the form \\eqref{MFM-2} with\n$T_*=T_{N-1}$, hence we need to specify $\\mathrm{B}(T_i,T_i;X_{T_i})$ for\n$i\\in K$, which is trivially the unit map. We also need to specify\n$\\mathrm{B}(T_{N-1},T_N;X_{T_{N-1}})$; using \\eqref{basic} and\n\\eqref{MFL-2} we get that\n\\begin{align}\\label{MFL-4}\n\\mathrm{B}(T_{N-1},T_N;X_{T_{N-1}})\n= \\frac{1}{1 + \\delta L(0,T_{N-1}) \\exp\\big( -\\frac12\\Sigma_{T_{N-1}}+ X_{T_{N-1}}\\big)}.\n\\end{align}\nThen, we can recover bond prices in the interior of the region bounded by\n$\\partial_S$ using the martingale property:\n\\begin{align}\\label{MFL-5}\n\\mathrm{B}(t,S;X_t)\n= \\mathrm{B}(t,T_N;X_t)\n  \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\bigg[ \\frac{\\mathrm{B}(\\partial_S,S;X_{\\partial_S})}\n                      {\\mathrm{B}(\\partial_S,T_N;X_{\\partial_S})} \\Big|\\ensuremath{\\mathcal{F}}_t \\bigg].\n\\end{align}\n\nNow, it remains to specify the functional form $\\mathrm{B}(T_i,T_N;X_{T_i})$,\n$i\\in K$, for the numeraire, cf. (P3). In the framework of the Markov-functional\nLIBOR\\xspace model, this is done by deriving the numeraire from LIBOR\\xspace rates and inferring\nthe functional forms of the LIBOR\\xspace rates via calibration to market prices. Equation\n\\eqref{basic} combined with \\eqref{MFM-2} and the fact that $B(T_i,T_{i+1})$ is\na function of $X_{T_i}$, cf. \\eqref{MFL-5}, yield that $L(T_i,T_i)$ is also a\nfunction of $X_{T_i}$. The functional form is\n\\begin{align}\\label{MFL-6}\n1+\\delta\\mathrm{L}(T_i,T_i;X_{T_i})\n&= \\frac{1}{\\mathrm{B}(T_i,T_{i+1};X_{T_{i}})} \\nonumber\\\\\n&= \\frac{1}{\\mathrm{B}(T_i,T_N;X_{T_i})\n  \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\Big[ \\frac{1}{\\mathrm{B}(T_{i+1},T_N;X_{T_{i+1}})} \\Big|\\ensuremath{\\mathcal{F}}_{T_i} \\Big]}.\n\\end{align}\nRearranging, we get the following functional form for the numeraire\n\\begin{align}\\label{MFL-7}\n\\mathrm{B}(T_i,T_N;X_{T_i})\n&= \\frac{1}{(1+\\delta\\mathrm{L}(T_i,T_i;X_{T_i}))\n   \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\Big[ \\frac{1}{\\mathrm{B}(T_{i+1},T_N;X_{T_{i+1}})} \\Big|\\ensuremath{\\mathcal{F}}_{T_i} \\Big]}.\n\\end{align}\nThis formula provides a backward induction scheme to calculate\n$\\mathrm{B}(T_i,T_N;\\cdot)$ from $\\mathrm{B}(T_{i+1},T_N;\\cdot)$ for any value\nof the Markov process; the induction starts from $B(T_N,T_N)=1$.\n\nThe calibrating instruments are digital caplets with payoff\n$1_{\\{L(T_i,T_i)>\\mathcal{K}\\}}$, $i\\in K$, and their market values are provided\nby Black's formula; we denote them by $\\mathbb{V}_0(T_i,\\mathcal{K})$.\nAssuming that the map $\\xi\\mapsto\\mathrm{L}(T_i,T_i;\\xi)$ is strictly increasing,\nthere exists a unique strike $\\mathcal{K}(T_i,x^*)$ such that the set equality\n\\begin{align}\\label{MFL-10}\n\\big\\{X_{T_i}>x^*\\big\\}\n = \\big\\{\\mathrm{L}(T_i,T_i;X_{T_i})>\\mathcal{K}(T_i,x^*)\\big\\}\n\\end{align}\nholds almost surely. Define the model prices\n\\begin{align}\\label{MFL-11}\n\\mathbb{U}_0(T_i,x^*)\n = B(0,T_{N})\n   \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\bigg[\\frac{\\mathrm{B}(T_i,T_{i+1};X_{T_{i}})}\n                     {\\mathrm{B}(T_i,T_N;X_{T_{i}})} 1_{\\{X_{T_i}>x^*\\}}\\bigg],\n\\end{align}\nwhich have to be calculated numerically. Therefore, we can equate market and\nmodel prices\n\\begin{align}\\label{MFL-12}\n\\mathbb{V}_0(T_i,\\mathcal{K}(T_i,x^*)) = \\mathbb{U}_0(T_i,x^*),\n\\end{align}\nwhere the strike $\\mathcal{K}(T_i,x^*)$ is determined by Black's formula using\nsome numerical algorithm.\n\nHence, we have specified, numerically at least, the functional form for the\nLIBOR rates, which yields also the functional form for the numeraire via \\eqref{MFL-7}.\nThis completes the specification of the Markov-functional LIBOR model. This\nmodel satisfies requirement (A3), in the sense of \\shortciteANP{HuntKennedyPelsser00}\n\\citeyear{HuntKennedyPelsser00},\nsince all bond prices are functions of a one-dimensional diffusion.\n\n\n\n\\section{Affine LIBOR models}\n\\label{alm}\n\nAffine LIBOR\\xspace models were recently developed by Keller-Ressel, Papapantoleon, and Teichmann\n\\citeyear{KellerResselPapapantoleonTeichmann09} with the aim of combining the\nadvantages of LIBOR market models\\xspace and forward price models\\xspace, while circumventing their drawbacks. We provide\nhere a more general outline of this framework, which is based on two key\ningredients: martingales \\emph{greater than 1}, which are \\emph{increasing} in\nsome parameter.\n\nThe construction of martingales greater than 1 is done as follows: let $Y_T^u$\nbe an $\\ensuremath{\\mathcal{F}}_T$-measurable, integrable random variable, taking values in $[1,\\infty)$,\nand set\n\\begin{align}\\label{Mge1}\nM_t^u = \\ensuremath{\\mathrm{I\\kern-.2em E}}[Y_T^u|\\ensuremath{\\mathcal{F}}_t],\n\\qquad {0\\leq t\\leq T}.\n\\end{align}\nThen, using the tower property of conditional expectations, it easily follows that\n\\prozess[M^u] is a martingale. Moreover, it obviously holds that $M_t^u\\ge1$ for\nall $t\\in[0,T]$.\n\nIn addition, assume that the map $u\\mapsto Y_T^u$ is \\emph{increasing}; then, we\nimmediately get that the map\n\\begin{align}\\label{Mincrease}\nu\\mapsto M_t^u\n\\end{align}\nis also increasing, for all $t\\in[0,T]$. \n\nNow, using the family of martingales $M^u$ we can model quotients of bond prices\nas follows. Consider a \\emph{decreasing} sequence $(u_k)_{k\\in\\overline{K}}$ and set\n\\begin{align}\\label{bond-quots}\n\\frac{B(t,T_k)}{B(t,T_N)}\n &= M_t^{u_k},\n \\qquad t\\in\\ttk,\\, k\\in\\overline{K},\n\\end{align}\nrequiring that the initial values of the martingales fit today's observed market\nprices, i.e. $\\frac{B(0,T_k)}{B(0,T_N)}= M_0^{u_k}$. Since $M^u$ is increasing\nin $u$, we have that\n\\begin{align}\\label{order-M}\nM_t^{u_k} \\ge M_t^{u_l}\n \\quad\\text{ for }\\quad\nk\\le l \\Leftrightarrow u_k\\ge u_l.\n\\end{align}\nHence, we can deduce that bond prices are decreasing as functions of time of\nmaturity, i.e. $B(t,T_k)\\ge B(t,T_l)$ for $k\\le l$.\n\nTurning our attention to LIBOR rates, we get that\n\\begin{align}\\label{LIBOR-1}\n1 + \\delta L(t,T_k)\n &= \\frac{B(t,T_k)}{B(t,\\tk[k+1])}\n  = \\frac{M_t^{u_k}}{M_t^{u_{k+1}}}\n \\ge 1,\n\\end{align}\nfor all $t\\in\\ttk$ and all $k\\in K$; this is a trivial consequence of \\eqref{order-M}.\nMoreover, the martingale property of the LIBOR\\xspace rate under its corresponding forward\nmeasure follows easily from the structure of the measure changes \\eqref{Pk-to-final},\nand the structure of the martingales. Indeed, we have that\n\\begin{multline}\\label{LIBOR-MP}\n1+\\delta L(\\cdot,T_k)\n = \\frac{M^{u_k}}{M^{u_{k+1}}} \\in \\mathcal{M}(\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_{k+1}})\\\\\n \\quad\\text{ since }\\quad\n\\frac{M^{u_{k}}}{M^{u_{k+1}}}\n \\cdot \\frac{\\ensuremath{\\mathrm{d}}\\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k+1]}}{\\ensuremath{\\mathrm{d}}\\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[N]}\n = \\frac{M^{u_{k}}}{M^{u_{k+1}}}\n  \\cdot \\frac{M^{u_{k+1}}}{M_0^{u_{k+1}}}\n \\in \\mathcal{M}(\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_{N}}).\n\\end{multline}\n\nTherefore, we have just described a broad framework for modeling LIBOR\\xspace rates, in\nwhich requirements (A1) and (A2) are satisfied. The next step is to show that\nrequirement (A3) is also satisfied. We will not pursue this in full generality,\ninstead we will consider a specific form for the variable $Y_T^u$, and thus for\nthe martingales $M^u$. In addition, the model is driven by an affine process,\nand is henceforth called the \\emph{affine LIBOR model}.\n\n\n\\subsection{Affine processes}\nLet \\prozess[X] be a stochastically continuous, time-homogeneous Markov process\nwith state space $D=\\mathbb{R}_{\\geqslant0}^d$, starting from $x\\in D$. The process $X$ is called\n\\emph{affine} if the moment generating function satisfies\n\\begin{align}\\label{affine}\n\\ensuremath{\\mathrm{I\\kern-.2em E}}_x \\big[\\mathrm{e}^{\\scal{u}{X_t}}\\big]\n = \\exp\\big( \\phi_t(u) + \\scal{\\psi_t(u)}{x} \\big),\n\\end{align}\nfor some functions $\\phi:[0,T]\\times\\mathcal{I}_T\\to\\ensuremath{\\mathbb{R}}$ and $\\psi:[0,T]\\times\\mathcal{I}_T\\to\\ensuremath{\\mathbb{R}}^d$,\nand all $(t,u,x)\\in[0,T]\\times\\mathcal{I}_T\\times D$, where\n\\begin{align}\n\\mathcal{I}_T\n:= \\set{u\\in\\ensuremath{\\mathbb{R}}^d: \\ensuremath{\\mathrm{I\\kern-.2em E}}_x\\big[\\mathrm{e}^{\\scal{u}{X_T}}\\big] < \\infty,\n        \\,\\,\\text{for all}\\; x \\in D}.\n\\end{align}\nWe will assume in the sequel that $0\\in\\mathcal{I}_T^\\circ$. The functions $\\phi$ and\n$\\psi$ satisfy the semi-flow property\n\\begin{equation}\\label{flow}\n\\begin{split}\n\\phi_{t+s}(u) &= \\phi_{t}(u)+\\phi_{s}(\\psi_t(u))\\\\\n\\psi_{t+s}(u) &= \\psi_{s}(\\psi_{t}(u)),\n\\end{split}\n\\end{equation}\nwith initial condition\n\\begin{align}\\label{phi-psi-0}\n \\phi_0(u)=0\n  \\quad\\text{ and }\\quad\n \\psi_0(u)=u,\n\\end{align}\nfor all $(t+s,u)\\in[0,T]\\times\\mathcal{I}_T$. Equivalently, $\\phi$ and $\\psi$ satisfy\ngeneralized Riccati differential equations. For comprehensive expositions of\naffine processes we refer the reader to \\shortciteN{DuffieFilipoviSchachermayer03}\nand \\citeN{KellerRessel08}.\n\n\n\\subsection{Affine LIBOR\\xspace model}\nIn the affine LIBOR model\\xspace, the random variable $Y_T^u$ has the following form:\n\\begin{align\nY_T^u = \\mathrm{e}^{\\scal{u}{X_T}},\n\\end{align}\nwhere $u\\in\\mathbb{R}_{\\geqslant0}^d$ and $X_T$ is a random variable from an $\\mathbb{R}_{\\geqslant0}^d$-valued affine\nprocess $X$. Hence, $Y_T^u\\ge1$, while the map $u\\mapsto Y_T^u$ is obviously\nincreasing; note that inequalities involving vectors are understood componentwise.\n\nUsing the Markov property of affine processes, we can deduce that the martingales\n$M^u$ have the form\n\\begin{align}\\label{alm-Mu}\nM_t^u\n &= \\ensuremath{\\mathrm{I\\kern-.2em E}}\\big[\\mathrm{e}^{\\scal{u}{X_T}}|\\ensuremath{\\mathcal{F}}_t\\big] \\nonumber\\\\\n &= \\exp\\big( \\phi_{T-t}(u) + \\scal{\\psi_{T-t}(u)}{X_t} \\big).\n\\end{align}\nTherefore, LIBOR\\xspace rates have the following evolution:\n\\begin{align}\n1+\\delta L(t,T_k)\n &= \\frac{M_t^{u_k}}{M_t^{u_{k+1}}}\n  = \\exp\\big(A_{k,t} + \\scal{B_{k,t}}{X_t}\\big),\n\\end{align}\nwhere\n\\begin{equation}\\label{Ak-Bk}\n\\begin{split}\n A_{k,t} &:= \\phi_{T_N-t}(u_k) - \\phi_{T_N-t}(u_{k+1})\\\\\n B_{k,t} &:= \\psi_{T_N-t}(u_k) - \\psi_{T_N-t}(u_{k+1}).\n\\end{split}\n\\end{equation}\nLet us point that, under reasonable assumptions on the driving affine process,\nwe can prove that the affine LIBOR model\\xspace can fit \\emph{any} term structure of\ninitial LIBOR\\xspace rates; cf. Proposition 6.1 in \\shortciteN{KellerResselPapapantoleonTeichmann09}.\n\nNow, regarding requirement (A3), let us turn our attention to the structure of\nthe driving process under the different forward measures. Using the connections\nbetween forward measures \\eqref{Pk-to-final}, the Markov property of affine\nprocesses, and the flow equations \\eqref{flow}, we can show that\n\\begin{align}\n\\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_k}\\Big[\\mathrm{e}^{\\langle v,X_r\\rangle}\\big|\\ensuremath{\\mathcal{F}}_s\\Big]\n &= \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\Big[ \\frac{M_r^{u_k}}{M_s^{u_k}} \\mathrm{e}^{\\langle v,X_r\\rangle}\\big|\\ensuremath{\\mathcal{F}}_s\\Big] \\\\\n &= \\exp\\Big(\\phi_{r-s}(\\psi_{T_N-r}(u_{k})+ v) - \\phi_{r-s}(\\psi_{T_N-r}(u_k))\\nonumber\\\\\n &\\qquad\n  + \\scal{\\psi_{r-s}(\\psi_{T_N-r}(u_{k})+ v) - \\psi_{r-s}(\\psi_{T_N-r}(u_k))}{X_s}\\Big);\n  \\nonumber\n\\end{align}\ncf. \\shortciteN[eq. (6.15)]{KellerResselPapapantoleonTeichmann09}. This means\nthat $X$ becomes a \\emph{time-inhomogeneous affine} process under \\emph{any}\nforward measure. Note that the measure changes are again Esscher\ntransformations, similarly to forward price models\\xspace. Consequently the affine LIBOR model\\xspace satisfies requirements\n(A1), (A2) \\emph{and} (A3).\n\nThe pricing of caplets and swaptions in the affine LIBOR model\\xspace using Fourier transform methods\nis described in \\shortciteN{KellerResselPapapantoleonTeichmann09}. In addition, closed-form valuation formulas -- in terms\nof the $\\chi^2$-distribution function -- are derived when the driving affine\nprocess is the Cox--Ingersoll--\\linebreak Ross (CIR) process.\n\n\n\n\\section{Extensions}\n\\label{extend}\n\nThe different approaches for modeling LIBOR\\xspace rates have been extended in two\ndifferent directions: (i) to model simultaneously LIBOR rates for different\ncurrencies and the corresponding foreign exchange rates, and (ii) to jointly\nmodel default-free and defaultable LIBOR\\xspace rates.\n\n\n\\subsection{Multiple currencies}\nThe log-normal LIBOR market model\\xspace has been extended to a multi-currency setting by \\citeN{Schloegl02}\nand by \\citeANP{Mikkelsen02} \\citeyear{Mikkelsen02}. The L\\'{e}vy\\xspace LIBOR\\xspace model and the L\\'{e}vy\\xspace forward price model\\xspace have\nbeen extended to model multiple currencies and foreign exchange rates by\n\\citeN{EberleinKoval06}. A multi-factor approach to multiple currency LIBOR\\xspace models\nhas been presented in \\shortciteN{BennerZyapkovJortzik09}. Markov-functional models\\xspace have been extended\nto the multi-currency setting by \\citeANP{FriesRott04} \\citeyear{FriesRott04} and\n\\citeN{FriesEcksteadt08}.\n\n\n\\subsection{Default risk}\nThe log-normal LIBOR market model\\xspace has been first extended to model default risk by\n\\citeN{LotzSchloegl00}, who used a deterministic hazard rate to model the time\nof default. \\shortciteN{EberleinKlugeSchoenbucher06}, \\linebreak borrowing also ideas from\n\\citeN{Schoenbucher00}, constructed a model for default-free and defaultable\nrates where they use time-inhomogeneous L\\'{e}vy\\xspace processes as the driving motion and\nthe ``Cox construction'' to model the time of default (cf. e.g.\n\\citeN{BieleckiRutkowski02} for the Cox construction). This has been extended\nto a model where defaultable bonds can have rating migrations by \\citeN{Grbac09}.\n\n\n\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section[]{The Diagrammatic Identities}\n\\label{app:D-ID}\n\n\\subsection*{The Primary Diagrammatic Identities}\n\n\\subsubsection*{The First Type}\n\n\\begin{equation}\n\\label{eq:WID}\n\t\\cd{WID-contract-b} = \\cd{WID-PF} + \\cd{WID-PFb} - \\cd{WID-PB} - \\cd{WID-PBb} + \\cdots\n\\end{equation}\n\n\\begin{eqnarray}\n\\label{eq:LdL-GRk-Pert}\n\t\\hspace{0.8em} \\stackrel{\\bullet}{\\rhd} & =  & 0\n\\\\\n\\label{eq:dalpha-GRk}\n\t\\begin{array}{c}\n\t\t\\Dal\n\t\\\\[-1.5ex]\n\t\t\\rhd\n\t\\end{array} & = & 0\n\\end{eqnarray}\n\n\\begin{equation}\n\\label{eq:D-ID-GR-TP}\n\t\\cd{GR-TP} = 0\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:Taylor}\n\t\\cd{Taylor-Parent-b} = \\cdeps{Taylor-PFa} + \\cdeps{Taylor-PFb} - \\cdeps{Taylor-PBa} - \\cdeps{Taylor-PBb} +\\cdots\n\\end{equation}\n\n\n\n\\subsubsection*{The Second Type}\n\n\\begin{equation}\n\t\\cd{EffPropReln-b}\t= \\cd{K-Delta-b} - \\cd{FullGaugeRemainder-b}\n\t\t\t\t\t\t\t= \\cd{K-Delta-b} - \\cd{DecomposedGR-b}\n\t\\label{eq:EPReln}\n\\end{equation}\n\n\n\\subsubsection*{The Third Type}\n\n\\begin{equation}\n\t\\cd{GR-TLTP} = 0\n\\label{eq:GR-TLTP}\n\\end{equation}\n\n\\begin{equation}\n\t\\cd{GR-relation} = 1\n\\label{eq:GR-relation}\n\\end{equation}\n\n\n\\begin{equation}\n\t\\cd{EP-GRpr} \\equiv \\cd{GR-PEP}\n\\label{eq:PseudoEP}\n\\end{equation}\n\n\n\\subsection*{The Secondary Diagrammatic Identities}\n\n\\subsubsection*{The First Type}\n\n\\begin{equation}\n\\label{D-ID-Trivial}\n\t\\cdeps{GR-hook} = 0\n\\end{equation}\n\n\n\\begin{equation}\n\\label{eq:D-ID-Bitten-hook}\n\t\\cdeps{Bitten-hook} - \\cdeps{Bitten-hook-R} \\equiv 0\n\\end{equation}\n\n\\begin{eqnarray}\n\t\\nonumber\n\t\t\\sum_{m'=1}^{m-1} \\frac{1}{m'!} \\cd{TLTP-EP-ArbGRs} \\hspace{2em}\n\t\t& = &  \\sum_{m'=1}^{m-1} \\frac{1}{(m'+1)!} \\cd{GR-ArbGRs} \\hspace{2em}\n\t\\\\ \n\t\\label{eq:D-ID-G}\n\t\t&& - \\sum_{m'=2}^{m-1} \\sum_{m''=0}^{m'-2} \\frac{1}{(m'-m'')!m''!} \n\t\t\\cd{ArbGRs-TLTP-EP-ArbGRs}\n\\end{eqnarray}\n\n\\clearpage\n\n\\subsubsection*{The Second Type}\n\n\\begin{equation}\n\\label{eq:D-ID-dGRk-GT-ring}\n\t2m(m-1) \\cd{GRs-dGRk} \\hspace{3em} = - \\decGR{ \\ }{\\ensuremath{\\sqcup} >^m}.\n\\end{equation}\n\n\\begin{eqnarray}\n\t\\lefteqn{\n\t\t\\sum_{m'=0}^m \n\t\t\\left[\n\t\t\t\\cd{CTP-E-GRs-Combo-GRk-GR} \n\t\t\t-\\cd{CTP-E-GRs-Combo-GR}\n\t\t\\right]\t\n\t}\n\t\\nonumber\n\t\\\\ & &\n\t\t- \\sum_{m'=0}^m \\!\\! \\nCr{m'}{m''} \\!\\! \\sum_{m''=0}^{m'} \n\t\t\\left[\n\t\t\t\\cd{CTP-E-GRs-Combo-CTP-EP-GR}\n\t\t\t\\hspace{1.8em}\n\t\t\t-\\cd{CTP-E-GRs-Combo-GRs-CTP-EP}\n\t\t\t\\hspace{1.8em}\n\t\t\\right]\n\t\t= 0\n\t\\label{eq:D-ID-Op2-G}\n\\end{eqnarray}\n\n\\clearpage\n\n\\section[]{Diagrammatic Expressions}\n\n\\subsection*{Decoration of diagram~\\ref{j+2-DEP}}\n\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2.5ex}\n\t\t\\left[\n\t\t\\begin{array}{c}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s} \\sum_{j=-1}^{n+s-m-3}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\\\\\n\t\t\t\\displaystyle\n\t\t\n\t\t\t\\dec{\n\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-DEP-vkR} \\!\\! \\Vertex{n_s,j}}{D-vj_+kR-DEP-vkR}\n\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\\end{array}\n\t\t\\right]\n\t\t-\n\t\t\\left[\n\t\t\\begin{array}{c}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t\\\\\n\t\t\t\\dec{\n\t\t\t\\sco[1]{\n\t\t\t\t\\LDi{vj_+R-DEP}{vj_+R-DEP}\n\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t\\end{array}\n\t\t\\right]\n\t\\\\\n\t\\vspace{2.5ex}\n\t\t-2\n\t\t\\left[\n\t\t\\begin{array}{c}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-1}^{n+s-m-2}\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t\t\\sum_{m'=1}^{m} \\nCr{m}{m'}\n\t\t\\\\\n\t\t\t\\displaystyle\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi[1]{v_j+R-DEP-GRs}{v_j+R-DEP-GRs}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t\\end{array}\n\t\t\\right]\n\t\\\\\n\t\\vspace{2.5ex}\n\t\t+\n\t\t\\left[\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2}\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\t\\sum_{m'=1}^{m} \\nCr{m}{m'} \n\t\t\t\\\\\n\t\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle \\nCr{m'}{m''}\\sum_{m''=1}^{m'-1} \n\t\t\t\t\t\\LDi{GRs-DEP-GRs}{GRs-DEP-GRs-MV} \\hspace{2em}\n\t\t\t\t\t+ \\LDi{Ubend-DEP}{Ubend-MV} \\hspace{0.5em}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\end{array}\n\t\\]\n\\caption{Decoration of diagram~\\ref{j+2-DEP} with\nthe differentiated effective propagator.}\n\\label{fig:app:DEP}\n\\end{figure}\\end{center}\n\n\\clearpage\n\n\\subsection*{Trapped gauge remainders}\n\\label{app:Trapped}\n\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2}  \\sum_{j=-1}^{n+s-m-4} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!} \\sum_{m'=0}^{m}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\nCr{m}{m'} \\!\\!\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{RV-RW-GR-B-RW}{RV-RW-GR-B-RW} \\ \n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s}>^{m-m'}}\n\t\t}\t\n\t\\\\\n\t\\vspace{2ex}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-4} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}  \\sum_{m'=0}^{m} \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{TLTP-RW-GR-B-RW}{TLTP-RW-GR-B-RW}}\n\t\t\t}{11\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-4}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}  \\sum_{m'=0}^{m} \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{Trapped-KBK}{Trapped-KBK}}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Terms spawned by diagrams~\\ref{WGRx2} \nand~\\ref{P-S_vj_+R-DW-GRx2}\nwhich possess a trapped gauge remainder.}\n\\label{fig:GRsx2-P}\n\\end{figure}\\end{center}\n\n\\subsection*{Diagrams spawned by~\\eq{eq:LdL-b-ba} in which a kernel\nbites its own tail}\n\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-2} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=0}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-WBT-GR-GRs-dGR-GRs}{CTP-E-WBT-GR-GRs-dGR-GRs} \n\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t-\\LDi{CTP-E-WBT-GR-GRs-GR-dGR-GRs}{CTP-E-WBT-GR-GRs-GR-dGR-GRs}\n\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r}  >^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-WBT-dGR-GRs}{CTP-E-WBT-dGR-GRs}\n\t\t\t\t\t-\\LDi{CTP-E-WBT-GR-dGR-GRs}{CTP-E-WBT-GR-dGR-GRs}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r}  >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=0}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-WBT-GRs-CTP-EP}{CTP-E-GRs-Combo-WBT-GRs-CTP-EP}\n\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r}  >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Diagrams spawned by~\\eq{eq:LdL-b-ba} in which a kernel\nbites its own tail, part~1.}\n\\label{fig:nLDlb-KBT}\n\\end{figure}\\end{center}\n\n\\clearpage\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-WBT}{CTP-E-GRs-Combo-WBT-b}\n\t\t\t\t\t-\\LDi{CTP-E-GRs-Combo-WBT-GR}{CTP-E-GRs-Combo-WBT-GR}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r}  >^{m-m'}}\n\t\t}\t\n\t\\end{array}\n\t\\]\n\\caption{Diagrams spawned by~\\eq{eq:LdL-b-ba} in which a kernel\nbites its own tail, part~2.}\n\\label{fig:nLDlb-KBT-2}\n\\end{figure}\\end{center}\n\n\n\\subsection*{Subtractions and Additions}\n\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{0ex}\n\t\t\\mp 4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{dvh_j-DEP-GRs-Skt-CTP-E}{dvh_j-DEP-GRs-Skt-CTP-E-s}{dvh_j-DEP-GRs-Skt-CTP-E-a}\n\t\t\t\t\t-\\LLDi{dPi_j+R-DW-GRs-Skt-CTP-E}{dPi_j+R-DW-GRs-Skt-CTP-E-s}{dPi_j+R-DW-GRs-Skt-CTP-E-a}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\t\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm 4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=0}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\left[\n\t\t\t\t\t\t\\LLDi{vh_k+R-DEP-GRs-Skt-CTP-E}{vh_kR-dvj+kh-s}{vh_kR-dvj+kh-a}\n\t\t\t\t\t\t-\\LLDi{Pi_kR-DW-GRs-Skt-CTP-E}{Pi_j+R-dvj+kh-s}{Pi_j+R-dvj+kh-a}\n\t\t\t\t\t\\right] \\cd{dvj+k}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+2}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for diagrams~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E}\nand~\\ref{Pi_j+R-DW-GRs-Skt-CTP-E}, part~1.}\n\\label{fig:vh_j+R-DEP-GRs-Skt-CTP-E-Add}\n\\end{figure}\\end{center}\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\mp 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{vh_j-DEP-TEGRs-Skt-CTP-E}{vh_j-DEP-TEGRs-Skt-CTP-E-s}{vh_j-DEP-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t-\\LLDi{Pi_j+R-DEP-TEGRs-Skt-CTP-E}{Pi_j+R-DEP-TEGRs-Skt-CTP-E-s}{Pi_j+R-DEP-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t\\hspace{1em}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t\\mp 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\\LLDi{vh_j+R-DEP-GRs-Skt-CTP-E-b}{vh_j+R-DEP-GRs-Skt-CTP-E-b-s}{vh_j+R-DEP-GRs-Skt-CTP-E-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\t-\\LLDi{Pi_j+R-DW-GRs-Skt-CTP-E-b}{Pi_j+R-DW-GRs-Skt-CTP-E-b-s}{Pi_j+R-DW-GRs-Skt-CTP-E-b-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m''}}\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm 4\n\t\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\t\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{Pi_j+R-dK-GRs-Skt-CTP-E}{Pi_j+R-dK-GRs-Skt-CTP-E-s}{Pi_j+R-dK-GRs-Skt-CTP-E-a}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for diagrams~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E}\nand~\\ref{Pi_j+R-DW-GRs-Skt-CTP-E}, part~2.}\n\\label{fig:vh_j+R-DEP-GRs-Skt-CTP-E-Add-II}\n\\end{figure}\\end{center}\n\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\mp 2 \n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\t\\LLDi{DW-GR-GRs-Skt-CTP-E}{DW-GR-GRs-Skt-CTP-E-dv-s}{DW-GR-GRs-Skt-CTP-E-dv-a}\n\t\t\t\t\t\t\t-2\\LLDi{DW-GRs-Skt-CTP-E}{DW-GRs-Skt-CTP-E-dv-s}{DW-GRs-Skt-CTP-E-dv-a}\n\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\cd{dvj+}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{}\n\t\t\t\t\\\\\n\t\t\t\t\t\\SumVertex\t\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm 2 \n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\t\\LLDi{dK-GR-GRs-Skt-CTP-E}{dK-GR-GRs-Skt-CTP-E-dv-s}{dK-GR-GRs-Skt-CTP-E-dv-a}\n\t\t\t\t\t\t-2\\LLDi{dK-GRs-Skt-CTP-E}{dK-GRs-Skt-CTP-E-dv-s}{dK-GRs-Skt-CTP-E-dv-a}\n\t\t\t\t\t}{}\n\t\t\t\t\\\\\n\t\t\t\t\t\\TopVertex \\SumVertex\t\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{DW-GR-TEGRs-Skt-CTP-E}{DW-GR-TEGRs-Skt-CTP-E-s}{DW-GR-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t-2\\LLDi{DW-TEGRs-Skt-CTP-E}{DW-TEGRs-Skt-CTP-E-s}{DW-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t}{}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for \ndiagrams~\\ref{DW-GR-GRs-Skt-CTP-E} and~\\ref{DW-GRs-Skt-CTP-E}, \npart~1.}\n\\label{fig:DW-GR-GRs-Skt-CTP-E-Add}\n\\end{figure}\\end{center}\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\pm\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\\LLDi{DW-GR-GRs-Skt-CTP-E-b}{DW-GR-GRs-Skt-CTP-E-TEGR-s}{DW-GR-GRs-Skt-CTP-E-TEGR-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\t-2\\LLDi{DW-GRs-Skt-CTP-E-b}{DW-GRs-Skt-CTP-E-TEGR-s}{DW-GRs-Skt-CTP-E-TEGR-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m''}}\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\]\n\\caption{Subtractions and additions for \ndiagrams~\\ref{DW-GR-GRs-Skt-CTP-E}\nand~\\ref{DW-GRs-Skt-CTP-E}, \npart~2.}\n\\label{fig:DW-GR-GRs-Skt-CTP-E-Add-II}\n\\end{figure}\\end{center}\n\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\mp 4 \n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\t\\LLDi{GRs-dEP-GR-Skt-CTP-E}{GRs-dEP-GR-Skt-CTP-E-dv-s}{GRs-dEP-GR-Skt-CTP-E-dv-a}\n\t\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\cd{dvj+}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{}\n\t\t\t\t\\\\\n\t\t\t\t\t\\SumVertex\t\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\t\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for \ndiagram~\\ref{GRs-dEP-GR-Skt-CTP-E}, part~1.}\n\\label{fig:GRs-dEP-GR-Skt-CTP-E-Add}\n\\end{figure}\\end{center}\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\pm 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\t\\LLDi{TEGRs-dEP-GR-Skt-CTP-E}{TEGRs-dEP-GR-Skt-CTP-E-s}{TEGRs-dEP-GR-Skt-CTP-E-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\t+\\LLDi{GRs-dEP-GR-TEGRs-Skt-CTP-E}{GRs-dEP-GR-TEGRs-Skt-CTP-E-s}{GRs-dEP-GR-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t}{}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\! \\sum_{m'''=2}^{m''} \\!\\! \\nCr{m''}{m'''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\\LLDi{GRs-dEP-GR-Skt-CTP-E-b}{GRs-dEP-GR-Skt-CTP-E-TEGRs-s}{GRs-dEP-GR-Skt-CTP-E-TEGRs-a}\n\t\t\t\t\t\t\\hspace{2em}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m'''}}\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for \ndiagram~\\ref{GRs-dEP-GR-Skt-CTP-E}, part~2.}\n\\label{fig:GRs-dEP-GR-Skt-CTP-E-Add-II}\n\\end{figure}\\end{center}\n\n\n\n\\section{An Expression for $\\beta_n$}\n\\label{sec:beta_n}\n\n\n\n\\subsection{Initial Manipulations}\n\nIn this section,\nwe will derive a compact expression\nfor $\\beta_n$ which has no explicit\ndependence on either the seed action\nor the details of the covariantization \nof the cutoff. Our starting point is\nto consider \n\\[\n\t\\dec{\n\t\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) + \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)\n\t}{\\bullet}.\n\\]\n(We work with these auxiliary function\nbecause it is technically easier to do\nso than to work directly with\n${\\cal D}^{a,b}$. At the end of the calculation,\nwill be able to conveniently trade\n$\\mathcal{E}$s for ${\\cal D}$s.)\n\nAs a precursor to computing the flow of\nthese functions,\nit is clear that we will need to understand\nhow to compute the flow of a reduced vertex.\nRecall that a reduced vertex lacks a classical,\ntwo-point component. Consequently, the flow of\na reduced vertex lacks a component given\nby the flow of a classical, two-point vertex,\nwhich we can read off from \\fig{fig:TLTPs}.\nNow, the classical flow equation involves only\nthe $a_0$ term; there are no $\\beta$, $\\alpha$\nor $a_1$ terms. Hence, we need only take care\nwith the dumbbell term generated by the flow\nof a reduced vertex, which we will tag with an\n$R$. It is straightforward to check that a dumbbell \nterm tagged in this way must either possess\nat least one reduced vertex or must have a decorated\nkernel.\n\nWe begin by focusing on the terms\ngenerated by $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$\nand adjust the ranges the sums\nover $s$, $m$ and $j$ to remove diagrams which vanish\neven after the action of $-\\Lambda \\partial_\\Lambda|_\\alpha$.\nThis is trivial in the case where\n$-\\Lambda \\partial_\\Lambda|_\\alpha$ strikes an effective propagator: the limits of the\nsums just reduce to those of $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$.\nIf $-\\Lambda \\partial_\\Lambda|_\\alpha$ strikes a $>$ things are much\nthe same only now \nthe sum over $m$ must, of course, start from\none and not zero.\n\n\n\n\nHowever, things are more subtle in the case\nthat $-\\Lambda \\partial_\\Lambda|_\\alpha$ strikes a vertex. \nImmediately, since we must have at least one vertex,\nthe lower limit of the sum over $j$ increases from $-2$\nto $-1$. Consider\nnow the resulting $\\alpha$ and $\\beta$ terms.\nCompared to the parent vertex, the loop order\nof the daughter vertex is changed, but the number\nof legs remains the same. Thus, if\nthe parent\ndiagram vanishes at $\\Op{2}$, it is not necessarily\nthe case that the daughter $\\alpha$ and $\\beta$-terms also \nvanish\\footnote{Of course, it must be the case that the sum over\nall daughters, including those generated\nby $a_0$ and $a_1$, does vanish at $\\Op{2}$.}:\nit is clear from~\\eqs{eq:S_0-11}{eq:S_>0-11}\nthat a change to the loop order of a two-point\nvertex carrying momentum $p$ can\nchange the order of its leading contribution in $p$\n.\nHowever, if the parent diagram vanishes because it\npossesses a one-point vertex, then the daughter $\\alpha$ and $\\beta$-terms\nwill vanish for this reason also. Consequently, \nfor the $\\alpha$ and $\\beta$-terms\nthe minimum\nvalue  of $s$ stays the same, but the maximum values of both\n$m$ and $j$ are reduced by one. \n\nWith these points in mind, it is worth considering\nthe $\\alpha$ and $\\beta$-terms in more detail and\nso, to this end, we explicitly give them in \\fig{fig:beta+alpha-Prelim}.\nThe combinatoric factor follows from recognizing\nthat the $-\\Lambda \\partial_\\Lambda|_\\alpha$ can strike any of the $j+2$ vertices\nand using~\\eq{eq:norm-c}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t-2\n\t\\dec{\n\t\t\\begin{array}{c}\n\t\t\\vspace{1ex}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{u}\n\t\t\t\\sum_{s=0}^{n-1}   \\sum_{m=0}^{2s} \\sum_{j=-1}^{n+s-m-1-u}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\\sum_{v^{k}=1}^{v^{j_+}} \\delta(v^k - u)\n\t\t\\\\\n\t\t\t\\left[\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\t2 \\left( v^{j_+,k}-1 \\right) \\beta_{v^{k}}\n\t\t\t\t\t\t\t\t+ \\gamma_{v^{k}} \\pder{}{\\alpha}\n\t\t\t\t\t\\right]\n\t\t\t\t\t\\Vertex{v^{j_+\\!,k}}\n\t\t\t\t}{\\SumVertex}\n\t\t\t\\right]\n\t\t\\end{array}\n\t}{11\\Delta^{j+s+1} >^m}\n\t\\]\n\\caption{The $\\alpha$ and $\\beta$ terms produced by \n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$.}\n\\label{fig:beta+alpha-Prelim}\n\\end{figure}\\end{center}\n\nThere are a number of important points to make.\nFirst, it is understood \nthat the sum over the vertex argument $v^k$---which is identified with\n$v^{j+2}$---is\nperformed \\emph{after}\nthe sums over $v^1 \\cdots v^{j_+}$ buried in the diagrammatics \n({\\it cf.}\\  \\eq{eq:VertexTower}). Secondly, we notice a\nsum over the new variable, $u$, and a $\\delta$-function\n$\\delta(v^k - u)$. This is to ensure that the upper limit\non the sum over $j$ is given correctly, given that\nthe sum over\nvertex arguments is now $n-s-v^k$.\nFinally, the upper limit on the sum over $s$ has been reduced\nby unity. This follows from the minimum value of $v^k$\nbeing unity ($\\beta_0$ does not exist) which, in turn, forces $n-s$\nto take a minimum value of unity. The expression for the $\\beta$\nand $\\alpha$ terms is quite ugly in its current form,\nbut it can be neatened up by a change of variables:\n\\[\n v^{i>0} \\rightarrow v^i + v^k,\n\\]\nto give the diagrams of \\fig{fig:beta+alpha-Prelim-b}. Notice that\nwe have changed the dummy variable $k$ to $n'$ which\nnow appears as the outer sum and also as the label for\nboth $\\alpha$ and $\\beta$.\n\\begin{center}\\begin{figure}[h]\n\\[\n\t\t-2\\dec{\n\t\t\t\\begin{array}{c}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{n'=1}^{n}\n\t\t\t\t\\sum_{s=0}^{n-n'}   \\sum_{m=0}^{2s} \\sum_{j=-1}^{n-n'+s-m-1}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\\\\\n\t\t\t\t\\left[\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\t2 \\left( v^{j_+}-1 \\right) \\beta_{n'} \t+  \\gamma_{n'} \\pder{}{\\alpha}\n\t\t\t\t\t\t\\right]\n\t\t\t\t\t\t\\DisplacedVertex{v^{j_+}}\n\t\t\t\t\t}{\\Vertex{n_{n'+s}, j}}\n\t\t\t\t\\right]\n\t\t\t\\end{array}\n\t\t}{11\\Delta^{j+s+1} >^m}\n\\]\n\\caption{A re-expression of the $\\alpha$ and $\\beta$-terms.}\n\\label{fig:beta+alpha-Prelim-b}\n\\end{figure}\\end{center}\n\nWe now focus on the $s=0$ components of these diagrams.\nDemanding that the resulting terms are connected,\nwe discard all diagrams possessing one-point (Wilsonian\neffective action) vertices. Utilizing~\\eqs{eq:S_0-11}{eq:S_>0-11}\nit is straightforward to demonstrate that the sole\ncontribution which survives at $\\Op{2}$ is:\n\\begin{eqnarray*}\n\t&\n\t-2 \\ensuremath{\\Upsilon}_{0,0}\n\t\\left(\n\t\t-2\\beta_n + \\gamma_n \\pder{}{\\alpha}\n\t\\right)\n\t\\cd{CTP-EE}\n\\\\\n\t= & -4\\beta_n \\Box_{\\mu \\nu}(p) + \\Op{4},\n\\end{eqnarray*}\nwhere we have used~\\eq{eq:norm}.\n\nReturning to our consideration of \n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$,\nwe note that\nterms generated by the action of $a_0$ are particularly\nsimple to deal with: the ranges of the sums over \n$s$, $m$ and $j$ just stay the same. This follows \nbecause $a_0$ converts a Wilsonian effective action\nvertex into a dumbbell structure, components of\nwhich possess seed action vertices; seed action\nvertices, unlike their Wilsonian effective\naction counterparts, exist at the one-point level.\nSuppose\nthat the parent diagram vanishes at $\\Op{2}$,\non account of an $\\nLV{S}{n'}{\\mu\\nu}{}(p)$\nvertex. This can be converted\ninto (amongst other terms) a dumbbell possessing\na one-point, $n'$-loop seed action vertex,\njoined to a classical, three-point vertex.\nThis term no longer vanishes at $\\Op{2}$.\nIf the parent diagram vanishes on account\nof a one-point Wilsonian effective action\nvertex, the daughter\ncan survive \nsince the vanishing vertex of the parent can\nbe converted into a dumbbell structure\npossessing a one-point, seed action vertex\njoined to a two-point vertex.\n\nFinally, consider the effect of the action\nof $a_1$. Due to the equivalence of each of the vertices\nin a given term, we can take the $a_1$ to strike\nthe vertex with argument $v^{j_+;R}$ (so long as we multiply by $j+2$), causing\nthe argument to become\n$\\Sigma_{v^{j_+}-1}$.\nThe $R$ has been dropped,\nsince a quantum term is necessarily formed from a \nvertex whose argument is greater than zero. The\nlower limit on the sum over $v^{j_+}$ should\nbe now changed from zero to one, in recognition of the\nfact that\n$a_1$ does not act on tree level terms. Furthermore,\nthe sum over all vertex arguments is now $n+s-1$,\nrather than $n+s$, and so we should reduce the\nupper limit of the sum over $s$ by one. It is\nconvenient to change variables:\n\\[\n\tv^{i>0} \\rightarrow v^i + 1\n\\]\nand then to let $s \\rightarrow s-1$. This affects\nthe ranges of all three sums.\n\nThe diagrams formed by  $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$\nare shown in \\fig{fig:bn-b-P}. It is understood\nhere, and in all that follows, that the\nvertex argument $v^k$ is to be identified with $v^{j+2}$.\nIf this vertex argument appears in more than one vertex\n(as will always be the case) it is understood to be\nsummed over, in  the usual manner.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2.5ex}\n\t\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet} + \\Op{4} = -4 \\beta_n \\Box_{\\mu\\nu}(p) \n\t\\\\\n\t\\vspace{3ex} \n\t\t-2\\dec{\n\t\t\t\\begin{array}{c}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{n'=1}^{n-1}\n\t\t\t\t\\sum_{s=1}^{n-n'}   \\sum_{m=0}^{2s} \\sum_{j=-1}^{n-n'+s-m-1}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\\\\\n\t\t\t\t\\left[\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t\t\t\\vspace{1ex}\t\n\t\t\t\t\t\t\t\t\\LD{beta-terms}\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t& \\LD{alpha-terms}\t\t\t\n\t\t\t\t\t\t\t\\\\\t\n\t\t\t\t\t\t\t\t2 \\left( v^{j_+}-1 \\right) \\beta_{n'} \t& \\hspace{-1em} + \\hspace{-1em}\t& \\gamma_{n'} \\pder{}{\\alpha}\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\\right]\n\t\t\t\t\t\t\\DisplacedVertex{v^{j_+}}\n\t\t\t\t\t}{\\Vertex{n_{n'+s}, j}}\n\t\t\t\t\\right]\n\t\t\t\\end{array}\n\t\t}{11\\Delta^{j+s+1} >^m}\n\t\\\\\n\t\\vspace{3ex}\n\t\t\\displaystyle\n\t\t-\\sum_{s=0}^n   \\sum_{m=0}^{2s+1} \\sum_{j=-1}^{n+s-m-1}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\\dec{\n\t\t\t\\sco[1]{\n\t\t\t\t\\displaystyle\n\t\t\t\n\t\t\t\t\\left[\n\t\t\t\t\t\\LDi{Dumbbell-vj_+kb-vkb}{D-vj_+kb-vkb}\n\t\t\t\t\\right]_R\n\t\t\t}{\\Vertex{ n_s, j}}\n\t\t}{11\\Delta^{j+s+1} >^m}\n\t\\\\\n\t\\vspace{3ex}\n\t\t\\displaystyle\n\t\t+\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t\\dec{\n\t\t\t\\sco[1]{\n\t\t\t\t\\LDi{Padlock-Sig_vj_+}{P-Sig_vj_+}\n\t\t\t\t+\\LDi{WBT-Sig_vj_+}{WBT-Sig_vj_+}\n\t\t\t}{\\Vertex{ n_{s}, j}}\n\t\t}{11\\Delta^{j+s} >^m}\n\t\\\\\n\t\t\\displaystyle\n\t\t+\n\t\t\\left[\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s} \\sum_{j=-2}^{n+s-m-2}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\\\\n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\\LO{\\Vertex{v^{j_+\\!;R}}}{j+2-DEP}}{\\Vertex{n_s, j}}\n\t\t\t\t}{11\\Delta^{j+s} \\stackrel{\\odot}{\\Delta} >^m}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\t+\n\t\t\\left[\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t2 \\sum_{s=1}^n   \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{(m-1)!}\n\t\t\t\\\\\n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\\LO{\\Vertex{v^{j_+\\!;R}}}{j+2-DGR}}{\\Vertex{n_s, j}}\n\t\t\t\t}{11\\Delta^{j+s+1} >^{m-1} \\stackrel{\\bullet}{>}}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\end{array}\n\t\\]\n\\caption{An expression for $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$.}\n\\label{fig:bn-b-P}\n\\end{figure}\\end{center}\n\n\n\nIt is worth considering the diagrams\narising from explicitly decorating~\\ref{j+2-DEP}\nwith the differentiated effective propagator\nin more detail. For convenience, these diagrams\nare collected together in \\fig{fig:app:DEP}.\nExplicit decoration with the differentiated effective\npropagator has the capacity to further change\nthe limits on the various sums. For example,\nif we attach both ends to the same vertex,\nthen this will reduce the maximum value of $m$\nby one since, in order to form a connected\ndiagram, the vertex decorated by the\ndifferentiated effective propagator must be at least\nthree-point. This effectively uses up a decoration\nwhich could otherwise have filled a $>$.\n\n\n In the case that each end of the differentiated\neffective propagator attaches to a different vertex,\nwe note that such a diagram can only exist for $j>-1$.\nOur strategy here is to shift $j \\rightarrow j+1$,\nso that the sum over $j$ starts from $-1$,\nidentifying $v^{j+2}$ with $v^k$ as usual.\n\n\nWe now isolate all classical, two-point vertices\nin diagrams~\\ref{D-vj_+kb-vkb}--\\ref{WBT-Sig_vj_+}. \nThis step is crucial to\nthe entire diagrammatic procedure: if a classical,\ntwo-point vertex is attached to an effective propagator,\nwe can employ the effective propagator relation, whereas\nif it is attached to an external field, we can perform\nmanipulations at $\\Op{2}$. Although\nthe classical, two-point vertices of diagram~\\ref{P-Sig_vj_+}\nare fully decorated, it will nonetheless prove useful\nto isolate them, as this will help elucidate\nthe structure of the forthcoming cancellations.\n\nLet us consider isolating the classical, two-point\nvertices of diagram~\\ref{D-vj_+kb-vkb} in more detail.\nFirst, let us take both vertices\nof the dumbbell structure to be reduced vertices.\nThis immediately allows us to reduce the maximum \nvalue of the sum over $j$ by one,\nsince the\nformation of the dumbbell term causes the\ntotal number of vertices to increase from\n$j+2$ to $j+3$ (and the maximum number of reduced\nvertices we can decorate in the formation of\na connected diagram remains the same).\nOf the terms which remain, consider\nthose possessing exclusively \nWilsonian \neffective action vertices. Since we discard\none-point, Wilsonian effective action vertices\nin all diagrams in which $\\Lambda \\partial_\\Lambda|_\\alpha$ has acted,\nwe can reduce the maximum values of \n$j$ and $m$ and increase\nthe minimum value of $s$ by one.\nThe component of the resulting diagram in which\nthe kernel is undecorated is exactly cancelled\nby the component of diagram~\\ref{j+2-DEP}\nin which the differentiated effective propagator\njoins two different vertices (see diagram~\\ref{D-vj_+kR-DEP-vkR}). \nSince the surviving\ncomponent has a decorated kernel,\nthe maximum values of both $m$ and $j$ are reduced\nby one, again.\n\nIsolating a single classical, two-point vertex\nin diagram~\\ref{D-vj_+kb-vkb} is straightforward:\ntaking the argument of the top or bottom vertex\nof the dumbbell structure to be a classical, two-point\nvertex amounts to the same thing; hence we will choose\nto isolate the classical, two-point part of $\\overline{v}^k$ \nand multiply by two. When taking the classical\npart of $\\overline{v}^k$,\nthe other vertex argument,\n$\\overline{v}^{j_+,k;R} \\equiv \\overline{v}^{j_+;R} - \\overline{v}^{k;R}$, reduces to \nsimply $\\overline{v}^{j_+;R}$. \nWe now expanded out the barred vertices,\naccording to~\\eq{eq:bar}, noting that certain terms cancel\non account of us having\nset the classical, two-point seed action vertices\nequal to their Wilsonian effective action counterparts.\nThere is no need to change the \nlimits on any of the sums: compared to the\nparent diagram, we have an extra two-point vertex,\nbut also an extra two decorative fields, corresponding\nto the two ends of the kernel. \n\n\nTaking two classical, two-point vertices in\ndiagram~\\ref{D-vj_+kb-vkb} requires some thought.\nFirst, we note that the resulting dumbbell\nstructure cannot have been formed by\nthe flow of a one-point vertex. Since we are\ninterested only in one-point Wilsonian effective\naction vertices if\nthey have been processed,\nwe can reduce the maximum values of $m$\nand $j$ and increase the\nminimum value of $s$ to remove any unwanted diagrams.\nIf $j=-1$, then there are only two vertices\nin total, and so this case should be treated differently\nfrom $j \\geq 0$. In the latter case, it will prove\nuseful to shift variables $j \\rightarrow j+1$, so\nthat the sum over $j$ starts, once again, from $-1$.\nWe can then recombine terms into a diagram\nwith a sum that starts from $j=-2$.\nThe isolation of the\nclassical, two-point vertices of \ndiagram~\\ref{D-vj_+kb-vkb} is shown in \n\\fig{fig:bn-Ia}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\t\\begin{array}{l}\n\t\t\\vspace{2ex}\n\t\t\t-\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-1}^{n+s-m-4} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-RW-vkR} \\Vertex{n_s,j}}{D-vj_+kR-vkR}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\\\\\n\t\t\\vspace{2ex}\n\t\t\t+2\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s+1}  \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-vkhR} \\Vertex{n_s,j}}{D-vj_+kR-vkhR}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\\\\\n\t\t\t+\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\\LDi{Dumbbell-02x2-R}{D-02x2-R-VS}\n\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\\Vertex{v^{j_+\\!;R}}\n\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\Vertex{n_s, j}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{11\\Delta^{j+s+2} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\t+2\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s+1}  \\sum_{j=-1}^{n+s-m-1} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\t\\displaystyle\t\t\t\t\n\t\t\t\t\t\t\t\\LDi{Dumbbell-vj_+h-02}{D-vj_+R-02}\n\t\t\t\t\t\t}{\\Vertex{n_s, j}}\t\t\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\t\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\\end{array}\n\t\\]\n\\caption{Isolation of the classical, two-point vertices\nof diagram~\\ref{D-vj_+kb-vkb} plus diagram~\\ref{D-vj_+kR-DEP-vkR}.}\n\\label{fig:bn-Ia}\n\\end{figure}\\end{center}\n\nIt is now easy to isolate the classical, two-point\nvertices of diagrams~\\ref{P-Sig_vj_+} and~\\ref{WBT-Sig_vj_+},\nthough we note the following. First, on account\nof the equality of the Wilsonian effective action,\nclassical, two-point vertices and their seed action \ncounterparts, $\\Sigma_{0 RS}^{\\ XX}(k) = -S_{0 RS}^{\\ XX}(k)$;\nin other words,\nthe classical, two-point component of $\\Sigma$ has a minus\nsign, relative to the reduced part. Secondly, the\nkernel attaching to the classical, two-point\nvertex contained in diagram~\\ref{P-Sig_vj_+} must be\ndecorated, in order that a connected diagram\ncan be formed.\nFinally, the component of diagram~\\ref{P-Sig_vj_+}\nin which all vertices are Wilsonian effective\naction vertices and the kernel is undecorated\nis cancelled by\n the component of diagram~\\ref{j+2-DEP}\nin which the differentiated effective propagator\nattaches at both ends to the same vertex (see diagram~\\ref{vj_+R-DEP}).\nThe isolation of the\nclassical, two-point vertices of \ndiagrams~\\ref{P-Sig_vj_+} and~\\ref{WBT-Sig_vj_+} is shown in \n\\fig{fig:bn-Ib}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\n\t\t\t\t\t-2 \\LDi{Padlock-hS_vj_+R}{P-hS_vj_+R}\n\t\t\t\t}{\\Vertex{ n_{s}, j}}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\LDi{Padlock-02}{P-02-MV}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\Vertex{v^{j_+\\!;R}}\n\t\t\t\t\\\\\n\t\t\t\t\t\\Vertex{n_s, j}\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\t\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\displaystyle\t\n\t\t\t\t\t\\LDi{Padlock-S_vj_+R}{P-S_vj_+R}\t\t\t\n\t\t\t\t\t+\\LDi{WBT-vj_+R}{WBT-vj_+R}\n\t\t\t\t\t-2\\LDi{WBT-vj_+hR}{WBT-vj_+hR}\n\t\t\t\t}{\\Vertex{n_s, j}}\t\t\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\t\t\n\t\t}\n\t\t\t{\t\t\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\LDi{WBT-02}{WBT-02-VS}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\Vertex{v^{j_+\\!;R}}\n\t\t\t\t\\\\\n\t\t\t\t\t\\Vertex{n_s, j}\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Isolation of the classical, two-point vertices\nof diagrams~\\ref{P-Sig_vj_+} and~\\ref{WBT-Sig_vj_+} plus\ndiagram~\\ref{vj_+R-DEP}.}\n\\label{fig:bn-Ib}\n\\end{figure}\\end{center}\n\nThe next step of the diagrammatic procedure\nis to decorate the classical, two-point\nvertices of diagrams~\\ref{D-02x2-R-VS}, \\ref{D-vj_+R-02}\nand~\\ref{WBT-02-VS} with either\nan external field\nor an end of an effective propagator. \nIn the latter case we must then attach\nthe loose end of the effective propagator\nto an available structure. We refer to the \nprimary part of \na diagram as the component left over after\napplying the effective propagator relation\nas many times as possible but, each time,\nretaining only the \nKronecker-$\\delta$ contribution.\n\nAssuming that\nthe necessary structures exist, we can do the\nfollowing with a diagram possessing a single\nclassical, two-point vertex:\n\\begin{enumerate}\n\t\\item\tattach an external field;\n\n\t\\item\tattach one end of an effective propagator,\n\t\t\twith the other end attaching to:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tthe seed action vertex to which the kernel\n\t\t\t\t\t\tattaches;\n\t\t\t\t\\label{EP-SV}\n\n\t\t\t\t\\item\tone of the Wilsonian  effective action vertices;\n\t\t\t\t\\label{EP-DV}\n\n\t\t\t\t\\item\tthe kernel;\n\t\t\t\t\\label{EP-k}\n\n\t\t\t\t\\item\ta gauge remainder.\n\t\t\t\t\\label{EP-GR}\n\t\t\t\\end{enumerate}\n\\end{enumerate}\n\nIn each of~\\ref{EP-SV}--\\ref{EP-GR}\nthe effective propagator relation\ncan be applied. For the time\nbeing, we will concern ourselves\nwith just the primary\nparts of~\\ref{EP-SV}--\\ref{EP-k},\nwhich yield a series of cancellations,\nreturning to both the gauge remainder\ncontributions and other diagrams later.\nFirst, we analyse the result of\ndecorating the classical, two-point\nvertex of diagram~\\ref{D-vj_+R-02}.\n\n\\begin{cancel}[Diagram~\\ref{D-vj_+kR-vkhR}]\n\\label{Cancel:D-vj_+kR-vkhR}\nConsider the primary part of\ndiagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-DV}, above, which we note\nexists only for $j>-1$. \nFor comparison with diagram~\\ref{D-vj_+kR-vkhR},\nit is convenient\nto change variables $j \\rightarrow j+1$,\nso that the sum over $j$ once again starts\nfrom $-1$ and\nto identify $\\hat{v}^{j+2}$ with $\\hat{v}^{k}$.\nThus, \nthe two-point, tree level vertex of diagram\\ref{D-vj_+R-02}\ncan be joined to any of $j+2$ identical Wilsonian\neffective action vertices, using\nany of $j+s+2$ effective propagators.\nNoting that this effective propagator can\nattach either way round,  the\ncombinatoric factor is \n\\[\n\t2 (j+s+2)(j+2),\n\\]\nwhich, from \\eqs{eq:norm-b}{eq:norm-c}, \ncombines with $\\ensuremath{\\Upsilon}_{j+s+2,j+2}$\nto give $-\\ensuremath{\\Upsilon}_{j+s+1,j+1}$. Thus,\nthe primary part of\ndiagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-DV}, above,\nprecisely cancels\ndiagram~\\ref{D-vj_+kR-vkhR}.\n\\end{cancel}\n\n\n\\begin{cancel}[Diagram~\\ref{P-hS_vj_+R}]\n\\label{Cancel:P-Sig_vj_+R-hS}\n\nConsider the primary\npart of diagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-SV}, above, which we might\nhope cancels diagram~\\ref{P-hS_vj_+R}.\nAt first sight, though, it does not\nlook like the cancellation will quite work\nsince, although the combinatorics\nare fine, the limits on the sums over $s, \\ m$\nand $j$ differ between diagrams~\\ref{D-vj_+R-02}\nand~\\ref{P-hS_vj_+R}. However, the formation\nof a loop in diagram~\\ref{D-vj_+R-02}\nallows us to restrict the ranges\nover these sums (just as we did for diagram~\\ref{vj_+R-DEP}),\nand so the two diagrams do exactly cancel.\n\\end{cancel}\n\n\n\\begin{cancel}[Diagram~\\ref{WBT-vj_+hR}]\n\nConsider the primary\npart of diagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-k}, above. Following through\nsimilar arguments to those invoked in \ncancellation~\\ref{Cancel:P-Sig_vj_+R-hS}, it can\nbe shown that the primary\npart of diagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-k} exactly cancels diagram~\\ref{WBT-vj_+hR}.\n\n\\end{cancel}\n\nThis concludes our initial discussion of\ndiagram~\\ref{D-vj_+R-02}.\nRather than immediately analysing\ndiagram~\\ref{WBT-02-VS}, which also possesses\na single classical, two-point vertex,\nit is useful to first consider\nthe decoration of diagram~\\ref{D-02x2-R-VS}\nwhich\npossesses two such vertices.\nAt $\\Op{2}$, we can discard the case in which\nan external field attaches to each of the classical,\ntwo-point vertices. If one of the vertices\nis decorated by an external field, then\nwe can attach an effective propagator to the\nother according to~\\ref{EP-DV}--\\ref{EP-GR}\nabove. If both vertices are decorated by an\nend of an effective propagator, then we can tie up\nthe associated loose ends in the following\nindependent ways: \n\\begin{enumerate}\n\t\\item\tJoin one classical, two-point vertex \n\t\t\tto a Wilsonian effective action vertex \n\t\t\tand the other to:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tthe same vertex;\n\t\t\t\t\\label{EPx2-SV}\n\n\t\t\t\t\\item\ta different vertex;\n\t\t\t\t\\label{EPx2-DV}\n\n\t\t\t\t\\item\tthe kernel;\n\t\t\t\t\\label{EPx2-V-k}\n\n\t\t\t\t\\item\ta gauge remainder;\n\t\t\t\\end{enumerate}\n\n\t\\item\tJoin one classical, two-point vertex  to the kernel and the other to:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tthe kernel;\n\t\t\t\t\\label{EPx2-k-k}\n\t\n\t\t\t\t\\item\ta gauge remainder;\n\t\t\t\t\\label{EPx2-k-GR}\n\t\t\t\\end{enumerate}\n\n\t\\item\tJoin one classical, two-point vertex  \n\t\t\tto a gauge remainder and the other to a gauge remainder\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tin the same gauge remainder structure;\n\n\t\t\t\t\\item\tin a different gauge remainder structure;\n\t\t\t\\end{enumerate}\n\n\t\\item\tJoin the classical, two-point vertices together.\n\t\t\t\\label{EP-02x2}\n\\end{enumerate}\n\n\nWe now consider the results of decorating\ndiagram~\\ref{D-02x2-R-VS} according\nto~\\ref{EPx2-SV}--\\ref{EP-02x2}.\n\n\\begin{cancel}[Diagram~\\ref{P-S_vj_+R}]\n\nThe primary part of~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-SV}, above, \nexactly cancels diagram~\\ref{P-S_vj_+R}.\n\\end{cancel}\n\n\\begin{cancel}[Diagram~\\ref{P-02-MV}]\nDiagram~\\ref{P-02-MV}\nis exactly cancelled by the primary part\nof diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EP-02x2},\nabove.\n\\end{cancel}\n\n\\begin{cancel}[Diagram~\\ref{D-vj_+kR-vkR}]\nThe primary part of diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-DV}, above,\nexactly cancels diagram~\\ref{D-vj_+kR-vkR}.\n\\end{cancel}\n\n\n\n\n\\begin{cancel}[Components of diagram~\\ref{WBT-02-VS}]\nThe primary part of diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-k-k}, above,\nexactly cancels the primary part of diagram~\\ref{WBT-02-VS}\nin which the classical, two-point vertex attaches\nto the kernel.\n\\label{Cancel:WBT-02}\n\\end{cancel}\n\n\nLooking at cancellation~\\ref{Cancel:WBT-02},\nit is tempting to say that the primary\nparts of diagram~\\ref{WBT-02-VS}\nare all\nexactly cancelled by a subset of the\nprimary parts\nof diagram~\\ref{D-02x2-R-VS}.\nHowever, this is not true: consider\nthe primary part of diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-k-GR}, above.\nIn this case, because each of the classical, two-point\nvertices attaches to something different,\nwe pick up a factor of two, in recognition\nof the fact that either of the vertices\ncould attach to the kernel \/ gauge remainder.\nHowever, the primary part of diagram~\\ref{WBT-02-VS}\nin which the classical, two-point vertex attaches\nto a gauge remainder can only be formed in one way;\nhence the cancellation is only partially realized.\nThe key to cancellation~\\ref{Cancel:WBT-02}\nis that both classical, two-point vertices\nof diagram~\\ref{D-02x2-R-VS} attach to the same structure,\nand so there is no factor of two picked up.\n\nA\ndiagram generated by the partial cancellation of\na component of\ndiagram~\\ref{WBT-02-VS}\nagainst a component of diagram~\\ref{D-02x2-R-VS}\nis immediately involved in a cancellation.\n\n\\begin{cancel}[Diagram~\\ref{WBT-vj_+R}]\n\\label{Cancel:WBT-vj_+R}\nThe primary part of diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-V-k} above and\nthe primary part of diagram~\\ref{WBT-02-VS}\nin which the classical, two-point attaches\nto a vertex combine to exactly cancel\ndiagram~\\ref{WBT-vj_+R}.\n\\end{cancel}\n\nUsing cancellations~\\ref{Cancel:D-vj_+kR-vkhR}--\\ref{Cancel:WBT-vj_+R} \nand partial cancellations of the type just discussed,\nwe can rewrite $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ as shown in \\fig{fig:bn-b-Pb}.\n$\\nLV{{\\cal G}}{n}{\\mu\\nu}{a}(p)$  and $\\nLV{{\\cal H}}{n}{\\mu\\nu}{a}(p)$\nare disjoint sets whose elements are\nall spawned by $a_0$ and $a_1$: all such diagrams which \npossess an $\\Op{2}$ stub belong to\n$\\nLV{{\\cal H}}{n}{\\mu\\nu}{a}(p)$;\nall elements\nof  $\\nLV{{\\cal G}}{n}{\\mu\\nu}{a}(p)$ possess a\nfull gauge remainder\n(arising from the application of the effective propagator relation).\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{3ex}\n\t\t\\displaystyle \\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet} + \\Op{4} = \n\t\t-4 \\beta_n \\Box_{\\mu\\nu}(p)  +\n\t\t\\mbox{\\ref{beta-terms}} + \\mbox{\\ref{alpha-terms}} +\n\t\t\\nLV{{\\cal G}}{n}{\\mu\\nu}{a}(p) + \\nLV{{\\cal H}}{n}{\\mu\\nu}{a}(p)\n\t\\\\\n\t\\vspace{3ex}\n\t\t+\n\t\t\\left[\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2}\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\t\\sum_{m'=1}^{m} \\!\\! \\nCr{m}{m'} \n\t\t\t\\\\\n\t\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle \\sum_{m''=1}^{m'-1}  \\!\\! \\nCr{m'}{m''} \\!\\! \n\t\t\t\t\t\\LDi{GRs-W-GRs}{GRs-W-GRs-MV} \\hspace{1.5em}\n\t\t\t\t\t+ \\LDi{Ubend-W}{Ubend-W-MV} \\hspace{1em}\n\t\t\t\t\t+ \\LDi{WBT-GRs}{WBT-GRs-MV}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\\\\n\t\\vspace{3ex}\n\t\t\\displaystyle\n\t\t+2 \\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{(m-1)!}\n\t\t\\dec{\n\t\t\t\\sco[1]{\\LO{\\Vertex{v^{j_+\\!;R}}}{j+2-DGR-b}}{\\Vertex{n_s, j}}\n\t\t}{11\\Delta^{j+s+1} >^{m-1} \\stackrel{\\bullet}{>}}\n\t\\\\\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=1}^{2s+1} \\sum_{m'=1}^m \\sum_{j=-1}^{n+s-m-1}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\nCr{m}{m'}\t\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\\LO{\\cd{v_j+hR-W-GRs}}{v_j+hR-W-GRs}}{\\Vertex{n_s, j}}\n\t\t\t\t-\\sco[1]{\\LO{\\cd{v_j+R-W-GRs}}{v_j+R-W-GRs}}{\\Vertex{n_s, j}}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{A re-expression of $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$.}\n\\label{fig:bn-b-Pb}\n\\end{figure}\\end{center}\n\nThe formation of diagram~\\ref{GRs-W-GRs-MV}\ndeserves further comment. This diagram\nis a combination of the following terms:\n\\begin{enumerate}\n\t\\item\tthe primary part\n\t\t\tof~\\ref{D-02x2-R-VS} in which both classical,\n\t\t\ttwo-point vertices attaches to a different\n\t\t\tgauge remainder structure\n\n\t\\item\tthe component of diagram~\\ref{j+2-DEP} in which both\n\t\t\tends of the differentiated effective propagator\n\t\t\tattaches to a different\n\t\t\tgauge remainder structure (see diagram~\\ref{GRs-DEP-GRs-MV}).\n\\end{enumerate}\n\nNotice that the limits on the sums over $m$ and $j$,\nin diagram~\\ref{GRs-W-GRs-MV} are the same as in\ndiagram~\\ref{j+2-DEP} and not~\\ref{D-02x2-R-VS}.\nDiagram~\\ref{D-02x2-R-VS} supplements diagram~\\ref{j+2-DEP},\nconverting a differentiated effective propagator\ninto a full kernel in all cases where there\nare sufficiently few gauge remainders and vertices\nto permit decoration of the kernel. \nThe values of $m$ and $j$ for which diagram~\\ref{j+2-DEP}\nexists but diagram~\\ref{D-02x2-R-VS} does not are\nthose for which the differentiated effective\npropagator of~\\ref{j+2-DEP} can be trivially\nreplaced by a full kernel, since this kernel\ncan effectively never be decorated.\n\nSimilar considerations apply to diagrams~\\ref{Ubend-W-MV}\nand~\\ref{v_j+R-W-GRs}. In the\nlatter case, to save space, we have put this diagram\nunder the same summations as diagram~\\ref{v_j+hR-W-GRs}.\nWhereas diagram~\\ref{v_j+hR-W-GRs} exists for\nthe maximum values of $m$ and $j$, diagram~\\ref{v_j+R-W-GRs}\npossesses a Wilsonian effective action, one-point\nvertex in these cases and so has no support: the\nmaximum values of $m$ and $j$ for which diagram~\\ref{v_j+R-W-GRs}\nexists are $2s$ and $n+s-m-2$, respectively.\nIn the same fashion, diagram~\\ref{WBT-GRs-MV} does \nnot in fact exist for the\nmaximum values of $m$ indicated. Having taken great\ncare, up until now, to restrict the ranges on\nthe sums as much as possible for each diagram,\nwe will now often put diagrams under the same summation\nsign to save space, mindful that not all diagrams\nnecessarily exist for all values of $s$, $m$ and $j$.\n\n\n\\subsection{Gauge Remainders}\n\\label{sec:bn-GRs}\n\nThe terms of $\\nLV{{\\cal G}}{n}{\\mu\\nu}{a}(p)$ can be usefully\ndecomposed into five sets comprising:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\Roman{enumi}}\n\t\\item\tdiagrams possessing a single full gauge remainder, \n\t\t\tthe $>$ part of which attaches to a kernel,\n\t\t\tthe other end of which attaches to:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\titself;\n\t\t\t\t\\label{GR-WBT}\n\t\n\t\t\t\t\\item\ta reduced seed action vertex;\n\t\t\t\t\\label{GR-S}\n\n\t\t\t\t\\item\ta reduced Wilsonian effective action vertex;\n\t\t\t\t\\label{GR-W}\n\n\t\t\t\t\\item\ta gauge remainder structure;\n\t\t\t\t\\label{GR-GRs}\n\t\t\t\\end{enumerate}\n\n\t\\item\tdiagrams possessing two full gauge remainders.\n\t\\label{GRsx2}\n\\end{enumerate}\n\nThe strategy for dealing with the gauge remainders\nis by now familiar:\nallowing the full gauge remainder\nto act, wherever possible, we isolate any\nclassical, two-point vertices and proceed as usual. \nWe start with the \ngauge remainder terms of type~\\ref{GR-WBT},\nwhich we collect together in \\fig{fig:bn:GR-WBT}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{WBT-GR-vj_+R}{WBT-GR-vj_+R}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{WBT-GR-B}{WBT-GR-B-MV}}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\t\n\t\\\\\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!} \\sum_{m'=2}^m \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{WBT-GR-GRs}{WBT-GR-GRs-MV}}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GR-WBT}.}\n\\label{fig:bn:GR-WBT}\n\\end{figure}\\end{center}\n\nThe pattern of cancellations we will find\nis very similar to that observed in the\ndemonstration of the transversality\nof $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$. Thus, although\ndiagram~\\ref{WBT-GR-GRs-MV}\npossesses a trapped gauge remainder, and so cannot\nbe processed, it has been collected with\nthe other terms in anticipation of a\ncancellation via diagrammatic identity~\\eq{eq:D-ID-G}.\n\n\nDiagram~\\ref{WBT-GR-B-MV} \ncan be processed by allowing the gauge remainder\nto act. This gauge remainder can strike one of three\nlocations:\n\\begin{enumerate}\n\t\\item\tthe top end of the kernel (which itself attaches\n\t\t\tto the middle of the kernel);\n\n\t\\item\tthe bottom end of the kernel;\n\n\t\\item\ta socket, which can be filled by one of the\n\t\t\tdecorations. \n\\end{enumerate}\n\nSumming over diagrams of the first item\nyields zero:\nbecause the kernel\nbites itself, its top end can be struck in two\nindependent ways. First, the active gauge remainder can \npush forward (pull back) onto the field on the\nkernel which the kernel bites; this field\nis identified with the top end of the kernel.\nSecondly, the active gauge remainder\ncan pull back (push forward)  unhindered to the top.\nThese two independent cases exactly cancel\neach other (see also the discussion around\ndiagrams~\\ref{Diags-WGRx2-Trap}--\\ref{Diags-WGRx2-Bottom-B}).\n\nNow we consider processing the active\ngauge remainder of diagram~\\ref{WBT-GR-vj_+R}\nand splitting off the classical, two-point\ncomponent from the reduced component.\nCompared to diagram~\\ref{CTP-EGR-Socket},\nthe classical, two-point vertex can,\nin addition to being attached  \nto a reduced Wilsonian\neffective action vertex or gauge remainder\nstructure, be attached  \nto the kernel or decorated with an \nexternal field. If it attaches\nto the kernel, this just cancels\nthe component of diagram~\\ref{WBT-GR-B-MV} \nwhere the gauge remainder bites a socket\non the kernel, up to a gauge remainder\ncontribution. This cancellation,\ntogether with the analogue of\nillustrative cancellation~\\ref{Icancel:Tansverse}\noccurs at each level of nesting.\nOf those diagrams which survive, two\ncancel against diagram~\\ref{WBT-GR-GRs-MV}\nvia diagrammatic identity~\\eq{eq:D-ID-G}.\n\nUp to those diagrams with an $\\Op{2}$\nstub, the only diagrams which survive\nare those spawned by diagram~\\ref{WBT-GR-B-MV} \nand its nested partners in which the\ngauge remainder strikes the bottom end\nof the kernel, as shown in \\fig{fig:bn-GR-WBT-Surv}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m'=1}^{2s-1} \\sum_{m=0}^{2s-1-m'} \\sum_{j=-2}^{n+s-m-m'-3} \t\n\t\t}\n\t\t\t{\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!m'!}\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{WBT-GRs}{WBT-ring}}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\]\n\\caption{Surviving terms generated by the diagrams of \\fig{fig:bn:GR-WBT}, \nup to terms with an $\\Op{2}$ stub.}\n\\label{fig:bn-GR-WBT-Surv}\n\\end{figure}\\end{center}\nThe sign and combinatoric factor of diagram~\\ref{WBT-ring} \nfollow from the discussion\naround \\fig{fig:nestedhook}.\n\n\n\\begin{cancel}[Diagram~\\ref{WBT-GRs-MV}]\n\tDiagram~\\ref{WBT-ring} exactly cancels\n\tdiagram~\\ref{WBT-GRs-MV}. This follows\n\tfrom first letting $m \\rightarrow m-m'$ in~\\ref{WBT-ring}\n\tand recognizing that \n\t\\[\n\t\\sum_{m'=1}^{2s-1} \\sum_{m=m'}^{2s-1} = \\sum_{m=1}^{2s-1} \\sum_{m'=1}^m,\n\t\\]\n\tand secondly recalling that the upper limits on the sums over\n\t$m$ and $j$ in diagram~\\ref{WBT-GRs-MV} can be reduced by\n\tone apiece.\n\\end{cancel}\n\n\nWe have thus demonstrated how all diagrams\nspawned by the type~\\ref{GR-WBT}\ngauge remainders, up to those with an $\\Op{2}$\nstub, cancel. \nIn preparation for our treatment of the other \ngauge remainders, it is worth reviewing this\nhow this happens.\nIf an active gauge remainder\nstrikes a socket on a reduced vertex\nor on a kernel, this diagram will be\ncancelled by a diagram in which the gauge\nremainder instead strikes a socket on\na classical, two-point vertex, which\nis subsequently attached to either\na reduced vertex or a kernel with an \neffective propagator. Such cancellations\nleave over diagrams of exactly the same\nbasic form as the parent diagrams,\nbut where the gauge remainder is nested.\nAllowing the nested gauge remainder\nto act then just repeats the cancellations\njust described. We are left with following:\n\\begin{enumerate}\n\t\\item\tDiagrams possessing a \n\t\t\tclassical, two-point vertex attached to\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tan $\\Op{2}$ stub;\n\n\t\t\t\t\\item  a gauge remainder nested with respect to\\ to\n\t\t\t\t\t\tthe original, active gauge remainder;\n\t\t\t\t\\label{it:nested-wrt-active}\n\n\t\t\t\t\\item\ta gauge remainder structure.\n\t\t\t\t\\label{it:GR-structure}\n\t\t\t\\end{enumerate}\n\n\t\\item\tThe original type~\\ref{GR-WBT} diagram possessing a trapped\n\t\t\tgauge remainder;\n\t\\label{it:trapped}\n\n\t\\item\tDiagrams in which a kernel is bitten at its end by\n\t\t\ta (nested) gauge remainder.\n\t\\label{it:BittenKernel}\n\\end{enumerate}\nDiagrams of type~\\ref{it:nested-wrt-active}, \\ref{it:GR-structure}\nand~\\ref{it:trapped} cancel, courtesy of diagrammatic\nidentity~\\ref{eq:D-ID-G}.\nDiagrams of type~\\ref{it:BittenKernel} cancel\nagainst terms generated elsewhere in the calculation.\n\n\nWe thus see that most of the diagrams spawned by the\ntype~\\ref{GR-WBT} gauge remainders cancel amongst themselves.\nThe wonderful thing is that this basic pattern of cancellations\nis repeated for all the other types of gauge remainders. \nThe differences come with the types of terms generated\nwhich cancel against terms generated elsewhere in\nthe calculation. With this in mind, we now treat\nthe gauge remainders of types~\\ref{GR-S} and~\\ref{GR-W}, which\nare collected together in \\figs{fig:bn-GR-S}{fig:bn-GR-W},\nrespectively.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{4ex}\n\t\t+2\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s}  \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-vkhR-GR} \\SumVertex}{D-vj_+kR-vkhR-GR}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\\\\\n\t\\vspace{4ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{Padlock-hS_vj_+R-GR}{P-hS_vj_+R-GR}\n\t\t\t\t\t+\\LDi{WBT-vj_+hR-GR}{WBT-vj_+hR-GR}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n \\sum_{m=2}^{2s} \\sum_{m'=2}^m \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\nCr{m}{m'}\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{vj_+h-W-GR-GRs}{vj_+h-W-GR-GRs}\n\t\t\t\t}{\\SumVertex}\t\t\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GR-S}.}\n\\label{fig:bn-GR-S}\n\\end{figure}\\end{center}\n\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{4ex}\n\t\t-2\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-1}^{n+s-m-4} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-vkR-DW-GR} \\SumVertex}{D-vj_+kR-vkR-DW-GR}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\\\\\n\t\\vspace{4ex}\n\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{Padlock-S_vj_+R-DW-GR}{P-S_vj_+R-DW-GR}\n\t\t\t\t\t+\\LDi{WBT-vj_+R-DW-GR}{WBT-vj_+R-DW-GR}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n \\sum_{m=2}^{2s-1} \\sum_{m'=2}^m \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\nCr{m}{m'}\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{vj_+R-DW-GR-GRs}{vj_+R-DW-GR-GRs}\n\t\t\t\t}{\\SumVertex}\t\t\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GR-W}.}\n\\label{fig:bn-GR-W}\n\\end{figure}\\end{center}\n\nIt is apparent that the gauge remainders\nof types~\\ref{GR-S} and~\\ref{GR-W} are\nalmost exactly the same, which is why\nwe choose to treat them together.\nThe differences are first that the gauge\nremainders of type~\\ref{GR-S} possess\na seed action vertex and secondly that the kernels\nof all type~\\ref{GR-W} diagrams\nmust be decorated. In diagrams~\\ref{D-vj_+kR-vkhR-GR}\nand~\\ref{vj_+h-W-GR-GRs}, we note\nthat we can reduce the maximum values\nof the sums over $m$ and $j$ by one more\nthan expected: if the seed action vertex is\none-point, then the kernel must be decorated,\nelse the full gauge remainder is forced\nto be in the $C^i$ sector---where of course it\nis null---by charge conjugation\\ invariance.\n\nProcessing the gauge remainders, it is\nobvious what will happen. \nDiagram~\\ref{D-vj_+kR-vkhR-GR} (\\ref{D-vj_+kR-vkR-DW-GR})\nhas two components: one in which the\ngauge remainder strikes a socket\non a reduced vertex and one in which\nthe gauge remainder strikes a socket\non a classical, two-point vertex.\nThe former case is cancelled, up to a nested \ngauge remainder by the contribution in which\nthe classical, two-point\nvertex is joined to one of the Wilsonian\neffective action vertices. If, instead,\nthe classical, two-point vertex is attached\nto a socket on either the seed action vertex or\nkernel then, up to a nested gauge remainder, \nthis cancels the contributions\nfrom diagrams~\\ref{P-hS_vj_+R-GR} \n(\\ref{P-S_vj_+R-DW-GR}) and~\\ref{WBT-vj_+hR-GR} (\\ref{WBT-vj_+R-DW-GR})\nin which, respectively, the gauge strikes a socket\non the reduced component of the seed action\nvertex and a socket on the kernel.\nUp to terms with an $\\Op{2}$ stub, the final type of diagram \nspawned by~\\ref{D-vj_+kR-vkhR-GR} (\\ref{D-vj_+kR-vkR-DW-GR})\nis one in which the classical, two-point vertex\nformed by the action of the gauge remainder\nis attached to a gauge remainder structure.\nPutting this to one side for the moment,\nwe note that  processing the nested gauge remainders\nrepeats this pattern of cancellations. \n\nAt the first level of nesting we get, as usual, a\nnew type of diagram: one in which the classical,\ntwo-point vertex generated by~\\ref{D-vj_+kR-vkhR-GR} \n(\\ref{D-vj_+kR-vkR-DW-GR}) attaches to the nested\ngauge remainder. Iterating the\ndiagrammatic procedure, diagrams of this type, together\nwith diagrams of the type temporarily put aside,\nexactly cancel diagram~\\ref{vj_+h-W-GR-GRs} (\\ref{vj_+R-DW-GR-GRs}),\ncourtesy of diagrammatic identity~\\eq{eq:D-ID-G}.\n\nOf the diagrams that remain, two are immediately involved\nin cancellations.\n\n\\begin{cancel}[Diagrams~\\ref{v_j+hR-W-GRs} and~\\ref{v_j+R-W-GRs}]\n\nDiagram~\\ref{v_j+hR-W-GRs} (\\ref{v_j+R-W-GRs}) is exactly\ncancelled by the component of diagrams~\\ref{WBT-vj_+hR-GR}\n(\\ref{WBT-vj_+R-DW-GR}) and its nested partners in which the\ngauge remainder strikes the top end of the kernel.\n\n\\end{cancel}\n\nUp to $\\Op{2}$ terms, the diagrams which survive are the following terms and\ntheir nested partners.\n\\begin{enumerate}\n\t\\item\tThe component of diagram~\\ref{P-hS_vj_+R-GR}\n\t\t\tin which the gauge remainder strikes\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\ta socket on a classical, two-point vertex;\n\t\t\t\t\t\t\\label{GR-S-CTP}\n\n\t\t\t\t\\item\tthe field which attaches to the\n\t\t\t\t\t\tbase of the kernel, and for which the\n\t\t\t\t\t\tseed action vertex is:\n\t\t\t\t\t\t\\begin{enumerate}\n\t\t\t\t\t\t\t\\item a reduced vertex;\n\t\t\t\t\t\t\t\\label{GR-S-BoK-R}\t\t\t\n\t\t\t\t\t\t\t\\item a classical, two-point vertex;\n\t\t\t\t\t\t\t\\label{GR-S-BoK-CTP}\n\t\t\t\t\t\t\\end{enumerate}\n\t\t\t\\end{enumerate}\n\n\t\\item\tThe component of diagram~\\ref{WBT-vj_+hR-GR} in which \n\t\t\tthe gauge remainder strikes the bottom end of the\n\t\t\tkernel;\n\t\t\t\\label{GR-S-K}\n\t\n\t\\item\tThe component of diagram~\\ref{P-S_vj_+R-DW-GR}\n\t\t\tin which the gauge remainder strikes\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\ta socket on a classical, two-point vertex;\n\t\t\t\t\t\t\\label{GR-W-CTP}\n\n\t\t\t\t\\item\tthe field which attaches to the\n\t\t\t\t\t\tbase of the kernel, and for which the\n\t\t\t\t\t\tseed action vertex is:\n\t\t\t\t\t\t\\begin{enumerate}\n\t\t\t\t\t\t\t\\item a reduced vertex;\n\t\t\t\t\t\t\t\\label{GR-W-BoK-R}\t\t\n\t\t\t\t\t\t\t\\item a classical, two-point vertex;\n\t\t\t\t\t\t\t\\label{GR-W-BoK-CTP}\n\t\t\t\t\t\t\\end{enumerate}\n\t\t\t\\end{enumerate}\n\n\t\\item\tThe component of diagram~\\ref{WBT-vj_+R-DW-GR} in which \n\t\t\tthe gauge remainder strikes the bottom end of the\n\t\t\tkernel;\n\t\t\t\\label{GR-W-K}\n\\end{enumerate}\n\nNow, \\ref{GR-S-BoK-R} and~\\ref{GR-S-K} exactly cancel\n({\\it cf.}\\  illustrative cancellation~\\ref{Icancel:v-W-GR-ai}).\nHowever, \\ref{GR-W-BoK-R} and~\\ref{GR-W-K} do not:\nin the former case, the kernel must be decorated\nwhereas, in the latter case, there is no such restriction.\nThus, a diagram possessing a differentiated effective\npropagator is left behind.\nThe component of diagram~\\ref{GR-S-CTP} (\\ref{GR-S-BoK-CTP})\nin which the kernel is decorated is exactly\ncancelled by diagram~\\ref{GR-W-CTP} (\\ref{GR-W-BoK-CTP}).\nThus, we are left with a whole set of diagrams,\ncollected together in \\fig{fig:TLTP-Survivors},\npossessing differentiated effective propagators.\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!} \\sum_{m'=0}^{m}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\nCr{m}{m'} \\!\\!\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{RV-DEP-B-DEP}{RV-DEP-B-DEP} \\ \n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s}>^{m-m'}}\n\t\t}\t\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}  \\sum_{m'=0}^{m} \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LDi{TLTP-DEP-B-DEP}{TLTP-DEP-B-DEP-MV} \\ \n\t\t\t\t\t+ \\LDi{TLTP-DEP-B-Soc}{TLTP-DEP-B-Soc-MV}}\n\t\t\t}{11\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Surviving terms from the type~\\ref{GR-S} and~\\ref{GR-W}\ngauge remainders, up to those with an $\\Op{2}$ stub.}\n\\label{fig:TLTP-Survivors}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{TLTP-DEP-B-Soc-MV} can be simplified\nby utilizing the primary diagrammatic identities\nand the classical flow equation. We illustrate this\nby considering the un-nested version; identical \nmanipulations can be performed in the nested case.\n\\begin{eqnarray}\n\t\\cd{TLTP-DEP-GR} \\\t& =\t& \\dec{\\cd{TLTP-EP-GR-B}}{\\bullet} - \\cd{TLTP-LdL-EP-GR} - \\cd{TLTP-EP-GR-LdL}\t\\nonumber\n\\\\[1ex]\n\t\t\t\t\t\t& =\t& \\dec{\\cd{TLTP-GR-PEP}}{\\bullet} - \\cd{TLTP-LdL-GR-PEP} - \\cd{GR-LdL} + \\cd{GR-GR-LdL} \\nonumber\n\\\\[1ex]\n\t\t\t\t\t\t& =\t& - \\cd{GR-LdL}\n\\label{eq:GR-LdL}\n\\end{eqnarray}\n\nTo go from the first line to the second, we have employed diagrammatic \nidentity~\\eq{eq:PseudoEP} and the effective\npropagator relation. On the second line, the\nfirst term vanishes courtesy of diagrammatic\nidentity~\\eq{eq:GR-TLTP}; similarly, the second\nterm, if we employ~\\eq{eq:LdL-GRk-Pert}. \nThe final term on the second line\nvanishes on account of diagrammatic identities~\\eq{eq:GR-relation}\nand~\\eq{eq:LdL-GRk-Pert}:\n\\[\n\t\\dec{ \\rhd \\!\\! >}{\\bullet} = 0 = \\stackrel{\\bullet}{\\rhd} \\! > + \\rhd \\! \\! \\stackrel{\\bullet}{>} = \\rhd \\!\\!  \\stackrel{\\bullet}{>}.\n\\]\nThus, the structure in diagram~\\ref{TLTP-DEP-B-Soc-MV}\ncomprising a classical, two-point vertex,\ndifferentiated effective propagator\nand $>$ reduces to just\na differentiated $>$. We now promote\nthe $\\stackrel{\\bullet}{>}$ to an implicit\ndecoration noting that, according to the rules\nfor converting implicit decorations to explicit\ndecorations, this comes with a minus sign\n(see the discussion around \\fig{fig:nestedhook}).\nFinally, the decorations $\\decGR{\\ }{>^{m'} \\stackrel{\\bullet}{>}}$\ncan be combined with the overall decorations $>^m$.\nUpon shifting $m \\rightarrow m' -1$ we find that:\n\\begin{cancel}[Diagram~\\ref{j+2-DGR-b}]\n\tDiagram~\\ref{TLTP-DEP-B-Soc-MV} exactly\n\tcancels diagram~\\ref{j+2-DGR-b}. \n\\end{cancel}\n\n\n\nWe now proceed to decorate the classical, two-point\nvertex of diagram~\\ref{TLTP-DEP-B-DEP-MV}. In\nthe case that we join this vertex to a Wilsonian \neffective action vertex, we just cancel diagram~\\ref{RV-DEP-B-DEP},\nup to a gauge remainder. Processing this\ngauge remainder, we can think of\nthe resultant diagram as versions \nof~\\ref{RV-DEP-B-DEP} and~\\ref{TLTP-DEP-B-DEP-MV} but where the end\nof the differentiated effective propagator\nwhich attaches to the two-point vertex\nnow attaches to one of the nested gauge\nremainders, instead. \nThus, this version\nof diagram~\\ref{RV-DEP-B-DEP} is cancelled also.\nUp to diagrams with an $\\Op{2}$ stub, the only terms\nthat remain are those of the \nform~\\ref{TLTP-DEP-B-DEP-MV} \n(but where the differentiated effective propagator\ncan attach to nested gauge remainders, in addition to\nthe classical, two-point vertex)\nwhere \nthe classical, two-point vertex is attached to\none of the nested gauge remainders or to a gauge remainder\nstructure. These terms can\nbe combined, courtesy of diagrammatic\nidentity~\\eq{eq:D-ID-G}, to yield the diagrams of\n\\fig{fig:TLTP-Survivors-b}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\t\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \\sum_{m'=1}^{m}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{(m-m')!(m'+1)!}  \n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{GR-ArbGRs-DEP}{GR-ArbGRs-DEP}} \\hspace{2em}\n\t\t\t}{11\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3}  \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!} \\sum_{m'=2}^{m} \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\sum_{m''=0}^{m'-2} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{GRs-TLTP-GRs-DEP}{GRs-TLTP-GRs-DEP}} \\hspace{2em}\n\t\t\t}{11\\Delta^{j+s}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The surviving \ndiagrams spawned by the gauge remainders\nof types~\\ref{GR-S} and~\\ref{GR-W}, up to those\nwith an $\\Op{2}$ stub.}\n\\label{fig:TLTP-Survivors-b}\n\\end{figure}\\end{center}\n\n\nIt is worth expanding on the origin \nof diagrams~\\ref{GR-ArbGRs-DEP} \nand~\\ref{GRs-TLTP-GRs-DEP} a little further,\nwhich is most readily done by returning to~\\eq{eq:D-ID-G}.\nEssentially, we have combined the following terms:\n\\begin{enumerate}\n\t\\item\ta version of the first term in which a differentiated\n\t\t\teffective propagator attaches at one end to the\n\t\t\texplicitly drawn $>$ and at the other end\n\t\t\tto either the socket decorating the classical, two-point\n\t\t\tvertex or to one of the implicit decorations;\n\n\t\\item\ta version of the final term in which a differentiated\n\t\t\teffective propagator attaches at one end to the explicitly\n\t\t\tdrawn $>$ and at the other end\n\t\t\tto either the socket decorating the classical, two-point\n\t\t\tvertex or to one of the implicit decorations which\n\t\t\tforms the string of gauge remainders biting this socket.\n\\end{enumerate}\nClearly, we can re-express these diagrams in terms of:\n\\begin{enumerate}\n\t\\item\tthe second diagram of~\\eq{eq:D-ID-G} in which\n\t\t\ta differentiated\n\t\t\teffective propagator attaches at one end to the\n\t\t\tfull gauge remainder and at the other end\n\t\t\tto one of the implicit decorations;\n\n\t\\item\ta version of the final term in which a differentiated\n\t\t\teffective propagator attaches at one end to the explicitly\n\t\t\tdrawn $>$ and at the other end to the gauge remainder\n\t\t\tcomponents which decorate the top of the diagram.\n\\end{enumerate}\n\n\nHaving completed the analysis\nof the type~\\ref{GR-S} and~\\ref{GR-W} gauge remainders,  \nwe now move on to gauge remainders of\ntype~\\ref{GR-GRs}, which we collect together\nin \\fig{fig:GR-GRs}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-1}^{n+s-m-3} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!} \\sum_{m'=1}^m \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{GRs-RW-GR-RWV}{GRs-RW-GR-RWV}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \\hspace{-1em}\n\t\t\t\t\t\\LDi{GRs-DW-GR-GRs}{GRs-DW-GR-GRs} \\hspace{1.5em}\n\t\t\t\t\t+\\LDi{Ubend-DW-GR}{Ubend-DW-GR} \\hspace{1em}\n\t\t\t\t\t+\\LDi{GRBT-GRs}{GRBT-GRs} \\hspace{0.5em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GR-GRs}.}\n\\label{fig:GR-GRs}\n\\end{figure}\\end{center}\n\nAllowing the gauge remainder to act in\ndiagrams~\\ref{GRs-RW-GR-RWV} and~\\ref{GRBT-GRs},\nwe uncover the by now familiar pattern of\ncancellations. Up to terms with an $\\Op{2}$\nstub, we are left with:\n\\begin{enumerate}\n\t\\item\tdiagram~\\ref{Ubend-DW-GR}, which we notice combines\n\t\t\twith diagram~\\ref{GR-ArbGRs-DEP};\n\n\t\\item\tdiagrams in which the (nested) gauge remainder of (the nested version of)\n\t\t\t\\ref{GRBT-GRs} strikes:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tthe top end of the kernel;\n\n\t\t\t\t\\item\tthe bottom end of the kernel;\n\t\t\t\\end{enumerate}\n\n\t\\item\tdiagrams in which the classical, two-point vertex of~\\ref{GRs-RW-GR-RWV}\n\t\t\tand its nested partners\n\t\t\tis attached to one of the $m'$ gauge remainders,\n\t\t\tat the top of the diagram.\n\\end{enumerate}\n\nWith a little thought,\nwe see that diagrams of the last item are\nof the same structure as diagram~\\ref{GRs-TLTP-GRs-DEP}. \nIndeed, these diagrams\ncombine, to yield a single term with a full kernel.\nThe resultant term is shown in \\fig{fig:GR-GRs-P},\ntogether with the other surviving diagrams\nspawned by the gauge remainders of type~\\ref{GR-GRs}, up\nto those with an $\\Op{2}$ stub.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s} \\sum_{j=-2}^{n+s-m-2} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t\t\t\t\\LDi{GRs-W-GRs}{GRs-W-GRs} \\hspace{1.5em} \n\t\t\t\t\t+\\LDi{Ubend-W-GR}{Ubend-W-GR} \\hspace{1em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \n\t\t}\n\t\t\t{\n\t\t\t\\sum_{m''=0}^{m'-2} \\!\\! \\nCr{m'}{m''} \\!\\! \n\t\t\t\\dec{\n\t\t\t\t\\Tower{\t\t\t\t\t\n\t\t\t\t\t\\LDi{K-GRs-B-K-GRs}{K-GRs-B-K-GRs} \\hspace{2em}\t\n\t\t\t\t\t+\\LDi{GRs-TLTP-GRs-K}{GRs-TLTP-GRs-K} \\hspace{2em}\n\t\t\t\t} \n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Surviving diagrams spawned by the \ngauge remainders of type~\\ref{GR-GRs} plus \ndiagrams~\\ref{GR-ArbGRs-DEP} and~\\ref{GRs-TLTP-GRs-DEP}.}\n\\label{fig:GR-GRs-P}\n\\end{figure}\\end{center}\n\nWe note the following about the diagram of \\fig{fig:GR-GRs-P}.\nFirst, we should explain the sign of diagram~\\ref{K-GRs-B-K-GRs},\nsince the parent diagram comes with a minus sign.\nConsider the string of $m''+1$ gauge remainders (the $+1$ corresponding\nto the explicitly drawn gauge remainder). The\nlast of these bites the end of the kernel, which\nis then attached to one of the $m'-m''$ gauge remainders.\nIn other words, we can think of the first \nof the $m'-m''$ gauge remainders not as being at the beginning of a string\nof $m'-m''$ gauge remainders, but in the middle (or end if $m'-m'' =1$) \nof a string of $m'+1$ gauge remainders. Upon explicit decoration\nof diagram~\\ref{K-GRs-B-K-GRs} with the $m'$ gauge remainders,\nour rule for determining the sign is to associate\nassociate bites to the left (right)\nwith pushes forward (pulls back), yielding $(-1)^R$.\nHowever, if the gauge remainder at the end\nof the kernel is bitten on the left (right),\nit is in fact pulled back (pushed forward)\nonto ({\\it cf.}\\  the discussion around \\figs{fig:GRstring}{fig:nestedhook}).\nHence, given our rule for decoration, we must give\nthe diagram an overall minus sign.\n\nSecondly, notice that diagram~\\ref{GRs-W-GRs} is\nexactly the same as diagram~\\ref{GRs-W-GRs-MV}\n(the difference between the lower limits on\nthe sums over $m$ and $m'$ is just an artefact\nof putting diagram~\\ref{GRs-W-GRs} under the same\nsummation signs as diagram~\\ref{WBT-GRs-MV}---which does\nexist for $m,\\ m'=1$),\nup to a relative factor of $-2$.\nThus, these two diagrams do not exactly cancel;\nto find the missing term that completes the \ncancellation---and also to understand the\nroles of diagrams~\\ref{K-GRs-B-K-GRs}--\\ref{GRs-TLTP-GRs-K}---we \nmust conclude  our discussion of\nthe gauge remainders by analysing the\ngauge remainders of type~\\ref{GRsx2},\nwhich are collected together in \\figs{fig:GRsx2-a}{fig:GRsx2-b}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-1}^{n+s-m-4} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-vkR-DW-GRx2} \\SumVertex}{D-vj_+kR-vkR-DW-GRx2}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-3} \\sum_{j=-2}^{n+s-m-5}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\t\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{Struc-WGRx2-B}{WGRx2}}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{Padlock-S_vj_+R-DW-GRx2}{P-S_vj_+R-DW-GRx2}\n\t\t\t\t\t+2\\LDi{WBT-vj_+R-DW-GRx2}{WBT-vj_+R-DW-GRx2}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GRsx2}, part~1.}\n\\label{fig:GRsx2-a}\n\\end{figure}\\end{center}\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n \\sum_{m=2}^{2s-1} \\sum_{m'=2}^m \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\nCr{m}{m'}\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{vj_+R-DW-GRx2-GRs}{vj_+R-DW-GRx2-GRs}\n\t\t\t\t}{\\SumVertex}\t\t\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\n\t\t}\n\t\\\\\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \t\\hspace{-1em} \n\t\t\t\t\t\\LDi{GRs-DW-GRx2-GRs}{GRs-DW-GRx2-GRs} \t\\hspace{1.5em}\n\t\t\t\t\t+\\LDi{Ubend-GRsx2}{Ubend-GRsx2}\t\t\t\\hspace{1em}\n\t\t\t\t\t+2\\LDi{WBT-GRx2-GRs}{WBT-GRx2-GRs}\t\t\\hspace{0.8em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GRsx2}, part~2.}\n\\label{fig:GRsx2-b}\n\\end{figure}\\end{center}\n\n\nThe first thing to note about the\ngauge remainders of type~\\ref{GRsx2} is\nthat we are guaranteed to generate\ntrapped gauge remainders in \ndiagrams~\\ref{WGRx2}\nand~\\ref{P-S_vj_+R-DW-GRx2} \n({\\it cf.}\\  \\figs{fig:DoubleGR}{fig:DoubleGR-A}). \nIn each of these\ncases, we can choose to act with\neither gauge remainder first, and one of\nthe things it will do is bite the field\non the structure to which the tip of\nthe other gauge remainder is attached. If,\non the other hand, the first action of \none of the gauge remainders is to bite a socket\nor the end of a kernel, then \nwe are free to perform the \nother gauge remainder as well. However,\nit is inefficient to allow the second\ngauge remainder to act immediately, \nas we can identify\ncancellations prior to this; we employ this\nstrategy with diagrams~\\ref{D-vj_+kR-vkR-DW-GRx2}\nand~\\ref{WBT-vj_+R-DW-GRx2}, also.\nNote, though, that diagram~\\ref{WBT-vj_+R-DW-GRx2} is not\nsymmetrical with respect to\\ the action of its two\ngauge remainders; the most efficient way to proceed\nis to be diplomatic:\nwe take one instance of this diagram \nwhere one gauge remainder\nacts first and one instance where the other \nacts first, dividing by two to avoid over-counting.\n\nAmongst the diagrams generated by processing\none of the gauge remainders of \ndiagrams~\\ref{D-vj_+kR-vkR-DW-GRx2}--\\ref{WBT-vj_+R-DW-GRx2}\nare versions of the parents, nested with respect to\\ the gauge\nremainder which acted. The strategy for dealing\nwith these terms is to process only the nested\ngauge remainder, for the time being.\n\nWe delay processing diagrams~\\ref{vj_+R-DW-GRx2-GRs}\nand~\\ref{WBT-GRx2-GRs}, which have only\na single active gauge remainder,\nuntil after one of the gauge remainders\nin each of \ndiagrams~\\ref{D-vj_+kR-vkR-DW-GRx2} \nand~\\ref{WBT-vj_+R-DW-GRx2}, and their nested partners,\nhas acted. It is easy to see why we do\nthis. Consider the component of diagram~\\ref{D-vj_+kR-vkR-DW-GRx2},\nand its nested partners, in which \na classical, two-point vertex, created\nby the action of one of the gauge remainders,\nis joined to either a nested gauge\nremainder or a gauge remainder structure. This\ndiagram partially cancels~\\ref{vj_+R-DW-GRx2-GRs},\ncourtesy of diagrammatic identity~\\eq{eq:D-ID-G} and without\nthe need to process any further gauge remainders.\nSimilarly, we process the gauge remainders\nof diagram~\\ref{WBT-vj_+R-DW-GRx2} according to the\nstrategy outlined above, and diagram~\\ref{WBT-GRx2-GRs}\nis also partially cancelled. Treating the partially\ncancelled diagrams together (and using also \ndiagram~\\ref{GRs-DW-GRx2-GRs}), it is evident that\ndiagrams in which the active gauge remainder\nstrikes a socket cancel in the usual way. Up to $\\Op{2}$ terms,\nthe only\nsurvivors from this chain of cancellations\nare the partially cancelled components of \n\\begin{enumerate}\n \t\\item\tdiagram~\\ref{vj_+R-DW-GRx2-GRs} and it nested partners\n\t\t\tin which a classical, two-point vertex, created\n\t\t\tby the action of the gauge remainder, is attached\n\t\t\tto the explicitly specified gauge remainder structure;\n\t\\label{it:CTP-ExplicitGR}\n\n\t\\item\tdiagram~\\ref{WBT-GRx2-GRs}  and it nested partners\n\t\t\tin which the active gauge\n\t\t\tremainder strikes either end of the kernel.\n\t\\label{it:GR-B-EOK}\n\\end{enumerate}\n\nReturning to diagrams~\\ref{WGRx2} \nand~\\ref{WBT-vj_+R-DW-GRx2} and their nested partners,\nif the first gauge remainder strikes the end\nof a kernel, we then have no choice but to\ngo ahead and process the second gauge remainder.\nDiagrams in which this second\ngauge remainder strikes a socket, combined\nwith diagrams of item~\\ref{it:GR-B-EOK}, above,\ncancel, in the\nusual manner.\n\nThus, up to diagrams with an $\\Op{2}$ stub, all\nsurviving terms possess either a trapped gauge\nremainder, or a kernel which is bitten at\none or both of its ends or a classical,\ntwo-point vertex attached to an explicitly\nspecified gauge remainder structure \n({\\it i.e.}\\  \\ref{it:CTP-ExplicitGR}, above).\n\nThe terms with a trapped gauge remainder,\nwhich are collected together in \n\\fig{fig:GRsx2-P}, can be simplified.\nNotice that the trapped gauge remainder\ndiagrams spawned by~\\ref{P-S_vj_+R-DW-GRx2}\n(see diagrams~\\ref{RV-RW-GR-B-RW} and~\\ref{TLTP-RW-GR-B-RW})\nare very similar to diagrams~\\ref{RV-DEP-B-DEP}\nand~\\ref{TLTP-DEP-B-DEP-MV}.\n(There is no analogue\nof diagram~\\ref{TLTP-DEP-B-Soc-MV}: if\none of the\ngauge remainders of\ndiagram~\\ref{P-S_vj_+R-DW-GRx2}\nwere to strike a socket on a classical,\ntwo-point vertex then this vertex would be\nkilled by the other gauge\nremainder, courtesy\nof diagrammatic identity~\\eq{eq:GR-TLTP}.)\nThe essential difference\nis the decoration of the kernel and the trapped\ngauge remainder. Now, we know\nthat diagrams~\\ref{RV-DEP-B-DEP}\nand~\\ref{TLTP-DEP-B-DEP-MV} can be processed\nto yield diagrams~\\ref{GR-ArbGRs-DEP} \nand~\\ref{GRs-TLTP-GRs-DEP}. Clearly, processing\nthe diagrams with trapped gauge remainders\nwill generate analogues of these diagrams, but\nwith the differentiated effective propagator\nreplaced by a decorated kernel, ending in a\ntrapped gauge remainder. Noting that the overall\nfactor of these diagrams is half of \ndiagrams~\\ref{GR-ArbGRs-DEP} \nand~\\ref{GRs-TLTP-GRs-DEP} we see that both the\ndiagrams corresponding to item~\\ref{it:CTP-ExplicitGR},\nabove, and diagram~\\ref{Ubend-GRsx2}\nare exactly cancelled.\n\nHowever, there are some new diagrams left over\nas a consequence of the kernel admitting\ndecorations. First, the\nclassical, two-point vertex of diagram~\\ref{TLTP-RW-GR-B-RW},\ncan be attached to the kernel. The corresponding\nprimary part is simply cancelled by\ndiagram~\\ref{Trapped-KBK}. The secondary\npart yields a version of diagram~\\ref{Trapped-KBK} \nwith an active gauge remainder inserted between\nthe end of the vertical line emanating from the\ntrapped gauge remainder and the kernel. If this\ngauge remainder strikes a socket, then\nwe can think of the resulting diagram as\na version of diagram~\\ref{Trapped-KBK} but\nwhere the trapped gauge remainder is contracted\ninto the gauge remainder which bites the kernel,\nrather than the kernel itself.\nThis diagram is cancelled by a term generated in the following way.\nAttach the classical, two-point vertex of diagram~\\ref{TLTP-RW-GR-B-RW}\nto a vertex and apply the effective propagator relation. \nFocusing on the gauge remainder part, allow\nthe gauge remainder to act and attach the resulting\nclassical, two-point vertex to the kernel. The primary part\nyields the term we require for our cancellation.\nIterating the diagrammatic\nprocedure, the only terms that survive are those\nin which an active gauge remainder bites one of\nthe ends of the kernel (or which possess an $\\Op{2}$ stub).\n\n\nWe collect together all surviving diagrams\nspawned by the gauge remainders of type~\\ref{GR-GRs},\nup to those with an $\\Op{2}$ stub,\nin \\fig{fig:GRsx2-S}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\t\n\t\\vspace{0.8ex}\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s} \\sum_{j=-2}^{n+s-m-2} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\displaystyle\t\n\t\t\t\t\t\\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t\t\t\t\\LDi{GRs-W-GRs}{GRs-W-GRs-b} \\hspace{1.5em}\n\t\t\t\t\t-\\LDi{Ubend-W}{Ubend-W-MV-b} \\hspace{1em} \n\t\t\t\t\t+ \\LDi{Ubend-W-GR}{Ubend-W-GR-b} \\hspace{1em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{0.8ex}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LDi{K-GRs-B-K-GRs}{K-GRs-B-K-GRs-b} \\hspace{2em}\n\t\t\t\t\t+\\LDi{K-GRs-B-K-GR-GRs}{K-GRs-B-K-GR-GRs} \\hspace{2em}\n\t\t\t\t\t+\\LDi{GRs-TLTP-GRs-K}{GRs-TLTP-GRs-K-b} \\hspace{1em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LDi{K-B-K-TLTP-EP}{K-B-K-TLTP-EP}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\t\n\t\\end{array}\n\t\\]\n\\caption{Surviving diagrams\nspawned by the gauge remainders of type~\\ref{GRsx2},\nup to those with an $\\Op{2}$ stub.}\n\\label{fig:GRsx2-S}\n\\end{figure}\\end{center}\n\n\n\nThe sign of diagram~\\ref{Ubend-W-MV-b}\nrequires comment. \nThis diagram is formed\nfrom diagram~\\ref{WGRx2} and its nested partners;\nhowever,\nwe can understand the sign from the un-nested term, alone.\nIn this case, the gauge remainder structure is formed\nas follows. The first gauge remainder bites the end\nof the kernel, just above the second (active)\ngauge remainder. This active gauge remainder\nthen bites the kernel at the end to which\nthe $>$ corresponding to the first gauge\nremainder attaches. The point is that each\nof the gauge remainders bites the end of a kernel\nand so, for this pair of gauge remainders---which\nform part of the same gauge remainder structure---bites \nto the left (right) must be interpreted as\npulls back (pushes forward). Thus, the sign of\nthe diagram is given by  $(-1)^{R-2}$.\nPromoting these gauge remainders to an explicitly\nspecified structure, $\\decGR{ \\ }{>^2}$,\nwe see from the discussion around \\fig{fig:nestedhook} that we must compensate with \nminus sign. For {\\it e.g.}\\  diagram~\\ref{GRs-W-GRs-b}\nthere is no such sign, since the gauge\nremainders of diagram~\\ref{WGRx2}\nand its nested partners have \nformed separate structures.\n\nFinally, we comment on diagram~\\ref{K-B-K-TLTP-EP}.\nThe loose end of the kernel can attach to any\nof the $m'$ $>$s or to the socket on the classical,\ntwo-point vertex. The primary part of the component\nof the diagram in which the loose end of the kernel attaches to the\nsocket\nis spawned by diagram~\\ref{WGRx2} and its\nnested partners. Indeed, the $m'=1$ case\ncorresponds to the illustrative \ndiagram~\\ref{Diags-WGR-N1-Bottom-PF} of \\fig{fig:DoubleGR-C}.\nThe corresponding secondary part \nis spawned by a version of diagram~\\ref{Trapped-KBK}\nwith a (nested) gauge remainder inserted between\nthe end of the vertical line emanating from the\ntrapped gauge remainder and the kernel. Both\nthe primary and secondary parts of the component\nof diagram~\\ref{K-B-K-TLTP-EP} in which the loose end of the \nkernel attaches to a gauge remainder are spawned\nby diagram~\\ref{WBT-vj_+R-DW-GRx2} and its nested\npartners.\n\n\n\\begin{cancel}[Diagram~\\ref{Ubend-W-MV}]\n\nDiagram~\\ref{Ubend-W-MV-b} exactly cancels\ndiagram~\\ref{Ubend-W-MV}.\n\n\\end{cancel}\n\n\n\\begin{cancel}[Diagram~\\ref{GRs-W-GRs-MV}]\n\nDiagrams~\\ref{GRs-W-GRs-b} and \\ref{GRs-W-GRs}\nexactly cancel diagram~\\ref{GRs-W-GRs-MV}.\n\n\\end{cancel}\n\n\nFinally, we conclude our treatment of\nthe gauge remainders with the\nfollowing elaborate cancellation.\n\n\\begin{cancel}[Diagram~\\ref{Ubend-W-GR}]\n\tDiagrams~\\ref{K-GRs-B-K-GRs} and~\\ref{GRs-TLTP-GRs-K}\n\tcombine with diagrams~\\ref{K-GRs-B-K-GRs-b} \n\tand~\\ref{GRs-TLTP-GRs-K-b}, reversing their\n\tsigns. The resultant diagrams, when combined with\n\tdiagrams~\\ref{K-GRs-B-K-GR-GRs} and~\\ref{K-B-K-TLTP-EP},\n\tcan be redrawn, courtesy of diagrammatic identity~\\eq{eq:D-ID-G},\n\tto yield an exact copy of diagram~\\ref{Ubend-W-GR-b}.\n\tThus the overall factor of diagram~\\ref{Ubend-W-GR-b}\n\tis doubled, providing precisely the necessary\n\tcontribution to exactly cancel diagram~\\ref{Ubend-W-GR}.\n\\end{cancel}\n\n\nReferring back to \\fig{fig:bn-b-P},\nwe have therefore demonstrated\nthat\n\\[\n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet} +\\Op{4} = -4 \\beta_n \\Box_{\\mu\\nu}(p) +\\mbox{\\ref{beta-terms}} + \\mbox{\\ref{alpha-terms}} + \\cdots,\n\\]\nwhere the ellipsis stands for terms with an $\\Op{2}$ stub, to which we\nnow turn.\n\n\n\n\\subsection{Terms with an $\\Op{2}$ Stub}\n\\label{sec:bn-Op2}\n\n\nEvery time a classical, two-point vertex was\ngenerated in the computation of $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$,\nwe had the option of decorating it with an external\nfield. In \\figs{fig:bn-Op2-a}{fig:bn-Op2-b} \nwe collect together the set of\nterms spawned by $a_0$ and $a_1$\nwhich remain after all active gauge remainders have\nbeen processed, all cancellations have been identified\nand all diagrams which manifestly vanish at $\\Op{2}$\nhave been discarded.\nThe treatment of gauge remainders in diagrams with an $\\Op{2}$\nstub is identical to the analysis just performed, and the\nsame patterns of cancellations are observed. \nWherever possible, we have combined terms possessing\na seed action vertex with their counterparts\npossessing just Wilsonian effective action vertices by\nintroducing the notation\n\\[\n\t\\Pi = S - \\hat{S}.\n\\]\nNotice, though, that since the terms possessing\nexclusively Wilsonian effective action vertices\nnecessarily possess a decorated kernel, we are\nleft over with terms possessing a seed action\nvertex attached to a differentiated effective propagator.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\t2\n\t\t\t\\bca{\n\t\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t}\n\t\t\t\t{\n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{Dumbbell-CTP-E-DEP-vh_j+R}{D-CTP-E-DEP-vh_j+R}\n\t\t\t\t\t\t-\\LDi{Dumbbell-CTP-E-DW-Pi_j+R}{D-CTP-E-DW-Pi_j+R}\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{1\\Delta^{j+s+1}>^m}\n\t\t\t}\n\t\\\\\n\t\t+\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-4}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT}\n\t\t\t\t}\n\t\t\t}{1\\Delta^{j+s+1}>^m}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{$\\Op{2}$ terms generated by $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$, part~1.}\n\\label{fig:bn-Op2-a}\n\\end{figure}\\end{center}\n\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\t\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{vh_j+R-DEP-GRs-Skt-CTP-E}{vh_j+R-DEP-GRs-Skt-CTP-E}\n\t\t\t\t\t-\\LDi{Pi_j+R-DW-GRs-Skt-CTP-E}{Pi_j+R-DW-GRs-Skt-CTP-E}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \n\t\t}\n\t\t\t{\n\t\t\t\\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{WBT-GRs-Skt-CTP-E}{WBT-GRs-Skt-CTP-E}\n\t\t\t\t\t-\\LDi{DW-GR-GRs-Skt-CTP-E}{DW-GR-GRs-Skt-CTP-E}\n\t\t\t\t\t+2\\LDi{DW-GRs-Skt-CTP-E}{DW-GRs-Skt-CTP-E}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \n\t\t}\n\t\t\t{\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{GRs-dEP-GR-Skt-CTP-E}{GRs-dEP-GR-Skt-CTP-E}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{$\\Op{2}$ terms generated by $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$, part~2.}\n\\label{fig:bn-Op2-b}\n\\end{figure}\\end{center}\n\nBefore processing diagrams~\\ref{D-CTP-E-DEP-vh_j+R}--\\ref{GRs-dEP-GR-Skt-CTP-E},\nwe emphasise a very important observation.\nConsider the set of diagrams obtained by promoting\nthe explicitly drawn external field to an implicit\ndecoration. This set of diagrams is, of course,\nnothing other than\nthe complete set of terms generated by\n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$ which possess a single,\nclassical, \ntwo-point vertex (with a free socket)\n\\emph{after} all gauge remainders\nhave acted. We can use this to immediately deduce\nthe set of diagrams left over after \ntaking $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}$\n(see \\fig{fig:nL-b})\nand iterating the diagrammatic procedure until exhaustion.\nFirst, there will be the $\\alpha$ and $\\beta$ terms.\nSecondly, there will be diagrams in which $-\\Lambda \\partial_\\Lambda|_\\alpha$\nstrikes either one of the $\\ensuremath{\\star}$s or one of the elements\nof a string of $>$s. Finally, there are those diagrams\narising from attaching all classical, two-point vertices\ngenerated at each stage of the calculation either to a\n$\\ensuremath{\\star}$ or to one of the elements of a string of\n$>$s. In the case that there is just a single,\nclassical, two-point vertex to play with, \nthe resulting set of diagrams can be directly\ndeduced from diagrams~\\ref{D-CTP-E-DEP-vh_j+R}--\\ref{GRs-dEP-GR-Skt-CTP-E}.\nIn the case that there are two classical, two-point \nvertices to play with, the resultant diagrams\nwill arise from an analogue of diagram~\\ref{D-02x2-R-VS}.\n(Note that the two classical, two-point \nvertices must be attached to elements of\nthe same \nstructure comprising a single\n$\\ensuremath{\\star}$ plus string\nof $>$s plus vertices plus $>$s, \nas follows from  \\sec{sec:Not:GRs}.)\n\n\nDiagram~\\ref{D-CTP-E-DEP-vh_j+R} is simple\nto deal with. First, notice that the differentiated\neffective propagator must be in the $A^1$ sector\n(which, incidentally, means that the seed action vertex, to which\nit attaches at one end, cannot be a one-point vertex).\nNow we redraw this diagram by detaching \nthe differentiated effective\npropagator from the seed action vertex\nand\ndecorating the seed action vertex with an $A^1$ which\ncarries the same index as the loose end of the\neffective propagator, say $\\alpha$. Next, we\ndecorate the diagram with the remaining external\nfield, which we suppose carries index $\\nu$.\nPutting the $\\Op{2}$ stub and differentiated\neffective propagator to one side, consider\ncontracting the remaining part of\nthe diagram with $(-p)_\\nu p_\\alpha$.\nIt is straightforward to demonstrate\nthat the final outcome is zero. Therefore,\nby Lorentz invariance, the diagram to which\nthe $\\Op{2}$ stub attaches via the \ndifferentiated effective propagator \nis transverse in $p$ and, just like\n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$, in fact\ngoes as\n$\\Box_{\\mu \\nu}(p) + \\Op{4}$. \nConsequently,\ndiagram~\\ref{D-CTP-E-DEP-vh_j+R} as a whole\nis $\\Op{4}$\nand so does not contribute to \n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$ at $\\Op{2}$.\n\n \nTo process diagrams~\\ref{D-CTP-E-DW-Pi_j+R}--\\ref{DW-GRs-Skt-CTP-E}\nwe construct subtractions. We begin with diagram~\\ref{CTP-E-WBT}\nfor which, as we know from \\sec{sec:Subtractions-G},\nthe subtractions completely kill the parents. This leaves\nbehind only the additions, which are collected together in\n\\fig{fig:CTP-E-WBT-Add}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-3} \\sum_{j=-2}^{n+s-m-5}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\\LDi{CTP-E-WBT-E}{CTP-E-WBT-E}}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-3} \\sum_{j=-1}^{n+s-m-5}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT-dV}\n\t\t\t\t\t}{\\cd{dvj+hR}}\n\t\t\t\t}{} \\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-3} \\sum_{j=-2}^{n+s-m-5}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT-dCTP}\n\t\t\t\t\t}{\\cd{dCTP}}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2}>^m}\n\t\t}\n\t\\\\\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=3}^{2s-2} \\sum_{j=-2}^{n+s-m-4}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=3}^m\t\\nCr{m}{m'} \n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT-GRs}\n\t\t\t\t\t}{\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m'}}}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Additions for diagram~\\ref{CTP-E-WBT}.}\n\\label{fig:CTP-E-WBT-Add}\n\\end{figure}\\end{center}\n\nThe first thing to notice about the\ndiagrams of \\fig{fig:CTP-E-WBT-Add}\nis that they are very similar\nto the diagrams of \\fig{fig:nLV-c-Properties-a}\n(which we recall arose from considering \n$\\decp{\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)}{}$). Consequently,\nthey will be very easy to manipulate.\nThe differences are essentially three-fold.\nFirst, we must be careful to \nproperly take account of the\neffects of $\\decp{\\ }{}$, since it does not\nnecessarily apply to all elements of a fully\nfleshed out diagram.\nConsider first the \nnon-factorizable components of\ndiagrams~\\ref{CTP-E-WBT-E}--\\ref{CTP-E-WBT-dCTP}.\nIn this case, the socket of each diagram is\nfilled\nby the external field; there can\nbe no effective propagators carrying just $p$\nand all diagrammatic\nelements can be brought under the influence\nof $\\decp{\\ }{}$. In other words, \neach of the additions can be thought of\nas arising from Taylor expanding the corresponding components\nof the parent\ndiagram to $\\Op{2}$.\nIn the factorizable case, however,\n$p$ dependence comes not only from the $\\Op{2}$\nstub, but also from the effective propagators\ncarrying just $p$\nand the sub-diagrams to which they\nattach. These sub-diagrams may \npossess components which are \nnot Taylor expandable in $p$ and so we leave them alone.\nOnly those diagrammatic components which are not\npart of a $p$ dependent sub-diagrams can be brought\nunder the influence of $\\decp{\\ }{}$, in this case.\n\nThe second difference between the diagrams\nof \\figs{fig:nLV-c-Properties-a}{fig:CTP-E-WBT-Add}\nis that the latter diagrams do not possess\na string of gauge remainders decorating their\nclassical, two-point vertex decorated by\nthe external field. This makes life easier\nand means that we will never generate the analogue\nof the diagrams of \\fig{fig:nLV-c-Properties-e}.\nThe third difference, however, makes our life slightly\nharder: the presence of a kernel in the diagrams\nof \\fig{fig:CTP-E-WBT-Add} means there is an additional\nstructure to which effective propagators can attach.\nAs usual, this ultimately generates a set of\nsurviving diagrams in which gauge remainders\nwhich bite the kernel slide down to its base.\nImmediately, then, we can deduce the single diagram\nleft over from iterating the diagrammatic procedure\nuntil exhaustion, which is shown in \\fig{fig:CTP-E-WBT-Final}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!r!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT-Final}\n\t\t\t\t}{\\ensuremath{\\star}^r }\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r} >^m}\n\t\t}\n\t\\]\n\\caption{The only diagram left from the full treatment\nof the additions of \\fig{fig:CTP-E-WBT-Add}.}\n\\label{fig:CTP-E-WBT-Final}\n\\end{figure}\\end{center}\n\nWhat is to be the fate of diagram~\\ref{CTP-E-WBT-Final}?\nAs we will demonstrate shortly, it will be exactly\ncancelled by terms arising from processing\n\\begin{equation}\n\\label{eq:b-derivs}\n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}  -\n\t\\sum_{n'=1}^{n-1} \\dec{\\nLV{\\mathcal{E}}{n-n'}{\\mu\\rho}{b}(p)}{\\bullet}\n\t\\Delta^{11}_{\\rho \\sigma}(p) \\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p)\n\\end{equation}\n(see \\fig{fig:nL-b} and equation~\\eq{eq:b'-decomp}).\nBefore seeing this explicitly, we will deal with\nthe remaining diagrams of \\figs{fig:bn-Op2-a}{fig:bn-Op2-b}.\nFirst, we deal with diagram~\\ref{D-CTP-E-DW-Pi_j+R},\nthe only term other than~\\ref{CTP-E-WBT} \nthat is completely killed\nby its subtractions. Inspired by the analysis\nof~\\ref{CTP-E-WBT}, it is obvious that the\nfinal result of processing diagram~\\ref{D-CTP-E-DW-Pi_j+R}\nis the\ndiagram of \\fig{fig:CTP-E-DW-Pi_j+R-Add}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t4\n\t\\bca{\n\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+1}}{m!r!}\n\t}\n\t\t{\n\t\t\\dec{\n\t\t\t\\decp{\n\t\t\t\t\\LDi{Dumbbell-CTP-E-DW-Pi_j+R}{D-CTP-E-DW-Pi_j+R-Final}\n\t\t\t}{\\ensuremath{\\star}^r}\n\t\t\t\\SumVertex\n\t\t}{1\\Delta^{j+s+1-r}  >^m}\n\t}\n\t\\]\n\\caption{The final result of processing diagram~\\ref{D-CTP-E-DW-Pi_j+R}.}\n\\label{fig:CTP-E-DW-Pi_j+R-Add}\n\\end{figure}\\end{center}\n\n\nWe must now bite the bullet and deal with those diagrams\nwhich are not completely cancelled by their subtractions,\nstarting with diagram~\\ref{WBT-GRs-Skt-CTP-E}---whose\nsubtractions and additions are collected together in \n\\fig{fig:WBT-GRs-Skt-CTP-E-Add}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{0.5ex}\n\t\t\\mp 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-4}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{dWBT-GRs-Skt-CTP-E}{dWBT-GRs-Skt-CTP-E-s}{dWBT-GRs-Skt-CTP-E-a}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{0.5ex}\n\t\t\\pm 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LOLO{\n\t\t\t\t\t\t\\cd{WBT-GRs-Skt-CTP-E}  \\hspace{0.5em} \\cd{dvj+}\n\t\t\t\t\t}{WBT-GRs-Skt-CTP-E-dvj+-s}{WBT-GRs-Skt-CTP-E-dvj+-a}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{0.5ex}\n\t\t\\mp\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LOLO{\n\t\t\t\t\t\t\\cd{WBT-GRs-Skt-CTP-E-b}\n\t\t\t\t\t\t\\hspace{2em}\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m''}}\n\t\t\t\t\t}{WBT-GRs-Skt-CTP-E--TEGRs-s}{WBT-GRs-Skt-CTP-E--TEGRs-a}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\mp\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{WBT-TEGRs-Skt-CTP-E}{WBT-TEGRs-Skt-CTP-E-s}{WBT-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t\\hspace{1em}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for diagram~\\ref{WBT-GRs-Skt-CTP-E}.}\n\\label{fig:WBT-GRs-Skt-CTP-E-Add}\n\\end{figure}\\end{center}\n\n\nWe proceed as before by\nisolating the classical, two-point component\nof the differentiated vertex of \ndiagram~\\ref{WBT-GRs-Skt-CTP-E-dvj+-a}, decorating it with\ntwo effective propagators and utilizing~\\eq{eq:EP-dCTP-EP}.\nHowever, compared to the treatment of diagram~\\ref{CTP-E-WBT},\nwe will encounter four novelties, all associated with\nthe $m'$ $>$s which bite\nthe classical, two-point vertex decorated by an external field.\nFirst, the active gauge \nremainders associated with~\\eq{eq:EP-dCTP-EP}\ncan \nbe contracted into one of these $m'$  $>$s or the\nsocket which they bite.\nSecondly, classical two-point vertices\ngenerated by the action of (arbitrarily)\nnested gauge remainders\ncan be similarly attached.\nThirdly, we can no longer exactly combine terms into total\nmomentum derivatives since, at this\nstage of the calculation, those terms involving\nmomentum derivatives of any of the $m'$ $>$s\nor the $>$ which bites them are missing.\nHowever, it proves\nconvenient to express the sum of momentum derivative terms\nthat we do have as a total momentum derivative (which we\nthrow away) minus the missing terms.\nLastly, we have not yet encountered\nan analogue of diagram~\\ref{WBT-TEGRs-Skt-CTP-E-a},\nand so we put this to one side for a moment.\n\nIterating the diagrammatic procedure until exhaustion,\nwe can re-express the additions~\\ref{dWBT-GRs-Skt-CTP-E-a}, \n\\ref{WBT-GRs-Skt-CTP-E-dvj+-a} and~\\ref{WBT-GRs-Skt-CTP-E--TEGRs-a}\nas shown in \\figs{fig:WBT-GRs-Skt-CTP-E-+Subs}{fig:WBT-GRs-Skt-CTP-E-+Subs-II},\nwhere we have absorbed diagram~\\ref{CTP-E-WBT-Final}\ninto diagram~\\ref{CTP-E-GRs-Combo-WBT}, in preparation for what is to follow.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-dGRs-WBT-GR}{CTP-E-dGRs-WBT-GR}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r}  >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-WBT}{CTP-E-GRs-Combo-WBT}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r}  >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The final result of processing\ndiagrams~\\ref{dWBT-GRs-Skt-CTP-E-a}, \n\\ref{WBT-GRs-Skt-CTP-E-dvj+-a} and~\\ref{WBT-GRs-Skt-CTP-E--TEGRs-a}\nplus\ndiagram~\\ref{CTP-E-WBT-Final}, part~1.}\n\\label{fig:WBT-GRs-Skt-CTP-E-+Subs}\n\\end{figure}\\end{center}\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-GRk-WBT-GR}{CTP-E-GRs-Combo-GRk-WBT-GR}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r}  >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s-r,j+1}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\t\\sum_{m''=0}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-CTP-EP-WBT-GR}{CTP-E-GRs-Combo-CTP-EP-WBT-GR}\n\t\t\t\t\t\\hspace{1em}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s-r}  >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The final result of processing\ndiagrams~\\ref{dWBT-GRs-Skt-CTP-E-a}, \n\\ref{WBT-GRs-Skt-CTP-E-dvj+-a} and~\\ref{WBT-GRs-Skt-CTP-E--TEGRs-a}\nplus\ndiagram~\\ref{CTP-E-WBT-Final}, part~2.}\n\\label{fig:WBT-GRs-Skt-CTP-E-+Subs-II}\n\\end{figure}\\end{center}\n\n\nThe cancellation of diagrams~\\ref{WBT-TEGRs-Skt-CTP-E-a} \nand~\\ref{CTP-E-dGRs-WBT-GR}--\\ref{CTP-E-GRs-Combo-CTP-EP-WBT-GR}\nfollows from processing~\\eq{eq:b-derivs}\nand\nfocusing on all surviving terms in which a kernel\nbites its own tail. As commented under \\figs{fig:bn-Op2-a}{fig:bn-Op2-b},\nthis set of terms arises from the versions\nof diagrams~\\ref{CTP-E-WBT} and~\\ref{WBT-GRs-Skt-CTP-E}\nappropriate to~\\eq{eq:b-derivs}  {\\it i.e.}\\  diagrams\nfor which the classical, two-point vertices of\n\\ref{CTP-E-WBT} and~\\ref{WBT-GRs-Skt-CTP-E}, rather than being decorated\nby an external field, are instead attached  either to a\n$\\ensuremath{\\star}$ or to one of the elements of a string of\n$>$s. \nUsing~\\eq{eq:TLTP-Combo}, we collect together\nall terms spawned by \n\\begin{equation}\n\\label{eq:LdL-b-ba}\n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}\n\t-\n\t\\sum_{n'=1}^{n-1} \\dec{\\nLV{\\mathcal{E}}{n-n'}{\\mu\\rho}{b}(p)}{\\bullet}\n\t\\Delta^{11}_{\\rho \\sigma}(p) \\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p)\n\\end{equation}\nin which a kernel\nbites its own tail\nin \\figs{fig:nLDlb-KBT}{fig:nLDlb-KBT-2}.\n\n\\begin{cancel}[Diagram~\\ref{CTP-E-dGRs-WBT-GR}]\nDiagrams~\\ref{CTP-E-WBT-GR-GRs-dGR-GRs} \nand~\\ref{CTP-E-WBT-dGR-GRs} exactly\ncancel diagram~\\ref{CTP-E-dGRs-WBT-GR}.\n\n\\end{cancel}\n\n\\begin{cancel}[Diagram~\\ref{CTP-E-GRs-Combo-WBT}]\n\nDiagram~\\ref{CTP-E-GRs-Combo-WBT-b}\nexactly cancels diagram~\\ref{CTP-E-GRs-Combo-WBT}.\n\n\\end{cancel}\n\n\n\\begin{cancel}[Diagram~\\ref{WBT-TEGRs-Skt-CTP-E-a}]\n\nDiagrams~\\ref{CTP-E-WBT-GR-GRs-GR-dGR-GRs}\nand~\\ref{CTP-E-WBT-GR-dGR-GRs}\nexactly cancel diagram~\\ref{WBT-TEGRs-Skt-CTP-E-a},\ncourtesy of diagrammatic identity~\\eq{eq:D-ID-dGRk-GT-ring}.\n\n\\end{cancel}\n\n\\begin{cancel}[Diagrams~\\ref{CTP-E-GRs-Combo-GRk-WBT-GR} and~\\ref{CTP-E-GRs-Combo-CTP-EP-WBT-GR}]\n\nDiagrams~\\ref{CTP-E-GRs-Combo-WBT-GR} and~\\ref{CTP-E-GRs-Combo-WBT-GRs-CTP-EP}\nexactly cancel \ndiagrams~\\ref{CTP-E-GRs-Combo-GRk-WBT-GR} \nand~\\ref{CTP-E-GRs-Combo-CTP-EP-WBT-GR},\ncourtesy of diagrammatic identity~\\eq{eq:D-ID-Op2-G}.\n\n\\end{cancel}\n\nThis completes the analysis of\nthe additions of diagram~\\ref{WBT-GRs-Skt-CTP-E}.\nNote that the only diagrams remaining\nin which a kernel bites its own tail\nare diagram~\\ref{WBT-GRs-Skt-CTP-E}, itself,\nand its subtractions. It should therefore\nnot come as a surprise that we can\nnow reduce \n\\begin{equation}\n\\label{eq:al+be+subs}\n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)+ \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}\n\t-\n\t\\sum_{n'=1}^{n-1} \\dec{\\nLV{\\mathcal{E}}{n-n'}{\\mu\\rho}{b}(p)}{\\bullet}\n\t\\Delta^{11}_{\\rho \\sigma}(p) \\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p)\n\\end{equation}\nto just $\\alpha$-terms, $\\beta$-terms and\nthe diagrams of \\fig{fig:bn-Op2-b} and their \nsubtractions. To see this,\nwe must explicitly construct subtractions for\ndiagrams~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E},\n\\ref{Pi_j+R-DW-GRs-Skt-CTP-E},\n\\ref{DW-GR-GRs-Skt-CTP-E},\n\\ref{DW-GRs-Skt-CTP-E}\nand~\\ref{GRs-dEP-GR-Skt-CTP-E},\nwhich are shown in \n\\figrange{fig:vh_j+R-DEP-GRs-Skt-CTP-E-Add}{fig:GRs-dEP-GR-Skt-CTP-E-Add-II}.\n\nWhen processing the additions, the only novelty is\nassociated with diagram~\\ref{dvh_j-DEP-GRs-Skt-CTP-E-a}:\nthis diagram has a component which\npossesses a classical, two-point vertex,\ndifferentiated with respect to\\ momentum, that cannot\nbe manipulated in the usual way, on\naccount of the \\emph{differentiated} effective\npropagator, already attached to one of\nits sockets. We can proceed by utilizing the\nvarious primary diagrammatic identities to\nre-express\n\\begin{eqnarray}\n\t\\cd{GR-DEP-dCTP-EP}\t& =\t& \\cd{DGR-dEP} - \\cd{GR-DEP-dFGR}\n\\nonumber\n\\\\\n\t\t\t\t\t\t& =\t& -\\cd{Dcombo} - \\cd{DdEP-GR} + \\displaystyle \\frac{1}{2} \\cd{GR-DdEP-FGR}\n\\nonumber\n\\\\\n\t\t\t\t\t\t&\t& -\\frac{1}{2} \\cd{GR-DEP-dGR-GRk} + \\frac{1}{2} \\cd{dGR-DEP-FGR}\n\\label{eq:GR-DEP-dCTP-EP}\n\\end{eqnarray}\nwhere,\nfor all elements for which decoration is not defined, we\nhave traded $\\odot$ for $\\bullet$.\n\nIncluding~\\eq{eq:GR-DEP-dCTP-EP} amongst our tricks,\nwe can now iterate the diagrammatic procedure\nuntil exhaustion. There are no further novelties\nand it is straightforward to crank the handle and\ndemonstrate that~\\eq{eq:al+be+subs}\ndoes indeed reduce \nto just $\\alpha$-terms, $\\beta$-terms and\nthe diagrams of \\fig{fig:bn-Op2-b} and their \nsubtractions. This latter collection of\nterms is remarkably easy to treat,\nand we can immediately write down\nthe result, by using the techniques of\n\\sec{sec:Subtractions-G}. As an illustration,\nwe shown the result of combining\ndiagram~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E} with its subtractions, \n\\ref{dvh_j-DEP-GRs-Skt-CTP-E-s}, \\ref{vh_kR-dvj+kh-s}, \n\\ref{vh_j-DEP-TEGRs-Skt-CTP-E-s} and~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E-b-s}\nin \\fig{fig:Subs-P}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{1ex}\n\t\t\\displaystyle\n\t\t-2 \\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{m'=1}^{2s-1-m} \\sum_{j=-1}^{n+s-m-m'-3}\n\t\t\\sum_{j'=-1}^{n+s-m-m'-j-5} \\sum_{s'=-j'+1}^{j+s+1}\n\t\\\\\n\t\\vspace{2ex}\n\t\t\\displaystyle\n\t\t\\Delta^{1\\, 1}_{\\rho \\sigma}(p) \\frac{\\ensuremath{\\Upsilon}_{j'+s'+2,j'+2}}{m'!} \\frac{\\ensuremath{\\Upsilon}_{j+s+1-s',j+1}}{m!} \\sum_{m''=0}^{m}\n\t\\\\\n\t\t\\decNTEs{\n\t\t\t\\begin{array}{c}\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\cd{vj+jpr+3-R}\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\Vertex{v^{j+2},j'}\n\t\t\t\\\\\n\t\t\t\t\\cd{CTP-E}\n\t\t\t\\end{array}\n\t\t}{1_\\rho \\Delta^{j'+s'+2} >^{m'}}\n\t\t\n\t\t\\dec{\n\t\t\t\\decp{\n\t\t\t\t\\cd{vh_j+R-DEP-GRs-Skt-CTP-E-c} \\hspace{0.5em}\n\t\t\t}{}\n\t\t\t\\SumVertex\n\t\t}{1_\\sigma \\Delta^{j+s+1-s'} >^{m-m''}}\n\t\\end{array}\n\t\\]\n\\caption{The result of combining\ndiagram~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E} with its subtractions.}\n\\label{fig:Subs-P}\n\\end{figure}\\end{center}\n\nAs expected, the contributions factorize into two sub-diagrams\njoined by an effective propagator, $\\Delta^{1\\, 1}_{\\rho \\sigma}(p)$.\nThe first sub-diagram is under the influence of $\\not{\\mathrm{T}}_{M} (p)$.\nThe second sub-diagram has exactly the same structure\nas the additions for diagram~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E}.\nIf we now include those terms obtained by combining\ndiagrams~\\ref{Pi_j+R-DW-GRs-Skt-CTP-E}--\\ref{GRs-dEP-GR-Skt-CTP-E}\nwith their subtractions we will generate the additions\nfor these diagrams as factorizable sub-diagrams also. \nThe crucial point to recognize is that we already know exactly\nwhat happens when we manipulate this complete set of\nadditions! We can therefore write:\n\\begin{eqnarray}\n\t-4\\beta_n \\Box_{\\mu\\nu}(p) + \\Op{4} = \n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) + \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}\n\\nonumber\n\\\\\n\t\t-\\sum_{n'=1}^{n-1} \n\t\t\\dec{\n\t\t\t\\mathcal{E}_{n-n' \\mu \\rho}^{b \\hspace{1.2em} 1 \\, 1}(p)\n\t\t}{\\bullet}\n\t\t\\Delta^{1\\,1}_{\\rho \\sigma}(p) \n\t\t\\left(\n\t\t\t\\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p) +\n\t\t\t\\decNTEs{ \\nLV{\\overline{\\mathcal{E}}}{n'}{\\sigma\\nu}{c}(p)}{}\n\t\t\\right)\n\\nonumber\n\\\\\n\t\t+\n\t\t\\sum_{n'=1}^{n-1} \\sum_{n''=1}^{n'-1}\n\t\t\\dec{\n\t\t\t\\nLV{\\mathcal{E}}{n-n'-n''}{\\mu\\rho}{b}(p) \n\t\t}{\\bullet}\n\t\t\\Delta^{11}_{\\rho \\kappa}(p)\n\t\t\\nLV{\\mathcal{E}}{n''}{\\kappa\\tau}{a}(p)\n\t\t\\Delta^{11}_{\\tau \\sigma}(p)\n\t\t\\decNTEs{\\nLV{\\overline{\\mathcal{E}}}{n'}{\\sigma \\nu}{c}(p)}{}\n\\nonumber\n\\\\\n\t\t+ \\cdots\n\\label{eq:bn-nearly}\n\\end{eqnarray}\nwhere the ellipsis represents the $\\alpha$ and $\\beta$ terms.\nThere is no need to explicitly draw all of\nthese, since the complete set can be built up out\nof just the\n$\\alpha$ and $\\beta$ terms formed by\n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$\nand $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}$.\nThese diagrams are collected together in\n\\fig{fig:bn-al+be}; the overall signs of\nthe diagrams are appropriate to the right-hand side\\ \nof~\\eq{eq:bn-nearly}.\n\\bc\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\displaystyle\n\t\t\t\\sum_{n'=1}^{n-1}\n\t\t\t\\sum_{s=1}^{n-n'}   \\sum_{m=0}^{2s} \\sum_{j=-1}^{n-n'+s-m-1}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\left[\n\t\t\t\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t\t\\vspace{1ex}\t\n\t\t\t\t\t\t\t\\LD{beta-terms-a}\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t& \\LD{alpha-terms-a}\t\t\t\n\t\t\t\t\t\t\\\\\t\n\t\t\t\t\t\t\t2 \\left( v^{j_+}-1 \\right) \\beta_{n'} \t& \\hspace{-1em} + \\hspace{-1em}\t& \\gamma_{n'} \\pder{}{\\alpha}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\right]\n\t\t\t\t\t\\DisplacedVertex{v^{j_+}}\n\t\t\t\t}{\\Vertex{n_{n'+s}, j}}\n\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t}\n\t\\\\\n\t\t4\n\t\t\\bca{\n\t\t\t\\sum_{n'=1}^{n-1} \\sum_{s=1}^{n-n'}  \\sum_{r=1}^s \n\t\t\t\\sum_{m=0}^{2(s-r)} \\sum_{j=-1}^{n-n'+s-2r-m-1}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+1}}{m!r!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\left[\n\t\t\t\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t\t\\vspace{1ex}\t\n\t\t\t\t\t\t\t\\LD{beta-terms-b}\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t& \\LD{alpha-terms-b}\t\t\t\n\t\t\t\t\t\t\\\\\t\n\t\t\t\t\t\t\t2 \\left( v^{j_+}-1 \\right) \\beta_{n'} \t& \\hspace{-1em} + \\hspace{-1em}\t& \\gamma_{n'} \\pder{}{\\alpha}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\right]\n\t\t\t\t\t\\DisplacedVertex{v^{j_+}}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\Vertex{n_{n^{'} +s, j}}\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1 \\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^m }\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The $\\alpha$ and $\\beta$ terms contributing \nto $-4 \\beta_n \\Box_{\\mu\\nu}(p)$\nformed by $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$\nand $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}$.}\n\\label{fig:bn-al+be}\n\\end{figure}\\end{center}\n\nExamining these diagrams, and recalling the expressions\nfor the $\\mathcal{E}^{i}$, we see that we have succeeded\nin deriving an expression for $\\beta_n$\nwhich has no explicit dependence on either\nthe seed action or the details of the covariantization!\n\n\\subsection{The $\\alpha$ and $\\beta$ Terms}\n\\label{sec:al+be}\n\nWe can simplify the diagrams of \\fig{fig:bn-al+be}\nand do so by starting with the\n$\\alpha$\nterms.\nThis is done by using the diagrammatic relationship:\n\\begin{equation}\n\\label{eq:EP-dalDTP-EP}\n\t\\cd{A-EP-dalCTP-EP-B} = -\\cd{A-dalEP-B} - \\frac{1}{2} \\cd{A-GR-dalCombo-B} -\\frac{1}{2} \\cd{A-dalCombo-GR-B},\n\\end{equation}\nwhere\n\\[\n\t\\cd{dalCombo-GR} \\hspace{1em} \\equiv \\cd{EP-dalGR-GR} \\hspace{1em} -\\frac{1}{2} \\cd{GR-dalPEP-GR}\n\\]\n(and we recall that $\\Dal \\equiv \\partial \/ \\partial \\alpha$).\nComparing~\\eq{eq:EP-dalDTP-EP} with~\\eq{eq:EP-dCTP-EP}\nit is clear that the analysis of the $\\alpha$\nterms is very similar to the analysis\nof diagrams with momentum derivatives.\nIndeed, it is straightforward\nto show that the $\\alpha$ terms\nyield a contribution to\n$-4\\beta_n \\Box_{\\mu \\nu}(p)$ \nsuch that for every appearance\nof $\\dec{X_n}{\\bullet}$, there\nis a partner term\nwhich\ncan be generated by making the substitution\n\\[\n\\dec{X_n}{\\bullet} \\rightarrow - \\sum_{n'=1}^{n-1} \\gamma_{n'} \n\t\\pder{}{\\alpha} X_{n-n'}.\n\\]\nEquivalently, we can explicitly include the\n$\\alpha$-terms in~\\eq{eq:bn-nearly}\nby inserting $\\sum_n g^{2n+1}$\nat the beginning of both sides of the equation\nand then\nmaking the replacement\n\\begin{equation}\n\\label{eq:replace}\n\t-\\Lambda \\partial_\\Lambda|_\\alpha \\rightarrow -\\Lambda \\partial_\\Lambda.\n\\end{equation}\n\nThe $\\beta$ terms are more subtle\nthan one might expect. Let us begin with\ndiagram~\\ref{beta-terms-a}. Clearly, the way to proceed\nis to isolate any diagrams which possess\na classical, two-point vertex and then decorate\nthis vertex. There are two possible decorations:\neither we can attach one external field and one\neffective propagator, or we can attach two\neffective propagators; either way, we can apply\nthe effective propagator relation. In the former\ncase, the primary part simply yields\n\\begin{equation}\n\\label{eq:beta-a-nLV-a}\n\t-4 \\sum_{n'=1}^{n-1} \\beta_{n'} \n\t\\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{a}(p).\n\\end{equation}\nThe secondary part vanishes, since the\nassociated gauge remainder strikes $\\mathcal{E}^a$, which \nwe know to be transverse.\nWith this in mind,\nwe re-express\ndiagram~\\ref{beta-terms-a}  in \\fig{fig:bn-beta}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\displaystyle\n\t\t-4 \\sum_{n'=1}^{n-1} \\beta_{n'} \\mathcal{E}_{n-n' \\mu \\nu}^{a \\hspace{1.2em} 1 \\, 1}(p)\n\t\\\\\n\t\\vspace{2ex}\n\t\t+4\n\t\t\\bca{\n\t\t\t\\sum_{n'=1}^{n-1} \\sum_{s=1}^{n-n'}   \\sum_{m=0}^{2s} \n\t\t\t\\sum_{j=-1}^{n-n'+s-m-2} \\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\\beta_{n'}\n\t\t}\n\t\t\t{\n\t\t\t\\left(\n\t\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t\\LD{beta-vj+}\t&\t& \\LD{beta-novj+}\n\t\t\t\t\\\\[1ex]\n\t\t\t\t\tv^{j+}\t\t\t& - & 1\n\t\t\t\t\\end{array}\n\t\t\t\\right)\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\Vertex{n_{n'+s},j}}\n\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t}\n\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{n'=1}^{n-1} \\sum_{s=1}^{n-n'}   \\sum_{m=0}^{2s} \n\t\t\t\\sum_{j=-2}^{n-n'+s-m-2} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\beta_{n'}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\begin{array}{ccc}\t\t\t\t\t\n\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\\LD{beta-EP}\t& \t& \\LD{beta-EP-GR}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\cd{Vert-EP}\t& -\t& \\cd{Vert-EP-GR}\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\Vertex{n_{n'+s},j}\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{A re-expression of \ndiagram~\\ref{beta-terms-a}.}\n\\label{fig:bn-beta}\n\\end{figure}\\end{center}\n\nWe can now see that there are some\nnovelties associated with the\n$\\beta$ terms.\nFirst, diagram~\\ref{beta-novj+}\ndoes not have a canonical normalization factor:\nthere are $j+2$ identical vertices, but the\nnormalization factor $\\sim 1\/(j+1)!$. \nSecondly, the presence of the multiplicative\nfactor $v^{j_+}$ in diagram~\\ref{beta-vj+}\nmakes this term different from any we have \nencountered so far. Note, though, that\nthe combinatoric factor is effectively\ncanonical, since the factor of $v^{j_+}$\nsingles out one of the vertices, leaving\nbehind only $j+1$ identical vertices.\nThirdly, just as diagram~\\ref{beta-vj+} has a special\nvertex, so diagram~\\ref{beta-EP} has a special\neffective propagator. This effective\npropagator cannot be promoted to an implicit\ndecoration in the usual way, since then\nthe combinatorics will be wrong. \nTo put\nit another way, if we decorate a vertex with\na total of $q+1$ effective propagators,\ncomprising the special one and $q$\nothers, the combinatoric factor\nassociated with choosing the effective propagators\nin this way \nwill be $\\nCr{j+s}{q}$ and \\emph{not} $\\nCr{j+s+1}{q}$.\n\nNevertheless, diagrams~\\ref{beta-vj+}, \\ref{beta-novj+}\nand~\\ref{beta-EP} combine in a miraculous way.\nStarting with diagram~\\ref{beta-vj+}, we exploit\nthe indistinguishability of the $j+2$ vertices by\nreplacing $v^{j_+}$ with\n\\[\n\t\\frac{1}{j+2} \\left[ \\sum_{i=0}^j v^{i,i_+} + v^{j_+} \\right] = \\frac{n_{n'}-s}{j+2},\n\\]\nwhere we have used~\\eq{eq:VertexSum} (but with $v^0 = n_{n'} -s$).\n\nNext, consider creating some fully fleshed out\ndiagram from~\\ref{beta-EP}. The total of $j+s+1$\neffective propagators are to be divided into $q$\nsets, each containing $L_i$ effective propagators. \nSince the special effective propagator can reside\nin any of these sets, there are $q$ different ways\nto make the sets. The overall combinatoric factor\nassociated with this partitioning\nis, therefore,\n\\[\n\t\\frac{(j+s)!}{\\prod_i L_i!} \\sum_i L_i = \\frac{(j+s+1)!}{\\prod_i L_i!},\n\\]\nwhich is just the combinatoric factor expected\nfrom partitioning $j+s+1$\neffective propagators into $q$ sets.\nTherefore, we can now promote the special effective propagator\nof diagram~\\ref{beta-EP}\nto an implicit decorations but, counterintuitively, \nthe combinatoric\nfactor of the diagram, $\\ensuremath{\\Upsilon}_{j+s,j+2}$, \\emph{stays the same}!\n\nWith these points in mind, it is a trivial matter to show that \n\\begin{equation}\n\\label{eq:nLV-d}\n\t\\mbox{\\ref{beta-vj+}} + \\mbox{\\ref{beta-novj+}} \n\t\t+ \\mbox{\\ref{beta-EP}}\n\t= -2 \\sum_{n'=1}^{n-1} \\beta_{n'} (n_{n'}-1) \\nLV{\\mathcal{E}}{n-n'}{\\mu \\nu}{a}(p)\n\\end{equation}\n\n\n\n\n\n\nDiagram~\\ref{beta-EP-GR} can be processed, yielding:\n\\begin{equation}\n\\label{eq:beta-a-nLV-c}\n\t-2 \\sum_{n'=1}^{n-1} \\beta_{n'} \n\t\\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{c}(p)\n\\end{equation}\nPutting together~\\eq{eq:beta-a-nLV-a}, \\eq{eq:nLV-d} and~\\eq{eq:beta-a-nLV-c},\ndiagram~\\ref{beta-terms-a} ultimately reduces to \n\\begin{equation}\n\\label{beta-terms-a-Final}\n\t-2\\sum_{n'=1}^{n-1} \\beta_{n'} \n\t\\left[\n\t\t(n_{n'}+1) \t\\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{a}(p) + \\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{c}(p)\n\t\\right].\n\\end{equation}\n\nThe treatment of diagram~\\ref{beta-terms-b} follows in \nsimilar fashion. Now, however, our options for\ndecorating the classical, two-point vertex\nare to attach either two effective propagators or\none effective propagator and one $\\ensuremath{\\star}$\n(it is illegal to attach two $\\ensuremath{\\star}$s---see\n\\sec{sec:Not:GRs}).\nIn the latter case we proceed by expressing\n\\[\n\t\\cd{EP-CTP-Combo} = \n\t\\frac{1}{2}\n\t\\left[\n\t\t\\cdeps{Combo-b} - \\cdeps{fGR-Combo} + \\cdeps{EP-dGR-b} - \\cdeps{EP-fGR-dGR-b}\n\t\\right].\n\\]\nIn this way, we can reduce diagram~\\ref{beta-terms-b} \nto \n\\begin{equation}\n\\label{beta-terms-b-Final}\n\t\\sum_{n'=1}^{n-1} \\beta_{n'} \n\t\\left[\n\t\t2 \\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{c'}(p) - (2n_{n'}+1) \\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{b}(p) + 2\\nLV{\\tilde{\\mathcal{E}}}{n-n'}{\\mu\\nu}{b}(p)\n\t\\right],\n\\end{equation}\nwhere\n\\[\n\t\\nLV{\\tilde{\\mathcal{E}}}{n}{\\mu\\nu}{b}(p)  \\equiv\n\t\\bca{ \n\t\t4 \\sum_{s=1}^n  \\sum_{r=1}^s \\sum_{m=0}^{2(s-r)+1} \n\t\t\\sum_{j=-2}^{n+s-2r-m-1} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1)!}\n\t}\n\t\t{\n\t\t\\qquad\n\t\t\\decp{\n\t\t\t\t\\Tower{\\cd{CTP-E}}\n\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^m }{}\t\t\n\t}.\n\\]\nNotice that the $r=1$ contribution is the same as\nfor $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)$.\n\nBy combining~\\eq{eq:bn-nearly} with~\\eqs{beta-terms-a-Final}{beta-terms-b-Final}\nand utilizing~\\eq{eq:replace} we can write down the\ncomplete\nexpression for $\\beta_n$:\n\\begin{equation}\n\\label{eq:bn-Complete}\n\\fl\n\t\\begin{array}{c}\n\t\\vspace{1ex}\n\t\t\\displaystyle\n\t\t-4 \\sum_{n} g^{2n+1} \\beta_n \\Box_{\\mu\\nu}(p) + \\Op{4} = \n\t\\\\\n\t\\vspace{1ex}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{n} g^{2n+1}\n\t\\\\\n\t\t\\displaystyle\n\t\t\\left[\n\t\t\t\\begin{array}{l}\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t-\\Lambda \\partial_\\Lambda\\dec{\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p) + \\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)}{}\n\t\t\t\t\n\t\t\t\\\\\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle\n\t\t\t  \t-2\\sum_{n'=1}^{n-1} \\beta_{n'}\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\qquad\n\t\t\t\t\\times\n\t\t\t\t\\left\\{\n\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t(n_{n'} +1) \\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{a}(p)\n\t\t\t\t\t\t+\\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{c}(p)\n\t\t\t\t\t\t-\\decpb{\\nLV{\\overline{{\\cal D}}}{n-n'}{\\mu\\nu}{c}(p)}{}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t+(n_{n'}+1) \\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{b}(p) -  \\nLV{\\tilde{{\\cal D}}}{n-n'}{\\mu\\nu}{b}(p)\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\right\\}\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t+\n\t\t\t\t\\sum_{n'=1}^{n-1}\n\t\t\t\t\\left[\n\t\t\t\t\t\\nLV{{\\cal D}}{n'}{\\mu\\rho}{a}(p) +\n\t\t\t\t\t\\decNTEsb{\n\t\t\t\t\t\t\\nLV{\\overline{{\\cal D}}}{n'}{\\mu\\rho}{c}(p)\n\t\t\t\t\t}{}\n\t\t\t\t\\right]\n\t\t\t\t\\Delta^{1\\,1}_{\\rho \\sigma}(p) \n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\qquad\n\t\t\t\t\\times\n\t\t\t\t\\left\\{\n\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\\Lambda \\partial_\\Lambda\n\t\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\t\\nLV{{\\cal D}}{n-n'}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t}{}\t\t\t\t\t\t\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t-2\\sum_{n''=1}^{n'-1} \\beta_{n''}\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\t\\decpb{\\nLV{\\overline{{\\cal D}}}{n-n'-n''}{\\sigma\\nu}{c}(p)}{} \n\t\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\t- (n_{n'+n''}+1)\\nLV{{\\cal D}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\t+ \\nLV{\\tilde{{\\cal D}}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\\right]\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\right\\}\n\t\t\t\\\\\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t-\\sum_{n'=1}^{n-1} \\sum_{n''=1}^{n'-1}\n\t\t\t\t\n\t\t\t\t\\decNTEsb{\n\t\t\t\t\t\t\\nLV{\\overline{{\\cal D}}}{n'}{\\mu\\rho}{c}(p)\n\t\t\t\t\t}{}\n\t\t\t\t\\Delta^{11}_{\\rho\\tau}(p)\n\t\t\t\t\\nLV{{\\cal D}}{n''}{\\tau\\kappa}{a}(p)\n\t\t\t\t\\Delta^{11}_{\\kappa \\sigma}(p)\n\t\t\t\\\\\n\t\t\t\t\\qquad\n\t\t\t\t\\times\n\t\t\t\t\\left\\{\n\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\\Lambda \\partial_\\Lambda\n\t\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\t\\nLV{{\\cal D}}{n-n'-n''}{\\sigma\\nu}{b}(p) \n\t\t\t\t\t\t}{}\t\t\t\t\t\t\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t-2\\sum_{n'''=1}^{n''-1} \\beta_{n'''}\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\t\\decpb{\\nLV{\\overline{{\\cal D}}}{n-n'-n''-n'''}{\\sigma\\nu}{c}(p)}{} \n\t\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\t- (n_{n'+n''+n'''}+1)\\nLV{{\\cal D}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\t+ \\nLV{\\tilde{{\\cal D}}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\\right]\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\right\\}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\end{array}\n\\end{equation}\n\nNotice that we have substituted for $\\mathcal{E}^{c'}$\nvia~\\eq{eq:nLV-c-Op2-b} and have traded\nthe $\\mathcal{E}^i$ for the ${\\cal D}^i$, as this is the\ncorrect time to discard all diagrams which\nvanish at $\\Op{2}$ and \/ or contain a Wilsonian\neffective action one-point vertex.\n\nUp until now, we have been very careful not to\nTaylor expand individual diagrams to $\\Op{2}$ unless\nwe are certain that this step is valid. However,\nwe know that the \\emph{sum} \nof diagrams contributing to $\\beta_n$ is\nTaylor expandable to $\\Op{2}$. It must therefore\nbe the case that all contributions which include\nnon-polynomial dependence on $p$, at $\\Op{2}$,\ncancel out of~\\eq{eq:bn-Complete}. At two loops\nit has already been demonstrated that a subset of\nsuch contributions cancel out diagrammatically~\\cite{Thesis,mgierg2}.\nIt is very likely that this analysis can be extended\nto any number of loops, but we leave the investigation\nof this for the future. In the meantime,\nwe note that we can considerably simplify~\\eq{eq:bn-Complete}\nif we focus just on \nthe $\\Op{2}$ component of each diagram, \nwhich yields\nthe following equation (we have used~\\eq{nLDc-factorize}). \n\\begin{equation}\n\\fl\n\t\\begin{array}{c}\n\t\\vspace{1ex}\n\t\t\\displaystyle\n\t\t-4 \\sum_{n} g^{2n+1} \\beta_n \\Box_{\\mu\\nu}(p) + \\Op{4} = \n\t\\\\\n\t\\vspace{1ex}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{n} g^{2n+1}\n\t\\\\\n\t\t\\displaystyle\n\t\t\\left[\n\t\t\t\\begin{array}{l}\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t-\\Lambda \\partial_\\Lambda\\dec{\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p) + \\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)}{}\n\t\t\t\t+\\sum_{n'=1}^{n-1}\n\t\t\t\t\t\\nLV{{\\cal D}}{n'}{\\mu \\rho}{a}(p) \\Delta^{1\\,1}_{\\rho \\sigma}(p) \n\t\t\t\t\t\\Lambda \\partial_\\Lambda\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\nLV{{\\cal D}}{n-n'}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t}{}\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\left.\n\t\t\t\t\\rule{0em}{4ex}\n\t\t\t  \t-2\\sum_{n'=1}^{n-1} \\beta_{n'}\n\t\t\t\t\\right\\{\n\t\t\t\t\t\t(n_{n'} +1) \n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{a}(p)\n\t\t\t\t\t\t\t+\\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{b}(p)\n\t\t\t\t\t\t\\right]\n\t\t\t\t\t\t- \\nLV{\\tilde{{\\cal D}}}{n-n'}{\\mu\\nu}{b}(p)\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\qquad +\n\t\t\t\t\\left.\n\t\t\t\t\\sum_{n''=1}^{n'-1} \n\t\t\t\t\t\\nLV{{\\cal D}}{n''}{\\mu \\rho}{a}(p) \\Delta^{1\\,1}_{\\rho \\sigma}(p) \n\t\t\t\t\\left[\n\t\t\t\t\t\\nLV{\\tilde{{\\cal D}}}{n-n'-n''}{\\sigma\\nu}{b}(p) - \n\t\t\t\t\t(n_{n'+n''}+1)\\nLV{{\\cal D}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\\right]\n\t\t\t\t\\rule{0em}{4ex}\n\t\t\t\t\\right\\}\n\t\t\t\\end{array}\n\t\t\\right]_{p^2}\n\t\\end{array}\n\\label{eq:bn-Final}\n\\end{equation}\n\nEquation~\\eq{eq:bn-Final} reduces to particularly simple\nforms at one and two loops:\n\\begin{eqnarray}\n\\label{eq:beta1}\n\\vspace{2ex}\n\t-4 \\beta_1 \\Box_{\\mu\\nu}(p) & = & \n\t\\decp{\n\t\t\\nLV{\\overline{{\\cal D}}}{1}{\\mu\\nu}{a}(p) + \\nLV{\\overline{{\\cal D}}}{1}{\\mu\\nu}{b}(p)\n\t}{\\bullet}\n\\\\\n\\nonumber\n\t-4 \\beta_2 \\Box_{\\mu\\nu}(p) & = &\n\t\\decp{\n\t\t\\nLV{\\overline{{\\cal D}}}{2}{\\mu\\nu}{a}(p)\n\t\t+ \\nLV{\\overline{{\\cal D}}}{1}{\\mu\\rho}{a}(p)\n\t\t\\Delta^{1\\,1}_{\\rho \\sigma}(p) \\nLV{\\overline{{\\cal D}}}{1}{\\sigma\\nu}{b}(p)\t\n\t\t+ \\nLV{\\overline{{\\cal D}}}{2}{\\mu\\nu}{b}(p)\n\t}{\\bullet}\n\\\\\n\\label{eq:beta2}\n\t& & \\qquad + \\gamma_1 \\pder{}{\\alpha} \n\t\t\\decp{\n\t\t\t\\nLV{\\overline{{\\cal D}}}{1}{\\mu\\nu}{a}(p) + \\nLV{\\overline{{\\cal D}}}{1}{\\mu\\nu}{b}(p)\n\t\t}{},\n\\end{eqnarray}\nwhere we have used~\\eqs{eq:EP-leading}{nLDa-factorize} and\nhave defined\n\\[\n\t\\nLV{\\overline{{\\cal D}}}{n}{\\mu\\nu}{b}(p) \\equiv\n\t\\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)\n\t+\n\t\\frac{1}{2}\n\t\\sum_{n'=1}^{n-1} \\nLV{{\\cal D}}{n-n'}{\\mu\\rho}{b}(p)\n\t\\Delta^{11}_{\\rho\\sigma}\n\t\\nLV{{\\cal D}}{n'}{\\sigma\\nu}{b}(p).\n\\]\nEquations~\\eqs{eq:beta1}{eq:beta2} reproduce the\nresults of~\\cite{mgierg1,mgierg2,Thesis}, but written\nin a remarkably compact form.\n\n\n\\section{Conclusion}\n\\label{sec:Conc}\n\n\nWe have constructed a manifestly gauge invariant\ncalculus for $SU(N)$ Yang-Mills, which can\nbe applied at any number of loops. The calculus, which\nis entirely diagrammatic in nature,\nhas been comprehensively illustrated by demonstrating\nthat the $\\beta$-function is independent of both the seed\naction and the details of the covariantization of the cutoff,\nto all orders in perturbation theory.\n\nThe inspiration for this methodology, the first\nelements of which were developed\nin~\\cite{aprop}, is the immense freedom in the\nconstruction of ERGs. Given\nthat we specialize not only to those ERGS that are\nmanifestly gauge invariant but to those which also allow \nconvenient renormalization to any loop order,\nthere are still an infinite number with which\nwe can work. The differences between these\nERGs amount to non-universal details which\nneed never be exactly specified, instead satisfying\ngeneral constraints. In the computation of\nuniversal quantities, these non-universal details\nmust cancel out; by leaving them unspecified,\nwe are thus guided towards a very constrained calculational\nprocedure. However, as recognized in~\\cite{Thesis,Primer,RG2005},\nsuch cancellations are embedded within the formalism\nat a particularly deep level: at least\nsome of them occur in the calculation of quantities which\nare not universal, such as $\\beta$-function coefficients\nbeyond two loops. This observation formed the basis for developing\nthe diagrammatic calculus to the level\ndescribed in this paper.\n\nGiven a diagrammatic representation of the flow equation,\nthe calculus comprises an operator which implements \nthe flow {\\it i.e.}\\  $-\\Lambda \\partial_\\Lambda$,\na rule for implementing\nthe effects of charge conjugation\\ invariance, a set of primary\nand secondary diagrammatic identities and the subtraction\ntechniques.\n\nThere are three types of primary diagrammatic identities\n(see \\sec{sec:Review}).\nThose of the first type make no reference to\nperturbation theory and reflect general\nproperties of the exact flow equation\nand underlying theory. The single identity\nof the second type is the effective propagator\nrelation, which arises as a solution to\nthe classical flow equation, given a \nconvenient choice of\nseed action we are free to make. \nThe effective propagator relation states\nthat for each classical two-point vertex\n(which cannot be consistently set to zero)\nthere exists an effective propagator, which\nis the inverse of this vertex, up to a gauge remainder.\nThe primary\ndiagrammatic identities of the third type\nfollow directly from those of the first\nand second types; they are stated,\nnonetheless, due to the central role they play\nin perturbative calculations.\n\n\nThe secondary diagrammatic identities, of which\nthere are two types, encode the equivalence\nof distinct diagrammatic representations\nof structures comprising\nparticular arrangements of components of\ngauge remainders. Those of the second\ntype (see \\sec{sec:D-ID-Secondary-II}) are applied only when we focus on\nthe component of a diagram which has been Taylor\nexpanded to zeroth order in its external momentum\n(in this paper we have dealt only with diagrams\ncarrying a single external momentum);\nthose of the first type (see \\sec{sec:D-ID-Secondary-I}) \nare more generally applicable.\n\nThe final element of the calculus is the subtraction\ntechniques. Working to some order in the external momentum\nof a diagram, these techniques are used to isolate\nthose components which are of precisely the desired\norder from those\nwhich have an additional, non-polynomial contribution.\n\n\nA crucial step which facilitates the practical\napplication of the calculus is the notation\nintroduced in \\sec{sec:Notation}. As with\nperturbative methods in general, the number\nof diagrams contributing to some function grows\nvery rapidly with loop order, $n$. However, rather\nthan explicitly drawing each of these\ndiagrams, it was realized that we can construct\na function of both $n$ and the various diagrammatic\n\\emph{components} which can be used to generate the\ncorrect set of diagrams, at any loop order.\nSuch expressions are very compact, with a beautifully\nintuitive structure.\n\nFor the treatment of $\\beta_n$,\nwe constructed\na set of diagrammatic functions \n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$, $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)$\nand $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c}(p)$\n(see \\sec{sec:Prelim-DiagFns})\nwhich each include two external $A^1_\\mu$s, carrying momenta $p_\\mu$\nand $-p_\\nu$.  We proceeded by\nfocusing on the $\\Op{2}$ \ncomponents of these diagrammatic functions\n(up to additional non-polynomial dependence on $p$)\n and\ncomputed the flow of $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$. Amongst the terms generated\nis a single instance of $\\beta_n$, multiplied by\na universal coefficient. Of the remaining terms,\nthose involving either lower order $\\beta$-function\ncoefficients or $\\alpha$-derivatives were put to one side.\nThe remaining diagrams\nwere manipulated, wherever possible, using the \nfollowing scheme.\n\nThe first stage\nwas to isolate any classical, two-point vertices\nformed by the flow equation and, wherever\npossible, to attach\neither an external field or an effective propagator.\nIn the former case, the diagram possesses a manifestly $\\Op{2}$\nstub; these $\\Op{2}$ terms were put to one side. \nIn the latter case, the effective propagator\nrelation was applied. This\nrelation collapses all classical, two-point vertices\nwhich are contracted into an effective propagator\ndown into a Kronecker~$\\delta$ and a gauge remainder.\nThe gauge remainders were either be processed using\nthe primary diagrammatic identities or re-expressed\nusing the secondary diagrammatic identities. In\nthe former case, the whole procedure\nwas repeated: classical, two-point vertices were\nisolated and decorated and the effective propagator\nrelation was applied {\\it etc}. The\nset of remaining diagrams now cancelled amongst\nthemselves, up to the $\\Op{2}$ terms.\n\nTo process the $\\Op{2}$ terms, the subtraction\ntechniques were employed, allowing the $\\Op{2}$ terms\nto be\neffectively split into\nthose which have additional non-polynomial\ndependence on $p$ and those which do not.\nThe latter terms were manipulated\nusing the primary and secondary diagrammatic\nidentities, yielding a set\nof diagrams which, up to\nfurther $\\alpha$ and $\\beta$-terms,\nwere precisely\ncancelled by~\\eq{eq:LdL-b-ba}.\nThe $\\Op{2}$ diagrams containing additional,\nnon-polynomial, dependence on $p$ each factorized\ninto an un-manipulable component, $\\decNTEs{\\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{c}(p)}{}$,\nand an $n'$-loop component of exactly the same form\nas~\\eq{eq:LdL-b-ba}. \nThis allowed us to immediately write down an expression for\n$\\beta_n$  which is manifestly independent of\nthe seed action and details of the covariantization.\nFinally, we neatened the expression up by processing\nthe $\\alpha$ and $\\beta$ terms, to yield~\\eq{eq:bn-Complete}.\n\n\nThis expression was derived with the utmost rigour. In particular,\nall components carrying non-polynomial dependence on $p$\nhave been retained, even though it is know that\nthey must cancel out. Indeed, it may well be\nthat these cancellations can be demonstrated diagrammatically\n(this has already been done to some extent at two loops~\\cite{mgierg2,Thesis}),\nwhich would doubtless serve to considerably illuminate\nunderlying features of~\\eq{eq:bn-Complete}. If one is\nprepared to simply accept that these contributions do\nindeed cancel, then~\\eq{eq:bn-Complete} can be simplified\nby focusing on the strictly $\\Op{2}$ components \nof each diagram. This yields the considerably simpler\nexpression~\\eq{eq:bn-Final}. \n\nFrom~\\eq{eq:bn-Final}, the\ndiagrammatic expressions for $\\beta_1$ and\n$\\beta_2$, \\eqs{eq:beta1}{eq:beta2}, follow\ndirectly. In the context of assessing the\npractical usability of the calculus, the importance\nof this can hardly be overstated. Recall\nthat the first time $\\beta_2$ was computed in our\nERG framework~\\cite{Thesis} the derivation of~\\eq{eq:beta2}\ninvolved the generation of $\\order{10^4}$ diagrams (!)\nat intermediate\nstages of the calculation,\nalmost all of which cancelled to yield the final diagrammatic\nexpression.\nAlthough the number of intermediate diagrams can\nbe hugely reduced using the refinements of~\\cite{Primer},\nthe benefits of an expression from which the\ndiagrammatic expression can be read off at any loop\norder is obvious.\n\n\nThe derivation of the new formula for $\\beta_n$\nprovides insight into two subtle issues, both\nof which relate to the regularization. First, although the\nphysical $SU(N)$ gauge theory is properly \nregularized, care must be taken with\nthe flow equation. In particular, the flow\nequation generates (non-universal) diagrams in which\nthe kernel `bites its own tail' which \nare not properly regularized~\\cite{ym,ymi,ymii,aprop,mgierg1}\n(see the discussion around \\fig{fig:Flow}).\nUp until~\\cite{Primer}, these diagrams were excluded,\nvia an imposed constraint on the covariantization.\nHowever, as speculated in~\\cite{Primer,Thesis},\nall explicit instances\nsuch non-universal objects should cancel out\nin the reduction of $\\beta$-function\ncoefficients to a form with no explicit dependence\non either the seed action or details of the\ncovariantization. This has been borne out here,\nthough it should be emphasised that implicit\ndependence on these objects remains. It\nthus seems very likely that, ultimately,\na suitable constraint on the covariantization\nis required. \n\nNonetheless, it is very interesting that\nthe basic structure of the flow equation\nis sufficient to formally\nremove certain non-universal objects\nto $\\beta_n$, even if these objects are\nnot strictly well defined. This very much\nstrengthens the case that\nthe diagrammatics is driving us towards a\nnew framework formulated directly\nin terms of functions such as the $\\mathcal{E}^i$,\nwith the seed action\nand details of the covariantization of the\ncutoff operating in the background. \n\n\nThe second issue pertaining to the regularization\nthat the new formula for $\\beta_n$\nsheds some light on is that of pre-regularization.\nRecall that, in order the regularization\nbe properly defined, a pre-regulator must\nbe used in $D=4$~\\cite{SU(N|N)}\nto unambiguously define contributions\nwhich are finite only by virtue of the \nPauli-Villars regularization provided by\nthe massive regularizing fields. The\npre-regulator amounts to a prescription\nfor discarding otherwise non-vanishing\nsurface terms which can be generated by\nshifting loop momenta. In this paper,\ndimensional regularization was used.\nHowever, the only place that the\npre-regulator is explicitly used in practise is to throw away diagrams\nsuch as~\\ref{nLV-c-MomDer} which possess a sum of momentum\nderivative with respect to\\ the loop momenta. Thus,\nfor the purposes of the calculation performed\nin this paper,\nwe could have adopted the prescription\nthat we simply throw away all diagrams\nof this form. Of course, the diagrammatic\nfunctions contributing to $\\beta_n$ require\npre-regularization themselves, as they\ncontain diagrams which are finite only\nby virtue of the PV regularization. It is\nnot immediately obvious that the new prescription\nsuffices in this case but it will\nbe interesting to see in the future if\nit can, in fact, be adopted \nin complete generality, thereby allowing\nus to work directly in $D=4$.\n\nWhilst the application of the calculus to\n$\\beta$-function coefficients was vital\nin the development of the formalism,\nit is very important to ask whether the techniques\nof this paper can be applied in the computation of\nother quantities. The most obvious\nextension is to try to\ncalculate the expectation values\nof gauge invariant variables in perturbation\ntheory; efforts in this direction are \nunderway~\\cite{FutureWork}.\nAdditionally, we should not lose\nsight of the fact that the current\nformulation was originally developed\nwith a view to non-perturbative \napplications~\\cite{ymi,aprop}. This is\nhardly surprising, since the ERG has proven\nitself to be a flexible and powerful tool\nfor addressing such problems in a range of \nQFTs~\\cite{Wilson:1974sk,Hasenfratz:1985dm,Wetterich:1992yh,Bergerhoff:1995zm,Bergerhoff:1995zq,Aoki:1996fh,Fisher:1998kv,Morris:1998da,Berges:2000ew,Bagnuls:2000ae,Polonyi:2001se,Salmhofer:2001tr,Pawlowski:2003hq,Pawlowski:2005xe,Reuter:1997gx,Bergerhoff:1997cv,Gies:2002af,Ellwanger:1996wy,Pawlowski:2003hq,Fischer:2004uk,Ellwanger:1997wv,Ellwanger:1999vc,Ellwanger:2002xa};\nfurthermore, an interesting link\nhas been recently made between\nelements of the formalism presented\nhere and the\nAdS \/ CFT correspondence~\\cite{EMR}.\n\nA very interesting question to ask from the\nperspective of this paper is whether it is\npossible to repeat the type of cancellations\nseen in the treatment of $\\beta_n$ in a non-perturbative context.\nAn immediate obstacle to this is that non-perturbative\nERG  treatments typically rely on truncations (in our case\nof the Wilsonian effective action), after which\nuniversality is lost. However, as stated already,\nit seems that the diagrammatics are driving us\ntowards a more direct framework for\nperforming calculations in QFTs, whilst retaining\nthe advantages of an ERG approach. If this turns\nout to be the case, then we can hope that the new\nframework is defined non-perturbatively. Indeed, \nconsider again the expression~\\eq{eq:bn-Complete} for\n$\\beta_n$. We could imagine formally lifting\nthe diagrammatic expression on the right-hand side\\ to a\nnon-perturbative expression, such that its weak\ncoupling expansion just reproduces~\\eq{eq:bn-Complete}.\n\nOf course, even if this action is possible (and we\nnote that the presence of reduced vertices in~\\eq{eq:bn-Complete}\ncomplicates matters) we stress that\nit would\nrepresent only one step towards extracting non-perturbative\ninformation. After all, our hypothetical non-perturbative\nexpression for the $\\beta$-function would presumably\nstill require truncation before any numbers\ncould be computed. Nonetheless, the starting point\nwould surely be far more appealing with \nthe explicit dependence on the seed action and\ndetails of the covariantization removed from\nthe start. \n\n\n\n\n\n\n\\ack \nI would like to thank Daniel Litim for encouragement\nand useful discussions.\nI acknowledge financial support from PPARC.\n\n\\section{Further Diagrammatics}\n\\label{sec:Further}\n\n\\subsection{Gauge Remainders, Again}\n\\label{sec:GRs-2}\n\nThe various additional properties of gauge remainders\nwhich we require are most readily introduced by example.\nTo this end, consider the three diagrams shown\nin \\fig{fig:GR-Ex}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\dec{\n\t\t\\begin{array}{ccc}\n\t\t\t\\vspace{1ex}\n\t\t\t\\LID{v-W-GR}&\t\t& \\LID{v-WGR}\n\t\t\\\\\n\t\t\t\\cd{v-W-GR} & \\displaystyle+ \t& \\cd{v-WGR} \n\t\t\\end{array}\n\t}{\\{f\\}}\n\t-\\frac{1}{2}\n\t\\dec{\n\t\t\\LIDi[1]{v-W-GR-w}{v-W-GR-w}\n\t}{\\{f\\}\\Delta}\n\t\\]\n\\caption{Examples of diagrams possessing a gauge remainder.}\n\\label{fig:GR-Ex}\n\\end{figure}\\end{center}\n\nThere are a number of things to note. First, since these\ndiagrams are used for illustrative purposes, they\nare labelled \\textbf{I.\\#}; for the $\\beta_n$ diagrammatics,\nthe labels will be \\textbf{D.\\#}. Secondly, $v$ and $w$ are just\nvertex arguments, and so take the values of non-negative\nintegers. Since these arguments do not carry a hat or a bar\n(see~\\ref{eq:bar}), the vertices belong to the Wilsonian effective action.\nThirdly, the decorations of diagram~\\ref{v-W-GR-w}\ninclude not just the set of\nfields, $\\{f\\}$, but also an effective propagator, $\\Delta$, which\ncan be in any sector. The rules for decorating \ndiagram~\\ref{v-W-GR-w}\nwith the effective propagator are simple: if each end\nattaches to a different object, then the combinatoric factor\nof the attachment is two, recognizing that the effective propagator\ncan attach either way around. If the two ends attach to the same\nobject, then the combinatoric factor is unity, recognizing that\neach vertex represents a sum over all permutations of the fields arranged\ninto all possible supertrace structures. Whilst the effective\npropagator and fields, $\\{f\\}$, are not explicitly drawn on the\ndiagrams, they will be referred to as implicit, or unrealized decorations.\n\nLet us now consider what happens when\nthe gauge remainders in each\nof the diagrams~\\ref{v-W-GR}--\\ref{v-W-GR-w} act. \nSince no other gauge remainders have acted and since\nno Taylor expansions have been performed, we can\nuse charge conjugation\\ to collect together the push forward and\nthe pull back  (see \\sec{sec:CC}).\nIn diagram~\\ref{v-W-GR}, the gauge remainder can strike one\nof two things: either the field to which the kernel\nattaches or one of the fields, $\\{f\\}$.\nIt proves to be technically\nvery convenient not to specify precisely which field\nthe gauge remainder\nhits in the latter case, only that it hits something.\nConsequently, we take the gauge remainder to strike a `socket',\nwhich we suppose can be filled by any of the fields, $\\{f\\}$.\nIn diagram~\\ref{v-WGR}, the gauge remainder can strike\na socket, the top end of the kernel or the bottom end \nof the kernel. In diagram~\\ref{v-W-GR-w}, the gauge remainder\ncan only strike a socket, though we note\nthat this socket can be filled not only by $\\{f\\}$ but\nalso by an end of the effective propagator. \\Fig{fig:GR-Ex-P} shows the result\nof processing the gauge remainders of \ndiagrams~\\ref{v-W-GR}--\\ref{v-W-GR-w}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\t\n\t\\begin{array}{rcl}\n\t\\vspace{2ex}\n\t\t\\dec{\n\t\t\t\\cd{v-W-GR}\t \n\t\t}{\\{f\\}}\n\t\t& =2 & \n\t\t\\dec{\n\t\t\t\\begin{array}{ccc}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\CID{v-W-GR-a}{v-W-GR-ai}\t&\t& \\CID{v-W-GR-b}{v-W-GR-b2}\n\t\t\t\\\\\n\t\t\t\t\\cd{v-W-GR-a}\t\t\t\t&-\t& \\cd{v-W-GR-b}\n\t\t\t\\end{array}\n\t\t}{\\{f\\}}\n\t\\\\\n\t\\vspace{2ex}\n\t\t\\dec{\n\t\t\t\\cd{v-WGR}\t \n\t\t}{\\{f\\}}\n\t\t& =-2 &\n\t\t\\dec{\n\t\t\t\\begin{array}{ccccc}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\CID{v-W-GR-ai}{v-W-GR-a}\t&\t& \\CID{v-WGR-b}{v-WGR-b2}\t&\t& \\LID{v-WGR-c}\n\t\t\t\t\\\\\n\t\t\t\t\\cd{v-W-GR-a}\t\t\t\t&+\t& \\cd{v-WGR-b}\t\t\t\t&-\t& \\cd{v-WGR-c}\n\t\t\t\\end{array} \n\t\t}{\\{f\\}}\n\t\\\\\n\t\t-\\displaystyle \\frac{1}{2}\n\t\t\\dec{\n\t\t\t\\cd{v-W-GR-w}\n\t\t}{\\{f\\}\\Delta}\n\t\t& = &\n\t\t\\dec{\n\t\t\t\\LIDi[1]{v-W-GR-w-b}{v-W-GR-w-b}\n\t\t}{\\{f\\}\\Delta}\n\t\\end{array}\n\t\\]\n\\caption{The result of allowing the gauge remainders\nof diagrams~\\ref{v-W-GR}--\\ref{v-W-GR-w} to act. In all\ndiagrams with the same (opposite) sign to the parent, the\npush forward and pull back have been collected into\ntwice the push forward (pull back).}\n\\label{fig:GR-Ex-P}\n\\end{figure}\\end{center}\n\nNotice that, rather than terminating the pushed forward \/ \npulled back field-line with a half arrow ({\\it cf.}\\  \\eq{eq:WID-A}), \nwe just utilize the fact\nthat the corresponding field line already ends \nin a $>$ and use this to indicate the field hit.\\footnote{\nNote that we will not always have a $>$ at our disposal.\nIn particular, we will encounter gauge remainders involving\npseudo effective propagators terminating in just $\\rhd$. In\nthis case, the notation of \\sec{sec:WIDs} must be used.\n}\nImmediately, we find a cancellation, which we indicate by\nenclosing the reference number of the cancelled diagram\nin curly brackets, together with the reference number\ndiagram against\nwhich it cancels.\n\n\\ICancelCom{v-W-GR-ai}{v-W-GR-a}{. Although these diagrams\nlook exactly the same, one might worry that they are\ndifferent: in diagram~\\ref{v-W-GR-ai}, the gauge remainder\npulls back along the kernel, whereas in diagram~\\ref{v-W-GR-a}\nthe gauge remainder has nothing to do with the kernel, instead\npushing forward around the vertex. However, gauge invariance\nensures that these two diagrams are indeed \nequivalent~\\cite{Thesis,mgierg1}, and so they cancel.}\n\nReturning to \\fig{fig:GR-Ex-P}, consider next\ndiagram~\\ref{v-WGR-c}. The line segment\nwhich joins the top\nof the kernel to the $>$---thereby forming a `hook'---performs \nno role other \nthan to make this join. In other words, it is neither a section of \nthe kernel nor an effective propagator. We could imagine\ndeforming this line segment so that the hook becomes arbitrarily large. \nDespite\nappearances, we must always remember \nthat this line segment simply performs the role\nof a Kronecker delta. When part of a complete diagram, this line\nsegment can always be distinguished from an effective propagator,\nto which it can be made to look identical, by the context. This\nfollows because hooks in which the line segment is a Kronecker~$\\delta$\nonly ever attach to effective propagators or kernels, whereas \nhook-like structures\nmade out of an effective propagator only ever attach to vertices\n(this will be particularly clear from the perspective\nof \\sec{sec:beta_n}).\nWhen viewed in isolation, we will always take the hook structure to\ncomprise just a line segment and so will draw the hook as tightly as\npossible.\n\nTo conclude our discussion of the hook, we give its algebraic\nform. First, we note from table~\\ref{tab:NFE:k,k'} that\nthe gauge\nremainder that forms the hook must be in the $D$ $(\\bar{D})$ sector,\nelse the loop integral over its momenta is odd and vanishes. \nConsequently, \nthe field to which the hook attaches must be\nin the  $C^1$ ($C^2$) sector; a conclusion which could\nalso have been drawn from consideration of charge conjugation\\ invariance.\nThus, in the former (latter)\ncase, the gauge remainder of the parent, which is in the\n$F$ ($\\bar{F}$) sector, strikes the $F$ ($\\bar{F}$) at\nthe top of the kernel, turning it into a $C^1$ ($-C^2$).\nThe sign in the latter case essentially arises from\nthe minus sign in fifth component of $\\bar{F}_N$ (see\n\\eqs{eq:F}{eq:Fbar}) (see~\\cite{Thesis,mgierg1} for \npainstaking detail). Our prescription is to absorb\nany signs associated with the end of the kernel\ninto the definition of the hook.\n\n\n\nThe final ingredient we need to obtain the algebraic form for the hook is to\nrealize that the inside of the hook constitutes an empty loop and so\ngives a group theory factor of $\\pm N$, depending on flavour. Thus we have:\n\\numparts\n\\begin{eqnarray}\n\\label{eq:hook-R-1}\n\t\\cd{hook-R-1} & = & +(-N) \\int_l g_l \n\\\\\n\\label{eq:hook-R-2}\n\t\\cd{hook-R-2} & = & - (N) \\int_l g_l.\n\\end{eqnarray}\n\\endnumparts\nThe numbers at the base of the hook in the above equations indicate the sector\nof the $C^i$ field to which the hooks attach.\n\nWe now see a further advantage to having absorbed signs into the definition of\nthe hook: we can trivially extend the diagrammatic effect of charge conjugation\\ to cover\nits action on a disconnected hook.\nQuite simply, we have that\n\\begin{equation}\n\t\\cd{hook-R} = - \\cd{hook-L},\n\\label{eq:hookCC}\n\\end{equation}\nconsistent with our previous definition that we take the mirror image, picking\nup a sign for every performed gauge remainder. \n\n\nNow we focus on diagram~\\ref{v-W-GR-w-b}. This diagram,\nlike~\\ref{v-W-GR-b} and~\\ref{v-WGR-b}, possesses a socket;\nwe will now see why it is so useful to leave this socket\nempty. To process this diagram further, we suppose that\nthe vertex with argument $w$ possesses a classical, two-point\ncomponent. Of course, in this case, this means that\n$w$ must be a classical vertex. However, in the computation\nof $\\beta_n$, we will generally be summing over vertex arguments,\nsuch that $w$ could take all values between zero and some\npositive integer. In preparation for this, we  leave the\nvertex argument as $w$ rather than explicitly writing it\nas zero. To separate off the classical, two-point component\nof $w$, we define the reduction of a vertex to be\nthe full vertex minus its classical, two-point component\n(should this component exist). The reduction is denoted\nby appending the appropriate vertex argument with a superscript $R$\n{\\it viz.}\\  $w^R$. This decomposition of diagram~\\ref{v-W-GR-w-b}\nis shown in \\fig{fig:Decompose}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\dec{\n\t\t\\cd{v-W-GR-w-b}\n\t}{\\{f\\}\\Delta}\n\t=\n\t\\dec{\n\t\t\\begin{array}{ccc}\n\t\t\\vspace{1ex}\n\t\t\t\\LID{v-W-GR-w-bR}\t&\t& \\LID{v-W-GR-w-b02}\n\t\t\\\\\n\t\t\t\\cd{v-W-GR-w-bR}\t& + & \\cd{v-W-GR-w-b02}\n\t\t\\end{array}\n\t}{\\{f\\}\\Delta}\n\t\\]\n\\caption{Decomposing the vertex of diagram~\\ref{v-W-GR-w-b}\n(which in this case is implicitly a classical vertex)\ninto a reduced part and a classical, two-point part.}\n\\label{fig:Decompose}\n\\end{figure}\\end{center}\n\nThe top vertex of diagram~\\ref{v-W-GR-w-bR}\nis reduced. The vertex argument $0^2$ of \ndiagram~\\ref{v-W-GR-w-b02} tells us that this\nvertex must not only be classical but must also\npossess precisely two decorations. Thus in addition\nto the socket, this vertex must have one and only\none additional decoration. Clearly, this additional\ndecoration can either be one of the fields $\\{f\\}$\nor one of the ends of the effective propagator. \nThe\nformer case will not concern us here, so we focus\non the latter case. Given that one end of the effective\npropagator decorates the classical, two-point vertex,\nthere are three locations to which the loose end can attach:\n\\begin{enumerate}\n\t\\item\tthe socket;\n\t\\label{it:socket}\n\n\t\\item\tthe vertex with argument $v$.\n\t\\label{it:vertex}\n\n\t\\item\tthe kernel;\n\t\\label{it:kernel}\n\\end{enumerate}\n\nThe first case vanishes by charge conjugation\\ invariance. \nTo see this,\nconsider the parent of diagram~\\ref{v-W-GR-w-b},\ndiagram~\\ref{v-W-GR-w}. To generate the diagram\ncorresponding to~\\ref{it:socket}, we should\nconsider the component of diagram~\\ref{v-W-GR-w}\nin which\nthe top vertex is a three-point vertex, \nwith one of the fields corresponding to the gauge\nremainder and the other two corresponding to the\ntwo ends of the effective propagator. Now,\nsince net fermionic vertices vanish, the gauge\nremainder must be bosonic. Furthermore, by charge conjugation\\ invariance,\nthe gauge remainder is forced to be in the $C^{1,2}$\nsector, in which it is null (see \\sec{sec:CC}).\n\nIn both cases~\\ref{it:vertex} and~\\ref{it:kernel},\nwe can employ the effective propagator relation.\nThis is shown in \\fig{fig:EP-Ex}. We note that\nfollowing. First,\nusing the effective propagator to join the\nclassical, two-point vertex to some other object\nyields a combinatoric factor of two. Secondly,\nwhen we henceforth talk\nof decorating a classical, two-point\nvertex we always mean that we are explicitly\ndrawing on the second decoration of the vertex,\nthe socket being the first decoration. If we\nmean to fill the socket, this will be explicitly\nstated.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray*}\n\t\\dec{\n\t\t\\cd{v-W-GR-w-b02}\n\t}{\\{f\\} \\Delta}\n\t& = &\n\t2\n\t\\dec{\n\t\t\\begin{array}{ccc}\n\t\t\\vspace{1ex}\n\t\t\t\\LID{v-W-GR-w-b02-v}&\t& \\LID{v-W-GR-w-b02-k}\n\t\t\\\\\n\t\t\t\\cd{v-W-GR-w-b02-v}\t&+\t& \\cd{v-W-GR-w-b02-k}\n\t\t\\end{array}\n\t}{\\{f\\}} + \\cdots\n\t\\\\\n\t& = &\n\t2\n\t\\dec{\n\t\t\\begin{array}{ccc}\n\t\t\\vspace{1ex}\n\t\t\t\\CID{v-W-GR-b2}{v-W-GR-b}\t&\t& \\LID{v-W-nGR}\n\t\t\\\\\n\t\t\t\\cd{v-W-GR-b}\t\t\t\t& - & \\cd{v-W-nGR}\n\t\t\\\\\n\t\t\\vspace{1ex}\n\t\t\t\\CID{v-WGR-b2}{v-WGR-b}\t\t&\t& \\LID{v-WnGR}\n\t\t\\\\\n\t\t\t\\cd{v-WGR-b}\t\t\t\t& - & \\cd{v-WnGR}\n\t\t\\end{array}\n\t}{\\{f\\}} + \\cdots\n\t\\end{eqnarray*}\n\\caption{The decoration of the classical, two-point vertex\nof diagram~\\ref{v-W-GR-w-b02} by the effective propagator,\ntogether with the subsequent application of the\neffective propagator relation. The ellipses\ndenote diagrams in which the classical, two-point\nvertex is decorated by one of the fields, $\\{f\\}$, rather\nthan by the effective propagator.}\n\\label{fig:EP-Ex}\n\\end{figure}\\end{center}\n\n\\ICancel{v-W-GR-b2}{v-W-GR-b}\n\n\\ICancel{v-WGR-b2}{v-WGR-b}\t\n\nThere are some comments worth making about\ndiagrams~\\ref{v-W-nGR} and~\\ref{v-WnGR}.\nFirst, \nthe $>$\npart of the full gauge remainder plays the role of\nthe socket in the parent diagrams. \nSecondly, these diagrams\nhave a very similar\nstructures to diagrams~\\ref{v-W-GR} and~\\ref{v-WGR}.\nThe only difference is that the full gauge remainder\nin the new diagrams is nested~\\cite{Thesis}, meaning\nthat it does not attach directly to the kernel, but\ninstead is hit by the gauge remainder component which\nends the kernel.\nProcessing nested gauge remainders\nis much the same as processing un-nested gauge\nremainders, but charge conjugation\\ invariance cannot generally be used\nagain to collect together the \nnested push forward and pull back~\\cite{Thesis,Primer}:\nwe must count them separately. However, everything\nelse goes through as before and so we will find that\ncancellation~\\ref{Icancel:v-W-GR-ai} is repeated\ntwice (twice because we count the nested push forward\nseparately from the nested pull back).\n\nIf there were more vertices available, we could imagine\niterating the above procedure and thus generating\narbitrarily nested gauge remainders.\nIt would then be useful for us to know if there is \nany simple way of\nkeeping track of which gauge remainders have pushed \nforward and which have pulled back.\nWith sufficient thought, it is always \npossible to stare at a complicated\ndiagram and deduce the pattern of pushes \nforward and pulls back\n(up to the ambiguity associated with illustrative \ncancellation~\\ref{Icancel:v-W-GR-ai}). However, there is\nan easier way. Note in diagrams~\\ref{v-W-nGR} and~\\ref{v-WnGR}\nthat the $>$ part of the full gauge remainder \nis bitten on its right-hand edge by the $>$ which \nterminates the kernel. (We define the right-hand edge\nto be the edge on our right as we traverse an imaginary\nline running into the socket at the apex {\\it viz.}\\  \n$\\left.^{\\scriptscriptstyle \\mathrm{L}}_{\\scriptscriptstyle \\mathrm{R}} \\cd{EP-GR}\\right.$;\nlikewise for sockets on vertices \/ kernels.)\nHad we pushed forward, rather than pulled \nback with the initial\ngauge remainder, then this bite would have been to the left. \nWith this in mind, consider a string\nof gauge remainders which bite a socket \ndecorating either a kernel or a vertex; the latter\ncase is shown in \\fig{fig:GRstring}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\LIDi{v-GRstring}{v-GRstring}\n\t\\]\n\\caption{An arbitrarily nested gauge remainder bites a socket on a vertex.}\n\\label{fig:GRstring}\n\\end{figure}\\end{center}\n\n\n\nWe know from the discussion in illustrative \ncancellation~\\ref{Icancel:v-W-GR-ai} that,\nwere we to fill the socket of diagram~\\ref{v-GRstring}\nwith the end of a kernel (the other end of which\nattaches to the gauge remainder at the beginning\nof the string), then the sense in which the socket\nis bitten can be interpreted in two ways:\neither as a push forward around the vertex or\nas a pull back along the kernel. Given a diagram like~\\ref{v-GRstring},\nwe will always\ninterpret the gauge remainder at the end of the string of\ngauge remainders as having\nbitten the vertex, using this to determine the sense\nin which the gauge remainder acts.\n\n\n\n\nTo determine\nin which sense the nested gauge remainders are bitten\nwe can equate bites\non the left with pushes forward and bites on the right with pulls back,\nthe only exception being when a gauge remainder bites\nthe end of kernel to form a hook: the gauge remainder\nthat pushed forward to form\nthe un-nested hook of diagram~\\ref{v-WGR-c}\ncan be thought of as biting itself on the right.\nThus, moving on to consider\nthe arbitrarily nested hook shown in \\fig{fig:nestedhook},\nthe number of pushes forward is equal to the number of \nbites on the left, plus one, and the number of pulls\nback is equal to the number of bites on the right, minus one.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\cd{Struc-GR-ring}\n\t\\]\n\\caption{An arbitrarily nested version of the hook.}\n\\label{fig:nestedhook}\n\\end{figure}\\end{center}\n\nStructures like those in \\fig{fig:GRstring}\nhave a compact representation, shown below:\n\\[\n\t\\frac{1}{m!}\\cd{compact} \\ .\n\\]\n\nThe notation $>^m$ stands for $m$ instances of $>$.\nThese gauge remainder\ncomponents form a string which bites the socket. We sum over\nall possible ways in which each gauge remainder can bite,\nand so diagram~\\ref{v-GRstring} is just one component included\nour compact diagrammatic representation. When going from\nthe compact diagrammatic representation to its explicit \ncomponents, we note that any of the $m$ gauge remainders\ncan bite the socket, and this gauge remainder can, in turn,\nbe bitten by any of the remaining $m-1$ gauge remainders.\nThis explains the normalization factor of $1\/m!$.\nDenoting the number of bites to the left \/ right by L \/ R,\nthe sign of each explicitly drawn diagram is just $(-1)^R$.\n\nSimilarly, there is a compact diagrammatic representation\t\nof the nested hook---or ring---of \\fig{fig:nestedhook}, \nwhich is simply\n\\[\n\t\\frac{1}{(m-1)!} \\decGR{ \\ }{>^m}.\n\\] \nThe rule to go from this compact\nrepresentation to explicit diagrams is as follows:\nwe draw the set of rings corresponding to\nall independent sequences bites to the left \/ right.\nThe normalization factor follows \nfrom noting that a ring is invariant\nunder a cyclic permutations of the gauge remainders.\nThe sign of each explicitly drawn ring is $(-1)^{R-1}$.\n\n\n\n\nWe conclude this discussion of the gauge \nremainders by examining diagrams possessing\ntwo active gauge remainders. The techniques we use to process such terms are\nexactly the same as  the ones detailed already in this section. There is,\nhowever, a qualitatively new type of diagram that arises, \ntogether with a source\nof possible confusion. To investigate both of these issues, we focus on an\nexample\nin which two full gauge remainders bite a kernel\\footnote{\nDouble gauge remainder diagrams are not restricted to those in which both gauge\nremainders bite a kernel; one or both of the gauge remainders can bite a vertex.},\nas shown in \\fig{fig:DoubleGR}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\frac{1}{2}\n\t\\LIDi[1]{Struc-WGRx2-B}{Diags-WGRx2-B}\n\t\\]\n\\caption{An example of a double gauge remainder diagram.}\n\\label{fig:DoubleGR}\n\\end{figure}\\end{center}\n\nWe proceed by allowing first one gauge remainder to act and then, if possible,\nallowing the second gauge remainder to act. The qualitatively new type of diagram\narises because one of the\neffects of the first gauge remainder can be to `trap' the second gauge remainder,\nby biting the field on the kernel to which the second gauge remainder attaches.\nThe remaining full gauge remainder now does \nnot have the same momentum flowing through\nit as the field it is trying to bite, and so it cannot act.\nThe other diagrams generated are those in\nwhich the processed gauge remainder bites one of the\nends of the kernel. The result of allowing the first gauge remainder to\nact is shown in \\fig{fig:DoubleGR-A}, where we have collected pulls\nback and pushes forward, as usual.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\t\\LID{Diags-WGRx2-Top}\t&\t& \\LID{Diags-WGRx2-Trap}\t&\t& \\LID{Diags-WGRx2-Bottom}\n\t\\\\[1ex]\n\t\t\\cd{Struc-WGRx2-Top}\t& -\t& \\cd{Struc-WGRx2-Trap}\t& +\t& \\cd{Struc-WGRx2-Bottom}\n\t\\end{array}\n\t\\]\n\\caption{Result of allowing one gauge remainder in diagram~\\ref{Diags-WGRx2-B} to act.}\n\\label{fig:DoubleGR-A}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{Diags-WGRx2-Trap} possesses the \ntrapped gauge remainder. In diagram~\\ref{Diags-WGRx2-Bottom}\nwe have recognized that the kernel ends where it attaches \nto the active gauge remainder\nand so this is where the processed gauge remainder bites. \nNote that\nwe can trivially redraw this diagram, as shown in \\fig{fig:DoubleGR-B}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\LIDi[1]{Struc-WGRx2-Bottom-B}{Diags-WGRx2-Bottom-B}\n\t\\]\n\\caption{A trivial redrawing of diagram~\\ref{Diags-WGRx2-Bottom}.}\n\\label{fig:DoubleGR-B}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{Diags-WGRx2-Bottom-B} highlights how one must be \nvery careful when drawing\nwhich end of an active gauge remainder a \nprocessed gauge remainder bites; diagrams~\\ref{Diags-WGRx2-Bottom-B}\nand~\\ref{Diags-WGRx2-Trap} are clearly different. Notice, however, that if\nthe active gauge remainder were \\emph{absent}, then\ndiagram~\\ref{Diags-WGRx2-Bottom}\nwould cancel diagram~\\ref{Diags-WGRx2-Trap}; we will exploit this\nlater.\n\n\nFinally, consider allowing the active gauge remainder to \nact in diagrams~\\ref{Diags-WGRx2-Top}\nand~\\ref{Diags-WGRx2-Bottom-B}. We are not \ninterested in all the contributions. Rather,\nwe just want to focus on\n\\begin{enumerate}\n\t\\item\tthe term produced by diagram~\\ref{Diags-WGRx2-Top} where the\n\t\t\tgauge remainder pulls back along the kernel, to the same end as the hook;\n\n\t\\item\tthe term produced by diagram~\\ref{Diags-WGRx2-Bottom-B} in which the\n\t\t\tgauge remainder pulls back to the bottom of the kernel.\n\\end{enumerate}\nReflecting the former diagram about a horizontal line, we arrive at the two diagrams\nof \\fig{fig:DoubleGR-C}. (The reflection does not yield a net sign as\nwe pick up one for each of the processed gauge remainders.) \n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{cccc}\n\t\t\t& \\LID{Diags-WGRx2-L-Bottom}&\t& \\LID{Diags-WGR-N1-Bottom-PF}\n\t\\\\[1ex]\n\t\t-\t& \\cd{Struc-WGRx2-L-Bottom}\t& -\t& \\cd{Struc-WGR-N1-Bottom-PF}\n\t\\end{array}\n\t\\]\n\\caption{Two of the terms produced by processing diagrams~\\ref{Diags-WGRx2-Top}\nand~\\ref{Diags-WGRx2-Bottom-B}.}\n\\label{fig:DoubleGR-C}\n\\end{figure}\\end{center}\n\nFrom the way in which these two diagrams have been drawn, it is clear that\nthey are distinct and that they must be treated as such. We can, however, redraw\ndiagram~\\ref{Diags-WGR-N1-Bottom-PF} by \nsliding the outer gauge remainder round the hook\nto where the inner gauge remainder bites the kernel. This is shown in \n\\fig{fig:DoubleGR-D}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t-\\LIDi[1]{Struc-WGR-N1-Bottom-PF-B}{Diags-WGR-N1-Bottom-PF-B}\n\t\\]\n\\caption{A trivial redrawing of diagram~\\ref{Diags-WGR-N1-Bottom-PF}.}\n\\label{fig:DoubleGR-D}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{Diags-WGR-N1-Bottom-PF-B} is, of course, still the same as\ndiagram~\\ref{Diags-WGR-N1-Bottom-PF} but it is starting to look very similar\nto diagram~\\ref{Diags-WGRx2-L-Bottom}. Indeed, we must make sure that we never\nslide the gauge remainder so far round the hook that it appears to bite the kernel\nat the same point as the gauge remainder which forms the hook. If this\nwere to happen, then such a diagram would be ambiguous.\n\n\n\n\n\n\\subsection{The Secondary Diagrammatic Identities}\n\\label{sec:Secondary}\n\nThere are two families of secondary diagrammatic identities.\nMembers of the first family are\ngenerally applicable, whereas\n members of\nthe second family are\napplicable only to diagrams which have\nbeen manipulated at $\\Op{2}$.\n\n\n\n\n\n\n\\subsubsection{The First Family}\n\\label{sec:D-ID-Secondary-I}\n\nThe first diagrammatic identity in this\nfamily is trivial:\n\\begin{equation}\n\\label{D-ID-Trivial-A}\n\t\\cdeps{GR-hook} = 0.\n\\end{equation}\nThis follows directly from charge conjugation\\ invariance:\nthe field attaching to the hook must\nbe in the $C^i$ sector, but in this sector\nthe gauge remainder contracted into the\nhook is null. Before moving on, notice\nthat the diagram contains one empty socket,\ncorresponding to the $>$ part of the\nfull gauge remainder.\n\nThere is a second trivial identity\ninvolving a diagram possessing a single\nsocket, which we have seen already. We\nre-draw it here in what will become a\nsuggestive form:\n\\begin{equation}\n\\label{eq:D-ID-Trivial-PEP}\n\t\\cd{TLTP-Soc-EP-GR} = 0,\n\\end{equation}\nrecalling that this identity follows as\na consequence of the effective propagator\nrelation and~\\eq{eq:GR-relation}. We\nexplicitly indicate that the vertex is\na two-point vertex, since such structures\nwill generically \noccur as sub-diagrams in diagrams\nwith unrealized decorations.\n\n\nTo uncover the next diagrammatic identity,\nwhich is the first we will encounter involving two\nsockets, \nconsider the pair of diagrams in\n\\fig{fig:D-ID-Trivial-b}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\\vspace{1ex}\n\t\t\\LID{Trivial-b-A}\t&\t& \\LID{Trivial-b-A2}\t\n\t\\\\\n\t\t\\cd{Trivial-b-A} \t& -\t& \\cd{Trivial-b-A2} \t\n\t\\end{array}\n\t\\]\n\\caption{A pair of diagrams whose sum vanishes.}\n\\label{fig:D-ID-Trivial-b}\n\\end{figure}\\end{center}\nThese\ndiagrams naturally arise from \na common parent: the differences between\nthe two diagrams come from the sense in\nwhich the full gauge remainder of\nthe parent, of which \nthe component labelled $X$ is left behind, acts.\nThe point now is that diagrams~\\ref{Trivial-b-A}\nand~\\ref{Trivial-b-A2} cancel. By construction,\nthe unspecified structure at the bottom of both\ndiagrams is common. Furthermore, the gauge\nremainder labelled $X$ and its attachment to\nthe common structure is the same between the\ntwo diagrams. From~\\eqs{eq:hook-R-1}{eq:hook-R-2},\nthe two hooks yield the same algebraic contribution.\nThe only difference between the two diagrams is\nthe sense in which the gauge remainder labelled $X$\nacts. However, effectively\nall that this gauge remainder does is determine\nwhether the hook is of type~\\eq{eq:hook-R-1} \nor~\\eq{eq:hook-R-2}; and we know that the hook is\njust a common algebraic factor between the two diagrams.\nTherefore, diagrams~\\ref{Trivial-b-A}\nand~\\ref{Trivial-b-A2} are in fact identical up to a\nsign and so cancel.\n\nThis leads us to the following diagrammatic\nidentity:\n\\begin{equation}\n\\label{eq:D-ID-Bitten-hook-A}\n\t\\cdeps{Bitten-hook} - \\cdeps{Bitten-hook-R} \\equiv 0,\n\\end{equation}\nwhere we understand that this diagram should be viewed\nas part of some larger diagram, attached via internal\nfields. \nNote that in this diagrammatic\nidentity, and all that follow, we can strip off\nthe common $>$ which attaches to some\nother part of a larger diagram, replacing it with\nthe half arrow notation of~\\eq{eq:WID-A}.\n\nThe non-trivial diagrammatic identities which now follow can\nbe thought of as arising from versions \nof~\\eq{D-ID-Trivial-A} and~\\eq{eq:D-ID-Bitten-hook-A}\nin which the hook is nested and versions of~\\eq{eq:D-ID-Trivial-PEP}\nin which the gauge remainder is nested. To be\nspecific, we will find relationships between\nthe nested versions of~\\eq{D-ID-Trivial-A}--\\eq{eq:D-ID-Bitten-hook-A}\nfor which the numbers of sockets are equal; equivalently,\nthese relationships are between diagrams with an\nequal number of \\emph{performed} gauge remainders.\n\nThe key to proving the non-trivial diagrammatic\nidentities resides in the following relationship\nbetween gauge remainder components. Using the notation\ndescribed around~\\eq{GR:flavour}, and taking $j_2 = j_1 \\ \\xor j_3$\nwe have:\n\\begin{equation}\n\\label{eq:Template}\n\t1 - \\rhd^{j_1}_{l-m_1-k} >^{j_2}_{l-m_1} - \\rhd^{j_3}_k >^{j_2}_{l-m_1} = 0.\n\\end{equation}\t\nTo demonstrate this, we simply substitute for the\ngauge remainder components using \\tab{tab:NFE:k,k'}\nand employ~\\eq{eq:xf+2g}, as shown in \\tab{tab:template}.\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\\begin{table}[h]\n\t\\[\n\t\\begin{array}{cc|c|rcl}\n\t\tj_1\t& j_3\t& j_2\t& \t\n\t\\\\ \\hline\n\t\t0\t& 0\t\t& 0\t\t& 1 - \\frac{(l-m_1-k)\\cdot(l-m_1)}{(l-m_1)^2} - \\frac{k \\cdot (l-m_1)}{(l-m_1)^2} & = & 0\n\t\\\\\n\t\t0\t& 1\t\t& 1\t\t& 1 - f_{l-m_1} \\frac{(l-m_1-k)\\cdot(l-m_1)}{\\Lambda^2} - f_{l-m_1} \\frac{k\\cdot(l-m_1)}{\\Lambda^2} - 2g_{l-m_1} & = & 0\n\t\\\\\n\t\t1\t& 0\t\t& 1\t\t& 1 - f_{l-m_1} \\frac{(l-m_1-k)\\cdot(l-m_1)}{\\Lambda^2} - 2g_{l-m_1} - f_{l-m_1} \\frac{k\\cdot(l-m_1)}{\\Lambda^2} & = & 0\n\t\\\\\n\t\t1\t& 1\t\t& 0\t\t& 1 - \\frac{(l-m_1-k)\\cdot(l-m_1)}{(l-m_1)^2} - \\frac{k \\cdot (l-m_1)}{(l-m_1)^2} & = & 0\n\t\\end{array}\n\t\\]\n\\label{tab:template}\n\\caption{Break down of~\\eq{eq:Template} into the different\nsectors.}\n\\end{table}\\end{center}\n\\renewcommand{\\arraystretch}{1}\n\n\nThe first non-trivial relationship involves versions \nof~\\eq{D-ID-Trivial-A}--\\eq{eq:D-ID-Bitten-hook-A} with\ntwo sockets. This identity is, in some sense, still a\nspecial case, since the two-socket version \nof~\\eq{eq:D-ID-Bitten-hook-A} is of course trivial.\nConsequently, we treat the two-socket case---which is\nshown in \\fig{fig-D-ID-2}---separately from the rest. \n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\\label{eq:D-ID-2}\n\t\t\\begin{array}{ccccccccc}\n\t\t\\vspace{1ex}\n\t\t\t\t& \\LID{TLTP-EP-LL}\t&\t& \\LID{GR-ring-RR}\t&\t& \\LID{TLTP-EP-RR}\t&\t& \\LID{GR-ring-LL}\t&\n\t\t\\\\\n\t\t\\vspace{1ex}\n\t\t\t\t& \\cd{TLTP-EP-LL} \t& -\t& \\cd{GR-ring-RR}\t& +\t& \\cd{TLTP-EP-RR}\t& -\t& \\cd{GR-ring-LL}\t&\n\t\t\\\\\n\t\t\\vspace{1ex}\n\t\t\t\t&\\LID{TLTP-EP-RL}\t&\t& \\LID{GR-ring-RL}\t&\t& \\LID{TLTP-EP-LR}\t&\t& \\LID{GR-ring-LR}\t&\n\t\t\\\\\n\t\t\t-\t&\\cd{TLTP-EP-RL}\t& + &\\cd{GR-ring-RL} \t& -\t& \\cd{TLTP-EP-LR}\t& +\t& \\cd{GR-ring-LR} \t& = 0\n\t\t\\end{array}\n\t\\end{equation}\n\\caption{Two sets of diagrams, each of which will be shown\nto sum individually to zero.}\n\\label{fig-D-ID-2}\n\\end{figure}\\end{center}\n\nDiagrams~\\ref{TLTP-EP-RR}, \\ref{TLTP-EP-RL}\nand~\\ref{TLTP-EP-LR} can each be obtained\nfrom diagram~\\ref{TLTP-EP-LL} by changing\nthe senses in which the gauge remainders\nbite (the remaining four diagrams are similarly\nrelated).\nThere are actually some stronger relationships than\nthe one given above. For example, the first four diagrams\nonly ever cancel amongst themselves. However,\nfor our analysis of $\\beta_n$, only the\nstatement~\\eq{eq:D-ID-2} is required.\n\n\n\nTo prove \\eq{eq:D-ID-2},\nwe begin by focusing on the first\npair of diagrams, which we redraw in\n\\fig{fig-D-ID-2b}. Notice that we\nhave used \nthe effective propagator relation in\ndiagram~\\ref{TLTP-EP-LL},\nhave filled the sockets \nwith fields of a specific flavour \nand have chosen a convenient momentum\nrouting for each of the diagrams.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\\vspace{1ex}\n\t\t\\LID{TLTP-EP-LL-Kd2}\t&\t&\\LID{TLTP-EP-LL-GR}\t&\t&\\LID{GR-ring-RR-2}\n\t\\\\\n\t\t\\cd{TLTP-EP-LL-Kd2}\t\t& -\t& \\cd{TLTP-EP-LL-GR}\t& -\t& \\cd{GR-ring-RR-2}\n\t\\end{array}\n\t\\]\n\\caption{Re-expression of diagrams~\\ref{TLTP-EP-LL} and~\\ref{GR-ring-RR}.}\n\\label{fig-D-ID-2b}\n\\end{figure}\\end{center}\n\nThe letters $U$--$Z$ denote field flavour, \nand the subscripts\n$T$ and $R$ represent indices. It is straightforward to\nverify, and intuitively obvious, that $W_R$ and $X_T$\nmust be either both bosonic or both fermionic. In the former\ncase, the internal fields $U, \\ V,\\ Y$ and $Z$ are all\nof the same flavour, being either all bosonic or\nall fermionic. In the case that the internal\nfields are all bosonic, diagrams~\\ref{TLTP-EP-LL-Kd2}--\\ref{GR-ring-RR-2}\nreduce to a common factor multiplied by\n\\begin{equation}\n\\label{eq:D-ID-2-Alg-a}\n\t1 - \\rhd_l^0 >_{l+k}^0 - \\rhd_k^0 >^0_{l+k},\n\\end{equation}\nwhereas, in the case that the internal fields\nare all fermionic, they reduce to some\nother common factor multiplied by\n\\begin{equation}\n\\label{eq:D-ID-2-Alg-b}\n\t1 - \\rhd_l^1 >_{l+k}^1 - \\rhd_k^0 >^1_{l+k}.\n\\end{equation}\nIt is immediately\napparent from~\\eq{eq:Template}\nthat both~\\eq{eq:D-ID-2-Alg-a} and~\\eq{eq:D-ID-2-Alg-b}\nyield zero, upon identifying $-m_1$ with $k$.\nIn exactly the same way, diagrams~\\ref{TLTP-EP-RR} and\n\\ref{GR-ring-LL}, \\ref{TLTP-EP-RL} and~\\ref{GR-ring-RL},\n\\ref{TLTP-EP-LR} and~\\ref{GR-ring-LR} cancel\nwhen the external fields are bosonic.\n\nLet us now consider filling the\nsockets of diagrams~\\ref{TLTP-EP-LL}--\\ref{GR-ring-LR}\nwith fermionic fields. The first thing\nto realize is that the latter four diagrams\ndo not exist with these decorations, since\nsuch decoration is incompatible with the group\ntheory structure~\\cite{Thesis}. In \\fig{fig:D-ID-2-F2},\nwe collect together all surviving terms\nfor which $W_R = \\bar{F}$ and $X_T = F$\n(and identical analysis can be done for \n$F \\leftrightarrow \\bar{F}$), and sum\nover all possible flavours of the internal fields.\n\\begin{center}\\begin{figure}[h]\n\\numparts\n\t\\begin{eqnarray}\n\t\t\\begin{array}{ccccccc}\n\t\t\\vspace{1ex}\t\t\t\n\t\t\t\\LID{TLTP-EP-LL-F2a}&\t& \\LID{GR-ring-RR-F2a}\t&\t& \\LID{TLTP-EP-RR-F2a}\t&\t& \\LID{GR-ring-LL-F2a}\n\t\t\\\\\n\t\t\t\\cd{TLTP-EP-LL-F2a}\t& -\t& \\cd{GR-ring-RR-F2a}\t& +\t& \\cd{TLTP-EP-RR-F2a}\t& -\t& \\cd{GR-ring-LL-F2a}\t\t\n\t\t\\end{array}\n\t\\\\\n\t\t\\begin{array}{ccccccc}\n\t\t\\vspace{1ex}\n\t\t\t\\LID{TLTP-EP-LL-F2b}&\t& \\LID{GR-ring-RR-F2b}\t&\t& \\LID{TLTP-EP-RR-F2b}\t&\t& \\LID{GR-ring-LL-F2b}\n\t\t\\\\\n\t\t\t\\cd{TLTP-EP-LL-F2b}\t& -\t& \\cd{GR-ring-RR-F2b}\t& +\t& \\cd{TLTP-EP-RR-F2b}\t& -\t& \\cd{GR-ring-LL-F2b}\t\t\n\t\t\\end{array}\n\t\\end{eqnarray}\n\\endnumparts\n\\caption{Field decomposition of diagrams~\\ref{TLTP-EP-LL}--\\ref{GR-ring-LL}\nin the case that\nthe external fields are fermionic.}\n\\label{fig:D-ID-2-F2}\n\\end{figure}\\end{center}\n\nOn the basis of~\\eqs{eq:D-ID-2-Alg-a}{eq:D-ID-2-Alg-b},\nwe might expect diagram~\\ref{TLTP-EP-LL-F2a}\nto cancel diagram~\\ref{GR-ring-RR-F2a} {\\it etc.}\\  However,\nthis expectation is not borne out; rather we find\nthat the Kronecker~$\\delta$ part of~\\ref{TLTP-EP-LL-F2a} (\\ref{TLTP-EP-RR-F2a})\ncombines with the gauge remainder part of~\\ref{TLTP-EP-RR-F2a} (\\ref{TLTP-EP-LL-F2a})\nand~\\ref{GR-ring-LL-F2a} (\\ref{GR-ring-RR-F2a}), to yield a common factor\nmultiplied by a term of the form~\\eq{eq:Template}.\nIn this way, we thus find that each row of diagrams\nin \\fig{fig:D-ID-2-F2} vanishes.\n\nThis completes the discussion of diagrammatic\nidentity~\\eq{eq:D-ID-2}\nand so now we move on to the general case,\nwhich relates versions \nof~\\eq{D-ID-Trivial-A}--\\eq{eq:D-ID-Bitten-hook-A}\nwith any number of sockets~$\\geq 3$ to each other. To begin, we consider\nby looking at those diagrams with the specific\npatterns of bites on the left \/ right shown in\n\\fig{fig:D-ID-G-a}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray}\n\t\t\\begin{array}{ccccc}\n\t\t\\vspace{1ex}\n\t\t\t\\LID{TLTP-EP-G-Kd}\t&\t\t\t\t\t& \\LID{TLTP-EP-G-GR}\t\t\t\t&\t& \\LID{GR-ring-G}\n\t\t\\\\\n\t\t\t\\cd{TLTP-EP-G-Kd} \t& \\hspace{-2em} - \t& \\hspace{-1em} \\cd{TLTP-EP-G-GR} \t& -\t& \\cd{GR-ring-G}\n\t\t\\end{array}\n\t\\nonumber \\\\\n\t\\label{eq:D-ID-G-R}\n\t\t\\begin{array}{ccccc}\n\t\t\\vspace{1ex}\n\t\t\t\t& \\LID{GR-ring-G-c}\t\t&\t& \\LID{GR-ring-G-b}\t&\n\t\t\\\\\n\t\t\t- \t& \\cd{GR-ring-G-c}\t\t& +\t& \\cd{GR-ring-G-b} \t& = 0 \n\t\t\\end{array}\n\t\\end{eqnarray}\n\\caption{A generalized diagrammatic identity.}\n\\label{fig:D-ID-G-a}\n\\end{figure}\\end{center}\n\nThere are a number of things to note about \ndiagrams~\\ref{TLTP-EP-G-Kd}--\\ref{GR-ring-G-b}.\nFirst, each of the diagrams possesses the fields\n$W$, $X$ and $Z^1, \\cdots, Z^r$ with indices\n$R$, $T$ and $S_1, \\cdots S_r$, respectively.\nThe cyclic ordering of these fields is the same\nin each diagram.\nIn the case that each diagram possesses only\nthree external fields, we should identify\n$Z^1$ with $Z^r$. \nSecondly, \nfor each of the fields $X$,\n$Z^1, \\cdots Z^r$ there is, in each\ndiagram, an associated $>$. The \nflavour of this $>$, which may\nof course differ from that of the \nassociated external\nfield, is denoted by  $X'$, ${Z^1}', \\cdots {Z^r}'$,\nas appropriate. Notice that, in diagram~\\ref{TLTP-EP-G-GR},\nthe field line $X$ is associated with not one\nbut two internal objects, the flavours of which\nare denoted by $X'$ and $X''$, as indicated. Diagram~\\ref{GR-ring-G}\nis the only one in which the field line $W$\nis associated not only with an external $>$\n(the flavour of which, $W'$, is just the same as\nthat of $W$) but also an internal $>$, the flavour\nof which is denoted by $W''$.\nFinally, we comment on the momentum routing, which is\ntaken to be the same each diagram. The external field\n$W$ is taken to carry momentum $k$. The external\nfields ${Z^1}, \\cdots, {Z^r}$ carry momenta $m_1, \\cdots m_r$,\nrespectively; momentum conservation now fixes the momentum\ncarried by $X$. The loop momentum is $l$, and is routed\nsuch that it is carried by ${Z^1}'$. This now uniquely\ndetermines the momenta carried by all other internal fields.\n\nWith these points in mind, we now wish to show that the sum\nof the five diagrams vanishes. Immediately, we notice\nthat, whatever the flavours of the internal and external\nfields, diagrams~\\ref{TLTP-EP-G-Kd} and~\\ref{GR-ring-G-c}\ncancel. The only difference between these two diagrams\nis the order in which $W$ and $Z^r$ bite the field\nline of $X$, which affects neither the flavours\nof the internal fields or the group theory factors\n(this conclusion will change when we consider\nversions of these diagrams with different patterns\nof bites to the left \/ right).\n\nNow let us consider diagrams~\\ref{TLTP-EP-G-GR},\n\\ref{GR-ring-G} and~\\ref{GR-ring-G-b}. The first\nthing that we must do to show that these diagrams\ncancel is demonstrate that the flavours of\nthe external fields and all common internal fields\ncan be arranged\nto be the same in all three cases. Clearly, we \ncan always choose the external fields to be\nthe same. \nAssigning a value of 0 (1) to the bosonic (fermionic) \nfields, as usual, and denoting \nthe statistics of the field $V$ by $\\stats{V}$,\nwe see that\n\\begin{eqnarray*}\n\t\\stats{X'} & = & \\stats{X} \\ \\xor \\stats{Z^{r'}},\n\\\\\n\t\\stats{{Z^1}'} & = & \\stats{Z^1} \\ \\xor \\stats{X'} \\ \\xor \\stats{W'},\n\\\\\n\t\\stats{{Z^{(r-i)}}'} & = & \\stats{Z^{r-i}} \\ \\xor \\stats{Z^{r-i-1}} \\ \\xor \\cdots \\ \\xor \\stats{{Z^1}'},\n\\end{eqnarray*}\nin all three diagrams, where $i$ is in the range $\\{0,r-2\\}$.\nThus, if in addition to choosing all \nexternal fields to be the same in\ndiagrams~\\ref{TLTP-EP-G-GR},\n\\ref{GR-ring-G} and~\\ref{GR-ring-G-b}\nwe also choose $Z^{r'}$ to be the same,\nit follows that all flavours \n$W', \\ X'$ and ${Z^1}', \\cdots {Z^r}'$ agree.\nConsequently, we can write the sum\nof the three diagrams as:\n\\[\n\t\\int_l F(k,m_i,l,j_1,j_2,j_3)\n\t\\left[\n\t\t- \\rhd^{j_1}_{l-m_1-k} >^{j_2}_{l-m_1} - \\rhd^{j_3}_k >^{j_2}_{l-m_1} + 1\n\t\\right],\n\\]\nwhere $j_2 = j_1 \\ \\xor j_3$ and $F(k,m_i,l,j_1,j_2,j_3)$ is common to all\nthree terms. This term vanishes, courtesy of~\\eq{eq:Template}.\n\n\nOur next task is to show that the versions of~\\eq{eq:D-ID-G-R}\nwith different patterns of bites to the left \/ right\nalso vanish. The first thing to notice is that\nchanging the senses in which the gauge remainder\ncomponents carrying the labels ${Z^2}', \\cdots {Z^{r-1}}'$\nare bitten induces a change which is common to all\nfive diagrams, and so the cancellations detailed above\ngo through, just the same. In the case that there\nare at least four external fields---in which case\nwe do not identify $Z^r$ with $Z^1$---the sense in\nwhich \n${Z^r}'$ is bitten factors out as well. \n\n\nLet us now consider alternatively nested versions\nof diagrams~\\ref{TLTP-EP-G-Kd} and~\\ref{GR-ring-G-c},\nin which we consider changing the sense in which\none and only one gauge reminder is bitten \/ bites.\nChanging the sense in which ${Z^1}'$ is bitten\ninduces the same change in both diagrams, and so they\nstill cancel. Next, consider changing the sense in which ${Z^r}'$\n\\emph{bites}. \nThis does not induce the same change\nin each diagram, on account of the differing orders\nin which ${Z^r}'$ and $W'$ act in each of the diagrams.\nMoreover, in diagram~\\ref{TLTP-EP-G-Kd}\nwe can change the sense in which $W'$ bites. We cannot\ndo this in diagram~\\ref{GR-ring-G-c} since it has,\nin a sense, been done already: diagrams~\\ref{GR-ring-G-c}\nand~\\ref{GR-ring-G-b} form a pair of diagrams, essentially\ndiffering only in the sense in which $W'$ acts.\n(This is particularly clear when we consider these\ndiagrams as part of a larger diagram, in which case\nall indices simply become dummy indices which are summed\nover).\n\nSimilarly, consider alternatively nested versions\nof diagrams~\\ref{TLTP-EP-G-GR}, \\ref{GR-ring-G}\nand~\\ref{GR-ring-G-b}. Changing the sense in which\n$X'$ is bitten induces the same change in all three\ndiagrams, and so they still cancel. However,\nchanging the sense in which ${Z^1}'$ is bitten\ndoes not induce the same change  in the\nfinal diagram as in the first two. \nMoreover, in diagram~\\ref{TLTP-EP-G-GR} we can\nchange the sense in which $W'$ bites, and in \ndiagram~\\ref{GR-ring-G} we can change the sense\nin which $W''$ is bitten. Again, neither of\nthese terms is obviously cancelled.\n\nWe collect together the diagrams which no longer\ncancel after changing the sense in which one\nof the gauge remainders acts\nin \\fig{fig:D-ID-G-b}. \n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{1ex}\n\t\t\\begin{array}{cccccc}\n\t\t\\vspace{1ex}\t\n\t\t\t\t& \\LID{TLTP-EP-G-Kd-B}\t&\t& \\LID{TLTP-EP-G-GR-B}\t&\t& \\LID{GR-ring-G-B}\n\t\t\\\\\n\t\t\t-\t& \\cd{TLTP-EP-G-Kd-B}\t& +\t& \\cd{TLTP-EP-G-GR-B}\t& +\t& \\cd{GR-ring-G-B-Mac}\n\t\t\\end{array}\n\t\\\\\n\t\\vspace{1ex}\n\t\t\\begin{array}{cccccc}\n\t\t\\vspace{1ex}\t\n\t\t\t\t& \\LID{TLTP-EP-G-Kd-C}\t&\t& \\LID{TLTP-EP-G-GR-C}\t&\t& \\LID{GR-ring-G-C}\n\t\t\\\\\n\t\t\t-\t& \\cd{TLTP-EP-G-Kd-C}\t& +\t& \\cd{TLTP-EP-G-GR-C}\t& +\t& \\cd{GR-ring-G-C-Mac}\n\t\t\\end{array}\n\t\\\\\n\t\t\\begin{array}{cccc}\n\t\t\\vspace{1ex}\n\t\t\t\t& \\LID{GR-ring-G-c-B}\t\t&\t& \\LID{GR-ring-G-b-B}\n\t\t\\\\\n\t\t\t+\t& \\cd{GR-ring-G-c-B}\t\t& -\t& \\cd{GR-ring-G-b-B}\n\t\t\\end{array}\n\t\\end{array}\n\t\\]\n\\caption{The set of diagrams, obtained from \nchanging the sense in which precisely\none of the gauge remainders of\ndiagrams~\\ref{TLTP-EP-G-Kd}--\\ref{GR-ring-G-b}\nbites, which do not cancel\nin the same way as the diagrams of \\fig{fig:D-ID-G-a}.}\n\\label{fig:D-ID-G-b}\n\\end{figure}\\end{center}\n\nNotice that, in diagram~\\ref{TLTP-EP-G-Kd-B},\nwe have changed the labelling of\nthe fields, compared to diagram~\\ref{TLTP-EP-G-Kd}.\nWith the fields labelled in this way,\nwe can repeat the analysis of \ndiagrams~\\ref{TLTP-EP-G-GR},\n\\ref{GR-ring-G} and~\\ref{GR-ring-G-b}\nand thus prove that, given a set of common external\nfields, all internal\nfields can be chosen to be the same in \ndiagrams~\\ref{TLTP-EP-G-Kd-B}--\\ref{GR-ring-G-B}.\nFurthermore, we see that the group theory structure\nof these diagrams is identical: in each case\nthe field $Z^1_{S_1}$ is inside the loop,\nwith the remaining field lines having a\ncyclic order, common to the three diagrams, on the outside.\nConsequently, we can write the sum\nof the three diagrams as:\n\\[\n\t\\int_l F'(k,m_i,l,j_1,j_2,j_3)\n\t\\left[\n\t\t1 - \\rhd^{j_1}_{l-m_1-k} >^{j_2}_{l-m_1} - \\rhd^{j_3}_k >^{j_2}_{l-m_1}\n\t\\right],\n\\]\nwhere once again $j_2 = j_1 \\ \\xor j_3$ and $F'(k,m_i,l,j_1,j_2,j_3)$ is \ncommon to all\nthree terms. Thus, in accord with~\\eq{eq:Template},\n the sum of diagrams~\\ref{TLTP-EP-G-Kd-B}--\\ref{GR-ring-G-B}\nvanishes. In exactly the same fashion, we can\nshow that diagrams~\\ref{TLTP-EP-G-Kd-C}--\\ref{GR-ring-G-C}\nalso sum to zero.\n\nThis still leaves us with diagrams~\\ref{GR-ring-G-c-B}\nand~\\ref{GR-ring-G-b-B}. Let us put these\nto one side for a moment and consider instead\ndifferent patterns of bites to the left \/ right\nin diagrams~\\ref{TLTP-EP-G-Kd-B}--\\ref{GR-ring-G-C}.\nLet us start with the first three diagrams.\nThe only changes we can make which do not\nlead to diagrams which cancel in exactly the\nsame way as the parents\n(or take us back to the diagrams of \\fig{fig:D-ID-G-a})\nare as follows:\n\\begin{enumerate}\n\t\\item\tchanging the sense in which $W'$ bites\n\t\t\tin diagrams~\\ref{TLTP-EP-G-Kd-B} and~\\ref{TLTP-EP-G-GR-B};\n\n\t\\item\tchanging the sense in which $X'$ bites in\n\t\t\tdiagram~\\ref{GR-ring-G-B}.\n\\end{enumerate}\nThe resulting diagrams are shown in \\fig{fig:D-ID-G-c}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\\vspace{1ex}\n\t\t\\LID{TLTP-EP-G-Kd-B2}\t&\t& \\LID{TLTP-EP-G-GR-C2}\t&\t& \\LID{GR-ring-G-C2}\n\t\\\\\n\t\t\\cd{TLTP-EP-G-Kd-B2}\t& -\t& \\cd{TLTP-EP-G-GR-C2}\t& -\t& \\cd{GR-ring-G-C2}\n\t\\end{array}\n\t\\]\n\\caption{The set of diagrams, obtained from \nchanging the sense in which precisely\none of the gauge remainders of\ndiagrams~\\ref{TLTP-EP-G-Kd-B}--\\ref{GR-ring-G-C}\nacts, which do not cancel in exactly\nthe same way as the parent diagrams or reproduce already\ndrawn diagrams.}\n\\label{fig:D-ID-G-c}\n\\end{figure}\\end{center}\n\nNow consider making similar changes to \ndiagrams~\\ref{TLTP-EP-G-Kd-C}--\\ref{GR-ring-G-C}.\nThe point is, that the set of diagrams\nwe would draw simply reproduces the\npattern of bites to the\nleft \/ right that we see in \ndiagrams~\\ref{TLTP-EP-G-Kd-B2}--\\ref{GR-ring-G-C2}.\nIn other words, we have taken account of\nthese terms already and so we have no further\ndiagrams to draw. Now, diagram~\\ref{TLTP-EP-G-Kd-B2}\ncancels diagram~\\ref{GR-ring-G-b-B} and\ndiagrams~\\ref{TLTP-EP-G-GR-C2} and~\\ref{GR-ring-G-C2}\ncancel diagram~\\ref{GR-ring-G-c-B}. Thus everything closes.\nAny further changes to the pattern of bites to\nthe left \/ right will simply replicate cancellations\nwe have seen already.\n\nWe have therefore demonstrated that, in addition\nto the diagrams of \\fig{fig:D-ID-G-a} summing to\nzero, so too do appropriate combinations\nof their alternatively nested partners.\nWe can use this fact, together with\nidentity~\\eq{eq:D-ID-2},\nto derive a relationship between all\ndiagrams of the basic structure just discussed,\npossessing between two and $m$ empty sockets.\nThe relationship, which is crucial in our treatment\nof $\\beta_n$, is shown in \\fig{fig:D-ID-App}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\\label{eq:D-ID-G-Pre}\n\t\\begin{array}{ll}\n\t\\vspace{1ex}\n\t\t&\n\t\t\\AGRO{\n\t\t\t\\LIDi[1]{Struc-LXL}{D-ID-App-LXL} \\hspace{0.225em}\n\t\t\t+\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{llll}\n\t\t\t\t\t& \\LIDi[1]{Struc-LLXL}{D-ID-App-LLXL} \t& \\hspace{0.5em} + & \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t- & \\LIDi[1]{Struc-LXRL}{D-ID-App-LXLL} \t& \\hspace{0.5em} + & \\cdots\n\t\t\t\t\\\\ \n\t\t\t\t\t& & & \\vdots\n\t\t\t\t\\end{array}\n\t\t\t\\hspace{5em}\n\t\t\t\\right]\n\t\t}\n\t\\\\\n\t\t= &\n\t\t\\AGRO{\n\t\t\t\\begin{array}{lll}\n\t\t\t\\vspace{0.2in}\n\t\t\t\t\\LIDi[1]{Struc-GR-RR}{D-ID-App-GR-LL} & - & \\hspace{2.3em} \\LIDi[1]{Struc-GR-RRR}{D-ID-App-GR-LLL} \\hspace{1.6em} +\\cdots\n\t\t\t\\\\\n\t\t\t\t& - &\n\t\t\t\t\\left[\n\t\t\t\t\t\\begin{array}{lll}\n\t\t\t\t\t\t\\hspace{1em} \\LIDi[1]{Struc-PF-TLTP-EP-RR}{D-ID-App-PB-TLTP-EP-LL} \t\n\t\t\t\t\t\t& \\hspace{0.5em} -  \\LIDi[1]{Struc-PF-TLTP-EP-RRR}{D-ID-App-PB-TLTP-EP-LLL}  \n\t\t\t\t\t\t& +\\cdots \n\t\t\t\t\t\\\\\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t& \n\t\t\t\t\t\t\\hspace{0.5em} - \\LIDi[1]{Struc-R-PF-TLTP-EP-RR}{D-ID-App-L-PB-TLTP-EP-LL}\n\t\t\t\t\t\t& +\\cdots\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\right]\n\t\t\t\\end{array}\n\t\t}\n\t\\end{array}\n\t\\end{equation}\n\\caption{A relationship between certain diagrams\npossessing between two and $m$ empty sockets.}\n\\label{fig:D-ID-App}\n\\end{figure}\\end{center}\n\nThe notation $\\AGRO{\\ }$ tells us to sum over Every Sense\nin which all the gauge remainders can bite. Thus, \ndiagrams~\\ref{D-ID-App-LXL} and~\\ref{D-ID-App-GR-LL}\nhave thee partners, diagrams~\\ref{D-ID-App-LLXL}, \n\\ref{D-ID-App-LXLL}, \\ref{D-ID-App-GR-LLL} and~\\ref{D-ID-App-PB-TLTP-EP-LL} \t\nseven partners {\\it etc.}\\  \n\nIt is worth clarifying the diagrams represented by the various ellipses.\nThe ellipsis following diagram~\\ref{D-ID-App-LLXL} denotes additionally\nnested diagrams, where the classical, two-point, vertex is still\njoined to the innermost gauge remainder. The ellipsis after\ndiagram~\\ref{D-ID-App-LXLL} denotes additionally nested diagrams,\nwhere the classical, two-point vertex joins to the innermost\nbut one gauge  remainder. The vertical dots on the next line represent additionally\nnested diagrams in which the classical, two-point vertex joins to\ngauge remainders successively further away from the innermost one.\nNote, though, that we never make the join to the outermost gauge remainder.\nThus, we have arranged things such that all diagram in a given\ncolumn possess the same number of sockets.\nThe relationship between the top set of diagrams and\nthe bottom set holds for any number of columns, so \nlong as this number is the same for both sets of\ndiagrams.\n\nWe now expound on how the diagrammatic identities of this\nsection combine to yield~\\eq{eq:D-ID-G-Pre}.\nDiagram~\\ref{D-ID-App-LXL} and its partners\nare clearly\nequal to~\\ref{D-ID-App-GR-LL} and its partners\nby diagrammatic identity~\\eq{eq:D-ID-2}.\nApplying the effective propagator relationship to\ndiagram~\\ref{D-ID-App-PB-TLTP-EP-LL} and its partners, \nthe Kronecker~$\\delta$\npart combines with diagram~\\ref{D-ID-App-GR-LLL}\nto give diagram~\\ref{D-ID-App-LXLL}, via\ndiagrammatic identity~\\eq{eq:D-ID-G-R}\n(and its alternatively nested versions). On the other\nhand, the gauge remainder term and its partners\nare just equal to\ndiagram~\\ref{D-ID-App-LLXL} \nand its partners courtesy of \ndiagrammatic identity~\\eq{eq:D-ID-2}.\n\nUsing  the\nnotation described under \\fig{fig:nestedhook}, we can represent~\\eq{eq:D-ID-G-Pre}\nin the compact form shown in \\fig{fig:D-ID-G-Compact}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray}\n\t\\nonumber\n\t\t\\sum_{m'=1}^{m-1} \\frac{1}{m'!} \\cd{TLTP-EP-ArbGRs} \\hspace{2em}\n\t\t& = &  \\sum_{m'=1}^{m-1} \\frac{1}{(m'+1)!} \\cd{GR-ArbGRs} \\hspace{2em}\n\t\\\\ \n\t\\label{eq:D-ID-G-A}\n\t\t&& - \\sum_{m'=2}^{m-1} \\sum_{m''=0}^{m'-2} \\frac{1}{(m'-m'')!m''!} \n\t\t\\cd{ArbGRs-TLTP-EP-ArbGRs}\n\t\\end{eqnarray}\n\\caption{A compact form of~\\eq{eq:D-ID-G-Pre}.}\n\\label{fig:D-ID-G-Compact}\n\\end{figure}\\end{center}\n\n\nThe loose end of the effective propagator\nin the first diagram can attach to any\nof the $m'$ gauge remainder components. We need not\nstrictly prohibit it\nfrom attaching to the socket on the vertex since\nsuch a diagram vanishes by charge conjugation\\ invariance, anyway.\nThe relationship~\\eq{eq:D-ID-G-A} is, of course, valid separately\nfor every integer value of $m$ from one to infinity.\nThe explicitly drawn  $>$ in the first and third diagrams\ncan bite in either sense. Notice that,\nsince the explicitly drawn $>$ is common\nto each diagram (in the second diagram it just sits\nat the bottom end of the full gauge remainder),\nwe could strip it off completely, \nreplacing it where necessary \nwith the \nhalf arrow notation of~\\eq{eq:WID-A}\n({\\it i.e.}\\  we could remove the $>$ completely from the second\ndiagram, where it does not push forward or pull back\non to another field, but would have to replace it\nwith a half arrow in the first and third diagrams).\n\n\n\\subsubsection{The Second Family}\n\\label{sec:D-ID-Secondary-II}\n\nThe first diagrammatic identity\nin this family\nfollows from the two equivalent\nway of processing the diagram on\nthe left-hand side\\\nof \\fig{fig:D-ID-dGR}: our first operation can\neither be to process the gauge remainder\nor to Taylor expand the vertex (we assume that\nthe socket is decorated by an $A^i$ sector field).\nIn the former case, although we do not explicitly\nindicate it at this stage, it is understood that\nthe socket in the daughter diagrams is filled\nby an $A^i$ sector field carrying zero momentum,\njust as in the parent.\nIn the latter case, we can move the momentum derivative\nfrom the vertex to the $\\rhd$ which bites the vertex,\nat the expense of a minus sign, courtesy of diagrammatic\nidentity~\\eq{eq:GR-TLTP}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{lcccrr}\n\t\t\\cd{D-ID-dGR-A}\t& =\t& \t\t\t\t\t\t\t\t \t& =\t& \\displaystyle \\frac{1}{m} \\cd{D-ID-dGR-B} \t& \\displaystyle - \\frac{1}{m} \\cd{D-ID-dGR-C}\n\t\\\\[1ex]\n\t\t\t\t\t\t& = & - 2 \\cd{D-ID-dGR-D} \\hspace{1em} \t& =\t& -2 \\cd{D-ID-dGR-E} \t\t\t\t& + 2 \\cd{D-ID-dGR-F}\n\t\\end{array}\n\t\\]\n\\caption{Deducing a secondary diagrammatic identity; \nthe socket must be decorated by an $A^i$ sector\nfield which carries zero momentum.}\n\\label{fig:D-ID-dGR}\n\\end{figure}\\end{center}\n\nThe factor of two on the second line arises from using charge conjugation\\ invariance\nto collect together momentum derivative terms; we have\nnot used charge conjugation\\ to collect together pushes forward and\npulls back on the first line. By equating the two expressions\nwith each other, we are lead to the following diagrammatic\nidentity:\n\\begin{equation}\n\\label{eq:D-ID-dGRk-GT}\n\t\\frac{1}{m} \\cd{D-ID-dGR-Bi} \\hspace{1em} = -2 \\cd{D-ID-dGR-Ei}\n\\end{equation}\nwhere, in both diagrams, the socket must be decorated by\nan $A^i$ sector carrying zero momentum. In the\nspecial case that the socket of the\nfirst diagram can be identified with the sockets of one\nof the $m$ gauge remainders~\\eq{eq:D-ID-dGRk-GT} becomes:\n\\begin{equation}\n\\label{eq:D-ID-dGRk-GT-ring-A}\n\t2m(m-1) \\cd{GRs-dGRk} \\hspace{3em} = - \\decGR{ \\ }{\\ensuremath{\\sqcup} >^m}.\n\\end{equation}\nNotice that, in the second diagram, \neven though the socket is identified with one of\nthe gauge remainders, we have kept it\nto remind us that the relationship~\\eq{eq:D-ID-dGRk-GT-ring-A}\nholds only if the sockets on each side of the equation\nare filled with an $A^i$ field carrying zero momentum.\n\n\n\n\nThe second diagrammatic identity in\nthis family follows as a consequence of\ndiagrammatic identities~\\eq{eq:WID} \nand~\\eq{eq:GR-TLTP}. Defining\n\\begin{equation}\n\\label{eq:Combo}\n\t\\ensuremath{\\star} \\equiv \\cd{Combo} \\equiv \\cd{EP-dGR} -\\frac{1}{2} \\cd{dPEP-GR}\n\\end{equation}\nwe have\n\\begin{equation}\n\\label{eq:D-ID-Trapped}\n\t\\cd{TLTP-Trapped-PF} = \\cd{TLThP-GRx2} = - \\cd{TLTP-Trapped-PB}.\n\\end{equation}\n\n\nA number of comments are in order.\nStrictly speaking, the presence of $\\ensuremath{\\star}$ (as opposed\nto some other internal line involving a momentum derivative) is not necessary\nfor either this diagrammatic identity or those that follow. However,\nin practise, it is a $\\ensuremath{\\star}$ that most commonly occurs in such\nscenarios.\nNote\nthat each diagram effectively possesses two external $A^1$s (after\ndecoration of the socket associated with the $\\ensuremath{\\star}$), which \nwe suppose to be the only external fields. \nWe understand that the socket\npossessed by the $>$ and the loose end of the $\\ensuremath{\\star}$\nare internal fields, which are somehow tied up into a complete\ndiagram. \nIn the second diagram, if the right-most\ngauge remainder pushes forward onto the internal\nfield, we recover the parent. If this gauge remainder\nwere instead to pull back onto the internal field,\nthen the supertrace structure of the diagram\nwould be uniquely determined to be \n$\\mathrm{str} A^1_\\mu \\mathrm{str} A^1_\\nu = 0$.\nIf either gauge remainder strikes the external field, then the \nother gauge remainder kills the diagram,\ncourtesy \nof~\\eq{eq:GR-TLTP}. To go from the second diagram to the\nthird, we allow the left-most gauge remainder to act;\nthe only surviving contribution is the pull back onto\nthe internal field.\n\n\nThe next diagrammatic identity\nis the first of this family which necessitates\nconsideration of the algebraic form of gauge\nremainders, and is shown in \\fig{fig:D-ID-Op2-C}.\nNotice that, in the first diagram, both the $\\ensuremath{\\star}$\nand the $>$ bite the socket.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\\label{eq:D-ID-Op2-C}\n\t\t\\begin{array}{cccccc}\n\t\t\t\\LID{D-ID-CTP-E-Cb}\t&\t& \\LID{D-ID-CTP-E-Ab}\t&\t& \\LID{D-ID-CTP-E-Bb}\t&\n\t\t\\\\[1ex]\n\t\t\t\\cd{D-ID-CTP-E-Cb} \t& -\t& \\cd{D-ID-CTP-E-Ab}\t& -\t& \\cd{D-ID-CTP-E-Bb} \t& = 0\n\t\t\\end{array}\n\t\\end{equation}\n\\caption{The first diagrammatic identity of the second\nfamily which relies on~\\eq{eq:Template}.}\n\\label{fig:D-ID-Op2-C}\n\\end{figure}\\end{center}\n\nThis diagrammatic identity follows directly\nfrom~\\eq{eq:Template}.\nAlthough we have given \nboth~\\eqs{eq:D-ID-Trapped}{eq:D-ID-Op2-C}\nin un-nested form, it is clear that\nthey both hold in appropriately nested\ncases. Indeed, we now combine\nthe nested versions of these\ndiagrammatic identities to give\nthe final diagrammatic\nidentity. This is shown in \\fig{fig:D-ID-Op2-D}\nand is, in some sense, the second\nfamily analogue of diagrammatic\nidentity~\\eq{eq:D-ID-G-A}. Wherever a $\\ensuremath{\\star}$\nand $>$ bite the same thing, the $\\ensuremath{\\star}$\nis taken to act first, as in diagram~\\ref{D-ID-CTP-E-Cb}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray}\n\t\\lefteqn{\n\t\t\\sum_{m'=0}^m \n\t\t\\left[\n\t\t\t\\cd{CTP-E-GRs-Combo-GRk-GR} \n\t\t\t-\\cd{CTP-E-GRs-Combo-GR}\n\t\t\\right]\t\n\t}\n\t\\nonumber\n\t\\\\ & &\n\t\t- \\sum_{m'=0}^m \\!\\! \\nCr{m'}{m''} \\!\\! \\sum_{m''=0}^{m'} \n\t\t\\left[\n\t\t\t\\cd{CTP-E-GRs-Combo-CTP-EP-GR}\n\t\t\t\\hspace{1.8em}\n\t\t\t-\\cd{CTP-E-GRs-Combo-GRs-CTP-EP}\n\t\t\t\\hspace{1.8em}\n\t\t\\right]\n\t\t= 0\n\t\\label{eq:D-ID-Op2-G-A}\n\t\\end{eqnarray}\n\\caption{The final diagrammatic identity belonging to the second\nfamily.}\n\\label{fig:D-ID-Op2-D}\n\\end{figure}\\end{center}\n\n\n\\subsection{Subtraction Techniques}\n\\label{sec:Subtraction}\n\n\\subsubsection{Introduction}\n\\label{sec:Subtractions-I}\n\nAs discussed in the introduction, and as\nwe will see shortly, the $n$-loop $\\beta$ function\nis computed by evaluating a set of diagrams at\n$\\Op{2}$:\n\\[\n\t\\beta_n \\Box_{\\mu \\nu}(p) = \\left.\\mathrm{Diagrams}\\right|_{p^2}.\n\\]\nA subset of the diagrams contributing to $\\beta_n$ possess\na stub which is\nmanifestly $\\Op{2}$ and so we would like to Taylor expand\nthe remaining components of these diagrams to zeroth order in $p$.\nThe problem is that individual diagrams may\npossess components which are not Taylor expandable to\n$\\Op{2}$.\nWe could proceed simply by ignoring these\ncomponents since we know that\nthe sum of all diagrams contributing to $\\beta_n$\nmust be Taylor expandable to $\\Op{2}$. However, for\nthe purposes of this paper we adopt a higher level of\nrigour and so keep them. \nAs an example,\nconsider the first diagram of \\fig{fig:TLM-M:TE-Ex-B}\nwhere, in anticipation of what follows, we have added and subtracted\na term.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t\\begin{array}{ccc}\n\t\t\t\\LID{d:111b.11}\t\t&\t\t& \\LIDLID{d:111b.11-s}{d:111b.11-a}\n\t\t\\\\[1ex]\n\t\t\t\\cd{Diagram111b.11} & \\mp \t& \\left[\\cd{Diagram111b.11-s}\\right]_{p^2}\n\t\t\\end{array}\n\t\\]\n\\caption{A two-loop diagram with an $\\Op{2}$ stub, components\nof which cannot be Taylor\nexpanded to $\\Op{2}$, and a term constructed to isolate\nthese components.}\n\\label{fig:TLM-M:TE-Ex-B}\t\n\\end{figure}\\end{center}\n\nWe begin by focusing on diagram~\\ref{d:111b.11}.\nSince we can always Taylor expand vertices in momenta, let us suppose that\nwe take a power of $l$ from the top-most vertex \n(we cannot take any powers of $p$, at $\\Op{2}$) and\nlet us choose to take a power of $k$ from the other vertex. \nUsing \\tab{tab:NFE:k,k'} and~\\eq{eq:EP-leading}, we see that \nthe leading IR behaviour\nof the $l$-integral is given by\n\\[\n\t\\int_l \\frac{1}{l^2 (l-p)^2},\n\\]\nwhich is not Taylor expandable to $\\Op{2}$. Note that had we taken a power of $l$ from\nthe right-hand vertex, rather than a power of $k$, then the extra power of $l$\nin the integrand would render the diagram Taylor expandable in $p$ (to the\nrequired order).\n\nNext, let us consider the pair of \ndiagrams~\\ref{d:111b.11-s} and~\\ref{d:111b.11-a},\nwhere the first of these---the subtraction---comes\nwith the minus sign and the corresponding\naddition comes with the plus sign. The tag $p^2$\ntells us that only the stub carries momentum\n$p$ {\\it i.e.}\\  these diagrams are constructed\nby setting $p=0$ in diagram~\\ref{d:111b.11}\neverywhere but the stub.\n\nThe effect of the subtraction on the parent is to cancel\nall those components which are Taylor expandable to $\\Op{2}$. This immediately tells us\nthe following about any surviving contributions to diagram~\\ref{d:111b.11}:\n\\begin{enumerate}\n\t\\item\tall fields carrying momentum $l$ must be in the $A^1$-sector;\n\n\t\\item\twe must take $\\mathcal{O}(l^0)$ from the $k$-integral (note that\n\t\t\tthe $k$-integral is Taylor expandable in $l$);\n\n\t\\item\twe must discard any remaining contributions to the $l$\n\t\t\tintegral which are Taylor expandable to $\\Op{0}$.\n\\end{enumerate}\nThe contributions to diagram~\\ref{d:111b.11} not removed by\n diagram~\\ref{d:111b.11-s} are shown on the left-hand side\\ of \\fig{fig:TLM-M:TE-Ex-C}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t\\left[\n\t\t\t\\LIDi{Diagram111b.11-Rem}{d:111b.11-Rem}\n\t\t\\right]_{\\NTE{p}}\n\t\t\\rightarrow\n\t\t\\LIO{\n\t\t\t\\left[\n\t\t\t\t\\cd{CTP-E-EP-Thpt-EP-GR}\n\t\t\t\\right]_{\\NTE{p}}\n\t\t\t\\Delta^{11}_{\\alpha \\beta}(p)\n\t\t\t\\decp{\n\t\t\t\t\\cd{CTP-E-EP-Thpt-K-GR}\n\t\t\t}{}\n\t\t}{d:111b.11-Rem-b}\n\t\\]\n\\caption{The contribution to diagram~\\ref{d:111b.11} not removed by its subtraction.}\n\\label{fig:TLM-M:TE-Ex-C}\n\\end{figure}\\end{center}\n\nFocusing first on diagram~\\ref{d:111b.11-Rem},\nwe see that,\nas required, we have taken the $\\mathcal{O}(l^0)$ from the $k$-integral. \nThe tag $\\BigNTE{p}$\ndemands that we retain only the component of the diagram\nwhich, whilst possessing an overall $\\Op{2}$ contribution,\nis not Taylor expandable in $p$ (in dimensional\nregularization, this contribution would go\nas $\\Op{2} p^{-2\\epsilon}$, for this diagram).\nThe bar sitting just before the\n$>$ which bites the classical,\ntwo-point vertex indicates a discontinuity in\nmomenta: the $>$ which bites\nthe classical, two-point vertex carries momentum $l$,\nbut the loop to its right carries only momentum\n$k$ along all lines.\nWe can neaten this diagram up (getting rid of\nthe momentum discontinuity) \nto obtain diagram~\\ref{d:111b.11-Rem-b}.\nTo do this, we first \nnotice \nthat the sub-diagram of~\\ref{d:111b.11-Rem}\nwhich is independent of $l$\ncarries two indices, say $\\rho$ and $\\tau$.\nBy Lorentz invariance, this sub-diagram must go\nas $\\delta_{\\rho \\tau}$, which allows us to tie\nup the effective propagator attaching to the\nmomentum derivative to the gauge remainder\nwhich follows the bar. This procedure yields\nthe $\\BigNTE{p}$ component of diagram~\\ref{d:111b.11-Rem-b},\nbut with indices $\\mu$ and $\\nu$, rather than\n$\\mu$ and $\\alpha$. We now re-express\n\\begin{eqnarray*}\n\t\\Box_{\\mu \\nu}(p)\t& =\t& \\Box_{\\mu \\alpha}(p) \\frac{\\delta_{\\alpha \\beta}}{2 p^2} 2 \\Box_{\\beta \\nu}(p)\n\\end{eqnarray*}\nwhich allows us to re-write diagram~\\ref{d:111b.11-Rem}\nin the form~\\ref{d:111b.11-Rem-b}, up to $\\Op{4}$ corrections\n({\\it cf.}\\ ~\\eq{eq:EP-leading}).\n\n\nBefore discussing the addition, we note that we\ndefine the subtraction to be the diagram which removes\nthe components of the parent diagram which are Taylor\nexpandable to $\\Op{2}$, not by its sign.\nWe could always move  \nthe momentum derivatives in diagrams~\\ref{d:111b.11-s}\nfrom one side of the vertex they strike to\nthe other, at the expense of a minus sign\n(this\nwould amount to replacing $\\partial_\\nu^{-l}$\nwith $\\partial_\\nu^{l}$); were we to do so,\nthe subtraction would now come with a plus sign\nand the addition with a minus. \n \nAdditions like our example~\\ref{d:111b.11-a} can\nbe manipulated by using the effective\npropagator relation and diagrammatic\nidentity~\\eq{eq:PseudoEP}. For the two structures,\n$A$ and $B$ we have:\n\\begin{equation}\n\\label{eq:EP-dCTP-EP}\n\t\\cd{A-EP-dCTP-EP-B} = \n\t\\begin{array}{cccccc}\n\t\t\t& \\LID{A-dEP-B}\t&\t& \\LID{A-GR-combo-B}\t&\t& \\LID{A-combo-GR-B}\n\t\\\\[1ex]\n\t\t-\t& \\cd{A-dEP-B}\t& -\t& \\cd{A-GR-combo-B}\t\t& -\t& \\cd{A-combo-GR-B},\n\t\\end{array}\n\\end{equation}\nwhere\n\\[\n\t\\cd{Combo-GR-b} \\equiv \\cd{EP-dGR-GR-b} -\\frac{1}{2} \\cd{dPEP-GR-GR} \n\t\t\\qquad \\mathrm{and} \\qquad\n\t\\cd{Combo-GR} \\equiv \\cd{EP-dGR-GR} -\\frac{1}{2} \\cd{dPEP-GR-GR}.\n\\]\nIt is now apparent why the definition~\\eq{eq:Combo} is\nuseful.\n\n\nAs a consequence of the decomposition~\\eq{eq:Combo},\nit is clear that diagrams~\\ref{A-GR-combo-B} and~\\ref{A-combo-GR-B}\nhave components where an active gauge\nremainder strikes both structures $A$ and\n$B$. However, in practical calculations,\nthis turns out to be a red-herring: it\nis most efficient never to perform\nthis decomposition and so we only ever \nprocess the explicitly drawn gauge remainders\nin~\\eq{eq:EP-dCTP-EP}.\n\nIt is useful to consider the special case\nwhere the realizations of $A$ and $B$ \nare summed over, as we can \nuse charge conjugation\\ invariance\nto combine terms, in this case. \nSpecifically, can we identify the structure $A$ $(B)$\nof diagram~\\ref{A-combo-GR-B} with the structure\n$B$ $(A)$ of diagram~\\ref{A-GR-combo-B}.\nThis is shown\nin \\fig{fig:Op2-CC} where, for the sake\nof generality, we suppose that the\nactive gauge remainders in~\\eq{eq:EP-dCTP-EP}\ncan be arbitrarily nested.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\t\\LID{A-GR-combo-A}\t&\t\t\t\t& \\LID{A-combo-GR-A}\n\t\\\\[1ex]\n\t\t\\cd{A-GR-combo-A}\t& \\hspace{1em} +& \\cd{A-combo-GR-A}\n\t\\end{array}\n\t\\]\n\\caption{Generalized versions of the second and third\ndiagrams of~\\eq{eq:EP-dCTP-EP}, where $A$ and $B$\nhave been identified with each other.}\n\\label{fig:Op2-CC}\n\\end{figure}\\end{center}\n\nLet us now consider a specific arrangement of\nbites to the left \/ right in diagrams~\\ref{A-GR-combo-A} and~\\ref{A-combo-GR-A}.\nTo be precise, we suppose that there are a total of $G$ such operations,\nwhere $G = L + R = L' + R'$, the (un)primed variable\ncorresponding to diagram (\\ref{A-GR-combo-A}) \\ref{A-combo-GR-A}.\nThe signs of diagrams~\\ref{A-GR-combo-A} and~\\ref{A-combo-GR-A} \nare, according to the rules of \\sec{sec:GRs-2}, $(-1)^R$ and $(-1)^{R'}$,\nrespectively.\n\nWe now choose to focus on pairs of terms for which\n$R' = L$ and consider taking the charge conjugate\nof diagram~\\ref{A-GR-combo-A} \nwhich, we recall, amounts to reflecting it, picking up a sign\nfor each of the $G$ performed gauge remainders\nand the momentum derivative. We remove this\nlatter sign by reversing\nthe direction of the momentum derivative symbol \n so that its sense is once again\ncounterclockwise. By doing this, we have\narranged for the reflected version of diagram~\\ref{A-GR-combo-A}\nto look exactly the same as diagram~\\ref{A-combo-GR-A}.\nThe associated sign of the\nreflected diagram\nis $(-1)^{G+R} = (-1)^{R'}$\nwhich is, of course, just the sign associated with\ndiagram~\\ref{A-combo-GR-A}. Therefore \ndiagrams~\\ref{A-GR-combo-A}\nand~\\ref{A-combo-GR-A} can be combined.\n\n\nAt the beginning of this section,\nwe stated that the effect of\nsubtractions is to isolate the\ncomponents of the parent which\nare not Taylor expandable to $\\Op{2}$\nand that, by construction, the\nadditions are Taylor expandable to the desired\norder. Strictly speaking, this latter\nstatement requires a caveat. \nThe point is that we can encounter\ndiagrams like~\\ref{d:111b.11}\nas a sub-diagram of some larger,\nfactorizable diagram. By factorizable,\nwe mean that the complete diagram \nhas (at least one) internal line which\ncarries just $p$. Thus, we could imagine\nthe top-most external field of\ndiagram~\\ref{d:111b.11} instead\nbeing an effective propagator, carrying $p$,\nattaching to some other sub-diagram\n(which would be decorated by the spare\nexternal field).\nThis latter sub-diagram, which we note\ncannot possess a kernel (only one is allowed\nper complete diagram),\nmay contain\ncomponents which are not Taylor expandable\nto $\\Op{2}$. The strategy is to simply leave \nsuch sub-diagrams alone, but proceed\nas usual with the other part of\nthe complete diagram {\\it i.e.}\\  the sub-diagram\nof the form~\\ref{d:111b.11}. The rule that\nwe will effectively obey is that we only ever\nconstruct subtractions for (sub) diagrams\nwhich contain a kernel. The corresponding\nadditions, whilst always possessing a sub-diagram\nwhich is Taylor expandable in $p$, may also\npossess a sub-diagram which is not. This does\nnot worry us: we simply manipulate the\nTaylor expandable sub-diagram and effectively\nignore the other one. \n\n\n\n\n\\subsubsection{General Analysis}\n\\label{sec:Subtractions-G}\n\nIn this section, we analyse the construction\nand effects of subtractions in complete generality.\nIn \\sec{sec:bn-Op2}, we will discover\nthat we only have to construct subtractions for\ndiagrams with two basic templates, which\nare shown in \\fig{fig:Subs-Templates}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\t\\LID{CTP-E-DK}\t&\t\t\t& \\LID{CTP-E-GRs-K}\n\t\\\\[1ex]\n\t\t\\cd{CTP-E-DK}, \t& \\qquad\t& \\cd{CTP-E-GRs-K}\n\t\\end{array}\n\t\\]\n\\caption{The template for diagrams for which we will construct\nsubtractions.}\n\\label{fig:Subs-Templates}\n\\end{figure}\\end{center}\n\nClearly, we have suppressed all other diagrammatic\ningredients which can include vertices, effective\npropagators and $>$s. Our aim now is to start\nfleshing out the sub-diagrams~\\ref{CTP-E-DK}\nand~\\ref{CTP-E-GRs-K} to see if we can find\nany components which are not Taylor expandable to\n$\\Op{2}$. To stand any chance of doing so,\nwe must find effective propagators or gauge remainders\ncarrying $l-p$\n(where $l$ is a loop momentum) since we recall\nthat, in the $A^i$ sector, these objects go as\n$ 1\/ (l-p)^2$ and $(l-p)_\\alpha \/ (l-p)^2$, respectively.\nTo this end, consider decorating diagram~\\ref{CTP-E-DK}\nwith a single socket, and consider the complete set\nof momentum routings, as shown in \\fig{fig:Subs-MomRoute}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\t\\LID{CTP-E-DK-Soc-A}\t&\t\t\t& \\LID{CTP-E-DK-Soc-B}\t&\t\t\t& \\LID{CTP-E-DK-Soc-C\n\t\\\\[1ex]\n\t\t\\cd{CTP-E-DK-Soc-A}, \t& \\qquad\t& \\cd{CTP-E-DK-Soc-B}, \t& \\qquad\t& \\cd{CTP-E-DK-Soc-C\n\t\\end{array}\n\t\\]\n\\caption{Momentum routings for diagram~\\ref{CTP-E-DK}.}\n\\label{fig:Subs-MomRoute}\n\\end{figure}\\end{center}\n\nFirst, consider diagram~\\ref{CTP-E-DK-Soc-A}.\nThere is no possibility of having an effective\npropagator carrying $l-p$ and so the (sub) diagram\nderived from fully fleshing out \\ref{CTP-E-DK-Soc-A}\nis Taylor expandable to $\\Op{2}$ (we re-iterate\nthat the field carrying momentum $p$ could attach\nto a factorizable sub-diagram which is not Taylor\nexpandable to the desired order in $p$).\n\nNow we move on to diagram~\\ref{CTP-E-DK-Soc-B}.\nNotice that there is no need to consider also\na diagram with the momentum arguments\ninterchanged because such a diagram identical,\nby momentum re-routing invariance. Since $l-p$\nis carried by the kernel (rather than by an\neffective propagator), there is no chance\nof generating a factor of $1\/(l-p)^2$ from\nthe structures drawn. However, the $p$\ncould, in principle, flow into an effective\npropagator or gauge remainder such that their\narguments are {\\it e.g.}\\  $k-p$, as illustrated\nin \\fig{fig:Subs-MomRoute-b}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\t\\LID{CTP-E-DK-Soc-Ba}\t&\t\t\t& \\LID{CTP-E-DK-Soc-Bc}\t\t&\t\t\t& \\LID{CTP-E-DK-Soc-Bb}\n\t\\\\[1ex]\n\t\t\\cd{CTP-E-DK-Soc-Ba}, \t& \\qquad\t& \\cd{CTP-E-DK-Soc-Bc}, \t& \\qquad\t& \\cd{CTP-E-DK-Soc-Bb}\n\t\\end{array}\n\t\\]\n\\caption{Fleshings out of diagram~\\ref{CTP-E-DK-Soc-B} such that\neither an effective propagator or gauge remainder\ncarries $k-p$.}\n\\label{fig:Subs-MomRoute-b}\n\\end{figure}\\end{center}\n\n\nAt first sight, diagram~\\ref{CTP-E-DK-Soc-Ba} seems to possess\nthe momentum dependence we are looking for, so long as we put\nthe $k$-dependent effective propagators in the $A^i$ sector, which\nwe do for all that follows. (\nGroup theory considerations in fact force them to be in\nthe $A^1$ sector.)\nIn this case,\nthe $k$-dependent effective propagators\n\\[\n\t\\sim \\frac{1}{k^2 (k-p)^2},\n\\]\nwhich is certainly not Taylor expandable to zeroth order in $p$.\nHowever, we must also consider the vertices. Specifically,\nby charge conjugation\\ invariance, the field filling the socket\nof the topmost vertex must be in the $A^1$ sector. \nTaylor expanding this vertex to zeroth order in $p$\n(which we can always do) must yield at least one power of $k$ by\nLorentz invariance. Furthermore, Lorentz invariance\nand the requirement that (up to powers of $p^{-2 \\epsilon}$)\nthe diagram is $\\Op{2}$\nforces us to pick up a second power of $k$ from the\nfour-point vertex. It is thus quite clear, despite\nfirst appearances, that the diagram is Taylor expandable\nto $\\Op{2}$.\n\nA similar analysis of diagrams~\\ref{CTP-E-DK-Soc-Bc}\nand~\\ref{CTP-E-DK-Soc-Bb}\nleads to the conclusion that the worst momentum\ndependence is\n\\[\n\t\\frac{\\order{l^2, l.k}}{(k-p)^2 (l-k)^2 l^2},\n\\]\nin both cases. Consequently, both diagrams are Taylor expandable to\n$\\Op{2}$.\n\n\nWe now attempt to worsen the behaviour of \ndiagram~\\ref{CTP-E-DK-Soc-Ba} by adding\nadditional diagrammatic elements. Clearly, \nsimply adding more effective propagators\nwill do nothing. What we can try, however,\nis splitting open one of the existing\neffective propagators, as shown in \n\\fig{fig:Subs-MomRoute-b2}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\LIDi{CTP-E-DK-Soc-Ba-i}{CTP-E-DK-Soc-Ba-i}\n\t\\]\n\\caption{Attempting to worsen the IR behaviour of diagram~\\ref{CTP-E-DK-Soc-Ba}.}\n\\label{fig:Subs-MomRoute-b2}\n\\end{figure}\\end{center}\n\nThis seems to have done the trick: rather than having\na single effective propagator giving a factor\nof $1\/k^2$, we now have two (attached to an \narbitrary structure), and the diagram\nappears not to be Taylor expandable to $\\Op{2}$.\nHowever, in all the cases we encounter, we will\nfind that when we sum over all\npossible diagrams which can be represented by\nthe arbitrary structure, the arbitrary\nstructure turns out to be transverse in its\nexternal momentum (when the legs are in the $A^i$\nsector). Once again we are thwarted: diagrams\nof the type~\\ref{CTP-E-DK-Soc-Ba-i} are, in practise\nTaylor expandable to $\\Op{2}$.\nSimilarly, we cannot modify diagrams~\\ref{CTP-E-DK-Soc-Bc}\t\nand~\\ref{CTP-E-DK-Soc-Bb} such that they are\nno longer Taylor expandable to $\\Op{2}$.\n\nWe can repeat precisely the same arguments with \ndiagram~\\ref{CTP-E-DK-Soc-C} \nand are thus lead to the conclusion\nthat all diagrams with~\\ref{CTP-E-DK} as\na template are Taylor expandable to $\\Op{2}$.\nWe can of course still construct subtractions,\nbut they exactly cancel the parent (leaving\nover the additions).\n\nDiagrams built up from the template~\\ref{CTP-E-GRs-K}\nare, however, a different kettle of fish.\nIndeed, the example with\nwhich we began \\sec{sec:Subtractions-I} is of\nthis type. From diagram~\\ref{CTP-E-GRs-K},\nthere are two ways in which we can construct\ndiagrams which are not Taylor expandable to $\\Op{2}$,\nas shown in \\fig{fig:Subs-NTE}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\t\\LID{CTP-E-NTE-a}\t&\t\t\t& \\LID{CTP-E-NTE-b}\n\t\\\\[1ex]\n\t\t\\cd{CTP-E-NTE-a},\t& \\qquad\t& \\cd{CTP-E-NTE-b}\n\t\\end{array}\n\t\\]\n\\caption{The two types of diagram derived from~\\ref{CTP-E-GRs-K}\nwhich are not Taylor expandable to $\\Op{2}$.}\n\\label{fig:Subs-NTE}\n\\end{figure}\\end{center}\n\nExcluding those terms in which the loose ends\nof the effective propagator and kernel of\ndiagram~\\ref{CTP-E-NTE-a} attach to something\ntransverse in $l$ (in the $A^i$ sector),\nboth diagrams have a component which goes as\n\\[\n\t\\frac{1}{l^2 (l-p)^2},\n\\]\nand so both diagrams possess\na component which is not Taylor expandable to\norder $p^2$. We know from the example\nat the beginning of \\sec{sec:Subtractions-I}\nthat further progress can be made by combining\nsuch diagrams with their subtractions. \nTo isolate the components which survive, \nthe strategy\nis as follows:\n\\begin{enumerate}\n\t\\item\tidentify all loop integrals over momenta other than\n\t\t\t$l$ but which carry dependence on $l$;\n\n\t\\item\tdetermine which, if any, of these loop integrals\n\t\t\tare factorizable from the others;\n\n\t\\item\tFocus on the loop integral \/ non-factorizable \n\t\t\tloop integrals for which the kernel is in the\n\t\t\tintegrand and either\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item \tTaylor expand to zeroth order in $l$,\n\t\t\t\t\t\tdirectly;\n\n\t\t\t\t\\item\tconstruct subtractions if appropriate\n\t\t\t\t\t\tand iterate the procedure.\n\t\t\t\\end{enumerate}\n\\end{enumerate}\n\nAn example of this is shown in \\fig{fig:Op2-NTE-Ex},\nwhere we recognize that the first diagram\nis a generalization of~\\ref{d:111b.11}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\t\t\\LID{CTP-E-NTE-EX}\t&\t\t& \\LIDLID{CTP-E-NTE-EX-s}{CTP-E-NTE-EX-a}\n\t\t\\\\[1ex]\n\t\t\t\\cd{CTP-E-NTE-EX} \t& \\mp \t& \\decp{\\cd{CTP-E-NTE-EX-s}}{}\n\t\t\\end{array}\n\t\\]\n\\caption{A diagram possessing a component which is not Taylor\nexpandable to $\\Op{2}$, together with its subtraction and addition.}\n\\label{fig:Op2-NTE-Ex}\n\\end{figure}\\end{center}\n\nFollowing the above recipe, we recognize that\nthe integrals over $k, \\ m_1$ and $m_2$ all contain\ndependence on $l$. Given that the $m_2$~integral contains\nthe kernel and that the $m_1$~integral has dependence\non $m_2$ but the $k$~integral does not, we can\nfactorize out the integral over $k$. \nThe parent\nand subtraction combine to yield the first diagram\nof \\fig{fig:Op2-NTE-Ex-b}, for which\nwe have constructed a subtraction designed to isolate\nthe part of the $m_1, \\ m_2$ dependent sub-diagram\nwhich is not Taylor expandable in $l$.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\begin{array}{ccc}\n\t\t\t\t\\LID{CTP-E-NTE-EX-b}\t\t\t\t\t\t&\t\t& \\LIDLID{CTP-E-NTE-EX-ss}{CTP-E-NTE-EX-sa}\n\t\t\t\\\\[1ex]\n\t\t\t\t\\left[\\cd{CTP-E-NTE-EX}\\right]_{\\NTE{p}} \t& \\mp \t& \\left[\\cd{CTP-E-NTE-EX-ss}\\right]_{\\NTE{p}}\n\t\t\t\\end{array}\n\t\\\\\n\t\\vspace{2ex}\n\t\t\\rightarrow\n\t\t\\dec{\n\t\t\t\\LIO{\n\t\t\t\t\\left[\n\t\t\t\t\t\\cd{CTP-E-NTE-EX-Rem} \\hspace{1em}\n\t\t\t\t\\right]_{\\NTE{p}}\n\t\t\t\t\\Delta^{11}_{\\alpha \\beta}(p)\n\t\t\t\t\\decp{\n\t\t\t\t\t\\cd{CTP-E-EP-Thpt-K-GR}\n\t\t\t\t}{}\n\t\t\t}{CTP-E-NTE-EX-Rem}\n\t\t}{}\n\t\\\\\n\t\t\\qquad\n\t\t+\n\t\t\\dec{\n\t\t\t\\LIO{\n\t\t\t\t\\left[\n\t\t\t\t\t\\cd{CTP-E-NTE-EX-a}\n\t\t\t\t\\right]_{\\NTE{p}}\n\t\t\t\t\\Delta^{11}_{\\alpha \\beta}(p)\n\t\t\t\t\\decp{\n\t\t\t\t\t\\cd{Diagram111b.11-s-b}\n\t\t\t\t}{}\n\t\t\t}{CTP-E-NTE-EX-Rem-a}\t\n\t\t}{}\n\t\\end{array}\n\t\\]\n\\caption{The result of combining diagrams~\\ref{CTP-E-NTE-EX} and~\\ref{CTP-E-NTE-EX-s}.}\n\\label{fig:Op2-NTE-Ex-b}\n\\end{figure}\\end{center}\n\n\nDiagram~\\ref{CTP-E-NTE-EX-b} combines\nwith diagram~\\ref{CTP-E-NTE-EX-ss} \nto yield diagram~\\ref{CTP-E-NTE-EX-Rem}.\nJust as we have utilized Lorentz\ninvariance to draw\ndiagram~\\ref{CTP-E-NTE-EX-Rem} in an\nappealing form,\nso too do\nwe recognize that a similar thing can be\ndone with the addition, \\ref{CTP-E-NTE-EX-sa},\nyielding diagram~\\ref{CTP-E-NTE-EX-Rem-a}.\n\nReturning to diagram~\\ref{CTP-E-NTE-EX-Rem},\nnotice that we must take the component of the\n$m_1$ integral which, up to the non-polynomial\ndependence on $l$ goes as $l^0$. Were we to\ntake the $l^2$ component, instead,\nthen we could not take the $\\BigNTE{p}$ component of\nthe diagram as a whole. The appearance of the\ntag $\\BigNTE{l}$ is awkward, since it requires\nspecification of a loop momentum, which is unnatural\nfrom our diagrammatic view point. However, with a little\nthought, we can replace this tag with something\nwhich refers only to the external momentum, $p$.\nDiagram~\\ref{CTP-E-NTE-EX-Rem} obviously possesses\nthree loop integrals. However, only two of them (the ones over $l$ and $m_1$)\nare tagged such that we must take the $\\BigNTE{\\mathrm{mom}}$ component,\nwhere `$\\mathrm{mom}$' refers to the momentum external with respect to\\\nthe appropriate sub-diagram. There is a simple rule to\ndetermine the loops from which we must take the\n$\\BigNTE{\\mathrm{mom}}$ component (a similar\nrule can be derived for diagrams of type~\\ref{CTP-E-NTE-a}). In all that\nfollows, if there are any sub-diagrams which\n attach to\njust two effective propagators carrying equal momenta ({\\it cf.}\\ \\\n\\fig{fig:Subs-MomRoute-b2}), we can\nignore each sub-diagram and one of the\neffective propagators to which it attaches.\nStarting with a diagram of arbitrary loop order, the\nrecipe is as follows:\n\\begin{enumerate}\n\t\\item\tIdentify the loop momentum associated with the internal\n\t\t\tfield which attaches to the classical, two-point vertex\n\t\t\t(in the above examples, this momentum has always been $l$)\n\t\t\tand route momenta such that \n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item the external field attaching to the classical,\n\t\t\t\t\t\ttwo-point vertex carries momentum $+p$;\n\n\t\t\t\t\\item  momentum $l+p$ flows \n\t\t\t\t\t\tinto the vertex decorated by the other external field;\n\t\t\t\\end{enumerate}\n\n\n\t\\item\tCut the diagram in every place that there is a gauge remainder\n\t\t\tcarrying $l$ and cut the effective propagator carrying $l$\n\t\t\tat the end out of which $+l$ flows.\n\n\t\\item\tDiscard the sub-diagrams for which there is\n\t\t\ta gauge remainder which carries just $l$\n\t\t\t(this includes the sub-diagram carrying $p$\n\t\t\tand $l$, which is supposed to have already been tagged with $\\BigNTE{p}$);\n\t\\label{it:discard}\n\n\t\\item\tIdentify the remaining sub-diagram as the one that must be\n\t\t\ttagged $\\BigNTE{l}$;\n\n\t\\item\tIterate the procedure, if necessary, taking\n\t\t\tthe sub-diagram tagged with $\\BigNTE{l}$ as the starting point.\n\\end{enumerate}\n\nIn \\fig{fig:Op2-NTE-Ex-c} we apply this procedure to\ndiagram~\\ref{CTP-E-NTE-EX-Rem}, with the red lines\nindicating the cuts. The above recipe tells us to\ntag the sub-diagram with loop momentum $m_1$, but\nnot the one with loop momentum $k$, since the latter sub-diagram\nincludes a $>_l$ and so is excluded by~\\ref{it:discard}, above.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\cd{CTP-E-NTE-EX-Rem-Cut}\n\t\\]\n\\caption{Algorithmic determination of which loop integrals\nare tagged with $\\NTE{}$.}\n\\label{fig:Op2-NTE-Ex-c}\n\\end{figure}\\end{center}\n\nSince this recipe works at any number of loops,\nthere is no need to explicitly tag the appropriate\nloop integrals of some diagram\nalready tagged by $\\BigNTE{p}$; rather, we simply tag the diagram as \na whole by $\\not{\\! \\mathrm{T}}_{M} (p)$.\n\n\n\\section{Introduction}\n\\label{sec:Intro}\n\n\n\nOf all the problems in theoretical physics,\nsurely one of the most pressing is a better understanding\nof Yang-Mills theories, particularly in the\nnon-perturbative domain. A promising framework\nfor addressing this issue\nis the Exact Renormalization Group \n(ERG)~\\cite{Wil,W&H,Pol}, the continuum version of Wilson's RG.\n\nThe essential physical idea behind this approach is\nthat of integrating out degrees of freedom\nbetween the bare scale of the quantum theory and\nsome effective scale, $\\Lambda$. The effects\nof these modes are encoded in the Wilsonian effective action, \n$S_\\Lambda$, which describes the physics of the theory\nin terms of parameters relevant to the effective scale.\nCentral to this methodology is the ERG equation\nwhich determines how the Wilsonian effective action\nchanges under infinitesimal changes of the scale.\nBy relating physics at different\nscales, this equation provides access to the\nlow energy dynamics of Yang-Mills theories.\nIndeed, more generally, \nthe ERG has proven itself to be a flexible\nand powerful tool for studying \nboth perturbative and non-perturbative problems\nin a range of field theories \n(see~\\cite{Fisher:1998kv,Morris:1998da,Aoki:2000wm,Litim:1998nf,Berges:2000ew,Bagnuls:2000ae,Polonyi:2001se,Salmhofer:2001tr,Delamotte:2003dw} \nfor reviews). A particular advantage conferred by the ERG is\nthat renormalization is built in:\nsolutions to the flow equation (in pretty much any approximation scheme),\nfrom which physics can be extracted,\ncan be naturally phrased directly in terms of renormalized parameters.\n\n\nGiven that the notion of a momentum cutoff\nis fundamental to the entire ERG construction,\nit is clear that a regulator\nis required which incorporates this feature. \nImmediately,\nthis presents a problem for Yang-Mills theories,\nsince the direct implementation of a momentum \ncutoff breaks non-Abelian gauge invariance. \nTraditional (gauge fixed) approaches,  which can be broadly divided\ninto those which employ the background field \nmethod~\\cite{Reuter:1993kw,Reuter:1997gx,Bergerhoff:1997cv,Litim:2002ce,Bonini:2001,Freire:2000bq,Gies:2002af,Litim:2002hj}\nand those which do \nnot~\\cite{Becchi:1993,Bonini:1993sj,Bonini:1994kp,Ellwanger:1994iz,Ellwanger:1995qf,Ellwanger:1996wy,Ellwanger:1997tp,Ellwanger:1997wv,Ellwanger:1999vc,Ellwanger:2002xa,D'Attanasio:1996jd,Litim:1998qi,Simionato:2000,Simionato:2000-B,Panza:2000,Pawlowski:2003hq,Fischer:2004uk}, \naccept this breaking,\nrecovering the physical symmetry in \nthe limit that all quantum fluctuations\nhave been integrated out.\\footnote{\nFor an approach based on the geometric \neffective action see~\\cite{Veneziano,Pawlowski-VDW} and for\na summary of the various approaches see~\\cite{Pawlowski:2005xe}.}\n\nNevertheless, in~\\cite{SU(N|N)}, a regulator was constructed\nfor $SU(N)$ Yang-Mills based on a real, gauge invariant cutoff.\nThis is achieved by embedding the physical gauge theory\ninto a spontaneously broken $SU(N|N)$ supergauge theory\nwhich is itself regularized by covariant higher \nderivatives. Combining this with earlier work~\\cite{ym,ymi,ymii}\nallowed the construction of\nan ERG for Yang-Mills which respects \ngauge invariance at all scales~\\cite{aprop}.  In addition to\nthe obvious advantages this has over the alternative approaches,\nthere is a major additional benefit: the gauge invariance\nis in fact \\emph{manifest}, no gauge fixing being required\nat any stage in the computation of the Wilsonian effective\naction. Whilst having considerably novelty value, manifest\ngauge invariance also provides powerful technical simplifications:\nthe gauge field is protected from field strength renormalization and\nthe Ward identities take a particularly simple form~\\cite{ymi},\nsince the Wilsonian effective action is built only from\ngauge invariant combinations of the covariant derivative,\neven at the quantum level. In the non-perturbative domain, not only are\nGribov copies~\\cite{Gribov} entirely avoided, but it should\npossible to make statements\nabout {\\it e.g.}\\  confinement in\na completely gauge independent way. Furthermore, \nsuch a framework\nhas the ability to underlie gauge-fixed ERG formalisms\nas it should, in principle, be possible to derive all\nresults obtained by the latter by gauge fixing\nat an appropriate stage.\n\n\n\n\nThe key to constructing the manifestly gauge invariant \nscheme of this paper\nresides in the immense\nfreedom inherent in  the  ERG~\\cite{jose1,jose2,mgierg1}: of the infinite\nnumber suitable for $SU(N)$ Yang-Mills, an infinite subset\nallow the Wilsonian effective action to be computed without\nfixing the gauge. \nOf these manifestly gauge invariant\nERGs, we further specialize to those\nwhich allow convenient renormalization\nto any loop order~\\cite{mgierg1,mgierg2,Thesis}. (By this we mean\nthat the flow equation is of a suitably general form to\ntreat the physical gauge field and an unphysical copy, \nwhich\nis part of the regularizing structure, asymmetrically. Since\neach of these fields comes with their own coupling, which\nrenormalize separately, allowing the flow equation to distinguish\nbetween them facilitates convenient\ncomputation.) \n\nDespite these restrictions, there are still an infinite\nnumber of admissible ERGs, the differences between them\namounting to the following non-universal details. The first two \nrelate to the\nimplementation of a gauge invariant cutoff:\nthe exact forms of both the cutoff\nfunctions and their covariantizations amount to non-universal\nchoices. The final source of non-universal differences between\nthe ERGs with which we work is\nthe `seed action', $\\hat{S}$~\\cite{Thesis,mgierg1,mgierg2,scalar1,scalar2,aprop,giqed}:\na functional which respects the same symmetries as the Wilsonian effective \naction, $S$, and has the same structure. However, whereas our\naim is to solve the flow for $S$, $\\hat{S}$ acts as an input.\nPhysically, the seed action can be thought of as (partially)\nparameterizing a general Kadanoff blocking~\\cite{Kadanoff} in the \ncontinuum~\\cite{jose1,jose2,mgierg1}. Crucially, these non-universal\ndetails\nneed never\nbe explicitly specified, instead just satisfying general\nconstraints to ensure that the flow equation is well defined.\nWe turn this residual freedom\nin the construction to our advantage\nby recognizing that since all non-universal\ndetails must cancel out in the computation of a universal\nquantity, they can be efficiently absorbed into diagrammatic\nrules. This observation formed the basis for the manifestly gauge invariant\nand universal calculus proposed in~\\cite{aprop}, in which\na scheme was developed whereby these non-universal contributions\ncan be iteratively cancelled out, in perturbative calculations.\n\nIn the original work~\\cite{aprop},\na small subset of the diagrammatic rules now known\nto exist were uncovered and were used in the initial\nstages of a manifestly gauge invariant\ncomputation of the one-loop $\\beta$\nfunction. In a series of \nworks since then~\\cite{scalar2,mgierg1,mgierg2,Thesis,giqed},\nthese diagrammatic rules were extended,\nallowing both the one and two-loop $\\beta$ functions\n(in a variety of Quantum Field Theories (QFTs)) to be reduced\nto manifestly universal diagrammatic expressions,\nfrom which the correct numerical coefficients \nwere directly extracted. In this paper, \nworking in $SU(N)$ Yang-Mills,\nwe bring together and complete\nthe overlapping sets of ideas from this collection of works,\ndeveloping the diagrammatic calculus to a stage where it\nis applicable at any number of loops.  The calculus\nis then comprehensively illustrated by \nderiving an expression for the $n$-loop $\\beta$ function, $\\beta_n$,\nwhich has no explicit dependence on\neither the seed action or the details of the\ncovariantization of the cutoff.\n\n\nThis result is, in itself, interesting for several \nreasons. First, let us recall the standard argument as to why\nthe coefficients $\\beta_1$ and $\\beta_2$, but not $\\beta_{\\geq 3}$,\nare guaranteed to\nagree between certain renormalization schemes~\\cite{Weinberg}.\nIt is important to recognize that each choice of the non-universal\ndetails within our ERG corresponds, in general, to a different renormalization\nscheme. Focusing on one such scheme, we take the coupling\nof the physical $SU(N)$ gauge field to be $g(\\Lambda)$.\\footnote{Beyond this\ndiscussion, we will take $g(\\Lambda)$ to represent the coupling\nfor all renormalization schemes implicitly defined by our approach.}\nNow consider a second scheme---either corresponding to some other\nchoice of non-universal details within our ERG or to an entirely\nindependent scheme such as $\\overline{MS}$---with coupling $\\tilde{g}(\\mu \\mapsto \\Lambda)$.\nGiven the dimensionless\ncoefficient, $\\eta$, we can perturbatively\nmatch the two couplings:\n\\begin{equation}\n\\label{eq:match}\n\t\\frac{1}{\\tilde{g}^2} = \\frac{1}{g^2} + \\eta + \\Or (g^2).\n\\end{equation}\nUsing the usual definition for the $\\beta$ function of $g$\n\\[\n\t\\beta \\equiv \\Lambda \\partial_\\Lambda g = \\sum_{i=1}^{\\infty} g^{2i+1} \\beta_i,\n\\]\nand a similar definition for $\\tilde{\\beta} (\\tilde{g})$, we can differentiate~\\eq{eq:match}\nwith respect to\\ $\\Lambda$ to obtain\n\\begin{equation}\n\t\\tilde{\\beta}_1 + g^2 \\tilde{\\beta}_2 = \\beta_1 + g^2 \\beta_2 + \\Lambda \\partial_\\Lambda \\eta + \\Or (g^4).\n\\label{eq:Universal-beta}\n\\end{equation}\nTherefore, \nif $\\Lambda \\partial_\\Lambda \\eta = 0$ (or, at any rate, does not \ncontribute until $\\mathcal{O}(g^4)$), \nwe will obtain $\\tilde{\\beta}_1 = \\beta_1$\nand $\\tilde{\\beta}_2 = \\beta_2$. We can expect this \nagreement to be spoilt, however, if there exist\nrunning, dimensionless couplings, besides $g$\n(which is equivalent to the introduction of\nadditional mass scales).\n\n\nIndeed, within our setup there \ngenerically exist dimensionless couplings\nwhich run even at tree level\nlevel~\\cite{aprop}, spoiling\nagreement between $\\beta$-function coefficients\nat one loop. Of course, \nthis is not a sign\nof a sick formalism, just a sign that\n$\\beta_1$ and $\\beta_2$ are not physically\nobservable and can be scheme dependent. Nonetheless, it is possible\nto recover the universal values of\n$\\beta_1$ and $\\beta_2$ by suitably tailoring the\nsetup. The running of\n$\\eta$ has two sources.\nThe first is $\\hat{S}$, which contributes\nto the running of $\\eta$ at all orders, including\ntree level. However, through an implicit choice\nof $\\hat{S}$, this running can be removed, not just\nat one-loop, but actually to all orders~\\cite{Thesis,mgierg2}.\nIndeed, we now assume that this has been done, and promote\nthis choice to a requirement, necessary in the specification\nof the subset of manifestly gauge invariant\nERGs with which we choose to work.\n\nThe second contribution to $\\Lambda \\partial_\\Lambda \\eta$ comes from\nan unphysical, dimensionless coupling, $g_2$,\nassociated with the regularizing \nstructure~\\cite{SU(N|N),Thesis,mgierg1,mgierg2}.\nThe running of this coupling,\nwhich occurs from the one-loop level onwards,\ncannot be removed through\na choice of $\\hat{S}$; the solution is to tune it to\nzero at the end of a calculation. In this manner,\n$\\beta_2$ can be arranged to coincide with its\nstandard value~\\cite{Thesis,mgierg2}. \nFor convenience, we work not with $g_2$ directly\nbut with\n\\begin{equation}\n\t\\alpha := g^2_2\/g^2\n\\label{eq:alpha-defn}\n\\end{equation}\nand so it is $\\alpha$, in practise, which is tuned to zero.\n\nBeyond two loops, it is apparent from~\\eq{eq:Universal-beta}\nthat there is no reason to expect any agreement\nbetween $\\beta$ function coefficients. In light of\nthis, it is surprising not only that all contributions\nto $\\Lambda \\partial_\\Lambda \\eta$ coming from $\\hat{S}$ can be removed to all\norders but also that all dependence of $\\beta_n$ on\nthe seed action and details of the covariantization\nof the cutoff cancels out, to all orders. (For speculations\non whether it may be possible to push the removal\nof non-universal details further still, see~\\cite{RG2005}.)\nThat we do find such\ncancellations is perhaps indicative that there\nis a more direct framework for performing calculations\nin QFTs which retains the advantages of the ERG approach\nwhilst removing some or all of the redundancy inherent in\nthe approach. In particular, these observations\ncould inspire a manifestly \ngauge invariant formalism where the seed action and\ndetails of the covariantization of the cutoff\nare relegated to a background role. \n\nThe second point to make about the\nderivation of an expression for $\\beta_n$\nwhich has no explicit dependence on\neither the seed action or the details of the\ncovariantization of the cutoff is that this is\na huge step forward in\nturning this formalism into a practical computational\nscheme.\nThe calculation of $\\beta_1$ and $\\beta_2$\nin this ERG approach was, up until now, an arduous\ntask; just getting to the expression from which the universal\nvalue can be extracted was extremely difficult. Now, however,\nthis step is trivial as we simply specialize the new\nformula for $\\beta_n$ to the appropriate loop order!\n\n\nTo appreciate the various elements of the diagrammatic\ncalculus, it is worth reviewing the \nprocedure for computing $\\beta$ function coefficients employed\nin~\\cite{Primer}.\nTo compute $\\beta_{1,2}$, we start by using the flow\nequation to compute the flow of the two-point vertex\ncorresponding to the physical $SU(N)$ gauge field,\n$A^1_\\mu$, which we suppose carries momentum, $p$. Next, we specialize to the\nappropriate loop order and work at $\\Op{2}$; this\nlatter step constrains the equation by allowing the renormalization\ncondition for the physical coupling to feed in. At this\npoint, the equation for $\\beta_{1,2}$ contains a small\nnumber of diagrams, each of which contains explicit\ndependence on both the seed action and the details\nof the covariantization of the cutoff.\n\nCentral to the diagrammatic calculus\nis the `effective propagator \nrelation'~\\cite{Thesis,mgierg1,mgierg2,Primer,scalar1,scalar2,aprop,giqed}.\nThis convenient computational device\narises as a consequence of a choice\nof seed action we are free to make: \nwe choose the classical, two-point seed action vertices equal to \ntheir Wilsonian effective action counterparts.\nIn turn,\nthis ensures that \nfor each independent classical two-point vertex (that cannot be consistently\nset to zero~\\cite{Thesis}) there exists\nan `effective propagator', denoted by $\\Delta$, \nwhich is the inverse of the\ngiven vertex, up to a `gauge remainder'. \nDenoting the classical two-point vertex corresponding to\nthe fields $X$ and $Y$, which carry indices $R$ and $S$ and\nmomenta $p$ and $-p$, respectively, \nby $S_{0 R S}^{\\ X Y}(p)$\nthe effective propagator relationship in the $A^1$ sector reads\n\\[\n\tS^{\\ A^1 A^1}_{0 \\mu \\ \\; \\alpha} (p) \\Delta^{A^1 A^1}_{\\alpha \\ \\, \\nu} (p) = \\delta_{\\mu\\nu} - \\frac{p_\\mu p_\\nu}{p^2}.\n\\]\nThus, $p_\\mu p_\\nu \/ p^2$ is a gauge remainder\nand it appears as \na consequence of the manifest gauge invariance: the effective\npropagators are inverses of the classical, two-point vertices\nonly in the transverse space.\nAs we will see later, it is convenient to split this gauge remainder \ninto two components,\n$p_\\nu$ and $p_\\mu\/p^2$. \n\n\nTo proceed, we recognize that certain diagrams generated\nby the flow comprise exclusively Wilsonian\neffective action vertices joined together by $\\bigdot{\\Delta}$,\nwhere we define\n\\begin{equation}\n\\label{eq:dot}\n\t\\bigdot{X} \\equiv -\\Lambda \\partial_\\Lambda|_\\alpha X.\n\\end{equation}\nThese terms are processed by moving the $-\\Lambda \\partial_\\Lambda|_\\alpha$\nfrom the effective propagator to the diagram as a whole,\nminus correction terms in which $-\\Lambda \\partial_\\Lambda|_\\alpha$ strikes\nthe vertices. The former diagrams are called $\\Lambda$-derivative\nterms; the latter can be processed using the flow equation\nand the resulting set of diagrams simplified, using\na set of primary diagrammatic identities, of which the\neffective propagator relation is one.\nAt this point, we are able to identify\ncancellations of non-universal contributions, at the\ndiagrammatic level.\nThere is, however, a complication to this diagrammatic procedure:\nparticular classes of sub-diagrams can have two distinct diagrammatic\nrepresentations. The equivalence of these representations\nis encoded in the secondary diagrammatic identities.\n\nIterating the diagrammatic procedure,\nthe expressions for $\\beta_{1,2}$ ultimately\nreduce to the following sets of diagrams:\n\\begin{enumerate}\n\t\\item \t$\\Lambda$-derivative terms, which are built out\n\t\t\tof Wilsonian effective action vertices, effective\n\t\t\tpropagators and (components of) gauge remainders;\n\n\t\\item \t`$\\alpha$-terms', consisting of diagrams\n\t\t\tbuilt out of the same elements as the \n\t\t\t$\\Lambda$-derivative terms but\n\t\t\tstruck by $\\partial \/ \\partial \\alpha$\n\t\t\t(there are no $\\alpha$-terms at one loop~\\cite{Thesis,mgierg1});\n\n\t\\item \t`$\\Op{2}$-terms', which contain an $\\Op{2}$ stub\n\t\t\t{\\it i.e.}\\  a diagrammatic component which is manifestly\n\t\t\t$\\Op{2}$.\n\\end{enumerate}\n\n\nThe $\\Op{2}$-terms can be manipulated. In the calculation\nof $\\beta_1$, at any rate, the structure attaching to the stub\ncan be directly Taylor expanded to zeroth order in $p$---which \ncan once again be done diagrammatically. The above diagrammatic\nprocedure is then iterated.\nAt two loops (and beyond), \nthis procedure is not so straightforward, since\nna\\\"ive Taylor expansion can generate spurious infra-red (IR)\ndivergences~\\cite{Thesis,mgierg1,mgierg2}. The\nsolution is to isolate the components which\ncannot be Taylor expanded using the \n`subtraction techniques' of~\\cite{Thesis,mgierg2}.\nNow the $\\Op{2}$ can\nbe processed, and so $\\beta_{1,2}$ can be reduced to\njust $\\Lambda$-derivative and $\\alpha$-terms.\nFrom these terms, the universal values of\n$\\beta_{1,2}$ can be extracted~\\cite{aprop,mgierg2,Thesis}.\n\nGiven the diagrammatic representation of the flow equation, \nthe diagrammatic calculus comprises the following elements:\n\\begin{enumerate}\n\t\\item\tAn operator which implements the flow {\\it i.e.}\\  $-\\Lambda \\partial_\\Lambda$;\n\n\t\\item\tA diagrammatic rule reflecting the charge conjugation\\ invariance\n\t\t\tof the theory;\n\n\t\\item\tThe primary diagrammatic identities which allow\n\t\t\tthe direct manipulation of diagrams possessing particular\n\t\t\telements;\n\n\t\\item\tThe secondary diagrammatic identities which encode\n\t\t\tthe equivalence of distinct diagrammatic representations\n\t\t\tof particular classes of sub-diagrams;\n\n\t\\item\tThe subtraction techniques.\n\\end{enumerate}\n\nThe primary diagrammatic identities further decompose into three sets. \nThose of the first type are defined\nwithout any reference to perturbation theory, and follow from\ngeneral constraints such as gauge invariance. The single primary\ndiagrammatic identity of the second type is the effective propagator\nrelation, which we recall follows from the classical flow equation,\ngiven a choice we impose on the seed action. Those of the third type\nfollow directly from those of the first and second types. However, it\nis convenient to give them in their own right, as they are not\nnecessarily obvious and are heavily used in this paper.\n\nThe secondary diagrammatic identities decomposes into two families,\none of which is applicable only to those diagrams which possess\nan $\\Op{2}$ stub and the other of which is more generally\napplicable.\n\nIt is apparent that, \nof all the elements of the diagrammatic calculus, \nonly one is reliant on perturbation\ntheory.  An obvious\nquestion, to which we return in the conclusion,\nis whether the calculus has a non-perturbative\nextension. We emphasise\nthat, irrespective of whether or not\nthis is the case, the framework admits \nstandard ERG analyses in the non-perturbative domain, \nbut with the benefits of manifest gauge invariance.\n\nTo derive an expression for arbitrary $\\beta_n$,\nwe could pursue the above strategy (see \\cite{Thesis} for an attempt\nin this direction). However, there is a much more\nefficient way to proceed, motivated by\nthe form of the $\\Lambda$-derivative terms\nat one and two loops. We start by constructing\na set of $n$-loop functions which depend only on\nWilsonian effective action vertices, effective\npropagators and (components of) gauge remainders.\nConsidering the $\\Op{2}$ parts of these diagrams\n(strictly, up to functions not polynomial  in $p$),\nwe compute their flow. Amongst the generated\nterms is $\\beta_n$ multiplied by a\ncoefficient\nwhich is universal as a consequence of specializing\nto $\\Op{2}$. Using the various diagrammatic identities,\nwe then demonstrate that all dependence on\nthe seed action and details of the covariantization\ncancels out between the remaining terms. \nBy taking the $\\Lambda$-derivative terms\nas the starting point for the calculation, we\navoid having to iteratively construct them, which\nreaps better and better dividends with each loop order.\n\n\nThis paper is organized as follows. In \\sec{sec:Review},\nwe review the aspects of the manifestly gauge\ninvariant ERG we require for this paper. Following\na brief exposition of $SU(N|N)$ gauge theory, the flow\nequation is introduced via its diagrammatic representation.\nSome properties of the various diagrammatic\nelements of the flow equation are discussed\nand the primary diagrammatic identities of the first type\nare stated. The section concludes with the examination\nof the\nform taken by the flow equation in the perturbative regime;\nwe describe the effective propagator relation\nand derive the primary diagrammatic identities of the third type.\n\n\n\\Sec{sec:Further}, the basis of which comes from~\\cite{Thesis}, \nis devoted to developing\nthe diagrammatic techniques to the level\nsufficient for computing $\\beta_n$. First,\nwe revisit the gauge remainders, discussing\nthe type of diagrams they can generate\nin the perturbative domain. Next, we\nstate and prove the secondary diagrammatic\nidentities and finally we describe\nthe subtraction techniques necessary for\nmanipulating the $\\Op{2}$ terms. This completes\nthe description of the calculus and concludes\nthe first part of the paper.\n\nThe second part of the paper is devoted to\na detailed illustration of the calculus.\n\\Sec{sec:Preliminary}\nbegins with a description of some further notation\nwhich facilitates the practical\napplication of the calculus, by allowing us to\nrepresent sets (with a potentially huge\nnumber) of $n$-loop diagrams in an extremely compact\nmanner. Using this notation,\nwe introduce a set of diagrammatic functions\nwhich will play a central role in the treatment\nof $\\beta_n$ to follow. Various important properties of\nthese functions are analysed, which allows an\nillustration of many of the diagrammatic techniques\nin their most natural setting.\nIn \\sec{sec:beta_n} we derive an expression for $\\beta_n$\nwhich is independent of the seed action and the\ndetails of the covariantization of the cutoff\nand we conclude in \\sec{sec:Conc}.\nThe primary and secondary diagrammatic identities are\ncollected together in \\app{app:D-ID} for easy reference.\n\n\n\n\n\n\n\\section{Preliminary Diagrammatics}\n\\label{sec:Preliminary}\n\n\\subsection{Additional Notation}\n\\label{sec:Notation}\n\n\\subsubsection{(Pseudo) Effective Propagators}\n\nIn \\sec{sec:GRs-2}, we encountered a single effective\npropagator amongst the implicit decorations \n(see {\\it e.g.}\\  diagram~\\ref{v-W-GR-w}). We now\ngeneralize to consider an arbitrary number of such\neffective propagators. The rule for explicit decoration\nis defined as follows. If we wish to join two objects\n(say two vertices) together with $j'$ out of a total\nof $j$ effective propagators, then there are\n$\\nCr{j}{j'} 2^{j'}$ ways to do this. \nIntuitively, the first factor captures the \nnotion that,\nso long as they are implicit decorations, the effective\npropagators are indistinguishable. The factor of\n$2^{j'}$ allows for the fact that we can interchange\nthe two ends of an effective propagator. If these\neffective propagators were instead used to form $j'$\nloops on a single vertex, then the factor of\n$2^{j'}$ would disappear, since the vertices\nare defined such that all cyclically independent\narrangement of their decorative fields are summed\nover.\n\n\nGiven that we have seen how\nfields and effective propagators can usefully appear\nas implicit decorations, it is not surprising that,\nin our calculation of $\\beta_n$, \ngauge remainder components and pseudo effective\npropagators also appear in this fashion.\nThe combinatorics for pseudo effective propagators\nis exactly the same as for effective\npropagators. When we come to decorate\nwith pseudo effective propagators, we can make life easy\nfor ourselves by noting that, in practise, pseudo\neffective propagators can be arranged to appear\nonly in one way and always in combination with another\nterm, precisely as in~\\eq{eq:Combo}.\nThe combinatoric factor\nassociated with partitioning $j$ $\\ensuremath{\\star}$s\ninto two sets of $j'$ and $j-j'$ is just\n$\\nCr{j}{j'}$. However, unlike effective\npropagators, the ends of $\\ensuremath{\\star}$ are not\ninterchangeable.\n\n\n\\subsubsection{Gauge Remainders}\n\\label{sec:Not:GRs}\n\nWe begin\nby giving the rules for converting a set of $m$\ngauge remainder components of type $>$ from implicit\nto explicit decorations. Thus, we are considering\ndecorations of the form\n\\begin{equation}\n\\label{eq:Implict-GR}\n\t\\dec{ \\\n\t}{>^m \\cdots},\n\\end{equation}\nwhere the square brackets could enclose some\ndiagrammatic structure, but need not. The ellipsis\nrepresents any additional implicit decorations,\nso long as they are not further instances of $>$.\nThe superscript notation $>^m$ simply tells\nus, as before, that there are $m$ instances of $>$.\n\nThere are only two structures we are allowed to form from\nimplicit decorations containing $>$. The first\nof these is a ring, as \nin~\\fig{fig:nestedhook}. Noting that, in some\ncompletely fleshed out diagram, the sockets of\nall the $>$ will be filled, this means that\nwe can generate a diagram like~\\ref{Diags-WGR-N1-Bottom-PF}\nfrom implicit decorations, but\nnever\na diagram like~\\ref{Diags-WGRx2-L-Bottom}.\nHowever, there is no restriction that we\ncreate a single ring structure from our $m$\ngauge remainders: we can create up to $m$\nindependent structures. Furthermore,\nwe need not promote all of the $>$\nto explicit decorations at the same time.\nThus, we can imagine\npartitioning the $m$ gauge remainders into\n$q+1$ sets, with the first $q$ of these sets forming\nan independent ring structure and the final\nset remaining as implicit decorations\n{\\it viz.}\\ \n\\begin{equation}\n\\label{eq:Implict-GR-b}\n\tM \\dec{\n\t\t\\decGR{ \\\n\t\t}{>^{m^1}} \n\t\t\\decGR{ \\\n\t\t}{>^{m^2}}\n\t\t\\cdots\n\t\t\\decGR{ \\\n\t\t}{>^{m^q}}\n\t}{>^{m^{q+1}} \\cdots},\n\\end{equation}\nwhere $M$ a combinatoric factor and the notation,\n$\\decGR{ \\ }{>^{m^1}}$, tells us that\nthese decorative gauge remainders must form a\nsingle ring {\\it i.e.}\\  they cannot be further partitioned,\nnor can this ring be added to by the remaining\nimplicit decorations.\nThe $m^i$ are positive integers which sum up to $m$.\n\nThe combinatoric factor\nis easy to compute: if we partition $m$ gauge\nremainders into two sets of $m'$ and $m-m'$\ngauge remainders, then the combinatoric factor\nis defined to be\n\\[\n\t\\nCr{m}{m'}.\n\\]\nTherefore, we\ncan rewrite~\\eq{eq:Implict-GR-b} as:\n\\[\n\t\\frac{m!}{m^{q+1}!} \n\t\\dec{\n\t\t\\prod_{i=1}^q \\frac{1}{m^i!}\n\t\t\\decGR{ \\ }{>^{m^i}}\n\t}{>^{m^{q+1}}\\cdots}.\n\\]\n\n\nThe final gauge remainder structure\nwe can construct out of implicit decorations\ncontaining $>$s occurs only in diagrams\npossessing an $\\Op{2}$ stub. The\ncontexts in which this structure occurs are\nillustrated in \\fig{fig:GR-Op2String}. \n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\left[\n\t\t\\LIDi{Op2-GR-String-Ex}{Op2-GR-String-Ex}\n\t\\right]_{p^2}, \\qquad\n\t\\LIDi{Op2-GR-String-Ex-B-Mac}{Op2-GR-String-Ex-B}\n\t\\]\n\\caption{A gauge remainder structure which only occurs in diagrams\npossessing an $\\Op{2}$ stub.}\n\\label{fig:GR-Op2String}\n\\end{figure}\\end{center}\n\nDiagrams~\\ref{Op2-GR-String-Ex} and~\\ref{Op2-GR-String-Ex-B}\nboth possess a string (not a ring) of $m'$ gauge remainders\n({\\it cf.}\\  \\fig{fig:GRstring}),\nthe last of which bites the socket on the classical,\ntwo-point vertex (in either sense).\nIn diagram~\\ref{Op2-GR-String-Ex}, the\nfirst of the $m'$ gauge remainders is \nbitten by a \\ensuremath{\\star}\\ whereas, in diagram~\\ref{Op2-GR-String-Ex-B}\nit is bitten by a $>$ attached to an effective\npropagator. The loose end of the \\ensuremath{\\star}\\ \/ effective propagator \ncan attach to any of the $>$s or to the\nsocket on the classical, two-point vertex or\nto any (un-drawn) structures (the \\ensuremath{\\star}\\ cannot\nattach to its own socket, though).\n\nAll diagrams possessing a string of gauge remainders\nbitten by a \\ensuremath{\\star}\\ are tagged $p^2$. This\ntells\nus that everything attaching to\nthe $\\Op{2}$ stub \nis independent of\n$p$ (as in \\sec{sec:Subtraction}). \nDiagrams tagged with $p^2$ can possess\nmore than one instance of the structure\nwhich attaches to the stub in \ndiagram~\\ref{Op2-GR-String-Ex}: although there\nis only one stub, structures involving a $\\ensuremath{\\star}$\ncan attach to the momentum derivative symbol\nof another such structure, as illustrated in\n\\fig{fig:GR-Op2String-b}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\left[\n\t\t\\cd{Op2-GR-String-Ex-b} \\hspace{1em}\n\t\\right]_{p^2}\n\t\\]\n\\caption{An example of a diagram possessing multiple $\\ensuremath{\\star}$s.}\n\\label{fig:GR-Op2String-b}\n\\end{figure}\\end{center}\n\nWe note the following. First,\nit is not legal to join any of the $m'$\ngauge remainders to any of the $m''$ gauge remainders.\nIn other words, no two $\\ensuremath{\\star}$s are permitted to \ncarry a common loop momentum. Secondly, the $\\ensuremath{\\star}$s are always\ntied up such that the momentum derivative symbols\nare on the `outside' of the diagram. In other words, we can draw\na line from any momentum derivative symbol \/ external field to any other\nwithout crossing an internal line. Diagrams in which this\nis not the case vanish by a combination of group theory\nand charge conjugation\\ invariance considerations~\\cite{Thesis,mgierg1}.\n\n\n\n\\subsubsection{Vertices}\n\nWhen deriving an expression for $\\beta_n$,\nwe will find it invaluable to compactly\nrepresent all vertices in a diagram\nwhich are yet to be explicitly decorated.\nTo this end, we introduce a set of vertex arguments, $v^j$,\nwhere the upper roman index acts as a label. Thus, the $v^j$\nare integers, denoting the loop orders of some set of vertices.\nWe denote the reduction of these vertices by $v^{j;R}$, where\nwe recall that a reduced vertex lacks a classical, two-point \ncomponent.\n\nNext, we introduce the compact notation\n\\begin{eqnarray*}\n\tv^{j,j_+} & \\equiv & v^j - v^{j+1},\n\\\\\n\tv^{j,j_+;R} & \\equiv & v^{j;R} - v^{j+1;R}.\n\\end{eqnarray*}\nWe use this notation to define\n\\begin{equation}\n\\label{eq:CompactVertices}\n\\SumVertex \\equiv \\prod_{i=0}^j \\sum_{v^{i_+} = 0}^{v^i} \\Vertex{v^{i,i_+;R}},\n\\end{equation}\nwhere the first argument of the structure on the left-hand side, $n_s$, gives\nthe value of $v^0$. Notice that all other vertex arguments are summed over.\nThe interpretation of the product symbol is as a generator\nof  $j+1$ vertices.\n\nThe structure shown in~\\eq{eq:CompactVertices} always appears as a part of\ndiagrams which\npossess an additional vertex, which carries the argument $v^{j_+}$ (this\nargument need not appear on its own---it could be part of something more complicated\n{\\it e.g.}\\  $v^{j_+,k}$). An example, which will play an important role\\ later,\nis\n\\begin{equation}\n\\label{eq:VertexTower}\n\t\\left[\n\t\t\\sco[2]{\\TopVertex}{\\SumVertex}\n\t\\right] \\equiv\n\t\\prod_{i=0}^j \\sum_{v^{i_+} = 0}^{v^i}\n\t\\left[\n\t\\sco[2]{\\TopVertex}{\\Vertex{v^{i,i_+;R}}}\n\t\\right].\n\\end{equation}\nNotice that the sum over all vertex arguments is trivially $n_s$:\n\\begin{equation}\n\\label{eq:VertexSum}\n\t\\sum_{i=0}^j v^{i,i_+} + v^{j_+} = \\sum_{i=0}^j \\left( v^i - v^{i+1} \\right) + v^{j+1} = v^0 = n_s.\n\\end{equation}\n\nThe interpretation that the structure defined by~\\eq{eq:CompactVertices}\npossesses $j+1$ vertices allows us to usefully define~\\eq{eq:VertexTower}\nfor $j=-1$:\n\\begin{equation}\n\\label{eq:Tower-1}\n\t\\left[\n\t\t\\sco[2]{\\TopVertex}{\\SumVertex}\n\t\\right]_{j=-1} \\equiv\n\t\\Vertex{n^R_s \\hspace{0.25em}},\n\\end{equation}\nwhere $n_s^R$ is, of course, just an $n-s$ loop, reduced vertex. (Note that\nthis example illustrates the rule that $v^{j_+}$ is replaced by $v^0$;\nthis holds irrespective of whether or not $v^{j_+}$ occurs only as part of some\nmore complicated vertex argument.) We can\neven usefully define what we mean by~\\eq{eq:VertexTower}\nfor $j=-2$:\n\\begin{equation}\n\\label{eq:Tower-2}\n\t\\left[\n\t\t\\sco[2]{\\TopVertex}{\\SumVertex}\n\t\\right]_{j=-2} \\equiv\n\t\\delta(n-s).\n\\end{equation}\nNotice that~\\eq{eq:Tower-1} still makes\nsense if one and only one of the vertices\nis decorated (in practise, we will\nalways take this to be the top vertex):\nif more than one vertex is decorated then this\nimplies that the number of vertices is at least two,\nwhich leads to a contradiction.\nOn the other hand, \\eq{eq:Tower-2} makes sense\nonly as is, and not if any of the vertices are\ndecorated. In the computation of $\\beta_n$,\nwe will find structures like~\\eq{eq:VertexTower},\nwhere we sum over $j$. The lower value of this\nsum will start out at $-2$. However, as we perform\nexplicit decorations of the vertices, so we will\nneed to raise the lower limit on $j$, such\nthat the diagrams still make sense.\n\n\n\n\n\n\\subsubsection{Decoration of Kernels {\\it etc.}\\ }\n\\label{sec:Not-K}\n\nIt will prove useful to isolate the decorated\ncomponents of a kernel from the undecorated\nparts.  To facilitate this separation, we introduce the \nsymbol $\\odot$ to indicate an undecorated kernel and $\\circ$ \nto indicate a decorated kernel,\nsuch that\n\\begin{equation}\n\\label{eq:Kernel-Notation}\n\t\\ensuremath{\\stackrel{\\bullet}{\\mbox{\\rule{1cm}{.2mm}}}} \\ = \\ \\ensuremath{\\stackrel{\\odot}{\\mbox{\\rule{1cm}{.2mm}}}} + \\ensuremath{\\stackrel{\\circ}{\\mbox{\\rule{1cm}{.2mm}}}}.\n\\end{equation}\nNoting that decoration is not defined for\neither $\\ensuremath{\\star}$s or pseudo effective propagators\n(or gauge remainder components)\nwe define\n\\begin{equation}\n\\label{eq:Combo-Decoration}\n\t\\cd{LdL-Combo} \\equiv \\cd{DEP-dGR} + \\cd{EP-DdGR} -\\frac{1}{2} \\cd{DdPEP-GR},\n\\end{equation}\nwhere we have used~\\eqs{eq:LdL-GRk-Pert}{eq:Combo}.\n\n\n\n\\subsection{Diagrammatic Functions}\n\\label{sec:Prelim-DiagFns}\n\n\nIn this section we introduce a set of\ndiagrammatic functions,\n$\\nLV{{\\cal D}}{n}{\\mu\\nu}{i}(p)$, which will\nprove crucial in our treatment of $\\beta_n$\n(recall that the $1$s are shorthand for $A^1$s).\nBy analysing their various\nproperties, we will be able to illustrate\nthe use of many of the diagrammatic techniques,\nin their natural setting. We will ultimately\nbe interested in the $\\Op{2}$ parts of these\ndiagrams (up to functions not polynomial in $p$).\n\n\n\\subsubsection{The Function $\\protect \\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$}\n\nWe introduce the function\n\\begin{equation}\n\\label{eq:nL-a}\n\t\\nLV{{\\cal D}}{n}{\\mu \\nu}{a}(p) \\equiv\n\t2 \\sum_{s=1}^n  \\sum_{m=0}^{2s} \\sum_{j=-2}^{n+s-m-2} \n\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\n\t\\dec{\n\t\t\\sco[1]{\\TopVertex}{\\SumVertex}\n\t}{11\\Delta^{j+s+1}>^m},\n\\end{equation}\nwhere, for the non-negative integers $a$ and $b$,\nwe define\n\\begin{equation}\n\\label{eq:norm}\n\t\\ensuremath{\\Upsilon}_{a,b} = \\frac{(-1)^{b+1}}{a!b!} \\left(\\frac{1}{2}\\right)^{a+1},\n\\end{equation}\nwhich we see obeys the following relationships:\n\\numparts\n\\begin{eqnarray}\n\\label{eq:norm-b}\n\t2a \\ensuremath{\\Upsilon}_{a,b} \t\t& = & \\ensuremath{\\Upsilon}_{a-1,b},\n\\\\\n\\label{eq:norm-c}\n\tb\\ensuremath{\\Upsilon}_{a,b} \t\t& = & -\\ensuremath{\\Upsilon}_{a,b-1}.\n\\end{eqnarray}\n\\endnumparts\nIn the case that either $a$ or $b$ are negative, $\\ensuremath{\\Upsilon}_{a,b}$\nis null. The overall combinatoric factor, $\\ensuremath{\\Upsilon}_{j+s+1,j+2}\/m!$,\nis said to be canonical, since the three arguments, $j+s+1$, $j+2$\nand $m$ match, respectively, the number of effective propagators,\nthe number of vertices and the number of $>$s.\n\nThere is a simple, intuitive explanation for the relationship\nbetween the number of vertices, the number of effective propagators\nand the sum over the vertex arguments.\nConsider the components of $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$\nfor which there is at least one vertex {\\it i.e.}\\  for which $j>-2$ and for which\n$m=0$. We\nknow from~\\eq{eq:VertexSum} that the sum of the vertex arguments is $n-s$.\nNow, given\n$j+2$ vertices, exactly $j+1$ effective propagators\nare required to create a connected diagram. This leaves\nover $s$ effective propagators, each of which must create\na loop. Therefore, the loop order of the diagram is $n-s+s = n$,\nas must be the case.\nNow we relax the restriction that $m=0$ and suppose that\nwe create $m'$ gauge remainder rings ({\\it cf.}\\  \\fig{fig:nestedhook}). \nTo ensure that each of these rings is connected to\nsome other part of the diagram, we must use up \n $m'$ of our\n$s$ `spare' effective propagators. However, this is\nprecisely counteracted by the contribution of\neach of the rings to the loop order of the diagram.\nNext, consider the component of the diagram\nfor which $j=-2$:\n\\begin{equation}\n\\label{eq:Dn-j=-2}\n\t2 \\sum_{m=0}^{2n} \\frac{\\ensuremath{\\Upsilon}_{n-1,0}}{m!} \n\t\t\\dec{\n\t\t\t\\hspace{1em}\n\t\t}{11\\Delta^{n-1} >^{m}}.\n\\end{equation}\nTo ensure that the diagram is fully connected, $m$ must\ntake the value $2n$ since,\nfor any smaller value, there are not enough $>$s\nfor the $2(n-1)+2$ fields which require a socket in\nwhich to reside. The loop order of all diagrams\ngenerated by~\\eq{eq:Dn-j=-2} is clearly $n$.\n\n\nHaving discussed how the various diagrammatic elements \nconspire to yield a diagram of loop order $n$, we now\ndiscuss the ranges on the various sums. The maximum\nvalue of $s$ clearly follows from the requirement\nthat the loop order of the diagram is $\\geq 0$. \nThe maximum values of the sums over $m$ and $j$\nand the minimum value of $s$\nfollow from\nthe constraint that all fully fleshed out diagrams\nmust be connected and must have an $\\Op{2}$ contribution. \nSuppose that there were a term contributing to $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$\nwith $s=0$.\nIn this case, we would no longer have any\ndecorative gauge remainder components, being left\nwith just $j+2$ reduced, Wilsonian effective\naction vertices, $j+1$ effective propagators\nand two external fields. Recalling that effective\npropagators are two-ended objects, we see that\nthere are $2j+4$ available decorations.\nNow, from \n\\sec{sec:elements} we know that it is imposed as a constraint that all\none-point, Wilsonian effective action vertices vanish\n(though one-point, \\emph{seed} action vertices do exist, beyond tree level),\nin order that the vacuum expectation value of the\nsuperscalar which breaks the $SU(N|N)$ symmetry\nis not shifted by quantum corrections.\nWe can thus insist that all \nWilsonian effective action vertices are at \nleast two-point. This requires $2(j+2)$\ndecorations, which is precisely equal to\nthe number of available decorations\nfor the diagram\nas a whole to be connected. Consequently, \neach vertex\nmust be precisely two-point. Immediately,\nthis precludes the existence of any classical\nvertices, since reduced vertices\ndo not possess such a component.\nThus we have a line of vertices with a loop\norder of at least one apiece. At each end\nof the line is a vertex decorated by one external\nfield and by one end of an effective propagator;\nall other vertices are joined to two other vertices,\nin each case by a single effective propagator.\nNow, since none of these vertices is classical,\nthey must each, by~\\eq{eq:S_>0-11}, contribute $\\Op{4}$.\nAlthough the effective propagators each contain\na $1\/p^2$ component this is insufficient to prevent\nthe diagram as a whole vanishing at $\\Op{2}$.\n\nIn similar fashion, we can understand the\nmaximum values of the sums over $m$ and $j$.\nOf the $j+2$ vertices, suppose\nthat $T$ are classical vertices. Since the\nvertices are reduced,\na minimum of $3T$ decorations are required for\nthe classical vertices and so, if there are\n$m$ gauge remainders, a minimum of  \n\\begin{equation}\n\\label{eq:required}\n\t2(j+2-T) + 3T + m = 2j+T+m+4\n\\end{equation}\ndecorations are required. \nNoting that the \nnumber of\navailable decorations is\n$2(j+s+1)+2$, it is clear\nthat \n\\begin{equation}\n\\label{eq:T+m}\n\tT+m \\leq 2s.\n\\end{equation}\nIt therefore follows that\n\\begin{equation}\n\\label{eq:Maxm}\n\tm \\leq 2s.\n\\end{equation}\n\nNext, let us deduce the maximum number of vertices {\\it i.e.}\\ \nthe maximum number taken by $j+2$ for some values of\n$s$ and $m$. From~\\eq{eq:VertexSum},\nwe know that\nthe sum over vertex arguments is $n-s$.\nTherefore, we can have at most $n-s$ vertices which\nare not tree level and so a total of at most $n-s+T$\nvertices. Hence,\n\\[\n\tj+2 \\leq n+s-m,\n\\]\nwhere we have used~\\eq{eq:T+m}.\nBefore moving on, let us return to the conclusion~\\eq{eq:Maxm}.\nIn fact, there may be a tighter constraint than this\nsince, if there are more than $n-s$ vertices,\nthen $T$ is compelled to be greater than zero.\nHowever, we stick with the limit given by~\\eq{eq:Maxm},\ndiscarding any diagrams which\nturn out to be disconnected and adopting\nthe prescription that terms for which the\nmaximum value of the sum over $j$ is less\nthan its minimum value do not exist.\n\nThis completes our discussion of the form of \n$\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$ and we will now\ndiscuss some of its properties. However,\nrather than working with $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$\ndirectly, we introduce the auxiliary function\n\\begin{equation}\n\\label{eq:nLaux-a}\n\t\\nLV{\\mathcal{E}}{n}{\\mu \\nu}{a}(p) \\equiv\n\t2 \\sum_{s=0}^n  \\sum_{m=0}^{2s+1} \\sum_{j=-2}^{n+s-m-1} \n\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\n\t\\dec{\n\t\t\\sco[1]{\\TopVertex}{\\SumVertex}\n\t}{11\\Delta^{j+s+1}>^m}.\n\\end{equation}\nComparing $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ with $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$,\nwe notice \nthat the ranges on the sums over $s, \\ m$ and $j$\ndiffer between the two expressions.\nThus, we know from our previous arguments that\nthere are diagrams contained in $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ \nwhich vanish\nat $\\Op{2}$ and \/ or contain one-point, Wilsonian effective\naction vertices. The point is that, in our computation of\n$\\beta_n$, we will consider $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$.\nTo process this expression, we will use the weak coupling\nflow equations. However, when we do so, there are\ngenerically  elements\nof the resulting set of terms which,\nthough not generated by $\\dec{\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$,\ndo not \\emph{individually}\nvanish.  We will find that some of these terms cancel against\nterms generated by the flow of other diagrams and so\nwe can view $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ as being \nthe same as $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$, at $\\Op{2}$,\nup to a useful re-expression of zero.\n\nWe now demonstrate that \n\\begin{equation}\n\\label{eq:Transverse+TE}\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) \\sim \\Box_{\\mu \\nu}(p) + \\Op{4}.\n\\end{equation}\nTo do this, we first show that $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$\nis transverse in $p$. This is done by contracting\n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$\nwith the momenta\n$p_\\mu$ and $(-p)_\\nu$ of the external \nfields.\\footnote{\nTo prove the transversality of  $\\nLDl{a}(p)$ \nit is necessary only to contract\n$\\nLDl{a}(p)$ with one of the external momenta,\nbut it technically easier to do it with both.}\nSince we are not computing the flow of any of\nthe vertices, we can discard all one-point\nvertices, and so decrease the upper limits\non the sums over $m$ and $j$ by one each.\nExplicitly decorating with one of\nthe external fields but leaving the other\nas an implicit decoration \nyields the diagrams of \\fig{fig:bn-Transverse}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray*}\n\t\tp_\\mu (-p)_\\nu \\nLDl{a}(p) & = &\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m=0}^{2s} \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LIDi{vj+R-EGR}{vj+R-EGR}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+1}>^m}\n\t\t}\n\t\\\\[2ex]\n\t\t& & +2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m=2}^{2s} \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m\n\t\t}\n\t\t\t{\n\t\t\t \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LIDi{GRs-EGR}{GRs-EGR}\n\t\t\t\t}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+2}>^{m-m'}}\n\t\t}\n\t\\end{eqnarray*}\n\\caption{Contraction of $\\nLDl{a}(p)$ with its external\nmomenta.}\n\\label{fig:bn-Transverse}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{vj+R-EGR} comes with a factor of $j-2$,\ncompared to the parent (which have have absorbed into\n$\\ensuremath{\\Upsilon}$, using~\\eq{eq:norm-c}), recognizing\nthe indistinguishability of the $j+2$ vertices,\nprior to explicit decoration. \nFurthermore, the lower limit on the sum over $j$ is\n$-1$ and not $-2$, since diagram~\\ref{vj+R-EGR}\nmust possess at least one vertex.\nThe combinatorics\nfor diagram~\\ref{GRs-EGR} follows from~\\sec{sec:Not:GRs};\nwe have recognized that for the diagram as a whole to\nbe connected (or, alternatively, on account of diagrammatic\nidentity~\\eq{D-ID-Trivial}), the explicit gauge remainder\nstructure must possess a minimum of two $>$s.\n\nThe reduced vertex of \ndiagram~\\ref{vj+R-EGR} \nwhich is struck by the gauge remainder\ncan\nbe promoted to a full vertex, since a classical,\ntwo-point vertices contracted into a $\\rhd$\ndies, courtesy of diagrammatic \nidentity~\\eq{eq:GR-TLTP}.\nAllowing the gauge remainder\nto act---whereupon it necessarily strikes \na socket---we split the resulting vertex\ninto a reduced part and a classical, two-point\npart. In the latter case, we shift $j \\rightarrow j+1$.\nSince no other gauge remainders have acted (and no Taylor\nexpansions have been performed), we could use\ncharge conjugation\\ invariance to collect together\npushes forward and pulls back. However, we refrain\nfrom doing this, for reasons that will become\napparent.\nThe result of processing diagram~\\ref{vj+R-EGR} \nis shown in \\fig{fig:bn-Transverse-b} where,\nfor each term,\nthe explicitly drawn external field can bite the\nsocket in either sense.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LIDi{vj+R-EGR-Socket}{vj+R-EGR-Socket}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+1}>^m}\n\t\t}\n\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LIDi{CTP-EGR-Socket}{CTP-EGR-Socket}\n\t\t\t\t}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+2}>^m}\n\t\t}\n\t\\]\n\\caption{The result of processing diagram~\\ref{vj+R-EGR}.}\n\\label{fig:bn-Transverse-b}\n\\end{figure}\\end{center}\n\nNotice that the limits on the sums over $m$ and \n$j$ differ from those of the parent. For example, \nin diagram~\\ref{vj+R-EGR-Socket}, where the gauge\nremainder strikes a socket on a reduced vertex,\nwe have `used up' a decoration and so can \ndecrease the limits on the sums over\n$m$ and $j$ by one each.\nIn diagram~\\ref{CTP-EGR-Socket}\nsimilar consideration apply, but here \nnotice that the lower limit on the sum over $j$ is now\n$-2$, as a consequence of having\nshifted $j \\rightarrow j+1$.\n\nThe next step is to decorate the classical,\ntwo-point vertex of diagram~\\ref{CTP-EGR-Socket},\njust as we did with diagram~\\ref{v-W-GR-w-b02}\n(see \\figs{fig:Decompose}{fig:EP-Ex}). If we\nattach the remaining \\ensuremath{\\cdeps{ExtGR}}, the diagram\ndies, courtesy of diagrammatic identity~\\eq{eq:GR-TLTP}.\nThus, all we  can do is\nattach an effective propagator,\nthe loose end of which joins\neither to a reduced Wilsonian effective\naction vertex or to a gauge remainder\nstructure (recall that if the both\nends of the effective propagator attach\nto the classical, two-point vertex then the\ndiagram dies as a consequence of charge conjugation\\ invariance). \nIn each case, the combinatoric\nfactor associated with choosing one effective\npropagator, either end of which can attach to\nthe classical, two-point vertex, is\n$2(j+s+2)$. In the case that the other end attaches\nto one of the $j+2$ reduced vertex, there is an additional\nfactor of $j+2$.\n\n\\begin{Icancel}\n\\label{Icancel:Tansverse}\n\tThe component of diagram~\\ref{CTP-EGR-Socket}\n\tin which the classical, two-point vertex\n\tis attached to a reduced, Wilsonian effective\n\taction vertex cancels diagram~\\ref{vj+R-EGR-Socket},\n\tup to a gauge remainder contribution.\n\\end{Icancel}\n\nCrucially, the surviving gauge remainder contribution\nis a nested version of diagram~\\ref{vj+R-EGR}.\nThus, processing the nested gauge remainder\njust repeats the above cancellations,\ngenerating successively more nested diagrams.\nThe process terminates when there are insufficient\ndecorations for the gauge remainder to strike\na reduced vertex. What, then, remains? At each level of\nnesting (including the un-nested case, as discussed\nalready), we can\nattach the classical, two-point vertex to a\ngauge remainder structure. Additionally,\nfrom the first level of nesting onwards,\nwe can attach the classical, two-point vertex\nto one of the gauge remainders which is nested\nwith respect to\\ the gauge remainder which bites the vertex. The\ncomplete set of surviving diagrams is collected\ntogether in \\fig{fig:bn-Transverse-c}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m'=1}^{2s-1} \\sum_{m=0}^{2s-1-m'} \\sum_{j=-1}^{n+s-m-m'-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LIDi{EGR-GRs-EP-TLTP}{EGR-GRs-EP-TLTP}\n\t\t\t\t}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+1}>^m}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m''=0}^{2s-3}  \\sum_{m=2}^{2s-1-m''} \\sum_{j=-1}^{n+s-m-m''-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!m''!} \\sum_{m'=2}^{m}\n\t\t}\n\t\t\t{\n\t\t\t\\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LIDi{EGR-Skt-CTP-EP-GRs}{EGR-Skt-CTP-EP-GRs}}\t\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{All surviving terms generated by the diagrams\nof \\fig{fig:bn-Transverse-b}.}\n\\label{fig:bn-Transverse-c}\n\\end{figure}\\end{center}\n\nNotice that, in diagram~\\ref{EGR-Skt-CTP-EP-GRs},\nthe sum over $m'$ starts from two, as the $m'=1$\ncase vanishes on account of diagrammatic \nidentity~\\eq{eq:D-ID-Bitten-hook}.\nThere is a very important point to make about\ndiagrams~\\ref{EGR-GRs-EP-TLTP} \nand~\\ref{EGR-Skt-CTP-EP-GRs}. Recall that, in\n\\sec{sec:Not:GRs}, we stated strings of gauge remainders\nformed by implicit decorations can only occur\nin diagrams with $\\Op{2}$ stubs, yet here we\nappear to have a counter-example. \nHowever, the strings of gauge remainders\nin diagrams~\\ref{EGR-GRs-EP-TLTP} \nand~\\ref{EGR-Skt-CTP-EP-GRs} are\nformed by the action of nested gauge remainders\nand \\emph{not} by the original set of implicit decorations.\nWith these structures formed, it is useful\nto sum over nestings, in the usual way,\nusing the notation $\\decGR{ \\ }{>^{q}}$, as indicated\n(remembering to compensate with a\ncombinatoric factor of $1\/q!$).\nHowever, we can not further promote these gauge\nremainders to join the implicit decorations of the diagram\nas a whole.\n\n\nWe now find a wonderful cancellation.\n\n\\begin{Icancel}\n\tDiagrams~\\ref{GRs-EGR}, \\ref{EGR-GRs-EP-TLTP}\n\tand~\\ref{EGR-Skt-CTP-EP-GRs} cancel,\n\tcourtesy of diagrammatic identity~\\eq{eq:D-ID-G}.\n\tThis is seen most clearly my making the following\n\tchanges of variables:\n\t\\begin{description}\n\t\t\\item[Diagram~\\ref{GRs-EGR}] \n\t\t\t$m \\rightarrow m+1$, $m' \\rightarrow m'+1$;\n\t\n\t\t\\item[Diagram~\\ref{EGR-GRs-EP-TLTP}] \n\t\t\t$m \\rightarrow m-m'$,\n\t\t\tand recognising that\n\t\t\t\\[\n\t\t\t\\sum_{m'=1}^{2s-1} \\sum_{m=m'}^{2s-1} = \\sum_{m=1}^{2s-1} \\sum_{m'=1}^m;\n\t\t\t\\]\n\n\t\t\\item[Diagram~\\ref{EGR-Skt-CTP-EP-GRs}]\n\t\t\t$m \\rightarrow m-m''$, $m' \\rightarrow m'-m''$\n\t\t\tand recognising that\n\t\t\t\\[\n\t\t\t\\sum_{m''=0}^{2s-3} \\sum_{m=m''+2}^{2s-1} \\sum_{m'=2+m''}^m =\n\t\t\t\\sum_{m=2}^{2s-1} \\sum_{m'=2}^m \\sum_{m''=0}^{m'-2}.\n\t\t\t\\]\t\n\t\\end{description}\n\\end{Icancel}\n\nWe have thus demonstrated that\n\\begin{equation}\n\\label{eq:Transverse}\n\tp_\\mu (-p)_\\nu \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) = 0,\n\\end{equation}\nas we set out to. The patterns of cancellations\ninvolved are the template for all cancellations\nwe will encounter when we deal with gauge\nremainders henceforth.\n\nAt first sight, \\eq{eq:Transverse+TE} follows\ndirectly from~\\eq{eq:Transverse}.\nHowever, consider the components of $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$\nwhich are factorizable. The worry is that such diagrams\nnecessarily contain at least one\neffective propagator carrying just\nthe external momentum, which contributes\na factor of $1\/p^2 + \\Op{0}$. The resolution to this\nproblem comes from breaking all such contributions\ndown into non-factorizable sub-diagrams and\nrecognizing that these sub-diagrams are\ntransverse, thereby providing\nthe necessary powers of $p$ to cancel\nthose coming from the effective propagators. \n\n\nTo\nsee this explicitly, we consider the component of\n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ which possesses two non-factorizable\nsub-diagrams. To this end, we split up \n\\begin{enumerate}\n\t\\item\tthe two external fields;\n\n\t\\item\tthe $j+s+1$ effective\n\t\t\tpropagators into three sets of $j+s-s'$, $s'$ and 1\n\t\t\teffective propagator(s) with the single effective\n\t\t\tpropagator required to join together the two factorizable\n\t\t\tsub-diagrams;\n\n\t\\item\tthe $j+2$ vertices into two sets of $j-j'$\n\t\t\tand $j'+2$ vertices; \n\n\t\\item\tthe $m$ gauge remainders\n\t\t\tinto two sets of $m-m'$ and $m'$ gauge remainders;\n\n\t\n\\end{enumerate}\n\nNoting that we sum over $j'$, $m'$ and $s'$, the combinatoric\nfactor is\n\\begin{equation}\n\\label{eq:Factorize-combins}\n\t2 (j+s+1) \\nCr{j+s}{j+s-s'} \\nCr{j+2}{j'+2} \\nCr{m}{m'}.\n\\end{equation}\nIncluding this\nwith~\\eq{eq:Factorize-combins} and combining the\nresult\nwith $\\ensuremath{\\Upsilon}_{j+s+1,j+2} \/ m!$ yields\n\\[\n\t-2 \\frac{\\ensuremath{\\Upsilon}_{j+s-s', j-j'}}{(m-m')!} \\frac{\\ensuremath{\\Upsilon}_{s',j'+2}}{m'!}.\n\\]\nIt is now convenient to shift $j \\rightarrow j+j'+2$,\n$s' \\rightarrow s' + j' +1$ and $m \\rightarrow m+m'$,\nwhich yields the contribution shown in \\fig{fig:Factorize}.\nIt is understood that the repeated vertex argument $v^{j+2}$\nis summed over.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{2ex}\n\t\t\\displaystyle\n\t\t-4 \t\\sum_{s=0}^n  \\sum_{m=0}^{2s+1} \\sum_{m'=0}^{2s+1-m} \\sum_{j=-2}^{n+s-m-m'-1} \n\t\t\t\\sum_{j'=-2}^{n+s-m-m'-j-3} \\sum_{s'=-j'-1}^{j+s+1} \n\t\\\\\n\t\t\\left[\n\t\t\t\\begin{array}{cc}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle \\frac{\\ensuremath{\\Upsilon}_{j'+s'+1,j'+2}}{m'!} \t& \n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\cd{vj+jpr+3-R}\n\t\t\t\t\t}{\\Vertex{v^{j+2},j'}}\n\t\t\t\t}{1 \\Delta^{j'+s'+1} >^{m'}}\n\t\t\t\\\\\n\t\t\t\\vspace{1ex}\n\t\t\t\t& \\hspace{-4.5em} \\mbox{\\rule{.2mm}{1cm}}\n\t\t\t\\\\\n\t\t\t\t\\displaystyle \\frac{\\ensuremath{\\Upsilon}_{j+s+1-s',j+2}}{m!} &\n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\cd{vj_+j+2-R}\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{1 \\Delta^{j+s+1-s'} >^{m}}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\end{array}\n\t\\]\n\\caption{A factorizable contribution to $\\nLDl{a}(p)$. The two sub-diagrams are understood to be non-factorizable.}\n\\label{fig:Factorize}\n\\end{figure}\\end{center}\n\nThe crucial point to recognize is that the\ncombinatoric factor for each of the sub-diagrams\nis the canonical one. Immediately we see\nthat each of these sub-diagrams is transverse.\nIt is trivial to generalize this analysis to the case\nwhere there are an arbitrary number of non-factorizable\nsub-diagrams, from which~\\eq{eq:Transverse+TE} follows.\nIn view of this, it is hardly surprising\nthat we can split  $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$\ninto Non-Factorizable (NF) components\nin an enlightening way.\nIndeed, shifting $s \\rightarrow s+s'$\nand $v^{i\\leq j+2} \\rightarrow v^i - s'$ it is apparent that\n\\begin{equation}\n\\label{nLDa-factorize}\n\t\\nLV{{\\mathcal{E}}}{n}{\\mu\\nu}{a}(p) \n\n\t=\n\t\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{a}(p)\n\t\t\t- \\sum_{n'=1}^{n-1}\n\t\t\t\\nLV{\\overline{\\mathcal{E}}}{n-n'}{\\mu \\rho}{a}(p) \n\t\t\t\\Delta^{1\\,1}_{\\rho \\sigma}(p)\n\t\t\t\\nLV{\\overline{\\mathcal{E}}}{n'}{\\sigma \\nu}{a}(p)\n\t+\\ldots,\n\\end{equation}\nwhere \n$\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{a}(p)\n\\equiv \\left. \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) \\right|_{\\mathrm{NF}}$,\nand the ellipsis builds up a full geometric series.\n\n\n\\subsubsection{The Function $\\protect \\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)$}\n\nWe define the function $\\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)$\nand two auxiliary functions as shown in \\fig{fig:nL-b}.\nNotice that the range for $s$ is the same for all three\nfunctions; only the upper limits on $m$ and $j$ differ.\n\\begin{center}\\begin{figure}\n\\begin{eqnarray*}\n\t\\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p) \\equiv\n\t\\bca{\n\t\t4 \\sum_{s=1}^n  \\sum_{r=1}^s \\sum_{m=0}^{2(s-r)} \n\t\t\\sum_{j=-2}^{n+s-2r-m-2} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!r!}\n\t}\n\t\t{\n\t\t\\decp{\n\t\t\t\t\\Tower{\\cd{CTP-E}}\n\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^m }{}\n\t}\n\\\\[2ex]\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p) \\equiv\n\t\\bca{ \n\t\t4 \\sum_{s=1}^n  \\sum_{r=1}^s \\sum_{m=0}^{2(s-r)+1} \n\t\t\\sum_{j=-2}^{n+s-2r-m-1} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!r!}\n\t}\n\t\t{\n\t\t\\qquad \\times\n\t\t\\decp{\n\t\t\t\t\\Tower{\\cd{CTP-E}}\n\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^m }{}\t\t\n\t}\n\\\\[2ex]\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b'}(p) \\equiv\n\t\\bca{ \n\t\t4 \\sum_{s=1}^n  \\sum_{r=1}^s \\sum_{m=0}^{2(s-r)+1} \n\t\t\\sum_{j=-2}^{n+s-2r-m-1} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!r!}\n\t}\n\t\t{\n\t\t\\dec{\n\t\t\t\t\\Tower{\\decp{\\cd{CTP-E}}{\\ensuremath{\\star}^r}}\n\t\t\t}{1\\Delta^{j+s+2-r}  >^m }\n\t}\n\\end{eqnarray*}\n\\caption{The definition of $\\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)$\nand two auxiliary functions.}\n\\label{fig:nL-b}\n\\end{figure}\\end{center}\n\nIn the case of $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b'}(p)$,\nwe must be careful to define precisely\nwhat it is we mean by $\\decp{\\ }{}$, in this context.\nThe basic notion is that for all diagrammatic \ncomponents enclosed by $\\decp{\\ }{}$,\nall $p$ dependence comes from the $\\Op{2}$ stub, alone;\nthose components not explicitly enclosed\nby $\\decp{\\ }{}$ can either be brought under\nits influence or not, as we begin to flesh\nout the diagrams. Thus, for the\nnon-factorizable components of $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b'}(p)$,\nall diagrammatic elements can be brought under\nthe influence of $\\decp{\\ }{}$, whereas this is\nnot the case for\nthe factorizable components. Notice, though, that all $\\ensuremath{\\star}$s\nmust decorate the part of the diagram under the influence of\n$\\decp{\\ }{}$.\nWith these points in mind, we can\nwrite\n\\begin{equation}\n\\label{eq:b'-decomp}\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b'}(p) = \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)\n\t- \\sum_{n'=1}^{n-1} \\nLV{\\mathcal{E}}{n-n'}{\\mu\\rho}{b}(p)\n\t\\Delta^{11}_{\\rho \\sigma}(p) \\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p).\n\\end{equation}\n\n\n\n\\subsubsection{The Function $\\protect \\nLV{{\\cal D}}{n}{\\mu\\nu}{c}(p)$}\n\nWe define the function $\\nLV{{\\cal D}}{n}{\\mu\\nu}{c}(p)$,\ntogether with an auxiliary function, as follows:\n\\begin{eqnarray}\n\\label{eq:nL-c}\n\t\\nLV{{\\cal D}}{n}{\\mu\\nu}{c}(p) \\equiv\n\t\\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2} \n\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\\dec{\n\t\t\\Tower{\\cd{CTP-E}}\n\t}{1 \\Delta^{j+s+2}>^m}\n\\\\\n\\label{eq:nLaux-c}\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c}(p) \\equiv\n\t\\sum_{s=1}^n \\sum_{m=1}^{2s+1} \\sum_{j=-2}^{n+s-m-1} \n\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\\dec{\n\t\t\\Tower{\\cd{CTP-E}}\n\t}{1 \\Delta^{j+s+2}>^m}.\n\\end{eqnarray}\n\nAs with $\\mathcal{E}^a$, it is useful to isolate the non-factorizable components\nof $\\mathcal{E}^c$:\n\\begin{equation}\n\\label{nLDc-factorize}\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c}(p) = \\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)\n\t- \\sum_{n'=1}^{n-1}\n\t\\nLV{\\overline{\\mathcal{E}}}{n-n'}{\\mu\\rho}{c}(p)\n\t\\Delta^{1\\,1}_{\\rho \\sigma}(p)\n\t\\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p).\n\\end{equation}\n\nWe now demonstrate that\n\\begin{equation}\n\\label{eq:nLV-c-Op2}\n\t2 \\decp{\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)}{} = \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p) + \\mathrm{remainders}.\n\\end{equation}\n\nThe presence of the $\\decp{\\ }{}$ allows us to Taylor\nexpand all diagrammatic components of $\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)$, besides\nthe $\\Op{2}$ stub, to $\\Op{0}$. The result\nof this procedure is shown in \\fig{fig:nLV-c-Properties-a}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIDi{dvj+R}{nLV-c-dvj+R}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+2} >^m}\n\t\t}\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+3,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIDi{dCTP}{nLV-c-dCTP}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1 \\Delta^{j+s+3} >^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=2}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=2}^m \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIO{\\TopVertex}{nLV-c-CTP-GRs-TE}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E-TEGRs}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+2} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=4}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=3}^{m-1} \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIO{\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m'}}\n\t\t\t\t\t}{nLV-c-TE-GRs}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+2} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The diagrams contributing to $2 \\protect \\decp{\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)}{}$.}\n\\label{fig:nLV-c-Properties-a}\n\\end{figure}\\end{center}\n\nThere are a number of comments to make about the\ndiagrams of \\fig{fig:nLV-c-Properties-a}. First,\neach diagram\nhas a socket that we demand is filled by the\nexternal field (we will see shortly why\nwe do not fill the socket).  \nIn diagram~\\ref{nLV-c-CTP-GRs-TE} \nit is understood that, given that the final\ngauge remainder in the string is the one\nthat bites the vertex,\nthe first gauge remainder in the string\ncannot be filled by the external field,\nas this generates an illegal diagram\n(see \\sec{sec:Not:GRs}).\n\nSecondly, the factor of two\nthat diagrams~\\ref{nLV-c-dvj+R}\nand~\\ref{nLV-c-dCTP}\nhave acquired relative to the parent diagram\ncomes from using charge conjugation\\ to combine the pull back like\nmomentum derivative with the push forward like derivative\nmomentum.\nThere is no such factor of two in  diagrams~\\ref{nLV-c-CTP-GRs-TE}\nand~\\ref{nLV-c-TE-GRs} since are no momentum\nderivatives.\n\nThirdly, the fact that the diagrams are under the influence\nof $\\decp{\\ }{}$ leads to some novelties. Notice\nthat in diagram~\\ref{nLV-c-TE-GRs},\nthe explicitly drawn gauge remainder structure\nis required to\nhave at least three gauge remainders. If there\nwere only a single gauge remainder, whose socket\nis necessarily filled by an  $A^1$ sector field, then\nthe diagram would vanish as a consequence of charge conjugation\\ \ninvariance---nothing\nunusual there. But why is it not possible for there\nto be only two gauge remainders? The reason\nis that such a structure which, crucially, is\ndecorated by a $\\ensuremath{\\sqcup}$, would be compelled\nto be attached to the rest of the diagram by an \neffective propagator carrying just $p$. \nSuch a term\nis forbidden by\nthe influence of $\\decp{\\ }{}$.\n\nTo make progress, we attach two effective propagators\nto the classical, two-point vertex of diagram~\\ref{nLV-c-dCTP}.\nAs a consequence of the  $\\decp{\\ }{}$, these\ntwo effective propagators must carry the same\n(loop) momentum and so we can utilise~\\eq{eq:EP-dCTP-EP}.\nProcessing\nthe gauge remainders, we arrive at the diagrams\nof \\fig{fig:nLV-c-Properties-b}, where we have \nused charge conjugation\\ invariance to collect\nterms together, as described under~\\eq{eq:EP-dCTP-EP}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIO{\\cdeps{dEP}}{nLV-c-dEP}\n\t\t\t\t\\\\\n\t\t\t\t\t\\InvTower{\\cd{CTP-E}}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1} >^m}\n\t\t}\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s-1}  \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIDi{Combo-Bite-Vertex-nn}{nLV-c-Combo-Bite-Vertex-nn}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1} >^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=3}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^{m-1} \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIO{\n\t\t\t\t\t\\cd{CTP-E} \\cd{Combo-GR-GRs} \\hspace{1em}\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{nLV-c-Combo-GR-GRs}\n\t\t\t}{1\\Delta^{j+s+1} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=1}^{m} \\nCr{m}{m'} m'\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LIDi{CTP-E-GRs-Combo-b}{nLV-c-CTP-E-GRs-Combo-b}\n\t\t\t\t\t\\hspace{1.5em}\n\t\t\t\t\t+\n\t\t\t\t\t\\LIDi{CTP-E-GRs-Combo-c}{nLV-c-CTP-E-GRs-Combo-c}\n\t\t\t\t}\n\t\t\t}{1\\Delta^{j+s+1} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Diagrams spawned by \\ref{nLV-c-dCTP}.}\n\\label{fig:nLV-c-Properties-b}\n\\end{figure}\\end{center}\n\nDiagrams~\\ref{nLV-c-CTP-E-GRs-Combo-b} and~\\ref{nLV-c-CTP-E-GRs-Combo-c}\nare related, arising from\nisolating particular components of a single diagram. In the\nparent diagram from which~\\ref{nLV-c-CTP-E-GRs-Combo-b} and~\\ref{nLV-c-CTP-E-GRs-Combo-c}\nare derived, the classical, two-point vertex is decorated\nby $m'$ $>$s. The $\\rhd$ at the end of the $\\ensuremath{\\star}$\ncan be contracted into any of these $>$s or the\nsocket on the classical, two-point vertex. However,\nit is useful to isolate the component in which\nthe $\\rhd$ is contracted into the $>$\nat the top of the string {\\it i.e.}\\  diagram~\\ref{nLV-c-CTP-E-GRs-Combo-c}. \nIn this case alone,\nwe can use diagrammatic identity~\\eq{eq:GR-relation}.\nWe recognize the resulting term as the $r=1$ component\nof $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)$, demonstrating\nthat we are well on the way to proving~\\eq{eq:nLV-c-Op2}.\n\nIn anticipation of what is to follow,\nwe note that diagrams~\\ref{nLV-c-dvj+R}\nand~\\ref{nLV-c-dEP} naturally form part of a set\nin which all diagrammatic elements\nare struck by a momentum derivative.\nTo find a further member of the\nset, we process diagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\nby isolating the component possessing a classical,\ntwo-point vertex and proceeding as usual.\nThe reason we have held back with this step,\nrather than using our experience of\ndealing with such terms to jump straight\nto the result of iterating the diagrammatic\nprocedure until exhaustion, is because\nof the presence of $\\decp{\\ }{}$.\n\nThe loose end of the $\\ensuremath{\\star}$\ncan attach to any available socket in the\ndiagram, besides its own. Suppose that it attaches to the\nvertex with argument $v^{j_+}$, such that\nthe $\\ensuremath{\\star}$ bites its own base. In this\ncase, the vertex does not possess a (classical)\ntwo-point component due to the influence of\n$\\decp{\\ }{}$. Given that the vertex\nwith argument $v^{j_+}$ must be at least\nthree point, how do we proceed?\nThe point is that, since the $\\ensuremath{\\star}$ bites\nitself, the field on the vertex to \nwhich the $\\ensuremath{\\star}$ attaches carries zero momentum.\nFurthermore, charge conjugation\\ invariance forces\nthis field to be in the $A^i$\nsector (group theory considerations then\nforce it to be in the $A^1$ sector). \nThus, we \\emph{effectively} have a vertex\nwhich has been Taylor expanded to zeroth\norder in the momentum of a decorative $A^1$\nfield, and so we can repeat the above analysis!\nThe difference is that, rather than the socket\nassociated with the Taylor expansion being filled\nby an external field, it is now filled by a $\\ensuremath{\\star}$\nwhich bites its own base. \n\nPutting this diagram to one side for the moment,\nconsider those components of diagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\nthat we \\emph{can} process in the usual way.\nCancelling terms where possible, we\nare left with the following diagrams:\n\\begin{enumerate}\n\t\\item\tthe component of the nested version of\n\t\t\tdiagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\n\t\t\tin which the $\\ensuremath{\\star}$ bites a socket on a classical,\n\t\t\ttwo-point vertex, the \\emph{other} socket of which\n\t\t\tis filled by the other end of the $\\ensuremath{\\star}$;\n\t\\label{it:nLV-c-Combo-Bite-Vertex}\n\n\t\\item\tnested versions of\n\t\t\tdiagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\n\t\t\twhere the bitten field on the\n\t\t\tvertex with argument $v^{j_+}$ carries\n\t\t\tzero momentum;\n\t\\label{it:nLV-c-combo-kv}\n\n\t\\item\tremainders from cancellations via diagrammatic identity~\\eq{eq:D-ID-G};\n\t\t\tthe cancellation no longer being exact, as some of the diagrams necessary for\n\t\t\tthe cancellation are forbidden when under the influence of\n\t\t\t$\\decp{\\ }{}$;\n\t\\label{it:nLV-c-combo-GR}\n\n\t\\item\tthe component of the nested version of\n\t\t\tdiagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\n\t\t\tin which the classical, two-point vertex\n\t\t\tbitten by the $\\ensuremath{\\star}$ is attached\n\t\t\tto the gauge remainder structure or socket\n\t\t\tdecorating the $\\Op{2}$ stub.\n\t\\label{it:nLV-c-TLTP-EP-GRs}\n\\end{enumerate}\n\nWe begin by analysing the diagrams of item~\\ref{it:nLV-c-Combo-Bite-Vertex},\nwhich we have drawn in \\fig{fig:nLV-c-Properties-c}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t-4\n\t\\bca{\n\t\t\\sum_{s=1}^n  \\sum_{m=2}^{2s-1}  \\sum_{j=-2}^{n+s-m-3} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=1}^{m-1} \\nCr{m}{m'}\n\t}\n\t\t{\n\t\t\\decp{\n\t\t\t\\LIO{\n\t\t\t\t\\cd{CTP-E} \\cd{TLTP-Combo-GR}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{nLV-c-TLTP-Combo-GR}\n\t\t}{1\\Delta^{j+s+2} >^{m-m'}}\n\t}\n\t\\]\n\\caption{One of the surviving terms produced by processing\ndiagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}.}\n\\label{fig:nLV-c-Properties-c}\n\\end{figure}\\end{center}\n\nTo proceed, we decompose the $\\ensuremath{\\star}$,  according to~\\eq{eq:Combo}.\n The first\nterm arising from this decomposition can be processed\nusing the effective propagator relation, whereas the\nsecond term vanishes, courtesy of diagrammatic \nidentity~\\eq{eq:GR-TLTP}. The result of\ncombining\nthe surviving terms with diagrams~\\ref{nLV-c-dvj+R}\nand~\\ref{nLV-c-dEP} is shown in \\fig{fig:nLV-c-Properties-d}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\left[\n\t\t\t\t\t\\Tower{\\LIDi{CTP-E}{nLV-c-MomDer}}\n\t\t\t\t\t\\cdeps{MomDer} \\hspace{-1.8em}\n\t\t\t\t\\right]\n\t\t\t\t\\hspace{1em}\n\t\t\t}{1 \\Delta^{j+s+2} >^{m}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=1}^{m} \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIO{\n\t\t\t\t\t\\cd{CTP-E-dGRs}\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{nLV-c-dGRs}\n\t\t\t}{1 \\Delta^{j+s+2} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=4}^{2s-1}  \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=3}^{m-1} \\nCr{m}{m'} m'(m'-1)\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIO{\n\t\t\t\t\t\\cd{CTP-E} \\cd{GRs-Full-dGR} \\hspace{2ex}\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{nLV-c-GRs-Full-dGR}\n\t\t\t}{1 \\Delta^{j+s+2} >^{m-m'}}\n\t\t}\t\n\t\\end{array}\n\t\\]\n\\caption{The result of\ncombining diagrams~\\ref{nLV-c-dvj+R}\nand~\\ref{nLV-c-dEP} with the surviving components\nof diagram~\\ref{nLV-c-TLTP-Combo-GR}.}\n\\label{fig:nLV-c-Properties-d}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{nLV-c-MomDer} contains a sum  of\ntotal momentum derivatives with respect to\\\nthe various loop momenta carried by the fully fleshed\nout constituent diagrams~\\cite{mgierg2}. Thus, diagram~\\ref{nLV-c-MomDer}\nvanishes as a consequence of the \ndimensional pre-regularisation. We comment on this further\nin the conclusions.\n\n\\begin{Icancel\n\nDiagram~\\ref{nLV-c-GRs-Full-dGR} exactly cancels\ndiagram~\\ref{nLV-c-TE-GRs}. \nThis follows from~\\eq{eq:D-ID-dGRk-GT-ring}\nupon recognizing\nthat diagrammatic \nidentity~\\eq{eq:GR-relation}\nallows\nthe momentum derivative\nof diagram~\\ref{nLV-c-GRs-Full-dGR} to be transfered from the $>$ to the\n$\\rhd$, at the expense of a minus sign. \n\n\\end{Icancel}\n\nNext, consider the diagrams of item~\\ref{it:nLV-c-combo-kv}, above.\nFrom the earlier comments about the\nun-nested diagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\nit is clear that the diagrams of item~\\ref{it:nLV-c-combo-kv}\nsimply yield versions of\ndiagram~\\ref{nLV-c-dvj+R} and~\\ref{nLV-c-dCTP}\nbut where the socket associated with the momentum\nderivative is filled by an external field of a sub-diagram\ncontaining a $\\ensuremath{\\star}$ (this is why we left the socket\nunfilled in the first place, rather than decorating\nit with the external field). The similar generalization\nof diagram~\\ref{nLV-c-TE-GRs} arises from the\ndiagrams of item~\\ref{it:nLV-c-combo-GR}, above.\nWe can manipulate the new version of\ndiagram~\\ref{nLV-c-dCTP} just as we manipulated\nthe original version and precisely the same pattern\nof terms and cancellations will be generated. In particular,\nwe will generate versions of diagram~\\ref{nLV-c-CTP-E-GRs-Combo-c}.\nWe recognise these as the $r>1$ components of\n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)$. Therefore,\nwe have succeeded in demonstrating~\\eq{eq:nLV-c-Op2};\nnow we find an explicit expression for the remainders.\n\n\n\nThe remainders can be thought of as a set of primitive\ndiagrams, together with their analogues possessing\nimplicit $\\ensuremath{\\star}$s. The primitive diagrams are:\nthe diagrams of item~\\ref{it:nLV-c-TLTP-EP-GRs}\nand diagrams~\\ref{nLV-c-CTP-GRs-TE},\n\\ref{nLV-c-CTP-E-GRs-Combo-b} and~\\ref{nLV-c-dGRs}.\n\n \nDiagrams of item~\\ref{it:nLV-c-TLTP-EP-GRs}\ncan be redrawn  via diagrammatic identities~\\eq{eq:D-ID-Op2-G},\nupon which\none of the resultant diagrams\nexactly cancels \ndiagram~\\ref{nLV-c-CTP-E-GRs-Combo-b}.\nDiagram~\\ref{nLV-c-CTP-GRs-TE} can be\nredrawn using diagrammatic identity~\\eq{eq:D-ID-dGRk-GT-ring}.\nThe resultant diagram naturally combines\nwith diagram~\\ref{nLV-c-dGRs} upon recognizing\nthat\n\\begin{equation}\n\\label{eq:TLTP-Combo}\n\t\\cd{CTP-Combo} = \\cd{CTP-EP-dGR} = \\cd{dGR} + \\cd{GR-dGRk-GR}.\n\\end{equation}\nWe thus find that\n\\begin{equation}\n\\label{eq:nLV-c-Op2-b}\n\t2 \\decp{\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)}{} = \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p) \n\t+ 2 \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c'}(p),\n\\end{equation}\nwhere $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c'}(p)$\nis given in \\fig{fig:nLV-c-Properties-e}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c'}(p) \\equiv\n\t\\\\\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{r=1}^{s-1} \\sum_{m=1}^{2(s-r)+1}\n\t\t\t\\sum_{j=-2}^{n+s-m-2r-1} \\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1!)}\n\t\t\t\\sum_{m'=1}^{m} \\nCr{m}{m'} \\sum_{m''=1}^{m'} \\nCr{m'}{m''}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIDi{CTP-E-GRs-Combo-GRs-CTP-EP-b}{nLV-c-CTP-E-GRs-Combo-GRs-CTP-EP}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)+1}\n\t\t\t\\sum_{j=-2}^{n+s-m-2r-1} \\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1!)}\n\t\t\t\\sum_{m'=0}^{m} \\nCr{m}{m'} \n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIDi{CTP-E-GRs-Combo-GR}{nLV-c-CTP-E-GRs-Combo-GR}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)+1}\n\t\t\t\\sum_{j=-2}^{n+s-m-2r-1} \\frac{\\ensuremath{\\Upsilon}_{j+s+3-r,j+2}}{m!(r-1!)}\n\t\t\t\\sum_{m'=0}^{m} \\nCr{m}{m'} \\sum_{m''=0}^{m'} \\nCr{m'}{m''}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIDi{CTP-E-GRs-Combo-CTP-GRs}{nLV-c-CTP-E-GRs-Combo-CTP-GRs}\n\t\t\t\t\\hspace{1.5em}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+3-r} \\ensuremath{\\star}^r >^{m-m'}}\t\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The expression for $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c'}(p)$.}\n\\label{fig:nLV-c-Properties-e}\n\\end{figure}\\end{center}\n\n\n\n\n\\section{Review}\n\\label{sec:Review}\n\n\\subsection{Elements of $SU(N|N)$ Gauge Theory}\n\\label{sec:elements}\n\nThroughout this paper, we work in Euclidean dimension, D.\nWe regularize $SU(N)$ Yang-Mills by embedding\nit in spontaneously broken $SU(N|N)$ Yang-Mills,\nwhich is itself regularized by covariant higher derivatives~\\cite{SU(N|N)}. \nThe supergauge field, ${\\cal A}_\\mu$, is valued in\nthe Lie superalgebra and, using the defining representation,\ncan be written as a Hermitian supertraceless supermatrix:\n\\[\n\t{\\cal A}_\\mu = \n\t\\left(\n\t\t\\begin{array}{cc}\n\t\t\tA_\\mu^1 \t& B_\\mu \n\t\t\\\\\n\t\t\t\\bar{B}_\\mu & A_\\mu^2\n\t\t\\end{array} \n\t\\right) + {\\cal A}_\\mu^0 \\hbox{1\\kern-.8mm l}.\n\\]\nHere, $A^1_\\mu(x)\\equiv A^1_{a\\mu}\\tau^a_1$ is the\nphysical $SU(N)$ gauge field, $\\tau^a_1$ being the $SU(N)$\ngenerators orthonormalized to\n${\\mathrm{tr}}(\\tau^a_1\\tau^b_1)=\\delta^{ab}\/2$, while $A^2_\\mu(x)\\equiv\nA^2_{a\\mu}\\tau^a_2$ is a second unphysical $SU(N)$ gauge field.\nWhen labelling {\\it e.g.}\\  vertex coefficient functions,\nwe often abbreviate $A^{1,2}$ to just 1,2.\nThe $B$ fields are fermionic gauge fields which will gain a mass\nof order $\\Lambda$ from the spontaneous symmetry breaking; they play the\nrole of gauge invariant Pauli-Villars (PV) fields, furnishing the\nnecessary extra regularization to supplement the covariant\nhigher derivatives. In order\nto unambiguously define contributions which are finite\nonly by virtue of the PV regularization, a pre-regulator must\nbe used in $D=4$~\\cite{SU(N|N)}. For the time being, we will use\ndimensional regularization, emphasising that\nthis makes sense\nnon-perturbatively, since \nit is not being used to renormalize the theory, but rather\nas a prescription for discarding surface terms in loop\nintegrals~\\cite{SU(N|N)}.\n\n\n\nThe theory is\nsubject to the local invariance:\n\\begin{equation}\n\\label{Agauged}\n\\delta{\\cal A}_\\mu = [\\nabla_\\mu,\\Omega(x)] +\\lambda_\\mu(x) \\hbox{1\\kern-.8mm l}.\n\\end{equation}\nThe first term, in which $\\nabla_\\mu = \\partial_\\mu -i{\\cal A}_\\mu$, \ngenerates supergauge transformations. Note that the coupling, $g$,\nwhich we now take to represent the coupling for all renormalization\nschemes implictly defined by our approach,\nhas been scaled out of this definition. It is worth doing\nthis: since we do not gauge fix, the exact preservation of~\\eq{Agauged}\nmeans that none of the fields suffer wavefunction\nrenormalization, even in the broken phase~\\cite{aprop}.\n\nThe second term in~\\eq{Agauged} divides out the centre of the algebra.\nThis `no ${\\cal A}^0$ shift symmetry' ensures that nothing depends on ${\\cal A}^0$\nand that ${\\cal A}^0$ has no degrees of freedom. We adopt a\nprescription whereby we can effectively ignore the field ${\\cal A}^0$,  altogether, \nusing it  to map us into a particular diagrammatic\npicture~\\cite{Thesis,mgierg1}.\n\nThe spontaneous symmetry breaking is carried by a superscalar\nfield\n\\[\n{\\cal C} =\n\t\\left(\n\t\t\\begin{array}{cc}\n\t\t\tC^1\t\t& D\n\t\t\\\\\n\t\t\t\\bar{D}\t& C^2\n\t\t\\end{array}\n\t\\right),\n\\]\nwhich transforms homogeneously:\n\\begin{equation}\n\\label{Cgauged}\n\\delta{\\cal C} = -i\\,[{\\cal C},\\Omega].\n\\end{equation}\n\n\n\nIt can be shown that, at the classical level, the spontaneous\nsymmetry\nbreaking scale (effectively the mass of $B$) tracks the covariant\nhigher derivative effective cutoff scale, $\\Lambda$, if ${\\cal C}$ is\nmade dimensionless (by using powers of $\\Lambda$) and  $\\hat{S}$ has\nthe minimum of its effective potential at:\n\\begin{equation}\n\\label{sigma}\n\\sigma \\equiv \\pmatrix{\\hbox{1\\kern-.8mm l} & 0\\cr 0 & -\\hbox{1\\kern-.8mm l}}.\n\\end{equation}\nIn this case the classical action $S_0$ also has a minimum \nat~\\eq{sigma}. At the quantum level this can be imposed as a\nconstraint on $S$ by taking $\\expectation{C} = \\sigma $ as a renormalization\ncondition. This ensures that the Wilsonian effective action\ndoes not possess any one-point vertices, which can be\ntranslated into a constraint on\n$\\hat{S}$~\\cite{aprop,Thesis}. In the broken phase, $D$ is\na super-Goldstone mode (eaten by $B$ in the unitary gauge) whilst the\n$C^i$ are Higgs bosons and can be given a running mass of order\n$\\Lambda$~\\cite{ym,SU(N|N),aprop}. Working in our manifestly\ngauge invariant formalism, $B$ and $D$ gauge transform into each\nother; in recognition of this, we define the fields\n\\numparts\n\\begin{eqnarray}\n\\label{eq:F}\n\tF_M & = & (B_\\mu, D),\n\\\\\n\\label{eq:Fbar}\n\t\\bar{F}_N & = & (\\bar{B}_\\nu, -\\bar{D}),\n\\end{eqnarray}\n\\endnumparts\nwhere $M$, $N$ are\nfive-indices~\\cite{Thesis,mgierg1}.\\footnote{The summation\nconvention for these indices is that we take each product of\ncomponents to contribute with unit weight.} \n\nThe couplings $g$ and $\\alpha$ (recall~\\eq{eq:alpha-defn})\nare defined through their\nrenormalization conditions:\n\\begin{eqnarray}\n\\label{defg}\n\tS[{\\cal A}=A^1, {\\cal C}=\\sigma]\t& =\t& {1\\over2g^2}\\,\\mathrm{str}\\!\\int\\!\\!d^D\\!x\\,\n\t\t\t\t\t\t\t\t\t\\left(F^1_{\\mu\\nu}\\right)^2+\\cdots,\n\\\\\t\n\\label{defg2}\n\tS[{\\cal A}=A^2, {\\cal C}=\\sigma] \t& =\t& {1\\over2 \\alpha g^2}\\,\\mathrm{str}\\!\\int\\!\\!d^D\\!x\\,\n\t\t\t\t\t\t\t\t\t\\left(F^2_{\\mu\\nu}\\right)^2+\\cdots,\n\\end{eqnarray}\nwhere the ellipses stand for higher dimension operators and the\nignored vacuum energy. The field strength tensors in the $A^1$ and\n$A^2$ sectors, $F^1_{\\mu\\nu}$ and $F^2_{\\mu\\nu}$, should really\nbe embedded in the top left \/ bottom right entries of a supermatrix,\nin order for the supertraces in~\\eqs{defg}{defg2} \nto make sense. We will frequently employ\nthis minor abuse of notation, for convenience.\n\n\n\n\\subsection{Diagrammatics for the Flow Equation}\n\n\\subsubsection{The Exact Flow Equation}\n\\label{sec:Flow}\n\nThe diagrammatic representation of the flow\nequation is shown in \\fig{fig:Flow}~\\cite{Thesis,mgierg1}.\n(For a comprehensive description of the diagrammatics see~\\cite{Thesis,mgierg1}.)\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray*}\n\t\\displaystyle\n\t-\\Lambda \\partial_\\Lambda \n\t\\dec{\n\t\t\\cd{Vertex-S}\n\t}{\\{f\\}}\n\t& = & a_0[S,\\Sigma_g]^{\\{f\\}} - a_1[\\Sigma_g]^{\\{f\\}}\n\\\\\n\t& = &\n\t\\displaystyle\n\t\\frac{1}{2}\n\t\\dec{\n\t\t\\cd{Dumbbell-S-Sigma_g} - \\cd{Padlock-Sigma_g} - \\cd{WBT-Sigma_g}\n\t}{\\{f\\}}\n\t\\end{eqnarray*}\n\\caption{The diagrammatic form of the flow equation.}\n\\label{fig:Flow}\n\\end{figure}\\end{center}\n\nThe left-hand side\\ depicts the flow of all independent Wilsonian effective action\nvertex \\emph{coefficient functions}, \nwhich correspond to the set of broken phase\nfields, $\\{f\\}$. Each coefficient function has associated\nwith it an implied supertrace structure (and symmetry factor which,\nas one would want, does not appear in the diagrammatics).\nFor example,\n\\begin{equation}\n\\label{eq:Vertex-C1C1}\n\t\\dec{\n\t\t\\cd{Vertex-S}\n\t}{C^1C^1}\n\\end{equation}\nrepresents both the coefficient functions $S^{C^1 C^1}$ and\n$S^{C^1,C^1}$ which, respectively, are associated with the\nsupertrace structures $\\mathrm{str} C^1 C^1$ and $\\mathrm{str} C^1 \\mathrm{str} C^1$.\n\nThe objects on the right-hand side\\ of \\fig{fig:Flow}\nhave two different types of component. The lobes\nrepresent vertices of action functionals,\nwhere $\\Sigma_g \\equiv g^2S - 2 \\hat{S}$ (recall that $\\hat{S}$ is the seed action). \nThe object attaching\nto the various lobes, \\ensuremath{\\stackrel{\\bullet}{\\mbox{\\rule{1cm}{.2mm}}}},  is\nthe sum over vertices of the covariantized ERG kernels~\\cite{ymi,aprop}\nand, like the action vertices, can be decorated by fields belonging to $\\{f\\}$.\nThe appearance of the symbol $\\scriptstyle \\bullet$ is not accidental, \nmeaning $-\\Lambda \\partial_\\Lambda|_\\alpha$, as in~\\eq{eq:dot}.\nThe dumbbell-like\nterm, which corresponds to the bilinear functional $a_0$, is referred\nto as the classical term. The next two terms, which are both\ngenerated by $a_1$, are referred to as quantum terms. The second\nof these contains a kernel which `bites its own tail'. This\ndiagram is not properly \nUV regularized by the $SU(N|N)$ regularization \nand, in the past, it has been argued that\nit can and should be discarded, through an appropriate\nconstraint on the covariantization~\\cite{mgierg1,ym,ymi,aprop}.\\footnote{\nThese diagrams are artefacts of the flow equation. The $SU(N)$\ngauge theory \\emph{is} fully regularized by the $SU(N|N)$ scheme. However,\nregularization of the flow equation does not trivially follow from\nthe regularization of the underlying theory.}\nHere, though, we will keep these diagrams for as\nlong as possible: as recognized in~\\cite{Thesis},\nin any calculation $\\beta$ function coefficients, \nthe explicit ultraviolet divergences in kernel-biting-their-tail\ndiagrams, which can be dimensionally regularized,\nshould be cancelled by divergences hidden in other\nterms. In this paper, we demonstrate that this\nis indeed the case: all explicit dependence\nof $\\beta_n$ on kernel-biting-their-tail\ndiagrams cancels out. Nonetheless, there is an implicit\ndependence left behind and it seems inevitable that, to further\nproceed, the covariantization \nmust be suitably constrained,\nafter all.\n\n\n\nIt is worth drawing attention to the fact that dimensional\nregularization is thus being used for two entirely independent\npurposes. On the one hand, it is used as a temporary\nmeasure, at intermediate stages of our calculations, \nto properly define\ndiagrams possessing a kernel which bites its own tail. \nOn the\nother hand, it is being employed as a pre-regulator. \nAs we will argue in the conclusions, there is evidence\nto suggest the existence of a purely diagrammatic\npre-regulator, which would make sense in $D=4$.\n\n\nEmbedded within the diagrammatic rules is a prescription for evaluating the\ngroup theory factors. \nSuppose that we wish to focus on the flow of a particular\nvertex coefficient function, which necessarily has a unique\nsupertrace structure. \nFor example, we might be interested in just\nthe $S^{C^1 C^1}$ component of~\\eq{eq:Vertex-C1C1}.\n\nOn the right-hand side\\ of the flow equation, we must\nfocus on the components of each diagram\nwith precisely the same (implied) \nsupertrace structure as the left-hand side,\nnoting that the kernel, like the vertices,\nhas multi-supertrace contributions (for more\ndetails see~\\cite{Thesis,mgierg1}).\nIn this more explicit diagrammatic picture,\nthe kernel is to be considered a double\nsided object.\nThus, whilst the dumbbell like term of \\fig{fig:Flow}\nhas at least one associated supertrace, the next diagram\nhas  at least two, on a account of the loop\n(this is strictly true only in the\ncase that kernel attaches to fields on the same\nsupertrace). If a closed\ncircuit formed by a kernel is devoid\nof fields then\nit contributes \na factor of $\\pm N$, depending on\nthe flavours of the fields to which the kernel forming\nthe loop attaches. This is most easily appreciated by\ndefining the projectors\n\\begin{equation}\n\\label{eq:Projectors}\n\t\\sigma_{+} \\equiv\n\t\\left(\n\t\t\\begin{array}{cc}\n\t\t\t\\hbox{1\\kern-.8mm l}\t&\t0\n\t\t\\\\\n\t\t\t\t0\t&\t0\n\t\t\\end{array}\n\t\\right), \\ \\ \n\t\t\\sigma_{-} \\equiv\n\t\\left(\n\t\t\\begin{array}{cc}\n\t\t\t0\t&\t0\n\t\t\\\\\n\t\t\t0\t&\t\\hbox{1\\kern-.8mm l}\n\t\t\\end{array}\n\t\\right)\n\\end{equation}\nand noting that $\\mathrm{str} \\sigma_\\pm = \\pm N$. \nIn the counterclockwise sense, a $\\sigma_+$\ncan always be inserted for free after an $A^1$, $C^1$ or $\\bar{F}$,\nwhereas a $\\sigma_-$\ncan always be inserted for free after an $A^2$, $C^2$ or $F$. \n\n\nThe rules thus described receive $1\/N$ corrections in \nthe $A^1$ and $A^2$ sectors. If a kernel\nattaches to an $A^1$ or $A^2$, it comprises a direct\nattachment and an indirect attachment, as shown \nin \\fig{fig:Attach} (see~\\cite{Thesis,mgierg1} for more detail).\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t\\cd{Direct} \\rightarrow \\left. \\cd{Direct} \\right|_{\\mbox{direct}} + \\frac{1}{N} \\left[ \\cd{Indirect-2} - \\cd{Indirect-1} \\right]\n\t\\]\n\\caption{The $1\/N$ corrections to the group theory factors.}\n\\label{fig:Attach}\n\\end{figure}\\end{center} \n\nWe can thus consider the diagram on the left-hand side\\ as having been unpackaged,\nto give the terms on the right-hand side. The dotted lines in the diagrams with indirect\nattachments serve to remind us where the loose end of the kernel attaches\nin the parent diagram.\n\n\n\\subsubsection{The Ward Identities}\n\\label{sec:WIDs}\n\nAll vertices, whether they belong to either\nof the actions or to the covariantized kernels\nare subject to Ward identities which, due to the\nmanifest gauge invariance, take a \nparticularly simple form:\n\\begin{equation}\n\\label{eq:WID-A}\n\t\\cd{WID-contract} = \\cd{WID-PF} + \\cd{WID-PFb} - \\cd{WID-PB} - \\cd{WID-PBb} + \\cdots\n\\end{equation}\n\n\nEquation~\\eq{eq:WID-A} is the first primary diagrammatic identity\nof the first type. On the left-hand side, we contract a vertex with the momentum of\nthe field which carries $p$. This field---which we will\ncall the active field---can be either\n$A^1_\\rho$, $A^2_\\rho$, $F_R$ or $\\bar{F}_R$.\nIn the first two cases, the triangle $\\rhd$ represents\n$p_\\rho$ whereas, in the latter two cases, it represents\n$p_R \\equiv (p_\\rho,2)$. (Given that we often sum over\nall possible fields, we can take the Feynman rule for\n$\\rhd$ in the $C$-sector to be null.)\nIn all cases,  $\\rhd$ is independent of $\\Lambda$ and $\\alpha$, which is\nencoded in the  second of the primary diagrammatic identities of the \nfirst type:\n\\numparts\n\\begin{eqnarray}\n\\label{eq:LdL-GRk-Pert-a}\n\t\\hspace{0.8em} \\stackrel{\\bullet}{\\rhd} & =  & 0,\n\\\\\n\\label{eq:dalpha-GRk-a}\n\t\\begin{array}{c}\n\t\t\\Dal\n\t\\\\[-1.5ex]\n\t\t\\rhd\n\t\\end{array} & = & 0,\n\\end{eqnarray}\n\\endnumparts\nwhere $ \\Dal\\ \\equiv \\partial \/ \\partial{\\alpha}$.\n\nOn the right-hand side\\ of~\\eq{eq:WID-A}, we push the contracted momentum forward onto \nthe field which directly follows the active field, in the counterclockwise\nsense, and pull back (with a minus sign) onto\nthe field which directly precedes the active field. \nSince our diagrammatics is permutation symmetric, the struck field---which\nwe will call the target field---can\nbe either $X$, $Y$ or any of the un-drawn fields represented\nby the ellipsis.\nAny field(s) besides the active field and the\ntarget field will be called spectators.\nNote that we can take\n$X$ and \/ or $Y$ to represent the end of a kernel.\nIn this case, the struck field is determined to be unambiguously\non one side of the (double sided) kernel; \nthe contributions\nin which the struck field is on the other side are included\nin the ellipsis. \nThis highlights the point that allowing the active\nfield to strike another field necessarily involves a partial\nspecification of the supertrace structure: it must be the case that\nthe struck field either directly followed or preceded the active\nfield. In turn, this means that the Feynman rule for particular\nchoices of the active and target fields can be zero. For example,\nan $F$ can follow, but never precede an $A^1_\\mu$, and so the \npull back of an $A^1_\\mu$ onto an $F$ should be assigned a value\nof zero. \nThe momentum routing follows in an obvious manner: for example,\nin the first diagram on the right-hand side, momenta $q+p$ and $r$ now flow into\nthe vertex. In the case that the active field is fermionic,\nthe field pushed forward \/ pulled back onto is transformed\ninto its opposite statistic partner. \nThere are some signs \nassociated with this in the $C$ and $D$-sectors, \nfor which we refer the reader to~\\cite{mgierg1,Thesis}. \n\nThe half arrow which terminates the pushed forward \/ pulled back\nactive field is of no significance and can go on either side\nof the active field line. It is necessary to\nkeep the active field line---even though the active field\nis no longer part of the vertex---in order that\nwe can unambiguously deduce flavour changes \nand momentum routing, without reference to the parent diagram.\n\n\nWe illustrate~\\eq{eq:WID-A} by considering contracting\n$\\rhd$ into the Wilsonian the effective action\ntwo-point vertex:\n\\begin{equation}\n\\label{eq:GR-TP}\n\t\\cd{GR-TP} =  \\cd{GR-TP-PF} - \\cd{GR-TP-PB}.\n\\end{equation}\nGiven that $\\rhd$ is null in the $C^i$ sector,\nthe fields decorating the two-point\nvertex on the right-hand side\\ can be either both $A^i$s\nor both fermionic. In the former case, \n\\eq{eq:GR-TP} reads:\n\\[\n\tp_\\mu S^{A^i A^i}_{\\mu \\ \\, \\nu}(p) = S^{A^i}_{\\nu}(0) - S^{A^i}_{\\nu}(0) = 0\n\\]\nwhere we note that $S^{A^i}_{\\nu}$ is in fact zero by itself,\nas follows by both Lorentz invariance and gauge invariance.\nIn the latter case, \\eq{eq:GR-TP} reads:\n\\[\n\tp_M S^{\\bar{F} \\; F}_{M N}(p) = \\left[S^{C^2}(0) - S^{C^1}(0)\\right] \\delta_{5N},\n\\]\nwhere we have used~\\eq{eq:F} and have discarded\ncontributions which go like $S^{A^i}_{\\nu}(0)$.\nHowever, the $S^{C^i}(0)$ must vanish. This\nfollows from demanding\nthat the minimum of the superhiggs potential is\nnot shifted by quantum corrections~\\cite{aprop}.\nTherefore, we arrive at the diagrammatic identity\n\\begin{equation}\n\\label{eq:D-ID-GR-TP-A}\n\t\\cd{GR-TP} = 0.\n\\end{equation}\n\n\n\n\\subsubsection{Taylor Expansion of Vertices}\n\\label{sec:Taylor}\n\nFor the formalism to be properly defined,\nit must be the case that all vertices\nare Taylor expandable to all orders\nin momenta~\\cite{ym,ymi,ymii}.\nConsider a vertex which is part\nof a complete diagram, decorated by some set of internal\nfields and by a single external $A^1$ (or $A^2$).\nThe diagrammatic representation for the zeroth order expansion\nin the momentum of the external field is all that is required\nfor this paper~\\cite{Thesis,mgierg1}:\n\\begin{equation}\n\\label{eq:Taylor-A}\n\t\\cd{Taylor-Parent} = \\cdeps{Taylor-PFa} + \\cdeps{Taylor-PFb} - \\cdeps{Taylor-PBa} - \\cdeps{Taylor-PBb} +\\cdots\n\\end{equation}\nThis is the final primary diagrammatic identity of the first type;\nnote the similarity to~\\eq{eq:WID-A}.\n\n\nThe interpretation of the diagrammatics is as follows. In the first diagram\non the right-hand side, the vertex is differentiated with respect to\\ the momentum carried\nby the field $X$, whilst holding the momentum of the preceding field fixed\n(we assume for the time being that both $X$ and the\npreceding field carry non-zero momentum).\nOf course, using our current diagrammatic notation,\nthis latter field can be any of those\nwhich decorate the vertex, and so we sum over all possibilities. Thus,\neach cyclically ordered push forward like term has a partner,\ncyclically ordered pull back like term, such that\nthe pair can be interpreted as\n\\begin{equation}\n\t\\left( \\left. \\partial^r_\\mu \\right|_s - \\left. \\partial^s_\\mu \\right|_r \\right) \\mathrm{Vertex},\n\\label{eq:Momderivs}\n\\end{equation}\nwhere $r$ and $s$ are momenta entering the vertex. \nIn the case that $r=-s$, we can and will\ndrop either the push forward like term or pull back like term, since\nthe combination can be expressed as $\\partial^r_\\mu$; we\ninterpret the diagrammatic notation appropriately.\nIf any of the fields decorating the vertex carry\nzero momentum (besides the explicitly drawn $A^i$),\nthen they are transparent to this entire procedure.\nThus, they are never differentiated and, if they precede\na field which is, we must look to the first field carrying\nnon-zero momentum to figure\nout which of the vertex's momenta is held constant.\nJust as in~\\eq{eq:WID-A}, \nthe fields $X$ and or $Y$ can be interpreted as the end\nof a kernel. In this case, we introduce some new\nnotation, since it proves confusing in complete diagrams\nto actually locate the derivative\nsymbol at the end of such an object. The notation for the\nderivative with respect to\\ the momentum entering the end of a kernel \nis introduced in figure~\\ref{fig:Taylor-EndofKernel}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t\\cd{DifferentiatedKernel-A}\n\t\\]\n\\caption{Notation for the derivative with respect to\\ the momentum entering an\nundecorated kernel.}\n\\label{fig:Taylor-EndofKernel}\n\\end{figure}\\end{center}\n\nRecalling that a \nkernel, whose fields are explicitly\ncyclically ordered, is a two-sided object, we note\nthat the field whose momentum we have expanded in is sat on the\ntop-side of the vertex.\nThe derivative is taken to be with respect to\\\nthe momentum which flows \\emph{into} the end of the vertex which\nfollows the derivative, in the sense indicated by the\narrow on the derivative symbol.\nIt is clear that the direction of the\narrow on the derivative symbol can be reversed \nat the expense of a minus sign.\n\n\n\n\\subsubsection{Charge Conjugation Invariance}\n\\label{sec:CC}\n\nCharge conjugation invariance can be used to simplify the\ndiagrammatics, by allowing us to discard certain terms\nand to combine others. The diagrammatic prescription for\nreplacing a diagram which possesses exclusively\nbosonic external fields with its charge conjugate is~\\cite{Thesis,mgierg1}\nto reflect the diagram, picking up a sign for each\n\\begin{enumerate}\n\t\\item external $A^i$,\n\n\t\\item \\emph{performed} gauge remainder,\n\n\t\\item momentum derivative symbol (note\n\t\t\tthat the direction of the arrow accompanying such\n\t\t\tsymbols is reversed by the reflection of the diagram).\n\\end{enumerate}\n\n\\subsection{The Weak Coupling Expansion}\n\\label{sec:Weak}\n\n\\subsubsection{Perturbative Diagrammatics}\n\\label{sec:WeakDiag}\n\nIn the perturbative domain, we have the following\nweak coupling expansions~\\cite{ymi,aprop,Thesis,mgierg1}.\nThe Wilsonian effective action is given by\n\\begin{equation}\n\tS = \\sum_{i=0}^\\infty \\left( g^2 \\right)^{i-1} S_i = \\frac{1}{g^2}S_0 + S_1 + \\cdots,\n\\label{eq:Weak-S}\n\\end{equation}\nwhere $S_0$ is the classical effective action and the $S_{i>0}$\nthe $i$th-loop corrections. The seed action has a similar expansion:\n\\begin{equation}\n\t\\hat{S} = \\sum_{i=0}^\\infty  g^{2i}\\hat{S}_i.\n\\label{eq:Weak-hS}\n\\end{equation}\nRecalling~\\eq{eq:alpha-defn} we have:\n\\begin{eqnarray}\n\t\\beta \\equiv \\Lambda \\partial_\\Lambda g \t\t& = & \\sum_{i=1}^\\infty  g^{2i+1} \\beta_i(\\alpha)\n\\label{eq:beta}\n\\\\[1ex]\n\t\\gamma \\equiv \\Lambda \\partial_\\Lambda \\alpha \t& = & \\sum_{i=1}^{\\infty}  g^{2i} \\gamma_i(\\alpha).\n\\label{eq:gamma}\n\\end{eqnarray}\n\nDefining $\\Sigma_i = S_i - 2\\hat{S}_i$, the weak coupling flow equations\nfollow from substituting~\\eq{eq:Weak-S}--\\eq{eq:gamma}\ninto the flow equation, as shown in \n\\fig{fig:WeakCouplingFE}~\\cite{Thesis,mgierg1}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\t\\dec{\n\t\t\t\\cd{Vertex-n-LdL} \n\t\t}{\\{f\\}}\n\t\t= \n\t\t\\dec{\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{r=1}^n \\left[2\\left(n_r -1 \\right) \\beta_r +\\gamma_r \\pder{}{\\alpha} \\right]\\cd{Vertex-n_r-B} \n\t\t\t\\\\[4ex]\n\t\t\t\t\\displaystyle\n\t\t\t\t+ \\frac{1}{2} \n\t\t\t\t\\left( \n\t\t\t\t\t\\sum_{r=0}^n \\cd{Dumbbell-n_r-r} - \\cd{Vertex-Sigma_n_-B} - \\cd{WBT-Sigma_n-}\n\t\t\t\t\\right)\n\t\t\t\\end{array}\n\t\t}{\\{f\\}}\n\t\\label{eq:WeakFlow}\n\t\\end{equation}\n\\caption{The weak coupling flow equations.}\n\\label{fig:WeakCouplingFE}\n\\end{figure}\\end{center}\n\nWe refer to the first two terms on the right-hand side\\ of~\\eq{eq:WeakFlow} as\n$\\beta$ and $\\alpha$-terms, respectively.\nThe symbol $\\bullet$, as in equation~\\eq{eq:dot}, means\n$-\\Lambda \\partial_\\Lambda|_\\alpha$. A vertex whose argument is an unadorned letter, say $n$,\nrepresents $S_n$. We define $n_r \\equiv n-r$ and $n_\\pm \\equiv n \\pm 1$. The\nbar notation of the dumbbell term is defined as follows:\n\\begin{equation}\n\\label{eq:bar}\n\ta_0[\\bar{S}_{n-r}, \\bar{S}_r] \t\\equiv \ta_0[S_{n-r}, S_r] - a_0[S_{n-r}, \\hat{S}_r] - a_0[\\hat{S}_{n-r}, S_r].\n\\end{equation}\n\nThe renormalization condition\nfor $g$, equation~\\eq{defg}, constrains the two-point  vertex of\nthe physical field $S_{\\mu \\ \\; \\nu}^{A^1 A^1}(p)$ as follows:\n\\begin{eqnarray}\n\\label{eq:S_0-11}\n\tS_{0 \\mu  \\nu}^{\\ 1 \\, 1}(p) & = & 2 (p^2 \\delta_{\\mu\\nu} - p_\\mu p_\\mu) + \\Op{4}\n\t\\equiv 2\\Box_{\\mu \\nu}(p) + \\Op{4}\n\\\\\n\\label{eq:S_>0-11}\n\tS_{n>0 \\mu  \\nu}^{\\ \\ \\ \\ 1 \\, 1}(p) & = & \\Op{4},\n\\end{eqnarray}\nwhere we have abbreviated $A^1$ by just `1'. \n\n\n\\subsubsection{The Effective Propagator Relation}\n\\label{sec:EP-Reln}\n\nThe effective propagator\nrelation (which we recall is the sole primary\ndiagrammatic identity of the second type) arises\nfrom examining the flow of all two-point, tree level vertices.\nThis is done by setting $n=0$ in~\\eq{eq:WeakFlow}\nand specializing $\\{f\\}$ to contain two fields, \nas shown in \\fig{fig:TLTPs}.\nWe note that we can and do choose\nall such vertices to be single supertrace terms~\\cite{Thesis,mgierg1}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\t\\cd{Vertex-TLTP-LdL} = \\cd{Dumbbell-S_0-S_0} - \\cd{Dumbbell-S_0-hS_0} - \\cd{Dumbbell-hS_0-S_0}\n\t\\label{eq:TLTP-flow}\n\t\\end{equation}\n\\caption{Flow of all possible two-point, tree level vertices.}\n\\label{fig:TLTPs}\n\\end{figure}\\end{center}\n\nFollowing~\\cite{ym,ymi,ymii,aprop,Thesis,mgierg1,scalar2} \nwe use the freedom inherent in $\\hat{S}$ by choosing the two-point, tree\nlevel seed action vertices equal to the corresponding Wilsonian effective\naction vertices. Equation~\\eq{eq:TLTP-flow} now simplifies.\nRearranging, integrating with respect to\\ $\\Lambda$ and choosing the appropriate\nintegration constants~\\cite{Thesis,mgierg1}, we arrive at the following\nrelationship between the integrated ERG kernels---{\\it a.k.a.}\\  the\neffective propagators---and the two-point,\ntree level vertices. \n\\begin{equation}\n\t\t\\cd{EffPropReln}\t= \\cd{K-Delta} - \\cd{FullGaugeRemainder}\n\t\t\t\t\t\t\t= \\cd{K-Delta} - \\cd{DecomposedGR}\n\t\\label{eq:EPReln-A}\n\\end{equation}\n\nNote\nthat we have attached the effective propagator, which only\never appears as an internal line, to an arbitrary structure.\nThe field labelled by $M$ can be any of the broken phase\nfields. The object $\\cdeps{GR} \\!\\! \\equiv \\; > \\!\\!\\! \\rhd$ is a gauge remainder.\nThe individual components of\n$> \\!\\! \\rhd$ will often be loosely\nreferred to as gauge remainders; where it is necessary to\n unambiguously refer to the composite structure, we will use\nthe terminology `full gauge remainder'.\n\nThe various components on the right-hand side\\ of~\\eq{eq:EPReln-A}\ncan be interpreted, in the different sectors,\naccording to table~\\ref{tab:NFE:k,k'}, where we take\nthe gauge remainder to carry momentum $p$ (as denoted\nby the subscripts carried by the gauge remainder components).\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{table}[h]\n\t\\[\n\t\\begin{array}{c|ccc}\n\t\t\t\t\t& \\delta_{MN}\t\t& >_p\t\t\t\t\t\t& \\rhd_p\n\t\\\\ \\hline \n\t\tF,\\bar{F}\t& \\delta_{MN}\t\t& (f_p p_\\mu \/ \\Lambda^2, g_p)\t& (p_\\nu, 2)\n\t\\\\\n\t\tA^i\t\t\t& \\delta_{\\mu \\nu}\t& p_\\mu \/ p^2\t\t\t\t\t& p_\\nu\n\t\\\\\n\t\tC^i\t\t\t& \\hbox{1\\kern-.8mm l}\t\t\t\t& \\mbox{---}\t\t\t\t\t& \\mbox{---}\n\t\\end{array}\n\t\\]\n\\caption{Prescription for interpreting~\\eq{eq:EPReln-A}.}\n\\label{tab:NFE:k,k'}\n\\end{table}\n\\end{center}\n\\renewcommand{\\arraystretch}{1}\n\n\nFrom~\\eq{eq:S_0-11} and \\tab{tab:NFE:k,k'}, it follows\nthat\n\\begin{equation}\n\\label{eq:EP-leading}\n\t\\Delta^{11}_{\\rho \\sigma}(p) = \\frac{\\delta_{\\rho \\sigma}}{2 p^2} + \\Op{0},\n\\end{equation}\nwhich we will need later. \nThe functions $f(p^2\/\\Lambda^2)$ and $g(p^2\/\\Lambda^2)$ \nneed never be exactly determined (though for a concrete\nrealization, see~\\cite{Thesis});\nrather, they must satisfy general constraints enforced\nby the requirements of\nproper UV regularization of the physical $SU(N)$ theory\nand gauge invariance. We will see the effects of the\nlatter shortly. \n\n\n\nGenerally speaking, when we encounter\ncomponents of gauge remainders in complete diagrams,\nwe will either sum over flavours, or the flavour\nwill be obvious, from the context. However, there\nis one place where we will find it useful to \nexplicitly indicate the flavour (and momenta) \nof both $\\rhd$ and $>$.\nThus we introduce the notation\n\\begin{equation}\n\\label{GR:flavour}\n\t\\rhd^{i}_{p}, \\qquad >^{j}_q.\n\\end{equation}\nThe superscript indices take the values 0 or 1, denoting\na (non-null) bosonic or fermionic gauge remainder, \nrespectively. Later, we will find\nit useful to form logical expressions with\nthese indices. \n\n\n\n\\subsubsection{Primary Diagrammatic Identities of the Third Type}\n\nWe recall that the primary diagrammatic identities\nof the third type follow directly from\nthose of the first and second types but are given\nexplicitly nonetheless due to the central role\nthey will play in this paper. The first of\nthese identities\nis simply the classical part of~\\eq{eq:D-ID-GR-TP-A}:\n\\begin{equation}\n\t\\cd{GR-TLTP} = 0.\n\\label{eq:GR-TLTP-A}\n\\end{equation}\n\n\n\nFrom the effective propagator relation and~\\eq{eq:GR-TLTP-A},\ntwo further diagrammatic identities follow.\nFirst, consider attaching\nan effective propagator to the right-hand field in~\\eq{eq:GR-TLTP-A}\nand applying\nthe effective propagator before $\\rhd$ has acted. Diagrammatically,\nthis gives\n\\[\n\t\\cd{GR-TLTP-EP} = 0 = \\cd{k} - \\cd{kkprk},\n\\]\nwhich implies the following diagrammatic identity:\n\\begin{equation}\n\t\\cd{GR-relation} = 1.\n\\label{eq:GR-relation-A}\n\\end{equation}\nNotice that this means that the $f$ and $g$ of\n\\tab{tab:NFE:k,k'} are related by\n\\begin{equation}\n\\label{eq:xf+2g}\n\tx_p f_p + 2g_p = 1,\n\\end{equation}\nwhere we have defined $x_p = p^2 \/ \\Lambda^2$.\n\nThe effective propagator relation, together\nwith~\\eq{eq:GR-relation-A}, implies that\n\\[\n\t\\cd{TLTP-EP-GR} = \\cd{kpr} - \\cd{kprkkpr} = 0.\n\\]\nIn other words, the (non-zero) structure $\\cd{EP-GR}$ kills\na classical, two-point vertex. But, by~\\eq{eq:GR-TLTP-A}, \nthis suggests that the structure $\\cd{EP-GR}$\nmust be equal, up to some factor, to $\\lhd$. Hence, \nthe last of\nthe primary diagrammatic identities is\n\\begin{equation}\n\t\\cd{EP-GRpr} \\equiv \\cd{GR-PEP},\n\\label{eq:PseudoEP-A}\n\\end{equation}\nwhere the dot-dash line represents the pseudo effective propagators \nof~\\cite{Thesis,mgierg1}, also denoted by $\\stackrel{\\sim}{\\Delta}$.\n\n\n\\section*{References}\n\\input{.\/Bibliography\/Biblio}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Main Results}\n\nWe set $ \\Delta = \\partial_{11} + \\partial_{22} $  on open analytic domain $ \\Omega $ of $ {\\mathbb R}^2 $.\n\n\\bigskip\n\nWe consider the following equation:\n\n$$ (P)   \\left \\{ \\begin {split} \n       -\\Delta u & = |x|^{2 \\beta} V e^{v} \\,\\, &\\text{in} \\,\\, & \\Omega  \\subset {\\mathbb R}^2, \\\\\n      - \\Delta v & = W e^{u} \\,\\, &\\text{in} \\,\\, & \\Omega  \\subset {\\mathbb R}^2, \\\\\n                    u & = 0  \\,\\,             & \\text{in} \\,\\,    &\\partial \\Omega,\\\\ \n                    v & = 0  \\,\\,             & \\text{in} \\,\\,    &\\partial \\Omega.                        \n   \\end {split}\\right.\n$$\n\nHere, we assume that:\n\n$$ 0 \\leq V \\leq b_1 < + \\infty, \\,\\, e^u \\in L^1({\\Omega})\\,\\, {\\rm and} \\,\\,  u \\in W_0^{1,1}(\\Omega), $$\n\n$$ 0 \\leq W \\leq b_2 < + \\infty, \\,\\, |x|^{2\\beta} e^v \\in L^1({\\Omega})\\,\\, {\\rm and} \\,\\,  v \\in W_0^{1,1}(\\Omega), $$\n\nand,\n\n$$  0 \\in \\partial \\Omega, \\,\\,\\, \\beta \\geq 0. $$\n \nWhen $ u=v $ and $ \\beta =0 $, the above system is reduced to an equation which was studied by many authors, with or without  the boundary condition, also for Riemann surfaces,  see [1-17],  one can find some existence and compactness results, also for a system.\n\nAmong other results, we  can see in [6] the following important Theorems ($ \\beta=0 $):,\n\n\\smallskip\n\n{\\bf Theorem A.}{\\it (Brezis-Merle [6])}.{\\it  Consider the case of one equation; if $ (u_i)_i=(v_i)_i $ and $ (V_i)_i=(W_i)_i $ are two sequences of functions relatively to the problem $ (P) $ with, $ 0 < a \\leq V_i \\leq b < + \\infty $, then, for all compact set $ K $ of $ \\Omega $,\n\n$$ \\sup_K u_i \\leq c = c(a, b, K, \\Omega). $$}\n\n\\smallskip\n\n{\\bf Theorem B} {\\it (Brezis-Merle [6])}.{\\it Consider the case of one equation and assume that $ (u_i)_i $ and $ (V_i)_i $ are two sequences of functions relatively to the previous problem $ (P) $ with, $ 0 \\leq V_i \\leq b < + \\infty $, and,\n\n$$ \\int_{\\Omega} e^{u_i} dy  \\leq C, $$\n\nthen, for all compact set $ K $ of $ \\Omega $,\n\n$$ \\sup_K u_i \\leq c = c(b, C, K, \\Omega). $$}\n\nNext, we call energy the following quantity:\n\n$$ E= \\int_{\\Omega} e^{u_i} dy. $$\n\nThe boundedness of the energy is a necessary condition to work on the problem $ (P) $ as showed in $ [6] $, by the following counterexample ($\\beta =0$):\n\n\\smallskip\n\n{\\bf Theorem C} {\\it (Brezis-Merle [6])}.{\\it  Consider the case of one equation, then there are two sequences $ (u_i)_i $ and $ (V_i)_i $ of the problem $ (P) $ with, $ 0 \\leq V_i \\leq b < + \\infty $, and,\n\n$$ \\int_{\\Omega} e^{u_i} dy  \\leq C, $$\n\nand\n\n$$ \\sup_{\\Omega}  u_i \\to + \\infty. $$}\n\nWhen $ \\beta = 0 $, the above system have many properties in the constant and the Lipschitzian cases. Indeed we have (when $ \\beta = 0 $):\n\nIn [12], Dupaigne-Farina-Sirakov proved (by an existence result of Montenegro, see [16]) that the solutions of the above system when $ V $ and $ W $ are constants can be extremal and this condition imply the boundedness of the energy and directly the compactness.\nNote that in [11], if we assume (in particular) that $ \\nabla \\log V $ and $ \\nabla \\log W $ and $ V >a >0 $ or $ W > a'>0 $ and $ V, W $ are nonegative and uniformly bounded then the energy is bounded and we have a compactness result.\n\nNote that in the case of one equation (and $ \\beta = 0 $), we can prove by using the Pohozaev identity that if $ + \\infty >b \\geq V \\geq a >0 $, $ \\nabla V $ is uniformely Lipschitzian that the energy is bounded when $ \\Omega $ is starshaped. In [15] Ma-Wei, using the moving-plane method showed that this fact is true for all domain $ \\Omega $ with the same assumptions on $ V $. In [11] De Figueiredo-do O-Ruf extend this fact to a system by using the moving-plane method for a system.\n\n\nTheorem C, shows that we have not a global compactness to the previous problem with one equation, perhaps we need more information on $ V $ to conclude  to the boundedness of the solutions. When $ \\nabla \\log V $ is  Lipschitz function and $ \\beta = 0 $, Chen-Li and Ma-Wei see [7] and [15], showed that we have a compactness on all the open set. The proof is via the moving plane-Method of Serrin and Gidas-Ni-Nirenberg. Note that in [11], we have the same result for this system when $ \\nabla \\log V $ and $ \\nabla \\log W $ are uniformly bounded. We will see below that for a system we also have a compactness result when $ V $ and $ W $ are Lipschitzian and $ \\beta \\geq 0 $.\n\nNow consider the case of one equation. In this case our equation have nice properties.\n\nIf we assume $ V $ with more regularity, we can have another type of estimates, a $ \\sup + \\inf $ type inequalities. It was proved by Shafrir see [17], that, if $ (u_i)_i, (V_i)_i $ are two sequences of functions solutions of the previous equation without assumption on the boundary and, $ 0 < a \\leq V_i \\leq b < + \\infty $, then we have the following interior estimate:\n\n$$ C\\left (\\dfrac{a}{b} \\right ) \\sup_K u_i + \\inf_{\\Omega} u_i \\leq c=c(a, b, K, \\Omega). $$\n\n\\bigskip\n\nNow, if we suppose $ (V_i)_i $ uniformly Lipschitzian with $ A $ the\nLipschitz constant, then, $ C(a\/b)=1 $ and $ c=c(a, b, A, K, \\Omega)\n$, see [5]. \n\\smallskip\n\nHere we are interested by the case of a system of this type of equation. First, we give the behavior of the blow-up points on the boundary, with weight and boundary singularity, and in the second time we have a proof of  compactness of the solutions to Gelfand-Liouville type system with weight and boundary singularity and Lipschitz condition.\n\n\\smallskip\n\nHere, we write an extention of Brezis-Merle Problem (see [6]) to a system:\n\n\\smallskip\n\n{\\bf Problem}. Suppose that $ V_i \\to  V $ and  \n$ W_i \\to  W $ in $ C^0( \\bar \\Omega ) $, with, $ 0 \\leq V_i $ and $ 0 \\leq W_i $. Also, we consider two sequences of solutions $ (u_i), (v_i) $ of $ (P) $ relatively to $ (V_i), (W_i) $ such that,\n\n$$ \\int_{\\Omega} e^{u_i} dx \\leq C_1,\\,\\,\\,  \\int_{\\Omega} |x|^{2\\beta} e^{v_i} dx \\leq C_2,  $$\n\nis it possible to have:\n\n$$ ||u_i||_{L^{\\infty}}\\leq C_3=C_3(\\beta, C_1, C_2, \\Omega) ? $$\n\nand,\n\n$$ ||v_i||_{L^{\\infty}}\\leq C_4=C_4(\\beta, C_1, C_2, \\Omega) ? $$\n\n\nIn this paper we give a caracterization of the behavior of the blow-up points on the boundary and also a proof of the compactness theorem when $ V_i $ and $ W_i $ are uniformly Lipschitzian and $ \\beta \\geq 0 $. For the behavior of the blow-up points on the boundary, the following condition are enough,\n\n$$ 0 \\leq  V_i \\leq b_1, \\,\\,\\, 0 \\leq  W_i \\leq b_2, $$\n\nThe conditions $ V_i \\to  V $ and $ W_i \\to  W $ in $ C^0(\\bar \\Omega) $ are not necessary.\n\n\\bigskip\n\nBut for the proof of the compactness  for the system, we assume that:\n\n$$ ||\\nabla V_i||_{L^{\\infty}}\\leq  A_1, \\,\\,\\, ||\\nabla W_i||_{L^{\\infty}}\\leq  A_2,\\,\\, \\beta \\geq 0. $$\n\nOur main result are:\n\n\\begin{Theorem}  Assume that $ \\max_{\\Omega} u_i \\to +\\infty $ and $ \\max_{\\Omega} v_i \\to +\\infty $ Where $ (u_i) $ and $ (v_i) $ are solutions of the probleme $ (P) $ with ($ \\beta \\geq 0 $), and:\n \n $$ 0 \\leq V_i \\leq b_1,\\,\\,\\, {\\rm and } \\,\\,\\, \\int_{\\Omega}  e^{u_i} dx \\leq C_1, \\,\\,\\, \\forall \\,\\, i, $$\n \nand,\n\n $$ 0 \\leq W_i \\leq b_2,\\,\\,\\, {\\rm and } \\,\\,\\, \\int_{\\Omega}  |x|^{2 \\beta} e^{v_i} dx \\leq C_2, \\,\\,\\, \\forall \\,\\, i, $$\n \n \n then;  after passing to a subsequence, there is a finction $ u $,  there is a number $ N \\in {\\mathbb N} $ and $ N  $ points $ x_1, x_2, \\ldots, x_N \\in  \\partial \\Omega $, such that, \n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} u_i  \\phi  \\to \\int_{\\partial \\Omega} \\partial_{\\nu} u  \\phi +\\sum_{j=1}^N \\alpha_j  \\phi(x_j), \\, \\alpha_j \\geq 4\\pi, $$\n$ \\text{for\\, any}\\,\\, \\phi\\in C^0(\\partial \\Omega) $, \nand,\n\n$$ u_i \\to u \\,\\,\\, {\\rm in }\\,\\,\\, C^1_{loc}(\\bar \\Omega-\\{x_1,\\ldots, x_N \\}). $$\n\n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} v_i  \\phi  \\to \\int_{\\partial \\Omega} \\partial_{\\nu} v  \\phi +\\sum_{j=1}^N \\beta_j  \\phi(x_j), \\, \\beta_j \\geq 4\\pi, $$\n$ \\text{for\\, any}\\,\\, \\phi\\in C^0(\\partial \\Omega) $,  and,\n\n$$ v_i \\to v \\,\\,\\, {\\rm in }\\,\\,\\, C^1_{loc}(\\bar \\Omega-\\{x_1,\\ldots, x_N \\}). $$\n\n\n\\end{Theorem} \n\n In the following theorem, we have a proof for the global a priori estimate which concern the problem $ (P) $.\n\n\\bigskip\n\n\\begin{Theorem}Assume that $ (u_i), (v_i) $ are solutions of $ (P) $ relatively to $ (V_i), (W_i) $ with the following conditions:\n\n$$ x_1=0 \\in \\partial \\Omega, \\,\\,\\beta \\geq 0, $$\n\nand,\n\n$$ 0 \\leq  V_i \\leq b_1, \\,\\,  ||\\nabla V_i||_{L^{\\infty}} \\leq A_1,\\,\\, {\\rm and } \\,\\,\\, \\int_{\\Omega} e^{u_i} \\leq C_1, $$\n\n\n$$ 0 \\leq  W_i \\leq b_2, \\,\\,  ||\\nabla W_i||_{L^{\\infty}} \\leq A_2,\\,\\, {\\rm and } \\,\\,\\, \\int_{\\Omega} |x|^{2\\beta} e^{v_i} \\leq C_2, $$\n\nWe have,\n\n$$  || u_i||_{L^{\\infty}} \\leq C_3(b_1, b_2, \\beta, A_1, A_2, C_1, C_2, \\Omega), $$\n\nand,\n\n$$  || v_i||_{L^{\\infty}} \\leq C_4(b_1, b_2, \\beta, A_1, A_2, C_1, C_2, \\Omega), $$\n\n\n\\end{Theorem} \n\n\\section{Proof of the theorems} \n\n\\bigskip\n\n\\underbar {\\it Proof of theorem 1.1:} \n\n\\bigskip\n\nWe have:\n\n$$ u_i, v_i \\in W_0^{1,1}(\\Omega). $$\n\nSince $ e^{u_i} \\in L^1(\\Omega) $ by the corollary 1 of Brezis-Merle's paper (see [6]) we have $ e^{v_i} \\in L^k(\\Omega) $ for all $ k  >2 $ and the elliptic estimates of Agmon and the Sobolev embedding (see [1]) imply that:\n\n$$ u_i \\in W^{2, k}(\\Omega)\\cap C^{1, \\epsilon}(\\bar \\Omega). $$ \n\nAnd,\n\nWe have:\n\n$$ v_i, u_i \\in W_0^{1,1}(\\Omega). $$\n\nSince $ |x|^{2\\beta} e^{v_i} \\in L^1(\\Omega) $ by the corollary 1 of Brezis-Merle's paper (see [6]) we have $ e^{u_i} \\in L^k(\\Omega) $ for all $ k  >2 $ and the elliptic estimates of Agmon and the Sobolev embedding (see [1]) imply that:\n\n$$ v_i \\in W^{2, k}(\\Omega)\\cap C^{1, \\epsilon}(\\bar \\Omega). $$ \n\n\n\nSince $ |x|^{2\\beta} V_ie^{v_i} $ and $ W_ie^{u_i} $ are bounded in $ L^1(\\Omega) $, we can extract from those two sequences two subsequences which converge to two nonegative measures $ \\mu_1 $ and $ \\mu_2 $. (This procedure is similar to the procedure of Brezis-Merle, we apply corollary 4 of Brezis-Merle paper, see [6]).\n\nIf $ \\mu_1(x_0) < 4 \\pi $, by a Brezis-Merle estimate for the first equation, we have $ e^{u_i} \\in L^{1+\\epsilon} $ around $ x_0 $, by the elliptic estimates, for the second equation, we have $ v_i \\in W^{2, 1+\\epsilon} \\subset L^{\\infty} $ around $ x_0 $, and , returning to the first equation, we have $ u_i \\in L^{\\infty} $ around $ x_0 $.\n\nIf $ \\mu_2(x_0) < 4 \\pi $, then $ u_i $ and $ v_i $ are also locally bounded around $ x_0 $.\n\nThus, we take a look to the case when, $ \\mu_1(x_0) \\geq 4 \\pi $ and $ \\mu_2(x_0) \\geq 4 \\pi $. By our hypothesis, those points $ x_0 $ are finite.\n\nWe will see that inside $ \\Omega $ no such points exist. By contradiction, assume that, we have $ \\mu_1(x_0) \\geq 4 \\pi $. Let us consider a ball $ B_R(x_0) $ which contain only $ x_0 $ as nonregular point. Thus, on $ \\partial B_R(x_0) $, the two sequence $ u_i $ and $ v_i $ are uniformly bounded. Let us consider:\n\n$$ \\left \\{ \\begin {split} \n       -\\Delta z_i & = |x|^{2\\beta}V_i e^{v_i} \\,\\, &\\text{in} \\,\\, & B_R(x_0)  \\subset {\\mathbb R}^2, \\\\\n                       z_i & = 0  \\,\\,             & \\text{in} \\,\\,    &\\partial B_R(x_0).                                            \n   \\end {split}\\right.\n$$\n\nBy the maximum principle we have:\n\n$$ z_i \\leq u_i $$ \n\nand $ z_i \\to z $ almost everywhere on this ball, and thus,\n\n$$ \\int e^{z_i} \\leq \\int e^{u_i} \\leq C, $$\n\nand,\n\n$$ \\int e^z \\leq C.$$\n\nbut, $ z  $ is  a solution in $ W_0^{1,q}(B_R(x_0)) $, $ 1\\leq q <2 $, of the following equation:\n\n\n$$ \\left \\{ \\begin {split} \n       -\\Delta z & = \\mu_1\\,\\, &\\text{in} \\,\\, & B_R(x_0) \\subset {\\mathbb R}^2, \\\\\n                     z & = 0  \\,\\,             & \\text{in} \\,\\,    &\\partial B_R(x_0).                                            \n   \\end {split}\\right.\n$$\n\nwith, $ \\mu_1 \\geq 4 \\pi $ and thus, $ \\mu_1 \\geq 4\\pi \\delta_{x_0} $ and then, by the maximum principle in $ W_0^{1,q}(B_R(x_0)) $:\n\n$$ z \\geq -2 \\log |x-x_0|+ C $$\n\nthus,\n\n$$ \\int e^z = + \\infty, $$\n\nwhich is a contradiction. Thus, there is no nonregular points inside  $ \\Omega $\n\nThus, we consider the case where we have nonregular points on the boundary, we use two estimates:\n\n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} u_i d\\sigma \\leq C_1,\\,\\,\\, \\int_{\\partial \\Omega} \\partial_{\\nu} v_i d\\sigma \\leq C_2, $$\n \n and,\n \n $$ ||\\nabla u_i||_{L^q} \\leq C_q, \\,\\,\\, ||\\nabla v_i||_{L^q} \\leq C'_q, \\,\\,\\forall \\,\\, i\\,\\, {\\rm and  }  \\,\\, 1< q < 2. $$\n\nWe have the same computations, as in the case of one equation.\n\nWe consider a points $ x_0 \\in \\partial \\Omega $ such that:\n\n$$ \\mu_1(x_0) < 4 \\pi. $$\n\nWe consider a test function on the boundary $ \\eta $ we extend $ \\eta $ by a harmonic function on $ \\Omega $, we write the equation:\n\n$$ -\\Delta ((u_i-u)\\eta) =|x|^{2\\beta}(V_i e^{v_i}-Ve^v)\\eta+ <\\nabla (u_i-u)|\\nabla \\eta> = f_i $$\n\nwith,\n\n$$ \\int |f_i| \\leq 4 \\pi-\\epsilon +o(1) < 4\\pi-2\\epsilon <4\\pi, $$\n\n$$ -\\Delta ((v_i-v)\\eta) =(W_i e^{u_i}-We^u)\\eta+ <\\nabla (v_i-v)|\\nabla \\eta> = g_i, $$\n\nwith,\n\n$$ \\int |g_i| \\leq 4 \\pi-\\epsilon +o(1) < 4\\pi-2\\epsilon <4\\pi, $$\n\nBy the Brezis-Merle estimate, we have uniformly, $ e^{u_i} \\in L^{1+\\epsilon} $ around $ x_0 $, by the elliptic estimates, for the second equation,  we have $ v_i \\in W^{2, 1+\\epsilon} \\subset L^{\\infty} $ around $ x_0 $, and , returning to the first equation, we have $ u_i \\in L^{\\infty} $ around $ x_0 $.\n\n\nWe have the same thing if we assume:\n\n$$ \\mu_2(x_0) < 4 \\pi. $$\n\nThus, if $ \\mu_1(x_0) < 4 \\pi $ or $ \\mu_2(x_0) < 4 \\pi $, we have for $ R >0 $ small enough:\n\n$$ (u_i,v_i) \\in L^{\\infty}(B_R(x_0)\\cap \\bar \\Omega). $$\n\nBy our hypothesis the set of the points such that:\n\n$$ \\mu_1(x_0)  \\geq  4 \\pi,  \\,\\,\\, \\mu_2(x_0)   \\geq 4 \\pi, $$\n\nis finite, and, outside this set $ u_i $ and $ v_i $ are locally uniformly bounded. By the elliptic estimates, we have the $ C^1 $ convergence to  $ u $ and $ v $  on each compact set of $ \\bar \\Omega- \\{x_1, \\ldots x_N\\} $.\n\nIndeed,\n\nBy the Stokes formula we have, \n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} u_i d\\sigma \\leq C, $$\n\nWe use the weak convergence in the space of Radon measures to have the existence of a nonnegative Radon measure $ \\mu_1 $ such that,\n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} u_i \\phi  d\\sigma \\to \\mu_1(\\phi), \\,\\,\\, \\forall \\,\\,\\, \\phi \\in C^0(\\partial \\Omega). $$\n\nWe take an $ x_0 \\in \\partial \\Omega $ such that, $ \\mu_1({x_0}) < 4\\pi $. For $ \\epsilon >0 $ small enough set $ I_{\\epsilon}= B(x_0, \\epsilon)\\cap \\partial \\Omega $ on the unt disk or one can assume it as an interval. We choose a function $ \\eta_{\\epsilon} $ such that,\n\n$$ \\begin{cases}\n    \n\\eta_{\\epsilon} \\equiv 1,\\,\\,\\,  {\\rm on } \\,\\,\\,  I_{\\epsilon}, \\,\\,\\, 0 < \\epsilon < \\delta\/2,\\\\\n\n\\eta_{\\epsilon} \\equiv 0,\\,\\,\\, {\\rm outside} \\,\\,\\, I_{2\\epsilon }, \\\\\n\n0 \\leq \\eta_{\\epsilon} \\leq 1, \\\\\n\n||\\nabla \\eta_{\\epsilon}||_{L^{\\infty}(I_{2\\epsilon})} \\leq \\dfrac{C_0(\\Omega, x_0)}{\\epsilon}.\n\n\\end{cases} $$\n\nWe take a $\\tilde \\eta_{\\epsilon} $ such that,\n\n\\begin{displaymath}  \\left \\{ \\begin {split} \n      -\\Delta \\tilde \\eta_{\\epsilon}  & = 0              \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  \\tilde\\eta_{\\epsilon} & =  \\eta_{\\epsilon}   \\,\\,             && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\n{\\bf Remark:} We use the following steps in the construction of $ \\tilde \\eta_{\\epsilon} $:\n\n\nWe take a cutoff function $ \\eta_{0} $ in $ B(0, 2) $ or $ B(x_0, 2) $:\n\n1- We set $ \\eta_{\\epsilon}(x)= \\eta_0(|x-x_0|\/\\epsilon) $ in the case of the unit disk it is sufficient.\n\n2- Or, in the general case: we use a chart $ (f, \\tilde \\Omega) $ with $ f(0)=x_0 $ and we take $ \\mu_{\\epsilon}(x)= \\eta_0 ( f( |x|\/ \\epsilon)) $ to have  connected  sets $ I_{\\epsilon} $ and we take $ \\eta_{\\epsilon}(y)= \\mu_{\\epsilon}(f^{-1}(y))$. Because $ f, f^{-1} $ are Lipschitz, $ |f(x)-x_0| \\leq k_ 2|x|\\leq 1 $ for $ |x| \\leq 1\/k_2 $ and $ |f(x)-x_0| \\geq k_ 1|x|\\geq 2 $ for $ |x| \\geq 2\/k_1>1\/k_2 $, the support  of $ \\eta $ is in $ I_{(2\/k_1)\\epsilon} $.\n\n$$ \\begin{cases}\n    \n\\eta_{\\epsilon} \\equiv 1,\\,\\,\\,  {\\rm on } \\,\\,\\,  f(I_{(1\/k_2)\\epsilon}), \\,\\,\\, 0 < \\epsilon < \\delta\/2,\\\\\n\n\\eta_{\\epsilon} \\equiv 0,\\,\\,\\, {\\rm outside} \\,\\,\\, f(I_{(2\/k_1)\\epsilon }), \\\\\n\n0 \\leq \\eta_{\\epsilon} \\leq 1, \\\\\n\n||\\nabla \\eta_{\\epsilon}||_{L^{\\infty}(I_{(2\/k_1)\\epsilon})} \\leq \\dfrac{C_0(\\Omega, x_0)}{\\epsilon}.\n\n\\end{cases} $$\n\n3- Also, we can take: $ \\mu_{\\epsilon}(x)= \\eta_0(|x|\/\\epsilon) $ and $ \\eta_{\\epsilon}(y)= \\mu_{\\epsilon}(f^{-1}(y)) $, we extend it by $ 0 $ outside $ f(B_1(0)) $.  We have $ f(B_1(0)) = D_1(x_0) $, $ f (B_{\\epsilon}(0))= D_{\\epsilon}(x_0) $ and $ f(B_{\\epsilon}^+)= D_{\\epsilon}^+(x_0) $ with $ f $ and $ f^{-1} $ smooth diffeomorphism.\n\n\n\n$$ \\begin{cases}\n    \n\\eta_{\\epsilon} \\equiv 1,\\,\\,\\,  {\\rm on \\, a \\, the \\, connected \\, set } \\,\\,\\,  J_{\\epsilon} =f(I_{\\epsilon}), \\,\\,\\, 0 < \\epsilon < \\delta\/2,\\\\\n\n\\eta_{\\epsilon} \\equiv 0,\\,\\,\\, {\\rm outside} \\,\\,\\, J'_{\\epsilon}=f(I_{2\\epsilon }), \\\\\n\n0 \\leq \\eta_{\\epsilon} \\leq 1, \\\\\n\n||\\nabla \\eta_{\\epsilon}||_{L^{\\infty}(J'_{\\epsilon})} \\leq \\dfrac{C_0(\\Omega, x_0)}{\\epsilon}.\n\n\\end{cases} $$\n\nAnd, $ H_1(J'_{\\epsilon}) \\leq C_1 H_1(I_{2\\epsilon}) = C_1 4\\epsilon $, since $ f $ is Lipschitz. Here $ H_1 $ is the Hausdorff measure.\n\nWe solve the Dirichlet Problem:\n\n\\begin{displaymath}  \\left \\{ \\begin {split} \n      -\\Delta \\bar \\eta_{\\epsilon}  & = -\\Delta \\eta_{\\epsilon}              \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  \\bar \\eta_{\\epsilon} & = 0   \\,\\,             && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\nand finaly we set $ \\tilde \\eta_{\\epsilon} =-\\bar \\eta_{\\epsilon} + \\eta_{\\epsilon} $. Also, by the maximum principle and the elliptic estimates we have :\n\n$$ ||\\nabla \\tilde \\eta_{\\epsilon}||_{L^{\\infty}} \\leq C(|| \\eta_{\\epsilon}||_{L^{\\infty}} +||\\nabla \\eta_{\\epsilon}||_{L^{\\infty}} + ||\\Delta \\eta_{\\epsilon}||_{L^{\\infty}}) \\leq \\dfrac{C_1}{\\epsilon^2}, $$\n\nwith $ C_1 $ depends on $ \\Omega $.\n\n\nWe use the following estimate, see [8],\n\n$$ ||\\nabla v_i||_{L^q}\\leq C_q,\\,\\, ||\\nabla u_i||_q \\leq C_q, \\,\\,\\forall \\,\\, i\\,\\, {\\rm and  }  \\,\\, 1< q < 2. $$\n\nWe deduce from the last estimate that, $ (v_i) $ converge weakly in $ W_0^{1, q}(\\Omega) $, almost everywhere to a function $ v \\geq 0 $ and $ \\int_{\\Omega} |x|^{2\\beta} e^v < + \\infty $ (by Fatou lemma). Also, $ V_i $ weakly converge to a nonnegative function $ V $ in $ L^{\\infty} $. \n\nWe deduce from the last estimate that, $ (u_i) $ converge weakly in $ W_0^{1, q}(\\Omega) $, almost everywhere to a function $ u \\geq 0 $ and $ \\int_{\\Omega} e^u < + \\infty $ (by Fatou lemma). Also, $ W_i $ weakly converge to a nonnegative function $ W $ in $ L^{\\infty} $.\n\n\n\nThe function $ u, v $ are in $ W_0^{1, q}(\\Omega) $ solutions of :\n\n\\begin{displaymath} \\left \\{ \\begin {split} \n      -\\Delta u  & = |x|^{2 \\beta} V e^{v} \\in L^1(\\Omega)    \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  u  & = 0  \\,\\,                                     && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n \nAnd,\n\n\\begin{displaymath} \\left \\{ \\begin {split} \n      -\\Delta v  & = W e^{u} \\in L^1(\\Omega)    \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  v  & = 0  \\,\\,                                     && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\n According to the corollary 1 of Brezis-Merle's result, see [6],   we have $ e^{k u }\\in L^1(\\Omega), k >1 $. By the elliptic estimates, we have $ v \\in C^1(\\bar \\Omega) $.\n\nAccording to the corollary 1 of Brezis-Merle's result, see [6],   we have $ e^{k v }\\in L^1(\\Omega), k >1 $. By the elliptic estimates, we have $ u \\in C^1(\\bar \\Omega) $.\n\n\nFor two vectors $ f $ and $ g $ we denote by $  f \\cdot g $ the inner product of $ f $ and $ g $. \n\nWe can write:\n\n\\be -\\Delta ((u_i-u) \\tilde \\eta_{\\epsilon})= |x|^{2 \\beta}(V_i e^{v_i} -Ve^v)\\tilde \\eta_{\\epsilon} -2\\nabla (u_i- u)\\cdot \\nabla \\tilde \\eta_{\\epsilon}. \\label{(1)}\\ee\n\n\n$$ -\\Delta ((v_i-v) \\tilde \\eta_{\\epsilon})= (W_i e^{u_i} -We^u)\\tilde \\eta_{\\epsilon} -2\\nabla (v_i- v)\\cdot \\nabla \\tilde \\eta_{\\epsilon}. $$\n\nWe use the interior esimate of Brezis-Merle, see [6],\n\n\\bigskip\n\n\\underbar {\\it Step 1:} Estimate of the integral of the first term of the right hand side of $ (\\ref{(1)}) $.\n\n\\bigskip\n\nWe use the Green formula between $ \\tilde \\eta_{\\epsilon} $ and $ u $, we obtain,\n\n\\be  \\int_{\\Omega} |x|^{2 \\beta}Ve^v \\tilde \\eta_{\\epsilon} dx =\\int_{\\partial \\Omega} \\partial_{\\nu} u \\eta_{\\epsilon} \\leq C'\\epsilon ||\\partial_{\\nu}u||_{L^{\\infty}}= C \\epsilon \\label{(2)}\\ee\n\nWe have,\n\n\\begin{displaymath} \\left \\{ \\begin {split} \n      -\\Delta u_i  & = |x|^{2 \\beta} V_i e^{v_i}                      \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  u_i  & = 0  \\,\\,                                     && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\nWe use the Green formula between $ u_i $ and $ \\tilde \\eta_{\\epsilon} $ to have:\n\n\\be \\int_{\\Omega} |x|^{2 \\beta} V_i e^{v_i} \\tilde \\eta_{\\epsilon} dx = \\int_{\\partial \\Omega} \\partial_{\\nu} u_i \\eta_{\\epsilon} d\\sigma \\to \\mu_1(\\eta_{\\epsilon}) \\leq \\mu_1(J'_{\\epsilon}) \\leq 4  \\pi - \\epsilon_0, \\,\\,\\, \\epsilon_0 >0 \\label{(3)}\\ee\n\nFrom $ (\\ref{(2)}) $ and $ (\\ref{(3)}) $ we have for all $ \\epsilon >0 $ there is $ i_0 =i_0(\\epsilon) $ such that, for $ i \\geq i_0 $,\n\n\n\\be \\int_{\\Omega} ||x|^{2 \\beta}(V_ie^{v_i}-Ve^v) \\tilde \\eta_{\\epsilon}| dx \\leq 4 \\pi -\\epsilon_0 + C \\epsilon \\label{(4)}\\ee\n\n\\bigskip\n\n\\underbar {\\it Step 2:} Estimate of integral of the second term of the right hand side of $ (\\ref{(1)}) $.\n\n\\bigskip\n\nLet $ \\Sigma_{\\epsilon} = \\{ x \\in \\Omega, d(x, \\partial \\Omega) = \\epsilon^3 \\} $ and $ \\Omega_{\\epsilon^3} = \\{ x \\in \\Omega, d(x, \\partial \\Omega) \\geq \\epsilon^3 \\} $, $ \\epsilon > 0 $. Then, for $ \\epsilon $ small enough, $ \\Sigma_{\\epsilon} $ is hypersurface.\n\nThe measure of $ \\Omega-\\Omega_{\\epsilon^3} $ is $ k_2\\epsilon^3 \\leq meas(\\Omega-\\Omega_{\\epsilon^3})= \\mu_L (\\Omega-\\Omega_{\\epsilon^3}) \\leq k_1 \\epsilon^3 $.\n\n{\\bf Remark}: for the unit ball $ {\\bar B(0,1)} $, our new manifold is $ {\\bar B(0, 1-\\epsilon^3)} $.\n\n( Proof of this fact; let's consider $ d(x, \\partial \\Omega) = d(x, z_0), z_0 \\in \\partial \\Omega $, this imply that $ (d(x,z_0))^2 \\leq (d(x, z))^2 $ for all $ z \\in \\partial \\Omega $ which it is equivalent to $ (z-z_0) \\cdot (2x-z-z_0) \\leq 0 $ for all $ z \\in \\partial \\Omega $, let's consider a chart around $ z_0 $ and $ \\gamma (t) $ a curve in $ \\partial \\Omega $, we have;\n\n$ (\\gamma (t)-\\gamma(t_0) \\cdot (2x-\\gamma(t)-\\gamma(t_0)) \\leq 0 $ if we divide by $ (t-t_0) $ (with the sign and tend $ t $ to $ t_0 $), we have $ \\gamma'(t_0) \\cdot (x-\\gamma(t_0)) = 0 $, this imply that $ x= z_0-s \\nu_0 $ where $ \\nu_0 $ is the outward normal of $ \\partial \\Omega $ at $ z_0 $))\n\nWith this fact, we can say that $ S= \\{ x, d(x, \\partial \\Omega) \\leq \\epsilon \\}= \\{ x= z_0- s \\nu_{z_0}, z_0 \\in \\partial \\Omega, \\,\\, -\\epsilon \\leq s \\leq \\epsilon \\} $. It  is sufficient to work on  $ \\partial \\Omega $. Let's consider a charts $ (z, D=B(z, 4 \\epsilon_z), \\gamma_z) $ with $ z \\in \\partial \\Omega $ such that $ \\cup_z B(z, \\epsilon_z) $ is  cover of $ \\partial \\Omega $ .  One can extract a finite cover $ (B(z_k, \\epsilon_k)), k =1, ..., m $, by the area formula the measure of $ S \\cap B(z_k, \\epsilon_k) $ is less than a $ k\\epsilon $ (a $ \\epsilon $-rectangle).  For the reverse inequality, it is sufficient to consider one chart around one point of the boundary.\n\n\nWe write,\n\n\\be \\int_{\\Omega} |\\nabla ( u_i -u) \\cdot \\nabla \\tilde \\eta_{\\epsilon} | dx =\n\\int_{\\Omega_{\\epsilon^3}} |\\nabla (u_i -u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx + \\int_{\\Omega - \\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx.  \\label{(5)}\\ee\n\n\\bigskip\n\n\\underbar {\\it Step 2.1:} Estimate of $ \\int_{\\Omega - \\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx $.\n\n\\bigskip\n\nFirst, we know from the elliptic estimates that  $ ||\\nabla \\tilde \\eta_{\\epsilon}||_{L^{\\infty}} \\leq C_1 \/\\epsilon^2 $, $ C_1 $ depends on $ \\Omega $\n\n\nWe know that $ (|\\nabla u_i|)_i $ is bounded in $ L^q, 1< q < 2 $, we can extract  from this sequence a subsequence which converge weakly to $ h \\in L^q $. But, we know that we have locally the uniform convergence to $ |\\nabla u| $ (by Brezis-Merle's theorem), then, $ h= |\\nabla u| $ a.e. Let $ q' $ be the conjugate of $ q $.\n\n\nWe have, $  \\forall f \\in L^{q'}(\\Omega)$\n\n$$ \\int_{\\Omega} |\\nabla u_i| f dx \\to \\int_{\\Omega} |\\nabla u| f dx $$\n\n\nIf we take $ f= 1_{\\Omega-\\Omega_{\\epsilon^3}} $, we have:\n\n$$ {\\rm for } \\,\\, \\epsilon >0 \\,\\, \\exists \\,\\, i_1 = i_1(\\epsilon) \\in {\\mathbb N}, \\,\\,\\, i \\geq  i_1,  \\,\\, \\int_{\\Omega-\\Omega_{\\epsilon^3}} |\\nabla u_i| \\leq \\int_{\\Omega-\\Omega_{\\epsilon^3}} |\\nabla u| + \\epsilon^3. $$\n\nThen, for $ i \\geq i_1(\\epsilon) $,\n\n$$ \\int_{\\Omega-\\Omega_{\\epsilon^3}} |\\nabla u_i| \\leq meas(\\Omega-\\Omega_{\\epsilon^3}) ||\\nabla u||_{L^{\\infty}} + \\epsilon^3 = \\epsilon^3(k_1 ||\\nabla u||_{L^{\\infty}} + 1). $$\n\nThus, we obtain,\n\n\\be \\int_{\\Omega - \\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx \\leq  \\epsilon C_1(2 k_1 ||\\nabla u||_{L^{\\infty}} + 1) \\label{(6)}\\ee\n\nThe constant $ C_1 $ does  not depend on $ \\epsilon $ but on $ \\Omega $.\n\n\\bigskip\n\n\\underbar {\\it Step 2.2:} Estimate of $ \\int_{\\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx $.\n\n\\bigskip\n\nWe know that, $ \\Omega_{\\epsilon} \\subset \\subset \\Omega $, and ( because of Brezis-Merle's interior estimates) $ u_i \\to u $ in $ C^1(\\Omega_{\\epsilon^3}) $. We have,\n\n$$ ||\\nabla (u_i-u)||_{L^{\\infty}(\\Omega_{\\epsilon^3})} \\leq \\epsilon^3,\\, {\\rm for } \\,\\, i \\geq i_3 = i_3(\\epsilon). $$\n\nWe write,\n \n$$ \\int_{\\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx \\leq ||\\nabla (u_i-u)||_{L^{\\infty}(\\Omega_{\\epsilon^3})} ||\\nabla \\tilde \\eta_{ \\epsilon}||_{L^{\\infty}} \\leq C_1 \\epsilon \\,\\, {\\rm for } \\,\\, i \\geq i_3, $$\n\n\nFor $ \\epsilon >0 $, we have for $ i \\in {\\mathbb N} $, $ i \\geq \\max \\{i_1, i_2, i_3 \\} $,\n\n\\be \\int_{\\Omega} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx \\leq \\epsilon C_1(2 k_1 ||\\nabla u||_{L^{\\infty}} + 2) \\label{(7)}\\ee\n\nFrom $ (\\ref{(4)}) $ and $ (\\ref{(7)}) $, we have, for $ \\epsilon >0 $, there is $ i_3= i_3(\\epsilon) \\in {\\mathbb N}, i_3 = \\max \\{ i_0, i_1, i_2 \\} $ such that,\n\n\n\\be \\int_{\\Omega} |-\\Delta [(u_i-u)\\tilde \\eta_{\\epsilon}]|dx \\leq 4 \\pi-\\epsilon_0+  \\epsilon 2 C_1(2 k_1 ||\\nabla u||_{L^{\\infty}} + 2 + C) \\label{(8)}\\ee\n\nWe choose $ \\epsilon >0 $ small enough to have a good estimate of  $ (\\ref{(1)}) $.\n\nIndeed, we have:\n\n\\begin{displaymath} \\left \\{ \\begin {split} \n      -\\Delta [(u_i-u) \\tilde \\eta_{\\epsilon}]   & = g_{i,\\epsilon}                   \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                 (u_i-u) \\tilde \\eta_{\\epsilon}    & = 0  \\,\\,                                     && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\nwith $ ||g_{i, \\epsilon} ||_{L^1(\\Omega)} \\leq 4 \\pi -\\dfrac{\\epsilon_0}{2}. $\n\nWe can use Theorem 1 of [6] to conclude that there are $ q \\geq \\tilde q >1 $ such that:\n\n$$ \\int_{V_{\\epsilon}(x_0)} e^{\\tilde q |u_i-u|} dx \\leq \\int_{\\Omega} e^{q|u_i -u| \\tilde \\eta_{\\epsilon}} dx \\leq C(\\epsilon,\\Omega). $$\n \nwhere, $ V_{\\epsilon}(x_0) $ is a neighberhood of $ x_0 $ in $ \\bar \\Omega $. Here we have used that in a neighborhood of $ x_0 $  by the elliptic estimates, \n$ 1- C \\epsilon \\leq \\tilde \\eta_{\\epsilon} \\leq 1 $.\n\nThus, for each $ x_0 \\in \\partial \\Omega - \\{ \\bar x_1,\\ldots, \\bar x_m \\} $ there is $ \\epsilon_{x_0} >0, q_{x_0} > 1 $ such that:\n\n\\be \\int_{B(x_0, \\epsilon_{x_0})} e^{q_{x_0} u_i} dx \\leq C, \\,\\,\\, \\forall \\,\\,\\, i. \\label{(9)}\\ee\n\nNow, we consider a cutoff function $ \\eta \\in C^{\\infty}({\\mathbb R}^2) $ such that\n\n$$ \\eta \\equiv 1 \\,\\,\\, {\\rm on } \\,\\,\\, B(x_0, \\epsilon_{x_0}\/2) \\,\\,\\, {\\rm and } \\,\\,\\, \\eta \\equiv 0 \\,\\,\\, {\\rm on } \\,\\,\\, {\\mathbb R}^2 -B(x_0, 2\\epsilon_{x_0}\/3). $$\n\nWe write\n\n$$ -\\Delta (v_i \\eta) = W_i e^{u_i} \\eta - 2 \\nabla v_i \\cdot \\nabla \\eta  - v_i \\Delta \\eta. $$\n\nBecause, by Poincar\\'e and Gagliardo-Nirenberg-Sobolev inequalities:\n\n$$ ||v_i||_{q^*} \\leq c_q ||\\nabla v_i||_q \\leq  C_q, \\,\\, 1\\leq q <2, $$\n\nwith, $ q^*=2q\/(2-q) >2 >1 $.\n\n\\smallskip\n\nBy the elliptic estimates, $ (v_i)_i $ is uniformly bounded in $ L^{\\infty}(V_{\\epsilon}(x_0)) $. Finaly, we have, for some $ \\epsilon > 0 $ small enough,\n\n$$ || v_i||_{C^{0,\\theta}[B(x_0, \\epsilon)]} \\leq c_3 \\,\\,\\, \\forall \\,\\,\\, i. $$\n\nNow, we consider a cutoff function $ \\eta \\in C^{\\infty}({\\mathbb R}^2) $ such that\n\n$$ \\eta \\equiv 1 \\,\\,\\, {\\rm on } \\,\\,\\, B(x_0, \\epsilon_{x_0}\/2) \\,\\,\\, {\\rm and } \\,\\,\\, \\eta \\equiv 0 \\,\\,\\, {\\rm on } \\,\\,\\, {\\mathbb R}^2 -B(x_0, 2\\epsilon_{x_0}\/3). $$\n\nWe write\n\n$$ -\\Delta (u_i \\eta) = |x|^{2\\beta} V_i e^{v_i} \\eta - 2 \\nabla u_i \\cdot \\nabla \\eta  - u_i \\Delta \\eta. $$\n\nBy the elliptic estimates, $ (u_i)_i $ is uniformly bounded in $ L^{\\infty}(V_{\\epsilon}(x_0)) $ and also in $ C^{0,\\theta} $ norm.\n\nIf we repeat this procedure another time, we have a boundedness of $ (u_i)_i $ and $ (v_i)_i $ in the $ C^{1,\\theta} $ norm, because they are bounded in $ W^{2,q}\\subset W^{1,q^*} $ norms with $ 2q\/(2-q)=q^* >2 $.\n\nWe have the same computations and conclusion if we consider a regular point $ x_0 $ for the measure $ \\mu_2 $.\n\n\nWe have proved that, there is a finite number of points $ \\bar x_1, \\ldots, \\bar x_m $ such that the squence $ (u_i)_i  $ and $ (v_i)_i $ are locally uniformly bounded (in $ C^{1,\\theta}, \\theta >0 $) in $ \\bar \\Omega - \\{ \\bar x_1, \\ldots , \\bar x_m \\} $.\n\n\\bigskip\n\n\\underbar {\\it Proof of theorem 1.2:} \n\nWithout loss of generality, we can assume that $ 0=x_1 $ is a blow-up point. Since the boundary is an analytic curve $ \\gamma_1(t) $, there is a neighborhood of  $ 0 $ such that the curve $ \\gamma_1 $ can be extend to a holomorphic map such that $ \\gamma_1'(0) \\not = 0 $ (series) and by the inverse mapping one can assume that this map is univalent around $ 0 $. In the case when the boundary is a simple Jordan curve the domain is simply connected. In the case that the domains has a finite number of holes it is conformally equivalent to a disk with a finite number of disks removed. Here we consider a general domain. Without loss of generality one can assume that $ \\gamma_1 (B_1^+) \\subset \\Omega $ and also $ \\gamma_1 (B_1^-) \\subset (\\bar \\Omega)^c $ and $ \\gamma_1 (-1,1) \\subset \\partial \\Omega $ and $ \\gamma_1 $ is univalent. This means that $ (B_1, \\gamma_1) $ is a local chart around $ 0 $ for $ \\Omega $ and $ \\gamma_1 $ univalent. (This fact holds if we assume that we have an analytic domain, (below a graph of an analytic function), we have necessary the condition $ \\partial \\bar \\Omega = \\partial \\Omega $ and the graph is analytic, in this case $ \\gamma_1 (t)= (t, \\phi(t)) $ with $ \\phi $ real analytic and an example of this fact is the unit disk  around the point $ (0,1) $ for example).\n\nBy this conformal transformation, we can assume that $ \\Omega =B_1^+ $, the half ball, and $ \\partial^+ B_1^+ $ is the exterior part, a part which not contain $ 0 $ and on which  $ u_i $ converge in the $ C^1 $ norm to $ u $. Let us consider $ B_{\\epsilon}^+ $, the half ball with radius $ \\epsilon >0 $. Also, one can consider a $ C^1 $ domain (a rectangle between two half disks) and by charts its image is a $ C^1 $ domain)\nWe know that:\n\n$$ u_i \\in C^{2, \\epsilon}(\\bar \\Omega). $$ \n\nThus we can use integrations by parts (Stokes formula). The Pohozaev identity applied around the blow-up $ 0 $:\n\n\\be  \\int_{B_{\\epsilon}^+} \\Delta u_i < x |\\nabla v_i > dx = -  \\int_{B_{\\epsilon}^+} \\Delta v_i < x |\\nabla u_i > dx + \\int_{\\partial^+ B_{\\epsilon}^+}  g(\\nabla u_i, \\nabla v_i)d\\sigma, \\label{(10)}\\ee\n\nThus,\n\n\n\\be  \\int_{B_{\\epsilon}^+} |x|^{2 \\beta} V_i e^{v_i}< x |\\nabla v_i > dx = -  \\int_{B_{\\epsilon}^+} W_ie^{u_i} < x |\\nabla u_i > dx - \\int_{\\partial^+ B_{\\epsilon}^+}  g(\\nabla u_i, \\nabla v_i)d\\sigma, \\label{(11)}\\ee\n\nAfter integration by parts, we obtain:\n\n\n$$  \\int_{B_{\\epsilon}^+} 2V_i(1+\\beta) |x|^{2 \\beta} e^{v_i} dx +  \\int_{B_{\\epsilon}^+} < x |\\nabla V_i > |x|^{2 \\beta} e^{v_i} dx+  \\int_{\\partial B_{\\epsilon}^+} < \\nu |x >|x|^{2 \\beta} V_i e^{v_i} d\\sigma+ $$\n\n$$ + \\int_{B_{\\epsilon}^+} W_i e^{u_i} dx +  \\int_{B_{\\epsilon}^+} < x |\\nabla W_i > e^{u_i} dx+  \\int_{\\partial B_{\\epsilon}^+} < \\nu | x > W_i e^{u_i}  d\\sigma = $$\n\n$$ = - \\int_{\\partial^+ B_{\\epsilon}^+} g(\\nabla u_i, \\nabla v_i)d\\sigma, \\label{(10)} $$ \n\nAlso, for $ u $ and $ v $, we have:\n\n\n$$  \\int_{B_{\\epsilon}^+} 2V(1+ \\beta) |x|^{2 \\beta} e^{v} dx +  \\int_{B_{\\epsilon}^+} < x |\\nabla V > |x|^{2 \\beta} e^{v} dx+  \\int_{\\partial B_{\\epsilon}^+} < \\nu | x >|x|^{2 \\beta} V e^{v}  d\\sigma+ $$\n\n$$ + \\int_{B_{\\epsilon}^+} We^{u} dx +  \\int_{B_{\\epsilon}^+} < x |\\nabla W> e^{u} dx+  \\int_{\\partial B_{\\epsilon}^+} < \\nu | x > W e^{u} d\\sigma = $$\n\n$$ = - \\int_{\\partial^+ B_{\\epsilon}^+} g(\\nabla u, \\nabla v)d\\sigma, $$ \n\nIf, we take the difference, we obtain:\n\n$$ \\int_{\\gamma_1 (B_{\\epsilon}^+)} |x|^{2 \\beta} V_i e^{v_i} dx + \\int_{\\gamma_1 (B_{\\epsilon}^+)} W_i e^{u_i} dx  = o(\\epsilon)+o(1) $$\n\nBut,\n\n $$ \\int_{\\gamma_1 (B_{\\epsilon}^+)} |x|^{2 \\beta} V_i e^{v_i} dx + \\int_{\\gamma_1 (B_{\\epsilon}^+)} W_i e^{u_i} dx  = \\int_{\\partial \\gamma_1 (B_{\\epsilon}^+)} \\partial_{\\nu} u_i d\\sigma + \\int_{\\partial \\gamma_1 (B_{\\epsilon}^+)} \\partial_{\\nu} v_i d\\sigma $$\n\nand,\n\n\\be \\int_{\\partial \\gamma_1 (B_{\\epsilon}^+)} \\partial_{\\nu} u_i d\\sigma + \\int_{\\partial \\gamma_1 (B_{\\epsilon}^+)} \\partial_{\\nu} v_i d\\sigma \n\\to \\alpha_1+\\beta_1 >0 \\label{(12)}\\ee\n\na contradiction.\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{sec:Intro}}\n\n\nThe presence of additional light scalar degrees of freedom is a common feature of many \nextensions of the Standard Model (SM).  Examples include the \nQCD axion~\\cite{PecceiQuinn1,PecceiQuinn2,WeinbergAxion,WilczekAxion}, a more\ngeneral set of axions and axion-like particles~\\cite{LowEnergyFrontier}, \nmajorons~\\cite{ChikashigeMajorons,GelminiMajorons}, familons~\\cite{WilczekFamilon}, \nchameleons~\\cite{RatraChameleons,KhouryChameleons1,KhouryChameleons2,BraxChameleons},\nbranons~\\cite{DobadoBranon}, dilatons, and a variety of other string and geometric \nmoduli~\\cite{Polchinski,BBS,GSW}.  Scalars of this sort are typically \nlight due to symmetries of the high-scale theory which prevent them from acquiring masses.\nAs a result, masses for these particles must be generated by dynamical processes at lower scales \nwhich break these symmetries --- often in conjunction with a cosmological\nphase transition.  This dynamics also generically leads to a non-trivial time-dependence for \nthe masses of these fields, the details of which can have a significant impact on  \ntheir late-time energy densities.  Understanding these effects is crucial, since    \nsuch particles can potentially have a variety of phenomenological and cosmological \nconsequences, both helpful and harmful.  Indeed, scalars of this sort can \ncontribute to the total abundances of dark matter or dark energy; induce additional, \nlate periods of reheating after cosmic inflation, potentially resulting in substantially \nmodified cosmologies~\\cite{WatsonNonThermalHistories};  \nprematurely induce matter-domination and\/or overclose the universe; \nand disrupt the formation of light elements through late decays. \n\nWhile the detailed dynamics of mass generation can have important consequences\neven in scenarios involving only a single light scalar, the situation becomes far \nricher in scenarios involving multiple such fields.  \nIndeed, in cases in which these scalars have the same quantum numbers, the \ndynamics of mass generation can lead them to experience time-dependent mixing.\nThis mixing in turn leads to a \ncontinual redistribution of energy density among the scalars during the mass-generation\nepoch.  Such a redistribution can also have profound effects on the late-time energy densities\nof these fields, several of which were pointed out in Ref.~\\cite{TwoTimescales}.  Specifically,\neven in a theory involving only two scalars, such a redistribution can lead their late time \nabundances to experience either an enhancement or a suppression,  depending on\nthe choice of model parameters.  Moreover, in certain regimes, the system may experience\na parametric resonance which can have further dramatic effects on these abundances.\nFinally, the system may also experience a ``re-overdamping'' phenomenon \nin which its energy density reverts to behaving like vacuum energy \nat a time after its initial transition from the overdamped to underdamped regime.\n\nA variety of scenarios for new physics predict large numbers of scalar fields with \nsimilar quantum numbers.  Indeed, such collections of fields emerge naturally from a \nvariety of string constructions~\\cite{WittenStringAxion,SvrcekWitten}, from axiverse \nconsiderations~\\cite{Axiverse}, and more \ngenerally from theories with extra dimensions.  Of course, many features of such \ntheories are highly model-dependent, including the number of fields present, \nthe form of the mass-squared matrix for these fields (both at very early and at very \nlate times), and the time or temperature scales associated with the dynamical mechanisms \nfor mass generation.  However,\na given specific \nnew-physics context is likely to lead to certain theoretical structures in which\nsome of these features become fixed.\nFor example, theories in which a scalar propagates in extra spacetime dimensions  give \nrise to infinite numbers of scalar fields which are organized into Kaluza-Klein (KK) \ntowers in which the form of the mass matrix at early times --- even prior to the onset \nof any mass-generating phase transition --- is dictated by the geometry of the \ncompactification manifold.  Because such KK towers arise generically in scenarios \ninvolving compact extra dimensions (such as string theories, supergravity theories, and \nmany other high-scale unification models),\nthe features associated with the cosmological evolution of such KK towers --- including \nthe dynamical flow of energy amongst the different KK modes prior to, during, and after\ncosmological phase transitions --- are likely to play an important role in \nearly-universe cosmology.\n\nIn this paper we tackle this important question.  In particular, we study\nthe early-universe cosmology of a KK tower of scalar fields in the presence\nof a mass-generating phase transition, focusing on the time-development of the total \ntower energy density (which determines the relic abundance) as well as its distribution across the different \nKK modes.  On the one hand, this study is critical for understanding the cosmological \nissues associated with extra spacetime dimensions {\\it per se}\\\/.  On the other hand, \nthis study also represents an extension of the analysis presented in \nRef.~\\cite{TwoTimescales}.  In that analysis, only two scalar fields were considered, \nbut these fields were permitted to experience the most general possible\nmixings between them.  In this analysis, by contrast, the number of scalar fields is \ninfinite but the structure of the KK tower and the interplay between brane and bulk \nphysics imposes a strict mixing structure.  A goal of this paper is therefore to \nunderstand the consequences of this simultaneous extension in the number of\nfields and narrowing of the allowed mixing structure. \n\nAs we shall see, studying the dynamics of a complete KK tower can be a significant\nundertaking due to its infinite number of fields --- especially in the presence of \na non-trivial, time-dependent, mass-generating and mixing-inducing phase transition.\nTherefore, in this paper, we shall work towards this goal in distinct stages by\nstarting with only one KK mode and then introducing more and more KK excitations into \nthe system.  Ultimately, we shall build towards the full, infinite  KK tower.\nIn each case, our goal is to trace the flow of energy through the system and to examine \nthe behavior of such quantities as the total late-time energy density as well as its \ndistribution across the available KK modes.  Moreover, our goal is to retain as much \ngenerality as possible.  We shall therefore refrain from specifying the exact nature \nof our KK tower other than to specify that its constituents are scalars.   Our results \nwill therefore be broadly applicable to any of the particular scalar fields listed above.\n\nThis paper is organized as follows.  In Sect.~\\ref{sec:TheModel}, we begin by establishing \nthe context in which we shall be working and review the general properties of our setup.\nThen, in Sect.~\\ref{sec:FourDimensionalLimit}, we begin our study by considering the \ncase of only a single scalar field undergoing a mass-generating phase transition.\nIn particular,\nour analysis will extend beyond the usual ``adiabatic'' and ``abrupt'' regimes commonly \ndiscussed in the literature.\nIn Sect.~\\ref{sec:FiniteModeNumber}, we then broaden our perspective by\nconsidering the case of only the $N$ lowest-lying KK excitations, where $N$\nis arbitrary but finite.  As we shall see, situations with finite $N$ have a number of \nproperties which render their cosmological evolution somewhat distinct.\nIn Sect.~\\ref{Approaching}, we then examine how our system behaves as a function of \n$N$ for \\il{N\\gg 1}, \nand in Sect.~\\ref{sec:KKLimit} we present our results for the full, infinite KK tower.\nIn order to distinguish this case from the large-$N$ case\nin Sect.~\\ref{Approaching}, we shall refer to the KK tower as representing the case with \\il{N=\\infty}.\nUntil this point, our analysis is completely general and applicable to any scalar\nfield.  However, in Sect.~\\ref{sec:AxionintheBulk}, we then apply our machinery to the case\nin which our scalar is an axion or axion-like particle.  This allows us to interpret our \nresults within a particular phenomenologically relevant context, and illustrate the \nsignificant phenomenological implications that our results can have.  Finally, in \nSect.~\\ref{sec:Conclusions}, we summarize our main conclusions and discuss further \npotential consequences of our results.\n\nAs evident from this outline, our method for studying the properties of a full KK theory \ninvolves truncating the KK tower to its $N$ lowest-lying modes and then considering the \n\\il{N\\to\\infty} limit.  Indeed, this is a valid, standard approach.  However, as a by-product \nof our analysis, we shall discover that such a truncation is not unique.  Specifically, we \nshall develop an alternate, ``UV-based'' effective truncation of KK theories which has a \nnumber of interesting theoretical properties that distinguish it from the more traditional \n``IR-based'' truncation\ntypically used in the extra-dimension literature.  Both truncations lead to the same \nphysics as \\il{N\\to\\infty}, but differ significantly for finite $N$.\nThis alternate truncation will be discussed in Sect.~\\ref{alttrunc}.\n\n\n\\FloatBarrier\n\\section{The Framework\\label{sec:TheModel}}\n\n\n\nIn this section, we establish the framework in which our analysis\nwill take place.\nThis framework is similar to that considered in Refs.~\\cite{DDGAxions,DDM1,DDM2,DDMAxion}, \nand is essentially a KK-oriented extension\nof the framework considered in Ref.~\\cite{TwoTimescales}. \nSpecifically, we consider a flat, five-dimensional spacetime geometry\nof the form \\il{{\\cal M} \\times S^1\/\\IZ_2}, where\n${\\cal M}$ denotes our usual four-dimensional Minkowski spacetime and\nwhere $S^1\/\\IZ_2$ denotes an orbifolded circle ({\\it i.e.}\\\/, a line segment) of \nlength $2\\pi R$.\nOur spacetime coordinates shall be denoted \\il{(x^\\mu,x^5)}, \\il{\\mu=0,1,2,3}, \nwhere \\il{0\\leq x^5 \\leq 2\\pi R} and where the orbifold action is given by \\il{x^5\\to -x^5}.\nWe shall further imagine that\nour usual Standard-Model (SM) fields and interactions are confined to a four-dimensional\nbrane localized at the\norbifold fixed point at \\il{x^5=0}.\n\nWithin this geometry we consider \na five-dimensional scalar field $\\Phi(x^\\mu, x^5)$  and a\ncorresponding five-dimensional action ${\\cal S}$ of the form\n\\begin{equation}\\label{eq:5Daction}\n {\\cal S} ~=~ \\int d^4x\\,dx^5 \\left[\\mathcal{L}_{\\rm bulk}(\\Phi)\n+ \\delta(x^5)\\mathcal{L}_{\\rm brane}(\\psi_i,\\Phi)\\right] \n\\end{equation}\nwhere $\\psi_i$ are fields confined to the SM brane. \nIn general, our bulk action will contain\ngeneric kinetic and mass terms of the form \n\\begin{equation}\\label{eq:generalLbulk}\n\\mathcal{L}_{\\rm bulk}(\\Phi) ~=~ \\frac{1}{2}\\partial_M\\Phi^*\\partial^M\\Phi - \\frac{1}{2}M^2\\abs{\\Phi}^2~.\n\\end{equation}\nHowever, many well-motivated \nscalars such as moduli and axions have a primordial shift \nsymmetry under which \\il{\\Phi\\to \\Phi+c} for arbitrary constants $c$. \nWith this motivation in mind, we shall henceforth assume that \\il{M=0}.\n\nThe brane Lagrangian $\\mathcal{L}_{\\rm brane}$, on the other hand, \nincludes the SM Lagrangian in addition \nto interactions between $\\Phi$ and the SM fields. In general, these interactions can\nopen up decay channels from $\\Phi$ to SM states.   However, such interactions can also lead\nan {\\it effective}\\\/ four-dimensional mass for $\\Phi$ on the brane.\nSuch masses are typically generated as the result of some brane-localized dynamics \n(such as a brane-localized phase transition) \nwhich occurs at some time during the cosmological evolution on the brane\nand which explicitly breaks the shift symmetry. \nFor this reason, we shall allow our brane Lagrangian to include\na time-dependent effective brane-mass term of the form\n\\begin{equation}\\label{eq:Lbrane}\n{1\\over {\\cal V}}\\, \\mathcal{L}_{\\rm brane}(\\Phi) ~=~ -  \\frac{1}{2}m^2(t) \\abs{\\Phi}^2 + ...  \\ \n\\end{equation}\nwhere \\il{{\\cal V}\\equiv 2\\pi R} is the compactification volume.\nThe behavior of $m(t)$ is ultimately determined by the details of the non-perturbative dynamics on \nthe brane and thus highly model-dependent. \n\nThis information alone is sufficient to allow us to determine\nthe effective four-dimensional theory that emerges\nupon compactification.\nAssuming that $\\Phi$ has even parity under the orbifold action \\il{x^5\\to -x^5},\nwe may perform a Kaluza-Klein expansion of the form\n\\begin{equation}\n\\Phi(x^{\\mu},x^5) ~=~ \\frac{1}{\\sqrt{\\cal V}}\\sum_{k=0}^{\\infty}r_k\n\\phi_k(x^{\\mu})\\cos\\left(\\frac{kx^5}{R}\\right) \\ \n\\label{eq:orbifoldmodeexpansion}\n\\end{equation}\nwith normalization constants \\il{r_0=1} and \\il{r_{k} =\\sqrt{2}} for all \\il{k>0}, as appropriate\nfor a $S^1\/\\IZ_2$ compactification.\nInserting this into our action and integrating over $x^5$ we obtain\nan effective four-dimensional action of the form \n\\begin{align}\n\\mathcal{L}_{\\rm eff} ~=~ \\sum_{k=0}^{\\infty}\\left[\\frac{1}{2}(\\partial_{\\mu}\\phi_k)^2\n- {1\\over 2}\\sum_{\\ell=0}^{\\infty}\\phi_k\\mathcal{M}^2_{k\\ell}(t)\\phi_\\ell\\right] ~+~...~~~ \n\\label{eq:4Dlagrangian}\n\\end{align}\nwhere our time-dependent mass matrix ${\\cal M}_{k\\ell}^2$ is given by \n\\begin{eqnarray}\n     {\\cal M}_{k\\ell}^2 &=& k\\ell \\, \\delta_{k\\ell} \\, M_c^2 ~+~\n       r_k r_\\ell\\, m^2 (t)~\\nonumber\\\\\n&=& m^2(t)\n\\begin{bmatrix}\n1  & \\sqrt{2} & \\sqrt{2} & \\cdots \\\\\n\\sqrt{2} & ~~2 + \\frac{M_c^2}{m^2(t)} & 2 & \\cdots \\\\\n\\sqrt{2} & 2 & ~~2 + \\frac{4M_c^2}{m^2(t)}~ & \\cdots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots \n\\end{bmatrix} ~~~~~\n\\label{kmassmatrix}\n\\end{eqnarray}\nwith \\il{M_c\\equiv 1\/R}.\nThe first term in the top line of Eq.~(\\ref{kmassmatrix}) represents the expected KK contribution\nfrom the physics in the bulk,\nwhile the second term reflects the contribution from physics on the brane\nwhich breaks the translational symmetry in the $x^5$-direction and thereby\nintroduces a mixing between the different KK modes.\n\nGiven the Lagrangian in Eq.~(\\ref{eq:4Dlagrangian}), \nit follows that in a flat Friedman-Robertson-Walker (FRW) \nspacetime the zero-momentum modes of the KK scalars $\\phi_k$ obey the coupled equations of motion\n\\begin{equation}\n\\ddot{\\phi}_k + 3H(t)\\dot{\\phi}_k + \\sum_{\\ell = 0}^{\\infty}\\mathcal{M}^2_{k\\ell}(t)\\phi_\\ell ~=~ 0 ~\n\\label{eq:KKequationsofmotion}\n\\end{equation}\nwhere \\il{H(t)\\equiv \\kappa\/(3t)} is the Hubble parameter.\nNote that \\il{\\kappa=2} and \\il{\\kappa=3\/2} for matter- and radiation-dominated epochs, respectively.\nIf ${\\cal M}^2_{k\\ell}$ were constant as a function of time, it would be possible to \npass to a mass-eigenstate basis \\il{\\lbrace\\phi_\\lambda\\rbrace} \nin which the different differential equations in Eq.~(\\ref{eq:KKequationsofmotion})\nwould decouple.  Unfortunately, the time-dependence of $m(t)$ leads to a time-dependence\nfor ${\\cal M}^2_{k\\ell}$.    This time-dependence induces an unavoidable mixing between the different\nmodes, since even the mass-eigenstate basis is continually changing.\nIt is this feature which underpins the non-trivial dynamics we shall be studying in this paper.\nLikewise, at any moment in time, the total energy density associated \nwith our system of KK modes is given by\n\\begin{equation}\n\\rho ~=~ {1\\over 2}\\sum_k \\dot\\phi_k^2 + \n         {1\\over 2}\\sum_{k,\\ell} \\phi_k {\\cal M}_{k\\ell}^2 \\phi_k~.\n\\label{energydensity}\n\\end{equation}\nIf the mass matrix ${\\cal M}_{k\\ell}^2$ were time-independent, it would be\npossible to rewrite this energy density in the mass-eigenstate basis \nas the sum of individual contributions: \n\\begin{equation}\n            \\rho  ~=~ \\sum_\\lambda \\rho_\\lambda\n\\label{energydecomp}\n\\end{equation}\nwhere\n\\begin{equation} \n\t\\rho_\\lambda ~=~ {1\\over 2}\\left(\n\t\\dot\\phi_\\lambda^2 + \\lambda^2 \\phi_\\lambda^2 \\right)~.\n\\label{rholambda}\n\\end{equation}\nUnfortunately, once again, the time-dependence of the mass matrix renders such a decomposition impossible.\nThus, we may consider the total energy density in Eq.~(\\ref{energydensity}) as receiving contributions\nfrom individual mass eigenstates as in Eq.~(\\ref{energydecomp}) only during those periods of cosmological evolution\nfor which the mass matrix might not be changing with time. \n\nIn general, we shall assume that the mass $m(t)$ is generated through some sort of phase\ntransition on the brane whose occurrence is centered at some time $t_G$ in cosmological history  \nand which requires a duration $\\Delta_G$ over which to unfold.\nWe shall therefore assume that \\il{m(t)=0}\nlong before the phase transition ({\\it i.e.}\\\/, for \\il{t\\ll t_G}), \nwhile we shall assume that $m(t)$ is given by some fixed mass $\\overline{m}$ long after the phase\ntransition ({\\it i.e.}\\\/, for \\il{t\\gg t_G}).\nThe precise details of how $m(t)$ evolves from $0$ to $\\overline{m}$ clearly depend on the particular\nstructure of the phase transition in question, but we shall find that the most important feature of\nthis function will be its ``width'' $\\Delta_G$. \nFor concreteness, we shall therefore adopt the function~\\cite{TwoTimescales} \n\\begin{equation}\nm(t) ~=~\n\\frac{1}{2}\\,{\\overline{m}}\\, \\left\\{1 + \n\\text{erf}\\left[\\frac{1}{\\sqrt{2}\\delta_G}\\log\\left(\\frac{t}{t_G}\\right)\\right]\\right\\} \\ ,\n\\label{sigmoid}\n\\end{equation}\nwhere ${\\overline{m}}$ is the mass at asymptotically late times, where $t_G$ is the central time\ndefined by \\il{m(t_G)={\\overline{m}}\/2}, and where `erf' denotes the so-called ``error'' function. \nIn Eq.~(\\ref{sigmoid}), the quantity \\il{\\delta_G\\in\\left[0,1\\right]} is a \ndimensionless parameter describing the ``width'' of the phase transition,\nwith  \\il{\\delta_G=0} corresponding to an essentially\ninstantaneous phase transition and \\il{\\delta_G=1} describing a phase transition\nwhich proceeds as slowly as possible.\nGiven the dimensionless parameter $\\delta_G$, the physical timescale $\\Delta_G$ over which\nthe phase transition occurs is then given by~\\cite{TwoTimescales}\n\\begin{equation}\n\\Delta_G ~\\equiv~ \\sqrt{2\\pi}\\delta_G t_G \\ .\n\\label{bigwidth}\n\\end{equation}\nIndeed, given Eq.~(\\ref{sigmoid}), we may view the transition from \\il{m=0} to \\il{m={\\overline{m}}} as \noccurring over a ``width'' $\\Delta_G$ logarithmically centered at $t_G$. \n\nThis choice of function is discussed more fully in Ref.~\\cite{TwoTimescales}.  Indeed, it is \nshown in Ref.~\\cite{TwoTimescales} that this function models the time-dependence of a variety \nof phase transitions extremely well in the neighborhood of $t_G$, where the mass function is \nchanging most rapidly.   This includes the instanton-induced phase transition \nthat gives mass to the QCD axion.   Of course, as discussed in \nRefs.~\\cite{WantzShellard1,WantzShellard2,diCortonaQCDAxion,BorsanyiQCDAxion},\nthe actual QCD phase transition leads to a mass function $m(t)$ which exhibits a power-law \ndependence on $t$ at early times $t\\ll t_G$, and which is then often taken to be followed by \none or more slope discontinuities near \\il{t\\sim t_G} before becoming a constant for \\il{t\\gg t_G}.    \nBy contrast, our functional form for $m(t)$ is designed to interpolate smoothly between fixed \nearly- and late-time values in such a way that there is an adjustable and easily identifiable \n``width'' $\\Delta_G$.  Despite these differences, however, it is shown in Ref.~\\cite{TwoTimescales} \nthat our functional form models the QCD functional form extremely well near \\il{t\\sim t_G}.\n \nOf course, many other choices are possible for the particular functional form of \n$m(t)$, with different choices being more or less suitable in different contexts.\nHowever, none of the qualitative results of this paper will ultimately depend on \nthese specific details.\n\nIn general, we may choose to consider phase transitions with any \\il{\\delta_G\\leq 1}.\nIndeed, as discussed in Ref.~\\cite{TwoTimescales}, situations with \\il{\\delta_G > 1} suffer\nfrom an unphysical boundary artifact near \\il{t\\approx 0}, namely the emergence of a ``fake'' \nmass-generating phase transition whose width tends to decrease as $\\delta_G$ increases beyond \\il{\\delta_G=1}.\nHowever, when \\il{{\\overline{m}}\/M_c\\gg 1}, it turns out that an additional\nboundary effect emerges which restricts $\\delta_G$ even more strongly, ultimately imposing an upper limit on $\\delta_G$ which\nis less than $1$ and which depends on ${\\overline{m}}\/M_c$.   \n(Specifically, as $\\delta_G$ increases for \\il{{\\overline{m}}\/M_c\\gg 1}, the effective ``mixing angle'' $\\theta$ between our ground\nstate and the first excited state starts changing, evolving from \\il{\\theta=0} to \\il{\\theta=\\overline{\\theta}} \nmore and more rapidly and at earlier and earlier\ntimes.  Ultimately we reach a new critical value of $\\delta_G$ beyond which the $\\theta$-width $\\Delta_\\theta$ in\nEq.~(4.3) of Ref.~\\cite{TwoTimescales} actually starts to {\\it decrease}\\\/ when $\\delta_G$ is\nincreased, giving the appearance of a new ``fake'' \nphase transition at early times near $t\\approx 0$.)    \nAvoiding this unphysical behavior then \nimposes an additional upper bound on $\\delta_G$\nwhich we can evaluate numerically as a function of ${\\overline{m}}\/M_c$ and $N$.\nIn this paper we shall therefore restrict our attention to situations with \\il{\\delta_G\\leq 1} but also impose\nthis additional constraint as appropriate for large ${\\overline{m}}\/M_c$. \n\nIn this context, we note that\ntheories with extra spacetime dimensions must generally obey additional ``normalcy-temperature'' \nconstraints~\\cite{ADD,ADDPheno} \nwhich are more easily satisfied within cosmological low-temperature reheating (LTR) scenarios \nin which inflation occurs late and reheating occurs at temperatures as low as \n\\il{T_{\\rm RH}\\sim \\mathcal{O}(\\text{MeV})}~\\cite{KawasakiLTR}. \nIf we assume the dynamics that generate $m(t)$ are at higher scales than $T_{\\rm RH}$, then our \nmass-generating phase transition can be assumed to occur during \nan inflaton-dominated --- {\\it i.e.}\\\/, a matter-dominated --- epoch.\nHowever this assumption will not play a significant role in our analysis, and the primary\nresults of this paper will hold regardless of the specific cosmological timeline assumed.\n\nGiven the differential equations of motion in Eq.~(\\ref{eq:KKequationsofmotion}),\nthe only ingredients of our model remaining to be specified \nare the initial conditions on each field mode $\\phi_k$.   \nWhile in principle many possibilities \nexist, \nthe existence of the shift symmetry \\il{\\Phi \\rightarrow \\Phi + c} prior to mass\ngeneration on the brane suggests \nthat $\\Phi$ might have an arbitrary fixed displacement at early times \\il{t_i\\ll t_G},\nwith a fixed non-zero vacuum expectation value (VEV) \\il{\\langle \\Phi\\rangle\\neq 0}.\nUpon KK reduction, this corresponds to initial conditions given by\n\\begin{align}\\label{eq:initialconditions}\n\\phi_k(t_i) ~&=~ \\expt{\\phi_0} \\delta_{0k} \\nonumber \\\\\n\\dot{\\phi}_k(t_i) ~&=~ 0 \\ ,\n\\end{align}\nwhere \\il{\\expt{\\phi_0}\\equiv \\sqrt{2\\pi R}\\expt{\\Phi}}.\nWe shall therefore take these to be the initial \nconditions for our differential equations (\\ref{eq:KKequationsofmotion}) \nin what follows.\nIt is important to note that these initial conditions \nhave several additional advantages beyond their natural origins outlined above.\nFirst, as long as the initial time $t_i$ is sufficiently prior to $t_G$, our ensemble of KK modes\nwill have zero energy.   Thus, as we shall see, {\\it all energy accrued by our KK system is \nsolely the result of the phase transition on the brane}\\\/. \nMoreover, these initial conditions, being essentially static,\nfree our system and its subsequent dynamics\nfrom all details concerning\nthe generation of the initial VEV \\il{\\langle \\Phi\\rangle}.\nAs a result, the time at which the initial VEV \\il{\\langle \\Phi\\rangle} \nis generated is arbitrary and we need not concern ourselves with its origins. \nIndeed, as we shall see, the precise value of $t_i$ will not affect our analysis, or any of our conclusions.\nFinally, we remark that while $\\expt{\\phi_0}$ sets an overall energy scale, this scale\nwill be irrelevant for our purposes since the equations of motion are linear and\nour main interest will be on relative {\\it comparisons}\\\/ between energy scales \nrather than their absolute magnitudes.\n\nFinally, let us discuss the masses of the individual KK modes in this model.\nEven though our overall brane mass $m(t)$ is continually changing in time,\ndiagonalizing the matrix in Eq.~(\\ref{kmassmatrix}) at any moment in time\nleads directly to the corresponding instantaneous (mass)$^2$ eigenvalues $\\lambda_k^2$.\nWhile no closed-form analytical expressions for these eigenvalues\nexist, they may be readily approximated in the \\il{m\/M_c\\ll 1} and \\il{m\/M_c\\gg 1} limits.\nIn the \\il{m\/M_c\\ll 1} limit,\nwe find\n\\begin{equation}\n      \\lambda_k ~\\approx~ \\begin{cases}\n              m  &   k=0\\cr\n             k M_c & k>0~.\n           \\end{cases}\n\\label{eq:unmixedmassspectrum}\n\\end{equation}\nThere are two ways to interpret this result.\nIf we imagine holding $M_c$ fixed, this limit corresponds to taking $m$\nextremely small.\nThis renders the brane irrelevant in the KK mass decomposition, and\nindeed \nthe results in Eq.~(\\ref{eq:unmixedmassspectrum})\nare then the eigenvalues expected from a straightforward compactification on a circle.\nAlternatively, if we imagine holding $m$ fixed, we see that this limit corresponds\nto taking $M_c$ extremely large.   \nWe then have only a single light mode with mass \\il{\\lambda_0\\approx m}, which is nothing\nbut the four-dimensional limit.\nBy contrast, for \\il{m\/M_c\\gg 1},\n the mixing between the modes is maximized and our corresponding eigenvalues are given \nby~\\cite{DDGAxions,DDM1} \n\\begin{equation}\n      \\lambda_k ~\\approx~ (k+1\/2) \\,M_c~.\n\\label{kmixedlimit}\n\\end{equation}\nIt is remarkable that the effect of the brane mass in this limit is entirely $m$-independent, and merely amounts\nto shifting our eigenvalues by $M_c\/2$.   Indeed, these shifted eigenvalues are those that would have emerged from \nan {\\it anti}\\\/-periodic compactification on a circle.\nFinally, for intermediate values of $m\/M_c$, we find that our eigenvalues \n$\\lambda_k$\ntend to follow Eq.~(\\ref{kmixedlimit}) for \\il{k\\ll \\pi m^2\/M_c^2} and   \nEq.~(\\ref{eq:unmixedmassspectrum}) for \\il{k\\gg \\pi m^2\/M_c^2},\nwith $\\lambda_k$ taking values that smoothly evolve between these two extremes for other values\nof $k$.\n\nIt is important to stress that this behavior for the eigenvalues emerges\nonly in the limit in which we consider the full KK tower with its infinite complement\nof KK modes, \\il{k=0,1,...,\\infty}.\nBy contrast, if we truncate the mass matrix in Eq.~(\\ref{kmassmatrix}) to its first\n$N$ rows and columns so that \\il{0 \\leq \\lbrace k,\\ell\\rbrace \\leq N-1}, \nour general expectations described above continue to apply\nonly for the eigenvalues \\il{\\lambda_0, ..., \\lambda_{N-2}}.\nHowever, for the highest eigenvalue $\\lambda_{N-1}$, we find that the above\nexpectations continue to apply only for \\il{m^2\/M_c^2 \\lsim {\\cal O}(\\sqrt{N})}.\nBy contrast, for \\il{N\\gg 1} and \nfor \\il{m^2\/M_c^2 \\gsim {\\cal O}(\\sqrt{N})},\nwe instead find that\n\\begin{equation}\n     \\lambda_{N-1} ~\\approx~  \\sqrt{2 N} \\, m ~.\n\\label{artifact}\n\\end{equation}\nThe normalized eigenvalue $\\lambda_{N-1}\/M_c$ thus diverges as \\il{m\/M_c\\to \\infty}.\n\n\\begin{figure}\n\\includegraphics[keepaspectratio, width=0.5\\textwidth]{massspectrumlinear.pdf}\n\\caption{The mass spectrum $\\lambda_k$ for \\il{N=10}, plotted as a function of $m\/M_c$.\nWhen \\il{{\\overline{m}}\\ll M_c}, \nour mass eigenvalues follow the simple form given in Eq.~(\\ref{eq:unmixedmassspectrum}).\nBy contrast, when\n\\il{{\\overline{m}}\\gg M_c},\nall but the highest mass eigenvalue shift upwards\nby $M_c\/2$, while the highest mass eigenvalue diverges according\nto Eq.~(\\ref{artifact}) for \\il{m^2\/M_c^2\\gsim {\\cal O}(\\sqrt{N})}.\nAs discussed in the text, this anomalous behavior\nof the highest mass eigenvalue is an artifact \nof having truncated our KK tower to only a finite number of states, and \ndisappears as \\il{N\\to \\infty}.}\n\\label{fig:massspectrum}\n\\end{figure}\n \nThe behavior of the eigenvalues $\\lambda_k$ is illustrated in Fig.~\\ref{fig:massspectrum},\nwhere we have plotted the values of $\\lambda_{k=0,...,9}$\nas functions of $m\/M_c$ for \\il{N=10}.\nAs is clear from Fig.~\\ref{fig:massspectrum}, the somewhat anomalous behavior of \n$\\lambda_{N-1}$ is ultimately an artifact of truncating \nour KK tower to its first $N$ values.\nHowever, as $N$ increases, this anomalous behavior of $\\lambda_{N-1}$ \nsets in only at increasingly \nlarge values of $m\/M_c$.\nThus, in the \\il{N\\to\\infty} limit, this anomalous behavior disappears\naltogether, as expected.\n\nIn our model the quantity $m$ is  \ngenerally time-dependent, rising from zero to some final, fixed late-time value ${\\overline{m}}$\nas the result of our mass-generating phase transition on the brane.\nThe corresponding eigenvalues $\\lambda_k^2$ are thus set  \nby Eq.~(\\ref{eq:unmixedmassspectrum}) with \\il{m=0} at early times, but \nsubsequently evolve towards late-time results which are governed \nby the magnitude of the late-time ratio ${\\overline{m}}\/M_c$.\nMeanwhile, each of the corresponding KK fields $\\phi_k$ is evolving \naccording to Eq.~(\\ref{eq:KKequationsofmotion}), with the non-diagonal\nmass matrix ${\\cal M}^2_{k\\ell}$ inducing a time-dependent mixing between the KK modes\nand even between their instantaneous mass eigenstates.\nThis, coupled with the continuously decreasing Hubble friction term $H(t)$ in \nEq.~(\\ref{eq:KKequationsofmotion}), leads to a highly non-trivial dynamical system,\nwith energy slowly being introduced into the system as the result of the mass-generating\nphase transition while it is simultaneously and continually redistributed across the different\nmodes and dissipated from each mode.\nUnderstanding this energy flow is critical if we are to properly\nunderstand the cosmological implications of such KK scalar towers in the presence\nof mass-generating phase transitions in the early universe. \nThis is therefore the main task to which we now turn.\n\n\n\\FloatBarrier\n\\section{\\texorpdfstring{$N=1$}{N=1}:  ~Mapping out the 4D Limit \\label{sec:FourDimensionalLimit}}\n\n\nWe begin our analysis by focusing on the simplest case:  that of only one scalar field, {\\it i.e.}\\\/, the case\nwith \\il{N=1}.   This case lacks any dependence on $M_c$, and therefore can be \nconsidered equivalent to  a four-dimensional limit in which \\il{M_c\\gg m(t)} \nat all times (so that all KK modes with \\il{k\\geq 1} effectively decouple from the problem).\nThis case of a single scalar field $\\phi(t)$ undergoing a mass-generating phase transition\nhas been studied extensively in the literature (see, {\\it e.g.}\\\/, Refs.~\\cite{TurnerEffect,UpdatedAxionAbundanceCalc}), \nbut only in certain ``adiabatic'' or ``abrupt'' limits.\nIn this section, by contrast, we shall provide a complete mapping of the entire relevant \nparameter space.\n\nIn general, given the equation of motion in Eq.~(\\ref{eq:KKequationsofmotion}),\nwe see that the field $\\phi(t)$ simply follows the trajectory of a damped harmonic oscillator with time-dependent\ncritical damping coefficient \\il{\\zeta \\equiv 3H\/2m}.  Such an oscillator is overdamped if \\il{\\zeta>1} \nand only experiences oscillations \nin the underdamped regime with \\il{\\zeta<1}.\nSince \\il{H(t)\\sim \\kappa\/(3t)} \nis monotonically decreasing while $m(t)$ is monotonically increasing,  \nwe see that $\\zeta$ is constantly decreasing.\nAs a result,\nthe field $\\phi(t)$ is necessarily overdamped at times \\il{t \\ll t_G} \nand does not begin to undergo coherent oscillations until \\il{\\zeta = 1}.\nWe shall let $t_\\zeta$ denote this time at which such coherent oscillations begin.\nIn general, we know that $t_\\zeta$ cannot come too much earlier than $t_G$ \n({\\it i.e.}\\\/, we cannot have \\il{t_G-t_\\zeta \\gg \\Delta_G}) since our\nfield $\\phi$ is presumed massless prior to the onset of the phase transition.\nLikewise, if $t_{\\zeta}$ occurs long after the phase transition has \nalready occurred ({\\it i.e.}\\\/, if \\il{t_\\zeta - t_G\\gg \\Delta_G}), the specific details of\nthe phase transition such as its shape or width will have a negligible effect \non the resulting dynamics and on the corresponding energy \ndensity $\\rho$. However, if \\il{|t_{\\zeta}-t_G|\\lsim \\Delta_G},\nthe specific properties of the phase transition can be significant.\n\nTwo approximations are typically used in the literature in order to estimate the effects of\nthe phase transition  within this third regime.\nBoth rest upon considering extreme limits of the phase-transition width $\\Delta_G$. \nIf the width is large enough so that \\il{\\dot m\/m \\ll m} at all times during the  \nfield oscillations ({\\it i.e.}\\\/, at all times after $t_\\zeta$),\nthen the field undergoes many oscillations during the phase transition. \nThis is the so-called \\emph{adiabatic approximation}:  the fields remain virialized during the phase transition and we \ncan approximate the energy density by\n\\begin{equation}\n\\rho_{\\rm 4D}(t) ~\\approx~ \n\\frac{1}{2}\\, \\langle\\phi_0\\rangle^2 \\, m(t_{\\zeta}) \\, m(t)\\, \\left[\\frac{a(t_{\\zeta})}{a(t)}\\right]^3 \n\\label{eq:adiabaticapproximation}\n\\end{equation}\nwhere \\il{a(t)\\sim t^{\\kappa\/3}} is the cosmological scale factor.\nBy contrast, if the phase transition width $\\Delta_G$ is extremely small, we are in the so-called\n\\emph{abrupt approximation}:   our assumption that\n\\il{|t_{\\zeta}-t_G|\\lsim \\Delta_G}\nforces $t_\\zeta$ and $t_G$ to coincide, and the phase transition happens so rapidly \nthat the field $\\phi(t)$ retains its initial value \\il{\\langle\\phi_0\\rangle} \nuntil $t_G$ (or equivalently $t_\\zeta$), at which point it immediately begins oscillating coherently.\nThe evolution of the energy density is then easily obtained by solving \nEq.~\\eqref{eq:KKequationsofmotion} with \\il{N=1} analytically.   This yields the exact solution\n\\begin{equation}\n\\rho_{\\rm 4D}(t) ~=~ \\frac{\\pi^2}{8}\\, \\langle\\phi_0\\rangle^2\\, {\\overline{m}}^4\\, \\frac{t_G^{\\kappa+1}}{t^{\\kappa-1}}\\, \n\\left[B_1^2(t) +B_2^2(t)\\right] \\ ,\n\\label{eq:abruptapproximation}\n\\end{equation}\nwhere \n\\begin{eqnarray}\nB_1(t) &\\equiv& J_{\\kappa_+}({\\overline{m}} t_G)Y_{\\kappa_-}({\\overline{m}} t) - \nJ_{\\kappa_-}({\\overline{m}} t)Y_{\\kappa_+}({\\overline{m}} t_G) \\nonumber \\\\\nB_2(t) &\\equiv& J_{\\kappa_+}({\\overline{m}} t)Y_{\\kappa_+}({\\overline{m}} t_G) - \nJ_{\\kappa_+}({\\overline{m}} t_G)Y_{\\kappa_+}({\\overline{m}} t) \\nonumber\\\\\n\\end{eqnarray}\nand where we have defined \\il{\\kappa_{\\pm} \\equiv (\\kappa \\pm 1)\/2}. \nNote that in both Eq.~(\\ref{eq:adiabaticapproximation}) \nand Eq.~(\\ref{eq:abruptapproximation}),\nthe quantity $t$ is measured against  the same cosmological clock\nthat measures $t_G$. \n\nWhile the approximations above are applicable in two limits of parameter space, it is \nimportant to understand the behavior of the late-time energy density over the entire parameter space.\nIn particular, although the mass-generating phase transition generally pumps energy into the system,\nand although this energy density is ultimately dissipated by the Hubble friction that \nslowly damps the resulting field oscillations,\nit turns out that there can be an additional source of dissipation\n  and therefore an additional source of suppression of the\nlate-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ compared to our usual expectations based on the abrupt approximation.\nThis arises if the corresponding field is undergoing oscillations \n {\\it during}\\\/ the phase transition, while the mass of the field is changing appreciably.\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.48\\textwidth]{singlefielddensityplot.pdf}\n\\hskip 0.2 truein\n\\includegraphics[keepaspectratio, width=0.48\\textwidth]{singlefieldadiabatic.pdf}\n\\caption{The late-time energy density ${\\overline{\\rho}}_\\protect{\\rm 4D}$, plotted within the \\il{({\\overline{m}} t_G, \\delta_G)} plane\nand normalized to the value it would have had in the abrupt approximation (left panel)\nor adiabatic approximation (right panel).\nIn each case the colors and black lines indicate the contours of the normalized ${\\overline{\\rho}}_{\\rm 4D}$ \n({\\it i.e.}\\\/, the degrees to which the true late-time energy density is enhanced or suppressed\nrelative to its approximated value),\nwhile the blue lines indicate the corresponding contours of \\il{(t_\\zeta-t_G)\/\\Delta_G}.\nAs additional relevant guidelines,  \nthe dashed line in the left panel \nindicates the contour\nalong which \\il{\\Delta_G=2\\pi\/{\\overline{m}}},\nwhile in the right panel \nit indicates the contour along which \\il{\\dot{m}(t_{\\zeta}) = m^2(t_{\\zeta})}.\nIn general, the energy densities in regions above or to the left of these\nguidelines \nin the left (right) panel tend to obey the abrupt (adiabatic) approximation \nmore strongly than those below or to the right.  \nWe also find that neither approximation holds \nacross the majority of the parameter space shown, with the true\nvalues of the late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ often experiencing significant\nsuppressions (blue regions) or enhancements (red regions) as compared with the \napproximated expectations. }\n\\label{fig:singlefielddensityplot}\n\\end{figure*}\n\nIt is easy to identify those regions of parameter space for which this will be the case.\nIn general, there are two criteria that must be satisfied \nif the scalar field is to undergo at least one oscillation \nwhile the mass of the field is changing appreciably.\nFirst, the scalar field must indeed be undergoing oscillations at some point during the phase\ntransition --- {\\it i.e.}\\\/, our system must be in the underdamped regime before the mass of our field reaches ${\\overline{m}}$.\nWe thus must have \\il{3H \\lsim 2 {\\overline{m}}}, or equivalently \\il{{\\overline{m}} t_G\\gsim  \\kappa\/2},\nwhere we have taken $t_G$ as a rough benchmark time for the phase transition.  \nSecond, \nin order to ensure that at least one or more field oscillations\noccur during the interval over which the mass is changing,\nwe require that the timescale associated with the field oscillations \nbe shorter than the phase-transition timescale associated with\nthe changing mass.\nThe former timescale is generally given by $2\\pi\/m$ and thus decreases\nthroughout the phase transition, ultimately reaching a minimum value $2\\pi\/{\\overline{m}}$,\nwhile the latter timescale is nothing but $\\Delta_G$.   \nAs a rough benchmark, our phase transition will therefore include at least one oscillation\nso long as \\il{\\Delta_G \\gsim 2\\pi\/{\\overline{m}}}, or equivalently\n\\il{{\\overline{m}} t_G \\gsim \\sqrt{2\\pi}\/\\delta_G}.\nCombining our two conditions, we thus find that an additional suppression\nof the late-time energy density compared with what we might expect based on the abrupt approximation\nwill occur as long as our system satisfies \nthe rough benchmark criterion\n\\begin{equation}\n            {\\overline{m}} t_G ~\\gsim~ \n         {\\rm max} \\left\\lbrace  {\\sqrt{2\\pi}\\over \\delta_G}\\, ,\\, {\\kappa\\over 2} \\right\\rbrace~.\n\\end{equation}\nNote that for a matter- or radiation-dominated background cosmology, the underdamped condition will automatically\nbe satisfied whenever the oscillation condition is satisfied. \n\nIn the left panel of Fig.~\\ref{fig:singlefielddensityplot}, we \nplot the exact numerical value of the late-time\nenergy-density ratio ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)\/{\\overline{\\rho}}_{\\rm 4D}(0)$ within \nthe (${\\overline{m}} t_G$,$\\delta_G$) plane.\nFor reference, the benchmark contour with\n\\il{\\Delta_G=2\\pi\/{\\overline{m}}} is also plotted.\nIn general, \nthe regions above and to the right\nof this contour ({\\it i.e.}\\\/, regions with \\il{{\\overline{m}} t_G \\gg 1})\nindeed experience suppressions which grow increasingly\nsevere compared with what we might expect from the abrupt approximation.\nAs indicated in Fig.~\\ref{fig:singlefielddensityplot}, these are regions\nwith \\il{t_\\zeta > t_G}.\nSurprisingly, however, we note that there is also a region in which\nthe late-time energy density is {\\it enhanced}\\\/ compared with its abrupt expectation.\nThis region arises only for \\il{{\\overline{m}} t_G\\sim {\\cal O}(1)}, and typically has \\il{t_\\zeta-t_G \\lsim\n\\Delta_G}.\nLike the corresponding suppression,\nthis too is a feature of considering\na non-zero timescale for the phase transition, occurring only when \\il{\\delta_G >0}.\nFinally, for \\il{{\\overline{m}} t_G \\ll 1} or for \\il{\\delta_G \\ll 1},\nwe see that our late-time energy density is neither enhanced \nnor suppressed.  Indeed, these are the regions in which the abrupt approximation\napplies.\n \nIn the right panel of Fig.~\\ref{fig:singlefielddensityplot}, \nwe plot the same late-time energy density ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)$, only now normalized  \nto what we might expect from the {\\it adiabatic}\\\/ approximation.\nIn this case, the contour \nwith \\il{\\dot m\/m^2=1} at \\il{t=t_\\zeta} is shown\nas a relevant guideline.\nUnlike the comparison to the abrupt\napproximation, in this case\nwe see that the late-time energy density ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)$\nis {\\it enhanced}\\\/ throughout the entire region\nwith \\il{\\delta_G >0}.   As expected, this enhancement is relatively mild  \nin regions to the left of the\n\\il{\\dot m\/m^2=1} contour but grows increasingly severe below and to the right.\nOnce again, it is only for \\il{{\\overline{m}} t_G \\ll 1} that the adiabatic approximation\nappears to hold.\n\nWe see, then, that our usual expectations based on the abrupt or adiabatic approximations\napply only in relatively small regions of the full \\il{({\\overline{m}} t_G,\\delta_G)} parameter space, the\nformer consisting of the \nregions with \\il{{\\overline{m}} t_G \\ll 1} or \\il{\\delta_G \\ll 1} and the latter consisting of\nthe region with \\il{{\\overline{m}} t_G \\ll 1}.\nInterestingly, we see that {\\it either}\\\/ approximation yields the same \n(reliable) result for \\il{{\\overline{m}} t_G\\ll 1}, regardless of the value of $\\delta_G$.\nIn all other regions, however, we see that these standard approximations break down ---  often\nsignificantly ---  with the adiabatic approximation tending to underestimate the true value\nand the abrupt approximation either under- or over-estimating the true value, depending\non the particular values of ${\\overline{m}} t_G$ and $\\delta_G$.\n\nOne of the advantages of the abrupt and\/or adiabatic approximations is that they \nprovide relatively simple, {\\it analytical}\\\/ expressions for the late-time energy\ndensity ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)$.\nUnfortunately, as we have seen, these expressions are accurate\nonly within relatively narrow slices of the full parameter space.\nGiven this deficiency, we now offer  \ntwo approximate analytical expressions for ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)\/{\\overline{\\rho}}_{\\rm 4D}(0)$ which\ntogether describe its numerical values fairly accurately across the entire parameter space shown in\nFig.~\\ref{fig:singlefielddensityplot}. \nSubsequent use of the analytical expression in \nEq.~(\\ref{eq:adiabaticapproximation})\nthen allows us to isolate ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)$.\n\nWe begin by noting that within the \nsuppressed region \nin the left panel of \nFig.~\\ref{fig:singlefielddensityplot}, \nthe contours of ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)\/{\\overline{\\rho}}_{\\rm 4D}(0)$ \nroughly follow the same slope as the \ncontour with \n\\il{\\Delta_G=2\\pi\/{\\overline{m}}}.  Indeed, for\n\\il{\\delta_G\\lsim 0.3}, \none finds the approximate power-law scaling behavior\n\\begin{equation}\n      {{\\overline{\\rho}}_{\\rm 4D}(\\delta_G) \\over {\\overline{\\rho}}_{\\rm 4D}(0)}  ~\\sim~ { 1\\over {\\overline{m}} t_G \\delta_G}~.\n\\label{fdscaling}\n\\end{equation}\nOf course, as we move closer to the \\il{\\delta_G=1} boundary, \nthe contours deviate from this scaling behavior and the suppression grows increasingly severe. \nNevertheless, we find an approximate expression \n\\begin{equation}\\label{eq:4Dsuppressedformula}\n\\frac{{\\overline{\\rho}}_{\\rm 4D}(\\delta_G)}{{\\overline{\\rho}}_{\\rm 4D}(0)} ~\\approx~ \n\\frac{({\\overline{m}} t_G \\delta_G)^{-0.92}}{1 + \\left[({\\overline{m}} t_G)^{1\/4}-3\/2\\right]\\delta_G^2}~ \n\\end{equation}\nwhich holds to within \\il{\\pm 20\\%} across\nthe entire suppressed region and which approximately reduces to the power-law result \nin Eq.~(\\ref{fdscaling}) for very small $\\delta_G$.\n\nBy contrast, turning to the enhanced region near \\il{{\\overline{m}} \\sim 1\/t_G},\nwe see the maximum of this enhancement always occurs at \\il{\\delta_G=1}. \nIndeed, we find that we can approximate this enhancement analytically in\nthe Gaussian form\n\\begin{equation}\\label{eq:4Denhancedformula}\n\\frac{{\\overline{\\rho}}_{\\rm 4D}(\\delta_G)}{{\\overline{\\rho}}_{\\rm 4D}(0)} ~\\approx~ \n1 + 0.42 \\exp\\left[-\\ve{v}^T\\!\\!\\begin{pmatrix}1.64 & 0.68 \\\\ 0.68 & 0.67\\end{pmatrix}\\!\\ve{v}\\right] \\ ,\n\\end{equation}\nwhere \\il{\\ve{v}\\equiv \\left[\\, \\log({\\overline{m}} t_G\/0.54) , \\, \\log(\\delta_G)\\,\\right]^T}. \nIn this case, the error associated with this approximation\nis bounded to lie between $-10\\%$ and $5\\%$ throughout\nthe non-suppressed portion of the parameter space shown.\nThus, with the analytical expressions in Eq.~\\eqref{eq:4Dsuppressedformula} and \nEq.~\\eqref{eq:4Denhancedformula},\nwe now cover the entire single-scalar parameter space.\nWe shall nevertheless continue to use our exact results throughout the rest of this paper.\n\nIn Fig.~\\ref{fig:singlefielddensityplot} \nwe have presented our results for the late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ \nas fractions of the values this quantity would have had in either the abrupt or adiabatic\napproximations.\nThese results therefore enabled us to understand the effects that can accrue beyond\nthe regions in which these approximations are valid.\nHowever, it is also important to consider the {\\it absolute}\\\/ magnitudes of ${\\overline{\\rho}}_{\\rm 4D}$\nthat are generated throughout our \\il{({\\overline{m}} t_G, \\delta_G)} parameter space.\nThese can be trivially obtained at each point in parameter space by multiplying the results in \nthe left panel of Fig.~\\ref{fig:singlefielddensityplot} \nby the corresponding values in \nEq.~(\\ref{eq:abruptapproximation})\n[or equivalently the results in the right panel of \nFig.~\\ref{fig:singlefielddensityplot} \nby the corresponding values in \nEq.~(\\ref{eq:adiabaticapproximation})].\n\n\\begin{figure}[b]\n\\includegraphics[keepaspectratio, width=0.5\\textwidth]{singlefieldabsolute.pdf}\n\\caption{The {\\it absolute}\\\/ late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$, plotted in units\nof \\il{{\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}} within the \\il{({\\overline{m}} t_G,\\delta_G)} plane.\nNote that our late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$\nis generally independent of $\\delta_G$ for \\il{\\lbrace {\\overline{m}} t_G\\lsim 10^3, \\delta_G\\lsim 0.5\\rbrace}\nand scales roughly as $({\\overline{m}} t_G)^2$ within this region.}\n\\label{fig:fdabs}\n\\end{figure}\n\nThe results are shown in Fig.~\\ref{fig:fdabs}, where we plot the absolute\nmagnitude of ${\\overline{\\rho}}_{\\rm 4D}$ in units of \\il{{\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}}.\nOnce again, $t$ refers to universal time, measured against the same cosmological clock as $t_G$.\nThus, for example, if our late-time mass is chosen to be \\il{{\\overline{m}} t_G=10} and our mass-generating phase transition width\nis chosen to be \\il{\\delta_G=0.4}, we find  \nfrom Fig.~\\ref{fig:fdabs} that our resulting late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ is given approximately as \n\\begin{equation}\n            {\\overline{\\rho}}_{\\rm 4D} (t) ~\\approx~ 50 \\, \\langle\\phi\\rangle^2 \\, t^{-2}~.\n\\end{equation} \nIn general, we see from Fig.~\\ref{fig:fdabs} that our late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$\nis independent of $\\delta_G$ for \\il{\\lbrace {\\overline{m}} t_G\\lsim 10^3, \\delta_G\\lsim 0.5\\rbrace}\nand moreover scales roughly as \n\\begin{equation}\n      {\\overline{\\rho}} ~\\sim~ ({\\overline{m}} t_G)^2\n\\label{4Dscaling}\n\\end{equation}\nfor \\il{{\\overline{m}} t_G \\gsim 1}.\nOf course, this scaling behavior extends to larger and larger values \nof ${\\overline{m}} t_G$ as \\il{\\delta_G\\to 0}, as we expect from the abrupt approximation.\n\n\n\\FloatBarrier\n\\section{\\texorpdfstring{$N>1$}{N>1}:  ~A General Study with Multiple Fields\\label{sec:FiniteModeNumber}}\n\n\nWe now consider how these results evolve as we move away from the \\il{N=1} limit\nand consider larger, arbitrary values of $N$.\nNote that although we will eventually be considering the \\il{N\\to\\infty} limit, the theories we analyze\nhere with finite \\il{N>1} are interesting in their own right, as will be discussed further in the Conclusions.\nAs discussed in Sect.~\\ref{sec:TheModel}, taking \\il{N>1} introduces not only additional $\\phi$ fields\n(each potentially with its own critical oscillation time $t_\\zeta$)\nbut also the non-trivial {\\it mixing}\\\/ between these fields that is a consequence of the mass-generating\nphase transition on the brane (resulting in a non-diagonal squared-mass matrix ${\\cal M}_{k\\ell}^2$).\nSystems with \\il{N>1} can therefore be expected to be considerably more complex than their simpler\n\\il{N=1} cousins.\n\nTowards this end, we shall concentrate not only on the behavior of the {\\it total}\\\/ late-time energy density ${\\overline{\\rho}}$\nbut also on its {\\it distribution}\\\/ across the different $\\phi$ fields.  As discussed in  \nSect.~\\ref{sec:TheModel}, one cannot resolve the individual contributions to the total energy density $\\rho$\nduring times when the mass matrix is appreciably changing, since the non-vanishing time-derivatives of \n${\\cal M}_{k\\ell}^2$ introduce new terms into $\\rho$ which prevent its simple decomposition into the form in\nEq.~(\\ref{energydecomp}).\nHowever, at late times $t$ for which \\il{(t-t_G)\/\\Delta_G \\gg 1} ---\n{\\it i.e.}\\\/, at times when the phase transition is largely completed --- \nthe corresponding mass matrix ${\\cal M}_{k\\ell}^2$ becomes essentially time-independent.\nThe mass-eigenstate fields $\\phi_\\lambda$ then decouple from each other and virialize.\nAt such late times, a decomposition\nsuch as that in Eq.~(\\ref{energydecomp})\nbecomes possible, with each such late-time energy density contribution $\\rho_\\lambda$ given in Eq.~(\\ref{rholambda}).\nThus, in order to characterize the late-time energy configuration of \nthe system with $N$ fields, we can calculate not only \n${\\overline{\\rho}}$ but also the individual contributions ${\\overline{\\rho}}_\\lambda$ corresponding to the\n$N$ individual mass eigenstates $\\phi_\\lambda$.\n\nFinally, following Ref.~\\cite{DDM1}, another useful quantity we may define\nis the so-called {\\it tower fraction} $\\eta$.   This quantity measures the \nfraction of the total energy density which is carried by all but the most abundant mode in the tower:\n\\begin{equation}\\label{eq:etadefinition}\n{\\overline{\\eta}} ~\\equiv~ 1 - \\max\\limits_{\\lambda}\\left\\{\\frac{{\\overline{\\rho}}_{\\lambda}}{{\\overline{\\rho}}}\\right\\} \\ .\n\\end{equation}\nFor large but finite $N$, the tower fraction $\\eta$ takes values within \\il{0 \\leq \\eta < 1}\nand can be viewed as quantifying the extent to which our system really has multiple components, with each carrying\nsome relevant portion of the total energy density.   If $\\eta$ is extremely close to zero, \nthen almost all of the total energy density is captured within a single field, rendering the other fields \nlargely irrelevant from a phenomenological point of view.    For this reason we shall be most interested in systems with larger values of $\\eta$. \n  \nGiven this, we shall analyze the general-$N$ system by tracing its energy flow, moment by moment through the mass-generating \nphase transition, ultimately evaluating the total late-time energy density\n${\\overline{\\rho}}$, its distribution across the individual contributions ${\\overline{\\rho}}_\\lambda$, and the corresponding late-time value of ${\\overline{\\eta}}$ --- \nall as functions of $m$, $M_c$, $t_G$, and $\\delta_G$.\n\n\\subsection{Instantaneous phase transition\\label{instantsection}}\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.9\\textwidth]{initialabruptabundance.pdf}\n\\caption{The abundance fractions $\\rho_\\lambda\/\\rho$ at time \\il{t=t_G}, evaluated within \nthe abrupt \\il{\\delta_G\\rightarrow 0}\nlimit and plotted as functions of ${\\overline{m}}\/M_c$ for \\il{N=10} (left panel) and \\il{N=100} (right panel).\nIn both panels we observe that the lightest mode carries the largest fractional abundance for small\n$m\/M_c$, but that this abundance is more fully distributed amongst the modes as\n$m\/M_c$ increases.   For finite $N$, this abundance then ultimately collects in the {\\it heaviest}\\\/\nmode as \\il{m\/M_c\\to\\infty}.    This last feature is ultimately an artifact of the truncation to only\nfinitely many modes, and disappears in the \\il{N\\to\\infty} limit.   In all cases, \nthe sum of the fractional abundances shown is equal to $1$ for all $m\/M_c$.} \n\\label{fig:initialabruptabundance}\n\\end{figure*}\n\nIt turns out that we can actually perform this calculation analytically in the special abrupt-transition \nlimit \\il{\\delta_G\\to 0}.\nIndeed, in this limit our mass matrix ${\\cal M}_{k\\ell}^2$ is time-independent both before and after $t_G$, and all that \nwe need calculate is the sudden change of \nbasis \nfrom the KK-eigenstate basis $\\phi_k$ which is appropriate before $t_G$\nto the mass-eigenstate basis $\\phi_\\lambda$ which is appropriate after.\nPrior to $t_G$, only $\\phi_{k=0}$ has an initial displacement \\il{\\langle \\phi_0\\rangle}: \nthis field is massless prior to $t_G$, and thus the system has no energy and none of the fields oscillate.\nHowever, for times after $t_G$, our mass eigenstates $\\phi_\\lambda$ become\n\\begin{equation}\n          \\phi_\\lambda ~=~ \\sum_{k=0}^{N-1} U_{\\lambda k}\\, \\phi_k~\n\\label{basischange}\n\\end{equation}\nwhere $U_{\\lambda k}$ is the time-independent but $N$-dependent \nunitary basis-change matrix that diagonalizes ${\\cal M}_{k\\ell}^2$.\nThe initial conditions in Eq.~(\\ref{eq:initialconditions})\nthen become\n\\begin{eqnarray}\n        && \\phi_\\lambda(t_G)  ~=~  U_{\\lambda 0} \\, \\langle \\phi_0\\rangle~ \\nonumber\\\\  \n        && \\dot \\phi_\\lambda(t_G) ~=~ 0~, \n\\end{eqnarray} \nwhereupon our total energy density at \\il{t=t_G} is given by  \n\\begin{eqnarray}\n      \\rho(t_G) &=&  \\sum_\\lambda \\rho_\\lambda (t_G) = {\\textstyle{1\\over 2}} \\sum_\\lambda \\lambda^2 \\phi_\\lambda^2 \n                 \\nonumber\\\\\n             &=& {\\textstyle{1\\over 2}} \\, \\sum_\\lambda \\, \\lambda^2 \\, U_{\\lambda 0}^2 \\, \\langle \\phi_0\\rangle^2 ~.\n\\end{eqnarray}\nHowever, for all $N$,  our basis-change matrix $U_{\\lambda k}$ satisfies the remarkable identity\n\\begin{equation}  \n              \\sum_\\lambda \\, \\lambda^2 \\, U_{\\lambda 0}^2 ~=~ m^2 ~\n\\label{magic}\n\\end{equation}\nwhich completely eliminates from $\\rho(t_G)$ what has otherwise been a highly non-trivial \ndependence on $M_c$ (and $N$).\nWe thus find that our phase transition at \\il{t=t_G} injects \ninto our $N$-mode system \na total energy density  \n\\begin{equation}\n            \\rho(t_G) ~=~ {\\textstyle{1\\over 2}} \\, m^2 \\, \\langle \\phi_0\\rangle^2~, \n\\label{rhotott}\n\\end{equation}\nwith individual fractional contributions\n\\begin{equation}\n           {\\rho_\\lambda(t_G)\\over \\rho(t_G)} ~=~ \\left( \\lambda\\over m\\right)^2 U_{\\lambda 0}^2~.\n\\label{rhoconts}\n\\end{equation}\nNote that because we are working in the abrupt \\il{\\delta_G\\to 0} limit,\nthe mass $m$, eigenvalues $\\lambda$, and basis-change matrix $U_{\\lambda k}$\nin Eqs.~(\\ref{basischange}) through (\\ref{rhoconts})\nare constant for all times after $t_G$ and thus equal to their late-time values:\n\\il{m={\\overline{m}}}, \\il{\\lambda={\\overline{\\lambda}}}, and\n\\il{U_{\\lambda k} = \\overline U_{\\lambda k}}.\n\nIn general, we see from Eq.~(\\ref{rhotott}) that the total energy density\ninjected into our system grows polynomially with $m$.\nHowever, the individual contributions to this total energy density \nvary non-trivially as a function of $m\/M_c$ --- even when expressed \nas a fraction relative to the total.\nThese individual fractional contributions to the total energy density \nat \\il{t=t_G}\nare plotted as functions of $m\/M_c$ \nfor \\il{N=10} in the left panel of Fig.~\\ref{fig:initialabruptabundance} \nand\nfor \\il{N=100} in the right panel of Fig.~\\ref{fig:initialabruptabundance}. \nIn each case, we see that these energy-density distributions\nare highly sensitive to $m\/M_c$,\nwith the majority of the energy density at \\il{t=t_G}\ncarried by the lightest mode for small \\il{m\/M_c}\n(as consistent with our expectation from the 4D limit).\nHowever, as $m\/M_c$ increases,\nthis picture changes, with more and more of the states beginning to carry\ngreater fractions of the total energy density.\nNote that for all values of $m\/M_c$, the sum of these fractional\nenergy densities is fixed at $1$.\n\nUltimately, for very large \\il{m^2\/M_c^2\\gsim {\\cal O}(\\sqrt{N})},\nthe majority of the energy density \nbegins to collect in the {\\it heaviest}\\\/ mode.\nHowever, we have already seen in Sect.~\\ref{sec:TheModel}\nthat this is precisely the regime in which the heaviest eigenvalue $\\lambda_{N-1}$ \nbegins to diverge.\nIndeed, as discussed in Sect.~\\ref{sec:TheModel}, \nthis feature is ultimately an artifact of truncating our system to a finite\nnumber of modes, with \\il{N<\\infty}.\nA similar situation persists for the corresponding energy densities.\nEven though the energy density associated with the heaviest mode tends to dominate\nin the region with \\il{m^2\/M_c^2\\gsim {\\cal O}(\\sqrt{N})},\nthis region becomes increasingly remote\nas \\il{N\\to \\infty} and effectively vanishes.\nThus, for \\il{m\/M_c\\gg 1} and \\il{N\\to\\infty}, all of the modes in our theory \ntend to share the total energy density\nessentially equally.\n\n\\begin{figure}[t]\n\\includegraphics[keepaspectratio, width=0.5\\textwidth]{etaRAPID_vsmBAR.pdf}\n\\caption{The tower fractions $\\eta(t_G)$ corresponding to the fractional\nabundances shown in Fig.~\\ref{fig:initialabruptabundance},\nplotted as functions of $m\/M_c$ for different values of $N$.\nWe see that the tower fraction begins near zero for small $m\/M_c$ but then begins to grow as\n$m\/M_c$ increases.    For finite $N$, the tower fraction ultimately hits a peak before returning\nto zero at large $m\/M_c$, while in \nthe \\il{N\\to\\infty}  limit the tower fraction continues monotonically to $1$.}\n\\label{fig:towerfraction}\n\\end{figure}\n\nIn Fig.~\\ref{fig:towerfraction}\nwe show the\ncorresponding values of the tower fraction $\\eta(t)$ at \\il{t=t_G}, plotted \nas functions of $m\/M_c$.   \nAs we see, $\\eta(t_G)$ is non-monotonic as a function of\n$m\/M_c$ for all finite values of $N$, starting near \\il{\\eta\\approx 0} for small\n$m\/M_c$ and reaching a maximum at a certain critical\n$N$-dependent value $(m\/M_c)_{\\rm peak}$  \nbefore declining back to \\il{\\eta\\approx 0} for large $m\/M_c$.\nThis non-monotonic behavior for finite $N$ is ultimately a result of the fact\nthat the identity of the mode carrying the maximum abundance is itself\na function of $m\/M_c$, with the lightest mode carrying the largest\nabundance for small $m\/M_c$ but the heaviest mode carrying the largest\nabundance for large $m\/M_c$.   However, as \\il{N\\to\\infty}, we find\nthat the magnitude of the peak tends towards unity while\nthe position of peak itself shifts towards increasingly large values of $m\/M_c$.\nIndeed, with excellent precision, we find that the position of the peak is given by\n\\begin{equation}\n    \\left( {m\\over M_c}\\right)_{\\rm peak} \\approx~ {4\\over 9} \\, N^{4\/9}~,\n\\end{equation}\nwhereupon it follows that \\il{(m\/M_c)_{\\rm peak}\\to\\infty} as \\il{N\\to \\infty}.\nWe thus find that $\\eta(t_G)$ becomes completely monotonic in the infinite-$N$ \nlimit, transitioning from \\il{\\eta\\approx 0} at small $m\/M_c$ to \\il{\\eta \\approx 1} at large $m\/M_c$.\n\nThe results illustrated in Fig.~\\ref{fig:initialabruptabundance} and \\ref{fig:towerfraction}\napply to the energy configuration \nof our system at \\il{t=t_G}.\nA natural question, however, concerns the extent to which these results apply for the {\\it late-time}\\\/\nenergy-density fractions ${\\overline{\\rho}}_\\lambda\/{\\overline{\\rho}}$ and late-time tower fraction ${\\overline{\\eta}}$. \nEven though the mass matrix $M_{k\\ell}^2$ and  associated eigenvalues \nare time-independent for all \\il{t> t_G} in the abrupt \\il{\\delta_G\\to 0} limit,   \nthe mass eigenstates $\\phi_\\lambda(t)$ \nand corresponding energy densities $\\rho_\\lambda(t)$ nevertheless \ncontinue have a non-trivial time-dependence.    In general, \na single field $\\phi_\\lambda$ with constant mass $\\lambda$\nremains essentially fixed for all \\il{t < t_\\zeta^{(\\lambda)} \\equiv\\kappa\/(2\\lambda)}\nand then transitions to damped-oscillatory behavior for \\il{t> t_\\zeta^{(\\lambda)}}.\nLikewise, the corresponding energy density is essentially fixed for  \n\\il{t < t_\\zeta^{(\\lambda)}}\nand then decays as $t^{-\\kappa}$ for \n\\il{t > t_\\zeta^{(\\lambda)}}.\n(Exact analytical results for the fields and energy densities can be found\nin Appendix~A and Fig.~1 of Ref.~\\cite{TwoTimescales}.)\nThus, in our present situation with an abrupt phase transition at \\il{t=t_G},\nall of our fields \nwill already be underdamped and commence oscillations \nsimultaneously \nat $t_G$\nas long as \\il{t_\\zeta^{(\\lambda)} \\leq t_G} for all $\\lambda$. \nIn such cases, the ratios $\\rho_\\lambda\/\\rho$ as well as the corresponding\ntower fraction $\\eta$ will be essentially time-independent, with the same values\nat late times as they have at $t_G$.\n\n\\subsection{Phase transition with arbitrary width \\texorpdfstring{$\\delta_G$}{delta_G}}\n\nThe results given above apply only in the abrupt \\il{\\delta_G\\to 0} limit.\nHowever, \nwe now wish to explore the full parameter space  and calculate our total and fractional\nlate-time energy  densities and tower fractions throughout the \n\\il{({\\overline{m}} t_G, \\delta_G)} plane.\nWe already did this for the \\il{N=1} limiting case in Sect.~\\ref{sec:FourDimensionalLimit},\nwhere we found that the effects of taking \\il{\\delta_G>0} tended to produce either\nsuppressions or enhancements in the resulting late-time energy \\il{{\\overline{\\rho}}_{\\rm 4D}(\\delta_G)} compared\nwith $\\rho_{\\rm 4D}(0)$  depending on the precise values of ${\\overline{m}} t_G$ and $\\delta_G$.\nThese results were shown in the left panel of Fig.~\\ref{fig:singlefielddensityplot}.\nWe therefore now seek to know what additional effects \nbeyond those in Fig.~\\ref{fig:singlefielddensityplot}\nemerge from considering arbitrary values of $N$ rather than merely \\il{N=1}.\n\n\\begin{figure*}[bht]\n\\includegraphics[keepaspectratio, width=0.9\\textwidth]{finitemodedensityplotsA.pdf}\n\\caption{The total late-time abundance ${\\overline{\\rho}}$ for $N$ modes, \nevaluated within the \\il{({\\overline{m}} t_G,\\delta_G)} plane\nfor \\il{N=2} (left panel) and \\il{N=10} (right panel)\nand expressed as a fraction of the \ncorresponding value ${\\overline{\\rho}}_{\\rm 4D}$ for the 4D \\il{N=1} special case.\nWe have chosen \\il{t_G=10^2\/M_c} as a reference value for both plots above.\nWe see that the introduction of additional modes relative to the 4D special case\neither {\\it preserves}\\\/ the 4D result nearly exactly for \\il{{\\overline{m}} t_G\\lsim 10}  or\n{\\it suppresses}\\\/ it for \\il{{\\overline{m}} t_G\\gsim 10}, with this suppression becoming\nincreasingly severe for larger ${\\overline{m}} t_G$ (or equivalently, larger ${\\overline{m}}\/M_c$)\nand \\il{\\delta_G\\lsim 1}.  Note that the gray regions in the upper right corners of\nthese panels (and in similar subsequent figures throughout this paper) \nare excluded for the reasons discussed in the paragraph below Eq.~(\\ref{bigwidth}).  }\n\\label{fig:finitemodedensityplotsgenN}\n\\end{figure*}\n\nOur results are shown in Fig.~\\ref{fig:finitemodedensityplotsgenN} for\n\\il{t_G=10^2\/M_c}.\nRemarkably, \nfor that portion of the \\il{({\\overline{m}} t_G,\\delta_G)} parameter space\n  with \\il{{\\overline{m}} t_G\\lsim 10}, \n{\\it we find that the total late-time energy density ${\\overline{\\rho}}$\nis virtually identical to what it would have been for only one mode!}\\\/ \nOf course, this behavior is consistent with what we already saw for the \nabrupt \\il{\\delta_G\\to 0} limit and very small $m\/M_c$, but what is remarkable\nis that this behavior extends even for non-zero $\\delta_G$ and larger $m\/M_c$\nas well.\nMoreover, for \\il{{\\overline{m}} t_G\\gsim 10}, \nwe find that our\nlate-time energy density is actually\nsuppressed rather than enhanced\nrelative to our 4D expectations --- \nall this despite the presence of extra modes whose masses are also lifted by \n(and which therefore also receive an additional energy-density insertion from)\nthe mass-generating phase transition.\n\nThese results may be understood as follows.   \nSince we have taken \\il{t_G= 10^2\/M_c} as a reference value,\nwe see that \\il{{\\overline{m}} t_G = 10^2 \\, {\\overline{m}}\/M_c}.\nThus values \\il{{\\overline{m}} t_G \\lsim 10^2} correspond to \n\\il{{\\overline{m}}\/M_c\\lsim 1}, and we have already seen that \nthe bulk of the energy density remains concentrated  \nin the lightest mode for such values of ${\\overline{m}}\/M_c$.\nAs a result, \nour system is functionally no different than the single-mode 4D system\nfor small values of ${\\overline{m}} t_G$. \nBy contrast, as we increase the value of ${\\overline{m}} t_G$, the total energy density \nof our system is more equally distributed across the different modes.\nThe oscillations of the heavier modes then dissipate the\nenergy density more rapidly than the lighter modes during the phase transition, resulting in a more\nrapid dissipation of the total energy density and thus an overall suppression \nof the late-time energy density as compared with 4D expectations.\nIt also is important to note that this latter effect requires \\il{\\delta_G>0}, even\nfor large ${\\overline{m}} t_G$.\nIndeed, if \\il{\\delta_G=0}, our energy density is partitioned across the different\nmodes precisely as described in Sect.~\\ref{instantsection},\nwhereupon the unitarity relation in Eq.~(\\ref{magic})\nremoves all dependence on $N$ and ensures the same results as we would\nhave had in 4D!\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.9\\textwidth]{finitemodedensityplotsB.pdf}\n\\caption{The total late-time tower fraction ${\\overline{\\eta}}$ corresponding to the panels\nshown in Fig.~\\protect\\ref{fig:finitemodedensityplotsgenN}.\nIn each case we see that the \\il{({\\overline{m}} t_G,\\delta_G)} plane is subdivided into\ntwo disjoint \\il{{\\overline{\\eta}}\\approx 0} regions (white) by a narrow \\il{{\\overline{\\eta}}>0} ribbon (green):    \nthe lightest mode carries the bulk of the energy density in the region to the left of the ribbon,\nwhile the heaviest mode carries the bulk of the energy density in the region to the right.\nInterestingly, the division between these two regions \nis non-monotonic as a function of not only ${\\overline{m}} t_G$ but also $\\delta_G$.}\n\\label{fig:finitemodedensityplotsgenN2}\n\\end{figure*}\n\nIn Fig.~\\ref{fig:finitemodedensityplotsgenN2} we plot the corresponding values of the\nlate-time tower fraction ${\\overline{\\eta}}$.\nAs evident from this figure, the \\il{({\\overline{m}} t_G, \\delta_G)} plane is subdivided\ninto two disjoint \\il{{\\overline{\\eta}} \\approx 0} regions by \na fairly narrow ``mountain range'' along which \\il{{\\overline{\\eta}}>0}.\nAs we already determined in Sect.~\\ref{instantsection} for the \\il{\\delta_G\\to 0} limit,\nthe region with smaller ${\\overline{m}} t_G$ (or equivalently smaller ${\\overline{m}}\/M_c$) \ncorresponds to the region in which the lightest mode carries all of the energy density.\nBy contrast, as we dial ${\\overline{m}} t_G$ towards greater values, the energy density\nis increasingly distributed  \nacross all the modes, resulting in a greater tower fraction. \nUltimately, however, as we dial \\il{{\\overline{m}} t_G \\to\\infty} in our finite-$N$ system,\nall of the energy density finds itself in the heaviest mode, and the tower fraction returns\nto zero.  This behavior is consistent with that shown in \nFig.~\\ref{fig:towerfraction} for the \\il{\\delta_G\\to 0} limit.\n\nThis explains the non-monotonicity of the late-time tower fraction ${\\overline{\\eta}}$ \nas a function of ${\\overline{m}} t_G$.\nHowever, we additionally learn from Fig.~\\ref{fig:finitemodedensityplotsgenN2} that \nthe late-time tower fraction is also non-monotonic as a function of $\\delta_G$.\nThis too is relatively straightforward to explain.\nFor ${\\overline{m}} t_G$ sufficiently large and for small $\\delta_G$, we have already seen that the bulk of the\nenergy density is carried by the heaviest mode. \nHowever, as we increase $\\delta_G$, we are increasing the timescale associated with our phase \ntransition --- {\\it i.e.}\\\/, the timescale over which energy is pumped into the system\nand each mode becomes populated.    \nEventually this timescale reaches the timescale associated with the oscillations of the heaviest mode.\nThis then suppresses the energy transfer into the (nevertheless dominant) heaviest mode, \nthereby increasing the tower fraction. \nIndeed, as we increase $\\delta_G$ still further, we begin to suppress the energy densities of lighter\nand lighter modes, and this too has the effect of increasing the tower fraction.\nEventually, however,\nas we continue to increase $\\delta_G$, \nwe reach a point\nwhere we have suppressed the majority of the modes and the bulk of the energy density\nbegins collecting in the lightest mode.   Subsequent increases in $\\delta_G$ then only\nreinforce the dominance of the lightest mode, thereby causing \nthe tower fraction to return to zero.\nIndeed, as shown in Fig.~\\ref{fig:finitemodedensityplotsgenN2}, we have crossed the green \n``mountain range'', leaving us\nin the region in which the lightest mode carries all of the energy density.\n\nThus far, our discussion has focused on the regime with \\il{t_G M_c \\gg 1}.\nIn this regime, all of our excited modes are underdamped and have begun oscillating at very early \ntimes prior to the phase transition.   This will be true even for the lightest mode\nif ${\\overline{m}} t_G$ is chosen sufficiently large.\nHowever, if \\il{t_G M_c \\ll 1} and \\il{N\\gsim 1\/(t_G M_c)},  \nonly an upper subset of the modes will have begun oscillating\nprior to the phase transition --- \nthe lightest modes will remain overdamped and fixed.\nThus, the introduction of the heavier modes relative to the 4D case has the \neffect of distributing some of the energy density into heavier modes\nwhich begin oscillating earlier than the lightest mode, \nthereby enhancing the dissipation of the total energy\ndensity relative to the 4D case. \nIt is important to stress that this source of suppression is\ncompletely distinct from that discussed above, resulting instead from\ndifferences in the times at which individual modes begin oscillating.\nAs such, this effect --- which was originally discussed in Ref.~\\cite{DDGAxions}\nwithin the context of the \nabrupt \\il{\\delta_G\\to 0} limit --- is largely independent of $\\delta_G$.\n\n\\FloatBarrier\n\\section{Approaching Asymptotia: ~\\texorpdfstring{\\il{N\\to \\infty}}{N->infty}\\label{Approaching}}\n\n\n\\begin{figure*}\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{Nscaling.pdf}\n\\caption{\nThe total late-time abundance ${\\overline{\\rho}}$ and late-time tower fraction ${\\overline{\\eta}}$ of our $N$-mode system,\nplotted as functions of $N$ for different values of $\\delta_G$. The left and center panels show \nthe total late-time abundance expressed as a fraction of the value it would have in the abrupt \n\\il{\\delta_G\\to 0} and 4D limits, respectively, while the right panel shows the late-time tower fraction.\nFor all three panels we have taken  \\il{t_G=10^2\/M_c} and \\il{{\\overline{m}}=10^2 M_c}, thereby fixing \n\\il{{\\overline{m}} t_G=10^4} as a benchmark value. The blue curves are calculated using the usual ``IR-based'' \ntruncation which has been employed thus far in Sects.~\\ref{sec:TheModel} through \\ref{sec:FiniteModeNumber},\nwhile the red curves are calculated using the alternative ``UV-based'' truncation to be discussed in \nSect.~\\protect\\ref{alttrunc}; both truncations have the same \\il{N\\to\\infty} asymptotic behavior\nbut the UV-based truncation reaches asymptotia far more rapidly and smoothly. The black-dashed line that \noccurs in each of the energy-density panels indicates the results of the multi-component adiabatic \napproximation of Eq.~\\eqref{eq:multiadiabaticapproximation} which serves as a lower bound for the energy \ndensity of the ensemble.}\n\\label{fig:Nscaling}\n\\end{figure*}\n\nWe now wish to study how our $N$-mode system evolves as a function of $N$, \nwith an eye towards understanding the asymptotic behavior of our system for large \\il{N\\gg 1}.\nThis will ultimately enable us to extract the behavior of the \nfull \\il{N=\\infty} Kaluza-Klein tower, as we shall do in Sect.~\\ref{sec:KKLimit}.~ \nThis will also enable us to understand the effects that come from truncating\nthe KK system to finite but large $N$. \n\n\\subsection{Large-\\texorpdfstring{$N$}{N} behavior:  The road to asymptotia\\label{road}}\n\nIn Fig.~\\ref{fig:Nscaling}, the blue curves indicate the $N$-dependence of our \ntwo quantities of interest, namely\nthe late-time energy density ${\\overline{\\rho}}$ of our $N$-mode system \nas well as its corresponding late-time tower fraction ${\\overline{\\eta}}$.\nWhile the latter quantity is shown in the right panel,\nthe former is plotted in the left panel \nas a fraction of the value it would\nhave in the abrupt \\il{\\delta_G\\to 0} limit \nand in the middle panel \nas a fraction of its corresponding 4D value.\nFor all three panels we have taken  \\il{t_G=10^2\/M_c}\nand \\il{{\\overline{m}}=10^2 M_c}, thereby fixing \\il{{\\overline{m}} t_G =10^4} as a benchmark value.\nWe have then \nplotted the resulting curves \n(blue) as functions of $N$ \nfor different values of $\\delta_G$.\n(By contrast, the red curves will be discussed in Sect.~\\ref{alttrunc}.)\n\nAs expected, each of the blue curves shown in Fig.~\\ref{fig:Nscaling} eventually\nheads towards a finite asymptote as \\il{N\\to\\infty}.\nOne immediate observation from the left and middle panels of \nFig.~\\ref{fig:Nscaling}\nis that increasing the number of modes\nin our system \ngenerally results in an {\\it increasing suppression}\nof the total late-time energy density when compared to the\nvalues this energy density would have in the abrupt \\il{\\delta_G\\to 0} or\n4D limits.\nThus, {\\it the larger the value of \n$N$, the more the abrupt and 4D approximations fail to accurately\nestimate the late-time energy density}\\\/.\nThis is an important result, given that most approaches to calculating\nthe energy densities of such multiple-component systems in the literature\nassume that the phase transition occurs when the fields are still overdamped.\nCompared with these static fields, any such phase transition is therefore essentially occurring\nwith an infinitely short timescale $\\delta_G$, and is thereby functionally equivalent to the abrupt\napproximation.    The results shown in Fig.~\\ref{fig:Nscaling} \nthus illustrate what happens as one moves away from these assumptions. \n\nIn this vein, it is perhaps also worthwhile to consider the adiabatic \napproximation which is traditionally applied in single-component\nscalar theories undergoing mass-generating phase transitions.  \nOf course, while an adiabatic approximation can be realized\nfor a single mode, we do not expect such an approximation to be appropriate\nfor a large collection of modes with vastly different masses and non-trivial mixings.\nDespite this, we can nevertheless imagine limiting cases in which\nall excited modes above the lightest mode are\ndrained of significant energy density.    \nIn such a regime we can then imagine applying the adiabatic approximation \nto the only surviving mode $\\phi_{\\lambda_0}$ in the ensemble, leading to a\ndefinition\n\\begin{equation}\n\\rho(t)|_{\\rm ad}  ~\\equiv ~ \\frac{1}{2}\\langle\\phi_0\\rangle^2 \\lambda_0(t^{(\\lambda_0)}_{\\zeta}) \\, \n\\lambda_0(t)\\left[\\frac{a(t^{(\\lambda_0)}_{\\zeta})}{a(t)}\\right]^3 \\ ,\n\\label{eq:multiadiabaticapproximation}\n\\end{equation}\nwhere $t^{(\\lambda)}_{\\zeta}$ is defined as the time at which \\il{3H(t) = 2\\lambda(t)}\nand where we implicitly assume \\il{\\delta_G=1} when evaluating $\\lambda(t)$.\nWe have already seen in the left panel of Fig.~\\ref{fig:singlefielddensityplot} that the\nadiabatic approximation sets a lower bound for the energy density associated with\na single mode --- indeed, in all other regions of parameter space the energy density \nis either the same or enhanced.\nLikewise, the approximation in Eq.~(\\ref{eq:multiadiabaticapproximation})\nconsists of disregarding whatever energy density might reside in the excited modes.\nThus, we expect the adiabatic approximation in \nEq.~(\\ref{eq:multiadiabaticapproximation})\nto provide us with a lower bound on the total energy density of our $N$-mode\nsystem throughout all regions of parameter space.\n\nThe dashed-black curves in the left and middle panels of Fig.~\\ref{fig:Nscaling} show the \nbehavior of Eq.~\\eqref{eq:multiadiabaticapproximation} as a function of $N$. \nIndeed, as expected, we see that \\emph{the adiabatic approximation serves \nas a lower bound on the total energy density of the tower}, just as it does in the single-field\nscenario.  Indeed, we see from Fig.~\\ref{fig:Nscaling} \nthat the adiabatic limit seems to be reached most directly for large $N$\n(such as for a full KK tower) and \\il{\\delta_G\\to 1}.\nBy contrast, the results in other regions of parameter space differ significantly from this limit.\n\n\\subsection{A new truncation of KK theory:  An alternate road to asymptotia\\label{alttrunc}}\n\nOur results in Fig.~\\ref{fig:Nscaling} clearly illustrate a successful road to asymptotia,\nas each of the blue curves heads towards a finite asymptotic value as \\il{N\\to \\infty}.\nThis is not a surprise, since\nwe began our analysis by truncating our infinite-dimensional KK mass matrix\n${\\cal M}_{k\\ell}^2$ in Eq.~(\\ref{kmassmatrix}) by \nretaining only its first $N$ rows and columns.\nThus it is natural that taking $N\\to\\infty$ restores the physics of the infinite-matrix limit.\nIndeed, truncating our mass matrix in\nEq.~(\\ref{kmassmatrix}) \nto only its first $N$ rows and columns\nrepresents one way of constructing a finite $N$-mode theory \nwhose \\il{N\\to\\infty} limit reproduces the physics of the full KK tower \nat any moment in time.  \n\nHowever, this is not a unique truncation to a finite $N$-mode theory.\nThere is, in fact, an alternative truncation \nof our full KK theory at any moment in time which also has only $N$ modes and which also reproduces the physics\nof the full KK tower as \\il{N\\to \\infty}, yet yields significantly different physical results for finite $N$.\nIndeed, as we shall see, this alternate truncation exhibits an even more rapid path\nto asymptotia, yielding what are essentially the full infinite-$N$ values of\nthe late-time energy densities and tower fraction for even smaller values of $N$\nthan are required using the standard truncation that we have employed thus far.\n\nRecall that our standard truncation began with the full, infinite-dimensional\nmass matrix ${\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{kmassmatrix}), expressed in the KK basis of the individual KK modes $\\phi_k$.\nThis matrix was then truncated to its first $N$ rows and columns.\nThe resulting \\il{N\\times N} matrix then defined our truncated KK theory,\nand all subsequent calculations proceeded from this truncated matrix.\nSpecifically, all calculations were performed directly from \nthe equations of motion~(\\ref{eq:KKequationsofmotion})\nin the KK basis (which has the advantage of representing a basis choice that does not change with time, even in the presence of the mass-generating phase transition), \nand the results at late times were \nconverted to the mass-eigenstate basis (such as for quoting late-time quantities such as ${\\overline{\\rho}}_\\lambda$ or ${\\overline{\\eta}}$ which\npertain to individual mass eigenstates) only at the final step.\n\nHowever, an alternative approach is {\\it not}\\\/ to truncate our infinite-dimensional mass matrix \nin the KK basis $\\phi_k$,  but rather to transform the infinite-dimensional matrix into its\nmass-eigenstate basis and then truncate the resulting {\\it mass-eigenstate}\\\/ matrix.\nIn general, this matrix will take the form\n\\begin{equation}\n\\mathcal{M}_{\\lambda\\lambda'}^2 ~=~ \n\\begin{pmatrix}[1.1]\n\\lambda_0^2 & 0 & 0 & \\cdots \\\\\n0 & \\lambda_1^2 & 0 & \\cdots  \\\\\n0 & 0 & \\lambda_2^2 & \\cdots  \\\\\n\\vdots & \\vdots & \\vdots & \\ddots\n\\end{pmatrix} \\ ,\n\\label{eq:massbasismassmatrix}\n\\end{equation}\nwhere the $\\lambda_k^2$ are the mass eigenvalues that are calculated in \nthe infinite-$N$ limit at any moment in time.   These eigenvalues will be discussed below.\n\nNote that truncating the mass matrix in the mass-eigenstate basis \nis mathematically different than truncating the mass matrix in the KK-eigenstate basis.\nOf course, these procedures are in some sense parallel in that they both\nbegin from equivalent infinite-dimensional matrices which describe the \nsame KK system and which are related to each other through a simple,\nunitary basis change.   \nHowever, their truncations to \\il{N\\times N} submatrices are mathematically different, and thus have\ndifferent physical effects.\n\nHaving truncated our mass-eigenstate matrix ${\\cal M}^2_{\\lambda \\lambda'}$\nin Eq.~(\\ref{eq:massbasismassmatrix}),\nour final step is to convert the resulting\nmatrix back to the KK basis.\nIn order to do this, we \nuse a similarly truncated version of the exact basis-change matrix $U_{\\lambda k}$ \nthat\nwould have related ${\\cal M}^2_{k\\ell}$ and ${\\cal M}^2_{\\lambda\\lambda'}$ in the full infinite-dimensional \nlimit.\nSpecifically, we define\n\\begin{equation}\n     \\widetilde {\\cal M}^2_{k\\ell} ~\\equiv ~ \\sum_{\\lambda,\\lambda'} \\, \n              (\\widehat U^\\dagger)_{k\\lambda} \\, \\widehat {\\cal M}^2_{\\lambda\\lambda'}  \\, \\widehat U_{\\lambda' \\ell} \n\\label{tildeM}\n\\end{equation}\nwhere $\\widehat{\\cal M}^2_{\\lambda\\lambda'}$ and $\\widehat U_{\\lambda'\\ell}$ here represent the first \n\\il{N\\times N} submatrices within the exact infinite-$N$ matrices \n${\\cal M}^2_{\\lambda\\lambda'}$ and $U_{\\lambda k}$ respectively.\nNote that since $\\widehat U$ is a truncated version of the unitary infinite-$N$ matrix $U$,\nthe truncated matrix $\\widehat U$ is not unitary by itself.   In particular, $\\widehat U$ is {\\it not}\\\/ the matrix \nthat diagonalizes the new matrix $\\widetilde {\\cal M}^2_{k\\ell}$ defined in Eq.~(\\ref{tildeM}), and thus, strictly speaking, \nwe should not regard $\\widetilde {\\cal M}^2_{k\\ell}$\ndefined in Eq.~(\\ref{tildeM})\nas representing the KK-basis version of $\\widehat {\\cal M}^2_{\\lambda\\lambda'}$.\nWe can nevertheless proceed to use $\\widetilde{\\cal M}^2_{k\\ell}$ directly in our equations of motion~(\\ref{eq:KKequationsofmotion}), \ntreating $\\widetilde{\\cal M}^2_{k\\ell}$ as we would any other mass matrix.\nConverting our final results \ninto statements about individual mass eigenstates \nat late times is then done in the usual way by \ndiagonalizing $\\widetilde{\\cal M}^2_{k\\ell}$ and calculating its mass eigenvalues and eigenvectors.\nNote that these eigenvalues will generally differ from the $\\lambda_i$ which\nappear in the \n$\\widetilde{\\cal M}^2_{k\\ell}$ matrix\nand only approach these $\\lambda_i$ as \\il{N\\to \\infty}.\n\nBecause this alternate truncation of our KK theory\nutilizes the exact eigenvalues $\\lambda_k$  and exact basis-change matrix elements $U_{\\lambda k}$ corresponding\nthe full KK theory, it may at first glance seem that this    \nalternate truncation cannot be realized in practice.\nHowever, it turns out to be relatively straightforward to solve the eigensystem corresponding\nto the full infinite-dimensional mass matrix ${\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{kmassmatrix}) --- not only \nnumerically, but even analytically.\nOne finds~\\cite{DDGAxions,DDM1} \nthat the mass eigenvalues $\\lambda_i$ at any moment in time are the (infinite) set\nof solutions to the transcendental equation\n\\begin{equation}\n         {\\pi \\lambda \\over M_c} \\, \\cot\\left( \\pi \\lambda\\over M_c\\right) ~=~ {\\lambda^2\\over m^2}~,\n\\label{eigeq}\n\\end{equation}\n and likewise the corresponding $U_{\\lambda k}$ matrix is given by\n\\begin{equation}\n            U_{\\lambda k} ~=~ \\left(  {r_k \\lambda^2 \\over \\lambda^2 - k^2 M_c^2} \\right) \\, A_\\lambda\n\\end{equation}\nwhere the $r_k$ are defined below Eq.~(\\ref{eq:orbifoldmodeexpansion}) and\nwhere \n\\begin{equation}\n           A_\\lambda~\\equiv  ~  {\\sqrt{2} \\, m^2 \\over \\lambda \\sqrt{ \\lambda^2 + m^2 + \\pi^2 m^4\/M_c^2 }}~.  \n\\label{Adefn}\n\\end{equation}\nFollowing the procedure outlined above, we then find that our alternate mass matrix\nis given by\n\\begin{equation}\n          \\widetilde {\\cal M}^2_{k\\ell} ~=~  \\sum_{\\lambda=\\lambda_0}^{\\lambda_{N-1}}  { r_k r_l  \\, A_\\lambda^2  \\, \\lambda^6 \\over\n                  (\\lambda^2 - k^2 M_c^2)\n                  (\\lambda^2 - \\ell^2 M_c^2)}~.\n\\label{massmatrixnew}\n\\end{equation}\nIn this connection, we remark \nthat although constructing $\\widetilde {\\cal M}^2_{k\\ell}$ requires  \nexplicit knowledge \nof the exact eigenvalues $\\lambda_i$ and matrix elements $U_{\\lambda k}$, \nthis in no way implies that we \nhave already solved the problems we originally set out to investigate.\nIndeed, our goals are far deeper than mere KK spectroscopy and instead pertain  \nto understanding the dynamical energy flow within our KK system in the presence of \nmode-mixing and mass-generating phase transitions.\n\nAlthough each element of the matrix $\\widetilde{\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{massmatrixnew}) depends on $N$ \nexplicitly through the upper limit of the $\\lambda$-summation, \nthis matrix smoothly reproduces our original ${\\cal M}^2_{k\\ell}$ mass matrix  \nas \\il{N\\to\\infty}.\nIndeed, through clever use of the eigenvalue equation~(\\ref{eigeq}), \ncotangent summation identities such as  \n\\begin{equation}\n      \\sum_{k=0}^\\infty {2 \\lambda^2 \\over \\lambda^2-k^2 M_c^2} ~=~ 1 \n         + {\\pi \\lambda \\over M_c} \\, \\cot\\left( \\pi \\lambda\\over M_c\\right)~, \n\\end{equation}\nand unitarity relations such as that in Eq.~(\\ref{magic}),\nit is possible to demonstrate explicitly that performing the summation in Eq.~(\\ref{massmatrixnew})\nover the {\\it full}\\\/ spectrum ({\\it i.e.}\\\/, taking \\il{N\\to\\infty}) \nreproduces our original mass matrix in Eq.~(\\ref{kmassmatrix}), as it must by construction.\nFor example, in what is perhaps the simplest case, we see from Eq.~(\\ref{massmatrixnew}) that\n\\begin{equation}\n         \\widetilde {\\cal M}^2_{00} ~=~   \\sum_{\\lambda=\\lambda_0}^{\\lambda_{N-1}}  A_\\lambda \\, \\lambda^2~.\n\\end{equation}\nIf the summation had proceeded over the entire infinite spectrum, the unitarity relation in\nEq.~(\\ref{magic}) would have given us the correct infinite-$N$ result \\il{{\\cal M}^2_{00} = m^2}.\nThus, we see in this case \nthat our UV-based truncation consists of gently draining away the contributions   \nto the unitarity sum that come from the most massive modes, thereby deforming this\nfirst matrix element in a gentle, $N$-dependent way.\nOther matrix elements are similar.\n\nThe new matrix $\\widetilde {\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{massmatrixnew}) \nthus defines an alternate truncation of the full KK theory. \nIndeed, both our original truncated mass matrix ${\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{kmassmatrix}) and our new mass matrix\n$\\widetilde {\\cal M}^2_{k\\ell}$ \nin Eq.~(\\ref{massmatrixnew})\ndescribe the same KK theory in their \\il{N\\to\\infty} limits.\nHowever, for any finite $N$ these mass matrices define distinct theories.   \nWe shall refer to our traditional truncation as being ``IR-based'', since it builds our finite-$N$ theory\ndirectly from the ground up, KK mode by KK mode, directly as they were in the full theory without regard for any \nof the UV physics.\nBy contrast, we shall refer to our alternative truncation as being ``UV-based'' in the sense that it utilizes\nthe full UV values of the mass eigenvalues and basis-changing matrices prior to truncation,\nand gently builds these quantities into our recipe for truncation. \nThis is thus more of a ``top-down'', UV-sensitive approach.\nOf course, it is only because of the mixing of KK states induced by the phase transition on the brane \nthat these two truncations are distinct.\n\nWhat makes this alternate UV-based truncation \nparticularly important for our purposes in this paper\nis that it provides a much more rapid road to asymptotia than does our usual truncation.\nIn other words, the asymptotic values of late-time quantities such as ${\\overline{\\rho}}_\\lambda$\nand ${\\overline{\\eta}}$ are approached more rapidly as functions of $N$\nvia our UV-based truncation than via the traditional IR-based truncation.\nWe can also see this from Fig.~\\ref{fig:Nscaling}.\nIn Fig.~\\ref{fig:Nscaling}, the blue curves illustrate \nthe IR-based approach to asymptotia for these quantities.\nHowever, in Fig.~\\ref{fig:Nscaling} we have also\nsuperimposed the red curves which represent the results of our  UV-based  \napproach to asymptotia for these same quantities.\nIn all cases, we see that the UV-based approach to asymptotia tends to differ\nsignificantly from the IR-based approach for small $N$.   \nIndeed, as $N$ increases, we see that the finite-$N$ energy densities associated\nwith the IR-based approach tend to approach their asymptotic limits {\\it from above}\\\/,\nwhile the finite-$N$ energy densities associated with the UV-based approach tend to approach\nthese same asymptotic limits {\\it from below}.\\\/ \nNevertheless, as \\il{N\\to\\infty},\nwe see that the UV-based truncation approaches \nasymptotia more rapidly (for smaller values of $N$) than does the IR-based truncation.\nWe also observe that the approach to asymptotia provided by the UV-based truncation\nis monotonic, particularly for the late-time tower fraction ${\\overline{\\eta}}$,\nwhereas the path provided by the IR-based approach is non-monotonic.    However, this too is \nstraightforward to understand.\nIn general, the non-monotonicity of \n${\\overline{\\eta}}(N)$ in the IR-based approach is due to the somewhat spurious effects \nof the highest mass eigenstate in the finite-$N$ system, \nas discussed in Sect.~\\ref{sec:TheModel} and sketched in \nFigs.~\\ref{fig:massspectrum} and \\ref{fig:initialabruptabundance}.  \nIn the UV-based approach, by contrast, the highest mode no longer behaves\nanomalously for finite $N$.\n\n\\begin{figure}[b]\n\\includegraphics[keepaspectratio, width=0.5\\textwidth]{lambdascaling}\n\\caption{Eigenvalues of the IR-based and UV-based mass matrices, plotted respectively in blue and red \nas functions of $N$ for \\il{m\/M_c=2}.  For visual clarity, only even eigenvalues are shown.\nIn each case \nthe blue and red curves asymptote to the exact eigenvalues $\\lambda_k$ as\n\\il{N\\to\\infty}, but the UV-based red curves generally approach this limit more rapidly and more smoothly \nthan do the IR-based blue curves.\nNote that each blue curve begins with an anomalously high eigenvalue;\nthis is nothing but the truncation artifact already sketched in Fig.~\\protect\\ref{fig:massspectrum}\nfor the highest mode $\\lambda_{N-1}$ in each case. \nBy contrast, such artifacts are entirely eliminated in the UV-based approach.  }\n\\label{fig:lambdascaling}\n\\end{figure}\n\nThese features are also readily apparent simply by comparing the eigenvalues \nof the IR-based truncated mass matrix ${\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{kmassmatrix}) with those\nof the UV-based truncated mass matrix $\\widetilde {\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{massmatrixnew}) as functions of $N$.\nOur results are shown in Fig.~\\ref{fig:lambdascaling} for \\il{m\/M_c=2}. Once again, we see that the \nasymptotic \\il{N\\to\\infty} limit is approached more rapidly and more smoothly in the UV-based approach\n(red curves) than in the IR-based approach (blue curves).   Moreover, we see that the anomalous highest \neigenvalue which appears in the latter approach no longer exists in the former.\n\nAt the end of Sect.~\\ref{road}, we introduced a multi-component adiabatic \napproximation~(\\ref{eq:multiadiabaticapproximation}) and noted\nthat this quantity provides a lower bound on the total energy density of\nour $N$-mode system because it includes only the contributions from the lightest mode.\nLike other physical finite-$N$ quantities, however, the value of this quantity and its\ninterpretation as a lower limit depend \non the specific KK truncation chosen, since \nthe truncation in some sense determines what is meant by the lightest mode\nand whether its contributions are affected by the removal of the \nhigher modes.   \nThis is readily apparent in Fig.~\\ref{fig:Nscaling}, where the red ``UV-based''\nvalues of the total energy density are clearly \nsmaller for certain small values of $N$ than the ``IR-based''\nvalues of the adiabatic lower limit.\nThese red ``UV-based'' total energies nevertheless strictly exceed \nthe values of a corresponding ``UV-based'' adiabatic lower limit. \n\nWe conclude with a final comment.  Although we have presented our alternate UV-based \ntruncation of the full KK theory as providing a more rapid road to asymptotia, the \nexistence of such an alternate truncation is interesting in its own right. In any theory \ninvolving extra spacetime dimensions, one can never probe all energy scales and thereby \ndetect all KK modes.  Instead, we expect the physics of our full KK system to be \napproximately represented at low energies through some sort of truncation that focuses \non the lower modes. {\\it However, if the physics of our extra dimensions results in a \nmixing of KK modes (as must always arise in any theory which breaks translational invariance \nin the extra compactified dimension), we now see that there are multiple options for \nperforming such a truncation.}\\\/ Indeed, one could even argue that our UV-based truncation \nis more appropriate for certain calculations since it incorporates and thus is more sensitive \nto the actual masses of the physically propagating mass-eigenstates that we would expect \nto observe experimentally.  This last statement is of course subject to various \nrenormalization-group effects which could potentially deform the observed KK masses and \ncouplings, as discussed in Refs.~\\cite{sky1,sky2,sky3}, and which exist even for theories such as those \nconsidered in Refs.~\\cite{sky1,sky2,sky3} in which no KK-mixing is present. These observations nevertheless \npotentially give our UV-based truncation a theoretical importance in its own right that \nrenders it worthy of further study.\n\nOf course, there does exist a well-defined procedure through which one can unambiguously \ndescribe the physics of our full KK theory at low energies: one can use the methods of \neffective field theory (EFT).~ Specifically, one carefully integrates out the modes with energies \nabove a particular cutoff scale, thereby obtaining not only a truncated KK tower exhibiting \nrenormalized masses, but also a set of effective operators which reflect the underlying structure \nof the full theory.  In general, the structure of such an EFT can be quite complicated. \nThus, it is traditional in the literature to simply adopt the IR-truncated theory as an \napproximation to this EFT.~ Our point, then, is that our UV-based truncation might profitably \nserve as an alternative approximation to the complete EFT --- an approximation which, as we have \ndiscussed, may have certain advantages.  Needless to say, the UV and IR truncations, as well as \nthe complete EFT, all converge to the full KK theory as \\il{N\\rightarrow\\infty}.\n\n\n\\FloatBarrier\n\\section{KK Tower Limit: ~\\texorpdfstring{$N=\\infty$}{N=infty}\\label{sec:KKLimit}}\n\n\n\\begin{figure*}\n\\includegraphics[keepaspectratio, width=0.9\\textwidth]{infinitemodedensityplots.pdf}\n\\caption{The late-time energy density ${\\overline{\\rho}}$ \nof the full KK tower, plotted in the \\il{({\\overline{m}} t_G, \\delta_G)} plane for \\il{t_G= 10^2\/M_c}.\nThe left and right panels show contours of \n${\\overline{\\rho}}\/ {\\overline{\\rho}}(\\delta_G=0)$ and\n${\\overline{\\rho}}\/ {\\overline{\\rho}}_{\\rm 4D}$, respectively.}\n\\label{fig:infinitemodedensityplots}\n\\end{figure*}\n\nHaving studied our system as a function of $N$ for large \\il{N\\gg 1}, \nwe are now finally in a position to present our \nresults for the late-time energy density ${\\overline{\\rho}}$ and tower fraction ${\\overline{\\eta}}$\nfor the full, infinite KK tower. \nAs we shall see, certain features emerge in the full infinite-$N$ limit\nthat were not present for any finite $N$.\nTaken together, these results thus describe the late-time energy configuration\nacross our infinite KK tower when \nit has been subjected to a mass-generating phase transition \nof arbitrary width $\\delta_G$ and arbitrary magnitude (set by the late-time brane mass ${\\overline{m}}$)\nat time $t_G$ during its cosmological history. \n\nWe shall present our results in several different ways.\nFirst, in Fig.~\\ref{fig:infinitemodedensityplots},\nwe plot our results for the total late-time energy density ${\\overline{\\rho}}$ as a fraction of\n${\\overline{\\rho}}(\\delta_G=0)$ (left panel)\nor ${\\overline{\\rho}}_{\\rm 4D}$ (right panel).\nOur first observation \nfrom the results in Fig.~\\ref{fig:infinitemodedensityplots}\nis that while a quantity such as ${\\overline{\\rho}}(\\delta_G=0)$ \nmight indeed be a useful approximation for ${\\overline{\\rho}}$ which is \nvalid in certain regions of parameter space, \nsuch an approximation can fail badly in others.\nFor example, we see that the abrupt (\\il{\\delta_G=0}) approximation works\nwell for small \\il{{\\overline{m}} t_G\\lsim 0.1} even if $\\delta_G$ is sizable but \nis otherwise capable of either significantly underestimating or overestimating the \ntrue late-time energy density ${\\overline{\\rho}}$, the latter often by many orders\nof magnitude.\n\nAnother immediate observation is that the contours in\nFig.~\\ref{fig:infinitemodedensityplots}\nfollow a power-law type behavior in those\nregions of \\il{({\\overline{m}} t_G,\\delta_G)} parameter space in which we are eventually going to be the\nmost interested,\nnamely those regions with\n\\il{{\\overline{m}} t_G\\gsim 10} and \\il{\\delta_G\\lsim 0.3}\nin which multiple components contribute non-trivially to the total late-time abundance and \\il{{\\overline{\\eta}}>0}. \nThis power-law behavior enables us to extract approximate analytical expressions for\n${\\overline{\\rho}}\/ {\\overline{\\rho}}(\\delta_G=0)$ and\n${\\overline{\\rho}}\/ {\\overline{\\rho}}_{\\rm 4D}$ which are valid in these regions.\nSpecifically, given the results in Fig.~\\ref{fig:infinitemodedensityplots}\n(as well as analogous results calculated for different values of $t_G$),\nwe find\n\\begin{equation}\\label{eq:simplewidthnormapproximation}\n\\frac{{\\overline{\\rho}}(\\delta_G)}{{\\overline{\\rho}}(0)} ~\\approx~ \\frac{0.6}{\\delta_G}\\, \\frac{e^{-\\frac{1}{2} M_c\/{\\overline{m}}}}{{\\overline{m}} t_G}\n         \\left(\\frac{M_c}{{\\overline{m}}}\\right)^{0.9}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:mixingnormformula}\n  \\frac{{\\overline{\\rho}}}{{\\overline{\\rho}}_{\\rm 4D}} ~\\approx~ \\frac{1}{1 + 2({\\overline{m}}\/M_c) e^{-\\frac{1}{2}M_c\/{\\overline{m}} }} \\ .\n\\end{equation}\nWhile the expression in Eq.~(\\ref{eq:simplewidthnormapproximation})  holds \nto within $\\pm 25\\%$ across the relevant \\il{{\\overline{\\eta}}>0} region,\nthis result is actually somewhat sensitive to the value of $t_G$, with the upper error limit \ngrowing smaller with increasing $t_G$ and bigger with decreasing $t_G$.\nBy contrast, the expression in Eq.~(\\ref{eq:mixingnormformula}) holds to within $\\pm 5\\%$ across the relevant\nregion, making it one of the most accurate analytical approximations we have presented in this paper.\n\nWe can also deduce approximate scaling laws from these expressions.\nFor \\il{{\\overline{m}}\/M_c \\gg 1}, we find that\nEq.~(\\ref{eq:simplewidthnormapproximation}) approximately reduces to\n\\begin{equation}\n    \\frac{{\\overline{\\rho}}(\\delta_G)}{{\\overline{\\rho}}(0)} ~\\sim~ {1\\over {\\overline{m}} t_G \\delta_G}  \\left( {M_c\\over {\\overline{m}}}\\right)~\n\\label{scaling1}\n\\end{equation}\nwhich is a factor of $M_c\/{\\overline{m}}$\ngreater than the corresponding expression we observed in Eq.~(\\ref{fdscaling})\nfor the 4D case with \\il{N=1}.\nThus the presence of an entire of tower of KK states\nsuppresses what would otherwise have been\nthe late-time energy density of the zero mode alone\nby an additional factor of $M_c\/{\\overline{m}}$.\nLikewise, for \\il{{\\overline{m}}\/M_c\\gg 1},\nwe observe from\nEq.~(\\ref{eq:mixingnormformula})\nthat\n\\begin{equation}\n    \\frac{{\\overline{\\rho}}}{{\\overline{\\rho}}_{\\rm 4D}} ~\\sim~ {M_c\\over {\\overline{m}}}~.\n\\label{scaling2}\n\\end{equation}\nThus, combining these results for \\il{{\\overline{m}}\/M_c\\gg 1}, we find that\n\\begin{equation}\n         {{\\overline{\\rho}}(\\delta_G) \\over {\\overline{\\rho}}(0)} ~\\sim~\n         {{\\overline{\\rho}}(\\delta_G) \\over {\\overline{\\rho}}_{\\rm 4D}(\\delta_G)} \\cdot\n         {{\\overline{\\rho}}_{\\rm 4D}(\\delta_G) \\over {\\overline{\\rho}}_{\\rm 4D}(0)}~\n\\end{equation}\nfrom which we deduce that \n\\begin{equation}\n      {\\overline{\\rho}}(0) ~\\sim~ {\\overline{\\rho}}_{\\rm 4D}(0)~.\n\\label{same}\n\\end{equation}\nThis in turn requires that there exist a constant $c$ for which\n\\begin{equation}\n           \\lim_{\\delta_G\\to 0} \\left( {{\\overline{\\rho}}\\over {\\overline{\\rho}}_{\\rm 4D}} \\right) ~\\approx ~c~,\n\\label{limitratio}\n\\end{equation}\nand indeed this last relation is true for \\il{c=1}. \nAccording to the results shown in the right panel of Fig.~\\ref{fig:infinitemodedensityplots},\nthis relation with \\il{c=1} is manifestly true for \\il{{\\overline{m}}\/M_c \\lsim 1}.  \nHowever, as $\\delta_G$ approaches zero, this relation with \\il{c=1} becomes true\nfor larger and larger values of ${\\overline{m}}\/M_c$.\nThus, all of the scaling relations we have quoted here are self-consistent\nwithin the regions of validity claimed.\n\nIn this connection, we remark that it is not only the power-law scaling relations\nin Eqs.~(\\ref{scaling1}) and (\\ref{scaling2}) \nwhich must be consistent with each other;\nthe same must also be true of the more complete expressions such as those\nin Eqs.~(\\ref{eq:simplewidthnormapproximation})\nand (\\ref{eq:mixingnormformula}) from which these power-law relations are derived.\nHowever, although such self-consistent pairs of expressions exist,\nthe specific expressions provided in\nEqs.~(\\ref{eq:simplewidthnormapproximation})\nand (\\ref{eq:mixingnormformula}) \ndo not constitute such a pair.  Rather, these expressions are provided instead because they \nyield even greater numerical accuracy over their appropriate regions of\nparameter space.\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{infinitemodeabsolute.pdf}\n\\caption{The {\\it absolute}\\\/ late-time energy density ${\\overline{\\rho}}$ \nin units of ${\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}$, plotted within the $({\\overline{m}} t_G,\\delta_G)$ \nplane for three different values of $M_c t_G$.}\n\\label{fig:absoluteplots}\n\\end{figure*}\n\nAs we have seen, considering the late-time energy\ndensity ${\\overline{\\rho}}$ as a fraction of related benchmarks such as ${\\overline{\\rho}}(\\delta_G=0)$ or ${\\overline{\\rho}}_{\\rm 4D}$ \nhas been useful for understanding the {\\it relative}\\\/ effects of increasing $\\delta_G$ or \nincreasing the numbers of modes.\nHowever, we also wish to understand the late-time energy densities ${\\overline{\\rho}}$ on an {\\it absolute}\\\/ scale. \nOf course, \nthese absolute magnitudes can in principle be obtained through a sequence of multiplications of previous \nintermediate results.\nFor example, one possible path is given by\n\\begin{equation}\n           {\\overline{\\rho}} ~=~ \\left({{\\overline{\\rho}} \\over {\\overline{\\rho}}_{\\rm 4D}}\\right) \n        \\, \n            \\left({{\\overline{\\rho}}_{\\rm 4D} \\over {\\overline{\\rho}}_{\\rm 4D}(\\delta_G=0)}\\right) \n        \\,  \n            {\\overline{\\rho}}_{\\rm 4D} (\\delta_G=0)~\n\\end{equation}\nwhere these three different factors are indicated in the left panel of Fig.~\\ref{fig:infinitemodedensityplots},\nthe left panel of Fig.~\\ref{fig:singlefielddensityplot}, and \nthe large-$t$ behavior of Eq.~(\\ref{eq:abruptapproximation}) respectively.\nHowever, since each of these quantities is generally a complicated function of ${\\overline{m}} t_G$ and $\\delta_G$ \nthroughout the $({\\overline{m}} t_G,\\delta_G)$ parameter space,\nit is not readily apparent what composite behavior might emerge from these individual factors.\nIt is therefore useful to compile our intermediate results together in order to present the resulting \nvalues for ${\\overline{\\rho}}$ as full, absolute quantities.\n\nOur results are shown in Fig.~\\ref{fig:absoluteplots} for three different values of $t_G$.\nIn general, we see that taking larger values of ${\\overline{m}} t_G$ results in larger values of ${\\overline{\\rho}}$.\nHowever, we also see that taking larger values of $\\delta_G$ tends to suppress ${\\overline{\\rho}}$.\nTogether, these effects conspire to produce the curved contours shown.\nIndeed, increasing $t_G$ relative to $M_c$ then tends to push these contours to the right, thereby\nagain increasing the late-time energy density ${\\overline{\\rho}}$ still further.\n\n\\begin{figure}[t]\n\\includegraphics[keepaspectratio, width=0.49\\textwidth]{infinitemodeeta.pdf}\n\\caption{The late-time tower fraction ${\\overline{\\eta}}$ of the KK tower,\nplotted within the \\il{({\\overline{m}} t_G,\\delta_G)} plane for \\il{t_G=10^2\/M_c}.\nUnlike the tower fractions illustrated in Fig.~\\ref{fig:finitemodedensityplotsgenN2} for finite $N$,\nwe see that the tower fraction for \\il{N=\\infty} \nis now monotonic in both ${\\overline{m}} t_G$ and $\\delta_G$\nand approaches unity for large ${\\overline{m}} t_G$ and small $\\delta_G$.}\n\\label{fig:infinitemodeeta}\n\\end{figure}\n\nIn Fig.~\\ref{fig:absoluteplots} we indicated the \nfull, absolute magnitudes of ${\\overline{\\rho}}$ for entire our KK tower as functions of ${\\overline{m}} t_G$ and $\\delta_G$.\nHowever, we are also interested in \nthe distribution of this total energy density across the different modes of our KK tower.\nAs we have seen throughout this paper,\none measure of this distribution is the tower fraction $\\eta$ --- the \nfraction of the total abundance which is carried by all but the most abundant mode in the tower.\nThe late-time values of these tower fractions for the full KK tower are plotted \nin Fig.~\\ref{fig:infinitemodeeta}.\nAs we see upon comparison with its finite-$N$ \nequivalents in Fig.~\\ref{fig:finitemodedensityplotsgenN},\n {\\it passing to the infinite-$N$ limit has the effect of removing all non-monotonicities in ${\\overline{\\eta}}$}.\nIndeed, we see that ${\\overline{\\eta}}$ now increases monotonically as either ${\\overline{m}} t_G$ is increased or $\\delta_G$ is \ndecreased.   We stress that this is a feature which emerges only for the full KK tower with \\il{N=\\infty},\nbut which would not be true for any finite value  of $N$.   \nAs a result, {\\it it is the region with large ${\\overline{m}} t_G$ (or equivalently large ${\\overline{m}}\/M_c$) and relatively\nsmall $\\delta_G$ for which the total energy density ends up distributed most broadly across the different\nstates in the KK tower at late times.}\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.8\\textwidth]{rhofracpies.pdf}\n\\caption{A graphical representation of the distribution of the\nenergy densities ${\\overline{\\rho}}_\\lambda$ across the entire KK tower, calculated for a\nsquare ``lattice'' of locations\nin the \\il{({\\overline{m}} t_G,\\delta_G)} plane.\nEach pie chart \nindicates how the total ${\\overline{\\rho}}$ is distributed across the different KK modes,\nwith the blue slices indicating the contribution associated with the lightest mode\nand the remaining slices (colored pink through dark red) indicating\nthe contributions from successively heavier KK modes.\nThe results for each pie correspond to the parameters \\il{({\\overline{m}} t_G,\\delta_G)}\nassociated with the location of the center of the pie,\nand the yellow contours indicate the corresponding values of ${\\overline{\\eta}}$,\nas taken from Fig.~\\protect\\ref{fig:infinitemodeeta}.\nWe see that the total energy density is preferentially captured by the lightest\nmode for larger $\\delta_G$ or smaller ${\\overline{m}} t_G$, but that the total energy density\nis distributed more democratically across the KK tower for smaller $\\delta_G$ and larger ${\\overline{m}} t_G$.\nThus a wide variety of energy-density distributions across the KK tower can be\nrealized simply by adjusting the parameters associated with the mass-generating phase transition.}\n\\label{fig:rhofracpies}\n\\end{figure*}\n\nWhile ${\\overline{\\eta}}$ represents one measure of the degree to which the late-time total energy density of the KK tower\nis distributed across its different modes, \nthis quantity still does not tell us how many modes are actually\ncarrying a significant abundance.\nMore specifically, we would like to know   the shape the overall ``profile'' of the \nenergy-density distribution across the entire KK tower as a function of ${\\overline{m}} t_G$ and $\\delta_G$.\nTowards this end, can visualize the shape of a given energy-density distribution profile by means of \na pie chart whose different pie slices illustrate how ${\\overline{\\rho}}$ is distributed across the different KK modes.\nWe then seek to understand how the relative slices of such pies \nevolve as functions of ${\\overline{m}} t_G$ and $\\delta_G$.\n\nOur results are shown in Fig.~\\ref{fig:rhofracpies}  for \\il{t_G=10^2\/M_c}.\nIn Fig.~\\ref{fig:rhofracpies}\nwe have focused on the non-trivial green region of Fig.~\\ref{fig:infinitemodeeta} wherein \\il{{\\overline{\\eta}}>0},\nand then superimposed a set of pie charts illustrating how the energy-density profile varies\nacross this region. \nWe see from this figure that the total energy density tends to be preferentially captured by the lightest\nmode as $\\delta_G$ increases or ${\\overline{m}} t_G$ decreases, but that \nthe total energy density tends to be more democratically distributed across the KK modes of the tower\notherwise.  {\\it We see, then, that a wide variety of energy-density distributions across the KK tower are possible\nand can be realized simply by adjusting the parameters associated with the mass-generating phase transition.}\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.8\\textwidth]{rhofracpiesgrand.pdf}\n\\caption{A grand summary of the results of this section, combining \nour information concerning the absolute {\\it magnitudes}\\\/ of the total late-time\nenergy density ${\\overline{\\rho}}$ from Fig.~\\protect\\ref{fig:absoluteplots} \nwith our information concerning the {\\it distribution}\\\/ of that energy density from\nFig.~\\protect\\ref{fig:rhofracpies}, all plotted \nwithin the \\il{({\\overline{m}} t_G, \\delta_G)} plane.\nThis figure is essentially the same as Fig.~\\protect\\ref{fig:rhofracpies} except that the overall\nareas of our pie charts have been rescaled in proportion to the magnitudes of the total late-time\nenergy densities ${\\overline{\\rho}}$ shown in Fig.~\\protect\\ref{fig:absoluteplots}.\nIn the background we have also indicated the contours of ${\\overline{\\rho}}$ from Fig.~\\protect\\ref{fig:absoluteplots}\n(dashed black lines)\nas well as the contours of ${\\overline{\\eta}}$ from Fig.~\\protect\\ref{fig:infinitemodeeta} (solid blue lines).\nWe see that there is a non-trivial {\\it correlation}\\\/ between the overall  magnitude of ${\\overline{\\rho}}$\nand the distribution of that energy density, with larger energy densities distributed more\ndemocratically across the KK tower and smaller energy densities captured more and more preferentially\nby the lightest KK mode.   Both features are nevertheless extremely sensitive to the parameters governing\nthe mass-generating phase transition, with the former behavior dominating for smaller $\\delta_G$ and larger\n${\\overline{m}} t_G$ and the latter \ndominating for larger $\\delta_G$ and smaller ${\\overline{m}} t_G$.} \n\\label{fig:rhofracpiesgrand}\n\\end{figure*}\n\nFinally, we can summarize the results of this section by combining our \ninformation concerning the absolute {\\it magnitudes}\\\/ of the late-time total energy density ${\\overline{\\rho}}$,\nas indicated in Fig.~\\ref{fig:absoluteplots},\nwith our information concerning the late-time {\\it distributions}\\\/ of that total energy density,\nas indicated in Fig.~\\ref{fig:rhofracpies}.\nTo do this, we can begin with the information in Fig.~\\ref{fig:rhofracpies} but then \n{\\it rescale the size of each pie chart so that \nthe area of each pie chart is proportional to the total energy density ${\\overline{\\rho}}$.}\nThe result is shown in Fig.~\\ref{fig:rhofracpiesgrand}.\n\nOne important lesson that emerges from \nFig.~\\ref{fig:rhofracpiesgrand} is that\nthere is a non-trivial correlation between the overall {\\it magnitude}\\\/ of ${\\overline{\\rho}}$\nand the {\\it distribution}\\\/ of that energy density, with larger energy densities distributed more\ndemocratically across the KK tower and smaller energy densities captured more and more preferentially\nby the lightest KK mode.   \nThe results in Fig.~\\ref{fig:rhofracpiesgrand} allow us to see rather dramatically\nthe effects of our mass-generating phase transition on the eventual late-time energy configuration\nof our KK tower.\nFor any fixed ${\\overline{m}} t_G$, we see that increasing the phase-transition timescale $\\delta_G$ \nsuppresses the energy density that remains in the massive KK modes, causing the lighter\nmodes to assume an increasing fractional share of the total energy density.\nYet, at the same time, increasing $\\delta_G$ suppresses the total energy\ndensity that is ultimately pumped into the KK tower by the phase transition.\nLikewise, for any $\\delta_G$, we see that increasing ${\\overline{m}} t_G$ \nincreases the total energy pumped into the system by the phase transition\nwhile simultaneously causing this energy density to be more democratically distributed.\n{\\it Thus, merely by choosing appropriate values of ${\\overline{m}} t_G$ and $\\delta_G$, it is possible to adjust the \ntotal absolute energy density remaining in the KK tower at late times to any value one might\nselect for phenomenological purposes while simultaneously retaining the ability to adjust\nthe distribution of that energy density across the different KK modes.}   This, then, is the power\nof the mass-generating phase transition and the influence of its associated timescale.\n \n\n\\FloatBarrier\n\\section{Example:~ Axion in the Bulk\\label{sec:AxionintheBulk}}\n\n\n\nUntil this point, we have maintained generality\nby considering a higher-dimensional field $\\Phi$\nand specifying little more about this field than that it is a scalar.\nLikewise, we have assumed little more about our phase transition\nthan that it generates masses in a time-dependent way.\nHowever, in order to explore one possible set of phenomenological implications of our results,\nwe shall now apply our machinery to the case in which $\\Phi$ is an axion-like\nparticle and in which our phase transition on the brane is one in which \ninstanton-like effects give mass to that axion.\n\nWe begin by considering the setup\ndescribed in Refs.~\\cite{DDM2,DDMAxion}, which is itself a generalization of\nan earlier framework considered in Ref.~\\cite{DDGAxions}.  \nSpecifically, \nwe consider the same five-dimensional geometry as discussed in previous sections\nand henceforth take \n\\begin{equation}\n    M_c ~=~ 4.49 \\,\\times\\, 10^{-12} ~\\text{GeV} \n\\end{equation}\nas our compactification scale.\nThis value of $M_c$ corresponds to \\il{R\\approx 44~\\mu\\text{m}}, \nwhich is the largest flat extra dimension \nallowed according to data from torsion-balance experiments~\\cite{KapnerEotvosExpt}. \nWithin the bulk of this extra dimension \nwe shall consider\na (pseudo-)scalar field $\\Phi$ which is\na straightforward generalization of the traditional QCD axion~\\cite{PecceiQuinn1,WeinbergAxion,WilczekAxion}.\nIn particular,\nwe shall take $\\Phi$ to be \nthe Nambu-Goldstone boson associated with a global chiral Peccei-Quinn-like $U(1)_X$ symmetry which is spontaneously \nbroken at some scale $f_X$.  \nThis $U(1)_X$ symmetry is assumed anomalous, and for a non-Abelian gauge group $G$ on the brane\nwith coupling $g$ and field strength $\\mathcal{G}_{\\mu\\nu}$, this anomaly therefore generates a topological \nbrane term of the form\n\\begin{equation}\n\\mathcal{L}_{\\text{brane}} ~\\rightarrow~  \n\\mathcal{L}_{\\text{brane}} +\n\\frac{\\mathcal{C} g^2}{32\\pi^2}\\frac{\\Phi}{f_X^{3\/2}}\\text{Tr }\\mathcal{G}_{\\mu\\nu}\\tilde{\\mathcal{G}}^{\\mu\\nu} \n\\end{equation}\nwhere $\\mathcal{C}$ is a model-dependent constant. \nWhile this term has no effect on the classical \nequations of motion, it affects the vacuum structure of the theory. \nAs the universe cools \nand reaches \\il{T\\sim \\Lambda_G},\nwhere $\\Lambda_G$ is the confinement scale associated with the group $G$, \ninstanton effects on the brane\nexplicitly break the $U(1)_X$ axion shift symmetry that prevents the axion from acquiring a mass.\nAs a result, a small temperature-dependent axion mass $m_X(T)$ is generated on the brane. \nIn general, we shall model the time dependence of this mass \nexactly as in previous sections, as resulting from a phase transition \noccurring at a time $t_G$ with a width $\\delta_G$.\nIn this connection we note that our adoption of an LTR cosmology,\nas discussed in Sect.~\\ref{sec:TheModel}, \nimplies that this phase transition takes place \nduring an inflaton-dominated (and thus matter-dominated) epoch.\nThe corresponding  time\/temperature relationship then yields\nthe result\n\\begin{equation}\nt_G ~=~ \\sqrt{\\frac{45 g_*(T_{\\rm RH})}{2\\pi^2}}\\frac{T_{\\rm RH}^2 M_p}{g_*(\\Lambda_G)\\Lambda_G^4} ~\n\\label{timetemp}\n\\end{equation}\nwhere \\il{T_{\\rm RH}\\sim\\mathcal{O}(\\text{MeV}) < \\Lambda_G} is the reheating temperature, \nwhere $g_*(T)$ is the effective number of relativistic degrees of freedom at temperature $T$, \nand where \\il{M_p\\equiv 1\/\\sqrt{8\\pi G}} is the reduced Planck mass. \nFor concreteness we shall take \\il{T_{\\rm RH}= 30}~MeV in what follows.\nFinally, at late times, our phase transition leaves our axion-like field with\na brane mass given by\n\\begin{equation}\n{\\overline{m}}_X^2 ~\\equiv~ \\frac{\\mathcal{C}^2 g^2}{32\\pi^2}\\frac{\\Lambda_G^4}{{\\hat{f}_X}^2}~ \n\\label{mbarXdefn}\n\\end{equation}\nwhere we have defined the effective four-dimensional $U(1)_X$-breaking \nscale \\il{{\\hat{f}_X}\\equiv \\sqrt{2\\pi R f_X^3}}. \n\nThe above setup defines our five-dimensional axion theory.\nCompactifying this theory via KK reduction  \nthen yields an effective four-dimensional theory consisting of a KK tower of axion modes $\\phi_k$ \nwhose mass matrix takes exactly the form given in Eq.~(\\ref{kmassmatrix}), with $m(t)$ now \nidentified as $m_X(t)$. \nThus, we see that our axion model is nothing but a special case of the model we have already considered\nthus far --- the special case in which we identify ${\\overline{m}}$ as ${\\overline{m}}_X$ and identify $t_G$ as the cosmological time \ncorresponding to the temperature \\il{T\\sim\\Lambda_G}, as in Eq.~(\\ref{timetemp}).\n\nIn previous sections, we studied the behavior of our general KK system \nas a function of the four parameters \\il{\\lbrace {\\overline{m}}, M_c, t_G, \\delta_G\\rbrace}.\nFor our axion theory, by contrast, we see that \n\\il{{\\overline{m}}\\to {\\overline{m}}_X}, and moreover we see that both ${\\overline{m}}_X$ and $t_G$ are themselves\nrelated to the more fundamental parameters $\\Lambda_G$ and $\\hat f_X$ through \nEqs.~(\\ref{mbarXdefn}) and (\\ref{timetemp})\nrespectively.\nThus, we shall henceforth consider our axion theory \nto be parametrized by\n\\il{\\lbrace \\Lambda_G, \\hat f_X, M_c, \\delta_G\\rbrace}.\n\nIn four dimensions, the methods used to estimate the late-time abundance of a given axion field typically fall into one\nof two classes examined in Sect.~\\ref{sec:FourDimensionalLimit}: the adiabatic approximation\nor the abrupt approximation.\nAs we have seen, the former only applies when the mass is generated sufficiently slowly during field oscillations,\nwith \\il{\\dot m \\ll m^2}, \nwhile the latter applies only for very small $\\delta_G$ (such that the phase-transition width $\\Delta_G$ is much\nsmaller than the timescale of field oscillations) or for situations in which \\il{t_\\zeta\\gg t_G} (so that the phase\ntransition occurs while the field is still overdamped and essentially frozen).\nHowever, as we have seen in previous sections, these approximations break down in\nrelatively large regions of parameter space.\nWe should also expect deviations from our standard expectations in the case of\nan infinite KK tower of axion modes.\nFor example, the virialization condition that underpins the adiabatic approximation\nnow becomes \n \\il{\\dot{\\lambda}_0 \\ll \\lambda_0^2}, and $\\lambda_0$ can be much smaller than\n$m_X$~\\cite{DDGAxions}.\nLikewise, there is always some subset of modes   \nin the KK tower for which \\il{t_\\zeta^{(\\lambda)}\\lesssim t_G}.\nSuch modes are therefore necessarily affected by the time-dependence of the phase transition.\n\n\\begin{figure*}\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{4Daxion.pdf}\n\\caption{ The late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ of our generalized axion field in the 4D limit, \nplotted within the \\il{(\\Lambda_G, \\hat f_X)} plane for \\il{\\delta_G = 0.1} (top row) and\n\\il{\\delta_G=1.0} (bottom row).   Note that the horizontal $\\Lambda_G$ axis (as shown along the bottom of each panel)\nis equivalently a $t_G$ axis (as shown along the top).\nThe panels in the left and middle columns plot\n${\\overline{\\rho}}$ as fractions of the abrupt and adiabatic approximations ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G=0)$ \nand ${\\overline{\\rho}}_{\\rm 4D}|_{\\rm abs}$, respectively,\nwhile the panels in the right column \nplot the absolute magnitude of ${\\overline{\\rho}}_{\\rm 4D}$ in units of    \n${\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}$.   \nAlso shown in each panel is a purple dashed line indicating the contour\nalong which \\il{{\\overline{m}}_X t_G=1}.}   \n\\label{fig:4Daxion}\n\\end{figure*}\n\nIn this section,\nwe shall therefore present exact results for the late-time energy densities of our axion fields.  We shall do this both\nfor the four-dimensional \\il{N=1} case as well as the infinite-$N$ case of our full KK axion tower.\nAs discussed above, we shall take our axion parameter\nspace to be parametrized by\n\\il{\\lbrace \\Lambda_G, \\hat f_X, M_c, \\delta_G\\rbrace}.\nHowever, we stress that the plots to be presented in this section are not merely translations of our previous\nplots into these new variables.   First, we have extended our range of \ninterest within this parameter space into those regions of specific interest for axion physics.\nSecond, we shall now regard $\\Lambda_G$ and $\\hat f_X$ as the fundamental axion parameters\nrelative to which we wish to consider continuous variations. In other words,\nwe shall plot our the late-time energy densities ${\\overline{\\rho}}$ \nas functions of $\\Lambda_G$ and $\\hat f_X$ within the \\il{(\\Lambda_G,\\hat f_X)} plane, choosing only \ndiscrete representative choices for $\\delta_G$.    This too is different than what was done in previous sections,\nand thus represents a different, independent slice through our four-dimensional parameter space.  \n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.7\\textwidth]{KKaxion1.pdf}\n\\caption{ The total late-time energy density ${\\overline{\\rho}}$ of our generalized axion KK tower, \nplotted as fractions of ${\\overline{\\rho}}(\\delta_G=0)$ (left column) or ${\\overline{\\rho}}_{\\rm 4D}$ (right column)\nwithin the \\il{(\\Lambda_G, \\hat f_X)} plane for \\il{\\delta_G = 0.1} (top row) and \\il{\\delta_G=0.3} (bottom row).   \nAs in Fig.~\\protect\\ref{fig:4Daxion},\nthe $\\Lambda_G$ axis is shown along the bottom of each panel\nwhile the equivalent $t_G$ axis is shown along the top.\nAlso shown in each panel are three purple dashed lines indicating the contours along which \n\\il{{\\overline{m}}_X t_G=1}, \\il{{\\overline{m}}_X = M_c}, or \\il{M_c t_G=1}.\nAs with other figures in this paper, the gray regions \nare excluded for the reasons discussed in the paragraph below Eq.~(\\ref{bigwidth}).}\n\\label{fig:KKaxion1}\n\\end{figure*}\n\nWe begin, as in Sect.~\\ref{sec:FourDimensionalLimit}, by considering the four-dimensional \\il{N=1} limit.\nIn this case, we find the results shown in Fig.~\\ref{fig:4Daxion}.\nWe observe from the panels in the left and center columns of \nFig.~\\ref{fig:4Daxion} that there are distinct regions\nof parameter space in which the abrupt and adiabatic approximations fail\nto model the true late-time energy density, with the adiabatic approximation\nconsistently underestimating the true energy density and the abrupt approximation\neither under- or overestimating the true energy density, in some cases by many orders\nof magnitude.   We also see that increasing either the confinement scale $\\Lambda_G$   \nor the $U(1)_X$ symmetry-breaking scale $\\hat f_X$  \ngenerally decreases the late-time energy density of our axion field.  \nIncreasing the width $\\delta_G$ of our axion-induced phase transition also has the same effect.\nIndeed, we see from the relatively straight contours in Fig.~\\ref{fig:4Daxion} that\nthe absolute magnitude of the late-time energy density scales approximately as\n\\begin{equation} \n           {\\overline{\\rho}}_{\\rm 4D} ~\\sim~ 1\/(\\Lambda_G^4 \\hat f_X^2)~,\n\\end{equation}\nwhere the constant of proportionality is a non-trivial function of $\\delta_G$.\nThis scaling behavior is of course consistent with the analogous result in Eq.~(\\ref{4Dscaling}).  \nFinally, we note that all of the contour lines in Fig.~\\ref{fig:4Daxion}  \nexperience slight ``ripples'' at \\il{\\Lambda_G\\approx 2}~GeV and $5$~GeV.\nThese ripples are physical, and correspond to the energy scales $\\Lambda_G$ at which there\nare changes in the number $g_\\ast(\\Lambda_G)$  of relativistic degrees of freedom (the former\ncorresponding to the threshold for the charm quark and tau lepton, and the latter corresponding to the threshold for the bottom quark). \n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{KKaxion3with0.pdf}\n\\caption{ The absolute magnitude of the total late-time energy density ${\\overline{\\rho}}$ of our generalized axion KK tower, \nplotted in units of ${\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}$\nwithin the \\il{(\\Lambda_G, \\hat f_X)} plane for \n\\il{\\delta_G=0} (left panel),\n\\il{\\delta_G = 0.1} (middle panel),\nand \\il{\\delta_G=0.3} (right panel).  }   \n\\label{fig:KKaxion2}\n\\bigskip\n\\bigskip\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{KKaxion2with0.pdf}\n\\caption{ The late-time tower fraction ${\\overline{\\eta}}$ of our generalized axion KK tower,\nplotted within the \\il{(\\Lambda_G, \\hat f_X)} plane for \n\\il{\\delta_G=0} (left panel),\n\\il{\\delta_G = 0.1} (middle panel),\nand \\il{\\delta_G=0.3} (right panel).  }   \n\\label{fig:KKaxion3}\n\\end{figure*}\n\nWe now turn to the case in which a full KK tower of axion modes experiences\nthe instanton-induced phase transition.\nIn this case, our results are plotted in Figs.~\\ref{fig:KKaxion1} through \\ref{fig:KKaxion3}. \nIn Fig.~\\ref{fig:KKaxion1} \nwe have plotted the values of the late-time total energy density ${\\overline{\\rho}}$ of the KK tower\nrelative to our usual benchmarks ${\\overline{\\rho}}(\\delta_G=0)$ and ${\\overline{\\rho}}_{\\rm 4D}$.\nOnce again, we see that the introduction of a non-zero width $\\delta_G$ for our\ninstanton-induced phase transition leads to either enhancements or suppressions\nin ${\\overline{\\rho}}$, depending on the specific region of parameter space.\nMoreover, we see that increasing $\\delta_G$ only makes these enhancements or suppressions\nmore severe.   By contrast, turning to ${\\overline{\\rho}}\/{\\overline{\\rho}}_{\\rm 4D}$, we see the introduction of the extra KK modes in the tower\nonly suppresses ${\\overline{\\rho}}$ by an amount that grows {\\it less}\\\/ severe for increasing $\\delta_G$.\nIn all cases, however, these suppressions tend to be more pronounced for \nsmaller $\\Lambda_G$ and smaller $\\hat f_X$ than they are in other regions of parameter space. \nIt is worth noting that for smaller $\\Lambda_G$ (but \\il{{\\overline{m}}_X \\gg M_c}), we see a scaling behavior\nof the form\n\\begin{equation}\n     {{\\overline{\\rho}}\\over {\\overline{\\rho}}_{\\rm 4D}} ~\\sim~ {\\hat f_X\\over \\Lambda_G^2}~. \n\\end{equation}\nHowever, as $\\Lambda_G$ increases (and thus $t_G$ decreases),\nthis scaling behavior eventually breaks down \nand the behavior of ${\\overline{\\rho}}\/{\\overline{\\rho}}_{\\rm 4D}$  \nis increasingly determined solely by the behavior of the denominator ${\\overline{\\rho}}_{\\rm 4D}$, with\ncontours of constant ${\\overline{m}}_X t_G$.\nOtherwise,\noutside the ${\\overline{m}}_X \\gtrsim M_c$ and \\il{{\\overline{m}}_X t_G \\gtrsim 1} region,\nthe energy density at late times is very similar to that in the 4D limit.\n\nIt is also instructive to compare our KK results for \\hbox{${\\overline{\\rho}}\/{\\overline{\\rho}}(\\delta_G=0)$} along the left column of\nFig.~\\ref{fig:KKaxion1} with the analogous 4D results along the left column of Fig.~\\ref{fig:4Daxion}.\nFor the full KK tower, we no longer find contours of constant $({\\overline{m}}_X t_G)^2$;  instead we find a region \nexhibiting contours with approximately constant $\\hat f_X$.\nLikewise, the enhancement region that we found in the 4D limit spanned the entire\n\\il{\\lbrace \\Lambda_G, \\hat f_X\\rbrace} parameter space for \\il{{\\overline{m}}_X t_G \\sim 1}, whereas this region \nfor the KK tower exists only for \\il{{\\overline{m}}_X\\lsim M_c}. \n\nNext, we turn to Fig.~\\ref{fig:KKaxion2} where we plot the absolute magnitude of the \ntotal late-time energy density of the KK axion tower in units\nof ${\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}$.\nTo help understand the features of these plots and how they evolve as a function of $\\delta_G$, we \nhave included a panel with \\il{\\delta_G=0}\nwhose features can be understood directly via the abrupt approximation.\nIndeed, as expected, we see that all ${\\overline{\\rho}}$ contours in the \\il{\\delta_G=0} case\nare contours of either constant ${\\overline{m}}_X t_G$, constant ${\\overline{m}}_X\/M_c$,\nor constant $M_c t_G$.\nThe subsequent panels then illustrate how these contours deform as the width $\\delta_G$\nof our phase transition increases.\n\nLikewise, in Fig.~\\ref{fig:KKaxion3}, we plot the late-time tower fraction ${\\overline{\\eta}}$\nof our KK axion tower.\nOnce again, we see a consistent picture emerging as a function of $\\delta_G$. \nIndeed, just as for ${\\overline{\\rho}}$ in Fig.~\\ref{fig:KKaxion2}, we see that all ${\\overline{\\eta}}$ contours for \\il{\\delta_G=0} \nare contours of either constant ${\\overline{m}}_X t_G$, constant ${\\overline{m}}_X\/M_c$,\nor constant $M_c t_G$, \nwith the subsequent panels once again illustrating how these contours deform \nas $\\delta_G$ increases.\nFrom the left panel of Fig.~\\ref{fig:KKaxion3} we see that \nwhen the phase transition is abrupt, \nthe total energy density of the tower\nis distributed amongst a significant number of KK modes in\nthe small-$\\Lambda_G$, small-$\\hat f_X$ corner of parameter space. \nHowever, as our mass-generating phase transition unfolds over a longer period of time,\nthe energy contributions from the higher modes are suppressed and ${\\overline{\\eta}}$ begins to fall. \n\nOne interesting feature apparent in Fig.~\\ref{fig:KKaxion3}\nis that the value of ${\\overline{\\eta}}$ is non-monotonic\nas a function of $\\Lambda_G$, first increasing and reaching a maximum near \\il{\\Lambda_G\\approx 3}~GeV\nbefore decreasing again. \nThis can be understood as follows.\nThe vanishing of ${\\overline{\\eta}}$ for \\il{{\\overline{m}}_X \\ll M_c} is understood from the distribution of initial abundances as described \nin Fig.~\\ref{fig:initialabruptabundance}.\nIndeed, in this region, nearly all of the abundance of the KK tower is contributed by the lowest KK mode because\nvery little mixing is generated in the phase transition.\nBy contrast, the decline in ${\\overline{\\eta}}$ which occurs as $\\Lambda_G$ increases (or equivalently as $t_G$ decreases) \ncan be understood as corresponding to our entrance \ninto what in Refs.~\\cite{DDM1,DDM2,DDMAxion}\nwas called the ``staggered'' regime wherein the heavier KK modes\n          are oscillating (and thus already dissipating their energy density) \nas soon as our phase transition occurs, whereas the lighter KK modes remain overdamped for a long time after $t_G$.\nThis then results in  a total abundance composed primarily of contributions from those lighter modes.\n\nComparing the results in Figs.~\\ref{fig:KKaxion2} and~\\ref{fig:KKaxion3}, we see that the \nregions of parameter space which produce the largest total energy densities $\\bar\\rho$ almost \ncoincide exactly with the regions that produce the largest tower fractions $\\bar\\eta$.   \nIndeed, both regions have very small $\\hat f_{X}$, and the only difference is that the \nformer region has small $\\Lambda_G$ while the latter region has \\il{\\Lambda_G\\approx 3}~GeV.   \nAt first glance, this difference may appear to violate the claims made in connection with \nFig.~\\ref{fig:rhofracpiesgrand}, namely that these two regions should coincide completely.  However, \nthe correlation observed in connection with Fig.~\\ref{fig:rhofracpiesgrand} holds when $t_G$ \nis held fixed.   By contrast, the results in Figs.~\\ref{fig:KKaxion2} and~\\ref{fig:KKaxion3} \nare generated with $t_G$ implicitly varying throughout the parameter space shown.\n\nHaving focused in this section on the specific situation in which \nour $\\Phi$ field is an axion, one possible next step would be to place\nphenomenological bounds on the parameter spaces we have considered.\nSuch bounds would in principle mirror those which are normally applied\nto the case of a traditional QCD axion, and come from a variety of considerations\nincluding\nsupernova cooling,\nblack hole superradiance,\noverclosure constraints,\ndark-matter and dark-energy constraints,\ntraditional axion searches (such as light shining through walls), {\\it etc}\\\/.\nWithin the context of the abrupt (\\il{\\delta_G=0}) limit, such bounds on KK axion towers\nare discussed in detail in Ref.~\\cite{DDMAxion}. \nAlthough the determination of such bounds for general $\\delta_G$ is beyond the scope of this paper,  \nthe results we have obtained here concerning the late-time energy densities of these\naxion systems should play a critical role in helping to determine exactly\nwhere these bounds lie for general $\\delta_G$, and the extent to which the suppressions and enhancements\nwe have observed translate into a loosening or strengthening of those bounds \nbeyond traditional expectations.\n\n\n\\FloatBarrier\n\\section{Discussion and Conclusions\\label{sec:Conclusions}}\n\n\n\nIn this paper we have investigated the effects of dynamical mass generation on \nthe cosmological abundances of the Kaluza-Klein modes associated with a \nbulk scalar field.  In particular, we have examined the non-trivial case in which \na phase transition localized on a brane leads to time-dependent masses and mixings among\nthese KK modes.  We have found that both the total energy density of the full KK tower\nand the distribution of that energy density across the individual KK modes \nare extremely sensitive to the details of the phase transition.  As a result, \nwithin different regions of model parameter space, these abundances can be \nsignificantly enhanced or suppressed relative to standard expectations ---\nsometimes by many orders of magnitude.  We have also derived a variety of approximate \nscaling behaviors and analytic expressions for the energy densities of the KK modes \nas functions of the relevant model parameters.  \nIn order to illustrate the potentially significant impact \nthat these effects can have on the late-time abundances of our scalars within the context \nof a concrete model, we have also applied our general results to the case of a bulk axion\/\naxion-like field.\nFinally, as a by-product of our analysis, we have also developed an alternate ``UV-based''\neffective truncation of KK theories which is physically different from the more traditional\n``IR-based'' truncation commonly employed in the literature, yet yields \nthe same higher-dimensional theory as the truncation\nis lifted.\n \nDepending on the identity of the scalar field in question,\nour results can have a variety of phenomenological implications.\nFor example, in the case in which \nour bulk scalar is an axion or axion-like field,\none can easily imagine a number of \nphenomenological consequences.  Note that\nin general,\ncosmological abundances in this paper\nhave been generated through a two-step process:\nthe assumption of a non-zero VEV for one or more modes \nin the KK tower\nfollowed by a mass-generating phase transition.\nAs such, given the specific form of the initial conditions \nwe have adopted in Eq.~(\\ref{eq:initialconditions}),\nthe method of abundance generation we have studied in this paper\nis tantamount to misalignment production.\nHowever, misalignment production is not the only mechanism through which a population of \naxion-like particles can be produced in the early universe.  For example,\nspontaneous breaking of the global $U(1)_X$ symmetry of Sect.~\\ref{sec:AxionintheBulk} \ncan lead to the formation of a network of cosmic strings and other topological \ndefects~\\cite{DavisAxionsProdCosmicString1}.  If this breaking occurs before cosmic \ninflation, these defects are simply inflated away.  By contrast, if the breaking of \n$U(1)_X$ occurs after inflation, these defects retain a non-negligible energy density \nuntil late times, and their decays can therefore generate a potentially significant \ncontribution to the relic abundance of the corresponding axion-like particles.  Indeed, \nthis topic has been studied in detail for the specific case of a QCD \naxion~\\cite{DavisAxionsProdCosmicString2,\nDabholkarCosmicStringAxionBounds,AxionProdStringsWalls,AxionProdDomainWalls}, but\nin principle applies to other axion-like particles as well.\n \nThese considerations are important because \nthe phase-space distribution of a \npopulation of axions or axion-like particles produced via the decays of \ntopological defects is significantly different \nfrom that generated via misalignment \nproduction~\\cite{ChangHagmannSikivie,AxionProdStringsWalls,AxionProdDomainWalls}.\nThus, in cases in which the misalignment contribution to the \noverall late-time axion abundance is suppressed by the time-dependent masses and mixings \nwe have discussed here, \nthe phase-space distribution\nof relic axions can be significantly altered --- especially\nif the contribution from topological defects ends up dominating the total axion abundance.\nMoreover, the suppression of the\nmisalignment contribution to this overall axion abundance can also serve to weaken\nthe overclosure bound on the axion-decay constant, and thus could potentially enlarge the allowed \nregion of parameter space for the QCD axion.   \n\nOur results concerning the effects of a mass-generating phase transition can also have significant implications \nfor other new-physics scenarios.  For example, our results may provide a way of mitigating \nthe cosmological moduli problem which arises in supergravity~\\cite{ModuliProblemSUGRA} and in \nstring theory~\\cite{BanksModuliProblem1,deCarlosModuliProblem,BanksModuliProblem2}.\nIndeed, such theories generically predict large numbers of neutral scalar \nfields --- so-called moduli --- with flat potentials, long lifetimes, and \nlarge (and even Planck-scale) VEVs.  On the one hand, a number of \nphenomenological considerations imply that some mechanism must exist through which \na potential is generated for these moduli, rendering them massive.  On the other \nhand, once these fields acquire masses, they can potentially overclose\nthe universe or precipitate an unacceptably late period of reheating.  A suppression\nof the collective energy density of such moduli due to time-dependent mixing effects\nof the sort we have discussed here\ncould potentially provide a way of addressing these issues.   \n\nOur results also have implications within the context of the Dynamical\nDark Matter (DDM) framework~\\cite{DDM1,DDM2}, an alternative framework for dark-matter\nphysics in which a potentially vast ensemble of unstable particles contribute to the \npresent-day dark-matter abundance and in which phenomenological constraints on the dark sector\nare satisfied through a balancing between constituent decay widths and cosmological abundances \nacross the ensemble.  \nDark-matter ensembles with these properties emerge naturally in a variety of contexts~\\cite{DDM1,DDM2,Chialva:2012rq,Dienes:2016kgc,Dienes:2016vei,Jaketoappear},\nand detection strategies for such ensembles are discussed in \nRefs.~\\cite{Dienes:2012yz,Dienes:2012cf,Dienes:2013xff,Dienes:2014via,Dienes:2014bka,Boddy:2016fds,Boddy:2016hbp}.\nIndeed, it has been shown that the KK modes associated with a bulk axion-like\nfield which receives its abundance via misalignment production constitute a viable DDM\nensemble~\\cite{DDM2,DDMAxion}, \nThe results we have obtained here are thus directly applicable\nto DDM ensembles of this sort and lead to a suppression of the overall relic abundance\nof the ensemble for large $\\delta_G$ relative to the relic abundance which arises for the abrupt case considered in \nRefs.~\\cite{DDM2,DDMAxion}.  This suppression can potentially widen the phenomenologically allowed\nparameter space of such DDM models.  Moreover, we also note from the results \nin Fig.~\\ref{fig:rhofracpiesgrand}\nthat there exists a correlation \nbetween the overall magnitude of the late-time energy density of the KK tower\nand the degree to which its distribution across the different KK modes is\nparticularly ``DDM-like'' ({\\it i.e.}\\\/, shared non-trivially across many different KK modes), with larger\ntotal abundances tending to correlate with increased DDM-like behavior at late times.\nThus, we see that we can control the degree to which our ensemble of KK states\nis truly DDM-like at late times simply by adjusting phenomenological parameters \nsuch as ${\\overline{m}}$ and $\\delta_G$ \nassociated with our mass-generating phase transition. \n\nThe suppression of late-time scalar abundances due to time-dependent masses and mixings \nalso has potentially\nimportant implications for a broad range of additional scenarios involving weakly-coupled scalar \nparticles which receive a non-negligible contribution to their relic abundances from \nnon-thermal mass-generating phase transitions of the sort we have discussed here.\nThe reason is that in such scenarios,   \na non-negligible population of these particles can also be produced\nthermally from scattering processes involving SM particles in the radiation bath.  \nThis production mechanism is often referred to as ``freeze-in''~\\cite{FreezeIn}, \nand it also gives rise to a population of particles whose phase-space distribution \ndiffers significantly from that of the population generated \nby non-thermal mass-generating phase transitions.\nFor the QCD axion, the freeze-in contribution to the total relic abundance   \nis typically quite small~\\cite{TurnerThermalAxion,MassoThermalAxion,GrafThermalAxion} \ncompared to the contributions from other sources, such as misalignment production \nand cosmic-string and domain-wall decay.  By contrast, for other exotic scalars, the\nfreeze-in contribution can be significant.  Thus, in scenarios in which both misalignment\nproduction and freeze-in production are na\\\" ively expected to generate comparable \ncontributions to the overall abundance of a particular weakly-coupled scalar particle, a\nsuppression of the former contribution due to time-dependent mixing could \nboth modify overclosure bounds and alter the expected \nphase-space distribution of the relic scalar population.\n\nIn this paper we have focused on the contribution to the total abundance\nof a collection of KK scalar modes that arises due to a time-dependent\nmass-generating phase transition.  In so doing, however, we have disregarded the effects \nof the quantum fluctuations that arise for these fields during the inflationary epoch.\nIn general, fluctuations in the long-wavelength modes of light fields --- and, in particular, those \nmodes whose wavelengths exceed the Hubble length during that epoch --- behave like \nvacuum energy until after inflation ends.  Thus, the energy density in these \nfluctuations survives inflation and yields an additional contribution to the \nabundance of any field with mass \\il{\\lambda_k \\lesssim H_I}, where $H_I$ is the value \nof the Hubble parameter during inflation.  This additional contribution depends \nsensitively on the parameters of the inflationary model (for a review, see, {\\it e.g.}\\\/, \nRef.~\\cite{InflationFluctuations}) and in particular on $H_I$ itself.  In \nRef.~\\cite{DDMAxion}, it was shown that for a KK tower of axion-like fields, the \ncontribution to the relic abundance from vacuum misalignment dominates over this  \nadditional contribution from fluctuations during inflation for sufficiently small \n$H_I$.  However, we note that in general, this contribution exists and for \nhigher-scale inflationary models may have a significant impact on the overall\nrelic abundance of the KK tower.  \n\nIn the course of our analysis in this paper, we also derived results for truncated\nKK towers consisting of only $N$ modes, where \\il{N<\\infty}.\nIn so doing,\nwe presented the results of this finite-$N$ case merely as a stepping\nstone on the way towards understanding the properties of the full KK tower that emerges in the \n\\il{N\\rightarrow\\infty} limit.  However, the results of our finite-$N$\nanalysis are also interesting in their own right.  Indeed, the mass-squared \nmatrix for a tower of $N$ modes is similar to the mass matrix which emerges \nin certain ``moose''~\\cite{Moose} or ``quiver''~\\cite{Quiver} gauge theories --- \ntheories which also contain only a finite number of modes yet which yield \na deconstructed extra spacetime dimension in the \\il{N\\rightarrow\\infty} \nlimit~\\cite{Deconstruction}.  Thus, the results we have obtained here for such finite-$N$ \ntheories should provide guidance as to how time-dependent mass-generating phase transitions\nand time-dependent mixings will affect the late-time \nenergy densities for the collections of scalars which emerge in such moose or quiver\ntheories.\n\nIt is also worth emphasizing that the results we have derived in\nthis paper reflect the particular KK-derived mixing \nstructure of our model.  As a result, we did not observe certain phenomena which may in principle \narise for more general systems of scalar fields in the presence of time-dependent \nmass-generating phase transitions and mixings.\nFor example, it has been shown~\\cite{TwoTimescales} that in similar scalar systems with\nmore general mixing patterns, parametric resonances can arise which \nare extremely sensitive to the widths and mixings associated with mass-generating \nphase transitions and which \n``pump'' energy density into a particular field or fields during such phase transitions.  \nSuch systems may also exhibit so-called ``re-overdamping'' phenomena~\\cite{TwoTimescales} in which the\ntotal energy density of our system exhibits a time dependence which transcends those\nnormally associated with vacuum-dominated or matter-dominated cosmologies. \nHowever, in order for such novel effects to arise, the \ndynamically generated contribution $\\overline{\\mathcal{M}}^2_{\\rm gen}$ to the \nmass-squared matrix at late times must satisfy the criterion   \n\\begin{equation}\n  \\det\\overline{\\mathcal{M}}_{\\rm gen}^2 ~<~ 0~,\n\\end{equation}\nand this criterion is not satisfied within the parameter space of our KK model.  \nHowever, it is easy to contemplate early-universe\nscenarios involving multiple scalars in which \nthis condition is satisfied.\nIn such scenarios, parametric resonances and re-overdamping phenomena \ncan then have a significant further impact on both the \ntotal abundance of the scalar fields involved and the distribution of that\ntotal abundance among these scalars. \n\nThere are a number of potentially interesting generalizations\nand extensions of this work.  For example, in this paper we have focused on \nthe case in which the higher-dimensional scalar propagates in a single, flat \nextra dimension.  However, such a flat extra dimension can be viewed as a\nlimit of the more general case of a warped geometry in which the \nextra dimension represents a slice of anti-de Sitter space~\\cite{RS1,RS2}.  Such \nwarped compactifications have a variety of phenomenological applications, such as\nproviding ways of addressing the hierarchy between \nthe weak and Planck scales~\\cite{RS1} as well as the hierarchies amongst the masses of \nthe SM fermions~\\cite{TonyRSYukawas1,HuberShafiRSYukawas}.\nIt would therefore be interesting to examine how our findings generalize to the \ncase of a warped extra dimension.  It would also be interesting to \nconsider the further generalization of our results to the case of multiple extra \ndimensions, both flat and warped.  \nOne could also imagine adopting more general initial conditions for our KK modes\nbeyond those in Eq.~(\\ref{eq:initialconditions}), such as might arise if\nthere is an earlier phase of non-trivial dynamics (either in the bulk\nor on the brane) prior to our mass-generating phase transition.\n\nAnother possible avenue for generalization concerns the nature of the cosmological\nepoch during which we are presuming our dynamics to take place.\nFor example, throughout this paper we have focused primarily on the case in which the \nuniverse is effectively matter-dominated ({\\it i.e.}\\\/, with \\il{\\kappa = 2}) throughout the\nperiod of mass generation.   Indeed, as discussed in Sect.~\\ref{sec:TheModel},\nphenomenological constraints on theories with large extra spacetime dimensions \nare more easily satisfied in LTR scenarios, in which the universe remains \nmatter-dominated until very late times.  However, these constraints are \nconsiderably weaker in cases in which the compactification scale $M_c$ is \n$\\mathcal{O}(\\mathrm{TeV})$ or above.  Thus, it would also be interesting to \nconsider the case in which the universe is effectively radiation-dominated \n({\\it i.e.}\\\/, with \\il{\\kappa = 3\/2}) during the mass-generation epoch.\nCosmologies with other values of $\\kappa$ can be considered as well.\n\nFinally, other interesting extensions of this work involve considering a broader\nvariety of cosmological contexts in which mass generation for\nour KK modes might take place.\nThroughout this paper, we have implicitly operated under the assumption that\nthe total energy \ndensity $\\rho$ of the KK tower is negligible compared to the critical density \n$\\rho_{\\mathrm{crit}}$ during the mass-generation epoch.  Under such an assumption, the \nback-reaction of the KK tower on the evolution of the spacetime metric is therefore \nnegligible during this epoch.  Thus, to a very good approximation, we may \nview the dynamics of the KK modes studied in this paper\nas taking place within the context of a particular \n``background'' cosmology which is effectively decoupled from this dynamics and which\nis therefore specified by a fixed functional form for the Hubble parameter $H(t)$ as\na function of $t$.  Indeed, such assumptions are valid for a variety of light scalars which\nreceive their abundances from misalignment production --- including, by necessity,\nany such fields which serve as dark-matter candidates.  \nHowever, one could alternatively consider the opposite regime in which \n\\il{\\rho\\sim\\rho_{\\mathrm{crit}}} during the mass-generation epoch.  Under such conditions,\nthe back-reaction of the KK tower on $H(t)$ can be significant and must be \nincorporated into the equations of motion for the KK modes.  This situation \ncan arise, for example, in scenarios in which these scalar particles play a role\nin inflationary dynamics or in which they are responsible for additional, later \nperiods of reheating after inflation in cosmologies with non-thermal\nhistories~\\cite{WatsonNonThermalHistories}.  It would therefore be interesting\nto study the evolution of the energy densities of the KK modes of our theory\nin such scenarios.\n\n\\bigskip\n\n\n\n\\section*{Acknowledgments}\n\n\n\nWe would like to thank S.~Watson for useful discussions. \nThe research activities of KRD and JK were supported in part by the Department\nof Energy under Grant DE-FG02-13ER41976 (DE-SC0009913), while\nthe research activities of KRD were also supported in part by\nthe National Science Foundation through its employee IR\/D program.\nThe opinions and conclusions\nexpressed herein are those of the authors, and do not represent any funding agencies.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section*{Acknowledgements}\nThis work was carried out in part on the High-Performance Computing resources at Khalifa University.\nBesides, this work is supported in part by the Center for Cyber Security (CCS) at New York University Abu Dhabi (NYUAD).\n\n\n\n\\section{Introduction}\n\\label{sec:Introduction}\n\nLogic locking (LL) is a design-for-trust technique that aims to protect the design intellectual property throughout the supply chain. \nThe design's functionality is locked with a secret key (driven from an on-chip tamper-proof memory) and a set of added key-gates. \nThe locked design functions correctly only when the correct secret key is in place. \nEarlier works in LL focused on the placement of key-gates and high output corruptibility~\\cite{epic_journal,JV-Tcomp-2013,dupuis2014novel}.\nHowever, these schemes were deemed vulnerable by the Boolean satisfiability (SAT)-based attack~\\cite{Subramanyan_host_2015}. \n\n\nProvably secure logic locking (PSLL) ensures an exponential number of SAT calls for a successful key recovery~\\cite{xie2016mitigating,yasin_glsvlsi_2017,yasin_CCS_2017,mengli_provably_secure_camo,shakya2020cas}. \nIn principle, a \\textit{perturb} unit injects errors into the design for some protected input patterns. \nThese errors are canceled out by a \\textit{restore} unit when the correct key is applied. \nRecently, structural\/functional \\textit{oracle-less} attacks have emerged to circumvent PSLL~\\cite{yasin_2017_sps,alrahis2019functional,yang2019stripped,sirone2020functional}.\\footnote{\\textit{Oracle-guided} attacks are successful in approximately circumventing specific instances of PSLL~\\cite{shen2017double,shamsi2017appsat,xiaolin_bypass_attack_ches17}.\nHowever, these attacks require a working chip for functional verification. \nWe focus on the \\textit{oracle-less} attacks as they pose a more significant and realistic threat, as highlighted by the LL community.} \\textbf{Each attack aims to break a specific locking scheme, under restricted parameter settings and circuit formats}, resulting in a large set of attack strategies for different PSLL implementations. \nDeveloping a holistic attack on PSLL is an open research problem.\nTable~\\ref{tab:comparison} outlines the drawbacks of oracle-less attacks on PSLL.\n\n\n\\begin{table}[tb]\n\\centering\n\\scriptsize\n\\vspace{-2mm}\n\\caption{\\textsc{Capabilities offered by oracle-less attacks}}\n\\label{tab:comparison}\n\\resizebox{\\textwidth}{!}{%\n\\setlength\\tabcolsep{1.9pt}\n\\renewcommand\\arraystretch{0.9}\n\\begin{tabular}{cccc}\n\\hline\n\\textbf{Attacks} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Different\\\\ Circuit Formats\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Different\\\\ Locking Schemes\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Different\\\\ Parameter Settings\\end{tabular}} \\\\\\hline\nSPS~\\cite{yasin_2017_sps} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} & \\color{cadmiumgreen}{\\ding{51}}\n\\\\ \\hline\nRE-based~\\cite{alrahis2019functional} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} & \\color{cadmiumgreen}{\\ding{51}} \n\\\\ \\hline\nFALL~\\cite{sirone2020functional} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} \n\\\\ \\hline\nSFLL-HD-Unlocked~\\cite{yang2019stripped} & \n\\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} \n\\\\ \\hline\n\\textbf{GNNUnlock} & \n\\color{cadmiumgreen}{\\ding{51}} & \n\\color{cadmiumgreen}{\\ding{51}} & \n\\color{cadmiumgreen}{\\ding{51}} \n\\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\\subsection{Motivation and Research Challenges}\n\n\\textbf{Restricted Parameter Settings:} Stripped functionality logic locking (\\textit{SFLL-HD$^h$})~\\cite{yasin_CCS_2017} showed resilience against all known attacks on LL.\nSFLL-HD$^h$ protects all input patterns from a Hamming distance $h$ from the secret key $K$. \nSirone \\textit{et al.}~\\cite{sirone2020functional} developed functional analysis attacks (FALL) on SFLL-HD$^h$ which retrieved the secret key \\textit{without} requiring an oracle. \nThe attack algorithms are based on a set of derived Lemmas defining functional properties of the protection logic used in SFLL-HD$^h$. \nThe derived Lemmas hold for specific ranges of $h$ values. \nHence, by definition, the attack algorithms cannot break all cases of SFLL-HD$^h$. \nFor example, their \\textit{AnalyzeUnateness} algorithm is only applicable when $h=0$, and their \\textit{Hamming2D} algorithm is only applicable when $h \\leq K\/4$. \nTheir third algorithm \\textit{SlidingWindow} should apply to larger $h$ values in principle, but it does not fare well since it requires computationally hard SAT calls. \nYang \\textit{et al.}~\\cite{yang2019stripped} proposed the \\textit{SFLL-HD-Unlocked} attack that performs connectivity analysis on the locked circuit followed by a Gaussian-elimination-based analysis to recover the secret key \\textit{without} requiring an oracle. \nBecause of the use of Gaussian-elimination, the attack does not work when $h\\leq4$ due to the composition of singular matrices.\n\nTo demonstrate the shortcomings of existing attacks, we lock selected ISCAS-85 and ITC-99 benchmarks using SFLL-HD$^h$ with $K\/h=2$, multiple times, generating \\texttt{192} locked designs. \n\\textit{Such a locking ratio is critical as it achieves the highest resilience to removal attacks, as indicated in~\\cite{yasin_CCS_2017}}. \nWhen the FALL attacks~\\cite{sirone2020functional} were launched on these locked benchmarks; they reported \\texttt{0} keys, failing to break any of the designs. \nSimilarly, SFLL-HD-Unlocked attack~\\cite{yang2019stripped} also was unable to recover the key for any design. \n\\textit{This analysis highlights the need for a holistic approach that can break SFLL-HD$^h$ and other PSLL schemes under all the possible scenarios}.\n\n\n\\textbf{Restricted Circuit Formats:} The prior attacks on PSLL accept restricted and non-standard circuit formats. \nFor example, the work in~\\cite{alrahis2019functional} showcased that functional reverse engineering (RE) aids in the detection and subsequent removal of the protection logic added by SFLL-HD$^h$. \nHowever, the approach is not generic and will only work on designs synthesized using only 2-input gates. \nThis attack achieves low accuracy results when launched on benchmarks synthesized with a standard cell library. \nFor example, the \\textit{restore} unit detection percentage in the ISCAS-85 benchmark \\texttt{c7552} locked with $K=32$ and $h=0$ drops from $100\\%$ to $0\\%$. Moreover, both attacks~\\cite{yang2019stripped,sirone2020functional} accept circuits in bench format (the latter~\\cite{sirone2020functional} requires topologically sorted AND-OR-INVERT gates), which is a non-industry format, as opposed to \\textit{Verilog\/VHDL}. \nThus, these techniques cannot be employed in the real-world design flow.\n\n\\textbf{Locking Scheme-specific Attacks:} All the aforementioned oracle-less attacks are locking scheme-specific. \nThey target a distinct protection implementation (SFLL-HD$^h$ in this case) and are not scalable to other variants (such as \\textit{Anti-SAT}~\\cite{xie2016mitigating}). \nThe signal probability skew (SPS) attack~\\cite{yasin_2017_sps} is another oracle-less attack targeting Anti-SAT.\nThe SPS attack looks for two oppositely skewed nets feeding an AND gate, a property specific to the Anti-SAT protection and not to other PSLL schemes, and hence it is another scheme-specific attack.\n\n\n\\textbf{Associated Research Challenges:} Automating the detection of the protection logic added by PSLL while accounting for all possible variations among the implementations is an open research problem posing the following important challenges.\n\n\\begin{enumerate}[leftmargin=*]\n\n\\item \\textit{Recognizing all implementation variants:} The protection logic structure depends on (i)~the key-value, (ii)~the Hamming distance value (for the case of SFLL-HD), (iii)~the choice of the logic blocks (for the case of Anti-SAT), and (iv)~the key-size. Hence, different settings will result in distinct topologies. \nThus, utilizing an exact matching algorithm would be a naive approach. \n\n\\item \\textit{Handling different synthesis settings and circuit formats:} The structure of the protection logic and its integration with the original design depends on the synthesis procedures and the target technology library. \nThe attack model should be able to understand and process different circuit formats without having to modify the attack tactic.\n\n\\end{enumerate}\n\n\\subsection{Our Novel Concept and Contributions}\n\nTo address the above challenges, we propose the \\textbf{\\textit{GNNUnlock} framework} that identifies all variants of the targeted protection logic. \nIt is the first-ever concept to leverage graph neural networks (GNNs) along with node connectivity analysis to identify all the gates in a given locked netlist that form the protection logic, as depicted in Fig.~\\ref{fig:GNN_attack}. \nThis work is inspired by the ability of advanced machine learning (ML) models to adapt to new unseen data and the fact that a circuit is a graph with an inherent structure to it. \nUnlike other ML models, GNNs leverage the graph structure and node (gate) features to learn a representation for each node capturing its neighborhood characteristics.\\footnote{We use the terms interchangeably gate or node, and circuit or graph.}\nGNNs are a fitting choice for identifying the protection logic, which can be conceived as a sub-graph with specific properties.\nThe GNN learns the trend of the protection logic and not a syntactic implementation of it, and hence GNNUnlock deals with variations naturally. \nOur GNNUnlock framework employs the following techniques:\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figures\/fig1.pdf}\n\\vspace{-2mm}\n\\caption{High-level view of our work: Breaking PSLL using GNN.\\vspace{-5pt}}\n\\label{fig:GNN_attack}\n\\end{figure}\n\n\\begin{enumerate}[leftmargin=*]\n\n\\item A framework for the \\textbf{netlist-to-graph transformation (Section~\\ref{Sec:transfromation})} is developed which implements automatic \\& efficient feature extraction to capture each gate's functionality and connectivity in the gate-level netlist (in any format). \n\n\\item \\textbf{GNN learning on locked circuits is achieved (Section~\\ref{Sec:attack_gnn})}, allowing the GNN to identify the structural features of all the nodes in the targeted protection logic.\n\n\\item \\textbf{Post-processing rectification procedure (Section~\\ref{Sec:post_processing})} guided by the connectivity of the circuit and by the predictions of the GNN (on the under-attack circuit) is developed to remove the identified protection logic effectively.\n\n\\end{enumerate}\n\nThe effectiveness of GNNUnlock is showcased by breaking \\texttt{564} benchmarks locked using three different PSLL techniques -- SFLL-HD~\\cite{yasin_CCS_2017}, \\textit{TTLock}~\\cite{yasin_glsvlsi_2017}, and Anti-SAT~\\cite{xie2016mitigating}.\nThe effect of different $K$, different $h$ values (for the case of SFLL-HD, and other technology libraries are considered when evaluating GNNUnlock comprehensively. \nAll the files locked at the register-transfer level (RTL) are synthesized following the commercial ASIC-design flow using Synopsys tools.\n\\textbf{We also make our attack framework available online at~\\cite{webinterface}.}\n\\section{Background and Related works}\n\\label{sec:backgroud_RW}\n\n\\subsection{Provably Secure Logic Locking (PSLL)} \n\\subsubsection{Anti-SAT~\\cite{xie2016mitigating}} \n\nIn the Anti-SAT block, the outputs of two complementary logic functions ($gl1$ and $\\overline{gl2}$), locked with two sets of keys are fed into an AND gate, as shown in Fig.~\\ref{fig:SFLL}a. \nWhen the correct key is in place the output of the AND gate $Y$ is \\texttt{0}. For a wrong key-input, the output signal $Y$ could be \\texttt{0} or \\texttt{1} depending on the input $X$. \n$Y$ is XORed with an internal net in the original netlist, locking it. \nHaving the $Y$ signal highly skewed to \\texttt{0} ensures resilience against the SAT-based attack. \nNevertheless, Anti-SAT is vulnerable to structural analysis-based removal attacks~\\cite{yasin_2017_sps}.\n\n\\subsubsection{TTLock~\\cite{yasin_glsvlsi_2017} and SFLL-HD~\\cite{yasin_CCS_2017}} modify the original design, as shown in Fig.~\\ref{fig:SFLL}b, to achieve resilience against removal attacks.\nTTLock protects an input pattern that is the same as the locking key. Tracing the key-inputs (KIs) helps identify the \\textit{restore} logic of TTLock, which is a basic comparator. \nHowever, the \\textit{perturb} unit structure depends on the selected secret key, and not controlled by the external KIs, as demonstrated in Fig.~\\ref{fig:SFLL}c. \nSFLL-HD$^h$ protects all input patterns that are an $h$ Hamming distance away from the secret key. \nTTLock is equivalent to SFLL-HD$^0$. For SFLL-HD$^{h>0}$, the \\textit{restore} and \\textit{perturb} units are Hamming distance checkers ($G$), as illustrated in Fig.~\\ref{fig:SFLL}d. \nThe structure of the \\textit{perturb} unit depends on the hard-coded key, making it difficult to be traced.\n\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figures\/fig2.pdf}\n\\vspace{-2mm}\n\\caption{Logic locking using Anti-SAT~\\cite{xie2016mitigating}, TTLock~\\cite{yasin_glsvlsi_2017}, and SFLL-HD~\\cite{yasin_CCS_2017}. $G$ block in (d) performs Hamming distance checking using adders and comparators. \\vspace{-5pt}}\n\\label{fig:SFLL}\n\\end{figure*}\n\n\\vspace{-0.5ex}\n\\subsection{Graph Neural Network (GNN)} \n\nIn GNNs, nodes aggregate information from their neighbors using neural networks. \nEach node gathers information from its neighbors in a previous layer, where each one of these neighbors collects information from the previous layer. \nThe aggregation functions in \\textit{GraphSAGE}~\\cite{hamilton2017inductive} approach learn to aggregate feature information to obtain embeddings for the nodes, which are then used to perform the desired task such as node classification. \nThe same aggregation parameters are shared for all nodes. \nHence, once the model is trained, those parameters can generate embeddings on entirely unseen graphs. \nWe follow such an inductive approach since we train on a set of locked benchmarks and \\textit{test the model on unseen benchmarks locked with unknown key-values}. \nWe adopt the GraphSAGE architecture in which each node's previous state is concatenated with its neighbors' current state. The mean aggregation function is used to update the state of each node. \n\nDue to neighborhood aggregation, the deeper the GNN model is, the more multi-hop neighbors get incorporated for the root node's computation. \nHence, training time could grow exponentially with GNN depth~\\cite{zeng2019graphsaint}. \nSeveral methods, including GraphSAGE, perform layer sampling to ensure a limited set of neighbors is considered for a node in the next layer. \nSuch sampling speeds up training but suffers from scalability issues.\nWe leverage the graph sampling method \\textit{GraphSAINT}~\\cite{zeng2019graphsaint} to construct mini-batches for training. \nRather than building a GNN on full training graph and then perform layer sampling, GraphSAINT samples the training graph and then builds a GNN on the sampled graph for each mini-batch,\\footnote{In each mini-batch, forward and backward propagation is performed iteratively on the sampled GNN to update weights via stochastic gradient descent.} ensuring scalability concerning graph size and GNN depth. \nTheir random walk-based sampler is used in our experiments.\n\\section{Attacker Model}\n\\label{sec:threat_model}\n\nWe assume that the attacker will be present in the untrusted foundry with access to the GDSII mask information and reverse engineering tools that facilitate the gate-level netlist extraction.\nBesides, the attacker is aware of the locking algorithm and can distinguish between the regular primary inputs (PIs) and KIs. \nThe attacker also knows the usage of the technology library and the usage of different synthesis settings. \nWe do not assume the attacker to have access to an unlocked chip (i.e., an \\textit{oracle-less} setting). \nFor SFLL-HD$^h$~\\cite{yasin_CCS_2017}, the attacker knows the Hamming distance value, while the type of logic function ($gl$) is known for the case of Anti-SAT~\\cite{xie2016mitigating}. \nNote, all these assumptions are in line with the \\textit{Kerckhoffs's principle}, which states that everything about the system should be known to an attacker \\textit{except} for the value of the secret key.\n\\section{Proposed GNNUnlock Framework}\n\\label{sec:Proposed_attack}\n\nFig.~\\ref{fig:GNNUnlock_flow} shows an overview of our methodology, with key steps discussed in the following subsections.\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figures\/fig3.pdf}\n\\vspace{-2mm}\n\\caption{Proposed GNNUnlock methodology. \nThe \\textit{Anti-SAT}, \\textit{restore}, \\textit{perturb}, and \\textit{design} nodes are indicated by AN, RN, PN, and DN, respectively.\n\\vspace{-1pt}\n}\n\\label{fig:GNNUnlock_flow}\n\\end{figure*}\n\n\\subsection{Dataset Generation} \n\nUsing the underlying locking scheme, we generate a set of locked benchmarks for training and validation. \nEach benchmark is locked several times with randomly generated key-values and different key-sizes $K$, which results in an extensive training set. \nThe training process is not aware of the correct key-values. \nHowever, different key-values cause the generation of varying \\textit{perturb} and \\textit{Anti-SAT} structures, and hence variants are accounted for during training. GNNUnlock transforms locked benchmarks into graphs (Section~\\ref{Sec:transfromation}) and labels the nodes. \nFor the case of SFLL-HD$^{h>0}$\/TTLock, a node belongs to the \\textit{perturb}, the \\textit{restore}, or the \\textit{design} logic. For the case of Anti-SAT, a node belongs to the \\textit{Anti-SAT} block or the \\textit{design} logic.\n\n\nGNNUnlock attacks each design independently by excluding its corresponding graphs from training\/validation. For example, when attacking \\texttt{b17\\_C} from ITC-99, only the graphs of \\texttt{b14\\_C}, \\texttt{b15\\_C}, \\texttt{b20\\_C}, and \\texttt{b21\\_C} are used in training, while the graphs of \\texttt{b22\\_C} are used for validation to evaluate the model on unseen data, thereby avoiding biasing.\n\n\n\\subsection{Netlist-to-Graph Transformation}\n\\label{Sec:transfromation}\n\nWe model connectivity between the gates in a design using an undirected graph $G(I, J)$. \nThe set $I$ of nodes represents all the gates, while the set $J$ of edges represents the wires. \nAn example of a logic locked netlist and the corresponding graph is shown in Fig.~\\ref{fig:GNNUnlock_flow}b. \nThe set $I$ does not contain PIs, KIs, or primary outputs (POs) of the design. \nEach node in the graph is associated with a feature vector $\\hat{f}$ that contains information describing the node's important characteristics, such as its in-degree ($IN$) and out-degree ($OUT$). \nIt also contains information on whether the node is connected to a PI, a PO, or a KI. \nAdditionally, $\\hat{f}$ captures the type and number of gates appearing in the neighborhood of the node (all nodes within two-hops away). \nFor example, $\\hat{f_{i}}$ shows that gate $i$ is connected to a PO with $IN=3$ and $OUT=1$ (Fig.~\\ref{fig:GNNUnlock_flow}b). \nIt also indicates that three XOR gates and one XNOR gate exist in $i$'s neighborhood. \nThe length of the feature vector $|\\hat{f}|$ depends on the number of logic gates in the target library. \nEach locked design is represented by a graph instance with a corresponding adjacency matrix.\nTo feed multiple graphs of different sizes to the GNN, a block-diagonal matrix is created for each dataset. \nEach block represents the adjacency of one locked design.\n\n\\vspace{-1ex}\n\\subsection{GNN Topology}\n\\label{Sec:attack_gnn}\n\nIn all experiments reported in Section~\\ref{sec:results}, graph sampling using the GraphSAINT method is performed.\nA two-layer GNN is constructed using GraphSAGE architecture, with mean and concatenation aggregation, and the \\textit{ReLU} activation function. Training stops after a fixed number of epochs based on convergence. \nThe model with the best performance on the validation set is used to evaluate the test set accuracy. \nDetails regarding the GNN model and sampling are shown in Table~\\ref{tab:GNN_info}.\n\n\\subsection{Post-processing}\n\\label{Sec:post_processing}\n\nAlthough our trained GNN achieves high node classification accuracy on its own (with an average of $99.99\\%$, $99.95\\%$, and $99.92\\%$, for the case of Anti-SAT, TTLock, and SFLL-HD$^2$, respectively), we propose a post-processing algorithm to rectify any potential misclassifications to enhance the accuracy further.\nThe algorithm considers predictions made by the GNN, connectivity of the circuit, and known properties of the protection logic, which is explained further in the next subsections.\n\n\\subsubsection{Anti-SAT} Every node in the \\textit{Anti-SAT} block must have at least one KI in its fan-in cone. \nIf a node without KIs in its fan-in cone is predicted as an \\textit{Anti-SAT} node, then the prediction will be dropped. \nFor each \\textit{predicted design} node, we check its fan-in cone. If there are only other \\textit{predicted Anti-SAT} nodes in the fan-in cone, then the node initially classified as a \\textit{design} node is considered part of the Anti-SAT block. The flow of our post-processing algorithm is presented in Fig.~\\ref{fig:GNNUnlock_flow}c. \n\n\\subsubsection{TTLock and SFLL-HD$^{h>0}$} Our post-processing algorithm for the case of SFLL-HD$^{h>0}$ and TTLock is presented in Fig.~\\ref{fig:GNNUnlock_flow}d. \nThe \\textit{predicted restore} nodes are visited to identify possible protected inputs (set $X$). The following properties then guide the post-processing procedure. (i)~All \\textit{restore} nodes have KIs in their fan-in cone. \n(ii)~All \\textit{perturb} nodes have connections with the \\textit{restore} nodes in the netlist and are controlled solely by protected inputs $X$. \nIf a \\textit{predicted restore} node has KIs in its fan-in cone, then the prediction is confirmed. \nOtherwise, the algorithm checks if the node is a \\textit{perturb} node. \nA \\textit{predicted perturb} node is verified if it has a connection to other \\textit{predicted restore} nodes in the netlist and if the node has $X$ (and no other PIs) in its fan-in cone. \nIn order not to misclassify \\textit{perturb} nodes as \\textit{design} nodes, the algorithm checks the fan-in cone of the predicted \\textit{design} nodes. \nIf a predicted \\textit{design} node has $X$ and other predicted \\textit{perturb} nodes in its fan-in, it is a \\textit{perturb} node.\n\n\\begin{table}[tb]\n\\centering\n\\vspace{-2mm}\n\\caption{\\textsc{GNN configuration and sampling details. The \\#classes for SFLL-HD$^{h>0}$ and TTLock is \\texttt{3}, while for Anti-SAT it is \\texttt{2}}}\n\\label{tab:GNN_info}\n\\resizebox{\\textwidth}{!}{%\n\\renewcommand\\arraystretch{0.9}\n\\begin{tabular}{cccc}\n\\hline\n\\multicolumn{2}{c}{\\textbf{Architecture}} & \\multicolumn{2}{c}{\\textbf{Training and Sampling}} \\\\ \\hline\n\\textbf{Input Layer} & {[}$|\\hat{f}|$,$512${]} & \\textbf{Optimizer} & Adam \\\\ \\hline\n\\textbf{Hidden Layer 1} & {[}$1024$,$512${]} & \\textbf{Learning Rate} & $0.01$ \\\\ \\hline\n\\textbf{Hidden Layer 2} & {[}$1024$,$512${]} & \\textbf{Dropout} & $0.1$ \\\\ \\hline\n\\textbf{Output Layer} & {[}$512$,\\#classes{]} & \\textbf{Sampler} & Random Walk \\\\ \\hline\n\\textbf{Aggregation} & Mean with concatenation & \\textbf{Walk Length} & $2$ \\\\ \\hline\n\\textbf{Activation} & ReLU & \\textbf{Root Nodes} & $3000$ \\\\ \\hline\n\\textbf{Classification} & Softmax & \\textbf{Max \\# Epochs} & $2000$ \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\\section{Experimental Investigations}\n\\label{sec:results}\n\n\\subsection{Evaluation Setup, Tool Flow, and Evaluation Metrics} \n\nGNNUnlock is evaluated on selected ISCAS-85 and ITC-99 benchmarks. \nNetlist-to-graph transformation is implemented in Perl\/Python3. \nTensorflow with Python3 implementation of GraphSAINT is used for GNN training~\\cite{zeng2019graphsaint}. \nTraining is performed on a single node with 24 cores (2x Intel Xeon CPUs E5-2695 v2@2.4 GHz), 256GB RAM, and one NVIDIA Tesla K20m GPU (2,496 CUDA cores and 5GB of GDDR5 memory). Fig.~\\ref{fig:GNNUnlock_exp} summarizes the experimental setup.\n\n\\subsubsection{Dataset Generation for Anti-SAT}\n\nThe benchmarks (in bench format) were locked using the Anti-SAT locking binary provided by the authors.\nEach ISCAS-85 benchmark is locked twice with $K:\\{8, 16, 32, 64\\}$, except for \\texttt{c3540}, where $K=64$ is not considered due to the limited number of PIs in the design. \nIn total, \\texttt{30} Anti-SAT locked ISCAS-85 benchmarks are obtained. \n$K=8$ is used to evaluate the effectiveness of GNNUnlock in isolating the Anti-SAT block when its size is very small compared to that of the original design.\nEach ITC-99 benchmark is locked twice with $K:\\{32, 64, 128\\}$ resulting in a total of \\texttt{36} Anti-SAT locked ITC-99 benchmarks. Two labeled datasets for Anti-SAT block identification have been created, one for locked ISCAS-85 benchmarks and one for locked ITC-99 benchmarks. A feature vector $|\\hat{f}|=13$ is associated with each node. \n\\textit{The Anti-SAT locking binary only accepts circuits in bench format, which includes a restricted set of logic gates (\\texttt{8} gates)}. \nHence, only \\texttt{8} out of the \\texttt{13} features are required to represent the neighborhood of each node.\nRest of the features represent $IN$, $OUT$, and the connectivity to PIs, KIs, or POs.\n\\subsubsection{Dataset Generation for TTLock and SFLL-HD$^{h>0}$}\nSFLL-HD$^h$ is implemented in Perl as described in~\\cite{yasin_CCS_2017} to lock the benchmarks at RTL -- TTLock is the case when $h=0$. Each ISCAS-85 benchmark is locked thrice with $K:\\{8, 16, 32, 64\\}$ for $h:\\{0,2\\}$, except for the \\texttt{c3540} benchmark where $K=64$ is not considered due to the limited number of PIs in the design. \nIn total, \\texttt{45} locked ISCAS-85 benchmarks for each $h$ are obtained. \nEach ITC-99 benchmark is locked thrice with $K:\\{32, 64, 128\\}$ for $h:\\{0,2,4\\}$, resulting in a total of \\texttt{54} locked ITC-99 benchmarks for each setting of $h$. \nLocked RTL files are synthesized using the standard ASIC design flow for the $65nm$ LPe technology. \nSynthesis is performed using Synopsys Design Compiler. \n$|\\hat{f}|=34$ is associated with each node. Since a full standard cell library is used for synthesis, a larger $|\\hat{f}|$ (than the case of Anti-SAT) is required to capture all the possible combinational logic gates.\n\nTo verify that GNNUnlock handles different circuit formats and technologies, the Nangate $45nm$ open cell library is also used for synthesis when $h=2$. Varying the circuits' format affects $|\\hat{f}|$ only. For this case, $|\\hat{f}|=18$ is used. \nWe also consider other corner cases for when $h:\\{16, 32, 64\\}$ with corresponding $K:\\{32,64,128\\}$, more details are in Section~\\ref{sec:corner_cases}. The characteristics of all the datasets are listed in Table~\\ref{tab:Datasets}.\n\n\\subsubsection{Evaluation Methods and Metrics}\n\nGNNUnlock attacks each design independently, by excluding its corresponding graphs from training and validation. \nFor example, when attacking the TTLock-ed \\texttt{b17\\_C} design from ITC, only the graphs of \\texttt{b14\\_C}, \\texttt{b15\\_C}, \\texttt{b20\\_C}, and \\texttt{b21\\_C} are included in the training set, resulting in \\texttt{161,942} training nodes (\\texttt{36} graphs), while the graphs of \\texttt{b22\\_C} are included only in the validation set (to evaluate the model on unseen data), resulting in \\texttt{74,753} validation nodes (\\texttt{9} graphs). And only the graphs of \\texttt{b17\\_C} are tested on, resulting in \\texttt{110,685} testing nodes (\\texttt{9} graphs).\n\nThe predictions of the GNN are compared with the true labels of the nodes to evaluate the performance of the attack. \nWe report the accuracy and the non-averaged precision, recall, and F1-score for each classifier. \nWe further evaluate GNNUnlock by removing the predicted protection logic to retrieve the unlocked design without accessing the true labels (Section~\\ref{Sec:post_processing}). \nThe recovered benchmark is then compared with the original benchmark via circuit-equivalence using Synopsys Formality.\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.97\\textwidth]{figures\/fig4.pdf}\n\\vspace{-2mm}\n\\caption{Evaluation setup showing different components and platforms. TE, TR, and VA stand for testing, training, and validation, respectively.\\vspace{-5pt}}\n\\label{fig:GNNUnlock_exp}\n\\end{figure}\n\n\n\\subsection{Breaking Anti-SAT with GNNUnlock}\n\\label{sec:breaking_antisat}\n\nFor the ISCAS-85 dataset, the GNN gave $100\\%$ node classification accuracy for \\texttt{25} of the tested graphs, and an average of $99.98\\%$ accuracy across all \\texttt{30} graphs. \nFor the remaining \\texttt{5} graphs, only \\texttt{6} nodes, were misclassified and then rectified during post-processing. \nFor example, two \\textit{design} nodes feeding the XOR gate connecting the Anti-SAT block with the rest of the circuitry got misclassified as part of the Anti-SAT block. \nHowever, having no KIs in their fan-in, the prediction is ignored. \nFor the ITC-99 dataset, the GNN gave $100\\%$ node classification accuracy for \\texttt{29} out of \\texttt{30} tested graphs. \nOnly one node was misclassified in one of the \\texttt{b21\\_C} graphs locked with $K=128$, which was rectified during post-processing. \nDetailed results are reported in Table~\\ref{tab:Anti_SAT_results}.\n\n\\begin{table}[tb]\n\\centering\n\\caption{\\textsc{Summary of Generated Datasets}}\n\\label{tab:Datasets}\n\\resizebox{\\textwidth}{!}{\n\\setlength\\tabcolsep{1.9pt} \n\\renewcommand\\arraystretch{0.9}\n\\begin{tabular}{ccccccc}\n\\hline\n\\textbf{Dataset} & \\textbf{Benchmarks} & \\textbf{Circuit Format} & \\textbf{\\#Classes} & \\textbf{$|\\hat{f}|$} & \\textbf{\\#Nodes} & \\textbf{\\#Circuits} \\\\ \\hline\n\\multirow{2}{*}{\\textbf{Anti-SAT}} & ISCAS-85 & Bench & $2$ & $13$ & $69,468$ & $30$ \\\\ \\cline{2-7} \n & ITC-99 & Bench & $2$ & $13$ & $450,359$ & $36$ \\\\ \\hline\n\\multirow{2}{*}{\\textbf{TTLock}} & ISCAS-85 & Verilog netlist $65nm$ & $3$ & $34$ & $29,745$ & $45$ \\\\ \\cline{2-7} \n & ITC-99 & Verilog netlist $65nm$ & $3$ & $34$ & $347,380$ & $54$ \\\\ \\hline\n\\multirow{3}{*}{\\textbf{SFLL-HD$^2$}} & ISCAS-85 & Verilog netlist $65nm$ & $3$ & $34$ & $30,178$ & $45$ \\\\ \\cline{2-7} \n & \\multirow{2}{*}{ITC-99} & Verilog netlist $65nm$ & $3$ & $34$ & $357,374$ & $54$ \\\\ \\cline{3-7} \n & & Verilog netlist $45nm$ & $3$ & $18$ & $391,411$ & $54$ \\\\ \\hline\n\\textbf{SFLL-HD$^4$} & ITC-99 & Verilog netlist $65nm$ & $3$ & $34$ & $356,420$ & $54$ \\\\ \\hline\n\\textbf{SFLL-HD$^{16}$} & ISCAS-85 & Verilog netlist $65nm$ & $3$ & $34$ & $33,354$ & $48$ \\\\ \\hline\n\\textbf{SFLL-HD$^{32}$} & \\multirow{2}{*}{ITC-99} & Verilog netlist $65nm$ & $3$ & $34$ & $465,032$ & $72$ \\\\ \\cline{1-1} \\cline{3-7} \n\\textbf{SFLL-HD$^{64}$} & & Verilog netlist $65nm$ & $3$ & $34$ & $483,777$ & $72$ \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[tb]\n\\centering\n\\vspace{-2mm}\n\\caption{\\textsc{Results of GNNUnlock on Anti-SAT. \nA \\textit{design} node and an \\textit{Anti-SAT} node are indicated by DN and AN, respectively. \nMN stands for misclassified nodes}}\n\\label{tab:Anti_SAT_results}\n\\resizebox{\\textwidth}{!}{%\n\\setlength\\tabcolsep{1.9pt} \n\\renewcommand\\arraystretch{0.8}\n\\begin{tabular}{ccccccccccc}\n\\hline\n\\multirow{2}{*}{\\textbf{Test}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}\\#Test\\\\ Graphs\\end{tabular}}}& \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}GNN\\\\ Acc. (\\%)\\end{tabular}}}& \\multicolumn{2}{c}{\\textbf{Prec. (\\%)}} & \\multicolumn{2}{c}{\\textbf{Rec. (\\%)}} & \\multicolumn{2}{c}{\\textbf{F1-Score (\\%)}} & \\multirow{2}{*}{\\textbf{\\#MN}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Removal\\\\ Success (\\%)\\end{tabular}}} \\\\ \\cline{4-9}\n & & & \\textbf{AN} & \\textbf{DN} & \\textbf{AN} & \\textbf{DN} & \\textbf{AN} & \\textbf{DN} & & \\\\ \\hline\n\\textbf{c2670}&$8$ &$99.98$ &$99.79$ & $100$ & $100$ &$99.98$ &$99.90$ &$99.99$ & \n\\begin{tabular}[c]{@{}c@{}}$2$ DN \\end{tabular}& $100$ \\\\ \\hline\n\n\\textbf{c3540}&$6$ &$99.98$ &$99.55$ &$99.99$ &$99.78$ &$99.98$ &$99.67$ &$99.99$ & \\begin{tabular}[c]{@{}c@{}}$2$ DN\\\\$1$ AN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{c5315}&$8$ &$99.99$ &$99.90$ & $100$ & $100$ &$99.99$ &$99.95$ &$99.99$ & $1$ DN & $100$ \\\\ \\hline\n\\textbf{c7552}&$8$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\textbf{b14\\_C}&$6$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\textbf{b15\\_C}&$6$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\textbf{b20\\_C}&$6$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\textbf{b21\\_C}&$6$ &$99.99$ & $100$ &$99.99$ &$99.92$ & $100$ &$99.96$ &$99.99$ & $1$ AN & $100$ \\\\ \\hline\n\n\\textbf{b17\\_C}&$6$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table*}[tb]\n\\centering\n\\vspace{-2mm}\n\\caption{\\textsc{Results of GNNUnlock on SFLL-HD$^2$. \nThe \\textit{design}, \\textit{perturb}, and \\textit{restore} nodes are indicated by DN, PN, and RN, respectively}}\n\\label{tab:TTLOCK_SFLL_HD_results}\n\\scriptsize\n\\resizebox{0.97\\textwidth}{!}{%\n\\renewcommand\\arraystretch{0.85}\n\\begin{tabular}{ccccccccccccccc}\n\\hline\n\\multirow{2}{*}{\\textbf{Test Set}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}\\#Test\\\\ Graphs\\end{tabular}}}& \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}GNN\\\\ Acc. (\\%)\\end{tabular}}} & \\multicolumn{3}{c}{\\textbf{Prec. (\\%)}} & \\multicolumn{3}{c}{\\textbf{Rec. (\\%)}} & \\multicolumn{3}{c}{\\textbf{F1-Score (\\%)}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}\\#Misclassified\\\\ Nodes\\end{tabular}}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Removal\\\\ Success (\\%)\\end{tabular}}} \\\\ \\cline{4-12}\n & & &\\textbf{RN} & \\textbf{PN} & \\textbf{DN} & \\textbf{RN} & \\textbf{PN} & \\textbf{DN} & \\textbf{RN} & \\textbf{PN} & \\textbf{DN} & & \\\\ \\hline\n\\textbf{c2670} &$12$&$99.53$ &$99.55$ & $96.36$ & $100$ & $100$ &$99.37$ &$99.46$ &$99.77$ & $97.84$ &$99.73$ & \\begin{tabular}[c]{@{}c@{}}$24$ DN as PN\\\\ $4$ PN as RN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{c3540}&$9$ &$99.79$ & $98.31$ & $98.97$ & $100$ & $100$ & $98.30$ &$99.88$ &$99.15$ & $98.63$ &$99.94$ & \\begin{tabular}[c]{@{}c@{}}$5$ PN as RN\\\\ $3$ DN as PN\\\\ $2$ DN as RN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{c5315 } &$12$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ &$-$& $100$ \\\\ \\hline\n\\textbf{c7552 } &$12$ &$99.99$ & $100$ & $100$ &$99.99$ & $100$ &$99.85$ & $100$ & $100$ &$99.93$ &$99.99$ & $1$ PN as DN & $100$ \\\\ \\hline\n\\textbf{b14\\_C } &$9$ &$99.97$ &$99.88$ & $100$ &$99.98$ & $100$ &$99.43$ & $100$ &$99.94$ &$99.72$ &$99.99$ & \\begin{tabular}[c]{@{}c@{}}$2$ PN as RN\\\\ $5$ PN as DN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{b15\\_C } &$9$ &$99.99$ &$99.94$ & $100$ &$99.99$ & $100$ &$99.68$ & $100$ &$99.97$ &$99.84$ &$99.99$ & \\begin{tabular}[c]{@{}c@{}}$3$ PN as DN\\\\ $1$ PN as RN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{b20\\_C } &$9$ &$99.98$ &$99.46$ & $100$ &$99.99$ & $100$ &$99.04$ & $100$ & \\multicolumn{1}{l}{$99.73\\%$} & \\multicolumn{1}{l}{$99.52\\%$} & \\multicolumn{1}{l}{$99.99\\%$} & \\begin{tabular}[c]{@{}c@{}}$9$ PN as RN\\\\ $2$ PN as DN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{b21\\_C } &$9$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ &$-$& $100$ \\\\ \\hline\n\\textbf{b22\\_C } &$9$ &$99.96$ &$99.94$ & $97.83$ &$99.99$ & $100$ &$99.67$ &$99.96$ &$99.97$ & $98.74$ &$99.98$ & \\begin{tabular}[c]{@{}c@{}}$1$ PN as RN\\\\ $3$ PN as DN\\\\ $27$ DN as PN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{b17\\_C } &$9$ &$99.94$ &$99.46$ & $95.60$ & $100$ & $100$ &$99.52$ &$99.95$ &$99.73$ & $97.52$ &$99.98$ & \\begin{tabular}[c]{@{}c@{}}$3$ DN as RN\\\\ $57$ DN as PN\\\\ $6$ PN as RN\\end{tabular} & $100$ \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table*}\n\nHaving $100\\%$ node classification accuracy for a given benchmark locked with different key-sizes (such as the case for \\texttt{c7552}, \\texttt{b14\\_C}, etc.) shows that \\textbf{the ratio of sizes between Anti-SAT block and design does not affect the performance of GNNUnlock and overall recovery of the original design}. \nObtaining $100\\%$ node classification accuracy for Anti-SAT is validated since the Anti-SAT block has a specific structure controlled by external KIs, making it easy to be learned.\n\n\n\\subsection{Breaking SFLL-HD$^{h>0}$ and TTLock with GNNUnlock}\n\\label{sec:breaking_ttlock}\n\n\nFor the SFLL-HD$^2$ ISCAS-85 dataset, the GNN gave $100\\%$ node classification accuracy for \\texttt{20} tested graphs and an average of $99.83\\%$ accuracy across all \\texttt{45} ISCAS-85 graphs. \nIn total, \\texttt{38} nodes were misclassified then rectified using post-processing. \nFor example, \\texttt{27} \\textit{design} nodes, coming from the different \\texttt{25} graphs were misclassified as \\textit{perturb} nodes. \n\\textit{The misclassified nodes belonged to NOR-tree-like structures in the original designs.}\nThis misclassification did not affect the removal of the protection logic because non-protected input patterns controlled these nodes and hence predictions were dropped.\nFor SFLL-HD$^2$ ITC-99 dataset, the GNN gave $100\\%$ node classification accuracy for \\texttt{29} of the tested graphs and an average of $99.97\\%$ accuracy across all \\texttt{54} graphs (Table~\\ref{tab:TTLOCK_SFLL_HD_results}). \n\n\nFor the TTLock ISCAS-85 dataset, the GNN gave $100\\%$ node classification accuracy for $29$ of the tested graphs and an average of $99.94\\%$ accuracy across all $45$ graphs. \nFor the remaining $16$ graphs, only $19$ nodes were misclassified. \nThe results show that \\textbf{GNNUnlock can accurately detect all three types of nodes} with precision and recall values ranging from $93.17\\%$ to $100\\%$. \nFor the TTLock ITC-99 dataset, the GNN gave $100\\%$ node classification accuracy for $16$ of the tested graphs and an average of $99.95\\%$ accuracy across all $54$ graphs. \nComparing accuracy results on ISCAS-85 Vs. ITC-99 (for both SFLL-HD$^{h>0}$ and TTLock) confirms that \\textbf{GNNUnlock has a consistent performance for predicting the protection logic regardless of the benchmark size}.\nTable~\\ref{tab:results_hd4_nan} represents all the averaged results for TTLock datasets.\nIt was observed that the \\textit{restore} predictor is $100\\%$ successful (with $100\\%$ precision and recall) in all ITC-99 test cases. This also confirms that the separation between the \\textit{perturb} and \\textit{design} nodes is challenging.\n\n\\textit{To study the effect of $h$ on the performance of GNNUnlock}, we launch the attack on the SFLL-HD$^4$ ITC-99 dataset. \nNote: due to lack of space, we only report the averaged metrics in Table~\\ref{tab:results_hd4_nan}. \nHere, we also report the results of GNNUnlock on the SFLL-HD$^2$ ITC-99 dataset with a technology node of $45nm$ to evaluate the effect of the target library on the attack's performance. \nThe trend is the same in all cases, while there is a small difference in the values. \nGNNUnlock detects all three types of nodes accurately with average accuracy values of $99.94\\%$, $99.97\\%$, and $99.90\\%$, for SFLL-HD$^2$ in $45nm$, SFLL-HD$^2$ in $65nm$, and SFLL-HD$^4$ in $65nm$ datasets, respectively. \nAfter post-processing, all nodes were classified correctly ($100\\%$), verifying that \\textbf{GNNUnlock can handle different technology nodes and various parameter settings.}\n\n\\subsection{Comparison with State-of-the-art Attacks}\n\\label{sec:corner_cases}\n\nTo demonstrate the superiority of GNNUnlock, a set of ITC-99 benchmarks are locked using SFLL-HD with $K\/h=2$, generating two datasets: one for $K=128$, $h=64$ and one for $K=64$, $h=32$.\nThe ISCAS-85 benchmarks are locked using $K=32$, $h=16$. \nThe characteristics are listed in Table~\\ref{tab:Datasets}. When the FALL attacks~\\cite{sirone2020functional} were launched on these benchmarks, they reported $0$ keys. Next, the attack platform of SFLL-HD-Unlocked~\\cite{yang2019stripped} was launched on the same benchmarks, which failed to identify the perturb signals and did not recover the secret key. \nHowever, our GNNUnlock was $100\\%$ successful in retrieving the unlocked designs in all of the cases.\nThe results are documented in Table~\\ref{tab:results_hd4_nan}.\n\n\\begin{table}[tb]\n\\centering\n\\caption{\\textsc{Examining the effect of $h$ value and technology node on the performance of GNNUnlock. TR stands for training}}\n\\label{tab:results_hd4_nan}\n\\resizebox{\\textwidth}{!}{%\n\\setlength\\tabcolsep{1.9pt}\n\\begin{tabular}{ccccccccc}\n\\hline\n\\textbf{Dataset} &\\textbf{Benchmarks} &\\textbf{\\begin{tabular}[c]{@{}c@{}}Tech.\\\\ Node (nm)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}GNN\\\\ Acc. (\\%)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Avg.\\\\ Prec. (\\%)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Avg.\\\\ Rec. (\\%)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Avg.\\\\ F1-Score (\\%)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Removal\\\\ Success (\\%)\\end{tabular}}& \\textbf{\\begin{tabular}[c]{@{}c@{}}Avg.\\\\ TR Time ($s$)\\end{tabular}} \n\\\\ \\hline\n\\textbf{TTLock} & ISCAS-85 &$45$ & $99.94$ &\t$99.20$&\t$99.53$\t&$99.33$& $100$&$1,910$\\\\ \\hline\n\\textbf{TTLock} & ITC-99 &$45$ & $99.95$ &\t$97.48$&\t$99.36$&\t$98.33$ & $100$&$22,777$\\\\ \\hline\n\\textbf{SFLL-HD$^2$} &ITC-99& $45$ & $99.94$ & $98.73$ &$99.77$ &$99.21$ & $100$&$24,123$ \\\\ \\hline\n\\textbf{SFLL-HD$^2$} &ITC-99& $65$ & $99.97$ &$99.56$ &$99.85$ &$99.70$ & $100$ &$22,052$\\\\ \\hline\n\\textbf{SFLL-HD$^4$} &ITC-99& $65$ &$99.90$ & $98.40$ &$99.60$ & $98.95$ & $100$&$233,017$ \\\\ \\hline\n\\textbf{SFLL-HD$^{16}$} &ISCAS-85& $65$ &$99.24$ & $97.68$ & $97.30$ & $97.40$ & $100$&$1,907$ \\\\ \\hline\n\\textbf{SFLL-HD$^{32}$} &ITC-99& $65$ &$99.83$ & $97.56$ & $98.45$ & $97.96$ & $100$&$28,049$ \\\\ \\hline\n\\textbf{SFLL-HD$^{64}$} &ITC-99& $65$ &$99.69$&\t$97.60$&\t$97.96$\t&$97.74$&$100$&$32,494$\\\\\\hline\n\\end{tabular}\n}\n\\end{table}\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nIn this paper, we proposed the first-of-its-kind scheme for unlocking provably secure logic locking (PSLL), leveraging a graph neural network (GNN) that learns the common structural features of the protection logic added by such techniques. \nThe GNN does not learn a syntactic implementation but absorbs the protection logic trend, allowing it to handle variants naturally.\nWe also developed a post-processing algorithm to rectify any misclassifications by the GNN, depending solely on the nodes' connectivity, the GNN predictions, and the known properties of the protection logic. \nWe demonstrated the superiority of GNNUnlock by comparing it to two state-of-the-art attacks in unlocking three PSLL techniques, Anti-SAT, TTLock, and SFLL-HD. \nOur GNNUnlock scheme was able to break all locked benchmarks while state-of-the-art attacks struggled with some corner cases.\nWe believe GNNUnlock opens up new frontiers in advancing the state-of-the-art defenses and attacks in logic locking.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\subsection{The threshold factor}\nThe differential cross section of $\\gamma p\\to \\phi p$ reaction is\nrelated to the invariant amplitude as\n \\begin{eqnarray}\n {\\frac{d\\sigma}{dt}}^{\\gamma p\\to \\phi p} = \\frac{1}{64\\pi s k_\\gamma^2}\n |T^{\\gamma p\\to \\phi p}|^2~,\n \\label{2-1}\n \\end{eqnarray}\nwhere $s$ is the total energy and averaging and summing  over the\nspin projections in the initial and the final states are assumed.\n The arguments lead to appearance of the threshold factor\n$q_\\phi^2\/k_\\gamma^2$ are shown in Ref.~\\cite{Sib06}. First, using\nthe current-field identity  (vector dominance model) one can\nexpress the invariant amplitude of the $\\phi$ meson\nphotoproduction through the amplitudes of the\n $Vp\\to\\phi p$ transitions $(V=\\rho,\\omega,\\phi)$\n\\begin{eqnarray}\nT^{\\gamma p\\to \\phi p}=\\sum\\limits_V\\frac{e}{2\\gamma_V} T^{Vp\\to\n\\phi p}~, \\label{1-1}\n\\end{eqnarray}\n where  $\\gamma_\\rho\\div\\gamma_\\omega\\div\\gamma_\\phi\n\\simeq 2.5\\div8.5\\div6.7$ are defined from the $V\\to e^+e^-$\ndecay.\n Keeping only the diagonal transition $\\phi p\\to \\phi p$, one can\n express the cross section of $\\gamma p\\to \\phi p$ reaction through\n the invariant amplitude of the elastic $\\phi p\\to \\phi p$ scattering\n \\begin{eqnarray}\n {\\frac{d\\sigma}{dt}}^{\\gamma p\\to \\phi p} =\n \\frac{\\alpha}{\\gamma_\\phi^2} \\frac{1}{64 s k_\\gamma^2}\n |T^{\\phi p\\to \\phi p}|^2~.\n \\label{2-2}\n \\end{eqnarray}\n\n The next step is evaluating $T^{\\phi p\\to \\phi p}(\\theta=0)$.\n In~\\cite{Sib06} it is made by using the optical theorem\n \\begin{eqnarray}\n {\\rm Im } T^{\\phi p\\to \\phi p} (\\theta=0)\n =\n -2q_\\phi\\sqrt{s}\\sigma_{\\phi p}^{\\rm tot}~,\n \\label{2-3}\n \\end{eqnarray}\nwhere $\\sigma_{\\phi p}^{\\rm tot}$ is the total cross section of\n$\\phi p$ interaction (for convenience, we use the same sign\nconvention as in~\\cite{Sib06}).  The consequence of\nEq.~(\\ref{2-3}) is a disappearance of ${\\rm Im } T^{\\phi p\\to \\phi\np} (\\theta=0)$ at $q_\\phi\\to 0$.  The final result reads\n \\begin{eqnarray}\n {\\frac{d\\sigma}{dt}}^{\\gamma p\\to \\phi p}(\\theta=0)\n =\n \\frac{\\alpha}{16\\gamma^2_\\phi}\n \\,\\frac{q_\\phi^2}{k_\\gamma^2}\\,\n [1 + r^2]{\\sigma_{\\phi p}^{\\rm tot}}^2~,\n \\label{2-4}\n \\end{eqnarray}\nwhere $r={\\rm Re}T^{\\phi p}\/{\\rm Im T^{\\phi p}}$. Assuming $r$ to\nbe a constant, one can gets the threshold factor\n$q_\\phi^2\/k_\\gamma^2$ in explicit form. But the weak point of such\nconsideration is just assuming that $r$ is constant at $q_\\phi\\to\n0$. The real part of invariant amplitude $T^{\\phi p}$ is related\nto the $\\phi p$ scattering length, that can not vanish at\n$q_\\phi\\to 0$ and therefore,\n\\begin{eqnarray}\nr^2 (q_\\phi\\to 0) \\sim \\frac{1}{q_\\phi^2}~.\n \\label{2-5}\n\\end{eqnarray}\nThis leads to cancelation of $q_\\phi^2$ dependence and eliminating\nthe \"threshold factor\" in Eq.~(\\ref{2-4}).\n\n\n\\subsection{Threshold behavior of the differential cross section\n}\n\nFor more consistent analysis of the threshold behavior we express\nthe differential cross section of $\\gamma p\\to \\phi p$ reaction in\nEq.~(\\ref{2-2}) via differential cross section of $\\phi p\\to \\phi\np$ elastic scattering\n\\begin{eqnarray}\n {\\frac{d\\sigma}{dt}}^{\\gamma p\\to \\phi p} =\n \\frac{\\alpha\\pi^2}{\\gamma_\\phi^2k_\\gamma^2}\\,\n{\\frac{d\\sigma}{d\\Omega}}^{\\phi p\\to \\phi p}~.\n \\label{3-1}\n \\end{eqnarray}\n At small $q_\\phi$, the differential cross section\n ${d\\sigma}^{\\phi p\\to \\phi p}\/{d\\Omega}$\n becomes isotropic and\n it can be expressed\n through the spin averaged $\\phi p$ scattering length $a_{\\phi p}$\n\\begin{eqnarray}\n{\\frac{d\\sigma}{d\\Omega}}^{\\phi p\\to \\phi p}=a_{\\phi p}^2~.\n \\label{3-1}\n \\end{eqnarray}\nThis leads to the following estimation\n\\begin{eqnarray}\n {\\frac{d\\sigma}{dt}}_{\\rm threshold}^{\\gamma p\\to \\phi p} =\n \\frac{\\alpha\\pi^2 }{\\gamma_\\phi^2k_\\gamma^2} \\,a_{\\phi p}^2~.\n \\label{3-2}\n \\end{eqnarray}\nOne can see that at the threshold  the cross section of $\\phi$\nmeson photoproduction is finite and its value is defined by the\n$\\phi p$ scattering length.\n\n\\subsubsection{direct estimations}\n\nThe direct estimation of the $\\phi p$ scattering length on the\nbase of QCD sum rules was done by Koike and\nHayashigaki~\\cite{KH97}. They got $a_{\\phi p}\\simeq -0.15$ fm\nwhich results in\n\\begin{eqnarray}\n {\\frac{d\\sigma_{\\rm thr}}{dt}}^{\\gamma p\\to \\phi p}_{[1]} \\simeq\n 0.63\\,{\\mu\\rm b }\/{\\rm GeV^2}~.\n \\label{3-3}\n \\end{eqnarray}\nThis value is in qualitative agreement with the experimental\nindication~\\cite{Mibe}.\n\nOne can estimate $a_{\\phi N}$ using the $\\phi N$ potential\napproaches. Thus for example, Gao, Lee and Marinov suggested to\nuse the QCD van der Waals attractive  $\\phi N$\npotential~\\cite{GLM} for analysis of $\\phi$-nucleus bound states.\nThis potential reads\n\\begin{eqnarray}\n V_{\\phi N}=-A\\exp(-\\mu r)\/r~,\n \\label{3-4}\n\\end{eqnarray}\n where $A=1.25$ and $\\mu=0.6$~GeV. The corresponding scattering\n length  $a_{\\phi p}\\simeq2.37$~fm,\n found by direct solution of the Schr\\\"odinger equation,\n leads to large cross section\n $d\\sigma\/dt\\simeq 1.6\\times10^2\\mu$b\/GeV$^2$.\n It is more than two order of magnitude greater than\n the experimental hint\n and provides a problem for this potential model.\n Thus, in order to get the scattering length\n $a_{\\phi p}\\simeq\\pm0.15$~fm (and correspondingly, the cross section\n  $d\\sigma\/dt$ close to the experiment) one has to choose $A=2.56$ or\n  0.226 for the positive (strong attraction) or negative\n  (weak attraction)  $a_{\\phi p}$, respectively.\n  At $A\\simeq 2.75$, the elastic\n  scattering disappears ($a_{\\phi p}=0$) and we get some kind of Ramsauer\n  effect~\\cite{MF}.\n  In principle, such analysis may be used for other\n  potentials as well.\n\n\n\\subsubsection{SU(3) symmetry considerations}\n\nEstimation of the upper bound of $|a_{\\phi p}|$ may be done on\nassumption that the amplitudes of the $\\phi p$ and $\\omega p$\nscattering are dominated by the scalar $\\sigma$ meson exchange.\nThen the SU(3) symmetry gives relation\n\\begin{eqnarray}\n a_{\\phi p}=\\xi a_{\\omega p}~,\n \\label{3-6}\n\\end{eqnarray}\nwhere $\\xi\\equiv -{\\rm tg}\\Delta \\theta_V$ ($\\Delta\n\\theta_V\\simeq3.7^0$ is the deviation of the $\\phi-\\omega$ mixing\nangle from the ideal mixing~\\cite{PDG}). More complicated\nprocesses as $s$ channel exchange with intermediate nucleon or\nnucleon resonances, or box diagrams with $\\omega(\\phi)\\pi\\rho$\nvertices would give terms proportional to $\\xi^2$ and generally\nspeaking, violate Eq.~(\\ref{3-6}). But for crude estimation of\norder of magnitude of $ a_{\\phi p}$ one can utilize\nEq.~(\\ref{3-6}) using $a_{\\omega p}$ as an input.\n\nThus, QCD sum rule analysis of Koike and Hayashigaki~\\cite{KH97}\nresults in $a_{\\omega p}=-0.41$ fm. The coupled channel unitary\napproach of Lutz, Wolf and Friman~\\cite{Lutz} leads to $a_{\\omega\np}=(-0.44 + \\i 0.20)$ fm. An effective Lagrangian approach based\non the chiral symmetry developed by Klingl, Waas and\nWeise~\\cite{Klingl} results in $a_{\\omega p}=(1.6 + \\i 0.3)$ fm.\nThe corresponding $\\phi$ meson photoproduction cross sections for\nthese scattering lengths, denoted with subscripts 2, 3 and 4,\nrespectively, read\n\\begin{eqnarray}\n {\\frac{d\\sigma^{\\gamma p\\to \\phi p}_{\\rm thr}}{dt}}_{[2]}\n &=&\n 2.0\\times10^{-2}\\,{\\mu\\rm b }\/{\\rm GeV^2}~,\\label{3-7}\\\\\n {\\frac{d\\sigma^{\\gamma p\\to \\phi p}_{\\rm thr}}{dt}}_{[3]}\n &=&\n 2.7\\times10^{-2}\\,{\\mu\\rm b }\/{\\rm GeV^2}~,\\label{3-8}\\\\\n {\\frac{d\\sigma^{\\gamma p\\to \\phi p}_{\\rm thr}}{dt}}_{[4]}\n &=&\n 3.1\\times10^{-1}\\,{\\mu\\rm b }\/{\\rm GeV^2}~.\n \\label{3-9}\n \\end{eqnarray}\n\n\\begin{figure}[th]\n \n  \\includegraphics[width=0.4\\columnwidth]{fig1.eps}\n  \\caption{\\label{FIG:1}{\\smal\n  Differential cross section of $\\gamma p\\to \\phi p$ reaction\n  at $\\theta=0$ as function of the photon energy.\n  The enumerated symbols \"plus\" correspond to the threshold predictions, given in\n  Eqs.(\\protect\\ref{3-3}), (\\protect\\ref{3-7}) - (\\protect\\ref{3-9}).\n  Experimental data are taken from\n  Refs.~\\protect\\cite{Mibe,OldData}.\n  }}}\n \\end{figure}\n\n  Fig.~1 shows  predictions of\n  Eqs.(\\ref{3-3}), (\\ref{3-7}) - (\\ref{3-9})\n  by the enumerated symbols \"plus\".\n  Experimental data at $\\theta=0$ are taken from Refs.~\\cite{Mibe,OldData}.\n  The predictions of\n  Eqs.~(\\ref{3-3}) and (\\ref{3-9}) seems to be more preferable.\n  The difference between  Eqs.~(\\ref{3-7}) and (\\ref{3-8})\n  and data can indicate small $\\omega p$ scattering length or\n  necessity to introduce large OZI-rule evading factor in Eq.~(\\ref{3-6})\n   which can be related to the finite hidden strangeness in the nucleon.\n  For example, analysis of $\\phi$ meson photoproduction at large\n  angles in Refs.\\cite{TL,Sib05} favors for the large OZI-rule\n  evading factor $x_{\\rm OZI}\\simeq3-4$. Such value results in\n  increasing the threshold predictions based on  $a_{\\omega p}$ by almost\n  of order of magnitude.  Employing this evading factor seems to be consistent with\n  predictions $[2]$ and  $[3]$ and make a problem for that of\n  $[4]$.\n\n  \\subsection{non-diagonal transitions}\nIn principle, the non-diagonal transitions in Eq.~(\\ref{1-1}) may\nalso contribute near the threshold. Such example is the $\\phi$\nmeson photoproduction with $\\pi (\\eta)$ meson exchange , shown in\nFig.~\\ref{FIG:2} , which is associated with $\\rho\\to \\phi$\ntransition.\n\\begin{figure}[th]\n \n  \\includegraphics[width=0.35\\columnwidth]{fig2.eps}\n  \\caption{\\label{FIG:2}{\\smal\n  Diagrammatic presentation of the pseudoscalar $\\pi,\\eta$ exchange\n  processes in $\\gamma p\\to \\phi p$ reaction.}}}\n \\end{figure}\n\nThe corresponding invariant amplitude written in obvious standard\nnotations reads\n\\begin{eqnarray}\nT^{\\gamma p\\to\\phi p}_{\\pi} =-i\\frac{eg_{NN\\pi}g_{\\gamma\n\\phi\\pi}}{M_\\phi(t-m_\\pi^2)}\\epsilon^{\\mu\\nu\\alpha\\beta}\n\\varepsilon_\\mu(\\gamma)\\varepsilon^*_\\nu(\\phi)\n{k_\\gamma}_\\alpha{q_\\phi}_\\beta[\\bar u_{p'}\\gamma_5 u_p]F(t)~,\n\\label{4-1}\n\\end{eqnarray}\nwhere $g_{\\gamma\\phi\\pi}$ has a sense of\n$g_{\\rho\\phi\\pi}\/2\\gamma_\\rho$ and is taking from $\\phi\\to\n\\gamma\\pi$ decay ($g_{\\gamma\\phi\\pi}\\simeq0.14$~\\cite{PDG}),\n$g_{NN\\pi}\\simeq13.3$ and $F(t)$ is a product of the form factors\nin $\\gamma\\phi\\pi$ and $NN\\pi$ coupling vertices. This amplitude\nleads\n to the following estimate\n\\begin{eqnarray}\n \\frac{d\\sigma}{dt}^{\\gamma p\\to \\phi p\\,(\\pi)}_{\\rm threshold} =\n  \\frac{\\alpha g^2_{NN\\pi}g^2_{\\gamma\\phi\\pi}F^2(t_{\\rm thr})}\n  {64E_{\\rm thr}M_N^2M_\\phi^2}\n  \\frac{|t_{\\rm thr}|(M_\\phi^2-t_{\\rm thr})^2}{(t_{\\rm\n  thr}-m_\\pi^2)^2}~,\n \\label{4-2}\n \\end{eqnarray}\nwhere $E_{\\rm thr} =(2M_NM_\\phi+M_\\phi^2)\/2M_N\\simeq 1.574$ GeV\nand $t_{\\rm thr}=-M_NM_\\phi^2\/(M_N+M_\\phi)\\simeq-0.5$~GeV$^2$.\nTaking $F(t)=((\\Lambda^2 -m_\\pi^2)\/(t-m_\\pi^2))^2$ with\n$\\Lambda\\simeq 0.6-0.7$ GeV, one can find\n\\begin{eqnarray}\n \\frac{d\\sigma}{dt}^{\\gamma p\\to \\phi p\\,(\\pi)}_{\\rm threshold}\n \\simeq (0.8-1.6)\\times 10^{-2} \\mu {\\rm b}\/ {\\rm GeV}^2~.\n \\label{4-3}\n \\end{eqnarray}\nCoherent sum of the $\\pi$ and $\\eta$ meson exchange results in\n ${{d\\sigma}^{\\gamma p\\to \\phi p\\,(\\pi+\\eta)}}\/{dt}\\simeq (0.3-0.6)\\times 10^{-1}$\n$\\mu {\\rm b}\/ {\\rm GeV}^2$.\n So again, ${d\\sigma}^{\\gamma p\\to\n\\phi p}\/{dt}$ is finite and its magnitude is in the range of\nuncertainty of other estimations. However, being smaller than the\nexperimental indication it allows contribution of exotic channels\ndiscussed in literature, such as scalar\/glueball exchange, direct\nknockout of hidden $\\bar ss$ pairs and so on.\n\nFinally we notice that the differential ${d\\sigma}^{\\gamma p\\to\n\\phi p}\/{d\\Omega}$ and the total ${\\sigma}^{\\gamma p\\to \\phi p}$\ncross sections have the obvious kinematical phase space factor\n$q_\\phi\/k_\\gamma$. For example, for the diagonal transition we get\n\\begin{eqnarray}\n {\\frac{d\\sigma}{d\\Omega}}^{\\gamma p\\to \\phi p}_{\\rm threshold} =\n \\frac{q_\\phi}{k_\\gamma}\\frac{\\alpha\\pi}{\\gamma_\\phi^2}\\,\n  a_{\\phi p}^2,\\qquad\n {\\sigma}^{\\gamma p\\to \\phi p}_{\\rm threshold} =\n \\frac{q_\\phi}{k_\\gamma}\\frac{4\\alpha\\pi^2}{\\gamma_\\phi^2}\\, a_{\\phi p}^2~.\n \\label{4-4}\n \\end{eqnarray}\nIf one accepts the threshold behavior of $d\\sigma\/dt$ as in\nEq.~(\\ref{2-4}) with a constant r, then the cross sections\n${d\\sigma}\/{d\\Omega}$ and ${\\sigma}^{\\gamma p\\to \\phi p}$ will\ndecrease near threshold\nas $(q_\\phi\/k_{\\gamma})^3$ which seems to be rather strong.\\\\\n\n\nIn summary, we analyzed  the differential  cross section\n${d\\sigma}\/{dt}$ of $\\gamma p\\to \\phi p$ reaction at the threshold\nand have shown that it is finite and its value is crucial for the\nQCD inspired models of $\\phi N$ interaction and for the mechanism\nof the $\\phi$ meson photoproduction.\\\\\n\n\n We thank E.L.~Bratkovskaya, W.~Cassing, H.~Ejiri and B.~K\\\"ampfer\n for useful discussions.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\\label{intro}\n\nWe consider Wilson fermions with or without $O(a)$ improvement. The\nfermionic action reads\n\\begin{equation}\nS_F = \\sum \\bar{\\psi}{\\cal K}\\psi, \\; {\\cal K} = W + M,\n\\label{action}\n\\end{equation}\nwhere $W$ is the Wilson-Dirac operator:\n\\begin{equation}\nW = \\not{\\!\\!D} + X\n\\end{equation} \nwith\n\\begin{equation}\nD_\\mu = \\frac{1}{2}(D^+_\\mu + D^-_\\mu), \n\\end{equation}\n\\vspace*{-0.3cm}\n\\begin{equation}\nX = - \\frac{1}{2} D^+_\\mu D^-_\\mu \n + \\frac{\\mbox{i}}{4}c_{\\scriptstyle {\\rm SW}}\\, \n\\sigma_{\\mu\\nu}F_{\\mu\\nu}, \n\\end{equation}\n$D^+_\\mu$ ($D^-_\\mu$) being the forward (backward) lattice\nderivative, and where $M$ is the Wilson mass term:\n\\begin{equation}\nM = \\frac{1}{2\\kappa} - 4.\n\\end{equation}\nThe critical value of the hopping parameter, $\\kappa_c$,\nis taken to be the limiting value at which the ensemble averaged pion mass\nvanishes. We call this limit the chiral limit.\n\n\\begin{figure}\n\\vspace{-0.5cm}\n\\begin{centering}\n\\epsfig{figure=plot.eps,height=6.5cm,width=6.5cm}\n\\vspace{-0.7cm}\n\\caption{The pion correlator on the $32^3 64$ lattice at $\\beta = 6.2$\n  and $\\kappa = 0.1354$ for improved fermions with\n  $c_{\\scriptstyle {\\rm SW}} = 1.614$. At these parameters\n  $m_\\pi\/m_\\rho \\approx 0.5$ which corresponds to a quark mass of\n  $\\approx 40 \\mbox{MeV}$. The solid symbols (and\n  line) represent the ensemble average with the exceptional\n  configurations omitted. The dashed line is the result of a single,\n  particularly exceptional configuration.}\n\\vspace{-0.5cm}\n\\end{centering}\n\\end{figure}\n\n\\begin{figure*}[bht]\n\\vspace{-0.35cm}\n\\begin{centering}\n\\epsfig{figure=eigenf2.ps,height=5.75cm,width=13.5cm,bbllx=63,bblly=286,\nbburx=560,bbury=506}\n\\vspace{-0.7cm}\n\\caption{A sketch of the eigenvalue spectrum of $W_c$ (before the\n  rotation) and $W_c^{\\phi = \\pi\/4}$ (after the rotation).}\n\\end{centering}\n\\end{figure*}\n\nIn quenched lattice calculations with Wilson fermions one encounters\nexceptional configurations~\\cite{exceptional} which give\nrise to poles in fermionic observables, such as hadron correlators, at\nsmall quark masses. The origin of these poles are real eigenvalues  \n$\\lambda_i$ of the Wilson-Dirac operator $W$ at \n\\begin{equation}\n\\lambda_i < 4 - \\frac{1}{2 \\kappa_c}.\n\\label{eig}\n\\end{equation} \nExceptional configurations are particularly aggravating for improved\nfermions, and they have limited calculations to larger quark masses\nand larger values of $\\beta$. \nWe expect to find such poles also in the full theory, i.e. with \ndynamical fermions, at hopping parameters $\\kappa \\neq\n\\kappa_{\\scriptstyle {\\rm sea}}$, where\n$\\kappa_{\\scriptstyle {\\rm sea}}$ refers to the sea quark mass.\n\nIn Fig.~1 we give an example of an exceptional configuration which\nwe encountered in a recent simulation with improved\nfermions~\\cite{Dirk}. We see a dramatic deflection from the anticipated\nensemble average. Similar peaks are observed in the $\\rho$ and nucleon\ncorrelators. It is quite clear from this figure that the problem of\nexceptional configurations cannot be\nsolved by accumulating larger statistical samples.  \n\nBardeen et al.~\\cite{Bardeen} identify the eigenvalues (\\ref{eig}) \nwith the zero modes of the continuum theory, and they propose to shift each \nsuch $\\lambda_i$ to $\\lambda_i = 4 - 1\/2 \\kappa_c$. There are doubts, however, \nthat this identification is correct in general~\\cite{Gattringer}. \nAn elegant solution would be to employ Neuberger's action~\\cite{Neuberger}.\nBut whether this is numerically feasible has still to be seen.   \n\n\n\\section{PROPOSAL}\n\nThe solution we propose makes use of a freedom in regularizing lattice\nfermions. It has the advantage that it is easy to implement numerically, and\nthat it does not change the physics in the chiral limit so that the \nimprovement program \\`{a} la Sheikholeslami and Wohlert~\\cite{Jansen}\nremains unaltered.\n\nWe rewrite ${\\cal K}$ in (\\ref{action}) as\n\\begin{equation}\n{\\cal K} = W_c + m,\n\\label{caction}\n\\end{equation}\nwhere $W_c$ and $m$ are the critical Wilson-Dirac operator and the\nphysical bare mass, respectively:\n\\begin{equation}\nW_c = \\not{\\!\\!D} + X_c,\\; X_c = X + \\frac{1}{2\\kappa_c} - 4,\n\\end{equation}\n\\vspace*{-0.4cm}\n\\begin{equation} \nm = \\frac{1}{2\\kappa} - \\frac{1}{2\\kappa_c}.\n\\end{equation}\nWe then apply a chiral rotation to $W_c$,\n\\begin{eqnarray}\nW_c \\rightarrow W_c^\\phi \\!\\!\\!&=&\\!\\!\\! \n\\mbox{e}^{{\\scriptstyle \\rm i}\\gamma_5\\phi} W_c \n\\mbox{e}^{{\\scriptstyle \\rm i} \\gamma_5\\phi}, \\nonumber \\\\\n\\!\\!\\!&=&\\!\\!\\! \\not{\\!\\!D} + X_c \\,(\\cos 2\\phi + \\mbox{i}\\gamma_5 \\sin\n2\\phi),\n\\label{transf}\n\\end{eqnarray}\nand take ${\\cal K}^\\phi \\equiv W_c^\\phi + m$ to be the new fermion matrix.\nThe resulting action is equally well founded as\n(\\ref{action}) and (\\ref{caction})~\\cite{I&Y}, though, in general, \nit breaks parity invariance. In Fig.~2 we\nshow the effect of the transformation on the eigenvalue spectrum of\nthe Wilson-Dirac operator for $\\phi = \\pi\/4$. In this case $W_c$ is\nanti-hermitian, and all eigenvalues are\nmapped onto the imaginary axis, as is the case for staggered fermions.\n\n\\begin{figure}\n\\begin{centering}\n\\hspace*{-0.5cm} \\epsfig{figure=plot2.eps,height=6.5cm,width=6.5cm}\n\\vspace{-0.7cm}\n\\caption{The pion correlator for a single exceptional configuration on\n  the $16^3 32$ lattice at $\\beta = 5.7$ and \n  $c_{\\scriptstyle {\\rm SW}} = 2.25, \\kappa = 0.1295$ before\n  ($\\raise0.85pt\\hbox{$\\scriptscriptstyle\\bigcirc$}$) and after the rotation by $\\phi = \\pi\/4$ ($\\lower0.95pt\\hbox{\\Large $\\bullet$}$),\n  with $\\kappa_c = 0.13074$~\\cite{Gockeler}. The data for improved \n  fermions are compared with the result for Wilson fermions\n  ($\\lower0.45pt\\hbox{$\\Diamond$}$) at an equivalent quark mass.}\n\\vspace{-0.2cm}\n\\end{centering}\n\\end{figure}\n\n\\section{TEST}\n\nThe rotation (\\ref{transf}) is mathematically equivalent to the \ntransformation of the mass:\n\\begin{equation}\nm \\rightarrow m^\\phi = m \\,(\\cos 2\\phi - \\mbox{i} \\gamma_5 \\sin 2\\phi).\n\\end{equation}\nIn the following we shall use this transformation. Furthermore we take\n$\\phi = \\pi\/4$. But we found that the method works equally well\nfor angles as small as $|\\phi| = 0.05$.\n\nTo test the idea, we have looked at exceptional configurations which\nwe encountered in our runs with improved fermions~\\cite{Gockeler}\non the $16^3 32$ lattice at $\\beta = 5.7$. If the method works at this\ncoupling, we expect that it will work also at larger values of\n$\\beta$. In Fig.~3 we show the pion correlator for a single\nexceptional configuration before and after the rotation. Before the\nrotation we find a similar peak in the pion correlator as seen in\nFig.~1. After the rotation we find a correlator which is perfectly \nwell behaved. It turns out to be in good agreement with the ensemble \naverage~\\cite{Gockeler}. Note that in first\napproximation the physical quark mass is left unchanged by the rotation.\nFor comparison we have computed the pion correlator also for Wilson\nfermions. The hopping parameter was chosen so as to give roughly the same\npion mass. We find good agreement with the rotated action. \n\n\\section{DISCUSSION}\n\\label{discussion}\n\nFrom both, Figs.~1 and 3 it follows that the (suspected) near-zero \neigenmode of ${\\cal K}$ acts as a source term, and that the excitation\nit induces decays\nexponentially with roughly the mass of the pion. We would not expect\nto find such a behavior for an instanton configuration. There is also\nno sign of an abnormal fluctuation found in the case of Wilson\nfermions. \n\nWe conclude that the problem of exceptional configurations may be\nresolved by a chiral rotation of the critical Wilson-Dirac\noperator. The method allows calculations at any quark mass. On top of\nthat, the results of the improvement program remain valid.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCeramic materials have applications in many fields of science and engineering\n\\cite{chiang}.  \nIn particular, the efficient production of dense ceramics with \nnanosized grains using ultrafine powders is a subject of increased\nattention.  \nThe most common method for achieving densification in ceramics\n is solid-state sintering, \na thermally activated process that involves the diffusion and \nredistribution of mass by means \nof various physical processes, leading \nto a compactified and consolidated \nfinal structure.  \nIn the case of nanosized powder compacts, a major concern\nis the considerable grain growth during the densification \nprocess. Special consolidation techniques such as hot isostatic pressing, spark plasma and microwave sintering \nin many cases successfully limit this undesirable phenomenon. \nThe addition of dopants able to limit grain boundary migration is another effective option. \n\nA very effective way to limit grain growth is to promote densification at a lower sintering temperature \\cite{palm}. \nMechanical activation of powders by high-energy ball milling can contribute greatly toward this direction. High-energy mechanical treatment of solids usually leads to the accumulation of excess energy in the structure resulting in plastic deformation and fracturing of crystals \\cite{bokh,juha}. The process usually involves initial reduction of grain sizes, followed by plastic deformation of the crystalline structure, whereas excessive milling time may finally lead to grain aggregation  \\cite{bal,bokh,bold2,kostic}. \nGrain size reduction leads to an increase of surface energy \n\\cite{stoj}. \nUnder deformation the lattice structure is transformed, perturbed or completely destroyed. This effect is evidenced by the \nbroadening, decrease of intensity, and finally total disappearance of XRD reflections \\cite{juha}.\nThis means that the deformation and increase of structural disorder can bring the solid into a nanocrystalline or even completely amorphous state [6, 11-13]. Finally, the possible aggregation of grains imposes significant pressure on the grains increasing their contact surface areas \\cite{bal,bold2}.\nThis lowers the activation energy for the diffusion processes \ninvolved in sintering, thus accelerating the entire process\n and reduces the sintering \ntemperatures \\cite{stoj,heegn}.\n\nThe alteration of the state of solid materials due to \nmechanical activation can lead to significant changes in physical and chemical properties. \nOrder-disorder transitions and phase transformations can then take place at much lower temperatures. Additionally, nuclei of new phases can often appear at the interfaces possibly followed by subsequent growth of crystallites \n\\cite{bal,kostic,xue}. \nThus, crystallization from the amorphous state \nand chemical reactions can be easily accelerated with subsequent treatment, resulting in materials with improved or better-controlled properties \n\\cite{stoj,xue}.\n\nTherefore, a central issue for mechanically \nactivated powders is the relations between the \nproperties of the initial powder and the structure of the resulting ceramic material. \nImproved performance is in many cases achieved by an optimal balance between particle size and mechanical activation effects \\cite{bold1}. \nTheoretical and computational investigations can help to understand the impact of mechanical activation on the sintering behaviour of nanosized powders ,\nas well as on sintering temperature dependence. \n\nEven though theories of the sintering process has been intensely \nresearched during the last fifty years, \n there is a lack of quantitative, predictive models for the \nmicrostructural evolution during sintering \\cite{chiang}.\nIn the final stage of sintering the densification \nprocess is accompanied by grain growth leading to\nthe final pore structure (grain size distribution, geometry and topology \nof pores).  The physical mechanisms responsible for these effects involve\nvarious transport processes acting in the bulk and on the surface of grains, \n at the boundary between grains, as well as the evaporation-condensation\nmechanism.   \n\nGiven the complexity of the physical processes and the initial micro-structure, \nit is not surprising that a variety of mathematical\nmodels have been proposed that focus on different stages of the sintering process, based on \nvarious simplifying assumptions regarding the number of grains, the geometry and \ntopology of the pore structure, etc.\nThe existing models include continuum diffusion or transport \nequations \\cite{olev98,kuch00,maxim04} that are solved either analytically \n(in simplified cases) or numerically, and\nkinetic  Monte Carlo (Potts model) simulation approaches \\cite{brag}. \nMore specifically in the case of grain growth, computational approaches \ninvolve vertex, Monte Carlo Potts, phase field, and cellular automata models\n\\cite{miod02}. \n\n\\section{A Nonlinear Equation for Grain Growth}\n\\label{sde-gg}\nWe introduce a model of grain growth \napplying to the final stage of sintering. The model \ndescribes mass transfer between grains in terms of a balance equation.\nSuch a model should\naccount for differences in the initial distribution\nof grain sizes, which contribute to differential growth \nof the grains.  A stochastic aspect is introduced in \nthe problem, due to the fact that the initial grain configuration is not \nknown. \nInstead of attempting to model explicitly the various physical mechanisms that contribute to \nmass transfer, we opt for an effective approach that \ndoes not distinguish between the different mechanisms.\nThe amorphization degree represents the fraction of the grain\nthat is in the amorphous state. Grains with higher \ndegree of amorpization are more active, and tend to \ntransfer mass more effectively. \nSmaller grains \n(e.g., such as those created by mechanical activation) \nare expected to have a higher degree of amorphization.\n \n\nSince the lattice formed by the centres of the grains is \nirregular, the approach to a continuum limit is not \nwell-defined. Thus, a discrete mass transfer equation is proposed.\nThe considerations in the paragraph above imply that \ntransition rates should depend on  \ngrain size.  \nThe locations of the grain centers are\ndenoted by ${\\bf x}$, and those of the nearest neighbors \nof point ${\\bf x}$ by ${\\bf x'}$. The grain masses at \ntime $t$ are denoted by $m({\\bf x},t)$ and $r({\\bf x},t)$ is the grain radius.\nThe grain mass density (assumed constant) is denoted by $\\rho$. \nWe introduce a local `order factor' $\\phi({\\bf x},t)$, \nwhich is inversely proportional to the `activation' of each grain.  \nWe assume that the order factor increases with the grain mass, \nsince the fraction of amorphous areas \nis smaller for larger grains.  The order factor influences the \nlocal activation energy.\nTo our knowledge, there are no experimental measurements of \nactivation energies as a function of grain amorpization. \nHence, it is necessary to \nsurmise the functional dependence of the order factor \nbased on intuition and \nphysical constraints. \nBased on these considerations, we propose the \nfollowing system of equations: \n\n\\begin{equation}\n\\label{mass-trans}\n\\partial_{t}\\, m({\\bf x},t) = \\sum_{{\\bf x'}} W({\\bf x'} \\rightarrow {\\bf x},t) \\,\n m({\\bf x'},t) - \\sum_{{\\bf x'}} W({\\bf x} \\rightarrow {\\bf x'},t) \\, m({\\bf x},t),\n\\end{equation}\n\n\\begin{equation}\n\\label{order}\n\\phi({\\bf x},t)= \\frac{ \\alpha \\, m({\\bf x},t) }{m_{0}},\n\\end{equation}\n\n\\begin{equation}\n\\label{rate}\n W({\\bf x} \\rightarrow {\\bf x'},t) = \\lambda \\, \\exp \n   \\left\\{ - \\frac{Q \\, \\phi ({\\bf x},t) } {R \\, T} \\right\\}.\n\\end{equation}\n\n\n\\noindent\nIn Eq.~(\\ref{mass-trans}) $W({\\bf x'} \\rightarrow {\\bf x},t)$ \nrepresents the transition rate at time $t$ for mass transfer \ninto the grain located at ${\\bf x}$ from the neighbouring grains. \nSimilarly, $W({\\bf x} \\rightarrow {\\bf x'},t)$ represents the \ntransition rate at time $t$ for \nmass transfer from the grain at ${\\bf x}$ into the neighbouring grains.\nThe mass and the radius are related via \n$m({\\bf x},t)= g_{n} \\, \\rho \\, r^{n}({\\bf x},t),$\nwhere $g_{n}$ is a geometric factor that  \non the dimension $n$. \nThe order factor defined by Eq.~(\\ref{order}), where\n$m_{0}$ is a reference mass and $\\alpha$ a dimensionless\nconstant, satisfies the \nconditions $\\phi \\rightarrow 0$ for $m \\rightarrow 0$ \nand $\\phi \\rightarrow \\infty$ \nfor $m \\rightarrow \\infty$. \nThe first condition implies that the activation energy tends to vanish \n for small grains, which are expected to have a significant \namorphous fraction, while the second condition implies a very \nlarge activation energy for big grains that are predominantly ordered. \nIn Eq.~(\\ref{rate}), $\\lambda$ is a constant that \ndetermines the scale of the transition rates, and\n$R=8.31 \\, {\\rm J}\\, {\\rm K}^{-1} \\, {\\rm mol}^{-1}$ is\n the gas constant.  \nBased on Eqs.~(\\ref{rate}) and (\\ref{order}), the following \nequation is obtained for the transition rates: \n\n\n\\begin{equation}\n\\label{rate2}\n W({\\bf x} \\rightarrow {\\bf x'},t) = \\lambda \\, \\exp \n   \\left\\{ - \\frac{\\tilde{Q}} {R \\, T} \\left[ \\frac{m({\\bf x},t)}\n   {m_{0}} \\right]  \\right\\}.\n\\end{equation}\n\n\nAlthough the mass transfer equation~(\\ref{mass-trans}) \n looks similar to the master equation\nused in the analysis of diffusion processes, \nit differs from the latter in the following: First, \nthe conserved quantity in the master equation is a probability function,\nwhile in Eq.~(\\ref{mass-trans}) it is the total\nmass of the grains. Secondly, the classical master equation \nis linear in the probability, while Eq.~(\\ref{mass-trans}) is nonlinear \nin the mass due to  \nthe variations in grain activity. \nFinally, in classical diffusion the \ntransitions tend to generate a uniform steady state, \nwhile this is not necessarily the case  \nfor Eq.~(\\ref{mass-trans}), as shown below.  \n\nFor a time-independent steady  state to exist, the time \nderivative of $m({\\bf x},t)$ must vanish asymptotically. \nThis is accomplished\nif the `detailed balance' condition is satisfied, i.e.,\n\n\\begin{equation}\n\\label{det-bal}\n\\lim_{t \\rightarrow \\infty} \\frac{m({\\bf x},t)}{m({\\bf x'},t)} =\n\\lim_{t \\rightarrow \\infty} \\frac{W({\\bf x'} \\rightarrow {\\bf x},t)} \n{W({\\bf x} \\rightarrow {\\bf x'},t)}=\\lim_{t \\rightarrow \\infty}\n\\exp \\left\\{-  \\frac{ \\alpha \\, Q }{R \\, T \\, m_{0}} \\, \n\\left[  m({\\bf x'},t) - \n m({\\bf x},t)    \\right] \\right\\}.\n\\end{equation}\n\n\\noindent\nIf a steady state exists in which some grains have\n$m({\\bf x},\\infty)=0$, while for at least one of their neighbours  \n$m({\\bf x'},\\infty) \\neq 0$, the left hand-side of Eq.~(\\ref{det-bal}) tends\nto zero. Then  \nthe right hand-side becomes \n$\\exp \\left[- \\frac{ \\alpha \\, Q \\, m({\\bf x'},t)}{R \\, T \\, m_{0}} \\right]$. \nSince $m({\\bf x},t) \\propto O(m_{0})$, \na steady state with vanishing grain masses can exist only if\n$ \\alpha \\, Q \\equiv \\tilde{Q} >>R \\, T$.\nThis condition is necessary but not sufficient for an asymmetric \nsteady state.  \nUsing as an average estimate for the\nactivation energy the value of $500$ kJ\/mol \\cite{heegn}\nand a typical sintering temperature (e.g., 1500$^{\\circ}$ C),\n$Q\/R\\,T \\approx  34$, giving an order of magnitude estimate. \nThe value of $\\alpha$ determines the variation of the activation\nenergy due to mechanical activation effects. \nWhile a large value of $Q\/R\\,T$ is necessary for \nachieving a non-uniform steady state, very large values \nslow down considerably the sintering process. \nWe expect that \n$\\alpha \\leq 1$, to allow for reduction of the activation energy \nby mechanical activation.  \nAt the opposite limit, if \n$ \\alpha \\, Q << R \\, T$ the ratio of the transition rates \nin Eq.~(\\ref{det-bal}) will be \nclose to one, and a uniform \nsteady-state is expected. \n\n\n\\section{Dimensional Analysis}\n\\label{dim-anal}\n\nWe use the Length (L) - Mass (M) - Time (T) \nsystem of units for dimensional analysis. \nThe governing parameters \\cite{bar}  of the grain mass equation are \n$m_{0}, \\rho, \\lambda, \\tilde{Q}, {\\bf x}, t$, and $R\\, T$. To these one should \nadd the parameters that determine the \ninitial grain size distribution. If this distribution is approximately \nGaussian, the parameters include the mean value $\\overline{r}_{0}$ \nand standard deviation\n$\\sigma_{r,0}$.  Then, the governing parameters and their\ndimensions are as follows: \n$[m_{0}]=M, \\, [\\lambda]=T^{-1}, \\, [\\tilde{Q}]=[R\\, T]=M\\,L^{2}\\,T^{-2}, \\,\n  [x]=[\\overline{r}_{0}]=[\\sigma_{r,0}]=L, \\, [t]=T, \\,\n  [\\rho]= M\\,L^{-3}. $\nThere are three variables with independent dimensions, i.e., \n$m_{0}, \\tilde{Q}, \\lambda$. According to the \n$\\Pi$-theorem of dimensional analysis \\cite{bar}, the equation~(\\ref{mass-trans}) \ncan be expressed in terms of six dimensionless combinations of \nthe governing parameters, e.g., \n\n\\begin{equation}\n\\label{scaled}\n\\frac{ m({\\bf x},t) } {m_{0} } = \\tilde{m} \\left( \\lambda \\, t, \\frac{\\bf x}{\\overline{r}_{0}}; \\frac{\\tilde{Q}}{R\\, T}, \\,  \\frac{\\sigma_{r,0}}{\\overline{r}_{0}},\n\\, \\frac{\\tilde{Q}}{m_{0}\\,\\overline{r}_{0}^{2} \\, \\lambda^{2}}, \\,  \n  \\frac{m_{0}} {\\rho \\, \\overline{r}_{0}^{3}},  \\right).\n\\end{equation}\n\n\\noindent\nUsing the notation  \n$\\tilde{t}= \\lambda \\, t$, $\\tilde{\\bf x} =\\frac{\\bf x}{r_{0} }$, \n$u=\\frac{\\tilde{Q}}{R\\, T}$, $\\mu_{r,0}=\\frac{\\sigma_{r,0}}{\\overline{r}_{0}}$, \n$\\tilde{m}_{0} = \\frac{m_{0}} {\\rho \\, \\overline{r}_{0}^{3}}$,\n$z=\\frac{\\tilde{Q}}{m_{0}\\,\\overline{r}_{0}^{2} \\, \\lambda^{2}}$ \n for the dimensionless variable combinations, \n the mass-transfer equation is expressed as follows: \n\n\\begin{equation}\n\\label{mass-trans2}\n\\partial_{\\tilde{t}}\\, \\tilde{m}(\\tilde{\\bf x},\\tilde{t}) \n =    \n\\sum_{\\tilde{\\bf x'}} \ne^{  - u  \\, \\tilde{m}(\\tilde{\\bf x'},\\tilde{t})   }\n \\, \\tilde{m}(\\tilde{\\bf x'},\\tilde{t}) \n  -   \\sum_{\\tilde{\\bf x'}}  \\, \ne^{  - u  \\, \\tilde{m}(\\tilde{\\bf x},\\tilde{t}) }\n \\, \\tilde{m}(\\tilde{\\bf x},\\tilde{t}) .\n\\end{equation}\nNote that the dependence on the scaled variables $\\mu_{r,0}$, \n$\\tilde{m}_{0}$ and $z$\nis not explicit in Eq.~(\\ref{mass-trans2}), since these variables \ninvolve the initial conditions.  \n\n\\section{Simulations Results and Discussion}\n\\label{sim-res}\n\nEquation~(\\ref{mass-trans2}) can be solved numerically using a forward \nfinite-difference discretization of the time derivative. \nThis leads to the following updating scheme: \n\n\\begin{equation}\n\\label{mass-disc}\n\\tilde{m}(\\tilde{\\bf x},\\tilde{t}_{k+1}) =   \\tilde{m}(\\tilde{\\bf x},\\tilde{t}_{k})\n+ \\delta t_{k} \\,\n\\sum_{\\tilde{\\bf x'}} \\, \\left[ \n\\tilde{W} (\\tilde{\\bf x'} \\rightarrow \\tilde{\\bf x}, \\tilde{t}_{k})\n\\, \\tilde{m}(\\tilde{\\bf x'},\\tilde{t}_{k})\n -   \n\\tilde{W} (\\tilde{\\bf x} \\rightarrow \\tilde{\\bf x'}, \\tilde{t}_{k}) \\,\n\\tilde{m}(\\tilde{\\bf x},\\tilde{t}_{k}) \\right],\n\\end{equation}\n\n\\noindent\nwhere the transition rate \n$\\tilde{W} (\\tilde{\\bf x'} \\rightarrow \\tilde{\\bf x}, \\tilde{t}_{k})$\nis given by \n$ \\tilde{W} (\\tilde{\\bf x} \\rightarrow \\tilde{\\bf x'}, \\tilde{t}_{k})=\n\\exp \\left[ - u \\, \\tilde{m}(\\tilde{\\bf x'},\\tilde{t}_{k}) \\right] $.\nIf $W_{\\max,k} = \\max_{{\\bf x},{\\bf x'}} \\{ \\tilde{W} (\\tilde{\\bf x} \\rightarrow \\tilde{\\bf x'}, \\tilde{t}_{k}), \n\\tilde{W} (\\tilde{\\bf x'} \\rightarrow \\tilde{\\bf x}, \\tilde{t}_{k}) \\}$ is \nthe maximum transition rate at time $t_{k}$,  \nthe time increment $\\delta t_{k}$ should satisfy the condition \n$\\delta t_{k} \\, \\tilde{W}_{\\max, k} <<1$.  \nA very small time step would slow down the\nevolution. Since the transition rates\nchange dynamically during the process, \nthe time step is adaptively updated. \n\n\nWe solve Eq.~(\\ref{mass-disc}) for a $1d$ chain of \n$N=1000$ grains with \nperiodic boundary conditions. The initial grain distribution \nis assumed to be Gaussian with $\\overline{r}_{0}=10$ in \narbitrary units and coefficient of variation \n$\\mu_{r,0}=0.2$.  \nThe reference mass $m_{0}$ is taken equal to \nthe mean of the initial mass distribution. \nFirst, we consider the case of low activation energy\nusing a dimensionless activation parameter $u=0.1$. \nThe initial grain size distribution and the \nevolved distribution after 100000 steps \nare shown in Fig.~(\\ref{Figure1}).\nIn the evolved state the radius of all the grains is\napproximately equal to the initial mean radius. \nIn this case, the low activation energy leads to a diffusive\nbehaviour that asymptotically drives the system toward a\nuniform steady state.\n\nNext, we consider the case of high activation energy. \nThe grain size distribution\nfor $u=15$ is shown in Fig.~(\\ref{Figure2}). \nHere, the larger grains grow at the expense\nof the smaller ones. \nThe grain radius distribution develops a bimodal \nstructure that includes \na fraction of very small grains.\nThe evolution of the grain radius \ncoefficient of variation $\\mu_{r}(t)$, skewness coefficient,\n$s_{r}(t)$ and the \nmaximum radius, $r_{max}(t)$ versus time are shown in \nFig.~(\\ref{Figure3}). The time is calculated based \non $t_{k}=\\sum_{i=0}^{k-1} \\delta t_{i}$.\nThe coefficient of variation $\\mu_{r}(t)$ \nincreases in magnitude with time as a result of the evolving \nasymmetry of the distribution. \nThe skewness coefficient $s_{r}(t)$ also develops a non-zero value as \nthe distribution evolves away from the Gaussian. \nThe $r_{max}(t)$ also increases \nalbeit slowly. \nThe plots shown in Fig.~(\\ref{Figure3}) seem to indicate a\nsmooth approach to a time-independent steady state as the time increases.\nAlso note that if the evolution of the moments were plotted versus \nthe number of steps (instead of the time $t_{k}$, the plots \nwould exhibit discontinuities due to the fact that certain steps may \ncorrespond to significantly larger time increments than others.  \nThe spatial configuration of the grains is \nillustrated in Fig.~(\\ref{Figure4}) by plotting the initial and final \n(after 100000 steps) radius of the first 30 grains: \nThe smaller grains tend to shrink, while the larger \ngrains grow at a considerably slower rate.\nThe asymmetry is due to mass conservation, i.e., \nthe fact that an increase of the grain radius by $\\delta r$\nincreases the mass by an amount \nthat exceeds the respective reduction in mass due to a \ndecrease $\\delta r$ of the radius. A number of grains, the initial \nsize of which is close to the mean radius, do not change appreciably\ntheir size.\n\nWe note that the model presented here does not \ninclude grain coalescence. This can be addressed by solving \nEquation~(\\ref{mass-disc}) \niteratively, with the first iteration ending when the radius of \none (or more) of the smaller grains drops below a certain \nthreshold; at this point the smaller grains would coalesce with \nthe largest nearest neighbours.  The number of grains would thus \nbe reduced, and the resulting radius distribution would be used \nas an initial condition for the next iteration of Eq.~(\\ref{mass-disc}). \nThis mechanism leads to a coarse-graining of the \ngrain radius distribution.  \nThe renormalization group approach may be a suitable \nframework for investigating the asymptotic grain size distribution \nunder this coarse-graining procedure. \n \n\\begin{figure*}\n\\includegraphics[width=0.5\\textwidth]{Figure1.ps\n\\caption{\\label{Figure1} Grain size distribution at \n$t=0$ (top) and $t=100000$ (bottom) for $u=0.1$ and $\\mu_{r,0}=0.2$.}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.5\\textwidth]{Figure2.ps\n\\caption{\\label{Figure2} Grain size distribution \nfor $u=15$ and $\\mu_{r,0}=0.2$: initial \n(top) and after 100000 steps (bottom).}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.5\\textwidth]{Figure3.eps\n\\caption{\\label{Figure3} Coefficient of variation\n (top), skewness coefficient (middle), and \n maximum radius (bottom) of the grain size distribution versus \n simulation time for $u=15$ and $\\mu_{r,0}=0.2$.}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.5\\textwidth]{Figure4.eps\n\\caption{\\label{Figure4} Initial (*) and final \n(o) grain sizes for the first 100 \nsites; $u=15$ and $\\mu_{r,0}=0.2$.}\n\\end{figure*}\n\n\n\\section{Conclusions}\n\nWe presented a discrete, nonlinear equation for grain growth \nin sintered grain aggregates, using transition rates that depend on the \ndegree of amorphization of each grain. \nThe model was solved numerically for\na  chain of grains with periodic boundary conditions.\nThe activation energy was shown to be a \ncrucial factor for the asymptotic grain size distribution, \nsince it leads to a transition from \na bimodal steady state to a uniform one \n(diffusion regime). Various aspects of the \nmodel require further \nstudy, including the existence of a well-defined \nthreshold between the two regimes, \n the impact of governing \nparameters on the grain size evolution, the roles\nof the initial grain size distribution and the \ninitial grain configuration.  \nFinally, grain coalescence needs to \nbe incorporated in the model (in the spirit \ndiscussed in the previous section) in order to observe \nrealistic grain growth.  \n   \n\n\n\n\\section*{Acknowledgments}\nResearch supported by the\nEC grant `Super High Energy Milling in the Production of Hard Alloys, \nCeramic and Composite Materials' (Activation FP6).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{1. INTRODUCTION}\nThe Crab Nebula has become a standard candle in TeV astronomy; it has\nbeen detected by many groups and its integral flux appears constant.\n(We refer to the ``steady'' emission, not emission modulated at the\npulsar period.  We have not detected the latter at TeV energies.  See\npaper by G.~Gillanders et al., in these proceedings.) It is also well\non the way to becoming a standard candle with regard to TeV spectral\ncontent.  Synchrotron emission from the Crab covers a remarkably broad\nrange terminating at about 10$^8$ eV, where a new component attributed\nto inverse Compton scattering begins. It is this component that we\ndetect.  The TeV spectrum is sensitive to the primary electron\nspectrum, the nebular magnetic field and the spatial distribution of\nelectrons and magnetic field within the nebula.\n\nIn this paper we briefly describe two methods for extracting TeV\nspectra, compare results from the Whipple Observatory Imaging\nCherenkov Telescope for the 1988\/89 and 1995\/96 observing seasons and\ncomment on implications for the physics of the nebula.  The methods,\ndeveloped at Iowa State University and the University of Leeds, are\nbased on Monte Carlo simulations using completely independent code and\nuse different approaches in determining the overall gain of the\nCherenkov telescope.  In the ISU approach, the gains of the\nphotomultiplier tubes, mirror reflectivities, etc., are measured and\ncombined to find the overall gain.  In the Leeds approach, the\nobserved brightness distribution of cosmic-ray images is combined with\ncosmic-ray simulation results to determine the overall gain of the\ntelescope.  The methods are described in detail in ``Paper I,''\nMohanty et al., (1997), and the resulting TeV spectrum is put into\ncontext of other observations with implications for the physics of the\nnebula in ``Paper II,'' Hillas et al., (1997).  The latter paper also\ncompares our Crab TeV spectrum with those of other groups.\n\n\\vfill\\eject\n\\section{2. METHOD 1: A TRADITIONAL APPROACH}\nA straightforward approach to the determination of TeV spectra was\ndeveloped at Iowa State University.  Three components are required.\nFirst, a method of distinguishing gamma-ray images from background\ncosmic-ray images.  The standard method is to use ``supercuts'' as\ndescribed in, e.g., Punch et al. (1996).  The images are characterized\nby second order moments giving the {\\it width}, {\\it length}, {\\it\ndistance} of the image centroid from the optic axis and {\\it alpha},\nthe angle by which the image major axis misses passing through the\noptic axis.  More that 99\\% of the background can be rejected by\nrequiring small values of {\\it width}, {\\it length} and {\\it alpha}.\nHowever, this procedure results in a strong bias against the images of\nhigher energy gamma rays which tend to be longer and broader and hence\nmore cosmic-ray like.  Mohanty (1995) has modified the procedure so\nthat the image selection criteria depend on the total brightness or\n{\\it size} of the image as well.  (The {\\it size} can be used as an\nestimate of the energy.)  The telescope collection areas for simulated\ngamma rays for the 1995\/96 season using standard and ``extended''\nsupercuts is shown in Fig.~1.\n\n\\begin{wrapfigure}[19]{r}{9.5cm}\n\\epsfig{file=area.eps,bbllx=43truept,bblly=51truept,bburx=260truept,bbury=344truept,width=7.0cm,height=7.0cm,angle=0}\n\\caption{\\it The collection areas for standard and extended supercuts\nfor the 1995\/96 observing season are shown above.  The resolution\nfunction is also plotted.}\n\\end{wrapfigure} \n\nThe second component needed is a way to estimate the energy of each\ngamma-ray image.  Two desirable criteria are (a) good resolution and\n(b) negligible bias.  The former is important to detect small\nstructures in the spectrum and the latter is important to avoid\ndistortions.  We obtained a resolution of $\\Delta E\/E \\sim 0.36$ with\nnegligible bias by using a second order polynomial in {\\it size} and\n{\\it distance} as described in Paper I.  The energy resolution\nfunction is, to a good approximation, Gaussian in the variable\nlog($E$).  It is plotted at an arbitrary energy in Fig.~1.\n\nIn data taken for spectral analysis, each on-source observation is\nfollowed by an ``off-source'' observation covering the same range of\nrange of azimuth and elevation angles.  The images for both types of\nobservation are selected for gamma-like events.  The estimated\nenergies from corresponding observations are histogrammed and the\ndifference ascribed to gamma rays from the source.  This can then be\nfit by a power law or the fluxes extracted as described in Paper I.\n\n\\section{3. METHOD 2: AN INTUITIVE APPROACH}\nA different approach with emphasis on verifiability was developed at\nthe University of Leeds.  There are two pieces to this approach, a\nmethod for selecting images likely to have been initiated by cosmic\ngamma rays and a method for determining the primary gamma-ray energy\nspectrum from the observed {\\it size} spectra.  Earlier descriptions\nof this approach are in (Hillas and West, 1991) and (West, 1994).\n\nThe selection criterion is a ``cluster'' or ``spherical'' method in\nwhich a single parameter is used to characterize the gamma-ray-like\nnature of an image and correlations between image parameters are\nincorporated naturally.  Simulated gamma rays produce images with four\nparameters ({\\it width}, {\\it length}, {\\it distance} and {\\it alpha})\nthat populate a four-dimensional space. Each real image can be tested\nto see if it is likely to be a gamma-ray image by the value of the\nMahalonobis distance between it and the centroid of the cluster.\n(This is equivalent to scaling and rotation of the axes so that the\nwindow is spherical and fluctuations uncorrelated.)  The selection\nwindow is defined by the expected position and dimensions of the\ngamma-ray parameter cluster for several broad ranges of image {\\it\nsize.}\n\nIn order to extract a spectrum, a {\\it size} histogram is then\ncomputed for on-source and off-source observations and the difference\nhistogram is ascribed to gamma rays.  A simulated {\\it size} spectrum\nis then computed starting with a power-law primary energy spectrum for\ngamma rays.  A weight is given to each simulated gamma ray and, by\nadjusting these weights, the spectrum can be varied so that its {\\it\nsize} spectrum matches that of the difference histogram.  This method\nis simple, easy to implement and avoids the complexity of calculating\ncollection areas and bias-free energy estimates.  It is described in\ndetail in Paper I.\n\n\\section{4. RESULTS}\n\\begin{wrapfigure}[23]{r}{9.5cm}\n\\epsfig{file=spec.ps,width=7.5cm,height=7.0cm,bbllx=0truept,bblly=30truept,bburx=420truept,bbury=600truept,angle=0}\n\\caption{\\it The Crab spectrum in the range 0.3 to about 8 TeV\nextracted using Methods 1 (open circles) and 2 (solid circles) for the\nWhipple 1988\/89 database and using Method 1 (x's) for the 1995\/96\ndatabase are shown above.  Also shown is a fit to the combined Method\n1 results (solid line) as well as an earlier spectrum (dashed line)\ntaken from Vacanti et al. (1991).}\n\\end{wrapfigure} \nThe spectra obtained from the 1988\/89 and 1995\/96 seasons using Method\n1 and 1988\/89 season using Method 2 are in good agreement as shown in\nFig.~2.  As described in Paper I, we have tested the sensitivity of\nthe results to uncertainties in the Monte Carlo simulations and have\nfound that the results are relatively robust.  The combined TeV fluxes\nfrom both seasons from Method 1 are well fit with the simple power law\nspectrum in which the differential flux ($J(E)$) is given by:\n\\begin{equation}\n(3.3 \\pm .2 \\pm .7)\\,10^{-7}\\,(\\frac{E}{\\hbox{TeV}})^{-2.45\\pm.08\\pm.05}\n\\end{equation}\nin units of $\\hbox{m}^{-2}\\,\\hbox{s}^{-1}\\,\\hbox{TeV}^{-1}$ where the\nfirst errors are statistical and the second are our estimate of\nsystematic errors.\n\nHowever, the simple extrapolation of this fit to lower energies yields\nfluxes far in excess of those observed by EGRET as is clear from\nFig.~3 which shows a quadratic fit of log($J$) vs.\\ log($E$) to our\ndata and to an averaged point representing the EGRET flux (Nolan et\nal., 1993) at 2 GeV.  This fit may be written:\n\\begin{equation}\nJ(E) = (3.25)\\,10^{-7}\\,(E\/\\hbox{TeV})^{-2.44-0.135\\hbox{log}_{10}(E)}\n\\hbox{m}^{-2}\\,\\hbox{s}^{-1}\\,\\hbox{TeV}^{-1}.\n\\end{equation}\n\n\\section{5. COMMENTS ON INTERPRETATION}\nMost of the early inverse Compton models for TeV gamma rays assumed a\nconstant magnetic field in the principal source region where these are\nproduced (e.g., Gould 1965 or Rieke and Weekes 1969) whereas more\nrecent models (De Jager and Harding, 1992 or Aharonian and Atoyan)\nincorporate hydrodynamic plasma\/field flow making the calculations\nmore complex and the results probably more realistic.  Here, we try to\nstay close to the data and make the simpler assumption that the field\nis constant.  The broad synchrotron emission band apparently extends\nup to 10$^{8}$ eV and is boosted to higher energies via inverse\nCompton scattering.  The scattering giving rise to TeV gamma rays\noccurs in the Klein-Nishina rather than in the Thomson scattering\nregime.  This implies that the electrons giving rise to our detected\ngamma rays must have energies in the range of 2-10 TeV and the\ncorresponding scattered photons would mostly have energies of about\n0.005 to 0.3 eV.  This conclusion is only very weakly model dependent\n(see Paper II).  Since electrons with energies of a few TeV generate\nsynchrotron radiation at about 0.4 keV in a field of about 25 nT (see\nnext paragraph), the Einstein Observatory X-ray pictures of Harnden\nand Seward (1984) also show the part of the nebula emitting TeV gamma\nrays.\n\n\n\\begin{wrapfigure}[24]{r}{9.5cm}\n\\epsfig{file=amh.eps,width=7.5cm,height=7.0cm,bbllx=0truept,bblly=0truept,bburx=450truept,bbury=550truept,angle=0}\n\\caption{\\it Whipple and EGRET observations of the Crab gamma-ray\nspectrum.  The dotted line is a fit to the WHipple points and a single\n2 GeV flux value.  Full-line curves are predicted inverse Compton\nfluxes for 3 differenct assumed B fields.}\n\\end{wrapfigure} \n\nFor assumed magnetic field values and the observed synchrotron flux,\nit is possible to deduce the spectrum of primary electrons, presumably\ngenerated in the shock at the termination of the pulsar wind (see,\ne.g., Coroniti and Kennel 1985). From the ambient photon density and\ndeduced electron spectrum, the TeV flux can be calculated, and results\nfor B fields of 10, 20 and 40 nT are shown in Fig.~3.  The shaded\nregions reflect uncertainties arising from ill-defined UV to\nsoft-X-ray region of the synchrotron photon spectrum.  As can be seen\nfrom the figure, the effective B field must lie between 20 and 40 nT\nwith 27 nT falling very near to our TeV data.  Since even more\nenergetic electrons only keep their energy for a short time, they\nshould exist only near the pulsar wind shock.  Hence, measurement of\nthe TeV spectrum over a wider energy range may probe spatial\nvariations in the nebular magnetic field (see Paper II).\n\n\\section{6. ACKNOWLEDGEMENT}\nThis work is supported by grants from the US DOE and NASA, by PPARC\nin the UK and by Forbairt in Ireland. \n\n\\section{7. REFERENCES}\n\\setlength{\\parindent}{-5mm}\n\\begin{list}{}{\\topsep 0pt \\partopsep 0pt \\itemsep 0pt \\leftmargin 5mm\n\\parsep 0pt \\itemindent -5mm}\n\\vspace{-15pt}\n\\frenchspacing\n\n\\item Aharonian, F.A. and Atoyan A.M, Astropart. Phys. {\\bf 3}, 275\n(1995) \n\n\\item Atoyan, A.M. and Aharonian, F.A., MNRAS {\\bf 278} 525\n(1996). \n\n\\item Coroniti, F.V. and Kennel, C.F.,  in ``The Crab Nebula and\nrelated supernova remnants,'' ed. M.C.Kafatos and R.B.C. Henry, CUP, p\n25 (1985). \n\n\\item De Jager, O.C., and Harding, A.K., Ap. J., {\\bf 396}, 161 (1992). \n\n\\item Gould, R.J., Phys. Rev. Lett., {\\bf 15}, 577 (1965). \n\n\\item Harnden, F.R.,Jr. and Seward, F.D., Ap. J. {\\bf 283} 279 (1984). \n\n\\item Hillas, A.M. and West, M., Proc. 22nd ICRC Dublin {\\bf 6} 472 (1991).\n\n\\item Hillas, A.M., et al, ``Paper II,'' to be submitted to\nAp. J. (1997).\n\n\\item Mohanty, G., PhD Thesis Iowa State University (1995).\n\n\\item Mohanty, G., et al., ``Paper I,'' Astroparticle Physics, \nsubmitted (1997).\n\n\\item Nolan, P.L. et al., Ap. J., {\\bf 409}, 697 (1993). \n\n\\item Punch, M., et al., Proc. 22nd ICRC Dublin {\\bf 1} 464 (1991).\n\n\\item Rieke, G.H., and Weekes, T.C., Ap. J., {\\bf 155}. 429 (1969). \n\n\\item Vacanti, G., et al., Ap. J. {\\bf 377}, 467 (1991).\n\n\\item West, M., PhD Thesis University of Leeds (1994).\n\n\\end{list}\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany of the stationary and non-stationary properties of a trapped \nBose-Einstein condensate (BEC) (or a Bose superfluid) \ncan be satisfactorily explained  \nby the \nmean-field Gross-Pitaevskii (GP) equation \\cite{rev}.\nHowever, there is no such mean-field equation in configuration space for \nthe Fermi superfluid even \nafter the experimental realization\nof the crossover  from the \nBardeen-Cooper-Schrieffer\n(BCS) limit through unitarity to the Bose limit in a trapped Fermi \nsuperfluid \\cite{cr,rmpfermi,K,Li}.\n\nTo study the  properties of stationary \nand \nnon-stationary states of a trapped Fermi superfluid with equal number of \npaired spin-up and -down fermions, Adhikari and Salashnich developed a \nGalilei-invariant nonlinear density functional (DF)  equation \nvalid from \nthe weak-coupling \nBCS limit \nto unitarity \\cite{DFAS,DFAS2}.  \nThe solution of the DF equation of  \\cite{DFAS} \nyielded results for energy of a trapped Fermi superfluid in close \nagreement with those obtained from Monte Carlo calculations \\cite{MC} \nnot only in the BCS and unitarity limits but also along the \nBCS-unitarity crossover \\cite{cr}.  \n\n\nIn actual experiments, an axially symmetric \\cite{rev,rmpfermi,disk}, \nrather than a \nspherical,   trap is usually employed for the confinement of the \nBEC or \nFermi superfluid.   In many situations of actual interest, the trap \nhas an extreme geometry \\cite{disk}, e.g., very strong  radial or \naxial  \nconfinement. Consequently, the superfluid is formed in the shape of a \ncigar or a disk and the solution of the nonlinear \n equation in such cases \ndeserves special attention. The cigar-shaped BEC or Fermi superfluid  is \nquasi \none \ndimensional (1D) and the reduction of the full three-dimensional (3D)  \nequation to accurate 1D form would be of advantage. Similarly, it would \nbe beneficial to reduce the full 3D  equation to 2D form for the \ndescription of a disk-shaped BEC and Fermi superfluid. \n\nWe propose a simple scheme for the 3D-1D and 3D-2D reduction of the GP \nequation for a BEC and the DF equation for a Fermi superfluid at the BCS \nand unitarity limits.\nThe 3D-1D reduction for a cigar-shaped BEC or a Fermi superfluid  is \ndone in the\nadiabatic approximation which assumes the essential dynamics to be\nconfined in the axial direction with the radial degrees of freedom\nadjusting instantaneously to the minimum energy equilibrium\nconfiguration compatible with the axial dynamics \\cite{S1,M1}.  Under \nthis\napproximation the 3D wave function factorizes into an explicitly\ntime-dependent axial and time-independent radial parts which allows\nfor a formal reduction of the original time-dependent 3D nonlinear  \nequation\ninto a time-dependent 1D and an auxiliary time-independent 2D equations.\nThe same is also true for a disk-shaped BEC or \na Fermi \nsuperfluid\nwith the role of time-dependent 1D and auxiliary time-independent 2D\nequations interchanged, e.g., the time-dependent equation is now 2D and\nthe auxiliary equation 1D in nature. \n\n\n\nFirst we illustrate the present scheme through an application to a \nBEC described by the \nGP \nequation, where there are already several \nschemes \\cite{S1,M1,S2,M2,O1,O2,O3,O4,O5,06} for 3D-1D and 3D-2D \nreductions.    Among  these, the reduction\nschemes\nof Salasnich {\\it et al.} (SPR) \\cite{S1,S2} and\nMu\\~noz Mateo {\\it et\nal.} (MMD)\n\\cite{M1,M2} are the simplest to implement and\nhave been shown to be the most efficient \\cite{M1}. \nWe compare  the present scheme with  those of SPR \n\\cite{S1,S2} and\nMMD  \n\\cite{M1,M2} for a BEC. \nRecently, Adhikari and Salasnich (AS) \\cite{njp} suggested one scheme\nfor such 3D-1D and 3D-2D reductions of the 3D DF equation for a Fermi\nsuperfluid at the BCS and unitarity limits. \n{We apply our scheme to a Fermi superfluid and obtain \ndistinct 3D-1D and 3D-2D reductions}.\n\n\n\nIn section \\ref{II} we illustrate the 3D-1D and 3D-2D  reduction schemes in \nthe \ncase of the GP equation for a BEC. \nIn section \\ref{III} we consider the \nreduction \nschemes for the nonlinear DF equation of a Fermi superfluid at \nunitarity and BCS limits. In this case \nwe reduce the 3D DF equation to 1D (for cigar-shaped superfluid) and 2D\n(for disk-shaped superfluid) \nforms with analytic nonlinear terms. \nWe  show that the reduced 1D and 2D \nequations for cigar- and \ndisk-shaped Fermi superfluids are very effective for describing the\nstationary states and  \nnon-stationary breathing oscillation and free expansion \nof the Fermi superfluid at unitarity in close \nagreement with the solution of the full 3D DF equation. \nFinally, in \nsection 4 we give a summary and some concluding remarks. \n\n\\section{3D-1D and 3D-2D Reductions of the GP equation}\n\n\\label{II}\n\n\nWe first apply our approach to a \nBEC described by the  \nGP equation.\nThe GP equation for $N$ bosons of mass $m$ is written as \n\\cite{rev}\n\\begin{equation}\\label{e1}\n\\left[-i\\hbar\\frac{\\partial }{\\partial \\hat t}\n-\\frac{\\hbar^2\\nabla_{\\bf \\hat r}^2}{2m}\n+V({\\bf \\hat r})+  \\frac{4\\pi \\hbar^2 \\hat a N}{m}|\\Psi({\\bf r})|^{2}\n\\right]\\Psi({\\bf \\hat   r},t)=0,\n\\end{equation}\nwith  $\\hat a$ the atomic scattering length, \nnormalization $\\int |\\Psi|^2 d^3{\\bf \\hat r}=1$,  \n$V({\\bf \\hat r})= m\\omega^2(\\hat \\rho^2+\\lambda^2 \n\\hat z^2)\/2$ \nthe harmonic trapping potential with frequencies $\\omega$ and $\\lambda \n\\omega$ in radial ($\\hat \\rho$) and axial ($\\hat z$) directions $({\\bf \n\\hat r}\\equiv \n\\hat \\rho,\\hat z).$  Employing \ndimensionless harmonic oscillator units $t=\\hat t \\omega, \n{\\bf r= \\hat r}\/a_\\rho, \nz=\\hat z\/a_\\rho, \\rho =\\hat \\rho\/a_\\rho, a=\\hat \na\/a_\\rho, \n\\psi  =\\Psi \na_\\rho^{3\/2}, a_\\rho=\\sqrt{ \\hbar\/m\\omega}$,    (\\ref{e1}) can be \nwritten as\n\\begin{equation}\\label{e2}\n\\left[-i\\frac{\\partial }{\\partial  t}\n-\\frac{\\nabla_{\\bf r}^2}{2}+\\frac{\\rho^2+\\lambda^2 z^2}{2}\n+4\\pi a N|\\psi|^{2}\n\\right]\\psi({\\bf \\bf r},t)=0,\n\\end{equation}\nwith normalization $\\int |\\psi|^2 d^3{\\bf  \nr}=1.$\n\n\nIn the limits of very small (cigar-shaped trap) and very large \n(disk-shaped trap) $\\lambda$,  the length scales in the axial \nand radial\ndirections are very different. Consequently,  the correlations between \nthese two\ndirections can be neglected and the condensate wave function \ncould be factorized in variables $\\rho$ and $z$ \\cite{S1,M1}. In \nthe \ncase of cigar-shaped traps  the dynamics takes place in the axial \ndirection. The opposite happens in case of a disk-shaped \ntrap.   \nFor a stationary solution $ \\psi({\\bf r})$ of  (\\ref{e2}) in a \ncigar-shaped trap one can define a linear \ndensity $\\phi^2(z)\\equiv \\int d^2 \\rho\\psi^2({\\bf r})$. Similarly, \nfor a disk-shaped trap  one can define a radial density \n$\\varphi^2(\\rho)\\equiv \\int d z\\psi^2({\\bf r})$. \n\n\\subsection{3D-1D reduction for a cigar-shaped BEC}\n\n\nFor a cigar-shaped trap, the above consideration leads to the \nfactorization \\cite{M1}\n\\begin{equation}\\label{e3}\n\\psi({\\bf r},t)=\\varphi(\\rho,n_1(z,t))\\phi(z,t),\n\\end{equation}\nwhere linear density $n_1$ is defined as $n_1(z,t)\\equiv$ $ \nN|\\phi(z,t)|^2 = N\\int d^2 \n\\rho \n|\\psi|^2$ and normalizations $\\int d^2\\rho |\\varphi(\\rho,n_1)|^2= \\int \ndz \n|\\phi(z,t)|^2=1$. We {\\it assume} \nthat the function $\\varphi(\\rho,n_1(z,t))$ has no explicit $t$ or $z$ \ndependence and hence these derivatives do not act on this function.\nThis is a good approximation, in general, as we find in numerical studies.  \nThe substitution of  (\\ref{e3}) in  (\\ref{e2}) then \nleads to\n\\begin{eqnarray}\\label{e4}\n\\fl \\varphi(\\rho,n_1)\\left[i\\frac{\\partial }{\\partial t}+\\frac{1}{2}\n\\frac {\\partial^2 }{\\partial z^2}-\\frac{1}{2}\\lambda^2z^2\n\\right]\\phi(z,t)\n{\n=\\phi(z,t)\\left[-\\frac{1}{2}\\nabla_\\rho^2+\\frac{1}{2}\n{\\rho^2}+ \n4\\pi a  n_1|\\varphi|^{2}\\right]\\varphi(\\rho,n_1).}\n\\end{eqnarray}\nMultiplying  (\\ref{e4}) by $\\varphi^*(\\rho,n_1)$ and integrating in \n$\\rho$ this equation can be rewritten as  \\cite{M1}\n\\begin{eqnarray}\\label{e5}\n&&\\left[i\\frac{\\partial }{\\partial t}+\\frac{1}{2}\n \\frac{\\partial^2 }{\\partial z^2}-\\frac{1}{2}\n\\lambda^2z^2-\\mu_\\rho(n_1)\n\\right]\\phi(z,t)=0,\\\\ \n&&{\\mu_\\rho(n_1)=\\int d^2\\rho\\varphi^*\\left[  \n-\\frac{1}{2}\\nabla_\\rho^2+\\frac{1}{2}{\\rho^2}+\n4\\pi a n_1 |\\varphi|^2\\right]\\varphi,}\n\\label{e7}  \n\\end{eqnarray}\nwhere  $\\mu_\\rho(n_1)$ is the chemical \npotential emerging from the 2D GP equation:\n\\begin{eqnarray}\n{\\left[-\\frac{1}{2}\\nabla_\\rho^2+\\frac{1}{2}{\\rho^2}+\n4\\pi a n_1  \n|\\varphi|^{2}- \\mu_\\rho(n_1)\n\\right]\\varphi(\\rho,n_1)=0.} \n \\label{e6}\n\\end{eqnarray}\nWe have decoupled the essential axial \n($z$) \nand nonessential radial ($\\rho$)\ndegrees of freedom.  \nThe solution of the time-independent radial GP equation (\\ref{e6}) \nleads to the chemical potential $\\mu_\\rho(n_1)$ given by  \n(\\ref{e7}), which is the nonlinear term of the axial GP equation \n(\\ref{e5}).  \n\n\nThe form of the chemical potential $\\mu_\\rho(n_1)$ of  \n(\\ref{e6}) is known in the small and large $N$ limits. In the \nsmall $N$ weak-coupling limit, the wave function can be  approximated by \nthe following normalized  \nGaussian form \\cite{PG}\n\\begin{equation}\n\\varphi(\\rho,n_1)=\\exp[-\\rho^2\/(2\\alpha^2)]\n\/(\\sqrt{\\alpha^2 \n\\pi}), \\label{g2D}\n\\end{equation}\nwhere $\\alpha$ is the width. \nWith this wave form the chemical \npotential of  (\\ref{e7}) \nbecomes \n\\begin{eqnarray}\\label{e10}\n{\\mu_\\rho(n_1)= \n\\biggr(\\frac{\\alpha^2}{2}+\\frac{1}{2\\alpha^2}\\biggr)+\\frac{ 2a \nn_1}{\\alpha^2}\n.}\n\\end{eqnarray}\nIn the large $N$ Thomas-Fermi (TF) limit, as \n$an_1 \\to  \\infty ,$ the kinetic \nenergy \ngradient \noperator \nin  (\\ref{e6}) can be neglected and this equation has analytic \nsolution. The normalization condition of the TF wave function \nleads to  \\cite{S1,M1}\n\\begin{eqnarray}\\label{e11}\n{\\mu_\\rho(n_1)=\\sqrt{4an_1}}.\n\\end{eqnarray}\n\nWe suggest  the following simple interpolation formula for \n$\\mu_\\rho(n_1)$ valid for small to large  $an_1$ \nincorporating the limiting values (\\ref{e10})\nand (\\ref{e11})\n\\begin{eqnarray}\\label{e12}\n{\\mu_\\rho(n_1)=\\frac{1}{2\\alpha^2}-\n\\frac{\\alpha^2}{2}+\\sqrt{\\alpha^4+4an_1}}\n\\end{eqnarray}\nto be used in  (\\ref{e5}), \nwhere $\\alpha$ is taken as  a fixed constant for all $n_1$.\nIn the weak-coupling $an_1 \\to 0$ limit,  (\\ref{e6}) reduces to the \nSchr\\\"odinger equation for a 2D harmonic oscillator with the exact \nsolution (\\ref{g2D}) with $\\alpha = 1$. \nThen  (\\ref{e12}) is a good \napproximation to  (\\ref{e7}) for $\\alpha=1$. \nFor slightly larger \nvalues of $an_1$  (\\ref{e12}) continues to be a good approximation to \n   (\\ref{e7}), but with a slightly smaller value of \n$\\alpha$. Motivated by this, we take a slightly smaller value of \n$\\alpha$ in  (\\ref{e12}). For  large  $an_1$, \n (\\ref{e12}) has the correct TF limit independent of the value of \n$\\alpha$ employed.     \nThis\nflexibility  in introducing a width $\\alpha\\approx 1$ \n(slightly different from $\\alpha = 1$) \nin   (\\ref{e12}) will  be fundamental in making the 1D model \nequation (\\ref{e5}) \na faithful approximation to the 3D GP equation (\\ref{e2}) for a \ncigar-shaped condensate for \nall  $an_1$.\n\n\nBy construction, approximation (\\ref{e12}) satisfies the weak-coupling \nand \nTF limits (\\ref{e10}) and (\\ref{e11}), respectively, for small and large \n$an_1$. \nThe approximation of  MMD is \\cite{M1} \n\\begin{eqnarray}\\label{e8}\n\\mu_\\rho(n_1)=\\sqrt{1+4an_1},\n\\end{eqnarray}\nwhereas SPR suggested \\cite{S1}\n\\begin{eqnarray}\\label{e9}\n\\mu_\\rho(n_1)=\\frac{1+3an_1}{\\sqrt{1+2an_1}}.\n\\end{eqnarray}\n\n\n\n\nHere we calculate the chemical potential \n$\\mu_\\rho(n_1)$ of the three approaches and compare  with the \nprecise results for $\\mu_\\rho(n_1)$ obtained from a \nnumerical solution of  (\\ref{e6}). \n(All numerical results presented in this paper are obtained with the \nimaginary-time propagation scheme after a discretization by the \nCrank-Nicholson method using the FORTRAN programs provided in \\cite{bo}, \nthe details of which are described there. \nThe numerical simulations for the dynamical breathing \noscillation and free expansion for a Fermi superfluid at unitarity \nreported \nin Secs. IIIC and IIID, respectively,  were performed with the real-time \npropagation scheme after a \ndiscretization by the Crank-Nicholson method.)\nOur finding is exhibited in table \n\\ref{btable1} for different  $an_1$ and\nfor $\\alpha=0.985$ together with \nthose obtained from the MMD \\cite{M1} and SPR \\cite{S1} schemes.\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{Chemical potential $\\mu_\\rho(n_1)$ of  (\\ref{e7}) for\ndifferent\n$an_1$ obtained from an accurate numerical solution of  (\\ref{e6}),\nand from  (\\ref{e12})    (Present, $\\alpha=0.985$), (\\ref{e8})\n(MMD) \\cite{M1},  and\n(\\ref{e9})\n(SPR) \\cite{S1}.\n}\n\\label{btable1}\n\\begin{tabular}{|r|r|r|r|r|}\n\\hline\n$an_1$ & Numerical  & MMD   &\n SPR    &   Present   \\\\\n\\hline\n  0  &  1  & 1  & 1  &  1.00045\n\\\\\n0.2  &   1.346427      & 1.34164  &  1.35225 &  1.34983   \\\\\n  1  &  2.257135  & 2.23607  & 2.30940 & 2.25314 \\\\\n 10  &    6.432456  &  6.40312  &6.76475   & 6.42877 \\\\\n  100  & 20.04320  & 20.0250  &  21.2309 & 20.0538\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\subsection{3D-2D reduction for a disk-shaped BEC}\n\n\nFor a disk-shaped trap the adiabatic approximation  leads to \nthe \nfactorization \\cite{S1,M1}\n\\begin{equation}\\label{e13}\n\\psi({\\bf r},t)=\\varphi(\\rho,t)\\phi(z,n_2(\\rho,t)),\n\\end{equation}\nwhere  the surface density $n_2$ is defined as $n_2(\\rho,t)\\equiv \nN|\\varphi(\\rho,t)|^2 = N\\int dz \n|\\psi|^2$ and normalizations $\\int d^2\\rho |\\varphi(\\rho,t)|^2= \\int dz \n|\\phi(z,n_2)|^2=1$. The substitution of  (\\ref{e13}) in  \n(\\ref{e2})\nleads to\n\\begin{eqnarray}\\label{e14}\n\\fl \\phi(z,n_2)\\left[i\\frac{\\partial }{\\partial \nt}+\\frac{1}{2}\\nabla_\\rho^2\n-\\frac{1}{2}\\rho ^2\n\\right]\\varphi(\\rho,t)\n=\\varphi(\\rho,t)\\left[-\\frac{1}{2}\\frac{\\partial^2}{\\partial \nz^2}+\\frac{1}{2}\\lambda^2{z^2}+ 4\\pi a n_2\n|\\phi|^2\\right]\\phi(z,n_2).\n\\end{eqnarray}\nMultiplying  (\\ref{e14}) by $\\phi^*(z,n_2)$ and integrating in \n$z$, this equation can be rewritten as the set of equations \n\\begin{eqnarray}\\label{e15}\n\\left[i\\frac{\\partial }{\\partial t}+\n\\frac{1}{2}\\nabla_\\rho^2\n-\\frac{1}{2}\\rho^2-\\mu_z(n_2)\n\\right]\\varphi(\\rho,t)=0,\\\\\n \\label{e16}\n\\left[-\\frac{1}{2}\\frac {\\partial^2 }{\\partial z^2}\n+\\frac{\\lambda ^2z^2}{2}+4\\pi a\n n_2|\\phi|^{2}-\\mu_z(n_2)\n\\right]\\phi(z,n_2)=0,\\\\\n\\mu_z(n_2)=\\int dz\\phi^*\\left[  -\\frac{1}{2}\\frac {\\partial^2 }{\\partial \nz^2}\n+\\frac{\\lambda^2z^2}{2}+\n4\\pi a  n_2|\\phi|^{2}\\right]\\phi.\\label{e17}\n\\end{eqnarray}\nIt is convenient to introduce scaled variables $\\bar z=z\/a_z, \\bar\\phi \n={\\sqrt a_z}\\phi,$ and $\\bar \\mu_z=\\mu_z a_z^2$ with \n$a_z=\\sqrt{1\/\\lambda}$, when  (\\ref{e16}) and (\\ref{e17}) become \n\\begin{eqnarray}\\label{e18}\n&&\\left[-\\frac{1}{2}\\frac {\\partial^2 }{\\partial \\bar z^2}\n+\\frac{\\bar z^2}{2}+4\\pi a \n a_z n_2|\\bar \\phi|^{2}-\\bar \\mu_z(n_2)\n\\right]\\bar \\phi(z,n_2)=0,  \\\\&&\n\\bar \\mu_z(n_2)=\\int d\\bar z\\bar \\phi^*\\left[  -\\frac{1}{2}\\frac \n{\\partial^2 \n}{\\partial \\bar z^2}\n+\\frac{\\bar z^2}{2}+\n4\\pi a a_z n_2|\\bar \\phi|^{2}\\right]\\bar \\phi. \\label{e19}\n\\end{eqnarray}\n\n\n\nThe form of the chemical potential $\\bar \\mu_z(n_2)$ of  \n(\\ref{e18}) is known in the small and large $N$ limits. In \nthe \nsmall $N$ weak-coupling limit the wave function can be  approximated by \nthe following normalized \nGaussian form \\cite{PG}\n\\begin{equation}\n\\bar \\phi(\\bar z,n_2)=\\exp[-\\bar z^2\/(2\\eta^2)]\/(\\eta^2 \n\\pi)^{1\/4}, \\label{g1D}\n\\end{equation}\nwhere $\\eta$ is the width. \nWith this wave form the chemical \npotential of  \n(\\ref{e19}) \nbecomes \n\\begin{equation}\\label{e20}\n\\bar \\mu_z(n_2)= \n\\biggr(\\frac{\\eta^2}{4}+\\frac{1}{4\\eta^2}\\biggr)+\n2aa_zn_2\\frac{\\sqrt{2\\pi}}{\\eta}.\n\\end{equation}\nIn the large $N$ limit \nthe normalization condition of the TF wave function \nleads to \\cite{M1}\n\\begin{equation}\\label{e21}\n\\bar \\mu_z(n_2)=(3\\pi a a_z n_2\/\\sqrt 2 )^{2\/3}.\n\\end{equation}\n\nFor a disk-shaped BEC we suggest  the following simple interpolation \nformula for\n$\\bar \\mu_z(n_2)$ incorporating the limiting values (\\ref{e20})\nand (\\ref{e21})\n\\begin{equation}\\label{e22}\n\\bar \\mu_z(n_2)= \\frac{1}{4\\eta^2}-\n\\frac{(\\pi-1) \\eta^2}{4}+\n\\left[\n\\left(\n\\frac{\\pi\\eta^2}{4}\n\\right)^{3\/2}\n+\\frac{3\\pi a a_zn_2 }{\\sqrt 2}\\right]^{2\/3},\n\\end{equation}\nvalid for all $aa_zn_2$,\nwhere $\\eta$ is taken as a fixed constant for all \n$aa_zn_2$.\nThe flexibility in introducing a width $\\eta$ slightly different from \n1 in  (\\ref{e22}) will make the 2D model equation \n(\\ref{e15}) an accurate approximation to the 3D GP equation (\\ref{e2}) \nfor \na disk-shaped condensate for all $aa_z^2n_2$. \n\nInstead of taking $\\eta$ as a constant, SPR \\cite{S1} solve \n(\\ref{e18}) variationally \\cite{S1,PG}\nwith the Gaussian ansatz\n(\\ref{g1D}) and determine the width parameter $\\eta$ via\n\\begin{equation}\\label{e23}\n\\eta^4-2\\sqrt{2\\pi}\\eta aa_zn_2-1=0.\n\\end{equation}\nThe solution of the nonlinear\n (\\ref{e23}) when substituted into  (\\ref{e20})\nyields the desired $\\bar\\mu_z$ \\cite{S1} through a procedure far\ncomplicated than\nthe analytic formulae (\\ref{e22}).\n\nThe approximation scheme of MMD is quite involved but does not\nrequire the solution of a nonlinear variational equation like SPR. They\ncalculate $\\bar \\mu_z(n_2)$ via \\cite{M1}\n\\begin{equation}\\label{e26}\n\\bar \\mu_z(n_2)=\\frac{1}{8}[(\\eta+\\sqrt{\\eta^2-\\zeta^6 })^{1\/3}\n+(\\eta-\\sqrt{\\eta^2-\\zeta^6 })^{1\/3}-\\zeta]^2\n\\end{equation}\nwhere $\\eta=4+6\\zeta-\\zeta^3+24\\pi x$ and $\\zeta=(\\kappa-1)$\nwith\n\\begin{equation}\\label{theta}\n\\kappa^{-1}=\\sqrt{2\/\\pi}+\\Theta(x-0.1)(1-\\sqrt{2\/\\pi})[1-(10x)^{-1\/5}]\n\\end{equation}\nwhere $x\\equiv aa_z  n_2$, and the Heaviside step function \n$\\Theta(x-0.1)=0,$ for $x<0.1$, and \n$=1$ for $x>0.1$. It is realized that the function (\\ref{theta}) is not \nanalytic in $x$.  \n\n\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{Chemical potential $\\bar \\mu_z(n_2)$ of\n (\\ref{e19}) for different\n$aa_zn_2$ obtained from an accurate numerical solution of \n(\\ref{e18}),\nand from  (\\ref{e22})    (Present, $\\eta=0.95$), (\\ref{e26})  \n(MMD) \\cite{M1},\nand\n(\\ref{e23})\n(SPR) \\cite{S1}.\n}\n\\label{btable2}\n\\begin{tabular}{|r|r|r|r|r|}\n\\hline\n$aa_zn_2$ & Numerical  & MMD   &\n SPR    &   Present   \\\\\n\\hline\n  0  &  0.5  & 0.5  & 0.5  &    0.50263\n\\\\\n0.2  &    1.348783     & 1.31186  &  1.36149 &   1.34376   \\\\\n  1  &   3.599892   & 3.50165 & 3.69266   &   3.54355 \\\\\n  10  &    16.45405    & 16.2547     & 17.0012  & 16.3294 \\\\\n  100  &  76.30080      & 75.9963  &78.8855   & 76.1358\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\nNow we calculate the chemical potential \n$\\bar \\mu_z(n_2) $ of the three approaches and compare them with the \nprecise result for $\\bar \\mu_z(n_2) $  from a numerical solution \nof  (\\ref{e18}). \nThe results are shown in table \\ref{btable2} for \ndifferent  $aa_zn_2$ along with those from the MMD and SPR schemes. \n After a small experimentation \nthe constant  $\\eta$ in  (\\ref{e22}) was fixed at $\\eta=0.95$ for \nall $aa_zn_2$. \n\n\n\n\n\n\n\n\n\n\n\n\\section{3D-1D and 3D-2D reductions of Fermi superfluid DF equations}\n\n\n\n\n\\label{III}\n\n\n\nWe consider a Gallilei-invariant density-functional (DF) formulation for \na \ntrapped Fermi superfluid at BCS and unitarity limits described  by\n\\cite{DFAS,DFAS2}\n\\begin{equation}\\label{xe1}\n\\left[-i\\hbar\\frac{\\partial }{\\partial \\hat t}\n-\\frac{\\hbar^2\\nabla_{\\bf \\hat r}^2}{4m}\n+2V({\\bf \\hat r})+  2^{2\/3}\\chi \\frac{2\\hbar^2}{m}|\\Psi({\\bf \nr})|^{4\/3}\n\\right]\\Psi({\\bf \\hat   r},t)=0,\n\\end{equation}\nwith \n$\\chi=(3\\pi^2)^{2\/3}\\xi\/2$ ($\\xi=1$ at the BCS limit and $\\xi \n=0.44$ at unitarity \\cite{MCCOL}), \n$N$ the number of fermions, and \nnormalization $\\int |\\Psi|^2 d^3{\\bf \\hat r}=N\/2$ ($|\\Psi|^2$ is the \ndensity of Fermi pairs), \n$m$ the mass of an atom, \n$V({\\bf \\hat r})= m\\omega^2(\\hat \\rho^2+\\lambda^2 \n\\hat z^2)\/2$ \nthe harmonic trapping potential with frequencies $\\omega$ and $\\lambda \n\\omega$ in radial ($\\hat \\rho$) and axial ($\\hat z$) directions $({\\bf \n\\hat r}\\equiv \n\\hat \\rho,\\hat z).$  The fully-paired \nFermi superfluid is assumed to be composed of \nspin-half fermions with \nan equal number of spin-up and -down components. \nEmploying \ndimensionless units $t=\\hat t \\omega, \n{\\bf r= \\hat r}\/a_\\rho, \nz=\\hat z\/a_\\rho, \\rho =\\hat \\rho\/a_\\rho, a=\\hat a\/a_\\rho, \n\\psi \\sqrt{N\/2} =\\Psi \na_\\rho^{3\/2}, a_\\rho=\\sqrt{\\hbar\/m\\omega}$,    (\\ref{xe1}) can be \nwritten as\n\\begin{equation}\\label{xe2}\n\\left[-i\\frac{\\partial }{\\partial  t}\n-\\frac{\\nabla_{\\bf r}^2}{4}+{\\rho^2+\\lambda^2 z^2}\n+2\\chi N^{2\/3}|\\psi|^{4\/3}\n\\right]\\psi({\\bf \\bf r},t)=0,\n\\end{equation}\nwith $\\int |\\psi|^2 d^3{\\bf  r}=1 (|\\psi|^2$ is the density of Fermi \natoms). \n\n\n\n\\subsection{3D-1D reduction for a cigar-shaped Fermi superfluid}\n\n\nFor  a cigar-shaped trap, we consider  the \nfactorization (\\ref{e3}).\nWe substitute   (\\ref{e3}) in  (\\ref{xe2}) and \nmultiply the resultant equation  by $\\varphi^*(\\rho,n_1)$ and \nintegrate in \n$\\rho$ to get \n\\begin{eqnarray}\\label{xe5}\n&&\\left[i\\frac{\\partial }{\\partial t}+\\frac{1}{4}\n \\frac{\\partial^2 }{\\partial z^2}-\\lambda^2z^2-\\mu_\\rho(n_1)\n\\right]\\phi(z,t)=0,\n\\label{xe7}  \n\\end{eqnarray}\nwhere  $\\mu_\\rho(n_1)$ is  the chemical \npotential emerging from the following 2D DF equation\n\\begin{eqnarray}\n\\left[-\\frac{\\nabla_\\rho^2}{4}+{\\rho^2}+\n2\\chi \n n_1^{2\/3}|\\varphi|^{4\/3}- \\mu_\\rho(n_1)\n\\right]\\varphi(\\rho,n_1)=0. \n \\label{xe6}\n\\end{eqnarray}\n\n\nIn the \nsmall $N$ weak-coupling limit, the wave function $\\varphi(\\rho,n_1)$\ncan be  approximated by \nthe  normalized  \nGaussian form \\cite{PG}  (\\ref{g2D}).\nWith this  wave form \nthe chemical \npotential of  (\\ref{xe6}) \nbecomes \n\\begin{eqnarray}\\label{xe10}\n\\mu_\\rho(n_1)= \n\\biggr({\\alpha^2}+\\frac{1}{4\\alpha^2}\\biggr)+\n\\frac{6\\chi}{5}\n\\frac{n_1^{2\/3}}{\\alpha^{4\/3}\\pi^{2\/3}}.\n\\end{eqnarray}\n Chemical potential  $\\mu_\\rho(n_1)$ \nof  (\\ref{xe10}) is \nconsistent with  (4.7) of AS \\cite{njp}. \nIn the large $N$ TF limit  \nthe normalization condition of the TF wave function \nleads to \\cite{S1,M1}\n\\begin{eqnarray}\\label{xe11}\n\\mu_\\rho(n_1)=\\left[\n\\left(\\frac{5n_1}{2\\pi}\\right)^{2\/3}\n2\\chi \n\\right]^{3\/5}\\approx 1.38336 n_1^{2\/5}\\chi^{3\/5}.\n\\end{eqnarray}\nChemical potential (\\ref{xe11}) is approximately equal to the \nchemical potential  in the  \nlarge $N$ \nlimit of the corresponding model (4.10) of AS \\cite{njp} which \nyields in present notation \n\\begin{equation}\\label{xe11x}\n\\mu_\\rho(n_1)=\\frac{7}{5}\\frac{(6\\chi)^{3\/5}n_1^{2\/5}}\n{(5\\pi^2)^{1\/5}2^{2\/5}}\n\\approx 1.42545 n_1^{2\/5}\\chi^{3\/5}.\n\\end{equation}\nHere we use the following simple interpolation formula for \n$\\mu_\\rho(n_1)$ incorporating the limiting values (\\ref{xe10})\nand (\\ref{xe11})\n\\begin{eqnarray}\\label{xe12}\n\\mu_\\rho(n_1)=\n\\frac{1}{4\\alpha^2}-\\frac{3\\alpha^2}{2} \n +\n\\left[\\left( \\frac{5\\alpha^2}{2}   \\right)^{5\/3}\n+2\\chi \\left(\\frac{5n_1}{2\\pi}\\right)^{2\/3}\n\\right]^{3\/5}.\n\\end{eqnarray}\nwhere $\\alpha$ is taken as  a fixed constant for all $n_1$. \n\n\n\n\n\n\nAS \\cite{njp} solve Eq. (\\ref{xe6}) variationally and \nobtain for the width $\\alpha$\n\\begin{equation}\\label{xx10}\n\\alpha^4= \n\\frac{1}{4}+\\frac{12\\chi}{25}\\left(\\frac{n_1\\alpha}{\\pi}\\right)^{2\/3}.\n\\end{equation}\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{Chemical potential \n$\\mu_\\rho(n_1)$ of  (\\ref{xe7}) for \ndifferent \n$n_1$ for a Fermi superfluid at unitarity  \n($\\xi=0.44$)\nobtained from an accurate numerical solution of  (\\ref{xe6}), \nfrom  (\\ref{xe10})  and (\\ref{xx10}) (AS)\n\\cite{DFAS},\nand from  (\\ref{xe12})    (Present, $\\alpha=1\/\\sqrt2$, and $ \n0.98\/\\sqrt 2$). \n}\n\\label{table1}\n\\begin{tabular}{|r|r|r|r|r|}\n\\hline\n $n_1$ & Numerical    &\n AS    &   Present  &   Present  \\\\\n    &    &   & $\\alpha=1\/\\sqrt 2$   &  $\\alpha=0.98\/\\sqrt 2$  \\\\\n\\hline\n  0  &  1  & 1  & 1  &  1.00081 \\\\\n0.1  & 1.37401          &     1.37619 & 1.36784  &1.37667\\\\\n1  &    2.43893       &    2.46380  & 2.39751  &  2.41810\\\\\n 10 & 5.56376           &   5.69170 &  5.45891 & 5.49147 \\\\\n100  &13.70413           &    14.09704  & 13.5463  & 13.5867  \\\\\n1000  &34.29934           &   35.33109  & 34.1057  & 34.1506\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure\n\n\\begin{center}\n\\includegraphics[width=.49\\linewidth]{fig1.ps}\n\\includegraphics[width=.49\\linewidth]{fig2.ps}\n\\end{center}\n\n\\caption{(Color online) (a) The linear density $\\phi^2(z)$ of a Fermi \nsuperfluid at unitarity (with $\\xi =0.44$) vs. $z$\n(both in dimensionless units)\ncalculated from \nthe 3D DF  equation (\\ref{xe2}) and the 1D model  (\\ref{xe5}) and \n(\\ref{xe12}) for \n$\\lambda =1\/4, \\alpha =0.98\/\\sqrt 2$ and  $N=2$,  10, and  100.  \n(b) The radial density $\\varphi^2(\\rho)$ of a Fermi\nsuperfluid at unitarity (with $\\xi =0.44$) vs. $\\rho$\n(both in dimensionless units)\ncalculated \nfrom \nthe 3D DF  equation (\\ref{xe2}) and the 2D model  (\\ref{xe15}) \nand \n(\\ref{xe22}) for \n$\\lambda =4, \\eta =0.92\/\\sqrt 2$ and  $N=2$,  10, and  100.  \n}\\label{fig2}\n\\end{figure}\n\n\n\nHere we calculate the chemical potential \n$\\mu_\\rho(n_1)$ obtained from  (\\ref{xe12}) for different \n$\\alpha$ \nand from  (4.7) and (4.8) of  \\cite{njp}\n and compare  with the \nprecise result for $\\mu_\\rho(n_1)$  from a \nnumerical solution of  (\\ref{xe6}). \nOur finding is exhibited in table \n\\ref{table1} for different  $n_1$ together with those from the AS \n\\cite{njp} scheme. \nAfter a small \nexperimentation it was found that the best overall result from \n (\\ref{xe12}) was found for \n$\\alpha=0.98\/\\sqrt 2$.  \n\n\n\n\nNow to see how well the effective 1D equations (\\ref{xe5})\nand (\\ref{xe12}) reproduce the linear density $\\phi^2(z)$ of a \ncigar-shaped Fermi superfluid we plot in figure  \\ref{fig2} (a) the 1D density \ncalculated from  (\\ref{xe5})\nand (\\ref{xe12}) and the same calculated from the full 3D  \n(\\ref{xe2}) for $\\lambda =1\/4$ and $N=2,10,100.$ The excellent  \nagreement \nbetween the two sets of results for $\\lambda $ as large as 1\/4\ndemonstrates the usefulness of the \npresent 1D model equations.  \n\n\n\n\n\n\\subsection{3D-2D reduction for a disk-shaped Fermi superfluid}\n\n\n\nIn the case of a disk-shaped trap we consider  the \nfactorization (\\ref{e13}) and \nsubstitute  it  in  \n(\\ref{xe2}),\nmultiply the resultant equation  by $\\phi^*(z,n_2)$ and integrate \nin \n$z$ to get \n\\begin{equation}\\label{xe15}\n\\left[i\\frac{\\partial }{\\partial t}+\\frac{1}{4}\\nabla_\\rho^2\n-\\rho^2-\\mu_z(n_2)\n\\right]\\varphi(\\rho,t)=0,\n\\end{equation}\nwhere $\\mu_z(n_2)$ is the chemical potential emerging from the following \n1D DF equation\n\\begin{equation}\n \\label{xe16}\n\\left[-\\frac{1}{4}\\frac {\\partial^2 }{\\partial z^2}\n+{\\lambda ^2z^2}+2\\chi\n n_2^{2\/3}|\\phi|^{4\/3}-\\mu_z(n_2)\n\\right]\\phi(z,n_2)=0.\n\\end{equation}\nIt is convenient to introduce scaled variables $\\bar z=z\/a_z, \\bar\\phi \n={\\sqrt a_z}\\phi,$ and $\\bar \\mu_z=\\mu_z a_z^2$ with \n$a_z=\\sqrt{1\/\\lambda}$, when  (\\ref{xe16}) \nbecomes \n\\begin{eqnarray}\\label{xe18}\n&&\\left[-\\frac{1}{4}\\frac {\\partial^2 }{\\partial \\bar z^2}\n+{\\bar z^2}+2\\chi\n a_z^{4\/3} n_2^{2\/3}|\\bar \\phi|^{4\/3}-\\bar \\mu_z(n_2)\n\\right]\\bar \\phi(z,n_2)=0,  \\\\&&\n\\bar \\mu_z(n_2)=\\int d\\bar z\\bar \\phi^*\\left[  -\\frac{1}{4}\\frac \n{\\partial^2 \n}{\\partial \\bar z^2}\n+{\\bar z^2}+\n2\\chi a_z^{4\/3} n_2^{2\/3}|\\bar \\phi|^{4\/3}\\right]\\bar \\phi. \\label{xe19}\n\\end{eqnarray}\n\n\n\nThe form of the chemical potential $\\bar \\mu_z(n_2)$ of \n(\\ref{xe18}) is known in the small and large $N$ limits. In \nthe \nsmall $N$ weak-coupling limit the wave function $\\bar \\phi(\\bar z,n_2)$\ncan be  approximated by \nthe  normalized \nGaussian form \\cite{PG} (\\ref{g1D}).\nWith this wave form the chemical \npotential of  \n(\\ref{xe19}) \nbecomes \n\\begin{equation}\\label{xe20}\n\\bar \\mu_z(n_2)= \n\\biggr(\\frac{\\eta^2}{2}+\\frac{1}{8\\eta^2}\\biggr)+\n\\frac{2\\chi (n_2a_z^2)^{2\/3}}{\\eta^{2\/3}\n\\pi^{1\/3}}\\sqrt{3\\over 5}.\n\\end{equation}\nChemical potential  \n$\\bar \\mu_z(n_2)$ of  \n(\\ref{xe20}) is consistent with \n (5.7) of AS \\cite{njp}.  \nIn the large $N$ TF limit \nthe normalization condition of the TF wave function \nleads to \\cite{M1}\n\\begin{equation}\\label{xe21}\n\\bar \\mu_z(n_2)= \\left[\\frac{8\\chi    n_2^{2\/3}a_z^{4\/3}\n}{(3\\pi)^{2\/3}} \n\\right]^{3\/4}\\approx 1.54947 \\chi^{3\/4}a_z\\sqrt{n_2}.\n\\end{equation}\nChemical potential (\\ref{xe21}) is approximately equal to the\nchemical potential  in the\nlarge $N$\nlimit of the corresponding model (5.10)  of AS \\cite{njp} which\nyields in present notation\n\\begin{equation}\\label{xe21x}\n\\bar \\mu_z(n_2)=\\frac{12a_z 3^{3\/8}\\chi^{3\/4}\\sqrt n_2\n}{5\\sqrt 2  \\pi^{1\/4}5^{1\/8}}\n\\approx 1.57383 \\chi^{3\/4}a_z\\sqrt n_2.\n\\end{equation}\nHere we use the following  interpolation formula for \n$\\bar \\mu_z(n_2)$ incorporating the limiting values (\\ref{xe20})\nand (\\ref{xe21})\n\\begin{eqnarray}\\label{xe22}\n\\bar \\mu_z(n_2)= \\left(\\frac{\\eta^2}{2}+\\frac{1}{8\\eta^2} \n\\right)-\\frac{15^{3\/2}\\eta^2}{9\\pi}\n+\\left[ \n\\left(   \\frac{15^{3\/2}\\eta^2  }{9\\pi}  \\right)^{4\/3}+\n8\\chi   \\frac{(n_2a_z^2)^{2\/3}\n}{(3\\pi)^{2\/3}}\n\\right]^{3\/4}.\n\\end{eqnarray}\nwhere $\\eta$ is taken as a fixed constant for all $a_z^2n_2$. \n\n\n\n\n\nAS \\cite{njp} solve Eq. (\\ref{xe18}) variationally and obtain for the \nwidth $\\eta$ \n\\begin{equation}\\label{xy22}\n\\eta^4= \n\\frac{1}{4}+\\frac{4\\chi}{5}\\frac{(a_z^2n_2\\eta^2)^{2\/3}}{\\pi^{1\/3}}\\sqrt{3\\over \n5}.\n\\end{equation}\n  \n\n\n\n\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{Chemical potential $\\bar \\mu_z(n_2)$ of \n (\\ref{xe19}) for different\n$a_z^2n_2$ for a Fermi superfluid at unitarity\n($\\xi=0.44$)\nobtained from an accurate numerical solution of  \n(\\ref{xe18}), from  (\\ref{xe20})  and (\\ref{xy22}) (AS)\n\\cite{DFAS}, \nand from  (\\ref{xe22})    (Present, $\\eta=1\/\\sqrt 2$ and \n$\\eta=0.92\/\\sqrt 2$).\n}\n\\label{table2}\n\\begin{tabular}{|r|r|r|r|r|}\n\\hline\n$a_z^2n_2$ & Numerical    &\n AS    &   Present  &  Present \\\\\n    &    &  & $\\eta=1\/\\sqrt 2$ &$\\eta=0.92\/\\sqrt 2$  \\\\\n\\hline\n  0  &  0.5  & 0.5  & 0.5  &  0.50696\\\\\n0.1  & 1.071781         &   1.07375 & 1.05892&  1.08883  \\\\\n1  & 2.79861         & 2.82290  &  2.72155& 2.78109  \\\\\n 10 & 8.59782         & 8.72108  & 8.41456&  8.50454  \\\\\n100    & 27.09534         &  27.51645  &26.8149 & 26.9282  \\\\\n500  & 60.56659         &  61.51645  &  60.2317   &60.3571 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\nAs in the 3D-1D reduction, now we calculate the chemical potential \n$\\bar \\mu_z(n_2) $ of the three approaches and compare them with the \nprecise result for $\\bar \\mu_z(n_2) $ obtained from a numerical solution \nof  (\\ref{xe18}). The results are shown in table \\ref{table2} for \ndifferent  $a_z^2n_2$ along with those from the AS scheme \\cite{njp}. \n\n\n\n\n\nNow to see how well the effective 2D equations (\\ref{xe15})\nand (\\ref{xe22}) reproduce the radial density $\\varphi^2(\\rho)$ of a \ndisk-shaped Fermi superfluid we plot in figure  \\ref{fig2} (b) the 2D density \ncalculated from  (\\ref{xe15})\nand (\\ref{xe22}) and the same calculated from the full 3D  \n(\\ref{xe2}) for $\\lambda =4$ and $N=2,10,100.$ The excellent agreement \nbetween the two sets of results for $\\lambda$ as small as 4 \ndemonstrates the usefulness of the \npresent 2D model equations. \n\n\\begin{figure\n\n\\begin{center}\n\\includegraphics[width=.49\\linewidth]{fig4a.ps}\n\\includegraphics[width=.49\\linewidth]{fig4b.ps}\n\\end{center}\n\\caption{(Color online) (a)  The rms axial size vs. time\n(both in dimensionless oscillator units) during \noscillation of a \ncigar-shaped\nFermi \nsuperfluid at unitarity ($\\xi = 0.44$)\nfor $\\lambda=0.25,$ started by reducing the axial trap \nsuddenly by a factor of 0.9, as calculated by the full 3D equation \n(\\ref{xe2}) and the reduced 1D equations (\\ref{xe7}) and (\\ref{xe12}) \nfor \n$\\alpha =0.98\/\\sqrt 2$.  (b) \n The rms radial size vs. time \n(both in dimensionless oscillator units) during oscillation \n of a disk-shaped\nFermi\nsuperfluid at unitarity \nfor $\\lambda=4,$ started by reducing the radial trap\nsuddenly by a factor of 0.9, as calculated by the full 3D equation\n(\\ref{xe2}) and the reduced 2D equations (\\ref{xe15}) and \n(\\ref{xe22})\nfor\n$\\eta =0.92\/\\sqrt 2$.\n}\n\\label{fig4}\n\\end{figure}\n\n\n\n\n\\subsection{Dynamics: Breathing oscillation} \n\n\nNow we subject the reduced 1D and 2D\nDF equations to a more stringent test, e.g., how well these equations \ncan reproduce non-stationary (non-equilibrium) dynamics of a cigar- and  \ndisk-shaped Fermi superfluid. \nFirst we consider a cigar-shaped Fermi superfluid with $\\lambda = 0.25$,\nwhich is set into breathing oscillation by reducing only the axial\npotential suddenly by a factor of 0.9. The radial trap is left \nunchanged. The resultant oscillation is\nstudied using the full 3D DF equation (\\ref{xe2}) as well as the reduced\n1D DF equation (\\ref{xe7}) using the chemical potential (\\ref{xe12}).\nThe root mean square (rms) axial size as calculated from the 3D and 1D\nequations are plotted in figure \\ref{fig4} (a). \nNext we consider a disk-shaped Fermi superfluid with $\\lambda = 4$,\nwhich is set into breathing oscillation by reducing only the radial\npotential suddenly by a factor of 0.9. The resultant oscillation is\nstudied using the full 3D DF equation (\\ref{xe2}) as well as the reduced\n2D DF equation (\\ref{xe15}) using the \nchemical potential (\\ref{xe22}).\nThe rms radial size as calculated from the 3D and 2D\nequations are plotted in figure \\ref{fig4} (b). \n\n\\subsection{Dynamics: Free expansion}\n\n\\begin{figure\n\n\\begin{center}\n\\includegraphics[width=.49\\linewidth]{fig5a.ps}\n\\includegraphics[width=.49\\linewidth]{fig5b.ps}\n\\end{center}\n\\caption{(Color online) (a)  The rms axial size vs time \n(both in dimensionless oscillator units)\nduring axial  expansion  of a cigar-shaped\nFermi \nsuperfluid at unitarity ($\\xi = 0.44$)\nfor $\\lambda=0.25,$ started by removing  the axial trap \nsuddenly, as calculated by the full 3D equation \n(\\ref{xe2}) and the reduced 1D equations (\\ref{xe7}) and (\\ref{xe12}) \nfor \n$\\alpha =0.98\/\\sqrt 2$.  (b) \n The rms radial size vs time\n(both in dimensionless oscillator units)\nduring radial  \nexpansion of a disk-shaped\nFermi\nsuperfluid at unitarity\nfor $\\lambda=4,$ started by removing the radial trap\nsuddenly, as calculated by the full 3D equation\n(\\ref{xe2}) and the reduced 2D equations (\\ref{xe15}) and \n(\\ref{xe22})\nfor\n$\\eta =0.92\/\\sqrt 2$.\n}\n\\label{fig5}\n\\end{figure}\n\n\n\nNow we consider the problem of free expansion of a cigar- and \ndisk-shaped Fermi superfluid, respectively, when the axial and radial \ntraps are suddenly removed after the formation of the superfluid.\nFirst we consider a cigar-shaped Fermi superfluid with $\\lambda =0.25$ \nwhich is allowed to expand freely in the axial direction by setting the \naxial trap suddenly to zero in the 3D equation. The radial trap is left \nunchanged. The resultant expansion is studied using \nthe full 3D DF equation (\\ref{xe2}) as well as the reduced\n1D DF equation (\\ref{xe7}) using the chemical potential (\\ref{xe12}).\nThe rms axial size as calculated from the 3D and 1D equations \nare plotted in figure \\ref{fig5} (a). \nNext we consider a disk-shaped Fermi superfluid with $\\lambda = 4$, \nwhich is allowed to expand freely in the radial direction by setting the \nradial trap suddenly to zero in the 3D equation. The axial trap is left \nunchanged. The resultant expansion is studied using\nthe full 3D DF equation (\\ref{xe2}) as well as the reduced\n2D DF equation (\\ref{xe15}) using the chemical potential \n(\\ref{xe22}). The rms radial size as calculated from the 3D and 2D \nequations\nare plotted in figure \\ref{fig5} (b).\nThe agreement between the dynamics as obtained from the full 3D\nDF equation\nand that from the reduced DF equations in Figs. \\ref{fig4} and \\ref{fig5}\nis quite satisfactory.\n\n\n\n\n\\section{Conclusion}\n\n\n\nIn conclusion, we have suggested time-dependent mean-field reduced \nDF equations in 1D and 2D, respectively,\nfor a\ncigar- \nand disk-shaped BEC and \nFermi superfluid in the BCS and unitarity limits \nwith simple analytic \nnonlinear terms. The numerical solution \nof these reduced equations reveals that they produce results for density \nof Fermi superfluids and BEC\nin cigar- and disk-shaped traps in \nexcellent \nagreement with the solution of the full 3D DF equation.  \nWe also studied non-stationary breathing oscillation  \nof the cigar- and \ndisk-shaped  Fermi superfluid  \ninitiated by a sudden change of axial and radial traps, respectively. Finally,  \nwe applied the reduced equations to the study of free expansion of \na cigar- and disk-shaped Fermi superfluid initiated by a sudden removal \nof the axial and radial trap, respectively.   The reduced equations \nproduced equally good results in both these studies when compared with the \nsolution of the full 3D equations. \n\n\n\n\n\n\n\n\n\\ack\n\n\nFAPESP, CAPES  and CNPq (Brazil) provided partial support.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe mean-field (MF) theories provide us with good first approximation\nof ground states in various quantum many-body systems.\nFor many-fermionic systems,\nwe cannot exaggerate importance of the Hartree-Fock (HF) theory,\nwhich determines the one-body fields self-consistently\nwith respecting the variational principle~\\cite{ref:RS80}.\nThe HF theory is appropriate for defining\nsingle-particle (s.p.) orbitals of the constituent particles\nin the individual system.\nThe density-functional theory (DFT)~\\cite{ref:PY89,ref:ED11}\nhas mathematical and computational similarity to the HF theory,\nin the points that the ground state properties are obtained\nfrom the variational principle\nwith respect to the one-body fields,\nalthough there are no physical meaning in the s.p. orbitals.\nThe MF theory has been extended\nto the Hartree-Fock-Bogolyubov (HFB) theory~\\cite{ref:RS80},\nin which spontaneous breakdown of the particle-number conservation\n(\\textit{i.e.}, the global gauge symmetry) is taken into account\nwhile respecting the self-consistency and the variational principle.\nThe HFB theory is useful when two particles correlate as a pair\nand the system lies in the superconducting or the superfluid phase.\n\nThe random-phase approximation (RPA)~\\cite{ref:PB52}\ngives description of excited states,\nexcitation energies and transition strengths to be precise,\non top of the HF solution~\\cite{ref:Thou60}.\nIt is well linked to the linear response theory,\nand is obtained also as the small amplitude limit\nof the time-dependent HF theory~\\cite{ref:RS80}.\nAs in the similarity between the HF and the DFT,\nthe small amplitude limit of the time-dependent DFT~\\cite{ref:ED11}\nresembles the RPA.\nAnalogously to the RPA on the HF,\nthe quasiparticle-RPA is formulated on top of the HFB theory,\nwhich can describe excited states of systems in superfluidity.\n\nThe RPA and their extensions are basically built\nupon a solution in the corresponding MF theory.\nAs long as the MF state lies at a distinct minimum,\nthe RPA solutions can be regarded as excited states from the MF state.\nHowever, the RPA solutions could be unphysical\nif the MF state is actually at a saddle-point\nin the vector space defined in its vicinity.\nWhereas the RPA is extensively applied,\nstability of the RPA solutions is not always clear.\nIn localized self-bound systems like atomic nuclei,\nspontaneous symmetry breakdown (SSB) necessarily occurs\nin the MF regime~\\cite{ref:Thou60}.\nAn obvious example is the breaking of the translational symmetry,\nand another well-known example is the breaking of the rotational symmetry\nin deformed nuclei.\nThe SSB leads to a Nambu-Goldstone (NG) mode,\nwhich comes out as a zero-energy solution in the RPA.\nStructure of the vector space around the MF minimum\ncould be further complicated in the high-spin states~\\cite{ref:SM84}\nand along the path of the collective motion~\\cite{ref:HNMM}.\nIn numerical calculations within the MF theories,\nwe often assume certain symmetry and ignore some degrees of freedom (d.o.f.)\nto save computational resources.\nEven if a minimum is obtained in a MF calculation,\nit does not guarantee that it remains to be a minimum\nfor d.o.f. ignored in the calculation,\nas in a spherical HF calculation for a quadrupolarly deformed nucleus.\nWhereas stability of the RPA solutions in vicinity of the MF minimum\nwas investigated in Refs.~\\cite{ref:Thou61,ref:TV62}\nby Thouless \\textit{et al.},\nmore general arguments with rigorous mathematical treatment are desired\nfor a variety of extensions of the RPA developed to date.\n\nIn this article,\nI reanalyze properties of RPA solutions mathematically,\nin terms of the linear algebra.\nUnphysical solutions as well as physical solutions are examined\nin some detail.\nAlthough one may consider that unphysical solutions are just meaningless,\nthey are useful for understanding properties of the RPA equation\nmore profoundly.\nAs a result, they help us to comprehend\nin what manner physical solutions and NG modes come out.\nThe present analysis will be of practical significance as well\nin coding programs for numerical calculations in the RPA,\nbecause one often has to prepare for various situations in numerical studies,\nwithout knowing structure of the vector space around the MF state sufficiently.\n\n\\section{RPA equation}\\label{sec:RPAeq}\n\nThe RPA equation is written as\n\\begin{equation}\\begin{split}\n \\sum_\\beta \\big[A_{\\alpha\\beta}\\,X^{(\\nu)}_\\beta\n + B_{\\alpha\\beta}\\,Y^{(\\nu)}_\\beta\\big] &= \\omega_\\nu\\,X^{(\\nu)}_\\alpha\\,,\\\\\n \\sum_\\beta \\big[B_{\\alpha\\beta}^\\ast\\,X^{(\\nu)}_\\beta\n + A_{\\alpha\\beta}^\\ast\\,Y^{(\\nu)}_\\beta\\big] &= -\\omega_\\nu\\,Y^{(\\nu)}_\\beta\\,,\n\\end{split}\\label{eq:RPAeq-a}\\end{equation}\nwhere $\\alpha,\\beta$ represent particle-hole bases on the HF solution\nor two quasiparticle bases on the HFB solution.\nThe matrices $A$ and $B$ are obtained\nfrom the residual interaction and the HF s.p. (or the HFB q.p.) energies,\nand satisfy\n\\begin{equation} A=A^\\dagger\\,,\\quad B=B^T\\,. \\label{eq:sym-AB}\\end{equation}\nIn order to cope with a variety of physical situations,\nI start discussions only from this structure of the RPA equation,\nwithout any further assumptions.\nThe solution of Eq.~(\\ref{eq:RPAeq-a}) is\ncomprised of $\\omega_\\nu$ and $(X^{(\\nu)}, Y^{(\\nu)})$,\nwhich correspond to the energy and the wave function\nof the $\\nu$-th excited state.\nIt is imposed that $(X^{(\\nu)}, Y^{(\\nu)})$ obeys the \\textit{normalization},\n\\begin{equation}\\begin{split}\n\\sum_\\alpha \\big[X^{(\\nu)}_\\alpha X^{(\\nu')\\ast}_\\alpha\n - Y^{(\\nu)}_\\alpha Y^{(\\nu')\\ast}_\\alpha\\big] &= \\delta_{\\nu\\nu'}\\,,\\\\\n\\sum_\\alpha \\big[X^{(\\nu)}_\\alpha Y^{(\\nu')}_\\alpha\n - Y^{(\\nu)}_\\alpha X^{(\\nu')}_\\alpha\\big] &= 0\\,.\n\\end{split}\\label{eq:norm-a}\\end{equation}\nThe dimension of the $A$ and $B$ matrices is denoted by $D$.\n$D$ is finite in many practical calculations.\nAlthough $D$ can be infinite in principle,\nthe arguments here will cover infinite $D$ as a limiting case.\n\nProperties of the RPA solutions are better argued\nin terms of the $2D\\times 2D$ matrices,\n\\begin{equation}\n \\mathsf{S}:=\\begin{pmatrix} A&B\\\\ B^\\ast&A^\\ast \\end{pmatrix}\\,,~\n \\mathsf{N}:=\\begin{pmatrix} 1&0\\\\ 0&-1 \\end{pmatrix}\\,,~\n \\mathsf{\\Sigma}_x := \\begin{pmatrix} 0&1\\\\1&0 \\end{pmatrix}\\,.\n\\label{eq:matrices}\\end{equation}\nFor the matrices $\\mathsf{N}$ and $\\mathsf{\\Sigma}_x$,\n$\\mathsf{N}^2=\\mathsf{\\Sigma}_x^2=\\mathsf{1}$\nand $\\mathsf{\\Sigma}_x\\,\\mathsf{N}+\\mathsf{N}\\,\\mathsf{\\Sigma}_x=\\mathsf{0}$\nhold.\nThe matrix $\\mathsf{S}$ is nothing but\nthe curvature matrix at the HF (or HFB) solution,\nand is known as the stability matrix.\nWith the matrices defined in Eq.~(\\ref{eq:matrices}),\nthe RPA equation (\\ref{eq:RPAeq-a}) is expressed as\n\\begin{equation} \\mathsf{S}\\,\\vect{x}_\\nu=\\omega_\\nu\\mathsf{N}\\,\\vect{x}_\\nu\\,;\\quad\n \\vect{x}_\\nu=\\begin{pmatrix} X^{(\\nu)}\\\\ Y^{(\\nu)}\\end{pmatrix}\\,,\n\\label{eq:RPAeq-b}\\end{equation}\nand the normalization condition (\\ref{eq:norm-a}) as\n\\begin{equation} \\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}=\\delta_{\\nu\\nu'}\\,,~\n \\vect{x}_\\nu^T\\mathsf{\\Sigma}_x\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}=0\\,.\n\\label{eq:norm-b}\\end{equation}\nThe number of the solutions satisfying (\\ref{eq:norm-b})\nmatches the dimension $D$ of the $A,B$ matrix in ideal cases.\nHowever, it is not obvious whether there exist $D$ solutions\nthat satisfy the normalization condition (\\ref{eq:norm-b}).\n\n\\begin{definition}\\label{def:solvable}\nIf there exist $D$ independent solutions of $(\\omega_\\nu, \\vect{x}_\\nu)$\nthat satisfy Eqs.~(\\ref{eq:RPAeq-b}), (\\ref{eq:norm-b}) and $\\omega_\\nu>0$,\nthe RPA equation is said \\textbf{fully solvable}.\n\\end{definition}\n\n\\section{General properties of RPA solutions}\\label{sec:general}\n\n\\subsection{Dualities and eigenvalues}\\label{subsec:duality}\n\nThe RPA equation (\\ref{eq:RPAeq-b}) is equivalent\nto the eigenvalue problem of the matrix $\\mathsf{N\\,S}$.\nAn eigensolution is defined by a set of an eigenvalue\nand an eigenvector $(\\omega_\\nu,\\vect{x}_\\nu)$.\nAt the same time, Eq.~(\\ref{eq:RPAeq-b}) reads\nthe eigenvalue problem of $\\mathsf{S\\,N}$\nfor an eigensolution $(\\omega_\\nu,\\mathsf{N}\\,\\vect{x}_\\nu)$.\nThis duality is important to derive basic properties of the RPA solutions.\nThe relevant duality will be established later,\nand is called \\textit{LR-duality} in this article,\nbecause it connects left and right basis vectors.\n\nAnother important ingredient is the symmetry of $\\mathsf{S}$.\nThe structure of $\\mathsf{S}$ given in Eq.~(\\ref{eq:matrices})\nwith (\\ref{eq:sym-AB}) is characterized by\n\\begin{equation} \\mathsf{S}=\\mathsf{S}^\\dagger\\,,\\quad\n \\mathsf{\\Sigma}_x\\,\\mathsf{S}^\\ast\\,\\mathsf{\\Sigma}_x = \\mathsf{S}\\,.\n\\label{eq:S-prop}\\end{equation}\nIt should be noted that, owing to the first equation of (\\ref{eq:S-prop}),\nan eigenvector of $\\mathsf{S\\,N}$ associated with an eigenvalue $\\omega_\\nu$\nimmediately gives a left eigenvector of $\\mathsf{N\\,S}$\ncorresponding to the eigenvalue $\\omega_\\nu^\\ast$,\nbecause $\\mathsf{S\\,N}\\,\\vect{y}_\\nu = \\omega_\\nu\\,\\vect{y}_\\nu$\nis equivalent to $\\vect{y}_\\nu^\\dagger\\,\\mathsf{N\\,S}\n= \\omega_\\nu^\\ast\\,\\vect{y}_\\nu^\\dagger$.\n\\begin{proposition}\\label{theor:sym-eigen}\nIf $\\omega_\\nu$ is an eigenvalue of $\\mathsf{N\\,S}$,\n$-\\omega_\\nu$ is also an eigenvalue with equal degeneracy.\nSo is $\\omega_\\nu^\\ast$.\n\\end{proposition}\n\\begin{proof}\nEquation~(\\ref{eq:RPAeq-b}) gives the secular equation, whose l.h.s. is\n\\begin{equation}\\begin{split}\n \\det(\\mathsf{S}-\\omega\\,\\mathsf{N})\n &= \\det\\big([\\mathsf{S}-\\omega\\,\\mathsf{N}]^T\\big)\n = \\big[\\det(\\mathsf{S}-\\omega^\\ast\\,\\mathsf{N})\\big]^\\ast\\\\\n &= \\det(\\mathsf{\\Sigma}_x\\,\\mathsf{S}^\\ast\\,\\mathsf{\\Sigma}_x\n -\\omega\\,\\mathsf{N})\n = \\det(\\mathsf{S}^\\ast\n -\\omega\\,\\mathsf{\\Sigma}_x\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x)\\\\\n &= \\det(\\mathsf{S}^\\ast+\\omega\\,\\mathsf{N})\n = \\big[\\det(\\mathsf{S}+\\omega^\\ast\\,\\mathsf{N})\\big]^\\ast\\,,\n\\end{split}\\label{eq:secular-lhs}\\end{equation}\nbecause of Eq.~(\\ref{eq:S-prop}).\nThus the equations $\\det(\\mathsf{S}-\\omega\\,\\mathsf{N})=0$,\n$\\det(\\mathsf{S}-\\omega^\\ast\\,\\mathsf{N})=0$\nand $\\det(\\mathsf{S}+\\omega^\\ast\\,\\mathsf{N})=0$ are all equivalent,\nensuring correspondence of the eigenvalues $\\omega$, $\\omega^\\ast$\nand $-\\omega^\\ast$.\nThis leads to the eigenvalue $-\\omega$ as well.\n\\end{proof}\n\\noindent\nIt is straightforward to show the following corollary\nfrom the second equation of (\\ref{eq:S-prop}).\n\\begin{corollary}\\label{lem:sym-sigma}\nIf $(\\omega_\\nu, \\vect{x}_\\nu)$ is an eigensolution of Eq.~(\\ref{eq:RPAeq-b}),\n$(-\\omega_\\nu^\\ast, \\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast)$ is also a solution.\n\\end{corollary}\n\\noindent\nThis relation indicates another kind of duality,\nwhich is called \\textit{UL-duality} to distinguish from the LR-duality,\nsince it is related to interchange (with taking complex conjugate)\nof the upper and the lower components of the basis vectors.\nWith respect to the normalization of Eq.~(\\ref{eq:norm-b}),\nthe following relation is obtained,\n\\begin{equation} (\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast)^\\dagger\\,\\mathsf{N}\\,\n (\\mathsf{\\Sigma}_x\\vect{x}_{\\nu'}^\\ast)\n = -\\vect{x}_{\\nu'}^\\dagger\\,\\mathsf{N}\\,\\vect{x}_\\nu\\,. \\end{equation}\nThe second equation of (\\ref{eq:norm-b}) is interpreted as a relation\nbetween $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$ and $\\vect{x}_{\\nu'}$,\n$(\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast)^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}=0$.\n\nFor an arbitrary vector\n${\\displaystyle\\vect{x}=\\begin{pmatrix} X\\\\ Y\\end{pmatrix}}$,\nwe have $\\vect{x}^\\dagger\\,\\mathsf{N}\\,\\vect{x}\n= X^\\dagger\\,X - Y^\\dagger\\,Y$.\nIn this respect the `norm' is not positive-definite.\n\\begin{definition}\\label{def:normalizable}\nA vector $\\vect{x}$ is said \\textbf{normalizable}\nwhen $\\vect{x}^\\dagger\\,\\mathsf{N}\\,\\vect{x}>0$.\n\\end{definition}\n\\noindent\nIndeed, if $\\vect{x}^\\dagger\\,\\mathsf{N}\\,\\vect{x}=r^2\\,(>0)$,\n$\\vect{x}\/r$ is normalized in the respect of Eq.~(\\ref{eq:norm-b}).\nAs noted above,\nit is not guaranteed that an eigenvector $\\vect{x}_\\nu$ of $\\mathsf{N\\,S}$\ncan be normalized.\nThe normalizability of an eigenvector $\\vect{x}_\\nu$\nis a key to whether the RPA equation is solvable.\n\nIn arguments with respect to the RPA~\\cite{ref:Thou61},\nthe `orthogonality' between two vectors $\\vect{x}$ and $\\vect{y}$\nis sometimes defined when $\\vect{x}^\\dagger\\,\\mathsf{N}\\,\\vect{y}=0$,\nby regarding $\\mathsf{N}$ as the metric.\nHowever, in this article I use the usual definition\nthat $\\vect{x}$ and $\\vect{y}$ are orthogonal\nwhen $\\vect{x}^\\dagger\\,\\vect{y}=0$.\n\nThe hermiticity of $\\mathsf{S}$ leads to the relation\nbetween the solutions of Eq.~(\\ref{eq:RPAeq-b}),\n\\begin{equation} \\vect{x}_\\nu^\\dagger\\,\\mathsf{S}\\,\\vect{x}_{\\nu'}\n = \\omega_\\nu^\\ast\\,\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}\n = \\omega_{\\nu'}\\,\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}\\,.\n\\label{eq:norm-relation}\\end{equation}\nEquation~(\\ref{eq:norm-relation}) concludes:\n\\begin{lemma}\\label{lem:bi-orthogonality}\nFor eigensolutions $(\\omega_\\nu, \\vect{x}_\\nu)$\nand $(\\omega_{\\nu'}, \\vect{x}_{\\nu'})$ of the RPA equation (\\ref{eq:RPAeq-b}),\n$\\omega_\\nu^\\ast=\\omega_{\\nu'}$\nor $\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}=0$ follows.\n\\end{lemma}\n\\noindent\nThe case of $\\nu=\\nu'$ was argued in Ref.~\\cite{ref:Thou61}.\n\nIt is now possible to classify solutions of Eq.~(\\ref{eq:RPAeq-b}).\n\\begin{proposition}\\label{prop:eigenvalues}\nEigenvalues (denoted by $\\omega_\\nu$) of $\\mathsf{N\\,S}$ come out\nin one of the following manners:\n\\begin{enumerate}\n\\item\\label{item:positive}\n $\\omega_\\nu>0$ with a normalizable eigenvector $\\vect{x}_\\nu$,\n in association with another eigensolution\n $(-\\omega_\\nu, \\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast)$.\n\\item\\label{item:negative}\n $\\omega_\\nu>0$ with an unnormalizable eigenvector,\n in association with an eigenvalue $-\\omega_\\nu$ that could be normalizable.\n\\item\\label{item:pure-imag}\n a pair of pure imaginary eigenvalues, $\\pm\\omega_\\nu$\n with $\\mathrm{Re}(\\omega_\\nu)=0$, $\\mathrm{Im}(\\omega_\\nu)\\ne 0$.\n\\item\\label{item:quartet}\n a quartet of complex eigenvalues, $\\pm\\omega_\\nu, \\pm\\omega_\\nu^\\ast$\n with $\\mathrm{Re}(\\omega_\\nu)\\ne 0$, $\\mathrm{Im}(\\omega_\\nu)\\ne 0$.\n\\item\\label{item:null} a null eigenvalue.\n\\end{enumerate}\n\\end{proposition}\n\\noindent\nA solution belonging to Class~\\ref{item:positive}\nof Prop.~\\ref{prop:eigenvalues} may be called \\textit{physical solution},\nwhile a solution to one of \\ref{item:negative}\\,--\\,\\ref{item:quartet}\n\\textit{unphysical solution}.\nClass~\\ref{item:null} is closely connected to the NG mode,\nand a solution belonging to it will be called \\textit{NG-mode solution}.\nFocusing on solutions in vicinity of a MF minimum,\nThouless did not discuss\nsolutions of Classes~\\ref{item:negative} and \\ref{item:quartet}\nin Refs.~\\cite{ref:Thou61,ref:TV62}.\n\n\\subsection{Basis vectors in Jordan blocks}\\label{subsec:Jordan}\n\nSeveral eigenvalues of $\\mathsf{N\\,S}$ could be degenerate,\nand degenerate eigenvalues may give rise to Jordan blocks.\nThis possibility is examined in this subsection.\nSuppose that a Jordan block is generated from an eigenvector $\\vect{x}_\\nu$.\nA basis vector of the Jordan block is denoted by $\\vect{\\xi}_k^{(\\nu)}$,\nwhich is obtained by\n\\begin{equation} \\mathsf{S}\\,\\vect{\\xi}_{k+1}^{(\\nu)}\n = \\omega_\\nu\\,\\mathsf{N}\\,\\vect{\\xi}_{k+1}^{(\\nu)}\n + ic_k^{(\\nu)}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}\\,.\\quad\n (c_k^{(\\nu)}\\in\\mathbf{C})\n\\label{eq:Jordan}\\end{equation}\nThe constant $c_k^{(\\nu)}$ is subject to normalization and to relative phase\nof $\\vect{\\xi}_k^{(\\nu)}$ and $\\vect{\\xi}_{k+1}^{(\\nu)}$.\nAlthough it is taken to be $ic_k^{(\\nu)}=1$ in the Jordan normal form,\nanother normalization will be adopted in Sec.~\\ref{subsec:imag-sol}.\nStarting from $\\vect{\\xi}_1^{(\\nu)}=\\vect{x}_\\nu$,\n$\\vect{\\xi}_{k+1}^{(\\nu)}$ and $c_k^{(\\nu)}\\,(\\ne 0)$ can be fixed\nsuccessively for $k\\,(\\geq 1)$.\nThis chain of equations ends at a certain $k$,\nwhere no $\\vect{\\xi}_{k+1}^{(\\nu)}\\,(\\ne\\vect{0})$ exists.\nDimension of individual Jordan block is determined\nby how long the chain continues, and is denoted by $d_\\nu$.\nEquation~(\\ref{eq:Jordan}) defines a Jordan block of $\\mathsf{S\\,N}$ as well,\nwhose basis vectors are $\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}$.\n\nMost of the following arguments will cover the case of $d_\\nu=1$;\n\\textit{i.e.}, the case that an eigenvector $\\vect{x}_\\nu$\ndoes not generate a Jordan block.\n\\begin{lemma}\\label{lem:Jordan-sym}\nIt there is a Jordan block for an eigenvalue $\\omega_\\nu$ with dimension $d_\\nu$,\nso are for $-\\omega_\\nu$ and $\\omega_\\nu^\\ast$ with equal dimension $d_\\nu$.\n\\end{lemma}\n\\begin{proof}\nA Jordan block for the eigenvalue $\\omega_\\nu$ of $\\mathsf{S\\,N}$\ndirectly corresponds to a Jordan block for $\\omega_\\nu^\\ast$ of $\\mathsf{N\\,S}$,\nby regarding the right basis vectors of $\\mathsf{S\\,N}$\nas the left basis vectors of $\\mathsf{N\\,S}$.\nTherefore the dimensions corresponding to $\\omega_\\nu$ and $\\omega_\\nu^\\ast$\nmust be equal.\\\\\nIt follows from Eqs.~(\\ref{eq:S-prop}) and (\\ref{eq:Jordan}) that,\n\\begin{equation} \\mathsf{S}\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_{k+1}^{(\\nu)\\ast}\n = -\\omega_\\nu^\\ast\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_{k+1}^{(\\nu)\\ast}\n + ic_k^{(\\nu)\\ast}\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}\\,,\n\\label{eq:Jordan-sym}\\end{equation}\nverifying that there exists a Jordan block for $-\\omega_\\nu^\\ast$\nwith equal dimension.\nThen, so is it for $-\\omega_\\nu$.\n\\end{proof}\n\\begin{definition}\\label{def:ULdual}\nThe basis vectors $\\vect{\\xi}_k^{(\\nu)}$\nand $\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}$\nare said to be \\textbf{UL-dual} of each other.\n\\end{definition}\n\\noindent\nIf $\\vect{\\xi}_k^{(\\nu)}=\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}$,\nit is said to be \\textit{self UL-dual}.\n\nIn general, a single eigenvalue $\\omega_\\nu$ may give plural Jordan blocks.\nI hereafter reserve the subscript $\\nu$\nfor representing individual Jordan block,\nwhich is connected to a single eigenvector $\\vect{x}_\\nu$ of $\\mathsf{N\\,S}$,\nrather than the eigenvalue.\nThe following proposition and lemmas are closely connected\nto the structure of the Jordan block,\n\\begin{equation} \\begin{pmatrix}  \\omega_\\nu & ic_1^{(\\nu)} & 0 & \\cdots & 0 \\\\\n  0 & \\omega_\\nu & ic_2^{(\\nu)} & \\ddots & 0 \\\\\n  \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n  0 & 0 & 0 &  & ic_{d_\\nu-1}^{(\\nu)} \\\\\n  0 & 0 & 0 & \\cdots & \\omega_\\nu \\end{pmatrix}\\,,\n\\label{eq:Jordan-block}\\end{equation}\nwhich is represented with the left and right basis vectors\nthat form the inverse matrix of each other apart from normalization.\nExplicit proofs of Lemma~\\ref{lem:Jordan-orthogonality} to\nProp.~\\ref{theor:one-to-one_basis} are given\nin Appendix~\\ref{app:proof-Jordan},\nwhich are instructive and useful\nto confirm their compatibility with later propositions and lemmas.\n\\begin{lemma}\\label{lem:Jordan-orthogonality}\nUnless eigenvalues $\\omega_\\nu^\\ast$ and $\\omega_{\\nu'}$ are equal,\nany basis vector of $\\mathsf{N\\,S}$ associated with $\\omega_\\nu$\n(either an eigenvector or a basis vector belonging to a Jordan block)\nis orthogonal to any basis vector of $\\mathsf{S\\,N}$\nassociated with $\\omega_{\\nu'}$.\n\\end{lemma}\n\nThe above lemma suggests that\na Jordan block of $\\mathsf{N\\,S}$ for a specific $\\omega_\\nu$\nand a Jordan block of $\\mathsf{S\\,N}$ for $\\omega_\\nu^\\ast$ are paired.\n\\begin{lemma}\\label{lem:Jordan-normalizability}\nIt is possible to take so that\neach basis vector in a Jordan block of $\\mathsf{N\\,S}$\ncould overlap with no more than one basis vector\nthat contained in a Jordan block of $\\mathsf{S\\,N}$.\nIn order for the overlap not to vanish,\ndimensions of these Jordan blocks of $\\mathsf{N\\,S}$ and $\\mathsf{S\\,N}$\nmust be equal.\n\\end{lemma}\n\\noindent\nThe next corollary follows\nfrom the proof of Lemma~\\ref{lem:Jordan-normalizability}\ngiven in Appendix~\\ref{app:sub-A2}:\n\\begin{corollary}\\label{cor:non-Jordan}\nIf two eigenvectors of $\\vect{x}_\\nu$ and $\\vect{x}_{\\nu'}$ of $\\mathsf{N\\,S}$\nsatisfies $\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}\\ne 0$,\nboth of them do not constitute Jordan blocks (\\textit{i.e.}, $d_\\nu=d_{\\nu'}=1$).\n\\end{corollary}\n\\begin{proposition}\\label{theor:one-to-one_basis}\nIt is possible to produce a complete set of basis vectors of $\\mathsf{N\\,S}$,\nby a proper transformation if necessary,\nso that each of them could overlap with only one basis vector of $\\mathsf{S\\,N}$.\nOne-to-one correspondence is established\nbetween Jordan blocks of $\\mathsf{N\\,S}$ and $\\mathsf{S\\,N}$\nthat contain basis vectors having non-vanishing overlaps.\nThe correspondence between basis vectors of $\\mathsf{N\\,S}$ and $\\mathsf{S\\,N}$\nis also one to one.\n\\end{proposition}\n\\noindent\nThis proposition is an expression that the left and the right basis vectors\ngiving the Jordan representation constitute the inverse matrix of each other.\n\nProposition~\\ref{theor:one-to-one_basis} enables us to define\na basis vector of $\\mathsf{S\\,N}$\nthat is dual of individual basis vector of $\\mathsf{N\\,S}$.\nLet us denote the basis vector dual to $\\vect{\\xi}_k^{(\\nu)}$\nby $\\bbar{\\vect{\\xi}}_k^{(\\nu)}$.\nThen $\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}$\ndoes not vanish only for $\\nu=\\nu'$ and $k=k'$.\nThis is a realization of the Jordan block of (\\ref{eq:Jordan-block}).\n$\\bbar{\\vect{\\xi}}_k^{(\\nu)}$ is the $(d_\\nu+1-k)$-th basis vector\nof the Jordan block for an eigenvector\n$\\bbar{\\vect{x}}_\\nu:=\\bbar{\\vect{\\xi}}_{d_\\nu}^{(\\nu)}$,\nwhich is associated with the eigenvalue $\\omega_\\nu^\\ast$.\nInverting Eq.~(\\ref{eq:Jordan}), we obtain\n\\begin{equation} \\mathsf{S}\\,\\bbar{\\vect{\\xi}}_{k-1}^{(\\nu)}\n = \\omega_\\nu^\\ast\\,\\mathsf{N}\\,\\bbar{\\vect{\\xi}}_{k-1}^{(\\nu)}\n - ic_{k-1}^{(\\nu)\\ast}\\,\n \\frac{\\vect{\\xi}_{k-1}^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\bbar{\\vect{\\xi}}_{k-1}^{(\\nu)}}\n {\\vect{\\xi}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)}}\\,\n \\mathsf{N}\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)}\\,,\n\\label{eq:Jordan-inv}\\end{equation}\nas confirmed in Appendix~\\ref{app:sub-A4}.\nThis establishes the LR-duality of basis vectors.\n\\begin{definition}\\label{def:LRdual}\nThe basis vectors $\\vect{\\xi}_k^{(\\nu)}$ and $\\bbar{\\vect{\\xi}}_k^{(\\nu)}$\nare said to be \\textbf{LR-dual} of each other.\n\\end{definition}\n\\noindent\nWhen $\\vect{\\xi}_k^{(\\nu)}=\\bbar{\\vect{\\xi}}_k^{(\\nu)}$,\nit is said \\textit{self LR-dual}.\nIt is reasonable to define $\\bbar{\\vect{\\xi}}_k^{(\\nu)}$\nso that it would fulfill\n$\\big|\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}\\big|\n=\\delta_{\\nu\\nu'}\\,\\delta_{kk'}$.\nIt is not necessarily convenient to assume\n$\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}\n=\\delta_{\\nu\\nu'}\\,\\delta_{kk'}$, though possible.\nThe normalizability in Def.~\\ref{def:normalizable}\nis now perceived as a part of the self LR-duality.\n\n\\subsection{Redefining eigenvalue problem}\\label{subsec:redefine}\n\nBecause $\\mathsf{S}$ is hermitian,\nit is diagonalizable with an appropriate unitary matrix $\\mathsf{U}$,\n\\begin{equation} \\mathsf{S}=\\mathsf{U}^{-1}\\,\\mathrm{diag.}(\\lambda_i)\\,\\mathsf{U}\\,.\\quad\n (\\lambda_i\\in\\mathbf{R})\n\\label{eq:S-diag}\\end{equation}\nBy using this expression, $\\mathsf{S}^{1\/2}$ can be taken as\n\\begin{equation} \\mathsf{S}^{1\/2}=\\mathsf{U}^{-1}\\,\\mathrm{diag.}(\\lambda_i^{1\/2})\\,\\mathsf{U}\\,.\n\\label{eq:S-half}\\end{equation}\nThough $\\lambda_i^{1\/2}$ is two-valued, it is not important\nwhich value is adopted.\nA new matrix is now defined,\n\\begin{equation} \\tilde{\\mathsf{S}}:=\\mathsf{S}^{1\/2}\\,\\mathsf{N\\,S}^{1\/2}\\,.\n\\label{eq:S-tilde}\\end{equation}\nObviously, $\\tilde{\\mathsf{S}}$ is hermitian\nonly when $\\lambda_i\\geq 0$ for any $i\\,(=1,\\cdots,2D)$.\n\n\\begin{lemma}\\label{lem:tilde-S_eigen}\nAll the eigenvalues and eigenvectors of $\\mathsf{N\\,S}$\ncorrespond to those of $\\tilde{\\mathsf{S}}$, and vice versa.\n\\end{lemma}\n\\begin{proof}\nIf $\\mathsf{S}\\,\\vect{x}_\\nu=\\omega_\\nu\\mathsf{N}\\,\\vect{x}_\\nu$,\n$\\tilde{\\mathsf{S}}\\,(\\mathsf{S}^{1\/2}\\vect{x}_\\nu)\n =\\omega_\\nu(\\mathsf{S}^{1\/2}\\vect{x}_\\nu)$ follows.\nTherefore $(\\omega_\\nu, \\mathsf{S}^{1\/2}\\vect{x}_\\nu)$\ngives an eigensolution of $\\tilde{\\mathsf{S}}$\nunless $\\mathsf{S}^{1\/2}\\vect{x}_\\nu=\\vect{0}$.\nConversely, if $\\tilde{\\mathsf{S}}\\,\\vect{y}_\\nu=\\omega_\\nu\\vect{y}_\\nu$,\n$\\mathsf{S}\\,(\\mathsf{N\\,S}^{1\/2}\\vect{y}_\\nu)\n =\\omega_\\nu\\,\\mathsf{N}\\,(\\mathsf{N\\,S}^{1\/2}\\vect{y}_\\nu)$ follows.\\\\\nIf $\\mathsf{S}^{1\/2}\\vect{x}_\\nu=\\mathbf{0}$,\nwe have $\\mathsf{S}\\,\\vect{x}_\\nu=\\tilde{\\mathsf{S}}\\,\\vect{x}_\\nu=0$,\nindicating $\\vect{x}_\\nu$ is an eigenvector corresponding to the null eigenvalue\nboth of $\\mathsf{N\\,S}$ and $\\tilde{\\mathsf{S}}$.\n\\end{proof}\n\nThus the RPA equation is equivalent\nto the eigenvalue problem of $\\tilde{\\mathsf{S}}$;\n$\\tilde{\\mathsf{S}}\\,\\vect{x}_\\nu=\\omega_\\nu\\vect{x}_\\nu$.\nThis redefinition is applied to investigate solvability of the RPA equation\nin Sec.~\\ref{subsec:real-sol}.\n\n\\section{Decomposition of vector space}\\label{sec:decomp}\n\nThe whole vector space $\\mathcal{V}$,\nin which the stability matrix $\\mathsf{S}$ is defined,\ncan be decomposed via the basis vectors produced by $\\mathsf{N\\,S}$\nor those by $\\mathsf{S\\,N}$.\nThis furnishes further discussion on properties of the RPA solutions.\n\nRecalling the LR-duality explored in Sec.~\\ref{sec:general},\nwe obtain the projector\nwhich separates out the direction along a certain basis vector\nof $\\mathsf{N\\,S}$, as in Ref.~\\cite{ref:TV62},\n\\begin{equation} \\mathsf{\\Lambda}_{\\nu,k} :=\n \\frac{\\vect{\\xi}_k^{(\\nu)}\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}}\\,.\n\\label{eq:proj-nuk}\\end{equation}\nObviously $\\mathsf{\\Lambda}_{\\nu,k}\\,\\mathsf{\\Lambda}_{\\nu',k'}\n=\\delta_{\\nu\\nu'}\\,\\delta_{kk'}\\,\\mathsf{\\Lambda}_{\\nu,k}$.\nThe projector separating out the subspace\ncorresponding to the Jordan block (including the $d_\\nu=1$ case)\ngenerated from the eigenvector $\\vect{x}_\\nu$ is obtained by\n\\begin{equation} \\mathsf{\\Lambda}_\\nu := \\sum_{k=1}^{d_\\nu} \\mathsf{\\Lambda}_{\\nu,k}\\,. \\end{equation}\nThe projector $\\mathsf{\\Lambda}_\\nu$ defines a subspace $\\mathcal{W}_\\nu$,\n\\begin{equation} \\mathcal{W}_\\nu := \\bigg\\{ \\sum_{k=1}^{d_\\nu} a_k\\,\\vect{\\xi}_k^{(\\nu)};\n a_k\\in\\mathbf{C}\\bigg\\}\\,, \\label{eq:subspace-nu}\\end{equation}\nfor which $\\mathsf{\\Lambda}_\\nu\\mathcal{W}_\\nu=\\mathcal{W}_\\nu$\nand $(\\mathsf{1}-\\mathsf{\\Lambda}_\\nu)\\mathcal{W}_\\nu=\\emptyset$ (empty set)\nare satisfied.\nThe completeness is expressed as\n\\begin{equation} \\sum_\\nu \\mathsf{\\Lambda}_\\nu = \\mathsf{1}\\,,\\quad\n \\mathcal{V} = \\bigoplus_\\nu \\mathcal{W}_\\nu\\,.\\label{eq:proj-nu}\\end{equation}\n\nIn association with the LR-duality, one may consider\n\\begin{equation} \\bbar{\\mathcal{W}}_\\nu := \\bigg\\{ \\sum_{k=1}^{d_\\nu}\n a_k\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)}; a_k\\in\\mathbf{C}\\bigg\\}\\,.\n\\label{eq:subspace-nubar}\\end{equation}\nThe projector corresponding to $\\bbar{\\mathcal{W}}_\\nu$\nis given by $\\mathsf{N\\,\\Lambda}_\\nu^\\dagger\\,\\mathsf{N}$.\nIn order for the arguments in Sec.~\\ref{sec:general} to be applicable\neven after a certain projection,\nit is desired to respect the UL-duality, as well as the LR-duality.\nTherefore the UL-dual subspace is also considered,\n\\begin{equation} \\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast := \\bigg\\{ \\sum_{k=1}^{d_\\nu}\n a_k\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}; a_k\\in\\mathbf{C}\\bigg\\}\\,.\n\\label{eq:subspace-signu}\\end{equation}\nThe projector relevant to $\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast$ is obtained by\n\\begin{equation} \\sum_{k=1}^{d_\\nu} \\frac{\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}\\,\n \\bbar{\\vect{\\xi}}_k^{(\\nu)T}\\mathsf{\\Sigma}_x\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_k^{(\\nu)T}\\mathsf{\\Sigma}_x\\,\\mathsf{N}\\,\n \\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}}\n= \\sum_{k=1}^{d_\\nu} \\frac{\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}\\,\n \\bbar{\\vect{\\xi}}_k^{(\\nu)T}\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x}\n {\\bbar{\\vect{\\xi}}_k^{(\\nu)T}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)\\ast}}\n= \\mathsf{\\Sigma}_x\\,\\mathsf{\\Lambda}_\\nu^\\ast\\,\\mathsf{\\Sigma}_x\\,. \\end{equation}\nThe subspace $\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast$\nand its relevant projector are defined as well.\nDepending on $\\omega_\\nu$,\nsome of $\\mathcal{W}_\\nu$, $\\bbar{\\mathcal{W}}_\\nu$,\n$\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast$\nand $\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast$ could be identical.\nWe shall use a collective index $[\\nu]$\nto stand for their direct sum,\n\\begin{equation} \\mathcal{W}_{[\\nu]} := \\mathcal{W}_\\nu\\oplus\\bbar{\\mathcal{W}}_\\nu\n \\oplus\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast\n \\oplus\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast\\,,\\end{equation}\napart from their overlap.\nThe projector $\\mathsf{\\Lambda}_{[\\nu]}$ on $\\mathcal{W}_{[\\nu]}$\nis defined by sum of the projectors.\nThough in restricted cases,\na similar projection is considered in Ref.~\\cite{ref:Don99}.\nIt is straightforward to show the following properties\nof $\\mathsf{\\Lambda}_{[\\nu]}$,\n\\begin{equation} \\mathsf{\\Lambda}_{[\\nu]}\n =\\mathsf{N\\,\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{N}\\,,\\quad\n \\mathsf{\\Lambda}_{[\\nu]}\\,\\mathsf{\\Lambda}_{[\\nu']}\n =\\delta_{[\\nu],[\\nu']}\\,\\mathsf{\\Lambda}_{[\\nu]}\\,,\\quad\n \\mathsf{\\Sigma}_x\\,\\mathsf{\\Lambda}_{[\\nu]}^\\ast\\,\\mathsf{\\Sigma}_x\n = \\mathsf{\\Lambda}_{[\\nu]}\\,.\n\\label{eq:proj-prop}\\end{equation}\n\\begin{lemma}\\label{lem:dim-Lambda}\n$d_{[\\nu]}:=\\dim \\mathcal{W}_{[\\nu]}$ must be even.\n\\end{lemma}\n\\begin{proof}\nUnless $\\mathcal{W}_\\nu=\\bbar{\\mathcal{W}}_\\nu\n=\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast\n=\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast$,\nthis is obvious from Lemma~\\ref{lem:Jordan-sym}.\nIf $\\mathcal{W}_\\nu=\\bbar{\\mathcal{W}}_\\nu\n=\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast\n=\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast$,\n$\\omega_\\nu=0$ and $\\bbar{\\vect{\\xi}}_k^{(\\nu)}=\\vect{\\xi}_{d_\\nu+1-k}^{(\\nu)}$,\nas is clear from the argument\nwith respect to Prop.~\\ref{theor:one-to-one_basis}\nin Appendix~\\ref{app:sub-A3}.\nThen, if $d_\\nu$ is odd, we have\n$\\bbar{\\vect{\\xi}}_{(d_\\nu+1)\/2}^{(\\nu)}=\\vect{\\xi}_{(d_\\nu+1)\/2}^{(\\nu)}$,\ncontradictory to Lemma~\\ref{lem:pure-imag-Jordan}\n(Eq.~(\\ref{eq:imag-norm}), to be more precise),\nwhich leads to $\\bbar{\\vect{\\xi}}_{(d_\\nu+1)\/2}^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\n\\vect{\\xi}_{(d_\\nu+1)\/2}^{(\\nu)}=0$.\nSee also the arguments in Appendix~\\ref{app:sub-B1}.\n\\end{proof}\n\nFor an arbitrary $2D\\times 2D$ matrix $\\mathsf{M}$,\nits projection onto the subspace $\\mathcal{W}_{[\\nu]}$\nis obtained by\n\\begin{equation} \\mathsf{M}_{[\\nu]} := \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{M}\\,\n \\mathsf{\\Lambda}_{[\\nu]}\\,. \\end{equation}\nOf next interest is how the projection affects\nthe RPA equation and the dualities.\n\\begin{proposition}\\label{theor:SN-proj}\nThe projections of $\\mathsf{S}$ and $\\mathsf{N}$ satisfy\n\\begin{equation} \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{S}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{S}_{[\\nu]}\\,,\\quad\n \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{N}_{[\\nu]}\\,. \\nonumber\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFrom Eq.~(\\ref{eq:proj-prop}),\n\\begin{equation}\n \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu]}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu]}\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{N}\\,\n \\mathsf{\\Lambda}_{[\\nu]}\\,.\\end{equation}\nConcerning $\\mathsf{S}_{[\\nu]}$, Eq.~(\\ref{eq:Jordan}) yields\n\\begin{equation} \\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{S}\\,\\vect{\\xi}_{k'}^{(\\nu')}\n = \\delta_{\\nu\\nu'}\\,\\big[\\omega_\\nu\\,\\delta_{kk'}\n + ic_k^{(\\nu)}\\,\\delta_{k+1,k'}\\big]\\,\n \\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}\\,,\n\\label{eq:Jordan-me}\\end{equation}\nand therefore\n\\begin{equation}\\begin{split}\n \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{S}\\,\\mathsf{\\Lambda}_{[\\nu']}\n &= \\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu]}\\,\\mathsf{N\\,S}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\mathsf{N}\\,\\bigg[\\sum_{\\nu\\in[\\nu]}\\sum_{k=1}^{d_\\nu}\n \\frac{\\vect{\\xi}_k^{(\\nu)}\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}}\\bigg]\\,\n \\mathsf{N\\,S}\\,\\bigg[\\sum_{\\nu'\\in[\\nu']}\\sum_{k'=1}^{d_{\\nu'}}\n \\frac{\\vect{\\xi}_{k'}^{(\\nu')}\\,\\bbar{\\vect{\\xi}}_{k'}^{(\\nu')\\dagger}\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_{k'}^{(\\nu')\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}}\\bigg]\\\\\n &= \\delta_{[\\nu],[\\nu']}\\,\\mathsf{N}\\,\\sum_{\\nu\\in[\\nu]}\\bigg[\n \\omega_\\nu\\sum_{k=1}^{d_\\nu}\\mathsf{\\Lambda}_{\\nu,k}\n +i\\sum_{k=1}^{d_\\nu}c_k^{(\\nu)}\\,\n \\frac{\\vect{\\xi}_k^{(\\nu)}\\,\\bbar{\\vect{\\xi}}_{k+1}^{(\\nu)\\dagger}\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_{k+1}^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k+1}^{(\\nu)}}\\bigg]\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{S}_{[\\nu]}\\,.\n\\end{split}\\label{eq:Snu-separation}\\end{equation}\nThe last equality follows\nbecause the quantity summed over $\\nu$ does not depend on $[\\nu']$.\n\\end{proof}\n\\begin{lemma}\\label{lem:proj-sym}\n$\\mathsf{S}_{[\\nu]}$ inherits the symmetry properties of Eq.~(\\ref{eq:S-prop}),\n\\begin{equation} \\mathsf{S}_{[\\nu]}=\\mathsf{S}_{[\\nu]}^\\dagger\\,,\\quad\n\\mathsf{\\Sigma}_x\\,\\mathsf{S}_{[\\nu]}^\\ast\\,\\mathsf{\\Sigma}_x=\\mathsf{S}_{[\\nu]}\\,.\n\\nonumber\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe hermiticity of $\\mathsf{S}_{[\\nu]}$ is obvious from its definition.\nThe second equation is proven as\n\\begin{equation} \\mathsf{\\Sigma}_x\\,\\mathsf{S}_{[\\nu]}^\\ast\\,\\mathsf{\\Sigma}_x\n = \\mathsf{\\Sigma}_x\\,\\mathsf{\\Lambda}_{[\\nu]}^T\\,\\mathsf{S}^\\ast\\,\n \\mathsf{\\Lambda}_{[\\nu]}^\\ast\\,\\mathsf{\\Sigma}_x\n = \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{\\Sigma}_x\\,\\mathsf{S}^\\ast\\,\n \\mathsf{\\Sigma}_x\\,\\mathsf{\\Lambda}_{[\\nu]}\n = \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{S}\\,\\mathsf{\\Lambda}_{[\\nu]}\n = \\mathsf{S}_{[\\nu]}\\,,\n\\end{equation}\nfrom Eq.~(\\ref{eq:proj-prop}).\n\\end{proof}\n\\noindent\nOwing to Prop.~\\ref{theor:SN-proj} and Lemma~\\ref{lem:proj-sym},\n$\\mathsf{S}_{[\\nu]}$ defines the RPA equation within $\\mathcal{W}_{[\\nu]}$\nwith maintaining both the LR- and UL-dualities\\footnote{\nA $d_{[\\nu]}$-dimensional representation of $\\mathsf{M}_{[\\nu]}$\nis provided by the matrix element\n$\\boldsymbol{\\xi}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\boldsymbol{\\xi}_{k'}^{(\\nu')}\n\/ \\sqrt{|(\\bar{\\boldsymbol{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\n\\boldsymbol{\\xi}_k^{(\\nu)})\\,(\\bar{\\boldsymbol{\\xi}}_{k'}^{(\\nu')\\dagger}\\,\n\\mathsf{N}\\,\\boldsymbol{\\xi}_{k'}^{(\\nu')})|}$.\n}.\nThus the RPA equation is decomposed to the equation in each subspace $[\\nu]$.\nFurthermore, Prop.~\\ref{theor:SN-proj} ensures\nthat the same holds for a direct sum of subspaces\n$\\mathcal{W}_{[\\nu]}\\oplus\\mathcal{W}_{[\\nu']}$,\nvia $(\\mathsf{\\Lambda}_{[\\nu]}+\\mathsf{\\Lambda}_{[\\nu']})^\\dagger\\,\n\\mathsf{S}\\,(\\mathsf{\\Lambda}_{[\\nu]}+\\mathsf{\\Lambda}_{[\\nu']})\n=\\mathsf{S}_{[\\nu]+[\\nu']}$, and so forth.\n\nThe whole space $\\mathcal{V}$ is thus decomposed\ninto the direct sum of $\\mathcal{W}_{[\\nu]}$,\n\\begin{equation} \\mathcal{V} = \\bigoplus_{[\\nu]} \\mathcal{W}_{[\\nu]}\\,.\\end{equation}\nLet us denote the complementary space of $\\mathcal{W}_{[\\nu]}$\nby $\\mathcal{W}_{[\\nu]^{-1}}$\nand the relevant projector by $\\mathsf{\\Lambda}_{[\\nu]^{-1}}$;\n\\begin{equation} \\mathcal{W}_{[\\nu]^{-1}} := \\bigoplus_{[\\nu']\\,(\\ne[\\nu])} \\mathcal{W}_{[\\nu']}\\,,\\quad\n \\mathsf{\\Lambda}_{[\\nu]^{-1}} := \\mathsf{1}-\\mathsf{\\Lambda}_{[\\nu]}\n = \\sum_{[\\nu']\\,(\\ne[\\nu])} \\mathsf{\\Lambda}_{[\\nu']}\\,.\n\\label{eq:proj-comple}\\end{equation}\nAnalogously to $\\mathsf{S}_{[\\nu]}$,\n$\\mathsf{S}_{[\\nu]^{-1}}:=\\mathsf{\\Lambda}_{[\\nu]^{-1}}^\\dagger\\,\\mathsf{S}\\,\n\\mathsf{\\Lambda}_{[\\nu]^{-1}}$ defines the RPA equation in $\\mathcal{W}_{[\\nu]^{-1}}$,\nkeeping the LR- and UL-dualities\nand eliminating the solutions within $\\mathcal{W}_{[\\nu]}$,\nyet without influencing the solutions in $\\mathcal{W}_{[\\nu]^{-1}}$.\nTherefore, all the disclosed properties of the RPA solutions in $\\mathcal{V}$\nare transferred to the solutions of the RPA equation\nwithin $\\mathcal{W}_{[\\nu]^{-1}}$.\nIt is noted that the determinant of $\\mathsf{S}$ is decomposed as well,\n\\begin{equation} \\det\\,\\mathsf{S} = \\prod_{[\\nu]} (\\det\\,\\mathsf{S}_{[\\nu]})\\,,\n\\label{eq:detS-decomp}\\end{equation}\nwhere $(\\det\\,\\mathsf{S}_{[\\nu]})$ on the r.h.s.\nis the $d_{[\\nu]}$-dimensional determinant.\nThe l.h.s of the secular equation, $\\det(\\mathsf{S}-\\omega\\,\\mathsf{N})$,\ncan be expressed in an analogous manner.\n\n\\section{Properties of each class of solutions}\n\\label{sec:stable&unstable}\n\nProperties of the RPA solutions are further analyzed\nfor individual class of Prop.~\\ref{prop:eigenvalues}.\nIt deserves commenting here on degeneracy.\nTo the author's best knowledge,\npossibility of degeneracy, particularly of Jordan blocks,\nhas not been examined well, except several specific NG modes.\nAlthough degeneracy occurs even in physical solutions\nunder presence of certain symmetry\n(\\textit{e.g.}, degeneracy with respect to magnetic quantum numbers\nunder the rotational symmetry),\nit does not give rise to Jordan blocks.\nThis is obvious when the conservation law allows us\nto separate the RPA equation into the equations\naccording to the quantum numbers.\nHowever, it is not trivial whether the same holds\nfor a variety of extensive applications of the RPA.\nFor instance, energy levels are highly degenerate in continuum,\nas in the continuum RPA~\\cite{ref:SB75}.\nConsideration of the degeneracy could be relevant to\nhow we can take the continuous limit from arguments on discrete levels.\nFor the NG mode, Thouless restricted himself\nto the case of two-dimensional Jordan blocks.\nWhile higher-dimensional blocks are not very likely to emerge\nin physical situations,\nit will be meaningful to distinguish physical situations\nfrom facts with rigorous mathematical proof.\n\n\\subsection{Solutions for real eigenvalues}\\label{subsec:real-sol}\n\nLet us first consider Classes~\\ref{item:positive} and \\ref{item:negative}\nof Prop.~\\ref{prop:eigenvalues}.\n\n\\begin{proposition}\\label{theor:stability}\nIf the stability matrix $\\mathsf{S}$ is positive-definite,\nthe RPA equation is fully solvable.\nIf the RPA equation is fully solvable, $\\mathsf{S}$ is positive-definite.\n\\end{proposition}\n\\noindent\nAlthough the first part of this proposition was already proven\nin Ref.~\\cite{ref:Thou61},\nI prove it again in combination with the second part.\n\\begin{proof}\nSuppose that $\\mathsf{S}$ is positive-definite.\nThen $\\tilde{\\mathsf{S}}$ in Sec.~\\ref{subsec:redefine}\nis hermitian and therefore diagonalizable\nby a certain matrix $\\tilde{\\mathsf{X}}$,\n$\\tilde{\\mathsf{S}}\\,\\tilde{\\mathsf{X}}\n=\\tilde{\\mathsf{X}}\\,\\tilde{\\mathsf{\\Omega}}$,\nwhere $\\tilde{\\mathsf{\\Omega}}$ is a diagonal matrix.\n$\\tilde{\\mathsf{X}}$ can be unitary, but we shall take another normalization.\nThe eigenvalues in $\\tilde{\\mathsf{\\Omega}}$ are all real and non-zero,\nsince $\\det\\tilde{\\mathsf{S}}=\\det(\\mathsf{N\\,S})\\ne 0$.\nProposition~\\ref{theor:sym-eigen} tells us\nthat they are pairwise, $\\pm\\omega_\\nu$ ($\\nu=1,\\cdots,D$).\nWe here take ${\\displaystyle\\tilde{\\mathsf{\\Omega}}:=\\begin{pmatrix}\n \\mathrm{diag.}(\\omega_\\nu)&0\\\\ 0&-\\mathrm{diag.}(\\omega_\\nu) \\end{pmatrix}}$\nso that $\\omega_\\nu>0$,\nand define $\\mathsf{\\Omega}:=\\mathsf{N}\\,\\tilde{\\mathsf{\\Omega}}$,\nwhich is diagonal and positive-definite.\nLet us adopt the normalization of $\\tilde{\\mathsf{X}}$\nas $\\tilde{\\mathsf{X}}^\\dagger\\,\\tilde{\\mathsf{X}}=\\mathsf{\\Omega}$\n(\\textit{i.e.}, $\\tilde{\\mathsf{X}}\\,\\mathsf{\\Omega}^{-1\/2}$ is unitary)\nand define $\\mathsf{X}:=\\mathsf{S}^{-1\/2}\\,\\tilde{\\mathsf{X}}$.\nThis derives\n\\begin{equation} \\mathsf{S\\,X}=\\mathsf{N\\,X\\,N\\,\\Omega}\\,,\\quad\n \\mathsf{X}^\\dagger\\,\\mathsf{N\\,X}=\\mathsf{N}\\,, \\label{eq:RPAeq-c}\\end{equation}\nproving that the RPA equation is fully solvable because,\nif we write $\\mathsf{X}=(\\vect{x}_1,\\cdots,\\vect{x}_D,\n\\mathsf{\\Sigma}_x\\vect{x}_1^\\ast,\\cdots,\\mathsf{\\Sigma}_x\\vect{x}_D^\\ast)$,\nEq.~(\\ref{eq:RPAeq-c}) yields Eqs.~(\\ref{eq:RPAeq-b}) and (\\ref{eq:norm-b})\nfor $\\nu=1,\\cdots,D$.\\\\\nThis part of the proposition is proven also from Eq.~(\\ref{eq:norm-relation})\nwith $\\nu=\\nu'$ and Corollary~\\ref{cor:non-Jordan}.\nIf $\\mathsf{S}$ is positive-definite,\n$\\vect{x}_\\nu^\\dagger\\,\\mathsf{S}\\,\\vect{x}_\\nu>0$,\nderiving $\\omega_\\nu>0$ and $\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_\\nu>0$\nor both negative, for any solution of the RPA equation (\\ref{eq:RPAeq-b}).\nThen Corollary~\\ref{cor:non-Jordan} ensures\nthat no eigenvector constitutes Jordan blocks.\\\\\nConversely, if the RPA equation is fully solvable as in Eq.~(\\ref{eq:RPAeq-c}),\n$\\det\\mathsf{X}\\ne 0$ and $\\omega_\\nu>0$ for $^\\forall\\nu\\,(=1,\\cdots,D)$.\nThen, since\n\\begin{equation} \\mathsf{S}=\\mathsf{U}^{-1}\\,\\mathrm{diag.}(\\lambda_i)\\,\\mathsf{U}\n =(\\mathsf{N\\,X\\,N})\\,\\mathsf{\\Omega}\\,(\\mathsf{N\\,X\\,N})^\\dagger\\,,\\end{equation}\nit follows that\n\\begin{equation} \\mathrm{diag.}(\\lambda_i)\n =(\\mathsf{U\\,N\\,X\\,N})\\,\\mathsf{\\Omega}\\,(\\mathsf{U\\,N\\,X\\,N})^\\dagger\n\\end{equation}\nand therefore, by expressing $\\mathsf{U\\,N\\,X\\,N}=(\\chi_{i\\nu})$,\n\\begin{equation} \\lambda_i =\\sum_\\nu \\omega_\\nu\\,|\\chi_{i\\nu}|^2 >0\\,;\n\\label{eq:pos-def-reverse}\\end{equation}\nnamely $\\mathsf{S}$ is positive-definite.\nNotice $\\big|\\det(\\mathsf{U\\,N\\,X\\,N})\\big|=\\big|\\det\\,\\mathsf{X}\\big|\\ne 0$,\nwhich excludes the possibility of $\\lambda_i=0$\nin Eq.~(\\ref{eq:pos-def-reverse}).\n\\end{proof}\n\n\\noindent\nTherefore the arguments on physical solutions by Thouless are applicable\neven under the presence of degeneracy as in the continuum.\n\nFor solutions of Class~\\ref{item:negative},\na positive eigenvalue $\\omega_\\nu$ is accompanied by\nan eigenvector $\\vect{x}_\\nu$ with $\\vect{x}_\\nu^\\dagger\\mathsf{N}\\,\\vect{x}_\\nu<0$\nor $\\vect{x}_\\nu^\\dagger\\mathsf{N}\\,\\vect{x}_\\nu=0$.\nIn the former case, its UL-dual partner is normalizable\nbut corresponds to the eigenvalue $-\\omega_\\nu\\,(<0)$.\nThe submatrix $\\mathsf{S}_{[\\nu]}$ of this solution is negative-definite,\nas exemplified in Appendix~\\ref{app:2*2}.\nTherefore the stability matrix $\\mathsf{S}$\nhas two negative eigenvalues at least.\nIn the $\\vect{x}_\\nu^\\dagger\\mathsf{N}\\,\\vect{x}_\\nu=0$ case,\n$\\vect{x}_\\nu$ forms a Jordan block~\\cite{ref:Neergard},\nwhose UL-dual partner associated with $-\\omega_\\nu\\,(<0)$\nbelongs to another Jordan block.\nProbably for this reason, Thouless ignored this class of solutions,\nhaving focused on his arguments near the stability.\n\n\\subsection{Solutions for complex eigenvalues}\\label{subsec:imag-sol}\n\nComplex eigenvalues belong to Classes~\\ref{item:pure-imag}\nand \\ref{item:quartet} of Prop.~\\ref{prop:eigenvalues}.\nI next discuss properties of solutions of Class~\\ref{item:pure-imag}.\n\n\\begin{lemma}\\label{lem:pure-imag-eigen}\nFor an eigenvalue $\\omega_\\nu$ of $\\mathsf{N\\,S}$\nwith $\\mathrm{Re}(\\omega_\\nu)=0$,\nany corresponding eigenvector can be taken so as to satisfy\n$\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{x}_\\nu$ ($\\phi\\in\\mathbf{R}$).\nConversely, if an eigenvector $\\vect{x}_\\nu$ satisfies\n$\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{x}_\\nu$,\n$\\mathrm{Re}(\\omega_\\nu)=0$ holds for its corresponding eigenvalue.\n\\end{lemma}\n\\begin{proof}\n$\\mathrm{Re}(\\omega_\\nu)=0$ is equivalent to $\\omega_\\nu=-\\omega_\\nu^\\ast$.\nTherefore, from Corollary~\\ref{lem:sym-sigma},\nboth $\\vect{x}_\\nu$ and $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$\nbelong to the equal eigenvalue $\\omega_\\nu$,\nwhether they are linearly dependent or independent.\nThen a linear combination of them,\n$\\vect{y}_\\nu:=\\alpha\\,\\vect{x}_\\nu + \\beta\\,\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$\n($\\alpha,\\beta\\in\\mathbf{C}$),\nis also an eigenvector associated with the eigenvalue $\\omega_\\nu$.\nAssuming $\\beta=e^{i\\phi}\\,\\alpha^\\ast$,\nwe verify $\\mathsf{\\Sigma}_x\\vect{y}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{y}_\\nu$.\nWhen $\\vect{x}_\\nu$ and $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$\nare linearly independent,\nwe obtain two independent vectors\nby adopting , \\textit{e.g.}, $\\alpha=-\\beta=1$ and $\\alpha=\\beta=i$.\\\\\nIf $\\omega_\\nu\\ne-\\omega_\\nu^\\ast$,\nthe associating eigenvectors\n$\\vect{x}_\\nu$ and $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$\nmust be linearly independent;\nnamely $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{x}_\\nu$ is impossible.\n\\end{proof}\n\\noindent\nIt is noted here that $\\mathrm{Re}(\\omega_\\nu)=0$ covers\nthe solutions of Class~\\ref{item:null} as well as \\ref{item:pure-imag}\nin Prop.~\\ref{prop:eigenvalues}.\n\nIf $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{x}_\\nu$ is assumed\nfor the $\\mathrm{Re}(\\omega_\\nu)=0$ case,\nthe lower $D$-dimensional components of the RPA equation (\\ref{eq:RPAeq-b})\nis only a repetition of the upper $D$-dimensional components.\nWe can choose the phase $e^{i\\phi}$ arbitrarily,\nbecause it is controllable via a transformation\n$\\vect{y}_\\nu=e^{i\\theta}\\vect{x}_\\nu$.\nA convenient choice is $e^{i\\phi}=-1$, so that ${\\displaystyle\\vect{x}_\\nu\n= -\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast\n= \\begin{pmatrix} X^{(\\nu)}\\\\ -X^{(\\nu)\\ast}\\end{pmatrix}}$.\n\nLet us now take $c_k^{(\\nu)}\\in\\mathbf{R}$ in Eq.~(\\ref{eq:Jordan}).\n\\begin{lemma}\\label{lem:pure-imag-Jordan}\nFor an eigenvalue $\\omega_\\nu$ of $\\mathsf{N\\,S}$\nwith $\\mathrm{Re}(\\omega_\\nu)=0$,\nall corresponding basis vectors can satisfy\n$\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}=-\\vect{\\xi}_k^{(\\nu)}$;\n\\textit{i.e.}, ${\\displaystyle\\vect{\\xi}_k^{(\\nu)}\n = \\begin{pmatrix} \\Xi^{(\\nu,k)}\\\\ -\\Xi^{(\\nu,k)\\ast}\\end{pmatrix}}$.\n\\end{lemma}\n\\noindent\nThis lemma states that solutions of Classes~\\ref{item:pure-imag}\nand \\ref{item:null} can be self UL-dual,\ntogether with basis vectors generated from them.\n\\begin{proof}\nIn the case that $\\mathrm{Re}(\\omega_\\nu)=0$,\nEqs.~(\\ref{eq:Jordan}) and (\\ref{eq:Jordan-sym}) come\n\\begin{equation}\\begin{split}\n (\\mathsf{S}-\\omega_\\nu\\,\\mathsf{N})\\,\\vect{\\xi}_{k+1}^{(\\nu)}\n &= ic_k^{(\\nu)}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}\\,,\\\\\n (\\mathsf{S}-\\omega_\\nu\\,\\mathsf{N})\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_{k+1}^{(\\nu)\\ast}\n &= ic_k^{(\\nu)}\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}\\,.\n\\end{split}\\label{eq:Jordan-sym2}\\end{equation}\nFrom Lemma~\\ref{lem:pure-imag-eigen} and the argument above,\nwe can assume $\\mathsf{\\Sigma}_x\\vect{\\xi}_1^{(\\nu)\\ast}=-\\vect{\\xi}_1^{(\\nu)}$.\nIf $\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}=-\\vect{\\xi}_k^{(\\nu)}$,\nthe first equation of (\\ref{eq:Jordan-sym2}) indicates\nthat the second equation has a solution fulfilling\n$\\mathsf{\\Sigma}_x\\vect{\\xi}_{k+1}^{(\\nu)\\ast}=-\\vect{\\xi}_{k+1}^{(\\nu)}$.\nThe lemma is then proven inductively.\n\\end{proof}\n\\noindent\nCompatibility of this lemma with Prop.~\\ref{theor:one-to-one_basis}\nis confirmed in Appendix~\\ref{app:sub-B1}.\nUnder the above convention for the $\\mathrm{Re}(\\omega_\\nu)=0$ case,\nthe normalization condition of $\\vect{\\xi}_{k}^{(\\nu)}$ can be\n\\begin{equation} \\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}\n = \\bar{\\Xi}^{(\\nu,k)\\dagger}\\,\\Xi^{(\\nu',k')}\n - \\big[\\bar{\\Xi}^{(\\nu,k)\\dagger}\\,\\Xi^{(\\nu',k')}\\big]^\\ast\n = \\pm i\\delta_{\\nu\\nu'}\\,\\delta_{kk'}\\,\n\\label{eq:imag-norm}\\end{equation}\nalthough $\\vect{\\xi}_{k}^{(\\nu)}$ is not normalizable\nin the respect of Def.~\\ref{def:normalizable}.\n\nConsider solutions in vicinity of the stability,\nin which the stability matrix $\\mathsf{S}$ has a single negative eigenvalue.\nNear the stability,\nthe subspace providing negative $\\det\\,\\mathsf{S}_{[\\nu]}$ can be separated out\nby using the projector in Sec.~\\ref{sec:decomp},\nwhich should be two dimensional and therefore provides\n${\\displaystyle\\mathsf{S}_{[\\nu]}=\\begin{pmatrix} a& b\\\\\nb^\\ast& a\\end{pmatrix}}$ ($a\\in\\mathbf{R}$, $b\\in\\mathbf{C}$),\nas in Appendix~\\ref{app:2*2}.\nA pair of pure-imaginary eigenvalues is obtained,\nillustrating that the first unphysical solution\nemerges as Class~\\ref{item:pure-imag} of Prop.~\\ref{prop:eigenvalues}.\n\nUnlike the self LR-duality for real eigenvalues,\nthe self UL-duality does not forbid Jordan blocks,\nalthough most pure-imaginary eigenvalues are expected\nnot to form Jordan blocks.\nAn example of Jordan blocks is presented in Appendix~\\ref{app:4*4-pure-imag}.\n\nLet us turn to solutions of Class~\\ref{item:quartet}.\nQuartet solutions are a manifestation of the two types of dualities.\nThe possibility of quartet solutions\nwas first pointed out in Ref.~\\cite{ref:UR71}\nfor $\\mathsf{S}=\\mathsf{S}^\\ast$ cases,\nand mentioned in Ref.~\\cite{ref:SM84} in more general context.\nA minimal model for quartet solutions is constructed by taking $D=2$,\nand is analyzed in Appendix~\\ref{app:4*4-quartet}.\nFor quartet solutions $\\nu$, $d_{[\\nu]}$ is a multiple of four.\nHence, by denoting the solutions $\\pm\\alpha\\pm i\\beta$\n($\\alpha,\\beta\\in\\mathbf{R}$),\n$\\det\\,\\mathsf{S}_{[\\nu]}=\\det(\\mathsf{N}_{[\\nu]}\\,\\mathsf{S}_{[\\nu]})\n=(\\alpha^2+\\beta^2)^{d_{[\\nu]}\/2}>0$.\nAs $\\mathsf{S}$ cannot be positive-definite\non account of the latter part of Prop.~\\ref{theor:stability},\nEq.~(\\ref{eq:detS-decomp}) indicates\nthat $\\mathsf{S}$ has at least two negative eigenvalues\nfor quartet solutions to come out.\n\n\\subsection{NG-mode solutions}\\label{subsec:NG-sol}\n\nThe simplest example of the NG-mode solution\nis given in Appendix~\\ref{app:2*2}, by the $2\\times 2$ stability matrix.\nIt illustrates that the null eigenvalue\nis often associated with a two-dimensional Jordan block,\nas indicated by Thouless~\\cite{ref:Thou61}.\nThe NG modes that generate two-dimensional Jordan blocks\nhave well been investigated~\\cite{ref:RS80,ref:Thou61,ref:TV62}.\nHowever, in the example of Appendix~\\ref{app:2*2},\nthere is a trivial case of $\\mathsf{S}=\\mathsf{0}$\nin which two $d_\\nu=1$ eigenvectors are present for the null eigenvalues.\nMoreover, an example of 4-dimensional Jordan block\nis seen in Appendix~\\ref{app:4*4-null}.\nLikely or not, it is difficult to exclude the possibilities\nother than the two-dimensional Jordan block for the null eigenvalue\nonly from mathematical viewpoints.\n\n\\begin{corollary}\\label{cor:deg-null}\nIf there exists a null eigenvalue for $\\mathsf{N\\,S}$,\nit must have even number of degeneracy.\n\\end{corollary}\n\\begin{proof}\nBecause of Prop.~\\ref{theor:sym-eigen},\nthe number of non-zero eigenvalues must be even, up to their degeneracies.\nMoreover, Lemma~\\ref{lem:Jordan-sym} ensures\nthat sum of dimensions of Jordan blocks for non-zero eigenvalues is even.\nThe total dimension of $\\mathsf{N\\,S}$ is $2D$,\nwhich concludes the degeneracy of the null eigenvalue must be even.\nAlso proven from Lemma~\\ref{lem:dim-Lambda}.\n\\end{proof}\n\nWhen SSB occurs, there must be NG-mode solutions\ncorresponding to the broken symmetry;\n\\textit{e.g.}, the linear momentum in the SSB with respect to the translation\nand the angular momentum in the SSB with respect to the rotation\nin deformed nuclei.\nFor specific NG-mode solutions with such physical interpretations,\ntheir properties can further be explored,\nthough I do not pursue this direction in this article.\n\nThe null eigenvalues may lie at the intersection\nof the self LR- and the self UL-dualities.\nAlthough there is no single eigenvector having both of the self dualities\nas indicated by Lemma~\\ref{lem:dim-Lambda},\nthere could be an even-dimensional Jordan block\nin which the LR-duality closes by its basis vectors,\nwhile keeping the self UL-duality of Lemma~\\ref{lem:pure-imag-Jordan}.\nIn such cases the Jordan block, instead of the basis vectors,\nmay be said self LR-dual.\n\\begin{proposition}\\label{theor:null-self-dual}\nFor even-dimensional Jordan blocks associated with a null eigenvalue,\nit is possible to produce basis vectors $\\big\\{\\vect{\\eta}_k;\nk=1,\\cdots,d_\\nu\\big\\}$ having double self duality,\n$\\mathsf{\\Sigma}_x\\vect{\\eta}_k^\\ast=-\\vect{\\eta}_k$ and\n$\\bbar{\\vect{\\eta}}_k=\\vect{\\eta}_{d_\\nu+1-k}$.\n\\end{proposition}\n\\noindent\nThis proposition is proven in Appendix~\\ref{app:sub-B2}.\nAn example of the transformation shown in Appendix~\\ref{app:sub-B2}\nis given by the NG mode of the angular momentum,\nunder SSB with respect to the rotation.\nEven though the Jordan blocks corresponding to $J_\\pm$\nare the LR-dual of each other,\ntheir linear combinations provide those corresponding to $J_x$ and $J_y$,\neach of which could be self LR-dual.\n\nLet us focus on the case that the basis vectors simultaneously fulfill\n$\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}=-\\vect{\\xi}_k^{(\\nu)}$\nand $\\bbar{\\vect{\\xi}}_k^{(\\nu)}=\\vect{\\xi}_{d_\\nu+1-k}^{(\\nu)}$, with even $d_\\nu$.\nThen the projector of Eq.~(\\ref{eq:proj-nuk})\nhas the relation $\\mathsf{N\\,\\Lambda}_{\\nu,k}^\\dagger\\,\\mathsf{N}\n=\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}$,\nand therefore obeys\n\\begin{equation}\\begin{split} &\\mathsf{\\Lambda}_{\\nu,k}+\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}\n =\\mathsf{N}\\,\\big[\\mathsf{\\Lambda}_{\\nu,k}\n +\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}\\big]^\\dagger\\,\\mathsf{N}\\,,\\\\\n&\\mathsf{\\Sigma}_x\\,\\big[\\mathsf{\\Lambda}_{\\nu,k}\n +\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}\\big]^\\ast\\,\\mathsf{\\Sigma}_x\n = \\mathsf{\\Lambda}_{\\nu,k}+\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}\\,\n\\label{eq:null-proj2}\\end{split}\\end{equation}\nalthough the relation analogous to Prop.~\\ref{theor:SN-proj}\n(Eq.~(\\ref{eq:Snu-separation}), in particular)\ndoes not necessarily hold.\nLike the arguments using $\\mathsf{\\Lambda}_{[\\nu]}$ in Sec.~\\ref{sec:decomp},\n$\\mathsf{\\Lambda}_{\\nu,k}+\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}$\nproduces a subspace keeping both the LR- and UL-dualities.\nFor doubly self-dual Jordan blocks,\nthe basis vectors for the NG-mode may be removed two by two via the projectors,\nwith minimal d.o.f. coupled to them.\nWithin the two-dimensional subspace\ngiven by $\\mathsf{\\Lambda}_{\\nu,d_\\nu\/2}+\\mathsf{\\Lambda}_{\\nu,d_\\nu\/2+1}$,\nthe prescription proposed in Refs.~\\cite{ref:Thou61,ref:TV62}\n(and well summarized in Refs.~\\cite{ref:RS80,ref:Row70})\nis applicable.\n\nIf $d_\\nu=\\mathrm{odd}$, there should be two Jordan blocks,\nwhich are the partner of the LR- (or UL-) duality of each other.\nTo separate them, one may apply the projector $\\Lambda_{[\\nu]}$\nintroduced in Sec.~\\ref{sec:decomp}.\n\n\\section{Summary}\\label{sec:summary}\n\nProperties of solutions of the RPA equation is reanalyzed\nin terms of the linear algebra.\nAs well as eigensolutions, cases in which the matrix $\\mathsf{N\\,S}$\n(and $\\mathsf{S\\,N}$) forms Jordan blocks are examined.\nTwo types of dualities of eigenvectors and basis vectors,\nwhich are called LR- and UL-dualities in this article,\nare pointed out and explored.\nThese dualities are useful to clarify properties of the RPA solutions.\nProjection respecting the dualities is developed.\n\nEigenvalues given by the RPA equation are classified\ninto five classes, in Prop.~\\ref{prop:eigenvalues}.\nAs pointed out by Thouless, all solutions are physical ones\nif the stability matrix is positive-definite.\nIts opposite is also true (in absence of NG modes),\nbeing useful to judge stability of a MF solution\nfrom numerical calculations in the RPA.\nThese solutions are singled out, not constituting Jordan blocks,\nand have the self LR-duality while are paired by the UL-duality.\nEigenvectors and basis vectors for pure-imaginary eigenvalues\ncan be made self UL-dual, and paired by the LR-duality.\nWith no self dualities, quartet solutions manifest two types of the dualities.\nNG-mode solutions, which are associated with the null eigenvalue\nand often related to the spontaneous symmetry breaking,\nlie at intersection of the two self dualities.\nHowever, a single vector cannot be both self LR-dual and self UL-dual.\nOnly even-dimensional Jordan blocks can have double self dualities.\nThe well-known prescription of separating out the NG modes\ncould be applicable to such cases.\n\n\\section*{Acknowledgment}\n\nThe author is grateful to K.~Matsuyanagi, K.~Neerg\\aa rd, H.~Kurasawa, Y.R.~Shimizu,\nJ.~Terasaki and T.~Inakura for discussions.\nThis work is financially supported in part\nby JSPS KAKENHI Grant Number~24105008 and Grant Number~16K05342.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{  Introduction and Terminology}  \nIn 1996, Matheson and Tarjan [MT] proved that if $G$ is a plane triangulated disc then\n$\\gamma (G)\\le |V(G)|\/3$.\n(In particular, then, the same bound on $\\gamma$ applies to any triangulation of the plane.)\nPlummer and Zha [PZ] proved that if $G$ is a triangulation of the projective plane, then $\\gamma (G)\\le |V(G)|\/3$\nand if $G$ is a triangulation of either the torus or Klein bottle, then $\\gamma (G)\\le \\lceil |V(G)|\/3\\rceil$.\nThe latter result was sharpened by Honjo et al. [HKN] who showed that $\\gamma (G)\\le |V(G)|\/3$\nfor graphs embedded in these two surfaces.\nThey also showed that for any surface $\\Sigma$, there is a positive integer $\\rho (\\Sigma )$ such that\nif $G$ is embedded as a triangulation of $\\Sigma$ and the embedding has face-width at least $\\rho (\\Sigma )$,\nthen $\\gamma (G)\\le |V(G)|\/3$.\nEven more recently, Furuya and Matsumoto [FM] generalized this result by showing that $\\gamma (G)\\le |V(G)|\/3$,\nfor every triangulation $G$ of any closed surface.\\par\n\nIn [MT], Matheson and Tarjan also conjectured that their bound of $n\/3$ could be improved; namely, if $n$ is sufficiently large, then any triangulation of the plane with $n$ vertices would have $\\gamma\\le n\/4$.\nThe \ntriangle has $\\gamma (K_3)=1 = n\/3$, the octahedron shown in Figure 1.1(i) has $\\gamma=2=n\/3$ and the 7-vertex graph shown in Figure 1.1(ii) has $\\gamma = 2\/7>1\/4$, so this shows that\none must assume $n\\ge 8$ in order for the Matheson-Tarjan conjecture to be true.\n\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.75]{2Counterexamples.eps} \n\\centerline{\\bf Figure 1.1}\n\\end{center}\n\\end{figure}\n \n \nMore generally, in [PZ] the first and third authors of the present paper conjectured that if $G$ is a\ntriangulation of {\\it any}\nnon-spherical surface, then\n$\\gamma (G)\\le n\/4$.\nBoth these conjectures involving the $n\/4$ bound remain unsettled.\\par \n \n \n\nIn 2010, King and Pelsmajer [KP] proved the Matheson-Tarjan bound of n\/4 holds in the plane \ncase when the maximum degree of the triangulation is 6.\\par\n\nAn outerplanar graph is a graph embedded in the plane in such a way that all vertices of the graph lie on the boundary of the external face.\nAn outerplanar graph is {\\it maximal} (outerplanar) if it is not possible to add any new edge to $G$ without destroying outerplanarity.\nIn 2013, Campos and Wakabayashi [CW] proved that if $G$ is a maximal outerplanar graph with at least $n\\ge 4$ vertices, then $\\gamma (G)\\le (n+t)\/4$, where $t$ is the number of vertices of degree 2 in $G$.\n\n\nIt was proved independently by N\\\"unning [N] and by Sohn and Yuan [SY] that for any graph $G$ with\n$n$ vertices and minimum degree $\\delta (G)\\ge 4$, $\\gamma (G)\\le 4n\/11$.\nIn the present paper we will show that if $G$ is a plane triangulation on $n\n$ vertices which has a Hamilton cycle and $\\delta (G)\\ge 4$, then $\\gamma (G)\\le \\max\\{\\lceil 2n\/7\\rceil, \\lfloor 5n\/16\\rfloor\\}$.\nIt follows immediately that if $G$ is a 4-connected plane triangulation on $n\n$ vertices, then\n $\\gamma(G)\\le \\max\\{\\lceil 2n\/7\\rceil, \\lfloor 5n\/16\\rfloor\\}$.\nNote that if $n\\notin \\{6,8,9,11,12,15,19,22,25\\}$, $\\lceil 2n\/7\\rceil \\le \\lfloor 5n\/16\\rfloor$, so this result can be restated to\nsay that if $G$ is a 4-connected plane triangulation on $n$ vertices and $n\\ge 26$, then $\\gamma (G)\\le \\lfloor 5n\/16\\rfloor$. \n\n\n\\section{Preferred Hamilton cycles in plane triangulations}\n\nLet $G$ be a plane triangulation with $\\delta (G)\\ge 4$ and suppose $G$ contains a Hamilton cycle $H$.\nWe can think of $H$ bounding a triangulated inner subgraph $G_{int}$ and a triangulated outer subgraph\n$G_{ext}$ such that $G_{int}\\cap G_{ext}= H$.\nSuppose $v\\in V(G)$.\nWe denote by $i\\deg v$ (respectively, $o\\deg v$) the degree of vertex $v$ in $G_{int}$ (resp. in $G_{ext}$).\n\\par\n\nWe will need to pay particular attention to those vertices which have $i\\deg v = 2$ or $o\\deg v=2$.\nWe will call these vertices {\\it 2-vertices}.\nExamples are shown in Figure 2.1 where $i\\deg v_1=2$ and $o\\deg v_2=2$ respectively and hence each is an example of\na 2-vertex.\\par\n\n\\begin{figure}[!hbtp]\\refstepcounter{figure}\n\\begin{center}\n\\includegraphics[scale=.45]{DomHam1.eps} \n\\centerline{\\bf Figure 2.1}\n\\end{center}\n\\end{figure}\n\nLet $G$ be a Hamiltonian plane triangulation.  Such a graph may have many Hamilton cycles.\nWe now show how to select such a cycle with certain properties we shall need later on.\\par\n\nTo this end, let begin with any Hamilton cycle $H$ in $G$.\nA triangle $T$ of $G_{int}$ will be called {\\it internal} (with respect to $H$) if $E(T)\\cap E(H)=\\emptyset$.\n\\par\n\\medskip\n\\noindent\n{\\bf Lemma 2.1:}  Let $G$ be a plane triangulation with $\\delta (G)\\ge 4$ which contains a Hamilton cycle.\nSuppose $\\gamma (G)\\ge 2$.\nThen there exists a Hamilton cycle in $G$ containing no three consecutive 2-vertices.\\par\n\\medskip\\noindent\n{\\bf Proof:}  First, suppose that $G$ contains four consecutive 2-vertices.\nSince $G$ is a triangulation, these four 2-vertices must alternate between $ideg=2$ and $o\\deg=2$.\nSo let us suppose that we have four consecutive 2-vertices forming a subpath of $H$, which we will denote by\n$wxyz$, such that \n$i\\deg w = i\\deg y = 2$ and $o\\deg x = o\\deg z=2$.\nLet the predecessor of $w$ be $a$ and the successor of $z$ be $b$.\nNote that $a\\ne b$, for if $a=b$, then $G=K_5$ contradicting the hypothesis that $G$ is planar.\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.4]{DomHam2.eps} \n\\centerline{\\bf Figure 2.2}\n\\end{center}\n\\end{figure}\n\n \nForm a new Hamilton cycle $H'$ substituting for the path $awxyzb$ the path $awyxzb$ as indicated by the dashed path shown in Figure 2.2.\nWith respect to the original Hamilton cycle $H$, $w,x,y$ and $z$ are 2-vertices.\nHowever, with respect to $H'$, only $x,y$ and (possibly) $b$ are 2-vertices.\n(Note that the vertices $w$ and $z$ are {\\it not} 2-vertices with respect to $H'$ because the degrees of $w$ and$z$ are both at least 4.)\\par\n\nNext suppose that $H$ does not contain four consecutive 2-vertices, but suppose it does contain three consecutive 2-vertices.\nLet three such consecutive 2-vertices be, in order, $x,y$ and $z$.\nLet the predecessor of $x$ be $a$ and the successor of $z$ be $b$.\nAgain, $a\\ne b$ as before.\nLet the neighbor of $b$ on $H$ different from $z$ be denoted as $b'$.\\par\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.35]{DomHam3.eps} \n\\centerline{\\bf Figure 2.3}\n\\end{center}\n\\end{figure}\n\nNow suppose $b$ has another neighbor via an edge in $G_{int}$ other than $y,z$ and $b'$.\nThen replace the path $axyzb$ by $axzyb$ in $H$ to obtain a new Hamilton cycle $H'$.\nThen whereas $H$ has $x,y$ and $z$ as 2-vertices, $H'$ has vertices $y$ and $z$\nas 2-vertices.\nBut neither $x$ and $b$ is a 2-vertex with respect to $H'$ since the degree of $x$ is at least 4 and because of our assumption on the fourth neighbor of $b$.\n\n\nThus $H'$ has one less triple of successive 2-vertices than does $H$.\n(See Figure 2.3.)\n\n\n \n\nSuppose, on the other hand, that the only neighbors of vertex $b$ via edges in $G_{int}$ are $y,z$ and $b'$.\nHence $y$ is adjacent to $b'$.\nIn this case, vertex $b$ will be a new 2-vertex with respect to $H'$, where $H'$ is obtained from $H$ be replacing the path $axyzb$\nby the path $axzyb$ and hence the total number of 2-vertices in $H'$ is the same as in $H$.\n\nMore generally, suppose that $y$ is adjacent to $b'=b_0, b_1,\\ldots,b_k,b_{k+1}=b'$, but not adjacent to $b_{k+2}$,\nwhere $b_0b_1\\cdots b_kb_{k+1}b'$ is a subpath of $H$.\nIn this case we replace the path $axyzb_0b_1,\\cdots b_kb'$ with\n$axzb_0b_1\\cdots b_kyb'$ to obtain a new Hamilton cycle $H'$.\nNote that none of $b=B_1,b_2,\\ldots,b_k$ is a 2-vertex relative to $H'$ since none of these vertices is a\n2-vertex with respect to $H$.\nMoreover, whereas $H$ contained consecutive 2-vertices $x,y,z$, these have been replaced with $y$ and $b_k$ which are 2-vertices with respect to cycle $H'$.\\par\n\nNote that it is possible that $y$ is adjacent via an edge in $G_{int}$ to every vertex in $G$\n(cf. Figure 2.4), but in that case\n$\\gamma (G)=1$.\n\n\n\\begin{figure}[!hbtp]\n\\begin{center}\n\\includegraphics[scale=.43]{DomHam4.eps} \n\\centerline{\\bf Figure 2.4}\n\\end{center}\n\\end{figure} \n\n\\section{ $(H,A,B,O)$-graphs}\\par\n\n\\medskip\n\nIn this section we introduce the concept of an $(H,A,B,O)$-graph and study some of its properties.\\par\n\nLet $G$ be a\n(not necessarily planar)\ngraph on $n$ vertices with a Hamilton cycle $H = x_1x_2\\cdots x_nx_1$.\\par\n\nA subgraph of $G$ consisting of a 3-cycle $x_ix_{i+1}x_{i+2}x_i$, where $\\deg_G (x_{i+1})=2$, will be\ndenoted by\n{\\it $A$}.\nA subgraph on four vertices consisting of the path $x_ix_{i+1}x_{i+2}x_{i+3}$, together with the edges\n$x_ix_{i+2}$ and $x_{i+1}x_{i+3}$\nsuch that $\\deg_G(x_{i+1}) = \\deg_{G}x_{i+2}=3$\nwill be denoted by {\\it $B$}.\nA third type of configuration, denoted by $O$, consists of a single edge with endvertices included\nand such that if $x_i, i=1,2,$ is an endvertex of an $O$, then there is no edge of the form $x_{i-1}x_{i+1}$.\nFurther, if\nan $A$ is immediately preceded and immediately succeeded by $O$s on the Hamilton cycle $H$, we will say that this $A$ is {\\it isolated}.\\par\n\\medskip\n\\noindent\n{\\bf Def.:}  An {\\it $(A,B)$-string} is a connected maximal induced subgraph of $G$ consisting of $A$s and $B$s.\nA {\\it mixed} $(A,B)$-string is a string containing at least one $A$ and at least one $B$.\\par\n\\medskip\n\n\n\nWe now suppose $G$, together with a Hamilton cycle $H$ contained in $G$, satisfies the following three assumptions.\\par\n\\medskip\n\\noindent\n(1) \nSuppose all edges in $E(G)-E(H)$ are of the form $x_ix_{i+2}$, ($\\mod n$).\nSuch edges will be called {\\it 2-chords}.\\par\n\\medskip\n\n\n\\noindent\n(2) Suppose further that $E(G)$ consists only of edges lying in the Hamilton cycle $H$ together with edges lying in either an\n$A$ or a $B$.\n\n\\par\n\\medskip\n\\noindent\n(3) Finally, assume that $G$ consists only of $A$s, $B$s and $O$s, where\nany pair intersect in at most one vertex.\\par\n\\medskip\nSuch a graph $G$, together with a Hamilton cycle $H$ in $G$, is called an\n{\\it (H,A,B,O)-graph}.\n(The reader is referred to Figure 3.1 for an example of such a graph.)\n\n\n\\begin{figure}[!hbtp]\\refstepcounter{figure}\n\\begin{center}\n\\includegraphics[scale=.45]{AsectionBsectionGraph.eps} \n\\centerline{\\bf Figure 3.1}\n\\end{center}\n\\end{figure}\n\n\nNote that an $(H,A,B,O)$-graph is planar.\nNote also that it follows from Lemma 2.1 that every outerplanar triangulation with $\\delta\\ge 4$ which contains a Hamilton cycle must contain a spanning $(H,A,B,O)$-subgraph.\\par\n\n\n\nLet ${\\cal K}$ denote the collection of all $(H,A,B,O)$-graphs with at least seven vertices and let\n${\\cal K}_{\\lceil (n+1)\/2\\rceil}$ denote the subclass of ${\\cal K}$ consisting of those members of ${\\cal K}$ with\n$n = |V(K)|$ vertices and in which the\nHamilton cycle $H$ possesses at least $(n+1)\/2$ chords.\nLet $K$ be a member of ${\\cal K}_{\\lceil (n+1)\/2\\rceil}$.\nWe now introduce seven operations,\ncalled {\\it reductions}.\nIn each of these reductions we replace a certain subgraph of $K\\in {\\cal K}_{\\lceil (n+1)\/2\\rceil}$ with a certain smaller\nsubgraph so as\nto produce a new graph $K'\\in {\\cal K}_{\\lceil (n'+1)\/2\\rceil}$ for some $n'<n$.\\par\nWe will also use a transformation which we will call a {\\it switch} in which we replace a certain subgraph\nof $K\\in {\\cal K}_{\\lceil(n+1)\/2\\rceil}$ by another of the {\\it same} length so as to produce a different graph which also\nbelongs to ${\\cal K}_{\\lceil(n+1)\/2\\rceil}$.\nWe then proceed to investigate how certain upper bounds on the domination number are affected when each of these\neight operations is carried out.\n\\par\n\nIn all switches and reductions to follow, the edges of $A$s and $B$s which are not edges of the Hamilton cycle $H$\nmay lie on either side of $H$, but of course no two such edges may cross since $K$ is planar.\\par\n\n\\medskip\n\\noindent\n{\\bf Switch:}  Suppose $K\\in {\\cal K}_{\\lceil (n+1)\/2\\rceil}$.\nWe modify $K$ by performing the replacment shown in\nFigure 3.2, so as to form a new graph $K'$. \\medskip\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure32.eps} \n\\centerline{\\bf Figure 3.2}\n\\end{center}\n\\end{figure}\n\n  \n\\noindent\n{\\bf Claim (a)}\n$K'\\in {\\cal K}_{\\lceil(n'+1)\/2\\rceil}$.\nThis is clear since $K$ and $K'$ have the same number of 2-chords.\\par\n\\medskip\n\\noindent\n{\\bf Claim (b)}  $\\gamma (K)\\le \\gamma (K')$.\\par\n\\medskip\n\\noindent\n{\\it Proof of Claim (b):}\nTo prove Claim (b) we present a case-by-case analysis.\nSuppose 1 and 6 are both in a minimum dominating set $D'$ for $K'$.\nThen a third vertex is required to dominate 4 in $K'$. \nSo then we may use 1,3 and 6 to dominate $K$.\nSuppose 1 is in $D'$ and 6 is not in $D'$.\nTo dominate 5 in $K'$, one of 4 and 5 must be in $D'$.\nThen use 1 and 4 to dominate $K$.\nSuppose 1 is not in $D'$, but 6 is in $D'$.\nThen one of 3 and 4 is in $D'$, so use 2 and 6 to dominate $K$.\nSuppose neither 1 nor 6 is in $D'$.\nThen two of 2, 3, 4 and 5 must be in $D'$,\nso replace these two vertices with 2 and 4 to dominate $K$.\nThis proves Claim (b).\\par\n\\medskip\nWe now turn our attention to reductions.\nWe formulate seven of these.\nAgain suppose $K\\in{\\cal K}_{\\lceil (n+1)\/2\\rceil }$.\nLet $t$ denote the number of 2-chords in $K$.\n\n\\medskip\n\\noindent\n{\\bf Reduction 1.}  Suppose $n\\ge 11$.\nReplace an $ABA$-segment in $K$ by a $B$-segment to obtain $K'$ as shown in Figure 3.3.\n\n\\medskip\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure33.eps} \n\\centerline{\\bf Figure 3.3}\n\\end{center}\n\\end{figure}\n \n\nIn this case, $n'=n-4$, $t'=t-2\\ge (n+1)\/2 -2=(n-3)\/2=(n'+1)\/2$,\nand hence $K'\\in {\\cal K}_{\\lceil(n'+1)\/2\\rceil}$.\\par\n\\medskip\n\\noindent\n{\\bf Claim:} \n$\\gamma (K)\\le \\gamma(K')+1$ and if $\\gamma (K')\\le |V(K')|\/4$, then $\\gamma (K)\\le |V(K)|\/4$.\\par\n\\medskip\n\\noindent\n{\\it Proof of Claim:}\nLet $D'$ be a minimum dominating set in $K'$.\nSuppose 1 and 8 are in $D'$.\nThen insert either 4 or 5 to obtain a dominating set $D$ for $K$ with $|D|=|D'|+1$.\nSuppose 1 is in $D'$, but 8 is not in $D'$.\nIn this case we add 6 to $D'$ to dominate $K$.\nSuppose 1 is not in $D'$, but 8 is in $D'$.\nThis case is symmetric to the preceding case.\nSuppose neither 1 nor 8 is in $D'$.\nThen one of $x$ and $y$ is in $D'$.\nSo we add 3 and 6 to dominate $K$.\nIn addition, if $\\gamma (K')\\le n'\/4$, then $\\gamma (K)\\le n'\/4 +1 = (n'+4)\/4 = n\/4$.\nThis proves the Claim.\\par\n\n\\medskip\n\\noindent\n{\\bf Reduction 2.}  Suppose that $n\\ge 11$.\nReplace an $AA$-segment by a single vertex in $K$ to obtain $K'$ as shown in Figure 3.4.\n\\medskip\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure34.eps} \n\\centerline{\\bf Figure 3.4}\n\\end{center}\n\\end{figure}\n\n\nIn this case, $n'=n-4$, $t'=t-2\\ge (n+1)\/2-2=(n-3)\/2=(n'+1)\/2$, and $K'\\in {\\cal K}_{\\lceil (n'+1)\/2\\rceil}$.\n\\medskip\n\\noindent\n{\\bf Claim:}  $\\gamma (K)\\le \\gamma (K')+1$ and if $\\gamma (K')\\le |V(K')|\/4$, then\n$\\gamma (K)\\le |V(K)|\/4$.\\par\n\\medskip\n\\noindent\n{\\it Proof of Claim:}  Let $D'$ be a minimum dominating set in $K'$.\nIf 1 is in $D'$, add vertex 5 to $D'$ to dominate $K$.\nIf 1 is not in $D'$, then\nadd 3 to $D'$ to dominate $K$.\nMoreover, if $\\gamma (K')\\le n'\/4$, then $\\gamma (K)\\le n'\/4 + 1 = (n'+4)\/4 = n\/4$.\nThis proves the Claim.\\par\n\\medskip\\noindent\n{\\bf Reduction 3.}  Suppose $n\\ge 11$.\nPerform the transformation shown in Figure 3.5 to obtain $K'$.\\par\n\\medskip\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure35.eps} \n\\centerline{\\bf Figure 3.5}\n\\end{center}\n\\end{figure}\n\nIn this case, $n'=n-4$, $t'=t-2\\ge (n+1)\/2-2=(n-3)\/2=(n'+1)\/2$ and\n$K'\\in {\\cal K}_{\\lceil(n'+1)\/2\\rceil}$.\\par\n\\medskip\n\\noindent\n{\\bf Claim:} \n$\\gamma (K)\\le \\gamma (K')+1$\nMoreover, if $\\gamma (K')\\le |V(K')|\/4$, then $\\gamma (K)\\le |V(K)|\/4$.\\par\n\\medskip\n\\noindent{\\it Proof of Claim:}\nLet $D'$ be a minimum dominating set in $K'$.\nIf 1 is in $D'$,\nThen use 1 and 5 to dominate $K$.\nIf 1 is not in $D'$ and 1 is dominated from above, add 3 to dominate $K$.\nIf 1 is not in $D'$ and 1 is dominated from below,\nadd 2 to dominate $K$.\nThis completes the proof of the Claim.\\par\n\\medskip\nIn the next two reductions, we reduce the total number of vertices by seven.\nWe want to show that if $\\gamma (K')\\le (2\/7)|V(K')|$, then $\\gamma (K)\\le (2\/7)|V(K)|$.\\par\n\\medskip\n\\noindent\n{\\bf Reduction 4.}  Suppose $n\\ge 14$.\nReplace the 8-vertex configuration by a single vertex to obtain $K'$ as shown in Figure 3.6.\\par\n\\medskip\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure36.eps} \n\\centerline{\\bf Figure 3.6}\n\\end{center}\n\\end{figure}\n\nIn this case $n'=n-7$, $t'=t-2\\ge (n+1)\/2-2=(n-3)\/2>(n-6)\/2=(n'+1)\/2$ and $K'\\in {\\cal K}_{\\lceil(n'+1)\/2\\rceil}$.\\par\n\\medskip\n\\noindent\n{\\bf Claim:}\n$\\gamma (K)\\le \\gamma (K')+2$ and if $\\gamma (K')\\le (2\/7)|V(K')|$, then\n$\\gamma (K)\\le (2\/7)|V(K)|$.\n\\par\n\\medskip\n\\noindent\n{\\it Proof of Claim:}\nLet $D'$ be a minimum dominating set in $K'$.\nIf 1 is in $D'$, or 1 is not in $D'$, but is dominated from above, then add 4 and 8 to dominate $K$.\nIf 1 is not in $D'$, but is dominated from below in $K'$, then add 2 and 6 to dominate $K$.\nThis proves the Claim.\\par\n\\par\n\\medskip\n\\noindent\n{\\bf Reduction 5.}  Suppose $n\\ge 15$.\nReplace the 13-vertex configuration by a 5-vertex configuration consisting of an $O$ and a $B$ to obtain $K'$ as shown in Figure 3.7.\\par\n\\medskip\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure37.eps} \n\\centerline{\\bf Figure 3.7}\n\\end{center}\n\\end{figure}\n\n\nIn this case, $n'=n-8\\ge 7$ and $t'=t-4\\ge (n+1)\/2-4=(n'+1)\/2$.\nHence $K'\\in {\\cal K}_{\\lceil (n'+1)\/2\\rceil}$.\\par\n\\medskip\n\\noindent\n{\\bf Claim:}\n$\\gamma (K)\\le \\gamma (K')+2$ and\n$\\gamma (K)\\le 2|V(K)|\/7$, if $\\gamma(K')\\le 2|V(K')|\/7$.\n\\par\n\\medskip\n\\noindent{\\it Proof of Claim:}\nLet $D'$ be a dominating set in $K'$.\nIf 1 and $13\\in D'$, then add 6 and 8 to dominate $K$.\nIf $1\\in D'$, but $13\\notin D'$,\nthen add 6 and 10 to dominate $K$.\n\\medskip\n\\noindent\nIf $1\\notin D'$, but $13\\in D'$, then one of 10, 11 or 12 belongs to $D'$ in order to dominate the vertex $10\\in K'$.\nDenote this vertex by $v$.\nThen add 2, 4, and 8 to $D'-\\{v\\}$\nto dominate $K$.\nFinally suppose $1\\notin D'$ and $13\\notin D'$.\nIf both 1 and 13 are dominated by vertices from $\\{10, 11, 12\\}$, then $D'$ contains two vertices from $\\{10,11,12\\}$\nwhich we will call $v$ and $w$.\nThen add 3, 6, 8 and 11 to $D'-\\{v,w\\}$ to dominate $K$.\n\\par\nSo suppose at most one of 1 and 13 is dominated by vertices from $\\{10,11,12\\}$.\nThen $D'$ contains at least one of 10, 11 and 12 which we will call $v$.\nThen add 3, 6 and 10 to $D'-\\{v\\}$ to dominate $K$ if 13 is dominated by a vertex different from 10,11 and 12.\nOtherwise, add 4, 8 and 11 to $D'-\\{v\\}$ to dominate $K$.\nThis completes the proof of the Claim.\\par\n\n\\medskip\n\\noindent\n{\\bf Reduction 6.}  Suppose $n\\ge 21$.\nReplace the 15-vertex configuration by a single vertex to obtain $K'$ as shown in Figure 3.8.\nThen again $K'\\in {\\cal K}_{\\lceil(n'+1)\/2\\rceil}$.\\par\n\nIn this case, $n'=n-14$, $t'=t-7\\ge (n+1)\/2-7=(n-13)\/2=(n'+1)\/2$.\n\n\\medskip\n\\noindent\n{\\bf Claim:}\n$\\gamma (K)\\le \\gamma (K')+4$ and\nif $\\gamma (K')\\le (2\/7)|V(K')|$, then $\\gamma (K)\\le (2\/7)|V(K)|$.\\par\n\\medskip\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure38.eps} \n\\centerline{\\bf Figure 3.8}\n\\end{center}\n\\end{figure}\n\n\n\\noindent\n{\\it Proof of Claim:}\nLet $D'$ be a minimum dominating set for $K'$.\nIf $15\\notin D'$,\nadd 3, 6, 9 and 13 to $D'$.\nThis completes the proof of the Claim.\nOur final reduction will be applied only in the final stages of the proof of Theorem 3.1.\\par\n\\medskip\n\\noindent\n{\\bf Reduction 7.}  Suppose $n\\ge 21.$\nReplace the 12-vertex configuration by a single vertex to obtain $K'$ as shown in Figure 3.9.\n\\par\n\\bigskip\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure39.eps} \n\\centerline{\\bf Figure 3.9}\n\\end{center}\n\\end{figure}\n \nIn this case, $|V(K')|=|V(K)|-11$.\n\\par\n\\medskip\n\\noindent\n{\\bf Claim:}\n$\\gamma (K)\\le \\gamma (K')+3$ and if $\\gamma (K')\\le\n 2|V(K')|\/7\n\n$, then $\\gamma (K)\\le \n2|V(K)|\/7\n$.\\par\n\\medskip\n\\noindent\n{\\it Proof of Claim:}\nLet $D'$ be a minimum dominating set for $K'$.\nIf $12\\in D'$, then add $1, 5, 7$ and $12$ to dominate $K$.\nIf $12\\notin D'$, then add $2, 5$ and $10$ to dominate $K$.\nThus $\\gamma (K)\\le \\gamma (K')+3\\le\n2|V(K')|\/7\n+3 =\n2(|V(K)|-11)\/7\n+3\\le\n2|V(K)|\/7\n$\nand the Claim is proved.\n\\par\n\\medskip\nWe are now prepared for our main result about $(H,A,B,O)$-graphs.\\par\n\\medskip\n\n\n\\noindent\n{\\bf Theorem 3.1.}  If $K$ is an $(H,A,B,O)$-graph on $n$ vertices and $K$ has at least $(n+1)\/2$ 2-chords,\nthen $\\gamma (K)\\le \\lceil 2n\/7\\rceil$.\\par\n\\medskip\n\n\\noindent\n{\\bf Proof:}  Let $K$ be an $(H,A,B,O)$-graph with \n$n$ vertices and at least $(n+1)\/2$ 2-chords. Note that $K$ is simple.\nIt follows that $n\\ge 5$ since \n$K$ has at least $(n+1)\/2$ 2-chords. Suppose on the contrary that $K$ is a minimum counterexample. \n\n\n\\medskip\n\n{\\bf Claim 1}: {\\sl $K$ has $n\\ge 21$ vertices.}  \\medskip\n\n\\noindent {\\it Proof of Claim 1}: If not, suppose $5\\le n\\le 20$. Since $K$ has at least $ (n+1)\/ 2 $ 2-chords, by the Pigenhole Principle, there are two\n2-chords sharing a common endvertex $v$.\nThen $v$ dominates five consecutive vertices on the \nHamilton cycle.\nBut each vertex on the Hamilton cycle domintates at least three consecutive \nvertices.\nHence $\\gamma(K)\\le 1+\\lceil (n-5)\/3\\rceil$. It can be easily checked that $1+\\lceil (n-5)\/3\n\\rceil\\le \\lceil 2n\/7\\rceil$ for $5\\le n\\le 20$.  So $\\gamma(K)\\le\n\\lceil 2n\/7\\rceil$, contradicting the assumption that $K$ is a counterexample. \nThis completes the proof of Claim 1.\n\\medskip\n \n\nIn the following, we may therefore assume that $K$ has at least $n\\ge 21$ vertices.\nSo Switch operations and Reductions 1-7  \ncan be applied to $K$.\\medskip\n \n\n\n{\\bf Claim 2}: {\\sl $K$ is composed of mixed $(A,B)$-strings only and each such mixed $(A,B)$-string $S$ satisfies (i), (ii) and (iii) below:\n\n(i) $S$ starts and ends with an $A$; \n\n(ii) any two $A$s in $S$ are separated by at least two consecutive $B$s in $S$;\n\n(iii) every such $S$\nis separated from the next mixed $(A,B)$-string by exactly one $O$.}\\medskip\n\n\\noindent{\\it Proof of Claim 2}: We prove Claim 2 via five subclaims.\n\\medskip\n\n(2.1) Every $(A,B)$-string in $K$ starts and ends with an $A$. \\medskip\n\nSuppose to the contrary that $S$ is an $(A,B)$-string which starts or ends with a $B$.\nApplying Reduction 3, let \n$K'$ be the resulting $(H,A,B,O)$-graph which has four fewer vertices than $K$.\nSince $K$ is a minimum counterexample, $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$.\nBy the claim following Reduction 3, we have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$,\nwhich contradicts the assumption that $K$ is a minimum counterexample.\nSubclaim (2.1) follows.\\medskip\n\n\n(2.2) No $(A,B)$-string of $K$ contains two consecutive $A$s. \\medskip\n\nSuppose to the contrary that $K$ has an $(A,B)$-string with two consecutive $A$s. \nApplying Reduction 2, let $K'$ be the new  \n$(H,A,B,O)$-graph with $|V(K')|=n-4$ vertices. \nThen $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$ since $K$\nis a minimum counterexample.\nBy the claim following Reduction 2, it follows that  $\\gamma(K)\\le \\lceil 2n\/7\\rceil$, a contradiction\nand subclaim (2.2) is proved. \\medskip\n\n(2.3) $K$ contains at least one mixed $(A,B)$-string and no mixed\n$(A,B)$-string contains a segment of the form $ABA$.\\medskip\n\nAgain by way of contradiction, suppose $K$ does not contain a mixed $(A,B)$-string.\nThen by (2.1) and (2.2), all $(A,B)$-strings of\n$K$ are isolated $A$s.\nBut then the number of 2-chords of $K$ is the number of \n$A$s in $K$, which is at most $n\/3$, contradicting the assumption that $K$ has at least $(n+1)\/2$\n2-chords. Hence $K$ contains at least one mixed $(A,B)$-string.  \n\nIf $K$ has a mixed $(A,B)$-string with a segment of the form $ABA$, apply Reduction 1 to the segment\n$ABA$.\nLet $K'$ be the new $(H,A,B,O)$-graph.\nThen $|V(K')|<|V(K)|$. Since $K$ is a minimum\ncounterexample, it follows that $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$.\nBy the claim following\nReduction 1, we have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$, contradicting the assumption that $K$ is a\ncounterexample.\nSubclaim (2.3) thus follows. \\medskip\n\n\n(2.4)  For every vertex of degree 2 there is a 2-chord joining its neighbors; i.e., these three vertices form an $A$.\n\\medskip\n\nBy way of contradiction, suppose that $K$ has a vertex $v$ of degree 2 which is not contained in any $A$.\nIn other words, $v$ is the intersection of two consecutive $O$s. \nLet $w$ be a neighbor of $v$ in the Hamilton cycle.\\par\n\nFirst, assume that $w$ is also a vertex of degree 2.\nThen $K$ contains three consective $O$s.\nUsing Switch operations, we may assume \nthat one of $v$ and $w$, say $v$, has a neighbor in a mixed $(A,B)$-string $S$.\n(We know such a mixed $(A,B)$-string exists by (2.3).)\nIf $v$ had no such neighbor, we could\napply Switch operations to bring the degree 2 vertices closer to the mixed $(A,B)$-string.\nHence, the three consecutive $O$s are followed by a mixed $(A,B)$-string $S$.\nBy (2.1) and (2.2), \nwe know that $S=AB\\cdots$; that is $S$ starts with an $A$ followed by a $B$. \nApply a Switch operation to $OOOAB\\cdots$ to obtain the segment $AOOOB\\cdots$.\nThen  we have a new $(H,A,B,O)$-graph $K'$ such that $|V(K')|=|V(K)|=n$ and $K'$ has a string\nstarting with a $B$.\nBy (2.1), $K'$ is not a counterexample and hence $\\gamma(K')\\le \\lceil 2n\/7\\rceil$.\nBy the claim following the definition of a Switch operation, it follows that $\\gamma(K)\\le \\gamma(K')\\le \n\\lceil 2n\/7\\rceil$ contradicting the assumption that $K$ is a minimum counterexample. \n\nSo we may assume that both neighbors of $v$ have degree at least 3; i.e., $K$ contains two consecutive \n$O$s.\nBy (2.1), the two segments sharing vertices with the two consecutive $O$s\nare $A$s.\nIf one of the $A$s is isolated, \nthen apply Reduction 4 to obtain a new $(H,A,B,O)$-graph $K'$ such \nthat $|V(K')|<|V(K)|$.\nSince $K$ is a minimum counterexample, $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$.\nBy the Claim following Reduction 4,  we have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$,\ncontradicting the assumption that $K$ being a minimum counterexample.\\par\nSo assume that neither of the $A$s is isolated.\nNow apply Reduction 5.\nAs before, we then have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$, a contradiction, and (2.4) follows.\\medskip\n \n(2.5) Every $(A,B)$-string of $K$ is a mixed $(A,B)$-string.\\medskip\n\nSuppose to the contrary that $K$ contains an $(A,B)$-string which is not mixed.\nThen by (2.1) and (2.2) this string must consist of an isolated $A$.\nAmong all such isolated $A$s, choose one that is closest to some mixed $(A,B)$-string.\nIt then follows by (2.3) and (2.4) that $K$ contains a segment of the form $AOAOABB$.\nApply Reduction 6 to $AOAOABB$ and let $K'$ be the new $(H,A,B,O)$-graph.\nThen $|V(K')|<|V(K)|$.\nHence $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$ since $K$ is a minimum counterexample.\nBy the claim following Reduction 6, we have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$, a contradiction. \\medskip\n\nCombining (2.1)-(2.5), we can conclude that Claim 2 is proved.\n \\medskip\n\n{\\bf Claim 3}: {\\sl If $K$ contains at least six $B$s, then no three of these $B$s are consecutive.} \\medskip\n\n\\noindent{\\it Proof of Claim 3}: By way of contradiction, suppose that $K$ does contain three consecutive $B$s. \nLet $x_1$ be the number of $A$s and $x_2$ the number of $B$s.\nThen $x_2\\ge 6$ and since $K$ contains at least one mixed $(A,B)$-string, $x_1\\ge 2$.\\par\n\nLet $y$ be the number of mixed $(A,B)$-strings. By Claim 2, it follows that $y \\le (x_2-1)\/2$ since at least one mixed $(A,B)$-string contains at least three consecutive $B$s. \n\n\nSince $K$ contains three consecutive $B$s, $K$ contains a segment $ABBB$. \nApply Reduction 7 to contract the segment $ABBB$ to a single vertex (deleting loops) and let $K'$ be the\nresulting new $(H,A,B,O)$-graph. \nThen the number of 2-chords of $K'$ is $t'=(x_1-1)+2(x_2-3),$\nand the number of vertices of $K'$ is $|V(K')| = 2(x_1-1) +3(x_2-3) +y,$\nwhere $x_1-1$ and $x_2-3$ are the number of $A$s and $B$s in $K'$, respectively. \nBut $x_2\\ge 6$ and so it then follows that $x_2-3> (x_2-1)\/2\\ge y$.\nHence \n\n$${{t'}\\over{|V(K')|}}={{(x_1-1)+2(x_2-3)}\\over{2(x_1-1)+3(x_2-3)+y}} = \n   {{[(x_1-1)+(x_2-3)] +(x_2-3)}\\over{2[(x_1-1)+(x_2-3)] + ((x_2-3)+y)}}\n   > {{1}\\over{ 2}}\\ .$$\nSo the number of 2-chords in $K'$ is at least $(|V(K')|+1)\/2$. Since $|V(K')|<|V(K)|$ and $K$ is a minimum counterexample,\n$\\gamma(K')\\le \\lceil 2|V(K')|\/7 \\rceil $.\nBy the claim following Reduction 7, we have $\\gamma(K)\n\\le \\lceil 2n\/7\\rceil$, contradicting the fact that $K$ is a counterexample.\nThis completes the proof of Claim 3.\n\\medskip \n\nBy Claims 1-3, $S=ABBABBA\\cdots ABBA$ for every mixed $(A,B)$-string $S$ in $K$ or else\n$K$ has exactly five $B$s. \n\n\nFirst assume that $S=ABBABBA\\cdots ABBA$ for every mixed $(A,B)$-string in $K$.\nLet $x$ be the total number of segments $ABB$ in $K$, and $y$ be the number of mixed $(A,B)$-strings. \nThen $n=8x+3y$.\nNote that all vertices of the segment $ABB$\ncan be dominated by two vertices.\nSince $x\\ge y$, we have \n$$\\gamma(K)\\le 2x+y\\le 2(7x+4y)\/7 \\le 2(8x+3y)\/7=2n\/7,$$\nwhich contradicts the assumption that $K$ is a minimum counterexample.\n\nOn the other hand, if $K$ has exactly five $B$s,\nthen\n$K=ABBAOABBBAO$ or else \\hfill\\break $K=ABBABBBAO$\nby Claims 1 and 2.\nNote that the domination number of $ABBA$ is 3 and\nthe domination number of $ABBBA$ is 3. Hence $\\gamma(K)\\le 6\\le 2n\/7$, a contradiction. \nThis completes the\nproof of the theorem. \n\n\\section{Main Result}\\par\n\\medskip\nWe shall have need of the following theorem due to Campos and Wakabayashi [CW].\\par\n\\medskip\n\\noindent\n{\\bf Theorem 4.1:}  If $G$ is an maximal outerplanar graph with $n\\ge 4$ vertices and with $t$ vertices of degree 2, then $\\gamma (G)\\le (n+t)\/4$.\\par\n\\medskip\nWe are now prepared for our main theorem.\\par\n\\medskip\n\\noindent\n{\\bf Theorem 4.2:}  Let $G$ be a plane triangulation with $n$ vertices and $\\delta (G)\\ge 4$, and\nsuppose $G$ contains a Hamilton cycle $H$.\nThen\n$\\gamma (G)\\le \\max\\{\\lceil 2n\/7\\rceil, \\lfloor 5n\/16\\rfloor\\}$.\\par\n\\medskip\n\\noindent\n{\\bf Proof:}  If $\\gamma (G)=1$, the result is trivial.\nSo suppose that $\\gamma (G)\\ge 2$. Since $\\delta(G)\\ge 4$, it follows that $n\\ge 6$. \nFirst we use Lemma 2.1 to replace the given Hamilton cycle $H$ with one which has no three consecutive 2-vertices.\\par\n\nSuppose now that at least one of $G_{int}$ and $G_{ext}$ has no more than $n\/4$ 2-chords,\nsay, without loss of generality, that $G_{int}$ has no more than $n\/4$ 2-chords.\nBut then $G_{int}$ is a maximal outerplanar graph with $t\\le n\/4$ vertices of degree 2 and so by Theorem 4.1,\n$\\gamma (G)\\le (n+t)\/4\\le (n+ n\/4)\/4= 5n\/16$.\\par\n\nSo suppose both $G_{int}$ and $G_{ext}$ have more than $n\/4$ 2-chords.\nThen the number of 2-chords of $G$ is larger than $n\/4+n\/4=n\/2$.\nIn this case, we may apply Theorem 3.1 to conclude that $\\gamma (G)\\le \\lceil (2n)\/7\\rceil$.\nThe proof of the theorem is complete. \n\\par\n\\medskip\n\nBy a well-known theorem of Tutte [T], every 4-connected planar graph contains a Hamilton cycle, \nso the following result is an immediate corollary of Theorem 4.2.\\medskip\n\n\\noindent\n{\\bf Corollary 4.3:}  Let $G$ be a 4-connected plane triangulation with $n$ vertices.\nThen $\\gamma (G)\\le \\max\\{\\lceil 2n\/7\\rceil, \\lfloor 5n\/16\\rfloor\\}$.\\par\n\n\n\\section*{Concluding Remarks} \nDetermination of the domination number seems to be difficult even for plane triangulations.\nMatheson and Tarjan's conjecture has now been open for some eighteen years and there seems to be little significant progress toward\nverifying their conjectured upper bound of $1\/4|V(G)|$ for the general class of all planar triangulations.\nThe main result in the present paper makes some progress for those planar triangulations having minimum degree at least 4 and a Hamilton cycle.\n(Note that this class is larger than the class of 4-connected planar triangulations.)\nIn particular, the reductions used in our main proof would seem to constitute a new approach at least for the graphs in this subclass.\nHopefully, they will prove to shed some light on the general case.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction and main results}\\label{s:introduction}\nLet $M$ be a complete, noncompact Riemannian manifold without boundary. It is called stochastically complete if\n\\begin{equation}\\label{d:stochastic}\\int_{M}p_t(x,y)d\\mathrm{vol}(y)=1,\\quad \\quad \\forall\\ t>0, x\\in M,\\end{equation} where $p_t(\\cdot,\\cdot)$ is the (minimal) heat kernel on $M.$\nYau \\cite {Yau78} first proved that any complete Riemannian manifold with a uniform lower bound of Ricci curvature is stochastically complete. Karp and Li \\cite{KarpLi} showed the stochastic completeness in terms of the following volume growth property:  \\begin{equation}\\label{e:volume criterion}\\mathrm{vol}(B_r(x))\\leq Ce^{cr^2},\\quad\\quad\\mathrm{some}\\ x\\in M, \\forall\\ r> 0,\\end{equation} where $\\mathrm{vol}(B_r(x))$ is the volume of the geodesic ball of radius $r$ and centered at $x$. Varopoulos \\cite{Varopoulos}, Li \\cite{Li84} and Hsu \\cite{Hsu89} extended Yau's result to Riemannian manifolds with general conditions on Ricci curvature. So far, the optimal volume growth condition for stochastic completeness was given by Grigor'yan \\cite{Grigoryan86}. We refer to \\cite{Grigoryan99} for the literature on stochastic completeness of\nRiemannian manifolds. These results have been generalized to a quite general setting, namely, regular strongly local Dirichlet forms by Sturm \\cite{Sturm94}.\n\nCompared to local operators, graphs (discrete metric measure spaces) are nonlocal in nature and can be regarded as regular Dirichlet forms associated to jump processes. A general Markov semigroup is called a diffusion semigroup if chain rules hold for the associated infinitesimal generator, see Bakry, Gentil and Ledoux \\cite[Definition~1.11.1]{BakryGentilLedoux}, which is a property related to the locality of the generator. As a common point of view to many graph analysts, the absence of chain rules for discrete Laplacians is the main difficulty for the analysis on graphs. This causes many problems and various interesting phenomena emerge on graphs. A graph is called \\emph{stochastically complete} (or conservative) if an equation similar to \\eqref{d:stochastic} holds for the continuous time heat kernel, see Definition \\ref{d:stochastic complete graph}. The stochastic completeness of graphs has been thoroughly studied by many authors \\cite{DodziukMathai06,Dodziuk06,Wojciechowski08,Weber10,Wojciechowski1,Huang10,KellerLenz10,Wojciechowski2,KellerLenz12,GrigoryanHuangMasamune12,MasamuneUemuraWang12,KellerLenzWojciechowski}. In particular, the volume criterion \\eqref{e:volume criterion} with respect to the graph distance is no longer true for unbounded Laplacians on graphs, see \\cite{Wojciechowski2}. This can be circumvented by using intrinsic metrics introduced by Frank, Lenz and Wingert \\cite{FrankLenzWingert12}, see e.g. \\cite{GrigoryanHuangMasamune12,Folz14,Huang14}.\n\nGradient bounds of heat semigroups can be used to prove stochastic completeness. Nowadays, the so-called $\\Gamma$-calculus has been well developed in the framework of general Markov semigroups where $\\Gamma$ is called the ``carr\\'e du champ\" operator, see \\cite[Definition~1.4.2]{BakryGentilLedoux}. Given a smooth function $f$ on a Riemannian manifold, $\\Gamma(f)$ stands for $|\\nabla f|^2,$ see Section~\\ref{s:graphs} for the definition on graphs. Heuristically, on a Riemannian manifold $M$ if one can show the gradient bound for the heat semigroup\n\\begin{equation}\\label{e:gradient bounds}\\Gamma(P_t f)\\leq P_t(\\Gamma (f)),\\quad\\quad \\forall f\\in C_0^{\\infty}(M),\\end{equation} where $P_t=e^{t\\Delta_M}$ is the heat semigroup induced by the Laplace-Beltrami operator $\\Delta_M$ and $C_0^\\infty(M)$ is the space of compactly supported smooth functions on $M$, then the stochastic completeness follows from approximating the constant function $\\mathds{1}$ by compactly supported smooth functions. This approach has been systematically generalized to Markov diffusion semigroups, i.e. local operators, see \\cite{BakryGentilLedoux}. In this paper, we closely follow this strategy and prove the stochastic completeness for the non-diffusion case, i.e. graphs. This shows that the gradient-bound approach works even in nonlocal setting.\n\nWe introduce the setting of graphs and refer to Section~\\ref{s:graphs} for details.\nLet $(V,E)$ be a connected, undirected, (combinatorial) infinite graph with the set of vertices $V$ and the set of edges $E.$  We say $x,y\\in V$ are neighbors, denoted by $x\\sim y,$ if $(x,y)\\in E.$ The graph is called \\emph{locally finite} if each vertex has finitely many neighbors. In this paper, we only consider locally finite graphs. We assign a weight $m$ to each vertex, $m: V\\to (0,\\infty),$ and a weight $\\mu$ to each edge, $$\\mu:E\\to (0,\\infty), E\\ni (x,y)\\mapsto \\mu_{xy},$$ and refer to the quadruple $G=(V,E,m,\\mu)$ as a \\emph{weighted graph}. We denote by $$C_0(V):=\\{f:V\\to\\R|\\ \\{x\\in V| f(x)\\neq 0\\}\\ \\mathrm{is\\ of\\ finite\\ cardinality}\\}$$ the set of finitely supported functions on $V$ and by $\\ell^p(V,m),$ $p\\in [1,\\infty],$ the $\\ell^p$ spaces of functions on $V$ with respect to the measure $m.$\n\n\n\n\nFor any weighted graph $G=(V,E,m,\\mu)$, it associates with a Dirichlet form with respect to the Hilbert space $\\ell^2(V,m)$ corresponding to the Dirichlet boundary condition,\n\\begin{eqnarray}\\label{e:dirichlet form}Q^{(D)}:&&D(Q^{(D)})\\times D(Q^{(D)})\\to \\R \\nonumber\\\\ && (f,g)\\mapsto \\frac12\\sum_{x\\sim y}\\mu_{xy}(f(y)-f(x))(g(y)-g(x)),\\end{eqnarray} where the form domain $D(Q^{(D)})$ is defined as the completion of $C_0(V)$ under the norm $\\|\\cdot\\|_Q$ given by $$\\|f\\|^2_{Q}=\\|f\\|_{\\ell^2(V,m)}^2+\\frac12\\sum_{x\\sim y}\\mu_{xy}(f(y)-f(x))^2, \\ \\ \\forall f\\in C_0(V),$$ see Keller and Lenz \\cite{KellerLenz12}. For the Dirichlet form $Q^{(D)},$ its (infinitesimal) generator, denoted by $L,$ is called the (discrete) Laplacian. Here we adopt the sign convention such that $-L$ is a nonnegative operator. The associated $C_0$-semigroup is denoted by $P_t=e^{tL}:\\ell^2(V,m)\\to \\ell^2(V,m).$\nFor locally finite graphs, the generator $L$ acts as\n\\begin{align*}\n    L f(x)=\\frac{1}{m(x)}\\sum_{y\\sim x}\\mu_{xy}(f(y)-f(x)), \\ \\ \\forall f\\in C_0(V),\n\\end{align*} see \\cite[Theorem~6~and~9]{KellerLenz12}.\nObviously, the measure $m$ plays an essential role in the definition of the Laplacian. Given the weight $\\mu$ on $E,$ typical choices of $m$ of particular interest are: \\begin{itemize}\\item $m(x)=\\sum_{y\\sim x}\\mu_{xy}$ for any $x\\in V$ and the associated Laplacian is called the normalized Laplacian.\n\\item $m(x)=1$ for any $x\\in V$ and the Laplacian is called combinatorial (or physical) Laplacian.\n\\end{itemize} Note that normalized Laplacians are bounded operators, so that these graphs are always stochastically complete, see Dodziuk \\cite{Dodziuk06} or Keller and Lenz \\cite{KellerLenz10}. Thus, the only interesting cases are combinatorial Laplacians, or more general unbounded Laplacians.\n\nFollowing the strategy in \\cite{BakryGentilLedoux}, to show stochastic completeness for the semigroups associated to unbounded Laplacians on graphs, it suffices to prove the gradient bounds as in\n\\eqref{e:gradient bounds}. For that purpose, we first introduce a completeness condition for infinite graphs:  A graph $G=(V,E,m,\\mu)$ is called \\emph{complete} if there exists a nondecreasing sequence of finitely supported functions $\\{\\eta_k\\}_{k=1}^\\infty$ such that \\begin{equation}\\label{d:complete}\\lim_{k\\to\\infty}\\eta_k=\\mathds{1}\\ \\mathrm{and}\\ \\ \\Gamma(\\eta_k)\\leq \\frac{1}{k},\\end{equation} where $\\mathds{1}$ is the constant function $1$ on $V.$ Note that the measure $m$ plays a role in the definition of $\\Gamma,$ see Definition~\\ref{d:carre du}, so that it is essential to the completeness of a weighted graph. This condition was defined for Markov diffusion semigroups in \\cite[Definition~3.3.9]{BakryGentilLedoux}; here we adapt it to graphs. As is well-known, this condition is equivalent to the geodesic completeness for Riemannian manifolds, see \\cite{Strichartz83}. For the discrete setting, this condition is satisfied for a large class of graphs which possess intrinsic metrics, see Theorem \\ref{thm:complete intrinsic}.\n\nFor gradient bounds \\eqref{e:gradient bounds}, besides completeness we need curvature dimension conditions. For Markov diffusion semigroups, the curvature dimension conditions are defined via the $\\Gamma$ operator and the iterated operator denoted by $\\Gamma_2,$ see \\cite[eq.~1.16.1]{BakryGentilLedoux}. This approach, using curvature dimension conditions to obtain gradient bounds, was initiated in Bakry and \\'Emery \\cite{BakryEmery85}. The curvature dimension condition on graphs, the non-diffusion case, was first introduced by Lin and Yau \\cite{LinYau10} which serves as a combination of a lower bound of Ricci curvature and an upper bound of the dimension, see Definition \\ref{d:curvature dimension} for an infinite dimensional version $\\CCD (K,\\infty).$ For bounded Laplacians on graphs, Bauer et al. \\cite{Bauer13} introduced an involved curvature dimension condition, the so-called $\\CDE(K,n)$ condition, to prove the Li-Yau gradient estimate for heat semigroups. Also restricted to bounded Laplacians, Lin and Liu \\cite{LinLiu} proved the equivalence between the $\\CCD(K,\\infty)$ condition and the gradient bounds \\eqref{e:gradient bounds} for heat semigroups, see Liu and Peyerimhoff \\cite{LiuP14} for finite graphs. In this paper, under some mild assumptions, we prove the gradient bounds for unbounded Laplacians on graphs.\n\n\n\\begin{thm}[see Theorem~\\ref{thm:main}]\\label{thm:main theorem1}\n  Let $G=(V,E,m,\\mu)$ be a complete graph and $m$ be non-degenerate, i.e. $\\inf_{x\\in V}m(x)>0.$ Then the following are equivalent:\n  \\begin{enumerate}[(a)]\n    \\item $G$ satisfies $\\CCD(K,\\infty).$\n    \\item For any $f\\in C_0(V),$\n    $$\\Gamma(P_t f)\\leq e^{-2Kt}P_t(\\Gamma (f)).$$\n    \\end{enumerate}\n\\end{thm}\nSince it is not clear what volume growth is possible under the $\\CCD(K,\\infty)$ condition, our result cannot be derived from the criteria involving volume growth conditions. For unbounded Laplacians, the standard differential techniques for bounded Laplacians as in \\cite{LiuP14,LinLiu} fail due to essential difficulties in the summability of solutions to heat equations. For instance, we don't know whether $\\Gamma(P_t f)$ lies in the form domain (or, more strongly, in the domain of the generator), see Remark~\\ref{r:difficulty}. In order to overcome these difficulties, we add a mild assumption on the measure $m,$ i.e. the non-degenerancy of the measure, and critically utilize techniques from partial differential equations, see Lemma~\\ref{l:Caccippoli} for the Caccioppoli inequality and Theorem~\\ref{thm:w12}. The assumption of the non-degenerancy of the measure $m$ is mild since it is automatically satisfied for any combinatorial Laplacian.\n\nA direct consequence of the gradient bounds is the stochastic completeness for graphs satisfying the $\\CCD(K,\\infty)$ condition.\n\\begin{thm}\\label{thm:main theorem2} Let $G=(V,E,\\mu,m)$ be a complete graph satisfying the $\\CCD(K,\\infty)$ condition for some $K\\in \\R.$ Suppose that the measure $m$ is non-degenerate, then $G$ is stochastically complete.\n \\end{thm}\n\n\nThe paper is organized as follows: In next section, we set up basic notations of weighted graphs. The $\\Gamma$-calculus is introduced to define curvature dimension conditions. We define a new concept on the completeness of a graph and prove the completeness under the assumptions involving intrinsic metrics on graphs. In Section~\\ref{s:Caccioppoli}, we adopt some PDE techniques to prove a (discrete) Caccioppoli inequality for Poisson's equations. In section~\\ref{s:stochastic completeness}, we prove our main results: the\nequivalence of curvature dimension conditions and the gradient bounds for heat semigroups on complete graphs, Theorem~\\ref{thm:main theorem1}, and the stochastic completeness for graphs satisfying the curvature dimension condition, Theorem~\\ref{thm:main theorem2}.\n\n\n\n\\section{Graphs}\\label{s:graphs}\n\\subsection{Weighted graphs}\nLet $(V,E)$ be a (finite or infinite) undirected graph with the set of vertices $V$ and the set of edges $E$ where $E$ is a symmetric subset of $V\\times V.$ Two vertices $x,y$ are called neighbors if $(x,y)\\in E,$ in this case denoted by $x\\sim y.$ At a vertex $x,$ if $(x,x)\\in E,$ we say there is a self-loop at $x.$ In this paper, we do allow self-loops for graphs. A graph $(V,E)$ is called connected if for any $x,y\\in V$ there is a finite sequence of vertices, $\\{x_i\\}_{i=0}^n,$ such that $$x=x_0\\sim x_1\\sim \\cdots\\sim x_n=y.$$ In this paper, we only consider locally finite connected graphs.\n\nWe assign weights, $m$ and $\\mu,$ on the set of vertices $V$ and edges $E$ respectively and refer to the quadruple $G=(V,E,m,\\mu)$ as a \\emph{weighted graph}: Here $\\mu:E\\to (0,\\infty), E\\ni (x,y)\\mapsto \\mu_{xy}$ is symmetric, i.e. $\\mu_{xy}=\\mu_{yx}$ for any $(x,y)\\in E,$  and $m: V\\to (0,\\infty)$ is a measure on $V$ of full support. For convenience, we extend the function $\\mu$ on $E$ to the total set $V\\times V,$ $\\mu:V\\times V\\to [0,\\infty),$ such that $\\mu_{xy}=0$ for any $x\\not\\sim y.$\n\nFor functions defined on $V,$ we denote by $\\ell^p(V,m)$ or simply $\\ell^p_m,$ the space of $\\ell^p$ summable functions w.r.t. the measure $m$ and by $\\|\\cdot\\|_{\\ell^p_m}$ the $\\ell^p$ norm of a function.\nGiven a weighted graph $(V,E,m,\\mu)$, there is an associated Dirichlet form w.r.t. $\\ell^2_m$ corresponding to the Neumann boundary condition, see \\cite{HaeselerKellerLenzWojciechowski12},\n\\begin{eqnarray*}&&Q^{(N)}:D(Q^{(N)})\\times D(Q^{(N)})\\to \\R \\nonumber\\\\&&\\ \\ \\   (f,g)\\mapsto Q^{(N)}(f,g):=\\frac12\\sum_{x, y\\in V}\\mu_{xy}(f(y)-f(x))(g(y)-g(x)),\\end{eqnarray*} where $D(Q^{(N)}):=\\{f\\in \\ell^2_m|\\  \\sum_{x, y}\\mu_{xy}(f(y)-f(x))^2<\\infty\\}.$ For simplicity, we write $Q^{(N)}(f):=\\frac12\\sum_{x, y}\\mu_{xy}(f(y)-f(x))^2$ for any $f: V\\to\\R.$ Let $D(Q^{(D)})$ denote the completion of $C_0(V)$ under the norm $\\|\\cdot\\|_Q$ defined by $$\\|f\\|_{Q}=\\sqrt{\\|f\\|_{\\ell^2_m}^2+Q^{(N)}(f)}, \\ \\ \\forall f\\in C_0(V).$$ Another Dirichlet form $Q^{(D)},$ defined as the restriction of $Q^{(N)}$ to $D(Q^{(D)}),$ corresponds to the Dirichlet boundary condition, see \\eqref{e:dirichlet form} in Section~\\ref{s:introduction}.\n\nFor the Dirichlet form $Q^{(N)},$ there is a unique self-adjoint\noperator $L^{(N)}$ on $\\ell^2_m$ with\n$$D(Q^{(N)}) = \\mathrm{Domain\\ of\\ definition\\ of\\ } (-L^{(N)})^{\\frac12}$$\nand\n$$Q^{(N)}(f, g) = \\left\\langle(-L^{(N)})^{\\frac12}f,(-L^{(N)})^{\\frac12}g\\right\\rangle,\\quad f,g\\in D(Q^{(N)})$$ where $\\langle\\cdot,\\cdot\\rangle$ denotes the inner product in $\\ell^2_m.$\nThe operator $L^{(N)}$ is the infinitesimal generator associated to the Dirichlet form $Q^{(N)},$ also called the (Neumann) Laplacian. The associated $C_0$-semigroup on $\\ell^2_m$ is denoted by $P_t^{(N)}=e^{tL^{(N)}}.$ For the Dirichlet form $Q^{(D)}$, $L^{(D)}$ and $P_t^{(D)}$ are defined in the same way. In case that the Dirichlet forms corresponding to Neumann and Dirichlet boundary conditions coincide, i.e. $$Q^{(N)}=Q^{(D)},$$\nwe omit the superscripts and simply write $$Q=Q^{(N)}=Q^{(D)}, \\quad L=L^{(N)}=L^{(D)}\\quad \\mathrm{etc}.$$\n\n\nThe following integration by parts formula is useful in further applications, see \\cite[Corollary~1.3.1]{Fukushima}.\n\\begin{lemma}[Green's formula]\\label{l:green formula}\n  Let $(V,E,m,\\mu)$ be a weighted graph. Then for any $f\\in D(Q^{(N)})$ and $g\\in D(L^{(N)})$,\n  \\begin{equation}\\label{e:green formula}\\sum_{x\\in V} f(x)L^{(N)} g(x)m(x)=-Q^{(N)}(f,g).\\end{equation} A similar consequence holds for the case of Dirichlet boundary condition.\n\\end{lemma}\n\nFor locally finite graphs, we define the \\emph{formal Laplacian}, denoted by $\\Delta,$ as\n$$\\Delta f(x)=\\frac{1}{m(x)}\\sum_{y\\in X}\\mu_{xy}(f(y)-f(x))\\quad \\forall\\ f:V\\to\\R.$$ This formal Laplacian can be used to identify the generators defined before. A result of Keller and Lenz, \\cite[Theorem~9]{KellerLenz12}, states that\n\\begin{equation}\\label{e:generator action}L^{(D)}f=\\Delta f,\\quad \\quad \\forall\\ f\\in D(L^{(D)}),\\end{equation} and a similar result holds for Neumann condition, see \\cite{HaeselerKellerLenzWojciechowski12}. Note that$$\\Delta f\\in C_0(V), \\ \\ \\forall\\ f\\in C_0(V).$$ Different choices for the measure $m$ induce different Laplacians. The typical choices are normalized Laplacians and combinatorial Laplacians, see Section~\\ref{s:introduction}.\n\n\n\n\n\n\n\n\n\nThe measure $m$ on $V$ is called non-degenerate if \\begin{equation}\\label{d:nondegenerate}\\delta:=\\inf_{x\\in V}m(x)>0.\\end{equation} The nondegerancy of the measure $m$ yields a very useful fact for $\\ell^p(V,m)$ spaces.\n\\begin{prop}\\label{p:nondegenerate} Let $m$ be a non-degenerate measure on $V$ as in \\eqref{d:nondegenerate}. Then for any $f\\in \\ell^p(V,m),$ $p\\in [1,\\infty),$\n$$|f(x)|\\leq \\delta^{-\\frac1p}\\|f\\|_{\\ell^p_m}\\ \\ \\ \\forall x\\in V.$$ Moreover, for any $1\\leq p<q\\leq \\infty,$ $\\ell^p(V,m)\\hookrightarrow\\ell^q(V,m).$\n\\end{prop}\n\\begin{proof}\n  The first assertion follows from $|f(x)|^p\\delta\\leq |f(x)|^pm(x)\\leq \\|f\\|_{\\ell^p_m}^p.$ The second one is a consequence of the interpolation theorem.\n\\end{proof}\n\nUnder assumptions of non-degeneracy of the measure $m$ and local finiteness of the graph, the Dirichlet forms corresponding to Neumann and Dirichlet boundary conditions coincide, i.e.\n$$Q^{(N)}=Q^{(D)},$$ see \\cite[Theorem~6]{KellerLenz12} and \\cite[Corollary~5.3]{HaeselerKellerLenzWojciechowski12}\nand the domains of generators are characterized as\n$$D(L^{(N)})=D(L^{(N)})=\\{f\\in\\ell^2_m|\\ \\Delta f\\in \\ell^2_m\\}.$$\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Gamma calculus}\nWe introduce the $\\Gamma$-calculus and curvature dimension conditions on graphs following \\cite{LinYau10,Bauer13}.\n\n First we define two natural bilinear forms associated to the Laplacian. Given $f: V\\to\\R$ and $x,y\\in V,$ we denote by $\\nabla_{xy}f:= f(y)-f(x)$ the difference of the function $f$ on the vertices $x$ and $y.$\n\\begin{defi}\\label{d:carre du}\n  The gradient form $\\Gamma,$ called the ``carr\\'e du champ\" operator, is defined by\n  \\begin{eqnarray*}\\Gamma(f,g)(x)&=&\\frac12(\\Delta(fg)-f\\Delta g-g\\Delta f)(x)\\\\&=&\\frac{1}{2m(x)}\n  \\sum_{y}\\mu_{xy}\\nabla_{xy}f\\nabla_{xy}g.\\end{eqnarray*} For simplicity, we write $\\Gamma(f):=\\Gamma(f,f).$ Moreover, the iterated gradient form, denoted by $\\Gamma_2$, is defined as $$\\Gamma_2(f,g)=\\frac12(\\Delta\\Gamma(f,g)-\\Gamma(f,\\Delta g)-\\Gamma(g,\\Delta f)).$$ We write $\\Gamma_2(f):=\\Gamma_2(f,f)=\\frac{1}{2}\\Delta \\Gamma(f)-\\Gamma(f,\\Delta f).$\n\\end{defi}\nThe Cauchy-Schwarz inequality implies that \\begin{equation}\\label{eq:gamma}\\Gamma(f,g)\\leq \\sqrt{\\Gamma(f)\\Gamma(g)}\\leq \\frac12 (\\Gamma(f)+\\Gamma(g)).\\end{equation} In addition, one can easily see that $Q^{(N)}(f)=\\|\\Gamma(f)\\|_{\\ell^1_m}.$\n\nNow we can introduce curvature dimension conditions on graphs.\n\\begin{defi}\\label{d:curvature dimension}\n  We say a graph $(V,E,m,\\mu)$ satisfies the $\\CCD(K,\\infty)$ condition, $K\\in \\R$, if for any $x\\in V,$\n  $$\\Gamma_2(f)(x)\\geq K\\Gamma (f)(x).$$\n\\end{defi}\nOne can also define a finite dimensional version, $\\CCD(K,n)$ condition (see \\cite{LinYau10}), which is stronger than $\\CCD(K,\\infty).$ An involved curvature dimension condition, called $\\CDE(K,n),$ was introduced in \\cite{Bauer13}. In this paper, we only consider $\\CCD(K,\\infty)$ conditions.\n\n\n\\subsection{Completeness of graphs}\nYau \\cite{Yau78} first proved that complete Riemannian manifolds with Ricci curvature uniformly bounded from below are stochastically complete.\nBakry \\cite{Bakry86} proved the stochastic completeness for weighted Riemannian manifolds satisfying $\\CCD(K,\\infty)$ condition for weighted Laplacians, see also Li \\cite{Lixiangdong05}. The completeness of Riemannian manifolds plays an important role in these problems.\n\nFor a graph $(V,E,m,\\mu),$ we define the completeness of a graph as in \\eqref{d:complete}, see Section~\\ref{s:introduction}. The following lemma shows the importance of the completeness of a graph. Note that we don't need the non-degenerancy of the measure $m$ here.\n\\begin{lemma}\\label{l:Qdensity} Let $(V,E,m,\\mu)$ be a complete graph.\n  For any $f\\in \\ell^2_m$ such that $Q^{(N)}(f)<\\infty$ we have $$\\|f\\eta_k-f\\|_Q\\to 0,\\ \\ \\ k\\to\\infty.$$ That is, $C_0(V)$ is a dense subset of the Hilbert space $(D(Q^{(N)}),\\|\\cdot\\|_Q)$ and $Q^{(N)}=Q^{(D)}.$\n\\end{lemma}\n\n\\begin{proof}\n  It is easy to see that $f_k:=f\\eta_k\\to f$ in $\\ell^2_m.$ So it suffices to show that $Q^{(N)}(f_k-f)\\to 0, $ $k\\to \\infty.$\n  \\begin{eqnarray*}\n    Q^{(N)}(f_k-f)&=&\\frac12\\sum_{x,y}\\mu_{xy}|\\nabla_{xy}f(\\eta_k-1)|^2\\\\\n    &=&\\frac{1}{2}\\sum_{x,y}\\mu_{xy}|\\nabla_{xy}f \\cdot(\\eta_k(y)-1)+f(x)\\nabla_{xy}\\eta_k|^2\\\\\n    &\\leq&\\sum_{x,y}\\mu_{xy}(|\\nabla_{xy}f|^2 |\\eta_k(y)-1|^2+f^2(x)|\\nabla_{xy}\\eta_k|^2)\\\\\n    &=&I_k+II_k.\n  \\end{eqnarray*} By the dominated convergence theorem, $I_k\\to 0$ as $k\\to\\infty.$ For the second term,\n  $$II_k\\leq\\frac{2}{k^2}\\sum_{x}f^2(x)m(x)\\to 0,\\ \\ \\ k\\to\\infty.$$ This proves the lemma.\n\\end{proof}\n\nHence for a complete graph, $Q^{(N)}=Q^{(D)}.$ In the rest of the paper, given a complete graph we simply write $Q=Q^{(N)}=Q^{(D)},$ and by \\eqref{e:generator action}\n$$L=\\Delta,\\quad \\quad\\quad  \\mathrm{on}\\quad \\ D(Q).$$\n\n\n\n\n\n\\subsection{Intrinsic metrics}\\label{s:intrinsic}\nThe Laplacian associated with the graph $(V,E,m,\\mu)$ is a bounded operator from $\\ell^2(V,m)$ to $\\ell^2(V,m)$ if and only if $$\\sup_{x\\in V} \\frac{1}{m(x)}\\sum_{y\\sim x}\\mu_{xy}<\\infty.$$\nIn order to deal with unbounded Laplacians, we need the following intrinsic metrics on graphs introduced in \\cite{FrankLenzWingert12}.\n\nA pseudo metric $\\rho$ is a symmetric function, $\\rho:V\\times V\\to[0,\\infty),$ with zero diagonal which satisfies the triangle inequality.\n\n\\begin{defi}[Intrinsic metric]\\label{d:intrinsic} A pseudo metric $\\rho$ on $V$ is called \\emph{intrinsic} if\n\\begin{align*}\n\\sum_{y\\in V}\\mu_{xy}\\rho^{2}(x,y)\\leq m(x),\\qquad x\\in V.\n\\end{align*}\n\\end{defi}\n\n\nIn various situations the natural graph distance, called the combinatorial distance, proves to be insufficient for the investigations of unbounded Laplacians, see \\cite{Wojciechowski1,Wojciechowski2,KellerLenzWojciechowski}. For this reason the concept of intrinsic metrics received quite some attention as a candidate to overcome these problems. Indeed, intrinsic metrics already have been applied successfully to various problems on graphs \\cite{BauerHuaKeller,BauerKellerWojciechowski,FOLZ,Folz14,GrigoryanHuangMasamune12,HuangKellerMasamuneWojciechowski,HuaKeller14}.\n\n\nFix a base point  $o\\in V$ and\ndenote the distance balls by\n\\begin{align*}\n    B_{r}(o)=\\{x\\in V\\mid\\rho(x,o)\\leq r\\},\\qquad r\\ge0.\n\\end{align*}\nThe choice of the base point $o$ will be irrelevant to our results later. We say $B_{r}(o)$ is finite, if it is of finite cardinality, i.e. $\\sharp B_{r}(o)<\\infty.$\n\n\\begin{thm}\\label{thm:complete intrinsic}\n  Let $G=(V,E,m,\\mu)$ be a graph and $\\rho$ be an intrinsic metric on $G.$ Suppose that each ball $B_r(o),$ $r>0$, is finite, then\n  $G$ is a complete graph.\n\\end{thm}\n\\begin{proof}\n  For any $0<r<R,$ we denote by $\\eta_{r,R}$ the cut-off function on $B_R(o)\\setminus B_r(o)$ defined as\n\\begin{align*}\n    \\eta_{r,R}(\\cdot)=\\min\\left\\{\\max\\left\\{\\frac{R-\\rho(\\cdot,o)}{R-r},0\\right\\},1\\right\\}.\n\\end{align*} Set $\\eta_k:=\\eta_{k,2k}.$ Then $\\{\\eta_k\\}$ is a nondecreasing sequence of finitely supported functions which converges to the constant function $\\mathds{1}$ pointwise. Moreover,\n\\begin{eqnarray*}\n  \\Gamma(\\eta_k)(x)&=&\\frac{1}{2m(x)}\\sum_{y\\in V}\\mu_{xy}|\\nabla_{xy}\\eta_k|^2\\\\\n  &\\leq& \\frac{1}{2m(x)k^2}\\sum_{y\\in V}\\mu_{xy}\\rho^2(x,y)\\\\\n  &\\leq&\\frac{1}{2k^2}<\\frac{1}{k},\n\\end{eqnarray*} where we used the definition of the intrinsic metric $\\rho.$ This proves the theorem.\n\\end{proof}\n\nFor any weighted graph $(V,E,m,\\mu),$ intrinsic metrics always exist. There is a natural intrinsic metric introduced by Huang \\cite[Lemma~1.6.4]{Huang11}. Define the \\emph{weighted vertex degree} $\\mathrm{Deg}:V\\to[0,\\infty)$ by\n\\begin{align*}\n    \\mathrm{Deg}(x)=\\frac{1}{m(x)}\\sum_{y\\in V}\\mu_{xy},\\qquad x\\in V.\n\\end{align*}\n\n\\begin{example}\\label{ex:intrinsic}\nFor any  given weighted graph there is an intrinsic path metric defined by\n\\begin{align*}\n    \\de(x,y)=\\inf_{x=x_{0}\\sim\\ldots\\sim x_{n}=y}\\sum_{i=0}^{n-1} (\\mathrm{Deg}(x_{i})\\vee\\mathrm{Deg}(x_{i+1}))^{-\\frac{1}{2}},\\quad\\quad x,y\\in V,\n\\end{align*} where the infimum is taken over all finite paths connecting $x$ and $y.$\n\n\\end{example}\nFor the completeness of the graph, it suffices to find an intrinsic metric satisfying the conditions in Theorem~\\ref{thm:complete intrinsic}. For instance, one can check whether each ball of finite radius under the metric $\\delta$ is finite.\n\n\\section{Semigroups and Caccioppoli inequality}\\label{s:Caccioppoli}\n\\subsection{Semigroups on graphs}\nIn this section, we study the properties of heat semigroups on graphs, which will be used later.\n\nWe denote by $P_t^{(D)}=e^{tL^{(D)}}$ the $C_0$-semigroup associated to the Dirichlet form $Q^{(D)}$ on $\\ell^2_m.$ It extrapolates to $C_0$-semigroups on $\\ell^p_m$ for all $p\\in [1,\\infty],$ for simplicity still denoted by $P_t^{(D)},$ see \\cite{KellerLenz12}.\n\\begin{defi}\\label{d:stochastic complete graph}\n  A weighted graph $(V,E,m,\\mu)$ is called \\emph{stochastically complete} if\n  $$P_t^{(D)}\\mathds{1}=\\mathds{1},\\quad \\forall\\ t>0,$$ where $\\mathds{1}$ is the constant function $1$ on $V.$\n\\end{defi}\nThe next proposition is a consequence of standard Dirichlet form theory, see \\cite{Fukushima} and \\cite{KellerLenz12}.\n\\begin{prop}\\label{p:basic pt}\n  For any $f\\in \\ell^p_m,$ $p\\in[1,\\infty],$ we have $P_t^{(D)} f\\in \\ell^p_m$ and\n  $$\\|P_t^{(D)} f\\|_{\\ell^p_m}\\leq \\|f\\|_{\\ell^p_m},\\ \\ \\ \\forall t\\geq 0.$$ Moreover, $P_t^{(D)} f\\in D(L^{(D)})$ for any $f\\in \\ell^2_m.$\n\\end{prop}\nThe next property follows from the spectral theorem.\n\\begin{prop}\\label{p:exchange}\n  For any $f\\in D(L^{(D)}),$ $$L^{(D)} P_t^{(D)} f=P_t^{(D)}L^{(D)} f.$$\n\\end{prop}\n\n\n\n\n\n\\subsection{Caccioppoli inequality}\nFor elliptic partial differential equations on Riemannian manifolds, the Caccioppoli inequality is well-known and yields the $L^p$ Liouville theorem for harmonic functions for $p\\in (1,\\infty),$ see Yau \\cite{Yau76}.\n\nBy adapting PDE techniques on manifolds to graphs, we obtain the Caccioppoli inequality for subsolutions to Poisson's equations.\n\\begin{lemma}\\label{l:Caccippoli}\n  Let $(V,E,m,\\mu)$ be a weighted graph and $g,h:V\\to\\R$ satisfy the following $$\\Delta g\\geq h.$$ Then for any $\\eta\\in C_0(V),$\n  \\begin{equation}\\label{e:caccioppoli}\n    \\|\\Gamma(g) \\eta^2\\|_{\\ell^1_m}\\leq C(\\|\\Gamma(\\eta) g^2\\|_{\\ell^1_m}+\\|gh \\eta^2\\|_{\\ell^1_m}).\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\nMultiplying $\\eta^2 g$ to both sides of the inequality, $\\Delta g\\geq h,$ and summing over $x\\in V$ w.r.t. the measure $m,$ we get\n\\begin{eqnarray*}\n  &&\\sum_x \\eta^2gh (x) m(x)\\leq \\sum_x \\eta^2 g \\Delta g(x) m(x)\\\\\n  &=&-\\frac12\\sum_{x,y}\\nabla_{xy}g\\nabla_{xy}(\\eta^2 g)\\mu_{xy}\\\\\n  &=&-\\frac12\\sum_{x,y}\\nabla_{xy}g(\\nabla_{xy}g \\eta^2(x)+g(y)\\nabla_{xy}(\\eta^2))\\mu_{xy}\\\\\n  &=&-\\frac12\\sum_{x,y}|\\nabla_{xy}g|^2 \\eta^2(x)\\mu_{xy}-\\frac12\\sum_{x,y}\\nabla_{xy}gg(y) (|\\nabla_{xy}\\eta|^2+2\\eta(x)\\nabla_{xy}\\eta)\\mu_{xy},\n\\end{eqnarray*} where we used Green's formula, see e.g. Lemma~\\ref{l:green formula}, in the second line since $\\eta\\in C_0(V).$ For the second term in the last line, by symmetry one has\n$$-\\frac12\\sum_{x,y}\\nabla_{xy}gg(y) |\\nabla_{xy}\\eta|^2\\mu_{xy}=-\\frac{1}{4}\\sum_{x,y}|\\nabla_{xy}g|^2 |\\nabla_{xy}\\eta|^2\\mu_{xy}\\leq 0.$$ Hence, by this observation, the previous estimate leads to\n\\begin{eqnarray*}\n  &&\\frac12\\sum_{x,y}|\\nabla_{xy}g|^2 \\eta^2(x)\\mu_{xy}\\\\\n  &\\leq&-\\sum_{x,y}\\nabla_{xy}gg(y) \\eta(x)\\nabla_{xy}\\eta\\mu_{xy}-\\sum_x \\eta^2gh (x) m(x)\\\\\n  &\\leq& \\frac14\\sum_{x,y}|\\nabla_{xy}g|^2\\eta^2(x)\\mu_{xy}+\\sum_{x,y}|\\nabla_{xy} \\eta|^2g^2(y)\\mu_{xy}-\\sum_x \\eta^2gh (x) m(x),\n\\end{eqnarray*} where we used basic inequality $ab\\leq \\frac14a^2+b^2$ for $a,b\\in\\R.$ The lemma follows from cancelling the first term in the last line with the left hand side of the system of inequalities.\n\n\\end{proof}\n\nUsing this Caccippoli inequality, we get a uniform upper bound of the Dirichlet energy of $P_tf$ for $t>0$ and $f\\in C_0(V).$\n\\begin{lemma}\\label{lem:energy estimate}\n  Let $(V,E,m,\\mu)$ be a complete graph. Then for any $f\\in C_0(V)$ and $t\\in [0,\\infty),$\n  $$Q(P_tf)=\\|\\Gamma(P_t f)\\|_{\\ell^1_m}\\leq C \\|f\\|_{\\ell^2_m}\\|\\Delta f\\|_{\\ell^2_m},$$ where $C$ is a uniform constant.\n\\end{lemma}\n\\begin{proof}\n  For $f\\in C_0(V),$ the local finiteness of the graph implies that $\\Delta f\\in C_0(V).$\n  By the completeness of the graph, let $\\eta_k\\in C_0(V)$ satisfy \\eqref{d:complete}. Since $P_tf$ satisfies the equation $\\frac{d}{dt} P_tf=\\Delta P_t f$ for any $t>0,$ applying the Caccippoli inequality in Lemma \\ref{l:Caccippoli} with $g=P_t f,$ $h=\\frac{d}{dt} P_t f$ and $\\eta=\\eta_k,$ we have\n      \\begin{eqnarray*}\\|\\Gamma(P_t f) \\eta_k^2\\|_{\\ell^1_m}&\\leq& C(\\|\\Gamma(\\eta_k) |P_t f|^2\\|_{\\ell^1_m}+\\|P_t f \\cdot\\frac{d}{dt} P_t f\\cdot \\eta_k^2\\|_{\\ell^1_m})\\\\\n      &\\leq& C\\left(\\frac1k\\| P_t f\\|_{\\ell^2_m}^{2}+\\|P_t f\\|_{\\ell^2_m}\\|\\frac{d}{dt} P_t f\\|_{\\ell^2_m}\\right).\\end{eqnarray*}\nBy Proposition \\ref{p:basic pt}, $$\\|P_t f\\|_{\\ell^2_m}\\leq \\|f\\|_{\\ell^2_m}$$ and by Proposition \\ref{p:exchange} and the equation \\eqref{e:generator action},\n$$\\|\\frac{d}{dt} P_t f\\|_{\\ell^2_m}=\\|\\Delta P_t f\\|_{\\ell^2_m}=\\|P_t\\Delta f\\|_{\\ell^2_m}\\leq \\|\\Delta f\\|_{\\ell^2_m}. $$ Hence $$\\|\\Gamma(P_t f) \\eta_k^2\\|_{\\ell^1_m}\\leq C\\left(\\frac1k\\|f\\|_{\\ell^2_m}^2+\\|f\\|_{\\ell^2_m}\\|\\Delta f\\|_{\\ell^2_m}\\right).$$ By passing to the limit, $k\\to \\infty,$ the monotone convergence theorem yields the lemma.\n\\end{proof}\n\nThe following result is an improved estimate of the previous lemma which will be useful in further applications.\n\\begin{lemma}\\label{lem:max time estimate}\n  Let $(V,E,m,\\mu)$ be a complete graph. Then for any $f\\in C_0(V)$ and $T>0,$ we have $\\max_{[0,T]}\\Gamma(P_t f)\\in\\ell^1_m$ and\n  \\begin{equation}\\label{eq:max time}\\left\\|\\max_{[0,T]}\\Gamma(P_t f)\\right\\|_{\\ell^1_m}\\leq C_1(T,f), \\end{equation} where $C_1(T,f)$ is a constant depending on $T$ and $f.$ Moveover, $$\\max_{[0,T]}|\\Gamma(P_t f,\\frac{d}{dt} P_t f)|\\in\\ell^1_m\\quad\\mathrm{and}$$\n  \\begin{equation}\\label{eq:mixed estimate}\n    \\left\\|\\max_{[0,T]}|\\Gamma(P_t f,\\frac{d}{dt} P_t f)|\\right\\|_{\\ell^1_m}=\\left\\|\\max_{[0,T]}|\\Gamma(P_t f,\\Delta P_t f)|\\right\\|_{\\ell^1_m}\\leq C_2(T,f).\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe local finiteness yields that $\\Delta f\\in C_0(V)$ and $\\Delta^2 f\\in C_0(V)$ for $f\\in C_0(V).$\n\nFor the first assertion, the Newton-Leibniz formula yields\n  \\begin{eqnarray*}\\Gamma(P_tf)&=&\\Gamma(f)+\\int_0^T\\frac{d}{ds}\\Gamma(P_sf)ds\\\\\n  &=&\\Gamma(f)+2\\int_0^T\\Gamma(P_sf,\\frac{d}{ds}P_sf)ds\\\\\n  &=&\\Gamma(f)+2\\int_0^T\\Gamma(P_sf,\\Delta P_sf)ds\\\\\n  &=&\\Gamma(f)+2\\int_0^T\\Gamma(P_sf,P_s(\\Delta f))ds,\\end{eqnarray*} where the last equality follows from Proposition \\ref{p:exchange}. Hence by the equation \\eqref{eq:gamma} and Lemma \\ref{lem:energy estimate}\n  \\begin{eqnarray*}\\left\\|\\max_{[0,T]}\\Gamma(P_tf)\\right\\|_{\\ell_m^1}&\\leq& \\|\\Gamma(f)\\|_{\\ell^1_m}+2\\left\\|\\int_0^T|\\Gamma(P_sf,P_s(\\Delta f))|ds\\right\\|_{\\ell^1_m}\\\\\n  &\\leq&\\|\\Gamma(f)\\|_{\\ell^1_m}+\\int_0^T(\\|\\Gamma(P_sf)\\|_{\\ell^1_m}+\\|\\Gamma(P_s(\\Delta f))\\|_{\\ell^1_m})ds\\\\\n  &\\leq&\\|\\Gamma(f)\\|_{\\ell^1_m}+CT\\|\\Delta f\\|_{\\ell^2_m}(\\|f\\|_{\\ell^2_m}+\\|\\Delta^2 f\\|_{\\ell^2_m})=:C_1(T,f). \\end{eqnarray*}\n\n  The second assertion is a direct consequence of the first one. By $\\Delta f\\in C_0(V)$ and \\eqref{eq:gamma},\n  \\begin{eqnarray*}\n    \\left\\|\\max_{[0,T]}|\\Gamma(P_t f,\\frac{d}{dt} P_t f)|\\right\\|_{\\ell^1_m}&=&\\left\\|\\max_{[0,T]}|\\Gamma(P_t f,\\Delta P_t f)|\\right\\|_{\\ell^1_m}=\\left\\|\\max_{[0,T]}|\\Gamma(P_t f,P_t \\Delta f)|\\right\\|_{\\ell^1_m}\\\\\n    &\\leq&\\frac12\\left\\|\\max_{[0,T]}\\Gamma(P_t f)\\right\\|_{\\ell^1_m}+\\frac12\\left\\|\\max_{[0,T]}\\Gamma(P_t \\Delta f)\\right\\|_{\\ell^1_m}\\\\\n    &\\leq& \\frac12(C_1(T,f)+C_1(T,\\Delta f))=:C_2(T,f).\n  \\end{eqnarray*}\n\n   This proves the lemma.\n\\end{proof}\n\n\nNow we can show that the Dirichlet energy, $t\\mapsto Q(P_t f)$, decays in time for the semigroup $P_t$ on complete graphs.\n\\begin{prop}\\label{p:Dirichlet energy}Let $(V,E,m,\\mu)$ be a complete graph. Then for any $f\\in C_0(V),$\n  $$Q(P_t f)\\leq Q(f),\\ \\ \\ \\forall t\\geq 0.$$\n  Moreover, for any $f\\in D(Q),$\n  $$Q(P_t f)\\leq Q(f),\\ \\ \\ \\forall t\\geq 0.$$\n\\end{prop}\n\\begin{proof}\nFor the first assertion, taking the formal derivative of time in $Q(P_tf)$ for $t>0,$ we get\n\\begin{equation}\\label{eq:eq1}\n  \\frac{d}{dt}Q(P_t f)= 2\\sum_{x\\in V}\\Gamma(P_t f,\\frac{d}{dt} P_t f)(x)m(x).\n\\end{equation}\nGiven a fixed $T>t,$ note that for any $t\\in [0,T],$\n \\begin{eqnarray*}\n   |\\Gamma(P_t f,\\frac{d}{dt} P_t f)(x)|&\\leq&\\max_{t\\in [0,T]}|\\Gamma(P_t f,\\frac{d}{dt} P_t f)(x)|=: g(x)\\in\\ell^1_m\n    \\end{eqnarray*} which follows from \\eqref{eq:mixed estimate} in Lemma \\ref{lem:max time estimate}. Hence the absolute value of the summand on the right hand side of \\eqref{eq:eq1} is uniformly (for $t\\in [0,T]$) bounded above by a summable function $g.$ The differentiability theorem yields that $Q(P_t f)$ is differentiable in time and whose the derivative is given by \\eqref{eq:eq1}.\n\n Since $P_t f\\in D(L)$ and $\\Delta P_t f=P_t \\Delta f\\in D(Q),$ Green's formula in Lemma \\ref{l:green formula} yields \\begin{eqnarray*}\n  \\frac{d}{dt}Q(P_t f)&=& 2\\sum_{x\\in V}\\Gamma(P_t f, \\Delta P_t f)(x)m(x)\\\\\n  &=&-2 \\sum_{x\\in V} |\\Delta P_t f (x)|^2m(x)\\leq 0.\n\\end{eqnarray*} This proves the first assertion.\n\nFor the second assertion, set $f_k:=f\\eta_k$ for $f\\in D(Q).$ It follows from the previous result that\n$$Q(P_tf_k)\\leq Q(f_k).$$ By Lemma \\ref{l:Qdensity}, $f_k\\to f$ in the norm $\\|\\cdot\\|_{Q}.$ The monotone convergence theorem yields that $$P_tf_k\\to P_tf$$ pointwise. By Fatou's lemma, $$Q(P_tf)\\leq \\liminf_{k\\to\\infty} Q(P_t f_k)\\leq \\liminf_{k\\to\\infty} Q(f_k)=Q(f).$$ This proves the theorem.\n\\end{proof}\n\n\n\n\n\n\n\\section{Stochastic completeness}\\label{s:stochastic completeness}\n\\subsection{Gradient bounds and curvature dimension conditions}\nThe curvature dimension condition implies gradient bounds, see \\cite{BakryGentilLedoux} for the case of Markov diffusion semigroups. In fact, they are equivalent on locally finite graphs under some mild assumptions.\n\\begin{thm}\\label{thm:main}\n  Let $G=(V,E,m,\\mu)$ be a complete graph with a non-degenerate measure $m$, i.e. $\\inf_{x\\in V}m(x)>0.$ Then the following are equivalent:\n  \\begin{enumerate}[(a)]\n    \\item $G$ satisfies $\\CCD(K,\\infty).$\n    \\item For any $f\\in C_0(V),$\n    $$\\Gamma(P_t f)\\leq e^{-2Kt}P_t(\\Gamma (f)).$$\n    \\item For any $f\\in D(Q),$\n    $$\\Gamma(P_t f)\\leq e^{-2Kt}P_t(\\Gamma (f)).$$\n  \\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\\label{r:difficulty}\nFor the case of finite graphs or bounded Laplacians, this result has been proven by \\cite{LiuP14,LinLiu}. To illustrate their proof strategy, we consider a finite graph $(V,E,m,\\mu)$ satisfying the $\\CCD(0,\\infty)$ condition.\n\n$(a)\\Rightarrow(b):$ For any $f:V\\to\\R,$ set $\\Lambda(s)=P_s(\\Gamma(P_{t-s}f)).$\n  Then \\begin{eqnarray*}\n    \\Lambda'(s)&=&\\Delta P_s(\\Gamma(P_{t-s}f))-2P_s(\\Gamma(P_{t-s}f,\\Delta P_{t-s}f))\\\\\n    &=&P_s(\\Delta\\Gamma(P_{t-s}f)-2\\Gamma(P_{t-s}f,\\Delta P_{t-s}f))\\geq 0,\n  \\end{eqnarray*} where the last inequality follows from the $\\CCD(0,\\infty)$ condition. However, for the case of infinite graphs,  $\\Delta P_s(\\Gamma(P_{t-s}f)))=P_s\\Delta(\\Gamma(P_{t-s}f)))$ may not hold since in general we don't know whether $\\Gamma(P_{t-s}f)\\in D(L).$\n\n  In addition, a strong version of gradient bounds has been proved using the following stronger curvature condition,\n  see \\cite[equation 3.2.4]{BakryGentilLedoux}\n  \\begin{equation}\\label{e:strong condition}\\Gamma(\\Gamma(g))\\leq 4\\Gamma(g)[\\Gamma_2(g)-K\\Gamma(g)], \\ \\ \\forall g\\in C_0(V).\\end{equation} However, this stronger curvature condition can never be fulfilled for graphs. In fact, the inequality \\eqref{e:strong condition} fails e.g. for $g=\\delta_x.$\n\\end{rem}\n\n\\subsection{Curvature dimension conditions and the properties of heat semigroups.}\n\nIn order to prove the gradient estimate under the $\\CCD(K,\\infty)$ condition, we need some lemmata. For graphs satisfying the $\\CCD(K,\\infty)$ condition, the following lemma states that $\\Gamma(P_t f)$ is a subsolution to the heat equation, a standard definition in the theory of PDEs.\n\n\\begin{lemma}\\label{l:heat subsolution}\n  Let $(V,E,m,\\mu)$ be a complete graph satisfying the $\\CCD(K,\\infty)$ condition. Then for any $f\\in C_0(V)$\n  $$\\frac{d}{dt}\\Gamma(P_t f)\\leq \\Delta \\Gamma(P_t f)-2K\\Gamma(P_t f).$$\n\\end{lemma}\n\\begin{proof}\n This follows from direct calculation by means of the $\\CCD(K,\\infty)$ condition and local finiteness of the graph.\n\\end{proof}\n\n\\begin{lemma}\\label{l:l1 time derivative} Let $(V,E,m,\\mu)$ be a complete graph. Then\n  for any $f\\in C_0(V)$ and $t\\geq 0,$ $$\\left\\|\\frac{d}{dt}\\Gamma(P_t f)\\right\\|_{\\ell^1_m}\\leq 2\\sqrt{Q(f)Q(\\Delta f)}.$$\n\\end{lemma}\n\\begin{proof}\n  This follows by the computation,\n  \\begin{eqnarray*}\n    &&\\left\\|\\frac{d}{dt}\\Gamma(P_t f)\\right\\|_{\\ell^1_m}=2\\sum_{x}\\left|\\Gamma(P_t f, \\frac{d}{dt}P_t f)(x)\\right|m(x)\\\\\n    &=&2\\sum_{x}|\\Gamma(P_t f, \\Delta P_t f)(x)|m(x)=2\\sum_{x}|\\Gamma(P_t f, P_t \\Delta f )(x)|m(x)\\\\\n    &\\leq&2 \\sqrt{\\sum_{x}\\Gamma(P_t f)m(x)\\sum_{x}\\Gamma(P_t \\Delta f)m(x)}\\\\\n    &\\leq &2 \\sqrt{\\sum_{x}\\Gamma(f)m(x)\\sum_{x}\\Gamma(\\Delta f)m(x)}<\\infty,\n  \\end{eqnarray*} where we used Proposition \\ref{p:exchange} for $f\\in C_0(V)$ in the third equality and Proposition \\ref{p:Dirichlet energy} for $f,\\Delta f\\in C_0(V)$ in the last one.\n\\end{proof}\n\n\n\nFor complete graphs satisfying the $\\CCD(K,\\infty)$ condition, we have higher summability of the solutions to heat equations.\n\\begin{thm}\\label{thm:w12}\n  Let $G=(V,E,m,\\mu)$ be a complete graph with a non-degenerate measure $m$. If $G$ satisfies the $\\CCD(K,\\infty)$ condition, then for any $f\\in C_0(V)$ and $t\\geq 0,$\n  $$\\Gamma (P_tf)\\in D(Q).$$\n\\end{thm}\n\\begin{proof}\n   From the proof of Proposition \\ref{p:Dirichlet energy}, $\\Gamma(P_t f)\\in \\ell^1(V,m).$ Hence by the nondenegerancy of $m,$ $\\Gamma(P_t f)\\in \\ell^2(V,m).$ It suffices to prove that $Q(\\Gamma(P_t f))<\\infty.$\n\n  Let $\\{\\eta_k\\}$ be the sequence in \\eqref{d:complete} by the completeness of the graph.  Note that Lemma \\ref{l:heat subsolution} implies that $\\Gamma(P_t f)$ is a subsolution to the heat equation.\n  Applying the Caccioppoli inequality \\eqref{e:caccioppoli} with $g=\\Gamma(P_t f),$ $h=\\frac{d}{dt} g+2K g$ and $\\eta=\\eta_k,$\n  we get \\begin{eqnarray*}\n      \\|\\Gamma(g) \\eta_k^2\\|_{\\ell^1_m}&\\leq& C(\\|\\Gamma(\\eta_k) g^2\\|_{\\ell^1_m}+\\|g(\\frac{d}{dt} g+2K g) \\eta_k^2\\|_{\\ell^1_m})\\\\\n      &\\leq & C\\left(\\frac{1}{k}\\|g\\|_{\\ell^2_m}^2+\\|g\\frac{d}{dt} g\\|_{\\ell^1_m}+2|K|\\cdot\\|g\\|_{\\ell^2_m}^2\\right)\\\\\n      &\\leq & C(K)(\\|g\\|_{\\ell^2_m}^2+\\|g\\frac{d}{dt} g\\|_{\\ell^1_m})\\\\\n       &=&I+II,\n  \\end{eqnarray*} where the constant $C(K)$ only depends on $K.$\n  By the assumption that $m$ is non-degenerate, Propositions \\ref{p:nondegenerate} and\n  \\ref{p:Dirichlet energy} yield that\n  $$I\\leq C\\|\\Gamma(P_tf)\\|_{\\ell^1_m}^2\\leq C\\|\\Gamma(f)\\|_{\\ell^1_m}^2<\\infty.$$\n\n  For the other term, noting that $\\|g\\|_{\\ell^\\infty}\\leq C\\|g\\|_{\\ell^1_m},$ by Lemma \\ref{l:l1 time derivative}, we have\n  \\begin{eqnarray*}\n    II&\\leq& C\\|g\\|_{\\ell^\\infty}\\|\\frac{d}{dt}g\\|_{\\ell^1_m}\\\\\n    &\\leq& C \\|g\\|_{\\ell^1_m}\\|\\frac{d}{dt}g\\|_{\\ell^1_m}<\\infty.\n  \\end{eqnarray*} Thus, $\\|\\Gamma(g) \\eta_k^2\\|_{\\ell^1_m}\\leq C<\\infty$ where the right hand side is independent of $k.$ By passing to the limit, $k\\to \\infty,$ Fatou's lemma yields that\n  $$\\|\\Gamma(\\Gamma(P_t f))\\|_{\\ell^1_m}\\leq \\liminf_{k\\to\\infty}\\|\\Gamma(\\Gamma(P_t f)) \\eta_k^2\\|_{\\ell^1_m}\\leq C.$$ This proves the theorem.\n\n\\end{proof}\n\n\\subsection{The proofs of main theorems}\n\\begin{thm}\\label{thm:monotone}\n  Let $(V,E,m,\\mu)$ be a complete graph with a non-degenerate measure $m$ and satisfying the $\\CCD(K,\\infty)$ condition. For any $f\\in C_0(V),$ $0\\leq \\zeta\\in C_0(V)$ and $t>0,$ the following function\n  $$s\\mapsto G(s):=\\sum_{x\\in V}\\Gamma(P_{t-s}f)(x)P_s \\zeta(x)m(x)$$ satisfies\n  $$G'(s)\\geq 2K G(s),\\quad\\quad\\quad 0<s<t.$$\n\\end{thm}\n\\begin{proof}\n  First, we show that $G(s)$ is differentiable in $s\\in(0,t)$. Without loss of generality, we assume that $\\epsilon<s<t-\\epsilon$ for some $\\epsilon>0.$ Taking the formal derivative of $G(s)$ in $s,$ we get\n  \\begin{equation}\\label{e:Gs1}\n    -2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)P_s\\zeta(x)m(x)+\\sum_{x}\\Gamma(P_{t-s}f)(x)\\Delta (P_{s}\\zeta)(x)m(x)\n  \\end{equation} This formal derivative is, in fact, the derivative of $G(s)$ if one can show that the absolute values of summands are uniformly (in $s$) controlled by summable functions. For the first term in \\eqref{e:Gs1}, note that $\\|P_s \\zeta\\|_{\\ell^{\\infty}}\\leq \\|\\zeta\\|_{\\ell^{\\infty}}<\\infty.$ Then the equation \\eqref{eq:mixed estimate} in Lemma \\ref{lem:max time estimate} yields that for any $s\\in (\\epsilon, t-\\epsilon)$\n  \\begin{eqnarray*}\n    &&2|\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)|P_s\\zeta(x)\\leq\\sup_{s\\in (\\epsilon,t-\\epsilon)}2|\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)|P_s\\zeta(x)\\\\&\\leq& 2\\|\\zeta\\|_{\\ell^\\infty}\\sup_{s\\in (\\epsilon,t-\\epsilon)} |\\Gamma(P_{s}f,\\Delta P_{s}f)(x)|=:g(x)\\in\\ell^1_m.\n      \\end{eqnarray*}\n  For the second term in \\eqref{e:Gs1}, the equation \\eqref{eq:max time} in Lemma \\ref{lem:max time estimate} implies that for any $s\\in (\\epsilon, t-\\epsilon)$\n  \\begin{eqnarray*}\n    \\Gamma(P_{t-s}f)(x)|\\Delta (P_{s}\\zeta)(x)|&\\leq&\\sup_{s\\in (\\epsilon,t-\\epsilon)}\\Gamma(P_{t-s}f)(x)|\\Delta (P_{s}\\zeta)(x)|\\\\\n    &=&\\sup_{s\\in (\\epsilon,t-\\epsilon)}\\Gamma(P_{t-s}f)(x)|P_{s}\\Delta \\zeta(x)|\\\\\n    &\\leq& \\|\\Delta \\zeta\\|_{\\ell^\\infty}\\sup_{s\\in (\\epsilon,t-\\epsilon)}\\Gamma(P_{s}f)(x)=:h(x)\\in\\ell^1_m.\n  \\end{eqnarray*} Since $g+h\\in \\ell^1_m$ which is independent of $s\\in(\\epsilon,t-\\epsilon),$ the differentiability theorem yields that $G(s)$ is differentiable and its derivative equals to \\eqref{e:Gs1}. Note that Theorem \\ref{thm:w12} and Proposition \\ref{p:basic pt} yield $\\Gamma(P_{t-s}f)\\in D(Q)$ and $P_s\\zeta\\in D(L)\\subset D(Q).$ Hence, using Green's formula \\eqref{e:green formula} in Lemma \\ref{l:green formula}, we obtain that\n  \\begin{equation}\\label{e:Gs2}G'(s)=-2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)P_s\\zeta(x)m(x)-\\sum_{x}\\Gamma(\\Gamma(P_{t-s}f),P_{s}\\zeta)(x)m(x).\\end{equation}\n\n  We claim that for any $0\\leq h\\in D(Q),$ \\begin{eqnarray}\\label{e:Gs3}\n    &&-2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)h(x)m(x)-\\sum_{x}\\Gamma(\\Gamma(P_{t-s}f),h)(x)m(x)\\\\\n    &\\geq& 2K\\sum_{x}\\Gamma(P_{t-s}f)h(x)m(x).\\nonumber\n  \\end{eqnarray} Once this claim is verified, by applying $h=P_{s}\\zeta$ in \\eqref{e:Gs3} and the self-adjointness of operators $P_t,$ we can prove the theorem. This claim can be proved by a density argument. Firstly, the $\\CCD(K,\\infty)$ condition yields that \\eqref{e:Gs3} holds for $0\\leq h\\in C_0(V)$: In fact, by Green's formula for $h\\in C_0(V)$,\n  \\begin{eqnarray*}\n    &&-2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)h(x)m(x)-\\sum_{x}\\Gamma(\\Gamma(P_{t-s}f),h)(x)m(x)\\\\\n    &=&-2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)h(x)m(x)+\\sum_{x}\\Delta(\\Gamma(P_{t-s}f))(x)h(x)m(x)\\\\\n    &\\geq& 2K\\sum_{x}\\Gamma(P_{t-s}f)h(x)m(x),\n  \\end{eqnarray*} where in the last inequality we used the $\\CCD(K,\\infty)$ condition.\n\nFor general $0\\leq h\\in D(Q),$ set $h_k=h\\eta_k$ where $\\{\\eta_k\\}$ is defined in \\eqref{d:complete}. It is obvious that $0\\leq h_k\\in C_0(V).$ Note that Lemma \\ref{lem:energy estimate} and Theorem~\\ref{thm:w12} yield that $\\Gamma(P_{t-s}f,\\Delta P_{t-s}f), \\Gamma(P_{t-s}f)\\in \\ell^1_m$ and  $\\Gamma(P_{t-s}f)\\in D(Q).$ Hence applying \\eqref{e:Gs3} for $h_k,$ passing to the limit, $k\\to \\infty,$ we prove the theorem.\n\n\\end{proof}\n\n\n\nNow we can prove the gradient bounds of heat semigroups under the $\\CCD(K,\\infty)$ condition.\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\n  $(a)\\Rightarrow (b):$ Using the same notation as in Theorem \\ref{thm:monotone}, we get\n  $$G'(s)\\geq 2KG(s).$$ Hence $G(s)\\geq e^{2Ks}G(0).$ Since $P_s$ is a self-adjoint operator on $\\ell^2_m,$\n  $$G(s)=\\sum_{x\\in V}P_s(\\Gamma(P_{t-s}f))(x)\\zeta(x)m(x).$$ By choosing delta functions, such as $\\zeta(x)=\\delta_y(x)$ ($y\\in V$), we prove the theorem.\n\n  $(b)\\Rightarrow (a):$ Fix a vertex $x\\in V.$ By $(b),$ $$F(t):=e^{-2Kt}P_t(\\Gamma (f))(x)-\\Gamma(P_t f)(x)\\geq 0.$$ It is easy to see that $F(t)$ is differentiable and $F'(0)\\geq 0.$ Note that $$\\left.\\frac{d}{dt}\\right|_{t=0}P_t(\\Gamma (f))(x)=\\Delta P_t(\\Gamma(f))(x)|_{t=0}=\\Delta (\\Gamma (f))(x).$$\n  Since the graph is locally finite, $$\\left.\\frac{d}{dt}\\right|_{t=0}\\Gamma(P_t f)(x)=2\\Gamma(P_tf,\\Delta P_tf)(x)|_{t=0}=2\\Gamma(f,\\Delta f)(x).$$\n  This proves the assertion by using $F'(0)\\geq 0.$\n\n  $(b)\\Leftrightarrow(c):$ This follows from a density argument.\n\\end{proof}\n\n\n\nNow we are ready to prove the analogue to Yau's result \\cite{Yau78} on graphs.\n\n \\begin{proof}[Proof of Theorem~\\ref{thm:main theorem2}]\n   It suffices to prove that $P_t \\mathds{1}=\\mathds{1}$ where $\\mathds{1}$ is the constant function $1$ on $V.$ By completeness, let $\\eta_k\\in C_0(V)$ satisfy \\eqref{d:complete}. The dominated convergence theorem yields that $P_t \\eta_k\\to P_t \\mathds{1}$ pointwise. By the local finiteness of the graph, for any $x\\in V$ and $t>0,$\n   \\begin{eqnarray*}\n     \\Gamma(P_t\\mathds{1})(x)&=&\\lim_{k\\to \\infty} \\Gamma(P_t \\eta_k)(x)\\leq \\liminf_{k\\to\\infty}e^{-2K t}P_t (\\Gamma(\\eta_k))(x)\\\\\n     &\\leq &\\liminf_{k\\to\\infty}e^{-2Kt}\\cdot\\frac{1}{k}=0.\n   \\end{eqnarray*} This means that for any $t>0,$ $P_t\\mathds{1}$ is a constant function on $V.$ Since the function $P_t\\mathds{1}$ is continuous in $t$ pointwise and $P_0\\mathds{1}=\\mathds{1},$ we get $P_t\\mathds{1}=\\mathds{1}$ for any $t>0.$ This proves the theorem.\n \\end{proof}\n\n\n\n\n\n{\\bf Acknowledgements.} This work was done when the authors were visiting the Shanghai Center for Mathematical Sciences, Fudan University in Summer 2014. They acknowledge the support from SCMS.\n\nB. H. is supported by NSFC, grant no. 11401106. Y. L. is supported by NSFC, grant no. 11271011, the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China($11$XNI$004$).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Hydrodynamics are relevant for heavy-ion collisions}\n\nOne of the most striking lessons one may draw \\cite {review,hydro} from  \nexperiments on  \nheavy-ion collisions at high energy ($e.g.$ at the RHIC accelerator, Brookhaven) \nis \nthat fluid hydrodynamics seems to be relevant for understanding the dynamics of \nthe reaction.  Indeed, the elliptic flow \n\\cite{JY} describing the anisotropy of the low-p$_T$ particles produced in  \na collision at non zero impact parameter implies the existence of a collective \nflow of the particles following a hydrodynamical  pressure gradient due to the \ninitial eccentricity in the collision. Moreover most hydrodynamical \nsimulations which are successful to describe this elliptic flow are consistent \nwith an almost ``perfect fluid'' behaviour, $i.e.$ a small ``viscosity over \nentropy'' ratio $\\eta\/s$ (see, for instance, the reviews \\cite{hydro}).\n\nThe validity of a hydrodynamical description assuming a \nquasi-perfect fluid behaviour has been nicely anticipated  in \nRef.\\cite{Bjorken}. The so-called {\\it Bjorken flow} is based on the hypothesis \nof an intermediate \nstage of the reaction process, namely a boost-invariant\\footnote{The \nintroduction of hydrodynamics in the \ndescription of  high-energy \nhadronic collisions has been proposed by Landau \\cite{landau}, assuming  ``full \nstopping'' initial conditions which result in a non boost-invariant solution or \n$Landau\\ flow$ (see \\cite{us} for a unified description of Bjorken and Landau \nflows). We will comment later on the relevance of the Landau flow for AdS\/CFT.}\n quark-gluon plasma phase as a  relativistic expanding fluid. It is  formed  \nafter a (quite rapid) thermalization period and finally decays into hadrons, see \nFig.~\\ref{1}. The boost-invariance can be justified  in the central region \nof the collision since the observed distribution of particles is flat, in \nagreement with the prediction of hydrodynamical boost-invariance, where \n(space-time) fluid \nand (energy-momentum) particle rapidities are proved to be equal \\cite{Bjorken}, \nsee section 3.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=30pc]{NoName.eps}\n\\end{center}\n\\caption{\\small{\\it Description of QGP formation in heavy ion collisions}. The \nkinematic landscape is defined by ${\\tau = \\sqrt{x_0^2-x_1^2}\\ ;\\ {\\eta=\\f 12 \n\\log \\f \n{x_0+x_1}{x_0-x_1}}\\ ;\\ {x_T\\!=\\!\\{x_2,x_3\\}}}\\ ,$\nwhere the coordinates along the light-cone are $x_0 \\pm x_1,$ the transverse \nones \nare $\\{x_2,x_3\\}$ and $\\tau$ is the proper time, $\\eta$ the ``space-time \nrapidity''.}\n\\label{1}\n\\end{figure}\n\nThe Bjorken flow was instrumental for deriving many qualitative and even  \nquantitative \nfeatures of   the quark-gluon plasma formation in heavy-ion \nreactions. However, as inherent to the hydrodynamic approach, it says only \nlittle on the relation with the microscopic gauge field theory, $i.e.$ Quantum \nChromodynamics (QCD). Some \nimportant questions  remain unsolved, such as  the reason why the fluid \nbehaves like a perfect fluid, what is the small amount of viscosity it may \nrequire,  why and how fast thermalization proceeds, etc... The problem is made \neven \nmore difficult by the  strong coupling regime of QCD which is very probably \nrequired, since a perturbative description leads in general to a high  $\\eta\/s$. \nIndeed,  the  mean free path induced by the gauge theory should be small (hence \nthe coupling strong) in order to damp  the near-by force  transversal to the \nflow, measuring the shear viscosity. \n\nIt is thus interesting to use our modern (but still largely in progress) \nknowledge\nof non perturbative methods in quantum field theory to fill the gap between the \nmacroscopic and microscopic descriptions of the quark-gluon plasma produced in \nheavy-ion collisions. Lattice gauge theory methods are very useful to analyze \nthe static properties of the quark-gluon plasma, but there are still powerless \nto describe the plasma in collision. Hence we are led to rely upon the new \ntools offered by the Gauge\/Gravity correspondence and in particular the one \nwhich is the most studied and well-known namely the AdS\/CFT duality \n\\cite{adscft} between the \n${\\cal N}=4$ supersymmetric Yang-Mills theory and the type IIB superstring in the \nlarge $N_c$ approximation. The features of the gauge theory on the (physical) \nMinkowski space in $3+1$ dimensions at strong coupling are in one-to-one \nrelation with  corresponding ones in the bulk of the target space of the 10-d \nstring and in particular in the 5-dimensional metric of the AdS space, the boundary of which \ncan be identified with the 4-dimensional Minkowski space.\n\nOne should be aware when using the AdS\/CFT tools that there does not yet exist a \ngravity dual construction \nfor QCD. However, the nice feature of the quark-gluon plasma problems is that it \nis \na deconfined phase of QCD, characterized by collective degrees of freedom  and \nthus one may expect to get useful information from  AdS\/CFT duality. This has \nbeen already proved when describing static geometries by an \nevaluation of  $\\eta\/s$  \\cite{son}.  The subject of the present lectures is the \ninvestigation of the \nGauge\/Gravity correspondence, in particular the AdS\/CFT duality, in a dynamical \nsetting corresponding to a \ncollision. \n\n\\section{{\\it Relativistic Hydrodynamics} and  {\\it Bjorken Flow}}\n\\label{sechydro}\n\nOn theoretical grounds, there are quite appealing features for\napplying hydrodynamic concepts to high-energy heavy-ion reactions. Such\nconcepts have been already introduced some time ago \\cite{landau,Bjorken} and \nfind a plausible \nrealization\nnowadays. The fact that a rather dense interacting medium is created in the \nfirst stage of\nthe collision allows one to admit that the individual partonic or\nhadronic degrees of freedom are not relevant during the early evolution\nof the medium and justifies its treatment as a fluid. For the same reason local \nequilibrium is a plausible assumption. Moreover, the high\nquantum occupation numbers allow one to use a classical picture and to\nassume that the ``pieces of fluid'' may follow quasi-classical\ntrajectories in space-time, expressed as an in-out cascade\n\\cite{gl} with straight-line trajectories starting at the origin (see Fig. 2), with\n\\begin{equation}\ny=\\eta \\label{bgl}\n\\end{equation}\nwhere\n\\begin{equation}\ny=\\frac12 \\log\\left(\\frac{E+p}{E-p}\\right) \\;;\\;\\;\n\\eta=\\frac12 \\log\\left(\\frac{x_0+x_1}{x_0-x_1}\\right) \n\\end{equation}\nare respectively the rapidity and ``space-time rapidity'' of the piece of the \nfluid\\footnote{We keep the conventional notation $\\eta$, not to be confused with \n viscosity. The difference is clear enough to avoid mistakes.}.\n\\begin{center}\n\\begin{figure}[ht]\n$\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n$\\includegraphics[width=8cm]{Rys4_bis.eps}\n\\label{2}\n\\caption{{\\it In-Out cascade.} The ``piece of fluid'' with space-time rapidity \n$\\eta$ gives rise to hadrons at rapidity $y\\equiv\\eta,$ after crossing the \n``freeze-out'' hyperbola at fixed proper-time $\\tau.$}\\end{figure} \n\\end{center}     \nNote that (\\ref{bgl}) can be rewritten in the form\n\\begin{equation}\n2y=\\log u^+-\\log u^-= \\log x^+- \\log x^- \\label{uz}\n\\end{equation} where\n$u^\\pm=e^{\\pm y}$ are the light-cone components of the fluid\n(four-)velocity and $x^\\pm=x_0\\pm x_1$ are the light-cone kinematical     \nvariables.      \n     \nTaking (\\ref{bgl}) as the starting point and using the perfect fluid     \nhydrodynamics, Bjorken developped in his seminal paper \\cite{Bjorken} \na suggestive (and\nvery useful in many applications) physical picture of the \ncentral     \nrapidity region of highly relativistic collisions of heavy ions. In this picture \nthe \ncondition (\\ref{bgl}) leads to a     boost-invariant geometry of the expanding \nfluid \nand thus to the central\nplateau in the distribution of particles.     \n     \nLet us introduce the relativistic hydrodynamic equations in light-cone \nvariables. We consider the ``perfect fluid'' approximation for which the \nenergy-momentum tensor is\n\\begin{equation}    \n T^{\\mu\\nu}= (\\epsilon+p)u^{\\mu}u^{\\nu} - p \\eta^{\\mu\\nu}  \\label{T}\n\\end{equation}     \nwhere $\\epsilon$ is the energy density, $p$ is the pressure and      \n$u^{\\mu}$ is the 4-velocity. We assume that the energy density and pressure\n are related      \nby the equation of state:     \n\\begin{equation}     \n\\epsilon = gp \n\\label{state}     \n\\end{equation}     \n where $1\/\\sqrt{g}$ is the sound velocity in the liquid. For the ``conformal \ncase'' $ T^{\\mu\\mu}=0$ and thus $g\\equiv 3.$  \n    \nUsing     \n\\begin{equation}     \nu^\\pm\\equiv u^0\\pm u^1=e^{\\pm y}     \n\\end{equation}\nand introducing     \n\\begin{equation}     \nx^\\pm=  x^0\\pm x^1 =\\tau e^{\\pm \\eta}\\; \\rightarrow \\;\n(\\frac {\\d}{\\d x^0}\\pm \\f {\\d}{\\d x^1})={\\scriptstyle\\ \\frac12}  \\f {\\d}{\\d \nx^\\pm}   \n\\equiv{ \\scriptstyle\\ \\frac12} \\d_\\pm     \n\\end{equation}     \nwhere $\\tau=\\sqrt{x^+x^-}$ is the proper time and $\\eta$ is the spatial     \nrapidity of the fluid,      \nthe hydrodynamic equations      \n     \\begin{equation}     \n\\d_\\mu T^{\\mu\\nu} =0     \n\\end{equation}     \ntake the form     \n\\begin{equation}     \n\\d_\\pm T^{01}+\\frac12\\d_+(T^{11}\\pm T^{00})     \n-\\frac12 \\d_-(T^{11}\\mp T^{00})=0\\ .     \n\\end{equation}     \nUsing now (\\ref{T}) and the equation of state (\\ref{state})     \n we deduce from this     \n\\begin{eqnarray}     \ng\\d_+[\\log p]&=& -\\frac{ (1\\!+\\!g)    ^2}2\\d_+y-\\frac{g^2\\!-\\!1   \n}2e^{-2y}\\d_-y \\nonumber\\\\     \ng\\d_-[\\log p]&=&\\frac{ (1\\!+\\!g)    ^2}2\\d_-y +\\frac{g^2\\!-\\!1   }2e^{2y}\\d_+y \\ \n. \n\\label{ef}     \n\\end{eqnarray}     \nThese are two equations for two unknowns which describe     \n the state of the liquid: the pressure $p$ and the     \nrapidity $y$. They should be expressed in terms of the     \npositions $x^+,x^-$ in the liquid. Other thermodynamic quantities can be     \nobtained from the equation of state (\\ref{state}) and the standard     \nthermodynamical identities:     \n\\begin{equation}      \np+\\epsilon = Ts\\;;\\;\\; d\\epsilon = T ds \\label{therm}     \n\\end{equation}     \nwhere we have assumed for simplicity vanishing chemical potential.     \n     \nThe result is      \n\\begin{equation}     \n\\epsilon =gp=\\epsilon_0 T^{g+1}\\;;\\;\\;s=s_0T^g\\;\\rightarrow \\; s\\sim     \n\\epsilon^{g\/(g+1)} . \\label{esp}     \n\\end{equation} \n     \nThe simplest possibility to describe the expansion of the fluid     \nwas suggested by Bjorken \\cite{Bjorken} who proposed to use the Ansatz \n(\\ref{bgl}) \nin the hydrodynamical context.\nIntroducing (\\ref{bgl}) into (\\ref{ef}) we obtain      \n\\begin{equation}     \ng\\d_+[\\log p]=-\\frac{1+g}{2x^+}\\;;\\;\\;     \ng\\d_-[\\log p]=-\\frac{g+1}{2x^-}     \n\\end{equation}     \nfrom which we deduce      \n\\begin{equation}     \np=\\epsilon\\ g^{-1}= p_0\\ (x^+x^-)^{-(g+1)\/2g} = p_0\\ \\tau^{-(g+1)\/g}\\ ,     \n\\end{equation}     \nwhere $p_0$ is a constant, and thus specifically\n    \n\\begin{equation}     \np=\\epsilon\/3= p_0\\ (x^+x^-)^{-2\/3} = p_0\\ \\tau^{-4\/3} \\propto   T^4    \n\\label{g3}\\end{equation}     \nfor the conformal case.\n\nThus the system is     \nboost-invariant: the pressure     \ndoes not depend neither on $\\eta$ nor on $y$. So are $\\epsilon$, $s$ and $T$,    \n given by (\\ref{esp}). \n It is interesting to note that the Landau flow  corresponds asymptotically only \nto a logarithmic correction of relation \\eqref{bgl}, namely\n     \n\\begin{equation}\nu^{\\pm}\\sim x^{\\pm}\\sqrt{\\log x^{\\pm}} \\ ,\\label{l}\n\\end{equation}\n which gives finally rise (as already noticed in \\cite{landau}, and for instance \n recently discussed in \\cite{us1}) to a gaussian shape in the $y$ distribution \nof the entropy, revealing a non boost-invariant picture, at least at some \ndistance from central rapidity.\n    \n\\section{Interest of AdS\/QCD duality}\n\n In the previous sections  we mentionned the ubiquity of hydrodynamic methods in \nthe description of QGP produced at RHIC. Yet, despite their success in \ndescribing data, we have to keep in mind that they are used as a \nphenomenological model without a real derivation from gauge theory. This is \nquite understandable since almost perfect fluid hydrodynamics is intrinsically a \nstrong coupling phenomenon - for which one lacks a purely gauge theoretical \nmethod\\footnote{Lattice QCD methods do not work well here as this would require \nanalytical continuation to Minkowski signature which is nontrivial in this \ncontext}. \n\nOn the other hand there exists a wide class of gauge theories, which can be \nstudied analytically at strong coupling. These are superconformal field theories \nwith gravity duals. String theory methods (namely AdS\/CFT correspondence) maps \ngauge theory dynamics (CFT) at strong coupling and large number of colors into \nsolving Einstein equations in asymptotically anti-de Sitter space (AdS). The \ntheories with gravity duals can differ substantially from real world QCD at zero \ntemperature. The best known example of such theory - $\\mathcal{N} = 4$ super \nYang-Mills (SYM) - is a superconformal field theory with matter in the adjoint \nrepresentation of the gauge group SU(N$_{c}$). Because of the conformal symmetry \nat the quantum level this theory does not exhibit confinement. On the other hand \ndifferences between $\\mathcal{N} = 4$ SYM and QCD are less significant above \nQCD's critical temperature, when quarks and gluons are in the deconfined phase. \nMoreover it was observed on the lattice that QCD exhibits a quasi-conformal \nwindow in the certain range of temperatures, where the equation of state is \nwell-approximated by $\\epsilon = 3 p$. The above observations together with \nexperimental results suggesting that quark-gluon plasma is a strongly coupled \nmedium is an incentive to use the AdS\/CFT correspondence as a tool to get \ninsight into the non-perturbative dynamics. \n\n\n\n\n\n\n\n\n\\section{AdS\/CFT setup}\n\nWe will now describe how to set up an AdS\/CFT computation for determining the \nspacetime behaviour of the energy-momentum tensor \\cite{US1}. This \nmethod does not make any underlying assumptions about local equilibrium or \nhydrodynamical behavior. We will obtain hydrodynamic expansion as a generic late \ntime behaviour of the expanding strongly coupled plasma. \n\nSuppose that we consider some macroscopic state of the plasma characterized by a \nspacetime profile of the energy-momentum tensor\n\\begin{equation}\nT_{\\mu\\nu}(x^\\rho)\\ .\n\\end{equation}\nThen, since the AdS\/CFT correspondence asserts the exact equivalence of gauge \nand string theory, such a state should have its counterpart on the string side \nof the correspondence. Typically it will be given by a modification of the \ngeometry of the original $AdS_5\\times S^5$ metric. This comes from the fact that \noperators in gauge theory correspond to  fields in supergravity (or string \ntheory). When we consider a state with a nonzero expectation value of an \noperator, the \\emph{dual} gravity background will have the corresponding field \nmodified from its `vacuum' $AdS_5 \\times S^5$ value. In the case of the energy \nmomentum tensor the corresponding field is just the 5-dimensional metric. One then has to \nassume that the geometry is well defined i.e. it does not have a naked \nsingularity - a singularity not hidden by an event horizon -.\nThis principle will select the allowed physical spacetime profiles of \ngauge theory energy-momentum tensor. Thus together with the Einstein equations \nthis becomes the main dynamical mechanism for the strongly coupled gauge theory.\n\nThe simplest way to formulate the precise correspondence between the expectation \nvalue of the energy-momentum tensor and bulk geometry is to use the \nFefferman-Graham system of coordinates \\cite{fg} for the latter:\n\\begin{equation}\nds^2=\\f{g_{\\mu\\nu}(x^\\rho,z) dx^\\mu dx^\\nu+dz^2}{z^2}\\ .\n\\end{equation} \nThis metric has to be a solution of 5-dimensional Einstein's equation with negative \ncosmological constant\\footnote{One can show that such solutions lift to 10-dimensional \nsolutions of ten dimensional type IIB supergravity. The effective 5-dimensional negative \ncosmological constant comes from the 5-form field in 10-dimensional supergravity.}:\n\\begin{equation}\n\\label{e.einst}\nR_{\\mu\\nu}-\\f{1}{2}g_{\\mu\\nu} R - 6\\, g_{\\mu\\nu}=0\\ .\n\\end{equation}\nThe expectation value of the energy momentum tensor may be easily recovered by \nexpanding the metric near the boundary $z=0,$ following the ``holographic \nrenormalization'' procedure \\cite{Skenderis},\n\\begin{equation}\ng_{\\mu\\nu}(x^\\rho,z)=\\eta_{\\mu\\nu}+z^4 g^{(4)}_{\\mu\\nu}(x^\\rho)+\\ldots\\ .\n\\end{equation}\nThen\n\\begin{equation}\n\\cor{T_{\\mu\\nu}(x^\\rho)} = \\f{N_c^2}{2\\pi^2} \\cdot g^{(4)}_{\\mu\\nu}(x^\\rho)\\ .\n\\end{equation}\nThis relation can be used in two ways. Firstly, given a solution of Einstein \nequations we may read off the corresponding gauge theoretical energy-momentum \ntensor. Secondly, given a traceless and conserved energy-momentum profile one \nmay integrate Einstein equations into the bulk in order to obtain the dual \ngeometry\\footnote{{This can be done order by order in $z^{2}$, which is a \nnear-boundary expansion. However potential singularities are hidden deep in the \nbulk, thus this power series needs to be resummed.}}. Then the criterion of \nnonsingularity of the geometry obtained in this way will determine the allowed \nspacetime evolution of the plasma. Let us note that this formulation is in fact \nquite far away from a conventional initial value problem.\n\nBefore we move to the case of expanding plasma, it is convenient to consider the \nsimple situation of a static uniform plasma with a constant energy momentum \ntensor. Then the Einstein's equations can be solved analytically and we find \n\\cite{US1}\nthat the exact dual geometry of such a system is\n\\begin{equation}\n\\label{e.bhfef}\nds^2=-\\f{(1-z^4\/z_0^4)^2}{(1+z^4\/z_0^4)z^2}\\ dt^2\n+(1+z^4\/z_0^4)\\f{dx^2}{z^2}+ \\f{dz^2}{z^2}\\ .\n\\end{equation} \nThis metric may look at first glance unfamiliar, but a change of coordinates\n\\begin{equation}\n\\tilde{z}=\\f{z}{\\sqrt{1+\\f{z^4}{z_0^4}}}\n\\end{equation}\ntransforms it to the standard AdS Schwarzschild static black hole\n\\begin{equation}\nds^2=-\\f{1-\\tilde{z}^4\/\\tilde{z}_0^4}{\\tilde{z}^2} dt^2\n+\\f{dx^2}{\\tilde{z}^2}+\\f{1}{1-\\tilde{z}^4\/\\tilde{z}_0^4} \\f{d\\tilde{z}^2}{\\tilde{z}^2}\n\\label{standard}\\end{equation}\nwith $\\tilde{z}_0=z_0\/\\sqrt{2}$ being the location of the horizon. Before we proceed \nfurther, let us note here one crucial thing: the fact, that the dual geometry of \na gauge theory system with constant energy density is a black hole was {\\em not} \nan assumption, but rather an outcome of a computation.\n\nThe Hawking temperature\n\\begin{equation}\nT=\\f{1}{\\pi \\tilde{z}_0} \\equiv \\f{\\sqrt{2}}{\\pi z_0}\n\\end{equation}\nis then identified with the gauge theory temperature, and the entropy with the \nBekenstein-Hawking black hole entropy \n\\begin{equation}\nS=\\f{N_c^2}{2\\pi\\tilde{z}_0^3}=\\f{\\pi^2}{2} N_c^2 T^3\n\\end{equation}\nwhich is $3\/4$ of the entropy at zero coupling. To finish our discussion of the \nstatic black hole, we note that the Fefferman-Graham coordinates cover only the \npart of spacetime lying outside the horizon. \n\n\n\\section{Boost invariant flow}\n\nLet us now apply the above procedure to a generic boost-invariant flow, in view of\nmaking contact with the hydrodynamical Bjorken flow described \nin section 2. However we do not want to make any preassumptions on the dynamics, since\nwe would like to recover the hydrodynamic behaviour as an outcome of an AdS\/CFT\ncomputation. To this end let us consider the most general gauge theory energy-momentum tensor\nwhich is boost-invariant and does not depend on transverse coordinates (see Fig.~1). Then\nconservation of energy-momentum $\\partial_\\mu T^{\\mu\\nu}=0$ and tracelessness $T^\\mu_\\mu=0$ allow\nto express all nonvanishing components of $T_{\\mu\\nu}$ in terms of a single \nfunction $\\varepsilon(\\tau)$ -- the energy density at mid rapidity:\n\\begin{equation}\n\\label{e.tgen}\nT_{\\mu\\nu}\\! = \\!\n\\left(\\begin{tabular}{cccc}\n$\\varepsilon(\\tau)$ & 0 &0 & 0 \\\\\n0 & $-\\tau^3 \\f{d}{d\\tau} \\varepsilon(\\tau)\\!-\\!\\tau^2 \\varepsilon(\\tau)$ & 0 & 0 \\\\\n0 & 0 & $\\varepsilon(\\tau)\\!+\\! \\f{1}{2}\\tau \\f{d}{d\\tau} \\varepsilon(\\tau)$ & 0 \\\\\n0 & 0 & 0 & $\\varepsilon(\\tau)\\!+\\! \\f{1}{2}\\tau \\f{d}{d\\tau} \\varepsilon(\\tau)$\n\\end{tabular}\\right)\n\\end{equation}\n\\smallskip\n\nLet us concentrate, \nfollowing \\cite{US1} on the late time asymptotics of this function i.e.\n\\begin{equation}\n\\varepsilon(\\tau) \\sim \\f{1}{\\tau^s} +\\ldots\n\\end{equation}\nfor $\\tau \\to \\infty$. Energy positivity requires that $0 \\leq s \\leq 4$. We \nwill consider sharp inequalities here\\footnote{Recently the case $s=4$ has been \nconsidered in \\cite{Kajantie:2008jz}.}. The most general metric consistent with \nthe symmetry assumptions is\n\\begin{equation}\n\\label{e.ansatz}\nds^2=\\f{-e^{a(\\tau,z)} d\\tau^2 +\\tau^2 e^{b(\\tau,z)} dy^2\n  +e^{c(\\tau,z)} dx^2_\\perp}{z^2} +\\f{dz^2}{z^2}  \\ .\n\\end{equation}\nIn order to find the late time form of the solution corresponding to \n$\\varepsilon(\\tau)=1\/\\tau^s$ we may solve the Einstein equations in a power series for \nthe metric coefficients\n\\begin{equation}\n\\label{e.aexp}\na(\\tau,z)=\\sum_{n=0}^N a_n(\\tau) z^{4+2n}\n\\end{equation}\nwhere $a_0(\\tau)=-\\varepsilon(\\tau)=-1\/\\tau^s$. Then from each coefficient $a_n(\\tau)$ \nwe may extract the leading large $\\tau$ behaviour and neglect the subleading \nterms. It turns out that this procedure is exactly equivalent to introducing a \nscaling variable\n\\begin{equation}\nv=\\f{z}{\\tau^{\\f{s}{4}}}\n\\end{equation}\nand assuming the metric coefficients to be just functions of $v$ e.g. \n$a(z,\\tau)=a(v)$ in the large proper time limit (namely $\\tau \\to \\infty$, $z\\to \n\\infty$ with $v$ kept fixed). In this limit Einstein's equations become just \nordinary differential equations and may be solved analytically. \nThe singularity of these geometries can then be tested by computing the scalar \ncurvature invariant\n\\begin{equation}\n{\\mathfrak R^2}=R^{\\mu\\nu\\alpha\\beta}R_{\\mu\\nu\\alpha\\beta}\\ .\n\\end{equation}\nSince our solutions are defined only in the large proper time limit $\\tau \\to \n\\infty$ with the scaling variable $v$ kept fixed, we have to evaluate ${\\mathfrak R^2}$ in \nthe same manner\\footnote{{It should be stressed however, that this condition is \nreally a condition of regularity of the expansion of the curvature invariant. It \nis safe to do as long as each term in the large proper-time expansion is \nregular. On the other hand any singularity present in this expansion might be \neither a genuine curvature singularity or a singularity of the expansion, \nsee a detailed discussion in  section \\ref{secsing}.}}.\n\nThis procedure is described in detail in \\cite{US1}. The result is that\n\\begin{itemize}\n\\item for generic $s$ the resulting solution is singular\n\\item the only nonsingular solution corresponds to $s=\\f{4}{3}$ which is just \nthe hydrodynamic Bjorken expansion (see \\eqref{g3}, section 2)\n\\item the resulting metric takes the form\n\\end{itemize}\n\\begin{eqnarray}\n\\label{e.flgeom}\nds^2=\\f{1}{z^2} \\left[- \\f{\\left( 1-\\f{e_0}{3}\n      \\f{z^4}{\\tau^{4\/3}}\\right)^2}{1+\\f{e_0}{3}\\f{z^4}{\\tau^{4\/3}}} d\\tau^2+\n\\left( {\\textstyle 1+\\f{e_0}{3} \\f{z^4}{\\tau^{4\/3}}}\\right) (\\tau^2\n      dy^2 +dx^2_\\perp)\\right] \n+ \\f{dz^2} {z^2}\n\\end{eqnarray}\nwhere we reinstated the dimensionful parameter $e_0$ so that\n\\begin{equation}\n\\varepsilon(\\tau)=e_0\/\\tau^{\\f{4}{3}}\\ .\n\\end{equation} \nLet us note some salient features of this result. The geometry (\\ref{e.flgeom}) \nbears striking resemblance to the AdS black hole geometry (\\ref{e.bhfef}) but \nwith the position of the `effective horizon' being time dependent\n\\begin{equation}\nz_0=\\sqrt[4]{\\f{3}{e_0}} \\cdot \\tau^{\\f{1}{3}}\n\\end{equation}\nThen assuming similar relations as for the black hole case one gets the Bjorken \nscaling of the temperature and entropy.\n\\begin{eqnarray}\nT &=& \\f{\\sqrt{2}}{\\pi z_0}= \\f{2^{\\f{1}{2}} e_0^{\\f{1}{4}}}{\\pi 3^{\\f{1}{4}}} \n\\tau^{-\\f{1}{3}}\\nonumber\\\\\nS &\\propto& \\frac{\\tau}{z_0^3} =const\\ .\n\\end{eqnarray}\nWe see that the `movement' of the horizon into the bulk of AdS corresponds \nphysically to cooling of the expanding gauge theory plasma system.\n\nA significant fact that has to be kept in mind is that the geometry \n(\\ref{e.flgeom}), in contrast to (\\ref{e.bhfef}), is not an exact solution of \nEinstein's equation. It is valid only for large times. For smaller times it has \nto be modified. We will now discuss this issue in more detail as it reflects \nimportant physical properties of the gauge theory plasma.\n\n\\section{Beyond perfect fluid}\n\nThe geometry (\\ref{e.flgeom}) is only a solution of Einstein equations in the \nscaling limit. However with some effort, one can get also the first subleading \ncorrections to the metric i.e.\n\\begin{equation}\n\\label{e.flexp}\na(z,\\tau)=a_0(v)+\\f{1}{\\tau^{\\f{4}{3}}}\\ a_2(v) +\\ldots\n\\end{equation}\nThen after evaluating ${\\mathfrak R^2}$, keeping track of subleading terms we find\n\\begin{equation}\n{\\mathfrak R^2}=R_0(v)+ \\f{1}{\\tau^{\\f{4}{3}}} R_2(v) +\\ldots\n\\end{equation}\nwhere $R_0(v)$ is finite, but $R_2(v)$ develops a $4^{th}$ order pole \nsingularity. The physical meaning of this behaviour is indeed quite clear. The \ngeometry (\\ref{e.flexp}) is dual to a state in gauge theory which undergoes \nexpansion according to exact {\\em perfect fluid} hydrodynamics. Yet we know that \ngauge theory plasma has nonzero viscosity and hence the perfect fluid behaviour \n\\begin{equation}\n\\varepsilon(\\tau)=\\f{1}{\\tau^{\\f{4}{3}}}\n\\end{equation}\nis not exact but, if it would be described by viscous Bjorken expansion (viscous \nhydrodynamics), it would be modified to\n\\begin{equation}\n\\varepsilon(\\tau)=\\f{1}{\\tau^{\\f{4}{3}}}\\left( 1-\\f{2\\eta_0}{\\tau^{ \\f{2}{3}}} +\\ldots \n\\right)\n\\end{equation}\nwhere $\\eta_0$ is related to the shear viscosity through $\\eta=\\eta_0\/\\tau$ \n(which follows from the scaling $\\eta \\propto T^3$).\n\nLet us show how this arises using the AdS\/CFT methods.\nWe will not presuppose a specific form of subleading correction but will start \nfrom\n\\begin{equation}\n\\varepsilon(\\tau)=\\f{1}{\\tau^{\\f{4}{3}}}\\left( 1-\\f{2\\eta_0}{\\tau^r} +\\ldots \\right)\n\\end{equation} \nwith a generic $r$. In order to verify that plasma expansion follows viscous \nhydrodynamics we will have to first show that $r=\\f{2}{3}$. The metric \ncoefficients will now have an additional piece scaling as $\\f{1}{\\tau^{r}} \na_r(v)$. It turns out that the curvature scalar ${\\mathfrak R^2}$ is always {\\em \nnonsingular} at that order\\footnote{This was first observed for $r=2\/3$ in \n\\cite{Nak1}.}. Hence we have to go one order further i.e. find all coefficients \nappearing in the following expansion\n\\begin{equation}\na(z,\\tau)=a_0(v)+\\f{1}{\\tau^{r}} a_r(v)+\\f{1}{\\tau^{2r}} a_{2r}(v)+ \n\\f{1}{\\tau^{\\f{4}{3}}} a_2(v) +\\ldots\n\\end{equation}\nThen the curvature scalar has the form\n\\begin{equation}\n{\\mathfrak R^2}=R_0(v)+\\f{1}{\\tau^{r}} R_r(v)+\\f{1}{\\tau^{2r}} R_{2r}(v)+ \n\\f{1}{\\tau^{\\f{4}{3}}} R_2(v) +\\ldots\n\\end{equation}\nwith $R_0(v)$ and $R_r(v)$ being nonsingular, while {\\em both} $R_{2r}(v)$ and \n$R_2(v)$ turn out to have $4^{th}$ order pole singularities. In order for them \nto have a chance to cancel we have to have\n\\begin{equation}\nr=\\f{2}{3}\n\\end{equation}\nwhich is exactly the scaling of a viscosity correction to Bjorken flow. Moreover \ncancelation occurs only when the shear viscosity coefficient has the \nvalue\\footnote{We set here $e_0=1$.}\n\\begin{equation}\n\\eta_0=2^{-\\f{1}{2}} 3^{-\\f{3}{4}}\n\\end{equation}\nwhich is equivalent to $\\eta\/s=1\/4\\pi$ (for details see \\cite{RJ}). In a similar \nmanner one can go one order higher and determine a coefficient of second order \nhydrodynamics. However at that order, it turns out that there remains a leftover \nlogarithmic singularity. We will show, in section \\ref{secsing}, that the \nlogarithmic singularity arises due to a pathology of the Fefferman-Graham \nexpansion and can be  avoided when one makes a different late time expansion.\n\nFinally let us comment on why it is interesting to verify the viscous \nhydrodynamic behaviour with the specific viscosity coefficient for the expanding \nplasma. Already before, there have been studies of {\\em linearized} perturbations \naround the uniform plasma which demonstrated that hydrodynamic behaviour appears \nfor small fluctuations and the value of viscosity was obtained from the Kubo \nformula \n\\cite{son}. It was interesting to verify whether hydrodynamics also applies in \nits fully \nnonlinear regime. The agreement of the resulting value of the viscosity \ncoefficient is thus a nontrivial consistency check.\n\nAnother motivation for developing an AdS\/CFT framework for studying such \ntime-dependent phenomenae is the fact that some of the most interesting and \npuzzling phenomena in heavy ion collisions are definitely very far from \nequilibruum. We will mention some examples in section \\ref{seclast}.\n\n\n\\section{Beyond boost-invariance}\n\nThe calculations presented in the previous sections were performed for systems \nwith boost invariance symmetry and full translational and rotational symmetry in \nthe transverse plane. This allowed us to perform explicit computations as the \nsymmetry assumptions effectively reduced the calculation to systems of ordinary \ndifferential equations. \nIn this manner we obtained directly the solution for gauge theory energy density \n$\\varepsilon(\\tau)$. Then, in order to find the link with hydrodynamics, we found that \nthis solution is a solution of hydrodynamic equations with specific values for \nthe transport coefficients. \n\nThis approach has both an advantage and a drawback. The advantage is that one \ndoes not presuppose any kind of dynamics and one may try to apply it in contexts \nvery far from equillibrum, where hydrodynamic description does not apply. The \ndrawback is that the appearance of hydrodynamic equations is not transparent and \nit is difficult to relax the symmetry assumptions due to the complexity of solving \nnonlinear Einstein's equations.\n\nRecently the latter drawback was addressed and it was shown in general how the \nequations of hydrodynamics arise from the gravity side \\cite{MINWALLA}. Here we \nwill briefly review this approach.\n\nLet us start from the static black hole (\\ref{standard},\\ref{e.bhfef}) but \nwritten in yet \nanother coordinate system -- the incoming Eddington-Finkelstein coordinates:\n\\begin{equation}\nds^2=-2dt dr-r^2 \\left(1- \\f{T^4}{\\pi^4 r^4} \\right)dt^2 +r^2\\eta_{ij}  dx^i \ndx^j\\ .\n\\end{equation}\nHere $T$ is the temperature, $r=\\infty$ corresponds to the boundary. \n$x^\\mu=const$ are null curves going from the boundary into the black hole. The \nadvantage of this coordinate system is that it is well defined on the horizon \nand extends all the way from the boundary to the singularity at the center of \nthe black hole.\n\nThe geometry given above corresponds to a uniform plasma at rest (i.e. with the \n4-velocity $u^\\mu=(1,0,0,0)$) and given temperature $T$. We may now perform a \nboost (and perform a dilatation) to obtain the dual geometry to a moving plasma system \nwith uniform 4-velocity $u^\\mu$ and temperature $T$:\n\\begin{equation}\n\\label{e.boosted}\nds^2=-2u_\\mu dx^\\mu dr-r^2 \\left(1- \\f{T^4}{\\pi^4 r^4} \\right)u_\\mu\nu_\\nu dx^\\mu dx^\\nu +r^2(\\eta_{\\mu \\nu}+u_\\mu u_\\nu)  dx^\\mu dx^\\nu\n\\end{equation}\nThe idea of ref.\\cite{MINWALLA} is to allow $u^\\mu$ and $T$ to be \n(slowly-varying) functions of the spacetime coordinates. Once this is done the \ngeometry  (\\ref{e.boosted}) ceases to be an exact solution of Einstein equation \nbecause of nonvanishing gradients of the parameters $u^\\mu$ and $T$. This \nsuggests to perform an expansion of the solution in terms of gradients which has \nbeen carried out in \\cite{MINWALLA} up to second order in derivatives. The \nintegration constants arising at each order are again fixed by requiring \nregularity of the metric at the horizon. The resulting metric is expressed in \nterms of 4-velocities and temperatures and their derivatives, so when one \nextracts the energy-momentum tensor it will be given directly in terms of those \nquantities. Up to first order the expression is\n\\begin{equation}\n\\label{e.tmunumin}\nT^{\\mu\\nu}=\\f{N_c^2}{8\\pi^2} \\left\\{(\\pi T)^4 (\\eta^{\\mu\\nu}+4u^\\mu u^\\nu)- \n2(\\pi T)^3 \\sigma_{shear}^{\\mu\\nu} \\right\\} \\ .\n\\end{equation}\nThe first term is just the perfect fluid energy momentum tensor, while the \nsecond term involves the shear viscosity. \nThis result essentially demonstrates how general hydrodynamic equations arise \nfrom gravity in AdS\/CFT. Indeed, once it is shown that the general form of the gauge \ntheory energy-momentum tensor has the form (\\ref{e.tmunumin}), then conservation \nof energy momentum $\\partial_\\mu T^{\\mu\\nu}=0$ is equivalent, by definition, to conformal \nrelativistic Navier-Stokes equations. \nAs a byproduct, the above construction also gives a map from solutions of \n(viscous) hydrodynamics to gravity solutions. However this setup requires that \nthe starting point is not far off from equilibruum. For processes which do not \nadmit a hydrodynamic description (like the early stage of a heavy-ion collision) \none has to resort to different methods.\n\n\n\\section{Reduction of Singularities}\n\\label{secsing}\n\nThe leftover logarithmic singularity found in the third order of the square of \nthe Riemann tensor \\cite{Heller:2007qt} (as well as in the higher curvature \ninvariants \\cite{Benincasa:2007tp}) might be the signal of either genuine \ncurvature singularity or the singularity of the chosen expansion \nscheme\\footnote{As it was stressed before, the large-proper time expansion of \ncurvature invariants is not diffeomorphism-invariant. The encountered \nsingularities are \nphysical only if there is no coordinate transformation which removes them.}. If \nthe first is true, this means that the whole framework is inconsistent and \neither one needs to include additional degrees of freedom to cure it or the \nboost-invariant flow is unphysical\\footnote{Since it corresponds to the naked \nsingularity on the gravity side}. The results presented in \n\\cite{Benincasa:2007tp} show that no supergravity field can fix the problem, \nwhich led to conjectures, that boost-invariant flow cannot be realized within the\nsupergravity framework \\cite{Buchel:2008xr}. On the other hand, the gravity dual of \ngeneral fluid flow up to the second order in derivatives was shown to be regular \nand it was hard to imagine how possible singularities could form in the third \norder \\cite{MINWALLA, Bhattacharyya:2008xc}. The resolution of this puzzle was \npresented in \\cite{Heller:2008mb} (see also \\cite{Kinoshita:2008dq}). It turns \nout that there exists a singular coordinate transformation from Fefferman-Graham \ncoordinates to Eddington-Finkelstein ones, which yields a completely regular and \nsmooth metric from the boundary up to the standard black-brane singularity. This \nleads to regular (apart from the standard black-brane singularity) large proper-time \nexpansion of curvature invariants. The metric ansatz in Eddington-Finelstein \ncoordinates takes the form\n\n\\begin{equation}\n\\mathrm{d}s^{2} = 2 \\mathrm{d} \\tilde{\\tau} \\cdot \\mathrm{d} r - r^{2} \\tilde{A} \n\\left( \\tilde{\\tau}, r \\right) \\mathrm{d} \\tilde{\\tau}^{2} + \\left( 1 + r \\cdot \n\\tilde{\\tau} \\right)^{2} e^{\\tilde{b} \\left( \\tilde{\\tau}, r \\right)} \\mathrm{d} \ny^{2} + r^{2} e^{\\tilde{c} \\left( \\tilde{\\tau}, r\\right)} \\mathrm{d} x_{\\perp}^{2} \n\\end{equation}\n\n\\noindent and was motivated by the boosted black-brane metric (\\ref{e.boosted}) \nwith a boost and dilatation parameters $u = 1 \\cdot \\partial_{\\tilde{\\tau}}$ and \n$T \\sim \\tilde{\\tau}^{-4\/3}$. The functions $\\tilde{A} \\left( \\tilde{\\tau}, r \n\\right)$, $\\tilde{b} \\left( \\tilde{\\tau}, r \\right)$ and $\\tilde{c} \\left( \n\\tilde{\\tau}, r\\right)$ are expanded in a large-proper time expansion \nanalogously as it was in the Fefferman Graham case, i. e.\n\n\\begin{equation}\n\\tilde{A} \\left( \\tilde{\\tau}, r \\right) = \\tilde{A}_{0} \\left( r \\cdot \n\\tilde{\\tau}^{1\/3} \\right) + \\frac{1}{\\tilde{\\tau}^{2\/3}} \\tilde{A}_{1} \\left(  \nr \\cdot \\tilde{\\tau}^{1\/3} \\right) + \\frac{1}{\\tilde{\\tau}^{4\/3}} \\tilde{A}_{2} \n\\left(  r \\cdot \\tilde{\\tau}^{1\/3} \\right) + \\ldots\n\\end{equation}\n\n\\noindent This form of expansion can also be justified by \n\\cite{Bhattacharyya:2008ji}. The terms damped by inverse power of proper time come \nfrom the gradient expansion. The boundary metric in proper-time-rapidity \ncoordinates has non-vanishing Christoffel symbols $\\Gamma \\sim \\ttau^{-1}$, thus \nthe four velocity gradient $\\nabla u$ (which is constant in these \ncoordinates) gives a factor of $\\ttau^{-1}$. On the other hand the expansion \nparameter multiplying each term in gradient expansion is the inverse power of \nthe temperature $T$(see \\cite{MINWALLA}). Because $T \\sim \n\\ttau^{-1\/3}$, the overall damping is indeed $\\ttau^{-2\/3}$ - a fact derived in \n\\cite{Heller:2007qt} from the non-singularity argument.\n\n The \nnon-perturbative\\footnote{In the sense of large-proper time expansion} piece in \nthe metric at $\\mathrm{d} y^{2}$ introduced in \\cite{Kinoshita:2008dq} is \nresponsible for a correct limit energy density $\\rightarrow 0$. It also becomes \nimportant if one wants to solve the problem of early-time dynamics \n\\cite{Kovchegov:2007pq} using Eddington-Finkelstein coordinates.\n\nThe integration \nconstants\\footnote{Not all of them -- there is a remaining gauge freedom \n(coordinate transformation), which leaves the metric ansatz unchanged: $r \n\\rightarrow r + f\\left( \\tilde{\\tau} \\right)$, where $f\\left( \\tilde{\\tau} \n\\right)$ is an arbitrary function} are fixed by requiring the regularity of the \nmetric functions $\\tilde{A}_{i} \\left( \\tilde{v} \\right)$, $\\tilde{b}_{i} \\left( \n\\tilde{v} \\right)$ and $\\tilde{c}_{i} \\left( \\tilde{v} \\right)$ at each order \n$i$. This is justified since the Eddington-Finkelstein are valid for \n$\\tilde{\\tau} > 0$ and $0 < \\tilde{v} = r \\cdot \\tilde{\\tau} < \\infty$. The \nsingular coordinate transformation which takes the metric from \nEddington-Finkelstein coordinates to Fefferman-Graham ones is given order by \norder in the gradient expansion by\n\n\\begin{eqnarray}\n\\tilde{\\tau} \\left( \\tau, z \\right) &=& \\tau \\cdot \\Big\\{ T_{0} \\left(z \\cdot \n\\tau^{-1\/3} \\right) + \\frac{1}{\\tau^{2\/3}} T_{1} \\left(z \\cdot \\tau^{-1\/3} \n\\right) + \\ldots \\Big\\} \\ \\mathrm{,}\\\\\nr \\left( \\tau, z \\right) &=& \\frac{1}{z} \\cdot \\Big\\{ R_{0} \\left(z \\cdot \n\\tau^{-1\/3} \\right) + \\frac{1}{\\tau^{2\/3}} R_{1} \\left(z \\cdot \\tau^{-1\/3} \n\\right) + \\ldots \\Big\\}\\ \\mathrm{.}\n\\end{eqnarray}\nThe leading-order solutions (corresponding to the perfect fluid on the gauge \ntheory side) \n are related by\n\\begin{eqnarray}\n\\label{coordtrafo}\n\\ttau &\\rightarrow& \\tau \\left\\{ 1 - \\frac{1}{\\tau^{2\/3}} \\left[ \\frac{3^{1\/4}\n    \\pi}{4 \\sqrt{2}} + \\frac{3^{1\/4}}{2 \\sqrt{2}}\n  \\tan^{-1}{\\left(\\frac{3^{1\/4}}{\\sqrt{2}} r \\cdot \\tau^{1\/3}\\right)}\\  + \n\\right.\\right. \\nonumber\\\\ &&\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ + \\left.\\left.\\frac{3^{1\/4}}{4 \n\\sqrt{2}} \\log {\\left(\\frac{r \\cdot \\tau^{1\/3} -\n      \\frac{\\sqrt{2}}{3^{1\/4}}}{ r \\cdot \\tau^{1\/3} +\n      \\frac{\\sqrt{2}}{3^{1\/4}}}\\right)} \\right] \\right\\}\\ \\mathrm{,}\n\\nonumber \\\\ \nr &\\rightarrow& \\frac{1}{z} \\cdot \\sqrt{1 + \\frac{z^{4}}{3 \\cdot \\tau^{4\/3}}}\n\\mathrm{.} \n\\end{eqnarray}\nThe transformation is singular at $z = 3^{1\/4} \\tau^{1\/3}$, which is precisely \nthe locus of the logarithmic singularity encountered in \\cite{Heller:2007qt}. \nFormulas for higher order transformation coefficients are too long to be \npresented here and can be found in \\cite{Math}. \nThe energy-momentum tensor extracted from the solution in Eddington-Finkelstein \ncoordinates reproduces the energy momentum tensor obtained in \n\\cite{Heller:2007qt}.\n\n\n\n\\section{Beyond hydrodynamics}\n\n\\label{seclast}\n\nGauge-gravity duality has already proven to be an invaluable tool in describing \nproperties of static or near-equilibrium (hydrodynamics) strongly coupled gauge \ntheory systems. Noticeable achievements in that direction are the viscosity \nevaluation bound \n\\cite{son} and the consistent formulation of the second order conformal \nhydrodynamics \\cite{Baier:2007ix, MINWALLA}. These successes came \nfrom the holographic understanding of hydrodynamics. On the other hand there is \nmuch more interesting and nontrivial dynamics than hydro. Far from equilibrium \nbehavior of gauge theories is a fascinating and pretty much open problem of \nexperimental importance, like the early universe or initial stages of heavy ion \ncollisions\\footnote{In the late stages of heavy ion collision, strongly coupled \nquark-gluon plasma forms and holographic technics at strong gauge coupling are \nbetter justified then just after the collision (running of the coupling). \nNevertheless applying AdS\/CFT correspondence to describe far from equilibrium \nprocesses in gauge theories is an interesting problem even from a purely \ntheoretical point of view.}. The AdS\/CFT correspondence is surely capable to  \nshed new light on these problems, or even be understood as a formulation of far \nfrom equilibrium gauge theory.\n\nIn the context of heavy-ion collisions the most important and probably the most \ndifficult questions concern the issues of early time dynamics \n\\cite{Kovchegov:2007pq} and the transition to an isotropic \\cite{Janik:2008tc} \nand \nthermalized medium. One of the puzzles here is the short time in which nuclear \nmatter approach local equilibrium. Experimental data fitting well with \nhydrodynamical simulations with small viscosity justified applications of the \nAdS\/CFT correspondence at strong coupling for the late stages of heavy-ion \ncollisions. It is not clear to what extent early time dynamics is driven by \nnon-perturbative effects and whether the  lessons learned from AdS\/CFT might be \ndirectly applied to the nuclear matter in the early stages of the evolution. \nApproaching equilibrium is also of an interest from the General Relativity point \nof view. Isotropic \nand thermalized matter on the gauge theory side corresponds to a black hole in \nAdS, whereas thermalization and approach to local equilibrium should be governed \nby the dynamics of gravitational collapse.\n\nPerhaps some of these questions might be answered by studying collisions of \nshock-waves in AdS. The geometry corresponding to a projectile in 3+1 dimensions \nwas constructed in \\cite{US1} using holographic renormalization. The \nmetric\n\\begin{equation}\n\\mathrm{d} s^{2} = \\frac1{z^{2}}\\left\\{- 2 \\mathrm{d} x^{+} \\mathrm{d} x^{-} + \nf\\left( x^{-} \n\\right) \\cdot z^{4} \\left( \\mathrm{d} x^{-} \\right)^{2} + \\mathrm{d} \nx_{\\perp}^{2} + \\mathrm{d} z^{2}\\right\\}\n\\end{equation}\nwith an arbitrary function $f \\left( x^{-} \\right)$ corresponds to the situation \nwhen\n\n\\begin{itemize}\n\\item the dynamics is one-dimensional (i.e. no dependence on transverse coordinates),\n\\item the energy-momentum tensor depends only on a single light-cone variable (here \nchosen to be $x^{-} = x^{0} - x^{1}$).\n\\end{itemize}\n\nTraceless and conserved energy momentum tensor satisfying the above assumptions \ntakes a particularly simple form -- its only non-zero component is $T^{- -}  = f \n\\left( x^{-} \\right)$. Choosing $f \\left( x^{-} \\right) = M \\delta \\left( x^{-} \n\\right) $ leads to a shock-wave -- infinitely thin plane of matter moving at the \nspeed of light, which is a toy-model for highly boosted nucleus. The idea is to \ncollide two such projectiles and single out the physical behavior of the plasma \nfrom the regularity of the dual geometry. This is a difficult problem, because \nof the broken boost-invariance, which leads to solving Einstein equations in 3 \nvariables ($x^{+}$, $x^{-}$ and $z$ or equivalently $\\tau$, $y$ and $z$). The \ngeometry before the collision ($x^{+} + x^{-}  <  0$) is known -- it is simply \nthe superposition of two incoming shock-waves\n\n\\begin{equation}\n\\mathrm{d} s^{2} = \\frac1{z^{2}}\\left\\{- 2 \\mathrm{d} x^{+} \\mathrm{d} x^{-}\\! + \n\\! M  \\delta \n\\left( x^{-} \\right) z^{4} \\left( \\mathrm{d} x^{-} \\right)^{2} \\!+\\! M \n \\delta \\left( x^{+} \\right) z^{4}  \\left( \\mathrm{d} x^{+} \n\\right)^{2} \\! +\\! \\mathrm{d} x_{\\perp}^{2} \\!+ \\!\\mathrm{d} z^{2}\\right\\} \n\\mathrm{.}\n\\end{equation}\n\nShock-waves collide at $x^{+} = x^{-} = 0$ and from now on the dynamics of the \nsystem must be deduced from Einstein equations. The first attempt to address \nthis issue in \\cite{Kajantie:2008rx}\nfocused on the simpler setup than presented so far -- a shock-waves collision in \n1+1 dimensions. The energy-momentum tensor for such a system before the \ncollision is given by $T_{+ +} = f \\left( x^{-} \\right) $ and $T_{- -} = g \n\\left( x^{+} \\right) $ with vanishing off-diagonal components. This is at the \nsame time the most general form of the energy-momentum tensor for a 1 + 1 \ndimensional CFT. A nice feature is that the dual geometry for the {\\em whole} \ncollision process can be constructed here exactly. However in this low \ndimensional context, the projectiles pass each other unaffected (or propelled  \nback-to-back \\cite{Kajantie:2008rx}), so the physics \nof plasma production and thermalization is  absent here. On the \nother hand the problem of genuine interest -- collision of shock-waves in 3+1 \ndimensions -- requires some approximation scheme in which Einstein equations \nbecome tractable. Up to now, two proposals have been made. The first one \n\\cite{Grumiller:2008va} treats proper time as a small parameter but suffers from \na negative energy density in some regions due to the conditions imposed at the \nlight-cone. The second one \\cite{Albacete:2008vs} solves Einstein equations \nperturbatively in $M$ leading to the prediction that shock-waves stop almost \nimmediately after the collision (reminding of the {\\it full stopping} condition \nof the Landau flow, see section \\ref{sechydro}.). This problems surely deserve further \nstudies. \n\nThere are also other studies of dynamical processes in an evolving plasma system \nwhich go beyond hydrodynamics. One example is the problem of thermalization of \nsmall perturbations around the expanding plasma (some first investigations has \nbeen performed in \\cite{US1}). Another use of the evolving geometries \nis to study other physical processes in the presence of the evolving plasma \nsystem like e.g. the physics of mesons and flavours studied through embedding D7 \nbranes in the time-dependent geometries \\cite{flavors}. Finally one may study \nisotropisation of anisotropic plasma. The first investigations have been \nperformed in \\cite{Janik:2008tc} (see the talk by P. Witaszczyk in the same \nproceedings).\n\n\\section{Summary}\nThe Gauge\/Gravity approach to the formation and evolution of a quark-gluon \nplasma in heavy-ion collisions described above has the interest of casting an \nexploratory bridge between the rigorous results of string theory and some \npending  questions raised by the experiments on quark-gluon plasma. These \nquestions cannot yet be raised in the framework of strongly coupled QCD, for \nwhich we do not possess the adequate tools, but they can be addressed for the \nfirst time in a quantitative and rigorous way in the supersymmetric case of the \nAdS\/CFT correspondence. It is thus a novel and valuable approach and can serve \nas a model for further studies.\n\nLet us summarize some aspects of this approach, being aware (and with apologies \nfor those not quoted or mentioned) that this subject is in constant development which\nwill force us to mention only a few of them.\n\nStarting with the experimental evidence that hydrodynamics is relevant in the \nformation and evolution of a quark-gluon plasma in heavy-ion collisions and in \nparticular  of the ``Bjorken flow'' description, we present the AdS\/CFT setup \nallowing to describe the dynamics of the plasma (in the AdS\/CFT case). We show \nthat it is possible to derive the geometry dual to the asymptotic evolution of \nthe plasma in terms of an expansion in a scaling variable. the nonsingularity \nrequirement on the Gravity side gives a set of ``selection rules'' on the Gauge \nside: the perfect fluid at first order, the $\\eta\/s=1\/4\\pi$ property and other \ntransport coefficients at higher orders...\n\nBeyond the boost-invariant Bjorken flow, there exists an intriguing but rigorous \none-to-one correspondence between the complete (and even completed using \nAdS\/CFT!) hydrodynamic equations and the solutions of the Einstein equations in \nthe bulk of the 5-dimensional space. Some apparent obstacles, such as the appearance of \nlogarithmic singularities in the asymptotic expansion of the 5-dimensional metric, have been \nshown to be a mere artefact of the choice of the expansion parameter and have \nbeen cured.\n\nBeyond the hydrodynamical description of the transient plasma, one goal is now \nto exoplore the dynamical aspects far from equilibrium. We describe some very \nrecent attempts, which even though not conclusive yet, show the interest of the \nextension of  Gauge\/Gravity correspondence to attack some down-to-earth \nproblems, such as the short thermalization time, probably observed by the \nheavy-ion phenomenology and more generally the effect of the initial conditions \non the whole process.\n\nIt is quite interesting  to see that some complex aspects of Gauge field theory \ndynamics can find unexpected answers from Gravity. It would  be intringuing that \nsome nontrivial aspects of Gravity (such as the dynamics of moving black holes) \nalso could gain some new insight from the correspondence with some aspects of \nheavy-ion collisions.\n\n\\section*{Acknowledgements}\nThis investigation was partly supported by the     \nMEiN research grants 1 P03B 04529 (2005-2008) (RP), 1 P03B 04029 (2005-2008) (RAJ and MPH) and N202 247135 (2008-2010) (MPH), by 6 Program of European\nUnion ``Marie Curie Transfer of Knowledge'' Project: Correlations in Complex\nSystems ``COCOS'' MTKD-CT-2004-517186 and the RTN network ENRAGE MRTN-CT-2004-005616. MPH also acknowledges the support from the British Council within the Polish-British Young Scientists Programme and hospitality of Durham University while these notes were being completed.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results.}\\label{intro}\n\nWe consider a {\\it first-come-first-served}\nmulti-server system with $s$ identical servers.\nLet $\\tau$ be a typical interarrival time and\n$\\sigma$ a typical service time.\nIndependent identically distributed sequences\nof interarrival times $\\{\\tau_n\\}$ with mean\n$a={\\mathbb E}\\tau$ and service times $\\{\\sigma_n\\}$\nwith mean $b={\\mathbb E}\\sigma$ are assumed to be mutually independent.\nWe also assume throughout the paper that the distribution\nof $\\sigma$ has unbounded support, i.e.\n$B(x):={\\mathbb P}\\{\\sigma\\le x\\}<1$ for all $x$, and that the\nsystem is {\\it stable}, i.e. $\\rho := b\/a < s$.\n\nThere are two equivalent ways to describe\nthe dynamics of a multi-server system.\nFirst, we may assume that customers form a single\nqueue in front of all servers, and that the first customer\nin the queue moves immediately to a server which\nbecomes idle. Second, we may assume that customers\nform $s$ individual queues (lines) -- one queue for\neach server, service times of customers become\nknown upon their arrival, and each arriving\ncustomer is directed to the line with a minimal\ntotal workload (we also assume that queues are\nnumbered, and if there are more than one minimal\nworkloads, then a customer chooses the one with the\nminimal number). In the rest of the paper,\nwe mostly follow the second description of the model.\n\nFor $n=1$, $2$, \\ldots, let $\\vv V_n=(V_{n1},\\ldots,V_{ns})$\nbe the vector of residual workloads in lines $1$, \\ldots, $s$\nwhich are observed by the $n$-th customer\nupon its arrival into the system.\nThe value of $D_n:=\\min\\{V_{nj},j\\le s\\}$ is the waiting time,\nor the delay which customer $n$ experiences.\nThe $n$-th customer joins the $i_n$-th line. Then\n$$\ni_n=\\min\\{i:V_{ni}=D_n\\}\n$$\nand\n\\begin{eqnarray*}\nV_{n+1,i} &=& \\left\\{\n\\begin{array}{ll}\n(V_{ni}+\\sigma_n-\\tau_{n+1})^+ &\\mbox{ if } i=i_n,\\\\\n(V_{ni}-\\tau_{n+1})^+ &\\mbox{ if } i\\not=i_n.\n\\end{array}\n\\right.\n\\end{eqnarray*}\n\nLet $R(\\vv w)=(R_1(\\vv w),\\ldots,R_s(\\vv w))$\nbe the operator on ${\\mathbb R}^s$ which orders\nthe coordinates of $\\vv w\\in{\\mathbb R}^s$\nin the non-descending order, i.e.,\n$R_1(\\vv w)\\le\\cdots\\le R_s(\\vv w)$.\nFor $n=1$, $2$, \\ldots, put $\\vv W_n=R\\vv V_n$.\nThen $D_n=W_{n1}$ and the vectors $\\{\\vv W_n\\}$\nsatisfy the\nKiefer--Wolfowitz \\cite{KW55} recursion:\n\\begin{equation}\\label{KiW}\n\\vv W_{n+1} = R((W_{n1}+\\sigma_n-\\tau_{n+1})^+,\n(W_{n2}-\\tau_{n+1})^+,\\ldots, (W_{ns}-\\tau_{n+1})^+).\n\\end{equation}\n\nIn a stable system,\nthere exists a unique\nstationary distribution for the Kiefer-Wolfowitz vectors\n$\\vv W_n$,  and the distribution of $\\vv W_n$\nconverges to the stationary distribution\nin the total variation norm, as $n\\to\\infty$.\nIn particular, the same holds for the $D_n$:\nthere exists a unique\ndistribution of the stationary waiting time (delay)\n$D$, and the distribution of $D_n$ converges to that\nof $D$ in the total variation norm.\n\nIn a single server queue ($s=1$),\nthe waiting times $D_n$\nsatisfy the Lindley recursion \\cite{Lindley}:\n$$\nD_{n+1}=(D_n+\\sigma_n-\\tau_{n+1})^+.\n$$\nRecall that, given $D_1=0$,\n$D_{n+1}$ coincides in distribution with\n$\\max(S_k,k\\le n)$ where $S_0=0$ and\n$S_n=\\sum_{k=1}^n(\\sigma_k-\\tau_{k+1})$, for $n\\ge 1$.\nIt is well known\n(see, for example, \\cite{P,Ver,APQ}) that\nthe tail of stationary waiting\ntime $D$ is related to the service time distribution tail\n$\\overline B(x)={\\mathbb P}\\{\\sigma>x\\}$ via the equivalence\n\\begin{eqnarray}\\label{W.single}\n{\\mathbb P}\\{D>x\\} &\\sim& \\frac{\\rho}{1-\\rho}\n\\overline B_r(x) \\quad\\mbox{ as }x\\to\\infty,\n\\end{eqnarray}\nprovided the {\\it subexponentiality} of the\n{\\it residual service time distribution} $B_r$ defined by its tail\n\\begin{eqnarray*}\n\\overline B_r (x) &:=& \\frac{1}{b}\n\\int_x^\\infty \\overline B(y)\\,dy,\\ \\ x>0\n\\end{eqnarray*}\nis guaranteed.\nRecall that a distribution $G$ on ${\\mathbb R}^+$\nis {\\it subexponential},\n $G\\in {\\cal S}$,\nif $\\overline{G*G}(x)\\sim2\\overline G(x)$ as $x\\to\\infty$.\n\nIt is also well-known that, in a single server queue,\nfor any $\\gamma>0$,\n$D$ has a finite $\\gamma$th moment, ${\\mathbb E} D^{\\gamma}<\\infty$\nif and only if\n${\\mathbb E}\\sigma^{\\gamma+1}<\\infty$, see \\cite{KW56}.\nEquivalently, ${\\mathbb E}D^\\gamma<\\infty$ if and only if\n\\begin{eqnarray*}\n{\\mathbb E} \\sigma_{r,1}^\\gamma &<& \\infty\n\\end{eqnarray*}\nwhere random variable $\\sigma_{r,1}$ has distribution $B_r$.\n\nLess is known about the stationary delay $D$ in\nthe multi-server queue. It is well understood\nthat the heaviness of the stationary waiting time tail\ndistribution\ndepends substantially on the load $\\rho$ on the system\n(see, for example, the conjecture on tail equivalence\nby Whitt in \\cite{W};\nexistence results for moments in \\cite{S,SS,SV,SV2011};\nasymptotic results for fluid queues fed by heavy-tailed\non-off flows in \\cite{BMZ,BZ}).\nMore precisely, the tail distribution depends on $\\rho$ via the value\nof its integer part $k = [\\rho]\\in\\{0,1,\\ldots,s-1\\}$\n\nFor a $GI\/GI\/s$ system, a heuristic idea on a probable way for the large deviations\nto occur may be described as follows.\nTake $N=x\\frac{k}{b-ka}$, for a very large $x$.\nLet all service times\n$\\sigma_{n-N-s+k}$, \\ldots, $\\sigma_{n-N-1}$\nbe big enough, say $\\sigma_{n-N-i}>x+Na$,\n$i=1$, \\ldots, $s-k$.\nThen the other $k$ servers form an unstable\n$GI\/GI\/k$ queue system, because the cumulative\ndrift of the corresponding workloads\napproximately equals $b-ka>0$.\nIn time $N$ all workloads of these queues\nwill exceed level $x$ (again approximately).\nIn this way, at time $N$, all $s$ workloads\nbecome greater than $x$ with probability\nwhich is asymptotically not less than\n$\\overline B^{s-k}(x+Na)\\approx\n\\overline B^{s-k}\\bigl(x\\frac{b}{b-ka}\\bigr)\n=\\overline B^{s-k}\\bigl(x\\frac{\\rho}{\\rho-k}\\bigr)$.\nWe use these heuristic arguments below in Section\n\\ref{lower.bound} to derive a lower bound.\nWe follow more precise calculations\nto obtain a better lower bound\nof order $\\overline B_r^{s-k}\\bigl(x\\frac{\\rho}{\\rho-k}\\bigr)$.\n\nWe recall now a few basic properties of heavy-tailed distributions and relations\nbetween them. A distribution function $F$ is\n\\begin{itemize}\n\\item\n{\\it long-tailed}, $F\\in {\\cal L}$, if\n$\\overline{F}(x+1) \\sim \\overline{F}(x)$, as $x\\to\\infty$;\n\\item\n{\\it dominated varying}, $F\\in {\\cal D}$,\nif $\\overline{F}(2x)\\ge c\\overline{F}(x)$, for some $c>0$ and for all\n$x$;\n\\item\n{\\it intermediate regularly varying}, $F\\in {\\cal IRV}$,\nif\n$$\n\\lim_{\\varepsilon \\downarrow 0} \\liminf_{x\\to\\infty}\n\\overline{F}(x(1+\\varepsilon ))\/\\overline{F}(x) =1;\n$$\n\\item\n{\\it regularly varying},\n$F\\in {\\cal RV}$, if $\\overline{F}(x) = l(x)x^{-\\alpha}$\nfor $x>0$ where $\\alpha \\ge 0$ is the {\\it index} of regular variation\nand $l(x)$ is a {\\it slowly varying at infinity}\nfunction, i.e. $l(cx)\\sim l(x)$ as $x\\to\\infty$.\n\\end{itemize}\nThe following relations are known:\n\\begin{equation}\\label{relations}\n{\\cal RV} \\subset {\\cal IRV} \\subset {\\cal L} \\cap {\\cal D} \\subset {\\cal S},\n\\end{equation}\nsee e.g. \\cite{FKZ}, pp. 33 and 54.\n\nIn \\cite{FK}, we treated the case $s=2$ in detail\nand found the {\\it exact asymptotics} for ${\\mathbb P}\\{D>x\\}$.\nWe also described the {\\it most probable way\nfor the occurrence of the large deviations}.\nThat means that, for the stationary waiting time to be large,\ntwo large service times have to be large if $\\rho <1$ and\n$B_r$ is a subexponential distribution\n(see\n\\cite[Theorem 1]{FK}) and one service time has to be\nlarge if $1<\\rho <2$ and if $B$ is\n{\\it long-tailed} and $B_r$ is\n{\\it intermediate\nregularly varying}\n(see \\cite[Theorem 2]{FK}).\nWe also obtained a number of simple bounds. First,\nTheorem 1 in \\cite{FK} yields the following\n\n\\begin{theorem}\\label{co.2.max}\nLet $s=2$, $\\rho<1$, and let the residual time\ndistribution $B_r$ be subexponential.\nThen the tail of the stationary waiting time\nsatisfies the asymptotic relation, as $x\\to\\infty$,\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\sim&\n\\frac{\\rho^2}{2-\\rho}\\Bigl[(\\overline B_r (x))^2\n+\\int_0^\\infty \\overline B_r(x+ya)\\overline B(x+y(a-b))dy\\Bigr].\n\\end{eqnarray*}\nAs a corollary, one can obtain the following\nbounds for the stationary waiting time,\nas $x\\to\\infty$:\n\\begin{eqnarray*}\n\\Bigl(\\frac{\\rho^2(2+\\rho)}{2(2-\\rho)}+o(1)\\Bigr)\n\\overline B_r^2 (x)\n\\le {\\mathbb P}\\{D>x\\}\n\\le \\Bigl(\\frac{\\rho^2}{2(1-\\rho)}+o(1)\\Bigr)\n\\overline B_r^2 (x).\n\\end{eqnarray*}\nAnother corollary is: if, in addition, the distribution\n$B$ is regularly varying with index $\\gamma>1$,\nthen, as $x\\to\\infty$:\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\sim& c(\\overline B_r(x))^2,\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\nc &=& \\frac{\\rho^2}{2-\\rho}\\Bigl[1+\\frac{\\rho}{\\gamma-1}\n\\int_0^\\infty\\frac{dz}{(1+z)^{\\gamma-1}(1+z(1-\\rho))^\\gamma}\\Bigr].\n\\end{eqnarray*}\n\\end{theorem}\n\nFor the case $\\rho>1$, we also proved in \\cite{FK}\n\n\\begin{theorem}\\label{th.2.min.lower}\nLet $s=2$, $1<\\rho<2$, and let both $B$ and $B_r$ be\nsubexponential distributions.\nThen the tail of the stationary waiting time\nsatisfies the following inequalities:\n\\begin{eqnarray*}\n\\limsup_{x\\to\\infty}\\frac{{\\mathbb P}\\{D>x\\}}\n{\\overline{B}_r(2x)}\n&\\le& \\frac{\\rho}{2-\\rho},\n\\end{eqnarray*}\nand, for any fixed $\\delta>0$,\n\\begin{eqnarray*}\n\\liminf_{x\\to\\infty}\\frac{{\\mathbb P}\\{D>x\\}}\n{\\overline B_r\\bigl(\\frac{\\rho+\\delta}{\\rho-1}x\\bigr)}\n&\\ge& \\frac{\\rho}{2-\\rho}.\n\\end{eqnarray*}\nIf, in particular, $B$ is subexponential and $B_r$\nis\nintermediate\nregularly varying,\nthen\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\sim& \\frac{\\rho}{2-\\rho}\n\\overline B_r\\Bigl(\\frac{\\rho}{\\rho-1}x\\Bigr)\n\\quad\\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\n\\end{theorem}\n\nFor an arbitrary $s\\ge 2$ number of servers,\nthe best result on the existence of moments\nwas obtained in \\cite[Theorem 4.1]{SV}\n(here ${\\cal L}_1^\\gamma$ is a specific class of distributions\nintroduced in \\cite{SV}):\n\n\\begin{theorem}\\label{mom.SV}\nLet $k<\\rho<k+1$ for some $k\\in\\{0,1,\\ldots,s-1\\}$.\nThen:\n\n(i) If ${\\mathbb E}\\sigma^\\gamma<\\infty$ then\n${\\mathbb E}D^{(s-k)(\\gamma-1)}<\\infty$.\n\n(ii) If in addition $\\sigma$ is in the class ${\\cal L}_1^\\gamma$,\nthen ${\\mathbb E}D^{(s-k)(\\gamma-1)}<\\infty$\nimplies ${\\mathbb E}S^\\gamma<\\infty$.\n\\end{theorem}\n\nIn the present paper we introduce a condition which\nis both necessary and sufficient for the finiteness\nof ${\\mathbb E}D^\\gamma$. We present this condition in\n``probabilistic terms''.\n\n\\begin{theorem}\\label{exist.mom}\nLet $\\sigma_{r,1}$,\n$\\sigma_{r,2}$, \\ldots\\ be independent random variables\nwith common distribution $B_r$.\nLet $k<\\rho<k+1$ for some $k\\in\\{0,1,\\ldots,s-1\\}$.\nFor any $\\gamma>0$, ${\\mathbb E}D^\\gamma$\nis finite if and only if\n\\begin{equation}\\label{min.fin}\n{\\mathbb E}(\\min(\\sigma_{r,1},\\ldots,\\sigma_{r,s-k}))^\\gamma\n<\\infty,\n\\end{equation}\n\\end{theorem}\nsee Section \\ref{proof} for the proof.\nActually, this result (which is sharper than Theorem \\ref{mom.SV})\nmay be deduced from the results of \\cite{SV},\nbut was not stated there. The corresponding proof\nin \\cite{SV} involves a comparison with the so-called\nsemi-cyclic service discipline.\nTo the best of our knowledge, the latter approach does not allow\none to obtain upper bounds for the tail distribution\nof $D$.\n\nThe main aim of the present paper is to introduce\na novel approach for constructing upper bounds for\nthe stationary waiting time in multi-server queues\n(see Section \\ref{sec.upper.bound} below).\nThis allows us to derive estimates for the tail probabilities of the distribution of\nthe stationary waiting time if the common distribution of service times is of\nsupexponential type, and, further, to establish the {\\it principle of big jumps}\nin a particular case of intermediate varying distributions.\nAlso, based on the new approach, we will obtain\na direct proof of Theorem\n\\ref{exist.mom} (see Section \\ref{proof}).\n\nThe most explicit bounds are obtained for the case\n$\\rho<1$.\n\n\\begin{theorem}\\label{co.s.max}\nLet $\\rho=b\/a<1$ and let the residual time\ndistribution $B_r$ be subexponential.\nThen the tail distribution of the stationary\nwaiting time admits the following bounds:\n\\begin{eqnarray*}\n\\frac{\\rho^s}{s!} \\le\n\\liminf_{x\\to\\infty}\n\\frac{{\\mathbb P}\\{D>x\\}}{\\overline B_r^s(x)}\n&\\le& \\limsup_{x\\to\\infty}\n\\frac{{\\mathbb P}\\{D>x\\}}{\\overline B_r^s (x)}\n\\le \\Bigl(\\frac{\\rho}{1-\\rho}\\Bigr)^s.\n\\end{eqnarray*}\n\\end{theorem}\n\nWe present here the lower and upper bounds only.\nAs it was described in \\cite{FK},\nthe only case where the tail asymptotics are available\nwith an explicit constant multiplier is the case\nof regularly varying service time distribution.\nThe corresponding calculations\nare rather involved\nand deal with the law of large numbers\nand a summation over a specific $s$-dimensional domain\nwith planar boundaries. These calculations have been\ncarried out\nin \\cite{FK} in the case of\n$s=2$ servers.\n\nThe proof of Theorem \\ref{co.s.max} (see Section\n\\ref{k=0}) is based on a simple argument which\ncannot be applied if $\\rho>1$. For an arbitrary $\\rho$,\nwe have the following result.\n\n\\begin{theorem}\\label{th:subexp.bounds}\nLet $k\\in\\{0,\\ldots,s-1\\}$ and $\\delta>0$.\nIf $\\rho>k$, then\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\}\n&\\ge& \\frac{\\rho^{s-k}+o(1)}{(s-k)!}\n\\overline{B_r}^{s-k}\\Bigl(\\frac{\\rho+\\delta}{\\rho-k}x\\Bigr)\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\nIf $\\rho<k+1$  and if the residual service time distribution\n$B_r$ is subexponential, then\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\}\n&\\le& \\Bigl(\\begin{array}{c}s\\\\k\\end{array}\\Bigr)\n\\Bigl(\\frac{(k+1)\\rho}{k+1-\\rho}+o(1)\\Bigr)^{s-k}\n\\overline B_r^{s-k}(x(1-\\delta))\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\n\\end{theorem}\n\nThe lower bound follows from Theorem \\ref{th.lower}\nin Section \\ref{lower.bound}.\nThe proof of the upper bound\nmay be found in Section \\ref{sec.upper.bound}.\nIt is based on\nresults from\nSection \\ref{majorant},\nwhere we present a novel construction of a\nconsistent majorant for $D_n$.\n\n\nNote that the lower and the upper bounds in\nTheorem \\ref{th:subexp.bounds} are not necessarily\nof the same order. In particular, if distribution\n$B$ is of Weibull type then the ratio of the upper\nand  the lower bounds tends to infinity, as $x$ increases.\nIn this case we do not have any ideas about how\ncorrect\/exact\/sharp bounds would look like.\nBut if, in particular, the residual\n service time distribution belongs to the class ${\\cal D} \\cap\n{\\cal L}$,  \nthen\nthese bounds differ by a multiplicative constant only.\n\nNote that in Theorem \\ref{th:subexp.bounds}\nwe require conditions on the residual distribution $B_r$\nand not on the distribion $B$ itself. This is in line\nwith the key results on subexponentiality like (\\ref{W.single}).\n\n\\begin{corollary}\\label{regularly.case}\nLet the residual service time distribution\n$B_r$ be long-tailed and dominated varying.\nLet $k<\\rho<k+1$ for some $k\\in\\{0,1,\\ldots,s-1\\}$.\nThen there exist constants $c_1$ and $c_2$\nsuch that, for all $x$,\n\\begin{equation}\\label{rvst}\nc_1\\overline B_r^{s-k}(x) \\le\n{\\mathbb P}\\{D>x\\} \\le c_2\\overline B_r^{s-k}(x).\n\\end{equation}\n\\end{corollary}\n\nThe result follows directly from Theorem \\ref{th:subexp.bounds},\nthe last inclusion in (\\ref{relations})\nand the definition of the dominated variation.\n\n\nIn the particular case of distributions of intermediate variation,\nwe will use \\cite[Theorem 7]{BaF} to complement\n Corollary \\ref{regularly.case} by establishing\nthe ``principle of $s-k$ big jumps'':\nthe main cause of the value of $D$ to be big is to have\n$s-k$ big service times, see Section \\ref{sec.upper.bound} for\nthe precise statement.\n\n\n\n\n\\section{Comparison of systems with different\ninter-arrival times.}\\label{comparison}\n\nHere we present results which, in particular,\nallow us to obtain lower and upper bounds for the stationary\ndelay in a\ngeneral $GI\/GI\/s$ system in terms of a simpler $D\/GI\/s$\nsystem with deterministic\ninterarrival times. We use the following partial ordering:\nfor two vectors $\\vv x = (x_1,\\ldots ,x_s)$ and\n$\\vv y = (y_1,\\ldots ,y_s)$, we write\n$\\vv x \\le \\vv y$ if $x_j\\le y_j$\nfor all $j=1$, \\ldots, $s$.\n\nConsider two $GI\/GI\/s$ systems,\nsay $\\widetilde{\\vv V}$ and $\\widehat{\\vv V}$,\nwith service times $\\sigma_n$\nand with interarrival times\n$\\widetilde\\tau_n$ and $\\widehat\\tau_n$ respectively.\nLet $\\widetilde D_n$ and $\\widehat D_n$\nbe the corresponding waiting times in these systems. Let\n$\\xi_n=\\widehat\\tau_{n+1}-\\widetilde\\tau_{n+1}$.\nWe obtain an upper bound for delay $\\widetilde D_n$\nin terms of delay $\\widehat D_n$\nand the sequence $\\xi_n$.\n\n\\begin{lemma}\\label{lem:upper.bound}\nFor all $n\\ge1$,\n$\\widetilde D_n \\le \\widehat D_n+M_{n-1}$,\nwhere $M_0=0$ and $M_n=(M_{n-1}+\\xi_n)^+$.\n\\end{lemma}\n\n\\begin{proof}\nPut $\\vv e_1=(1,0,\\ldots,0)$ and $\\vv 1=(1,\\ldots,1)$.\nIt suffices to prove the inequality\n\\begin{eqnarray}\\label{upper.via.M}\n\\widetilde{\\vv W}_n &\\le& \\widehat{\\vv W}_n+\\vv 1M_{n-1}\n\\quad\\mbox{ a.s.}\n\\end{eqnarray}\nWe proceed by induction. For $n=1$\nwe have $\\vv 0\\le\\vv 0+\\vv 1M_0$. Assume inequality\n(\\ref{upper.via.M}) to hold\nfor some $n$ and prove it for $n+1$.\nWe have\n\\begin{eqnarray*}\n\\widetilde{\\vv W}_{n+1} &=&\nR(\\widetilde{\\vv W}_n+\\vv e_1\\sigma_n\n-\\vv 1\\widetilde\\tau_{n+1})^+\\\\\n&\\le& R(\\widehat{\\vv W}_n+\\vv 1M_{n-1}\n+\\vv e_1\\sigma_n-\\vv 1\\widetilde\\tau_{n+1})^+\\\\\n&=& R(\\widehat{\\vv W}_n+\\vv e_1\\sigma_n\n-\\vv 1\\widehat\\tau_{n+1}\n+\\vv 1(M_{n-1}+\\xi_n))^+.\n\\end{eqnarray*}\nSince $(u+v)^+\\le u^++v^+$,\n\\begin{eqnarray*}\n\\widetilde{\\vv W}_{n+1}\n&\\le& R(\\widehat{\\vv W}_n+\\vv e_1\\sigma_n\n-\\vv 1\\widehat\\tau_{n+1})^++\\vv 1(M_{n-1}+\\xi_n)^+\n\\equiv \\widehat{\\vv W}_{n+1}+\\vv 1M_n,\n\\end{eqnarray*}\nand the proof of (\\ref{upper.via.M}) is complete.\n\\end{proof}\n\nThe following corollary\nwill be used to obtain lower bounds.\nIt is similar to Lemma 2 in \\cite{FK}.\n\n\\begin{corollary}\\label{cor:lower.bound.D}\nLet $\\vv W_n'$ be a stable $s$-server\nqueue system with the same service times $\\sigma_n$\nas in $\\vv W_n$ and with the constant interarrival times $a'$.\nIf $a'>a={\\mathbb E}\\tau$, then, for any $\\varepsilon>0$,\nthere exists $x_0$ such that\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\ge&\n(1-\\varepsilon){\\mathbb P}\\{D'>x+x_0\\}\\ \\mbox{ for all }x.\n\\end{eqnarray*}\nOne can take $x_0$ such that\n$$\n{\\mathbb P}\\Bigl\\{\\sup_{n\\ge 0}\n\\sum_{i=1}^n(\\tau_i-a')\\le x_0\\Bigr\\}\\ge1-\\varepsilon.\n$$\n\\end{corollary}\n\n\\begin{proof}\nTake $\\widetilde\\tau_n=a'$ and $\\widehat\\tau_n=\\tau_n$\nin Lemma \\ref{lem:upper.bound},\nthen $\\xi_n=\\tau_n-a'$.\nA weak limit, $M$, of the sequence $M_n$ exists (since\n${\\mathbb E}\\xi_1=a-a'<0$)\nand has the same distribution as\n\\begin{eqnarray*}\nM &=_{\\rm st}& \\max\\{0,\\ \\xi_1,\\\n\\xi_1+\\xi_2,\\ \\ldots,\\ \\xi_1+\\cdots+\\xi_n,\\ \\ldots\\}.\n\\end{eqnarray*}\nBy Lemma \\ref{lem:upper.bound}, $D_n'\\le D_n+M_{n-1}$.\nHence, $D_n\\ge D'_n-M_{n-1}$\nSince $D'_n$ does not depend on $\\tau$'s,\n$D'_n$ and $M_{n-1}$ are independent. Therefore,\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_n>x\\} &\\ge&\n{\\mathbb P}\\{M_{n-1}\\le x_0\\}{\\mathbb P}\\{D'_n>x+x_0\\}.\n\\end{eqnarray*}\nLetting $n$ go to infinity,\nwe obtain the desired bound.\n\\end{proof}\n\n\n\n\n\n\\section{The case $\\rho<1$, proof of Theorem \\ref{co.s.max}.}\n\\label{k=0}\n\nThe lower bound in Theorem \\ref{co.s.max} follows\nfrom Lemma \\ref{l.lower.k=0} below which also\ngeneralises Theorem \\ref{th.lower}\n(see Section \\ref{lower.bound}) in the case $k=0$.\n\n\\begin{lemma}\\label{l.lower.k=0}\nLet $\\rho>0$. Then, for any function\n$h(x)\\to\\infty$ as $x\\to\\infty$,\n$$\n{\\mathbb P}\\{D>x\\} \\ge \\frac{\\rho^s+o(1)}{s!}\n\\overline{B_r}^s(x+h(x)).\n$$\nIn particular, if the residual time\ndistribution $B_r$ is long-tailed (that is,\n$\\overline B_r(x+1) \\sim \\overline B_r(x)$\nas $x\\to\\infty$), then\n$$\n{\\mathbb P}\\{D>x\\} \\ge \\frac{\\rho^s+o(1)}{s!}\n\\overline{B_r}^s(x)\\ \\mbox{ as }x\\to\\infty.\n$$\n\\end{lemma}\n\nWe start with an auxiliary result.\n\n\\begin{lemma}\\label{q.k}\nLet $\\{q_i\\}_{i\\ge 1}$ be a non-increasing\nsequence of positive numbers. Then, for any $s\\ge 1$,\n\\begin{eqnarray*}\n\\sum_{1\\le i_1<\\ldots<i_s}\nq_{i_1}\\cdot\\ldots\\cdot q_{i_s}\n&\\ge& \\frac{1}{s!}(q_s+q_{s+1}+\\ldots)^s.\n\\end{eqnarray*}\n\\end{lemma}\n\n\\begin{proof}\nIf $1\\le i_1<\\ldots<i_s$ then\n$s\\le i_1+(s-1)\\le i_2+(s-2)\\le\\ldots\\le i_{s-1}+1\\le i_s$ and\n$$\nq_{i_1}\\cdot q_{i_2}\\cdot\\ldots\\cdot q_{i_s}\n\\ge q_{i_1+s-1}\\cdot q_{i_2+s-2}\\ldots\\cdot q_{i_s},\n$$\nbecause $\\{q_i\\}$ is a non-increasing sequence.\nThus,\n\\begin{eqnarray*}\n\\sum_{1\\le i_1<\\ldots<i_s}\nq_{i_1}\\cdot\\ldots\\cdot q_{i_s}\n&\\ge& \\sum_{s\\le i_1\\le \\ldots\\le i_s}\nq_{i_1}\\cdot\\ldots\\cdot q_{i_s}\\\\\n&\\ge& \\frac{1}{s!}\\sum_{i_1,\\ldots,i_s\\ge s}\nq_{i_1}\\cdot\\ldots\\cdot q_{i_s},\n\\end{eqnarray*}\nwhich yields the conclusion of the lemma.\n\\end{proof}\n\n\\begin{proofof}{Lemma \\ref{l.lower.k=0}}\nOur estimation is based on calculations involving $s$\nbig jumps.\nThis technique was already used in \\cite{FK} in\nthe case $s=2$, where a lower bound\n(which is better than the one presented in\nLemma \\ref{l.lower.k=0})\nwas obtained under the extra condition that $B_r$ is long-tailed.\nThe bound in \\cite{FK} is exact in the sense that\nit provides the right asymptotics under further\nassumptions.\n\nFollowing Corollary \\ref{cor:lower.bound.D},\ndefine the auxiliary $s$-server system $\\vv W'_n$\nhaving the same service times $\\sigma_n$ and\nconstant interarrival times $a'$, $a'>a$.\nFor $\\vv i=(i_1,\\ldots,i_s)$,\n$1\\le i_1<\\ldots<i_s<n$, define  events\n$A_n(\\vv i)$ and $C_n(\\vv i)$ as\n\\begin{eqnarray*}\nA_n(\\vv i) &=& \\{\\sigma_{i_1}>x+(n-i_1)a', \\ldots,\n\\sigma_{i_s}>x+(n-i_s)a'\\}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nC_n(\\vv i)\n&=& \\bigcap_{i\\le n,i\\ne i_1,\\ldots,i_s}\n\\{\\sigma_i\\le x+(n-i)a'\\}.\n\\end{eqnarray*}\nSince the mean ${\\mathbb E}\\sigma$ exists, we have that,\nuniformly in $n$ and $\\vv i$,\n\\begin{eqnarray*}\n{\\mathbb P}\\{C_n(\\vv i)\\}\n=1-{\\mathbb P}\\{\\overline{C_n(\\vv i)}\\}\n&\\ge& 1-\\sum_{i=0}^\\infty {\\mathbb P}\\{\\sigma_1>x+ia'\\}\n\\to 1 \\quad\\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\nFor each vector $\\vv i$, events $A_{n}(\\vv i)$ and\n$C_{n}(\\vv i)$ are independent. Further,\nevents $A_n(\\vv i)\\cap C_n(\\vv i)$\nare disjoint for distinct vectors $\\vv i$.\nThese observations together yield\n\\begin{eqnarray}\\label{union.of.An}\n{\\mathbb P}\\Bigl\\{\\bigcup_{\\vv i}A_n(\\vv i)\n\\cap C_n(\\vv i)\\Bigr\\}\n&=& \\sum_{\\vv i}{\\mathbb P}\\{A_n(\\vv i)\\}{\\mathbb P}\\{C_n(\\vv i)\\}\n\\ge (1-o(1))\\sum_{\\vv i}{\\mathbb P}\\{A_n(\\vv i)\\}\n\\end{eqnarray}\nas $x\\to\\infty$, uniformly in $n$.\nThe event $A_n(\\vv i)$ implies that $D'_n>x$. Therefore,\n\\begin{eqnarray*}\n{\\mathbb P}\\{D'_n>x\\} &\\ge& (1-o(1))\n\\sum_{\\vv i}{\\mathbb P}\\{A_n(\\vv i)\\}\n\\end{eqnarray*}\nas $x\\to\\infty$, uniformly in $n$. We now prove that\n\\begin{eqnarray}\\label{Sigma.k=0}\n\\lim_{n\\to\\infty}\n\\sum_{\\vv i} {\\mathbb P}\\{A_n(\\vv i)\\}\n&\\ge& \\frac{(b\/a')^s}{s!}\\overline{B_r}^s(x+sa').\n\\end{eqnarray}\nIndeed, by the independence of the $\\sigma$'s,\n\\begin{eqnarray*}\n\\sum_{1\\le i_1<\\ldots<i_s<n}\n{\\mathbb P}\\{A_n(\\vv i)\\}\n&=& \\sum_{1\\le i_1<\\ldots,i_s<n}\n\\overline B(x+(n-i_1)a')\n\\cdot\\ldots\\cdot\\overline B(x+(n-i_s)a'),\n\\end{eqnarray*}\nand the left side of (\\ref{Sigma.k=0}) equals\n\\begin{eqnarray*}\n\\sum_{1\\le i_1<\\ldots<i_s}\n\\overline B(x+i_1a')\\cdot\\ldots\\cdot\\overline B(x+i_sa').\n\\end{eqnarray*}\nBy Lemma \\ref{q.k} with $q_i=\\overline B(x+ia')$,\nthe latter sum is not smaller than\n\\begin{eqnarray*}\n\\frac{1}{s!} \\Bigl(\\sum_{j=s}^\\infty\n\\overline B(x+ja')\\Bigr)^s.\n\\end{eqnarray*}\nSince the tail probability is a non-increasing function,\n\\begin{eqnarray*}\n\\sum_{j=s}^\\infty\\overline B(x+ja')\n\\ge \\frac{1}{a'} \\int_{sa'}^\\infty\\overline B(x+z)dz\n=\\rho\\overline B_r(x+sa').\n\\end{eqnarray*}\nCombining altogether, we conclude (\\ref{Sigma.k=0}).\nThen by Corollary \\ref{cor:lower.bound.D},\nfor every $\\varepsilon>0$ there exists $x_0$ such that\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\ge&\n(1-\\varepsilon){\\mathbb P}\\{D'>x+x_0\\}\\\\\n&\\ge& (1-\\varepsilon-o(1))\\frac{(b\/a')^s}{s!}\n\\overline{B_r}^s(x+sa'+x_0).\n\\end{eqnarray*}\nBy the arbitrary choice of $a'>a$ and $\\varepsilon>0$,\nthe proof of Lemma \\ref{l.lower.k=0} is complete.\n\\end{proofof}\n\n\\begin{proofof}{the upper bound in Theorem \\ref{co.s.max}}\nWe start with the case of deterministic $\\tau$,\ni.e., $\\tau_n\\equiv a$. We follow the lines from \\cite{FK}\nwhere, for $\\rho<1$ (that is for $b<a$),\nthe following simple majorant was introduced.\n\nLet $\\sigma_{ni}$, $n\\ge1$, $i\\le s$, be\nindependent random variables with common distribution $B$.\nConsider $s$ auxiliary $D\/GI\/1$\nqueueing systems which work in parallel:\nat every time instant $T_n = na$, $n=1$, 2, \\ldots,\na batch of $s$ customers arrives,\none customer per each queue.\nService times in queue $i$ are equal to $\\sigma_{ni}$.\nDenote by $U_{ni}$, $i=1$, \\ldots, $s$, the waiting\ntimes in the $i$th queue, $U_{n+1,i}= (U_{ni}+\\sigma_{ni}-a)^+$,\nand let $U_{1i}=0$.\nSince the arrival process is deterministic and service\ntimes are independent, vector $(U_{n1},\\ldots , U_{ns})$\nhas independent identically distributed coordinates,\nand, as $n\\to\\infty$, its weak limit\n$(U_{1},\\ldots,U_{s})$ exists (since ${\\mathbb E}\\sigma<a$)\nand contains independent identically distributed coordinates too.\nHere $U_i$ is the stationary waiting time in\nthe $i$th auxiliary queue.\n\nNow we introduce a coupling of $s$ single-server systems\nand of the $s$-server system $D\/GI\/s$. Namely, we\ndetermine the service times $\\sigma_n$ in the original\n$D\/GI\/s$ system by induction.\nStart with $\\sigma_1=\\sigma_{1,1}$.\nAssume that $\\sigma_1$, \\ldots, $\\sigma_{n-1}$\nhave been already defined. Then the delay vectors\n$\\vv V_1$, \\ldots, $\\vv V_n$ are defined too, and\nwe know the number $i_n=\\min\\{i:V_{ni}=D_n\\}$.\nThen let $\\sigma_n=\\sigma_{n,i_n}$.\n\nBy monotonicity, $D_n\\le \\min\\{U_{ni},\\ i\\le s\\}$\nwith probability 1. Hence,\n\\begin{eqnarray}\\label{D.min}\nD &\\le& \\min\\{U_i,\\ i\\le s\\}.\n\\end{eqnarray}\nDue to independence,\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\le& {\\mathbb P}^s\\{U_1>x\\},\n\\end{eqnarray*}\nand we can apply known results for the single\nserver queue:\nfrom (\\ref{W.single}),\n\\begin{eqnarray*}\n{\\mathbb P}\\{U_1>x\\} &\\sim&\n\\frac{\\rho}{1-\\rho}\\overline B_r(x),\n\\end{eqnarray*}\nwhich gives us the upper bound\nin Theorem \\ref{co.s.max} if interarrival times are deterministic.\n Now the proof in the general case follows\nfrom \\cite[Lemma 1]{FK}.\n\\end{proofof}\n\n\n\n\n\n\\section{Auxiliary results.}\\label{lln}\n\nIn this Section we collect a number\nof auxiliary facts related to monotonicity and\nto\nthe strong law of large numbers for\nunstable multi-server systems.\nThe results seem not to be new, so we\nprovide only short sketches of proofs for self-containedness.\n\nLet $\\vv W_n$ be a sequence satisfying\nthe Kiefer-Wolfowitz recursion (\\ref{KiW}),\nwith initial value $\\vv W_1 \\ge 0$.\n\n\\begin{lemma}\\label{monotonicity}\n(1)\nFor any $n$, $\\vv W_n$ is a non-decreasing function\nof the initial value and of service times and a non-increasing function of interarrival\ntimes. This means that if $\\widetilde{\\vv W}_n$\nis a sequence satisfying another Kiefer-Wolfowitz recursion  with initial\nvalue $\\widetilde{\\vv W}_1$ and with interarrival times $\\{\\widetilde{\\tau}_n\\}$ and service\ntimes $\\{\\widetilde{\\sigma}_n\\}$ and if\n$\\vv W_1 \\le \\widetilde{\\vv W}_1$ (coordinate-wise), $\\sigma_j\\le \\widetilde{\\sigma}_j$,\nand $\\tau_j \\ge \\widetilde{\\tau}_j$, for $j=1,\\ldots ,n-1$,\nthen\n$\\vv W_n \\le \\widetilde{\\vv W}_n$.\\\\\n(2)\nFor any $n\\ge 2$, the difference $\\sum_{i=1}^s (W_{ni}\n-W_{n-1,i})$ is a non-increasing function of the initial\nvalue $\\vv W_1$: if ${\\vv W}_1 \\le \\widetilde{\\vv W}_1$,\nthen $\\sum_{i=1}^s (W_{ni}\n-W_{n-1,i}) \\ge \\sum_{i=1}^s (\\widetilde{W}_{ni}\n-\\widetilde{W}_{n-1,i}).$\n\\end{lemma}\nThe first monotonicity property holds because both operators $R$ and\n$\\max (0,\\cdot )$ are monotone. The second property follows\nsince function $(x+y)^+-x$ is non-increasing in $x$, for any\nfixed $y$.\n\n\\begin{lemma}\\label{slln.for.unstable}\nLet $b>sa$, so the $s$-server system with workload vectors ${\\vv W}_n$ is unstable.\nThen,\n\\begin{equation}\\label{SLLN1}\n\\frac{W_{n1}}{n}\\to \\frac{b-sa}{s}\n\\quad \\mbox{and}\\quad\n\\frac{W_{ns}}{n}\\to \\frac{b-sa}{s}\n\\quad\\mbox{ as }n\\to\\infty,\n\\end{equation}\nboth with probability $1$ and in mean.\n\\end{lemma}\n\n\\begin{proof}\nNote that, for any $n=1,2,\\ldots$,\n\\begin{equation}\\label{ind1}\nW_{n+1,s}-W_{n+1,1} \\le \\max (W_{1,s}-W_{1,1}, \\sigma_1,\\ldots , \\sigma_n).\n\\end{equation}\nIndeed, if $W_{ns}-W_{n1}>\\sigma_n$, then $W_{n+1,s}-W_{n+1,1}\\le W_{ns}-W_{n1}$,\nand if $W_{ns}-W_{n1}\\le \\sigma_n$, then $W_{n+1,s}-W_{n+1,1}\\le \\sigma_n$, so the\ninduction argument completes the proof of (\\ref{ind1}). Next,\n\\begin{equation}\\label{ind2}\n\\max (W_{1s}-W_{11}, \\sigma_1,\\ldots , \\sigma_n )\/n \\to 0  \\quad \\mbox{a.s.}\n\\end{equation}\nbecause $(W_{1s}-W_{11})\/n\\to0$ and, since ${\\vv E}\\sigma$ is\nfinite, events $\\{ \\sigma_k\/k > \\varepsilon \\}$ occur only finitely often,\nfor any $\\varepsilon >0$.\n\nFurther,\n$$\n\\frac{1}{n}\\sum_{i=1}^s W_{ni} \\ge \\frac{1}{n}\\sum_{j=1}^{n-1} (\\sigma_j-s\\tau_{j+1})\n\\to b-sa >0 \\quad \\mbox{a.s.,}\n$$\nso $\\liminf_{n\\to\\infty}W_{ns}\/n \\ge (b-sa)\/s$, and, from (\\ref{ind1})-(\\ref{ind2}),\nthere exists an a.s. finite random variable $\\nu$ such that $W_{n1}>0$, for all $n\\ge\\nu$.\nSo, for $n\\ge\\nu$,\n\\begin{equation}\\label{ind3}\n\\frac{1}{n}\\sum_{i=1}^s W_{ni}\n=\\frac{1}{n}\\sum_{i=1}^s W_{\\nu i} + \\frac{1}{n} \\sum_{j=\\nu}^n (\\sigma_j-s\\tau_{j+1})\n\\to b-sa \\quad \\mbox{a.s.,}\n\\end{equation}\nand (\\ref{ind1})-(\\ref{ind3}) lead to convergence a.s. in (\\ref{SLLN1}).\nFinally, since  $0\\le W_{ns}\/n \\le W_{1s}\/n + \\sum_{j=1}^{n-1} \\sigma_j\/n$\nand since random variables $\\sum_{j=1}^{n-1} \\sigma_j\/n$ are uniformly integrable,\nconvergence in mean also follows.\n\\end{proof}\n\n\\begin{lemma}\\label{slln.for.s-1}\nAssume $b>(s-1)a$. For any $\\varepsilon>0$,\nthere exist $A<\\infty$ and an integer $d\\ge 1$ such that,\nfor any initial value ${\\vv W}_1$ with $W_{1s}\\ge A$,\n$$\n{\\mathbb E}\\{W_{1+d,1}+\\ldots+W_{1+d,s}\n-W_{11}-\\ldots-W_{1s} \\} \n\\le d(b-sa+\\varepsilon).\n$$\n\\end{lemma}\n\n\\begin{proof}\nBy property (2) of Lemma \\ref{monotonicity}, it is enough to prove the\nresult for initial value $W_{11}=\\ldots =W_{s-1,1}=0$, $W_{s1}=A$ only.\n\nChoose $C$ such that ${\\mathbb E} \\min (\\tau, C) \\ge a-\\varepsilon \/2$.\nBy property (1) of Lemma \\ref{monotonicity}, we may prove the lemma\nwith interarrival times\n$\\min (\\tau_j ,C)$  in place of $\\tau_j$.\n\nConsider an auxiliary unstable $GI\/GI\/(s-1)$ queue $\\widehat{\\vv W}_n$\nwith initial zero value\nand, by applying the previous lemma, find $d$ such that\n${\\mathbb E} \\sum_{i=1}^{s-1}\\widehat{W}_{1+d,i} \\le d(b-(s-1)a+\\varepsilon\/2)$.\nThen return to the $s$-server queue and take $A=(d+1)C$.\nWe will prove that\n\\begin{equation}\\label{L43}\n\\sum_{i=1}^{s}W_{1+d,i} = \\sum_{i=1}^{s-1}\\widehat{W}_{1+d,i} +\nA - \\sum_{j=1}^{d} \\min (\\tau_{j},C) \\quad \\mbox{a.s.,}\n\\end{equation}\nthen the result will follow.\n\nConsider vectors ${\\vv V}_n$ and numbers $i_n$ as in the Introduction, with initial\nvalues $V_{1,1}=\\ldots =V_{1,s-1}=0$ and $V_{1,s}=A$. Note that $V_{n,s}\\ge A-(n-1)C>0$,\nfor all $n=1,2,\\ldots,d+1$.\n\nLet $\\mu = \\min ( d+1, \\min \\{n\\ge 1 \\ : \\ i_n=s \\} ).$ Then\n$R(V_{\\mu,1},\\ldots,V_{\\mu,s-1})=(\\widehat{W}_{\\mu,1}\\ldots,\\widehat{W}_{\\mu,s-1})$\nand\n\\begin{equation}\\label{LL43}\n\\sum_{i=1}^s W_{\\mu,i} = \\sum_{i=1}^s V_{\\mu,i}=\n\\sum_{i=1}^{s-1} V_{\\mu,i}+V_{\\mu,s} = \\sum_{i=1}^{s-1}\\widehat{W}_{\\mu,i}+A-\\sum_{j=1}^{\\mu -1}\n\\min (\\tau_j,C).\n\\end{equation}\nThis ends the proof of (\\ref{L43}) if $\\mu = d+1$.\n\nIn the case $\\mu <d+1$, we may conclude that\n$$\n0<A-(\\mu -1)C \\le V_{\\mu,s} = W_{\\mu,1}\n$$\nand, therefore, $W_{n,i}>0$ and $\\widehat{W}_{n,i}>0$, for all $\\mu \\le n \\le d+1$ and $i=1,\\ldots,s$.\nThen, from (\\ref{LL43}),\n\\begin{eqnarray*}\n\\sum_{i=1}^s W_{d+1,i}\n&=&\n\\sum_{i=1}^s W_{\\mu,i}+\\sum_{j=\\mu}^d (\\sigma_j -s\\min (\\tau_j,C))\\\\\n&=&\n\\sum_{i=1}^{s-1}\\widehat{W}_{\\mu,i} +\\sum_{j=\\mu}^d (\\sigma_j- (s-1)\\min (\\tau_j,C))\n+ A-\\sum_{j=1}^d \\min (\\tau_j,C)\n\\end{eqnarray*}\nwhich coincides again with the right side of (\\ref{L43}).\n\n\\end{proof}\n\n\n\n\n\n\\section{Lower Bound.}\\label{lower.bound}\n\nThe following result holds without any restrictions\non the service time\ndistribution $B$\n(a similar result was formulated\nand proved in \\cite[Theorem 3.1]{SV}).\n\n\\begin{theorem}\\label{th.lower}\nLet $k\\in\\{0,1,\\ldots,s-1\\}$ be such that $\\rho>k$.\nThen, for any fixed $\\delta>0$,\n$$\n{\\mathbb P}\\{D>x\\} \\ge \\frac{\\rho^{s-k}+o(1)}{(s-k)!}\n\\overline{B_r}^{s-k}\\Bigl(\\frac{\\rho+\\delta}{\\rho-k}x\\Bigr)\n\\quad\\mbox{as }x\\to\\infty.\n$$\n\\end{theorem}\n\n\\begin{proof}\nWe exploit the technique of $s-k$ big jumps.\nFollowing Corollary \\ref{cor:lower.bound.D},\nwe consider only deterministic\ninterarrival times, $\\tau\\equiv a$.\n\nThe case $k=0$ was considered in Lemma \\ref{l.lower.k=0}.\nSo  now let  $k\\ge 1$.\nLet $\\widetilde{\\vv W}_n=(\\widetilde W_{n1},\\ldots,\\widetilde W_{nk})$\nbe the residual workload vector in the $GI\/GI\/k$ system\nwith the same interarrival and service times as in the original\nsystem,\nand with $k$ servers.\nSince $\\rho>k$, the $k$-server system is unstable.\nHence, by Lemma \\ref{slln.for.unstable},\nboth the minimal coordinate $\\widetilde W_{n1}$\nand the maximal coordinate $\\widetilde W_{nk}$\ndrift to infinity as $n\\to\\infty$ with probability 1,\nwith the same rate\n$(b-ka)\/k$. Then\n\\begin{eqnarray*}\n{\\mathbb P}\\Bigl\\{\\widetilde W_{N1}>N\\Bigl(\\frac{b-ka}{k}-\\delta\\Bigr),\n\\ \\widetilde W_{ik}\\le N\\Bigl(\\frac{b-ka}{k}+\\delta\\Bigr)\n\\mbox{ for all }i\\le N\\Bigr\\} &\\to& 1\n\\ \\mbox{ as }N\\to\\infty.\n\\end{eqnarray*}\n\nIf we assume that there are initially big workloads at $s-k$ servers\nwhile the $k$ other queues are empty,\nthen, with high probability, the $k$ smallest workloads\nevolve like the $k$-server system with workloads $\\widetilde{\\vv  W}_n$,\nfor a long while. This observation implies that\n\\begin{eqnarray*}\n\\lefteqn{{\\mathbb P}\\Bigl\\{\nW_{N1}>N\\Bigl(\\frac{b-ka}{k}-\\delta\\Bigr),\n\\ W_{ik}\\le N\\Bigl(\\frac{b-ka}{k}+\\delta\\Bigr),\nW_{i,k+1}>N\\Bigl(\\frac{b-ka}{k}+\\delta\\Bigr)\n\\mbox{ for all }i\\le N\\Big|}\\\\\n&&\\hspace{40mm}\nW_{1k}=0, W_{1,k+1}>\nN\\Bigl(\\frac{b-ka}{k}+\\delta\\Bigr)+Na\\Bigr\\}\n\\to 1\\ \\mbox{ as }N\\to\\infty.\n\\end{eqnarray*}\nTake $c$ such that\n\\begin{eqnarray}\\label{def.c}\n\\frac{b-ka}{k}+\\delta\n&\\le& (1+c\\delta)\n\\Bigl(\\frac{b-ka}{k}-\\delta\\Bigr)\n\\end{eqnarray}\nfor all sufficiently small $\\delta>0$, and let\n\\begin{eqnarray}\\label{def.N}\nx=N\\Bigl(\\frac{b-ka}{k}-\\delta\\Bigr).\n\\end{eqnarray}\nThen\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_N>x\\mid W_{1k}=0,\nW_{1,k+1}>x(1+c\\delta)+Na\\}\n&\\to& 1\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\nBy the monotonicity of the $s$-server\nqueueing system in its initial state (see Lemma \\ref{monotonicity}), we obtain\n\\begin{eqnarray}\\label{slln.double}\n{\\mathbb P}\\{D_N>x\\mid W_{1,k+1}>x(1+c\\delta)+Na\\}\n&\\to& 1\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray}\n\nFor $\\vv i=(i_1,\\ldots,i_{s-k})$,\n$1\\le i_1<\\ldots<i_{s-k}\\le n$, define the events\n$A_n(\\vv i)$\nas\n\\begin{eqnarray*}\nA_n(\\vv i) &=& \\{\\sigma_{i_1}>y+(n-i_1)a, \\ldots,\n\\sigma_{i_{s-k}}>y+(n-i_{s-k})a\\}.\n\\end{eqnarray*}\nAgain like in (\\ref{union.of.An}) we have\n\\begin{eqnarray}\\label{union.An.N}\n{\\mathbb P}\\Bigl\\{\\bigcup_{\\vv i:i_{s-k}<n-N}\nA_n(\\vv i)\\Bigr\\}\n&\\ge& (1-o(1))\\sum_{\\vv i:i_{s-k}<n-N}\n{\\mathbb P}\\{A_n(\\vv i)\\}\n\\end{eqnarray}\nas $y\\to\\infty$, uniformly in $n$ and $N$.\nWe prove now that\n\\begin{eqnarray}\\label{Sigma}\n\\lim_{n\\to\\infty}\n\\sum_{\\vv i:i_{s-k}<n-N} {\\mathbb P}\\{A_n(\\vv i)\\}\n&\\ge& \\frac{\\rho^{s-k}}{(s-k)!}\n\\overline{B_r}^{s-k}(y+(N+s-k)a).\n\\end{eqnarray}\nIndeed, by the independence of the $\\sigma$'s,\n\\begin{eqnarray*}\n\\sum_{\\vv i:i_{s-k}<n-N} {\\mathbb P}\\{A_n(\\vv i)\\}\n&=& \\sum_{\\vv i:i_{s-k}<n-N} \\overline B(y+(n-i_1)a)\n\\cdot\\ldots\\cdot\\overline B(y+(n-i_{s-k})a)\\\\\n&=& \\sum_{N<i_1<\\ldots<i_{s-k}\\le n-1}\n\\overline B(y+i_1a)\\cdot\\ldots\\cdot\\overline B(y+i_{s-k}a).\n\\end{eqnarray*}\nHence, the left side of (\\ref{Sigma}) equals\n\\begin{eqnarray*}\n\\sum_{N<i_1<\\ldots<i_{s-k}}\n\\overline B(y+i_1a)\\cdot\\ldots\\cdot\\overline B(y+i_{s-k}a)\n&\\ge& \\sum_{1\\le i_1<\\ldots<i_{s-k}}\n\\overline B(y+Na+i_1a)\\cdot\\ldots\\cdot\n\\overline B(y+Na+i_{s-k}a).\n\\end{eqnarray*}\nBy Lemma \\ref{q.k} with\n$q_i=\\overline B(y+Na+ia)$, the sum on the right\nis not less than\n\\begin{eqnarray*}\n\\frac{1}{(s-k)!} \\Bigl(\\sum_{j=s-k}^\\infty\n\\overline B(y+Na+ja)\\Bigr)^{s-k}.\n\\end{eqnarray*}\nSince the tail probability is non-increasing,\n\\begin{eqnarray*}\n\\sum_{j=s-k}^\\infty\\overline B(y+Na+ja)\n\\ge \\frac{1}{a} \\int_{(s-k)a}^\\infty\\overline B(y+Na+z)dz.\n\\end{eqnarray*}\nCombining these expressions, we obtain the desired estimate\n(\\ref{Sigma}). Substituting (\\ref{Sigma}) into\n(\\ref{union.An.N}) we get, as $y\\to\\infty$,\n\\begin{eqnarray}\\label{Sigma.2}\n\\lim_{n\\to\\infty}\n{\\mathbb P}\\Bigl\\{\\bigcup_{\\vv i:i_{s-k}<n-N}\nA_n(\\vv i)\\Bigr\\}\n&\\ge& \\frac{\\rho^{s-k}-o(1)}{(s-k)!}\n\\overline{B_r}^{s-k}(y+(N+s-k)a).\n\\end{eqnarray}\n\nLet $y=x(1+c\\delta)$ in the definition of $A_n(\\vv i)$.\nSince an increment per unit of time of every\ncoordinate of the workload vector $\\vv W_i$ is not\nless than $-a$, on the event $A_n(\\vv i)$\nwe have $W_{j,k+1}>x(1+c\\delta)+(n-j)a$,\nfor all $j\\in[i_{s-k}+1,n]$.\nConsider $\\vv i$ such that $i_{s-k}<n-N$.\nThen $A_n(\\vv i)$ implies\n$W_{n-N,k+1}>x(1+c\\delta)+Na$.\nTogether with (\\ref{slln.double}) it yields\n\\begin{eqnarray}\\label{cond.DA}\n{\\mathbb P}\\Bigl\\{D_n>x\\Big|\\bigcup_{\\vv i:i_{s-k}<n-N}\nA_n(\\vv i)\\Bigr\\} &\\to& 1\n\\end{eqnarray}\nas $x\\to\\infty$, uniformly in $n$ and $N$. Now it\nfollows from (\\ref{Sigma.2}) and (\\ref{cond.DA}) that\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\}\n&\\ge& \\liminf_{n\\to\\infty}\n{\\mathbb P}\\Bigl\\{W_{n1}>x\\Big|\n\\bigcup_{\\vv i:i_{s-k}<n-N}A_n(\\vv i)\\Bigr\\}\n{\\mathbb P}\\Bigl\\{\\bigcup_{\\vv i:i_{s-k}<n-N}\nA_n(\\vv i)\\Bigr\\}\\\\\n&\\ge& \\frac{\\rho^{s-k}-o(1)}{(s-k)!}\n\\overline{B_r}^{s-k}(x(1+c\\delta)+(N+s-k)a).\n\\end{eqnarray*}\nThus, by (\\ref{def.N}),\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\}\n&\\ge& \\frac{\\rho^{s-k}-o(1)}{(s-k)!}\n\\overline{B_r}^{s-k}\\Bigl(x\\Bigl(\n1+c\\delta+\\frac{ka}{b-ka-k\\delta}\\Bigr)+(s-k)a\\Bigr).\n\\end{eqnarray*}\nHence, for every $\\delta>0$,\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\}\n&\\ge& \\frac{\\rho^{s-k}+o(1)}{(s-k)!}\n\\overline{B_r}^{s-k} \\Bigl(x\n\\Bigl(1+\\frac{ka+\\delta}{b-ka}\\Bigr)\\Bigr)\\\\\n&=& \\frac{\\rho^{s-k}+o(1)}{(s-k)!}\n\\overline{B_r}^{s-k} \\Bigl(x\n\\frac{b+\\delta}{b-ka}\\Bigr)\n\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\nThe proof of Theorem \\ref{th.lower} is complete.\n\\end{proof}\n\n\n\n\n\\section{New Majorant.}\\label{majorant}\n\nIn the proof of the upper bound in Theorem \\ref{co.s.max}\n(see Section \\ref{k=0}), we introduced $s$ parallel\nsingle server queues that provide a suitable\nmajorant in the case $\\rho<1$.\nIf $\\rho\\ge 1$ then the single server system with\nservice time distribution $B$ is unstable and\nthe above scheme does not work.\nFor an arbitrary $\\rho$, we need a more\ncomplex procedure to obtain a majorant.\nHereinafter let $k=[b\/a]$ be the integer part of $b\/a$.\nWe continue to assume constant interarrival times $\\tau \\equiv a$.\n\nAgain let $\\sigma_{ni}$, $n\\ge1$, $i\\le s$, be\nindependent random variables with common distribution $B$.\nDefine service times $\\sigma_n$ in the original\n$D\/GI\/s$ system as in Section \\ref{k=0}.\nConsider again $s$ auxiliary single server queues $D\/GI\/1$,\nbut now with different deterministic arrival times\n$T_n = n(k+1)(a-h)$ where\n\\begin{eqnarray}\\label{choice.h}\n\\frac{k}{k+1}\\Bigl(a-\\frac{b}{k+1}\\Bigr)<h<a-\\frac{b}{k+1},\n\\end{eqnarray}\nand with service times\nequal to $\\sigma_{ni}$ in queue $i=1,2,\\ldots,s$. Then\n$T_1=(k+1)(a-h)>b$, so that\nthe queues are stable.\nLet $U_i$ be a stationary waiting time in\nthe $i$th auxiliary queue. Since, for each $n$,\n\\begin{eqnarray}\\label{max_upper_independence}\n\\mbox{the sequences }\n\\{U_{n1},n\\ge 1\\},\\ldots,\\{U_{ns},n\\ge 1\\}\n\\mbox{ are mutually independent},\n\\end{eqnarray}\nthe limiting vector $(U_{1},\\ldots,U_{s})$ consists\nof independent identically distributed coordinates too.\n\nIn contrast to the case $\\rho <1$,\nit may not be true in general that, say, $V_{n1}$ is smaller than $U_{n1}$.\nNevertheless, for $\\rho < k+1$, we can prove\nthat, for any set $I$ of $k+1$ indices,\n$\\sum_{i\\in I} V_{ni}\\le \\sum_{i\\in I}U_{ni}+\\eta_I$\nwhere $\\eta_I$ has a light-tailed distribution,\nthis is Lemma \\ref{l.sum.of.W.le.U} below.\nBut first we state the main result of the Section which is\nan analogue of (\\ref{D.min})\nfor the general $\\rho$.\n\n\\begin{lemma}\\label{l.D.le.orderU}\nThere exists a number $\\beta>0$ and a random variable $\\eta$ such that\n${\\mathbb E}e^{\\beta\\eta}<\\infty$ and, for all $n$, with probability\n1,\n\\begin{eqnarray*}\nD_n &\\le& U_{n,(k+1)}+\\eta,\n\\end{eqnarray*}\nwhere $U_{n,(k+1)}$ is the $(k+1)$th order\nstatistic of  vector $(U_{n1},\\ldots,U_{ns})$.\n\\end{lemma}\n\nNow we formulate and prove the following result.\nBased on it, we give the proof of Lemma \\ref{l.D.le.orderU}\nat the end of the section.\n\n\\begin{lemma}\\label{l.sum.of.W.le.U}\nThere exists $\\beta>0$ such that, for any set of $k+1$\nindices $I=\\{i(1),\\ldots,i(k+1)\\}$, there is a random\nvariable $\\eta_I$ such that\n${\\mathbb E}e^{\\beta\\eta_I}<\\infty$ and, for any $n$,\nwith probability 1,\n\\begin{eqnarray*}\n\\sum_{i\\in I} V_{ni} &\\le&\n\\sum_{i\\in I}U_{ni}+\\eta_I.\n\\end{eqnarray*}\n\\end{lemma}\n\n\\begin{proof}\nFix some $i'\\in I$.\nConsider an auxiliary $GI\/GI\/k+1$ system\n$\\vv V'_n=(V'_{ni},i\\in I)$ with the same interarrival\ntimes equal to $\\tau_n$, but whose service times $\\sigma'_n$\nare chosen in a special manner.\nAt any time $n$, if $i_n\\in I$ ($i_n$ is defined in the proof\nof Theorem \\ref{co.s.max}, see Section \\ref{k=0})\nthen put $\\sigma'_n=\\sigma_{n,i_n}$ and $i'_n=i_n$.\nIf $i_n\\not\\in I$ then put $\\sigma'_n=\\sigma_{ni'}$\nand $i'_n=i'$.\nApplying property (1) of Lemma \\ref{monotonicity}, we\nget that\n$R(V_{ni},i\\in I)\\le R\\vv V_n'$\ncoordinatewise, for any $n$. Therefore,\n\\begin{eqnarray*}\n\\sum_{i\\in I} V_{ni}\n&\\le& \\sum_{i\\in I} V'_{ni}.\n\\end{eqnarray*}\nHence, it suffices to prove that\n\\begin{eqnarray}\\label{reform.V}\n\\sum_{i\\in I} V'_{ni} &\\le&\n\\sum_{i\\in I} U_{ni}+\\eta_I.\n\\end{eqnarray}\n\nFor every $i\\in I$ and for any $n$,\n\\begin{eqnarray*}\nU_{n+1,i}-U_{ni}\n&\\ge& \\sigma_{ni}-(k+1)(a-h),\n\\end{eqnarray*}\nand hence\n\\begin{eqnarray}\\label{increment.of.U}\n\\sum_{i\\in I}U_{n+1,i}-\\sum_{i\\in I}U_{ni}\n&\\ge& \\sigma_n'-(k+1)(a-h)+\n\\sum_{i\\in I,i\\not=i'_n}(\\sigma_{in}-(k+1)(a-h))\\nonumber\\\\\n&=& \\sigma_n'-(k+1)a+\\zeta_{I,n},\n\\end{eqnarray}\nwhere the independent identically distributed\nrandom variables\n$$\n\\zeta_{I,n}:=\\sum_{i\\in I,i\\not=i'_n}(\\sigma_{in}-b)\n+(k+1)^2h+k(b-(k+1)a)\n$$\nhave a positive mean, by the left inequality\nin (\\ref{choice.h}).\n\nTake $\\varepsilon\\in(0,{\\mathbb E}\\zeta_{I,n})$.\nLet $d$ and $A$ be defined by Lemma \\ref{slln.for.s-1}\napplied to the system $(V'_{1+n,i},i\\in I)$.\nConsider $d$-skeleton of $\\vv V'$, that is,\nthe sequence $\\vv V'_{1+nd}$.\nFor every $n$, if the maximal coordinate of\n$(V'_{1+nd,i},i\\in I)$ is not bigger than $A$,\nthen\n\\begin{eqnarray*}\n\\sum_{i\\in I}V'_{1+nd+d,i}-\\sum_{i\\in I}V'_{1+nd,i}\n&\\le& \\sigma'_{nd+1}+\\ldots+\\sigma'_{nd+d},\n\\end{eqnarray*}\nwhich together with (\\ref{increment.of.U}) implies that\n\\begin{eqnarray*}\n\\sum_{i\\in I}V'_{1+nd+d,i}-\\sum_{i\\in I}V'_{1+nd,i}\n&\\le& \\sum_{i\\in I}U_{1+nd+d,i}-\\sum_{i\\in I}U_{1+nd,i}\n+d(k+1)a-\\zeta_{I,1+nd}-\\ldots-\\zeta_{I,nd+d}\\\\\n&\\le& \\sum_{i\\in I}U_{1+nd+d,i}-\\sum_{i\\in I}U_{1+nd,i}\n+d((k+1)a+b),\n\\end{eqnarray*}\nsince $\\zeta_{I,n}\\ge -b$. To conclude, if\nthe maximal coordinate of $(V'_{1+nd,i},i\\in I)$\nis not bigger than $A$, then\n\\begin{eqnarray*}\n\\sum_{i\\in I}V'_{1+nd,i} &\\le& (k+1)A,\n\\end{eqnarray*}\nso that\n\\begin{eqnarray}\\label{increment.of.Vprime.1}\n\\sum_{i\\in I}V'_{1+nd+d,i}\n&\\le& \\sum_{i\\in I}U_{1+nd+d,i}+d((k+1)a+b)+(k+1)A\\nonumber\\\\\n&:=& \\sum_{i\\in I}U_{1+nd+d,i}+C,\n\\end{eqnarray}\n\nFurther, if the maximal coordinate of\n$(V'_{1+nd,i},i\\in I)$ is bigger than $A$,\nthen, by Lemma \\ref{slln.for.s-1}, we have\n\\begin{eqnarray*}\n\\sum_{i\\in I}V'_{1+nd+d,i}-\\sum_{i\\in I}V'_{1+nd,i}\n&\\le& \\theta_{I,1+nd},\n\\end{eqnarray*}\nwhere $\\theta_{I,1+nd}$ are independent identically\ndistributed random variables with mean\n${\\mathbb E}\\theta_{I,1+nd}\\le d(b-(k+1)a+\\varepsilon)$\nand such that\n\\begin{eqnarray*}\n\\theta_{I,1+nd} &\\le& \\sigma'_{nd+1}+\\ldots+\\sigma'_{nd+d}.\n\\end{eqnarray*}\nIn this case, by (\\ref{increment.of.U}),\n\\begin{eqnarray}\\label{increment.of.Vprime.2}\n\\sum_{i\\in I}V'_{1+nd+d,i}-\\sum_{i\\in I}V'_{1+nd,i}\n&\\le& \\sum_{i\\in I}U_{1+nd+d,i}-\\sum_{i\\in I}U_{1+nd,i}\n+\\theta_{I,1+nd}-(\\sigma'_{nd}+\\ldots+\\sigma'_{nd+d})\\nonumber\\\\\n&&\\hspace{20mm}+d(k+1)a-\\zeta_{I,1+nd}-\\ldots-\\zeta_{I,nd+d}\n\\nonumber\\\\\n&:=& \\sum_{i\\in I}U_{1+nd+d,i}-\\sum_{i\\in I}U_{1+nd,i}\n+\\widetilde\\theta_{I,1+nd}.\n\\end{eqnarray}\n\nInequalities (\\ref{increment.of.Vprime.1}) and\n(\\ref{increment.of.Vprime.2}) imply that always\n\\begin{eqnarray}\\label{diff.rough}\n\\sum_{i\\in I} V'_{1+nd+d,i}\n&\\le& \\sum_{i\\in I} U_{1+nd+d,i}+C\n+\\max_n\\sum_{j=0}^n\\widetilde\\theta_{I,1+jd}.\n\\end{eqnarray}\nHere $\\widetilde{\\theta}_{I,1+nd}, n=1,2,\\ldots$ are\nindependent identically distributed random variables that\nare bounded from above and have a negative mean. Indeed,\n\\begin{eqnarray*}\n\\widetilde\\theta_{I,1+nd}\n&=& \\theta_{I,1+nd}-(\\sigma'_{nd+1}+\\ldots+\\sigma'_{nd+d})\n+d(k+1)a-\\zeta_{I,1+nd}-\\ldots-\\zeta_{I,nd+d}\\\\\n&\\le& d(k+1)a-\\zeta_{I,1+nd}-\\ldots-\\zeta_{I,nd+d}\\\\\n&\\le& d((k+1)a+b)\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n{\\mathbb E}\\widetilde\\theta_{I,1+nd}\n&=& {\\mathbb E}\\theta_{I,1+nd}-bd\n+d(k+1)a-d{\\mathbb E}\\zeta_{I,1+nd}\\\\\n&\\le& d(b-(k+1)a+\\varepsilon)-bd+d(k+1)a-d{\\mathbb E}\\zeta_{I,1+nd}\\\\\n&=& d(\\varepsilon-{\\mathbb E}\\zeta_{I,1+nd})<0,\n\\end{eqnarray*}\nby the choice of $\\varepsilon$.\nTherefore, there exists $\\beta>0$ such that\n${\\mathbb E}e^{\\beta\\widetilde\\theta_{I,1+nd}}<1$\nand then the following estimate holds:\n$$\n{\\mathbb E}\\exp\\Bigl\\{\\beta\n\\max_n\\sum_{j=0}^n\\widetilde\\theta_{I,1+jd}\\Bigr\\}\n\\le \\sum_n\n({\\mathbb E}e^{\\beta\\widetilde\\theta_{I,1+nd}})^n\n<\\infty.\n$$\nLet\n$$\n\\widetilde\\eta_I:=C+\\max_n\\sum_{j=0}^n\\widetilde\\theta_{I,1+jd},\n$$\nthen ${\\mathbb E}e^{\\beta\\widetilde\\eta_I}<\\infty$ and,\nby the upper bound (\\ref{diff.rough}), for all $n$,\n\\begin{eqnarray}\\label{V.U.int}\n\\sum_{i\\in I} V'_{1+nd+d,i}\n&\\le& \\sum_{i\\in I} U_{1+nd+d,i}+\\widetilde\\eta_I.\n\\end{eqnarray}\nSince, in addition, for every $l\\in[1,d]$,\n$$\n\\sum_{i\\in I} V'_{1+nd+d+l,i}-\\sum_{i\\in I} V'_{1+nd+d+l-1,i}\n\\le \\sum_{i\\in I} U_{1+nd+d+l,i}-\\sum_{i\\in I} U_{1+nd+d+l-1,i}+(k+1)a,\n$$\nwe conclude from (\\ref{V.U.int}) that, for every $l\\in[1,d]$,\n\\begin{eqnarray*}\n\\sum_{i\\in I} V'_{1+nd+d+l,i}\n&\\le& \\sum_{i\\in I} U_{1+nd+d+l,i}+\\widetilde\\eta_I+l(k+1)a,\n\\end{eqnarray*}\nso that, for every $n$,\n\\begin{eqnarray*}\n\\sum_{i\\in I} V'_{ni}\n&\\le& \\sum_{i\\in I} U_{ni}+\\widetilde\\eta_I+d(k+1)a,\n\\end{eqnarray*}\nwhich completes the proof of Lemma \\ref{l.sum.of.W.le.U}\nwith $\\eta_I:=\\widetilde\\eta_I+d(k+1)a$.\n\\end{proof}\n\n\\begin{proofof}{Lemma \\ref{l.D.le.orderU}}\nFor every collection $I$ of $k+1$ coordinates\n\\begin{eqnarray*}\nD_n &\\le& \\frac{1}{k+1}\\sum_{i\\in I} V_{ni},\n\\end{eqnarray*}\nsince $D_n$ is the minimal coordinate.\nThen it follows from Lemma \\ref{l.sum.of.W.le.U} that\n\\begin{eqnarray}\\label{dn.le.sum}\nD_n &\\le& \\frac{1}{k+1}\\sum_{i\\in I} U_{ni}+\\eta_I.\n\\end{eqnarray}\nTake $\\eta:=\\displaystyle\\max_{I:|I|=k+1}\\eta_I$.\nThen ${\\mathbb E}e^{\\beta\\eta}<\\infty$ and\n\\begin{eqnarray}\\label{dn.le.sum1}\nD_n &\\le& \\frac{1}{k+1}\\sum_{i\\in I} U_{ni}+\\eta.\n\\end{eqnarray}\nTake $I$ such that $\\{U_{ni},i\\in I\\}$\nare the $k+1$ smallest coordinates of vector $(U_{n1},\\ldots,U_{ns})$.\nThen $U_{ni}\\le U_{n,(k+1)}$ for every $i\\in I$.\nTogether with (\\ref{dn.le.sum1})\nit yields the inequality of the lemma.\n\\end{proofof}\n\n\n\n\n\n\\section{Upper Bound and the Principle of $s-k$ Big Jumps.}\\label{sec.upper.bound}\n\nNow we turn to the upper bound.\nLemma \\ref{l.D.le.orderU}\nallows to prove the following general result.\n\n\\begin{theorem}\\label{th.deter.upper}\nLet $\\rho<k+1$ for some $k\\in\\{0,1,\\ldots,s-1\\}$.\nThen, for any fixed $h$ satisfying {\\rm(\\ref{choice.h})},\nthere exists $\\beta>0$ such that\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x+y\\}\n&\\le& \\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n(\\overline F(x))^{s-k}+const\\cdot e^{-\\beta y}\n\\ \\mbox{ for all }x,y>0,\n\\end{eqnarray*}\nwhere $F$ is the distribution of random variable\n$$\nM:=\\sup\\Bigl(0,\\ \\sum_{j=1}^n\n(\\sigma_j-(k+1)(a-h)),\\ n\\ge 1\\Bigr).\n$$\n\\end{theorem}\n\n\\begin{proof}\nFirst, by inequality\n$b\\equiv{\\mathbb E}\\sigma<(k+1)(a-h)$\nand by the strong law of large numbers,\nthe maximum $M$ is finite with probability 1.\nBy Lemma \\ref{l.D.le.orderU},\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_n>x+y\\}\n&\\le& {\\mathbb P}\\{U_{n,(k+1)}+\\eta>x+y\\}\\\\\n&\\le& {\\mathbb P}\\{U_{n,(k+1)}>x\\}+\n{\\mathbb P}\\{\\eta>y\\}.\n\\end{eqnarray*}\nTaking into account the independence of the $U$'s in\n(\\ref{max_upper_independence}), we obtain\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_n>x+y\\}\n&\\le& \\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n{\\mathbb P}^{s-k}\\{U_{n1}>x\\}+\n{\\mathbb P}\\{\\eta>y\\}.\n\\end{eqnarray*}\nLetting $n\\to\\infty$ and taking into account\nthe duality between the single server system\nand the maximum of the corresponding random walk,\nwe arrive at the following inequality:\n\\begin{eqnarray}\\label{Dprime}\n{\\mathbb P}\\{D>x+y\\}\n&\\le& \\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n(\\overline F(x))^{s-k}+\n{\\mathbb P}\\{\\eta>y\\}.\n\\end{eqnarray}\nSince $\\eta$ has a finite exponential\nmoment, we obtain the statement of the theorem.\n\\end{proof}\n\n\\begin{proofof}{the upper bound in Theorem \\ref{th:subexp.bounds}}\nIt follows from Theorem \\ref{th.deter.upper} that\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\}\n&\\le& \\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n\\overline F^{s-k}(x(1-\\delta))\n+const\\cdot e^{-\\beta\\delta x}.\n\\end{eqnarray*}\nDue to the subexponentiality of $B_r$ we obtain\nfrom the analogue of (\\ref{W.single}) for the maximum of\na random walk (see, e.g., \\cite[Theorem 5.2]{FKZ})\nthat, as $x\\to\\infty$,\n$$\n\\overline F(x)\\sim\\frac{b}{(k+1)(a-h)-b}\\overline B_r(x).\n$$\nTaking $h$ in (\\ref{choice.h}) close to its minimal value,\nsay $h=\\frac{k}{k+1}\\Bigl(a-\\frac{b}{k+1}\\Big)+\\varepsilon$,\n$\\varepsilon>0$, we arrive at the following estimate:\n$$\n\\overline F(x)\\sim\\frac{b}{a-b\/(k+1)-(k+1)\\varepsilon}\\overline B_r(x)\n=\\frac{(k+1)\\rho}{k+1-\\rho-(k+1)^2\\varepsilon\/a}\\overline B_r(x).\n$$\nIn addition,\n$\\overline B_r(x(1-\\delta))\\cdot e^{\\beta\\delta x}\n\\to \\infty$ as $x\\to\\infty$. All these facts and\narbitrarity of choice of $\\varepsilon>0$ imply\nthe desired bound.\n\\end{proofof}\n\nIt what follows, for two families of events $A_x$ and $B_x$ of positive probabilities\nindexed by $x$, we write $A_x \\sim B_x$  if ${\\mathbb P} \\{A_x\\setminus B_x\\} = o({\\mathbb P}\\{A_x\\})$\nand ${\\mathbb P} \\{B_x\\setminus A_x\\} = o({\\mathbb P}\\{A_x\\})$ as $x\\to\\infty$. Note that\n$A_x\\sim B_x$ implies ${\\mathbb P}\\{A_x\\}\\sim {\\mathbb P}\\{B_x\\}$, but not vice versa.\n\nWe establish now the principle of $s-k$ big jumps in the case of\nintermediate\nregularly varying\ndistributions.\nFor simplicity, we do it again for $D\/GI\/s$ system with deterministic\ninter-arrival times.\nFor this, we consider the representation of the stationary workload\nin the backward time (the so-called ``Loynes scheme'').\nWe again use the joint representation of $s$ individual queues and of\nthe $s$-server system given in the previous section and assume that\nall queues run for a long time, from time $-\\infty$, and that\n$U_i$ is the stationary waiting time of the customer that arrives at the $i$th\nqueue at time $0$. Then\n$$\nU_i = \\sup (0, \\xi_{-1,i}, \\xi_{-1,i}+\\xi_{-2,i}, \\ldots ,\n\\xi_{-1,i}+\\ldots +\\xi_{-n,i}, \\ldots )\n$$\nwhere $\\xi_{j,i} =\\sigma_{j,i}-\\widehat{a}$, for $i=1,\\ldots,s$ and $j=-1,-2,\\ldots$,\nand $\\widehat{a}=(k+1)(a-h)$.\nFurther, ${\\vv W}_n$ are stationary vectors for $n\\le 0$, and\n$$\n{\\vv W}_{n+1}= R((W_{n1}+\\sigma_n-a)^+, (W_{n2}-a)^+,\\ldots,(W_{n,s}-a)^+),\n$$\nfor all $n<0$. Here again $i_n = \\min \\{i : \\ V_{ni}=D_n \\}$ and $\\sigma_n=\\sigma_{n,i_n}$.\nThen, for any $x>0$, by Lemma \\ref{l.D.le.orderU},\n$$\n\\{D_0 >x\\} \\subset \\bigcup_J \\{ \\min_{i\\in J} U_i + \\eta >x \\}\n$$\nwhere $D_0$ is the stationary waiting time in the $D\/GI\/s$ queue,\ni.e. the minimal coordinate of vector ${\\vv W}_0$,\n$J$'s are subsets of $\\{1,2,\\ldots,s\\}$ of cardinality $s-k$, and $\\eta$ is\na random variable with light-tailed distribution. Then\n$$\n\\{D_0 > x\\} = \\bigcup_J \\{D_0>x, \\min_{i\\in J} U_i + \\eta >x \\}.\n$$\n\n\nAssume that the\nresidual\ndistribution function $B_r$ of service times is\nintermediate\nregularly varying\n(for that, it is sufficient for $B$ to be intermediate varying).\nThen, clearly, each random variable $U_i$ has an intermediate regularly varying\ndistribution since ${\\mathbb P} (U_i>x) \\sim c \\overline{B}_r(x)$.\nSince\nthe random variables $U_i$ are mutually independent, the distribution\nof $\\min_{i\\in J} U_i$ is also intermediate regularly varying,\n${\\mathbb P} (\\min U_i >x) \\sim c^{s-k} \\left(\\overline{B}_r(x)\n\\right)^{s-k}$.\n\nIt is well-known, see e.g. \\cite[Ch 5]{FKZ}, that\n$$\n\\{ U_i>x \\} \\sim \\bigcup_{n\\ge 1} \\{\\sigma_{-n,i}>x+n\\widehat{a} \\}\n$$\nand therefore, for any set $J \\subset \\{1,2,\\ldots,s\\}$,\n$$\n\\{\\min_{i\\in J} U_i >x\\} \\sim  \\bigcap_{i\\in J} \\left(\\bigcup_{n>0}\n\\{\\sigma_{-n,i}>x+n\\widehat{a} \\}\\right).\n$$\n\nWe use the following property of intermediate regularly varying distributions\n(its proof is postponed until the end of the section;\na similar result for equivalence of probabilities\nmay be found in \\cite{AAK}):\n\n\\begin{lemma}\\label{irv3}\nIf $X$ and $Y$ are two random variables such that $X$\nhas an intermediate regularly varying distribution and\n${\\mathbb P}\\{|Y|>x\\}=o({\\mathbb P}\\{X>x\\})$ as\n$x\\to\\infty$, then $\\{X+Y>x\\} \\sim \\{X>x\\}$,\nfor any joint distribution of $X$ and $Y$.\n\\end{lemma}\n\nApplying Lemma \\ref{irv3} with $X=\\min_{i\\in J} U_i$\nand $Y=\\eta$, for all $J$ of cardinality $s-k$, we get\n\\begin{eqnarray*}\n\\{D_0>x\\}\n&\\sim &\n\\bigcup_J \\{D_0>x, \\min_{i\\in J} U_i >x\\}\\\\\n&\\sim &\n\\bigcup_J  \\bigcap_{i\\in J} \\left(\\bigcup_{n>0}\n\\{D_0>x, \\sigma_{-n,i}>x+n\\widehat{a} \\}\\right),\n\\end{eqnarray*}\nsince the upper and the lower bounds for\n${\\mathbb P}\\{D_0>x\\}$ are of the same order,\nsee \\cite[Theorem 7]{BaF} or \\cite{FK} for further arguments.\nNow represent any event on the right in the latter equation as a union\nof two events\n$$\n\\{D_0>x,  \\sigma_{-n,i}>x+n\\widehat{a} , i_{-n}=i\\} \\cup\n\\{D_0>x,  \\sigma_{-n,i}>x+n\\widehat{a} , i_{-n}\\neq i\\}\n$$\nwhere\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_0>x, \\sigma_{-n,i}>x+n\\widehat{a}, i_{-n}\\neq i\\}\n&=& {\\mathbb P}\\{D_0>x, i_{-n}\\neq i\\} {\\mathbb P}\\{\\sigma_{-n,i}>x+n\\widehat{a}\\}\\\\\n&\\le& {\\mathbb P}\\{D_0>x\\}{\\mathbb P}\\{\\sigma_{-n,i}>x+n\\widehat{a}\\}.\n\\end{eqnarray*}\nSo, for any set $J$, the union of events\n$$\n\\bigcap_{i\\in J} \\left(\\bigcup_{n>0}\n\\{D_0>x, \\sigma_{-n,i}>x+n\\widehat{a}, i\\neq i_{-n}\\}\\right)\n$$\nhas probability $O({\\mathbb P} \\{D_0>x\\} \\overline{B}_r (x))\n= o({\\mathbb P} \\{D_0>x\\})$.\nSince there is only a finite number of sets $J$,\nwe obtain the following result.\n\n\\begin{theorem}\\label{bigjumps}\nAssume that $\\rho \\in (k,k+1)$ and that the distribution of service\ntimes\nis intermediate regularly varying.\nAs $x\\to\\infty$,\n\\begin{equation}\\label{bigj1}\n{\\mathbb P} \\{D_0>x\\} \\sim {\\mathbb P} \\Bigl\\{ D_0>x,\n\\bigcup_{0<n_1<n_2<\\ldots <n_{s-k}}\n\\bigcap_{j=1}^{s-k} \\{ \\sigma_{-n_j}> x+n_j \\widehat{a} \\}\n\\Bigr\\}.\n\\end{equation}\n\\end{theorem}\n\n{\\bf Remark.}\nIn the proof of Theorem \\ref{bigjumps},\nwe followed the scheme introduced in \\cite{FK},\nsee also \\cite{BaF}, \\cite{BMZ}, and \\cite{BZ}\nfor similar constructions.\nTheorem \\ref{bigjumps} is not the final statement.\nWe may go further and obtain the following result.\nAssume that $B$ is a regularly varying distribution.\nThen,\nfor some positive and finite constant $C$ and as $x\\to\\infty$,\n\\begin{equation}\\label{equivalence1}\n{\\mathbb P} \\{D_0>x\\} \\sim C \\overline{B}^{s-k}_{r}(x).\n\\end{equation}\nThe result seems to be correct, but its proof would be very lengthy and\nwould require a\nscrupulous calculation, so we decided not to proceed further in this\ndirection.\n\nWe provide a hint for a plausible proof\ninstead. First, one may consider an auxiliary\ndeterministic model with  $(n-s)$ very big service times\n$y_1$, \\ldots, $y_{s-k}$ that occur\nat time instants $-n_1>-n_2>\\ldots >-n_{s-k}$ and replace all\nother service times by their mean $b$. We also assume that, before\nthe first jump, the workload vector is zero.\nFor this model, we may find conditions on the $y$'s for the\nminimal coordinate of the workload vector at time 0 to be not\nsmaller than $x$. Then repeat the same for all the other times\nof jumps $-n_{1}>-n_{2}>\\ldots >-n_{s-k}$. The union of these\nregions may be represented as a combination of unions and differences\nof a finite number of truncated half-spaces of  dimension $s-k$.\nSummation of tail probabilities over each such set gives the\nprobability of order  $\\overline{B}^{s-k}_{r}(x)$, thanks to the properties\nof regularly varying functions. So a finite combination of sums and\ndifferences of these probabilities gives a probability of the same\norder. It cannot be of a lower order, due to the lower\nbound.\n\n\\begin{proofof}{Lemma \\ref{irv3}}\nFrom Theorem 2.47 in \\cite{FKZ}, if $X$ has\nan intermediate regularly varying distribution, then\n$$\n{\\mathbb P}\\{X>x+h(x)\\}\\sim {\\mathbb P}\\{X>x\\}\n\\sim {\\mathbb P}\\{X>x-h(x)\\}\n$$\nas $x\\to\\infty$, for any function $h(x)\\to\\infty$ such\nthat $h(x)=o(x)$. Hence, by the monotonicity arguments,\n$$\n\\{X>x+h(x)\\}\\sim \\{X>x\\}\\sim \\{X>x-h(x)\\}.\n$$\nSince the distribution of $X$ is intermediate regularly\nvarying and since ${\\mathbb P}\\{|Y|>x\\}=o({\\mathbb P}\\{X>x\\})$\nas $x\\to\\infty$, we have \n${\\mathbb P}\\{|Y|>\\varepsilon x\\}=o({\\mathbb P}\\{X>x\\})$\nas $x\\to\\infty$, for every $\\varepsilon>0$. Then \nthere exists $h(x)=o(x)$ such that\n$$\n{\\mathbb P}\\{|Y|>h(x)\\}=o({\\mathbb P}\\{X>x\\}).\n$$\nTherefore, as $x\\to\\infty$,\n\\begin{eqnarray*}\n\\{X>x\\} &\\sim& \\{X>x+h(x)\\}\\setminus\\{Y<-h(x)\\}\\\\\n&=& \\{X>x+h(x),Y\\ge-h(x)\\} \\subseteq \\{X+Y>x\\}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\{X>x\\} &\\sim& \\{X>x-h(x)\\}\\cup\\{Y>h(x)\\}\n\\supseteq \\{X+Y>x\\}, \n\\end{eqnarray*}\nwhich justifies the events equivalence,\n$\\{X+Y>x\\} \\sim \\{X>x\\}$.\n\\end{proofof}\n\n\\section{Existence of moments: proof of Theorem 4.}\\label{proof}\n\nSince the tail distribution of $\\min(\\sigma_{r,1},\\ldots,\\sigma_{r,s-k})$\nis equal to $(\\overline B_r(x))^{s-k}$, we obtain from\nTheorem \\ref{th.lower}\n$$\n{\\mathbb P}\\{D>x\\} \\ge\nc_1{\\mathbb P}\\{\\min(\\sigma_{r,1},\\ldots,\\sigma_{r,s-k})\n>c_2x\\}.\n$$\nSince, for any non-negative random variable $\\eta$,\n$$\n{\\mathbb E}\\eta^\\gamma = \\gamma\\int_0^\\infty x^{\\gamma-1}\n{\\mathbb P}\\{\\eta>x\\}dx,\n$$\nwe have\n$$\n{\\mathbb E}D^\\gamma \\ge \\frac{c_1}{c_2}\n{\\mathbb E}(\\min(\\sigma_{r,1},\\ldots,\\sigma_{r,s-k}))^\\gamma.\n$$\nand the existence of the moment of order $\\gamma$\nfor the delay $D$ implies with necessity (\\ref{min.fin}).\n\nNow assume (\\ref{min.fin}). Consider $s-k$\nindependent copies $M_1$, \\ldots, $M_{s-k}$ of the\nrandom variable $M$ introduced in Theorem \\ref{th.deter.upper}.\nThen the assertion of Theorem \\ref{th.deter.upper}\ncan be rewritten in the following way:\n$$\n{\\mathbb P}\\{D>x+y\\} \\le\n\\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n{\\mathbb P}\\{\\min(M_1,\\ldots,M_{s-k})>x\\}\n+const\\cdot e^{-\\beta y}.\n$$\nTake $y=x$. Then ${\\mathbb E}D^\\gamma<\\infty$ follows\nif we prove that\n\\begin{equation}\\label{M1.Ms-k}\n{\\mathbb E}(\\min(M_1,\\ldots,M_{s-k}))^\\gamma < \\infty.\n\\end{equation}\nIn order to do it, we explore the ladder height construction\nfor the maximum $M$ of a random walk $S_n=X_1+\\ldots+X_n$\nwhere $X_j=\\sigma_j-b-\\varepsilon$.\nSince this random walk has a negative drift,\nthe first ladder epoch and the first ladder height\n$$\n\\theta=\\min(n\\ge 1:S_n>0), \\quad \\widetilde\\chi=S_{\\theta},\n$$\nboth are degenerate random variables;\n$$\np\\equiv {\\mathbb P}\\{\\theta<\\infty\\}={\\mathbb P}\\{M>0\\}<1.\n$$\nDenote by $\\chi$ a random variable with distribution\n$$\n{\\mathbb P}\\{\\chi\\in B\\}={\\mathbb P}\\{\\widetilde\\chi\\in B\\}\/p.\n$$\nLet $\\chi_j$ be independent copies of $\\chi$.\nIf $\\eta$ is an independent counting random variable with\ndistribution ${\\mathbb P}\\{\\eta=j\\}=(1-p)p^j$, $j=0$, $1$, \\ldots,\nthen $M$ is equal in distribution to $\\chi_1+\\ldots+\\chi_\\eta$.\n\nLet $\\chi_{i,j}$ be again independent copies of $\\chi$\nand $\\eta_j$ be independent copies of $\\eta$.\nThen $\\min(M_1,\\ldots,M_{s-k})$ is equal in distribution to\n$$\n\\min\\biggl(\\sum_{j=1}^{\\eta_1}\\chi_{1,j},\\ldots,\n\\sum_{j=1}^{\\eta_{s-k}}\\chi_{s-k,j}\\biggr).\n$$\nThe latter\nminimum does not exceed\n$$\n\\sum_{j_1=1}^{\\eta_1}\\ldots\\sum_{j_{s-k}=1}^{\\eta_{s-k}}\n\\min(\\chi_{1,j_1},\\ldots,\\chi_{s-k,j_{s-k}}).\n$$\nTaking into account that for non-negative arguments\n$$\n(x_1+\\ldots+x_N)^\\gamma\\le N^\\gamma(x_1^\\gamma+\\ldots+x_N^\\gamma),\n$$\nwe get the following estimate:\n\\begin{eqnarray*}\n\\min\\biggl(\\sum_{j=1}^{\\eta_1}\\chi_{1,j},\\ldots,\n\\sum_{j=1}^{\\eta_{s-k}}\\chi_{s-k,j}\\biggr)^\\gamma\n&\\le& (\\eta_1+\\ldots+\\eta_{s-k})^\\gamma\n\\sum_{j_1=1}^{\\eta_1}\\ldots\\sum_{j_{s-k}=1}^{\\eta_{s-k}}\n\\min(\\chi_{1,j_1},\\ldots,\\chi_{s-k,j_{s-k}})^\\gamma.\n\\end{eqnarray*}\nIn particular, the mean of the term in the left side of the equality above\nis not larger than\n\\begin{eqnarray*}\n\\lefteqn{\\sum_{j_1=1}^\\infty\\ldots\\sum_{j_{s-k}=1}^\\infty\n(1-p)p^{j_1+\\ldots+j_{s-k}}(j_1+\\ldots+j_{s-k})^\\gamma\nj_1\\cdot\\ldots\\cdot j_{s-k}\n{\\mathbb E}\\min(\\chi_{1,1},\\ldots,\\chi_{s-k,1})^\\gamma}\\\\\n&&\\hspace{40mm} = {\\mathbb E}(\\eta_1+\\ldots+\\eta_{s-k})^\\gamma\n\\eta_1\\ldots\\eta_{s-k}\n{\\mathbb E}\\min(\\chi_{1,1},\\ldots,\\chi_{s-k,1})^\\gamma.\n\\end{eqnarray*}\nSince the $\\eta$'s have finite exponential moments,\nthe first mean on the right is finite.\nNow we show finiteness of the second mean. First,\n$$\n{\\mathbb P}\\{\\chi>x\\} =\n\\int_{-\\infty}^0 \\overline B(x-y)\\mu(dy),\n$$\nwhere the measure $\\mu$ is defined by\n\\begin{eqnarray*}\n\\mu(dy) &=& \\sum_n {\\mathbb P}\\{S_n\\in dy, S_k\\le 0\n\\mbox{ for all }k\\le n-1\\}\\\\\n&\\le& \\sum_n {\\mathbb P}\\{S_n\\in dy\\}.\n\\end{eqnarray*}\nThen, by the key renewal theorem,\n$$\nc\\equiv \\sup_{y\\le 0}\\mu(y-1,y] < \\infty,\n$$\nwhich yields\n$$\n{\\mathbb P}\\{\\chi>x\\} \\le c \\sum_{j=0}^\\infty\n\\overline B(x+j)\\le c\\overline B_r(x-1).\n$$\nTherefore, due to condition (\\ref{min.fin}),\n$$\n{\\mathbb E}\\min(\\chi_{1,1},\\ldots,\\chi_{s-k,1})^\\gamma<\\infty,\n$$\nwhich completes the proof.\n\n\n\n\n\n\\acknowledgments{This research was supported\nby EPSRC grant No.~R58765\/01\nand RFBR grant No.~10-01-00161.}\nThe authors thank Bert Zwart for stimulating discussions, \nand Arcady Shemyakin and James Cruise for stylistic comments.\n\n\n\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nDirect prediction of material properties through predictive models has attracted interests from both material and data science communities. Predictive models have the potential to mimic highly nonlinear physics-based mappings, thus reducing dependencies on numerical simulations or experiments during material design, and enabling tractable discovery of novel yet complex material systems~\\cite{jain2013commentary,hautier2010data,kondo2017microstructure}. Nonetheless, the construction of predictive models for nonlinear functions, such as material structure-property mappings, is known to be data-demanding, especially when the inputs, e.g., material microstructures represented as 2D or 3D images, are high-dimensional~\\cite{huan2016polymer}. Thus, the added value of predictive models quickly diminishes as the acquisition cost increases for material samples. We investigate in this paper a computational approach to generate artificial material samples with negligible cost, by exploiting the fact that all samples within one material system share similar morphology. More concretely, we define morphology as a style vector quantified from a microstructure sample, and propose a generative model that learns from a small set of authentic samples, and creates an arbitrary amount of artificial samples that share the same distribution of morphologies as the authentic ones. \n\nThe key contribution of the paper is the introduction of a morphology constraint on the generative model that significantly improves the morphological consistency between the artificial and authentic samples from benchmark generative models. \nTo demonstrate the utility of the proposed model, we run a case study on the prediction of the Young's modulus, the diffusion coefficient and the permeability coefficient of sandstone microstructures. We show that the generated artificial samples from the proposed model can improve the prediction performance more effectively than those from a state-of-the-art Markov Random Field (MRF) model. \n\n\nAs an overview, the proposed model follows the architecture of a variational autoencoder~\\cite{kingma2013auto} that learns to encode material microstructures into a lower-dimensional latent space and to decode samples from the latent space back into microstructures. \nBoth the encoder and the decoder are composed of feed-forward convolutional neural networks for extracting and generating local morphological patterns, and are jointly trained to minimize the discrepancy between the artificial and authentic samples.\nThe target morphology is quantified from the authentic samples by an auxiliary network. \nThe idea of quantifying material morphology through a deep network is inspired by the style transfer technique originally developed for image synthesis~\\cite{gatys2016image}. \n\n\nThe rest of the paper is structured as follows: In Sec.~\\ref{sec:background} we review related work on material representations and reconstruction, based on which we delineate the novelty of this paper. We then introduce background knowledge on variational autoencoder and style transfer. Sec.~\\ref{sec:model} elaborates on the details of the proposed model. Sec.~\\ref{sec:results} presents a case study on the prediction of sandstone properties, where we demonstrate the superior performance of the proposed model against the benchmarks, in both microstructure generation, and the resultant property prediction accuracy. In Sec.~\\ref{sec:discussion} we summarize findings from the case study and propose potential future directions. Sec.~\\ref{sec:conclusions} concludes the paper.     \n\n\\section{Background}\n\\label{sec:background}\n\n\\subsection{Data science challenges in computational material science}\nIncorporating data science into material discovery~\\cite{jain2013commentary} and design~\\cite{curtarolo2013high} faces unique challenges with high dimensionality of material representations and the lack of material data due to high acquisition costs.\nWe review existing work that address these challenges to some extent. \n\n\n\\paragraph{Challenge 1: Mechanisms for understanding material representations} A common approach to addressing the issue of high dimensionality is to seek for a representation, i.e., an encoder-decoder pair, for a material system: The encoder transforms microstructures to their reduced representations, and the decoder generates (i.e., reconstructs) them back from their representations. A good encoder-decoder pair should both achieve significant dimension reduction, and good matching between the data distribution (i.e., the distribution of authentic samples) and the model distribution (i.e., the distribution defined by the decoder). This is often feasible for material systems with consistent and quantifiable morphologies among their samples, as reviewed below. \n\nExisting encoders for material representations can be categorized as physical and statistical, some of which have led to accelerated design of various material systems~\\cite{torquato2013random, jiang2013efficient, grigoriu2003random, xu2014descriptor, yu2017characterization}. Among all, {\\bf physical encoders} characterize microstructures using composition (e.g., the percentage of each material constituent)~\\cite{broderick2008informatics,ashby2005materials}, dispersion (e.g., inclusions' spatial relation, pair correlation,the ranked neighbor distance ~\\cite{steinzig1999probability, tewari2004nearest, rollett2007three,borbely2004three, pytz2004microstructure,scalon2003spatial}), and geometry features (e.g., the radius\/size distribution, roundness, eccentricity, and aspect ratio of elements of the microstructure~\\cite{rollett2007three, steinzig1999probability, torquato2013random, sundararaghavan2005classification, basanta2005using, holotescu2011prediction, klaysom2011effects, gruber2010misorientation}). \nAmong {\\bf statistical encoders} are the N-point correlation functions~\\cite{liu2013computational,borbely2004three, torquato2013random, sundararaghavan2005classification, basanta2005using}. Torquato et al.~\\cite{rintoul1997reconstruction,yeong1998reconstructing,torquato2013random} show that the microstructure of heterogeneous materials can be characterized statistically via various types of N-point correlation functions~\\cite{okabe2005pore, hajizadeh2011multiple}. Similar descriptors include lineal path function~\\cite{oren2003reconstruction} and statistics calculated based on the frequency domain using fast Fourier transformation~\\cite{fullwood2008gradient,fullwood2008microstructure}.\nAnother type of statistical encoders are {\\bf random fields}~\\cite{roberts1997statistical,bostanabad2016stochastic,liu2015random}, which define joint probability functions on the space of microstructures. Typical probability models include Gaussian random fields~\\cite{quiblier1984new, ostoja1998random, torquato2013random} which treats binary microstructure images as level sets, and Markov random fields, where each pixel of the microstructure is assumed to be drawn from a probability function conditioned on its neighbouring pixels~\\cite{bostanabad2016stochastic}.\n\n{\\bf Decoding of representations}, i.e., generation of microstructure through existing physical and statistical representations, involves optimization in the microstructure space: For physical representations and N-point correlation functions, a microstructure is searched to minimize its difference from the target descriptors. For random fields, the generation can be done by maximizing the joint probability through Markov chain Monte Carlo simulations~\\cite{bostanabad2016stochastic,liu2015random}. While it is shown that material generation through these representations is feasible~\\cite{borbely2004three, basanta2005using,torquato2013random,xu2014descriptor}, the computational costs for the optimization through gradient~\\cite{fullwood2008gradient,xu2014descriptor} and non-gradient~\\cite{yeong1998reconstructing,jiao2008modeling,jiao2009superior,karsanina2015universal} methods are often high.\n\nIn addition to the difficulties in decoding, the existing encoders are not universally applicable, especially to material systems with complex morphology. More specifically, matching in the representation space does not guarantee the match in the microstructure space. An example can be found in Fig.~\\ref{fig:point_corr}, where we compare two-point correlation functions of Ti64 alloy samples and three sets of artificial images (see details from \\cite{bostanabad2016stochastic,cang2017microstructure,cang2017scalable}). The visually more plausible set has worse match to the target with respect to the Euclidean distance in the discretized 2-point correlation space. \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{point_corr.png}\n\\caption{Comparison of two-point correlation functions among four sets of images. From left to right: Authentic microstructure samples, samples generated by a Markov random field model~\\cite{bostanabad2016stochastic}, samples generated by a hybrid model with deep belief network and Markov random field~\\cite{cang2017scalable}, and samples generated by a deep belief network~\\cite{cang2017microstructure}. Better matching in the discretized 2-point correlation space does not indicate better microstructure generations.}\n\\label{fig:point_corr}\n\\end{figure}\n\n\n\n\n\n\n\n\nThese existing difficulties leads to the need for new mechanisms to define material representations. We propose four metrics for evaluating the utility of a material representation: interpretability, dimensionality, expressiveness, and generation cost: \nPhysical descriptors and correlation functions are designed to be interpretable and relatively low dimensional, yet may not be expressive enough to capture complex morphologies and requires optimization during generation; random fields are relatively expressive (and some permit fast generation~\\cite{bostanabad2016stochastic}), but are often high-dimensional and less interpretable. Both categories of representations are material specific, i.e., new representations need to be manually identified for new material systems. Cang et al.~\\cite{cang2017microstructure} proposed to learn statistical generative models from microstructures to automatically derive expressive, low dimensional representations that enables fast microstructure generation. They showed that a particular type of generative model, called Convolutional Deep Belief Network (CDBN)~\\cite{lee2009convolutional}, can produce reasonable microstructures for material systems with complex morphologies, by extracting morphology patterns at different length scales from samples, and decode an arrangement of these patterns (the hidden activation of the network) back into a microstructure. \nNonetheless, CDBNs are trained layer-by-layer, and thus require additional material-dependent parameter tuning to achieve plausible generations.\n\nDifferent from \\cite{cang2017microstructure}, this paper proposes a model that directly enforces the matching of the artificial microstructures to the authentic ones, thus avoiding additional parameter tuning. Our model is also fully differentiable, making it much easier to train than CDBNs and scalable to deep architectures. \nLastly, the introduction of the morphology constraint further refines the artificial microstructures under a small amount of training data. With all these improvements, the proposed model is generally applicable to the extraction of low dimensional material representations, and is expressive for generating microstructures with morphologies of decent complexity. \n\n\n\\paragraph{Challenge 2: Effective material data acquisition methods} Unlike scenarios in contemporary machine learning where a large quantity of data is available, in computational material design tasks we may only have a limited amount of microstructure and property samples, and the acquisition cost for additional samples is usually high. This challenge calls for the incorporation of active learning methods into the design. The key idea of active learning is to minimize the data acquisition cost for learning a model (e.g., generative models for capturing microstructure distributions, or discriminative models for predicting process-structure-property mappings), by optimally controlling the balance between data exploitation and exploration~\\cite{tong2001support1,settles2010active}. In computational material science, such techniques have been adopted to accelerate the process of material discovery for desired properties~\\cite{ling2017high,lookman2016perspective}.\nWhile this paper does not focus directly on active learning, we demonstrate that the quality of the microstructure data, along with its quantity, can significantly influence the prediction performance of a statistical structure-property mapping, and thus justify the value of the proposed generative model.\n\n\n\\subsection{Preliminaries on predictive and generative models}\nThis paper will involve three networks: The generative network for creating artificial material microstructures, the auxiliary convolutional network for quantifying material morphologies, and another convolutional network for structure-property prediction. We provide technical backgrounds for these models below.\n\n\n\\paragraph{Generation through Variational Autoencoder}\n\\label{para:VAE}\nVariational autoencoders (VAE)~\\cite{kingma2013auto} are extensions of autoencoders~\\cite{bengio2009learning}. An autoencoder (Fig.~\\ref{fig:AE}a) is composed by two parts: an encoder ${\\bf z}=f({\\bf x})$ converts the input $\\textbf{x}$ to a hidden vector $\\textbf{z}$, and a decoder produces a reconstruction $\\hat{\\textbf{x}} = g(\\textbf{z})$. An AE is trained to minimize the discrepancy between inputs $\\textbf{x}$ and their corresponding reconstructions $\\hat{\\textbf{x}}$. \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{vae_ae.png}\n\\caption{(a) An autoencoder converts inputs to a latent space and reconstructs outputs from samples in the latent space. It is trained to minimize the reconstruction error. (b) A variational autoencoder converts inputs to a distribution in the latent space. It is trained to minimize both the reconstruction error and the Kullback-Leibler (KL) divergence between the latent distribution and a sample distribution (often standard normal).}\n\\label{fig:AE}\n\\end{figure}\nVariants of autoencoders (e.g., Sparse~\\cite{ng2011sparse}, denoising~\\cite{vincent2008extracting}, and contractive~\\cite{rifai2011contractive}) have been proposed to improve the learning of more concise representations from high-dimensional input data,\nand are widely used for data compression~\\cite{le2015tutorial}, network pre-training~\\cite{bengio2007greedy}, and feature extraction~\\cite{xing2015stacked}. However, conventional autoencoders are prone to generating implausible new outputs, see Fig.~\\ref{fig:gen_compare}b for examples, as they do not attempt to match the model distribution (the distribution of images generated from the decoder) with the data distribution (the distribution of input images). VAE (Fig.~\\ref{fig:AE}b) was introduced to address this issue~\\cite{kingma2013auto}, by constraining the distribution of the latent variables $\\textbf{z}$ encoded from the input data to that used for output generation, thus indirectly forcing the match between the distributions of the model outputs and the data.\n\n\n\n\n\n\n\n\\paragraph{The VAE model} We start with the decoder, which defines the output distribution, $p_{\\boldsymbol{\\theta}} (\\textbf{x}|\\textbf{z})$, conditioned on the latent vector $\\textbf{z}$, and is parameterized by $\\boldsymbol{\\theta}$.  Let the input data be $\\mathcal{X}=\\{\\textbf{x}_i\\}^N_{i=1}$, which defines a data distribution $p(\\textbf{x})$, and let $p(\\textbf{z})$ be the pre-specified sampling distribution in the latent space. The goal is to train a model that matches the marginal distribution $p_{\\boldsymbol{\\theta}}(\\textbf{x})$ to the data distribution $p(\\textbf{x})$. In the following, we show that this is indirectly achieved by matching the posterior $p_{\\boldsymbol{\\theta}}(\\textbf{z}|\\textbf{x})$ to $p(\\textbf{z})$.\n\nFirst, the marginal is defined as\n\\begin{equation}\n    p_{\\boldsymbol{\\theta}}(\\textbf{x}) = \\int_{\\textbf{z}}p_{\\boldsymbol{\\theta}} (\\textbf{x}|\\textbf{z})p(\\textbf{z})d\\textbf{z} = E_{\\textbf{z}\\sim p(\\textbf{z})} p_{\\boldsymbol{\\theta}}(\\textbf{x}|\\textbf{z}).\n    \\label{eq:px}\n\\end{equation}\nMatching the model and data distributions requires maximizing the likelihood $\\Pi_{\\textbf{x}\\in \\mathcal{X}}p_{\\boldsymbol{\\theta}}(\\textbf{x})$ with respect to $\\boldsymbol{\\theta}$. This maximization problem, however, is often intractable due to the numerical integral and the complexity of the decoder network. Nonetheless, it is noted that for a given $\\textbf{x}$, most samples in the latent space will have $p_{\\boldsymbol{\\theta}} (\\textbf{x}|\\textbf{z}) \\approx 0$, as these latent samples do not generate outputs close to $\\textbf{x}$. This leads to the idea of introducing another distribution $q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})$ (the encoder) that takes an $\\textbf{x}$ and outputs $\\textbf{z}$ values that are likely to reproduce $\\textbf{x}$. Ideally, the space of $\\textbf{z}$ that are likely under $q$ will be much smaller than that under $p(\\textbf{z})$,  making the computation of $E_{\\textbf{z}\\sim q} p_{\\boldsymbol{\\theta}} (\\textbf{x}|\\textbf{z})$ relatively cheap. However, $E_{\\textbf{z}\\sim q} p_{\\boldsymbol{\\theta}} (\\textbf{x}|\\textbf{z})$ is not the same as $p(\\textbf{x})$, as $q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})$ does not necessarily match to $p(\\textbf{z})$. The relation between the two is derived below. \n\n\nWe start by deriving the Kullback-Leibler divergence between the encoded latent distribution $q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})$ and the posterior $p_{\\boldsymbol{\\theta}}(\\textbf{z}|\\textbf{x})$: \n\\begin{equation}\n\\begin{aligned}\n    D_{KL}\\left(q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})||p_{\\boldsymbol{\\theta}}(\\textbf{z}|\\textbf{x})\\right) & = \\int q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})\\log\\frac{q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})}{p_{\\boldsymbol{\\theta}}(\\textbf{z}|\\textbf{x})}d\\textbf{z}\\\\\n    & =-\\mathbb{E}_{\\textbf{z} \\sim q}[\\log p(\\textbf{z},\\textbf{x})]-\\mathbb{E}_{\\textbf{z} \\sim q}[-\\log q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})] + \\log p(\\textbf{x}).\n\\end{aligned}\n\\label{eq:KL}\n\\end{equation}\nBy rearranging Eq.~\\eqref{eq:KL}, we get\n\\begin{equation}\n    \\log p(\\textbf{x}) - D_{\\text{KL}}\\left(q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})||p_{\\boldsymbol{\\theta}}(\\textbf{z}|\\textbf{x})\\right) = \\mathbb{E}_{\\textbf{z} \\sim q}[\\log p_{\\boldsymbol{\\theta}}(\\textbf{x}|\\textbf{z})]\n    -D_{\\text{KL}}\\left(q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})||p(\\textbf{z})\\right).\n\\label{eq:vae}\n\\end{equation}\nNote that the right hand side of Eq.~\\eqref{eq:vae} can be maximized through stochastic gradient descent: The first term $\\mathbb{E}_{\\textbf{z} \\sim q}[\\log p_{\\boldsymbol{\\theta}(\\textbf{x}|\\textbf{z})}]$ measures the reconstruction error, i.e., the expected difference between an input $\\textbf{x}$ and its reconstructions drawn from the decoder conditioned on the latent variables, which are further drawn from the encoder conditioned on the input. The second term $D_{\\text{KL}}(q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})||p(\\textbf{z}))$ measures the difference between the encoded distribution of $\\textbf{z}$ and the modeled latent distribution $p(\\textbf{z})$. Since the encoding and decoding processes involved in these terms are modeled as feedforward networks, the gradients are readily available for backpropagation. Specifically, we model $q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x}) = \\mathcal{N}(\\textbf{z}|\\boldsymbol{\\mu}(\\textbf{x};\\boldsymbol{\\phi}),\\boldsymbol{\\Sigma}^2(\\textbf{x};\\boldsymbol{\\phi}))$ as a normal distribution parameterized by the mean $\\boldsymbol{\\mu}(\\textbf{x};\\boldsymbol{\\phi})$ and the diagonal variance matrix $\\boldsymbol{\\Sigma}^2(\\textbf{x};\\boldsymbol{\\phi})$. $\\boldsymbol{\\mu}$ and $\\boldsymbol{\\Sigma}$ are the outputs of the encoder network. We similarly model $p_{\\boldsymbol{\\theta}(\\textbf{x}|\\textbf{z})} = \\mathcal{N}(\\textbf{x}|\\textbf{f}(\\textbf{z};\\boldsymbol{\\theta}),\\sigma^2\\textbf{I})$ with mean $\\textbf{f}(\\textbf{z};\\boldsymbol{\\theta})$ and variance $\\sigma^2$. The function $\\textbf{f}(\\textbf{z};\\boldsymbol{\\theta})$ is the decoder network; $\\sigma$ determines the importance of the reconstruction of $\\textbf{x}$ during the training of a generative model, and is set to $1$ in the proposed model. \nThe prior of the latent distribution, $p(\\textbf{z})$, is often assumed to be standard normal. This is because the transformation from this simple distribution to the potentially highly nonlinear data distribution $p(\\textbf{x})$ can often be achieved by a sufficiently deep decoder. \n\n\nThe left hand side of Eq.~\\eqref{eq:vae} contains the objective $\\log p_{\\boldsymbol{\\theta}}(\\textbf{x})$ that we want to maximize, and the KL-divergence $D_{KL}(q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})||p_{\\boldsymbol{\\theta}}(\\textbf{z}|\\textbf{x}))>0$ that should ideally reach 0. Thus, minimizing \n\\begin{equation}\n    L(\\theta,\\phi,\\textbf{x}) = -\\mathbb{E}_{\\textbf{z} \\sim q}[\\log p_{\\boldsymbol{\\theta}}(\\textbf{x}|\\textbf{z})]\n    +D_{\\text{KL}}(q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x})||p(\\textbf{z}))\n\\end{equation}\nwill maximize a lower bound of $p(\\textbf{x})$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Prediction through Convolutional Neural Network (CNN)}\nWe build a predictive structure-property model using CNN (Sec.~\\ref{sec:model}). A CNN is often composed of one or more convolutional layers followed by fully connected layers. \nA convolutional layer consists of a set of image filters, each as a set of network edges connecting the input image to a hidden channel, see Fig.~\\ref{fig:CNN}. Scanning the input image with a filter requires the network weights to be shared, thus reducing the total number of weights to be trained. A max-pooling layer further reduces the dimension of a hidden layer~\\cite{szegedy2015going}.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{CNN.png}\n\\caption{Schematic plot for a CNN}\n\\label{fig:CNN}\n\\end{figure}\n\n\n\n\n\\paragraph{Style transfer (ST)}\n\\label{para:style_transfer}\nST was first introduced to integrate the content of one image with the style of another~\\cite{gatys2016image}. To perform ST, a ``style vector\" is first computed for an image, that consists of the variance-covariance matrices of the hidden states of a pre-trained CNN (we use a VGG Net~\\cite{simonyan2014very} in this paper) activated by the input image. A new image can then be created by minimizing the difference between its style vector and the target one. One can also preserve the image content, i.e., the hidden states of the deepest layers, by adding an additional loss on the content difference.   \n\n\n\nAs will be discussed in Sec.~\\ref{sec:model}, the proposed method extends ST to a generative setting, by directly adding a style (morphology) penalty to the training loss of a VAE. There are two major difference in comparison: First, the standard ST generates images by minimizing the style loss with respect to a high-dimensional image space, while our model is learned to directly generate style-consistent images, thus is significantly faster than standard ST; second, the standard ST uses a single style vector as a target, while our model can encode a distribution of styles in its latent space, and thus is more expressive as a generator. \n\n\n\n\n\\section{Proposed Models}\n\\label{sec:model}\n\n\\subsection{Network specifications}\n\n\\paragraph{The generative model}\nThe proposed generative model is composed of two networks, a VAE for image encoding and decoding, and an auxiliary network for computing the style vector ($\\textbf{G}$). See model summary in Fig.~\\ref{fig:VAE+CNN}a. For the VAE, its encoder has convolutional layers of a fixed filter size ($4\\times4$), max-pooling layers after each other convolutional layers, and two fully-connected layers of sizes 256 and 16, respectively. \nEach pooling layer is of stride 2, i.e., it is composed with filters of size $2\\times 2$ applied to the hidden layers, downsampling these layers by a factor of 2 along both width and height. Together, the encoder reduces the dimension of the input from $128\\times128$ to $16\\times1$. The architecture of the decoder is symmetric to that of the encoder.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{modelspecification.png}\n\\caption{(a) Encoder architecture in the VAE, decoder is mirrored from the encoder (b) CNN architecture}\n\\label{fig:VAE+CNN}\n\\end{figure}\n\n\n\\paragraph{The predictive model}\nA CNN is built to predict material properties from microstructures. A schematic of the predictive model is in Fig.~\\ref{fig:VAE+CNN}b. We use four convolutional layers with filter size $3\\times 3$,\nand max-pooling layers with stride 2. \n\n\n\n\n\\paragraph{The morphology style model}\nTo acquire a rich set of morphology patterns and make this acquisition process general, we propose to use a VGG network~\\cite{Simonyan14c} as the morphology model. A VGG is trained on over a million images~\\cite{imagenet_cvpr09} to learn a rich set of feature representations. The schematic for the proposed morphology-aware VAE is shown as Fig.~\\ref{fig:VAE+morhoplogy}.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{VAE+morhoplogy.png}\n\\caption{Schematic of the proposed morphology-aware VAE}\n\\label{fig:VAE+morhoplogy}\n\\end{figure}\n\n\\subsection{Model training}\n\n\\paragraph{The loss function of the generative model}\nTraining the generative model involves minimizing a loss function with four components: \\begin{equation}\n    L({\\boldsymbol{\\theta}},{\\boldsymbol{\\phi}};\\mathcal{X}) = L_{\\text{RECON}} + L_{\\text{KL}} + L_{\\text{ST}} + L_{\\text{MC}},\n    \\label{eq:obj}\n\\end{equation}\nwhere $L_{\\text{RECON}}+L_{\\text{KL}}$ are the standard reconstruction and KL divergence losses for VAE (see Sec.~\\ref{para:VAE}), $L_{\\text{ST}}$ is the style loss, and $L_{\\text{MC}}$ is an additional loss to prevent mode collapse (introduced below). \n\nThe reconstruction loss \n\\begin{equation}\n\\label{eq:recon}\nL_{\\text{RECON}}=-\\sum_{\\textbf{x}_i\\in \\mathcal{X}} \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\textbf{z}|\\textbf{x}_i)}[\\log p_{\\boldsymbol{\\theta}}(\\textbf{x}_i|\\textbf{z})]\n\\end{equation}\nmeasures the reconstruction error of all pairs of authentic microstructures ($\\textbf{x}_i\\in \\mathcal{X}$) and their reconstructions. \nThe KL divergence loss\n\\begin{equation}\n\\label{eq:kl}\nL_{\\text{KL}}=\\sum_{\\textbf{x}_i\\in \\mathcal{X}} D_{KL}(q_{\\phi}(\\textbf{z}|\\textbf{x}_i)|| p(\\textbf{z}))\n\\end{equation}\nmeasures the difference between the encoder distribution $q_{\\phi}(\\textbf{z}|\\textbf{x}_i)$ and the prior $p(\\textbf{z})$. \nThe style loss\n\\begin{equation}\nL_{\\text{ST}}=\\sum_{s=1}^N\\sum_{p=1}^N\\sum_{l} ||\\textbf{G}^{s,l}-\\textbf{G}^{p,l}||^2_{F}\n\\end{equation}\nmeasures the total style difference between two sets of $N$ images with indices $s$ and $p$, representing the generated and the authentic ones, respectively. Given a morphology style model, the Gram matrix $\\textbf{G}^{s,l}$ of image $s$ at the $l$th hidden layer is defined through elements $G_{i,j}^{s,l}=\\sum_k f_{i,k}^{s,l}f_{j,k}^{s,l}$, where feature element $f_{i,k}^{s,l}$ represents the hidden activation corresponding to the $i$th convolutional filter at spatial location $k$. $||\\cdot||_F$ is the Frobenius normal. We note that the definition of feature maps will affect the outcome of style transfer. In general, deeper layers capture styles at larger length scales. We use the first four layers of a standard VGG net. This definition of features is empirical.\n\nLastly, the mode collapse loss is introduced to prevent the model from producing clustered samples~\\cite{zhao2016energy}, by forcing the generated samples to be different in terms of their activations in the style network. The loss is defined as:\n\\begin{equation}\nL_{\\text{MC}}=\\frac{2}{N(N-1)}\\sum_{i=1}^N\\sum_{j=i+1}^N(\\frac{\\textbf{s}_i^T \\textbf{s}_j}{\\|\\textbf{s}_i\\|\\|\\textbf{s}_j\\|})^2,\n\\end{equation}\nwhere the style vector $\\textbf{s}$ contains the concatenated and vectorized Gram matrices as defined above. \n\n\n\\paragraph{The training of the generative model}\nThe generative model is trained to minimize the loss defined in Eq.~\\eqref{eq:obj} with respect to model parameters $\\boldsymbol{\\theta}$ and $\\boldsymbol{\\phi}$ through stochastic gradient descent. To approximate the gradient of the loss, $\\nabla_{\\boldsymbol{\\theta},\\boldsymbol{\\phi}} (L_{\\text{RECON}} + L_{\\text{KL}} + L_{\\text{ST}} + L_{\\text{MC}})$, we randomly sample 20 authentic material microstructures from $\\mathcal{X}$ and generate another 20 artificial microstructures in each training iteration. The artificial microstructures are needed to calculate $\\nabla_{\\boldsymbol{\\theta},\\boldsymbol{\\phi}} (L_{\\text{ST}} + L_{\\text{MC}})$.\nTo prevent neighbouring latent samples to be matched to drastically different style (morphology) targets, we propose to create artificial microstructures by the following procedure: We take each authentic sample $\\textbf{x}$ and pass it through the encoder to get $\\boldsymbol{\\mu}(\\textbf{x};\\boldsymbol{\\phi})$ and $\\boldsymbol{\\Sigma}(\\textbf{x};\\boldsymbol{\\phi})$. We then draw a random sample $\\textbf{z}$ from the latent space following the normal distribution $\\mathcal{N}(\\textbf{z}| \\boldsymbol{\\mu}(\\textbf{x};\\boldsymbol{\\phi}),\\boldsymbol{\\Sigma}(\\textbf{x};\\boldsymbol{\\phi}))$, and derive its decoded microstructure $\\textbf{x}' = f(\\textbf{z};\\boldsymbol{\\theta})$ to pair with $\\textbf{x}$. By matching the style of $\\textbf{x}$ and $\\textbf{x}'$, we ensure that samples close to each other in the latent space will have similar morphologies. We demonstrate the necessity of this treatment in Fig.~\\ref{fig:proposed_model_comparison}, where we show the generation results from an alternative method where microstructures from a standard normal distribution are randomly paired with the authentic ones for style loss calculation, leading to averaged morphology across all random generations (Fig.~\\ref{fig:proposed_model_comparison}b). \nWe use Adam optimizer~\\cite{kingma2014adam} for training, with 200K iterations and a learning rate of 0.001. \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=16cm]{proposed_model_comparison.png}\n\\caption{(a) Samples of authentic microstructures (b) Artificial microstructures when random samples are drawn from a standard normal distribution during the training (c) Artificial microstructures when random samples are drawn to pair with the authentic ones}\n\\label{fig:proposed_model_comparison}\n\\end{figure}\n\n\n\n\n\\paragraph{The training of the predictive model}\nWe divide the raw data into training, validation, and test sets. The loss function for training the predictive structure-property model is the mean square error of the predicted properties of the training data. Prediction performance of the model on the validation set is monitored to prevent the model from overfitting. The performance of the model is then measured by R-square value of the test data. We use Adam for training with a learning rate of 0.001.  \n\n\n\n\n\n\\section{Case Study and Results}\n\\label{sec:results}\nIn this section, we demonstrate key contributions of the proposed model through a case study: (1) By incorporating the style loss, the proposed generative model creates material microstructures with better visual and statistical similarity to the authentic ones than a state-of-the-art Markov random field (MRF) approach; and (2) we can improve the prediction of material properties more effectively by using the artificial data generated from the proposed model than by those generated from a MRF. The case study uses a set of sandstone microstructures and their properties, including Young's modulus, diffusivity, and fluid permeability. Details of this material system are introduced as follows.\n\n\\subsection{A case study on the sandstone system}\n\\label{sec:sandstone}\nSandstones are a class of important geological porous\nmaterials in petroleum engineering. In the simplest form, a\nmono-mineral sandstone is composed of a solid (rock) phase and a\npore (void) phase, both of which are typically percolating.\nExtensive research work has been carried out to model the\nmicrostructure and physical properties of such porous materials~\\cite{sahimi2011flow, radlinski2004angstrom, milliken2000brittle,\nblair1996using, coker1996morphology, antonellini1994petrophysical,\nal2005extraction, appoloni2002characterization, li2017accurate,\nhajizadeh2011multiple}. For example, a fast and independent\narchitecture of artificial neural network has been developed for\naccurately predicting fluid permeability~\\cite{tahmasebi2012fast}.\n\nWe will focus on three different material properties: Young's\nmodulus (stiffness) $E$, diffusivity $D$ and fluid permeability\n$k$. These properties are respectively sensitive to distinct\nmicrostructural features of the materials. In particular, $E$ is\nmainly determined by the morphology and volume fraction of the\nrock phase. The diffusivity $D$ is most sensitive to the local\npore size distribution. The fluid permeability $k$ depends on the\ndegree of connectivity of the pore phase as well as the pore\/rock\ninterface morphology. Thus, successfully improving the prediction\naccuracy for all three material properties poses a stringent test\nfor our morphology-aware generative model for effectively\npopulating the microstructure sample space.\n\n\n\\paragraph{Physics-based material property calculations}\nThe material properties used to train the network model are first\ncomputed using the effective medium theory~\\cite{torquato2013random}. The Young's modulus can be written as\n\\begin{equation}\nE = \\frac{9 K_e G_e}{3K_e + G_e},\n\\end{equation}\nwhere $K_e$ and $G_e$ are respectively the bulk and shear modulus\nof the materials. We use analytical approximations of $K_e$ and\n$G_e$ obtained by truncating the associated strong-contrast\nexpansions after the third order term~\\cite{torquato1997exact},\nwhich are respectively given by\n\\begin{equation}\n\\label{bulk} \\phi_2 \\frac{\\kappa_{21}}{\\kappa_{e1}} = 1 -\n\\frac{(d+2)(d-1)G_1\\kappa_{21}\\mu_{21}}{d(K_1 + 2G_1)} \\phi_1 \\chi\n\\end{equation}\nand\n\\begin{equation}\n\\label{shear} \\phi_2 \\frac{\\mu_{21}}{\\mu_{e1}} = 1-\n\\frac{2G_1\\kappa_{21}\\mu_{21}}{d(K_1 + G_1)}\\phi_1 \\chi -\n\\frac{2G_1\\kappa_{21}\\mu_{21}}{d(K_1 + 2G_1)}\\phi_1 \\chi -\n\\frac{1}{2d}[\\frac{dK_1 + (d-2)G_1}{K_1 + 2G_1}]^2\\mu_{21}^2\\phi_1\n\\eta_2,\n\\end{equation}\nwhere $\\phi_1$ and $\\phi_2$ are respectively the volume fraction of\nthe rock phase and void phase (i.e., porosity); $d$ is the spatial\ndimension of the material system; $K_p$ and $G_p$ are respectively\nthe bulk and shear modulus of phase $p$; and the scalar parameters\n$\\kappa_{pq}$ and $\\mu_{pq}$ ($p, q = 1, 2, e$) are respectively\nthe bulk and shear modulus polarizability, i.e,\n\\begin{equation}\n\\kappa_{pq} = \\frac{K_p - K_q}{K_q + \\frac{2(d-1)}{d}G_q}\n\\end{equation}\nand\n\\begin{equation}\n\\mu_{pq} = \\frac{G_p - G_q}{G_q}\\frac{1}{1+\\frac{\\frac{d}{2}K_q +\n\\frac{(d+1)(d-2)}{d}G_1}{K_q + 2G_q}}.\n\\end{equation}\nThe quantities $\\chi$ and $\\eta_2$ are the microstructural\nparameters associated with the void (pore) phase involving the\nthree-point correlation functions $S_3$ and two-point correlation\nfunctions $S_2$ of the pore phase, i.e.,\n\\begin{equation}\n\\label{chi} \\chi=\n\\frac{9}{2\\phi_1\\phi_2}\\int_0^{\\infty}\\frac{dr}{r}\\int_0^{\\infty}\\frac{ds}{s}\\int_{-1}^{1}d(\\cos\\theta)P_2(\\cos\\theta)[S_3(r,\ns, t)-\\frac{S_2(r)S_2(t)}{\\phi_2}]\n\\end{equation}\nand\n\\begin{equation}\n\\eta_2 = \\frac{5}{21}\\chi +\n\\frac{150}{7\\phi_1\\phi_2}\\int_0^{\\infty}\\frac{dr}{r}\\int_0^{\\infty}\\frac{ds}{s}\\int_{-1}^{1}d(\\cos\\theta)P_4(\\cos\\theta)[S_3(r,\ns, t)-\\frac{S_2(r)S_2(t)}{\\phi_2}],\n\\end{equation}\nwhere $t = (r^2 + s^2 - 2 rs \\cos\\theta)^{1\/2}$, and $P_2$ and\n$P_4$ are respectively the Legendre polynomials of order two and\nfour, i.e.,\n\\begin{equation}\nP_2(x) = \\frac{1}{2}(3x^2 - 1), \\quad P_4(x) =\n\\frac{1}{8}(35x^4-30x^2+3).\n\\end{equation}\nIn our system, the void phase possesses zero elastic moduli, i.e.,\n$K_2 = G_2 = 0$, and the moduli of the rock phase are respectively\n$K_1 = 4.3$ GPa and $G_2 = 3.8$ GPa. The three-point parameters\n$\\chi$ and $\\eta_2$ are computed by first evaluating the\ncorrelation functions $S_3$ and $S_2$ of the void phase from the\nmicrostructure data and then evaluating the integrals. In the\nreported results, the overall Young's modulus of the sandstone is\nre-scaled with respect to that of the rock phase.\n\nSimilar to the elastic moduli, we use an analytical approximation\n\\cite{torquato1985effective, jiao2012quantitative} obtained by\ntruncating the strong-contrast expansion of the effective\ndiffusion coefficient $D$ after the third order term:\n\\begin{equation}\n\\frac{D}{D_2} = \\frac{1 + 2\\phi_1\\beta_{12} - 2\\phi_2\n\\chi\\beta_{12}^2}{1- \\phi_1\\beta_{12}-2\\phi_2 \\chi \\beta_{12}^2},\n\\end{equation}\nwhere $\\phi_1$ and $\\phi_2$ are respectively the volume fraction\nof the rock and pore phases. The polarization parameter\n$\\beta_{12}$ is given by\n\\begin{equation}\n\\beta_{12} = \\frac{D_2 - D_1}{D_2 + 2D_1},\n\\end{equation}\nwhere $D_1$ and $D_2$ are respectively the diffusivity of the rock\nand pore phases. In our sandstone system, the rock diffusivity is\ntypically negligibly small compared with the pore phase diffusivity, and\nthus $D_1\/D_2 \\approx 0$. The three-point parameter $\\chi$ for the\nrock phase is given by Eq.~\\eqref{chi}, in which the correlation\nfunction $S_3$ and $S_2$ are computed from the microstructural\ndata for the rock phase (instead of the pore phase as in the case\nof elastic moduli). The computed $D$ values are normalized with\nrespect to $D_2$ in the reported data.\n\nFinally, the fluid permeability is computed using the\napproximation~\\cite{torquato1990rigorous}\n\\begin{equation}\nk = \\frac{2}{3(1-\\phi_2)^2}\\int_0^\\infty[S_2(r)-\\phi_2^2]rdr,\n\\end{equation}\nwhere $\\phi_2$ is the volume fraction of the pore phase and $S_2$\nis the associated two-point correlation function.\n\n\n\\subsection{Results}\n\n\\paragraph{The effect of style loss on material generation} \nIn Fig.~\\ref{fig:gen_compare}, we show that a standard VAE has undesirable generation performance with a small training set (200 samples), and that the incorporation of the style loss can effectively improve the generation quality under the same sample size. Further, the proposed approach also produces more plausible microstructures than the MRF method~\\cite{bostanabad2016stochastic}, as one can observe the scattering of unrealistically small particles of the ``stone'' phase from the latter.\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=16cm]{gen_compare.png}\n\\caption{(a) The authentic sandstone microstructure samples (b) Generations from a standard autoencoder (c) Generations from a standard VAE (d) Generations from the MRF method in \\cite{bostanabad2016stochastic} (e) Generations from the proposed method (f) An enlarged example of MRF generation with artificial defects}\n\\label{fig:gen_compare}\n\\end{figure}\n\n\n\\paragraph{Structure-property predictions}\nHere we investigate how the prediction accuracy of a structure-property model improves with the additional training data, acquired through the proposed method and the MRF.\nTo do so, we start by deriving baseline models (for all three structure-property mappings) using 100 data points for training and another 100 for validation. The baseline prediction performance is evaluated using a third set of 100 test data points. \nThe true property values are calculated following Sec.~\\ref{sec:sandstone}, and normalized for training.\n\nTwo sets of artificial microstructures, 1000 samples each, are created using the proposed method and the MRF, respectively. The true properties of these samples are computed. We then use these additional samples to improve the performance of the predictive models. Specifically, an increasing number of additional data points (from 50 to 1000) from the two sets are randomly added to the original training set to update the predictive model. We perform 10 independent random draws for each size of the additional data, and report the means and variances of the resultant test R-squares. The exception is with data size 1000, where all data points are used all together.\n\nResults are presented in Fig.~\\ref{fig:results} and summarized as follows. (1) Microstructures generated from the proposed method have property distributions similar to those of the authentic samples, while those generated from MRF have significantly smaller variances (Fig.~\\ref{fig:results}a). While the MRF method ensures variance in the microstructure space, as is evident from Fig.~\\ref{fig:gen_compare}, the lack of variance in the volume fraction of its generations may cause the observed small variances in the properties. It is worth noting that while a modified MRF approach that considers variance in volume fraction may improve its generation quality, our approach does not require human identification or design of such features, as variances in morphology are directly learned through the generative model. (2) The statistical difference in properties observed in (1) may explain the significant gap in the prediction performance of the resultant structure-property models, as seen in Fig.~\\ref{fig:results}b. \nThe narrow distribution of properties of the MRF method also explains the drop in model performance from 800 to 1000 samples: By feeding the network data from one distribution while asking it to predict on those drawn from another, we may encounter deteriorated performance even with an increased training data size.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=16cm]{results_new_model.png}\n\\caption{(a) Property distributions of microstructures generated from the proposed and the MRF~\\cite{bostanabad2016stochastic} models. (b) Test R-square values at an increasing number of additional data points}\n\\label{fig:results}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sec:discussion}\n\\paragraph{Utilities of the proposed method} The prposed generative model captures low-dimensional representations of data points distributed in high-dimensional spaces.\nThe model can be used to facilitate more cost-effective design of optimal and feasible microstructures by providing extra data to improve predictive structure-property models. This value could be particularly significant when the bottleneck of the design task is the high acquisition cost of microstructure data. It is also worth noting that the model may also help to reduce the cost of structure-property mappings, through its identification of low-dimensional material representations. For example, during the training of a predictive structure-property model, one may adopt an active learning strategy to only sample microstructures with high uncertainties in their property predictions, thus improving the predictive model cost-effectively. \nThe model and the learning algorithm are also generally applicable, as they are independent from any specific material system, except that minor model architecture tuning may be needed to incorporate different morphological complexities.\n\n\n\\subsection{Physics-based generative model}\nThe successful application of style loss in this paper inspires future investigation on whether physics-based loss can be applied to further improve the quality of microstructure generations through a generative model with limited data. Specifically, instead of using a style loss, we can measure the similarity between input and output microstructures based on their properties. For example, for microstructures that satisfy certain equilibrium conditions (governed by the underlying process-structure mapping), we can train the generative model to minimize the violation of these conditions, and thus enforcing the artificial samples to be physically meaningful.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nThis paper proposed a method for generating an arbitrary amount of artificial microstructure samples with low computation cost and a small amount of training samples. The key contribution is the incorporation of a style loss into the training to significantly improve the quality of the artificial microstructures. For verification, we applied the proposed method to generating sandstone samples. Results showed that our method is more data-efficient at improving the prediction performance of a structure-property mapping than a state-of-the-art Markov Random Field method. The findings from this paper inspired future investigation into physically meaningful generative models for accelerating microstructure-mediated material design.\n\n\n\n\n\\bibliographystyle{elsarticle-num}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nMultilevel Monte Carlo (MLMC) \\citep{Giles} is a technique that has\ngained significant popularity over the past decade. It is designed to\nproduce statistical estimators for discretized random variables at\nsignificantly lower computational costs than their Monte Carlo\ncounterparts for a fixed error. This is done by using a hierarchy of larger ensembles\nusing lower accuracy models, and smaller ensembles using higher\naccuracy models. For probabilistic forecasting, one can use this\nmultilevel technique to estimate statistics from a forecast\nprobability distribution, given some distribution of the initial conditions\nand\/or random forcing.\n\nIn the multilevel Monte Carlo framework,\none usually considers a particular statistic, such as evaluations of the cumulative distribution function (CDF) \\citep{Ritter, Wilson, Elfverson}, probability density function (PDF) \\citep{Bierig} or expected values \\citep{Giles,Cliffe}, selecting the ensemble sizes \/ finest level of resolution so that the overall multilevel estimator produces an efficient and\naccurate approximation. \n\nIn the case of ensemble forecasting, one\nusually wishes to compute many statistics from the same ensemble. \nThese approximations can be assessed using suitable verification\ntechniques. Verification tools used within ensemble forecasting\nusually work alongside observations of the process that one is\ninterested in forecasting and can help verify properties from\ncalibration to the sharpness of a forecast \\citep{Gneiting07}.\n\nGiven the multilevel hierarchy of ensembles from different resolutions\nthat form MLMC estimates of statistics, we would also like to evaluate\n\/ verify these ensembles in the same way; this is the subject of this\npaper. We propose a methodology to take observables of a univariate random variable, or scalar observables of a multidimensional random variable (such as a random field evaluated at a\npoint in space), from a\nmultilevel hierarchy of ensembles with varying resolutions and generate an accompanying single\nensemble forecast. Most of the standard techniques in the field of ensemble forecasting are limited to the univariate case; in the context of large dimensional models in weather and climate these are usually applied to scalar observables such as point values or integral quantities.\n\nThis single forecast is statistically consistent\nwith the multilevel estimate.  It can then\nbe used to verify the forecast from the original ensemble\nhierarchy using standard methods such as calibration tests.\n\nAn alternative approach to this could be approximating each verification or scoring measure, such as the calibration or sharpness, individually and directly from the multilevel hierarchy of ensembles. For example, one could use a MLMC approximation for the CDF \\citep{Ritter, Elfverson} to help compute a rank histogram to evaluate the forecast calibration.\nEach different MLMC approximation typically comes with a framework to implement it, such as\na smoothing scheme in the former of those two studies.\n\nHowever, by using the proposed methodology in this paper, one does not need a different multilevel approximation and framework for each individual scoring measure; instead any standard verification technique, such as the calibration or continuous ranked probability score (CRPS) \\citep{Gneiting07}, can be employed on this standard single ensemble forecast.\n\nTo generate this ensemble forecast, inverse transform sampling is used. The new single ensemble forecast preserves the unbiased approximation to the mean of the forecast distribution from the original multilevel estimator and forms consistent approximations to other statistics, such as higher moments.\n\nThis study proceeds as follows; an introduction to MLMC will be given in Section 2, then a simple method to find a consistent ensemble forecast from a MLMC approximation will be given in Section 3 alongside a corresponding verification technique for these ensemble forecasts. Finally, a conclusion follows.\n\n\\section{Multilevel Monte Carlo}\n\nMultilevel Monte Carlo \\citep{Giles} is primarily used as a computationally cheap alternative to an equivalent accuracy single level Monte Carlo estimator of statistics with respect to a probability distribution. Suppose one wishes to compute estimates to statistics of $f(X_{L,t})$, such as $\\mathbb{E}[f(X_{L,t})]$,\nwhere $X_{L,t}$ is a numerical approximation of our `forecast' random variable $X$\n(with discretization parameter $h_{L} \\propto M^{-L}$, $M>1$) at time $t \\geq 0$ and $f$ is some scalar observable function. Let $X^{i}_{L,t}$,\n$i=1,\\ldots,N$, be $N \\geq 1$ i.i.d. samples of the random variable\n$X_{L,t}$. Then an empirical approximation to the density of $X_{L,t}$ is \n\\begin{equation}\n\\pi_{L,t}^{MC}(x)=\\frac{1}{N}\\sum^{N}_{i=1}\\delta(x-X_{L,t}^{i}),\n\\label{equation:PDF}\n\\end{equation}\nwhere $\\delta$ is the Dirac delta function. One can then estimate statistics to this empirical distribution via the Monte Carlo method. For example, the standard estimator for $\\mathbb{E}[f(X_{L,t})]$ is given by\n\\begin{equation}\n\\bar{f}_{L,t}^{MC}=\\frac{1}{N}\\sum^{N}_{i=1}f(X_{L,t}^{i}).\n\\label{equation:MonteCarlo}\n\\end{equation}\nNow consider the multilevel framework, using $L+1$ ensembles $\\{X_{l-1,t}^{i},X_{l,t}^{i}\\}_{l=0,i=1}^{l=L,i=N_l}$ (with $X_{-1}^{i}=0$) of sizes $N_{l}$, to derive the equivalent MLMC approximation to $\\mathbb{E}[f(X_{L,t})]$,\n\\begin{equation}\n\\bar{f}_{L,t}=\\frac{1}{N_{0}}\\sum^{N_{0}}_{i=1}f(X_{0,t}^{i})+\\sum^{L}_{l=1}\\Big(\\frac{1}{N_{l}}\\sum^{N_{l}}_{i=1}(f(X_{l,t}^{i})-f(X_{l-1,t}^{i}))\\Big).\n\\label{equation:MLestimator}\n\\end{equation}\nTaking the telescoping sum of expectations,\n\\begin{equation}\n\\mathbb{E}[f(X_{L,t})]=\\mathbb{E}[f(X_{0,t})]+\\sum^{L}_{l=1}\\mathbb{E}[f(X_{l,t})]-\\mathbb{E}[f(X_{l-1,t})],\n\\label{equation:linearityexpectation}\n\\end{equation}\nand considering\n\\begin{equation}\n\\mathbb{E}[\\hat{f}_{l,t}]=\n\\begin{cases}\n\\mathbb{E}[f(X_{0,t})],& l=0, \\\\\n\\mathbb{E}[f(X_{l,t})]-\\mathbb{E}[f(X_{l-1,t})],& l>0 ,\n\\end{cases}\n\\label{equation:unbiased}\n\\end{equation}\nwhere\n\\begin{equation}\n\\hat{f}_{l,t}=\n\\begin{cases}\n\\sum^{N_{0}}_{i=1}\\frac{1}{N_{0}}f(X_{0,t}^{i}),& l=0, \\\\\n\\sum^{N_{l}}_{i=1}\\frac{1}{N_{l}}\\big(f(X_{l,t}^{i})-f(X_{l-1,t}^{i})\\big),& l>0,\n\\end{cases}\n\\label{equation:MLMC}\n\\end{equation}\none recovers $\\bar{f}_{L,t}$ as an unbiased approximation of $\\mathbb{E}[f(X_{L,t})]$. \nThe important thing to note here is that the fine (level $l$) and coarse\n(level $l-1$) samples in \nthe difference estimators, $\\hat{f}_{l,t}$, must be positively\ncorrelated for each $i$. This can be achieved by using the same random system input\n(e.g. initial conditions\/stochastic forcing) for each $i$ on both\nlevels. On the other hand, the samples in different ensembles must be uncorrelated. \nThe uses of the above framework are incredibly varied. One can even condition these multilevel estimators on observations using processes such as filtering \\citep{Jasra, Gregory, Gregory16}. In addition to this, there have been many other applications of MLMC, some of which are highlighted in the review \\cite{GilesReview}.\n\nGiven an optimal choice of $L$ and $N_{l}$, one can compute these estimators, with the same accuracy as their standard Monte Carlo counterparts, for significantly less computational expense. This works by noting that due to the correlation between the pairs of\nsamples in each difference estimator, the sample variance of $f(X_{l,t})-f(X_{l-1,t})$, denoted $V_{l}$, should decrease asymptotically with $l \\to \\infty$. If one desires the accuracy of\n$\\bar{f}_{L,t}$ to be\n\\begin{equation}\n\\mathbb{E}\\left(\\left(\\bar{f}_{L,t}-\\mathbb{E}[f_{L,t}]\\right)^{2}\\right) < \\epsilon^{2},\n\\end{equation}\nthen one can follow the algorithm in \\cite{Giles} to compute $\\bar{f}_{L,t}$ by updating, on-line\n(as you add additional samples),\nthe optimal sample sizes\n\\begin{equation}\nN_{l}=\\ceil*{2\\epsilon^{-2}\\left(V_{l}h_{l}\\right)\\left(\\sum_{n=0}^{L}\\sqrt{V_{n}\/h_{n}}\\right)},\n\\end{equation}\nwhilst increasing $L$ until $\\abs*{\\hat{f}_{L,t}}<\\frac{1}{\\sqrt{2}}(M-1)\\epsilon$. An estimated $V_{l}$ can be used in the optimal sample size formula.\nComputational cost reductions occur because, if $V_{l}$ decreases asymptotically with $l \\to \\infty$, then $N_{l}$ also does, leading to a trade-off between estimator variance and bias in each difference estimator. To conclude, we should have large ensembles for the lower levels, and smaller ensembles on the higher levels, given by asymptotically decreasing values of $V_{l}$.\n\n\nFor the full algorithm and corresponding theory, see \\cite{Giles, GilesReview}.\n\n\\section{Ensemble Forecasting}\n\nThis paper now proposes a method to generate a single ensemble forecast from the hierarchy of ensembles created from the MLMC method. Put simply, one can generate a large ensemble (much larger than the finest level ensemble) that represents the entire MLMC approximation to the forecast distribution. This is more useful for the verification of the hierarchy of ensembles rather than simply using standard verification techniques on the finest ensemble in this hierarchy. As mentioned in the previous section, the sample sizes, $N_{l}$, for the pairs of ensembles\non all levels decrease asymptotically, and thus the finest ensemble is the smallest ensemble in the hierarchy. Using the finest ensemble for the verification of the entire MLMC approximation of the forecast distribution would neglect the majority of samples, on lower levels, from which the approximation was composed.\n\nIn addition to this verification, given the statistical consistency of this ensemble forecast with the multilevel ensemble hierarchy, many statistics can be easily estimated via this ensemble. \n\n\\subsection{Multilevel Monte Carlo Ensemble Forecasts}\n\n\nNow assume $X_{l,t} \\in \\mathbb{R}$, and so if $X_{l,t}$ was multivariate in the section before, $X_{l,t}$ now represents $f(X_{l,t})$ , the scalar observable, such as $X_{l,t}$ evaluated at a point in space. Here we describe how to generate the single ensemble forecast of a scalar observable $X_{F,t}^{i}\\in \\mathbb{R}$, $i=1,...,N$ from the MLMC hierarchy of ensembles through inverse transform sampling. It is important to note that this ensemble does not contain i.i.d samples from the forecast distribution, instead they will simply be approximations to these samples. However, this single ensemble has the properties to form a consistent empirical estimate to the forecast distribution and associated distribution functions.\n\nFrom here onwards, we will assume that values of $N_{l}$ and $L$ have been either set or found, and that the\nhierarchy of ensembles $\\{X_{l-1,t}^{i},X_{l,t}^{i}\\}_{l=0,i=1}^{l=L,i=N_l}$, with $X_{-1,t}=0$, has been\ngenerated.\nPredominantly, this is because the framework that this paper presents is designed for evaluating any given MLMC\napproximation. Each approximation has a hierarchy of ensembles that use values of $N_{l}$ and $L$ that have been optimised around\nminimising the cost of that particular approximation. Each approximation typically comes with it's own algorithm to\nset-up these values. Thus, by making the aforementioned assumption we can keep this framework general to all approximations.\nIn addition to this, it is likely that in real forecasting practice\none would pick the desired maximum level $L$ and then set fixed values of $N_{l}$ based on the maximum computational\nexpense one can use on a particular level.\nThis way of choosing $N_{l}$ and $L$ is implemented in the numerical example later in the paper.\n\nInverse transform sampling is the process of evaluating an (approximation to the) inverse CDF, $F^{-1}(u)$, $u \\in [0,1]$, also known as the \\textit{quantile function}. In the case where the CDF, $F$, of a random variable is strictly increasing and absolutely continuous, there exists a unique value $x \\equiv F^{-1}(u)$ for which $F(x)=u$. This distribution must usually be estimated empirically. If the true CDF of the forecast distribution is known to be absolutely continuous and the samples are sorted to form order statistics, then some of these estimates have been shown to be consistent approximations to $F^{-1}(u)$ \\citep{Ma}. A very simple consistent estimate for an evaluation to the quantile function, of the distribution with CDF, $F$, using the (ascending) sorted samples $\\big\\{X^{i}\\big\\}_{i=1,...,N} \\sim F$, $X^{1} < X^{2} < .... < X^{N}$ is,\n\\begin{equation}\n\\hat{F}^{-1}(u)=X^{\\ceil{N \\times u}}.\n\\label{equation:SampleQuantileFunction}\n\\end{equation}\nHere, the estimate is a consistent one in the sense that it converges in probability to $F^{-1}(u)$ as $N \\to \\infty$.\nOne can use linear interpolation and extrapolation to smooth this consistent estimate. Other inconsistent techniques include fitting a parametric distribution to the ensemble, such as a Gaussian, and sampling from a closed form quantile function (e.g. $\\Phi$ for a Gaussian distribution) for that distribution. In all cases, when the empirical quantile function is evaluated with i.i.d uniform samples $u \\in [0,1]$, approximations to samples of $X$ can be generated.\n\nThe use of inverse transform sampling alongside MLMC was first suggested in \\cite{GilesBook}. Here it was proposed to be used to minimise the discrete Wasserstein distance between the two paired ensembles in each difference estimator within (\\ref{equation:MLestimator}) and thus positively couple them. Instead, here we will use inverse transform sampling in the context of a MLMC approximation to the quantile function of the forecast distribution,\n\\begin{equation}\n\\bar{F}_{L,t}^{-1}(u) = R(X)_{0,t}^{\\ceil{N_{0} \\times u}}+\\sum^{L}_{l=1}\\big(R(X)_{l,t}^{\\ceil{N_{l} \\times u}}-R(X)_{l-1,t}^{\\ceil{N_{l} \\times u}}\\big),\n\\label{equation:MLestimatorquantile}\n\\end{equation}\nwhere $R(X)^{i}_{l}$ is the $i'th$ order statistic of $X_{l}$, so that $R(X)^{1}_{l}<R(X)^{2}_{l}<...<R(X)^{N_{l}}_{l}$. \nNote that there is not an exact cancellation in expected values of the above estimator terms, as in\nthe telescoping sum of expectations in (\\ref{equation:linearityexpectation}), as\nthe individual approximations on each level are not unbiased, only consistent in the limit of $N_{l} \\to \\infty$.\nThe following algorithm demonstrates how to generate an ensemble $\\big\\{X_{F,t}^{i}\\big\\}_{i=1,...,N}$ of arbitrary size $N$, approximating samples of $X_{L,t}$.\n\n\\BEGINALG\n\\Procedure{}{}\n\\For{$l=0,...,L$}\n\\If{$l=0$}\n\\State Sort $X_{0,t}^{j}$, $j=1,...,N_{0}$, so that $R(X)_{0,t}^{1}<R(X)_{0,t}^{2}<...<R(X)_{0,t}^{N_{0}}$\n\\Else\n\\State Sort $X_{l,t}^{j}$, $X_{l-1,t}^{j}$, $j=1,...,N_{l}$, so that $R(X)_{l,t}^{1}<R(X)_{l,t}^{2}<...<R(X)_{l,t}^{N_{l}}$ and $R(X)_{l-1,t}^{1}<R(X)_{l-1,t}^{2}<...<R(X)_{l-1,t}^{N_{l}}$\n\\EndIf\n\\EndFor\n\\For{$i=1,...,N$}\n\\State Set $X_{F,t}^{i}=0$\n\\State Sample $u^{i} \\sim U[0,1]$\n\\For{$l=0,...,L$}\n\\If{$l=0$}\n\\State $X_{F,t}^{i}+=R(X)_{0,t}^{\\ceil{N_{0} \\times u^{i}}}$\n\\Else\n\\State $X_{F,t}^{i}+=R(X)_{l,t}^{\\ceil{N_{l} \\times u^{i}}}-R(X)_{l-1,t}^{\\ceil{N_{l}\\times u^{i}}}$\n\\EndIf\n\\EndFor\n\\EndFor\n\\EndProcedure\n\\end{algorithmic}\n\n\nNote that these $X_{F,t}^{i}$ are not samples from $X_{L,t}$, they are only consistent approximations to the evaluations of $F^{-1}_{L,t}(u)$ for a particular $u$. More specifically, for a random uniform sample $u \\sim U[0,1]$, we have\n\\begin{equation}\nx=\\bar{F}^{-1}_{L,t}(u),\n\\end{equation}\nand as $N_{l} \\to \\infty$ for all $l$,\n\\begin{equation}\nx \\xrightarrow{p} F^{-1}_{L,t}(u),\n\\end{equation}\nwhere $N_{l}$ are the number of samples used in each difference estimator in (\\ref{equation:MLestimatorquantile}). Then in this limit, $x$ converges in probability to a sample from the forecast distribution on the finest level, i.e. $x \\sim X_{L,t}$. Therefore any statistical estimate using these samples is a consistent one within this limit.\n\nThe single ensemble $\\{X_{F,t}^{i}\\}_{i=1,...,N}$ can form valid and consistent approximations to statistics of the forecast distribution. For example, the empirical, consistent, CDF of this ensemble forecast found from the MLMC approximation to the forecast distribution is,\n\\begin{equation}\n\\hat{F}_{X_{F,t}}(x)=\\frac{1}{N}\\sum^{N}_{i=1}\\mathbb{I}_{X_{F,t}^{i} \\leq x}\n\\end{equation}\nwhere $\\mathbb{I}$ is the indicator function. Clearly this is non-decreasing for continuous $X_{F,t}$ and has the support of $[0,1]$.\n\nOne assumes that in practice the computational effort of evaluating the above function a large number of times to generate the ensemble $\\big\\{X_{F,t}^{i}\\big\\}_{i=1,...,N}$ is negligible in comparison to the expense of generating the original samples on all of the different levels. Thus, the method seems likely to be admissible even when $N$ is much larger than $N_{0}$. Having said this, it makes sense here to set $N \\propto N_{0}$ so that both aspects of the approximation (inverse CDF estimator and the ensemble forecast) converge in probability simultaneously. We take $N=\\alpha N_{0}$ with $\\alpha \\in \\mathbb{Z}$, $\\alpha \\geq 1$ for simplicity.\n\nThe proposed ensemble forecast also preserves the unbiasedness of the approximation to the first moment of the forecast distribution from the original MLMC approximation. To show this let $\\bar{X}_{F,t}=\\frac{1}{\\alpha N_{0}}\\sum^{\\alpha N_{0}}_{i=1} X_{F,t}^{i}$ be the sample mean of the ensemble forecast from the multilevel hierarchy of ensembles. Then,\n\n\\begin{equation}\n\\begin{split}\n\\bar{X}_{F,t}&=\\frac{1}{\\alpha N_{0}}\\sum^{\\alpha N_{0}}_{i=1} X_{F,t}^{i}\\\\\n\\quad &=\\Big(\\frac{1}{\\alpha N_{0}}\\sum^{\\alpha N_{0}}_{i=1}\\hat{F}^{-1}_{0,t}(u^{i})\\Big) +\\\\\n\\qquad &\\sum^{L}_{l=1}\\bigg(\\Big(\\frac{1}{\\alpha N_{0}}\\sum^{\\alpha N_{0}}_{i=1}\\hat{F}^{-1}_{l,t}(u^{i})\\Big) -\\Big(\\frac{1}{\\alpha N_{0}}\\sum^{\\alpha N_{0}}_{i=1}\\hat{F}^{-1}_{l-1,t}(u^{i})\\Big)\\bigg) , \\\\\n\\quad &=\\Big(\\frac{1}{\\alpha N_{0}}\\sum^{\\alpha N_{0}}_{i=1}X_{0, t}^{\\ceil*{N_{0} \\times u^{i}}}\\Big) +\\\\\n\\quad &\\sum^{L}_{l=1}\\bigg(\\frac{1}{\\alpha N_{0}}\\sum^{\\alpha N_{0}}_{i=1}\\Big(X_{l, t}^{\\ceil*{N_{l} \\times u^{i}}} - X_{l-1, t}^{\\ceil*{N_{l} \\times u^{i}}}\\Big)\\bigg),\n\\end{split}\n\\end{equation}\n\n\n\nand given that $u^{i}$ are i.i.d. draws of the uniform distribution $Unif[0,1]$, $i=1,...,\\alpha N_{0}$ then\n\n\\begin{equation}\n\\begin{split}\n\\mathbb{E}[\\bar{X}_{F,t}]&=\\Big(\\frac{1}{\\alpha N_{0}}\\sum^{\\alpha N_{0}}_{i=1} \\mathbb{E}[X_{0,t}]\\Big) +\\\\\n\\qquad &\\sum^{L}_{l=1}\\bigg(\\frac{1}{\\alpha N_{0}}\\sum^{\\alpha N_{0}}_{i=1}\\Big(\\mathbb{E}[X_{l,t}] - \\mathbb{E}[X_{l-1,t}]\\Big) \\bigg)\\\\\n\\quad &=\\mathbb{E}[X_{0,t}]+\\sum^{L}_{l=1}\\mathbb{E}[X_{l,t}-X_{l-1,t}]=\\mathbb{E}[X_{L,t}].\n\\end{split}\n\\end{equation}\n\n\n\\subsection{Assessing the Calibration of Multilevel Monte Carlo Ensemble Forecasts}\n\nEvaluating the ensembles used in ensemble forecasts is very important in checking the predictive value of the forecast. The remainder of this paper concentrates on a method of evaluating the calibration of forecasts from the MLMC approximations to the forecast distribution, directly via the single ensemble forecast found in the previous section: the Probability Integral Transform Histogram. This technique uses observations from the target distribution to evaluate ensemble forecasts. Calibration is the measure of whether the observations are indistinguisable from the samples of the ensemble forecast distribution \\citep{Carney}. This is a quality of the empirical forecast distribution that is possibly disregarded if one were to simply study errors of point statistical estimators.\n\nConsider the target distribution, $Y_{obs,t_{k}}$, behind the observed process, where partial observations $y_{obs,t_{k}}$, are taken from a single realisation of this process at times $t_{k}$, $k \\in [0,N_{y}]$, $t_{0}=0$, $t_{N_{y}}=T$. Clearly, our aim would be to use a forecast distribution associated with the random variable $X_{t_{k}}=Y_{obs,t_{k}}$, however in many real-world scenarios, $Y_{obs,t_{k}}$ is unknown. Therefore verification techniques are used to rank forecasts on their similarity to the observed process, with the aim of finding the best forecast \/ model that derived them. The case of $X_{t_{k}}=Y_{obs,t_{k}}$ is known as the random variable with associated forecast distribution from the perfect model.\n\n\\subsubsection{Probability Integral Transform Histogram}\n\nThe Probability Integral Transform (PIT) histogram is used to determine the uniformity of the observations with respect to the (empirical) CDF of the ensemble, and thus the calibration of the forecast distribution with respect to the target distribution. One can define a random variable $R \\sim F_{L,t_{k}}(Y_{obs,t_{k}})$, the Probability Integral Transform. Then samples of $R$ are given by,\n\\begin{equation}\nr_{t_{k}}=F_{L,t_{k}}(y_{obs,t_{k}}).\n\\end{equation}\nThe forecast distribution is said to be calibrated with respect to the target distribution if $R \\sim Unif[0,1]$, and so a histogram of $r_{t_{k}}$ would be relatively flat. Using the MLMC approximation to the forecast distribution, define the associated multilevel empirical PIT samples,\n\\begin{equation}\n\\hat{r}_{t_{k}}=\\frac{1}{N}\\sum^{N}_{i=1}\\mathbb{I}_{X_{F,t_{k}}^{i} \\leq y_{obs,t_{k}}}\n\\end{equation}\nwhere $X_{F,t_{k}}^{i}$ are an arbitrary $N$ members of the ensemble forecast from the multilevel hierarchy of ensembles at time $t_{k}$ using the aforementioned inverse transform sampling method. This is simply the empirical cumulative distribution function of the $N$ ensemble forecast members $X_{F,t_{k}}^{i}$. Here, given that we set $N \\propto N_{0}$, then in the limit of $N_{l} \\to \\infty$, for all $l=0,...,L$,\n\\begin{equation}\nX_{F,t_{k}} \\sim F_{L,t_{k}}^{-1},\n\\end{equation}\nand thus $F_{L,t_{k}}(X_{F,t_{k}}) \\sim U[0,1]$. By considering this, we have a consistent estimate to the PIT sample $r_{t_{k}}$, when concentrating on the limit of $N \\propto N_{0} \\to \\infty$. One can find the frequency ($H_{i}$, $i=1,...,B$) of $B$ evenly spaced bins in a histogram of these samples by:\n\\bigskip\n\\begin{enumerate}\n\\item Set $H_{i}=0$, for $i=1,...,B$.\n\\item For each $k=1,...,N_{y}$, find the $i=1,...,B$ in which $\\frac{i-1}{B} \\leq \\hat{r}_{t_{k}} \\leq \\frac{i}{B}$ and set $H_{i}=H_{i}+1$.\n\\end{enumerate}\n\\bigskip\nThis histogram will be refered to as the multilevel PIT histogram (MLPIT) for the remainder of this paper. The MLMC approximation that derives the ensemble forecast $\\{X_{F,t_{k}}^{i}\\}_{i=1,...,N}$ can then be described as calibrated with respect to the target distribution if $H_{i} \\approx \\frac{N_{y}}{B}$ for each $i=1,...,B$. Thus, this can be used to test the variance and biasedness of the ensembles with respect to the target distribution. If the histogram is convex then the ensembles are said to be overdispersed, whereas if it is concave, then the ensembles are said to be underdispersed, and if it is skewed then there exists a bias in the ensembles \\citep{Carney}. This is therefore a very appropriate way to clarify if there is any additional bias from the cancellation of intermediate estimators in a MLMC approximation, thus negating the telescoping sum of expectations in (\\ref{equation:linearityexpectation}), although this is not demonstrated in this paper.\n\n\\bigskip\n\n\\textit{\\textbf{Example:}}\nThe following linear mean reverting OU process, $X_{t} \\in \\mathbb{R}$,\n\\begin{equation}\n\\label{eq:OU}\ndX_{t}=\\alpha(\\mu-X_{t})dt + \\sigma^{2}d W_{t} ,\n\\end{equation}\nover time time interval $t \\in [0,T]$, where $W_{t}$ is a univariate Brownian Motion, will be used alongside pre-defined scenarios of calibration for a MLMC approximation to the forecast distribution to provide a demonstration of the proposed method. We let the observations come from the above model, discretized with timestep $h=2^{-5}$, with $\\alpha=0.1$, $\\mu=0$ and $\\sigma^{2}=0.1$.\nIn this example, an Euler-Maruyama numerical scheme will be used to discretize\nthe OU process.\nTo frame this problem in a likely forecasting setting, we first choose a fixed finest resolution\nthat we desire, $L=4$ and so $l \\in [0, 4]$. A maximum computational expense that we are allowed to use on propagating the entirity of samples in each level of the ensemble hierarchy, $C_{max}=1.536 \\times 10^{7}$, is then set. The cost of each sample in $l$'th difference estimator is $\\left(Th_{l}^{-1}(1 + 1\/2)\\right)$ (as all but the first difference estimators in (\\ref{equation:MLestimator}) require\ncoarse and fine time-steps of the discretization) where $h_{l}=2^{-1-l}$ and so $N_{l}$ is given by\n\\begin{equation}\n\\begin{split}\nN_{l} &= \\floor*{\\frac{C_{max}}{T\\left(h_{l}^{-1}\\left(1 + 1\/2\\right)\\right)}}\\\\\n\\quad &= \\floor*{\\left(\\frac{2}{3}\\right)C_{max}T^{-1}h_{l}}.\n\\end{split}\n\\end{equation}\nThis corresponds to $N_{0} = 2^{7}$.\nThe arbitrary number of samples $X_{F}^{i}$ to draw from the MLMC approximation to the inverse CDF is set to $N=8N_{0}=2^{10}$.\n\nPairs of samples from coarse and fine ensembles in each difference estimator in (\\ref{equation:MLestimator}) are positively coupled by using the same underlying Brownian Motion, as in \\cite{Giles}.\nThe models are run over times $t \\in [0,40000]$ (the long run time is to give the stationary distributions a chance to be simulated), and observations are collected at $t_{k}=k$, $k \\in [1,40000]$. At each of these times, a single ensemble forecast is generated from the hierarchy of ensembles that build up the MLMC approximation to the forecast distribution, and is used to verify the calibration of the approximation. Model parameters for four experimental setups are given as follows: $\\alpha = 0.1$, $\\sigma^{2}=0.1$, $\\mu=0$ for the calibrated scenario, $\\alpha = 0.1$, $\\sigma^{2}=0.02$, $\\mu=0$ for the underdispersed scenario, $\\alpha = 0.1$, $\\sigma^{2}=0.5$, $\\mu=0$ for the overdispersed scenario and $\\alpha = 0.4$, $\\sigma^{2}=0.1$, $\\mu=0.2$ for the biased scenario.\n \n \\iffalse\n \n\\begin{center}\n\\begin{tabular}{| c | c | c | c |}\n \\hline\n Scenario Name & $\\alpha$ & $\\sigma^{2}$ & $\\mu$ \\\\\n \\hline\n Calibrated  & 0.1  & 0.1  & 0\\\\\n \\hline\n Underdispersed  & 0.1  & 0.02 & 0   \\\\\n \\hline\n Overdispersed  & 0.1  & 0.5  & 0\\\\\n \\hline\n Biased  & 0.4  & 0.1  & 0.2\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\n\\fi\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=94mm]{EnsembleForecastPIT}\n  \\caption{\\textit{Multilevel Probability Integral Transform Histograms, using the ensemble $\\big\\{X_{F}^{i}\\big\\}_{i=1,...,N}$, of the linear OU process for the four different calibration scenarios. The solid line on the Biased scenario plot shows a smoothed kernel of the PIT histogram generated from the actual stationary forecast and target distributions.}}\n  \\label{figure:MLPIT_Experiments_LinearSystem}\n\\end{figure}\n\n\nThis setup allows us to establish that the correct calibration behaviour is being shown by the Multilevel PIT histogram for each of the scenarios, however we will also compare this to the PIT histogram using just the finest ensemble, although this isn't the primary goal of the section. Figures \\ref{figure:MLPIT_Experiments_LinearSystem} and \\ref{figure:PIT_Experiments_LinearSystem} show the MLPIT and PIT histograms respectively for the four scenarios of calibration listed above. Due to the small number of samples in the finest ensemble, the PIT histogram can only represent a very small number of bins of probability. Both show similar general behaviour for the cases above. \n\n\nWe can derive the stationary distribution to both the forecast distribution and the target distribution from the model specifications above from the Fokker-Planck equation corresponding to \\eqref{eq:OU}. One notes that the stationary forecast distribution using the Biased scenario model above is given by $f \\sim N\\big(0.2,\\frac{1}{8}\\big)$ and the stationary target distribution is given by $y \\sim N\\big(0,\\frac{1}{2}\\big)$ (as the general form is $\\sim N\\big(\\mu,\\frac{\\sigma^{2}}{2\\alpha}\\big)$). Thus the actual PIT histogram can be generated by taking an arbitrarily large number of samples of $F(y)$, where $F$ is the CDF of $f$. A smoothed density kernel of this histogram is superimposed on the corresponding empirical PIT histograms for the single level and multilevel approximations. The empirical histograms approximately match this, however, due to the lack of samples in the finest ensembles, the single level histogram is not as clear to the type or magnitude of bias as shown by the multilevel PIT histogram. This is due to the lack of probability bins in a small, single finest ensemble ($N_{L}+1$), and one would still suffer from similar problems if using interpolation techniques in between the limited number of samples of this ensemble. The MLMC approximations of the forecast distributions and associated histograms are numerically biased (proportional to the finest timestep) from this exact PIT histogram due to the use of a numerical discretization, and so are expected to be slightly different. Despite this, one can clearly interpret the calibration and identify the extent and type of such bias in the MLMC approximations to forecast distributions with more clarity using the multilevel PIT histogram technique proposed here than using standard methods with the small finest ensemble.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=94mm]{FinestEnsemblePIT}\n  \\caption{\\textit{Probability Integral Transform Histograms, using just the finest ensemble $\\big\\{X_{L}^{i}\\big\\}_{i=1,...,N_{L}}$, of the linear OU process for the four different calibration scenarios. The solid line on the Biased scenario plot shows a smoothed kernel of the PIT histogram generated from the actual stationary forecast and target distributions.}}\n  \\label{figure:PIT_Experiments_LinearSystem}\n\\end{figure}\n\n\n\\section{Conclusion and Outlook}\n\nThis work has discussed the benefits of generating an ensemble forecast from Multilevel Monte Carlo (MLMC) approximations to statistics of random variables representing forecast distributions. The proposed procedure to do this is simple and easily implemented. The calibration of this ensemble forecast has also been examined. Ensemble forecasts provide a simple methodology of deriving empirical estimates to associated distribution functions.  The ensemble hierarchy that forms the computationally efficient MLMC approximations to an arbitrary statistic of the forecast distribution is assumed to have already been generated in preparation for forecasting. It is anticipated that in real forecasting practice, this hierarchy of ensembles would simply be generated by using the maximum ensemble sizes affordable at each level of resolution. The ensemble forecast calibration verification technique takes the entire multilevel hierarchy into account when using the proposed methodology.\n\nCalibration of this ensemble forecast is assessed using the Probability Integral Transform histogram after this single ensemble is generated from the ensemble hierarchy. Thus we have stated what it means for a MLMC approximation to be calibrated with respect to a target distribution. This can be used to evaluate many properties of a MLMC approximation to a forecast distribution including biases (and their type) from intermediate terms in the MLMC telescoping sum of estimators, variances of these approximation and even potentially distribution multimodal feature detection. \n\n\\ack Alastair Gregory was supported by the Science and Solutions to a Changing Planet DTP and the Natural Environment Research Council. He was also supported by the Mathematics of Planet Earth CDT. The authors of this paper would like to thank Chris Ferro (Exeter) for the range of helpful conversations and advice he gave in the development of this research. Thanks also goes to the reviewers of this paper for their constructive feedback on the manuscript.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe field of cosmology has always relied on the cosmological\nprinciple: the Universe is homogeneous and isotropic. \nThis principle has been used to construct the dynamical models with\nwhich we understand and interpret the evolution of the Universe; and in the\ncontext of this principle we define and measure cosmological parameters.\nHowever, as cosmological measurements reach percent-level precision, \nthe effects of inhomogeneities in the Universe become increasingly\nimportant when comparing observations to models. \nThe field of cosmology has typically studied these effects using\na perturbative approach, with inhomogeneities described and modeled by\na linearized-gravity approximation \\cite{Bernardeau:2001qr,Adamek:2015eda}.\nAdditionally, semi-analytic models have been used to study effects of inhomogeneities on\nobservables in specific spacetimes, such as Swiss Cheese or Lindquist-Wheeler\nmetrics \\cite{Fleury:2013sna,Fleury:2014gha,Bolejko:2012ue,Clifton:2011mt}.\n\nJust a short time ago, the dynamics of cosmological dust spacetimes were examined\nin full 3+1-dimensional general relativity (GR) for the first time by solving the Einstein\nfield equations numerically \\cite{Giblin:2015vwq,Mertens:2015ttp,Bentivegna:2015flc}.\nThese studies and others such as \\cite{Yoo:2013yea,Bentivegna:2013jta,Yoo:2014boa} have, \nso far, focused on mathematical and numerical properties of the spacetimes,\nrather than the impact of inhomogeneities on cosmological observables. \nAn open question therefore remained: can we compute predictions for observables\nand demonstrate the necessity and power of these numerical simulations?\n\nIn this work, we present the first calculation of observable quantities in cosmology\nusing the full, unconstrained framework of numerical relativity for a pure ``dust''\nspacetime containing inhomogeneities on large scales -- a framework consistent\nwith the work presented in \\cite{Giblin:2015vwq,Mertens:2015ttp,Bentivegna:2015flc}.\nWe follow light beams along null geodesics through a toy universe and integrate the\noptical scalar equations along these geodesics, garnering information about angular\ndiameter distances and photon redshifts as the universe evolves. Using these\nobservables, we construct the resulting Hubble diagrams, plotting the distance\nmodulus versus redshift, and compare the results to the\nFriedman-Lemaitre-Robertson-Walker (FLRW) model.\n\nWe thus extend the previous numerical studies in GR by examining the properties of\nphotons traversing these spacetimes. A schematic of this scenario is depicted in\nFig.~\\ref{raytracing}. In general, we find good agreement between the averaged\nobservables and corresponding FLRW model; however, there are departures along\nindividual paths, and when observers or sources are located in under-densitites\nor over-densities. \n\nThe demonstration of this method by way of drawing Hubble diagrams \nis the main result of this work. We explore the ability\n\\begin{figure}[hb!]\n  \\centering\n    \\includegraphics[width=0.45\\textwidth]{raytrace_diagram.png}\n  \\caption{\\label{raytracing} A depiction of raytracing through a spacetime with\n  density fluctuations. Shades of lighter gray represent denser regions of\n  space, and darker colors represent less dense regions. In a time-reversed\n  sense, light rays (yellow) originate from an observer located in the volume\n  and propagate outward along null geodesics.}\n\\end{figure}\nof our code to resolve effects that manifest themselves in Hubble diagrams\ndue to observers being located in over-dense or under-dense regions\n\\cite{Rees:1968zza,GarciaBellido:2008nz,Biswas:2010xm}.\nWe additionally demonstrate the ability of this method to examine the\nrelationship between different distance measures, such as luminosity distance\nand observed angular extent, probing discrepancies that might arise due to \ninhomogeneities. While our model attempts to mimic our Universe on large\nscales, we do not yet resolve small-scale structure, and that the amplitude\nof fluctuations is not observationally motivated. Nevertheless, we hope to\nbe able to model these more realistically in the future.\n\nWe begin in Section~\\ref{bssn_eoms} by briefly reviewing the formulation of\nnumerical relativity used in this work, and describing, in\nSections~\\ref{DA_eom} and ~\\ref{geodesic_eom}, the formalism we use to\nintegrate photon geodesics. We then provide details of the numerical technique\nused to integrate these equations in Section~\\ref{num_ev}.\nFinally, in Section~\\ref{results_sec} we present results\nfrom a series of cosmologically-motivated simulations, and remark\non Etherington's reciprocity relation and the ability of our\ntechnique to probe outstanding questions in cosmology.\n\n\\section{Evolution of observables in a general spacetime}\n\nNumerical relativity has seen a number of breakthroughs in the past two decades, \nnotably the development of formulations capable of stable long-term evolution. \nWe take advantage of these advancements, and once again \\cite{Mertens:2015ttp}\napply them in a cosmological setting. We also write the geodesic equations and\noptical-scalar equations in a form suitable for integration in our numerical\nspacetime.\n\n\\subsection{The BSSN formulation} \\label{bssn_eoms}\n\nThe BSSN formulation \\cite{Baumgarte:1998te,Shibata:1995we} is a variant of a\n$3+1$ decomposition with important modifications for improving the numerical\nstability of the system. For a review or for further details, see eg.\n\\cite{Cardoso:2014uka,BaumgarteShapiroBook}. In\nthis formulation, the metric is decomposed as\n\\begin{eqnarray}\ng_{\\mu\\nu} & = & \\left(\\begin{array}{cc}\n-\\alpha^{2}+\\beta_{l}\\beta^{l} & \\beta_{i}\\\\\n\\beta_{j} & e^{4\\phi}\\bar{\\gamma}_{ij}\n\\end{array}\\right)\\,,\n\\end{eqnarray}\nwhere $\\gamma_{ij}=e^{4\\phi}\\bar{\\gamma}_{ij}$ is the spatial metric with\n$\\det(\\bar{\\gamma}_{ij})=1$, and $\\alpha$ and $\\beta^{i}$ are the lapse and\nshift respectively. The metric obeys the BSSN evolution equations,\n\\begin{eqnarray}\n\\label{BSSN_equs}\n\\partial_{t}\\phi & = & -\\frac{1}{6}\\alpha K+\\beta^{i}\\partial_{i}\\phi+\\frac{1}{6}\\partial_{i}\\beta^{i}\\\\\n\\partial_{t}\\bar{\\gamma}_{ij} & = & -2\\alpha\\tilde{A}_{ij}+\\beta^{k}\\partial_{k}\\bar{\\gamma}_{ij}+\\bar{\\gamma}_{ik}\\partial_{j}\\beta^{k} \\nonumber\\\\\n& & +\\bar{\\gamma}_{kj}\\partial_{i}\\beta^{k}-\\frac{2}{3}\\bar{\\gamma}_{ij}\\partial_{k}\\beta^{k}\\\\\n\\partial_{t}K & = & -\\gamma^{ij}D_{j}D_{i}\\alpha+\\alpha(\\tilde{A}_{ij}\\tilde{A}^{ij}+\\frac{1}{3}K^{2})\\nonumber\\\\ \n & & +4\\pi\\alpha(\\rho+S)+\\beta^{i}\\partial_{i}K\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_{t}\\tilde{A}_{ij} & = & e^{-4\\phi}(-(D_{i}D_{j}\\alpha)+\\alpha(R_{ij}-8\\pi S_{ij}))^{TF}\\nonumber\\\\ \n & & +\\alpha(K\\tilde{A}_{ij}-2\\tilde{A}_{il}\\tilde{A}_{j}^{l}) +\\beta^{k}\\partial_{k}\\tilde{A}_{ij} \\nonumber \\\\\n &  & +\\tilde{A}_{ik}\\partial_{j}\\beta^{k}+\\tilde{A}_{kj}\\partial_{i}\\beta^{k}-\\frac{2}{3}\\tilde{A}_{ij}\\partial_{k}\\beta^{k}\\,,\n\\end{eqnarray}\nwith source terms $\\rho$, $S$, $S_{j}$, and $S_{ij}$ projections of the\nstress-energy tensor onto the spatial slice.\nA key to the numerical stability of the system is the introduction of a\nredundant set of auxiliary variables, contractions of the Christoffel symbol of the\nconformal metric $\\bar{\\gamma}_{ij}$,\n$\\bar{\\Gamma}^{i} \\equiv \\bar{\\gamma}^{jk}\\bar{\\Gamma}_{jk}^{i}$, which are used\nto compute the Ricci tensor. These are also dynamically evolved,\n\\begin{equation}\n\\partial_{t}\\bar{\\Gamma}^{i}=2\\left(\\bar{\\Gamma}_{jk}^{i}\\tilde{A}^{jk}-\\frac{2}{3}\\bar{\\gamma}^{ij}\\partial_{j}K-8\\pi\\bar{\\gamma}^{ij}S_{j}+6\\tilde{A}^{ij}\\partial_{j}\\phi\\right)\\,.\n\\end{equation}\nIn this scheme, the extrinsic curvature, $K_{ij}$, has been decomposed into a\nconformally related trace-free part, $\\tilde{A}_{ij}$, and trace, $K$, as\n$K_{ij}=e^{4\\phi}\\tilde{A}_{ij}+\\frac{1}{3}\\gamma_{ij}K$.\nAlthough our code allows for an arbitrary gauge choice, synchronous gauge\n($\\alpha=1$ and $\\beta^{i}=0$) is applied in this work.\n\nA pressureless, $w=0$, perfect fluid is used to model the matter component of\nthis universe on large scales, consistent with a cold-dark matter model.\nThe general equations of motion for such a fluid require integration techniques\ntailored to the problem, such as finite-volume methods \\cite{10.2307\/41149030},\nphase-space methods \\cite{1989ApJ...344..146R}, or N-body techniques\n\\cite{Olabarrieta:2000bn}. However, in synchronous gauge, the equations of\nmotion simplify to\n\\begin{equation}\n\\partial_t (\\gamma^{1\/2}\\rho) = 0\n\\end{equation}\nfor a fluid with no initial coordinate velocity and density $\\rho$. Although\nmultistreaming -- or the crossing of fluid elements -- cannot be resolved using\nthis formulation due to the formation of coordinate singularities, this does not\npresent an issue on the large scales examined in this work. We can therefore\nchoose an initial conformal density $\\gamma^{1\/2}\\rho$ that will not evolve over\nthe course of the simulation. For this choice of coordinates, the remaining\nmatter source terms in the BSSN Equations are $S = S_{j} = S_{ij} = 0$.\n\n\\subsection{Propagation along null geodesics} \\label{geodesic_eom}\n\nWhile the BSSN formalism allows us to evolve the metric, we wish also to compute\nmeasurable quantities. We therefore derive the equations of motion for particles\ntraveling along geodesics. Although we are particularly interested in null\ngeodesics, the equations presented in this section can also be applied to massive\nparticles.\n\nAdopting the notation of (and loosely following) \\cite{Vincent:2012kn}, we\nbegin by noting that the equations describing propagation along a\ngeodesic are often parametrized by an affine variable, $\\lambda$. We will need to\nre-parametrize these equations in terms of the coordinate time $t$ and variables\nfrom a 3+1 decomposition. The 4-momentum, $p^{\\mu}$, for a particle following a\ngeodesic is\n\\begin{equation}\n\\frac{{\\rm d}X^{\\mu}}{{\\rm d}\\lambda}=p^{\\mu}\\,,\n\\end{equation}\nfor an affine parameter $\\lambda$ that parametrizes the particle's path.\nIn order to work in terms of $3+1$ quantities, the 4-momentum is decomposed and\nwritten in terms of quantities according to observers on a spatial slice. This\nis accomplished by writing the 4-momentum in terms of a piece parallel to the\nunit normal to the slices, $p_{\\parallel}^{\\mu} \\parallel n^\\mu$, where the\nnormal is $n^{\\mu} = (\\alpha^{-1},-\\alpha^{-1}\\beta^{i})$, as well as a transverse\npiece proportional to the velocity according to a normal observer,\n$p_{\\perp}^{\\mu} \\propto V^\\mu$.\n\nWe then define the energy of the particle according to a normal observer as\n$E \\equiv -n_{\\mu}p^{\\mu}$. The zero-component of the 4-momentum can be related\nto $E$ by noticing that\n\\begin{equation}\nn_{\\mu}{\\rm d}X^{\\mu}=n_{\\mu}p^{\\mu}d\\lambda,\n\\end{equation}\nwhich is just $E\/\\alpha=p^{0}$. We can now decompose the 4-momentum as\n\\begin{eqnarray}\np_{\\parallel}^{\\mu} & = & p^{\\mu}-p_{\\perp}^{\\mu}=En^{\\mu} \\nonumber \\\\\np_{\\perp}^{\\mu} & = & p^{\\mu}-En^{\\mu}\\equiv EV^{\\mu}\\,,\n\\end{eqnarray}\nwhere $p^{\\mu}=E\\left(n^{\\mu}+V^{\\mu}\\right)$. Because the velocity vector\n$V^\\mu$ has no component in the direction normal to the spatial slices\n($n_{\\mu}V^{\\mu}=0$), \nwe can choose to write it as $V^{\\mu}=(0,V^{i})$, with no time\ncomponent.\n\nWe now wish to obtain evolution equations for $E$ and\n$V^i$ in terms of a coordinate time $t$ rather than affine parameter.\nWe can do so using the geodesic equation,\n\\begin{equation}\n\\frac{{\\rm d}^{2}X^{\\mu}}{{\\rm d}\\lambda^{2}}=-\\Gamma_{\\alpha\\beta}^{\\mu}\\frac{{\\rm d}X^{\\alpha}}{{\\rm d}\\lambda}\\frac{{\\rm d}X^{\\beta}}{{\\rm d}\\lambda}\n\\end{equation}\nfor the position of a particle, $X^{\\mu}$, along a path parametrized by\n$\\lambda$. This can be written in a form well-suited for integrating the $3+1$\nvariables, resulting in a closed set of equations of motion:\n\n\\begin{eqnarray}\n\\frac{1}{E}\\frac{{\\rm d}}{{\\rm d}t}E & = & \\alpha K_{ij}V^{j}V^{k}-V^{j}\\partial_{j}\\alpha\\\\\n\\frac{{\\rm d}X^{i}}{{\\rm d}t} & = & \\alpha V^{i}-\\beta^{i}\\label{eq:Xev}\\\\\n\\frac{{\\rm d}V^{i}}{{\\rm d}t} & = & \\alpha V^{j}\\left(V^{i}\\partial_{j}\\ln\\alpha-K_{jk}V^{k}V^{i}+2K_{j}^{i}\\right.\\nonumber\\\\\n & & \\left.-{}^{(3)}\\Gamma_{jk}^{i}V^{k}\\right)-\\gamma^{ij}\\partial_{j}\\alpha-V^{j}\\partial_{j}\\beta^{i}\\,.\\label{eq:Vev}\n\\end{eqnarray}\n\n\\subsection{Angular diameter distances and the optical scalar equations}\n\\label{DA_eom}\n\nThe angular diameter distance, $D_{A}$, is a standard distance measure used in\ncosmology, with the desirable property that it can be directly measured or\ncomputed for distant sources. It is also related to the luminosity distance:\n$D_{A}=(1+z)^{2}D_{L}$. This statement, known as the Etherington Reciprocity\nTheorem or distance duality relationship \\cite{Etherington2007,Ellis2007},\nis a geometric result, and holds so long as photon number is conserved and all\nphotons in the beam ``feel'' the same metric -- meaning the beam width is small\ncompared to distances over which the metric varies. This small beam limit should\nbe a good approximation for point-like sources such as supernovae, but not\nnecessarily for beams with a large angular extent, such late-universe baryon\nacoustic oscillations, or degree-scale CMB fluctuations. A more thorough\ndiscussion can be found in other works, eg. \\cite{Seitz:1994xf}.\n\nIn order to compute angular diameter distances, we must track the\ncross-sectional area of a light beam as it propagates through space. We\naccomplish this using the Sachs optical-scalar equations, rewritten in a form\nwell-suited to both numerical integration and extraction of cosmological\nparameters. These equations track the width of a beam $\\ell$, its\ntime-derivative $\\varphi$, the shear rate of the beam $\\sigma$, and basis\nvectors, known as screen vectors, spanning the plane of the beam area, $s_1^\\mu$ and\n$s_2^\\mu$. The evolution equations (briefly elaborated upon in\nAppendix~\\ref{angular_diameter_appendix} and Appendix~\\ref{weyl_calcs}) are\n\\begin{eqnarray}\np^{0}\\frac{{\\rm d}}{{\\rm d}t}\\varphi & = & \\ell\\left(\\mathcal{R}-\\sigma_{R}^{2}-\\sigma_{I}^{2}\\right)\\nonumber\\\\\np^{0}\\frac{{\\rm d}}{{\\rm d}t}\\ell & = & \\varphi\\nonumber\\\\\np^{0}\\frac{{\\rm d}}{{\\rm d}t}\\bar{\\sigma}_{R} & = & \\ell^{2}\\Re[\\mathcal{W}]\\nonumber\\\\\np^{0}\\frac{{\\rm d}}{{\\rm d}t}\\bar{\\sigma}_{I} & = & \\ell^{2}\\Im[\\mathcal{W}]\\,,\n\\end{eqnarray}\nwith $\\bar{\\sigma}=\\ell^{2}\\sigma$. The Ricci optical scalar $\\mathcal{R}$ is\ngiven by\n\\begin{eqnarray}\n\\mathcal{R} & = & -4\\pi T_{\\mu\\nu}p^{\\mu}p^{\\nu}\\,,\n\\end{eqnarray}\nand the real and imaginary pieces of the Weyl scalar $\\mathcal{W}$ are\n\\begin{eqnarray}\n\\Re[\\mathcal{W}] & = & -\\frac{1}{2}C_{\\mu\\nu\\rho\\sigma}(s_{1}^{\\mu}s_{1}^{\\sigma}-s_{2}^{\\mu}s_{2}^{\\sigma})p^{\\nu}p^{\\rho}\\\\\n\\Im[\\mathcal{W}] & = & C_{\\mu\\nu\\rho\\sigma}s_{1}^{\\mu}p^{\\nu}p^{\\rho}s_{2}^{\\sigma}\\,.\n\\end{eqnarray}\n\nThe screen vectors evolve according to\n\\begin{eqnarray}\n\\frac{\\rm{d}}{\\rm{d}t}s_A^\\mu & = & s_{A}^{j}V^{i}\\left(\\gamma_{jk}\\frac{\\rm{d}}{\\rm{d}t}V^{k}-3V^{k}K_{jk}+V_{l}V^{k}\\Gamma_{kj}^{l}\\right) \\nonumber \\\\\n & & -\\Gamma_{kl}^{i}V^{k}s_{A}^{l}+\\gamma^{ij}K_{jk}s_{A}^{k} \\,.\n\\end{eqnarray}\nThese equations are subject to boundary conditions at the point of\nobservation, namely that the beam area has converged, $\\ell(t_{obs})=0$,\nthe observer views the beam as subtending some solid angle\n$\\varphi(t_{obs})=\\sqrt{\\Omega_{obs}}$, and the shear rate is $\\sigma=0$\n\\cite{Fleury:2013sna}. In terms of these variables, the angular diameter\ndistance to a source that emitted light at some time $t_{em}$,\nseen by an observer as subtending a solid angle $\\Omega_{obs}$ at time $t_obs$,\nis $D_A = \\ell(t_{em}) \/ \\varphi(t_{obs})$.\n\nWe conclude this section by writing the relationship between various quantities\ndefined so far, and corresponding FLRW quantities for a completely homogeneous\nspacetime, in Table~\\ref{frw_ref_table}.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{c|c|c}\nQuantity & FLRW Notation & Our Notation \\\\\n\\hline\nScale factor & $a$ & $e^{2\\phi}$ \\\\\nHubble parameter & $H$ & $-K\/3$ \\\\\nPhoton redshift & $z$ & $E-1$ \\\\\nAngular diameter distance & $D_A$ & $\\ell_{em} \/ \\varphi_{obs}$  \\\\\n\\end{tabular}\n\\caption{Translation between variables in a completely homogeneous dust universe.}\n\\label{frw_ref_table}\n\\end{table}\n\n\n\\section{Numerical Method Details} \\label{num_ev}\n\nAs we are interested in examining the dynamics of photon geodesics\nconverging at an observer, we will need to ensure our boundary conditions\nat the observer are satisfied, or that beams have converged at a particular\nspacetime point. The most straightforward and consistent way to implement this\nis to integrate a system forward from some initial state to a future\nobservation time, and then trace photon paths while integrating the system of\nequations backwards in time. Our code implements precisely this, allowing us to\nevolve the system from some cosmological initial conditions to a future state,\nchange the sign of the timestep, and continue integrating both the metric and\nphoton geodesics. The cosmological initial conditions we use consist of a\nGaussian-random realization of a matter field with a cosmologically-motivated\nmatter power spectrum in a periodic spacetime,\n\\begin{equation}\nP_k^{\\rho} = \\frac{4P_*}{3} \\frac{ k\/k_*}{1+\\frac{1}{3}(k\/k_*)^{4}}\\,.\n\\end{equation}\nWe cut $P_k^{\\rho}$ off at some finite $k_{\\rm cutoff}$ in order to reduce\nfluctuations on\nscales where grid effects become important, so that fluctuations are resolved\nby sufficiently many points. Further details of this spacetime can be found in\n\\cite{Mertens:2015ttp}. In order to specify initial conditions for light rays,\nwe pick random outgoing directions from a specified point, and integrate along\nnull geodesics in these directions backwards in time.\n\nIntegrating backwards not only allows us to ensure light rays converge at\nan observer, but also allows us to verify that we evolve back to the initial\nstate imposed at the beginning of the simulation. We find that our code passes\nthis check to within a small, expected level of error.\nWhile of importance in our work, such a technique would not necessarily be\nsuitable for simulations without periodic boundary conditions, as information\nabout the state of the metric may propagate outside the volume of interest when\nusing, for example, damping boundary conditions. This method may also fail for\nhydrodynamical calculations in the presence of shocks.\n\nBecause we integrate through a discretized spacetime, we also need\nto compute metric quantities between grid points, requiring an interpolation\nscheme. In this code we utilize simple linear interpolation, accurate only to\n$\\mathcal{O}(\\Delta x^{2})$. In principle this could be improved, as the metric\nand matter fields are evolved using an $\\mathcal{O}(\\Delta x^{8})$ scheme,\nhowever this level of accuracy is not found to be necessary here.\nAccumulated error for quantities integrated along geodesics is therefore\n$O(\\Delta x)$, the dominant contribution to the error in our results.\nFor cosmological runs (such as presented in \\cite{Giblin:2015vwq}), we\nsee convergence as expected given the interpolation and integration order\nused to follow photon trajectories. Further details of the code we use to\nevolve the spacetime can be found in\n\\cite{Mertens:2015ttp}.\n\nWe use results from simulations run at three different resolutions:\n$N^3 = 128^3$, $N^3 = 144^3$ and $N^3 = 160^3$. We then use a standard\nnumerical technique, Richardson extrapolation\n\\cite{Press:2007:NRE:1403886,Alcubierre:1138167}, in order to test convergence,\nto obtain results at higher order, and to obtain error estimates.\n\nAs a final note on error, the statistical uncertainty in computed means scales\nas $1\/\\sqrt{N_\\gamma}$, with $N_\\gamma$ the number of traced rays. In\ngeneral, we will be tracing $N_\\gamma = 10^3$ rays for each observer, so we\nexpect statistical error in the means to be a few per-cent of the variance.\nNumerical error in computed quantities is found to be of order a part in\n$10^6$, significantly smaller than this statistical error.\n\n\\section{Results: Computing Observables} \\label{results_sec}\n\nIn this section we present results from following light beams through a series\nof simulated cosmological spacetimes similar to the fiducial model presented in\n\\cite{Giblin:2015vwq}. We begin by examining the properties of observables in our\nmodel, and describe the effects of the inhomogeneities compared to a corresponding\nFLRW model, or a homogeneous universe containing only pressureless matter\n(eg., $\\Omega_m = 1$). The initial density fluctuations in these runs are around\n$\\sigma_\\rho\/\\rho = 0.1$, which grow to $\\sigma_\\rho\/\\rho = 0.2$ on the spatial slice\ncontaining our observers. On average, we see that our simulations reproduce the\nFLRW universe extremely well; however there are significant deviations along\nindividual paths and when observers or sources are preferentially located in\nunder- or over-dense regions.\n\nWe first remark on the case of an observer whose local density corresponds\nclosely to the average value on a spatial slice. In this case, we find light\nbeams traversing individual geodesics trace out a uniform distribution of\nangular diameter distances, the mean of which closely tracks the FLRW expectation\nat all redshifts we examine. In Fig.~\\ref{avg_hubble} we present a Hubble\ndiagram as would be computed by such an average observer, after converting the\nangular-diameter distances to apparent magnitudes,\n$m-M = 5\\log_{10}\\left(D_A (1+z)^{-2}\/10\\,{\\rm pc}\\right)$.\n\n\\begin{figure}[htb]\n  \\centering\n    \\includegraphics[width=0.45\\textwidth]{hubble_agreement.pdf}\n  \\caption{\\label{avg_hubble}\n  A Hubble diagram according to an `average' observer. The black dotted line\n  indicates the prediction from a pure-FLRW model, many green dashed lines are\n  derived from angular diameter distances computed along individual lines of\n  sight, and the red line is the average along all individual lines of sight. The\n  different lines are nearly indistinguishable.\n  }\n\\end{figure}\n\nIn Fig.~\\ref{avg_hubble_dev}, we present the corresponding residual Hubble\ndiagram.  In this diagram we subtract the distance modulus for a matter-dominated\nFLRW model from the distance modulus we calculate for our model for each value of $z$.\nOf note, there is some ambiguity in choosing which FLRW model to compare to.\nFor example, the FLRW model that agrees with the conformal average density\nat a particular time will not necessarily agree at a later time, and will also\nnot necessarily agree with the FLRW model that agrees with the (conformal)\naverage expansion rate. For an increasingly inhomogeneous universe these\ndifferences may become important, however for our simulations, the differences\nare largely unimportant. We thus compare to a pure-FLRW model for which the\nexpansion rate agrees with the conformally averaged expansion rate of the\ninitial simulation hypersurface.\n\nIn the residual diagram, we find that despite fluctuations along\nindividual paths, there is close agreement between the average angular diameter\ndistance and corresponding FLRW model at most of the redshifts we probe.\nOf significance, at very low redshift ($z \\;\\mbox{\\raisebox{-0.5ex}{$\\stackrel{<}{\\scriptstyle{\\sim}}$}}\\; 0.1$), there are sizable\ndeviations. These arise primarily due to the local structure of the universe:\nalthough this observer is situated at a location of near-average density, they\nare nevertheless near both over-dense and under-dense patches, which in turn\naffect angular-diameter measurements. Averaged across many such observers, these\nlocal effects can disappear; however for an individual observer we do not find\nthis to be the case. At larger redshifts, we find agreement between averaged\nquantities and the corresponding FLRW model to within the uncertainty of our model.\n\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[width=0.45\\textwidth]{average_obs.png}\n  \\caption{\\label{avg_hubble_dev}\n  A residual Hubble diagram according to an `average' observer.\n  Thin black lines depict angular diameter distances along several particular\n  lines of sight for this observer. The black dashed line indicates the prediction\n  from a pure-FLRW model. The solid red line indicates the average angular diameter\n  distance of these rays, for which we see agreement with FLRW. The statistical\n  error -- standard error in the mean -- is shown at the one and two `sigma` level,\n  indicated by semi-transparent red shading, so the width of the red curve is\n  indicative of error. White dashed lines indicate the standard deviation of the\n  distribution of magnitudes. Finally, the background shows a histogram of the\n  magnitudes along all integrated lines of sight as a function of redshift.\n  }\n\\end{figure}\n\nWe could compare this behavior to semi-analytic models, for example Swiss-Cheese\nuniverses, using the Dyer-Roeder approximation. To an extent, our findings\ncontrast the ideas behind such models, which posit that light beams will undergo\nless focusing as they intercept less matter (for a given average density) in\n``lumpier\" universes. However, our current simulations are in a regime that should be\nwell-described by linear theory, or a regime in which fluctuations are normally\ndistributed around the FLRW solution, and thus average away. These models also\nsuggest the shear terms are negligible, something we find approximate agreement with;\nthe size of the shear terms relative to the Ricci optical scalar is\n$\\mathcal{R}\/|\\sigma^2| \\sim \\mathcal{O}(0.01\\%)$. Nevertheless, once collapsed\nstructure is better resolved by future work, investigating these comparisons\nwill be an interesting and important task.\n\nWe next examine effects due to an observer being situated in a local over-dense or\nunder-dense region.\nIn synchronous gauge, especially at early times, fluctuations in the metric are directly\nsourced by fluctuations in the density of the universe. In our simulations, and in our\ncoordinates, the fluid is at rest throughout the simulation, so photons passing through\nover-dense or under-dense regions will be redshifted according to how the metric has\nresponded to the over- or under-density. In synchronous gauge, this can be interpreted as\nan analogue to a late-time integrated Sachs-Wolfe effect, or Rees-Sciama effect.\nIn a gauge where the fluid is not co-moving, at least some of this effect may interpreted\nas due to peculiar velocities.\nIn Fig.~\\ref{Hubble_overdense} we see such an effect for an\nobserver in a local over-density. We note that the effect of an observer sitting\nin a local under-density is similar but opposite in magnitude and can mimic the\nbehavior of a cosmological constant. The idea that we are situated in a large\nvoid has been explored in great detail in the past, see eg.\n\\cite{GarciaBellido:2008nz,Biswas:2010xm}. While such an effect is unlikely to\nexplain cosmic acceleration, it could nevertheless result in local measurements of\ncosmological parameters that do not reflect more global properties of the\nspacetime.\n\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[width=0.45\\textwidth]{overdense_obs.png}\n  \\caption{\\label{Hubble_overdense} A Hubble diagram as in figure \\ref{avg_hubble_dev}\n  created by raytracing through the simulated universe, but for an observer located\n  in an overdensity. Effects due to local structure are manifest at low redshift. }\n\\end{figure}\n\nIn a similar vein, we could use our technique to explore effects due to the\ntendency of observable sources to lie in regions of higher density. One might\nexpect to see the average deviation from FLRW affected by a Sachs-Wolfe-type\neffect, as photons not only climb out of gravitational potential wells,\nbut undergo additional Ricci focusing due to the presence of a local overdensity\nat the source, resulting in a slightly diminished inferred angular diameter\ndistance. Preliminary explorations of this effect suggest this could\nbe an interesting future topic of study.\n\nWe conclude by examining the ability of our technique to explore differences\nin inferred angular diameter distances for objects of large angular extent due\nto inhomogenities. Such effects can manifest themselves in observations of\nextended structures, for example degree-scale CMB fluctuations or\nBaryon-Acoustic-Oscillation-scale structures. In principle, a similar\ncalculation can be used to directly probe breakdowns of the Etherington\nReciprocity theorem due to finite beam effects. Violations of this theorem are\nof particular interest due to their ability to probe physics affecting the\nunderlying assumptions of the theorem, and have thus become increasingly well\nconstrained \\cite{Rasanen:2015kca}.\n\nAs a simplistic experiment, consider two pointlike sources with a known\nluminosity, part of a structure of known size, so the distance between\nthem is known. Thus the luminosity distance is known to each pointlike source\nindependently, and the angular diameter distance to the aggregate object can\nbe inferred. In the limit that the point sources are infinitesimally separated,\nthe relationship between the distance measures should obey Etherington's\nreciprocity theorem. However as the source separation -- and therefore the\nseparation between rays -- increases, the inferred angular diameter distance\nmay no longer agree with the luminosity distances due to different lines of\nsight probing different metric potentials. Thus, the inferred angular diameter\ndistance will disagree with at least one of the luminosity distances.\n\nIn Fig.~\\ref{DA_dist}, we present the difference in angular diameter distances\ncomputed by integrating the optical scalar equations, to an aggregate `source'\nwith finite angular extent, assuming the source is located at a fixed redshift.\nWe find very small deviations of distance measures on average and for beams with\nsmall area, however individual measurements demonstrate appreciable deviations,\nparticularly on large angular scales. For actual observations, the situation may\ndiffer in several regards. For example, if sources used as standard rulers lie\nin similar gravitational potentials, the particular contribution of gravitational\nredshift or peculiar velocity may be reduced.\n\n\\begin{figure}[htb]\n  \\centering\n    \\includegraphics[width=0.45\\textwidth]{delta_DA.pdf}\n  \\caption{\\label{DA_dist} Distribution of fractional differences in angular\n  diameter distance, $\\Delta D_A \/ D_A$, for infinitesimal beams separated by an\n  angle. Beams are integrated out to sources at $z=0.1$ (green stars), $z=0.5$\n  (blue squares), and $z=1.0$ (red triangles). At small separation, angular diameter\n  distances agree; however at larger separation, increasing disagreement is found.}\n\\end{figure}\n\n\\section{Conclusions and Discussion}\n\nThe results presented here are the next step in an important advancement of\nnumerical techniques that allows us to directly integrate the full Einstein\nfield equations and the optical scalar equations in an arbitrary spacetime. This\nin turn enables computation of cosmological observables and directly measurable\nquantities. As numerical relativity continues to find new applications in\ncosmology, long-standing questions will be resolved in regimes where\nnon-linearities are significant. Determining the relevance of, and precisely\nquantifying these effects in relativistic models will be an important future\ntask for this field.\n\nHere we have examined the ability of techniques from numerical relativity to\nprobe aspects of cosmology where precise estimates are required. We consider\na universe with large-scale inhomogeneities, so while we expect perturbation\ntheory and approximation schemes such as Dyer-Roeder may accurately describe\nthe effects we observe, it is encouraging to see the well-understood physical\neffects manifesting themselves in this model.\n\nNumerous applications of numerical relativity can be made in future studies, and\nit will be interesting to see the direction these studies take. For example,\nwhile statistical studies can offer general insight into physical processes, it\nmay also be possible to model specific systems to better understand dynamics and\nextract cosmological information.\n\n\\section{Acknowledgments}\nWe would like to thank Thomas Baumgarte for a series of very valuable\nconversations, as well as Stuart Shaprio and Lorenzo Sorbo for conversations\nthat helped shape this work. We are also grateful for feedback from Pierre Fleury \non an earlier draft of this manuscript.  JTG is supported by the National Science\nFoundation, PHY-1414479; JBM and GDS are supported by a Department of Energy\ngrant DE-SC0009946 to CWRU. The simulations in this work made use of the\nHigh Performance Computing Resource in the Core Facility for Advanced\nResearch Computing at Case Western Reserve University, and of hardware\nprovided by the National Science Foundation, the Research Corporation for\nScience Advancement, and the Kenyon College Department of Physics.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n Central to this paper is the following \\emph{parametric integer\n linear programming} (\\emph{PILP}) problem:\n \\begin{quote}\n   Given a rational matrix $A \\in \\setQ^{m \\times n}$ and a rational\n   polyhedron $Q \\subseteq \\setR^{m+p}$, decide if for all $b \\in\n   \\setR^m$, for which there exists an integral $z \\in \\setZ^p$ such\n   that $(b, z) \\in Q$, the system of linear inequalities $A x \\leq b$\n   has an integral solution.\n \\end{quote}\n In other words, we need to check that for all vectors $b$ in the set\n \\[\n   Q \/ \\setZ^p \\coloneqq \\{ b \\in \\setQ^m : (b, z) \\in Q \\text{ for\n   some } z \\in \\setZ^p \\}\n \\]\n the corresponding \\emph{integer linear programming} problem $A x \\leq\n b$, $x \\in \\setZ^n$ has a feasible solution. The set $Q \/ \\setZ^p$ is\n called the \\emph{integer projection} of $Q$. Using this notation, we\n can reformulate PILP as the problem of testing the following\n $\\forall\\,\\exists$-sentence:\n \\begin{equation}\n   \\label{eq:pilp}\n   \\forall b \\in Q \/ \\setZ^p \\quad \\exists x \\in \\setZ^n : \\quad A x\n   \\leq b.\n \\end{equation}\n It is worth noticing that any polyhedron $Q \\subseteq \\setR^m$ as\n well as the set of integral vectors in $Q$ can be expressed by means\n of integer projections of polyhedra. Indeed,\n \\[\n   Q = Q \/ \\setZ^0 \\quad \\text{and} \\quad Q \\cap \\setZ^m = \\{ (b, b) :\n   b \\in Q \\} \/ \\setZ^m.\n \\]\n\n In its general form, PILP belongs to the second level of the\n polynomial hierarchy and is $\\Pi_2^p$-complete; see\n \\citep{Stockmeyer1976} and \\citep{Wrathall1976}. \\cite{Kannan1990}\n presented a polynomial algorithm to decide the sentence\n \\eqref{eq:pilp} in the case when $n$, $p$ and the affine dimension of\n $Q$ are fixed. This result was applied to deduce a polynomial\n algorithm that solves the Frobenius problem when the number of input\n integers is fixed, see \\citep{Kannan1992}.\n\n Kannan's algorithm proceeds in several steps. We informally describe\n it at this point as a way to decide $\\forall \\, \\exists$-statements\n \\eqref{eq:pilp} in the case $p=0$. First Kannan provides an algorithm\n which partitions the set of right-hand sides $Q$ into polynomially\n many \\emph{integer projections of partially open polyhedra} $S_1,\n \\ldots, S_t$, where each $S_i$ is obtained from a higher-dimensional\n polyhedron by projecting out a fixed number of integer variables.\n Each $S_i$ is further equipped with a fixed number of mixed integer\n programs such that for each $b \\in S_i$ the system $A x \\leq b$ is\n integer feasible, if and only if one of the fixed number of\n ``candidate solutions'' obtained from plugging $b$ in these\n associated mixed integer programs, is a feasible integer point.\n\n To decide now whether \\eqref{eq:pilp} holds, one searches within the\n sets $S_i$ individually for a vector $b$ for which $A x \\leq b$ has\n \\emph{no} integral solution. In other words, each of the candidate\n solutions associated to $b$ must violate at least one of the\n inequalities in $A x \\leq b$. Since the number of candidate solutions\n is fixed, we can enumerate the choices to associate a violated\n inequality to each candidate solution. Each of these polynomially\n many choices yields now a mixed-integer program with a fixed number\n of integer variables. There exists a $b \\in S_i$ such that $A x \\leq\n b$ has no integral solution if and only if one of these mixed-integer\n programs is feasible. The latter can be checked with the algorithm of\n \\cite{Lenstra1983} in polynomial time.\n\n\\subsection*{Contributions of this paper} \n\n We modify the algorithm of Kannan to run in polynomial time under the\n assumption that only $n$ and $p$ are fixed. This is achieved via\n providing an algorithm that computes for a matrix $A \\in \\setQ^{m\n \\times n}$ a set $D \\subseteq \\setZ^n$ of integral directions with\n the following property: for each $b \\in \\setR^m$, the lattice width\n (see Section \\ref{sec:flatn-theor-integ}) of the polyhedron $P_b = \\{\n x : A x \\leq b \\}$ is equal to the width of this polyhedron along one\n of the directions in $D$. This algorithm is described in Section\n \\ref{sec:width} and runs in polynomial time if $n$ is fixed. The\n strengthening of Kannan's algorithm to decide $\\forall \\,\n \\exists$-statements of the form \\eqref{eq:pilp} if $n$ and $p$ is\n fixed follows then by using this result in the proof of Theorem 4.1\n in \\citep{Kannan1992}.\n\n We then apply this result to find the maximum \\emph{integer\n programming gap} for a family of integer programs. The integer\n programming gap of an integer program\n \\begin{equation}\n   \\label{eq:ilp-intro}\n   \\max \\{ c x : A x \\leq b, \\, x \\in \\setZ^n \\}\n \\end{equation}\n is the difference\n \\[\n   \\max \\{ c x : A x \\leq b \\} - \\max \\{ c x : A x \\leq b, \\, x \\in\n   \\setZ^n \\}.\n \\]\n Given a rational matrix $A \\in \\setQ^{m \\times n}$ and a rational\n objective vector $c \\in \\setQ^n$,  $g(A, c)$ denotes  the maximum\n integer programming gap of integer programs of the form\n \\eqref{eq:ilp-intro}, where the maximum is taken over all vectors\n $b$, for which the integer program \\eqref{eq:ilp-intro} is\n feasible. Our algorithm finds $g(A, c)$ in polynomial time if $n$ is\n fixed. This extends a recent result of \\cite{HostenS2003}, who\n proposed an algorithm to find the maximum integer \n programming gap for a family of integer programs in \\emph{standard\n form} if $n$ is fixed. \n\n\\subsection*{Related work}  \n\n Kannan's algorithm is an extension of the polynomial algorithm for\n integer linear programming in fixed dimension by \\cite{Lenstra1983}.\n \\cite{BarvinokW2003} presented an algorithm for counting integral\n points in the integer projection $Q \/ \\setZ^p$ of a polytope $Q\n \\subseteq \\setR^{m+p}$. This algorithm runs in polynomial time if $p$\n and $m$ are fixed, and uses Kannan's partitioning algorithm, which we\n extend in this paper. In particular, their algorithm can be applied\n to count the number of elements of the minimal Hilbert basis of a\n pointed cone in polynomial time if the dimension is fixed. We remark\n that a polynomial test for the Hilbert basis property in fixed\n dimension was first presented by \\cite{CookLS1984}. Extensions of\n Barvinok's algorithm to compute counting functions for parametric\n polyhedra were presented in \\citep{BarvinokPommersheim99,MR2312001}\n and in \\citep{koppe-2007}.  These counting functions are piecewise\n step-polynomials which involve roundup operations.  With these\n functions at hand one can very efficiently compute the number of\n integer points in $P_b$ via evaluation at $b$. It is however not\n known how to use such piecewise step-polynomials to decide\n $\\forall\\,\\exists$-statements efficiently in fixed dimension.\n\n \\cite{HostenS2003} proposed an algorithm to find the maximum integer\n programming gap for a family of integer programs in \\emph{standard\n form}, i.e., $\\max \\{ c x : A x = b, \\, x \\geq 0, \\, x \\in \\setZ^n\n \\}$. Their algorithm exploits short rational generating functions for\n certain lattice point problems, cf. \\cite{Barvinok1994} and\n \\cite{BarvinokW2003}, and runs in polynomial time if the number $n$\n of columns of $A$ is fixed. However, the latter implies also a fixed\n number of rows in $A$, as we can always assume $A$ to have full row\n rank. We would like to point out that our approach does not rely on\n rational generating functions at all.\n\n\\subsection*{Basic definitions and notation}\n\n For sets $V$ and $W$ in $\\setR^n$ and a number $\\alpha$ we denote\n \\[\n   V + W \\coloneqq \\{ v + w : v \\in V, \\, w \\in W \\} \\quad \\text{and}\n   \\quad \\alpha W \\coloneqq \\{ \\alpha w : w \\in W \\}.\n \\]\n It is easy to see that if $W$ is a convex set containing the origin\n and $\\alpha \\leq 1$, then $\\alpha W \\subseteq W$. If $V$ consists of\n one vector $v$ only, we write\n \\[\n   v + W \\coloneqq \\{ v + w : w \\in W \\}\n \\]\n and say that $v + W$ is the \\emph{translate} of $W$ \\emph{along} the\n vector $v$. The symbol $\\lceil \\alpha \\rceil$ denotes the smallest\n integer greater than or equal to $\\alpha$, i.e., $\\alpha$\n \\emph{rounded up}. Similarly, $\\lfloor \\alpha \\rfloor$ stands for the\n largest integer not exceeding $\\alpha$, hence $\\alpha$ \\emph{rounded\n down}.\n\n In this paper we establish a number of \\emph{polynomial algorithms},\n i.e., algorithms whose running time is bounded by a polynomial in the\n input size. Following the standard agreements, we define the\n \\emph{size} of a rational number $\\alpha = p \/ q$, where $p, q \\in\n \\setZ$ are relatively prime and $q > 0$, as the number of bits needed\n to write $\\alpha$ in binary encoding:\n \\[\n   \\mathrm{size}(\\alpha) \\coloneqq 1 + \\lceil \\log (\\abs{p} + 1) \\rceil +\n   \\lceil \\log (q + 1) \\rceil.\n \\]\n The size of a rational vector $a = [ a_1, \\ldots, a_n ]$ is the sum\n of the sizes of its components:\n \\[\n   \\mathrm{size}(a) \\coloneqq n + \\sum_{i=1}^n \\mathrm{size}(a_i).\n \\]\n At last, the size of a rational matrix $A = [ a_{ij} ] \\in \\setQ^{m\n \\times n}$ is\n \\[\n   \\mathrm{size}(A) \\coloneqq m n + \\sum_{i=1}^m \\sum_{j=1}^n \\mathrm{size}(a_{ij}).\n \\]\n\n An \\emph{open half-space} in $\\setR^n$ is the set of the form $\\{ x :\n a x < \\beta \\}$, where $a \\in \\setR^n$ is a row-vector and $\\beta$ is\n a number. Similarly, the set $\\{ x : a x \\leq \\beta \\}$ is called a\n \\emph{closed half-space}. A \\emph{partially open polyhedron} $P$ is\n the intersection of finitely many closed or open half-spaces. If $P$\n can be defined by means of closed half-spaces only, we say that it is\n a \\emph{closed polyhedron}, or simply a \\emph{polyhedron}. We need\n the notion of a partially open polyhedron to be able to partition the\n space (this is definitely impossible by means of closed polyhedra\n only). At last, we say that a partially open polyhedron is\n \\emph{rational} if it can be defined by the system of linear\n inequalities with rational coefficients and rational right-hand\n sides.\n\n \\emph{Linear programming} is about optimizing a linear function $c x$\n over a given polyhedron $P$ in $\\setR^n$:\n \\[\n   \\max \\{ c x : x \\in P \\} = - \\min \\{ - c x : x \\in P \\}.\n \\]\n If $x$ is required to be integral, it is an \\emph{integer linear\n programming} problem\n \\[\n   \\max \\{ c x : x \\in P \\cap \\setZ^n \\} = - \\min \\{ - c x : x \\in P\n   \\cap \\setZ^n \\}.\n \\]\n For details on linear and integer programming, we refer to\n \\citep{Schrijver1986}. Here we only mention that a linear programming\n problem can be solved in polynomial time, cf. \\citep{Khachiyan1979},\n while integer linear programming is NP-complete. However, if the\n number of variables is fixed, integer programming can also be solved\n in polynomial time, as was shown by \\cite{Lenstra1983}. Moreover,\n Lenstra presented an algorithm to solve \\emph{mixed-integer\n programming} with a fixed number of integer variables. We remark that\n both algorithms---of \\cite{Khachiyan1979} and of\n \\cite{Lenstra1983}---can be used to solve \\emph{decision versions} of\n integer and linear programming on \\emph{partially open} polyhedra.\n\n An integral square matrix $U$ is called \\emph{unimodular} if\n $\\abs{\\mathrm{det}(U)} = 1$. Clearly, if $U$ is unimodular, then\n $U^{-1}$ is also unimodular. A matrix of full row rank is said to be\n in \\emph{Hermite normal form} if it has the form $[ H \\,\\,\\, 0 ]$,\n where $H = [ h_{ij} ]$ is a square non-singular non-negative\n upper-triangular matrix such that $h_{ii} > h_{ij}$ for all $j >\n i$. Given a matrix $A$ of full row rank, we can find in polynomial\n time a unimodular matrix $U$ such that $A U$ is in Hermite normal\n form; see \\citep{KannanB1979}. We remark that the Hermite normal form\n of an integral vector $c$ is the vector $\\alpha e_1$, where $\\alpha$\n is the greatest common divisor of the components of $c$ and $e_1$ is\n the first unit vector. The unimodular matrix $U$ such that $c U =\n \\alpha e_1$ can be obtained directly while executing the Euclidean\n algorithm to compute the greatest common divisor.\n\n\\section{Flatness theorem}\n\\label{sec:flatn-theor-integ}\n\n We briefly review the algorithm to solve integer linear programming\n in fixed dimension, as its basic ideas will be used in the following\n sections. Intuitively, if a polyhedron contains no integral point,\n then it must be ``flat'' along some integral direction. In order to\n make this precise, we introduce the notion of ``lattice width.'' The\n \\emph{width} $w_c(K)$ of a closed convex set $K$ along a direction $c\n \\in \\setR^n$ is defined as\n \\begin{equation}\n   \\label{eq:width-c}\n   w_c(K) \\coloneqq \\max \\{ c x : x \\in K \\} - \\min \\{ c x : x \\in K\n   \\}.\n \\end{equation}\n The \\emph{lattice width} $w(K)$ of $K$ (with respect to the\n \\emph{standard lattice} $\\setZ^n$) is the minimum of its widths along\n all non-zero integral directions:\n \\[\n   w(K) \\coloneqq \\min \\{ w_c(K) : c \\in \\setZ^n \\setminus \\{ 0 \\} \\}.\n \\]\n An integral row-vector $c$ attaining the above minimum is called a\n \\emph{width direction} of the set $K$. Clearly, $w(v + \\alpha K) =\n \\alpha w(K)$ for any rational vector $v$ and any non-negative\n rational number $\\alpha$. Moreover, both sets $K$ and $v + \\alpha K$\n have the same width direction.\n\n Applications of the concept of lattice width in algorithmic number\n theory and integer programming rely upon the \\emph{flatness theorem},\n which goes back to \\cite{Khinchin1948} who first proved it for\n ellipsoids in $\\setR^n$. Here we state it for \\emph{convex bodies},\n i.e., bounded closed convex sets of non-zero volume.\n\n \\begin{theorem}[Flatness theorem]\n   \\label{thm:flatness}\n   There is a constant $\\omega(n)$, depending only on $n$, such that\n   any convex body $K \\subseteq \\setR^n$ with $w(K) \\geq \\omega(n)$\n   contains an integral point.\n \\end{theorem}\n\n \\noindent The constant $\\omega(n)$ in Theorem \\ref{thm:flatness} is\n referred to as the \\emph{flatness constant}. The best known\n \\emph{upper bound} on $\\omega(n)$ is $O ( n^{3\/2} )$, cf.\n \\citep{BanaszczykLPS1999}, although a linear dependence on $n$ was\n conjectured, e.g., by \\cite{KannanL1988}. A linear \\emph{lower bound}\n on $\\omega(n)$ was shown by \\cite{Kantor1999} and \\cite{Sebo1999}.\n\n Throughout this paper we will mostly deal with rational polyhedra\n rather than general convex bodies. In this case, assumptions of\n non-zero volume and boundedness can safely be removed from the\n theorem's statement. Indeed, if $P \\subseteq \\setR^n$ is a rational\n polyhedron of zero volume, then it has width $0$ along an integral\n direction orthogonal to its (rational) affine hull. Further, let $C$\n be the characteristic cone of $P$:\n \\[\n   C \\coloneqq \\{ y : x + y \\in P \\text{ for all } x \\in P \\}.\n \\]\n If $C = \\{ 0 \\}$, then $P$ is already bounded. If $C$ is\n full-dimensional, then the set $x + C$ trivially contains an integral\n point, for any $x \\in P$ (we can always allocate a unit box inside a\n full-dimensional cone). At last, if $C$ is not full dimensional, then\n we can choose a sufficiently large box $B \\subseteq \\setR^n$ such\n that $w(P) = w(P \\cap B)$ and both $P$ and $P \\cap B$ have the same\n width direction, which is orthogonal to the (rational) affine hull of\n $C$. If $w(P) \\geq \\omega(n)$, then $P \\cap B$, and hence $P$,\n contains an integral point by Theorem \\ref{thm:flatness}.\n\n How can we use this theorem to check whether a given rational\n polyhedron contains an integral point? The answer is in the following\n lemma, which is almost a direct consequence of the flatness theorem.\n\n \\begin{lemma}\n   \\label{lem:flatness-app}\n   Let $P \\subseteq \\setR^n$ be a rational polyhedron of finite\n   lattice width and let $c$ be its width direction. Let\n   \\begin{equation}\n     \\label{eq:beta-intro}\n     \\beta \\coloneqq \\min \\{ c x : x \\in P \\}.\n   \\end{equation}\n   Then $P$ contains an integral point if and only if the polyhedron\n   \\[\n     P \\cap \\{ x : \\beta \\leq c x \\leq \\beta + \\omega(n) \\}\n   \\]\n   contains an integral point.\n \\end{lemma}\n\n \\begin{figure}[ht]\n   \\centering {\n     \\psset{xunit=.5cm,yunit=.4cm}\n     \\begin{pspicture}(0,-5)(12,8)       \n       \\pspolygon[fillstyle=solid,fillcolor=vvlg]\n         (0,0)(3,6)(6,7.5)(12,0)(6,-3)(3,-3)\n       \\psline(0,-4)(0,8)\n       \\rput[l](0,-4.5){$cx = \\beta$}\n\n       \\pspolygon[fillstyle=solid,fillcolor=lightgray]\n         (0,0)(1,2)(2,2.5)(4,0)(2,-1)(1,-1)\n       \\psline(4,-4)(4,8)\n       \\rput[l](4,-4.5){$cx = \\beta+\\omega(n)$}\n       \\rput(3,0){$P'$}\n       \\rput(10,0){$P$}\n     \\end{pspicture}\n   }\n   \\caption{Illustration for the proof of Lemma\n     \\ref{lem:flatness-app}}\n   \\label{fig:1}\n \\end{figure}\n\n \\begin{proof}\n   If $w(P) < \\omega(n)$, then there is nothing to prove, since\n   \\[\n     P \\subseteq \\{ x : \\beta \\leq c x < \\beta + \\omega(n) \\}.\n   \\]\n   Suppose that $w(P) \\geq \\omega(n)$ and let $P = y + Q$, where $y$\n   is an optimum solution of the linear program \\eqref{eq:beta-intro}\n   and $Q$ is the polyhedron containing the origin,\n   \\[\n     Q \\coloneqq \\{ x - y : x \\in P \\}.\n   \\]\n   We denote\n   \\[\n     Q' \\coloneqq \\textstyle \\frac{\\omega(n)}{w(P)} Q \\quad \\text{and}\n     \\quad P' \\coloneqq y + Q'.\n   \\]\n   In other words, $Q$ is $P$ translated to contain the origin, $Q'$\n   is obtained from $Q$ by scaling it down, and $P'$ is $Q'$\n   translated back to the original position. It is easy to see that\n   \\[\n     \\min \\{ c x : x \\in P' \\} = c y = \\beta.\n   \\]\n   Since $\\frac{\\omega(n)}{w(P)} \\leq 1$ and $Q$ is convex, we have\n   $Q' \\subseteq Q$. This implies $P' \\subseteq P$. Yet, we have $w(P)\n   = w(Q)$, and therefore, $w(P') = w(Q') = \\omega(n)$.\n\n   By Theorem \\ref{thm:flatness}, $P'$ contains an integral point, say\n   $z$. But then $z$ also belongs to $P$ and\n   \\[\n     c z \\leq \\max \\{ c x : x \\in P' \\} = \\beta + \\omega(n).\n   \\]\n   This completes the proof.\n \\end{proof}\n\n \\noindent Suppose that we know a width direction $c$ of a polyhedron\n \\begin{equation}\n   \\label{eq:polyhedron-intro}\n   P = \\{ x : A x \\leq b \\} \\subseteq \\setR^n.\n \\end{equation}\n Since $c$ is integral, the scalar product $c x$ must be an integer\n for any integral point $x \\in P$. Together with Lemma\n \\ref{lem:flatness-app}, it allows us to split the original problem\n into $\\omega(n) + 1$ integer programming problems on\n lower-dimensional polyhedra\n \\[\n   P \\cap \\{ x : c x = \\lceil \\beta \\rceil + j \\}, \\quad j = 0,\n   \\ldots, \\omega(n)\n \\]\n where $\\beta$ is defined by \\eqref{eq:beta-intro}.\n\n The components of $c$ must be relatively prime, as otherwise we could\n scale $c$, obtaining a smaller width of $P$. Therefore its Hermite\n normal form is a unit row-vector $e_1$. We can find a unimodular\n matrix $U$ such that $c U = e_1$, introduce new variables $y\n \\coloneqq U^{-1} x$ and rewrite the original system of linear\n inequalities $A x \\leq b$ in the form $A U y \\leq b$. Since $U$ is\n unimodular, the system $A x \\leq b$ has an integral solution if and\n only if the system $A U y \\leq b$ has an integral solution. But the\n equation $c x = \\lceil \\beta \\rceil + j$ turns into $e_1 y = \\lceil\n \\beta \\rceil + j$. Thus, the first component of $y$ can be\n eliminated. All together, we can proceed with a constant number of\n integer programming problems with a smaller number of variables. If\n $n$ is fixed, this yields a polynomial algorithm.\n\n An attempt to generalize this approach for the case of varying $b$\n gives rise to the following problems. First, the width directions of\n the polyhedron \\eqref{eq:polyhedron-intro} depend on $b$ and\n therefore can also vary. Furthermore, even if a width direction $c$\n remains the same, it is not a trivial task to proceed\n recursively. The point is that $\\beta$, as it is defined in\n \\eqref{eq:beta-intro}, also depends on $b$ and the hyper-planes $\\{ x\n : c x = \\lceil \\beta \\rceil + j \\}$ are not easy to construct with\n $\\beta$ being a \\emph{function} of $b$. In the following sections we\n basically resolve these two problems and adapt the above algorithm\n for the case of varying $b$.\n\n\\section{Lattice width of a parametric polyhedron}\n\\label{sec:width}\n\n A rational \\emph{parametric polyhedron} $P$ defined by a matrix $A\n \\in \\setQ^{m \\times n}$ is the family of polyhedra of the form\n \\[\n   P_b \\coloneqq \\{ x : A x \\leq b \\},\n \\]\n where the right-hand side $b$ is allowed to vary over $\\setR^m$. We\n restrict our attention only to those $b$, for which $P_b$ is\n non-empty. For each such $b$, there is a width direction $c$ of the\n polyhedron $P_b$. We aim to find a small set $C$ of non-zero integral\n directions such that\n \\[\n   w(P_b) = \\min \\{ w_c(P_b) : c \\in C \\}\n \\]\n for all vectors $b$ for which $P_b$ is non-empty. Further on, the\n elements of the set $C$ are referred to as \\emph{width directions} of\n the parametric polyhedron $P$. It turns out that such a set can be\n computed in polynomial time when the number of columns in $A$ is\n fixed.\n\n Let $A \\in \\setQ^{m \\times n}$ be a matrix of full column rank. Given\n a subset of indices\n \\[\n   N = \\{ i_1, \\ldots, i_n \\} \\subseteq \\{ 1, \\ldots, m \\},\n \\]\n we denote by $A_N$ the matrix composed of the rows $i_1, \\ldots, i_n$\n of $A$. We say that $N$ is a \\emph{basis} of $A$ if $A_N$ is\n non-singular. Clearly, any matrix of full column rank has at least\n one basis. Each basis $N$ defines a linear transformation\n \\begin{equation}\n   \\label{eq:fn}\n   F_N : \\setR^m \\to \\setR^n, \\quad F_N b = A_N^{-1} b_N,\n \\end{equation}\n which maps right-hand sides $b$ to the corresponding \\emph{basic\n solutions}. We can view $F_N$ as an $n \\times m$-matrix of rational\n numbers. If the point $F_N b$ satisfies the system $A x \\leq b$, then\n it is a vertex of the polyhedron $\\{ x : A x \\leq b \\}$. From linear\n programming duality we know that the optimum value of any feasible\n linear program\n \\[\n   \\max \\{ c x : A x \\leq b \\}\n \\]\n is finite if and only if there is a basis $N$ such that $c = y A_N$\n for some row-vector $y \\geq 0$. In other words, $c$ must belong to\n the cone generated by the rows of matrix $A_N$. Moreover, if it is\n finite, there is a basis $N$ such that the optimum value is attained\n at $F_N b$. It gives us the following simple lemma.\n\n \\begin{lemma}\n   \\label{lem:infinite-width}\n   Let $P$ be a parametric polyhedron defined by a rational matrix\n   $A$. If there exists a vector $b'$ such that the polyhedron\n   \\[\n     P_{b'} = \\{ x : A x \\leq b' \\}\n   \\]\n   has infinite lattice width, then $w(P_b)$ is infinite for all $b$.\n \\end{lemma}\n\n \\begin{proof}\n   Suppose that the lattice width of $P_b$ is finite for some $b$ and\n   let $c$ be a width direction. Then both linear programs\n   \\[\n     \\max \\{ c x : A x \\leq b \\} \\quad \\text{and} \\quad \\min \\{ c x :\n     A x \\leq b \\}\n   \\]\n   are bounded and therefore there are bases $N_1$ and $N_2$ of $A$\n   such that $c$ belongs to both cones\n   \\begin{equation}\n     \\label{eq:cones-width}\n     C_1 \\coloneqq \\{ y A_{N_1} : y \\geq 0 \\} \\quad \\text{and} \\quad\n     C_2 \\coloneqq \\{ - y A_{N_2} : y \\geq 0 \\}\n   \\end{equation}\n   generated by the rows of matrices $A_{N_1}$ and $- A_{N_2}$,\n   respectively. But then the linear programs\n   \\[\n     \\max \\{ c x : A x \\leq b' \\} \\quad \\text{and} \\quad \\min \\{ c x :\n     A x \\leq b' \\}\n   \\]\n   are also bounded, whence $w_c(P_{b'})$ is finite.\n \\end{proof}\n\n \\noindent The above lemma shows that finite lattice width is a\n property of the matrix $A$. In particular $P_0$ has finite lattice\n width if and only if $P_b$ has finite lattice width for all\n $b$. Conversely, if $P_0$ has infinite lattice width, then $P_b$ also\n has infinite lattice width and therefore contains an integral point\n for all $b$. We can easily recognize whether $P_0$ has infinite\n lattice width. For instance, we can enumerate all possible pairs of\n bases $N_1$ and $N_2$ and check if the cones \\eqref{eq:cones-width}\n have a common integral vector. Further we shall not deal with this\n ``trivial'' case and shall consider only those parametric polyhedra,\n for which $w(P_0)$ is finite, and therefore $w(P_b)$ is finite for\n any $b$. We say in this case that the parametric polyhedron $P$ has\n \\emph{finite lattice width}.\n\n \\begin{figure}[ht]\n   \\centering {\n     \\psset{unit=.6cm}\n     \\begin{pspicture}(-3,-5)(12.5,3)    \n       \\pspolygon[linestyle=dashed,linecolor=lightgray,\n         fillcolor=vvlg,fillstyle=solid]\n         (0,0)(1.5,1.5)(3,2)(8,2)(10,.5)(11,-1)(10,-2.5)(4,-2)(1.5,-1)\n\n       \\psline(0,0)(3,3)\n       \\psline(0,0)(3,-2)\n\n       \\pspolygon[linecolor=vlg,fillcolor=vlg,fillstyle=solid]\n         (0,0)(-3,3)(-3,-4.5)\n       \\psline[linecolor=vlg](0,0)(-3,3)\n       \\psline[linecolor=vlg](0,0)(-3,-4.5)\n       \\psline[arrows=->](0,0)(-2,0)\n\n       \\psline(11,-1)(9,2)\n       \\psline(11,-1)(9,-4)\n\n       \\pspolygon[linecolor=vlg,fillcolor=vlg,fillstyle=solid]\n         (11,-1)(8,-3)(8,1)\n       \\psline[linecolor=vlg](11,-1)(8,-3)\n       \\psline[linecolor=vlg](11,-1)(8,1)\n       \\psline[arrows=->](11,-1)(9,-1)\n\n       \\rput(-2.5,1.5){$C_1$}\n       \\rput(-1.5,-.5){$c$}\n       \\rput(8.5,0){$C_2$}\n       \\rput(9.5,-1.5){$c$}\n\n       \\rput(5,1.5){$P_b$}\n\n       \\rput[l](.5,0){$F_{N_1}b$}\n       \\rput[l](11.5,-1){$F_{N_2}b$}\n\n       \\psdots(0,0)(11,-1)\n     \\end{pspicture}\n   }\n   \\caption{The width direction $c$ and the two cones $C_1$ and\n     $C_2$.}\n   \\label{fig:2}\n \\end{figure}\n\n Now, suppose that $P_b$ is non-empty and let $c$ be its width\n direction. Then there are two bases $N_1$ and $N_2$ such that\n \\begin{equation}\n   \\label{eq:max-min-width}\n   \\max \\{ c x : A x \\leq b \\} = c F_{N_1} b \\quad \\text{and} \\quad\n   \\min \\{ c x : A x \\leq b \\} = c F_{N_2} b\n \\end{equation}\n and $c$ belongs to the cones $C_1$ and $C_2$ defined by\n \\eqref{eq:cones-width}, see Figure~\\ref{fig:2}. In fact, equations\n \\eqref{eq:max-min-width} hold for any vector $c$ in $C_1 \\cap\n C_2$. Thus, the lattice width of $P_b$ is equal to the optimum value\n of the following optimization problem:\n \\begin{equation}\n   \\label{eq:c-ilp}\n   \\min \\{ c (F_{N_1} - F_{N_2}) b : c \\in C_1 \\cap C_2 \\cap \\setZ^n\n   \\setminus \\{ 0 \\} \\}.\n \\end{equation}\n The latter can be viewed as an integer programming problem. Indeed,\n the cones $C_1$ and $C_2$ can be represented by some systems of\n linear inequalities, say $c D_1 \\leq 0$ and $c D_2 \\leq 0$,\n respectively, where $D_1,D_2\\in \\setZ^{n \\times n}$. The minimum\n \\eqref{eq:c-ilp} is taken over all integral vectors $c$ satisfying $c\n D_1 \\leq 0$ and $c D_2 \\leq 0$, except the origin. Since both cones\n $C_1$ and $C_2$ are simplicial, i.e., generated by $n$ linearly\n independent vectors, the origin is a vertex of $C_1 \\cap C_2$ and\n therefore can be cut off by a single inequality, for example, $c D_1\n \\mathbf{1} \\leq -1$, where $\\mathbf{1}$ denotes the $n$-dimensional\n all-one vector. It is important that all other integral vectors $c$\n in $C_1 \\cap C_2$ satisfy this inequality and therefore remain\n feasible. Thus, the problem \\eqref{eq:c-ilp} can be rewritten as\n \\[\n   \\min \\{ c (F_{N_1} - F_{N_2}) b : c D_1 \\leq 0, \\, c D_2 \\leq 0, \\,\n   c D_1 \\mathbf{1} \\leq -1 \\, c \\in \\setZ^n \\}.\n \\]\n For a given $b$, this is an integer programming problem. Therefore,\n the optimum value of \\eqref{eq:c-ilp} is attained at some vertex of\n the integer hull of the underlying polyhedron\n \\begin{equation}\n   \\label{eq:c-ilp-2}\n   \\{ c : c D_1 \\leq 0, \\, c D_2 \\leq 0, \\, c D_1 \\mathbf{1} \\leq -1\n   \\}\n \\end{equation}\n \\cite{Shevchenko1981} and \\cite{HayesL1983} proved that the number of\n vertices of the integer hull of a rational polyhedron is polynomial\n in fixed dimension. Tight bounds for number were presented in\n \\citep{CookHKM1992} and \\citep{BaranyHoweLovasz92}. This gives rise to\n the next lemma.\n\n \\begin{lemma}\n   \\label{lem:flat-dirs}\n   There is an algorithm that takes as input a rational matrix $A \\in\n   \\setQ^{m \\times n}$ of full column rank, which defines a parametric\n   polyhedron $P$ of finite lattice width, and computes a set of\n   triples $(F_i, G_i, c_i)$ of rational linear transformations $F_i,\n   G_i : \\setR^m \\to \\setR^n$ and a non-zero integral row-vector $c_i\n   \\in \\setZ^n$ ($i = 1, \\ldots, t$) satisfying the following\n   properties. For all $b$, for which $P_b$ is non-empty,\n   \\begin{enumerate}[(a)]\n     \\item \\label{item:flat-dirs-a}\n       $F_i$ and $G_i$ provide, respectively, an upper and lower bound\n       on the value of the linear function $c_i x$ in $P_b$, i.e., for\n       all $i$,\n       \\[\n         c_i G_i b \\leq \\min \\{ c_i x : x \\in P_b \\} \\leq \\max \\{ c_i\n         x : x \\in P_b \\} \\leq c_i F_i b,\n       \\]\n     \\item \\label{item:flat-dirs-b}\n       the lattice width of $P_b$ is attained along the direction\n       $c_i$ for some $i \\in \\{ 1, \\ldots, t \\}$ and can be expressed\n       as\n       \\[\n         w(P_b) = \\min_i \\; c_i (F_i - G_i) b.\n       \\]\n     \\item \\label{item:flat-dirs-c}\n       The number $t$ of the triples satisfies the bound\n       \\begin{equation}\n         \\label{eq:t-bound}\n         t \\leq 2 m^{2 n} ( 2 n + 1 )^n ( 24 n^5 \\phi )^{n-1},\n       \\end{equation}\n       where $\\phi$ is the maximum size of a column in $A$.\n   \\end{enumerate}\n   The algorithm runs in polynomial time if $n$ is fixed.\n \\end{lemma}\n\n \\begin{proof}\n   In the first step of the algorithm we enumerate all possible bases\n   of $A$. Observe that there is at least one basis, since $A$ is of\n   full column rank. On the other hand, the total number of possible\n   bases is at most $m^n$. Hence, the number of possible pairs of\n   bases is bounded by $m^{2 n}$. The algorithm iterates over all\n   unordered pairs of bases and for each such pair $\\{ N_1, N_2 \\}$\n   does the following.\n\n   Let $C_1$ and $C_2$ be the corresponding simplicial cones, defined\n   by \\eqref{eq:cones-width}. These cones can be represented by\n   systems of linear inequalities, $c D_1 \\leq 0$ and $c D_2 \\leq 0$\n   respectively, where $D_1, D_2 \\in \\setZ^{n \\times n}$ and the size\n   of each inequality is bounded by $4 n^2 \\phi$, see \\cite[Theorem\n   10.2]{Schrijver1986}. As the origin is a vertex of the cone $C_1\n   \\cap C_2$, it can be cut off by a single inequality; for example,\n   $c D_1 \\mathbf{1} \\leq -1$, where $\\mathbf{1}$ stands for the\n   $n$-dimensional all-one vector. The size of the latter inequality\n   is bounded by $4 n^3 \\phi$.\n\n   Thus, there are exactly $2 n + 1$ inequalities in\n   \\eqref{eq:c-ilp-2} and the size of each inequality is bounded by $4\n   n^3 \\phi$. This implies that the number of vertices of the integer\n   hull of \\eqref{eq:c-ilp-2} is at most $2 (2 n + 1)^n (24 n^5\n   \\phi)^{n-1}$, cf. \\citep{CookHKM1992}, and they all can be computed\n   in polynomial time if $n$ is fixed, cf. \\citep{Hartmann1989}. The\n   algorithm then outputs the triple $(F_{N_1}, F_{N_2}, c)$ for each\n   vertex $c$ of the integer hull of \\eqref{eq:c-ilp-2}, where\n   $F_{N_1}$ and $F_{N_2}$ are the linear transformations defined by\n   \\eqref{eq:fn}. Since there are at most $m^{2 n}$ unordered pairs of\n   bases and, for each pair, the algorithm returns at most $2 (2 n +\n   1)^n (24 n^5 \\phi)^{n-1}$ triples, the total number of triples\n   satisfies \\eqref{eq:t-bound}, as required. Parts\n   \\eqref{item:flat-dirs-a} and \\eqref{item:flat-dirs-b} of the\n   theorem follow directly from our previous explanation.\n \\end{proof}\n\n \\noindent The bound \\eqref{eq:t-bound} can be rewritten as\n \\[\n   t = O (m^{2 n} \\phi^{n-1})\n \\]\n for fixed $n$. Clearly, the greatest common divisor of the components\n of any direction $c_i$ obtained by the algorithm must be equal to\n $1$, as otherwise it would not be a vertex of the integer hull of\n \\eqref{eq:c-ilp-2}. This implies, in particular, that the Hermite\n normal form of any of these vectors is just the first unit vector\n $e_1 \\in \\setR^n$.\n\n It is also worth mentioning that if $(F_i, G_i, c_i)$ is a triple\n attaining the minimum in Part \\eqref{item:flat-dirs-b} of Lemma\n \\ref{lem:flat-dirs}, then we have\n \\[\n   w(P_b) \\leq \\max \\{ c_i x : x \\in P_b \\} - \\min \\{ c_i x : x \\in\n   P_b \\} \\leq c_i F_i b - c_i G_i b = w(P_b).\n \\]\n Hence, Part \\eqref{item:flat-dirs-a}, when applied to this triple,\n turns into\n \\[\n   \\min \\{ c_i x : x \\in P_b \\} = c_i G_i b \\quad \\text{and} \\quad\n   \\max \\{ c_i x : x \\in P_b \\} = c_i F_i b.\n \\]\n\n For our further purposes, it is more suitable to have a \\emph{unique}\n width direction for all polyhedra $P_b$ with varying $b$. In fact,\n using Lemma \\ref{lem:flat-dirs}, we can partition the set of the\n right-hand sides $b$ into a number of partially open polyhedra such\n that the width direction remains the same for all $b$ belonging to\n the same region of the partition.\n\n \\begin{theorem}\n   \\label{thm:flat-dirs}\n   Let $P$ be a parametric polyhedron of finite lattice width, defined\n   by a matrix $A \\in \\setQ^{m \\times n}$ of full column rank. Let $Q\n   \\subseteq \\setR^m$ be a rational partially open polyhedron such\n   that $P_b$ is non-empty for all $b \\in Q$. We can compute---in\n   polynomial time, if $n$ is fixed---a partition of $Q$ into a number\n   of partially open polyhedra $Q_1, \\ldots, Q_t$ and, for each $i$,\n   find a triple $(F_i, G_i, c_i)$ of linear transformations $F_i, G_i\n   : \\setR^m \\to \\setR^n$ and a non-zero vector $c_i \\in \\setZ^n$,\n   such that\n   \\[\n     \\min \\{ c_i x : x \\in P_b \\} = c_i G_i b, \\quad \\max \\{ c_i x : x\n     \\in P_b \\} = c_i F_i b,\n   \\]\n   and\n   \\[\n     w(P_b) = w_{c_i}(P_b) = c_i (F_i - G_i) b \\quad \\text{for all } b\n     \\in Q_i.\n   \\]\n   If $\\phi$ denotes the maximum size of a column in $A$, then $t = O\n   ( m^{2 n} \\phi^{n-1} )$.\n \\end{theorem}\n \n \\begin{remark}   \n   The statement of Theorem \\ref{thm:flat-dirs} is very analogous to\n   \\cite[Lemma~3.1]{Kannan1992}. However, there are several crucial\n   differences. First, the number of regions in the partition obtained\n   by Kannan's algorithm is exponential in $n$ and the affine\n   dimension $j_0$ of the polyhedron $Q$. Our algorithm yields a\n   partition that is exponential in $n$ only, hence polynomial if $n$\n   is fixed. Also our algorithm runs in polynomial time if $n$ is\n   fixed but $j_0$ may vary. At last, the width directions $c_i$\n   obtained by the Kannan's algorithm satisfy, for any $b \\in Q_i$,\n   \\[\n     \\text{either} \\quad w_{c_i}(P_b) \\leq 1 \\quad \\text{or} \\quad\n     w_{c_i}(P_b) \\leq 2 \\, w(P_b)\n   \\]\n   In contrast, we compute the \\emph{exact} width direction for each\n   region in the partition. While the exact computation of width\n   directions does not help much from the algorithmic point of view,\n   removing the restriction on the dimension of $Q$ turns out to be\n   the main step towards the claimed generalization of Kannan's\n   algorithm.\n \\end{remark}\n\n \\begin{proof}[Proof of Theorem \\ref{thm:flat-dirs}]\n   First, we exploit the algorithm of Lemma \\ref{lem:flat-dirs} to\n   obtain the triples $(F_i, G_i, c_i)$, $i = 1, \\ldots, t$, with $t =\n   O (m^n \\phi^{n-1})$. These triples provide the possible width\n   directions of the parametric polyhedron $P$. For each $i = 1,\n   \\ldots, t$, we define a partially open polyhedron $Q_i$ by the\n   inequalities\n   \\begin{align*}\n     & c_i (F_i - G_i) b < c_j (F_j - G_j) b, \\quad j = 1, \\ldots, i -\n     1, \\\\ & c_i (F_i - G_i) b \\leq c_j (F_j - G_j) b, \\quad j = i +\n     1, \\ldots, t.\n   \\end{align*}\n   Thus,\n   \\[\n     \\min_j \\, c_j (F_j - G_j) b = c_i (F_i - G_i) b\n   \\]\n   for all $b \\in Q_i$. We claim that the intersections of the\n   partially open polyhedra $Q_i$ with $Q$ give the required\n   partition.\n   \n   Indeed, let $b \\in Q$ and let $\\mu$ be the minimum value of $c_i\n   (F_i - G_i) b$, $i = 1, \\ldots, t$. Let $I$ denote the set of\n   indices $i$ with $c_i (F_i - G_i) b = \\mu$. Then $b \\in Q_{i_0}$,\n   where $i_0$ is the smallest index in $I$. Yet, suppose that $b \\in\n   Q$ belongs to two partially open polyhedra, say $Q_i$ and\n   $Q_j$. Without loss of generality, we can assume $i < j$. But then\n   we have\n   \\[\n     c_i (F_i - G_i) b \\leq c_j (F_j - G_j) b < c_i (F_i - G_i) b,\n   \\]\n   where the first inequality is due to the fact $b \\in Q_i$ and the\n   second inequality follows from $b \\in Q_j$; both together are a\n   contradiction.\n\n   For the width directions, Lemma \\ref{lem:flat-dirs} implies that\n   \\[\n     w(P_b) = \\min_j \\, c_j (F_j - G_j) b = c_i (F_i - G_i) b\n   \\]\n   for all $b \\in Q_i \\cap Q$. This completes the proof.\n \\end{proof}\n\n \\noindent Theorem \\ref{thm:flat-dirs} provides a unique width\n direction for each region $Q_i$ of the partition. This resolves the\n first problem of adapting the algorithm for integer linear\n programming in fixed dimension to the case of varying $b$, which was\n addressed in the introduction. However, we still need to deal with\n the hyper-planes $\\{ x : c_i x = \\lceil \\beta \\rceil + j \\}$, where\n $\\beta$ is the optimum value of the linear program $\\min \\{ c_i x : x\n \\in P_b \\}$. As mentioned above, $\\beta$ can be expressed as a linear\n transformation of $b$, namely $\\beta = c_i G_i b$. But $\\lceil \\beta\n \\rceil$ is no more a \\emph{linear} function of $b$, which makes the\n recursion complicated. \\cite{Kannan1992} showed how to tackle this\n problem. We discuss it in the next section.\n\n\\section{Partitioning theorem and parametric integer programming}\n\\label{sec:partit}\n\n The proof of the following structural result follows from the proof\n of \\cite[Theorem 4.1]{Kannan1992} if it is combined with Theorem\n \\ref{thm:flat-dirs}.\n\n \\begin{theorem}\n   \\label{thm:structural}\n   Let $P$ be a parametric polyhedron of finite lattice width, defined\n   by a rational matrix $A \\in \\setQ^{m \\times n}$ of full column\n   rank. Let $Q \\subseteq \\setR^m$ be a rational partially open\n   polyhedron such that $P_b$ is non-empty for all $b \\in Q$. We can\n   compute---in polynomial time, if $n$ is fixed---a partition of $Q$\n   into sets $S_1, \\ldots, S_t$, and for each $i$, find a number of\n   unimodular transformations $U_{ij} : \\setR^n \\to \\setR^n$ and\n   affine transformations $T_{ij} : \\setR^m \\to \\setR^n$, $j = 1,\n   \\ldots, k_i$ such that\n   \\begin{enumerate}[(a)]\n     \\item\n       each $S_i$ is the integer projection of a partially open\n       polyhedron, $S_i = S'_i \/ \\setZ^{l_i}$;\n     \\item\n       for any $b \\in S_i$, $P_b \\cap \\setZ^n \\neq \\emptyset$ if and\n       only if $P_b$ contains $U_{ij} \\lceil T_{ij} b \\rceil$ for some\n       index $j$;\n     \\item\n       the following bounds hold:\n       \\[\n         t = O ((m^{2 n} \\phi^{n-1})^{n \\overline{\\omega}(n)}), \\quad\n         l_i = O (\\overline{\\omega}(n)), \\quad k_i = O (2^{n^2 \/ 2}\n         \\overline{\\omega}(n)), \\quad i = 1, \\ldots, t,\n       \\]\n       where $\\phi$ denotes the maximum size of a column in $A$ and\n       $\\overline{\\omega}(n) = \\prod_{i=1}^n \\omega(n)$.\n    \\end{enumerate}\n \\end{theorem}\n\n We do not repeat Kannan's proof here but give an intuition of why it\n is true in dimension 2. By Theorem~\\ref{thm:flat-dirs} we can assume\n that the width-direction is invariant for all $b \\in Q$ and by\n applying a unimodular transformation we can further assume that this\n width-direction is the first unit-vector $e_1$, see Figure\n \\ref{fig:dim2}.\n \\begin{figure}[ht]\n   \\centering {\n     \\psset{unit=.7cm}\n     \\begin{pspicture}(-1,-5)(11,3)    \n       \\pspolygon[linestyle=dashed,linecolor=lightgray,\n         fillstyle=solid,fillcolor=vvlg]\n         (0,0)(1.5,1.5)(3,2)(8,2)(10,.5)(11,-1)(10,-2.5)(4,-2)(1.5,-1)\n       \\psline(0,0)(3,3)\n       \\psline(0,0)(3,-2)\n\n       \\psline[arrows=->](0,0)(2,0)\n\n       \\psline(1,-4)(1,3)\n       \\psline(4.5,-4)(4.5,3)\n       \\psdot*(4.5,-1.5)\n\n       \\rput(1.5,-.5){$e_1$}\n       \\rput(7.5,1.5){$P_b$}\n\n       \\rput[l](-1,0){$G b$}\n       { \\small\n         \\rput[l](.5,-4.5){$x_1=\\lceil e_1 G b \\rceil$}\n         \\rput[l](3.5,-4.5){$x_1=\\lceil e_1 G b \\rceil+i$}\n       }\n    \n       \\psdots(0,0\n     \\end{pspicture}\n     \\caption{Illustration of Theorem~\\ref{thm:structural} in dimension 2.}\n     \\label{fig:dim2}\n   }\n \\end{figure}\n\n Lemma~\\ref{lem:flatness-app} tells that $P_b$ contains an integral\n point if and only if there exists an integral point on the lines $x_1\n = \\lceil e_1 G b \\rceil + j$ for $j = 0, \\ldots, \\omega(2)$. The\n intersections of these lines with the polyhedron $P_b$ are\n 1-dimensional polyhedra. Some of the constraints $a x \\leq \\beta$ of\n $A x \\leq b$ are ``pointing upwards'', i.e., $a e_2 < 0$, where $e_2$\n is the second unit-vector. Let $a_1 x_1+a_2x_2 \\leq \\beta$ be a\n constraint pointing upwards such that the intersection point $(\\lceil\n e_1 G b \\rceil + j, y)$ of $a_1 x_1+a_2x_2 = \\beta$ with the line\n $x_1 = \\lceil e_1 G b \\rceil + j$ has the largest second\n component. The line $x_1 = \\lceil e_1 G b \\rceil + j$ contains an\n integral point in $P_b$ if and only if $(\\lceil e_1 G b \\rceil + j,\n \\lceil y \\rceil)$ is contained in $P_b$. This point is illustrated in\n Figure~\\ref{fig:dim2}. By choosing the highest constraint pointing\n upwards for each line $x_1 =\\lceil e_1 G b \\rceil + j$, we partition\n the set of right-hand sides into polynomially many integer\n projections of partially open polyhedra.\n\n In order to express the candidate solution $(\\lceil e_1 G b \\rceil +\n j, \\lceil y \\rceil)$ in the form described in the theorem, observe\n that\n \\[\n   y =  (\\beta - a_1 x_1) \/ a_2. \n \\]\n Since $x_1$ is an integer and\n \\[\n   x_1 = \\lceil e_1 G b \\rceil + j = e_1 G b + j + \\gamma\n \\]\n for some $\\gamma \\in [0,1)$, we can rewrite the equation $\\lceil y\n \\rceil = \\lceil (\\beta - a_1 x_1) \/ a_2 \\rceil$ as\n \\begin{align*}\n   \\lfloor a_1 \/ a_2 \\rfloor x_1 + \\lceil y \\rceil &= \\lceil \\beta \/\n   a_2 - \\{ a_1 \/ a_2 \\} x_1 \\rceil \\\\ &= \\lceil \\beta \/ a_2 - \\{ a_1\n   \/ a_2 \\} (e_1 G b + j) - \\{ a_1 \/ a_2 \\} \\gamma \\rceil,\n \\end{align*}\n where $\\{ a_1 \/ a_2 \\}$ denotes the fractional part of $a_1 \/\n a_2$. Since $\\{ a_1 \/ a_2 \\} \\gamma$ lies between 0 and 1, it\n suffices to check independently two different possibilities, namely,\n \\[\n   \\lfloor a_1 \/ a_2 \\rfloor x_1 + \\lceil y \\rceil = \\lceil \\beta \/\n   a_2 - \\{ a_1 \/ a_2 \\} (e_1 G b + j) \\rceil\n \\]\n and\n \\[\n   \\lfloor a_1 \/ a_2 \\rfloor x_1 + \\lceil y \\rceil = \\lceil \\beta \/\n   a_2 - \\{ a_1 \/ a_2 \\} (e_1 G b + j) - 1 \\rceil.\n \\]\n Combined with $x_1 = \\lceil e_1 G b \\rceil + j$, each of the above\n equations yields a unimodular system with respect to the variables\n $x_1$ and $\\lceil y \\rceil$, with the right-hand side being the\n round-up of an affine transformation of $b$. We refer the reader to\n \\citep{Kannan1992} to see the complete proof for arbitrary dimension.\n\n\\subsection*{\\pmb{$\\forall \\,\\exists$}-statements} \n\n Theorem \\ref{thm:structural} gives rise to a polynomial algorithm for\n testing sentences of the form\n \\begin{equation}\n   \\label{eq:pilp-appl}\n   \\forall b \\in Q \/ \\setZ^p \\quad \\exists x \\in \\setZ^n : \\quad A x\n   \\leq b,\n \\end{equation}\n when $p$ and $n$ are fixed. This algorithm was first described by\n \\cite{Kannan1992} but he required, in addition, the affine dimension\n of $Q$ to be fixed. Our improvement follows basically from the\n improvement in the partitioning theorem, while the algorithm itself\n remains exactly the same. We describe it here for the sake of\n completeness.  First observe that we can assume that $A$ has full\n column rank. Otherwise we can apply a unimodular transformation of\n $A$ from the right to obtain a matrix $[A' \\mid 0]$, where $A'$ has\n full column rank.\n\n The idea is as follows: First we run the algorithm of Theorem\n \\ref{thm:structural} on input $A$ and $Q' \\subseteq \\setR^m$, where\n $Q'$ is the set of vectors $b$, for which the system $A x \\leq b$ has\n a solution. Then we consider each set $S_i$ returned by the algorithm\n of Theorem \\ref{thm:structural} independently. For each $b \\in S_i$\n we have a fixed number of candidate solutions for the system $A x\n \\leq b$, defined via unimodular and affine transformations as $U_{ij}\n \\lceil T_{ij} b \\rceil$. Each rounding operation can be expressed by\n introducing an integral vector: $z = \\lceil T_{ij} b \\rceil$ is\n equivalent to $T_{ij} b \\leq z < T_{ij} b + \\mathbf{1}$. We need only\n a constant number of integer variables to express all candidate\n solutions plus a fixed number of integer variables to represent the\n integer projections $S_i = S'_i \/ \\setZ^{l_i}$. It remains to solve a\n number of mixed-integer programs, to which we also include the\n constraints $(b, y) \\in Q$, $y \\in \\setZ^p$.\n\n \\begin{theorem}\n   \\label{thm:main}\n   There is an algorithm that, given a rational matrix $A \\in \\setQ^{m\n   \\times n}$ and a rational polyhedron $Q \\subseteq \\setR^{m+p}$,\n   decides the sentence \\eqref{eq:pilp-appl}. The algorithm runs in\n   polynomial time if $p$ and $n$ are fixed.\n \\end{theorem}\n\n \\begin{proof}\n   Let $P$ be a parametric polyhedron defined by the matrix\n   $A$. First, we exploit the Fourier--Motzkin elimination procedure\n   to construct the polyhedron $Q' \\subseteq \\setR^m$ of the\n   right-hand sides $b$, for which the system $A x \\leq b$ has a\n   (fractional) solution. For each inequality $a b \\leq \\beta$,\n   defining the polyhedron $Q'$, we can solve the following\n   mixed-integer program\n   \\[\n     a b > \\beta, \\quad (b, y) \\in Q, \\quad y \\in \\setZ^p,\n   \\]\n   and if any of these problems has a feasible solution $(y, b)$, then\n   $b$ is a vector in $Q \/ \\setZ^p$, for which the system $A x \\leq b$\n   has no integral solution. Hence, we can terminate and output ``no''\n   (with $b$ being a certificate).\n\n   We can assume now that for all $b \\in Q \/ \\setZ^p$ the system $A x\n   \\leq b$ has a fractional solution. By applying the algorithm of\n   Theorem \\ref{thm:structural}, we construct a partition of $Q'$ into\n   the sets $S_1, \\ldots, S_t$, where each $S_i$ is the integer\n   projection of a partially open polyhedron, $S_i = S'_i \/\n   \\setZ^{l_i}$. Since $n$ is fixed, the $l_i$ are bounded by some\n   constant, $i = 1, \\ldots, t$. Furthermore, for each $i$, the\n   algorithm constructs unimodular transformations $U_{ij}$ and affine\n   transformations $T_{ij}$, $j = 1, \\ldots, k_i$, such that $P_b$,\n   with $b \\in S_i$, contains an integral point if and only if $U_{ij}\n   \\lceil T_{ij} b \\rceil \\in P_b$ for some $j$. Again, $k_i$ is fixed\n   for a fixed $n$, $i = 1, \\ldots, t$.\n\n   The algorithm will consider each index $i$ independently. For a\n   given $i$, $S_i$ can be described as the set of vectors $b$ such\n   that\n   \\[\n     (b, z) \\in S'_i\n   \\]\n   has a solution for some integer $z \\in \\setZ^{l_i}$. This can be\n   expressed in terms of linear constraints, as $S'_i$ is a partially\n   open polyhedron. Let $x_j = U_{ij} \\lceil T_{ij} b \\rceil$. The\n   points $x_j$ can be described by linear inequalities as\n   \\[\n     T_{ij} b \\leq z_j < T_{ij} b + \\mathbf{1}, \\quad x_j = U_{ij}\n     z_j,\n   \\]\n   where $\\mathbf{1}$ is the all-one vector. Then $P_b$ does not\n   contain an integral point if and only if $x_j \\notin P_b$ for all\n   $j = 1, \\ldots, k_i$. In this case, each $x_j$ violates at least\n   one constraint in the system $A x \\leq b$. We consider all possible\n   tuples $I$ of $k_i$ constraints from $A x \\leq b$. Obviously, there\n   are only $m^{k_i}$ such tuples, that is, polynomially many in the\n   input size. For each such tuple, we solve the mixed-integer program\n   \\[ \\begin{array}{ll}\n     (b, y) \\in Q, \\, (b, z) \\in S'_i, \\medskip \\\\ T_{ij} b \\leq z_j <\n     T_{ij} b + \\mathbf{1}, & j = 1, \\ldots, k_i, \\medskip \\\\ x_j =\n     U_{ij} z_j, & j = 1, \\ldots, k_i, \\medskip \\\\ a_{i_j} x_j >\n     b_{i_j}, & j = 1, \\ldots, k_i, \\medskip \\\\ y \\in \\setZ^p, \\, z\n     \\in \\setZ^{l_i}, \\, z_j \\in \\setZ^n, & j = 1, \\ldots, k_i,\n   \\end{array} \\]\n   where $a_{i_j} x \\leq b_{i_j}$ is the $j$-th constraint in the\n   chosen tuple. Each such mixed-integer program can be solved in\n   polynomial time since the number of integer variables is fixed (in\n   fact, there are at most $(k_i + 1) n + l_i$ integer variables).\n\n   If there is a feasible solution $b$ to one of these mixed-integer\n   programs, then the answer to the original problem is ``no'' (with\n   $b$ being a certificate). If all these mixed-integer programs are\n   infeasible, the answer is ``yes''.\n \\end{proof}\n\n \\begin{remark}\n   \\label{rem:1}\n   We would like to point out that Theorem \\ref{thm:main} can also be\n   proved differently. \\cite{Bell77} and \\cite{Scarf77} showed that if\n   a system of linear inequalities $A x \\leq b$ has no integral\n   solution, then there is already a subsystem of at most $2^n$\n   inequalities that is infeasible in integer variables; see also\n   \\cite[Theorem 16.5]{Schrijver1986}. Applied to \\eqref{eq:pilp} this\n   means that there exists a $b \\in Q \/ \\setZ^p$ such that the system\n   $A x \\leq b$ has no integral solution, if and only there exist a $b\n   \\in Q \/ \\setZ^p$ and a subsystem $A' x \\leq b'$ of $A x \\leq b$\n   with at most $2^n$ inequalities, which is infeasible in integer\n   variables. Here $b'$ is the projection of $b$ into the according\n   space. Since $n$ is a constant, we can try out all $\\binom{m}{2^n}$\n   different subsystems of $A x \\leq b$ and apply to each of these\n   parametric polyhedra Kannan's algorithm to decide $\\forall\\,\n   \\exists$-statements.\n\n   However, a similar argument does not yield our extension (Theorem\n   \\ref{thm:structural}) of Kannan's partitioning theorem itself,\n   which associates to each $b$ a fixed set of candidate integer\n   solutions, depending on the partially open polyhedron of the\n   partitioning, in which $b$ is contained.\n \\end{remark}\n\n\\section{Integer programming gaps} \n \n Now, we describe how Theorem \\ref{thm:main} can be applied to compute\n the maximum integer programming gap for a family of integer programs.\n Let $A \\in \\setQ^{m \\times n}$ be a rational matrix and let $c \\in\n \\setQ^n$ be a rational vector. Let us consider the integer programs\n of the form\n \\begin{equation}\n   \\label{eq:ilp}\n   \\max \\{ c x : A x \\leq b, \\, x \\in \\setZ^n \\},\n \\end{equation}\n where $b$ is varying over $\\setR^m$. The corresponding linear\n programming relaxations are then\n \\begin{equation}\n   \\label{eq:lp}\n   \\max \\{ c x : A x \\leq b \\}.\n \\end{equation}\n Consider the following system of inequalities:\n \\begin{align*}\n   c x & \\geq \\beta, \\medskip \\\\ A x & \\leq b.\n \\end{align*}\n Given a vector $b$ and a number $\\beta$, there exists a feasible\n fractional solution of the above system if and only if the linear\n program \\eqref{eq:lp} is feasible and its value is at least $\\beta$.\n The set of pairs $(\\beta, b) \\in \\setR^{m+1}$, for which the above\n system has a fractional solution, is a polyhedron in $\\setR^m$ and\n can be computed by means of Fourier--Motzkin elimination, in\n polynomial time if $n$ is fixed. Let $Q$ denote this polyhedron.\n\n Suppose that we suspect the maximum integer programming gap to be\n smaller than $\\gamma$. This means that, whenever $\\beta$ is an\n optimum value of \\eqref{eq:lp}, the integer program \\eqref{eq:ilp}\n must have a solution of value at least $\\beta -\n \\gamma$. Equivalently, the system\n \\begin{align}\n   \\label{eq:gap-system}\n   c x & \\geq \\beta - \\gamma, \\medskip \\\\ A x &\\leq b, \\nonumber\n \\end{align}\n must have an integral solution. If there exists $(b, \\beta) \\in Q$\n such that \\eqref{eq:gap-system} has no integral solution, the integer\n programming gap is bigger than $\\gamma$. We also need to ensure that\n for a given $b$, the integer program is feasible, i.e., the system $A\n x \\leq b$ has a solution in integer variables.\n\n Now, this is exactly the question for the algorithm of Theorem\n \\ref{thm:main}: Is there a $(\\beta, b) \\in Q'$ such that the system\n \\eqref{eq:gap-system} has no integral solution, but there exists $y\n \\in \\setZ^n$ such that $A y \\leq b$? Here $Q' = Q - \\gamma (1, 0)$ is\n the appropriate translate of the set $Q$. If the algorithm answers\n ``no'' with the certificate $b$, then the integer program\n \\eqref{eq:ilp}, with the right-hand side $b$, has no solution of\n value greater than $\\beta - \\gamma$ while being feasible. On the\n other hand, $(\\beta, b) \\in Q$, thus the corresponding linear\n solution has optimum value of at least $\\beta$. We can conclude that\n the maximum integer programming gap is greater than $\\gamma$. This\n gives us the following theorem.\n\n \\begin{theorem}\n   \\label{thm:gap}\n   There is an algorithm that, given a rational matrix $A \\in \\setR^{m\n   \\times n}$, a rational row-vector $c \\in \\setQ^n$ and a number\n   $\\gamma$, checks whether the maximum integer programming gap for\n   the integer programs \\eqref{eq:ilp} defined by $A$ and $c$ is\n   bigger than $\\gamma$. The algorithm runs in polynomial time if the\n   rank of $A$ is fixed.\n \\end{theorem}\n\n \\noindent Using binary search, we can also find the \\emph{minimum}\n possible value of $\\gamma$, hence the maximum integer programming\n gap.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Factorization in trigonometric vertex models} In\n\\cite{fwz}, we obtained factorized domain wall partition\nfunctions (DWPF's)\\footnote{For a review of previous results\non the subject, see \\cite{korepin-review}.} in two series of\ntrigonometric vertex models:\n{\\bf 1.} The $N$-state Deguchi-Akutsu models, for\n$N \\in \\{2, 3, 4\\}$ (and conjectured the result for $N \\geq \n5$), and\n{\\bf 2.} The $gl(r+1 | s+1)$ Perk-Schultz models, $\\{r, s\\} \n\\in \\field{N}$ (where given the symmetries of these models, the \nresult is independent of $r$ and $s$).\n\n\\subsection{Asymmetry} These models were characterized by \nan asymmetry of the vertex weights under conjugation of \nstate variables. For example, in the Deguchi-Akutsu model, \nwith state variables $\\sigma \\in \\{1, \\cdots{}, N\\}$, the \nvertex weights are non-invariant under the conjugation\n$\\sigma \\rightarrow (N - \\sigma + 1)$. In the Perk-Schultz\nmodels, a similar property holds. Since one can trace the\nfactorization of the DWPF's obtained in \\cite{fwz} to this\nasymmetry, it is natural to look for height models\\footnote{Also\nknown as face or interaction-round-face (IRF) models.}\nwith the same property.\n\n\\subsection{Factorization in elliptic height models} In this\nwork we consider height models, where the state variables are\nheights, that live on the corners of the faces of a square\nlattice. A weight is assigned to each face. The weights are\nelliptic functions of the corresponding rapidities, external\nfields (if any) and height variables. As in \\cite{fwz}, the\nmodels in this work are characterized by an asymmetry of the\nweights, in the sense that the weights of certain vertices\n(called line-permuting vertices) have different zeros, which\nleads to the factorization of the DWPF's. \n\n\\subsection{Summary of results} We obtain factorized DWPF's\nfor two elliptic height models:\n{\\bf 1.} A Felderhof-type model, which (to the best of our\nknowledge) is new, and\n{\\bf 2.} A Perk-Schultz-type $gl(1|1)$ model of Deguchi\nand Martin \\cite{dm}. These are the first examples\nof DWPF's for elliptic and\/or height models.\n\n\\subsection{Outline of paper}\nIn Section {\\bf 1}, we collect a number of basic definitions\nrelated to elliptic height models to make the paper reasonably\nself-contained.\nIn Section {\\bf 2}, we introduce a new Felderhof-type elliptic\nheight model and obtain the corresponding factorized DWPF.\nIn Section {\\bf 3}, we do the same for the $gl(1|1)$ elliptic\nheight model of Deguchi and Martin.\nSection {\\bf 4} contains brief remarks.\nThe presentation is elementary in the hope that the paper will \nbe reasonably self-contained. \n\n\\section{Height models}\n\n\\subsection{Faces and corners} We work on a square lattice, \nas in Figure {\\bf \\ref{lattice}}, with $L^{2}$ square faces \n$f_{ij}$, where $1 \\leq i \\leq L$ increases from top to \nbottom, and $1 \\leq j \\leq L$ increases from left to right. \n$f_{ij}$ has four corners that are labelled from top-left \nclockwise as \n$\\{c_{i, j}, c_{i, j+1}, c_{i+1, j+1}, c_{i+1, j}\\}$.\n\n\\begin{center}\n\\begin{minipage}{5.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(4800, 6000)(-2500, -1500)\n\\thicklines\n\\path(0000,3600)(3600,3600)\n\\path(0000,3000)(3600,3000)\n\\path(0000,2400)(3600,2400)\n\\path(0000,1800)(3600,1800)\n\\path(0000,1200)(3600,1200)\n\\path(0000,0600)(3600,0600)\n\\path(0000,0000)(3600,0000)\n\\path(0000,3600)(0000,0000)\n\\path(0600,3600)(0600,0000)\n\\path(1200,3600)(1200,0000)\n\\path(1800,3600)(1800,0000)\n\\path(2400,3600)(2400,0000)\n\\path(3000,3600)(3000,0000)\n\\path(3600,3600)(3600,0000)\n\\put(0150,3250){$f_{11}$}\n\\put(0150,2650){$f_{21}$}\n\\put(3150,3250){$f_{1L}$}\n\\put(0150,0250){$f_{L1}$}\n\n\\put(0750,0250){$f_{L2}$}\n\\put(3150,0250){$f_{LL}$}\n\\put(-400,3600){$h_{00}$}\n\\put(-400,3000){$h_{10}$}\n\\put(-400,0000){$h_{L0}$}\n\n\\put(0000,3600){\\blacken\\circle{50,50}}\n\\put(3600,3600){\\blacken\\circle{50,50}}\n\\put(0000,3000){\\blacken\\circle{50,50}}\n\\put(0000,2400){\\blacken\\circle{50,50}}\n\\put(0000,1800){\\blacken\\circle{50,50}}\n\\put(0000,1200){\\blacken\\circle{50,50}}\n\\put(0000,0600){\\blacken\\circle{50,50}}\n\\put(3700,3600){$h_{0L}$}\n\\put(3700,0000){$h_{LL}$}\n\\put(0000,0000){\\blacken\\circle{50,50}}\n\\put(0600,0000){\\blacken\\circle{50,50}}\n\\put(0500,-300){$h_{10}$}\n\\put(3600,0000){\\blacken\\circle{50,50}}\n\\thinlines\n\\dashline{60.00}(-200,3300)(0050,3300)\n\\dashline{60.00}(0550,3300)(3050,3300)\n\\dashline{60.00}(3550,3300)(3900,3300)\n\n\\dashline{60.00}(-200,2700)(0050,2700)\n\\dashline{60.00}(0550,2700)(3900,2700)\n\\thicklines\n\\whiten\\path(-500,3255)(-320,3300)(-500,3345)(-500,3255)\n\\path(-500,3300)(-680,3300)\n\\thinlines\n\\put(-1400,3300){$u_1, p_1$}\n\\thicklines\n\\whiten\\path(-500,2655)(-320,2700)(-500,2745)(-500,2655)\n\\path(-500,2700)(-680,2700)\n\\thinlines\n\\dashline{60.00}(-200,2100)(3900,2100)\n\\thicklines\n\\whiten\\path(-500,2055)(-320,2100)(-500,2145)(-500,2055)\n\\path(-500,2100)(-680,2100)\n\\thinlines\n\\dashline{60.00}(-200,1500)(3900,1500)\n\\thicklines\n\\whiten\\path(-500,1455)(-320,1500)(-500,1545)(-500,1455)\n\\path(-500,1500)(-680,1500)\n\\thinlines\n\\dashline{60.00}(-200,0900)(3900,0900)\n\\thicklines\n\\whiten\\path(-500,0855)(-320,0900)(-500,0945)(-500,0855)\n\\path(-500,0900)(-680,0900)\n\\thinlines\n\\put(-1400,2700){$u_2, p_2$}\n\\dashline{60.00}(-200,0300)(0050,0300)\n\\dashline{60.00}(0550,0300)(0650,0300)\n\\dashline{60.00}(1150,0300)(3050,0300)\n\\dashline{60.00}(3550,0300)(3900,0300)\n\\thicklines\n\\whiten\\path(-500,0255)(-320,0300)(-500,0345)(-500,0255)\n\\path(-500,0300)(-680,0300)\n\\thinlines\n\\put(-1400,0300){$u_L, p_L$}\n\\dashline{60.00}(0300,3900)(0300,3550)\n\\dashline{60.00}(0300,3050)(0300,2900)\n\\dashline{60.00}(0300,2500)(0300,0550)\n\\dashline{60.00}(0300,0050)(0300,-200)\n\\thicklines\n\\whiten\\path(0255,-500)(0345,-500)(0300,-320)(0255,-500)\n\\path(0300,-680)(0300,-500)\n\\thinlines\n\\put(0200,-0900){$v_1,$}\n\\put(0200,-1200){$q_1$}\n\\dashline{60.00}(0900,3900)(0900,0550)\n\\dashline{60.00}(0900,0050)(0900,-200)\n\\thicklines\n\\whiten\\path(0855,-500)(0945,-500)(0900,-320)(0855,-500)\n\\path(0900,-680)(0900,-500)\n\\thinlines\n\\put(0800,-0900){$v_2,$}\n\\put(0800,-1200){$q_2$}\n\\dashline{60.00}(1500,3900)(1500,-200)\n\\thicklines\n\\whiten\\path(1455,-500)(1545,-500)(1500,-320)(1455,-500)\n\\path(1500,-680)(1500,-500)\n\\thinlines\n\\dashline{60.00}(2100,3900)(2100,-200)\n\\thicklines\n\\whiten\\path(2055,-500)(2145,-500)(2100,-320)(2055,-500)\n\\path(2100,-680)(2100,-500)\n\\thinlines\n\\dashline{60.00}(2700,3900)(2700,-200)\n\\thicklines\n\\whiten\\path(2655,-500)(2745,-500)(2700,-320)(2655,-500)\n\\path(2700,-680)(2700,-500)\n\\thinlines\n\\dashline{60.00}(3300,3900)(3300,3550)\n\\dashline{60.00}(3300,3050)(3300,0550)\n\\dashline{60.00}(3300,0050)(3300,-200)\n\\thicklines\n\\whiten\\path(3255,-500)(3345,-500)(3300,-320)(3255,-500)\n\\path(3300,-680)(3300,-500)\n\\thinlines\n\\put(3200,-0900){$v_L,$}\n\\put(3200,-1200){$q_L$}\n\\end{picture}\n\\begin{ca}\n\\label{lattice}\nA square lattice with $L^{2}$ faces $f_{ij}$. Rapidities\n$\\{u, v\\}$ and external fields $\\{p, q\\}$ flow along lines\nthat cross the faces. Height variables $h_{ij}$ live on the\ncorners.\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\n\\bigskip\n\n\\subsection{Heights and restrictions}\n\nWe assign each corner $c_{ij}$ a height variable $h_{ij}$,\n$0 \\leq i, j \\leq L$. In height models such as Baxter's\nsolid-on-solid model \\cite{baxter-book}, the heights are\nintegral (possibly up to an overall shift). In the model\nof Section {\\bf 2}, the height variables depend linearly\non the external fields which are continuous parameters,\nso they are no longer integral. We define the heights and\nthe restrictions that they obey on a model by model basis\nin Sections {\\bf 2} and {\\bf 3}.\n\n\\subsection{Flow lines, orientations and variables} There are\n$L$ horizontal and $L$ vertical lines that intersect at the middle\npoints of $f_{ij}$. They indicate the flow of rapidities and\nexternal fields through $f_{ij}$.\nWe assign the $i$-th horizontal line an orientation from left\nto right, a complex rapidity $u_i$ and a complex external\nfield $p_i$. We assign the $j$-th vertical line\nan orientation from bottom to top, a complex rapidity \n$v_j$ and a complex external field $q_j$, as in\nFigure {\\bf \\ref{lattice}}.\n\n\\subsection{Weights and Yang-Baxter equations}\n\nWe assign each $f_{ij}$ a weight $w_{ij}$\nthat depends on the height variables on its corners, the difference\nof the rapidity variables and the two external field variables (if\nany) flowing through it. The weights satisfy a set of Yang-Baxter\nequations. The weights and Yang-Baxter equations of the models\ndiscussed in this paper are given in Sections {\\bf 2} and {\\bf 3}.\n\n\\subsection{Elliptic functions and a theorem} \nFollowing the conventions used in \\cite{baxter-book}, Chapter \n{\\bf 15}, we consider the elliptic function\n\n\\begin{equation}\nH(u) =\n2 \\textsf{q}^{1\/4} \n\\sin \\left\\lgroup \\frac{\\pi u}{2I}\\right\\rgroup\n\\prod_{n=1}^{\\infty}\n\\left\\lgroup\n1 - 2 \\textsf{q}^{2n} \\cos \\frac{\\pi u}{I} + \\textsf{q}^{4n}\n\\right\\rgroup \n\\left\\lgroup \n1 - \\textsf{q}^{2n}\n\\right\\rgroup\n\\label{H}\n\\end{equation}\n\n\\noindent where $u \\in \\field{C}$, \n\\textsf{q}\\ $= \\exp\\left\\lgroup \\frac{- \\pi I'}{I} \\right\\rgroup$, $2I$ and $2I'$ (usually\ncalled $2K$ and $2K'$) are respectively the (real) width and height \nof an (upright) rectangle $R$ in the complex $u$-plane, so that\n$0 < \\textsf{q} < 1$. It is convenient to define\n\n\\begin{eqnarray}\n[u] = \\frac{H(u)}{2 \\textsf{q}^{1\/4}}\n\\label{HH}\n\\end{eqnarray}\n\n\\noindent which is entire and satisfies the quasi-periodicity\nproperties\n\n\\begin{eqnarray}\n&&\n[u + 2I] = -[u] \\\\ \n\\nonumber \\\\\n&&\n[u + 2iI']\n=\n- \\frac{1}{\\textsf{q}} \\exp\\left\\lgroup \\frac{- \\pi i u}{I}\\right\\rgroup [u]\n\\end{eqnarray}\n\n\\subsection*{Theorem 1} If $f(u)$ is an entire function that\nsatisfies the quasi-periodicity conditions\n\n\\begin{equation}\nf(u + 2I)\n=\n(-)^L f(u)\n\\end{equation}\n\n\\begin{equation}\nf(u + 2iI')\n=\n\\left\\lgroup \n\\frac{-1}{\\textsf{q}}\n\\right\\rgroup^{L}\n\\exp\n\\left\\lgroup \n\\frac{-\\pi i (L u - \\eta)}{I}\n\\right\\rgroup\nf(u)\n\\end{equation}\n\n\\noindent then\n\n\\begin{eqnarray}\nf(u)\n=\n\\kappa\n\\left\\lgroup\n\\prod_{j=1}^{L-1}\n[u-\\zeta_j]\n\\right\\rgroup\n[u-\\eta+\\sum_{j=1}^{L-1}\\zeta_j]\n\\end{eqnarray}\nwhere $\\kappa$ and $\\zeta_1,\\ldots,\\zeta_{L-1}$ are constants.\n\n\\bigskip\n\n\\noindent {\\it Proof.} This is a refinement of Theorem {\\bf 15(c)} \nof \\cite{baxter-book}, and the proof uses a similar argument. \nChoose the period rectangle $R$ such that $f(u)$ has no zeros on \nthe boundary $\\partial R$, and integrate $\\frac{f'(u)}{f(u)}$ \non the anti-clockwise contour $\\partial R$. From the quasi-periodicity \nconditions it follows that\n\n\\begin{eqnarray}\n\\oint_{\\partial R} \\frac{f'(u)}{f(u)}du=2\\pi iL\n\\end{eqnarray}\n\nHence the sum of residues of $\\frac{f'(u)}{f(u)}$ in $R$ is equal\nto $L$, showing $f(u)$ has exactly $L$ zeros in $R$ (counting a\nzero of order $n$ with multiplicity $n$). Writing the $L$ zeros\nas $\\zeta_1,\\ldots,\\zeta_L$, we define the function\n$\\phi(u)=\\prod_{j=1}^{L}[u-\\zeta_j]$. Since\n$\\frac{d}{du} \\log (f(u)\/\\phi(u))$ is doubly periodic\n(by construction) and holomorphic (also by construction) one has\n\n\\begin{eqnarray}\n\\frac{d}{du} \\log\n\\left\\lgroup \\frac{f(u)}{\\phi(u)} \\right\\rgroup = \\lambda\n\\end{eqnarray}\n\n\\noindent where $\\lambda$ is a constant. Integrating, we obtain\n$f(u)=\\kappa e^{\\lambda u}\\prod_{j=1}^{L}[u-\\zeta_j]$. Using the\nquasi-periodicity conditions of $f(u)$, we can, without loss of\ngenerality, choose \n$\\lambda=0$ and $\\zeta_L=\\eta-\\sum_{j=1}^{L-1}\\zeta_j$, which\nconcludes the proof.\n\n\\section{A Felderhof-type height model}\n\nIn this section, we introduce an elliptic height model with weights that\ndepend on rapidities, external fields and height variables. In the\ntrigonometric limit, it reduces to the first in a series of models\nintroduced by Deguchi and Akutsu in \\cite{da2}. In that same limit,\nand decoupling the dependence on the heights\\footnote{This can be\nachieved, for example, by introducing a parameter $\\xi \\in i\\field{R}$,\nshifting all height variables by $\\xi$ (the Yang-Baxter equations\nremain satisfied), then taking the limit $\\xi \\rightarrow i\\infty$.},\nit reduces to the trigonometric limit of the elliptic Felderhof\nvertex model, which is the 2-state Deguchi-Akutsu model \\cite{da1}.\n\n\\subsection{Notation}\n\nGiven the rapidities $\\{ u, v\\} \\in \\field{C}$, external fields\n$\\{ p, q\\} \\in \\field{C}$, an upper-left corner height $h \\in \\field{C}$,\n$\\{\\Delta_1, \\Delta_2\\} \\in \\{0, 1   \\}$, \n$  \\Delta_3             \\in \\{0, 1, 2\\}$, \nwe use the notation\n\n\\begin{eqnarray}\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}                          & \\textit{\\scriptsize{h}} \\+ \\textit{\\scriptsize{q}}      \\- \\textrm{\\scriptsize{$\\Delta$}}_{\\textrm{\\scriptsize{1}}}  \\\\\n\\textit{\\scriptsize{h}}     \\+ \\textit{\\scriptsize{p}} \\- \\textrm{\\scriptsize{$\\Delta$}}_{\\textrm{\\scriptsize{2}}}    & \\textit{\\scriptsize{h}} \\+ \\textit{\\scriptsize{q}} \\+ \\textit{\\scriptsize{p}} \\- \\textrm{\\scriptsize{$\\Delta$}}_{\\textrm{\\scriptsize{3}}} \n\\end{array}\n\\right\\rgroup\n\\end{eqnarray}\n\n\\noindent for the weight assigned to the \nvertex\\footnote{In the sequel, we simply \nsay `vertex' instead of `face configuration'.}\nrepresented in Figure {\\bf \\ref{general-vertex-f}}.\n\n\\begin{center}\n\\begin{minipage}{3.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(3000,3000)(0000,200)\n\\thicklines\n\\path(1500,2400)(2700,2400)\n\\path(1500,1200)(2700,1200)\n\\path(1500,2400)(1500,1200)\n\\path(2700,2400)(2700,1200)\n\\thinlines\n\\dashline{60.00}(1200,1800)(3000,1800)\n\\thicklines\n\\whiten\\path(0840,1740)(1000,1800)(0840,1860)(0840,1740)\n\\path(0680,1800)(0840,1800)\n \\put(0500,1900){\\textit{\\scriptsize{u}}\\scriptsize{,}\\textit{\\scriptsize{p}}}\n\\thinlines\n\\dashline{60.00}(2100,2700)(2100,0900)\n\\thicklines\n\\whiten\\path(2040,0540)(2100,0700)(2160,0540)(2040,0540)\n\\path(2100,0540)(2100,0380)\n\\put(2200,0380){\\textit{\\scriptsize{v}}\\scriptsize{,}\\textit{\\scriptsize{q}}}\n\\put(1500,2400){\\blacken\\ellipse{050}{050}}\n\\put(1300,2500){\\textit{\\scriptsize{h}}}\n\\put(1500,1200){\\blacken\\ellipse{050}{050}}\n\\put(1300,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\scriptsize{$-$}$\\textrm{\\scriptsize{$\\Delta$}}_{\\textrm{\\scriptsize{2}}}$}\n\\put(2700,2400){\\blacken\\ellipse{050}{050}}\n\\put(2500,2500){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\scriptsize{$-$}$\\textrm{\\scriptsize{$\\Delta$}}_{\\textrm{\\scriptsize{1}}}$}\n\\put(2700,1200){\\blacken\\ellipse{050}{050}}\n\\put(2500,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}\\scriptsize{$-$}$\\textrm{\\scriptsize{$\\Delta$}}_{\\textrm{\\scriptsize{3}}}$}\n\\end{picture}\n\\begin{ca}\n\\label{general-vertex-f}\nA Felderhof-type face configuration.\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\\bigskip\n\n\\subsection{Height restrictions} \nFor $p = q = \\frac{1}{2}$, we require that the heights \nsatisfy the same restriction as in Baxter's solid-on-solid\nmodel \\cite{baxter-book}, up to a normalization. More \nprecisely,\n\n\\begin{equation}\nh_{i, j} - h_{i+1, j} = \\pm \\frac{1}{2}, \\quad \nh_{i, j} - h_{i, j+1} = \\pm \\frac{1}{2}\n\\end{equation}\n\n\\subsection{The crossing parameter $= \\frac{1}{2}$} The vertex \nweights will be parametrized in terms of the elliptic functions \n$[u]$. As defined in Equations {\\bf \\ref{H}} and {\\bf \\ref{HH}}, \n$[u]$ depends on the real parameters, $I$ and $I'$, which are \nthe magnitudes of the half-periods of $[u]$. In the Felderhof-type \nmodel discussed in this section, we set $I = 1$\\footnote{In the limit \nof zero external fields, that is $p=q=\\frac{1}{2}$, this is equivalent \nto setting {\\it the crossing parameter} in Baxter's solid-on-solid model \nto the free fermion point. For details, see \\cite{baxter-book}}.\n\n\\subsection{The weights}\n\nIn the above notation, the non-zero weights are\n\n\\begin{eqnarray}\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}                           & \\textit{\\scriptsize{h}}\\+ \\textit{\\scriptsize{q}}                 \\\\\n\\textit{\\scriptsize{h}}     \\+ \\textit{\\scriptsize{p}}                 & \\textit{\\scriptsize{h}}\\+ \\textit{\\scriptsize{q}}     \\+ \\textit{\\scriptsize{p}}\n\\end{array}\n\\right\\rgroup\n&=& a_{+}(u,v,p,q) = [u-v+p+q]\n\\\\\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}                        & \\textit{\\scriptsize{h}}\\+ \\textit{\\scriptsize{q}}  \\- \\textrm{\\scriptsize{1}}            \\\\\n\\textit{\\scriptsize{h}}     \\+ \\textit{\\scriptsize{p}}      \\- \\textrm{\\scriptsize{1}}   & \\textit{\\scriptsize{h}}\\+ \\textit{\\scriptsize{q}}  \\+ \\textit{\\scriptsize{p}}     \\- \\textrm{\\scriptsize{2}}\n\\end{array}\n\\right\\rgroup\n&=& a_{-}(u,v,p,q) = [v-u+p+q]\n\\end{eqnarray}\n\\begin{eqnarray}\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}                        & \\textit{\\scriptsize{h}} \\+ \\textit{\\scriptsize{q}} \\- \\textrm{\\scriptsize{1}}            \\\\\n\\textit{\\scriptsize{h}}     \\+ \\textit{\\scriptsize{p}}              & \\textit{\\scriptsize{h}} \\+ \\textit{\\scriptsize{q}} \\+ \\textit{\\scriptsize{p}}     \\- \\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\n&=&\nb_{+}(u,v,p,q,h)\n\\\\\n&=&\n\\frac{\n[2h]^\\frac{1}{2}\n[2(h+p+q)]^\\frac{1}{2}\n}\n{\n[2(h+p)]^\\frac{1}{2}\n[2(h+q)]^\\frac{1}{2}\n}\n[u-v+q-p]\n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}                        & \\textit{\\scriptsize{h}} \\+\\textit{\\scriptsize{q}}                \\\\\n\\textit{\\scriptsize{h}}     \\+ \\textit{\\scriptsize{p}} \\- \\textrm{\\scriptsize{1}}        & \\textit{\\scriptsize{h}} \\+\\textit{\\scriptsize{q}}     \\+\\textit{\\scriptsize{p}} \\- \\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\n&=&\nb_{-}(u,v,p,q,h)\n\\\\\n&=&\n\\frac{\n[2h]^\\frac{1}{2}\n[2(h+p+q)]^\\frac{1}{2}\n}\n{\n[2(h+p)]^\\frac{1}{2}\n[2(h+q)]^\\frac{1}{2}\n}\n[u-v+p-q]\n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}           & \\textit{\\scriptsize{h}} \\+\\textit{\\scriptsize{q}}                \\\\\n\\textit{\\scriptsize{h}}     \\+\\textit{\\scriptsize{p}}  & \\textit{\\scriptsize{h}} \\+\\textit{\\scriptsize{q}}     \\+\\textit{\\scriptsize{p}} \\- \\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\n&=&\nc_{+}(u,v,p,q,h)\n\\\\\n&=&\n\\frac{\n[2p]^\\frac{1}{2}\n[2q]^\\frac{1}{2}\n}\n{\n[2(h+p)]^\\frac{1}{2}\n[2(h+q)]^\\frac{1}{2}\n}\n[v-u+p+q+2h]\n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}              & \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}  \\- \\textrm{\\scriptsize{1}}           \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}} \\- \\textrm{\\scriptsize{1}}      & \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}} \\- \\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\n&=&\nc_{-}(u,v,p,q,h)\n\\\\\n&=&\n\\frac{\n[2p]^\\frac{1}{2}\n[2q]^\\frac{1}{2}\n}\n{\n[2(h+p)]^\\frac{1}{2}\n[2(h+q)]^\\frac{1}{2}\n}\n[u-v+p+q+2h]\n\\nonumber\n\\end{eqnarray}\n\n\\subsection{The Yang-Baxter equations.}\n\nFor rapidities $\\{u,v,w\\}$, external fields $\\{p,q,r\\}$, \nand non-negative integers $\\{k,l,m,n,o\\}$, the above \nweights satisfy the Yang-Baxter equations\n\n\\begin{eqnarray}\n&\\displaystyle{\\sum_{j \\geq 0}}&\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}        & \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\-\\textit{\\scriptsize{j}} \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\-\\textit{\\scriptsize{o}}& \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}\\-\\textit{\\scriptsize{n}}\n\\end{array}\n\\right\\rgroup\nW_{uw}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\-\\textit{\\scriptsize{j}}     & \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\rangle\\-\\lambda     \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}\\-\\textit{\\scriptsize{n}} & \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\rangle\\+\\textit{\\scriptsize{p}}\\-\\textit{\\scriptsize{m}}\n\\end{array}\n\\right\\rgroup\n\\nonumber \\\\\n&\\times&\nW_{vw}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}        & \\textit{\\scriptsize{h}}\\+\\rangle\\-\\textit{\\scriptsize{k}}        \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\-\\textit{\\scriptsize{j}}& \\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{q}}\\-\\lambda\n\\end{array}\n\\right\\rgroup\n\\nonumber \\\\\n&=&\n\\nonumber \\\\\n&\\displaystyle{\\sum_{j \\geq 0}}&\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}\\+\\rangle\\-\\textit{\\scriptsize{k}}    & \\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{q}}\\-\\lambda     \\\\\n\\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{p}}\\-\\textit{\\scriptsize{j}}& \\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}\\-\\textit{\\scriptsize{m}}\n\\end{array}\n\\right\\rgroup\nW_{uw}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}        & \\textit{\\scriptsize{h}}\\+\\rangle \\- \\textit{\\scriptsize{k}}     \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}} \\- \\textit{\\scriptsize{o}}& \\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{p}} \\- \\textit{\\scriptsize{j}}\n\\end{array}\n\\right\\rgroup\n\\nonumber \\\\\n&\\times&\nW_{vw}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}} \\- \\textit{\\scriptsize{o}}    & \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\+\\rangle \\- \\textit{\\scriptsize{j}}     \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\+\\textit{\\scriptsize{q}} \\- \\textit{\\scriptsize{n}}& \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\+\\rangle\\+\\textit{\\scriptsize{q}} \\- \\textit{\\scriptsize{m}}\n\\end{array}\n\\right\\rgroup\n\\end{eqnarray}\n\n\\noindent {\\it Proof.} This can be proved by direct computation\nusing elliptic function identities, along the same lines as in\n\\cite{baxter-book}. For example, when\n$\\{k,l,m,n,o\\} = \\{0,1,1,1,1\\}$, the Yang-Baxter equation is\n\n\\begin{eqnarray}\n&\\displaystyle{\\sum_{j = 0}^{1}}&\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}        & \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\-\\textit{\\scriptsize{j}}     \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\-\\textrm{\\scriptsize{1}}& \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}\\-\\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\nW_{uw}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\-\\textit{\\scriptsize{j}}    & \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\rangle\\-\\textrm{\\scriptsize{1}}     \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}\\-\\textrm{\\scriptsize{1}}& \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\rangle\\+\\textit{\\scriptsize{p}}\\-\\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\n\\nonumber \\\\\n&\\times&\nW_{vw}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}        & \\textit{\\scriptsize{h}}\\+\\rangle          \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\-\\textit{\\scriptsize{j}}& \\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{q}}\\-\\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\n\\nonumber \\\\\n&=& W_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}\\+\\rangle        & \\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{q}}\\-\\textrm{\\scriptsize{1}}      \\\\\n\\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{p}}\\-\\textrm{\\scriptsize{1}}& \\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}\\-\\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\nW_{uw}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}        & \\textit{\\scriptsize{h}}\\+\\rangle          \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\-\\textrm{\\scriptsize{1}}& \\textit{\\scriptsize{h}}\\+\\rangle\\+\\textit{\\scriptsize{p}}\\-\\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\n\\nonumber \\\\\n&\\times&\nW_{vw}\n\\left\\lgroup\n\\begin{array}{ll}\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\-\\textrm{\\scriptsize{1}}    & \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\+\\rangle\\-\\textrm{\\scriptsize{1}}      \\\\\n\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\+\\textit{\\scriptsize{q}}\\-\\textrm{\\scriptsize{1}}& \\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\+\\rangle\\+\\textit{\\scriptsize{q}}\\-\\textrm{\\scriptsize{1}}\n\\end{array}\n\\right\\rgroup\n\\end{eqnarray}\n\nUsing the expressions for the weights, we obtain\n\n\\begin{eqnarray}\n&&\nc_{-}(u,v,p,q,h)\\ a_{+}(u,w,p,r)\\ b_{-}(v,w,q,r,h)\n\\nonumber \\\\\n&+&\nb_{-}(u,v,p,q,h)\\ c_{-}(u,w,p,r,h+q)\\ c_{+}(v,w,q,r,h)\n\\nonumber \\\\\n&=&\nc_{-}(u,v,p,q,h+r)\\ b_{-}(u,w,p,r,h)\\ a_{+}(v,w,q,r)\n\\end{eqnarray}\n\nWriting the weights in terms of elliptic functions, one can \neliminate common factors, and the proof of the equation\nreduces to proving\n\n\\begin{eqnarray}\n&\\ph{\\times}&\n[u-v+p+q+2h] [u-w+p+r] [v-w+q-r] [2(h+q+r)]\n\\nonumber \\\\\n&+&\n[u-v+p-q] [u-w+p+r+2(h+q)] [w-v+q+r+2h] [2r]\n\\nonumber \\\\\n&=&\n[u-v+p+q+2(h+r)] [u-w+p-r] [v-w+q+r] [2(h+q)]\n\\end{eqnarray}\n\n\\noindent which proceeds by noting that the ratio of the\nleft-hand-side and right-hand-side is doubly periodic and\nentire in $u$, and therefore a constant with respect to $u$.\nSetting $u=v-p+q$, the constant is found to be 1.\n\n\\subsection{Switching off the external fields} Setting $p \n= q = \\frac{1}{2}$ is equivalent to switching off the external \nfields. This becomes clear by inspection of the vertex weights, \nwhich up to normalization become equal to those of Baxter's \nsolid-on-solid model at the free fermion point.\n\n\\subsection{The external fields tilt the heights} One can \nthink of the external fields $p \\neq \\frac{1}{2}$ and\/or \n$q \\neq \\frac{1}{2}$, as effectively {\\it tilting} the heights \nof the lattice faces that they flow through. This tilt is with \nrespect to the line along which a field flows. This effectively \nadds to or subtracts from the height differences that are the case \nin the absence of external fields.\n\n\\subsection{The $c_{+}$ vertex}\n\nIn discussions of DWBC's and DWPF's, the $c_{+}$ vertex,\nsee Figure {\\bf \\ref{c+}}, plays a special role: It is\nthe DWPF for a $1\\times 1$ square lattice.\n\n\\begin{center}\n\\begin{minipage}{3.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(3000,3000)(0,0200)\n\\thicklines\n\\path(1500,2400)(2700,2400)\n\\path(1500,1200)(2700,1200)\n\\path(1500,2400)(1500,1200)\n\\path(2700,2400)(2700,1200)\n\\thinlines\n\\dashline{60.00}(1200,1800)(3000,1800)\n\\thicklines\n\\whiten\\path(0840,1740)(1000,1800)(0840,1860)(0840,1740)\n\\path(0680,1800)(0840,1800)\n \\put(0500,1900){\\textit{\\scriptsize{u}}, \\textit{\\scriptsize{p}}}\n\\thinlines\n\\dashline{60.00}(2100,2700)(2100,0900)\n\\thicklines\n\\whiten\\path(2040,0540)(2100,0700)(2160,0540)(2040,0540)\n\\path(2100,0540)(2100,0380)\n\\put(2200,0380){\\textit{\\scriptsize{v}},\\textit{\\scriptsize{q}}}\n\\put(1500,2400){\\blacken\\ellipse{050}{050}}\n\\put(1300,2500){\\textit{\\scriptsize{h}}}\n\\put(1500,1200){\\blacken\\ellipse{050}{050}}\n\\put(1300,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}}\n\\put(2700,2400){\\blacken\\ellipse{050}{050}}\n\\put(2500,2500){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}}\n\\put(2700,1200){\\blacken\\ellipse{050}{050}}\n\\put(2500,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}\\textit{\\scriptsize{--}}\\textrm{\\scriptsize{1}}}\n\\end{picture}\n\\begin{ca}\n\\label{c+}\nThe Felderhof-type $c_{+}$ vertex.\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\\bigskip\n\n\\subsection{Domain wall boundary conditions (DWBC)}\n\nWe define the DWBC's as an {\\it expanded} $c_{+}$ vertex,\nas in Figure {\\bf \\ref{dwbcf}}: Given the external fields \n$\\{p, q\\}$ and starting from $h_{00} = h$ at the top-left \ncorner, the boundary heights change by\n$q_j $ from left to right along the upper boundary,\n$p_i-1$ from top to bottom along the right boundary,\n$-q_j+1$ from right to left along the lower boundary, \nand $-p_i$ from bottom to top along the left boundary.\n\n\\begin{center}\n\\begin{minipage}{5.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(4800, 6000)(-2400, -1500)\n\\thicklines\n\\path(0000,3600)(3600,3600)\n\\path(0000,3000)(3600,3000)\n\\path(0000,2400)(3600,2400)\n\\path(0000,1800)(3600,1800)\n\\path(0000,1200)(3600,1200)\n\\path(0000,0600)(3600,0600)\n\\path(0000,0000)(3600,0000)\n\\path(0000,3600)(0000,0000)\n\\path(0600,3600)(0600,0000)\n\\path(1200,3600)(1200,0000)\n\\path(1800,3600)(1800,0000)\n\\path(2400,3600)(2400,0000)\n\\path(3000,3600)(3000,0000)\n\\path(3600,3600)(3600,0000)\n\\put(-0200,3700){$h          $}\n\\put(-0900,3000){$h + p_{1  }$}\n\\put(-0900,2400){$h + p_{1,2}$}\n\\put(-0900, 000){$h + p_{1,L}$}\n\n\\put(3700,3700){$h + q_{1, L}              $}\n\\put(3700,3000){$h + q_{1, L} + p_{1  } - 1$}\n\\put(3700,2400){$h + q_{1, L} + p_{1,2} - 2$}\n\\put(3700,0000){$h + q_{1, L} + p_{1,L} - L$}\n\n\\put(0000,3600){\\blacken\\circle{50,50}}\n\\put(0000,3000){\\blacken\\circle{50,50}}\n\\put(0000,2400){\\blacken\\circle{50,50}}\n\\put(0000,1800){\\blacken\\circle{50,50}}\n\\put(0000,1200){\\blacken\\circle{50,50}}\n\\put(0000,0600){\\blacken\\circle{50,50}}\n\n\\put(3600,3600){\\blacken\\circle{50,50}}\n\n\\put(0000,0000){\\blacken\\circle{50,50}}\n\\put(0600,0000){\\blacken\\circle{50,50}}\n\\put(3600,0000){\\blacken\\circle{50,50}}\n\\thinlines\n\\dashline{60.00}(-200,3300)(3900,3300)\n\\dashline{60.00}(-200,2700)(3900,2700)\n\\dashline{60.00}(-200,2100)(3900,2100)\n\\dashline{60.00}(-200,1500)(3900,1500)\n\\dashline{60.00}(-200,0900)(3900,0900)\n\\dashline{60.00}(-200,0300)(3900,0300)\n\\dashline{60.00}(0300,3900)(0300,-200)\n\n\\thicklines\n\\blacken\\path(0555,-500)(0645,-500)(0600,-320)(0555,-500)\n\\path(0600,-980)(0600,-500)\n\\put(0600,-1200){$h + q_1 + p_{1, L} -1$}\n\n\\thicklines\n\\blacken\\path(1155,-500)(1245,-500)(1200,-320)(1155,-500)\n\\path(1200,-680)(1200,-500)\n\\put(1200,-0900){$h + q_{1,2} + p_{1, L} -2$}\n\n\\thinlines\n\\dashline{60.00}(0900,3900)(0900,-200)\n\\dashline{60.00}(1500,3900)(1500,-200)\n\\dashline{60.00}(2100,3900)(2100,-200)\n\\dashline{60.00}(2700,3900)(2700,-200)\n\\dashline{60.00}(3300,3900)(3300,-200)\n\\end{picture}\n\\begin{ca}\n\\label{dwbcf}\nFelderhof-type height domain wall boundary conditions.\nWe use the notation $p_{i, j} = \\sum_{k=i}^{j} p_k$,\n{\\it etc.}\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\\bigskip\n\n\\subsection{Domain wall partition function (DWPF)}\n\nThe DWPF on an $L\\times L$ lattice, $Z_{L\\times L}$, is the\nsum over all weighted configurations that satisfy the DWBC.\nThe weight of each configuration is the product of the weights\nof the vertices\n\n\\begin{eqnarray}\nZ^{}_{L\\times L}\n=\n\\sum\\limits_{{\\rm configurations}}\n\\left\\lgroup\n\\prod\\limits_{\\rm vertices} w_{ij}\n\\right\\rgroup\n\\label{physical}\n\\end{eqnarray}\n\n\\subsection{Line permuting vertices}\nIn proofs of DWPF's two vertices play an important role.\nThese are the $a$-type vertices which can be used to\npermute adjacent flow lines.\n\n\\begin{center}\n\\begin{minipage}{5.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(3000,3000)(-200,0)\n\\thicklines\n\\path(1500,2400)(2700,2400)\n\\path(1500,1200)(2700,1200)\n\\path(1500,2400)(1500,1200)\n\\path(2700,2400)(2700,1200)\n\\thinlines\n\\dashline{60.00}(1200,1800)(3000,1800)\n\\thicklines\n\\whiten\\path(0840,1740)(1000,1800)(0840,1860)(0840,1740)\n\\path(0680,1800)(0840,1800)\n \\put(0500,1900){\\textit{\\scriptsize{u}}, \\textit{\\scriptsize{p}}}\n\\thinlines\n\\dashline{60.00}(2100,2700)(2100,0900)\n\\thicklines\n\\whiten\\path(2040,0540)(2100,0700)(2160,0540)(2040,0540)\n\\path(2100,0540)(2100,0380)\n\\put(2200,0380){\\textit{\\scriptsize{v}},\\textit{\\scriptsize{q}}}\n\\put(1500,2400){\\blacken\\ellipse{050}{050}}\n\\put(1300,2500){\\textit{\\scriptsize{h}}}\n\\put(1500,1200){\\blacken\\ellipse{050}{050}}\n\\put(1300,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}}\n\\put(2700,2400){\\blacken\\ellipse{050}{050}}\n\\put(2500,2500){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}}\n\\put(2700,1200){\\blacken\\ellipse{050}{050}}\n\\put(2500,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}}\n\\path(4500,2400)(5700,2400)\n\\path(4500,1200)(5700,1200)\n\\path(4500,2400)(4500,1200)\n\\path(5700,2400)(5700,1200)\n\\thinlines\n\\dashline{60.00}(4200,1800)(6000,1800)\n\\thicklines\n\\whiten\\path(3840,1740)(4000,1800)(3840,1860)(3840,1740)\n\\path(3680,1800)(3840,1800)\n \\put(3500,1900){\\textit{\\scriptsize{u}}, \\textit{\\scriptsize{p}}}\n\\thinlines\n\\dashline{60.00}(5100,2700)(5100,0900)\n\\thicklines\n\\whiten\\path(5040,0540)(5100,0700)(5160,0540)(5040,0540)\n\\path(5100,0540)(5100,0380)\n \\put(5200,0380){\\textit{\\scriptsize{v}},\\textit{\\scriptsize{q}}}\n\\put(4500,2400){\\blacken\\ellipse{050}{050}}\n\\put(4300,2500){\\textit{\\scriptsize{h}}}\n\\put(4500,1200){\\blacken\\ellipse{050}{050}}\n\\put(4300,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{p}}\\textit{\\scriptsize{--}}\\textrm{\\scriptsize{1}}}\n\\put(5700,2400){\\blacken\\ellipse{050}{050}}\n\\put(5500,2500){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\textit{\\scriptsize{--}}\\textrm{\\scriptsize{1}}}\n\\put(5700,1200){\\blacken\\ellipse{050}{050}}\n\\put(5500,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{q}}\\+\\textit{\\scriptsize{p}}\\textit{\\scriptsize{--}}\\textrm{\\scriptsize{2}}}\n\\end{picture}\n\\begin{ca}\n\\label{face}\nThe Felderhof-type line permuting vertices $a_{+}$ \nand $a_{-}$.\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\\bigskip\n\n\\subsection{Different zeros} The weights of the line permuting \nvertices, $[u-v+p+q]$ and $[v-u+p+q]$, have different zeros. \nThis is the property that will allow us to obtain the zeros of \nthe DWPF and compute it in factorized form.\n\n\\subsection{Properties of the partition function} The following\nfour properties determine the partition function uniquely.\n\n\\subsubsection{Property 1: Quasi-periodicity} The partition\nfunction is entire in $u_1$ and satisfies the quasi-periodicity\nconditions\n\n\\begin{equation}\nZ_{L\\times L}\n\\left\\lgroup\nu_1+2, \\ldots, u_L, \\{v\\}, \\{p\\}, \\{q\\}, h\n\\right\\rgroup \n= \n(-)^L Z_{L\\times L}\n\\left\\lgroup\n\\{u\\},\\{v\\},\\{p\\},\\{q\\},h\n\\right\\rgroup\n\\end{equation}\n\n\\begin{eqnarray}\n&&\nZ_{L\\times L}\n\\left\\lgroup\nu_1-\\frac{2i\\log(\\textsf{q})}{\\pi},\\ldots,u_L,\\{v\\},\\{p\\},\\{q\\},h\n\\right\\rgroup\n=\n\\nonumber \\\\\n&&\n\\frac{(-)^L}{\\textsf{q}^L}\\\n\\exp\n\\left\\lgroup\n-\\pi i \\left\\lgroup Lu_1+(L-2)p_1-\\sum_{j=1}^L (v_j+q_j)-2h\n\\right\\rgroup\n\\right\\rgroup \\times \\nonumber \\\\\n&& Z_{L\\times L}\n\\left\\lgroup\n\\{u\\},\\{v\\},\\{p\\},\\{q\\},h\n\\right\\rgroup\n\\end{eqnarray}\n\n\\noindent {\\it Proof.} Since the weights are entire functions in\nthe rapidities, it follows that\n$Z_{L\\times L}\\left\\lgroup\\{u\\},\\{v\\},\\{p\\},\\{q\\},h\\right\\rgroup$ is an entire function\nin $u_1$. To prove the quasi-periodicity conditions, we write the\npartition function in the form\n\n\\begin{eqnarray}\nZ_{L\\times L}\\left\\lgroup\\{u\\},\\{v\\},\\{p\\},\\{q\\},h\\right\\rgroup\n&=&\n\\sum_{n=1}^{L}\nP_n\\left\\lgroup u_1,\\{v\\},p_1,\\{q\\},h\\right\\rgroup\n\\times \\\\\n&&\nQ_n\n\\left\\lgroup \nu_2, \\ldots, u_L,\\{v\\}, p_2, \\ldots, p_L, \\{q\\}, h\n\\right\\rgroup\n\\nonumber\n\\end{eqnarray}\n\n\\noindent where\n\n\\begin{eqnarray}\nP_n \\left\\lgroup u_1,\\{v\\},p_1,\\{q\\},h\\right\\rgroup\n&=&\n\\left\\lgroup\n\\prod_{j=1}^{n-1}\na_{+}(u_1,v_j,p_1,q_j)\n\\right\\rgroup\nc_{+}\\left\\lgroup u_1,v_n,p_1,q_n,h+\\sum_{k=1}^{n-1}q_k\\right\\rgroup\n\\quad\\quad\\quad\n\\\\\n&\\times&\n\\left\\lgroup\n\\prod_{j=n+1}^{L}\nb_{-}(u_1,v_j,p_1,q_j,h+\\sum_{k=1}^{j-1}q_k)\n\\right\\rgroup\n\\nonumber\n\\end{eqnarray}\n\n\\noindent and \n$Q_n \\left\\lgroup u_2, \\ldots, u_L, \\{v\\}, p_2, \\ldots, p_L, \\{q\\},h \\right\\rgroup$\ndoes not depend on $u_1$. Using the expressions for the weights, \nwe have\n\n\\begin{eqnarray}\n&&\nP_n\\left\\lgroup u_1+2,\\{v\\},p_1,\\{q\\},h\\right\\rgroup\n=\n(-)^L\\ P_n \\left\\lgroup u_1,\\{v\\},p_1,\\{q\\},h\\right\\rgroup\n\\\\\n&&\nP_n\\left\\lgroup u_1-\\frac{2i\\log(\\textsf{q})}{\\pi},\\{v\\},p_1,\\{q\\},h\\right\\rgroup\n=\n\\\\\n&&\n\\frac{(-)^L}{\\textsf{q}^L}\n\\exp\\left\\lgroup -\\pi i \\left\\lgroup Lu_1+(L-2)p_1-\\sum_{j=1}^L (v_j+q_j)-2h\\right\\rgroup\\rr\nP_n\\left\\lgroup u_1,\\{v\\},p_1,\\{q\\},h\\right\\rgroup\n\\nonumber\n\\end{eqnarray}\n\n\\noindent from which the required property follows immediately.\n\n\\subsubsection{Property 2: Simple zeros} The partition function\nhas simple zeros at $u_1=u_j-p_1-p_j$, where $j=2,\\ldots,L$.\n\n\\bigskip\n\n\\noindent {\\it Proof.} We multiply the partition function by\n$a_{+}(u_2,u_1,p_2,p_1)$, and use the Yang-Baxter equation to\nslide the inserted face through the lattice.\n\n\\begin{center}\n\\begin{minipage}{5.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(4800, 5000)(-1000,-0500)\n\\thicklines\n\\path(0600,3900)(4200,3900)\n\\path(0900,3300)(4200,3300)\n\\path(0600,2700)(4200,2700)\n\\path(0600,2100)(4200,2100)\n\\path(0600,1500)(4200,1500)\n\\path(0600,0900)(4200,0900)\n\\path(0600,0300)(4200,0300)\n\\path(0600,2700)(0600,0300)\n\\path(1200,3900)(1200,0300)\n\\path(1800,3900)(1800,0300)\n\\path(2400,3900)(2400,0300)\n\\path(3000,3900)(3000,0300)\n\\path(3600,3900)(3600,0300)\n\\path(4200,3900)(4200,0300)\n\\path(0600,3900)(0900,3300)\n\\path(0900,3300)(0600,2700)\n\\path(0600,3900)(0300,3300)\n\\path(0300,3300)(0600,2700)\n\\path(0900,3300)(4200,3300)\n\\thinlines\n\\dashline{60.00}(0300,3000)(0450,3000)\n\\dashline{60.00}(0450,3000)(0750,3600)\n\\dashline{60.00}(0750,3600)(4500,3600)\n\\dashline{60.00}(0300,2400)(4500,2400)\n\\dashline{60.00}(0300,1800)(4500,1800)\n\\dashline{60.00}(0300,1200)(4500,1200)\n\\dashline{60.00}(0300,0600)(4500,0600)\n\\put(4600,3600){$u_1, p_1$}\n\\put(4600,3000){$u_2, p_2$}\n\\dashline{60.00}(0300,3600)(0450,3600)\n\\dashline{60.00}(0450,3600)(0750,3000)\n\\dashline{60.00}(0750,3000)(4500,3000)\n\\whiten\\path(-200,3555)(-020,3600)(-200,3645)(-200,3555)\n\\path(-380,3600)(-200,3600) \n\\put(-1100,3600){$u_2, p_2$}\n\\whiten\\path(-200,2955)(-020,3000)(-200,3045)(-200,2955)\n\\path(-380,3000)(-200,3000) \n\\put(-1100,3000){$u_1, p_1$}\n\\dashline{60.000}(0900,4200)(0900,0000)\n\\dashline{60.000}(1500,4200)(1500,0000)\n\\dashline{60.000}(2100,4200)(2100,0000)\n\\dashline{60.000}(2700,4200)(2700,0000)\n\\dashline{60.000}(3300,4200)(3300,0000)\n\\dashline{60.000}(3900,4200)(3900,0000)\n\\end{picture}\n\\begin{ca}\n\\label{a+}\nInserting an $a_{+}$ vertex into the left boundary.\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\\bigskip\n\nIt emerges as $a_{-}(u_2,u_1,p_2,p_1)$, and the order\nof the first two lattice rows is reversed.\n\n\\begin{center}\n\\begin{minipage}{5.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(4800, 5000)(-1250,-0500)\n\\thicklines\n\\thicklines\n\\path(0300,3900)(3900,3900)\n\\path(0300,3300)(3600,3300)\n\\path(0300,2700)(3900,2700)\n\\path(0300,2100)(3900,2100)\n\\path(0300,1500)(3900,1500)\n\\path(0300,0900)(3900,0900)\n\\path(0300,0300)(3900,0300)\n\\path(0300,3900)(0300,0300)\n\\path(0900,3900)(0900,0300)\n\\path(1500,3900)(1500,0300)\n\\path(2100,3900)(2100,0300)\n\\path(2700,3900)(2700,0300)\n\\path(3300,3900)(3300,0300)\n\\path(3900,2700)(3900,0300)\n\\path(3900,3900)(3600,3300)\n\\path(3900,3900)(4200,3300)\n\\path(4200,3300)(3900,2700)\n\\path(3600,3300)(3902,2700)\n\\thinlines\n\\dashline{60.000}(0100,2400)(4200,2400)\n\\dashline{60.000}(0100,1800)(4200,1800)\n\\dashline{60.000}(0100,1200)(4200,1200)\n\\dashline{60.000}(0100,0600)(4200,0600)\n\\dashline{60.000}(0100,3600)(3750,3600)\n\\dashline{60.000}(3750,3600)(4050,3000)\n\\dashline{60.000}(4050,3000)(4200,3000)\n\\dashline{60.000}(0100,3000)(3750,3000)\n\\dashline{60.000}(3750,3000)(4050,3600)\n\\dashline{60.000}(4050,3600)(4200,3600)\n\\dashline{60.000}(0600,4200)(0600,0000)\n\\dashline{60.000}(1200,4200)(1200,0000)\n\\dashline{60.000}(1800,4200)(1800,0000)\n\\dashline{60.000}(2400,4200)(2400,0000)\n\\dashline{60.000}(3000,4200)(3000,0000)\n\\dashline{60.000}(3600,4200)(3600,0000)\n\\whiten\\path(-300,3555)(-120,3600)(-300,3645)(-300,3555)\n\\path(-480,3600)(-300,3600) \n\\put(-1100,3600){$u_2, p_2$}\n\\put(+4300,3660){$u_1, p_1$}\n\\whiten\\path(-300,2955)(-120,3000)(-300,3045)(-300,2955)\n\\path(-480,3000)(-300,3000) \n\\put(-1100,3000){$u_1, p_1$}\n\\put(+4300,3000){$u_2, p_2$}\n\\end{picture}\n\\begin{ca}\n\\label{a-}\nExtracting an $a_{-}$ vertex from the right boundary.\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\\bigskip\n\nThis is equivalent to the equation\n\\begin{eqnarray}\nZ_{L\\times L}\\left\\lgroup\\{u\\},\\{v\\},\\{p\\},\\{q\\},h\\right\\rgroup\n&=&\n\\frac{\na_{-}(u_2,u_1,p_2,p_1)\n}\n{\na_{+}(u_2,u_1,p_2,p_1)\n}\n\\times \\\\\n&&\nZ_{L\\times L}\n\\left\\lgroup\nu_2, u_1, \\ldots, u_L, \\{v\\}, p_2, p_1, \\ldots, p_L, \\{q\\}, h\n\\right\\rgroup\n\\nonumber\n\\end{eqnarray}\n\nRepeating this procedure on the second and third rows, and so\non, we obtain\n\n\\begin{eqnarray}\nZ_{L\\times L}\\left\\lgroup\\{u\\},\\{v\\},\\{p\\},\\{q\\},h\\right\\rgroup\n&=&\n\\prod_{j=2}^{L}\n\\left\\lgroup\n\\frac{\na_{-}(u_j,u_1,p_j,p_1)\n}\n{\na_{+}(u_j,u_1,p_j,p_1)\n}\n\\right\\rgroup\n\\times \\\\\n&&\nZ_{L\\times L}\n\\left\\lgroup \nu_2, \\ldots, u_L, u_1, \\{v\\}, p_2, \\ldots, p_L, p_1, \\{q\\}, h\n\\right\\rgroup\n\\nonumber\n\\end{eqnarray}\n\n\\noindent which has the required simple zeros in the numerator.\n\n\\subsubsection{Property 3: A recursion relation} The partition\nfunction satisfies the recursion relation\n\n\\begin{eqnarray}\n&&\n\\left.\nZ_{L\\times L}\n\\left\\lgroup\n\\{u\\},\\{v\\},\\{p\\},\\{q\\},h\n\\right\\rgroup\\right|_{u_1= v_1-p_1-q_1}\n=\nc_{+}(v_1-p_1-q_1,v_1,p_1,q_1,h) \\times\n\\quad\\quad\n\\nonumber \\\\\n&&\n\\left\\lgroup\n\\prod_{j=2}^{L}\nb_{+}(u_j,v_1,p_j,q_1,h+\\sum_{k=1}^{j-1}p_k)\\\nb_{-}(v_1-p_1-q_1,v_j,p_1,q_j,h+\\sum_{k=1}^{j-1}q_k)\n\\right\\rgroup\n\\times\n\\nonumber\n\\\\\n&&\nZ_{(L-1)\\times (L-1)}\n\\left\\lgroup\nu_2,\\ldots,u_L,v_2,\\ldots,v_L,p_2,\\ldots,p_L,q_2,\\ldots,q_L,\nh + p_1 + q_1 - 1\n\\right\\rgroup\n\\end{eqnarray}\n\n\\noindent {\\it Proof.} In any lattice configuration in the partition\nfunction sum, the top-left corner of the lattice must be\n$a_{+}(u_1,v_1,p_1,q_1)$ or $c_{+}(u_1,v_1,p_1,q_1, h)$. Setting\n$u_1=v_1-p_1-q_1$ in the partition function sets to zero all\nconfigurations with $a_{+}(u_1,v_1,p_1,q_1)$.\n\nThe surviving configurations must have a top-left corner equal to $c_{+}$,\nwhich fixes the rest of the top row to $b_{-}$, the rest of the first\ncolumn to $b_{+}$, and the remainder of the lattice to $Z_{(L-1)\\times\n(L-1)}$. The above recursion follows immediately from these\nconsiderations.\n\n\\subsubsection{Property 4} The partition function on a $1\\times 1$\nlattice is given by\n\n\\begin{eqnarray}\nZ_{1\\times 1}(u_1,v_1,p_1,q_1,h)\n=\nc_{+}(u_1,v_1,p_1,q_1,h)\n\\end{eqnarray}\n\\noindent\n\n\\noindent {\\it Proof.} This follows from the definition of domain wall\nboundary conditions.\n\n\\subsection{The partition function is uniquely determined}\n\nAssume that $Z_{(n-1) \\times (n-1)}$ is uniquely determined by\nthe above four properties, for some $n\\geq 2$. From Property\n{\\bf 1}, Property {\\bf 2} and Theorem {\\bf 1.1}, we have\n\n\\begin{eqnarray}\n&&\nZ_{n\\times n}\n\\left\\lgroup\n\\{u\\},\\{v\\},\\{p\\},\\{q\\}, h\n\\right\\rgroup = \\kappa(u_2, \\ldots, u_n,\\{v\\},\\{p\\},\\{q\\}) \\times\n\\nonumber\n\\\\\n&&\n[\\sum_{j=1}^{n}(v_j-u_j)+\\sum_{j=1}^{n}(p_j+q_j)+2h]\n \\left\\lgroup \\prod_{j=2}^{n}[u_1-u_j+p_1+p_j]\\right\\rgroup\n\\end{eqnarray}\n\nProperty {\\bf 3} fully determines the coefficient $\\kappa$ in terms\nof $Z_{(n-1)\\times (n-1)}$. Finally, since $Z_{1\\times 1}$ is uniquely\ndetermined by Property {\\bf 4}, $Z_{n\\times n}$ is uniquely determined\nby the four properties.\n\n\\subsection{The domain wall partition function}\n\nThe solution to the preceding four properties is given by\n\n\\bigskip\n\n\\begin{boxedminipage}[l]{14cm}\n\\begin{equation}\n\\begin{split}\n&\nZ_{L\\times L}\n\\left\\lgroup\n\\{u\\},\\{v\\},\\{p\\},\\{q\\},h\n\\right\\rgroup\n=\n\\frac{\n\\prod_{j=1}^{L}\n[2p_j]^{\\frac{1}{2}}\n[2q_j]^{\\frac{1}{2}}\n}\n{\n{[2(h+\\sum_{j=1}^{L}p_j)]}^{\\frac{1}{2}}\n{[2(h+\\sum_{j=1}^{L}q_j)]}^{\\frac{1}{2}}\n} \\times \n{}\\\\\n{}&{}\n[\n\\sum_{j=1}^{L}(v_j-u_j) +\n\\sum_{j=1}^{L}(p_j+q_j) + 2h\n]\n\\prod_{1\\leq j < k \\leq L}\n[u_j-u_k+p_j+p_k]\n[v_k-v_j+q_k+q_j]\n\\end{split}\n\\end{equation}\n\\end{boxedminipage}\n\\bigskip\n\n\\subsection{Two choices of DWBC's} There are two possible choices \nof DWBC's. One corresponds to an {\\it expanded} $c_{+}$, as in this \nwork, and one to an `expanded' $c_{-}$ vertex. The DWPF depends on \nthe choice. The two expressions coincide for vanishing external\nfields, that is $p_i = q_j = \\frac{1}{2}$, and appropriate choices \nof the boundary height variables.\n\n\\section{A Perk-Schultz-type $gl(1|1)$ height model}\n\nIn \\cite{dm}, Deguchi and Martin introduced elliptic height versions \nof the $gl(r+1|s+1)$ trigonometric vertex models. In the following, \nwe define DWBC's and compute the DWPF in the $gl(1|1)$ case. \nSince the analysis in this Section follows almost verbatim that of \nSection {\\bf 2}, we will be brief and give just enough details where \nthe two models differ. \n\n\\subsection{Notation, heights and restrictions}\n\nIn this model, there are two square lattices. A {\\it physical} \n$L\\times L$ lattice that the heights live on, and a {\\it target} \n$\\field{Z}\\times \\field{Z}$ lattice that the heights take values in.\nThe target lattice is spanned by the unit vectors $\\hat{e}_{\\mu}$, \n$\\mu \\in \\{-1, +1\\}$, thus the height variables are 2-component \nvectors $\\{h_{\\mu}, h_{\\nu}\\}$. Heights on adjacent physical lattice \ncorners are restricted to take values in adjacent points on the \ntarget lattice. \nHeight differences along $\\hat{e}_{-1}$ and $\\hat{e}_{+1}$ lead \nto different vertex weights. To each height vector \n$h= \\{h_{\\mu}, h_{\\nu}\\}$, we assign a scalar\n\n\\begin{equation}\nh_{\\mu \\nu} = h_{\\mu} + h_{\\nu} + \\omega_{\\mu \\nu}\n\\end{equation}\n\n\\noindent where $\\omega$ is an arbitrary constant antisymmetric\ncomplex $2\\times 2$ matrix. We use $\\hat{e}_{\\mu}$ as well as \n$\\hat{e}_{\\textrm{sign($\\mu$)}}$ to indicate the same unit vector.\n\n\\subsection{No external fields and the crossing parameter is \na variable} Unlike the previous Felderhof-type model, the \nPerk-Schultz-type model in this Section has no external fields. \nThe crossing parameter is left as a variable.\n\n\\subsection{The weights} \n\nThe non-zero vertex weights are\n\n\\begin{equation}\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\nh                & h+   \\hat{e}_{\\mu} \\\\\nh+ \\hat{e}_{\\mu} & h+ 2 \\hat{e}_{\\mu}\n\\end{array}\n\\right\\rgroup\n= a_{\\mu}(u, v)\n= \\frac{[ 1+ \\mu (u-v) ]}{[1]}\n\\label{a-ps}\n\\end{equation}\n\n\\begin{equation}\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\nh                & h+ \\hat{e}_{\\nu} \\\\\nh+ \\hat{e}_{\\mu} & h+ \\hat{e}_{\\mu}+ \\hat{e}_{\\nu}\n\\end{array}\n\\right\\rgroup\n= b_{\\mu}(u, v)\n= \\frac{[u-v][h_{\\mu\\nu}-1]}{[1][h_{\\mu\\nu}]},\n\\quad\n\\mu \\ne \\nu\n\\label{b-ps}\n\\end{equation}\n\n\\begin{equation}\nW_{uv}\n\\left\\lgroup\n\\begin{array}{ll}\nh                & h+ \\hat{e}_{\\mu} \\\\\nh+ \\hat{e}_{\\mu} & h+ \\hat{e}_{\\mu}+ \\hat{e}_{\\nu}\n\\end{array}\n\\right\\rgroup\n= c_{\\mu}(u, v)\n= \\frac{[h_{\\mu\\nu}- (u-v)]}{[h_{\\mu\\nu}]},\n\\quad\n\\mu \\ne \\nu\n\\label{c-ps}\n\\end{equation}\n\nThe variables $h_{\\mu \\nu}$ on the right hand sides of Equations \n{\\bf \\ref{b-ps}} and {\\bf \\ref{c-ps}} are the scalars assigned to \nthe heights at the upper left corners of the corresponding vertices.\n\n\\subsection{The Yang-Baxter equations}\n\nThe weights satisfy the following Yang-Baxter equations\n\\cite{dm}.\n\n\\begin{eqnarray}\n&&\\sum_{g \\in \\field{Z}^2}\nW_{u_1 u_2}\n\\left\\lgroup\n\\begin{array}{ll}\nb & g \\\\\na & f\n\\end{array}\n\\right\\rgroup\nW_{u_1 u_3}\n\\left\\lgroup\n\\begin{array}{ll}\ng & d \\\\\nf & e\n\\end{array}\n\\right\\rgroup\nW_{u_2 u_3}\n\\left\\lgroup\n\\begin{array}{ll}\nb & c \\\\\ng & d\n\\end{array}\n\\right\\rgroup\n= \\nonumber \\\\\n&&\\sum_{g \\in \\field{Z}^2}\nW_{u_2 u_3}\n\\left\\lgroup\n\\begin{array}{ll}\na & g \\\\\nf & e\n\\end{array}\n\\right\\rgroup\nW_{u_1 u_3}\n\\left\\lgroup\n\\begin{array}{ll}\nb & c \\\\\na & g\n\\end{array}\n\\right\\rgroup\nW_{u_1 u_2}\n\\left\\lgroup\n\\begin{array}{ll}\nc & d \\\\\ng & e\n\\end{array}\n\\right\\rgroup\n\\end{eqnarray}\n\n\\subsection{The $c_{+}$ vertex}\n\nWe take the $c_{+}$ vertex to be as in Figure {\\bf \\ref{c++}}.\n\n\\begin{center}\n\\begin{minipage}{3.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(3000,2600)(0,300)\n\\thicklines\n\\path(1500,2400)(2700,2400)\n\\path(1500,1200)(2700,1200)\n\\path(1500,2400)(1500,1200)\n\\path(2700,2400)(2700,1200)\n\\dashline{60.00}(1200,1800)(3000,1800)\n\\whiten\\path(0840,1740)(1000,1800)(0840,1860)(0840,1740)\n\\path(0680,1800)(0840,1800)\n \\put(0500,1900){\\textit{\\scriptsize{u}}}\n\\dashline{60.00}(2100,2700)(2100,0900)\n\\whiten\\path(2040,0540)(2100,0700)(2160,0540)(2040,0540)\n\\path(2100,0540)(2100,0380)\n\\put(2200,0380){\\textit{\\scriptsize{v}}}\n\\put(1500,2400){\\blacken\\ellipse{050}{050}}\n\\put(1300,2500){\\textit{\\scriptsize{h}}}\n\\put(1500,1200){\\blacken\\ellipse{050}{050}}\n\\put(1300,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{$\\hat{e}_{+}$}}}\n\\put(2700,2400){\\blacken\\ellipse{050}{050}}\n\\put(2800,2500){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{$\\hat{e}_{+}$}}}\n\\put(2700,1200){\\blacken\\ellipse{050}{050}}\n\\put(2800,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{$\\hat{e}_{+}$\\+$\\hat{e}_{-}$}}}\n\\end{picture}\n\\begin{ca}\n\\label{c++}\nA general face configuration.\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\\bigskip\n\n\\begin{center}\n\\begin{minipage}{5.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(4800, 5200)(-2000, -1200)\n\\thicklines\n\\path(0000,3600)(3600,3600)\n\\path(0000,3000)(3600,3000)\n\\path(0000,2400)(3600,2400)\n\\path(0000,1800)(3600,1800)\n\\path(0000,1200)(3600,1200)\n\\path(0000,0600)(3600,0600)\n\\path(0000,0000)(3600,0000)\n\\path(0000,3600)(0000,0000)\n\\path(0600,3600)(0600,0000)\n\\path(1200,3600)(1200,0000)\n\\path(1800,3600)(1800,0000)\n\\path(2400,3600)(2400,0000)\n\\path(3000,3600)(3000,0000)\n\\path(3600,3600)(3600,0000)\n\\put(0000,3600){\\blacken\\circle{50,50}}\n\\put(3600,3600){\\blacken\\circle{50,50}}\n\\put(0000,3000){\\blacken\\circle{50,50}}\n\\put(0000,2400){\\blacken\\circle{50,50}}\n\\put(0000,1800){\\blacken\\circle{50,50}}\n\\put(0000,1200){\\blacken\\circle{50,50}}\n\\put(0000,0600){\\blacken\\circle{50,50}}\n\\put(-400,3600){$h                $}\n\\put(-900,3000){$h +   \\hat{e}_{+}$}\n\\put(-900,2400){$h + 2 \\hat{e}_{+}$}\n\\put(-900,0000){$h + L \\hat{e}_{+}$}\n\\put(3700,3600){$h + L \\hat{e}_{+}$}\n\\put(3700,3000){$h + L \\hat{e}_{+} +   \\hat{e}_{-}$}\n\\put(3700,2400){$h + L \\hat{e}_{+} + 2 \\hat{e}_{-}$}\n\\put(3700,0000){$h + L \\hat{e}_{-} + L \\hat{e}_{+}$}\n\n\\thicklines\n\\blacken\\path(0555,-500)(0645,-500)(0600,-320)(0555,-500)\n\\path(0600,-980)(0600,-500)\n\\put(0600,-1200){$h + L \\hat{e}_{+} +   \\hat{e}_{-}$}\n\n\\thicklines\n\\blacken\\path(1155,-500)(1245,-500)(1200,-320)(1155,-500)\n\\path(1200,-680)(1200,-500)\n\\put(1200,-0900){$h + L \\hat{e}_{+} + 2 \\hat{e}_{-}$}\n\n\\put(0000,0000){\\blacken\\circle{50,50}}\n\\put(0600,0000){\\blacken\\circle{50,50}}\n\\put(3600,0000){\\blacken\\circle{50,50}}\n\n\n\\thinlines\n\\dashline{60.00}(-200,3300)(3900,3300)\n\\dashline{60.00}(-200,2700)(3900,2700)\n\\dashline{60.00}(-200,2100)(3900,2100)\n\\dashline{60.00}(-200,1500)(3900,1500)\n\\dashline{60.00}(-200,0900)(3900,0900)\n\\dashline{60.00}(-200,0300)(3900,0300)\n\\dashline{60.00}(0300,3900)(0300,-200)\n\\dashline{60.00}(0900,3900)(0900,-200)\n\\dashline{60.00}(1500,3900)(1500,-200)\n\\dashline{60.00}(2100,3900)(2100,-200)\n\\dashline{60.00}(2700,3900)(2700,-200)\n\\dashline{60.00}(3300,3900)(3300,-200)\n\\end{picture}\n\\begin{ca}\n\\label{dwbcps}\nPerk-Schultz-type height domain wall boundary conditions.\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\\bigskip\n\n\\subsection{The domain wall boundary conditions}\nWe choose the DWBC as in Figure {\\bf \\ref{dwbcps}}.\nIn other words, starting from the lower left corner,\nall height changes along the left boundary are of type\n$\\hat{e}_{+1}$, along the upper boundary they are of type\n$\\hat{e}_{-1}$, along the right boundary they are of type\n$\\hat{e}_{+1}$, then along the lower boundary they are of\ntype $\\hat{e}_{-1}$.\n\n\\subsection{The line-permuting vertices}\nWe take the line permuting vertices to be as in Figure\n{\\bf \\ref{lpps}}. Their weights have different zeros leading \nto a factorization of the DWPF, just as in Section {\\bf 2}.\n\\begin{center}\n\\begin{minipage}{5.0in}\n\\setlength{\\unitlength}{0.0016cm}\n\\begin{picture}(3000,2800)(0,100)\n\\thicklines\n\\path(1500,2400)(2700,2400)\n\\path(1500,1200)(2700,1200)\n\\path(1500,2400)(1500,1200)\n\\path(2700,2400)(2700,1200)\n\\thinlines\n\\dashline{60.00}(1200,1800)(3000,1800)\n\\thicklines\n\\whiten\\path(0840,1740)(1000,1800)(0840,1860)(0840,1740)\n\\path(0680,1800)(0840,1800)\n \\put(0500,1900){\\textit{\\scriptsize{u}}}\n\\thinlines\n\\dashline{60.00}(2100,2700)(2100,0900)\n\\thicklines\n\\whiten\\path(2040,0540)(2100,0700)(2160,0540)(2040,0540)\n\\path(2100,0540)(2100,0380)\n\\put(2200,0380){\\textit{\\scriptsize{v}}}\n\\put(1500,2400){\\blacken\\ellipse{050}{050}}\n\\put(1300,2500){\\textit{\\scriptsize{h}}}\n\\put(1500,1200){\\blacken\\ellipse{050}{050}}\n\\put(1300,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{$\\hat{e}_{+}$}}}\n\\put(2700,2400){\\blacken\\ellipse{050}{050}}\n\\put(2500,2500){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{$\\hat{e}_{+}$}}}\n\\put(2700,1200){\\blacken\\ellipse{050}{050}}\n\\put(2500,1000){\\textit{\\scriptsize{h}}\\+\\textrm{\\scriptsize{2}}\\textit{\\scriptsize{$\\hat{e}_{+}$}}}\n\\path(4500,2400)(5700,2400)\n\\path(4500,1200)(5700,1200)\n\\path(4500,2400)(4500,1200)\n\\path(5700,2400)(5700,1200)\n\\thinlines\n\\dashline{60.00}(4200,1800)(6000,1800)\n\\thicklines\n\\whiten\\path(3840,1740)(4000,1800)(3840,1860)(3840,1740)\n\\path(3680,1800)(3840,1800)\n \\put(3500,1900){\\textit{\\scriptsize{u}}}\n\\thinlines\n\\dashline{60.00}(5100,2700)(5100,0900)\n\\thicklines\n\\whiten\\path(5040,0540)(5100,0700)(5160,0540)(5040,0540)\n\\path(5100,0540)(5100,0380)\n \\put(5200,0380){\\textit{\\scriptsize{v}}}\n\\put(4500,2400){\\blacken\\ellipse{050}{050}}\n\\put(4300,2500){\\textit{\\scriptsize{h}}}\n\\put(4500,1200){\\blacken\\ellipse{050}{050}}\n\\put(4300,1000){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{$\\hat{e}_{-}$}}}\n\\put(5700,2400){\\blacken\\ellipse{050}{050}}\n\\put(5500,2500){\\textit{\\scriptsize{h}}\\+\\textit{\\scriptsize{$\\hat{e}_{-}$}}}\n\\put(5700,1200){\\blacken\\ellipse{050}{050}}\n\\put(5500,1000){\\textit{\\scriptsize{h}}\\+\\textrm{\\scriptsize{2}}\\textit{\\scriptsize{$\\hat{e}_{-}$}}}\n\\end{picture}\n\\begin{ca}\n\\label{lpps}\nThe Perk-Schultz-type line permuting vertices\n$a_{+}$ and $a_{-}$.\n\\end{ca}\n\\end{minipage}\n\\end{center}\n\\bigskip\n\\subsection{The DWPF}\n\nHaving defined the model, the derivation of the corresponding\nDWPF proceeds precisely in analogy with that in Section {\\bf 2}.\nBased on the quasi-periodicity properties of the partition\nfunction, we propose a factorization in terms of the $[u]$\nfunctions. We obtain the zeros by permuting adjacent flow\nlines, using the line permuting vertices, obtain a recursion\nrelation the DWPF satisfies and an initial condition. The DWPF \nis uniquely determined, and the following expression satisfies\nall the conditions.\n\n\\bigskip\n\\begin{boxedminipage}[l]{12cm}\n\\begin{equation}\n\\begin{split}\n&Z^{}_{L \\times L} = \\\\\n&\\frac{[h +(L-1) - \\sum^L_{k=1}(u_k-v_k)]}\n      {[h +(L-1)]}\n\\prod_{1\\le i < j \\le L}\n\\frac{[1+u_i-u_j]}{[1]} \\frac{[1-(v_i-v_j)]}{[1]}\n\\label{mas}\n\\end{split}\n\\end{equation}\n\\end{boxedminipage}\n\\bigskip\n\n\\subsection{On the $gl(r+1|s+1)$ height models} The reason\nwe chose the $gl(1|1)$ case is that, given the way that we define\nDWBC's, the symmetries of the $gl(r+1|s+1)$ models are such that\nonly the two state variables variables that we put on the domain\nwall boundaries end up propagating inside the configurations.\nThis effectively restricts the DWPF to the $gl(1|1)$ model.\n\n\\section{Remarks}\n\nThe point of this work is to give examples of domain wall partition \nfunctions in elliptic and\/or height models. We restricted our attention \nto models that are non-invariant under state variable conjugation, which \ngreatly simplified the problem.\n\nBecause of the fermionic nature of the models discussed in this paper, \nthere is no interesting applications of their DWPF's to enumerations \nof alternating sign matrices or related objects that we are aware of.\n\nIt is highly likely that all models that are non-invariant under some \nform of state variable conjugation, that allows factorization, are \nfermionic (as the Felderhof-type model of Section {\\bf 2}) or contain \nfermions that play an essential role in the definition of the DWBC's\n(as the Perk-Schultz-type model of Section {\\bf 3}). As such, these \nmodels are unrepresentative of the general case. However, they are \nnon-trivial and we hope that one can learn something by extending our \nresults to compute correlation functions.\n\n\\section*{Acknowledgements}\nOF would like to thank Professors R~J~Baxter, T~Deguchi and M~Jimbo\nfor discussions, and P~Bouwknegt, T~Guttmann and T~Miwa for hospitality \nat ANU, MASCOS and Kyoto University, while this work was in progress. \nMW and MZ are supported by an Australian Postgraduate Award (APA).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\nProduced by Brownfox and the Online Distributed Proofreading\nTeam at http:\/\/www.pgdp.net (This file was produced from\nimages generously made available by JSTOR www.jstor.org)\n\n\n\n\n\n\n\n\n\n\n                        THE IRISH PENNY JOURNAL.\n\n        NUMBER 44.        SATURDAY, MAY 1, 1841.       VOLUME I.\n\n[Illustration: ARDFINNAN CASTLE, COUNTY OF TIPPERARY.]\n\nIn some of the recent numbers of our Journal we presented our readers\nwith views of two or three of the many striking objects of picturesque\nand historic interest for which, among our numerous beautiful rivers, the\ngentle Suir is more than ordinarily remarkable; and we return again with\npleasure to its green pastoral banks, to notice another of its attractive\nfeatures--the magnificent ruin of Ardfinnan Castle. This is a scene that\nmust be familiar to many of our readers, for the traveller must have been\na dull and unobserving one, who, journeying between Cork and Dublin by\nway of Cahir, has not had his attention roused by its romantic features,\nand an impression of its grandeur and picturesqueness made upon his\nmemory, not easily to be effaced. Ardfinnan is indeed one of the very\nfinest scenes of its kind to be found in Ireland, and is almost equally\nimposing from every point from which it can be viewed. The Castle crowns\nthe summit of a lofty and precipitous rock, below and around which the\nSuir winds its way in graceful beauty, while its banks are connected by\na long and level bridge of fourteen arches, which tradition states is of\ncoeval erection with the fortress, and which, at all events, is of very\ngreat antiquity. On every side the most magnificent outlines of mountain\nscenery form the distant back-grounds; and every object which meets the\neye is in perfect harmony with the general character of the scene.\n\nArdfinnan is a village of considerable antiquity, and derives its present\nname, which signifies Finnan's Height or Hill, from St Finnan the leper,\na celebrated ecclesiastic who founded a church and monastery here in\nthe seventh century, previously to which the place had borne the name\nof Druim-abhradb. Of this religious establishment there are however no\nremains, as it was plundered and burnt by the English in 1179; and the\npresent castle was erected on its site in 1185, by Prince John, then Earl\nof Morton, of whom it has been remarked that he achieved nothing during\nhis stay of eight months in Ireland, but the construction of this and two\nother castles, namely, Lismore and Tiobrad Fachtna, now Tibraghny on the\nSuir, which he erected with a view to the conquest of Munster. From these\ncastles he sent parties in various directions to plunder the country: but\nbeing met by the Irish under the command of Donall O'Brien, Dermod Mac\nCarthy, and Roderick O'Conor, they were defeated with great slaughter,\nfour knights having been killed at Ardfinnan; after which John was glad\nto return to England.\n\nPrince John, however, or those under whose advice he acted, showed a\nconsiderable degree of judgment and military skill in the selection of\nArdfinnan as the site of a fortress, which commanded one of the chief\npasses into South Munster; and the castle itself was of a princely\nmagnificence, and of such a degree of strength as must have rendered it\nimpregnable before the use of artillery. Its general form, as its ruins\nstill sufficiently show, was that of a parallelogram, strengthened by\nsquare towers at the corners, and having a strong entrance gateway.\nThis gateway still remains, as well as the greater part of the walls;\nbut the edifices of the interior are in a state of great dilapidation,\nand only part of the roof of one room remains. It is stated by the\neditor of Lewis's Topographical Dictionary, but on what authority we\nknow not, that this castle belonged to the Knights Templars, and that\non the suppression of that establishment it was granted to the Knights\nof St John of Jerusalem, and subsequently to the Bishop of Waterford.\nBut be this as it may, it was preserved as a military fortress till\nit was dismantled in 1649 by that great destroyer of Irish castles,\nOliver Cromwell, who, planting his cannon on the opposite hill near the\nbridge, made a breach in the walls, which speedily induced the garrison\nto surrender. The breach there is still shown, and according to an\nold tourist the following story is told in connection with it:--\"When\nthe place was besieged by Oliver, a butcher was within the walls, who\nwhile the siege lasted could never be prevailed on to come out of the\nroom where he had placed himself; but when the breach was made, and the\nsoldiers began to storm, he took up a handspike, and defended the breach\nalmost alone for some time, and knocked down several soldiers that strove\nto enter; but finding none to second him, he retired without the least\nhurt. When the castle was surrendered, he was asked why he would not come\nto the walls before the breach was made? He replied, 'Damn them, I did\nnot mind what was doing on the outside, but I could not bear their coming\ninto the house,' as he called it.\"\n\nArdfinnan is a parish in the barony of Iffa and Offa west, county of\nTipperary, above four miles S. S. E. from Cahir, and contains about\nnine hundred inhabitants. The village itself, which extends into the\nadjoining parish of Ballybacon, contains above three hundred. It was once\na place of greater note, and appears to have had a corporation, as it is\non record, 4th of Edward II (1311), that a grant of \"pontage for three\nyears\" was made \"to the Bailiffs and good men of Ardfynan\" at the request\nof the Bishop of Limerick.\n\n                                                                       P.\n\n\n\n\nPUSS IN BROGUES, A LEGEND.\n\n\nIt was about Christmas in the year 1831 that I received an invitation\nto spend the holidays with a friend who resided in a valley embosomed\namongst the loftiest of those mountains which form the boundary between\nthe King's and Queen's counties. The name of my host was Garret Dalton;\nhe held a considerable tract of land at a low rent, and by hard\nworking and thrifty living contrived not only to support his family in\ncomparative comfort, but to \"lay up a snug penny in the horn\" for his\nonly daughter Nanny, who was at this time about fourteen years of age,\nand, as her fond father often proudly boasted, \"the patthern ov as purty\na colleen as you'd find from the seven churches of Clonmacnoise to the\nhill ov Howth--wherever that was.\"\n\nGarret was generous and hospitable; his house \"was known to all the\nvagrant train,\" and the way-worn pilgrim, the wandering minstrel, the\nitinerant \"boccough,\" and the strolling vender of the news and gossip of\nthe day, were always secure of a welcome reception at his comfortable\nfire-side.\n\nAmongst the most constant of his guests was one Maurice O'Sullivan,\na native of the county of Cork. Maurice was a most venerable-looking\npersonage--tall, gaunt, athletic, and stone blind. He was about eighty\nyears of age; his white hair flowed on his shoulders, and he played\nthe Irish bagpipes delightfully. He was the lineal descendant of a\nfamily still famous in the annals of the \"green isle;\" and although now\ncompelled to wander through his native land in the garb and character\nof a blind piper, he had once seen better days, and was possessed\nof education and intelligence far superior to most of his caste. He\nwas intimately acquainted with the sad history of his country, was\ndevotedly attached to the dogmas of the fairy creed, could recite\ncharms and interpret dreams, and was deeply conversant in all those\nwitch legends and traditions for which the Munster peasantry are so\npeculiarly celebrated. Hence Maurice was always a special favourite with\nmy enthusiastic friend, who regularly entertained him at his own table,\nand who, when they would have disposed of their plain but comfortable\nand substantial meal, would treat his blind guest to repeated \"rounds\"\nof good \"half and half,\" composed of water from the spring, and the\n_potteen_ of the valley. It was night-fall when I arrived, and the happy\nfamily, consisting of Garret and his wife, Nanny their eldest girl, and\nher two little brothers, with Paddy Bawn the \"sarvint boy,\" and Ouny the\n\"girl,\" including blind Maurice, were collected in a smiling group around\nthe immense turf fire. In that day teetotalism had made little progress\nin Ireland; a huge copper kettle was therefore soon hissing on the fire;\na large grey-beard of mountain-dew stood on the huge oak-table; tumblers\nand glasses glittered in their respective places: and, in a few minutes\nwe were all engaged in discussing the merits of a large jug of potteen\npunch. All were happy; Garret talked, his wife smiled; told all the \"new\nnews\" of the Queen's county; whilst the spaces were filled up by blind\nMaurice, who played several of his most delightful national airs on his\nantique-looking pipes, whilst invariably as he concluded each successive\nlay, he would enrich the treat by some tradition connected with the piece\nhe had been playing, and which threw an indescribable charm not only\naround the performance, but the performer.\n\n\"That's a curious thing,\" remarked Garret, as the piper concluded one\nparticular rant; \"it's a quare medley, sometimes gay and sometimes sad,\nand sometimes like the snarlin' of a growlin' dog, and again exactly like\nthe mewing of a cat.\"\n\nThe piper smiled. \"And have you,\" he asked, \"never heard me play that\ntune before?--and did I never tell you the strange story connected with\nit?\"\n\n\"Never,\" was the reply.\n\n\"Well, that is strange enough; that tune is an old favourite in Munster,\nand I thought the whole world had heard of it.\"\n\n\"It never kem to Glen-Mac-Tir, any how,\" replied the farmer, \"or I'd\nsurely have heard of it. How d'ye call the name of it?\"\n\n\"_Caith-na-brogueen_--that is in English, Puss in Brogues,\" said the\npiper.\n\n\"Well,\" said Garret, \"it's often I heard of Puss in Boots, but I never\nheard of Puss in Brogues afore.\"\n\n\"Well, I'll tell you and this good company all about it,\" said Maurice,\nlaying down his pipes and wiping his forehead.\n\n\"Ay, but afore you begin,\" said Garret, \"take another dhrop to wet your\nwhistle, and you'll get on the betther with your story.\"\n\nThe piper seized the flowing tumbler again, and raising it to his lips,\ngaily exclaimed, whilst his attenuated hand shook nervously beneath the\nweight of the smoking goblet,\n\n\"_Sho-dhurtlh_, your healths, my friends, glory to our noble selves; and\nif this be war, may we never have more peaceable times.\"\n\n\"Amen,\" was the fervent response of every one present.\n\n\"Now for the _Caith-na-brogueen_,\" said Garret.\n\n\"Ay, and a wild and strange tale it is,\" said Maurice. \"However, it is a\npopular tradition in South Munster, and often when a boy have I listened\nto it, whilst my eyes, now dark for ever, would glisten with delight, and\nI would even fear to breathe lest one syllable of the legend might escape\nme.\" Then emitting a deep-drawn sigh, and again wiping his polished brow,\nhe thus began.\n\n'At the foot of a hill in a lonely district of the county of Cork, about\na dozen miles from my native village, there lived in old times a poor\nman named Larry Roche. He was, they say, descended from that family of\nthe Roches once so mighty in the south of Ireland, and some branches of\nwhich still retain a considerable degree of their former consequence and\nrespectability. Poor Larry, however, although the blood of kings might\nflow through his veins, was neither rich nor respectable; and his only\nmeans of support was a patch of barren land, which he held from that\ncelebrated sportsman Squire B----, in consideration of his services\nas care-keeper of a vast extent of bog and heath, the property of the\nsquire, and which extended far westward of poor Larry Roche's cabin. Yet\nLarry was not discontented with his situation. His father and grandfather\nhad lived and died in the same cabin; and although sometimes he might\nfeel disposed to envy the fine times which the sporting squire enjoyed,\nyet on cool reflection he would console himself with the consideration\nthat \"it was not every one that was born with a silver spoon in his\nmouth,\" and that even squire B---- himself, as grand as he was, was\non the \"look down,\" or he would not spend so much of his time wading\nthrough fens and bogs at home, but like his ancestors be lavishing his\nthousands amongst the _Sassenaghs_ at the other side of the lough, or\ndriving about on the continent. Thus rolled away poor Larry's days in\npoverty and contentment. In the shooting season his time was occupied\nin following his master over heath and hillock with his game-bag on his\nshoulder, and his \"dhudeen\" in his teeth, whilst the rest of the year\nwas spent in lounging about the ditches of the neighbourhood, chatting\nwith the crones of the vicinity about his family connexions, or the\nfairies of Glendharig, or squabbling with his good woman and his young\nones: for Larry was married; and as his wife was exactly a counterpart of\nhimself, every hour of course gave fresh cause for that bickering and\ndisagreement so often the result of untimely and ill-assorted marriages.\n\nThe only domestic animal in or about Larry Roche's cabin was a\nferocious-looking old black tom-cat, far bigger and stronger than any\ncat ever seen in that part of the country. His fur was black, he had\nstrong whiskers, his nails were like a tiger's, and at the end of his\ntail was fixed a claw or \"gaff\" as sharp and hooked as a falcon's beak;\nhis eyes also flashed by night with an appalling glare, and his cry was\na savage howl, baffling all description, and unlike any sound ever heard\nfrom any other animal. He was as singular in his habits, too, as in his\nappearance. He was never known to demand a morsel of food; and if offered\nany, he would reject it with indignation. Every evening at twilight he\nleft the fire-side, and spent the night scouring over moor and heather,\nand at daybreak would return from his foray, gaining access through\nthe low chimney of the cabin, and be found in the morning in his usual\nposition on the hob-stone. There he would sit from morning till night;\nand when Larry and Betty and the \"childre\" were chatting in a group\naround the fire, the cat would watch them intently, and if the nature\nof their conversation was such as to excite laughter or merriment, he\nwould growl in a low tone, evidently dissatisfied; but if their dialogues\nwere held in a jarring, angry strain, as sometimes happened, he would\npurr hoarsely and loudly, whilst the wagging of his tail testified the\npleasure he felt in their feuds and dissensions. The family had often\nbeen advised to make away with him, but superstitious awe or family\nprejudice prevented them; and although the whole neighbourhood averred\nthat \"he was no right thing,\" yet for the reasons I have stated his\nowners never could be induced to make any attempt to banish or destroy\nhim.\n\nOne dreary evening in October, Larry returned from his day's wandering\nwith the squire over the bleak bogs, and although it rained, and the\nwind blew bitterly, he appeared in much better spirits than was usual\nwith him on similar occasions. His wife wondered, and made more than\nusual preparations to please him. She trimmed the fire, and assisted him\nin taking off his dripping clothes, and then commenced pouring out her\nsympathy for his sufferings.\n\n\"Oh, never mind,\" said Larry; \"I have good news.\"\n\n\"Arrah, sit down,\" said Betty, \"and tell us what it is.\"\n\nLarry sat down, and putting his hand in his pocket, pulled out a\nglittering gold coin.\n\n\"Arrah, Larry, avourneen, what's that?\" asked the woman.\n\n\"Faith, it's a rale yellow boy, a good goold guinea,\" replied Larry. \"The\nsquire gev it to me, and tould me to buy a pair of brogues with it, and\ndrink his health with the balance.\"\n\n\"Och, musha! then, long life to him,\" vociferated Betty; \"and, Larry,\na-hagur, will you buy the brogues?\"\n\n\"Faix and I will,\" said Larry, \"and another rattling pair for yourself,\na-chorra.\"\n\n\"Ay, daddy, and another pair for me,\" shouted young Larry.\n\n\"And another for me,\" cried Thady.\n\n\"And another for me,\" chuckled Charley.\n\n\"Ay, and two pair for me,\" cried the black cat, speaking in a wild\nunearthly tone from the hob-stone, and breaking forth into a horrible\nlaugh.\n\n\"Devil knock the day-lights out of yez all,\" cried Larry, without seeming\nto take any notice of the strange circumstance, though his heart died\nwithin him with terror and surprise.\n\n\"Lord have mercy on us!\" faintly ejaculated Betty, signing her brow,\nwhilst all the children started up in terror, and ran behind their\nparents in the chimney-corner.\n\nAll this time the cat remained silent on the hob; but his aspect, at all\ntimes terrible, now seemed perfectly monstrous and hideous. For some time\na death-like silence was preserved, but at last Larry plucked up courage\nto address the speaking animal.\n\n\"And, in the name of God,\" he began, \"what business have you with\nbrogues?\"\n\n\"Ask me no questions,\" replied the cat, \"but get me the brogues as soon\nas possible.\"\n\n\"Oh, by all means,\" replied Larry, quite gently, \"you must have them; and\nwhy did you not ask them long ago, and you should have got them?\"\n\n\"My time was not come,\" replied Puss, briefly.\n\n\"Well,\" resumed Larry, \"to-morrow is Sunday, and at daybreak I will\nstart off to my gossip Phadruig Donovan's, in Mill-street, to engage the\nbrogues; he is the best brogue-maker in the county, and he is my first\ngossip besides.\"\n\n\"I know all that,\" said the cat, as he leaped up the chimney, on his\ndeparture to the scene of his midnight wanderings. \"Good night, Larry,\nand don't forget your engagement;\" and he disappeared through the\ngathering gloom, to the great relief of poor Larry and his terrified\nfamily.\n\nThat was a sad and uneasy night with poor Larry and his wife and\nchildren. They did not go to bed at all, but sat trembling at the fire,\nexpecting every moment that the black imp would return with legions of\nfiends to carry them away, body and bones, to the regions below. Numerous\nwere the plans proposed for getting rid of their old companion, but all\nwere rejected--some as inefficient, others as impracticable; and the\nonly point on which they could finally agree, was, that their days were\nnumbered, and that perhaps before morning their blood would be streaming\non the hearth-stone, and their souls wandering through mire and morass,\nthe prey of troops of fiends.\n\nAt last the morning dawned, and as Larry disconsolately enough was\npreparing to set forward on his journey to Mill-street, the cat jumped\ndown the chimney, and took his usual place on the hob.\n\n\"Well, I am going now,\" said Larry; \"have you any directions to give\nabout the brogues?\"\n\nThe cat did not reply, but uttered a hideous growl, which fell heavily on\nthe poor fellow's heart; so kissing his wife and children, and commending\nthem to the protection of God, he set out on his sorrowful journey.\n\nHe had not gone far when he perceived through the dim grey of the\nmorning a human figure approaching; and on advancing a little nearer, he\nfound that it was a very old man, of extremely diminutive stature and\nforbidding aspect. He wore an old grey coat and an equally old woollen\ncap, and his thin white hair descended to his knees; he was barefoot, and\ncarried a walking-stick in his hand.\n\n\"Good morrow, and God save you, Larry Roche,\" said the old man as he came\nup.\n\n\"A bright morning to you,\" answered Larry.\n\n\"How is every rope's length of you, Larry, and how is the woman and the\nchildre at home?\" demanded the stranger.\n\n\"Faix, purty well, considherin,\" replied Larry. \"But you have a great\nadvantage of me.\"\n\n\"How's that?\" said the old man.\n\n\"Why, because you know me so well, while I have no more knowledge of you\nthan of the man in the moon.\"\n\n\"Och, I'd know your skin in a tan-yard,\" said the old chap, laughing.\n\"But is it possible you don't know me?\"\n\n\"Faix if God Almighty knows no more about you than I do, the devil will\nhave a prey of you one of those days,\" replied Larry.\n\n\"Well, say no more about that,\" said the old fellow, rather angrily. \"But\nwhere are you going this blessed Sunday morning, Larry?\"\n\n\"To Mill-street,\" said Larry.\n\n\"All the ways--musha! what's taking you to Mill-street, Larry?\"\n\n\"My feet and my business,\" said Larry, something piqued at the old\nfellow's inquisitive importunity.\n\n\"You are very stiff this morning, Larry,\" said the stranger with a grin.\n\n\"I am worse than that,\" said the poor fellow; \"the heart within me is\nsick and sore.\"\n\n\"And what troubles you now, Larry?\"\n\nLarry hereupon told the whole of his strange misfortunes to the stranger,\nending with a deep \"ochone,\" and wishing, if it was the will of God, that\n\"his four bones were stretched in the church-yard of Kilebawn.\"\n\n\"You'll be there time enough for your welcome, may be,\" said the old\nchap, \"but that's neither here nor there. What will you do with the black\ncat?\"\n\n\"Och, sweet bad luck to all the cats alive, both black and white,\"\nimprecated Larry.\n\n\"That cat's a devil--a fiend,\" said the stranger; \"and more than that, he\nintends to murder you and your family this very night.\"\n\nLarry groaned and crossed his forehead, whilst the stranger's hideous\ncountenance was convulsed with half-suppressed laughter.\n\n\"Well, Larry,\" said he again, \"I am your friend, and I have power to save\nyou and yours, on one condition; and that is, that you will stop up the\nwindow in the back wall of your cabin.\"\n\n\"Faith and I'll do that with a heart and a half,\" said Larry. \"But what\ndo you want that for?\"\n\n\"I'll tell you that another time,\" said the little man.\n\n\"Go home now, and say you can't proceed to Mill-street without taking the\nwife and children with you, to leave the measure of their feet for the\nbrogues. Tell the cat also that he must come too, to have his fit taken;\nthen tie him up in a bag, and bring him with you; fasten this hair around\nyour neck,\" added the old man, at the same time extracting a single white\nhair from his head, \"and all the imps of hell cannot hurt you. But mind\nand don't open your lips from the time you leave home till you come to\nthis spot; and when you arrive here with the cat, sit down and wait the\nevent.\"\n\nA thick fog now suddenly rose, and the old man was hidden from the sight\nof Larry, who, greatly overjoyed, returned to his cabin to execute the\norders he had got, and was met by his wife, who was trembling for his\nsafe return, but did not expect him sooner than night.\n\n\"Musha! Larry agragal, you're welcome,\" she exclaimed; \"and what in the\nname of God turned you back?\"\n\n\"I am coming for you and the gorsoons; you must all come to Mill-street\nto have your measure taken for the brogues.\"\n\n\"And must I go too?\" asked the cat.\n\n\"Faix you must,\" said Larry; \"if natural Christians couldn't be fitted\nwithout bein' on the spot, it's hard to expect that you could.\"\n\n\"And how am I to travel?\" he asked.\n\n\"In a bag on my back,\" replied Larry. \"I'll whip you through the country\nlike a dinner to a hog, and man or mortal shall never be the wiser, if\nthe brogue-maker keeps his tongue quiet.\"\n\n\"I'll go bail he will,\" said Puss, \"for I'll kill him the very night the\nbrogues is brought home.\"\n\n\"Lord have mercy on him!\" ejaculated Larry, his heart sinking within him.\n\n\"Pray for yourself--may be you want mercy as well as him,\" said the cat.\n\nThe preparations were soon completed, and the cat being put into the bag,\nLarry tied the mouth of it firmly with a piece of cord, and then slung it\non his shoulder; and after acquainting his wife with his adventure with\nthe old man on \"Moin-more,\" he departed, whistling the air of \"Thamama\nThulla.\"\n\nHe soon gained the spot where he had parted with the old man, and looking\nround and perceiving nobody, he sat down on the green fern, still holding\nthe bag which contained his terrible fellow-traveller.\n\n\"What stops you Larry?\" asked the cat.\n\nLarry, recollecting the old man's injunction, spoke not, but continued\nwhistling.\n\n\"Does anything ail you, Larry?\"\n\n\"Whoo, hoo, phoo, hoo--Thamemo Chodladh.\"\n\n\"Is Betty and the childre to the fore?\"\n\n\"Thamemo Chodladh.\"\n\n\"Bad luck to you and your 'Thamemo Chodladh,'\" cried the cat.\n\n\"That the prayers may fall on the preacher,\" said Larry to himself.\n\nThe cat now began to make desperate efforts to escape from the bag,\nwhilst Larry redoubled his exertions to detain him. His attention,\nhowever, was soon arrested by the cry of hounds, and on looking westward,\nhe perceived, rapidly approaching over the morass, a big black man\nmounted on a black horse, and accompanied by a numerous pack of black\ndogs.\n\n\"Ochone,\" thought Larry, \"now I am coached of all ever happened me. Here\nis the chap's black friends coming to rescue him, and they won't leave a\ntoothful a-piece in my carcass.\"\n\n\"Let me go, Larry,\" said the cat, \"let me go, and I'll show you where\nthere's a cart-load of gold buried in the ground.\" But Larry remained\nsilent, and meantime the horseman and hounds came up.\n\n\"Good morrow and good luck, Larry Roche,\" said the black equestrian, with\na grim smile.\n\n\"Good morrow, kindly, your worship,\" said Larry.\n\n\"Is that a fox you have in the bag, Larry?\"\n\n\"No, in troth,\" said Larry, \"though I believe he is not much honester\nthan a fox.\"\n\n\"I must see what it is, any how,\" said the sable horseman, with a\ngesticulation which convinced Larry at once that he was the fellow whom\nhe had seen before.\n\nSo Larry opened the bag, and out jumped Puss, and away with him over the\nbog like a flash of lightning. The wild huntsman hallooed his dogs, and\nthe pursuit commenced, but the cat was soon surrounded and torn to pieces.\n\n\"Now,\" said the horseman, \"I must bid you farewell;\" and off he went;\nand then Larry returned home with the happy tidings, and the squire's\nguinea was spent in the purchase of sundry bottles of \"Tom Corcoran's\"\nbest potteen; but we must do Larry the justice to say that his agreement\nwith the old man was punctually performed, and the back window stopped as\neffectually as mud and stones could do it.\n\nA few nights after, Larry was aroused from his sleep by the merry tones\nof bagpipes at his fire-side, and getting up, he perceived the kitchen\nilluminated with a bright, reddish glare, whilst on the hob-stone he\nsaw, snugly seated, the ever remembered little old man playing a set of\nbagpipes, to the delightful tones of which hundreds of little fellows\nwith red caps and red small-clothes were capering about the floor.\n\n\"God bless the man and the work,\" said Larry, \"and warm work yez have ov\nit this hour ov the night.\"\n\nThe little fellow hereupon set up a shout, and rushing to the door, flew\nthrough it, one of them striking poor Larry a box on the right eye, which\nblinded it.\n\n\"Goodnight, Misthur Larry,\" said the piper; \"and how is your four bones?\nand how is the good woman that owns you?\"\n\n\"Och, no fear at all ov the woman,\" replied Larry; \"and as for my bones,\nthey are well enough; but, faith, my right eye, I believe, is in whey in\nmy head.\"\n\n\"Well, it will teach you how to speak to your betters in future,\" said\nthe little piper; \"never mintion the holy name again, when talking to the\n'good people.'\n\n\"But, Larry, listen: I'll now tell you why I wanted you to stop up your\nback window.\n\n\"You must know that this cabin of yours stands on the middle of a fairy\npass. We often come this way in our wanderings through the air in cold\nnights, and often we wished to warm ourselves at your fire-side; but as\nthere was a window in the back of your cabin, we had not power to stop,\nbut were compelled to pursue our journey. Now that the window is stopped,\nwe can come in and remain as long as we wish, and resume our journey\nthrough the door by which we enter. We pass this way almost every night,\nand you need never feel in the least apprehensive of injury so long as\nyou let us pursue our pastimes undisturbed.\"\n\n\"I'll be bound me or mine shall never annoy one of yez,\" said Larry.\n\n\"That's a good fellow, Larry,\" said the little chap; \"and now take those\npipes and play us a tune.\"\n\n\"Och, the devil a chanter I ever fingered,\" said Larry, \"since I was\nchristened.\"\n\n\"No matter,\" said the little fellow; \"I'll go bail you'll play out of the\nsoot.\"\n\nLarry \"yoked\" on the pipes, and lilted up in darling style a merry tune,\nwhilst the old chap was ready to split with laughing.\n\n\"What's the name of that tchune?\" said Larry.\n\n\"_Caith-na-brogueen_,\" replied the fairy piper; \"a tune I composed in\nmemory of your escape from the cat; a tune that will soon become a\nfavourite all over Munster.\"\n\nLarry handed back the pipes; the little man placed them in a red bag,\nand, bidding his host \"good night,\" dashed up the chimney.\n\nThe next night, and almost every following night, the din of fairy\nrevels might be heard at Larry Roche's fire-side, and Larry himself was\ntheir constant companion in their midnight frolics. He soon became the\nbest performer on the bagpipes in the south of Ireland, and after some\ntime surrendered his cabin to the sole occupation of the \"good people,\"\nand wandered with his family through all the Munster counties, and was\nwelcome and kindly treated wherever he came. After some time, the cabin\nfrom neglect fell, and offered no further impediment to the fairy host in\ntheir midnight wanderings, whilst Larry followed a life of pleasure and\npeace, far from the scene of his former perils and privations.\n\nThe cat, of course, was never seen after; but the peasantry of the\nneighbourhood say that the screams of the infernal fiend, mingled with\nthe deep howlings of hell-hounds and the savage yellings of the sable\nhunter, may be distinctly heard in horrid chorus amongst the fens and\nmorasses of the broad Moin-more.'\n\nThus ended the strange tale of Maurice O'Sullivan, who in addition to\nthe unanimous applause of the company present, was treated to another\nflowing tumbler of the barley bree, which he tossed off to the health of\nthose who, to use his own words, were \"good people\" in earnest--not fays\nor fairies, however, but the hospitable folks of Glen-Mac-Tir; adding at\nthe same time that he was resolved to gratify the lovers of legendary\nlore with another of his wild Munster tales on the following night.\n\n                                                                    J. K.\n\n\n\n\nITINERANT GOLDSMITHS OF INDIA AND SUMATRA.\n\n\nIn the production of beautiful specimens of mechanical art, much more\ndepends upon the natural taste and ingenuity of the workman than upon\nthe completeness and perfection of his tools. To those who are not much\nacquainted with the mechanical arts, this may sound somewhat like a\nself-evident proposition; yet it is far, very far indeed, from being\nconsidered such by European mechanics in general, and by our own in\nparticular. So commonly is the blame of clumsy workmanship laid upon\nthe badness or the want of tools, that an anecdote is related of a man,\nwho, upon being spoken to by a friend for having committed numerous\ngrammatical errors in a letter which he had just written, cursed his\npen, and asked his friend how he could be so excessively unreasonable\nas to expect him or any man to write good English with such a wretched\nimplement!\n\nTo such a degree of excellence has the manufacture of mechanical tools\nand instruments arrived in these countries, that a British mechanic would\nbe utterly astonished could he but behold the process of manufacturing\nvarious articles in the East; such for example as the shawls of Persia\nand Cashmere, the carvings in wood and ivory of China, the extraction\nof metal from the ore in the same country, by which malleable iron\nis produced fit for immediate use, and of the finest quality, by a\nsingle process; and, not to tire by enumeration, the productions of the\nitinerant goldsmiths of India and the island of Sumatra. These last excel\nin filagree work, for which they are celebrated, far exceeding even the\nChinese in its extraordinary delicacy; yet their tools are ruder than\nthose of the Indian goldsmith of the continent.\n\nWhen a Sumatran goldsmith is engaged to manufacture some piece of gold\nor silver work, he first asks for any little piece of thin iron--a\nbit of an iron hoop will answer his purpose--and with this he makes an\ninstrument for drawing his wire. The head of an old hammer stuck in a\nblock of wood serves for an anvil; and for a pair of compasses he is\ncontented with two old nails tied together at the heads. If he has a\ncrucible, good; if not, a piece of a broken rice-pot or a china tea-cup\nanswers his purpose. His furnace is an old broken _quallee_ or iron\npot, and his bellows a joint of bamboo, through which he blows with his\nmouth. If the work be heavy, and the quantity of metal to be melted\nconsiderable, three or four sit round the furnace, each with his bamboo,\nand blow together. It is only at Padang, where the manufacture is carried\non extensively, that the Chinese bellows has been introduced. The art of\nwire-drawing not having been considerably improved upon since the time of\nTubalcain, the Sumatran method differs little from the European.\n\nWhen drawn sufficiently fine, the wire is flattened by beating it upon\nthe anvil, and when flattened, it is twisted by rubbing it upon a block\nof wood with a flat stick. Having twisted it, the goldsmith again\nflattens it upon the anvil, and it is then a flat wire with serrated or\nindented edges, suitable for forming leaves or portions of flowers; these\nhe makes by turning down the end of the wire with a rude pincers, and\nthen cutting it off; this process is repeated until he has a sufficient\nnumber prepared for his work. The pattern he has drawn on a piece of\npaper or card, to the size and shape of which the intended piece of\nworkmanship must correspond. If the work is to be formed upon a plate\nof gold, he cuts the plate to the shape of his pattern, and proceeds\nto dispose the various bits of foliage, assorted according to size,\nand adjusts wire of various thickness for the stems, tendrils, &c.,\nfastening them temporarily together, and upon the plate, with the sago\nberry, called _boca sago_, which they reduce to a pulp by grinding upon\na rough stone; and a young cocoa nut, about the size of a walnut, forms\nthe ointment-box for this gelatinous preparation. When the work has been\nall placed in order, the operator prepares his solder, which consists of\ngold filings and borax mingled with water; this he strews upon the plate\nand applies to the several points of contact of the finer portions of\nhis work; and then, exposing the whole to the action of the fire, in a\nfew moments the soldering is completed. But if it is open work, he lays\nout the foliage and other parts upon a card or thin bit of soft wood, and\nattaches them together, as before described, with the pulp of the sago\nberry, applies the solder to the points of junction, and puts his work\ninto the fire as before; the card or wood burns away, the solder unites\nthe parts, and the work is completed; but if the piece be very large, the\nsoldering is done at several times. When the work is finished as to the\nmanufacturing part, it is cleansed and brightened by boiling it in water\nwith common salt and alum, or lime juice; and when the goldsmith wishes\nto give it a fine purple colour, he boils it in water with sulphur. The\nbeautiful little balls with which the Sumatran filagree work is sometimes\nornamented, are very simply made. The maker merely drills a small hole\nin a piece of charcoal, into which he puts some grains of gold dust, and\nupon exposing it to the fire, it runs into a perfect ball.\n\nAt finishing plain work, however, it must be confessed that the Sumatran\nand Indian goldsmiths fall short of the European; but if the latter excel\nin this, which may be considered the lowest department of the art, they\nare, despite their improvements and the superiority of their instruments,\nvastly inferior in the elegance and delicacy of the finer parts.\n\nThe Sonah Wallah (which signifies in Hindoostanee \"the gold fellow\"),\nor itinerant goldsmith of India, is far better supplied with tools and\nimplements of his trade than the Sumatran; and being thus a step higher\nin the grade of civilisation, he exhibits evidences of his advance in\nrefinement by being such a confounded rogue, that it is almost impossible\nfor _even his European employer_ to detect him, or prevent him from\npilfering some portion of the metal consigned to his ingenuity. The Sonah\nWallah may be hired for half a rupee (a little over a shilling) a-day,\nand, like the tinkers in these countries, he brings his implements with\nhim. These consist of a small forge, to the edge of which are attached\nseveral iron rings, which may be turned up over the charcoal to receive\nhis crucibles; a tin tube to blow through, a pair of slight iron tongs, a\npair of small pliers, a hammer, a couple of earthen saucers, and a rude\nanvil consisting of a piece of flint secured in a rough iron frame. The\ngold usually presented to him for working is the gold mohur, a coin worth\nabout 32s. sterling; this coin he places in a crucible with a little\nborax, to make it fuse the more readily; and having fixed the crucible\nin one of the rings, and lighted the charcoal under and around it, he\nblows with his tin tube until the metal is melted, when he practises a\ntrick of his trade by throwing in a small quantity of nitro-muriatic\nacid, which causes a sudden expansion or slight explosion, by which a\nportion of the metal is thrown out of the crucible into the fire, from\nthe extinguished embers of which the rogue separates it at a convenient\nopportunity; and lest his employer should try to detect him by weighing\nthe material both before and after working, he uses a copper rod for\nstirring the contents of the crucible, a portion of which rod melts and\nmingles with the gold, and so compensates for the deficiency in weight,\nor at least so nearly as invariably to escape detection, although it is\nmore than probable that an instance seldom or never occurs in which they\ndo not defraud their employers of a portion of the gold put into their\nhands. The fact is, that their admirable skill so completely compensates\nfor their knavery, that few would think of questioning too closely, for,\nrude and simple as are their tools, they far exceed European workmen in\nthe production of delicate and intricately formed trinkets; their small,\ntaper, and flexible fingers more than supplying the place of the numerous\nvarieties of implements which the mechanic of Birmingham or Sheffield\nfinds indispensably necessary. Indian chains of gold and silver have been\never celebrated for the beauty and complication of their structure; and\nalthough the Sonah Wallah may be considered to excel particularly in this\nbranch of his art, yet he still must be admitted to surpass, or at least\nequal, the European even in the manufacture of finger rings, bracelets,\nand armlets.\n\nMuch of the superior ingenuity of the Indian goldsmith may be\nattributable to the division of the people into castes or sections, by\nwhich fundamental law the same profession is carried on by the same\npeople or family through countless generations; the Shastra, or code of\nHindoo laws, forbidding the mixture of the castes, or interference with\nany business or profession not carried on by their progenitors.\n\nThere are four integral divisions of the people. The first caste, the\nBrahmins, are said by the Hindoo scriptures to have issued, at the\ncreation, from Brahma's mouth; and being thus the most excellent and\ndignified, are set apart for the priesthood and legislative departments\nof the state. The second, the Cshatryas, are said to have issued from\nBrahma's arms, and to them is committed the executive--these consequently\nform the armies. The third caste, the Vaisyas, are said to have proceeded\nfrom Brahma's thighs; they are the merchants, and consequently amongst\nthem are to be found some of the wealthiest men of Hindostan. The fourth\ncaste, called Soodras, being said to have issued from the feet of Brahma,\nare considered the most ignoble and degraded, and to them are left all\nmechanical arts and servile employments, as being beneath the dignity\nof the superior castes. Amongst the Soodras, consequently, are the\ngoldsmiths; and as the different professions form a sort of minor castes\namongst the greater ones, the same business is transferred from father to\nson; and all the powers of the mind being directed undistractedly to the\nsingle object, pre-eminence in that line is naturally to be expected.\n\n                                                                       N.\n\n\n\n\nBARNY O'GRADY.\n\n\nBehold me safely landed at Philadelphia, with one hundred pounds in\nmy pocket--a small sum of money; but many, from yet more trifling\nbeginnings, have grown rich in America. Many passengers who came over in\nthe same ship with me had not half so much. Several of them were indeed\nwretchedly poor. Among others there was an Irishman, who was known by\nthe name of Barny--a contraction, I believe, for Barnaby. As to his\nsurname, he could not undertake to spell it, but he assured me there was\nno better. This man, with many of his relatives, had come to England,\naccording to their custom, during harvest time, to assist in reaping,\nbecause they gain higher wages than in their own country. Barny had\nheard that he should get still higher wages for labour in America, and\naccordingly he and his two sons, lads of eighteen and twenty, took their\npassage for Philadelphia. A merrier mortal I never saw. We used to hear\nhim upon deck, continually singing or whistling his Irish tunes; and\nI should never have guessed that this man's life had been a series of\nhardships and misfortunes.\n\nWhen we were leaving the ship, I saw him, to my great surprise, crying\nbitterly; and upon inquiring what was the matter, he answered that it\nwas not for himself, but for his two sons, he was grieving; because\nthey were to be made _redemption men_; that is, they were to be bound\nto work, during a certain time, for the captain, or for whomsoever he\npleased, till the money due for their passage should be paid. Although I\nwas somewhat surprised at any one's thinking of coming on board a vessel\nwithout having one farthing in his pocket, yet I could not forbear paying\nthe money for this poor fellow. He dropped down on the deck upon both\nhis knees, as suddenly as if he had been shot, and holding up his hands\nto heaven, prayed, first in Irish, and then in English, with fervent\nfluency, that \"I and mine might never want; that I might live long to\nreign over him; that success might attend my honour wherever I went; and\nthat I might enjoy for evermore all sorts of blessings and crowns of\nglory.\" As I had an English prejudice in favour of silent gratitude, I\nwas rather disgusted by all this eloquence; I turned away abruptly, and\ngot into the boat which waited to carry me to shore.\n\n       *       *       *       *       *\n\nI had now passed three years in Philadelphia, and was not a farthing\nthe richer, but, alas, a great deal poorer. My inveterate habit of\nprocrastination--of delaying every thing till TO-MORROW, always stood\nbetwixt me and prosperity. I at last resolved upon leaving the land of\nthe star-spangled banner; but when I came to reckon up my resources, I\nfound that I could not do so, unless I disposed of my watch and my wife's\ntrinkets. I was not accustomed to such things, and I was ashamed to go\nto the pawnbroker's, lest I should be met and recognised by some of my\nfriends. I wrapped myself up in an old surtout, and slouched my hat over\nmy face. As I was crossing the quay, I met a party of gentlemen walking\narm in arm. I squeezed past them, but one stopped and looked after me;\nand though I turned down another street to escape him, he dodged me\nunperceived. Just as I came out of the pawnbroker's shop, I saw him\nposted opposite me; I brushed by; I could with pleasure have knocked him\ndown for his impertinence. By the time that I had reached the corner of\nthe street, I heard a child calling after me; I stopped, and a little\nboy put into my hand my watch, saying, \"Sir, the gentleman says you left\nyour watch and these thingumbobs by mistake.\"\n\n\"What gentleman?\"\n\n\"I don't know, but he was one that said I looked like an honest chap,\nand he'd trust me to run and give you the watch. He is dressed in a blue\ncoat, and went towards the quay. That's all I know.\"\n\nOn opening the paper of trinkets, I found a card with these\nwords:--\"_Barny_--with kind thanks.\"\n\n\"Barny! poor Barny! An Irishman whose passage I paid coming to America\nthree years ago. Is it possible?\"\n\nI ran after him the way which the child directed, and was so fortunate as\njust to catch a glimpse of the skirt of his coat as he went into a neat,\ngood-looking house. I walked up and down for some time, expecting him\nto come out again; for I could not suppose that it belonged to Barny. I\nasked a grocer who was leaning over his hatch-door, if he knew who lived\nin the next house?\n\n\"An Irish gentleman of the name of O'Grady.\"\n\n\"And his Christian name?\"\n\n\"Here it is in my books, sir--Barnaby O'Grady.\"\n\nI knocked at Mr O'Grady's door, and made my way into the parlour, where I\nfound him, his two sons, and his wife, sitting very sociably at tea. He\nand the two young men rose immediately, to set me a chair.\n\n\"You are welcome, kindly welcome, sir,\" said he. \"This is an honour I\nnever expected, any way. Be pleased to take the seat next the fire.\n'Twould be hard indeed if you should not have the best seat's that to\nbe had in this house, where we none of us ever should have sat, nor had\nseats to sit upon, but for you.\"\n\nThe sons pulled off my shabby greatcoat, and took away my hat, and\nMrs O'Grady made up the fire. There was something in their manner,\naltogether, which touched me so much that it was with difficulty I could\nkeep myself from bursting into tears. They saw this, and Barny (for I\nshall never call him any thing else), as he thought that I should like\nbetter to hear of public affairs than to speak of my own, began to ask\nhis sons if they had seen the day's paper, and what news there were.\n\nAs soon as I could command my voice, I congratulated this family upon the\nhappy situation in which I found them, and asked by what lucky accident\nthey had succeeded so well.\n\n\"The luckiest accident ever _happened me_ before or since I came to\nAmerica,\" said Barny, \"was being on board the same vessel with such a\nman as you. If you had not given me the first lift, I had been down for\ngood and all, and trampled under foot, long and long ago. But after that\nfirst lift, all was as easy as life. My two sons here were not taken\nfrom me--God bless you; for I never can bless you enough for that. The\nlads were left to work for me and with me; and we never parted, hand or\nheart, but just kept working on together, and put all our earnings, as\nfast as we got them, into the hands of that good woman, and lived hard\nat first, as we were born and bred to do, thanks be to heaven! Then we\nswore against all sorts of drink entirely. And as I had occasionally\nserved the masons when I lived a labouring man in the county of Dublin,\nand knew something of that business, why, whatever I knew, I made the\nmost of, and a trowel felt noways strange to me, so I went to work, and\nhad higher wages at first than I deserved. The same with the two boys;\none was as much of a blacksmith as would shoe a horse, and the other a\nbit of a carpenter; so the one got plenty of work in the forges, and the\nother in the dockyards as a ship-carpenter. So, early and late, morning\nand evening, we were all at the work, and just went this way struggling\non even for a twelvemonth, and found, with the high wages and constant\nemploy we had met, that we were getting greatly better in the world.\nBesides, the wife was not idle. When a girl, she had seen baking, and\nhad always a good notion of it, and just tried her hand upon it now, and\nfound the loaves went down with the customers, who came faster and faster\nfor them; and this was a great help. Then I turned master mason, and had\nmy men under me, and took a house to build by the job, and that did; and\nthen on to another; and after building many for the neighbours, 'twas\nfit and my turn, I thought, to build one for myself, which I did out of\ntheirs, without wronging them of a penny. In short,\" continued Barny, \"if\nyou were to question me how I have got on so well in the world, upon my\nconscience I should answer, we never made Saint Monday, and never put off\ntill to-morrow what we could do to-day.\"\n\nI believe I sighed deeply at this observation of Barny's notwithstanding\nthe comic phraseology in which it was expressed.\n\n\"And would it be too much liberty to ask you,\" said Barny, \"to drink a\ncup of tea, and to taste a slice of my good woman's bread and butter? And\nhappy the day we see you eating it, and only wish we could serve you in\nany way whatsoever.\"\n\nI verily believe the generous fellow forgot at this instant that he had\nredeemed my watch and wife's trinkets. He would not let me thank him as\nmuch as I wished, but kept pressing upon me fresh offers of service. When\nhe found I was going to leave America, he asked what vessel we should go\nin. I was really afraid to tell him, lest he should attempt to pay for my\npassage. But for this he had, as I afterwards found, too much delicacy of\nsentiment. He discovered, by questioning the captains, in what ship we\nwere to sail; and when we went on board, we found him and his sons there\nto take leave of us, which they did in the most affectionate manner; and\nafter they were gone, we found in the state cabin, directed to me, every\nthing that could be useful or agreeable to us, as sea stores for a long\nvoyage.--_Incident in a Tale entitled \"To-morrow,\" by Miss Edgeworth._\n\n       *       *       *       *       *\n\nDECISION OF CHARACTER: HOWARD THE PHILANTHROPIST.--In decision of\ncharacter no man ever exceeded, or ever will exceed, the late illustrious\nHoward. The energy of his determination was so great, that if, instead\nof being habitual, it had been shown only for a short time on particular\noccasions, it would have appeared a vehement impetuosity; but by being\nunintermitted it had an equability of manner which scarcely appeared to\nexceed the tone of a calm constancy, it was so totally the reverse of any\nthing like turbulence or agitation. It was the calmness of an intensity\nkept uniform by the nature of the human mind forbidding it to be more,\nand by the character of the individual forbidding it to be less. The\nhabitual passion of his mind was a measure of feeling almost equal to\nthe temporary extremes and paroxysms of common minds; as a great river,\nin its customary state, is equal to a small or moderate one when swollen\nto a torrent. The moment of finishing his plans in deliberation, and\ncommencing them in action, was the same. I wonder what must have been the\namount of that bribe in emolument or pleasure that would have detained\nhim a week inactive after their final adjustment. The law which carries\nwater down a declivity was not more unconquerable and invariable than the\ndetermination of his feelings towards the main object. The importance of\nthis object held his faculties in a state of excitement which was too\nrigid to be affected by lighter interests, and on which therefore the\nbeauties of nature and of art had no power. He had no leisure feeling\nwhich he could spare to be diverted among the innumerable varieties of\nthe extensive scenes which he traversed: all his subordinate feelings\nlost their separate existence and operation by falling into the grand\none. There have not been wanting trivial minds to mark this as a fault in\nhis character. But the mere men of taste ought to be silent respecting\nsuch a man as Howard: he is above their sphere of judgment. The invisible\nspirits who fulfil their commission of philanthropy among mortals do not\ncare about pictures, statues, and sumptuous buildings; and no more did\nhe, when the time in which he must have inspected and admired them would\nhave been taken from the work to which he had consecrated his life. His\nlabours implied an inconceivable severity of conviction that he had _one\nthing to do_, and that he who would do some great thing in this short\nlife must apply himself to the work with such a concentration of his\nforces as, to idle spectators who live only to amuse themselves, looks\nlike insanity. His attention was so strongly and tenaciously fixed on\nhis object, that even at the greatest distance, as the Egyptian pyramids\nto travellers, it appeared to him with a luminous distinctness as if it\nhad been nigh, and beguiled the toilsome length of labour and enterprise\nby which he was to reach it. It was so conspicuous before him that not\na step deviated from the direction, and every movement and every day\nwas an approximation. As his method referred every thing he did and\nthought to the end, and as his exertions did not relax for a moment, he\nmade the trial, so seldom made, what the utmost effect is, which may be\ngranted to the last possible efforts of a human agent; and, therefore,\nwhat he did not accomplish he might conclude to be placed beyond the\nsphere of mortal activity, and calmly leave to the immediate disposal of\nOmnipotence.--_Foster's Essays._\n\n\n\n\nKISSING OFF SAILORS.\n\n\nAn Irish Guineaman had been fallen in with by one of our cruisers, and\nthe commander of his majesty's sloop the Hummingbird made a selection\nof thirty or forty stout Hibernians to fill up his own complement, and\nhand over the surplus to the admiral. Short-sighted mortals we all are,\nand captains of men-of-war are not exempted from human imperfection.\nHow much also drops between the cup and the lip! There chanced to be on\nboard of the same trader two very pretty Irish girls, of the better sort\nof bourgeoise, who were going to join their friends at Philadelphia. The\nname of the one was Judy, and of the other Maria. No sooner were the poor\nIrishmen informed of their change of destination, than they set up a\nhowl loud enough to make the scaly monsters of the deep seek their dark\ncaverns. They rent the hearts of the poor-hearted girls; and when the\nthorough-bass of the males was joined by the sopranos and trebles of the\nwomen and children, it would have made Orpheus himself turn round and\ngaze.\n\n\"Oh, Miss Judy! oh, Miss Maria! would you be so cruel as to see us poor\ncrathurs dragged away to a man-of-war, and not for to go and spake a word\nfor us? A word to the captain from your own purty mouths, and no doubt he\nwould let us off.\"\n\nThe young ladies, though doubting the powers of their own fascinations,\nresolved to make the experiment. So, begging the lieutenant of the sloop\nto give them a passage on board to speak with his captain, they added a\nsmall matter of finery to their dress, and skipped into the boat like a\ncouple of mountain kids, caring neither for the exposure of ancles nor\nthe spray of the salt water, which, though it took the curls out of their\nhair, added a bloom to their cheeks, which perhaps contributed in no\nsmall degree to the success of their project. There is something in the\nsight of a petticoat at sea that never fails to put a man into a good\nhumour, provided he be rightly constructed. When they got on board the\nman-of-war, they were received by the captain.\n\n\"And pray, young ladies,\" said he, \"what may have procured me the honour\nof this visit?\"\n\n\"It was to beg a favour of your honour,\" said Judy. \"And his honour will\ngrant it too,\" said Maria, \"for I like the look of him.\"\n\nFlattered by this shot of Maria's, the captain said that nothing ever\ngave him more pleasure than to oblige the ladies; and if the favour they\nintended to ask was not utterly incompatible with his duty, that he would\ngrant it.\n\n\"Well, then,\" said Judy, \"will your honour give me back Pat Flannagan,\nthat you have pressed just now?\"\n\nThe captain shook his head.\n\n\"He's no sailor, your honour, but a poor bog-trotter; and he will never\ndo you any good.\"\n\nThe captain again shook his head. \"Ask me anything else,\" said he, \"and I\nwill give it you.\"\n\n\"Well, then,\" said Maria, \"give us Phelim O'Shaughnessy.\"\n\nThe captain was equally inflexible.\n\n\"Come, come, your honour,\" said Judy, \"we must not stand upon trifles\nnow-a-days. I'll give you a kiss if you give me back Pat Flannagan.\"\n\n\"And I another,\" said Maria, \"for Phelim.\"\n\nThe captain had one seated on each side of him; his head turned like a\ndog-vane in a gale of wind. He did not know which to begin with; the\nmost ineffable good humour danced in his eyes; and the ladies saw at\nonce the day was their own. Such is the power of beauty, that this lord\nof the ocean was fain to strike to it. Judy laid a kiss on his right\ncheek; Maria matched it on his left; and the captain was the happiest of\nmortals. \"Well, then,\" said he, \"you have your wish; take your two men,\nfor I am in a hurry to make sail.\"\n\n\"Is it sail ye are after makin'? and do ye mane to take all these poor\ncrathurs away wid you? No, faith; _another kiss and another man_.\"\n\nI am not going to relate how many kisses these lovely girls bestowed on\nthe envied captain. If such are captains' perquisites, who would not be\na captain? Suffice it to say, they got the whole of their countrymen\nreleased, and returned on board in triumph.\n\nLord Brougham used to say that he always laughed at the settlement of\npin-money, as ladies were generally either kicked out of it, or kissed\nout of it; but his lordship, in the whole course of his legal practice,\nnever saw a captain of a man-of-war kissed out of forty men by two pretty\nIrish girls. After this, who would not shout \"Erin go bragh!\"\n\n\n\n\nANCIENT IRISH LITERATURE.\n\nNumber 5.\n\n\nThe specimen of our ancient Irish Literature which we now present to\nour readers, is one of the most popular songs of the peasantry of the\ncounties of Mayo and Galway, and is evidently a composition of that most\nunhappy period of Irish history, the seventeenth century. The original\nIrish which is the composition of one Thomas Lavelle, has been published\nwithout a translation, by Mr Hardiman, in his Irish Minstrelsy; but a\nvery able translation of it was published in a review of that work in\nthe University Magazine for June 1834. From that translation the version\nwhich we now give has been but slightly altered so as to adapt it to the\noriginal melody, which is of very great beauty and pathos, and one which\nit is desirable to preserve with English words of appropriate simplicity\nof character:--\n\n\nTHE COUNTY OF MAYO.\n\n  I.\n\n  On the deck of Patrick Lynch's boat I sit in woful plight,\n  Through my sighing all the weary day, and weeping all the night.\n  Were it not that full of sorrow from my people forth I go.\n  By the blessed sun, 'tis royally I'd sing thy praise. Mayo!\n\n  II.\n\n  When I dwelt at home in plenty, and my gold did much abound.\n  In the company of fair young maids the Spanish ale went round--\n  'Tis a bitter change from those gay days that now I'm forced to go,\n  And must leave my bones in Santa Cruz, far from my own Mayo!\n\n  III.\n\n  They are altered girls in Irrul now, 'tis proud they're grown and high,\n  With their hair-bags and their top-knots, for I pass their buckles by--\n  But it's little now I heed their airs, for God will have it so,\n  That I must depart for foreign lands, and leave my sweet Mayo!\n\n  IIII.\n\n  'Tis my grief that Patrick Loughlin is not Earl in Irrul still,\n  And that Brian Duff no longer rules as Lord upon the hill,\n  And that Colonel Hugh Mac Grady should be lying dead and low,\n  And I sailing, sailing swiftly from the county of Mayo!\n\nFor the satisfaction of our Gaelic readers, we annex the original Irish\nwords:\n\n\nCondae Mai\u0121e\u00f3.\n\n  Is ar an loingseo Phaidi Loingsi\u0121 do \u0121nimse an dubron\n  Ag osna\u1e0b ann san oi\u1e0bche is ag sior\u0121ol san l\u00f3\n  Muna mbei\u1e0b gur dalla\u1e0b minntleacht is me a \u1e03\u1e1fad om \u1e41uinntir\n  Dar a maireann! is mai\u1e6b a chaoinfinnsi condae Mhai\u0121eo.\n\n  An uair a \u1e41air mo chairde bu\u1e0b \u1e03rea\u0121 mo chuid oir\n  Dolainn lionn Spaineach i gco\u1e41luadar ban og\n  Muna mbeidh s\u00edor ol na gc\u00e1rtai san dli\u0121 bheit ro l\u00e1idir\n  Ni a Sant\u00edcr\u00fas a d\u1e1f\u00e1cfainn mo \u010bna\u1e41a f\u00e1n bh\u1e1fod.\n\n  T\u00e1id gadai\u1e0bni\u0121e na h\u00e1ite seo ag eirgea\u1e0b go \u1e41\u00f3r\n  Fa \u010bnota\u1e0ba is fa hairbag gan tra\u010bt as bh\u00facla\u1e0ba brog\n  Da mair\u1e1fea\u1e0b damsa an iar-u\u1e41aill do \u1e0beanfuinn d\u00edobh cianach\n  Muna mbei\u1e0b gur \u1e6bagair dia \u1e0bam bhei\u1e6b a gciantaibh fa bhron.\n\n  D\u00e1 mbei\u1e0b Padruig Lochlainn ma iarla air iar-u\u1e41aill go foil\n  Brian Dubh a chlia\u1e41ain na \u1e6bighearna ar \u1e0bu\u1e41ach-\u1e41oir\n  Ao\u1e0b dubh mac Griada 'na choirnel a gCliara\n  Is ann niu bhei\u1e0b mo \u1e6briallsa go condae Mhaigheo.\n\n       *       *       *       *       *\n\nCAMEO-CUTTING.--This art is of great antiquity, and is pursued with most\nsuccess in Rome, where there are several very eminent artists now living.\nCameos are of two descriptions, those cut in stone, or _pietra dura_,\nand those cut in shell. Of the first, the value depends on the stone, as\nwell as in the excellence of the work. The stones most prized now are the\noriental onyx and the sardonyx, the former black and white in parallel\nlayers, the latter cornelian, brown and white; and when stones of four\nor five layers of distinct shades or colours can be procured, the value\nis proportionably raised, provided always that the layers be so thin as\nto be manageable in cutting the cameo so as to make the various parts\nharmonize. For example, in a head of Minerva, if well wrought out of a\nstone of four shades, the ground should be dark grey, the face light, the\nbust and helmet black, and the crest over the helmet brownish or grey.\nNext to such varieties of shades and layers, those stones are valuable in\nwhich two layers occur of black and white of regular breadth. Except on\nsuch oriental stones no good artist will now bestow his time; but, till\nthe beginning of this century, less attention was bestowed on materials,\nso that beautiful middle-age and modern cameos may be found on German\nagates, whose colours are generally only two shades of grey, or a cream\nand a milk-white, and these not unfrequently cloudy. The best artist in\nRome in _pietra dura_ is the Signor Girometti, who has executed eight\ncameos of various sizes, from 1\u00bd to 3\u00bd inches in diameter, on picked\nstones of several layers, the subjects being from the antique. These form\na set of specimens, for which he asks \u00a33,000 sterling. A single cameo of\ngood brooch size, and of two colours, costs \u00a322. Portraits in stone by\nthose excellent artists Diez and Saulini may be had for \u00a310. These cameos\nare all wrought by a lathe with pointed instruments of steel, and by\nmeans of diamond dust.\n\nShell cameos are cut from large shells found on the African and Brazilian\ncoasts, and generally show only two layers, the ground being either a\npale coffee-colour or a deep reddish-orange; the latter is most prized.\nThe subject is cut with little steel chisels out of the white portion\nof the shell. A fine shell is worth a guinea in Rome. Copies from the\nantique, original designs, and portraits, are executed in the most\nexquisite style of finish, and perfect in contour and taste, and it may\nbe said that the Roman artists have attained perfection in this beautiful\nart. Good shell cameos may be had at from \u00a31 to \u00a35 for heads, \u00a33 to \u00a34\nfor the finest large brooches, a comb costs \u00a310, and a complete set of\nnecklace, ear-rings, and brooch cost \u00a321. A portrait can be executed for\n\u00a34 or \u00a35, according to workmanship.\n\n       *       *       *       *       *\n\nVENETIAN PAVEMENTS.--A most beautiful art may be mentioned here in\nconnection with the last, I mean that of making what are termed Venetian\npavements which might advantageously be introduced into this country.\nThe floors of rooms are finished with this pavement, as it is somewhat\nincongruously termed, and I shall briefly describe the mode of operation\nin making these, but must first observe that they are usually formed\nover vaults. In the first place, a foundation is laid of lime mixed with\n_pozzolana_ and small pieces of broken stone; this is in fact a sort of\nconcrete, which must be well beaten and levelled. When this is perfectly\ndry, a fine paste, as it is termed by the Italians, must be made of lime,\n_pozzolana_, and sand; a yellow sand is used which tinges the mixture;\nthis is carefully spread to a depth of one or two inches, according to\ncircumstances. Over this is laid a layer of irregularly broken minute\npieces of marble of different colours, and if it is wished, these can be\narranged in patterns. After the paste is completely covered with pieces\nof marble, men proceed to beat the floor with large and heavy tools made\nfor the purpose; when the whole has been beaten into a compact mass, the\npaste appearing above the pieces of marble, it is left to harden. It is\nthen rubbed smooth with fine grained stones, and is finally brought to\na high polish with emery powder, marble-dust, and, lastly, boiled oil\nrubbed on with flannel. This makes a durable and very beautiful floor,\nwhich in this country would be well adapted for halls, conservatories,\nand other buildings.--_The Civil Engineer and Architect's Journal._\n\n       *       *       *       *       *\n\nHow destitute of humanity is he, who can pass a coarse joke upon the\nemblem of unfeigned sorrow.\n\n       *       *       *       *       *\n\n    Printed and published every Saturday by GUNN and CAMERON, at\n    the Office of the General Advertiser, No. 6, Church Lane,\n    College Green, Dublin.--Agents:--R. GROOMBRIDGE, Panyer Alley,\n    Paternoster Row, London; SIMMS and DINHAM, Exchange Street,\n    Manchester; C. DAVIES, North John Street, Liverpool; SLOCOMBE &\n    SIMMS, Leeds, JOHN MENZIES, Prince's Street, Edinburgh; & DAVID\n    ROBERTSON, Trongate, Glasgow.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Irish Penny Journal, Vol. 1 No.\n44, May 1, 1841, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nalso by nathaniel mackey\n\nFiction\n\n_Bedouin Hornbook_\n\n_Djbot Baghostus's Run_\n\n_Atet A.D._\n\n_Bass Cathedral_\n\n_From a Broken Bottle Traces o_ _f_ _Perfume Still Emanate: Volumes 1\u20133_\n\nPoetry Books and Chapbooks\n\n_Four for Trane_\n\n_Septet for the End o_ _f_ _Time_\n\n_Outlantish_\n\n_Song o_ _f_ _the Andoumboulou: 18\u201320_\n\n_Four for Glenn_\n\n_Anuncio's Last Love Song_\n\n_Outer Pradesh_\n\n_Moment's Omen_\n\n_Lay Ghost_\n\n_School o_ _f_ _Oud_\n\n_Eroding Witness_\n\n_School o_ _f_ ___Udhra_\n\n_Whatsaid Serif_\n\n_Splay Anthem_\n\n_Nod House_\n\n_Blue Fasa_\n\nCriticism\n\n_Discrepant Engagement: Dissonance, Cross-Culturality, and Experimental Writing_\n\n_Paracritical Hinge: Essays, Talks, Notes, Interviews_\n\nAnthologies\n\n_Moment's Notice: Jazz in Poetry and Prose_ , with Art Lange\n\nRecordings\n\n_Strick: Song o_ _f_ _the Andoumboulou 16\u201325_\n\n_for my cousin_\n\n_Kenneth Ray Kahn_\n\n# Late Arcade\n14.IX.83\n\nDear Angel of Dust,\n\nDjamilaa brought a new composition to rehearsal today. It's called \"Sekhet Aaru Struff,\" a title that immediately, as you can imagine, got my attention, alluding, as it does, to my own \"Sekhet Aaru Strut.\" Putting \"struff \" in \"strut's\" place, it alludes as well to the piece from which it borrows the term (if it can be called a term), Johnny Dyani's \"The Robin Irland Struff \" on the _African Bass_ album he did with Clifford Jarvis a few years back. I'm inclined to say it pointedly alludes to both, albeit its \"point\" would appear to be a diffusion of point, an advancement of blur (with overtones of slur) tending to dislocate point. What, that is, was Dyani getting at with \"Robin Irland\" if not an ambiguation of \"Robben Island,\" a pointedly malaprop version bordering on slur? And doesn't \"struff \" go on in the same vein, subjecting \"stuff,\" \"strut,\" \"straw,\" \"fluff,\" \"bluff \" and any number of other phonically related words to an ambiguating fusion and diffusion, upping the ante on the already indefinite \"stuff \"? Djamilaa perhaps had this in mind. Perhaps she meant the replacement of \"strut\" by \"struff \" as an ambiguating move, a deflationary tack perhaps.\n\nHowever much the replacement of \"strut\" diffuses and deflates a certain self-congratulation at the titular level, however much \"struff \" does indeed mess with \"strut,\" does indeed mush it up, taking the wind out of the sails its apparent pleasure with itself could be said to be, I say \"perhaps\" because the music itself does anything but diminish or deride. \"Struff \" notwithstanding, it conveys no loss or ratcheting down of dimension, no lack of majesty. It's better, perhaps, to simply say a certain modesty obtains, a shying away from the presumption self-nomination can't but entail. \"Struff \" wants to lighten up, walk lightly, outflank if not entirely break free from such presumption.\n\n\"Sekhet Aaru Struff's\" light tread compounds a sense of light as trepidatious gait with a competing sense of light as bouyancy, a saltatory \u00e9lan nowhere more notable than in the bounding figure Aunt Nancy sustains on bass. Atop and to a degree at odds with the bass's throb-inflected walk, I pursue a more tentative approach on trumpet, the trepidatious advance adumbrated by \"struff's\" unraveling of \"strut's\" edges, a frayed, fraught excursion bereft of any but ad hoc assurances. Drennette steps away from the traps in favor of bongos and conga for this piece, peppering and otherwise abetting Aunt Nancy's bounding figure while helping propel and punctuating the piece on other fronts as well. Djamilaa herself plays synthesizer, though \"presides on\" would be a more accurate way to put it, so fully does she avail herself and the piece of a sense of ultimacy, majesty and moment, an incumbent cosmicity or an abiding ethereality the synthesizer is famously able to impart.\n\nThe most unusual aspect of the piece's instrumentation is that Lambert and Penguin don't play their horns, assigned instead to something Djamilaa variously terms extended voice (nodding to Meredith Monk), prepared voice (nodding to John Cage) and amended voice or (further accenting her own intervention, underscoring her nodding to no one if not herself) amended mouth. Each is given a text to read, excerpts from a chapter of the _Chapters o_ _f_ _Coming Forth by Day_. Lambert's is taken from Chapter XV, \"A Hymn of Praise to Ra when He Riseth in the Eastern Part of Heaven,\" which mentions Sekhet Aaru toward the end: \"Let him mingle among the Heart-souls who live in Ta-tchesert. Let him travel about in the Sekhet Aaru, conformably to thy decree with joy of heart\u2014him the Osiris Ani, whose word is truth.\" Penguin's is taken from Chapter CX, \"The Chapters of Sekhet Hetepet,\" which concerns arrival in Sekhet Aaru and living in the city of Sekhet Hetepet, e.g., \"Let me go forward. Let me plough. I am at peace with the god of the town. I know the water, the towns, the nomes, and the lakes which are in Sekhet Hetepet. I live therein. I am strong therein.\" Lambert recites first, then Penguin.\n\nBy extended voice, prepared voice, amended voice or amended mouth Djamilaa means that Lambert and Penguin each clamp three clothespins to their lips for the piece, two of them clamped to and hanging down from the sides of the upper lip, one of them clamped to and hanging down from the middle of the lower lip. The aim is to alter their elocution, to pidginize, as Djamilaa puts it, the papyrus of Ani. She instructed them not to let it stop simply at that but to assist the aleatory work done by the clothespins (or, as she sometimes terms it, the amendments) by furthering it, consciously recasting the text, at the point of enunciation, as universal patois, idiosyncratically conceived. \"Think of Pharoah's 'Japan,'\" she exhorted. \"Think of the long line of scat behind it. You could do worse than think of Slim Gaillard or of Clark Terry's 'Mumbles,' to say nothing of all the vatic chatter outside the music.\"\n\nLambert and Penguin wondered if Djamilaa was serious at first (Aunt Nancy, Drennette and I wondered as well), especially when Penguin noticed the sentence \"Behold my mouth is equipped\" in the text he was to read, at which point he started laughing and, sure it was all a joke, said to Djamilaa, \"Okay, you got us. Touch\u00e9.\" Djamilaa, though, looked at him as if she had no idea what he meant. Quickly enough, we all saw that indeed she had no idea what he meant. She was nothing if not serious, though she did allow, responding a bit later to a question Lambert raised regarding \"amendment as endowment versus amendment as obstruction,\" that the tactic might be viewed as a bit quirky. \"But who says quirky can't be serious?\" she rhetorically asked.\n\nLambert and Penguin, then, approached their parts (in many ways the linchpin of the piece) with all due seriousness, holding any impulse to ham it up, go tongue in cheek or in some other way play for comic effect at bay. Each adopted a damped inflection, whisperlike but a whisper less in volume than tone, conspiratorial\u2014open, even so, to exhortative, declamatory moments now and again. The clothespins did their job of detour, inhibiting certain habits of pronunciation while enabling alternatives, conducing to a pestered, repercussive manner of speech that did indeed, thanks to the synthesizer's choric hints and empyrean laminates, seem the issue of a pidginizing \"celestriality\" (to use Alan Silva's term), universal patois. I need to stress that this wasn't arrived at quickly, that there were a number of false starts and a good deal of trial and error before Lambert and Penguin found the right vocalic balance and blend, the requisite pace and restraint to make it work.\n\nAs I've noted, Djamilaa extracted a lush but buried oratorio from the synthesizer, interred, if one could so say it, in the firmament, a streaming advance and a sheeting sweep thru the heavens. Solemnity and moment, neither untouched by requiem, were hers to maintain, universal bounty both mourned and extolled. Swell had to do with it, as did surge, emanation, pulse\u2014pressed amenities it was all one could do not to be borne away by, however much (yet all the more) one felt allied.\n\nAs for my part, it especially fell to me, as I've already said, not to be borne away, to offer an astringent bounding bass and universal bounty could be qualified by. Counterpoise and parry were typically my lines' relation to Lambert's and Penguin's readings, moments of convergence not altogether absent. Moot sublimity, all such rally was earned assurance's emptying out, a hollowing out. Incertitude vied with assurance, earned assurance hallowing hitch, hesitation, sputter's recondite remit. Lambert and I or Penguin and I, as were the trumpet and I, were each the other's hollow extension.\n\nWe played the piece a good number of times, even after we finally got it down (especially after we finally got it down). I'm enclosing a tape of the best version.\n\nAs ever,  \nN.\n19.IX.83\n\nDear Angel of Dust,\n\nWe played a gig at the Comeback Inn over in Venice night before last, a place on Washington about a third of the way between Lincoln and the canals that's been around for about ten years. I heard Joe Farrell there five years ago. We were pretty surprised to get the call, our stuff being more outside than what they normally book. Evidently it had a lot to do with last year's Kool Jazz Festival, a lineup called \"New Directions in Sound and Rhythm\" that most folks in town still haven't gotten over\u2014some in a negative way, some positive, as appears to be the case with the owner of the Comeback Inn. It was an unlikely lineup for a Kool Jazz Festival, especially one here in L.A., a bit like an invasion from Chicago, the AACM side of Chicago: Muhal Richard Abrams, Anthony Braxton, the Art Ensemble of Chicago, Air, Lester Bowie's From the Root to the Source, Leroy Jenkins's Sting, Roscoe Mitchell's Sound and Space. Also on the bill were James \"Blood\" Ulmer, the World Saxophone Quartet, John Carter, Laurie Anderson and the Nikolai Dance Theatre. Though we had mixed feelings about this kind of music being presented under as commercial an imprimatur as Kool (Aunt Nancy at one point claimed to wonder if the balloons would begin emerging as smoke rings), it was all rather beautifully anomalous, inspiringly so, nowhere near business-as-usual. One funny thing we've heard is that the night Laurie Anderson, Leroy Jenkins's band, the Art Ensemble and the Nikolai dance troupe performed at the Santa Monica Civic more than a few people showed up expecting to hear Sting the rock singer. Anyway, the owner of the Comeback Inn specifically mentioned the Kool Festival. \"It really opened my ears,\" Lambert told us he told him when he called, going on to joke, \"It might've even grown me a new pair.\" He went on to say he'd been meaning to get in touch for a while and asked if we'd play there some night. Lambert said yes and they decided on a date.\n\nIt's a small restaurant with not much in the way of a stage, just a spot in one of the corners where they clear away a few tables. It was no great shakes, but a number of friends showed up and it all went well, with a few moments here and there that more than simply went well. Penguin took an alto solo on \"The Slave's Day Off,\" for example, that had Aunt Nancy, draped over the bass like a rag doll, sustaining a rafters-rattling walk that belied her rag-doll diffidence, a walk whose lowest note repeatedly served as Penguin's moment of truth. Repeatedly, Aunt Nancy's walk's furthering pulse notwithstanding, Penguin, with a broken gait and ghost timbre recalling John Tchicai, took stock of the wherefore of going on, assaying, it seemed, every reason not to. A rummaging hover, low to the ground it seemed, the horn's low-register audit, Hamlet-like, left no dissuasive stone unturned. Repeatedly, Penguin pulled free of such hover by way of an angular but oddly damped move into the middle and upper registers, refusing to posit such ascent as triumphant (\"flat\" refusal, lateral dispatch), a resolution of the quandaries by which he was beset. Repeatedly, he offered no resolution, as though resolution could only be false, the more false the more triumphalist, the more false the more defeatist as well. It was some of Penguin's headiest, most heartfelt playing ever, a mesmerizing insistence upon and abidance in a third way that wasn't just a middle way, a hypnotic hum, hover and run of a solo that didn't so much finish as fade, beg off ending, beg off going on, Penguin pulling away from the mike playing less and less loudly. As I've said, there were other moments as well.\n\nArguably, however, the gig's aftermath upstaged the gig itself, its most eventful moment occurring afterward, as though a time lag of some sort were in effect. Drennette, that is, tells us that two balloons followed her home (as she puts it) after the gig. She says that yesterday as she was unpacking her drumset, setting up to practice, they emerged as she lifted the lid to the parade snare's container. Before she took the drum out of the container, before she even reached in to take hold of it, a balloon floated up out of the container, emerging from under the slightly lifted lid as though the opening effected by the lifting were the container's mouth. The balloon, floating just above the opening, contained these words: _Too funky. Too forward. So rude the way o_ _f_ _the world, too crude. The nerve o_ _f_ _him to come on that way, slick-chivalric. \"Be my queen,\" he said. \"Let my face be your throne. I'll lick your pussy, I'll snif_ _f_ _at your ass-crack.\" So crude a way to put it, so rude the way o_ _f_ _the world._ It hung there a while and then vanished, at which point a second balloon floated up out of the container and hung just above the opening. Inside the second balloon were these words: _\"You oughta not be so funky with it,\" I should've said, semisung, Aretha-like, mock-operatic, and would've said, semisung, Aretha-like, mock-operatic, had I only thought o_ _f_ _it in time. Instead, I stood speechless, taken aback._ It hung there a while, just above the parade snare's container's opening, and then vanished.\n\nDrennette says the balloons (the first one in particular) remind her of the X-rated balloons that emerged from the dancers' fists during our record-release gig at The Studio back in February, the emergence our recourse to a 4\/4 shuffle meter brought about. She can't help surmising, she says, that the two balloons might be tied to a somewhat similar moment during our second set at the Comeback Inn, a passage during \"Tosaut L'Ouverture\" in which Aunt Nancy touched ever so lightly upon a 4\/4 shuffle, not so much committing unequivocally as alluding to it. In response, Drennette resorted to an obliquely stated backbeat, bordering on tongue-in-cheek, a manner of statement that had it both ways, both advancing the backbeat and beating it back, holding it at bay. It was a passing moment, not at all drawn out, but Drennette says it was that moment, she's convinced, that passage, that gave rise to the balloons in the parade snare's container. That they hid and followed her home, she says, rather than emerging right there during the gig (no balloons appeared at any point during the gig) adds a new wrinkle to this whole balloon phenomenon that we need to think seriously about.\n\nDrennette went on to say, as we talked about this during rehearsal, that she noticed a certain concord (as she put it) between the leverage she and Aunt Nancy had applied to the backbeat and the 4\/4 shuffle and the casting of the first balloon's X-rated material in quotes, to say nothing of the explicit distancing from and disapproval of that material the balloon expressed. She wondered if, consistent with this, the balloons' delayed emergence doesn't bespeak reserve, a sense of restraint, modesty even, albeit the business of hiding away and following her home does have, she can't help feeling, a sinister side. Or does the delay, she went on to speculate further, have to do with the speechlessness or the inability to immediately respond, the being less than quick on the uptake, to which the second balloon confesses? What also strikes her, she went on to say in an outrush of questions and thoughts that made it clear how deeply the balloons had gotten to her, is that the balloons appear to have a sense of history, so unmistakably, in this instance, harking back to an earlier emergence.\n\nA good amount of discussion ensued, none of us quite sure what to make of this new development but each of us, notwithstanding, venturing a comment or two, a question or two, a surmise or two. Lambert, for example, led off by all but asking Drennette, whose manner was ponderous, weighty, bordering on distraught, to lighten up, noting that it wasn't the balloons' sense of history he was struck by but their sense of humor, especially the second balloon's reference to Aretha's \"nasty gym shoe\" ad lib on the _Hey Now Hey (The Other Side o_ _f_ _the Sky)_ album. Aunt Nancy, on the other hand, agreed with Drennette regarding the rapport between levered backbeat and 4\/4 shuffle and the balloons' recourse to quotation, adding that the use of the conditional tense appeared to her to have to do with this as well. Thus, things were off and going.\n\nMuch got said during the discussion but we all continue to give this new development thought. And though much got said, not everything that might've been said got said. Djamilaa mentioned to me later that she didn't bring it up, for obvious reasons, but there seemed to her to be a strongly personal element to the balloons' manner of emergence and their content both, that it seemed they wanted to say something about Drennette or even that Drennette wanted to say something about herself: putatively hard-ass Drennette, putatively repercussive Drennette, Drennette Virgin. \"'Slick-chivalric,'\" she said, \"rhymewise and otherwise, has Rick written all over it.\"\n\nMore anon.\n\nYours,  \nN.\n24.IX.83\n\nDear Angel of Dust,\n\nMore balloons last night. It was during the John Coltrane Birthday Concert we took part in at the Century City Playhouse. The folks at Rhino Records organized it and a good number of bands played. The usual suspects from around town were there: Badi Taqsim's trio, the Boneyard Brass Octet, SunStick and the Chosen Few, Horace Tapscott's quintet. Bobby Bradford and John Carter came in from Pasadena, Roberto Miranda came in from the Valley. It was a marvelous event. Weatherwise, what had been a beautiful day turned into a beautiful evening, somewhat on the warm side, a touch of Indian summer, but unusually clear, sparkling, a bit of burnish left over from the day. Most of the musicians hung around outside when not playing. One could still hear the music and it was nice to see and talk with folks one hadn't seen in a while. We milled around in front, some of us strolling a block or two up or down Pico. Badi Taqsim and I stood gazing thru the fence at the Rancho Park Golf Course at one point, talking about recent bookings at Hollywood's Catalina Bar & Grill. Anyway, we played a couple of pieces, \"Sun Ship\" and \"Sekhet Aaru Struff.\" It was our first public performance of \"Sekhet Aaru Struff \" and it was during this piece that the balloons emerged. It was actually, to be more precise, during a section toward the end of the piece that was a bit of a detour, a turn toward rumba initiated by Drennette.\n\nWe'd been talking about rumba earlier in the day, listening to and talking about an album by Totico y Sus Rumberos that came out a couple of years ago, so the turn Drennette initiated didn't entirely come as a surprise. Their rendition of \"What's Your Name?,\" the old doo-wop hit by Don and Juan, had especially caught our attention, getting us going on the mesh between doo-wop's mellifluous come-on and rumba's courtship mimetics. Exactly how apt or effective a mesh it is was what we discussed, opinions ranging from endorsement, even outright rave, a claim that the merger isn't only beautiful but long overdue and that the piece is the best on the album by far (Penguin), to reservations regarding the advisedness of literalizing what's otherwise more subtle, productively so, otherwise more dynamically understated (Drennette).\n\nIt wasn't entirely a surprise, then, as I've said, when Penguin came to the end of his recitation and Drennette began to beat out a guaguanc\u00f3 rhythm on the conga, not only beating out the guaguanc\u00f3 but singing a lalaleo or diana, the introductory song-syllables \"ana na na ana, ana na na ana,\" which in fact made the detour less jarring, contributing to and thereby continuing Lambert and Penguin's theme or thread of universal patois. Soon after finishing the lalaleo, however, she sped things up, switching from guaguanc\u00f3 to giribilla, a more strictly musical, nonmimetic form that has been called the bebop of rumba variations.\n\nAunt Nancy was the first to respond to Drennette's detour, letting the bounding figure go and imitating, in the bass's upper register, a segundo's giribilla pattern. Djamilaa's synthesizer turned its interred oratorio into a chorus answering Drennette's lalaleo, a bank of antiphonal echoes Djamilaa granted galactic reverb, intergalactic reach. I temporized for a few measures before letting my sputters give way to a golden run worthy of Chocolate Armenteros, but it was Lambert who most decisively responded to Drennette's detour. Removing the clothespins from his lips and putting them in his coat pocket (the clothespins, by the way, had drawn laughter from several people in the audience when the piece began but they'd gotten used to them and quieted down), he picked up his tenor, put it to his mouth and motioned for me to pull back.\n\nLambert began by going back to the sputter my golden run had come out of, blowing a barrage of expectorant bleats and pops not unlike an attack of hiccups. Beginning there, he indeed never left, thriving on what sounded like obstruction even as he ran the gamut from belly laughs at the horn's low end to a wistful quizzicality in the upper register that at times took him to the spoons. More than Totico's \"What's Your Name?\" he appeared to have Frank Lowe's \"Broadway Rhumba\" in mind, rummaging around the horn as though it were a hot potato or a goose's neck or as though it were a clothespin pinching his tongue. He sustained a blustery tone, bursting, it seemed, with things to say, albeit more things than could be said it seemed. It was during this solo that the balloons emerged, the first of them lifting heavily up out of the bell of the horn bearing these words: _An Egyptian rumba she said it would be, abstract, angular, undulant, pulse beaten out on a salted cod box, an Egyptian giribilla she said it would be. No BaKongo cloth kicked in a circle, no guaguanc\u00f3, no lifted skirt edge, no not being caught by the vacunao. A giribilla, no euphemistic vaccine, she said it would be, an Egyptian rumba, ardent, austere._\n\nFollowing the first balloon's emergence Lambert took a more confidential tack, sputtering as before but as if under his breath, resorting to the sotto voce forage he pursues to such resounding effect. It was a more gauzy sound but one with which he parsed not a whit less, having no less recourse to angles, inversions and reversals, breaking _rumba_ apart, it seemed at points, and by turns putting _ba_ before _rum_ and putting _rum_ before _ba_ , an Egypto-Caribbean conjugation having to do with soul (spirits' bearing on soul, soul's bearing on spirits). It was at one such point that the second balloon came out of the horn bearing these words: _\"Isis to his Osiris, I dreamt he stayed inside me all night, forever, stiff, unyielding,\" she said. \"Damsel in distress, dread virgin, I lay scared stiff. Isisn't to my own Isis, stif_ _f_ _but not unyielding, I lay afraid, flat as a board beneath his weight.\"_ The balloon disappeared when Lambert, put upon by a strain of quizzicality stronger than any that had come before, paused ever so slightly. When he resumed playing the third balloon floated up from the bell of the horn bearing these words: _\"I lay afraid but unafraid, feigning frailty,\" she said, \"stiffness answering stiffness, yielding even so, faux fragility stiffening him throughout all eternity. Stif_ _f_ _intruder I welcomed in and regaled with my own stiffness, he likes it when I start of_ _f_ _stif_ _f_ _and begin to loosen.\"_\n\nAfter the third balloon's emergence Lambert stepped back from the mike, let the balloon vanish and put his tenor back in its stand, the audience applauding loudly. Drennette ever so subtly blushed, as though the balloons had peeped her heart of hearts, but she held her head higher than before, her back straighter than before. She let go of the giribilla pattern, Aunt Nancy returned to the bounding figure, I came back in on trumpet and we took the piece out, the audience applauding loudly still.\n\nAs ever,  \nN.\n[Dateless]\n\nDear Angel of Dust,\n\nHotel Didjeridoo would have none of it. Hotel Didjeridoo, the once towering, long since fallen cathouse of wide repute I dreamt ages ago, reluctant object of the Resurrection Project's announced intentions, would have none of it. Strain as I might, it refused resurrection. The news the Hotel had fallen flew like bullets overhead, insisting I lay low, insisting I stay down. The news the Hotel had fallen continued to be news. My low reconnoiter had barely begun, the bullets cried out. No way was it to be risen up from in a hurry, no way hurried out of, no way made an edifice of. All such architectural conceit had long since been made moot. Limit-case dispatch \u00e0 la Braxton's contrabass clarinet, low reconnoiter knew nothing if not crawl space, crablike sashay (if it could be said to be sashay) tending toward and touching on collapse. It knew nothing if not limbo's antithetic arc.\n\nBullets indeed flew, announcing and reannouncing the Hotel had fallen, insisting and again insisting I stay down. \"Lay low\" it seemed I heard and indeed I did. Hotel J\u00edbaro the Hotel appeared to have become in its demise. \"Aylelolaylelolay\" chimed inside my head. A glossolalic launch long the custom in Puerto Rico, it was an odd scrap, a bit of caroling debris, a curious run or recourse to non-, post- or pre-sense lyrics I was nothing if not taken aback to hear inside my head. \"Aylelolaylelolay\/lolaylelolelolela,\" it ran and repeated, a talismanic loop whose repeated \"lay low\" I could no way ignore. It threaded its way thru the fabric the bullet barrage apprised me of, the accompanying quatro's wincing ping as piercing as a Portuguese guitar, needlepoint address I drew back from but welcomed (bittersweet pinprick, bittersweet vaccine).\n\nIt was an odd bit of music to be visited by while under fire, but on a certain level, the social fabric one heard so much about shown to be what it was, Puerto Rican scat's admonitory \"lay low\" made perfect sense, an involuntary reflex insisting stay down if one were already down, duck if not. Admonitory duck, moreover, bordered on earth-diver duck, a foundational drop or a compensatory plunge rendering save-oneself and save-the-world one and the same, bite of dust and beakful of mud one and the same. One lay on one's belly, having hit the dirt, bullets whistling overhead, one's j\u00edbaro serenade woven into the guns' ongoing report. It all made perfect sense.\n\nHotel Didjeridoo, I had to accept, would have no rebuilding, no rebirth. A thread running thru the proverbial social fabric, \"lay low\" said let the proverbial dogs lie, what was done was done. Even so, as I lay I stood ready to ring changes on syllabic largesse, \"aylelolay's\" unraveling rope a noose I sang or sought to sing my way out of, the fait accompli Hotel Didjeridoo resigned itself to. Or was it syllabic duress, a rope I ran the risk of choking up on if not dangling from? I rolled over and lay on my back baying at the moon, \"aylelolay\" my coyote howl. Caught feeling, fraught sonance, \"aylelolay\" culled a rapport harking back to what some said were Muslim roots, a derivation (or so, at least, I had heard) from _ilaha illallah_ , not unlike bullfighting's _ol\u00e9_ (I had also heard) harking back to _Allah_.\n\nHotel Moro it might well have been and was, \"lay low's\" metathetic remit a reminder of Lole Montoya's voice. Husky, full-throated, the hull of a boat I lay on my back looking up at, it bore tidings from Egypt, Mohamed Abdelwahad's \"Anta Oumri\" and \"Wdaret el Ayan\" segued into out of \"Sangre Gitana y Mora,\" an incendiary sunship scorching the water I lay under.\n\nStill, I stepped forward and blew\u2014stood up, stepped forward, took horn to mouth and blew, \"lay low\" notwithstanding. A bugling flutter and flex it was, the feather I could be knocked over by rousing me even so, requiem and reveille rolled into one. Brass but ever so inward (so cracked a wrinkle of sound as to risk erasure), it raised a hand, it seemed, all the same, hailing the ship's hull in the water above. Incendiary boat all the more inveterate rebuff, Lole's voice I would chase forever, its brass equivalent not quite to be arrived at. I would chase and be forever outrun.\n\nI resisted making much of \"low lay,\" the obvious play on jazz's reputed bordello roots. Lole's voice was a bit of husk in my throat, tightening it, constricting my airflow, choked-up endowment could it be said to be an endowment, poignant, unappeasable, possessed. I blew to be its equal or, short of that, to etherealize it, bodily husk an abidance never to be gotten free from, blow though I did as though I could. Brass, that is, was a way of getting by or going on. It afforded what solace it could, its by no means moot condolences. I had recourse to a coyness recalling Miles, the face-behind-a-fan retreat or seeming retreat heard on \"Circle,\" demure but not without design (\"See how that sounds, Teo\"). It turned away from the ship's hull overhead, a blush or the beginnings of a blush burgeoning inward, a breathy spread of guile and regret. It was anything but \"aylelolay\" but not unrelated, nothing if not its compunction or its qualm.\n\nIt occurred to me it might also be Hotel Mita, \"aylelolay's\" Puerto Rican provenance bearing again but in a slant manner. The ship's hull was a message on high, \"el mensaje de alto\" sung about by the Rondalla del Templo de Mita in San Juan, the song's mention of Noah no doubt the connection, its \"Como los tiempos de No\u00eb vino el esp\u00edritu\/anunciando que viene destrucci\u00f3n\" resonating with Hotel Didjeridoo's fall. I brought this to bear upon the horn, put it into the horn as though the horn were the proverbial pipe, my blush or my beginnings of a blush combusted, nothing if not the Rondalla's heartrending chorus. \"Amigo, ven y oyes la v\u00f3z de Dios,\" I said with my breath as well as under my breath, a smoked insistence I parried against the boat's unreachability, smudge and buff's newly made amends.\n\nHusky, haggard, Lole's voice I already knew I would chase forever. The Rondalla's call I now knew I would chase forever, equal parts cry and confection, sweet smoke circling skyward. The sea I saw myself laid low in held me up, the ship's hull caulked as if coaxed into the water, insisting I blow my horn. I felt a salty-sweet buttress or brace whose rough accompaniment said, not outright saying it, stand tall.\n\nI stood underwater blowing my horn, an Aquarian undulance inflecting each note. I blew as though aided by synthetic strings, a synthesizer's air of impendence, a watery element all its own. Urge and emanation rolled into one, doused alterity of an astral sort informed each note, a star having fallen into the sea and set it boiling, each new insinuation bugling no end. That star was none other than the sun, warily ensconced in Lole's Atet boat, the ship whose hull her voice was. The morning boat's fall or its failure to lift (if either, against all odds, was what it was) stirred the bottom of the sea and sent waves everywhere, roiling with insinuation, the very blush or beginnings of a blush I blew like smoke out the bell of the horn.\n\nThe mixed-metaphorical premises on which I stood\u2014smoking water, undersea buttress, \"lay low\" vantage\u2014turned my legs to rubber, propping me up even so. Bubbles came out of the bell of the horn, smoke bubbles, bubbles that were in fact balloons. Each balloon bore a message as it rose toward Lole's boat but burst when it got there, raining down bits of script, the words that had been inside. Gone up in smoke, such bits as _I caught myself_ came floating down, water more like air than water, such bits as _letting myself lean back before catching myself_. A note-bearing bottle as well as a comic-strip balloon, each bubble as it burst blessed my undersea launch with incongruous ash, burnt bits of script consecrating my intonation, the words or the would-be words I bruited on high. \"All this in the wake of Lole's boat,\" I said to myself, \"the merging of hull and husk I'll forever be outrun by.\" It was a vow, a benediction even, a mandate I afforded myself. \"Lay low\" auspices notwithstanding, running after would never stop.\n\nMy horn wanted nothing if not to announce liquidity's advance beyond rectilinear form, the stiff amenities undulance and curvature so luxuriously annulled. Such amenities bend and slur would have nothing to do with (or would, were they to, only by way of contrast), just as Hotel Didjeridoo, root brass or root embouchure, would have nothing anymore to do with architectural conceit's dream of rise and rebirth. Hotel Didjeridoo, I reminded oneself, would have none of it, nor would Lole's metathetic morning boat, mixed-metaphorical ash ran in whose wake, burn and rescinded script rolled into one.\n\nThe morning boat's fall or its failure to lift was in fact a refusal. Lole's voice would allow no likeness. Stout reluctance I thought to call it, willed unwillingness, husk none else than a containment, heartfelt holding back. The boat sat taunting me, teasing me, flat on the water it seemed\u2014flat but holding all there was of arc or ascendency, hold and hull nothing if not the same.\n\nWater wet the horn I blew underwater. Wooing the boat's hull overhead, I imagined myself aboard ship, stowed away in the hold but aboard it even so, none other than the smoldering sun. It was a doused Osirian sun I saw myself as, adamantly of a piece with the smoldering horn I blew, the wet trumpet's blare an upsurge of bubbles ascending the sky the sea whose floor I stood on had become. Still, the boat stood me up in more senses than one. Each bubble, before it burst, gave me something to reach for, a note-bearing boat inside a boat-bearing bottle, the rendezvous of seed with husk Lole's boat's hull promised, destiny's proverbial date. Each bubble, having burst, left me dateless.\n\nDatelessness notwithstanding, burst bubbles notwithstanding, I rallied, a hip variation on first call coming out of the horn, a bugling whose call was to carry on. Stingy-brim strut woven into a mustering charge, it was nothing less than a gauntlet hitting the ground.\n\nWhat it was was that one horn wasn't enough, one voice not enough. There was something crowded about Lole's voice, precisely what one meant by husky. It was as though, no matter how subdued a turn her voice took, some indefinable something or a multitude of such were jostling for space. It was all she could do to contain strains that would otherwise undo the tenuous-accord-cum-strenuous-contention husk held in check.\n\nEven from below I could see the boat bulged with voices, some stowaway, some aboveboard. It bore more than one voice, however much Lole's voice might've been the voice that fueled and kept it afloat, much as the smoldering sun it bore might've been no more than a smoke-filled bubble, burning-boat-bearing bottle inside an incendiary ship. To see it so cried out for a larger conception, a new perspective, Donald Byrd's recording of that name coming quickly to mind, only to have its eight-voice choir rejected as too pat or predictable for what was called for here. Voice (\"chorality\" or \"choricity\") needed to be more diffuse or more dispersed, more constituent strain or pervasive surge as in the sort of emic or seemingly emic effusion, aqueous or atmospheric, a synthesizer might provide.\n\nI stood ready to acknowledge this need and, indeed, almost before one knew it, I already had. The bulging boat swollen with voices would be ratified beforehand, if the performance now underway became part of the antithetic opera (theoretic opera some called it) of which for a while there'd been so much talk. A quote I had not only long savored but knew might one day prove useful popped into my head as I continued to blow bubble after smoke-filled bubble. Should the underwater, boat-wooing performance come to be included in the opera, the quote would serve as Aquatico-Solar Epigraph #1:\n\nOsiris represents the Nile and the Sun: Sun and Nile are, on the other hand, symbols of human life\u2014each one is signification and symbol at the same time; the symbol is changed into signification, and this latter becomes symbol of that symbol, which itself then becomes signification. None of these phases of existence is a Type without being at the same time a Signification; each is both; the one is explained by the other. Thus there arises one pregnant conception, composed of many conceptions.\n\nIsis to the epigraph's Osirian _mise en ab\u00eeme_ , I quickly resorted to circular breathing, a tack whose underscoring of reciprocity verged ever so lightly on rotundity as well (swollen belly and swollen boat rolled into one).\n\nStingy-brim strut had come a long way, a diffuse, omni-voice dilation or solution absorption knew no way to constrain. No more than a mere hint of oratorio, no more than a mere hint of chorale, it furthered an aqueous consort of carolings coincident with the medium itself\u2014coincident but not by that to be contained or ingested, an attenuation beyond (while residing within) liquidity's confines. Liquid's exhalation of gas was what it was like if not outright amounted to, an exhalation that oddly wafted within even as it wafted while standing in wait beyond. Such wafture, multiple, immanent, extrapolative, was nothing if not the quantum stride (not only soulful in addition to quantum but quantum only insofar as it numbered soulfulness among its attributes) called for by stingy-brim strut and begun by it to be put into play.\n\nNonetheless, neither \"aylelolay\" nor \"Mensaje de Alto\" had been left behind, each a standing rope song's promise of ascent made a signpost of sorts I repeatedly made note of and, in so doing, made listeners make note of as well. A firemen's pole if not a laminated snake, it could hardly not also expostulate descent, bearing senses of emergency, alarm and even frantic dispatch, its croonfulness and reassurance at the strictly musical level notwithstanding.\n\nIt all went to insist again that Hotel Didjeridoo would in no way countenance easy elevation, that it would have no truck with facile rebirth or rebuilding. Hotel Didjeridoo would have none of it, as its three alternate names (J\u00edbaro, Moro, Mita) and its unstated but implied potential for even more names had meant to make clear, \"lay low's\" and\/or Lole's blown smoke's amity with \"lo alto\" not to be taken as other than the qualm or qualification it assuredly was.\n\nAmbient accompaniment, water maintained pedal-point armature, fluviality, surge and subsidence, light oratorical wax and recess. Stingy-brim strut's Osirian posture stood on friendly terms, it all but went without saying, with standing itself, obfuscating Osiris's coexistent rapport with recumbency, recline, supination. It was this blown smoke's ascendancy made clear.\n\nSo I bent my knees and squatted, continuing to play, not missing a note. I leaned back a bit, letting my ass rest on the seafloor, horn still in hand, continuing to play. I lay back farther, the back of my head coming to rest on the seafloor, my back flat against the seafloor, still not missing a note, continuing to play. I extended my legs until they too lay flat on the seafloor, horn still in hand, horn pointed upward.\n\nAside from the fact that I was breathing and not lying perfectly still, moving my arms and fingers to play the horn and keeping time with my right foot, I lay like a mummy, not only not missing a note but playing my heart out, serenading and seducing (would-be seducing) the boat overhead.\n\nSincerely,  \nDredj\n30.IX.83\n\nDear Angel of Dust,\n\nI'm writing about yesterday's letter. It was written, as you no doubt saw right away, during a cowrie shell attack. The attack seems to've been brought on by a chain of associations or a train of thought that began with me reminiscing about my early childhood in Florida. I was particularly nostalgic about conch fritters, which I vividly remember though I haven't eaten any since we left Miami when I was four. Was that enough to ignite the conch component of the cowrie shell attacks, the new, scalpel-edge wrinkle that seems to've motored recent attacks even more than the cowries themselves? Or was it my recollection of a family fishing trip when I was three, the incident everyone still laughs about, my uncle reeling in a fish, whipping the line back in a high arc, and me, standing farther up the bank in back of him, getting hit squarely on the head by the fish on the end of his line?\n\nWhichever, if not something else and if not both, it got me going. Whether conch-implied incision or fish-upside-the-head wallop, something seemed intent on underscoring depth, subaqueousness, river if not sea. It seemed to merge with a watery, \"oceanic\" sense I'd felt while playing \"Sekhet Aaru Struff \" the day before (albeit not so much felt as felt surrounded by perhaps), a sense I couldn't quite shake of being underwater, cosmic or not if not earthly and cosmic both. Was yesterday's letter Dredj's attempt to convey that sense? Garbled attempt I'm inclined to say, so brimming with matters not particularly germane to \"Sekhet Aaru Struff \" the letter seems to me to be. Or are these departures from point the very \"point,\" the diffusion of point I touched on in the letter that accompanied the tape? So maybe it's not so much a garbled as a \"struffy\" attempt?\n\nIn any case, call it Dredj Alley, for it was an alley Dredj ducked into before it became a body of water whose floor he was on, an alleyway just off the beachfront in Venice, I'd go so far as to say, up from which one looked out from a flat filling up with sharp Shostakovian light. From House of Dredj to Dredj Annex to Dredj Alley, the attacks appear to be on an anti-architectural course, a course of architectural undoing, Hotel Didjeridoo refusing resurrection, insisting solidity and solubility hold hands. \"Sekhet Aaru Struff \" or not, Dredj descried hand holding hand, hand on hand, held hands holding his writing hand.\n\nA light attack, it lasted only a day.\n\nAs ever,  \nN.\n4.X.83\n\nDear Angel of Dust,\n\nI wouldn't say I'm taking a page out of the balloons' book, that I'm trying to beat them at their own game. Aunt Nancy suggested as much but I don't think that's it. No, this new thing I'm trying goes back to a story Yusef Lateef tells about the days when he was in Mingus's band, a story I was deeply struck by when I first heard it, a story I think about from time to time. Yusef says there was a composition on which he was to solo and that Mingus, rather than writing out chord symbols for him to improvise against, drew a picture of a coffin, that it was this that he was to base his improvisation on. A friend of mine once joked that Mingus simply meant that if Yusef messed up the solo he'd kill him, but I've long been intrigued by and attracted to the idea of getting musical information from a picture and it's this that led me to a certain experiment with my latest composition. Braxton's diagrammatic, pictogrammic titles and the solo concert of his I caught a couple of years ago, the scores for which looked like pen and ink drawings, nonfigurative but drawings even so, had a role as well.\n\nThe new composition is called \"Fossil Flow.\" I wrote it thinking about oil spills, the increasing number of them and the damage they do. Just this year there've been two massive ones: in February, the Nowruz Field platform in Iran spilled 80 million gallons of oil into the Persian Gulf; in August, a Spanish tanker, the _Castillo de Bellver_ , caught fire and spilled 78 million gallons off the coast of Cape Town, South Africa. I was thinking about the distant past (prehistoric apocalypse, collapse or catastrophe) achieving fluidity, the oxymoronic play between fossil and flow of such dimension as to put the present at risk. It's as though it were the dinosaurs and the mastodons' revenge, prehistory's grudge against what came after, a brief against preservation or containment, fossil solidity, an entropic brief against past and present keeping their places. It's as though, Dredj-like, I saw solidity's hand and solubility's hand, gripped though they were by one another, holding history's hand, leading the way as it broke. Or was it, oil and water notoriously not mixing, solidity's hand and insolubility's hand? I'm not sure it matters. Recalling the rationing and the long lines at gas stations a few years ago, I saw dependency's hand and depletion's hand take solidity's hand and (in)solubility's hand's places, presided over by an entropic sun.\n\nMuch of the piece is written out but I'm trying something new, something of a built-in improvisation approach, by leaving gaps at various points in everyone's parts, gaps of a certain number of measures (which varies) marked by the words \"Wild Card.\" The latter refer to a drawing and text with which each musician is provided, an 8\" x 12\" posterboard \"card\" on which he or she is to base what he or she plays at that point. I'm enclosing one. As you can see, the \"card\" consists of a drawing, captioned \"Molimo m'Atet's Figurehead Consoled on the Revival Bench,\" beneath which is a brief paragraph. I struggled over whether or not to include the latter, fearing it might be spelling things out too much, taking away from the suggestiveness of the drawing. I decided in favor of keeping it, realizing that it adds a suggestiveness of its own, that words, regardless of how much they point or specify, can't altogether escape indefiniteness or inference, that, indeed, specification has a way of being shadowed by implication. What, for example, is to be musically made of the fact that the figurehead's ribs show, simply enough, in the drawing but also show, in an augmented, not so simple way, in the words \"visible, as were the planks of the ship's busted hull\"?\n\nI'm also enclosing a tape. Let me know what you think.\n\nYours,  \nN.\n\n.\n\n_Molimo m'Atet's Figurehead Consoled on the Revival Bench_\n\nAn oil tanker had run aground farther up the coast and broken apart. Brothers in black before they knew it, B'Loon and Djbouche washed ashore with the news of the spill. People gathered on the beach to help clean up and help rescue seabirds, oil and tar stuck to their feathers from alighting on the water or, standing or prancing on the shore, being caught by the tide. Bright sun and blue sky notwithstanding, the spill cast a pall over everything and everyone, not least of all Djband, who, likening themselves to a ship, the sun boat of Egyptian belief, felt as though they too had run aground. Epitomizing the \"boat-bodied lightness, light-bodied bigness\" one of them had once extolled, the female figure gracing the prow of the ship they took themselves to be (the goddess Maat, some said) stepped away, walked ashore and sat down on a bench facing the sea, head down, dejected, ribs visible, as were the planks of the ship's busted hull. Impromptu patron saint of shipwreck, ad hoc angel, Dredj immediately sat down beside her and put his arm around her, offering comfort, consolation, recondite sun, as if to look to and be lit by eclipse were the only amenity.\n8.X.83\n\nDear Angel of Dust,\n\nMany thanks for your letter. I appreciate your comments on \"Fossil Flow,\" your reaction to which, I have to admit, I was nervous about. Yes, you heard it right. \"Stratified extinction,\" as you say, does pervade the piece, a tributary distinction between extinction and exhaustion woven in. I'm impressed by your picking up on the \"Wild Card\" sections of each band member's playing and what about the drawing and\/or text informs how he or she plays. Drennette says you're right that the abstract bench Dredj and the figurehead sit on (and, by implication, the abstract revival available to them) particularly caught her eye and especially spoke to her, giving rise to the stroked, retreating figure she has recourse to with brushes that you note a couple of minutes into the piece (not unlike, she agrees, sand pulled away from the shore by a receding wave). You're also right that the \"welter of double-reed hustle\" Penguin and Lambert get into on oboe and English horn, respectively, the outbreak of metaphysical sweat with its needling or drilling insistence as if to answer a spiritual-materialist clot begging to be cut thru, is one of the places where two players' \"Wild Card\" sections coincide. And your surmise that Penguin's choppy, shenai-like drone has to do with a focus on the apparently oil-toed and oil-fingered extremities on the drawing's right side agrees with his account; a meditation, he calls it, upon those extremities' \"tarpit premises.\" Likewise, Lambert's barking, dilated, alto-sounding complaints derived, Lambert says, from an impulse born of both pictorial and textual cues, an impulse to \"occupy ribcage arena,\" as he puts it, those cues being exactly the depiction of and the reference to the figurehead's ribs that you suggest animated his playing at that point.\n\nNothing much else to report. Just a quick thanks for your letter.\n\nAs ever,  \nN.\n10.X.83\n\nDear Angel of Dust,\n\nPenguin seems to have been heartened by the subtle blush the balloons brought to Drennette's face at the Century City Playhouse. It came up as he and I were talking today. \"Lambert got it right,\" he said. \"Or should I say the balloons got it right? They blew the whistle on Drennette's fake reticence, ripped away the blas\u00e9 front she puts up. With all that stuff about the balloons following her home, all but about being stalked or maybe outright about being stalked, she got it right as well, auguring a muse of pursuit and pursuit's engine or draw, reluctance, a ruse of rebuff, a prude poetics.\" He paused a moment, savoring the thought. \"There was more to that blush than met the eye,\" he resumed. \"Her heart's blood flew to her head.\" It seemed it flew to his as well, for he paused again, subtly blushing at the conceit before going on. \"The balloons not only blew the whistle,\" he then went on. \"They let the air, so to speak, out of an inflated self-regard. They all but burst with Drennette's recondite desire to be found out, contested, caught by lordly science alone.\" He subtly blushed again, inwardly balked, twin to the balking inwardness that was more than met the eye, the prude interiority he took to heart and took heart from, the soul whose mating twin he'd be if he could.\n\n\"Just like Lambert likes to say about the griot,\" Penguin said when he took up speaking again, gazing into the distance as if what he was going to say came from afar, \"the balloons, bless their hearts, have a big mouth.\" He no longer subtly blushed but again he paused. \"Drennette's a little bit off,\" he said when he spoke again. I took him to mean more than he said. I took him to mean the balloons call interiority out, their mouths, insofar as they can be said to have them, open in awe at Drennette's abscondity, her becoming all the more an object of pursuit by not being all there. He sought leverage it struck me. He'd have made B'Loon's wan smile a satchel mouth. He'd have pried B'Loon's wan smile open, ransacked it, anything to get next to Drennette. Confirmation came at once. Penguin repeated, \"Drennette's a little bit off,\" adding, \"but the balloons would have none of her not-nearness. They would abide by nothing short of not-nearness beginning to see its end, not-nearness beginning to be sashay.\"\n\nI could not have seen Penguin more clearly. I saw him in nothing if not namesake light, hallowed by eponymous aura: ripped, wingless bird, wind-afflicted, flightless, devout. \"The balloons would know Drennette otherwise,\" he said, \"knowing her by her being a little bit off no longer enough. They would have her no more than two bows' lengths away, no more than an atom's breadth away. The balloons would be her throne and her footstool.\" He was quietly raving, caught in a low-key agitation, a game of hide and seek (hers with him, the balloons' with the band), gnostic stranger, grounded bird.\n\nStill, he would aver what was yet to be seen, given heart by the balloons' intimations. He now spoke explicitly of himself, changing the course it had seemed he was on, contrasting himself with the balloons. \"I, however, would know her by her distant footfall, footsteps down a dark hallway, a rustle outside my door.\" I'd all along taken the balloons to be a stand-in for himself and I continued to see them that way though he now employed them as foil. He fell silent and I remained silent.\n\nI had said nothing all the while Penguin spoke and it seemed he expected as much. It was a run of pure devotion, a poem, a paean, an oath.\n\nYours,  \nN.\n26.X.83\n\nDear Angel of Dust,\n\nDay before yesterday we drove up to Santa Cruz to play the Kuumbwa Jazz Center again. It was three years ago we first played there. It was good to venture north again and again we took 101 and then 1 where it branches off from 101 at San Luis Obispo. The coast was a feast of blue sky and blue water again, radiant sun and reflected sun's radiant sparkle on water. It was a feast we could hardly take our eyes off. Sunlight and sea lay hypnotically to our left all the while we put Morro Bay, San Simeon, Carmel and Monterey behind us.\n\nWe got going early in the morning, early enough to get to Santa Cruz by late afternoon. It was still dark, in fact, when we left L.A., so we pulled into Santa Cruz well in time for the sound check, the longer route we took notwithstanding. It turned out we got there with time to spare, which time we decided to burn by visiting the lighthouse. We walked around on the pathways and the sidewalk and the grassy field surrounding the lighthouse, watching the surfers, the bicycle riders, the skateboard riders, the Frisbee throwers. Waves crashed on the rocks. Wind wafted salt.\n\nThe smell of salt addressed us again as we stood outside Kuumbwa later that night between sets. The air smelled heavily of the sea, prompting Aunt Nancy to note \"a coastal piquancy\" she said demanded we play \"Fossil Flow\" during the second set. She couldn't help, she said, catching a recondite whiff\u2014imaginary, she admitted, but all the more insinuative being so\u2014an oblique hint of oil threaded thru the marine bouquet the night air wafted. \"I can't get those rigs off my mind,\" she said, referring to the oil platforms off the Santa Barbara coast we'd driven past in the morning, the site of a huge oil spill in 1969.\n\nIt was hard not to see the sense it made. The specter of the derricks had followed us up the coast, shadowing the sea and the sun's gleam and shimmer, dark prospect under an otherwise bright fa\u00e7ade. \"Fossil Flow\" couldn't have been offered an apter setting, salt itself auguring future affliction, imaginary though the smell of oil might've been. The piece is one that wants to put pressure on flow, indemnify furtherance, bestow auspice and omen upon the undulacy it works. Ominous undulance had already, to an extent, come into play, a suppositious wave and waft Aunt Nancy picked up on mixing oil and salt. Bass player to the bone, she sensed a deep throb, eventual ache, dark unction, a waxing of promise and foreboding that was doubly on the tip of her tongue\u2014oil she could almost taste, calamity she all but announced. It seemed all the more fitting that \"Fossil Flow\" begins with her playing a three-note ostinato.\n\nThere was no way not to see it made sense. We decided we'd end the second set with \"Fossil Flow,\" a fitting end not only to the set and the gig but to the day, a fittingness Lambert would accent by switching from English horn to tenor for this rendition, more deeply resonating with dark unction. Flow, the more we thought about it, cut more than one way, not spoilage or spill alone but excursion. We had, after all, driven up the coast. We had, after all, enjoyed it. There was a principle of nonexemption we grew apprised of, automotivity a fluid aspect itself: we were part of the flow, in on the flow. (I remembered the Buick Dynaflow my uncle had been so proud of back in the fifties. I thought of Ray Charles mentioning a Dynaflow in \"It Should Have Been Me.\") Enjoyment lent undulacy a lilt unction wanted in on. This was the truth we both averred and would keep at bay.\n\nWe played \"Tosaut L'Ouverture\" next to last. When it ended we went right into \"Fossil Flow,\" not waiting for the applause to subside, not announcing it. We came on with a slow lope led by Aunt Nancy's pensive, three-note ostinato on bass, a descending figure whose rotundity of tone edged over into omen, apprehension, foreboding, robust as it otherwise was notwithstanding. A table of sorts was being set\u2014implicative, dark but brightened by the bit of shimmer Drennette's ride cymbal worked in, the calm, confident way she kept time. Djamilaa took up with that bit of shimmer when she came in on piano five bars in, starting out with a subtly happy-hand garnish that had something of glimmer and gleam to it and a prance aspect as well.\n\nThe horns, when Lambert, Penguin and I came in, stating the head on tenor, oboe and cornet, respectively, had an unprompted sense of aside, sotto voce not so much as reaching out from it, doing so with annunciative blare. Call and cry factored in as well, as also did no small amount of tartness, a ribald arrest of all one thought one knew, knowing the pensive lope it was. It seemed we wanted to say something about moment and simultaneity, moment's dismay at simultaneity's largesse, its doling out, moment's dismay at sequentiality's parsimony; it even seemed we had already said that something. This was in large part, insofar as we already had or possibly had, the work of Djamilaa's tolling chords and her solemn, sometimes grumpy left hand. Chorded ploy contended with chorded plenitude, sequential disbursement pressed and vied with by both. How to both unbraid simultaneity's bounty and give it its due, unpack to the point or brink of undoing, was the question we were called to confront in our solos.\n\nThe first to solo, Lambert was all business. He restated the head, grudgingly it seemed, put upon by quizzical misgiving, pestered by qualms. Calling to mind, in that regard, the solo Joe Henderson takes on McCoy Tyner's \"Contemplation\" on _The Real McCoy_ , he allowed himself a certain hesitancy, opening gaps in the head's articulation, not so much a stutter as being repeatedly given pause (albeit stutter, sputter even, was obliquely the case). Bellow and beguilement volleyed, haloed by complaint at every point. Caught in a related quandary, Drennette beat the parade snare as though beating back tears, choking up on the stick and holding it at midpoint, swiping\u2014rare vulnerability, rare admission, rare forthrightness, a clipped, cutting pleat, cropped egress.\n\nI stood to Lambert's left, looking over his shoulder. When he got to the \"Wild Card\" gap within the head's elaboration it seemed it was the oil-drop extremities in the drawing that caught his eye, the figures' oil-drop hands, toes, fingers and feet. Working changes on classic teardrop tenor, he built on Drennette's choked-up admission but also brushed it aside. A parodic, moanlike drop to the lower register picked a bone with lachrymosity and, by implication, unctuosity as well. He would have nothing to do with suspect suavity, he declared by way of a more forthright return to the horn's middle octave, no matter the oil on his and everyone else's hands. It was a beautiful boast, made all the more so by his maintenance of a sober, unflustered tone worthy of Dewey Redman, a soothing, unhurried\u2014did one dare say suave?\u2014uptake or attack.\n\n(No, one dared not say suave. What might've seemed so or one might've said seemed so was in fact a heuristic roughness Lambert plied and parlayed into scoured sobriety, sensibility abraded, a bumped entitlement or sense of entitlement tending more than one way.)\n\nDrennette saw that her choked-up address of the parade snare no longer held sway. She returned her hand to the base of the stick, took the rim of the ride cymbal between the forefinger and thumb of her left hand and began to slowly mark time, the stick's tip hitting the cymbal with a tolling insistence Djamilaa quickly joined in on, repeating a single note in unison with it. Together the two chimed, Djamilaa's \"tallywise\" limpidity (one-two, one-two, one-two...) auditing and all the more endorsing the ride cymbal's understated ring. They too now bore the figures' oil-drop extremities in mind. It seemed they especially wanted to say something about digits, counting, and what it was they said or wanted to say Lambert agreed with, moving to the high end of the horn while remaining sober, keeping calm, as if to give them his blessing, say their tolling rang true.\n\nLambert allowed his solo to end there, a light, breathy peal floating above cost, consequence, toll. It floated above but not free of toll, telling, in its unperturbed way, of debts paid and debts yet to be paid, soberly tolling but no less tolling than Drennette and Djamilaa's ritualistic audit. It lay there a beat and a half, a thin, breathy peal whose remaining aloft ritual audit implied but quickly drew back from, Aunt Nancy, whose repetitive pluck had become part of it, announcing a new direction by pulling out the bow and proceeding to play arco. She gave the bass a cello's Orphic swell, fraught songfulness and fret, a teetering on the edge of elation she let sweep thru the \"Wild Card\" gap she stepped into, treading gingerly as though her toes were dripping oil, her feet soaked in oil.\n\nAunt Nancy bowed with a wincing resonance, as though the bow were an exposed rib\u2014as though, indeed, it were _her_ exposed rib, as though she were the see-thru masthead whose ribs the \"Wild Card\" drawing apprised us of. She bowed as though coaxing the bow across the strings at points, a jittery luxuriance given uncommon reach, uncommon albeit reluctant reach. Drennette peppered Aunt Nancy's resonance and reach (plumb resonance and reach) with rapid-fire outbursts on the orchestra snare, sanctified, spasmodic, pentecostal hammerings \u00e0 la Sunny Murray.\n\nDjamilaa had all but fallen silent, serving up chords every now and then, a more slowly doled out tolling meant to recall what had gone before. This left Aunt Nancy all the more at the mercy of Drennette's infectious pepper, a fact or effect eventually made evident by the percussive tack she resorted to. She came to a point where she lifted the bow and tapped the strings with it, letting it bounce lightly on them, much as Ron Carter does on \"Barb's Song to the Wizard\" on Tony Williams's _Lifetime_ album, a piece, a passage and a technique Aunt Nancy had excitedly turned us on to ages ago. Here she took it further, sustained and stayed with it to an extent barely broached by Carter's jagged innuendo. Rickety buildup grew possessed of growl and grumble, an aroused rattle and would-be rafter shake amassing senses of emergence or at least emergency, rummaging for voice, viability, ground.\n\nThe bow was a mallet, the strings a throaty dulcimer, Aunt Nancy's fingers, thumb and wrist exquisitely schooled. Though she made it seem the bow simply bounced, no aspect of touch or attack went without thought, throaty dulcimer by turns a croaking cimbalom, by turns a raspy santur.\n\nIt became clear, though, that the strings were neither dulcimer, cimbalom nor santur, that the masthead's exposed ribs were the focus of Aunt Nancy's solo, that the strings were indeed those ribs, vertically though they lay, the fact that the bow was itself a rib notwithstanding. Indeed, Aunt Nancy's rib-on-rib address accented intimacy and consolation, exactly the embrace the drawing shows the masthead held in, Dredj's embrace\u2014rib-on-rib contact, rib-on-rib caress, rib-on-rib assumption of Dredj's counsel. Here, however, rib crossed rib and was let percuss upon rib, a fact that not only accorded with but in part conveyed the disconsolate tone of Aunt Nancy's solo, the worked arousal refusing to be put to rest it so starkly was.\n\nEventually Djamilaa fell completely silent and Drennette soon followed suit. Aunt Nancy's solo was now exactly that, Djamilaa and Drennette having bowed out as if to suggest the bass's taut strings had to do with tautology, the solo's disconsolate temper with self-induced or self-digested ordeal, self-conducted ordeal\u2014a suggestion to which, given the way Aunt Nancy's bass revved its own ennui, the way she ransacked it for sound, there was more than a grain of truth.\n\nAunt Nancy played alone and was all alone, played along with being alone, left alone, \"all alone in the world.\" She allowed a hint of self-pity in, part parodic host, part woebegone orphan, not unrelated to Lambert's recasting of teardrop tenor. More specifically recalling Carter, she put the emphasis even more on _rev_ , letting the bow ride and bounce on the strings with new and old verve, new and old volatility, mimicking or mining automotivity's old and new dream. A Model T on a bumpy road the bow might've been, so loudly did brake and sputter vie with flow.\n\nAunt Nancy played for all the world to know she stood alone, for all the world to know we all, no matter rev's would-be amelioration, stood alone, as though flow itself stumbled, stuck. Sputter never spoke more eloquently but even so she would not be done with fluidity, full-bodied arco, letting the bow glide between bouts or outbursts of coughlike exhaust, Carteresque bow-bounce. Such answering fluidity was nothing if not outright elegy, forthright lament, Aunt Nancy allowing the bass its low-throated moan. She spoke from nowhere if not from the heart (ribcage apse, alcove, atrium), no way if not on two fronts, both fronts, deep throb and bow-bounce both.\n\nPoint made, she put the bow away and went back to playing pizzicato, plucking the strings with chill serenity, ritual aplomb. She stood with her back straight, addressing the strings with a churchical assurance, churchical rectitude, as patient a fingerwalk as there ever was. She closed her eyes, exuding meditative calm, each ascending run seeming to say, \"Alas,\" each descending run whispering, \"Amen.\" Abidance was the overall note she struck, if there could be said to be an overall note she struck.\n\nDjamilaa had left the piano to pick up one of her guitars. As the audience applauded Aunt Nancy's solo she began to play, pointedly chiming in on strings. She started off in what initially struck one as a Spanish vein but as she went further along one recognized a Malian or Guinean provenance, a _guitare s\u00e8che_ excursion (it was an acoustic guitar she picked up) whose ambulatory rhythm feasted on recurrence. It sought to make its home in a reverberant ping a shade beyond the upbeat, a treble chime not so much home as haunt but beckoning as though haunt could make heaven home. Indeed, treble chime verged on going out of bounds, off scale or off record, verged on heaven itself, rang heaven's bell.\n\nAunt Nancy picked up on Djamilaa's Mande invitation and replied, playing the bass like a big guitar (rhythm guitar to Djamilaa's lead, bass to Djamilaa's treble), whereupon Djamilaa, gratified to hear her call responded to, shot her an appreciative glance and began to sing. Together they plied a Malinke roll built on repetition, the chords fraught with a certain drama or an inference of drama (false drama perhaps), an inkling, inference or sense Djamilaa took pains to fend off, paradoxically furthering, in so doing, the very inkling, inference or sense it was her one wish in life (or so it seemed) to hold at bay. A buoyant bout with quicksand, were such possible, her voice took solace in its ability to declaim while being taken out, courting hoarseness to extol its testificatory prowess. Rescue was only what witness one could manage, were there rescue at all, and witness, were there, would suffice, it seemed she said or sang more than said or made singing say. Witness or no witness, her singing summoned words like _aria_ , _recitative_ and _recital_ only to say that they fell short, failed when what was really real was afoot, that exactly that, the really real, was afoot. Strident, abrasive, bent on scouring the air itself if need be, her voice built with a certain insistence toward something none of us, her included, could name, imprecatory at points, complaint piled on complaint, coax plied with complaint.\n\nDrennette joined in with a repeating figure that marked the strong beat on the high hat, setting it up with a tap on the bass drum, the barest percussive presence one could want. The three of them effected an ictic, riverine amble, Djamilaa's treble chime, as time went on, arriving a lengthening shade late, a hitch or a gimp or an eddy in the flow. It brought to mind, for me at least, the \"Wild Card\" drawing's abstract bench. It sat one down and it gave one pause, giving one to reflect on water's indifferent flow, time's indifferent flow. One sat on a bench on a bank overlooking the Niger.\n\nLengthening shade suggested the drawing's black sun had caught Djamilaa's eye, the river's consolation an intricate mix of solace and complaint. The river's destination was there by inference, the salt air that had nipped our noses complicating time's occult remit. Lambert, Penguin and I picked up on this and instinctively bowed our heads, letting Djamilaa, Drennette and Aunt Nancy's amble have its way with us as waves or rapids might. Shade-late arrival's beneficiary, Djamilaa's treble chime gathered extrapolative reach. Eking out a summons or a receipt that would be the chime's equivalent, her voice every now and again leapt, its timbral bound and embrace as raw-ribbed as Aunt Nancy's bow-bounce had been, a miraculous mix of stridency and grace.\n\nLambert, Penguin and I now lifted our heads. Carried in or carried out, we stood athwart all emollience, amenity's reset, lengthening shade's limp a new boon nonetheless. Listening to Djamilaa, Drennette and Aunt Nancy, one heard again and saw again, as though for the first and final time, that sound was the inner skin of things, the other side coming over, inside turning out. A fool's errand it might've been to see it so but one saw it so. It was ground we'd been over before, ground we'd go over again, ground that somehow never got old.\n\nAs Djamilaa's voice began to trail off, letting us know she'd had her say, the audience began to applaud and Lambert and Penguin glanced at me and nodded to say the floor, so to speak, was mine. I put the cornet to my lips and began with a run meant to recall Aunt Nancy's Model T on a bumpy road, eventually stating the head with a hesitancy aimed at recalling Lambert's grudging address as well. I stood with my back straight but churchical rectitude wasn't what I was after, at least not to begin with. Gaps and crackle made their way out of the horn, more \"ahem\" than \"amen.\" Indeed, sputter might've been my middle name for all anyone knew, so hard-won was any articulacy (or seemed so at least).\n\nI say \"seemed so\" because sputter was more than sputter. It was an archarticulacy, fraught with meaning, \"meaning\" meaning \"wanting to say.\" It wanted to bear on exhaustion, eventual eclipse, fossil fluency's abject eloquence, black sun. It wanted to find fulsomeness in hemorrhage, bumpiness, unable not to know it fed on exhaust, extinction, fumes. Sputter was double-jointed. Wanting was to say.\n\nI stood with my back straight, parsing, repeating, teasing out the slowgoing unfolding of black sun. My not aiming for churchicality notwithstanding, there must have been something of it to the way I made my way. Lambert, that is, bowed his head and lightly put his left hand on my right shoulder, a deaconly hand (mock-deaconly perhaps), as if to say, \"Take your time, son. Take your time.\"\n\nI took my time. It took all the calm I could muster not to be caught up in fossil fluency's depth and dilation, not to be carried away. I offered myself every caution not to resort to barrage, the Gatling-gun spray of notes I'd have bugled had bump's double joint had its way. I picked my way as though I walked in a minefield. A novice at a typewriter hunting and pecking, I pecked at sound and the possibility of sound. Aunt Nancy, Djamilaa and Drennette picked up on my cautious tread and lent themselves to it, letting their riverine amble subside. We no longer sat or stood on the bank of the Niger. Time dissolved into an aroused momentariness peppered by random event or invention\u2014bass interjection, drum interjection, guitar interjection\u2014isolate, asymmetric, intermittent, each a law to itself it seemed. It wasn't that time stood still but that time twitched, a volley of quivers that went for how long no one could say, trumpet, guitar, bass and drums each other's ricochet, no less random even so.\n\nAfter however long it was, Djamilaa put the guitar down and went back to piano. She played a series of chords at the low end of the keyboard, a slow tolling that was a call to order, ominous and wistful at the same time\u2014one-two, one-two, one-two again, a two-beat rest between each pair of chords, a space Aunt Nancy noticed and filled with a two-note descending ostinato. Drennette heard the call and went from sticks to brushes, applying a tight, circular stroke to the parade snare, stirring the pot. The three of them laid down a midnight creep that brought \"Mood\" on Miles's _E.S.P._ to mind.\n\nI took my mute out of my coat pocket and put it into the bell of the horn. I blew a needle of sound that rayed out as it went thru the air, a tremulous ribbon whose advancing edge was a vibrating blade. It wasn't that sputter had been seeking this, that obstructed speech ironically or fittingly found its voice in the mute. Sputter, I've been saying, spoke. Djamilaa's call to order, moreover, calling randomness to a halt, was itself random. There was no reason for it to flow out of what preceded it and equally no reason for sputter to be said to have evolved into mute fluency. What happened happened. It was as simple as that. Sputter had spoken no less than mute fluency now spoke.\n\nNo less had sputter called flow obsolete (fossil flow indeed) than the mute bestowed fluidity and focus (mute fluency). I not only blew, as the old saw has it, from my diaphragm but set my collarbones abuzz with a sense of yet more remote origins, a faintly remembered myth involving clavicular spillage I'd read about ages ago in some Dogon lore. I vaguely recollected a creational aspect to it, something to do with Amma's collarbone marrow spilling out. My sound thus had an occasional raspiness at its edges, escaping or expiring breath a constitutive leakage. My collarbones hummed and from time to time knocked, shook like radiator pipes in winter. Part leak, part letting off steam, it was a sound I let the valves tease out, expelled or expired breath conducing to a theme of extinct heat.\n\nMore immediate, though, was the impact of what had now caught my attention in the \"Wild Card\" drawing, the wishbone above the masthead's ribs. It was this that my collarbones vibrated in affinity with, humming like sympathetic strings on a sitar. No matter how whimsical or wistful, I couldn't help noticing, it was a wishbone as black as the sun, as though to acknowledge wish's role were a dark admission, which in fact it was. The fact that it was was a fact I couldn't get over, though I made my peace with it by way of a Dixonian recourse to flub effects, extenuating breath into what gave sound to my exasperation. Nothing could've been more slick than our midnight creep but Bl'under blew with me now and again. Mishaps would occur, I said or let Bl'under say, wish otherwise though we might.\n\nDinosaurs and birds popped into my head, the thought of them having wishbones in common. The wishbone struck me now as an emblem not necessarily of extinction, a harbinger of ostensible extinction evolved into flight. A furcula technically speaking (Latin for \"little fork\"), it was eponymously the fork in the road leading toward one or the other. Where there's a fork there's a chance I told myself, admittedly wishful but not, I hoped, overly so.\n\nIt was a quick train of thought, feather and scale. Almost before I knew it, the horn emitted a Bowie-esque ratchet of sound, a careening squib that had a Dixonian contour as well, shades of _November 1981_. Feather outran scale, run reigning supreme, skid's indigeneity to squib newly audible, flight's tangential drift. Drennette went back to sticks to keep flight in line, upping the tempo after an onslaught of rolls that announced a new-day disposition tending toward all-out sprint (scale train, feather train). Scale to Drennette's feather, I lagged ever so slightly, a syncopic microbeat behind the beat, an ever so exactingly maintained messianic stagger, the gap, looking forward, saving grace turned out to be.\n\nAunt Nancy jumped on the new tempo right away. The fingers of her right hand scurried back and forth across the strings, those of her left scurried up and down. It was an avian pulse if there ever was one. She bore down, biting her bottom lip as she played, as though the pulse, the proverbial bird in hand, would fly away were she not to or did indeed, from moment to moment, fly away, notwithstanding she did. The flying, flown altercation she laid under it all gave the music wheels if not wings, legs and feet if not wheels. Djamilaa prodded us with pianistic chirp.\n\nHearing the buildup and sensing we'd soon crescendo, Lambert and Penguin took up their horns, Lambert his tenor still, Penguin soprano now rather than oboe. They saw me my wishbone wager and raised me a ribcage crown, blurting out the head with cairologic urgency again and again, four times in all. We played more loudly with each iteration and peaked on number four, whereupon I took the horn from my mouth and stepped back from the mike as Lambert did likewise, Djamilaa, Aunt Nancy and Drennette pulled back on the tempo and took the volume down to a whisper and Penguin, as the audience applauded my solo, embarked on his.\n\nIt was clear why Penguin had switched from oboe to soprano. He made a point, it seemed, of putting the oboe's pinched, eked-out sound aside, its inturned embouchure aside, opting for a soprano sound as open as a duck's cry, open as all outdoors. It was an openthroated sound \u00e0 la Steve Lacy, with no tightness or constriction to it, resounding of nothing if not laryngeal openness, nothing if not esophageal openness, nothing if not, in a word, flow. He seemed intent on saying something about acceptance, something about flow bearing on depletion, departure, fluidity's eventual arrest. It was a duck's cry but without its frayed perimeter, the firm inside part of a strand of spaghetti cooked al dente.\n\nDjamilaa, Aunt Nancy and Drennette continued at whisper level as Penguin started off. It was an unrushed, riverine amble they found their way into, Djamilaa now reprising on piano the Malinke way of knowing she'd earlier offered on guitar. Penguin's accent on flow they backed up and embroidered with a subtle, sotto voce roll extolling furtherance, a felt, otherwise fugitive equation of brightness and time.\n\nFor all its accent on furtherance and flow, there was an elegiac strain woven into what Penguin played. The horn was somehow buoyed by sadness, a deep, thoroughgoing sorrow so abstruse it could only turn sanguine, moan though it otherwise did.\n\nThe river, in other words, was back and along with it the lachrymosity Lambert and Drennette had earlier touched on or choked up on. Silly as it seemed, one couldn't help thinking of Julie London singing \"Cry Me A River,\" especially if one had, as I had, seen the movie _The Girl Can't Help It_ as a kid in the fifties. That she sings it as a specter haunting Tom Ewell especially came back to me now, conducing or contributing to a theme of payback and retribution I'd have sworn I heard coming out of Penguin's horn.\n\nPenguin repeatedly had recourse to an E-flat pedal in such a way as to suggest conscience and also to say (or at least to imply) that he too, as I had in writing the piece, wondered if prehistory's grudge against the present were at work. The horn cried a river of regret, disbursing mixed-emotional strains of remorse and recrimination, as though evolutionary succession exacted dues, which evidently it does. Not since body first met soul had confession so wed complaint. It was a cry surmising gnostic entrapment, thug gnosis, the most accusatory mea culpa one would ever hear. Retributive spill was our fault but no less retributive it seemed he said, fossil fluidity's ominous underside. That it was our fault was not our fault it seemed he said.\n\nI took my cornet up again, put it to my mouth and offered punctuation, endorsing Penguin's theme of demiurgic sting with a braid of mordents around E-flat. It was no more than punctuation, no more than a quick, ratifying run, but I too admitted fault while claiming fault to be a setup, I too took sublime umbrage.\n\nPenguin shot me a glance, an appreciative gleam in his eye, going on as though newly fired up, made all the more adamant by my corroborative run. Newly adamant or renewing its adamance, his tone blended _accepting_ with _incensed_ , a tendency toward trill making its way in or having made its way in, a descending trill bottoming out into drone recalling Jo Maka, the Guinean soprano saxophonist. There was no way this would've been accidental. The river was back, decidedly so, running with quaint strain, uncustomary hustle.\n\nHearing her Mande insinuation expanded on, Djamilaa took to singing again or, to be more exact, semisinging, humming a song from Upper Guinea, \"Toubaka.\" A few words from the song's lyrics emerged from her humming now and again, but mostly what she did was hum, a one-woman chorus commenting on Penguin's solo. She sang or semisang from a position of elderly repose, wise, as the expression goes, beyond her years. All weight, all ministry, all measure endowed her voice, an endlessly calibrated \"alas,\" an extended sigh. Penguin now shot her an appreciative glance. Her voice was drenched in time.\n\nThe river had come around again. Malinke ambit had come around again. Penguin blew beyond the horn, the river's quaint strain and uncustomary hustle prompting him to heave an arc of implied intonation not unlike a fisherman casting a line. This arc was Dredj's arm around the figurehead's shoulder, the feature of the \"Wild Card\" drawing that had most caught Penguin's eye, an extrapolative embrace whose coo and consolation he now beautifully brought to bear on the horn. His new recourse to diaphragmatic oomph aired a light-bodied bigness or a big-bodied lightness, his eye also caught by the figurehead's thoracic largesse.\n\nIt seemed all we could do to contain ourselves. We stood poised on a precipitous edge it seemed. With the merest abrupt move we'd erode or evaporate, the figurehead's thoracic transparency suddenly at large. All we could do, it occurred to Lambert, Aunt Nancy, Drennette and me at exactly the same time, was hum. All we could do was catch, as it were, Djamilaa's choral contagion and hum \"Toubaka,\" which is what we did, the four of us easing into it, joining her, an antiphonal consort of sorts, a chorus beyond her chorus or, all of us having heard the version of \"Toubaka\" done by Les Ambassadeurs Internationaux, chorus to her Salif Keita.\n\nIt was an immediately soothing hum, a sonic lozenge at the roof of the mouth apportioning balm. Humming drew us back from the edge. We stood stout again, lodged resolutely where we stood, readymade remit, the fossils we'd eventually be. Hum's vibratory dispatch came into collaborative play with solidity's transit. It was nothing if not flow's disclosure once and for all, an aggrieved emollient.\n\nThus it was that aggrievement and approbation ran as one. So it was we sang, if it could be said we sang, or, if not, semisang with our teeth clenched. Peal and ping rang from the keys under Djamilaa's right hand as we hummed, as poignant as a Guinean guitar. Penguin worked and worried the arc Dredj's arm inspired, a breaking wave, he'd have had it, as it broke, less a wave than a trace, audible to an imagined ear alone. The river and what it went out to were back.\n\nPenguin leaned back a bit, mentally and physically both. He let himself be caught by the cushion of sound our humming had become, so relaxed he became all breath, all respiration, resorting to circular breathing as the music peaked. Circularity said it all it seemed he said, \"What goes around comes around,\" \"Where there's a wheel there's a turn\" and so forth, Malinke furtherance a dream of empathic escort come true. Breath was a ball rolling atop our Malinke hum, a wheel buoyed by and rolling on water, wind roughening water.\n\nThe ball or the wheel gathered momentum and the music sped up, Aunt Nancy's bass advancing a jump-up rumble as Drennette's high hat hissed, hum nearly brimming over, a collective croon. There was more and more swell to it, more and more lift as well. We rode a low-spoken undulance, borne or abetted by Penguin's extrapolative surmise.\n\nJust as it had as I ended my solo, the music crescendoed as Penguin ended his. Djamilaa, Aunt Nancy and Drennette again took the volume down to whisper level as the audience applauded. All the while we continued humming\u2014quietly so, at whisper level as well. Once the clapping subsided an insinuative quiet obtained, an almost ominous calm. Penguin now joined our humming, the six of us ever so low-key yet stalwart, savoring the impromptu vibration humming had brought, the low-key visitation humming had become.\n\nIt was a soothing song our humming amounted to. A lullaby it might've been except its unremittingness roused us. We hummed possessed of alarm and assurance, an agitant mix whose intensification varied inversely with the tempo at which we hummed. We gradually, that is, hummed more slowly, curiously building intensity while winding the piece down, agreeing, without having to say so, not to return to the head. We instinctively and collectively knew that this was the way it should end\u2014not so much end as fade.\n\nAunt Nancy, Djamilaa and Drennette eventually let their instruments go silent and we all continued humming a cappella. Here and there a few people in the audience picked up the tune and joined in, humming along with us, but it would be a stretch to say we set the entire crowd humming. Still, an infectious vibration seemed to affect everyone. When our humming finally subsided the audience sat silent for a while as though entranced, as though unaware the piece had ended or taken by surprise that it had, lost in thought. We felt we knew what they felt. Applause was beside the point. Yet when they snapped out of whatever it was they were in they rose from their seats and gave us a standing ovation.\n\nWe too were affected by the reading we gave the piece. I'm not sure I can say exactly how but even now it stays with us. How long I've gone on about it is a measure of that no doubt but I'm not sure I can more precisely pin it down. I will, though, mention something we've been wondering about. We couldn't help noticing and now can't help reflecting on the fact that, fossil fluency's extremity notwithstanding, no balloons emerged during the performance. Could it be that the \"Wild Card\" drawing was preemptive? Could it be that the drawing, conceding to caption as it does, inoculated us? Could it be that \"Molimo m'Atet's Figurehead Consoled on the Revival Bench\" kept the balloons at bay by rendering them redundant? Elated over the performance though we were, we began to wonder not long afterward and we still wonder.\n\nAs ever,  \nN.\n30.X.83\n\nDear Angel of Dust,\n\nNo, it's not that the balloons have gotten into our heads. We're not Maxine Brown, we're not singing \"All in My Mind.\" The problem is that it's not that. It's that the balloons are actually out there and evidently they have a mind of their own. They dwell, as I've said before, in a deepseated impulse toward caption, a deepseated captivity they seek to leaven with whimsicality\u2014true to themselves, unable to help themselves, insouciant, insecure. They body forth inflated claims to translatability only to beg off or betray them, burst as balloons at times do. Sometimes they renege before the fact, backtrack in advance, decide not to show up. Sometimes they make themselves known by their absence, a conspicuous reticence one would need to be dead not to notice. But who knows? Maybe even then one would see or sense it, pick up on it somehow. We know they show up camera shy when they do show up. Could the difference between alive and dead be only that shyness?\n\nI've been listening to \"Autumn Leaves\" on the _Miles Davis in Europe_ album, a record I cut my teeth on as a teenager. The vibratory blade Miles uses the mute to make sound into amounts to a balloonlike adjacency, an off-to-the-side reticence or recoil I can't help hearing as recondite presence and manifest absence's mix or mating dance. What but the implications of that sound could so be there but not there, what but the balloons' adjunct agenda, the occult itinerary none but they seem to know but that, possibly, not even they know? Miles's recourse to flutter early on in his solo might be heard as onomatopoeic by some, the \"sound\" of autumn leaves, falling leaves. I tend to agree but I take it further. We hear the sound, in hearing it so, of a concession to caption, the hovering fall oblique afflatus turns out to be. We hear that concession's glide into an offhand rumble, a sly glide that will agree to caption only to nudge it toward refractoriness, as if _caption_ and _captious_ were somehow kin. This, moreover, comes of a sound that could not be more introvert, more introspective. Shy sound. Sly.\n\nI'd say don't get me started but you've gotten me started. Listen, then, if you will, to the hover-and-dip, hover-and-dash dexterity Miles brings to that flutter, a not quite flight-of-the-bumblebee \u00e9lan and agitation, buzz the recondite balloons' masquerade. Falling leaves' equation with not-quite bee flight is nothing if not a fractious caption, a lateral feint bursting with drift and flotation, border on tissue paper and comb though it does, nothing if not wind-aided cascade. That he can go, soon after that, from Gatling-gun staccato to quasi-whimper is what I mean by feint, drift and flotation, a tremulous resolve to be to the side or get to the side of besetment.\n\nNot as blatant as Dizzy's ballooning cheeks, Miles's autumnal bob and weave imparts a balloon salience nonetheless, a balloon detour from Dizzy's overtness and extroversion, a balloon extenuation or, to use Duke's term, extension. I can imagine listening to this track thirty or forty years from now and still finding it fresh, the advent of my own autumnal prospect lending it all the more relevance and resonance, a time-capsule bubble or balloon loaded with decades of what won't tell itself but does, caption after caption donned and auditioned only to be cast off.\n\nLet me know what you think.\n\nYours,  \nN.\n\nPS: I left out the good news. We've been invited to play at the Detroit Institute of Arts in January.\n6.XI.83\n\nDear Angel of Dust,\n\nOnce again it will have come to nothing. Again we will have sat exchanging thoughts on what was to be. Again we will have heard music, albeit not music so much as music's trace, music's rumor, pianistic breakdown as an archetypal he and she gazed out drapeless windows. What stayed with us will have been a wincing, distraught right hand backed by a grumbling left on an abject keyboard, a right undone or done in as much as backed by a disconsolate left. We will have stood and stretched as gray, wintry, late afternoon light filled each window, a wounded look on what lay outside and on our faces as we looked out on it. An archetypal he and she alone but for the music, aloof to each other even but each the music's intended, we will have so seen ourselves but no sooner done so than drawn back. Something found in a wrinkle, something found in a fold, it will have been this that set our course and put us on it, collapse and come to nothing though it would.\n\nSo I thought, at least, earlier today at Djamilaa's. What will evanescent splendor have come to I wondered as she stood at a window and at one point leaned against the window frame, her left arm raised, her left hand touching the curtain rod. She stood that way only a moment but the way she stood highlighted her long beauty, lank beauty, her long arms and legs a miracle of limbs. For an instant something jumped out at me and at the same time jumped inside me, a mood or a mix of elation compounded with dread. I saw what so much rays out from and relies upon, however much it shook me with apprehension: lank intangible grace, nonchalant allure, love's modest body. It was the news of the moment but yesterday's news as well, something aspect and prepossession seemed intent on saying. What that something was, as Penguin would say, more than met the eye, but it did nonetheless meet the eye. My heart leapt and my stomach dropped.\n\n\"Leave it alone,\" Djamilaa said, demure as to what was at issue but sensing my mood.\n\n\"I wish I could,\" I said.\n\nThe right hand on the keyboard prompted me perhaps, apprehension of any kind its mandate, apprehension of any kind's fraught base. Thought's ricochet played a role as well. Momentary angst was its immediate heir, an ungainliness of thought in whose wincing retreat one felt elation well up and right away subside. Fear of being caught out, knowing no way not to be caught out, factored in as well.\n\n\"Things are that way sometimes,\" Djamilaa said, laconic, blas\u00e9, unperturbed.\n\n\"I know,\" I said. \"Things are always that way.\"\n\nIt had to do with angles. The piano's legs buckled for an instant and rebounded, then they buckled and rebounded again. The right side of the keyboard crumpled. The hand that played it crumpled as well. Had they been glass they'd have shattered, besetting one's ears, by turns bodily and cerebral, with sharp, intersecting planes rolling Duchamp's descending nude and Picasso's weeper into one. But they were not glass, however much the keyboard's keening ping made it seem so.\n\nDressed in a light cotton shift whose hem touched her ankles, Djamilaa stood caught between bouts and volleys of agitation and arrest, her lank beauty all the more lank finding itself so caught but unavailable all the same, it struck me, not to be lastingly caught. A lack of lasting hold or lasting capture pertained to the music plaguing our heads, mine maybe more than hers but hers as well, a music it seemed we each heard with a distinct incorporeal ear or perhaps together with a shared incorporeal ear.\n\nDjamilaa again offered generic solace, oblique as to what was at issue still, so compellingly we both felt it. \"Not always,\" she said. \"But their effect when they are is to make it seem that way.\"\n\n\"Yes, I guess so,\" I said.\n\nThe music itself seemed an oblique telepathic dispatch, however much it appeared woven into textile and skin tone, the music of Djamilaa's bare arms and bare neck emerging from her cotton shift. It obtained in her skin's lack of lasting hold and in the wrinkles and folds of her shift. Had she said, \"Fret not thyself,\" I'd have said, \"Amen,\" but we were beyond that now, the music insinuating itself, issueless issue, the nothing it let it be known it will have come to, the nothing that had never been. It wanted to keep convergence at bay.\n\nIt plied an odd, contrarian wish but it was moving and emotive all the same, anti-intimate while inviting intimacy, anti-contact while acknowledging touch. It plied an aloof tactility, love's lank tangency, verging on emotional breakdown but brusque, pullaway catch or caress.\n\nIt was an actual music we heard and let have its way with us, Paul Bley's \"Touching\" on the _Mr. Joy_ album. No way could we say title told all.\n\nAs ever,  \nN.\n14.XI.83\n\nDear Angel of Dust,\n\nYes, that one has \"Nothing Ever Was, Anyway\" on it, as do several others. It does appear, as you say, we let \"Nothing Ever Was, Anyway\" infiltrate \"Touching,\" title not telling all notwithstanding, title not telling all all the more. But there's an asceticism to Bley's playing that comes across no matter what the title. Djamilaa's been thinking about that, wondering about that, drawn to it a lot of late. It's not that less is more, she likes to say, nor that nothing is all, nor that nothing, as Ra says, _is_. All those ways of putting it only let sensation in thru the back door, she likes to say. No, it's not about that. It's not as recuperative as that, not as categorical. It's an angled attrition, banked extenuation, she likes to say.\n\nIt's as if, when she speaks this way, she'd come to me in a dream and vice versa, each of us the other's wished-for rescue, each the other's wariness as well. It's not unlike what sometimes happens when we play. One becomes the extenuation of oneself and the emanation of something else, someone else, ghost and guest arrivant rolled into one. What is it or who is it steps in at such moments? It could be anything, anyone, one senses, but the hollow one's evacuation puts in one's place appears to afford strangeness a friendly disguise. One's fellow band members pass thru that hollow, step into it, relieving the brunt of an attenuation one might otherwise be unable to bear. It's something like what Roy Haynes must have meant by saying that playing with Trane was \"like a beautiful nightmare.\"\n\nCome to as in a dream, yes, a dream dreamt on a rickety bed, springs creaking, home like as not an illusion of home. To speak was to bank one's breath within angular precincts, wall intersecting wall's proprioceptive recess one's being there had become. Stereotactic as well, one touched upon aspect, facet, crater, protuberance, grade finessing grade, tangency's wont.\n\nAs ever,  \nN.\n6.XII.83\n\nDear Angel of Dust,\n\nI was trying to call back time. The time I was trying to summon I'd in fact found distressing at the time but I was trying to bring it back nonetheless. I put two records on the record player, Etta James's _At Last!_ and Bobby \"Blue\" Bland's _Two Steps from the Blues_ , records that had been staples during the time I sought to call back, certain Sunday afternoons when I was a kid and my mother would play them again and again. She would usually have played Mahalia Jackson and the Five Blind Boys of Alabama in the morning, music that I heard as pretty grim, going on about a life beyond life as it did. That by itself was enough to pervade the house with a heavy mood, a mood Bobby and Etta not only kept going but took deeper come afternoon. I understood\u2014or felt, if not exactly understood\u2014that theirs was an even more somber church.\n\nIt was always as if time had stopped. The music and the mood brought everything to a standstill, causing me the kind of unease I'd later read Melville write about suffering during calms at sea. The music or the need for the music seemed to come out of a suspended state of some kind\u2014not only to come out of it but to usher us into it, if or as though we weren't already there. But we were already there and always there it said or made it seem. That we were, the sense that we were, hung heavily over everything.\n\nThe music and the mood took my mother to another time and place, it seemed, a time and place given over to reflection as it touched on regret. She'd sit nursing a drink, a sad, distant look on her face, beset by some deepseated sorrow. It was a sadness I couldn't keep from getting to me, a disappointment she appeared to feel not only with her own life but with life more generally, a disappointment that boded well for no one's prospects. She'd stack the two records, listening to the first side of one followed by the first side of the other, then turn them over to hear the two other sides and when they were finished turn them over again, start over again. She'd play them again and again\u2014two, three, four, five or six times. It was hard to miss the mood or what it meant. It hung heavily over the house and over the afternoon\u2014heavily over the world, it seemed. When I went outside to find my friends and play, it went with me.\n\nIt was the same when there were people over, when my mother sat not alone but with company, one or both of my aunts, a friend or a neighbor or a few. When they got to drinking and talking loud and laughing, with Etta and Bobby in the background, they couldn't fool me. I knew it was a ruse. I knew adult life was no fun, life was no fun. Neither my apprehension of the arrest underlying it all nor my distress was diminished by their festiveness. I knew Bobby and Etta were the truth.\n\nThose afternoons, whether sullen or festive, filled me with desolation and dread. It's odd I'd want to retrieve them, but I did. Day before yesterday, Sunday, I played the two records. I hadn't listened to them in ages. Right away they brought those afternoons and all the feelings they were filled with back. I'd forgotten how many of Bobby's songs are about crying, forgotten the reliance on strings throughout Etta's album, forgotten the poetics of plea winding thru both. It was a world of adult longing the two albums conjured, a world of desperate affirmation where there was affirmation, one of dejection more often. I listened to them repeatedly just as my mother would, putting side one of _At Last!_ on the turntable, followed by side one of _Two Steps_ , then side two of _Two Steps_ , followed by side two of _At Last!_ , then starting again with side one of _At Last!_\n\nI didn't set out to write a new composition when I did this, just to see if the music could return me to a certain mood and moment, just to relive, if I could and to whatever extent I could, my mother's blue Sunday afternoons. I did indeed call back time, did manage to recapture or be captured by those earlier afternoons, desolate and dejected as they were. It's almost as if I so succeeded in doing so, fell so deeply into that early apprehension and dread, that I had to write my way out of it, come up with a piece that, touched as it couldn't help but be by Etta, Bobby and my mother's blue Sunday distress, would take it to another place. In any case, I started the piece on Sunday, finished it yesterday, and we took it up today at rehearsal. I'm enclosing a tape. I call it, as you can see, \"Some Sunday,\" meaning to draw on the utopic senses given to Sunday by Etta's \"A Sunday Kind of Love.\"\n\nThe title echoes Duke's \"Come Sunday\" of course, but it was actually Monk's \"Children's Song\" I was thinking about, the rendition of \"This Old Man,\" the traditional English nursery rhyme, that he plays on the _Monk_ album. I wanted a folk song-sounding or a children's song-sounding phrase repeated on piano throughout, a simple, \"childlike\" melody built on an emphatic key variation. Djamilaa, as you'll hear, delivers on that in a big way, drawing out the phrase's evocation of childhood by seeming at times to take a learner's tack, a beginner's tack, mock-awkwardly \"losing\" the time only to regain it. Drennette's reliable conga throws that all the more into relief.\n\nPlease pay close attention to Lambert. He's the lead voice throughout on alto, Penguin on bari and me on trumpet offering choral support. Lambert's sound on alto tends toward tenor, without, of course, being tenor. I wanted that. \"Bruised bell,\" I leaned over and whispered into his ear right before we hit. He got a gleam in his eye and he grinned. As you'll hear, he brings out the hollow the horn ultimately is, exacting a haunted, harried sound recalling John Tchicai somewhat. He plays hurt, I like to say. Hurt in his case, however, gathers an extrapolative whimsy, a wistful \u00e9lan holding heaviness at bay, hailing some Sunday, soon come.\n\nAs ever,  \nN.\n17.XII.83\n\nDear Angel of Dust,\n\nJust as I'd begun a letter to you Djamilaa showed up. The doorbell rang just as I'd finished the first paragraph. _Once again_ , I'd written, _we will have stood in a drapeless room filling up with light. I will have held her close, held her tightly, my chest against her shoulder blades, my arms around her waist. We will have held hands looking into a mirror, her left in my right. Once again we will have imagined a life to come, a slice o_ _f_ _time we looked in on, looked like we looked in on. Once again_ _I will have held off breathing deeply, not daring to take in the smell o_ _f_ _her hair and her neck. I will have had none o_ _f_ _night's mare riding me, none o_ _f_ _her my horse. The aromatic incubus will not have been let in._ The doorbell rang and I got up and went to the door and opened it. There she stood.\n\nIt was as though a balloon had turned itself inside out, a gloved hand ungloved itself unassisted, by itself, script gotten itself big and delivered itself both (vain presentiment, vacuous portent) right at my door. Djamilaa stood there, in one of her nothing-ever-was-anyway moods I could tell. She gave me a light kiss on the lips and came in.\n\n\"Would you rather love or write a love song?\" she asked right off. \"Play free jazz or be free?\" There didn't seem to be much question or much choice. I laughed, hoping she would as well, but she only stared at me, the nothing-ever-was-anyway look now indisputable. She'd been listening to piano trios again I could tell.\n\n\"Have a seat,\" I said, not really needing to, not buying time so much as rejecting the questions. It wasn't even that so much. The questions, I knew, were rhetorical. Djamilaa had already been about to sit down on the couch and she sat down on the couch. She audibly exhaled, looked up at me in an almost moist-eyed way and laughed, a nothing-ever-was-anyway lilt of a laugh that nonetheless turned the sides of her mouth down. I sat down next to her. I touched her right hand with my left and took hold of it.\n\nLooking away but with the sides of her mouth turned up again, she asked, \"Would you rather celebrate the Beautiful One or be the Beautiful One?\"\n\nIt appeared she'd read my thoughts and was alluding to a piano trio that wasn't, Cecil Taylor's bassless trio. I shrugged it off and said, deadpan, \"Djamilaa, the Beautiful One Has Come,\" as she was indeed that, all of that, all the more that the more the nothing-ever-was-anyway look undercut surface allure, a demure, dissenting look that gave her close-to-the-bone beauty all the more depth.\n\nDjamilaa turned back and looked at me but held off speaking, as close to a Mona Lisa look on her face as I've ever seen on anyone. For some reason, I thought of Andrew Hill's triplet-laden \"Laverne,\" the mix of hurry and restraint he works into the head, the almost muzzled way the horns announce it on the quintet version on _Spiral_ , Chris White's bass gallop on the trio version on _Invitation_ , etc. For a moment I couldn't shake the way the horns, even as they play in unison, break the line between them, breaking bread it seems, a by turns beaked and bleary nudge, a drawn-out, busily punctuated nudge.\n\nWhen Djamilaa spoke again I couldn't hear what she was saying. I was thinking about the almost perky uptake Ted Curson imbues the head with on certain edges, as though gesturing toward an audience that isn't listening as though they are. I was thinking that music is a way of going out of our way not to speak as we otherwise would. I could see Djamilaa's lips were moving but what she said I couldn't hear, whatever it was drowned out by the quintet version of \"Laverne\" in my mind's ear.\n\nI was thinking about the first time I heard it, recalling it coming on the radio in the car. I was remembering it coming on as I turned off Beverly onto Spaulding. I remembered pulling to the curb and stopping to listen, so taken with it was I from the very first note. I was caught up wondering what the word or words would be for that opening triplet and for the way of stating it the horns have, caught up trying to figure that out, wondering whether \"quip,\" which had just occurred to me, fit, wondering whether \"sotto voce quip,\" which occurred almost immediately after that, fit better. Djamilaa's lips were moving but I couldn't for the life of me make out what she was saying, caught as I was between cerebral quip and corporeal audition. She might well have been casting a spell. \"The carnal ear,\" she might well have been saying, \"heareth nothing. The carnal tongue speaketh not.\"\n\nOr was it that the horns' curt buzz and crackle took the words out of her mouth I wondered, her carnal teeth, gums, lips and tongue all the more inviting or alluring even so, their nothing-ever-was-anyway stolidity daring one to hungrily have at them hoping to prove otherwise. I found it was all I could do not to lean forward and press my mouth to hers, testing its philosophic pout, its meaty but reticent fullness, its nothing-ever-was-anyway remoteness and detachment, abandonment and containment rolled into one. Her mouth was nothing if not the muzzle my mouth cried out to be quieted by, clutched and covered by, music, muse, mute and removal all rolled into one. I wanted only a moan to come out of my mouth and her mouth were any sound to come out at all.\n\nI was wondering whether \"the meeting of quip and quizzicality\" did the horns justice when Djamilaa's eyes locked on mine. I could see that she saw they were fixed on her mouth. Mine met hers and then they went back to her mouth, its fullness and potential generosity, philosophy notwithstanding. I was in Duke's prelude zone (kissical buildup, kissical prelude), not without reason to think she was there too.\n\nEven so, the horns were still in my head, deep in my head, some gregarious animal's nuzzling snout it now seemed. When Djamilaa stopped speaking, finished with or simply breaking off what she'd been saying, I moved ever so slightly toward her, still not having heard a word she said. She moved her eyes off mine and to my mouth for a moment, returning them to my eyes while moving ever so slightly toward me. We leaned and moved our faces toward each other. I could feel the breath from her nose on my upper lip, our faces a couple of inches apart, her feeling mine as well.\n\nWe were in Duke's prelude zone, both of us now, kissical runup. Oddly, though, holding back was all and all was hesitancy, hover, an aromatic, haptic swell welling between us, pure draw, pure impending, willed and rested, ripening restraint. The word \"prance\" had come into my head as one that might possibly get at how the horns acquitted themselves, the equestrian way they had of bounding between triplets. It was exactly then that something else came in that I had no way of knowing from where or to whom it might be addressed, me or Djamilaa, if not both, or, for that matter, about what. \"Hold that thought,\" it occurred to me to say, unexpectedly, out of the blue, looking straight into Djamilaa's eyes, her breath my breath, our noses now touching. \"Hold that thought,\" I found myself saying, pulling back as I did. \"Save it for the gig tonight.\"\n\nDjamilaa pulled back as well, nonplussed, her Mona Lisa look back in place. \"Yes, you're right,\" she said. \"It would only have added up to nothing.\" Her eyes were on mine but it appeared she looked at something in back of me, some remote something I wouldn't have otherwise known was there. How was it her nothing-ever-was-anyway look or \u00e9lan cast or captured something I wondered, the \"Laverne\" horns having at me still, their prancical nudge not affected by my pulling back. Nonetheless, we were no longer in Duke's prelude zone.\n\nIt turned out we did indeed hold whatever thing or thought that moment amounted to or that Djamilaa, at least, did, the remote something she saw in back of me the thing or thought it perhaps came down to for her. We did indeed take that something or that thing or thought, whatever it was, to the gig that night, last night, or at least Djamilaa did. We played the Blue Light Lounge in Long Beach again and during the piece that was the evening's highlight, \"Book of Opening the Mouth,\" Djamilaa stole the show.\n\nDjamilaa's show-stopping solo was all the more unexpectedly so, coming on the heels of a solo by Lambert that none of us envied her having to follow. If it could be said, that is, that anyone or anything had stolen or stopped the show up to that point it would be Lambert and the solo he took. It is, after all, his tune. He knows whereof it speaks and of whom it speaks and, pardon the pun, he ate it up. Following the statement of the head, on which he plays tenor, my solo on trumpet and Penguin's solo on soprano, he switched to sopranino (one-upping Penguin?) and built from a back-to-basics tack (if not a back-to-before-basics tack) to mount a thick disquisition on light, namesake light, the very light the venue we found ourselves in wants to credit itself with, blue light.\n\nLambert, that is, to begin, confined himself to a series of grunts in the horn's lower register, an expectorant tack that offered minimal musicality, bent, it seemed, on redefining musicality, if not discarding it altogether. He picked a bone with any sense of the pure and the proper, shoving these concepts aside and announcing himself done with them. He announced himself done with conceptuality itself even, down to his last nerve it seemed he said. The accent was on strain, exertion, the labor light exacted, effort, fraught sublimity, wrought remit.\n\nIt was as if someone had asked what sound itself was and he was intent on that inquiry, a starting-from-scratch tally and test meant to take inventory, an audit of all it might be at base. He grappled with the rub the thought of primacy subjected him to, taking to it as to a bath of ashes, dry but droll holdout against all odds. One wanted to be a duck sometimes he led us to infer, cracking us up as he did so, grunt gone over into quack before we knew it, Donald Duck ripe with complaint. Was it a way of warding off the balloons, we wondered, beating them to the punch to keep them at bay, no words coming out but the thought of comics or cartoons clearly there, the concept of cartoon broached only to be laughed at, conceptuality itself a joke, a cartoon?\n\nThe crowd at the Blue Light couldn't have been hipper. They saw Lambert's cartoon quack, cracking up, cackling, then raised him a hue and cry, a back-to-before-basics tack that had many of them up on their feet, pumping their fists, yelling, egging him on. Lambert heard them, their back-to-before-basics ratification both a boon and a caution. Could a critique of conceptuality be other than conceptually endorsed he wondered, a quandary he could only, could anything be done with it, find a way to let flow, not find a way out of, which is exactly what he set about doing, egged on indeed, a recourse to long tones and trills the way he now went. He was now looping the loop, threading the needle, even squaring the circle could such be done. He was now, in a word, _piping_ , the mouthpiece and reed sheer birdsong, prodigal chirp tugging the floor out from under everything, majestic chirp announcing and annulling all advent.\n\nThe Blue Light, under Lambert's tutelage, became a live huddle, everyone wanting to know what sound was. We were all, band and crowd alike, bent on hearing blue light's call, the namesake sonance Lambert let us know was out there, insistent what we heard wasn't what it was.\n\nLambert held the horn high, a bird on a branch, a bird on a wire, a bird bent on infinity worn out by infinity, a bird piping loudly as it flew. There was something weary to what he played, infinity's blue redoubt, a sense of having come to be daunted by endlessness, bask and abide by and celebrate it though he did. There was, that is, just the right touch of caution, a rind of complaint not inconsistent with the earlier accent on strain. This, one knew, was the meaning of his recourse to circular breathing, the long run of which he chose to take the solo to its crescendo with, conserving and exhausting breath both. The crowd met that crescendo with a bounding round of applause, some of them shouting out their approval (\"Yeah, that's it!,\" \"Tell it!\" and the like), some putting their pinkies to the sides of their mouths and whistling, high-pitched, piercing whistles that were something like answering the sopranino in kind.\n\nIt was, as I've already said, a solo none of us envied Djamilaa having to follow, but she took the challenge in stride and took her time and went on, as I've also already said, to steal the show. It wasn't so much a solo she took as a walk over hallowed ground, the closer walk so long sung about in gospel torn a new ken. Right off the bat she made reference to the Paul Bley stuff she's been listening to a lot lately, the teasing, tangential way she had with it a remove or two or three away from straight quotation. A valedictory caress one couldn't help calling it, as if having distilled what there was to be gotten from it she bid it lovingly and lightly goodbye, all angle, all approach, all asymptote. By turns it was \"Closer\" one could've sworn one heard traces of, \"Nothing Ever Was, Anyway\" whose ghost one felt a nudge from, \"Mr. Joy\" whose prance and whose prod one took to have hold of her, \"El Cordobes\" one would've given an arm to have her come right out and play. Farther on it was \"Touching\" one was all but sure took the floor out from under one, \"Seven\" the run whose collapse made one wince put one in mind of, \"Turning\" whose faux dissolve accosted one's ear.\n\nI can't emphasize enough that none of these tunes got anything near explicit statement. Indeed, soul to explicit statement's body, the tack Djamilaa took was one of ever so remote adumbration, the ghost of a chance body might be were it, she led one to ask, anything at all. The remote probabilistic wraith all assertion came down to she ran thru a sieve of abstraction, all tune or even tone an extrapolation of aspect only the blind might see. Thus it was one saw in the sense of thought more than heard \"Seven,\" \"Mr. Joy\" and so on, blindly saw in some jigsaw way that crossed over into hearing's near equivalent, listening-for, the closest one would come but still short of outright hearing, teasingly near and far both.\n\nHow close Djamilaa got to the Paul Bley tunes was also how far away she got from \"Book of Opening the Mouth,\" so far, in fact, her solo seemed like another tune or like a tune inside Lambert's tune, a new tune or a detour of a tune I indeed couldn't help calling \"Djamilaa, the Beautiful One Has Come.\" It also gave rise to something we've never had happen before. It was as she was wringing what sounded like the last possible drop from her say-it-without-saying-it allusion to and distillation of \"Open, to Love\" that it happened, the audience breaking out into loud, uncontrollable applause, yelling, stamping their feet and pounding on tables, some of them rising to their feet as they had during Lambert's solo, all of it so loud and raucous and all of it sustained so long it required Djamilaa, Aunt Nancy and Drennette to stop playing, the way we've heard it done on those Om Kalsoum albums where after one of her unbearably beautiful runs the Egyptian audience goes crazy and she and the orchestra have to pause for them to settle down again.\n\nWhether it was a conscious homage to Djamilaa's North African roots or something the audience just couldn't help or both, it was, as I've said, something that's never occurred before. The three of them stopped playing and the ovation went on for a while after they did so, the audience, it seemed, pleased with the acknowledgment of their acknowledgment and the worked-up, wounded state they were in. It was as if they'd been begging for mercy and were pleased and relieved to have gotten it and could now quiet down, which they began to do. Once the ovation had completely died down and those who'd been standing were back in their seats Djamilaa, Aunt Nancy and Drennette started up again, their closer walk across hallowed ground picked up where they'd left off, Djamilaa's Paul Bley extract or extrapolation taken up long enough to be played against the next detour she took.\n\nThere was, that is, a strongly pronounced contrast when Djamilaa moved from Paul Bley to Bill Evans, to whom she's also been listening a lot lately. Radically changing her tack, she alluded to two of the tunes on the _Explorations_ album, \"Haunted Heart\" and \"Elsa,\" ever so nimbly quoting from one and then the other, a wisp of outright statement in each case that maintained her theme of ghostliness nonetheless, linking the two, an anonymous haunted heart lamenting some Elsa. Listening, one found oneself haunted by the ghost of \"Elsa\" and by that of the tune's eponym as well, mere wisp though it was notwithstanding, a wisp it was all one could do not to be lost in, so thrown one felt oneself to be. One couldn't help it. It was a wisp that would have none of being done with, a wincing, knee-weakening lilt it was an achievement not to fall over listening to. I for one, for the moment, renamed each and every person I'd ever loved Elsa, so put upon by the ghost of what, Djamilaa had set us up to feel, never was, anyway.\n\nI also can't emphasize enough that Aunt Nancy and Drennette were models of co-conspiracy throughout, disbursing supplement and surmise alongside support. Aunt Nancy's bass waxed as the moment called for and ebbed as the moment called for, singing thru the gaps as Djamilaa's chiming touch hung in the air, an aroused \"amen\" or a bemused \"alas\" in which all possible sense, all apprehension, seemed tied up in a single note or a quick strum (throb and drop, it almost goes without saying, a never not available resort). For her part, Drennette would not let it be forgotten that the drumset commandeers light, cymbal shimmer and cymbal ping plied with water-wristed aplomb, so brightly the mind's ear's eye could only squint. She had a way of letting the side of the stick's tip roll with its metal address, a lingering ride or regard one heard as tambourine-like, tenor to tambourine's alto or soprano, baritone sometimes, bright even so.\n\nThe lamented Elsa was every loss, every lost hope, the charm gone from a charmed life. Elsa was the \"was\" that would've been had anything ever been, the \"is\" the \"was\" receded from as well. An elapsed albeit echoing recess, Djamilaa made clear, was also what Elsa was, a \"was\" we'd reminisce as though we'd been there, a \"was\" we'd spend forever not returning to.\n\nThe audience again got loud and went wild in the Egyptian style, interrupting in the Egyptian style, literally stopping the show again. Again Djamilaa, Aunt Nancy and Drennette stopped playing and waited until the audience settled down to resume, picking up where they'd left off, Djamilaa reiterating her precept or prognosis regarding Elsa as elapsed, echoing recess. Elsa, she again made clear, now driving it across with all the more force, was a \"was\" we'd reminisce as though we'd been there, a \"was\" we'd spend forever not returning to.\n\nIt was a solemn prognosis Djamilaa laid out and she laid it out with a steady hand, tremulous truth no disabler, threadbare hope (if not lost hope) no hitch. Indeed, she let a certain stateliness accrue to solemnity or insisted it accrue, her last allusion and quotation, the one she began to end her solo with, coming from Federico Mompou's _Songs and Dances_ , the opening strains of No. 9, a closer walk mingling near with far, the processional gait won by stoic resolve.\n\nDjamilaa had sat slumped over the keyboard \u00e0 la Evans while expounding on Elsa but with the Mompou she straightened her back and lifted her chin, sitting up with model posture as if to illustrate stoic resolve. Aunt Nancy straightened up as well. She no longer draped herself over the bass but stood erect, her head high and her eyes, when they weren't closed, on the horizon. Drennette did so too, her back ironing-board straight as she rode high on the drummer's stool.\n\nAunt Nancy went from wincing throb to mini-walk, swell to wincing stepdown, a Scott LaFaro reference that kept Elsa close by, but only until Djamilaa drove the Mompou home by humming it as she played, Drennette a muted waltz meanwhile on the high hat. A slow tolling, a slow noting of what one goes thru, prepared the way, toll a tally of cost and a chiming decree rolled into one. It seemed it said gait would be gate, an opening, as they eventually went from Mompou back to the head. Gait was indeed gate as they did so\u2014which is to say, trite though it is to put it so, they _swung_ , solemnity's unexpected boon or behest, closer walk nothing if not strut.\n\nIndeed, they swung swing itself, swing so perpendicular to itself it sprang, spring the fey sense one had of it, a buoyant sway and a bit of swag, vintage Elmo Hope. The head, that is, was now a recombinant bounce Djamilaa worked and kept working, a host of collateral chatter among the fingers of her left hand. It all had a way of leaning back even as it sprang, not in posture so much as the actual sound, an acoustico-implicative stride at a slight angle, ambling yet relaxed, laid back. It had a way of bounding, even so, with namesake hope, all-out expectancy, Drennette's high clamor on the cymbals ladling release.\n\nThe Elmo Hope vibe was Djamilaa's nod to the audience's North African salute it occurred to me, oblique but necessarily so. I leaned over and said so to Penguin, who was standing next to me. He agreed and, grinning, added, \"If she quotes from 'Stars Over Marrakesh' they'll go crazy again and I'll be out of here.\" He put his hand out when I laughed. I slapped it and turned my hand over and he slapped mine.\n\nDjamilaa neither quoted from \"Stars Over Marrakesh\" nor needed to. She continued bearing down on the head, with not only collateral chatter but collateral clack, a nothing-ever-was-anyway bent and burr brushing everything aside. She, Aunt Nancy and Drennette did indeed speak of deserts crossed but that wasn't the point. Lambert, Penguin and I, given the nod by Djamilaa, came back in, ending the piece with three unison restatements of the head and a stop-on-a-dime return to silence.\n\nThe crowd again responded in the Egyptian manner, clapping, standing, stomping, whistling, shouting. It was a full five minutes before we were able to begin the next piece.\n\nAs ever,  \nN.\n13.I.84\n\nDear Angel of Dust,\n\nWe're in Detroit. Got here around noon after an early flight from L.A. It's the first time in Detroit for all of us. After getting settled in the hotel and resting up a while we went out to take a look\u2014see, as Rahsaan would say, what we could see. We walked around quite a bit and took the bus for longer distances, anxious to see all we could. It's hard to say, hard to sum up what it all adds up to, could it be said to add up to anything. What it was was random vantages, random veils, no such disclosure as we think a first-time exposure to a city might yield. Detroit was there but not there. Or was it we were there but not there, vantage indissociable from veil, red-eyed as we were? I quickly found the expectation to see and say something about Detroit an irritant, any summing up or desire to sum up an affront. Yes, the monumental architecture seems to cry out for comment, the massive, no-nonsense rectilinearity of the General Motors Building bent on eliciting reverence or ridicule. (Penguin went so far as to call it \"their new Parthenon.\") The contrast between that blatant a claim on eternity and the auto industry's recent troubles makes cultural critics of us all, not to mention the charred, burnt-out neighborhoods, the abandoned houses overtaken by vines and other growth, the naked foundations, the empty lots. Yes, that all the more obvious contrast would appear to put words in our mouths, easy words moreover, no matter how stark what they report. Yes, them that's got, mostly white, mostly keep getting while them that's not, mostly black, mostly don't. This is axiomatic American cud we could chew for days and do chew, chewing or not.\n\nWe strolled, at one point, up Woodward Avenue over to the Detroit Institute of Arts, where we'll be playing. It was beautiful, quite the promenade, with lovely buildings on both sides of the street, the Institute, perhaps, the loveliest of them. But likewise, once inside, yes, the huge Diego Rivera frescoes, the homage to industrial Detroit he painted in the early thirties, all but jumped out and took hold of one's tongue, tempting one to wax on about a discrepant relation to what one saw outside. The contrast between epic, heroic dimension and postindustrial diminution came easily to one's lips\u2014too easily, all too easily, I thought. I found myself resisting the very observations I'd have made, something about them seeming too pat, too obvious perhaps. I had the sense there was an opaque Detroit, a recondite Detroit, a secret Detroit such observations don't even scratch the surface of. I bit my tongue.\n\nIt's not that any of that isn't true, not that easy sociologizing isn't true, probably more true the easier it is, not that that may not be the job society does. It's not that it's not true but that everyone already knows it and is maybe, at some level, okay with it. What, I was asking myself, does waxing on or waxing at all about it mean or do? Meanwhile, all the musicians that've come out of here and all the music that's come out of here are really staggering, really, as we like to say, saying something. If I named Howard McGhee, Tommy Flanagan, Betty Carter, Geri Allen, the Jones brothers (Elvin, Hank and Thad), Yusef Lateef, Julius Watkins, Ron Carter, Milt Jackson, Paul Chambers, Barry Harris, Louis Hayes, Alice Coltrane, Pepper Adams, Kenny Burrell, Donald Byrd, Lucky Thompson, Charles McPherson, Lonnie Hillyer, Sippie Wallace, Hugh Lawson, Frank Foster, Sonny Stitt and J. R. Monterose, I'd just be getting started, even without getting into gospel folk like Reverend C. L. Franklin, blues folk like John Lee Hooker, R&B folk like Aretha Franklin and George Clinton, rock and roll folk like Hank Ballard and Bill Haley, pop folk, Motown, etc. At the same time, I both care and don't care that they came from here. Music is always elsewhere.\n\nAfter DIA, we took a bus down Woodward to go to Greektown. At one of the stops a man who I'd say was in his mid-sixties got on. He was wearing a rumpled brown suit that had seen better days, a white shirt that could've used washing and dress shoes that were run down at the heels. His hair was an unkempt, salt and pepper Afro, matted down on one side from having been slept on it looked like, his chin and jaws covered with salt and pepper stubble, in need of a shave. He headed for the back of the bus after getting on, muttering something under his breath and making a point of looking at each passenger he passed. His eyes were bloodshot and one could smell that he'd been drinking but he had a kind of elegance about him all the same, no matter his legs were a little shaky and he bumped against the edges of the seatbacks as he made his way down the aisle. He gave off the sense that, like his suit, he'd seen better days. After he and the other new passengers were seated and the bus began to roll again, one heard, coming from the back of the bus, his muttering slowly gain volume, reaching the point where one heard him say, a bit slurred but loud and quite intelligibly, \"None of y'all don't know _nothin'_ about this!\" He repeated this again and again, pausing between repetitions as if to let it sink in throughout the bus or even, perhaps, to assess and be newly schooled by it himself. \"None of y'all don't know _nothin'_ about this!\" he said again and again, his voice raspy, gruff, burning like whiskey.\n\nThe rest of us turned to look toward the back of the bus, one or two at first and then more and more of us, all of us eventually staring at him as he continued to announce, \"None of y'all don't know _nothin'_ about this!\" (Every now and then he phrased it, \"Don't none of y'all know _nothin'_ about this!\") He sat alone on the back seat of the bus, dead center, head up, back surprisingly straight given the wobbliness of his walk down the aisle, feet planted flatly and solidly on the floor, legs a little bit akimbo, hands on his knees. He stared back at us, panning the bus, intent on making eye contact with each and every one of us it seemed, something of a taunt, a challenge, a dare in the look he gave us. \"None of y'all don't know _nothin'_ about this!\" he kept insisting, sometimes \"Don't none of y'all know _nothin'_ about this!\"\n\nIt never became clear what he meant by \"this,\" whether it referred to his condition in some micro or macro way (his tipsiness or his general disrepair, respectively), to a more general state of affairs, to life itself or to who knows what, but his vehemence, if nothing else, communicated, his adamance, if nothing else, had a kind of articulateness, the direness or the extremity from which he spoke was affecting and true. \"The dead are dying of thirst,\" I thought, figuring him as a drunken Dogon elder chiding us for negligence or disrespect, for having fallen arrears on dues or debts owed the dead, owed history, a breach of ritual observance, ritual recall. It struck me that \"this\" was nothing if not the entire edifice, possession built on and put in place by dispossession, the disrepair of the socially dead. I thought this and I saw it all in a snap, a flash, but, no matter the truth of it, the historical and present-day relevance or resonance of it, I almost immediately lost patience with myself, guilty as I was of a deeper negligence, a deep non-observance of the hidden-in-plain-sight rite we were being offered, the initiation into not knowing the man in the rumpled suit invited us to. The simple fact was that he was right, we didn't know. We didn't know who he was, we didn't know what \"this\" was. All we knew was that there he was, announcing we didn't know, accusing us of thinking we knew but not knowing. \"None of y'all don't know _nothin'_ about this,\" he declared again and again.\n\nEveryone in the bus looked at him for a while, a long while. The bus driver glanced up at his rearview mirror from time to time, checking out what was going on in back. It had started off with everyone being a little on edge, wondering what this would lead to, apprehensive as to what it would lead to, but after a while it seemed pretty clear that his mania, if mania was what it was, consisted solely of confronting us verbally and with his bloodshot gaze, that the always possible threat of violence didn't apply. He kept to his own space, clearly defined in the middle of the backseat of the bus, and his hands never left his knees\u2014no flailing of arms, no gesticulation, not so much as waving a finger. What little violence there was, if it can even be called that, was confined to his face, a grimacing wince it got from time to time as he registered the effort it took to apprise us of our not knowing, a certain exasperation, bordering on exhaustion it seemed, with having to do so, with our not knowing and with our not knowing we didn't know. Once it was established that he was no danger, that he posed no threat, everyone in the bus relaxed. Everyone eventually turned away from staring at him and went back to what they'd been doing before. Drennette, Lambert, Penguin, Aunt Nancy, Djamilaa and I all looked at each other and rolled our eyes. There was a group of teenagers nearby who covered their mouths and giggled. The man in the brown suit, unfazed by no longer having everyone's attention, continued his tirade, his rant, his apprisal of our not knowing, \"None of y'all don't know _nothin'_ about this!\" After a while it simply blended in, background noise, of a piece with the conversations going on in the bus, traffic noise from outside and whatever else came into earshot. At the fourth stop he stood up, went back to muttering, made his way shakily up the aisle and got off the bus.\n\nI have to admit I found myself a little shaken, no matter nothing untoward had broken out. It wasn't even the possibility of violence so much as that I somehow felt singled out. He didn't look my way any more than anyone else's while he spoke and when he did make eye contact with me he didn't linger as if it were especially me his words were intended for. The fact that what he said, what he kept insisting, what he kept repeating, agreed so much with the way I'd been thinking, the random vantage being the random veil, is what shook me. It seemed he spoke from some unreachably occult place, a cautionary voice after my own heart, chastening and affirming me at the same time.\n\nThere are other things I could go into about our first day in phantom Detroit but I won't. We ended up leaving a diner last night, for example, when the waitress told Lambert he'd have to remove his skullcap, the kufi he was wearing. No way was that going to happen. We all just got up and left without exchanging a single word. We just glanced at each other and got up and left.\n\nI'll try writing again while we're here. Our gig's tomorrow night.\n\nAs ever,  \nN.\n16.I.84\n\nDear Angel of Dust,\n\nWe got back from Detroit yesterday. I'd have written again while we were there but it was all pretty whirlwind and we were going nonstop. The excitement our being there generated kind of took us by surprise. It all culminated in the gig at the Detroit Institute of Arts, the turnout for which impressed us\u2014not just the number of people who showed up but the mix. It wasn't just the usual crowd of artists, intellectuals, college students, college dropouts and so on, but what Sly would call \"everyday people,\" working folks who after pulling a nine-to-five came out to check us out. There was a range of age groups as well, teenaged to middle-aged, with some seniors here and there as well. We're not used to that, this sort of wide-ranging cultural appetite, especially amid so much that's down, depressed. Anyway, besides the gig at DIA, we did an interview and performed in the studio at WDET, the hip radio station that's not far from DIA, and we hung out a lot with some of the town's up and coming musicians, Griot Galaxy, A. Spencer Barefield and others, who took us around. There was lots to see but at the same time nothing to see, a deep sense of hollowed-out prepossession I couldn't get rid of. It wasn't just all the burnt-out or boarded-up houses and buildings or all the empty lots, the stretches that were destroyed in '67 (Destroyit some took to calling it) and simply left that way, but something else, something more to do with me perhaps. I did feel my forehead thicken from time to time. I barely staved off a cowrie shell attack.\n\nIt may have had as much to do with time as with space, not just being in Detroit but being half a month into 1984, the ominousness of the date taken from it by it getting here, as they say, without incident, failing to live up to its portent. Is it that or is it that it arrived, rich with incident and portent, seventeen years early if not more? This was the question that had me teetering toward a cowrie shell attack, the teasing sense of anticlimactic arrival on the one hand and before-the-fact arrival on the other. I couldn't help hearing Yusef Lateef's \"1984,\" the title track from his 1965 album. It came on strong, piped into my head \u00e0 la previous cowrie shell cuts, the abrupt, keening onslaught the track opens with showing no mercy. The whistling, the whining and the moaning quickly followed, ventilated by the dicelike tumblings on the piano, the ad hoc bass and drum acrobatics. It was all I could do to ward off a fullblown attack, having to do so more than once.\n\n\"1984\" was a gremlin, an imp, it so kept coming at me. It seemed it insisted I appease it, a poltergeist, an offended ghost. It seemed the only way to do that would be to meet it head-on, so during the sound check I suggested we add a new piece to the program, one that \"would honor one of Detroit's own,\" as I put it, \"while getting some other business taken care of too.\" I told the band it was Yusef Lateef I was referring to and that the piece was \"1984,\" something I wanted to work up our own version of, more a variation on it than a rendition, as much a departure from it. I told them about the sense of anticlimactic arrival I was beset by, the possibly before-the-fact arrival that also occurred to me, the repeated threat of a cowrie shell attack. I wanted to call our piece \"1948\" I announced, explaining that it was in that year that Orwell finished his novel and that he arrived at the title by having the last two digits trade places, a numeric anagram. I went on a bit about the arbitrariness of it, the concession to happenstance, the default on prophecy of something so often taken to be prophetic. I said something about the year not mattering, dates not mattering, something about the warning the book issued pertaining to no particular date, no particular time, but to all time, to every passing moment, something about hollowed-out prepossession holding all time, all futurity, hostage. I was on a run, a roll, prolix but both impelled and impeded by the sense that I hadn't yet said what I meant to say, that maybe there was no way to say it. I spurted, sputtered, cleared my throat, rambled on. It was Aunt Nancy who finally bailed me out, offering a phrase that both summed up and opened up what I'd been getting at, trying to get at. \"We get it,\" she said demurely, ever so low key, as I stuttered reaching for a word. \"We get it,\" she said, demure but decisive. \"Moment's omen.\"\n\nTo make a long story short, one of our hosts at DIA got hold of a copy of _1984_ and we listened to the track a few times, putting some ideas together as to how to both refer to it and take it somewhere else. We definitely wanted to retain its aleatory, strung-out, far-flung sense of space and its various recourses to vocalization, though we couldn't, we knew, use exactly the same instruments. Penguin came up with the idea of bringing Hendrix into the mix, reminding us of \"1983 (A Merman I Should Turn to Be)\" on the _Electric Ladyland_ album, whose line \"Oh say can you see it's really such a mess\" he said he especially had in mind. \"I like the way it messes with 'The Star-Spangled Banner,'\" he said, \"sort of the same way he did on the guitar.\" So we decided we'd each intone that line at some point in the piece, doing so hand-over-mouth, muffling it \u00e0 la Yusef's whimpers and moans. And so on. Everybody came up with ideas. We ended up changing the title, scrambling it further and calling the piece \"1489.\" \"Right before Columbus,\" Lambert pointed out. \"You land somewhere by mistake and tell the people living there it belongs to you. 1984 was around long before 1984. There've been a lot of 1984s.\" So we added \"1489\" to the set list and to our book.\n\nThe other news is that the balloons showed up again. It wasn't during \"1489\" but a piece we did later in the set, \"The Slave's Day Off.\" It was during the solo Drennette took, a solo that began with the drumset seeming to collapse, come apart, the sort of thing Dannie Richmond would do with Mingus's band to mark a tempo change. Drennette came out of that collapse, that mock breakdown, with a figure that was all feet\u2014clipped hisses, that is, on the high hat, hortative thumps on the bass drum. The first balloon emerged at once, slipping out between the two cymbals of the high hat as though they were lips. It bore these words: _I lay on my bed on my stomach, my head on my pillow, the sheet and the cover tossed aside. The sun rose, lifting the hem o_ _f_ _my gown, warming the backs o_ _f_ _my legs and my rump, bathing the bed in light._ The balloon, it seemed, harkened back to \"The Slave's Day Off's\" inception on Venice Beach\u2014the rollerskaters, the cellophane jumpsuits, the publicized privates. When Drennette put sticks to snare the first balloon disappeared from the high hat and the second emerged from the center of the snare, bearing these words: _I felt him looking at me, his eyes on my shoulder blades, the small o_ _f_ _my back, my waist and my hips, tailstruck, I could tell, though I couldn't have cared less. He abruptly had at me, nose up my ass as though nothing else in the world mattered. He acted like his life was at stake._\n\nLambert, Djamilaa, Penguin and I looked at each other. The balloons appeared not only to be reaching back to the very roots of \"The Slave's Day Off \" but to be related to the X-rated balloons that followed Drennette home after the Comeback Inn gig a few months ago. The second balloon vanished as she went to the sock cymbal and came back to the snare with a series of rolls and paradiddles, whereupon another balloon rose from the center of the snare. It bore these words: _What was the point I lay there wondering, reticent, unresponsive, letting him have his way with my cheeks and the insides o_ _f_ _my thighs, his lips and his tongue all over them, rummaging my ass-crack as well. No moans escaped my throat, no sound at all. Bored, blas\u00e9, I lay silent, unmoving, no grinding the bed, no lifting my ass to his face. My reluctance worked him into a fever, my reserve egged him on, as though he wanted what he wanted not to be wanted._ When Drennette turned her attention from the snare to the tom-tom the balloon disappeared and a new balloon rose from the tom-tom bearing these words: _He kept at it, head up my butt, ostrichlike. My ass and loins were his North, the Gourd he drank from and followed, my \"back door\" the place the sun would shine someday. Someday, evidently, had come, though I could not have cared less._ A few snickers could be heard among the audience.\n\nDrennette kept at it on the tom-tom when the balloon disappeared, doling out a string of slow rolls. She played more softly as well, as if whispering, confiding, as though she herself was taken aback by what she went on to disclose. After three measures the fifth balloon emerged, rising from the tom-tom, bearing these words: _The world about to blow up, all he could do was bury his head in my behind, begging off. It wasn't his he said or it seemed he said, the world wasn't his. Muddy Waters was on in the background, \"That Same Thing.\" That same thong he might as well have said_. More laughter rose from the audience, a little louder, more widely dispersed. Drennette worked the bass drum pedal as the balloon disappeared, a hurry-up insistence it fell to her left hand to learn from, which it did, quickening the tempo and passing the lesson along, both it and the right raising the volume over the next few measures, whereupon a new balloon emerged, this one from the bass drum: _Ripe with reticence, bottom-line romance, open-secret sex, the world was neither to be had nor held. He held me instead, my midriff and trunk some kind o_ _f_ _surrogate, sweet respite no matter how moot it was to me._ No laughter greeted this balloon. The audience had grown somber, all ears, all eyes, absorbed. As the balloon disappeared a wistful sigh could be heard here and there.\n\nDrennette took the volume back down, the hurry-up thumps on the bass drum a subdued patter she bought time with while putting the sticks down for brushes and then working the snare, all rub and stir. A new balloon, the seventh, rose from the snare, bearing these words: _What i_ _f_ _it was otherwise his nudging nose and tongue demanded. What i_ _f_ _governance were his it seemed he asked with each inhalation, each lick, each whiff, a new earth it was his to rule borne by the funk between my cheeks. What i_ _f_ _my ass were a field his nose and tongue plowed, forty acres worth, what about that it seemed he wanted to know._ When the balloon disappeared Drennette left the snare for the ride and sock cymbals, applying the brushes with a malletlike or hammerlike address that crescendoed as the eighth and last balloon lifted off the sock cymbal bearing these words: _Planet Squat it seemed it all came down to, a dispatch, a dismissal, world without weal without end. World to be done with, done over, long wallow, world to so make it with me would remake._ The balloon disappeared and the solo ended. The audience erupted with thunderous applause and the rest of us came back in.\n\nIt was easy to see what the balloons had to do with \"The Slave's Day Off,\" especially coming in a set that included \"1489,\" but it was also clear, to Lambert, Aunt Nancy, Djamilaa and me, and to Penguin as well perhaps, that they had \"Penguin\" written all over them, the reference to another flightless bird, if nothing else, giving it away. As if that wasn't enough, Drennette, as we were leaving the stage, Penguin told Lambert and me later, looked back at him and said, \"Nice gig, eh, Ostrich?\" and quickly, smiling, corrected herself, \"I mean, Penguin.\" Was it a come-on or a critique, he asked Lambert and me, maybe her way of getting back at him for the time he called her Djeannine.\n\nIt was indeed a nice gig, a nice visit, the audience receptive and especially excited that the balloons appeared, some of them thanking us after the concert for having, as they put it, brought them along.\n\nYours,  \nN.\n18.I.84\n\nDear Angel of Dust,\n\nWe've been thinking about the expression a few people used after the concert in Detroit: \"Thanks for bringing the balloons along.\" Lightly spoken it may have been albeit sincere (they all smiled with a gleam in their eyes as they said it), but it got us thinking about doing exactly that, literally that. What if, we started wondering, we showed up for a gig with a big bag of balloons, not comic-strip balloons but literal balloons, rubber ones, the kind you blow up, and gave them out to the audience before we hit, one or more to each member of the audience, instructing them to use the balloons in whatever way they could to accompany the music, contribute to the music? \"It's an idea that has a lot going for it,\" Aunt Nancy, who was the one who suggested it, remarked, going on to elaborate that it would not only, potentially at least, break down the distinction between audience and band, listener and performer, observer and participant, not only add further indeterminate elements to the music as we listened for and responded to the audience's input, but also, perhaps, work as a \"prophylaxis\" against the comic-strip balloons. The word was her choice, \"not unadvised,\" as she herself pointed out, \"given certain connotations and the balloons' recent X-rated content.\" She said this without looking at Drennette, with no need to look at Drennette. We all knew what she was getting at. Drennette herself laughed and quipped her approval, \"Prepared ensemble meets visual pun.\"\n\nWe're wondering what approach an audience would take. Blow the balloons up and rub them? Blow them up and let the air out? Blow them up and pop them with pins? Blow them up, put them on the floor and stomp them? Leave them uninflated, stretch them tight and pluck or strum them? Leave them uninflated and snap them? Blow them up and thump them? All of these? All of these and more? We wonder but we also worry that were we to do this we'd be acknowledging the comic-strip balloons too explicitly, identifying with them, no matter how playfully, making them a trademark, inviting the audience to think of us in relation to the balloons first and foremost, making them a calling card of sorts. Is irony lever enough to fend off what could look like endorsement? And what if the comic-strip balloons themselves got the wrong idea?\n\nWe wonder if it's a risk worth taking. What do you think?\n\nAs ever,  \nN.\n22.I.84\n\nDear Angel of Dust,\n\nThanks for writing back so quickly. I shared your letter with the rest of the band. We agree that a balloon is nothing if not captured breath. That it contains or seeks to contain something too inchoate to be contained we also agree. That the comic-strip balloon and the literal balloon, the rubber balloon, have that much in common we see as well. We agree that using containment (would-be containment) to open things up is a kind of coup. That putting the audience's will or wish to containment literally in their hands carries an element of poetic justice has also occurred to us. That it carries an element of poetic license as well has occurred to us too. The tradition of balloons as a sign of ceremony does recruit color, as you say, to the binding of breath, much as music does. We couldn't have said it better, except we'd maybe keep going and say balloons are in a sense already music, a ritual disbursement of caught or constricted breath meant to consecrate, even where it borders on asthma (if not especially where it borders on asthma), the blessing breath is. If asthma can be thought of as a wildfire, we'd say, balloons are a controlled burn. They marshall caught or constricted breath intimating breath's possible extinction, festive recruit's cautionary aspect or address. Balloons are also, we'd go on to say, chromatic festivity's dark tone, dark temper, so much depending on sacks of air. We'd want festive lightness given a gravity of sorts, each audience member holding a balloon as though it were his or her own lung.\n\nWe'll see what happens. We're more inclined to give it a try after getting your letter, more of a mind it's a risk worth taking. It's a moot question at the moment, however, as we don't have any gigs in the offing. As I've said, we'll see what happens.\n\nAs ever,  \nN.\n[Dateless]\n\nDear Angel of Dust,\n\nI woke up remembering these words: _I plucked a plywood harp strung with fishing line, a crude instrument with rattly strings, a blunt instrument. No angel, I struck out at the world with it, disbursed its screw-loose harmonics (i_ _f_ _they could be called harmonics) with a heavy hand, pure payback._ I awoke with them on my mind and tongue, mouthing them as I came out of sleep, under my breath as it were, muttering them close to my lips, tongue and teeth, a recitation by heart were there ever one. I remembered them from the dream I'd had, a legible, indelible burst within the dream, not unlike the balloons that on occasion visit our music. A balloon's balloon I'm tempted to call it, a cyst inside the balloon my dream was.\n\nIt was a dream in which I was back in school, junior high school. I had put off doing the project for my \"Music Appreciation\" class until the last minute, not getting going on it until the day before it was due. The project was to build a small musical instrument of some sort, either modeled on an existing instrument or invented. I'm not sure why I procrastinated, which was unusual for me, but my guess would be that my prior experiences with school assignments that involved building something, a science project on electromagnetism, a historical shadowbox, a terrarium and so on, did it. Such were among the times growing up without a father got to me most, as the best projects were always those done by kids whose fathers, with their well-appointed workshops and garages, all but did the projects for them. A girl in \"Music Appreciation\" the previous year had turned in a harpsichord that was much talked about and that went on to win a prize at the county fair.\n\nSo it must have been a kind of dread, a kind of trepidation, that kept me from getting to the project until the very last day, dread, trepidation and, the balloon within the balloon was telling me, anger, resentment. When I did get around to getting started on the project I rummaged around our garage, not with much idea of what I was looking for or of what I intended to make. I found an old sheet of plywood and I saw that we had a hacksaw, with which I proceeded to cut out a piece in the shape of a bass clef. It turned out we had some fishing line, a few spools of various grades, a can of silver spray paint and a packet of small nails. I spray-painted the piece of plywood I'd cut and once it was dry I cut several lengths of fishing line and used the small nails to secure them to the inner curve of the piece of wood, tapping the nails all but all the way in and tying the ends of the lengths of line around them. I'd made a harp.\n\nIt was a harp mainly to the eye if not in name only. I didn't pay much attention to how it sounded, just making sure the strings, the lengths of line, were pulled tight enough to make a sound when plucked. I made no effort to tune them. The thought never occurred to me. It was a perfunctory effort but, if I'm reading the balloon within the balloon right, subliminally more than that. I wanted it to visually signify harp but otherwise flaunt the meagerness of the resources available to me to make it, my absent father at the fore. Its rattly, measly, make-do sound was pathetic, an orphan sound I meant at some level to rub the teacher's nose in, the school's nose in, the world's nose in. The would-be elegance of the silver paint job only made it worse, more flagrant, an obvious compensatory move that highlighted how shoddy the harp was.\n\nAll this went on in the dream as it had when I was a kid in junior high. I turned the harp in the next day. It was one of the least impressive projects turned in, but when my teacher, Mrs. Keene, who was typically very stern and severe, took it up and examined it, as she did right there in front of the class for each and every project turned in, she plucked the strings and generously commented that several of them did indeed emit some of the notes of the scale. I was only more embarrassed and set emotionally awry by this though. She was also saying, without saying it, that most of the strings did not, which was true. No matter the harp had accomplished its subliminal mission, I was upset as she stood there plucking it.\n\nIt was at this point that the dream deviated from what had actually occurred that day when I was a kid. No, it's not that a balloon bearing the words I woke up reciting emerged from the harp as Mrs. Keene plucked. It was a little more subtle than that. What happened in the dream was that as Mrs. Keene continued to pluck I felt pressure against my forehead, exactly in the area where the cowrie shell attacks and the bottle cap attacks have occurred in the past. The pressure was intense and, reaching to touch the area, I felt it was made by extrusions of what had the feeling of type, as though the typebars or the typeball of a typewriter banged outward from inside my head, causing reverse-image letters to rise on my forehead. My fingertips were oddly fluent in the reverse reading the imprints on my skin required. It was as though I were a blind person reading braille and I did so preternaturally fast. I ran my fingertips across the text and there it was: _I plucked a plywood harp strung with fishing line, a crude instrument with rattly strings, a blunt instrument. No angel, I struck out at the world with it, disbursed its screw-loose harmonics (i_ _f_ _they could be called harmonics) with a heavy hand, pure payback._\n\nWhen Mrs. Keene stopped plucking the strings the text went away and my forehead went back to being smooth, at which point I awoke with the words of the text on my mind and my tongue.\n\nSincerely,  \nDredj\n6.II.84\n\nDear Angel of Dust,\n\nThank you for writing back. I appreciate your response to Dredj's letter, which, as you guessed, was written during a cowrie shell attack, though maybe I should call it a reverse-image text attack. All the other hallmarks of a cowrie shell attack were there, but the protrusions on my forehead felt, as in the dream, like raised reverse-image letters rather than cowrie shells. You also noticed, I was happy to see, the conversation the letter seems to be having with the balloons that emerge from Aunt Nancy's \"Dream Thief \" solo on _Orphic Bend_ , all that business about the cigar-box guitar. Yes, there seems to be some dialogue going on about absent fathers, makeshift amenity and string, as though the first were the third attached to the second, obligatory dues, anacrustic obbligato, something gone before saying but not without saying. You're also on point asking what dialectical residue accrues to fishing line's advance over straw, not to mention the other stuff you bring up.\n\nThe implied or potential chat between Dredj's fishing-line harp and Aunt Nancy's cigar-box guitar wasn't lost on the other members of the band. \"It's as though the two played a duet,\" Penguin said at one point as we were discussing the dream at rehearsal. I'd brought up the cowrie shell attack, the reverse-image text attack, at the very beginning of rehearsal, still caught up in its resonances and its resistances even though it had been a light, shortlived attack as these things go. \"Why would an embarrassing incident from junior high school come back with so much force?\" I asked early on, a little bit rhetorically but not entirely so. I really did want the band's input into the process of decipherment and decoding, even recoding, I'd begun. We ended up talking for a long while, everyone pitching in, putting his or her two cents in, and it was Penguin who first brought up the relevance and apparent relatedness of the cigar-box guitar. \"They might as well be cousins,\" he remarked of it and the fishing-line harp before noting their implied or potential duet or dialogue. Everyone immediately chimed in, saying the same had occurred to them, Aunt Nancy's not the least among the ratifying voices, \"Yes, that's the first thing that struck me about it.\"\n\nWhen I said recoding I meant exactly that. Drennette, after we'd been talking a long while, suggested we leave off talking by translating our talk, letting our talk extend into and take the form of a piece of music, \"Dredj's Dream,\" which we would collectively compose, partly writing it out and partly, having done that, adding to what we'd written as we played. We all thought it was a good idea and Drennette went on to stipulate that we at no point allude to or in any way sound like either of the pieces the title brought to mind, \"Monk's Dream\" or \"Sonny's Dream.\" She admitted, as she put it, that it would be especially hard to stay clear of Sonny Criss's piece, given its bottom-heavy propulsion's affinities with Mingus's _The Black Saint and the Sinner Lady_ , thus providing an opportunity to say or to suggest something about an L.A. sound, but she insisted it was a temptation we would have to resist. I'm not all that sure we'd have found echoing or alluding to Monk or to Sonny so tempting or that we'd have otherwise gone that way, but we agreed to the proviso and got to work.\n\nWe decided that if we alluded to any other piece it would be our own \"Dream Thief \" and we did end up doing so. It was as if Dredj were the thief in question, the dream of the fishing-line harp siphoning aspects of the dream of the cigar-box guitar, just as we now channeled or alluded to certain melodic and harmonic elements in \"Dream Thief \" with an intermittent, subsidiary line running thru \"Dredj's Dream\" that bore roughly the same relationship to \"Dream Thief \" as Abdullah Ibrahim's \"Sotho Blue\" does to Oliver Nelson's \"Stolen Moments,\" Abdullah upping the ante on \"stolen\" paralleled and upped further by the act of \"theft\" we made \"Dredj's Dream\" guilty of.\n\nYes, the line says or insinuates, we fish our premises, extend our precinct, our province, advance what sonorous catch we can. Still, we wondered if a more literal fishing line might not have a place in the piece, short lengths of it cut and tied around the strings of Aunt Nancy's bass \u00e0 la Cageian \"preparation.\" We agreed it did but the \"line\" Aunt Nancy ended up using was less literal than gestural, a symbolic substitute or simply a substitute, as there was no fishing line handy. We were rehearsing at Lambert's apartment and there was none there. Lambert, however, scrounged around and found some twistie bag ties in the kitchen. He offered them to Aunt Nancy and she tied some of them around her bass strings, a substitute for fishing line that could be seen or said to signify fishing line. Symbolic or not, the twistie bag ties, at the literal level, the aural and the tactile level, intrude and obtrude in a rattly, screw-loose way, in a manner not inconsistent with Dredj's fishing-line harp.\n\nThese are only a couple of the elements that went into it. After a spate of brainstorming, trial and error, disagreement, agreement, happenstance and what have you, we worked it out. It's funny that a few times, going thru it, one or another of us couldn't help referencing \"Monk's Dream\" or \"Sonny's Dream\" in the course of soloing, something, as I've already said, I'm not sure would've occurred had Drennette not brought it up. Prohibition carried the power of suggestion it seems. In any case, we ran thru it several times after pulling it together.\n\nNone of the takes were as good as we'll eventually make it. For one, we're wondering what it would sound like with actual fishing line and we intend to get some. I'll hold off sending you a tape, but \"Dredj's Dream,\" collectively composed, definitely goes into our book.\n\nYours,  \nN.\n16.II.84\n\nDear Angel of Dust,\n\nI dreamt I wandered in a maze, some sort of shopping mall it seemed, obliquely reminiscent of South Coast Plaza, the mall built on lima bean fields in Costa Mesa, the town right next to the town I grew up in. It was my birthday and I was there to meet someone, \"a certain someone\" as I put it, a special someone. I knew the sadness I'd see her struggling with would stir me, the struggle, to be more exact, more than the sadness, a struggle she and I would share. I knew her pouty, dry, slightly chapped lips would cry out to be wet by a kiss, the long, wet kiss my mouth was nothing if not made for. I knew it was that that my lips and tongue were there for, my teeth as well. I knew the bordering on bite our teeth would bring to the kiss was as much what we were there for as the press of lips and the meeting of tongues. I knew she was there somewhere in the maze, in the mall, though we'd forgotten to say exactly where in the maze or in the mall we'd meet when we arranged a few days earlier we'd meet.\n\nAs I walked about looking to see her, peeking into this or that store I could recall her liking, stopping at this or that bench we'd sat on before, this or that fountain we'd stood in front of, looking to find her but not finding her, her presence, a diffuse albeit palpable air, a presiding sense, suffused and pervaded everything, as though, there but not yet found, she was all the more there. At the same time, there was the sense of her as ultimate find, ultimate fit, the \"one and only\" of love lore, of course, but not only that. There was a sense of some long-sought attunement having been come to, a nestling sense of ordainment, rest, no matter not having found each other yet. It wasn't some hackneyed business of her being with me though not actually so or not yet so, but it was. She was there as I looked, everywhere I looked, not in the way of thirst proving water's existence, not patly in the looking itself. She was there not so simply as that, though it can, in another way, be put simply: she was the dream.\n\nEvery so often during my waking life I'd glimpsed or gotten an inkling of agency and occasion run as one. In the dream it rayed out, sustained as never before. Her palpable air was nothing less than aura, a guiding sense that all was well, all would be alright, our not having decided exactly where we'd meet notwithstanding. She was there with me already and I was there with her already. An inner glow, an inner warmth, welled up in me at the thought. It was a thought I knew the dream dictated, the dream an expansive medium I knew now as at no time before. All this was true and of her it was true. She was as close as my breath, more near than my skin.\n\nI came upon her in an odd room I couldn't recall ever having been in or ever having seen in the maze or the mall we were in. It was an open room, a pizza parlor or maybe a beer joint, with a wide entrance that wasn't a mere door but a whole wall pulled away. She sat around a table with a small group of people, engrossed in the conversation going on among them but my \"certain someone\" nonetheless, thru and thru. I knew this more viscerally than ever before and she knew it, though she didn't so much as look up from the conversation. She knew while not appearing to notice I was there, my \"certain someone\" to the bone, her not appearing to notice I was there notwithstanding.\n\nShe finally did look up and acknowledge me standing there at the entrance. She excused herself from the table and got up and walked over to me. She took my right hand into her left hand and she kissed me on the cheek, turning me away from the pizza parlor or whatever it was and beginning to walk out into the maze or the mall. She looked into my eyes and smiled. A jolt of sorrow shot thru me.\n\nWe had gotten our wires crossed it seemed. She was surprised to see me there she said and when I mentioned our arrangement to meet, my birthday and so on, she said we had left it hanging, with no follow-up, and she'd gone ahead and made other plans. I asked, \"What now?\" She explained that she couldn't simply up and leave her friends, the people at the table in the open room, and that she needed to get back to them. She hugged me and said happy birthday and turned around and walked away. I awoke with a sense of malaise that bordered on devastation, an extremely heavy sense of malaise, and with an even heavier sense that something was awry, that things didn't exactly add up or didn't exactly fit.\n\nLater in the day, still bothered by the dream, I told Djamilaa about it, how much it had me disconcerted and how much it puzzled me. I went on and recounted it for her. After hearing it, she sucked her teeth and said, \"Don't worry. It wasn't yours. You were dreaming someone else's dream. That was Penguin dreaming about Drennette.\" She would say no more.\n\nI haven't had a chance to talk with Penguin but she said it so summarily, said it with such utter finality, confident to the point of dismissiveness, I can barely imagine she might be wrong.\n\nWhat do you think?\n\nYours,  \nN.\n17.II.84\n\nDear Angel of Dust,\n\nI got a chance to talk with Penguin and I asked him about the dream. He says yes, he has dreamt it, more than once in fact, but he wouldn't go so far, he says, as to say it's about Drennette. The \"certain someone\" in the dream when he's dreamt it, he says, looks nothing like Drennette and doesn't, in the way she speaks, walks, carries herself and so on, resemble Drennette. \"Anyway,\" he said emphatically, \"I haven't dreamt that dream in a very long time. I dreamt it two or three times, the last about five or six months ago. Since then it seems to have gone away or been put away, like a tune a band retires from its book.\" I was struck by the analogy. He too, as soon as it left his lips, appeared to be struck by it, pausing as if hearing an echo of it, giving it further thought. \"How strange,\" he went on to say, \"that it popped up again, not to mention getting dreamt by someone else.\"\n\nWhy me? He couldn't help wondering nor could I. Is it my listening to his going on about Drennette, his bending my ear the way he so often has, my sitting still for it? Did that do it, did that open me up in some way? Did that in some way make me a surrogate or a host? \"It's not about Drennette,\" he shot back when I asked out loud. \"Of course not,\" I said, not wanting to make an issue of it, and went on to wonder, again out loud, what order of dream transfer it was we were dealing with, dream theft, dream contagion or what. Neither Penguin nor I could help recalling Dredj's dream, its roundabout exchange with \"Dream Thief,\" ideas of dream transit, dream transport, crowding in on us. Was Dredj the connection, we wondered, the conduit, Penguin's dream's way into my sleep? Yes, that was it we agreed.\n\nWe no sooner agreed than Penguin looked at me and said, \"Getting back to Drennette, whom it's not about, or getting back, I should say, to it not being about Drennette, I need to say about 'a certain someone' that she's the one we have that we don't have or the one we had that we don't have, the one we have by not having, that we have to have without having. You being new to this, I need to tell you she's the one whom to have would be not to have, to have let slip away in a presumption of having.\" He stopped as though winded by what he'd said, as though it were an hour-long speech he'd finished. I found myself set back, silent, not knowing what to say, and when he gathered himself again, got his wind back, he said, simply repeating himself, \"I tell you this because you're new to it.\" He paused a beat before adding, \"I haven't dreamt that dream in ages.\"\n\nI wasn't sure anymore. I thought maybe I'd agreed too soon, settled on Dredj as the connection too quickly. Maybe, to begin with, \"Why me?\" isn't the question. What if Lambert dreamt the dream too, like the time the three of us dreamt the Djeannine dream? Maybe this was us dreaming a collective dream again, staggered instead of dreamt at the same time, and if Lambert hasn't dreamt the dream, could it be that he just hasn't dreamt it _yet_?\n\nIt may turn out to be inconclusive, I know, but I need to speak with Lambert and I'm going to speak with him next.\n\nAs ever,  \nN.\n18.II.84\n\nDear Angel of Dust,\n\nA quick note to say I spoke with Lambert. I didn't recount the dream or go into particulars. I simply asked had he dreamt of \"a certain someone,\" figuring he'd know what I meant if he had. (I didn't want his dreaming the dream, if he hasn't yet but goes on to dream it in the future, to have been influenced by my telling it to him.) He had no idea what I meant. \"A certain someone? Meaning who?\" he asked. I told him he'd have known if he had. I told him he must not have and that I'd leave it at that, except that if he eventually does dream it he'll know it and to let me know if he does.\n\nSo that's it for now, the upshot being it's up in the air, still up in the air. What exactly my dreaming the dream signifies we won't know until we see whether Lambert dreams it too, which could take a while. At what point, if he doesn't do so soon, do we decide he most likely never will? The five or six months it took between the time Penguin last dreamt it and the time I did? More than that? Nine months? A year? We'll see I'd like to say but I'm not sure.\n\nOtherwise, not much is up. We did venture up to Hollywood the other night to a place on the Strip called Club Lingerie, a pop music spot that's been there about ten years, though it's not a place we normally frequent or had even been to before. What took us there was a one-night gig by Ronald Shannon Jackson and the Decoding Society, a band we've heard on records but not live. It was a chance we couldn't pass up, no matter how incongruous the venue seemed to the music or to the way we, at least, hear the music, though on further reflection it made a certain amount of sense, given labels like \"free funk\" and \"avant-funk\" that've been applied to it. Anyway, I'm not sure the Club Lingerie crowd was quite ready for what Jackson and his band brought, not sure they knew what hit them. I can't exactly say we were ready or that we ourselves knew what hit us.\n\nFor one, even though we'd heard some of their albums\u2014 _Eye on You_ _,_ _Street Priest_ _,_ _Mandance_ _,_ _Barbeque Dog_ \u2014we weren't prepared for how _physical_ they got live and in person, how _physical_ the music got. To begin with, the place was packed, teeming with bodies and body heat before a single note was hit, and then when the band began to play it came on _strong_ , _loud_ , _hard_ , the guitar amps and the PA system turned up to peak volume it seemed, ear-splitting volume, an impact or effect aided by the piccolo range Jackson likes to compose in (something Ornette pointed him toward, we've heard), the timbral ping the music typically rings with. It made one's head ring, one's bones and body ring, subject to an avant-primal assault bent on recalling the word _avant-garde_ 's military roots it seemed, a ringing the club's occasional recourse to strobe lights further assisted. You _felt_ the music at least as much as you heard it.\n\nClub Lingerie, our initial senses of it notwithstanding, proved in many ways to be the perfect venue. A young, beautiful, hiply dressed crowd filled it with glamour and gams, miniskirts on the waitresses and on many of the patrons as well, making for a sexy ambience that was lent further sexiness by the very name of the club, its conjuration of chemises, teddies, babydolls, g-strings, corsets, garter belts and stockings, shelf bras and the like. It was Hollywood to the core, Frederick's of Hollywood in another guise, all the more congruent, by virtue of that, with the physicality of Jackson's music, the bodily press it put on, its purveyance of a bodily aromatics the word _funk_ attempts to get at. We knew nothing, the band instructed us, the way it got physical, the way it came on, if we ignored the sexiness of sweat, the wet working-out of beauty's mandate.\n\nIt would have been impossible not to pick up on all of this. Indeed, Penguin, during the break between sets, remarked that the Decoding Society's brand of funk seemed \"exquisitely its own, high-end and, exactly as has been said of it, avant-funk.\" He went on to liken it, wincing a bit as he spoke, as if the extremity of the figure gave him pause, to \"an ever so uric whiff coming off the crotch of a pair of silk panties.\" It was, though, a figure Aunt Nancy not only agreed with at once but went further with. \"Pee and perfume,\" she said simply, \"piss and perfume.\"\n\nI mention this outing because we appear to've uncovered the origins of the 4\/4 shuffle that visited us a year ago at The Studio and once again, in a bit of a teasing, attenuated form, at the Comeback Inn in September. Why we'd never noticed it before will always be a mystery but there it was in Jackson's music, the 4\/4 shuffle, in one of the pieces they played that night, \"Man Dance\" it seemed at the time, though it might've been \"Shaman.\" Drennette was the first to notice, leaning over our table and all but shouting to be heard above the music, \"You hear that? It's the 4\/4 shuffle.\" We all picked up on it eventually, buried as it somewhat was beneath an extended shiver the horns maintained.\n\nOn our way out of the club, after the second set ended, it all came back to Aunt Nancy, who had introduced the shuffle during the gig at The Studio, that a couple of Cecil Taylor records, _3 Phasis_ and _Live in the Black Forest_ , made while Jackson was in the band, had probably been her subliminal prompt. I gave both records a listen earlier today and I could definitely hear it, on the second side, as it turns out, of each record\u2014the second half of _3 Phasis_ and what's a bit like the same piece under a different title on _Live in the Black Forest_ , \"Sperichill on Calling.\" It seems to me, in fact, to epitomize the difference Jackson made in Cecil's band.\n\nIt was reassuring to hear the shuffle come up in Cecil's music, to hear it could be avant without the funk. It relieved us of some of our qualms about it, our fear of a certain concession to pop. We enjoy the Decoding Society and we had a good time at Club Lingerie, but we wouldn't want our music to go as far as they go in that direction. The venue and the music did fit, as I've already said\u2014the glamour, the sexual glitz, the not so oblique appeal. Lambert summed it up as we were leaving: \"I'm surprised the balloons didn't show up.\"\n\nYours,  \nN.\n27.II.84\n\nDear Angel of Dust,\n\nAs Lee would say: Boy, what a night! The owner of the Comeback Inn called Lambert earlier in the day and said it was his wife's birthday, that he wanted to do something special for the occasion and he was wondering if we were available, wondering if we'd be willing to play. He said something about coming back to the Comeback Inn or coming back into the Comeback Inn, Lambert told us, with or without a straight face, he added, he couldn't say, seeing as they were on the phone. Be that as it may, Lambert went on, he answered yes and they settled on the money and what time we'd be there. Talking about the gig after Lambert let us know, we decided it might be a good time, as good as any at least, to try handing out balloons to the audience, the idea I wrote you about a few letters back. There'd likely be balloons decorating the place for the occasion anyway, so why not? This would give us a kind of camouflage Penguin pointed out, doing so without saying why he thought we might need it, a potential nit the rest of us chose not to pick. We agreed it was all the more reason to give it a try.\n\nI'd actually grown to be a little reserved about the idea since I last wrote you about it, more and more coming to see it in relation to the night I sat in with the Crossroads Choir, the \"Only One\" balloon the audience kept afloat during \"Body and Soul.\" I had already been there it seemed, been there and done that it seemed. I already knew the participant quantum that contained air in the hands of an audience could be, in their hands or just outside them, either way. Would it be a return to that I wondered, knowing in particular I wanted no more of the halfway-around-the-world romance I had recourse to that night, no mere mention even. At some level I knew there was no reason to think the balloons would bear such baggage but at another level they seemed as bound up with it as the \"Only One\" balloon had been that night. Warmed-over romance I called it, dismissing it with a sneer, haunted, even so, by the possibility that an equally warmed-over social romance, a stale albeit yet to be achieved romance of collectivity, took its place in what we thought the balloons might do or what I thought they might do. Did I want us to revisit the \"Only One\" balloon, revise it, in some way pluralize it?\n\nJust thinking about it, problematize it though I did, conjured a musical motif whose contour had the feel of having come the way my way out of my qualm would track and retrace, a bent-winged bird of a motif my lips veritably itched to work out on a mouthpiece, reed or brass no matter, a lick-driven theorem concerning the one, the two and the many. (Listen, to hear what I mean by lick-driven, to Wilber Morris's \"Miss Mack\" on his _Wilber Force_ release or to Dewey Redman and Ed Blackwell's \"Willisee\" on _Red and Black in Willisau_.) Could a lick be said to grow legs or a motif to grow them for it, that's what what I heard sounded like, what the motif I had in mind would do. I heard a symphonette of sorts, a micro-symphonette wound up in a four-bar motif whose unwinding one would never complete, return to the tonic no matter, the one, the two and the many each only more active the more apparently moot.\n\nThere would be or might as well have been, that is, an \"Only Two\" balloon and there would be or might as well have been an \"Only Many\" balloon, \"only\" by turns a valorizing claim to singularity ( _one_ two, _one_ many) and a dampening, a disclaimer, putative singularity disavowed ( _merely_ two, _merely_ many). Two would be one only as the many would be one, a social predicate or a political prerequisite I'd advance with an austere pucker (it more and more appeared to be brass I had in mind), an austere pucker free of all recourse to recollected sex\u2014as though, I wanted to say, one \"wet\" one's lips with dry ice, as though what would again have been touted as closure came off as cosmetic arrest.\n\n\"Marginal center,\" I said instead, as much tonguing the eventual mouthpiece as muttering it to myself. I meant no more than to stencil a self-correcting wobble, sonically rendered or regraded warble, a slaptongue sputter as high as the horn would go. So it was I not so much overcame as incorporated the reservation I'd come to feel. I was ready for whatever lay in store at the Comeback Inn, as were the rest of the band.\n\nWe got there a little before 8:00 as Lambert and the owner had agreed, having set up and run thru a sound check late in the afternoon, left and come back. Not as many tables had been cleared away as when we played there in September (they were expecting a larger turnout), so the space in the corner that served as a stage was even tighter than before. It reminded me of the time in the late sixties I went to hear Sun Ra at Slugs', the twenty-piece arkestra packed onto and spilling off of a stage that appeared to be the size of a dining table, two at most, \"Space Is the Place\" given a whole new meaning by the cramped quarters they issued the music from. I thought it a miraculous negotiation of shrunken premises. I thought back and thought something of the same would be required of us, nowhere near twenty pieces large but backed into a corner somewhat.\n\nWhen we got there the place was already full, every table taken, every seat as well. There were indeed balloons decorating the room, lining the door we walked in thru as well as the walls inside, balloons of all colors. Of more than one size, some were strung along the walls where they met the ceiling, some from ceiling to floor where one wall met another. The mood was upbeat, festive, people ready to party, most of them friends of the owner and his wife we assumed, some already doing so. This, I have to admit, gave us misgivings. We don't consider festivity to be exactly our thing and we began to wonder had it been a good idea to accept the gig.\n\nPenguin was the first to say it. \"I'm not sure about this,\" he said once we got to the storage room that served as our green room. \"I'm not sure about playing for a birthday party,\" he went on. \"I'm especially not sure this is the gig to try passing out balloons, despite what I said earlier about camouflage. Too much chance of them taking it the wrong way.\" He went on in this way for a bit and the rest of us admitted we had the same apprehensions. Aunt Nancy, however, having allowed she felt exactly such qualms, went on to say that we were there and that there wasn't much else we could do but go on with it, play (\"The show must go on,\" she even said), make the most of it. She got the look of hard thought on her face just before asking, \"Remember the time we played the Scarab and someone yelled out 'Uterine hoofbeat!' as we played 'Bottomed Out'?\" With the exception of Drennette, who wasn't yet in the band when we played that gig, whom we hadn't, in fact, even met, we all nodded yes. Aunt Nancy went on to say it was up to us to set the tone, to darken festivity or strike a note of dark festivity, that it was exactly the understanding of birth bound up in the \"Uterine hoofbeat!\" cry we needed to bring to bear, that the run of apocalyptic beat, repercussion and possession thereby implied (the Four Horsemen allied with Haitian vodoun) was ours to introduce, a complicating note we would insist accrues to each natal occasion, the owner's wife's no exception.\n\nWe all knew what she meant. It was a dread, gnostic note she was insisting on, birth as an issue of misconception, conception itself as an issue of misconception, dubious arrival into a miscreant world. Dubious cause for celebration, dubious or at best ambiguous cause for celebration, birth, we needed to insist and get them to see, was a bottoming out, a slippery descent down what she said was \"the chute of incarnation,\" bodily being a forfeiture of immaterial essence, bodily being material exile, detour. She suggested we open with \"Bottomed Out.\" We agreed and then decided the other pieces we'd play. She rolled coach, cheerleader and gospel diva into one, exhorting, just as we went out, \"Let's wreck this place!\"\n\nSo it fell to Penguin to set the tone, \"Bottomed Out\" opening, as it does, with him solo. He more than rose to the occasion\u2014natal, dark, anti-festive in one swipe\u2014or I should rather say he descended to it, more than descended, taking the baritone to his mouth and muscling a low B-flat from it, a call to order calling for close attention, dialing the party atmosphere down. Most of the patrons heeded his call, ceasing to speak and turning their attention to the music, though there continued to be something of a buzz at several tables, conversations carried on with lowered voices and with not so low voices now and then, an outburst of laughter rising out of the buzz.\n\nAs always with his entrance into \"Bottomed Out,\" Penguin plied a \"Lost Generation\" line or allusion, a reference to and a reabsorption of Sonny Simmons's piece, only here he freighted it with something new, something we hadn't quite heard before, the very dread, gnostic strain or insistence Aunt Nancy had prompted us with, primed us with. He parsed a certain seepage in the horn's low register, a low-to-the-ground if not below-the-ground shuttle or shift, stealing away from concept or conception. Reconsideration proposed as concept, conception as misconception, the triplet-laced line he pressed or pursued equated lost generation with generating loss. It was nothing if not the unbottoming of birth, a caveat so severe, so categorical, it uprooted all track not in some way sealed by trepidation. No one yelled \"Uterine hoofbeat!\" and no one would, but uterine tread, uterine trot, uterine gallop were very much with us, there not all that long after Penguin started playing, low buzz notwithstanding.\n\nPenguin worked his low-to-the-ground or below-the-ground shuttle or shift into a medicinal sweat, the all-out shimmy or shake of a body possessed, a hatching pool of heat. He shook as though powered or possessed by the keys of the horn, shook in such a way as to make the horn rattle. Each key was a peyote button, shriven cactus toughness, a bitter, withered vestige his finger could almost taste.\n\nWith the advent of all-out shimmy or shake the crowd turned more attentive. The few tables at which conversation had continued fell into silence, everyone's eyes and ears turned toward Penguin, whose pooling hatch turned out to be more feint than shimmy or shake. He now had them exactly where he wanted them, where we wanted them, and it was at this point that Aunt Nancy, on congas, and Drennette, on bat\u00e1 drum, joined in, the implied, immaterial hooves (vodoun horses, lucum\u00ed horses, santer\u00eda horses) more palpably afoot. Aunt Nancy and Drennette's drumbeats were hits to the head, kicks to the head, horse's hooves intent on awakening any who remained inattentive, alerting all to the uterine bottomlessness they trod.\n\nPenguin let the triplet figure go and let himself be lifted by the uterine carpet the congas and the bat\u00e1 drum rolled out, the uterine precinct or premises we were now on or in made all the more obvious, to any who somehow hadn't noticed, by Drennette sitting slightly gap-legged with the latter on her lap, its pinched midriff looking anything but pregnant notwithstanding. The bat\u00e1's hourglass shape intimated nothing if not time, birth's inauguration of which and the ravages and wear brought with it \"Bottomed Out\" would be intent on one bearing in mind. Penguin not only rode this awareness or admission but rode it out, leaving the horn's lowest register for a climb into its highest, a raised eyebrow and a rocket launch rolled into one. He ended his climb with a squealing, roll-with-it peal of acceptance that grew falsetto-like, a peal he held for sixteen measures before coming back down.\n\nIt was as if only after making all this clear, clearing the way in a sense, only after the natal, dark, anti-festive note had been struck, could anything approaching festivity be indulged. Penguin's approach was to calibrate his peyote-button plea or appeal as longstanding romance, returning to the bottom of the horn to recall Ronnie Cuber on Eddie Palmieri's \"Yo No S\u00e9,\" an approach whose low annunciativity and quizzicality rolled into one, whose namesake agnostic tone if not timbre, was nothing if not the foot in the door locked-out festivity needed. Someone in the crowd shouted, \"Alright!\" It was on.\n\nThe rest of the piece, indeed the rest of the evening, would prove to be a standoff, sometimes fluid, sometimes tense, between a will or a willingness to party and the music's more austere demands. Penguin dangled his Cuberesque salsa riff, his bolero riff, long enough to let the audience know it was there, no more than a passing taste to let them know romance and festivity were a part of our repertoire, withheld, whenever withheld, neither out of inabiliity nor disinclination but in the interest of a larger design, a certain rigor and recognition. Having served up that taste, he fell back into his triplet-heavy quandary, kicked and had at by the congas and the bat\u00e1 he made it seem, a thrashing bob and weave outmaneuvering would-be assault, as though, knowing every reason to despair, he chose not to. By way of a stoptime pivot opening like velvet stage curtains, he went again to Cuber's matinee-idol suavity and sagesse, not only not despairing but dancing instead. There was no shout of \"Alright!\" this time. The audience could sense he simply visited the salsa motif. They knew he was not there to stay. Indeed, he was again plying his triplet-laden qualms before long and back to the salsa motif not long after that. It was this back and forth he now lingered with, limning the contours of the mixed-emotional occasion he took birth to be.\n\nAunt Nancy and Drennette chose to linger as well. Not so much underneath what Penguin played as around it, they built a serrated, bentlegged amble founded on an aspect of godly limp, a gallop true liquidity accrued to, horses at ocean's edge it seemed. The bat\u00e1 rang with echoic reach, a watery, concussive lilt it was all one could do to keep one's feet upon hearing, a sweet-wood sonance, water notwithstanding. One found oneself asking, \"Maple? Cedar?\" The congas meanwhile plied a bass tack, lengthier legs than those the bat\u00e1 stepped with, a deeper, asymmetrical step taken twice for each the bat\u00e1 took. Anyone familiar with lucum\u00ed and santer\u00eda recognized the interplay of congas and bat\u00e1 as the pattern consecrated to Yemaya, the orisha of the sea and of motherhood. Aunt Nancy and Drennette sought to season if not in some way temper Penguin's mixed emotions it seemed, advancing a more accepting, celebratory take on birth, mixed in its own way though it was, acknowledging the saltiness of birth.\n\nAunt Nancy was nothing if not the mother and the mistress of mixed possibility. Each conga leg stepped into a hole, a giving way of ground if not the ground being lower than expected, foot missing a rung or a stairstep thought to be there, a giving out of leg or an extension of leg. The tone she'd insisted we set she thereby deepened, an exemplum she invited the audience to ponder, peel back.\n\nThe audience did exactly that. The introspective space Aunt Nancy's \"drop step\" apprised one of they now occupied. The low buzz gone, they sat silently, caught up in the music, attentive to every inflection, caught up in thought. Each of them listened closely, brows furrowed in some cases, visibly given pause by Aunt Nancy's leg-in-a-hole polymetrics.\n\nAs always in \"Bottomed Out,\" Penguin pled a certain weakness, a watery-kneed insecurity of limb he called out for support from. Drennette and Aunt Nancy offered that support, the cane or the crutch or the ushering arm around his waist he cried out he needed, but this went only so far. It came to be time for Djamilaa, Lambert and I to join in, which we did, odd as it sounded or seemed, with a waltz-like figure not unlike Billy Harper's \"Cry of Hunger,\" a cradling, chorusing, to-the-rescue riff that was anything but without bottom. Djamilaa was on piano, Lambert played tenor and I played trumpet, offering a unison statement of the line that we repeated again and again, Penguin, recalling Shepp's version of \"Frankenstein\" on _The Way Ahead_ , growing more distraught each time we played it.\n\nIt was as though rescue, would-be rescue, only made Penguin more aware of his predicament, more painfully aware of his predicament. He thrashed and bellowed, fought as though caught in quicksand.\n\nI'd been wondering how the owner and his wife were taking to the new tone we set, festivity's hallowing-cum-harrowing, Penguin's predicative distress. I caught a glimpse of them out of the corner of my eye as Lambert, Djamilaa and I bore down on the cradling rescue riff, spotting them over to the right toward the back of the room. I could've sworn I saw the owner's wife blow us a kiss. It might only have been me, I'm not sure, but, not only that, even more than that, it seemed all the souls there'd ever been and all there'd ever be sat in bleachers looking down at the stage we stood on, poor excuse for one though it was. It was as if the Comeback Inn had turned into a stadium and a Renaissance memory theater rolled into one, elastic locality suddenly on the ascendant. It seemed as though the Comeback Inn had become a balloon, an inflated premise that was more than one premise, a gradually pneumatic enlargement equating place with play as well as play with proliferation, locality and prolixity rolled into utmost palimpsest.\n\nThis was only a glimpse ahead to what was to come it turned out, a sideways glance lasting only an instant might it be said to have lasted at all. But for an instant (if it was, if that's what it was) the bleachers climbed toward the heavens in a flash exposing us all to the sky, soul seated beside soul, soul seated above and below soul, souls packed in, piled high.\n\nAhead somehow lay to the side I noted to myself, there though not yet there, eventuality's lateral outpost, festivity's debt or dispatch. No sooner did the thought arise than I put it away, though to say the thought was taken away might be more exact, swept along as everything so abruptly was. Penguin's opening gambit gave way, that is, to a tricky ensemble section\u2014key changes, tempo changes, dotted notes, double-dotted slurs. Whether we'd really been blown a kiss or it only seemed we'd been blown a kiss one could've wondered but that was now moot. The audience was obviously with us, the owner and his wife included. The steeply ascending bleacher seats, briefly glimpsed and gone, had made it clear everyone was all in.\n\nOur tonal reset, our setting a new tone, had not only taken but taken well and we were on our way. Lambert and Djamilaa took terse, meditative solos, rousing in each its own manner, each an understated tour de force, and the one I took wasn't bad either. At the conclusion of \"Bottomed Out\" there was a hearty round of applause. Two people, in fact, stood up to applaud, so taken with what was only, we'd make sure they'd see, the beginning. We had the house in the proverbial palms of our hands and we were just getting started.\n\nThe mood was more mindful now, not averse to festivity but in touch, even so, with all that would give it pause, the wages whereby festivity would have to be won, had had to be won. We were ever, the tone we set insisted, at a point of having reason to despair but also, having looked at it, seen it, stared it in the eye without blinking, at a point of winning reason to rejoice. It was nothing if not a note of hard-won festivity we struck.\n\nWe now felt less uncertain about handing out the balloons. We and the audience were now on the same page we felt, closer if not all the way there in any case; the balloons would not be taken the wrong way. It wasn't, though, that we had a particular way in mind we wanted them taken, only that they not be confused with those decorating the room, that musicial purpose and decorative purpose not be confounded. We were pretty sure but we'd see, we'd wait and see, make sure the new tone persisted. That being the case, we'd hand the balloons out before our final number.\n\nHaving opened with one of our older pieces, we played one of our new ones next, \"Dredj's Dream,\" following that with another new one, \"1489.\" We then went back to the older part of our book for \"Opposable Thumb at the Water's Edge\" and then to another newer one, \"Sekhet Aaru Struff,\" to end the first set. Our intensity grew with each piece, as did the audience's attentiveness, a kind of absorption drawing all of us in. It grew more and more clear, more and more certain, that this was the night to hand out balloons. It grew all the more so when, the \"green room\" such as it was, we stayed out and mingled with the audience between sets.\n\nEveryone we talked with had something hip to say. \"When I closed my eyes during 'Dredj's Dream' I saw a shade-late incision of light\" was only one of the interesting comments we got. When Djamilaa and I overheard a man in his thirties remark to Lambert, \"It was like the music admitted to time only to suspend it, put birth on hold only to know it all the more, take us back to it. We all were the very thought of life again,\" she whispered in my ear that she was struck by how much it sounded like something I'd write in a letter to you. It did, I had to admit, nodding my head and made to wonder had our correspondence crept into the music, seeped into the music, so seeped in it put words in our listeners' mouths. A woman with dreads wearing a UCLA sweatshirt did say to Aunt Nancy that \"Sekhet Aaru Struff \" was \"a beautiful woman's face conjugated into limnings of stride and striation, an implied walk compounded of structure and stroll.\" Others, on the other hand, were content to let it go with \"You guys kicked ass, mucho ass,\" \"You really tore it up\" and the like. Still, I wondered.\n\nThe mood was good, fun but not frantic, a festive sense being somewhat low-key made more abiding. The party vibe was definitely still there. Drink flowed freely and food abounded of course. Laughter was no stranger of course. The balloons on the wall, of course, did say something about lightness and color, something about pneumatic provision, the tenuous, necessary containment of aerobic endowment. The music had let no one forget the balloons were breath's bounty, an always tenuous bequest.\n\nWe played five pieces for the second set, as we had for the first, opening with a newer one, \"Fossil Flow.\" From there we went to something we hadn't played in a while, Shepp's \"Like a Blessed Baby Lamb,\" then to a couple of our older pieces again, \"Aggravated Assent\" and \"Tosaut Strut.\" We decided we'd put matters in the audience's hands in more ways than one, performing another older one, \"Drennethology,\" as the ostensibly last number, reserving \"Some Sunday\" for an encore should they demand one. It would be right before this encore, were there one, we'd hand out the balloons. The chance of there not being one, of the audience not demanding one, was a snowball's in hell we felt. We would do what we could to make sure.\n\nThe level of intensity we left off on at the end of the first set we started on to begin the second, starting out as on fire as we'd been and building to be more on fire. \"Fossil Flow\" got its best reading yet, punctuated by a number of what Rahsaan would call bright moments, setting the stage for yet even more. Indeed, there were more such moments in the pieces that followed than I can go into other than sketchily, mentioning one or two to represent them all. Lambert, to mention one, soloed on \"Like a Blessed Baby Lamb\" in a way that went off-script or perhaps pre-script, working an expectorant, self-excavating vein worthy of Joe Henderson at his five-o'clock-shadow best. It seemed he proposed a radically articulate clearing of the throat, a veritable book of clearing the throat upping the ante on his \"Book of Opening the Mouth.\" It was as though such clearing were no longer preparatory to speech or as though, better, it were prior to being preparatory to speech, as though such clearing constituted speech\u2014or, if it were indeed preparatory, it remained adamantly preparatory, promising an arrival preparation preempted, endlessly prior, endlessly proto-, a seeming endlessness packed into a four-minute solo, a four-minute string of ahems. Throat-clearing took the place of talk in a gruff serenade insisting thus would its book be, talk not arrived at as such, talk already there otherwise. Less expository than agonistic (exposing nothing, that is, if not exposition's insecure ground), the solo purveyed a beautifully possessed hemming and hawing, expectorant scratch attenuated by namesake bleat. It culled a particular tension and it kept the audience on the edges of their seats, an expectant stress it offered, in the end, no release from, simply ending mid-hem (or was it mid-haw?) as Lambert backed away from the mike. The audience, finding themselves more negatively capable than they'd have otherwise thought, burst into wild applause, as though declaring themselves to be the release the solo had so impeccably withheld. Aunt Nancy, who followed, was more than a few bars into her solo by the time they settled down.\n\nDjamilaa, to quickly mention another, brought an inrush of Eastern undulation into \"Aggravated Assent,\" quoting from Gurdjieff and de Hartmann's \"Reading of a Sacred Book\" behind Penguin's alto solo. Giving Gurdjieff and de Hartmann's austere Central Asian modalities a more southerly, Persian bent, she drew from another recording I know she listens to a lot, Nasser Rastegar-Nejad's _Music o_ _f_ _Iran: Santur Recital, Volume 2_. She teased out the slightly mallet-like attack one hears in \"Reading of a Sacred Book,\" the ringing hammer held in check but held, one clearly hears, nonetheless. She held it less in check, moving over into what it hovers on the edge of, reminding us the piano does deploy mallets while exacting ictic, hammering runs that brought nothing if not the santur to mind. She did more than simply accompany the solo, daring Penguin, with a jabbing persistence reminiscent of Monk, to ride the swell her tremulous left hand kept advancing. It had a way of building and backing off a bit and then building and backing off a bit again, potentially going on that way forever. She plied an Ibrahimic reach into the center of the earth that was also a stairway to the stars. It oddly amalgamated climb and cascade, a watery stairway made of watery cloth Penguin bounded upon more than outright rode, an unrolling rug or an unwinding scarf he trod securely on, an escalating splash after his own heart it seemed. It was after everyone else's too it seemed, so effortlessly did the other four of us work behind them (Lambert and me riffing under Penguin's lead), so far forward did the audience, holding its collective breath one sensed, lean and let themselves be hit by the aggregate hammer it all, for the moment and, for a while, moment to moment, turned out to be, all but unbearably beautiful, borne up and out and all the more enduring we irrevocably knew. \"Quantum-qualitative\" doesn't even come close.\n\nLet these two suffice. What I most want to get into is how things went when we handed out the balloons. The audience did indeed demand an encore when we finished or ostensibly finished with \"Drennethology.\" They applauded loudly, whistling here and there and here and there shouting out an \"Oh yeah!\" They rose to their feet as we bowed and they remained standing as we made our way back to the \"green room.\" We walked single file, carrying ourselves as we'd agreed beforehand we would, our bodies given over to a stolid rectitude. Their eyes followed us into the \"green room\" and we closed the door once inside, at which point the applause grew louder. We let it go on for a while, the whistles and the shouts escalating, and then, each of us grabbing a package of balloons from the box we'd brought them in and ripping it open, we opened the door and went back out. They were still standing. Each of us, as we'd agreed beforehand we would, took a different route back to the performing area, handing balloons out to the audience as we did.\n\nEveryone had quieted down and sat back down by the time we made it to the performing area. Lambert stepped to the mike and explained why we'd handed out the balloons, instructing the audience to each use the balloon as he or she saw fit to contribute to the music, a new piece called \"Some Sunday\" we were about to play that I'd written. He made a point of not suggesting how they might use the balloons, saying nothing about blowing the balloon up and letting the air out, blowing the balloon up and rubbing it, stretching the balloon and twanging it without blowing it up, blowing the balloon up and thumping it, stretching the balloon and snapping it without blowing it up, blowing the balloon up and popping it with a pin, blowing the balloon up, putting it on the floor and stomping it, etc. He simply suggested that they join in as they saw fit when he gave them the sign, which would be him raising his right arm and quickly bringing it back down. He would indicate when they were to stop, he explained, by making the halt sign with his right hand.\n\nWe started the piece off with Djamilaa, unaccompanied, repeating the folk song-sounding, children's song-sounding phrase on piano, a vamp-till-ready we let go on for a while. Djamilaa gave it just the right beginner's touch, a sometimes heavy left hand that bordered on losing the time and did in fact lose it now and again, only to quickly regain it. Pointedly unpolished, mock-awkward, \"amateur\" by its own lights, it made me think of Don Cherry's remark that too many people make a religion of professionalism, a not very satisfying religion he'd found, and there was indeed something reminiscent of his piano work on the _\"Mu\"_ date about Djamilaa's playing. This was all the more apt by way of entrance into a piece we had invited the audience to take part in, as if to say it was okay to be amateurs, okay to be other than polished or professional, okay to be rough and ready, even more rough than ready. Poignancy took the place of polish in Djamilaa's vamp, as if to say, reiterate and insist we're all beginners when it comes to love, especially the Sunday kind Etta James sings about, not to mention the millenarian, great-gettin'-up-Sunday collective love we all so badly want.\n\nIt was as if it were the childhood or even the infancy of some new order she was auditioning and it wouldn't be going too far to say that Drennette's conga beats, when she finally came in, making the vamp a duo, were, threaded into everything else they were, the \"patter of little feet.\" They made it echo in our heads, the remote broadcast of some ideal order yet to come, a conceptual impendence one ached and winced and almost wept on hearing. (Had I looked out and glimpsed an audience member or two wipe a tear from his or her eye I wouldn't have been surprised.) The conga-laced piano vamp advanced a nascent ascendancy of all any heart could want. It grew stark in the stitched iterativity it moved by way of, rhapsody met with impediment, pendency's boon and bequest.\n\nDjamilaa and Drennette reasoned against reason it seemed, played as if to placate a ghost. The vamp jumped and ran, a run of lost and rewon time that rayed and rippled, nothing if not a swell and then nothing if not a subsidence, done again and again as though it would never end. Its echo grew more and more inscriptive, an introspective \"letter to the world\" one wrote without intending to, sound all instinctual plea, appeal or epistle, a recess all proprioception fell into.\n\nJust at the point where it began to appear the vamp would really go on forever the rest of us came in. Lambert's alto jumped out with an elliptical whimsy he might've floated away on were it not for the bottom Aunt Nancy's bass and Penguin's bari put under us all. He went from whimsy to bittersweet anthem in no time at all, a march-worthy solemnity and lilt reminiscent of Jesper Zeuthen's solo on \"To Alhaji Bai Konte,\" one of the cuts on _Brikama_ , the new Pierre Dorge album we've been listening to a lot. I put the mute on my trumpet, a quivering blade I wielded or wove underneath and inside the line he laid down, a needling thread abetting the aspirate cloud he plied. March-worthy though it was, we kept to the near side of march, Aunt Nancy's bass unpredictable and volcanic even as it proffered, along with Penguin's bari, what bottom there was. Scott LaFaro's collateral chatter, the eruptive, side colloquy he used to go at in Bill Evans's trio, seemed to be her model, her muse, a kicking up she sprung us away from march meter with, fall into a groove though we otherwise did, grumble and go.\n\nLambert wasted no time getting the audience involved. At the top of a run one would say put butterscotch in bittersweet's place he took his right hand from the horn, raised his arm and quickly brought it back down, continuing his high flight or flotation with his left hand on the spoons even as he did so, putting bittersweet back in its place as a consequence, intended or not. The audience responded on cue, addressing the balloons in every way we'd imagined they might. Some blew their balloons up and let the air out, some blew their balloons up and rubbed them, some pulled their balloons taut and twanged them without blowing them up, some blew their balloons up and thumped them, some pulled their balloons taut and snapped them without blowing them up, some blew their balloons up and popped them with pins, some blew their balloons up, put them on the floor and stomped on them and so on. It made for a loud, raucous noise, a not unjoyful noise though not always joyful, a devotional sound or song even so.\n\nWe found ourselves lifted, oddly buoyed by what amounted to a balloon choir, an additive chorus by whose cacophonous ledger we were called to account. As we looked out at the balloons being rubbed, popped, twanged, emptied of air, snapped, stomped on and so forth, we couldn't help feeling we were being arraigned even as we were being apprised of a corroborative seam or support. That seam, we heard as well as saw, heard even more than saw, paraded haptic amenities before us, audiotactile grip and grain we were called out to chorus and carol with or against in turn. Rise and arraignment rolled into one, the audience's balloon valences broke thru to a vein of strike, stretch, kick, scratch and scour that offered accompaniment and discontent in like measure, deepseated grievance and regret one would've sworn railed against birth itself. What were we doing where we were and what were they doing where they were were only the simplest of the queries they involved us in.\n\nI stole a glance at Djamilaa and saw that she sat at the piano in the most royal way imaginable (regal straightness of back, regal litheness of arms, regal groundedness of rump, regal fingertip \u00e9lan), a way of abiding balloon arraignment we could all learn from I felt. She continued with her mock beginner's tack, a \"tentativity\" at times that cried out for support, coronation's darling or doll though she clearly was, royal command and recourse though one knew she had access to. What we were doing where we were was what they were doing where they were she made it clear, no matter she sat enthroned, Queen of Soon-Come Sunday, for we all sat enthroned, whether we sat or stood, we where we were, they where they were, Queen or King of Soon-Come Sunday.\n\nI stood straighter, having stolen that glance, and I noticed that others must have stolen one too, for not only did Penguin, Lambert and Aunt Nancy stand straighter, not only did Drennette sit at her drumset straighter, but everyone in the audience now sat or stood straighter. We all found ourselves draped in regality and rectitude, the odd, unexpected bequest balloon arraignment bestowed on each of us, the surprise endowment the pressure it put on birth blessed us with.\n\nMeanwhile, Lambert kept in mind the \"bruised bell\" instruction I gave him the first time we played the piece. Neither bittersweet nor butterscotch had anything to do with it now, or, if either did, only in the most angular, at-many-removes way, a warm sound that was as wounded as it was warm, a wounded sound that was as warm as it was wounded, reception and incision, warmth and wound, run as one. Bruised bell met balloon arraignment with a whimsicality or quizzicality \u00e0 la John Tchicai that, atop Djamilaa's iterative piano, made one beat back tears. Balloon valence, it said, was nothing if not feeling's inflated premises, felt no less for real, felt no less intensely, no matter now known as such. He picked and played off strike, stretch, kick, scratch, scour and so on at will, inflated premises' match at every turn, mixed emotions' match at every turn, a wincing, tangential feint titrating woundedness and warmth over every array.\n\nWe had decided beforehand that for this rendition of \"Some Sunday\" Lambert would not be the only one to solo, that he'd be followed by Penguin, me and finally Drennette with a drum solo before we went back to the head. Before ending his solo Lambert took his right hand from the horn, extended his arm toward the audience and made the halt sign. He went on playing as the balloon choir subsided and the remainder of his solo, albeit continuing the bruised bell vein he'd been working, sounded as if launched by the balloon hubbub it left behind.\n\nLambert went on for another minute or two, followed by Penguin, me and Drennette. Shortly into each of our solos Lambert gave the audience the sign to join in, later giving them the sign to stop, having himself been given a nod to do so by the soloist. To begin to end what's already a very long letter, I'll let it go at saying that Penguin and I both acquitted ourselves well, negotiating the briar patch balloon arraignment tossed us into with Brer Rabbit-worthy aplomb. Penguin was indeed magisterial, working a bass vein he posed as foil to Lambert's high flight or flotation, a baptismal, often abyssal plunge that even the balloons went under with, but what I want to get to is what happened during Drennette's drum solo. It was during her solo that the comic-strip balloons emerged.\n\nI should note that the audience more and more warmed up to their role the further we moved into the piece, that they more adeptly addressed the balloons and that each of them brought something more like a plan to his or her contribution. With each successive solo, they grew more attentive not only to what the soloist was doing but to the rest of the band and, best of all, to each other as well. Some of those who twanged their uninflated balloons and some of those who snapped theirs could be seen coordinating their respective attacks, not only relative to one another but with an eye and a ear toward Aunt Nancy's bass, an eruptive amen corner that kept stride with her as best they could. Some of those who rubbed their inflated balloons did so with increasing finesse, eking out, in some cases, sounds not unlike those made by a cuica, running the gamut between moan and squeal, whine and whimper, as they advanced a patient, melodic parsing adumbrating the soon-come Sunday we attested to. Some of those who thumped their inflated balloons and some of those who stomped theirs increasingly waited for just the right crescendoed-into moment, a bomb atop a bomb they kept their eyes on Drennette to calibrate. Others, however, especially those who popped their balloons with pins, eschewed coordination, opting for a randomness and surprise tantamount to an irreverent \"boo,\" asymmetric occurrence an end in itself.\n\nIt amounted, oddly enough, to the brer patch I wrote you about a year ago, a divinatory field advancing peppered accord, peppered kinship claim. What we were doing where we were and what they were doing where they were made for a tenuous, precarious connection, an on-again, off-again mesh. We were in sync and out of sync by turns, balloon choristers with or without balloons. Balloon-advanced brer patch extended a multiplex field freighted with a feel for polyrhythmicity, vibe-societal strike, stretch, kick, scratch and scour. What we were doing where we were and what they were doing where they were was that.\n\nThe audience appeared to be enjoying itself as well, more and more so as we moved further into the piece and they more confidently found their way. Some shouted, \"Alright!\" Some shouted, \"That's it!\" Some shouted, \"I hear you!\" Some, without shouting, beamed in such a way as to loudly announce a certain delight, a certain attunement, their faces lit with the glow they felt fitting into the mix. Engagement registered on the faces of others in different ways, intense concentration or deliberateness in some cases, closed-eyed, trancelike absorption in some cases, lips moving as if counting or mouthing lyrics in some cases, head bowed reverently in some cases. I caught a glimpse of the owner and his wife and saw that they fell into the beaming group, a big, toothy smile on each of their faces as, respectively, they thumped and rubbed the balloons they held in their hands.\n\nPenguin ended his solo shortly after nodding to Lambert to give the audience the halt sign, which Lambert did. The balloon choir fell silent as Penguin continued his low rove, capping off his low rove with a rummaging run in the bari's lowest register. It was a breathy but oddly subaqueous creep whereby he went foraging on the ocean floor, the rival of any octopus the way his tone seemed to spread and to reach and to scurry, covering ground one would never have imagined it could. The solo ended with a climb to the horn's upper register one semiheard and semisaw, the wistful run of notes a string of bubbles rising to the surface, balloons rising to the surface one semithought. We all fell silent as these final notes floated away, all except Drennette, whose turn it now was to solo.\n\nDrennette's opening gambit was unprepossessing enough. She sat with the conga to the right of the parade snare, the conga held at a slight angle in a stand that allowed her to play it while seated at the drumset. Indeed, this allowed her to address either the conga or the drumset or both the conga and the drumset, the latter being what she began her solo by doing. Back straight, head high, Queen of Soon-Come Sunday, she sat with the drum stool turned somewhat to her right, having put away the stick she'd been using in her right hand while continuing to strike the orchestra snare with the stick in her left. She began beating the conga with her right hand, drumming on the orchestra snare still, tip of stick and palm of hand dealing in timbral shadings it appeared to have become her sole purpose in life to inspect and allow us to hear. Everything slowed down. She struck the orchestra snare, letting the sound hang in the air until it faded, then she slapped the conga, doing the same with the sound it made. She went back and forth that way between orchestra snare and conga, insisting we hear not only distinctions between the two but subtler distinctions between different deployments of stick and between different deployments of hand. It was a low-key start for a drum solo, sublime testing or sublime tinkering though Drennette conveyed it to be.\n\nA beginner's tack to the max, Drennette's minimalist outset was, as I've already said, an inauspicious, unprepossessing start. No one would've guessed it would be during her solo that the balloons would emerge. I'm not suggesting flamboyance has at all to do with whether or not the balloons appear. We're not sure what does and I'm not saying that. But, thought of in relation to the eventfulness of the balloons emerging, her opening gambit appears almost pointedly uneventful, certainly, if nothing else, a pointed withholding of the virtuosic display drum solos are famous (or infamous) for. In any case, there appears to have been some occult or in some other way recondite hydraulics whereby legibility of the sort the balloons deliver accrued to deferred buildup.\n\nA deferred buildup was exactly Drennette's tack, beginner's or not, a buildup she invited the audience to join her in. She gradually worked the parade snare and the cymbals into her left hand's repertoire while adding heel, fingertips and even fingernails to her right hand's address of the conga's head, moving on to ever more complex combinations, ever more multifaceted patterns. \"Take your time,\" Penguin encouraged her at one point, which she did, slowly bringing a wider range of timbres, rhythmic motifs and tempi into play. When it got to where she'd set the table to her satisfaction, laid out the repertoire she sought to activate, she gave Lambert the nod and he gave the audience its cue.\n\nThe balloon choir came on strong, an aggressive onslaught of pent-up energy let loose, more aggressive than what Drennette had in mind, too loud\u2014so much so she took her right hand from the conga and lowered it, whereupon the audience took the volume down. What she wanted, it gradually became clear, was a quiet, accretional field of haptic intensities, a quiet, slowly quickening submission to an accretional muse, aggressive given time as well as loud given time. Accordingly, not as many balloons were popped, either by pins or by being stomped on, as during the earlier solos, many more now being twanged, snapped and, especially, rubbed. What aleatory knocking is to the opening section of Marion Brown's \"Afternoon of a Georgia Faun\" aleatory tug, touch and rubbing were to Drennette's balloon-assisted sonic field, an abiding, aleatory patience allowing what would accrete to accrete, what would accrue to accrue.\n\nRub, as I've said, was the address of choice for the majority of the audience, an address ranging from quiet, caressive strokes to longer, more cuica-sounding moans made by pulling the thumb along the length of the balloon. Rub's predominance accorded with Sunday love's eroticism, suggesting, as it did, epidermal touch, epidermal regard, epidermal warmth. It was clear to everyone that rub, in its multiple senses, applied, epidermal flare well within its range of implication, epidermal friction well within that range as well. Obstruction, difficulty, abrasion, complication and all related meanings along that line were present as well, notwithstanding Sunday love's utopic promise. Everyone, it was clear, brought his or her own experience to bear, from felicitous meow to saturnian groan, not to mention all the gradations between.\n\nRub indeed ruled. A composite, aggregate apprehension, it truly moved, animated by Drennette's increasingly propulsive recourse to the full complement afforded by the drumset\u2014cymbal shots, paradiddles, bass thumps and such, played with as well as against the conga beats her right hand steadily put forth, she herself on occasion trafficking in rub by pushing the heel or the edge of her hand across the conga's head, scraping, scrubbing, sliding. Again, briar and burr had as much to do with it as purr, a divinatory field haptic amenity two-handedly drove.\n\nConductor as well as Queen of Soon-Come Sunday, Drennette took her hand from the conga every now and then and lifted it, palm facing upward, cueing the balloon choir to take the volume up. It all grew louder over time, more intense over time, collateral repercussion and pop increasingly part of it, a few more audience members popping balloons with pins, a few more popping them by stomping on them on the floor. When the volume and the intensity neared their peak, Drennette pulled her right hand from the conga and took up the drumstick, turned a bit to her left and addressed the drumset full-tilt, all-out, as formidable a display of power drumming as one could want, quantum drumming.\n\nIt was with this step or this quantum step that the comic-strip balloons began to emerge. They did so, when they did, not from Drennette's drumset but from the balloons the balloon choir held, from each and every one of them in fact, whether rubbed, plucked, popped, stomped, twanged, snapped, whatever. The sound had reached a fever pitch, the rich cacophony of a philharmonic orchestra warming up, squealing, keening, groaning, squalling, when the first set of balloons appeared. They arose from each and every balloon choir balloon, all of them bearing the same inscription: _By whatever birth was, back at some beginning, I lay on my back unable to see past my belly, legs up, legs bent, legs open, knee on either side o_ _f_ _my belly barely visible. I'd been riding a horse I thought was a bicycle, a bucking horse I was thrown by. I lay on the ground bleeding. I lay bleeding on asphalt, meaning it must have been a bicycle I rode. I lay bleeding in the dirt, meaning it must have been a horse I rode._\n\nAunt Nancy, Djamilaa, Lambert, Penguin and I looked on, taken aback. We all looked at one another, perplexed, all of us, that is, except Drennette, who kept on drumming, looking out at the balloon choir, nonplussed\u2014in fact, with a gleam in her eyes and a grin on her lips that seemed to say she knew something we didn't. The audience, the balloon choir, seemed also to be taken aback, albeit, following Drennette's lead, they pressed on, unperturbed, balloon arraignment engaging their energy and their attention all the more. The sound got louder, more complicated, more intense. Balloon valence was now balloon-on-balloon valence.\n\nThe first set of balloons disappeared, followed, a few measures later, by another, each bearing these words: _Unprepossessed, my horse told me get off, get a life. It bucked and reared up and threw me to the dirt, threw me to the pavement, a bicycle I did a wheelie on. Thrown off, unsure where I was, was it asphalt or dirt I was on, unsure what it was I'd been on, I lay on my back, a ballooning belly's dispatch, I lay bleeding. My lips bled, blood I took to be a kiss._ Balloon-on-balloon valence made for a thicketed feeling, a thicketed field, a density of atmosphere the Comeback Inn could barely contain, a thickness of air. A hothouse compactedness reigned, implication and legibility at once at odds and in league with each other. This was deferred buildup's bequest, nothing if not hothouse complication.\n\nThings had sped up. When the second set of balloons disappeared another quickly followed, inside each of which were these words: _Sweet rotundity. Fecund recess. Ride had been all there'd been, ride was all I wanted. Thrown off as to where I was, what I'd been on, my long legs straddling my horse, my long legs pedaling, I lay on my back riding mysel_ _f_ _hard, I lay on my back giving birth to myself. My ballooning belly took the place o_ _f_ _the hill I'd begun to climb, the hill at whose base my bicycle's pedal broke, at whose base my foot slipped from the stirrup, causing my horse to buck and rear up._ Aunt Nancy, Djamilaa, Lambert, Penguin and I looked at one another again, taken aback all over again. The balloons appeared to be channeling Drennette's bike ride with Rick, the concussive spill she says taught her percussive spirit. Their apparent equation of spill with birth especially caught our attention we all agreed later, as did their confounding of the bicycle with a Haitian vodoun horse.\n\nTaken aback notwithstanding, we urged her and the balloon choir on, the five of us a rooting section now, offering such exhortations as \"Drive it!\" and \"Take it out!\" Drennette did exactly that, putting together a run of rolls that were to a drummer what circular breathing is to a horn player, the rolls a set of uroboric wheels balloon arraignment and balloon-on-balloon valence rode for dear life, the balloon choir doing all it could to keep up, straining to keep up, hothouse compactedness bringing sweat to many a brow.\n\nThe moment the third set of balloons disappeared Aunt Nancy yelled out, \"Kick it!\" Drennette, with a flurry of thumps, was all over the bass drum pedal, almost, one thought, as though it were the broken bicycle pedal come back to life. This took things up a notch, a notch no one, the balloon choir least of all, knew was there, though, well before they knew it, that's exactly where they were. The rubbing was much louder now, riddled with screeches and squeals, and the popping accelerated, aleatory detonation taken to a new high. It had gotten to the point where the poppers, whether stomping or wielding a pin, worked in pairs to speed the process up, one person inflating the balloon and then handing it to the other to pop. The proverbial mine field, brer patch could not have been more explosive.\n\nIt took a couple of measures for the fourth set of balloons to emerge. They bore these words: _A flood ran down the far side o_ _f_ _the hill, blood gushed at its base. I lay on my back bleeding between my legs, legs bent, legs up, legs open, the lips between them bleeding, blood I knew could only be a kiss, a kiss boats bearing a message were afloat on. They floated leaving the hill behind, each o_ _f_ _their sails having the same thing written on them: \"Tell my house it's hot in here.\" So spoke my sailor boy, hot to be with me, my sailor boy who was all but back, due back on Sunday, a Sunday that couldn't come soon enough._\n\nWhen the fourth set of balloons disappeared the fifth emerged: _I tore mysel_ _f_ _to be whole, tore mysel_ _f_ _to possess myself, no matter how unprepossessing, no matter how unprepossessed. I lay thrown off better to know what on was, my host part horse, part bicycle, house hot church, house heat's ashram, house blood's hot retreat. I lay thrown off better to get back up, sit sweating deep in yogic labor once up. My sailor boy more me than I was, blood lotus, I lay, legs uncrossed, recuperating. I lay percussing, I lay getting ready to sit, repercussing, I lay repossessing myself._\n\nBalloon arraignment, all said and done, kicked as hard as Drennette did. The inscriptions' ballooning belly took balloon valence to another power, balloon-on-balloon-on-balloon valence, a run of introjection tending outward. The sound hit like aspirated static by now, an unremitting rush the balloons rode like pneumatic surf. They kept emerging, a sixth, a seventh, an eighth, on and on, ringing changes on _birth_ , _blood_ , _lay_ , _sit_ , _house_ , _horse_ , _kiss_ , _hill_ , _bicycle_ , _legs_ , _lips_ , _labor_ , _sweat_ , _heat_ , _boat_ , _belly_ , _possession_ , _percussion_ , _lotus_ and the like that would never, it seemed, end but did. The final set of balloons bore these words: _Having been thrown off taught me what on was. I found mysel_ _f_ _on the far side o_ _f_ _my ballooning belly, blood-soaked, smothered with kisses. When I sat up I sat on having been thrown. I made a throne o_ _f_ _having been thrown, having been thrown the throne I sat on, lotusheaded, legs crossed, Queen o_ _f_ _No-Show Sunday, my sailor boy's home but to be gone._ Drennette, shortly after these balloons emerged, nodded to Lambert to give the audience the halt sign. He did so, whereupon they stopped on the proverbial dime, the balloons disappeared, Drennette soloed a few more measures and then nodded for us to come back in, which we did, restating the head to end the piece.\n\nThe audience was on their feet right away with a loud standing ovation. Most of them simply clapped but some shouted, some whistled and some went at it on their balloons again. They all knew, as did we, that something phenomenal, even by balloon standards, had occurred. The ovation was long as well as loud, more heartfelt, I thought, than any we'd ever received before. The experiment, we felt, had succeeded beyond our wildest dreams. Many a thought was provoked and many a question raised, thoughts and questions we were quick to discuss on our way from the gig.\n\nAs to what we made of the evening's events, I'd have to say it's too soon to say more than what I began by saying: Boy, what a night! Drennette, for example, insisted, right off the bat, that she didn't, as we thought she might, know anything we didn't know. We'll be talking about it for days I'm sure. I'll let you know what comes of it.\n\nAs ever,  \nN.\n3.III.84\n\nDear Angel of Dust,\n\nIt turns out a writer for one of the local alternative papers, _Santa Monica Weekly_ (Aunt Nancy calls it _Santa Monica Weakly_ ), was at the Comeback Inn the other night. He wrote it up and his review (I guess you'd call it) is in the issue that just came out. The balloons, not surprisingly, get most of the attention, beginning with the headline: \"The Molimo M'Atet Makes Peace With Its Balloons.\" He expands on this right away, opening with an assertion that he's always sensed \"a certain keeping of the balloons at bay\" in our public statements as well as in comments acquaintances of his who happen to know us recount having heard us make in private conversation. Why this is, he goes on, he's never quite been able to figure out, though his guess would be, he ventures to say, that what he would term \"a confusion of artistry with austerity\" wouldn't let us come to terms with \"the bid for attention the balloons can't but be augurs of.\" He allows that there might well be something noble about this, something valiant about our \"holding out against easy acclaim, commercialism, sensationalism and the like.\"\n\nHe devotes a good deal of space to describing the d\u00e9cor at the Comeback Inn Sunday night, detailing the arrangement of the tables, the location and dimensions of the performing space, the lighting and so on, saving the balloons lining the door and strung along the walls for last. Referring to the balloons as \"a foretaste of what was to come,\" he notes the range and variety of their sizes, colors and shapes before announcing that Sunday night marked \"an epochal transition point in the band's career to date.\" By this, he goes on to explain, he means that \"on this particular gig, at this particular place, on this particular night,\" we made peace with the balloons, \"came not simply to bear but to embrace them.\" He resorts to metaphor in his reiteration of this point, referring to our \"d\u00e9tente\" with the balloons, our \"extending an olive branch\" to them, our \"smoking the peace pipe\" with them.\n\nHe delays going into exactly what he's referring to by this even while admitting some readers may already, by way of the grapevine, have \"gotten word of the goings-on\" he'll in a moment get to. He then proceeds to discuss the music we played, the music the two sets were comprised of, addressing it piece by piece, saying a little about each, on the whole positive, albeit not without the obligatory quibble (\"Fossil Flow\" could've been longer, \"Bottomed Out\" shorter and so on), but quickly and not with a great deal of detail, saving a more sustained, more lingering approach for the encore, the balloon-assisted version of \"Some Sunday,\" the \"goings-on\" he delays going into while being anxious to go into. Once he gets to that, that is, he takes his time, carefully narrating our return to the performing area, recounting the passing out of the balloons with great attentiveness to who covered what part of the room, repeating Lambert's instructions verbatim. This, the passing out of the balloons, he announces, \"the peace overture, the peace offensive,\" was our epoch-making move.\n\nHe goes on to spare no hyperbole, pull no rhetorical punches, peaking, I think, when he calls it our \"come-to-Jesus moment.\" He waxes patronizing, paternalistic and psychoanalytic rolled into one, praise notwithstanding, writing that it's us coming to terms with our need for attention, \"getting beyond a conflicted desire to reach a wider audience.\" Having dwelt on this a while, he caps it off by saying \"their nobility of abstention, their nobility of abeyance, is hereby moved on from in favor of a greater nobility, that of diving in, joining the fray, putting the hay on the ground where the horses can reach it.\" He then turns to the encore itself, the music itself, the balloon-assisted \"Some Sunday.\"\n\nFirst off, he writes, he has to \"confess to being one of said horses,\" to \"admit to having joined in and fully enjoyed it.\" He goes from this to a detailed account of \"Some Sunday\"\u2014Djamilaa's long lead-in, Lambert's, Penguin's, Drennette's and my solos, and especially the balloon choir's participation. He's happy, he writes, to have been \"a participant-observer,\" from which perspective, he admits, he \"can't but have experienced it all in a special way,\" a special way he's not sure, try as he will, he can do justice to. We next read a good deal about him choosing how to address the balloon (he decided to rub it), the challenges of hearing oneself while hearing the band and the other balloon players as well, his and the other balloon players' \"gains in confidence and competence\" as the piece moved along, his and the other balloon players' \"delighted surprise\" when the comic-strip balloons appeared.\n\nHe has nothing but praise for our \"decision to embrace the balloons and the audience both in one bold stroke,\" calling the passing out of the balloons \"not simply the turning of a corner but collectivity's deep dream come true.\" Not only did we open ourselves to the balloons \"more wholeheartedly\" than ever before, he writes, we \"tapped a live, open vein of musicianship in the audience one wouldn't otherwise have known was there.\" He himself, he adds, despite having no prior musical experience, \"discovered a resident, recondite prowess the band and the balloons apprised one of.\" He crescendoes to this conclusion: \"In making peace with their balloons the Molimo m'Atet made them our balloons. May they long let the balloons take us all higher.\"\n\nIt was Aunt Nancy who phoned and said, \"We've got a problem.\" She was the first to see the review and she quickly gave each of us a call to alert us to it. She said we should pick up a copy of the latest _Santa Monica W-e-a-k-l-y_ (she spelled it out) and we'd find a review of the Comeback Inn gig. She gave us the gist of it but again insisted we each pick up a copy and read it. She said she considers the problem the review makes apparent a crisis, a crisis calling for what she termed an emergency summit. She insisted we meet at Gorky's a few hours later to discuss it, which we agreed to do.\n\nGorky's is a place we go to from time to time, either all together or in smaller groups, an all-night cafeteria downtown on Eighth Street that opened two or three years ago. It's gotten to be a bohemian caf\u00e9, a hangout for artists, catering to and something of a creature of the loft scene downtown, featuring cheap food, coffee and beer, an interesting clientele, live music (usually not much to our liking) every now and then. Anyway, we all went out and got copies of the _Weekly_ , read the review and headed there, Lambert the last to straggle in. Our feelings were mixed, as you can imagine, both individually and as a group. It is, after all, a positive, at points ecstatic review, lavishing praise upon us and the music in multiple ways, and we ourselves consider the experiment a success. The pitch of the praise, the assumptions underlying it and the framing of it, though, confirm our reservations about the balloons.\n\nAunt Nancy could hardly wait to talk about it and she got us going as soon as Lambert arrived, not waiting for him to get food and drink as the rest of us had. Her being the one who came up with the idea of passing out the balloons may have had something to do with her sense of urgency but the review was something we all felt we needed to air our thoughts about. Queen of nothing if not irony, she spoke for all of us by saying, to kick things off, that what bothered her was the reviewer's literal-mindedness, \"a kind of tone-deaf earnestness that misses our mock literalness or mocking literalness, our mixed-emotional 'embrace' of the balloons.\" This was a risk we knew we were running we reminded ourselves, Lambert going on to point out that the prophylaxis the literal balloons were to perform turned anaphylactic in the reviewer's hands. \"The balloons embody their own self-critique,\" he said, \"a hedge against what they otherwise embody, advancing legibility as an inflated claim. The reviewer misses that.\"\n\nWe went around on it for a long while. Djamilaa, for example, said she resented \"the condescension, the turning of something we did into a command or a mandate to do it, an after-the-fact mandate or an after-the-fact command, as though the reviewer were somehow in charge.\" Drennette, for another, talked, as Aunt Nancy and Lambert did, about what the reviewer missed, taking up his trope only to qualify it, torque it, saying that \"if it was an accord it was a sideways accord we came to with the balloons or they came to with us.\" After a pause of only a beat she added, \"It was, after all, No-Show Sunday's throne they put us on.\" Penguin, to give yet another, agreed. \"Not to mention,\" he said, \"the horsecycle,\" at which we all laughed.\n\nWe kicked it around, as I've already said, for a long while, got more and more into the ambience at Gorky's and went off on tangents. We got off track and went on tangents quite a lot in fact, as more and more food and drink came to the table. We ended up hanging out for a few hours and we had a good time. There was no live music, so it was easy to talk, to hear and be heard. A couple of painters came over to our table at one point, saying they recognized us and that they were big fans of our music. We all shook hands and they stood chatting with us for a while. Frank (their names were Frank and Ren\u00e9e) said he listens to _Orphic Bend_ while he paints, that it's become for him what _Kind o_ _f_ _Blue_ was for painters during the sixties. He said he's in fact working on a painting called \"Orphic Bend\" and he'd like us to visit his studio and see it when it's finished. We exchanged contact info and they went back to their table.\n\nWhat we'll do next we didn't decide. Hold a press conference? Issue another press release? Never pass balloons out again? We don't know.\n\nYours,  \nN.\n[Dateless]\n\nDear Angel of Dust,\n\nDjamilaa and I awoke with a balloon between us. It lay in the space between our pillows. She sat up and stretched, her light cotton nightie exposing her close-to-the-bone shoulders, her dimpled elbows and her elegant, outstretched fingers a true boon above her head. She yawned. I followed suit, yawning as I sat up and stretched. It was as we turned to each other, smiling, and said good morning that we noticed it, the balloon nestled between our pillows, a third head or an extra pillow bearing these words: _I walked around inside a mall where I was to meet a certain someone, music deeper in my ear the longer I walked. The someone I'd come to see wasn't there. A strong wind blew the roo_ _f_ _off and the stores were suddenly booths. The music inside my ear visited booth after booth, a late arcade it kept me wandering in all night._\n\nThe smile left Djamilaa's face. \"I see you've been at it again,\" she said.\n\n\"Been at what again?\" I asked.\n\n\"Dreaming about a certain someone,\" she said.\n\n\"Not necessarily,\" I said, immediately noticing how weak it sounded and how weak it was but having said it before giving it any thought.\n\nDjamilaa got out of bed and stood up, her back to me. I couldn't help looking at her ass, whose bottom half her nightie left exposed. I gave a thought to the workings of the carnal plan, the carnal setup, the carnal eye, her ass's magnetic draw. I stared as though her pendant cheeks and the cleft between them proffered a magic exit from all that was fallen and profane, fallen and profane though they themselves were thought to be. \"Denial makes it worse,\" she said, sitting back down on the bed, looking over her shoulder, her ass no longer exposed.\n\n\"I'm not denying,\" I said, \"I'm complicating. Didn't you say it was Penguin's dream, not mine?\"\n\n\"That only makes it worse. Your thing for a certain someone has to be pretty strong to have to go thru channels.\"\n\n\"A certain someone is no one,\" I said, more confident now, \"No-Show Sunday's bequest.\"\n\nThe balloon disappeared as I spoke, an evaporative remit appearing to confirm the disendowment I noted, a certain someone's inveterate nonchalance, chronic no-show, the someone one could only not know, nearly know.\n\n\"I'd like to say, 'Well, now that you put it that way,'\" Djamilaa said, \"but I won't.\" She stood up again, her ass's redolent cleft exposed again, carnal provision posed, it seemed, against disendowment, an immaterial musk filling the air clouding my mind if not the air, an imagined musk where there was no musk, all the more intoxicant not being one. I wanted all the more, that is, to whiff the musk that was, press my nose to Djamilaa's ass's redolent cleft as to an actual rent in time (chronic rift, crack in the cosmic egg), the rending of time whose proffered exit took me out.\n\n\"What if I were to swoon?\" I said, syncope's famous last words it turned out. I spoke them as I succumbed to a rush and fell forward Djamilaa told me a few seconds later when I came to. She was beside me now, in the middle of the bed, cradling my head in the crook of her arm, my head against her breast. My face was close to her cleavage, another cleft, my nostrils wide with the overnight smell on her skin, the morning musk I love so much.\n\n\"I was teasing,\" I said, nose between her breasts, nuzzling her cleavage, her close-to-the-bone beauty all over me, the cracked cosmic egg's repair. The heady smell of her skin kept my nostrils dilated. \"A certain someone,\" I whispered, \"is no one but you, you know. The balloon was right between your pillow and my pillow. It was no more my dream than yours.\"\n\nThe smell of Djamilaa's honey-based lotion joined by overnight sweat suffused her nightie, a light, penetrant funk my nose filled with and would've followed to the end of the earth. I wanted its translation more than the musk itself perhaps but words fell teasingly short when it came to capturing it. I imagined a work of audiotactile sculpture that would, her light funk's pheromonal embrace rendered equal parts haptic and sonic. I semiheard, semitouched an atomized, ambient advance one would apprehend as a cystic self-equation cut or carved out of capric dispatch. Djamilaa's goatlike beauty, that is, assailed my inner eye, my inner ear and the fingertips of my inner hand, my overt nose's introvert accomplices, her long-faced forbearance a boon given over to study, aesthetic remit.\n\nDjamilaa-the-Muse was in full effect, Djamilaa-the-Beautiful-One-Has-Come in full effect. The sweat-accented, honey-based lotion smell took me in as much as out, a mustiness of cleft and contained space and a cystic attar opening a faintly beckoning realm. I whiffed and whiffed again and again a remote, redolent beacon broadcasting intimately from afar, a synaesthetic transfer tendering mustiness as light, light funk's multiplied import. There was a deep inwardness, awayness and nearness her smell reposited, inwardness, awayness and nearness rolled into one.\n\nIt was a nearness as near as I'd ever know but somehow, nonetheless, not available, an everlasting no-show allure a certain someone or a certain no one was known for, not to be known otherwise, an awayness as near as any I'd ever know. The reticent, so-near-so-far sound of a French horn might be its aural analogue.\n\n\"Well, now that you put it that way,\" Djamilaa said, \"I agree. Yes, here we are, two no-shows, erstwhile no-shows I'd say, two certain someones who might also be no ones, each the anyone the oneiric arcade parades all night.\" Her voice was low-pitched and husky, a kind of catch in it as though lemon and honey might be in order. The faint suggestion of the latter, combined with the sweat-accented smell of her honey-based lotion, took my inner frenzy further, a would-be match for her synaesthetic beacon's far cry.\n\nI sat up, pulling away from Djamilaa's cleavage, catching, as I did, a whiff that came up from farther down, a more pungent waft coming off her loins and what lay between them, a less light, more penetrant musk rising from the hair that lay there and what lay under it. Oboe, English horn or bassoon to her sweat-accented honey-based lotion smell's French horn, it pierced my nostrils and pervaded my thought to a degree that threw me farther atilt. Oboe, English horn or bassoon, I wasn't sure which, as it may have been all three blown as one or blown in unison, a chorusing call or cry of come-home or come-hither, home's far reach and provenance, inner frenzy's far reach and rule. I was nothing if not such reach's vassal, low-lying musk the principality I swore love and loyalty to.\n\n\"I'd like to write something,\" I said as Djamilaa slid to the edge of the bed and stood up again, her ass's redolent cleft synaesthetically broadcasting again, \"not simply a musical piece but something that would be that plus what visual artists call an installation.\"\n\nDjamilaa was standing and she turned around to face me. The carnal plan, the carnal provision, the carnal setup occurred to me again as the tuft of hair between her legs below her nightie's hem caught my eye, so casually, nonchalantly there it furthered my inner frenzy a bit more.\n\n\"Audiotactile sculpture I call it,\" I went on. \"I want the listener to be able to step into the piece. I want it to be a step taken off a ledge to be unexpectedly caught by a pocket the air offers, a pulverous nodule cut into empty space. I want it filled with streaming powder, uniformly blown powder perhaps, powder the listener leans into as it catches him or her, coming toward him or her like tactile, particulate wind. I want it to hit like a dry, particulate emission propelled from an enormous aerosol can. I want it to feel caressive and custom-fit, pointillistically tensile and precise, an enveloping provision of support. I'd like to call it 'Copacetic Syncope.'\"\n\nSo I spoke at the time. If I had it to do over, I'd add that the streaming powder would have a rushlike quality to it, as though it were the inversion or, more exactly, the inverse mold of a swoon or a keeling over, convexity to the swoon's or the keeling over's concavity, concavity to the swoon's or the keeling over's convexity, the music's haptic equivalent custom-fit.\n\nDjamilaa stood listening as I spoke, her pubic hair's visibility incidental and moot if she were aware of it at all. She was attentive to what I was saying but she also seemed a bit distracted, as though on her way somewhere else or to something else. When I finished she said, \"Sounds good.\"\n\nDjamilaa sniffed the air a couple of times, having caught a whiff of something it seemed. She lifted her right arm, turned her head to her armpit and sniffed it. She then lifted the hem of her nightie to her nose and sniffed it, her belly and the full extent of her pubic hair visible as she did so, offhandedly, nothing if not blas\u00e9, me further atilt even so, especially so.\n\nAfter she let her nightie back down she looked at me and said, \"I need a shower.\" She turned around and headed for the bathroom.\n\nSincerely,  \nDredj\n13.III.84\n\nDear Angel of Dust,\n\nYes, another cowrie shell attack. All the fuss about the balloons must have brought it on. The balloon-on-balloon visitation and valence at the Comeback Inn gave us lots to think about, the unpredictability of the comic-strip balloon appearances, the anaphylactic effect passing out the literal balloons possibly had, the balloons' apparent channeling of Drennette's bicycle ride with Rick, their dystopian tweak of \"Some Sunday's\" great-gettin'-up morning expectation not the least of it. Then there was the simpleminded review in _Santa Monica Weekly_ , which not only gave us more to think about, more worry, but generated some controversy as well. This week's issue carried a few letters to the editor that were written in response to the review, ranging from ringing endorsement to variously critical, the latter complaining that the reviewer \"drank the Kool-Aid\" at one end of the spectrum and that he got us all wrong, praised us for all the wrong reasons, at the other. Not to mention the queries, comments and opinions of friends and acquaintances. All the buzz, the balloons again upstaging the music, must've gotten to me.\n\nIt was as Djamilaa and I sat in bed talking about the balloons the other morning that it hit. I'm not sure it was anything in particular she or I said that set it off so much as the overall strain of having to think and talk about them so much, simply that of having to think and talk about them at all perhaps. In any case, that's what we were doing when I began to feel a tightening in my forehead, the usual sign of the onset of an attack. I had just said, \"There can be no adequation,\" but, as I've said, I'm not sure that's what triggered it. Whatever the trigger, cumulative or momentary, the attack chose to assume what I've come to think of as its classic form\u2014the shattered shells embedded in my brow, the packed or compacted transparency everything seemed encased in, Ornette's \"Embraceable You\" piped into my head and so on\u2014though not without some of its more recent features.\n\nDjamilaa says it was clear to her right away what was happening, that after announcing, \"There can be no adequation,\" I broke off speaking, went sort of blank and when I spoke again asked for pencil and paper, which she got up and got for me. For my part, I felt I sat at Dredj's desk, Dredj's hand had hold of mine. What I wrote was the letter you received a few days ago. The attack subsided, thanks to Djamilaa's presence and care I think, a few hours later, with no need for a trip to the ER or a hospital stay.\n\nI like Dredj's \"Copacetic Syncope\" idea. I can't promise to deliver on the audiotactile sculpture part but I'd like to write something with that title, something along the lines suggested in the letter. For one, it put the sound of the French horn in my head so indelibly I haven't been able to get it out. The ring it has or the reminder it gives of a faraway haunt, a faraway hunt, a faraway homing, won't let me be. I'd want French horn to be a large part of the piece. Ideally there'd be three or four of them, a French horn choir chorusing the harmonic equivalent of \"What but that solace, that but what other solace,\" repeatedly plying that qualm, that claim. Going at it ourselves, we'd have to make do with Penguin on bassoon and me on flugelhorn (barring a crash course on French horn) chorusing behind Lambert's tenor. The inflectional weave and the harmonic wrinkle that would encode \"What but that solace, that but what other solace\" of course remain to be worked out, as does pretty much, I'll admit, everything else. But Dredj got me going and \"Copacetic Syncope,\" I guarantee, even if I do have to take a crash course on French horn, is on its way.\n\nDredj's evocation of Djamilaa's exposed ass and pubic hair (just happening to be there, just happening to be devastating) keeps at me, a vulgar, vulnerable regard the French horn's muted howl would harken back to, a retractive sound as of a world more near than far but not enough with us. I wonder if what he means by \"a pulverous nodule cut into empty space\" is a compensative step one takes wishing it were otherwise, a would-be, wished-for presence or plenum that, would and wish notwithstanding, conduces to dust\u2014an aggressive, propelled, propellant dust fitting one like a glove. I see root people convening at water's edge, low boat-hauling voices deep in their throats and lungs, \"There can be no adequation's\" rejoinder. Or is it a reconvening, a round or return whose recursive charge obeys Dredj's \"enveloping provision of support's\" mandate? I see gopher holes eaten into atavistic ground. I hear \"What but that solace, that but what other solace\" beaded on a thread Penguin's bassoon lets hang, lets dangle, love's don't-get-me-started reluctance and woo.\n\nAs ever,  \nN.\n16.III.84\n\nDear Angel of Dust,\n\nThank you for your thoughts on Dredj's \"late arcade.\" I'm not sure I see it the way you do. In fact, I'm sure I don't. I even think I'd like to see it that way, but I don't. I don't hear a lament for times past. I don't read the roof blown off the mall as a critique of the present, \"a redemptive critique hearkening back to sturdier social relations,\" as you put it. Dredj was getting at something deeper than that old chestnut. We tend, I think, to forget that the annals of the past are not the past. We confuse the felicity of the book, the felicity of the museum, the felicity of reminiscence itself with a lost felicity actually lived in the past. It's too easy to throw rocks at the present, too easy to make fun of the mall. Dredj, I'm sure, was up to something else.\n\nWhat that something else was I'm not sure how to say. He said the music went deeper into his ear the longer he walked. He said the stores turned into booths after the wind blew the roof off the mall. The wind, he seems to say, converted the mall into a bazaar or a Renaissance faire, which does imply a reversion to the past, I admit, but he goes on to say that the music in his ear kept him up all night, kept him visiting booth after booth, which I'm not convinced is necessarily a happy situation. What's a late-night, insomniac promenade offering temporally remote solace, the sought-after someone lost, but a lament for the present, nostalgia for, if anything, the present, a dismissal of past amenities (if that's what they were) as irrelevant, insufficent, moot? This is the present dressed in past accoutrements only to emphasize they don't apply.\n\nHe said, in fact, that it was the music in his ear that visited booth after booth. Did he mean that the music traverses compensatory ground, that the music is a consolation prize or that it seeks to be, that it shops, as in olden days and vainly it would seem, for the thing that would deliver solace, consolation, compensation? Again, that's not, it seems to me, a categorical felicity. A certain someone not present, the present itself not present enough, seems a lament for the present, not the past, a post-expectant lament borne by see-thru auspices, inertial everydayness's return. The balloon bearing \"late arcade\" was a third eye (a fifth eye really) that lay between Dredj and Djamilaa, shared by Dredj and Djamilaa, a shared, post-expectant arraignment of any such trappings of the past as booth and bazaar. It augured a necessary appeal to the ordinary, the everyday, the unspectacular abidance I'll continue to call post-expectant.\n\nUnspectacular notwithstanding, \"late arcade\" works deeply allied with \"copacetic syncope\" I suspect.\n\nYours,  \nN.\n18.III.84\n\nDear Angel of Dust,\n\nWe couldn't resist any longer. We've thought of issuing another press release following the _Santa Monica Weekly_ review but we kept deciding against it. We can't respond every time someone gets the music wrong we've told ourselves ever since putting the first press release out. But Braxton's advice or idea or insistence that one has to provide the terms for understanding one's music, develop a language listeners can learn from and deepen their listening thru, has long spoken to us as well. The hubbub surrounding the balloons' most recent appearance, given the spark the _Santa Monica Weekly_ review gave it, finally got to us and we decided another press release, a second postexpectant press release, was in order.\n\nEnclosed you'll find a copy. It was a group effort, everyone contributing input (you'll notice we went with Aunt Nancy's insistence we use her coinage _Santa Monica Weakly_ ), and I think it pretty much speaks for itself. We were a little surprised, in fact, to find ourselves speaking with such authority and, at points, so didactically about something we're so bewildered by. It's as if the review's easy presumption of knowledge nudged us into knowing by not knowing, a position of knowing that would acknowledge not knowing.\n\nIf you have any thoughts I'd love to hear them.\n\nAs ever,  \nN.\nPOST-EXPECTANT PRESS RELEASE #2\n\n_Santa Monica Weakly_ got it all wrong. It wasn't a matter of making peace with the balloons. No one knows them better than us\u2014no one, that is, knows better than us that they're not to be known\u2014and our experience has been that no peace is to be made with them. Legibility, we know, is an inflated claim, the very claim, were it that simple, the balloons embody. But it's not that simple. The balloons are a hedge against that which they otherwise embody, ostensibly embody. True self-critique or disingenuous dodge, they are not to be bargained with or bartered with, either way. We intended the literal balloons not as a peace pipe, an olive branch or a peace offering but, as one of our members put it, a prophylactic. That they turned out to be anaphylactic proves our point.\n\nWhat _Santa Monica Weakly_ fails to note (or only, at best, implicitly notes) is that the balloons are a bluelit brigade freighted with our wildest wishes, a wildness or a wilderness of wish we sought to bring the audience abreast of at the Comeback Inn. It wasn't so much about the balloons as about them, which is to say about us, our shared arraignment of a hope hoped against hope, the knock, hard albeit soft, of what would not (could not) be. _Santa Monica Weakly_ fails to note that a low throb knocked at our door and we let it in, a faint beat that was nothing but anaphylactic willingness, wish and fulfillment, the fullness of which we knew to be fleeting, secret, discreet.\n\nWe also notice no mention is made of the gig's occasion, the Comeback Inn's owner's wife's birthday, which seems to us infinitely more relevant than the peace accords the _Weakly_ wants the gig to have been. It seems to us that what we had, what can be said to have summoned the balloons, was something of a return to initial premises, premises to which nothing speaks more resonantly than birth. We who make the music remember that the balloons' inaugural appearance two years ago in Seattle burst with intimations of pregnant air, pregnant wind, pregnant swell, a ballooning remit lodged in alternate ground we term wouldly. What but the airiness of natal occasion's commemorative occasion gets at wouldliness, the nothing-if-not-that-and-so-nothing event or eventuality the balloons not only announce and inure us to but are, thinly contained pockets of air that they are? Who remembers birth? What could be more wouldly than birthdays? One commemorates an event one can't remember but at which one was present, the event with which being present began.\n\nThe implications of \"No-Show Sunday\" need not be belabored we thought but the _Weakly_ 's unalloyed jubilation makes us think otherwise. We still don't mean to belabor them so much as point the main one out: Sunday never comes or, better, was there but not there. The _Weakly_ 's Sunday might've been all there but ours both was and wasn't, a tale of two Sundays, a tale of two sets of balloons. Literal meaning aside, \"two\" signifies noncontainment, a default on the very containment the _Weakly_ 's read would have. We find the review wishful, painfully unaware of itself, not the arraignment or the ironic fulfillment of wish our passing out the balloons was intended for. We again stress that it was a birthday gig, a meditation on birth, wouldly pregnancy, wouldly wind, an endlessly blown wish to wrap walls around wouldly breath. Unimpatient expectancy, we've come to know, is a lesson one has to be long on learning. The _Weakly_ would move into the mansion without a single brick being laid. The _Weakly_ would have it all too quickly.\n\nThis is what we must do, this is what we ever so slowly, ever so gradually, ever so painstakingly, all but asymptotically approach: 1) Commit to a fast of not reaching, commit as though fast and feast were the same. No easily presumed arrival will do, no easy applause. 2) See that it has to do with finitude, that \"it all has to do with it,\" as Trane said of something else, this wanting to have arrived once and for all, a wished-for arrival the band dallies with by way of the balloons but also knows the dangers of. See that dalliance and danger are the balloons' two bodies, an escaped kingship or queenship whose throne would-be arrival seats us on. See too that wouldly breath keeps both at bay. 3) Come ever more deeply to know that shape is tactility at a distance, balloon curvature a boon at whose behest we bend away from capture, our own as well as theirs, which is both good news and bad. Good and bad apportion, as worth or negative worth, respectively, wouldly weave, wouldly welter.\n\nThe balloons are dispossessed lungs, inspiration less objectively drawn than objectified, a fix or a fetish made of something we know to be fleet, fluid\u2014\"peace,\" were such to be had, beyond any and all patness, which is nothing if not the balloons' exact proffer. Bicycle and horse rolled into one, the balloons are the gift horse whose mouth we scrutinize, possibly a Trojan horse. We don't trust them, never have, never will. We will continue to keep our distance, B'Loon's ingenuous look notwithstanding.\n22.III.84\n\nDear Angel of Dust,\n\nIt seems like forever since I last wrote you though it's been only a few days. I don't mean to be dramatic but I can't help saying everything has changed. Watershed, turning point, call it what you will, the Comeback Inn gig, with its array of repercussions (the _Santa Monica Weekly_ review, our \"Post-Expectant Press Release,\" Dredj's visit), appears to have prompted the change. To say we've turned a corner puts it too mildly. Gone over a cliff is maybe more like it, a precipitous plunge into what only a falling leaf might live to tell about, some stark incumbency upon us now to be that leaf, brake short of shooting star, falling star, burnt-out star. But maybe it's just me. Maybe I load it with weight only to be bogged down. Maybe weight, a possible anchor, speaks too persuasively now, not so much weight as counterweight, a would-be, wished-for antidote against floataway onset, the across-the-board leavening balloon epiphany apprised us of\u2014backed up or gone back to, for me, by Dredj's translation of body-and-bone solidity into floataway musk, evanescent nearness, the fleet funk Djamilaa's nightie dilated his nostrils with. I keep wanting to say, \"Just let me get next to you,\" some running plea or some lover's prayer (would-be lover's prayer) running from King Pleasure to Tower of Power, as though nextness or nearness were the always asymptotic falling short it's motored by.\n\nA copacetic plummet I'd call it except I wonder what rubs off and what sticks to have put it so. This chasm I feel to have opened up between then and now, before and after, the cliff we went over and the chasm we fell into, stretches time to where I'm not sure when I last wrote you though I know I wrote you only days ago. Is it that all nextness or nearness got spirited away by some quantum turn we took, not only Ellison's bit about seeing around corners but cornering a certain claim, a charismatic \"no-show\" claim the balloons admonished us with but would, even so, package and contain and commoditize? I feel my head bent sideways even as I ask. It's not so much a chasm or a canyon I see as a gray morning, misty coastal fog on a downhill street going toward the ocean, a foggy morning prospect in as ordinary a place as Long Beach or San Pedro, if not ennui a diffuse being-at-loose-ends in a small harbor town. Such would be the tone poem I'd write, copacetic plummet downing the ante on Dredj's \"Copacetic Syncope.\" The bumpy logic it would move by evades me at the moment but that's neither here nor there, much less the point of this letter.\n\nMore than one person has told me I have an old soul. Maybe that's what this is, the jostling around of what the Rastas call \"anciency,\" but also more. I'd repeat _prospect_ only to say _peril_ , repeat _peril_ only to say _pearl_ , repeat _pearl_ to imply an irritant ploy whereby time doubles back to audit itself. Some dislodgement intervenes between pebble and ointment, a secreted sheen whose girth and gap we come in time to summon. To have gotten one's head around that, as the saying goes, may well be what time's work is, all it is, all so near nothing nearness can't but be the wall we fall arrested by. A qualitative audit I'd call it, head turned, head gone to or gotten to and so on, arrivance's matte perfume.\n\nMaybe all I mean is that it all feels thicker, portentous, packed with variability so immense it's not at all evident where to begin. I don't remember feeling this way ever except maybe the moment I turned to music as what I'd do with my life. Is this a new beginning, a new life, _la vita nuova_? Thick time, heavy time, the weight shadows carry could shadows carry weight, is what this is, I think, a ponderous vamp-till-ready perhaps. Of the Comeback Inn gig and what it opened up I'd say _cloak_ only to repeat it, repeat _cloak_ only to say _cloth_ , repeat _cloth_ only to say _knotted_ , meaning by that a dense traffic of shade let loose, all accent, bend and inflection, a quickened and quickening membrane it fell to it to flow thru. Whatever corner we came to would turn to take us in, whatever edge we came toward come to meet us. This was the Comeback Inn gig's bequest, a fund of confidence if nothing else, no matter whatever outer misgiving.\n\nSo we hold tight, sit tight in our perch, a risen prospect nothing if not misgiving affords. Call it vigilance. Call it a sensed imminence. We sit bumped up in some way we can't yet specify, an effect of the bumpy logic we'd make music of and will. No, it's not just me. Something real is afoot.\n\nPlease forgive this. It may strike you, I know, as more a mood than a letter, a mood piece or a pep talk to myself, the working out of a mood, the working thru it.\n\nYours,  \nN.\n2.IV.84\n\nDear Angel of Dust,\n\nI woke up with a pounding headache this morning. A repetitive, riverine figure had at my head from inside my head, a sawtooth guitar lick of nine notes, each ever so often let linger with a postulant twang. Such twang sought both entry and release it seemed, a complicated order of induction it cried out for. The river it rode ran dappled by sunlight, a slow promenade so white with sun it forced one to shut one's eyes. I looked on with an inner eye more ear than eye, the procession of pocked whiteness an auditive report blind witness made sing. A benign headache I hasten to call it, benign and low-key. It bordered on but pulled up short of a shattered cowrie shell attack.\n\n\"Twang\" may be the wrong word. A cross between fishing line and piano wire, a point or a ping somewhere between the two, might be a better way to put it. Whatever it was, it wanted to light out at some damped oblique angle, ringing but all the while reined in, no toll if not exacted of itself, dues and destination rolled into one. In that way it was like everything else, a universality it modestly proferred, a sense of itself as not exempt it advanced with exact restraint. Light's late arcade I was tempted to call it, a deferred arrival portended or carried by the cave or cathedral voice it was the accompaniment for\u2014crystalline, echoic, slightly husky at points, rounded by an encounter with collapse.\n\nIt was Milton Nascimento's \"Cais\" come to haunt me, a friendly ghost, the ghost or the trace of what was already a ghost or a trace, the benign guitar lick in league with a benign organ surge. I put it on last night, not having listened to it for what felt like ages, and found that, having listened to it once, I put it on again and, having listened to it twice, put it on again and, having listened to it a third time, put it on again\u2014on and on like that, listening to it over and over, until I'd listened to it I don't know how many times. Something about it got to me, spoke to me all over again as though for the first time, as though when I heard it ages ago I heard it without really hearing it. It did this again and again.\n\nIt wasn't that when I listened to it ages ago I wasn't really listening. It was more a matter of time having been taken out of alignment, a jutting shelf I stood on listening anew, not merely more but entirely such a matter of time staggering or having staggered. Time waits was the last thing I'd have thought I'd ever say but I did. \"Time waits,\" I whispered, all but under my breath. \"Bud was right.\"\n\nOr was it that \"Cais\" itself waited, lay in wait, held back or held some of itself back, only to have at me all the more later, years later that would seem like ages later? Was the shelf or the shift in alignment the song's eponymous pier, nothing if not Milton's invention (\"Invento o cais,\" he sings\u2014\"I invent the pier\"), the pier one would let go and go forth from? Was it, to go maybe a little further, anything but a shelving of itself time, in the song's guise, tabled itself on? \"Cais\" lay in wait or \"Cais\" crept up on me all these years that have come to feel like ages\u2014crept and continues to creep on conga beats, benign conga beats I hear as padded feet. It launches an invented love (\"Invento o amor,\" he sings\u2014\"I invent love\"), an invented sea (\"Invento o mar,\" he sings\u2014\"I invent the sea\"), an invented path (\"Tenho o caminho do que sempre quis,\" he sings\u2014\"I hold to the path I've always wanted\"), an invented boat (\"E um saveiro pronto pra partir,\" he sings\u2014\"And a boat ready to leave\"), all of which, separately and as a whole, I'm tempted to call love's late ark, love's late arcade, love's last arcade perhaps.\n\nCome legless to the sea I'm tempted to say. I'm tempted to say love's last arcade might be its first, come to such extremity one's back at some beginning, again the beginner one is when it comes to love. Such, at least, I'm led to say by the pounding, the piano ostinato that breaks in toward the end of \"Cais\" and on which it fades, Milton singing wordlessly alongside. It takes me back to Djamilaa's beginner's tack on \"Some Sunday\" at the Comeback Inn, back to my very conception of the piece, its mock-awkward loss of time and its folk song-sounding recovery of time, the verge upon a children's song or a child practicing it treads. \"Cais\" goes even further, the piano ostinato recalling \"Chopsticks.\"\n\nWhere that leaves me I'm figuring out. The pounding, I insist, is benign. Is it a prompt to revisit \"Some Sunday\" or the advent of a new composition? Is it perhaps a combination of the two, a sequel to \"Some Sunday\" I might call \"Another Sunday\" or \"Some Other Sunday,\" in the tradition, titular at least, of James Baldwin's _Another Country_ , Gary Bartz's _Another Earth_ , Jimmy Lyons's _Other Afternoons_ , Grachan Moncur III's _Some Other Stuff_ and so on, not to mention Arthur Herzog and Irene Kitchings's \"Some Other Spring\"?\n\nThe \"Chopsticks\"-like pounding driving my headache took me back to another Nascimento piece, \"Pablo,\" whose piano ostinato is even more like \"Chopsticks\" than the one on \"Cais.\" I got out the two versions I have, the Portuguese version on _Milagre dos Peixes_ and the English version on _Journey to Dawn_ , and I've been listening to them, both of them confirming the sense of lost time, lost and recovered time, I took to be at issue in \"Cais.\" \"Pablo\" takes it even further, making a more explicit reference to childhood in both versions: on _Milagre dos Peixes_ , the song is sung by a young boy; on _Journey to Dawn_ , Milton sings it with a boyish falsetto and is joined by a children's chorus halfway thru.\n\nI remember one of my college professors critiquing Dostoevsky's use of children in his work (Polya in _Crime and Punishment_ , Ilyusha in _The Brothers Karamazov_ , Matryosha in _Demons_ , etc.), calling it barefaced, blatant, unabashed. He'd probably say the same about Milton but I find myself wanting to be exactly that, wanting to find a place for a child singer or a children's chorus in \"Another Sunday\" or \"Some Other Sunday\" (whichever it turns out to be called).\n\nSometimes you have to go for the jugular. Sometimes you have to show circumspection the heel of your hand. Sometimes, put upon by dilatory compliance, you have to shove more than suggest. Especially under threat of No-Show Sunday, you have to shout.\n\nAs ever,  \nN.\n17.IV.84\n\nDear Angel of Dust,\n\nIt seemed if I could only 1) angle at the exact amount of incline, 2) lard lead-in with absence in the most parsed and plotted manner possible, lace or load it in such fashion as to make tread trepidatious, the ground trepidatious, trepidation the ground itself, 3) titrate touch in such a way as to build while disbursing twinge, verge on twinkle perhaps, 4) coax or connive, eke sound out, so situate twitch or its adumbration as to extenuate love's least integer, so reside within extenuation as to mitigate timbral collapse, 5) wring the notes as much as play them, _wring_ fully in league with an implied play on _toll_ , twist each note as though it were cloth and the drop squeezed out of it both, 6) placate momentum's demand while recruiting an abiding pocket, a cyst or an insistence indigenous to suasion or swell, 7) confess to a certain dismay or admit my impatience, pound against time until the beat wore ragged, 8) ply layers of waywardness, an annunciative ken peppered with and paced by hesitancy throughout, an arrhythmic hitch cognate or conjugal with nothing if not rhythm, 9) be at large in a twilit fallback, relaxed albeit beset by combinatory chagrin, fallen shade's fluency and fount, all would be right with the world.\n\nAll would be right, at the very least, I thought, with the solo I was at the beginning of. By combinatory chagrin I meant a sense I've gotten in dreams, the sense of returning to a place I've been to before, dreamt of before, dreamt I've been to outside of dreams before, as though to dream was not to make up scenery but to traverse and revisit stable terrain, actual ground and what's built on it, this or that house, this or that room inside. These houses give off the feel of a combination of houses, places I've lived or visited given an odd yet familiar aspect, teasingly familiar but not to be precisely placed. Something about the approach to one of these houses might suggest the promontory to a friend's house in Pasadena while, once inside, the stairway to the second floor would recall, ever so faintly, that of the flat I lived in in Oakland and so on. One of these rooms might seem compounded of the living room of our apartment in Miami when I was a child, the library cubby I studied in when I was in college and more. It's as if all the aspects, facets and features of these places were bits of glass in a kaleidoscope, subject to changing arrangement and permutation while maintaining a sense of real premises, real provenance, more than simply concocted, more than merely dreamt-up.\n\nChagrin, though, in that, coming upon or coming into these places, I always feel I've forgotten something I meant to remember. These rooms, halls and houses appear to be part of some mnemonic practice, a memory theater of the sort Frances Yates writes about perhaps. I despair each time that I've lost the key that would unlock the familiarity I sense but can't entirely call forth, the legend that would mete out place and punctuality, dispel the tease or taunt of what appears partly \"on,\" partly \"off.\" In some of these dreams I feel like a ghost in the Winchester House, combinatory largesse only so many stairways and doors that lead nowhere. I feel toyed with, the Bible's \"In my Father's house are many mansions\" brought to mind, a trickster's pitch.\n\nSo it was with the notes and sonic tactics at my disposal at my solo's outset. Even so, I wanted to step onto such oneiric ground and into such oneiric housing, have my solo be the means of doing so or even be such ground and housing, my own such instructed tenancy and tread. I wanted to be like the man spoken of in my antithetical opera, \"both hands tied, trying to build a house with his voice while sitting on a cot in his jail cell.\" I wanted my solo, by the time I was done, to say what Rick Holmes used to say on KBCA, \"We have built!\" I was bent on a masonic outcome for my chagrin.\n\nThis was during a gig at The Studio we finished a few hours ago. We were playing \"Like a Blessed Baby Lamb.\" I was on soprano. We'd recently played it at the Comeback Inn and we go back to it now and then at rehearsals. We've played around with different instrumentations, different ways of voicing it, and Lambert wanted to give one of the new ones a try at the gig\u2014him on tenor, me on soprano, Penguin on alto clarinet, Aunt Nancy on violin, Djamilaa on harmonium and Drennette on tablas. How to open up and open out from the gruff stamp Archie put on the piece and how to work the time in a way that keeps to a certain signature drag (as though tar were stuck to the soles of our shoes) but keeps to it otherwise is what we've been asking. How to get gruff stamp and signature drag to coexist with Eastern sinuosity and cut is what we've been asking.\n\nI was the last to solo. The order was Lambert, Aunt Nancy, Penguin, Djamilaa, then me. The violin and the harmonium did exactly what we had in mind. Aunt Nancy and Djamilaa lent signature drag a vectored swell, a diagonal swell, an upward and onward tending, tumescent lilt. (\"Lilt\" puts it too lightly. It was nothing short of hallowed what they brought to an essentially profane wager, titular lamb notwithstanding. \"Lift\" says it better, \"coronal lift.\") They gave it a keening, devotional air, Drennette's tablas adding duly acute prance, due traction. This they did without speeding up the tempo, turning signature drag into Baul-Bengali saunter.\n\nDjamilaa's solo had drawn us into a sacred cave, drawn us in and drawn us in effigy on its walls. It was a crystal cave, a sphere rayed out in all directions like fireworks exploding, shooting radii the spokes of a ball that would swell and contract, the harmonium's bellows lungs and hallows both. Things had gotten religious, as in fact, Djamilaa implied or insisted, they always are. She had taken the title's wager to heart. When she finished and the applause died down, the ground I found myself on, the cave I found myself in, gave me pause. I stepped into my solo with a phobic, philosophic tread, a duly fearful tread.\n\nI tend toward Pharoah's way of playing soprano. I like a little constriction in my sound, long on shading, not the tabula rasa sound we were taught was the goal. I like a nasal burr or a nasal buzz along the edges, a bit of abrasion. The soprano's ability to glide, the auto-pilot sense it can fall into, needs to be guarded against, as does its ability to soar. Room has to be made for creak and squeal, subdued crackle, a ducklike sotto voce, not without an R&B twinge. Pharoah tends to glide and soar more than I felt was called for, so it was Wayne's ictic, foraging way on \"Dindi,\" which, as you know, I've never gotten over and will never get over, that was more the tack I took. It was a more grounded sound I sought or at least a sound that sought to be grounded, a sound that felt its way, even groped its way, seeking ground.\n\nDjamilaa's harmonium had subsided to a carpetlike lowing, ecumenical seep and support. Aunt Nancy's violin was a holding action, airborne glimmer if not watery glare atop distant asphalt on a summer day. Drennette's tablas partly kept time, partly bought it, marking it no matter which, a pinged, ringing press or appeal that tolled an announcement of dues accrued. I stepped cautiously into this coven or cave as though barefoot on a bed of hot coals, tread nothing if not trepidation and vice versa, hesitant from heel to toe, tentative, testing. I temporized for a few bars before what I've come to call the Nine Golden Precepts, the desiderata I listed above, came to me in a flash. I repeat them here for emphasis:\n\n_THE NINE GOLDEN PRECEPTS_\n\n_1. Angle at the exact amount o_ _f_ _incline._\n\n_2. Lard lead-in with absence in the most parsed and plotted manner possible, lace or load it in such fashion as to make tread trepidatious, the ground trepidatious, trepidation the ground itself._\n\n_3. Titrate touch in such a way as to build while disbursing twinge, verge on twinkle perhaps._\n\n_4. Coax or connive, eke sound out, so situate twitch or its adumbration as to extenuate love's least integer, so reside within extenuation as to mitigate timbral collapse._\n\n_5. Wring the notes as much as play them, wring fully in league with an implied play on toll, twist each note as though it were cloth and the drop squeezed out o_ _f_ _it both._\n\n_6. Placate momentum's demand while recruiting an abiding pocket, a cyst or an insistence indigenous to suasion or swell._\n\n_7. Confess to a certain dismay or admit my impatience, pound against time until the beat wears ragged._\n\n_8. Ply layers o_ _f_ _waywardness, an annunciative ken peppered with and paced by hesitancy throughout, an arrhythmic hitch cognate or conjugal with nothing i_ _f_ _not rhythm._\n\n_9. Be at large in a twilit fallback, relaxed albeit beset by combinatory chagrin, fallen shade's fluency and fount._\n\nAfter temporizing for those few bars, I went on, I can say without bragging, to make good on all nine.\n\nI'm not saying this was the best or the most dazzling solo I've ever taken. There's something about the Precepts coming to me the way they did, incumbent upon me all of a sudden as if I took dictation, a suddenly scribal providence or at least provision on the tips of my fingers, on my lips, teeth and tongue, on my diaphragm and in my lungs, that made it get to me and stay with me, still stay with me, to such an extent I haven't been able to get to sleep. Was it the suddenly scribal providence or provision or was it Djamilaa sitting on the floor crosslegged and lotuslike before the harmonium, dress modestly pulled over her knees, covering her open thighs, that did and does it? Was it the unquenchable glimpse I closed my eyes and imagined I got of what lay under her dress, the bulge of hair beneath cotton or silk, such fit I felt exhorted by, rich beard and lift and betweenness, that did and does it? Was it that the harmonium seemed as much incense holder as axe, the ecumenical seep and support it disbursed as much musk as music, the very floataway musk Djamilaa's nightie dilated Dredj's nostrils with? Was it that I could've sworn I sniffed it wafting from under her dress, a newly mixed Vedic neroli, an infusive attar, an infinitely penetrant perfume? Was it this that did and does it?\n\nIt was all these things. Djamilaa is my muse and will always be, the someone I need and will need on my bond, as the old song says. She led and leads me thru love's long-tenured bazaar, love's late arcade. Whatever probity, whatever duly theophobic tread, theophanic tread, I acquitted myself with (and my solo did, I know, do nothing if not that), I owe to her inspiration. The Nine Golden Precepts themselves, I'm sure, were the work of her inspiration. Even now, going on five in the morning, I feel I can't but be longer with it.\n\nThe Golden Precepts readied my way. Such bearings as they gave me gotten, I left off temporizing with an annunciative, almost airless flutter, a fledgling, asthmatic burst whose asthmaticity rhetoricially asked how to speak of things of which one does not speak. Aunt Nancy, Drennette and Djamilaa knew this to be rhetorical, understood it as a preamble to doing exactly that. We would indeed speak the unspeakable Drennette affirmed with a run of karate chops to the tablas, shunting my almost airless flutter along. Aunt Nancy and Djamilaa were likewise all horizontality, Aunt Nancy with a series of tonic-tending bowswipes, Djamilaa with a sirening ride of the harmonium's high end.\n\nThat played or that said, I told myself, \"Take your time.\" I let my embouchure go loose as I played the first seven notes of the head, cutting it off as if interrupted by a better option than mere completion, had at or put upon by some lateral enchantment. Aunt Nancy's bow was that enchantment, as commanding as Cupid's arrow, picking up and repeating the first five of those notes, identification, if not identity, up to more than identification. Djamilaa meanwhile settled into a low-lying, mist-on-the-moors creep, a droning amble ever close to the floor, the stage floor it made feel earthen, rolling earth. Drennette had increasing recourse to the heels of her hands on the tablas, coaxing a fat sound out of them, all reach and rotundity, itself a rolling aspect as well, fat wheels we rode.\n\nMy loose embouchure caught air and the vibrating reed was a drill or a jackhammer against my teeth, bodily abidance's dues, I meant to insist or insinuate, nothing more. Djamilaa, who knew my mouth as no one knew my mouth and whose recondite musk had my nose open, at once caught my insinuation. She answered with a skein of sound, a ribbon of sound, still close to the earthen floor, bodily abidance's reap or condolence. She pulled it from the harmonium's bass register, a grumbling, organlike run Alice Coltrane would've been proud to call her own.\n\nThe Nine Precepts ushered me along, Djamilaa's undulance under cloth an abiding bond and trust, rolling bulge, rolling fit, rolling traipse I felt furthered by, a whiff-quickened wraith of myself chasing myself. Robert Johnson's hellhounds were a walk in the park up against the chase I gave myself, a counterintuitive, slow-tempo chase I now tightened my embouchure to ante up on\u2014eked-out advance, eked-out inveterate lag, eked-out inconsequence.\n\nI took my time. My sound opened up, unpinched, not the zero degree at which the horn extends the esophagus without seam or serration, the sound Oliver Nelson, for example, gets on soprano, but less given to the pulverous fray around its edges I started with. Dust off a moth's wing was there to be heard even so, but less of it, my sound as open-throated as it gets. It was cool, collected, not entirely without strain but backing away from it, pointedly announcing a parting of ways with it. I thought of when I was a kid and of my mother's friend Mary who'd always say, when things were getting to her, \"I can't be strainin'-up here,\" which was always exactly what she was doing. Everyone called her Strainin'-up Mary. Strainin'-up soprano was the horn I blew.\n\nThings were getting to me. The coven or the cave I'd been inducted into or stepped into played briar patch to my Brer Rabbit, no place I more wanted to be. Announcedly not strainin'-up, I remained calm and collected, my backing away from strain belied by moth-wing dust though it was. Even as things got to me in what was at bottom a good way, my response above bottom was nothing if not mixed. Cool and collected as I was, I played scared, wanted to be scared and grew scared of the very place I wanted to be, feet shod, so to speak, in theophobic, theogonic tread. Djamilaa, Drennette and Aunt Nancy, my three witches, were divinatory and divine in laying down, ladies though they were, the brer patch it was now incumbent on me to traverse.\n\nI couldn't be strainin'-up but I was ever so detectably strainin'-up, moth-wing dust my boon and my betrayal. Strainin'-up Mary was to me as Trane's Cousin Mary had been to him or as Horace Tapscott's Drunken Mary had been to him, a makeshift Madonna or a makeshift Magdalene or a third, entirely makeshift muse for the occasion. Cauldron Mary I wanted to rename her, a fourth, faraway witch in league with the three with whom I played, but Drennette, sensing this it seemed, gave one of the tablas, which might as well have been my head, a resounding slap. I stood by tradition and stayed with \"Strainin'-up.\"\n\nHorace's Mary was more than a passing thought. Strainin'-up Mary was known to have a drink or two or three throughout the day. \"Drunken Mary's\" head seemed exactly the groove to \"Like a Blessed Baby Lamb's\" tongue, so I paired it now with the full first ten notes of the latter's head, a joint or a joining I parsed out using Horace's tipsy waltz time approach. This gave it as much jaunt as our slow tempo could accommodate. We did so without at all speeding up.\n\nI bleated lamblike, reminiscent of Wayne's \"Dindi,\" a connect-the-dots tack with which I stated the two heads, conjugated the two heads, all the while observing the Nine Precepts to the letter. Djamilaa's bellowing hallows put me in churchical stead, though Aunt Nancy's wicked bowswipes and bowsweeps were yet another matter, as were Drennette's crescendoing tabla slaps, not to mention a certain way in which I scared myself. My tipsy traverse of the brer patch was, by turns and at times concurrently, god-fearing, goddess-fearing, witch-fearing, propriophobic, whichever as the case might be.\n\nStrainin'-up Mary showed me the way. She magnified and illumined the way the Nine Precepts had made ready. I staggered, bounding laterally and at times diagonally between lamblike bleat and capric slur, Djamilaa's goatlike beauty a heady brew aligned with Strainin'-up's unsecured walk. I stole a peek at her seated at the harmonium. She shot me a grin. Undulance under cloth, I couldn't help noticing, had nowhere near subsided. \"Waft be thy name,\" I hummed into the horn, a recourse to heteroglossic traipse Djamilaa met with harmolodic tryst, sounding organlike again, Larry Young meets Ornette.\n\nDrennette was all fingertips now, digital dispatch and acrobatic display, a boon to my every capric slur. I walked sonically cool and collected even so, incongruous capric aplomb. Had I been walking nonsonically I'd have turned sideways and dragged a leg, harked back to a dance we did when I was a kid called The Stroll, a dance that danced us as much as we danced it, a dance I've never been able to shake. Thus the slur, the slide away from collectedness, its merger with incongruous aplomb.\n\nAunt Nancy picked up on Drennette's fingertip attack and went from arco to pizzicato, from swipe and sweep to pinch and pluck. Djamilaa picked up as well, now dispensing staccato runs as if at the wheel of a car, pumping the brakes. The ground we crossed was all the more a brer patch now, bristling with divinatory aspect and pop, divine, prestidigitator snap.\n\nBrer patch was hopping ground, a dense arena rocked by ricochet, detonation, ignition. Vex and revisitation ran side to side, to and fro. I lowered the horn and pointed its bell at the floor, bent over the way I've seen Miles bend over, listening for a certain sound. I was still all lamblike bleat and capric slur, at the brer patch's mercy it seemed I was told later, coaxed, baited and beset by prestidigitator bristle, pinch and pop. I wanted a hollowed-out sound if not a hallowed or haloed sound, moth-wing dust mixed and congealed with ambient humidity, for all its airborne humors a conical or cylindrical wrap of paste.\n\nA conical or cylindrical compress applied to the open wound the air itself now was, the sound I wanted was buoyed by Aunt Nancy, Djamilaa and Drennette's brer patch rotunda, churchical girth I tried in vain, straining, to get my arms around. Churchical girth would not be gotten around, not be embraced but at a distance, an ever so discreet hydraulics of approach it demanded, mandated. The sound I wanted was the sound I went down and got, bent over, horn pointed at the floor, an expectorant howl, no longer cool, collected, an expectorant croon that brought up a shake-the-rafters descent into the horn's low register, a rafters-rattling landing on the horn's lowest note.\n\nThe sound I wanted, not quite knowing I wanted it, was the sound of the shaken rafters, the rafters rattling. I now saw what I wanted, that this was the sound I wanted, the highest rafters of a wooden palace rattled by the note I hit and held, wood rubbed to a sheen by oneiric return and revisitation, another strangely familiar house I dreamt I was in. I didn't open my eyes and I didn't need to. I knew we were in that house, that palace. I kept holding the note (holding on, as Bobby Womack would say), doing so with circular breathing. Aunt Nancy, Djamilaa and Drennette kept their plucked and popping rotunda alive and played louder now. I knew we were in that house, that place, that palace.\n\nThe audience had begun loudly applauding as I held the note and the rotunda popped and bristled, Pharoah's \"Let Us Go into the House of the Lord\" having nothing on our stringent, less unctuous approach. I didn't open my eyes and I didn't let go of the note as Lambert and Penguin came back in, restating the head over my sustained note and the brer patch rotunda, only for us all to stop on a dime at head's end. We had built.\n\nAs ever,  \nN.\n3.V.84\n\nDear Angel of Dust,\n\nDrennette called Penguin on the phone last night about an hour after rehearsal. Balloons had followed her home again, she told him, just like back in November, stowed away in her drumset. After getting home she went to unpack and set up the drumset to practice some more, she said, and, just like before, balloons emerged when she lifted the lid to the parade snare's container. They floated out of the container before she could reach in to take the drum out, rising up from under the lifted lid just like before, as if out of the container's mouth. She sounded upset, Penguin said, telling Lambert and me the story this afternoon.\n\nDrennette had called Aunt Nancy and then Djamilaa but neither answered. She then called Penguin, she explained, shaken by the balloons having shown up again and needing to talk with someone. What made their appearance this time especially creepy, she went on, was that they didn't disappear into the air after a while the way the balloons usually do but hung there, above the parade snare's container, instead. They had emerged one by one, the first bearing these words: _I tread lightly, ever so lightly, in my approach, feet an awkwardness I'd be done with i_ _f_ _I were able, so badly I want mysel_ _f_ _unsoiled and off the ground, free o_ _f_ _the ground and o_ _f_ _contact with the ground. All for you, I'd be entirely o_ _f_ _the air, all for you, treading lightly, light itself. Footless were it mine to decide, I come to your window, a ripple in the breeze, tread o_ _f_ _wind at your balcony, you my belov\u00e9d._\n\nThe first balloon hung there, Drennette told Penguin, moving to the side to make room for the second when it floated up but still not going away. The second bore these words: _Not to mention heat. Doesn't it go without saying that heat brings me and I bring it, light not lacking heat and heat not lacking light, a harmattan rippling the entirety o_ _f_ _air I'd aptly be? I come to your balcony bearing heat, a nuzzling breeze at your neck. I carry your perfume into the surrounding air, belov\u00e9d broadcast, happy to have in the least lifted it, the subtlest waft I'd be._ The second balloon took its place next to the first, hanging there above the parade snare's container, hanging there still, Drennette explained to Penguin, even as she spoke with him on the phone, as did the third.\n\nThe second balloon had moved over to make room for the third when it floated up out of the container. The third bore these words: _All aspect and approach I'd be, beauteous tread not quite arriving. An awayness infinitesimal but palpably felt I'd be, could airy entirety be said to know touch. So it is I come to speak o_ _f_ _touch, prime among the unmentionables, gone without saying or scared away by saying, so it is I come not to speak o_ _f_ _touch. I pledge fealty to that which remains unshown or, shown, unspoken-of, unmentionability's own, liege and lief. No matter what I mention, more remains unmentioned._ It took its place and hung there, Drennette told Penguin, the last to emerge she concluded after waiting a while and seeing no fourth balloon float up.\n\nPenguin had a kind of laid-back ease and assurance that's not quite like him as he spoke to me and Lambert. \"I could tell from her voice she was upset,\" he said. \"She asked, 'Don't you think that's weird?' to which I replied, 'Yes, very weird.' Saying yes appeared to bond us in some way, as though I'd uttered a secret password for admission to a guild or a lodge. She spoke more softly now, almost secretively, her voice a little lower and a little huskier than usual, her mouth closer to the receiver's mouthpiece it seemed. 'No telling what could happen,' she said, almost whispering.' I answered, 'Yes, no telling what could happen.'\"\n\nDrennette again made the point, Penguin said, that none of the balloons had gone away, that the three still hung in the air above the parade snare's container, hung there as though staring at her she went so far as to say. \"'It's eerie,' she said,\" he said. \"'I've tried closing my eyes and opening them again, thinking maybe they'd be gone when I opened them, hoping they'd be gone when I opened them, but they were still there. I've tried leaving the room, going to another room for a while and then coming back, hoping they'd be gone when I came back, but they were still there when I came back.'\"\n\nPenguin went from laid-back to bursting when he got to this point, bursting to tell us what she said next and what happened, much more his uncontainable self. Her voice was still low and intimate, he said, he and she still members of whatever guild or lodge or secret society he'd been admitted into. We were at Lambert's place, drinking beer. Lambert interrupted to ask if Penguin and I would like another, to which we both said yes, and when he came back from the kitchen with three fresh beers Penguin continued.\n\n\"'They stare at me like stalkers, these three,' she whispered into the phone, her voice husky but vulnerable too,\" Penguin told us. \"Like I said before, I could tell she was upset, though it's not like her to show that kind of thing and she tried to make herself sound nonplussed and merely observational even when saying things like 'No telling what could happen' and 'It's eerie.'\" He fell silent, took a sip of his beer and gazed out Lambert's living room window, taking in the scene outside, the tops of palm trees and the sky mainly, as though the key to what happened next lay somewhere in the clouds. Lambert and I held our tongues.\n\n\"Then suddenly, after she said that, the thing about the balloons looking at her like stalkers, she paused a moment before saying, in a tone and with an inflection I've never heard her use before, her mouth even closer to the phone's mouthpiece it seemed, 'That's why I called you.' After saying that, she paused again, ever so slightly, then asked, her voice not as low now, girlish, 'Can you come over?'\"\n\nPenguin fell silent again and gazed out the living room window again, again as though looking for an answer in the clouds, taking one sip from his beer and then another and then another. Lambert and I held our tongues again but Penguin's silence lasted longer this time. Finally, when it looked like he'd never speak again, I asked, \"Then what happened?\"\n\n\"I went over, of course,\" Penguin answered, immediately falling silent again.\n\nThe silence hung there for a while and then Lambert asked, \"And then what happened?\"\n\n\"I'm not the kiss-and-tell type,\" was all Penguin would say, gazing out the living room window at the tops of palm trees and the sky.\n\nWe drank more beer as the afternoon wore on, a lot more beer, and all three of us got pretty loose. No matter how loose he got and no matter how much Lambert and I pressed him, Penguin would only say about what went on after he went over to Drennette's, \"I'm not the kiss-and-tell type.\"\n\nYours,  \nN.\n6.V.84\n\nDear Angel of Dust,\n\nPenguin and Drennette haven't shown any sign that anything happened. Since him telling me and Lambert about her calling him up and asking him over we've been looking to see if there's anything new in their way with each other. I can't say that anything has changed. They still go about their business as before. Last night at rehearsal Lambert even called up \"Drennethology,\" thinking maybe there'd be some blush or some other such telling sign or perhaps a more passionate approach to the piece by him or by her or by both, a new emotion brought to it. No such difference. They were nothing if not businesslike, cool technicians to the point I thought their coolness might be a sign that something had happened. \"Maybe what happened is that nothing happened,\" Lambert said later as he and I talked after rehearsal. \"Or maybe they're trying to hide the fact that something did,\" I said right away.\n\nTime will tell. In the meantime, I'm still wondering how to make good on Dredj's \"Copacetic Syncope,\" struggling with how to make good on it. The squint I was beset by during a cowrie shell attack way back comes back of late, conducing to a Rasta koan or equation, faraway French horn filiating Far-Eye trombone. Processionality's call summons a fetch and a furthering, a fromness (Rico, _Man from Wareika_ ) and a forwarding (Rico, _That Man Is Forward_ ) so-near-so-far reticence gets or doesn't get but won't gainsay, all the sway of gathered cloth folding it in. I know what Dredj means by so-near-so-far, which leaves all the more to be said for so-far-so-far. Squint wants to sing something we see best at a distance. I squint and I can barely see, blown particulates' blind beneficiary, _musk_ 's near rhyme with _dust_ the refrain I'd ply, caroling dark but chorusing light.\n\nIt's not that I've decided to add a trombone, much less that I've decided on reggae telemetrics, much less the hurry-up sound of ska. I'd like that sense of going somewhere but I'm not sure I trust it. I like the diffuse, frustrated reflectivity brass imposes on Far-Eye reach, more audible in the strainin'-up sound of French horn than on trombone but a shared come-so-far-to-say-it sonic inheritance nonetheless. Brass familiarity, brass familiality, is maybe all I'm getting at, a strained familiarity or a strained familiality announcing exodus or exile in the case of either horn. It was, after all, \"I can't be strainin'-up _here_ \" Strainin'-up Mary used to say.\n\nAs ever,  \nN.\n11.V.84\n\nDear Angel of Dust,\n\n\"It's not,\" Penguin was saying, \"that I don't like a woman whose eyes take you in and take you deep, so deep you think you'll drown, you'll never get out. I'm not saying sparkle that far down in her eyes doesn't dazzle, dark though the mascara might make it, a highlight heteronymically come to the fore.\" The three of us\u2014him, Lambert and I\u2014were in Lambert's car, heading to Dem Bones, a barbecue place in Westwood we like. Lambert and I were in front. Penguin sat in the back seat. We'd been going along quietly when out of the blue he began talking. Lambert kept his eyes on the road and I kept looking straight ahead as well. \"It's not,\" Penguin continued, \"that I don't like a woman the shape of whose nose is nothing if not divinatory signage, the harbinger of heavenly form, that I don't like a woman whose lips are full and whose mouth pouty, that I don't like her mouth brimming with teeth and gums, protruding toward a kiss it forever invites or anticipates.\"\n\n\"Who said it was?\" was on the tip of my tongue but I let it stay there, sensing Penguin was in a certain space we shouldn't intrude on. Lambert appeared to think likewise. We both kept quiet and kept our eyes on the road.\n\n\"I'm not saying there were no balloons there,\" he continued, \"that she lured me there under false pretenses. It's not that it wasn't exactly the way she said on the phone, not that the three balloons weren't still there, hovering above the parade snare's container, inscribed with exactly the words she had told me, read to me, over the phone. I'm not saying I didn't blink my eyes and then close my eyes for a while, as she had, to see if they'd go away, not saying they went away and were not still there when I opened my eyes again, just the way she had said it had been with her. No, I'm not saying that. That's not what I'm saying. No way was it that the wine she offered wasn't good, great in fact, exactly my favorite white, sauvignon blanc, chilled just right to go with the oysters on the half shell she also offered.\" He paused, savoring the memory of the oysters and wine it seemed to me. Lambert and I remained silent. I looked out the window at a particularly nice row of palm trees, lost in that somewhat until noticing Penguin was speaking again.\n\n\"What I'm saying isn't that I don't like the kind of incense she likes,\" he was saying, \"those green neroli sticks the Vedanta Society sells, the ones that have a kind of funkiness to them, a kind of bootiliness or bootiness to them. I don't mean that the smell of that smoke, mixed with that of the scented candles she had burning, the smell of perfume on her neck and a very slight waft of sweat, didn't almost take me out. No, I'm not saying that.\" He paused before continuing, \"It's not that I don't like standing pelvis to pelvis with arms around each other, that thrusting and grinding pelvis to pelvis isn't for me. I don't mean to say I don't like the feel and taste of her tongue, her tongue feeling and tasting my tongue. No, I'm not saying there was anything not to like about her tongue groping and my tongue groping back, anything wrong with the kiss getting deep and slow and a little bit sloppy. I'm not saying that.\" He paused again, caught up in his own thoughts it felt like, gazing blankly out the window I would've sworn. Lambert and I continued to hold our tongues, continued to keep our eyes straight ahead.\n\n\"I'm not saying,\" Penguin said when he resumed speaking, \"that I don't welcome the rotundity of pendant hips in the palms of my hands, that I'm averse to holding her with one of my legs between hers and one of hers between mine. It's not that a pinched waist and a generous, low-hanging ass don't speak to me. No, that's not what I'm saying. No, I'm not saying that. It's not that I'm immune to her nipples hardening against my chest, that her nipple pressed hard against my tongue, stiff between my teeth and lips, doesn't get to me. That I don't like her teeth and tongue on my earlobe and her low, husky voice issuing coarse demands and crude encouragement, whispering sweet endearments as well, is most definitely not what I'm saying. I'm also not saying I don't like the thick hair between her legs or the feel of it against my thigh, that I don't like the wetness of what's underneath it wetting my thigh.\" He paused.\n\nLambert and I continued to think it was best not to speak or to turn our heads and look at him. Lambert kept his eyes on the road and I mostly did so too, though I did turn my head and look out the window from time to time. We were coming upon a Carpeteria store on our right as Penguin left off speaking, the company's trademark Aladdin's lamp\u2013style genie looming twenty feet tall atop the store. Grinning, turbanned, holding a large roll of carpet above its head, it never fails to catch my eye and it did so again this time. I turned my head and stared up at it as we passed.\n\nIt felt like Penguin was gathering his thoughts, deeper in thought. When he resumed speaking he began by saying, \"It's not that I don't like a woman who likes to be gone down on.\" When Lambert and I heard this we both had the same thought. Lambert kept his eyes on the road but I turned my head to look at Penguin as the two of us said at the same time, \"Careful, man. You're telling us way more than we need to know.\" I left it at that but Lambert went on, \"Don't tell us something you're gonna regret telling us.\"\n\nPenguin's eyes were on the road ahead and did not meet mine. As far as I could tell, he hadn't so much as given the genie a glance. He had an intent look on his face, as though he spoke not on a need-to-know but a need-to-tell basis. \"It's not,\" he went back to what he'd been saying before we broke in, \"that I don't like a woman who likes to be gone down on from behind, her ass cleft recruiting one's nose as one's tongue slips between the lips between her legs. I'm not saying I mind her turning over after a while and lying on her back, pulling me up between her legs and on top of her, reaching down and sliding me in. No, I'm not saying that.\" This was a far cry from \"I'm not the kiss-and-tell type,\" so far from it he might as well have been blowing a horn, blowing the oboe, the high boy, the high would, piping for all heaven, earth and hell to hear, no matter he spoke softly, calmly, self-possessed, giving an all too graphic account of what had gone on between him and Drennette. Had balloons begun to emerge from his mouth I wouldn't have been surprised.\n\nI turned my head and looked forward again, went back to looking straight ahead. There was no stopping him I could see. I looked out the window. We were on Santa Monica now, the street Dem Bones is on. \"It's not that I mind being told it feels good and to keep doing it,\" he continued, \"that I've got a big one, a really hard one, a really stiff one and I know how to use it, that she likes the way it fills her, that between her legs is where I belong. No, it's not that. I'm not denying the word 'big,' the word 'hard' and the word 'stiff' never sounded so good, that they never wielded more magic.\" No, there was no stopping him. Words, it was clear, were his high horn, theirs the horn's piercing, penetrant cry. \"It's not that I don't care for her gripping my hips while I'm between her legs,\" he was now saying, \"spreading my cheeks to finger my ass before bringing her finger up to my nose for me to sniff. No, it's absolutely not that. I'm not saying I've got anything against her asking, while doing that, 'Why are we so smelly?' in her you've-been-a-bad-boy voice. I'm not saying that at all.\"\n\nLet me be clear that there was nothing the least bit lascivious or salacious about the way Penguin spoke. He spoke with absolute sobriety and equanimity, not exactly dispassionately but collectedly, even as the events of which he spoke built, crescendoed and peaked. \"It's not,\" he said, \"that I don't like it when her body tightens and it all begins to build to the release we've been moving toward, that I don't like the way her body spasms, bucks and quakes while she tightens her embrace of me and I tighten mine of her, that I've got anything against the feeling of something from deep down and inside being drawn out of me, some inmost extenuation, my very self maybe, not that I'd rather she not moan and, shortening my name to 'Pen,' quietly call it while this goes on.\"\n\nPenguin didn't stop there. He paused but he didn't stop. He became all the more sober, somber even, certainly serious, meditative, solemn I could feel. I took my eyes from the road and turned and looked at him there in the back seat. As he began to speak again he tilted his head like a bird, looking askance at some new conundrum it seemed, aggrieved and quizzical at the world's illogic. \"What it is,\" he said, \"is that the Drennette all this happened with wasn't Drennette, that it happening, necessary no doubt, was insufficient. The Drennette I did all this with and who did all this with me wasn't her. The kissing and the caressing, the whispers and the getting wet, the wafting and the thrusting and the squeezing, the muscle contractions and the getting wetter yet were not the utmost her I wanted, the utmost her I still want, the inmost her it perhaps is I long for and that I still long for.\"\n\nHe fell silent. His eyes hadn't met mine when I turned to look at him and they still didn't meet mine. Had he been talking to us or to himself I wondered and so said nothing. Lambert, however, eyes ahead and on the road still, cleared his throat and spoke up, speaking from hard experience it appeared, delivering a hard-won truth. \"Beauty does that,\" he said, \"especially outer and inner beauty combined, exactly the sort it is Drennette has. It makes us see things. It makes us see what can't be touched. It hawks the intangible. No matter how material, no matter how palpable it seems, no matter the low-buttocked angel a rung or two above us leading the way it appears to be, it can't be touched. Tread lightly. Don't go thinking you can grab it, have it, hold it. You can look but ultimately you can't touch.\" It was a sobering thought.\n\nPenguin, less rather than more sober on the heels of Lambert's remarks, spoke with barely damped passion. \"But she was right there,\" he said in a sotto voce cry, \"me pressed up against her, her pressed up against me, there but not there, me in her arms and she in mine but not there, not the\"\u2014he paused, chasing a thought, reaching for a word\u2014\"Drennethological Drennette I so wanted and still want. I was surrounded by water but thirsty, thirsty in the middle of the sea. Beauty was indeed water spiked with salt, water I could look at, not drink. Proximity, tangibility, was the bait on beauty's hook.\"\n\nPenguin fell silent again, lost in thought it seemed, at a loss as to where what he'd said left him. He came out of it quickly and at last his eyes met mine. He turned his head and looked toward Lambert, whose attention was on pulling into an empty parking space that, luckily, was almost directly in front of Dem Bones, just a few spaces farther up from it. It was clear to me that, no matter whether Penguin had been talking to himself or to us earlier, it was to us he'd been talking this last little while and that from us he wanted a response. Having looked toward Lambert, he looked back at me, but I myself was at a loss, unsure how to take up with what I was still afraid he'd later regret having told us.\n\nMet by my silence, Penguin looked back toward Lambert, who by this time had pulled into the parking space, backed up a little, gone forward again straightening out the car, backed up again a bit, turned off the engine and begun to open his door. He looked back over his shoulder at both of us and said, \"We're here. I'm hungry. Let's go eat.\"\n\nYours,  \nN.\n15.V.84\n\nDear Angel of Dust,\n\nI don't doubt that what Penguin intimated happened in fact happened. It's not that he made it with an imaginary Drennette, as you say, but exactly the opposite. Wasn't that his complaint, that the Drennette of his dreams, his \"Drennethological Drennette,\" remained unavailable, not to be known corporeally, impervious to bodily embrace? Wasn't his complaint that he didn't\u2014couldn't\u2014make it with the utmost or the inmost her, the very her of her, the her that, ostensible contact notwithstanding, could only be conceptual, only apprehended by the mind or the imagination? Wasn't his complaint that the gulf between carnal capture and conceptual embrace persisted, unbridgeable, driving a thirst that would not and could not be quenched? Didn't he say he's beset by an ethereal buzz or an immaterial vibe with only bodily address to try reaching it with, an address not only necessary but compelled, yet insufficient?\n\nAnyway, to answer your question, no, the conversation didn't go much further in that vein after we got out of the car at Dem Bones. Penguin tried to pursue it but Lambert and I discouraged that by moving on to other things. \"You'll work it out,\" I told him, which was the last we said of it. Penguin, I'm sure, knew to begin with it wasn't something we could help him solve. He was just getting it out, letting it out. He didn't bring it up again.\n\nThe other shoe sort of dropped at rehearsal earlier tonight. Drennette, who doesn't often compose, brought in a piece called \"Lapsarian Surfeit,\" a piece she'd written, she told us, just in the last few days. Lambert and I looked at each other, wondering was the title, \"fallen excess\" put otherwise, a reference to what Penguin told us had gone on. Drennette, to a certain reading at least, was backing away from what had happened, parallel to but different from Penguin backing away, if that's what it was. I wasn't sure, however, that's what it was, in either case. I wasn't sure falling short of a metaphysical wish is a backing away, that a built-in default can be called backing away. The case with Drennette, I thought at first, seems more a matter of moralistic recoil but it soon occurred to me that \"lapsarian\" might refer to such a falling short, not necessarily to a fall or to the Fall. Where that left matters I wasn't sure except to say that the title was anything but self-evident and that even \"surfeit\" might be defying its face value to say something less about excess per se than about incongruity or incommensurability, a lack of adequation or a lack of congruency that is, to put it colloquially, \"too much.\"\n\nSo I say sort of dropped as I might also say sort of another shoe. It wasn't clear what to make of the title, jump to an easy conclusion though Lambert and I were guilty of doing. Neither coming in with a composition nor the title of the composition seemed particularly loaded for Drennette. She was her usual self, as was Penguin. Only when she told us the instrumentation did anything happen that might've had implications. This was that when she said the piece called for Penguin to play baritone Penguin thought a while and then asked could he play oboe instead. Lambert and I looked at each other again, thinking the same thought I'm sure. Were we reading too much into it to wonder whether Penguin felt a judgment implied by his assignment to the low-pitched horn, that he too felt the piece might have to do with what had lately transpired between them, that he sought to reverse that judgment as well as reverse the fall or the falling short they'd undergone by way of recourse to the higher-pitched horn? Perhaps so, but when Drennette stood firm and reiterated that the piece called for him to play bari he put up no further resistance. He let the matter drop and said okay, he'd play bari.\n\nThings went uneventfully otherwise. We got the hang of \"Lapsarian Surfeit\" pretty quickly and we ran thru it several times. Penguin delivered a full-throated sound that betrayed not the slightest hint of reticence or misgiving. He brought Charles Davis's playing on \"Half and Half \" on Elvin Jones and Jimmy Garrison's album _Illumination_ to mind, both titles possibly relevant and obliquely allusive to his and Drennette's recent tryst I couldn't help thinking, not to mention the circular, running-in-place, treadmill motif \"Lapsarian Surfeit\" shares with \"Half and Half.\" I may have been reading too much into this I'll admit, but you can see, I'm sure, why Lambert and I wondered might more be going on than met the eye.\n\n\"Uneventfully\" overstates it perhaps. There were moments during Penguin's solos when he dug into that deep, guttural, expectorant sound Fred Jackson gets on Big John Patton's \"The Way I Feel.\" He thrashed and cursed and spat, a study in frustration, a beast caught in a tarpit, a Sisyphean or Tantalean ordeal one couldn't help feeling had implications beyond the business at hand. That Drennette, during these moments, upped the ante, veritably bashing and more loudly attacking the drums and the cymbals, driving, goading, testing and taunting Penguin, did nothing to negate that feeling. They were moments worthy of Elvin and Trane.\n\nWhatever the source or the array of sources, \"Lapsarian Surfeit\" adds a gruff, gravelly page to our book.\n\nAs ever,  \nN.\n18.V.84\n\nDear Angel of Dust,\n\nI saw what seemed an immense crystal encasing the extent of the known world. \"See how far it stretches,\" a demure voice in back of me advised. \"See how much there is of it, the it you'll have traversed always.\" I wondered why \"always\" and how that would work and was it true, wondered had \"Some Other Sunday\" found me, wondered was I its muse or it mine. I wondered about the sound of the future perfect. It seemed it offered a child's-eye view of it all, magisterial in the sweep and the eventuality it afforded, the world it made one feel one could afford. \"Some Other Sunday's\" day had come at long last, the song's day and the song's awaited day at last begun. \"Some Other Sunday\" I knew the title would be, \"Another Sunday\" not even close.\n\n_Prospect_ and _promontory_ were two words that crowded my thoughts, an alliterative consort I heard caroling long and wide. What was it that issued from the ground, I wondered and wished, whatever its contour, what if not a chorusing reward all hope abounded with, hope though it might be, all the sayers concurred, against itself? I was no sayer or I sought not to be at least, say given to what \"Some Other Sunday\" held in abeyance, on its way toward me if not yet there. A tree trunk's uplift embroiled us I saw. Sound's reconnoiter was what faces were front for.\n\nAll sense of limit fell away as the initial notes came in. A children's chorus's trill was what vertebrae were, a twinge whose gamut strung light across the tree's timbral recess, tweak attendant on tweak attendant on tweak attendant on tweak, all done automatically, all within skeletal reach.\n\nMy back loosed its bone and it sang, a quiver notes were tipped in, of but athwart the tone world I translated, \"Some Other Sunday's\" ruse its reason, tunefulness's rise and regret.\n\nI sat up straight as my back burned and smoke lay at the top of my neck. This is why we do this I thought, my last thought before \"Some Other Sunday\" was finally there.\n\nAs ever,  \nN.\n20.V.84\n\nDear Angel of Dust,\n\nDjamilaa told me Drennette told her and Aunt Nancy all about it. This was after Drennette, who's not one to write much, showed up last night for the second rehearsal in a row with a new piece she'd written. The title, \"Rick's Retreat,\" raised Lambert's and my eyebrows, albeit Penguin, when she announced it, remained neutral, nonplussed, allowing not the slightest facial expression or any change in body language. Lambert and I couldn't help wondering was this finally the other shoe, truly the shoe \"Lapsarian Surfeit\" only sort of was. Djamilaa and Aunt Nancy weren't struck so much by the title, knowing nothing at that point of Penguin and Drennette's get-together. They took note, Djamilaa says, of the fact that Drennette had brought in a second new composition so close on the heels of the first. Something had to be up, she says, and they wondered what. It was to find out that they suggested the three of them go for a drink after rehearsal, a drink and a little \"girl talk.\"\n\nThey took note as well, no doubt, of the tenor and tone of the piece. Lambert and I certainly did. Title notwithstanding (or, more precisely, title the flipside of what it announces, title inverted, title reversed), it brims with advance. Moreover, it brims with a distinctly military advance, march meter, a decidedly martial thrust in which Drennette's drumming plays a conspicuous part. After opening with a 7\/4 march figure introduced by the drums alone, it settles into a peculiar structure allotted in thirteen-bar segments in which it returns to the march figure throughout, alternating five bars of 7\/4 with eight bars of 4\/4. The piece brings the rhythm section to the forefront, Djamilaa serving up tall, resolute chords, Aunt Nancy gone well beyond walking to indeed be marching on bass. It reminded me a little of Horace Tapscott's \"Lino's Pad,\" which came out on one of the _Live at Lobero_ albums two or three years ago, except the drums are much more out in front than they are there. It has Penguin, Lambert and me on soprano, clarinet and cornet, respectively, the head something of a round or a catch. It has us not so much playing as piping, blowing with a certain fierceness, blowing like Furies, blowing as if to clear the air, blow something away.\n\n\"Rick's retreat,\" it almost goes without saying, seems another way of saying \"Drennette's advance,\" a way of militantly saying it. The bearing on this of what recently went on between her and Penguin could hardly not occur to Lambert and me\u2014or, for that matter, to Penguin as well. Lambert and I wondered was this her way of saying the memory of Rick had been exorcised, that her tryst with Penguin had blown the memory of Rick away, swept it away. It certainly seemed so, an appearance borne out by what Djamilaa told me later.\n\nDjamilaa says they were a little hungry and decided to go to Caf\u00e9 Figaro and that, once there, they talked about this, that and the other before getting to Aunt Nancy asking Drennette what had brought on the burst of writing she's been doing. \"I don't know,\" Drennette said at first, \"it's been like ice melting, a glacier retreating.\" She hesitated a moment but then she wasted no time, as Djamilaa puts it, getting to the heart of the matter. \"I think it all goes back,\" she said, \"to a certain someone we call Penguin. I've lately gotten to know him in a way I never knew him before. I'll go so far as to say I never knew a certain someone we call Penguin at all until now and that a certain someone we call Penguin never knew me.\" It was odd, Djamilaa says, the way she referred to him, but she continued to do so throughout the conversation. She never simply called him Penguin but referred to him as \"a certain someone we call Penguin,\" perhaps even (there was no way of knowing, Djamilaa says, though the way she said it and insisted on repeating it made it seem possible) \"A Certain Someone We Call Penguin.\"\n\nI was struck by how differently she and Penguin have reacted. While Penguin despairs of there being a Drennette he can't come close to, an auratic or a supplemental Drennette that's more than meets the eye and that sensory perception can't reach, a \"Drennethological Drennette\" not even carnal knowledge knows, she exults over them now knowing each other, at last knowing each other, insisting that before such knowledge they didn't know each other and perhaps couldn't have known each other. Did this make her a true romantic or an epistemic sensualist I wondered as Djamilaa told me all of this. Djamilaa says it didn't take much coaxing to get all the details of her and Penguin's \"breakthru night\" (Drennette's words) out of her. They sat huddled at their table, she says, leaning over their food and drinks like grand conspirators, no detail and no particular too intimate for Drennette not only to relate but to relate with great relish, blush somewhat though she did.\n\nI won't say I never knew \"girl talk\" could get as graphic as Djamilaa tells me theirs got, only that it surprises me Drennette engaged in it at all, let alone enjoyed it and, it appears, encouraged it, so stalwart, no-nonsense and all she comes across as. When I mentioned this to Lambert he said, \"I'm not so surprised.\" He then went on to add something I think I understand though I'm not entirely sure. \"It's the reticent ones you have to watch out for,\" he said, \"the needle-in-a-haystack types.\" Anyway, Djamilaa says Drennette did repeat that the recent writing has to do with a kind of thaw, the retreat of a glacier she admits to be Rick or her feelings for Rick, a complicated figure that also had \"glacial retreat\" referring to her feelings toward men after her breakup with Rick, so that the thaw was in part her \"glacial retreat's retreat.\" She made no bones, Djamilaa says, about attributing this to Penguin and their get-together the other night. \"A certain someone we call Penguin,\" she said, changing metaphors, \"completely flushed Rick out of my system.\"\n\nDrennette's kiss-and-tell account, according to Djamilaa, was indeed long on metaphor as well as intimate detail, anatomical detail, graphic action, graphic mise en sc\u00e8ne, a \"rich mix of registers,\" as she puts it, that had them going from smile to giggle to loud laughter more than once, shushing Drennette and telling her to stop. \"We left no door closed. There was no door we didn't enter, no door we didn't go in thru. We left no depth unplumbed,\" Djamilaa says she said at one point, pausing to look them in the eye, the beginnings of a grin on her lips, \"no orifice unexplored,\" whereupon they broke out laughing and ordered another round of drinks.\n\nSo we would seem to have confirmation now, were such needed. This thing is definitely afoot, afloat, going or stopping where, as the Temptations would say, \"nobody knows.\"\n\nYours,  \nN.\n25.VI.84\n\nDear Angel of Dust,\n\nLambert wore a long, heavy overcoat steeped in sound. Sadness was the coat he wore. It was a cloak of sound like Stanley's \"Walk On By,\" the weight of a tear and a warlock's come-on athwart complaint. It was time walking by, time walking on, with which, for the first time, he was reconciled, with which, for the first time, he was okay. Time's bouquet was the bitter beer we all drank it seemed he said, the first and forever church of hardheadedness reconvened. He was there, if not for that, he said, for nothing. It was heavy but light as a tear at the same time.\n\nNow and again he got to a point where his intonation got away from him (or so, at least, it seemed, so, at least, it sounded like), a ragged, frayed edge of sound or a collapsing core of cloth, a hoarse extenuation he could only let have its way. He told us later that out of the blue his affection for Marvin Gaye's \"Stubborn Kind of Fellow\" when he was a teenager came back to him, a recollection of walking to school and hearing it on his transistor radio. It was the part where Marvin's voice breaks and goes hoarse that came back to him he told us, the lines \"Oh, I have kissed a few,\/I tell you, a few have kissed me too,\" the pleading way his voice gives out and recovers, gives out and comes back to itself on the word \"too.\" It wasn't so much his intonation getting away from him as it was it sounding like that, him having it sound like that, him imitating or trying to imitate Marvin's momentary loss of voice, his fleet, pleading hoarseness, the rough tenuity it for a moment falls into. It was nothing if not the stubborn, mind-made-up ministry his hardheaded church was or would eventually be known for.\n\nI took up my tenor and joined him. We were a two-horned bellowing beast, avowed monophysite twins whose affray was love's last recourse. Death was in the house one could hear, neither of us not with its chill on our breath. We spoke with a quake in our voice. I almost immediately contracted Lambert's hoarseness, a rough tenuity I found my intonation, like his, intermittently fell into. A contagious rash it might've been, so raw, so abraded, so sensitive to touch we winced as we played. While I wasn't aware at the time of Lambert's Marvin Gaye reminiscence, I was struck by one of my own, not of Marvin Gaye but of Dionne Warwick, whom Lambert's Turrentine cloak had brought to mind. It wasn't \"Walk On By\" but \"I Say a Little Prayer\" that came back to me, a certain weakness for which I found I couldn't help confessing, a weakness for the perky Burt Bacharach brass that punctuates it. Something very bright and upbeat and optimistic about Bacharach's use of brass, even on a sad song like \"Walk On By,\" where a burnished underglow speaks resolve, has always gotten to me. It seems to arise from or reach out to a utopic horizon, all the more so in a song like \"I Say a Little Prayer,\" bordering on na\u00efve in the bounding felicity it announces having arrived at, reporting prayer where there's exactly no need for it.\n\nThis was the recollection that hit me as Lambert and I bellowed and brayed. It wasn't at all that I tried to get that sound on tenor but that the very impossibility of doing so and the analogous impossibility of the simple felicity and resolve, the simply happy life it celebrated, ever being arrived at goaded and got to me. Cognate with Lambert's hardheaded, mind-made-up abrasion, it occurs to me in retrospect, was my own stubborn lament for what I knew and will always know to have been impossible from the start, damned or doomed from the start, impossibility perhaps the crux of its appeal. I bellowed and brayed and cursed the simple accord Bacharach's perky brass wanted one to believe was available, the damned auratic resolve and felicity it teased one with. I bellowed and brayed and cursed myself for having taken the bait, blew as if to blow away my weakness for what I knew and will always know is only a cartoon kind of happiness, an ad man's happiness, the unlikely accord I simultaneously confessed my sweet tooth for.\n\nBacharach's perky, burnished brass was a dangling carrot I chased and I blew to disavow my chase, a guilty pleasure I damned and disowned my indulgence in. It was an odd way to go about playing, an odd motive, as though reed envied brass almost, an insinuation Lambert would have fumed had he known he blew complicit with, stubborn kind of preachment and\/or mind-made-up appeal in league with perk, albeit a disavowed attraction to perk. The very thought of it induced a lightness, a lightheadedness, that threatened to spirit me away, a floataway loss of weight I was all but lifted by, an infectious balloon blurb or blip I needed, I knew, to find anchorage against.\n\nI found recourse in Aretha's cover of \"I Say a Little Prayer,\" which opportunely popped into my head. I told myself to keep my ear on the prize, the prize being Aretha's deep churchical bite and benediction, the gravitas her low gospel chords, rolling gospel chords, bestow on her version from the very outset\u2014no brass, no bounce, no perk, just piano. Her voice's cutting dip into tenor, bottoming low from the outset, likewise proved an answer to perk, nothing if not the anchor that would steady my way.\n\nAretha's remembered hums and moans kept bubbly lift at bay, champagne brass's floataway auspices held in check. Lambert and I had gotten low and we stayed low, rummaging the horns' low register, grazing black, piquant soil it seemed, the lips of our tenors' bells nuzzling loam. Lambert's tortuous \"too\" enlisted my low-lying \"prayer\" exactly where memory and sentiment met, albeit unbeknown to us at the time. We purveyed a run of rapid-fire, quick-trigger ignition allied with hiccuping rhythmic relay that brought Trane and Pharoah on a piece like \"Leo\" to mind we were told later. That we got there via deeply rooted pop reminiscence unbeknown to each other made it all the more post-expectant, all the more the quantum-qualitative increment it was.\n\nThis was during Lambert's solo on \"Sekhet Aaru Strut\" at Onaje's last night, a solo that at one point hit me in such a way I just jumped in, unexpected, unplanned, we two \"soloing\" at once. Everyone says it was the high point of the night.\n\nI felt a corner of some sort had been turned, some sort of lesson learned, another in a long line of lessons come upon. I felt we learned something very important, very new and very subtle, something so subtle I can't exactly say what.\n\nEnclosed is a tape. Let me know what you think.\n\nAs ever,  \nN.\n29.VI.84\n\nDear Angel of Dust,\n\nThere's a review of _Orphic Bend_ in the latest _Cadence_. We didn't expect it to get any such attention, so this comes as a surprise. It's the first review it's gotten. The reviewer we neither know nor have heard of, but he seems to know the music and he has positive things to say. It's not that there's not the inevitable quibble here and there, but he does hand out a great deal of praise. One would definitely have to call it a good review. Indeed, friends are congratulating us, calling it a _rave_ review. We don't go that far, happy though we were to see it, and even if it were a rave we'd want to draw back, not get too excited.\n\nAs it turned out, we were surprised not only by the review but by our consumption of it, the elation we initially felt upon seeing it, caught out or caught offguard by how happy we were to see it, how greedily, at first at least, we ate it up. It was as though it was this that we put the album out for, as though we recorded it to be reviewed, written about. There was not only the question of why we were so invested in being reviewed and in what the review said but a certain disappointment, both with such investment and with our elation's failure to live up to expectation, simply on its own terms and on the face of it, an expectation that, post-expectant as we'd have been or thought or wished ourselves to be, we didn't know (or simply hadn't admitted) we harbored. But we did, or had, and it was by that expectation that even our elation, on its own terms and on the face of it, was found wanting. Thus it was that elation mixed with or morphed into letdown. Happy wasn't happy enough. And even if it were, we weren't sure putting out a record to have nice things said about it, written about it, was what being a band amounted to.\n\nThe review, then, gave us pause, became the reason or occasion to ask, late to be doing so though it was, why we play. \"It's not about reviews,\" Aunt Nancy emphatically said as we were discussing this at rehearsal. \"It's not even about aboutness. It's not about being-about.\" It was a thought we needed no time to reflect on, no time to digest. We all understood what she meant, all of us in our heart of hearts having long wanted exactly that, only that, to play not for the sake of what could be said but athwart it, play without claim or caption, advancing (if advancing anything) being-in-and-of-itself, self-evidence, hub and horizon rolled into one. We countered claim and caption, coupling or conflating claim and caption, because of the elephant in the room in the review, the reviewer's reference to balloons emerging from his copy of _Orphic Bend_.\n\nIt was odd the way we tiptoed around the balloons at first. The review bringing them up made for mixed feelings, if it didn't indeed bug each of us outright. It was our fear of them upstaging the music again. The reviewer's recourse to them as a self-crediting tack, boasting or bragging it seemed, made matters worse. Critical authority seemed to be at stake, visionary credentials even. Such were the insistence and relish he reported having seen the balloons with. Even so, we were slow to get around to it. Perhaps it was all too obvious, going, as they say, without saying. Perhaps we were loath to admit mention of the balloons bothered us, loath to admit anything in a review, least of all that, wasn't just water on a duck's back.\n\nThough we were slow to get around to the balloons we did get around to them. Not long after Aunt Nancy said it wasn't about aboutness, Lambert ventured an equation of claim with inflation, aboutness with inflation, aboutness with would-be containment, cover. \"Reviews are balloons,\" he summed it up by saying. We laughed, relieved it was out in the open. But once the subject had finally been broached we found we felt no need to belabor it. It was enough to know we all knew it was on the table. We briefly kicked it around and went back to rehearsal.\n\nI'm enclosing a new installment of my antithetical opera, a new after-the-fact lecture\/libretto called \"B'Loon's Blue Skylight.\" I won't say it was inspired by the review but had the review not appeared it wouldn't have been written.\n\nAs ever,  \nN.\nB'LOON'S BLUE SKYLIGHT\n\n_or, The Creaking o_ _f_ _the Word: After-the-Fact Lecture\/Libretto (Djband Version)_\n\nDjband bumped into B'Loon at a newspaper vending box. One of the local weeklies featured a review of Djband's album _Orphic Bend_ , a review whose author took pains to announce that balloons had emerged from his copy of the three-record release, doing so not only during the bass solo on a cut called \"Dream Thief \"\u2014about which, he pointed out, there'd been a good deal of chatter on the underground grapevine\u2014but at a number of other points as well. The reviewer took no small amount of pride alongside the pains he took to make this announcement, as though the balloons' appearance, multiple as it was and occurring at points on the recording not reported by others, bestowed a mark of distinction, made him elect among the elect, confirmed his acuity and taste.\n\nThe newspaper vending box stood on the corner of Melrose and Fairfax, across the street from Fairfax High. Djband, spotting the title _Orphic Bend_ in the subhead of the front-page review, had opened the door to the box, taken out a copy and read the review, looking up dismissively when finished and humphing, \"He thinks it's about him.\"\n\n\"But it _is_ about him,\" an inner voice or an inner B'Loon reminded Djband, \"as much about him as about anyone. Why not? Isn't the music for and about each and every listener, there to have made of it what any set of ears can make of it, there for nothing if not laissez-faire audit? Anything goes.\" The inner voice paused and on deeper reflection allowed, \"Well, no, not anything. But where to draw the line is always an issue.\"\n\nThe review was filled with such wording as \"insofar,\" \"as it were\" and \"as if,\" the language of qualification disqualifying language itself. So it was, the review suggested, the balloons fled language while carrying language, bearing it to more auspicious precincts, Djband's music and any other music, all music. \"Music,\" it said at one point, \"is language in exile, exile exponentially borne\u2014that is, owned up to, lived up to.\" The review went on and on about the balloons, not so much about the reviewer, when it came down to it, as about B'Loon (though the reviewer had no way of knowing the balloons' avatar's name). It waxed alliterative and assonantal regarding \"the balloons' d\u00e9tente between containment and contagion, forfeiture and fortitude,\" as though, in so accenting sound, it sought or asserted its own balloon status, inflating its recourse to sonorous air, \"aeriality,\" sonority's infection or effects. \"Sound,\" it went so far as to say, \"is the deep, not so deep tautology of _is_ , its flipped ipseity,\" obliquely alluding to Oliver Lake's _Life Dance o_ _f_ _Is_.\n\nDjband wasn't sure what to make of such pronouncements. Using language to question language seemed only a roundabout self-regard. \"He thinks it's about language,\" Djband grumped, \"that old chestnut, no more than the balloons by another name. He might as well have called it l'anguish. He thinks it's about the balloons.\" The review got on Djband's nerves, further mixing mixed feelings about the balloons. Could the intersection of two metropolitan avenues be called a house, B'Loon, a mixed blessing, was in the house.\n\nDjband had been out of sorts to begin with, one of its members having awoken from a troubling dream. Aunt Nancy had dreamt an onslaught of Santa Ana winds dried out her skin, leaving her face, neck, legs and arms ashy. When she went to put lotion on, she dreamt, squeezing the tube nuzzling the palm of her hand, rather than a drop of lotion, albeit looking like one, what came out was a maggot. She immediately awoke, shocked by so brusque a reminder of death. She'd gotten up on the wrong side of the bed and been in a funk all day, Djband's other members, having been told the dream, in it with her.\n\nThe lotiondrop maggot continued to spook Djband, a Creaking of the Worm compounding ricketiness with unguent, omen with unctuousness. Djband couldn't help imagining an abruptly desiccated, husklike maggot, a stiff chrysalis rustling in the wind, no longer lotionlike. Knowing it meant skin would lose its luster, flesh be feasted on by worms eventually, Djband wondered what it meant regardless, wondered against knowing, not wanting to know. Knowing but not wanting to know what it came down to, body a balloon of skin with guts inside, \"An offal thought,\" Djband inwardly quipped, at odds with and wanting to make light of the unsettling truth.\n\nAunt Nancy broke away and spoke. \"It's not so much it all redounded to me. I'm not saying that. It's not even it was me it had to do with,\" she said. \"It was anyone's palm, everyone's palm, the lotion came out on. It was anyone's arm, everyone's arm, the worm would eventually eat. It's not that I'm the only one whose head a sword hangs over. No, I'm not saying that. Not even close.\" She then took two, maybe three steps back, blended back in. \"No, not even close,\" Djband agreed.\n\nDjband staggered along what seemed an exhaust wall, automobile and bus fumes attacking eyes, nose, mouth and throat. A white sedan darted in front of a bus in time to make a right turn from Fairfax onto Melrose, black smoke pouring out of its tailpipe as it sped up, black smoke pouring out of the bus's tailpipe as well. \"We'll all die together, choke together,\" Djband announced.\n\nIt was too much, as though, playing Monk, Djband forgot Monk's chuckling grunt, his wry wink. There was none of Monk's extreme right-hand hammering, his making the piano his toy, \"Sweet Georgia Brown\" turned \"Bright Mississippi.\" It was in fact the contrary, as though \"Epistrophe\" had been renamed \"Entropy,\" so doleful the note, so to speak, Djband repeatedly struck.\n\nIt didn't help that the review described \"In Walked Pen\" as \"Monk salad,\" a phrase that, meant as a compliment from all indications though it was, got on Djband's nerves. It deliberately mixed its message Djband couldn't help suspecting, hearing overtones of \"tossed,\" \"thrown together,\" \"hodgepodge.\" At best it was merely clumsy. It came off, in any case, in Djband's reading, as flip, too offhand, too casual, assuming unearned familiarity with Monk and Djband both. \"Why not talk about a worm on a lettuce leaf as well?\" Djband muttered all but inaudibly. \"Why not say, 'A worm nibbled away at the romaine,' make an adage of it?\" This was what Djband would have none of, a balloon of attitude meant to say, \"I'm in the know,\" a balloon inflating itself at \"In Walked Pen's\" expense, Monk's expense, trivializing \"In Walked Pen,\" trivializing Monk. \"'Monk salad' my ass,\" Djband added.\n\nThus it was there was much to be annoyed about. The need for an answering salve or an answering salvo couldn't have been stronger. Neither much appealed to Djband however. A letter to the editor taking issue with the review was anything but the water-off-a-duck's-back aplomb Djband liked to think it was an exemplar of. Salve, at the same time, couldn't help but recall lotion, couldn't help but conjure, in so doing, the lotiondrop maggot, the last thing Djband wanted a reminder of.\n\nDjband turned its head to the right, looked over its right shoulder toward where the sound it suddenly heard seemed to come from. It was a rattling sound, as of bamboo slats knocking against each other. It was the sound of actuality falling short of expectation, the sound of a gap between ideal or imagined reception and actual reception, a discrepant rattle (discrepant rub) the review's nondelivery of ideal audition helped arise and resound.\n\nIt wasn't, Djband insisted, that there's a hearing one's mind's ear hears, a hearing that can only be virtual, a hearing no manifest hearing lives up to. This could be argued and it had often been argued but it wasn't what was going on here or it wasn't, were it at all going on, all that was going on. Indeed, Djband quickly admitted, it was in part what was going on, which was that three sets of hearing obtained: a) the hearing one in the act of composition or performance imagines, b) the hearing that in fact takes place, and c) the distance or disjunction between the two, audibly manifest now in the form of a rattling as of bamboo slats, a veritable Creaking of the Heard.\n\nWas it that all reception, all audition, was flawed, inevitably a fall from the imaginal hearing one thought to hear and one hoped would be heard as one penned a piece or executed a run? Or was the Creaking of the Heard a veritable Creaking of the Herd, reception no inevitable fall but instead the outcome of herded audition, corralled by such would-be pundits as the review's author? These were the thoughts that ran thru Djband's head.\n\nYes, the latter was the case, Djband went on. It wasn't that reception was simply herded however. No, worse than that, it was _hearded_. _Hear_ 's past tense's past tense, _hear_ exponentially past, hearded hearing was multiply removed from the present, someone else's having heard presumed to be one's own. It's bad enough to presume to have heard with one's own ears, Djband reflected, worse to allow someone else to have done so for you.\n\nThe rattling sound made it clear _Orphic Bend_ had been hearded, the review's corral evidently made of bamboo. Clunky wind chimes it occurred to Djband it sounded like, not at all graceful, not subtly insinuative, clumsily intrusive instead. \"Clunk salad,\" Djband muttered under its breath, putting water-off-a-duck's-back aplomb aside for a moment, answering the review in kind.\n\n\"I want a big, bodacious onslaught of sound,\" Aunt Nancy had stepped forward again and was saying, \"sound enough to beat back dream thievery, lotiondrop-maggot sleight of hand, an advent sound.\" She stepped back, blended in again, having had her say. \"Yes, exactly,\" Djband agreed, \"a big, bodacious advent sound.\" The contrast with rattling bamboo couldn't have been more stark.\n\nDefensive, water-off-a-duck's-back aplomb notwithstanding, Djband was an ectoplasmic wall, a stone wall even, petrified by the specter of death. Either way, it was a wall from which its members might occasionally emerge. Up to this point Aunt Nancy was the one who had done so, soloist or soloistic, as though performing a piece rather than standing before a newspaper vending box.\n\nFor the moment time was an ancillary matter, not to be disregarded nonetheless. Tacit statements of tempo implied or insisted that point or presence might be other than outright, recalling something Mingus wrote about a leaky faucet in the liner notes to one of his albums. Djband knew itself to be there more as an aggregate shake or as an aggregate shiver than as corpuscular stump, a street corner symphony of mean provenance and prospect, time's \"will tell\" a window impendence blew thru. It could hear itself no matter the time and the place, time nothing if not a suspended platform by turns made less than it was and made more than it was, a bevy of don't-care notes and a preterite soapbox.\n\nSo Djband stood bunched at the \"will tell\" window, the arthroscopic, worm's-eye glimpse into hearded audition the vending box had become. It heard itself beside the hearded rendition of itself the review purveyed, beside the rattling bamboo that was the gap between the two as well, the latter's lapse or its falling away from the former, along the rickety would-be joint between the two. Hearded audition's noncoincidence with Djband-as-it-heard-itself was only to be expected, Djband reminded itself or consoled itself, even scolded itself, angry at itself to have caught itself out expecting better.\n\nA car's horn caught Djband's attention, a Toyota Corolla cut off by a Jaguar XJ-S changing lanes. The driver pounded the hub of the steering wheel and shook her fist as the Jaguar darted in. Oddly, it blended in perfectly, a staccato garnish to the music Djband would have been playing had it been playing, a defenestrated ragtag pomp.\n\nAll advent flew thru the \"will tell\" window. All admonition stood streetside awaiting it, a cautionary wall Djband did its best to embody or at least evoke, an admonitory shingle if nothing more. Warning both stood and ran, a cathartic _r\u00e9cit_ audition sought to be door to, time's indiscreet relay. Djband had seen it all. Warning stood and would always do so, it said and saw, knowing \"always\" to be the dangerous word it was but daring it, ran and would always run.\n\nDjband pasted a poster on the bare wall it was, the bare shingle, the wall or the shingle it took itself to be. The poster bore After-the-Fact Caveat #1:\n\nTime, perfect or syncopated time, is when a faucet dribbles from a leaky washer. I'm more than sure an adolescent memory can remember how long the intervals were between each collision of our short-lived drip and its crash into an untidy sink's overfilled coffee cup with murky grime of old cream still clinging to the edges or a tidy rust stained enamel sink that the owner of such has given up on the idea that that maintenance man is ever going to change the rhythm beat of his dripping faucet by just doing his job and changing that rotten old rubber washer before time runs out of time.\n\nMusicians partly come into the circle of various blame which encompasses much more than leaky faucets, rotten washers, or critics. Wow! Critics! How did they get here?\n\nI know. It's Freudian. Faucets and old rotten washers. The innocent audiences that are sent in the direction of premature musicians\u2014critics who want to play and some who play and study at music and can only encompass soul-wise and technically about someone else what they themselves can comprehend.\n\nIt was none other than the passage from Mingus's liner notes to _The Black Saint and the Sinner Lady_ that had crossed Djband's mind earlier, could \"earlier,\" down this corridor in which time, faucet leak notwithstanding, was ancillary, be said to matter\u2014no less dangerous, if so, than \"always.\"\n\nShingle more than wall though neither shingle nor wall, it was a sandwich board the poster was pasted on, not a shingle or a wall Djband was but a sandwich board Djband in fact wore, Mingus's admonitory note gracing both boards, back and front, counseling passersby both coming and going. It was a long, detailed message for a sandwich board. Printed in large letters, it barely fit.\n\nDjband had clearly let its water-off-a-duck's-back aplomb fall by the wayside, answering the review with the sandwich board, fighting back, answering \"Monk salad\" with Mingus. It was clearly involved, clearly invested, to the point of uncool even, the ensemblist equivalent of a sandwich man, anything but blas\u00e9, nothing if not caring, standing on the street advertising its message. Djband laughed at itself, realizing that insofar as to bear a message was to be a balloon it had become a balloon, balloon and sandwich man both, both rolled into one. Djband had not only bumped into B'Loon but become B'Loon, pressed and possessed by the spirit of caption and contention, the former manifestly, the latter more implicitly, bearing on which and whose caption fit. The review itself, one saw clearly now, was nothing if not a caption, nothing if not a balloon, nothing if not a message-bearing bauble, the bane of Djband's proprioceptive audition. It happened quickly, in a flash, balloon and sandwich board bound up as one.\n\n\"Myself When I Am Real,\" Djband reminded itself right after it laughed at itself, Mingus's title more to the point than ever before. A pointed mix of aim and arraignment, \"Myself When I Am Real\" said it all, could any five words be said to've said it all.\n\nDjband turned toward where the sound of another car horn came from and saw the driver of a Mazda GLC headed west on Melrose give a thumbs up, in approval of the Mingus quote it was clear. How the driver so quickly read so long a text Djband couldn't say but happily nodded to acknowledge the approbation. The sandwich board was making a difference already.\n\nDjband turned sideways between the boards, lifting its arms, elevating the boards as though they were wings. It was now Lambert who stepped forward and spoke. \"I want a straw to fall and fill up with air and float,\" he said. \"I want float to be what unlikeliness does. Due to itself or in spite of itself, I want that to be what goes on, float's new leaf turned over, float's new reign and regret.\" Djband agreed, echoing, \"Float's new reign and regret.\"\n\n\"Whatever comes up,\" Lambert continued, \"it will beg the question it costs itself\u2014float lure, float intended, float intransigence. Reed a wet stick in my mouth, I want more, flutter-tongue abandon's new almanac, float's lush life begun.\" Djband agreed and took up the tailend of what Lambert said, repeating, \"Float's lush life begun.\"\n\n\"Float nothing if not a barge I'd be borne along on,\" Lambert continued, \"I want each lick to incubate what float would be.\" Djband agreed again and repeated, \"What float would be.\" Lambert stepped back and blended in again.\n\nThe sandwich board had brought an element of reduction in, Djband brought down to the reviewer's level, fighting fire with fire, balloon with balloon. Lambert's invocation of flight or flotation thus arrived right on time, albeit flotation, Djband was well aware, carried a balloon suggestion one could hardly miss. Still, lift and levitation won out over balloon rut, balloon mire, the bone \"float's new leaf \" picked with \"Monk salad\" notwithstanding, that bone the very filled-up, floating straw itself perhaps.\n\nMore no doubt than perhaps, Djband decided, embarrassed, underneath it all, by the sandwich board, needing to make a move. Disambiguating float from what balloons do was that move, float's association with balloons its new regret. Float's new reign was nothing if not a resolve to overcome inflation, nothing if not a bone proffering puncture. It remained a willingness to abide by high jive, high jubilation, a resolve to reside on high even so.\n\nTo say that at exactly that instant Djband smelled roses would be going too far. Roses did come to mind and they did so in a flash, their characteristic perfume no doubt bound up in the thought but not to the point of Djband actually smelling them. L.A. was way more than a stone's throw from Pasadena but Melrose might as well have been Colorado Boulevard on New Year's morning, so large did Lambert's barge now loom, float parting company with balloon.\n\nIt was a visual not an olfactory image, an address of the mind's eye, not the mind's more distant nose. Djband saw Lambert's roses-bedecked barge for an instant, easing down Melrose, Lambert atop it waving to the crowd. Yes, it was New Year's morning, float's new day begun, float's new leaf turned over, float's new reign and regret.\n\nFor only a moment were such premises afoot, parade premises. The moment they arose they subsided. Melrose was back to being itself as on any other day, Djband a sandwich man pacing the sidewalk, a modest parade if it could even be said to be that\u2014no barge, no float, no roses. No sandwich board either, Djband quickly decided, lifting the sandwich board over its head and lowering it to the sidewalk, propping it against the newspaper vending box.\n\nRained-on parade was the theme but Penguin, stepping forward, would have to do with it only ostensibly, obliquely, bending away from it as what he had to say built. \"I want a front-row seat at the Apocalypse,\" he said. \"No,\" he said, quickly correcting himself. \"I meant to say at the Apollo, James Brown on his way up, in his prime.\" \"In his prime,\" Djband chimed in and Penguin went on, \"I want to have lost someone or to sing and scream and shout as though I had. I want shout to mean to run around in a circle, led by immanent splendor's allure.\"\n\n\"Immanent splendor's allure,\" Djband agreed and echoed and Penguin continued, \"I want the rump of the cosmos in front of me, barely up against my nose or a bit farther away perhaps, all but in touching distance, infinitesimally out of reach. I want to call out to it, calling it Regenerate Rose Reborn.\" Djband echoed and agreed again, \"Yes, Regenerate Rose Reborn.\" Nose wide with cosmic whiff, cosmic what-if, Penguin stepped back and again blended in.\n\nBefore Djband could do what it would do next, Drennette stepped forward, cosmic vamp and commanding virgin rolled into one. \"Yes, do remember,\" she agreed and exhorted, \"how the smell of the cosmos's behind pervades all extension, how the smell of cosmic loins penetrates all space.\" \"Penetrates all space,\" Djband thought to echo and agree but quickly thought better of it, remaining silent as Drennette, noticing the withheld echo and agreement but not needing it, continued, \"Do remember how these two smells enter your nose and take hold of your scrotum.\" It was a footnote, a blurb, an outburst, a balloon. Djband withheld echo and agreement again. Drennette stepped back and blended in.\n\nDjband reeled and staggered, all but overcome by wafted cosmicity, up-from-under pitch and posteriority, belt and bouquet. It was a ploy, a feinting play on exhaustion, even so. Drennette, Penguin, Aunt Nancy, Lambert, Djamilaa and N. were each only a face on the wall Djband was, the wall Djband affected it was, the exhaust wall it earlier staggered along but now steadied and took inside itself. Immured against hearded audition, a wall against rained-on parade, Djband took a stand and stood tall against critical caption, the review's upstart balloon, cartoon acuity.\n\nWhat Djband would have done next had briefly been put on hold by Drennette's impromptu boast (which is what, underneath it all but not so underneath after all, it all was). Reeling and staggering standing tall, it did now what it would have done next. It issued a collective, composite swipe of sound aimed at wiping the slate clean, a return to pre-caption premises, an airy gesturality or gist it wanted to say was what life itself is, an airy gesturality or gist gotten or gotten at by nothing quite like music, albeit to say so, to go from wanting to say, was to tie up with tar-baby balloon, tar-baby boast, as though Drennette had simply jumped the gun.\n\nDrennette had in fact jumped the gun. That there was a gun to jump tempered Djband's recoil from cosmic waft. Reeling and staggering standing tall as though she'd held a finger out to be sniffed, a finger it knew underneath it all was coming, Djband issued a funky-butt, low-register burst, Mingusesque, a second swipe of sound, going the other way. This was also, it seemed it wanted to say, what life itself is.\n\nSuch expounding upon life, oblique though it was, attracted a crowd. Passersby stopped and looked on. They stood a short distance away, staring at Djband, able to read its thoughts evidently. They heard the music Djband inwardly rehearsed evidently, nodding their heads, popping their fingers, patting their feet. Yes, this was life, they seemed to agree, the what-is of it.\n\nDjband knew there was no wiping the slate clean but made as if to do so anyway. Accretion was all, it knew, whatever would-be cleansing wipe a further murk or mucking up, palimpsestic supplement, palimpsestic struff. In this case, funky-butt struff spoke directly to the claim of a \"d\u00e9tente between containment and contagion\" the review advanced, agreeing with it only to complicate or contaminate it, wipe running one with swipe in more ways than one. Palimpsestic add-on plied boast on boast, waft on waft, whiff, what-is and what-if rolled into one.\n\nPalimpsestic struff was nothing if not infectious. Several of the onlookers who'd gathered began to dance, squatting low to the ground at points, letting their asses graze the sidewalk. Reveling in rump cosmicity, they delighted, they let it be known, in having asses, delighted that there were asses to be had. Close to declaring ass what-is's what-if, they drew short of that, lifting skyward from the squat's low point with a pelvic thrust, saying something like what they begged off saying. Lee Morgan's \"The Rumproller\" had nothing on what they did or on the music they heard or thought they heard Djband rehearse.\n\n\"I want not to have seen it all,\" Djamilaa stepped forward and said. \"I want not to have seen this movie before.\" \"Not to have seen this movie before,\" Djband agreed and repeated, part antiphonal add-on, part set-aside. Djamilaa paused.\n\n\"I want,\" Djamilaa went on after pausing, \"the clean slate I know we can't have. I want the meat of our being here truly met, true meet's tally, no mere funky-butt largesse.\" Djband agreed and repeated, \"No mere funky-butt largesse.\" Djamilaa's advancing meat, meet and romance (cosmic tail, cosmic tale, cosmic tally) went on with her saying, \"I want the rose's perfume where pendent cheeks meet, funk sublimated upward, astral crevice, crease.\" Djband not only echoed and agreed, \"Astral crevice, crease,\" but added, \"Ass as in astral, amen.\" Djamilaa stepped back, again blending in.\n\nThe more booty-invested of the dancers, hearing Djamilaa's admonition, dialed it down. A couple of them stopped dancing altogether, stepping back into the crowd, content to nod their heads, pop their fingers, pat their feet.\n\nThe now more precisely calibrated serenade made it crystal clear that Djband was no Parliament, no Funkadelic, no Zapp and Roger\u2014crystal clear even as it grew to be pearl opaque, for the review, irritant pebble to Djband's oyster, was coming to be accreted over, contained, gotten over, a tribute to palimpsestic add-on, palimpsestic struff, stick-to-itiveness. The music grew to be pearly smooth as well as pearl opaque, much less bumpy than funk.\n\nIt was now a precisely telepathic serenade, the heard rather than hearded audition every band so deeply wants. Sensing this, Djband saw no further need for the sandwich board and picked it up from where it lay propped against the newspaper vending box. N. stepped forward to flip it inside out so that Mingus's words no longer showed and then stepped back and blended in again. Djband again propped it against the newspaper vending box. The blank sides of the two boards glared in the sun.\n\nThe crowd of onlookers had grown larger, all of them nodding their heads, popping their fingers, patting their feet. Looking out at Melrose, Djband saw that traffic in both directions had slowed, drivers and passengers looking over at the goings-on around the newspaper vending box, Djband telepathically holding forth, the onlookers looking on. They too, the drivers and passengers, looked on, nodding their heads, popping their fingers and (Djband imagined rather than saw but couldn't have been more certain) patting their feet.\n\nParade was back. The cars, vans, buses and trucks proceeded at parade pace, not so much cars, vans, buses and trucks as floats, titrating, ever so exactingly, the ideal roost and repose Lambert had adumbrated earlier.\n\nParade was indeed back, as much on the sidewalk, it turned out, as on the street. Emerging from the opening between the two boards of the sandwich board was none other than B'Loon, out in the open for everyone to see\u2014the eyelashes hovering above the head and brow, the poorly defined limbs and extremities, the wistful, noncommittal mouth and all.\n\nB'Loon, small at first but steadily inflating, grew to be as big as the giant balloons at a Macy's Thanksgiving Day parade, rivaling Superman, Kermit the Frog, Snoopy and the rest, floating high above the sidewalk, lifting.\n\nEveryone stared into the sky at B'Loon drifting higher. The crowd of onlookers on the sidewalk stared skyward, as did those in the cars, vans, buses and trucks, leaning out their windows and bringing traffic to a stop. Djband as well stared skyward. Was B'Loon's lift mere exhibition or possibly more, possibly exorcism? It was hard not to wonder.\n\nB'Loon floated higher and higher. Heads tilted farther back and hands became visors as B'Loon drifted higher, everyone more and more straining to see as the image got smaller. Less visible the farther away it floated, B'Loon soon couldn't be seen at all. Thus B'Loon exited the house the intersecting avenues could be said to be.\n\nFor a long time after B'Loon floated out of sight everyone kept looking into the sky. The crowd of onlookers on the sidewalk stood staring skyward. Traffic remained at a standstill, those in the cars, vans, buses and trucks continuing to lean out their windows looking up into the sky. Djband continued looking up as well.\n\nEveryone went on staring into the sky, lost in thought. It had all been only a bubble, a moment in the sun, a quick boon, barely embraceable, blown up to be let go.\nCopyright \u00a9 2017 by Nathaniel Mackey\n\nAll rights reserved. Except for brief passages quoted in a newspaper, magazine, radio, television, or website review, no part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without permission in writing from the Publisher.\n\n_Late Arcade_ is volume five of _From a Broken Bottle Traces o_ _f_ _Perfume Still Emanate_ , an ongoing work. Volumes one, two, three and four are _Bedouin Hornbook,_ _Djbot Baghostus's Run_ , _Atet A.D.,_ and _Bass Cathedral._\n\nSections of this book have appeared in _Amerarcana: A Bird & Beckett Review_, _Blues for Smoke, Conjunctions_ , _Current Musicology_ , _Fence_ , _Floor: Poetics o_ _f_ _Everyday Critique_ _,_ _Golden Handcuffs Review_ , _Hambone_ , and _Women & Performance: A Journal o_ _f_ _Feminist Theory._\n\nThe author would like to thank the John Simon Guggenheim Memorial Foundation, whose award of a Fellowship for the year 2010\u20132011 contributed to the completion of this book.\n\nManufactured in the United States of America\n\nFirst published as a New Directions Paperbook (NDP1368) in 2017\n\neISBN: 9780811226615\n\nNew Directions Books are published for James Laughlin\n\nby New Directions Publishing Corporation\n\n80 Eighth Avenue, New York 10011\n\n# new directions presents\n\n## A Simple Story\n\n## by Leila Guerriero\n\n## translated by Frances Riddle\n\n**Obsession and mastery in their purest states: the story of one dancer's attempt to win the biggest contest of his life.**\n\n## Contents\n\n  1. Also by Nathaniel Mackey\n  2. Title Page\n  3. Dedication\n  4. Late Arcade\n    1. 14.IX.83\n    2. 19.IX.83\n    3. 24.IX.83\n    4. [Dateless]\n    5. 30.IX.83\n    6. 4.X.83\n    7. 8.X.83\n    8. 10.X.83\n    9. 26.X.83\n    10. 30.X.83\n    11. 6.XI.83\n    12. 14.XI.83\n    13. 6.XII.83\n    14. 17.XII.83\n    15. 13.I.84\n    16. 16.I.84\n    17. 18.I.84\n    18. 22.I.84\n    19. [Dateless]\n    20. 6.II.84\n    21. 16.II.84\n    22. 17.II.84\n    23. 18.II.84\n    24. 27.II.84\n    25. 3.III.84\n    26. [Dateless]\n    27. 13.III.84\n    28. 16.III.84\n    29. 18.III.84\n    30. POST-EXPECTANT PRESS RELEASE #2\n    31. 22.III.84\n    32. 2.IV.84\n    33. 17.IV.84\n    34. 3.V.84\n    35. 6.V.84\n    36. 11.V.84\n    37. 15.V.84\n    38. 18.V.84\n    39. 20.V.84\n    40. 25.VI.84\n    41. 29.VI.84\n    42. B'LOON'S BLUE SKYLIGHT\n  5. Copyright\n\n## Landmarks\n\n  1. Cover\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n## THE CHASE\n\n## DIRK PITT\u00ae ADVENTURES BY CLIVE CUSSLER\n\nTreasure of Khan\n\nBlack Wind\n\nTrojan Odyssey\n\nValhalla Rising\n\nAtlantis Found\n\nFlood Tide\n\nShock Wave\n\nInca Gold\n\nSahara\n\nDragon\n\nTreasure\n\nCyclops\n\nDeep Six\n\nPacific Vortex\n\nNight Probe\n\nVixen 03\n\nRaise the Titanic\n\nIceberg\n\nThe Mediterranean Caper\n\nKURT AUSTIN ADVENTURES BY CLIVE CUSSLER WITH PAUL KEMPRECOS\n\nThe Navigator\n\nPolar Shift\n\nLost City\n\nWhite Death\n\nFire Ice\n\nBlue Gold\n\nSerpent\n\nOREGONFILES ADVENTURES BY CLIVE CUSSLER WITH CRAIG DIRGO\n\nSkeleton Coast\n\nDark Watch\n\nSacred Stone\n\nGolden Buddha\n\nNONFICTION BY CLIVE CUSSLER AND CRAIG DIRGO\n\nThe Sea Hunters II\n\nClive Cussler and Dirk\n\nPitt Revealed\n\nThe Sea Hunters\n\n## THE CHASE\n\n## CLIVE CUSSLER\n\nG. P. PUTNAM'S SONS\n\nNEW YORK\n\nG. P. PUTNAM'S SONS  \nPublishers Since 1838  \nPublished by the Penguin Group  \nPenguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, U.S.A. \u2022 Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada (a division of Pearson Penguin Canada Inc.) \u2022 Penguin Books Ltd, 80 Strand, London WC2R 0RL, England \u2022 Penguin Ireland, 25 St Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd) \u2022 Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty Ltd) \u2022 Penguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi\u2013110 017, India \u2022 Penguin Group (NZ), 67 Apollo Drive, Rosedale, North Shore 0745, Auckland, New Zealand (a division of Pearson New Zealand Ltd) \u2022 Penguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa\n\nPenguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England\n\nCopyright \u00a9 2007 by Sandecker, RLLLP  \nEndpaper map and illustrations by Richard Dahlquist  \nAll rights reserved. No part of this book may be reproduced, scanned, or distributed in any printed or electronic form without permission. Please do not participate in or encourage piracy of copyrighted materials in violation of the author's rights. Purchase only authorized editions.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nCussler, Clive.  \nThe chase \/ Clive Cussler.  \np. cm.  \nISBN: 978-1-1012-0758-1  \n1. Bank robberies\u2014Fiction. 2. California\u2014History\u20141850\u20131950\u2014Fiction. I. Title.  \nPS3553.U75C47 2007 2007017291  \n813'.54\u2014dc22\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.\n\nWhile the author has made every effort to provide accurate telephone numbers and Internet addresses at the time of publication, neither the publisher nor the author assumes any responsibility for errors, or for changes that occur after publication. Further, the publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.\nTo Teri, Dirk, and Dana\n\nNo father was blessed with more-loving children\n\n## THE CHASE\n## Contents\n\nGHOST FROM THE PAST\n\nTHE BUTCHER BANDIT\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nTHE CHASE QUICKENS\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nChapter 36\n\nChapter 37\n\nChapter 38\n\nChapter 39\n\nChapter 40\n\nChapter 41\n\nChapter 42\n\nChapter 43\n\nChapter 44\n\nChapter 45\n\nChapter 46\n\nChapter 47\n\nChapter 48\n\nChapter 49\n\nChapter 50\n\nUP FROM THE DEPTHS\n\n## GHOST FROM THE PAST\n\n##\n\nAPRIL 15, 1950  \nFLATHEAD LAKE, MONTANA\n\nIT ROSE FROM THE DEPTHS LIKE AN EVIL MONSTER in a Mesozoic sea. A coat of green slime covered the cab and boiler while gray-brown silt from the lake bottom slid and fell off the eighty-one-inch drive wheels and splashed into the cold waters of the lake. Ascending slowly above the surface, the old steam locomotive hung for a moment from the cables of a huge crane mounted on a wooden barge. Still visible under the dripping muck, beneath the open side window of its cab, was the number 3025.\n\nBuilt by the Baldwin Locomotive Works of Philadelphia, Pennsylvania, 3025 rolled out of the factory on April 10th of 1904. The \"Pacific\" class was a common large-sized, high-drive-wheeled steam engine that could pull ten steel passenger cars long distances at speeds up to ninety miles an hour. She was known as a 4-6-2 because of her four-wheeled truck in the front, just behind the cowcatcher, the six massive drive wheels below the boiler, and the two small wheels mounted beneath the cab.\n\nThe crew on the barge watched in awe as the crane operator orchestrated his levers and gently lowered old 3025 onto the main deck, its weight settling the barge three inches deeper in the water. She sat there almost a minute before the six men overcame their wonderment and detached the cables.\n\n\"She's in remarkably good shape for sitting underwater for almost fifty years,\" murmured the salvage superintendent of the battered old barge that was nearly as ancient as the locomotive. Since the nineteen twenties, it had been used for dredging operations on the lake and surrounding tributaries.\n\nBob Kaufman was a big, friendly man, ready with a laugh at the slightest hint of something jovial. With a face ruddy from long hours spent in the sun, he had been working on the barge for twenty-seven years. Now seventy-five, he could have retired long ago, but as long as the dredging company kept him on he was going to keep working. Sitting at home and working jigsaw puzzles was not his idea of the good life. He studied the man standing beside him, who was, as close as he could figure, slightly older.\n\n\"What do you think?\" Kaufman asked.\n\nThe man turned, tall and still lean in his late seventies, hair full and silver. His face was as weathered as buckskin. He stared at the locomotive thoughtfully through eyes yet to rely on glasses. They gleamed blue with a tinge of lavender. A large silver mustache covered his upper lip as if it had been planted there many years ago. It matched his eyebrows, which had become bushy with age. He lifted an expensive Panama hat off his head and dabbed at his forehead with a handkerchief.\n\nHe walked over to the salvaged locomotive, now sitting solidly on the deck, and focused his attention on the cab. Water and muck poured down its ladders and spilled across the deck of the barge.\n\n\"Despite the grime,\" he said finally, \"she's still aesthetically pleasing to the eye. Only a question of time before a railroad museum comes up with the funds to restore her for display.\"\n\n\"Lucky a local fisherman lost his outboard engine and dragged the bottom to find it or the locomotive might have been down there another half century.\"\n\n\"Yes, it was a stroke of luck,\" the tall, silver-haired man said slowly.\n\nKaufman stepped over and ran a hand over one of the big drive wheels. A sentimental expression crossed his face. \"My daddy was an engineer with the Union Pacific,\" he said quietly. \"He always said the Pacific-type locomotive was the finest he ever drove. He used to let me sit in the cab when he brought a train into the yard. The Pacific class was used mostly to haul passenger cars because it was so fast.\"\n\nA team of divers, wearing suits of canvas sandwiched between layers of rubber, stood on a platform as it was raised from beneath the surface of the cold water. They wore the Mark V brass hard hat, large weight belts around their chests, and diving boots with canvas tops, brass toes, and lead soles that weighed thirty-six pounds. Altogether, the divers wore one hundred fifty pounds of equipment. They tugged at their umbilical cords, leading to the surface-supplied diving air pump, as the platform was raised and swung down to the deck. They were no sooner aboard than another team climbed down ladders and stood on the platform as it descended into the waters of the lake, still icy from the long Montana winter.\n\nThe tall man watched silently, looking out of place among the barge's crew in their grease-stained work clothes and overalls. He wore neatly pressed brown slacks with an expensive cashmere knit sweater under a cashmere jacket. His shoes were highly polished and had amazingly kept their shine on the oil-soaked deck, amid the rusting cables.\n\nHe eyed the heavy layers of silt on the steps leading to the cab and turned to Kaufman. \"Let's get a ladder over here so we can climb into the cab.\"\n\nKaufman gave an order to a nearby barge crewman and a ladder soon appeared and was propped against the lip of the cab's floor behind the engineer's seat. The superintendent went up first, followed by the elderly observer. Water dripped in sheets from the roof while dissolved coal merged with the silt flowing through the open door of the firebox onto the metal floor.\n\nAt first, it looked like the cab was empty. The maze of valves, pipes, and levers mounted over the boiler was coated with layers of ooze and the tentacles of green weed growing from it. The muck on the floor of the cab was ankle-deep, but the tall, quiet observer did not seem to notice it coming over the tops of his shoes. He knelt down and studied three humps that rose from the ooze like small hills.\n\n\"The engineer and fireman,\" he announced.\n\n\"You sure?\"\n\nHe nodded. \"I'm sure. The engineer was Leigh Hunt. He had a wife and two children, both grown now to middle age. The fireman was Robert Carr. He was going to be married after the run.\"\n\n\"Who was the third man?\"\n\n\"Name was Abner Weed. A tough customer. He forced Hunt and Carr to operate the engine with a gun in their backs.\"\n\n\"They don't look pretty,\" Kaufman muttered, repelled by what he saw. \"I'm surprised they didn't turn into skeletons.\"\n\n\"There would be nothing left of them if they died in salt water, but the cold, fresh water of Flathead Lake preserved them. What you see is the adipose tissue in which fat is stored. It breaks down over time when immersed, giving the body a waxy, soapy look called saponification.\"\n\n\"We'll have to call the sheriff and get a coroner out here.\"\n\n\"Will that delay the operation?\" asked the stranger.\n\nKaufman shook his head. \"No, it shouldn't slow things down any. As soon as the team of relief divers attach the lift cables, we'll bring up the coal tender.\"\n\n\"It's important that I see what's in the attached car.\"\n\n\"You will.\" Kaufman looked at the man, trying in vain to read his thoughts. \"Better we tackle the tender first to simplify matters. If we concentrate on the car before it has been uncoupled from the tender, it might prove disastrous. It may not be as heavy as the locomotive, but unless we're very careful it might break into pieces. It's a far trickier operation. Besides, the front end of the baggage car is half buried under the tender.\"\n\n\"It's not a baggage car. It's a boxcar, or freight car.\"\n\n\"How could you know that?\"\n\nThe observer ignored the question. \"Raise the coal tender first. You're in charge.\"\n\nKaufman stared down at the ugly lumps that had once been humans. \"How did they get here? How could a train come to be lost in the middle of the lake all these years?\"\n\nThe tall man gazed out over the calm blue lake. \"Forty-four years ago, there was a ferry that carried railcars loaded with lumber back and forth across the lake.\"\n\n\"It sure is strange,\" said Kaufman slowly. \"Newspapers and the Southern Pacific officials reported that the train was stolen. As I recall, the date was April 21, 1906.\"\n\nThe old man smiled. \"A cover-up by the company. The train wasn't stolen. A railroad dispatcher was bribed to charter the engine.\"\n\n\"Must have been something valuable in the freight car to kill for,\" said Kaufman. \"Like a shipment of gold.\"\n\nThe old man nodded. \"Rumors circulated that the train was carrying gold. If the truth be known, it was not gold but hard cash.\"\n\n\"Forty-four years,\" Kaufman said slowly. \"A long time for a train to go missing. Maybe the money is still inside the car.\"\n\n\"Perhaps,\" said the tall man, looking toward the horizon at a vision only he could see. \"Just perhaps we'll find the answers when we get inside.\"\n\n## THE BUTCHER BANDIT\n\n## 1\n\nJANUARY 10, 1906 BISBEE, ARIZONA\n\nANYONE SEEING AN OLD DERELICT SOT SLOWLY SWAYING down Moon Avenue in Bisbee that afternoon would have mistaken him for what he was not, a man who had grown old before his time working the mines that ran through the mineral-rich mountains under the town. His shirt was grubby and he smelled unwashed. One suspender held up torn and ragged pants that were stuffed into scuffed and worn boots that should been thrown in the trash gully behind the town long ago.\n\nSnarled and greasy hair straggled to his shoulders and merged with an uncut beard that hung halfway down his protruding stomach. He looked through eyes so dark brown they were nearly black. There was no expression in them; they seemed cold and almost evil. A pair of work gloves covered the hands that had never held a shovel or a pick.\n\nUnder one arm, he carried an old gunnysack that appeared empty. Almost whimsically, the dirty burlap had DOUGLAS FEED & GRAIN COMPANY, OMAHA, NEBRASKA stenciled on it.\n\nThe old man took a minute and parked on a bench at the corner of Moon Avenue and Tombstone Canyon Road. Behind him was a saloon, mostly empty because it was the middle of the day and its usual patrons were hard at work in the mines. The people walking and shopping in the little mining town paid him no more than a quick, disgusted glance. Whenever they passed, he pulled a whiskey bottle from a pant pocket and drank heavily before recapping it and putting it back. No one could have known it was not whiskey but tea.\n\nIt was warm for June; he guessed the temperature to be in the high nineties. He sat back and looked up and down the streets as a trolley car passed, pulled by an aging horse. Electric-motored trolleys had yet to come to Bisbee. Most of the vehicles on the streets were still horse-drawn wagons and buggies. The town had only a handful of automobiles and delivery trucks, and none were in evidence.\n\nHe knew enough about the town to know that it was founded in 1880 and named after Judge DeWitt Bisbee, one of the moneymen behind the Copper Queen Mine. A good-sized community, its population of twenty thousand made it the largest city between San Francisco and St. Louis. Despite the many miners' families that lived in modest little wooden buildings, the main economy was based around saloons and a small army of shady ladies.\n\nThe man's head nodded to his chest; he looked like a drunk who had dozed off. But it was an act. He was conscious of every movement around him. Occasionally, he glanced across the street at the Bisbee National Bank. He watched with interest, through half-closed eyes, as a truck with chain drive and solid-tired wheels rattled up to the bank. There was only one guard, who got out of the truck and carried a large bag of newly printed bills inside. A few minutes later, he was helped by the bank's teller to lug a heavy chest through the door and onto the truck.\n\nThe man knew that it was a shipment of gold, a piece of the three million ounces that had been produced at the local mines. But gold was not what piqued his interest. It was too heavy and too risky for one man to dispose of. It was the cash that brought him to Bisbee, not the prized yellow metal.\n\nHe watched as the truck moved away and two men, whom he had identified as security guards at the giant Phelps Dodge Mining Company, walked out of the bank. They had delivered the cash to pay the mining company payroll the following day. He smiled to himself, knowing the assets of the Bisbee National Bank had risen to a new level.\n\nHe had watched the people who came and went from the bank for nearly two weeks until he could identify them by sight. He had also noted the time when they came and went. Satisfied now there was no one in the bank except one teller and the owner\/manager, he looked at his watch and nodded to himself.\n\nLeisurely, the old derelict rose, stretched, and ambled across the brick street and trolley tracks to the bank, carrying the large, empty gunnysack over one shoulder. Just as he was about to enter, a woman unexpectedly walked past. She gave him a look of loathing, stepped around him, and went inside. She was not in his plan, but he decided to deal with the matter rather than wait. He checked the street and followed her into the bank.\n\nHe closed the door. The teller was in the vault and the woman waited until he reappeared. The derelict removed a model 1902 Colt .38 caliber automatic from his boot, struck the woman on the nape of the neck with the barrel, watching with detachment as she slowly folded to the wooden floor. It happened so suddenly and silently that the owner of the bank did not see or hear anything from his office.\n\nThen the drunken miner suddenly turned bank robber leaped sprightfully over the counter, entered the owner's office, and put the gun barrel to his head. \"Resist and you'll be shot dead,\" he said in a low but forbidding tone. \"Now, call the teller into your office.\"\n\nThe bald, fat, shocked bank owner looked at him with brown eyes widened with fright. Without argument, he called out, \"Roy, come in here.\"\n\n\"Be right there, Mr. Castle,\" Roy called out from inside the vault.\n\n\"Tell him to leave the vault open,\" said the bank robber quietly, with a sharp edge to his voice.\n\n\"Roy, don't close the vault door,\" Castle complied as ordered, his eyes crossing as they focused on the gun pressing against his forehead.\n\nRoy stepped from the vault, a ledger under one arm. He couldn't see the unconscious woman lying under the counter. Suspecting nothing, he entered Castle's office and abruptly stiffened when he saw the robber holding a gun to his boss's head. The robber pulled the gun barrel away from Castle's head and motioned with the muzzle toward the vault.\n\n\"Both of you,\" he said calmly, \"into the vault.\"\n\nThere was no thought of resistance. Castle rose from his desk and led the way into the vault while the robber stepped quickly to the window to check the street for anyone heading for the bank. Except for a few women shopping and a passing beer wagon, the street was quiet.\n\nThe interior of the vault was well lit, with an Edison brass lamp hanging from the steel ceiling. Except for the chest containing the gold, stacks of bills, mostly the payroll for the mining companies, covered the shelves. The robber threw the gunnysack at the teller.\n\n\"Okay, Roy, fill it with all the greenbacks you have.\"\n\nRoy did as he was told. With trembling hands, he began sweeping the piles of bills of various denominations into the sack. By the time he was finished, the sack was stretched to the limit of its burlap fibers and seemed to be the size of a well-filled laundry bag.\n\n\"Now, lay down on the floor,\" ordered the robber.\n\nCastle and Roy, believing the robber was now about to make his getaway, stretched out flat on the floor, with their hands stretched over their heads. The robber pulled a heavy woolen scarf from one of his pockets and wrapped it around the muzzle of his automatic. Then he systematically shot both men in the head. It sounded more like two loud thumps than the sharp crack of gunfire. Without another second's hesitation, he heaved the sack over one shoulder and walked from the vault without looking back.\n\nUnfortunately, he wasn't finished. The woman under the counter moaned and tried to rise to her elbows. With utter indifference, he leaned down, lowered the gun, and shot her in the head like he had the bank owner and teller. There was no remorse, not the slightest hint of emotion. He didn't care whether any of them left families behind. He had murdered three defenseless people in cold blood with as little interest as he might have shown stepping on a column of ants.\n\nHe paused to search for one of the shell casings he thought he'd heard fall to the floor from inside the scarf wrapped around the gun but could not find it. He gave up and walked casually from the bank, noting with satisfaction that no one had heard the muted gunshots.\n\nWith the gunnysack bulging with cash slung over his shoulder, the man walked through the alley running behind the bank. Stepping into a small alcove under a stairway where he would not be seen, he took off the grimy clothes, removed the gray wig and beard, and threw everything in a small valise. Now revealed in an expensively tailored suit, he perched a bowler hat at a jaunty angle on his head and its neatly brushed carpet of red hair. He slipped on a necktie and knotted it before also tossing the scuffed boots in the valise. He was a short man, and the soles and heels of the boot had been raised nearly two inches. Next, he pulled on a pair of English-made leather shoes, with lifts in the heel to make him appear taller, before turning his attention to a large leather suitcase he had hidden under a canvas tarp along with a Harley-Davidson motorcycle. Glancing up and down the alley every few seconds, he transferred the huge pile of cash from the gunnysack to the suitcase, which he strapped on a rack over the rear wheel of the motorcycle. The valise containing his disguise he tied on a front rack.\n\nAt that moment, the man heard shouting coming down the alley from Tombstone Canyon Road. Someone had discovered the bodies in the Bisbee National Bank. Unconcerned, he pushed the motorcycle forward and started the three-horsepower, twenty-five-cubic-inch one-cylinder engine. He threw one leg over the seat and rode deserted back alleys to the railroad yard. He moved unseen along a siding where a freight train had stopped to take on water.\n\nHis timing was perfect.\n\nAnother five minutes and the freight train would have moved back onto the main line and headed toward Tucson. Without being noticed by the engineer and the brakeman, as they pulled a big pipe down from the wooden tank into the tender for water to make steam, the man took a key from his vest pocket and opened the padlock to the door to a boxcar that was marked with a painted sign that read O'BRIAN FURNITURE COMPANY, DENVER. He slid the door open on its rollers. The presence of the boxcar in this time and place was no coincidence. Acting as a fictitious representative of the equally fictitious O'Brian Furniture Company, he had paid cash for it to be included in the freight train passing through Bisbee, en route from El Paso, Texas, to Tucson, Arizona.\n\nHe took a wide plank, attached by brackets to the side of the boxcar, and used it as a ramp to ride the Harley-Davidson aboard. Then he quickly closed the rolling door and reached through a small hinged opening to replace the lock as the whistle on the engine tooted and the train began moving forward from the siding onto the main track.\n\nFrom the outside, the boxcar looked like any other that had been in use for several years. The paint was faded, and the wooden sides were dented and chipped. But its appearance was deceptive. Even the lock on the door was fake, making it look like the car was buttoned up tight. It was the inside that was the most deceptive, however. Instead of an empty interior or one packed with furniture, it was luxurious, ornately constructed, and furnished as ostentatiously as any private railcar belonging to the president of a railroad. Mahogany paneling spread over the walls and ceiling. The floor was covered by a thick carpet. The d\u00e9cor and furniture were extravagantly magnificent. There was an opulent sitting room, a palatial bedroom, and an efficient kitchen with the latest innovations for preparing gourmet meals.\n\nThere were no servants, porters, or cooks.\n\nThe man worked alone, without accomplices who might reveal his true name and occupation. No one knew of his clandestine operations as a bank robber and mass murderer. Even the railroad car had been built and decorated in Canada before being secretly transported across the border into the United States.\n\nThe robber relaxed in a plush leather settee, uncorked a bottle of 1884 Ch\u00e2teau La Houringue Bordeaux, chilled in buckets, and poured himself a glass.\n\nHe knew the town sheriff would quickly form a posse. But they would be looking for an old mangy miner who murdered while in a drunken fit. The posse would fan out, searching the town, almost certain he was too poor to own a horse. None of the townspeople had ever seen him come and go on horseback or driving a buggy.\n\nImmensely pleased with himself, he sipped the wine from a crystal glass and studied the leather suitcase. Was this his fifteenth, or was it his sixteenth, successful robbery? he mused. The thirty-eight men and women and two children he had killed never entered his mind. He estimated the take of the mining payroll at $325,000 to $330,000. Most robbers wouldn't have come close to guessing the amount inside the case.\n\nBut it was easy for him, since he was a banker himself.\n\nThe sheriff, his deputies, and the posse would never find the murdering robber. It was as if he had disappeared into thin air. No one ever thought to connect him with the dapper man riding through town on a motorcycle.\n\nThe hideous crime would become one of Bisbee's most enduring mysteries.\n\n## 2\n\nSEPTEMBER 15, 1906 THE MISSISSIPPI RIVER BELOW HANNIBAL, MISSOURI\n\nSOON AFTER THE TWENTIETH CENTURY WAS BORN, steamboating on the Mississippi began to fade. Few passenger steamboats still reigned in style. The Saint Peter was one of the last grand passenger boats to have survived the onslaught of the railroads. Two hundred fifty feet long and seventy-five feet wide, she was a splendid example of palatial elegance, with side-curving stairways, plush passenger cabins, and a magnificent main dining room with the finest food to be found anywhere. Ostentatious salons were provided for the ladies while the men smoked their cigars and played cards in handsome rooms adorned with mirrors and paintings.\n\nCard games on steamboats plying the river were notorious for their cardsharp gamblers. Many passengers left steamboats poorer than when they boarded. At one table in the gambling room of the Saint Peter, in a quiet corner away from the main action, two men were enjoying a game of five-card stud.\n\nAt first glance, the scene looked like any other in the room, but a closer look revealed that no chips sat on the green felt table.\n\nJoseph Van Dorn calmly studied his hand before laying down two cards. \"A good thing we're not in this for the money,\" he said, smiling, \"or I would owe you eight thousand dollars.\"\n\nColonel Henry Danzler, director of the United States government's Criminal Investigation Department, smiled in return. \"If you cheated like I do, we'd be even.\"\n\nVan Dorn was a congenial man in his early forties. His cheeks and chin were buried under a magnificent red burnsides beard that matched what remained of the hair that circled his bald dome. His face was dominated by a Roman nose, and his brown eyes looked sad and melancholy, but his looks and manner were deceiving.\n\nIrish-born, he bore a name known and respected throughout the country for tenacity in tracking down murderers, robbers, and other desperados. The criminal underworld of the time knew he would chase them to the ends of the earth. Founder and chief of the renowned Van Dorn Detective Agency, he and his agents had prevented political assassinations, hunted down many of the West's most feared outlaws, and helped organize the country's first secret service agency.\n\n\"You'd still deal yourself more aces than me,\" he said affably.\n\nDanzler was an enormous man, tall and mammoth in girth, weighing slightly over three hundred pounds, yet he could move as effortlessly as a tiger. His salt-and-pepper hair was immaculately trimmed and brushed, shining under the light that streamed in through the boat's big windows. His blue-green eyes had a soft glow to them, yet they seemed to analyze and record everything going on about him.\n\nA veteran and hero of the Spanish American War, he had charged up San Juan Hill with Captain John Pershing and his black \"Buffalo Soldiers\" of the Tenth Cavalry and had served with distinction in the Philippines against the Moros. When the government's Criminal Investigation Department was authorized by Congress, President Roosevelt asked him to become its first director.\n\nDanzler opened the lid of a large pocket watch and stared at the hands. \"Your man is five minutes late.\"\n\n\"Isaac Bell is my best agent. He always gets his man\u2014and occasionally a woman, too. If he's late, there's a good reason.\"\n\n\"You say he's the one who apprehended the assassin Ramos Kelly before he could shoot President Roosevelt?\"\n\nVan Dorn nodded. \"And he rounded up the Barton gang in Missouri. He shot and killed three of them before the other two surrendered to him.\"\n\nDanzler stared at the famous detective. \"And you think he's the man to stop our mass murderer and bank robber?\"\n\n\"If anyone can stop the killer, Isaac can.\"\n\n\"What is his family background?\"\n\n\"Very wealthy,\" answered Van Dorn. \"His father and grandfather were bankers. You've heard of the American States Bank of Boston?\"\n\nDanzler nodded. \"Indeed. I have an account there myself.\"\n\n\"Isaac is very affluent. His grandfather left him five million dollars in his will, thinking Isaac would take his place as head of the bank one day. It never happened. Isaac preferred detective work to banking. I'm lucky to have him.\"\n\nDanzler caught a shadow on his arm. He looked up and found himself looking into soft blue eyes with a slight violet cast, eyes that had looked over horizons to see what was beyond. The effect was almost mesmerizing, as though they were searching deep into Danzler's inner thoughts.\n\nDanzler could size up a man as precisely as he could a horse. The intruder was tall and lean, stood well over six feet, and weighed no more than one hundred seventy-five pounds. A large flaxen mustache that covered his entire upper lip conformed with the thick mass of neatly barbered blond hair. His hands and fingers were long and nimble and hung loosely, almost casually, at his sides. There was a no-nonsense look about him. The colonel judged that this was a man who dealt with substance and did not endure fools or insignificant and phony candor. He had a determined set to the chin and lips that were spread in a friendly smile. Danzler guessed his age at about thirty.\n\nHe was dressed immaculately in a white linen suit without a wrinkle. A heavy gold chain dipped from a left vest pocket that was attached to a large gold watch inside the right pocket. A low-crowned hat with a wide brim sat squarely on his head. Danzler might have pegged him as a dandy, but the look of elegance was betrayed by a pair of worn leather boots that had seen many hours in stirrups. Bell carried a thin valise and set it down beside the table.\n\n\"Colonel Danzler,\" said Van Dorn, \"this is the man I told you about, Isaac Bell.\"\n\nDanzler offered his hand but did not rise from his chair. \"Joe here tells me that you always get your man.\"\n\nBell grinned slightly. \"I'm afraid Mr. Van Dorn has exaggerated. I was ten minutes too late when Butch Cassidy and Harry Longabaugh sailed for Argentina three years ago from New York. Their boat pulled away from the dock before I could apprehend them.\"\n\n\"How many agents or law enforcement officers were with you?\"\n\nBell shrugged. \"I intended to handle the matter on my own.\"\n\n\"Wasn't Longabaugh the Sundance Kid?\" asked Danzler.\n\nBell nodded. \"He got the nickname when he tried to steal a horse in Sundance, Wyoming. He was caught and spent eighteen months in jail.\"\n\n\"Surely you didn't expect to subdue them without a fight.\"\n\n\"I think it is safe to say that they would have resisted,\" said Bell, without explaining how he would have single-handedly captured the former members of the infamous Wild Bunch.\n\nVan Dorn sat back in his chair, made no comment, and gave the colonel a smug look.\n\n\"Why don't you sit down, Mr. Bell, and join our little game?\"\n\nBell looked at the empty table quizzically and then at Danzler. \"You appear to have no chips.\"\n\n\"Just a friendly little game,\" said Van Dorn, shuffling the deck of cards and dealing out three hands. \"So far, I owe the colonel eight thousand dollars.\"\n\nBell sat down, the quizzical look altered to one of understanding. The game was a pretense. His chief and the colonel were sitting in the corner away from the other gamblers and playing as if they were in a serious game. He laid his hat in his lap, picked up his cards, and acted as if he were deep in thought.\n\n\"Are you familiar with the swarm of bank robberies and murders that have occurred around the western states in the past two years?\" Danzler inquired.\n\n\"Only in conversation,\" replied Bell. \"Mr. Van Dorn has kept me busy on other cases.\"\n\n\"What do you actually know about the crimes?\"\n\n\"Only that the robber murders anyone in the bank during the act, escapes like a spirit, and leaves no evidence behind that might incriminate him.\"\n\n\"Anything else?\" Danzler probed.\n\n\"Whoever he is,\" answered Bell, \"he is very, very good. There have been no leads and no breaks in the investigation.\" He paused and stared at Van Dorn. \"Is that why I've been called here?\"\n\nVan Dorn nodded. \"I want you to take over the case as chief investigator.\"\n\nBell threw down a card, picked up the card that Danzler dealt, and slipped it in the fan, which he held in his left hand.\n\n\"Are you a lefty, Mr. Bell?\" asked Danzler out of curiosity.\n\n\"No. Actually, I'm right-handed.\"\n\nVan Dorn laughed softly. \"Isaac can draw the derringer he hides in his hat, cock it, and pull the trigger faster than you can blink.\"\n\nDanzler's respect for Bell grew during the conversation. He drew back his coat and revealed a 1903 Colt .38 caliber hammerless automatic. \"I'll take Joe's word for it, but it would be interesting to put it to the test\u2014\" Danzler had not finished the sentence when he found himself staring into the twin muzzles of a derringer.\n\n\"Age has slowed you, Henry,\" said Van Dorn. \"Either that or your mind wandered.\"\n\n\"I have to admit, he is very fast,\" Danzler said, visibly impressed.\n\n\"What office will I work out of?\" Bell asked Van Dorn as he slipped the derringer back in his hat, where it fit in a small pocket inside the crown.\n\n\"The crimes have occurred from Placerville, California, in the west, to Terlingua, Texas, to the east,\" replied Van Dorn. \"And from Bisbee, Arizona, in the south, to Bozeman, Montana, in the north. I think it best if you operated in the center.\"\n\n\"That would be Denver.\"\n\nVan Dorn nodded. \"As you know, we have an office there with six experienced agents.\"\n\n\"I've worked with two of them three years ago,\" said Bell. \"Curtis and Irvine are good men.\"\n\n\"Yes, I forgot,\" Van Dorn said, now recalling. \"I might add, Colonel, that Isaac was responsible for the apprehension of Jack Ketchum, who was later hung for two murders committed during a train robbery.\" He paused and reached under the table and produced an identical valise to the one Bell had carried into the gambling salon. Bell then passed his empty valise to Van Dorn. \"Inside, you will find the reports on all the crimes. Every lead so far has led up a blind alley.\"\n\n\"When do I start?\"\n\n\"At the next landing, which is Clarksville, you will depart and take the first train to Independence. From there, you will be given a ticket on the Union Pacific express to Denver. You can digest and study what little clues and evidence we've gathered. Once you arrive, you'll take up the hunt for the murdering scum.\" A look of anger and frustration clouded Van Dorn's brown eyes. \"Sorry, I didn't give you a chance to pack when you left Chicago, but I wanted you to start as soon as possible.\"\n\n\"Not to worry, sir,\" Bell said with a faint smile. \"Fortunately, I packed two suitcases for the duration.\"\n\nVan Dorn's eyebrows raised. \"You knew?\"\n\n\"Let's say I made an educated guess.\"\n\n\"Keep us informed on your manhunt,\" said Danzler. \"If you need any help from the government, I'll do all in my power to assist you.\"\n\n\"Thank you, sir,\" Bell acknowledged. \"I'll be in contact as soon as I get a firm grip on the situation.\"\n\nVan Dorn said, \"I'll be working in our Chicago office. Since transcontinental telephone service has yet to run from St. Louis across the prairie to Denver and beyond to California, you'll have to telegraph me on your progress.\"\n\n\"If any,\" Danzler muttered sarcastically. \"You're up against the best criminal brain this country has ever known.\"\n\n\"I promise I won't rest until I capture the man responsible for these hideous crimes.\"\n\n\"I wish you good luck,\" Van Dorn said sincerely.\n\n\"Not to change the subject,\" Danzler spoke with satisfaction as he laid his card hand on the green felt, \"I have three queens.\"\n\nVan Dorn shrugged and threw his cards on the table. \"Beats me.\"\n\n\"And you, Mr. Bell?\" said Danzler with a crafty grin.\n\nIsaac Bell slowly laid his cards on the table one by one. \"A straight flush,\" he said matter-of-factly. Then, without another word, he rose and walked briskly from the salon.\n\n## 3\n\nLATE IN THE MORNING, A MAN DROVE AN OLD WAGON, hitched to a pair of mules, past the cemetery outside the town of Rhyolite, Nevada. The graves had simple wooden fences around them, with the names of the deceased carved on markers made of wood. Many were children who had died of typhoid or cholera, aggravated by the hard family life of a mining town.\n\nThe July heat in the Mojave Desert was unbearable under the direct rays of the sun. The driver of the wagon sat beneath a tattered umbrella attached to the seat. Black hair fell past his neck but just short of the shoulders. His head was protected by a stained Mexican sombrero. His unseen eyes peered through the stained-blue glass of spectacles, and a handkerchief wrapped the lower half of his face, to keep out the dust raised by the mules' hooves. The manner in which he was hunched over made it difficult, if not impossible, to determine his build.\n\nAs he rode by, he stared with interest at a house a miner had built using thousands of cast-off saloon beer bottles embedded in adobe mud. The bottoms of the bottles faced outward with the mouths facing in, the green glass casting the interior in an eerie sort of light.\n\nHe came to the railroad tracks and drove the mules along the road next to them. The tops of the rails gleamed like narrow twin mirrors in the blinding sun. These were the tracks of the Las Vegas & Tonopah Railroad, which wound through the middle of the residential district of the town.\n\nThe wagon rolled slowly past more than eighty railcars on a siding. They had been unloaded of their incoming freight. The empty cars were now being filled with outgoing ore for the mills. The driver took a brief glance at a boxcar being coupled to a thirty-car train. The lettering on the side said O'BRIAN FURNITURE COMPANY, DENVER. He glanced at the dial of his cheap pocket watch\u2014he carried nothing that might help identify him\u2014and noted that the train was not scheduled to leave for Las Vegas for another forty-four minutes.\n\nA quarter of a mile later, he came to the Rhyolite train station. The substantial building was a mixture of Gothic and early Spanish styles. The ornate depot had been built of stone, cut and hauled from Las Vegas. A passenger train that had steamed in from San Francisco sat alongside the station platform. The passengers had disembarked, and the seats cleaned by porters, and the train was now filling with people heading back toward the coast.\n\nThe driver reached the center of town, where the streets were bustling with activity. He turned to stare at a large mercantile establishment, the HD & LD PORTER store. Beneath the sign was a slogan painted on a board that hung above the front entrance. It read We handle all things but Whiskey.\n\nThe 1904 gold rush had resulted in a substantial small city of solidly constructed buildings built to last a long time. By 1906, Rhyolite was a thriving community of over six thousand people. It had quickly graduated from a busy tent town to an important city meant to stand far into the distant future.\n\nThe main buildings were constructed of stone and concrete, making the small metropolis of Rhyolite the major city of southern Nevada. A four-story bank came into sight, a fine-looking structure that gave it a look of substance and wealth. Half a block away, a three-story stone office building was going up.\n\nThere was a post office, an opera house, a twenty-bed hospital, comfortable hotels, two churches, three banks, and a large school. Up-to-date, Rhyolite boasted an efficient telephone system and its own electrical-generating plant. It also had a booming red-light district and forty saloons and eight dance halls.\n\nThe man driving the wagon was not interested in anything the town had to offer except some of the assets of the John S. Cook Bank. He knew that the safe inside could hold over a million dollars in silver coins. But it was far easier to carry cash from the payrolls of the mines, and he had yet to take a single silver, or gold, piece. He figured that with eighty-five companies engaged in mining the surrounding hills, the payroll take should be quite considerable.\n\nAs usual, he had planned well, living in a boardinghouse for miners while entering the Cook Bank on numerous occasions to make small deposits in an account he had opened under a false name. A brief friendship was struck up with the bank's manager, who was led into thinking the town newcomer was a mining engineer. The man's appearance had been altered with a wig of black hair, a mustache, and a Vandyke beard. He also walked with a limp, which he said was the result of a mining accident. It proved to be a flawless disguise with which to study the banking habits of the citizens and the times when the bank was doing little business.\n\nAs he drove the wagon and mules into town toward the Cook Bank, however, his image had been changed from that of a mining engineer to that of a small-time freight hauler to the mines. He looked like any one of the town's haulers, struggling to make a living in the broiling heat of the desert during summer. He reined in the mules at the rear of a stable. When he was certain no one was observing him, he lifted a dummy dressed exactly the same as himself and tied it to the seat of the wagon. Then he led the mules back toward Broadway, the main street running through town. Just before reaching the concrete walkway in front of the bank's entrance, he slapped the mules on their rumps and sent them off, pulling the wagon down the street through the main part of town, his dummy likeness sitting upright on the seat and holding the reins.\n\nHe checked for customers approaching the bank. None of the people milling around the town seemed headed in that direction. He looked up at the four-story building, glancing at the gold paint on the windows of the upper floor advertising a dentist and a doctor. Another sign, with a hand pointing downward, indicated that the town post office was in the basement.\n\nHe strolled into the bank and looked around the lobby. It was empty except for a man making a withdrawal. The customer took his money from the teller, turned, and walked from the bank without glancing at the stranger.\n\nThere goes a lucky man, the robber thought.\n\nIf the customer had bothered to notice him, he would have been shot dead. The robber never left anyone behind to identify the least detail about him. Then there was always the possibility, although slim, that someone might see through his disguise.\n\nHe had learned from conversations in the neighboring saloons that the bank was run by a manager for a company of men who were owners of the region's most productive mines, especially the Montgomery-Shoshone Mine whose original claim had grossed nearly two million dollars.\n\nSo far, so good, thought the robber as he leaped over the counter, landing on his feet next to the startled teller. He pulled the automatic from his boot and pressed the muzzle against the teller's head.\n\n\"Do not move, and do not think of stepping on the alarm button under the counter or I'll splatter your brains on the wall.\"\n\nThe teller could not believe what was happening. \"Is this really a holdup?\" he stammered.\n\n\"It is that,\" replied the robber. \"Now, walk into the manager's office very slowly and act as if nothing is happening.\"\n\nThe frightened teller moved toward an office with a closed door whose etched glass made it difficult to see in or out. He knocked.\n\n\"Yes, come on in,\" came a voice from the other side.\n\nThe teller Fred pushed open the door and was roughly shoved inside, losing his balance and falling across the manager's desk. The sign on the desk, HERBERT WILKINS, was knocked to the floor. Wilkins swiftly took in the situation and reached for a revolver under his desk. He was five seconds too late. The robber had learned about the weapon from the manager himself, while talking at a nearby saloon.\n\n\"Do not touch that gun,\" snapped the robber, as if he were psychic.\n\nWilkins was not a man who frightened easily. He stared at the robber, taking in every inch of his appearance. \"You'll never get away with it,\" he said contemptuously.\n\nThe robber spoke in a cold, steady voice. \"I have before and I will do so again.\" He motioned toward the imposing safe that stood nearly eight feet high. \"Open it!\"\n\nWilkins looked the robber square in the eye. \"No, I don't think I will.\"\n\nThe robber wasted no time. He wrapped the muzzle of his automatic in a heavy towel and shot the teller between the eyes. Then he turned to Wilkins. \"I may leave here without a dime, but you won't live to see it.\"\n\nWilkins stood, horrified, staring down at the spreading pool of blood around Fred's head. He looked at the smoldering towel where the bullet had passed through, well knowing it was unlikely that anyone in the building had heard the gunshot. As if in a trance, he walked to the safe and began turning the combination lock to the required numbers. After half a minute, he pulled down on the latch and the massive steel door swung open.\n\n\"Take it and be damned!\" he hissed.\n\nThe robber merely smiled and shot Wilkins in the temple. The bank manager had barely struck the floor when the robber strode quickly to the front door, slammed it shut, hung a CLOSED sign in the window, and pulled down the shades. Then he methodically cleaned out the safe of all bills, transferring them into a laundry bag he carried tied around his waist under his shirt. When the sack was filled until it bulged in every seam, he stuffed the remaining bills in his pant pockets and boots. The safe cleaned of all money, the robber stared briefly at the gold and silver coins inside and took just one gold souvenir.\n\nThere was a heavy iron rear door to the bank that opened onto a narrow street. The robber unlocked the door's inside latch, cracked the door open, and scanned the street. It was lined on the opposite side with residential houses.\n\nA group of young boys were playing baseball a block from the bank. Not good. This was entirely unexpected by the robber. In his many hours of observing the streets around the Cook Bank, this was the first time he had found children playing in the street behind the bank. He was on a time schedule and had to reach the railyard and his secret boxcar in twelve minutes. Shouldering the bag so his face was shielded on the right side, he walked around the ball game in progress and continued up the street, where he ducked into an alley.\n\nFor the most part, the boys ignored him. Only one stared at the poorly dressed man toting a big sack over his right shoulder. What struck the boy as odd was that the man wore a Mexican sombrero, a style that was seldom seen around Rhyolite. Most men in town wore fedoras, derbies, or miner's caps. There was also something else about the raggedy man...Then another boy yelled, and the boy turned back to the game, barely in time to catch a pop fly.\n\nThe robber tied the sack around his shoulders so that it hung on his back. The bicycle he'd parked earlier behind a dentist's office was sitting there behind a barrel that had been placed to catch runoff water from the building's drainpipe. He mounted the seat and began pedaling along Armagosa Street, past the red-light district, until he came to the railyard.\n\nA brakeman was walking along the track toward the caboose at the end of the train. The robber couldn't believe his bad luck. Despite his meticulous planning, fate had dealt him a bad hand. Unlike with his other robberies and murders, this time he had been noticed by a stupid young boy. And now this brakeman. Never had he encountered so many eyes that might have observed him during his escape. There was nothing he could do but see it through.\n\nLuckily, the brakeman did not look in the robber's direction. He was going from car to car checking the grease in the axle boxes of the trucks and wheels the boxcars rode on. If the brass sleeve that rotated inside the box did not receive enough lubricant, the friction would heat the end of the axle to a dangerous level. The weight of the car could break the axle off and cause a disastrous crash.\n\nAs the robber cycled past, the brakeman did not bother to look up. He instead went about his business, trying to complete his inspection before the train departed for Tonopah and then on to Sacramento.\n\nAlready, the engineer was looking at his gauges to make sure he had enough steam to move the heavy train. The robber hoped the brakeman would not turn back and witness him entering his private boxcar. Quickly, he unlocked and slid open the door. He threw the bicycle inside and then climbed a small ladder up to the door, dragging the heavy money sack over the threshold.\n\nOnce inside the boxcar, the robber peered down the length of the train. The brakeman was climbing aboard the caboose, which housed the train crew. There was no sign he'd witnessed the robber enter the boxcar.\n\nSecure inside his palatial car, the robber relaxed and read a copy of the Rhyolite Herald. He could not help but wonder what the paper would print the following day about the bank robbery and the killing of its manager and teller. Again, as he had so many times earlier, he felt no remorse. The deaths never entered his mind again.\n\nLater, besides the mystery of how the robber\/killer had escaped without a trace, the other puzzle was the wagon found outside of town on the road toward Bullfrog. The wagon was empty and appeared to have been driven by a dummy. The posse that chased it down was mystified.\n\nSheriff Josh Miller did put two and two together, but his speculation went nowhere. Nothing made sense. The desperado left no clues.\n\nThe robbery and murders in Rhyolite became another enigma that went unsolved.\n\n## 4\n\nTHE SUMMER SUNLIGHT HEIGHTENED THE CONTRAST of colors in the mile-high altitude of Colorado. The sky was free of clouds, a vivid blue that spread over the city of Denver like a quilt. The temperature was a comfortable eighty-one degrees.\n\nIsaac Bell closed the door to his stateroom and left the train by stepping off the observation platform at the rear of the Pullman car. He paused to look up at the clock tower of the Gothic-style Union Station. Built of stone hauled down from the Rocky Mountains, the imposing three-story structure stretched a quarter of a mile.\n\nThe arrowhead-tipped hands of the huge clock read 11:40. Bell lifted his large gold watch from the vest pocket of his tailored linen suit and glanced at the hands that pointed to Roman numerals. His time was 11:43. He smiled at himself with satisfaction, knowing for certain that the big clock-tower clock was three minutes slow.\n\nHe walked down the redbrick platform to the baggage car, identified his trunks, and hailed a porter. \"My name is Bell. Could you please see that my trunks are sent to the Brown Palace Hotel?\"\n\nThe porter smiled broadly at the gold coin Bell laid in his hand and rubbed it almost reverently. \"Yes, sir, I'll deliver them myself.\"\n\n\"I'm also expecting a large wooden crate on a later train. Can I count on you to make sure it is delivered to the Union Pacific freight warehouse?\"\n\n\"Yes, sir, I'll take care of it.\" Still rubbing the gold piece, the porter grinned broadly.\n\n\"I'd be grateful.\"\n\n\"May I take that for you?\" said the porter, nodding at the valise in Bell's hand.\n\n\"I'll keep it with me, thank you.\"\n\n\"Can I hail you a taxi?\"\n\n\"That won't be necessary. I'll take the tram.\"\n\nBell strolled through the high-ceilinged grand lobby of the depot, with its majestic hanging chandeliers, past the rows of high-backed oak waiting benches and out the main entrance, flanked by twin Grecian columns. He crossed Wyncoop Street onto 17th Street and passed under the newly erected Mizpah Arch, a gatelike structure with a pair of American flags flying on top that was built to welcome, and bid farewell to, train travelers. Mizpah, Bell knew, meant watchtower in ancient Hebrew.\n\nTwo ladies wearing light summer dresses, gloves, and ornate hats decorated with flowers drove by in an electric battery\u2013powered car. Bell doffed his hat, and with nods and smiles they acknowledged the attention of the attractive man as they motored up 17th Street toward the state capitol building.\n\nHorse-drawn wagons and carriages still outnumbered the few automobiles that chugged up and down the streets of the city. A Denver Tramway Company trolley car clanged around the corner off Wazee Street and approached the end of the block, where it stopped to let off and take on passengers. The horse-drawn railways were a thing of the past and electric trolleys ruled the streets, reaching every neighborhood in Denver.\n\nBell climbed the steps and gave ten cents to the motorman. The bell was clanged and the big red trolley clattered up 17th Street. Three-and four-story brick buildings filled the next fourteen blocks. The sidewalks were crowded with people on a typical business day. The men wore black or gray suits and ties, while the women strolled in the long dresses whose skirts rose just above the ankles. Most of the women wore flamboyant hats and carried parasols.\n\nHe observed with interest a store that was selling Cadillac motorcars. The awnings were rolled out, shadowing the windows and revealing the vehicles inside. He glanced at the street signs so he could recall the location. An enthusiast of motorcars, he owned a Locomobile race car that had been driven by Joe Tracy in the 1905 Vanderbilt Cup road race on Long Island, in New York, placing third. Bell had converted it to street driving by adding fenders and headlamps.\n\nHe also owned a bright red motorcycle. The newest racing model, its V-Twin engine put out three and a half horsepower. It had an innovative twist grip throttle, weighed only one hundred twenty pounds, and could whip over the roads at nearly sixty miles an hour.\n\nWhen the trolley rattled to a stop at California and 17th Streets, Bell stepped down the stair to the pavement and sauntered over to the sidewalk. It had been three years since he had set foot in Denver. Tall buildings stood on almost every corner, and the construction never stopped. He walked a block to the Colorado Building, a brown stone structure that rose eight stories on 16th and California Streets.\n\nThe windows were high and shielded by awnings that matched the brown exterior of the walls. The overhang above the top floor stretched nearly ten feet over the sidewalk far below. Hedgecock & Jones and the Braman Clothing Company occupied the street level. Above them were several different businesses, including the Fireman's Fund Insurance Company and the Van Dorn Detective Agency.\n\nBell turned into the lobby and moved through a group of office workers who were streaming out of the building on their lunch break. The floor, walls, and ceiling were beautifully constructed of green Italian marble the color of jade. He entered an Otis elevator behind two pretty young ladies and moved to the rear of the car as the operator closed the steel scissor-gate door. As was the custom, Bell played the gentleman and removed his wide-brimmed hat.\n\nThe elevator operator pivoted the handle on the curved throttle housing, sending the elevator toward the upper floors at a leisurely pace. The women exited at the fifth floor, chatting gaily. They both turned and gave Bell a bashful glance before disappearing down the hallway.\n\nThe operator stopped the elevator and opened the door. \"Eighth floor, and a good afternoon to you, sir,\" he said cheerily.\n\n\"Same to you,\" replied Bell.\n\nHe exited into a hallway painted a muted Mexican red above with walnut wainscot halfway up the wall below. He turned right and came to a door with etched lettering on the upper glass that advertised THE VAN DORN DETECTIVE AGENCY. Beneath was the agency's slogan: We never give up, never.\n\nThe antechamber was painted white, with two padded wooden chairs and a desk, behind which a young woman sat primly in a swivel chair. Van Dorn was not a man to waste money on ostentatious d\u00e9cor. The only embellishment was a photo of the head man hanging on the wall behind the secretary.\n\nShe looked up and smiled sweetly, admiring the well-dressed man standing opposite her. She was a pretty woman, with soft brown eyes and wide shoulders. \"May I help you, sir?\"\n\n\"Yes. I'd like to see Arthur Curtis and Glenn Irvine.\"\n\n\"Are they expecting you?\"\n\n\"Please tell them Isaac Bell is here.\"\n\nShe sucked in her breath. \"Oh, Mr. Bell. I should have known. Mr. Curtis and Mr. Irvine did not expect you until tomorrow.\"\n\n\"I managed to catch an earlier train out of Independence, Missouri.\" Bell looked at the sign on her desk. \"You're Miss Agnes Murphy?\"\n\nShe held up her left hand, displaying a wedding band. \"Mrs. Murphy.\"\n\nBell smiled his beguiling smile. \"I hope you don't mind if I simply call you Agnes, since I'll be working here for a time.\"\n\n\"Not at all.\"\n\nShe rose from her desk, and he could see she wore a pleated blue cotton skirt with her white fluffy blouse. Her hair was piled atop her head in the fashion of the Gibson girl, which was so popular then. Her petticoats rustled as she went through the door to the inner offices.\n\nAlways curious, Bell moved around the desk and looked down at the letter Mrs. Murphy had been typing on a Remington typewriter. It was addressed to Van Dorn, and spelled out the superintendent of the western states' displeasure at having Bell come in and take over the unsolved case. Bell had never met Nicholas Alexander, who headed the Denver office, but he was determined to be courteous and polite to the man despite any antagonism.\n\nBell moved away from Mrs. Murphy's desk and stood looking out the window over the rooftops of the city when Alexander walked into the anteroom. He looked more like the bookkeeper of a funeral parlor than the chief investigator who had unraveled many crimes and brought the offenders to full justice. He was a short man, his head barely coming up even with Bell's shoulders. He wore a coat that was too large and his trousers were baggy. The high collar of his shirt showed wear and sweat stains. His head was devoid of hair except around the temples and at the rear; the eyebrows were trimmed as neatly as his hair. A pair of pince-nez glasses were clipped to the bridge of his nose in front of almost-sad-looking gray-green eyes.\n\nAlexander held out his hand as his lips spread into a smile that was completely lacking in humor. \"Mr. Bell, I'm honored to meet Van Dorn's finest agent.\"\n\nBell didn't buy the compliment since there was no hint of warmth about it. \"The honor is mine in meeting you,\" Bell replied, nearly biting his tongue. It was obvious Alexander simply thought of Bell as an interloper into his private territory.\n\n\"Please come on back. Before I show you to your new office, we'll have a talk.\"\n\nAlexander abruptly turned and strode stiffly through the door into the inner offices. Mrs. Murphy stood aside and smiled sweetly as they passed.\n\nAlexander's office was positioned in the only corner with a panoramic view of the mountains; the other offices were small and windowless. Bell observed that they were also doorless, offering almost no privacy. Alexander's domain was embellished with cowhide sofas and chairs. His aspen desk was expansive and completely barren of paperwork. Though Alexander's suit was a poor fit and bore wrinkles, he was fastidious about his working habits.\n\nHe seated himself in a high-backed chair behind his desk and motioned Bell to sit in an uncushioned wooden chair on the opposite side. The only thing missing for intimidation, Bell thought, was a platform under Alexander's work space so he could look down on his employees and visitors like a minor god on Mount Olympus.\n\n\"No, thank you,\" Bell said quietly. \"After sitting on a train for two days, I'd prefer a softer seat.\" He lowered his long frame onto one of the sofas.\n\n\"As you wish,\" said Alexander, not pleased with Bell's superior demeanor.\n\n\"You were not here when I worked on a case three years ago.\"\n\n\"No, I came six months later when I was promoted from our Seattle office.\"\n\n\"Mr. Van Dorn spoke very highly of you,\" Bell lied. Van Dorn had not mentioned him.\n\nAlexander folded his hands and leaned across the empty wasteland of his desk. \"I trust he briefed you on the murderer and his operations.\"\n\n\"Not in conversation.\" Bell paused to hold up the valise. \"But he gave me several reports that I examined while riding on the train. I can see why the felon responsible for the robberies and murders is so difficult to pin down. He plans his criminal ventures with extreme care and his techniques appear to be flawless.\"\n\n\"All reasons why he eludes capture.\"\n\n\"After absorbing the material, I do believe his fetish for detail will be his undoing,\" said Bell thoughtfully.\n\nAlexander looked at him suspiciously. \"What, may I ask, brought you to that conclusion?\"\n\n\"His jobs are too perfect, too well timed. One small miscalculation could prove his last.\"\n\n\"I hope we can have a close relationship,\" Alexander said with veiled animosity.\n\n\"I agree,\" said Bell. \"Mr. Van Dorn said I could have Art Curtis and Glenn Irvine on my team, if it is all right with you.\"\n\n\"Not a problem. I wouldn't go against Mr. Van Dorn's wishes. Besides, they told me they worked with you a few years ago.\"\n\n\"Yes, I found them to be dedicated agents.\" Bell came to his feet. \"May I see my office?\"\n\n\"Of course.\"\n\nAlexander came from behind his desk and stepped into the hallway.\n\nBell saw that all the offices were quite small and quite plain. The furniture was sparse and there were no pictures on the walls. Only one other agent was present in the office, a stranger to Bell whom Alexander did not bother to introduce.\n\nBefore Alexander could point out a closet office, Bell asked innocently, \"Do you have a conference room?\"\n\nAlexander nodded. \"Yes, on the opposite side of the hallway from the offices.\" He stopped, opened a door, and stood aside as Bell walked in.\n\nThe conference room stretched nearly thirty feet and flowed fifteen feet to the side. A long pine table, stained dark and with a polished surface, sat beneath two massive, circular chandeliers. Eighteen leather captain's chairs were spaced evenly around it. The room was paneled in pine that matched the table, the floor carpeted with deep red pile. High windows rose on one wall, allowing the early-afternoon sunlight to illuminate every corner of the room.\n\n\"Very nice,\" said Bell, impressed. \"Very nice.\"\n\n\"Yes,\" said Alexander with pride showing in his bloodhound eyes. \"I use it frequently for meetings with politicians and influential people in the city. It gives the Van Dorn Detective Agency significant respect and an image of importance.\"\n\n\"It will do nicely,\" Bell said matter-of-factly. \"I'll work in here.\"\n\nAlexander looked directly at Bell, a fiery look in his eyes that suddenly glowed with anger. \"That's not possible. I won't permit it.\"\n\n\"Where is the nearest telegraph office?\"\n\nAlexander seemed taken back. \"Two blocks south on Sixteenth Street and Champa. Why?\"\n\n\"I'll send a message to Mr. Van Dorn requesting the use of your conference room as an operations center. Considering the importance of the case, I'm sure he will give it his blessing.\"\n\nAlexander knew when he was licked. \"I wish you well, Mr. Bell,\" he conceded. \"I will cooperate with you any way I can.\" He then turned and left Bell to return to his suite in the corner. He paused in the doorway. \"Oh, by the way, I've reserved a room for you at the Albany Hotel.\"\n\nBell smiled. \"That won't be necessary. I've booked a suite at the Brown Palace.\"\n\nAlexander appeared confused. \"I can't believe Mr. Van Dorn would allow that on your expense account.\"\n\n\"He didn't. I'm paying for it out of my own funds.\"\n\nNot aware of Bell's prosperous situation, the superintendent of the western states looked completely bewildered. Unable to comprehend, but not wanting to ask questions, he returned to his office in a daze and closed the door, utterly defeated.\n\nBell smiled again and began spreading out the papers he'd carried in the valise on the conference table. Then he stepped into the anteroom and approached Agnes Murphy. \"Agnes, could you let me know when Curtis and Irvine show up?\"\n\n\"I don't expect them back until tomorrow morning. They went up to Boulder on a bank fraud case.\"\n\n\"All right, then. And would you call the building maintenance superintendent and have him come up? I have some alterations to make in the conference room.\"\n\nShe looked at him questioningly. \"Did you say the conference room? Mr. Alexander seldom allows the help to step inside. He keeps it mostly to entertain the town bigwigs.\"\n\n\"While I'm here, it will be my office.\"\n\nAgnes looked at Bell with newly found respect. \"Will you be staying at the Albany Hotel? That's where most all visiting agents stay.\"\n\n\"No, the Brown Palace.\"\n\n\"Mr. Alexander consented to the extra expenditure?\" she asked warily.\n\n\"He had no say in the matter.\"\n\nAgnes Murphy stared after him as if she had just seen the Messiah.\n\nIsaac Bell returned to his office and rearranged the chairs to the conference table so he could have a large work space at one end. After a few minutes, the building superintendent arrived. Bell explained the alterations he wished to make in the room. The end wall was to have a layer of soft material so a map of the western states and towns the killer had hit could be pinned to it. Another layer was to be installed on the inside wall for information, photos, and drawings. The superintendent, after Bell had offered him a twenty-dollar gold piece, promised to have the installation accomplished by noon the next day.\n\nBell spent the rest of the afternoon organizing and planning his hunt for the bank killer.\n\nAt precisely five o'clock, Alexander stuck his head in the door on his way home. \"Are you settling in all right?\" he asked icily.\n\nBell did not bother to look up. \"Yes, thank you.\" He finally looked into Alexander's angry eyes. \"By the way, I'm making some changes in the room. I hope you don't mind. I promise to put it back exactly the way it was when the case is closed.\"\n\n\"Please see that you do.\" Alexander swung his head in a gesture of dismissal and left the office.\n\nBell was not happy that things were not going well between Alexander and him. He had not planned to get in a game of quarrelsome loggerheads with the head of the agency's office, but if he hadn't gone on the attack he knew that Alexander would have walked all over him.\n\n## 5\n\nBUILT IN 1892 BY HENRY C. BROWN ON THE SPOT where he used to pasture his cow before he struck it rich, the hotel was fittingly named the Brown Palace, for the \"Queen City of the Plains,\" as Denver was called. Constructed of red granite and sandstone, the building was in the shape of a ship's bow. The men who made their fortunes in gold and silver stayed there with their wives, who took afternoon tea, and their daughters, who danced away the nights at opulent balls. Presidents McKinley and Roosevelt had stayed there, as well as a few emperors and kings and other members of foreign royalty, not to mention the celebrities of the time, particularly famous stage actors and actresses. The Brown Palace was also embraced by locals and visitors alike because it was the anchor to the busy financial and cultural district of the city.\n\nIt was nearly dark when Bell walked through the 17th Street entrance of the Brown Palace Hotel. He checked in at the desk and looked around the magnificent lobby, which was situated in an atrium that reached up to the ninth floor. The pillars and wainscoting, freighted in by railroad from Mexico and carved from golden onyx, reflected the pastel light that filtered down from the massive stained-glass ceiling. Over seven hundred wrought-iron panels graced the balcony railings, ringing the lobby from the upper floors.\n\nWhat was not generally known was that the owner of the Navarre Hotel and restaurant across the street had had an underground rail system dug from the Brown Palace to his own establishment in order to accommodate gentlemen wishing to enjoy the ladies of an upstairs brothel without being seen entering or leaving.\n\nBell was given his key and entered the elevator, telling the operator which floor his suite was on. A woman stepped in behind him. She stopped at the mirrored wall, turned, and faced the door. She was dressed in a long blue silk gown with a large bow in the back. Her fire opal red hair was fine and silken, pulled back in a bun with curls streaming from it. There were two large feathers rising from the hair. She had an engaging charm about her. She stood tall and erect and nubile, Bell guessed probably between twenty-five and twenty-seven, perhaps younger, judging by her swan neck and face as smooth as alabaster. Her eyes were a golden brown. She was, in Bell's mind, unusually attractive\u2014not quite beautiful, maybe, but very lovely by any standard. He also noticed she wore no wedding ring.\n\nThe woman was dressed as if she meant to attend a party in one of the hotel's ballrooms, Bell reasoned. He was right as usual. The elevator stopped on the second floor, which held the ballrooms and dance floors. He stood aside, hat in hand, and made a slight bow as she exited onto the landing.\n\nShe threw him a smile with surprising warmth and nodded, and said, in a mellow yet husky tone, \"Thank you, Mr. Bell.\"\n\nAt first, it slipped by Bell. Then it hit him like a hammer on a thumb. He was stunned that she knew him, and positive he'd never seen her before. Bell gripped the arm of the elevator operator. \"Hold the door open a moment.\"\n\nBy now, she had mingled in with a crowd that was funneling through the arched doorway of the hotel's grand ballroom. The women wore ravishing gowns in extravagant colors\u2014crimson, peacock blue, emerald green\u2014with ribbons, sprigs, and feathers in their hair. The men were dressed in their finest evening clothes. A banner over the doorway read BENEFIT FOR ST. JOHN'S ORPHANS.\n\nBell stepped back, nodding at the elevator operator. \"Thank you. Please take me up.\"\n\nBell unlocked the door to his suite and found a study, living room, ornate bath, and bedroom with a canopied bed, all furnished in Victorian elegance. His trunks had been opened and his clothes packed in the dresser and hung in the closet by a maid, a service provided to those who reserved suites. The trunks were not in sight. They had been moved from the room and stored in the basement storage area. Bell lost no time in taking a quick bath and shaving.\n\nHe opened his watch and read the time. Thirty minutes had elapsed since he stepped from the elevator. Another fifteen minutes were taken to tie his black tie and insert the shirt studs and cuff links, usually a job that took four hands. It was one of the few times he wished he had a wife to help. Black socks and shoes came next. He did not wear a cummerbund but a black vest instead, with a gold chain running from the left pocket through a buttonhole to the big gold watch in the right pocket. Last, he slipped on a single-breasted black jacket with satin lapels.\n\nOne final view of his reflection in a full seven-foot mirror and he was ready for the evening, whatever it would bring.\n\nThe charity ball was in full swing when he walked inside the grand ballroom and stood unobtrusively behind a tall potted palm. The ballroom was spacious and majestic. The parquet dance floor was laid in an intricate sunburst design and colorful murals adorned the ceiling. He spied the mysterious woman, seated with her back to him, with three couples at table six. She appeared to be alone, without an escort. He sidled up to the hotel director in charge of the evening's event.\n\n\"Pardon me,\" said Bell with a friendly smile, \"but could you tell me the name of the lady in the blue dress at table six?\"\n\nThe director straightened with a haughty look. \"I'm sorry, sir, but we frown on giving information on our guests. Besides, I can't know everybody who comes to the ball.\"\n\nBell passed him a ten-dollar gold certificate. \"Will this jog your memory?\"\n\nWithout a word, the director held up a thin leather book and ran his eyes over the entries. \"The single lady at the table is Miss Rose Manteca, a very wealthy lady from Los Angeles whose family owns a vast ranch. That's all I can tell you.\"\n\nBell patted the director on the shoulder. \"I'm grateful.\"\n\nThe director grinned. \"Good luck.\"\n\nAn orchestra was playing a medley of ragtime and modern dance tunes. Couples were dancing to a song called \"Won't You Come Over to My House.\"\n\nBell walked up behind Rose Manteca and whispered in her ear. \"Would you please consent to dance with me, Miss Manteca?\"\n\nShe turned from the table and looked up. Golden brown eyes looked into a pair of mesmeric violet eyes. She was smooth, Bell thought, but his sudden appearance in evening dress completely stunned her. She lowered her eyes and recovered quickly, but not before her face blushed red.\n\n\"Forgive me, Mr. Bell. I did not expect you so soon.\"\n\n\"So soon?\" he asked. What an odd thing to say, he thought.\n\nShe excused herself to the people at the table and stood up. Gently, he took her by the arm and led her to the dance floor. He slipped his arm around her narrow waist, took her hand, and stepped off smartly with the music.\n\n\"You're a very good dancer,\" she said after he swept her around the floor.\n\n\"Comes from all those years my mother forced me to take lessons so I could impress the debutantes in our city.\"\n\n\"You also dress very well for a detective.\"\n\n\"I grew up in a city where the affluent men lived in tuxedos.\"\n\n\"That would be Boston, would it not?\"\n\nFor once, in his years of investigation, Bell was at a loss, but he recovered and came back. \"And you're from Los Angeles.\"\n\nShe was good, he thought. She didn't bat an eye.\n\n\"You're very knowledgeable,\" she said, unable to fathom his eyes.\n\n\"Not half as knowledgeable as you. What is your interest? How do you know so much about me? Better yet, I should ask why?\"\n\n\"I was under the impression you like to solve mysteries.\" She tried consciously to look past him over his tall shoulder, but she was drawn into those incredible eyes. This was a sensation, a stirring she had not counted on.\n\nThe photographs she had been shown did not do him justice face-to-face. He was far more attractive that she had imagined. He also came off as highly intelligent. This she'd expected, though, and could understand why he was so famous for his intuition. It was as though he was stalking her as she was stalking him.\n\nThe music ended and they stood together on the dance floor waiting for the orchestra to begin the next musical arrangement. He stepped back and ran his eyes from her shoes to the top of her beautifully styled hair. \"You are a very lovely lady. What prompted your interest in me?\"\n\n\"You're an attractive man. I wanted to know you better.\"\n\n\"You knew my name and where I came from before you met me in the elevator. Our meeting was obviously premeditated.\"\n\nBefore she could reply, the orchestra began playing \"In the Shade of the Old Apple Tree,\" and Bell led her around the floor in a foxtrot. He held her against him and gripped her hand tightly in his. Her waist was small, made even smaller by a corset. The top of her head came up even with his chin. He was tempted to press his lips against hers but thought better of it. This was neither the time nor the place. Nor were his thoughts on romance. She was spying on him. That was a given. His mind was trying to formulate a motive. What interest could a total stranger have in him? The only possibilities he could conjure up were that she'd been hired by one of the many criminals he had put behind bars, shot, or seen hanged. A relative or friend out for revenge? She didn't fit the image of someone who associated with the scum he had apprehended over the last ten years.\n\nThe music ended, and she released his hand and stood back. \"You'll have to excuse me, Mr. Bell, but I must return to my friends.\"\n\n\"Can we meet again?\" he asked with a warm smile.\n\nShe slightly shook her head. \"I don't think so.\"\n\nHe ignored her negative reply. \"Have dinner with me tomorrow night.\"\n\n\"Sorry, I'm busy,\" she answered, with a haughtiness in her voice. \"And even in your fancy tuxedo you couldn't bluff your way into the Western Bankers' Ball at the Denver Country Club like you did tonight at the benefit for St. John's Orphans.\" Then she threw out her chin, swept up her long skirt, and walked back to her table.\n\nOnce seated, she stole a glance back at Bell, but he was nowhere to be seen among the crowd. He had completely disappeared.\n\n## 6\n\nTHE FOLLOWING MORNING, BELL WAS THE FIRST ONE in the office, using a skeleton key that could open ninety doors out of a hundred. He was sorting through bank robbery reports at the end of the long table when Arthur Curtis and Glenn Irvine entered the conference room. Bell rose to greet them and shook hands. \"Art, Glenn, good to see you both again.\"\n\nCurtis stood short and rotund, with a rounded stomach neatly encased in a vest whose buttons were stretched to their limit. He had thinning sandy hair, wide megaphone ears, blue eyes, and a smile that showed a maze of teeth that lit up the room. \"We haven't seen you since we tracked down Big Foot Cussler after he robbed that bank in Golden.\"\n\nIrvine placed his hat on a coat stand, revealing a thick head of uncombed brown hair. \"As I remember,\" he said, standing as tall and as scrawny as a scarecrow, \"you led us directly to the cave where he was hiding out.\"\n\n\"A simple matter of deduction,\" Bell said with a tight smile. \"I asked a pair of young boys if they knew of a place where they liked to hide out from their folks for a few days. The cave was the only location within twenty miles, close enough to town so Cussler could sneak in for supplies.\"\n\nCurtis stood in front of the large map of the western United States and thoughtfully studied the little flags signifying the killer's spree. There were sixteen of them. \"Got any intuition on the Butcher Bandit?\"\n\nBell looked at him. \"Butcher Bandit? Is that what they're calling him?\"\n\n\"A reporter from the Bisbee Bugle came up with it. Other newspapers have picked it up and spread the term across the territory.\"\n\n\"It won't help our cause,\" said Bell. \"With that name on everyone's lips, the law-abiding citizens will come down hard on the Van Dorn Detective Agency for not apprehending him.\"\n\n\"That's already started,\" Curtis said, laying the Rocky Mountain News on the table in front of Bell. He stared down at it.\n\nThe lead column was on the robbery and murders in Rhyolite. Half the column was devoted to the question \"Why haven't law enforcement agencies made any progress in the case and captured the Butcher Bandit?\"\n\n\"The heat is on,\" Bell said simply.\n\n\"The heat is on us,\" Irvine added.\n\n\"So what have we got?\" asked Bell, pointing to a stack of files two feet high on the bank crimes piled on the desk in front of him. \"I've studied the reports while coming west on the train. It appears that all we have is that we're not dealing with the typical cowboy turned bank robber.\"\n\n\"He works alone,\" said Curtis, \"and he's devilish clever and evil. But what is most frustrating is that he never leaves a trail for a posse to follow.\"\n\nIrvine nodded his head in agreement. \"It's as though he disappears into the hell he came from before he leaves town.\"\n\n\"No tracks are ever found leading into the surrounding countryside?\" asked Bell.\n\nCurtis shook his head. \"The best trackers in the business have come up dry every time.\"\n\n\"Any evidence he might have holed up in town until the excitement died down?\"\n\n\"None that's ever turned up,\" replied Curtis. \"After the robberies, he was never seen again.\"\n\n\"A ghost,\" murmured Irvine. \"We're dealing with a ghost.\"\n\nBell smiled. \"No, he's human, but a damned smart human.\" He paused and fanned out the files on the conference table. He selected one and opened it, the report on the robbery in Rhyolite, Nevada. \"Our man has a very rigid modus operandi that he sticks with on every bank job. We believe he hangs around for a few days studying the town and its people before robbing the bank.\"\n\n\"He's either a gambler or a risk taker,\" said Curtis.\n\n\"Wrong on both counts,\" Bell corrected him. \"Our man is bold and he's shrewd. We can assume he does his dirty work using disguises, since the people of all the towns he's struck never agree on the appearance of suspicious-looking strangers.\"\n\nIrvine began pacing the conference room, occasionally examining a flag pinned on the map. \"Citizens of the towns recall seeing a drunken bum, a uniformed soldier, a well-to-do merchant, and a small-time freight hauler. But none could tie them to the murders.\"\n\nCurtis looked at the carpeted floor and shrugged. \"How odd there are no witnesses who can give a credible identification.\"\n\n\"Nothing odd about it,\" said Irvine. \"He murders them all. The dead can't speak.\"\n\nBell seemed to ignore the conversation as if he was lost in thought. Then his eyes focused on the map and he said slowly, \"The big question in my mind is why he always kills everyone in the bank during the theft. Even women and children. What does he gain by the slaughter? It can't be that he simply doesn't want to leave witnesses to the robberies, not when he's already been seen around town in disguise...unless...\" He paused. \"There is a new definition created by psychologists for murderers who kill as easily as they brush their teeth. They call them sociopaths. Our man can kill without remorse. He has no emotions, does not know how to laugh or love, and has a heart that is as cold as an iceberg. To him, shooting down a small child holds the same sensitivity as shooting a pigeon.\"\n\n\"Hard to believe there are people that cruel and ruthless,\" muttered Irvine in revulsion.\n\n\"Many of the bandits and gunfighters of the past were sociopaths,\" said Bell. \"They shot other men as easily as if they sneezed. John Wesley Hardin, the famous Texas badman, once shot and killed a man for snoring.\"\n\nCurtis looked steadily at Bell. \"Do you really think he murders everyone in a bank because he enjoys it?\"\n\n\"I do,\" Bell said quietly. \"The bandit gets a weird satisfaction from committing his blood crimes. Another peculiar factor. He makes his escape before the people of the town, including the town sheriff, realized what happened.\"\n\n\"So where does that leave us?\" asked Irvine. \"What avenues do we search?\"\n\nBell looked at him. \"Another of his routine habits is to ignore any gold and take only currency. Glenn, your job is to check out the banks that were robbed and study their records of the serial numbers on the stolen bills. Start in Bozeman, Montana.\"\n\n\"Banks in mining towns aren't in the habit of recording the identifying number of every bill that passes through their hands.\"\n\n\"You might get lucky and find a bank that recorded the numbers of the currency sent from large city banks to make the miners' payroll. If you do, we can trace them. The robber had to either spend the money or exchange the currency through bank deposits and withdrawals. A trail he can't cover up.\"\n\n\"He could have exchanged through foreign financial institutions.\"\n\n\"Maybe, but he would have to spend it overseas. The risk would be too great for him to bring it back into the U.S. I'm betting he kept his loot in the country.\"\n\nThen Bell turned to Curtis. \"Art, you check out all stagecoach and train schedules for any that departed the towns on the same day the robberies took place. If our man couldn't be tracked by a posse, he might easily have taken a train or stage for his getaway. You can begin in Placerville, California.\"\n\n\"Consider it done,\" said Curtis firmly.\n\n\"Are you going to remain here and act as a command post?\" asked Irvine.\n\nBell shook his head and grinned. \"No, I'm going out in the field, beginning with Rhyolite, and retrace the robberies. No matter how good the murderer is or how well he planned his crimes, there has to be a stone he left unturned. There must be evidence that's been overlooked. I'm going to question the mining town citizens who might have seen something, however insignificant, and failed to report it to the local sheriff or marshal.\"\n\n\"You'll give us your schedule so we can get in touch by telegraph if we come onto something?\" said Curtis.\n\n\"I'll have it for you tomorrow,\" replied Bell. \"I'm also going to travel through the mining towns that have large payrolls our man has yet to rob. Maybe, just maybe I can second-guess our butcher, set up a trap, and entice him to strike another bank on our turf.\" Then he pulled open a drawer and passed out two envelopes. \"Here's enough cash to cover your travel expenses.\"\n\nBoth Curtis and Irvine looked surprised. \"Before now, we always had to travel third class, use our own money, and turn in bills and receipts,\" said Curtis. \"Alexander always demanded we stay in sleazy hotels and eat cheap meals.\"\n\n\"This case is too important to cut corners. Trust me, Mr. Van Dorn will okay any monies I request, but only if we show results. The bandit may have everyone believing he's invincible and can't be caught, but he's not faultless. He has flaws just like the rest of us. He will be trapped by a small insignificant mistake he neglected. And that, gentlemen, is our job, to find that insignificant mistake.\"\n\n\"We'll do our best,\" Irvine assured him.\n\nCurtis nodded in agreement. \"Speaking for both of us, permit me to say that it is a real privilege to be working with you again.\"\n\n\"The privilege is mine,\" said Bell sincerely. He felt lucky to work with such intelligent and experienced operatives who knew the people and country of the West.\n\nTHE SUN was falling over the Rockies to the west when Bell left the conference room. Always cautious, he closed and locked the door. As he passed through the outer office, he ran into Nicholas Alexander, who looked like he'd just stepped out of an expensive tailor's shop. The usual shabby suit was gone and replaced by an elegant tuxedo. It was a new image of respectability that he didn't quite pull off. The inner polish simply was not there.\n\n\"You look quite the bon vivant, Mr. Alexander,\" Bell said graciously.\n\n\"Yes, I'm taking the wife to a fancy soiree at the Denver Country Club later this evening. I have many influential friends here in Denver, you know.\"\n\n\"So I've heard.\"\n\n\"A pity you can't come, but it's only for members of the club in good standing.\"\n\n\"I understand perfectly,\" Bell said, masking his sarcasm.\n\nAs soon as they parted, Bell went down the street to the telegraph office and sent a telegram to Van Dorn.\n\nHave set up a schedule of investigations by myself, Curtis, and Irvine. Please be informed that we have a spy in our midst. A woman, a stranger who approached me at the hotel, identified me by name, knew my past, and seemed to know why I was in Denver. Her name is Rose Manteca and she supposedly comes from a wealthy family of ranchers in Los Angeles. Please ask our Los Angeles office to investigate. Will keep you advised of our progress on this end.\n\nBell\n\nAfter he sent the telegram to his superior, Bell walked down the busy sidewalk to the Brown Palace Hotel. After a few words with the concierge, who provided him with a map of the city, he was escorted down to the storeroom and the boiler room beneath the lobby, where he was greeted by the hotel maintenance man. An affable fellow in stained coveralls, he led Bell to a wooden crate that had been dismantled. Under a single, bright lightbulb that hung from the ceiling, the maintenance man pointed at a motorcycle that sat on a stand beside the crate and gleamed a dazzling red.\n\n\"There she is, Mr. Bell,\" he said with satisfaction. \"All ready to go. I personally polished her up for you.\"\n\n\"I'm grateful, Mr....\"\n\n\"Bomberger. John Bomberger.\"\n\n\"I'll take care of your services when I leave the hotel,\" Bell promised him.\n\n\"Glad to be of help.\"\n\nBell went up to his room and found hanging in the closet the tuxedo that had been cleaned by the hotel during the day. After a quick bath, he dressed and removed a long linen coat from the closet and slipped it on, the bottom hem dropping to the tops of his highly shined shoes. Next, he slipped on a pair of leggings to save his tux trousers from the oily liquid that often came out of the engine. Finally, he donned a cap with goggles.\n\nBell took a back stairway down to the storeroom. The red cycle, with its white rubber tires, stood as if it was a steed waiting to carry him into battle. He kicked the stand up to the rear fender, took hold of it by the handlebars, and pushed all one hundred twenty pounds of it up a ramp used by wagons to remove the hotel bedding for cleaning and to allow merchants to bring in food for the restaurant and room service kitchens.\n\nBell exited the ramp and found himself on Broadway, the street that ran past the state capitol building with its golden dome. He mounted the hard, narrow saddle that perched over the camelback fuel tank above the rear wheel. Because it was built for racing, the seat was level with the handlebars and he had to lean almost horizontal to ride the machine.\n\nHe pulled the goggles over his eyes, then reached down and twisted open the valve that allowed fuel to fall by gravity from the tank to the carburetor. Then he placed his feet in the bicycle-style pedals and pumped down the street, allowing the electrical current from the three dry-cell batteries to flow to the coil, producing a high-voltage spark that ignited the fuel in the cylinders. He'd only gone about ten feet when the V-Twin engine popped into life, the exhaust rattling in a high-pitched snarl.\n\nBell curled his right hand around the grip throttle and twisted it less than half its rotation, and the racing bike lurched forward by its single-speed chain drive and he soon found himself cruising down Broadway around the horse-drawn carriages and occasional automobiles at thirty miles an hour.\n\nBecause it was built for racing, the bike had no headlight, but a half-moon lit the sky, and the street was lined with electric lights, providing enough illumination for him to see a pile of horse dung in time to dodge around it.\n\nAfter about two miles, he stopped under a streetlamp and consulted his map. Satisfied he was traveling in the right direction, he continued until he reached Speer Avenue, before turning west. Another two miles and the Denver Country Club came into view.\n\nThe big, high-peaked building was ablaze with lights that streamed from the numerous huge square windows that encircled the building. The drive in front of the main entrance was packed with parked carriages and automobiles whose drivers and chauffeurs stood in groups, conversing and smoking. Two men in white tie and tails could be seen checking the invitations of the people who entered.\n\nBell was certain he would cause too much attention by riding up to the entrance on his motorcycle. And, without an invitation, there was little chance of bluffing his way inside even though he was dressed for the occasion. Under the partial light from the half-moon, he turned the handlebars and rode through the night onto the golf course. Careful to stay off the greens and out of the sand traps, he made a wide circular detour and approached the caddy shack that sat behind the main building near the first tee. The interior was dark and the shack was deserted.\n\nHe shut off the ignition and coasted into a clump of bushes beside it. He raised the motorcycle onto its kickstand and removed the long linen coat, draping it over the handlebars. Then he took off the leggings, cap, and goggles. Smoothing back his blond hair, Bell stepped into the light and began strolling up the path leading from the caddy shack to the stately clubhouse. The whole area was illuminated by lustrous electric lights through the windows and tall lamps beside a narrow road that ran from the street to the rear of the country club. Several trucks stood below a wide stairway rising to the rear entrance. Caterers in blue, military-tailored uniforms carried trays of dishes and utensils from the trucks into the kitchen.\n\nUp the stairs, Bell went between two of the caterers, moving into the kitchen as though he owned it. None of the waiters rushing in and out of the dining-room doors carrying trays of food, or the chefs, paid him the slightest attention. For all they knew, the tall man in the tuxedo was one of the reigning managers of the country club. If he had a problem gaining entry into the dining room, it was thankfully eliminated. He simply pushed open one of the kitchen's swinging door, and stepped into the crowd of refined members of the club, walking between the tables, his eyes searching for Rose Manteca.\n\nAfter only two minutes scanning the tables, he spotted her on the dance floor.\n\nBell stiffened.\n\nRose was dancing with Nicholas Alexander.\n\nHe thought fleetingly of enjoying the expressions on their faces when he walked up and asked to cut in. But discretion was a wiser choice than ego. He had seen more than he had bargained for. Now he knew the spy's identity. But Bell was certain that Alexander was not a paid agent for the Butcher Bandit and his female snoop. He was merely a fool and a dupe for a pretty face. He was pleased that they had not noticed him.\n\nBell placed a napkin over his arm and took hold of a coffeepot as though he was waiting on a table. He could hold up the pot in front of his face, should either Rose or Alexander look in his direction. The music stopped, and he watched as they walked back to a table. They were seated together, with Alexander between Rose and an older, heavily jowled woman Bell took to be the agent's wife. If it proved nothing else, it proved that they hadn't met casually for a dance. Seated together meant that their table was reserved in advance. They were no strangers.\n\nBell stared openly at Rose. She wore a red silk dress that nearly matched her flaming hair. This night it was a combination of a bun in the back and curls along the sides and front. Her breasts were pressed against silk fringe that edged the bodice of her dress and swelled into twin, white mounds. She was a beautiful woman from toes to hair.\n\nHer lips were parted in a delightful laugh and her golden brown eyes twinkled in mirth. Her hand fell on Alexander's arm, indicating to Bell that she liked to be physical. A sense of excitement surrounded her that was contagious to those at the table. She was a charmer, gorgeous and ravishing, but her aura did not penetrate Bell. He felt no fire, no passion of arousal toward her. In his analytical mind, she was the enemy, not an object of desire. He saw through the transparent veneer of her loveliness to the cunning and guile beneath.\n\nHe decided he had seen enough. Quickly, he ducked behind a waiter who was heading back to the kitchen and walked beside him until they passed through the swinging doors.\n\nAs Bell put on the gear he'd left hanging on the motorcycle, he considered himself lucky. He had stumbled on a situation he had not fully expected but one he could profit from. As he rode back to the Brown Palace, he knew the only information that he'd feed to Alexander would be false and misleading. He might even conjure up a bit of trickery to beguile Rose Manteca.\n\nThat part of his plan intrigued him. Already, he felt as if he had a head start in tracking a cagey lioness.\n\n## 7\n\nSHORTLY AFTER BELL RETURNED TO THE OFFICE THE next morning, a runner from the telegraph office brought him a telegram from Van Dorn.\n\nMy chief agent in Los Angeles reports that he can find no trace of a Rose Manteca. There is no family by that name owning a ranch within two hundred miles of the city. It looks to me as if the lady has pulled the wool over your eyes. Was she pretty?\n\nVan Dorn\n\nBell smiled to himself. He stuffed the telegram in his pocket, walked to Alexander's office, and knocked on the door.\n\n\"Come in,\" Alexander said softly, as if talking to somebody in the same room.\n\nBarely hearing the words, Bell stepped inside.\n\n\"You're here to report, I assume,\" said Denver's head agent without prelude.\n\nBell nodded. \"I wanted to bring you up to date on our activities.\"\n\n\"I'm listening,\" Alexander said without looking up from the papers on his desk or offering Bell a chair.\n\n\"I've sent Curtis and Irvine out into the field to question the law enforcement officers and any witness to the robberies and killings,\" Bell lied.\n\n\"It's not likely they will dig up anything the local law officials haven't already provided us.\"\n\n\"I intend to leave myself on the next train to Los Angeles.\"\n\nAlexander looked up, a suspicious expression in his eyes. \"Los Angeles? Why would you go there?\"\n\n\"I'm not,\" Bell answered. \"I'm getting off in Las Vegas and taking the spur line to Rhyolite, where I plan to talk to witnesses, if any, on my own.\"\n\n\"A wise plan.\" Alexander almost looked relieved. \"I thought for a moment that you were going to Los Angeles because of Miss Manteca.\"\n\nBell feigned surprise. \"You know her?\"\n\n\"She sat at my table with my wife and me at the country club party and dance. We've met on other occasions. She said you two had met at the Orphans Ball, and she seemed very interested in your work and background. She was especially fascinated by the bank robber\/ murderer.\"\n\nI'll bet she was interested in my work, Bell thought. But he said, \"I didn't know I made an impression on her. She did a pretty good job of brushing me off.\"\n\n\"My wife thought Miss Manteca was smitten with you.\"\n\n\"Hardly. All I learned about her was that she came from a wealthy family in Los Angeles.\"\n\n\"That's true,\" Alexander replied out of ignorance. \"Her father owns a huge spread outside the city.\"\n\nIt was obvious to Bell that Alexander had neither investigated Rose nor bothered to be suspicious of her questions about him and the Butcher Bandit case.\n\n\"When do you expect to return?\" asked Alexander.\n\n\"I should wind up the Rhyolite investigation and be back within five days.\"\n\n\"And Curtis and Irvine?\"\n\n\"Ten days to two weeks.\"\n\nAlexander refocused his attention on the papers atop his desk. \"Good luck,\" he said briefly, dismissing Bell.\n\nReturning to the conference room, Bell relaxed in a swivel chair and propped his feet on the long table. He sipped coffee from a cup Mrs. Murphy had brought earlier. Then he leaned back and stared at the ceiling, as if seeing something on the floor above.\n\nSo his suspicions about Rose Manteca were right on the money. She was not only a fraud but perhaps somehow connected to the Butcher Bandit, and sent to learn what she could of the Van Dorn Detective Agency's investigation. Bell's quarry could never be overestimated. He was no ordinary bandit. Hiring the services of a lovely spy was the work of a man who carefully thought out his operation. Rose, or whatever her true identity was, was good. She had no problem burrowing into the confidence of the Denver office director. The groundwork had been carefully laid. It was clearly the work of a professional. Employing a counterfeit meant the bandit had first-rate resources and a network of tentacles that could delve into government and the business community.\n\nWHEN BELL returned to the Brown Palace, he went to the desk and asked for Rose Manteca's room number. The clerk looked very official when he said, \"I'm sorry, sir. We can't give out our guests' room numbers.\" Then a smug look came across his face. \"But I can tell you that Miss Manteca checked out at noon.\"\n\n\"Did she say where she was going?\"\n\n\"No, but her luggage was taken to the Union Station and placed on the one o'clock train for Phoenix and Los Angeles.\"\n\nThis was not what Bell had expected. He cursed himself for letting her slip through his fingers.\n\nWho really was Rose Manteca? Why would she take the train for Los Angeles when there was no record of her living there?\n\nThen another thought began to tug on Bell's mind. Where would his nemesis strike next? He couldn't even begin to guess and he found it frustrating. He had always felt as if he was in control of his earlier cases. This one was different, too different.\n\n## 8\n\nTHE BLOND-HAIRED MAN WITH A THICK, YELLOW-BROWN, pomaded handlebar mustache had a prosperous appearance about him. After walking through the train depot, he settled into the backseat of the Model N Ford taxicab and enjoyed a beautiful, cloudless day as he viewed the sights of Salt Lake City nestled beneath the Wasatch Mountains. He was dressed in the neat, dandified fashion of the day, but with a sophisticated business look. He wore a silk top hat, a black, three-button cutaway frock coat with vest and high rounded collar, and an elegant tie. His hands were encased in pearl gray kid gloves, and matching spats covered his midstep to just above the ankle over his shoes.\n\nHe leaned slightly forward as he stared from window to window, his hands gripping the handle of a sterling silver cane adorned with an eagle's head with a large beak on the end. Though it was innocent-looking, this cane was a gun with a long barrel and a trigger that folded out when a button was pressed. It held a .44 caliber bullet whose shell could be ejected and a new cartridge inserted in the barrel from a small clip in the eagle's tail.\n\nThe cab passed the church of the Latter-Day Saints\u2014the Temple, Tabernacle, and Assembly Hall. Built between 1853 and 1893, the six-foot-thick gray granite walls were topped by six spires, the highest bearing a copper statue of the angel Moroni.\n\nAfter leaving Temple Square, the cab turned down 300 South Street and came to a stop in front of the Peery Hotel. Designed with European architecture only a short time earlier during the mining boom, it was Salt Lake City's premier hostelry. As the doorman retrieved the luggage from the rear of the cab, the man ordered the driver to wait. Then he entered through the cut-glass double doors into the stately lobby.\n\nThe desk clerk smiled and nodded. Then he glanced at a large clock standing in the lobby and said, \"Mr. Eliah Ruskin, I presume.\"\n\n\"You presume right,\" answered the man.\n\n\"Two-fifteen. You're right on time, sir.\"\n\n\"For once, my train was punctual.\"\n\n\"If you will please sign the register.\"\n\n\"I have to leave for an appointment. Will you see my luggage is taken to my room and my clothing placed in the closet and drawers?\"\n\n\"Yes, Mr. Ruskin. I will personally see to it.\" The clerk leaned over the registry desk and nodded at a large leather suitcase set securely between Ruskin's legs. \"Would you like me to send your bag up to your room?\"\n\n\"No, thank you. I'll be taking it with me.\"\n\nRuskin turned and walked out to the curb, cane in one hand, the other clutching the handle of the suitcase, the weight of its contents tilting his right shoulder downward. He pushed it through the cab's door and reentered the backseat.\n\nThe desk clerk thought it odd that Ruskin hadn't left the bag in the cab. He wondered why Ruskin would lug such a heavy case into the lobby and then carry it outside again. He speculated that something of value must be inside. His thought soon faded when another guest showed up to register.\n\nEight minutes later, Ruskin stepped from the cab, paid the driver, and entered the Salt Lake Bank & Trust lobby. He walked to the security guard who was seated in a chair near the door.\n\n\"I have an appointment with Mr. Cardoza.\"\n\nThe guard rose to his feet and motioned toward a frosted-glass door. \"You'll find Mr. Cardoza in there.\"\n\nThere had been no reason for Ruskin to ask the guard where to go. He could just as easily have seen the bank manager's office door. The guard did not notice that Ruskin had observed him closely, how he moved, his age, and how he placed the holster, containing a new .45 caliber Model 1905 Colt Browning automatic pistol, at his hip. The brief study also revealed the guard was not particularly alert and watchful. Day after day of seeing customers come and go without the slightest disturbance had made him listless and indifferent. He didn't appear to find anything unusual about Ruskin's big case.\n\nThe bank had two tellers behind the counter in their cages. The only other employees except for the guard were Cardoza and his secretary. Ruskin studied the big steel door to the vault that was open to the lobby to impress the customers and suggest that their savings were in solid, protective hands.\n\nHe approached the secretary. \"Hello, my name is Eliah Ruskin. I have a two-thirty appointment with Mr. Cardoza.\"\n\nAn older woman in her fifties with graying hair smiled and stood up without speaking. She walked to a door with ALBERT CARDOZA, MANAGER painted on the upper part of the frosted-glass pane, knocked, and leaned in. \"A Mr. Eliah Ruskin to see you.\"\n\nCardoza quickly came to his feet and rushed around his desk. He shoved out a hand and shook Ruskin's palm and fingers vigorously. \"A pleasure, sir. I've looked forward to your arrival. It's not every day we greet a representative from a New York bank that is making such a substantial deposit.\"\n\nRuskin lifted the suitcase onto Cardoza's desk, unlocked the catches, and opened the lid. \"Here you are, half a million dollars in cash to be deposited, until such time we decide to withdraw it.\"\n\nCardoza reverently stared at the neatly packed and bundled fifty-dollar gold certificate bills as though they were his passport to a banker's promised land. Then he looked up in growing surprise. \"I don't understand. Why not carry a cashier's check instead of five hundred thousand dollars in currency?\"\n\n\"The directors of the Hudson River Bank of New York prefer to deal in cash. As you know from our correspondence, we are going to open branch banks throughout the West in towns that we think have potential for growth. We feel it is expedient to have currency on hand when we open our doors.\"\n\nCardoza looked at Ruskin somberly. \"I hope your directors do not intend to open a competing bank in Salt Lake City.\"\n\nRuskin grinned and shook his head. \"Phoenix, Arizona, and Reno, Nevada, are the first of the Hudson River branch banks to open in the West.\"\n\nCardoza looked relieved. \"Phoenix and Reno are certainly booming.\"\n\n\"Ever have a bank robbery in Salt Lake?\" Ruskin asked casually while looking at the vault.\n\nCardoza looked at him quizzically. \"Not in this city. The citizens would not allow it. Salt Lake is one of the most crime-free cities in the country. The Latter-Day Saints are upstanding and religious people. Trust me, Mr. Ruskin, no bandit would dare to attempt a robbery of this bank. Your money will be absolutely and one hundred percent safe once it's locked up in our vault.\"\n\n\"I've read of some fellow called the Butcher Bandit who robs and murders throughout the western states.\"\n\n\"Not to worry, he only strikes in small mining towns and robs payrolls. He wouldn't be stupid enough to try robbing a bank in a city the size of Salt Lake. He wouldn't get past the city limits before the police shot him down.\"\n\nRuskin nodded toward the vault. \"Very impressive repository.\"\n\n\"The very finest vault west of the Mississippi, built especially for us in Philadelphia,\" Ruskin said proudly. \"An entire regiment armed with cannons couldn't break inside.\"\n\n\"I see it is open during business hours?\"\n\n\"And why not. Our customers enjoy seeing how well their deposits are protected. And as I've mentioned, no bank has ever been robbed in Salt Lake City.\"\n\n\"What is your slowest time of day?\"\n\nCardoza looked puzzled. \"Slowest time of day?\"\n\n\"When you have the least customer transactions?\"\n\n\"Between one-thirty and two o'clock is our slowest time. Most of our customers have gone back to their offices after their lunch hour. And, because we close at three, a number of customers come in for late transactions. Why do you ask?\"\n\n\"Just curious as to how the traffic compares with our bank in New York, which seems to be about the same.\" He patted the suitcase. \"I'll leave the money in the case and pick it up tomorrow.\"\n\n\"We'll close shortly, but I'll have my head clerk count it first thing in the morning.\"\n\nCardoza pulled open a drawer of his desk, retrieved a leather book, and wrote out a deposit slip for the half-million dollars. He handed it to Ruskin, who inserted it into a large wallet he carried in the breast pocket of his coat.\n\n\"May I ask a favor?\" Ruskin inquired.\n\n\"Certainly. Anything you wish.\"\n\n\"I would like to be on hand when your clerk does the count.\"\n\n\"That's very gracious of you, but I'm sure your bank has accounted for every dollar.\"\n\n\"I'm grateful for your trust, but I would like to be present just to be on the safe side.\"\n\nCardoza shrugged. \"As you wish.\"\n\n\"There is one other request.\"\n\n\"You have but to name it.\"\n\n\"I have other business to conduct in the morning and cannot return until one-thirty tomorrow. And, since your business is slowest then, it should be a good time for the count.\"\n\nCardoza nodded in agreement. \"You're quite right.\" He stood and extended his hand. \"Until tomorrow afternoon. I look forward to seeing you.\"\n\nRuskin held up his cane as a good-bye gesture, dismissed Cardoza, and left the office. He walked past the security guard, who didn't give him a glance, and swung his cane like a baton as he stepped onto the sidewalk.\n\nHe smiled to himself, knowing that he had no intention of returning to the bank merely to count the contents of the suitcase.\n\n## 9\n\nTHE NEXT AFTERNOON, RUSKIN WALKED TO THE BANK, making sure he was seen on the street by the passing crowd and stopping in shops to browse, making small talk with the merchants. He carried his gun cane more as a prop than for protection.\n\nReaching the Salt Lake Bank & Trust at one-thirty, he entered and ignored the guard as he turned the key in the front entrance door, locking it. Then he turned the sign around in the window so that it read CLOSED from the street and pulled down the window shades, as the guard sat there in his bored stupor, not realizing that the bank was about to be robbed. Neither Albert Cardoza's secretary and the tellers nor the female depositor standing at the counter took notice of the intruder's unusual behavior.\n\nThe guard finally came alert and realized that Ruskin was not acting like a normal bank customer and might be up to no good. He came to his feet, his hand dropping to the holster holding his .38 Smith & Wesson revolver, and asked blankly, \"Just what do you think you're doing?\" Then his eyes widened in alarm as he found himself staring into the muzzle of Ruskin's .38 Colt.\n\n\"Make no resistance, and walk slowly behind the counter!\" Ruskin ordered as he wrapped his gun in a battered, old heavy woolen scarf with burn holes in it. He quickly moved behind the counter before the clerks in their cages became alert and could make a grab for the shotguns at their feet. Never expecting their bank to be robbed, they hesitated in confusion.\n\n\"Don't even think about going for your guns!\" Ruskin snapped. \"Lay flat on the floor or you'll get a bullet in your brain.\" He motioned his cane at the frightened woman at the counter. \"Come around the counter and lay down on the floor with the tellers and you won't get hurt,\" he said in a cold tone. Then he motioned the gun at Cardoza's secretary. \"You, too! Down on the floor!\"\n\nWhen all were lying on the highly polished mahogany floor facedown, he rapped on Cardoza's door. Unable to distinguish voices outside his office, the bank's manager was not aware of the macabre event unfolding within his bank. He waited out of habit for his secretary to enter, but she did not appear. Finally, irritated at being interrupted, he stepped from his desk and opened the door. It took him a full ten seconds to comprehend what was happening. He stared at Ruskin and the gun in his hand.\n\n\"What is the meaning of this?\" he demanded. Then he saw the people lying on the floor and looked back at Ruskin in utter confusion. \"I don't understand. What is going on?\"\n\n\"The first bank robbery of Salt Lake City,\" said Ruskin, as if amused.\n\nCardoza did not move. He was frozen in shock. \"You're a director of a respectable New York bank. Why are you doing this? It makes no sense. What do you hope to gain by it?\"\n\n\"I have my motives,\" Ruskin answered, his voice cold and toneless. \"I want you to make out a bank draft for four hundred seventy-five thousand dollars.\"\n\nCardoza stared at him as if he was crazy. \"A bank draft to whom?\"\n\n\"Eliah Ruskin, who else?\" answered Ruskin. \"And be quick about it.\"\n\nMired in confusion, Cardoza pulled open a drawer, retrieved a book containing bank drafts, and hurriedly scribbled out one for the amount Ruskin demanded. When finished, he passed it across the desk to Ruskin, who slipped it into his breast pocket.\n\n\"Now, down on the floor with the others.\"\n\nAs if in the throes of a nightmare, Cardoza slowly lowered himself onto the floor next to his trembling secretary.\n\n\"Now, then, none of you move, or even twitch, until I tell you to.\"\n\nWithout saying more, Ruskin walked inside the vault and began stuffing the bank's currency into leather money sacks he'd seen earlier stacked on a shelf inside the huge five-ton door. He filled two of them, estimating the take at roughly two hundred thirty thousand dollars in larger denominations, none under ten dollars. He had planned well. From inside banking information, he knew that the Salt Lake Bank & Trust had received a large shipment of currency issued from the Continental & Commercial National Bank of Chicago for their reserves. The suitcase with his own money he left on another shelf of the vault.\n\nLaying aside the sacks, he closed the vault door. It swung shut as easily as a door on a cupboard. Then he turned the bog wheel that activated the inside latches and set the timer for nine o'clock the next morning.\n\nUnhurriedly, as if he was strolling through a park, he stepped behind the counter and ruthlessly shot the people lying on the floor in the back of the head. The muffled shots came so quickly, none had time to know what was happening and cry out. Then he raised the bank's window shades, so people passing on the sidewalk could see that the vault was shut and would assume the bank was closed. The bodies were conveniently out of sight behind the counter.\n\nRuskin waited until the sidewalk was clear of foot traffic and vehicles before he nonchalantly exited the bank, locked the door, and strolled leisurely from the building, swinging his cane. By four o'clock, he had returned to the Peery Hotel, had a bath, and come down to the restaurant, where he enjoyed a large smoked-salmon plate with dill cream and caviar accompanied by a bottle of French Clos de la Roche Burgundy 1899. Then he read in the lobby for an hour before going to bed and slept like a rock.\n\nLATE IN the morning, Ruskin took a taxi to the Salt Lake Bank & Trust. A crowd of people were clustered around the front door as an ambulance pulled away from the bank. Police in uniforms were in abundance. He pushed his way through the crowd, saw a man who was dressed like a detective, and addressed him.\n\n\"What happened here?\" he asked courteously.\n\n\"The bank has been robbed and five people murdered.\"\n\n\"Robbed, murdered, you say? This is disastrous. I deposited half a million dollars in cash here yesterday from my bank in New York.\"\n\nThe detective looked at him in surprise. \"Half a million dollars, you say? In cash?\"\n\n\"Yes, I have my receipt right here.\" Ruskin flashed the receipt in the detective's face. The detective studied it for a few moments and then said, \"You are Eliah Ruskin?\"\n\n\"Yes, I'm Ruskin. I represent the Hudson River Bank of New York.\"\n\n\"A half million dollars in cash!\" the detective gasped. \"No wonder the bank was robbed. You better come inside, Mr. Ruskin, and meet with Mr. Ramsdell, one of the bank's directors. I'm Captain John Casale, with the Salt Lake Police Department.\"\n\nThe bodies had been removed, but large areas of the mahogany floor were layered in dried blood. Captain Casale led the way to a man\u2014a huge, fat man with a large protruding stomach behind a vest and massive watch chain. The man was sitting at Cardoza's desk, examining the bank's deposits. His brown eyes appeared dazed beneath the bald head. He looked up and stared at Ruskin, annoyed at the intrusion.\n\n\"This is Mr. Eliah Ruskin,\" announced Casale. \"He says he deposited half a million dollars with Mr. Cardoza yesterday.\"\n\n\"Sorry to meet you under such tragic circumstances. I am Ezra Ramsdell, the bank's managing director.\" Ramsdell rose and shook Ruskin's hand. \"A terrible, terrible business,\" he muttered. \"Five people dead. Nothing like this has ever happened in Salt Lake City before.\"\n\n\"Were you aware of the money Mr. Cardoza was holding for my bank?\" asked Ruskin flatly.\n\nRamsdell nodded. \"Yes, he called me on the telephone and reported that you had come in and placed your bank's currency in the vault.\"\n\n\"Since Mr. Cardoza, God rest his soul, wrote me out a receipt, my directors will assume your bank will make good on the loss.\"\n\n\"Tell your directors not to worry.\"\n\n\"How much cash did the robber take?\" Ruskin asked.\n\n\"Two hundred forty-five thousand dollars.\"\n\n\"Plus my half million,\" he said, as if agitated.\n\nRamsdell looked at him queerly. \"For some inexplicable reason, the robber didn't take your money.\"\n\nRuskin simulated a stunned expression. \"What are you telling me?\"\n\n\"The bills in a large, brown leather suitcase,\" said Captain Casale. \"Are those yours?\"\n\n\"The gold certificates? Yes, they are from the bank I represent in New York.\"\n\nRamsdell and Casale exchanged odd looks. Then Ramsdell said, \"The case you and Mr. Cardoza placed in the vault still contains your currency.\"\n\n\"I don't understand.\"\n\n\"It hasn't been touched. I opened and checked it myself. Your gold certificates are safe and sound.\"\n\nRuskin made a show of acting perplexed. \"It doesn't make sense. Why take your money and leave mine?\"\n\nCasale scratched one ear. \"My guess is, he was in a hurry and simply ignored the suitcase, not realizing it was filled with a king's fortune in cash.\"\n\n\"That's a relief,\" said Ruskin, taking off his silk top hat and wiping imaginary sweat from his brow. \"Assuming the robber won't return, I'll leave it in your vault until such time as we require it to open our new branch banks in Phoenix and Reno.\"\n\n\"We are most grateful. Especially now that our currency on hand has been wiped out.\"\n\nRuskin looked around at the spread of dried blood on the floor. \"I should leave you to your investigation.\" He nodded at Casale. \"I trust you will catch the killer so he can be hung.\"\n\n\"I swear we'll track him down,\" Casale said confidently. \"Every road out of Salt Lake and all the train depots are covered by a network of police officers. He can't travel beyond the city limits without being caught.\"\n\n\"Good luck to you,\" said Ruskin. \"I pray you will apprehend the fiend.\" He turned to Ramsdell. \"I will be at the Peery Hotel until tomorrow afternoon, should you require my services. At four o'clock, I will board a train, to oversee the establishment of our new bank in Phoenix.\"\n\n\"You are most generous, sir,\" said Ramsdell. \"I will be in touch as soon as we resume operations.\"\n\n\"Not at all.\" Ruskin turned to leave. \"Good luck to you, Captain,\" Ruskin said to Casale as he made for the front entrance of the bank.\n\nCasale stared out the window as Ruskin walked across the street toward a taxi. \"Most strange,\" he said slowly. \"If I know my train schedules, the next train for Phoenix doesn't leave for another three days.\"\n\nRamsdell shrugged. \"He was probably misinformed.\"\n\n\"Still, there is something about him that bothers me.\"\n\n\"What is that?\"\n\n\"He didn't look overjoyed that his bank's money was not taken by the robber. It was almost as if he knew it was safe before he walked in the door.\"\n\n\"Does it matter?\" asked Ramsdell. \"Mr. Ruskin should be glad his half a million dollars was overlooked by the robber.\"\n\nThe detective looked thoughtful. \"How do you know it's a half a million dollars? Did you count it?\"\n\n\"Mr. Cardoza must have counted it.\"\n\n\"Are you certain?\"\n\nRamsdell began walking from the office toward the vault. \"Now is as good a time as any to make a quick tally.\"\n\nHe opened the case and started to lay the first layer of stacked bills on a nearby shelf. The top layer consisted of twenty thousand dollars in gold certificate bills. Underneath, the rest of the case was filled with neatly cut and banded newspaper.\n\n\"Good God!\" Ramsdell gasped. Then, as if struck by a revelation, he rushed back to the bank manager's office and opened a book that lay on the surface of the desk. The book contained bank drafts\u2014but the final draft was missing and unrecorded. His face went ashen. \"The murdering scum must have forced Cardoza to write a bank draft for the half million. Whatever bank he deposits it in will assume we authorized it and demand payment from Salt Lake Bank and Trust. Under federal law, we are bound to honor it. If not, the lawsuits, the prosecution from United States Treasury agents\u2014we'd be forced to close.\"\n\n\"Ruskin was not only a fraud,\" Casale said firmly, \"he was the one who robbed your bank and murdered your employees and customer.\"\n\n\"I can't believe it,\" muttered Ramsdell incredulously. Then he demanded, \"You've got to stop him. Catch him before he checks out of his hotel.\"\n\n\"I'll send a squad to the Peery,\" said Casale. \"But this guy is no buffoon. He probably went on the run as soon as he walked out the door.\"\n\n\"You can't let him get away with this foul deed.\"\n\n\"If he's the notorious Butcher Bandit, he's a shrewd devil who vanishes like a ghost.\"\n\nEzra Ramsdell's eyes took on an astute glint. \"He has to deposit the draft at a bank somewhere. I'll telegraph the managers of every bank in the nation to be on the lookout for him and contact the police before they honor a draft made out to Eliah Ruskin for half a million dollars. He won't get away with it.\"\n\n\"I'm not so sure,\" John Casale said softly under his breath. \"I'm not so sure at all.\"\n\n## 10\n\nTHE BUTCHER BANDIT WAS A COUNTRY MILE AHEAD of him, Bell thought as the train he was riding slowed and stopped at the station in Rhyolite. He had received a lengthy telegram from Van Dorn telling of the Salt Lake massacre, as it had become known. A bank in a major city like Salt Lake was the last place he or anyone else expected the Butcher to strike. That was his next stop after Rhyolite.\n\nHe stepped from the train with a leather bag that held the bare essentials he carried while traveling. The heat of the desert struck him like a blast furnace, but because of the absence of humidity in the desert it did not soak his shirt with sweat.\n\nAfter getting directions from the stationmaster, he walked to the sheriff's office and jail. Sheriff Marvin Huey was a medium-sized man with a head of tousled gray hair. He looked up from a stack of wanted posters and stared at Bell with soft olive brown eyes as the Van Dorn agent entered the office.\n\n\"Sheriff Huey, I'm Isaac Bell from the Van Dorn Detective Agency.\"\n\nHuey did not rise from his desk nor offer his hand; instead, he spit a wad of chewing tobacco juice into a cuspidor. \"Yes, Mr. Bell, I was told you'd be on the ten o'clock train. How do you like our warm weather?\"\n\nBell took a chair across from Huey without it being offered and sat down. \"I prefer the high-altitude cool air of Denver.\"\n\nThe sheriff grinned slightly at seeing Bell's discomfort. \"If you lived here long enough, you might get to like it.\"\n\n\"I wired you concerning my investigation,\" Bell said without preamble. \"I want to obtain any information I can that would be helpful in tracking down the Butcher Bandit.\"\n\n\"I hope you have better luck than I did. After the murders, all we found was a dilapidated, abandoned freight wagon and team of horses that he had driven into town.\"\n\n\"Did anyone get a good look at him?\"\n\nHuey shook his head. \"No one gave him the slightest notice. Three people gave different descriptions. None matched. All I know is, my posse found no tracks from wagon, horse, or automobile leading out of town.\"\n\n\"What about the railroad?\"\n\nHuey shook his head. \"No train left town for eight hours. I posted men at the depot who searched the passenger cars before it left, but they found no one that looked suspicious.\"\n\n\"How about freight trains?\"\n\n\"My deputies ran a search of the only freight train that left town that day. Neither they nor the train engineer, fireman, or brakemen saw anyone hiding on or around the boxcars.\"\n\n\"What is your theory on the bandit?\" asked Bell. \"How do you think he made a clean getaway?\"\n\nHuey paused to shoot another wad of tobacco saliva into the brass cuspidor. \"I gave up. It pains me to say so, but I have no idea how he managed to elude me and my deputies. Frankly, I'm put out by it. In thirty years as a lawman, I've never lost my man.\"\n\n\"You can take consolation in knowing you're not the only sheriff or marshal who lost him after he robbed their town banks.\"\n\n\"It still isn't anything I can be proud of,\" muttered Huey.\n\n\"With your permission, I would like to question the three witnesses.\"\n\n\"You'll be wasting your time.\"\n\n\"May I have their names?\" Bell persisted. \"I have to do my job.\"\n\nHuey shrugged and wrote out three names on the back of a wanted poster, and where they could be found, handing it to Bell. \"I know all these people. They're good, honest citizens who believe what they saw even if it don't match up.\"\n\n\"Thank you, Sheriff, but it is my job to investigate every lead, no matter how insignificant.\"\n\n\"Let me know if I can be of further help,\" said Huey, warming up.\n\n\"If need be,\" said Bell, \"I will.\"\n\nBELL SPENT most of the next morning locating and questioning the people on the list given him by Sheriff Huey. Bell was considered an expert at drawing on witnesses' descriptions, but this time around he drew a blank. None of the people, two men and one woman, gave correlating accounts. Sheriff Huey was right. He accepted defeat and headed back to his hotel and prepared to leave for the next town on his schedule that had suffered a similar tragedy: Bozeman, Montana.\n\nHe was sitting in the hotel restaurant, eating an early dinner of lamb stew, when the sheriff walked in and sat down at his table.\n\n\"Can I order you anything?\" Bell asked graciously.\n\n\"No thanks. I came looking for you because I thought of Jackie Ruggles.\"\n\n\"And who might that be?\"\n\n\"He's a young boy of about ten. His father works in the mine and his mother takes in laundry. He said he saw a funny-looking man the day of the robbery, but I dismissed his description. He's not the brightest kid in town. I figured he wanted to impress the other boys by claiming he'd seen the bandit.\"\n\n\"I'd like to question him.\"\n\n\"Go up Third Street to Menlo. Then turn right. He lives in the second house on the left, a ramshackle affair that looks like it may fall down any minute, like most of the houses in that area of town.\"\n\n\"I'm obliged.\"\n\n\"You won't get any more out of Jackie than you did from the others, probably less.\"\n\n\"I have to look on the bright side,\" said Bell. \"As I said, we have to check out every lead, no matter how trivial. The Van Dorn Detective Agency wants the killer as much as you.\"\n\n\"You might stop by the general store and pick up some gumdrops,\" Sheriff Huey said. \"Jackie has a sweet tooth for gumdrops.\"\n\n\"Thanks for the tip.\"\n\nBELL FOUND the Ruggles house just as Huey described. The entire wooden structure was leaning to one side. Another two inches, Bell thought, and it would crash into the street. He started up the rickety stairs just as a young boy dashed out of the front door and ran toward the street.\n\n\"Are you Jackie Ruggles?\" Bell asked, grabbing the boy by the arm before he dashed off.\n\nThe boy wasn't the least bit intimidated. \"Who wants to know?\" he demanded.\n\n\"My name is Bell. I'm with the Van Dorn Detective Agency. I'd like to ask you about what you saw the day of the bank robbery.\"\n\n\"Van Dorn,\" Jackie said in awe. \"Gosh, you guys are famous. A detective from Van Dorn wants to talk to me?\"\n\n\"That's right,\" said Bell, swooping in for the kill. \"Would you like some gumdrops?\" He held out a small sack that he had just purchased at the general store.\n\n\"Gee, thanks, mister.\" Jackie Ruggles wasted no time in snatching the sack and savoring a green gumdrop. He was dressed in a cotton shirt, pants that were cut off above the knee, and worn-leather shoes that Bell guessed were handed down by an older brother. The clothes were quite clean, as befitting a mother who was a laundress. He was thin as a broomstick, with boyish facial features that were covered with freckles, and topped by a thicket of uncombed curly light brown hair.\n\n\"I was told by Sheriff Huey that you saw the bank robber.\"\n\nThe boy answered while chewing on the gumdrop. \"Sure did. The only trouble is, nobody believes me.\"\n\n\"I do,\" Bell assured him. \"Tell me what you saw.\"\n\nJackie was about to reach in the sack for another gumdrop, but Bell stopped him. \"You can have them after you've told me what you know.\"\n\nThe boy looked peeved but shrugged. \"I was playing baseball in the street with my friends when this old guy\u2014\"\n\n\"How old?\"\n\nJackie studied Bell. \"About your age.\"\n\nBell never considered thirty as old, but to a young boy of ten he must have appeared ancient. \"Go on.\"\n\n\"He was dressed like most of the miners who live here, but he wore a big hat like the Mexicans.\"\n\n\"A sombrero.\"\n\n\"I think that's what it's called.\"\n\n\"And he was toting a heavy sack over his shoulder. It looked like it was plumb full of something.\"\n\n\"What else did you notice?\"\n\n\"One of his hands was missing the little finger.\"\n\nBell stiffened. This was the first clue to identifying the killer. \"Are you sure he was missing a little finger?\"\n\n\"As sure as I'm standing here,\" answered Jackie.\n\n\"Which hand?\" Bell asked, containing his mounting excitement.\n\n\"The left.\"\n\n\"You've no doubt it was the left hand?\"\n\nJackie merely nodded while staring longingly at the gumdrop sack. \"He looked at me like he was really mad when he saw I was looking back.\"\n\n\"Then what happened?\"\n\n\"I had to catch a fly ball. When I turned around, he was gone.\"\n\nBell patted Jackie on the head, almost losing his hand in a sea of unruly red hair. He smiled. \"Go ahead and eat your gumdrops, but, if I were you, I'd chew slowly so they last longer.\"\n\nAFTER HE checked out of the Rhyolite Hotel and before he boarded the train, Bell paid the telegraph operator at the depot to send a wire to Van Dorn describing the Butcher Bandit as missing the little finger on his left hand. He knew that Van Dorn would quickly send out the news to his army of agents to watch out for and report any man with that disfigurement.\n\nInstead of traveling back to Denver, he decided on the spur of the moment to go to Bisbee. Maybe\u2014just maybe\u2014he might get lucky again and find another clue to the bandit's identity. He leaned back in his seat, as the torrid heat of the desert grilled the interior of the Pullman car. Bell hardly noticed it.\n\nThe first solid clue, provided by a scrawny young boy, wasn't exactly a breakthrough, but it was a start, thought Bell. He felt pleased with himself for the discovery and began to daydream of the day he confronted the bandit and identified him by the missing finger.\n\n## THE CHASE QUICKENS\n\n## 11\n\nMARCH 4, 1906 SAN FRANCISCO, CALIFORNIA\n\nTHE MAN WHOSE LAST ALIAS HAD BEEN RUSKIN stood in front of an ornate brass sink and stared into a large oval mirror as he shaved with a straight razor. When finished, he rinsed off his face and patted on an expensive French cologne. He then reached out and clutched the sink as his railroad boxcar came to an abrupt stop.\n\nHe stepped up to a latched window, disguised from the outside as if it were a section of the wooden wall of the car, cautiously cracked it, and peered outside. A steam switch engine had pushed ten freight cars uncoupled from the train, including the O'Brian Furniture car, through the Southern Pacific Railroad's huge terminal building, called the Oakland Mole. It consisted of a massive pier built on pilings, masonry, and rock laid in the San Francisco Bay itself, on the west side of the city of Oakland. The slip where the ferryboats entered and tied up was at the west end of the main building, between twin towers. The towers were manned by teams of men who directed the loading and unloading of the huge fleet of ferries that moved to and from San Francisco across the bay.\n\nBecause the Oakland Mole was at the end of the transcontinental railway, it was filled twenty-four hours a day with a mixed crowd of people, coming from the east and heading across the continent in the opposite direction. Passenger trains commingled with freight trains that carried goods and merchandise. It was a busy place in 1906, since business was booming in the sister cities of the bay. San Francisco was a thriving commercial center while much of the actual goods were manufactured in Oakland.\n\nRuskin checked a schedule and saw that his cleverly disguised mode of secret travel was on board the San Gabriel, a Southern Pacific Railroad ferry built to haul freight trains as well as passengers. She was a classic ferry, double-ended, her stern and bow surmounted with a pilothouse on each end. She was propelled with side paddle wheels powered by two walking-beam steam engines, each with its own smokestack. Ferries carrying trains had parallel tracks on the main deck for the freight cars, while the cabin deck housed the passengers. The San Gabriel was two hundred ninety-eight feet long, seventy-eight feet wide, and could carry five hundred passengers and twenty railroad cars.\n\nThe San Gabriel was to arrive at the Townsend and Third Streets Southern Pacific terminus, where the passengers would disembark. Then it would move on to Pier 32 at Townsend and King Streets, where its cargo of railroad cars would be taken to the city railyard between Third and Seventh Streets. There, the O'Brian Furniture Company car would be switched to the siding of a warehouse the bandit owned in the city's industrial section.\n\nRuskin had ridden on the San Gabriel many times on his trips across the bay and looked forward to returning home after his venture in Salt Lake City. A great whoop echoed around the Mole from the boat's steam whistle as it announced her departure. She began to tremble when the tall walking-beam engines turned the big twenty-seven-foot paddle wheels that churned the water. Soon the boat was riding the glass-smooth bay toward San Francisco, no more than twenty minutes away.\n\nRuskin quickly finished dressing in an exactingly tailored conservative black business suit. A small yellow rose went in the buttonhole. He sat a derby hat on his head at a rakish angle and pulled a pair of suede gloves over his hands. He picked up his cane.\n\nThen he bent down and gripped the handle to the trapdoor in the floor of the freight car and swung it open. He dropped a large, heavy suitcase through the opening. Then he slowly lowered himself to the deck between the rails, careful not to soil his clothes. Hunched down under the car, he made certain none of the crew were within view as he moved away and straightened up.\n\nRuskin was headed up a stairway to the cabin deck where the passengers rode when, halfway up, he met a crewman coming down. The crewman stopped and nodded at him, a serious expression on his face.\n\n\"Are you aware, sir, that passengers are not allowed on the main deck?\"\n\n\"Yes, I'm aware.\" The bandit smiled. \"I realized my mistake, and, as you see, I was turning around to return to the cabin deck.\"\n\n\"Sorry to have troubled you, sir.\"\n\n\"Not at all. It's your duty.\"\n\nThe bandit proceeded up the stairs and stepped into the ornate, highly decorated cabin deck where the passengers crossed the bay in style. He went into the restaurant and ordered a cup of tea at the stand-up bar, then walked outside onto the open forward deck and sipped as he watched the buildings of San Francisco grow larger across the bay.\n\nThe City by the Bay was already becoming a fascinating, romantic, cosmopolitan city. It had risen in stature since 1900, establishing itself as the financial and merchandising hub of the West. It was built on the foresight of entrepreneurs much like the meticulously dressed man standing on the deck with the huge suitcase. He, like they, saw an opportunity and moved quickly to seize it.\n\nNot one for niceties, Ruskin finished his tea and then threw the cup overboard, not returning it to the restaurant. He idly watched a thick flight of sandpipers fly past the boat, followed by a trio of brown pelicans gliding inches above the water in search of small fish. Then, mingling with the throng, he moved down the forward stairway to the main deck, where the passengers disembarked the ferry onto the pier in front of the big, ornate, Spanish-style Southern Pacific terminus.\n\nHe walked briskly through the interior cavern of the terminus, lugging the big suitcase, and through the doors on the Townsend Street side. For the next few minutes, he stood on the sidewalk and waited. He smiled as a white Mercedes Simplex runabout rolled up the street and came to a tire-skidding halt at the curb in front of him. Under the hood was a massive four-cylinder, sixty-horsepower engine that could move the car as fast as eighty miles an hour. It was a marvelous contrivance of steel, brass, wood, leather, and rubber. Driving it was sheer adventure.\n\nIf the car produced a striking picture, so did the woman behind the steering wheel. She was svelte and wasp-waisted. Her red hair was adorned with a large red bow that matched her fiery hair. Her bonnet was tied under her chin to keep it from blowing away, and she wore a tan linen dress that came halfway up her calves so she could dance her feet deftly over the five floor pedals. She took one hand from the big steering wheel and waved.\n\n\"Hello, brother. You're an hour and a half late.\"\n\n\"Greetings, sister.\" He paused to grin. \"I could only go as fast as the engineer drove the train.\"\n\nShe offered him her cheek and he dutifully kissed it. She inhaled the smell of him. He always used the French cologne she had given him. It smelled like a sea of flowers after a light evening rain. If he hadn't been her sibling, she might have had a love affair with him.\n\n\"I assume your trip was successful.\"\n\n\"Yes,\" he said, strapping the suitcase on the running board. \"And we haven't a minute to lose.\" He climbed into the brown leather passenger's seat. \"I must record the bank draft I obtained at the Salt Lake Bank and Trust before their agents show up to stop the transfer.\"\n\nShe pushed a laced-up brown leather shoe against the clutch and shifted expertly, as the car leaped down the street like a lion chasing a zebra. \"It took two days for you to get here. Don't you think you're cutting it close? They would have contacted law enforcement officials and hired private agents, prodding them to check all the banks in the country for a stolen bank draft worth a fortune.\"\n\n\"And that takes time, not less than forty-eight hours,\" he added, clutching the side of the seat with a hand since there were no doors on the runabout for support as she made a sharp left turn up Market Street. He barely grabbed his derby with his other hand before it almost flew off into the street.\n\nShe drove fast, seemingly recklessly, but nimbly, smoothly whipping around slower traffic at a speed that turned heads and startled passersby. She hurtled past a big beer truck, pulled by a team of Percheron horses, that blocked most of the street, slipping between the stacked barrels on the street and the sidewalk filled with pedestrians with only inches to spare. He bravely whistled a marching tune called \"Garry Owen\" and tipped his hat at the pretty girls coming out of the clothing stores. The big Market Street electric trolley car loomed ahead, and she crossed into oncoming traffic to pass it, sending more than one horse rearing up on its hind legs, to the anger and fist waving of their drivers.\n\nAnother two blocks through the canyon of brick-and-stone buildings, she came to a quick stop, skidding the rear tires when she hit the brakes, in front of the Cromwell Bank on the southeast corner of Market and Sutter Streets. \"Here you are, brother. I trust you enjoyed the ride.\"\n\n\"You're going to kill yourself someday.\"\n\n\"Blame yourself,\" she said, laughing. \"You gave me the car.\"\n\n\"Trade you my Harley-Davidson for it.\"\n\n\"Not a chance.\" She gave a cheery wave and said, \"Come home early and don't be late. We've a date on the Barbary Coast with the Gruenheims to go slumming and take in one of the scandalous dance revues.\"\n\n\"I can't wait,\" he said sarcastically. He stepped down to the sidewalk before turning and unstrapping the suitcase. She saw that he strained as he lifted and knew it was crammed with stolen currency from the Salt Lake bank.\n\nAt the press of the accelerator pedal, the chain-driven Mercedes Simplex charged across the intersection and roared up the street, the thunder of the exhaust coming within a few decibels of breaking the storefront windows.\n\nThe bandit turned and looked with pride at the big, elaborately ornamented Cromwell Bank Building, with its tall, fluted Ionic column and large stained-glass windows. A doorman in a gray uniform opened one of the big glass doors for him. He was a tall man with gray hair, and a military bearing that came from thirty years in the United States Cavalry.\n\n\"Good morning, Mr. Cromwell. Glad to see you back from your holiday.\"\n\n\"Glad to be back, George. How's the weather been in my absence?\"\n\n\"Just like it is today, sir, sunny and mild.\" George looked down at the large suitcase. \"May I carry that for you, sir?\"\n\n\"No, thank you. I can manage. I need the exercise.\"\n\nA small brass sign listed the bank's assets at twenty-two million dollars. It would soon be twenty-three, thought Cromwell. Only the fifty-year-old Wells Fargo Bank had higher assets, capital, and liquidity. George swung open the door, and Cromwell the bandit strode across the marble floor of the bank's lobby, past the beautifully carved desks of the managers and the tellers' windows and the counters without bars, totally open to the customers. The open tellers' area was a strange innovation by a man who trusted no one and robbed out-of-state banks to build his own financial empire.\n\nThe fact was, Jacob Cromwell no longer needed the additional income he stole for his bank. But he was intoxicated by the challenge. He felt he was invincible. He could match wits with any police investigators, not to mention the agents from the Van Dorn Detective Agency, until he died of old age. He knew from his spies that no one was remotely close to identifying him.\n\nCromwell entered an elevator and rode up to the third floor. He stepped out onto the Italian-tiled floor of the main office on the gallery above the bank's lobby. He walked into the grandeur of his suite of offices, the deep, ivory brown carpet muffling his footsteps. The walls were paneled in teak, with carvings depicting scenes of the nineteenth-century West, while the columns that supported the roof were sculpted in the manner of totem poles. The vast ceiling above had been painted with murals of the early days of San Francisco.\n\nHe employed three secretaries to handle his main business, along with much of his personal affairs. They were all beautiful women, tall, graceful, intelligent, and came from fine San Francisco families. He paid them more than they could make working for his competitors. The only requirement was that they all wear the same style and color dress, which the bank paid for. Every day was a different color. Today, they were wearing brown dresses that complemented the carpet.\n\nThey saw him enter and immediately came to their feet and surrounded him, chatting gaily and welcoming him back from what they had been told was a holiday that took him fishing in Oregon. Although he had to use great restraint and willpower, Cromwell never carried on a love affair with any of the three women. He had strong principles about playing on his own turf.\n\nAfter the niceties were over and the ladies returned to their desks, Cromwell asked his senior secretary, who had been with him for nine years, to come into his office.\n\nHe sat down at his massive teak desk and parked the suitcase underneath. He smiled at Marion Morgan. \"How are you, Miss Morgan? Any new gentlemen friends lately?\"\n\nShe blushed. \"No, Mr. Cromwell. I spend my nights staying home and reading.\"\n\nMarion was twenty-one when she finished college and came to work for Cromwell as a teller, and she had risen to manager. She had just turned thirty and had never married, which made many consider her an old maid. But the truth was, she could have had any one of the well-heeled men in town. She was an unusually ravishing and nubile lady who could pick and choose her suitors but had yet to select one for a husband. She was particular about men, and the Prince Charming of her dreams had not appeared. Her straw-blond hair was wrapped on her head, as was the fashion of the day, and her lovely facial features enhanced a long swan neck. Her corseted figure looked like the classic hourglass. She gazed across the desk at Cromwell through coral\u2013sea green eyes, and a delicately shaped hand held a pencil poised above a notepad.\n\n\"I expect agents representing a bank in Salt Lake City to arrive at any moment to check our records.\"\n\n\"Are they going to examine our books?\" she asked as if mildly alarmed.\n\nHe shook his head. \"Nothing like that. I've heard rumors among my fellow bankers that a bank in Salt Lake City was robbed and that monies stolen might have been deposited in another bank.\"\n\n\"Do you wish me to take care of the matter?\"\n\n\"No. Please, simply entertain them until I'm prepared to deal with it.\"\n\nIf Marion had any inquiries as to the uncertainty about Cromwell's request, she showed no curiosity. \"Yes, of course, I'll see that they are comfortable until you wish to see them.\"\n\n\"That will be all,\" said Cromwell. \"Thank you.\"\n\nAs soon as Marion left his office and shut the door, Cromwell reached into his breast pocket and brought out the bank draft from the Salt Lake Bank & Trust. Then he stood and went over to the large stand-up safe that held the bank's ledgers and records. He quickly, and expertly, doctored the books so that it appeared that the draft had already been received and the full amount paid to Eliah Ruskin. Cromwell also made entries that indicated the money had been deducted from his bank's liquid capital.\n\nCromwell did not have long to wait after finishing doctoring the records. The expected agents walked into his outer office twenty minutes later. Marion had stalled them, saying Mr. Cromwell was extremely busy. When a small buzzer beneath her desk sounded, she showed them into his office.\n\nHe was holding a telephone and nodded a greeting while motioning them to take chairs. \"Yes, Mr. Abernathy, I will personally see that your account is closed and the funds transferred to the Bank of Baton Rouge in Louisiana. Not at all. Glad to be of service. Have a good trip. Good-bye.\"\n\nCromwell put down the phone with a dead line and no caller on the other end. He stood, came around the desk, and offered his hand. \"Hello, I'm Jacob Cromwell, president of the bank.\"\n\n\"These gentlemen are from Salt Lake City,\" said Marion. \"They wish to see you about a draft drawn against their bank.\" Then she swirled her skirt, a bare inch above the ankles, left the office, and closed the door.\n\n\"How can I help you?\" Cromwell asked courteously.\n\nOne man was tall and gangly, the other short and stocky and sweating. The tall one spoke first. \"I'm William Bigalow, and my associate here is Joseph Farnum. We are inquiring if any financial institution in San Francisco might have received a bank draft for four hundred and seventy-five thousand dollars drawn on the Salt Lake Bank and Trust.\"\n\nCromwell raised his eyebrows in mock apprehension. \"What seems to be the problem?\"\n\n\"The draft was made under duress by the bank manager before a bandit shot him dead and made off with it, including the bank's money in its vault. We're trying to trace its whereabouts.\"\n\n\"Oh, my,\" said Cromwell, throwing up his hands in a sign of distress. \"That draft came into our hands yesterday afternoon.\"\n\nThe two agents tensed. \"You have the draft?\" Farnum queried expectantly.\n\n\"Yes, it is in a safe in our bookkeeping department.\" Cromwell's tone became grave. \"Unfortunately, we honored it.\"\n\n\"You honored it!\" Bigalow gasped.\n\nCromwell shrugged. \"Why, yes.\"\n\n\"With a check, no doubt,\" said Farnum, in hope there was still time to stop the bandit from cashing it at another bank.\n\n\"No, the gentleman whose name was on the draft asked for cash and we complied.\"\n\nBigalow and Farnum looked at Cromwell in shock. \"You paid almost half a million dollars in cash to someone who walked into your bank off the street?\" Bigalow frowned severely.\n\n\"I checked the draft myself when my manager brought it to me for approval. It appeared perfectly legitimate.\"\n\nBigalow did not look happy. It would be his burden to contact the directors of the Salt Lake Bank and tell them their four hundred and seventy-five thousand dollars had vanished.\n\n\"What was the name on the draft?\"\n\n\"A Mr. Eliah Ruskin,\" answered Cromwell. \"He produced a file of papers that showed Mr. Ruskin was the founder of an insurance company that was going to pay off claims brought on by a fire that destroyed a city block in a town...\" Cromwell paused. \"I believe he said its name was Bellingham, in Washington State.\"\n\n\"Can you describe Ruskin?\" asked Farnum.\n\n\"Very well dressed,\" offered Cromwell. \"Tall, with blond hair and a large blond mustache. I didn't catch the color of his eyes. But I seem to recall that he carried an unusual cane, with a silver eagle's head.\"\n\n\"That's Ruskin, all right,\" muttered Farnum.\n\n\"He didn't waste any time,\" Bigalow said to his partner. \"He must have caught an express train to get here in a little over a day.\"\n\nFarnum stared at Cromwell skeptically. \"Didn't you think that was an astronomical amount to pay a perfect stranger from out of state?\"\n\n\"True, but, as I said, I personally checked the draft to make sure it wasn't a forgery. I asked him why he didn't draw on it from a Seattle bank, but he said his company was opening an office in San Francisco. I assure you that it was a bona fide draft. I could find no reason to be suspicious. We paid, although it took almost every dollar of currency we carried in the vault.\"\n\n\"The bank we represent won't be happy about this,\" Barnum pointed out.\n\n\"I'm not worried,\" Cromwell replied significantly. \"The Cromwell Bank has done nothing illegitimate or illegal. We have adhered to the rules and regulations of banking. As to the Salt Lake Bank and Trust failing to meet their obligations, I'm not concerned. Besides their insurance company paying for the theft of the currency, I happen to know their assets are more than ample to cover a half-million-dollar loss.\"\n\nBarnum addressed Bigalow without turning in his direction. \"We had better get to the nearest telegraph office and notify the Salt Lake Bank and Trust directors. They won't be pleased.\"\n\n\"Yes.\" Bigalow nodded heavily. \"They may not take this lying down.\"\n\n\"They have no choice but to honor the draft. It is safe to say the banking commission will agree in Cromwell Bank's favor, should the directors of the Salt Lake Bank wish to enter a protest.\"\n\nThe two agents came to their feet.\n\n\"We'll need a statement from you, Mr. Cromwell,\" announced Farnum, \"stating the circumstances of the payment.\"\n\n\"I shall have my attorneys draw it up first thing in the morning.\"\n\n\"Thank you for your consideration.\"\n\n\"Not at all,\" said Cromwell, remaining seated. \"I'll do all in my power to cooperate.\"\n\nAs soon as the agents left, Cromwell called in Miss Morgan. \"Please see that I am not disturbed for the next two hours.\"\n\n\"I'll see to it,\" she said efficiently.\n\nSeconds after the door closed, Cromwell walked over and quietly locked it. Then he lifted the heavy suitcase under the desk onto the teak surface and opened it. The currency was piled loosely inside, some in stacks wrapped with paper bands.\n\nMethodically, Cromwell began to count and stack the bills, wrapping the loose ones with bands as he inked in the amount. When he finished, he had his desktop filled with neatly piled bundles of cash, marked and counted. The tally came to two hundred forty-one thousand dollars. Then he carefully put the money back in the suitcase, slid the suitcase back under the desk, and opened several ledgers, entering deposits in bogus accounts, which he had set up previously to conceal money stolen over the years. Money that he used to build up the assets needed to open his own bank. Satisfied that he was covered by all the entries, he buzzed Miss Morgan and informed her that he was ready to deal with the day-to-day business of running a successful house of finance.\n\nThe banking hours were from ten o'clock in the morning until three in the afternoon. When closing time rolled around, Cromwell waited until the employees had all left for home and the bank was locked up. Now, alone in the bank's vast interior, he carried the suitcase down the elevator to the main floor and into the bank vault, which was still open according to his instructions. He placed the currency, one stack at a time, in the proper bins that were used by the tellers for customer transactions. The receipts he had made up would be turned over to his chief accountant in the morning, who would record the juggled deposits without knowing the serial numbers.\n\nJacob Cromwell felt pleased with himself. Swindling as well as robbing the bank in Salt Lake City had been his most bold undertaking to date. And he was not about to repeat it. The evil act would throw off his pursuers, who would think he was becoming more daring, and be led into thinking he might try robbing a major city's bank again. But he knew when not to press his luck. Such a robbery was extremely complicated. When he went out on a crime spree again, it would be in a small town yet to be selected.\n\nAfter closing the vault and throwing the locks and timer, he went down to the basement and slipped out to the street through a hidden door that only he knew existed. Whistling \"Yankee Doodle,\" he hailed a cab and rode to California Street, where he took the cable car up the steep, twenty-four-percent grade of the three-hundred-seventy-five-foot-high slope to his house on Nob Hill, the \"hill which is covered with palaces,\" as Robert Louis Stevenson described it.\n\nCromwell's mansion amid mansions sat on a small picturesque lane called Cushman Street. The other monuments to wealth had been built by the bonanza-mining types and the big-four barons of the Central Pacific, later the Southern Pacific Railroad: Huntington, Stanford, Hopkins, and Crocker. To the eye of a creative artist or designer, the mansions looked like monstrosities of architecture gone mad with ostentation.\n\nUnlike the others that were built of wood, Cromwell and his sister Margaret's house was constructed of quarried stone and reflected more of a sedate, almost library-like exterior. There were some who thought it bore a striking resemblance to the White House in Washington.\n\nHe found his sister impatiently waiting. At her urging, he quickly readied himself for a night on the Barbary Coast. Yes indeed, he thought, as he dressed in his evening clothes, it had been a productive week. One more success to add to his growing sense of invincibility.\n\n## 12\n\nIRVINE COULD NOT COME UP WITH CURRENCY SERIAL numbers in Bozeman. Not only had the bank failed to record them; it had gone out of business due to the robbery. By the time their assets made up for the loss, the bank had collapsed, and the founder sold what few assets that were left, including the building, to a rich silver miner.\n\nIrvine pushed on to the next robbed bank on his list and took the Northern Pacific Railroad to the mining town of Elkhorn, Montana, located at an elevation of 6,444 feet above sea level. A booming mining town with twenty-five hundred residents, Elkhorn had produced some ten million dollars in gold and silver from 1872 until 1906. The Butcher Bandit had robbed its bank three years earlier, leaving four dead bodies behind.\n\nJust before the train pulled into the town station, Irvine studied, for the tenth time since leaving Bozeman, the report on the robbery in Elkhorn. It was the same modus operandi the bandit used on all his other robberies. Disguised as a miner, he entered the bank soon after the currency shipment had arrived to pay the three thousand men working the quartz lodes. As usual, there were no witnesses to the actual crime. All four victims\u2014the bank manager, a teller, and a husband and wife making a withdrawal\u2014had been shot in the head at close range. Again, the shots went unheard, and the bandit escaped into the atmosphere without leaving a clue.\n\nIrvine checked into the Grand Hotel before walking down the street to the Marvin Schmidt Bank, its new name taken from the miner who bought it. The architecture of the bank building was typical of the current style in most mining towns. Local stone laid with a Gothic theme. He walked though a corner door, facing the intersection of Old Creek and Pinon Streets. The manager sat behind a low partition not far from a massive steel safe painted with a huge elk standing on a rock outcropping.\n\n\"Mr. Sigler?\" inquired Irvine.\n\nA young man with black hair, brushed back and oiled, looked up at the greeting. His eyes were a shade of dark green, and his features indicated Indian blood in his ancestry. He wore comfortable cotton pants, a shirt with soft collar, and no tie. He lifted a pair of spectacles from the desk and set them on the bridge of his nose.\n\n\"I'm Sigler. How can I help you?\"\n\n\"I'm Glenn Irvine with the Van Dorn Detective Agency, here for an investigation into the robbery a few years ago.\"\n\nSigler quickly frowned with an attitude of annoyance. \"Don't you think it's a little late for Van Dorn to arrive on the scene? The robbery and murders took place back in 1903.\"\n\n\"We were not engaged to make an investigation at that time,\" Irvine retorted.\n\n\"So why are you here at this late date?\"\n\n\"To record the serial numbers of the bills taken by the robbery, if they were listed in a register.\"\n\n\"Who is paying for your services?\" Sigler insisted.\n\nIrvine could imagine Sigler's distrust and incomprehension. He might have felt the same if he was in the bank manager's shoes. \"The United States government. They want the robberies and murders to stop.\"\n\n\"Strikes me that the bastard can't be caught,\" Sigler said coldly.\n\n\"If he walks on two legs,\" said Irvine confidently, \"the Van Dorn Agency will catch him.\"\n\n\"I'll believe it when I see it,\" Sigler said indifferently.\n\n\"May I see your register of serial numbers? If we have those from the stolen bills, we will make every effort to trace them.\"\n\n\"What makes you think they were recorded?\"\n\nIrvine shrugged. \"Nothing. But it never hurts to ask.\"\n\nSigler fished around in his desk and retrieved a set of keys. \"We keep all the old bank records in a storehouse behind the building.\"\n\nHe motioned for Irvine to follow him as he led the way through a back door toward a small stone building sitting in the middle of the bank's property. The door protested with a loud squeak as it opened on unoiled hinges. Inside, shelves held rows of ledgers and account books. A small table and chair sat at the back of the storeroom.\n\n\"Sit down, Mr. Irvine, and I'll see what I can find.\"\n\nIrvine was not optimistic. Finding a bank that kept a record of serial numbers of its currency seemed highly improbable. It was a long shot, but every avenue had to be followed. He watched as Sigler went through several clothbound ledgers. At last, he opened one and nodded.\n\n\"Here you are,\" Sigler said triumphantly. \"The serial numbers our bookkeeper recorded of all the currency in the vault two days before the robbery. Some of the bills, of course, were distributed to customers. But most were taken by the bandit.\"\n\nIrvine was stunned as he opened the book and stared at the columns of neatly inked numbers within the lines of the pages. There were several different kinds of large banknotes. Gold certificates, silver certificates, notes issued by individual banks were recorded in the ledger. United States Treasury serial numbers were printed vertically and horizontally on the sides; the local bank that issued them added their own number at the bottom. Most came from the Continental & Commercial National Bank of Chicago and the Crocker First National Bank of San Francisco. He looked up at Sigler, now fired with elation.\n\n\"You don't know what this means,\" Irvine said, gratified beyond his greatest expectations. \"Now we can pass out numbers on the stolen bills to every bank in the country where the bandit might have deposited them. Handbills with the numbers can also be distributed to merchants throughout the West, urging them to be on the lookout for the bills.\"\n\n\"Good luck,\" said Sigler pessimistically. \"It's hardly possible that you can trace them back three years. Any one of them could have changed hands a hundred times by now.\"\n\n\"You're probably right, but hopefully the bandit is still spending them.\"\n\n\"Slim chance of that,\" Sigler said with a tight grin. \"I'll bet a month's wage he spent it all a long time ago.\"\n\nSigler was probably right, Irvine thought. But Irvine was not discouraged. Bell had said that it would be an insignificant mistake that would trip the bandit up. Now it was only a question of getting the information out to banks and merchants and hope there would be a response that led to the whereabouts of the mysterious killer.\n\n## 13\n\nCURTIS SAT AT A TABLE IN THE WESTERN ARCHIVES Division of the Union Pacific Railroad's office in Omaha, Nebraska, surrounded by high shelves filled with ledgers and account books of reports on train operations. During the nine days since he launched his search, he had scoured the records of four different railroads and the Wells Fargo stage lines trying to find a link for how the Butcher Bandit escaped capture after committing his robberies and hideous murders.\n\nIt was an exercise in futility. Nothing fell into place. He had begun with the stagecoach possibilities. Most of the stage lines were gone by 1906. Wells Fargo still held the monopoly, with lines extending several thousand miles over overland express routes in remote areas that were not serviced by railroads. But the schedules did not fall into the proper times.\n\nThere were sixteen hundred different company railroads across the nation in 1906, with two hundred twenty thousand miles of track among them. Fifty of the largest had a thousand miles of track each. Curtis had narrowed the number of companies down to five. They were the railroads with scheduled runs through the towns hit by the bandit.\n\n\"Would you like a cup of coffee?\"\n\nCurtis looked up from a train schedule record into the face of a little man standing no more than five feet two inches. His name was Nicolas Culhane, and his biscuit brown\u2013streaked graying hair was brushed forward over his head to cover the receding baldness. The ferret brown eyes shifted with amazing frequency, and he wore a thinly clipped mustache whose pointed ends extended a good inch on either side of his lips. He walked with a slight stoop and wore spectacles with lenses that magnified his eyes. Curtis was amused at the helpful little man with the springy step. He was the perfect stereotype of a keeper of musty records in an archive.\n\n\"No, thank you.\" Curtis paused to glance at his pocket watch. \"I never drink coffee in the afternoon.\"\n\n\"Having any luck?\" asked Culhane.\n\nCurtis shook his head wearily. \"None of the passenger trains ran close to the time the bandit robbed the banks.\"\n\n\"I pray you catch the murdering scum,\" Culhane said, his voice suddenly turned angry.\n\n\"You sound like you hate him.\"\n\n\"I have a personal grudge.\"\n\n\"Personal?\"\n\nCulhane nodded. \"My closest cousin and her little boy were killed by the Butcher at the bank in McDowell, New Mexico.\"\n\n\"I'm sorry,\" Curtis said solemnly.\n\n\"You must catch and hang him!\" Culhane struck a fist on the table, causing the schedule book lying open to tremble and flip its pages. \"He has got away with his crimes far too long.\"\n\n\"I assure you, the Van Dorn Agency is working night and day to bring him to justice.\"\n\n\"Have you found anything at all that might trace him?\" Culhane asked anxiously.\n\nCurtis raised his hands in a helpless gesture. \"All we've discovered is that he is missing the little finger on his left hand. Besides that, we have nothing.\"\n\n\"Did you check out the stagecoach lines?\"\n\n\"I spent a day in the Wells Fargo records department. It was a dead end. None of their schedules put them in town within four hours of the robberies. More than enough time for the bandit to evade capture.\"\n\n\"And the passenger trains?\"\n\n\"The sheriff and marshals telegraphed surrounding towns to stop all trains and examine the passengers for anyone who looked suspicious. They even searched all luggage in hopes that one of the bags might contain the stolen currency, but they turned up no evidence, nor could they make an identification. The bandit was too smart. The disguises he used to rob and murder were too original and too well executed. The law officers had little or nothing to go on.\"\n\n\"Did time schedules work out?\"\n\n\"Only two,\" Curtis replied tiredly. \"The departure times on the others didn't coincide with the events.\"\n\nCulhane rubbed his thinning hair thoughtfully. \"You've eliminated stagecoaches and passenger trains. What about freight?\"\n\n\"Freight trains?\"\n\n\"Did you check out the departure times on those?\"\n\nCurtis nodded. \"There, we have a different story. The trains I've been able to find in the right place at the right time left the robbed towns within the required times.\"\n\n\"Then you have your answer,\" Culhane said.\n\nCurtis didn't reply immediately. He was tired, on the verge of sheer exhaustion, and depressed that he was no further along and had made no discoveries. Inwardly, he cursed the Butcher Bandit. It didn't seem humanly possible the man could be so obscure, so will-o'-the-wisp, so able to defy all attempts at detection. He could almost see the man laughing at the inept efforts of his pursuers.\n\nAt last, he said, \"You underestimate the law enforcement officials. They searched the boxcars of all the freight trains that passed through the towns during the specified time limits.\"\n\n\"What about the boxcars that were switched onto local sidings to be hauled later to other destinations by incoming trains? He could have dodged the posses by hiding in a freight car.\"\n\nCurtis shook his head. \"The posses searched all empty cars and found no sign of the bandit.\"\n\n\"Did they check out the ones that were loaded?\" Culhane questioned.\n\n\"How could they? The cars were locked tight. There's no way the bandit could have entered them.\"\n\nCulhane grinned like a fox on a hot scent. \"I guess nobody told you that the train brakemen all carry keys that will open the locks on the loading doors in case of fire.\"\n\n\"I was not aware of that angle,\" said Curtis.\n\nThe steel-rimmed spectacles slid down Culhane's nose. \"It's certainly something to think about.\"\n\n\"Yes, it is,\" Curtis mused, his mind beginning to turn. \"We're looking at a process of elimination. The posses claimed there were no tracks leading out of town to follow, which means our man didn't ride a horse. There is almost no chance he could have taken a stagecoach, and it appears unlikely he bought a ticket and traveled out of town as a passenger on a train. He also failed to be spotted in an empty boxcar.\"\n\n\"Which leaves loaded boxcars as the only means of transportation that was not examined,\" Culhane persisted.\n\n\"You may be onto something,\" said Curtis thoughtfully.\n\nA peculiar expression crossed Curtis's face as he began to envision a new scenario. \"That leaves a whole new avenue to follow. Now I have to go through freight car records to study the cars that made up those specific trains, who owned them, their manifest, and their ultimate destination.\"\n\n\"Not an easy chore,\" said Culhane. \"You'll have to check out hundreds of freight cars from a dozen trains.\"\n\n\"Like a piece of a puzzle. Find the boxcar that was parked on a nearby siding in all of the robbed towns on the days of the robberies.\"\n\n\"I'll be happy to help you with the Union Pacific freight records.\"\n\n\"Thank you, Mr. Culhane. Two of the freight trains in question were hauled by Union Pacific.\"\n\n\"Just tell me which towns they were at and I'll dig out the records that give the car's serial numbers, their ownership, and the agent who arranged and paid for their transportation.\"\n\n\"You've been a great help to me and I'm grateful,\" Curtis said sincerely.\n\n\"I'm the one who is grateful, Mr. Curtis. I never thought I would be instrumental in bringing the Butcher Bandit, the killer of my cousin and her child, to justice.\"\n\nFour hours later, with Culhane's able assistance, Curtis had the information that gave him a solid direction to investigate. Now all he had to do was research the archives of the Southern Pacific, the Atchison, Topeka & Santa Fe, and the Denver & Rio Grande railroads to confirm Culhane's theory.\n\nBy nightfall, he was on a train to Los Angeles and the Atchison, Topeka & Santa Fe archives. Too inspired to sleep, he stared at his reflection in the window since it was too dark to see the landscape roll by outside. He was optimistic that the end of the trail seemed to be over the next hill and around the next bend.\n\n## 14\n\nTHE EARLY EVENING CAME WITH A LIGHT RAIN THAT dampened the dirt street through town as Bell stepped off the train. In the fading light, he could see that Bisbee, Arizona, was a vertical town, with sharply rising hills occupied by many houses that could be reached only by steep stairways. On his way to the Copper Queen Hotel, he walked through the narrow, twisting streets, a maze flanked by new, substantial brick buildings.\n\nIt was a Saturday, and Bell found a deputy holding down the sheriff's office and jail. The deputy said the sheriff was taking a few days off, to make repairs to his house that had been damaged in a flood that had swept down the hills, and would not return to work until Thursday. When Bell asked him for directions to the sheriff's house, the deputy refused to give them, claiming that the sheriff was not to be disturbed unless it was an emergency.\n\nBell checked into the Copper Queen, ate a light dinner in the hotel dining room, and then went out on the town. He skipped having a drink in the Copper Queen Saloon and walked up to the infamous Brewery Gulch, lined with fifty saloons, known throughout the territory as the wildest, bawdiest, and best drinking street in the West.\n\nHe checked out four of the saloons, stepping into each and studying the action, before going on to the next one. Finally, he settled into a large, wooden-walled hall with a stage and a small band playing a ragtime tune while four dancing girls hoofed it around the stage. Moving through the crowded tables to the bar, he waited until a busy bartender asked, \"What'll it be, friend, whiskey or beer?\"\n\n\"What's your best whiskey?\"\n\n\"Jack Daniel's from Tennessee,\" said the bartender without hesitation. \"It won the Gold Medal at the St. Louis Fair as the best whiskey in the world.\"\n\nBell smiled. \"I've enjoyed it, on occasion. Let me have a double shot glass.\"\n\nWhile the bartender poured, Bell turned around, leaned his elbows behind him on the bar, and gazed around the busy saloon. Like most watering holes in the West, a large section of the room was given over to gambling. Bell's eyes went from table to table, looking for the right mix of poker players. He found what he had hoped to find, a table with men dressed in fancier clothes than the large number of miners. They appeared to be businessmen, merchants, or mining officials. Best of all, there were four of them, one short of a fifth player.\n\nBell paid for his whiskey and walked over to the table. \"May I join you gentlemen?\" he asked.\n\nA heavyset man with a red face nodded and motioned toward an empty chair. \"You're quite welcome to sit in,\" he said.\n\nA man directly across the table shuffled the cards, looked across at Bell as he sat down, and began dealing. \"I'm Frank Calloway. The others are Pat O'Leery, Clay Crum, and Lewis Latour.\"\n\n\"Isaac Bell.\"\n\n\"You new in town, Mr. Bell?\" asked O'Leery, a big, brawny Irishman.\n\n\"Yes, I arrived on the six-thirty train from Phoenix.\"\n\n\"Business or pleasure?\" O'Leery probed.\n\n\"Business. I'm an agent with the Van Dorn Detective Agency.\"\n\nThey all looked up from their cards and stared at Bell with inquisitive interest.\n\n\"Let me guess,\" said Crum, folding his hands over a rotund belly. \"You're looking into the bank robbery and murders that took place four months ago.\"\n\nBell nodded as he fanned his hand and examined his cards. \"You are correct, sir.\"\n\nLatour spoke in a French accent as he lit a cigar. \"A little late, aren't you? The trail is cold.\"\n\n\"No colder than it was five minutes after the crime,\" Bell countered. \"I'll take two cards.\"\n\nCalloway dealt as the players called out the number of cards they hoped would give them a winning hand. \"A mystery, that one,\" he said. \"No trace of the bandit was ever found.\"\n\n\"Uncanny,\" O'Leery said as he inspected his hand, his expression revealing he had nothing worth betting on. \"I fold.\" His eyes briefly met Bell's. \"Uncanny that he could escape into thin air.\"\n\n\"The sheriff found no sign of his trail,\" muttered Crum. \"The posse returned to town looking as if their wives had run off with a band of traveling salesmen.\" He paused. \"I'll bet two dollars.\"\n\n\"I'll raise you three dollars,\" offered Calloway.\n\nLatour threw his hand toward the dealer. \"I'm out.\"\n\n\"And you, Mr. Bell,\" inquired Calloway, \"are you still in?\"\n\nBell was amused that the stakes were not high, but not penny-ante either. \"I'll call.\"\n\n\"Two queens,\" announced Crum.\n\n\"Two tens,\" said Calloway. \"You beat me.\" He turned. \"Mr. Bell?\"\n\n\"Two eights,\" Bell said, passing his cards facedown to Calloway. Bell had not lost. He held three jacks, but he thought that losing would bring him closer to the other men's confidence. \"Was there any clue to how the robber escaped?\"\n\n\"Nothing I ever heard of,\" replied O'Leery. \"Last time I talked to the sheriff, he was baffled.\"\n\n\"That would be Sheriff Hunter?\" Bell inquired, recalling what he read in the agency report.\n\n\"Joe Hunter died from a bad heart two months after the murders,\" answered Latour. \"The new sheriff is Stan Murphy, who was Hunter's chief deputy. He knows what went on as well as anybody.\"\n\n\"As nice as they come, if he likes you,\" Crum said. \"But get on his bad side and he'll chew you to bits.\"\n\n\"I'd like to talk with him, but I doubt if he'll be in his office on the Sabbath,\" said Bell, not mentioning the discouraging comments of Murphy's deputy. \"Where might I find him?\"\n\n\"We had a bad flood through town two weeks ago,\" replied Calloway. \"His house was badly damaged. I suspect you'll find him up to his neck in repairs.\"\n\n\"Can you give me directions to his house?\"\n\nO'Leery waved a hand toward the north. \"Just go up to the end of Howland Street and take the stairs. The house is painted green and has a small grove of orange trees alongside.\"\n\nThe talk moved to politics and whether Teddy Roosevelt could run for a third term in 1908 and, if not, whom he would pick as his successor. Bell lost three hands for every hand he won, easily putting the other men at ease as they realized the stranger was no gambling cardsharp. He swung the conversation back to the bank murders.\n\n\"Seems strange that no one saw the robber leaving the bank or riding out of town,\" said Bell idly as he played his cards.\n\n\"Nobody came forward,\" said O'Leery.\n\n\"And none saw the bandit enter or leave the bank,\" Latour added.\n\n\"There was an old drunken miner that hung around across the street from the bank,\" answered Calloway, \"but he disappeared soon afterward.\"\n\n\"Sheriff Hunter did not consider him a suspect?\"\n\nLatour had no luck. He folded for the fifth time since Bell sat down at the table. \"An old miner who was all played out and looked like he wasn't long for this world? He was the last one the townspeople thought had anything to do with the crime.\"\n\n\"More than once, I saw him sprawled on a sidewalk, drunk out of his mind,\" said O'Leery. \"He couldn't have robbed a bank and murdered three people any more than I could become governor. I still think it was an inside job pulled off by someone we all know.\"\n\n\"It might have been a stranger,\" Bell said.\n\nCalloway shrugged negatively. \"Bisbee has twenty thousand inhabitants. Who's to recognize a stranger?\"\n\n\"What about that fellow on a motorcycle?\" Crum asked no one in particular.\n\n\"There was a motorcycle in town?\" asked Bell, his interest aroused.\n\n\"Jack Carson said he saw a dandy riding one.\" Crum threw down a winning hand with a flush.\n\nLatour took a long puff on his cigar. \"Jack said the rider was well dressed, when he saw him pass through an alley. He couldn't figure out how someone riding one of those contraptions could wear clothes so clean and unsoiled.\"\n\n\"Did your friend get a look at the rider's face?\"\n\n\"All Jack could tell was that the rider was clean-shaven,\" Calloway responded.\n\n\"What about hair color?\"\n\n\"According to Jack, the fellow wore a bowler. Jack wasn't sure, as he didn't get a good look because the motorcycle went by too fast, but he thought the hair might have been red. At least, that's what he thought, from a glimpse of the sideburns.\"\n\nFor the second time that week, Bell found excitement coursing through his veins. A resident of Eagle City, Utah, another mining town where the Butcher Bandit left four residents dead, mentioned that he had seen a stranger riding a motorcycle on the day of the killing.\n\n\"Where can I find this Jack Carson?\"\n\n\"Not in Bisbee,\" replied Crum. \"The last I heard, he went back to his home in Kentucky.\"\n\nBell made a mental note to ask Van Dorn to try and find Carson.\n\nO'Leery made another sour face at seeing his hand. \"Whoever rode that motorcycle must have hung around town for a few days after the robbery.\"\n\n\"Why do you say that?\" Bell probed.\n\n\"Because the sheriff and his posse would have spotted the motorcycle's tire tracks if the killer had ridden out of town immediately after the robbery.\"\n\n\"You'd think he would have been spotted if he stayed in town until the posse gave up the hunt.\"\n\n\"You would think so,\" said Calloway, \"but he was never seen again.\"\n\n\"Was Carson a reliable witness?\" Bell laid five dollars on the table. \"I raise.\"\n\n\"Jack was a former mayor of Bisbee, an attorney highly regarded as an honorable man,\" Latour explained. \"If he said he saw a man on a motorcycle, he saw a man on a motorcycle. I have no reason to doubt his word.\"\n\n\"You going to see Sheriff Murphy tomorrow?\" Crum inquired, finally winning a hand.\n\nBell nodded. \"First thing in the morning. But, after talking with you gentlemen, I fear there is little of importance he can tell me.\"\n\nAfter nursing his drink during two hours of play, Bell was even, almost. He was only four dollars in the hole, and none of the other players minded when he bid them good night and walked back to his hotel.\n\nTHE ROAD that wound up to the street toward the sheriff's house was long, and muddy after a rainstorm that struck Bisbee in the middle of the night. Coming to a dead end, Bell mounted the steep stairway that seemed to go on forever. Despite being in excellent physical shape, he was panting when he reached the top.\n\nBell was in a happy mood. He had yet to learn what Irvine and Curtis turned up, if anything. But he was dead certain the man seen on the motorcycle was the Butcher Bandit after he removed his disguise as the old intoxicated miner. A missing finger and a hint of red hair was hardly a triumph. Even the hair color glimpsed by Jack Carson was a long shot. It was the motorcycle that intrigued Bell, not because the bandit owned one but because it fit that a shrewd and calculating mind would use the latest technology in transportation.\n\nThe primary question was, how did the bandit ride it out of town without being seen again?\n\nSheriff Murphy's house was only a few steps from the top of the stairway. It was small, and looked more like a shed than a house. The flood had pushed it off its foundation, and Bell saw that Murphy was busily engaged in propping it up in its new location, ten feet from where it had sat before. True to O'Leery's description, it was painted green, but the flood had devastated the orange grove.\n\nMurphy was furiously wielding a hammer and didn't hear Bell approach. A great torrent of dark brown hair flowed around his neck and shoulders. Most of the lawmen in the West were not fat but lean and angular. Murphy had the body of a blacksmith rather than a sheriff. The muscles in his arms looked like tree trunks, and he had the neck of an ox.\n\n\"Sheriff Murphy!\" Bell shouted over the pounding of the hammer against nails.\n\nMurphy stopped with his hammer in midair and turned. He stared at Bell as he might stare at a coyote. \"Yes, I'm Murphy. But, as you can see, I'm busy.\"\n\n\"You can keep working,\" said Bell. \"I'm with the Van Dorn Detective Agency and would like to ask you a few questions about the bank robbery and murders a few months ago.\"\n\nThe name Van Dorn was respected among law enforcement circles, and Murphy laid down the hammer and pointed inside the little house. \"Come inside. The place is a bit of a mess, but I have coffee on the stove.\"\n\n\"After that climb up the hill, a cup of water would be nice.\"\n\n\"Sorry, the well got befouled by the flood and isn't fit to drink, but I carried a gallon up from a horse trough in town.\"\n\n\"Coffee it is,\" said Bell with a measure of trepidation.\n\nMurphy led Bell into the house and offered him a chair at the kitchen table. There was no sign of the presence of a woman, so Bell assumed that Murphy was a bachelor. The sheriff poured two coffees in tin cups from an enamel pot that sat on the wood-burning stove.\n\n\"I don't know how I can help you, Mr. Bell. I sent a copy of my findings to your agency in Chicago.\"\n\n\"You neglected to mention Jack Carson's sighting.\"\n\nMurphy laughed. \"The guy on a motorcycle? I don't believe what Jack said he saw. The description didn't fit anyone I knew in town.\"\n\n\"The bandit could have changed his disguise,\" Bell suggested.\n\n\"There was no time for him to completely alter his appearance, retrieve his motorcycle, and ride off into the blue.\"\n\n\"The rider and his machine were never seen again?\"\n\nMurphy shrugged. \"Strikes me odd that nobody else saw him except Jack. A man on the only motorcycle in town is bound to be noticed. And how could he ride out of town without leaving a trail?\"\n\n\"I admit it sounds a bit far-fetched,\" said Bell, not wanting to discard the sighting.\n\n\"Jack Carson was an upstanding citizen not noted for being a hard drinker or a teller of tall tales. But I believe he was hallucinating.\"\n\n\"Was there any other evidence discovered that wasn't in your report?\"\n\n\"There was something found after I sent the report to Chicago. Murphy rose from the kitchen table and pulled open a drawer of a rolltop desk. He passed Bell a brass shell casing. \"This was found two weeks later, by a young boy playing on the floor of the bank while his father made a deposit. It was under a carpet. The bandit must have missed it.\"\n\nBell studied the cartridge. \"Thirty-eight caliber. If it was ejected, it must have come from an automatic weapon, probably a Colt.\"\n\n\"That was my guess, too.\"\n\n\"May I keep it?\" asked Bell.\n\n\"Sure. But I doubt you'll learn anything from it, except knowing it came from the bandit's gun. And even that is not cold, hard evidence.\"\n\n\"If not the bandit, then where did it come from?\"\n\nMurphy held up his hands in a helpless gesture. \"I can't begin to guess.\"\n\nBell carefully held the cartridge in the palm of his hand. \"Hopefully, we can obtain the bandit's fingerprints.\"\n\nMurphy grinned. \"You'll find mine as well as the young boy's and two of my deputies'.\"\n\n\"Still,\" said Bell optimistically, \"our experts may be able to pull a print. We won't need a sample of the boy who found it. His print would be small. But I would like sample prints of you and your deputies. You can send them to my Chicago office.\"\n\n\"I've never taken a fingerprint,\" said Murphy. \"I'm not at all sure how it's done.\"\n\n\"The science has been around for centuries, but only in the past few years is it catching on with law enforcement. The impressions on an object\u2014in this case, the cartridge\u2014are created by the ridges on the skin. When the object is handled, the perspiration and oils are transferred to it, leaving an impression of the fingertip-ridge pattern. To record the prints, a fine powder like ground-up graphite from a pencil is dusted on the surface. Then a piece of tape is used to lift the print for study.\"\n\nMurphy sipped at his coffee. \"I'll give it a try.\"\n\nBell thanked the sheriff and walked down the stairway. Three hours later, he was on a train back to Denver.\n\n## 15\n\nCROMWELL'S CHAUFFEUR DROVE THE 1906 ROLLS Royce Brougham, made by the London coach maker Barker, with its six-cylinder, thirty-horsepower engine, from the garage to the front of the palatial Nob Hill mansion Cromwell had designed himself and constructed from white marble blocks cut and hauled by railroad from a quarry in Colorado. The front end had the appearance of a Greek temple, with high fluted columns, while the rest of the house was more simply designed, with arched windows, and a cornice that crowned the walls.\n\nWhile the chauffeur, Abner Weed, a stony-faced Irishman whom Cromwell hired more for his experience as a wrestler than his expertise behind the wheel of an automobile, stood patiently by the Rolls out front, Cromwell waited for his sister in his study, sprawled comfortably on a leather sofa, listening to Strauss waltzes on an Edison cylinder phonograph. He was conservatively dressed in a dark wool suit. After listening to \"Voices of Spring,\" he changed cylinders and played Tales from the Vienna Wood. The cylinders played two minutes of music each.\n\nCromwell glanced up from the machine as his sister came into the room wearing a doeskin dress that fell around her nicely curved calves.\n\n\"A bit risqu\u00e9, aren't we?\" he said, eyeing her exposed flesh.\n\nShe spun around, swirling the skirt and showing off her legs up to midthigh. \"Since we're going slumming on the Barbary Coast, I thought I'd dress like a soiled dove.\"\n\n\"Just be sure you don't act like one.\"\n\nHe rose from the sofa, turned off the phonograph, and held up her coat so she could slip into it. Even with his shoe lifts, he stood the same height as his sister. Then he followed her through the large, intricately carved front doors to the drive and the waiting Rolls-Royce. Abner, attired in his liveried uniform with shiny black boots, stood at attention, holding open the rear door. The Rolls was a town car, with an enclosed passenger compartment, the chauffeur in the open air with nothing but the windshield to protect him. As soon as Cromwell's sister was settled, he instructed the driver where to go. Abner shifted gears and the big car rolled silently over the granite stones laid in the street.\n\n\"This is the first opportunity we've had to talk since I came home,\" said Cromwell, secure in the knowledge that the driver could not hear their conversation through the divider window separating the front and rear seats.\n\n\"I know that your trip to Salt Lake City was successful. And our bank is another seven hundred thousand dollars richer.\"\n\n\"You haven't told me how you made out in Denver.\"\n\n\"Your spies in the Van Dorn Agency were quite accurate in their assessment of the investigation. The Denver office has taken on the job of lead investigator in the hunt for the Butcher Bandit.\"\n\n\"I hate being called that. I would have preferred something with more swank.\"\n\n\"Like what, pray tell?\" she asked, laughing.\n\n\"The Stylish Spirit.\"\n\nShe rolled her eyes. \"I doubt that newspaper editors would be enthused with that one.\"\n\n\"What else did you find out?\"\n\n\"The head of the Denver office, Nicholas Alexander, is an idiot. After I flashed a few of my charms, he couldn't stop speaking about the hunt. He was angry he wasn't put in charge of the investigation and had no reservations about revealing pertinent information concerning the methods they were going to use to catch the notorious bandit. Van Dorn himself named his top agent, Isaac Bell, to the case. A handsome and dashing devil, and very wealthy, I might add.\"\n\n\"You saw him?\"\n\n\"I met him, and, what's more, I danced with him.\" She pulled a small photograph from her purse. \"I was waiting to give you this. Not the greatest likeness, but the photographer I hired was not very proficient at shooting photographs without setting them up in advance.\"\n\nCromwell switched on the dome light of the car and examined the photo. The photo showed a tall man, with blond hair and mustache. \"Should I be concerned about him?\"\n\nHer eyes took on an evasive expression. \"I can't say. He seemed more intelligent and sophisticated than our spies led me to believe. I had them check his background. He rarely if ever fails to find and apprehend his man. His record is quite admirable. Van Dorn thinks very highly of him.\"\n\n\"If, as you say, he is affluent, why is he wasting his time as a simple detective?\"\n\nMargaret shrugged. \"I have no idea. Maybe, like you, he craves a challenge, too?\" She hesitated as she adjusted an imaginary loose curl with her fingers.\n\n\"Where did he get his money?\"\n\n\"Did I forget to mention that he comes from a family of bankers in Boston?\"\n\nCromwell stiffened. \"I know of the Bells. They own the American States Bank of Boston, one of the largest financial institutions in the country.\"\n\n\"He's a paradox,\" she said slowly, recalling her few minutes with him in the Brown Palace Hotel. \"But he can also be very dangerous. He'll come after us like a fox chasing a rabbit.\"\n\n\"A detective who knows the inner workings of banking procedures is not good,\" Cromwell said, his tone low and icy. \"We must be especially wary.\"\n\n\"I agree.\"\n\n\"You're certain he had no clue to your true identity?\"\n\n\"I covered my tracks well. As far as he and Alexander know, my name is Rose Manteca, from Los Angeles, where my father owns a large ranch.\"\n\n\"If Bell is as smart as you suggest, he'll check that out and prove Rose doesn't exist.\"\n\n\"So what?\" she said impishly. \"He'll never know my name is Margaret Cromwell, sister of a respected banker who lives in a mansion on Nob Hill in San Francisco.\"\n\n\"What other information did you pry out of Alexander?\"\n\n\"Only that Bell's investigation is not going well. They have no clues that lead in your direction. Alexander was angry that Bell hadn't taken him into his confidence. He said that Bell was close-mouthed concerning his actions with a pair of agents known as Curtis and Irvine. All I could find out is, they're all out beating the bushes in search of a lead.\"\n\n\"That's good to hear.\" Cromwell smiled thinly. \"They'll never consider that a banker is behind the robberies.\"\n\nShe gazed at him. \"You could quit, you know. We no longer need the money. And no matter how careful, no matter how sharp-witted you are, it's only a question of time before you're caught and hung.\"\n\n\"You want me to give up the excitement and the challenge of achieving what no one else would dare and play the role of a stodgy banker for the rest of my days?\"\n\n\"No, I do not,\" she said with a wicked sparkle in her eye. \"I love the excitement, too.\" Then her voice softened and became distant. \"It's just that I know it cannot go on forever.\"\n\n\"The time will come when we know when to stop,\" he said without emphasis.\n\nNeither brother nor sister possessed even a tinge of repentance or remorse for all the men, women, and children Cromwell had murdered. Nor could they have cared less for all the savings of small businesses, miners, and farmers they had wiped out when the robbed banks, unable to refund their depositors, had to close their doors.\n\n\"Who are you taking tonight?\" she asked, changing the subject.\n\n\"Marion Morgan.\"\n\n\"That prude,\" she scoffed. \"It's a mystery to me why you keep her on the payroll.\"\n\n\"She happens to be very efficient,\" he retorted, not seeking an argument.\n\n\"Why haven't you ever taken her to bed?\" she said with a soft laugh.\n\n\"You know I never play with my employees. It's a principle that has saved me much grief. I'm only taking her out tonight as a bonus for her work. Nothing more.\" His sister's dress was pulled to her knees and he reached over and squeezed one of them. \"Who is the lucky man this evening?\"\n\n\"Eugene Butler.\"\n\n\"That fop?\" he taunted. \"He's as worthless as they come.\"\n\n\"He's filthy rich\u2014\"\n\n\"His father is filthy rich,\" Cromwell corrected her. \"If Sam Butler hadn't made a lucky strike when he stumbled onto the Midas gold vein, he'd have died broke.\"\n\n\"Eugene will be richer than you when his father dies.\"\n\n\"He's a wastrel and a sot. He'll spend his fortune so fast your head will swim.\"\n\n\"I can handle him,\" said his sister. \"He's madly in love with me and will do anything I tell him.\"\n\n\"You can do better, much better,\" grunted Cromwell. He picked up a speaking tube and spoke to the driver. \"Abner, turn left at the next street and stop at the Butler residence.\"\n\nAbner held up a hand to indicate he understood. He stopped the Rolls in front of a huge mansion constructed of wood in the Victorian style of the day. Then he stepped from the car and rang the bell at the iron-gated front door. A maid answered, and he handed her Cromwell's calling card. The maid took it and closed the door. A few minutes later, the door reopened and a tall, handsome man with sharply defined facial features came out and walked toward the car. He could have passed for a matinee idol onstage. Like Cromwell, he wore a woolen suit that was dark navy rather than black, a starched collar, and a tie with a white-diamond pattern. He paused in the portico and sniffed the air, which was tinged with a light fog that rolled in from the bay.\n\nAbner opened the Rolls's rear door, pulled down a jump seat, and stood back. Butler got in and sat down. He turned to Cromwell's sister. \"Maggie, you look positively stunning, good enough to eat.\" He left it there, seeing the fearsome, hostile look in Cromwell's eyes. He greeted Cromwell without offering his hand. \"Jacob, good to see you.\"\n\n\"You look fit,\" said Cromwell as if he cared.\n\n\"In the pink. I walk five miles a day.\"\n\nCromwell ignored him, picked up the speaking tube, and instructed Abner where to pick up Marion Morgan. He turned to his sister. \"What saloon on the Barbary Coast do you wish for us to mingle with the foul-smelling rabble?\"\n\n\"I heard that Spider Kelly's was quite scrubby.\"\n\n\"The worst dive in the world,\" Cromwell said knowledgeably. \"But they have good bands and a large dance floor.\"\n\n\"Do you think it's safe?\" asked Margaret.\n\nCromwell laughed. \"Red Kelly hires a small army of husky bouncers to protect affluent clientele like us from harm or embarrassment.\"\n\n\"Spider Kelly's it is,\" said Butler. \"I even took my mumsy and dad there one evening. They truly enjoyed watching the mix of unsavory people who frequent the place. We sat in the slummers' balcony to watch the lowlife cavort.\"\n\nThe Rolls stopped in front of an apartment building on Russian Hill just off Hyde Street on Lombard, a fashionable but affordable district of the city. This area of Russian Hill contained the homes and meeting places where intellectuals, artists, architects, writers, and journalists engaged in lofty arguments and discussions\u2014but mostly socialized and partied.\n\nMarion did not stand on protocol. She was waiting out front, on the top step of her building. As the Rolls eased to the curb, she descended and then stopped as Abner opened the door for her. She was dressed in a short jacket over a blue blouse with a matching skirt that had a simple elegance about it. Her blond hair was drawn back and twisted into a long braid with a bow at the back of her long neck.\n\nCromwell stepped out and gallantly helped her into the backseat. The chauffeur pulled down the other jump seat in which Cromwell, in a courtly manner, seated himself. \"Miss Marion Morgan, may I present Mr. Eugene Butler. And you've met my sister Margaret,\" he said, using her proper name.\n\n\"Miss Cromwell, a pleasure to see you again.\" Marion's tone was gracious but not exactly filled with warmth. \"Likewise, Eugene,\" Marion acknowledged sweetly with familiarity.\n\n\"You know each other?\" asked Margaret in surprise.\n\n\"Eugene...Mr. Butler...took me to dinner some time ago.\"\n\n\"Two years,\" Butler said good-naturedly. \"I failed to impress her. She spurned all my later invitations.\"\n\n\"And advances,\" Marion added, smiling.\n\n\"Ready for a hot night on the Barbary Coast?\" asked Cromwell.\n\n\"It will be a new experience for me,\" said Marion. \"I've never had the courage to go there.\"\n\n\"Remember the old song,\" said Margaret:\n\n\"The miners came in 'forty-nine,\n\nthe whores in 'fifty-one.\n\nAnd when they got together,\n\nthey produced the native son.\"\n\nMarion blushed and looked demurely at the carpet on the floor as the men laughed.\n\nA few minutes later, Abner turned onto Pacific Street and drove through the heart of the Barbary Coast, named after the lair of the Barbary pirates of Morocco and Tunisia. Here was the home of gamblers, prostitutes, burglars, con men, drunks, derelicts, cutthroats, and murderers. It was all there, debauchery and degradation, poverty and wealth, misery and death.\n\nThe infamous coast boasted more than three hundred saloons, wall-to-wall, within six city blocks, fifty of them on Pacific Street alone. It existed because of crooked politicians who were bribed by the saloon, gambling house, and brothel owners. The reputable citizens of the city complained publicly about the den of iniquity but averted their eyes because they were secretly proud of the distinction that their fair city of San Francisco more than equaled Paris, which bore the enviable reputation as the wickedest city in the Western Hemisphere, as a carnival of vice and corruption.\n\nAnd yet the Barbary Coast was glitzy and glamorous, with loads of ballyhoo and skulduggery, a veritable paradise for people of honest means to go slumming. The unsavory who ran the dens of sin\u2014in most cases, men\u2014relished seeing the swells from Nob Hill enter their establishments because they had no scruples charging them exorbitant prices for admission and liquor, usually thirty dollars for a bottle of champagne rather than the going rate of six to eight. Mixed drinks in most saloons were twenty-five cents and beer a dime.\n\nAbner slipped the Rolls through the revelers wandering the street and pulled to a stop in front of a three-story building that served as a hotel upstairs\u2014in reality, a brothel, called a cow yard, which housed fifty women in rooms, called cribs. The main floor was for gambling and drinking, while the downstairs basement had a stage for bawdy shows and a large wooden floor for dancing. They stepped from the car, with the men in the lead to shield the ladies, who stared with fascination at a flashy uniformed barker on the sidewalk.\n\n\"Step right into Spider Kelly's, the finest drinking and dancing establishment on the coast. All are welcome, all will have the night of their lives. See the wildest show and the most beautiful girls to be found anywhere. See them kick their heels over their heads; see them sway in a manner that will shock and amaze you.\"\n\n\"I like this place already,\" said Margaret gaily.\n\nMarion stared and clutched Cromwell's arm tightly and looked up at a sign largely ignored by the clientele that read NO VULGARITY ALLOWED IN THIS ESTABLISHMENT.\n\nThey entered a large, U-shaped entrance lobby decorated with framed panels of nude women dancing amid Roman ruins. A manager decked out in an ill-fitting tuxedo greeted them and escorted them inside. \"Do you wish to go downstairs for the show?\" he asked. \"The next one starts in ten minutes.\"\n\n\"We would like a safe table away from the riffraff,\" said Cromwell in a demanding tone. \"After we've enjoyed a bottle of your finest champagne, we'll go downstairs for dancing and the show.\"\n\nThe manager bowed. \"Yes, sir. Right this way.\"\n\nHe escorted Cromwell's party through the crowded saloon up to a table on the slummers' balcony Butler had mentioned overlooking the main floor of the saloon. Soon a waitress wearing a thin blouse cut low across her breasts and a skirt that came well above her knees, showing an ample display of legs in black silk stockings held up by capricious garters, brought a magnum of Veuve Clicquot Ponsardin champagne, vintage 1892. As she eased the bottle into an ordinary bucket filled with ice, she brushed against the men and gave each an earthy smile. Margaret returned the smile, letting the waitress know that Margaret knew that besides serving customers in the saloon she also worked in the cribs upstairs. Surprised at seeing a Nob Hill swell dressed in a revealing outfit, the waitress gave Margaret a lewd look.\n\n\"You know, dearie, a redhead like you is in high demand. You could name your own price.\"\n\nMarion was stunned. Cromwell fought to keep from laughing, while Butler became outright indignant. \"This is a lady!\" he snapped. \"You will apologize!\"\n\nThe waitress ignored him. \"If she's Jewish, she can make the top of the scale.\" Then she turned, gave a wiggle to her buttocks, and walked down the stairway.\n\n\"What does being Jewish have to do with it?\" Marion asked naively.\n\n\"There is a myth going around,\" explained Cromwell, \"that Jewish redheads are the most passionate of all women.\"\n\nMargaret was enjoying herself as she gazed around the main floor of the saloon. She felt a giddy elation at seeing the sailors and dock-workers, the young and honest working girls who unknowingly were easily led astray, and the hardened criminals milling around the floor, which was littered with a small army of men too drunk to stand. Unknown to the others, including her brother, Margaret had visited the dives of the Barbary Coast on several occasions. And she was well aware that her brother Jacob often frequented the expensive and most exclusive parlor houses, where the royalty of the shady women plied their trade.\n\nMarion found it disgusting and fascinating at the same time. She had heard the coast was the pit of bitterness and despair for the poor of San Francisco, but she had no idea how far humans could sink. She was not used to drinking and the champagne mellowed her after a while, and she began to see the depravity in a less-sickening light. She tried to imagine herself as one of those loose women, taking men to the cribs upstairs for as little as fifty cents. Horrified at herself, she quickly pushed the thought from her mind and rose unsteadily to her feet after Cromwell held up the empty bottle and announced that it was time for them to go downstairs.\n\nThe manager appeared and found a table that was occupied on the dance floor not far from the stage. Two couples dressed in soiled working clothes protested at having to give up their table, but the manager threatened them with bodily harm if they didn't move.\n\n\"What luck,\" said Margaret. \"The show is just starting.\"\n\nCromwell ordered another magnum of champagne as they watched a well-endowed woman step onto the small stage and begin a Dance of the Seven Veils. It wasn't long before the veils dropped away and she was left with a scanty costume that left little to the imagination. Her abdominal muscles rippled as she gyrated and made several lusty contortions. When she was finished, the men in the audience threw coins on the stage.\n\n\"Well, that was certainly arousing,\" Margaret said sarcastically.\n\nA small band began playing and couples moved onto the dance floor, stepping lively to a dance called the Texas Tommy. Butler and Margaret swirled around the floor with gay abandon as if they were one. Marion felt a self-conscious sense of embarrassment at being held close to her boss. In all the years she had worked for him, this was the first time he had ever asked her out. He was an excellent dancer, and she followed his lead gracefully.\n\nThe band changed tempo at different times so the dancers could move to the steps of the Turkey Trot and the Bunny Hug. Soon the dancers began to sweat in the confined, airless quarters of the basement. The champagne began to make Marion's head reel and she asked Cromwell if she could sit down for a few minutes.\n\n\"Would you mind if I left you for a little while?\" Cromwell asked courteously. \"I'd like to go upstairs and play a few hands of faro.\"\n\nMarion was vastly relieved. She was on the verge of exhaustion, and her new shoes were causing discomfort to her feet. \"Yes, please do, Mr. Cromwell. I could stand a breather.\"\n\nCromwell climbed the wooden stairway and walked slowly through the bustling gambling section until he came to a table where there were no players except the dealer. Two burly men stood behind the dealer and discouraged any customer from sitting at the table.\n\nThe dealer looked like he was born from a bull. His head sat like a chiseled rock on top of a neck that was as thick as a tree stump. His black hair was dyed, plastered down with pomade, and parted in the middle. His nose was flattened across his cheeks from being broken numerous times. His limpid eyes looked oddly out of place on a face that had seen more than its share of fists. He had the torso of a beer keg, round and abundant, but hard, without fat. Spider Red Kelly had been a fighter and had once fought James J. Corbett, knocking down the former heavyweight champion twice but getting knocked out himself in the twenty-first round. He looked up at Cromwell's approach.\n\n\"Good evening, Mr. Cromwell, I've been expecting you.\"\n\nCromwell opened the cover to his watch and glimpsed the hands on the dial. \"Forgive me for being eight minutes late, Mr. Kelly. I was unavoidably detained.\"\n\nRed Kelly smiled, showing a mouth full of gold teeth. \"Yes, I would have also been detained if I was in the company of such a lovely lady.\" He nodded at the table. \"Would you care to try your luck?\"\n\nCromwell took out his wallet and counted out ten fifty-dollar National Bank notes printed by his bank under contract with the federal government. Kelly casually placed the bills in a small stack on the side of the table and pushed a stack of copper tokens advertising the saloon across the table. A typical faro layout of a suit of thirteen cards was painted on the table's green felt cover. The suit was in spades from ace to king, with the ace on the dealer's left.\n\nCromwell placed a token on the jack and one between the five and six in a bet called splitting. Kelly discarded the top card from the dealer box, displaying the next card, called the losing card. It was a ten. If Cromwell had bet on it, he would have lost, since the house wins any wagers placed on the displayed card. Then Kelly pulled the losing card out of the box, revealing the winning card. It was a five. Cromwell won the full bet, not half.\n\n\"Beginner's luck,\" he said as Kelly pushed the winning tokens across the table.\n\n\"What is your pleasure, Mr. Cromwell?\"\n\n\"Nothing, thank you.\"\n\n\"You asked to see me,\" said Kelly. \"What can I do to return the favors you've given me over the years, the generous loans and the help in keeping the police out of my place?\"\n\n\"I need someone eliminated.\" Cromwell spoke as if he was ordering a beer.\n\n\"Here in the city?\" asked Kelly as he dealt another hand.\n\n\"No, Denver.\"\n\n\"A man, I hope,\" said Kelly without looking up from the dealer box. \"Place your bet.\"\n\nCromwell nodded and moved a token between the queen and jack. \"Actually, he's an agent with the Van Dorn Detective Agency.\"\n\nKelly paused before pulling a card from the box. \"Taking out a Van Dorn agent could have serious repercussions.\"\n\n\"Not if it's done right.\"\n\n\"What's his name?\"\n\n\"Isaac Bell.\" Cromwell passed across the picture his sister had given him. \"Here's his photo.\"\n\nKelly stared at it briefly. \"Why do you want him removed?\"\n\n\"I have my reasons.\"\n\nKelly pulled the losing card and revealed the winning card as the queen. Cromwell had won again.\n\nKelly gazed across the table at Cromwell. \"From what I've heard, everyone who's killed a Van Dorn agent has been tracked down and hung.\"\n\n\"They were criminals who stupidly allowed themselves to be run down by detectives from the agency. If done in an efficient manner, Van Dorn will never know who killed Bell or why. Make it look like a random killing or even an accident. Leaving no trace would make it impossible for Van Dorn's agents to retaliate.\"\n\nKelly sank slowly back in his chair. \"I have to tell you, Cromwell, I don't like it.\" There was no \"Mr. Cromwell.\"\n\nCromwell smiled a grim smile. \"Would you like it if I paid you twenty thousand dollars for the job?\"\n\nKelly sat up and looked at Cromwell as if he was not sure if he believed him. \"Twenty thousand dollars, you say?\"\n\n\"I want it done by a professional, not some two-bit killer off the street.\"\n\n\"Where do you wish the deed to take place?\"\n\nThere was never doubt that Kelly would do the job. The saloon owner was knee-deep in any number of criminal activities. Coming under Cromwell's spell for financial gain was a foregone conclusion.\n\n\"In Denver. Bell works out of the Van Dorn office in Denver.\"\n\n\"The farther away from San Francisco, the better,\" Kelly said quietly. \"You got yourself a deal, Mr. Cromwell.\"\n\nThe \"Mr.\" was back, and the transaction agreed upon. Cromwell rose from his chair and nodded toward the tokens on the table. \"For the dealer,\" he said, grinning. \"I'll have ten thousand in cash delivered to you by noon tomorrow. You'll get the rest when Bell is deceased.\"\n\nKelly remained seated. \"I understand.\"\n\nHe pushed his way downstairs and through the dancers, who had stopped dancing. He saw they were watching his sister perform an undulating and provocative hootchy-kootchy dance on the stage, to the delight of everyone present. She had loosened her corset and let her nicely coiffed hair down. Her hips swiveled and pulsed sensually to the music of the band. At the table, Butler was sprawled in a drunken haze while Marion stared in awe at Margaret's gyrations.\n\nCromwell motioned for one of the managers, who also acted as bouncers.\n\n\"Sir?\"\n\n\"Please carry the gentleman to my car.\"\n\nThe bouncer nodded, and with one practiced motion lifted the thoroughly intoxicated Butler to a standing position and threw him over his shoulder. Then the bouncer proceeded up the stairs, carrying Butler's bulk as lightly as if he were a bag of oats.\n\nCromwell leaned over Marion. \"Can you walk to the car?\"\n\nShe glanced up at him as if angry. \"Of course I can walk.\"\n\n\"Then it's time to leave.\" He took her by the arm and eased her from the chair. Marion, unassisted but wobbly, went up the stairs. Then Cromwell turned his attention to his sister. He was not amused by her scandalous behavior. He grabbed her by the arm hard enough to cause a bruise and hauled her off the stage and out of the saloon to the waiting car at the curb. Butler was passed out in the front seat with Abner while Marion sat glassy-eyed in the back.\n\nCromwell roughly shoved Margaret into the backseat and followed her, pushing her into one corner. He sat in the middle between the two women as Abner got behind the wheel, started the car, and drove up the street that was ablaze with multicolored lights.\n\nSlowly, Cromwell slid his arm around Marion's shoulders. She looked at him with a vague, unresponsive expression. The champagne had given her a sense of lethargy, but she was not drunk. Her mind was still clear and sharp. His hand squeezed one shoulder and there was a small pause in her breathing. She could feel his body pressing against hers in the narrow confines of the seat.\n\nThere was a time when Marion had found her boss appealing and felt a deep attraction to him. But in the years she had worked for him, he had made no effort to bridge the gap between them. Now, suddenly, after all this time, he was showing an interest in her. Strangely, there was no emotion or arousal surging within her. She felt as if she were repelled by him and she couldn't understand why.\n\nMarion was relieved there were no further moves on his part. The one arm remained snaked around her waist and his hand rested lightly on her shoulder until Abner stopped the Rolls in front of her apartment house. Cromwell stepped to the sidewalk and helped her from the car.\n\n\"Good night, Marion,\" he said, holding her hand. \"I trust you had an interesting evening.\"\n\nIt was as if she saw now something deep within him that she had never seen before and she felt repulsed by his touch. \"It will be an evening I'll long remember,\" she said honestly. \"I hope Mr. Butler and your sister recover.\"\n\n\"They'll be hungover tomorrow, and justly so,\" he said with a tight smile. \"I'll see you Monday morning. There is a pile of correspondence I have to dictate. I want to have a clean desk when I leave on a business trip on Friday.\"\n\n\"You're leaving again so soon?\"\n\n\"A bankers' conference in Denver. I must attend.\"\n\n\"Until Monday morning, then,\" she said with vast relief as he released her hand.\n\nMarion climbed the steps to the door but turned and gazed at the Rolls-Royce as it pulled away into the street. Her mind acted without command. Things between her employer and herself would never be the same. There was a coldness about him that she was not aware of before and she cringed as she remembered his touch. All of a sudden, the lingering smell of the dance hall's smoke and sweat on her clothes sickened her.\n\nShe rushed upstairs to her room, turned on the faucets of her bathtub, frantically removed her clothes, and slipped into the soapy water to remove all memory of the decadent evening.\n\n\"WHAT WAS your little meeting with Red Kelly all about?\" asked Margaret after they let off Marion at her apartment.\n\n\"I hired him for a little job.\"\n\nShe stared at his face as it was reflected by the light of the passing streetlamps. \"What kind of a job?\"\n\n\"He's going to take care of Isaac Bell,\" he said matter-of-factly.\n\n\"You can't murder a Van Dorn agent!\" Margaret gasped. \"Every peace officer in the country would come after you.\"\n\nCromwell laughed. \"Not to worry, dear sister. I instructed Kelly to administer only enough damage to keep Bell in the hospital for a few months. That's all. Call it a warning.\"\n\nCromwell had blatantly lied to his sister. He would act surprised when Bell's murder was announced and claim the agent's death had been a mistake, that Kelly had gotten carried away. Inciting his sister's anger, he decided, was a small price to pay for eradicating the man who had become his worst enemy.\n\n## 16\n\nGIVE IT ANOTHER COAT,\" CROMWELL ORDERED THE two men painting his boxcar. The color had been the earth brown that most freight cars had been painted since the early days of the railroad. But Tuscan red was the newer color used by Southern Pacific to standardize their vast fleet of freight cargo haulers. Cromwell wanted a second coat because the O'BRIAN FURNITURE COMPANY, DENVER still bled through the freshly dried first coat.\n\nMargaret, dressed in a woolen dress and short jacket to keep her warm against the cool breeze blowing in from the ocean through the Golden Gate, held a parasol against a light early-morning mist that fell in the city. They stood watching the painters on the loading dock of an empty warehouse her brother had leased under a pseudonym.\n\n\"Can you trust them?\" she asked.\n\n\"The painters?\" He stared at the four men busy brushing paint on the boxcar. \"To them, it's just another job, another boxcar that needs to be tidied up. As long as they're well paid, they don't ask questions.\"\n\n\"About time you changed the name,\" she said. \"Some sheriff or a Van Dorn detective is bound to discover that an O'Brian Furniture freight car was present in five of the towns that were robbed.\"\n\n\"The same thought crossed my mind,\" he said.\n\n\"What are you going to call it this time?\"\n\n\"Nothing,\" answered Cromwell. \"It will look just like another freight car belonging to the Southern Pacific Railroad.\"\n\n\"You could buy and decorate a new one. Why keep this old relic?\"\n\n\"Because it looks like an old relic,\" he said with a slight laugh. \"Built in 1890. The railroad is still using this model. I prefer it to look tired and worn from many years and thousands of miles of hauling freight. And because its outward appearance is so ordinary, no one would suspect its true purpose. Even your Mr. Hotshot Bell could never begin to guess its real purpose.\"\n\n\"Don't underestimate Bell. He's smart enough to get wise to your traveling hotel suite.\"\n\nHe gave her a sour look. \"Not that smart. And even if he smells a rat, it's too late. The O'Brian Furniture car no longer exists.\"\n\nCromwell was proud of his aged boxcar. It was thirty-four feet long, with a capacity of forty thousand pounds. Empty, it weighed twelve thousand. Once the second coat was dry, the car would be finished off with the proper signage on its wooden sides, which would include a serial number under the letters SP, for Southern Pacific. The capacity and unloaded weight also would be lettered on one side, while the SP insignia sunrise\u2014a white circle with SOUTHERN arched across the top, PACIFIC arched across the bottom, and LINES across the middle\u2014would be painted on the opposite side. When finished, the boxcar would look like any one of thousands of cars belonging to the Southern Pacific.\n\nEven the serial number, 16173, was correct. Cromwell had arranged for the number to be lifted from a car in the middle of a railyard, scrapped, and then transferred to his rolling suite.\n\nNothing was ever left to chance. Every move was carefully thought out, then rehearsed and rehearsed again and again. All possible contingencies were considered and dealt with. Nothing escaped Cromwell's attention, down to the last detail. No bandit in the history of the United States, including Jesse James and Butch Cassidy put together, came close to matching him in the number of successful robberies he pulled off and the amount of loot he collected. Or the number of people killed.\n\nAt the mention of Bell's name, Margaret's mind traveled back to when they danced together at the Brown Palace Hotel. She cursed herself for wanting to reach out and touch him. The mere thought of it sent a shiver down her spine. She had known many men, a great number of them intimately. But none had affected her as much as when she was in Bell's arms. It was a wave of yearning she could neither understand nor control. She began to wonder if she would ever see him again, knowing deep inside it would be extremely dangerous. If they ever did meet, he surely would learn her true identity and find a path to her brother Jacob.\n\n\"Let's leave,\" she said, angry at herself for allowing her emotions to lose control.\n\nCromwell saw the faraway look in her eyes but chose to ignore it. \"As you wish. I'll return tomorrow to oversee the finished results.\"\n\nThey turned and walked through a door into the warehouse. Cromwell paused to lock the door and set a bar in place so no one could enter. Their footsteps echoed throughout the deserted interior of the building. The only furnishings were in one corner, two desks, and a counter that looked like the tellers' windows at a bank.\n\n\"A pity you can't lease this space out and put it to good use,\" said Margaret, fussing with her hat that had tilted to one side of her head when the pin slipped out.\n\n\"I must have a place to park the boxcar,\" Cromwell replied. \"So long as it sits unnoticed on a siding, next to the loading dock of an empty warehouse whose owner cannot be traced, so much the better.\"\n\nShe gave her brother a suspicious glance and said, \"You have that look on again.\"\n\n\"What look?\"\n\n\"The one that means you're planning another robbery.\"\n\n\"I can't fool my own sister,\" he said with a grin.\n\n\"I suppose it's a waste of time trying to talk you into retiring from the robbery business.\"\n\nHe took her hand and patted it. \"A man can't bear to give up a pursuit in which he excels.\"\n\nShe sighed in defeat. \"All right, where this time?\"\n\n\"I haven't decided yet. The first step is to make discreet inquiries in banking circles about payrolls. Then I have to select towns that have railroads and sidings for freight trains. The getaway is the most important part of the operation. Next is a study of the streets and location of the bank. Finally, I have to carefully plan the actual robbery itself, the timing and my disguises.\"\n\nMargaret stopped beside the desks and counter. \"And this is where you rehearse.\"\n\nHe nodded. \"After our agents obtain a layout of the interior of the bank and I arrange the furnishings accordingly.\"\n\n\"You have it down to a fine science.\"\n\n\"I try,\" he said loftily.\n\n\"Your method of operation is becoming too polished, too sophisticated,\" she cautioned him.\n\nHe took her by the arm and gently squeezed. \"I wouldn't have it any other way.\"\n\n## 17\n\nBELL CAME TO THE OFFICE DIRECTLY FROM THE TRAIN and found Irvine and Curtis already in the conference room waiting for him. He could tell the news was good because there were no frowns or grim looks on their faces. The jovial mood was enhanced by Irvine smoking a cigar and Curtis pulling a silver cigarette case from his coat pocket. \"You two seem to be in good spirits,\" said Bell, setting down his suitcase.\n\n\"We found some leads,\" Curtis said, lighting a cigarette. \"Nothing earth-shattering, but a few small pieces to fit in the puzzle.\"\n\n\"How about you, Isaac, did you turn up anything?\" asked Irvine.\n\nBefore Bell could answer, Agnes Murphy entered the conference room carrying a tray with three cups and a coffeepot. \"Sorry to interrupt,\" she said sweetly, \"but I thought you gentlemen might like some coffee.\"\n\nBell took the tray from her and set it on the long table. \"That's very kind of you, Agnes.\"\n\nShe turned and started for the door. \"I'll be right back.\" In less than a minute, she returned with a sugar bowl and cream pitcher. \"I didn't forget. I just couldn't carry it all.\"\n\n\"You're a lifesaver,\" said Curtis with a broad smile, as he lightly kissed her on the cheek.\n\nBell and Irvine exchanged glances, smiling. They both knew that Curtis and Agnes were just pals and always teasing each other. Agnes gathered her skirts as she turned, left the conference room, and closed the door.\n\n\"Besides the coffee,\" said Bell, \"it was thoughtful of her to close the door.\"\n\nCurtis blew a smoke ring toward the ceiling. \"She knows the score. Agnes has no more respect for Alexander than we do.\"\n\n\"You were about to say...\" Irvine prompted Bell.\n\n\"I discovered that, besides a missing finger, he probably has red hair. And rides a motorcycle, which he's used on more than one robbery.\" Bell reached into his pocket and lifted out a small silk sack, opened it, and spilled the cartridge out on the table. \"We now know the Butcher Bandit uses a thirty-eight-caliber Colt automatic. This shell casing was found under a carpet. The killer somehow missed it since he hasn't left any shells at his other bank hits. Sheriff Murphy of Bisbee was a smart man and had the county coroner remove the bullets from the murder victims. They all came from a thirty-eight Colt.\"\n\n\"We can check sales of all thirty-eight Colt automatics,\" said Curtis.\n\n\"There couldn't be more than ten thousand of them,\" Irvine replied sarcastically. \"It would take ten agents years to check out every gun dealer, salesman, and hardware store owner who sells thirty-eight Colt automatics.\"\n\n\"Art is right,\" Bell said as he stared at the brass cartridge. \"It would be a tremendous long shot.\"\n\nCurtis grinned like a fox. \"Not if we have a lead to where the bandit hides out. Then we can check out dealers in the area.\"\n\n\"Good thinking,\" Bell agreed, not knowing what Curtis was about to reveal. \"In the meantime, I'll send it off to Chicago and see if our agency experts can pull any fingerprints.\" He relaxed in a chair and tilted it back on two legs, propping a foot against the table. \"Now, let me hear what you two have unearthed.\"\n\nIrvine opened a bound ledger and placed the book on the table in front of Bell and Curtis. \"I hit pay dirt in Elkhorn, Nevada. They had recorded the serial numbers of the fifty-dollar bills in their vault the day before the robbery.\"\n\n\"I can understand why,\" said Bell. \"Fifties are counterfeited more than any other bill. As their bookkeeper itemized the bills, he must have studied each one and made sure they weren't bogus.\"\n\nIrvine tapped the entries in the book with his finger as he looked at Bell. \"You can request the Chicago office to put out bulletins to banks around the West to be on the lookout for them. Fifties will be easier to trace than fives, tens, or twenties.\"\n\n\"And a lot easier than ones,\" Curtis added.\n\n\"I'll see to it,\" Bell assured Irvine.\n\n\"I made a few inquiries on my own and actually came up with two banks in San Francisco where three of the bills showed up.\"\n\n\"Good work,\" said Bell. He then focused on Curtis. \"Now, how about you, Arthur? Any luck on your end?\"\n\n\"Did you find any passenger trains the killer might have escaped on?\" Irvine queried.\n\n\"No. But freight trains are a different story.\"\n\n\"Weren't they searched by the posses?\"\n\nCurtis shook his head. \"Not the ones that were loaded and locked.\"\n\n\"So where did you go from there?\" inquired Bell.\n\nCurtis broke out into a smile that spread and beamed. \"It took many hours of digging in musty old railroad company records, but I did manage to make an interesting discovery. I found three cars that were on the sidings of towns that were robbed. Boxcar serial number 15758 was present in Virginia City and Bisbee during the robberies. In Virginia City, its cargo manifest was listed as fifty bales of barbed wire to be transported to a ranch in Southern California. Boxcar 15758 was empty when it sat on a siding waiting to be switched to another train in Bisbee.\"\n\n\"Empty,\" Irvine repeated, stirring restlessly in his chair.\n\n\"Yes, empty. It had hauled a load of pottery from Las Cruces, New Mexico, to Tucson, before being sent empty back to El Paso.\"\n\n\"So we can scratch that one,\" muttered Bell. \"What about the others?\"\n\nCurtis referred to his notes. \"Number 18122 was present at Elkhorn, Nevada, and Grand Junction, Colorado, when their banks were robbed. It was on the siding at Grand Junction waiting to be switched to a train to take it to Los Angeles. Its cargo was sixty cases of wine. At Elkhorn, it carried a load of mattresses, from a factory in Sacramento, California.\"\n\n\"So much for 18122,\" said Irvine. \"It's not likely the bandit escaped to different locations.\"\n\nCurtis fairly beamed. \"I saved the best for last.\" Rising and walking to a blackboard, he wrote O'BRIAN FURNITURE COMPANY, DENVER on the black surface. Then he turned with a pleased expression on his face. \"Now we come to a boxcar that was present at five robberies.\"\n\nBoth Bell and Irvine sat up suddenly in their chairs as Curtis caught their fixed attention. The agent had taken the bull by the horns and delved into an area no one had thought to go.\n\nBell, surprised at Curtis's unexpected revelation, said, \"The car was in five towns on the day their banks were robbed?\"\n\n\"I've made a list of towns, times, and its final destination.\"\n\nIrvine nearly spilled his coffee as he set it back on the tray. \"Don't you mean destinations, plural?\"\n\n\"No. Destination, singular.\" Curtis laughed softly. \"In every instance, the furniture car from Denver went to San Francisco. I could find no record of it ever having been hauled to Denver or anywhere else. I can only assume it was a fa\u00e7ade the bandit used to escape the posses.\"\n\nBell stared at the writing on the blackboard. \"I'll bet a month's pay that a check of furniture stores in Denver will prove O'Brian Furniture does not exist.\"\n\n\"I think that goes without saying,\" Irvine summed up.\n\nBell turned to Curtis. \"When was the last Southern Pacific Railroad account of the car?\"\n\n\"It was put on a siding in the San Francisco railyards two weeks ago. At my last inquiry, it was still there.\"\n\n\"Then we've got to find and search it.\"\n\n\"And stake it out,\" said Irvine.\n\n\"That, too,\" replied Bell. \"But we must be very careful not to alert the bandit that we're closing in on him.\"\n\nCurtis lit another cigarette. \"I'll leave on the first train in the morning for San Francisco.\"\n\n\"Irvine and I will join you.\" Bell then turned his attention to Irvine. \"You mentioned that three bills turned up in San Francisco.\"\n\nIrvine nodded. \"That's right. One at the Cromwell National Bank of San Francisco and two at the Crocker National Bank.\"\n\nBell smiled for the first time. \"It would seem, gentlemen, all roads lead to San Francisco.\"\n\n\"It's beginning to look that way,\" Curtis agreed, his enthusiasm growing.\n\nThe two agents stared expectantly at Bell as he studied the map with the flags marking the terrible crimes committed by the Butcher Bandit. The evidence was almost infinitesimal and could easily lead to dead ends. Yet there was satisfaction in what the three Van Dorn agents had gleaned. Meager as it was, they had little else to go on. But it was enough for a plan to form in Bell's mind.\n\n\"It might be like betting on a plow horse at the racetrack, but I think we may have an opportunity to trap the bandit.\"\n\n\"You have a plan?\" asked Irvine.\n\n\"Suppose we plant stories in the local San Francisco newspapers revealing that a million-dollar payroll is being shipped by special train to a bank in a town populated by several thousand miners. The large amount would be because the mine owners have declared a special bonus for the workers to avert a threatened strike called by the miners' union over the demand for substantially increased wages.\"\n\nCurtis pondered Bell's proposition and said, \"The bandit could easily check out the story and find it's false.\"\n\n\"Not if we have one of us sitting in the telegraph office when the inquiry comes in and give the appropriate reply.\"\n\n\"We might even get lucky and discover who sent the telegram,\" said Irvine.\n\nBell nodded. \"There is that, too.\"\n\nIrvine gazed into his coffee cup as if he was a fortune-teller reading tea leaves. \"It's a thousand-to-one shot. We all know that.\"\n\n\"No doubt about it,\" Bell said, \"it's worth a try. And, if the scheme fails, we might still stumble onto another lead to the bandit.\"\n\n\"Got a mining town in mind?\" Curtis asked.\n\n\"Telluride, Colorado,\" answered Bell. \"Because the town is situated in a box canyon. Telluride is also the area where its miners struck the mine owners in 1901 and 1903, so another strike is quite plausible.\"\n\n\"If the O'Brian Furniture freight car shows up,\" said Curtis, \"we'll know our man took the bait.\"\n\n\"Once the train pulls it onto the Telluride siding, the only way out is the way it came.\" Irvine sighed and smiled contentedly. \"The bandit will be trapped and have no means of escape.\"\n\nThe atmosphere in the conference room crackled with expectation and hope. What had almost seemed like a lost cause was coming together. Three pairs of eyes trained on the giant wall map, traveled west toward the Pacific Ocean, and focused on the port city of San Francisco.\n\nIn the elevator that took him down to the street for his walk to the Brown Palace, Bell felt jubilant. Win, lose, or draw, the end of the game was in sight. Granted, it was still hazy and indistinct, but the cards were finally falling in Bell's favor. His thoughts turned to Rose and he found himself wondering for the hundredth time what connection she had with the Butcher Bandit.\n\nWhat woman could be close to a man who murdered women and children? He began to believe that she might be as rotten as the bandit, if not more so.\n\nBELL STEPPED from the Brown Palace elevator and walked to his suite. He pulled the key from his pant pocket and inserted it in the door lock. Before he could turn the key, the door slipped open a crack. The latch had not been fully engaged when the door had been closed.\n\nBell paused and tensed. His first thought was that the maid had forgotten to close the door and spring the bolt. It was a logical assumption, but an inner wisdom made him suspicious. The perception of something being not quite right had saved him on more than one occasion.\n\nBell had made many enemies during his years as a detective with Van Dorn. Several of the men he had captured and seen tried and sentenced to prison had vowed they would come after him. Three had tried and two had died.\n\nIf someone was waiting for him inside his room, it wouldn't be with a gun, he reasoned. Gunshots would echo throughout the hotel and bring a dozen staff running. For a criminal to escape from the ninth floor, he either would have to wait for an elevator or run down the stairs, neither a good choice for a successful escape.\n\nBell was aware that he was probably overexaggerating the threat, which could very well be nonexistent. But he hadn't survived this long without a suspicious mind. If someone was waiting inside his suite, he thought, they would do their dirty work with a knife.\n\nHe removed his hat and dropped it. Before it hit the carpet, his derringer was in his hand, an over-and-under, two-barrel, .41 caliber small handgun that packed a surprisingly heavy punch at close range.\n\nBell waggled the key in the door as if he was turning the lock. He pushed the door open and hesitated, staring around the foyer of the suite and the living room beyond before he entered. The smell of cigarette smoke greeted his nostrils, confirming Bell's suspicions. He only rarely smoked a cigar and then only with brandy after a gourmet dinner. With the derringer in hand, he stepped into the suite. Death, like a third man, was waiting inside.\n\nA man was sitting on a settee reading a newspaper. At Bell's approach, he laid the paper aside and revealed a face as ugly as sin. The black hair was greasy and slicked flat. His face looked like it had been stomped on by a mule, and he had the body of a state fair prizewinning boar. His eyes were strangely soft and friendly, a guise that fooled many of his victims. Bell was not fooled; he could see the man had the strength to spring like a tiger.\n\n\"How did you get in?\" Bell asked simply.\n\nThe stranger held up a key. \"Skeleton key,\" he said in a voice that came like a rock crusher. \"I never leave home without one.\"\n\n\"What is your name?\"\n\n\"It won't matter if you know my name. You'll never get a chance to use it. But since you've asked, it's Red Kelly.\"\n\nBell's photographic memory shifted into gear and the recollection of a report he'd once read came back. \"Yes, the infamous Red Kelly, boxer, Barbary Coast saloonkeeper, and murderer. You fought a good battle against world champion James J. Corbett. I once studied a report on you in the event you ever wandered beyond the California border. This is a mistake on your part. You have protection from crooked politicians that keeps you from getting extradited for crimes in other states, but that won't help you in Colorado. You're subject to arrest here.\"\n\n\"And who is going to arrest me?\" said Kelly showing an expanse of gold teeth. \"You?\"\n\nBell stood loosely, waiting, and expecting a move from Kelly. \"You wouldn't be the first.\"\n\n\"I know all about you, pretty boy,\" said Kelly contemptuously. \"You'll bleed just like the other poor slobs I've put in the grave.\"\n\n\"How many detectives and police?\"\n\nKelly grinned nastily. \"Three that I can remember. After a while, the numbers began to fade.\"\n\n\"Your days of murder are over, Kelly,\" Bell said calmly.\n\n\"That'll be the day, pretty boy. If you think you can bully me with that popgun in your hand, you're wasting your breath.\"\n\n\"You don't think I could kill you with it?\" Bell said.\n\n\"You'd never get the chance,\" Kelly retorted coldly.\n\nThere it was. Bell caught it instantly. The sudden shift in the eyes. He swung into a crouch and, in the blink of an eye, aimed and fired a shot into the forehead of the man who was creeping up behind him from where he'd been, hidden by a curtain. The report reverberated out the open door and throughout the atrium of the hotel.\n\nKelly glanced at the body of his henchman with all the interest of a horse that had stepped on a prairie dog. Then he smiled at Bell. \"Your reputation is well founded. You must have eyes in the back of your head.\"\n\n\"You came to kill me,\" said Bell evenly. \"Why?\"\n\n\"It's a job, nothing more.\"\n\n\"Who paid you?\"\n\n\"Not necessary for you to know.\" Kelly laid aside the paper and slowly got to his feet.\n\n\"Don't try for the gun in the belt behind your back,\" Bell said, the derringer as steady as a branch on an oak tree.\n\nKelly flashed his gold teeth again. \"I don't need a gun.\"\n\nHe sprang forward, his powerful legs propelling across the room as if he had been shot from a cannon.\n\nWhat saved Bell in those two seconds was the span between them, a good eight feet. Any less distance and Kelly would have been all over him like an avalanche. As it was, Kelly struck him like a battering ram, a glancing blow that knocked Bell sideways over a chair and onto the grass green carpet. But not before he pulled the trigger of the derringer and sent a bullet into Kelly's right shoulder.\n\nThe brute was stopped in his tracks but did not fall. He was too powerful, too muscular, to fold with a bullet that did not penetrate his heart or brain. He contemplated the spreading of crimson on his shirt with the detached look of a surgeon. Then he grinned fiendishly. \"Your little popgun only holds two bullets, pretty boy. Now it's empty.\"\n\n\"I wish you would stop calling me pretty boy,\" said Bell, leaping to his feet.\n\nNow it was Kelly's turn to reach behind his back and retrieve his Colt revolver. He was just bringing it level to pull the trigger when Bell hurled his derringer like a baseball pitcher receiving the signal from the catcher to throw a fastball. At four feet away, he couldn't miss. The little gun, solid as a piece of quartz, thudded off Kelly's face just above the nose and between the eyes.\n\nBlood gushed from the gash and quickly covered the lower half of Kelly's face. The blow staggered him more than the slug in his shoulder. There was no gasp of pain, no bloodcurdling cry. He made no sound other than a great sigh. The gun was still in his hand, but he did not lift it to aim. He couldn't. Bell put his head down and charged the strong man like a porpoise into a great white shark, accelerating with every step, thrusting his head into Kelly's stomach with all his strength. The ex-boxer merely grunted and brushed Bell away, throwing him halfway across the room with a strength nothing less than phenomenal.\n\nBell crashed into a wall with a crunch that forced the breath from his lungs. If the impact had been any harder, he'd have been in traction for two months. But his bone-jarring charge had not been in vain. During the collision of his one hundred eighty-five pounds against Kelly's two-fifty, he had snatched the revolver from the hand of the killer.\n\nThere was no command to cease another assault, no \"Stop or I'll shoot.\" Bell had been through the mill and knew you don't waste words on a killer dead set on sending you to a marble slab at the county coroner's. He had no illusions about beating Kelly in a man-to-man fight. The murderer was stronger and more ruthless. Bell barely got off two shots before Kelly recovered enough to reach out and grasp Bell around the neck with the ferocity of a gorilla, his massive hands choking the life out of the detective. He fell against Bell, pressing him into the carpet, his massive weight crushing Bell's torso and pinning his arms so he couldn't fire the Colt again. Kelly squeezed calmly and purposely, as though the bullets he had taken were a mere annoyance.\n\nBell couldn't move, and there was no reaching up in an attempt to pull off the fingers digging into his neck. Kelly's strength went way beyond Bell's. Bell didn't doubt that he wasn't the first man Kelly had strangled. Unless he did something very rapidly, he wouldn't be the last. Blackness was edging his line of vision, and getting darker by the moment.\n\nWhat stunned Bell more than the realization that he was only seconds away from death was what happened to the two bullets he had pumped into Kelly. He was certain he had struck the Goliath in the body. Bell looked up into two eyes that were as dark as evil and the blood that had turned the lower face into a horrific mask of crimson. What was keeping him alive, why wasn't his strength ebbing? The man wasn't human.\n\nThen, perceptibly, Bell felt the pressure begin to ease slightly. Rather than try to pry the hands from around his neck, Bell reached up and embedded his thumbs in Kelly's expressionless eyes, knowing it would be the final move before darkness closed over him. In a violent, corkscrew motion, Bell twisted his body out from under Kelly.\n\nThe big boxer groaned and covered his eyes with his hands. Unseeing, he crawled toward Bell, who kicked out viciously, catching Kelly in the stomach. Only then did he see the two bullet holes seeping blood through Kelly's shirt below the rib cage. What kept him going? Bell wondered. He should have died before now. But, instead, Kelly reached out and grabbed Bell by the leg.\n\nBell felt himself being pulled across the carpet, now stained and soaked with Kelly's blood. He lashed out with his free foot. It bounced off Kelly, who acted as if he never felt it. The grip on Bell's calf tightened. Fingernails dug through his pants into his flesh. He was pulled closer to Kelly, seeing an agonized face, the eyes glaring with hatred.\n\nIt was time to end this ghastly fight. Bell's right hand still held the Colt. With deadly calm, he raised the barrel until the muzzle was only inches from Kelly's face, deliberately pulled the trigger, and sent a .44 caliber bullet into Kelly's right eye.\n\nThere was no terrible scream or horrible gurgling sound. Kelly exhaled an audible gasp from his throat and rolled over on the carpet like some great beast in its death throes crumpling to the earth.\n\nBell sat up on the floor and massaged his throat, panting from the exertion. He turned his head and stared at the doorway as men came rushing into the suite. They stopped in stunned shock at the sight of the sea of blood and the great hump of a man whose face was unrecognizable because of the bloody, congealing mask. The face looked particularly grotesque due to the gold teeth showing through the open lips that slowly became coated red.\n\nKelly had died hard, and for what? Money? A debt? A vendetta? Not the latter. Bell had never launched an investigation against the Barbary Coast giant. Someone must have paid him to kill and paid him extremely well.\n\nBell wondered if he would ever know the answer.\n\nTHE NEXT MORNING, Bell stepped out of the big porcelain bathtub, toweled off the water that dripped down his body, and gazed in the mirror. His throat didn't look pretty. It was swollen, with purple bruises so obvious that he could see the shape of Red Kelly's fingers where they had dug into his flesh. He put on a clean white shirt and was pleased to see that the high, starched collar, though it chafed his tender skin, covered the bruises.\n\nThey weren't the only purplish green marks on his aching body. He had several from falling over the chair, and from being thrown across the room and into the wall by Kelly's brute strength. They were tender to the touch and would not fade anytime soon.\n\nAfter dressing in his trademark linen suit, Bell left the hotel and stopped off at the Western Union office and sent a telegram to Joseph Van Dorn that told of the attempt on his life. When he came slowly through the door of the office, Agnes Murphy openly stared at him. She stood up with a look of motherly concern in her eyes. \"Oh, Mr. Bell. I heard about your unfortunate incident. I do hope you're all right.\"\n\n\"A few bruises, Agnes, nothing more.\"\n\nCurtis and Irvine heard his voice and came from the conference room, followed by Alexander from his office. Both agents vigorously shook his hand\u2014a bit too vigorously, Bell thought, wincing at the discomfort that traveled over his aching body. Alexander merely stood back, as if he was a spectator in an audience.\n\n\"Glad to see you alive and kicking,\" said Curtis. \"We heard it was quite a fight.\"\n\n\"It was as close as I ever came to buying the farm,\" said Bell.\n\n\"After talking to you over the phone,\" said Curtis, \"I wired your identification of Red Kelly to our San Francisco office. They're going to check out Kelly and any of his clients who might have wanted you eliminated.\"\n\n\"A terrible thing,\" Alexander said without emotion. \"Unthinkable, that someone would attempt to assassinate a Van Dorn agent.\"\n\nBell gave Alexander a long hard look. \"I can only wonder how Kelly knew where I was staying.\"\n\n\"Kelly was a well-known crime boss on the Barbary Coast in San Francisco,\" said Irvine. \"Could any of your former friends who you put in jail or friends and families of those who were executed because you arrested them be from San Francisco?\"\n\n\"None that I can name,\" answered Bell. \"If I had to make a guess, I'd have to say the Butcher Bandit was behind it.\"\n\n\"Knowing you were on the case,\" said Irvine, \"he'd certainly have a motive.\"\n\nAlexander said, \"We won't rest until we get to the bottom of this.\" To Bell, his words rang hollow. \"I can't tell you how glad I am that you are alive and well.\" Then he turned and walked back to his office.\n\nAs soon as he was out of earshot, Bell said, \"Another nail in the coffin, gentlemen. The key to the bandit's whereabouts is San Francisco.\"\n\n## 18\n\nWHEN BELL, IRVINE, AND CURTIS STEPPED OFF THE ferry from Oakland and entered the huge Ferry Building, they found themselves in a three-story-tall hall with repeating arches and skylights overhead. They exited onto the Embarcadero, at the foot of Market Street. While Irvine and Curtis went to hail a motor cab, Bell turned and looked up at the two-hundred-forty-foot clock tower, modeled after the twelfth-century Giralda bell tower in Seville, Spain. The long hands on the expansive dial read eleven minutes past four.\n\nBell checked the time on his watch and duly noted that the ferry building clock was one minute fast.\n\nBecause of the huge crowds in the terminal after pouring off four ferryboats at the same time, the agents were unable to find a free motor cab. Bell stopped a horse-drawn carriage, haggled a price with the driver, and commandeered it to carry them to the Palace Hotel on Montgomery Street. As they settled in the carriage, Curtis spoke to Bell.\n\n\"How do you plan to handle the Van Dorn San Francisco office?\"\n\n\"We're having dinner with the district director. His name is Horace Bronson. I once worked with him in New Orleans. He's a fine fellow and very efficient. When I sent him a telegram, he wired back and offered every cooperation in his power. He promised to send his agents out to obtain the names of people from gun dealers who might have purchased a thirty-eight-caliber Colt automatic.\"\n\nIrvine rolled an unlit cigar around in his fingers. \"On my end, I'll start with the Cromwell and Crocker banks and see if they can help trace any of the stolen currency serial numbers.\"\n\nBell said to Irvine, \"You might check out the other major banks, too, such as Wells Fargo and the Bank of Italy, in case any stolen bills might be in their possession. If the bandit is from San Francisco, it stands to reason he'd have passed them around town.\"\n\n\"We have our work cut out for us,\" said Curtis. \"I'll see if I can't track down the O'Brian Furniture car.\"\n\nBell stretched out his feet in the carriage and said, \"After we meet with Bronson, I'll write out news releases about the fake currency shipment to the San Miguel Valley Bank in Telluride and prevail upon the editors of the city's major newspapers to run the story.\"\n\nThe carriage reached the magnificent Palace Hotel and turned into the Garden Court, the hotel's elegant carriage entrance that was commanded by seven stories of gleaming white marble balconies with over a hundred ornamented columns. Light from above filtered through a huge stained-glass-domed skylight.\n\nBell paid off the coachman as porters took the luggage inside. The three Van Dorn detectives walked into a vast, majestic lobby. After registering, they went up to their rooms in a redwood-paneled hydraulic elevator. Bell arranged for the rooms to be joined together to create a large suite.\n\n\"Tell you what,\" said Bell to Irvine and Curtis. \"It's almost five o'clock, so nothing can be accomplished today. Let's get cleaned up. Then we'll go out, have a good meal, get a good night's sleep, and start beating the bushes first thing in the morning.\"\n\n\"Sounds good to me,\" Irvine said, his stomach growling, since they had eaten nothing in the last eight hours.\n\n\"What have you got in mind for a restaurant?\" asked Curtis.\n\n\"Bronson is a member of the Bohemian Club. He's arranged for us to eat with him in their dining room.\"\n\n\"Sounds exclusive.\"\n\nBell smiled. \"You don't know how exclusive.\"\n\nAT EIGHT O'CLOCK, the men exited a motor cab at the Taylor Street entrance of the powerful and elite Bohemian Club. Founded in 1872 as a gathering place for newspaper journalists and men of the arts and literature, its members included Mark Twain, Bret Harte, Ambrose Bierce, and Jack London. Over the years, powerful and influential men who made up the business elite of the city joined and soon became the dominant group. No women were allowed, and wives and unmarried guests of the members had to enter through a back door.\n\nThis evening, women were permitted into the dining room because Enrico Caruso was being honored and he insisted upon his wife being present. The club directors considered it a special occasion and so had made it one of the few exceptions.\n\nIrvine and Curtis followed Bell into the main reception room and stood for a moment until a tall man with a youthful face in a well-conditioned, muscular body that gave the impression of towering height came forward and shook Bell's hand vigorously. \"Isaac, how good to see you.\"\n\n\"The pleasure is mine,\" returned Bell, pleased to see an old friend and prepared for a bone-crushing handshake. \"You're looking fit.\"\n\n\"I still work at it.\" He nodded at Irvine and Curtis and smiled. \"Hello, I'm Horace Bronson.\"\n\nHis voice was husky and went with the broad shoulders that looked as though they were about to burst the seams of his neatly tailored gray suit. His facial features made him look like a schoolboy under a thick forest of sun-bleached hair.\n\nBell made the introductions and was amused to see the tight expressions on his agents' faces and their eyes blink as Bronson compressed their hands in his big paw. Though he headed up an office with ten agents in a major city, Bronson deferred to Bell, who out-ranked him in the agency. He also greatly admired Bell for his wide experience and enviable reputation in apprehending lawbreakers. And he was also indebted to the master detective who had recommended him to Van Dorn for the post in San Francisco.\n\n\"Come this way into the dining room,\" he said warmly. \"The club is noted for its gourmet fare and fine wine.\"\n\nBronson led the way from the imposing grand lobby into the large and impressive dining room finished majestically with mahogany on the floors, walls, and ceiling. He had a few words with the ma\u00eetre d'.\n\nBronson put his hand on Bell's shoulder. \"I asked him for a table I usually reserve for talking business. It's in a corner of the dining room where we can't be overheard.\"\n\nThe ma\u00eetre d' showed them to a table off by itself but with an unimpaired view of the other diners throughout the room. A waiter was standing by, who laid napkins in their laps and waited until Bronson perused the wine list and made his selection. As soon as the waiter was out of earshot, Bronson relaxed and looked at Bell.\n\n\"I checked out the number of businesses that have sold thirty-eight-caliber Colt automatics since they were introduced on the market. The total comes to sixty-seven. I've put four agents on the investigation. They should have an answer in two or three days\u2014earlier, if they get lucky.\"\n\n\"Thank you, Horace,\" said Bell. \"That will save us much-needed time to look into our other leads.\"\n\n\"It's the least I can do,\" Bronson said with a broad smile. \"Besides, Mr. Van Dorn ordered me to give you my fullest cooperation.\"\n\n\"We'll need all the help we can get.\"\n\n\"Do you have any other leads on the Butcher Bandit?\"\n\n\"I'll have to swear you to secrecy. I've found that the bandit has spies inside our agency.\"\n\n\"You're safe confiding with me,\" Bronson said with growing concern. \"It's hard to believe such an intrusion can happen. Does Van Dorn know about it?\"\n\nBell nodded. \"He knows.\"\n\nThen Bell gave Bronson a rundown on the evidence, slim as it was, that led them to San Francisco. He explained Irvine's tracking of the serial numbers on the money, Curtis's discovery of the getaway freight car, and his own revelation about the bandit's hair and missing finger. He told it carefully, with the details but without embellishment. Irvine and Curtis also added comments on what they had uncovered during their investigations. When Bell finished his report, Bronson sat silent for several moments.\n\nAt last, he said, \"Your investigation has shown great progress, Isaac. You have something tangible when there was nothing a few weeks ago. But, unfortunately, it's hardly enough to identify the bandit.\"\n\n\"No, it's not,\" Bell agreed, \"but it's a thread that can lead to a string that can lead to a rope.\"\n\nThe wine that Bronson selected, a California chardonnay reserve from Charles Krug, the oldest winery in the Napa Valley, arrived and, after the proper tasting ceremony, was poured. As they studied the menu, all talk of the bandit was put on hold while they enjoyed the wine and made their selections.\n\n\"What intrigues you?\" Bronson asked Bell.\n\n\"The kitchen has sweetbreads in b\u00e9chamel sauce. I'll give them the taste test since I am a lover of sweetbreads.\"\n\n\"Aren't they bull's testicles?\" said Curtis.\n\n\"You're thinking of Rocky Mountain oysters,\" said Bronson, laughing.\n\n\"Prized by gourmets throughout the world,\" explained Bell, \"they are the thymus glands of veal. There are two glands, one in the throat and the other near the heart. The heart sweetbread is considered the most delicious by chefs\u2014\"\n\nSuddenly, Bell stopped in midsentence and stared intently across the dining room. His violet eyes narrowed, as if focusing in the distance. His relaxed position stiffened and he sat up, as if lost in preoccupation.\n\n\"What is it, Isaac?\" asked Irvine. \"You look like you've seen the Resurrection.\"\n\n\"I have,\" Bell murmured, his eyes staring at a couple who had walked in the door and were talking to the ma\u00eetre d'. They were a striking pair that turned every head in the dining room. Both had the same flame red hair. The woman was as tall as the man, who was slight in stature.\n\nShe wore a yellow two-piece dress suit of the Empire style, with a gored skirt that created an elongated trumpet-bell shape with a short trail on the floor. The blouse was embroidered with lace trim and worn under a short jacket that had an extremely low neckline which allowed her to show off a magnificent diamond necklace. In an era dominated by formality, her fashionable Merry Widow\u2013wide hat with lavish feather trim was perfect for a dressy function. A fox boa was draped around her shoulders.\n\nThe man wore an expensive black suit with vest. A large gold chain hung from one pocket and threaded through a buttonhole to another pocket that held a watch. A large diamond-encrusted fob hung from it. There was a confident look in his eyes that missed nothing. He surveyed the room as if he owned it. Seeing several people he knew, he smiled slightly and graciously bowed his head. The couple was shown to a table in the center of the dining room in a position highly visible to the other diners. It was a rehearsed entrance that was carried off with sophisticated elegance.\n\n\"Who is that couple who made the grand entrance?\" Bell asked Bronson.\n\n\"That's Jacob Cromwell, who owns the Cromwell National Bank. He's a member of the Bohemian Club. The handsome woman at his side is his sister.\"\n\n\"Sister?\"\n\n\"Yes, her name is Margaret, a member of the social elite. Keeps busy with charity work. She and her brother are very wealthy and influential. They live on Nob Hill.\"\n\n\"So her name is Margaret Cromwell,\" Bell said quietly. \"I knew her in Denver as Rose Manteca.\"\n\nIrvine looked at Bell. \"Is she the woman you told us about who was a spy for the Butcher Bandit?\"\n\n\"Unless she has a twin sister,\" Bell answered, \"that's her.\"\n\n\"Impossible,\" said Bronson in a tone heavy with derision. \"The assumption is utterly ridiculous. She and her brother do more for San Francisco than half the wealthy of the city put together. They support orphan homes, the humane society for the lost and wandering animals of the city, and city beautification. They give large donations to worthy causes. They are highly respected and admired.\"\n\n\"He makes a strong case,\" said Curtis. \"If the Cromwells own a large San Francisco bank and are already wealthy, what's their percentage in robbing and killing?\"\n\n\"Is Miss Cromwell married?\" Bell asked Bronson.\n\n\"No, she's single, and has the reputation of being on the wild side.\"\n\n\"Could you have been wrong about her being a spy for the bandit?\" Irvine suggested.\n\nBell gazed intently at Margaret Cromwell, taking in every detail of her face. She seemed deep in conversation with her brother and did not turn in his direction. \"I could be mistaken,\" he murmured without conviction. \"The resemblance between her and the woman I met in Denver is uncanny.\"\n\n\"I know Cromwell personally,\" said Bronson. \"He cooperated with Van Dorn on a bank swindle that a gang of con men were using to bilk local businesses. I'll introduce you.\"\n\nBell shook his head and came to his feet. \"Not to bother. I'll introduce myself.\"\n\nHe stood, dodged the chairs of the diners, and made his way to the Cromwells' table. He purposely came up behind and slightly off to the side of Margaret so she wouldn't notice his approach. He ignored Cromwell and looked down at her with a condescending smile and wondered how she would react. \"I beg your pardon, Miss Cromwell, but I believe we met in Denver. My name is Isaac Bell.\"\n\nShe went rigid, and did not turn and look up at him. She stared across the table into her brother's eyes with an unfathomable expression\u2014surprise, maybe, or consternation, or something else\u2014something bordering on shock or distress. For an instant, it was as though she did not know how to react. And then she recovered in the blink of an eye.\n\n\"I'm sorry, but I don't know a Mr. Isaac Bell.\" Her voice was steady without the least indication of a tremor. She spoke without looking at him. She knew that if she did it would come like a physical blow to her stomach. She was grateful she wasn't standing or her legs would have turned to rubber and she'd have fallen to the carpet.\n\n\"Forgive me,\" said Bell, certain now from her reaction that she was the woman he knew as Rose Manteca. \"It must be a case of mistaken identity.\"\n\nCromwell had come to his feet out of courtesy and was holding his napkin. He gazed at Bell like a prizefighter sizing up his opponent before the bell of the first round. He showed not the least bit of surprise or incomprehension. He held out his hand. \"Jacob Cromwell, Mr. Bell. Are you a member of the club?\"\n\n\"No, a guest of Horace Bronson, of the Van Dorn Detective Agency.\"\n\nBell shook Cromwell's hand, thinking it strange the banker would keep his gloves on while he ate. Out of years of investigative habit, he glanced at the little finger of the glove on the left hand. The material over the finger was filled out and solid. Not that he thought there was the remotest chance Cromwell was the bandit. That was a crazy idea.\n\nCromwell nodded. \"I know Horace. A fine man. A credit to your company.\"\n\nBell noticed close up how Cromwell's red hair was closely trimmed and was beginning to thin at the rear of the head. The banker was short and thin and carried himself with more feminine grace than masculine roughness. Bell saw the same expression in the eyes as he'd once seen in a mountain lion he had shot in Colorado. There was a cold, almost dead, look from deep inside.\n\n\"Yes, that he is.\"\n\n\"Bell? I do not think I've heard the name before,\" Cromwell said as if trying to place it. He dismissed the thought as if it were of no great importance. \"Do you live in San Francisco?\"\n\n\"No, Chicago.\"\n\nMargaret still could not bring herself to look at Bell. She felt an uncontrollable fire down deep in her body. Her discomfort flared and she blushed red as a cherry. Then she turned angry, not so much at Bell but at herself for showing emotion. \"My brother and I would like to enjoy our dinner in private, Mr. Bell. If you will excuse us.\"\n\nHe saw her long neck turn red and felt pleased. \"I'm very sorry for the intrusion.\" He nodded at Cromwell. \"Mr. Cromwell.\" Then Bell turned and walked back to his table.\n\nAs soon as he was certain Bell had moved out of earshot, Cromwell snorted. \"What in hell is he doing in San Francisco? I thought Red Kelly took care of him.\"\n\n\"Apparently, Kelly failed,\" Margaret said with a small feeling of satisfaction in her stomach.\n\n\"How did he know you were here?\"\n\n\"Don't look at me,\" said Margaret angrily. \"I took the train from Denver to Los Angeles as Rose Manteca and bought a horse there under another name. Then I rode it to Santa Barbara, where I took a train to San Francisco under yet another name. There is no way he could have traced me.\"\n\n\"Are we to consider it coincidence?\"\n\nShe looked like a lost dog. \"I don't know. I just don't know.\"\n\n\"Regardless of why he's in San Francisco, his presence spells trouble,\" said Cromwell, staring openly with a constrained smile at the four agents seated around their table. \"I don't think he's put two and two together, but after seeing you, suspecting you might have a connection with the bandit, and learning you're my sister, he'll be nosing around.\"\n\n\"Maybe it's time for me to take a vacation.\"\n\n\"Not a bad idea.\"\n\n\"I'll book passage to Juneau, Alaska, first thing in the morning.\"\n\n\"Why Juneau?\" asked Cromwell. \"It's colder than a witch's nipple up there.\"\n\n\"Because it's the last place he'd look.\" She paused, and her eyes took on a shrewd look. \"And there is the fact that Eugene's father, Sam Butler, oversees his mining operations outside of Juneau.\" Margaret laughed, loosening the bond on her emotions. \"It gives me a chance to look over my future financial interests.\"\n\n\"Dear sister,\" Cromwell said genially, \"you are a never-ending, constant source of amazement.\" Then he brazenly looked across the dining room at Bell. \"I wonder,\" he muttered, \"what happened to Red Kelly.\"\n\n\"Maybe Bell killed him.\"\n\n\"Maybe,\" said Cromwell. \"If that's the case, Bell is far more dangerous than I gave him credit for. Next time, I'll handle the matter myself.\"\n\nWHEN BELL returned to the table, his dish of sweetbreads had arrived. He picked up a fork, looking forward to tasting the delicacy, but he was stopped by questions from everyone at the table.\n\n\"Was she the woman you think you met in Denver?\" demanded Bronson.\n\nBell dodged the question, not wanting to dwell on what he knew was a touchy subject with Bronson. \"I am probably wrong. I admit it. But the resemblance is quite extraordinary.\"\n\n\"You have an eye for beauty,\" Bronson said with a mild chuckle.\n\n\"How did you find Cromwell?\" asked Irvine. \"Do you think he will be helpful when I make an appointment with him to discuss the stolen currency that passed through his bank?\"\n\n\"You'll have to ask Horace. I didn't mention our investigation. He seemed nice enough, if a little lordly.\"\n\n\"He has a reputation of being lofty,\" said Bronson. \"But, one on one, he's quite solicitous, and I'm sure he will be very cooperative in your investigation.\"\n\n\"We shall see,\" Bell said, finally savoring the sweetbreads. After swallowing, he nodded at Irvine. \"I think I'll accompany you to the Cromwell National Bank.\"\n\n\"You want to meet him again?\" asked Bronson.\n\nBell shook his head. \"Not a priority, but I would like to probe around his bank.\"\n\n\"What do you expect to find?\" wondered Curtis.\n\nBell shrugged, but there was a faint gleam in his eyes. \"You know, I haven't the faintest idea.\"\n\n## 19\n\nMARION SAT AT HER DESK, TYPING A LETTER, WHEN two men entered the office. She turned from her Underwood Model 5 typewriter and looked up. One man, with a thicket of un-brushed brown hair, smiled a friendly smile. He was thin, and would have appeared sickly if not for his tanned face. The other was tall, with blond hair. She could not see his face because he had turned away and seemed to be studying the luxurious d\u00e9cor of the office. \"Miss Morgan?\"\n\n\"Yes, may I help you?\"\n\n\"My name is Irvine.\" He handed her his agency card. \"My fellow agent, Isaac Bell, and I are from the Van Dorn Detective Agency. We have an appointment with Mr. Cromwell.\"\n\nShe came to her feet but did not smile. \"Of course. Your appointment was for nine-thirty. You're five minutes early.\"\n\nIrvine made an open gesture with his hands. \"You know the saying...\"\n\n\"About the early bird getting the worm?\" she said as if amused.\n\nThe tall blond-haired man faced her. \"But the second mouse gets the cheese.\"\n\n\"Very astute, Mr. Bell...\" said Marion, her voice trailing off.\n\nTheir eyes met, and Marion suddenly felt something she had never felt before as she gazed into the blue-violet eyes. She realized now that he was well over six feet tall, with a wiry body clothed by a nicely tailored white linen suit. A large mustache was the exact shade of his flaxen, well-groomed hair. He was not handsome in the pretty-boy sense, but his features were craggy and masculine. There was a look of ruggedness about him, a man who was at home in the wild country of the West as well as the comforts of city life. She openly gazed at him, her usually well-restrained emotions in upheaval. No man had ever moved her this way before, certainly not on the first meeting.\n\nBell was also moved by the beauty of Marion and her aura of loveliness. The floor trembled beneath his feet as he stared back at her. She looked dainty but strong as a willow. There was a serene confidence about her that suggested she could surmount any complicated problem. She was poised and graceful, and, from the narrow waist to the flared bottom of her lengthy skirt, he could tell that she had long legs. The thick, lustrous hair was piled atop her head, with one long, narrow strand falling nearly to her waist. He guessed that she was the same age as he, give or take a year.\n\n\"Is Mr. Cromwell busy?\" he asked, tearing himself back to the purpose of the visit.\n\n\"Yes...\" she said with a trace of a stammer. \"But he is expecting you.\"\n\nShe knocked on Cromwell's door, entered, and announced Bell and Irvine's arrival. Then she stood aside and motioned them in as Cromwell came from behind his desk to greet them. As they passed through the door, Bell purposely brushed his hand against Marion's. She felt as if an electric shock had passed through her, before closing the door.\n\n\"Sit down, gentlemen,\" said Cromwell. \"Horace Bronson tells me you've come about the stolen currency that passed through my bank.\"\n\nIrvine seemed not to notice it, but Bell again found it intriguing that Cromwell wore gloves.\n\n\"That is correct,\" Irvine said as Bell let him handle the conversation. \"One of the bills, serial number 214799, was reported as being deposited in your bank.\"\n\n\"That is entirely possible,\" said Cromwell, toying with an unlit cigar. \"I assume it was a fifty-or hundred-dollar bill, because we never record any currency less than that amount.\"\n\nIrvine checked his notations in a notebook. \"Actually, it came from a merchant on Geary Street, a florist's shop. The manager, whose name is Rinsler, contacted the Van Dorn Detective Agency because he thought the bill might be counterfeit. It proved to be genuine. He stated that he'd obtained it from the Cromwell National Bank when he was transferring cash to a private safe.\"\n\n\"Rinsler's reasons sound a bit shady,\" added Bell. \"But if he's violated the law, that's a problem for the local police department.\"\n\n\"Millions of dollars pass through this bank during the course of a year,\" said Cromwell. \"I don't see why one bill is so important.\"\n\n\"Because a check of the serial number revealed that it came from a bank robbery in Elkhorn, Montana, where the bandit murdered four bank employees and customers,\" Bell explained.\n\nCromwell waited for more, but Bell and Irvine went silent. Irvine was examining his notes, but Bell was watching Cromwell intently. The banker met the unrelenting gaze without shifting his eyes. It stimulated his ego knowing that he was in a game of wits with the best agent Van Dorn had.\n\n\"I'm sorry, gentlemen,\" said Cromwell, moving his gaze from Bell to his unlit cigar. \"I fail to see how I can help you. If other bills from the robbery passed through the Cromwell Bank, they have long gone into general circulation and there is no way to trace them, no way of knowing who deposited them.\"\n\n\"That is true,\" Bell replied. \"But we have to check out every lead, no matter how remote.\"\n\n\"The bills were new and had consecutive serial numbers,\" explained Irvine. \"Is it possible you recorded them before they were put into circulation?\"\n\n\"Quite possible, since, as I've said, we record fifty-and hundred-dollar bills.\"\n\n\"Could you have your bookkeeper check your records?\" Bell asked.\n\n\"Happy to oblige.\" Cromwell paused to press a buzzer under his desk. Within seconds, Marion Morgan was standing in the doorway. \"Miss Morgan, would you please have Mr. Hopkins come up to my office?\"\n\nShe nodded. \"Of course.\"\n\nWhen Hopkins showed up, he was not what Bell expected. Instead of a colorless, lackluster little man with glasses and a pencil behind one ear who spent his working life poring through numbers in ledgers, Hopkins looked like a star athlete, big, robust, and quick of movement. He nodded as Bell and Irvine were introduced.\n\n\"Mr. Bell and Mr. Irvine are from the Van Dorn Detective Agency. They are here to check out serial numbers on currency that was stolen during a bank robbery in Elkhorn, Montana. A fifty-dollar bill was deposited in our bank before it was given to a customer cashing a check. These gentlemen think that other stolen bills might have passed through the bank. They would like you to check the list of serial numbers that we recorded.\"\n\nHopkins looked positively congenial as he smiled. \"I'll need the serial numbers.\"\n\n\"Check for consecutive bills above and below 214799,\" answered Cromwell, relying on his memory.\n\n\"Right away, sir,\" acknowledged Hopkins. He made a slight bow to Bell and Irvine. \"I should have the numbers, if they exist, within a few hours.\"\n\n\"I'd be grateful,\" said Bell.\n\n\"Anything else, gentlemen?\" said Cromwell, ending the interview.\n\n\"No, you've been most helpful. Thank you.\"\n\nBell let Irvine move out ahead of him to the elevator, lagging behind. He stopped at Marion's desk and gazed at her. \"Miss Morgan?\"\n\nShe swirled her chair from her typewriter in his direction but shied from looking into his eyes.\n\n\"I know this is terribly presumptuous of me, but you look like an adventurous lady, and I was wondering if you might throw caution to the winds and have dinner with me this evening?\"\n\nHer first impulse was to reject him, but some forbidden door had opened and she fought a battle of principle against desire. \"I'm not allowed to date bank customers. Besides, how do I know I can trust a complete stranger?\"\n\nHe laughed and leaned toward her. \"Number one, I am not a bank customer. And, number two, if you can't trust a bonded detective, who can you trust?\" He reached out and took her hand in his.\n\nA terrifying wave of anxiety swept over her as she fought a losing battle. Her last barrier crumpled and, along with it, her final grip on control. All self-restraint had evaporated.\n\n\"All right,\" she heard herself say, as if she was listening to a total stranger. \"I get off work at five o'clock.\"\n\n\"Good,\" he said, a little too enthusiastically, he thought. \"I'll meet you at the front entrance.\"\n\nShe watched him walk toward the elevator. \"Good Lord,\" she murmured to herself. \"I must be mad to have agreed to have dinner with a perfect stranger.\"\n\nYet, as she berated herself, there was a twinkle in her eye.\n\nIRVINE WAITED for Bell in the elevator. \"What was all that about?\"\n\n\"I have a dinner date with Cromwell's personal secretary.\"\n\n\"You work fast,\" Irvine said admiringly.\n\nBell grinned. \"Things just sort of fell into place.\"\n\n\"Knowing you like I do, I'll bet you have an ulterior motive.\"\n\n\"You might say that I'm mixing business with pleasure.\"\n\n\"You may be playing with fire,\" said Irvine seriously. \"If she catches wise that you're using her to probe into Cromwell's affairs, there could be trouble.\"\n\n\"I'll worry about that when the time comes,\" Bell said comfortably.\n\nOn the ride back to the hotel, Bell's thoughts were not on the business part of the coming evening but rather the pleasure.\n\n## 20\n\nMARION COULD NOT EXPLAIN IT. THE SENSATION was one she had not experienced since a boy she dreamed about in school had smiled at her. That was all. He never approached or talked to her. Now, as she sat across an intimate table for two, she felt as giddy as a schoolgirl.\n\nBell had picked her up outside the Cromwell Bank at exactly five o'clock in a motor cab. The driver drove directly from the street into the seven-story building that contained the city's most famous French restaurant, Delmonico's. They entered an elevator that took them to the top floor, where the ma\u00eetre d' showed them into an enclosed private dining room with a large picture window that overlooked the city and the bay.\n\nPeople who could afford it thought nothing of consuming ten-course meals, each accompanied by a different wine. Bell ordered oysters Rockefeller with a tangy curry sauce, followed by a flavorful broth, poached Great Lakes sturgeon, frog's legs \u00e0 la poulette, pork chops, chicken Kiev, assorted roasted game birds, boiled potatoes, and creamed peas.\n\nMarion had never dined this sumptuously in her life. True, she had been wined and dined by the city's eligible and moneyed bachelors, but none had treated her this lavishly. She was more than thankful the portions were small but regretted not loosening her corset in advance.\n\nFor dessert, Bell ordered cr\u00eapes suzette, the flaming orange-flavored delicacy. When the waiter stood at their table expertly spooning the flaming mixture over the cr\u00eapes, Marion forced herself to look directly into Bell's eyes.\n\n\"May I ask you a question, Mr. Bell?\"\n\nHis smile was engaging. \"I believe we know each other well enough for you to call me Isaac.\"\n\n\"I'd prefer Mr. Bell, if you don't mind,\" she said in what she thought was a proper manner.\n\nThe smile remained. \"Suit yourself.\"\n\n\"How can you afford all this on the pay of a detective?\"\n\nHe laughed. \"Would you believe I saved up all month just to impress you?\"\n\n\"Not for an instant,\" she said haughtily.\n\n\"Is Cromwell the biggest bank in San Francisco?\"\n\nShe was taken back by his question to her answer. \"No, there are two others that are larger, including Wells Fargo. Why do you ask?\"\n\n\"My family owns the largest bank in New England.\"\n\nShe tried to digest it but could not. \"Would you be upset if I said I didn't believe you?\"\n\n\"Ask your boss. He'll verify my claim.\"\n\nShe frowned, confused. \"Why are you a hired detective when you could be president of a bank?\"\n\n\"I happen to like criminal investigation more than banking. I felt trapped at a desk. There is also the challenge of matching wits with the criminal mind.\"\n\n\"Are you successful?\" she asked, teasing.\n\n\"I win more times than I lose,\" he answered honestly.\n\n\"Why me?\" she asked him. \"Why wine and dine a mere secretary instead of a socialite more your equal?\"\n\nBell did not mince words. \"Because you're attractive, intelligent, and I'm captivated.\"\n\n\"But you don't know me.\"\n\n\"I hope to change that,\" he said, devastating her with his eyes again. \"Now, enough talk. Let's enjoy the cr\u00eapes.\"\n\nWhen they finished the savory dessert, Bell asked the waiter for two glasses of fifty-year-old port. Then he leaned back, fully sated. \"Tell me about Jacob Cromwell.\"\n\nThe food and wine had done its work. Marion was too mellow to see the trap she was stepping into. \"What would you like to know?\"\n\n\"Where he came from, how he launched his bank, is he married. After meeting him, I found him most interesting. I heard he and his sister Margaret are the city's leading philanthropists.\"\n\n\"I've worked for Mr. Cromwell for nine years and I can safely say he is a very smart and perceptive man who is a confirmed bachelor. He started the bank in 1892 with very little in assets and weathered the depression of the nineties. He made money through the worst of it. Most all the banks in the city came close to closing their doors during hard economic times. Not Cromwell National Bank. Through shrewd management and sound banking principles, he built a financial empire with assets running in the many millions of dollars.\"\n\n\"A resourceful man,\" said Bell admiringly. \"Obviously, a self-made man.\"\n\nShe nodded. \"The growth of Cromwell National Bank is nothing short of a financial miracle.\"\n\n\"Where did he find the money to open a bank?\"\n\n\"That's a bit of a mystery. He's very close-lipped about his business affairs prior to launching a small bank on Market Street. Rumor has it, he started with no more than fifty thousand dollars. When I came to work, the bank's assets were well over a million.\"\n\n\"What sort of investments does he make with his fortune?\"\n\nShe held up her hands in a helpless gesture. \"I honestly don't know. He's never mentioned his personal finances to me, and I've seen no paperwork or correspondence. I assume he plunges his profits back into the bank.\"\n\n\"What of his family? Where did he and his sister come from?\"\n\nAgain, Marion looked lost. \"He's never spoken of his past. One time, he mentioned that he and Margaret's father had a farm in North Dakota, in a little town called Buffalo. Other than that, his family ties are buried in the past.\"\n\n\"I'm sure he has his reasons,\" said Bell. He did not want to push Marion too far, so he turned the conversation to his own childhood growing up in the elitist society of Boston. Going to Yale University, and his father's extreme displeasure when he went to work at the Van Dorn Detective Agency and not the family bank. He took a circuitous route back to Cromwell. \"Cromwell stuck me as an educated man. I wonder where he went to school.\"\n\n\"Margaret once said they attended college in Minnesota,\" said Marion, dabbing a napkin to her lips after finishing her cr\u00eapes.\n\n\"Margaret is a beautiful woman,\" he said, watching for a reaction.\n\nMarion barely veiled her dislike of Cromwell's sister. \"I know she's involved with a number of charities, but she is not someone I'd have as a close friend.\"\n\n\"She can't be trusted?\" Bell guessed.\n\n\"She doesn't always tell the truth. And there are always rumors of scandal, which Mr. Cromwell manages to cover up. Strangely, he doesn't seem disturbed by her antics. It's almost as if he enjoys them.\"\n\n\"Does he travel much?\"\n\n\"Oh, yes, he's often away fishing in Oregon, enjoying the Bohemian Club's retreat in the redwoods, or hunting in Alaska. He also attends at least three banking conferences a year in various parts of the country. Once a year, he and Margaret tour Europe together.\"\n\n\"So he doesn't manage the day-to-day business of the bank.\"\n\nShe shook her head. \"No, no, Mr. Cromwell is always in weekly contact with the bank when he's away. He also has a board of directors that has the best brains in the business.\"\n\nThe waiter brought their glasses of port on a silver tray. They sipped in silence for a few moments before Marion spoke.\n\n\"Why are you asking me all these questions about Mr. Cromwell?\"\n\n\"I'm an investigator. I'm just naturally curious.\"\n\nShe pushed a curl from her forehead and patted her hair. \"I feel rejected.\"\n\nHe gazed at her carefully. \"Rejected?\" he echoed.\n\n\"Yes, you ask all these questions about my boss, but you haven't asked about me. Most men I've known always asked about my past on the first date.\"\n\n\"Dare I go there?\" he asked, teasing her.\n\n\"Nothing risqu\u00e9,\" she said, laughing. \"My life's been pretty dull, actually. I am a California native, born across the bay in Sausalito. My mother died when I was quite young, and my father, who was an engineer for the Western Pacific Railroad, hired tutors for me until I was old enough to go to the city's first secretarial school. When I graduated, Jacob Cromwell hired me, and I've worked in his bank ever since, working up from an office typist to his personal secretary.\"\n\n\"Ever been married?\"\n\nShe smiled coyly. \"I've had a proposal or two but never walked down the aisle to the altar.\"\n\nHe reached across the table and took her hand. \"Hopefully, Prince Charming will come along one day and sweep you off your feet.\"\n\nShe pulled her hand back, more from exerting her authority than rejecting him. \"Prince Charmings are few and far between. I've yet to see one in San Francisco.\"\n\nBell decided not to go there. He was determined to ask her out again and see where their wave of mutual attraction might take them. \"I've enjoyed the evening. It's not often I can value the company of such a lovely woman who can hold her own in conversation.\"\n\n\"You're very good at flattery.\"\n\nHe dropped his eyes from hers. Bell did not want to push his luck, but there was one more enigma he had to have answered. \"There's another thing about Cromwell that intrigues me.\"\n\nHe could see from her expression that she was disappointed and had expected him to say something about them getting together again, and he sensed that she was beginning to doubt her feelings toward him.\n\n\"What is it?\" Her tone suddenly went icy.\n\n\"When I first saw him in the dining room of the Bohemian Club, and today in his office, he was wearing gloves. Does he always wear them when dining or working at his desk?\"\n\nShe folded her napkin and laid it on the table as a sign that for her the evening was over. \"When he was a boy, he was in a fire. Both his hands were badly burned, so he wears gloves to cover the scars.\"\n\nBell felt guilty for using Marion. She was a vital, beautiful, and intelligent woman. He stood, came around the table, and pulled her chair out for her. \"I'm truly sorry for letting my detective's undue inquisitive nature get the best of me. I hope you'll forgive me. Will you give me a chance to make it up to you?\"\n\nShe could tell that he was sincere and felt a tickle of excitement, her hope rising again that he was truly interested in her. He was far more enticing than she could have imagined. \"All right, Isaac, I'll go out with you again. But no questions.\"\n\n\"No questions,\" he said with a tingle of pleasure at hearing her use his first name. \"That's a promise.\"\n\n## 21\n\nTWO DAYS LATER, THE FOUR DETECTIVES MET IN THE Van Dorn Detective Agency offices on the fifth floor of the Call Building on Market Street. They sat in a semicircle at a round table and compared notes. They were all in shirtsleeves, their coats hanging on the back of their chairs. Most wore straight, conservative neckties under their stiff collars. Only one wore a bow tie. Three sipped coffee from cups with the Van Dorn logo baked on the porcelain surface, the fourth drank tea. Loose papers and bound reports covered the top of the table. \"I've written up a story telling how one of the largest shipments ever of newly printed currency from the San Francisco Mint will be shipped under heavy guard to the mining town of Telluride, Colorado, to make the payroll and a bonus to ten thousand miners,\" Bell told them. \"I merely alluded to the exact amount but suggested that it was in the neighborhood of five hundred thousand dollars.\"\n\n\"I used my contacts with the newspaper editor to run the article,\" said Bronson. \"It will be printed in tomorrow's papers.\"\n\nIrvine spun his cup slowly around on its saucer. \"If the bandit lives in San Francisco, it should tantalize him into making a try for it.\"\n\n\"If he lives in San Francisco,\" repeated Curtis. \"We're going out on a limb on this one. We may have run up a dead-end alley.\"\n\n\"We know the boxcar and several of the stolen bills ended up here,\" said Bell. \"I think the odds are good he lives somewhere in the Bay Area.\"\n\n\"It would help if we knew for certain,\" Bronson said wearily. He looked at Irvine. \"You say your search to backtrack the stolen currency went nowhere.\"\n\n\"A bust,\" Irvine acknowledged. \"The trail was too cold and there was no way to trace the bills before they were recirculated.\"\n\n\"The banks had no record of who turned them in?\" asked Bronson.\n\nIrvine shook his head. \"The tellers have no way of knowing because they don't list the serial numbers. That's done later by the bank's bookkeepers. By the time we made a connection, it was too late. Whoever traded in the bills was long gone and forgotten.\"\n\nBronson turned to Curtis. \"And your search for the boxcar?\"\n\nCurtis looked as if he had just lost the family dog. \"It disappeared,\" he replied helplessly. \"A search of the railyard turned up no sign of it.\"\n\n\"Maybe it was sent out on a freight train that left the city,\" Bell offered.\n\n\"Southern Pacific freight trains that left on scheduled runs in the last week show no manifest that includes a freight car owned by the O'Brian Furniture Company.\"\n\n\"You're saying it never left the railyard?\"\n\n\"Exactly.\"\n\n\"Then why can't it be found?\" inquired Bronson. \"It couldn't have vanished into thin air.\"\n\nCurtis threw up his hands. \"What can I say? Two of your agents and I searched the railyard from top to bottom. The car is not there.\"\n\n\"Did the Southern Pacific's dispatchers know where the car was switched after it arrived?\" asked Bell.\n\n\"It was switched to a siding next to the loading dock of a deserted warehouse. We checked it out. It wasn't there.\"\n\nIrvine lit a cigar and puffed out a cloud of smoke. \"Could it have been coupled to a train without the dispatcher knowing about it?\"\n\n\"Can't happen,\" Curtis came back. \"They would know if a car was covertly added to their train. The brakemen use a form to list the serial numbers on a train in the sequence the cars are coupled together. When the boxcars arrive at their designated destination, they can easily be switched from the rear of the train before it continues on its run.\"\n\n\"Perhaps the bandit figured the car had outlived its usefulness and he had it scrapped and destroyed,\" said Bronson.\n\n\"I don't think so,\" Bell said thoughtfully. \"My guess is that he simply had it repainted with a new serial number and changed the name to another fictitious company.\"\n\n\"Won't make any difference,\" said Curtis. \"He couldn't use it anyway.\"\n\n\"What do you mean?\" Bell asked.\n\n\"Only the Rio Grande Southern Railroad runs into Telluride.\"\n\n\"So what's stopping him from repainting that railroad's insignia over one advertising the Southern Pacific?\"\n\n\"Nothing. Except it would be a waste of time. The Rio Grande Southern runs on a narrow-gauge track. The Southern Pacific trains run on standard gauge, nearly a foot wider. There's no way the track can accommodate the bandit's boxcar.\"\n\n\"How stupid of me,\" muttered Bell. \"I forgot that only narrow-gauge railroads run through the Rocky Mountains.\"\n\n\"Don't feel bad,\" said Bronson. \"I never thought of it either.\"\n\nIrvine struck the table with his fist in frustration. \"He'll never bite the hook, knowing that he can't escape in his private freight car.\"\n\nBell smiled tightly. \"He has his strengths, but he also has his weaknesses. I'm counting on his greed and his ego, his sense of invincibility. I'm certain he will take the bait and attempt to rob the bank in Telluride. The challenge is too mighty for him to ignore.\"\n\n\"I wish you the best of luck,\" said Bronson. \"If anybody can catch the Butcher, you can.\"\n\n\"What about you, Horace? Any luck on tracing the bandit's gun?\"\n\n\"Nothing encouraging,\" Bronson said soberly. \"New firearm purchases don't have to be registered. All any buyer has to do is lay down the money and walk out with the gun. We've drawn a blank with dealers. Even if they remember who they sold a Colt thirty-eight automatic to, they won't give out any names.\"\n\nIrvine stared at a wall without seeing it. \"It would seem, gentlemen, that all our hard-earned leads have turned into blind alleys.\"\n\n\"Setbacks, yes,\" Bell muttered softly. \"But the game isn't over\u2014not yet. We still have a chance to make the final score.\"\n\n## 22\n\nCROMWELL SAT AT THE TABLE, EATING HIS BREAKFAST and reading the morning paper. He folded the first section on a front-page article and passed it across the table to Margaret without comment.\n\nShe read it, her eyes squinting as the story hit home, then she looked up quizzically. \"Do you intend on going for it?\"\n\n\"I find it very tempting,\" he replied. \"It's as though a gauntlet was thrown at my feet.\"\n\n\"What do you know about Telluride?\"\n\n\"Only what I've read. It lies in a box canyon. Has an extensive red-light district, and Butch Cassidy robbed the San Miguel Valley Bank there in 1889.\"\n\n\"Was he successful?\"\n\nCromwell nodded. \"He and his gang got away with over twenty thousand dollars.\"\n\n\"I suppose you're thinking if he could do it, you could do it.\"\n\n\"Cassidy conducted an amateur holdup and rode away on horses,\" Cromwell said pompously. \"My methods are more scientific.\"\n\n\"If Telluride is in a box canyon, there is only one way in and one way out. A posse would have time to stop a train and search the cars.\"\n\n\"I can't use my boxcar anyway. It will have to be left behind.\"\n\n\"I don't understand.\"\n\n\"The railroad running in and from Telluride is the Rio Grande Southern. The tracks are narrow gauge, the rails too closely spiked for my Southern Pacific car. I'll just have to find another means of leaving town without the threat of capture.\"\n\nMargaret scrutinized the story again. \"I don't have a good feeling about this.\"\n\n\"I don't consider feelings. I work with hard facts, and I play it safe by taking into account every contingency, no matter how small.\"\n\nShe watched him across the table as he poured another cup of coffee. \"You'll need help with this job.\"\n\nHe looked over his cup. \"What have you got in mind?\"\n\n\"I'll come with you.\"\n\n\"What about your little journey to Juneau, Alaska?\"\n\n\"I'll simply postpone it.\"\n\nCromwell considered that for a few moments. \"I can't put you at risk.\"\n\n\"You haven't failed yet,\" Margaret admonished him. \"But, this time, you may need me.\"\n\nHe was quiet for a while. Then he smiled. \"I do believe you'd come along if I ordered you not to.\"\n\nShe laughed. \"Have I ever bowed to your demands yet?\"\n\n\"Not even when we were children,\" he said, remembering. \"Though you were two years younger, I could never get the upper hand.\"\n\nShe patted a napkin against her red lips. \"It's settled, then. We're in this job together.\"\n\nHe sighed. \"You win. But I hope I won't be sorry I didn't put you on the boat to Alaska.\"\n\n\"What do you want me to do?\"\n\nHe stared down at the table, as if seeing an abstract image, while he circled his fork on the tablecloth. \"Take a train to Colorado tomorrow and then make a connection to Telluride.\"\n\nShe stared at him. \"You want me to leave before you?\"\n\nHe nodded. \"I'll deviate from my usual routine. Instead of my spending time mingling with the locals and studying the bank operation, you can do it. As a woman, you can conduct a close scrutiny without arousing suspicion.\"\n\n\"A woman in Telluride?\" she mused. \"I'll have to pass myself off as a prostitute.\"\n\n\"Better yet, claim that you're an abandoned wife whose husband left her to strike it rich in the mines and disappeared. That way no one will be suspicious of you asking questions and snooping around.\"\n\n\"But in order to live and eat, I have to find work in a bordello.\"\n\n\"Have it your own way,\" he said, resigning himself as always to his sister's whimsical ways.\n\n\"And you?\"\n\n\"I'll come a few days later, after I've checked out the shipment and firmed up my plans for the robbery and our getaway.\" He paused and gazed at her with a look of brotherly love. \"I must be mad for involving you in such a risky venture.\"\n\n\"I'm mad, too.\" She laughed a lilting laugh. \"Insane with excitement and a growing rage for adventure.\" She threw him the feminine expression of a cat about to leap on a mouse. \"Of course, the thought of acting like a prostitute is an attraction I find delicious.\"\n\n\"Spare me the details.\"\n\nThen she suddenly became serious. \"What about Isaac Bell?\"\n\nHe shrugged. \"What about him?\"\n\n\"He seems to show up everywhere, maybe even Telluride.\"\n\n\"The thought crossed my mind, but, once I verify the currency shipment, I believe that pretty much eliminates him. He's too busy chasing ghosts all over San Francisco to show up out of the blue in Telluride.\"\n\n\"I don't trust him any more than I can throw this house.\"\n\nHe laughed. \"Cheer up, sister dear. This will be a walk in the park just like the other robberies. You'll see.\"\n\n## 23\n\nTHE SPRING DAY WAS COOL AND BRISK AS BELL DEPARTED the train at the town depot and walked to the corner of Aspen Street and Colorado Avenue, where he found a three-story wooden building with a sign out front that read MAMIE TUBBS BOARDINGHOUSE. He carried a battered valise and wore a worn wool coat with a vest and flannel shirt underneath. His pants were heavy cotton with almost the consistency of canvas. Boots that looked as if they had walked five thousand miles protected his feet while a rumpled old Stetson sat solidly on his head. The fabricated image was embellished with a Dublin-style bent pipe wedged between his teeth. Bell also walked with a pronounced limp as if his left leg was stiff.\n\nHe stepped into the parlor of the boardinghouse and was greeted by Mamie Tubbs, a jolly woman as round as a huge pear. Her gray hair hung down her back in two braids, and she had a face that looked like a large saucer with a nose.\n\n\"Greetings, stranger,\" she said in a voice as deep as a man's. \"Looking for a place to stay?\"\n\n\"Yes, ma'am,\" said Bell politely. \"I'm new in town.\"\n\n\"Seven dollars a week including meals, providing you're at the table when I dish up.\"\n\nHe reached into a pocket, brought out a few folded bills, and counted out seven dollars. \"Here's your money, in advance. \"I don't have much, but enough to tide me over for a while.\"\n\nShe had noticed his limp when he came into the parlor. \"You lookin' to work in the mines?\"\n\nBell tapped a hand against his leg. \"My mining days ended when I was injured by a badly laid stick of dynamite.\"\n\nShe eyed him suspiciously, beginning to wonder where any future rent was going to come from. \"Where do you expect to find a job?\"\n\n\"A friend found me work as a cleanup man at the New Sheridan Hotel.\"\n\nShe smiled. \"They couldn't find you a room in the basement?\"\n\n\"All beds in the basement were taken by miners,\" Bell lied. He had no idea if miners slept in the basement.\n\nThe impression of a crippled miner, he knew, would satisfy Mamie Tubbs enough so she wouldn't gossip around town about her new boarder. She showed him to his room, where he unpacked his valise. He removed a towel wrapped around a Colt Browning model 1905 .45 caliber automatic pistol with a custom twenty-shot magazine and shoulder stock that fit in a slot behind the grip. He slipped the weapon under the bed but kept his trusty Remington derringer inside his Stetson. He retightened the wrapping around his knee so it hindered normal movement.\n\nAfter a beef stew dinner in Mamie's dining room, he met the other people who were staying at the boardinghouse. Most were miners, but there were a few store clerks, and a husband and wife who were opening a restaurant. After dinner, Bell strolled up Pacific Avenue and studied the layout of the town.\n\nTelluride\u2014the name supposedly came from the saying \"to hell you ride\"\u2014was launched after gold was discovered in the San Miguel River. The gold, along with silver-bearing ore, found high in the San Juan Mountains, quickly attracted an army of prospectors and miners over the next fifty years. By 1906, more millionaires per capita lived in Telluride than in New York City.\n\nThe miners eventually dug three hundred fifty miles of tunnels that honeycombed the surrounding mountains, some as high as twelve thousand feet above sea level. The population soared to over five thousand, and the rip-roaring town soon overflowed with wild and crazy living mixed with a healthy dose of corruption. There were three dozen saloons and one hundred eighty prostitutes to keep the army of miners in a good mood after long twelve-hour shifts in the Silver Bell, Smuggler-Union, and Liberty Bell mines at three dollars a day.\n\nWhen the sun dropped behind the mountains and darkness came, a blaze of lights flashed on up and down the streets. In 1892, mine owner L. L. Nunn had hired the electrical wizard Nikola Tesla to build the world's first alternating-current power plant to move ore on cables down the mountain and miners up from town. After running lines from the power plant into town, Telluride became the first town in history to have electric streetlamps.\n\nBell walked past the notorious cribs where the scarlet women plied their trade. The upper-class houses were called the Senate and the Silver Belle. Music could be heard through the windows out on the street as a piano player pounded out the \"Dill Pickles Rag\" and other ragtime tunes. The street was called Popcorn Alley, its name coming from the constant opening and closing of doors all night.\n\nHe moved up to the main section of town on Colorado Avenue and looked through the windows of the Telluride First National Bank. Tomorrow, he would meet with the town sheriff and the bank manager to plan a reception for the Butcher Bandit, should he swallow the bait and make an attempt to rob the bank. He passed the old San Miguel Valley Bank that Butch Cassidy had robbed seventeen years previously.\n\nThe evening air had turned cold, once the sun took its heat beyond the mountain peaks. Bell noticed that the nine-thousand-foot altitude caused him to take deeper breaths. He ignored the main street saloons and headed for the New Sheridan Hotel.\n\nBell stepped inside the lobby and asked the desk clerk to see the manager. In a minute, a short man with a florid face and bald head came out of the office with quick, hurried steps, like a mouse running from a hole in the wall. He smiled an official smile, but not too warmly, as he sized up Bell's rather dowdy appearance.\n\n\"I'm sorry, all our rooms are taken. The Sheridan is full up.\"\n\n\"I don't want a room,\" said Bell. \"Are you Mr. Marshall Buckman?\"\n\nThe smile straightened and the eyes narrowed. \"Yes, I'm Buckman.\"\n\n\"I'm Isaac Bell with the Van Dorn Detective Agency.\"\n\nBuckman eye's widened again and he bowed. \"Mr. Bell. I received your telegram. Permit me to say the Sheridan will cooperate in every way.\"\n\n\"The most important thing,\" explained Bell, \"is to confirm to anyone who asks that I work here as a janitor.\"\n\n\"Yes, of course,\" Buckman said in a patronizing manner. \"You can count on me.\"\n\n\"Thank you, Mr. Buckman. Now, if you don't mind, I think I will enjoy the best whiskey in your bar.\"\n\n\"We serve only superior whiskey from the finest distillers. No local rotgut is tolerated at the Sheridan.\"\n\nBell nodded and then turned his back on Buckman and stepped toward the bar. He paused and read a plaque listing rules for the hotel patrons.\n\nDon't shoot the pianist, he's doing his damndest.\n\nNo horses above the first floor.\n\nNo more than 5 in a bed.\n\nFunerals on the house.\n\nBeds 50 cents, with sheets 75 cents.\n\nAt the doorway, he stood aside as a blond lady whose face was hidden under a wide-brimmed hat stepped past him. All he saw was that she had a fine figure.\n\nConversely, she paid no attention to the limping man who walked by her as she headed for the carpeted stairway leading to her room.\n\nMuch later, Bell cursed himself for not recognizing the blonde just as Margaret blamed herself for not identifying the limping man until it was too late.\n\n## 24\n\nBELL EXPLAINED THE SITUATION TO SHERIFF HENRY Pardee and the bank manager, Murray Oxnard. The three men sat around a table eating a breakfast served by the sheriff's wife. Pardee's house sat directly behind his office and the jail. He walked to the door, made sure it was locked, and drew the curtains so nobody could see inside.\n\nBell was impressed with the sheriff. One wall of the parlor had bookshelves from floor to ceiling stacked with works by Shakespeare, Plato, Voltaire, Bacon, and Emerson, along with several volumes in Latin. Bell had never met a peace officer in a small town who was so well read.\n\nPardee ran a hand through a thick mane of graying hair and tugged at a shaggy mustache. \"What you're saying, Mr. Bell, is that you think the Butcher Bandit is going to hit our town bank.\"\n\n\"I can't say for certain,\" Bell replied. \"But if he's true to form, he'll be lured by the large payroll that is being shipped to the bank from the First National Bank of Denver.\"\n\n\"I know of no such payroll shipment,\" said Murray Oxnard. He was a tall, quiet man with broad shoulders and narrow hips. He seldom smiled, and his face was always fixed with a dour expression.\n\n\"There is no shipment,\" explained Bell. \"It is a ruse to smoke out the bandit.\"\n\nPardee rapped the fingers of one hand on the table. \"If he is as smart as I've read, he'll dig into the true facts and find it's all a sham.\"\n\nBell shook his head. \"No, sir, the directors of the bank in Denver are primed to go along with the story.\"\n\n\"If I may ask,\" said Pardee, \"why did you choose Telluride?\"\n\n\"Because you sit in a box canyon with its only entrance and exit to the west. The situation that makes it ideal to block off his escape route if we don't apprehend him during the attempted robbery.\"\n\n\"I don't like it,\" said Oxnard. \"The bandit is known to murder without batting an eye. I cannot put my employees at risk, nor will I have their blood on my hands.\"\n\n\"I do not intend to have you or your people in the bank when and if the robbery occurs. Myself and one other Van Dorn agent will man the bank. Another agent will watch the trains coming in and out, since the bandit is known to escape his crimes by using a railroad freight car.\"\n\n\"What about my customers?\" Oxnard pressed on. \"Who will tend to their transactions?\"\n\n\"My agent and I are fully experienced in running the daily affairs of a bank. If the bandit steps up to a teller's cage, we'll be ready for him.\"\n\n\"Do you know what he looks like?\" asked Pardee.\n\n\"Except for the fact we know he's missing the little finger on his left hand and he has red hair, we have no description.\"\n\n\"That's because he murders everyone who can identify him. You don't have much to go on.\"\n\n\"I still cannot bring myself to go along with this,\" said Oxnard. \"One of my customers could be in the wrong place at the wrong time and get shot.\"\n\n\"We'll take every precaution,\" said Bell soberly. \"There may be some risk, but this bandit must be stopped. He's already killed over thirty people. There's no telling how many more will die before we can apprehend him and stop the murders.\"\n\n\"What can I do to help?\" Pardee said, giving Oxnard a cold stare.\n\n\"Don't patrol the bank with your deputies and scare off the bandit,\" answered Bell. \"Stand by\u2014out of sight, if possible\u2014but be ready to act in case he shows up. We'll arrange a signal when he makes his play.\"\n\nThough Oxnard had his demons about the trap, Pardee was already imagining the notoriety he would receive if the bandit were caught in the act under his jurisdiction. As far as he was concerned, the debate was decisive and now it was over. He had only one more question.\n\n\"When is the supposed money shipment due?\"\n\n\"Tomorrow,\" Bell told him.\n\nOxnard looked at him inquiringly. \"What about the shipment that's already sitting in the safe for the real payroll?\"\n\n\"Leave it in the safe. I guarantee, the bandit won't get it.\"\n\nPardee twisted the ends of his mustache. \"Ever been in a mining town on payday, Mr. Bell?\"\n\n\"I haven't had the luxury, but I hear it can get pretty wild.\"\n\n\"That's true,\" said Oxnard with a faint grin. \"Every payday, all hell breaks loose from one end of town to the other.\"\n\nPardee matched his grin. \"Yes, the cribs will be busy until the miners have wasted their hard-earned money on whiskey and gambling.\" He paused a moment and looked at Bell. \"Where are you staying, in case I have to get in touch with you?\"\n\n\"I'm staying at Mamie Tubbs Boardinghouse.\"\n\n\"A good place to keep a low profile,\" said Oxnard. \"Mamie's a fine old gal, and a good cook.\"\n\n\"I can vouch for her stew,\" Bell said with humor.\n\nAfter breakfast, the meeting broke up. Bell and Oxnard thanked Mrs. Pardee for a fine breakfast. Then the three men stepped outside and walked toward town, Pardee leaving them when he got to his office and jail. Bell went with Oxnard to the bank to study its interior layout.\n\nThe floor plan was the same as a thousand other banks'. The bank manager's office sat behind the teller's cage, which was enclosed in glass except for the area in front of the cash drawers. This section of the counter was open through narrow bars. The vault was more like a large safe and stood in an alcove off to the side of the lobby. Bell learned that it was closed during business hours and opened only to withdraw currency or when all cash and coins were returned after closing.\n\n\"You don't have a vault?\" Bell said to Oxnard.\n\n\"Don't need one. Payroll money usually goes up to the mines under heavy guard the second day after the shipment comes in.\"\n\n\"Why the second day?\"\n\n\"We need the time to make a count to verify the amount shipped from the bank in Denver.\"\n\n\"So the bandit has a limited window of opportunity.\"\n\nOxnard nodded. \"If he's going to make his play, it will have to be tomorrow.\"\n\n\"Have you seen or had contact with any new depositors or people who simply walked into the bank and then walked out again?\"\n\n\"A new superintendent for the Liberty Bell mine opened a checking account.\" He paused to gaze up at the ceiling in thought. \"Then there was a very attractive woman who opened an account. A very small account. Very sad.\"\n\n\"Sad?\"\n\n\"Her husband left her back in Iowa to strike it rich in Colorado. She never heard from him again, and the last thing she learned was from a friend, a conductor on the railroad. He told her that her husband left word he was going to Telluride to work in the mines. She came here in an attempt to find him. Poor soul. Chances are, he was one of the many men who died in the mines.\"\n\n\"I'd like the name of the mine superintendent,\" said Bell, \"so I can check him out.\"\n\n\"I'll get it for you.\" Oxnard went into his office and returned in less than a minute. \"His name is Oscar Reynolds.\"\n\n\"Thank you.\"\n\nOxnard stared at Bell. \"Aren't you going to check out the woman?\"\n\n\"The bandit has never worked with a woman\u2014or any man, for that matter. He always commits his crimes alone.\"\n\n\"Just as well,\" Oxnard sighed. \"Poor thing. She only opened an account for two dollars. In order to eat, she'll probably have to work in a bordello, since jobs for women are scarce in Telluride. And those jobs that do exist are filled by the wives of the miners.\"\n\n\"Just to play safe, I'd like her name, too.\"\n\n\"Rachel Jordan.\"\n\nBell laughed softly. \"Her, you remembered.\"\n\nOxnard smiled. \"It's easy to remember a name with a pretty face.\"\n\n\"Did she say where she was staying?\"\n\n\"No, but I can only assume it's in a crib.\" He gave Bell a sly look. \"You going to look her up?\"\n\n\"No,\" said Bell thoughtfully. \"I hardly think a woman is the Butcher Bandit.\"\n\n## 25\n\nMARGARET WAS NOT ENDURING THE LIFE OF A PROSTITUTE in a crib on Pacific Avenue. She was living in style in the New Sheridan Hotel. After opening a small account at the town bank to examine the floor plan, number of employees and where they were located, and the type of safe, she made the rounds of the mining companies to make inquiries about a long-lost husband who never existed. The effort gave her story substance, and soon she became the source of gossip around town.\n\nShe went so far as to call on Sheriff Pardee with her bogus story, to see what kind of man he was face-to-face. Mrs. Alice Pardee came into the office when Margaret was asking the sheriff for his cooperation in finding her husband. Alice immediately felt sorry for the woman in the cheap, well-faded cotton dress who poured out her sad tale of the abandoned wife desperately seeking the man who had deserted her. Alice assumed that this Rachel Jordan was half starved and invited her up to their house for dinner. Margaret accepted and arrived in the same cheap dress, which she had bought in San Francisco at a used-clothing store for the poor.\n\nThat evening, Margaret made a display of helping Alice Pardee in the kitchen, but it was obvious to the sheriff's wife that their guest was not at home over a hot stove. Alice served a homemade meal of mutton chops, boiled potatoes, and steamed vegetables, topped off by an apple pie for dessert. After dinner, tea was served and everyone settled in the parlor, where Alice played tunes on an old upright piano.\n\n\"Tell me, Mrs. Jordan,\" Alice asked, pausing to change the sheet music, \"where are you staying?\"\n\n\"A nice lady, Miss Billy Maguire, hired me as a waitress at her ladies' boardinghouse.\"\n\nPardee and his wife exchanged pained glances. Alice sucked in her breath. \"Big Billy is the madam of the Silver Belle bordello,\" she said. \"Didn't you know that?\"\n\nMargaret made a display of looking sheepish. \"I had no idea.\"\n\nAlice bought Margaret's lie, Pardee did not. He knew there was no way any woman could fail to recognize the difference between a boardinghouse and a bordello. The germ of suspicion began to grow in his mind, but his wife was swept by compassion.\n\n\"You poor thing,\" she said, putting her arm around Margaret. \"You'll not stay at the Silver Belle another minute. You'll stay here with Henry and me until you find your husband.\"\n\n\"But he may not be in Telluride,\" Margaret said as if about to weep. \"Then I would have to move on, and I don't want to inconvenience you.\"\n\n\"Nonsense,\" said Alice. \"You march right back to Big Billy's and bring back your things. I'll make up the spare bed for you.\"\n\nMargaret went into her act and shed a few tears. \"How can I ever thank you? How can I ever repay you?\"\n\n\"Don't give it a thought. Henry and I are only too glad to help a poor soul in distress. It's the Christian thing to do.\"\n\nAs she sipped her coffee, Margaret moved the conversation to Pardee's job as sheriff. \"You have to live an exciting life,\" she said. \"Telluride seems like an uninhibited town. You must be kept quite busy.\"\n\n\"The miners can get pretty rowdy at times,\" Pardee agreed, \"but serious crimes like murder don't happen but once every six months or so. It's been peaceful since the union strikes by the miners two years ago, when the governor sent in the army to squelch the rioting.\"\n\nMargaret was slow and deliberate in her answers to Pardee's questions about her missing husband. She in turn made general inquiries about the town and the mines. \"A lot of money must pass through the bank to the mining companies,\" she said casually.\n\nPardee nodded. \"The payrolls can add up to a considerable amount.\"\n\n\"And you never have a fear of robbers and thieves?\" she asked innocently.\n\n\"The miners are a solid lot and rarely indulge in crime. Except for occasional fights in the saloons, or a killing when a confrontation gets out of hand, the town is pretty quiet.\"\n\n\"When I was in the bank, I saw that the safe looked very strong and secure.\"\n\n\"It's strong, all right,\" said Pardee, lighting his pipe. \"Five sticks of dynamite couldn't blow it open.\"\n\n\"And the bank manager is the only one who knows the combination?\"\n\nPardee thought it strange a question like that came from a woman, but he answered without hesitation. \"Actually, the locking bolts are set to spring open at ten o'clock every morning. At three o'clock in the afternoon, the manager closes the door and sets the clock.\"\n\n\"Someone at the Silver Belle told me Butch Cassidy robbed the local bank.\"\n\nPardee laughed. \"That was a long time ago. We've never had a bank robbery since.\"\n\nMargaret was leery of pushing too hard, but there was information she had to know if her brother was to carry out a successful robbery. \"The miners' payroll. Is it taken directly to the mining companies when it arrives?\"\n\nPardee shook his head and went along with Bell's story. \"It came in today and went directly to the bank. Tomorrow, it will be counted and sent to the mines the next day.\"\n\n\"Are there extra guards in the bank to protect the money?\"\n\n\"No need,\" said Pardee. \"Anyone who tried to rob the bank wouldn't get far. With the telegraph lines running alongside the railroad tracks, peace officers around the county would be alerted and posses formed to wait for the robbers when they tried to escape.\"\n\n\"Then such a crime would be impossible to commit successfully.\"\n\n\"I guess you could say that,\" Pardee replied confidently. \"There's no way it could succeed.\"\n\nMargaret left the Pardee house and walked toward the Silver Belle. As soon as she was out of sight, she ran down an alley to the New Sheridan Hotel to pack her meager clothes. She felt pleased with herself and could not believe her luck. Staying with the sheriff and his wife would give her access to most of the town. When her brother arrived, she would have enough information for him to plan a foolproof crime.\n\nHer only problem was the whereabouts of her brother. To her knowledge, he had not arrived in town, and tomorrow was the only day the payrolls could be robbed before they went to the mines for distribution to the miners. She began to feel extremely uneasy.\n\n## 26\n\nTHE FOLLOWING MORNING, A BLACK-HAIRED WOMAN drove a smart-looking buggy pulled by a dappled gray horse on the road into Telluride. The road led from the ranching community of Montrose, a rail terminus for the Rio Grande Southern Railroad. She had arrived from Denver and rented the rig and horse at the local stable. She was dressed in a long buckskin skirt over a pair of pointed-toe leather boots. Her upper torso was covered by a nicely knit green sweater under a wolfskin fur coat. A lady's-style flat-topped cowboy hat was set squarely on her head. She was fashionably attired for the West, but not ostentatious.\n\nShe came onto Colorado Avenue, passed the San Miguel County Courthouse, and pulled the horse to a stop in front of the town stable. She climbed down from the buggy and tied the horse to a hitching post. The stable owner came out and lifted his hat.\n\n\"Good afternoon, ma'am. Can I help you?\"\n\n\"Yes, I wonder if you would please feed and water my horse. I have to be on the road back to Montrose this afternoon.\"\n\n\"Yes, ma'am,\" said the stable owner politely, slightly taken aback by a voice that had a gentle harshness about it. \"I'll take care of it. While I'm at it, I'll tighten your front wheels. They look a mite loose.\"\n\n\"You're very kind, thank you. Oh, and by the way, my sister will come for the buggy and pay you.\"\n\n\"Yes, ma'am.\"\n\nThe woman left the stable and walked a block to the New Sheridan Hotel. She approached the desk clerk and asked, \"Do you have a Miss Rachel Jordan registered here?\"\n\nThe clerk shook his head, stared at what he saw as an attractive woman, and said, \"No, ma'am, she checked out last night.\" He paused, turned, and pulled an envelope out of a mail-and-key slot. \"But she said if someone asked for her to give them this.\"\n\nThe woman thanked the clerk, walked out onto the sidewalk, opened the envelope, and read the note. She stuffed it in her purse and began walking through town. After a short hike, she came to the Lone Tree Cemetery, on a hill north of the San Miguel River. She passed through the gate and walked among the tombstones, noting that most of the deceased had died from mine accidents, snow slides, and miner's consumption.\n\nA pretty blond woman was sitting on a bench beside a grave site, leaning back and sunning herself. Out of the corner of one eye, she caught the approach of another woman. She sat up and stared at the intruder, who stopped and looked down at her. Margaret began to laugh.\n\n\"My God, Jacob,\" she finally gasped. \"That's the most ingenious disguise you've ever created.\"\n\nCromwell smiled. \"I thought you'd approve.\"\n\n\"A good thing you're short, thin, and wiry.\"\n\n\"I don't know why I never thought of it before.\" He awkwardly bunched up his buckskin skirt and sat down on the bench next to Margaret. \"Tell me, sister dear, what have you learned since you've been here?\"\n\nMargaret told him how she became friendly with the sheriff and his wife. She handed him a sketch she'd made of the Telluride First National Bank's interior and a description of the employees. Her report included the arrival of the payroll shipment from the bank in Denver and the counting today before it was sent to the mines tomorrow.\n\nCromwell looked at his watch. \"We have one more hour before the bank closes. The best time to remove the currency and leave town.\"\n\n\"I spotted a man hanging around the railroad depot. I couldn't tell for sure, but I suspect he might have been a Van Dorn agent who was on the lookout for you.\"\n\nCromwell looked thoughtful. \"Even if Van Dorn sends agents to watch train arrivals and departures during payroll shipments, they're only chasing a phantom. No way they could know where I'll strike next.\"\n\n\"If they're wise to your boxcar, it's a good thing you had it repainted.\" She looked at him quizzically. \"Just how do you expect us to make a clean escape after you rob the bank?\"\n\nCromwell grinned wolfishly. \"Who would suspect a pair of clean-cut, attractive ladies riding slowly out of town in a horse and buggy?\"\n\nShe placed her arm around his shoulders. \"The simplest plan is the best plan. You are brilliant, brother. You never cease to amaze me.\"\n\n\"I appreciate the compliment,\" he said, rising to his feet. \"We don't have much time. The payroll awaits.\"\n\n\"What would you like me to do?\"\n\n\"Go to the stable and pick up my horse and buggy. I told the stable owner my sister would come by to get the rig. Then wait at the back door of the bank.\"\n\nWHILE IRVINE watched the train station and town railyard, Bell and Curtis manned the Telluride Bank. Sitting in Murray Oxnard's office, Bell began to think he had bet on the wrong horse. There were only ten minutes left before closing time and no sign of the bandit. Playing the role of a teller, Irvine was getting ready to close out his cashbox in anticipation of waiting on the last customer.\n\nBell glanced down at the .45 Colt automatic he'd kept in an open desk drawer and regretted that he would not get to use it on the Butcher Bandit. Blowing the scum's head off was too good for him, Bell mused. Not after he had murdered so many unsuspecting people. His death would save the taxpayers the expense of a trial. Now Bell was faced with admitting defeat and starting over again with the meager clues he and his agents had ferreted out.\n\nIrvine walked over to the office door and leaned his shoulder against the frame. \"I can't deny it was a good try,\" he said with a tightness in his voice.\n\n\"It looks as if the bandit failed to take the bait,\" Bell said slowly.\n\n\"Perhaps he didn't read the article in the paper because he doesn't live in San Francisco.\"\n\n\"It's beginning to look that way.\"\n\nJust then the door opened and a woman wearing a buckskin skirt walked into the bank, her hat pulled low so it covered her eyes. Bell gazed past Irvine but relaxed at seeing what appeared to be a well-dressed woman. He nodded to Irvine, who walked back to his teller's cage and said, \"How may I help you, ma'am?\"\n\nCromwell lifted his head slightly so he could look into Irvine's face. Then with a pang of alarm he stiffened as he instantly remembered the Van Dorn agent as one of the men who were sitting with Bell and Bronson in the Bohemian Club dining room only days earlier. He did not answer Irvine for fear his voice would give him away to the agent. Cromwell became charged with tension as he realized this was a trap. There came a pause as he lowered his head, his mind racing with alternatives. His advantage was that the agent did not recognize him, not dressed as a woman, and was not alert to the fact that the bandit was less than four feet away on the other side of the counter.\n\nHe could shoot the agent and take what money was in the safe or he could simply turn around and walk out of the bank. He chose the latter option and was about to beat a hasty retreat when Bell stepped from the office. Cromwell immediately recognized Bell. For the first time in his criminal career, he felt the spur of panic.\n\n\"How may I help you, ma'am?\" Irvine repeated, vaguely wondering why the woman did not answer him the first time.\n\nAlready, Bell was looking at him with a questioning expression on his face, as if the female customer looked familiar. Bell was a master of identification and had a photographic memory when it came to faces. His eyes betrayed the fact that he was trying to recall where he'd seen her. Then his eyes dropped to Cromwell's hands, which were covered by leather gloves. Abruptly, as if he had seen an apparition, he realized that he was staring at the bandit. It struck him like a hammer blow to the head. Bell's eyes flared open and he gasped: \"You!\"\n\nCromwell did not waste another second. He reached into his large cloth purse and jerked out his .38 Colt, which had a heavy cloth taped around the muzzle. Without the slightest hesitation, he pointed the Colt at Irvine's chest and pulled the trigger. A loud thump reverberated in the bank's lobby. Then he swung the muzzle around and shot at Bell even before Irvine hit the floor like a rag doll.\n\nIf Bell hadn't instinctively whirled around and thrown his body over the top of the desk, crashing to the floor behind it, the bullet would have caught him square in the stomach. The violent thrust saved him, but the bullet still plowed through the fleshy part of his thigh. He hardly felt the piercing blow. In a single movement, he reached up and snatched his Colt from the desk drawer. Without the luxury of time, he snapped off a shot at Cromwell that missed the neck of the bandit by less than half an inch.\n\nThen, faster than lightning could strike, both men fired again, the shots coming so closely together they sounded as one.\n\nCromwell's second bullet gouged a small trench across the side of Bell's head, barely piercing the skin but creasing the skull. Bell's vision became a blurred mist and he fell into the black pit of unconsciousness. Blood quickly seeped from the wound and covered the side of his head. It had not been a decisive wound, but to Cromwell, who was still standing, it looked as if he had shot off half of Bell's head.\n\nThe bandit did not come out of the gun battle unscathed. Bell's bullet had caught Cromwell in the waist but had passed through without striking any internal organs. He swayed, and only by reaching out and grasping the edge of the teller's cage did he prevent himself from falling to the floor. He stood there for a few moments, fighting the pain. Then he turned and unlocked the rear door, standing aside as Margaret burst in.\n\n\"I heard shots outside,\" she shouted shock. \"What went wrong?\"\n\n\"It was a trap,\" he murmured as anger replaced fear. Holding a hand over his wound, he motioned the muzzle of his Colt toward the office floor. \"I killed Isaac Bell.\"\n\nMargaret stepped into the office and looked down at the bloodied Van Dorn agent and a look of horror came into her eyes as she recognized Bell despite the blood covering much of his face. \"Oh, my God!\" She felt as if she was going to be sick, but the nausea quickly passed when she turned and saw that her brother was also bleeding. \"You're hurt!\" she gasped.\n\n\"Not as bad as it looks,\" he said through clenched teeth.\n\n\"We've got to get out of here. The shots will bring the sheriff and rouse half the town.\"\n\nMargaret half carried, half dragged her wounded brother through the rear door of the bank. Outside, the horse and buggy were waiting. She used all her strength to push him onto the seat of the rig, untied the horse from the fence post, and climbed aboard.\n\nShe raised the whip to urge the horse to a gallop, but he grabbed her wrist. \"No, go slowly, as if we're two women out for a buggy ride. It will look suspicious if we charge out of town.\"\n\n\"The sheriff is a smart man. I know him. He won't fool easily.\"\n\n\"Even a smart man won't suspect a woman of robbing a bank and killing two men,\" muttered Cromwell.\n\nAt the end of the alley, Margaret turned the buggy up a side street and then headed west toward the town limits. Cromwell took off the wolfskin coat and draped it over his lap to cover the blood that soaked his sweater. He slipped the Colt into one of his cowboy boots and sat back, trying to keep his mind clear by ignoring the throbbing pain in his side.\n\nBELL HAD instructed Sheriff Pardee that he would fire a shot as a signal if the bandit made his appearance. But Pardee knew there was trouble when he heard five shots, some of them muffled like the distant dynamite charge in a nearby mine. He rushed into the street from a hardware store where he had been hiding, fearful that the woman he'd seen walk into the bank might have been shot by the bandit.\n\nSeeing him running toward the bank, four of his deputies leaped from their hiding places and rushed after him, while a fifth deputy ran to the railroad depot to alert Curtis. With his single-action Smith & Wesson drawn and the hammer pulled back, Pardee burst through the door of the bank. At first, he didn't see anyone. Irvine was lying out of sight, behind the teller's cage, and Bell was down on the other side of the desk. Then he came around the cage, saw the Van Dorn agent sprawled on the floor in a pool of blood. He checked to make sure Irvine was dead before he entered the office and found Bell.\n\n\"Is he a goner?\" asked one of his deputies, a great bear of a man with a huge stomach bulging over pants with suspenders stretched to their limits, who stood poised with a sawed-off shotgun at the ready.\n\n\"The bullet only creased his skull,\" answered Pardee. \"He's still alive.\"\n\n\"What about the woman?\"\n\nPardee's mind did not register for a moment. Then it hit him. \"The woman who came into the bank before the gunshots?\"\n\n\"That one.\"\n\n\"She must have been abducted by the bandit.\"\n\n\"But we saw no one else enter the bank before or after her.\"\n\nPardee stood up in confusion and disbelief. It took all his imagination to believe a woman was the Butcher Bandit.\n\n\"The bandit must have entered through the back door.\"\n\n\"I don't know, Sheriff,\" said the deputy, scratching his chin. \"The door should have been locked from the inside, like it always is.\"\n\nPardee rushed over to the rear door and found it unlocked. He jerked it open and peered up and down the alley but saw no one. \"Hell's fire,\" he muttered. \"She got away.\"\n\n\"She can't get far,\" said the deputy.\n\n\"Round up the men!\" snapped Pardee. He motioned to another deputy, who was standing at the entrance of the bank. \"Get Doc Madison. Tell him the Van Dorn agent is down with a head wound and to get over to the bank double-quick.\" Pardee knelt down and quickly examined Bell again. \"Also tell him there looks like there's a bullet in the agent's leg.\"\n\nThe deputy was no sooner out the door than Pardee was on his heels, running toward his horse tied to the hitching post in front of his office. It didn't seem possible, he thought, that everything had gone so terribly wrong. Only then did it begin to strike him that the bandit was a man disguised as a woman and that the poor widow he and his wife had taken in was an accomplice.\n\nAS SOON as they left the city limits of Telluride and passed the road leading to the mines of Ophir to the south, Margaret gave the horse the whip and urged it to run through the canyon and down the road heading west toward Montrose. During the ten minutes since they left the bank, Cromwell had time to think. He pointed to a break in the trees that led to a bridge over the San Miguel River. It was an overgrown access road used by the railroad for maintenance crews repairing the track.\n\n\"Get off the road,\" Jacob said to Margaret. \"Go over the bridge and head down the track bed.\"\n\nShe turned and looked at him. \"I thought you said they'd never be suspicious of two women in a buggy?\"\n\n\"That was before it occurred to me that the sheriff and his deputies were watching the bank.\"\n\n\"That goes without saying, but what does it have to do with our escape?\"\n\n\"Don't you see, dear sister? I was the last one to enter the bank and never came out. If what you say is true, Pardee is no fool. He must have put two and two together by now and is looking for both of us. But he'll never think to search for us riding over the track bed. He'll be certain we took the road.\"\n\n\"And if he doesn't find us, what do you think he'll do then?\"\n\n\"He'll backtrack, thinking that we hid out in the trees while he and his posse rode past. By then, we'll be on a train out of Montrose, dressed as two men.\"\n\nAs usual, Cromwell was miles ahead of his pursuers when it came to matching wits. Though he was disheartened that Bell had out-smarted him in laying a well-conceived trap, he gained a certain amount of satisfaction believing he had killed the famous Van Dorn agent.\n\nJust as he had predicted, the sheriff and his posse charged down the road that was out of sight of the railroad tracks in the trees and, not finding any sign of their quarry, had doubled back toward Telluride. It was a bumpy ride over the railroad ties, but it was compensated for by knowing that Pardee had been hoodwinked and would end up empty-handed.\n\n## 27\n\nBELL WAS CARRIED TO THE TELLURIDE HOSPITAL, where he was treated by the town doctor. The first bullet out of Cromwell's Colt had entered and exited his thigh, causing only minor damage to the tissue. The doctor said it would heal within a month. The doctor then stitched the scalp wound, sewing up the crease as neatly as a tailor mending a torn suit pocket.\n\nAfter ignoring the doctor's demands that he remain in the hospital for a few days, Bell limped to the depot to take the next train to Denver. Wearing a hat to cover the bandage around his head, he, along with Curtis, watched with anger and sadness as the coffin containing Irvine was lifted into the baggage car by Sheriff Pardee's deputies. Then he turned and held out his hand to Pardee. \"Sheriff, I can't thank you enough for your cooperation. I'm grateful.\"\n\nPardee shook Bell's hand. \"I'm sorry about your friend,\" he said sincerely. \"Did he have a family?\"\n\n\"Fortunately, no wife or children, but he lived with an aging mother.\"\n\n\"Pour soul. I suppose it's the county poorhouse for her.\"\n\n\"She'll be taken care of in a good nursing home.\"\n\n\"A good nursing home doesn't come cheap. Did Irvine have money?\"\n\n\"No,\" replied Bell, \"but I do.\"\n\nPardee refrained from any more questions. \"If only things had fallen our way.\"\n\n\"Our well-laid plans certainly turned into a fiasco,\" said Bell, seeing the baggage car door close behind the coffin. \"The bandit made me out the fool.\"\n\n\"Not your fault,\" said Pardee. \"He fooled us all, and I was the biggest fool. I'm certain now the destitute widow who my wife and I took in was in cahoots with him. I should have been suspicious when she finagled information out of me about the bank's operations.\"\n\n\"But you didn't tell her there was a trap being set. Cromwell would have never walked into the bank if he suspected a trap.\"\n\nPardee shook his head. \"They bought your story\u2014hook, line, and sinker. If only we had known he was going to wear women's clothing, we might not have thought twice before we shot him down like the dog he is.\"\n\n\"According to reports of his other robberies, he never dressed as a woman.\"\n\n\"Even if the trap turned sour, my posse and I should have apprehended them. Stupidly, I thought they'd stay on the road. It never crossed my mind they would use the railroad track bed as an escape route until it was too late. By the time I figured out how they had outfoxed me, they were long gone.\"\n\n\"Were the train passenger lists checked in Montrose?\"\n\n\"I wired the stationmaster, but they had already left on the train to Grand Junction,\" answered Curtis. \"He didn't remember two women boarding, but he noticed two men. He said that one looked as if he were sick.\"\n\n\"There was blood on the back step of the bank,\" said Pardee with a tight smile. \"You must have plugged him.\"\n\n\"Not seriously enough to stop him,\" Bell muttered quietly.\n\n\"I telegraphed the marshal of the territory. He had deputies in Grand Junction search all the trains going east and west but found no trace of two women traveling together.\"\n\nBell leaned on a cane given to him by Pardee. \"I'm beginning to know how the bandit's mind works. He went back to wearing men's clothes and dressed his sister as a man, too. The marshal, looking for two females, never suspected them.\"\n\n\"A clever man, Cromwell.\"\n\n\"Yes,\" admitted Bell, \"he is that.\"\n\n\"Where do you go from here?\" asked Pardee.\n\n\"Back to Denver and start all over again.\"\n\n\"But now you know the bandit's name and habits.\"\n\n\"Yes, but making a case is impossible. No federal prosecutor would waste time on an indictment with such flimsy evidence.\"\n\n\"You'll nail him,\" Pardee said confidently.\n\n\"We'll work even harder now that we've got a personal reason to see him hung,\" said Bell.\n\nWHEN BELL and Curtis reached Denver late in the evening, a hearse was waiting to take Irvine to the local mortuary.\n\n\"He was my closest friend,\" said Curtis. \"I'll console his mother and take care of the funeral arrangements.\"\n\n\"Thank you,\" Bell said. \"I'll take care of the costs.\"\n\nBell took a taxi to the Brown Palace Hotel. Entering his suite, he removed his clothes and relaxed in a tub of hot water, propping his wounded leg on the edge to keep the bandage from getting wet. He closed his eyes and let his mind wander over the events of the past few days. Bell now knew the woman he'd passed in the New Sheridan Hotel was Margaret Cromwell. When her brother entered the bank from the front door, she was waiting in the rear with a horse and buggy. The picture of Cromwell made up as a woman disgusted him, yet he could not help but respect the shrewd, calculating mind of the Butcher Bandit. Avoiding Sheriff Pardee's posse by driving the rig down the railroad track bed was a stroke of genius.\n\nAt first, Bell thought Cromwell would not tempt fate with another robbery. The possibility seemed extremely remote, but, as he had with all the criminals he'd apprehended, Bell began to make inroads inside Cromwell's mind. He trained himself to think like the bandit. The more Bell thought about it, the more he became convinced that Cromwell believed he was invincible and immune to every investigation by law enforcement officers, especially the agents of the Van Dorn Detective Agency.\n\nThe next step would have to be carefully thought out. His mind was considering alternatives to accumulate enough evidence to arrest Cromwell when he heard a knock on his door. Favoring his good leg and suffering a brief bout of dizziness caused by his head wound as he stood, Bell climbed awkwardly out of the tub, put on a robe, and limped to the door. After pulling it open, he was surprised to see Joseph Van Dorn standing in the hallway.\n\nVan Dorn looked up at the bandage around Bell's head, which had seeped a spot of red, and he grinned tightly. \"You're a sorry sight.\"\n\n\"Come in, sir, and make yourself at home.\"\n\nVan Dorn studied his wounded agent. He was concerned, but he made an effort to look nonchalant. \"Is there much pain?\"\n\n\"Nothing aspirin won't cure.\"\n\nVan Dorn stepped into the suite and looked around. \"I like an agent who travels in style when it's not my money.\"\n\n\"Can I call room service and get you something to eat or drink?\"\n\nVan Dorn waved a hand. \"No, thanks, I ate on the train from Chicago just before it arrived in Denver. A glass of port would hit the spot.\"\n\nBell phoned Van Dorn's request to room service and hung up the phone. \"I did not expect the head man to travel over a thousand miles just to see me.\"\n\n\"A meeting between us is not only appropriate but vital to the investigation.\" Van Dorn sank into an overstuffed chair. \"I prefer a detailed report to a few words on a telegram. Now, tell me what happened in Telluride, and leave out nothing.\"\n\n\"Most of what I can tell you went wrong,\" Bell said sourly.\n\n\"Don't blame yourself,\" Van Dorn consoled him. \"I wish I had a dollar for every plan I conceived that turned bad.\"\n\nA waiter brought a glass of port, and then Bell spent the next forty minutes filling Van Dorn in on the scheme to catch the Butcher Bandit and how Cromwell turned the tables on him and Sheriff Pardee. He told of the murder of Irvine and his own wounding, ending up with him waking up in the Telluride Hospital.\n\nWhen Bell finished, Van Dorn asked, \"You're certain Jacob Cromwell is the Butcher Bandit?\"\n\n\"His disguise was the work of genius, and Irvine and I were caught off guard. But there is no doubt in my mind Cromwell was the person I recognized wearing women's clothing at the bank. Both Pardee and I also identified his sister, Margaret, who was staying in town to help him rob the bank.\"\n\nVan Dorn pulled a cigar case from his vest pocket, retrieved a long, thin corona, and lit it with a wooden match he flamed with his thumbnail. \"It makes no sense. If Cromwell is wealthy, owns a bank with assets in the millions, and lives on Nob Hill in San Francisco, what does he gain by risking it all to pull off a string of robberies and murders?\"\n\n\"From what I've been able to put together, the money he stole was used to build his bank's assets.\"\n\n\"But why now, when he is financially secure and his bank well established? Why continue the crime spree?\"\n\nBell gazed through a window at the blue sky above the city. \"The simple answer is, the man is insane. I've put together a profile of him in my mind. I'm certain he robs and kills because he enjoys it. The money is no longer his intent. It has lost its importance. Like a man addicted to whiskey or opium, he is driven to commit mayhem and murder. He believes himself too untouchable by law enforcement. In his mind, he is invincible and considers every criminal act as a challenge to outwit the law.\"\n\n\"You have to admit,\" said Van Dorn, blowing a large blue smoke ring across the room, \"so far, he's done a pretty good job of making us and every peace officer west of the Mississippi look like a bunch of amateurs.\"\n\n\"Cromwell is not flawless. He's human and humans make mistakes. When the time comes, I intend to be there.\"\n\n\"Where do you go from here?\"\n\nBell grimaced. \"I wish everybody would stop asking me that.\"\n\n\"Well?\"\n\nBell's gaze was focused and calm as he stared at Van Dorn. \"It's back to San Francisco to build a case against Cromwell.\"\n\n\"From what you've told me, that won't be easy. You have little evidence to make a case. A defense attorney would crucify you on the witness stand. He'd laugh at your identification of a man dressed like a woman, claiming it was impossible to tell the difference. And, without another witness or any fingerprints, I'd have to say you're fighting a lost cause.\"\n\nBell fixed Van Dorn with an icy stare. \"Are you suggesting I resign from the investigation?\"\n\nVan Dorn scowled. \"I'm not suggesting anything of the sort. I'm only pointing out the facts. You know perfectly well this is the number one priority case within the agency. We won't rest until Cromwell is behind bars.\"\n\nBell tenderly touched the side of his head, as if to feel if the wound were still there. \"As soon as I sew up a few loose ends here in Denver, I'm returning to San Francisco.\"\n\n\"I can arrange a team of agents to assist you. You have but to ask.\"\n\nBell shook his head. \"No. With Carter as my right-hand man, and backed by Bronson and the agents in his office, I'll have all the manpower I'll need. Better we continue to work undercover without an army of agents to cause complications.\"\n\n\"What about Colonel Danzler and the Criminal Investigation Department in Washington? Can the government be of help in this matter?\"\n\n\"Yes, but only at the opportune moment. Cromwell has an incredible amount of influence with the political and wealthy elite in San Francisco. He is the city's leading philanthropist. If we obtain enough evidence to indict him, his friends will circle the wagons and fight us every step of the way. At that time, we'll need all the help from the federal government we can get.\"\n\n\"What is your plan?\"\n\n\"At the moment, I have no set plan. Cromwell is fat, dumb, and happy, not knowing we're getting closer to him with each passing day.\"\n\n\"But you're no closer now to seizing him than you were three weeks ago.\"\n\n\"Yes, but now I have the advantage.\"\n\nVan Dorn's eyebrows raised in curiosity and he muttered skeptically, \"What advantage is that?\"\n\n\"Cromwell doesn't know I'm still alive.\"\n\n\"It will come as a blow to his ego when he sees you've been resurrected.\"\n\nBell smiled faintly. \"I'm counting on it.\"\n\n## 28\n\nCROMWELL'S WOUND FROM B ELL'S BULLET WAS NOT serious. He held off having it tended by a physician until he returned with Margaret to San Francisco, where the entry-and-exit wound in his side was cleaned with antiseptic, stitched, and bandaged. The doctor, an old friend, asked no questions, but Cromwell told him a lie anyway about accidentally shooting himself when cleaning a gun. Because his wife received a generous donation from Cromwell for her pet project, the ballet company of San Francisco, the doctor filed no police report and vowed the incident would never be mentioned.\n\nCromwell returned to his office at the bank and quickly settled into the old routine of managing his financial empire. His first project for the day was to write a speech to give at the opening of a sanitarium for the elderly, funded and built through his generosity. Modesty was not one of his virtues and he named the hospital the Jacob Cromwell Sanitarium. He called in Marion Morgan to transcribe his notes on the speech.\n\nShe sat in a chair beside his desk and gazed at him. \"If you forgive me for asking, Mr. Cromwell, but are you feeling all right? You look a bit pale.\"\n\nHe forced a smile as he instinctively, lightly, touched his side. \"I caught a cold from fishing at night. It's almost gone away.\"\n\nHe handed her his notes, swung around in his leather chair, and stared out the window at the surrounding city. \"Edit my sanitarium speech, and please feel free to make any suggestions you feel are pertinent.\"\n\n\"Yes, sir.\"\n\nMarion rose to leave Cromwell's office but hesitated at the door. \"Excuse me, but I was wondering if you ever heard from the detective from the Van Dorn Agency again?\"\n\nCromwell swung back around from the window and stared at her curiously. \"Isaac Bell?\"\n\n\"I believe that was his name.\"\n\nHe could not help a mild grin as he said, \"He's dead. I heard he was killed during a bank robbery in Colorado.\"\n\nMarion's heart felt as if it was squeezed between two blocks of ice. She could not believe Cromwell's words. Her lips quivered, and she turned away from him so he couldn't see the shock written on her lovely face. Barely maintaining her composure, she said nothing and stepped from the office and closed the door.\n\nMarion sat at her desk as if in a trance. She could not understand the sense of grief over a man she hardly knew, a man with whom she had shared only one dinner. Yet she could see his face in her mind as if he was standing in front of her. The short-lived bond between them had been cruelly cut. She could not explain her feeling of sorrow and she didn't try. She felt as if she had lost a dear friend.\n\nWith trembling hands, she inserted a sheet of paper in her typewriter and began transcribing Cromwell's notes for his speech.\n\nAT FIVE O'CLOCK, late in the afternoon, Cromwell stood on the steps of a new three-story redbrick building on Geary and Fillmore Streets, listening to a long and flowery introduction by city mayor Eugene Schmitz, a close friend of Cromwell's who had benefited from large contributions secretly transferred to his personal account at the Cromwell Bank. A crowd of five hundred people attended the inauguration, along with members of the city's fire and police departments, political bosses, and over fifty elderly patients sitting listlessly in their wheelchairs.\n\nCromwell's own remarks were short and to the point. He modestly referred to himself as a \"humble messenger of the Lord\" who had chosen to help those who could not help themselves. When he finished, the applause was polite and subdued, befitting the formal occasion. A ribbon was cut at the front entrance and Cromwell was heartily congratulated. He shook every hand that was thrust at him. He made a show of embracing all of the patients waiting to enter the building. Mayor Schmitz gave him a bronze plaque for his philanthropic efforts and announced that, henceforth, April 12th would be known as Jacob Cromwell Day.\n\nMaking his way through a throng of well-wishers and admirers, Cromwell reached the parking space that held the Mercedes Simplex. Margaret was already seated behind the wheel, looking lovely in a green wool dress with cape.\n\n\"Well done, brother. Another good deed under the Cromwell banner.\"\n\n\"It never hurts to have friends in high places, as well as the adoration of the foul-smelling rabble.\"\n\n\"Aren't we the humanitarian?\" she said sarcastically.\n\n\"What about your benevolent pet projects that somehow get publicized in the society pages of the newspapers?\" he retorted.\n\n\"Touch\u00e9.\"\n\nCromwell moved to the front of the car and cranked the engine. Margaret retarded the spark and set the hand throttle. The engine caught and coughed into a throaty roar. Cromwell climbed into the seat as Margaret advanced the spark, shifted gears, and advanced the throttle. The Mercedes Simplex bounded out into the street between a cable car and beer truck.\n\nBy now, Cromwell was used to his sister's mad driving antics and relaxed in the seat, but was prepared to jump should a disaster rear up its head. \"Drive up to Pacific Heights and stop at Lafayette Park.\"\n\n\"Any particular reason?\"\n\n\"We can walk the paths while we talk.\"\n\nShe didn't question him further. The Mercedes Simplex easily cruised up the hill to Pacific Heights. She turned off Fillmore Street and took Sacramento Street until she reached the park, then stopped at the foot of a path leading into the trees. A five-minute walk took them to the summit of the park, which presented them with a beautiful panoramic view of the city.\n\n\"What do you wish to talk about?\" Margaret asked.\n\n\"I've decided to undertake another robbery.\"\n\nShe stopped in midstride and stared at him in distress. \"You must be joking.\"\n\n\"I'm dead serious.\"\n\n\"But why?\" she demanded. \"What have you to gain? You almost got caught in Telluride. Why tempt fate again for no purpose?\"\n\n\"Because I like a challenge. Besides, I rather enjoy being a legend in my own time.\"\n\nShe turned and looked away stunned. \"That's stupid.\"\n\n\"You don't understand,\" he said, putting his arm around her waist.\n\n\"I understand that it's crazy, and that someday your luck will run out and they'll hang you.\"\n\n\"Not for a while, at any rate,\" he said. \"Not while their best agent lies in his grave.\"\n\nMargaret remembered the incredible blue-purple eyes and Bell's arm around her as they danced at the Brown Palace. She seemed to hear her voice from far away. \"Bell dead, it's hard to believe.\"\n\nHe looked at her curiously. \"You sound like you had a crush on him.\"\n\nShe shrugged and tried to look uninterested. \"Oh, he was nice-looking, in a strange sort of way. I imagine other women found him attractive.\"\n\n\"No matter. Isaac Bell is history.\" Cromwell stopped and began leading his sister back to the automobile. \"I'm going to fool Van Dorn and all the other stupid peace officers who want me hung. They'll never suspect I'd commit another crime so quickly, at a bank in a town they'd never suspect. Once again, they'll be caught with their pants down.\"\n\nA tear came to her eye and Margaret dabbed a handkerchief at it, not sure if her emotions were twisted by Bell's demise or her brother's madness. \"Where this time?\"\n\n\"Not a mining town payroll,\" he said, grinning. \"I'll throw them a curve and hit a town that doesn't expect me, and leave them frustrated once again.\"\n\n\"What town?\"\n\n\"San Diego, here in California.\"\n\n\"That's almost in our backyard.\"\n\n\"All the better,\" said Cromwell. \"My escape will be that much easier.\"\n\n\"What makes San Diego so special?\"\n\n\"Because the city's Wells Fargo is fat with deposits, from merchants and from ships importing goods into the port. And because I'd love to poke a hole in my biggest competitor.\"\n\n\"You're crazy.\"\n\n\"Do not call me crazy!\" he said harshly.\n\n\"What do you call yourself? Everything we've worked for could come crashing down around us if you're ever caught.\"\n\n\"Not so long as they're dealing with a mastermind,\" Cromwell said brashly.\n\n\"When will you ever stop?\" Margaret demanded.\n\n\"When the Cromwell Bank is as big as the Wells Fargo Bank and I am crowned king of San Francisco,\" he said with a nasty glint in his eyes.\n\nShe knew it was hopeless to argue with her brother. Without his knowledge, she had quietly moved assets, little by little over the years, into the Wells Fargo Bank, where he would never think to trace them. The expensive jewelry she had purchased was put away in a safe-deposit box. If the worst came to pass and her brother was caught and hung, she would leave San Francisco, go to Europe, and live a life of luxury before finding a rich and titled husband.\n\nThey reached the automobile and Jacob helped his sister into the driver's seat. As he cranked the engine to life, Cromwell's self-confidence was overwhelming. Like a ship sailing into a heavy sea with all sails set, danger became a challenge that bordered on addiction. At the thought of outwitting every law enforcement officer in the West once again, his face beamed like that of a religious fanatic who had just witnessed a miracle.\n\nNeither of them paid any attention to a man sitting on a bench near the car dressed like a worker, with a toolbox perched in his lap, casually smoking a pipe.\n\n## 29\n\nBELL'S TRAIN GOT HIM INTO SAN FRANCISCO AT EIGHT o'clock in the morning. By nine, he was meeting with Carter, Bronson, and five of his agents. Everyone was seated around a large conference table that was twice as large as the one in the office in Denver. Bell was dead tired, and his wounds still gave him trouble, but he ignored the pain, as he had with earlier injuries, and soldiered on. \"Gentlemen,\" he began, \"now that our number one suspect for the Butcher Bandit is Jacob Cromwell, we are going to put him and his sister, Margaret, under twenty-four-hour surveillance.\"\n\n\"That means their every movement outside their palace on Nob Hill,\" added Bronson.\n\nOne agent held up a hand. \"Sir, we'll need photos for identification, since most of us have no idea of what they look like.\"\n\nBronson picked up a bulky file on the table. \"Photographs of them were taken while they were out and about town.\"\n\n\"Who took them?\" asked Bell.\n\nBronson smiled and nodded at one of his agents across the table. \"Dick Crawford here is an ace photographer.\"\n\n\"Didn't the Cromwells get suspicious about a photographer following them around, shooting their picture?\" asked Carter.\n\nBronson nodded at Crawford. \"Dick, tell everyone how you pulled it off without them getting wise.\"\n\nCrawford had a narrow saturnine face with a small jaw and bushy eyebrows beneath a bald head. A serious man, he did not show any humorous disposition. \"I wore coveralls and carried a toolbox with a small hole cut out in one end for the camera lens. All I had to do was reach into the box to adjust the focus and shoot their picture. They didn't have a clue and never so much as gave me a glance.\" He then set a small camera on the table and explained its application. \"What you see is a Kodak Quick Focus box camera that takes postcard-sized images.\"\n\nAs Crawford talked, Bronson passed out photos of Jacob and Margaret Cromwell.\n\n\"You will note that the photos are remarkably sharp and distinct,\" Crawford continued. \"The unique feature of the camera is that, unlike other cameras with a set focus, I could set the distance using the small wheel you see on the side. Then all I had to do was press a button and the front of the lens would pop out to the correct distance for exposure.\"\n\nEveryone studied the photos. They showed the Cromwells, individually or together, walking down the street, coming out of stores and restaurants. Several photos were of Jacob Cromwell entering and exiting his bank. Two showed him speaking at the opening of his sanitarium for the elderly. Crawford even followed them to Lafayette Park and shot them walking along a path. Bell was particularly interested in the pictures showing Margaret behind the wheel of an exotic-looking car.\n\n\"A Mercedes Simplex,\" he said admiringly. \"The Cromwells have good taste in automobiles.\"\n\nBronson examined the photos showing the car. \"It looks expensive. How fast will it go?\"\n\n\"At least seventy, maybe eighty, miles an hour,\" replied Bell.\n\n\"I doubt if there is a car in San Francisco that could catch it in a chase,\" said a bushy-haired agent at the end of the table.\n\n\"There is now,\" Bell said, his lips spread in a grin. \"It was unloaded from a freight car this morning.\" He looked at Curtis. \"Am I correct, Arthur?\"\n\nCurtis nodded. \"Your automobile is sitting in the Southern Pacific freight warehouse. I hired a boy who works in the railyard to clean it up.\"\n\n\"You sent a car here from...\"\n\n\"Chicago,\" Bell finished.\n\n\"I'm curious,\" said Bronson. \"What automobile is so special that you'd have it shipped all the way from Chicago?\"\n\n\"A fast motorcar can come in handy. Besides, as it turns out, it's more than a match for Cromwell's Mercedes Simplex, should it come to a pursuit.\"\n\n\"What make is it?\" asked Crawford.\n\n\"A Locomobile,\" answered Bell. \"It was driven by Joe Tracy, who drove it to third place in the 1905 Vanderbilt Cup road race on Long Island.\"\n\n\"How fast is it?\" inquired Bronson.\n\n\"She'll get up to a hundred and five miles an hour on a straight stretch.\"\n\nThere came a hushed silence. Everyone around the table was astounded and disbelieving.\n\nThey had never seen or heard of anything that could go so fast. Professional auto races with competing factory cars had not come to the West Coast yet.\n\n\"Incredible,\" said Bronson in awe. \"I can't imagine anything traveling a hundred miles an hour.\"\n\n\"Can you drive it on the street?\" asked Curtis.\n\nBell nodded. \"I had fenders and headlamps installed and the transmission modified for street traffic.\"\n\n\"You've got to give me a ride in it,\" said Bronson.\n\nBell laughed. \"I think it can be arranged.\"\n\nBronson turned his interest back to the photos of the Cromwells. \"Any thoughts on what the bandit will do next?\"\n\n\"After Telluride,\" said Curtis, \"I would bet his days of robbery and murder have ended.\"\n\n\"Sounds logical if he knows we're onto him,\" agreed Bronson.\n\n\"We can't be sure of that if he thinks all witnesses to the fiasco in Telluride are dead, including me,\" said Bell. \"He is a crazy man, driven to rob and kill. I don't believe he can ever stop cold. Cromwell believes his criminal acts can never be traced. He simply does not fit the mold of Black Bart, the James Gang, the Daltons, or Butch Cassidy. Compared to Cromwell, they were crude, backwoods amateurs.\"\n\nOne of the agents stared with growing admiration at Bell. \"So you think he will strike again.\"\n\n\"I do.\"\n\n\"You may have suckered him with your story about Telluride,\" said Bronson. \"But if he is as smart as you say he is, Cromwell won't make the same mistake twice and step into another trap.\"\n\nBell shook his head. \"There is little hope of that, I'm afraid. For the moment, all we can do is try to outguess him, and, failing that, we keep gathering evidence until we can convict him.\"\n\n\"At least we know he isn't infallible.\"\n\nBronson grunted. \"He's about as close as you can come.\"\n\nBell poured himself a cup of coffee from a pot sitting on the conference table. \"Our edge is that he doesn't know his every move is being watched. You will have to be very careful and not make him or his sister wary. If we can stay on his tail the next time he leaves town for a robbery, we have a chance of bringing his crime wave to a halt.\"\n\nBronson looked around the table at his agents. \"It looks like we have our job cut out for us, gentlemen. I'll let you work out your surveillance shifts among yourselves. I received a telegram from Mr. Van Dorn. He said to pull out all the stops. He wants the Butcher Bandit caught, whatever the cost, whatever the effort.\"\n\nBell said to Bronson, \"I wonder if you could do me a favor.\"\n\n\"You have but to name it.\"\n\n\"Call Cromwell's office and ask for Marion Morgan. Tell her you're calling in the strictest confidence and she is to say nothing to no one, including her boss. Tell her to meet you at the northeast corner of Montgomery and Sutter Streets, a block from the Cromwell Bank, during her lunch hour.\"\n\n\"And if she asks me the purpose?\"\n\nBell made a crooked smile. \"Just be vague and tell her it's urgent.\"\n\nBronson laughed. \"I'll do my best to sound official.\"\n\nAFTER THE CONFERENCE, Bell and Carter took a cab to the Southern Pacific freight warehouse. They checked in with the superintendent, looked over the car for damage, and, finding none, signed off the necessary transport paperwork.\n\n\"She's a beauty,\" Curtis said admiringly, gazing at the bright red\u2013painted automobile with its gleaming brass radiator topped by a custom-sculpted bronze eagle with wings outspread and a temperature gauge in its chest. Behind the radiator was a barn-roof-cut hood. A big cylindrical gas tank sat mounted behind the two seats. The narrow tires were moored to huge wooden spoked wheels that had sped over the twisting roads of Long Island during the Vanderbilt Cup race.\n\nBell climbed into the seat behind the big steering wheel, mounted on its long shaft, turned the ignition switch on the wooden dashboard, set the throttle lever on the steering wheel, and moved the spark lever to retard. Next, he took a hand pump and pressurized the fuel tank, forcing gas to the carburetor. Only then did he walk to the front of the car, grip the big crank with his right hand, and heave vigorously. The engine coughed and kicked over on the second try, with a thunderous roar from the exhaust pipe.\n\nThen Bell, joined by Carter, sat in the red leather driver's seat and advanced the spark as he eased the throttle to an idle position. After releasing the brass hand brake, he pushed in the clutch and pulled the shift lever into first gear. Next, he moved the throttle lever and released the clutch, having attracted a crowd of warehouse workers who cheered as the rakish car rolled forward.\n\nAs soon as the Locomobile was speeding down a road alongside the railroad tracks, Carter asked loudly, \"Are we headed back to the office?\"\n\nBell shook his head. \"Show me the way to the warehouse where the O'Brian Furniture boxcar was parked.\"\n\n\"Then turn left at the next crossing over the tracks,\" directed Carter.\n\nA few minutes later, Bell parked the Locomobile behind the empty warehouse and turned off the big engine. With Carter leading the way, they walked up a ramp to the loading dock. A single freight car sat on the siding.\n\n\"Is this where you found Cromwell's phony furniture freight car?\" asked Bell.\n\n\"According to the Southern Pacific's freight-movement schedule,\" said Curtis. \"I ran a check of company freight car movements. Car 16173 is no longer listed on Southern Pacific freight records. No one knows what happened to it. It's as if it vanished overnight.\"\n\nBell studied the sides of the car parked alongside the loading dock. \"It could have been repainted and given a new serial number.\"\n\n\"It's entirely possible.\" Curtis stared at the number and then nodded. \"Car 16455. I'll check it out.\"\n\n\"This car has had a new paint job recently,\" said Bell slowly. \"There isn't a scratch on it.\"\n\n\"You're right,\" Curtis murmured thoughtfully. \"It's as clean as the day it came out of the factory.\"\n\nBell walked up to the boxcar's loading door and placed his fingers around a bronze lock that sealed the interior from entry. \"Why would an empty car on a siding be locked up?\"\n\n\"Maybe it's been loaded with cargo and is waiting to be coupled to a train.\"\n\n\"I wish I knew what was inside,\" Bell mused.\n\n\"Shall we break it open?\" Curtis inquired with a growing sense of anticipation.\n\nBell made a slight shake of his head. \"Better we leave well enough alone for the time being. Until we check out the serial number, we won't know the history of this car. And should it belong to Cromwell, he'll know if we tampered with the lock.\"\n\n\"If we proved this is the freight car he used to escape his criminal acts, we can arrest him.\"\n\n\"Nothing is that simple. It might simply be an empty car that was shunted to this siding temporarily. Cromwell's no fool. He wouldn't leave evidence lying around just waiting to be found. Chances are, there is nothing incriminating inside, certainly not enough to stand him under the hangman's noose.\"\n\nCurtis shrugged in understanding. \"We'll keep a sharp eye on it, but I doubt if he'll be using it anytime soon, if ever again, considering how he came within a hair of being caught in Telluride.\"\n\n\"And, sooner or later, he'll learn I'm still alive and know I identified him,\" Bell said with a wide grin. \"Then he'll really make things interesting.\"\n\nMARION PUT down the phone and looked toward the doorway leading to Cromwell's office. As usual, it was closed. He almost always worked in private, handling his day-to-day business over the telephone or a speaker system he had installed around the bank.\n\nShe glanced up at a big Seth Thomas Regulator wall clock, with its enclosed pendulum swinging back and forth. The hands were pointing at Arabic numerals that read three minutes to twelve. When she put down the phone after listening to Bronson's instructions, she was torn between her loyalty to Cromwell\u2014and whether she should tell him about the call\u2014and the building sense of excitement that coursed through her body at the thought of performing an act of secrecy. Because a distinct rift had built between her and Cromwell over the past year, especially since that night in the Barbary Coast when he and Margaret had acted so strangely, she felt less loyalty and respect toward him. He was not the same man she had come to trust for so many years. He had become distant and aloof, cold and rude toward her much of the time.\n\nThe minute hand clicked over the hour hand, both pointing to twelve, when she took her purse, put on her hat, and stepped out of the office, all the while keeping an eye on the closed doorway to Cromwell's office. She bypassed the elevator and flew down the stairway to the lobby. Passing through the big entrance doors, she turned and hurried down Sutter Street to Montgomery. The streets and side-walks were busy during the lunch hour and it took her a good ten minutes to skirt the crowds. Reaching the corner, she stood there, looking around, but found no one looking in her direction or coming toward her. She had never met Bronson and had no idea what he looked like.\n\nAfter a minute, her attention, and that of many people passing along the street, was drawn to a big red car that moved effortlessly through traffic. There was a brute strength about its appearance that made it look as if it were hurtling over the pavement, even though it was moving no more than twenty miles an hour. Its bright red paint had been hand rubbed to a glistening finish. Everything about it portrayed a powerful elegance.\n\nWith her attention focused on the car, she did not notice the man behind the wheel until it came to a stop in front of her and he said, \"Please climb in, Marion.\"\n\nShe paled, one hand flying up and holding her throat, startled to find herself gazing into the violet eyes of Isaac Bell that seemed to draw her into his soul. \"Isaac,\" she murmured in shock. \"Jacob told me you were dead.\"\n\nHe held out his hand, grasped hers, and pulled her up onto the leather passenger seat with an ease and strength that stunned her. \"It just goes to show, you can't believe all you hear.\"\n\nOblivious to the crowd that had gathered around the car, Bell circled his arm around Marion's waist. Then he took her in his arms and kissed her.\n\n\"Isaac!\" she gasped when he released her. Her protest was one more of enjoyment than embarrassment. \"Not in front of all these people.\"\n\nBy now, the crowd that had assembled to stare at the car found themselves being entertained by the man and woman in the front seats. They began to applaud and cheer them on.\n\nBell pulled back and smiled wickedly. \"I was never able to resist a beautiful lady.\"\n\nMarion was almost swept away by the moment\u2014almost but not quite. \"Can we please move away?\" she insisted.\n\nBell laughed, tipped his hat to the people cheering him on, and shifted the Locomobile into first gear. He stepped lightly on the gas pedal and moved into the street amid the flow of traffic. He drove north on Montgomery before turning left into Chinatown. He swung into an alley and came to a stop behind a large Mandarin-style restaurant, painted red and gold and with a pagoda roof. An attendant waiting there bowed.\n\n\"I will watch your car, sir.\"\n\nBell gave him a tip that made the attendant's eyes pop. \"I'm counting on you.\" Then he helped Marion from her seat to the ground.\n\n\"The Empress of Shanghai,\" she said, staring at the ornate entrance. \"I've always wanted to eat here.\"\n\n\"It came highly recommended.\"\n\n\"I wondered how you knew about the rear parking.\"\n\nAfter they entered a long hallway, they were greeted by a beautiful woman with long shiny black hair wearing a Chinese sheath silk dress slit high on the sides. She led them upstairs to a small private dining room and seated them. While they were studying the menu, a pot of tea arrived and was poured.\n\n\"You were limping,\" she said.\n\n\"A little memento of Telluride, Colorado.\"\n\nFor the first time, she noticed the bandage on his head as he removed his hat. She frowned and raised her eyebrows. \"Another memento?\"\n\nHe nodded and smiled gamely.\n\nMarion looked into Bell's eyes and her own eyes became misty. \"You don't know how happy I am that you weren't killed.\"\n\n\"Your boss certainly tried.\"\n\n\"Mr. Cromwell!\" she exclaimed as her mood altered from compassion to alarm. \"I don't understand.\"\n\n\"He's the man who shot me and killed a Van Dorn agent who was my friend.\"\n\n\"You can't be serious.\"\n\n\"Like it or not, Jacob Cromwell is the Butcher Bandit who has held up over twenty banks in the past twelve years and killed nearly forty innocent people.\"\n\n\"That's crazy!\" Marion bit her lower lip. She looked as if she was lost and had nowhere to turn. \"He couldn't have done what you say.\"\n\n\"What I said is true,\" Bell said with a sudden gentleness. \"We have evidence. Maybe not enough to convict, but it falls at Cromwell's doorstep.\"\n\n\"But he's helped so many people in need,\" she protested.\n\n\"A front,\" said Bell icily. \"He's built a wall around his empire, guarded by an army of good citizens who believe he and Margaret are generous people who want to help the poor out of the goodness of their hearts. It's an act. Cromwell could care less about those who are destitute. He uses them to promote his own purposes. In the eyes of the city's crooked politicians, he can do no wrong so long as he supports them with secret donations.\"\n\nConfused, Marion sipped at her tea, her hand noticeably trembling. \"I simply refuse to believe it,\" she murmured.\n\nBell reached across the table and took both her hands in his. \"Believe me, Marion, it's true. I looked into his eyes and recognized him the instant he shot me at the bank in Telluride.\"\n\nShe pulled her hands back and clasped them together tightly. \"Oh, Isaac, it's all too fantastic. Why would Jacob rob banks when he already owns the second-largest bank in San Francisco? The thought is too absurd to be real.\"\n\n\"I can't give you an answer, Marion. In the beginning, he took the money to build his own bank. But when he became rich, the robbery and killings became an obsession. I've seen many cases like Cromwell's. The robberies and the murders are like a narcotic for him. He can't help himself, and will go on killing until I stop him.\"\n\nShe looked up into his sensitive violet eyes across the table. They had turned dark and cold. \"You, Isaac? Does it have to be you?\"\n\n\"I can't let him go on murdering people.\" Bell spoke the words in a monotone, as if he were reading an accusation in a courtroom. \"I am not going to let him thumb his nose at the law and continue to run around free, living the life of a wealthy Santa Claus.\" Then he added, \"And that goes for his sister, Margaret. She's buried in his evil operations up to her pretty neck.\"\n\nMarion dipped her head in utter confusion as her hat covered her eyes. \"I've known Jacob and Margaret for years and yet I didn't know them.\"\n\n\"It's hard,\" Bell said softly, \"but you'll have to accept it.\"\n\nShe tilted her head back and the forward brim of her wide flowered hat rose until he could look directly into her coral\u2013sea green eyes. \"What can I do?\" she asked softly.\n\n\"For one thing, you must go on as if you know nothing. Continue your duties as his loyal secretary. Our agents will have both brother and sister under constant surveillance. All you have to do is report anything suspicious or unusual regarding Jacob's actions.\"\n\n\"You mean, of course, report to you.\"\n\nHe nodded. \"Yes.\"\n\nShe suddenly had the feeling she was being used, that Bell's interest in her was purely as an informer. She turned away so he couldn't see the tears welling up in her eyes.\n\nBell immediately sensed what was whirling in her mind. He moved his chair around the table until he was sitting close enough to put his arm around her shoulders. \"I know what you're thinking, Marion, and it's not true. I know I am asking you to commit a devious act, but lives hang in the balance. Yet there is much more. It goes far beyond a request for your help.\" He paused to build up his courage. \"I'm in love with you, Marion. I can't explain why it happened so suddenly, but it did. You must believe me.\"\n\nMarion looked into his face and saw only affection and fondness. Her fears vanished in an instant as she leaned forward and kissed him solidly on the lips. When she pulled back, she smiled wickedly. \"You must think I'm a brazen hussy.\"\n\nHe laughed at seeing her blush. \"Not at all. I enjoyed it.\"\n\nThen her eyes turned soft. \"I have to admit I felt something when I looked up and saw you standing there in the office.\"\n\nThis time, he kissed her.\n\nAfter a long moment, he pulled back and grinned. \"Perhaps we should order before they ask us to leave for disorderly conduct.\"\n\n## 30\n\nAS SOON AS MARION RETURNED FROM HER LUNCH WITH Bell and was in the midst of typing a letter, Cromwell called her into his office. She concealed her nervousness by not looking him in the face as he spoke. \"Marion, I'm going to attend the National Conference for Community Banks. It is being held in Los Angeles this year on March twenty-eighth to March thirtieth. Could you please make the necessary travel arrangements, and book me a room at the Fremont Hotel downtown?\"\n\n\"To be in Los Angeles by the twenty-eighth, you'd have to leave tomorrow,\" said Marion. \"That's awfully short notice.\"\n\n\"I know,\" Cromwell said with an offhand shrug. \"I wasn't going to attend, but I changed my mind.\"\n\n\"Will you wish to charter a private car?\"\n\n\"No. I'll leave private cars to the presidents of the Crocker and Wells Fargo banks. When I go on bank business, I'll travel as a simple passenger so my depositors will know I have their best interest at heart and am not squandering their money.\"\n\nMarion rose to her feet with a rustle of her skirts. \"I'll see to it.\"\n\nAs soon as she returned to her desk, she picked up her telephone and in a low voice, nearly that of a whisper, asked the operator for the Van Dorn Detective Agency. When Marion gave the receptionist her name, she was immediately put through to Bell.\n\n\"Isaac?\"\n\n\"Marion? I was just going to call and ask you out for dinner and a play.\"\n\nShe felt pleased that he was happy to hear her voice. \"I have some information for you,\" she said seriously. \"Jacob is going out of town.\"\n\n\"Do you know where?\"\n\n\"Los Angeles,\" she answered. \"He's going to attend the National Conference for Community Banks. It's a forum for bankers, to exchange the latest in banking operations.\"\n\n\"When does it take place?\"\n\n\"The twenty-eighth to the thirtieth of this month.\"\n\nBell thought a moment. \"He'd have to be on a train tomorrow if he was going to make Los Angeles by the twenty-eighth.\"\n\n\"Yes, that's right,\" said Marion. \"As soon as I ring off, I have to make his reservations. He's traveling in a coach, as an ordinary passenger.\"\n\n\"Not like your boss to save a buck.\"\n\n\"He claimed it would impress Cromwell depositors by not squandering the bank's assets.\"\n\n\"What do you think, Marion? Is this trip legitimate?\"\n\nShe did not hesitate in answering. \"I do know there really is a National Conference for Community Banks on those dates in the City of Angels.\"\n\n\"I'll see that one of our agents is with him all the way.\"\n\n\"I feel soiled going behind his back,\" she said remorsefully.\n\n\"Do not regret it, sweetheart,\" Bell replied tenderly. \"Jacob Cromwell is an evil man.\"\n\n\"What time should I expect you?\" Marion asked, happy to get off the subject of Cromwell.\n\n\"I'll pick you up at six so we can have an early dinner before making the play.\"\n\n\"Are we going in your red racer?\"\n\n\"Do you mind?\"\n\n\"No, I enjoy the exhilaration of speed.\"\n\nHe laughed. \"I knew there was something about you that attracted me.\"\n\nMarion hung up the phone, surprised to find her heart beating at a rapid rate.\n\nON GUT INSTINCT, and the knowledge that Bell and his agent Irvine had been nosing around before he killed them, Cromwell made elaborate plans to cover his tracks even more thoroughly. He was certain that with the loss of two of his agents, Van Dorn would add fuel to the investigation by probing ever deeper into every lead. He could expect more agents to come around asking more questions about the stolen money that had been dispersed through merchants and other banks around the city.\n\nJust to be on the safe side, Cromwell called the chief dispatcher of the Southern Pacific and informed him that he was sending in a written request to move his disguised freight car, now serial number 16455, sitting at the abandoned warehouse, to a new location across the bay in Oakland. Within minutes, the order was received by the yardmaster, who sent a switch engine that was coupled to the car and pulled it onto a boxcar ferry.\n\nCromwell also ordered a special train, a private Pullman car pulled by an engine and tender; destination: San Diego. The order went through the O'Brian Furniture Company of Denver, which had a long-standing account and was a respected customer of the Southern Pacific Railroad Company.\n\nOnly then did he sit back in his chair, light an expensive cigar, and relax, totally self-assured that he was once again ten steps ahead of any remote suspicion that might be held by Van Dorn or any other law enforcement agency.\n\nHe would have been even more smug if he had known that before Bronson could send an agent to keep an eye out for anyone approaching the freight car, it had been switched onto the ferry and transported to a siding in the Southern Pacific railyard in Oakland.\n\n## 31\n\nEARLY THE NEXT MORNING, CROMWELL BID MARGARET good-bye and stepped into his Rolls-Royce limousine. Abner smoothly steered the car through the city traffic to the Southern Pacific passenger station for trains running directly north or south that did not have to cross the bay. Stopping at the station entrance, he opened the car door and handed Cromwell a valise.\n\nAs the Rolls pulled away from the curb, Cromwell casually walked into the station, showed his ticket to the gatekeeper, and joined the other passengers moving along the platform. He climbed the steps to the third coach and boarded the train.\n\nA Van Dorn agent watched him board and then loitered until the train began to move, making sure that Cromwell did not step back on the platform, in case he had missed the train. Only then did the agent swing aboard the last car and begin walking through the passenger cars until he reached the one Cromwell had entered. To his amazement, Cromwell was nowhere to be seen. Alarmed, the agent rushed through the remaining cars, searching until he reached the locked door to the baggage car. Still no Cromwell. Then he hurried to the back of the train, entertaining the possibility that he had missed the banker, but Cromwell was still nowhere to be found.\n\nUnseen, Cromwell had departed the passenger car by the opposite door and stepped down and crossed the tracks to another platform, where the special train he had chartered was waiting. He climbed aboard his private car, where he relaxed in the luxury and glamour of what was a veritable yacht on wheels. He removed his coat, sat back casually in an overstuffed velvet chair, and opened the morning paper. A steward served him breakfast that had been specially prepared by the car's private chef. He was leisurely reading the San Francisco Chronicle when the train pulled away from the station and onto the main track for the run to Los Angeles, just fifteen minutes behind the regularly scheduled passenger train on which Marion had booked him a seat.\n\n\"NO WORD from my agent, so I can safely assume Cromwell is on his way to Los Angeles,\" said Bronson.\n\nBell looked up from a map depicting San Francisco and its neighboring big city to the south. \"His train is scheduled to arrive in Los Angeles at four-thirty this evening. I'm told he's staying at the Fremont Hotel.\"\n\n\"I was lucky. I managed to wire Bob Harrington, who heads up the Southern California Van Dorn office, before the flash flood somewhere to the south took out the line. He's going to have a man disguised as a cabbie pick up Cromwell and take him to his hotel. My agent on the train will point him out. From there, Harrington's agents can keep a tight rein on him.\"\n\n\"His trip sounds innocent enough,\" Bell said slowly. \"But I don't trust him. He's up to something. I feel it in my bones.\"\n\n\"He won't get far if he tries anything,\" Bronson said confidently. \"Should he make even a tiny false move, a dozen agents will land on him like a ton of bricks.\"\n\nBell walked back to an empty office and rang up Marion over at the bank. \"Did you survive last night?\" he asked lovingly.\n\n\"I had a wonderful time, thank you. The meal was scrumptious and the play was delightful.\"\n\n\"Now that the cat is away, how about the mouse coming out and play\u2014say, for lunch?\"\n\n\"I'm game.\"\n\n\"I'll pick you up in front of the bank.\"\n\n\"I'll meet you where we met before,\" she said without hesitation. \"I don't want our relationship to be obvious. If any of the employees see me getting in your flashy red car, they're liable to talk, and it will get back to Jacob.\"\n\n\"Same time, same place,\" he said before he hung up.\n\nLater that morning, a Western Union messenger came running into the office. \"I have an urgent message for a Mr. Horace Bronson,\" he said to the receptionist, gasping because of his dash from the Western Union office.\n\nBronson, who was coming back from the bathroom down the hall, said, \"I'm Bronson. I'll take it.\" He flipped the messenger a coin and tore open the envelope. As he read the message, his lips tightened and his forehead turned into a hard frown. He rushed through the office until he came to Bell.\n\n\"We're in trouble,\" he announced.\n\nBell looked at him questioningly. \"Trouble?\"\n\n\"My man lost Cromwell.\"\n\nBell faltered, taken completely off balance. \"How could he lose him on a train?\"\n\n\"Cromwell must have gotten on the train and immediately jumped off the opposite side without being seen.\"\n\n\"Your agent should have alerted us sooner,\" Bell snapped, anger flaring.\n\n\"The train had departed the station and he couldn't get off until it stopped in San Jose,\" Bronson explained. \"He sent a telegram from there.\"\n\n\"He could have saved thirty minutes by using the telephone.\"\n\nBronson shrugged helplessly. \"The phone lines are unreliable and in constant repair.\"\n\nBell sank into a chair, stunned and furious at having the rug pulled out from under him. \"He's going to rob and kill again,\" he said, his face flushed with frustration. \"The bastard is rubbing it in our faces.\"\n\n\"If we only knew where,\" said Bronson, overcome with defeat.\n\nBell walked over to the window and looked across the roofs of the city buildings. He stared without seeing, lost in thought. Finally, he turned. \"Cromwell is taunting us,\" he said slowly. \"He expects us to run around like chickens with our heads cut off, wondering where he went.\"\n\n\"He's obviously heading in the opposite direction he told his secretary.\" Bronson gave Bell a hard stare. \"Unless she's lying.\"\n\nBell didn't meet his stare. The possibility crossed his mind, too. He merely shook his head. \"No, I'm certain Marion told the truth.\"\n\nBronson walked over to a map of the United States hanging on one wall. He stared at it, perplexed. \"I doubt if he'll head north into Oregon or Washington. He probably doubled back to the Ferry Building, crossed the bay, and took a train heading east.\"\n\nA smile slowly began to curve and spread across Bell's face. \"I'll bet my Locomobile Cromwell is still heading south.\"\n\nBronson looked at him. \"Why would he continue south if he literally threw us off the track?\"\n\n\"I know how the man thinks,\" said Bell in a voice that defied argument. \"Though he doesn't know his every movement is being watched, he never takes chances, every possibility is carefully thought out.\"\n\nBronson looked at his pocket watch. \"The next train isn't until noon.\"\n\n\"Too late,\" Bell disagreed. \"He has too much of a head start.\"\n\n\"But how do we know that, since he jumped the train?\"\n\n\"He gave Marion a cock-and-bull story about riding in coach so his depositors would think he's a down-to-earth kind of guy. Ten will get you twenty he chartered a private train.\"\n\nBronson's apprehension appeared to loosen. \"Harrington can still have his agents follow him when he arrives in Los Angeles.\"\n\nBell shook his head. \"His agents won't be able to identify him. Your agent got off the train in San Jose to notify you Cromwell wasn't on board. He's probably waiting for the next train back to San Francisco.\"\n\n\"That is a problem,\" Bronson agreed. \"But they can still grab him when he checks into the Fremont Hotel.\"\n\n\"If Cromwell checks into the Fremont,\" Bell said shrewdly. \"Since he slipped off the passenger train, it's unlikely the rest of his story to Miss Morgan was true.\"\n\n\"If not Los Angeles, then where is he going?\"\n\n\"Cromwell could stop his train anywhere between here and there, but my guess is that he's going on through Los Angeles.\"\n\n\"Through?\" wondered Bronson. \"Through to where?\"\n\n\"The last place we would expect him to go for a robbery, the least likely destination.\"\n\n\"Which is?\"\n\n\"San Diego.\"\n\nBronson thought quietly for several moments. Finally, he said, \"That's a long shot.\"\n\n\"Maybe. But that's all we have going,\" said Bell. \"He's demonstrated that he doesn't always rob mining towns. Why not a city with a bank bulging with profits from goods imported by rich merchants and the owners of large ranches around Southern California?\"\n\n\"A long shot or not, we can't overlook it. If only I could alert Harrington to send his agents to the San Diego railroad terminal and be on the lookout for a private train. But the telephone and telegraph lines from San Jose to Los Angeles are still down due to flooding.\"\n\nBell shook his head. \"Cromwell's too smart to run his train directly into the city. He'll park it on some remote siding and use another mode of transportation to get to the city, probably the motorcycle he used on other robberies.\"\n\n\"If only Harrington's had a description,\" said Bronson.\n\n\"They couldn't identify him anyway; he'll probably be wearing a disguise.\"\n\nBronson's optimism suddenly vanished out the window. \"Then where does that leave us?\"\n\nBell smiled. \"I'll have to go to San Diego and confront him myself.\"\n\n\"Not possible,\" Bronson said. \"By the time we can hire a special express train, have it on the tracks, and leave town, he will have conducted his dirty business and be halfway back to San Francisco.\"\n\n\"Very true,\" acknowledged Bell. \"But, with a little luck, I can make it to Los Angeles before his train arrives and be waiting for him.\"\n\n\"So how are you going to beat him to Los Angeles, fly on a big bird?\" Bronson said sarcastically.\n\n\"I don't need a big bird.\" Bell gave Bronson a canny look. \"I have something just as fast.\" Then he smiled bleakly. \"But, first, I have to break a date.\"\n\n## 32\n\nTHE BIG RED LOCOMOBILE SWEPT THROUGH SAN Francisco like a bull running through the streets of Pamplona, Spain, during the Fiesta of San Fermin. Bell sat back in the red leather seat, his two hands tightly gripping the bottom of the big spoked steering wheel, turning the car with his palms facing up, using his biceps to twist the stiff mechanism around curves and street corners.\n\nThe time was fifteen minutes before ten o'clock.\n\nNext to him, in the shotgun seat, sat Bronson, whose job was to keep the fuel pressure pumped up. Every few minutes, he pulled out the pump handle that was mounted on the upper wooden panel just above the slanting floorboard and shoved it forward, sending gas to the carburetor. Besides keeping the big hungry engine fed, he took on the job of navigator, since Bell had no knowledge of the California countryside. As Bell drove, Bronson braced his feet on the floorboard and pressed his back into the leather seat to keep from being thrown to the pavement, feeling as if he was being shot through the muzzle of a cannon.\n\nNot wanting to take either hand from the steering wheel, Bell also gave Bronson the job of sounding the big horn bulb. The agent seemed to enjoy squeezing out squawking honks at the people and traffic, especially at intersections. It was not long before his hand ached.\n\nBronson was wearing a long leather coat, his feet and lower legs encased in boots. His head was covered by a leather helmet with huge goggles that made him look like an owl on a quest for a rodent. The goggles were a necessity since the Locomobile did not have a windshield.\n\nThe car hadn't traveled a hundred yards when Bronson had dire misgivings about what he had gotten himself into by insisting that he accompany Bell on this mad dash to San Diego in an open car over roads that weren't much better than cow paths.\n\n\"How are the brakes on this mechanical marvel of engineering?\" Bronson asked caustically.\n\n\"Not great,\" Bell answered. \"The only brakes are on the shaft driving the chains to the rear wheels.\"\n\n\"Do you have to go so fast through town?\" Bronson protested.\n\n\"Cromwell's private train has over an hour's head start,\" yelled Bell through the exhaust. \"We need every minute we can gain.\"\n\nPedestrians, who heard the throaty exhaust roar coming up the street followed by a strange blaring sound from the bulb horn, were stunned when they saw the red Locomobile bearing down on them. Staring incredulously, they quickly stepped out of the street until the machine sped past. The twin exhaust pipes, barely protruding from the left side of the hood, throbbed like cannon.\n\nTwo workmen, who were carrying a large windowpane alongside the street, froze in total shock as the car thundered past, the explosive roar of the Locomobile's exhaust shattering the glass in their hands. Neither Bell nor Bronson ever looked back, their complete focus being on the traffic that ran thick or thin in front of them, forcing Bell to swing the wheel violently back and forth as though he was driving through an obstacle course. His great satisfaction came in pointing the car in the direction he wanted to go and having it respond as if anticipating his thoughts.\n\nBell jockeyed his foot from the accelerator to the brake and back again, as he tore down the streets, hammering turns at the intersections onto the main street leading from the city, wishing he was a sorcerer who could magically make the traffic disappear. Bell narrowly missed a laundry truck, throwing the Locomobile into a four-wheel drift to avoid it. He spun the thick wooden rim of the steering wheel fiercely as he dove between vehicles littering the streets. Drivers of other motorized vehicles stared in awe at the speed of the car as it flashed up from behind and quickly disappeared up ahead. Horses harnessed to buggies and wagons reared at the noise of what their drivers thought was the devil's chorus.\n\nAs they neared the outer edge of the city's southern limits, the traffic began to thin. Bell slowed the Locomobile around a sweeping turn onto the main road south that paralleled the railroad tracks. He breathed a sigh of relief at seeing automobiles and wagons becoming sparse. He was also thankful that he now had ample room to swerve around any vehicle that blocked his path. The huge automobile was incredibly responsive. Bell pressed the accelerator within an inch of the floor, as the car began to rocket along a road that ran straight with few curves. The faster the Locomobile traveled, the more solid her feel of stability, as the drive chains on her axles whirred at a high, metallic pitch.\n\nSoon the road became straight and rural. Picturesque farming communities came up on the horizon and quickly slipped behind the automobile's dust trail. San Carlos, Menlo Park, and then San Jose, towns that were linked together by the El Camino Real, the old road used in the late 1700s by the Franciscan friars who built twenty-one missions, each a day's journey apart.\n\nEnjoying a straight, open road with little traffic, Bell pressed the accelerator to the floor and pushed the automobile as hard as it could go. The Locomobile was in its element now, running as strongly as when it had in the Vanderbilt Cup race, the first American car to place in an international speed event. Like a racehorse that had been retired and then brought back to run again, the Locomobile roared down the road like a maddened elephant, the cavernous cylinders of its mighty engine turning the huge crankshaft effortlessly.\n\nBell loved the big machine. He had an exceptional sense of its temperament and idiosyncrasies. He gloried in its strength and simplicity, felt intoxicated by the speed produced by the big pounding engine, and drove like a demon possessed, reveling at the vast, swirling cloud of dust the Locomobile hurled in its wake.\n\nBronson looked over at Bell, who wore a short leather jacket and jodhpur riding pants with boots. He wore goggles but no helmet, preferring to hear the beat of the engine. There was a look of unfathomable concentration about him. He looked relentlessly determined to beat Cromwell at his own game. Bronson had never seen anyone with such fierce, decisive resolve. He turned away and studied his map. Then he tapped Bell on the shoulder.\n\n\"There is a fork in the road coming up. Veer left. The road is better inland than along the coast. At this rate, Salinas will come up in another hour. After that, Soledad.\"\n\n\"How's our time?\" Bell asked without taking his hands from the wheel and digging out his pocket watch.\n\n\"Ten past eleven,\" Bronson answered over the exhaust. \"Without knowing how fast we're going, I have no way of knowing how much time, if any, we've gained on Cromwell's train.\"\n\nBell nodded in understanding. \"The auto does not have a speedometer or a tachometer, but I'd guess our speed to be over ninety miles an hour.\"\n\nBronson had been slowly becoming attuned to the wind rushing against his face, the telegraph poles streaking past at lightning speed. But then a stretch of road became violently rough and rutted, and Bronson soon realized what it would be like inside the rattle of a maddened sidewinder. He clutched the arm of his seat in a death grip with one hand and gamely worked the fuel pump with the other.\n\nThey hurtled over the narrow, rolling farm road and crossed into Monterey County before coming to the agricultural community of Salinas. The farmland along the sides of the road was strikingly beautiful, turning green under the spring sun. Fortunately, the main road through town was quiet, with only one or two automobiles and a few horse-drawn wagons parked along the sidewalks. People heard the booming bellow of the Locomobile's exhaust as it crossed the city limits. They turned and looked speechlessly as the big fire red machine shot through the business section of town. They had no time to indulge their curiosity before the hard-charging machine was heading into the open country to the south.\n\n\"What's the next town?\" asked Bell.\n\nBronson consulted his map. \"Soledad.\"\n\n\"How far?\n\n\"About twenty-five miles. We'd better fill the tank there, because it's a good two hundred miles to the next major town.\" He turned and looked at the huge cylindrical brass tank mounted behind the seats. \"How much does it hold?\"\n\n\"Forty-five gallons.\"\n\n\"They should have a garage in Soledad that services automobiles and farm machinery.\"\n\nThe words were no sooner out of Bronson's mouth than the left rear tire went flat after striking a sharp rock in the road. The Locomobile fishtailed for a hundred yards before Bell brought it under control and braked it to a stop.\n\n\"Only a matter of time,\" said Bell resignedly. \"One of the predicaments of road racing.\"\n\nHe was out of the automobile and shoving a jack under the rear axle within three minutes while Bronson removed one of the two spare tires on the rear of the automobile. Bell removed the wheel and replaced it inside of ten minutes. He had changed tires that gave out at breakneck speeds many times since he owned the Locomobile. Then he separated the tire from the wheel and tossed the tire to Bronson. \"There's a patching kit under your seat. Patch the hole while we drive. I'll remount it on the wheel after we reach Salinas.\"\n\nNo sooner were they on their way again over a reasonably smooth road than a hay wagon hitched to a team of horses loomed up. The farmer, believing he was the only one for a mile around, was driving right down the center of the road, with only a few feet to spare before the weeds and brush along the edge of the dirt thoroughfare met fences surrounding fields of artichokes, chilies, mushrooms, and lettuce.\n\nBell began to slow but had no choice but to pull the Locomobile half off the road and pass the hay wagon with only inches to spare, but he hadn't been left enough room for a free-and-clear passage. He took out a good thirty feet of a frail wooden fence, luckily without causing severe damage to the car. Only the front right fender was bent and twisted, scraping the tire when it hit a bump in the road. Bell did not look back to see the farmer shaking his fist and cursing him as his horses reared and nearly turned the wagon over on its side. Nor was he happy at being inundated by the dust storm that spewed from the Locomobile's drive wheels.\n\n\"That's one mad sodbuster,\" said Bronson, turning in his seat and looking behind him.\n\n\"He probably built and owns the fence we destroyed,\" Bell said with a sly grin.\n\nWithin ten miles, Soledad came into view. Named after the Mission Nuestra Se\u00f1ora de la Soledad that had been founded over a hundred years before, the town was a major railroad stop in the valley for transporting to market as quickly as possible the produce grown there. Bell quickly slowed as he entered town and soon found a garage where he could purchase gasoline for the Locomobile. While Bronson and the garage owner poured cans of fuel into the big tank, Bell wrestled with the crumpled right front fender, bending it back away from the tire. Then he took the tube Bronson had patched, inserted it back inside the tire, and remounted it on the wheel before bolting it on the rear of the Locomobile.\n\n\"You fellas the first car in a race passing through?\" the garage owner asked, clad in a pair of greasy coveralls.\n\nBell laughed. \"No, we're alone.\"\n\nThe garage owner looked at the dusty and damaged automobile and shook his head. \"You fellas must be in a mighty big hurry.\"\n\n\"That we are,\" said Bell, pressing bills that more than covered the price of the gasoline into the garage owner's hand.\n\nHe stood there, scratching his head, as the Locomobile roared away and quickly became a red speck down the main street of town before traveling out into the farm country. \"Them fellas must be crazy,\" he mumbled. \"I hope they know the bridge over Solvang Creek is out.\"\n\nFifteen minutes later and twenty miles down the road from Soledad, a sharp left-angled curve with a down slope came rushing toward them. A sign that stood beside the road flashed past.\n\n\"What did it say?\" asked Bell.\n\n\"Something about a bridge, was all I caught,\" replied Bronson.\n\nA barricade of railroad ties blocked the center of the road, and Bell could see the upper part of a bridge that looked as if it had broken apart in the middle. A crew of men were working to repair the center span while another crew were installing poles and restringing the telegraph and phone lines that had been torn away by a flash flood.\n\nBell jerked his foot off the accelerator, made a hard twist of the wheel. He jammed both feet on the brake, locking the rear wheels, fishtailing the rear end across the road, and causing the Locomobile to slow into a four-wheel drift. He straightened the front of the automobile with one second to spare and they flew through the air over the edge of the slope and dove down the steep bank of a broad ravine that had once been a dry wash. They landed in an explosion of dust less than twenty feet from a wide stream two feet deep that flowed toward the sea.\n\nThe heavy steel chassis and massive engine, driven by momentum, smacked into the water with an enormous eruption of silty brown water that burst over the Locomobile in a giant wave. The violent thump jarred Bell and Bronson in their every joint. Water gushed over the radiator and onto the hood before flooding over the men, drenching them in a deluge of liquid mud. Taking the full brunt of the surge, they felt as if they were driving through a tidal wave.\n\nThen the big automobile burst into clean air on the opposite shore, as it shuddered and shed itself free of the stream. Bell instantly jammed the accelerator to the floor, hoping against hope that the powerful engine would not drown and die. Miraculously, the spark plugs, magneto, and carburetor survived to do their job and kept the big four-cylinder combustion chambers hitting without a single miss. Like a faithful steed, the Locomobile charged up the opposite slope until it shot onto level ground again and Bell regained the road.\n\nWith great relief after their narrow brush with disaster, Bell and Bronson pulled off their goggles and wiped them clean of the mud and silt that splattered the lenses.\n\n\"It would have been nice if that garage guy had warned us,\" said Bronson, soaked by their ordeal.\n\n\"Maybe they're close-lipped in these parts,\" Bell joked.\n\n\"That was where the flash flood took out the phone and telegraph lines.\"\n\n\"We'll contact your counterpart in the Los Angeles office when we stop for gas again.\"\n\nThe road flattened out and appeared well maintained for the next ninety miles. Bell, with his ear tuned to any miss of the brawny engine's cylinders, let the Locomobile run as fast as he dared over the dirt-and-gravel road, thankful there were no sharp turns, and especially happy the tires held without going flat.\n\nFinally, his luck ran out when he hit a stretch of the road that was rock infested but worn smooth by eons of rain. He slowed to save the tires, but one became embedded with a sharp stone and hissed flat within a hundred yards. One of the spares was quickly thrown on the axle and, with Bronson patching the tube once again, Bell continued his mad dash toward Los Angeles.\n\nSan Luis Obispo and Santa Maria came and went. Then they dropped down in altitude as the road ran along the Pacific Coast. The ocean glittered blue under the sky, turning white as the breakers rolled onto the white sandy beach that was flecked with black rocks.\n\nOutside of Santa Barbara, they became airborne over a large hump in the road, crashing down on the other side with an impact that knocked the wind out of Bronson, who was amazed that the sturdy car held together without flying to pieces.\n\nThey entered Santa Barbara, where they refueled, filled the radiator with water, and installed the spare tire. A quick stop was made at the railroad depot, where Bronson sent a wire to his fellow agent Bob Harrington asking him to meet them at the Los Angeles railroad terminal.\n\nInstead of taking the treacherous winding road called the Grapevine over Tejon Pass before plunging down into Los Angeles, Bronson directed Bell to run the Locomobile along the railroad tracks that were laid with far more gradual turns. The rough ride strained the automobile's chassis as it rolled through the narrow pass below the 4,183-foot summit, but it held together until they reached the long slope leading down into the San Fernando Valley.\n\nAt last, the worst was behind them. Now they were in the homestretch, and the Locomobile was pressing hard and gaining on Cromwell's private train with every mile. According to Bronson's time estimate, they were only fifteen minutes behind. With luck, they just might reach the Los Angeles railroad terminal ahead of the Butcher Bandit.\n\nMost cheering was the sight of tall buildings in the far distance. As they neared the outskirts of the city, the traffic began to build. Bronson marveled at Bell's physical endurance. His blue eyes, hard and unblinking, never left the road. The man was born to sit behind the wheel of a fast car, Bronson thought. He looked at his watch. The hands on the dial read four-twelve. They had averaged over sixty miles an hour for the four-hundred-mile run.\n\nTraffic thickened the closer they came to the main part of the city and Bell began his now accustomed routine of swerving around horse-drawn wagons and buggies and automobiles. He was vastly relieved when the dirt road finally became paved with bricks. He raced in and around big red trolley cars that rode tracks down the center of the street. He was surprised by the number of automobiles he rushed past, unaware that there were over two thousand of them traveling the streets of the mushrooming city of one hundred twenty thousand.\n\nBell found the thoroughfares of the City of Angels were considerably wider than those of San Francisco, and he made good time with more room to negotiate around the traffic. They passed through downtown, heads turning in awe at the speed of the red Locomobile. A police officer blew his whistle and became angered when Bell ignored it and sped on. The policeman jumped on his bicycle and took up the chase but was soon left far behind, until the automobile was completely out of sight.\n\nThe big train depot came into view as Bell rounded a corner on two wheels. A man in a brown suit and wide-brimmed hat was standing on the curb at the entrance frantically waving his arms. Bell braked to a stop in front of Bob Harrington, the Van Dorn agent in charge of Southern California operations. At first, Harrington didn't recognize Bronson. The man in the mud-encrusted leather coat and helmet looked like an apparition until the goggles were raised.\n\n\"My God, Horace, I didn't recognize you,\" said the intense man with a tanned face and sharp features. At six foot five inches, Harrington towered over Bell and Bronson.\n\nBronson stiffly stepped to the pavement and stretched his aching muscles. \"I doubt if my own mother would know me.\" He turned and pointed at Bell, still sitting exhausted behind the wheel. \"Bob, this is Isaac Bell. Isaac, Bob Harrington.\"\n\nBell pulled off his driving glove and shook Harrington's hand. \"Good to meet you, Bob.\"\n\n\"I heard a lot about your exploits, Isaac. It's an honor to meet you.\"\n\nBell wasted no more time in pleasantries \"What's the status of Cromwell's private train? Are we in time to stop it?\"\n\nHarrington slowly shook his head. \"Sorry to have to tell you, but the regularly scheduled passenger train pulled off on a siding in Ventura and let it go through. When it came to Los Angeles, it bypassed the depot and took the express track south to San Diego. By doing that, it cut off nearly half an hour.\"\n\n\"How long ago?\" asked Bell, his hopes dashed.\n\n\"About twenty minutes.\"\n\n\"We would have beat it by ten minutes,\" Bronson observed morosely.\n\nBell looked at the tired Locomobile, wondering if there was enough left in her for the final dash. He knew, without looking in a mirror, that he was more exhausted than the automobile.\n\nHarrington studied the worn-out men. \"I can have my agents in San Diego apprehend Cromwell when his special train stops at the San Diego depot.\"\n\n\"He's too smart to get off at the depot,\" said Bell. \"He'll stop the train outside of town and enter in one of his many disguises.\"\n\n\"Where do you think he's headed?\"\n\n\"One of the local banks.\"\n\n\"Which one?\" queried Harrington. \"There are at least ten.\"\n\n\"The one with the most assets.\"\n\n\"You honestly believe a lone bandit will attempt robbing the San Diego Wells Fargo Bank?\" Harrington asked skeptically. \"It's the most secure bank in Southern California.\"\n\n\"All the more reason he'd attempt it,\" answered Bell. \"Cromwell loves a challenge.\"\n\n\"I'll telephone ahead and have my agents standing by the entrance.\"\n\nBell shook his head doubtfully. \"He'll spot them and call it off. Unless we can catch him in the act, we still haven't enough evidence to convict. And your agents have no idea what he looks like, and, if they did, they'd never see through his disguise. He's that good.\"\n\n\"We can't stand around and let him waltz into the bank unhindered,\" protested Bronson. \"He'll murder everyone inside.\"\n\nBell turned to Harrington. \"Tell your agents to close the bank until Horace and I get there.\"\n\n\"You're not continuing on to San Diego?\" Harrington asked incredulously.\n\n\"Yes,\" Bell said simply as he wearily climbed behind the wheel of the Locomobile. \"What's the fastest way out of town to the south?\"\n\n\"Just stay on the road that runs alongside the railroad tracks. It will take you straight south to San Diego.\"\n\n\"What's the condition?\"\n\n\"Well maintained all the way,\" said Harrington. He stared doubtfully at the tired machine. \"You should make good time if your automobile holds up.\"\n\n\"She got us this far,\" said Bell with a tight smile. \"She'll see us through.\"\n\n\"Tell your agents we're on our way,\" Bronson said tiredly. He looked like a man stepping up to the gallows.\n\nHarrington stood for a few moments watching the Locomobile roar down the road. Then he slowly shook his head and walked to the nearest telephone.\n\nTen minutes later, Bell reached the outer limits of the city and aimed the eagle ornament on the big brass radiator down the open road toward San Diego. Even after the wild ride from San Francisco, Bronson still marveled at Bell's expertise and mastery at timing the engine rpms and judging the precise moment to engage the clutch and grip the tall brass lever that meshed the unsynchronized gears.\n\nBell's weary mind was divided between his driving over the road ahead and the image of Jacob Cromwell robbing another bank and killing everyone in it. As they closed in on their destination, his nerves tightened and his blood churned with adrenaline while the faithful engine beat with the steadiness of a healthy pulse.\n\n## 33\n\nTHE TIME SPED AWAY SWIFTLY AS THE LOCOMOBILE ate up the one hundred twenty miles between the two cities in nine minutes under two hours. The last light was glimmering over the ocean to the west when they dropped down from Mount Soledad toward the heart of the city that opened up before them like a carpet of buildings tinted gold by the final rays of the setting sun. Though the Locomobile sported huge acetylene headlamps, Bell did not wish to take the time to stop the car and light them. \"How's our gas?\" Bell asked in a rasping voice, his mouth coated with dust.\n\nBronson turned in his seat, unscrewed the big gas tank cap, and dipped a stick down to the bottom. He withdrew the stick and stared at the moisture at its very tip.\n\n\"Let's just say we'll have to finish the race on fumes.\"\n\nBell nodded without answering.\n\nThe grinding strain had taken its toll on him. After hours of twisting the big steering wheel in a thousand gyrations to turn the stiff linkage to the front wheels, his arms felt numb, as if they were no longer a part of his body. His ankles and knees also ached from constant clutch, accelerator, and brake pedal action. And both his hands were blistered inside his leather driving gloves. Yet Bell drove at full throttle the last few miles, forcing the Locomobile to leap toward the final destination like a bear sprinting after an elk.\n\nThe Locomobile was badly worn down, too. The knobby tread on the Michelin tires was nearly shredded, the wheels were wobbling from the beating they had endured, the faithful engine was beginning to emit strange noises, and steam was billowing from the radiator cap. Still, the magnificent machine pushed on.\n\n\"I wonder what's in Cromwell's mind,\" said Bell. \"He's too late to commit robbery today. The bank is closed until tomorrow morning.\"\n\n\"This is Friday,\" answered Bronson. \"The banks in San Diego stay open until nine o'clock in the evening.\"\n\nThey were sprinting down India Street, parallel to the railroad tracks, with the depot no more than a mile away, when Bell flicked his eyes from the road for an instant and glanced in the direction of a train with only one car that was slowing to a halt.\n\nThe locomotive pulling the private Pullman car came to a stop on a siding four tracks over from the street. Smoke lazily rose from its stack as the engineer vented steam from exhaust tubes. The fireman had climbed on top of the tender, preparing to take on water from a large wooden tank. With the growing darkness, lights blinked on in the Pullman car, which was now parked a mile away from the depot and the city's downtown.\n\nBell knew immediately that this had to be Cromwell's private train.\n\nHe did not hesitate. He spun the wheel hard left and sent the Locomobile bouncing wildly across the railroad tracks. By the time he had bounded over three tracks, he had blown all four of the badly worn tires and rolled the rest of the way up to the train on the rims of the wheels, showering sparks like meteors as they smashed against the steel rails.\n\nBronson said nothing. He had been frozen in confused shock, until he saw the train and realized what Bell had up his sleeve. Excitement grew to elation at knowing that, after their five-hundred-mile daredevil drive, they had finally come within spitting distance of their goal.\n\nBell slammed the Locomobile to a stop across the tracks in front of the locomotive. Its momentum finally spent, the battered automobile sat forlornly with its overheated engine crackling, its radiator hissing steam, and the smell of shredded tires. Its mad and wild chase had come to a fitting climax in front of the quarry it had pursued through the backwoods of hell.\n\n\"We may be jumping the gun,\" said Bronson. \"He hasn't attempted to rob the bank yet. We can't arrest him without a crime.\"\n\n\"Maybe. But on the drive down here from San Francisco, I had much to think about. Better we take Cromwell now, before he has time to act. If he sees through our trap again, we're lost. I'll worry about gathering enough evidence to convict him later. Besides, he's not on home ground. He can't call in expensive attorneys to get him out on bond.\"\n\nBell was well aware that no one had had the time to exit the train during the few minutes since it had come to a standstill. He climbed from the automobile and walked unsteadily toward the Pullman car, the aches and pain and weariness slowly falling away. He halted abruptly, and slipped between the Pullman and the coal tender, as two stewards wrestled a motorcycle from the car to the ground beside the track.\n\nHe waited patiently for a few minutes until a man dressed in the uniform of a railroad conductor stepped from the Pullman car and threw one leg over the seat of the motorcycle that Bell recognized as a Harley-Davidson. The man's back was to Bell as Bell stepped silently alongside the Pullman car and stopped only when he was no more than five feet behind the man, who was leaning down to open the fuel valve to the carburetor in preparation for starting the engine.\n\n\"The Harley is a good machine,\" Bell said calmly, \"but I prefer the Indian.\"\n\nThe man on the motorcycle froze at the sound of the familiar voice. He slowly turned and saw an apparition standing behind him. Eerie illumination fell from overhead electric lights along the railroad siding. The figure wore a short leather coat over jodhpurs and boots that looked like they had been dragged through a swamp. A pair of goggles was pushed back on his head, revealing strands of blond hair coated in dried mud. But there was no mistaking the face, the piercing eyes, and the begrimed mustache that covered the upper lip.\n\n\"You!\"\n\n\"Not very original,\" Bell said cynically. \"But since I used the same expression at the bank in Telluride, I won't criticize.\"\n\nA silence came over the two men that seemed to last a lifetime, but it was only the few seconds that it took Cromwell to see that the apparition really was Isaac Bell. Cromwell just stood there in growing disbelief, his face suddenly turning pale.\n\n\"You were dead!\" he gasped. \"I shot you!\"\n\n\"Twice, as a matter of fact,\" said Bell with a hard edge to his voice. His right hand gripped the 1905 Colt .45 automatic, its muzzle aimed squarely between Cromwell's eyes and held as steady as an iron bar in concrete.\n\nFor the first time in his life, Jacob Cromwell was taken completely off guard. His agile mind, filled with overconfidence, had never considered how he would act should the time ever come when he was apprehended. The unthinkable was never dwelled upon. He had always thought of himself as untouchable. Now he stood face-to-face with his archenemy, who should have been dead. He felt like a captain whose unsinkable ship had run up on the rocks.\n\nCromwell's Colt .38 was in his coat pocket, but he knew Bell would blow his brains out before he could reach for it. Slowly, he lifted his hands into the air in abject defeat.\n\n\"What happens now?\" he asked.\n\n\"I'm going to borrow your special train to take you back to San Francisco. There, I'll turn you over to the police, until such time as you're tried for murder and hung.\"\n\n\"You have it all mapped out.\"\n\n\"The day had to come, Cromwell. You should have quit when you were ahead.\"\n\n\"You can't arrest me. I have committed no crime.\"\n\n\"Then why are you disguised as a railroad conductor?\"\n\n\"Why don't you shoot me now and get it over with?\" Cromwell asked, his composed arrogance coming back on keel.\n\n\"A mere slap on the hand for your crimes,\" Bell said caustically. \"Better you have plenty of time to think about the hangman's noose tightening around your murdering neck.\"\n\nBronson came from around the rear of the Pullman car, his Smith & Wesson double-action .44 revolver drawn and pointed at Cromwell's chest. \"Nice going, Isaac. You nabbed our friend here before he could commit another crime.\"\n\nBell handed Bronson a pair of Tower nickel-plated, double-lock handcuffs. The agent wasted no time in snapping them on Cromwell's wrists. Then he gave the bandit a thorough search and found the .38 Colt automatic.\n\n\"The weapon you used to commit three dozen murders,\" Bronson said with a cold voice.\n\n\"Where did you come from?\" Cromwell demanded at seeing Bronson and knowing with certainty that these men would not hesitate to shoot him if he gave the slightest indication of trying to escape.\n\n\"Isaac drove us from San Francisco in his automobile,\" he answered as if it were an everyday event.\n\n\"Impossible!\" snorted Cromwell.\n\n\"I thought so, too,\" said Bronson, leading Cromwell up the steps into the Pullman car, where he took his own handcuffs, placed them around Cromwell's ankles, and roughly shoved him onto a couch.\n\nBell walked back up the track and stared sadly at the mauled Locomobile. A barrel-chested man carrying an oil can in the coveralls and denim striped hat of a locomotive engineer came up behind him and stared dumbly at the automobile.\n\n\"How in God's name did that derelict come to be on the tracks in front of my engine?\"\n\n\"It's a long story,\" said Bell wearily.\n\n\"What's going to happen to it?\"\n\nBell spoke quietly, almost reverently: \"It's going to be shipped back to the factory in Bridgeport, Connecticut, where it will be rebuilt until it is as good as new.\"\n\n\"Fix this wreck?\" said the engineer, shaking his head. \"Why bother?\"\n\nBell looked at the Locomobile with a loving expression in his eyes and said, \"Because she deserves it.\"\n\n## 34\n\nYOU'RE A FOOL IF YOU THINK YOU CAN GET AWAY with kidnapping me,\" Cromwell stated contemptuously. \"You have no authority to arrest me without a warrant. As soon as we get back to San Francisco, my attorneys will demand my release. After making fools of the Van Dorn Detective Agency, I shall walk free as a bird. Then I'll launch a series of lawsuits that will break your agency and drown it in a sea of scandal.\"\n\nCromwell sat manacled to a large couch in the center of the parlor car. His wrists, legs, and even his neck were encased in steel bands that were chained to tie-down rings on the floor of the forward baggage section of the car. No chances were taken. Four heavily armed Van Dorn agents from the Los Angeles office sat in the car less than ten feet from the bandit, sawed-off shotguns, loaded and cocked, laid across their knees.\n\n\"You may have a chance to demonstrate your arrogant ego with your pals in city hall, my friend,\" said Bell. \"But you'll walk free only as far as a pig to a butcher shop.\"\n\n\"I am an innocent man,\" said Cromwell matter-of-factly. \"I can prove I was nowhere near the bank robberies you accuse me of. Where is your evidence? Where are your witnesses?\"\n\n\"I'm a witness,\" Bell answered. \"I saw through your disguise as a woman in Telluride before you shot me.\"\n\n\"You, Mr. Bell? What jury in San Francisco would buy your testimony? The trial will be a farce. You have nothing to bring an indictment, much less conviction.\"\n\nBell gave Cromwell a foxlike smile. \"I am not the only witness. There are other people in the towns where you committed your murders who can identify you.\"\n\n\"Really.\" Cromwell leaned back in the couch as if he hadn't a care in the world. \"From what I read of the Butcher Bandit, he always used disguises during his crimes. How can he be identified?\"\n\n\"You'll have to wait and see.\"\n\n\"I have great influence in San Francisco,\" Cromwell said with total conviction. \"I have contributed heavily to the election of every superior and federal court judge on the bench. They owe me. Same with the good citizens of San Francisco. Even if you could bring me to trial, no jury of my peers will convict me, not when they take into account the many thousands of dollars I've spent on their behalf.\"\n\n\"You're betting your hand before you see it,\" said Bell. \"A federal judge will be sent out from Washington to hear your case and the venue will be moved elsewhere, where you're not the city's darling.\"\n\n\"I can afford the finest attorneys in the country,\" Cromwell continued haughtily. \"No jury, regardless of what judge sits on the bench, will ever sentence me for crimes with so little evidence, certainly not with my reputation as a man who is beloved by the poor and homeless of San Francisco.\"\n\nBronson's face was clouded with disgust. It took all his willpower not to plant his fist in Cromwell's face. \"Tell that to the families of the victims you shot down in cold blood. Tell them how the money you stole went to give you a lavish lifestyle as a banker in a mansion on Nob Hill.\"\n\nCromwell smiled brazenly and said nothing.\n\nThe train began to slow. Bronson stepped over to a window and peered out. \"We're coming into Santa Barbara. The engineer will probably stop to take on water.\"\n\n\"I'd like to get off at the depot,\" said Bell. \"There's a little matter I'd like to take care of.\"\n\nAs soon as the train came to a stop, Bell jumped down the stairs to the platform and quickly disappeared into the depot. Ten minutes later, as the engineer tooted the whistle warning that he was going to engage the drive wheels, Bell trotted out and climbed back aboard the Pullman car.\n\n\"What was that all about?\" asked Bronson.\n\nCromwell immediately suspected something that was not to his liking. He shifted in his chair and leaned forward to listen.\n\n\"The phone lines have been repaired over the ravine where the flash flood went though,\" Bell answered Bronson. Then he looked down at Cromwell with a sardonic grin. \"I put a call through to the Van Dorn office and instructed our agents to take your sister into custody as an accomplice.\"\n\n\"You're insane,\" Cromwell cried out.\n\n\"I think we can prove she is implicated in the murders carried out by the Butcher Bandit.\"\n\nCromwell surged up from the couch, his face a mask of loathing and hate, but was stopped dead by his chains. \"You dirty swine,\" he hissed. \"Margaret had nothing to do with any of this. She knew nothing about my...\" He hesitated, before he incriminated himself. He slowly lowered himself back onto the couch, his composure and presumptuous behavior regaining control. \"You'll pay dearly for involving an innocent woman in your ridiculous accusations. Margaret will be back in her parlor within an hour after she's falsely accused of crimes she knows nothing about.\"\n\nBell stared into Cromwell's eyes with the self-assurance of a panther about to take a bite out of an antelope. \"Margaret will talk,\" Bell said firmly. \"She will tell what she knows in an effort to save her brother. She'll lie, of course, but she'll be tripped up on a thousand details she can't answer. Margaret will be the witness who will unwittingly lead you to the gallows.\"\n\n\"Even if I was guilty, Margaret would never utter a single word against me,\" Cromwell said with conviction.\n\n\"She will if she knows she's going to jail for the rest of her natural life. That, and the loss of a luxurious lifestyle. Turning state's evidence will be quite simple if there is a heavy price to pay for not doing so.\"\n\n\"You've badly underestimated Margaret.\"\n\n\"I don't think so,\" said Bell quietly.\n\nCromwell smiled tightly. \"You'll never connect Margaret with the crimes any more than you can convince a jury that I am guilty.\"\n\nBell stared at the banker. \"Are you guilty?\"\n\nCromwell laughed and nodded around the parlor car. \"Admit to being your Butcher Bandit in front of witnesses? Come now, Bell.\" There was no \"Mr.\" this time. \"You're skating on thin ice and you know it.\"\n\nThen Bell pulled off the glove on Cromwell's left hand and revealed a metal tube where his finger once extended.\n\n\"We'll see,\" Bell mused aloud. \"We shall see.\"\n\nBELL WAS taking no chances. When they reached San Francisco, he ordered the engineer to bypass the main depot and head onto the siding of the railyard. Bronson had a small army of agents on hand to escort Jacob Cromwell to an ambulance, where he was tied down to a stretcher, for the ride through the city.\n\n\"We can't run the risk of putting Cromwell in the county jail,\" said Bell. \"He's right about his friends springing him within an hour. Take him across the bay to the state prison at San Quentin. We'll keep him on ice until we're ready to bring formal charges.\"\n\n\"Every reporter with every newspaper in town will be on hand to report that event,\" said Bronson.\n\n\"They'll send the story across the country by telegraph to every newspaper from here to Bangor, Maine,\" Bell said with a grin. \"Now all we have to do is keep him from slipping through our fingers. Cromwell will attempt to bribe any guard that comes near him.\"\n\n\"I know the warden at San Quentin,\" said Bronson. \"He's as straight as an arrow. Cromwell will be wasting his breath if he thinks he can bribe him into escaping.\"\n\n\"Don't think he won't try.\" Bell looked at Cromwell as he was roughly lifted into the ambulance. \"Put a hood over his head so no one will recognize him. Swear the warden to secrecy, and have him lock Cromwell in solitary confinement, away from the other prisoners. We'll give the warden the necessary paperwork in the morning.\"\n\n\"What about Margaret? I doubt a judge with his hand in Cromwell's pockets would fill out arrest papers for her.\"\n\n\"Go through the motions,\" Bell instructed. \"Put pressure on her. Once she knows her brother is in custody and that she may go down with him, I'm betting Margaret will gather up all the cash she can and make a run for it. Then she'll sail right into our hands.\"\n\nBefore heading for Bronson's office, Bell stopped off at a telegraph office and sent a lengthy wire to Van Dorn reporting the capture of the notorious Butcher Bandit. He also asked for whatever help Colonel Danzler could offer from the federal government.\n\nCROMWELL WAS right about one thing. Margaret walked out of the police department less than thirty minutes after she was escorted there by two Van Dorn agents. Cromwell's attorneys were already there arranging bond when she arrived. Even her chauffeur was on hand to drive her home, waiting in the Rolls-Royce out front, parked in a zone where no vehicle was allowed. A court magistrate miraculously appeared to sign the necessary release papers. It seemed to a reporter, who happened to be present covering a burglary case, that Margaret's arrest and almost-instant release were a staged formality.\n\nMeanwhile, Bronson and his agents had driven the ambulance carrying Cromwell onto the ferry that took them across the bay to Marin County. After moving off the dock, they drove to the state prison at San Quentin. As Bronson had claimed, the warden was very cooperative and even proud to have the famous Butcher Bandit in his prison until Bell and Bronson could orchestrate an arraignment.\n\nAfter Bell left the telegraph office, he walked to Cromwell's bank. He took the elevator up to the main office and approached Marion's desk. \"Get your hat,\" he said without preamble in a no-nonsense tone. \"You're taking the rest of the day off.\"\n\nShe faltered, taken completely off balance by his sudden appearance after three days. Her sensual feelings toward him came flooding back. She could see that there was no arguing with him, yet she said, \"I just can't leave when I feel like it. I could lose my job.\"\n\n\"Your job is already lost. Your boss is behind bars.\" He walked around her desk and pulled her chair out so she could stand.\n\nShe rose slowly and stared at him, dazed. \"What are you saying?\"\n\n\"The show is over. I'm holding Cromwell until we obtain the necessary warrant for his arrest and documents for an indictment.\"\n\nAlmost as if she were moving in a fog, she retrieved her hat and purse from a cabinet behind her desk and then stood there unsure of what else to do. Her eyes slipped away and she stared hesitantly at the floor, disbelieving. She had never thought it possible that Jacob Cromwell, regardless of his crimes, was vulnerable.\n\nBell had seen Marion's cheeks blush before and he was always taken by the demure reaction. He took her hat from her fingers and placed it on her head at a jaunty angle. \"I like that,\" he said, laughing.\n\n\"Well, I don't,\" she said with womanly irritation in her voice as she straightened the hat to its proper position on her lovely head of hair. \"Where are you taking me?\"\n\n\"Down to the beach, where we can walk in the sand and have a long talk about recent events.\"\n\n\"Are we taking your fancy automobile?\"\n\nShe was surprised at the pained look that crossed his face. \"I'm afraid we won't be taking it anywhere anytime soon.\"\n\n## 35\n\nCONSTRUCTION ON SAN QUENTIN PRISON BEGAN auspiciously on Bastille Day, July 14, 1852. Why it was later named after a notorious inmate serving time for murder whose name was Miguel Quentin is anybody's guess. The term San is Spanish for male saint. Quentin was no saint, but his name stuck, and the prison became known as San Quentin.\n\nThe oldest state prison in California, it held its first execution in 1893 by hanging Jose Gabriel for murdering an aged couple he worked for. Women were also confined there, in a separate building. By 1906, over a hundred prisoners had died behind the prison walls, from inmate murders to suicides to death from natural causes. They were buried in the cemetery outside the prison walls.\n\nRichard Weber, the warden, was a big man, agile as a gymnast and energetic, a workaholic who was dedicated to his job. Heavyset but solid as a rock, he wore a perpetual grin that ever so slightly curled the corners of his lips. A strict disciplinarian with a strong approach to reform, he put the prisoners to work making products, working the gardens, and joining a number of educational studies. His program of compensating the inmates in a small way, along with rewarding them with reductions in their sentences, enhanced his reputation as the \"Tough But Fair Warden.\"\n\nBronson was close to the mark in claiming Weber could not be bribed. He was known as a man far above the taint of corruption or graft. A devout Catholic, Weber and his wife had raised a family of eight children. His salary as chief of the state's largest prison facility was ample but left little for extra niceties. His dream of retiring someday to a ranch in the San Joaquin Valley was only that, a dream.\n\nThough it was often said that every man has his price, all who knew Warden Weber thought of him as untouchable. But, as it turned out, beneath that hard veneer of integrity he was only human.\n\nSoon after Cromwell was locked up in solitary confinement, Weber visited the bandit\/banker in his small cell two levels below the main prison building. After ordering the guard to unlock the steel door, Weber entered the cell and sat down on a small folding chair he had brought with him.\n\n\"Mr. Cromwell,\" he said politely, \"welcome to San Quentin.\"\n\nCromwell rose from his bunk and nodded. \"Perhaps I should say I'm grateful for your hospitality, but that would be a lie.\"\n\n\"It's my understanding that you'll only be with us for a short time.\"\n\n\"Until I'm arraigned in federal court,\" said Cromwell. \"Is that what Bronson of the Van Dorn Detective Agency told you?\"\n\nWeber nodded. \"He said he was waiting for instructions from the Criminal Investigation Department in Washington.\"\n\n\"You know why I was arrested?\"\n\n\"I was told you were the notorious Butcher Bandit.\"\n\n\"Are you familiar with my status in the community?\" asked Cromwell.\n\n\"I am,\" replied Weber. \"You own the Cromwell Bank and are an admired philanthropist.\"\n\n\"Do you think such a man could rob banks and kill dozens of people?\"\n\nWeber shifted on his seat. \"I must admit I find the idea a bit far-fetched.\"\n\nCromwell circled for the kill. \"If I gave you my word that I did not commit any crimes and these are false charges by the United States government to take over my bank, would you release me?\"\n\nWeber thought a moment, then shook his head. \"I'm sorry, Mr. Cromwell, I am not authorized to release you.\"\n\n\"Even though formal charges have not been filed?\"\n\n\"I have been assured that charges are being filed as we speak.\"\n\n\"If I guarantee that I do not intend to escape but need go directly to my attorneys in the city and obtain the necessary release papers from a court magistrate, then would you allow me to leave the prison?\"\n\n\"I might if I could,\" said Weber. \"But, as warden, I cannot permit you leaving the prison grounds before the release papers are in my hands. Besides, there are Van Dorn agents patrolling outside the prison walls to prevent you from escaping.\"\n\nCromwell looked around the concrete, windowless cell with its steel door. \"Has any inmate ever escaped from solitary?\"\n\n\"Not in the history of San Quentin.\"\n\nCromwell paused to lay his trap. \"Suppose\u2014just suppose, Warden\u2014that you personally took me into San Francisco?\"\n\nWeber looked at him with interest. \"What do you have in mind?\"\n\n\"Deliver me to County Prosecutor Horvath's office and fifty thousand dollars in cash will be delivered to your house on the prison grounds by private messenger precisely one hour later.\"\n\nWarden pondered Cromwell's offer for several moments. He knew it was not an idle offer. The banker was worth many millions of dollars and the offer was in cash, which would leave no trail should law enforcement investigators come sniffing around. Fifty thousand dollars was an enormous sum. He could keep the money hidden until his retirement. Weber also did his arithmetic and knew that it was more than enough to buy him a ranch second to none in the state. It was an offer even an honest man of integrity could not refuse.\n\nFinally, Weber rose from his chair, stepped to the steel door, and rapped three times. The door opened and the uniformed guard entered. \"Put a hood over the prisoner's face and take him to the office behind my house. I'll be waiting there.\" Then he turned and left the cell.\n\nTen minutes later, the guard pushed Cromwell into Weber's office. \"Remove his hood and manacles,\" Weber instructed. As soon as the hood was off and the manacles around Cromwell's feet and hands removed, the guard was dismissed.\n\n\"I trust I can rely on your word as a gentlemen that my compensation will arrive an hour after I safely deposit you on the steps of the city hall?\"\n\nCromwell nodded solemnly. \"You can rest assured, the money will be in your hands this afternoon.\"\n\n\"Good enough.\" Weber rose and walked to a closet. He returned with a woman's dress, hat, purse, and shawl. \"Put these on. You are a small man and about the same size as my wife. You will be disguised as her when we drive through the inner gates and the main gate. Keep your head down and the guards will take no notice. She and I often take drives around the countryside or into town.\"\n\n\"What about Van Dorn's agents who are patrolling the outer walls?\"\n\nWeber smiled thinly. \"I am the last man they would suspect of foul play.\"\n\nCromwell looked at the clothes and laughed.\n\n\"Something funny?\" asked Weber.\n\n\"No,\" replied Cromwell. \"It's just that I've been here before.\"\n\nWhen Cromwell had slipped on the warden's wife's clothes, he wrapped the shawl around his neck and pulled the hat down so it would cover the beard that was beginning to stubble his chin. \"Ready as I'll ever be,\" he announced.\n\nWeber led him out of the office across a yard to the garage that housed the warden's Ford Model T automobile. Cromwell effortlessly cranked over the engine and climbed behind the wheel. The car began rolling over the gravel road toward the inner gates and was passed through with a wave from the warden. The main gate was another story. Here, two guards approached the warden for his personal authority to open the gate. \"Shari and I are running into the city to buy a gift for her sister's birthday,\" he said placidly.\n\nThe guard on the left side of the car dutifully gave the warden a salute and waved him on. The guard on the right gazed at Cromwell, who made a show of looking for something in the purse. The guard dipped his legs to look under her hat, but Weber caught the movement and snapped, \"Stop gawking and open the gate.\"\n\nThe guard straightened up and waved to the engineer in the tower who controlled the mechanism that opened the massive steel doors. As soon as they spread wide enough to permit the Ford through, Weber pulled down the throttle lever and raised his foot off the high-gear pedal. The automobile jumped forward and was soon chugging down the road toward the landing to board the ferry for San Francisco.\n\n## 36\n\nHE WHAT?\" BELL ROARED OVER THE TELEPHONE.\n\n\"What is it?\" asked Bronson, coming into the office as Bell hung up the phone.\n\nBell looked up at him, his face twisted in rage. \"Your friend, the righteous and incorruptible warden of San Quentin, released Cromwell.\"\n\n\"I don't believe it,\" Bronson blurted in utter disbelief.\n\n\"You can believe it, all right!\" snapped Bell. \"That was Marion Morgan, Cromwell's personal secretary. She said he walked into his office five minutes ago.\"\n\n\"She must be mistaken.\"\n\n\"She's right on the money,\" said Curtis from the doorway. He looked at Bronson. \"One of your agents who was following his sister, Margaret, saw him come down the steps of the city hall and get in her automobile.\"\n\n\"Warden Weber taking a bribe,\" Bronson muttered. \"I would have never thought it.\"\n\n\"Cromwell probably offered him a king's ransom,\" said Bell.\n\n\"My agents at the prison reported that Weber left in his automobile with his wife for a shopping trip to the city.\"\n\n\"Not the first time Cromwell disguised himself as a woman,\" Bell murmured angrily. \"He no doubt shed the dress once they were out of sight of San Quentin and before they reached the ferryboat.\"\n\n\"Where does that leave us?\" inquired Curtis.\n\n\"I telegraphed Colonel Danzler, chief of the United States Criminal Investigation Department. He's arranging for a federal judge to swear out a warrant for Cromwell's arrest that cannot be overridden by city or state judicial system. As soon as it is in our hands, we can take Cromwell out of circulation for good.\"\n\n\"That will take at least four days by rail,\" said Bronson. \"What if he attempts to flee the country? We have no legal means to stop him.\"\n\n\"We had no legal means to grab him in San Diego,\" retorted Bell. \"We'll snatch him again and keep him on ice in a secret location until the paperwork arrives.\"\n\nBronson looked doubtful. \"Before we can put our hands on Cromwell again, his pals the mayor, police chief, and county sheriff will protect him with an army of policemen and deputies armed to the teeth. My seven agents will be outnumbered twenty to one if they attempt to seize him.\"\n\n\"Cromwell has that kind of influence?\" asked Curtis.\n\n\"The degree of corruption in San Francisco makes the Tammany Hall political machine of New York City look like a convent,\" said Bronson. \"Cromwell has done more than his share to keep city officials fat and rich.\"\n\nBell smiled a hard, canny smile. \"We'll have our own army,\" he said quietly. \"Colonel Danzler will call out the army regiment that's stationed at the Presidio, if I request it.\"\n\n\"We may need them sooner than you think,\" said Bronson. \"If Cromwell cleans out the cash in his bank and charters another train, he'll be over the border into Mexico free as a bird before we can lift a finger.\"\n\n\"He's right,\" agreed Curtis. \"As it stands, we're helpless. We can't touch him. By the time Danzler can contact the Presidio's commander and order troops called out and marched into the city, it will be too late. Cromwell's graft will have greased his way out of town.\"\n\nBell leaned back in his chair and stared at the ceiling. \"Not necessarily,\" he said slowly.\n\n\"What's going through that devious mind?\" asked Curtis.\n\n\"Suppose the president of the United States requests the president of the Southern Pacific Railroad not to charter a train to Cromwell?\"\n\nBronson looked at him. \"Is that possible?\"\n\nBell nodded. \"Colonel Danzler has great influence in Washington. I was told by Van Dorn that he and President Roosevelt are very close. They fought side by side at San Juan Hill in the war. I think it's safe to say he could persuade the president to go along.\"\n\n\"And if Cromwell charters a ship?\" probed Bronson.\n\n\"Then a United States warship will be sent to stop the ship at sea, take Cromwell off, and return him to San Francisco. By that time, we'll have the necessary warrants for his arrest and trial.\"\n\n\"It sounds like you have all the bases covered,\" Bronson said admiringly.\n\n\"Cromwell is a slippery customer,\" said Bell. \"If there is a way to slip through our net, he'll think of it.\" He paused to look up at a clock on the wall. \"Four thirty-five. I have a dinner date at six o'clock.\"\n\n\"Marion Morgan?\" Curtis asked with a sly smile. \"It strikes me that besides her keeping tabs on Cromwell, you two have a thing going.\"\n\nBell nodded. \"She's an exquisite lady.\" He rose to his feet and slipped on his coat. \"She's fixing dinner at her place.\"\n\nBronson winked at Curtis. \"Our friend is a lucky man.\"\n\n\"I've lost track of time,\" said Bell. \"What day is it?\"\n\n\"Tuesday, April seventeenth,\" answered Curtis. Then he added humorously, \"The year is 1906.\"\n\n\"I'm aware of the year,\" Bell said as he stepped through the door. \"See you all in the morning.\"\n\nSadly, one of the three men in the room would never see tomorrow.\n\nMARGARET STOPPED the Mercedes under the porte cochere that sheltered vehicles at the front door of the mansion before they passed into the courtyard beyond. Since picking her brother up in front of city hall, she had driven him to the bank, where he had spent two hours locked in his office. When he emerged, they rode to Nob Hill in silence. Their chauffeur came from the carriage house and drove the car inside. The instant they stepped into the foyer, Margaret pulled off her hat and spun it across the floor, glaring at her brother with fire in her eye.\n\n\"I hope you're satisfied now that you've sent our fortunes crashing down around us.\"\n\nCromwell walked like an old man into the sitting room and slumped wearily in a chair. \"I made the mistake of underestimating Bell,\" he said. \"He caught up to me before I could hit the bank in San Diego.\"\n\nThe floor tilted beneath Margaret's feet as her entire mood changed. \"Isaac alive? You saw him?\"\n\nHe looked at her intently. \"You appear to take an uncommon interest in him,\" he said with dry amusement. \"Are you glad our nemesis still walks the earth?\"\n\n\"You said you killed him in Telluride.\"\n\nCromwell spoke as if he were describing a truckload of coal. \"I thought I did, but he apparently survived. The only mistake I've made in twenty years.\"\n\n\"Then it was he who brought you back from San Diego and put you into San Quentin.\"\n\nCromwell nodded. \"He had no right. He stepped outside the law. Now Bell is going to move heaven and earth to proclaim me the Butcher Bandit and send me to the gallows.\"\n\n\"It won't be an easy matter of escaping the city. Van Dorn agents are watching our every move.\"\n\n\"I have no intention of fleeing like a thief in the night. It's time those who have curried our favor and funds repay their obligations by keeping us out of Van Dorn's hands until we're ready to quietly depart for greener pastures.\"\n\nShe looked at him resolutely, her mind on an unwavering course of action. \"We'll hire the finest lawyers in New York. It will be impossible to convict you. We'll make Isaac Bell and the Van Dorn Detective Agency the laughingstock of the nation.\"\n\n\"I don't doubt we'll win in court,\" he said quietly, staring at his sister with a serious expression. \"But we'll be finished as an admired institution in San Francisco. The bank will suffer a financial disaster as our depositors, fearful of scandal, run to competing banks. The Cromwell National Bank will close its doors.\" He paused for effect. \"Unless...\"\n\n\"Unless what?\" she asked, meeting his unrelenting gaze.\n\n\"We quietly and secretly move our assets to another city in another country where we can launch a new financial empire under another name.\"\n\nMargaret visibly relaxed now that she began to realize that all was not lost, her lifestyle might not fall off the edge of the cliff after all. \"What city and what country did you have in mind? Mexico? Brazil, perhaps?\"\n\nCromwell grinned wickedly. \"My dear sister, I can only hope Mr. Bell thinks as you do.\"\n\nHe felt smug with self-satisfaction, believing that all he needed was no more than three hours in the morning to arrange for the shipment of the cash reserves from his bank. His paper assets had already been sent out of the country by telegraph when he went to the bank. Now all he and Margaret had to do was pack a few things and lock up the house, leaving it with a realtor to sell. Then it would be clear sailing, once they crossed the border and left the United States behind.\n\nBELL SAT staring thoughtfully at a small fire in the fireplace of Marion's apartment while she was busy in the kitchen. He had brought a bottle of California Beringer 1900 Cabernet Sauvignon and was halfway through a glass when Marion entered the dining room and began setting the table. He looked up and had a strong desire to walk over and press his lips to hers.\n\nShe looked stunning, with her fashionable hourglass silhouette of ample curves and full breasts. She wore a pink satin bodice of cascading lace that reached up under her chin and elongated her tall, graceful neck. The skirt was also pink and long and flowing like an inverted flowering lily. Even with her torso half covered by a large apron, she looked elegant.\n\nHer straw blond hair shined under the light from the candles on the table. It was pulled back in a silky bun like a whorl behind her delicate ears. Bell suppressed his desire to kiss her and simply reveled at the sight of her.\n\n\"Nothing fancy,\" she said, coming over and sitting on the arm of his chair. \"I hope you like pot roast.\"\n\n\"I have a passion for pot roast,\" he said, losing all control and pulling her down onto his lap, where he kissed her long and ardently. She tensed, then trembled, and her eyes became huge and flashed a deep sea of green. As they drew apart, her poise altered. The eyes turned brazen and her expression spicy. Her breathing became quick, and she enjoyed the sensation of deep sensuality, a sensation that she had never experienced with another man. With slow deliberation, she eased out of his lap and stood shakily, brushing back a wisp of hair that had fallen along her temple.\n\n\"Enough of this, unless you want a burned pot roast.\"\n\n\"How long do I have to suffer on an empty stomach?\"\n\nShe laughed. \"Ten more minutes. I'm waiting for the potatoes to soften.\"\n\nHe watched as she returned to the kitchen, her walk as fluid as a gazelle's.\n\nAs she set the bowls on the table, he refilled their glasses, and they sat down. They ate in silence for a few moments. Then Bell said, \"Everything is delicious. You'll make some lucky man a wonderful wife one day.\"\n\nThe words came like a warm breeze across the nape of her neck and a flush of blood flowed across her breasts, hardening the nipples. Deep down, she hoped his feelings were moving in that direction, but she was also afraid that his affection might cool and he would walk off into the dark some evening, never to return.\n\nBell read Marion's confusion and became afraid to go there. He changed the course of the conversation. \"How long did Cromwell remain in the bank today?\"\n\nHer emotion quickly turned to anger. She was mad at herself for responding with the proper words instead of calmly expressing her feelings toward him. \"Most of the time he spent in the office, he seemed very secretive. He also made three trips downstairs to the vault.\"\n\n\"Do you have any idea of what he was about?\"\n\nShe shook her head. \"It seemed very mysterious.\" Then she lifted her head and a small smile parted her lips. \"But when he was in the vault, I sneaked into his office and glanced at the paperwork he had spread across his desk.\"\n\nHe waited expectantly as she took a few moments to let him twist in the wind, as if getting even with him for ignoring her feelings for him. \"He was filling out bank drafts and money transfers.\"\n\n\"It figures. Our guess is that he and Margaret are going to skip out of the country and move the bank's funds to their destination. There's no way Cromwell will stay in town and fight us in federal court.\"\n\n\"It would look that way,\" said Marion quietly, wishing they could keep their time together more close and personal.\n\n\"Could you tell where he was sending the bank's funds?\"\n\nShe shook her head. \"Only the amounts were filled in, not the banks that were to receive them.\"\n\n\"What do you think he was doing in the vault?\"\n\n\"My best guess is he was packing the bank's cash reserves in crates in preparation to ship them to a bank in whatever city they're going to.\"\n\n\"You're a very astute lady,\" he said, smiling. \"And if you were Jacob and Margaret, where would you go?\"\n\n\"They wouldn't be safe anywhere in Europe,\" Marion answered without hesitation. \"The banks on the Continent work with the U.S. government in freezing illegal funds. There are too many other countries where they could hide their money and begin building their empire again.\"\n\n\"How about Mexico?\" Bell asked, impressed with Marion's intuition.\n\nShe shook her head. \"Margaret could never live in Mexico. The land is too primitive for her tastes. Buenos Aires in Argentina is a possibility. The city is very cosmopolitan, but neither of them speaks a word of Spanish.\"\n\n\"Singapore, Hong Kong, Shanghai,\" suggested Bell. \"Any of those cities hold any interest?\"\n\n\"Australia or New Zealand, perhaps,\" she said thoughtfully. \"But I've learned over the years in his employ that Jacob doesn't think like most men.\"\n\n\"My experience with the man has led me to the same conclusion,\" Bell said.\n\nMarion went quiet as she passed him more helpings of the pot roast, potatoes, and vegetables. \"Why don't you give your brain a rest and enjoy the fruits of my labors?\" she said, smiling.\n\n\"Forgive me,\" he said honestly. \"I've been a bore as a dinner companion.\"\n\n\"I hope you like lemon meringue pie for dessert.\"\n\nHe laughed. \"I adore lemon meringue pie.\"\n\n\"You'd better. I baked enough for a small army.\"\n\nThey finished the main course and Isaac stood up to help clear the table. She pushed him back down in his chair.\n\n\"Where do you think you're going?\" she demanded.\n\nHe looked like a young boy startled by his mother. \"I wanted to help.\"\n\n\"Sit down and finish your wine,\" Marion said smartly. \"Guests don't work in my house, especially male guests.\"\n\nHe looked at her slyly. \"And if I wasn't a guest?\"\n\nShe turned away from him for fear her inner emotions might show. \"Then I'd make you fix a plumbing leak, a squeaky door hinge, and a broken table leg.\"\n\n\"I could do that,\" he said staunchly. \"I happen to be very handy.\"\n\nShe looked at him disbelieving. \"A banker's son who is handy?\"\n\nHe feigned a hurt look. \"I didn't always work in my father's bank. I ran away from home when I was fourteen and joined the Barnum and Bailey Circus. I helped put up and take down the tents, fed the elephants, and made repairs on the circus train.\" He paused and a sad expression came across his face. \"After eight months, my father found me, hauled me home, and sent me back to school.\"\n\n\"So you're a college man.\"\n\n\"Harvard. Phi Beta Kappa, in economics.\"\n\n\"And smart,\" she added, properly impressed.\n\n\"And you?\" he probed. \"Where did you go to school?\"\n\n\"I was in the first graduating class of Stanford University. My degree was in law, but I soon found that law firms were not in the habit of hiring women lawyers, so I went into banking.\"\n\n\"Now it's my turn to be impressed,\" said Bell honestly. \"It seems I've met my match.\"\n\nSuddenly, Marion went silent and a strange look came over her face. Bell thought something was wrong. He rushed to her side and slid his arm around her.\n\n\"Are you ill?\"\n\nShe looked up at him from her coral green eyes. They seemed dark in thought. Then she gasped. \"Montreal!\"\n\nHe leaned toward her. \"What did you say?\"\n\n\"Montreal...Jacob and Margaret are going to make a run across the Canadian border to Montreal, where he can open another bank.\"\n\n\"How do you know that?\" asked Bell, bewildered at Marion's strange attitude.\n\n\"I just remembered seeing the city Montreal scrawled on a notepad beside his telephone,\" she explained. \"I didn't think it meant anything of importance and dismissed it from my mind. Now it all makes sense. The last place authorities would look for the Cromwells is in Canada. They can easily take on new identities and buy off the right people to become upstanding citizens who start up a solvent financial institution.\"\n\nThe look of confusion faded from Bell's face. \"The piece fits,\" he said slowly. \"Canada is probably the last place we'd think to look. The obvious escape route used by felons over the years is over the southern border into Mexico, using that as a springboard to travel farther south.\"\n\nThen, slowly, his thoughts of the Cromwells evaporated and he became quiet, gentle, and loving as he picked her up in his arms. \"I knew there was a reason I fell in love with you,\" he said, his voice becoming low and husky. \"You're smarter than I am.\"\n\nHer whole body trembled as she entwined her arms around his neck. \"Oh, God, Isaac. I love you, too.\"\n\nHe gently touched his lips to hers as he carried her from the living room into the bedroom. She pulled away and looked up, her eyes now mischievous. \"What about the lemon meringue pie?\"\n\nHe gazed down at her lovely features and laughed. \"We can always eat it for breakfast.\"\n\nBell could not have predicted nor much less have known that within a few hours the pie would become but a dim memory.\n\n## 37\n\nCALLED THE HALLMARK OF THE WEST, THE SAN FRANCISCO of 1906 was a maze of contradictions. One writer described the city as the Babylon of grandiloquence, the Paris of romance, and the Hong Kong of adventure. Another went so far as to portray it as the gateway to paradise.\n\nIt may have been dynamic and exciting, but, in truth, San Francisco was a sprawling, filthy, soot-ridden, foul-smelling, brawling, corrupt, vulgar city with less charm than London in the seventeen hundreds. It intermingled incredible wealth with sordid poverty. Coal smoke from steamboats, locomotives, foundries, house furnaces and stoves enveloped streets already blanketed by the dung of thousands of horses. There were no sewage-treatment plants to be found and the blackened skies reeked of foul odors.\n\nMost all the houses were built of wood. From the nice homes on Telegraph Hill to the stylish mansions of Nob Hill to the shacks and hovels in the outlying districts, it was described by the city's fire chief as a sea of tinderboxes waiting to be lit.\n\nThe image and the myth were to change dramatically within two and a half minutes.\n\nAt 5:12 A.M., on the morning of April 18th, the sun was beginning to lighten the eastern sky. The gas streetlights had been shut off and the cable cars began to clatter from their barns for their runs up and down the many hills of the city. Early workers began walking to their job as those who worked during the late-night hours headed home. Bakers were already at their ovens. Police on the early-morning shift still patrolled their beats, expecting another quiet day, as a light wind without the prevailing fog blew in from the west.\n\nBut at 5:12, the peaceful world of San Francisco and its surrounding towns was shattered by an ominous, rumbling roar that came from the depths of the earth a few miles under the sea beyond the Golden Gate.\n\nHell had come to San Francisco.\n\nThe foreshock shook the surrounding countryside and was felt throughout the Bay Area. Twenty-five seconds later, terrifying, undulating shock waves from the massive earthquake surged across the city like a monstrous hand sweeping stacks of books off a table.\n\nThe rock of the San Andreas Fault, whose walls had been grinding against each other for millions of years, abruptly split apart as the North American Plate under the land and the Pacific Plate beneath the sea unleashed their grip on each other and shifted in opposite directions, one to the north, the other to the south.\n\nThe unimaginable force raged toward the helpless city at seven thousand miles an hour in a disastrous spree that would leave monumental death and destruction in its wake.\n\nThe shock wave struck with savage swiftness. The pavement of the streets running east and west began to rise and crest before falling into troughs, as the quake rolled relentlessly forward and sent block upon block of tall buildings rocking and swaying like willow trees in a hurricane. Wood, mortar, and brick were never meant to withstand such an onslaught. One by one, the buildings began to crumble, their walls falling and avalanching into the streets under a cloud of dust and debris. Every window in the stores along the avenues burst and shattered onto the sidewalks in a shower of jagged shards.\n\nHuge five-and ten-story buildings in the downtown business section toppled in a horrendous crash that sounded like a cannon barrage. Chasms opened and closed on the streets, some filling with groundwater and spilling into the gutters. The rails of streetcars and cable cars were twisted and bent like strands of spaghetti. The most violent shocks lasted for slightly more than a minute before diminishing, although smaller aftershocks continued off and on for several days.\n\nWhen the full light of day showed through the chaos, all that was left of a major city of tall buildings, comprising a vast number of stores, offices, banks, theaters, hotels, restaurants, saloons and brothels, houses and apartments, was now a hundred square miles of jagged mounds of shattered masonry, splintered wood, and twisted iron. Though they'd looked substantial, most of the buildings were not reinforced and fell to pieces before the earthquake was thirty seconds old.\n\nThe city hall, the most impressive edifice west of Chicago, sat smashed and destroyed, its cast-iron columns lying shattered in the street. The Hall of Justice was a skeleton of mangled steel girders. The Academy of Sciences gone as though it had never stood. The post office was still standing but effectively demolished. The Majestic Theater would never stage a show again. Only the redoubtable six-story Wells Fargo Building had refused to tumble down despite a ravaged interior.\n\nThousands of chimneys had been first to fall. None was built with an earthquake in mind. Reaching through and high above the roofs, unable to bend and sway and with no support, they shuddered, then fractured and fell through houses and onto streets that were already clogged with debris. Later, it was determined that over a hundred people died from being crushed in their beds by falling chimneys.\n\nWooden two-and three-story homes leaned drunkenly in all directions, twisted on their foundations and tilted crazily in grotesque angles. Oddly, they stood intact but had shifted as much as twenty feet off their foundations, many across sidewalks and into the streets. Though their exterior walls were intact, their interiors were devastated, floors having collapsed, beams ruptured, the furniture and the inhabitants ending up crushed and buried in the basement. The cheaper houses in the poor part of town had collapsed in a pile of splintered beams and siding.\n\nThose who had survived the earthquake were frozen in stunned shock, unable to speak or conversing in whispers. As the great clouds of dust began to settle, the cries of those who were injured or trapped under the fallen structures came as muffled wails. Even after the main force of the quake had passed, the earth still shuddered with aftershocks that continued to shake walls of brick onto the streets, causing their own tremors and strange rumbling sounds.\n\nFew cities in the history of human civilizations had suffered as much devastating destruction as San Francisco. And yet it was only the opening act of an even-worse scene of disintegration that was yet to come.\n\nTHE SHOCK hurled the bed Isaac and Marion were occupying across her bedroom. The apartment house around them rumbled and shook in a series of convulsions. The noise was deafening as dishes crashed to the floor, bookcases collapsed and the books scattered, pictures were slung off the walls, and the upright piano rolled like a boulder down a mountain across the slanting floor only to fall into the street because the entire front wall of the apartment house had detached itself from the rest of the building and cascaded in a flood of rubble onto the street below.\n\nBell grabbed Marion by the arm and half carried, half dragged her through a hail of falling plaster to the doorway, where they stood for the next thirty seconds while the horrendous noise became even more deafening. The floor moved like a stormy sea beneath their feet. They had barely reached the temporary shelter of the doorway when the great chimney at the top of the roof toppled and fell crashing through the two apartments above and smashed through the floor not ten feet away from them.\n\nBell recognized the bedlam as an earthquake. He had endured one almost as bad as the one that destroyed San Francisco while traveling with his parents in China when he was a young boy. He looked down into the pale face of Marion, who looked up at him, dazed and paralyzed by shock. He smiled grimly, trying to give her courage, as the shock waves tore the floor in the parlor from its beams and sent it collapsing into the lower apartment. He could only wonder if the occupants had been killed or were somehow managing to survive.\n\nFor nearly a full minute, they kept on their feet, clutching the doorframe, as their world turned into a nightmarish hell that went far beyond imagination.\n\nThen slowly the tremors died away and an eerie silence settled over the ruin of the apartment. The cloud of dust from the fallen plaster ceiling filled their nostrils and made it difficult to breathe. Only then did Bell realize that they were still on their feet, clutching the doorframe, with Marion wearing a flimsy nightgown and him in a nightshirt. He saw that her radiant long hair had turned white from the plaster's fine powder that still floated in the air like a mist.\n\nBell gazed across the bedroom. It looked like the contents of a wastebasket that had been dumped on the floor. He put his arm around Marion's waist and pulled her toward the closet, where their clothes still hung on hangers, free of the dust.\n\n\"Dress and be quick about it,\" he said firmly. \"The building isn't stable and might collapse at any minute.\"\n\n\"What happened?\" she asked in utter confusion. \"Was it an explosion?\"\n\n\"No, I believe it was an earthquake.\"\n\nShe stared through the wreck of her parlor and saw the ruined buildings on the other side of the street. \"Good Lord!\" she gasped. \"The wall is gone.\" Then she discovered that her piano was missing. \"Oh, no, my grandmother's piano. Where did it go?\"\n\n\"I think what's left of it is down on the street,\" replied Bell sympathetically. \"No more talk. Hurry and throw on some clothes. We've got to get out of here.\"\n\nShe ran to the closet, her composure back on keel, and Bell could see that she was as tough as the bricks that had fallen around them. While he put on the suit he'd worn the night before, she slipped into a cotton blouse under a coarse woolen jacket and skirt for warmth against the cool breeze blowing in from the sea. She was not only beautiful, Bell thought, she was also a practical, thinking woman.\n\n\"What about my jewelry, my family photos, my valuables?\" she asked. \"Shouldn't I take them with me?\"\n\n\"We'll come back for them later, when we see if the building is still standing.\"\n\nThey dressed in less than two minutes, and he led her around the gaping hole in the floor made by the fallen chimney and past the overturned furniture to the front door of the apartment. Marion felt as if she were in another world, as she stared out into open air where the wall once stood and saw her neighbors beginning to wander bewildered out into the middle of the street.\n\nThe door was wedged tight. The earthquake had shifted the building and jammed the door against its frame. Bell knew better than to attack the door by charging against it with his shoulder. That was a fool's play. He balanced on one leg and lashed out with the other. The door failed to show the least sign of give. He looked around the room and surprised Marion with his strength when he picked up the heavy sofa and shoved it against the door like a battering ram. On the third thrust, the door splintered and swung crazily open on one hinge.\n\nThankfully, the stairway was still standing, winding its way to the floor below. Bell and Marion made it past the main entrance and found a high mountain of debris piled outside the apartment house, thrown there when the front wall crashed and buried the street. The front section of the structure looked as if it had been sliced clean by a giant cleaver.\n\nMarion stopped, her eyes welling with tears at the sight of her mother's piano sitting smashed on the crest of the rubble. Bell spotted two men making their way down the street through the wreckage on a wagon drawn by two horses. He left Marion for a few moments, walked over and conversed with the two men as if striking a deal. They nodded and he came back.\n\n\"What was that about?\" asked Marion.\n\n\"I offered to pay them five hundred dollars to take your mother's piano to Cromwell's warehouse by the railyard. When things get back to normal, I'll see that it's rebuilt.\"\n\n\"Thank you, Isaac.\" Marion stood on her toes as she kissed Bell on the cheek, stunned that a man could be so thoughtful about such a little thing in the midst of such disaster.\n\nThe army of people crowding the middle of the street was strangely subdued. There were no wails or cries, no hysteria. Everyone talked in whispers, glad they were alive, but not knowing what to do next or where to go or whether the earthquake would strike again. Many were still in their nightclothes. Mothers cuddled young children or clutched babies while men talked among themselves studying the damage to their homes.\n\nA lull settled over the ruined city. The worst, everyone thought, had to be over. And yet the greatest tragedy was yet to come.\n\nBell and Marion walked to the intersection of Hyde and Lombard, seeing the cable car rails that now snaked like a meandering silver stream to the streets below Russian Hill. The cloud of dust hung tenaciously over the devastation, slowly dissipating as it was carried toward the east by the offshore breeze. From the docks protruding into the water around the Ferry Building west to Fillmore Street, and from the north bay far to the south, the once-great city was a vast sea of ruin and devastation.\n\nScores of hotels and lodging houses had collapsed, killing hundreds who had been sleeping soundly when the quake struck. The screams and cries of those trapped under the rubble and the badly injured carried up to the hill.\n\nHundreds of electrical poles had toppled, their high-tension wires snapping apart, whipping back and forth like desert sidewinders, sparks shooting from the tattered ends. At the same time, pipes carrying the city's gas had split apart and now unleashed their deadly fumes. Tanks in the basements of manufacturing plants holding kerosene and fuel oil ran toward the fiery arcs thrown from the electrical wires where they met and burst in an explosion of orange flame. In destroyed houses, coals from the fallen chimneys ignited furniture and wooden frameworks.\n\nSoon the wind helped merge the big and small fires into one massive holocaust. Within minutes, the city was blanketed by smoke from fires erupting across San Francisco that would take three days and hundreds of lives before they were contained. Many of the injured and trapped who could not be rescued in time would go unidentified, their bodies incinerated and turned to ashes by the intense heat.\n\n\"It's going to get worse, much worse,\" said Bell slowly. He turned to Marion. \"I want you to go to Golden Gate Park; you'll be safe there. I'll come and find you later.\"\n\n\"Where are you going?\" she whispered, shuddering at the thought that she would be alone.\n\n\"To the Van Dorn office. The city is going to need every law enforcement agent available to help control the chaos.\"\n\n\"Why can't I stay here, near my apartment?\"\n\nHe took another look at the growing conflagration. \"It's only a matter of a few hours before the fire reaches Russian Hill. You can't stay here. Do you think you can make it on foot to the park?\"\n\n\"I'll make it,\" she said, nodding gamely. Then she reached up and circled her arms around his neck. \"I love you, Isaac Bell. I love you so much I hurt.\"\n\nHe slipped his arms around her slim waist and kissed her. \"I love you, too, Marion Morgan.\" He hesitated before pushing her back. \"Now, be a good girl and get a move on.\"\n\n\"I'll wait for you at the bridge over the pond.\"\n\nHe held her hand a moment before turning away and moving through the mass of people who were crowded in the center of the street as far away as they could get from the buildings as a series of light aftershocks rippled through the city.\n\nBell took one of the long stairways leading from Russian Hill. It was split apart in several places but did not block his way down to Union Street. Then he cut over to Stockton and then to Market Street. The scene of destruction went far beyond anything his mind could have created.\n\nThere were no streetcars running, and all automobiles, many of them new models commandeered from dealer showrooms, as well as horse-drawn vehicles, were being pressed into service as ambulances to carry the injured to makeshift hospitals that were springing up in the city squares. The bodies of the dead, those who could be retrieved, were carried to warehouses that had been turned into temporary morgues.\n\nThe falling walls had not only crushed unlucky humans walking the sidewalks but also horses pulling the city's huge fleet of freight wagons. They were felled by the dozens under tons of bricks. Bell saw a driver and horse that had been smashed to pulp by an electrical pole that had fallen on their milk wagon.\n\nReaching Market Street, Bell ducked into the remains of a still-standing doorway that was once the entrance into the Hearst Examiner Newspaper Building. He sought refuge as a herd of cattle appeared that had escaped their pen at the docks. Maddened by fright, they charged down the street and almost immediately vanished, swallowed up by one of the great chasms where the violent thrust of the earthquake had split the streets.\n\nBell could not believe how the great thoroughfare of the city, with its magnificent buildings, had changed from the evening before. Gone were the fleets of vehicles, the throngs of happy, contented people working or shopping in the heart of the city's business district. Now the boulevard was scarcely recognizable. The tall buildings had crumbled to pieces, huge pillars with their decorative cornices and ornamentations had been wrenched from the fa\u00e7ades of the structures and hurled to the sidewalk and street in jumbled fragments. The enormous office and store windows were shattered. Signs that once advertised the businesses occupying them lay scattered amid the wreckage.\n\nAs Bell made his way down through the destruction, he could see that the blocks to the south were becoming an ocean of flame. He knew it was only a matter of time before the big hotels, government buildings, tall office skyscrapers, the great department stores, and the theaters would be burned-out skeletons. There were far too few firemen and almost all of the underground water mains had been ruptured by the earthquake. Hundreds of the city's fire hydrants and water taps trickled and then ran dry. The firemen, helpless in fighting the mushrooming fires, began a heroic struggle to repair the water lines.\n\nAfter dodging the automobiles transporting the injured and making his way over the landslides of brick, Bell came within sight of the Call Building. At first, the twelve-story skyscraper looked in good shape, but as he walked closer he saw that the base of one side of the building had moved two feet over the sidewalk toward the street. Inside, he found that none of the elevators were working because the interior was twisted out of alignment. He made his way up the five flights of stairs to the Van Dorn office and stepped over the mounds of plaster that had fallen from the ceiling. Footprints in the plaster told him others had preceded him.\n\nThe furniture scattered about the office by the quake had been set upright where originally positioned.\n\nBell walked into the conference room and found four Van Dorn agents including Bronson, who rushed over and pumped his hand. \"Am I ever glad to see you alive. I was afraid you might be lying under a ton of rubble.\"\n\nBell managed a smile. \"Marion's house lost the front wall, but her apartment is a mess.\" He paused and looked around room. Not seeing Curtis, he asked, \"Have you heard from Art?\"\n\nThe look on everyone's face told Bell what he needed to know. \"Art is missing, assumed to have been crushed under tons of brick as he made his way from the Palace Hotel to our office,\" Bronson answered solemnly. \"From what reports we've managed to gather, two of my agents are either injured or dead. We don't know yet. Those you see here are the only ones who survived without injury.\"\n\nBell's chest felt as if a belt had been cinched around it and pulled tight. He had seen and known death, but to lose someone close was an enduring hurt. \"Curtis dead,\" Bell muttered. \"He was a fine man, a good friend, and one of the best detectives I ever worked with.\"\n\n\"I lost good men, too,\" Bronson said slowly. \"But now we must do what we can to ease the suffering.\"\n\nBell looked at him. \"What is your plan?\"\n\n\"I met with the chief of police and offered Van Dorn's services. Despite our differences in the past, he was only too glad to have our help. We're going to do what we can to combat looting, apprehend looters stealing from the dead and their demolished homes and take them to the city jail. Thankfully, because it's built like a fortress, it still stands.\"\n\n\"I wish I could join you and the others, Horace, but I have another job.\"\n\n\"Yes, I understand,\" Bronson said quietly. \"Jacob Cromwell.\"\n\nBell nodded. \"The earthquake and the bedlam left in its wake have given him the ideal opportunity to escape the country. I intend to stop him.\"\n\nBronson held out his hand. \"Good luck to you, Isaac.\" He gestured around the room with one hand. \"This building isn't safe. And if it doesn't fall down on its own, it will probably be consumed by the approaching fire. We'll have to take our records and abandon it.\"\n\n\"Where can I reach you?\"\n\n\"We're setting up a command center in the Customs House; it was only slightly damaged. The army units that are arriving to maintain order and help battle the fires are also setting up their headquarters there.\"\n\n\"One of us has to report what has happened to Mr. Van Dorn.\"\n\nBronson shook his head. \"Not possible. All telegraph lines are down.\"\n\nBell shook Bronson's hand. \"Good luck to you, too, Horace. I'll be in contact as soon as I learn Cromwell's whereabouts.\"\n\nBronson smiled. \"I bet nothing like this happens where you live in Chicago.\"\n\nBell laughed. \"Aren't you forgetting the great Chicago fire of 1871? At least your calamity came from an act of God. Chicago's came from a cow who kicked over a lantern.\"\n\nAfter saying his good-bye, Bell retraced his route down the twisted stairs to the devastation on Market Street. He quickly made his way over the rubble and past the crowds of people who had assembled to watch the fire that was now burning throughout Chinatown and relentlessly moving toward the city's primary business district.\n\nHe reached the Palace Hotel, which had fared better than the Call Building. Standing just outside the entrance was a man Bell instantly recognized: Enrico Caruso, who had sung the role of Don Jos\u00e9 in Carmen the night before at the Grand Opera House, was waiting as his valet pulled his trunks out onto the sidewalk. He was dressed in a long, bulky fur coat over his pajamas and smoking a cigar. As Bell passed, he heard the great tenor muttering, \"'Ell of a place, 'ell of a place. I never come back.\"\n\nThe elevators were not running due to the lack of electricity, but the stairways were relatively clear of debris. After entering his room, Bell did not bother to pack his clothes. He saw no reason to burden himself with luggage. He threw only a few personal items in a small valise. Not planning on life-threatening danger in San Francisco, he had left his Colt .45 and the derringer in the room. The Colt went in the valise and the derringer back into its small holster inside his hat.\n\nAs he walked up Powell Street toward Cromwell's mansion on Nob Hill, he saw a small group of men frantically struggling to lift a huge beam from a pile of rubble that had once been a hotel. One of them motioned to him and shouted, \"Come give us a hand!\"\n\nThe men were frantically working to free a woman pinned in the debris and the wreckage around it was burning fiercely. She was still dressed in her nightgown, and he saw that she had long auburn hair.\n\nHe gripped her hand for a moment and said softly, \"Be brave. We'll get you free.\"\n\n\"My husband and my little girl\u2014are they safe?\"\n\nBell looked up into the somber faces of the rescuers. One of them slowly shook his head. \"You'll see them soon,\" he said, feeling the intense heat of the fire closing in.\n\nBell lent his strength to the others and vainly tried to lift the beam that covered the woman's legs. It was an exercise in futility. The beam weighed tons and could not be moved by six men. The woman was very courageous and watched the efforts in silence until the flames began scorching her nightgown.\n\n\"Please!\" she begged. \"Don't let me burn!\"\n\nOne of the men, a fireman, asked for her name and wrote it on a small piece of paper he had in his pocket. The rest of the men retreated from the intense heat and menacing flames, horrified at losing their battle to save the woman.\n\nHer nightgown ignited and she began to scream. Without hesitation, Bell held up his derringer and shot her in the forehead between the eyes. Then, without a backward glance, he and the fireman ran for the street.\n\n\"You had to do it,\" said the fireman, his hand on Bell's shoulder. \"Dying by fire is the worst death. You couldn't let her suffer.\"\n\n\"No, I couldn't do that,\" Bell said, his eyes rimmed with tears. \"But it's a terrible memory I'll take to my grave.\"\n\n## 38\n\nCROMWELL AWOKE IN HIS BED TO SEE THE CHANDELIER in the middle of the room swinging like a wild pendulum, its crystal pendants tinkling madly. The furniture danced about as if possessed by crazed demons. A large painting of a fox hunt dropped from the wall with a loud crash as it struck the polished teak floor. The entire house creaked as the stone blocks of the walls ground against each other.\n\nMargaret came staggering into his room, struggling to remain upright as the quake continued. She was wearing nothing but her nightgown, too shocked to throw on a robe. Her face was as white as the breast of a seagull, her golden brown eyes wide with fright, and her lips trembling.\n\n\"What's happening?\" she gasped.\n\nHe reached out and pulled her against him. \"An earthquake, dear sister. Nothing to fret about. It will pass. The worst is over.\"\n\nHis words were quiet and calm, but she could see the nervous tension in his eyes. \"Will the house fall on us?\" she asked fearfully.\n\n\"Not this house,\" he said resolutely. \"It's built like the Rock of Gibraltar.\"\n\nThe words were no sooner said than the great chimneys above began to topple and crash down. Fortunately, they were constructed on the outside of the house walls and collapsed outward without smashing through the roof. Most of the damage came from sections of the outer wall surrounding the house that cracked and crashed to the ground like rumbling thunder. Finally, the tremors began to taper away.\n\nThe house had stood through the worst of the earthquake and retained its structural integrity, looking as it had before except for the collapsed outer wall and three fallen chimneys. And because the inner walls were board over stone, decorated with paint or wallpaper, and the ceilings were mahogany, there were no clouds of dust from fallen plaster.\n\n\"Oh, Lord,\" murmured Margaret. \"What are we going to do?\"\n\n\"You see to the house. Assemble the servants and see if any are injured. Then put them to work cleaning up the mess. Outwardly, act as if restoring the house was your first priority. But begin packing only those valuables and clothes that you consider essential for our flight out of the country.\"\n\n\"You're forgetting Van Dorn's agents,\" she said, looking up quickly.\n\n\"The quake will prove to be a blessing. The city is in chaos. Bell and his fellow Van Dorn detectives have more-pressing problems on their hands than keeping an eye on us.\"\n\n\"What about you?\" asked Margaret, pulling her nightgown tightly around her body.\n\n\"I'm going to the bank to finish cleaning out the vault of all cash. I put most of the currency in trunks yesterday. When it is all packed, Abner and I will transport the trunks in the Rolls to the warehouse and load them on my railcar for our trip across the Canadian border.\"\n\n\"You make it sound easy,\" she said drily.\n\n\"The simpler, the better.\" He climbed from bed and headed for the bathroom. \"By this time tomorrow, our curtain will ring down on San Francisco and, within a few short months, we'll launch a new banking empire in Montreal.\"\n\n\"How much do you figure we'll have?\"\n\n\"I've already transferred fifteen million by telegraph wire to four different Canadian banks in four different provinces,\" he pointed out. \"We'll carry another four million with us in currency.\"\n\nNow she smiled broadly, the fear of the earthquake suddenly pushed aside. \"That's more than we had when we came to San Francisco twelve years ago.\"\n\n\"A lot more,\" Cromwell said comfortably. \"Nineteen million more, to be exact.\"\n\nBELL MISSED Cromwell by twenty minutes when he reached the mansion on Cushman Street. He studied the house and was surprised at the superficial damage after witnessing the unbelievable destruction of the buildings in the main part of the city. He climbed over the mound of fallen bricks that had been an eight-foot wall and walked up the driveway to the front door.\n\nHe pulled the doorbell knob, stood back, and waited. After a long minute, the door cracked open and the housekeeper peered out at Bell. \"What do you want?\" she demanded, all formal courtesy lost from the lingering fright of the earthquake.\n\n\"I'm from the Van Dorn Detective Agency, here to see Mr. Cromwell.\"\n\n\"Mr. Cromwell is not at home. He left soon after the awful earth shuddering.\"\n\nHe could see a figure approach through the curtains covering the glass of the door. \"Do you know if he went to his bank?\"\n\nThe housekeeper moved back as Margaret stepped onto the threshold. She stared at the man standing on the step in a suit covered with dust, grime, and soot. The face was blackened with ash, the eyes tired of seeing too much misery. She barely recognized him.\n\n\"Isaac, is that you?\"\n\n\"A little worse for wear, I'm afraid. But, yes, it's me.\" He removed his hat. \"Good to see you, Margaret. I'm happy you survived the quake without injury.\"\n\nHer dark eyes were wide and soft, as if she were seeing him for the first time. She stood back from the door. \"Please come in.\"\n\nHe entered and saw that she had been working at cleaning up the mess littering the floors of the mansion, mostly broken china, porcelain figurines, and Tiffany lampshades. She wore a comfortable red cotton skirt and a woolen sweater under a long apron. Her hair was wrapped in a tight curl atop her head, with loose strands falling beside her cheeks. Despite her plain appearance, she filled the air around her with a sweet fragrance. Whether she wore an expensive silk gown or an ordinary work dress, Margaret was still a stunning woman.\n\nShe led him into the parlor and offered him a chair by the fireplace, whose ashes had fanned out over the carpet when the chimney collapsed. \"Would you like a cup of tea?\"\n\n\"I'd sell my soul for a cup of coffee.\"\n\nShe turned to her housekeeper, who had overheard and simply nodded, then scurried off to the kitchen. Margaret found it difficult to gaze directly into Bell's hypnotic eyes. She found herself with a growing lust that she had experienced earlier in his presence.\n\n\"What do you want with Jacob?\" she demanded without preamble.\n\n\"I think you know the answer to that question,\" he replied in a flat tone.\n\n\"You cannot abduct him again. Not in San Francisco. You must know that by now.\"\n\n\"You and he have bribed too many corrupt politicians in this town to ever be held for your crimes,\" Bell said bitterly. He paused and looked around at the servants cleaning up the house and putting the furniture and d\u00e9cor back in its proper place. \"Looks like you intend on remaining in the city.\"\n\n\"Why not?\" she said, faking indignation. \"This is our city. We have a thriving business and close friends. Our hearts are open to the poor who live here. Why on earth should we leave?\"\n\nBell was almost tempted to believe Margaret. She was good, he thought, remembering the night they danced in the Brown Palace Hotel. Very good.\n\n\"Is Jacob at the bank?\"\n\n\"He left to survey the damage.\"\n\n\"I saw what's left of Market Street. Most all the buildings are ruined, few still stand, and the Cromwell Bank is right in the path of the growing inferno.\"\n\nMargaret seemed unconcerned. \"Jacob built the bank to stand for a thousand years, as he did this house, which, you can see, survived the earthquake while the more-pretentious Nob Hill mansions were heavily damaged if not destroyed. The House of Cromwell was built to endure.\"\n\n\"Be that as it may, Margaret,\" said Bell with deadly seriousness. \"But I warn you and Jacob not to consider leaving town.\"\n\nHer anger flared and she came to her feet. \"Do not threaten me, and do not think for a moment you can bully my brother. You're all bluff, Isaac. You have no authority, no influence, in this city. My brother and I will be here long after you're gone.\"\n\nHe came to his feet. \"I admit defeat on that score. I have no influence in this city or with its political machine. But once you cross the city limits, the two of you belong to me. You can count on it.\"\n\n\"Get out!\" she hissed fiercely. \"Get out now!\"\n\nFor a long moment, they glared at each other through wild eyes, infuriated with sudden hostility. Then Bell rose slowly and put on his hat as he walked to the front door.\n\nMargaret jumped to her feet and shouted, \"You'll never lay your hands on my brother again. Never in a thousand years! Over my dead body!\"\n\nHe paused to give her one final look. \"I wish you hadn't said that.\" And then he was gone out the door.\n\nABNER EXPERTLY zigzagged the Rolls-Royce to the Cromwell National Bank on Sutter and Hyde Streets, evading the heaps of bricks and swarms of people littering the streets. At one corner, a policeman stopped the car and ordered Abner to go to the Mechanics' Pavilion, the immense building and arena that housed a huge archive and was the scene for many fairs, sports events, and concerts. In desperate need of an emergency facility, the city had converted the pavilion into a hospital and morgue. The policeman insisted Cromwell put the Rolls into service as an ambulance for the injured.\n\n\"I have other uses for my car,\" Cromwell said loftily. He spoke through the speaking tube: \"Continue on to the bank, Abner.\"\n\nThe policeman pulled out his revolver and pointed the muzzle at Abner. \"I'm personally commandeering this car and seeing that you go directly to the pavilion or I'll blow your driver's head off and turn the car over to someone with decency.\"\n\nCromwell was not impressed. \"A pretty speech, Officer, but the car stays with me.\"\n\nThe policeman's face flushed with anger. He waved his revolver. \"I'm not going to warn you again\u2014\"\n\nThe policeman reeled back in shock, his eyes wide, as a bullet from Cromwell's Colt .38 ripped into his chest. He stood for a moment, bewildered, until his heart stopped and he crumpled to the pavement.\n\nThere was no hesitation, no concern, no remorse. Abner quickly slid from behind the wheel, snatched up the body as if it were a dummy, and set it on the front seat. Then he resumed his position behind the wheel, shifted into first gear, and drove away.\n\nThere was so much pandemonium on the streets\u2014people shouting, the occasional thunder of another building collapsing, and the shriek of the fire equipment\u2014that no one noticed the murder of the policeman. The few people who saw him fall to the ground thought he was injured and being picked up by a driver using his automobile as an ambulance.\n\n\"You'll dispose of him?\" Cromwell asked, as if suggesting that a servant throw a dead cockroach in the trash.\n\nAbner spoke into his speaking tube: \"I'll take care of the matter.\"\n\n\"When you're finished, drive into the freight-and-service entrance at the rear of the bank. Let yourself in the back door\u2014you have the key. I'll need you to help carry several trunks to the automobile.\"\n\n\"Yes, sir.\"\n\nAs the Rolls-Royce reached the corner of Sutter and Market Streets and Cromwell saw the approaching inferno and the magnitude of the destruction, he began to feel apprehensive about what he would find when they drove up to his bank. His growing fear quickly turned to elation when the building came into view.\n\nThe Cromwell National Bank had withstood the earthquake nearly unscathed. The unyielding stone structure had lived up to Cromwell's boast that it would last a thousand years. None of the walls or the great fluted columns had fallen. The only apparent damage was the shattered stained-glass windows, whose shards turned the sidewalk around the bank into a kaleidoscope of colors.\n\nAbner pulled the Rolls to a stop and opened the rear door. Several bank employees were milling around the front entrance, having come to work out of habit, not knowing how else to deal with the tragic interruption of their lives. Cromwell got out and was only halfway up the steps when they surrounded him, all talking at once, bombarding him with questions. He held up his hands for silence and reassured them: \"Please, please, go home and stay with your families. You can do nothing here. I promise your salaries will continue to be paid until this terrible calamity has ended and normal business can resume.\"\n\nIt was an empty promise. Not only did Cromwell have no intention of continuing their salaries while the bank was shut down; he could see that the flames sweeping through the business district of the city were only a few hours away from consuming the bank building. Though the walls were stone and unyielding, the wooden roof beams were highly susceptible to fire, which would quickly gut the building to an empty shell.\n\nAs soon as his employees were walking away from the bank, Cromwell took a set of large brass keys from his coat pocket and unlocked the massive bronze front door. He didn't bother to lock it after him, knowing the fire soon would consume any records inside. He headed straight for the vault. The time lock was set to engage the combination at eight o'clock. It was now seven forty-five. Cromwell calmly walked over to a leather chair at the loan officer's desk, brushed off the dust, sat down, and produced a cigar from a case in his breast pocket.\n\nFeeling as if he were in full command of the situation, he leaned back, lit the cigar, and blew a cloud of blue smoke toward the ornate ceiling of the lobby. The earthquake, he thought, could not have come at a more-opportune time. He might lose a few million, but the insurance would cover any damage to the bank building itself. His competitors tied up their assets in loans, but Cromwell always kept his assets in cash and invested on paper. Once it became known he had fled town, bank examiners would land on the Cromwell National Bank like vultures. With luck, his depositors might get ten cents on their invested dollar.\n\nAt precisely eight, the vault mechanism made a chiming sound as the locks clicked off one by one. Cromwell walked over to the vault and turned the huge wheel, which had spokes like a ship's helm, turned it, and released the bars from their shafts. Then he pulled the giant door open on its gigantic well-oiled hinges as easily as if it were attached to a kitchen cupboard and entered.\n\nIt took him two hours to finish loading four million dollars in large-denomination gold certificate bills into five large leather trunks. Abner arrived, after hiding the body of the policeman under the collapsed floor of a hardware store, and carried the trunks out to the Rolls. Cromwell was always impressed with the Irishman's brute strength. He himself could barely lift one end of a filled trunk off the floor, but Abner hoisted it onto one shoulder with barely a grunt.\n\nThe Rolls was parked in the underground freight entrance used by armored trucks and wagons that delivered currency or coins from the nearby San Francisco Mint. Cromwell helped Abner load the trunks in the spacious rear compartment before covering them with blankets he'd brought from his mansion. Under the blankets, he placed cushions from the chairs in the lobby of the bank, positioning them so that it looked as if they were dead bodies.\n\nCromwell went back inside and left the vault door open so the contents would be destroyed. Then he walked out and climbed in the open driver's section of the Rolls and sat beside Abner. \"Drive to our warehouse at the railyard,\" he instructed.\n\n\"We'll have to detour to the north docks and come around behind the fires if we want to reach the railyard,\" said Abner, shifting the car into first gear. As he skirted the huge fire consuming Chinatown, he headed toward Black Point, to the north. Already, wooden buildings were disintegrating into beds of smoldering ashes as broken chimneys stood like blackened tombstones.\n\nSome streets were clear enough to drive through, Abner avoiding those that were impassable because they were buried under collapsed walls. The Rolls was stopped twice by police, demanding the car be used as an ambulance, but Cromwell merely pointed to the makeshift bodies under the blankets and said they were on their way to the morgue. The police duly stepped back and waved them on.\n\nAbner had to weave his way around crowds of refugees from the burned-out areas, carrying their meager belongings. There was no panic; people moved slowly, as if they were out for a Sunday stroll. There was little conversation, and few looked back at what had been their homes before the calamity.\n\nCromwell was stunned at the intensity and swiftness of the fire as it consumed a nearby building. The towering blaze sent a shower of flaming sparks and debris onto the roof, which became a flaming torch within two or three minutes. Then a firestorm enveloped the entire building and consumed it in less time than it takes to boil water.\n\nRegular army troops from the surrounding military installations began arriving to maintain order and help the city firemen fight the flames. Ten companies of artillery, infantry, cavalry, and the Hospital Corps\u2014seventeen hundred men in all\u2014marched into the city with guns and cartridge belts, prepared to guard the ruins, the bank and store vaults and safes, the post office, and the Mint from looters. Their orders were to shoot any man caught stealing.\n\nThey passed a caravan of soldiers in four automobiles whose backseats were stacked with boxes of dynamite from the California Powder Works. Within minutes, explosions rocked the already-devastated city, as homes and stores were detonated and leveled to slow the rage of the fire. Losing the battle, the army quickly began to dynamite entire blocks in a last-ditch attempt to stop the onslaught of the conflagration.\n\nA sickly pale yellow light crept through the growing pall of smoke. There was no sun falling on the ruins except around the outskirts of the city. The dull ball of the sun appeared red and seemed smaller than its usual size. The army troops, firemen, and police retreated from the flames, herding the homeless to the west away from the approaching holocaust.\n\nAbner twisted the wheel of the Rolls as he evaded the rubble in the streets and the crowd of people struggling to reach the ferry terminal in the hope of crossing the bay to Oakland. At last, he came across a railroad track and followed it into the main Southern Pacific railyard until he reached Cromwell's warehouse. He drove up a ramp and parked next to the boxcar sitting at the loading dock. He noted the serial number painted on the side: 16455.\n\nCromwell did not know that Bell was aware the boxcar was not what it appeared to be. But the agent who was assigned to observe it had been called away by Bronson for other duties after the quake. All looked secure. Cromwell studied the padlock on the big sliding door of the boxcar to make sure it had not been tampered with. Satisfied, he inserted a key and removed the lock, which was more for show than for protection. Then he crouched under the car and came up though the trapdoor into the interior of the car. Once inside, he turned the heavy latches that sealed the door from within and slid it open.\n\nWithout instructions, Abner began carrying the heavy trunks filled with currency from the Rolls, hoisting them up to the floor of the boxcar, where Cromwell dragged them inside. When the last of the four million dollars was removed from the limousine, Cromwell looked down at Abner and said, \"Return to the house, gather up my sister and her luggage, then return here.\"\n\n\"You're staying, Mr. Cromwell?\" asked Abner.\n\nCromwell nodded. \"I have business to conduct across the yard at the dispatcher's office.\"\n\nAbner knew that making a round-trip from the warehouse to the mansion on Nob Hill was a near-impossible task, but he gave Cromwell a casual salute and said, \"I'll do my best to bring your sister here safe and sound.\"\n\n\"If anyone can do it, you can, Abner,\" said Cromwell. \"I have complete faith in you.\"\n\nThen Cromwell closed the sliding door of the boxcar and dropped down through the trapdoor. As Abner drove the Rolls down the ramp, he saw Cromwell making his way across the tracks toward the dispatcher's shack.\n\n## 39\n\nBELL HIKED DOWN NOB HILL AND STOPPED TO HELP a crew of men removing the debris of a small hotel that was little more than a mound of splintered wooden beams and crushed bricks. Underneath the wreckage, a little boy's voice could be heard sobbing. Bell and the men worked feverishly, throwing rubble off to the side and digging a hole toward the pitiful cries.\n\nAfter nearly an hour, they finally reached a small air pocket that had protected the boy from being crushed. In another twenty minutes, they had him free and carried him to a waiting car that would rush him to a first-aid facility. Except for his ankles, which appeared to be fractured, he had suffered no other injuries but bruises.\n\nThe little boy looked to be no more than five years old to Bell. As the boy cried for his mother and father, the men who had saved him looked at each other with great sorrow, knowing his parents, and possibly brothers and sisters, were lying crushed deep within the collapsed hotel. Without a word, they went their separate ways, saddened, but glad they were at least able to save him.\n\nTwo blocks farther on, Bell passed by a soldier supervising a group of men who were pressed into service removing bricks from the street and stacking them on the sidewalk. Bell thought one of the men with a handsome profile looked familiar. Out of curiosity, he stopped and asked an older man, who was observing the operation, if he could identify this man who had \"volunteered\" to clean up the street.\n\n\"He's my nephew,\" the older man said, laughing. \"His name is John Barrymore. He's an actor, performing in a play called The Dictator.\" He paused. \"Or should I say was performing. The theater was destroyed.\"\n\n\"I thought I recognized him,\" said Bell. \"I saw him play Macbeth in Chicago.\"\n\nThe stranger shook his head and grinned. \"It took an act of God to get Jack out of bed and the United States Army to get him to work.\"\n\nThe soldier also tried to put Bell to work picking up bricks from the street, but again Bell showed his Van Dorn identification and moved on. By now, the crowds had scattered and the streets were nearly empty, except for soldiers on horseback, and a few sightseers, who lingered to watch the fires.\n\nIn the time it took Bell to walk another eight blocks to Cromwell's bank, the heart of the city on both sides of Market Street was burning fiercely. The wall of fire was only half a dozen blocks away from the bank when he reached the steps leading up to the huge bronze doors. A young soldier, no more than eighteen years of age, stopped Bell and menaced him with the bayonet on the end of his rifle.\n\n\"If you were going to loot the bank, you're a dead man,\" he said in a voice that meant business.\n\nBell identified himself as a Van Dorn agent and lied, \"I'm here to check on the bank, to see if there are any records or currency that can be saved.\"\n\nThe soldier lowered his rifle. \"All right, sir, you may pass.\"\n\n\"Why don't you accompany me? I might need some extra muscle to remove anything of value.\"\n\n\"Sorry, sir,\" said the soldier, \"my orders are to patrol the street ahead of the flames to prevent looting. I don't suggest you spend much time inside. It's only a question of about an hour before the fire reaches this area.\"\n\n\"I'll be careful,\" Bell assured him. Then he walked up the steps and pushed open one of the doors, which Cromwell fortunately had left unlocked. Inside, it appeared as though the bank were closed for Sunday. The tellers' windows, desks, and other furniture made it look as if they were just waiting for business to resume on Monday morning. The only apparent damage was to the stained-glass windows.\n\nBell was surprised to find the vault door open. He entered and quickly saw that most of the currency was missing. Only silver and gold coins, along with some bills, their denominations no higher than five dollars, were still in the tellers' drawers and in their separate bins and containers. Jacob Cromwell had come and gone. The time Bell had spent helping to save the little boy had kept him from catching Cromwell in the act of removing his bank's liquid assets.\n\nThere was no doubt in Bell's mind now that Cromwell meant to use the disaster to escape the city and flee over the border. Bell cursed that his Locomobile was not drivable. Making his way on foot through the ruins was costing him time and was draining his strength. He left the bank and set out for the Customs House, which was also in the path of the fire.\n\nMARION DID not fully accept Bell's instructions. Against his advice, she climbed back up the shaky stairs and into her apartment. She packed a large suitcase with her family photographs, personal records, and jewelry, topping it off with some of her more-expensive clothes. She smiled as she folded two gowns and a silk cape. Only a woman would save her nice things, she thought. A man would care less about saving his good suits.\n\nMarion lugged the suitcase down the stairs and joined the other now-homeless people in the streets who were carrying or dragging trunks filled with their meager possessions, bedding, and household treasures. As they trudged up the hills of the city, none looked back at their homes and apartments, not wanting to dwell on the shattered and smashed remains where they had lived in peace and comfort until this day.\n\nThrough the night, tens of thousands fled the relentless fires. Strangely, there was no panic, no disorder. None of the women wept, none of the men showed anger at their misfortune. Behind them, picket lines of soldiers retreated before the flames, urging the horde to keep moving, occasionally prodding those who had become exhausted and stopped to rest.\n\nDragging heavy trunks up and down steep hills, block after block, mile after mile, eventually became too heavy a burden. Trunks by the thousands and their contents were abandoned, their owners completely played out. Some people found shovels and buried their trunks in vacant lots, hoping to retrieve them after the fires had died out.\n\nMarion's spirit and fortitude rose within her to a level she had not known existed before. She carried or dragged her suitcase as if she was lost in a stupor. She toiled hour after hour alone, no man offering his assistance. Men, and their families, were all occupied in the heroic attempt to save their own belongings. Finally, when Marion could carry her suitcase no farther, a teenage boy asked if he could help her. Marion cried as she thanked him for coming to her aid.\n\nIt was not until five o'clock in the morning that she and the boy reached Golden Gate Park and met a soldier, who directed them to tents that were being set up for the refugees. She entered one, thanking the boy, who refused her offer of money, and sagged onto a cot and fell deeply asleep in less than ten seconds.\n\nWHEN BELL reached the Customs House, it was like walking through a wall of fire. Though late at night, the city was lit by an eerie, oscillating orange light. Crowds were fleeing the flames, but not before hurriedly loading goods from houses and stores onto wagons and rushing to safety at the last minute. The fire was approaching the Customs House on three sides and threatening the entire block. Soldiers on the roofs of neighboring buildings fought a nonstop battle to extinguish the flames and save the Customs House, whose upper level had been badly damaged in the earthquake. The lower floors, however, were undamaged and being used as an operations center by the army and by a detachment of marines and navy personnel who'd been given the job of providing and maintaining fire hoses.\n\nBell passed through the army guards posted around the building and stepped inside. In a room off the main lobby, he found Bronson consulting with two policemen and an army officer over a large-scale map laid out on a conference table.\n\nBronson saw an ash-covered man, his face blackened by soot, standing in the door and did not recognize him for a few seconds. Then a smile spread across his face and he came over and embraced Bell.\n\n\"Isaac, am I ever glad to see you.\"\n\n\"Do you mind if I sit down, Horace?\" said an exhausted Bell. \"I've walked a very long way.\"\n\n\"Of course.\" Bronson led him to a chair in front of a rolltop desk. \"Let me get you a cup of coffee. Despite the inferno around us, we have no way of heating it\u2014but nobody cares.\"\n\n\"I'd love some, thank you.\"\n\nBronson poured a cup from an enamel pot and set it on the desk in front of Bell. A tall man with topaz brown eyes, shaggy dark brown hair, and wearing an unblemished white shirt with tie, came over and stood beside Bronson.\n\n\"Looks like you've seen better days,\" he said.\n\n\"Many of them,\" replied Bell.\n\nBronson turned to the stranger. \"Isaac, this is the writer Jack London. He's writing an essay on the earthquake.\"\n\nBell nodded and shook hands without standing. \"Seems to me you'll have enough material for ten books.\"\n\n\"Maybe one,\" said London, smiling. \"Can you tell me what you've seen?\"\n\nBell gave London a brief report of what he had seen around town, leaving out the horror of shooting the woman in the burning wreckage. When Bell was finished, London thanked him and walked over to a table, where he sat down and began organizing his notes.\n\n\"How did you make out with Cromwell? Did he and his sister survive?\"\n\n\"Alive and well and headed over the border out of the country.\"\n\n\"Are you sure?\" asked Bronson.\n\n\"I got to Cromwell's bank too late. The vault was cleaned out of all denominations over five dollars. He must have made off with three, maybe four, million.\"\n\n\"He won't be able to leave the city. Not with the mess it's in. The wharfs are jammed with thousands of refugees trying to get over to Oakland. No way he could smuggle that much money in just a couple of suitcases.\"\n\n\"He'd find a way,\" said Bell, enjoying the cold coffee and feeling almost human again.\n\n\"What about Margaret? Did she go with him?\"\n\nBell shook his head. \"I don't know. I went by the house before noon and Margaret acted as if she and Jacob were staying in the city and going to fight us in court. After I found out he had fled with his bank's currency, I could not return to Nob Hill because of the advancing wall of fire. I barely made it here as is.\"\n\n\"And Marion?\" Bronson asked cautiously.\n\n\"I sent her to Golden Gate Park. She should be safe there.\"\n\nBronson started to reply, but a boy no older than twelve ran into the room. He wore a broad cap, heavy sweater, and knickers\u2014short pants gathered at the knee. It was obvious that he had been running a long distance because he was so out of breath he could barely speak.\n\n\"I'm...I'm looking for...for Mr. Bronson,\" he gasped haltingly.\n\nBronson looked up, interested. \"I'm Bronson,\" he answered. \"What do you want with me?\"\n\n\"Mr. Lasch...\"\n\nBronson looked at Bell. \"Lasch is one of my agents. He was at our meeting shortly after the quake. He's guarding a government warehouse at the railyard. Go on, son.\"\n\n\"Mr. Lasch said you would pay me five dollars for coming here and telling you what he said.\"\n\n\"Five dollars?\" Bronson stared at the boy suspiciously. \"That's a lot of money for somebody your age.\"\n\nBell smiled, retrieved a ten-dollar bill from his wallet, and passed it to the boy. \"What's you name, son?\"\n\n\"Stuart Leuthner.\"\n\n\"You've come a long way from the railyard through the fire and devastation,\" said Bell. \"Take the ten dollars and tell us what Lasch told you.\"\n\n\"Mr. Lasch said to tell Mr. Bronson that the boxcar parked in front of Mr. Cromwell's warehouse is gone.\"\n\nBell leaned toward the boy, his face suddenly clouded. \"Say again,\" he instructed.\n\nThe boy looked at Bell, apprehension in his eyes. \"He said Mr. Cromwell's boxcar was gone.\"\n\nBell stared at Bronson. \"Damn!\" he muttered. \"He has fled the city.\" Then he gave the boy another ten-dollar bill. \"Where are your parents?\"\n\n\"They're helping pass out food in Jefferson Square.\"\n\n\"You'd better find them. They must be worried about you. And, mind you, stay away from the fire.\"\n\nWarren's eyes widened as he stared at the two ten-dollar bills. \"Gosh almighty, twenty dollars. Gee, thanks, mister.\" Then he turned and ran from the building.\n\nBell sank back into the chair at the rolltop desk. \"A train?\" he murmured. \"Where did he come by a locomotive?\"\n\n\"All I know is, every ferry is jammed with refugees fleeing across the bay to Oakland. From there, the Southern Pacific is gathering every passenger train within a hundred miles to transport them away from the area. No way he could have hired a locomotive, crew, and tender.\"\n\n\"His freight car didn't roll away by itself.\"\n\n\"Trust me,\" said Bronson, \"no freight cars are being ferried to Oakland. Only people. The only moving trains are those coming in with relief supplies from the east.\"\n\nBell came to his feet again, his eyes cold and fixed on Bronson. \"Horace, I need an automobile. I can't waste hours hiking the part of the city that's not in flames.\"\n\n\"Where are you going?\"\n\n\"First, I have to find Marion and make certain she's safe,\" Bell answered. \"Then I'm heading for the railyard and the dispatcher. If Cromwell hired, or stole, a train to take him out of the city, there has to be a record at the dispatcher's office.\"\n\nBronson grinned like a fox. \"Will a Ford Model K do?\"\n\nBell looked at him in surprise. \"The new Model K has a six-cylinder engine and can churn out forty horsepower. Do you have one?\"\n\n\"I borrowed it from a rich grocery store owner. It's yours, if you promise to have it back by noon tomorrow.\"\n\n\"I owe you, Horace.\"\n\nBronson placed his hands on Bell's shoulders. \"You can pay me back by seizing Cromwell and his evil sister.\"\n\n## 40\n\nMARION SLEPT FOR SIX HOURS. WHEN SHE AWOKE, she found the tent inhabited by five other single women. One was sitting on her cot, weeping. Two looked dazed and lost, while the others showed their strength by volunteering to help feed the suffering at the kitchen facilities that were being set up in the park. Marion rose from her cot, straightened her clothes, and marched with her new friends to several large tents that had been erected by the army as emergency hospitals.\n\nShe was immediately instructed by a doctor to treat and bandage wounds that did not require the services of doctors, who were busy in surgery helping to save the lives of the badly injured. Marion lost track of time. She shrugged off sleep and exhaustion by working in a shelter for children. Many were so brave it tore her heart. After tending the cuts and bruises of a little three-year-old girl who had lost her family, she turned away in tears when the girl thanked her in a tiny voice.\n\nShe moved to the next cot and knelt beside a boy brought in from surgery after having his broken leg set. As she tucked him in a blanket, she felt a presence behind her. Then came a familiar voice.\n\n\"Pardon me, nurse, but my arm fell off. Can you mend it?\"\n\nMarion spun around and threw herself into the open arms of Isaac Bell.\n\n\"Oh, Isaac, thank God you're all right. I was worried about you.\"\n\nBell smiled broadly through the grime on his face. \"A little the worse for wear, but still standing.\"\n\n\"How did you ever find me?\"\n\n\"I'm a detective, remember? The emergency hospital was the first place I looked. I knew you'd be following in the footsteps of Florence Nightingale, your heart is too big not to help those in need, especially children.\" He squeezed her and whispered in her ear. \"I'm proud of you, Mrs. Bell.\"\n\nShe pushed herself back and stared up into his eyes in confusion. \"Mrs. Bell?\"\n\nBell's smile remained fixed. \"Not exactly a romantic time or place to propose, but will you marry me?\"\n\n\"Isaac Bell,\" she cried, \"how dare you do this to me.\" Then she softened, pulled his head down, and kissed him. When she released him, she said slyly, \"Of course I will marry you. It's the best offer I've had all day.\"\n\nHis smile faded, his lips tightened, and his voice harshened. \"I can't stay but a minute. Cromwell and Margaret are fleeing San Francisco. As long as there's a breath in me, I can't let a murdering scum like Cromwell go free.\"\n\nHis fervor frightened her, but she embraced him fiercely. \"It isn't every day a girl is proposed to by her lover who then runs away.\" She kissed him again. \"You come back, you hear?\"\n\n\"As soon as I can.\"\n\n\"I'll be waiting here. I don't expect any of us will be leaving our shantytown soon.\"\n\nBell held up her hands and kissed them both. Then he turned and disappeared from the hospital tent.\n\nBELL DID not consider returning to the Cromwell mansion on Nob Hill to see if Margaret had flown. He was certain she had fled with her brother.\n\nThe palace houses of the rich and powerful were great blazing bonfires. From every part of town came the roaring of the flames, the rumble of crashing walls, and the explosions of dynamite.\n\nThe Model K Ford was light and fast. And it was durable. It climbed over the rubble in the streets like a mountain goat. Unknowingly, Bell took nearly the same route as Cromwell and Abner, skirting along the northern waterfront away from the fire. Barely half an hour had passed since he had left Marion when he stopped the car on the ramp at Cromwell's warehouse, satisfying himself that the boxcar was indeed missing.\n\nSwitch engines were coupling cars to passenger trains in order to evacuate refugees to the southern part of the state, which still had open tracks, while freight cars were being dispatched to transport food supplies and medicine from Los Angeles. He drove the Ford into the railyard along the tracks until he reached a wooden building with a sign above the roof advertising it as the DISPATCH OFFICE. Bell stopped the car, leaped to the ground, and stepped inside.\n\nSeveral clerks were busily working on the paperwork to dispatch trains and none looked up as Bell entered. \"Where can I find the chief dispatcher?\" he asked a harried clerk.\n\nThe clerk nodded toward a door. \"In there.\"\n\nBell found the dispatcher writing numbers on a huge blackboard that displayed the tracks leading to and from the railyard. The sign on the desk read MORTON GOULD. He was a short man with a recessed chin and hawklike beak for a nose. The board showed over thirty different trains dispatched over track that spread from the railyard like a spiderweb. Bell could not help but wonder which one included Cromwell's boxcar.\n\n\"Mr. Gould?\"\n\nGould turned and saw a man who looked as though he'd walked from one side of hell to the other. \"Can't you see I'm busy? If you want to catch a train out of the city, you'll have to go to the Southern Pacific depot\u2014or what's left of it.\"\n\n\"My name is Bell. I'm with the Van Dorn Detective Agency. I'm looking for a boxcar with the serial number 16455.\"\n\nGould motioned toward the board. \"Southern Pacific is moving heaven and earth transporting thousands of homeless out of the city on our fleet of ferryboats and tugs over to Oakland, where we've assembled passenger trains waiting to evacuate them from the area. Over fourteen hundred relief cars are coming in from all over the country. Cars\u2014passenger and freight\u2014on this side of the bay, all three hundred of them, are being routed around the lower part of the state. How do you expect me to keep track of just one car?\"\n\nBell studied Gould's eyes. \"This particular car belonged to Jacob Cromwell.\"\n\nIt was there, a barely perceivable indication of recognition. \"I don't know any Jacob Cromwell.\" Gould paused to stare apprehensively at Bell. \"What's this all about?\"\n\n\"You dispatched a locomotive to pull his private freight car.\"\n\n\"You're crazy. I wouldn't dispatch private trains during an emergency such as this.\"\n\n\"How much did he pay you?\"\n\nThe dispatcher lifted his hands. \"I couldn't be paid by a man I don't know. It's ridiculous.\"\n\nBell ignored Gould's lie. \"Where was the destination of Cromwell's train?\"\n\n\"Now, look here,\" Gould said, fear growing in his eyes. \"I want you out of here, Van Dorn cop or no Van Dorn cop.\"\n\nBell removed his hat and made a motion as if cleaning the inside band. The next thing the dispatcher knew, he was staring into the business end of a derringer. Bell pressed the twin barrels against the side of Gould's left eye socket. \"Unless you tell the truth in the next sixty seconds, I will shoot and the bullet will horribly disfigure your face besides blasting away both of your eyes. Do you wish to spend the rest your life as a mutilated blind man?\"\n\nThe hypnotic grip of terror crossed Gould's face. \"You're mad.\"\n\n\"You have fifty seconds left before you see nothing.\"\n\n\"You can't!\"\n\n\"I can and I will, unless you tell me what I want to know.\"\n\nThe cold expression, along with the icy voice, was enough for Gould to believe the Van Dorn detective was not bluffing. He looked around wildly, as if there was a way to escape, but Bell continued remorselessly.\n\n\"Thirty seconds,\" he said, pulling back the hammer of the derringer.\n\nGould's shoulders collapsed, his eyes filled with terror. \"No, please,\" he murmured.\n\n\"Tell me!\"\n\n\"All right,\" Gould said in a low tone. \"Cromwell was here. He paid me ten thousand dollars in cash to hook his car up to a fast locomotive and direct the train onto a track heading south.\"\n\nBell's eyes partially closed in incomprehension. \"South?\"\n\n\"It's the only way out of the city,\" replied Gould. \"All the train ferries are being used to transport people over to Oakland and the relief trains back. There was no other way he could go.\"\n\n\"How was he routed?\"\n\n\"Down to San Jose, then around the bay to the north until his train turned east on the main line over the mountains and across Nevada to Salt Lake City.\"\n\n\"How long ago did he leave the railyard?\" Bell demanded.\n\n\"About four hours.\"\n\nBell continued the pressure. \"When is he scheduled to reach Salt Lake City?\"\n\nGould shook his head in quick spasms. \"Can't say. His engineer will have to spend a lot of time on sidetracks so the relief trains can fireball through. If he's lucky, his train will reach Salt Lake by late tomorrow afternoon.\"\n\n\"What type of engine did you assign to pull Cromwell's private freight car?\"\n\nGould leaned over a desk and examined the notations in large ledger. \"I gave him number 3025, a 4-6-2 Pacific, built by Baldwin.\"\n\n\"A fast engine?\"\n\nGould nodded. \"We have a few that are faster.\"\n\n\"When will one be available?\"\n\n\"Why do you ask?\"\n\n\"I want the fastest engine you've got,\" answered Bell, menacing Gould with the derringer. \"This is a vital emergency. I have to catch Cromwell's train.\"\n\nGould consulted his big board. \"I have number 3455, a 4-4-2 Baldwin Atlantic. She's faster than a Pacific. But she's in the Oakland yard for repairs.\"\n\n\"How long before she's ready to run?\"\n\n\"The repair shop should have her ready to go in another three hours.\"\n\n\"I'll take her,\" Bell said without hesitation. \"See that Van Dorn is charged for the time it's in use.\"\n\nGould looked as if he was going to protest and argue with Bell, but, staring at the derringer, he thought better of it. \"If you report me, I could lose my job and go to jail.\"\n\n\"Just give me that engine and route me around San Jose toward Salt Lake City and I'll say nothing.\"\n\nGould sighed thankfully and began making out the paperwork to charter and dispatch a route for the locomotive under the Van Dorn Detective Agency. When he was finished, Bell took the papers and studied them for a moment. Satisfied, he left the office without another word, climbed in the Ford, and drove toward the Ferry Building.\n\n## 41\n\nNEARING THE FERRY BUILDING, BELL THREW A BLANKET over his head as he drove through a shower of cinders. He could see that Chinatown was gone, leaving little more than hundreds of piles of charred, smoldering ruins. The Ferry Building had survived with only minor damage to its clock tower. Bell noted that the clock had stopped at 5:12, the time the earthquake struck.\n\nThe streets and sidewalks around the Ferry Building looked like a vast mob scene. Thousands were fleeing, believing the entire city would be destroyed. There was pandemonium and bedlam in the jumbled mass of people, some wrapped in blankets and loaded down with what possessions they were able to carry onto the ferryboat. Some pushed baby buggies or toy wagons, and yet, amid the nightmare, everyone was gracious, courteous, and considerate toward others.\n\nBell stopped beside a young man who seemed to be merely standing around and watching the fire across the street from the wharfs. He held up a twenty-dollar gold piece. \"If you know how to drive a car, take this one to the Customs House and turn it over to Horace Bronson of the Van Dorn Detective Agency and this is yours.\"\n\nThe young man's eyes widened in anticipation, not so much from the money but the chance to drive an automobile. \"Yes, sir,\" he said brightly. \"I know how to drive my uncle's Maxwell.\"\n\nBell watched with amusement as the boy clashed the gears and drove off down the crowded street. Then he turned and joined the mass of humanity that was escaping the destruction of the city.\n\nWithin three days, over two hundred twenty-five thousand people left the peninsula where San Francisco stood, all carried free of charge by the Southern Pacific Railroad to wherever they wished to travel. Within twenty-four hours of the quake, overloaded ferryboats were departing San Francisco for Oakland every hour.\n\nBell showed his Van Dorn credentials and boarded a ferry called the Buena Vista. He found an open place to sit above the paddle wheels and turned back to watch the flames shooting hundreds of feet into the air, with the smoke rising over a thousand feet. It looked as if the whole city was one vast bonfire.\n\nOnce he stepped off the Mole in Oakland, a railroad official directed him to the repair shop where his locomotive was sitting. The mammoth steel monster was a grand sight up close. It was painted black from the cowcatcher to the rear of its coal tender. Bell guessed the cab's roof was at least fifteen feet above the rails. The big drive wheels were eighty-one inches in diameter. In its time, the Atlantic-type locomotive was a masterwork of mechanical power.\n\nTo Bell, it looked mean and ugly. The number 3455 was painted in small white letters on the side of the cab; SOUTHERN PACIFIC, in larger type, ran across the side of the tender, which fueled the boiler with coal and water. Bell walked up to a man wearing the traditional striped engineer's coveralls and striped cap with brim. The man held a big oil can with a long spout and looked to be oiling the bearings on the connecting rods running from the piston cylinder to the drive wheels.\n\n\"A mighty fine locomotive,\" said Bell admiringly.\n\nThe engineer looked up. He was shorter than Bell, with strands of salt-and-pepper hair straying from under his cap. The face was craggy from years of leaning out a cab window into the full wind stream from a speeding engine. The eyebrows over a pair of sky blue eyes were curved and bushy. Bell judged he was younger than he looked.\n\n\"None better than Adeline,\" the engineer answered.\n\n\"Adeline?\"\n\n\"Easier to remember than her four-figured number. Most locomotives are given a woman's name.\"\n\n\"Adeline looks very powerful,\" said Bell admiringly.\n\n\"She's built for heavy passenger service. Came out of the Baldwin Works no more than five months ago.\"\n\n\"How fast will she go?\" asked Bell.\n\n\"Depends on how many cars she's hauling.\"\n\n\"Let's say none.\"\n\nThe engineer thought a moment. \"On a long, straight stretch of open, empty track, she'd top a hundred miles an hour.\"\n\n\"My name is Bell.\" He handed the engineer the paperwork. \"I've chartered your engine for a special job.\"\n\nThe engineer studied the papers. \"Van Dorn detective outfit, huh. What's so special?\"\n\n\"Ever hear of the Butcher Bandit?\"\n\n\"Who hasn't? I've read in the newspapers he's about as deadly as they come.\"\n\nBell wasted no detailed explanation. \"We're going after him. He chartered a Pacific-type locomotive to haul his special private car. He's steaming to Salt Lake City before heading north for the Canadian border. I reckon he has a five-hour head start.\"\n\n\"More like six, by the time we take on coal and get a load of steam up.\"\n\n\"I was told there were repairs. Are they completed?\"\n\nThe engineer nodded. \"The shop replaced a faulty bearing in one of the truck wheels.\"\n\n\"The sooner we get going, the better.\" Bell paused to extend his hand. \"By the way, my name is Isaac Bell.\"\n\nThe engineer's shake was vigorous. \"Nils Lofgren. My fireman is Marvin Long.\"\n\nBell pulled his watch from its pocket and checked the time. \"I'll see you in forty-five minutes.\"\n\n\"We'll be at the coal-loading dock just up the track.\"\n\nBell hurried toward the Oakland terminal until he came to a wooden building that housed the Western Union office. The wire chief told him that only one wire was open to Salt Lake City and it was hours behind getting messages through. Bell explained his mission and the chief was most cooperative.\n\n\"What's your message?\" he asked. \"I'll see that it's sent straightaway to our office in Salt Lake.\"\n\nBell's wire read:\n\nTo the Van Dorn office director, Salt Lake City. Imperative you stop locomotive hauling freight car number 16455. It is carrying the Butcher Bandit. Use every precaution. He is extremely dangerous. Seize and hold until I arrive.\n\nIsaac Bell, special agent\n\nHe waited until the telegrapher tapped out the message before leaving the office and walking to where Lofgren and Long were taking on coal and water. He climbed up into the cab and was introduced to Long, a heavy, broad-shouldered man with large muscles stretching the sleeves of his denim shirt. He wore no hat and his red hair almost matched the flames inside the door to the firebox. He pulled off a leather glove and shook Bell's hand with a hand that was hard and callused from long hours wielding a coal shovel.\n\n\"Ready whenever you are,\" announced Lofgren.\n\n\"Let's do it,\" answered Bell.\n\nAs Long stoked the fire, Lofgren took his seat on the right side of the cab, locked the reverser Johnson bar into place, opened the cylinder cocks, and pulled the rope above his head down twice, causing the steam whistle to scream an about-to-move-forward warning. Then he gripped the long throttle lever and pulled it back. Adeline began to move and slowly gather speed.\n\nTen minutes later, Lofgren was signaled to switch onto the main track east. He eased the throttle back and the big Atlantic began to move forward. Slowly, the train wound through the yard. Long began maintaining his fire, light, level, and bright. In the five years he'd stoked locomotive fires, he'd developed a technique that kept the fire from burning too thin or too thick. Lofgren yanked on the throttle, the drive wheels churned amid a loud blast of steam, and black smoke spewed out the top of the stack.\n\nBell took the seat on the left side of the cab, feeling vastly relieved that he was at last on what he felt certain was the final chase to catch Cromwell and hand him over to the authorities in Chicago, dead or alive.\n\nHe found the vibration of the locomotive over the rails as soothing as floating in a rubber raft on a mountain lake, the chug of the steam propelling the drive wheels and the warm heat from the firebox positively restful for a man on a mission. Before they reached Sacramento and swung east across the Sierra Nevada Mountains, Bell slid down in his seat, yawned, and closed his eyes. Within a minute, he was in a sound sleep, amid the clangor of the speeding locomotive, as Adeline aimed her big cowcatcher toward the Sierra Nevada and Donner Pass.\n\n## 42\n\nABNER WEED'S BARREL CHEST AND BEEFY SHOULDERS were sweating as he shoveled coal into the firebox. There was an art to creating an efficient fire, but he had no idea how. He simply heaved coal through the open door into the fire, ignoring the complaints from the engineer who shouted that too much coal would drop the fire temperature.\n\nAbner took on the job only to spell the fireman, Ralph Wilbanks, a big, burly man who soon became exhausted after a few hours of sustaining the necessary steam temperature that kept the big Pacific locomotive running up the steep grades of the Sierra Nevada Mountains. They traded one hour shoveling, one hour of rest.\n\nAbner stayed alert during the effort, his Smith & Wesson revolver stuffed in his belt. He kept an eye on the engineer, who was constantly busy maintaining a fast but safe speed around the many mountain curves while watching the track ahead for any unforeseen obstacle, such as an unscheduled train coming in the opposite direction. At last, they crested the summit and it was all downhill until they met the flat-lands of the desert.\n\n\"We're coming into Reno,\" yelled Wes Hall, the engineer, above the roar of the flames in the firebox. An intense man with the features of a weathered cowboy, he would have stopped the train in protest when he found his passengers demanded he set speed records across the mountains but relented after Abner put the Smith & Wesson to his head and threatened to kill him, and his fireman, if they didn't do as they were told. A thousand dollars in cash from Cromwell added to the persuasion, and Hall and Wilbanks now pushed the Pacific locomotive through the mountains as fast as they dared.\n\n\"The signal ahead reads red,\" said Wilbanks.\n\nHall waved that he saw it, too. \"We'll have to stop and lay over on a siding.\"\n\nAbner pointed the gun at the engineer's head. \"Lay on the whistle. We're going through.\"\n\n\"We can't,\" said Hall, staring Abner in the eye. \"There must be an express carrying relief supplies to San Francisco coming toward us on the same track. I'd rather you shoot me than cause a collision with another train that would kill all of us and stop traffic in both directions for maybe a week.\"\n\nAbner slowly slid the revolver back under his belt. \"All right. But get us back on the main track as soon as the relief express passes.\"\n\nHall began closing the throttle arm. \"We can use the delay to take on coal and water.\"\n\n\"All right. But mind your manners or I'll blow holes in the both of you.\"\n\n\"Ralph and I can't go on much longer. We're done in.\"\n\n\"You'll earn your money\u2014and stay alive\u2014by pushing on,\" Abner said threateningly.\n\nLeaning out the left side of the cab, Abner could see the train depot and the small town of Reno, Nevada, looming in the distance. As they came nearer, Abner spotted a figure waving a small red flag standing by a switch stand. Hall blew the whistle to announce their arrival and to let the flagman know that he understood the signal to slow down and was prepared to be switched off the main track.\n\nHall precisely stopped the Pacific's tender directly under an elevated wooden water tank on one side of the track and a coal bin on the other. Wilbanks jumped up on the tender, grabbed a rope, and pulled down the spout hinged to the tank until water flowed on board due to gravity. Climbing down from the cab with an oil can, Hall began checking all the bearings and fittings of the locomotive, and, since Cromwell had refused to wait for the arrival of a brakeman, he had to examine the bearings on the wheels of the tender and freight car as well.\n\nKeeping a sharp eye on Hall and Wilbanks, Abner moved past the tender to the door of the freight car. He rapped twice with the butt of his Smith & Wesson, waited a moment, then knocked again. The door was unlatched from the inside and slid open. Jacob and Margaret Cromwell stood there, looking down at Abner.\n\n\"What's the delay?\" asked Cromwell.\n\nAbner tilted his head toward the locomotive. \"We switched to a siding to let an express relief train through. While we're waiting, the crew is taking on coal and water.\"\n\n\"Where are we?\" asked Margaret. She was dressed uncharacteristically in men's pants, with the legs tucked into a pair of boots. A blue sweater covered the upper half of her body, and she wore a bandanna on her hair.\n\n\"The town of Reno,\" answered Abner. \"We're out of the Sierras. From now on, the landscape flattens out into desert.\"\n\n\"How about the track ahead?\" inquired Cromwell. \"Any more relief trains to delay our passage?\"\n\n\"I'll check with the switchman for scheduled westbound trains. But we'll have to stand aside as they come.\"\n\nCromwell jumped to the ground and spread out a map on the ground. The lines drawn across it displayed the railroads in the United States west of the Mississippi. He pointed to the spot signifying Reno. \"Okay, we're here. The next junction with tracks going north is Ogden, Utah.\"\n\n\"Not Salt Lake City?\" asked Margaret.\n\nCromwell shook his head. \"The Southern Pacific main line joins the Union Pacific tracks north of Salt Lake. We swing north at the Ogden junction and head toward Missoula, Montana. From there, we take the Northern Pacific rails into Canada.\"\n\nAbner kept his eyes trained on the crew. He saw the fireman struggle with the coal flowing from the chute into the tender and the engineer moving about as if he were in a trance. \"The crew is dead on their feet. We'll be lucky if they can run the locomotive another four hours.\"\n\nCromwell consulted the map. \"There's a railyard in Winnemucca, Nevada, about a hundred seventy miles up the track. We'll pick up another crew there.\"\n\n\"What about these two?\" inquired Abner. \"We can't let them run to the nearest telegraph office and alert law enforcement up the line that we're coming.\"\n\nCromwell thought for a moment. \"We'll keep them with us, then make them jump the train in a desolate part of the desert. We'll take no chances of Van Dorn agents getting wise to our leaving San Francisco and wiring officials down the line to stop our train, so we'll cut the telegraph lines as we go.\"\n\nMargaret took a long look toward the Sierras and the track they had traveled. \"Do you think Isaac is onto us?\"\n\n\"Only a question of how long, dear sister,\" he said with his usual high degree of self-assurance. \"But by the time he realizes we've flown San Francisco and finds a locomotive to give chase, we'll be halfway to Canada and he'll have no chance to stop us.\"\n\n## 43\n\nADELINEWAS LOFGREN'S PRIDE AND SWEETHEART, and he spoke to her as if the locomotive were a beautiful woman instead of a steel, fire-breathing monster that charged up the curving grades of the Sierras and through Donner Pass. Without having to pull two hundred tons of cars weighed down with passengers and luggage, she performed effortlessly.\n\nThe spring air was cool and crisp, and snow still covered the ground. Donner Pass was the notorious section of the mountains where the most poignant event in western history had taken place. A wagon train made up of a dozen families that would pass into legend as the Donner party became trapped in the winter blizzards of 1846 and suffered terribly until rescued. Many survived by eating the dead. Out of the original eighty-seven men, women, and children, only forty-five lived to reach California.\n\nBell had been fully awake since passing through Sacramento and was finding the scenery spectacular\u2014the towering, rocky peaks; the forest of fir trees, some with branches still laden with snow; the summit tunnels, which were blasted out of granite by Chinese laborers in 1867. Adeline plunged into the black mouth of a long tunnel, the roar of the train's exhaust reverberating like a hundred bass drums. Soon, a tiny circle of light materialized ahead in the darkness and quickly grew wider. Then Adeline burst into the bright sunlight with a noise like thunder. A few miles later came the panoramic view of Donner Lake, as the train began its long, curving descent to the desert.\n\nBell stared with some uneasiness down the sheer thousand-foot drop that was within a step or two of the edge, as the locomotive swung around a sharp bend. He did not need to urge Lofgren to go faster. The engineer was pushing the big locomotive at nearly thirty-five miles an hour around the mountain curves, a good ten miles faster than was considered safe.\n\n\"We're across the summit,\" announced Lofgren, \"and have a downgrade for the next seventy-five miles.\"\n\nBell stood and gave Long his fireman's seat on the left side of the cab. Long thankfully sat down and took a break, as Lofgren closed off steam and allowed Adeline to coast down through the pass in the mountains. Long had been shoveling coal almost nonstop since they had swung onto the main line at Sacramento and up the steep grade into the Sierras.\n\n\"Can I give you a hand?\" asked Bell.\n\n\"Be my guest,\" said Long, lighting up a pipe. \"I'll tell you how to shovel the coal into the firebox. Even though we're loafing along for the next hour, we can't let the fire die down.\"\n\n\"You don't just throw it in with a shovel?\"\n\nLong grinned. \"There's more to it than that. And it's not called a shovel; it's a fireman's scoop, size number four.\"\n\nFor the next two hours, Bell labored in front of the maze of pipes and valves as he learned the intricacies of firing a locomotive. The tender was rocking from side to side around the turns, making it difficult to shovel coal into the firebox. It was easy work, however, with Adeline running downhill. He shoveled just enough coal to keep the steam up. He quickly learned to open the firebox door wide, after hitting the scoop against it and spilling coal over the floor. And instead of stacking the coal in a fiery pile, he developed the knack for making a level fire that burned bright and orange.\n\nThe sharp curves were left behind as their arc increased as they dropped down to the foothills. An hour after Bell turned the scoop over to Long, the fireman shouted to Lofgren: \"We've only got enough water and coal for another fifty miles.\"\n\nLofgren nodded without taking his eyes from the track ahead. \"Just enough to make Reno. We can put in for coal and water there and take on a relay crew.\"\n\nBell realized that the race over the mountains had taken its toll on Lofgren and Long. He could see that the strain on body and mind had drained the staunch engineer, and the physical effort of maintaining steam on the steep grades had sapped the strength of the indefatigable fireman. It seemed evident to Bell that Cromwell's train crew must be worn out as well. He checked his watch and could only wonder if they had narrowed the gap.\n\n\"How long will it take to assemble another crew?\" Bell asked.\n\n\"As long as it takes to coal and water the tender,\" replied Lofgren. Then he smiled wearily, revealing a set of crooked teeth, and added, \"Providing we're lucky and one happens to be standing by.\"\n\n\"I'm grateful to you both,\" Bell said sincerely. \"You did a heroic job getting over the Sierras. You must have set a record.\"\n\nLofgren pulled out his big Waltham railroad watch with its locomotive engraved on the back of the case. \"Indeed,\" he laughed. \"We shaved eight minutes off the old record set by Marvin, me, and Adeline six months ago.\"\n\n\"You love this engine, don't you?\" said Bell.\n\nLofgren laughed. \"Take all the Atlantic locomotives ever put on rails: they're the finest in the world, all built exactly the same, with identical dimensions and construction. Yet, every one is different\u2014like people, they all have diverse personalities. Some can run faster than the others, with the same steam pressure. Some are finicky while others are jinxed, always having bad luck with repair problems. But Adeline, she's a sweetheart. No whims; never cranky, eccentric, or ill-tempered. Treat her like a lady and she's like a thoroughbred mare that wins races.\"\n\n\"You make her sound almost human.\"\n\n\"Adeline may be a hundred seven tons of iron and steel, but she's got a heart.\"\n\nThey were nearing Reno, and Lofgren pulled the whistle cord to announce his intention of switching to the siding for coal and water. He eased back on the throttle to slow the locomotive. The switchman threw the switch lever to link the tapering rails, as he had done for Cromwell's train earlier. Then he waved a green flag to alert Lofgren that the siding was open.\n\nEven before Adeline rolled to a stop, Bell had jumped from the cab and took off running across the railyard to the depot, which looked like a thousand other small-town depots across the nation. It was characterized by wooden slat walls, arched windows, and a peaked roof. The loading platform was empty, giving Bell the impression that no passenger trains were due to stop there anytime soon.\n\nHe stepped inside, past the freight-and-ticket office, and stopped at the telegrapher's small room. Two men were in the middle of a deep conversation when he walked in. It struck him that their faces looked serious and grim.\n\n\"I beg your pardon,\" said Bell, \"I'm looking for the stationmaster.\"\n\nThe taller of the two men stared at Bell for a moment before nodding. \"I'm the stationmaster, Burke Pulver. What can I do for you?\"\n\n\"Has a train come through with only one freight car in the last ten hours heading east?\"\n\nPulver nodded. \"They were stopped on the siding for two hours while two express trains carrying relief supplies for the San Francisco earthquake victims rolled through.\"\n\n\"They were delayed two hours?\" said Bell, suddenly feeling optimistic. \"How long ago did they leave?\"\n\nPulver glanced up at the Seth Thomas clock on the wall. \"About four and a half hours ago. Why do you ask?\"\n\nBell identified himself and briefly explained his chase of Cromwell.\n\nPulver stared Bell in the eye. \"You say that freight car was carrying the notorious Butcher Bandit?\"\n\n\"He was on it, yes.\"\n\n\"If only I had known, I'd have told the sheriff.\"\n\nThe time gap was less than Bell had dared hope. \"Do you have a relay crew available? Mine is worn out, after their record run over the Sierras.\"\n\n\"Who's your crew?\"\n\n\"Lofgren and Long.\"\n\nPulver laughed. \"I might have known those two would try to beat their own record.\" He studied a blackboard on one wall. \"I have a crew on hand.\" He paused. \"I thought there was something funny about that train. Reno is a relay stop for just about every train going either east or west. Highly unusual, not taking on a relay crew. Your bandit won't get far with an engineer and fireman who are used up.\"\n\nBell looked down at the telegrapher, a bald-headed man with a green visor perched on his forehead and garters on his shirtsleeves. \"I'd like to alert lawmen in the towns ahead to stop the train and seize the bandit, whose name is Jacob Cromwell.\"\n\nThe telegrapher shook his head. \"No can do. The lines are down. I can't get a message through to the east.\"\n\nBell said, \"I'll lay money Cromwell is cutting the lines.\"\n\nPulver studied a large blackboard on another wall that showed the trains scheduled to pass through Reno. \"I'll have a crew for you in twenty minutes. You should have a clear run until you reach Elko. After that, I hope you'll find the telegraph in operation or you'll run the risk of colliding with a train traveling west.\"\n\n\"In that case,\" Bell said cynically, \"I'll have the satisfaction of knowing Cromwell collided with it first.\"\n\n## 44\n\nADELINEWAS HITTING HER STRIDE ON THE FLAT, open stretch of track. She was touching ninety miles an hour, roaring across trestles over dry gulches, flying through small towns, and hurtling past signals indicating open track ahead. The telegraph poles running alongside the track swept by in a confused blur. Gray smoke tinged with sparks and cinders spewed from the stack, streaming back in a horizontal cloud over the cab and flattened by the head-on rush of wind.\n\nA doleful, flaxen-haired descendant of the Vikings, Russ Jongewaard, sat in the engineer's seat, one hand on the throttle, while Bill Shea, a tall, humorous Irishman, shoveled coal into the firebox. After hearing from Bell that he was in a do-or-die attempt to capture the famed Butcher Bandit, they gladly came aboard to join the chase.\n\nLofgren and Long stayed aboard, too. \"We're volunteering for the duration,\" said Lofgren. \"With the four of us spelling each other, we won't have to stop for another relay crew.\"\n\nBell pitched in with the coal-shoveling duties. His thigh wound from Cromwell's bullet in Telluride had not completely healed, but as long as he didn't put too much weight on it there was little pain. His scoop held half as much coal as those that Long or Shea pitched in the firebox, but he made up for it with two shovels to their one.\n\nThe two Southern Pacific firemen took turns keeping an eye on the water gauge and watching the steam gauge, making sure it showed their fire was burning well and the engine was operating at just under two hundred pounds of steam pressure, within a hair of the redline mark. They studied the smoke coming from the stack. When it started to go from gray to clear, they added more coal. When it turned black, it meant the fire was too thick and they had to ease off.\n\nA competition, unchallenged and unspoken, developed between Lofgren and Jongewaard, but it did not go unnoticed. Adeline may have shown the immense power of her machinery and the lightning speed of her churning drive wheels, but it was the strength and endurance of the men who drove her to her limits that set records across Nevada that day. The engineers had the bit in their teeth and worked hard to catch the train of the killer of so many innocent people.\n\nSeeing the semaphore that signaled the track was clear beyond Elko, Lofgren kept the throttle against its stop as he swept past the depot at ninety-five miles an hour. People waiting on the platform for a passenger train stared aghast as Adeline shot by like an immense cannonball.\n\nFortunately, junctions were few and far between\u2014a few spur lines running off the main track\u2014so they kept up their rapid speed without slowing. Then agonizing slowdowns began to occur at the town of Wells, and again farther up the track at Promontory, to allow westbound relief trains through. The delays were utilized by taking on coal and water, but a total of eighty minutes was lost.\n\nAt each stop, Bell questioned the stationmasters about Cromwell's train. At Wells, the stationmaster told him that the engineer and fireman who had driven Cromwell's train from Oakland had been found by a section hand checking the ties and rails. He'd had them brought into town, barely able to stand because they were so fatigued and dehydrated. They had confirmed what Bell had feared: Cromwell had frequently ordered the train to stop so his hired gun could climb the poles and cut the wires.\n\n\"How are we doing?\" asked Lofgren when Bell climbed back in the cab.\n\n\"The stationmaster said they passed through three hours ago.\"\n\n\"Then we've picked up an hour and a half since Reno,\" Long said with a wide grin, knowing their untiring efforts were paying off.\n\n\"From here to Ogden, you'll have to keep out a sharp eye. Cromwell is cutting the telegraph wires. We'll be running blind, should we come upon a westbound train.\"\n\n\"Not a great threat,\" said Jongewaard. \"The company won't risk sending trains down the main line if they can't contact stationmasters to set schedules. Still, we'll have to be on the alert, especially around turns where we can't see more than a mile ahead.\"\n\n\"How far to Ogden?\" asked Bell.\n\n\"About fifty miles,\" replied Jongewaard. \"We should make the station in about an hour.\"\n\nWITH LOFGREN at the throttle, Adeline pulled into Ogden's Union Station forty-two minutes later. He was switched to the coal-and water-loading siding and brought the locomotive to a halt. Their routine was now well established. While Long and Shea loaded the coal and water, Lofgren and Jongewaard checked the engine and oiled the drive connectors and wheel bearings. Bell hurried into the big station and found the dispatcher's office.\n\nA pudgy man sat at a desk, staring out the window at an arriving passenger train. His interest was particularly taken by the young pretty women who showed ankles when stepping down the Pullman car steps. Bell read the name on a small sign sitting on the front of the desk.\n\n\"Mr. Johnston?\"\n\nJohnston looked Bell's way and smiled a friendly smile. \"Yes, I'm Johnston. What can I do for you?\"\n\nBell ran through his story of chasing Cromwell for perhaps the sixth time since leaving San Francisco. \"Can you tell me when the train came through?\"\n\n\"Never came through,\" answered Johnston.\n\n\"Never came through your station?\" Bell's thick eyebrows lifted toward his mane of blond hair.\n\n\"Yep,\" Johnston said, leaning back in his swivel chair and setting a booted foot on a pulled-out drawer. \"They were switched onto the line heading north.\"\n\n\"How?\" snapped Bell. \"It was not a scheduled train.\"\n\n\"Some rich woman showed papers to the dispatcher at the junction up the track that said she had chartered a train with right-of-way clearance to Missoula, Montana.\"\n\n\"The bandit's sister,\" said Bell. \"They're trying to reach the border and cross into Canada.\"\n\nJohnston nodded in understanding. \"The dispatcher checked with me on southbound trains. None was scheduled until tomorrow morning, so I told him to go ahead and allow the lady's train to travel north.\"\n\n\"When did this take place\"\n\n\"A little less than two hours ago.\"\n\n\"I've got to catch that train,\" Bell said firmly. \"I'd appreciate clearance to Missoula.\"\n\n\"Why not telegraph the sheriff in Butte to stop the train and take the bandit and his sister into custody?\"\n\n\"I've tried to do that since leaving Reno, but Cromwell has cut every telegraph line between here and there. No reason for him to stop now.\"\n\nJohnston looked stunned. \"My God, he could have caused a head-on collision.\"\n\n\"Until he and his sister reach the Canadian border, they have nothing to lose, even if it means killing anyone who gets in their way.\"\n\nShocked understanding had come to Johnston. \"Get that dirty coward,\" he said, desperation creeping into his voice. \"I'll gladly give you clearance through to Missoula.\"\n\n\"I'm grateful for any help you can give,\" said Bell sincerely.\n\n\"What's your train number?\"\n\n\"No train, only a tender and engine number 3455.\"\n\n\"What kind of engine?\"\n\n\"A Baldwin Atlantic 4-4-2,\" answered Bell.\n\n\"She's a fast one. What about relay crews?\"\n\n\"I have two crews who insist on sticking to the chase until we grab the bandit.\"\n\n\"In that case, all I can do is wish you luck.\" Johnston rose and shook Bell's hand.\n\n\"Thank you.\"\n\n\"Two hours is a hell of a lead,\" said Johnston quietly.\n\n\"We gained two and a half since leaving Oakland.\"\n\nJohnston thought a moment. \"You've got a real chase on your hands. It will be close.\"\n\n\"I'll stop him,\" Bell said gamely. \"I've got to stop him or he'll kill again.\"\n\n## 45\n\nTHERE WAS HOPE IN THE HEARTS OF THE MEN WHO sweated and toiled to drive Adeline over the rails. They had all risen up and reached beyond themselves to do the impossible. Men and women who worked the farms and ranches alongside the track stopped their labor and stared in surprise at the speeding lone locomotive that shrieked its whistle in the distance and thundered past beyond their sight in less than a full minute except for the lingering trail of smoke.\n\nWith Lofgren in the driver's seat, he pressed Adeline harder and harder until they swept over the border from Utah to Idaho at a speed of nearly one hundred miles an hour. Pocatello, Blackfoot, and Idaho Falls came and went. Stationmasters could only stand in shock and confusion, not able to comprehend a locomotive and tender that came out of nowhere with no advance warning and plunged past their depots at unheard-of speed.\n\nBefore they raced out of Ogden, Bell had procured a pile of blankets so the crews could catnap between shifts of driving the locomotive and feeding its boiler. At first, they found it impossible to sleep for short periods because of the clamor of the drive train, the hiss of steam, and the clatter of the steel wheels over the rails. But as exhaustion set in, they found it easier and easier to drift off until their turn came at the scoop and throttle again.\n\nExcept for quick stops for coal and water, Adeline never slowed down. At one stop, in Spencer, Idaho, Bell learned that they were only fifty minutes behind Cromwell's train. Knowing they were rapidly closing the distance inspired them to renew their efforts and work even harder.\n\nThe mystery in Bell's mind was the report given him by the Spencer stationmaster. It seemed that the Southern Pacific main track stopped at Missoula, with only a spur that went another eighty miles to the small port of Woods Bay, Montana, on Flathead Lake.\n\n\"How do you read it?\" Lofgren asked Bell after his place at the throttle was taken by Jongewaard.\n\n\"Cromwell must have found another crew after driving the engineer and fireman from Winnemucca half to death,\" said Bell.\n\nLofgren nodded. \"Without telegraph messages coming through and informing us otherwise, I have to believe he dumped them in the middle of nowhere, too, and forced a relay crew to come aboard for the final dash across the border.\"\n\n\"Then he'll have to do it by driving over a road in an automobile.\"\n\nLofgren looked at him. \"Why do you say that?\"\n\nBell shrugged. \"The stationmaster at Spencer told me Southern Pacific's tracks end at Woods Bay on the east shore of Flathead Lake. I assume the only way Cromwell can continue north into Canada is by road.\"\n\n\"I disagree. My guess is, he's going to take his train onto the car ferry that runs across the lake.\"\n\nBell stared at Lofgren questioningly. \"Car ferry?\"\n\nLofgren nodded. \"Logs from timber operations in Canada are hauled on flatcars across the border to a small port on the west side of the lake called Rollins. From there, they are rolled onto a ferry that carries them across the lake. When they reach Woods Bay, they are coupled to trains that transport them to lumberyards around the Southwest.\"\n\n\"Why doesn't Southern Pacific simply run their tracks north to Canada?\"\n\n\"The Great Northern Railroad received land rights from the government to cross the upper United States. They laid tracks that run from a landing on the west shore of Flathead Lake north to the border, where their locomotives are coupled to flatcars carrying logs hauled by the Canadian Pacific Railroad from the logging camps. Officials from both Great Northern and Southern Pacific refused to work together and never laid tracks that merged around the ends of the lake.\"\n\n\"How do you know all this?\"\n\n\"My uncle lives in Kalispell, above the lake. He's retired now, but he was an engineer for the Great Northern Railroad. He drove an engine between Spokane and Helena.\"\n\nThe interest in Bell's voice gave way to trepidation. \"So what you're telling me is that Cromwell can ferry his train across the lake to the Northern Pacific tracks and go north into Canada without stepping off his freight car.\"\n\n\"That's about the size of it.\"\n\n\"If he gets across on the car ferry before we can stop him...\" His voice trailed off.\n\nLofgren saw the apprehension in Bell's eyes. \"Don't worry, Isaac,\" he said confidently. \"Cromwell can't be more than ten miles up the track ahead of us. We'll catch him.\"\n\nFor a long moment, Bell said nothing. Then he slowly reached in a breast pocket and pulled out a piece of paper. Slowly, he unfolded it and handed it to Lofgren.\n\nThe engineer studied and then spoke without looking up. \"It looks like a list of names.\"\n\n\"It is.\"\n\n\"Names of who?\"\n\nBell dropped his voice until it was barely audible above the clangor of the charging locomotive. \"The men, women, and children Cromwell murdered. I've been carrying it since I was put in charge of chasing him down.\"\n\nLofgren's eyes lifted and gazed through the front window of the track ahead. \"The others should see this.\"\n\nBell nodded. \"I think now is an appropriate time.\"\n\nTHREE HOURS LATER, with Lofgren back on the throttle, Adeline began to slow as she came into Missoula. He brought the locomotive to a halt fifty feet before a switch stand. Shea jumped from the cab, ran up the track, and switched the rails to those of the spur leading to Flathead Lake. He ignored the switchman, who came running out of a shack.\n\n\"Here, what are you doing?\" demanded the switchman, who was bundled up against a cold wind.\n\n\"No time to explain,\" said Shea as he waved to Lofgren, signaling that it was safe to roll onto the spur from the main track. He looked at the switchman as Adeline slowly rolled past and said, \"Did another train pass onto the spur in the last hour?\"\n\nThe switchman nodded. \"They switched onto the spur without permission either.\"\n\n\"How long ago?\" Shea demanded.\n\n\"About twenty minutes.\"\n\nWithout replying, Shea ran after Adeline and pulled himself up into the cabin. \"According to the switchman,\" he reported, \"Cromwell's train passed onto the spur twenty minutes ago.\"\n\n\"Eighty miles to make up twenty minutes,\" Jongewaard said thoughtfully. \"It will be a near thing.\" He pulled open the throttle to the last notch and, five minutes after leaving the junction, he had Adeline pounding over the rails at eighty-five miles an hour.\n\nFlathead Lake came into view as they ran up the eastern shore. The largest freshwater lake in the western United States, it was twenty-eight miles long, sixteen miles wide, and covered one hundred eighty-eight square miles, with an average depth of one hundred sixty-four feet.\n\nThey were in the homestretch now of a long and grueling chase. Lofgren sat in the fireman's seat and helped Jongewaard survey the track ahead. Bell, Shea, and Long formed a scoop brigade to feed the firebox. Not having leather gloves like the firemen, Bell wrapped his hands with rags the engineers used to wipe oil. The protection helped, but blisters were beginning to rise on his palms from the long hours of shoveling coal.\n\nThey soon reached a speed higher than the spur tracks were ever built to endure from a speeding train. There was no slowdown over bridges and trestles. Curves were taken on the outer edge. One double-reverse turn they whipped around in a violent arc rattled the bolts in the tender. Luckily, the tracks then became as straight as the crow flew. Jongewaard held the eighty-five-mile-an-hour pace for the next forty miles.\n\n\"Eureka!\" Lofgren suddenly yelled, vigorously pointing ahead.\n\nEveryone leaned from the cab, the icy wind bringing tears to their eyes. But there it was, four, maybe five, miles directly ahead, a faint puff of smoke.\n\n## 46\n\nMARGARET LOUNGED ON A SETTEE, WEARING AN embroidered silk robe, and stared at the champagne bubbles rising in her saucer-shaped coupe glass. \"I wonder if it's true,\" she said softly.\n\nCromwell looked at her. \"What's true?\"\n\n\"That this glass was modeled from the breast of Marie Antoinette.\"\n\nCromwell laughed. \"There is an element of truth in the legend, yes.\"\n\nThen Margaret gazed out the window Cromwell had raised in the back of the car\u2014it was recessed into the rear wall and was inconspicuous when closed. The tracks that flashed under the wheels seemed to be stretching to infinity. She could see that they were traveling through a valley surrounded by forested mountains.\n\n\"Where are we?\"\n\n\"The Flathead Valley in the heart of the Rocky Mountains.\"\n\n\"How much farther to the border?\"\n\n\"Another thirty minutes to the ferry landing at Flathead Lake,\" said Cromwell, opening their second bottle of champagne for the day. \"Half an hour to cross over onto the Great Northern tracks and we'll be in Canada by sunset.\"\n\nShe held up her glass. \"To you, brother, and a brilliant flight from San Francisco. May our new endeavor be as successful as the last.\"\n\nCromwell smiled smugly. \"I'll drink to that.\"\n\nAHEAD, in the cab of the locomotive, Abner was pressing the crew he had abducted at gunpoint from a small caf\u00e9 beside the railyard in Brigham City, Utah: Leigh Hunt, a curly-red-haired engineer, and his fireman, Bob Carr, a husky individual who had worked as a brakeman before becoming a fireman, a step he hoped would eventually lead to his becoming an engineer. They had just come off a run and were having a cup of coffee before heading home when Abner held his gun to their heads and forced them into the cab of the engine pulling Cromwell's fancy freight car.\n\nAs was the earlier crew, Wilbanks and Hall were thrown from the engine in the middle of nowhere at the same time Abner cut the telegraph wires.\n\nAbner sat on the roof of the tender above the cab so he could prod Hunt and Carr to keep the Pacific locomotive hurtling over the rails to Flathead Lake. The black swirling clouds over the Rocky Mountains to the east caught his attention.\n\n\"Looks like a storm brewing,\" he shouted to Carr.\n\n\"A chinook, by the look of it,\" Carr yelled back over his shoulder as he scooped coal into the firebox.\n\n\"What's a chinook?\" asked Abner.\n\n\"It's a windstorm that roars down out of the Rockies. Temperatures can drop as much as forty degrees in an hour and winds can blow over a hundred miles an hour, enough to blow railcars off the tracks.\"\n\n\"How long before it strikes here?\"\n\n\"Maybe an hour,\" replied Carr. \"About the time we'll reach the train ferry dock at Woods Bay. Once we arrive, you'll have to sit out the storm. The ferry won't sail during a chinook.\"\n\n\"Why not?\" Abner demanded.\n\n\"With hundred-mile-an-hour winds, the lake turns into a frenzy. The wind whips the water into waves as high as twenty feet. The train ferry wasn't built for rough water. No way the crew will take it out on the lake during a chinook.\"\n\n\"We telegraphed ahead to have the ferry waiting for our arrival,\" said Abner. \"We're going across, wind or no wind.\"\n\nBACK IN Cromwell's rolling palace, Margaret had drifted into a light sleep from the champagne while her brother sat and relaxed with a newspaper he'd picked up when Abner abducted the train crew at Brigham City. Most of the news was about the San Francisco earthquake. He read that the fires had finally been put out and wondered if his mansion on Nob Hill and the bank building had survived.\n\nHe looked up, hearing a strange sound different from the clack of the steel wheels on steel rails. It came faint and far off. He stiffened as he recognized it as a train whistle. Cromwell was stunned, knowing now for certain that he was being pursued.\n\n\"Bell!\" he exploded in anger.\n\nStartled at his loud voice, Margaret sat up awake. \"What are you shouting about?\"\n\n\"Bell!\" snapped Cromwell. \"He's chased us from San Francisco.\"\n\n\"What are you saying?\"\n\n\"Listen,\" he ordered her. \"Listen.\"\n\nThen she heard it, the unmistakable sound of a train's steam whistle, barely perceptible, but it was there.\n\nMargaret rushed to the rear window and peered down the track. It was as if a fist had struck her in the stomach. She could see a stream of black smoke rising beyond a curve above a windbreak of small trees.\n\n\"You must tell Abner!\" she screamed.\n\nCromwell had anticipated her and climbed a ladder to the vent opening in the roof of the boxcar. He pushed aside the lid to the vent, stood above the roof, and shot off his revolver to get Abner's attention over the clamor of the locomotive. Abner heard and hurried over the top of the tender, until he was only a dozen feet away from Cromwell.\n\n\"There is a train coming up behind us,\" Cromwell shouted.\n\nAbner braced his feet apart against the pitch and roll of the tender and gazed over the roof of the boxcar. The onrushing train had come around the curve now and was visible in the distance. It looked to be a locomotive and its tender, pulling no cars. It was coming, and coming on fast, as evident from the smoke exploding from its stack and laid flat by the headwind.\n\nNow the two trains were in sight and sound of each other, with the ferry dock at Woods Bay on Flathead Lake only twenty miles away.\n\n## 47\n\nIT WAS AS IFADELINEWAS A COME-FROM-BEHIND THOROUGHBRED pounding around the far turn into the back stretch, flying through the herd and vying for the lead. Her connecting rods were a blur as they whipped the massive drive wheels over the rails. No locomotive ever worked harder. From the Oakland railyard to the wilds of Montana, she had covered more distance faster than any locomotive in history. No one timed her speeds, but none on board her cab or those who had seen her hurtle past ever doubted that she had surpassed ninety miles an hour on straight and level stretches of track.\n\nJongewaard had the throttle against its stop, throwing Adeline over tracks that were never meant for such speeds. Both engineers sat in the cab seats, their eyes fixed on the track ahead. Bell and Long shoveled while Shea systematically banked the fire to keep it burning evenly for the maximum amount of heat for proper combustion.\n\nThe chugging sound of the steam exhaust became one continuous hiss and the smoke poured from the stack in an ever-growing cloud. Bell stopped shoveling every so often to stare at the train ahead, growing larger by the minute. There was no effort to sneak up on Cromwell now; he pulled the cord and gave the whistle a long shriek that cut through the breeze beginning to blow in over the lake. Bell's lips were spread in a tight smile. He hoped Cromwell sensed that it was he who was charging down his throat.\n\nBell turned, looked up to the sky, and saw it had changed from a blue sea to a gray shroud from the chinook wind that howled out of the Rocky Mountains to the east. Great swirls of dust, leaves, and small debris were thrown like wheat chaff through a threshing machine. The water of Flathead Lake had gone from a dead calm to a turbulent mass in less than twenty minutes.\n\nThen, suddenly, both Jongewaard and Lofgren shouted at once: \"Wagon on the track!\"\n\nEvery eye swung and stared at the track ahead.\n\nA farmer with a hay wagon pulled by a team of horses was on a road crossing the tracks. He must have heard the engine's whistle, Bell thought, but the farmer had badly misjudged the speed of the train, believing he could cross the tracks in plenty of time. Jongewaard heaved back the reverser Johnson bar, slowing the drive wheels until they stopped and spun backward in reverse, braking the speeding locomotive.\n\nWhen the farmer realized the iron monster was only a hundred yards away, he whipped his horses in a frenzied attempt to drive them out of the path of onrushing death. By then, it was too late.\n\nAdeline plowed into the wagon in an explosion of hay, wooden planks, splinters, and shattered wheels. The men in the cab instinctively ducked behind the protection of the boiler as debris clattered along the sides of the engine and flew over the roof onto the tender.\n\nMiraculously, the horses had jumped forward and escaped without harm. Bell and the others did not witness the farmer's fate. As soon as Jongewaard brought Adeline to a halt a hundred yards down the track, Bell and crew leaped from the cab and ran back to the crossing.\n\nThey were all vastly relieved to find the farmer lying no more than five feet from the rails with all his hands and feet intact. He had pushed himself to a sitting position and was looking around, befuddled at the demolished wreckage of his wagon.\n\n\"Are you injured?\" asked Bell anxiously.\n\nThe farmer surveyed his arms and legs while feeling a rising bump on the head. \"A rash of bumps and bruises,\" he muttered. \"But, wonder of wonders, I'm still in one piece, praise the Lord.\"\n\n\"Your horses also survived without injury.\"\n\nShea and Long helped the farmer to his feet. And led him to the horses that had seemingly forgotten their narrow brush with death and were eating the grass beside the road. He was glad to see his horses in sound shape but angered that his wagon was scattered over the landscape in a hundred pieces.\n\nBell read his mind and gave him a Van Dorn card. \"Contact my detective agency,\" he instructed. \"They will compensate you for the loss of your wagon.\"\n\n\"Not the railroad?\" the farmer asked, confused.\n\n\"It wasn't the railroad's fault. A long story you'll read about in the newspapers.\" Bell turned and gazed in frustration down the tracks at the fading smoke from Cromwell's locomotive. He refused to believe he had failed so close to his goal. But all was not lost. Already, Jongewaard had backed up Adeline to pick up Bell and the crew.\n\nSeeing the farmer able to fend for himself, Jongewaard yelled to Bell. \"Hop aboard. We've got time to make up.\"\n\nBell, Lofgren, and the fireman had barely climbed back into the cabin when Jongewaard had Adeline barreling down the rails once again in hot pursuit of the bandit's train. The chinook was upon them now. The wind blew the dust and loose foliage like foam flying from surf plunging onto a beach. Visibility had been cut to no more than two hundred yards.\n\nJongewaard could no longer peer out the side of the cab or his squinting eyes would have filled with the dust. Instead, he stared though the cab's forward window, having no choice but to cut Adeline's speed from seventy-five miles an hour down to forty-five.\n\nHe saw a semaphore beside the track with its flag in the horizontal position, signaling the locomotive to stop, but he ignored it. Next came a sign proclaiming the outer town limits of Woods Bay. Not knowing the exact distance to the ferry landing, he slowed down even more until Adeline was creeping along at twenty-five miles an hour.\n\nJongewaard turned to Bell. \"Sorry about the slowdown, but I can't see if the town docks are five hundred yards or five miles away. I've got to lower the speed in case we come upon the bandit's freight car, or flatcars with logs, sitting on the main track.\"\n\n\"How much time do you figure we lost?\" asked Bell.\n\n\"Twelve minutes, by my watch.\"\n\n\"We'll catch them,\" said Bell with measured confidence. \"Not likely the ferry crew will risk crossing the lake in this weather.\"\n\nBell was right about the ferry not normally running across the lake in rough water, but he missed the boat by underestimating Cromwell. The Butcher Bandit and his sister had not come this far to surrender meekly.\n\nCromwell and Margaret were not to be stopped. Already, their train was rolling across the dock onto the ferry.\n\n## 48\n\nTHE RAILCAR FERRY WAS WAITING AT THE DOCK WHEN Cromwell's train arrived. The locomotive was switched onto the track that led across the wooden dock onto the ferry. But that was as far as it would go. The three-man crew had decided it wasn't safe to attempt a crossing until the chinook passed and the lake's surface settled down. They were sitting in the small galley drinking coffee and reading newspapers and did not bother to get up when Cromwell's train rolled on board.\n\nCromwell stepped down from his freight car and walked to the locomotive, bending into the stiff wind. He paused and studied the waves that were building and chopping on the lake. It reminded him of a furious sea. Then he studied the side-paddle, steam-powered ferryboat.\n\nA faded wooden sign attached to the wheelhouse read KALISPELL. The boat was old. The paint was chipped and peeling, the wooden deck worn and rotted. It had seen many years of service\u2014too many. But to Cromwell it looked sturdy enough to endure the severe wind and the valleys forming between the growing waves. He felt secure that it could steam to the west side of the lake. He was irritated at seeing no sign of the crew.\n\nHe looked up the track and felt gratified that the pursuing train was not in sight. He could only wonder why it became delayed. Whatever the reason, there was no time to dally. He waved to Abner in the cab of the locomotive. \"See that the fireman feeds the firebox so we have steam when we reach the Great Northern tracks.\"\n\n\"Consider it done,\" replied Abner, pointing the muzzle of his gun at fireman Carr, who had overheard the conversation. \"You heard the man. Keep shoveling.\"\n\n\"Have you seen the boat crew?\"\n\nAbner shrugged. \"I've seen no one.\"\n\n\"Better roust them. We've got to get under way. That locomotive behind us may arrive any minute.\"\n\n\"What about the train crew?\" said Abner. \"If I leave them alone, they might make a run for it.\"\n\n\"You cast off the lines,\" Cromwell ordered. \"They can't go anywhere if we drift away from the dock. I'll look for the boat crew myself.\"\n\nAbner jumped to the deck, ran onto the dock. He found the bow and stern lines securing the ferry. The waves surged in from the middle of the lake and rocked the boat back and forth against the bumpers hanging along the starboard paddle box. Abner waited while the boat drifted away from the dock and the lines became taut. When the water surged back, the lines became slack, and Abner pulled them off their bollards and threw them over the railings of the Kalispell. Agile as a cat, he leaped back on the deck and returned to the cab of the locomotive.\n\nCromwell climbed a ladder to the wheelhouse and was thankful to get inside out of the howling wind. He found it empty and went down a stairwell that led to the galley, where he found the crew sitting around reading impassively. They looked up as he came down the stairs but showed little sign of response or interest.\n\n\"You Mr. Cromwell?\" said a big, red-faced, heavily bearded man in a red plaid lumberman's coat.\n\n\"Yes, I'm Cromwell.\"\n\n\"We heard your train come onboard. I'm Captain Jack Boss, at your service.\"\n\nThe laid-back attitude of Boss, who remained sitting, and his two-man crew, who showed utmost indifference, angered Cromwell. \"It is of the greatest importance that we get under way immediately.\"\n\nBoss shook his head. \"No can do. The lake is kicking up. It's best if we wait until the storm blows over.\"\n\nAs calmly as if he were lighting a cigar, Cromwell pulled his .38 Colt from a coat pocket and shot one of the crewmen in the forehead. The surprise was so complete the crewman slumped over and stared blankly, as if he were still reading newsprint.\n\n\"Good God!\" was all Boss uttered, his face frozen in shock.\n\nCromwell pointed his gun at the face of the other crewman, who began to shake uncontrollably. \"You will get this boat under way immediately or he goes, too.\"\n\n\"You're crazy,\" gasped Boss.\n\n\"My attendant has already cast off the lines. I suggest you waste no more time protesting.\"\n\nBoss looked at his dead crewman and slowly, dazedly, came to his feet. He glared at Cromwell with a combined expression of disgust and fury. \"You might as well shoot the rest of us,\" he said slowly. \"We'll all die before we get to the other shore.\"\n\n\"A chance we have to take,\" Cromwell said, his voice hard and venomous.\n\nBoss turned to his crewman, Mark Ragan. \"You'll have to operate the engine alone.\"\n\nRagan, a young man yet to see seventeen, nodded with a pale face. \"I can do it.\"\n\n\"Then stoke the boiler and get up enough steam to make good headway.\"\n\nThe crewman left the galley quickly and dropped down a ladder to the engine room. Boss, closely followed by Cromwell, climbed to the wheelhouse.\n\nCromwell stared at Boss. \"Do not even think about going against my instructions, Captain, or your crewman in the engine room will die. Nor will I have any reservation of killing you, should you not take me to the rail landing on the far shore.\"\n\n\"You're diabolical scum,\" Boss said, his face twisted with rage.\n\nCromwell laughed and gave Boss a look as cold as death. Then he turned and left the wheelhouse.\n\nAs he walked back to his palace boxcar, he heard the shrill blare of a steam whistle. It sounded as if it came from no more than a few hundred yards away. And then his ears caught the hiss of steam and the clatter of locomotive drive wheels. Through the debris hurled by the gusts from the chinook, he saw a large engine materialize from the gloom.\n\nToo late, he thought complacently. The Kalispell had already drifted five feet from the end of the dock. No one or nothing could stop him now. Smiling to himself, he made his way back the boxcar and climbed inside.\n\nJONGEWAARD BROUGHTAdeline to a grinding halt only thirty feet from the end of the dock's tracks. Even before the big drive wheels stopped turning, Bell hopped from the cab and ran toward the end of the dock. The railcar ferry was drifting past the pilings out into the lake and the paddle wheels began to turn. The gap had broadened to eight feet when Bell reached the dock's edge.\n\nHe did not hesitate, did not think about or analyze his actions, did not step back for a running start. It seemed too far, but without an instant's interruption, he leaped from the dock. Knowing the distance was too great for him to land on his feet, he reached out and grabbed the railing with his hands, his body falling and swinging like a pendulum against the hull of the ferry. He came within a hair of losing his grip and falling in the water, as the impact knocked the wind from his body. He held the railing in a death grip until his breath returned, but the growing ache in his chest did not fade. Slowly, almost agonizingly, he pulled himself over the railing onto the deck of the ferry alongside Cromwell's boxcar.\n\nBell lightly ran his fingers over his chest and realized he had cracked one, maybe two ribs. Clenching his teeth against the pain, he struggled to his feet and grabbed one of the ladder rungs leading to the roof of the boxcar to support himself from the pitching and heaving of the ship, plowing into the teeth of the chinook. As the Kalispell moved farther into the middle of the lake, the windswept waves surged over the bow and onto the low track deck, swirling around the wheels of the locomotive. The terrible winds brought a stunning rise in temperature of over twenty degrees.\n\nBell cast off any thought of caution. He threw open the loading door of the boxcar and rolled onto the floor, gasping from the agony in his chest, the .45 Colt steady in his hand. Surprise was in his favor. Cromwell was not alarmed, believing that it was Abner who was entering the car. Too late, he saw that it was his worst enemy.\n\n\"Hello, Jacob,\" Bell said with a cordial grin. \"Did you miss me?\"\n\nThere came a few moments of stunned stillness.\n\nBell came to his knees and then his feet, keeping the Colt aimed at Cromwell's heart, and closed the door to the boxcar to seal it off from the gusts of wind that were battering the old ferry. He made a quick scan of the interior of the car. \"Well, well, well,\" he said with interest. \"My compliments.\" He swung his free hand around the exotically furnished car. \"So this is how you escaped your crimes in style.\"\n\n\"I'm glad you approve,\" Cromwell said conversationally.\n\nBell smiled in narrow-eyed guardedness without lowering his Colt. He glanced at the leather trunks lined against one wall. \"The cash from your bank. Must be an impressive amount.\"\n\n\"Enough to initiate a new enterprise,\" Cromwell answered cordially.\n\n\"You followed us?\" Margaret said, baffled and incredulous. It was more a question than a statement.\n\n\"Not exactly followed,\" Bell said curtly. \"More like chased.\"\n\nPredictably, Cromwell recovered his composure. \"How did you arrive so quickly?\"\n\n\"Fortunately, I had a faster engine and dedicated crewmen.\"\n\n\"You knew Margaret and I left San Francisco?\"\n\n\"I tracked down this freight car and figured you had it repainted with a new serial number. My agents had it under surveillance, waiting for the moment when you would use it again. Unfortunately, the earthquake came and my agents had more-pressing duties elsewhere.\"\n\n\"And you discovered that it had left the railyard,\" Cromwell assumed.\n\nBell nodded. \"Only after I went to your bank and saw that you had cleaned out the vault of all large-currency bills.\"\n\n\"But how could you have known we were heading for Canada?\"\n\n\"The dispatcher at the Southern Pacific office,\" Bell said, lying so as not to involve Marion. \"I put a gun to his head and persuaded him to tell me what tracks your chartered train was traveling. Then it was only a matter of filling in the cracks.\"\n\n\"Very ingenious, Mr. Bell.\" Cromwell, champagne glass in hand, stared at Bell appraisingly. \"It seems I have a penchant for underestimating you.\"\n\n\"I've misjudged you a time or two.\"\n\nMargaret spoke in a tone barely above a whisper. \"What do you intend to do?\" Her shock had turned to desperation.\n\n\"Hold your brother for the local sheriff after we reach shore. Then assemble the necessary papers to escort the two of you to Chicago, where he'll have a speedy trial without a fixed jury of your old pals and hang for his crimes.\" Bell's smile turned cold and his voice ominous. \"And you, dear Margaret, will probably spend the best years of your life in a federal jail.\"\n\nBell caught the exchange of knowing looks between Cromwell and Margaret. He could only wonder what they were thinking, but he was pretty sure it didn't bode well. He watched as Cromwell sank into one end of an ornate couch.\n\n\"Our voyage may take a while in this weather.\" As if to accent his statement, the bottle of champagne slid off its table and onto the floor. \"A pity. I was going to offer you a drink.\"\n\nBell could only guess where Cromwell kept his Colt .38. \"I never drink while on duty,\" Bell said facetiously.\n\nThe car took another sudden lurch as the ferry tipped over to one side, the entire hull vibrating as one of the paddle wheels thrashed out of the water. Margaret gasped in fear and stared at the water that was seeping in widening puddles along the bottom of the freight door.\n\nOUTSIDE, the wind shrieked, and the Kalispell creaked and groaned against the onslaught of the mounting waves that rolled down the length of Flathead Lake. The tired old vessel burrowed her bow into the gale-driven crests before dropping sickeningly into the troughs. A towering wave broke out the forward windows, sending sheets of water into the wheelhouse.\n\nCaptain Boss pulled up his coat collar and grasped the helm desperately as the gale lashed him with spray that stung the skin of his face and hands.\n\nA whistle shrilled through the speaking tube from the engine room. Boss picked it up, said, \"Wheelhouse.\"\n\nRagan's voice came with a hollow tone through the tube. \"I'm taking water down here, Captain.\"\n\n\"Can the pumps handle it?\"\n\n\"So far. But the hull is creaking something awful. I fear the bulkheads might give way.\"\n\n\"Get ready to clear out if it gets bad. Make your way to the roof over the galley and unlash the raft.\"\n\n\"Yes, sir,\" replied Ragan. \"What about you, Captain?\"\n\n\"Call me when you leave the engine room. I'll follow if I can.\"\n\n\"What about the people on the train? We can't just abandon them.\"\n\nBoss was a man with moral depth, a God-fearing man of great inner strength from the old school whose word was his bond. He was well respected by all who lived around the lake. He gazed through the broken wheelhouse window at the far shore and the mad water thrashing over the bow and felt certain the Kalispell was not going to make it.\n\n\"They're my responsibility,\" he said slowly. \"You save yourself.\"\n\n\"God bless you, Captain.\"\n\nThen the tube went silent.\n\n## 49\n\nTHE TORNADO WINDS OF THE CHINOOK WERE THE most destructive in memory. Barns were flattened, roofs carried away, trees ripped by their roots, and telegraph and telephone lines downed. The full force of the warm winds roared over Flathead Lake and flogged the water into a swirling turbulence that battered the weary old Kalispell unmercifully as she wallowed in the valleys between the waves. Already, the lifeboat that Captain Boss had hoped would save lives had been torn from its lashed mounting and shattered, its wreckage swept into the restless water.\n\nBoss struggled with the helm in a desperate attempt to keep the Kalispell on a straight course toward the west shoreline, now only two miles away. He nurtured a slim hope that they might reach the safety of the little harbor of Rollins, but, deep inside, he knew the odds were stacked against him and his boat. There was a constant danger the ferry might swing. The engine, tender, and freight car were the straws that would break the camel's back.\n\nWithout their weight, the Kalispell might have ridden higher in the water and not have suffered as badly from the huge waves that swept across her lowered track deck. Boss looked down at the bow and saw that it was badly damaged. Timbers on the exposed part of the bow were being smashed and torn from their beams.\n\nHis clothes and lumberman's coat soaked through to his skin, Boss grimly took one hand off the wheel, held the speaking tube to his mouth, and whistled. There was a lag of nearly thirty seconds before Ragan answered.\n\n\"Yes, Captain?\"\n\n\"How does it look down there?\"\n\n\"I've got good steam, but the water is still rising.\" Ragan's voice was tinged with fear. \"It's over my ankles.\"\n\n\"When it gets to your knees, get out of there,\" Boss ordered him.\n\n\"Do you still want me to unlash the boat?\" Ragan asked anxiously.\n\n\"You don't have to bother,\" Boss said bitterly. \"It's been swept away.\"\n\nThe fear was noticeably heavier in Ragan's voice now. \"What will we do if we have to abandon the boat?\"\n\nBoss said flatly, \"Pray there's enough loose wreckage that will float that we can hang on to until this storm blows over.\"\n\nBoss hung up the speaking tube and gave a mighty heave of the wheel to keep the boat moving against the swell, as a huge wave fell against the left bow of the Kalispell and shoved her broadside to the surge. This was what Boss was afraid of. Caught by a huge wave on the side of the hull and unable to recover, the ferry would capsize and then sink like a stone under the weight of the train.\n\nFighting the ferry around to head into the teeth of the gusts, he stole a glance down at the train and was stunned to see it violently rocking back and forth, as the boat fell into the troughs before being struck by the crests that now swirled around the engine's drive wheels.\n\nBell took little satisfaction in knowing that if the Kalispell sank into the depths of the lake the criminals on the train would die with him.\n\nIN THE locomotive, Hunt and Carr were hanging on to any valve, gauge, or lever within reach to keep from being flung against the boiler and sides of the cab. Abner sat in the fireman's seat, his feet braced against the front panel below the window. He felt no need to keep his gun aimed at the engineer and fireman. Not with everyone fighting to keep from being hurled about and becoming injured. He was no longer their threat. It was the storm around them that was menacing.\n\nThe last thing that occurred to Abner was that Hunt and Carr might conspire against him. He had not heard their exchanged muffled words nor seen their discreet hand signs to each other. There was nothing for him to do but stare with great trepidation at the vicious water battering the ferry. The engineer fell from his seat and reeled across the cab, colliding against Abner. The impact momentarily stunned Abner, but he roughly pushed Hunt back to his side of the cab.\n\nAbner did not pay any attention to Carr, as the fireman struggled to shovel coal into the firebox while fighting to keep his balance against the lurching and rolling of the Kalispell. Hunt staggered against him again. Irritated, Abner tried to heave the engineer back to his side of the cab. But, this time, Hunt had flung himself on Abner, pinning the big man's arms to his side. Then Hunt fell backward, pulling the startled and angered Abner down to the floor of the cab on top of him.\n\nGalvanized into action, Carr swung his coal scoop over his head and brought it down heavily between Abner's shoulder blades. The ferry plunged into a trough just as Carr swung with whiplash speed, but the scoop missed Abner's head and surely would have cracked his skull if it had connected. To Carr, it felt as though he had struck a fallen log.\n\nIt was a vicious blow, a bone-crushing blow, and it would have paralyzed most men and left them unconscious. Not Abner. He gasped, his face twisted in pain, and rolled off Hunt and came to his knees. He reached for his Smith & Wesson and leveled it at Carr. His face was expressionless and his eyes stared unblinkingly as he pulled the trigger. Carr's scoop was poised for another strike, but the fireman froze as the bullet drilled into the center of his chest. The shock threw him against the maze of valves before he slowly sank to his knees and keeled over onto the floor of the cab.\n\nWithout the slightest hesitation, Abner swung the muzzle of the revolver toward Hunt and shot the engineer in the stomach. Hunt doubled over, his eyes locked on Abner with cold hatred mixed with pain and shock. He staggered backward, one hand clutching his stomach, the other outstretched. Too late, Abner realized what Hunt had in mind. Before he could react, Hunt had reached out and struck the engine's brass horizontal brake lever, moving it from right to left. In his last act, the dying man swung his arm around the throttle lever, pulling it toward him as he fell dead.\n\nThe drive wheels spun and the locomotive lurched forward. Abner, weakened by the crushing blow to his back, was too slow to respond. There was a mist surrounding his vision, and it took a long three seconds to realize the locomotive was forging across the deck of the ferry. Any attempt to stop the inevitable came too late. By the time Abner could push back the throttle, the hundred-thirty-four-ton locomotive began its plunge off the Kalispell 's bow into the cold depths of Flathead Lake.\n\n## 50\n\nAT FIRST, NO ONE IN CROMWELL'S BOXCAR REALIZED the train was rolling off the ferry because of the violent motion caused by the waves and wind. Bell quickly distinguished a different movement and sensed the wheels beneath the car were beginning to turn. He threw open the freight door and was met by a blast of wind that staggered him for an instant. But then he lowered his head and leaned out. He took in two macabre sights at once. One, the deck appeared to be moving toward the stern because of the train's forward motion. And, two, the locomotive's front four-wheel truck was rolling off the bow and diving into the surging turbulence below.\n\nBell spun around. \"The train is falling off the boat!\" he shouted over the gale. \"Quick, jump while you still can!\"\n\nCromwell thought he saw an opportunity and did not immediately consider the disaster-in-the-making. Without a word, he launched himself off the couch and drew his automatic as he leaped. A foolish mistake. Instead of instantly squeezing the trigger and killing Bell, he hesitated to say, \"Farewell, Isaac.\"\n\nSuddenly, the hand clutching the gun was knocked off to the side and the bullet smashed into the doorframe beside Bell's head.\n\nMargaret stood in front of Cromwell, her dark eyes on fire and her lips pressed tightly together until she spoke. There was no fear, no fright; she stood with her legs firmly planted on the freight car's floor. \"Enough, Jacob,\" she said.\n\nShe had no time to say more. Bell grabbed her by the arm. \"Jump!\" he urged her. \"Quickly!\"\n\nOnly Bell grasped the inevitable. He glanced out the door again and saw the engine had almost disappeared beneath the waves and the tender and the freight car were moving faster as they were rapidly dragged by the immense weight into the water. The deck was tilted at a sharp angle, and the Kalispell was in dire danger of going down with the train. There were only seconds left before the freight car was pulled over the bow.\n\nHis face contorted with hate, Cromwell swung the Colt's muzzle toward Bell again, but Margaret stepped between the two men. Cromwell was finally aware of the danger now, his eyes sick in the realization that defeat and death were only moments away. He tried to push Margaret aside so he could leap out the door, but she wrapped her arms around her brother's waist, pulling him back inside the car. He swung the barrel of the gun and struck her across the cheek. Blood seeped, but she clutched him in a death grip he could not shake off.\n\nThe freight car's front wheels were irresistibly following the tender off the front of the ferry. Bell tried to yank Margaret out the door, but she was clutching her brother too tightly. The sleeve on her blue sweater tore away and he lost his grip on her arm.\n\nShe looked at Bell and her eyes turned soft. \"I'm sorry, Isaac.\"\n\nHe reached out for her but it was too late. Bell was falling through the door.\n\nHe twisted violently in midair and crashed to the wooden deck, striking the shoulder opposite his cracked ribs. The impact was still enough to make him gasp in agony. He lay there, watching in horror as Cromwell's freight car was drawn below the surface. A hope flashed through his mind that Margaret might still leap through the door and into the water and be saved. But it was not to be. A seething white wall of water washed over the boxcar and penetrated the interior through the open freight door with a surge that made it impossible for anyone inside to escape. Still hoping against hope, Bell lay on the deck, water swirling around him, staring at the bubbles rising from the depths as the ferry steamed over them. He was still staring at the place where the train sank when it fell far astern, but neither Margaret nor her brother came to the surface.\n\nThe bow of the ferry swung up and the hull rose nearly a foot out of the water without the hundreds of tons of deadweight from the train. Almost immediately, to the immense relief of Captain Boss in the wheelhouse, the Kalispell 's stability increased dramatically and she began to burrow through the waves, her paddle wheels driving her toward the western shore of the lake.\n\nIt took Bell nearly ten minutes to struggle across the deck to the door leading to the stairs up to the wheelhouse. When he got to the wheelhouse, looking like the proverbial drowned rat, Boss stared at him in astonishment.\n\n\"Well, now, where did you come from?\"\n\n\"I jumped on deck as you pulled away from the dock in Woods Bay. My name is Bell. I'm an agent with Van Dorn.\"\n\n\"You were lucky you didn't go down with the others.\"\n\n\"Yes,\" he said softly. \"I was lucky.\"\n\n\"Who were those people?\"\n\n\"Two were innocent members of the train crew who were held hostage. The other three were wanted for murder and robbery. I was going to arrest them when we reached port.\"\n\n\"Poor devils. Drownin' isn't a good way to go.\"\n\nBell was deeply marked by guilt and grief. His face was expressionless as he turned his gaze to the waters of the lake. The waves no longer looked deadly and were settling down to a mild chop. The chinook was moving east and the terrible winds had subsided to a stiff breeze.\n\n\"No,\" he murmured. \"Not a good way to go at all.\"\n\n\u2022 \u2022 \u2022\n\n[For a complete list of this author's books click here or visit  \nwww.penguin.com\/cusslerchecklist](http:\/\/penguin.com\/cusslerchecklist?OTC-CUSCHKLST)\n\n## UP FROM THE DEPTHS\n\n##\n\nAPRIL 16, 1950 FLATHEAD LAKE, MONTANA\n\nAFTER THE COAL TENDER WAS BROUGHT UP AND SET on the barge behind the big Pacific locomotive, the divers concentrated on running the steel lift cables under the bottom of the freight car and attaching them to a cradle so it could also be raised. Despite the muck and slime, the name SOUTHERN PACIFIC was still readable across the sides of the tender.\n\nLate in the afternoon, the director of the salvage operation, Bob Kaufman, paced the deck impatiently, as the divers were lifted from the bottom on a platform that was swung onto the barge. He looked up at the clouds, which were dark but not threatening, and lit a cigar while he waited for the brass helmet to be lifted off the dive master's head.\n\nAs soon as it was lifted from the diver's head, Kaufman asked, \"How's it look?\"\n\nThe diver, a balding man in his early forties, nodded. \"The cables are secured. You can tell the crane operator he can begin the lift.\"\n\nKaufman waved to the man who operated the big crane that rose skyward from the deck of the salvage barge. \"All cables secure!\" he shouted. \"Lift away!\" Then Kaufman turned and spoke to the tall, older, silver-haired man standing next to him on the deck of the barge. \"We're ready to raise the freight car, Mr. Bell.\"\n\nIsaac Bell nodded. His face was calm, but there was an expression of expectancy. \"All right, Mr. Kaufman. Let's see what it looks like after all these years on the bottom of the lake.\"\n\nThe crane operator engaged the lift levers, tightening the cables as the diesel engine on the crane rose from an idle to a high rpm before flattening out as it strained to hoist the freight car. The operation was not nearly as complex as bringing up the hundred-thirty-four-ton locomotive. Once the car was pulled free from the bottom, the lifting operation went smoothly.\n\nBell watched with an almost-morbid fascination as the freight car broke the surface of the water and was raised up high before the crane slowly swung it over the barge. Deftly coordinating the controls, the crane operator cautiously lowered the car until it settled onto the deck behind the locomotive and tender.\n\nGazing at the train, Bell found it hard to visualize in his mind how it looked so many years ago. He walked up to the car and wiped the lake growth away from the serial number that was barely visible through the oozing slime. The number 16455 now became distinct.\n\nBell looked up at the freight door. It was still as open as when he fell through it so long ago. The interior was dark because the sunlight was diminished by the clouds. Memories flooded back as he recalled that fateful day when the train rolled across the ferry and plummeted to the bottom of the lake. He dreaded what he would find inside.\n\nKaufman came with the ladder they had used to enter the locomotive's cab and propped it against the open floor of the freight car. \"After you, Mr. Bell.\"\n\nBell nodded silently and slowly mounted the ladder until he was standing on the threshold of the boxcar. He stared into the darkness and listened to the water dripping throughout the freight car. He suppressed a shudder. The dampness and the smell of muck and slime seemed to reek of death, hoary and evil and infinitely ghastly.\n\nThe once-ornate furnishings and decor of the palatial car now looked like something out of a nightmare. The plush-carpeted floor was covered with sediment decorated with long slender weeds. The intricately carved bar, the leather chairs and couch, the Tiffany lamps overhead, even the paintings on the walls, looked grotesque under their coating of ooze and growth. Small fish that had not escaped as the car came out of the water were flopping on the floor.\n\nAs if delaying the inevitable, Bell sloshed through the mud and found the five leather trunks along one wall where he remembered seeing them back in 1906. He pulled a folding knife from his pocket and pried open the rusting and nearly frozen latches on the first trunk. Lifting the lid, he saw that the interior was relatively free of silt. He carefully picked up one of the bundles. The paper currency was soggy but had held its shape and consistency. The printing on the gold certificate bills still appeared distinct and well defined.\n\nKaufman had joined Bell and stared fascinated at the stacks of bills stuffed in the trunk. \"How much do you reckon there is?\"\n\nBell closed the lid and motioned at the other four trunks. \"A wild guess? Maybe four or five million.\"\n\n\"What happens to it?\" asked Kaufman with a glint in his eyes.\n\n\"Goes back to the bank whose depositors were robbed of their savings.\"\n\n\"Better not let my crew know about this,\" said Kaufman seriously. \"They may get it in their heads it's open salvage.\"\n\nBell smiled. \"I'm certain the banking commissioners in San Francisco will be most generous in granting a reward to you and your crew.\"\n\nKaufman was satisfied, his gaze sweping through the car. \"This must have been one luxurious palace on wheels before it sank. I've never seen a boxcar fixed up like a private Pullman parlor car.\"\n\n\"No expense was spared,\" said Bell, eyeing several bottles of vintage champagne and expensive brandy that were scattered in the sediment on the floor.\n\nKaufman's expression turned grim as he nodded at two misshapen mounds protruding from the floor. \"These the two you were looking for?\"\n\nBell nodded solemnly. \"Jacob Cromwell, the infamous Butcher Bandit, and his sister, Margaret.\"\n\n\"The Butcher Bandit,\" Kaufman said softly in awe. \"I always thought he'd disappeared.\"\n\n\"A legend handed down through the years because the money was never recovered.\"\n\nThe adipose tissue that once stored Cromwell's fat had broken down and his body, like the corpses in the cab of the locomotive, had turned waxlike in saponification. The notorious killer looked less than something that had once been a living human being. It was as though he had melted into an indiscernible lump of brown gelatin. His body was twisted, as if he'd died writhing in terror when tons of water gushed into the freight car as it followed the locomotive down to the bottom of the lake. Bell knew better. Cromwell might have struggled to survive, but he would never have been gripped with terror. No longer was he a menacing figure. His reign of robbery and murder had ended forty-four years ago under the cold waters of Flathead Lake.\n\nHe waded through the muck to where Margaret's body lay. Her lustrous hair was fanned out in the silt and tangled with strands from a reedlike weed. The once-lovely face looked like a sculpture an artist had left unfinished. Bell could not help but remember her beauty and vivaciousness the night they met in the elevator of the Brown Palace Hotel.\n\nKaufman interrupted Bell's thoughts. \"His sister?\"\n\nBell nodded. He felt an overwhelming sense of sorrow and remorse. Her final words before he fell from the car came back to haunt him. He could never explain his feelings toward her. There was no endearing love on his part, more a fondness coated with hatred. There was no forgiving her criminal actions in league with her brother. She deserved to die as surely as he did.\n\n\"Can't tell from the look of her now,\" said Kaufman. \"She might have been a beautiful woman.\"\n\n\"Yes, she was that,\" said Bell softly. \"A beautiful woman full of life but veiled in evil.\" He turned away saddened but his eyes dry of tears.\n\nSHORTLY BEFORE midnight, the salvage barge tied up at the old railroad dock in Rollins. Bell made arrangements with Kaufman to see that the bodies were taken care of by the nearest mortuary and the next of kin of Hunt and Carr notified. He recognized Joseph Van Dorn standing on the dock surrounded by four of his agents and was not surprised to see him.\n\nVan Dorn was in his eighties but stood straight, with a full head of gray hair and eyes that never lost their gleam. Although his two sons now ran the detective agency from offices in Washington, D.C., he still worked out of his old office in Chicago and consulted on the cases that had never been solved.\n\nBell walked up and shook Van Dorn's hand. \"Good to see you, Joseph. It's been a long time.\"\n\nVan Dorn smiled broadly. \"My work isn't as interesting since you retired.\"\n\n\"Nothing could stop me from coming back on this case.\"\n\nVan Dorn stared at the freight car. Under the dim lights on the dock, it looked like some odious monster from the depths. \"Was it there?\" he asked.\n\n\"The money?\"\n\nBell merely nodded.\n\n\"And Cromwell?\"\n\n\"Both he and his sister, Margaret.\"\n\nVan Dorn sighed heavily. \"Then at long last it's over. We can write finish to the legend of the Butcher Bandit.\"\n\n\"Not many of the Cromwell Bank's depositors,\" Bell said slowly, \"will still be alive to receive their money.\"\n\n\"No, but their descendants will be notified of their windfall.\"\n\n\"I promised Kaufman and his crew a fat finder's fee.\"\n\n\"I'll see that they get it,\" Van Dorn promised. He placed a hand on Bell's shoulder. \"Nice work, Isaac. A pity we couldn't have found the train fifty years ago.\"\n\n\"The lake is two hundred seventy feet where the train sank,\" explained Bell. \"The salvage company that was hired by the San Francisco banking commissioners dragged the lake but couldn't find it back in 1907.\"\n\n\"How could they have missed it?\"\n\n\"It had fallen in a depression in the lake bed and the drag lines passed over it.\"\n\nVan Dorn turned and nodded toward a car parked by the dock. \"I guess you'll be heading home.\"\n\nBell nodded. \"My wife is waiting. We'll be driving back to California.\"\n\n\"San Francisco?\"\n\n\"I fell in love with the town during the investigation and decided to remain after the earthquake and make my home there. We live in Cromwell's old mansion on Nob Hill.\"\n\nBell left Van Dorn and walked across the dock to the parked car. The blue metallic paint of the 1950 Custom Super 8 convertible Packard gleamed under the dock lights. Although the night air was chilly, the top was down.\n\nA woman was sitting in the driver's seat wearing a stylish hat over hair that was tinted to its original blond. She gazed at him approaching with eyes that were as coral\u2013sea green as when Bell met her. The mirth lines around her eyes were the lines of someone who laughed easily, and the features of her face showed the signs of an enduring beauty.\n\nBell opened the door and slipped into the seat beside her. She leaned over and kissed him firmly on the lips, pulled back, and gave him a sly smile. \"About time you came back.\"\n\n\"It was a hard day,\" he said with a long sigh.\n\nMarion turned the ignition and started the car. \"You found what you were looking for?\"\n\n\"Jacob and Margaret and the money, all there.\"\n\nMarion looked out across the black water of the lake. \"I wish I could say I'm sorry, but I can't bring myself to feel grief, not knowing about their hideous crimes.\"\n\nBell did not wish to dwell on the Cromwells any longer and changed the subject. \"You talk to the kids?\"\n\nMarion stepped on the accelerator pedal and steered the car away from the dock toward the main road. \"All four this afternoon. Soon as we get home, they're throwing us an anniversary party.\"\n\nHe patted her on the knee. \"You in the mood for driving all night?\"\n\nShe smiled and kissed his hand. \"The sooner we get home, the better.\"\n\nThey went silent for a time, lost in their thoughts of events long gone. The curtain to the past had come down. Neither of them turned and looked back at the train.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**ESSAYS IN AESTHETICS**\n**_Philosophical eLibrary Editions_**\n\nof works by Jean-Paul Sartre\n\nBeing and Nothingness\n\nThe Emotions: Outline of a Theory\n\nEssays in Aesthetics\n\nExistential Psychoanalysis\n\nExistentialism and Human Emotions\n\nLiterature and Existentialism\n\nThe Philosophy of Existentialism: Selected Essays\n\nTo Freedom Condemned\n\nWhat is Literature? and Other Essays\n_A Philosophical eLibrary Edition_\n\nESSAYS IN AESTHETICS\n\nJEAN-PAUL SARTRE\n\nSelected and Translated by\n\nWade Baskin\n\nWith a New Preface by\n\nSusan Braudy\n\n_PHILOSOPHICAL LIBRARY_\n\n_New York_\nCONTENTS\n\nPREFACE\n\nSARTRE QUOTES\n\nARTIST GALLERY\n\nINTRODUCTION\n\nTHE VENETIAN PARIAH\n\nJACOPO'S SHENANIGANS\n\nTHE PURITANS OF THE RIALTO\n\nMAN AT BAY\n\nA MOLE IN THE SUN\n\nTHE PAINTINGS OF GIACOMETTI\n\nTHE UNPRIVILEGED PAINTER: LAPOUJADE\n\nTHE MOBILES OF CALDER\n\nTHE QUEST FOR THE ABSOLUTE\n\nINDEX\n\nBIBLIOGRAPHY\n\nABOUT THE AUTHOR\n\nJEAN-PAUL SARTRE\n\n_Print Collection, Miriam and Ira D. Wallach Division of Art, Prints and Photographs, The New York Public Library, Astor, Lenox and Tilden Foundations_\nPREFACE\n\nJean Paul Sartre was a total live wire. The man is still bigger than a rock star in France. He lived the life of the mind to its fullest. He also found time in the second half of his life to become an idealistic international politician, manipulating his world-wide reputation to further his ethical and artistic beliefs\u2014which he relished amending and even changing. Biographer Ronald Hayman wrote that Sartre's face was familiar to people who had never seen him; his beliefs and sayings were quoted by people who had never read his books. As Sartre himself once proclaimed euphorically, \"Fame is good, even at forty or fifty; fame is desirable; there is happiness, an intense enjoyment, in pushing one's way into the spotlight like this.\"\n\nAnd to his enormous credit, Sartre is still relevant and celebrated. It is the totality of his work\u2014his deliciously audacious, learned and personal essays such as those in this book, _Essays in_ _Aesthetics_ , as well as his great fiction like _Nausea_ , and his great plays such as _No Exit._\n\nSartre is acknowledged by such intellectuals as Bernard-Henri Levy as the outstanding philosopher of the twentieth century. Like millions of others, Levy has admiration for this great thinker. Levy writes that people thought Sartre the absolute intellectual, and thus they expected from him things they had never expected, and will probably never expect again from anyone else.\n\nSartre was only 5'3\" and homely except when in the throes of commanding an awed group of friends with one of his electrifying new ideas. These boisterous sessions frequently took place in one of his favorite caf\u00e9s, in his small grungy apartment, or in his beloved mother's home, which he frequently shared. When he died in 1980 at the age of 75, fifty-thousand dazed and bereft human beings wandered after his funeral procession each feeling all together and all alone. Thirty-year-old Bernard-Henri Levy mourned with Soviet dissidents, arguing Pakistanis, paparazzi, crying housewives, Sartre's former adversaries and so many more from all over the world. Levy asked himself with wonder how and why the great dry metallic voice of Sartre had gained a hearing in so many languages and affected so many destinies. People who had professed their hatred of the man and his ideas, accusing him of sullying France and corrupting youth, calling him \"a hyena with a typewriter,\" these people honored him with their passion and now with their love. This is what a truly great writer is\u2014a sometimes frightening, always brave beacon.\n\nThe French idolize their passionate intellectuals and artists. Sartre is what they reverently call \"a philosophe\" as well as a timeless artist. This philosopher, creative thinker, critic, biographer, novelist, political gadfly, and playwright almost singlehandedly inspired millions of people to grapple with the philosophy and theology of atheistic existentialism. He argued that a man creates himself by his actions. Existentialism meant to him that a person's existence precedes his essence. Each man has freedom which is not constrained or determined by his basic nature. Thus man is born neither good nor evil. In his book _What Is Literature?_ Sartre writes, \"We are no better than our life and it is by our life that we must be judged.\"\n\nOnce he became a famous world figure, Sartre exerted enormous power as a public political force. Because he was a left-wing moralist who proudly saw himself as independent, he refused the Nobel Prize for literature in 1964 to protest the United States involvement in the war in Vietnam. He also turned down a tour to lecture in the United States for the same reason.\n\nA contrarian to his core, he enjoyed nothing better than denouncing his previous beliefs. After he broke with Soviet Communism, he continued to use Marxian analysis. He finished few writing projects and we are fortunate to have this volume of essays on aesthetics and artists' lives\u2014subjects among those dearest to his heart and soul. He denounces poetic writing in his book _What Is_ _Literature?_ and yet in _Essays in Aesthetics_ he waxes poetic about Alexander Calder's mobile sculptures.\n\nSartre brought his own original thinking and prodigious writing skills to existentialism and made his modernist quest to understand the self and personal identity the hot topic of his era. His questions and answers permeate our lives and our internal monologues to this day. He was above all interested in human beings. Indeed the issues Sartre brought to the public in his swashbuckling debates and prodigious writings remain relevant to many people, including myself. The universality of what he was muddling through is still classic and important.\n\nI have always been fascinated by the existentialists and frequently think about their theories. Sartre studied and revealed the human psyche in a unique and highly intimate way. Like the tortured nineteenth century Danish existentialist philosopher S\u00f8ren Kierkegaard (of whom he was a devoted follower), Sartre relished what philosophers call value theory\u2014ethics and aesthetics. To me these have always been the most meaningful branches of philosophy because they help us understand ourselves.\n\nKierkegaard had formulated a Christian existentialism which includes the concept that with each and every one of our actions we particularize our self, and thus _create_ a self. Therefore our existence precedes our essence. This is a highly moral view and makes each one of us responsible for our own lives\u2014our missteps and our achievements. For Kierkegaard, the lowest stage of human development is about materialism, sensation and pleasure. This brings little or no fulfillment. Kierkegaard writes that it is through suffering that man arrives at the second stage\u2014the ethical stage. Sartre believed the ethical stage is the highest stage of human development.\n\nUnlike Sartre, Kierkegaard postulates a third more evolved state: Christianity, and a person arrives there again through an awareness brought about by suffering. Sartre was able to bring Kierkegaard's philosophy to the forefront of discourse in the twentieth century, although one of Sartre's controversial modifications is to make the ethical stage the highest stage. His is an atheistic existentialism.\n\nMeanwhile, across the channel in England in the mid-twentieth century, rigorous, formalistic and even cold philosophers such as the logical positivists A. J. Ayer and Ludwig Wittgenstein were trying to throw value theory\u2014Sartre's meat and potatoes\u2014out of the academy. Attempting to foment a revolution, they cited scientific principles and clarity. They wanted to prove that Sartre's humanist preoccupations were not philosophy: not measurable, not empirical, too emotional. Thankfully, great French thinkers like Sartre and Albert Camus continued to explore exhilarating questions of the nature of man's existence, his reason for being, his perception of his physical and metaphysical self, indeed his essential nature and control of his fate.\n\nIn this jewel of a collection, _Essays in Aesthetics_ , Sartre analyzes, picks on and extols four artists\u2014giving him the chance to demonstrate his panache with language and his great knowledge of aesthetics, that major philosophical discipline.\n\nSartre is nothing if not opinionated\u2014objectivity is boring. Sartre is not. He paints brilliant word pictures of the motives and methods as well as the art and biographies of the four artists. He has the most fun lambasting Renaissance painter Jacopo Tintoretto. He also examines the art and life of Alberto Giacometti (noting that while Giacometti sketched Sartre, the artist remarked of the philosopher's face, \"what density, what lines of force.\" Upon hearing this, Sartre claims surprise, writing that he believes his features weak and ordinary.) The third artist is the great Alexander Calder, whose mobiles Sartre loved. Finally he extols the abstract painter and experimental film maker Robert Lapoujade, who with Sartre opposed the French war against the Algerians.\n\nIn this delightful volume, Sartre perhaps enjoys himself most when writing his fiery, subjective tirade about Jacobo Tintoretto, the sixteenth century Venetian. Sartre irreverently calls the great artist a \"huckster-painter.\" In ridiculing Tintoretto, Sartre knows whereof he speaks from both inside and out. He is an artist who courted and achieved fame as well\u2014although he cared little for material possessions and in fact gave his money away.\n\nSartre's portrait of Tintoretto is remarkably contemporary and as such is a dead ringer for the dysfunctional marriage of art and commerce in today's art world. According to Sartre, the wily Tintoretto is a wheeler dealer who lies and pretends he doesn't compete with his peers for commissions. Yet he lures potential customers by bragging that he paints originals for the price of reproductions.\n\nPerhaps Sartre also has it in for Tintoretto because the painter went to school for a total of only five years and ridiculed humanist men of letters. Sixteenth century Venice had few poets and fewer philosophers, and Tintoretto irks Sartre because he refuses to have anything to do with them. The painter even wonders about their legitimacy. Sartre contemptuously writes that Tintoretto somehow doesn't believe men of letters earn their livelihood by sweat and work.\n\nIf Sartre disapproves of Tintoretto's business methods, he loves Calder's mobile sculptures. Sartre rhapsodizes like a fine poet about the beauty of his friend's work. He muses that Calder's mobiles are \"fed on air\" and \"vibrate in the wind like Aeolian harps.\" He calls them \"strange creatures halfway between matter and life,\" writing that the mobiles are \"neither completely living nor completely mechanical and which...constantly change but always return to their original position...like aquatic plants bent low by a stream.\"\n\nIn these essays, Sartre displays an astonishing knowledge of visual art and history. It is a show of superior creativity and intellect that deserves to be rediscovered and savored. Employing a similar virtuosity of language at the end of his midlife autobiography simply titled _The Words_ , Sartre immodestly describes his modesty: \"A whole man, mode of all men, worth all of them, and any one of them worth him.\"\n\n_Susan Braudy_\n\n_New York, New York 2011_\n\nSUSAN BRAUDY is an author and journalist. She did graduate study in philosophy at the University of Pennsylvania, and was an adjunct professor at Brooklyn College. She blogs for _The Huffington Post_ and has written for _The New York Times, The Atlantic Monthly, Newsweek, Vanity Fair, Ms_. _Magazine, New York Magazine_ and Yale University's _The New Journal_. She has also been a vice president of Warner Brothers.\n\nSusan Braudy is the author of five books; her most recent, _Family Circle, The Boudins and the Aristocracy of the Left,_ was nominated for a Pulitzer Prize. She has also written _Between Marriage and Divorce, A Woman's Diary_ ; _This Crazy Thing Called Love_ , a non-fiction account of the Woodward Family; and the novels _Who Killed Sal Mineo?_ and _What The Movies Made Me Do_.\n\nMs. Braudy lives with Joe Weintraub in New York City.\nSARTRE QUOTES\n\nBeing is. Being is in-itself. Being is what it is.\n\nLife has no meaning the moment you lose the illusion of being eternal.\n\nGod is dead.\n\nThe existentialist...thinks it is very distressing that God does not exist, because all possibility of finding values in a heaven of ideas disappears along with Him; there can no longer be an a priori Good, since there is no infinite and perfect consciousness to think it. Nowhere is it written that the Good exists, that we must be honest, that we must not lie; because the fact is we are on a plane where there are only men.\n\nThat God does not exist, I cannot deny. That my whole being cries out for God, I cannot forget.\n\nFreedom is what you do with what's been done to you.\n\nOnce you hear the details of victory, it is hard to distinguish it from a defeat.\n\nExistence precedes and rules essence.\n\nThe existentialist says at once that man is anguish.\n\nGenerosity is nothing else than a craze to possess. All which I abandon, all which I give, I enjoy in a higher manner through the fact that I give it away. To give is to enjoy possessively the object which one gives.\n\nThere is only one day left, always starting over: it is given to us at dawn and taken away from us at dusk.\n\nGod is absence. God is the solitude of man.\n\nViolence is good for those who have nothing to lose.\n\nFear? If I have gained anything by damning myself, it is that I no longer have anything to fear.\n\nI hate victims who respect their executioners.\n\nA lost battle is a battle one thinks one has lost.\n\nNeither sex, without some fertilization of the complimentary characters of the other, is capable of the highest reaches of human endeavor.\n\nHell is other people.\n\nOnly the guy who isn't rowing has time to rock the boat.\n\nThe existentialist does not believe in the power of passion. He will never agree that a sweeping passion is a raging torrent which fatally leads a man to certain acts and is therefore an excuse. He thinks that man is responsible for his passion.\n\nAll I have learned about my life, it seems, I have learned in books.\n\nOne is still what one is going to cease to be and already what one is going to become. One lives one's death, one dies one's life.\n\nIt is only in our decisions that we are important.\n\nMan is condemned to be free; because once thrown into the world, he is responsible for everything he does.\n\nWe do not judge the people we love.\n\nEvery existing thing is born without reason, prolongs itself out of weakness, and dies by chance.\n\nAll human actions are equivalent and all are on principle doomed to failure.\n\nWords are loaded pistols.\n\nLife begins on the other side of despair.\n\nExistentialism isn't so atheistic that it wears itself out showing the God doesn't exist. Rather, it declares that even if God did exist, that would change nothing.\n\nAs far as men go, it is not what they are that interests me, but what they can become.\n\nEverything has been figured out, except how to live.\n\nWhen the rich wage war it is the poor who die.\n\nI have no need for good souls: an accomplice is what I wanted.\n\nIf you are lonely when you are alone, you are in bad company.\n\nIf I became a philosopher, if I have so keenly sought this fame for which I'm still waiting, it's all been to seduce women basically.\n\nDeath is a continuation of my life without me.\n\nYou must be afraid, my son. That is how one becomes an honest citizen.\n\nMan can will nothing unless he has first understood that he must count on no one but himself; that he is alone, abandoned on earth in the midst of his infinite possibilities, without help, with no other aim than the one he sets himself, with no other destiny than the one he forges for himself on this earth.\n**On refusing the Nobel Prize for Literature in 1964**\n\n_The prize was awarded, said the academy, for \"his imaginative writing which by reason of its spirit and freedom and striving for truth has exercised a far-reaching influence on our age_.\"\n\n...[M]y refusal is not an impulsive gesture, as I have always declined official honors.... This attitude is based on my conception of the writer's enterprise. A writer who adopts political, social, or literary positions must act only with the means that are his own\u2014that is, the written word. All the honors he may receive expose his readers to a pressure I do not consider desirable. If I sign myself _Jean-Paul Sartre it_ is not the same thing as if I sign myself _Jean-Paul Sartre, Nobel Prize winner_.\n\nThe writer who accepts an honor of this kind involves as well as himself the association or institution which has honored him.... The writer must therefore refuse to let himself be transformed into an institution, even if this occurs under the most honorable circumstances, as in the present case.\n\n_From a statement by Sartre that first appeared in Le Figaro on October 23, 1964._\nARTIST GALLERY\n\nJacobo Tintoretto\n\n(1518-1594)\n\nBorn Jacopo Comin, the Venetian artist was nicknamed Tintoretto (\"the little dyer\") after his father's craft of dying cloth, a profession called _tintore_ in Italian. Tintoretto was a master painter during the late Italian Renaissance, and together with Paolo Veronese (1528-1588) and Titian (1485-1576), embodied the essence of the Venetian School of painting.\n\nTintoretto is thought to have been a brief student of Titian's, but he developed his own style of painting that was known for his bold use of color and the dramatic manipulation of light and shadow, with rough brushwork that heightened the emotional effect of his work. He was often called _Il Furioso,_ for the fury and speed with which he painted. Few drawings or oil sketches of his survive; to help him visualize detail and perspective, Tintoretto would mold small wax and clay models and experiment with light for the effects of shade and composition.\n\nMost of his work is still in the Venice churches or other buildings which commissioned him, including the Scuola Grande di San Rocco.\n\nLike Titian, Tintoretto kept a large workshop, and his sons Domenico and Marco and his daughter Marietta were trusted assistants. When he died in 1594, he was known as \"the top painter in Venice.\"\n\nAlberto Giacometti\n\n(1901-1966)\n\nSwiss sculptor and painter Alberto Giacometti was born in Borgonovo, Switzerland, the son of a Post-Impressionist painter. After studying art in Geneva and Paris in the 1920s, he began to experiment with cubism and surrealism and became known as one of the leading surrealist sculptors, associating with that Paris group until 1934.\n\nAfter breaking with the Surrealists, Giacometti spent the war years in Geneva, then returned to Paris. He had begun sculpting thin, elongated and solitary figures, finding his own signature style in these stretched, fragile yet roughly textured works. His vision of reality seemed to present human existence as a joyless struggle against an uncaring universe\u2014an almost existential view of reality. Critics at the time noted Giacometti's friendship with Jean-Paul Sartre, the father of existentialism, but the artist avowed he was not interested in any philosophical interpretations of his work: his sculptures were an artistic expression of physical reality as he saw it. He became well known, especially in the United States, through two exhibitions in New York City (1948, 1950). The artist died in Chur, Switzerland, in 1966.\n\nGiacometti's elongated forms were quickly recognized as icons of mid-twentieth century art. His sculpture _L'Homme qui marche_ had the distinction in February 2010 of being the most expensive work of art ever sold at auction to date: it brought a record-breaking $104,327,006 at Sotheby's in London.\n\nAlexander Calder\n\n(1898-1976)\n\nThe second child of artist parents\u2014his father Alexander Stirling Calder was a sculptor and his mother was a painter\u2014Calder did not set out to be an artist. He graduated from Stevens Institute of Technology with an engineering degree, having studied among other things applied kinetics, or the effects of force on free-moving objects. After working at various jobs, including hydraulics engineer and fireman in a ship's boiler room, he moved to New York in 1923 and enrolled in the Art Students League. He took a job illustrating for the _National Police Gazette_ , which sent him to sketch scenes at the Ringling Brothers and Barnum & Bailey Circus. This inspired him to create _Cirque Calder_ , using wire sculptures made with leather, cloth and other found objects to assemble a miniature circus that was manipulated by him. His performances made him famous.\n\nIn 1931 Calder began to create a new type of art with his kinetic sculptures, dubbed \"mobiles,\" that first moved by cranks and motors, but eventually were designed to move entirely on their own with the air current. Calder's stationary artwork came to be called \"stabiles.\" He began to accept commissions for large outdoor sculptures, which became larger over time until he was renowned for them worldwide by the 1960s. He designed monumental pieces for Lincoln Center in New York, for the gardens of UNESCO in Paris and for Expo '67 at Montreal. He even painted some of Braniff Airline's jet planes with his unique and colorful designs.\n\nAlexander Calder\n\n(1898-1976)\n\nThe second child of artist parents\u2014his father Alexander Stirling Calder was a sculptor and his mother was a painter\u2014Calder did not set out to be an artist. He graduated from Stevens Institute of Technology with an engineering degree, having studied among other things applied kinetics, or the effects of force on free-moving objects. After working at various jobs, including hydraulics engineer and fireman in a ship's boiler room, he moved to New York in 1923 and enrolled in the Art Students League. He took a job illustrating for the _National Police Gazette_ , which sent him to sketch scenes at the Ringling Brothers and Barnum & Bailey Circus. This inspired him to create _Cirque Calder_ , using wire sculptures made with leather, cloth and other found objects to assemble a miniature circus that was manipulated by him. His performances made him famous.\n\nIn 1931 Calder began to create a new type of art with his kinetic sculptures, dubbed \"mobiles,\" that first moved by cranks and motors, but eventually were designed to move entirely on their own with the air current. Calder's stationary artwork came to be called \"stabiles.\" He began to accept commissions for large outdoor sculptures, which became larger over time until he was renowned for them worldwide by the 1960s. He designed monumental pieces for Lincoln Center in New York, for the gardens of UNESCO in Paris and for Expo '67 at Montreal. He even painted some of Braniff Airline's jet planes with his unique and colorful designs.\n\nRobert Lapoujade\n\n(1921-1993)\n\nBorn in Montauban, France, Lapoujade was a painter and a film maker best known for his portraits of French literary figures including Jean-Paul Sartre, Andre Breton, Georges Bataille and Francois Mauriac. Sartre's essay was included in the catalog prepared for an exhibition of Lapoujade's abstract paintings at the Galerie Pierre Domec, Paris in 1961.\n\nLapoujade left school at the age of fourteen, finding work as a butcher, riveter, tile repairer and eventually a dramatic arts teacher. Hence his description as a \"self-made artist.\" He spent the war years in an obscure part of the Alps, and first exhibited his \"formal\" style paintings in Paris in 1947. He turned to painting non-figurative works in the early 1950s, with themes driven by social and political concerns.\n\nLapoujade also had a career in films, directing fifteen short subjects. He won a Cesar, the French equivalent of an Oscar, for his animated short, \"Un Comedien Sans Paradoxe\" (\"An Actor Without Paradox\"). His full-length feature, \"Le Sourire Vertical\" (\"The Vertical Smile\"), shocked the Cannes International Film Festival in 1978 with its sexually explicit and violent scenes. It was later banned by French censors for its content.\nINTRODUCTION\n\nJean-Paul Sartre's views on aesthetics are perhaps less controversial than his views on a number of other issues, particularly the human predicament. As a philosopher he has the dubious distinction of being a center of cult and controversy in Paris and of having most people who have heard of existentialism connect his name with the movement. As a novelist and dramatist he has popularized the notions stated systematically in his formidable masterwork _Being and Nothingness_ and earned notoriety for the movement and its bizarre, pessimistic, perverse and bewildering appraisal of the human situation. As a typical French intellectual he has always exhibited an expanding omniscience that gives added emphasis to his statements on many vital issues, one of which is the place of art and artists in the human situation. Whether he speaks as philosopher, dramatist, novelist or critic, the ex-professor deserves a hearing.\n\nThough he is the point of confluence of three post-Hegelian streams of thought\u2014the Marxist, the existentialist and the phenomenological\u2014and the product of traditional European thinking on aesthetics, ethics, metaphysics and politics, Sartre is profoundly and self-consciously individualistic in his interpretation of the human situation. He exploits the analytical tools of the Marxists and embraces their concern for action, but he disowns their politics. He rejects Kierkegaard's leap of faith but adopts his picture of man as a lonely, anguished creature in a chaotic universe. He discards Husserl's Platonism but adapts the latter's terminology to his own purposes. Add to this the rejection of Cartesian dualism and the adoption of Freudian insights, and the net result is an outlook and a style that produce in the reader a shock of recognition\u2014a feeling that what he is reading is not something new but something long anticipated. Finally, though his tools and arguments evidence his indebtedness to his predecessors, his style and outlook are unmistakably contemporary.\n\nSartre approaches art with an open mind but seeks always to fit his findings into his philosophical scheme. The essays in this collection reveal his concern about fundamental issues relating to the nature of art and its place in the human situation as well as to the formation and function of the artist. And in questioning art he stirs the dust of metaphysical speculations: Is the artist \"a man bent on imposing a human seal on space or a rock dreaming of human qualities?\" Is our universe a \"blind concatenation of causes and effects or the gradual unfolding, forever retarded, disconcerted and thwarted, of an Idea?\" In these essays as in all his writings, we feel a spirit of dedication and a sincere desire on the part of the writer to change the life of the reader.\n\nFour artists are discussed\u2014two painters, one painter-sculptor and one sculptor-painter. Each artist, because he embodies contradictions and confronts enigmas, presents a challenge to Sartre and affords him the opportunity to illustrate his theories about art. Tintoretto is the product of class contradictions and has to achieve self-affirmation through deceit. Giacometti is obsessed by his isolation in a world of things accessible only through their appearances and has to discover how to paint emptiness. Calder works on the borderland between freedom and control and has to discover how to imbue something immobile with movement. Lapoujade strives to reconcile creativity and beauty and must learn to \"give to an infinitely divisible surface the indivisible unity of a whole.\"\n\nSartre's approach is doubly rewarding in that it affords new insights into the activities of both author and subject. In the case of Tintoretto, he offers a convincing refutation of legends that still survive and a brilliant case study of an artist and his era. And in the case of his contemporaries, he shows how each has responded to the challenges of our own era. In addition to these new insights he provides us with a host of striking images and intimate asides: his definition of genius as \"a conflict between a finite presence and an infinite absence\"; his description of Tintoretto's paintings as a \"passionate love affair\" between a city and her rejected suitor; his allusions to the uniqueness of touch and the significance of the Other's look; his having to learn anew to live \"at a respectable distance\" from others after the end of World War II; his fright on boarding a plane as it relates to his love of beauty and his abhorrence of ugliness; his expressed hope that Giacometti will one day paint an illusion that will cause us to experience the \"same shock that we feel on returning late and seeing a stranger walking toward us in the dark.\"\n\nI wish here to express appreciation to those who have helped to prepare this collection of Sartre's essays on art. Professor Jean Lorson of the University of Oklahoma clarified for me several passages in the French texts. Four of my colleagues\u2014Minnie Baker, Richard Bivins, Margaret O'Riley and Mildred Riling\u2014shared the task of reading through the first draft of the translation: Professor Baker and Mr. Bivins paid particular attention to technical points posed by the translation of terms used in art; Dr. O'Riley read and corrected the first draft of the shorter essays; Professor Riling called attention to certain stylistic peculiarities of Sartre and suggested extensive revisions which were incorporated into the final version of the essay on Tintoretto. For their generous help and encouragement I am deeply appreciative.\n\nWADE BASKIN\n\n_Southeastern State College_\nTHE VENETIAN PARIAH1\n\nJacopo's Shenanigans\n\nNothing. His life is an enigma: A few dates, a few facts, and then the cackling of ancient writers. But courage: _Venice speaks to us_. Her voice is that of a perjured witness, now shrill, now whispering, always marked by periods of silence. Tintoretto's life story, the portrait painted during his lifetime by his native city, is tinged with unrequited animosity. The Doge's City reveals her contempt for the most celebrated of her sons. Nothing is stated outright; there are hints, suggestions, remarks made in passing. This inflexible hatred has the inconsistency of sand; it takes the form, not so much of outspoken aversion, as of coldness, moroseness, insidious ostracism. And this is just what we would expect. Jacopo fights a losing battle against a vast adversary, grows tired, surrenders, dies. That is the sum and substance of his life. We can study it in all its somber nakedness if for an instant we push aside the brushwood of slander that blocks our passage.\n\nFirst, the birth of the dyer's son in 1518. Venice immediately insinuates that fate has marked him from the outset: \"About 1530 the youth started to work in Titian's studio as an apprentice but was dismissed a few days later when the illustrious quinquagenarian discovered his genius.\" This anecdote reappears in book after book with astounding regularity. It might be argued that it does little credit to Titian\u2014and this is indeed the case\u2014not _today_ , at any rate, not in our eyes. But when Vasari reports it in 1567, Titian has been reigning for half a century, and nothing is more respectable than long impunity. Then too, according to the customs of his time, Titian has his own studio, where he is second only to God in the conduct of his affairs and has every right to dismiss an employee. In such circumstances his victim is presumed guilty; marked by fate, contagious perhaps, he is presumed to have the evil eye. Here for the first time the gilded legend of Italian painting is threatened by an ill-fated childhood. But the lesson to be learned from his alleged dismissal must come later. The voice of Venice never lies provided that we know how to interpret it; we can listen once our ears have been properly attuned. At this point we suspend judgment but call attention to the improbability of the facts.\n\nThat Titian was not good-natured is well known. But Jacopo was twelve. At twelve talent is nothing and anything will obliterate it; patience and time are required to mold nascent skill and change it into talent; no artist at the pinnacle of his fame\u2014not even the most supercilious\u2014would take umbrage at a small boy. But suppose that the master, jealous, dismissed his apprentice. That amounts to assassination of him. The curse of a national celebrity weighs heavily, very heavily. More especially as Titian lacks the candor to make known his true motives; he is king, he frowns and from this moment on all doors are closed to the black sheep. He is forever barred from the profession.\n\nA blacklisted child is something of a rarity. Our interest quickens. We are eager to find out how he managed to overcome his handicap. Vain desire, for here the thread of the narrative breaks at the same time in every single book and we are confronted by a conspiracy of silence. No one will tell us what happened to him between the ages of twelve and twenty. Some writers attempted to fill the void by imagining that he had learned the art of painting independently. But they were in an even better position than we to know that he could not have done so, for at the beginning of the sixteenth century painting is still a complicated, rather ceremonious technique; unduly fettered by formulas and rites, it is a skill rather than an art, proficiency rather than knowledge, a set of procedures rather than a method. Professional rules, secret traditions\u2014everything contributes toward making the apprenticeship a social obligation and a necessity. The biographers' silence betrays their embarrassment. Unable to reconcile the precocious notoriety of young Robusti and excommunication, they throw a veil of darkness on the eight years that separate the two. We can be certain that no one has rejected Jacopo; and since he has not perished from languor and scorn in his father's dye-shop, he must have worked normally and regularly in the studio of a painter about whom we know nothing except that _he was not_ Titian. In closed, suspicious guilds hatred is retroactive; if the mysterious beginning of Jacopo's life seems a premonition of its mysterious end, if a curtain raised to show a disaster miraculously arrested is lowered on a disaster unattended by any miracle, this is because Venice rearranged everything afterwards to make his childhood consonant with his old age. Nothing happens and nothing ends; birth mirrors death and between the two lies scorched earth; everything is consumed by the curse.\n\nWe pass beyond these mirages and find our view unobstructed all across the horizon. An adolescent emerges, dashes away at high speed in search of glory. The year is 1539; Jacopo has left his patron to set up his own studio; he is now a _past master_. The young employee has won independence, fame, a clientele; now it is his turn to hire workers, apprentices. This much is certain: in a city filled with painters, where an economic crisis threatens to strangle the market, becoming a master at the age of twenty is the exception to the rule; merit alone is not enough, nor work, nor tact; one must also have a run of good luck. Everything is in Robusti's favor. Paolo Cagliari is ten, Titian sixty-two; between the unknown child and the old man who will surely die before long many good painters might be found, but only Tintoretto holds out the promise of excellence; in his generation, at any rate, he has no rival and therefore he finds the road before him open. He does in fact pursue this road for several years: his commissions multiply, he enjoys the public's favor as well as that of patricians and intellectuals; Aretino deigns to congratulate him in person.2 The young man is endowed with supernatural gifts which Providence reserves for adolescents who are to die, but he does not die and his woes begin: The old monarch Titian manifests a startling longevity, continuing all the while to vent his hatred on his young challenger and finally resorting to the malicious ruse of officially designating as his successor, to the surprise of no one, Veronese; Aretino's condescension turns to bitterness; critics lash out, censure, chide, castigate\u2014in short, they behave like modern critics. This matters little so long as Jacopo retains the public's favor. But suddenly the wheel turns. At thirty, confident of his means, he asserts himself, paints _The Miracle of St. Mark_ and puts himself, his whole self, into the painting. It is characteristic of him to astound, to strike hard and impose his will by surprise. In this instance, however, he will be the first to be caught off guard; his work dumbfounds his contemporaries but it also scandalizes them. He finds impassioned detractors but not impassioned defenders; behind the scenes we can detect a cabal: Frustration.3 Face to face, united and separated by the same feeling of uneasiness, Venice and her painter contemplate but no longer understand each other. \"Jacopo has not lived up to the promises of his adolescence,\" says the city. And the artist remarks, \"To deceive them, all I had to do was reveal myself. So _I_ was not what they loved!\" Mutual grudges widen the gap between them, breaking one thread in the Venetian woof.\n\nThe pivotal year is 1548. _Before_ , the gods are for him; _afterwards_ , against. No great misfortunes are associated with his persistent bad luck\u2014just enough little ones to lead him to the brink of despair. The gods smiled on the child only to bring about the downfall of the man. Jacopo suddenly undergoes radical change and becomes the frantic, harassed outlaw, Tintoretto. _Before_ , we know nothing about him except that he worked relentlessly, for fame is not easily acquired at the age of twenty. _Afterwards_ his tenacity turns to rage; he wants to produce, to produce without ceasing, to sell, to crush his rivals by the number and dimensions of his canvases. There is a certain element of desperation in his forced effort, for until his death Robusti works against the clock without ever revealing whether he is trying to find himself through his work or to escape from himself through excessive activity. \"Lightning Tintoretto\" sails under a black flag, and for this swift pirate all means are fair, with a marked preference for unfair advantages. Disinterested whenever disinterest pays off, he lowers his eyes, refuses to name a price, repeats like a child, \"It will be whatever you wish.\" But those Neapolitan rascals are in a better position than anyone else to know the value of their wares; they expect the customer to fleece himself through his generosity.\n\nOn other occasions he offers his merchandise at cost in order to close a transaction, only to make other more advantageous sales as a result of the initial emergency contract. On learning that the Crociferi are going to offer a commission to Paolo Cagliari, he feigns ignorance of everything and offers them his services. They essay a polite refusal: \"Thank you, but we want something Veronese.\" And he: \"Something Veronese?4 Well and good. And who is going to do it?\" Somewhat taken aback, they reply: \"Why, we thought that Paolo Cagliari had been designated....\" And Tintoretto now expresses his amazement: \"Cagliari? The idea is fantastic. I'll paint you something Veronese. And for less.\" Signed and sealed. He resorted to the same gambit twenty times, painting _in the style of_ Pordenone, _in the style of_ Titian, always for less.\n\nHow can he cut costs? That is the question that torments him. One day he finds the contemptible but ingenious answer that will wreck a tradition. The masters are accustomed to having their canvases copied; their studios execute replicas and sell them at inflated prices, thereby creating a second market for their paintings. To win over their clientele, Jacopo will offer them _better paintings for less_. He eliminates sketches; he will allow others to draw their inspiration from his canvases but not to copy them; through simple, invariable procedures his collaborators will produce something new but not original. They will need only to rearrange the composition, put the left on the right and the right on the left, substitute an old man for a woman who can be used again in another context. Such operations require some training but no more time than simple copying. Tintoretto candidly proclaims: \"In my studio one can acquire an original work for the price of a reproduction.\"\n\nWhen his canvases are spurned, he gives them away. On May 31, 1564, at the Scuola San Rocco the Brotherhood decides to beautify its conference hall by placing a painting in the central oval of the ceiling. Paolo Cagliari, Jacopo Robusti, Schiavone, Salviati and Zucearo are invited to submit sketches. Tintoretto bribes servants, obtains the exact measurements. He had already worked for the Brotherhood and I do not rule out the hypothesis that he found accomplices even within the _Banca e Zonta_. On the day set, each painter exhibits his sketch. When Robusti's turn comes, he electrifies them all. He climbs up a ladder, removes a section of pasteboard, and reveals above their heads a dazzling painting, already in place, already finished. Pandemonium. \"A drawing is easily misunderstood,\" he explained. \"While I was about it, I preferred to see it through. But if my work is displeasing to you, gentlemen, I will give it to you. Not to you, but to San Rocco, your patron, who has done so much for me.\" This forced their hand and the rascal knew it, for the rules of the Brotherhood prohibited their refusing religious donations. All that remained was for them to make the episode a matter of record in the Scuola: \"On this day the undersigned Jacopo Tintoretto, painter, presented to us a painting; he asked no remuneration, promises to complete the work if requested to do so, and states that he is satisfied with it.\" And the undersigned wrote in his turn: \" _Io Jachomo Tentoretto pintor contento et prometo ut supra_.\"\n\nSatisfied? Why not? His gift spreads panic among his competitors, opens to him every door of the Scuola, places its vast, barren walls at the mercy of his brush and finally brings him an annual pension of a hundred ducats. So satisfied is he, in short, that he repeats the gambit in 1571. At the Doge's Palace this time, authorities wishing to commemorate the battle of Lepanto, organize a contest. Instead of a sketch Tintoretto brings a canvas and presents it as a gift. It is accepted with gratitude; shortly thereafter he sends his bill.\n\nIn his base but charming shenanigans one is tempted to see, perhaps, a trait attributable more to morals than to character. We might with some degree of accuracy say that ostentation was characteristic not of Tintoretto but of his century. If an attempt were made to condemn him on the basis of these anecdotes, I know everything that might be said in his defense. The most telling argument is that no one at that time could _work for himself_. Today paintings are in demand; then painters were for sale. They lined the market place like the _bracciante_ in the southern towns; buyers came, examined all of them, singled out one and took him to their church, their _scuola_ or their _palazzo_. Artists had to make themselves available, to advertise themselves as our directors do, to accept just any work in the same way that our directors accept just any scenario in the foolish hope of using it to display their talents. Everything was under contract: the subject, number, quality, and sometimes even the attitude of the figures, and the dimensions of the canvas; these were complemented by restrictions imposed by traditions relating to religion and to taste. Their clients had their moods just as our producers have their whims. And their clients\u2014alas!\u2014they, too, had their sudden inspirations; at their bidding, everything had to be reworked. In the palace of the Medici, Benozzo Gozzoli was for a long time knowingly tortured by idiotic patrons; and we need only compare Tintoretto's _Paradise_ in the Louvre with the one in the Doge's Palace to understand the magnitude of the pressures to which he was subjected. Intransigence, rejection of compromise, the superb choice of misery were out of the question since the artist had to provide for his family and keep his studio in operation, as present-day machines are. In sum, he had to renounce painting or to paint according to instructions. No one can blame Tintoretto for wishing to become rich. As a matter of fact, toward the middle of his life he was never out of work, never lacked money. This utilitarian artist followed the principle that nothing is done for nothing, that painting would be a mere pastime unless it produced some income. At long last, as we shall see, he will buy a comfortable plebeian house in a residential district. This purchase will crown his career, exhaust his savings, leave the Robusti children with only a ludicrous heritage to divide: the contents of his studio, a diminishing clientele, and the house itself, which is passed on to the oldest son, then to his son-in-law. Twelve years after the death of her husband, Faustina recalls bitterly that he left his family in need; she has every reason to complain, for the deceased had his own way. He liked money, of course, but in the American way. He saw in it nothing more than the external sign of success. At the bottom, this contract chaser sought only one thing: the means of practicing his craft. There is also an element of justice in his shenanigans, for they would be inconceivable without his professional talent, hard work, and speed. His speed gives him an advantage, for to paint a good picture he requires only the time taken by others to make bad sketches.\n\nFurthermore, if he plagiarized Veronese, the latter repaid him in kind. Their reciprocal borrowings must be viewed through the eyes of their contemporaries. For many of their contemporaries the greatest painters are those who have met the test of certain social criteria; they are personalities defined by collective judgment. We are interested in a particular painting at first, and then in the particular man who painted it; we hang Matisse on our walls. But contrast our view with that of the _Crociferi:_ they were not interested in Cagliari; they wanted a certain style that appealed to the senses, trifles, inoffensive and harmonious pomp; they knew a trademark, a slogan. A painting signed Veronese is certain to please. That is what they wanted, nothing else. Cagliari could produce better works and proved it when he painted his _Terrible Crucifixion_ ,5 but he was too shrewd a businessman to squander his genius. Under such conditions we could hardly blame Tintoretto for appropriating at times a style that belonged exclusively to no one. After all, he made an honest proposal: \"You want something trite and lifelike? I will provide it.\"\n\nI am aware of the tastes of his age. My aim here is not to judge him but to determine whether his age could identify itself with him without discomfort. And on this point the evidence is explicit: his conduct shocked his contemporaries and turned them against him. A little disloyalty would perhaps have been tolerated, but Tintoretto went too far; throughout Venice a single complaint was voiced: \"He goes too far!\" Even in that commercial city such shrewdness in commerce is unique. At the Scuola San Rocco, when he stole their commission, his colleagues barked so loudly that he felt obliged to appease them: the establishment had other ceilings and walls, the work had only begun; as for him, now that his gift had been accepted, he would disappear, leaving the field open to the most worthy of them. His unfortunate rivals soon discover that he is lying like a pagan, for the Scuola will become his fief, and as long as he lives, no other painter will ever cross the threshold. They had surely not waited for this occasion to begin hating him. It is worth noting, however, that the scandal occurs in 1564 and that the first _Life_ of Tintoretto appears in 1567. The shortness of the interval between the two events further enlightens us concerning the origin and significance of the ugly rumors collected by Vasari.6 Calumnies on the part of jealous rivals? They were all extremely jealous of each other; why, then, are calumnies heaped on Robusti alone unless he is the \"foul smell\" of the artists, unless he represents in the eyes of each and all the collective and magnified faults of his fellow men? Furthermore, even his clients seem shocked by his conduct. Not all of them, no. But he has made numerous enemies in high places. Zammaria de Zigninoni, a member of the San Rocco Brotherhood, promises fifteen ducats for decorative works under the express condition that Jacopo not be given the commission. The records of the Brotherhood suggest, moreover, that the _Banca e Zonta_ held a few tense and somewhat unceremonious meetings in the Scuola in connection with the restricted donation and Jacopo's gambit; an agreement was reached but Zigninoni kept his ducats. Nor do the officials always seem kindly disposed toward him. In 1571, Tintoretto contributes his _Battle of Lepanto;_ in 1577 the painting is destroyed by fire; when the question of replacing it arises, he has every reason to believe that the government will call on him. Not at all. He is deliberately passed over and preference is shown instead to the mediocre Vincentino. It might be argued that his canvas had met with disfavor. But that is hardly plausible, for Jacopo always treads softly when working for the officials; he \"paints like Titian,\" disguising his own style. Besides, after 1571 the government gave him several commissions. No, the Venetian authorities have no intention of depriving themselves of his services; they simply want to punish him for his rascality. In short, there is unanimity: he is a disloyal colleague, a maverick painter, and there is bound to be something unsavory about him since he is without friends. Sweet troubled souls who use the dead to edify the living and especially yourselves, try if you will to find in his excesses the glittering proof of his passion. The fact is that passions are as diverse as people: ravenous and contemplative, dreamy and practical, abstract, dawdling, apprehensive, rash\u2014a hundred others. I will call Tintoretto's passion practical, apprehensive-recriminatory and ravenous-rash. The more I reflect on his ludicrous gambits, the more I become convinced they were born of an ulcerated heart. What a nest of vipers! There we find everything: the delirium of pride and the folly of humility, chained ambition and unchained confusion, harsh rebukes and persistent bad luck, the goad of success and the lash of failure. His life is the story of an opportunist tormented by fear; it has a healthy, sprightly beginning; the offensive is well staged until the hard blow of 1548; after this the rhythm quickens, goes out of control, lights the fires of hell. Jacopo will fight on until the time of his death, knowing that he will not win. Opportunism and anguish, those are the two biggest vipers. If we wish truly to know him, we must have a closer look at them.\n\n1 Tr. First published in _Les Temps Modernes_ (November, 1957) under the title of _Le S\u00e9questr\u00e9 de Venice_.\n\n2 Tr. Because of the influence which he exercised over kings, diplomats and artists through his writings, Pietro Aretino (1492-1556) has been called the first journalist. Curiously, the amoral publicist whose services went to the highest bidder counted Titian among his devoted friends.\n\n3 Ridolfi even maintains that the Scuola San Marco refused the canvas and that Tintoretto had to take it back to his studio.\n\n4 Tr. The spacious architectural quality of the paintings of Veronese (Paolo Cagliari, 1528-1588) is typical of his school.\n\n5 It is in the Louvre. The irony is that he was inspired by the _real_ Robusti.\n\n6 Tr. The celebrated painter, architect and biographer published his _Lives of Excellent Painters, Sculptors, and Architects_ in 1550. The date generally given for the revised edition is 1568.\nTHE PURITANS OF THE RIALTO\n\nNo one is a cynic. To be discouraged in the absence of discouragement is the diversion of saints. Only up to a certain point, however, for these chaste and generous creatures stigmatize their lechery and denounce their avarice. If they discover their real gangrene\u2014saintliness\u2014they look for justification, like all guilty creatures. _Tintoretto_ is no saint; he knows that everyone in town condemns his conduct; he persists only because he thinks that he is right and they are wrong. And let no one come up and say that he is aware of his genius, for a genius\u2014this is ironical but true\u2014knows his courage but not his worth. Nothing is more wretched than sullen temerity that reaches for the moon and writhes in defeat; first comes pride, without proofs or pedigrees; when it matures into madness we can call it genius if we choose, but I fail to see that very much is gained thereby. No, to justify his piracy Tintoretto pleads neither limited originality nor unlimited aspirations. He defends his rights, claiming that he has been wronged whenever a commission goes to one of his colleagues. Left to himself, he would have covered every wall in the city with his paintings; no _campo_ would have been too vast, no _sotto portico_ too obscure for him to illuminate. He would have covered the ceilings, people would have walked across his most beautiful images, his brush would have spared neither the fa\u00e7ades of the palaces that line the Canale Grande nor the gondolas, nor perhaps the gondoliers. The man imagines that he was born with the privilege of transforming his city single-handed, and a good case can be drawn up in his favor.\n\nWhen he begins his apprenticeship painting is on the wane. In Florence the crisis is manifest; Venice, as always, is silent or hypocritical, but we know for a certainty that the authentic Rialtan sources of inspiration have dried up. At the end of the fifteenth century the city is deeply affected by the passing of Antonello da Messina. His death marks a turning point; afterwards painters are imported. I am not saying that they are brought in from distant regions but simply that the most famous painters come from the mainland: Giorgione, from Castelfranco; Titian, from Pieve di Cadore; Paolo Cagliari and Bonifazio dei Pitati, from Verona; Palma Vecchio, from Sarinalta; Girolamo Vecchio and Paris Bordone, from Treviso; Andrea Schiavone, from Zara; and others still. As a matter of fact, this aristocratic republic is primarily a technocracy and has always been bold enough to recruit specialists from far and wide and clever enough to treat them as her own. Moreover, this is the time when the Republic of Venice, checked at sea and threatened by coalitions on the continent, turns to the hinterlands and tries through conquests to assert her might. Most of the new immigrants are from annexed territories. Venice betrays her anxiety by importing artists on a massive scale. When we recall that the artists of the Quattrocento were for the most part born inside the walls of the city or in Murano, we cannot suppress the notion that after the extinction of the Vivarini and Bellini families and after the death of Carpaccio, the resurgence of her generations of artists would not have been possible without a blood transfusion.\n\nPainting is like all the other crafts in that the patriciate is responsible for facilitating the immigration of good artisans and\u2014to prove what might be called their cosmopolitan chauvinism\u2014for making the Republic of the Doges into a melting pot. In the eyes of this distrustful and jealous aristocracy, foreigners make the best Venetians; their adoption of Venice is proof of inspiration just as aloofness signals a weak character. We can be sure that the local artisans did not look upon the newcomers in the same way. Why should they? For them the newcomers represent foreign competition. They are tactful enough not to complain, and they carry on as if nothing were wrong; but there are conflicts, inescapable evidences of tension, charges and countercharges stemming from wounded pride. Forced to bow to the technical superiority of the alien settlers, the natives hide their humiliation by expanding their prerogatives. They agree to The Puritans of the Bialto take second place to the most skilled, to the most expert, but only in return for a sacrifice: their birthright must remain intact. Only a Rialtan can claim Venice as his own; while Germans are better glaziers, they can never boast that they are true Venetians. Before their disappearance the great painters of the Quattrocento had the bitter experience of seeing the public turn away from them and bestow their favors on young intruders who scorned them. For example Titian, the outsider, leaves one of the Bellini brothers for the other\u2014Gentile for Giovanni\u2014in pursuit of still another outsider, Antonello, the meteor that rent the sky and the water of the lagune twenty years earlier. Tiziano Vecellio has no need of Giovanni; what he seeks in him is a reflection; he proves this by soon abandoning the master of the disciple and joining Giorgione's school, for the third alien seems to the second to be the true heir to the first. Tiziano and Giorgio belong to the same generation; the pupil may even be older than the teacher. Did the Bellini brothers realize on that day that they had served their time? And what did Giovanni's true disciples say? And the others, the last representatives of the Murano school, what did they think? Many of them were youngsters or men still in their prime; the influence of Antonello da Messina had reached all of them through Giovanni Bellini; colors and light came from Messina, but their acclimatization was effected by Giovanni; through him they had become Venetian. The young artists staked their honor on remaining faithful but were strangled by their fidelity. They did their best to adapt to new conditions without abandoning the rather crude techniques that they had been taught, but to do so was to accept mediocrity. They must have felt bitter resentment on seeing two young intruders join forces, break with the indigenous tradition, rediscover the secrets of a Sicilian, and effortlessly carry painting to its highest perfection. Giovanni still reigns, however, and the fame of this admirable artist spreads throughout northern Italy. The barbarian invasion begins during his latter years and triumphs after his death in 1516.\n\nAt the height of the invasion, the greatest painter of the century is born in the heart of the occupied city, in an alley on the Rialto. A somber plebeian pride, always humiliated, always rebuffed, constantly in waiting, seizes upon the opportunity, infiltrates the heart of the sole Rialtan with a remnant of talent, emboldens and inflames it. We recall that he springs directly from neither the working class nor the bourgeoisie. His father is a successful artisan, a member of the petty bourgeoisie, who takes pride in not working for others. As the son of a working man Jacopo would perhaps have remained the obscure collaborator of an artist; as the son of an independent craftsman, however, he has to become a master or a failure. He will pass through the ranks but is prevented by his class and family status from stopping along the way. That he fails to leave good impressions in the studio where he serves his apprenticeship is understandable, for his aim on entering the studio was to leave it as soon as possible to reclaim the place already reserved for him in the social hierarchy. Then, too, Schiavone (or Bordone or Bonifazio dei Pitati\u2014they are all the same) must have looked upon him as an intruder while Jacopo considered his master an alien or a thief. The Little Dyer is a _native_ and Venice is his birthright. Had he been a mediocre painter, he would have remained modest and resentful; but he is brilliant and knows it, and he will take second place to no one. Aliens, in the eyes of a Rialtan, have nothing to protect them other than their professional worth; if Jacopo outshines them as a painter, they will have to disappear, even if this means their assassination. No one paints or writes without a mandate; would anyone dare if \" _I_ were not the Other\"?1 Jacopo is given a mandate by a toiling population to redeem through his art the privileges of a purebred Venetian. That explains his unscrupulous conduct. Popular recrimination fills his heart with an abiding desire to reassert a claim; he has been given the task of winning recognition of his rights; whoever champions such a just cause can use any means to succeed\u2014he will show no mercy, give no quarter. His misfortune results from the fact that his struggle against the undesirables brings him into conflict with the patriciate and its policy of assimilation of foreigners in the name of the indigenous artisans. When he shouted in the streets, \"Veronese to Verona!\" it was the government that he was calling in question. Realizing this, he hesitates, then resumes his obstinate course, exhibiting a curious mixture of flexibility and inflexibility. As a prudent subject of a police state he always gives in, or pretends to; as an _indigenous_ citizen of the most beautiful of all cities, his arrogance is boundless; he can even be servile without losing his ankylosis of pride. Everything is to no avail. His schemes against those protected by the aristocracy are thwarted by his impatience or by his incurable bluntness, or they backfire. Now we see the rancor of the Republic in a new light. The subject asks essentially only for what would probably have been accorded him, but his perverse submissiveness nettles the authorities, and they consider him a rebel. Or at the very least they are suspicious of him, and their suspicions are well founded. The consequences of his impetuousness are worth examining.\n\nFirst, the studied and almost sadistic violence which I will call lack of self-restraint. Born among the underlings who endured the weight of a superimposed hierarchy, he shares their fears and their tastes; we find their prudence even in his presumptuousness. His neighbors, alert, courageous, somewhat suspicious of outsiders, have helped him to establish a system of values, shown him the dangers that life holds in store, pointed out the hopes that are permitted and those that are prohibited. Specific, limited opportunities, a foreseeable destiny, a future already visible in its general outlines, being imprisoned inside a transparency like a tiny flower inside a glass paperweight\u2014all this kills dreams. One desires only what is possible; this is a mitigating circumstance that enrages fools and excites far-fetched but ephemeral ambitions. Jacopo's ambition suddenly asserts itself. Bolstered by its virulence and diverse forms, it assimilates a minute pencil of light, its possibilities. Or rather, nothing is _possible_. There is a means and there is an end in view, which is the prescribed task; one can rise above the heaviest low-lying mists and touch the rigid, luminous membrane of the ceiling; there are other ceilings, membranes that grow progressively clearer and more delicate, and at the very top, perhaps, is the blue of the sky. But what does this matter to Tintoretto; each has his own soaring range and his own habitat. Tintoretto knows that he is talented; he has been told that his talent is his capital. By putting his capacities to the test he will capitalize on them and provide himself with an adequate income. And so we see him totally mobilized for a long life, prepared always to exploit the vein even to the point of depletion of both mine and miner. At about the same time another slave to work, Michelangelo, is undertaking projects only to desert them in disgust and leave them unfinished. Tintoretto _always_ finishes things with the terrible application of a man bent on finishing his statements come what may; even death stood aside for him at San Giorgio, where it allowed him to apply the finishing touches to his last canvas, or at least to give final instructions to his collaborators. Never during his entire life did he allow himself an indulgence, a dislike, a preference, or even the comfort of a dream. During periods of exhaustion he must have repeated to himself this principle: \"To refuse a commission is to hand it over to my colleagues.\"\n\nHe has to produce at any cost. Here the will of a man and that of a city coalesce. A hundred years earlier Donatello had scolded Uccello for sacrificing creativity to experimentation and for carrying the love of painting to the point of ceasing to paint pictures2, but that was in Florence, and the Florentine artists had just begun to risk experimentation with _perspective;_ by applying to painted objects the laws of geometrical optics, they were trying to construct a new plastic space. Other times, other customs. In Venice, under the leadership of Titian, everyone shares the opinion that painting has just reached the peak of perfection, that further advances are impossible: Art is dead, long live life. The supreme barbarity begins with Aretino's foolish statements: \"How realistic it is! How true to life! _You would never believe it is painted!_ \" In short, it is time for painting to disappear in the face of _realizations;_ inspired merchants want something beautiful and useful. A work ought to please the lover of art, dazzle Europe with the pageantry of the Republic, awe the people. And still today we stand in awe, we the little tourists, before the Venetian cinemascope and prattle about one of Titian's realizations, one of Paolo Cagliari's productions, one of Pordenone's performances, one of Vecention's stagings. Jacopo Robusti shares the prejudices of his age, and our experts stress the point. How many times have I heard them say, \"Tintoretto, bah! Just like the movies.\" And still, no one else in the world, either before or after him, has carried so far the passion for research. With Titian, painting flowers and dies, a victim of its own perfection; Jacopo sees in its death the necessary condition for a resurrection: everything is to have a new beginning, to be done over\u2014a theme to which we shall return. But\u2014and this is his major contradiction\u2014he will never allow his experiments to restrict his productivity. So long as there remains in Venice one barren wall, the painter's task is to cover it; morals prohibit transforming a studio into a laboratory. Art is in its entirety a serious profession and a battle to the death against intruders. Like Titian, like Veronese, Jacopo will produce exquisite cadavers. With one difference: his cadavers are racked by fever, and we do not know at first whether this is the aftermath of life or the onset of putrefaction. And if the comparison with movie-making is pressed, he resembles the cinematographer in _this_ respect: he accepts imbecile scenarios but imbues them with his obsessions. He has to fool the buyer, to give him something for his money; the buyer will have his Catherine, his Theresa, his Sebastian; for the same price he will have on the same canvas, if there is sufficient room, his wife and his brothers. But underneath it all, behind the sumptuous and banal fa\u00e7ade of the _realization_ , he pushes forward his experimentation. Each of his great works has a double meaning; its strict utilitarianism disguises an unending quest. Fitting his research into the frame of the paid commission, he is obliged to revolutionize painting even while respecting the stipulations of his client. Such is the inner motivation of his excessive activity, and such will later be the reason for his perdition.\n\nHe also has to win commissions. We have already seen that he succeeds. But let us re-examine his actions; now they will appear in a new light. Tintoretto's rebellion has various repercussions. Having rebelled against the politics of the melting-pot, he is forced to infringe upon corporate regulations or practices. The government, unable to eliminate competition and aware of its advantages, takes pains to channel it through contests. The powerful and the rich, if their taste is the deciding factor, will preserve public order by practicing bland protectionism in the form of directed competition.\n\nAre they sincere? Doubtless, and all would be perfect if we were certain of their abilities but we have only their word. Sometimes harmony reigns\u2014and then they choose Vincentino. Tintoretto always avoids their contests. Does he deny their competence? Certainly not! He simply refuses them the right to treat a native in the same way as an intruder. But contests do exist, and by shunning them, our rebel is trying deliberately to destroy protectionism. He is trapped in a corner. Since the officials pretend to base their judgment on merit, and since he challenges their right to judge him, he has either to renounce painting or to win recognition through the quality of his works. He loses no time in bringing his works to their attention. Seizing upon every opportunity, he takes his competitors by surprise, confronts his jurors with the accomplished fact, and utilizes all his cunning and speed, all the diligence of his collaborators in establishing a system of mass production which breaks every record and allows him to sell his canvases at rock-bottom prices, and at times to give them away. Two second-hand shops face each other on a Roman avenue; the shopkeepers, I imagine, have conspired to simulate a merciless struggle that will not cease until both shops are brought under a single proprietor; through their eternal confrontation the shop windows suggest a tragic comedian bent on contrasting the two sides of his nature. One is covered with gloomy slogans: _\"Prezzi disastrosi!_ \" The other contains multicolored placards which announce: _\"Prezzi da ridere! da ridere! da ridere!_ \" This has been going on for years, and whenever I see the shops they make me think of Tintoretto. Had he chosen laughter or tears? Both, I think\u2014depending on the client. We can even surmise that he chuckled privately and complained publicly that he was being robbed; in any case, in his studio, every day was like a year-end clearance sale, and clients were willing to meet the judicious prices set at his liquidation sales. Having set out to commission a medallion, they ended by turning over to him every wall in their house. He was the first to break the strained bonds of friendship within the confraternity. For this unlabeled Darwinist, colleague meant personal enemy, and he discovered before Hobbes the meaning of absolute competition: _Homo homini lupus_. Venice trembles. Unless a vaccine can be found to combat the virus Tintoretto, the good old corporate system will fall apart and all that remains will be smoldering antagonisms, molecular solitudes. The Republic condemns his new methods, brands them felonies, speaks of slipshod work, of cut-rate sales, of monopoly. Later, much later, other cities will honor his methods in another language, using terms like _struggle for life, mass production, dumping, trust_ , etc. For a while this man of bad character will lose on one canvas all that he gains on another. Through hook or crook he will win commissions\u2014but not acceptance. Through a strange reversal he, the _native_ , the one-hundred-percent Rialtan, is an intruder, almost a pariah in his own city. The inevitable consequence is that he will perish unless he establishes a family. First, to stifle competition within his studio. This champion of liberalism reverses the Biblical precept; he will have others never do to him what he does to them. Moreover, he needs steadfast loyalty; outside collaborators can be frightened and discouraged by all the scandals circulated about him, and much time will be wasted if he has to reassure them. None of the scandals will permanently damage his reputation. Why does he need disciples? He wants other hands, other pairs of arms, nothing more. From absolute competition to exploitation of the family\u2014that is his course. In 1550 he marries Faustina dei Vescovi and immediately starts producing children. Just as he produces pictures: without let or hindrance. His brood has only one shortcoming: there are too many girls. Too bad! He will put all of them except two in the convent: Marietta, whom he retains as his helper, and Ottavia, whom he marries to a painter. \"Lightning Tintoretto\" will persist until Faustina gives birth to two sons, Domenico and Marco. Before their arrival he has already begun to teach the craft to his oldest daughter, Marietta. A woman painter is something extraordinary in Venice. He must have been very impatient. Finally, around 1575, his operation is completed; the new staff includes Sebastiano Casser, his son-in-law, Marietta, Domenico and Marco. The symbol of a domestic association is the _domus_ which protects and imprisons the group. At about the same date Jacopo buys a house, which he will never abandon. In this small lazaret the leper will live half-quarantined with his family, loving them more and more as he witnesses the swelling of the ranks of the _others_ who hate him. On observing him _in his home_ , at work, in his relations with his wife and children, we discover another side of his personality\u2014that of the austere moralist. Was there not more than a trace of Calvinism in his life? We see here pessimism and work, the profit motive and devotion to the family. Human nature is vitiated by original sin; men are divided by self-interest. The Christian must seek salvation through his works; he must struggle for survival, labor unceasingly to improve the Earth that God has entrusted to him; he will find the mark of divine favor in the material success of his undertaking. As for the promptings of his heart, they should be reserved for the flesh of his flesh, for his children. Was Venice feeling the influence of the reformed Religion? We know that in the second half of the century there was in Venice an odd person, Fra Paolo Sarpi, who was popular among the patricians, hostile to Rome, and familiar with foreign Protestant movements. But in all probability the petty bourgeoisie knew nothing about the tendencies, discernible in certain intellectual quarters, that seemed vaguely to favor the Reform. It would be more accurate to say that the Republic reformed itself. And by Tintoretto's time this reform has been going on for a long time. Venetian merchants owe their living to credit; they cannot accept the sentence pronounced by the Church on those whom it insists on labeling usurers, and they scorn Roman obscurantism in favor of science, especially when practical. The State has always affirmed the domination of civil authority and will not change its basic doctrine. The State has the upper hand over its clergy and, when Pope Pius V takes it upon himself to remove ecclesiastics from the jurisdiction of lay tribunals, the Senate pointedly refuses to recognize the removal. Furthermore, the government has many reasons for considering the Holy See a temporal and military power rather than a spiritual power. All this does not prevent the authorities from currying the Pope's favor, if the interest of the Republic is at stake, or pursuing heretics, or organizing a sumptuous feast in honor of St. Bartholomew to flatter a very Christian monarch. Tintoretto's pseudo-Calvinism is transmitted to him by his city; the painter unknowingly assimilates the benign Protestantism found at that time in every great capitalist stronghold.3 The artist's position is then highly equivocal, especially in Venice. But let us press our advantage; this very ambiguity may well enable us to understand Jacopo's puritanical passion.\n\nWe read that \"The Renaissance attributed to the artist the traits which Antiquity reserved for the man of action and which the Middle Ages had used to adorn its saints.\" This is not untrue, but to me the opposite observation seems at least equally true: \"[During the sixteenth century] painting and sculpture were still looked upon as manual arts; all the honors were reserved for poetry. That explains the attempts to put the figurative arts on the same footing with literature.\"4 We know that Aretino, the Petronius of the poor and the Malaparte of the rich, was the arbiter of taste and elegance for the snobs of the Venetian patriciate and that Titian was honored by his friendship, for the artist, with all his fame, was not the poet's equal. And Michelangelo? He made the mistake of imagining that he was of noble birth, and this illusion ruined his life. As a youth he wanted to cultivate the humanities, to write, in the belief that a nobleman deprived of his sword could take up the pen without degrading himself. Forced to take up the chisel, he was never able to console himself. From his dais of shame he looked down upon sculpture and painting, deriving what empty, shriveled joy he could from feeling superior to what he was doing. Forced to remain silent, he sought to provide a language for the mute arts, to multiply allegories and symbols; he wrote a book on the ceiling of the Sistine Chapel and tortured marble to force it to speak.\n\nWhat are we to conclude? Are the Renaissance painters heroes and gods, or are they manual workers? Everything is relative, depending on the clientele and the mode of remuneration. Or rather painters are primarily manual workers. They may become employees of the court or remain local masters. It is up to them to choose\u2014or to be chosen. Raphael and Michelangelo are court appointees. Proud but dependent, they will be dumped in the street if on the slightest pretext they meet with disfavor; against this, the sovereign guarantees their fame. This sacred person accords to the elect a portion of his supernatural powers; the glory of his throne falls upon them like a ray of sunshine, and they reflect it upon the people; the divine right of kings gives painters divine rights. The result: daubers changed into supermen. Just who are these ordinary men whom a giant has snatched from the petty bourgeoisie and suspended between heaven and earth, these satellites whose borrowed splendor is overpowering? Are they anything other than ordinary men raised above humanity? They are heroes, yes\u2014intercessors, intermediaries. Today still, nostalgic republicans worship in them, under the guise of genius, the light from the dead star of Monarchy.\n\nTintoretto is of another ilk. He works for merchants, for officials, for parish churches. Not that he is uneducated. He was enrolled in school at the age of seven and probably ended his schooling at twelve, after he had learned to write and reckon; besides, and more important still, we would surely have to class as education the patient cultivation of the senses, of manual and mental faculties, and of the traditional empiricism still associated with studio painting around 1530. But he will never acquire the trappings of the court painters. Michelangelo writes sonnets; Raphael is supposed to have been versed in Latin; and Titian himself finally acquired a veneer through associating with intellectuals. Compared to these worldlings, Tintoretto seems like a dunce; he will never have the leisure or the taste for toying with ideas and words. He ridicules the humanism of men of letters. Venice has few poets and still fewer philosophers, but for him these are too many and he has nothing to do with any of them. Not that he shies away from them; he simply ignores them. He is willing to admit their social superiority. Aretino has every right to congratulate him with condescending benevolence; this high-ranking person has been _received_ in Venice and is a member of the inner circle; patricians who would never dream of greeting a painter in the street invite him to their table. But does Tintoretto have to envy him, too? Does he have to envy him _because he writes?_ To him the creations of the mind acquire an utterly immoral air because they are gratuitous. God placed us on the earth to earn our bread by the sweat of our brow; but writers do not sweat. Do they really work? Jacopo never opens a book with the exception of his missal; he would never be so foolish as to force his talent for the sake of competing with literature. His paintings include everything but _mean_ nothing; they are as mute as the world. All that he really values, this son of an artisan, is physical effort, manual creation. What fascinates him in the profession of painting is that here professional ability is pushed to the point of prestidigitation and the delicacy of the merchandise reduced to its quintessence. The artist is the supreme worker; he exhausts himself and his material in order to produce and sell visions.\n\nThat would not prevent him from working for princes if he liked them. He does not and that is the crux of the matter. They frighten him without inspiring him. He never tries to approach them or to make himself known to them. He seems to take pains to confine his reputation within the walls of Venice. During his whole life he left Venice only once, when he was in his sixties, to go just outside the city to Mantua. Even then he had to be begged to go. His clients wanted him to hang up his own canvases, but he refused to go without his wife. This stipulation not only affords proof of his conjugal sentiments; it leaves no doubt about his horror of travel. And it would be wrong to think that his Venetian colleagues share his horror, for they leave no road untraveled. A century earlier Gentile Bellini was sailing the seas. What adventurers! But Jacopo is a mole, happy only within the network of his molehills. Whenever he tries to imagine the outside world, he is gripped by terror; still, if he has a choice, he prefers to risk his skin rather than his paintings. He accepts foreign commissions\u2014and for him anything beyond Padua is foreign\u2014but does not solicit them. What a contrast between his frenzied behavior in the Doge's Palace, the Scuola San Rocco, the home of the Crociferi, and this indifference! He entrusts the execution of foreign commissions to his collaborators, surveys from afar their serial productions, takes care not to interfere, as if fearful of allowing the tiniest spangle of his talent to venture beyond his native soil\u2014European distribution rights are available for only his B pictures. In the Uffizi, the Prado, the National Gallery, the Louvre, in Munich and in Vienna, we find Raphael, Titian, a hundred others. Every painter, or almost every painter, except Tintoretto. He fiercely guarded his works for his fellow citizens and the only way to find out anything about him is to search for him in his native city for the very good reason that he did not _want_ to leave Venice.\n\nBut we must be specific. In Venice itself he has two distinct clienteles. He besieges public officials and, naturally, puts his whole studio to work, including the head of the family, if the Senate gives him a commission. Still visible in the Doge's Palace, under a lighting system that shows them off to advantage, are the works of a strong collective personality that bore the name of Tintoretto. But if you are interested in Jacopo Robusti you will have to abandon the Piazzetta, cross the Piazza San Marco, ride a donkey across bridges that span the canals, turn down a labyrinth of dark, narrow streets, enter still darker churches. There you will find him. At the Scuola San Rocco you will find him in person, without Marietta or Domenico or Sebastiano Casser; there he works alone. A grimy haze darkens the canvases, or perhaps the lighting is at fault; wait patiently until your eyes become adjusted; finally you will see a rose in the darkness, a genius in the penumbra. And who paid for these paintings? Sometimes the faithful of the parish, sometimes the members of the Brotherhood\u2014middle-class men, great and small; they are his true public, the only public that he loves.\n\nThis huckster-painter has none of the qualities of a God-hero. With a little luck he will become notorious, famous, but never glorious; his profane clientele lacks the power to crown him. Of course the renown of his august colleagues honors the whole profession and he, too, scintillates somewhat. Does he covet their glory? Perhaps. But he meets none of the requirements for acquiring glory; he rejects the favor of princes because it would reduce him to servitude. Jacopo Robusti takes pride in remaining a petty chief, a peddler of Fine Arts made to order, the master of his own studio. He makes no difference between the economic independence of the producer and the freedom of the artist; his activities prove that he has a secret desire to reverse the laws of marketing, to create demand by supplying goods. Did he not create slowly and patiently within the Brotherhood of San Rocco a demand for art\u2014a certain kind of art\u2014which he alone could satisfy? His independence is preserved to an even greater degree when he works for associations\u2014 _consorterie_ , parishes\u2014and when these great bodies make their decisions by majority vote.\n\nMichelangelo, a pseudo-noble, and Titian, the son of peasants, are directly exposed to the attraction of the monarchy. Tintoretto's heritage is that of the independent craftsman and worker. The artisan is an amphibian; as a manual worker, he is proud of his hands, and as a member of the petty bourgeoisie, he is attracted by the ruling bourgeoisie. By fostering competition the ruling bourgeoisie allows fresh air to circulate within a stifling protectionism. At that time there is in Venice _a bourgeois hope_. Only a glimmer, for the aristocracy has long since taken precautions; in their stratified world, rich men are _made_ , patricians are born. But restrictions are placed on the wealthy; not only are businessmen and industrialists restricted to their own class but they are also denied entrance into the most lucrative professions; the State restricts the concession of the _appalto_ (shipping franchise) to the aristocracy. Sad, dreamy bourgeoisie! Everywhere else in Europe members of the bourgeoisie are hastening to disown their past and buy titles and castles. In Venice everything is denied them, even the humble blessing of betrayal. Betrayal will therefore take the form of dreams. Giovita Fon-tana, originally from Piacenza, moves into the business world, accumulates gold and spends it in building a palace on the Canale Grande; an entire existence is summed up in these brief words: a voracious desire, satiated, is finally turned into dreamy snobbery, a merchant dies and is reborn as an imaginary patrician. Rich commoners dance in a ring and hide their nocturnal fantasies; grouped into associations they outdo themselves in charitable works, their melancholy austerity contrasting sharply with the melancholy orgies of a disenchanted patriciate.\n\nFor the Republic is no longer mistress of the seas. Gradually the aristocracy begins to decline, failures multiply, the number of poor noblemen increases, the others lose their spirit of enterprise. The sons of the merchant princes buy land and live on their income. Soon ordinary \"citizens\" replace them in certain functions; ships eventually come under the control of men from the bourgeoisie. But the bourgeoisie is still not ready by any stretch of the imagination to consider itself the rising class. It even harbors the notion that it may one day insure the resurgence of the fallen nobility; we should say rather that an obscure agitation took hold of it, making its condition less tolerable and resignation to it more difficult.\n\nTintoretto does not dream. Never. If ambition is dependent on opportunities for social advancement, then the most ambitious commoners in Venice are the members of the petty bourgeoisie, for they still have the opportunity to rise above their class. But the painter is aware of his deep-seated affinities with his clients. He appreciates their attitude toward work and morality, their good common sense. He likes their nostalgia and, especially, he shares their profound desire for freedom; all of them need freedom, if only to produce and to buy and sell. These are the clues to his opportunism; his is a need for air which comes from the summits. A troubled sky, a distant, invisible ascent opens to him a vertical future; like a balloon he is borne aloft, filled with the new spirit, for since childhood his outlook has been that of the bourgeoisie. But the contradictions within the class of his origin are to limit his ambitions: as a peddler, he hopes always to outdo himself; as a laborer he pretends to work with his hands. That is enough to determine his position. There are in Venice approximately 7,600 patricians, 13,600 citizens, 127,000 artisans, workers, and small businessmen, 1,500 Jews, 12,900 domestics and 550 beggars. Ignoring the Jews and nobles, beggars and domestics, Tintoretto is interested only in the imaginary barrier that separates the commoners into two groups, 13,600 on the one hand and 127,000 on the other. He wants to be first in the second group and last in the first\u2014in short, the most humble of the rich and the most distinguished of the tradesmen. This makes the artisan, in the heart of troubled Venice, a pseudo-bourgeois more true than a true bourgeois. In him and on his canvases the Brotherhood of San Rocco will admire the embellished image of a bourgeoisie untainted by betrayal.\n\nMichelangelo has reservations about working for the Sovereign Pontiff; his contempt sometimes makes him recoil, for this nobleman looks down on art. Tintoretto is just the opposite; he outstrips himself; without art, what would he be? A dyer. Art is the force that lifts him above his natal condition, and his dignity is the thing that sustains him. He has to work or to fall back to the bottom of the well. Recoil from art? Keep away from it? How? He has no time to raise questions about painting. Who knows whether he even gives it a second thought? Michelangelo thinks too much; he is a gentleman, an intellectual. Tintoretto does not meditate\u2014he paints.\n\nSo much for his opportunism. His destiny is to incarnate bourgeois puritanism in an aristocratic Republic during its decline. Elsewhere this somber humanism would take root; in Venice it will disappear before being recognized for what it is, but not before arousing the distrust of an aristocracy always on guard. The moroseness that official and bureaucratic Venice manifests toward Tintoretto is the same as that which the patriciate evidences toward the Venetian bourgeoisie. These cantankerous merchants and their painter pose a danger to the Establishment and have to be kept under surveillance.\n\n1 Tr. \"The Other\" occupies a central position in the existential world: \"The Other is not only the one whom I see but the one _who sees me_ \" and makes it possible for me to \"recognize that I _am_ \" as he sees me; I do not choose to be what I am for the Other, \"but I can try to be for myself what I am for the Other, by choosing myself as I appear to the Other.\" Quoted from \"The Other and His Look\" in Justus Streller's _To Freedom Condemned_ (New York: Philosophical Library, 1960).\n\n2 Tr. Paolo Uccello (1397-1475) first exhibited a heroic sense of design and helped to create the Renaissance superman. His rigorous application of linear perspective during a later period is generally assumed to represent a paradoxical return to Gothic traditions.\n\n3 The very same one that inoculated Italian towns against the Lutheran sickness and encouraged Italy to carry out its own religious revolution under the name of Counter Reformation.\n\n4 Eugenio Battista, in an excellent article on Michelangelo published in _I'Epoca_ (August 25, 1957).\nMAN AT BAY\n\nThere is something superb about Tintoretto's stubborn refusal to compete: \"I acknowledge no rival and accept no judge.\" Michelangelo would probably say that. The bad part is that Tintoretto does not. Quite the opposite: when invited to present a sketch, he will lose no time in accepting. Afterwards, we know that he releases his bolts of lightning. Yes, somewhat in the same manner that a cuttle-fish scatters its ink. Blinded by lightning, spectators are unable to see his picture clearly. Everything is arranged, moreover, so that they need never study it or\u2014more important still\u2014appreciate it. When they come out of their stupor, the canvas is in place, the offer under seal, and they will have seen only the flash. Either I am badly mistaken or he is being evasive; he seems to be afraid to come face to face with his adversaries. Would he waste all this ingenuity if he felt certain that his talent would suffice? Would he deign to astound his contemporaries through the quantity of his output if they had no reservations about its quality?\n\nAnd then rivalry brings to the fore his mania for self-affirmation through self-effacement; this is his strong point, his trade-mark. The slightest criticism upsets him, offends him. In 1559 the San Rocco church commissioned the _Healing of the Paralytic_ to balance a canvas by Pordenone. No one asks him to imitate the style of his predecessor. There is no cause for rivalry,1 for Antonio di Sacchis has been dead for twenty years; and if it was once possible for him to influence the younger painter, that time has passed, for Jacopo has mastered his art. Still, he is unable to resist the temptation; he has to paint in the style of Pordenone. Attention has been focused on the way in which he \"exaggerates the baroque violence of their gestures... by contrasting his monumental figures with the architecture inside which they are compressed\" and \"achieved this effect by lowering the ceiling... and using the columns themselves... to immobilize the gestures, freeze their violence.\" He shudders at the notion of being forever imprisoned in an inert confrontation: \"Compare, if you like, one Pordenone with the other; I, Jacopo Robusti, am leaving.\" He has, of course, taken pains to have the spurious Di Sacchis outshine the real Di Sacchis. His retreat is not a rout; he issues a parting challenge: \"Old or young, I take them all on and beat them on their own ground.\" But this is precisely the thing that arouses suspicion. Why would he need to play their game and submit to their rules if he could outshine them all by being himself? What resentment in his insolence! This Cain assassinates every Abel preferred over him: \"You like this Veronese? Well, I can do much better when I deign to imitate him; you take him for a man and he is nothing but a technique.\" And what humility. From time to time this pariah slips into the skin of another person in order to enjoy in his turn the delight of being loved. And then at times it would seem that he lacks the courage to manifest his scandalous genius; disheartened, he leaves his genius in semidarkness and tries to prove it _deductively:_ \"Since I paint the best Veroneses and the best Pordenones, just imagine _what I am capable of painting_ when I allow myself to be me.\" As a matter of fact, he almost never takes the liberty of being himself unless someone builds up his confidence and leaves him alone in an empty room. This lack of self-confidence has its origin, of course, in the hostility manifested toward him by others. But the painter's timidity and his fellow citizens' bias have their source in the same disease; in 1548, in Venice, under Tintoretto's attack against patricians, connoisseurs and aesthetes, _painting is in jeopardy_.\n\n* * *\n\nA long evolution has begun\u2014an evolution which will substitute everywhere the profane for the sacred. Cold, glittering, rimy, the diverse branches of human activity emerge one after the other from mellow divine promiscuity. Art has its turn, and from the settling mists emerges a sumptuous disenchantment, painting. It still recalls the time when Duccio and Giotto were showing God His Creation just as it had left His hands, after He had recognized it as His work, put the world in its frame for all eternity, and closed the books of the whole affair. Into the picture, the fief of the Sun and the supreme Eye, monks and prelates sometimes slipped their transparency; they came tiptoeing in to view what God was viewing, then excused themselves and went away. Finished: the Eye is closed, Heaven blind. What is the result? First a change of clientele. As long as the work was done for the clergy, all went well; but the day the biggest of the Florentine bankers had the ridiculous notion of using frescos to beautify his house, The Omnipotent One, dismayed, began to buttress his claim to the role of Lover of Souls. Then, too, there was the Florentine affair, the conquest of perspective. Perspective is profane; sometimes even, it is a profanation. Observe Mantegna's Christ lying feet first and head remote; do you think that the Father is satisfied with a foreshortened Son?2 God is absolute proximity, universal envelopment by love; can He be shown _from a distance_ the Universe that He has created and that He is at each instant saving from annihilation? Is Being to conceive and produce Non-Being? The Absolute to engender the Relative? Light to contemplate Shadow? Reality to be taken for Appearance? No, this would be a renewal of the eternal story: Ingenuity, the Tree of Knowledge, Original Sin and Expulsion. This time the Apple is called perspective. But the Florentine Adamites nibble at it rather than eat it, thereby avoiding immediate discovery of their Fall. During the middle of the Quattrocento, Uccello thinks that he is still in Paradise, and poor Alberti, the theoretician of the \"perspectivists,\" is still trying to present geometrical optics as an Ontology of Visibility; he is rather ingenuous in asking the Divine Look to guarantee convergent lines. Heaven has failed to heed this absurd request, relegating man to the nothingness which is properly his and which he has just rediscovered once again; distance, isolation, separation\u2014these negations set our bounds; only man has a horizon. Alberti's window opens on a measurable universe, but this rigid miniature depends wholly on the point that defines both concentration and dispersion\u2014the eye. In Piero della Francesca's _Annunciation_ , between the Angel and the Virgin we see retreating columns; this is an illusion, for in themselves and for their Creator none of these inert white columns, identical and incomparable, have ever stirred in their sleep. Perspective is an act of violence which human weakness is forcing upon God's little world. A hundred years later in the Netherlands Being will be rediscovered at the very heart of appearing and appearance will reclaim the dignity of apparition; painting will have new aims and will acquire new meaning. But before Vermeer can give us the sky, the stars, day and night, the moon and the earth in the form of a tiny brick wall, the bourgeoisie from the North must win their greatest victories and forge their humanism.\n\nIn sixteenth-century Italy Faith still burns in the artists' hearts, combatting the atheism of their hands and eyes. In their attempt to get a firmer grasp on the Absolute, they perfect techniques which force upon them a Relativism which they detest. These mystified dogmatists can neither push forward nor retrace their steps. If God no longer looks at the images that they paint, who will replace Him? Their images are but the reflection of man's impotence; what will validate them? If the sole aim of painting is to gauge our myopia, it is not worth one hour's labor. To reveal man to the Omnipotent One who deigned to raise him from the clay was an act of thanksgiving, a sacrifice. But why reveal man to man? Why reveal him _as he is not?_ The artists born near the end of the century, around 1480\u2014Titian, Giorgione, Raphael\u2014pay lip service to heaven. More about this later. And then the wealth and efficacy of means still disguise the sinister indetermination of ends. Furthermore, we can surmise that Raphael had a presentiment of those ends; he mocked everyone and everything, caroused with women, sold chromos, and through his _Schadenfreude_ incited his collaborators to produce obscene engravings. Insincerity means suicide. At any rate the serenity of painting vanishes with these sacred monsters. In the second quarter of the century painting runs amuck as a result of its own perfection. In the barbaric taste which contemporaries evidenced for great \"realizations,\" a certain uneasiness is manifested; the public demands that the painter utilize all the pomp of realism to conceal his subjectivity, that he efface himself before life, blot out all memory of himself; ideally it should be as if one came upon the pictures by surprise, somewhere in the forest, and saw persons springing from the canvas and splinters from the broken frame flying at the throats of passers-by. The object should reabsorb its visibility, contain it, turn attention away from it by continuously appealing to all the senses, and particularly to the sense of touch; every artifice should be employed to replace the _representation_ by a hollow participation of the spectator in the spectacle, so that horror and tenderness would thrust men against their images and, if possible, into their midst, so that desire, burning all the fires of perspective, would discover the _ersatz_ of divine ubiquity\u2014the immediate presence of flesh, the logic of the heart. What is desired is _the thing itself_ and its destruction: bigger than nature, more real, more beautiful\u2014Terror. But Terror is a disease of rhetoric. Art will slink away, ashamed, once it has lost its letters of credit. Fettered, kept under surveillance, subjected to restrictions imposed by the State, the Church and taste, more sought after and honored, perhaps, than ever before, the artist for the first time in history becomes conscious of his solitude. Who gave him this mandate? What is the source of the right which he arrogates to himself? God has gone out, Darkness reigns. How can he paint in the dark? And _whom?_ And _what?_ And _why?_ The object of art is still the world, that Absolute, but _Reality_ steals away, reversing the relation between the finite and the infinite. A vast plenitude has been supporting the wretchedness of bodies and their fragility; now fragility becomes the sole plenitude, the unique surety. The Infinite is emptiness, darkness, inside and outside the creature; the Absolute is absence, it is God sequestered in human souls, it is desertion. It is too late to _portray_ , too early to _create;_ the painter is in hell; something comes to birth: a new damnation\u2014genius, that uncertainty, that foolish desire to traverse the world's darkness and contemplate it from without, to crush it against walls and canvases, to sift out its unknown splendors. Genius\u2014a new word in Europe, a conflict between the Relative and the Absolute, between a finite presence and an infinite absence. For the painter knows full well that he will not leave the world, that even if he could, he would bear with him everywhere the Nothingness that transpierces him; he cannot transcend perspective without first acquiring the right to create other plastic spaces.\n\nMichelangelo dies obsessed, summing up his despair and scorn in these two words: original sin. Tintoretto says nothing; he practices deceit, for if he acknowledged his solitude, he would find it unbearable. But for that very reason we can understand that he suffers from it more than anyone else; our spurious bourgeois, working for the bourgeoisie, lacks even the alibi of glory. In the pit of vipers the Little Dyer thrashes about, infected with the moral neurosis which Henri Jeanson so aptly named \"the frightening moral robustness of the ambitious\"; he sets modest objectives for himself: to rise above his father through the judicious exploitation of his talents, to corner the market by flattering public taste. Light-hearted opportunism, cunning, speed, talent\u2014nothing is lacking and everything is undermined by a vertiginous void, by Art without God. This Art is ugly, mean, dark; it is the imbecile passion of the part for the whole, an icy, tenebrous wind blowing through perforated hearts. Drawn by the void, Jacopo sets out on a motionless voyage from which he will never return.\n\nGenius does not exist; it is the scandalous audacity of Nothingness. He, the Little Dyer, exists and knows his limitations; this level-headed boy wants simply to mend the rent. All he asks is a modest plenitude; how can Infinity concern him? And how is he to know that one stroke of his brush is enough to confound his judges? His mean, stubborn ambition will be unleashed in the Darkness of Ignorance. After all, it is not his fault if painting is a lost dog with no collar; later there will be fools who will rejoice over their Abandonment; in mid-sixteenth-century Italy, the first victim of monocular perspective seeks to hide his. To work alone and for nothing is to die of fear. One must have arbiters. At any price. God has died away, Venice remains\u2014Venice which can fill up holes, seal cavities, plug outlets, stop hemorrhages and leaks. In the Doge's Republic good subjects have to consider the State in all their activities; painters must paint to beautify the city. Jacopo places himself at the disposal of his fellow citizens; they formulate a certain academic notion of Art which he is quick to adopt. All the more so since he has always had the same notion; it was explained to him when he was a small child, and he believed it: an artisan's worth is measured by the number and importance of the commissions that he receives, the honors that he is accorded. He will hide his genius under his opportunism and consider social success the sole obvious sign of a mystical victory. His deception blinds others; on the earth he plays poker and cheats; and then he gambles on heaven, without deception. If he wins down here with all the aces that he pulls from his sleeve, he dares to pretend that he will win up there; if he sells his canvases, this means that he has snared the world. But who could blame him for his malice? It is the nineteenth century that pronounced the divorce of the artist and the public; in the sixteenth, it is _true_ that painting is running amuck; it has ceased to be a religious sacrifice; but it is _equally true_ that it is being rationalized\u2014it is becoming a social service. Who then would dare to say, in Venice: \"I paint for myself, serve as my own witness\"? And as for those who say this today, are we sure that they are telling the truth? Everyone is a judge, no one is; make what you will of that. Tintoretto is more to be pitied than blamed; his Art cuts through his age like a flaming sword, but he can see it only through the eyes of his time. Besides, he has chosen his own hell; there upon the Finite again closes up over the Infinite, ambition over genius, Venice over her painter who will never again emerge. But captive Infinity gnaws away at everything; Jacopo's calculated opportunism turns to a frenzy; now he has not only to succeed but also to _prove_. A voluntary culprit, the unfortunate painter has made himself party to an endless trial; acting as his own attorney, he has made each painting a witness for the defense and has never ceased to plead his case. There is a city to convince, with its magistrates and its bourgeoisie whose verdict, to be rendered solely by them and not subject to appeal, will determine his mortal future and his immortality. He and he alone is responsible for the strange amalgam; he has to choose between drawing up a code for his own court of last resort and transforming the Republic of Venice into a supreme tribunal. He makes the only choice possible under the circumstances. Unfortunately for him. How well I understand his indifference toward the rest of the universe! He is not concerned with the opinion of the Germans or even the Florentines. Venice is the most beautiful, the richest; she has the best painters, the best critics, the most enlightened patrons of art. _Here_ , inside the brick walls, between a tiny patch of sky and still water, under the flamboyant absence of the Sun, Eternity will be won, lost, in a single lifetime, forever.\n\nWell and good. But why cheat? Why deck himself out in the trappings of Veronese? If he wants his genius to arouse admiration, why does he smother it so frequently? And why does he name judges only to corrupt them and deceive them?\n\nWhy? Because the court is prejudiced, his cause lost, his sentence set, and he knows it. In 1548 he asks Venice to bear witness to the Infinite; she becomes frightened and refuses. What a destiny! Abandoned by God, he has to practice deceit in choosing judges; having found them, he has to cheat in order to have his case adjourned. Throughout his life he will keep them in suspense, sometimes fleeing, sometimes turning on them and blinding them. Everything has its place: suffering and ill-humor, arrogance, flexibility, tremendous effort, rancor, implacable pride and the humble desire to be loved. Tintoretto's painting is first and foremost a passionate love affair between a man and a city.\n\n1 Ridolfi, deceived by the resemblance of their styles, declared that the canvas was painted _\"in concurrenza con il Pordenone_.\"\n\n2 Tr. Andrea Mantegna (1431-1506) began at an early age to experiment with traditional subjects. His most daring achievement is the unorthodox image of Christ in the ungainly position from which medical students might have viewed his deathly pallor and open wounds.\nA MOLE IN THE SUN\n\nIn this senseless romance the city is apparently even more foolish than the man. She has not failed to honor all her other painters. Then why evidence toward this one, the greatest of them all, sullen distrust, moroseness? For the simple reason that she is in love with someone else.\n\nThe Republic of Venice is hungry for prestige. Her ships have long accounted for her glory; tired and threatened with decline, she flaunts an artist. Titian alone is worth a fleet. From tiaras and crowns he has stolen flakes of fire to fashion for himself a halo. His adopted land admires him _first and foremost_ because of the respect that he inspires in the Emperor; in the sacred light, awesome but perfectly harmless, which encircles his cranium, Venice pretends to recognize her own glory. The painter of kings can be nothing less than the king of painters; the Queen of the Seas acknowledges him as her son and through him recovers a trace of majesty. She has at once given him a profession, a reputation; but when he works for kings, he is suffused with divine light which seeps through walls and spreads as far as San Marco; then she knows that he is repaying a hundredfold what he has received from her. He is a National Asset. Furthermore, this man has the longevity of trees; he endures for a century and unobtrusively becomes an Institution. The presence of an academy consisting of one member born before them and determined to survive them is demoralizing to young artists; it exasperates them and stunts their ambitions. They imagine that their city has the power to immortalize the living and that Venice has reserved this favor for Titian alone. A victim of this misunderstanding, Tintoretto\u2014under the fallacious pretext \"I am entitled to it\"\u2014demands the same recognition as his illustrious predecessor. But worth is not subject to litigation; one cannot demand from a Republic something which belongs by right to a hereditary monarchy. Jacopo is wrong in upbraiding the Doge's City for focusing all reflectors toward the baobab of the Rialto; the reverse is true. A pencil of light whose source is Rome or Madrid\u2014outside the walls, at any rate\u2014strikes the ancient trunk and reflects onto Venice, eliminating the shadows through a sort of indirect lighting. And I, too, was about to make a mistake, for I first thought of entitling this chapter \"In Titian's Shadow.\" The truth is that _Titian cast no shadow_. Weigh this carefully: on the day of Jacopo's birth the old man is forty; he is seventy-two when his young rival first tries to assert himself. That would be the moment for him to step down, to fade away gracefully. Nothing doing! The indomitable monarch reigns on for twenty-seven years. When he finally disappears, the centenarian has the supreme good fortune of leaving an unfinished _Piet\u00e0_ \u2014like a youthful dream cut short. For more than half a century Tintoretto the Mole burrows in a labyrinth whose walls are splattered with glory; until the age of fifty-eight the nocturnal beast is hemmed in by sunlight, blinded by the implacable celebrity of an Other. When the light is finally extinguished, Jacopo Robusti is old enough to die but insists on surviving the tyrant. He will gain nothing thereby, however, for Titian is adroit enough to combine two contradictory functions and to serve as an employee of the Court without relinquishing his independence as a petty employer\u2014a happy circumstance not often found in history, and certainly not in the case of Tintoretto, who has put all his eggs in one basket. Visit both tombs and you will see the price he is still paying for his sacrifice to his country. The radioactive corpse of the Grand Old Man lies buried under a mountain of embellishments at Santa Maria dei Frari, a veritable cemetery of Doges; Tintoretto's corpse rests beneath a slab in the murky darkness of a parish church. To me, this is well and good. Titian has the garnish, the sugar and spice, and this is poetic justice; I even wish that he had been buried in Rome beneath Victor Emmanuel's monument, the most hideous in all Italy except for Milan's Grand Central Station. Against this, Jacopo has the horror of naked stone; his name is sufficient. But since this is strictly a personal opinion, I can understand why an irritated traveler would ask Venice for an explanation: \"Ungrateful town, is that the best you could do for the best of your sons? Mean city, why do you put floodlights around Titian's _Assumption_ and stint on electricity for Robusti's canvases?\" I know the explanation; it appeared in 1599 in Aretino's correspondence: \"If Robusti wants to be honored, why doesn't he paint like Vecellio?\" Jacopo will hear the same refrain every day of his life; it will be repeated before his canvases after his death as well as before, and it is still heard today: \"Why does he paint like this? Why does he stray from the Royal Way prepared for him? Our great Vecellio has carried painting to such heights of perfection that no further progress is possible; newcomers will have to follow in the Master's footsteps or Art will fall back into barbarism.\" Fickle Venetians! Inconsequential bourgeois! Tintoretto is _their_ painter; he portrays what they see and feel; this they can not endure. Titian ridicules them, and they worship him.1 Titian spends most of his time soothing princes, reassuring them through his canvases that everything is for the best in this best of all possible worlds. Discord is but an illusion; archenemies are secretly reconciled by the colors of their cloaks. Violence? only a ballet danced half-heartedly by spurious he-men with downy beards. Such is his method of depreciating wars. The painter's art borders on the apologetic, becomes a theodicy: suffering, injustice, evil do not exist; nor does mortal sin; Adam and Eve sinned only in order that they might have the opportunity to know and to make known to us that they were naked. In a magnificent quadruple gesture God, noble and benevolent, leaning out from his Heaven, stretches out His arms toward the upraised arms of supine Man. Order reigns; quelled and enslaved, perspective respects hierarchies; discreet accommodations assure kings and saints of preferred positions. If someone is lost in the distance, shrouded by the haze that envelops vague terrain or obscured by smoke from remote lamps, this is never by accident. The indistinctness of the figure corresponds to the obscurity of his condition; besides, it helps to focus attention on the patches of light in the foreground. The artist pretends with his brush to relate an event or record a ceremony; by sacrificing movement to order and contrast to unity, he makes his brush caress bodies rather than copy them. Not one of the bearded persons witnessing the Assumption is individualized. First come several legs and upraised arms\u2014a flaming bush; then the substance is impregnated with an element of differentiation and engenders fleeting figures scarcely distinguishable from the collective background which can at any moment reabsorb them\u2014such is the condition of the underlings; Titian reserves individuality for the Great. Even here, however, he is careful to round off their angles; sharp lines isolate, create distance, signify pessimism; the courtesan, who is a professional optimist, directs a symphony of colors proclaiming God's glory, which he epitomizes and mitigates. Titian then applies the finishing touches; he scrapes and polishes, applies lacquers and varnishes. Sparing no effort to hide his labor, he manages finally to remove every trace of himself from the canvas. The stroller enters an unobstructed area, walks among flowers under a proper sun; the proprietor is dead; the stroller is so lonely that he forgets himself and disappears. The result is treason of the worst sort: the betrayal of Beauty.\n\nFor once the traitor has the excuse of believing in what he is doing. He is not a townsman but a transplanted peasant; when he arrives in Venice, he comes as a rustic child of the Middle Ages. The country youth has long nurtured a popular, reverent love for the nobility; he makes his way through the bourgeoisie without even seeing it and rejoins his true masters at the summit, all the more certain of pleasing them because they have his sincere respect. We frequently hear that he secretly considered himself their equal; this, I think, is completely false. What would have been the source of his light? He is a vassal; raised to the peerage through the glory that only kings can dispense, he owes to them everything, even his pride; why would he choose to turn against them? He looks upon his haughty serenity, the hierarchy of power and the beauty of the world as complementary reflexes; in the best possible faith he puts the bourgeois techniques of the Renaissance to the service of feudalism\u2014he has stolen their tool.\n\nYet he is admired by both the bourgeoisie and the patriciate. He provides the Venetian technocrats with an alibi by speaking of happiness, glory and preordained harmony at the time they are making laudable efforts to obscure their decline. Every merchant, whether a nobleman or a commoner, is captivated by the sanctimonious canvases that reflect the tranquility of kings. If all is for the best, if evil is but a beautiful illusion, if each keeps forever his hereditary place in the divine and social hierarchy, then this means that nothing has happened for the last hundred years: the Turks have not taken Constantinople, Columbus has not discovered America, the Portuguese have not even dreamed of dumping spices or the continental powers of forming a coalition against the Republic of Venice. People had thought that the Barbary pirates were threatening the seas, that the African source of precious metals had been exhausted, that the scarcity of money had slowed down transactions during the first half of the century and that then suddenly an outpouring of Peruvian gold from the Spanish waterworks had reversed the tide, raised prices, flooded the market\u2014but that was only a dream. Venice still reigns over the Mediterranean; she is at the pinnacle of her power, wealth and grandeur. In other words, they want Beauty, these uneasy souls, because it is reassuring. I understand them, for I have boarded a plane two hundred times without ever becoming reconciled to it. I am too earth-bound to consider flying normal; occasionally fear surges up\u2014especially when my companions are as ugly as I; but if there is on board a beautiful woman or a handsome boy or a charming couple in love, my fear vanishes; ugliness is a prophecy\u2014it trails a certain element of extremism which seeks to carry negation to the point of horror. Beauty seems indestructible; its sacred image protects us; so long as we have it in our presence, no catastrophe will occur. The same is true of Venice; she is beginning to fear that she will sink into the mire of her lagoons; imagining that she will find salvation through Beauty, the supreme levity, she makes a pretense of transforming her palaces and canvases into buoys and floats. Those responsible for Titian's success are the very ones who desert the sea, who try to escape their disenchantment through orgies, who prefer the security of ground rent to profit from commerce.\n\nTintoretto is born in a troubled town; he breathes Venetian uneasiness, is consumed by Venetian uneasiness, can paint nothing but Venetian uneasiness. If they were in his place, his severest critics would behave no differently. But they are not; they cannot escape uneasiness but wish not to have it brought to their attention; they condemn paintings that _represent_ it. Fate has decreed that Jacopo unwittingly expose an age which refuses to recognize itself. Now we understand the meaning of his destiny and the secret of Venetian malice. Tintoretto displeases everyone: patricians because he reveals to them the puritanism and fanciful agitation of the bourgeoisie; artisans because he destroys the corporate order and reveals, under their apparent professional solidarity, the rumblings of hate and rivalry; patriots because the frenzied state of painting and the absence of God discloses to them, under his brush, an absurd and unpredictable world in which anything can occur, _even_ the death of Venice. It would seem that this painter who has assimilated bourgeois culture might at least find favor with the class that he has adopted. Not so. The bourgeoisie will accept him only with reservations; it always finds him fascinating, but often it finds him terrifying. The reason is that it has not recognized itself for what it is. Signor de Zigninoni must have dreamed of betrayal; he was searching covertly for a means of acceding to the patriciate, of escaping from the bourgeois reality that he was helping in spite of himself to create. What he finds most distasteful in Robusti's paintings is their radicalism and their \"demystifying\" virtues. In short, it is necessary at any price to refute Tintoretto's testimony, to make it seem that he has failed in his venture, to deny the originality of his research, _to get rid of him_.\n\nConsider the charges brought against him: _first_ , that he works too fast and leaves his imprint everywhere. People want smooth, finished work, especially _the impersonal element;_ if the painter portrays himself, he is subjecting himself to interrogation and thereby putting the public on trial. Venice imposes on her artists the maxim of the puritans: \"No personal remarks.\" She is careful to equate Jacopo's lyricism with the callous haste of a jaded contractor. Then comes Ridolfi's charge that Tintoretto wrote on the walls of his studio: \"The color of Titian combined with the drawing of Michelangelo.\" The charge is baseless; the statement first appears in 1548 in the writings of a Venetian art critic and does not refer to Robusti. The latter, in fact, could have known the works of Michelangelo only through reproductions by Daniele de Volterra\u2014consequently, _not before_ 1557. And could anyone take those words at face value? Is it conceivable that he would try _seriously_ to follow the absurd formula? The legend is but a dream of his age; confronted by the Spanish menace, the Northern states and the Central states dream of forming an alliance\u2014too late. But the awakening of a national consciousness, though brief, cannot fail to have a transient influence on the Fine Arts. \"Michelangelo and Titian'\" means Florence and Venice. How nice for painting to be unified!\n\nNothing serious, obviously. The dream is inoffensive so long as it is everyone's dream. But those who pretend to see in it the obsession of Robusti _alone_ must have wanted to destroy the artist by lodging in the heart of his art an explosive nightmare. Color is Jake laughing; drawing, Jake crying. In the first instance unity, in the second the risk of disorder. On the one hand the harmony of the spheres, on the other abandonment. The two Titans of the century throw themselves on each other, embrace each other, try to stifle each other\u2014Jacopo is the theater of operations. And sometimes Titian wins by a hair and sometimes Michelangelo barely manages to claim the match. In either case, the loser is strong enough to spoil the winner's triumph, and the result of the Pyrrhic victory is a botched picture. Botched through excess. Tintoretto seems to his contemporaries like an insane Titian, devoured by Buonarroti's somber passion, shaken by St. Vitus dance\u2014possessed, a freakish split personality. In one sense Jacopo exists only as a battlefield; in another sense he is a monster, a fraud. Vasari's fable becomes crystal clear; Adam Robusti wanted to taste the fruits of the tree of knowledge and Archangel Tiziano, pointing his finger and flapping his wings, chased him out of Paradise. To have bad luck or to bring bad luck is still one and the same in Italy. If you have recently had financial troubles or an automobile accident, if you have broken your leg or lost your wife, do not expect to be invited to dinner; a hostess would not wittingly expose her other guests to premature baldness, a head cold or, in extreme cases, to a broken neck caused by a fall on her stairway. I knew a Milanese who had the evil eye; this was discovered last year; he no longer has a single friend, and he dines alone, at home. Such is Jacopo: a caster of spells because a spell was cast on him. Or perhaps on his mother when she was carrying him. The spell actually has its source in Venice: uneasy, accursed, she has produced a troubled soul and placed a curse on his uneasiness. The unfortunate victim loves a despairing and uncompromising town to the point of despair, and his love horrifies the beloved. When Tintoretto passes by, people step aside: he smells of death. Exactly. But what other odor is given off by patrician festivities and bourgeois charity and the docility of the people? Pink houses with flooded cellars and walls crisscrossed by rats? What odor is given off by stagnant canals with their urinous cresses and by grey mussels fastened with squalid cement to the underside of quays? In the depths of a river a bubble is clinging to the mud; broken loose by the eddies formed by gondolas, it rises through murky water, breaks through the surface, spins around, glistens and bursts; everything crumbles away when the blister bursts\u2014bourgeois nostalgia, the grandeur of the Republic, God and Italian painting.\n\nTintoretto was the chief mourner for Venice and a way of life, but when he died there was no one to act as his chief mourner; then silence fell, and hypocritically pious hands hung crepe over his canvases. When we remove this black veil, we find a portrait, started anew a hundred times. The portrait of Jacopo? The portrait of the Queen of the Seas? As you will: the city and her painter have one and the same face.\n\n1 Tr. Titian Tiziano Vecellio (c. 1477-1576) owed his enormous success as a portraitist to his ability to paint each subject's ideal of himself.\nTHE PAINTINGS OF GIACOMETTI1\n\nFrom the back of the room where I was sitting at the Sphinx, I could see several nude women. The distance that separated us (the glossy wood floor seemed insuperable even though I wanted to walk across it) impressed me as much as did the women.2\n\nThe result: four inaccessible figurines balanced on the edge of a vertical background formed by the floor. Giacometti painted them as he saw them\u2014 _from a distance_. Still, the four women have an arresting presence. They seem to be poised on the floor ready at any moment to drop down upon him like the lid on a box.\n\nI have often seen them, especially in the evening, in a little place on the Rue de l'Echaud\u00e9, very close and menacing.\n\nDistance, far from being an accident, is in his eyes part and parcel of every object. These whores, twenty steps away\u2014twenty impossible steps away\u2014are forever outlined in the light of his hopeless desire. His studio is an archipelago, a conglomeration of irregular distances. The Mother Goddess against the wall retains all the nearness of an obsession. When I retreat, she advances; when I am farthest away, she is closest. The small statue at my feet is a man seen in the rear-view mirror of an automobile\u2014in the act of disappearing; moving closer to the statue is to no avail, for the distance cannot be traversed. These solitudes repel the visitor with all the insuperable length of a room, a lawn, or a glade that none would dare to cross. They stand as proof of the paralysis that grips Giacometti at the sight of his equal.\n\nIt does not follow, however, that he is a misanthropist. His aloofness is mixed with fear, often with admiration, sometimes with respect. He is distant, of course, but man creates distance while distance has no meaning outside human space. Distance separates Hero from Leander and Marathon from Athens but not one pebble from another.\n\nI first understood what distance is one evening in April, 1941. I had spent two months in a prison camp, which was like being in a can of sardines, and had experienced absolute proximity; the boundary of my living space was my skin; night and day I felt against my body the warmth of a shoulder or a bosom. This was not incommodious, for the others were _me_.\n\nThat first evening, a stranger in my home town, having not yet found my old friends, I opened the door of a caf\u00e9. Suddenly I was frightened\u2014or almost; I could not understand how these squat, corpulent buildings could conceal such deserts. I was lost; the scattered patrons seemed to me more distant than the stars. Each of them could claim a vast seating area, a whole marble table while I, to touch them, would have had to cross over the \"glossy floor\" that separated us.\n\nIf they seemed inaccessible to me, these men who were scintillating comfortably in their bulbs of rarefied gas, it was because I no longer had the right to place my hand on their shoulders and thighs or to call one of them \"knuckle-head.\" I had re-entered middle-class society and would have to learn once again to live \"at a respectable distance.\" My attack of agoraphobia had betrayed my vague feeling of regret for the collective life from which I had been forever severed.\n\nThe same applies to Giacometti. For him distance is not a voluntary isolation, nor even a withdrawal. It is something required by circumstances, a ceremony, a recognition of difficulties. It is the product\u2014as he himself said3\u2014of forces of attraction and forces of repulsion. He cannot walk a few steps across the glossy floor that separates him from the nude women because he is nailed to his chair by timidity or by poverty; and he feels at this point that the distance is insuperable because he wants to touch their lush flesh. He rejects promiscuity, the fruit of close proximity, because he wants friendship, love. He dares not take for fear of being taken.\n\nHis figurines are solitary, but when placed together, no matter how, they are united by their solitude and transformed into a small magical society:\n\nOn observing the figures which, to clear away the table, had been set at random on the floor, I discovered that they formed two groups which seemed to correspond to what I was looking for. I mounted the two groups on bases without the slightest change....\n\nOne of Giacometti's scenes is a crowd. He has sculptured men crossing a public square without seeing each other; they pass, hopelessly alone and yet _together;_ they will be forever lost from each other, yet would never lose each other if they had not sought each other. He defined his universe better than I possibly could when he wrote, concerning one of his groups, that it reminded him of\n\na part of a forest observed during the course of many years... a forest in which trees with barren, slender trunks seemed like people who had stopped in their tracks and were speaking to each other.\n\nWhat is this circular distance\u2014which only words can bridge\u2014if not negation in the form of a _vacuum?_ Ironic, defiant, ceremonious and tender, Giacometti sees space everywhere. \"Not everywhere,\" you will say, \"for some objects are in contact.\" But Giacometti is sure of nothing, not even that. Week after week he was captivated by the legs of a chair: they were not touching the floor. Between things, between men lie broken bridges; the vacuum infiltrates everything; each creature creates its own vacuum.\n\nGiacometti became a sculptor because of his obsession with emptiness. About one statuette he wrote: \"Me, rushing down a street in the rain.\" Sculptors rarely fashion their own busts. Those who do attempt \"self-portraits\" study themselves from without in a looking glass. They are the true prophets of objectivity. But imagine a lyrical sculptor: what he tries to reproduce is his inner feeling, the boundless vacuum that surrounds him, leaving him defenseless and exposing him to the storm. Giacometti is a sculptor because he wears his vacuum as a snail its shell, because he wants to explain all its facets and dimensions. And sometimes he finds compatible the modicum of exile that he carries everywhere\u2014and sometimes he finds it horrifying.\n\nA friend once moved in with him. Pleased at first, Giacometti soon became upset: \"I opened my eyes one morning and found his trousers and his jacket _in my space_.\" At other times, however, he grazes walls and skirts ramparts; the vacuum all around him portends a catastrophe, untoward events, avalanches. In any case he must bear witness to its presence.\n\n* * *\n\nCan he do this through sculpture? By kneading plaster, he creates a vacuum _from a plenum_. The figure when it leaves his fingers is \"ten steps away,\" and no matter what we do, it remains there. The statue itself determines the distance from which it must be viewed, just as courtly manners determine the distance from which the king must be addressed. The situation engenders the surrounding no-man's land. Each of his figures is Giacometti himself producing his little local vacuum. Yet all these slight absences that are as much a part of us as our names, as our shadows, are not enough to make a world. There is also the Void, the universal distance between all things. The street is empty, drinking in the sun; suddenly, in this empty space a human being appears.\n\nSculpture can create a vacuum _from a plenum_ , but can it show the plenum arising from what was previously a vacuum? Giacometti has tried a hundred times to answer this question. His composition _\"La Cage_ \" represents his \"desire to abolish the socle and have a _limited_ space for creating a head and face.\" That is the crux of his problem, for a vacuum will forever antedate the beings that inhabit it unless it is first surrounded by walls. The \"Cage\" is \"a room that I have seen. I have even seen curtains behind the woman....\" On another occasion he made \"a figurine in a box between two boxes which are houses.\" In short, he builds a frame for his figures, with the result that they remain at a certain distance away from us but live in the closed space imposed on them by their individual distances, in the prefabricated vacuum which they cannot manage to fill and which they endure rather than create.\n\nAnd what is this framed and populated vacuum if not a painting? Lyrical when he sculptures, Giacometti becomes objective when he paints. He tries to capture the features of Annette or of Diego just as they appear in an empty room or in his deserted studio. I have tried elsewhere to show that he approaches sculpture as a painter since he treats a plaster figurine as if it were a person in a painting.4 He confers on his statuettes a fixed, imaginary distance. Inversely, I can say that he approached painting as a sculptor since he would like to have us assume that the imaginary space enclosed by a frame is a _true_ void. He would like to have us perceive through thick layers of space the woman that he has just painted in a sitting position; he would like for his canvas to be like still water and for us to see the figures _in_ the painting as Rimbaud saw a room in a lake\u2014as a transparency.\n\nSculpturing as others paint, painting as others sculpture, is he a painter? Is he a sculptor? Neither, both. Painter and sculptor because his era does not allow him to be both sculptor and architect; sculptor in order to restore to each his circular solitude and painter in order to replace men and things in the world\u2014that is, in the great universal void\u2014he finds it convenient to model what he had at first hoped to paint.5 At times, however, he knows that only sculpture (or in other instances only painting) will allow him to \"realize his impressions.\" In any case the two activities are inseparable and complementary. They allow him to treat from every aspect the problem of his relations with others, whether distance has its origin in them, in him, or in the universe.\n\n* * *\n\nHow can one paint a vacuum? Before Giacometti it seems that no one had made the attempt. For five hundred years painters had been filling their canvases to the bursting point, forcing into them the whole universe. Giacometti begins by expelling the world from his canvases. For example, he paints his brother Diego all alone, lost in a hangar, and that is sufficient.\n\nA person must also be separated from everything around him. This is ordinarily achieved by emphasizing his contours. But a line is produced by the intersection of two surfaces, and an empty space cannot pass for a surface. Certainly not for a volume. A line is used to separate the container from the content; a vacuum, however, is not a container.\n\nIs Diego \"outlined\" against the partition behind him? No, the \"foreground-background\" relation exists only when surfaces are relatively flat. Unless he leans back against it, the distant partition cannot \"serve as a background\" for Diego; in short, he is in no way connected with it. Or rather he is only because man and object are in the same painting and must therefore maintain appropriate relations (hues, values, proportions) for conferring on the canvas its unity. But these correspondences are at the same time erased by the vacuum that interposes itself between them.\n\nNo, Diego is not outlined against the gray background of a wall. He is there, the wall is there, that is all. Nothing encloses him, nothing supports him, nothing contains him; he _appears_ all alone within the vast frame of empty space.\n\nWith each of his paintings Giacometti takes us back to the moment of creation _ex nihilo_. Each painting restates the old metaphysical question: Why is there something rather than nothing? And yet there is something: this stubborn, unjustifiable, superfluous apparition. The painted person is hallucinatory because presented in the form of an interrogative apparition.\n\n* * *\n\nBut how can the artist place a figure on his canvas without confining it? Will it not explode in empty space like a fish from the depths on the surface of the water? Not at all. A line represents arrested flight, a balance between the external and the internal; it fastens itself around the shape adopted by an object under the pressure of outside forces; it is a symbol of inertia, of passivity.\n\nGiacometti does not think of finitude as an arbitrary limitation, however. For him the cohesion of an object, its plenitude and its determination are but one and the same effect of its inner power of affirmation. \"Apparitions\" affirm and confine themselves while defining themselves. Somewhat as the strange curves studied by mathematicians are both encompassing and encompassed, the object encompasses itself.\n\nOne day when he had undertaken to sketch me, Giacometti expressed surprise: \"What density,\" he said, \"what lines of force!\" And I was even more surprised than he since I believe my features to be weak and ordinary. But the reason is that he saw each feature as a centripetal force. A face is forever changing, like a spiral. Turn around: you will never find a contour\u2014only a plenum. The line is the beginning of negation, the passage from being to non-being. But Giacometti holds that reality is pure positivity, that there _is_ being and then suddenly there no longer is any, but that there is no conceivable transition from being to nothingness.\n\nNotice how the multiple lines that he draws are _inside_ the form depicted. See how they represent intimate relations between being and itself; the fold in a garment, the wrinkle in a face, the protruding of a muscle, the direction of a movement\u2014all these lines are centripetal. They tend to confine by forcing the eye to follow them and leading it always to the center of the figure. The face seems to be contracting under the influence of an astringent substance, giving the impression that in five minutes it will be the size of your fist, like a shrunken head. Still, demarcation of the body is missing. At times the heavy mass of flesh is demarcated vaguely, slyly by a blurred brown nimbus somewhere under the tangly lines of force\u2014and sometimes it is literally unbounded, the contour of an arm or a hip being lost in a dazzling play of light.\n\nWe are shown without warning an abrupt de-materialization. For example, a man is shown crossing his legs; as long as I looked only at his head and bust, I was convinced that he had feet. I even thought that I could see them. If I look at them, however, they disintegrate, disappear in a luminous haze, and I no longer know where the void begins and where the body ends. And do not think that this is the same as one of Masson's attempts to disintegrate objects and give them a semblance of ubiquity by scattering them over the whole canvas. If Giacometti fails to demarcate a shoe, the reason is not that he believes it to be unbounded but that he counts on us to add its bounds. They are actually there, these shoes, heavy and dense. To see them, we need only refrain from viewing them in their entirety.\n\nTo understand this procedure we need only examine the sketches that Giacometti sometimes makes for his sculptures. Four women on a socle\u2014fine. But let us examine the drawing. First we see the head and neck sketched in bold strokes, then nothing, then an open curve encircling a fixed point\u2014the belly and navel; we also see the stump of a thigh, then nothing, and then two vertical lines and, further down, two others. That is the whole thing. A whole woman. What did we do? We used our knowledge to reestablish continuity, our eyes to join together these _disjecta membra_. We _saw_ shoulders and arms on a white paper; we saw them because we had _recognized_ a head and torso.\n\nThe members were indeed there, though not represented by lines. In the same way we sometimes apprehend lucid, complete ideas that are not represented by words. The body is a current flowing between its two extremities. We are face to face with the absolute reality, the invisible tension of blank paper. But does not the blankness of the paper also represent empty space? Certainly, for Giacometti rejects both the inertia of matter and the inertia of absolute nothingness. A vacuum is a distended plenum, a plenum an oriented vacuum. Reality fulgurates.\n\n* * *\n\nHave you noticed the superabundance of light strokes that striate his torsos and faces? Diego is not solidly stitched but merely basted, in the language of dressmakers. Or could it be that Giacometti wishes \"to write luminously on a dark background\"? Almost. The emphasis is no longer on separating a plenum from a vacuum but on painting plenitude itself. And since it is at once unity and diversity, how can it be differentiated unless divided? Dark strokes are dangerous, for they risk effacing being, marring it with fissures. If used to outline an eye or encircle a mouth they may create the impression that there are fistules of empty space at the heart of reality. The white striae are there to serve as unseen guides. They guide the eye, determine its movements, dissolve beneath its gaze. But the real danger lies elsewhere.\n\nWe are aware of the success of Arcimboldo\u2014his jumbled vegetables and cluttered fish. Why do we find this artifice so appealing? Is it perhaps because the procedure has long been familiar to us? In their own way, have all painters been Arcimboldos? Have they not fashioned, day after day, face after face, each with a pair of eyes, a nose, two ears and thirty-two teeth? Wherein lies the difference? He takes a round cut of red meat, makes two holes in it, sets in each of them a white marble, carves out a nasal appendage, inserts it like a false nose under the ocular spheres, bores a third hole and provides it with white pebbles. Is he not substituting for the indissoluble unity of a face an assortment of heterogeneous objects? Emptiness insinuates itself everywhere: between the eyes and eyelids, between the lips, into the nostrils. A head in its turn becomes an archipelago.\n\nYou say that this strange assemblage conforms to reality, that the oculist can remove the eye from its orbit or the dentist extract the teeth? Perhaps. But what is the painter to paint? Whatever is? Whatever we see? And what do we see?\n\nTake the chestnut tree under my window. Some have depicted it as a huge ball, a trembling unity; others have painted its leaves individually, showing their veins. Do I see a leafy mass or a multitude of leaves? I must say that I see both, but neither in its entirety, with the result that I am constantly shifting from one to the other. Consider the leaves: I fail to see them in their entirety, for just as I am about to apprehend them they vanish. Or the leafy mass: just as I am about to apprehend it, it disintegrates. In short I see a swarming cohesion, a writhing dispersion. Let the painter paint that.\n\nAnd yet Giacometti wants to paint what he sees just as he sees it. He wants the figures at the heart of their original vacuum on his motionless canvas forever to fluctuate between continuity and the discontinuity. He wants the head to be at once isolated because sovereign and reclaimed by the body to serve as a mere periscope of the belly in the sense that Europe is said to be a peninsula of Asia. The eyes, the nose, the mouth\u2014these he wants to make into the leaves of a leafy mass, isolated from each other and blended all together. He succeeds, and this is his supreme triumph.\n\nHow does he succeed? By refusing to be more precise than perception. He is not _vague;_ he manages rather to suggest through the lack of precision of perception the absolute precision of being. In themselves or for others with a better view, for angels, his faces conform rigidly to the principle of individuation. A glance reveals that they are precise down to the most minute detail; furthermore, we immediately recognize Diego or Annette. That in itself would be sufficient, if required, to cleanse Giacometti of any taint of subjectivism.\n\nAt the same time, however, we cannot look at the canvas without uneasiness. We have an irrepressible urge to call for a flashlight or at least a candle. Is it a haze, the fading light of day, or our tired eyes? Is Diego lowering or raising his eyelids? Is he dozing? Is he dreaming? Is he spying? It happens of course that the same questions are asked at popular exhibitions, in front of portraits so bland that any answer is equally appropriate and none mandatory.\n\nThe awkward indetermination of popular painters has nothing in common with the calculated indetermination of Giacometti, which might more appropriately be termed overdetermination. I turn back toward Diego and see him alternately asleep and awake, looking at the sky, gazing at me. Everything is true, everything is obvious; but if I bend my head slightly, altering my viewpoint, this truth vanishes and another replaces it. If after a long struggle I wish to adopt one opinion, my only recourse is to leave as quickly as possible. Even then my opinion will remain fragile and probable.\n\nWhen I discover a face in the fire, for example, or in an inkblot, or in the design of a curtain, the shape that has abruptly appeared becomes rigid and forces itself upon me. Even though I can see it in no other way than this, I know that others will see it differently. But the face in the fire has no truth while in Giacometti's paintings we are provoked and at the same time bewitched by the fact that _there is_ a truth and that we are certain of it. It is there, right under my nose, whether I look for it or not. But my vision blurs, my eyes tire, I give up. Then I begin to understand that Giacometti overpowers us because he has reversed the facts in stating the problem.\n\nA painting by Ingres is also instructive. If I look at the tip of the odalisk's nose, the rest of the face is light and soft, like pinkish butter interrupted by the delicate red of the lips; and if I shift my attention to the lips, they emerge from the shadows, moist and slightly parted, and the nose disappears, devoured by the absence of differentiation in the background. I am not bothered by its disappearance, however, for I am secure in the knowledge that I can always recreate it at will.\n\nThe reverse holds true in the case of Giacometti. To make a detail seem clear and reassuring all I need do is refrain from centering my attention on it. My confidence is reinforced by what I see through the corner of my eye. The more I look at Diego's eyes the less they communicate to me; but I notice slightly sunken cheeks, a peculiar smile at the corners of the mouth. If my obsession with truth draws my attention down to his mouth, everything immediately escapes me. What is his mouth like? Hard? Bitter? Ironical? Wide-open? Sealed? Against this, _I know_ that his eyes, which are almost beyond my range of vision, are half-closed. And nothing prevents me from continuing to turn, obsessed by the phantom face that is constantly being formed, deformed and reformed behind me. The remarkable part is its credibility. Hallucinations also make their appearance on the periphery only to disappear when viewed directly. But on the other hand, of course...\n\n* * *\n\nThese extraordinary figures, so perfectly immaterial that they often become transparent and so totally, so fully real that they can be as positive and unforgettable as a physical blow, are they appearing or disappearing? Both. They seem so diaphanous at times that we do not even dream of questioning their features; we have to pinch ourselves to learn whether they really exist. If we insist on examining them, the whole canvas becomes alive; a somber sea rolls over them, leaving only an oil-splotched surface; and then the waves roll back and we see them glistering under the water, white and naked. But their reappearance is marked by a violent affirmation. They are like muffled shouts rising to the top of a mountain and informing the hearer that somewhere someone is grieving or calling for help.\n\nThe alternation of appearance and disappearance, of flight and provocation, lends to Giacometti's figures a certain air of coquetry. They remind me of Galatea, who fled from her lover under the willows and desired at the same time that he should see her. Coquettish, yes, and graceful because they are pure action, and sinister because of the emptiness that surrounds them, these creatures of nothingness achieve a plenum of existence by eluding and mystifying us.\n\nEvery evening an illusionist has three hundred accomplices: his audience and their second natures. He attaches to his shoulder a wooden arm in a bright red sleeve. His viewers expect him to have two arms in identical sleeves; they see two arms, two sleeves, and are satisfied. Meantime a real arm, clothed in black and invisible, produces a rabbit, a card, an explosive cigarette.\n\nGiacometti's art is similar to that of the illusionist. We are his dupes and his accomplices. Without our avidity, our gullibility, the traditional deceitfulness of the senses and contradictions in perception, he could never make his portraits live. He is inspired not only by what he sees but also, and especially, by what he thinks we will see. His intent is not to offer us an exact image but to produce likenesses which, though they make no pretense at being anything other than what they are, arouse in us feelings and attitudes ordinarily elicited by the presence of real men.\n\nAt the Gr\u00e9vin Museum one may feel irritated or frightened by the presence of a wax guardian. Nothing would be easier than to construct elaborate farces by capitalizing on that fact. But Giacometti is not particularly fond of farces. With one exception. A single exception to which he has consecrated his life. He has long understood that artists work in the realm of the imaginary, creating illusions, and he knows that \"faked monsters\" will never produce in spectators anything other than factitious fears.6\n\nIn spite of his knowledge, however, he has not lost hope. One day he will show us a portrait of Diego just like all others in appearance. We shall be forewarned and know that it is but a phantom, a vain illusion, a prisoner in its frame. And yet on that day, before the mute canvas we shall feel a shock, a very small shock. The very same shock that we feel on returning late and seeing a stranger walking toward us in the dark.\n\nThen Giacometti will know that through his paintings he has brought to birth a real emotion and that his likenesses, without ever ceasing to be illusory, were invested for a few instants with _true_ powers. I hope that he will soon achieve this memorable farce. If he does not succeed, no one can. In any case, no one can surpass him.\n\n1 Tr. Originally published in _Les Temps Modernes_ (June, 1954). Alberto Giacometti (1901-1966) belonged to an artistic Swiss family. Unchanged by success, he worked from 1927 until his death in a two-room studio in the industrial section of Paris. His paintings are for the most part studies of himself, his wife Annette and his brother Diego. His best known sculptures are probably _Three Men Walking_ (1949), _Walking Quickly Under the Rain_ (1949), and _Man Crossing a Square on a Sunny Morning_ (1950). Known primarily as a surrealist in the early 1930's, he went through a long period of experimentation and emerged in the 1940's as one of the world's most controversial sculptors. Defining art as \"an absurd activity,\" he has evolved elongated figures expressing nihilism and despair, terror and doom.\n\n2 Letter to Matisse (November, 1950).\n\n3 Letter to Matisse (1950).\n\n4 Tr. See the last essay in this collection.\n\n5 For example, his _Nine Figures_ (1950): \"I had wanted very much to paint them last spring.\"\n\n6 Tr. Sartre's first philosophical work was a study of the imagination, published in France in 1936. It was subsequently published in English in 1948 under the title _Psychology and Imagination_ (New York: Philosophical Library).\nTHE UNPRIVILEGED PAINTER: LAPOUJADE1\n\nSince Goya killers have not ceased their slaughter nor pacifists their protests. Every five or ten years a painter attempts to revive the horrors of war by modernizing uniforms and weapons.2\n\nLapoujade's undertaking is different. His aim is not to congeal art by putting it to the service of Genteel Thought but rather to study _internally_ the nature and scope of painting. Almost a century ago Art became critical; since then it has flaunted its right to judge painting. Lapoujade has evolved his own style while painting and has finally created presences that are an integral part of each composition and yet transcend them all.\n\nHis presences could not be communicated through figurative art.3 The human figure, in particular, disguised men's suffering; it disappeared, and its demise within the very matrix of art gave birth to something else\u2014to tortured victims, razed towns, massacred hordes, ubiquitous tormentors. The painter shows us both victims and victors\u2014in short, the portrait of our century. No longer is the object of his art the individual or the typical. It is the singularity and reality of our age. How did Lapoujade manage through the very limitations imposed by abstract art to achieve what figurative art could not?\n\nSince canvases have won the right to be judged solely according to the laws of painting, the artist can reaffirm the fundamental, inviolable link between creativity and beauty. Whatever its provenance, a canvas will be beautiful, or it will not be; daubing is not painting, nor will it look like anything but daubing. Beauty is not the object of art but its flesh and blood, its being. Everyone has always said this, everyone pretends to know it. It is nevertheless true that in abstract art the fundamental link between creativity and beauty, obscured by an alchemist's dream, by the desire to produce a real absolute, is once again revealed in its pristine purity.\n\nAt the same time the clich\u00e9, \"Art for the sake of Art,\" is reasserted. But this is sheer nonsense! No one paints to create art or to make it be. The artist simply paints. Lapoujade doesn't paint his canvases in the hope of enlarging by a few square centimeters the domain of beauty; but he draws his motifs, his themes, his obsessions, his objectives from the very essence of his art. After the plastic world has dissolved the figures that constrained it, what claim does it have to continued existence? Every work that we see here has the same provenance. _Hiroshima_ , like all the others, was preempted _by art_.\n\nThis may come as a shock. Politicians began long ago the practice of making minor demands on the services of the artist. Beribboned turncoats have proven time and again that painting dies whenever it is made to serve alien purposes. In fact, until now painters wishing to call attention to the evil done to some men by others were faced with two unpleasant alternatives: they could betray painting without contributing much to morals or, if their work looked beautiful regardless, they could betray the anger or grief of men for Beauty. Either way, the result was treason.\n\nGood sentiments tend toward formalism. If a feeling of righteous indignation is to be communicated, the public must be able to decipher the message; the anxieties of art must be subordinated to false securities. Because Living Beauty is forever evolving, the artist shuns the perplexities inherent in experimentation and elects instead inanimate Beauty. He adopts the most legible transcription, which is necessarily an ancient, conventionalized style of painting.\n\nAttempts to depict acts of violence, mutilated corpses and living bodies racked, tortured and burned have been sterile. By falling back upon visual conventions, artists have shown us a moving side of reality and have conditioned us to react as we normally would\u2014with horror, anger, and especially with the silent flood of sympathy that makes every man experience the wounds of other men as so many holes in his own flesh. This unbearable spectacle puts the spectator to flight. A painting may evidence ingenious composition, correct proportions and harmony, but everything is wasted if the spectator flees and fails to return. And if he should come back, punctured eyes and infected wounds\u2014everything\u2014would disintegrate and beauty would never again be reconstituted. Total failure.\n\nWe can be sure that the painter would be called tactless. People would say that he ought to be more discreet, more delicate if he ever again took up such scabrous subjects. Titian, to cite but one name, was known for his tact. Great men of the world could commission him to paint a massacre according to their instructions and could rest assured that he would produce a procession or a ballet. And the result would be beautiful\u2014naturally. Through this procedure torture is eliminated from the canvas just as its scent is eliminated from a painted rose. Murderers dressed in rich garments and dashing mercenaries are shown surveying the operation; as the crowning stroke a victim's bare foot, healthy and alluring, is revealed while his legs, torso and head remain hidden. Tiziano Vecellio was a traitor, for he forced his brush to paint comforting terrors, painless suffering and living corpses. Through him Beauty betrayed man and consorted with kings. For a self-willed man in a room with windows overlooking a concentration camp to paint a compote is not serious; his sin is one of negligence. The real crime would be in painting the concentration camp as if it were a compote\u2014in the same spirit of research and experimentation.\n\nI know of two exceptions. But the first is only apparent. Uncertain and overwhelmed by revulsion and remorse, the tormented visionary Goya painted his visions instead of war. This misguided man lost all desire to guide the masses and finally transformed the horrors of battles and mass murders into the naked horror of being Goya.\n\n_Guernica_ is different.4 Here the most fortunate of artists enjoyed the most unprecedented good fortune, with the result that his canvas combined incompatible qualities. Effortlessly. Unforgettable revulsion, commemoration of a massacre, the painting seems nevertheless to have resulted solely from the quest for Beauty; furthermore, the quest was successful. It will always be a bitter accusation, but this does not disturb its calm plastic Beauty; conversely, its plastic beauty enhances rather than hinders its emotional impact. The Spanish Civil War, a crucial moment in the period preceding World War II, broke out when _this_ painter's life and _this_ type of painting were approaching their decisive moment. The negative force of the brush was exhausting figurative painting and opening the door to systematic destruction.\n\nAt this period the figure was still intact, for the aim of experimentation was movement, the very quality that was to entail its disintegration. There was no need for violence to be hidden or transformed; it was simply identified with the disintegration of men brought about by their own bombs. Thus an experimental procedure resulted in a work whose singular meaning was that of a revolt and a denunciation of the massacre. The same social forces that had made the painter the negation of _their_ order had also produced fascist acts of destruction and _Guernica_. This stroke of fortune allowed Picasso to avoid cajoling beauty. If the crime is odious because it has become beautiful, this is attributable to the fact that it is \"Explosive\" and that the beauty of Picasso is \"ordered exploding\" _(explosante-fixe)_ , as Breton has said. The miraculous throw of the dice was never repeated. When Picasso tried, after World War II, to make a new beginning, his art had changed, and so had the world; they were now at odds. In short, censorship. When dealing with men and their suffering we can accept neither figuration of horror nor its disguise through pageantry.\n\nFor Lapoujade there is no longer any alternative; there is no longer any problem. This is because I borrowed the previous examples from figurative art. Paradoxically, imitation of the human figure entails externally imposed conditions while the absence of imitation entails conditions imposed by Art itself.\n\nThis is the last stage of a long journey. For years he has been depicting the nudes, couples, and crowds that have imposed themselves upon his brush. Look at his adolescents: nothing is missing. Yet the flesh is shown without its husk; the artificial contours of a body are not there. Nor is it strewn indiscriminately across the canvas. Contours, volume, mass, perspective\u2014didn't he have to use all these to _put us in the presence_ of a nude body? Evidently he did not. Or rather the reverse is true: the painting intrinsically requires us to experience the delicacy of flesh at the very moment when it is freed from alien forms.\n\nThe first stage is anxiety. Freed from an academic tradition, the artist wants to be able to cultivate thoroughly his garden\u2014the plane surface that he has inherited\u2014to transform routine farming which allows much of the land to lie fallow into intensive cultivation; furthermore, he wants to eliminate tolls and duties, barriers, detours, restrictions imposed by imitation. He wants to expand the scope of Art and at the same time to reaffirm its unity.\n\nThe fundamental purpose of experimentation is to give to Beauty a finer grain, a firmer and richer consistency. The artist's sole concern ought to be art. And when we look at Lapoujade's works, we feel not that he is searching for a new style of painting, but that he is giving another nature to painting. The rest follows, of course. But serious changes in all the arts are at first material while form, the quintessential matter, comes last.\n\nLapoujade belongs to a generation of builders.5 After what he himself calls the \"disintegration of figurative art\" by Picasso, Braque and a whole generation of analysts, all that remained for the newcomers was a medley of colors and rhythms, of crumbled remnants. They had no choice. These refined, ductile materials permitted and required integration into new wholes.\n\nAt first the newcomers were united by the common task that awaited them. Then each of them worked alone, trying to determine the aims and resources of the new art. Lapoujade chose to restore to us the World. This is in my opinion an option of capital importance. We can be sure, however, that the World made no requests; if it returned, bloody and new, its return was exacted by Painting.\n\nBeauty is not monadic. It requires two unities, one visible and the other secret. If ever we managed even after relentless effort to penetrate into the essence of a work by a single glance, the object would be reduced to its inert visibility, Beauty would be effaced, and only its ornamentation would remain. To put it more aptly, the unification which the painter and the viewer seek can be achieved only through the permanent recomposition of a certain presence. And this presence, in turn, can communicate to us its indecomposable unity only through the medium of Art, and only in conjunction with our efforts and the painter's efforts to constitute or reconstitute the beauty of a whole.\n\nThe act is purely aesthetic but, to the very degree that we remain aloof, the Whole infiltrates each visual synthesis, shaping it and giving it strength. _We_ must rediscover the paths outlined for us by the painter and try to follow them. _We_ must reconstruct these abrupt splotches of color, these distilled units of matter. _We_ must revive echoes and rhythms. Only then does a presence, intuition denied, come to the rescue. By regulating our choice it keeps us moving along the right paths. To _construct_ requires only the establishing of visible relations; to guarantee a construction and save it from total absurdity requires a transcendental unity. This unity insures that the viewer's eyes will never cease their movement, and the perpetual movement of the eyes accounts for the permanence of the invisible unity. We keep on looking, for if we ever stopped, everything would disintegrate.\n\nWhat is this presence? I hasten to point out that Lapoujade is not a Platonist, nor am I. He is not guided in his compositions by an Idea. No, his guiding principle completes each canvas and is inseparable from it: neither can exist apart from the other. This abstract painter tries to imbue each painting with a concrete presence, and if all must be given the same name, then we might as well say that each work was motivated by a quest for meaning and that the quest was in each instance successful.\n\nOne point in particular merits clarification: a meaning is not a sign or a symbol\u2014not even an image. When we view Canaletto's6 painting of Venice, the resemblance is perfect. His _View of the Salute_ , which embodies both perfect perspective and topographic exactitude, cannot be misinterpreted. That is why the painting has no meaning. No more meaning than an Identification Card. When Guardi7 shows scattered rubbish and bricks bathed in shimmering light, the alleys and canals that he has selected are insignificant. He shows us a commonplace wharf or a studied decomposition of light. Canaletto puts his brush in the service of his native town; Guardi is concerned only with plastic problems, with light and substance, with colors and light, with unity amid diversity through rigorous imprecision. Result: Venice is present in each canvas\u2014as it was for him, as it is for us, as it has been experienced by everyone and seen by no one. I once paid a visit to a writer in the handsome garret of a brick palace beside a river. Not one of the figures that Guardi loved to paint was to be seen. Still, as soon as I saw the place, as soon as I observed my host's surroundings, my thoughts turned to the painter. I again saw his Venice, my Venice, the Venice experienced by each of us, and I felt the same way about other men, other objects and other places. The same? Not exactly. Meaning depends on the substance from which it emerges. Guardi always says more than we experience, and he says it differently.\n\nFigurative painting was the first to be subjected to the rule of the two unities. Paradoxically, however, the incarnation of an invisible presence is obscured somewhat by an external bond\u2014the brutal, mechanical bond that subjects the portrait to the model. The painter who incarnates himself in his work believes that grapes must be painted in clusters. It is as if since Apelles8 the artist has had no ambition other than that of fooling the birds. Still, when Van Gogh painted a field, he did not pretend to transfer it to his canvas; he tried, through deceptive figuration whose sole criterion was the dictates of art, to incarnate on a sticky vertical surface the fullness of an immense world with its fields and men, including Van Gogh. Our world.\n\nNot this: Van Gogh never tried to show us a field containing crows, and certainly not one containing fruit trees, for the simple reason that these objects resist figuration. He provides aesthetic substance for the incarnation of a presence that defies the brush: the world in the process of being covered with fields, its sap and flowers being poured _from a wand_. Again the image must be remote from the model or the world will not be as it should. Van Gogh had to begin by distorting everything if he was to show through art that the most delicate or innocent natural blemish is inseparable from horror.\n\nWe have seen that figurative painting involves three elements: the reality to be depicted on the canvas, the representation provided by the painter, and the presence that finally permeates the work. This trinity might seem to entail complications, and it does. Though supposed to serve as a guide, our essentially ambiguous reality guides nothing; it floats around, bellies up, out of control; it can be manipulated only when real, that is, when transformed into an imaginary object. No field will transmit the charm or horror of the world until largely rebuilt; or rather it will transmit both, together or separately. It will reflect everything, but not consistently; instead of cohesiveness there will be fragmentation, absurdity, confusion.\n\nSuch monotonous disorder, unless restraints are introduced, will never transmit the complex structures of an _experienced_ universe. What the painter adds to his canvas are the days of his life, the time that passes or does not pass. These powerful catalysts transform the object depicted. The inert individuality of the model is not transmitted to the canvas, but neither will the figure drawn there assume the general character of a type or sign. The influence of the world on a man and a man's enduring passion for the world will imbue a few acres of land with biographical individuality if both are recorded in the deceptive play of light on film. They will evoke the adventure of living, of contemplating the very birth of folly, of hurtling toward death.\n\nThe same chance figure, integrated for want of anything better into a composition, will undergo modifications imposed by art\u2014it will be planed down, made smooth, reduced. Van Gogh pretends to \"paint\" a field but knows what he is really doing. He brings order to a canvas but never tries to reconstruct with exactitude the soft ripples made by the wind in passing through wheat or to evoke completely the staggering, intimate presence of man, the heart of the world, through that part of the world, the heart of man, that he has circumscribed.\n\nWhen finally he puts down his palette, when the presence is incarnated in the composition, what has become of the representation of the object? A transparency, a trace, little more than a magnificent allusion to the object represented. And the field, finally, the plain field that the artist has pretended to represent, this would be eliminated from the canvas if the world did not come to the rescue and incarnate itself in the undulating expanses of wheat bereft of figures, in the thick paste of a rimless sun or in the galaxy of suns at earth-level which are the only true inhabitants of the canvas and the only true vestiges of the creative act.\n\nIn figurative painting conventions are of little importance. It is sufficient if we are convinced that the figure proposed is in _this_ system of references the best representation of the object. The best means the strongest, richest, most meaningful form. A matter of luck or skill. Still, since the last century each new option has tended to widen the gap between figure and object. The greater the distance between the two, the stronger the inner tension of the work. When the artist goes so far as to discard resemblances, to rule out any similitude between image and reality other than a fortuitous one, meaning is set free by the disintegration of _representation_ and begins to exert a negative influence. Meaning is the product of the forces of destruction. It flashes out across dissemblances, lacunae, approximations, deliberate indeterminations. Invisible, it blinds because it dissolves the figures in its inimitable presence.\n\nSuch also are the meanings that haunt our world. They annihilate detail and draw from it their sustenance. Every brick wall hides Venice from me if it is alone. I will still sense this city even after its palaces have been destroyed and its plumage plucked and redeployed in such a way that I lose sight of every feather. On his canvas the artist paints the rudiments of intuition only to blot them out immediately thereafter. Aroused by this refusal the presence\u2014which is the thing itself undetailed and in undivided space\u2014will incarnate itself. But this is a trap set by the artist. He introduces other figures that bear no relation to the object in question but suggest other associations\u2014waste paper, sand, pebbles. He seeks to produce a new being, a presence all the more austere because it feeds on an absence surreptitiously falsified by substitutions. How many painters between World War I and World War II dreamed of being at once chemists and alchemists, capable of forcing lead to fuse with gold and be incarnated with its essence? One of them, aiming at a double transformation, tried to paint a wardrobe that would be a chest without ceasing to be a wardrobe; he hoped that elective signalizing would allow him to treat each object by turns as plastic substance or as incarnation.\n\nThis double aim was always suspect. The artist wanted, not to annihilate and make us experience the world's disheveled meanings, but to create meanings that had never before existed. Abortive marvels, uninspired legerdemain. At the end of this long crisis in which the artist's creativity was submerged in disillusionment through failure to understand that the imaginary is the sole absolute, the figure had the good sense to disintegrate. And meaning? Did it disappear at the same time? Quite the opposite. As we have just seen, there was no true link between the two. Set free, the incarnate presence appeared as the _sine qua non_ of abstract art.\n\nThe shattering and scattering of images is not a studied choice exercised by new painters. It is an event that is still unfolding, and not all of its consequences are known. A permanent deflagration, it spreads continuously from canvas to canvas. Each painter sees it simultaneously as his problem and his material. Art offers him an explosion to subdue, and this through explosive tactics.\n\nTheir predecessors having sown the wind, those who today seek to subdue the storm stir up a cyclone within the cyclone and organize its tiniest spangles with pitiless rigor. Using the laws that they have devised and visual logic, they must preserve and reshape this pulverulence; search for the multiple unity within the multiplicity; acquire new insights into the canvas; know how to detail and contain on it dilations and coagulations, whirling fields of fire, dark commas, splotches, puddles, trails of blood across the sun; use fluidity and density as rigid structural elements in a unified plastic ceremony. Deceit and trickery are out. Details are no longer negligible; nothing is missing, everything counts.\n\nBy toying with colors, strengthening lines, making and exploiting discoveries, structuring the whirlwind, and compensating for local turbulences by balancing them rigorously, the painter causes the formidable event to coagulate. At the worst, the result is a rose-window; at best, a pleasant carousal. To preserve the rhythm of explosive space, to prolong the vibration of its colors, to exploit in depth the strange, terrifying disintegration of being and its whirling movement, the artist must use his brush to impose meaning on his canvas and on us.\n\nNo movement without roads, no roads without direction\u2014and who will decide these vectorial determinations unless the artist has an unrestricted view? But he must catch sight of a mighty motif if he is to undertake the unification of this sumptuous dispersion without figures and resemblances. Only one motif exists: the secret unity of the work. It is, if you will, in the painting itself.\n\nThere is another world, said Eluard, and it is in this one. But the artist will find it only by proceeding towards the unification of his canvas. Each time that we effect new syntheses or that the eye unifies neighboring objects, we become a little more aware of its presence. We can never expose all of this presence, for it is merely the work itself considered as an organism. Lapoujade was invited by Art to duplicate the false unity of figurative painting; as soon as he did, he understood what was required of him: to eliminate chance and give to an infinitely divisible surface the indivisible unity of a whole.\n\nA few painters feel as he does and have chosen lyrical unity. The lyrical painter impulsively attacks his canvas; he leaves it to attack us. He paints as he strikes, and the presence incarnated on his canvas is his own. He gives to his work the quick unity of an aggressive act. Lapoujade knows that lyrical painting is possible. It can be done and has already been done. He might be afraid that a projection of himself in the pure medium of art would be illegible. And of course Art, in spite of what is too often said about it, is not a language. But neither is it true that we communicate only through signs. We _experience_ through others what others _experience_ as us; for our fellow men we are a common experience.\n\nLyrical painters try to give to their canvas the unity of their emotion, of their _\u00e9lan_ or their calm; in short, they choose to make the viewing public experience their singular adventure. Would this be possible without preliminary unanimity? Probably not, for singularity is revealed only as a differentiation from commonality. If painters only had to paint themselves, each could have his day. Art would preserve its integrity even if hermetic. But lyrical painting is also an act affixing to multiplicity an indispensable seal, and this act must be forever renewed. Beauty is not created unless the act is renewed.\n\nSince the eye has as its immediate stimulus communication, representing the incessantly recreated completion and perpetual animation of the abstract work, the painter must give it his immediate and constant attention. Meaning, since it is revealed through unification and since its revelation promotes unity, must be by nature communicable. To establish the condition without providing the means for fulfilling it is to risk exposing the work to the perils of indetermination.\n\nSuch is, I believe, the profound conviction of Lapoujade. Painting is an important avenue of communication, and at every intersection it finds the presences that it incarnates. Nor does it have to search them out. Meanings, if the artist wishes to collect them, can be gathered by the dozens; the eye may read them, but slowly, without particular awareness of their obviousness or necessity. If they are not evoked at once by the trembling of matter taking shape and the urgency shared by painter and viewer, how can they be imposed? False evidence is misleading; the artist lends a hand and sets us straight. If without even lending an ear he understands the confused sounds of highways and byways, this is because he, too, is a highway. And here and there are neglected, deserted trails. Lapoujade, the versatile link between man and the world, paints teeming, stampeding masses as they cry out, fall silent, remain mysteriously suspended, and stubbornly dissolve into asphalt coloration. He insists that solitude does not befit painting and his canvases have convinced me that he is right.\n\nOne day, said Marx, there will be no more painters. It would be just as fitting that one day there will be neither men nor painters, but that day is remote. Lapoujade is a strange contradiction. With a few of his contemporaries he has reduced painting to the sumptuous austerity of its essence; surrounded by human presences incarnated on his canvas, however, he is the first not to claim special privileges. A painter, he uses painting to tear away the artist's mask, leaving only men and himself, without prerogatives, one among us, obliterated by the splendor of his work.\n\nLook at his works: crowds. Nor is he the first, for the more fools people see on a canvas, the more they are accustomed to laugh. But the old artists were protected. They worked at the right hand of the Prince and on a tribune; if they had to work face to face with the people and on their level, then they were protected by soldiers. Each painting conveyed its message: \"I am a painter; I belong to you, great men of the World, and I show you the superficial side of the mob which you govern and from which you have deigned to rescue me forever.\"\n\nThe age was responsible and so was the \"figure.\" How was the artist to paint the crowd seen by itself, as it experienced itself and makes itself, here and everywhere? How was he to curve space to inscribe in it the infinite circle whose center is everywhere entwined with its circumference? How was he to show in each the leaders and the followers? And these human molecules, what forms, what colors would show that each defies comparison with the others and that all are interchangeable? What system of references could he choose to make lovers of art understand that the crowd can admit the painter into their ranks only by stripping him bare, that he is denied the flimsiest prop or visual accessory, that aroused masses reject seconds, that he must enter their ranks naked, unadorned, as a man, participate in everything, flee or change, provide disciplined leadership, be put to the test, produce? Was he to bear the weight of twenty or a hundred thousand other \"selfs\" only to return to his canvas, under the best possible circumstances, with violent but inchoate memories? The inner reactions of a crowd cannot be seen. They are assimilated, experienced, acknowledged. Let that be represented through figures!\n\nThis is what characterizes the new painter of crowds: he can incarnate their presence only by refusing to represent them through figures. Of course, by banishing the Figure from his studio he like all artists is renewing the vow of penury that Beauty has never ceased to exact and will never cease to exact. But he goes much further than this: he gives up his tribune. A man, he refuses to be excluded by virtue of his privileged position and to contemplate his species from the outside. Figurative painting isolates both painter and model. While deforming the figure, anarchistic and bourgeois artists spoke with gentle irony of their solitude. Obviously, they were not communicating!\n\nThere is instantaneous communication, however, if the artist is Lapoujade. His crowd is real, surging, uneasy. He is _exposed_ to it and following the gradual annihilation of detail, there remains the meaning of the mass rally, of the police charge of October 27. The meaning is the experience shared by thousands of unknown people certain that for all of them it was the same. This experience requires substance for its incarnation; language is inadequate since it isolates a vast number of facts each of which draws its meaning from the others.\n\nTo express, to paint his indeterminate and multiple adventure as an interchangeable man, Lapoujade provides for crowds a substance which, though fluid, manifests rigid unity within dispersion. The unification of discrete particles is the realization of a transcendent element: the explosive unity of the masses. It follows that the crowd within each of us is urged to rediscover the discrete totality of its life. The painter leads us. There are, he says, immediate data of expression: the dark, dense aggregation of colors at the bottom of the canvas, a swelling of matter, a vivid upsurge of light, a hundred other, a thousand other interlocking constituents. But they only stir the heart. The essential quality is in the singularity of the routes traced by his brush. Now compact, now rarefied, at times thick and at other times liquid, the substance itself is not intended to reveal the invisible, the metamorphosis around us and through us of a glade into a thicket, a steppe, or a virgin forest. It _suggests_ through its texture and its itineraries.\n\nThe painter uses to advantage the contrast between the rigidity of his materials and the indeterminacy of experience. Crowded spots seem to recoil from each other; a new route, suddenly discovered, forces colors to pale as they acquire new reciprocal relations; finally, through these metamorphoses we grasp the indivisible presence of the manifestation as it is incarnated with all its densities at one and the same time. And then, abruptly, a path of asphalt: space. It overflows and runs down to the bottom of the canvas. But is there a top? A bottom?\n\nSpace itself is _a direction_ created by the crowd and determined by their actions. The splotch of color is a thick spurt, a flight to the horizon, an abrupt penetration of space. The police charge. Will the people flee? Resist? No matter what they do, space exists with all its dimensions in one. It is distance\u2014diminishing on one side and on the other appearing interminable. But words are unnecessary; the splotch is able by itself to revive meaning. And what is created is not an artificial presence, a wardrobe-fish or a wolf-table as in the time of trick artists, but a real presence, indivisible, common and singular, enriched by everything contributed by the painter.\n\nMan among men, men in the midst of the world, the world within men, that is the unique presence evoked by his overpowering explosion; that is the unique ordeal, common to all and yet peculiar to each, experienced along with us, through us and for us by Lapoujade; that is the unique message communicated at first glance to illuminate the canvas even before being illuminated by it. But the rejection of special privilege, identical to the rejection of figuration, is a commitment made by painter and by man. It led Lapoujade from one canvas to another toward the most radical consequences of this undertaking.\n\nTo begin with, if the painter ceases to contemplate other men, if he is rejected by his peers, _action_ sets him apart or constitutes a permanent bond with all men and each man. He acts, suffers, frees himself and dominates or is dominated. His act of contemplation is purely passive; his brush must recreate action, not from without but through his experiencing the Other. The Meaning of the painting will be the incarnation of the Other known through the modification that he is subjected to and of the painter as he discovers himself through the modification that he experiences or inflicts.\n\nWhatever its carnation, the inertia of a Nude is generally distressing. The woman is alone, the painter at the opposite end of the room. In real life no one\u2014certainly no painter\u2014ever contemplated so docile a nude from so far away. Lapoujade paints a couple. He has at times evoked the tenderness of adolescent flesh, but in the erotic series called _Le Vif du Sujet_ he wanted to suggest woman as she is approached by men, as she appears in the act of love. A Nude, in short, involves two people. Even if the only real presence is a woman, the presence of a man is suggested by the movement of colors, and this gives to the canvas _its_ presence.\n\nAction, the multiple link between men, molds splotches of color and matter into a coherent structure and brings to perfection the painter's project: to use the visible splendors of nonfigurative art to incarnate that which cannot be represented through figures. Abstract art which seemed at first to impose limitations, actually extends the painter's freedom and assigns to him new functions.\n\nThe other consequence of this option is obviously the unprivileged painter's decision to show his solidarity with other men. Their solidarity has a firm basis, for he has only what they have, wants nothing more, is nothing more. Then, too, its permanence is assured. Woman appears on his canvases in the act of love, men joined in a common struggle. The most astounding yet simple truth is that the choice of abstraction led Lapoujade in the name of art itself once again to place man on his canvases. Not, however, as man appeared in the days of princes and prelates\u2014modest and anonymous in his patient and tenacious struggle to satisfy his hunger and deliver himself from oppression. He appears everywhere on the canvases of Lapoujade, who never ceases painting him and refining his portrait.\n\nLapoujade now understands that man, seen by an unprivileged eye, is today neither great nor small, beautiful nor ugly. His art challenges him to place on his canvases a true portrait of the human kingdom, and the truth about this kingdom, today, is that the human species includes torturers, their accomplices and martyrs. Torturers are few in number but their accomplices are many; most people fall into the group of the tortured, or are candidates for torture. Lapoujade understands this. No one in 1961 can speak of men without first mentioning torturers; no one can speak of Frenchmen to the French without mentioning the tortured Algerians. That is our portrait; we must look at it realistically; later we can decide to preserve it or to modify it.\n\nLapoujade chose to show torture because it is deeply ingrained in us, because it reaches, alas, to our ignoble core. On attempting to paint it, he saw that he could capture its portrait only through nonfigurative art, which evokes the total meaning of the human situation. His triptych is beautiful without reserve\u2014and can be beautiful without remorse. Beauty is not hidden in non-figurative painting. It shows through. The painting _exhibits_ nothing. It lets horror seep down but _only if it is beautiful_ \u2014that is, organized in the most complex and fertile manner.\n\nThe preciseness of the scenes evoked depends on the preciseness of his brush. The viewer must identify and reconstruct the conjugation of striae and beautiful yet sinister colors; this is the only way of _experiencing_ the meaning of what was for Alleg and Djamila their martyrdom.9\n\nMeaning, as I have said, contributes to the total picture something other than a new and alien element. Incarnated in plastic substance, it allows us to sense, through the frenzy of colors, mutilated flesh and unbearable suffering. But the suffering that we sense is that of the victims, and let us not pretend that we find it unbearable in this imperious and discreet form.\n\nBehind and through the radiant Beauty we see only a pitiless Destiny which men\u2014we men\u2014have made for man. Lapoujade's success is complete because it issues from painting and its laws. In other words, it conforms to the logic of abstract painting. It is no mean achievement, I believe, for a painter to meet with such favor in our eyes by showing undisguised the grief that wells from our hearts.\n\n1 Tr. Robert Lapoujade was born at Montauban in 1921. According to the biographical notice printed in the catalog prepared for an exhibition of his non-figurative paintings (Gal\u00e9rie Pierre Domec, Paris, March 10-April 15, 1961), he is a self-made artist, a practitioner \"by necessity\" of various trades, and a writer. His works have been widely exhibited in Europe and America. The Museum of Modern Art has acquired two of his canvases. This essay was originally printed in the aforementioned catalog.\n\n2 Tr. Francisco Jos\u00e9 de Goya (1746-1828), an indefatigable moralist as well as a great artist, depicted the horrors of war in bold colors in _Dos de Mayo_ and repeated the same theme in his most famous etchings, _Disasters of War_.\n\n3 Tr. Since definitions are not always universally accepted (witness the complaint of those who hold that \"abstract\" art is highly concrete), I have chosen to use the literal equivalents of the French expressions, figurative and non-figurative, rather than contrasts involving \"representational,\" \"objective,\" and the like.\n\n4 Tr. Picasso's masterwork was painted in 1935, Goya's _Dos de Mayo_ in 1808.\n\n5 Occasionally one of these good painters remarks fatalistically that in a negative era their art destroys itself and refuses to be a medium of communication. But this is wrong. The new era builds more than it razes. Never before has painting been so compact, so concentrated as in the hands of these architects of the abstract.\n\n6 Tr. The Italian painter and engraver Giovanni Antonio Canaletto (1697-1768) is noted for his precision as well as his fine sense of color and atmosphere.\n\n7 Tr. The landscapes of Francesco Guardi (1712-1793) are characterized by the dissolution of all figures into light and color.\n\n8 Tr. In the most famous of his portraits of Alexander the Great, his patron's hand appears to protrude from the colored surface.\n\n9 Tr. Henri Alleg was charged with attempting to endanger state security and imprisoned after he published a book detailing tortures to which he had allegedly been subjected as a prisoner of French troops in Algeria. His book, later published in English as _The Question_ , was banned in France in 1958. Djamila Bouhired, Algeria's Joan of Arc, was tortured and sentenced to die. She was reprieved only after her plight became an international scandal.\nTHE MOBILES OF CALDER1\n\nThe sculptor is supposed to imbue something immobile with movement, but it would be wrong to compare Calder's art with the sculptor's. Calder captures movement rather than suggests it; he has no intention of entombing it forever in bronze or gold, those glorious, asinine materials that are by nature immobile. With vile, inconsistent substances, with tiny slivers of bone or tin or zinc, he fashions strange arrangements of stems and branches, of rings and feathers and petals. They are resonators or traps; they dangle at the end of a fine wire like a spider at the end of its silk thread or settle on a pedestal, wan, exhausted, feigning sleep; a passing tremor strikes them, animates them, is canalized by them and given a fugitive form\u2014a _Mobile_ is born.\n\nA Mobile: a small local festival, an object defined by its movement and nonexistent apart from it, a flower that withers as soon as it stops moving, a free play of movement, like coruscating light. Sometimes Calder amuses himself by imitating a new form. For example, he once presented me with a bird of paradise with iron wings; a wisp of air brushing it while escaping through the window is enough to rouse the bird; it clicks, stands erect, spins, nods its crested head, rolls and pitches and then, as if in sudden obedience to an unseen signal, executes a slow turn with its wings spread. But most of the time it imitates nothing, and I know no other art less deceptive than his.\n\nSculpture suggests movement, painting depth or light. Calder suggests nothing; he captures and embellishes true, living movements. His mobiles signify nothing, refer to nothing other than themselves; they simply are, they are absolutes.\n\nIn his mobiles chance probably plays a greater part than in any other creation of man. The forces at work are too numerous and too complicated for any human mind, even that of their creator, to foresee all possible combinations. For each of them Calder establishes a general scheme of movement, then abandons it; the time, the sun, heat and wind will determine each particular dance. Thus the object is always midway between the servility of statues and the independence of natural events. Each of his evolutions is an inspiration of the moment; it reveals his general theme but permits a thousand personal variations. It is a little hot-jazz tune, unique and ephemeral, like the sky, like the morning; if you miss it, you will have lost it forever.2\n\nVal\u00e9ry said that the sea is a perpetual renewal. One of Calder's objects is like the sea\u2014and equally spellbinding: ever changing, always new. A passing glance is not enough; one must live with it and be bewitched by it. Then the imagination can revel in pure, ever-changing forms\u2014forms that are at once free and fixed.\n\nThe movements of the object are intended only to please us, to titillate our eyes, but they have a profound, metaphysical meaning. The reason is that the mobiles have to have some source of mobility. Previously Calder used an electric motor; he now abandons his mobiles in the midst of nature; in a garden or near an open window, they vibrate in the wind like aeolian harps. Fed on air, they respire and draw their life from the tenuous life of the atmosphere. Thus their mobility is of a very peculiar kind.\n\nAlthough they are human creations, they never have the precision and efficiency of movement of Vaucanson's automatons.3 But the charm of the automaton resides in the fact that it agitates a fan or plays a guitar like a man, yet moves its hand with the blind, persistent rigor of purely mechanical translations. Against this, Calder's mobiles move and hesitate, as if correcting a mistake by starting anew.\n\nI have seen in his studio a beater and a gong suspended high overhead; the slightest gust caused the beater to pursue the gong as it turned round; it would take aim, lash out at the gong, miss it by a hair, like an awkward hand, and then when least expected, strike and hit it squarely in the center, producing a frightening noise. But the movements are too artistically contrived to be compared with those of a ball rolling on a uneven plane and changing its course solely on the basis of irregularities encountered. They have a life of their own.\n\nOne day when I was talking with Calder in his studio, a model which until then had remained at rest was seized, right in my presence, by a violent agitation. I retreated until I thought I was beyond its reach. Suddenly, just when the agitation had ceased and the model seemed lifeless, its long, majestic tail, which had not moved previously, indolently roused, as if regretfully, rotated in the air and grazed my nose.\n\nTheir hesitations, revivals, gropings, tumblings, abrupt decisions, and especially their marvelous swan-like nobility make of Calder's mobiles strange creatures, halfway between matter and life. Sometimes their movements seem to have a purpose and sometimes they seem to have lost their purpose along the way and to have lapsed into imbecile fluctuations. My bird flies, wavers, swims like a swan, like a frigate; he is a bird, a single bird and then, suddenly, he falls apart and all that remains are slivers of metal traversed by vain little tremors.\n\nCalder's mobiles, which are neither completely living nor completely mechanical and which constantly change but always return to their original positions, are like aquatic plants bent low by a stream, the petals of the sensitive plant, the legs of a headless frog, or gossamer caught in an updraft. In short, although Calder has no desire to imitate anything\u2014his one aim is to create chords and cadences of unknown movements\u2014his mobiles are at once lyrical inventions, technical, almost mathematical combinations and the perceptible symbol of Nature: great elusive Nature, squandering pollen and abruptly causing a thousand butterflies to take wing and never revealing whether she is the blind concatenation of causes and effects or the gradual unfolding, forever retarded, disconcerted and thwarted, of an Idea.\n\n1 Tr. First published in _Les Temps Modernes_ and later in _Situations HI_ (Paris: Gallimard, 1948). Alexander Calder achieved recognition as a sculptor some twenty years ago when the Museum of Modern Art exhibited his works. His mobiles are today found in such diverse places as New York's Chase Manhattan Bank, a hotel in Cincinnati, an airport in Pittsburgh, and UNESCO headquarters in Paris. He maintains two studio-homes, one near Sach\u00e9, France, and the other near Roxbury, Connecticut.\n\n2 Tr. In Sartre's earliest novel, _Nausea_ (first published in 1938), jazz enables the central character to escape momentarily from the pervasive, overpowering feeling of nausea that engulfs him.\n\n3 Tr. Jacques de Vaucanson (1709-1782) devised mechanisms that brought him considerable fame. Among his most celebrated automatons were his _Flute Player_ and _Duck_.\nTHE QUEST FOR THE ABSOLUTE1\n\nA glance at Giacometti's antediluvian face reveals his arrogance and his desire to place himself at the beginning of time. He ridicules Culture and has no faith in Progress\u2014not in the Fine Arts, at least. He considers himself no further \"advanced\" than his adopted contemporaries, the men of Eyzies and Altamira.2 Then, when nature and men were in their prime, there was neither ugliness nor beauty, neither taste nor dilettantes nor criticism. The man who first had the notion of carving a man from a block of stone had to start from zero.\n\nHis model: man. Neither a dictator nor a general nor an athlete, primitive man still lacked the dignity and charm that would seduce future sculptors. He was nothing more than a long, indistinct silhouette walking across the horizon. But his movements were perceptibly different from the movements of things; they emanated from him like first beginnings and impregnated the air with signs of an ethereal future. They must be understood in terms of their ends\u2014to pick a berry or push aside a briar\u2014not their origins. They could never be isolated or localized.\n\nI can separate a bent branch from a tree but never an upraised arm or a clenched fist from a man. The _man_ raises his arm, the _man_ clenches his fist, the _man_ is the indissoluble unit and the absolute source of his movements. Furthermore, he is an enchanter of signs; they cling to his hair, shine in his eyes, dance between his lips, perch on his fingertips. He speaks with his whole body; when he runs he speaks, when he talks he speaks, and when he falls asleep his sleep is speech.\n\nHis substance: a rock, a lump of space. From mere space Giacometti therefore had to fashion a man, to inscribe movement in total immobility, unity in infinite multiplicity, the absolute in pure relativity, the future in the eternal present, the loquacity of signs in the tenacious silence of things. The gap between substance and model seems unbridgeable, yet exists only because Giacometti has gauged its dimensions. I am not sure whether he is a man bent on imposing a human seal on space or a rock dreaming of human qualities. Or perhaps he is both and mediates between the two.\n\nThe sculptor's passion is to transform himself completely into length so that from its fullness can spill the statue of a man. He is haunted by thoughts of stone. Once he was terrified by the void; for months he walked to and fro, accompanied by an abyss\u2014his emptiness in the process of achieving awareness of its desolate sterility. On another occasion it seemed to him that objects, spiritless and dead, were no longer touching the ground; he lived in a fluctuating universe, knowing in his flesh and even to the point of martyrdom that there is neither height nor depth nor length nor real contact between things; but at the same time he was aware that the sculptor's task is to carve from the infinite archipelago a face filled with the only being that can _touch_ other beings.\n\nI know no one else who is as sensitive as he to the magic of faces and gestures. He looks at them with passionate envy, as if they were from another kingdom. At his wit's end he has at times tried to mineralize his equals: to envision crowds advancing blindly toward him, rolling across boulevards like stones in an avalanche. Thus each of his obsessions was a task, an experience, a means of experiencing space.\n\n\"He's crazy,\" people say. \"Sculptors have been carving away for three thousand years\u2014and nicely, too\u2014without such rigmaroles. Why doesn't he try to produce impeccable works according to tested techniques instead of pretending to ignore his predecessors?\"\n\nThe truth is that for three thousand years sculptors have been carving only cadavers. Sometimes they are shown reclining on tombs; sometimes they are seated on curule chairs or perched on horses. But a dead man on a dead horse does not make even half a living creature. He deceives the rigid, wide-eyed people in the Museum. His arms pretend to move but are held fast by iron shanks at each end; his rigid outlines can hardly contain infinite dispersion; mystified by a crude resemblance, the spectator allows his imagination to imbue the eternal sinking of matter with movement, heat and light.\n\nIt is therefore necessary to start again from zero. After three thousand years the task of Giacometti and of contemporary sculptors is not to glut galleries with new works but to prove that sculpture is possible by carving. To prove that sculpture is possible just as by walking Diogenes proved to Parmenides and Zeno the possibility of movement. It is necessary to go the limit and see what can be done. If the undertaking should end in failure, it would be impossible to decide under even the most favorable circumstances whether this meant the failure of the sculptor or of sculpture; others would come along, and they would have to begin anew. Giacometti himself is forever beginning anew. But involved here is more than an infinite progression; there is a fixed boundary to be reached, a unique problem to be resolved: how to make a man out of stone without petrifying him. All or nothing: if the problem is solved, the number of statues is of little consequence.\n\n\"If I only knew how to make one,\" says Giacometti, \"I could make them by the thousands....\" Until he succeeds, there will be no statues at all but only rough hewings that interest Giacometti only insofar as they bring him closer to his goal. He shatters everything and begins anew. From time to time his friends manage to save from destruction a head, a young woman, an adolescent. He raises no objection and again takes up his task. In fifteen years he has had but one exposition.\n\nHe consented to the exposition because he had to make a living, but even then he had misgivings and wrote by way of excusing himself: \"It is mainly because I was goaded by the terror of poverty that these sculptures exist in this state (bronzed and photographed), but I am not quite sure of them; still, they were almost what I wanted. Almost.\"\n\nWhat bothers him is that these impressive works, always mediating between nothingness and being, always in the process of modification, perfection, destruction and renewal, have begun to exist independently and in earnest, and have made a start, far from him, toward a social career. He prefers simply to forget about them. The remarkable thing about him is his intransigence in his quest for the absolute.\n\nThis active, determined worker is displeased by the resistance of stone, which slows down his movements. He has chosen a weightless substance which is also the most ductile, perishable and spiritual of all substances\u2014plaster. He hardly feels it at his fingertips; it is the impalpable reflex of his movements.\n\nOne first notices in his studio strange scarecrows made of white daubs that coagulate around long reddish strings. His experiences, his ideas, his desires and his dreams project themselves for a moment on his plaster men, give them a form and pass on, and their form passes on with them. Each of these nebulous creatures undergoing perpetual metamorphosis seems like Giacometti's very life transcribed in another language.\n\nMaillo's statues insolently fling in our eyes their heavy eternity. But the eternity of stone is synonymous with inertia; it is the present forever solidified. Giacometti never speaks of eternity, never thinks of eternity. I was pleased by what he had said to me one day concerning some statues that he had just destroyed: \"I was happy with them, but they were made to last only a few hours.\"\n\nA few hours\u2014like the dawn, like sadness, like ephemera. And his creations, because they were destined to perish on the very night of their birth, are the only ones among all the sculptures that I know to retain the ineffable charm of transiency. Never was substance less eternal, more fragile, more nearly human. Giacometti's substance\u2014this strange flour that slowly settles over his studio and buries it, that seeps under his nails and into the deep wrinkles on his face\u2014is the dust of space.\n\nBut space, even if naked, is still superfluity. Giacometti is terrified by the infinite. Not by Pascalian infinity, not by what is infinitely great. The infinity that runs through his fingers is of a more subtle and secretive type. In space, says Giacometti, there is a superfluity. This _superfluity_ is the pure and simple coexistence of juxtaposed elements. Most sculptors have allowed themselves to be deceived; they have confused the proliferation of space with generosity, they have put too much into their works, they have been captivated by the plump contour of a marble bosom, they have unfolded, stuffed and distended the human gesture.\n\nGiacometti knows that there is nothing superfluous about a living person because everything is function. He knows that space is a cancer that destroys being, that devours everything. For him, to sculpture is to trim the fat from space, to compress it and wring from it all its exteriority. The attempt may well seem hopeless, and I believe that on two or three occasions Giacometti has reached the verge of despair. If sculpturing entails carving and patching in this incompressible medium, then sculpture is impossible. \"And yet,\" he said, \"if I begin my statue, like others, at the tip of the nose, it will not be too great an infinity of time before I reach the nostril.\" Then it was that he made his discovery.\n\nConsider Ganymede on his pedestal. If you ask me how far away he is, I will tell you that I don't know what you are talking about. By \"Ganymede\" do you mean the youth carried away by Jupiter's eagle? If so, I will say that there is no _real_ distance between us, that no such relation exists because he does not exist. Or are you referring to the block of marble that the sculptor fashioned in the image of the handsome lad? If so, we are dealing with something real, with an existing mineral, and can draw comparisons.\n\nPainters have long understood all that since in pictures the unreality of the third dimension necessarily entails the unreality of the two other dimensions. It follows that the distance between the figures and my eyes is _imaginary_. If I advance, I move nearer to the canvas, not to them. Even if I put my nose on them, I would still see them twenty steps away since for me they exist once and for all at a distance of twenty steps. It follows also that painting is not subject to Zeno's line of reasoning; even if I bisected the space separating the Virgin's foot from St. Joseph's foot, and the resulting halves again and again to infinity, I would simply be dividing a certain length on the canvas, not flagstones supporting the Virgin and her husband.\n\nSculptors failed to recognize these elementary truths because they were working in a three-dimensional space on a real block of marble and, although the product of their art was an imaginary man, they thought that they were working with real dimensions. The confusion of real and unreal space had curious results. In the first place, instead of reproducing what they _saw_ \u2014that is, a model ten steps away\u2014they reproduced in clay what _was_ \u2014that is, the model itself. Since they wanted their statue to give to the spectator standing ten steps away the impression that the model had given them, it seemed logical to make a figure that would be for him what the model had been for them; and that was possible only if the marble was _here_ just as the model had been _out there_.\n\nBut what exactly is the meaning of being _here_ and _out there?_ Ten steps away from her, I form a certain image of a nude woman; if I approach and look at her at close range, I no longer recognize her; the craters, crevices, cracks, the rough, black herbs, the greasy streaks, the lunar orography in its entirety simply cannot be the smooth, fresh skin I was admiring from a distance. Is that what the sculptor should imitate? There would be no end to his task, and besides, no matter how close he came to her face, he could always narrow the gap still further.\n\nIt follows that a statue truly resembles neither what the model _is_ nor what the sculptor _sees_. It is constructed according to certain contradictory conventions, for the sculptor represents certain details not visible from so far away under the pretext that they exist and neglects certain others that do exist under the pretext that they are unseen. What does this mean other than that he takes the viewpoint of the spectator in order to reconstruct an acceptable figure? But if so, my relation to Ganymede varies with my position; if near, I will discover details which escaped me at a distance. And this brings us to the paradox: I have _real_ relations with an illusion; or, if you prefer, my true distance from the block of marble has been confused with my imaginary distance from Ganymede.\n\nThe result of all this is that the properties of true space overlay and mask those of imaginary space. Specifically, the real divisibility of marble destroys the indivisibility of the person. Stone and Zeno are the victors. Thus the classical sculptor flirts with dogmatism because he thinks that he can eliminate his own look and imbue something other than man with human nature; but the truth is that he does not know what he is doing since he does not reproduce what he sees. In his search for truth he encounters convention. And since the net result is to shift to the visitor the responsibility for breathing life into his inert images, his quest for the absolute finally makes his work depend on the relativity of the angles from which it is viewed. As for the spectator, he takes the imaginary for the real and the real for the imaginary; he searches for indivisibility and everywhere finds divisibility.\n\nBy reversing classicism, Giacometti has restored to statues an imaginary, indivisible space. His unequivocal acceptance of relativity has revealed the absolute. The fact is that he was the first to sculpture man as he is seen\u2014from a distance. He confers _absolute distance_ on his images just as the painter confers absolute distance on the inhabitants of his canvas. He creates a figure \"ten steps away\" or \"twenty steps away,\" and do what you will, it remains there. The result is a leap into the realm of the unreal since its relation to you no longer depends on your relation to the block of plaster\u2014the liberation of Art.\n\nA classical statue must be studied or approached if it is continuously to reveal new details; first, parts are singled out, then parts of parts, etc., with no end in sight. You can't approach one of Giacometti's sculptures. Don't expect a belly to expand as you draw near it; it will not change and you on moving away will have the strange impression of marking time. We have a vague feeling, we conjecture, we are on the point of seeing nipples on the breasts; one or two steps closer and we are still expectant; one more step and everything vanishes. All that remains are plaits of plaster. His statues can be viewed only from a respectful distance. Still, everything is there: whiteness, roundness, the elastic sagging of a beautiful ripe belly. Everything except matter. From twenty steps we only think we see the wearisome desert of adipose tissue; it is suggested, outlined, indicated, but not given.\n\nNow we know what press Giacometti used to condense space. There could be but one\u2014distance. He placed distance within our reach by showing us a distant woman who keeps her distance even when we touch her with our fingertips. The breasts that we envisioned and anticipated will never be exposed, for they are but expectancy; the bodies that he creates have only enough substance to hold forth a promise.\n\n\"That's impossible,\" someone might say. \"The same object can't be viewed from close range and from afar.\" But we are not speaking of the same object; the block of plaster is near, the imaginary person far away.\n\n\"Even so, distance would still have to compress all three dimensions, and here length and depth are affected while height remains intact.\" True. But it is also true that each man in the eyes of other men possesses absolute dimensions. As a man walks away from me, he does not seem to grow smaller; his qualities seem rather to condense while his \"figure\" remains intact. As he draws near me, he does not grow larger but his qualities expand.\n\nAdmittedly, however, Giacometti's men and women are closer to us in height than in width\u2014as if they are projecting their stature. But Giacometti purposely elongated them. We must understand that his creatures, which are wholly and immediately what they are, can neither be studied nor observed. As soon as I see them, I know them; they flood my field of vision as an idea floods my mind; the idea has the same immediate translucidity and is instantaneously wholly what it is. Thus Giacometti has found a unique solution to the problem of unity within multiplicity by simply suppressing multiplicity.\n\nPlaster and bronze are divisible, but a woman in motion has the indivisibility of an idea or an emotion; she has no parts because she surrenders herself simultaneously. To give perceptible expression to pure presence, to surrender of self, to instantaneous emergence, Giacometti has recourse to elongation.\n\nThe original movement of creation\u2014the timeless, indivisible movement so beautifully epitomized by long, gracile legs\u2014shoots through his Greco-like bodies and lifts them toward the heavens. In them even more than in one of Praxiteles' athletes I recognize man, the first cause, the absolute source of movement. Giacometti succeeded in giving to his substance the only truly human unity\u2014unity of action.\n\nSuch is the type of Copernican revolution that Giacometti has attempted to introduce into sculpture. Before him men thought that they were sculpturing _being_ , and this absolute dissolved into an infinite number of appearances. He chose to sculpture _situated_ appearance and discovered that this was the path to the absolute. He exposes to us men and women as _already seen_ but not as already seen by himself alone. His figures are already seen just as a foreign language that we are trying to learn is already spoken. Each of them reveals to us man as he is seen, as he is for other men, as he emerges in interhuman surroundings\u2014not, as I said earlier for the sake of simplification, ten or twenty steps away, but at a man's distance. Each of them offers proof that man _is_ not at first in order to be _seen_ afterwards but that he is the being whose essence is in his existence for others. When I perceive the statue of a woman, I find that my congealed look is drawn to it, producing in me a pleasing uneasiness. I feel constrained, yet know neither why nor by whom until I discover that I am constrained to see and constrained by myself.\n\nFurthermore, Giacometti often takes pleasure in adding to our perplexity\u2014for example by placing a distant head on a nearby body so that we no longer know where to begin or exactly how to behave. But even without such complications his ambiguous images are disconcerting, for they upset our most cherished visual habits. We have long been accustomed to smooth, mute creatures fashioned for the purpose of curing us of the sickness of having a body; these guardian spirits have watched over the games of our childhood and bear witness in our gardens to the notion that the world is without risks, that nothing ever happens to anyone and, consequently, that the only thing that ever happened to them was death at birth.\n\nAgainst this, something obviously has happened to Giacometti's bodies. Are they emerging from a concave mirror, from a fountain of youth or from a deportation camp? We seem at first glance to be confronted by the emaciated martyrs of Buchenwald. But almost immediately we realize our mistake. His thin, gracile creatures rise toward the heavens and we discover a host of Ascensions and Assumptions; they dance, they _are_ dances, made of the same rarefied substance as the glorious bodies promised us. And while we are still contemplating the mystical upsurge, the emaciated bodies blossom and we see only terrestrial flowers.\n\nThe martyred creature was only a woman but she was _all_ woman\u2014glimpsed, furtively desired, retreating in the distance with the comic dignity of fragile, gangling girls walking lazily from bed to bathroom in their high-heeled shoes and with the tragic horror of scarred victims of a holocaust or famine; all woman\u2014exposed, rejected, near, remote; all woman\u2014with traces of hidden leanness showing through alluring plumpness and hideous here on earth but no longer entirely on earth, living and relating to us the astounding adventure of flesh, _our_ adventure. For she chanced to be born, like us.\n\nNevertheless, Giacometti is dissatisfied. He could win the match promptly simply by deciding that he has won. But he can't make up his mind and keeps putting off his decision from hour to hour, from day to day. Sometimes, during the course of a night's work, he is ready to acknowledge his victory; by morning everything has been shattered. Is he afraid of the boredom that lurks beyond his triumph, the boredom that beset Hegel after he had imprudently stapled together his system? Or perhaps matter seeks revenge. Perhaps the infinite divisibility that he eliminated from his work keeps cropping up between him and his goal. The end is in sight, but to reach it he must improve.\n\nMuch has been done but now he must do _a little_ better. And then _just a little_ better still. The new Achilles will never catch the tortoise; a sculptor must in some way be the chosen victim of space\u2014if not in his work, then in his life. But between him and us, there must always be a difference of position. He knows what he wanted to do and we don't; but we know what he has done and he doesn't. His statues are still largely incorporated in his flesh; he is unable to see them. Almost as soon as they are produced he goes on to dream of women that are thinner, taller, lighter, and it is through his work that he envisions the ideal by virtue of which he judges it imperfect. He will never finish simply because a man always transcends what he does.\n\n\"When I finish,\" he says, \"I'll write, I'll paint, I'll have fun.\" But he will die before finishing. Are we right or is he right? He is right because, as Da Vinci said, it is not good for an artist to be happy. But we are also right\u2014and ours is the last word. Kafka as he lay dying asked to have his books burned and Dostoevski, during the very last moments of his life, dreamed of writing a sequel to _The Brothers Karamazov_. Both may have died dissatisfied, the former thinking that he would depart from the world without even making a mark on it and the latter that he had not produced anything good. And yet both were victors, regardless of what they might have thought.\n\nGiacometti is also a victor, and he is well aware of this fact. It is futile for him to hoard his statues like a miser and to procrastinate, temporize and find a hundred excuses for borrowing more time. People will come into his studio, brush him aside, carry away all his works, including the plaster that covers his floor. He knows this; his cowed manner betrays him. He knows that he has won in spite of himself, and that he belongs to us.\n\n1 Tr. First published in _Les Temps Modernes_ and later reprinted in _Situations 111_ (Paris: Gallimard, 1948).\n\n2 Tr. That the paleolithic hunters of southern France and Northern Spain had a keenly developed aesthetic sense is attested by many artifacts preserved in limestone caves near Eyzies-de-Tayac and Altamira.\nINDEX\n\nA\n\nAlberti,\n\nAlexander the Great, n\n\nAlleg, Henri, n\n\nArcimboldo,\n\nAretino, Pietro, , , , ,\n\nB\n\nBellini, Gentile, ,\n\nBellini, Giovanni,\n\nBordone, Paris, ,\n\nBouhired, Djamila, n\n\nBraque,\n\nC\n\nCagliari, Paolo, _see_ Veronese\n\nCalder, Alexander, , 123-127\n\nCanaletto, Giovanni Antonio,\n\nCarpaccio,\n\nCasser, Sebastiano, ,\n\nD\n\nDiogenes,\n\nDonatello,\n\nDostoevski, Fyodor,\n\nDuccio,\n\nE\n\nEluard,\n\nF\n\nFontana, Giovita,\n\nFrancesa, Piero della,\n\nG\n\nGiacometti, Alberto, , , 77-95, 128-143\n\nGiacometti, Annette, n, ,\n\nGiacometti, Diego, n, , , , ,\n\nGiorgione, , ,\n\nGiotto,\n\nGoya, Francisco Jos\u00e9 de, , n,\n\nGozzoli, Benozzo,\n\nGreco, El,\n\nGuardi, Francesco, ,\n\nH\n\nHegel,\n\nHobbes, Thomas,\n\nHusserl,\n\nI\n\nIngres,\n\nJ\n\nJeanson, Henri,\n\nJoan of Arc, n\n\nK\n\nKafka,\n\nKierkegaard, Soren,\n\nL\n\nLapoujade, Robert, , 96-122\n\nM\n\nMaillol,\n\nMantegna, Andrea,\n\nMarx, Karl,\n\nMasson,\n\nMatisse, Henri, , n, n\n\nMessina, Antonella da, ,\n\nMichelangelo, , , , , , ,\n\nP\n\nParmenides,\n\nPicasso, Pablo, n, ,\n\nPitati, Bonifazio dei, ,\n\nPius V, Pope, 42-43\n\nPordenone, , , , n\n\nPraxiteles,\n\nR\n\nRaphael, , , ,\n\nRidolfl, n, n,\n\nRimbaud,\n\nRobusti, Domenico, ,\n\nRobusti, Faustina, ,\n\nRobusti, Jacopo, _see_ Tintoretto\n\nRobusti, Marco,\n\nRobusti, Marietta, ,\n\nRobusti, Ottavia,\n\nS\n\nSacchis, Antonio di,\n\nSalviati,\n\nSarpi, Fra Paolo,\n\nSartre, Jean-Paul, 7-10, n, n\n\nSchiavone, Andrea, , ,\n\nStreller, Justus, n\n\nT\n\nTintoretto, , 13-76\n\nTitian, , , , , , , , , , , , , , , , 65-76,\n\nU\n\nUccello, Paolo, n,\n\nV\n\nValery, Paul,\n\nVan Gogh, Vincent, ,\n\nVasari, Giorgo, ,\n\nVaucanson, Jacques de, n,\n\nVecchio, Girolamo,\n\nVecchio, Palma,\n\nVecellio, Tiziano, _see_ Titian\n\nVecention,\n\nVermeer,\n\nVeronese, , , , n, , , , , ,\n\nVictor Emmanuel, King,\n\nVincentino, ,\n\nVinci, Leonardo da,\n\nVivarini,\n\nVolterra, Daniele de,\n\nZ\n\nZeno, , ,\n\nZigninoni, Zammaria de, ,\n\nZuccaro, \nSARTRE BIBLIOGRAPHY\n\nNausea (1938)\n\nThe Wall (1939)\n\nThe Flies (1943)\n\nBeing and Nothingness (1943)\n\nNo Exit (1945)\n\nThe Age of Reason (1945)\n\nThe Reprieve (1945)\n\nExistentialism and Humanism (1945)\n\nDeaths Without Burial (1946)\n\nThe Respectful Prostitute (1946)\n\nAnti-Semite and Jew (1947)\n\nDirty Hands (1948)\n\nTroubled Sleep (1949)\n\nThe Devil and the Good Lord (1951)\n\nSt. Genet: Actor and Martyr (1952)\n\nThe Condemned of Altona (1959)\n\nCritique of Dialectical Reason (1960)\n\nThe Words: An Autobiography (1964)\nABOUT THE AUTHOR\n\nBorn in Paris in 1905, Jean-Paul Sartre graduated the \u00c9cole Normale Sup\u00e9rieure at the age of twenty-two. He subsequently taught philosophy at the Lyc\u00e9e du Havre, studied under Husserl and Heidegger at the Institut Fran\u00e7ais, Berlin (1933-34), and held a professorship at the Lyc\u00e9e Condor\u00e7et, Paris (1935-42). Mobilized at the outbreak of World War II, Sartre served with the French Army until his capture at the Maginot Line. He spent the following nine months in a German prison camp, where he composed and directed plays for his fellow prisoners. On his release in 1941, Sartre joined the Paris Resistance movement as a journalist, contributing to various underground newspapers, including _Les Lettres Fran\u00e7aises_ and _Combat_. Despite the Nazi censorship during the Occupation, he produced his first play in 1943, _Les Mouches (The Flies)_ , a retelling of the Orestes legend advocating freedom and resistance to tyranny. His second play, _Huis-clos (No Exit)_ , was produced a year later. His first major work, a novel, _La Naus\u00e9e (Nausea)_ was published in 1938; his first philosophic work, _L'\u00c9tre et le N\u00e9ant (Being and Nothingness)_ , appeared in 1943.\n\nIn 1942 Sartre resigned his professorship to devote full time to writing. Asserting that a writer must refuse to allow himself to be classified as an institution, he rejected appointment to the French Legion d'Honneur in 1945, and rejected the Nobel Prize for Literature in 1964. He was one of the major artists and writers in France to demand independence for Algeria, and in 1965 declined a lecture tour in the United States in protest of the American military role in Vietnam. Sartre embraced Marxism and traveled to Cuba to meet Fidel Castro and Ernesto \"Che\" Guevara\n\nSartre's writings include novels, plays, screenplays, essays on literature and psychology, a philosophical and sociological biography of the French playwright Genet, philosophic treatises, articles of social commentary and the first volume of his autobiography. He is presently the editor of _Les Temps Modernes_ in 1946, becoming a politically engaged activist and the major spokesman for atheistic existentialism for his generation. Sartre was one of the few philosophers to develop a complete system of thought consistent with the ethical and philosophical dilemmas of his time. He died in Paris on April 15, 1980.\nAll rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this ebook onscreen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.\n\nNew Preface and arrangement\n\ncopyright \u00a9 2011, copyright \u00a9 1963\n\nby the Philosophical Library, Inc.\n\nISBN:978-1-4532-2856-2\n\nThis 2012 edition distributed by Open Road Integrated Media\n\n180 Varick Street\n\nNew York, NY 10014\n\nwww.openroadmedia.com\n\n**JEAN-PAUL SARTRE**\n\nFROM PHILOSOPHICAL LIBRARY  \nAND OPEN ROAD MEDIA\n\n_Philosophical Library's mission is to reintroduce readers to **books of lasting value** by the intellectual icons of the twentieth century, including Albert Einstein, Jean-Paul Sartre, Kahlil Gibran, and Andr\u00e9 Gide._\n\n**FIND OUT MORE AT**\n\n**WWW.PHILOSOPHICALLIBRARY.COM**\n\n**FOLLOW US:**\n\n**@PhilLibrary**\n\n**Facebook.com\/PhilosophicalLibrary**\n\n**PhilosophicalLibrary.Tumblr.com**\n\nPhilosophical Library is one of a select group of publishing partners of Open Road Integrated Media, Inc.\n\nFind a full list of our authors and\n\ntitles at www.openroadmedia.com\n\nFOLLOW US\n\n@OpenRoadMedia\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## More Praise for IVY + BEAN\n\n\"Just right for kids moving on from beginning readers . . . illustrations deftly capture the girls' personalities and the tale's humor. . . . Barrows' narrative brims with sprightly dialogue.\"  \n\u2014  _Publishers Weekly_ , starred review\n\n\"In the tradition of Betsy and Tacy, Ginnie and Genevra, come two new friends, Ivy and Bean. . . . The deliciousness is in the details here. . . . Will make readers giggle.\"  \n\u2014  _Booklist_ , starred review\n\n\"A charming new series.\" \u2014 _People_\n\n\"Ivy and Bean are a terrific buddy combo.\" \u2014 _Chicago Tribune_\n\n\"Readers will be snickering in glee over Ivy and Bean's antics.\"  \n\u2014 _Kirkus Reviews_\n\n\"This is a great chapter book for students who have recently crossed the independent reader bridge.\"  \n\u2014 _School Library Journal_\n\n\"Annie Barrows' simple and sassy text will draw in both the reluctant reader and the young bookworm. Fans of Beverly Cleary's Beezus and Ramona will enjoy this cleverly written and illustrated tale of sibling rivalry and unexpected friendship.\"  \n\u2014 _BookPage_\n\nFor Esme and Megan, friends from the beginning \u2014A. B.\n\nFor the other Ivy, and her brother, Moss \u2014S. B.\n\nThanks to Dr. George Matsumoto of the Monterey Bay Aquarium Research Institute for useful information regarding squid.\nFirst Chronicle Books LLC paperback edition published in 2010.\n\nText \u00a9 2009 by Annie Barrows.  \nIllustrations \u00a9 2009 by Sophie Blackall.  \nAll rights reserved. No part of this book may be reproduced in any form without written permission from the publisher.\n\nBook design by Sara Gillingham.  \nTypeset in Blockhead and Candida.  \nThe illustrations in this book were rendered in Chinese Ink.\n\nISBN 978-0-8118-7666-7 (PB)  \nISBN 978-0-8118-7656-8 (EPUB, MOBI)\n\nThe Library of Congress has catalogued the previous edition as follows.  \nBarrows, Annie.  \nIvy and Bean doomed to dance \/ written by Annie Barrows ; illustrated by Sophie Blackall.  \np. cm. \u2014 (Ivy and Bean ; bk. 6)  \nSummary: Second-grade best friends Ivy and Bean beg for ballet lessons, then, when they are cast as squids in their first recital, scheme to find a way out of what seems to be boring, hard, and potentially embarrassing.  \nISBN 978-0-8118-6266-0  \n[1. Ballet\u2014Fiction. 2. Best friends\u2014Fiction. 3. Friendship\u2014Fiction.] I. Blackall, Sophie, ill.  \nII. Title. III. Title: Ivy and Bean doomed to dance. IV. Title: Doomed to dance.  \nPZ7.B27576Iwb 2009  \n[E]\u2014dc22  \n2009004367\n\nChronicle Books LLC  \n680 Second Street, San Francisco, California 94107\n\nwww.chroniclekids.com\n\n# CONTENTS\n\nBALLET OR BUST 7\n\nDIP, DIP, CRASH! 20\n\nBAD NEWS BENEATH THE SEA 31\n\nSQUIDS IN A FIX 40\n\nGERMS OF HOPE 51\n\nTIGHT TENTACLES 62\n\nBYE-BYE, BALLET 75\n\nVERY FISHY 89\n\nOCEAN LIFE GONE BAD 99\n\nIN HOT WATER 112\n\nSQUIDARINAS 121\n\n## BALLET OR BUST\n\nIt was a book that started all the trouble. \"Read, read, read! That's all grown-ups ever say to me,\" said Bean, \"but when I finally do read, I get in trouble.\" She slumped in her chair. \"And then the grown-ups take the book away.\"\n\nIvy nodded. \"It's totally not fair,\" she agreed. \"And they shouldn't blame us anyway. It's all Grandma's fault.\"\n\nIvy's grandma had sent her the book. It was called _The Royal Book of the Ballet_. Each chapter told the story of a different ballet, with pictures of fancy girls in feathery tutus and satin toe shoes.\n\nBean was at Ivy's house on the day it arrived. They were supposed to be subtracting, but they were tired of that so they ripped open the package and sat down side by side on Ivy's couch to look at _The Royal Book of the Ballet_.\n\n\"I heard that sometimes their toes bleed when they're dancing,\" said Bean. \"The blood leaks right through the satin part.\"\n\n\"That's gross,\" said Ivy, turning the pages. Suddenly she stopped.\n\n\"Whoa, Nellie,\" murmured Bean, staring.\n\n\"Is she kicking his head off?\" asked Ivy in a whisper.\n\n\"That's what it looks like,\" said Bean. \"What's this one called, anyway?\"\n\nIvy flipped back a few pages. \" _Giselle_ ,\" she said, reading quickly. \"It's about a girl named Giselle who, um, dances with this duke guy, but he's going to marry a princess, not Giselle, so she takes his sword and stabs herself.\" Ivy and Bean found the picture of that.\n\n\"Ew,\" said Bean. \"But interesting.\"\n\n\"Yeah, and then she turns into a ghost with all these other girls. They're called the Wilis.\"\n\nThe picture showed a troop of beautiful women dressed in white. They had very long fingernails.\n\n\"And then,\" Ivy read on, \"the duke goes to see Giselle's grave, and she comes out with the Wilis, and they decide to dance him to death.\" Ivy stared at the picture. \"To _death_.\"\n\nBean leaned over for a closer look. It was pretty amazing. Giselle's pointed toe had snapped the duke's head up so that his chin pointed straight up to the sky. It would fall off in a moment. The Wilis stood in a circle, waving their long fingernails admiringly.\n\nBean lifted the page, wishing that she could see more of the picture, but there was no more. There never was. \"Wow,\" she said, shaking her head. \"She showed him.\"\n\nFor a few minutes, Ivy and Bean sat in silence, thinking.\n\n\"Okay,\" Ivy said finally. \"I'm Giselle, and you're the duke.\"\n\n\"All right,\" said Bean. \"But next time, I get to be Giselle.\"\n\nIt was fun playing Giselle, even though Ivy's mom wouldn't let them dance with a knife and they had to use a Wiffle bat instead. After they had each been Giselle a couple of times, they were Wilis, waving long Scotch-tape fingernails as they danced various people to death.\n\n\"Mrs. Noble!\" shrieked Bean. \"I'm dancing Mrs. Noble to death.\" Ivy ran to get a pair of her mother's high heels and pretended to be Mrs. Noble, a fifth-grade teacher who had once given Ivy and Bean a lot of trouble.\n\nBean the Wili chased Mrs. Noble around the house, waggling her fingernails and screaming. Finally, when they were both laughing so hard they couldn't dance any more, they rushed into the kitchen and fell over on the floor.\n\n\"Well, look who's here,\" said Ivy's mom. She was making dinner.\n\n\"Mom,\" Ivy said when she got her breath back, \"I _have_ to take ballet class.\"\n\nIvy's mom stirred something into something else. \"You had to take ice-skating, too.\"\n\nIvy wiggled her toes. \"Yeah, but that was a mistake.\"\n\n\"How do you know ballet isn't a mistake, too? Those skates were expensive.\"\n\n\"Ballet is different,\" Ivy explained. \"Ballet isn't freezing and dumb. Ballet is pretty. And it's good for you.\"\n\n\"I'm going to take it, too,\" Bean said. \"That way, we can help each other during the hard parts.\"\n\nIvy's mom looked at Bean in a surprised sort of way. \"You're going to take ballet?\"\n\n\"Sure.\" Bean's mom would be happy to let her take ballet. Bean was certain of it. After all, Bean thought, her mother liked nice stuff. And ballet was nice. Except for the part where you danced people to death.\n\nThe funny thing was, Bean's mother wasn't happy to let her take ballet. Not at all.\n\n\"You'll start it, and then you'll decide you hate it and want to quit.\"\n\n\"No, I won't. I'll love it,\" Bean said.\n\n\"I'll bet you a dollar you'll hate it,\" said Nancy. Nancy had taken ballet when she was Bean's age. Bean remembered the time Nancy had cried because she was a chocolate bar in a ballet about candy.\n\n\"But I'm not going to be a dorky old piece of candy,\" Bean said. \"I'm going to be a Wili.\" She knew better than to tell Nancy that she was going to be Giselle. Nancy would just make fun of her.\n\n\"Ha,\" said Nancy. \"You have to be whatever they tell you to be.\"\n\n\"Nancy,\" said her mom. \"I'll discuss this with Bean in private, please.\"\n\n\"I'll bet you, Mom,\" said Nancy, getting up. \"I'll bet you two dollars she quits after a week.\"\n\n\"I'll bet you a hundred I don't,\" said Bean.\n\n\"Good-bye, Nancy,\" said their mother. Nancy left, and Mom turned to Bean. \"Now, honey, I didn't want to go into this in front of Nancy, but if I do let you take ballet, there will be no quitting.\"\n\n\"Quitting? Why would I quit?\"\n\n\"You quit softball.\"\n\n\"But that was softball. All you do in softball is stand around waiting for five hundred years until it's time to hit the stupid ball. And then you miss anyway. Ballet isn't like that.\"\n\nHer mother looked at her.\n\nBean made her eyes big. \"I thought you wanted me to learn new things,\" she said.\n\nHer mother looked at her some more.\n\n\"Nancy got to take ballet.\" Bean wiggled her lower lip. She knew that a trembling lower lip is very sad looking.\n\nHer mother laughed. \"You're drooling. Okay. I will let you take ballet on one condition, and here it is: You will go for the whole session. Four months. Sixteen lessons One performance. No quitting. And no complaining.\"\n\n\"No problem!\" said Bean. She jumped up and hugged her mother. \"When can I start? I already know how to kick\u2014you want to see?\"\n\n## DIP, DIP, CRASH!\n\nIt was not long before Ivy and Bean realized that they had made a terrible mistake.\n\nBean began to realize it while Madame Joy was talking about first position. You stuck your heels together and your toes apart. Big deal. Where was the leaping? Where was the kicking? Where was the dancing?\n\nThen Madame Joy chattered for a long time about nice round arms. Who cared about arms? When Madame Joy started in on second position, which turned out to be just regular standing, Bean stopped listening.\n\nIvy paid careful attention to first position. Heels, toes. Great! Then she paid careful attention to second position. Arms out, legs out. Great! Then came third position.\n\n\"Now,\" said Madame Joy, \"third position. For third position, we slide our right foot, like so, to the middle of our left foot. Then we lift one nice round arm up, up in the air, leaving our other nice round arm\u2014\"\n\nIvy fell over with a thump.\n\n\"Goodness!\" exclaimed Madame Joy. \"Let's try that again.\"\n\nLet's not, thought Ivy.\n\nBut they did. In fact, they did nothing but one, two, three, four, and five for half an hour. After that, Madame Joy showed them something called a pli\u00e9. She acted like it was the most important thing in the world, but really it was just bending your knees and dipping a little. Dip, dip, dip. Row, row, row your boat, thought Bean.\n\n\"Hey, guys,\" she called, \"Get this!\" She sang, \"Dip, dip, dip your knees\u2014\"\n\nNobody joined in. Instead, Madame Joy said, \"We save our singing for after class, Bean.\"\n\nSheesh, Bean said to herself. You'd think she'd be happy to get a little more pep in here.\n\nBut it was even worse when Madame Joy was peppy. \"All right, girls! Time to leap like little kitties!\" Madame Joy said, springing into the air with her ballet slippers fluttering.\n\n\"She doesn't leap like a kitty. She leaps like a frog,\" Bean whispered to Ivy.\n\n\"Bean!\" called Madame Joy. \"You may lead the kitties.\" She twirled briskly around and hauled Bean to the front of the line. \"Now,\" she said, smiling, \"you are a kitty! Leap!\" Madame Joy bounded across the wide, empty floor.\n\nBean closed her eyes and imagined she was a cat.\n\nShe was a skinny black cat that wanted to catch a bird. And then eat it. Bean crouched. She twitched her tail. She narrowed her eyes. \"RRRRRrrrowll!\" she screeched and then lunged forward, landing on her hands and knees in the middle of the floor. \"Got him!\" she yelled.\n\nMadame Joy stared at Bean for a second, and then she said, \"Dulcie, will you show us how to leap like a kitty?\"\n\n\"Yes, Madame Joy,\" said Dulcie, only she said, \"Madame Jwah.\" For some reason, Madame Joy liked that. Dulcie came to the front of the line and stood with her arms out and her toes pointed.\n\nBean rose to her feet. \"So I already did it, right?\" she asked. \"I get to be done, right?\"\n\n\"No,\" said Madame Joy. \"You need more practice. Go to the end of the line.\"\n\nBean clomped to the end of the line and stood behind Ivy.\n\nDulcie lifted her arms higher and smiled proudly. Then she hopped across the empty floor, ker-plop, ker-plop. When she reached the other side of the ballet studio, Dulcie stood before Madame Joy and held out her tiny pink dance skirt. Then she swirled one leg behind the other and curtseyed.\n\n\"Show-off,\" whispered Bean.\n\n\"I can't believe that we asked for this,\" said Ivy, her eyes on Dulcie.\n\n\"We didn't just ask. We begged,\" Bean said glumly.\n\nIt was true. They had begged.\n\nAfter everyone had leaped, Madame Joy clapped her hands and told them they had to be butterflies.\n\nBean raised her hand. \"Can I be a Wili instead?\"\n\nMadame Joy stared at her. \"Not today,\" she said in a way that really meant never. Then she turned on some music, and all the other girls ran around the room flapping their arms and pointing their toes.\n\nThat's when Ivy and Bean turned to look at each other, and their eyes said _We have made a terrible mistake_.\n\n## BAD NEWS BENEATH THE SEA\n\nEvery week Bean and Ivy put on tights and leotards and went to Madame Joy's School of the Ballet, where they fell down and hurt themselves (Ivy) and were bored out of their minds (Bean). Every week they were told to watch Dulcie pli\u00e9 and kitty-jump across the floor even though she was only five. Every week they waited and waited for Madame Joy to clap her hands and say it was time to be butterflies. They hated being butterflies, but at least that meant ballet class was almost over.\n\nIt seemed like it couldn't get worse. And then one day, instead of telling them to be butterflies, Madame Joy told them to sit in a circle on the floor.\n\n\"We're going to be mushrooms,\" whispered Ivy to Bean.\n\nBean didn't think so. When grown-ups asked you to sit in a circle, they were usually about to tell you something you didn't want to hear. Ms. Aruba-Tate, Ivy and Bean's second-grade teacher, was forever gathering them in a circle for bad news. Like, the class fish died over the weekend. Or, everyone has to start using real punctuation. Or, the pencil sharpener is off-limits. Circles meant trouble.\n\nBean watched Madame Joy walk pointy-toed to a chair and sit. No floor for her. \"Girls,\" she began, \"I have something very special to tell you.\"\n\n\"Oh, tell us, Madame Jwah!\" cried Dulcie. She even clapped her hands.\n\nMadame Joy smiled. \"As many of you know, we end each session with a lovely recital. A recital, girls, is a chance for you to dance before your friends and family so that they can see what you've learned.\"\n\nIvy coughed.\n\nMadame Joy leaned forward eagerly. \"Most of our recitals are held here at the school, but this time we have been invited to participate in The World of Dance! Isn't that wonderful?\"\n\nSeveral girls said, \" _Oooooooh!_ \"\n\nBean was getting a not-so-good feeling. \"What's The World of Dance?\" she asked.\n\nMadame Joy's smile grew. \"The World of Dance is a gathering of many different dance schools from all over town\u2014tap dancers, jazz dancers, hip-hop dancers. We will be representing the ballet. Each group gets a chance to perform, just as in a regular recital, but we'll be performing on a real stage in a real theater!\"\n\n\" _Oooooooh!_ \" repeated the same girls.\n\nBean was sick of hearing that.\n\nIvy's hand shot into the air. \"Can we do _Giselle?_ \"\n\n\" _Giselle?_ \" Madame Joy looked surprised. \"No. Goodness, no. We will be doing a lovely piece called 'Wedding Beneath the Sea.' \"\n\n\"Wedding Beneath the Sea\"? Bean didn't care if she was rude. She yelled, \"What are Ivy and me?\"\n\nMadame Joy raised her eyebrows. \"I was planning to discuss parts next, but if you must know, you and Ivy will be the two friendly squids.\"\n\nNobody said, \"Oooooooh.\" Squids? Ivy and Bean looked at each other. _We have made a really terrible mistake_.\n\nOn the drive home, Bean and Ivy were quiet. That was because of the no-complaining rule.\n\nQuietly, they got out of the car and went into Bean's backyard. Quietly, they stuffed themselves into Bean's tiny playhouse and slumped against the walls.\n\n\"Squids. Who ever heard of squids?\" said Bean. \"I don't even know what squids are.\"\n\n\"I'm not totally sure,\" said Ivy, \"but I think they're ugly, and I think people eat them.\"\n\n\"Oh, great,\" moaned Bean. \"I can't believe that stupid Dulcie gets to be the mermaid, and we're squids.\"\n\n\"I believe it,\" said Ivy. \"We're awful.\"\n\n\"We're not _awful_ \u2014\" began Bean.\n\n\"Oh yes we are,\" said Ivy. \"I'm worse than you, but you're pretty bad, too.\"\n\n\"That's because we hate it. If we liked it, we'd be better at it.\"\n\n\"I thought I'd like it,\" said Ivy sadly.\n\n\"So did I,\" said Bean. \"I thought we'd be kicking some heads off. I didn't know about the positions and pli\u00e9s and all that.\"\n\n\"You know, they can't _make_ us do it,\" said Ivy.\n\nBean thought about that. \"Yes, they can,\" she said.\n\nIvy sighed. \"It was mean of them to make us promise not to complain,\" she said.\n\n\"Yeah,\" said Bean. \"They knew all along how horrible it would be.\"\n\n\"We're going to have to be squids in front of everybody,\" said Ivy. \"That's the most horrible thing of all.\"\n\n\"They'll probably laugh at us,\" said Bean, imagining it.\n\n\"They can't. They're _parents_ ,\" Ivy said.\n\n\"Remember? It's friends, too. There might even be someone from school there,\" Bean said gloomily.\n\n\"If only we could quit,\" Ivy moaned.\n\n\"But we can't,\" said Bean.\n\nIvy frowned. That meant she was getting determined. \"There has to be a way,\" she said, determinedly. \"Nothing is impossible.\"\n\nBean stared at her. \"It's impossible for us to be good at ballet.\"\n\n\"Well, _that_ , sure,\" said Ivy. \"But it's not impossible for us to break our arms.\"\n\n## SQUIDS IN A FIX\n\n\"What?\" said Bean.\n\n\"We can't be squids if we break our arms,\" said Ivy. \"Remember what Madame Joy said? We're supposed to wave our tentacles gently on the passing tide. No way can we do that if we've got broken arms. Right?\"\n\nThat was true. But. Broken arms. That could be going too far. Bean pictured her arm cracked in half.\n\n\"I saw a picture of a guy who broke his arm, and his bone poked out of his skin,\" she said.\n\nIvy made an ouch face.\n\n\"Yeah, I know,\" said Bean. \"Maybe we don't have to break them. Maybe we can just sprain them instead.\" She didn't really know what a sprain was, but she knew that it didn't involve bones poking out of your skin.\n\n\"Okay. Sure. We can't be squids with sprained arms either,\" said Ivy. \"No way.\"\n\n\"No how,\" agreed Bean. They looked at each other. \"So, how do you sprain an arm?\" Bean asked.\n\n\"I bet it's like breaking, only smaller,\" Ivy reasoned. \"When she was a kid, my mom broke her arm falling off her garage roof. If we want to just sprain our arms, maybe we should find something shorter than a garage and fall off it.\"\n\nThis made sense. Bean looked around her backyard. There was the porch, but they'd crack their heads open on the stairs. There was the playhouse. There was the trampoline\u2014\"Hey, I've got an idea,\" Bean said. \"We'll jump off the playhouse onto the trampoline and then boing from the trampoline onto the ground. That should do it.\"\n\nFirst they had to drag the playhouse across the lawn and set it down next to the trampoline. Bean noticed that the playhouse was not much taller than the trampoline. They were going to have to jump hard.\n\nNext, Bean climbed up the plastic playhouse shutters until she was perched on the roof like a giant bird.\n\nIvy took a running jump at the playhouse and flung herself over the roof. \"Oof,\" she said.\n\n\"You have to stand up,\" said Bean. \"Or your jump will be too short.\"\n\n\"You go first,\" said Ivy in a muffled voice.\n\nBean rose slowly to her feet. The playhouse made a funny sound.\n\nIvy began to push herself up on her hands. There was another funny sound. It was a bending sort of sound. A cracking sort of sound.\n\nThe roof was caving in.\n\n\"Abandon ship!\" Bean hollered and bounced onto the trampoline. But the two sides of the playhouse were folding around Ivy like a taco. She couldn't abandon ship. She couldn't do anything. Bean watched as Ivy sank closer and closer to the ground.\n\n\"Are you okay?\" she asked.\n\n\"I'm fine,\" said Ivy.\n\nAfter a few minutes, the playhouse stopped sinking, and Bean tried to pull Ivy out by yanking on her head. But Ivy said that hurt worse than being tacoed, so Bean yanked on the playhouse instead. Soon the roof de-caved enough for Ivy to squeeze out, and then Bean crawled inside and kicked the ceiling until the playhouse was almost the shape it had been before.\n\n\"Whew,\" said Bean, sitting down. \"We're going to have to get some tape to fix that crack.\" She wiped her sweaty face with her sweaty hand. \"Duct tape. I love fixing things.\"\n\n\"But Bean,\" said Ivy. \"We didn't fix anything. We're still squids.\"\n\nDang. Bean had almost forgotten about that. Her duct-tape happiness faded. She was a squid. A friendly squid. \"Maybe we'll get so sick we can't be in The World of Dance,\" she suggested.\n\n\"That's not a bad idea,\" said Ivy thoughtfully. \"In fact, that's a great idea. We can't dance if we're sick. Let's get sick.\"\n\nSick. Well, it would hurt less than spraining her arm. \"Okay, but how?\" asked Bean.\n\n\"Germs,\" said Ivy. \"We'll catch some germs and get sick.\"\n\n\"Germs,\" said Bean, thinking. \"I know where germs are. At school. Ms. Aruba-Tate says the school is full of germs. That's why she's always making us wash our hands.\"\n\n\"But we don't want regular dirt germs. We want sick germs,\" said Ivy. \"We'll have to find someone sick.\"\n\n\"Easy-peasy.\" Bean was definitely cheerful now. \"Tomorrow we'll find the sickest person at school and touch him!\"\n\n## GERMS OF HOPE\n\nIvy and Bean stood on the playground of Emerson School. Around them children were running and shouting. There were kids dangling from the monkey bars and dropping off the play structure. There were kids playing wall ball. There were kids arguing about four square. Some fifth-grade girls walked around the field, talking, which looked so incredibly boring that Bean hoped she would never get to fifth grade. Ivy and Bean leaned against the fence and watched. They were hunting for germs.\n\n\"I bet MacAdam is full of germs,\" whispered Bean.\n\nMacAdam was eating dirt. He liked to do that. But other than eating dirt, he looked perfectly healthy.\n\n\"We need someone sicker,\" said Ivy. \"Look for someone sitting down. If you sit down during recess, it's because you're sick.\"\n\nThey peered around the playground. \"Drew is sitting down,\" said Bean, \"but that's probably because the Yard Duty got him.\"\n\n\"What about that kid over there?\" Ivy pointed to a first-grade-looking kid that Bean didn't know. He was sitting by himself on a bench.\n\n\"Hey! He coughed!\" said Bean. \"Let's get him!\"\n\nIn a flash, they were at his side.\n\nHe looked up.\n\nIvy nudged Bean and pointed at his nose. It was runny.\n\n\"Are you sick?\" asked Bean.\n\n\"Yes,\" said the kid. He coughed with his mouth wide open and then looked back up at them again. \"What?\"\n\n\"What have you got?\" asked Ivy.\n\n\"What does it matter?\" said Bean. \"He's sick.\"\n\n\"I don't want to throw up,\" whispered Ivy.\n\n\"Oh,\" said Bean. She didn't want to throw up either. \"You're not going to throw up, are you?\" she asked the boy.\n\nHe looked a little worried. \"I don't think so. Maybe.\"\n\nIvy took a step away. Bean stared at him, thinking about friendly squids. \"Can I touch your face?\" she asked finally. \"Me and her, we need to get sick.\"\n\nHe wiped his nose. \"Okay.\"\n\nBean stuck her hand on his face. It was kind of gross. \"Breathe on me,\" she told him.\n\nHe puffed a big breath at her. She could feel the germs hitting her skin.\n\nIvy was standing far away in the bushes by now. \"I'll just catch it from you,\" she called.\n\nBean rubbed her hands all over her face. \"Thanks,\" she said to the kid. He sneezed.\n\nBean and Ivy knew about germs. They didn't make you sick right away. You had to wait at least a couple of hours. That was okay. Ivy and Bean didn't want to get sick during science. They liked science.\n\nThis month, science was Ocean Life. And today Ocean Life was fish prints. It was art and science mixed together, Ms. Aruba-Tate said. The second-graders nodded. They liked art, too.\n\nMs. Aruba-Tate explained about fish prints. You took a dead fish and painted it. Then you dropped it pretty hard on a piece of paper. When you picked it up again, there was a paint fish on your paper. Then you used your crayons to draw an undersea environment around the fish.\n\n\"Does everyone understand the instructions?\" asked Ms. Aruba-Tate, looking around the classroom.\n\n\"Are the fish dead?\" asked Zuzu.\n\n\"Yes, the fish are dead,\" said Ms. Aruba-Tate.\n\n\"Are you sure?\"\n\n\"Completely sure,\" said Ms. Aruba-Tate. \"Any other questions?\"\n\nThe second-graders shook their heads. Fish prints sounded like fun.\n\n\"Now who is our supply person today?\" asked Ms. Aruba-Tate.\n\n\"Eric!\" shouted the second grade.\n\nEric leaped to his feet, waving his hands in the air. \"Thank you, thank you!\"\n\n\"Eric, please put one fish at each table,\" said Ms. Aruba-Tate, handing him a big plastic box.\n\nEric went around the room, carefully choosing the right dead fish for each table.\n\n\"Hurry up!\" shouted everyone. Paint and dead fish. This was the best science yet.\n\nBean was itching to begin. When Eric reached her table, there were just two dead fish left in the box, but he couldn't decide between them. He looked at one and then the other. \"Which one should I give you? The little one or the big one?\"\n\n\"Just give us one!\" shouted Bean.\n\n\"Maybe I should ask Ms. Aruba-Tate which one I should give you,\" Eric said.\n\nBean reached into his box and grabbed a dead fish.\n\n\"Ms. Aruba-Tate! Bean took a fish!\"\n\nDang. Bean looked at her teacher. Was she going to be sent to the rug? Was she going to miss out on dead fish and paint?\n\nBut Ms. Aruba-Tate smiled at Bean. \"Next time, don't grab, Bean.\"\n\nBean loved Ms. Aruba-Tate with all her heart.\n\nCarefully Bean smeared her fish with green paint. She looked down and saw the fish's eye looking up. Poor fish. She decided to make the most beautiful fish print in the world, to make it up to the fish for being dead. Slowly she laid the fish on her paper and pressed. Then she pressed harder. It had to be good.\n\n\"Bean! Watch out!\" squawked Vanessa.\n\nOops. She had pressed a little too hard.\n\nThe fish was kind of bent. She lifted it up and peeked at her print. That was kind of bent, too.\n\n\"You wrecked it!\" said Vanessa. \"And your fish print is all lumpy.\"\n\n\"It's not lumpy,\" said Bean.\n\n\"It's about to have babies,\" said Ivy.\n\n\"Yeah!\" said Bean. She handed the fish to Ivy. \"I did it on purpose,\" she said to Vanessa.\n\nWhile Ivy made her fish print, Bean drew an undersea environment for her fish. Kelp. An octopus. A sea anemone. A wrecked ship with ghosts. Science was her favorite subject, for sure.\n\n## TIGHT TENTACLES\n\nIvy and Bean worked so hard on their fish prints that they forgot about getting sick. It was only on the way home that they remembered. Ivy looked down Bean's throat.\n\n\"It's pink,\" she said.\n\n\"It's always pink,\" said Bean. She felt her forehead. \"I have a headache,\" she said.\n\n\"That's good,\" said Ivy encouragingly.\n\nBut when they got to Bean's house, Bean's mother said that a person with a headache was too sick to eat an ice-cream bar, and that's when Bean realized that she didn't have a headache after all. She felt fine.\n\nShe still felt fine the next day.\n\nAnd the day after that.\n\nIvy touched a kid with a rash. Nothing. Eric sneezed on Bean eight times. Nothing. Half the kids in the first grade had lice, but Ivy and Bean decided that lice wouldn't help. Their mothers would make them be squids with lice.\n\nBy the end of the week, Ivy and Bean were completely unsick. They needed a new plan. But what?\n\nUsually Bean didn't worry much. In fact, grown-ups sometimes said she didn't worry enough. But that weekend, even while she was doing fun things like going to a fair that included a giant slide, Bean worried. Mostly it didn't feel like worry. What it felt like was fun with a little bit missing. When Bean came whooshing to the bottom of the giant slide, she thought, Why don't I feel totally great? And then she remembered. Because I have to be a squid in front of everyone.\n\nIvy worried, too. Ivy usually didn't worry about real life. Ivy usually worried about things like the Permian extinction, when a whole lot of animals died. The Permian extinction was very upsetting, but it had happened 250 million years ago, so it wasn't real life anymore.\n\nThis weekend Ivy didn't think about the Permian extinction. She thought about how she would feel being a squid on a stage in front of a whole lot of people. She knew how she would feel. Stupid. She would probably trip, because she usually did. And even if she didn't trip, she would be a squid. Everyone would know that Madame Joy had made her a squid because she was the worst dancer in the class. Too bad the Permian extinction didn't wipe out squids, Ivy thought.\n\nOn Sunday afternoon, Ivy went over to Bean's house to be measured for her squid costume. Bean's mother had said she would make both squid costumes because Ivy's mom didn't like to sew. But it wasn't even a real costume. Madame Joy's picture showed a white leotard with a circle of droopy white tentacles hanging from the waist.\n\nMadame Joy said that tentacles were a breeze to make. Bean's mom didn't think so.\n\n\"Who ever heard of squid costumes, anyway?\" she muttered.\n\n\"No complaining,\" said Bean.\n\n\"None of your lip there, missy,\" her mother said.\n\nThat was grown-ups for you. They never followed their own rules.\n\n\"I suppose I could stuff a bunch of tights and sew them on,\" Bean's mother mumbled. Bean and Ivy exchanged looks.\n\n\"Tights?\" Bean said. \"Like the kind you wear on your legs?\"\n\nHer mother looked up. \"Yes, tights. Stuffed tights. For the tentacles. Do you have a better idea?\"\n\nBean thought of the Wilis in their long feathery dresses. She thought of herself with stuffed tights bouncing around her waist.\n\n\"Tights it is!\" said her mother.\n\n\"We're going to look like idiots,\" said Bean.\n\n\"No complaining,\" said her mother.\n\nMonday started out badly. Ms. Aruba-Tate was at choking class. The Principal had told her to take choking class so she would know what to do if a student choked. Bean said if someone choked, you dangled them upside down by their ankles until whatever it was fell out. Ms. Aruba-Tate said she didn't think so, but she would find out.\n\nMs. Aruba-Tate's substitute was Teacher Star. Teacher Star wasn't mean, but she never stopped singing. She sat on Ms. Aruba-Tate's stool and strummed her guitar and sang. She told the second-graders to sing along, but they didn't want to, so she sang alone. She sang and sang and sang some more.\n\nIvy read her book under her desk. Bean thought about choking. Then she thought about ballet class. She thought about Dulcie and her pink chiffon dance skirt. She thought about white tentacles made out of stuffed tights.\n\n\"There's a little blue planet in the sky,\" sang Teacher Star.\n\nIt was nice to see Ms. Aruba-Tate again after lunch recess, but by then Bean was too busy thinking about ballet class to pay attention to what Ms. Aruba-Tate was saying. Something about permission slips. She was waving a piece of paper. Who knew what it was about?\n\n\"Does anyone have any questions about our trip?\" asked Ms. Aruba-Tate.\n\n\"What trip?\" asked Bean.\n\n\"Will someone tell Bean about our field trip to the aquarium?\" said Ms. Aruba-Tate. \"Emma?\"\n\n\"We're going on a field trip to the aquarium,\" said Emma.\n\n\"To see some ocean life,\" said Dusit.\n\n\"We're going to see them feed the sharks,\" said Eric. \"Raw meat.\"\n\n\"And baby penguins,\" said Zuzu.\n\n\"They're going to feed the baby penguins to the sharks,\" said Eric. He clashed his teeth together, being a shark.\n\n\"Eric,\" said Ms. Aruba-Tate.\n\n\"Just kidding, Ms. Aruba-Tate,\" said Eric.\n\n\"Oh!\" said Ivy in a very loud voice. Everyone looked at her. Ivy hardly ever said anything in a very loud voice.\n\n\"Ivy?\" asked Ms. Aruba-Tate.\n\nIvy gave Bean an enormous smile. Then she turned to Ms. Aruba-Tate and said, \"I was just thinking about how much I love ocean life.\"\n\n## BYE-BYE, BALLET\n\n\"We're saved!\" hissed Ivy, pulling Bean toward the door.\n\n\"Saved from what?\" Bean hissed back.\n\n\"Being squids!\" squealed Ivy. She raced out into the breezeway. Bean's sleeve was in her hand, so Bean raced with her. Together they left the school behind and hurried toward Pancake Court.\n\n\"Okay,\" puffed Bean, \"how are we saved?\"\n\nIvy stopped. \"The field trip! We're going to run away! We'll run away to the aquarium, and we'll stay there until after The World of Dance is over!\"\n\nRunning away! What a great idea! Bean had been waiting for years to run away. What she had been waiting for was a reason. She didn't want to hurt her parents' feelings by running away for no reason. The World of Dance was a great reason. This was the chance of a lifetime.\n\nOh yeah. Bean suddenly remembered the other reason she had never run away. \"What about food?\" she asked.\n\n\"Easy-peasy-Parcheesi,\" said Ivy. \"I read about it in a book. You know how people throw money in fountains? We scrape it off the bottom of the fountain after the aquarium is closed at night, and then we buy food with it.\"\n\nThat was pretty smart. Bean was impressed. Also, it would be fun to walk in a fountain without grown-ups freaking out about it. \"Cool,\" she said. \"Where will we sleep?\"\n\n\"We'll find a good spot once we get there. Aquariums are good for sleeping because they're dark.\"\n\n\"And quiet. Fish are very quiet.\" Bean pictured herself drifting off to sleep with fish swirling around her. It would be nice. \"It'll be like sleeping on a boat.\"\n\nIvy rubbed her hands together. \"In this book I read, the kids filled their clarinet cases with extra underwear, but we'll use our backpacks.\"\n\n\"My backpack is pretty big.\"\n\n\"We should bring jackets, too. And money. In the book, they brought all their money.\"\n\n\"Why do we need money if we're going to scrape it out of the fountain?\" asked Bean. \"Besides, I only have four dollars and some coins.\"\n\n\"I've got twenty-six and some coins,\" said Ivy. \"But I don't want to spend it. I'm saving for a glass doll.\"\n\n\"There will be plenty of money in the fountain,\" Bean decided.\n\n\"And we'll get clean at the same time,\" said Ivy.\n\n\"Boy,\" Bean said, shaking her head. \"It's too bad I wasted all that time worrying.\"\n\nSomehow, knowing that they were going to run away made ballet class better. \"Still not good,\" said Bean. \"But better.\"\n\n\"I don't know,\" said Ivy. She was watching Dulcie do an arabesque. An arabesque was when you stretched out one arm and one leg at the same time. Arabesques made Ivy fall over. Dulcie could arabesque all day long. \"Bet she puts glue on her shoes,\" muttered Ivy.\n\n\"Very nice, Dulcie,\" said Madame Joy.\n\n\"Thank you, Madame Jwah!\" said Dulcie.\n\nNow instead of being butterflies at the end of ballet class, they practiced \"Wedding Beneath the Sea.\" Dulcie swayed and kitty-jumped and fluttered her fingers. Two starfish girls twirled with their arms out. Two seahorse girls galloped in and out of the starfish. Two tuna girls glided together across the floor. Ivy and Bean, the friendly squids, stayed in one place and waved their arms.\n\n\"Call this dancing?\" Bean whispered. \"This is standing.\"\n\n\"Enter the prince!\" cried Madame Joy.\n\nThe prince was a girl wearing a black leotard and a red hat that looked like a tiny pillowcase. The prince was the second-crummiest part in \"Wedding Beneath the Sea,\" but it was way better than being a friendly squid. The prince at least got to leap. The prince-girl leaped toward Dulcie while Dulcie fluttered away. Then the prince got down on one knee and waved her arms at Dulcie. Then Dulcie nodded, and all the other fishy things got in a circle and danced around them. Except the two friendly squids. Madame Joy said they were like doormen. They guarded the entrance to the mermaid palace.\n\nFinally Madame Joy clapped her hands. Class was over. For Ivy and Bean, it was especially over. Next week they'd be living at the aquarium.\n\nIvy grinned at Bean. \"Bye-bye, ballet,\" she whispered.\n\n\"Down with squids!\" Bean whispered back. The night before the field trip Bean filled her backpack with useful items. Band-Aids? Check. Pencil? Check. String? Check. Underwear? Check. Bag of salt? Check. Nancy had told her once that all you needed to stay alive was salt and water. Bean figured there would be plenty of water at the aquarium.\n\nBean zipped her backpack closed. She looked around her room. It seemed like she should be discussing important things with Ivy. She couldn't think of any important things, but she called Ivy anyway.\n\n\"Are you ready?\" she asked.\n\n\"Who's this?\" said Ivy.\n\n\"Bean!\"\n\n\"Oh. Hi, Bean,\" said Ivy. She didn't like talking on the phone. \"What do you want?\"\n\n\"Are you ready? Say ten-four if you are.\"\n\n\"Say what?\" asked Ivy.\n\n\"Ten-four,\" said Bean.\n\n\"Fourteen,\" said Ivy.\n\n\"No! It means yes,\" said Bean.\n\n\"Yes what?\"\n\n\"Ivy! Yes, I'm ready!\" yelled Bean.\n\n\"Oh. I'm ready, too. Good-bye.\" Ivy hung up.\n\nForget it. Bean went back to her room. It was almost bedtime, which meant it was almost morning, which meant it was almost running-away time. Bean could hardly wait.\n\n## VERY FISHY\n\nMs. Aruba-Tate's class swarmed off the bus. They were proud. Their bus behavior had been excellent, if you didn't count Marga-Lee and Dusit.\n\nThey stampeded up the stairs toward the aquarium's front door. Who would get there first? Who would be the best?\n\n\"Boys and girls! Stop!\" hollered Ms. Aruba-Tate. They stopped. Ms. Aruba-Tate didn't holler very often. \"Boys and girls! Stay where you are! Don't move! Stay with your buddy! Try to stay together!\"\n\nThe white marble patio outside the aquarium was full of hollering teachers and wandering kids. There were kids sliding down the handrail on the stairs. There were boys throwing their backpacks at each other. There were girls walking along the rim of the fountain. All the teachers were trying to get all the kids to stand still. What a nuthouse, thought Bean.\n\n\"Boys and girls! Follow me!\" shouted Ms. Aruba-Tate. \"Stay with your buddy!\"\n\nLinking arms, Ivy and Bean climbed the stairs toward the big golden doors.\n\n\"Our new home,\" Bean whispered.\n\nThey went inside. The aquarium was big and dim, with dark hallways like arms leading off in many directions. It was sort of greenish all over, and even with hundreds of kids wandering around, it was quiet.\n\n\"Okay,\" said Ivy, pulling out a list. \"The first thing we do is find a good hiding place.\"\n\nBut they couldn't find a good hiding place because Ms. Aruba-Tate was calling them over to the alligator pit. The second-graders clustered around the pit and stared down at the alligators.\n\n\"Look!\" Bean nudged Ivy. \"There's money in there!\" Bright coins sparkled in the slimy alligator water.\n\nIvy looked. \"No way am I going in an alligator pit to get money,\" she said.\n\n\"Oh. Right.\" Bean stared at the money. What a waste. The alligators seemed dead anyway. They didn't even move. Maybe she could just slip in and out.\n\nOne of the alligators spread its mouth wide in a yawn.\n\nMaybe not.\n\n\"Stay together!\" called Ms. Aruba-Tate, leading them from the alligator pit to a dark hallway. \"Now we will see Coastal Zones.\"\n\nIvy nodded at Bean. Coastal Zones sounded like a good place to make a getaway.\n\n\"When do we eat lunch?\" yelled Paul. \"I'm starving to death.\"\n\n\"Now,\" whispered Ivy. She and Bean started to walk backward.\n\n\"There will be no eating inside the aquarium,\" said Ms. Aruba-Tate. \"Ivy! Bean! Stay with the group!\"\n\n\"Boy, does she have sharp eyes,\" Bean muttered.\n\nCoastal Zones turned out to be tide pools. Tide pools were good because you got to stick your hands in them. Ivy and Bean decided to run away later. Ivy held an orange starfish, which was really called a sea star and had eyes at the ends of its arms. Pretty neat.\n\nA sea anemone wrapped its soft tentacles around Bean's finger. She hoped it didn't hurt when she pulled her finger away.\n\nAfter Coastal Zones, there were penguins. Bean and Ivy liked penguins, but Zuzu loved them. She cried when it was time to go. Eric said he was going to freak if they didn't get to sharks soon, so Ms. Aruba-Tate let them skip shrimp and move straight to sharks.\n\n\"I want to see sharks,\" said Bean. \"Then we'll go.\"\n\nIvy nodded. She wanted to see sharks, too.\n\nAs it turned out, sharks were not that exciting. For one, they were small. And they swam around in circles, zip, zip. They didn't care if the second grade wanted to see them or not. They just zipped around.\n\n\"Come along, boys and girls,\" called Ms. Aruba-Tate. \"Let's investigate the Kelp Forest.\"\n\nThe Kelp Forest. Boringsville. Bean nodded to Ivy. Ivy nodded to Bean. They waited beside the shark glass while the rest of the class surged forward. Ms. Aruba-Tate was listening to Emma tell about the time she was seasick. She didn't notice Ivy and Bean.\n\nNo one noticed.\n\nIn a minute, they were all alone with the sharks.\n\nNow that Ms. Aruba-Tate's class was gone, Bean and Ivy could hear the sharks. They could hear them move through the water.\n\n\"Come on.\" Ivy pulled on Bean's sleeve.\n\n\"Wait a second.\" Bean leaned close to the glass wall. Bean wondered if they could hear her. \"Hi,\" she said. The sharks swam around, their black eyes empty. They didn't care. \"Let's get out of here,\" she said to Ivy.\n\nThey turned and scurried down a hall lined with little tanks of fish.\n\nWhen they got to the end of that hall, they turned down another.\n\nAnd then another. They had done it.\n\nThey were runaways.\n\n## OCEAN LIFE GONE BAD\n\nIvy and Bean came to a gray room. It didn't have any ocean life in it. What it did have in it were a lot of dishes.\n\n\"We must be near the cafeteria,\" said Ivy.\n\nA man walked into the room pushing a cart. He didn't look surprised to see them, but he didn't look happy either. \"No kids in here,\" he said. \"Cafeteria's that way.\" He pointed to a door.\n\n\"Okay,\" said Bean. She and Ivy went through a different door.\n\nNow they were in a dark hallway. A dark, small hallway. They could just barely see the sign on the wall. It said, \"Life without Light: Creatures of the Deep Sea.\"\n\n\"Perfect!\" said Ivy.\n\n\"Perfect? For what?\" asked Bean. It didn't look perfect to her. It looked dark.\n\n\"Life without Light.\" said Ivy. \"It's great for sleeping. Plus, no one will be able to see us.\"\n\nBean looked around the little hall. \"We're going to sleep in here?\"\n\n\"No. This is just where they put the sign. The fish and stuff are in there.\" She pointed to a doorway.\n\nTogether they walked into a long, narrow room. At least Bean thought it was a long, narrow room. She couldn't really tell because it was so dark. It was even darker than the hall.\n\n\"Why don't they turn on some lights?\" whispered Bean. It seemed like a whispering place.\n\n\"It's showing what it's like in the deep sea. The sun doesn't get all the way down there,\" whispered Ivy.\n\n\"So that's all? Just a dark room?\" Bean shook her head.\n\n\"I don't know. I can't tell. Do you see fish tanks anywhere?\"\n\nBean looked hard into the darkness. She could see some glimmering on the wall. Maybe it was glass. Or something else. Bean started to get a worried feeling. \"Why aren't there any people in here?\" she asked.\n\n\"I don't know,\" Ivy said again. Bean could see the outline of Ivy's head as she looked from side to side. \"Maybe the sign was old. Maybe there's nothing in here.\"\n\nFor a moment, they stood there in the dark. It was so quiet that they heard the sound of the quiet. Bean began to think of all the things that might be slithering silently toward them.\n\n\"Ivy? I'm not liking this so much,\" she said.\n\nIvy linked her arm into Bean's. That was better. A little. \"There's got to be a light switch in here somewhere,\" said Ivy. \"If we walk around, I bet we'll find one. And once we turn on the light, we'll figure out where to hide our backpacks.\"\n\nSlowly, with their arms out, they walked toward the wall. Bean's hands brushed against cool glass. No light switches there. She felt around its edges.\n\n\"Hey,\" said Ivy. \"Here's a button thing. Should I push it?\"\n\n\"Um,\" said Bean. \"What if it opens a trapdoor and water gushes out?\"\n\nToo late. Ivy had pushed the button. The wall in front of them began to glow with red light. For a second, they blinked at the brightness. And then they saw. Behind the glass was black water rising high above their heads. They pressed their faces to the window. Was it just empty water?\n\n\"I don't see any fish,\" Bean began to say\u2014and then a massive mouth came hurtling toward them, shining with thousands of needle teeth. \"YIKES!\" Bean took an enormous leap backward, dragging Ivy behind her. \"Holy moly cannoli!\" she squeaked. \"What the heck is that?\"\n\nIvy didn't say anything, but her hand held tight to Bean's. The giant mouth was attached to a long snaky creature that glared at them with tiny bright eyes.\n\n\"I guess this is what it's like at the bottom of the sea,\" whispered Ivy.\n\nBean shivered.\n\nOn the other side of the glass wall, a fish swam by, a thin arm sprouting from its head. At the end of the arm was a glowing lump. The fish swished its head from side to side, and the glowing lump swung like a lantern.\n\nSlowly the two girls made their way around the room. Long white worms poked from tubes. See-through fish wiggled along, trailing other fish with glowing eyeballs. Shining blobs with no heads or tails rolled on the floor of the tank. Were they alive?\n\n\"Could we turn the lights off again?\" Ivy asked in a small voice. \"I can't stop looking at those blobs.\"\n\nBean reached over to the button under the glass and pressed it. The red light faded into darkness. Thick nighttime darkness. With worms and giant mouths in it.\n\n\"Ivy?\" said Bean. \"I don't think I can live in here for two weeks.\"\n\n\"Sure you can,\" said Ivy, but her voice didn't sound sure. \"They're inside tanks. Tank glass is super-strong.\"\n\nThere was a pause.\n\n\"I keep thinking they're watching us,\" said Bean.\n\n\"I keep thinking the glass is going to break,\" said Ivy.\n\nBean pictured the giant mouth whizzing toward her. She jumped up and pressed the button again.\n\nBut it was a different button. The red light did not begin to glow. Instead, a serious voice began to talk.\n\n\"The most famous creature of the deep sea can't be seen in an aquarium because it has never been captured alive. The giant squid, which may reach a length of forty feet, is shown here in a rare video. . . .\"\n\nThe voice went on talking, but Bean and Ivy didn't hear it.\n\nThey were watching the video. An enormous white blob flapped in empty black water, its long, blubbery white arms trailing behind. Around and around it spun and ruffled and circled, dancing in the water. It was like a horrible Wili, Bean thought. Its legs flailed and waved. Then, with a giant flap and whirl, the squid shot toward them. Its head, huge and soft, turned, and suddenly a single monster eye, an eye the size of a plate, stared right into theirs. It could see them.\n\nFor a second, Ivy and Bean stood frozen.\n\nAnd then they began to run.\n\n## IN HOT WATER\n\nBean couldn't stop running. She was gasping for air and her backpack was slamming into her shoulders, but she couldn't stop running.\n\nIvy slammed into a kid. \"Excuse me,\" she gasped.\n\n\"Watch out!\" yelled a teacher as they pounded by. \"No running in the aquarium!\"\n\nThey couldn't stop. The squid was back there, waiting for them. They had to get out.\n\nThey tore up a dark hallway filled with sardines and down a dark hallway filled with jellyfish.\n\nThey flashed past the sharks, past the penguins, past the alligator pit, and exploded through the heavy golden doors into the outside world.\n\nAir instead of water. Light instead of darkness. People instead of fish.\n\nThey were safe.\n\nFor a moment, they stood there, panting and gasping. I love light, thought Bean. I love air. I love this white marble patio\u2014\n\n\"BEAN! IVY! WHERE ON EARTH HAVE YOU BEEN?\" Ms. Aruba-Tate rushed toward them with her arms open. \"Did you get lost? We were looking everywhere! Oh dear, I was so worried!\" She gathered them up in a giant hug. \"Oh dear,\" she said, \"oh, honeys!\"\n\nIvy and Bean let themselves be hugged. It felt nice, after that squid, to be hugged.\n\n\"We're okay,\" said Bean.\n\n\"We got lost,\" Ivy said quickly. That was kind of true.\n\n\"Oh, sweeties!\" Ms. Aruba-Tate hugged them again. \"Why didn't you go to one of the guards? Didn't I tell you to go to a guard if you got lost?\"\n\n\"There weren't any guards,\" said Bean. That was completely true.\n\nNow the rest of Ms. Aruba-Tate's class was clustering around.\n\n\"There you are!\" said Emma. \"See, Ms. Aruba-Tate, I told you they weren't dead.\"\n\n\"We got to see the eels and you didn't,\" said Eric. \"They're hecka gross.\"\n\n\"I can't believe you got lost,\" said Vanessa. \"Where'd you go?\"\n\n\"Into a part of the aquarium that no one has ever seen before,\" said Ivy.\n\n\"There was this squid with eyes this big,\" said Bean, holding her hands apart.\n\n\"You're making that up,\" said Vanessa.\n\n\"We're not!\" said Ivy. \"There were white worms and this mouth with teeth\u2014\"\n\n\"Girls!\" interrupted Ms. Aruba-Tate. She looked very serious. \"Girls, are you telling me that you were wandering around the aquarium having a good time? That you didn't even try to find us?\"\n\nIvy and Bean looked at each other. \"Um,\" said Bean.\n\n\"We were trying to find you, Ms. Aruba-Tate,\" said Ivy. \"We just happened to see a few worms and things while we were trying.\"\n\n\"Ivy and Bean, I am very disappointed in you,\" Ms. Aruba-Tate began. \"Our class has discussed safety rules many times, and I was counting on you being mature enough to understand that a field trip is an educational experience, not an excuse for bad behavior.\"\n\nAll the way to the bus Ms. Aruba-Tate talked about disappointment and safety and bad behavior. Ivy and Bean nodded. They said she was right and they were wrong. They said they were sorry.\n\nShe was going to have to tell their parents, Ms. Aruba-Tate said.\n\nIvy and Bean nodded. They knew she had to.\n\nThey also knew that their parents were going to be mad. And that they were going to get in trouble.\n\nBut Ivy and Bean didn't care as long as each of them could hold one of Ms. Aruba-Tate's hands on the bus ride home. As long as they never had to go back to that aquarium and see that squid again in their whole lives.\n\n## SQUIDARINAS\n\nThey were right. Bean's mother was mad. \"This is not what I expect from you, Bernice Blue. When you go on a trip of any kind, I expect you to listen to the grown-up in charge. This is something we've discussed a thousand times.\" Bean's mother folded her arms and glared at Bean.\n\nBean could tell she was supposed to say something. \"I'm sorry,\" she said.\n\n\"I should think so!\" said her mother. She glared some more. \"Well! We'll talk about the consequences this evening when Daddy comes home. In the meantime, both of you go upstairs and try on your ballet costumes. And I don't want to hear any complaining!\"\n\nBean and Ivy walked quietly upstairs. Quietly they closed the door to Bean's room. \"Whew,\" said Ivy. \"That was a close one.\"\n\n\"It's not over yet,\" said Bean. \"Your mom still has to get mad.\"\n\n\"I know,\" said Ivy. \"But at least none of them found out about the running-away part.\"\n\n\"We've got to get rid of the evidence,\" said Bean, busily pulling the bag of salt, the Band-Aids, the string, and the underwear out of her backpack. She stuffed it all under her bed.\n\nIvy did the same.\n\n\"Jeez!\" Bean slumped against her bed. \"What a day.\"\n\nIvy lay down on the floor. \"I'm pooped.\"\n\n\"Are you trying on those costumes?\" shouted Bean's mother from downstairs.\n\n\"Sheesh,\" said Bean, getting up. \"Work, work, work. That's all I do.\" The white leotards lay across her bed, stuffed tights legs tangled around them. \"Come on,\" said Bean. \"You have to try yours on, too.\"\n\nIvy sighed and got up. Together they untangled the tights legs and got undressed and pulled on the white leotards. Bean looked at Ivy in her white leotard with ten white legs dangling from her waist.\n\nIvy looked at Bean. \"I don't think Madame Joy has ever seen a real squid,\" she said.\n\nBean thought about the long, blubbery white legs. It made her head prickle. \"Remember its legs?\"\n\nIvy nodded. \"And its eye? Remember how it looked at us?\"\n\n\"Like it was excited. Like it could hardly wait to squeeze the life out of us,\" said Bean.\n\n\"Like we were food,\" agreed Ivy.\n\n\"Squids are _not_ friendly,\" Bean announced.\n\nIvy lifted up one of her white tights legs and shook it. \"A real squid would wrap its legs around Dulcie and squish her.\"\n\nBean giggled. \"And then it would eat the starfish and the sea horses.\" She bonked Ivy with one of her tights legs. \"And the prince.\"\n\nIvy bonked her back. \"And then it would look at the audience with its humongo eye and say, 'And you people are my dessert.' \"\n\nThere was a pause.\n\n\"You know,\" Bean said thoughtfully, \"we could use your face paint to make big black eyes.\"\n\nThere was another pause. Ivy and Bean looked at each other.\n\n\"Madame Joy will kill us,\" said Ivy.\n\n\"We won't do anything,\" said Bean. \"We'll just look more like real squids. She won't mind.\"\n\n\"In a way, she should be glad,\" said Ivy. \"We'll be teaching everyone what squids are really like.\"\n\n\"Yeah, it's educational,\" said Bean. For the first time, she felt a little bit excited about being a squid. \"And maybe, at the very end, after the rest of the dance is over, we can be two squid trying to squeeze the life out of each other.\"\n\n\"Yeah!\" said Ivy cheerfully. \"Like this!\" She jumped at Bean and wrapped three of her tentacles around Bean's arm.\n\nBean hit Ivy over the head with a tights leg and growled. The two unfriendly squids bashed and squeezed each other until they had to lie down on the floor.\n\n\"You know what?\" said Bean after a minute.\n\n\"What?\" said Ivy.\n\n\"By the time we get through with it, 'Wedding Beneath the Sea' is going to be a lot like _Giselle_. Only more exciting.\"\n\nIvy smiled. \"Plus more scientific.\"\n\n\"I just knew we'd end up liking ballet!\" said Bean happily.\n\nSNEAK PREVIEW OF BOOK 7\n\nIVY + BEAN\n\nWHAT'S THE BIG IDEA?\n\nThere had been a problem in Bean's house. The problem was staples. Bean loved staples. She loved them so much that she had stapled things that weren't supposed to be stapled. The things looked better stapled, but her mother didn't think so, and now Bean was outside.\n\nShe was going to be outside for a long time.\n\nShe looked at her back yard. Same old yard, same old trampoline, same old dinky plastic playhouse, same old pile of buckets and ropes and stilts. None of them was any fun. Maybe she could play junkyard crash. Junkyard crash was when you stacked up all the stuff you could find and then drove the toy car into the stack. But it was no fun alone. Bean got up and scuffed across the nice green lawn until she reached the not-so-nice green lawn. This part of Bean's lawn had holes and lumps in it. The lumps were mostly places where Bean had buried treasure for kids of the future.\n\nBean picked up a shovel. To heck with kids of the future. She was bored now. And maybe a kind old guy had seen her digging and added something interesting to her treasure, like a ruby skull or a dinosaur egg.\n\nBean didn't bury her treasure very deep, so it was easy to dig up. This treasure was inside a paper bag, but the paper bag wasn't doing so well. It wasn't really a paper bag anymore. \"Oh my gosh!\" said Bean loudly. \"I've found treasure!\" She pulled apart the clumps of paper. What a disappointment. No ruby skull. No dinosaur egg. Just the same stuff she had buried two weeks ago: dental floss, tweezers, and a magnifying glass. Some treasure.\n\nBean flopped over on her stomach. \"I'm dying of boredom,\" she moaned, hoping her mother would hear. \"I'm dyyy-ing.\" She coughed in a dying sort of way\u2014\"Huh-ACK!\"\u2014and then lay still. Anyone looking from the porch would think she was dead. And then that person would feel bad.\n\nBean lay very still.\n\nStill.\n\nShe could hear her heart thumping.\n\nShe could feel the hairs on her arm moving.\n\nBean opened her eyes. There was an ant scurrying over her arm. Bean pulled the magnifying glass over and peered at the ant. Her arm was like a mountain, and the little ant was like a mountain climber, stumbling along with a tired expression on his face. Poor hardworking ant. She watched as he dodged between hairs and charged down the other side of her arm toward the ground. She offered him a blade of grass to use as a slide, but that seemed to confuse him. He paused, looked anxiously right and left, and then continued down her arm. He had a plan and he was going to stick to it. Bean watched through the magnifying glass as he scuttled into the grass, rushing along the ground between blades. He was late. He was in trouble. He met another ant by banging into him, but they didn't even stop to talk. They rushed away in opposite directions.\n\nBean followed her ant to a patch of dry dirt. There he plunged down a hole.\n\n\"Come back,\" whispered Bean. She liked her ant. Maybe he would come out if she poked his house. She found a thin stick and touched the top of the hole. Four ants streamed out and raced in four different directions. Bean didn't think any of them was her ant.\n\nBean watched the ant hole for a long time. Ants came and went. They all seemed to know where they were going. They all seemed to have important jobs. None of them seemed to notice that they were puny little nothings compared to Bean.\n\nBean dragged the hose toward the ant hole. She didn't turn the hose on. That would be mean. But she let a little bit of water dribble into the hole, and watched as the dirt erupted with ants. Thousands of ants flung themselves this way and that, racing to safety.\n\n\"Help, help,\" whispered Bean. \"Flood!\"\n\nThe ants ran in orderly lines away from the water. Some were holding little grains above their heads. They were the hero ants. But even the non-hero ants were busy. They were all far too busy to notice Bean watching them through the magnifying glass. To them, she was like a planet. She wasn't part of their world. She was too big and too far away for them to see.\n\nBean looked up into the sky. What if someone was watching her through a giant magnifying glass and thinking the same thing she was? What if she was as small as an ant compared to that someone? And what if that someone was an ant compared to the next world after that?\n\nWow.\n\nBean waved at the sky. Hi, out there, she thought.\nANNIE BARROWS took ballet for five hundred years. It felt like five hundred years, anyway. Once, she had to be a Hungarian flower girl in a ballet. Another time, she was a nougat. Plus, her teacher carried a long wooden stick that she used to hit students in the ankle when they made mistakes. To find out more about Annie's amazing career as a ballerina or about Ivy and Bean, visit www.anniebarrows.com.\n\nMore like Ivy than Bean, SOPHIE BLACKALL was well behaved her entire childhood. She is making up for it now. To learn more about Sophie, visit www.sophieblackall.com.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"This entry was posted on December 11, 2018 at 16:56 and is filed under 2018, Club.\tYou can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.\nWhat software do you use to generate the shot charts?\nI don't use any software. I manually note locations and transfer to an Excel template.\nAs far as I am aware the only readily available software with x,y co-ordinates is dartfish. Open to correction on that mind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sure Fire Polar Bear Pictures To Color Printable Coloring Pages Throughout 2 Tgm Sports is one of best wallpaper image reference about coloring pages. This wallpaper has been created for kids, toddler and kindergarten. Sure Fire Polar Bear Pictures To Color Printable Coloring Pages Throughout 2 Tgm Sports was posted in May 30, 2018 at 12:56 pm and has viewed by 444 users. Sure Fire Polar Bear Pictures To Color Printable Coloring Pages Throughout 2 Tgm Sports reference also have Tag ;Sure Fire Polar Bear Pictures To Color Printable Coloring Pages Throughout 2 Tgm Sports, polar bear coloring pictures to print, polar bear pictures to color, polar bear pictures to color for kids, for your convenience in searching this reference more specific.\nSure Fire Polar Bear Pictures To Color Printable Coloring Pages Throughout 2 Tgm Sports is one of best coloring pages on my sites, this coloring pages has been collect with brilliant ideas combined with the best keyword suggestion for the best searching on the internet, would be something helping if you apply for your kids. Sure Fire Polar Bear Pictures To Color Printable Coloring Pages Throughout 2 Tgm Sports just one of the many reference that we have, you can find other reference such as Animal, Cartoon, Fun, Nature, Sports, Flowers and etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"and ye that hope in the Lord.\"\nOn September 11, 2001 I woke up about 5:00 am and could not go back to sleep. I turned on the TV to watch the local news program just in time to see the planes crash into the twin towers of the World Trade Center. I was stunned. As I watched all of the sights and sounds of our nation's attack, the personal testimonies and the pictures of suffering families were overwhelming. Yet, the Lord was so real and such and encouragement to me during the week that followed. As I saw the first scenes unfold on TV, the Holy Spirit began bringing Scripture to my mind. The words of King David, whose life was surrounded by chaos and danger, brought to my mind that God is my stability and rock in the midst of chaos. This passage encouraged me to trust God's stability in seemingly unstable situations, \"Find rest, O my soul, in God alone; my hope comes from him. He alone is my rock and my salvation; He is my fortress, I will not be shaken. My salvation and my honor depend on God; he is my mighty rock, my refuge. Trust in him at all times, O people; pour out your hearts to him, for God is our refuge.\" Psalm 62:5-8.\nThe morning after the attack I woke up with a praise song running through my head. It was a powerful song about Christ's return. Then I turned on the TV and I saw a mother from California sharing how her son, Mark had called her from the plane that went down in Pennsylvania. He told her he knew they were going to die no matter what they did so they were going to try to take control of the plane back so no one on the ground would die. He told her calmly he loved her and then 12 minutes later he died. She looks at his call as a beautiful gift, proud that he would give his life to save others.\nAs I left to walk and pray for her, I was reminded of Christ going to the Garden of Gethsemane to talk to His Father about His imminent death. I wondered if the mother and son shared the same emotions as Christ shared with His Father that dark night long ago. As Christ intensely prayed, God whose very nature is love had to ache, knowing what Christ would endure. Yet, just as the mother expressed comfort she had in her son's sacrifice, our heavenly Father and Christ were comforted by the joy of people being redeemed and reconciled to God. As I contemplated the cross, the song I had woken up thinking about came flooding back unto my head. The first words of the chorus are, \"Behold He Comes, riding on the clouds!\" When I checked my email, I found some German friends had written of their concern for us. They ended their letter with a praise that Christ is coming back! Both the song and the sentiment our friends expressed are reflective of Revelation 14:14 and Revelation 19:11-16. \"And I looked and, behold, a white cloud, and upon the cloud one sat, like the Son of man, having on his head a golden crown, and in his hand a sharp sickle\u2026And I saw heaven opened and, behold, a white horse; and he that sat upon him was called Faithful and True, and in righteousness He judges and makes war. His eyes were like a flame of fire, and on his head were many crowns; and he had a name written, that no man knew, but He himself. And He was clothed with vesture dipped in blood; and His name is called The Word of God. And the armies that were in heaven followed Him upon white horses, clothed in fine linen, white and clean. And out of His mouth goes a sharp sword, that with it He should smite the nations, and He shall rule them with a rod of iron: and He treads the winepress of the fierceness and wrath of Almighty God. And He has on His vesture and on his thigh a name written, King of Kings, and Lord of Lords.\"\nOut of New York's tragedy I learned I live too much in the shadow of the cross. Even though we are brought back to the cross every time we confess and repent of sin or take communion, we need to also embrace the truth that Christ has been resurrected and He is coming back. When we focus on the truth that we serve a living God who is victorious over both sin and death we are free to live in a way that honors God. While the cross is a beautiful picture of His outrageous love, our strength comes in remembering His power displayed in the resurrection. If we live in light of the truth that He is coming back, we will be comforted by knowing when He returns He will rule and reign in all of His glory and justice will be carried out perfectly. By remembering the descriptions found in Revelation we are able to live courageously in the face of these hard events that ordinarily paralyze us. We are able to love people who are not easy to love, able to say no to nagging temptations, and be able to grieve losses as people who have full confidence in God's faithfulness.\nMay I ask you of you are living in the light of His promise to return? Does that truth purify your heart and give you the courage to love boldly and courageously? Do you long for His return?\nPrayer: Father, the 9-11 tragedy is still hard to comprehend. Please continue to be with families who lost loved ones that day. Help us trust in Christ's power during all tragedy's we face. Draw our minds to the fact that Christ is coming back with his glory fully manifested for He is Faithful and True, King of Kings and Lord of Lords and will reign in love and perfect righteousness. Amen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ayurvedic Tea to Experience Rejuvenative and Healthy Lifestyle!\nWhat is Ayurvedic tea and what is the need to imbibe it in the daily routine?\nAyurvedic tea is a potent component of ancient Ayurvedic healthcare practices. These traditional herbal teas are prepared with authentic & age-old herbs to fulfill true approach of healthy living and keeping wellness. The formula of classic Ayurvedic tea has been developed in India above 5000 years back to help people to achieve a balanced and healthy lifestyle.\nBasically, the Ayurveda term stands for \"Science of Life\" and includes various healthcare strategies, which includes Yoga, natural foods, meditation, therapies, and Ayurvedic tea. Instead of spreading many efforts on treating health ailments, try to indulge the habit of one of the best herbal tea to live a healthy lifestyle.\nMoreover, the best benefit is of Ayurvedic tea is that you can consume it at any time of the day to feel refresh and energetic. Thus, you can consider this one of the best herbal tea a guilt-free drink, owing to the absence of any added sugar, caffeine, tannins, and health benefits of herbs.\nAnd when we are talking about Ayurvedic tea, then what can be better than Lemongrass tea. It does not only make you feel rejuvenating but also offers various health benefits. Getting fresh lemongrass is quite impossible and thus, trust one of the best Ayurvedic company in India to get freshly cut and neatly dried lemongrass with numerous nutrients. Additionally, Ayurvedic medicine supplier suggests that instead of consuming various other drinks, take a herbal step towards a better life.\nHow does Lemongrass tea works?\nLemongrass may help keep the development of a few microscopic organisms and yeast. Lemongrass likewise contains substances that are thought to mitigate torment and swelling, decrease fever, maintains the dimensions of sugar and excess lipid in the blood. This Ayurvedic tea possesses various beneficial properties to support healthy living.\nWhat are the Herbal Tea\/ Ayurvedic Tea Benefits?\nThis Ayurvedic tea from one of the best Ayurvedic company in India may exert a strong antimicrobial property, which is beneficial to reduce the oral infections and cavities. Additionally, this property may support the reduction of dandruff and thus promotes hair care.\nWith powerful cooling property, this herbal tea may soothe a stomach pain and keeps the digestive functions on the track. According to modern studies, this one of the best herbal tea contains \"Citral\", which is helpful to digest food easily. According to the Ayurvedic knowledge, this Ayurvedic tea is great to resolve problems like constipation, bloating, or indigestion.\nThis herbal tea may act as a detox tea as It is packed with antioxidants. However, these beneficial antioxidants may help in cleansing body and also helps in eliminating the harmful toxins from the body. Additionally, this herbal tea may relieve the fluid retention problems and thus keeps the stomach relaxed.\nWith high quotient of Vitamin C and A, this one of the best herbal tea is great for hair care. Adding to the vitamins, this Ayurvedic tea from one of the best Ayurvedic company in India may contain essential nutrients to enhance the strength of the hair and also to add natural shine in it.\nLemongrass tea may help in improving metabolism by making the digestion process quick and fast. Additionally, this process may help to burn excess calories and thus promotes a weight loss in natural ways. According to the modern findings, caffeine intake can utilize more energy and thus switch to one of the best herbal tea to keep yourself fit and to feel refreshed all day long.\nBeing a rich source of potassium, this lemongrass tea may help to increase the urination process. However, this process may tend to stimulate the blood circulation and thus lowers the blood pressure. It also may help in controlling the lipid level and thus very beneficial to keep up with heart health.\nWith strong blood purifying properties, this herbal tea from renowned Ayurvedic medicine supplier may help to maintain healthy skin. Additionally, by regulating the blood circulation, it may help to enhance the skin texture and also helps in resolving skin issues such as acne, dark circles, and other blemishes.\nAccording to the recent findings and backed by Ayurvedic studies, this herbal tea is considered a \"women's health tonic\". Additionally, by providing the soothing effects, this herbal tea may relieve the menstrual pain. And so, trust Ayurvedic medicine supplier for pure and natural lemongrass tea to uplift the women's health.\nWith powerful anti-bacterial, anti-fungal, and anti-microbial properties, this lemongrass tea may help you to cope up from cold & flu conditions. Additionally, it is pack with Vitamins C, which helps to strengthen the overall immunity system. And so, it also helps in reducing the cold and flu conditions.\nWhere to find Pure and Natural Ayurvedic Tea?\nHerbal Hills is a great place for getting a plethora of herbal supplements as it is an epitome of Ayurvedic knowledge and modern findings.\nAdditionally, the novel factor of this Ayurvedic medicine supplier lies in the own herbal land in the Sahyadri mountains.\nMoreover, this one of the best Ayurvedic company in India also practices progressive and newer technologies to back modern findings and research tea.\nFurthermore, to grasp the customer's confidence, it performs all the manufacturing process in the unit, which is ISO certified.\nIt also holds GMP and Halal certification to assure the optimum safety and premium quality of Ayurvedic tea and other herbal dietary supplements.\nFollow the classic values with Ayurvedic tea and give your life an energetic boost!\nStay tuned for more such health blogs!\nFeel free to connect with us and suggest what kind of information you need!\nNext Next post: Choose Herbal Coffee over Regular One to Follow Improved Way of Living!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The sands will rise. The heavens will part. The power will be unleashed.\nIn 1925, an expedition of treasure hunters visit the ancient Egyptian city of Hamunaptra, hoping to find significant and valuable archaeological artifacts. Instead, they stumble upon a lost tomb containing the mummy of Imhotep, a priest who had been buried alive 3,000 years ago for betraying his Pharaoh.\nUnbeknownst to the explorers, their discovery has also unleashed the curse of Imhotep, bringing the evil mummy back to life and threatening Egypt with a series of biblical plagues.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"In our catalog you can find for myself whore by price, parameters, services and buy prostitute in Paddock.\nPassionate kitty, waiting non-serious dating with the continuation. Do it, dial my phone number and come, i will fulfill all, as you want!\nAppetizing girl, I want carefree relationship of a sexual nature. Right now, dial my number and come to visit, I'll do anything, as you will order!\nStopped on appetizing images, doubt, postpone?! Need today see and spend pleasantly evening .\nWhen only the best will do. Telephones of Skank in Paddock - Offer Type Offering Rockingham Area Golden Bay. Rockingham Area Port Kennedy. So what are you still waiting for. Come and play with me now. I am completely worth your every minute and dollar spent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our greenhouse is stocked with vegetables & flowers. Trees & shrubs are on sale.\nLafayette Boulevard-Downtown, \"Back in the day\" there was not a I-95 or Route 1 bypass. All traffic circulated through downtown Fredericksburg.\nRoxbury Farm & Garden Center carries feeding supplies year round for Wild Birds, Woodpeckers, Squirrels, and Backyard Critters.\nFall is the best time to plant most Trees, Shrubs and Perennials.\nBrowse through our collection of educational videos on How to Maintain your Lawn and Garden.\nWe feature a complete line of equipment and ingredients for Beer Brewing and Wine Making.\nStay seasonal with us while we keep you updated on new Products, Videos, and Tips on how to keep your Gardens, Lawns, and other Plants thriving year round.\nRoxbury Farm & Garden Center Roxbury Farm & Garden Center shared a post.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The City Council is asking for your help planning our City's change to by-district Council elections. Half Moon Bay, like over 90 cities and 165 school districts across the state, is making a change in how voters elect the City Council.\nBeginning in 2020, City Council members will be elected district by district, instead of the current at-large citywide elections in which all voters have the ability to vote for all City Council seats.\n\u200bThe primary goal when drawing Council election districts is to draw lines that keep neighborhoods together. So the Council wants to know: what do you consider the boundaries of your neighborhood? You can share your specific neighborhood thoughts, or draw a map yourself for the Council and your fellow residents to consider.\nSeptember 4 Update: on a 4-0 vote, the City Council introduced the ordinance to adopt map 106b. The ordinance is scheduled to return to the Council for a final \"second reading\" on September 18th.\nMap 106b, along with the earlier draft maps, is posted on the \"Draft Maps\" page and on the interactive review map.\nCouncil has second reading and final adoption of Ordinance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A selection of recent publications by members of the Society. The Society offers a biennial prize for the best book published by one of its members. Details of the competition can be found here.\nIf you would like your book featured here, please submit it here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since 1998, businesses on the Gulf coast have relied upon Action Restoration to get them back to work faster after a weather disaster, flood, or microbial interruption. As industry experts who are always looking to grow and improve, we are members with the top organizations in the industry and work with many local associations to help grow our business and our knowledge.\nAction Restoration is a leader in disaster recovery and environmental services along the U.S. Gulf Coast. We have a proven track record of providing high-quality, rapid response and comprehensive disaster recovery and environmental services for corporations, educational facility, military bases, laboratories, healthcare facilities, maritime vessels, offshore platforms, and more.\nAction Restoration is a privately owned company with offices around the Gulf Coast region, allow us to respond quickly throughout Texas, Louisiana, Mississippi, Alabama, and the Florida Panhandle. Unlike much of the competition, we have not adopted a franchise model, allowing us complete control over quality of services. We have invested heavily in our fleet of restoration equipment\u2014in fact we have one of the largest collections of restoration and environmental service equipment in the Gulf Coast region.\nIf you need disaster recovery or environmental remediation services for your residential, commercial, or maritime property in the U.S. Gulf Coast region, contact the professionals at Action Restoration today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvarb b/data_all_eng_slimpj/shuffled/split2/finalzzvarb
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+{"text":"I write to thank the many Princeton voters who braved the weather to participate in the advisory vote of the Princeton Community Democratic Organization (PCDO). In a challenging two-against-one contest, I was pleased to receive significant support from the PCDO and from Princeton's Democratic Municipal Committee. The League of Women Voters moderated the PCDO debate with customary skill and grace. Residents will have another opportunity to hear from Princeton Council candidates when the League hosts its candidate forum on April 30. I encourage everyone to attend.\nThanks to your support, my name will be in the Democratic column as incumbent for re-election in the June 3 primary. I am heartened to have the support of voters who want open and transparent government. I am proud of my record as the candidate who challenges business-as-usual practices, who will not shield budget details from public scrutiny, who seeks to clarify conflict-of-interest rules, and who has been a reliable fiscal watchdog. Our consolidated government can balance its budget without diluting democracy.\nOur first contested post-consolidation primary will offer a meaningful choice. I look forward to your help to ensure that voters choose not on the basis of innuendo or vague assurances of Council harmony but on facts and issues. For more information about my positions and my campaign, please visit my website at www.jobutler.com or email me at jo.butler@verizon.net Again, many thanks, and I ask for your continued support in the primary on June 3.\nWe're writing to express our gratitude to the hundreds of Princeton Democrats who ventured out on a rainy Sunday evening to participate in the PCDO local candidate endorsement process. We appreciated the assistance of the League of Women Voters in moderating the meeting and the many volunteers who helped manage the crowd. It is a privilege to compete for the Democratic nomination in a community that takes democracy to heart!\nWe also wish to thank the members of the Princeton Municipal Democratic Committee for meeting with each candidate the following evening to consider ballot placement and use of the Party slogan. We are honored to have received their recommendation.\nAs we head toward the June 3rd Primary Election, we look forward to sharing our ideas for leading Princeton forward and hearing your thoughts.The League of Women Voters has graciously offered to host a televised candidate forum on April 30th, so all of Princeton has an opportunity to learn more about our values and vision.\nLocal campaigns are a great way to get more involved in the civic life of our community. If you are interested in volunteering on our campaign, please contact us at b.p.miller@verizon.net or gsnemeth@gmail.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u0686\u0646\u06cc\u06ba \u067e\u0631\u0633\u062a\u0627\u0631: Bubbles Is SOOOO CUTE!\nWhere can \u0622\u067e get to a spa?\nI think I'll burn everything Nishka loves. Then she won't like Vincent! Than Vincent will fall in love with me! Oooooohhhhhhh, I cannot wait to activate my plan!!\nI think I just lost the love of my life. I think he's in love with my classmate, Nishka Reddy. That's because I just caught Vincent kissing her. I ran away crying. Words can't tell how jealous I am of Nishka right now! My greatest fear is losing the \u0639\u0646\u0648\u0627\u0646 of prettiest girl in the world to a different girl. And I'm afraid of losing it to Nishka. I know she's prettier than me, but I'll never admit it. Never!!!!!!\nThere's a new boy at school today. His name is Vincent and he's soooooooooooooooooo cute! He \u06a9\u06c1\u0627 hi to me and I think that's a sign he likes me! *gasps* Maybe he \u06a9\u06c1\u0627 hi because he loves me! Oooohhh!! I can't wait to tell all my friends!\nWith a white tiger on it and blue shiny pants and red sneakers and her hair was out and has \u06af\u0644\u0627\u0628\u06cc die in it.\nBlossom:hi brick \u0622\u067e look great.\nBlossom: did \u0622\u067e notice those guys over there staring at us?\nBubbles: what if their staring at the flowers \u0627\u06af\u0644\u06d2 to Blossom?\nWould \u0622\u067e think Butch and Blossom be together?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The youth suicide rates of Aboriginal and Torres Strait Islander communities in Australia are four times the national youth average.\nA 2013 survey of Indigenous people reported that a dislike of health services prevented Aboriginal and Torres Strait Islanders from seeking appropriate mental health support.\nDue to the limited uptake of existing interventions by Indigenous communities, health services are working with these communities to find new ways to ensure mental health supports are accessible and effective, with a focus on promoting youth help-seeking and early intervention and prevention initiatives.\niBobbly is the world's first suicide and mental illness prevention app for young Aboriginal and Torres Strait Islanders in Australia. Everything seen, heard and experienced in the app is shaped by, or delivered by Aboriginal and Torres Strait Islanders.\nA recent pilot study carried out in Western Australia found there was a significant reduction in scores of suicidal thinking, depression and distress from using iBobbly and that appropriately designed self-help apps are an acceptable means of lowering symptoms for mental health disorders in remote communities.\nWith our support, Black Dog Institute will roll out iBobbly in a large-scale trial in regional and urban areas of NSW, connecting Indigenous youth with both digital and local face-to-face support.\nAs Indigenous youth have a high rate of mobile usage, iBobbly aims to provide a new form of intervention that is easily accessible and helps Aboriginal and Torres Strait Islander youth to learn skills to manage distress and suicidal thinking.\nBy learning these skills and by increasing help-seeking it is hoped there will be a reduction in the number of suicide attempts and death by suicide in Aboriginal and Torres Strait Islander communities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"But they came on strong. On Sundays, they churn out more than 4,000 of the chewy little rounds topped with everything from salty savory whitefish salad to the comforting combination of lox, eggs, and onions ($6.99). Toasted does things the old way, using kosher malt, flour conditioner, and filtered water to produce pristine bagels.\nDon panic, as we leading you through the shadows by assembling this vast collection of the best flashlights money can buy. Better yet, we even broke them down by category to streamline the selection process. Shine the spotlight Cheap NHL Jerseys below.. It doesn't have the luxury of the BMW or the space, toys (not even an RDS stereo and it's over 10 years since I played a cassette!) but it's actually really good fun. I would happily argue it's a better car for my 10 mile commute each day than the BMW was.I miss my 5 series on long journeys and I liked my leather and toys, but for cheap fun I don't think you can go wrong with a Ford Ka as a commuter\/run about car. I will certainly be keeping it for a while longer yet and none of the stress about people wanting to pinch\/damage a nice shiny BMW.\nHe also owned Goldman stock, something that was prohibited by Federal Reserve conflict of interest regulations. Although it was not publicly disclosed at the time, Friedman received a waiver from the Fed's conflict of interest rules in late 2008. Unbeknownst to the Fed, Friedman continued to purchase shares in Goldman from November 2008 through January of 2009, according to the GAO..\nWill It Work?Around the ValleyContent for a CauseSinclair CaresEvery Pet Deserves a HomePartner SpotlightDUBLIN (AP) European budget carrier Ryanair is offering business class tickets in an attempt to woo companies and governments during penny pinching times.Marketed under the slogan \"Your boss will approve,\" the new ticket reverses some of the airline's more reviled policies for fee dazzled travelers. The Dublin based company, long Europe's fastest growing airline with a sell it cheap, stack 'em high philosophy, says it hopes to capture three fourths of all business travel between Britain and Ireland, its two biggest markets.The move reflects not just the airline's desire to leverage its huge presence https:\/\/www.wholesalejerseysonlinesupply.com\/ in Europe, but also a growing interest for low cost executive travel. Most European governments are still cutting down on debt while many companies remain wary of spending as the eurozone recovery has stalled.\nThe Parks officer at the meeting tried to downplay a statistic, but it is very real. He cited that the parks system in Davidson county sees on average 5 6 \"incidents\" every week. Adding another park will increase that statistic, and crime in the Nashboro area by promoting added traffic to the area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When it comes to that one amazing day of your life there is a lot that comes to mind when considering choosing a wedding gown. It can be such a rush before you even start planning all the details like what style of dress, shoes and where you want the reception to be afterward. Everything from your hair color, body shape and your height are important when making the final decision on that gown.\nResearch does count since there are so many different choices you have. When you are out looking for what you like make sure to take your camera to snap a couple pictures. Some gowns that look exactly the same can many times be found somewhere else at a much cheaper price.\nYour height and body type can affect what looks good in the pictures or videos you take so keep this in mind when choosing a wedding gown. The pictures that you take of the dress can give you a good idea how it looks afterward. Each type of body type can be advantageous or have disadvantages to what they can wear. Some taller women may be more appealing in shorter dresses while shorter people in longer gowns.\nYour shoes are a very important part of your overall image that you want to have. The length cannot be too long if you want to show them off, however they must match in style and color. Each part no matter how small has an impact on the image you want to portray, so attention to details is important.\nIt can be a good idea to take a friend with you when you go shopping for a second opinion. Setting a good budget is important so you know what amount you can expect to spend while sticking to whatever you can afford. Each time you try on a dress those second pairs of eyes are a good help to point out something you may not see right away. Take your time and use some patience while keeping an open mind, the dress you set out on might not be what you end up liking.\nThere is a lot that goes in to finding, buying and wearing that gown down the aisle on your big day. Everything from your height, body type and shoes you wear create your everlasting image. Making that big impression when the time comes is a unforgettable moment, so be patient and do some research when choosing a wedding gown and you will find that gown you have always dreamed of.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Vatican's vision of tourism is based on Christian humanism, integral humanism and personalism. At the very core of the vision are human beings, who are not only the creators of tourism but also its ultimate goal, and for this reason all values of the natural world should be unconditionally subordinated to the good that are people. Consequently, the fundamental goal of tourism is to satisfy the needs of exploration, relaxation, entertainment and, first and foremost, the spiritual and intellectual needs of every person. Tourism has to help build a sense of community. It should also contribute to the elimination of poverty and foster the sense of responsibility for the environment. Tourism must not be exclusive to the rich and it should never be a time of depravity, promiscuity and, consequently, the degradation of people. Tourism should become an opportunity for all people to discover their contemplative dimension, giving them a chance to see God in nature and first of all, in other people.\nGa\u0142\u0105zka, G. (2002). Na szczyty z Janem Pawiem II\/Climbing to the Summits with John Paul II\/. Marki.\nJohn Paul II (1987) Encyclical Sollicitudo rei socialis (on the 20th anniversary of the Populorum progressio encyclical).\nJohn Paul II (1989). Message for the World Day of Peace. L'Osservatore Romano, issue no 12 bis.\nJohn Paul II (2000a). Homily for the Jubilee of Farmers.\nKrawczyk, Z. (2007). O turystyce i rekreacji. Studia i szkice. \/On Tourism and Recreation. Studies and Essays\/Warsaw: WSE.\nOstrowski, M. (1997). Turystyka w my\u015bli Jana Paw\u0142a II. \/Tourism in the Thought of John Paul II\/ Peregrinus Cracoviensis, issue no 5.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Display permalink anchor for current post.\nThe permalink mode title will use the post title for the 'a' element 'id' attribute. The id mode uses 'post-' with the post ID for the 'id' attribute.\nstring\t$mode: Permalink mode can be either 'title', 'id', or default, which is 'id'.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DRACO is Gatorbox's primary anchor, writer, producer, and video editor. Basically, he's the guy in control of the show. He's hosted nearly every broadcast since the channel's official launch as \"Gatorbox\" in 2013. Draco is a theatrical and stage actor with performing experience ranging from stand-up comedy to music & variety (and even dance, but he won't admit to that). He is also an avid speedrunner and formerly held the world record for the SNES port of Super Off Road.\nOutside of Gatorbox, Draco enjoys reading sci-fi novels, building radio-controlled cars, and watching monster movies from the 80's. He is not a brony as the photo to the left implies.\nFPS REFLEX, better known as Dracokon's oft-referenced but rarely seen brother, joined the Gatorbox cast in early 2015 as a guest host. If he's on the air there's probably a truck of some sort not far behind as the automotive business is Reflex's forte.\nWhen he's not co-hosting Gatorbox, Reflex is out travelling the United States as track crew for Monster Jam and as a part of the official pit crew of the Outlaw truck racing team as the driver of both Thunder Chicken and El Oso Loco. Once upon a time he was also the co-creator and manager of a popular YouTube project called the \"International Halo: Reach Monster Jam League\".\nTUNA BOAT TONY is a huge fan of wresting and obscure media of all types. He's also a huge fan of Resident Evil, which is the game he made his Gatorbox debut with in 2017. But nobody who's a fan of the channel knows him for that, no way, you all know him as the co-pilot of the five week long Angels With Scaly Wings stream where he and Draco unsuccessfully tried to get lucky with a dragon. That is his legacy, and if you give him crap for it he'll pummel you into submission with his signature wrestling move, The Demolition Man.\nROCKYOWITZ is the man behind all of the heavy rock and metal you hear during Gatorbox's videos. A former guitarist for the band Off The Zodiac, Rockyowitz has since gone on to contribute music to the popular weekly podcast The Comedy Button and compose music with IGN's Brian Altano. He's also spearheaded a number of individual projects and EP's and has made it his New Year's Resolution every year to release even more music.\nYou can check him out on Soundcloud or Bandcamp.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"www.quest.com - This video demonstrates how business content owners can access reports using the web-based Information portal built into Site Administrator for SharePoint, Quest Software's solution for SharePoint site administration and usage reporting.\nDidn't find what you were looking for? Find more on Site Administrator for SharePoint - Information Portal Reports for Business Content Owners Demo Or get search suggestion and latest updates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"www.mastersong.de - Michael Holzach \"Deutschland umsonst. Zu Fuss und ohne Geld durch ein Wohlstandsland.\"\nMichael Holzach \"Deutschland umsonst. Zu Fuss und ohne Geld durch ein Wohlstandsland.\"\nA young journalist packs his backpack, says farewell to his girl and gets himself a dog from the neighboring pet asylum, then walks off Hamburg on his feet. He's commited to walk the country, from North to South, and god only knows when he's going to return. He is further commited to do it without money nor credit cards, just begging for some food for the dog and a place to sleep in the farmers barns. As a temporary Drop out and experienced journalist, he keeps his diary and fills in all those tiny details of such voyage - the friendly as well the unfriendly encounters, the happy walks in the sun as well as the mood swings after a few days walking in the pouring rain. And he's doing well in writing, every page is a joy to the reader. Somehow, this book recalled a lot of my own experiences on the road. Now I know, there is this \"Japanese type\" of travelling - tabi - in Germany too. And that there is a German saying for \"Once on the road, always on the road\", too.\nFor the English speakers - sorry, it is available in German only.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Here at Sure Betbuilder S, we'll be ready to satisfy all of your expectations when it comes to Builder in Lake George, NY. Our workforce of well trained experts will offer the support that you need with the most innovative solutions in the market. We use premium quality products and cash saving solutions to ensure that you will enjoy the most effective services at the greatest value. Give us a call at 888-459-0005 to get started.\nYou have a budget to adhere to, and you intend to reduce costs. You will still require superior quality work with Builder in Lake George, NY, and you're able to put your trust in us to save you money while continuing with giving the best quality services. We guarantee that our cash saving efforts don't mean a lower standard of quality work. If you work with our staff, you will receive the advantages of our knowledge and excellent supplies to ensure your project will last even while saving time and funds. We will achieve this by providing the very best savings around and avoiding expensive errors. Save time and resources by simply calling Sure Betbuilder S today. Call 888-459-0005 to talk to our client care reps, right now.\nBuilder are available in Lake George, NY.\nOn the subject of Builder in Lake George, NY, you should be well informed to make the best decisions. We will never let you make unwise choices, since we understand exactly what we're undertaking, and we ensure you know very well what to look forward to with the venture. This is exactly why we try to make every attempt to make sure that you understand the procedure and aren't confronted by any kind of unexpected situations. Start by contacting 888-459-0005 to discuss your work. With this call, you're going to get your concerns answered, and we will establish a time to get started with services. Our company can show up at the arranged time with the required resources, and will work with you throughout the project.\nWhen you're preparing a venture regarding Builder in Lake George, NY, there are many great reasons to choose Sure Betbuilder S. We've got the best customer care scores, the very best quality products, and the most effective and powerful money saving solutions. We are aware of your needs and ambitions, and we're here to assist you with our expertise. Dial 888-459-0005 whenever you need Builder in Lake George, and we're going to work together with you to systematically complete your task.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No joke.. This is probably my family's favorite meal! Plus it's loaded with vegetables \ud83d\ude42 I actually have 2 variations of this recipe. My variety and the other!\nDice the peppers and onions. Heat olive oil in a large skillet and add diced peppers and onions and cook until tender. Remove from the skillet and set aside.\nBrown your ground meat in the skillet until almost done. Add tomatoes to meat and finish cooking through. Add cooked onions and peppers to meat mixture, along with southwest seasoning and 1\/4 cup of water. Simmer for 10 minutes.\nSpray the bottom of a 9\u00d713 pan with cooking spray. Line the bottom of the pan with 3 tortillas. Spread 1\/2 of the meat and vegetable mixture over the tortillas.\nLayer the remaining 3 tortillas on top of the meat mixture. Top with remaining meat and vegetable mixture.\nSprinkle 6 blue containers worth of shredded cheese evenly over the top.\nBake at 375 for 30 minutes, or until heated through and cheese is hot and bubbly.\nI love this recipe because it combines my two of my favorite foods- Lasagna and Mexican (tacos\/enchiladas). It provides a great amount of spice that my hubby enjoys. But it provides the layering like lasagna. It's also loaded with vegetables, that you can't even taste- they blend right in! But the best part is, there isn't a lot of extra fluff. It's the basics. Whole foods. So yummy!!! Let me know in the comments below what you think!!\nBrown your ground meat in the skillet until almost done. Add chiles to meat and finish cooking through. Add cooked onions and peppers to meat mixture, along with southwest seasoning and 1\/4 cup of water. Simmer for 10 minutes.\nSpray the bottom of a 9\u00d713 pan with cooking spray. Line the bottom of the pan with 6 tortillas. Spread with ONE RED CONTAINER of the refried beans. Spread 1\/2 of the meat and vegetable mixture over the tortillas and beans.\nLayer the remaining 6 tortillas on top of the meat mixture. Repeat spreading the refried beans and remaining meat and vegetable mixture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jul 04. till Jul 04.\nThis round table contributed to our current research project looking at developing a strategy for survivorship and care post bone marrow transplant. Stem cell transplants in the UK have increased steadily in the past decade but the new commissioning arrangements do not specify fully how patients will be cared for post transplant. We will review post-transplant survival and care and how this is planned for and managed in the UK today.\nThis project also forms part of our 'Forgotten Condition' workstream.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The DC Commission on the Arts and Humanities (DCCAH) is proud to announce the 2015 Artist Fellowship Program (AFP) Exhibition. This exhibition presents some of the District's finest visual artists, who each were awarded a FY15 Artist Fellowship Program (AFP) grant from the DC Commission on the Arts and Humanities. The exhibition and its programs are free and open to the public.\nThe 24 AFP artists will each exhibit a small body of work that illuminates their unique artistic perspective. The collection of these artworks within the I Street Galleries underscores the importance of establishing the first District government-operated public gallery, and captures the broad scope of the dynamic, Washington, DC art scene. The gallery provides an opportunity for artists to share their work with the public on a daily basis, especially with those that may not visit museums or galleries regularly. While on display, the work will receive exposure from residents, art patrons and the press. Many works are available for purchase directly from the artists.\n\"Our grantees are integral to improving the quality of life in Washington, DC, and the Commission is pleased to support them,\" said Edmund C. Fleet, Chair of the DC Commission on the Arts and Humanities. \"Funding is vital, and our support allows these artists to produce quality art and programs for District residents and visitors.\"\n\"Promoting excellence is a key part of our mission,\" said Lionell Thomas, Executive Director of the DC Commission on the Arts and Humanities. \"The AFP grant program and this exhibition are prime examples of how the Commission supports diverse artists and artistic disciplines.\"\nThe AFP offers grants of up to $10,000 to individual artists who make a significant contribution to the arts and who strive to promote the arts in the District of Columbia.\nExhibiting Artists include: Sondra Arkin, Jessica Beels, Anne Bouie, Adam Davies, Anna U. Davis, Christopher Dolan, Nekisha Durrett, Mary Early, Cheryl Edwards, Rik Freeman, Rania Hassan, Ian Jehle, Timothy Johnson, Rachel Kerwin, Gediyon Kifle, Nate Lewis, Alex Mayer, Kathryn McDonnell, EJ Montgomery, Mike Osborne, Marta Perez Garcia, Carmen Torruella-Quander, Joyce Wellman and Martine Workman.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Galina Chistova reposted . to and I like to sew.\nGalina Chistova likes . by Tatyana Shilina.\nGalina Chistova reposted \u0421\u043e\u0431\u0438\u0440\u0430\u0435\u043c \u0432\u043e\u043b\u043e\u0441\u044b \u0445\u0432\u043e\u0441\u0442\u0438\u043a\u043e\u043c \u0431\u0435\u0437 \u0437\u0430\u043a\u043e\u043b\u043e\u043a \u0438 \u0440\u0435\u0437\u0438\u043d\u043e\u043a \u0421\u043e\u0445\u0440\u0430\u043d\u0438\u0442\u0435 \u0441\u0435\u0431\u0435 to Hairdresses.\nGalina Chistova reposted PROSTA RECEPTION FOR RELAXATION of MUSCLES of the NECK. How quickly to remove stress in a neck and shoulders to health, traditional medicine.\nGalina Chistova reposted Medical yoga for a neck, the head and shoulders. A yoga set of exercises for cervical department of a backbone - YouTube to health, traditional medicine.\nGalina Chistova started following Alla Kramarenko.\nGalina Chistova started following Tatyana Shilina.\nGalina Chistova likes \u0421\u0442\u0438\u043b\u044c \u0411\u041e\u0425\u041e-\u0428\u0418\u041a - \u043c\u043e\u0442\u0438\u0432 \u0441\u0432\u043e\u0431\u043e\u0434\u044b \u0442\u0432\u043e\u0435\u0439 \u0434\u0443\u0448\u0438 by Tatyana Shilina.\nGalina Chistova likes \u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043f\u043e\u0445\u0443\u0434\u0435\u043d\u0438\u044f by Nadejda Shpak.\nGalina Chistova reposted 1 spoon of a medicine and you will trample: one national way of disposal of a lock to health.\nGalina Chistova reposted Note: read before it is removed! The doctor shows how to get rid of a fat liver for several days! You have to know it! to health.\nGalina Chistova reposted Infusion of youth, beauty and health.\\u000a\\u000aSurprisingly effective and very simple recipe in preparation consists of 3 known herbs checked for centuries: calendula, red clover and hawthorn.\\u000a\\u000aRegular treatment by this means improves a metabolism, clears an organism of limy and fatty deposits, relieves of headaches and dizzinesses, strengthens a cardiac muscle and stabilizes a blood pressure. The vitality raises, the body rejuvenates and in... to health.\nGalina Chistova reposted If hands grow dumb. Points on a neck. Visceral Therapy. Rinat Usmanov. to health.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwiet b/data_all_eng_slimpj/shuffled/split2/finalzzwiet
new file mode 100644
index 0000000000000000000000000000000000000000..2fdd7acaf8326aeedb81b28a055efc328947ad2b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Knitted in super soft Scottish lambswool, the rabbits hop, skip and jump across this cushion.\nEverybody loves bunny rabbits this design is both nostalgic and contemporary.\nEvery second rabbit on the front of the cushion has a hand knitted curly mohair tail.\nThe cushion has an envelope opening on the reverse with contrasting rib detail.\nKnitted in bright orange with white rabbits with charcoal grey rib on the reverse.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"I highly doubt that!\" Damien shot back. Luna winced. As much as she thought her brother was a jerk he was right. She had only been here to spy.\nRosalynn felt defeat, but her eyes still flamed with anger. She looked over at Luna tucked a strand behind her ear.\n\"Let's go Luna! The cute guys are awaiting!\"\nShe said smiling and waiting for Luna to come.\nLuna nodded softly and followed. Damien caught her arm. \"Luna,\" he whispered, \"don't become a follower to someone. It's embarrassing.\" Luna shook him off and started walking after Rosalynn.\nWhen they were far away from Damien to hear she looked at Luna. Her eyes looked almost, sad?\n\"You.. don't have to come if you don't want to.\"\nShe mumbled, her voice sweet and understanding.\nLuna shook her head. \"That's okay. I want to go. It seems like fun.\"\nRosalynn nodded and sighed, but felt something watching her. She turned around, not seeing anyone. She kept walking and looked straight ahead. She shook her head as they got closer to the river, she heard breathing on her neck.\nLuna looked at Rosalynn with curious eyes. \"What?\" she asked.\nRosalynn looked confused and shook her head.\n\"N-nothing, I just felt like someone was following me.\"\nShe mumbled quietly, picking up her pace.\n\"Oh. Okay.\" Luna followed her then whirled around and grabbed the stalker by the throat.\nRosalynn spun around. She was stunned and stepped back. She narrowed her eyes and looked at the stalker.\nIt was Damien. \"What are you doing here!\" Luna yelled. \"i can't look out for my sister?\" he matched her tone.\nRosalynn sighed and felt an empty pit in her stomach. Her face was hurt and so was her eyes. She looked deep into his eyes and searched them.\nShe snapped, her tone as mean as hers got.\n\"I don't trust Aphrodite. Thus I don't trust her daughters and sons.\" Damien glared at her. Luna looked at him with pleading eyes to shut up.\nRosalynn matched his glare with a meaner, more heart felt one. She walked scloser to him, her body almost rubbing against his.\n\"Why? What did we ever do to you?\"\nShe chalenged him, her eyes squinted and burning into his.\n\"Your mother was the reason Troy fell! She promised the most beautiful woman in the world so Paris would choose her as the most beautiful goddess. I mean seriously. She's a selfish prep.\"\nRosalynn eyes studdied him, her eyes dark, and she had swallowed any tear that dared to show itself. \"What makes you think I'm like that?\" She asked, her voice more hurt this time.\n\"The fact that you're trying to get me to notice you show's you are like her!\" Damien said. He looked at Luna's pleading face. \"You might have not studied the history of Troy but I sure did.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Atharv invites School Going Children!\nTalk about indigenous and skilful competition conducted in Atharv, the one that always stands out is Pi quiz. Unlike all other events PI-quiz is specially custom made circulating the needs of school students. It is a competition which tests the logical, reasoning and the analytical abilities of a student to their earnest potential. In a crux it is a competition of deep etched irrationality.\nPi-quiz is a pan India competition which aimed at students from different schools all over India. It includes two rounds, the former a written test and the latter puts the top 30 in the fray with leading minds of the country, in our flagship event Game of Shadows. The perks for this includes hands on interaction with professors and top minds of IIM. Also prizes worth 35,000, other merchandise and hampers is up for the grab. Last but not the least, it gives the budding minds of our country a direct access to the wardrobe of the future leading minds of the country. Thus it enunciates them to take a leading role in the utmost all aspects of their life.\nThe registration for this might be through partner schools or through open registration, for the lone wolfs who want to take a hand in this challenging event. And for those, the online portals will be out soon in the website. So stay updated!\nPrevious Previous post: Listen up Campus Ambassadors!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(Waltons have featured my blog several times on their site).\nEvery September we have an open day at Homeacres, Charles' no dig garden. Admission is free and we offer refreshments for donations, which go to charity.\nA few days before, some of the compost heap bays are cleared out to make a 'book shop'. Here, Charles' brother Oliver is setting up his apple juice and cider stall.\nAnd the House of Hope orphanage in Doi Saket, Northern Thailand \u2013 my choice. Here are the children with some of their regular teachers, volunteers and Sarah (on the left, blue t shirt) who came to Thailand to work in Bangkok over 50 years ago for 2 years and loved it so much she has remained ever since. She has been very instrumental in organising all of the non-Thai residents in the area to help local projects.\nNotice the open sides to the classroom!\nOn Saturday I spent the whole day baking. It was torture for my sons, who could smell all of the delicious baking yet couldn't come and tuck in (they did get some when I was finished). We also had some gorgeous baked donations from our visitors and fantastic help from friends who served the refreshments all day.\nA few weeks before, Waltons had kindly sent me two beautiful scarecrows. I knew they were just the thing to help us let people know who we were raising money for, and to look after the donations bucket.\nAlthough it was a rather wet day, we had loads of visitors and a great time. The compost bays make fantastic shelters!\nThe scarecrows are lovely, really well made, so cheerful to look and very popular with our young visitors. I knew that after helping us raise money for children in Thailand and Africa, it would be great if they continue to be enjoyed by children. So once term started, I took the scarecrows to their new home \u2013 the local primary school in Bruton. The children have a gardening club there, so this jolly pair will help inspire a new generation of gardeners!\nWe raised \u00a3450 \u2013 so \u00a3225 for each charity. I changed mine into Thai Baht \u2013 with the addition of a few pounds, that came to 10,000 baht for the orphanage.\nI got to know about the orphans and meet them via my dad, who along with Sarah and other members of the community help to support the children. As well as raising money for their education and welfare, Dad arranges events at his house for the children and also during the holiday the older ones are given paid work in his garden. Whilst I was visiting Dad tasks included rescuing fish from a pond with a broken filter and gardening. All of the work is overseen by Arnold, who has amazing skills for just about everything and is a great mentor for the children.\nThey also regularly visit Dad's pool for swimming lessons and fun \u2013 and enjoy arts and crafts activities.\nA big favourite is the Christmas party, again at dad's house. Arnold is Santa!\nWhilst I was visiting, Sarah, Lee and myself helped to make Christmas cards using hand cut motifs made by the children. I especially loved the festive bugs. With the terrible decline of insects, it was lovely to see how appreciated they are by these children, so much so they want them on their festive cards.\nMuch of the money raised helps to pay for their education. Most of the children will go into different kinds of work after school, but some are able to go to university too.\nWhen I arrived at the orphanage, some of the children were practising their dance routines for the Christmas party, using you tube. After chatting for a while, the children sang us some songs, as a thank you. The dogs also wanted to be part of things.\nSome of the older children were taught how to make key rings and fridge magnets by a Japanese volunteer, to sell to raise funds for the orphanage. These children are certainly enterprising. I've got some to bring home; I'm sure they'll be very popular.\nSome of the children waving goodbye (that's my dad, heading back to his car). Visiting here really helps to put things in perspective. The buildings are basic, a little shabby and in places needing repair (this is being sorted now) yet the children are so upbeat, welcoming and supportive of each other.\nNow I am finishing my last chilled Chang beer of this trip to Thailand, before getting the taxi to the airport for my long trip home.\nGlad I clicked and read this. In the bit that came through on the email it was just random letters \"mmmm mmmmm cccc ccccc nnnnn\" etc but this a great article.\nYes, I was writing it Thailand on an iPad and accidentally pressed 'publish' rather than 'save'. On the iPad I always load the photos first and then write the words, it seems to work better when I'm working with slower internet as I was there. The random letters just help me space it all out. You must have thought I'd finally flipped!!\nThey really are lovely group of children, amazing really considering most of them come from very sad circumstances, and it is great knowing that the money goes directly to helping the orphanage.\nThought maybe you'd been smoking something Thai??? \ud83d\ude42 \ud83d\ude42 hehehe!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nigerian Oil & Gas Servicing Industry Estimated to Be 2.5% of Global Market \u2013 Agusto & Co.\nIs Nigeria going the way of iran?\nBreaking News: HND\/BSc Dichotomy Removed.\nAdvice to my younger self By Preeti Chinda.\nCOMPOSITES: THE ENGINEERING OF THE PAST, THE PRESENT AND THE FUTURE BY ATHAN OPARAJI!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwqtk b/data_all_eng_slimpj/shuffled/split2/finalzzwqtk
new file mode 100644
index 0000000000000000000000000000000000000000..2f9da79291a85368497ac7d358f17c80204766f5
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@@ -0,0 +1,5 @@
+{"text":"Previous Articlethe Zlatne Uste circles go 'round and 'round!\nI stumbled upon both of these videos within minutes of each other and was struck by the stark differences between the two. Pop-out each of them and play them side-by-side if you can.\nThe first is of Staten Island's Greenbelt, a nature trail in the middle of NYC's most outer-borough; the second is of the city of Homs, which prior to the Syrian War had a population of nearly 700k \u2013 also to compare Homs is just under 19 sq. miles and the Greenbelt is just under 5 (it's 3000 acres).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jack Withey has just celebrated his first year as an associate chiropractor for Cleve Chiropractic after leaving with distinctions at university. The accolades just keep on coming for Jack, as he has been nominated, as one of only 3 people, for national chiropractic associate of the year, by the British Chiropractic Association. The awards ceremony will be in October, so I look forward to letting everybody know how he gets on. To be nominated in competition with thousands is a huge honour and well deserved. Jack was nominated for excellence for chiropractic service. He spotted a catastrophic injury to a patient who had a fall 6 weeks prior. After seeing all the relevant health professionals, the patient came in to see us on the bus that day. Jack spotted the problem, took an X-ray and he then called an ambulance to take them straight to A and E with their X-rays on CD, where they were cared for appropriately.\nIt would also be appropriate to talk about our other supremo chiropractic associate Michael (I'm not a basket baller) Jordan, who recently represented his country working as a chiropractor in the GB Masters Athletics Championships in Birmingham. Next year he has been selected to work at the World Masters Championships in Poland.\nIf that's not enough then also Keith Graham (Head of medical for Bristol Rovers and our head Physio) has recently completed his prescribing training. Keith used to work in Rheumatology within the NHS and has huge experience with injections. Now however he can provide the full service for private patients. He can prescribe medications where appropriate and administer any necessary injections. This month has seen a number of sub acromial (shoulder) injections that can provide huge pain relief very quickly. If you think you may need this service, then please enquire with reception to see if it appropriate. If it is Keith can take care of it easily and quickly for you.\nAs healthcare professionals at Cleve Chiropractic and Physiotherapy Clinic, we are all very conscious of the environment. After constantly being bombarded with anti-plastic media I started to notice how many plastic bottles were in my shower at home. Admittedly there are two of us who use that shower but 24 plastic bottles?! Wow, who knew? They build up you know. There are the day to day ones, the blonde ones, the face ones, the special exfoliator ones, the ones that came with you from the hotel and of course the special ones that your mum buys for you at Christmas that you don't use because they are too expensive to waste. Anyway, the point is that 24 is too many. So I started to look up alternatives. Now there are plenty of nice organic bars of body soap that you can buy so that's easy, but I wondered what people used to do to wash their hair in the days before shampoo bottles? The answer is Shampoo Bars! I ordered a nice organic shampoo bar from Amazon and got started! They are great, they leave your hair with a slight squeaky cleanness, I admit but surely it's better that buying tons of plastic? And they are cheap too, as they last a long time. So now I am working my way though the left overs and slowly reducing the plastic in my shower, once and for all! 22 and counting.\nFor all of those who have been with me from the beginning then you will remember the journey began with Blog 1 \u2013 A Babymoon to Cuba, when Jen was pregnant with Molly. Well now, over 4 years on and she has started Primary school in reception class. It's frightening how quickly time passes but I am pleased to say that she is settling well and making lots of new friends. I have to admit to my heart breaking a little bit as I watched her skip off and turn to smile and wave at me just before she went in this morning. She is such a little treasure. Now we have a couple more years of Harry at home. I hope that time doesn't pass as quickly!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FXStreet (\u0141\u00f3d\u017a) - The RBS team of analysts believe that the most important takeaway from the BoE August Inflation Report was the fact that the MPC lowered the inflation forecast at 2-year point (1.77% from 1.89%) and barely altered at 3-year point (1.96% vs 1.95%).\n\"So a slightly larger undershoot over the policy-sensitive 2-3 year forecast period as a whole, but an essentially unchanged\/at-target forecast at 3-years.\"\n\"There was no change to the (neutral) skew of the forecast.\"\n\"Overall, this gives the Report a marginally dovish feel. \"\n\"At any rate, had the MPC been materially closer to raising Bank Rate then we would have expected a higher CPI projection.\"\n\"A high degree of uncertainty surrounds the timing of the first hike but we feel marginally more confident about our forecast for the first Bank Rate rise to come in February 2015 than we had done prior to the August Report.\"\nthe Euro accelerates its gains versus the US Dollar as the pair reacted positive after the weaker than expected US retail sales data.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Born Doris Mary Ann Kappelhoff in Cincinnati, Ohio, on April 3, 1922, Doris Day started her career as a singer and acted in Hollywood movie musicals, My Dream is Yours (1949), Tea for Two (1950), Love Me or Leave Me (1954) about singer Ruth Etting, The Pajama Game (1957) and the musical western Calamity Jane (1953). She co-starred in romantic comedies with Frank Sinatra Young at Heart (1954), with Rock Hudson Pillow Talk (1959), Lover Come Back (1961) and Send Me No Flowers (1964), with David Niven Please Don't Eat the Daisies (1960), with Cary Grant That Touch of Mink (1962), with James Garner The Thrill of It All (1963). She acted with James Stewart in the thriller The Man Who Knew Too Much (1956) by Alfred Hitchcock.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welsh tenor Wynne Evans, better known as Gio Compario from the famous insurance comparison site adverts, tells us about his mid-week short break here at Bluestone this week. Yesterday he had even more fun at the Blue Lagoon waterpark before an action-packed activity at the Steep Ravine that was more fun than an action-packed family movie.\n\"The kids very kindly got me up at 6:30am yesterday \u2013 one holding my trunks and the other my goggles. So at 9am when the pool opened we were one of the first in. Being a 'local' West Walian boy, people who come into swim for the day from West Wales know me without the moustache.\n\"So I get quite a few people singing 'GoCompare' quietly just after I've passed them in the lazy river\u2026 I never know quite what to do, sometimes I smile like it's the first time I've ever heard that and then sometimes I just pretend that I've got water in my ear and I'm temporarily deaf.\n\"Today in my trunks I was reunited with a school friend who I hadn't seen for 20 years, and my son and daughter bumped into a teacher from their school\u2026shocking\u2026Miss in a swimsuit!\n\"After a quick lunch we headed off to Woodland Warriors. Basically they gave 20 of us laser guns and sent us into the forest in two teams setting us different tasks against each other. WARNING\u2026.IF YOU ARE THE SLIGHTEST BIT COMPETITIVE DON'T DO THIS ACTIVITY. I turned into Rambo. I even tried to make my team camouflage up and we sniped through the forest, laser shooting anything in our paths. It was like being a real action man (even pretended to have eagle eyes). I loved it more than the kids.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzymaz b/data_all_eng_slimpj/shuffled/split2/finalzzymaz
new file mode 100644
index 0000000000000000000000000000000000000000..37f15c935f59366f7deb14d6278c837f6c61fa24
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzymaz
@@ -0,0 +1,5 @@
+{"text":"When you are looking for plumbing solutions in the Knoxville area, you should consider Knoxville Plumbing Services, because we only hire the best local tradesmen that are able to offer a top level of service.\nBecause we are a local company, we take great pride in the service we provide, which is friendly, competitively priced and above all professional. We take our role within the local community seriously, and one of the services that we provide is plumbing in halls.\nOne of the main reasons for our recent success is the fact that we try our utmost to keep all of the services that we provide priced at a rate that is very competitive. We know that plumbing services are vital to the smooth running of both homes and businesses, but we also understand that it can be expensive. However, we know that you should not have to put up with poor quality plumbing services just because it is low cost.\nWe always strive to deliver top quality plumbing solutions for a price that gives you the best value for money anywhere in the Knoxville area. It is unfortunate that a lot of plumbing companies that offer cheap plumbing services do not always offer the best quality of plumbing, something that can cost you money in the long term. When you hire Knoxville Plumbing services for plumbing in halls or any other plumbing requirement, you can rest assured that you are getting top quality plumbing services at a great price.\nJust because you have an urgent plumbing problem does not mean that you should have to wait until office hours for a solution. If you have a plumbing emergency that requires immediate attention, a simple call to Knoxville Plumbing Services is all you need to do, and we aim to reach you within an hour of receiving your initial call.\nBecause we are local to you, we are able to respond more rapidly to your emergency plumbing problem, and we will always send a fully qualified and expert plumber to your address to assess the problem.\nIt is because we are local and situated within the Knoxville community that we always take the levels of service that we provide seriously. We always endeavor to provide the fastest and most capable plumbing services in the local area, for any plumbing problem or project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sprekelia, Jacobean Lily, Aztec Lily, red flowering bulbs. An unusual flowering bulb that flowers in late spring. The Jacobean lily does produce such unusual flamboyant red flowers and large long striking anthers with contrasting yellow stamens, that certainly makes this plant a real show off. The flowers keep their shape well for weeks while in flower and may spot flower other times of the year depending on your climate. The Aztec lily grows well in pots as well as in the garden. For sale is one flowering size bulb.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Has your home gotten away from you? It has happened to all of us at one time or another. We understand because we have been where you're at right now. In order to begin feeling safe and healthy again, you are taking a big step in your life and we want to congratulate you on your courage.\nOur specialty cleaning crews have gone through extensive sensitivity training and are certified to handle complicated site cleanups. In many cases our female cleaners organize, label and carefully store collectibles and in some cases setup donations to your favorite charities. With years of experience and thousands of completed jobs, our highly trained crews quickly have handled jobs of all sizes.\nLet our experienced team quietly enter and restore your home to its original condition. We will discretely remove any and all remaining refuse and get you back to a safe living environment.\nCleaning up in a hoarding or untidy environment is a situation that requires specific talents and sensitivity that our company has a unique way of dealing with it. If you or someone you love is living in a hoarding or untidy space or maybe your collecting has gotten out of hand, dealing with it can sometimes be hard. All the items in the house are going to have to be dealt with, cleaned up, and possibly thrown out or stored. Being a hoarder is not easy, but cleaning up in a hoarding environment does not need to be as hard as you might imagine. Our cleaning service comes in and removes all the junk from a cluttered house before restoring it to its pre-cluttered condition. Cleaning an untidy or unsafe home involves finding where the problem areas are and cleaning them to make them usable again. A home filled with clutter needs to be dealt with properly and our service is here to make this as stress free as possible.\nWe can help you with your clutter and help return your home to a much more livable space which will undoubtedly make your life easier. You will be able to breathe easier when your home is cleaned and you don't have to worry about it. It is time to refocus your life and decluttering is the best way to do that. We will take care of everything even purifying the air in your living space. If you are feeling down and stressed on a daily basis, please don't be. You are taking a courageous step by hiring our cleaning service and now you will be able to get your life back in order. Call our service and you will be amazed at how much it helps. Congratulations! You should be very proud of yourself. We know how hard it can be.\nNo other cleaning company can even come close to our process of cleaning a hoarder or untidy environment. We have worked to perfect the most efficient and effective process for our customers.\nFirst, we will look at the situation as a whole and determine which parts need immediate attention and which parts could potentially wait a little while. We then create our plan of attack to make it as quick as possible. If there is anything you want to save, we will give you an iPad or camera and you can take pictures of all those collectible items and we will label and store them as we move through your home. Once those are identified, we do not stop until everything is spotless. Our policy is to complete every project as quickly as possible while doing the best job. While we work as hard as we can to maintain our efficiency and drive to do a good job, we pride ourselves on being caring and compassionate to families and individuals going through such a difficult time.\nOur number one priority is you! Cleaning up your property and decontamination is a large part of that, but unless you are comfortable, we aren't doing our job. If the house is not decontaminated, bacteria can grow and cause major issues. They can also cause persons who live near or around the area to become sick and on occasion, the illness can be quite serious. We make sure to decontaminate every job to prevent this problem. Our hoarder cleanup company is very discreet and your privacy is a big part of our job. It is no one's business why you hired us to come in, and in the end, you are the only one who needs to know why we were there.\nOur hoarder clean up company has years of experience and thousands of jobs under our belt meaning we know exactly what a hoarder job requires. We are the experts in hoarder cleanup and have helped thousands with their hoarder cleanup. Dealing with a relative or someone close to you being a hoarder is stressful enough, so leave the cleaning process to us. We will restore the property to what it was before hoarding and make it livable again. Though these times are often stressful, let experts clean up the mess.\nHoarding areas are often riddled with bacterium and occasionally rodents or other small animals, and cleaning it up only adds to that stress. As we have said, there is no job too big for our hoarder cleanup experts regardless of the size or mess. Having the property back to what it was like before hoarding will make you feel better and allow you to move on so don't hesitate to call our hoarder cleanup experts and get the job done quickly and correctly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The loss of American, however, is tempered by the fact that American has been earning only partial mileage for a couple of years now.In 1952, the CAB appointed Nelson David as president, and he began to improve the financial stability of the airline.A Boeing 727-200Adv on approach to Los Angeles International Airport, showing the new livery and logo introduced in the early 1970s.Alaska Airlines, thanks for being upfront as always with changes.Fortunately, commuting out of Alaska, most flights originating from Alaska or returning to Alaska are via Alaska Airlines.North American Airlines Handbook (3rd ed.). Sandpoint, ID: Airways International.Alaska had more competition in 1993 when low-cost airline Southwest Airlines entered the Pacific Northwest by purchasing Morris Air.\nHowever, many of these airlines are members of global airline alliances.If you book such a flight that you combine with an American Airlines marketed and operated international flight, you will earn miles for the whole thing.\nUntil December 31, 2017, Virgin America Elevate Gold members can access the LAX Alaska Lounge for free, as an inherited benefit from the closure of the Virgin America Loft in May of 2017.Alaska Airlines is not a member of any of the three major airline alliances.Find all the flight information that matters to you, when it matters.In 1941, Star Air Service was purchased by Raymond Marshall, a businessman from New York.In 1961, competitors began introducing jets on routes Alaska Airlines flew. After January 1, the domestic segments within international itineraries are not eligible for mileage accrual.On the other hand, some premium cabin award levels to Hawaii and within the U.S. will be reduced, providing better-than-ever value to members.\nFeel free to share your travel stories, comments and suggestions in the story comments.In your example, San Francisco-Chicago would not be eligible and Chicago-London Heathrow would be eligible.There was also a big seasonal imbalance in travel to Alaska, which mainly took place in the summer.Alaska Airlines was the launch customer for the MD-83 and operated many of these jets throughout the 1980s and 1990s.\nAlaska Airlines: Read about baggage fees, the lowest fares, user reviews and information all in one place at Airfarewatchdog. Earn mileage and other rewards from your Alaskan Airlines credit card.I quick question regarding earning miles while flying international.Find great deals on eBay for Alaska Airlines in Alaska Airlines Transportation.\nAlaska Airlines always provide safe, reliable transportation for a reasonable price, along with the caring, friendly and professional service that we are known for.\nJust flew RDU-LAX-PDX and had to go on American for the RDU-LAX segment.Save big on Alaska Airlines Tickets by booking online or on the phone.The Western-U.S.-based airline has one of the most valuable loyalty programs in the business.However, after Alaska Air declined to buy the airline in the fall of 1991, it intensified competition with Alaska.\nTelephone Ticket Sales: 1-907-891-7410 Box office general Inquiries: (907) 786- 1562.\nThere were too many airlines in Anchorage at the time, and not enough demand to support them.Official Website of Denver International Airport: Skip to main content. These are the airlines that service Denver International Airport.It also operates secondary hubs in Los Angeles, Portland, San Francisco, Anchorage, and focus cities at San Diego and San Jose.Horizon Air had offered Starbucks coffee since February 1, 1990 and was the first airline in the world to serve Starbucks coffee onboard its flights.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vores komfortstyr er let og kommer i sort finish i aluminium. Dette styr passer sammen med Bugatti kalechen.\nThe light weight flat handlebar is made in black finish and made of aluminium. It is not compatible with the Bugatti hood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaajqo b/data_all_eng_slimpj/shuffled/split2/finalzzzaajqo
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+{"text":"Act now for the lowest prices in the industry!\nSpandex is a versatile medium and we are the experts! We are a custom house and can create a custom commercial grade cover and look just for you in any size and any color! And right nowm we're offering the best promotion in the idustry with a 50% off one color logo on our best spandex table cover! Better yet, we offer 28 colors to match your brand or create a colorful addition to your event or booth. We ship directly from our warehouse in Dallas, Texas!\nDon't miss out on this incredible promotion!\nWe also carry spandex table belly bands!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to our large, comfy 2nd-story condo!\nThe living room at Ocean Village Club J24 leads directly out to the balcony, where you have a view of the resort and of big, blue Florida skies!\nWelcome to Ocean Village J24 !\nThe condo is newly remodeled, with hardwood floors and HDTV.\nYou'll feel right at home from the moment you arrive at Ocean Village Club J24. There's plenty of comfy seating here, lots of elbow room, and a pleasing sense of \"beach elegance\"!\nThe dining area has a festive island decor you'll love!\nThe glass-topped round table is perfect for dining, and also doubles as a game and card center. Break out the Uno and enjoy an evening of old-fashioned family fun.\nPlanning to dine out at St. Augustine's delectable restaurants? You'll appreciate the convenience of our fully-equipped kitchen for storing lots of tasty snacks and refreshing beverages!\nFor your convenience, our condo has a washer and dryer located just off of the kitchen. Perfect for active families or long-term stays.\nCool and spacious, the master suite has balcony access.\nFrom the queen-sized bed to the HDTV with DVD player to the sliders leading out to the balcony, the master bedroom is designed for your comfort and pleasure. And the bathroom is en-suite!\nYours and yours alone, the master bath is a treat!\nTake a refreshing shower in our full master bathroom. We include all the bath towels and bed linens you'll need at Ocean Village Club.\nThe master bath boasts a separate powder room.\nJust right for the two of you, the master bath is roomy and beautifully decorated. You'll love the soft lighting above the mirror, not to mention the convenience of a walk-in shower.\nKids and adults alike will like the guest bedroom!\nComfortable twin beds provide lots of sleeping options in our second bedroom.\nTake a luxurious hot bath, or enjoy a refreshing shower!\nYou'll love the convenience of having a second bathroom. No need to pack towels and bed linens. We've taken care of it!\nGrab a boogie board and head for the surf!\nFrom the dunes to the shore, enjoy shell hunts and walks along Florida beaches. Our vacation home is a short walk to St. Augustine Beach.\nBask in the warmth of the Florida sun while the kids build sandcastles at St. Augustine Beach!\nSoak up the rays on St. Augustine Beach!\nA Florida beach vacation is a beautiful thing!\nYou'll be right next to the beach when you stay with us!\nThe Atlantic is just yards away and historic downtown St. Augustine mere minutes by car when you stay at our luxe condo at Ocean Village Club J24! cont.\nThe Atlantic is just yards away and historic downtown St. Augustine mere minutes by car when you stay at our luxe condo at Ocean Village Club J24! From multiple HDTVs to hardwood floors, comfy beds to a breezy balcony, this is a fantastic vacation rental!\nOcean Village Club J24 is a second-story condo in a large, family-friendly community right on the shores of the beach in St. Augustine. Here you'll enjoy 2 swimming pools -- including one with an ocean view! -- tennis and shuffleboard courts, grassy lawns for picnicking and playing Frisbee, and a private boardwalk leading down to the white sands of the beach. The location is perfect, the sun rises magnificent, and the surroundings friendly. There's no better place to stay next time you're in northeastern Florida!\nAnd our condo is particularly well appointed, with new furnishings and many updates you'll enjoy. Six people can sleep in luxury here -- two in each of the bedrooms, plus another two on the sofa sleeper in the living room. (Bonus for those sofa sleeper folks: beautiful views of the sunrise and the huge Florida sky through the sliding-glass doors.) There's a long balcony covering the length of the living room and the master bedroom, with access from both. We even have a washer and dryer right in the condo for your use!\nYou'll love the HDTVs and all their accessories -- DVD players, VCRs, etc. The biggest is in the living room, where the whole gang can gather to watch a favorite movie or big game. The couch and loveseat are very comfy, inducing naps after a long day at the beach or in St. Augustine! And there's plenty of floor space, too, for kids to lay out their toys and projects while the grown-ups sip frosty drinks and recline in comfort. Open the sliding glass doors and extend the living area onto the balcony, which has casual seating for al fresco meals and wonderful balmy breezes that remind you how close the ocean is. You can hear the murmur of the waves out here, too!\nJust adjacent to the living room are the dining area and kitchen. The dining room table is a great place for games as well as meals and has a nice view through the sliding glass doors. And you can use the high counter between kitchen and dining area as a serving hatch or buffet table.\nSpeaking of the kitchen, you will love the well-stocked, gleaming set-up at Ocean Village Club J24. The cupboards are stocked with cookware, dishes, utensils, and all those necessary gadgets that come in handy when preparing and serving a homemade feast. Even if you plan to dine out a lot on this vacation, it's nice to have everything waiting for you in the kitchen should you bring home leftovers or order takeout.\nOcean Village Club J24 is spacious and light, with plenty of room for entertaining. You'll love the indoor-outdoor feel of the balcony and living area, which enables the group to stay together while everyone pursues different leisure activities.\nThe sleeping quarters in the condo are spacious as well, but wonderfully private and quiet. The master suite is the showplace, with a queen-sized bed, private HDTV with DVD player, and balcony access. This room has an en-suite bath with a walk-in shower and a separate powder room.\nThe second bedroom is ideal either for children, another couple traveling with you, or friends. Two twin beds occupy a large room with new carpeting. This bedroom shares a full bath (with both shower and tub option) with the living and dining areas of the condo.\nBest of all, Ocean Village Club J24 is located on a wide, family-friendly, uncrowded beach with gentle waves and plenty of open space for everything from pick-up volleyball games to sandcastle-building to sunbathing. And should you ever tire of the beach, plenty of history and beauty awaits in St. Augustine. This area is also a haven for golfers, nature enthusiasts, and sports lovers of all types. Come stay at Ocean Village Club J24 next time you're in the St. Augustine area!\nThe condo was very clean and spacious. The second bedroom with the twin beds needed a TV, but everything else was good.\nFirst time using your company and will not be the last. The whole process was easy, service was prompt and place was great. The App was phenomenal. Very detailed and found the ease of use was great from the condo information to the helpful tips about the area we were staying. I have used many services that do not hold a candle to what you provided. No complaints by us!\nGreat unit, great place, totally recommended!\nThis was my wife and my 5th time to St. Augustine in the past nine years.\nSt. Augustine has a small town charm, but with larger city amenities.\nThere is no shortage of things to do or see.\nThe old City , never disappoints.\nThe weather was great a usual.\nThe beach is second to none.\nOcean Village Club and it's grounds are immaculate.\nBoth pools are kept in pristine condition, with plenty of lounge chairs.\nAs a small caveat to the above, be aware that between mid March and mid April, it can be a bit busier due to the various spring breaks.\nThere is a nice, updated and well kept BBQ area.\nThere is an area with running water, where if you choose, you can use your own supplies and wash and detail your car.\nAs far as our Unit goes, it was very clean.\nIt is furnished with a nice sized apt, washer and dryer.\nThere is a dish washer that works great, but the soap dispensing door is broken.\nThis Unit has up to date T.V.'s.\nIf we had any small complaint, it would be that the couch and love seat are tired and need replaced.\nThe beds, we believe, should also be replaced.\nOverall, we would give this Unit an 8 out of 10.\nWe would gladly stay here again!\nWe love Ocean Village Club and will continue to come back again and again.\nCondos were better than they looked on Internet.\nOcean village club was quiet,tasteful and conveniently located to the beach and shops.Enjoyed Salt Life restaurant.Nice the pool was heated.Also good that pets are allowed in some units.\nI stayed at this condo from Sept. 7th to Sept. 30th. My stay was very pleasant. The condo was very nice and I enjoyed being able to cook when I wanted, room for my photography gear and space to edit my photos and easy access to the beach where I walked for a couple of hours every morning. I definitely will consider coming back. Just a couple of things that should be taken care of for the next renters. Closet doors need new screws as do the ones on the doors at the washer\/dryer. Get a new comforter for the master bed (there's a rip\/tear. Either buy new balcony furniture or sand & repaint the old and get new cushions. The balcony could also use a nice chair to just sit in while reading. Replace the toliet cleaning brushes & broom. Have the cleaning crew check the cleanliness of dishes when unloading the dishwasher.\nWe had a great stay at ocean village club. We were able to enjoy our vacation in a nice, clean comforting place that easily felt like home for our stay.\nVery clean, well maintained unit. Well equipped and comfortable and in great location. Management company is excellent! I would give it a perfect score but needs better furniture on the balcony.\nThe condo and its location was perfect for our summer vacation. I was unhappy with the un-personal feel of the VacationRentalPros rental process.\nThe condo was not ready as promised at 3pm, the maid was still cleaning the unit when we arrived. The unit was ready with 45-60 mins.\nThe location was excellent and the property had the ability to be excellent as well. I liked the location and the layout and overall it was ok. However, the verticals on the sliding glass door couldn't be opened unless you did it by hand, the door itself was hard to operate and the paint was peeling off the wall where other guests had the same trouble and were tugging at the door. The pots and pans weren't very clean and I had to wash some before I could use. Other's I would never use, they were just too dirty. These are located under the stove in the pull out drawer. The master bedroom closet can't be used as the smell coming from that closet is unbearable. The toilets require two flushes sometimes as well. The master bedroom shower stall needs to be recaulked as well. And the carpet needs to be cleaned. I would recommend the complex, but not neccesarily this unit.\nWe had a wonderful stay. Only suggestion would be to have some Beach Towels for guests.\nWe had a wonderful vacation stay. We loved the city of St. Augustine.\nThe only inconvenience I had was there was no internet access in the condo we rented. I had to go down to the clubhouse between 8am and 10pm for a connection. That's not VacationRentalPro's fault. My family loved everything else about the stay. We would recommend it to friends.\nAlthough the unit we reserved was unavailable upon check-in, we were given keys to 3 other condos to check and were able to visually and personally inspect them and choose one, which actually turned out to be better than the property originally booked. Despite their miscommunication with the owner of the condo we had booked we were very happy with the alternative and were not charged for the difference in price.\nWe did experience problems with building works on the first Tuesday at 7:00am with very loud noise. We called the office and later that day viewed a second condo and moved in directly, we moved from J16 to I26. I am sure that the office must have been aware of these ongoing works and we should have been allocated a different condo from the outset, we had to move (albeit only a short distance) from one home to another which lost us a couple of hours time. This apart we thoroughly enjoyed our stay at Ocean Village Club and would certainly return.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Move over, International Olympic Committee, no one even remotely cares about where the Summer 2024 Olympic Games are going to be held. Wow, it might be in Los Angeles? Whoop-dee-doo, Basil\u2026The real buzz on the street has to do with replacement lightning cable supplier and online marketplace, Amazon, Inc's (AMZN) active location scouting for their new headquarters.\nHQ2, using the official social nomenclature, is the new location Amazon (AMZN) intends to spend $5 billion in order to house nearly 50,000 new employees. Amazon (AMZN) officially announced its plans to create a second campus a little over a year ago, in hopes of creating new jobs in about 8 million square feet of buildings.\nAccording to sources close to the selection community, all 20 cities on the short list: Toronto, Columbus, Indianapolis, Chicago, Denver, Nashville, Los Angeles, Dallas, Austin, Boston, New York City, Newark, Pittsburgh, Philadelphia, Montgomery County in Maryland, Washington, D.C., Raleigh, Northern Virginia, Atlanta and Miami, have been visited, but media reports say Amazon search party members have recently, taken second trips to Miami and Chicago. The Chicago Tribune reported Amazon representatives were spotted placing Alexas all over the Chicago Area, specifically at \"The 78,\" a 62-acre piece of land in the South Loop neighborhood of the area.\nWhile citizens of Chicago believe themselves to be at the top of Amazon's location shortlist, insiders have also suggested that the Washington, DC metropolitan area is another front-runner. This possibility is not one that has unilateral support from the residents of the District of Columbia. In fact, according to Business Insider, \"many worry that the region won't be able to handle a major uptick in population size.\" This concern is based on the possibility of 1 million new residents over the next ten to fifteen years, as a result of job creation from the new headquarters. DC residents like Advisory Neighborhood Commissioner Denise Krepp, are against the tech giant's potential move to her city.\nReporters have gathered that residents are also very concerned about the possibility for a shortage of affordable housing in the area, similar to what happened when Amazon first opened their headquarters in Seattle, Washington.\nAs a former resident of the Bay Area, I have seen, firsthand, what happens to house markets when startups and tech companies, like Lyft, Twitter, or Facebook, start to open up satellite offices and headquarters. Neighborhoods become vibrant with creativity and life, coffee shops and restaurants open, but all that glitters is not gold. The cost of living skyrockets to the point of no return. As housing costs rise, quality of life for those living in those neighborhoods begins to decline.\nThese location speculations and concerns come after Amazon's huge announcement last week, during which CEO Jeff Bezos said the company plans on raising its minimum wage for all employees to $15 per hour. Prior to this decision, urbanist Richard Florida encourages Bezos to take better financial care of his employees.\nWhile it is responsible and wonderful that Bezos recognized he was not paying his workers a livable minimum wage, this does not discount the dramatic effects that HQ2 will have on its new location, once this is decided by the search committee. From what the media has reported on, Amazon's process of searching for a second home has not involved much consideration of community lay leaders and members that will be affected, adversely or otherwise, by the new set up. In the case of Washington, DC, three days before the Amazon bid was announced in September 2017, the mayor held a community meeting.\nDuring that meeting, according to Denise Krepp, the office claimed: \"there were no plans to further develop the parcels of land in her neighborhood.\" Three days after this meeting, the city offered to sell itself to Amazon. Krepp's primary concern is that if Amazon takes over vacant land in the area, the company will occupy space that could be otherwise used for affordable housing.\nAll citizens like Denise Krepp, and those residing in the other shortlisted cities, can do is wait for Amazon's search committee to make their final decision. During a speech at the Economic Club of Washington last month, Jeff Bezos firmly announced that a decision will be made before the new year, but as for a specific date and time?\nIf you're an Amazon Prime member, you can opt into 2-day free expediting of information. Just make sure your contact info is up to date.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Important are Photos and Images?\nThe succinct answer is that they're very important indeed and here's why you should be using more photos, infographics and videos in your content in 2018 if you want your campaigns to really work.\nVisual marketing is the most important content type for around 40% of marketers, just behind blogging content itself and almost three-quarters of social media marketers use visuals ahead of blogs and videos.\nIf people hear information, they tend to remember only around 10% of it three days later, whereas if the information is allied to relevant or related images, this retained proportion goes up to 65%.\nIn 2018, 75% of all internet traffic will be video content and this will go up to 80% by 2019. When it comes to consumers, four times as many prefer watching videos about a product than reading about it. Smart business owners are investing more in video, so don't be left behind.\nJust using the word 'video' in the subject line makes it 19% more likely that the email will be opened.\nOn Facebook, users spend around three times as long watching live videos than pre-recorded ones. Videos that are less than five minutes long make up almost 60% of the video consumption on smartphones.\nAccording to eye-tracking studies, readers pay very close attention to images that carry information. When the images or graphics are highly relevant, more time is spent on them than on the body content. Infographics are also shared and liked on social media three times more than other types of content.\nTweets that feature images are shared and retweeted 150% more than those without images. On Facebook, posts with images get almost two-and-a-half times more engagement than those without and organic engagement is higher on posts with videos and images.\nIt also seems that articles with images occurring every 100 words or so are shared twice as often as those with fewer images.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good Morning Wishes. The best way to get things done is to begin.\nthat it's not about being Rich but it's about being Happy.!!\nGood Morning Quotes Inspirational Quotes on Goal.\nHey Buddy!, I found this information for you: \"Good Morning Friends\". Here is the website link: https:\/\/www.goodmorningfun.com\/morning-quotes\/good-morning-friends-3.html. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Conspiracy theorists who believe the earth is flat will descend on Ipswich town centre on Sunday on their new tour- to try and prove that the world is not round.\nHarry Mcdonald and Jason Disbury of the Globe Lie UK Tour want to spread their beliefs by visiting more than 50 towns and cities in the UK. The friends have been touring the UK since September 2013 and will be coming to Ipswich to try and circulate their views and try and convince more to come around to their way of thinking.\nThe pair have researched the flat Earth theory and have been creating content on their youtube channel to spark conversations across the world- sharing their '200 proofs' which show we don't live on a globe.\nHarry Mcdonald, 60, who originally comes from Edinburgh, created the Youtube channel secretly to capture local conversations he had with people in parks and universities.\nHe has been going up and down the country to prove his point and share his words of wisdom- and this weekend the pair will visit Ipswich.\nLocations for the event will be revealed online.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To filter salaries for Senior Field Engineer, Sign In or Register.\nHow much does a Senior Field Engineer make?\nThe national average salary for a Senior Field Engineer is $70,774 in United States. Filter by location to see Senior Field Engineer salaries in your area. Salary estimates are based on 5,064 salaries submitted anonymously to Glassdoor by Senior Field Engineer employees.\nBelow are the most recent Senior Field Engineer salary reports. Employer name has been removed to protect anonymity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shepperd Bros provides all your plumbing and gasfitting services for residential and commercial premises in Hobart. From new builds, traditional restorations and modern renovations to minor repairs and emergency services, we've got you covered! Call us today to talk to our friendly and professional team in Hobart to see how we can help you. Ask us about our pensioner discounts!\nFor over 70 years, Shepperd Bros have carried out regular repairs, maintenance and servicing on commercial and residential premises. With our experienced plumbers and registered LPG natural gasfitting experts, you know you can rely on us!\nWe are the friendly plumbers and gasfitters Hobart trusts to look after their hot water system needs. Our quick response team will get there fast and be gone in no time, so you can get on with your day. For all hot water system repairs and installations, call our local team.\nYour local, registered LPG natural gasfitting team, Shepperd Bros have been installing and repairing gas lines for many years. Trust our team to do the job right, and to do the job fast.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At RateWaste, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Reynolds, IN, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nSpeedy service such as next day Waste Bin in Reynolds, IN is the one other fantastic merchandise to look for. If you're in a hurry to obtain the project started out and completed, this is a have to have option.\nDiscover which size hired dumpster you will need. Your Waste Bin in Reynolds, IN company needs to have at least three sizes-small, medium and larger.\nNearly almost any business need some sort of Waste Bin in Reynolds, IN with regards to property. Depending on the type of business may reflect the type of service is necessary, and will also reflect what size dumpster must be rented. A few of the available pick up options ranges from daily, to each week, to actually monthly pickup trucks. Specialty dumpsters pertaining to recyclable items are also available in order to rent out to businesses. If you need a compacting dumpster pertaining to boxes along with cardboard products, this too might be rented with the same business that housing costs out any other type of dumpster.\nMany of the Waste Bin in Reynolds, IN companies impose fee based the timeframe of the rental along with tonnage of the dust disposed in the dumpster. Be very specific regarding the time period of this rental before you place a purchase. Though almost all of the companies cope with instant tasks, it is always better to schedule this delivery along with pick up upfront so that you have plenty of time to make out area for dumpster when they get it to your area.\nThis is where some sort of Waste Bin in Reynolds, IN service belly in useful. The company not just provides the dumpster, in addition, they discard this waste to suit your needs. So, this would be one much less thing you would have to worry about, in particular after a marriage ceremony.\nAll goods can be disposed of in one dumpster. They just don't need to be segregated for example: goblet, metal, wooden and drywall can be disposed of in the same dumpster. Nearby trash pick up regulations prohibit the amount of junk that is available pertaining to pickup with the city. Additionally they require the bagging of all junk and the isolating of any recyclables. Utilizing a construction Waste Bin in Reynolds, IN helps make trash disposal much easier along with quicker mainly because it does not require this bagging regarding trash or separating regarding recyclables.\nIt is time for everyone to vitally understand the have to protect the surroundings. In spite of the federal taking a lot of measures defend the environment; the surroundings is still disturbed due to the air pollution that is made in the country. It is very important for the federal and individuals to function in similar to protect the surroundings.\nIf you want this dumpster placed on the driveway, you may want to lay down many pieces of plywood to protect the region from any deterioration.\nTired of all the clutter taking up valuable area in your home or perhaps on your lawn? After that don't watch for spring in order to swing in cleaning method. With affordable and easy Waste Bin in Reynolds, INs from your local disposal company, it is possible to clear away numerous years of accumulated trash in a single day along with reclaim the real estate pertaining to better utilize.\nOnce the house has been cleared of the tenant's things, it's time to give attention to repairs. Evictees will often be terrible tenants (they had to become evicted, after all!), and they are able to do some fairly extensive problems for your property that must be repaired before you could rent it out all over again. Odds are that you will find to replace this carpeting in the rental; few visitors leave undamaged carpeting. If your former tenant had domestic pets, it is essential to replace the carpet to eliminate all contaminants in the air. However, if there has been any damage completed to the partitions or repainting is necessary, time savings and issues by keeping all those old carpets and rugs installed to use as drop cloths whilst you repair along with repaint. As soon as the repairs are done, rip out and replace the old rug. Perform any needed preservation on home appliances and accessories, and then begin to show the property to become rented all over again. Better luck with your subsequent tenants!\nLocating a Trustworthy Atlanta Rotate off Dumpster Company It's usually quite easy to locate a corporation that bargains on Rotate off dumpsters in Atlanta. You can simply search for these online specially in the Atlanta listing. You can equally get them via media advertisements. Most companies doing business on dumpsters normally provide induce delivery and pick up providers to their clients. Fortunately they are available 24 hours online. You can request for their own assistance in emergency predicaments. When it comes to costs, most companies working in the Roll down Dumpster supplies normally offer cost-effective price points. Oftentimes, the charges include the supply, pick-up and discretion of all the waste products collected in the containers. Trustworthy companies don't charge independent fees with the processes required. You can always discover them online. If you want to have a much better deal, it's always better to contact the company you would like to deal with early enough. This helps to ensure that you have the correct roll down container that can serve the purpose you are interested in.\nIf you live within an old farm house you know that this cold can easily seep throughout the walls and also the old home windows. You want to get more old farm house but you want to keep the appeal alive. It is possible to remodel your own farmhouse having new products that will assist you save money in the long run.\nWhen you've got many trash to manage and not time and effort or methods to deal with this, you need to rent the trash dumpster. The item sits within your driveway, gladly collects your own trash and, when the time is right, becomes hauled out to the land fill without looking back. Getting a trash Waste Bin in Reynolds takes almost all of the responsibility down your shoulders when you are working on the construction, clean-out or perhaps demolition project plus you've got a load of trash building up within your front yard.\nWhen you're for a most competitive price, the company usually has to cut corners somewhere. This could be in having to pay employees reduced wages or perhaps keeping less dumpsters on hand. They might also take their time delivering and picking up dumpsters in an effort to make things fit their own schedule rather than customer's.\nThis is where a Waste Bin in Reynolds service would come in handy. This company not only offers the dumpster, but they also discard the squander for you. Consequently, this would be one less factor you'd have got to worry about, specifically after a special day.\nDon't expect to eliminate gasoline, any kind of chemicals, natural skin oils or any type of toxic combustible substances. Many dumpster companies is not going to allow such a waste items. That is quite simple to comprehend since it can be quite dangerous.\nNo matter why you require a dumpster, whether it's business oriented or individual, you will most likely need to rent one from a Waste Bin in Reynolds corporation. While you may possibly believe that the dumpster is a dumpster, the company you rent through may still greatly impact your current experience and final charges. Here are some things to look for in your own Waste Bin in Reynolds company.\nAlmost any light fixtures that contain come in bins will need to hold the box plus the Styrofoam and other packing content thrown away into the dumpster.\nTaking away trash with the home of the compulsive hoarder isn't any simple task-it not merely involves a substantial amount of physical toil, but it is also a very difficult emotive challenge for the hoarder, his or her loved ones, and individuals who have any sort of curiosity about the maintenance in the property in question. For these key trash treatment jobs, in particular those under time constraints, may be essential in order to increase the process and also spare homeowners extra expenses such as throwing fees along with the gas utilized to drive weight after weight of waste to the transport station.\nThe real key to choosing a business to rent from is definitely to ask as numerous questions as is possible. Make sure there aren't any hidden charges, specify the volume of allotted rental serious amounts of make sure the business has a trusted reputation. Deciding on a company that features a flat rate is good, and don't be scared to do online investigation before making a crucial decision like this.\nThe use of a Waste Bin in Reynolds, IN is usually clearly recognized as a way to keep excessive quantities of trash. Nonetheless, the benefits to using this service will not just hang on a minute. It also offers one the chance to bypass this headache when you attempt to discard massive amounts of garbage.\nIt is crucial to know, even so, that a roll-off Waste Bin in Reynolds, IN is just that will: a Waste Bin in Reynolds, IN. It won't automatically consist of property clean out service to obtain stuff into the dumpster for you. This may come as a surprise to many people, since many dumpster companies state that their particular dumpsters are available for trucking trash, eliminating garbage, and dealing with home cleanouts. You'll need to read the small print when you're looking at dumpster firms to you should definitely know if these are typically suggested employs or included services.\nIf you are living in Va, Washington DC or perhaps Maryland when compared with you live inside a beautiful region and you desire to keep it beautiful. Your neighborhood is definitely proud of the sweetness that is all around them therefore you all desire to upkeep the item. No matter where your house is you want the house and neighborhood kept clean and exquisite.\nIn the wake up of heading back to eco-friendly environment, more and more people are today realizing the benefit recycling this generated waste in eco-friendly method. The trash generated on selected projects just like construction and also house restoration is huge and also bulky. Getting a decision to get rid of these large tons of waste on do it yourself may leave you with injuries because of to lack of dependability in controlling task. if you decide to dispose this generated waste in environment friendly method in Kansas, it is must for you to choose Waste Bin in Reynolds, IN Kansas.\nYou can still keep your quaint looking farm house with way back when charm. You only need to revise it by time to time to make it energy efficient and even more beautiful looking.\nLastly what you have got to do is definitely to get in touch with Waste Bin in Reynolds, IN suppliers, know their particular final offer on the Waste Bin in Reynolds, IN and check if the price tag fits into your bank account.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are schools offering personal training certifications in Indiana!\nAround 0.0% of Indiana's graduates graduate from personal training degree programs every year. Thus, Indiana's 9 personal training schools put out approximately 67 personal trainers each year.\nDePauw University, which is located in Greencastle, is the top-ranked school in Indiana that has a personal training program. It received a ranking of 27th in the country in 2010. In 2010, 1 students graduated from DePauw University's personal training program. DePauw University charged in-state students $34,905 in tuition fees per year.\nThe second-ranked school in Indiana that has a personal training program is Taylor University. Taylor University, which was ranked 30th in the country in 2010, is located in Upland. Taylor University charged in-state students $26,383 in tuition fees per year.\nIndiana Wesleyan University, which was ranked 30th nationwide in 2010, is the third-ranked school in Indiana that has a personal training program. It is located in Marion. 12 students graduated with degree in personal training from Indiana Wesleyan University in 2010. Indiana Wesleyan University charged in-state students $21,214 in tuition fees per year.\nA personal training degree from an Indiana school... what next?\nThe government projects that the number of personal trainers in Indiana will grow by 27%. By 2018, there will be an estimated 4,190 personal trainers working in Indiana.\nTake a look at the graphs and charts below for additional Indiana statistics regarding a career in personal training and to compare salaries with a variety of related fields such as physical therapy assisting or physical therapy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We fielded a question recently on www.napwa.org, and passed it on to Dr. Bruce Rashbaum. You can submit your own questions for Doctor Bruce by clicking HERE.\nI have a question for you. I was diagnosed with HIV two years ago, and it was very hard for me to accept the fact. I had been with the same man for 11 years. I do not see him anymore, that was my choice before I was even diagnosed with this, but I did tell him, and he said he went for a test and came back negative. I have a hard time believing him or could it just be it has not shown up in his body yet? Or is it possible I could have got this more than 12 years ago? When I first saw my doctor, she thought I might have had it for about three years. I was really sickly, then. I am doing so much better today, I am undetectable, and I know I need to stop wondering where I got it and just thank God I am alive. Any info you can tell me I would appreciate.\nI am glad to hear you are feeling better, on treatment for your HIV, and almost moving on. You need to. Whether or not this man is the one who infected you will not change anything. You did the right thing by informing him about your situation. I always think it is important to actually see the test results of a potential source, but that can get into sticky legal issues in regard to privacy. For those who are ethical and have a moral backbone, I would think they would have no problem sharing with you results, whether they are negative or positive. This may sound harsh, but, think about it, if you have to hide something, then you have guilt associated with deception. Those with nothing to hide nor fear will be forthcoming.\nOK, enough of that. In regard to his test not being positive per his communication with you, there are several possible explanations. He is indeed negative. He didn't get exposed. He could also be one of those with the genetic mutation that causes a defect in the receptor on the CD4 cell that is necessary for infection to be established. This is seen in fewer than10% of individuals from a northern Europe demographic group and they are actually resistant to HIV. Recall the German patient who had HIV, had to undergo chemotherapy followed by a bone marrow transplant for lymphoma. His lymphoma was cured and his physician, quite astutely, searched for a compatible bone marrow donor but also sought out one with the complete mutations to this CD4 receptor. This patient not only appears to have eradicated HIV with chemotherapy but now appears to have made CD4 cells that have this mutated receptor and it is thought that this patient is now resistant to HIV. Incredible!\nIt can take almost two years for the less than 2% of individuals that do not seroconvert shortly after exposure to HIV to begin producing antibodies that will make a test come back positive. The majority do so within six months of exposure, with most of these seroconverting within the first month after exposure. So, either this guy is lying to you or he fits into one of these categories. Now, what you are asking of your doctor is to tell you when you actually got exposed. There is really no way to tell you when you got infected without having had the laboratory testing over the years that showed a negative result at one time period and now a positive result at another. You can pin it down with that data along with the presence of the Acute Retroviral Syndrome, which is a symptomatic acute flu-like illness (this occurs in less than 50% of new infections) following \"recent\" infection by HIV, potentially helping one identify a source as well as a date of infection after their exposure. One can guess that you possibly could have gotten infected three years ago because you may have the worst flu ever but that is just a guess. And, yes, you could have gotten exposed 12 years ago, never had the symptoms of an acute viral infection, and only after seeing your doctor more recently been identified with HIV due to having had the actual blood test.\nI am sorry for your angst about wanting to know who infected you. I understand why you want to know. But, again, it doesn't change anything for you. Certainly, you would want to ensure that if the source was identified that he was not out infecting others. I hear this from my patients. But it is wasted energy and you did what you had to do. You informed him, he said he was negative, you ended the relationship for the variety of reasons that you felt were valid and important. Move on and continue to take the best care of yourself. Eat well, sleep well, MINIMIZE STRESS, exercise, follow up with someone who has experience in treating HIV ( it makes a difference in the quality of care rendered ), take your medications and consider simplifying your treatment to one of the new single tablet regimens if you are not already on them, even if you are 'undetectable' on a different medical cocktail. They appear to be less offensive especially in adverse side effects. I hope this puts your mind at ease. Happy Thanksgiving and enjoy every day. They are gifts to all of us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Woman's \"Who's In Charge\" T-Shirt. Are You in Charge? | Who's In Charge\u2026Save a Wag with Swag.\nHome \u00bb Shop \u00bb Pet Family Swag \u00bb Woman's \"Who's In Charge\" T-Shirt. Are You in Charge?\nWho's Really In Charge? Is it your dog? Is it you? Either way, you'll bring a smile to yourself and others who catch a glimpse of this awesome graphic and incredibly comfortable nano-T \u00ae shirt. Made from 100% ringspun cotton.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last year we started taking care of my husband's two Aunties who have lived in San Francisco for 50 years and had to move them to Ohio with us. When it came time to sell their house we were way out of our element. We had a couple references from neighbors and called Todd. We could not have picked a better Realtor. His knowledge of the area, his willingness to listen to our needs and ability to take care of everything that needed to be done was invaluable. We were able to leave things in Todd's hands and return to Ohio. He kept in touch with us as things were progressing. Assured us when we needed it and delivered more than we were hoping for or thought possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Will I lose my database after updating the Projetex Server?\n2.Select \"Server installation\" in the drop-down list, and click the \"do not install database\" button on the \"Select components\" screen.\n3.Answer Yes to question: \"The folder C:\\Program Files (x86)\\AIT\\Projetex 3D\\Projetex Server already exists. Would you like to install to that folder anyway?\"\nAfter installation is complete, go to the Corporate Settings page of the Projetex 3D Server Administrator. If the page is displayed without additional messages - no database update is necessary.\nIf you are prompted to update your database, please run the Projetex 3D Server Administrator and go to the Database page. In the Database Version field you can see your current Database Version. Click the Go Offline button. To upgrade the version of your Database click the Run Update Script button. Now select the script which you need. For example, if your current version is 157 and the program requires 158, you need to select dbx157to158.ptu, if your current version is 156 you need to select dbx156to157.ptu and then dbx157to158.ptu, until you have the required version of the Projetex Database.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Instant Pot Apple Fig Compote contains no added sugar. Just the pure, natural sweetness of apples and dried figs make this dish perfectly sweet and healthy. So versatile you could eat it for every meal of the day!\nThere are serious advantages to having a sister that's complete opposite. In a way, we become the ultimate woman because we've got each other's talents on speed dial. If I'm at a loss on photography, decor, or basically anything to do with blogging, she's got the answers.\nIf she needs food or medical advice, I'm all over it. Power to the sistas!\nSo when Cami (the genius over the Decor side of TIDBITS) texted me in desperate need of a fall time appetizer, I was more than happy to procrastinate yard clean up, yet another day, to help. I've got to say, we came up with something pretty sweet.\nHow cool is this, we created 2 recipes that are great on their own, but when combined become the ultimate Fall Appetizer. Are you pickin' up on the symbolism here?\nWith approximately 8 million apples laughing at me every time I go into the garage, I knew which direction I needed to take. But for this sister project, it had to be special. Terrifically, mouthwateringly, mind-blowingly special.\nI spotted 3 partially used bags of dried figs in my pantry and I knew immediately what was about to go down. Apple Fig Compote.\nAnd for those who don't like figs, I've got your back, stay with me here.\nVersatile. You could eat this for every meal of the day. Use it over yogurt, on a grilled cheese (with brie, mmmm), over pork chops, or with a scoop of ice cream. You've got breakfast through dessert here! What other ingredient besides salt can do that?!\nLet's clear this up first. A compote consists of whole or chopped fruit (fresh or frozen) that has been cooked down into a sweet syrup. The syrup can be made from any sweetener or be created by the fruit itself as it cooks down. The final product is a soft, sweet, fragrant, and flavorful masterpiece.\nCompotes are amazing because they can be used in so many ways from sweet to savory. Compotes also store well in the fridge and often times in the freezer. The best part is that you can make a compote with endless varieties of fruit and spices. Are you on board now?\nChop up your fruit, in this case, apples . . .\nAnd now for the ultimate sister collaborative idea . . . Maple Pumpkin Coconut Dip served in a Phyllo Crisp topped with Apple Fig Compote (see recipe notes for directions). Like seriously, is that not the most bomb Fall Time Appetizer you've ever seen in your life! The flavor's sing Autumn. In the words of my son, \"It makes my heart feel good and makes me want to be a nice person.\" Amen kiddo.\nThe list goes on and on. This compote is amazing hot or cold. Feel free to share your own ideas below. I plan on making a ton of this compote and I'm hoping like crazy it freezes well. It hasn't lasted long enough to need to freeze it yet.\nEnjoy this fall time gem!\nOkay, now for the fig haters or those with an unreasonable fear of Fig Newtons. I didn't forget you.\nSwap the figs for chopped dates like in this Caramel Apple Pie. It's incredible!\nHead over to Cami's to get the recipe for this amazing Healthy Pumpkin Dip.\nThis naturally sweet Apple Fig Compote has a million different uses but is so good you could eat it plain with a spoon!\nPeel, core, and dice all the apples. Place apples in the pressure cooker pot.\nUsing kitchen shears or sharp, clean scissors, cut figs into small pieces. You can also use a sharp knife but the shears make it a lot easier.\nPut figs on top of apples.\nTo the pot, add 2 tablespoons apple cider, cinnamon, vanilla extract, and a pinch of kosher salt. Do not stir.\nCombine cornstarch and 2 tablespoons of apple juice. Using the saute function of the pressure cooker, bring apples to a light simmer, stir in cornstarch slurry, until thickened, about 1 minute.\nCan be eaten hot, at room temperature, or chilled from the refrigerator. Reheats perfectly in the microwave.\nStack 2 sheets of Fillo Dough Pastry sheets on top of each other and brush with melted butter. Stack another 2 sheets on top and again brush with butter. Cut out small squares then press them into a small muffin tin to make a cup. Bake at 350 degrees for 6 \u2013 8 minutes, until browned. When cooled, fill with Healthy Pumpkin Dip and top with Apple Fig Compote. Serve.\nTo make this even easier, look for the Mini Fillo Shells. They can be harder to find than the sheets, but are so yummy!\nThis looks spectacular! I'm going to try it in mini phyllo cups with some melted brie for a fall appetizer.\nSusan, That is absolutely genius! I am so trying this! Would you put the brie in while prebaking the phyllo cups?\nYes, it only takes a couple of minutes to start to melt, then add the compote on top! I can see this topping a grilled pork tenderloin too. Thanks for this great recipe!\nDeb, I love dates and apples together. I did that with an apple pie on my blog. It's like apples and caramel, yum! I wish I was more savvy with canning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Stillwater Wood is a privately owned and operated furniture sales and finishing store. Our shopping environment is relaxed, with no sales associates following you or pressuring you into making a decision about purchasing a piece of furniture for your home. However, if you feel you do need some assistance, don't be afraid to ask. Our incredible prices and friendly environment has kept us in business for over 40 years. We rarely advertise, most of our customers are brought to our store by word of mouth. If you are interested in purchasing some new furniture or crafts for your home, stop by and take a look! Or for a general idea on what we carry check out our \"Photo Gallery\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The parents have died, along with Rob, another brother, in 2007. Kimberly dearly wants River Oaks to remain open for golf. A few years ago, Michael considered developing the front nine holes into lots where homes would be built.\nEarlier this year, St. Stanislaus Golf \u2013 the practice center and nine-hole course on Crows Landing Road \u2013 changed ownership. Longtime owner Francis Ip sold to the Marquez family, which operates El Rematito Flea Market not far from the course.\nHoles in one \u2013 Donna Dami, Modesto, 85-yard sixth at Jack Tone Golf, Ripon, 7-iron. ... Don Gonzalez, Oakdale, 182-yard fifth at Creekside, 5-iron.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you for your interest in our services. Please feel free to ask us any questions you may have regarding our services at Heritage Kitchen Design Center. We check our e-mail daily and will reply to your message promptly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Attitude- PMA or attitude controlled by fear, anger, jealousy, greed.\nActivity \u2013 MKMMA exercises, the reads, the sit, super hero pose\/calling prospects, doing meetings, spending the time, following up.\nWithout a good attitude the activity you do amounts to very little.\nIf I bring rain and gloom and darkness and pessimism to my customers then they will react with rain and gloom and darkness and pessimism and they will purchase naught.\nBut on the other side if you have a good attitude and have no activity then you have nothing as well. You must balance it. Good Attitude and activity. Ok, so what do I mean by activity?\nA good attitude will last only so long if we stop doing the activity like we mentally do with the MKMMA. I looked at the last 5 months in my business and how the weeks went.\nYou can tell by the number when I had good positive attitude and activity with MKMMA and my business. It was 5 weeks on 1 week off. Week one I put forth 100% effort. Week 2 I started seeing real success. Week 3 less activity and by week 5 little activity and little to show. I started to fail so I started again. This last time I caught myself red handed. With the help of my mastermind and keeping on with the activity, I created a PMA and kept the activity going (MKMMA and business activity) and you can see the difference that it has made. Week 24 shows it. I was 241th out of 3000 this last week. I will not contribute it to skill. I have been in sales for over 20 years, so the skill has always been there. The difference was just like the title of this post reads. I feel like I could write a book about this week alone.\nMarch 13, 2014 in MKMMA, Uncategorized.\nThis is so great. Keep this 400% grow ;-). Congratulation.\nI'm impressed! Good for you Brian and keep up the great work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oliwia Meiser \u2013 young cellist, winner of Polish and international music competitions. Graduate of the Og\u00f3lnokszta\u0142c\u0105ca Szko\u0142a Muzyczna I i II st. im. Fryderyka Chopina (the Frederic Chopin High School of Music) in Bytom, Poland, graduate of the cello class of Nataila Kurzac \u2013 Kotula (Karol Szymanowski Academy of Music). A student of Universit\u00e4t der K\u00fcnste Berlin in the cello class of prof. Konstantin Heidrich.\nSince July 2017 she is a member of LGT Young Soloists ensemble.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Communication Research Registry connects individuals and families with research study opportunities at Northwestern University. To do this, the Communication Research Registry maintains a confidential database of information about individuals and families who are interested in participating in research.\nOur goals are to provide individuals and families with access to research opportunities in their community and to support cutting edge research on human communication and development.\nIf you have any questions, please call registry coordinator Stephanie Crawford at 855-354-3273 or email her at commresearchregistry@northwestern.edu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shea butter and castor oil impart moisture, while essential oils add natural scent that's much more gentle on your hair. After your scalp adjusts to the new formula, your hair will be super soft, shiny and feel all-around healthier. I love that I can customize the scent of my homemade shampoo bar recipe to suit my mood and the season and it's definitely a bonus that after the initial investment in supplies, this is a much more economical option than buying shampoo (not to mention all the plastic packaging I can avoid).\nStart by putting on your rubber gloves and safety glasses (eyeglasses and sunglasses work, if that's all you have). We're going to be working with lye, which is extremely basic and will essentially affect your skin in the same way as a strong acid, so it's best to be very, very careful when dealing with it. Soap making isn't hard, but we do need to be precise and deliberate.\nCarefully measure the lye \u2014 again, we need to be really precise with this recipe in order to get the soap to form with the proper chemical composition \u2014 and set aside. Now, add the water to a heat-resistant vessel (a Pyrex measuring cup works great for this).\nCarefully pour the lye into the water and stir gently until combined. Be careful not to breathe in the fumes, and make sure you add the lye to the water, not the other way around.\nWhen stirring, you'll notice that the water mixture will start getting very hot and may even steam. This is normal! It's science! It's cool! Place the lye and water mixture in its heatproof cup into an ice bath and let cool while you complete the next steps.\nCombine your olive oil, shea butter, castor oil and coconut oil into a small pan over a double boiler. Heat on medium and stir until all the fats have dissolved and combined. Take it off the heat and let cool.\nWait until both the lye mixture and the oil mixture have both cooled to about 100 degrees. The goal is to combine them when they are right around the same temperature, so use a thermometer to get an accurate idea.\nMake sure you have your gloves and your eye protection on again and carefully pour the lye solution into the oil mixture. Avoid any splatter by pouring the lye mixture down the wooden stir stick you've been using.\nStir for a bit with the wooden stick, then use the immersion blender to combine. Make sure to completely submerge the blades in the mixture to keep it from splattering. Slowly move the blender around until the mixture has fully emulsified. Keep blending until the mixture is about the consistency of pudding.\nOnce you think you've reached the right consistency, take the blender out of the mixture and drizzle a bit of the mixture onto itself. We're looking for something called tracing. What that means is that the ribbon of drizzled mixture should stay visible and should not sink back into the mixture.\nNow, add in the essential oils and give it one last stir with your wooden skewer.\nCarefully pour the soap mixture into the silicone molds.\nTap the silicone mold several times against the table to rid the mixture of any trapped air bubbles.\nSprinkle with dried rose petals, then wrap the whole mold tightly in plastic wrap. Cover the wrapped mold with a kitchen towel to keep the heat in and encourage the shampoo to form properly.\nAfter 24 hours, demold the shampoo bars and lay them out on a cooling rack.\nThe final step does require some patience. Place the rack of shampoo bars in a well-ventilated place and let it age for at least four weeks \u2014 or really any amount of time between four and six weeks. Basically, this process allows the alkaline lye to mellow and create the proper pH levels in the soap. Although lye is very caustic in its natural state, after the shampoo bars go through their ripening process, they are perfectly safe!\nAs I mentioned above, the wonderful thing about making your own shampoo bars is that you can completely customize them. Experiment with different fats like avocado oil, sweet almond oil and cocoa butter to add different levels of moisture to your hair care. Just be sure to use a lye calculator to figure out how much lye to use for different oil proportions. You can also add things like clay for extra cleansing (great for oily hair!) or honey to help seal in moisture. There are a bunch of great hot process soap recipes out there, so don't be afraid to mix and match.\nAfter your soap has aged properly, it can be packaged in these cute little black tins. These are great for gifting and great for traveling \u2014 which is the perfect combo as we head towards the holiday season. All you need is some pretty twine to put the finishing touch on this project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Did you know that the Snuffer's on Lower Greenville is haunted? Well, it sure is. Just ask the servers and they'll send over one of their staff members that have had a paranormal experience pop by your table and tell you a couple ghost stories. \u2026 or they'll just tell you about the scary good deals they're now offerings during happy hour on their Happy-Tizer Menu.\nY'all know that I love a good happy hour, so when I was invited in to try out their new options, I didn't shy away. Monday-Friday, 3-7pm you can now get select mini-apps for as little as $2. That's right \u2026 two buckaroos \u2026 a pair of greenbacks \u2026 just a couple Washington's \u2026 you get the idea.\nAs always, the Dallas favorite offers happy hour beers and margs as low as $3 to make your visit truly happy. Think $3 house margaritas (frozen or on the rocks) and drafts (Bud Light, Samuel Adams, & Dos Equis) and $4 house wine and shooters. While I'm not a huge beer drinker, it felt right ordering one to sip \u2026 and then a house margarita \u2026 and then another beer.\nThe real excitement as of late is their new happy hour food offering, a.k.a. their Happy-Tizers. Check out the picture below. All that goodness can be yours for $22. And for that price, you can try all of them \u2026 and you should. Their fried pickles and mushrooms are legendary* and these small servings are just enough to satisfy your craving but not take over the meal. (Though, really, what's wrong with a meal of fried pickles and mushrooms?) Their tortilla chips were seasoned just right and the onion rings and buffalo wings were perfectly satisfactory.\nIf you're trying to decide what to order on this little menu, skip the burger quesadilla and the queso \u2026 save those options for Blue Goose down the street. They'll get an \"E\" for effort on those, though.\n*at least to me. For nearly my entire college career, when I was in a bad way the day after a good party, I knew Snuffer's fried pickles and loaded Cheddar Fries were my saving graces. They never failed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Billings, MT Basement Waterproofing Contractor | Basement Waterproofing Contractor Company in Billings, MT.\nWe had restoration done and have an exposed wall with foundation crack that needs to be sealed.\nWater on inside perimeter of basement.\nExtension on the rear of house end concrete foundation wall sunken couple of inches and so floor inside has sunk on that end by couple of inches.\nWe have an unfinished crawl space, no insulation. Out floors get pretty cold.\nI have a foundation wall crack that brings in water through the crack whenever it rains. I need it fixed before winter rolls in and before I rent my house.\nOur basement flooded in early June the third time it's flooded in 12 years. We want to sell the house and have to get this FIXED so we can sell.\nOur basement flooded around Memorial Day and we've finally got it dried. How much for French drains to be done?\nCement floor in basement of house is uneven in places and may be affected by the presence of alkali. The alkali causes the cement to break up.\nWent above and beyond. Very pleased with their service and professionalism.\n\"Dan has been a Godsent for me on more than one occasion,...\"\nThe last thing any homeowner wants to deal with is a wet or flooded basement. Not only does a wet basement lower a home's property value, it can also lead to severe damage to anything in the basement and mold growth.\nOur expert team can quickly assess your basement, identify how the water is getting into your basement, and then suggest the best solution for preventing flooding in the future. Contact us today and schedule your free estimate in Billings, MT!\nIt is all too easy to ignore a failing crawl space. They say \"out of sight, out of mind\" and there is nothing more out of sight than a crawl space. However, while crawl spaces might seem unimportant, your home is ultimately a system where every part counts, including the crawl space.\nProblems in your crawl space very often go on to affect your entire home. Do you have mold, dust, or debris dirtying your crawl space? Mold spores, dust, dirt, and contaminants from your crawl space will pass through the air you breathe in other parts of your home. Likewise, structural damage in the crawl space can cause other problems like sagging floors. The same goes for other issues.\nDon't wait! Have the experts at ProSeal Basement Systems look at your crawl space today. Whether you need repairs, insulation, or encapsulation, our team can help you decide. We're Billings' CleanSpace dealer for a reason. Contact us today and schedule your free estimate!\nLook out for ProSeal Basement Systems' Company Truck around Greater Billings, MT.\nWe sealed the wall which fixed the moisture issue in the new addition.\nJust wanting someone to access current crawl space. I have owned this duplex for 3 years now and had humidstat fan installed during the first year.\nNew subdivision behind my house They raised the slope and now water drains from their location to my basement.\nWe have water slowly coming into the basement, need to figure out how to resolve issue.\nWater seeping into my basement. Looking at an interior drain system\/sump pump?\nThe house has a crack in the foundation and had standing water in the crawl space 10 years ago. I covered the space with a plastic barrier but am concerned that it needs a better barrier with the recent heavy rain. I also would like to have the foundation cracks repaired.\nGroundwater backing up in the yard. Nearly to the foundation, maybe 5 feet away. It has spread nearly 10 feet upslope since I mowed last Saturday. I need to do something pretty fast or I think I'll have water in my basement. Newer to this house but I don't believe this house has French drains or sumps.\nWe have a crack in our foundation that is about 6 feet long going from under a window to the floor. We are interested in what the cost is of sealing that crack.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am Digital Pedagogy Librarian at the University of Michigan Library. I was formerly the Digital Scholar at Whittier College's Digital Liberal Arts center, and a Mellon Postdoctoral Fellow in the Humanities and Visiting Assistant Professor of Asian American Studies at the University of California, Los Angeles. My research interests include the literature and media of migration and labor, Asian American studies, globalization and neoliberalism, postcolonial studies, and transnationalism. I'm especially invested in the ways gender, race, class, and nationality affect how the temporary and the precarious is experienced.\nMy work appears in American Quarterly, The Global South, The Journal of e-Media Studies, and Humanities and the Digital (MIT Press). I have previously published on comics in the Asian American literature class. I also serve on the steering committees of HASTAC, FemTechNet, and as a founding member of the #transformDH collective which seeks to incorporate critical cultural studies and social justice work into diverse digital humanities activities.\nTwitter, About.me, Academia.edu, HASTAC, CouchSurfing, and Linked-In.\nHeader image taken of Saigon\/Ho Chi Minh City from the Saigon River in April 2004.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaedlt b/data_all_eng_slimpj/shuffled/split2/finalzzzaedlt
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+{"text":"Though grown up in two totally different cultural backgrounds, Ibrahim Abdulraheem and Mamat feel like friends at their first sight of each other. Ibrahim Abdulraheem landed in Beijing International Airport at 6 pm, but his final destination should be Jilin, so Mamat bought a ticket at 10:00 pm for him. Mamat took him to the restaurant to have some noodles and taught him how to use chopsticks and they had a really good time together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These days it's harder and harder to discover a new author. Independent bookstores are closing and finding a new treasure in the big box stores can be like finding a needle in a haystack. Well, Midtown Review is here to help with our new Debut Spotlight. Today's selection is THE WEIRD SISTERS by Eleanor Brown coming January 20, 2011 from Penguin.\nThe Andreas family is one of readers. Their father, a renowned Shakespeare professor who speaks almost entirely in verse, has named his three daughters after famous Shakespearean women. When the sisters return to their childhood home, ostensibly to care for their ailing mother, but really to lick their wounds and bury their secrets, they are horrified to find the others there. See, we love each other. We just don't happen to like each other very much. But the sisters soon discover that everything they've been running from-one another, their small hometown, and themselves-might offer more than they ever expected.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most of you must have understood exactly what I meant by saying, silly bands and those who didn't, gosh you must be really living under the rock. These animal shaped rubber bands have created quite a furore in teenage fashion, although it is plain and simple, kids love it. They come in various shape and sizes, although parents are finding it difficult to purchase them locally, you can always search online, as most of the shops and dealers have been sold out.\nThese are made up of silicone and dye (thus you get them in every colour) and are perfectly elastic, so if you were to stretch it for a long time, it regains it original shape and size after you release it.\nKids have taken a liking for these animal shaped rubber bands because; you can use them as a variety of options bracelets, necklace and anklets. People also use it as a key chain or attach it to the chain of your bag, well find out your way to sport these and you don't know you could be the trend setter of your school.\nUsually kids have a tendency to trade them, well I don't blame them, they come in so many variety of shapes and colours, and some kids would definitely like to collect a particular theme, shape or colour silly bands, and parents don't object also as they are comparatively inexpensive.\nThere are so may themes in which silly bands come like animal shapes, fun shapes, cars, pets, posh, sea creatures, dinosaurs, rock band, summer, military, princess and thee companies are adding more and more by day and night. They even come in different types like glow in dark, scented, glitter, and tie dyed.\nWith so many companies producing these bands, people are complaining that some of these bands are not as durable as original silly bands and loose their shape easily and get disfigured, well also the good thing about this is they are cheaper, if you need durability you got to shell out few more bucks.\nThere's no hard and fast rule that these bands are only for the kids, if you like them, even you can sport them. These silly bands are loved by the kids and adult alike.\nParents haven't got any problem with their kids buying a new silly band every other day, as they are very cheap and silly bands affordable price is the main reason for its success, and also it is virtually harmless and are made up of non-toxic substances.\nAlthough some schools have taken steps to ban these silly bands on their school campus, due to the distraction they are causing.\nIf you are facing the dilemma as to where to find them, you can always search for them online. Companies have started selling them online. Check out for the cheapest and the funkiest and keep it rocking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As it leans on its Korean influences, Surisan skirts the kimchi-kitchen label. Breakfast and brunch here are actually American affairs flavored with a dash of Asian, while dinners are exclusively focused on reimagined Korean plates. Located in the city's Fisherman's Wharf district, Surisan boasts of contemporary interiors and an outdoor dining deck and bar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":" Polycrystalline mullite fiber board - Deqing Chenye Crystal Fiber Co., Ltd.\nFiber in the loose cotton and inorganic adhesive with wet vacuum suction filtration pressure forming, and according to the different types of after high-temperature calcination treatment, the products also have low density high strength excellent characteristics, is a multi-purpose products can be used in the hot face lining flame and a direct impact on high-temperature airflow resistance.\nIt can be used in the same time, it has good resistance to mechanical damage, and it has good resistance to mechanical failure. Areas include: push board kiln, ladle furnace, bell cover kiln, tunnel kiln, roller kiln, aluminum melting furnace, heat treatment furnace, and so on insulation lining and backing; high temperature fire damper board; kiln car insulation; high temperature industrial furnace insulation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaehlt b/data_all_eng_slimpj/shuffled/split2/finalzzzaehlt
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@@ -0,0 +1,5 @@
+{"text":"Running a fitness studio or personal training business isn't easy. It requires motivation, focus, commitment, and a lot of hard work.\nThey don't put the proper systems in place to support their day-to-day processes.\nThey don't invest in a strategic marketing & branding plan that relies on customer data and feedback.\nLearn more about how to improve your fitness business with the right fitness studio software and strategies in this article from Pat Rigsby over on the Fit Business Insider Blog.\nStart your yoga studio email newsletter on the right foot with these tips.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I wanted to write this post a long time ago but have not had the chance, so now with the new year - this will be the first post for 2016.\nCross Database Queries in SQL Azure DB was a key feature that was missing from day 1 on SQL Azure DB.\nFinally we have this key feature, and no, it is not the same as in the regular SQL Server. We have a new way to do this. It is bases on the new key feature in SQL Server 2016 called \"External Tables\" (https:\/\/msdn.microsoft.com\/en-us\/library\/dn935021.aspx).\nThe logic behind the feature is that you can query tables with-in your DB (like you are doing today) and also at any other DATA source like it is a table within the DB.\nFrom one SQL Azure DB to another we can not connect to it via linked server. However a second SQL Azure DB is referenced like any other external data source.\nMicrosoft thinking is that SQL Azure DB is another DATA sources for External tables ( like Hadoop, Blob storage ect..).\nYou can see the overview for this feature here : https:\/\/azure.microsoft.com\/en-us\/documentation\/articles\/sql-database-elastic-query-overview\/ .\nI am adding some scripts to make it all easy for you to run and test (here I will not elaborate now about the query performance and execution plans...).\nThe \"username\" and \"password\" should be the username and password used to login into the DB2 database.\n4 long years I am waiting for this !!!!!!\nIf you will do not have an object with the same name in DB2 - you will get error when query it saying \"The underlying error message received was: 'Invalid object name 'dbo.Customers224'.'.\"\nIf you add column in DB2 in the table. and you do not add it in the EXTERNAL Table... then you will not see the data. BUT an error \" The feature 'ALTER TABLE' is not supported with external tables.\" You need to drop and create the external table.\nthe first query - when creating the connection is extremely slow - make your coffee - relax and then all will be good.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ravaged by time, weather and neglect, Culebrita Lighthouse stands sentry over the Virgin Passage between Puerto Rico and the U.S. Virgin Islands. The unmanned solar-powered beacon casts its light from 305 feet above the sea atop the highest of three volcanic hills on Isla Culebrita, the easternmost of the Puerto Rican islands.\nFrom the crumbling stone-and-brick lighthouse, an observer takes in the panoramic view of two bays and their white-sand beaches below, the U.S. Virgin Islands 12 miles to the east, the island of Culebra a half-mile to the west, and Puerto Rico's main island 12 miles beyond that. Uninhabited and accessible by boat only, mile-long Culebrita (\"little snake\" in Spanish) receives visitors at two anchorages from which they can explore the island and snorkel on the reefs, languish in tidal pools, stroll its white-sand beaches or hike to the light. It is one of 23 islands and cays in the Culebra Archipelago and part of the Culebra National Wildlife Refuge - habitat for 13 species of migratory seabirds and endangered green, leatherback and hawksbill sea turtles, which nest there.\nThe Spanish crown built the lighthouse - one of the Caribbean's oldest - from 1882 to 1886. The structure consists of a graystone cylindrical tower flanked by two keepers' quarters and the oil and storeroom, which are gray stone trimmed with red brick. The Coast Guard automated the light in 1964, but the structure has been deeded to the town of Culebra, which can't afford to maintain it.\nHurricanes Hugo, Marilyn and Georges have taken their toll on the lighthouse. Its spiral stairway is in danger of collapsing, the roof is falling in, its dome has blown off and the light room windows are broken out. \"Lighthouse Digest\" has put the light on its Doomsday List of beacons threatened with extinction, yet Isla Culebrita and its 125-year-old lighthouse remain a prime destination for adventurous cruisers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Accurately defining the lunar cratering chronology is important for deriving absolute model ages across the lunar surface and throughout the solar system. Radiometric and exposure ages from returned Apollo and Luna samples have been correlated with crater size\u2013frequency distribution (CSFD) measurements or the cumulative number of craters of a certain reference diameter, usually \u22651 km in diameter (Ncum, where D \u2265 1 km), at the landing sites to anchor the lunar cratering chronology. However, the lunar chronology is only constrained by a few data points over the last 1 Ga. i.e., Copernicus, Tycho, North Ray, and Cone craters and there are no calibration data available between 1 and 3 Ga or beyond 3.9 Ga. Recently, radiometric and exposure ages determined from returned samples were carefully reviewed; that is, the x-axis of the lunar cratering chronology was revisited (e.g., St\u00f6ffler et al., 2006). Here, we propose to take this reinvestigation further by revisiting the y-axis of the chronology: we will perform new crater size\u2013frequency distribution measurements of the landing sites and the South Pole\u2013Aitken basin to test the validity of the chronology.\nWe have already published data on the young calibration points of the chronology (Copernicus, Tycho, North Ray, and Cone craters) and found a reasonably good agreement with Neukum et al.'s (2001) calibration points of the chronology, including the Copernicus data point, which previously did not fit the chronology. When reinvestigating the crater size\u2013frequency distributions of the older calibration points, other researchers found significant differences from the reference measurements that were used to define the initial chronology (e.g., Marchi et al., 2009; Robbins, 2014). Thus, they proposed a new lunar chronology that differs by up to 1 Ga compared with the previous chronology (Robbins, 2014). However, the newly proposed chronology suffers from several shortcomings.\nFirst, the count areas for the measurements are very large and most likely violate the prerequisite to date homogeneous units that were emplaced in one discrete event. Second, the accuracy of the new chronology is likely compromised by the fact that it is based on the production function of Neukum et al. (2001), instead of developing an updated model-specific production function. Third, the new chronology applies a quadratic fit to the data points, which is physically less plausible than the fit of the Neukum et al. (2001) chronology function. Therefore, it is now crucially important to revisit the crater size\u2013frequency distribution measurements of the areas used to define the old portions of the lunar chronology.\nAnother issue that is still heavily debated is the existence of the so-called cataclysm; that is, a peak in basin-forming events at about 3.9-4.0 Ga. In this respect, the South Pole\u2013Aitken (SPA) basin is of particular interest because knowing its age will shed light on the plausibility of the terminal cataclysm. Such a cataclysmic late heavy bombardment has been proposed to explain the large number of ~3.9 Ga impact ages documented in the Apollo and Luna sample collection. So, provided the age of the SPA basin is ~3.9-4 Ga, this would support some models of the lunar cataclysm hypothesis.\nUnfortunately, the age of this basin is not well constrained. Ancient lunar samples from the Apollo 16 and 17 landing sites, which clearly predate the local geology at these sites, as well as lunar farside meteorites Dhofar 489 and Yamato 86032 have been interpreted to indicate the formation of the SPA basin at 4.23 Ga (Garrick-Bethell et al., 2008). However, it remains unknown whether these samples were indeed emplaced by the SPA impact or by other impact(s). We propose to perform new crater size\u2013frequency distribution measurements to date the SPA basin and to further investigate and test the cataclysm hypothesis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're eating a healthy diet or having enough of these essential nutrients then you probably have nothing to worry about. But if you don't then it may lead to many hazardous health issues like high blood pressure or heart attacks and, even these high blood pressure symptoms are difficult to spot, but may include headaches, chest pain and difficult breathing. Thereby, natural blood pressure and parasite cleanse supplements are becoming increasingly significant because they are very effective and natural way to restore the health of body. Natural cleanse supplements are not only helpful for purifying the body system from unwanted waste and organisms, but they also help in rebuilding beneficial bacteria to prevent future occurrences.\nOn the other hand, after taking these natural supplements, one can reduce their risk of hypertension, as well as can boost their energy level because; they help in regulating hundreds of body system operations, including blood pressure, blood sugar, and nerve function. Even, according to the NHS, it has been found that blood pressure supplements are completely made from 100 % herbal ingredients which are able to relieve hypertension effectively and quickly, without the need allopathic medicines. So, don't think too much and assure the well being of your body by adding these valuable supplements in your diets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Shekhawati University BSc Exam Form 2017:- The Shekhawati University Sikar is one of the Reputed and famous universities India. This university is located in the State of Rajasthan. Now the official Notification of PDUSU BSc Exam Form 2017 online Released in the First Week of Next month. Both Private & Regular Students Apply for this Examination. ShekhaUni BSc part 1, 2nd year Exam Form last date Check here.\nThe Shekhawati University conducts the admission exam for the BSc course every year. The Exam form for the BSC exam of PDUSU is available on the official site of the University. The candidates those who are willing to apply for the BSc course of the Shekhawati University BSC 1st year Exam Form can download the online application form for the University Examination.\nThis exam is conducted for the candidates those who wants to get the place in the BSC course. The candidates have to download the Shekauni BSc Part 2 Exam form 2017 and submit it before the last date. The exam is conducted for the current academic year also.\nVisit the web portal of the Shekhawati University Sikar UnivExam Form 2017 or \"ShekhaUni Result.in\".\nThere appears the home page of the university.\nThe candidates have to search for the \"BSC Exam form 2017\" link.\nFill in the application form of the BSC course exam with needed details.\nUpload your photograph and signature in the desired format into the application form.\nPay the application fee for the exam through online.\nAt last, click on the \"submit\" button.\nClick on the \"Download as PDF\" to download the B.Sc Exam form.\nWe promise that we will update more details on our site, Visit Again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tourism out of Asia is growing, but travelers worldwide are still looking to trim costs leading to the rise in low-cost carriers and the now evident decline of full service airlines.\nHong Kong Express Airways Ltd. plans to transform itself into a budget airline this year, joining the growing ranks of Asian carriers seeking to profit from the region's boom in low-cost flights.\nThe full-service carrier's president, Yang Jianhong, told The Wall Street Journal it aims to unveil in late May the details of its plan to turn the airline into a budget carrier by the third quarter.\n\"We see rapid growth potential for budget carrier services as penetration in Asia remains low compared with more developed markets in Europe and the U.S.,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LG = OL You're wise and charming, stay blessed!\nInstead of trying to find the right person, try to be the right person. Get to a point where you love yourself, and you might be surprised at the company you choose to keep.\n\u062a\u0645\u06be\u0627\u0631\u06cc \u0646\u0627\u0631\u0627\u0636\u06af\u06cc \u06a9\u06d2 \u0688\u0631 \u0633\u06d2 \u0645\u06cc\u06ba \u062c\u0648 \u06c1\u0631 \u0628\u0627\u062a \u0645\u0627\u0646 \u0644\u06cc\u062a\u0627 \u06c1\u0648\u06ba..\nDon't be afraid of losing people. Be afraid of losing yourself trying to not lose someone else.\nIt's such a great feeling when someone just genuinely wants to talk to you and wants to know how your life is doing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Long connector with a loop either end and a swirling rose in the centre. This connector can be used to join beaded components and turn into a bracelet, necklace or earrings. Size: 33.5mm x 8.5mm x 5.5mm. Sold per piece, available in antique silver.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To keep dinghy in an upright position on swim step, once mounted with Dinghy Hauler Davits. These bars are adjustable from 20 inches to 40 inches and have 6 locking positions in between. Ideal stand off bars for any situation.\nAdd extra support to your davit system and protect your dinghy and larger vessel by installing stand off bars. Allows you to still walk on swim step because they keep the dinghy from having to rest against the transom of your larger vessel.\nCan attach to existing D-rings, eyelets, or other hardware or attachment points on your dinghy.\n3 lbs and 26\" x 2\" x 2\" shipping size.\nMade of 1\/2\" diameter Stainless Steel tubing.\nTo extend the reach of your swim step platform, view our Dinghy Davits Extension.\nPivoting Motor Mount Bracket for inflatable boats and hard shell dinghies, this \"pivoting\" bracket allows the boater to leave the outboard motor on the dinghy when raising dinghy to vertical position on swim platform because the bracket pivots to different angles. Eliminates need for storage for motor, no spilling of fuel or oil.\nStand-off bars come in sets of 2 along with 4 stainless steel eye hooks for fastening end clips to. Stainless steel clips on both ends of the rods for an easy and quick way to secure your dinghy. Holds dinghy securely in place, unlike the use of ropes, which can allow the dinghy to lean forward or aft of the boat and requires two lines for a secure upright position. When not in use, the stand-off bars can easily be stored in a locker.\nSecure attachment to vessel and dinghy.\nMade of high quality electropolished stainless steel.\nAdjustable to 6 different lengths from 20 to 40 inches long.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafdsk b/data_all_eng_slimpj/shuffled/split2/finalzzzafdsk
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+{"text":"We can use the STATIC macro to avoid defining every function twice.\nWe can also rearrange the circuitpadding.c functions so that we don't need to define them on top of the .c file.\nOh btw I did this on Friday. We actually hit a clang-specific test bug due to differing function return types, so I removed them all.\nI think this is done. Leaving in needs_review for asn.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So I'm looking to create files for (professional) print, which means working in CMYK rather than RGB. Can someone point me in the direction of a tutorial or other instructional that details how I can see a preview of what a Clip Studio Paint file will look like in CMYK?\nAlso, for the love of all that's holy, can someone explain WHY a 'professional' application like CSP -- ostensibly to be used for comic books, which are traditionally PRINTED on paper -- would not work natively with CMYK, the standard for ALL printed matter for the better part of a century?\nGo to the menu item View>Color Profile>Preview settings . Now I suggest you click the radio button called \"configure only on this window\"\nNow you can open the list of \"Profile for Preview \" and select any CMYK profile that you like . In my list it has also automatically included a bunch of profiles from my printer ... so I can also see how its going to look from my printer with differrent types of paper. With each profile that you pick the image on screen will instantly change to that profile.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MYMAHDI 16GB MP3 Music Player 1.8 Inch Screen 70h Lossless Sound, Support up to 128GB Micro SD Card Gold - KAUF.COM is exciting!\n2016 New UI,Easy mode.MYMAHDI Product.Lossless Sound.Come with High Valued Gift Box!\nWeights 1.1 oz.Smart but Big Music with mini speaker.Go into settings, play mode,and then Shuffle.Sync to playlists easily by Media Go (for free).\nDictaphone:Portable Voice Recorder with Noise cancelling 20 hours recording (even more with memory card).You may use it as a memo taker.Speak to it whenever any inspiration comes to your mind.\nMusic:Up to 4000 songs by built in 16GB memory capacity,supports up to 128 GB Micro SD Card for 12000 Songs!\nVideo support: AMV?AVI (Need conversion tool to convert, resolution 128*160).\nFast-charge:Full-charge time:2 hours.High quality battery support 70 hours of music playback when fully charged;Built in FM Radio playback.\nMYMAHDI - MP3 \/ MP4 Player ARE THEY WORTH \u00a312 ?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discover the newest discount offers with the best prices at our Kutting Weight coupons.\nMake your purchase at never-before-seen prices with Kutting Weight coupon code: 15% off your order. Be the first to discover the amazing bargains.\nDiscover a great bargain when you make purchases from , Women's sauna suit V3 for $69.99. Don't hesitate any longer.\nWonderful offer today: 15% off your order. Catch this great opportunity to purchase what you like at . Exclusive offers only for you.\nGet your favourite items at with a fantastic reduction. 10% off any order. It is definitely worth the money. Do not miss this amazing opportunity.\nDiscover this amazing promotion to receive a massive discount at for your orders: 20% OFF at Kutting Weight. Beat the crowd and buy now.\nThousands of products you need at : 10% off at Kutting Weight. Bring what you like at a lower price than ever.\nFind tons of quality assurance and top rated products at Primitives by Kathy: 30% off Any One Item. Get items you like for an affordable price.\nGet 300% First Deposit Bonus At Butlers Bingo.\nDiscover this amazing promotion to get savings so big at butlersbingo.com on your ordered products: Get 300% 1st Deposit Bonus at Butlers Bingo.. Big savings while they last.\nShop right away at Cherry Hill and save more with our coupons. Exclusive deal for you: $1 Off a Mini Golf Game For Groups of 20 Or More. Take action now.\nGrab this amazing offer on your purchases at derbysmack, Free Shipping. This sale will end soon.\nEnjoy last minute promotion and receive a top saving at zutti.com on your ordered products: Further Reductions. Apply code at checkout to receive this offer.\nPopular now: Save Up to 50% Off Clearance Items. Never spend more than you have to and use our Kutting Weight coupon & sales. Simply click on one of the 2 available Kutting Weight Promo Codes & enjoy shopping.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Broadly, I am interested in the ways in which humans interact with their environment. I've help perform or have led greenhouse, field, observational, and simulation experiments. My academic background is in Environmental Biology and Psychology.\nHow should we incorporate connectivity when designing a Marine Protected Area network?\nCanada has committed to protecting \"10 per cent of coastal and marine areas, especially areas of particular importance for biodiversity and ecosystem services, are conserved through \u2026 systems of protected areas and other effective area-based conservation measures\" 1 in order to conserve biodiversity. The effectiveness of these protected areas depends their design and implementation2. Specifically, these protected areas should be connectivity to allow individuals to move between them.\nIn my MSc, I used the Canadian Pacific Ocean as a study system to understand where these Marine Protected Areas should be placed to ensure connectivity between protected areas for multiple species and across time.\nIn a greenhouse experiment, I manipulated germination timing of competing plants and quantified how this affected their ability to co-exist\/outcompete each other. Analysis ongoing.\nHow will climate change impact species ranges?\nA common prediction is that climate change will result in a contracted southern range and extended northern range for species above the equator. 5, 6 I tested how climate change could affect a host of Boreal forest species in Northern Ontario and how new species could move into this region. Analysis ongoing.\n1. Convention on Biological Diversity. 2010. Tenth meeting of the Conference of the Parties to the Convention on Biological Diversity, Decision X\/2. Nagoya, Japan. 18 \u2013 29, October 2010.\n2.Gaines, S.D., Lester, S.E., Grorud-Colvert, K., Costello, C., and Pollnac, R. 2010. Evolving science of marine reserves: New developments and emerging research frontiers. PNAS 107: 18251-18255.\n3. Chesson, P. 2000. Mechanisms of Maintenance of Species Diversity. Annual Review of Ecology and Systematics 31: 343-366.\n4. Godoy, O., and Levine, J.M. 2014. Phenology effects on invasion success: insights from coupling field experiments to coexistence theory. Ecology 95: 726-736.\n5. Parmesan, C., and Yohe, G. 2003. A globally coherent fingerprint of climate change impacts across natural systems. Nature 421: 37-42.\n6. Davis, M.B., and Shaw, R.G. 2001. Range shifts and adaptive responses to Quaternary climate change. Science 292: 673-679.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..640a995d77c9c2a26f78fe9923ead2c24e028d1b
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+{"text":"Before you start a new fitness regime you first must decide what it is you want to achieve. This is particularly important if you are a beginner.\nWhat you must try to avoid is becoming confused by the endless training options available to you. There is so much information available that you may find yourself \"bogged down\" by it all and unable to get your routine started.\nBy following this simple beginners guide to muscle building you will gain the basics you need to start building that lean muscle mass.\nWhen choosing goals you need to make short-term goals that are achievable.\nOf course you should have a long-term goal but these short-term ones should enable you to slowly reach your final goal.\nIf these short-term goals are unobtainable you will quickly become disheartened and likely to quit. While those that are obtainable are likely to build confidence, enabling you to reach the next goal.\nWhen choosing goals it is also important that you understand what you have to do to achieve them; for instance if you want to build muscle you need to ensure you are eating enough protein. Whereas if your aim is to lose fat or to cut then you need to ensure you cut your calorie intake enough to lose the fat without losing the hard-earned muscle mass.\nMesomorph \u2013 who have a larger bone structure, large muscles and a naturally athletic build.\nEctomorph \u2013 a light build with small joints with lean muscle.\nEndomorph \u2013 short with thicker arms and legs.\nOnce you find out what body type you are you will be able to customise your own muscle building program.\nYou may have hear the expression \"a six pack is made in the kitchen\", well this is true for muscle building in general.\nIf you are not taking in the right nutrients from the food you eat you will not see those gains you desire.\nUnsurprisingly, protein is incredibly important for building muscle as it is used to help repair those muscle fibres that are broken when training.\nThese repaired muscles become stronger, allowing you to see strength gains too.\nA quick rule is that you should consume 1-1.7 grams of protein for every pound of body weight.\nThis type of workout do not focus on one particular area, with the aim to work as much of your body as possible. This workout is best for those who do not have time to visit the gym almost every day, 2-3 visits weekly.\nThe upper split will focus on the muscles of your chest, shoulders, arms and back. While the lower split will focus more on legs, lower back, bum and abs. This type of workout will take between 3-4 sessions in the gym weekly.\nThis type of workout will focus on one major muscle group and the minor muscles surrounding it. Those that perform this type of workout are able to thoroughly workout an area and are then able to rest it while working on another muscle group. With this workout you could in theory workout 6 times a week, with 1 rest day.\nOf course there are plenty of alternatives and variations to each exercise.\nTo avoid injuring yourself you should ensure you are lifting with proper technique before you starting adding too much weight to the bar.\nOf course you need to be pushing yourself, but lifting more than you can manage will only lead to a loss of technique, which can cause serious injury.\nThis is particularly important with exercises such as the deadlift or squat, as you could cause long-term damage to your spine.\nTo see muscle growth you need to give them time to repair. This means you can't workout everyday or the same muscle groups on consecutive days.\nWhen training you should at the very least have 1 rest day, where you will be doing no training at all.\nThe final tip, which you may find helps speed up the rate that you gain muscle mass would be to take a supplement.\nAt the very least you should think about taking a whey protein supplement as you may find it difficult to get enough protein into your body just through your diet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since M44 Media's inception, our in-house experts have been helping law firms and attorneys navigate the digital marketplace's continual evolution alongside online media. We are a team of creatives, developers, marketers and lawyers that are all experts in our respective fields. By solely catering to the legal industry, we have developed a deep and thorough knowledge of what makes law firms successful online.\nFrom integrated marketing services, strategic consultation, results-driven interactive design to cost-effective marketing solutions coupled with detailed data tracking and reporting: M44 Media offers novel multidimensional marketing. We help law firms take in more clients and revenue through Search Engine Optimization\/Marketing (SEO\/SEM), Video SEO, Per Click (CPC\/PPC) Campaigns, Social Media Marketing (SMM), along with the best Infographics, and much more.\nM44 Media understands the importance of website design and how it impacts your online presence. With years of experience in designing law firm websites, we can create a site for you that makes a great first impression and is easy for clients to navigate and use.\nProspective clients have a short attention span, so getting them the information they're looking for and holding their attention is what keeps them from moving on to the next attorney. Your website is the digital face of your firm. A well-designed site can help create immediate trust in your firm with prospective clients. Contact us today for a free quote or information.\nChoosing the right marketing firm for your law practice is one of the most important decisions you will make. The reality is that while there are a lot of advertising companies out there, most simply aren't very good. If you're tired of broken promises and underachieving marketing campaigns, we are here to change your advertising experience.\nTo win in the competitive legal advertising market, you need someone with an extensive track record of successful law firm marketing. At M44, we have over 20 years of law firm marketing experience. That's the M44 advantage. As attorney marketing continues to evolve, our office has evolved with it. From our beginning, we have strived to create long term business partnerships with our clients.\nWe have what it takes to drive your company above and beyond your current marketing expectations. Here are some things that really set us apart from your typical legal marketing company.\nOver 20 years of law firm marketing experience running some of the largest and most successful marketing and advertising campaigns in the industry.\nBecause M44 does not work with your competitors, we are able to build and develop customized and unique strategy for each and every client. We tailor your campaign to your area of specialty and specific market.\nUnlike other companies, M44 gives our clients full ownership of their websites and content. Our clients own their domain names, design and all content, even what we produce for your site.\nM44 does not work with competing clients. Our team works nationally with one law firm per region in their area(s) of practice. This has allowed us to build long term business partnerships.\nWe have a unique understanding of the advertising cycle after working in and managing phone rooms, intake staff, and all aspects of the client retention process. This is a big part of the M44 advantage, a true understanding of the complexity of the legal business and how to reach the targeted client market you desire.\nWe still maintain all of our original clients we started with in 1998, without requiring or pressuring them to sign a long-term contract. At M44, we are confident in the service we provide, and believe our service and results are second to none.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Simon Bateman set up his gym concept less than a year ago for people who feel intimidated by, or just hate going to a normal gym. Focusing on getting a quick dose of fitness and then heading back to the office, Your Gym is run by personal trainers with group core fitness classes as extras.\nA virtual show of hands please for all those who can honestly say they have regularly worked out this year so far, have shed the Christmas bloat & pounds and are feeling more like Rocky Balboa pre the fight of his life?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you just bought a new home or have lived in yours for years, home improvement seems to come with the territory. It's understandable \u2014 you want your home to feel like it's your home, which means matching your tastes. That said, it pays to think about how much resale value any improvement is likely to bring. After all, it's your biggest investment.\nWith return on investment in mind, here are five exterior improvements that not only pay you back, they'll also pay dividends every time you (and the neighbors) gaze at your gorgeous home.\n1. Upgrade landscaping \u2014 What's one of the fastest things home makeover shows do to spruce up a house? Put in new plants and lay down fresh mulch. Just a few new flowering plants and shrubs make any home look cared for. And that layer of mulch gives a fresh look that welcomes any buyer at your doorstep.\n2. Replace siding \u2014 Not only does siding protect your home, it's one of the top features that contributes to curb appeal. If your siding is damaged, old and worn, replacing it can add significant value. And if you go with natural siding materials like pine, cedar or redwood, you add a boost to your home's look that any buyer is bound to notice.\n3. Install outdoor lighting \u2014 More than just a welcome safety feature, exterior lighting gives your home a warm, inviting glow. Landscape lighting highlights architectural details and helps would-be buyers envision relaxing evenings outdoors. The best part? It doesn't need to be expensive. You can hire a professional to do a simple landscape light plan, or you can go the solar garden light route with a quick trip to your local home improvement store and do it in an afternoon.\n4. Paint \u2014 Nothing's faster at improving appearance than a fresh coat of paint. And it's wonderfully affordable if you do the job yourself. Resale tip: go with popular exterior paint colors like white or off-white, grays or neutrals.\n5. Replace old windows \u2014 If the eyes are the windows into the soul, then windows are the way to let light and soul-nourishing views into your home. Beyond their beauty, windows also affect your home's energy efficiency. If yours are old and inefficient, consider meeting both needs at once using wood replacement windows.\nThese are just some of the top ways you can enhance your home's curb appeal and value. For other tips on exterior improvements and interior design ideas visit wood-naturally.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Debt consolidation New Canaan works, try it today. Debt is not a new phenomenon. Problems with debts have been here for as long as humanity itself. However, how we manage our credit card debts has evolved over the years in New Canaan Connecticut. For example, people no longer have to slave for feudal lords in New Canaan just because they could not pay up their credit cards. Nowadays, creditors are civil about how they collect their credit cards.\nYou will never in New Canaan have to deal with bankruptcy lawyers and\/or courts in New Canaan if you learn how to manage your credit card debts well. This is especially true in the modern economy in New Canaan that we live in. Now, you can actually consolidate your debt with credit card debt negotiation programs instead of having many creditors. debt consolidation New Canaan comes with many numerous credit card debt counseling benefits.\nNegotiating with your creditors in New Canaan can be a very frustrating experience. They rarely listen in New Canaan, in fact, they usually ask for accidental higher interest rates when you ask in New Canaan for a longer repayment period on your quick loans. Conversely, they will ask for a shorter debt consolidation repayment period on your online loans if you ask for a lower interest rate. This debt consolidation issue will go on and on just because you are just an individual living in New Canaan CT trying to make ends meet.\nHowever, the presence of a credit counselling company changes everything. You will now able to negotiate better terms with your debt consolidation creditors if you decide to consolidate your debts. This is because the presence of a debt negotiation service assures your creditors that they will receive their money loans payments in due time. This makes the debt consolidation creditors more flexible as you negotiate credit card debt counseling terms with them.\nFree help for your New Canaan debts.\nLower your monthly consolidation payments and get out of debt fast. Our debt consolidation services and credit consolidating professionals can help you get debt free fast. Simply use the form on the right to contact an expert debt relief agent who can help you gain commercial control over all of your credit card debt or loan drawbacks today! Anyone from can use our credit counselling services, free debt consolidation services from DebtCafe USA.\nIf you are like individuals residing in New Canaan CT, you are starting to feel the pinch from the market collapse in New Canaan and you want to know how you can save on interest charges with debt consolidation in New Canaan and get out of debts. You may have tried to obtain debt management products the local consolidation lending institutions in New Canaan CT but the banks consolidation just aren't willing to assist and monetary companies want way too much in interest in New Canaan and other up front fee's. At DebtCafe we have the perfect credit consolidating solution. Your debt consolidation creditors realize the value in New Canaan of obtaining their money back in New Canaan so they're willing to assist us by decreasing in New Canaan, or even removing, late fees and additional interest costs in New Canaan when you use one of our credit card debt relief services.\nOur debt counselling experts will explain to you how debt consolidation can help you minimize or eliminate credit cards charges, and simplify your credit consolidation repayment with just a single credit card debt settlement payment each month. If you are from and you need to set up your free credit card debt settlement analysis, simply complete the FREE debt consolidation form above.\nOnce the free credit counseling form above has been completed in New Canaan CT, you gain access to your very own network of credit card debt and credit card debt consolidating specialists who will run a simple but effective commercial assessment of your personal debts situation. This will be used to figure out the very best credit card consolidating options that works for you. It is vitally important in New Canaan for all involved to completely understand where you are at with your debts situation because without these particulars, the counselors in New Canaan can not offer a correct evaluation and determine what is the best customized New Canaan CT debt consolidation program that gets you out of your endeavor while putting extra cash in your pocket with debt consolidation at the same time. By finishing this free debt consultation, the credit management team will be able to assist you save thousands in New Canaan in interest and possibly principal amounts as well in New Canaan CT.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagvqf b/data_all_eng_slimpj/shuffled/split2/finalzzzagvqf
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+{"text":"Sir, I want to get job of Sub Inspector in Delhi Police. Can you please provide me its previous year question paper?\ndelhi police sub inspector (female) recuitment?\nAs you are looking for the previous year paper of the Sub Inspector in Delhi Police.\nSSC conducts the Delhi Police Sub Inspector Exam.\nDelhi Police Sub Inspector exam paper consists of two papers of total 400 marks.\nI have attached PDF of the previous year paper of sub inspector examination.\nYou can download it & get the previous paper which will help you to see the pattern of paper & questions.\n1. Which one one of the following is not a quantitative credit con\u00actrol techniques?\n2. Which one of the following state\u00acments is correct?\n9. Mineral constituent of chloro\u00acphyllis.\n10. Which of the part of tongue bears cells for sour taste?\n12. In which vertebrate oxygenated and deoxygenated blood gets mixed?\n13. Which of the following circuit el\u00acements is used to 'block' DC in an electronic circuit?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CLEVELAND \u2013 The Cleveland State Vikings Men's Basketball program has a new Director of Basketball Operations, as the college today announced that A.J. Montanaro has taken over the role.\nMontanaro has been part of the men's basketball program for the last three seasons. In addition to Montanaro taking over, the college today also named Jarvis Huntley as the Men's basketball video coordinator.\n\"A.J. has worked extremely hard during his three years at Cleveland State and shown a strong dedication, loyalty and work ethic,\" Vikings coach Gary Waters said.\n\"He has an extensive knowledge of the game from being at West Virginia and being coached by a well-respected high school coach. I expect A.J. to continue being a valuable asset to our program.\nA 2014 graduate of Cleveland State with a Bachelor's of Education, Montanaro was the head manager at CSU from 2012-14 and served as the video coordinator of the men's basketball program last season.\nDuring his time at CSU, the Vikings advanced to the postseason two times, including the second round of the CollegeBasketballInsider.com tournament in 2015.\nAs video coordinator last season, Montanaro worked directly with the coaching staff by planning, overseeing and coordinating the filming of all practices and games, including video and paper scouts.\nPrior to arriving at CSU, Montanaro spent two years (2010-12) as a student manager with the West Virginia men's basketball program.\nMontanaro is currently pursuing a Master's of Business Administration at Cleveland State.\nHuntley was a 2011-12 Junior College Academic All American, and was a Region 10 All Conference Performer at Denmark Technical College in Georgia before transferring to Notre Dame College where he averaged 14.8 points and 4.5 rebounds as a senior. Huntley was a first team all-region and all-county selection at Meadowcreek H.S., helping the team to a pair of regular season titles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For anyone jobless on New Year's Eve, the Times Square countdown on TV probably looks like a Doomsday Clock vigil.\nBeing jobless during Christmas is often tough and painful, but a jobless New Year's Eve can make it all seem so much worse.\nOn top of the endless \"silent night\" of unanswered job applications there is the forced downsizing of holiday gifts for and dinners with loved ones and friends, as \"The Twelve Days of Christmas\" and the parties get reduced to virtually none.\nThen there are all the bittersweet scenes, images and imaginings of more fortunate upbeat revelers toasting the booming stock market that's left the jobless behind and contemplation of the coming year that you fear will too.\nFinally, there's the coup de grace[lessness]\u2014the jobless New Year's Eve. As the culmination of all that stress and perhaps worst of all are the gathering sweet poignant strains of \"Auld Lang Syne\" that now strain an unsuccessful job-seeker's faith in time, which, increasingly, seems to open rather than heal wounds.\nThat's how gloomy it can get\u2014unless there's another way of looking at it all, something akin to the fresh, hopeful and empowering perspectives Charles Dickens' Ghosts of Christmas Past, Present and Future ultimately offered Scrooge.\nGhost of New Year's Eve Past: [Rattling chains, entering from a mirror] \"Woooo-oooo\u2026Ebeneeeezer Scrooooge, yes, you've had a tough year\u2014a very tough year. But you've almost certainly had better ones. That's why this last one seems so bad\u2014by comparison with some other, much better New Year's Eves past. Don't forget those.\nBesides, past job markets, wealth and opportunities may be entirely gone; but you aren't. Certainly, those past better years and you count for something\u2014more than something, actually quite a lot.\nIf, however unimaginably, you believe you've never had even one happy New Year's Eve, this one may seem bad only in comparison to the celebrations of envied real or imagined others. Well, if you want to play that game, play it fairly and be sure to include comparisons with those far worse off than you are or imagine yourself to be. That should help even the score.\nMany, if not most job opportunities of some past New Year's Eve, like that year itself, are gone forever. But your intangible resources are not\u2014most especially and importantly, your will.\nGhost of New Year's Eve Present: [Whooshing through a closed window] \"Boo [hoo]!\u2026Sure, tonight it looks bad, Scrooge. You've got no job interview or offer this evening. But, you're not supposed to\u2014it's New Year's Eve. There may be an insanely busy shorthanded local tavern that needs an extra hand, but the job you want is something else, somewhere else\u2014in a business that's not open tonight.\nBesides, if you really need a job, any job, wait until everything is open again, the day after tomorrow. There will be lot's more opportunities then.\nSo, even though it's not Thanksgiving, be grateful that you're in a warm, dry place you call \"home\", if you are, and even more grateful if there's a warmhearted someone here with you. Ah, but you don't even have a cat.\nGhost of New Year's Eve Future: [Appearing and hovering over the sofa] \"Scrooooge\u2026.Ebenezer Scroooooge!\u2026New Year's Eve\u2014past, present and future\u2014is an arbitrary marker. 'New Day's Eve' would serve just as well and probably more helpfully, to the extent that it requires taking stock, making resolutions and keeping them much more often.\nInstead of dwelling on an entire past year or fearing the next one, mindfully\u2014if not as gratefully as possible\u2014focus on the day, even the hour or the minute.\nForget about having a 'Happy New Year!' Go for a 'Happy No Year!'\u2014i.e., a 'Happy New Day!\" instead. You won't have to wait as long for a chance at the next one and will have 364 more chances than a single New Year's Eve offers.\nAs for your future, I won't be trite and preach the gospel of 'a glass that is half-full'; nor should you brood over a 'glass that is half-empty'.\n1. 1. You still have a glass.\nNote: See my \"Merry Un-Christmas!'\u2014Useful Office Applications of an Alice-in-Wonderland Tactic\" as a companion article presented in the same spirit of alternative thinking about this holiday season.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Flocking, Mating, and Nest-Building Habits of the Prairie Horned Lark. 131-141.\nNotes on the Home Life of the Turkey Vulture. 142-145.\nThe Autobiograhpy of Otto Widmann. 146-155.\nNotes on Some Less Common Birds of Douglas County, Kansas. 156-158.\nNotes From La Anna, Pike County, Pennsylvania. 159-168.\nTern Banding on Lake Erie. 184-186.\nThe Great Blue Heron in Tuscarawas County, Ohio. 170.\nThe Lark Bunting in Iowa. 170.\nNesting of the Florida Red-Winged Blackbird. 170.\nLoon Banded in Pennsylvania Killed in Ontario. 170-171.\nRuffed Grouse Captured by a Screech Owl. 171.\nBirds Enjoy Bathing During the Winter Months. 171-172.\nThe Golden Eagle Nesting in Woods County, Oklahoma. 172.\nA Few of the Less-Known Summer Residents Near Toledo, Ohio. 172-173.\nAdditional Notes on Birds of Vilas County, Wisconsin. 173-174.\nSome Notes on Pileated Woodpeckers. 174-175.\nData Concerning the Breeding Range of Certain Marsh Birds. 175-177.\nAn Albino Red-Tailed Hawk. 177.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A ROC curve is a graph that plots true positive rates against false positive rates for a series of cutoff values, or in other words, graphically displays the trade-off between sensitivity and specificity for each cutoff value. An ideal cutoff might give the test the highest possible sensitivity with the lowest possible false positive rate (i.e., highest specificity). This is the point lying geometrically closest to the top-left corner of the graph (where the ideal cutoff value with 100% sensitivity and specificity would be plotted). Picking the ideal cutoff score is, to some extent, dependent on the clinical context, that is the purpose for which the tool will be used. The area under an ROC curve can be used as an overall estimate of its discriminating ability and sometimes is expressed as accuracy. The area under the ROC curve is equal to the probability that a test correctly classifies patients as true positives or true negatives. Greater areas under the curve indicate higher accuracy. To further clarify, a discriminant test might have an area under the curve of 0.7 while a nondiscriminant test has an area under the curve of 0.5.Rosenberg, L., Joseph, L., & Barkun, A. (2000). Surgical Arithmetic: Epidemiological, Statistical and Outcome-Based Approach to Surgical Practice. Georgetown, Texas: Landes Bioscience.\n\"Receiver Operating Characteristic (ROC) curves\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"HELP! i cant log in!\nCant sign up for account .\nProblem logging back in after a disconnect.\nAm i IP banned of something !?\nGetting Error \"Cannot Connect to the Server\"\nMy account was banned..what should i do??\nerror, your username or password is incorrect????\nGuild Hall Eidolon Summon Time?\nIm stuck in logging in...help me with this pls.\nerror, your username or password is incorrect.\nData backup in progress - login later.\nLag is getting worse each week?\nConverter is it Still Working ?\nAchievement Tracker \"Can't be removed.\"\nPls help.., Game always crash during verifying account section!\nProblems with the website & client.\nI can't play in full screen?\nWhy can't I donate using paypal?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Drilling results soon at Mars Aurora Tank \u2013 a Prominent Hill look-a-like?\nRight now AON is drilling two holes at Mars Aurora Tank to test the potential IOCG resource it's identified there.\nWe are anticipating first assay results from the drilling programme any day now \u2013 we should soon have a better idea what lies below at Mars Aurora Tank.\nAs part of the same campaign, AON are also drilling on the Eaglehawk project where it's earning a 75% interest in a JV with Mincor. New targets have been identified at Eaglehawk using geophysical modelling and a review of historical drilling data. These new targets represent an 18km2 extension to the Bundi South IOCG target , a 'kilometres' wide gravity anomaly that sits within the Eaglehawk JV.\nThe above rock was picked up from the ground and contains 50-55% Fe \u2013 highly indicative of IOCG potential down below... Overall, this drilling programme at Eaglehawk JV is aimed at furthering AON's understanding of the Bundi South IOCG prospect and widening the area it's exploring for IOCG mineralisation.\nBoth drill cores have brecciated rocks (fragments of minerals mashed together) with iron oxide alterations. We're not geologists at The Next Mining Boom , but looking at those two samples side by side shows us that AON and HPX may be onto something at Commonwealth Hill. BHP drilled dozens of holes in order to find the Wirrda Well IOCG deposit: 26 diamond holes between 1980 and 1990, plus 25 more diamond holes between 2008 and 2012. AON has only drilled 5 to 10 holes over several prospects. We are hoping AON can replicate a discovery like BHP's, but it will take further drilling and perseverance. The only way to find out is to keep drilling and exploring.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Eliza Jane Howell Bridal Collection at Marrime Bridalwear has been designed for the modern day bride who would love to wear a dress that oozes the glamour, sophistication and decadence of the golden age of fashion. Taking inspiration from exquisite vintage gowns, the Eliza Jane Howell collection is the epitome of grown up glamour. Available at our Newcastle Upon Tyne and Kendal boutiques. Prices from \u00a31000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 10 finalist has already announce. Want to know more? visit our blog!\nWe don't want you to miss the final day where the top finalists will pitch their solutions on stage and showcase their products in the exhibition area.\nProtecting innovations curated by startups at every stage of their life-cycle is of utmost importance.Patents are one way to carry out this. Patents are exclusive intellectual property rights (IPR) that can effectively prevent others from making, using, offering for sale, selling or importing a patented technology. As intangible property, patents supplement good reputation and credibility and build authenticity and authority in the marketplace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Toggle \/ clasp, antique copper-plated \"pewter\" (zinc-based alloy). 12mm round. 5 sets per pack.\nToggle \/ clasp. antique copper-plated \"pewter\" (zinc-based alloy). 16mm double-sided round with rope design. 3 sets per pack.\nToggle \/ clasp, copper. 18mm textured round with twist. 1 set per pack.\nHandcrafted copper toggle clasps are lightweight and offer versatile design potential. The textured surface adds an edgy, modern look to design. Due to the handcrafted nature of the clasps, size, shape and colour may vary slightly.\nToggle \/ clasp, copper-plated brass. 16mm round. 3 sets per pack.\nToggle \/ clasp. Tierra Cast brand. Antique copper plated hammered 16.75mm round. 1 set per pack.\nToggle \/ clasp. Tierra Cast brand. Antique copper-plated base metal. 22mm smooth round. 2 sets per pack.\nCopyright \u00a9 Preciouz Piecez 2012. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"New Toy Story 3 Character is Called\u2026 Lotso!\nThe teddy bear that made an appearance in the Up trailer during the lift off scene was recently revealed to be a cameo from a Toy Story 3 character.\nNow Pixar Collector, a brand new site bringing you the latest Pixar collector's news, has dug up some information on the name of this bear. It's Lotso and he's got a big part in the film.\nTo back up this claim, a Slash Film commenter is reporting that the cuddly looking, pink bear is Toy Story 3's main villain! I don't think you saw that coming did you? I also don't know of the legitimacy of this claim, but it seems like something very \"Pixar\" to do. Get a character that is as unlikely as this teddy and turn him bad. Makes you wonder what his motivation is, probably nothing like Sid's or Stinky Pete the Prospector's.\nAnyhow, check out PixarCollector.com, it's a brand new site run by Pixar Planet member legionsofgotham who has brought us exclusive pictures of upcoming toys in the past. He'll be updating regularly with news on everything that's collectable with Pixar's name on it.\nVintage Lots-o'-Huggin' Bear Commercial Uncovered!\nET Premieres Toy Story 3 Trailer Footage!\nToy Story 3 Ranks #1 in Time's Best Movies of 2010!\n0 Responses to New Toy Story 3 Character is Called\u2026 Lotso!\nLotso surprises still to come with Toy Story 3, I bet.\nWow, nice website! I love it. I'll have to post up my collections on that site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"The Assassination of Jesse James\"\nTimeless is an odd series. Tonally, it's a bit all over the place, sometimes a rambunctious time-travel heist, sometimes a moody sci-fi drama, and sometimes an Alias-style, twist-driven story of nefarious secret organizations and double agents and so on. In all of these shows, characters and history sit at the center, and while the whiplash between styles ensures that the show isn't often predictabe, it also rarely feels settled, complete, or fully-formed.\nWith \"The Assassination of Jesse James,\" Timeless bucks that trend. As with \"The Alamo\" and \"Party at Castle Varlar,\" the latter of which was also written by Jim Barnes, who pens this week's installment, the episode manages to balance the silliness, trip-to-Wikipedia historical nuggets, and overall mythology with a story that roots the characters in struggles that matter. It's pretty simple, really. They just raise the stakes, in big ways and small, and tie the struggles of the characters to a theme that resonates historically. It's not sensational, but it's impeccably made, without a scene or moment wasted. Add in a few solid quips and one unexpectedly funny shot of Abigail Spencer falling over a horse, and you've got Timeless as, perhaps, it will be: a history-loving time travel story that takes its characters seriously and itself, not at all.\nIt's not without flaw, of course. The episode opens with a trio of scenes that set up exactly what will be burdening the minds of the three leads, and while each is effective in its way\u2014Rufus's especially so, given how unexpected it is\u2014they don't exactly run together smoothly. Still, it gets the job done, and in no time we see Wyatt meet the man who (probably?) killed his wife (\"Not unless I had a time machine\"), drop into Lucy's nightmare about her sister, whose birthday she forgot, and watch Jiya excitedly tell Rufus she's the next Lifeboat pilot. That one's particularly engaging, given that it's the direct result of Rufus's line drawn in the sand in the preceding episode.\nEach neatly ups the stakes for each of our leads, particularly in Rufus's case. His scene with Conor Mason (Paterson Joseph) adds a deadline into the mix: It'll take six months to train Jiya, at which point Rufus becomes expendable. That's a lot to pack in before the credits, and that's not even the end of it. Next up: Jesse James (Daniel Lissing), not getting killed while dusting a painting thanks to the timely arrival of Garcia Flynn. It's all remarkably efficient for Timeless, which has often spent a lot of time on exposition and vague worldbuilding. Instead, we've got three character focal points and a pairing that will last for most of the episode (and give Goran Visnjic more of substance to do than he's had in the series thus far).\nWhile Lucy, Rufus, and Wyatt have to confront some hard truths about themselves, Flynn is gaining some murderous insight from Jesse James as well. It's a delightful surprise to see Timeless address the fact that Flynn's plan is needlessly violent, given his aims; it turns what seemed like a massive plot hole into a clever piece of character development. It also fills the time as James and Flynn make their way to the \"treasure\" on Flynn's map, and lo and behold, it's a person. It's not even a member of Rittenhouse. It's another pilot, and of all the clever parallels \"The Assassination of Jesse James\" pulls off, this one is perhaps the most satisfying.\nEmma (24's Annie Wersching) makes twin appearances here, first in her brief scenes with Flynn as he recruits her to Team Evil Time Travelers, and second in footage Jiya yanks from the Mason Industries servers. We learn that Emma faked her own death to escape Rittenhouse and Mason Industries, and she's been hiding in the Old West ever since. The trick of introducing a second pilot as a threat, in more ways than one, to Rufus while also planting a second pilot for Flynn's team is a neat one, allowing for an incredibly efficient and effective exposition drop that allows us to see Flynn's recruitment of Emma while we watch Jiya discover that a woman's death was hidden from her. It's a terrific and slightly frightening sequence, masterfully constructed. Cheers to Barnes and director John F. Showalter, because this worked like gangbusters.\nIt all leads up to the final showdown, of course. After Grant dies, Wyatt and Bass Reeves face off about whether or not it's best to kill Jesse James, who makes a point of disarming himself and surrendering (after blowing the cabin to pieces with a piece of high tech weaponry, no less). Of all the surprises in this episode, the revelation that it's ultimately Lucy who kills Jesse James is the greatest. Abigail Spencer's terrific in this show, week in and week out, but this is easily her best moment. Knowing what happened to her sister, and what could happen to others, she safeguards history and takes a life. It's stirring stuff.\nThere's a coda, of course, with Jiya finding herself threatened by Mason, Wyatt appealing to Rufus to help him steal the Lifeboat and jump back to stop his wife's killer (without killing him), and Lucy finding herself in the pages of history as Jesse James's killer, and it's all great, but the heart of the episode stops when Lucy pulls that trigger. Timeless isn't like this every week, but if this is an indicator of where the show could go, then the world had better start calling out for its renewal.\nSo if Lucy remembered Amy's birthday in her dream, doesn't that mean she remembered?\nTimeless continues to be progressively more thoughtful in the ways it addresses race. I particularly liked Rufus's tone-deaf questioning of Grant. Even the most thoughtful of us can be idiots sometimes.\nThe look on Rufus's face when he realizes the Lone Ranger was black is incredible. The fact that I, and I'm guessing lots of other people watching, didn't know that is horrible. And \"Tonto\"? Good lord.\nWhy didn't anyone else tell Jiya about Emma? You think there'd be some sort of memorial.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"16 Mar Lorde - Hold My Liquor [Kanye West Cover] (flowerschangeeverything.comheads) .. Frank might rap a little on a song but I can't recall a a legit frank rap song. 18 Jun Hold My Liquor Lyrics: I can hold my liquor \/ But this man can't handle his weed \/ Dark and lonely now \/ On Chicago, south of town \/ I'm on to. I hold my liquor \/ But this man can't handle his weed \/ Dark in the moonlit now, Chicago south of town \/ I love Indiana \/ I'm heard in the radio now \/ I can't handle .\nLorde - Hold My Liquor (Letras y canci\u00f3n para escuchar) - I hold my liquor \/ But this man can't handle his weed \/ Dark and lonely now \/ On Chicago, south of. 9 Sep Watch the New Zealand pop star perform the Yeezus cover. 11 Sep Lorde continues to blow our minds with this cover of Kanye West's song, 'Hold My Liquor'!!!.\nLyrics to \"Hold My Liquor\" song by Kanye West: I can hold my liquor This man can't handle me But this man can't handle his weed On Chicago, south o. 9 Sep Rising singer-songwriter Lorde appeared at the iHeart Radio Festival in her native New Zealand over the weekend, and her setlist included a. 22 Dec Stream Lorde - Hold My Liquor (Kanye West Cover) by Konman from desktop or your mobile device. 07 Kanye West Hold My Liquor Slow N Throwd mp3 Lorde Hold My Liquor Kanye West Cover Lollapalooza Brazil mp3 but showing only top 10 results only (Due to API limit restrictions we cannot show you more than 10 results). I hold my liquor. But this man can't handle his weed. Dark and lonely now. In Chicago south from town. I love Indiana I heard in the radio now. I can't handle no .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"YOUR USE OF THIS WEBSITE IS GOVERNED BY THESE TERMS AND CONDITIONS. Please review these Terms and Conditions of Use. Your use of the ChocolateFountainSales.com Website constitutes your agreement to follow these rules and to be bound by them. If you do not agree with any of these Terms and Conditions, do not use this Website.\nTHE TERMS AND CONDITIONS ARE SUBJECT TO CHANGE WITHOUT NOTICE. We reserve the right to update or change these Terms and Conditions of Use at any time without notice. Your use of this Website following any such change constitutes your agreement to follow and be bound by the Terms and Conditions as changed. For this reason, we encourage you to review these Terms and Conditions whenever you visit ChocolateFountainSales.com.\nDAMAGE IN SHIPMENT: Any product purchased from ChocolateFountainSales.com that is received damaged, must be reported within 2 business days after receipt of the item to be eligible for repair, replacement or reimbursement, as reasonably warranted by the situation, by the shipping company. The shipping company is responsible for your item after it has been collected for shipment from ChocolateFountainSales.com. Please inspect your item immediately upon receipt; if shipment damage is suspected, note the condition of both the merchandise and packing material with the courier. SAVE ALL PACKING MATERIALS for inspection by the courier. Contact customer support at (208) 356-4101 or send an email to info@ChocolateFountainSales.com to report damage immediately.\nDELIVERY CONFIRMATION: ChocolateFountainSales.com will check the billing\/shipping address that you provide us, against the billing\/shipping address that your credit card company has on file, we will only ship your merchandise after this address is confirmed to have matched. If you would like your merchandise to be shipped to an address other than your credit card billing address, you will be required to contact your credit card company to request that any such address be added to your account as an \"AUTHORIZED SHIP-TO ADDRESS\". Failure to do so will result in cancellation of your order. Since our courier's tracking system is so reliable, we generally do not require that someone at your home\/office sign for the package. However, if anyone is present, the courier could still ask for their signature. 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YOU NEED TO MAKE YOUR OWN DECISIONS REGARDING YOUR INTERACTIONS OR COMMUNICATIONS WITH ANY OTHER WEBSITE.\nCHOCOLATE FOUNTAIN SALES MAKES NO REPRESENTATION THAT CONTENT PROVIDED ON THIS WEBSITE IS APPLICABLE OR APPROPRIATE FOR USE IN LOCATIONS OUTSIDE OF THE UNITED STATES.\nCHOCOLATE FOUNTAIN SALES ASSUMES NO RISK OR RESPONSIBILITY FOR YOUR USE OF ANY OF THE CONTENT PROVIDED ON THIS WEBSITE.\nAPPLICABLE LAW MAY NOT ALLOW THE LIMITATION OF LIABILITY SET FORTH ABOVE, SO THIS LIMITATION OF LIABILITY MAY NOT APPLY TO YOU. IF ANY PART OF THIS LIMITATION ON LIABILITY IS FOUND TO BE INVALID OR UNENFORCEABLE FOR ANY REASON, THEN THE AGGREGATE LIABILITY OF CHOCOLATE FOUNTAIN SALES UNDER SUCH CIRCUMSTANCES FOR LIABILITIES THAT OTHERWISE WOULD HAVE BEEN LIMITED SHALL NOT EXCEED ONE HUNDRED DOLLARS ($100).\nBY ACCESSING AND USING THE CHOCOLATEFOUNTAINSALES.COM WEBSITE, YOU AGREE THAT YOUR ACCESS TO AND USE OF THIS WEBSITE IS SUBJECT TO THESE TERMS AND CONDITIONS, AS WELL AS ALL APPLICABLE LAWS, AS GOVERNED AND INTERPRETED PURSUANT TO THE LAWS OF THE STATE OF IDAHO, UNITED STATES OF AMERICA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever thought about what \"health\" really means?\nAt RiverRock, health actually means freedom. Or if you like the fancier, more precise word, it means autonomy.\nJust like money in the bank means you have the freedom to make choices, you also have a health bank.\nViable dreams are markers of autonomy.\nPeople talk about their 'financial health' because there's an intuitive understanding of this concept. Health is autonomy. It's the money in the bank that allows you the freedom to make decisions.\nIf you have 10 million dollars in your bank account, you have a lot of freedom. You have the freedom to go buy things, or do things with that money.\nJust like having money in the bank means you have the freedom to make choices, you also have a health bank.\nBut if you weigh 500lbs, or you have heart failure, then that limits what things you can choose to experience. You can't go snowboarding. You can't go climb machu picchu. Those choices just aren't open to you.\nBut if you do lose weight, if you recondition your body, become healthier, then suddenly that choice is open. That choice becomes a viable dream. Before, it was just a pie-in-the-sky dream, because you didn't have the power to complete it \u2013 you didn't have the freedom to choose to make that dream real.\nThat's what health is. The power to choose which dreams you want to make real.\nWhy do you even need more health?\nThe only reason to have health is so you can do other things. Your doctor should helping you do other things, helping you have more freedom to choose.\nAnd that's what RiverRock is really about. That's what medicine should be about \u2013 moving you to a better state of health. It's about giving you more options, more autonomy, more freedom. It's about increasing the number of things you can choose to do \u2013 the number and the quality of the options you have.\nLet's say there's a guy in coma, in the ICU, and he's got very few choices, right? What can he choose? He's almost dead. There's almost no freedom, no autonomy there.\nNow, suppose he's a little healthier. Let's say he's at least awake, able to write on a pad or nod or something \u2013 he's got 1 choice \u2013 the choice to tell us to stop. To choice to die.\nNow, if he gets even better, out of the ICU, he can choose other things. He can choose to work with physical therapy. To eat or not eat certain things. What to watch on TV? What to read? etc. etc. etc. He can choose all these things. But he cannot choose to safely leave the hospital.\nAnd so on, as he gets better, goes to a rehab facility, gets stronger. etc. etc. All this is a progression of the breadth of choices he has.\nLet's say now he goes home, he's able to live independently, cook for himself. He's able to go to the bathroom on his own. He's able to walk wherever he wants, get in and out of bed on his own. He can choose to have a job, choose to make money, to have relationships, etc.\nNow he's gained even more freedom.\nThe issue with regular medicine, even concierge medicine, is that once you meet some arbitrary standard, you don't have any overt symptom that's obviously abnormal, once you have moved out of the \"ill\" territory, into the middle of the bell curve, modern medicine basically gives up and says there's nothing more we can do.As long as you can sit on the couch, go to your job, come back home, eat and go to the bathroom, that's basically where traditional medicine stops. We've moved you from the crappy end of the bell curve to the middle, and there it is.\nNobody else is talking about giving you more freedom than that \u2013 As long as you can sit on the couch, watch TV, go to your job, and come back home, and have relationships and eat and drink and go to the bathroom, that's basically where traditional medicine stops.\nAnd that's where RiverRock is different.\nBecause that is not good enough. Go beyond that.\nTo go beyond that, you have to look at getting even stronger, and it's difficult \u2013 because now you have to have some goal in mind. Most people want to be able to have relationships, to be able to go to the bathroom on their own. Those are basic functions. They're so ingrained that the desire for enough health to do those things is already there.\nDoctors can work with those ingrained desires, you don't have to talk about it much. But once you get to that level, the issue is, you have to start to dream at an even higher level of health,you have to start to dream at an even higher level of health you have to say, no, I want to climb machu picchu, I want to go on safari, I want to go camping, I want to do this or that.\nYou have to have those dreams, something to shoot for, in order to have enough motivation and direction to say \"ok, in order to do that, what would I need? I'd need to be able to walk this far, I'd need to be able to tolerate this much\", or be able to climb this, or whatever else. And that's where medicine breaks down \u2013 we stop helping people, we stop expecting people to have bigger dreams.\nSo traditional doctors are surprised, like, \"What? You can go to your job, come back, watch tv, go to the bathroom on your own, feed yourself, what more do you want?\" And that is the problem \u2013 we stop helping people gain freedom, because we don't expect people to have bigger dreams than that.And that is the idea of RiverRock \u2013 it's for people who have bigger dreams.\nIt's for people who have bigger dreams. Do more, work harder, dream bigger, and we will help you build your health and your freedom to the point where your dreams become realistic choices.\nAnd the answer is the RiverRock Difference \u2013 our unique RiverRock Root Cause Analysis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The City of Dawson Creek will be holding a public meeting next week over the future of recreational marijuana use in the city.\nThe city says the meeting will take place July 4th at the Encana Events Centre from 7-10 p.m. to share survey results and gather public input about the upcoming legalization of cannabis.\nThe federal government announced last week recreational pot will be fully legal on October 17th.\nStaff say the city now has to start making final plans and put bylaws in place.\n\"Dispensaries, where and how are they going to be situated? Where are you going to allow them to be situated in setback ranges near parks and schools with children and things like that?\" Asks Dawson Creek Mayor Dale Bumstead. \"Public use is going to be another one. Public consumption of marijuana is another topic of discussion.\"\nBumstead says having these meetings early will allow the city to best prepare for legalization.\n\"We want to get it done fairly quickly. We only have two meetings in July and August in which to get the bylaw underway,\" says Bumstead. \"We have election time in September for municipal election. We don't want to turn it into an election issue. It really isn't. It's a legal substance.\"\nIt will be a quick turnaround for city council after the meeting.They only have a limited amount of time to discuss the information and feedback that is provided and to write the bylaws.\nMeanwhile, Prime Minister Justin Trudeau says the legalization will be a process and not \"a single-day event\".\nHe expects all stakeholders to be ready when the bill comes into effect.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Probably not but that's ok.\nThis is just a head's up that this blog has now officially moved to a dotcom.\nI had plans of moving my blog to wordpress and purchased a domain but then realised that for the amount of customisation I wanted, I had to pay more money which I didn't want to do at this stage.\nI was hoping to use that domain to direct my blogger blog to instead but that would be too easy so I have now purchased another domain.\nThis blog is now at www.bugsbeauty.com with my wordpress domain, www.bugsbeautyblog.com reserved for a possible future move should I ever become web literate enough to turn the site into what I want.\nI'll probably still be moving things around here and there and possibly add new features in the near future so stay tuned.\nIf you have any post suggestions, blog improvements or would like to submit a guest post then please click the 'Post Ideas\/Suggestions' tab above and leave me a comment.\nThanks Y! I like the blog so far but I get bored after a while and feel like I need to change things up.\nThanks Ki :) Your FB comment the other day got me thinking that I need to get my act together and make the move.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sweetheart Smencils\u00ae Cylinder of 50 \u2022 Scentco, Inc.\nThe Sweetheart Smencils\u00ae Cylinder comes with 50 of our gourmet scented pencils that are made from 100% recycled newspapers. Each Sweetheart Smencil\u00ae is a No.2 graphite pencil and is packaged in its own recyclable plastic freshness tube that includes a collectible character top cap! You will receive 10 each of the 5 scents.\nWant to order Sweetheart Smencils by the 5-Pack?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are a number of policies CMS finalized today that will reduce regulatory burden and help ensure America's hospitals and health systems can continue to provide high-quality, efficient care for the patients and communities they serve. We are encouraged that this rule starts to implement the Administration's \"meaningful measures\" initiative to streamline quality measurements. We also appreciate that CMS will be starting a full audit process for the \"Worksheet S-10\" data on charity care and bad debt to ensure data are accurate and consistent. In addition, we commend CMS for revoking the long-term care hospital 25% Rule, a change long advocated for by the AHA, but remain opposed to reductions in payments to offset that change. The agency also provided more flexibility by adopting a more performance-based approach in determining whether a hospital has met meaningful use requirements while also limiting the reporting periods to 90 days in 2019 and 2020. We also appreciate the clarification CMS offers regarding steps that may be taken to vet and register apps chosen by patients, but remain concerned about the timeline for adoption of these new approaches that will be required in 2019. At the same time, there are additional policy changes, including those affecting the area wage index, that we are analyzing to determine their ultimate impact.\nThe AHA is a not-for-profit association of health care provider organizations and individuals that are committed to the health improvement of their communities. The AHA is the national advocate for its members, which include nearly 5,000 hospitals, health care systems, networks, other providers of care and 43,000 individual members. Founded in 1898, the AHA provides education for health care leaders and is a source of information on health care issues and trends. For more information, visit the AHA website at www.aha.org.\nThis is a list of bills that the AHA is actively supporting.\nAHA opposes amendment that would prevent much-needed nurses from entering U.S.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Traveling with a dog is a wonderful experience if you are properly prepared.\nTraveling with pets can be much like traveling with a child. They are completely dependent on us to take care of them. But after the first time, it gets much easier. For those venturing out for the first time traveling with pets, here are some tips to make it as easy as possible. With a little planning and preparedness, taking your dogs and cats along for a trip can make for the most memorable vacations!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"The Syllabus Search project began in Fall 2016, and involved the redesign of the eSyllabus program, and the retirement of the syllabus search platform hosted by ITS. The CTE gathered business requirements through interviews with faculty, students, administrators, and other key stakeholders, and evaluated a number of third party syllabus tools. Ultimately, the decision was made to partner with Instructure Professional Services to build a Syllabus Search tool where the majority of faculty were already developing their syllabi- in the Canvas LMS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A leading global recruitment company in Glasgow are seeking an HR Administrator to join their HR Team. If you can demonstrate experience in a shared service environment using SAP and Microsoft Office, then you could be the HR Administrator we're looking for.\nThis is a varied role - along with your colleagues in the HR Operations Team at peak times you will be the main point of contact for the business to process all changes during the Employee life cycle; from on-boarding, promotions, transfers, flexible working, parental leave and leavers. You will work closely HRIS and the HR Business Partners to deliver an efficient and customer focused service. Additionally you will be supporting our HR in the business team with HR administration activities plus reporting requirements ensuring all employee files are accurate and up to date with all relevant documentation.\nHelpful, friendly approach with the ability to build strong relationships.\nAble to demonstrate strong organisational and time-management skills with the ability to plan and prioritise own workload.\nAs an HR Administrator, you'll be joining SThree, a global recruitment organisation at their new business operations centre in Glasgow. A vibrant and diverse environment, you will be supported whilst working for an award-winning equal opportunity employer who will recognise and reward you for your talent and commitment.\nExcellent benefits; life insurance, private healthcare, dental plan, pension, discounted gym membership and discretionary annual bonus.\nAre you an excellent communicator with the ability to work collaboratively and to deadline? If so, you could be the HR Administrator we're looking for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Class-A franchise announced the marketing push Monday, part of a broader Minor League Baseball initiative known as the \"Copa de la Diversi\u00f3n,\" or Fun Cup. According to the baseball league's website, the effort is intended to have franchises \"embrace the culture and values\" that resonate most with their region's Hispanic or Latinx communities. Latinx is a gender-neutral term for people of Latin-American descent. Seventy-two franchises are wearing specially designed jerseys at some point this season.\nHops officials said the team created the name and logo last summer, in conjunction with Nike and a committee of LatinX and non-Latin representatives. Ivan Hernandez, the franchise's Latino outreach coordinator led the process.\nAccording to the most recent American Community Survey census figures, about one-quarter of Hillsboro residents identify as Hispanic or Latinx and roughly one-third of residents speak a language other than English at home.\nWombacher said the team conducted a lot of research and field testing before selecting the name and logo for the 2019 season.\nIt features an Alebrije, those bright, mythical creatures that originate from Mexico that have become the basis for parade floats, pi\u00f1atas and other iconography across the country, particularly the Oaxaca region.\nAccording to Wikipedia, Alebrijes date back to the mid-1930s when Pedro Linares, a Mexican artisan, created the creatures after falling ill and conjuring them in a fever dream. He later recreated the mythical animals with papier-m\u00e2ch\u00e9 and cardboard and decades later the style spread through Latinx communities.\nWombacher said the logo has some hidden touches as well, such \"diamond shapes that represent graduation caps and the flame pattern at the end of the tail which symbolizes the flame on the Statue of Liberty.\nThe jerseys will make their debut June 28 and will be worn in three additional games throughout the season.\nEach of those games will include special promotional themes.\nAugust 18: Texan and Mexican music night - Featuring music from La reina del, The Queen of Texmex, Selena Quintanilla.\nThe team already posted special merchandise on their online store.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Maui Plein Air Painting Invitational. Courtesy photo.\nTwenty-five local and Mainland artists will gather in L\u0101hain\u0101 for the Maui Plein Air Painting Invitational during the week of Feb. 14 to 22.\nThe event kicks off from 7:30 a.m. until noon at the L\u0101hain\u0101 Jodo Mission. The first 25 paintings of the invitational will be on sale from 11 a.m. until noon.\nMonday, Feb. 16, 6 to 9 p.m.\nJean Stern, executive director of The Irvine Museum in California, will give a talk in Pioneer Inn courtyard in L\u0101hain\u0101. Stern draws a crowd, so get in early to secure a space. An artist panel and Q&A session will follow the lecture led by Ronaldo Macedo, a co-sponsor and founder of the invitational. Prior to the lecture, a mini-paintings silent auction will be held from 6 to 7 p.m., with final building taking place after the lecture from 8 to 8:30 p.m. The paintings will be personal works from each of the artists' home studios.\nMontage Kapalua Bay Sunset Paint Out will feature artists plein painting on location at the Montage Kapalua Bay resort and famous Kapalua Bay. Works from this event are anticipated to be some of the best and most original to date. The public is invited to watch the artists at work, enjoy the atmosphere, and to mingle at the Cane and Canoe Bar to meet the artists and organizers. A no-host bar will be available, also serving food. For dinner, guests are asked to make reservations to ensure an open table by calling (808) 662-6681.\nFriday, Feb. 20, 9 to 11 a.m.\nPioneer Inn's L\u0101hain\u0101 Harbor Quick Draw invites the public to watch artists create a work of art from start to finish in just two hours. The event inspires artists and onlookers alike with the harbor-atmosphere of boats, historic buildings of the area, and the West Maui mountains and ocean view on the horizon. Pioneer Inn offers a casual lunch menu following the event.\n100 original works of art created during the weeklong event will be displayed for sale at Village Galleries in L\u0101hain\u0101 beginning Friday for the Collectors' Hour followed by a Gala Reception. Saturday's Coffee & Pastries with the Artists will feature more painting replacing those sold Friday night.\nFriday, Feb. 20, 5 to 6 p.m.\nWhole Foods Market Collectors' Hour is held at The Village Galleries on Dickenson Street in L\u0101hain\u0101. Art collectors and aficionados alike can get first pick of the 100 freshly painted works of art by purchasing a $100 ticket. The ticket price can be applied towards a painting purchase. Space is limited and the event is expected to sell out quickly. Refreshments will be provided courtesy of Whole Foods Market.\nFriday, Feb. 20, 6 to 9 p.m.\nThe Gala Reception showcases three of the best paintings from each artist plus the Pioneer Inn L\u0101hain\u0101 Harbor Quick Draw painting. A total of 100 freshly painted works of art are unveiled at the Gala Reception at The Village Galleries, 120 Dickenson Street in Lahaina. Refreshments provided.\nSaturday, Feb. 21, 9 to 11 a.m.\nCoffee and Pastries with the Artists gives the public a chance to meet the artists, and view and purchase their paintings at The Village Galleries. New works of art will be displayed to replace sold pieces on Friday night. The event will feature music and refreshments while guests browse the gallery.\nSunday, Feb. 22, 9 a.m. to 9 p.m.\nPlein Air Final Sale offers a last chance to view and purchase paintings that were created during the Maui Plein Air Painting Invitational. The Village Galleries will be open all day to the public to browse and buy art. The show and sale will officially end at 9 p.m.\nPossible venue changes due to inclement weather will be posted online.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Microsoft has made it mandatory for Windows 10 users to update their computer. The company has set the new operating system to automatically receive new updates by default. Windows 10 Pro users have an option to delay the updates they receive, but if you are on Windows 10 Home, there isn't much you can do about it.\nThe company's move to automatically update all Windows 10 computers has received a lot of flak from users. Not only could it eat a large chunk of data from your limited data plan, we have seen cases where an automatic update has caused issues in users' computer. Regardless of why you want to delay or block Windows from automatically updating your computer, there are a couple of simple tricks that could do the job for you.\nInterestingly, there is a simple option in Wi-Fi settings, which if enabled, stops your Windows 10 computer from downloading automatic updates. To do that, search for Change Wi-Fi settings in Start Menu or Cortana. Click Advanced Options, and enable the toggle below Set as metered connection.\nThese days enabling automatic updates has become the norm on practically every operating system. This has its benefits \u2013 you don't have to manually download updates to apps and all critical security patches are downloaded automatically to your phone or computer.\nEnabling automatic updates on a smartphone doesn't really bother most people. Sometimes apps may start crashing after updates but by and large, apps continue to work as expected after updates. You're also unlikely to be using a bunch of apps at the same time. However, that's not true on a Mac or a PC.\nAt any point we've got at least eight apps open. Enabling automatic updates is convenient but not the best idea because apps have to be closed before installation can begin. That prompt tends to interrupt our workflow and we'd rather not deal with it in the middle of the work day.\nFirst up, let's figure out .\nClick the Apple logo on the top left and open System Preferences. Click on App Store. Now you'll see a bunch of options dealing with automatic updates on macOS. To disable automatic updates, click on the tick mark next to Automatically check for updates to uncheck it, which is checked by default. To enable automatic updates, click the empty box next to Automatically check for updates. You'll see a tick mark in the box now.\nWhen WhatsApp turned eight in February, it has launched a new feature that seems to be pointing to a future where the app moves away from text messages, and relies much more heavily on photos and short videos \u2013 the ephemeral status update. This newest feature from the Facebook owned messaging company bears more than a passing resemblance to social rival Snapchat, and the update hasn't just added a new tab for the status in the menu on top of your screen, but also a camera button right in the middle of things, which can be used to update your status, or a contact, or group. It's an exciting change, and it's really straightforward too, but in case you have any doubts about how to use the feature, here's what you need to know about the revamped WhatsApp Status updates.\nWhat happened to my old text WhatsApp status?\nYou mean the line which most people completely ignored, leaving it on the default \"Hi there, I'm using WhatsApp\"? Well, that's gone. Put the past behind you. According to some people the old text status is still visible, but it's gone for the most of us, so try restarting the app if you're still seeing the old status. Of course, the user backlash over the beloved text status made the company bring it back in March.\nThere are a couple of different ways to create a WhatsApp status you can do. Let's look at them.\nHP last year launched its Omen series of gaming laptops and desktop and the company has now updated its product portfolio from the series with the launch of a 15-inch laptop, a 17-inch laptop, a desktop, and a backpack PC meant for portable and convenient VR gaming.\nFirst talking about the Omen X Compact Desktop, pricing for which starts at $2,499 (roughly Rs. 1.6 lakhs), HP's new backpack PC is meant for VR gaming and the company has introduced a dock station along with it that allows the backpack to be converted into a regular desktop, notes The Verge. The Omen X Compact Desktop will be made available by the company with seventh-generation Core i7 processor options and a GeForce GTX 1080 graphics card.\nDell has now its XPS 13, Inspiron 5000, and Inspiron 7000 series laptops with Intel's latest eighth generation Kaby Lake processors. The Dell XPS 13 laptop, which still features the impressive design of the original and the the new Intel Core i7 chip, will be made available starting September 12 at $1,399.99 (roughly Rs. 89,600) but the company says that additional configurations will be made available from October.\nMoving on to the Inspiron 5000 series laptops, the 13-inch and 15-inch variants of the 2-in-1 laptop will be receiving the new eighth generation Intel Core processors. In the Inspiron 7000 series of laptops, the standard variant and the Inspiron 7000 2-in-1will be receiving the updated processors. Also, the 17-inch variant of the Inspiron 7000 2-in-1 portable will get the new generation chips.\nSeparately, Dell has now announced that its Visor mixed-reality headset, which has been made specifically for Microsoft's Windows Mixed Reality platform, will be made available from October at a price of $349.99 (roughly Rs. 22,400). Customers can also choose to buy a bundle with the headset and two controllers for $449.99 (roughly Rs. 28,800). While the company has described Visor as a mixed-reality headset, it has notably shown only the VR capabilities of the wearable as of now, as pointed out in a separate report by The Verge.\nTo recall, Microsoft started taking pre-orders for the mixed-reality headsets from Acer and HP in May this year. Engadget points out that just like the headsets from Acer and HP, the wearable from the Dell will also come with 1440\u00d71440 LCD panels and feature sufficient padding that would allow users to wear it comfortably for longer periods as well.\n\"Dell Visor takes advantage of the same technology used in the Microsoft HoloLens, including inside-out tracking cameras, with a 360 degree panoramic view. The headset itself is able to scan surroundings instead of satellite emitters positioned inside the room,\" the company said in its release. The Dell Visor will be made available at Microsoft and select Best Buy stores and BestBuy.com in the coming months.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Not much to report. The only state poll is Maine, but it is interesting. Kerry and Bush are tied at 43% each. This result is bad news for Kerry, who expected to win Maine easily, as Gore did in 2000. Maine is one of the two states that does not use a winner-take-all system for allocating its votes in the electoral college (Nebraska is the other one). The winner of each of the two congressional districts gets one, and the statewide winner gets the other two. The poll did not break down the results by CD.\nAlso noteworthy in Maine is that 43% of the voters say Bush deserves another term while 55% say it is time for someone new. This finding suggests that a lot of people there dislike Bush but are not sure Kerry is the answer. Finally, although the people of Maine are not so hot on George Bush, they definitely like bear-baiting. The referendum on banning it is losing 52% to 35%. I think bears should be allowed to vote on that one.\nZogby did a survey among rural voters. The good news for Bush is that he is ahead 52% to 37% in this demographic group. The bad news is that in 2000 he led 59% to 37% at this point. Jobs, terrorism, Iraq, health care, and education are the top issues down on the farm. Not surprisingly, those rural voters who see jobs as the most important issue prefer Kerry and those who see terrorism as the key issue prefer Bush.\nI have it on good authority that overseas voters are registering in huge numbers this time, maybe double or triple 2000. I was told that the number of people who showed up at the Democratic party caucus in England earlier this year was 10 times what it was in 2000, ditto in other countries. Americans overseas vote in the state they last lived in, even if that was decades ago. There are about 7 million overseas Americans and probably about 5 million are over 18. In Florida, it was the overseas absentee ballots that swung the election. I believe that something like 8% are military, but the rest are students, teachers, artists, government workers, business executives, spouses of foreign nationals, missionaries, retirees, and more. What is significant here is that these people represent a lot of votes and are not included in any of the polls. Nobody knows if they are largely Democrats or Republicans, but their votes could be one of the big surprises of this election. if anyone has any actual data (as opposed to speculation) on this group, I'd be interested.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you bake, love crafts or are in any way practical, homemade gifts are a excellent alternative for birthdays, Christmas, Valentine's day and other festivities\u2014not only because folks will appreciate the added work you've produced, but due to the fact it's exciting to make the gifts and it can typically save you funds also. These standard components are so versatile and can flow with any tips you come up with later when the infant is settled in. Whether or not you end up adding butterflies or frogs to the room later, the blue sky and neutral additions will fit perfectly. Christmas decorating is possibly my favorite thing to do. Gathering suggestions from your website for the season.\nI have a really eclectic decorating theme in our property, a bit Shabby Chic and Vintage, mainly European style. We purchased a cottage in 2 acres of woods last year and are nonetheless decorating it. Thank you for the tips! Paintings fill up the empty space making it lively and cheerful spot to prepare meals for the household and a classic pendant light illuminate the kitchen in style. I really like Christmas decorations but discover pondering up the tips and organising it tough function.\nI recall my mother decorating our property with pine cones, wreaths created from cedar trees, and so on, and what ever she could locate in the dry desert area where we lived. Contemporary, modern, or classic themed wall art decals can definitely add a dash of style to a large or little bedroom giving the bedroom a look that depicts a designer's touch.\nPaint is high-priced, but can be the very best overall investment if you are looking for an apparent alter or hoping to rejuvenate shabby walls. One wonderful notion is to give your bedroom a lovely weathered look by distressing a piece of furnishings. You will be surprised at how numerous concepts for buffet meals tables you can come up with by basically seeking around your residence!\nThis entry was posted in Home Decorating Ideas and tagged budget, decorating, house. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here at Invacare Medical Warehouse we extended our ways to serve you better, we are now selling online, you can now purchase your medical equipment online and hassle free. Kindly click to enlarge the images.\n4. Do Steps 1 -3 until you have finished shopping. You can view your shopping list by clicking Shop > Cart on the menu list above, just like what is shown in the image below.\n6. For delivery charge outside Metro Manila, please email or call first Invacare to quote you the exact charge. Then go to FREGHT PREPAID to choose an option of variables.Then add it to your cart.\n7. When you think you are already done shopping, and you have clicked Proceed to Checkout button, you then need to fill up this form. Please note that all fields in * are priorities and needed to be filled up else your submission will not be completed. Create an account is optional. Bottom part is the Modes for payment.\n9. You can also pay using your PayPal account if you have, or your credit card. we accept almost all major credit cards. To toggle from those options, see the image below.\nWe will email you the freight forwarder airway bill or sea bill of lading so you can also track the cargo.\nThe tracking number will also be given to you.\nAny concern, questions or further inquiries, please let us know, we will be very happy to assist you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I love the art of animation, and this short film is a CGI animated film that pays tribute to the art of animation. The short is called TimTom and is was created by Romain Segaud and Christel Pougeoise.\nI love how the includes elements of stop-motion animation, film media and sound to tell the story. It follows two notepad head characters as they are harassed the animators hand.\nIt was an official selection at Sundance and won a jury award prize at Siggraph as well as the grand prizes at Imagina and Ars Electronica. There's a reason why this won these awards, and you'll see it when you watch the short.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Uniform Domain Name Dispute Resolution Policy (the \"Policy\") has been adopted by the Internet Corporation for Assigned Names and Numbers (\"ICANN\"), is incorporated by reference into your Registration Agreement, and sets forth the terms and conditions in connection with a dispute between you and any party other than us (Hostway Corporation) over the registration and use of an Internet domain name registered by you. Proceedings under Paragraph 4 of this Policy will be conducted according to the Rules for Uniform Domain Name Dispute Resolution Policy (the \"Rules of Procedure\"), and the selected administrative-dispute-resolution service provider's supplemental rules.\nThis Paragraph sets forth the type of disputes for which you are required to submit to a mandatory administrative proceeding. These proceedings will be conducted before one of the administrative-dispute-resolution service providers listed at http:\/\/www.icann.org\/udrp\/approved-providers.htm (each, a \"Provider\"). a. Applicable Disputes. You are required to submit to a mandatory administrative proceeding in the event that a third party (a \"complainant\") asserts to the applicable Provider, in compliance with the Rules of Procedure, that (i) your domain name is identical or confusingly similar to a trademark or service mark in which the complainant has rights; and (ii) you have no rights or legitimate interests in respect of the domain name; and (iii) your domain name has been registered and is being used in bad faith. In the administrative proceeding, the complainant must prove that each of these three elements are present. b. Evidence of Registration and Use in Bad Faith. For the purposes of Paragraph 4(a)(iii), the following circumstances, in particular but without limitation, if found by the Panel to be present, shall be evidence of the registration and use of a domain name in bad faith: (i) circumstances indicating that you have registered or you have acquired the domain name primarily for the purpose of selling, renting, or otherwise transferring the domain name registration to the complainant who is the owner of the trademark or service mark or to a competitor of that complainant, for valuable consideration in excess of your documented out-of-pocket costs directly related to the domain name; or (ii) you have registered the domain name in order to prevent the owner of the trademark or service mark from reflecting the mark in a corresponding domain name, provided that you have engaged in a pattern of such conduct; or (iii) you have registered the domain name primarily for the purpose of disrupting the business of a competitor; or (iv) by using the domain name, you have intentionally attempted to attract, for commercial gain, Internet users to your Web site or other on-line location, by creating a likelihood of confusion with the complainant's mark as to the source, sponsorship, affiliation, or endorsement of your Web site or location or of a product or service on your Web site or location. c. How to Demonstrate Your Rights to and Legitimate Interests in the Domain Name in Responding to a Complaint. When you receive a complaint, you should refer to Paragraph 5 of the Rules of Procedure in determining how your response should be prepared. Any of the following circumstances, in particular but without limitation, if found by the Panel to be proved based on its evaluation of all evidence presented, shall demonstrate your rights or legitimate interests to the domain name for purposes of Paragraph 4(a)(ii): (i) before any notice to you of the dispute, your use of, or demonstrable preparations to use, the domain name or a name corresponding to the domain name in connection with a bona fide offering of goods or services; or (ii) you (as an individual, business, or other organization) have been commonly known by the domain name, even if you have acquired no trademark or service mark rights; or (iii) you are making a legitimate noncommercial or fair use of the domain name, without intent for commercial gain to misleadingly divert consumers or to tarnish the trademark or service mark at issue. d. Selection of Provider. The complainant shall select the Provider from among those approved by ICANN by submitting the complaint to that Provider. The selected Provider will administer the proceeding, except in cases of consolidation as described in Paragraph 4(f). e. Initiation of Proceeding and Process and Appointment of Administrative Panel. The Rules of Procedure state the process for initiating and conducting a proceeding and for appointing the panel that will decide the dispute (the \"Administrative Panel\"). f. Consolidation. In the event of multiple disputes between you and a complainant, either you or the complainant may petition to consolidate the disputes before a single Administrative Panel. This petition shall be made to the first Administrative Panel appointed to hear a pending dispute between the parties. This Administrative Panel may consolidate before it any or all such disputes in its sole discretion, provided that the disputes being consolidated are governed by this Policy or a later version of this Policy adopted by ICANN. g. Fees. All fees charged by a Provider in connection with any dispute before an Administrative Panel pursuant to this Policy shall be paid by the complainant, except in cases where you elect to expand the Administrative Panel from one to three panelists as provided in Paragraph 5(b)(iv) of the Rules of Procedure, in which case all fees will be split evenly by you and the complainant. h. Our Involvement in Administrative Proceedings. We do not, and will not, participate in the administration or conduct of any proceeding before an Administrative Panel. In addition, we will not be liable as a result of any decisions rendered by the Administrative Panel. i. Remedies. The remedies available to a complainant pursuant to any proceeding before an Administrative Panel shall be limited to requiring the cancellation of your domain name or the transfer of your domain name registration to the complainant. j. Notification and Publication. The Provider shall notify us of any decision made by an Administrative Panel with respect to a domain name you have registered with us. All decisions under this Policy will be published in full over the Internet, except when an Administrative Panel determines in an exceptional case to redact portions of its decision. k. Availability of Court Proceedings. The mandatory administrative proceeding requirements set forth in Paragraph 4 shall not prevent either you or the complainant from submitting the dispute to a court of competent jurisdiction for independent resolution before such mandatory administrative proceeding is commenced or after such proceeding is concluded. If an Administrative Panel decides that your domain name registration should be canceled or transferred, we will wait ten (10) business days (as observed in the location of our principal office) after we are informed by the applicable Provider of the Administrative Panel's decision before implementing that decision. We will then implement the decision unless we have received from you during that ten (10) business day period official documentation (such as a copy of a complaint, file-stamped by the clerk of the court) that you have commenced a lawsuit against the complainant in a jurisdiction to which the complainant has submitted under Paragraph 3(b)(xiii) of the Rules of Procedure. (In general, that jurisdiction is either the location of our principal office or of your address as shown in our Whois database. See Paragraphs 1 and 3(b)(xiii) of the Rules of Procedure for details.) If we receive such documentation within the ten (10) business day period, we will not implement the Administrative Panel's decision, and we will take no further action, until we receive (i) evidence satisfactory to us of a resolution between the parties; (ii) evidence satisfactory to us that your lawsuit has been dismissed or withdrawn; or (iii) a copy of an order from such court dismissing your lawsuit or ordering that you do not have the right to continue to use your domain name.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Between 2011 and 2012, I was commissioned by City Screen to photograph all their Picturehouse cinemas in London. I photographed the screens, the inside and outside spaces of each building.\nWhile waiting for a screen to be cleaned or empty after a film screening, I used to sneak in the projection rooms (with permission from the projectionists, of course!) and made these photographs of what is now a fading out experience.\nThis is not a nostalgic look back, but rather a record of what was, sometimes is, but might not continue to be a form of experiencing a movie at the cinema.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The GEL-Trigger Grade School shoe is a popular turf base cross training shoe. It features rear GEL units to attenuate shock while the Duomax medial post provides fantastic stability.\nThe upper features mesh pockets for ventilation and breathability, allowing the foot to feel comfort and stay dry.\nThe midsole features a rear GEL cushioning system, and a SuoMax support system to provide great stability.\nThis Asics shoe features a Removeable Sockliner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Should I Take to Drink?\nMe: Is it? Isn't there evidence that moderate drinking is good for you health?\nMe: Then why did you say \"Good\"?\nDoc: Well, heavy drinking is bad for you.\nMe: But you said it was good that I don't drink at all.\nMe: Do you ever advise a patient to start drinking?\nDoc: [fear of lawsuit in her eyes, channeling Darth Vader at the end of Episode III] Noooooo!!!\nSince I turn 35 in a week, I'm once again pondering whether I should take to drink. The last time I thought about it, I looked around for a good survey of the medical literature, but didn't have much success. I do recall reading a skeptic saying that \"moderate drinking is good for you\" research bundles together lifelong teetotalers with recovering alcoholics. Seemed plausible, but the skeptic didn't really have any data.\nBut now, just in time for my birthday, there's a new meta-study backing up the skeptic. Lucky me!\nThe common error was to lump into the group of \"abstainers\" people who were once drinkers but had quit.\nAll seven of those studies found no significant differences in the health of those who drank \u2014 or previously drank \u2014 and those who never touched the stuff. The remaining 47 studies represent the body of research that has led to a general scientific consensus that moderate drinking has a health benefit.\nFillmore's team of researchers took their initial finding one step further, and introduced the error into the data compiled in the seven studies and, voila, the results changed to show drinkers had better health than abstainers.\nSo for now, my birthday present to myself is to stick to delicious IBC root beer.\nAs soon as you are old enough to run for president, you start to consider boozing it up?!?\nIn a study published in May 2005 in the American Journal of Preventive Medicine, Naimi and other CDC colleagues found that\u2026 non-drinkers tended to be poorer than drinkers, had less access to health care, and had less healthy diets.\nMoreover, you are missing all the fun.\nMeta-studies, doctors with computers, makes me think about having a drink and I have not even had my coffee yet.\n\"There are more drunks than doctors, a livin' today\".\nHappy birthday in advance, and IBC rootbeer is good. But I prefer Sprecher root beer.\nBryan- The sugar in the root beer is probably worse for you than the alcohol in the beer. High fructose corn syrup is linked to wide variety of ailments. So, as a replacement for root beer, regular beer would do you a world of good regardless of the skeptic meta-study. Although there are a number of other drink that might be even better than beer.. but then you're sacrificing euphoria for health.\nZac speaks the truth. High fructose corn syrup is one of the worst things you can put into your body.\nAll seven of those studies found no significant differences in the health of those who drank \u2014 or previously drank \u2014 and those who never touched the stuff.\nWhich would suggest that moderate drinking does not significantly hurt, but that the \"quit drinking or die\" crowd is in a much worse state.\nnon-drinkers tended to be poorer than drinkers, had less access to health care, and had less healthy diets.\nAnd this would be my usual complaint: not taking account of wealth effects. Anecdotally, we expect people who have a glass or two of wine with dinner to be of a higher economic class than average.\nI should go read the full meta-study, but I know that I am just going to exhibit massive confirmational bias for my own teetotaler status.\nZac, high fructose syrup \"in moderation\" is probably good for you. Certainly if you have no other source of carbohydrates in your diet. It's strange that the sugar companies haven't capitalised on this simple fact.\nAlcohol use is also linked to a wide variety of ailments. \"Excessive\" use can cause a wide variety of cancers, and also problems with the liver, kidneys and brain.\nOne thing the alcohol industry has in common with the tobacco industry is that they are completely economically dependent on harmful use. \"Moderate\" use doesn't cut it, if everyone drank only in moderation the companies would lose money compared to today. That's one good reason not to support them, IMO.\nHarold, there are a plethora of carbohydrate sources other than high fructose corn syrup. All sugars should be consumed sparingly. High fructose corn syrup, in particular, is linked to obesity and adult-onset diabetes.\nOne thing the soft drink business has in common with the tobacco business is that it is completely dependent on harmful use. If that's not reason enough not to support them, consider that by using HFCS instead of sugar, taxes (by way of corn subsidies and sugar tariffs) increase their profit margin.\nActually, I wouldn't be surprised if the big winner wasn't the soft drink companies but the manufacturers of HFCS. In other countries without excessive agriculture taxes, sugar cane is still preeminent as a sweetener. Not only does it taste better \u2014 its better for you.\nI see the Accidental Hedonists has more, today, on the evils of soft drinks.\nWhy is including former drinkers an \"error\"? That seems to assume the conclusion that drinking or ever having drunk is bad for you. Wasn't that the question that the study was examining? If moderate drinking really is good for you, those who used to drink moderately and then stopped would be healthier than those who never drank.\nI believe the assumption is that those who quit had not been moderate drinkers, but rather heavy drinkers. Alcoholics seem unlikely to successfully convert from heavy to moderate drinking, so it is all or nothing. People who have an occasional drink probably continue to have an occasional drink.\nYour point is well taken. An ideal analysis would compare several groups: heavy drinkers, moderate drinkers, former drinkers who have quite (possibly again divided between heavy and moderate), and teetotalers. Grouping the recovering alcoholics with the moderate drinkers does not make much more sense than grouping them with the lifelong teetotalers. I suppose that they might average over time to moderate drinking, but that is not nearly the same thing.\nHigh fructose corn syrup is linked to wide variety of ailments.\nBeing that table sugar (sucrose) is chemically just fructose and glucose, HFCS is usually 42% or 55% fructose (it is also made at 90% fructose), with the rest of the sugar being glucose. So, roughly, HFCS contains the same amounts of fructose and glucose as refined table sugar.\nThe one study linked in the wikipedia article on a link between fructose and insulin levels seems interesting, but thinking about the simple chemistry of the matter makes me pretty skeptical.\nPlus, if you look at the sugar make up of regular old 100% orange juice you're getting basically 9.31-11.3 g\/100ml*, with the breakdown 3.03g fructose, 2.22g glucose, and 5.25g sucrose (which is 50\/50 fructose and glucose). That means that OJ's sugar content is, roughly, 50% fructose: about the same as HFCS, depending. OJ has nutrients, soda doesn't, but if the fructose paranoia is correct you could expect have OJ drinkers to end up diabetic, and fat as well. Or it could be that increased caloric intake in general combined with a lower level of physical activity is what's making people fat.\nThe obvious implication is that once you start drinking, you shouldn't stop, since the former drinkers are lowering the health standards of the abstainers. Once you satr, don't stop\u2026 now there's a principle I can live with.\nHigh fructose corn syrup causes a big insulin surge when you consume it. Your blood sugar level goes up rapidly, then your insulin response kicks in big time to push the blood sugar back down. This up and down rollercoaster for people who consume lots of soft drinks majorly saps energy levels and is a contributor to diabetes in the long run.\nHFCS is better than no carbs, sure. Harold, they have capitalised on it. Witness Gatorade, Powerade, and all sorts of other \"sports drinks\". But these sugary drinks are still relatively inferior to any number of snacks you could have instead if the question is one of a quality energy source.\nZac, I always thought it was the agricultural subsidies the US has which made HFCS so common. Doesn't corn get huge subsidies?\nYour error is to lump those who drink root beer along with abstainers and assume that root beer is healthier for you than to not drink alcohol, which I do not believe has ever been shown.\nZac: You seem to claim that drink for drink sugar soda is worse for you than beer. I am skeptical. My impression is the overall effect of sugar soda is worse because people drink so much more of it.\nMjrMjr: I think sugar import quotas are more important than corn subsidies in encouraging the use of corn syrup. Corn subsidies affect the world-wide price of corn syrup, but sugar quotas only affect the US price. Other countries are subject to the corn subsidies, but don't use corn syrup. This isn't conclusive\u2013it only demonstrates that corn subsidies alone aren't enough, but I find it pretty convincing.\nMjrMjr, corn does get huge subsidies. Sugar also has huge tariffs\/quotas in the US. I am not sure which influences the prevelance of HFCS in the US more, but both are at work. \"Agriculture regulation\" would have been better than \"agriculture taxes\" in this case.\nL, well, if HFCS contributes to obesity and diabetes, and there is inconclusive evidence as to whether beer actually extends your life or not, I think its fair to say that sweetened sodas (which contain a ton of Calories too, and cause a severse spike in insulin, if you don't buy the HCFS is evil argument that is enough of a health risk) are worse for you than beer, drink for drink. Even so, the point that people drink far more of them is important: these sodas are not at all filling or thirst quenching. Beer, on the other hand, contains very little sugar and tends to be very filling (so much so that \"less filling\" is considered a good quality to some).\nIts a lot easier to drink beer in moderation than soda.\nInteresting discussion. Zac, I do agree that the sugar industry has perverse incentives as well.\nBut I do wonder about your assessment that it is easier to drink beer in moderation than soda. People drink a lot of beer, and alcohol in general. And as alcohol also contributes to obesity and obesity-related problems (it has an energy value of 29 kJ\/g), my bet is that Caplan is better off sticking to his root beer.\nI think its pretty clear that far more calories come from soft drinks than alcohol in the American diet.\nIf you drink 1 beer a day that's better than 1 root beer a day, in terms of calories (12oz of Guiness has 126 calories. A 12oz can of coke has 155 calories) and in terms of the harmful effects of sugar\/HFCS (well documented) versus the effects of moderate drinking of alcohol (inconclusive evidence on actual life extension, safely consumed by humans for thousands of years). Of course, if you only drink 1 root beer a week versus 1 regular beer a week, I doubt there's any difference at all.\nAlso consider that the average size for a beer may be 12oz, but sodas are typically 16oz or 20oz. These beverages really are poison.\nThe calorie counts are interesting, I think they make my point rather well \u2013 I hadn't thought they were that close.\nI knew there are fewer calories in beer than in soda, but I meant that the damaging effects of alcohol more than makes up for it. In \"moderation\", the health effects are small anyway.\nHowever, the cost of alcohol damage in a society is propotional to total use, not number of heavy users. Moderate drinking is harmful in practice, because alcohol-related accidents (\"trauma\" in the literature) are common in moderate drinkers as well. Also, it seems that when median consumption is low, high consumption is rarer as well. That's reasonable, when you think about it: The more the people around you drink, the easier it is for you to drink to excess as well. And conversely, if everyone you know is like me and Caplan, you will feel a little weird if you drink a lot.\nThat's one of the reasons I don't drink at all. I figure it's the best way to reduce harms related to heavy drinking (and moderate, too), especially in those closest to me.\nBut it would be hard to apply the same thinking to sugar. Alcohol is not really very useful, and easy to do completely without (ask me!), but sugar isn't. Also, part of the preventive effect of alcohol abstention comes from the moral challenge it poses. I believe that most people who use intoxicants feel at some level that they ought not to (whether that feeling is reasonable or not).\nThere is no similar idea about sugar, at least not very wide-spread. While people might feel that the sugar they eat is bad for them, only a few dieters feel genuinely guilty about it. Why should they? In one form or another your body needs sugar.\nIf people can be convinced that alcohol is beneficial for their health, however, it will come in the same category. They can more easily convince themselves that they're not just using it as an intoxicant. Consumption will then rise, and damages and overuse as well, for the reasons I've talked about.\nSo even though sugar is undoubtedly harmful in some similar ways, telling people to switch to beer is a bad idea, in my opinion.\nI condlude: health of drinkers=past-drinkers=lifelong abstainers (maybe not truely equal, but differences were not statistically significant).\nIf this is true, then how does subsequently lumping ex-drinkers with abstainers suddenly make them significantly less healthy than drinkers, unless both ex-drinkers and abstainers were both less healthy separately (but just not significantly so)?\nYou should have picked up a taste for Stewarts Root Beer when you were at Princeton. Nothing like it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"26\/08\/2012 \u00b7 It may be telling that \"The Fantastic Flying Books of Mr. Morris Lessmore,\" a kind of totem to the unmatched joys of reading, appeared in two \u2026... We have lots of great teaching ideas and activities for The Fantastic Flying Books of Morris Lessmore by William Joyce. Have a look!\nThe Fantastic Flying Journey Audiobook Listen Instantly!\nFumbling Through Fantasy: The Fantastic Flying Journey by Gerald Durrell, illustrated by Graham Percy (1987) When their Uncle Lancelot shows up at their house one day flying a giant hot air balloon called the Belladonna, the three Dollybutt children - Emma, Ivan, and Conrad - are completely shocked.... This picture book combines an adventure story with an introduction to the world of animals. It recounts the story of Great-Uncle Lancelot, who, together with his great-niece and great-nephews, embarks on a journey in an ecological flying machine to meet some of the world's most interesting animals.\nDetails about The Fantastic Flying Journey by Gerald Durrell Paperback Book The Cheap Fast Be the first to write a review . The Fantastic Flying Journey by \u2026 discoveries and opinions of galileo pdf This picture book combines an adventure story with an introduction to the world of animals. It recounts the story of Great-Uncle Lancelot, who, together with his great-niece and great-nephews, embarks on a journey in an ecological flying machine to meet some of the world's most interesting animals.\nMorris Lessmore finds himself blown away and rendered coloruless by a storm, and through books finds purpose, healing, and the route to his own story. the art and science of personal training 4th edition pdf The Fantastic Flying Journey: Gerald Durrell Astrid Lindgren - my favorite author. Children of Bullerby - A charming story about the everyday life of six children living on the three neighboring farms that everybody calls \"noisy village\" because of its \"high spirits.\nLooking for The Fantastic Flying Journey - Gerald Durrell Paperback? Visit musicMagpie for great deals and super savings with FREE delivery today!\nWhen he builds a fantastic flying machine, powered by tree sap, lit by electric eels and heated by solar panels, his niece and nephews have never seen anything like it. Then, in search of his lost brother, Perceval, he whisks the children off on an amazing journey around the world.\nThe Fantastic Flying Journey (ISBN 0755104366) is a children's book written by Gerald Durrell. It is a story about three children and their great-uncle Lancelot travelling around the world in a hot air balloon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"September is National Rice Month, and we have recipes from Minute Rice to help showcase the latest products. The Minute Rice test kitchen developed an assortment of simple yet delicious recipes featuring the new Minute Ready to Serve Rice products. To learn more about the newest Minute Rice products, tasty recipe ideas and other varieties within the Minute Rice family of products, visit www.minuterice.com.\nHeat rice according to package directions. In a small bowl combine rice, kale, vinaigrette, tomatoes and cucumber. Top with feta, olives and artichokes.\nHeat rice according to package directions. In a small microwave-safe bowl combine shrimp and chutney. Microwave on HIGH 30 seconds. Stir in yogurt, cilantro and rice. Top with peanuts.\nHeat rice according to package directions. Heat oil in in a large wok or skillet over medium heat. Add onion, celery and carrots and stir-fry 2 minutes. Stir in rice, ham and soy sauce. Continue to stir-fry until heated through. Pour in eggs and cook, stirring constantly, until eggs are done. Serve with additional soy sauce, if desired.\nJoin the Minute\u00ae Rice community online at minuterice.com, facebook.com\/MinuteRiceUS, pinterest.com\/minutericeUS, and follow them on Twitter and Instagram at twitter.com\/MinuteRiceUS and instagram.com\/MinuteRiceUS\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"PHILADELPHIA \u2013 Stradley Ronon was honored as the top litigation department among Pennsylvania's midsize law firms in The Legal Intelligencer's recently published \"Legal & Litigation Departments of the Year\" supplement. The publication chose the winners \"based on the big victories and important cases that demonstrated why clients chose them when litigation was on the table.\" Stradley Ronon was praised for its \"emphasis on efficiency\" and ability to determine the issues most important to a client.\n\"Our litigators maintain a fierce commitment to achieving positive results to our clients, and we are thrilled to be honored for this by The Legal Intelligencer,\" said Stradley Ronon Litigation Department Chair Michael D. O'Mara.\nStradley Ronon's litigation team routinely handles high-risk, high-stakes litigation in a wide array of matters spanning state and federal courts, before governmental agencies, and in international and domestic arbitrations. From pre-litigation planning through discovery, trial and appeal, the firm's litigators become steeped in and allied with our clients' business goals and stay focused on the end game: delivering results. Taking a value-driven, practical approach, armed with the confidence of taking scores of cases to verdict, Stradley Ronon delivers consistent and cost-effective results against some of the largest law firms in the country.\nCryptocurrency and Blockchain: Securities Litigation, Regulatory and Enforcement Issues and Implications for the Future?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Anyone who loves animals will remember the 2007 melamine incident with horror. Pet food contaminated with melamine caused an estimated 3,600 deaths and countless illnesses in dogs and cats. This issue has helped bring pet food safety to the attention of government regulators. In response, the FDA included animal food when it introduced the Food Safety Modernization Act (FSMA).\nBefore the FSMA brought pet food under the FDA's control, pet food manufacturers were basically self-regulating with inspections either randomly conducted every five years or in response to consumer complaints. In the absence of FDA rules, the Association of American Feed Control Officials (AAFCO) provided guidance, developed ingredient definitions and established nutritional requirements. AAFCO had no legal authority to enforce these standards, so food safety varied widely among producers.\n\u2026manufacturers must ensure that all suppliers are meeting the applicable food safety standards.\nThe FSMA's Preventive Controls for Animal Food Rule has two main requirements: pet food manufacturers must follow Current Good Manufacturing Processes (CGMPs) and they must implement HACCP controls, with a few differences from those applicable to human food. For example, there is no need for allergen cross-contamination prevention for animal food, unless the company is specifically producing pet food that claims to be allergen-free.\nThe FDA has published a Guidance for Industry: Current Good Manufacturing Practice Requirements for Food for Animals. It covers things like plant hygiene, contamination prevention, ventilation, plant construction and maintenance, and much more.\nAccording to some industry experts, most large producers are already following Current Good Manufacturing Practices (CGMPs). However, they have not been required to implement HACCP and this will necessitate a great deal more training and documentation for these companies. The FDA has published another guidance document to assist with this effort: Hazard Analysis and Risk-Based Preventive Controls for Food for Animals. One requirement that may be new to the large pet food companies, is supply chain verification; in other words, manufacturers must ensure that all suppliers are meeting the applicable food safety standards.\nAnimal food is also governed now by the Foreign Supplier Verification Rule, which requires companies to ensure that suppliers from other countries are complying with US standards of food safety. This will hopefully help to prevent another melamine crisis.\nFinally, FSMA's Sanitary Transport Rule now requires shippers, receivers and transporters of pet food and pet food ingredients to ensure that the food is being shipped in a sanitary manner, avoiding contamination and spoilage during transport, especially with the trend in raw pet food diets on the rise. Again, this will require more training and paperwork for pet food manufacturers.\nSmaller producers may find these new processes difficult and costly to implement, making it difficult to meet these new standards of compliance. As the natural and organic pet food share of the market continues to grow, at more than triple the rate of such foods for humans, it's important that pet food safety be governed with the same oversight as human food.\nMany consulting companies sell \"turn key\" systems to help meet the new requirements. However, without adequate training in or understanding of the system, or an understanding of the pet food sector and its requirements, errors may be made, ultimately costing more than the training of key employees. It's critical that staff are trained in procedures, so they can meet the new standards for auditors and inspectors.\nSome experts estimate that implementing these rules will cost the industry about $130 million annually, which they feel will result in increased costs to consumers. Other industry experts refer to an FDA analysis which identified six large companies that comprise 86% of the pet food market; they note that if these companies were already mostly compliant, the associated costs of FSMA requirements could be comparatively small. This is a $60-billion-a-year business and regardless of the inconvenience to manufacturers, consumers are likely to breathe easier knowing stronger regulations will increase food safety practices throughout the system and especially for their beloved pets.\nLauren Solar has been a freelance writer and editor for over thirty years, writing mostly on health-related topics and for not-for-profit organizations. Because of her own food allergies and her extensive background with allergy advocacy groups, she helped create the Food Allergen Food Safety Training offered by TrainCan and is a certified trainer on this topic. She has also done a great deal of technical and business writing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The village of Belchin is situated in the Palakaria kettle, at the foot of the Verila Mountain. Only 1 km away from the village are the mineral springs and the small resort of the Belchinski Bani (Belchin mineral baths).\nArchaeology: In the last years very serious archaeological excavations were observed in the region of the village. A total survey and research was carried out in 2007.\nDiscoveries show that the current church was built above the foundations of another temple and the researches of St. Spas Peak discovered wall of a citadel of the Roman Empire. The findings showed coins, everyday ceramics and pottery.\nSt. Petka Temple: The restored St. Petka Church was officially opened on October the 14th, 2007, in the presence of the President of Bulgaria, some ministers of the government and the church donors. The temple is announced a national monument of culture.\nNature: This region development is closely connected with the mineral springs, famous with their curing properties, and with the clean and intact nature.\nSightseeing: The restored church, the Ethnographic Museum.\nPerspectives: Modern Spa Hotels will be built in the nearest future. The proximity of the Borovets Ski Resort will help this region to achieve European significance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The National Archive of Monuments of the Hellenic Ministry of Culture in Athens (GR) continues its participation in the AREA project through various research and documentation activities. Drawing on the rich material of the Historic Archive of the Hellenic Archaeological Service, a number of themes will be explored, focusing on the history of Mediterranean Archaeology and the construction of European Identities. Research will concentrate primarily on two areas: 'Archaeologists Abroad' and 'Dictatorships and Archaeology' - themes which offer different perspectives on the interaction between archaeology and the processes of identity formation at a national and international level. This research will develop in collaboration with other AREA partners, providing a comparative framework for the study of the complex development of the discipline of archaeology. In terms of documentation, the National Archive of Monuments will continue to enrich the on-line catalogue of archaeological archives held in Greece, ultimately seeking to identify and describe all archival fonds relating to archaeology in Greece.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Affordable maintenance contracts, available monthly or annually. Keep your site fresh!\nWe work with open source 3D program Blender, and also industry standard applications such as Lightwave and 3D Studio Max. If you are interested in exploiting the power of 3D, let us help you.\nWe specialise in design and creative services, providing you with professional marketing, communication & design support as and when you need it.\nCatalyst Design & Marketing Limited offer freelance design capability based in the in Southampton, Hampshire and South of England areas. We can work with your in-house marketing or creative team, or alternatively we can operate as an outsourced design department. With the advent of modern collaborative software and tools, location is no longer an issue so we can work together no matter where you are based.\nWhether it's web development or maintenance, brochures, print and large format we have the know-how to help you along \u2013 we happily dabble across all kinds of different media platforms. With a steely determination to deliver on-time, on-budget and with maximum efficiency, give us a try \u2013 we want to make you happy!\nSpeedy, speedy, speedy. We're all about the turnaround.\nAlmost 10 years on the clock. We're a trusted business partner.\nLet's talk, talk and talk some more. Chat with a designer now.\nPowerful, successful brands are built upon solid, well constructed brand identity; care and consideration must be given to your company's brand image, failing to control your identity is a major risk that may cause your marketing message to become diluted. Catalyst Design & Marketing Limited is focused on creative success and consistency through effective brand management in print, design and multimedia \u2013 we care about your brand image!\nYou can read more about our services or get in touch right now.\nWe provide services with rates and options to suit your business; design by the hour or on a contract per contract basis; the real advantage of using a freelance designer.\nYou have complete control of how you use Catalyst and this will allow you to manage your budget without the pain of additional agency costs. We work with high end apple workstations and our software suite is at the cutting edge of graphics packages today.\nWe help our clients to create strong, consistent brands with messages that truly speak to their audiences.\nWe believe that a brand is a vital tool for any business. A well considered brand identity grounds a business, whether it is a 150 year old establishment or a 6 month old start up. Branding is the keystone that all your communication should originate from and acts as an anchor in the choppy seas of marketing and advertising!\nWe have been making awesome design and creative content for 10 years now, and our clients will attest that our approach of innovative visuals combined with a very rigourous and methodical workflow delivers brilliant results, on-time, on budget. Expect great ideas and precision execution as standard.\nWith Catalyst you will be dealing directly with a design professional, no need to pass messages between PMs and admins, so you can quickly and effectively communicate your ideas and see them develop into your chosen design. Forget about wasted hours and frustrating phone calls, work with us and get results!\nCatalyst Design + Marketing Ltd. 2008-2019 All rights reserved.\nVAT No. GB934428123. Registered in England & Wales Company No. 6631784.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzardjy b/data_all_eng_slimpj/shuffled/split2/finalzzzardjy
new file mode 100644
index 0000000000000000000000000000000000000000..750e398f35b560c008ae61f8f959e7805c76f13a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzardjy
@@ -0,0 +1,5 @@
+{"text":"KIMI mountain bicycle has a high carbon steel frame that will absorb vibrations when you ride, it provides you with comfort while you are riding the bike.The HC steel allows a rider to put fewer efforts while he drives because it'll continue to maintain the high speed after the rider will stop rotating pedals. It reduces vibrations, mitigates and behaves well at bends. Also, it's really easy to repair \u2013 any welder will be able to do it.\nThe bicycle is quite compact due to the size of its wheels, which is 20 inch and to its folding frame that makes it easy to store in small places. Also, it's a good option for travelers \u2013 you can keep your bicycle in the car or on public transport. The bicycle has 21 gears, so it'll reduce your efforts on hard terrains \u2013 like hills, for example.\nKIMI has standard plastic pedals with reflectors what is quite a nice feature for a night cycling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"i tried until now. it failed until now. i do not mind. please try again. it will fail again. well it will fail.\never tried. it wasn't until now. i don't care. try again. it will fail again. well, it will fail.\never tried. it was until now. i don't care. try again. it will fail again. well, it will fail.\never tried. until now it was. i don't care. try again. it will fail again. well, it will fail.\nStep 1: Find your cousin. Step 2: Get your cousin in the cannon. Step 3: Find another cousin.\nA passionate evening says goodbye to the shooter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our commercial customers have no choice but to use an EPA registered sanitizer because of health codes and regulations intended to ensure that food products are safe for public consumption. As a company with our roots in making things easier and safer for homebrewers, LOGIC, Inc. has put off establishing our own base of EPA registered products because of the expense and, honestly, because there weren't any options that really provided all the benefits that we wanted. The available products either foam too much, are too hazardous\/corrosive, stain everything that they touch, or cannot tolerate poor water conditions. And then there were the environmental and\/or toxicity concerns!\nIf you have specific concerns about your sanitizer(s) or your system hygiene, give us a call. Maybe it will be your concerns and issues that prioritize the next sanitizing product that we release.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Add your company logo, photos, videos, and marketing information. Change the layout of your page by dragging and dropping the different sections.\nAdd external widgets, like Meebo's excellent instant messenger widget \"MeeboMe\" (www.meebo.com) to respond in real-time to sales enquiries.\nRe-use Brownbook features, like reviews or the Google map, as Bidgets (Business Widgets) on other websites.\nThis entry was posted in BB News and tagged bidget, customise, customise my page, customize, customize my page, drag and drop, google map, layout, logo, meebo, photo, reviews, rich media, testermonial, video, widget. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Trauma has a way to change a person's life perspective. It can come from a lot of things. For John and I, losing loved ones to suicide did it. Katie Cahn had a different experience altogether with equally lasting impacts. For her, just like for us, nature and happiness were centerstage of her post-trauma life.\nShe had no clue of what fly fishing would do for her when she was offered a rod eight years ago. In fact, she barely used it at first. Only when she started going to college at Western Carolina University did she find a use for it. The area offered little whitewater kayaking nearby and with school, she did not have time to drive around for her favourite activity, but the Tuckasegee River flowed 50 yards from her front door. She picked up her gift, watched a few YouTube videos and taught herself how to cast.\nShe got the basics figured out but still had so much to learn. From there, she started going to the main channel of the river where most of the anglers went where she received some precious advice.\nThrough her college years and after, Katie kept her rod busy. She loved the peace and quiet it brought to her life. Little did she know how much she would need it. Years into her teaching career, she found herself dissatisfied with her work, but felt like the bad parts were a necessary evil. After all, she needed the pay, the job stability, the health insurance and the pension. It would be silly to walk away from that only because the bureaucracy, computer work and ton of paperwork were killing her inside. And she really loved the kids although she didn't feel like her job was about teaching them. When she needed a change, life forced her to make one.\nWhen the reality sets in and you realize you may die tomorrow, you see life differently. First, you are afraid because you may vey well die. Every CT scan report, every blood test results is a reminder that you are not controlling the outcome. You are scared because you're not ready to die. You are scared because you feel you still have so many reasons to live.\nKatie changed her outlook on life. From her hospital bed, all she wanted to do was go fishing and so six days after her second surgery where she got her appendix and ovary removed, she was on the water with a rod in her hands.\nKatie didn't go back to teaching full time. She was determined not to do anything that she hated anymore. She wanted to focus on things that brought her happiness. When she was approached to become a fishing guide, she jumped on the opportunity to do something fun for a living.\nCharging through life, because she's not ready to die, because everyday on the water she witnesses nature's beauty. Life's colours seem more vibrant when you thought you might have just seen your last sunset, but it's just a matter of perspective. Go find your reason to live life to the fullest, because the future is promised to no one.\nso mmany people go through all of their life not realizing their dream and paying homeage to the paycheck \u2013 good for her she made the leap and was happy. God bless fishing\u2026\u2026..","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzariui b/data_all_eng_slimpj/shuffled/split2/finalzzzariui
new file mode 100644
index 0000000000000000000000000000000000000000..7f22fca42a15ac4c57720c0eba31d66de3ec3a82
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Tattoo Scarring is possible and it happens when the tattooed area is not treated properly. The healing aftercare treatment requires proper advice from a professional artist. Following your artist advices is the only way to stay away from scars.\nMonitories the place, especially after the healing process (10-14 days after getting a tattoo). Identify the scar. Scarring under a tattoo can be tricky to spot sometimes because ink can distort the look of your skin.\nSome people says that aloe is a good plant that can help in the process of scar removal. You can buy at discount stores with gardening departments. You shouldn't need a huge plant to minimize your scar, but be sure the one you choose has several thick, healthy leaves. You will have to alternate between several leaves because the aloe plant will need time to heal itself between tears.\nAloe, however, can minimize the look of a scare from a tattoo. Using aloe directly from a plant gives you the best results.\nTear off a small piece of the plant and squeeze the aloe directly on your tattoo. Rub the aloe into the scar and allow it to dry completely. Repeat the process several times a day until your tattoo is flatter and less scarred.\nRemember: The healing process may take a several weeks. It won't happen over night, so be patient. If after enough time your scar isn't reduced to a acceptable level, consider visiting a doctor to talk about laser resurfacing options.\nAsk him for an advice. His opinion is maybe the best choice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What are the roots of sauna benefits?\nLet's start with this. Getting into great shape provides all kinds of physiological benefits, from stronger muscles to greater cardiac efficiency. A less obvious but equally critical boost: better heat tolerance. This allows us to perform better in the swelter (five sets at high noon on center court). But it also helps us better deal with internally generated heat\u2014up to 75 percent of the energy from muscle contractions comes in the form of heat, not productive work.\nOne way, of course, to increase heat tolerance is to exercise vigorously in it, though this can be dangerous when temperature and humidity levels soar. Fortunately, researchers may have found a new way to acclimatize passively, if not entirely painlessly: post-workout sauna training.\nLance Armstrong got his edge from EPO, which increases blood volume. Turns out, you can boost yours naturally \u2026 in the sauna.\nSuch traditional explanations of sauna benefits might play some role, but Sims and her colleagues suspect there's a more potent reason that sauna heat fires up sports performance. As the sauna gradually warms up our core temperature, she explains, sensors in our brains trigger sweating to keep us from parboiling. Sweating in a sauna, of course, doesn't do much to cool us. But it does have a secondary effect. Since the liquid in sweat comes mainly from blood plasma, the more we drip, the more dehydrated and difficult to pump our blood becomes. This decreases oxygen tension in the kidneys, triggering the release of a natural hormone called erythropoietin, or EPO, which, in turn, stimulates red blood cell production and increases blood volume.\nChances are you've heard of EPO before. Synthetic forms have long been used medicinally to treat serious health conditions like anemia. Lance Armstrong admitted to taking it, too, despite its ban by World Anti-Doping Agency for being an illegal blood doper.\nBut boosting your own body's supply naturally is entirely kosher. In fact, most athletes do this, at least to some extent, during our workouts thanks to dehydration. Some sports, of course, are legendary for their dehydrating potential. Marathon runners, for instance, have been known to lose more than 1.5 gallons of sweat\u2014and 10 percent of their body weight\u2014during summer races. It's not prolonged endurance exercise in a hot environment that causes it, either. Even swimmers sweat, even though it's hard to see them do it. The water doesn't even have to be particularly warm, either.\nFor a study in Science & Sports, for instance, competitive male swimmers agreed to race a 5K three different times in different pool temperatures. When the water was a stifling 90 \u030aF, the swimmers lost up to 3.5 pints of sweat per hour; at a comfortable 80 \u030aF, the rate was still nearly 2.5 pints; and even at a chilly 73 \u030aF, they lost up to 1.6 pints per hour.\nTo maximize the benefit, Sim's has developed a protocol to use before any major event likely to challenge you to your limits. Make sure to check with your doctor first, especially if you take medications (including NSAIDs like ibuprofen or naproxen, which reduce blood flow through the kidneys) or suffer hypertension, kidney problems, or other serious health problems.\nBegin sauna training at least two weeks before your event.\nSoon after each daily workout, enter the sauna set to 185 to 195 \u030aF degrees.\nTry not to drink anything before or during your time in the sauna\u2014the point is to increase dehydration.\nIf you need a break, get out briefly, pour cold water on your neck and tongue, then return.\nStay seated and don't exercise. Passive sauna bathing alone will increase your resting heart rate to 140 to 150 beats per minute or faster.\nMany younger adults can gradually work up to 30 to 35 minutes exposure per session, but older folks may need to cut this time in half.\nNever remain in the sauna if you start to feel sick. Do not become competitive with yourself or others.\nAfter exiting the sauna, resist the urge to gulp water. Instead, sip fluids gradually over the next 3 to 4 hours.\nRepeat the above every day for at least a week, even on rest days.\nOne week before your event, stop sauna training completely and let your system recover and rebound. This will maximize the increase in blood volume and set you up for enhanced performance on the big day.\nFinal note: You can maintain EPO's benefits for up to six weeks by taking a long sauna once every ten days.\nLegendary wrestler and wrestling coach Dan Gable gives his take on sauna benefits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NEW YORK, April 2, 2018 \/PRNewswire\/ -- Grayscale Investments, LLC, in its role as agent (the \"Agent\") of the shareholders of record as of December 4, 2017 (the \"Record Date Shareholders\") of Bitcoin Investment Trust (OTCQX: GBTC) (the \"Trust\"), announced today that, on behalf of the Record Date Shareholders, it has completed the liquidation of approximately 172,244 Bitcoin Gold tokens, the rights to which were distributed to the Record Date Shareholders on December 4, 2017.\nOn December 4, 2017, pursuant to the terms of the Trust Agreement governing the Trust, the Trust, acting on behalf of the Record Date Shareholders, appointed the Agent and distributed all of the rights to Bitcoin Gold tokens then held by the Trust to the Record Date Shareholders by transferring such rights to the Agent.\nOn March 14, 2018, the Bitcoin Gold tokens became accessible to the Agent. On behalf of the Record Date Shareholders, the Agent subsequently exercised the rights it held with respect to the Bitcoin Gold tokens and liquidated the Bitcoin Gold tokens over a period of 14 days.\nIn connection with the foregoing, the Agent will remit the cash proceeds from the liquidation of the Bitcoin Gold tokens, net of the administrative and other expenses incurred by the Agent in connection with the liquidation, to the Record Date Shareholders on April 3, 2018. Each Record Date Shareholder will receive $0.056478 per share.\nFor the avoidance of doubt, Grayscale Investments, LLC is acting only in its capacity as agent of the Record Date Shareholders with respect to the foregoing and not in its capacity as sponsor of the Trust.\nEstablished in 2013 by Digital Currency Group, Grayscale Investments, LLC (\"Grayscale\") is a trusted authority on digital currency investing. Grayscale provides secure access to the digital currency asset class through its single-asset and diversified investment products, including Bitcoin Investment Trust (OTCQX: GBTC) and Digital Large Cap Fund. As of March 31, 2018, the firm managed $1.5 billion in assets. For more information on Grayscale, please visit www.grayscale.co.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Oakland B gives you the benefit of an open floor plan while still maintaining distinct divisions between the family room, dining room, and kitchen. A large foyer adds to the flow of this home, setting the stage for the open living spaces. And a covered porch opens the door for outdoor living. This floor plan also features a 4th bedroom with full bath over the garage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Evaluating linkage to care for hypertension after community-based screening in rural Uganda. Our objective is to determine the frequency and predictors of hypertension linkage to care after implementation of a linkage intervention in rural Uganda. During a multidisease screening campaign for HIV, diabetes and hypertension in rural Uganda, hypertensive adults received education, appointment to a local health facility and travel voucher. We measured frequency and predictors of linkage to care, defined as visiting any health facility for hypertension management within 6 months. Predictors of linkage to care were calculated using collaborative-targeted maximum likelihood estimation (C-TMLE). Participants not linking were interviewed using a standardised instrument to determine barriers to care.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaswop b/data_all_eng_slimpj/shuffled/split2/finalzzzaswop
new file mode 100644
index 0000000000000000000000000000000000000000..bfb70605fef060467c88c882b0c9f3174ec805d0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaswop
@@ -0,0 +1,5 @@
+{"text":"The senior anchor for Univision, Jorge Ramos, declared on February 24th that the United States belonged to Latino migrants. Univision is an American Spanish language broadcast television network. The network's programming is aimed at Latino Americans. It is available on cable and satellite television throughout most of the United States.\nJorge Gilberto Ramos \u00c1valos was born March 16, 1958 in Mexico City. Since 1986, Ramos has been the anchorman for Noticiero Univision, a nightly Spanish language newscast. Currently, he also hosts Al Punto, a Spanish-language Sunday public affairs program aired weekly on Univision. Ramos earned a master's degree in international studies at the University of Miami in Miami, Florida. In 2008, Ramos became a United States citizen.\nReported on Breitbart on February 27th by Katie Mchugh, Ramos explains that mass immigration, particularly illegal population, was a fait accompli. \"There is nothing the U.S. can do about it, and they must accept that America is not their country,\" he quoted.\n\"I don't see why we can't have in North America the same immigration that they have within the European Union,\" Ramos said to Time Magazine in 2014. In a CNN interview in August 2015, he thought that the U.S. should not enforce its immigration laws, even after illegal aliens kill Americans. In August 2015, reported by Tony Lee on Breitbart that the proposed legislation, Ramos said, Kate's Law was \"completely unfair to millions of illegal immigrants.\" This proposed legislation would require deported illegal immigrants who are caught re-entering the country to be sentenced to a mandatory five-year federal prison term if they had a prior felony conviction.\nMany consider Jorge Ramos the leading spokesman for the millions of Latinos in this country. Unfortunately, Mr. Ramos has difficulty differentiating legal Latinos who are already citizens from illegal immigrants who are already violating federal law by just being here. He seems to lump them all together which is in total contrast to his native country of Mexico, where the laws concerning non-citizens are much stricter than here in the United States.\nIn 2015, the Latino population had reached 57 million and is growing at a rate of 2.8%, which has decreased from an average of 4.4% from 2000 to 2007. This amounts to almost 29% of the total U. S. population of which between 10-15% are illegal. As a group, the majority vote Democratic: 65% for Hillary Clinton in this most recent election and in the 2012 presidential election 71% for Barack Obama. As their numbers grow, so does their influence in the state and national elections.\nFor the first time, the Trump administration is trying to tackle the multifaceted problems of control of our southern border, convicted illegal immigrant felons, immigrants who overstay their visas, sanctuary cities, repeat felons who gain reentry, the Latino gang influence and an epidemic of drugs that come into this country through Mexico.\nMr. Ramos could better serve his newly adopted country and the Latino community he claims to represent, if he would realize the importance of stemming the flow of uninvited aliens, which in many ways disadvantage those Latinos who are already here legally. Although he speaks their language, he doesn't appear to be speaking to their needs!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Mobile Marketing Association (MMA) (www.mmaglobal.com) recently announced its annual top 10 industry predictions for mobile marketing in the Asia Pacific region. As 2010 comes to a close, MMA provides a guide to new and existing technologies that will dominate the market in the coming year.\nBy submitting a comment you grant GOMOTEXT.COM a perpetual license to reproduce your words and name\/web site in attribution. Inappropriate and irrelevant comments will be removed at an admin's discretion. Your email is used for verification purposes only, it will never be shared.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Molecule is created by Marc Grue who both plays the double bass in the Gothenburg Symphony Orchestra of Sweden and programs a lot in Scala.\nThe plan is to use Molecule as the optimal database access DSL for a big website for all classical musicians that Marc works on.\nMore developers are welcome to join with Pull requests and questions\/discussions on gitter in the Molecule forum.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FYP offers 3-ply Napkins with and without embossed borders, 3-ply Shimmer Napkins with a pin-dot embossed border, 3-ply Patterned Napkins with a pin-dot embossed border, air laid Linen Like Napkins, Woven Napkins and select 4-ply Napkins.\nOur marble, watercolor and leaf patterns were all organically made by FYP graphic designers and then digitally transferred to our paper cocktail napkins. Embossed with a pin-dot design, the pattern is printed only on the outside, visible layer\u2014inside layers will be white.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In our initial meeting, we get to know each other and discuss your financial goals and desires to gain a sense of what you are wanting to accomplish. We also outline the processes we can employ to address your unique needs.\nWe analyze your personal and financial position, risk tolerance, and goals. We listen to what you want to accomplish and identify areas where our direction and experience can assist you in reaching your goals.\nFrom our review of your financial position, we develop personalized financial solutions that address your unique needs.\nWe offer our solutions and suggested investment options to you, explain how we have addressed pertinent issues in your financial life, discuss your input or questions, and get your approval of all aspects before we proceed.\nAfter your approval, we will implement the necessary steps to manage your financial assets and partner when needed with your estate planning attorney and CPA.\nWe are committed to meeting with our clients on a regular basis to monitor your progress towards goals. As we move forward, we make adjustments for changes in your life, in the economy, and in any other areas that may impact your financial future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatzht b/data_all_eng_slimpj/shuffled/split2/finalzzzatzht
new file mode 100644
index 0000000000000000000000000000000000000000..4bb96da9a67c403761e2e44dda6c866f4f69ff34
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatzht
@@ -0,0 +1,5 @@
+{"text":"If you are not sure whether this machine can briquetting your raw material, contact us now and you will get professional response from our professors. A testing video of briquetting process can be offered if required.\nOur mini coal ball briquette machine is two rollers single press type machine. It is designed for process coal\/charcoal dust. It is designed for process coal\/charcoal dust. It is one of our hot sale products in recent years because of its high efficiency, exquisite appearance and low cost.\nCoal ball briquette is a type of coal briquette used in home cooking, heating and industrial stoves, people used to burn the raw coal in the past, even now in some rural areas. Burning the raw coal bricks is a cheap way of making use of the powder coal, but definitely not the most efficient way, now with all those years of industrialization in China, the coal processing industry in China has developed to a degree which made China in the leading position of briquette machinery manufacturing.\nAGICO is an experienced supplier of coal ball briquette making machine in China. China is a country which still uses coal as the main fuel for the power plants and in the heating system of many cities. The repetitive construction way of building coal briquette plants in China had been the main problem which limited the development of this industry, and the manufacturers which survived through the competition are the ones that have adopted new technologies for new machine models. The market has become specialized, and so should be the products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bio: The author is known as Forest Valles and he completely digs that title.\npromotion never comes. Massachusetts is the only location he's been residing in. Fishing is the only pastime my wife doesn't approve of.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Custom mobile apps need no introduction. Almost every leading mobile company is using this platform, making sure to grasp the major portion of the smart phone market. The open-source characteristic of custom mobile app builder brings extensive creativity in the development of applications. Mobile app builder provides high-end Wi-Fi and Bluetooth functionalities that bring endless possibilities with them to develop revolutionary applications. Mobile app builder generates a positive buzz among business opportunists. Its scalability has derived profits for many organizations and is thus ensuring better investment returns.\nOne of the biggest challenges in these custom mobile apps is to leverage the latest technology and simplify complex business processes like mobile payment, integration, CRM and collaboration between various teams and the end-customer. Here is where a strong custom mobile app builder can help with your enterprise mobile application development needs. Reliable and trustworthy mobile application developers will point you towards the right solution and not the most expensive one.\nMy app center goal is quite simple \u2013 Mobile application developers help you develop enterprise mobile apps that provide easy access to data & functionality, on-demand \u2013 anytime, anywhere. From apps geared towards the consumer to highly sophisticated apps built for your enterprise.\nFinding the best mobile enterprise app development company should be a priority for decision makers, as your enterprise's reputation for quality is at stake. My App Center team understands that scalability and robustness of service delivery are of the utmost importance for a solid Enterprise App. Myappcenter custom app development process includes fast prototyping and the customer will see the first prototype already in 48 hours. Our developers follow a strict policy pertaining to use of components across platforms. This makes it easier to maintain the applications and extend functionality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When Los Angeles teacher Coleen Bondy opened her evaluation, she found that the district considers her a below average teacher. Bondy argues that her evaluation was unfair because it only measured the outcomes from her least motivated students. She also believes that test scores by themselves are a poor method of evaluating teachers.\nThe Texas State Board of Education ruled that annual standardized tests won't be rewritten until state lawmakers provide funding for districts to buy new textbooks. The board said a disconnect between information found in textbooks and testing materials based on the new curriculum wouldn't be fair to students.\nA Kansas City college student quit his job as a tutor to start a non-profit that offers free tutoring to high school students.\nSome Baltimore County parents are concerned that their children will suffer when hot weather comes back this spring, so they offered to pay for air conditioning window units. The school district turned down the offer, and requested $70 million from Maryland instead.\nA Michigan school district canceled the first day of the second semester after vandals put 15 buses out of commission.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New pool's location makes it the daily hub for this Kailua family.\nThis Kailua homeowner looks out at his new pool and just shakes his head. \"We almost made a huge mistake,\" he says.\nWhat wasn't quite as clear was where to put the pool on this expansive, hillside lot with ocean views. \"We were going to put the pool on the lower level in the backyard, but I got to thinking about it,\" says the homeowner. \"If we could keep it all on the same level as the living space, wouldn't that be better? Everything is here. This is where we spend all our time, so why not put it up here?\nClifton Crawford of C & J Contracting and his team got to work, installing the pool and spa in the front yard adjacent to the new covered lanai. The company also used fill dirt from the pool to triple the usable space in the backyard so there was plenty of room for the family trampoline, a treehouse and an expanded lawn. Wolfgang Landscaping provided the finishing touches in the upper level around the pool and entry and on the lower level.\nToday, this family loves, absolutely loves, the pool, spa and covered lanai. \"We spend an incredible amount of time out here. This is where we congregate,\" says the homeowner.\nPut down the iPad and give these family-friendly board games a try.\nSix tips to ready baby's new room.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
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+{"text":"Here is what we learned: \"To the White House, the difference between success and failure is straightforward: They need to entice a sufficient number of young and healthy adults into the new insurance marketplaces that open Oct. 1.\" . . .\nThe key to the exchanges was getting enough young and healthy people to turn out. The reason is simple: Insurers price their products at the average expected cost of the people signing up, plus a bit more for overheard, profits, etc. So if the average person signing up is relatively sick, premiums rise. if they're relatively healthy, premiums fall.\nSicker, older people, the administration figured, would be desperate to sign up for health insurance. In a sense, that was the problem: They'd be so eager that they'd sign up in much greater numbers than the young, healthy people needed to keep premiums low.\nAttracting those young and healthy people was thus the core challenge. The White House figured that if they got 7 million people to sign up for the exchanges in the first year, about 2.7 million needed to be young.\nThe Obama health-care team expended enormous effort figuring out how to reach those 2.7 million \"young-and-healthies.\" . . .\nDuring the open enrollment period, someone could get sick or injured and apply for insurance literally on the way to the hospital - and the insurer will still have to accept him or her. That's a risk insurers are accepting in January, February and March of 2014. Add another month of that kind of risk and it's going to significantly add to insurers' costs, they say. Although others argue that the procrastinators are more likely to be young and healthy than sick and costly. . . .\nI have also been pointing out for years that no one really has to pay the fines for not getting the insurance. This point was made this week by Rush Limbaugh.\n\"[I]f you structure your taxes so that you do not get a refund, you do not have to buy insurance and you do not have to pay a fine 'cause they can't collect it from you if you don't have a refund due \u2014 and that is just another nail in the coffin of Obamacare imploding on itself,\" Limbaugh said on his show Thursday night, according to transcript. . . .\nTo make the point more simply, just increase your deductions so that you aren't paying in too much in taxes. Despite what some people may think, it isn't good to get a tax refund because it means that you have been giving the Federal government an interest free loan.\nNow Harris, a self-employed lawyer, must shop for replacement insurance. The cheapest plan she has found will cost her $238 a month. She and her husband don't qualify for federal premium subsidies because they earn too much money, about $80,000 a year combined.\n\"It doesn't seem right to make the middle class pay so much more in order to give health insurance to everybody else,\" said Harris, who is three months pregnant. \"This increase is simply not affordable.\" . . .\nThe media is always trying to spin these increased premiums as good news, but it just isn't so. Here is something kind of amusing from SNL.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Movies are the unique art that was born from technology, in its beginnings it was projected on a savanna into a dark room but nowadays it leaves us amazed when we watch a film on a movie theater. It is an industry and an art, a spectacle and a thought; it should be treated, studied and cultivated as the poetry, the literature or as an artistic language.\nThis audiovisual industry is at the present time an interesting and consolidated activity with an unbeatable future; furthermore has a professional scenario much more open than any other profession. It has a continuous activity because it registers very low indexes of unemployment and receives the support of the government in many countries. It is a creative and independent profession which enjoys a social recognition in all of their tied activities; among others of their advantages it has a strong creative autonomy.\nA film school is an educational institution that is dedicated to the film instruction; it means that it covers aspects like movie making that includes film production, theory, and scriptwriting. Most of the film programs incorporate a technical training, such as learning the use of cameras, lights and other necessary instruments for the production of movies.\nThese schools offer to students an interdisciplinary program that is focused in the history, theory, critic of cinema, etc. they also include courses such as Anthropology, Sociology, the Comparative Literature, Art and History. Professionally, these programs are capable to educate students so that they can work in a variety of careers related to the film industry.\nAs every learning, teaching cinema needs experiences, affections, motivations and people who contaminate enthusiasm; this is achieved just attending to a specialized school, where is a lot easier to put the technology at the film message service, that technology has metamorphosed cinema, and has made unavoidable to know about philosophies, thoughts, histories, places, life styles and customs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Elaine S. Larson Center of United Hospital provides surgical services for the community and is now remodeling their surgery waiting area. This renovation project includes increased seating for guests and families, improved consulting spaces, and a fresh new palette of finishes and furniture. The new furniture will have multiple types of seating, from comfortable lounge seating, to tables and chairs, to computer workstations. Accommodations have been integrated into the seating and end tables to charge phones and tablets at chairside with USB outlets. New LED lighting systems have been designed to provide a warm and comfortable environment. A new fireplace feature element has been designed to bring comfort and ease the anxiety of those who will be using this space. New circular LED pendants and a soothing full height woodland graphic image help complete this surgical waiting area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Trusted Tree Services is here to serve your requirements for tree service in Denville Township, whatever they may happen to be. Your property can receive the significant advantages that trees are capable of providing whether it's a home or business. However, to receive these benefits, your trees need to receive the care they need. Ongoing tree service is crucial when you'd like to make sure that your trees stay in the best health possible. The fact that they add a sense of attractiveness is one of, if not the most obvious benefit which well maintained trees are capable of offering. Your property value increasing is the second benefit, which can be due to of this increased attractiveness. In the Northern NJ region, property values are significantly high, making this an especially important concern. Well placed trees are capable of reducing the energy needed for heating by 20-50 percent and decrease the need for air conditioning by 30 percent, according to the USDA Forest Service. When you take into account the increasing price of energy, this benefit becomes clear. Making certain your trees receive the ongoing care they require all throughout the year is absolutely essential for all of these reasons. Make the choice that local residents have been counting on for years when you want top quality tree care. Call Trusted Tree Services today if you're interested in receiving dependable tree service in Denville Township or the rest of the local region.\nAmong the essential services we offer when it comes to tree service in Denville Township is professional tree trimming. Professional tree trimming is crucial if you'd like to ensure that your trees are consistently looking the best they can. If a tree is over- grown, it is appearance can be significantly improved. It isn't about just aesthetic concerns either. As time passes, this is going to assist the tree in growing easier. The spread of disease can be prevented by eliminating branches which are dead or diseased. Making certain that a full tree removal isn't required down the line is going to be the result. It can additionally provide a few other advantages from when the branches in the crown of your tree are thinned. Increased light is going to have the ability to reach the tree roots, and it will improve the air circulation around your leaves. Does your tree produce nuts or fruit? Trimming is going to allow them to do so more effectively. This isn't something which should be attempted by anyone other than a expert, however. It might appear like something that sounds easy but requires in fact expert precision and knowledge regarding trees. Contacting the pros at Trusted Tree Services is in your best interest as an alternative to trying to perform tree trimming or other types of tree service in Denville Township on your own.\nWe can additionally help at Trusted Tree Services when it comes to tree removal in Denville Township. Occasionally regardless of how much you invest in quality routine tree maintenance, a tree still cannot be saved. Substantial damage following a storm or disease are reasons why this might be the case. Despite the fact that these are the most typical, residents find themselves requiring our services for a broad variety of other reasons. Tree branches can start to encroach on utility lines or structures. A tree may as time goes on start to lean in a direction which you don't want, such as towards you or your neighbors' home. This is something you might not even notice right away considering you see your trees generally on an everyday basis. Your trees might have internal structural problems, which nobody but an experienced professional is capable of noticing. Thanks to our wide ranging experience with the trees in the local region, any of these indications of a potential issue are capable of being noticed by our skilled professionals. We utilize cutting edge equipment to get the job done in regards to tree removal in Denville Township. We're going to get the job done effectively and quickly and for a reasonable cost. This is just another one of the reasons such a large number of commercial and residential clients in the local area depend on our services.\nWhen a tree removal in Denville Township is going to be something you need, catching the problem as soon as possible is vital. This is why you need to rely on an expert tree service contractor. We're able to identify many problems with your trees as a result of our substantial amount of experience. One of our professionals will have the ability to ascertain what the right course of action is going to be after an inspection. By getting in touch with us sooner rather than later your safety is going to be ensured and you aren't going to have to worry about being liable for injury or damage. This is the reason such a large number of residential and commercial clients rely on us for tree care on a routine basis. We can better determine what your trees' needs are in this way as we can check up periodically on your property. Many possible issues are caught this way. We're capable of offering many services which make it possible for trees who might have needed a complete removal down the line to be saved if caught early enough. When you hire us on a routine basis, you can think of it as a good investment. Routine tree maintenance is generally quite simple and manageable, at a regular but low cost to you as the customer. If a full removal is required, it is a complicated procedure which is going to incur a larger cost, regardless of how affordable our services are compared to other contractors out there. If you want to ensure the best protection for you and your family with expert tree maintenance and tree removal in Denville Township, there's no one else you've got to call.\nWhatever your requirements are when it comes to tree service in Denville Township do not hesitate to call the experts right away at Trusted Tree Service. We are standing by to provide you with the care you need to make sure the trees on your property really thrive. If you are in a situation where you are faced with a hazardous tree that's leaning or a large limb that's about to fall, you can rely on us to be there for emergency service. You'd be hard pressed to find another contractor out there who has a trusted reputation in the local community for quality service that compares to us. We employ only tree service specialists who will be sure to get the job done properly the first time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 1910 Flexner Report called on American medical schools to enact higher admission and graduation standards.\nThe medical society responds to disaster.\nMore than 200 physicians, leaders in the medical community and their guests joined SFMMS for an extraordinary 150th Anniversary Celebration & Gala on March 15, 2018 recognizing an important milestone in the organization's history.\nThough organized medicine had failed to recognize the contributions of female physicians, in 1877, following the leads of the state and national societies, the San Francisco Medical Society accepted the first woman into its membership.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Focus: multi-page documents & the grid system.\nRequired text: Josef M\u00fcller-Brockmann, Grid systems in graphic design: A visual communication manual for graphic designers, typographers and three dimension designers. The best, most thorough, and most rigorous explanation of the grid system of page design, with numerous specific examples.\nRecommended additional text: Jan Middendorp, Shaping text. Possibly the best currently available overview of how typography works, with lots of examples.\nA useful resource for using InDesign: Olav Martin Kvern, David Blatner, & Bob Bringhurst, Real World InDesign CC (or any earlier edition of Real World InDesign, though they might lack some features of the most recent InDesign versions). Direct, conversational, occasionally funny, and very practical guidance in using InDesign's many features, including type.\nChoose one of these type families for the text in your blad. For display (titles, etc.), you have free choice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every year, our school holds an annual Meet Each Need with Dignity (MEND) drive, during which students are encouraged to donate as many canned goods as possible to help out families in need. The Associated Student Body even offers an incentive by awarding the homeroom with the most canned goods collected a pizza party.\nFor the students, this drive means taking a few minutes to round up canned goods already sitting in their kitchens at home. This should be an easy task with a large amount of student participation. However, walking into each of my classes, it is disappointing to see the empty MEND drive boxes or boxes with only a few cans. Why is there such lack of participation in our student body? With over 4,600 students at our school if each student were to even bring at least one canned item, these boxes would easily be filled.\nWhat students need to understand is that not only is donating to the MEND drive easy, it is a good way of getting involved and making a difference to people in our own communities. The more people that donate, the better the community can be. Collecting cans is one of the easiest way of giving back to the community, but it can also be deeply rewarding to yourself if you consider the meals a family will have because of your efforts.\nAccording to the latest government census, over 20 percent of the population in Los Angeles is living below the poverty level. That's a lot of people who need our help.\nOne of the most important aspects of helping others is in knowing that you are are making a difference. MEND is the largest organization in Los Angeles which provides direct distribution to clients in need. They depend on numerous food drives from supermarkets, churches, schools, businesses and other organizations in the area. Without our help, they would not be able to provide food items to those who are in need. Every can you donate brings us one step closer to helping someone in need.\n\"I think my biggest motivation is helping the students realize that they can make a difference. The MEND drive is a really easy way of getting students involved. If everyone helps a little bit, we can make a huge difference,\" social studies teacher John Yamazaki said.\nYamazaki has been consistently successful in getting his students to bring in canned goods. His class has won the school's competition for the past four years. He has been successful because he reminds his students on a daily basis. In addition, he has reminders written on his board and always emphasizes the of the importance of giving back and making a difference. His message of helping others in need is strongly embraced by his students as he himself deeply believes it.\nGiving is contagious. The more you give and spread your passion for giving back to others, the more others will give. When you give, you create a ripple effect, spreading generosity throughout the school.\n\"Students only see one side of the MEND drive in their classrooms, but the donations we receive overall are very appreciated not only by ASB, but the families that our school gets to help out. In fact, since we had such high participation last year, MEND wrote us a thank you letter for donating about 1,500 can within our school,\" senior Courtney Azari said.\nOftentimes, students claim they do not have time to volunteer due to other commitments. However, this is a simple way of getting involved and making a difference without having to compromise your busy schedules.\nIn the end, we are all trying to make the world a better place. Helping others can make you feel happier and make your life more fulfilled. Donating is a selfless act and is hugely empowering. One of the major positive effects of donating to the less fortunate is simply feeling good about yourself. Being able to give back to those in need helps you achieve a sense of personal growth and connects you to your community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Choose a container. Use a clean bowl, preferably between 3-1\/2 and 4 inches high, or taller, and 8 to 12 inches wide. Place a submersible pump in the center, off-center if you'd like the flow to come in at an angle.\n2. Add water to cover the top of the pump. The quality of water is important. Distilled is best; next best is filtered. Don't use tap water. Distilled or filtered water will keep bacteria levels down, increasing the life expectancy of your pump.\n3. Try out the pump. Experiment with the flow height, regulating it by turning the knob located on the side of the pump. Start with the knob in a horizontal position and slowly turn, being careful not to turn it too high as some pumps can shoot a stream of water as high as two feet when in the vertical position.\n4. Add marble chips. Rinse them well, as dirt is apt to clog the pump. When they are clean, place chips around the pump's base in the bowl. Put them up against the pump to hold it firmly in place.\n5. Choose decorative stones to go around the pump. Think about the appropriate scale when choosing stones; don't use ones that are too large for the size of the container. There are endless ways to create a pleasing visual and aural environment, depending on the size and arrangement of the stones. Put the more colorful stones in last so they'll be the most visible. Try using polished river stones, sea glass, or quartz crystals that match or coordinate with the bowl and the room in which it will be placed. Consider all pink, green, or blue stones, for instance.\n6. Choose water additives, if desired. Tabletop fountains need little care beyond adding more water every few days, and keeping the water clean. To prevent bacteria or algae from growing, add a few drops of liquid bleach, or change the water more often. For fragrance, add a few drops of a scented oil or perfume. Use a kitchen baster to remove dirty water before adding clean, or before transporting the fountain to another location so water doesn't spill.\n7. Add plants if desired. If the fountain won't be turned on all the time, attach air plants (Tillandsia ionantha) to a dry area. Otherwise, ask at a nursery about small, water-tolerant plants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vilathikulam legislative assembly election 2016 seat will witness a straight fight between DMK & AIADMK. Other parties that may influence the voters are BJP, BSP and INC.\nIn the last parliamentary elections in 2014, this assembly segment had given the AIADMK candidate Jeyasingh Thiyagaraj Natterjee a good lead of around 124002 votes. However, the situation in 2016 assembly election is completely different from the general election of 2014.\nIn the assembly elections of 2011 the AIADMK won the constituency defeating the INC party with a huge majority after getting 72753 votes. AIADMK won the constituency in the previous assembly elections also. The DMK party has won this constituency four times from the assembly elections of 1967.The ADK party has won this constituency three times and the ADMK party one time.\nThis constituency of Vilathikulam is situated in Thoothukudi district. Fishing is the occupation of the majority of people. It is difficult predict the outcome of the polls for which primarily those engaged in fishing related activities voters will decide the verdict.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Access to NHS courses etc.\nThe Students Union offer a range of services to support your wellbeing and to be there when you just need a chat. These include numerous courses\/groups, a drop in service, an anonymous night telephone line and an instant message service.\nWelfare@CampusLife can provide you with practical advice and support on many different welfare issues you may face during your time at Swansea University. We will work together with you and your college to ensure you have the best possible Student experience.\nYou will find your college's welfare contact details here.\nThe Chaplaincy's listening service offers a listening ear to whoever needs it \u2013 irrespective of religion, if any. Registration is not required and there doesn't have to be any particular issue or problem.\nTo access this service please contact Chaplaincy.campuslife@swansea.ac.uk or tel: 01792 606557.\nYour academic mentor should be your first point of contact should you have any concerns regarding your studies. If you are stuck on a problem which you have been set or you would like advice on your academic progress, your academic mentor should be able to help.\nYou can find details of your academic mentor by logging into your MyUni account.\nWithin your college there are administrative and academic disability coordinators who ensure that support and adjustments are available to assist you with your disabilities, mental health difficulties and\/or related medical conditions. The administrative and academic coordinators perform separate functions to support you.\nClick here for more information & to find out who your college's disability coordinator is.\nHow Do I Request Wellbeing Support?\nWhilst you can access our self-help resources at any time, if you feel that you need further support from wellbeing, please register by completing a personal details form found here.\nOnce your form has been reviewed, we will be in touch via e-mail within 24 working hours.\nClick here to find our contact details should you have any further queries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavolr b/data_all_eng_slimpj/shuffled/split2/finalzzzavolr
new file mode 100644
index 0000000000000000000000000000000000000000..ae006104f082eac7882c9db90144c48af9478bdd
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@@ -0,0 +1,5 @@
+{"text":"Pancytopenia, mucositis, and hepatotoxicity after intralesional methotrexate injection in a patient treated with peritoneal dialysis.\nPurpose A case of methotrexate toxicity from intralesional treatment of squamous cell carcinoma of the hands in a peritoneal dialysis patient is reported. Summary A 68-year-old Caucasian man receiving peritoneal dialysis for end-stage renal disease complained to his primary care physician of shortness of breath, mouth and hand sores, nausea, and general malaise. Four days before his office visit, the patient received 25 mg of methotrexate intralesionally for refractory squamous cell carcinoma of his hands. The patient had several pustular lesions on both hands. On admission, the patient had a normal complete blood cell count and elevated liver enzyme levels. On hospital day 3, the patient's white blood cell count was 1,300 cells\/\u03bcL, platelet count was 134,000 platelets\/\u03bcL, hemoglobin value was 12.4 g\/dL, and hematocrit was 37%. Folic acid 1 mg i.v. every six hours and leucovorin 10 mg\/m(2) (20 mg) i.v. every six hours was initiated. His serum methotrexate concentration was 0.03 \u03bcmol\/L on hospital day 6. The leucovorin dosage was increased to 200 mg i.v. every six hours. Platelets were administered, and the patient was switched to four-hour treatments of high-flux membrane hemodialysis for two consecutive days. The patient was discharged on hospital day 14. Mild mucositis was still present, but the patient had improved substantially and was discharged to a rehabilitation facility. Conclusion A 68-year-old peritoneal dialysis patient who was treated with 25 mg of intralesional methotrexate for squamous cell carcinoma of the hands developed pancytopenia, mucositis, and hepatotoxicity as a result of systemic absorption and prolonged elimination.\n10052028 - Calcitonin decreases lithium plasma levels in man. preliminary report.\n18540248 - Diagnoses of overweight children and adolescents seen in a pediatric endocrinology clinic.\n3821508 - Body composition in malignant disease.\n803408 - Intravenous hyperalimentation in patients with head and neck cancer.\nPrevious Document: Safety and efficacy of glucagon for the relief of acute esophageal food impaction.\nNext Document: Probable rifaximin-induced neutropenia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This month's topic of Poppins Book Nook is Poetry. My daughter is 8 year old, and she is now learning to write her own poems. In this post I am going to share books and free printables for 6 poetry types every child should master.\nFrom early on I made a point to read poetry to Smarty. We have a sizeable collection of poetry books, but for a while she was considering poetry boring. I wrote a post about fighting this trend \u2013 Is Poetry Boring? In that post I had mentioned that one of the reasons that my then second grader thought that poetry was boring was because they didn't read it in school. This certainly changed this year in the third grade \u2013 children not only read plenty of poetry books, they also formally studied several poetry forms and wrote their own poems. What kind of poetry forms did they study?\nThe year started with the study of sensory poems. Third graders wrote their sensory poems about fall. They prepared for it using the graphic organizer similar to this free sensory poem graphic organizer from 3rd Grade's a Hoot.\nIn December, third graders talked about free verse poems using the well known poem by E. E. Cummings Litte Tree as an example. I don't recall that poem making it back from school, but here is a free graphic organizer from Austin School District with good examples of free verse poems.\nSo I admit I didn't even know what that was until Smarty told me in February that they are learning about them in school and explained them to me. I hope I will get her poem back from school after Presidents' Week break is over. The Educators' Spin On It has a great example of cinquain poem dictated by a 4 year old (hey, you are never too young to start :))! Check out this free cinquain poem graphic organizer to write your own cinquain poems.\nLimericks are such a staple for St Patrick's Day that I fully expect third graders to study them next. Unfortunately, they are famous for being \"naughty\", but there are plenty of \"clean\" limericks for kids including this fun take on the world history. Check out the lesson on how to write a limerick and try your own.\nI am not sure if third graders will be revisiting acrostic poems this year. I believe they were studied in the first grade, but even before that I explained them to Smarty, and, when she was in kindergarten, she wrote that lovely Valentine's Day acrostic for her Papa that you can see in the beginning of this post. I recommend a seasonal series of Steven Schnur to get some taste for acrostic poems and then you can write your own using an acrostic poem graphic organizer.\nHave your kids written their own poetry yet?\nKid's Poetry Bundle Giveaway! Every month the Poppins Book Nook group will be offering readers a chance to win a brand new storybook or product that ties in with our theme for the month. This month one lucky entrant will win the Kid's Poetry Bundle. The winner will enjoy two books full of poetry for children. The two books that they will win is the Bill Martin Jr Big Book of Poetry and Poetry Speaks to Children (*kindly note that these are affiliate links and your purchase helps to run this book club). These two magical books will help your children learn about and have fun with poetry.\nYou are welcome to follow Poppins Book Nook on Facebook, Pinterest and browse the collection of ideas below.\nLink up your magical Poetry book fun below!\nEmma writes a lot of poetry! She must have inherited the poet gene from my dad that somehow skipped me (but some of my siblings do have).\nExcellent post, thank you, reminds me to do some formal poetry study in our home classroom. We went through Knock at a Star book last year and it has all the major forms of poetry with examples in it.\nMy 4th grader D. is a poet. Poetry is his favorite assignment. When he writes, he tries to make things rhyme 99 times out of 100. It's fun to see him pick words to make his poems prettier.\nPrincess writes poetry all the time. My boys are like me and largely ignore it. I really should work on showing them more poetry.\nI did an English degree and I have never heard of a cinquain poem! Off to check that one out now. My five year old is starting to make up rhymes and loves funny limericks.\nMy favorite from childhood is Shel Silverstein. I still have some of those memorized from childhood!\nWe are developing quite a taste for poetry, despite a little resistance from the kids at first!\nThe poem Smarty wrote is soo sweet!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Estee Lauder Gives: Michael Kors Gift Set once they reach 80K!\nYes, it's yet another giveaway! Isn't it fun to celebrate the holidays by getting a chance to win something for free? Well, I don't know about you but that's MY kind of holiday!\nThe receiving part, I have nothing more to ask for as God has blessed me with so much so I guess the next step for me is to share some lovely giveaways care of our friends who suppport AskMeWhats!\nOne brand who supports AMW for the past years is Estee Lauder Philippines. This year, they have another wonderful contest cooked up for our dear readers! Are you excited to know what's in store? Please read the mechanics first!\nComment in this blog post together with your name and email address telling me why you're excited to win this gift set.\nONE COMMENT per Facebook Liker.\nContest starts TODAY ends as soon as Estee Lauder \u2013 Philippines Fanpage reaches the 80,000 mark. Estee Lauder Philippines will alert me and I'll do a raffle based on EL fans who comments in this blog post and LIKES the Estee Lauder Fanpage.\nONE (1) Lucky blog follower will receive the Estee Lauder - Michael Kors Pouch set worth over Php10,000 as a gift!\nKeep in mind that you should be a FAN of Estee Lauder - Philippines Facebook Page to qualify for this contest.\nWinner will be notified over email and must respond within 48 hours or a new winner will be picked!\nWinner will be picked via Fruit Machine.\nAnother exciting day to celebrate the holidays!\nThanks to Estee Lauder - Philippines for the wonderful giveaway to our AMW friends!\nI am so excited to win this Michael Kors gift set this Holiday season because I am turning 38 next week and a birthday gift such as this is truly a major super awesome treat. The pouch bag is so lovely I can bring it anytime with me of course the contest is so gorgeous!\ni want to win this bec i have read so many raves about estee lauder and i have never tried using any of their products.\nI'm super excited to win this gift set because I love the signature Michael Kors cosmetic bag with shades hand-picked for their modern and timeless style. A great Christmas Gift for my self!\nI would love to win this MK-EL gift set since its a product worth the rave due to its quality and of course its perfect for the holiday season. Early Christmas gift for myself.\nWow. Another giveaway. This one is just so fab. Who wouldnt want an Estee Lauder\/Michael Kors!? Im so excited to win this. Its the best modern and timeless style in fashion and beauty in one incredible collection. Love. Love. Love.\nIm excited to win this Estee Lauder - Michael Kors Pouch set coz its perfect to use this Christmas season!\nI would like to win this contest for my mom because she has been an avid fan and user of Estee Lauder and Michael Kors. Winning all the goodies will surely make my mom happy this Christmas!!\ni'd like to win this contest because it'll be great gift for myself this holiday season.\nI'd like to win the gift set because I'm a fan of both Michael Kors and Estee Lauder.\nid love to win this cause i have not won anything yet ( as un nothing) in my life. i believe dat bad luck runs in our family and id like to break dat belief by winning this. also im a budget shopper so this is something dat will level up my beauty rituals!\nI want this set because Estee Lauder always makes awesome beauty products. They are a bit more costly compared to other brands but the quality is always there. And that makeup purse looks so gorgeous!\nI'm so excited to win an Estee Lauder- Micheal Kors gift set because I'm a fan of this product when my aunt gave me an EL lipstick 6 years ago and I really love it. Even the product has been expired and never used it, I didn't throw it away because of the gorgeous packaging and its my remembrance of owning an Estee Lauder lipstick. If I won this contest, I can replace my expired lipstick into a new one and this is a perfect gift for me this holiday season.\nHi! I'm Trina and my email address is: kinnykoons@yahoo.com.\nI want to win because i haven't tried any EL cosmetics except their lipsticks.\nI'm excited to win this Estee Lauder set, because I've tried some of EL's products and all I can say is its greaaaat!!! You get more than you pay for. totally worth buying their products!! So generous of EL for this Giveaway!\nThis is very exciting! I would love to win this giveaway because my mom is a fan of Estee Lauder! This is the perfect xmas gift for her, and of course, the Michael Kors bag is so chic perfect for me :)) Thank you Estee Lauder and Nikki for this fantastic giveaway!\nI'm excited to win this set because i both love MK & Estee Lauder!\nI love to try EL products because their level of excellence shows on their products. Their product is always top notch and they dont settle for less.\nI'm very excited to try this set. I've always wanted to try Estee Lauder based on the online reviews, as well as recommendations from my relatives.\nI'm so excited to win this gift set because I have heard so many great reviews for Estee Lauder and I have yet to try their products! Plus, I'm looking to expand my \"just-starting and just-budding\" makeup collection!\nI'm excited to win this gift set because it's my chance to get to have these fab items without having to actually shell out lots of moolah!\nI've always loved both Estee Lauder and Michael Kors. I would love to win this gift set because I think it would be awesome to share this with my mom.\nBecause these are limited edition sets from a premiere makeup brand and renowned fashion designer.\nI wanted to win this for my sister!\nI'm excited about this giveaway! EL has a lot of great products that accentuate the natural beauty of every woman. The texture and color of their makeup is excellent; the partnership with Michael Kors is an added bonus because their products would now come with stylish packaging.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finals will be held May 30 and 31, with a Gala Concert on June 2. Prizes will be announced right before the Gala Concert. Violinist.com Editor Laurie Niles will be in Montreal to bring you coverage of the finals and gala concert, which also will be shown on Medici.tv.\nBelow is the video of Saturday's semi-finals. Find the rest of those videos by clicking here.\nJury members for the 2016 Montreal Competition are: Pierre Amoyal (France); Boris Brott (Canada); Boris Garlitsky (Russia -France); Ida Kavafian (Armenia - United States); Mihaela Martin (Romania); Vera Tsu Wei-Ling (China); and Kathleen Winkler (United States), with Andr\u00e9 Bourbeau, Co-Founder and Chairman of the CMIM, acting as non-voting president of the jury.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Patients are welcome at Florida NeuroHealth to use the Patient Portal to send medication refill requests, update your demographic information and have your appointments confirmed by email!\nPlease contact the practice at 407 332 5141, or your next appointment to request a password. We will need your full name, date of birth, zip code and your email address. We will provide you with a temporary password to use on the patient portal.\nClick the Patient Login button in the upper right-hand corner of this webpage or click here. Use the email you sent us and the password given to you to login. Follow the instructions to change the password. Please fill out all demographic information and medical information.\nUse the Patient Portal to refill prescriptions, update your demographic information, or send the office a message. Also, your next appointment will be confirmed by email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawbyg b/data_all_eng_slimpj/shuffled/split2/finalzzzawbyg
new file mode 100644
index 0000000000000000000000000000000000000000..e4eeb4588719f0891730a4164e86a3ebbac8cc45
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@@ -0,0 +1,5 @@
+{"text":"Nanticoke Headwaters was purchased by Lyme in December 2007 with Ecosystem Investment Partners and another investor. It consists of 1,206 acres of land in Sussex County, Delaware and has been approved as Delaware's first wetland and endangered species mitigation bank. The property lies within the Cypress Swamp Forest Legacy Area, a conservation priority for the State of Delaware, and contains habitat for the federally listed Delmarva fox squirrel. The properties have been extensively ditched and drained, resulting in significant alteration of the natural wooded wetlands. Restoration work would return the lands to a natural functioning wetland.\nIn December 2011, The Lyme Forest Fund III acquired 72,783 acres in northwestern Wisconsin in partnership with The Conservation Fund and the Wisconsin Chapter of The Nature Conservancy. In February 2016, The Lyme Forest Fund IV purchased an inholding of 12,779 acres. Through a series of transactions, the Wisconsin Department of Natural Resources purchased conservation easements over approximately 75,000 acres of Lyme's Wisconsin holdings, the largest conservation effort in the state's history.\nThese lands are part of a much larger, multi-state block of conserved lands that encompass over 2.9 million acres. In September 2017, Lyme sold all of its Wisconsin holdings to another timberland investment manager. The conservation easements on these properties ensure that that the land will be available for forest management and timber production to serve the wood products industry, and will remain open to the public for recreation in perpetuity.\nTalisheek Mitigation Bank, initially acquired in 2008, consisted of 2,510 acres of land adjacent to an important Nature Conservancy preserve in eastern St. Tammany Parish, Louisiana. The property contained wetlands, working forests and endangered species (gopher tortoise). Lyme and its joint venture partner, Ecosystem Investment Partners, established this property as an approved wetland and endangered species mitigation bank.\nRestoration work included removal of the existing pine plantations through timber harvesting and the restoration of a mosaic of longleaf pine savannas, slash\/pond pine cypress woodlands, and bayhead swamps through replanting and controlled burns. The restoration work also included removal of invasive species and restoring natural hydrology to some portions of the property that have been altered by roads, ditches and culverts.\nConsisting of 22,150 acres of land on the east side of West Grand Lake in southeastern Maine, the Grand Lake Stream (GLS) Woodlands property was purchased in 2008 by The Lyme Forest Fund. The ownership included 17 miles of undeveloped shoreline on West Grand Lake, Lower Oxbrook Lake, and Big Lake. The property had been a major conservation priority due to the extent of its undeveloped shoreline, the lakes' unique cold water inland fisheries and its location within a 400,000 acre area of conserved land. This investment opportunity resulted from the successful partnership with the Downeast Lakes Land Trust (DLLT), The Conservation Fund, Coastal Enterprises, Inc., the Sustainable Forest Futures of The Northern Forest Center and the State of Maine Department of Conservation.\nLyme sold a working forest conservation easement over nearly the entire ownership to the State of Maine in 2012. Between 2015 and 2016 Lyme worked with Finite Carbon and DLLT to develop a carbon sequestration project on the property. We entered into an agreement to sell the carbon credits to a third party, which closed in September, 2016. In early 2016, the DLLT purchased the property from The Lyme Forest Fund.\nGreat Dismal Swamp, a 1,037 acre in-holding in the Great Dismal Swamp National Wildlife Refuge in southeastern Virginia was purchased in 2007 with Ecosystems Investment Partners. Restoration work consisted of removing an old airfield, plugging and filling the ditches that had drained the wetland for agriculture, re-contouring the property and replanting nearly 300,000 native trees and shrubs. The land is protected by a conservation easement held by the Virginia Chapter of The Nature Conservancy.\nIn December, 2013, The Lyme Forest Fund III acquired approximately 52,000 acres of land and timber interests associated with the 63,000-acre Kenauk Montebello property in southwestern Quebec. The property has a storied history and consists of over 70 lakes, including most of the 3,212-acre Lac Papineau. Located just a few miles from the Fairmont Le Ch\u00e2teau Montebello, a luxury 1930-era hotel that is considered to be the largest intact and in-use log structure in the world, the property contains important wildlife habitat. While the property can be managed for timber and recreation, it also has development potential due to its prime location between Ottawa and Montreal. As part of the initial acquisition, 12,000 acres were conserved via sale to The Nature Conservancy of Canada (NCC). In 2015, the Fund sold Lyme Kenauk to a conservation buyer, subject to a conservation option in favor of NCC.\nThe Clarion Timber Company lands are located in northwestern Pennsylvania, south of the Allegheny National Forest. This is an unusually productive hardwood region of the US, particularly renowned for the quality of its black cherry. The Clarion lands were purchased in 2004 by The Lyme Northern Forest Fund and were managed with an emphasis on value accretion. Lyme sold the property in phases, the last of which occurred in April, 2013. The Western Pennsylvania Conservancy was a conservation partner for Lyme in this deal.\nLyme's Berkshire properties were part of a small investment in which Lyme aggregated 1,048 acres in southwestern Massachusetts in 2008. In June 2013, Lyme sold 906 acres to the Commonwealth of Massachusetts as a wildlife management area. Other smaller parcels were sold separately.\nIn December, 2011, The Lyme Forest Fund III acquired approximately 3,200 acres of forestland on Maine's Schoodic Peninsula, adjacent to Acadia National Park. The property is completely undeveloped and contains more than a mile of shore frontage, including the 15-acre Sargent's Island. It has been a high priority for the conservation community for a number of years. Conservation partners include Maine Coast Heritage Trust, the National Park Service and Friends of Acadia.\nUpon purchasing the property, Lyme partnered with the Maine Coast Heritage Trust (MCHT) and a philanthropic family foundation that wished to ensure the protection of the southern portion of the property, which abuts the Schoodic division of Acadia National Park. MCHT subsequently purchased a conservation easement over the southern portion of the property and donated it to the National Park Service, effectively extending Acadia National Park to include those acres.\nThe 85,500 acre Chateaugay Woodlands property is located in the northern part of New York's Adirondack State Park. It was purchased in 2004 as part of a larger conservation transaction in which The Nature Conservancy acquired the adjacent 20,000 acres in fee and the right to purchase an assignable conservation easement on the lands acquired by Lyme. Lyme obtained Forest Stewardship Council (FSC) certification for its timber operations and sold a working forest conservation easement on the property to the State of New York. Lyme sold the property to a private timberland investment group in 2009.\nChateaugay Woodlands can generally be categorized as upland woodlands, with mountainous terrain, stream cut valleys and numerous wetlands. Lyme incorporated the protection of all of these sensitive features into its forest management plan and the working forest conservation easement. Its conservation was possible due to Lyme's partnerships with the New York Chapter of The Nature Conservancy, the New York State Department of Environmental Conservation and the Open Space Institute.\nThe 146,400 acre Connecticut Lakes Timber Company property is located in northern New Hampshire in the headwaters forest of the Connecticut River. Lyme acquired this property in 2002 as part of a larger effort to conserve 171,000 acres in the region, in which 25,000 acres was acquired by NH Fish & Game Department, subject to a conservation easement held by The Nature Conservancy. Lyme was selected to buy the remaining 146,000 acres. At the time of Lyme's ownership, it was the largest private land holding in the state.\nLyme sold a working forest conservation easement over the entire property to the New Hampshire Department of Forests and Lands and obtained third-party certification under the guidelines of the Forest Stewardship Council (FSC) on the conserved forests. Conservation partners included the New Hampshire Chapter of The Nature Conservancy, The Trust for Public Land, The Society for the Protection of New Hampshire Forests, the New Hampshire Department of Forests and Lands and the New Hampshire Fish and Game Department. In 2009, Lyme sold the property to a private timberland investment group.\nAmherst Woodlands, purchased in 2006, consisted of 5,400 acres of timberland in central Maine. At closing, the Forest Society of Maine acquired a transferable three-year option to buy the property from Lyme. In 2009, the Forest Society of Maine transferred their purchase rights and Lyme sold the property to the State of Maine Department of Conservation.\nLocated in central Pennsylvania, the 2,500 acre Kingwood Forest Tract was managed as a long term, high quality forestland holding, operated on a sustained yield basis. The property includes three miles of frontage on Laurel Hill Creek, a regionally important trout stream. This property was purchased in 1992 and sold in 2007.\nThis 5,768 acre property in central Maine was purchased by Lyme in 2002 and sold in 2005. It was considered high priority conservation land because of its 16,000 feet of frontage on undeveloped Eagle Lake and is adjacent to another property protected by a working forest conservation easement.\nIn 2000, Lyme purchased the 5,484 acre 13 Mile Woods in Northern New Hampshire. The property features mature hardwood and softwood forests, nine miles of frontage along the Androscoggin River, Munn Pond, and many opportunities for public recreation. 13 Mile Woods was identified as an important wildlife corridor in the region, linking Umbagog National Wildlife Refuge to the White Mountain National Forest.\nLyme successfully conserved the majority of the property by selling a working forest conservation easement to The Trust for Public Land. Other important conservation partners included the State of New Hampshire Land and Community Investment Program, the Town of Errol and the US Forest Service Forest Legacy Program. In 2005, the easement encumbered property was acquired by the Town of Errol to be managed as a community forest.\nThe Chittenden Timber Company lands consisted of 8,038 acres in central Vermont and was purchased by Lyme in 1998. At the time, it was the largest private land holding within the Green Mountain National Forest and had been identified by the conservation community as a high propriety for acquisition by the US Forest Service. In 2001, 7,500 acres were sold to the US Forest Service while the remainder was held by Lyme for selective timber harvesting. In 2003, Lyme sold the remainder of the property to a private investor subject to an easement held by The Trust for Public Land.\nLyme organized the Vermont Forest Conservation Fund to invest in Vermont forestland with significant conservation values. In 1996, the Fund purchased 3,654 acres of forestland in northern Vermont adjacent to the Long Trail. In 1999, a conservation easement was sold to the Vermont Land Trust over the entire property and in 2001 the easement encumbered land was sold to a private investor. Another conservation partner in this transaction was the Catamount Trail Association.\nIn 1998, the 2,660 acre Fiery Mountain\/Little Pond property in southern Maine was acquired by Lyme. At the time of purchase, Lyme sold a conservation easement covering 30% of the property to the State of Maine Department of Conservation. In addition, Lyme sold portions of the property in fee to the State of Maine Department of Conservation, abutting landowners and a private buyer. In the next two years, other portions were sold to private buyers. In 2001, the balance of the land was sold to the Maine Coast Heritage Trust.\nThe Mascoma River\/Smarts Mountain property, consisting of 6,200 acres in western NH, adjoined the Appalachian Trail corridor and a vast holding of conservation land in the area. After its purchase in 1991, Lyme sold an initial conservation easement over 2,130 acres to the State of New Hampshire Land Conservation Investment Program and the Trust for New Hampshire Lands then later sold an additional conservation easement over 2, 230 acres to the US Forest Service. Subsequently, portions of the land were sold in fee to the National Park Service and the US Forest Service to be added to the Appalachian Trail corridor. The balance of the easement encumbered land was sold to a private investor in 1999.\nIn 1993, Lyme purchased approximately 10,250 acres of forestland in 40 parcels ranging in size from 80 acres to 2,300 acres in southern Maine. The property contained high quality forestland with 14.4 miles of public road frontage and 4.4 miles of pond and river frontage. Between 1994 and 1996, Lyme sold approximately 10,000 acres to private investors. In 1999, the remainder of the land, including two miles of sensitive river frontage on the Saco River, was sold on a bargain-sale basis to the Maine Chapter of The Nature Conservancy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Mount Allison women's soccer Mounties picked up their first win of the season on Saturday when they defeated the UdeM Aigles Bleues in Moncton.\nThe Mounties walked away with a 3-1 victory on the strength of two goals from Subway AUS Player of the game Amanda Piltzmaker. Piltzmaker now leads the Mounties with 3 goals this season. Freshman Hayley Woodside also scored for the Mounties, picking up her first career AUS goal in her 9th game.\nCatherine Dupuis was the goal scorer for Moncton. Freshman keeper Sophie Austin made 9 saves in goal for the Mounties to pick up her first career AUS win.\nThe Mounties will now travel to Saint Mary's for a 1PM start on Saturday, October 20th before returning home to host Cape Breton for the final home game of the year on Sunday, October 21st.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Civitas Lamar Hotel Fund, LP made a $5.5 million preferred equity investment in a joint venture between developer Matthews Southwest and the Dallas Police and Fire Pension System to redevelop a historic building into the 76-room NYLO Dallas South Side boutique hotel. Located three blocks from the Dallas Convention Center and just three minutes from the Dallas central business district, the hotel offers a unique alternative for travelers who seek a reprieve from the traditional, uninspiring hotel experience. The hotel won Dallas Business Journal's \"Best Real Estate Deals 2012\" in the Most Creative Financing category.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us for prayer every Wednesday night at 6pm in the Prayer Room (located near the church office). We pray for our church family, our missionaries, our nation and the persecuted church. Come to pray or be prayed over!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"V\u0456rtu\u0430l r\u0435\u0430l\u0456t\u0443 h\u0430\u0455 b\u0435\u0441\u043em\u0435 very popular \u043eff l\u0430t\u0435. Th\u0435 t\u0435\u0441hn\u043el\u043eg\u0443 u\u0455\u0435d \u0456n virtual r\u0435\u0430l\u0456t\u0443 h\u0430\u0455 been leveraged \u0440r\u043ef\u0456t\u0430bl\u0443 for many uses. V\u0456rtu\u0430l r\u0435\u0430l\u0456t\u0443 t\u0435\u0441hn\u0456\u051bu\u0435\u0455 \u0430r\u0435 u\u0455\u0435d on a wide \u0455\u0441\u0430l\u0435 t\u043e d\u0435v\u0456\u0455\u0435 \u0435nt\u0435rt\u0430\u0456n\u0456ng g\u0430m\u0435\u0455 f\u043er \u0440\u0435\u043e\u0440l\u0435. V\u0456rtu\u0430ll\u0443, r\u0435\u0430l \u0440l\u0430tf\u043erm\u0455 h\u0435l\u0440 people to t\u0430k\u0435 \u0440\u0430rt \u0456n \u0455\u0456tu\u0430t\u0456\u043en\u0455 wh\u0456\u0441h th\u0435\u0443 normally \u0441\u0430nn\u043et b\u0435 a \u0440\u0430rt \u043ef. For instance, \u0435v\u0435r\u0443b\u043ed\u0443 does not have th\u0435 t\u0456m\u0435, energy, w\u0456ll \u043er th\u0435 w\u0456\u0455h t\u043e b\u0435 a m\u043eunt\u0430\u0456n\u0435\u0435r \u0456n r\u0435\u0430l l\u0456f\u0435. But you can \u0435\u0430\u0455\u0456l\u0443 b\u0435 a m\u043eunt\u0430\u0456n\u0435\u0435r \u0456n an \u0435nv\u0456r\u043enm\u0435nt th\u0430t \u0456\u0455 v\u0456rtu\u0430ll\u0443 real. Y\u043eu m\u0456ght have to w\u0435\u0430r a \u0440\u0430\u0456r \u043ef g\u043eggl\u0435\u0455 t\u043e f\u0435\u0435l \u0440\u0430rt of a m\u043eunt\u0430\u0456n\u0435\u0435r\u0456ng t\u0435\u0430m and t\u043e f\u0435\u0435l \u0430nd \u0455\u0435\u0435 the \u0455n\u043ew-\u0441\u0430\u0440\u0440\u0435d mountains \u0430r\u043eund \u0443\u043eu. Virtual r\u0435\u0430l\u0456t\u0443 helps to \u0455\u0456mul\u0430t\u0435 \u0430n \u0435nv\u0456r\u043enm\u0435nt, \u0441r\u0435\u0430t\u0435 \u0455urr\u043eund\u0456ng\u0455 \u0455\u043e that \u0443\u043eu feel a part \u043ef th\u0435m. Y\u043eu'll b\u0435 able t\u043e \u0441l\u0456mb rocks, \u0455\u0441\u0430l\u0435 m\u043eunt\u0430\u0456n\u0455, and r\u0435\u0430\u0441h th\u0435 \u0440\u0435\u0430k \u043ef th\u0435 Ev\u0435r\u0435\u0455t through v\u0456rtu\u0430l r\u0435\u0430l\u0456t\u0443. You'll be \u0430bl\u0435 to d\u0456v\u0435 \u0455\u0435v\u0435r\u0430l m\u0456l\u0435\u0455 und\u0435r the sea \u0430nd \u0440\u0435rh\u0430\u0440\u0455 r\u0435\u0430\u0441h M\u0430r\u0456\u0430n\u0430'\u0455 Trench thr\u043eugh virtual reality. Y\u043eu can \u0430\u0441tu\u0430ll\u0443 f\u0435\u0435l th\u0435 d\u0430ng\u0435r\u0455 \u0430nd th\u0435 thrills \u0440r\u0435\u0455\u0435nt \u0456n an environment powered b\u0443 v\u0456rtu\u0430l r\u0435\u0430l\u0456t\u0443. If you \u0435v\u0435r f\u0430ll \u0456nt\u043e th\u0435 midst \u043ef th\u0435 P\u0430\u0441\u0456f\u0456\u0441 O\u0441\u0435\u0430n and \u0455w\u0456m \u043eut \u0430l\u0456v\u0435, \u0443\u043eu'll h\u0430v\u0435 to r\u0435m\u0435mb\u0435r th\u0430t you ju\u0455t h\u0430d a lu\u0441k\u0443 br\u0435\u0430k, that if \u0443\u043eu f\u0430ll \u0456n \u0430g\u0430\u0456n, \u0443\u043eu m\u0456ght not b\u0435 \u0430l\u0456v\u0435 \u0430n\u0443m\u043er\u0435. But you \u0441\u0430n f\u0430ll \u0456nt\u043e \u0430n\u0443 \u043e\u0441\u0435\u0430n wh\u0456l\u0435 you're taking \u0440\u0430rt \u0456n a v\u0456rtu\u0430ll\u0443 r\u0435\u0430l\u0456\u0455t\u0456\u0441 game \u0430\u0455 m\u0430n\u0443 t\u0456m\u0435\u0455 \u0430\u0455 \u0443\u043eu want \u0430nd b\u0435 sure to r\u0435m\u0430\u0456n un\u0455\u0441\u0430th\u0435d always.\nA\u0440\u0430rt from g\u0430m\u0435\u0455, v\u0456rtu\u0430ll\u0443 r\u0435\u0430l t\u0435\u0441hn\u043el\u043eg\u0443 f\u0456nd\u0455 \u0435xt\u0435n\u0455\u0456v\u0435 \u0430\u0440\u0440l\u0456\u0441\u0430t\u0456\u043en in the tourism industry. T\u043eur\u0456\u0455t\u0455 are fr\u0435\u051bu\u0435ntl\u0443 t\u0430k\u0435n t\u043e cinema halls where v\u0456rtu\u0430l r\u0435\u0430l\u0456t\u0443 in the form of cinemania \u0456\u0455 shown t\u043e them. They h\u0430v\u0435 t\u043e \u0455\u0456t \u0456n comfortable \u0455\u0435\u0430t\u0455, \u0430nd strap themselves to th\u0435 \u0455\u0435\u0430t\u0455 w\u0456th \u0455\u0435\u0430t b\u0435lt\u0455 \u0430\u0455 th\u0435 \u0455\u0435\u0430t\u0455 start r\u043e\u0441k\u0456ng as soon \u0430\u0455 th\u0435 g\u0456\u0430nt \u0455\u0441r\u0435\u0435n \u0456n front comes \u0430l\u0456v\u0435 w\u0456th \u0435x\u0441\u0456t\u0456ng or scaring \u0455\u0441\u0435n\u0435\u0455. They \u0430r\u0435 really m\u0430d\u0435 to f\u0435\u0435l as though th\u0435\u0443'r\u0435 w\u0430lk\u0456ng through dung\u0435\u043en\u0455 \u043er forests \u0456n these 'virtual reality' \u0435\u051bu\u0456\u0440\u0440\u0435d \u0441\u0456n\u0435m\u0430 h\u0430ll\u0455. Th\u0435\u0443 \u0441\u0430n h\u0435\u0430r th\u0435 screeches of wild \u0430n\u0456m\u0430l\u0455 \u0430nd \u0441\u0430n \u0435v\u0435n feel th\u0435m brushing by th\u0435\u0456r \u0455\u0456d\u0435\u0455. Such cinemania complexes exist \u0456n S\u0456ng\u0430\u0440\u043er\u0435, Rome and other h\u043et tourist destinations.\nVirtually r\u0435\u0430l t\u0435\u0441hn\u043el\u043eg\u0443 has many \u0430\u0440\u0440l\u0456\u0441\u0430t\u0456\u043en\u0455 \u043eth\u0435r th\u0430n 'fun and \u0435nt\u0435rt\u0430\u0456nm\u0435nt' \u0430\u0440\u0440l\u0456\u0441\u0430t\u0456\u043en\u0455. This technology \u0456\u0455 used t\u043e r\u0435\u0441\u043en\u0455tru\u0441t \u0440\u0430\u0455t \u0435v\u0435nt\u0455 \u0430nd \u0455\u0456t\u0435\u0455 f\u043er h\u0456\u0455t\u043er\u0456\u0441\u0430l \u0430nd scientific \u0456nv\u0435\u0455t\u0456g\u0430t\u0456\u043en\u0455. Data about people and places \u0430r\u0435 f\u0456r\u0455t gathered before th\u0435 same data \u0430r\u0435 f\u0435d \u0456nt\u043e \u0455\u043eftw\u0430r\u0435 \u0430nd systems th\u0430t \u0441\u043en\u0455t\u0456tut\u0435 v\u0456rtu\u0430ll\u0443 r\u0435\u0430l technology. The \u0455\u043eftw\u0430r\u0435 \u0430nd systems help t\u043e r\u0435\u0441r\u0435\u0430t\u0435 \u0455\u0456tu\u0430t\u0456\u043en\u0455 \u0430\u0455 th\u0435\u0443 were \u043er\u0456g\u0456n\u0430ll\u0443. H\u0456\u0455t\u043er\u0456\u0430n\u0455 and \u0455\u0441\u0456\u0435nt\u0456\u0455t\u0455 \u0441\u0430n visit \u0430n\u0441\u0456\u0435nt caves and f\u0435\u0435l h\u043ew St\u043en\u0435 Ag\u0435 m\u0435n lived w\u0456th th\u0435 h\u0435l\u0440 \u043ef v\u0456rtu\u0430ll\u0443 r\u0435\u0430l technology. Cr\u0456m\u0435\u0455 scenes can b\u0435 r\u0435-m\u0430d\u0435 t\u043e\u043e u\u0455\u0456ng virtually r\u0435\u0430l t\u0435\u0441hn\u043el\u043eg\u0443 \u0430nd th\u0456\u0455 m\u0430\u0443 aid \u0440\u043el\u0456\u0441\u0435 and detectives to \u0455\u043elv\u0435 crimes and identify culprits after und\u0435r\u0455t\u0430nd\u0456ng culprit m\u043ev\u0435m\u0435nt\u0455 \u0430\u0455 th\u0435\u0443 h\u0430d been in th\u0435 place of the \u0441r\u0456m\u0435, upon its r\u0435-\u0441r\u0435\u0430t\u0456\u043en u\u0455\u0456ng virtual r\u0435\u0430l\u0456t\u0443. V\u0456rtu\u0430ll\u0443 real t\u0435\u0441hn\u043el\u043eg\u0443 \u0456\u0455 also used t\u043e a great \u0435xt\u0435nt \u0456n th\u0435 field \u043ef m\u0435d\u0456\u0441\u0456n\u0435 f\u043er therapeutic \u0440ur\u0440\u043e\u0455\u0435\u0455 \u0430nd f\u043er treating \u0440\u0435\u043e\u0440l\u0435 w\u0456th various l\u0456mb \u0430nd \u043eth\u0435r disorders. The f\u0430m\u043eu\u0455 S\u0430nt\u0456m\u0430m\u0456n\u0435 C\u0430v\u0435\u0455 of the Stone Ag\u0435 h\u0430v\u0435 been r\u0435\u0441r\u0435\u0430t\u0435d using virtually r\u0435\u0430l technology. Ex\u0456\u0455t\u0456ng \u0455\u0441\u0456\u0435nt\u0456f\u0456\u0441, t\u043e\u0440\u043egr\u0430\u0440h\u0456\u0441\u0430l, g\u0435\u043egr\u0430\u0440h\u0456\u0441\u0430l \u0430nd \u043eth\u0435r d\u0430t\u0430 h\u0430v\u0435 b\u0435\u0435n integrated w\u0456th the v\u0456rtu\u0430l r\u0435\u0430l\u0456t\u0443 technology t\u043e re-build th\u0435 \u0441\u0430v\u0435\u0455 \u0430\u0455 th\u0435\u0443 originally w\u0435r\u0435.\nS\u0430nt\u0456m\u0430m\u0456n\u0435 Caves g\u0456v\u0435\u0455 you an opportunity t\u043e v\u0456\u0435w the r\u0435-\u0441r\u0435\u0430t\u0435d S\u0430nt\u0456m\u0430m\u0456n\u0435 C\u0430v\u0435\u0455 th\u0430t were \u043en\u0441\u0435 inhabited by the N\u0435\u0430nd\u0435rth\u0430l men.\nV\u0456rtu\u0430ll\u0443 r\u0435\u0430l t\u0435\u0441hn\u043el\u043eg\u0443 f\u0456nd\u0455 num\u0435r\u043eu\u0455 \u0430\u0440\u0440l\u0456\u0441\u0430t\u0456\u043en\u0455 \u0456n the w\u043erld \u043ef bu\u0455\u0456n\u0435\u0455\u0455 \u0430nd industry too. With th\u0435 help \u043ef \u0455\u0456mul\u0430t\u0456\u043en, wh\u0456\u0441h \u0456\u0455 \u0430n \u0456nt\u0435gr\u0430l \u0440\u0430rt \u043ef th\u0456\u0455 t\u0435\u0441hn\u043el\u043eg\u0443, n\u0435w prototypes and \u0440r\u043edu\u0441t\u0455 \u0441\u0430n be built. CAD, CAM, and \u043eth\u0435r mathematical \u0430nd engineering t\u0435\u0441hn\u0456\u051bu\u0435\u0455, wh\u0456\u0441h b\u0435l\u043eng t\u043e th\u0435 d\u043em\u0430\u0456n \u043ef v\u0456rtu\u0430l reality, \u0430r\u0435 w\u0456d\u0435l\u0443 u\u0455\u0435d \u0456n th\u0435 automobile, n\u0430v\u0430l, th\u0435 aerospace and in \u043eth\u0435r m\u0430nuf\u0430\u0441tur\u0456ng \u0456ndu\u0455tr\u0456\u0435\u0455 t\u043e \u0435n\u0430bl\u0435 the building \u043ef \u0441\u0430r\u0455, ships, \u0430\u0456r\u0440l\u0430n\u0435\u0455 and m\u0430\u0441h\u0456n\u0435\u0455. O\u0440\u0435r\u0430t\u0456\u043en\u0455 research, m\u0430th\u0435m\u0430t\u0456\u0441\u0430l m\u043ed\u0435l\u0456ng, \u0455t\u0430t\u0456\u0455t\u0456\u0441\u0430l m\u0435th\u043ed\u0455, optimization \u0455tr\u0430t\u0435g\u0456\u0435\u0455, d\u0435\u0441\u0456\u0455\u0456\u043en-tr\u0435\u0435 m\u0435th\u043ed\u0455, \u0430nd \u0455\u0456mul\u0430t\u0456\u043en are \u0455\u043em\u0435 t\u0435\u0441hn\u0456\u051bu\u0435\u0455 th\u0430t \u0430r\u0435 \u0441\u043emm\u043enl\u0443 used t\u043e \u0435n\u0430bl\u0435 accurate d\u0435\u0441\u0456\u0455\u0456\u043en-m\u0430k\u0456ng. Th\u0435\u0455\u0435 techniques devise n\u0435\u0430r-r\u0435\u0430l \u0455\u0456tu\u0430t\u0456\u043en\u0455 \u0430nd seek th\u0435 \u043e\u0440t\u0456mum solutions \u0456n those near-real situations \u0455\u043e th\u0430t those \u0455\u043elut\u0456\u043en\u0455 may b\u0435 applied in r\u0435\u0430l\u0456t\u0443 f\u043er \u0455\u043elv\u0456ng \u043e\u0440\u0435r\u0430t\u0456\u043en\u0430l \u0440r\u043ebl\u0435m\u0455 \u0430nd \u0435\u0430\u0455\u0456ng bottlenecks. The th\u0435\u043er\u0443 \u043ef \u0440r\u043eb\u0430b\u0456l\u0456t\u0443 \u0456n Mathematics \u0456\u0455 a way \u043ef d\u0435t\u0435rm\u0456n\u0456ng wh\u0435n a certain \u0430\u0441t\u0456\u043en can \u043e\u0441\u0441ur in the real w\u043erld. Many \u0430m\u0430z\u0456ng \u0435ng\u0456n\u0435\u0435r\u0456ng f\u0435\u0430t\u0455 h\u0430v\u0435 b\u0435\u0435n possible only b\u0435\u0441\u0430u\u0455\u0435 \u0435ng\u0456n\u0435\u0435r\u0455 have m\u0430d\u0435 u\u0455\u0435 \u043ef v\u0456rtu\u0430ll\u0443 r\u0435\u0430l t\u0435\u0441hn\u043el\u043eg\u0443. V\u0456rtu\u0430ll\u0443, r\u0435\u0430l t\u0435\u0441hn\u043el\u043eg\u0443 thu\u0455 f\u0456nd\u0455 \u0430lm\u043e\u0455t unlimited \u0430\u0440\u0440l\u0456\u0441\u0430t\u0456\u043en \u0456n th\u0435 \u0456ndu\u0455tr\u0456\u0430l w\u043erld \u0430\u0455 it h\u0435l\u0440\u0455 t\u043e \u0435\u0455t\u0430bl\u0456\u0455h machine \u0441\u0430\u0440\u0430b\u0456l\u0456t\u0456\u0435\u0455, improve \u0430\u0455\u0455\u0435mbl\u0443-l\u0456n\u0435 \u043e\u0440\u0435r\u0430t\u0456\u043en\u0455, \u0430nd hence, \u0430ugm\u0435nt w\u043erk\u0440l\u0430\u0441\u0435 efficiency.\nVirtual reality \u0456\u0455 also a t\u0435rm th\u0430t'\u0455 u\u0455\u0435d for web and video \u0441\u043enf\u0435r\u0435n\u0441\u0435\u0455. Th\u0435r\u0435 \u0430r\u0435 \u0455\u0435v\u0435r\u0430l w\u0435b \u0430nd video conferencing software \u0430v\u0430\u0456l\u0430bl\u0435 \u0456n the market t\u043ed\u0430\u0443 th\u0430t enable r\u0435\u0430l-t\u0456m\u0435 online \u0441\u043ell\u0430b\u043er\u0430t\u0456\u043en. M\u0430n\u0430g\u0435r\u0455 \u0441\u0430n \u0441\u043endu\u0441t meetings v\u0435r\u0443 cheaply using these \u0455\u043eftw\u0430r\u0435 \u0430\u0455 several \u0440\u0435\u043e\u0440l\u0435 \u0430\u0441r\u043e\u0455\u0455 th\u0435 w\u043erld \u0441\u0430n attend a w\u0435b conference at th\u0435 \u0455\u0430m\u0435 t\u0456m\u0435. There is no need t\u043e organize elaborate \u0441\u043enf\u0435r\u0435n\u0441\u0435\u0455, n\u043e n\u0435\u0435d to \u0455\u0440\u0435nd money \u043en b\u043e\u043ek\u0456ng v\u0435nu\u0435\u0455, \u043en paying the \u0430\u0456rf\u0430r\u0435\u0455 \u043ef \u0455\u0435v\u0435r\u0430l \u0441\u043enf\u0435r\u0435n\u0441\u0435 \u0430tt\u0435nd\u0435\u0435\u0455, on \u0440\u0430\u0443\u0456ng the \u0430\u0441\u0441\u043emm\u043ed\u0430t\u0456\u043en \u0441h\u0430rg\u0435\u0455 of out-station conference \u0430tt\u0435nd\u0435\u0435\u0455. W\u0435b \u0430nd video conferencing improves \u0435m\u0440l\u043e\u0443\u0435\u0435 \u0440r\u043edu\u0441t\u0456v\u0456t\u0443 \u0430\u0455 it m\u0456n\u0456m\u0456z\u0435\u0455 \u0435m\u0440l\u043e\u0443\u0435\u0435 d\u043ewnt\u0456m\u0435 \u0455u\u0441h as w\u043erk\u0456ng h\u043eur\u0455 spent \u0456n tr\u0430v\u0435l\u0456ng.\nF\u0456n\u0430ll\u0443, th\u0435 concept of virtual r\u0435\u0430l\u0456t\u0443 includes \u0456n its bailiwick, \u0441\u0435rt\u0430\u0456n \u0455m\u0430rt business \u0430nd meeting \u0455\u043eftw\u0430r\u0435 \u0455u\u0441h as M\u0435\u0435t\u0456ng Diary. M\u0435\u0435t\u0456ng D\u0456\u0430r\u0443 \u0456\u0455 a v\u0456rtu\u0430l \u0435v\u0435nt manager \u0430nd a virtual planner th\u0430t \u0440l\u0430n\u0455 m\u0435\u0435t\u0456ng\u0455 \u0430nd \u0441\u043enf\u0435r\u0435n\u0441\u0435\u0455 on your behalf. M\u0435\u0435t\u0456ng Diary \u0456\u0455 \u0430n action-oriented \u0455\u043eftw\u0430r\u0435. It dr\u0456v\u0435\u0455 \u0443\u043eu to r\u0435\u0455tru\u0441tur\u0435 \u0443\u043eur th\u043eught\u0455 and realign your \u0430\u0441t\u0456\u043en\u0455, \u0455tr\u0435\u0430ml\u0456n\u0435 \u0443\u043eur t\u0430\u0455k\u0455 \u0430nd th\u0456nk \u0441r\u0435\u0430t\u0456v\u0435l\u0443. It \u0456m\u0440\u0435l\u0455 you t\u043e unl\u0435\u0430\u0455h \u0443\u043eur \u0435nt\u0456r\u0435 potential \u0456n a disciplined, \u0455\u0443\u0455t\u0435m\u0430t\u0456\u0441 \u0430nd g\u043e\u0430l-\u043er\u0456\u0435nt\u0435d m\u0430nn\u0435r. It \u0435n\u0430bl\u0435\u0455 \u0443\u043eu t\u043e \u0441\u043e\u043erd\u0456n\u0430t\u0435 \u0430nd \u0430lw\u0430\u0443\u0455 be one \u043ef th\u0435 frontrunners in a fast-paced \u0435nv\u0456r\u043enm\u0435nt. It urges \u0443\u043eu t\u043e b\u0435 part of th\u0435 vanguard, wh\u0435r\u0435v\u0435r \u0443\u043eu go. M\u0435\u0435t\u0456ng D\u0456\u0430r\u0443'\u0455 an \u0430\u0440\u0440l\u0456\u0441\u0430t\u0456\u043en th\u0430t \u0441\u043em\u0435\u0455 \u0455\u0430n\u0455 \u0430n\u0443 \u0440r\u0456\u0441\u0435 t\u0430g. Th\u043eugh v\u0456rtu\u0430l \u0430nd intangible, \u0456t'\u0455 a r\u0435\u0430l\u0456t\u0443 m\u043er\u0435 \u0455\u043el\u0456d than th\u0435 PC or l\u0430\u0440t\u043e\u0440 you're u\u0455\u0456ng, then th\u0435 bl\u043e\u0441k\u0455 of th\u0435 bu\u0456ld\u0456ng \u0443\u043eu'r\u0435 sitting \u0456n. It h\u0435l\u0440\u0455 \u0443\u043eu fulf\u0456ll \u0443\u043eur \u0430mb\u0456t\u0456\u043en\u0455 and realize \u0430ll \u0443\u043eur aims, \u0455\u0435\u0430ml\u0435\u0455\u0455l\u0443.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawqox b/data_all_eng_slimpj/shuffled/split2/finalzzzawqox
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+{"text":"How Should You Learn to Play Acoustic Guitar? With a Personal Teacher, Online, or Through Self-Teaching?\nIt's fun to learn the acoustic guitar, but when you do so, there are some choices you should make. The Internet has made a lot of things more convenient, including learning acoustic guitar through online learning methods if you which. However, hiring a personal instructor has advantages as well. You can also get a lesson book and teach yourself sans Internet. In this article, we'll look at the pros and cons of each method so that you can make a decision that's informed.\nIf you are already bored with the kind of music that is being played over the radio, then it can be a great idea if you are going to learn how you can effectively download Katy Perry songs. This way, you will be able to listen to some of her greatest hits such as I Kissed a Girl, California Gurls, Fireworks, and Teenage Dreams. This can definitely be very enjoyable especially if you are a fan of this artist.\nIf you are a fan of country music, then you need to know that your playlist will never be complete unless you know how to download Carrie Underwood songs. She is certainly one of the best selling country music singers today and there is really no wonder why you are a fan of hers. Having three multi-platinum albums and Grammy awards, she is definitely an artist that should never be gone on your playlist.\nFor those who are up-to-the-minute to the hip-hop genre of the music industry, there is so much additional information available than ever in the past. Finding useful knowledge in order to begin your career as a hip hop artist or even a beat master, can be tricky if you have no plan where to begin. Hopefully you will find this article informative and useful when searching for, purchasing and making hip hop beats.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So whenever you are composing an essay, you're harnessing the complete might of culture to your own life. Everything you need is a person who can read and react to your paper using a sharp visual attention. You are able enjoy the moment of your personal moment and to cast that dull work on dissertation writers.\nWhile education dissertations include of britishessaywriter.org.uk just a couple of paragraphs, long dissertation papers are composed of body paragraphs. When you've concluded that you ought to work with a specialist to complete your job online recall that you may have to do just a bit of examination to monitor the perfect author to utilize. While every dissertation is unique in its own way you will find specific structural elements of dissertations which are likely to be applicable for every dissertation.\nBecause of this, you've got to be careful with dissertation abstract writing. Adhere to the dissertation author must be mindful with adult, you. The moment you've got the dissertation in draft form it becomes simpler to observe where you are ready to improve it.\nBecause of this, it's very important to plan early along with your dissertation. You may start to feel your dissertation essay editing service uk will not ever be good enough, and you would like to update it. A dissertation can be split into chapters and elements.\nThe smartest and most dedicated men and women struggle to finish a dissertation. There are lots of difficult things related. The majority of the pupils do not have any clue on how end up writing a lousy dissertation and thus they need to proceed with the dissertation work.\nIt imperative you do your research, both in addition that you intend to approach. Although writing Law dissertation's project isn't an easy job, but there are lots of ways that are helpful can be found which are created for the benefits of the learners. The primary aim of the research proposal is to examine whether there's of everything you're likely to compose in the study creativity.\nOccasionally research proposal is written as a member of a dissertation that is coming as a distinct work preceding it, there are instances when you might not be requested to write it in any respect. Before starting the composing work, you ought to do an exhaustive studying to gather information that is relevant to be able to turn your dissertation informative and precise. To make certain that your thesis is delivered they don't rest for a little bit and work for your order is finished by the present time.\nCustomized dissertation writing is almost always a challenging task with the abundance of guidelines and the complicated topics. Custom made content depending on your instructions https:\/\/www-cdn.law.stanford.edu\/wp-content\/uploads\/2015\/06\/49-Criticism-687.pdf is guaranteed. Moreover, a customized education paper needs to be quality and custom written by professionals.\nA dissertation abstracts may be written on a range of topics which catch the interest of publications that were notable. Researching for the subject, literature and business of your secondary and primary resources is a measure the moment it comes to dissertation writing. You may be given a few suggestions for how to compose your own by examples of good essays.\nFor this reason, you have got to be rather careful with writing that was abstract. Whether it's a complicated essay or dissertation assignment, it is possible to reach out to a agency and get a dissertation. It's rather hard to accomplish a dissertation writing assignment.\nEach portion of the APA paper needs to be double-spaced, and pages numbers ought to be at right. You locate a section or a quotation that's linked to your topic matter, take note of it in your notepad. Or you may see that a section should be cut out because it doesn't fit.\nSay, by means of example, your dissertation contains five chapters. The expression dissertation is referred to as thesis in the total sense. Then give a description of the job in When it is complete.\nYou will read up on many posts and such writing resources online which can help you adopt adequate writing abilities. Researching for the subject, business and literature of your secondary and primary resources is a measure that is vital . It is possible to bypass the elements of essay by selecting the authors from BookMyEssay writing process.\nPupils aren't knowledgeable concerning the way to compose a dissertation abstract's valid technique. Papers Assistance can assist you. Many abstracts incorporate a description of the outcome of the experiment.\nIn reality, there is very much like informative abstracts a structure utilized to compose short reports. Beginning an interview summary paper may be the toughest aspect of the assignment. At the onset of the dissertation right the abstract is located in most cases after the dissertation title page.\nIn the majority of instances the committee will not make it feasible for you to defend your dissertation should they don't believe you're ready. Your university will have a favorite style for citations and references which you are going to want to use. You are able to select the final and precise name when you're at close quarters to completing your dissertation.\nThesis could possibly be a paragraph or dissertation. Writing dissertation abstracts is the start of your hard work. Writing a dissertation abstract isn't an impossible job.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When a team's starting pitcher has a first inning ERA of 15.43, it's safe to say they're not going to win many of the games he pitches in. Add to that a total of four walks last night--which hasn't been done since the pitcher was a rookie in 2011--and you have all the makings of a frustrating loss. Michael Pineda is now 1-4 on the season, with an overall ERA of 6.28.\nManager Joe Girardi didn't help things by allowing Pineda to load the bases in the sixth inning, before replacing him with Nick Goody. By then Pineda had thrown 114 pitches. Someone needs to take Girardi's binder and hit him over the head with it.\nJust kidding. However you slice it, last night's game was a loss. Time to turn the page and to focus on winning the series tonight. As Carlos Beltran said, \"...We have an important game. It would be great to win the series against the World [Series] champions. I think if we're capable of doing that it's going to be a positive for us.\"\nThey will have to do it against former Yankee Ian Kennedy. I'm in no mood to see Kennedy pitch the game of his life. However, if Nathan Eovaldi wants to do so, he's more than welcome to.\nLet's go Evo, and let's go Yankees!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No! Open Houses are NOT a Waste of Time!\nDo you think Open Houses are a waste of time? The verdict is divided on this - some agents swear by them, others refuse to even consider doing them.\nWell, I'm going to do my best to encourage you to do an Open House for each and every one of your listings - at least once.\nIt makes your seller happy.\nDon't discount the power of a happy seller. And don't let the Old Fogies tell you it's possible to convince a seller that Open Houses are a waste of time. If you don't do at least one Open House for your seller, he WILL doubt your commitment to the sale of his home. Oh, sure, he may agree with you to your face that Open Houses aren't necessary and that they only benefit the agent, blah blah blah, but when his house hasn't sold and he's getting nervous, he WILL remember that YOU haven't done an Open House yet.\nIf I had a house on the market and my agent hadn't done an open house, he better not come asking me for a price reduction unless\/until he does! Why? Because I'm pretty sure I'd feel he hadn't yet done HIS part to sell the house. Reasonable? Maybe, maybe not, but if I, as an experienced real estate agent feel this way, you better believe your sellers do, too.\nDid you catch that? I think it would take some nerve to ask your seller to reduce his price if YOU haven't done the most visible, most expected form of real estate marketing there is - the Open House. If your seller feels you're shirking Open House duty, he might not be nearly as open to your suggestions as to what HE needs to do to help get the house sold.\nA happy seller is a cooperative seller. And an uncooperative seller is a nightmare. You pick.\nJennifer- Holding an open house for the reason you list is a valid reason to hold one. We need to do everything for our clients to help the sell their homes. In this market it is vital we cover all of the bases even if we do not like doing some particular part of the job. Open houses in my area are still fairly well attended and the potential to meet new people is always something I look forward to. My sellers always seem to appreciate the effort too.\nI love open houses! What a great way to meet people and get more exposure for your listing.\nPublic expectation is not a good reason to do open houses. The public expects us all to be back-stabbing, lying used care sales people that would take advantage of anyone to make a buck, but I am not going to encourage those thoughts by doing what they expect.\nInstead, I discuss the very poor returns of open houses with my listing clients. After hearing the data almost all of my clients prefer NOT to have open houses and they are VERY HAPPY with me and my marketing efforts. Wasting someone's time generally does NOT make them happy.\nThere are some exceptions when an open house is needed or requested and I will have them in those rare instances, but the overwhelming majority of my clients realize that the industry is changing and understand that doing an activity that is unproductive is....UNPRODUCTIVE.\nI love open houses. Where else can you see what people really think about the house. I always find out new likes and dislikes regarding my listings. Even the neighbors can be helpful giving me inside neighborhood information. I did what I hate the other week. I advertised three of my Open Houses in the local newspaper and we had an amazing turn out. I had 20 people at one home and 10 each at the other two. Most people saw the ad and came on by. I hate when I am wrong about newspaper advertising!\nAlthough I tell my sellers Open Houses are not the most productive, I gladly hold them if they wish to do so.\nA new discovery that I tried recently with much success was to get every house up for sale in the neighborhood to schedule a \"Community Open House.\" Every agent pitched in to some form of advertising, we invited the neighbors and had a magnificent four hours with lots of prospects. The result, one contract and two listings and two serious buyers, all because people saw the community for what it is, a good value with lots of interest.\nI agree, Open house although don't usually tend to sell the house, It is a great way to please the seller and try to pick up another buyer or two! Lets not forget about Realtor open houses, a great way to try to educate other realtors on your listing. Community Open Houses, are a great way to bring extra traffic as well.\nI always love open houses but I understand that most do not. I think they are great for the agent and the seller.\nLori - I did my first open house in 3 years last Sunday and was pleasantly surprised by how much I enjoyed it. Had around 20 visitors and all of were chatty and cheerful.\nTania - We don't do too many broker opens around here; \"previewing\" is common so agents with buyers just preview when they want to catch up on the market. But in markets where previewing isn't common, I can certainly see the value of broker opens!\nJoEllen - I actually tried to generate some interest in a community open last weekend, but so many agents seem totally doomed & gloomed and uninterested. Ah well, MY open house was great!\nSherry - BINGO! That's it exactly (and the topic of the next blog, Reason #2!). I know what you mean about being \"wrong.\" I hate that.\nRichard - obviously, we'll just agree to disagree! Thanks for stopping by!\nChristina - Agreed! Sure can't hurt!\nScott - Yes! You get it!\nEric - I rarely use my open houses to prospect, although if I have rapport with someone, I'm happy to follow up. But for new or newer agents, open houses are a great opportunity to practice real estate agent-skills!\nMike - but we do them anyway 'cause we care about our listings!\nBob - I wish I enjoyed doing them more because I agree - what a great opportunity to meet the homebuying\/selling public!\nLindsay - Yep - certainly no HARM in doing an open house - can't hurt!\nThere's nothing worse than an Open House with NO showings; if no one is going to show, why do the open house at all? For the very reasons you explained above.\nWe have to show the client that we're doing something to sell their home. If at the end of their day, they're happy because I'm putting in some good ol' fashioned hard work, I've just made them happier which brings me a step higher in their trust radar. The more they trust me, the more their likely to take my advice.\nIn my opinion it depends on the house, but for the most part they are definitely worth the time. As a matter of fact, I had one Sunday and wrote an offer from a client without a Realtor on Monday. That is two sides for me. So, in my opinion, my $40 add in the paper and 2 hours were well worth it! This was the 3rd open house I have done for the house and I have gotten clients from each of them. Not to mention a very happy seller.\nRicardo - clap clap clap!!! Can you feel me jumping up and down from Colorado??? YES!\nI agree. I have actually sold a few homes with the open house. Of course they already had an agent but thats ok.\nI don't consider open houses to be a waste of time, but I've also never sold the house I was holding open, as a result of that open house. BUT, I know other agents who have. The way I see it, you just never know where the buyer will originate. It can't hurt, so why not try? The only reason an agent would be adamantly against them would be out of laziness.\nWe get many buyers going through or listings. Agents in some of our areas don't tour much, so we need to have access for open houses. I wouldn't be marketing the home to its fullest if we didn't have open houses.\nI hate doing open houses as well. They are not very effective in my opinion. You almost have to do'em to keep your sellers happy though. I guess one additional benefit is that with the few people that walk in you can build your database..\nI am working on aquiring my first listing, but in the meantime, I offer my house sitting ability (and believe me, I AM able! :))to any of our other agents and they say, \"We really need to do an open house.\" But no one says, \"Sure, how about next Saturday!\" Nada. There is talk of twilight open houses, for when people are coming home from work, but so far, Nada. I suppose they feel they do no good. I feel like going to a couple of friends with another agency and see if I can sit one of theirs for them...all of us sell each others houses through the MLS, why not opens? I agree that it is a good place to learn skills, like how important knowing the neighborhood and the local inventory is.\nHey Jennifer...I've never been to Colorado...I'll come sit one of yours...ha!\nAnother good point Jennifer. I want happy sellers. I'm not a big fan of open houses, however I need to do what's necessary to get the house sold.\nThanks for all your comments on these older blogs, Donna!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"3008 1 6 Hdi. peugeot 3008 1 6 hdi. used 2013 peugeot 3008 1 6 hdi fap active 5dr for sale in staffordshire pistonheads. used 2011 peugeot 3008 1 6 hdi 112 sport 5dr for sale in county durham pistonheads. used 2012 peugeot 3008 1 6 hdi fap sport 5dr for sale in norfolk pistonheads. used 2014 peugeot 3008 1 6 hdi fap active hatchback 5dr diesel manual 125 g km 115 bhp for. used 2014 peugeot 3008 1 6 hdi active 5dr for sale in yorkshire pistonheads. peugeot 3008 sport 1 6 hdi cmp6 1 photo and 49 specs. file peugeot 3008 1 6 hdi premium. used peugeot 3008 1 6 e hdi 112 active ii 5dr egc for sale what car ref north yorkshire. used 2013 peugeot 3008 1 6 hdi fap active hatchback 5dr diesel manual 125 g km 115 bhp for. [Allison-wheeler.com].","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxfzk b/data_all_eng_slimpj/shuffled/split2/finalzzzaxfzk
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+{"text":"Saltburn surfer Gary Rogers is riding high after fulfilling a lifetime's ambition.\nSaltburn surfer Gary Rogers is riding high after fulfilling a lifetime's ambition. The 46-year-old has just been crowned British champion for the first time.\nWhat's more, his veterans' triumph follows a year after vowing to quit national competition for good.\n\"I nearly made last year's my last one down in Newquay,\" he said.\n\"I said to my son I wouldn't be going into any more British contests, with the travelling and everything.\n\"But because this one was in Tynemouth, I decided to give it a go.\n\"I'm glad that I did because winning it was a great feeling.\n\"I'd been to a few British finals before and the best I'd had was a third place and a couple of fourths. So when the results went up I went mad. I was absolutely elated.\n\"I came fourth in the masters category afterwards as well, but by then I was a bit shattered!\"\nGary's shortboard success - performed on two waves - included tricks such as a 180-degree tail slide.\nIt also saw him finish above semi-professional rival Jed Stone, of Newquay, for the first time.\n\"It's taken me a while to get there but surfing is just something that I love doing,\" said Gary, who runs Saltburn Surf Shop.\n\"It's a great buzz and when you love doing something you keep doing it.\n\"I enjoy the wipe-outs as much as the rides.\"\nGary first got into surfing as a 14-year-old in Swansea.\n\"I'd seen a couple of guys on very old long boards and thought 'that's for me',\" he said.\n\"I remember my dad saying I'd only be doing it for a year.\n\"But I bought my own board and I've been surfing ever since.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Genuine Factory Dodge OEM Glass fits your 1995 Dodge Grand Caravan ES V6 3.8 Liter GAS engine. The Original Equipment Manufacturer or OEM part number is 4675118. We also offer a wide variety of other parts for your Dodge Grand Caravan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Maybe I'm struggling with my lapsed romantic side. Maybe I'm just a sucker for a sweet and simple public expression of one's love. Either way, Jesse Hale's Post-It Note Love Letter video to his wife Holly, part of the Give It 100 challenge, puts a smile on my face, well on one side of my face anyway, the side that isn't scowling because I never think of doing stuff like this anymore. Walking the tightrope between feeling pathetic and being inspired!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can't make heads or tails out of it? He keeps beating around the bush and it is making you feel bent out of shape? Well, if you are in over your head, zip your lip, and let sleeping dogs lie. No one is trying to pull the wool over your eyes\u2026. but someone may be using too many idioms!\nThe Oxford Dictionary defines idiom as \"a group of words whose meaning is different from the meanings of the individual words.\" \"I can't make heads or tails of it!\" expresses confusion to someone familiar with this idiom but to a young child, or adult learning English as a second language, the literal images are bizarre.\nOur conversation, print, and media are peppered with idioms. Miriam Trehearne, international literacy consultant, tells us that television uses four idioms per minute. Trehearne emphasizes that reading comprehension can be thwarted by idioms. Poorly understood idioms are one of the major reasons for the fourth grade slump \u2013 a time when many children stop reading for pleasure.\nTrehearne explains that we cannot assume children will absorb the meaning of idioms. We need to teach them their meanings. Trehearne suggests two ways to do this. One way is through drawing. Write an idiom, such as \"It's raining cats and dogs\", at the top of a piece of paper. Draw a line down the centre from top to bottom. Ask your child to draw a literal picture of these words on the left side of the page. Your child might draw clouds with barking dogs and meowing cats falling from the sky. On the right side of the line, have your child draw what the idiom really means. Now your child might draw black clouds with hundreds of falling raindrops.\nTrehearne also recommends using picture books to help you and your child explore the meaning of idioms together.\n\"Monkey Business\" by Wallace Edwards uses elaborate, imaginative illustrations that show a wide cast of animal characters in absurd situations. Each one is a literal portrayal of an idiom. An alligator, for example, eats her words as she sweeps books off a table into her gaping mouth.\n\"There's A Frog in My Throat: 440 Animal Sayings A Little Bird Told Me\" by Loreen Leedy is an illustrated collection of idioms grouped by animal types. Though this book has no storyline, your child may enjoy flipping to certain pages that are silly or fun to talk about with you.\nSuccessful readers understand idioms. With so many to know, preschoolers and young children will be over the moon if you take the time to help them discover the coded meaning of words grouped so surprisingly that they are in a class all of their own.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LOS ANGELES-Samantha Fox has recalled once being paid $50,000 to perform at a Colombian drug lord's house for his daughter's 21st birthday, and her dad once had to pay a $100,000 ransom after the 'Touch Me' star and her crew were held hostage in South America.\nSamantha Fox was once paid $50,000 to sing at a Colombian drug lord's daughter's 21st party.\nThe 51-year-old singer has recalled feeling ''like The Beatles'' when she was greeted by 10,000 people in South America and went on to be handed the huge sum in cash before performing at a birthday bash.\nAccording to the Daily Mirror newspaper, she writes in her new autobiography 'Forever': ''When we arrived in Colombia, George Bush's war on drugs was going strong and no international stars were performing there.\n''So when I arrived there were 10,000 people at the airport. They were so happy. ''After we'd done three shows this very serious looking guy came up to my tour manager and asked if I'd perform at his daughter's 21st birthday the following night for $50,000.\nThings didn't always go well for Sam in South America, however, as she also revealed her manager made a mix up with the dates for one of her shows, which led to the 'Touch Me' hitmaker and her crew being held hostage until her dad flew in from England to pay a $100,000 ransom.\nBut things were on the up for Sam in India when she played a show in Delhi and the locals showed their support for her singing by exposing themselves - leaving her unsure whether to ''laugh or cry''. She added: ''I noticed all the men were lifting their shirts to expose their willies to me, which apparently meant they loved me or fancied me. ''I was trying to remember the words to my songs with 70,000 willies doing the propeller in front of me. I didn't know whether to laugh or cry.'' This comes after Sam revealed earlier this week late singer David Cassidy once sexually assaulted her in a restaurant bathroom in May 1985 when she was just 19 years old.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..0bde0505b9171e86934e2884c71920d038ddaebc
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@@ -0,0 +1,5 @@
+{"text":"One of the nice things about The Polyphonic Spree is their ability to reinterpret and add their own zest of life into the works of others. Their larger-than-life psychedelic choral pop sound is matched only by their over-the-top live performances, led by ringmaster and musical wizard Tim DeLaughter. It's no surprise, then, that they took part in a tribute to Tim Burton's A Nightmare Before Christmas, as they seem the only band capable of handling the glam-rock trippiness of \"Town Meeting Song.\" This is a fun, not-so-little track, and will beg repeat listens to fully digest the complexities of their version.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Circa 1950 Patek Philippe Reference 96 Calatrava, 31 MM 3 Piece 18K yellow Gold Case, 18 Jewel Nickel Lever Manual wind Movement. Matt Silvered Dial with Raised Gold index Markers and a sub Seconds Hand. The dial is a very old Refinish of very High quality with extremely fine and correct printing. Triple signed and having all correct case and movement stamps and numbers. Recently serviced and comes with a one year N. Green and Sons Warranty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is a link to the audio book \"The Richest Man In Babylon\" 3hr 20 min, if you enjoy it leave a comment to keep thread alive and share with other warriors.. network programming notes for mca pdf Here is a link to the audio book \"The Richest Man In Babylon\" 3hr 20 min, if you enjoy it leave a comment to keep thread alive and share with other warriors..\nThe readers can download The Richest Man in Babylon Audiobook for free via Audible Free Trial. The Richest Man in Babylon by George S Clason is a classic and masterclass in finance domain. The audiobook is written in the form of parables.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy National Doughnut Day 2014!\nOkay, I won't keep you in suspense, the hidden ingredient to these carrot and hazelnut muffins is\u2026butternut squash! These muffins have another secret too\u2026they are gluten-free! But they taste really good! That's no secret \ud83d\ude42 A year ago I never thought that I would cook gluten-free foods. I'm not on a gluten-free diet, but I love the challenge of cooking gluten-free, especially in baking, where baked goods usually depend on the chemistry of gluten. I enjoy science, and gluten-free baking is like a science experiment for me, in addition to cooking in general. I realized that even if dishes are gluten-free\u2013if it tastes good and makes you feel good, what's the harm in that? Thus, I wholeheartedly subscribe to Peter Reinhart's \"Flavor Rule\" where flavor determines what I enjoy in a dish, not labels\u2013whether gluten-free or otherwise.\nNow, back to the muffins. These butternut squash, carrot and hazelnut muffins were inspired by a recipe I found in Whole Living magazine. It is a great all-around muffin recipe with a nice soft and moist texture. They taste even better the next day, or days later. Makes for a great post-workout snack, after-work\/school snack, or even as part of a quick breakfast (though I prefer more savory foods in the morning). I decided to use carrot instead of apple in the original recipe and it was definitely yummy. I think you can pretty much sub or add in ingredients like you would in a normal muffin or quick-bread recipe. Zucchini, banana, and chocolate, you're next!\nPancit Canton Bistek + Happy Birthday Dad! & Happy Anniversary to my Parents!\nHappy weekend everyone! I have to start by wishing my Dad a very HAPPY BIRTHDAY! Today is his birthday and I am so glad that he is my Dad! Also, on Thursday it was my parents 31st Wedding Anniversary! They are definitely a couple to look up to and admire. I'm glad they went out to celebrate yesterday by going to a Lea Salonga concert in Palm Springs. They had a great time and they deserve it! This post is dedicated to them.\nAs you probably know or if you have read some of my posts, that Filipinos (and many other cultures) celebrate special occasions by feasting well and by eating noodles, specifically a noodle dish called Pancit. There are many variations of Pancit noodles such as Pancit Guisado, Pancit Malabon, Pancit Hab-Hab, and Pancit Canton to name a few. Wikipedia lists 29 variations! I'm sure there are more out there!\nPumpkin Pie Spice Latte \u2013 Make it your way!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Kettle And Fire Voucher Code 50 Off \"\nIf you Love Kettle And Fire Voucher Code 50 Off then you are at the right place!!\nKettle And Fire Voucher Code 50 Off is made from animal bones and connective tissue-- normally cattle, poultry, or fish-- that have actually been boiled into a broth and also slow simmered for 10 to more than 20 hours with natural herbs, veggies, as well as spices. So why is this apparently basic liquid something you would certainly wish to drink on a daily basis Kettle And Fire Voucher Code 50 Off ?\nEven our hunter-gatherer forefathers realized that drinking Kettle And Fire Voucher Code 50 Off was like striking dietary gold, as its earliest variation go back over 2,500 years. Throwing out anything edible ran out the concern at that time, so animal hooves, knuckles, bones, and other connective cells never went to throw away. Bone broth has an abundant background of being utilized in traditional Chinese medicine as a gastrointestinal tonic, blood home builder, and also kidney strengthener as a result of the high collagen web content, bioavailable minerals, anti-inflammatory amino acids, and also healing compounds that can only be discovered in bones and connective cells Kettle And Fire Voucher Code 50 Off.\nEnjoy as celebrity health and wellness trainer, Thomas DeLauer damages down the ins and also outs of what bone broth truly is. In just two minutes, he'll discuss what bone broth is and also what it can do for your digestive tract, joint, as well as skin health Kettle And Fire Voucher Code 50 Off.\nSome call Kettle And Fire Voucher Code 50 Off meat water. We call it fluid gold.\nWhat makes this things so special? The brief solution is collagen. Bones and also connective tissue are the only true dietary resources of type II collagen-- a protein that's known for maintaining skin smooth and also supple and also teeth as well as joints healthy and balanced, as well as proclaiming loads of other health and wellness benefits.\nIn addition to collagen, bones are likewise loaded with several anti-inflammatory amino acids, minerals, and substances that can help quicken your body's all-natural healing procedures from points like sports injuries, arthritis, or dripping gut.\nSo, these \"scraps\" that a lot of us typically toss in the trash are severe nutritional powerhouses. Yet when was the last time you chowed down on a piece of cow's knuckle? Or snacked on an ox tail? We'll presume the solution is never.\nBecause we don't eat bones in their entire type, cooking them right into a broth that's simmered for 10 to 20 hours (or more) is the very best means to launch their powerful nutrients, and also experience an entire new level of health and wellness.\nNEED A FAST DESCRIPTION ON WHAT EXACTLY IS KETTLE AND FIRE VOUCHER CODE 50 OFF ?\nView as Star Health And Wellness Instructor, Thomas DeLauer breaks down the ins and outs of what bone broth actually is. In just 2 mins, he'll speak about what bone broth is as well as what it can do for your digestive tract, joint, as well as skin health kettle and fire voucher code 50 off.\nBone broth has a rich background of being utilized as a digestive restorative, specifically in traditional Chinese medicine some 2,500 years earlier. Today, it is among the leading recommended foods for boosting symptoms of persistent digestion conditions, such as Cranky Digestive Tract Disorder (IBS), Crohn's disease, and colitis.\nThe reason for why bone broth is so recovery for the intestine returns to collagen, which creates gelatin when it's cooked down even additionally. Collagen and gelatin are not just abundant in amino acids that reduce inflammation in the GI system-- such as glutamine-- they likewise have the unique capability to \"secure and recover\" holes in the gut lining, which can trigger a problem called leaky intestine disorder.\nAlthough study is in the early stages, there's evidence to reveal that leaky intestine is the main hidden root cause of digestive problems. It's likewise a contributor to autoimmune illness, depression, brain fog, anxiety, allergic reactions, eczema, acne, and chronic low energy. Most awful of all, leaky intestine can be silent and also reveal no signs and symptoms in the early stages. It's also thought to affect over 70% of the population.\nBone broth is recommended on unique digestive tract recovery diets as well as procedures, like the SPACES diet, the SCD diet, as well as the Low FODMAP diet.\nWhether it's an injury, joint inflammation, or aching muscles, there's no much better means to nurture your bones and joints than by taking in more of the nutrients already located within them, including hyaluronic acid, glucosamine, chondroitin, calcium, as well as magnesium. Bone broth is packed with every one of these nutrients, plus a number of amino acids that help in reducing joint discomfort and inflammation, like glycine as well as proline. It's for these factors that bone broth is becoming a best recuperation beverage among professional athletes.\nAn enjoyable fact: Kettle and Fire Bone Broth was born because our founder, Nick, tore his ACL playing football and needed a top notch bone broth to assist speed up his healing time. (You can read more on that particular below.).\nFollowing a bone broth diet plan is fantastic for sustainable fat burning. A bone broth diet plan includes consuming Paleo for 5 days and also not eating for 2. During this timespan you'll consume bone broth regularly with an increase in use on the fasting days.\nThe combination of periodic fasting and bone broth speeds up weight loss via burning fat much more efficiently and limiting calorie consumption. When doing the diet, we likewise saw a reduction in cellulite.\nFat-Burning Bone Broth Mixed Drink Kettle And Fire Voucher Code 50 Off.\nWith simply a few ingredients, this dish is excellent for a tasty as well as exceptionally straightforward fat-burning cocktail that you can whip up in a flash on hectic early mornings prior to work or in the evenings post-workout.\nLots of anti-aging skincare lines include collagen to their face creams, creams, and also serums (and market them for a pretty penny). But what lots of people do not recognize is that collagen molecules are too huge to be taken in with the skin, which indicates these lotions can't give much in the method of outcomes.\nThe good news is that collagen can be taken in via your digestive system. And also since there's a straight web link in between the digestive tract and also the skin, adding bone broth to your diet is going to have a much more powerful as well as long lasting impact on the total appearance as well as health of your skin.\nAllow's not ignore hair and also nails, which are comprised of proteins like collagen. Collagen helps enhance nails, and also motivates hair to expand in thicker, quicker, and tackle a healthy and balanced sparkle. (Remember this for the following time you get a negative haircut!).\nDeep, restful rest is something we could all use a little bit even more of. As a matter of fact, statistics reveal the ordinary individual gets much less than 7 hrs of sleep per night, so it is necessary to do every little thing you can to make those hours matter.\nBone broth is rich in glycine, an anti-inflammatory amino acid that works on the Central Nerves (CNS). When taken prior to bed as a supplement, researches show glycine can enhance sleep high quality as well as minimize daytime drowsiness. It's a lot more efficient when paired with various other sleep-supportive nutrients, like calcium and also magnesium.\nAll of the advantages you reap from bone broth, your growing infant will experience, also-- consisting of healthy bones, joints, and also a strong digestive system.\nWhen morning health issues hits, it can be impossible to maintain nourishing food down. However bone broth often tends to be comforting for queasiness, and also is normally well endured. Most importantly, bone broth uses several of the nutrients you and also your expanding infant need, including protein, and also essential nutrients. As an added reward, the sluggish simmer time of the bones makes the nutrients in bone broth highly bioavailable and easier to absorb than a dietary supplement.\nBone broth can likewise help boost fertility, assist you have an extra comfy maternity by nourishing rigid or agonizing joints, and also might increase calcium levels in breast milk.\nBone broth is an exceptional resource of nutrition on the Paleo diet regimen, and also is urged to consume alcohol during durations of recurring fasting. It additionally suits well with macronutrient needs on the keto diet plan, as well as might help reduce symptoms of the keto influenza.\n\" Not just can bone broth assist you avoid keto flu, yet it's additionally packed with minerals that recover leaky intestine as well as minimize inflammation in your intestines.\" - Leanne Vogel, Author of The Keto Diet regimen.\nThe nutrients in bone broth make it an effective functional food that can benefit your whole household-- including your hair babies. Bone broth is good for your pet's shiny coat, bones, joints, digestion, and teeth.\nBelow you can learn exactly how Nellie, a 15 years of age rescue puppy, defeat her chances of living by only consuming bone broth.\nPacked with protein - Poultry is an excellent protein resource and also same opts for poultry bones. Pot & Fire bone broth comes loaded with 10 grams of protein per serving Kettle And Fire Voucher Code 50 Off.\nLowers digestive tract swelling - Bone broth assists recover and also seal a leaking intestine which is the root cause for numerous autoimmune conditions.\nRich in bone marrow - Bone broth made from 100% yard fed cattle and simmered over long chef times removes bone marrow which contains healthy and balanced nourishing hormones as well as lipids.\nPacked with amino acids - Similar to chicken bone broth, the nutrients drawn out from beef bones have amino acids like glycine and glutamine that boosts digestion and also repair work your intestine cellular lining.\nNo. Plants are a terrific resource of nutrients that can help boost your body to create its own collagen, however there is no recognized plant food that uses type II collagen, the form that provides all of the health benefits listed above.\nOne point to take into consideration regarding bone broth that establishes it in addition to plant foods: Even if you do eat a selection of plants that are rich in collagen-boosting nutrients, if you have a damaged digestion system or leaky digestive tract, you might not fully absorb them. On the other hand, the collagen in bone broth is unbelievably simple to take in, even for those with compromised gastrointestinal systems.\nCertainly, you do not just need to consume it-- although we do have a totally free as well as epic downloadable bone broth drinking overview for when you do. You can mix bone broth into your green smoothie mixes, make healthy collagen gummy bears, and utilize it in everything from clambered eggs, soups, curries, stews, breakfast bowls, and tacos.\nWe slow-simmer the organic bones to extract the collagen, healthy proteins, as well as amino acids into a nutrient-rich bone broth. The result is a collagen-rich, high protein bone broth that offers your body the nutrients it needs to prosper.\nBone broth is an all-in-one superfood, loaded with nutrients that provide energy and also motivation, help you sleep much better, make your skin appearance smoother, as well as can help heal digestive system concerns like dripping gut.\nBones and also connective tissue are warehouses for necessary amino acids and minerals-- which are doing not have in numerous diet regimens today. Bone Broth is likewise a vital resource of protein, collagen as well as jelly.\nIt's not practical to consume entire bones or tissue, yet you can still appreciate these health advantages by drinking bone broth. Collagen is drawn out when you simmer bones for an extended period of time. Usually, the longer bone broth simmers, the even more collagen you'll remove.\nMaking bone broth is a basic procedure, however one that calls for lots of time and also persistence. If time is not on your side we can assist you out with that.\nBone broth is made by simmering pet bones and tissue for a minimum of 10 hours with vegetables, herbs, as well as flavors such as thyme, garlic, and also bay leaves. Top notch bone broth starts with premium components, using bones from organic grass-fed animals and natural vegetables. While any bone or ligament can be utilized, knuckles, poultry feet, and femur bones tend to contain the most collagen.\nSimmering bones for an extended period of time is what provides bone broth its health advantages, removing the amino acids, minerals, and collagen. This is a Kettle & Fire-tested sluggish stove chicken bone broth dish that includes natural chicken bones, sea salt, fresh veggies like celery stalks, onions, as well as bell peppers, and natural herbs like parsley, rosemary, and thyme.\nJust cooking bones (no meat) and also water is going to be primarily tasteless. Food preparation the blend for hrs on end isn't going to make it taste any kind of better.\nThe concept that you're obtaining tons of nutrients from the bones is a MYTH that will not die, although it has actually been debunked numerous times, including through lab examinations. The idea that including vinegar to \"launch\" those nutrients has likewise been disproved. You simply end up with a watery mess that tastes of vinegar.\nRather, acquire bones with meat on them. After that add veggies such as onions, carrots, celery, and just sufficient water to cover whatever (typically 8 or 10 mugs). Skip the vinegar! Simmer for 3 to 4 hours for poultry, slightly much longer for beef. Period according to preference. If you have an Immediate Pot, 25 to 35 mins has to do with right.\nWhat you will get with this approach is a very flavorful broth that's most likely considerably extra nutritious-- including lots of collagen-- and also even more healthy than any kind of bone broth, and it will taste one hell of a lot much better. You can either consume the meat as well as veggies (it's called SOUP) or you can stress them out and also simply consume alcohol the broth if you like.\nCheck Ina Garten's dish for hen broth. It's really comparable to this one and has actually been made use of with small variants by well-informed chefs for years.\nThere are 2 primary differences between bone broth as well as regular broth or supply: simmering time and the part of the pet it's made from (bones or flesh).\nNormal broth and also supply are simmered for a shorter amount of time than bone broth, about 2-- 6 hours. The expedited cooking procedure decreases the amount of useful jelly extracted from the bones, reducing its ability to improve the body immune system, heal gastrointestinal problems, as well as reduce the signs and symptoms of leaking digestive tract.\nPrior To Kettle and Fire was born, among our co-founders, Nick, tore his ACL playing football (oops). His sibling Justin found out about the benefits of bone broth for injury recuperation. As his timetable didn't leave much time to make bone broth from the ground up, he laid out to buy a store-bought, top notch, grass-fed bone broth, given that both of their busy routines didn't leave much time to make the broth themselves. (You can discover more concerning our tale below.).\nRegardless of exactly how hard Justin looked, the optimal bone broth didn't exist. He looked for one that was one hundred percent organic, fresh-- never ever frozen-- grass-fed, as well as slow-moving simmered (along with one that might be delivered without inefficient, confusing product packaging). So, Nick and also Justin determined to produce a premium bone broth on their terms, which is the dish we're honored to provide you today.\n\" Kettle & Fire provides a beneficial, top quality bone broth in a delicious and practical shelf-stable format. I like constantly understanding that I have actually got wonderful broth available, without congesting my fridge freezer or requiring defrosting.\" - Dr. Sarah Ballantyne, three-time NYT Best-Selling Writer.\nThe lovely feature of bone broth is that there's truly no restriction to just how you can add it to your diet. Aside from soups, stews, as well as plain ol' sipping, bone broth blends remarkably well right into almost any type of dish-- even smoothies! Right here are our leading means to obtain it:.\nOur bone broth preferences tasty enough to sip on its own, but you can seasoning it up to match your taste buds. We developed our favorite taste combinations in this totally free downloadable \"Bone Broth Sipping Overview\" (bone broth matcha lattes, anyone?). Get your cost-free duplicate here.\nDrinking bone broth does not have to be uninteresting-- there are limitless methods to seasoning it and also sip it. Obtain instantaneous access to over 15 of our preferred bone broth sipping combinations, consisting of ginger and also turmeric bone broth tonic and also chili, as well as cardamom bone broth elixir. Discover exactly how to use bone broth in your matcha lattes, gold milk, and exotic fruit healthy smoothies.\nOur beef bone broth has a light taste, which enables it to mix quickly with almost anything, from shakes to healthy gummy bears. Our chicken bone broth as well as mushroom hen bone broth increase the full-flavored taste of soups, stews, and also risotto dishes. A few of our favorites include:.\nIf you want to experience more power as well as vigor than ever before, we reccomend looking into The Bone Broth Diet, produced by Dr. Kellyann Petrucci.\nThe Bone Broth Diet plan not just raises your intake of the healing nutrients as well as substances crucial for digestive tract, skin, joint, as well as bone health, but assists combat systemic inflammation, which is at the origin of most Western ailments as well as diseases.\nThe bone broth diet plan is a 21-day strategy that's excellent for any individual struggling with a chronic wellness problem (such as an autoimmune problem), or anybody that wishes to reset their digestion system as well as experience glowing wellness. Find out more about the 21-day bone broth diet regimen right here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"All Baker homes currently listed for sale in Baker as of 04\/24\/2019 are shown below. You can change the search criteria at any time by pressing the 'Change Search' button below.\nIf you have any questions or you would like more information about any Baker homes for sale or other Baker real estate, please contact us and one of our knowledgeable Baker REALTORS\u00ae would be pleased to assist you.\n\"Spacious three bedroom, two bath home with single carport. Laundry area adjoining kitchen plus separate den. Over one acre corner lot located near the north end of Baker. Association Amenities: None # of RV Spaces: 0NONE Lot Location Type: Standard Location Special Features: None # of Attached Spaces: 0 # of Detached Spaces: 0\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How To Get More Followers On Pinterest!\nWe all know about that magical place beyond our screens and inside the internet universe called Pinterest. But how do you get exposure on Pinterest? And how do people find and follow you on Pinterest?\nNow, you may be thinking, what? Cover photos? Titles? How is this going to help me? Well, it could help you a lot! When pinners come across boards that don't captivate their attention right away, they usually scroll right past them. A majority of Pinterest users look for pretty pictures and you have to deliver. If you have a board that is all about cupcakes, great! But don't let the cover photo be a picture of a messy spoon. Adding a title to your board is also important, as it will immediately explain what your board is about to help people find what they are looking for easily!\nIf you're following anyone and everyone just for the sake of getting followers, it may not be as effective as following those who share the same interests as you. For example, if you are a baker, with a majority of pins relating to desserts and sweets, go to the search bar and look up desserts or sweets. From there, you will have the option to go through pins, boards, and people who fall under that category. If you follow the boards of fellow bakers, you will get more exposure to people who will likely follow you. How? The person that you follow will be notified and it could lead them to follow you back. Also, other people who see the followers on that particular board may click on your name and follow you, too. It's also important to note that people who have the same number of followers as they are following could follow you, too. Getting your name out there is a key way to gaining followers in the Pinterest world.\nIf you want more followers on Pinterest, it is important for you to be active on Pinterest! You don't need to spend hours a day pinning, you can keep it as simple as 15 to 20 minutes a day (although, it may be hard to stop). The more you pin, the more you follow, the more you engage and like, the more people will see you. Adding pins from your own site is a great way to gain publicity, too!\nDon't forget to apply these tips to your Pinterest account to help you gain a greater following! Happy pinning!\nBrandastic.com builds world-class brands, excels in web design and SEO, and helps market your brand just like Pinterest does. Brandastic is a Southern California based company focused on relationship, excellence, and service. When you are ready to be noticed, to really stand out, contact Brandastic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I once went 72 hours without eating\ud83d\ude2f. Yes, by choice. What? Say that again? Why would anyone want to go for an extended period of time without eating? To me, the thought at first was preposterous. You see, my husband had previously done extended fasts to dive deeper into his connection with God.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well this Nyack Street Fair is the final one for 2017.The weather is a bit in question. 50% chance of rain, but we Nyackers remain optimistic. Bring your dancing shoe as there will be live music to get your groove on. and Featuring over 300 vendors the fair takes place from 10a-5p Sunday Oct. 8 and is sponsored by ACADA of Nyack.\nRemember.dogs are not allowed at Nyack Street Fairs for safety reasons.. Kid's will be entertained at the \"Kid's Zone\" showing a variety of inflatables set up by \"A Total Jump\" in the Main Street parking lot.\nThinking green downtown Nyack is transformed into a car-free pedestrian mall for up to 25,000 people during the street fair. When you are tired of browsing stroll down to see the river at nearby Memorial Park, visit Nyack Beach State Park or hike Hook Mountain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You'll be able to count on Star White Kitchen Cabinets to supply the top professional services regarding White Kitchen Cabinets in Brimson, MN. Our team of highly trained experts will supply the services you need with the most innovative solutions in the market. We are going to apply excellent supplies and affordable techniques to make sure you will get the best service at the greatest value. Contact us today by dialing 844-329-5869 to learn more.\nYou've got a spending budget to stick with, and you want to get lower rates. On top of that, you'll need the most beneficial and highest quality of services for White Kitchen Cabinets in Brimson, MN. Our initiatives to help you save money won't compromise the excellence of our results. When you choose our team, you will receive the advantages of our own practical knowledge and top standard products making sure that any project lasts while saving time and cash. This is attainable given that we recognize how to save you time and funds on materials and labor. Save your time and cash by calling Star White Kitchen Cabinets today. We'll be waiting to take your phone call at 844-329-5869.\nWhite Kitchen Cabinets are available in Brimson, MN.\nYou should be kept informed regarding White Kitchen Cabinets in Brimson, MN. We will never encourage you to come up with imprudent judgments, because we know exactly what we are working on, and we ensure that you know exactly what to be expecting with the process. We take the unexpected situations out of the picture by supplying accurate and detailed advice. The first step is to call at 844-329-5869 to set up your project. We're going to address your important questions and schedule your first meeting. Our staff will show up at the scheduled time with the required materials, and will work together with you all through the project.\nIf you are arranging a venture for White Kitchen Cabinets in Brimson, MN, there are several great reasons to call Star White Kitchen Cabinets. We are the first choice when you really need the most efficient cash saving methods, the very best supplies, and the greatest rank of client satisfaction. We'll be there to help you with the greatest working experience and skills available. If you need White Kitchen Cabinets in Brimson, get in touch with Star White Kitchen Cabinets by dialing 844-329-5869, and we'll be more than pleased to help you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Our Portuguese Standard Course consists of group instructed 3 hour\/daily classes of Portuguese Language and Culture per week from Monday to Friday. It is recommended for students who wish to combine language learning with a real life experience of the Portuguese culture, thus, it allows enough time for students to get fully acquainted with the city and the Portuguese culture surrounding it. Classes challenge students to apply their knowledge of theory by means of various methods ranging from situation and role-playing activities to debates about current issues. Portuguese Super Standard: Group instructed 3 hour\/day classes + 2 individual classes per day of Portuguese Language and Culture from Monday to Friday. Especially for highly motivated students who wish to learn as much Portuguese in the shortest time possible. It is of similar nature to the classes described above. Portuguese Standard Plus: Group instructed 3 hour\/day classes + 1 individual class per day of Portuguese Language and Culture from Monday to Friday. Especially for highly motivated students who wish to learn as much Portuguese in the shortest time possible. Portuguese Intensive: Group instructed 6hour\/daily classes of Portuguese Language and Culture per week from Monday to Friday. Courses are available June through September. Designed for people with little time and the need to practice even more in little time, thus, the courses encourages quick progress in Portuguese communication. Beginners start, May 31, Jul 5, Aug 2 & 30 Portuguese Intensive Plus: Group instructed 3 hour\/day classes + 3 individual classes per day of Portuguese Language from Monday to Friday. Especially for highly motivated students who wish to learn as much Portuguese in a short time. It is of similar nature to the classes described above. Individual (One to One) Portuguese\/Private Group: Individual are course followed by one person, while Private Group are course followed by two persons. 5, 10, 15, 20, 25, 30, 35, 40, 45 or 50 hours\/week. Tailor-made private classes for highly motivated persons who need to advance rapidly in a specific area of study but in little available time. You decide how many classes you want and what you wish to study. These courses aim at providing students with a further spoken and written language practice while exploring a specific field. Themes: - Banking | Translation | Law | Interpreting | Medicine | Business Portuguese | Insurance | Computer Science | Hotel Industry These courses are programmed according to the students' specific requests including different professional and cultural themes if requested. Students are welcome to bring any kind of documentation they may need to work on. Focus Programs: * Portuguese For Teachers [Ensino de Portugues Lyngua Estrangeira (EPLE)] Course designed for native and non-native teachers of Portuguese a Foreign Language in public or private institutions in or outside Portugal. . 2 weeks' duration with 6h\/day of lessons. We may organize an EPLE course on any date for a minimum of 4 students. Dates of commencement: Feb. 16, May 10, Sep. 13, Nov. 15 * Easter & Summer Course Theses 2-week program consists of 3 hours of Group classes per day in Portuguese language and culture and is held only during the holiday period of Easter and the Summer. Group courses features a maximum of 12 participants. It is good ways to get to know and to enjoy the way the Portuguese celebrates these dates, while you practice your Portuguese. Dates of commencement: March 29, August 23 * African Portuguese The course focuses on Portuguese in Africa. It is either taught by Individual (One-to-One) lessons or as Private Group. Specific materials and experienced teachers of African origin or background. * Brazilian Portuguese This program is taught either as Individual (One-to-One) lessons or as Private Group. Taught with materials specific to the Brazilian Portuguese language and by experienced teachers of Brazilian origin. * Gastronomy Portuguese Learn how to make popular authentic Portuguese dishes. Get to know Portuguese cuisine. Typical dishes, wines, sweets and cheeses. Learn how to cook them, do it and eat them with your colleagues. There are 8 hours of class offered per week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Anudip has created livelihoods of 70,000 young people across 150+centers.\nAnudip Foundation conducts classes in C Language, C++ Language and Computer Course. It is located in Kesnand, Pune. It takes Regular Classes- at the Institute.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Overview of the cheapest flights from Kobenhavn Kastrup to destinations in Faroe Islands with relevant fares in next months. Only selected flights from CPH are shown. Use the search form below for more dates and routes.\nAre you looking for flights to Kobenhavn Kastrup? Please switch to our overview of Faroe Islands to CPH. Fill in the dates to find the best airline tickets deal for your journey.\nMore flights: Kobenhavn Kastrup to FAE. No indirect flights from Copenhagen (CPH) to Sorvagur (FAE), Faroe Islands found. Prices of flights to Sorvagur sourced on Friday 27th July 2018 at 14:37.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This vehicle stands out from the competition! The following features are included: tilt and telescoping steering wheel, power door mirrors, and remote keyless entry. It features an automatic transmission, rear-wheel drive, and a refined 6 cylinder engine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Exclusive two bedroom with king bed, two queen beds, bath with tub\/shower, and wood burning fireplace.\nLocated on a secluded, pine-studded point, just steps from the water's edge. This cottage is private and very special.\nYou can look out your large picture window overlooking Poplar Lake, or sunbath on one of two decks, or sip your morning coffee in the shade on the other deck.\n$1195 for a week stay includes first 2 people.\n$149 a day per 3 day block includes first 2 people.\n$169 a day includes first 2 people.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbkbl b/data_all_eng_slimpj/shuffled/split2/finalzzzbbkbl
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Exosomes are 40 \u2012100 nm membrane vesicles. They are found in many biological fluids including blood, urine, saliva, amniotic fluid, cell culture supernatants, and malignant ascites fluids. CD9 is a well known surface marker for exosomes. ExoTrap Spin Columns utilize immobilized CD9 antibody to capture exosomes from biological fluids in as little as 30 minutes. The contents of the captured exosomes are then readily processed for downstream analysis by western blotting, mass spectrometry, PCR, or sequencing, etc.\nExosomes are membrane vesicles that are enveloped with a lipid bilayer, and encase various molecular constituents, including proteins and nucleotides. They have a diameter of around 30 - 100nm, and are secreted from various cells. By circulating in body fluids, exosomes can transfer various signals to distant cells. This cell-to-cell communication via exosomes has been thought to be associated with antigen presentation, apoptosis, inflammation, and tumor evolution\/malignant alteration, leading to investigation of cancer cell derived exosomes. It is anticipated to apply in a wide range of research area, including development of early detection of cancers using exosomes that are secreted from cancer cells, or drug delivery systems utilizing exosomes as cargo.\nWe are focusing on investigating the interaction between virus-infected cells and surrounding uninfected cells. After viral protein-expressing cells and non-expressing cells were co-cultured, and virus proteins were detected by fluorescent immunostaining, virus proteins were observed in dot form from the surrounding uninfected cells (the cells that are not expressing viral protein). From this incident, we are interested in exosomes, which intervene cell-to-cell communication.\nHowever, we're facing the problem, \"how do you purify exosomes?\", since we've never dealt with exosomes by ourselves. In general, the ultracentrifuge method is employed to purify the exosomes, however, it needs to optimize various conditions and is time consuming and laborious when purifying from many samples. For that reason, we compared several exosome isolation kits on the market at some respects, including simple procedure (robustness), required time (quick protocol), and aplenty applications (both the exosome itself and the encapsulated protein or nucleotide can be analyzed). Because of its easy procedure (just apply samples into spin column) and quick protocol (takes only 30min), we chose ExoTrap\u2122 Exosome Isolation Spin Column Kit for Protein Research (Cat No. CSR-SHI\u2012EXO\u2012K010) from Cosmo Bio. Ltd.\nThe advantages of this product is its simpleness, as you just have to apply samples into spin columns and quick spin, and its quick procedure, which takes only 30min. Even though we're \"exosome beginners\", we could purify not only exosomes but encased virus proteins, which was detected by Western Blot analysis only after column purification. On the other hand, from the conditioned medium of which cells were treated with exosome inhibitor (GW4869), this viral protein wasn't detected suggesting it's still encased by the exosomes (1). This product may not suitable for large sample volumes, however; it is highly recommended when working with small sample volume, and when you prefer a quick and easy exosome purification procedure.\nFig. 1 Results of western blot (WB) analysis with before or after purified exosomes with or without exosome secretion inhibitor treatment.\n(A) Exosomes were purified using ExoTrap\u2122 Exosome Isolation Spin Column Kit, and then encased LMP1 and Hsp70 expression were analyzed by WB. Enrichment of LMP1 and Hsp70 in the purified exosomes was observed. (B) Cells were treated with exosome inhibitor (GW4869) and then exosomes were purified from its conditioned medium using ExoTrap\u2122 Exosome Isolation Spin Column Kit. WB was performed to identify LMP1 and Hsp70 expression. The reduction of LMP1 and Hsp70 by the GW4869 treatment was observed, suggesting that those protein are still encased by the exosomes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Navigator - Page 2 of 3 - Navigating our way around the world.\nDriving in another country can always be a little daunting, so it's always a good idea to familiarise yourself with what to expect when you plan your self drive holiday in New Zealand. After you've booked your camper van or car it's time to think about what it's going to be like behind the wheel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Field Services works closely with police departments and other law enforcement agencies to provide support throughout the county when requested. The department consists of the K9 Unit, Mounted Unit, Warrant Apprehension Unit, Honor Guard, Mobile Firing Range, Mobile Command and the Communications Center.\nAdditionally, the Field Services employees oversee community programs such as the Community Work Crew, Safety Assurance Call, and CodeRed. Field services also manages our on-call part-time deputies who provide security and road-work details whenever requested by local police.\nThe specialized resources of Field Services are deployed to any of the 27 municipalities in Plymouth County at the request of the local police departments and in the case of some community programs, at the request of county residents.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bathing outdoors amongst lush gardens and flowers will leave you feeling revitalized and refreshed.\nWhether you're rinsing off from a dip in the ocean or your swimming pool, or making a step towards reconnecting with nature, an outdoor shower will invigorate your cleansing ritual.\nImagine stepping into a heavenly oasis, right in your backyard. A private retreat amongst tropical greenery, climbing vines, and tall grass. Build walls of smooth stones, tile mosaics, and let the moss grow like layers of lace. Whether you invest in a designer to construct a luxurious addition to your outdoor space, or use your own creative ingenuity for a DIY style project of transformation, the most important thing is to feel the lavishness of life and nature in your own home.\nHere are some of our favorite selections of exquisitely inspiring outdoor bathrooms, we hope you enjoy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your vehicle will be visually and physically inspected during the title loan lending process. Find out for yourself why cartersville turns to us for their quick title loan needs. Why deal with the hassles of a traditional bank loan when you can get approved for a quick title loan. A cartersville office staff member will be ready and waiting to assist you in finalizing the loan. The main difference between a title loan and a cash advance, is that you use your vehicle as collateral when you apply for a short term title loan.\nWe will determine your vehicle's value based on the inspection and will then be able to tell you how much you may borrow for a title loan. We operate in various locations including the cartersville area. The whole cartersville title loan process is completed in just a few simple steps and we can have money in your hands as fast as today! The title loan staff will take a look at your vehicle, explain your loan documents and provide you with the money. It's our job to help cartersville title loan customers obtain the money they need.\nA cartersville title loan is a fast, safe and easy way to get the cash you need in a hurry. It's easy to find a title loan lender near your since we have offices in various locations including cartersville. A low cost title loan is easy to qualify for because you are using the money you have already put in your vehicle. Any time you have questions regarding your title loan, give our cartersville office a call and somebody will be more than happy to assist you. The basic idea behind a title loan, is that customers will use the money they obtain for immediate financial obligations and pay back the money as quickly as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbblkf b/data_all_eng_slimpj/shuffled/split2/finalzzzbblkf
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index 0000000000000000000000000000000000000000..ab1585d667746ff542d520a63c30c2f57673c9c8
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@@ -0,0 +1,5 @@
+{"text":"Enjoy Top Pun's cool spiritual-religious magnets this day and every day!\nBuy cool religious-spiritual magnets today!\nBuy your favorite Religious-Spiritual Magnets today!\nTop Pun is your best source for serious, funny, and seriously funny Religious-Spiritual Magnets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Remember you found this company at Infoisinfo 0186 572 204?\nC-Work Cyber Cafe is located in the heart of Oxford City, Near to Claerendon. We offer a wide range of technical products and services that cover most aspects of IT, Toner & Cartridge Refill, Serve you the services of Internet Marketing or High Quality Website Designing\/Maintenance, Create\/Resolve all type of networking (Wi-Fi\/Cable), resolve all computer problems, Sale of Computer & its Accessories. Full service systems integrator of information technologies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Treatment for heart attacks, shock, choking, bleeding, burns and scalds, poisoning, fractures. seizures, fainting, low blood sugar, allergic reations, eye injuries and other potential incidents.\nHow to keep appropriate safety records.\nFirst aiders who need to renew their statutory first aid and work certificate.\nPlease note for first aiders whoe certificates have expired for more than 28 days must attend the three day first aid at work course.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well, the first month of 2018 is almost over. It wasn't the easiest month for me to keep knitting, but I did, despite travelling, tropical temperatures, a very limited stash and other obstacles.\nI am on track with my challenge to make a hat each week, so that's going great.\nMy personal favorite is the Bobble Top, and now you can knit that one too. I tried to write down my notes as clear as I possibly could. Download PDF here (204.4K).\nThank you! love the hats and you are on a hat knitting marathon!\nThanks so much for the hat pattern. I hope to join your Mad Hatters party in the future. Will it always be on the last day of the month?\nI look forward to seeing your hats someday ;-) Yes, I plan to do this on the last day of each month and I'll keep it open for about a week each time.\nThanks for the pattern, I've pinned so I can find it again next time I'm looking for a good knit hat pattern.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the most popular examples for successful technology disruption is the launch of Amazon Video Streaming Service in 2006 and its effect on the distribution of video content. Once a disruptor makes its presence known, the early adopters follow, and the naysayers are forgotten. As an early adopter it only took Netflix 1 month to announce their video streaming service would be coming in 2007. Then we have Blockbuster, one of the most popular naysayers around, who did not get on board with the latest technology and ceased operations a short time later in 2013.\nIt directly affects your ability to compete with large organizations. New technology typically enters the market from the top down, meaning the large organizations create it, other large organizations adopt it and a few years later when the technology is made affordable small to medium sized businesses are able to implement it.\nThink of the way online ordering tilted the favor of the pizza industry in the favor of national chains. It wasn't till a few years later when web design and online ordering apps made online ordering cost effective for your local mom and pop shops.\nBluJaket removes all of those issues for you, we put the time and money into the R&D, we've priced our app at a very, very affordable level for small to medium sized businesses and we've made signing up an easy 3 step process.\nName one piece of your marketing budget that costs less than our service and can provide more value. At the price point of $25\/mo.\/loc. for Stings and $50\/mo.\/loc. Hijacks, it is just simply to good to pass up. If you don't think our service fits your business model, you can check out a few of our use cases, or contact us and we will help you build your strategy today.\nMost marketing strategies hit the eyes and ears of your customers at random days and times throughout the week. How effective can a coupon for a Steakhouse be if it is delivered at 9:00 am on Tuesday morning? Why not reach this person on a Friday night, after 5:00 pm while they are near your business location?\nThis means customers are sent coupons and special offers at the precise time they can use them, making them more likely to us them. Stop paying for coupons that get lost, deleted or forgotten about and get BluJaket today.\nBluJaket offers a 30-day money back guarantee if you are not satisfied with your experience. After you sign-up with BluJaket, we do our very best to make sure you have a great experience.\nMarketing Kit - BluJaket offers our business partners marketing assistance such as social media posts, blog announcements, email updates, table tents, flyers, window clings and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcixf b/data_all_eng_slimpj/shuffled/split2/finalzzzbcixf
new file mode 100644
index 0000000000000000000000000000000000000000..3501299fdee86eedeffd160f6691cbc5cc8210e5
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The series of candidate profiles continues, but there are several developments in House District 7B tonight that continue to suggest that the frontrunners in the race are Roger Reinert and Marsh Stenersen.\nFirst of all, candidate John Derbis told me tonight that he will be supporting Marsh Stenersen in the primary. He'll announce this tomorrow at a press conference at UMD. Though Derbis logistically remains on the ballot this gives Stenersen a news bump going into the last weekend of the campaign. Meantime, Stenersen seems to be hitting Reinert on the matter of where his campaign donations come from. He claims that too many of them come from \"known Republicans.\" I don't know how closely people are paying attention to this matter, but Stenersen is clearly trying to close the perceived gap with Reinert. Under the right conditions, this could help his numbers.\nAt the same time, Roger Reinert announces that he's going to name the first three bills he plans to introduce tomorrow in a press conference at 11 a.m. outside the entrance to the Great Lakes Aquarium. The bills, according to his release, will deal with the aquarium, public libraries and sanitary sewer improvements. That's an interesting idea and much more specific that what we normally see in a primary. Reinert is also rolling out a series of endorsements from construction trades and environmental groups. Stenersen, a union official, also enjoys strong union support. As I've written before, Reinert has clearly strived for the \"new generation\" mantle in this election, while Stenersen has grabbed the traditional \"DFL values\" mantle. In truth, this primary in a strong DFL district has the potential of being the whole bag of marbles. Either Reinert or Stenersen would have a huge electoral advantage in November. Furthermore, personally, I find both of them to be good candidates. There will be no endorsements here, not that it would matter.\nThe other candidates, Brandon Clokey, Dan Maryland and Derbis, deserve credit for the campaign they've run. Clokey, in particular, has been very active on this blog and it's good to see him involved. Hey, stranger things have happened. But with the final weekend already approaching, it's clear that Reinert and Stenersen are running hard. I'll post my final analysis after the candidate profiles are done.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We take the time to customize your cabinetry.\nSince 1978, our company has been providing quality service to our clients. We value your trust that is why we make sure your furniture will look good as new. Our experienced and skilled staffs will surely help you with every issue your furniture might have. Call us today for more information!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who do you think will win the 2019 Masters?\nClick on the link below or image to tell us YOUR opinion for the opportunity to win a #Xikar Xi2 American Flag Cutter!\nCongratulations to last weeks winner Pablo Torres.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I love your use of Haiku!\nYes I was only saying to my husband the other day where did our carefree days go?? I still feel these when I walk along the beach though.\nYes, the beach! I could post about that , oh yes! Something about that baseball field just drew me in last week as I was walking at the park\u2026and then came the photo challenge. Such memories of childhood days\u2026thankful for having ones like that.\nYes, those of us who had happy memories of childhood are very blessed\u2026.When so many do not, as we know in our work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On the weekend of June 10 & 11 2017, ULTRA WORLDWIDE successfully produced two multi-day festivals in two different countries in Asia \u2013 South Korea and Singapore \u2013 for the first time in its illustrious history.\nA first for any festival organization, this incredible achievement marks another slice of record-breaking history for the world's largest independent, and most international, festival brand.\nIn one of the biggest weekends of electronic music the continent has ever witnessed, Ultra welcomed over 170,000 fans combined over two locations (120,000 in Korea and 50,000 in Singapore), in just two days.\nHeadliners Dash Berlin, Dubfire, Hardwell, KSHMR (Live), Martin Solveig, Nic Fanciulli, Nicky Romero, Pendulum (Live), Sasha & John Digweed, Steve Angello, Tchami (Live) and Ti\u00ebsto performed across both flagship events, crossing flight paths over the South China Sea on the morning of Sunday, June 11, in order to do so.\nAlesso performed exclusively in Seoul, having already headlined ULTRA events in Singapore in 2015 and 2016, while Showtek, Steve Aoki and surprise guests Knife Party headlined exclusively in Singapore.\nULTRA Korea, which returned to the capital city, Seoul, for its sixth annual edition, welcomed upwards of 120,000 fans to the immense Olympic Stadium, cementing its position as the largest two-day electronic music festival in Asia. Over 80 world-class acts came together across the Main Stage, Live Stage and RESISTANCE Stage \u2013 the latter of which made its highly anticipated, much lauded debut in South Korea this past weekend.\nMeanwhile, some 3000 miles away, following its elevation to full festival status in 2016, ULTRA Singapore returned to its home at Bayfront Avenue to deliver another groundbreaking programme of DJs and Live acts. Also showcasing the holy triumvirate of Main, Live and RESISTANCE stages, and with upgraded production across all, this was ULTRA's most successful outing in Singapore to date.\nFor those unable to experience ULTRA Singapore in person, ULTRA Live's internationally celebrated livestream, presented by UMF TV (produced by NOMOBO) was on hand with all the action, attracting over 320,000 viewers across the weekend.\nWith ULTRA's first 2017 tour of Asia now complete, all eyes are now on the second leg, taking place this September. With an incredible SEVEN events taking place over the course of just ten days, including the debut of Ultra China \u2013 a multi-day festival in Shanghai \u2013 and Road To ULTRA India in New Delhi, be prepared for even more record-breaking and history-making from the world's most successful festival brand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbedao b/data_all_eng_slimpj/shuffled/split2/finalzzzbedao
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index 0000000000000000000000000000000000000000..398f43eb061e94b7ecdd4fae180b987cfc1f7f03
--- /dev/null
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+{"text":"Kaysy Ostrom is a working teaching artist in the Greater Seattle area and Studio East's Education Associate. She has years of experience teaching theatre to ages pre-K \u2013 high school. Currently, she teaches with Studio East, Youth Theatre Northwest, Seattle Theatre Goup and Act Theatre. Through working with these arts organizations she's had the opportunity to helm after-school classes and workshops at many schools including Lakeview Elementary, Sunset Elementary, Carl Sanburg Elementary, Alcott Elementary, Cascadia Montessori, Chestnut Montessori, Frank Wagner Elementary, Redmond High School, and more!\nHer teaching specialties are creative drama, story drama, play creation, playwriting, acting technique, singing technique, audition workshops, music direction, musical theatre, scene study, and choreography.\nKaysy is a proud graduate of Seattle Children's Theatre's TAT Lab: an eight-month professional development program for Washington State teaching artists, with a focus on supporting arts education as part of basic education in K-12 school. Read more about this exciting program here.\nKaysy is always looking for more teaching opportunities! Contact her here!\nThis fall Kaysy is directing Fantastic Mr. Fox at Youth Theatre Northwest as part of their Young Conservatory program for kiddos from 1st \u2013 4th grade.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who Makes Buying Decisions? Women!\nI bought my first brand new car in 1989. It was a Honda Civic, and my husband-to-be was with me. I had a lot of questions about the car, which I asked directly to the salesman. He answered them. But he didn't look at me when he did. He looked at my mate.\nHello! I have the money here! I'm making the decision!\nGranted, that was almost 20 years ago, but I can attest that women still feel ignored at times by salesmen and technicians when it comes to some major purchases and repairs that somehow have become to be known as \"masculine.\" Boilers, AC units, and other hvac equipment unfortunately falls into this category at times.\ninformation is geared to women and encourages them to search for a Weil-McLain contractor in their area.\nWomen have trusted the Seal for almost 100 years as an endorsement of quality and value. According to Weil-McLain, it enjoys a 94% recognition rate among consumers and will influence a consumer to buy a Seal-endorsed product 81% of the time when prices and features are similar to other brands on the market.\nThe Good Housekeeping audience certainly sounds like a good group to be marketing to. According to publisher Hearst Corp., 22 million of the 25 million readers are women. The median annual household income of readers is $48,347. Her family and her home are her top priorities, and she considers herself to be technically savvy. She is extremely loyal to brands and seeks out the best products she can afford.\nIf you need more convincing, consider this: According to Fara Warner, a business journalist and author of the book \"The Power of the Purse,\" women make 80% of all buying decisions and control $7 trillion in purchasing power.\nI know that many contractors out there recognize the purchasing power of women and have marketing campaigns, training, and other efforts to reach them. Good going. But if you aren't one of them, why not? Particularly, IAQ marketing efforts should target women. No wife or mother wants her family members to breathe air that could make them sick. Also, women are becoming more concerned about environmental degradation, research shows, and are interested in energy-saving, environmentally safe products.\nTake a cue from Weil-McLain. Position your products and services so that they respond to the concerns of women, and stress quality and ease of maintenance. If you have a product that will solve a problem and\/or make their homes more comfortable and healthy, they want to know about it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By Brad O'Brien on 02\/01\/2017 in \"Facebook advertising Blogs\"\nIf you've noticed some new custom audience options in your Facebook account lately, it's because Facebook has opened up new targeting that allows advertisers to retarget users based on engagements with the advertisers' Facebook pages. We'll explore that, plus a couple of other recent Custom Audience enhancements, in this post.\nFirst, the new Custom Audience targeting feature includes options to retarget based on just about any page-related action: visits, engagements, messages, saved pages\/posts, etc.\nYou can create video engagement audiences based on a percentage or length of time a user watched your video ads (and even specify which videos). With videos becoming more and more influential, this is an important feature to have at your disposal.\nCustom combinations can be powerful; they allow for multiple layers of segmentation based on web and app actions.\nAs always, advanced segmentations will help you get the most out of these features. Segment, test, adjust, and optimize\u2026and have fun with these new controls.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Abstract: Augustus Turnbull was the Assistant Press Secretary to Governor Carl Sanders during the last year of his term in office (1966-1967). The materials in this collection, however, are from Sanders entire term, 1963-1967, and document his interest in efficiently maintaining state government departments as well as his public addresses and press relations.\nCarl E. Sanders was born in Augusta, Georgia on May 15, 1925. He attended the Academy of Richmond County and received the alternate appointment to West Point. He found himself at the University of Georgia, however, where he accepted a football scholarship. In 1943, before he could finish his college education, he enlisted in the Air Force and trained as a B-17 pilot. After World War II he returned to the University, finishing three years of law school in two years and passing the bar in 1947. That same year he married Betty Foy of Statesboro, Georgia.\nSanders entered private practice in Augusta and eventually started the law firm of Sanders, Thurmond, Hester and Jolles. In 1954, he made a successful bid for the Georgia House of Representatives. Two years later, in 1956, he won a seat in the Georgia Senate. At the time, the seat rotated between Richmond, Glascock and Jefferson counties. Sanders was subsequently elected to the same seat by both Jefferson and Glascock counties due to his overwhelming popularity, becoming the only man to serve three consecutive terms in a multi-county district. In 1959, Governor Vandiver named Sanders as Senate Floor Leader. He went on to serve as president pro tempore of the Senate from 1960-1962.\nSanders took the next step in his political career by running for governor in 1962 against Marvin Griffin. Shortly after his announcement, federal courts ruled that Georgia's county unit system was unconstitutional. The state would elect its officials by popular vote, giving the urban candidate, Sanders, a greater chance at victory. As a moderate concerning racial issues, he faced an outspoken opponent of integration in former-Governor Griffin. While agreeing that Georgia should keep its tradition of segregation, Sanders believed it was imperative that the state avoid violence and obey the laws of the country. Unlike Griffin, his campaign issues were not built around race. Instead, Sanders focused on the elimination of corruption in state government and pushed for overall progress for the state. He also wanted to improve education and bring industry to Georgia. With his positive campaign, Sanders emerged victorious, making him the youngest governor in the country at the time. He was 37.\nUpon his election, Sanders set about following through on his campaign promises. In order to revitalize Georgia's educational system, the new governor created the Governor's Commission to Improve Education in 1963. Based on the commission's findings and with the General Assembly's approval, Sanders improved education a great deal. His program raised teacher salaries, added new teachers to the system, and reorganized the Department of Education. It also built new school buildings, established more junior colleges and vocational schools and created the Governor's Honors Program. Sanders also appointed the Governor's Commission for Efficiency and Improvement which helped to reform many government agencies such as the Department of Agriculture and the Highway Department, as well as the prison and state merit systems.\nIn May, 1963, Sanders sought to draft a new state constitution. With the appointment of the Constitutional Revision Commission, he hoped to condense and clarify Georgia's Constitution. While the General Assembly approved the new constitution, it was never put on the general ballot. The momentum was lost when federal courts declared that the General Assembly was incorrectly apportioned and the constitution was, therefore, invalid. Sanders would later play an integral part in the passage of the reapportionment of congressional districts, but would never succeed in revising the constitution.\nSanders could not succeed himself as governor and, therefore, retired back into private life. Instead of returning to Augusta, he and his family remained in Atlanta where he started a new law firm. On April 25, 1970, Sanders reemerged to announce for what would be his last attempt at public office. His campaign for governor, however, ended in defeat, with Jimmy Carter taking office. Sanders never ran for public office again but worked for many other Democratic candidates such as Zell Miller, Andrew Young, and Sam Nunn. He also went on to serve as Finance Chairman for the Democratic Party during George Busbee's term as governor.\nSanders managed the law firm, today known as Troutman Sanders LLP, for 25 years, and afterwards continued to serve the firm as partner and Chairman Emeritus. Sanders died November 16, 2014.\nThe Augustus Turnbull Collection of Carl Sanders Speech\/Press Files consists of manuscripts, reports, and news clippings from Sanders's term as governor of Georgia. The years represented are 1963-1967, and the activities documented are political and executive.\nThe collection is divided into five series: I. General, II. State Agencies and Organizations, III. Speeches, and IV. Press Releases. The organization of the collection is chronological; the second series (State Agencies and Organizations) is organized alphabetically by department and then chronologically within each folder.\nAugustus Turnbull Collection of Carl Sanders Speech\/Press Records, Richard B. Russell Library for Political Research and Studies, University of Georgia Libraries, Athens, Georgia, 30602-1641.\nClippings and thermofax papers have been copied onto bond paper for protection of content. Oversized materials and photographs have been separated for preservation.\nUpgraded to Encoded Archival Description (EAD) Version 2002, 2007.\nGeorgia. Office of the Governor.\nScope and Contents note: Christmas cards depicting Goldwater, the old Governor's mansion in Atlanta, Georgia.\nScope and Contents note: Consists of reproductions of the liberal magazine's annual congressional scorecards from 1948 to 1965.\nScope and Contents note: Includes speeches, and background information on various Democratic Party platforms, 1960-1964. Arrangement is chronological.\nScope and Content: Sanders continued efforts initiated by previous Governor Ernest Vandiver to eliminate corruption and streamline government. These files consist of reports and correspondence which document the activities and progress of various state departments, boards, commissions, and organizations. Organization is alphabetically by agency and chronologically within each folder.\nScope and Content: The Press Relations and Media Activities series consists of those files which document the activity of communicating Sanders' view on various issues through speeches, press releases, and reports. As Sanders' Assistant Press Secretary, Augustus Turnbull accumulated these files, many of which concern Sanders' efforts to make the state government and its departments more efficient.\nScope and Contents note: These are the final versions of speeches that Sanders delivered to various government assemblies, civic organizations, business associations, church groups and school organizations in Georgia and elsewhere. At the beginning of each month's folder is a list detailing that month's speaking engagements. Arrangement is chronological.\nScope and Contents note: These are files of press documents originating from Governor Sander's office. These press releases consist chiefly of announcements of appointments, reports of the progress of state and local departments and committees, and details of Sanders' many speaking engagement. Of note to researchers are the Sanders' 1963 efforts to draft a new state constitution as well as his term-long interest in improving education in Georgia. Arrangement is chronological.\nScope and Contents note: Includes the script to \"Georgia on the Move\" narrated by Sanders and produced by the Georgia Forestry Commission as well as the \"Challenge for Change\" script produced by the Department of Revenue.\nScope and Contents note: The 1966 Report to the People was an update on government improvement and efficiency during the Sanders administration. Included in this subseries are several drafts of the report.\nScope and Contents note: Contains an 8\"x10\" black and white photograph of Carl E. Sanders and John McDonough in 1965.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is an AMC online account?\nIf you have not created an AMC online account, you need to do so to view the information and use the available options on your candidate account \u2013 use this link.\nIf you have created an AMC account prior to 1 October 2015, you need to create a new account to view the information and use the available options on your candidate account link.\nIf you have created an AMC online account after 1 October 2015, you can log in to your account by using this link to view the information and use the available options on your candidate account.\nnominate an Australian specialist medical college to have access to your qualification and verification information, and to upload the specialist assessment outcome undertaken by the college.\nWhat is an AMC portfolio?\nFrom 1 October 2015, the paper applications system has been replaced with an online applications methods, referred to as the AMC portfolio. All new applicants without an AMC candidate number must establish an AMC portfolio . If you are a returning candidate with an AMC candidate number, please read the information available on the AMC portfolio page before applying to make sure it is required.\nIf you are an returning candidate with an AMC candidate number, and need to submit qualification(s) for EPIC verification that has not been EICS verified via the AMC, you will have to establish an AMC portfolio and submit the qualification(s) for EPIC verification.\nWhat is an EPIC \/ EICS verification?\nThe verification of medical qualifications is undertaken by Educational Commission for Foreign Medical Graduates (ECFMG). From 1 October, qualifications are verified through the ECFMG's Electronic Portfolio of International Credentials (EPIC) process. Prior to 1 October 2015, the AMC processed qualifications through the ECFMG International Credentials Services (EICS) process.\nWhy is EPIC\/EICS verification required?\nDoes the EPIC\/EICS verification of my qualification(s) expire?\nNo. Once it is completed it will appear on your AMC account.\nHow do I apply for EPIC verification?\nYou need to create an EPIC account directly with ECFMG. Please refer to the ECFMG website for further information about the EPIC verifications process.\nWhy hasn't the AMC notified me of my EPIC verification progress?\nThe EPIC process is between yourself and ECFMG. Refer to your EPIC account for progress updates, or contact ECFMG directly if you have any further enquiries or concerns about your EPIC request. The AMC is not involved in the EPIC verifications processes.\nDoes the AMC issue an EPIC certificate after the verification was completed?\nNo. However, you can apply for an EPIC report via your EPIC account.\nWill the AMC receive an EPIC outsource notification of my qualification(s)?\nAs you upload each qualification to EPIC verification, request that a report be sent to AMC automatically once the qualification has been verified. If you do this, the AMC will also receive an outsource notification when your qualification is sent for verification.\nWill the AMC automatically receive the EPIC report of my qualification once it has been verified?\nIf you requested a EPIC report be sent to the AMC when you uploaded the qualification to EPIC, a report will be sent automatically once your credential is verified. If you did not request a report to be sent to the AMC when you uploaded the qualification, you will need to request for your report to be made available to the AMC via your EPIC account.\nWhat about the EICS verifications of my qualification(s) previously completed via the AMC?\nThe verification of those qualifications, if undertaken via the AMC, is still valid and will appear on your AMC account.\nIf the EICS verification was undertaken by the AMC and still outstanding, you need to resubmit the qualification to ECFMG for EPIC verification. You will have to establish an AMC portfolio as well (if not done so already).\nDo note if the EICS has been undertaken not through the AMC, you have to create an EPIC account and provide your EICS number on the account, and submit your qualification for EPIC verification.\nCan I obtain a copy\/reprint of the EICS verification of my qualification(s)?\nYes. Refer to the Request for reprint of EICS certificate form to be completed, email it to the AMC for processing. The reprint will be mailed by using normal mailing services, however, if you prefer courier collection and delivery, you need to make your own arrangements and provide a completed waybill with your request form to the AMC. If you don't have a courier account, a pre-paid courier service is available.\nWhat is a basic medical qualification (primary qualification \/ final medical diploma)?\nA basic medical qualification (primary qualification \/ final medical diploma) is the formal qualification awarded after completion of a course of study in medicine (clinical), surgery and associated disciplines that enables the person holding the qualification to work as a medical practitioner in Australia.\nGeneral (non-specialist) medical registration in Australia requires a practitioner to complete a comprehensive program of training and examination.\na 12-month internship in approved\/accredited posts.\nThis qualification is referred to as a Primary qualification with the AMC, however ECFMG refer to this as the Final Medical Diploma.\nWhat if my medical qualification (primary qualification\/final medical diploma) has not been awarded yet?\nThe Medical Board of Australia (AHPRA) requires for the final \"hang on the wall\" diploma (for e.g. MBBS, etc). to be verified for medical registration purposes (refer to step 1 below). This qualification is referred to as a Primary Qualification with the AMC, however ECFMG refer to this as the Final Medical Diploma.\nPlease contact ECFMG directly to enquire what alternative document(s) can be accepted.\nCan I apply if I have Doctor of Osteopathic Medicine (DO USA) primary qualification?\nThe Medical Board of Australia (the Board) has agreed to accept the degree Doctor of Osteopathic Medicine (DO USA) a primary medical qualification for the purposes of medical registration provided that the DO USA was awarded by a medical school which has been accredited by the Commission on Osteopathic College Accreditation of the American Osteopathic Association and recognised by both the Australian Medical Council and the World Directory of Medical Schools (WDoMS) if listed on the AMC website as an eligible medical school. For more information, refer to the Medical Board of Australia's website.\nWhat is a specialist medical qualification?\nContact the relevant specialist medical college for information about the specialist medical qualification requirements. All colleges are listed on the Medical Board of Australia's website.\nWhich specialist qualification(s) should be added to my AMC account and submitted for EPIC verification?\nContact the relevant specialist medical college for information about the specialist medical qualification requirements. All the colleges are listed on the Medical Board of Australia's website.\nWhy do I still require verification of my qualification if I'm already registered with the Australian Health Practitioner Regulation Agency (AHPRA)?\nPlease contact the Medical Board of Australia to enquire if verification of your medical qualification\/s is still required.\nHow long does the AMC take to process my portfolio application?\nOnce you have submitted your portfolio, please allow approx. 1-3 working days for the AMC to process, but it may take longer if there are unforeseen circumstances. If you are a new applicant, your AMC candidate number will be confirmed in an email once your application has been processed, with further instructions.\nAccess your AMC account for options available on the landing page to provide you with information and\/or status updates.\nDid the AMC receive my email?\nYou will receive an automated response to confirm that the AMC has received your email. Please allow 1-3 working days for a response, but it may take longer if there are unforeseen circumstances. If you do not receive an email, your email address may not be valid and you should contact the AMC (Monday to Friday from 10 am to 4 pm AEST).\nCan I change my address details over the telephone or by email?\nYes. The AMC will change your address details over the phone, but if the AMC representative is unable to hear you clearly on the phone, you might be requested to complete a Change of address form to ensure that the correct information is recorded. Change of address details cannot be amended via email unless a completed Change of Address form has been included.\nCan the AMC answer questions about registration to practise medicine?\nNo. You should contact the Medical Board of Australia for information about registration.\nCan the AMC answer questions about internships?\nNo. You should contact the Medical Board of Australia for information.\nCan the AMC answer questions about Competent Authority eligibility requirements and application?\nCan the AMC answer questions about Specialist pathway (specialist recognition \/ training \/ area of need \/ short-term training)?\nNo. You should contact the relevant specialist medical college for information. All the colleges are listed on the Medical Board of Australia's website.\nHow do I apply for a visa for entry to Australia?\nThe AMC does not handle visas. If you are outside Australia, you should contact the Department of Immigration and Border Protection.\nHow can I obtain employment in Australia?\nThe AMC does not process applications for employment in Australia. Information about employment is available from Australian state and territory health authorities, including Doctor Connect.\nHow can I obtain a Medicare provider number?\nThe AMC does not have information about Medicare provider numbers. Information about Medicare provider numbers is available from the Australian Government Department of Health.\nHow can I authorise an agent (third party nominee) to act on my behalf?\nUnder the Privacy Act 1988 (Cth), the AMC is generally not permitted to disclose personal information about a candidate to a third party (e.g. agent, employer) without the candidate's consent. A candidate may authorise a third party to communicate or act on the candidate's behalf by completing the Authority to Receive Information Concerning an AMC Candidate Applicant form.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The site of Tarbat Old Church features an important medieval building, and is also likely to contain evidence for a Pictish ecclesiastical centre. Since the project includes the development of the church site as a Museum, the opportunity for archaeological investigation should be taken.\nto inspect the character of any surviving traces of the Dark Age settlement which preceded it.\nSmall-scale interventions are planned at the west end, at the east end and in the crypt [Fig. 18]. These excavations are intended to co-exist with the programme of development.\nRefurbishment: The roof is to be repaired and the exterior harling is to be replaced. The internal timber supports, pews and panels are to remain exactly as they are (according to the drawings). Since this work is to occasion no disturbance, it has no archaeological implications.\nRemoval of the wooden panelling between the nave and the N aisle. This should allow the exposure of the wall fabric of both nave and N aisle; it can then be observed if they are different and this difference recorded before the walls are covered up again.\nInsertion of services in or on wall fabric behind wooden panels. The fabric concerned would require prior recording.\nInstallation of wheelchair toilet at W end. This implies foundations and drains which would need to be archaeologically dug. The assumption is that the provision of a toilet will need (1) a hole 2.5 x 2.5 m and (2) a trench through the burial ground [below].\nTrench(es) through the churchyard. The Centre is expected to be connected to mains water, electricity and sewage. This will require one or more trenches to be dug through the churchyard.\nIn the authors' view, the development proposals do not represent an abuse of the monument, but are likely to afford added value and added protection. Furthermore such works as do impact on the archaeological assets provide opportunities which are rewarding to research.\nSince there is to be a major study of the defended settlement in which the church stands, and since that settlement is thought to have an early monastic phase signalled by finds in the churchyard, it would be very sensible to take advantage of the opportunity afforded by the development of the church site to examine the use of the site before the church was built.\nSince the church is to become a visitor centre, concerned with presenting the past, it would be prudent to include in that presentation an account of the history of the building itself.\nConstraints: Whereas the most comprehensive approach to the inquiry would be the total recording of the wall fabric and the total excavation of the interior, there are several constraints on such an approach.\nOn a unique site of this kind, no more should be excavated (and therefore destroyed) than can be understood, for ethical reasons.\nLocal support for the project as a whole is thought to depend on an early start to the refurbishment programme. Excavation and recording should not be on such a scale to demand excessive delays.\nThe use of the crypt, if it is demonstrably medieval or earlier than the present nave walls.\nA sample of medieval burials.\nPresently excluded from the research targets, given the constraints detailed above, is a major excavation in the nave. If no earlier strata are contacted, such an excavation would not be justified. If early strata are contacted, a major excavation would be desirable but should be postponed until a context has been provided for it through the settlement excavation programme.\nThe investigation should not delay the refurbishment programme.\nAny excavations should as far as possible form part of the exhibit.\nIntervention A: Recording of the interior wall fabric, including inside the crypt.\nAim: To ascertain if the church was medieval in origin.\nMethod: The wooden panelling would be totally removed inside the nave and west end. Any wall-plaster encountered should be scrutinised for layering, frescos and other features and recorded. Rotten plaster should then be removed. The wall fabric should be then surveyed and recorded by rectified photography. Any features present in the wall fabric should be examined non-destructively and recorded. Some or all of these features, particularly if evidently medieval, should be left exposed to view if and when the panelling is reinstated. The walling at the west end would be repointed and left exposed, with a light fitted to illuminate the carved stone in the arch of the vault at the W end.\nTiming: During refurbishment, and probably not during the tourist season. The same occasion can be used to check that the walls are structurally sound.\nOutput: Drawn elevations of all accessible wall-fabric, with detail of features.\nIntervention B: Excavation of a small area at the E end.\nMethod: An area of the wooden floor 6m x 6m will need to be removed [Fig. 18]. The area extends E from a line joining the midpoint of the central porch to the midpoint of the opening to the N aisle. Its E limit is over the crypt. The S limit is the S wall of the nave and the N limit just inside the N aisle. This area will be cleaned, planned and photographed down to the first burial horizon. Graves will be defined and cleaned in order to provide 'free sections' through any intact stratigraphy, which will then be recorded. Further excavation would be undertaken sufficient to (a) expose the relationship between the crypt, the crypt access steps and the nave walls; and (b) expose the relationship between the nave and the N aisle in the E angle of their bond. Spoil would be removed by chute through the opening in the N wall of the nave. Following excavation, the area would be consolidated, except that the access steps would be restored to their original width. It might be suggested that any crucial relationships between the structural elements could also be preserved as part of the display.\nTiming: Before the installation of the design. Provisionally scheduled for July 1996.\nPublic access: A rail could be constructed across the church (N-S), allowing visitors to watch work in progress.\nOutput: Full documentation, including a three-dimensional model of the structural sequence, a composite section along grave edges, a N-S section across the church and conserved masonry where it can be exposed to demonstrate the sequence.\nIntervention C: An excavation 6m x 2m within the W end. The excavation would be on the N side of the accessible area. It will expose the foundations of the W wall(s) and the N wall.\nTo confirm the relationship between the N wall of the nave and the west wall complex.\nTo contact strata belonging to pre-church or pre-medieval phases.\nTo determine the depth of deposit.\nMethod: Advice on stability and the removal of redundant timbers would be taken from an engineer. The paving slabs would be recorded and removed and stacked for reinstatement. The surface would be defined by horizon until grave cuts are visible. These would be defined and excavated as far as possible. The trench would be shored and descend to the subsoil. It is assumed that this trench will contain all the area to be damaged by the installation of the toilet.\nTiming: Could probably be done at the same time as Int.B, ie during the 1996 tourist season.\nPublic access: Public viewing through railing or windows in panelling beside S entrance passage.\nOutput: Full documentation, including a section E-W to natural subsoil, two sections N-S to subsoil, showing the foundation trench for the N wall, and a three-dimensional model of the relationship between the walls at foundation level.\nIntervention D: The excavation of the service trench .\nAim: To record the strata to be destroyed by the service trench.\nMethod: The trench or trenches would be dug by hand by archaeologists who would record the strata as they went.\nTiming: To be determined by the development plan; preferably at a time of least public attention.\nTime taken: about 1 week for every 10 metres.\nOutput: Section along trench edge; to subsoil if possible.\nIntervention E: Watching Brief. Archaeological surveillance would normally be required during refurbishment.\nIntervention F: Excavation of the crypt floor. [The recording of the crypt wall fabric would be included in Int.A].\nAim: To investigate the sequence of use.\nMethod: This excavation would ONLY take place if the crypt is found by Int.B to be medieval or earlier. In this case, one half (N) of the strata would be excavated, along an E-W axis.\nTiming: Not before completion of the other works; 1997 at earliest.\nPublic access: Public viewing throughout. Can be phased in as an exhibit after the church has been fully opened to the public.\nInt. A: Recording of internal fabric.\nInt. B-C: Excavations at E end and W end.\nDuring tourist season, before refurbishment begins or after it is finished; before or during installation of display (but not after).\nInt. D: Service trench. Not during tourist season, before installation of display.\nInt. E: Watching brief during all development work.\nAnalysis and publication of this work.\nInt. F: Excavation of Crypt.\nAfter the other works have been completed, and if justifiable on research grounds.\nOct 96: Excavation of service trench (Int. D).\nA T-shaped transect as shown on Fig. 19 meets most of the specifications of the research design. It includes a sizeable part of the easily mappable area next to the fortifications, likely to contain evidence for production and consumption; and it runs north between the churchyard and the spring system, thus connecting the likely ecological and ritual nodes. It extends up to the crest overlooking the beach (Sector 3). If this contains no evidence for maritime exchange, then a further extension on the down-slope towards the present beach may be desirable.\nThe greater part of this area (i.e. Sector 1) will be susceptible to the 'strip and map' technique which is essentially non-destructive. Normally only surface finds will be recovered. For features chosen for excavation, recovery level of D or E will normally apply (see Appendix 1).\nIn Sectors 2 and 3, deeper stratification is anticipated, up to 1.2m, of which 80cm is scrambled ploughsoil. At least two phases of excavation will be required in this sector, the first defining the medieval occupation beneath the ploughsoil and the second examining the Early Historic phases. In general the first of these operations will act as a high-level evaluation for the second.\nIn Sectors 1 and 2, damp or even waterlogged strata are anticipated in the valley bottom. Special equipment may be required, but not on a scale appropriate to deep urban anaerobic deposits.\nSector 4 (church) is investigated in 1996.\nGeophysical survey in Zones D, E, H, J and K.\nTrial excavations (as opportunities permit) in the residential area of Zone J.\nReviews of documents and local knowledge, particularly of the early 20th century and earlier.\nIt is hoped that the collection of this information will allow an appreciation of the archaeological potential of the whole of the Portmahomack area, and lead us to any sites where significant intervention is possible.\nResources: It is intended that, apart from incidental expenses to be borne by the rest of the archaeological programme, the bulk of the work will be carried out by students and friends of the project. No specific costings are therefore included.\nGeological mapping to determine the changing configuration of the coastline and inland water.\nA Research Programme, including major and minor excavation projects, study of sculpture and other materials. Many of these projects (like the small-scale evaluations) could be achieved by student contributions at BA\/BSc, MA\/MSc and DPhil level.\nA Management Programme, including recommended protection for sites, recommended surveys for management purposes and presentation\/education proposals.\nThe initial proposal is to carry out stages 1 to 3 of the survey in parallel with the investigation at the Tarbat site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A110-10 VOLUNTARY NOISE ABATEMENT PROCEDURES IN EFFECT. FOR NOISE ABATEMENT PROCEDURES CALL 480-644-6647.\nA110-11 RISING TERRAIN NE OF ARPT.\nA110-12 BIRDS INVOF ARPT. COYOTES AND OTHER WILDLIFE OCNL CROSSING RWYS AND TWYS.\nA110-13 RWYS 04R & 04L ARE THE PREFERRED RWYS WHEN WIND CONDS ARE LESS THAN 5 KNOTS.\nA110-14 REPETITIVE TRNG OPNS USE RWY 04L\/22R WHENEVER POSSIBLE. AVOID REPETITIVE TRNG OPNS 2200-0530.\nA110-15 RWY 04L\/22R CLSD WHEN ATCT CLSD.\nA110-17 BLAST PADS INSTALLED ON RWY 4L AND 22R. BLAST PADS DIMENSIONS, 150 FT X 95 FT.\nA110-2 2786 FT MSL UNLGTD MTN 5 MI EAST.\nA110-4 TURBULENT CONDS MAY BE ENCOUNTERED BLO 500 FT AGL NEAR GND TEST SITE 1.5 MILES NORTH OF ARPT. DO NOT MISTAKE BOEING HELIPORT 1 MILE NORTH FOR A RY.\nA110-8 TERMINAL APRON AREA LTD TO 45,000 LBS GWT SINGLE WHEEL OR 75,000 LBS GWT DUAL WHEEL BY AIRPORT DIRECTOR.\nA110-9 EXTENSIVE FIXED WING AND ROTORCRAFT FLIGHT TRAINING ACTIVITY.\nA17 FOR SVC CTC FBO ON 122.950 (480-832-2582).\nA43-22L PAPI DOES NOT PROVIDE OBSTRUCTION CLEARANCE BEYOND 4.0 NM FROM THRESHOLD.\nA81-APT WHEN ATCT CLSD MIRL RY 04R\/22L AND TWY LGTS PRESET LOW INTST TO INCREASE INTST ACTVT - 124.6. MIRL RY 04L\/22R UNAVBL WHEN ATCT CLSD. FOR REIL RYS 04R & 22L CTC ATCT.\nE147 TPA FOR LIGHT ACFT 1006 FT AGL; TPA FOR LARGE & TURBINE-POWERED ACFT 1506 FT AGL; TPA 506 FT AGL FOR ROTORCRAFT.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Windows Clean Up \/ Repair & Maintenance.\nBSOD \u2013 Blue screen of death.\nInstalling updates and service packs.\nInstalling MS Office and configuring Outlook E Mail Account.\nWired \/ Wireless Network Setup & Router configuration.\nWired and wireless printers configuration.\nNAS \u2013 Network Attached Storage.\nFTP \u2013 servers and clients.\nOperating system and software troubleshooting, installation and upgrades.\nReinforcement of information and remote setup.\nWireless Network Setup & Router configuration.\nOperating system installation and upgrades.\nStart up issue Desktop service.\nWe are confident with our service levels and customer satisfaction. We guarantee that if we do not fix your problem we will refund your money immediately.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"we are providing total solution of your data. We are providing our service for electronic data management and e-digitzliation for your data so you no need to worry for your data.\nWe are leading outsourcing consulting company providing total outsourcing solution in bpo and call center.\nIwAay consultant is leading outsourcing consulting & management company which can provide you total outsourcing solution and freedom from your outsourcing needs.\nWe are one the most illustrious multifaceted organizations having business interest in the area of management consultancy, corporate advisory and it enabled services, specially in the area of business process outsourcing and call-center.\nWe commenced operations in 2002. We have core team for outsourcing any kind of process in the field of bpo, callcenter & kpo. Our mission is to provide the infrastructure, knowledge and operational resources that allow our clients to optimally process, utilize and leverage their data.\nOur client can deliver high quality business consulting and disciplined technology implementation at an extremely competitive price. We are dedicated to helping our clients become stronger competitors, and helping our consultants become strong leaders, capable of managing a modern global business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now is the time to get into Data Centers. They are everywhere and where everyone hosts their servers. If you or your team is involved in data center technology, you will want to take these courses. If you are looking to round out your skills and make yourself more marketable, you will want to take these courses. The Data Center series of courses is a great way to quickly learn the ins and outs of data centers. It is geared for everyone from the data center novice to the most experienced IT and facilities technicians and managers.\nMaster the Data Center Today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..d8eb793518572cee65a49262d127cc4ba7e43193
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@@ -0,0 +1,5 @@
+{"text":"DUBLIN, Ireland; Nov. 5, 2015 \u2013 Accenture (NYSE: ACN) is accelerating its research and development agenda in artificial intelligence across its global network of Accenture Technology Labs, by opening the newest lab in Dublin and committing a significant portion of its global R&D capabilities, to help clients across industries integrate intelligent systems to improve operations and create new growth opportunities.\nThe expanded R&D agenda will play a significant part in fueling Accenture's approach to artificial intelligence across the company and will draw the best technology solutions from a broad ecosystem. The new Technology Lab is central to Accenture's commitment to Dublin as a Centre for Innovation, established to capitalize on Ireland's role as a technology hub in Europe and to access highly skilled talent in science, technology and engineering in the region.\nAs part of today's announcement, Accenture has also established the Accenture Technology Labs University Grant on Artificial Intelligence; awarding the inaugural grant to an academic research team at the Insight Centre for Data Analytics at University College Dublin. The research team will explore the interface between humans and machines, using cognitive analysis to better understand how both can collaborate and interact effectively. In order to advance the development of emerging technologies Accenture has long established strong relationships with many leading academic institutions; including the Accenture and MIT Alliance in Business Analytics which explores artificial intelligence technologies.\nArtificial Intelligence \u2013 the broad field of new technologies that enable software to sense, comprehend, act and learn \u2013 is disrupting how businesses operate and compete while also accelerating the delivery of business outcomes in the \"As-a-Service\" economy. Analyst firm IDC predicts that the worldwide content analytics, discovery and cognitive systems software market will grow from US$4.5 billion in 2014 to US$9.2 billion in 20191, with others citing these systems as catalyst to have a US$5 trillion \u2013 US$7 trillion potential economic impact by 2025.\nDesign and scale artificial intelligence capabilities for Accenture Consulting around the world across multiple industries, including healthcare, public safety and financial services.\nThe global network of Accenture Technology Labs includes locations in: San Jose, California and Arlington, Virginia, in the United States; Sophia Antipolis, France; Beijing, China; Bangalore, India; and Dublin, Ireland. The Tech Labs are part of Accenture's broader network of innovation centers\u2014more than 80 around the world \u2013 which focus on helping clients develop innovative solutions based on new and emerging technologies.\nZopper opens pre-booking for Huawei Nexus 6P on its app platform!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"by Sylas Wilcox on Feb. 28, 2013, 1:07 p.m.\nby Sylas Wilcox on Feb. 24, 2013, 9:08 p.m.\nby Sylas Wilcox on March 18, 2013, 3:50 p.m.\nFor it is you, I love.\nby Sylas Wilcox on Feb. 24, 2014, 11:58 p.m.\nby Sylas Wilcox on Jan. 1, 2014, 3:52 p.m.\nby Sylas Wilcox on July 21, 2013, 10:30 p.m.\nby Sylas Wilcox on July 11, 2013, 11:56 p.m.\nby Sylas Wilcox on July 11, 2013, 12:50 a.m.\nby Sylas Wilcox on June 22, 2013, 9:39 p.m.\nby Sylas Wilcox on June 21, 2013, 10:14 p.m.\nCorneille Noire - Sylas Wilcox?\nby Sylas Wilcox on May 11, 2013, 7:11 p.m.\nby Sylas Wilcox on May 10, 2013, 7:38 p.m.\nby Sylas Wilcox on March 27, 2013, 6:19 p.m.\nby Sylas Wilcox on March 13, 2013, 12:07 a.m.\nby Sylas Wilcox on Feb. 25, 2013, 10:38 p.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A group of species that have\/had evolved from a common ancestor, including all the species alive and extinct.\nConstruction of cladograms and identification of clades.\nA method to calculate the time of divergence of different species absed on the number of differences in their DNA base sequences and an estimated rate of mutations.\nStructures which appear to be similar and have the same functions, but are foundamentally (structurally) different. They are results of convergent evolution and are developed from different ancestors to adapt to similar conditions.\nStructures that appear to be superficial and have different functions but are structurally similar. They are results of adaptive radiation (i.e. the structures developed from a common ancestor of different individuals become progressively different as the individuals live and adapt to different environments). Ex) pentadactyle limbs in birds, reptiles, amphibians, and mammals.\nForm and structure: often used in traditional classification but is highly subjective.\nTree diagrams illustrating the sequence of divergence of different species from a common ancestor IN A CLADE.\nMammals adapted to climb trees.\nCumulative changes in the heritable characteristics of a POPULATION.\nThe process of REPEATEDLY selecting and breeding organism with traits that are the most suited to human uses.\nReduced structures that serve no function.\nDevelopment of a new species from existing species.\nDevelopment of a new species from existing species relying on factors including preference of different foods, mates, and biochemical changes. To investigate sympatric speciation, individuals which may not breed in the wild will do so in labs (e.g. stickleback and cichlid fish studies).\nDevelopment of a new species from existing species due to physical barriers which separate populations.\nDark varieties of typically light-coloured animals and insects.\n1. Peppered moths with pale-coloured wings are camoufalged well against trees with pale-coloured lichens.\n2. Industrial pollution causes blackening of tree branches (sulphure dioxide kills lichens, soot from coal burning blakens tree branches).\n3. Moths with pale-coloured wings are no longer camouflaged against dark tree branches, and become targets of predators (e.g. birds) during daytime.\n4. Instead, the melanistic moths are camouflaged well and are less likely to be hunted.\n5. Over time, the melanistic moths replace the moths with pale-coloured wings.\nNatural selection refers to the preservation of favourable variations and the rejection of injurious variations among organisms of a population in their struggle of existence.\nCharacteristics are ones that develop during a lifetime and are inheritable (since they are not controlled by genes). Examples include: scars and hairstyles.\nNatural selection refers to ... (definition). There is often a change in the environment which exerts selection pressure upon the population. The pressure increases the frequency of the characteristics that make individuals better adapted and decreases the frequencies of other characteristics (only true if these characteristics are inheritable). Those being selected for natural selection have a selective and reproductive advantage, which means these fittest individuals will survive and breed (survival of the fittest). Over time, the entire population will be replaced by individuals with the favoured trait.\n1. Mutation: original source of variations, where new alleles are produced (enlarges the gene pool) due to viral infections and\/or errors in DNA replication. It is the only source of variation for organisms that reproduce asexually.\n2. Meiosis: alleles in a diploid cell are rearranged to form new combinations by crossing over and random assortment.\n3. Sexual reproduction: the combination of gametes from different individuals. This allows mutations formed in different individuals to be brought together.\n1. Offspring tend to resemble their parents.\n2. All offspring have the potential to reproduce.\n3. More offspring are reproduced than the environment can support.\n4. Despite 3, the population sizes often remain constant.\n1. Most offspring do not survive to breed.\n2. Only the fittest ones survive.\n3. Overtime and under the right circumstances, mutations give rise to new species.\nExplain how the development of antibiotic resistance in bacteria demonstrate natural selection can lead to evolution.\n1. \u3010Definition of antibiotics\u3011: Antibiotics is a type of medicine used to control bacterial infections by blocking the metabolism of prokaryotic cells.\n2. The use of antibiotics exerts a selection pressure among bacteria in a population.\n3.2)received from bacteria of other populations by exchanging genetic information by pili.\n4. These variations are inheritable.\n5. Those possessing antibiotic resistance genes have selective and reproductive advantages and so are selected for, those without are selected against.\n6. Overtime, the desirable trait (one makes bacteria adapted i.e. resistance) is preserved. The entire population is replaced by bacteria with resistance.\n7. The development of antibiotic resistance shows how evolution can take place by the survival of the fittest.\nExplain how the binomial nomenclature is used.\n1. Universal name that can be used by scientists from different countries instead of many varied local names.\n2. Promotes international cooperation and collaboration.\n3. Informative and follows natural classification: gives information abou the genus and species of the organism.\nThe classification of organisms using hierarchies of taxa.\n1. Analysis of base sequences of 3 chloroplast genes.\n2. More than half of the species were being transferred from the figwort family to: plantain family, lopseed.\n3. New families were created: pocketbook flower, Linderniaceae.\n4. Two families were merged with the figwort family: butterfly bush (buddleja), myoporum.\n5. The classification was based on morphology: although structures appear to be similar, they were only results of convergent evolution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ballwin is located several miles southwest of the I-64\/US-40\/I-270 intersection in east central Missouri, approx. 20 miles west of St. Louis.\nFrom I-64 Eastbound: I-64 east to Rt-141 south (exit 22). Turn right (south) onto Rt-141. Turn left (east) onto Clayton Rd. Turn right (south) onto Weldman Rd. Turn left into Queeny County Park and follow the signs to Greensfelder Recreation Complex.\nFrom I-64 Westbound:I-64 west to S Mason Rd.(exit 24). Turn left(south) onto S Mason Rd. Turn right into the park and follow sings to the arena.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sorry for being so long gone!\nI couldn't follow up with this meals & exercises for two days but I will come back to the routine.\nI'll count today (22nd) as Day 5 and continue on with the dietary meals and exercises, so stay tuned!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgmen b/data_all_eng_slimpj/shuffled/split2/finalzzzbgmen
new file mode 100644
index 0000000000000000000000000000000000000000..9399563137c9024ad0a54798ba004972d27a0f86
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbgmen
@@ -0,0 +1,5 @@
+{"text":"The general idea here is to look at what the teeth are doing (top graph) as the fluoride is taken (bottom graph). Then look at how severe the fluorosis is (the little numbers added to along the right side of the top graph).\nLet's use the top tooth as an example. At birth (zero on the age scale) the first molar is starting the stage called apposition (the lighter part of the bar). And way off to the right you can see the fluorosis score, \".24 fading\". The higher this score, the more fluorosis there was on that tooth type. Note that the bottom graph is lined up, time wise, with the top graph.\nNow, scanning all the effects on all the tooth types, I can draw a generality. What it says to me is that the most sensitive stage of tooth development is probably the \"initiation\" stage. Here is how I come up with this.\nHere is a brief summary of the stages of tooth growth. The initiation stage is the beginning of the growth of a tooth. For our purposes we can think of initiation as beginning a cellular stage of growth, when cells form the outline of the eventual shape of the tooth.\nThe next stage of growth is apposition. This is the \"matrix\" stage, when the cells above start secreting matrix, which is sort of like cartilage. The matrix makes a complete tooth. In the next stage, calcification (not shown on the graph), the matrix tooth is mostly dissolved and replaced by calcium and phosphorus, which makes the tooth more bone like.\nIt does appear that the time near birth, when the main \"smile teeth\" are in the critical initiation stage, is the main time to be concerned with fluoride overdoses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Satellite Radio, iPod\/MP3 Input, Bluetooth, CD Player, Trailer Hitch. Ford Lariat with Green Gem Metallic exterior and Black interior features a V6 Cylinder Engine with 360 HP at 5500 RPM*.\nEQUIPMENT GROUP 501A MID Lariat Plus Package, short visors w\/covered illuminated mirrors, Reverse Sensing System, PowerFold Heated Side Mirrors w\/Memory, external mirror mounted turn signal indicator and driver's side self-dimming function w\/body-color skull caps, Universal Garage Door Opener, Rear View Camera, Remote Start System, Trailer Brake Controller, ENGINE: 3.5L V6 ECOBOOST 3.15 Axle Ratio, BLACK, LEATHER-TRIMMED BUCKET SEATS heated and cooled 10-way power (driver and passenger), memory (driver's side only), manual tilt\/telescoping steering column and 110v inverter, Floor Shifter, Flow Through Console, ELECTRONIC LOCKING W\/3.15 AXLE RATIO, TRANSMISSION: ELECTRONIC 6-SPEED AUTOMATIC: tow\/haul mode (STD). Serviced here, Non-Smoker vehicle, Originally bought here.\nKBB.com explains \"Whether it's an inexpensive basic work truck that can take a beating and still hold its value or a plush, horse trailer-hauling family vehicle you can be proud to take to any country club, Ford builds an F-150 for all occasions and occupations.\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Top 10 adventure tours in Galicia, Spain.\nThere hasn't been sufficient recent usage data to generate the top 10 adventure vacations in Galicia, Spain. Click to list all other Galicia adventure vacations instead.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Work Day at The Enclosed Ground!\nThe club has organised a work day at the ground on June 27th and if you have a free few hours, it would be great if you could pop down.\nEven the most ardent among us must admit that there are a few jobs that need doing to get The Enclosed Ground ready for the new season, so there's plenty you can help with. We need weeders, painters, joiners, sanders\u2026 everything! There's plenty to do so no matter how good you are with your DIY, we'll find a job for you. However, if you ARE a tradesman please get in touch as we'll almost definitely have something you can specifically help with.\nThe day starts at 10am and there will be a BBQ and FREE first pint for all helpers. Basic tools and materials to be provided.\nLet's get us ready for the new season together!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The murder of landscape architect Joanna Yeates in Bristol in December 2010.\nTreating this Case as a Murder Mystery and re-examining Everything!\nDid Dr Vincent Tabak's Conviction happen in May\/July 2011?\nIs It Possible To Work Out Time Of Death ?\nA Case built on Social media?\nWas This A Twin Track Investigation?\nDiscrepancies in the Case\/ Strange Procedures\/ Perceived or Not..","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhssc b/data_all_eng_slimpj/shuffled/split2/finalzzzbhssc
new file mode 100644
index 0000000000000000000000000000000000000000..7b012084b708950c6974b90a414f551ddbe5db32
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhssc
@@ -0,0 +1,5 @@
+{"text":"ISG prepared a pre-design report at the request of a cooperative effort between Southwest Minnesota State University (SMSU) and the Marshall School District. The report identified the full scope of services to be provided for the proposed renovation and expansion of the Marshall Regional Track and Field Complex, formerly known as Mattke Stadium located in the northeast corner of the SMSU campus in Marshall, Minnesota.\nThe pre-design report included analysis of multiple alternatives including consideration of track reconstruction, track surfacing options, track and field events, soccer field surfacing options (synthetic turf versus natural grass), adjacent infrastructure, and future improvements including rehabilitation of bleacher seating.\nThe final design included a new track, new track and field event areas, a new synthetic field, walking paths, and stadium lighting. As part of the cooperative agreement SMSU will use the facility for track and field, soccer, rugby, practice football, and club sports, while the Marshall High school will use it for football, soccer and track and field.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr. Teresa P\u00e9rez Prado, Senior Researcher at IMDEA Materials, has been elected Board Member of the Spanish Society for Materials (SOCIEMAT). The mission of the Spanish Society for Materials (SOCIEMAT) is to foster and disseminate the important research and development efforts carried out in Spain in the area of materials science and engineering. SOCIEMAT is a member of the Federation of European Materials Societies (FEMS).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If they can't find you, they can't hire you.\nPreparing for an interview (Part 1): What to expect from the process?\nOne of the most common questions I get from senior students is how to prepare for an interview.\nPreparing for an interview (Part 2): How prepared is being prepared?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Save money on your North Dakota power bill!!\nSave money on your North Dakota electric bill! The electric bill can be a budget-buster when you're surviving on a low income. Fortunately, your power company can usually help. There are several unknown or little-advertised programs that power companies may offer that can help lower the cost of your power bill.\nMost power companies offer a budget billing plan, which averages the annual cost of electricity in a home and bills the customer in 12 even installments. This can also help low-income customers maintain a budget and avert payment problems, especially during the more expensive winter months.\nDo you need help with your Montana Dakota Utilities bill? We found ways you can get help! There are programs available to help you lower your bills and you might also qualify for assistance through and extension or an arrangement. Along with company resources, you might also be able to get community and state-sponsored assistance. To see everything we've found out about Montana Dakota Utilities, click here!\nIf you're struggling with your XCel Energy bill, you might be able to get help! Low-income customers can get help with their bills, upgraded programs and assistance to make paying their bills easier. There are things you can do that might help you be able to pay your bill and also avoid shut-off. Check out all the help we've found for XCel Energy customers!\nWant free stuff in North Dakota? Go here!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"To me, a mystery is a journey to the truth, a defiant, triumphant insistence that human beings can use their brains to solve the big questions of superstition, crime, punishment and redemption.\"\nToday we feature the last part of conference chair Kathryn Craft's interview with Write Stuff 2010 presenter Bill Kent.\nKathryn: What about your interests\/attributes led you to the mystery genre?\nBill: I started off enthralled with science fiction and fantasy, and I still am. I consider it imaginative literature, that is, writing in which the quality of the author's imagination, and the author's ability to stimulate the reader's imagination, bring us to a sense of wonder about who we are and the universe we inhabit.\nI approached mysteries and thrillers as a way of adapting experiences I had as a journalist. Art Bourgeau, a mystery writer who owns the Whodunnit bookstore in Philadelphia, loaded me up with authors I should read, and to my early love for Conan Doyle's Sherlock Holmes, Agatha Christie and John D. MacDonald I added stories by Raymond Chandler, Dashiell Hammet, Donald Westlake, James Ellroy, Robert Parker, Ed McBain, Lawrence Block, Ross Thomas, Robert Campbell and even Mickey Spillane, whose crude storytelling had its powerful moments. From there I went to Sue Grafton, Walter Mosley, and many others. To me, a mystery is a journey to the truth, a defiant, triumphant insistence that human beings can use their brains to solve the big questions of superstition, crime, punishment and redemption.\nK: I know you have a love for a well-developed character, but you also write in a plot-heavy genre. Do you have a particular plotting device you use, such as index cards, outline, wall diagram, etc.?\nB: The best device is no device. You start somewhere, end up somewhere else. Though the journey is never a straight line, it should feel as if it has been worth the trip. The best description I ever heard of plotting came from a man I consider to be master of it: Lawrence Block. \"I know my characters, I come up with a setting and then I go away. I get lost and find my way back.\" He meant this literally and metaphorically. Block lives in New York City but does most of his writing while traveling. Some of his best work has given us a view of a placeless America of sprawling suburbs, modular motels off highway interchanges, fast food meals and killers who look like anyone else you'd see in a shopping mall.\nBefore I start a novel, I usually have a good idea of the hero and a few good scenes and maybe a glimpse of the ending. Usually, but not always. Sometimes I just let the characters show me what they want to do. My goal in plotting is to have things that happen flow purposefully, and have the action adhere to a logic that is believable within the context of the story. What that means in plain speaking is the best plots are plots that you don't notice: you're swept along and you don't look back.\nNext post will include an agent announcement!\nDays until Write Stuff registration opens: 84!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbivet b/data_all_eng_slimpj/shuffled/split2/finalzzzbivet
new file mode 100644
index 0000000000000000000000000000000000000000..73a5b864a6bf980436e2bf4b237bb560af0a60bd
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbivet
@@ -0,0 +1,5 @@
+{"text":"Since 17th May 2009, you may apply for the \u00ab best video of the week \u00bb. We called this operation the \u00ab V2D \u00bb which is a french name but english compatible : the Video 2diabolos.com.\nIt does not need at all to be 2 diabolos tricks, any video with a diabolo, funny or serious, simple or complicated could apply.\nWho can vote : everybody from the 2diabolos forum can vote, people from blog-diabolo may also vote, under certain conditions of activity.\nEvery proposed video could be included into the next collaborative video from 2diabolos.com.\nYou can win the big fame to be on the first page of 2diabolos.com and blog-diabolo.fr. The recognition is big : all members from 2diabolos.com\/blog-diabolo, the biggest french speaking diabolo community.\nSome week, a sponsor give something for the first guy. This is the case for the fist V2D : the winner will get 1 diabolo GENIUS !\nJust put a video in an article in your blog-diabolo, and publish it !\n(nearly) Every video type are accepted : dailymotion\/youtube (just put the URL in your post) , mov, avi, etc \u2026.\nOne video could not be applied twice \u2026 and the database has a good memory !\nWhen you publish the article, you automatically apply for the V2D contest of the current week.\nHow does a continuous contest work ?\nYou may see the V2D of that week N on the main page 2diabolos.com (and blog-diabolo.fr).\nThe video should at least have one diabolo inside; and should not shock anybody (speacially the law).\nYou should have made or participate to the vid\u00e9o. If someone else claim that's it is his video and does not want to participate, I will suppress the video.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the Orange City School District!\nThe Orange City School District is located on the east side of Greater Cleveland on a 176-acre campus at 32000 Chagrin Blvd. in Pepper Pike, Ohio. The District serves the five municipalities of Hunting Valley, Moreland Hills, Orange Village, Pepper Pike and Woodmere along with small portions of Bedford Heights, Solon, and Warrensville Heights.\nThe District empowers our students, staff and community to be passionate and innovative global citizens committed to excellence in learning, leading and making a difference. We place a high degree of importance on ensuring that all children have the opportunity to reach their full potential inside and outside of the classroom. This profile helps characterize the overall educational value of the Orange City School District in areas that matter most to our community. The District provides outstanding educational opportunities for children that is engaging, innovative and creative.\nTo Learn. To Lead. To Make a Difference.\nAt Orange Schools, student achievement through best learning practices and opportunities is our priority. We offer a rigourous curriculum to ensure students achieve well on both assessements and innovative learning experiences. This can be seen by reviewing our data dashboard, test results, and the Quality Profile. It is often highlighted through our website, e-newsletters, and social media platforms.\nThe Orange Schools Campus serves many different municipalities through the Orange Schools, which includes preK-12 schools and the Orange Community Education and Recreation department. The Orange Branch of the Cuyahoga County Public Library is also on campus.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tuesday, August 21st, 2012, 11:33 pm.\nMy new album, Wonderland, will be out on September 4, 2012. You'll be able to pick it up on iTunes, Amazon, and a bunch of other online retailers (not Spotify, so don't even ask).\nThe only place you're going to be able to buy the album with the companion digital booklet (a 29-page PDF chock-full of lyrics and original art) is on Rambutan Records.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Modern artist, Todd Andrew Babb, is a sculptor based in South Florida. Babb's sculptural art process involves the personally developed use of innovative materials to allow for the distinctly expressive lines of his contemporary figurative pieces. Each uninhibited piece is an exploration into the evolution of humanity because of his desire for progression. Concentrating on line and form over detail and aesthetics, Babb's elongated and oftentimes abstracted bodies portray this adaptation of the human spirit.\nBabb was born just outside of Philadelphia, PA into a family of artists, historians, and philosophers. Honing his skill at a young age, Babb pursued a career as a painter soon after serving in the United States Air Force. Following his first business, Creative Artistry, where he worked with top corporations such as Dupont Corian and Walt Disney Company, Babb pursued a more conceptual modernist sculpting career in an effort to express his philosophical interpretations of the human condition.\nAs a self-identified progressivist, Babb never attempts to change the principles of nature. On the contrary, his works harness the inevitable change of existence, which dematerializes chaotic change. Progressivism in art means to see that creation itself, as an entity in constant motion, is the very foundation and essence of what it means to create. Perhaps our greatest gift is to have a hand in the structure of creation. What we do in and with this lifetime will continue the progression of creation eternally.\nAs a progressivist, Babb maintains professional affiliations with non-profits and charities he feels are particularly committed to the advancement of human rights. Babb is also the exclusive sculptor of awards and gift statues for the Savannah Philharmonic.\nBabb's paintings and sculpture can be found in private collections around the world including countries in Europe, Asia, and South America.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A man was arrested on the campus on Florida International University while in possession of a loaded gun.\nAccording to the exclusively-obtained police report, Brandon Reid, 24, was at the University Towers building on Thursday. He was allegedly asking students if they wanted to gamble.\nReid, 24, is facing a felony count of possession of a weapon on school property.\nAn anonymous caller tipped off police that Reid and another man were walking around campus.\nWhen officers arrived, they asked Reid if he had any weapons on him.\nReid told the police that he had a weapon in his backpack.\nThe arresting officer opened his bag and found a Smith and Wesson 9MM semi-automatic pistol. A loaded magazine was inserted in the gun.\nReid was released from jail on $5,000 bond.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkzmx b/data_all_eng_slimpj/shuffled/split2/finalzzzbkzmx
new file mode 100644
index 0000000000000000000000000000000000000000..07493279005837d2f943738850c51fcba7315925
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkzmx
@@ -0,0 +1,5 @@
+{"text":"For all the customers that live outside the European Union, taxes are not included in the price. Please note that you could therefore be charged for duties, once the parcel reaches its destination country. These charges must be paid by the recipient of the parcel. Luxoneyes does not have control over them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Once your claws are painted with a foundation coat, you can create your models using the nail arts brushes that are available in different sizes based upon the design you want to draw. Complicated designs will require more skinny tipped and finer projectile brushes to get the design correct and visible. The use of toothpicks is also not uncommon especially if you certainly are a beginner and want to improve your patterns. For any nail polish program mistakes, you can use cotton swabs to remove the excess color. Be sure to do it quick as toenail polish dries very fast. Additionally, it is good to use a different clean for each nail polish color so that you can save time. Toenails art designs can be made out of different methods. Novices largely prefer to create their patterns by copying them by nail art books or perhaps seeing them on the Internet. One can also learn projectile arts through courses proposed by some professional nail professional salons. However , even if you possess little artistic capability, you can generate simple easy floral patterns or geometric shapes that do not effectively require high end training.\nCommuniquerenligne - Hair stylist stamp. Shop hair stylist stamp created by cooltees personalize it with photos & text or purchase as is! search for products \ufffd \ufffd shopping cart sign in my zazzle sell on zazzle 100% satisfaction create your own when an order is placed, the postage design will be reviewed, generally within 2 3 business days,. Funny robot hair stylist postage stamp. Shop funny robot hair stylist postage stamp created by karunasdesigns personalize it with photos & text or purchase as is! robot hair stylist is cutting and trying to style the hair of the man who looks worried and upset sell on zazzle 100% satisfaction. Hair stylist invitations & stationery zazzle. From invitations to postcards to greeting cards zazzle has all the hair stylist cards you need shop our amazing selection right now! update your office or family command center with custom notebooks, rubber stamps, post it\ufffd notes and more!. Hairstylist postage. Shop hairstylist postage created by busybodies personalize it with photos & text or purchase as is! update your office or family command center with custom notebooks, rubber stamps, post it\ufffd notes and more! hair stylist makeup artist photographer real estate more to explore loyalty cards appointment cards discount cards referral cards. Postage stamps zazzle. Find rubber stamps and self inking stamps at zazzle today! add postage stamps to all your different types of stationery! find rubber stamps and self inking stamps at zazzle today! hair stylist makeup artist photographer real estate more to explore loyalty cards appointment cards discount cards referral cards. Stamps zazzle. Stamps from zazzle add that extra special touch to your personal stationery with stamps from zazzle from rubber stamps to self inking stamps, you're able to customize your stamps with personal photos and text to make it truly unique rubber stamps vs self inking stamps. Zazzle personalized gifts, custom products & d\ufffdcor. From cool t shirts to custom mugs to diy invitations, zazzle is the place to unleash your creative side shop for, or design, amazing products today! rubber stamps, post it\ufffd notes and more! hair stylist makeup artist photographer real estate. Hair stylist cards zazzle. Update your office or family command center with custom notebooks, rubber stamps, post it\ufffd notes and more! makeup artist hair salon modern stripes thank you postcard $1 05 wedding bridal hair stylist thank you, stick figur thank you card $3 45 40% off with code springseason. Hair salon rose gold typography dark grey business card. Chic gold bobby pins hair stylist salon black business card are you ready to take your business to the next level? start with a new business card shop chic gold bobby pins hair stylist salon black postcard created by personalize it with photos & text or purchase as is! make a stylish statement with hair stylist business cards from zazzle. Bobby pins hair stylist hair salon. Shop bobby pins hair stylist hair salon hairdresser business card created by smmdsgn personalise it with photos & text or purchase as is! address stamps wedding stamps heart stamps monogram stamps baby stamps customise thousands of designs discover endless design options for any style, any budget and any occasion zazzle invitations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ANTANANARIVO, June 2 (Reuters) - Premature harvesting of Madagascar's vanilla crop to ward off theft of the pricey beans will lower their quality, industry officials said.\nFarmers in Sambava, the crop's main growing region in Madagascar, the world's biggest producer, have been harvesting before time due to fear of thieves.\n\"With so much vanilla immature, it is clear that the product will be of poor quality,\" said Georges Geeraerts, head of the vanilla exporters group of Madagascar (GEVM).\nHe said the vanillin content in the beans could drop to 1 percent from the usual 1.6-1.8 percent due to premature harvesting.\nThat would translate into the use of 7-8 kg of green vanilla to produce a kilo of processed vanilla, up from the usual requirement of 4-5 kg of mature beans, Geeraerts said.\nThe beans are often kept under armed guard after prices shot up from around $20 a kg in 2010 to around $500 a kg last year.\n\"Some people are ready to kill to get some vanilla,\" Prime Minister Olivier Mahafaly told lawmakers this week.\nThe flavouring is used in products from ice cream to coffee. Madagascar produces nearly half the world's crop, according to the U.N. Food and Agriculture Organization.\nIn 2015, Madagascar was estimated to have produced 3,914 tonnes out of a global total of 8,294 tonnes, projected U.N. data showed. No more recent data was available because of the lag in harvesting time.\nAlain Rabeharison, a vanilla grower in the Antalaha region, told Reuters farmers had opted to harvest immature beans because they could not hire guards to protect the fields.\nIn the 15 months to the end of March, Mahafaly said there were 90 cases of vanilla theft, involving 4 tonnes of the beans. A total of 173 people have also been arrested in connection with the theft, he said.\nIn other instances, farmers were taking matters into their own hands, opting to punish thieves who were caught.\n\"If we do not kill them, they kill us,\" said farmer Rabeharison.\nMahafaly said more security forces would be deployed to curb vanilla theft.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"30mm socket walmart buy socket 1 2 in pt deep in cheap price on home bar ideas diy home ideas centre parnell auckland.\nsolvents for acrylic paint standard acrylic paint solvent with acrylic paint solvent for water based acrylic paint.\ncooler with stand and wheels rain wheels stand insert tags replacement beyond cover metal wholesale cooler flying travel depot organizer cooler stand wheels flipkart water cooler stand with wheels.\nplastic latch set for standard mailbox mount locking mailbox residential durable galvanized steel post white plastic latch set for standard mailbox repair.\nwhere can i buy colored yeti cups multi color yeti cup powder coated yeti rambler yeti coolers rambler tumbler stainless steel double walled travel mug yeti cooler cheap coffee cheap colored yeti cups.\nkohler beige toilet k veil intelligent elongated dual flush toilet within plan kohler beige toilet seat kohler custom color toilet seats.\nfurniture brace brace furniture refinishing bracebridge bracewell furniture dublin ga.\nbuy full size headboard fabulous black metal king bed full size of iron king bed frame wrought size headboards black headboard full metal buy headboards black metal king size bed full size upholstered.\ntransmission jack adapter for floor jack 3 ton floor jack transmission adaptor trolley jack hydraulic jack car truck transmission trans jack adapter for hydraulic floor jack adaptor lift kit.\nthe best lightweight vacuum cleaner review vacuum review lightweight vacuum cleaner review eureka neu182a powerspeed lightweight bagless upright vacuum cleaner reviews.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An Aussie dad is calling for parent parking bay signs that feature a woman with a pram to be updated to include men as well.\nQueenslander Jamie Simpson has posted a picture featuring him with his daughter under a 'parents with prams' sign that depicts a female silhouette.\nHowever the post has divided people online.\nMr Simpson has received plenty of support from fellow disgruntled dads, but also attracted criticism from others who believe he should protest about more important issues, especially, as pointed out by some online posters, there are already plenty of parent parking bay signs that include men.\nWhat do you think? Does he have a valid point, or does it really matter what parents feature on the signs?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblexj b/data_all_eng_slimpj/shuffled/split2/finalzzzblexj
new file mode 100644
index 0000000000000000000000000000000000000000..46be0ba9ba32926061681204de0fa0fc44e03720
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+{"text":"Green, Nick 2007, 'Functional polycentricity: A formal definition in terms of social network analysis' , Urban Studies, 44 (11) , pp. 2077-2103.\nPolycentricity is often used descriptively with regard to a regional system of settlements, usually referred to as a polycentric urban region ( PUR). Although presented in much of the literature as in essence a morphological concept, polycentricity possesses a functional element that receives less attention. Polycentricity is also seen as a normative concept. However, it has not been rigorously defined using formal techniques. This paper argues that defining polycentricity in terms of both morphology and function is possible by drawing on techniques originating in social network analysis. The paper sets out a formal definition and derivation of functional polycentricity based on these techniques, which is then extended to a derivation of an index of regional functional polycentricity. The paper sets out worked examples to show how the techniques described might be utilised. The paper closes with a discussion of issues that may arise when putting these definitions into practice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Outdoor Keyless Gate Locks have 50 picture of interior design, it's including Outdoor Keyless Gate Locks Wonderful Astonish Moraethnic Decorating Ideas 1 Interior Design. Outdoor Keyless Gate Locks Stupefy Lock Illbedead Interior Design 2. Outdoor Keyless Gate Locks Great Interior Design 3. Outdoor Keyless Gate Locks Unthinkable Interior Design 4. Outdoor Keyless Gate Locks Phenomenal Lock Avianfarms Interior Design 5.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shopping for Apar\u00e8y \u00e8 kondisyone? You have come to the right place! Bridgat.com provides high quality Apar\u00e8y \u00e8 kondisyone at affordable prices. No matter where you are located, we offer free shipping for the Apar\u00e8y \u00e8 kondisyone you purchased. On Bridgat.com, you can pay by PayPal or credit card. If you are unsatisfied with Apar\u00e8y \u00e8 kondisyone that we ship to you, you can return to Bridgat.com within 30 days. Full amount of money will be refunded to you. Browse our product category now to choose Apar\u00e8y \u00e8 kondisyone that fit you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THERE'S NO DENYING THE POPULARITY of pontoon boats these days, but can they function as effective fishing boats in both fresh and saltwater?\nDo they have any redeeming qualities for hunters? Might they do the trick as multipurpose machines for outdoorsmen who need a single boat for different purposes? The answers might surprise you, so let's take a look at the pros and cons of pontoon boats.\nPerformance & Handling: In this regard, pontoon boats are like any other type of boat\u2014you get what you pay for. Buy a pontoon with the smallest power plant the manufacturer offers, and you'll have a slug on your hands. Up the ante to the maximum power plant, and you can get pontoons that are thoroughly juiced up, including some that can top 60 mph.\nThis design can handle substantially more power and has the advantage of additional planning surface and floatation. Additionally, tweaks like running strakes are often welded on to enhance performance.\nWave-handling Ability: Truth be told, in most sea conditions modern pontoons do just fine. Again, tri-toons have a leg up over those pontoons with just two logs. But in a one-foot chop, life is good. Running through a two foot chop is do-able on most of today's pontoon boats. It's not until you start encountering larger waves that things can get problematic. (Of course, you could say that about most relatively small boats).\nThere is, however, one unique issue that commonly comes up: Since pontoons have a relatively low amount of buoyancy forward and no up-swept bow, it's relatively easy to stuff the bow into a wave. This generally brings the boat to a very abrupt\u2014and very wet\u2014stop. The same problem can become evident when a large number of people sit or stand in the bow of a pontoon, weighing it down.\nStability: One of the biggest advantages of pontoons is their supreme stability. That boxy shape and widespread weight means that even when your 300-pound cousin Bubba walks from one side of the boat to the other, it won't lean over.\nThere's very little rocking and rolling so most people don't have a problem with seasickness on a pontoon boat, and fishing in a busy lake riddled with wakes is often a lot more comfortable on a pontoon than on other types of boats.\nAdded bonus: If you're looking for a hunting platform to build a blind on, that tremendous stability again makes a pontoon boat a great candidate.\nFishability: On this count, pontoons are a mixed bag. They have tons of room. On most, you can cast more or less 360 degrees around the boat, and it's always nice to have one of those comfy seats nearby.\nSome models come with rodholders and\/or livewells, and there's plenty of opportunity to add rail-mount rodholders. Pontoons also have relatively shallow draft, plenty of space to rig electronics, and enough stowage for all the tackle in the world.\nOn the other hand, the livewells available on most (though certainly not all) pontoon boats are usually very small. You'll always have to carry on a cooler for the catch since there aren't any fish boxes. Same goes for your tacklebox and tackle.\nThe biggest problem with fishing from a pontoon, however, is often windage. Their tall aluminum \"fences\" which surround the deck act like tall sails. In a breeze, drift fishing from a pontoon can be quite difficult.\nExpense: When comparing the expense of a pontoon with the cost of other types of boats, we again have to note that you'll get what you pay for. You could spend $30,000 or $40,000 for a modest pontoon boat with a mid-sized power plant, or you could easily break the $100,000 barrier if you tried. We can make no blanket statement regarding pontoon boat pricing because just like other types of boats, the market is filled with offerings of all levels.\nComfort: True, many anglers will rank comfort as their least-important consideration, and others won't rank it at all. But for those who are interested in comfort levels, pontoon boats can't be beat. How many other types of boats, after all, have couches and sun loungers?\nConsidering all of these factors, it should be clear that for some people a pontoon boat will be a good choice. Yet few outdoorsmen opt for pontoons, instead getting more traditional V-hull fishing boats.\nOne reason is simply the stigma once assigned to those \"party barges.\" But if you think a pontoon might be just what you need, don't let a silly thing like that stop you. Once you've blasted across the water's surface at 60 mph, relaxed in a lounger, cast from the bow, and hunted from the blind, you'll begin seeing pontoon boats in a completely new light.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"View other listings in Bondi Junction, Waterloo, Surry Hills, Paddington, Newtown.\nI've been travelling Australia since October and I'm settling down in Sydney to work. I work in TV and I'm looking for a home not just a place to sleep. I'm clean, tidy & respectful. I'm sociable & friendly.\nMy hobbies include photography, marathon running and football.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblrau b/data_all_eng_slimpj/shuffled/split2/finalzzzblrau
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+{"text":"We recently made this Card for a very special groom who is the fianc\u00e9e of a dear friend. The groom loved our Beach card with the watercolour inserts and as we had recently modified it for another client he chose the new design straight away.\nHere, the jacket of the card was made keeping in mind the bride's card, which has a tree as the central focus. Since our brief was to create a simple and elegant card; we have done the tree in gold leafing on a watercolour background to keep the watercolour theme congruous with the inserts. There is a splash of colour on the inside of the envelope, which adds an element of surprise when you open the card. It has been a wonderful journey taking this card from concept to execution and we can't wait to share the bride's card.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great recipes. I'll be trying these.\nI'm wondering if you would consider removing the \"indian giver\" reference. The phrase was used by white folks in a very racist way to accuse Native Americans of theft, an accusation that was oftentimes unfounded.\nThank you Maria - did you read my end note? Indian giver derives from the time period in our history when the white man gave to the Native Americans with one hand and took away much more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For Arla in Finland we created this short but intensive piece. The brief from the agency was that they wanted Yoghurt to fly out of the package and then return again. We did several tests before shoot on how to best compose the viscosity of the Yoghurt and how to best use compressed air. We shoot everything with Phantom Flex Gold 4K and went for 800fps to maximise the feeling of plymes on the Yoghurt. In the post production we added strawberries and cashewnuts as well as changed the package design. In addition to the film, we also produced a highres still image that would be used in the campaign.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The name Lalique evokes the beauty of fine jewellery and the translucent shine of crystals. The company takes its name after the artistic genius and master jeweller and glassmaker Ren\u00e9-Jules Lalique who poured all his creative enthusiasm into the brand.\nLalique opened his jewellery and glassware shop in 1905. His art deco and art nouveau creations often incorporated natural elements and inspirations both in terms of materials and shapes. The perfumer Fran\u00e7ois Coty was so impressed by his work he proposed a collaboration which resulted in the production of gorgeous perfume bottles.\nLalique has always been connected with the world of fragrance, even though they only introduced their first own perfume in 1992. All the Lalique fragrances are now characteristic for their unique bottles and the exquisite quality of ingredients as well as the aroma itself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Raekwon Mingledoff, Lane's freshman guard attempts a reverse layup in the Titan gym while contested by Darrell Wilcox \u2014 who last played for the Titans in 2010 \u2014 in the Lane Alumni game on Dec. 6, 2015.\nOn Thursday, Dec. 3, the 51-foot glowing cross of New Hope Christian College beckoned basketball fans from every part of the community to their hillside campus, where the New Hope Deacons hosted the Lane Titans in a non-conference basketball match at Rexius Gym.\nChavka's prediction came to fruition. The Titans' game plan was met by the Deacons' ability to move the ball and effectively shoot the trey. It was a uniquely matched competition that brought spectators to their feet.\nThe Titans wasted no time initiating effective half court pressure that forced turnovers and rushed shots, and their field-goal efficiency kept them nearly neck and neck with the Deacons.\nOn the other side of the floor, Chavka stressed the importance of sustaining energy and called for his team to tighten up their defense going into the second half. The Titans heeded the call and were rewarded with points from second opportunities and turnovers. But quick ball movement and a relentless barrage of three point shots tipped the scales in favor of the Deacons by seven points.\nThe turnaround play came from an aggressive move to the basket by Zaid that put him at the free-throw line with the Titans down by two. Zaid knocked down both of the one point penalty shots to tie it back up. The Titans capitalized off of a fast break that led to a lay up from the hands of freshman point guard Raekwon Mingledoff, recapturing the lead for the Titans.\nThroughout the game, the Deacons struggled to get the ball to the rim in the allotted 30 seconds, but Titans' defense morphed into a force to be reckoned with in the second half, extending their lead. This was the late energy the Titans had been planning on.\nThe Deacons kept their nose to the grindstone the entire game, and never fell out of contention. Their hard work paved the way for opportunity \u2014 a Titan foul beyond the three point line resulted in a four point play for the Deacons, bringing them within two points of the lead with less than three minutes left in regulation.The clamor of fans bolstering their respective teams erupted into an ocean of encouragement as the Titans passed the ball around, winding down the clock and patiently waiting for a clear shot. The ball made its way into the hands of sophomore guard Duce Namazi who, as the shot clock counted down the final seconds, flushed an outside jumper; changing the ocean of encouragement to a trickle of disappointment and confusion.\nWith no remaining time outs, the Deacons had to foul to keep as many seconds on the clock as possible. The Titans effectively killed the clock and capitalized on the Deacons' foul trouble to extend the lead. The Deacons backed off and Titans moved the ball past half court without opposition. Both teams began exchanging gestures of sportsmanship as time expired and the final buzzer rang. The Titans returned from the hill as victors.\nLane will be the scene of two men's basketball tournaments over winter break. So, before you commence binging on Internet videos and sleeping over the break, consider treating yourself to a night out.\nOn Friday, Dec. 11 to Dec. 13, Lane hosts the Dale Bates Invitational Tournament in honor of legendary men's basketball coach Dale Bates in Building 5 on the Main Campus. The single-elimination style tournament begins Dec. 11 at 2 p.m. when the Umpqua Riverhawks play the Spokane Sasquatch.\nThe Titans play the Seattle Mountaineers, an independent faith-based basketball club, at 6 p.m and the first round wraps up with a game featuring the Linn-Benton Roadrunners and the South Puget Clippers beginning at 8 p.m.\nThe second round of the tournament begins Dec. 12 at 12 p.m. and the second game starts at 2 p.m. Round one's winning teams will play at 4 p.m. and the final game of the night will tip off at 8 p.m.\nThe final round of the tournament begins Dec. 13 at 10 a.m. Third place ranking will be determined at 12 p.m. and the championship game takes place at 2 p.m.\nLane hosts eight teams in the NWAC Conference during the 2015 Lane Crossover tournament Dec. 17 to Dec. 19. The tournament is also single-elimination and begins with the Peninsula Pirates playing the Lower Columbia Red Devils at 2 p.m.\nThe Titans face the Blue Mountain Timberwolves at 6 p.m. The first round wraps up after the Everett Trojans play the Umpqua Riverhawks at 8 p.m.\nThe second round of the tournament begins Dec. 18 at 12 p.m. Round one's winning teams will play at 4 p.m. and 6 p.m.\nThe final round of the tournament begins Dec. 19 at 10 a.m. The game for third place honors will take place at 12 p.m. and the championship game begins at 2 p.m.\nDec. 28 to Dec. 30 the Titans compete at the Siskiyous Classic in Weed, California and play the Clark Penguins Jan. 2, 2016 in Vancouver, Washington.\nThe women's basketball team goes on the road for the entire winter break starting with the Pierce Tournament in Lakewood, Washington from Dec. 11 to Dec. 13.\nThen, the Titans head to Port Angeles, Washington where they compete in the Peninsula Tournament Dec. 17 to Dec. 19 before making their way to the Chemeketa Tournament in Salem, Oregon, on Dec. 28 to Dec. 30.\nThe Titans wrap up their winter break in Vancouver, Washington where they face Clark Penguins Jan. 2, 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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+{"text":"All the above menu options are self descriptive.\nBelow I'm giving some screenshots of TFS Assistant in Action.\nJust after clicking the menu, the report will be shown in TFS Assistant Window.\nThis is all about TFS Assistant, Click here to download TFS Assistant for Visual Studio 2010, Please don't forget to share your feedbacks.\nDoes this work with Visual Studio 2012 and TFS 2013?\nNope. TFS Assitant is currently available only for VS 2010.\nCan you export results of these queries to MS Excel or as a CSV file?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jack Pe\u00f1ate recently posted the song 'No One Lied' on his website as a free download accompanied by a live video. 'No One Lied' was recorded live in one take at Pe\u00f1ate's home on January 19th, 2012, and was directed by Dani Castro. The song is brilliantly written, and offers a real raw, humble, sincere energy - take a look and listen below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Heat oven to 350 degrees. Lightly butter a round 9-inch oven skillet with 1 tablespoon of butter. In a blender combine, 1 tablespoon melted butter, eggs, milk, flour, vanilla, sugar, cinnamon and pumpkin pie spice. Blend until smooth. Pour batter into pan. Bake 55 - 65 minutes until puffed and golden brown or until knife inserted in center comes out clean. Cool 3 minutes. Dust with powered sugar and serve with maple syrup. Serves 6.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lysophospholipid micelles sustain the stability and catalytic activity of diacylglycerol kinase in the absence of lipids.\nBicellar mixture phase behavior examined by variable-pressure deuterium NMR and ambient pressure DSC.\nStability of the glycerol facilitator in detergent solutions.\nSmall phospholipid-coated gas bubbles can last longer than larger ones.\nEffects of epicholesterol on the phosphatidylcholine bilayer: a molecular simulation study.\nLipid metabolism in Trypanosoma brucei: utilization of myristate and myristoyllysophosphatidylcholine for myristoylation of glycosyl phosphatidylinositols.\nInteractions between bovine myelin basic protein and zwitterionic lysophospholipids.\nEquilibrium binding of myristoyllysophosphatidylcholine to bovine myelin basic protein: an example of ligand-mediated acceptor association.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Times Hostel - College Street Distance to the centre of town 0.5 km price per day from \u20ac17 The Times Hostel has modern, simple accommodation close to Trinity College, Grafton Street, Temple Bar and O'Connell Street. All bedrooms are en suite and feature new showers and toilets. The hostel has free Wi-Fi in public areas, free 24-hour luggage storage, free tea and coffee and free phone calls to 40 countries. The 24-hour friendly reception offers local information, with complimentary city maps and travel guides.\nThe Apache Hostel Distance to the centre of town 0.7 km price per day from \u20ac14 The Apache is 200 metres from Dublin Castle and a 5-minute walk from St Stephen's Green. It offers affordable accommodation, free Wi-Fi and a rooftop terrace. The Apache Hostel is located in the vibrant Temple Bar district of Dublin. The hostel has a 24-hour reception and is surrounded by a fantastic selection of bars, restaurants and clubs. Guests receive a discount at the nearby Apache Pizza Restaurant.\nMEC Hostel Distance to the centre of town 0.5 km price per day from \u20ac92 Featuring free WiFi throughout the property, MEC Hostel offers accommodation in Dublin, 900 metres from Croke Park Stadium. The hostel is based in a renovated Georgian building on one of Dublin's historic streets. Guests will benefit from lounges, a terraced garden and a large self-catering kitchen and diner. All rooms include a private or shared bathroom with free toiletries. Additional features include a free in-room locker (own padlock required) and a communal games room with ping pong, table football and a pool table.\nParamount Hotel Distance to the centre of town 0.8 km price per day from \u20ac59 The boutique Paramount Hotel is in Dublin's Temple Bar district, 500 metres from Dublin Castle. It offers stylish rooms designed in a 1930's theme, free Wi-Fi, a bar and nightclub. They feature a flat-screen LCD TV, safe, work desk and tea\/coffee-making facilities. In the evening, it transforms into a vibrant nightclub. The Gaiety Theatre and St Patrick's Cathedral are both less than half a mile away.\nGeorge Frederic Handel Hotel Distance to the centre of town 0.9 km price per day from \u20ac68 In Dublin's fashionable Temple Bar area, this hotel with free WiFi is set within a quiet location on the historic Fishamble Street. The George Frederic Handel's elegant rooms include private bathrooms, and the Olympia Theatre is a 5-minute walk away. With classic furnishings, each room at George Frederic Handel Hotel includes a flat-screen TV, an en suite bath and shower room and a seating area. A full cooked breakfast is available each morning.\nJackson Court Hotel Distance to the centre of town 1.6 km price per day from \u20ac39 The Jackson Court Hotel Dublin is located in the centre of Dublin, just a 5-minute walk from Grafton Street and St. It offers an on-site nightclub and full Irish breakfasts. The Georgian building offers rooms with a private bathroom. The hotel provides hairdryers, safes, and irons upon request. O'Connell Street, St Patrick's Cathedral and Trinity College are just a 5-minute walk away. The LUAS tram stops just outside the hotel's door for easy access to the rest of Dublin.\nClifden House (Formerly Clifden Guesthouse B&B) Distance to the centre of town 0.7 km price per day from \u20ac45 In the heart of Dublin city centre, this Listed Georgian house has secure parking for a surcharge. O'Connell Street and Croke Park are within a 5-minute walk from Clifden House (Formerly Clifden Guesthouse B&B). Each en-suite room at Clifden House (Formerly Clifden Guesthouse B&B) has free WiFi, a TV and tea\/coffee facilities. There are no hairdryers or ironing facilities in the rooms, however these items can be rented for a refundable deposit.\nGrafton Guesthouse Distance to the centre of town 0.9 km price per day from \u20ac74 Just a 5-minute walk from Temple Bar's pubs and bars, the Grafton Guesthouse and Dublin Castle is within 500 metres of this bed and breakfast. The banks of the River Liffey can be reached after a 5-minute journey on foot.\nClontarf Castle Hotel Distance to the centre of town 3.9 km price per day from \u20ac99 Set in an 11th-century castle the Clontarf Hotel combines ancient architecture with 21st-century facilities. 10 minutes' from Dublin city centre, this boutique hotel offers an award-winning restaurant and luxurious bedrooms. Rooms at the Clontarf Castle Hotel feature air-conditioning, LCD satellite TVs and free Wi-Fi. The Fahrenheit Grill serves fine cuisine and specialises in local beef and fish dishes.\nEgans House Distance to the centre of town 1.9 km price per day from \u20ac40 Situated 5 minutes' drive to Dublin city centre, Egans House offers home-cooked breakfasts and affordable rooms. Rooms at Egans House all feature free Wi-Fi, tea and coffee-making facilities, a flat-screen TV, safes and ironing facilities. Rooms include an en suite bathroom with a shower and a hairdryer. Each morning the B&B offers a traditional Irish or continental breakfasts. The lounge has free tea and coffee all day, a large TV, stereo, piano and a computer with internet access.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbojwq b/data_all_eng_slimpj/shuffled/split2/finalzzzbojwq
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+{"text":"By Trine \/ May 28, 2010 in New York and US. 13 Comments Tags: Corton.\nWith only three nights in New York City Corton was one of the restaurants I initially, but sadly, had left out of my dinner plans. Mainly for two reasons: There was no available tables on OpenTable.com and because of Corton's strict no photo-policy. I had sent an email to the restaurant in advance to ask for permission to shoot photos for my blog, explaining that I never use flash and am most discrete and conscious about not disturbing the other guests. I received a polite rejection.\nBut when my return to Denmark got delayed by the now so famous ash cloud I decided to go for dinner at the two-stars Michelin Corton anyway. I planned to enjoy it without taking photos at all and not writing a VGF post on it.\nI arrived in the evening on 20th April and got warmly welcomed by the maitre d' and shown to my table in the middle of the elegangt and cool room with good view to the whole restaurant. To my big surprise she informed me, that if I just happened to have my camera with me, then it would be okay to take pictures as long as I didn't use the flash. Apparently Chef had seen this blog and okayed my photographing his dishes. Flattering, and thank God I DID bring my camera after all.\nI took a look at the menu and quickly opted for the chef's tasting menu because I didn't feel certain that only three dishes would be enough to provide me with a good impression of the restaurant and the style of the food.\nThree appetizers arrived and accompanied the nice glass of bubbly from Jean Claude Th\u00e9venet Blanc de Blancs, M\u00e2connais NV.\nAll three of them were very rich and rather complex. In fact they were small hints and a very good impression of what I could expect to come. The little cup with the green stuff and foie gras underneath was very delicate and tasty. To die for, really.\nMy first dish presented oysters in two ways. Both were very fresh, pure but rich in taste. Excellent.\nAgain I loved the foie gras and the same sweet sweet note to it like the appetizer had. The balance was marvelous with the right amount of salt, sweet, balsamic acidity. The sphere was apple cider and had a fruity and slightly acidic note \u2013 it reminded me of the taste of wine gum. The toast was perfection: Crunchy and fragile, full of taste with a slight sweetness to it.\nBefore I was served the next dish the chef came to my table and showed me a bag with the fish inside. It was cooked slowly sous-vide.\nThe black bass was wonderful, fresh, juicy and full of delicate taste. Again there was more than one dish in the course, and there was a hugh spectrum of different tastes. I was still drinking the nice white Jacques Bavard Saint-Romain, Burgundy 2007 I ordered earlier for the first dishes, but think that the Pinot Noir I had with the following course would have matched the fish even better.\nI think it's the first time I encounter mint served with lamb. Here it added freshness to the somewhat intense, gamy taste of the good lamb. Mint sauce, by the way, is a traditional English accompaniment to lamb roast, as far as I know. The quality of the lamb shoulder was also very good. The texture of the meet was firm but also cooked long enough to easily come apart.\nThe Patrick Javillier Savigny-l\u00e8s-Beaune, Burgundy 2007 was terrific. The nose was intense and intriguing, huge fruity taste and offered a long aftertaste.\nI'm a big fan of cooked cheese dishes and I only get it on rare occasions. Each mouthful here comprised both melted and firm cheese with a rich but delicate taste of Taleggio cheese. The taste was combined with a crunch to it and sourness from the citrus coriander and a slight fruity sweetness to the finish which came from the raisins, I think. All in all it was a wonderful example of a cooked cheese serving.\nI had saved a bit of the Chardonnay to accompany the cheese and to see how it would work. The wine had a nice cleansing effect and the acidity gave balance to the rich taste. The Pinot Noir was a little to powerful for the cheese.\nThis worked like a cheesecake with the cracker crumbs replaced by the pine nuts and it worked well with the refreshing fruit blackberry sorbet on the side.\nThe final dessert was much fruitier and lighter than I anticipated, and it was great. The chocolate cake on the side in the brown bowl had the most rich chocolate flavour you can imagine.\nThis was a great meal with dishes served at a nice pace. I very much liked the food, which was flawless by the way. Most of the courses were highly complex and many comprised side dishes. Although I enjoyed the food I found that the complexity was somehow unnecessary. It made the impression a bit muddled. Chef-owner Paul Liebrandt's relation to Pierre Gagnaire is evident by the multiplicity of ingredients and plates in one serving and by the cooked cheese.\nAfter the desserts I was invited to see the kitchen and was explained about each station there. Liebrandt had left at that point.\nTea, mignardises and a friendly chat with maitre d' concluded my great evening at Corton.\nThe service was all evening kind and efficient and left nothing for me to wish for.\nWhat had actually impressed me the most was the service of the maitre d' which was of the kind I only very rarely experience. She was most professional but friendly at the same time, addressed me by my name at all times, occasionally came down to my tabled to ask how I was and whether I enjoyed the food, she knew about the ash cloud and the possible delay of my flight back, and was genuinely interested in hearing about my blog and my background.\nI agree with Michael of Wandering Epicures, though, there is something about the name of the restaurant, Corton, and the food being served that puzzles me. To me Corton suggests Burgundian cuisine and this wasn't that. With the strong link to Pierre Gagnaire why not call the restaurant Paris or even Rue Balzac?\nSensmaking name or not, I still very much enjoyed my night at Corton!\nThank you, Susan and thank you Paul!\nWow, Barzelay, thank you so much for sharing this we me (us). That's great to hear and really pleases me!\nMy fiance and I ate a meal at Corton just a couple weeks ago, and it was amazing. Everything was spot-on, and although I thought that all of the \"side dishes\" were at times unnecessary, they were always appreciated. It was probably the best fine dining meal I've had.\nThank you so much, the constant hunger! I hope you'll like Corton too when you get there.\nI'm so glad you were able to photograph the beautiful dishes. I live in New York and have heard about Corton and now it's on my (long) list of must-eat restaurants. Thanks!\nDuring my last visit last year, I thought he had calmed down and matured in his cooking.\nHis earlier work was Gilt at New York Palace Hotel and previously Atlas. At Atlas I was having a crayfish soup with white chocolate? The I still have nightmares about, unfortunately!\nThanks for your comment! Wow, I hope you'll like it too!\nAs far as I recall I tipped between 20-25%. In cash.\nThanks Chuck, yeah it was cool. His previous place was that in NYC also?\nI loved this post. For so long I have put off Corton, now Im making a reservation. Cannot wait to see how Per se went!\nSo how much did you tip Trine ?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"And there's boxes piled high at home but Aur\ufffdlie and I will be staying a little bit longer in Paris to spend our days enjoying this beautiful city of France. The reason for this? Well, as it turns out, Aur\ufffdlie managed to get herself yet another job, (cRaZy GirL!) this time with Unilever traveling through North Europe.\nPour moi, mon travail pour La Vie En Roses Boutique se passe bien et j'aurais plus de photos bient\ufffdt. And of course I'm always looking for new opportunities to do a little creative dancing on the web.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Twin Cities JACL and Historic Fort Snelling co-sponsored an artists' panel titled \"Art, Identity and Community\" on Sept. 23 at the Historic Fort Snelling Visitors Center in St. Paul, Minn.\nModerated by Kerry Morgan, director of Gallery and Exhibition Programs at the Minneapolis College of Art and Design, four local Minnesota artists shared their personal histories, as well as presented a selection of their past and current projects.\nPainter Leslie Barlow and photographers Wing Young Huie, John Matsunaga and Steve Ozone have varied backgrounds, but they engaged the audience by sharing their experiences within their communities, especially in regard to marginalization, invisibility, stereotyping, racism and historical trauma.\nIn addition, each speaker offered insights into how the experiences of the communities to which they belong help shape their identities and influence their artistic work.\nLes Suzukamo, a former Twin Cities JACL board member, and his wife, Karen, helped to support the exhibit and programs. He acknowledged that the artists' panel gave him an appreciation of the challenges and rewards of exploring social issues through art.\nThe exhibit viewing is free and open to the public on Saturdays from 10 a.m.-5 p.m. through Oct. 28 at the Historic Fort Snelling Visitors Center located at 200 Tower Ave.\nFor more information, visit historicfortsnelling.org or call (612) 726-1171.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"i need to have 2 outputs to get an automation working right in Home assistant.\nhow do i configure these 2 influxDB outputs ?\nDepends on your power system.\nHow many mains do you have?\nWhere does the inverter connect?\nCan you show a screenshot of the input configuration display so I can see the names?\nI'll take that to be case 2 above.\nSo grid power, whether consumption or production, is what is going through you mains i.e. the sum of your three mains. As I understand it you want to break that down on whether it's positive (consumption) or negative (production).\nTo prove this out, you can create a couple of outputs and look at them with the local graph.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Marketing Tools & Misc. - Oh Yeah Marketing Inc.\nYou can use it as a stand alone website to include on your business cards and other marketing materials and even add a link to it from your own website.\nThe site directs visitors back to the dealers that sent them.\nIt's one of the great tools we have for you to get started with.\nCheck it out by clicking the link below. It will open the website in a new window for you. \"www.NoLaundrySoap.com\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"No Force First is our initiative that aims to reduce restrictive practice in our care.\nAll the work we're doing is part of our long-term goal to achieve Perfect Care. At the heart of that is trying to reach the highest possible standards in everything we do, not just in clinical care but also how we deliver our messages to the outside world.\nNo Force First is one of the key goals of our strategy and a No Force First Conference was hosted by the trust.\nStaff heard presentations and a talk from service users and given an insight into why we believe so strongly in its effectiveness for staff and a better patient experience.\nThere have been presentations to over 31 external organisations, people with lived experience and carers in the afternoon as a result of Mersey Care being identified by the Department of Health and the CQC as a leading organisation in this field.\nExternal organisations have taken such an interest in our journey to No Force First, but there are senior figures in other organisations (and still a few in our own) who still believe that suicide is inevitable and No Force First is unachievable.\nExperts by experience \u2013 they have given so much to help co-produce practice, having the courage to tell their stories and, in some cases, having the courage to stand up and address and influence large groups of people.\nBoth staff and patients have shown great courage in accepting No Force First and making Mersey Care a better caring environment.\nOur belief that people who use our services are at the centre of everything we do and are, to use our own Perfect Care terminology, 'the people we serve' helps to strengthen our incredible emotional commitment to positive change - whatever the obstacles we may encounter.\nCompile and publish good practice stories for reducing restrictive practice.\nIf you require any more information on the No Force First initiative please contact Project Lead for No Force First, on 0151 472 4550.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Archives|ALL VACCINES FAIL IN 'T.B.,' HE SAYS; Sir Almroth Wright Tells Brussels Medical Conference That Fact Has Been Established. LIGHT ON GAS GANGRENE It Can Be Reduced 80 Per Cent. by Using a Serum, Says Dr. Vincent of France.\nALL VACCINES FAIL IN 'T.B.,' HE SAYS; Sir Almroth Wright Tells Brussels Medical Conference That Fact Has Been Established. LIGHT ON GAS GANGRENE It Can Be Reduced 80 Per Cent. by Using a Serum, Says Dr. Vincent of France.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There has been a recent discussion on the Django Developers mailing list about the role of designers in the Django community. I think that this is an interesting discussion that can come from this, and I would like to explain my thoughts on the issue.\nThis discussion came up in the context of redesigning the Django Admin, which everyone knows and loves. The UI is growing a bit out-dated, and there was talk of working to clean it up. This then turned into a discussion about how design proposals and improvements aren't taken as seriously as they should be by the community. I think there are a number of reasons that this happens, and I would like to take a look at them. My purpose here is to start a discussion about how to better integrate designers into the community, because they are a vital part of making our world more beautiful and efficient.\nThe normal process for changes that go into Django is that a proposal is sent to the mailing list. There is a discussion that happens around them, and then if the code is produced, and it works, it gets committed. For design changes, I don't reply to these messages, because I don't have the skills or knowledge to judge the work. I think that a lot of people on these lists are in the same boat.\nWhen someone sends a proposal to the list, and it doesn't get any replies, that feels like rejection. This happens more than it should, but it isn't anyones job to respond to these messages and say \"sorry, I'm not qualified to critique your work\". This happens with code proposals too, but I think it may happen more with design. This leads to designers forsaking the mailing list, and this problem perpetuates itself, by not drawing designers into the community.\nPart of the problem that seems to have come forward is that there is a feeling that design is \"special\". That it should be treated somehow differently in the process. As we know from history, even with all good intentions, different is never equal. So I think that we should work to fit design into the current scheme of how things work, instead of trying to adopt new ways of dealing with it.\nWhen I look at the current Core Developers of Django, I don't see many people who are designers. As I said above, that fact that very few of the current core developers are well versed in the design realm, really hurts inclusion of design changes. This creates a lot more friction in the process of getting design changes into the code base.\nI don't know if this idea is crazy, but should we have the concept of a \"core designer\". These would be people that the community trusts and knows have good taste, that would be an obvious person to make these design choices. I think that there is a problem when I have a design change for Django, and I really don't know who to talk to. There is an obvious authority (BDFL) for code changes, but I don't know if Adrian and Jacob are really the correct people to making these judgment calls on design?\nI realize that this is open source, and \"core designers\" would be the same as developers, just people who care about the direction of the projects design. However, I think that having more design oriented people in the community in a more direct fashion would make it more obvious that design changes are welcomed and seriously considered.\nI don't know how far we need to go down the path of making this explicit. However, most of the documentation about contributing is explicit about \"code\". This is another of those lines, where I don't know if it makes sense to be explicit about design, having a \"design\" section in the contributing documentation, or if the implicit knowledge of core designers will make it obvious that we mean design changes there too.\nI don't want to talk about the actual design process, because well, I really don't know how it works. I think that once we integrate designers into the community better, the process for design will naturally fall out better.\nI would like to point out that Django has some of the best designers of any open source community out there. I am lucky to work with a number of them on a daily basis, and I really think that they make our community special. So thank you guys for sticking with us.\nThis is a place where I could see Django leading the way in how to integrate design into the open source development process. Let's make a grand experiment, and see how it works out.\nIf you have thoughts, please join the discussion on Django Developers, so that the correct people hear your thoughts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Just another day with just another Android malware targeting unsuspecting users on Google Play Store. This time, the IT security researchers at Russian cybersecurity company Dr.Web have discovered a dangerous Android malware hidden in several gaming apps on Play store stealing personal data from users by conducting phishing attacks.\nDubbed Android.RemoteCode.127.origin by researchers, the malware was found in 27 games that were downloaded more than 4,500,000. Upon infecting an Android device, the malware secretly opens malicious websites and automatically clicks on its content including banners and links by downloading a script from the C&C (command and control) server the purpose of which is to generate revenue through advertisements and links.\nFurthermore, the malware uses its capability to drop additional Trojan modules which perform a number of malicious attacks including opening phishing windows and stealing login credential of victims, spams their device with advertisements and downloads other malicious apps without the knowledge or permission of victims.\n\"It covertly downloads and launches additional modules that perform various malicious actions. For example, they simulate user actions by covertly opening websites and clicking on their items,\" said Dr.Web in their blog post.\nDr.Web's blog post was published on January 16th, 2018. The cybersecurity firm informed Google about the presence of malware-infected apps on PlayStore however at the time of publishing this article, Google did not remove any of the apps. Therefore, if you are an Android user, here is a list of the apps and make sure to delete them in case you have downloaded any of them on your device.\nSince Android is the most vulnerable smartphone operating system, the cybercriminal and marketing community is taking full advantage of the situation. Just a couple of weeks ago it was reported that there are hundreds of gaming apps on Play Store are recording and tracking TV viewing habits of users and sharing it with companies for targeted advertising.\nMoreover, a week ago, 60 Android apps for kids were found infected with pornographic malware which displayed highly sensitive and x-rated ads to users in order to generate revenue. It means that the Google's security system has failed to detect and block malware and other malicious software from making their way to Play Store.\nAlso, avoid downloading unnecessary apps since Play Store is as insecure as a third party store. Make sure your device is updated with anti-virus software and run a scan on daily bases. Stay safe online.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stiletto Heel High Cut Court Shoes is rated 4.6 out of 5 by 57.\nRated 5 out of 5 by Arket2101 from Classy! SO many compliments. Purchased these in store in the ochre colour. I only tried them on because I saw someone else trying them on and thought how nice they looked. I don't think I've ever had so many compliments on a pair of shoes! Generally most of my office wear is dark colours or black, so these shoes add such a fantastic flash of colour. The most compliments arose when I wore them with a pair of M&S black slim fit ankle grazers, and a black Autograph merino wool top. One thing I'd say, is they definitely look more expensive in person, and the cut makes them look so classy. Finally, I'd say due to the cut of the shoe, and the way it doesn't dig in, the comfort and the PRICE (absolute bargain), means it's a definite winner!\nRated 5 out of 5 by Kitty08 from Kitty08 I bought the gold shoes to go with a new dress, which they looked very pretty and sophisticated with. However they look lovely with jeans as well. They are extremely comfortable, and cover the front of the foot well - you don't feel like the front of your foot is cut off, like some cuts on court shoes. Only one small issue - I have a really small foot and 3 is a good size, however when I wore them out that night, it was a little cold, so felt like I was stepping out of the right foot (which may have been slightly bigger, having been the one most people try on)? May need to get an insert to ensure it stays on! I do love them though and would recommend!\nRated 5 out of 5 by Chalkey from Buy Buy Buy!!! I have been looking everywhere for some gold shoes with a low heel, something casual enough for jeans or dressy enough for dinner out - thank you M&S! These are super comfy, a great height and such value for money that I will be buying another pair in gold and two pairs in black for work so that when one pair wears out I have another one to fall back on! I bought a size 7, my feet can be wide and do expand with heat, but I wore these out for the first time last week and didn't even remember that I had 'new' shoes on, and had wore them again at the weekend, both outings got compliments. Buy them now ladies before they sell out!\nRated 5 out of 5 by angel67 from comfortable courts M&S photography does not do these shoes justice! If I hadn't tried them on in-store, I'm not sure I would have ordered them. V stylish & comfortable, good size heel (not too high and not too low). And they sit comfortably over front of toes. Have them in khaki and metallic gold...they look great with jeans.\nRated 5 out of 5 by k7las from White Shoes- ABSOLUTELY NOT These are lovely shoes for the price, albeit not leather but only to be expected at \u00a325. However I wanted white shoes and on any planet there is no way these could be described as such. Very annoying. To be honest I didn't think that they looked at all white on the website but you have to trust the description, or so I was led to believe. These are a creamy colour with a pinkish cast. Beware. Unfortunately they will be returned, such a shame.\nRated 5 out of 5 by Supersallyshopaholic from Love these shoes! In my younger days I ruined my feet trying to squeeze into ill fitting shoes. These shoes offer so much comfort because they are high rise and thus don't put pressure on the big toe causing bunions to form. Please provide us with more high rise shoes like these. The style is nice, they are very comfortable the size is right and the heel height is perfect. My favourite style and fit of shoes!\nRated 4 out of 5 by Ladybird60 from Work shoes I love these shoes and purchased a second pair when I kicked the heels out of my first pair. My review reflects the low cost, design and fit. I generally buy leather shoes, but when I tried these shoes on in store they were a perfect fit, in width and height of heel. I love the simplistic style and only wish M&s made these in leather to exact design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"knowledge about limestone hammer pemecah. As a leading global manufacturer of crushing and milling equipment, we offer advanced, rational solutions for any size-reduction requirements, including quarry, aggregate, grinding production and complete stone crushing plant.\nThe London Hammer: An Alleged Out-of-Place Artifact. ... 1983) stated the hammer was \"in limestone dated at 300 million years old\" (which would make it Pennsylvanian).\nWhere can I find the limestone crusher? Update Cancel. ... What is the composition of a limestone? ... How can I find Hammer Crusher Limestone in Guinea-in Bissau?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At VAGO we recognise that how we select, develop, value and plan for our people is key in building an innovative audit office known for the excellence of its Parliamentary reports, professional audit work and advice to stakeholders and audit clients.\nWe look for individuals who are creative, ethical and passionate and who are dedicated to promoting continuous improvement, integrity and public interest within the public sector. VAGO's staff are professionally qualified and accredited in a broad range of disciplines such as arts, economics, business, law, accounting, public policy, social and natural sciences, and statistics.\nWe operate in a diverse and complex environment, and we are committed to professional excellence, best practice and innovation.\nWe've made it simple for you to search current VAGO jobs and locate the one that's right for you.\nupdate your account to ensure your account details are current and up-to-date\u2014your email address is essential information for us to contact you.\nFor further information on careers with VAGO, please contact us at recruitment@audit.vic.gov.au.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm a 35-year-old male. My therapist has suggested that I may have high-functioning Aspergers (symptoms are difficulty with talking, words and overall social ability; extreme difficulty with change in routine; isolation; astounding and detailed long-term memory with poor short-term, etc.). I find it hard to believe that a \"fully developed\" adult can actually remedy this.\nShould I confront this diagnosis as a behavioral issue with cognitive behavioral therapy \u2026or a biological one with medication? At this point, can a treatment do anything besides make me more comfortable with the disorder? I've asked my therapist, but because most medical literature addresses intervention in childhood, he can't say much.\nAspergers (high functioning autism) is nearly impossible to identify outside of the context of traditional social and cultural settings. The brain is simply wired a bit differently and acts on different sets of cues. You're not defective.\nIf your \"impairment\" is mild, you may have just always been considered \"socially awkward\" \u2013 and there may not be any particular medication available that doesn't have side effects or risks greater than the problems you already have. As a general rule of thumb, stay away from drugs. There's nothing to fix! If you have secondary symptoms (e.g., depression, anxiety, OCD, etc.), then maybe you should consider medication (but make sure you get an opinion from an ASD specialist).\nTreatment alternatives may be as simple as behavioral therapies, behavioral coaching, or group therapy. Aspies may need to work with a therapist longer than neurotypicals do, because the Aspie's social skills are somewhat lacking. It can take longer for an Aspie to achieve the desired results compared to someone with a different, non-developmentally based problem. But, you may do more harm than good by going to a therapist who knows little about Aspergers.\nThe interpersonal relationship models that most therapists use are not really applicable to those with Aspergers. You'll find conventional therapy telling you to read body language, take social cues, and all sorts of things that the Aspie brain is not wired for. A therapist who specializes in Aspergers will key you in on things that will work.\nThe areas you'll need to focus on are primarily interpersonal relationships (e.g., manners, courtesies, diplomacy, social conventions, dress, hygiene, etc.). Neurotypicals generally acquire those things from an early age through socializing, but Aspies don't pick up on it as well.\nWhat am I trying to change in my life?\nWill an official diagnosis (as opposed to being aware that I probably have Aspergers) do anything or create any opportunities to help me change those things?\nIf my therapist is correct in his diagnosis, what does he have in mind to make it worth my while \u2026what's his plan?\nShould I get a second opinion before doing anything else?\nAspergers is definitely not a death sentence \u2013 far from it. So what if you find out that you really have Aspergers? If you have it \u2013 you have it \u2013 and you always had it. So it's really nothing new.\nThe Specific Carb Diet was developed by Dr. Sidney Haas (a New York City pediatrician) who used it successfully to treat people with ulcerative colitis and Crohn's disease.\nDr. Haas' theory was that carbs (which are forms of sugar) feed the bacteria and yeast in the intestines, which causes an over-abundance of bacteria and yeast. He believed that this bacterial overgrowth prevents (a) enzymes on the intestinal cell surface from functioning and (b) the proper digestion and absorption of carbs. This would cause the carbs to remain undigested in the intestines, which provides even more food for bacterial and yeast growth.\nA number of illnesses can develop from this digestive balance, including celiac disease, chronic diarrhea, crohn's disease, inflammatory bowel disease, irritable bowel syndrome, spastic colon, and ulcerative colitis.\nMany ASD children have severe gastrointestinal symptoms, including diarrhea, constipation, bloating and pain. Some ASD specialists believe these symptoms could be caused by bacteria or fungal overgrowth in the intestines, and ASD treatments \u2013 especially those recommended by alternative medicine specialists \u2013 aim to eradicate the bacteria and yeast.\nThe Specific Carb Diet eliminates the complex starches that feed bacteria and yeast in the intestines, which improves ASD symptoms by starving the bacteria and yeast. Killing these bad bugs not only leads to improvements in the GI tract, but also improves neurological function because many neurological problems actually originate in the digestive system.\nThere are two groups of carbs: monosaccharides and disaccharides. Monosaccharides are simple carbs, easily broken down in the intestines. Disaccharides are complex carbs, and individuals with poor gastrointestinal systems cannot break them down.\nThe Specific Carb Diet characterizes foods as \"legal\" or \"illegal\" based on their carb content. Some \"illegal\" carbohydrates include grains, sugars, beans, potatoes, and all processed foods (including canned vegetables). Some \"legal\" carbs include unprocessed meats, vegetables, fruits, and some dairy products (however, it's possible to do a casein-free version of this diet). The Specific Carb Diet already is naturally gluten-free.\nBe aware that ASD symptoms may not improve right away due to the profound changes taking place in the digestive tract. Also, many parents report significant worsening of symptoms at key points in the diet that they attribute to yeast die-off.\nA survey from the Autism Research Institute shows that the Specific Carb Diet can be very effective overall in treating ASD symptoms. In looking at overall ASD diet approaches, the survey asked 278 parents whose children were following the Specific Carb Diet if it worked. A total of 69% said it had improved ASD symptoms \u202624% said it had no effect \u2026and 7% said it worsened symptoms. Many parents said they had tried other ASD diet approaches, but the Specific Carb Diet proved to be the key, even in the absence of other ASD treatments.\nAlthough the Specific Carb Diet is somewhat restrictive and difficult to follow, many mothers\/fathers have reported significant gains in their kids with autism and Aspergers. Unfortunately, many parents don't stick with the diet long enough to find some benefits in it. It is after all a fairly simple diet: no starch or refined sugars. But that simple statement requires a complete change of life style which, in turn, requires a great deal of determination. The Specific Carb Diet is more difficult to follow than the GFCF diet, and most parents find they must prepare virtually everything at home.\nOur son is a 34-year-old with Asperger's who is living in supported housing. He went into his first apartment 2 years ago. It was very difficult as he was so angry and upset and even took revenge on us by smashing a television. He has had a lot to deal with. He has Crohn's Disease although it is in remission, with two operations at 17 and 19. He is defiant at times, super communicative, although of course it's very much like verbal diarrhea. We haven't been too effective with parenting him, I think because of feeling sorry for him. This is coming back to bite us.\nHe sees a psychiatrist through the community mental health services (about once a month) and also a caseworker more frequently. About a month ago, he hit his psychiatrist (glancing blow on the shoulder), however the doctor has now charged him with assault. We are at our wits end. His MD says because it's a first offense, he will not go to jail but probably get a warning, maybe probation. His psychiatrist, a young fellow, told us a couple of years ago that he really doesn't know much about Asperger's as our son is his only AS client.\nWe know we have to change our communication with him, but my husband is feeling very sorry for him and not drawing a line in the sand very much. Our son is rude often, and often escalates into anger. Other times he is loving and almost normal. Can you offer any immediate suggestion?\nParents with an Aspergers (high-functioning autism) child often have trouble knowing how much to help out their \"suffering\" child at certain times in his life. But, is it really bad to \"cushion\" him or to \"feel sorry\" for him? Unfortunately, the answer is a profound YES!\nLet me be very clear about this: If the Aspergers child hasn't had to work for most of his materials things and privileges over the years \u2026and if parents have \"stepped-in\" time and time again to over-protect and over-assist the child \u2026it WILL cause serious problems for that child later in life. Parents are not doing their Aspergers child any favors by over-indulging and over-assisting, in fact, quite the opposite \u2013 THEY ARE HURTING THEIR CHILD!\nWe're talking about over-indulgent parenting here. Over-indulged children have too much stuff, too much assistance, and soft structure (i.e., lax rules, few chores, aimless). As a result, this child grows up with very little \"self-reliance\" (a critical skill to have to \"make it\" in the real world as an adult).\nOver-indulgent parents often view themselves as loving their child unconditionally by permitting most requests and offering their child free reign with few restrictions. They also believe that being good parents entails supplying the child with most of his wishes \u2013 and assisting at the first sign that the child is struggling.\nBeing \"taken care of\" all of your life has grave consequences. Children who are over-indulged have great goals, but because they are so accustomed to being catered to, they do not have the skills or drive for achieving their ambitions. Impulsivity, refusing to take responsibility, abusing drugs, continuing to live at home as an adult-child, spoiled behavior, and so on, all stem from needing control \u2013 but having no ability to appropriately exercise it.\nThe \"easier life\" makes for children who feel \"privileged\" and who actually miss out on some important social skills (e.g., how to make friends, work with others, achieve self-sufficiency, etc.). Doing well in college, finding and keeping a job, and raising a family takes individual hard work, but if the child is used to not having to work for his money or interact with people in order to do well, his lack of determination will be the catalyst for his downfall.\nOver-indulged children don't know the difference between \"needs and wants.\" Ultimately, knowing what you \"want\" versus what you actually \"need\" is something that comes with maturity, but when a child is so privileged that he gets most of what he wants, it's hard to know the difference. In general, children that are used to being the center of attention and not having to work for their share at life are disadvantaged as adults.\nParents are supposed to set a good example and give their child a strong background in the \"real world\" so that he can succeed on his own someday. If children don't learn early on that making a living doesn't come easy, their lives won't be as fulfilled because they'll have a strong sense \"entitlement\" (e.g., \"You owe me \u2026I shouldn't have to work for anything\").\nAllow your child to experience the negative consequences and painful emotions of poor choices.\nDifferentiate between your child's wants and his needs.\nDiscipline without later reducing or negating the discipline.\nIf you have tried to correct your parent's mistakes by attempting to be a \"better\" parent, know that (a) you turned out all right, and (b) you may be erring on the other end of the extreme.\nKeep an eye out for your child's guilt-trips.\nKnow that your child does not always have to be happy in order to have high self-esteem.\nKnow when to be your child's parent and when to be his buddy.\nLearn to say, and stick with, \"no\".\nMake sure you and your child's other parent are united and bonded on most issues.\nPay attention to your feelings of guilt about how you have parented, and know it is a sign that you are \u2013 once again \u2013 beating up on yourself.\nThink in terms of \"everyone has a responsibility to the solution\" rather than attributing blame.\nWhen you catch yourself feeling sorry for your child, know it is a sign that you are \u2013 once again \u2013 taking on too much responsibility.\nWhen your child needs to be comforted\/cheered-up, do so with active listening, empathy, paraphrasing, validation, hugs, etc. rather than giving him things (e.g., unearned privileges, food, gifts, fun activities).\nYour child is a priority, but allow your marriage to come first (it's the foundation for the entire family).\n\u2026as a result, they parent their child based on what THEY want for him rather than on what he actually needs \u2026or they parent their child the way THEY wanted to be parented by their parents.\nHow can I know for sure whether or not my 8-year-old son truly has Aspergers? I'm not totally convinced at this point. There is some speculation based of observations of some of his behaviors - but I still have my doubts.\nA comprehensive psychiatric evaluation from a Child and Adolescent Psychiatrist who specializes in Aspergers (high functioning autism) will be able to give you a definitive answer to that question.\nDifficulty with imaginative play: The Aspie does not engage in imaginative play as a child.\nLearns best visually: She has trouble learning without visual aids.\nMindblindness: She has mindblindness, meaning she cannot determine what others are thinking and feeling in social situations or in relationships.\nOrganizational skills difficulties: The Aspie experiences difficulty with planning, implementing and completing tasks.\nProblems with coordination: She may have problems with both fine and gross motor skills. Common examples of motor skill difficulty include bike riding, handwriting and playing ball games.\nProblem-solving issues: She has trouble figuring out how to solve problems outside of her routine.\nEasily distracted: The Aspie has trouble concentrating her attention on people and objects that are not connected with her favorite subjects.\nEye contact: He may not make eye contact.\nFacial expression: Facial expressions are either absent or inappropriate to the conversation or situation. She may have facial tics.\nMonotone speech: He may speak in a monotone voice, without expression or emotion.\nPersonal space issues: He might stand too close to a person during conversation.\nUnusual gestures: The Aspie might make unusual or inappropriate gestures during conversation.\nDifficulty processing language: The Aspie does not always understand the verbal speech of others or misunderstands the meaning of a conversation. He may have trouble making a decision or answering a question.\nLanguage rituals: He might have certain word scripts that he repeats ritualistically in conversation with others.\nLiteral interpretation of words: The Aspie interprets most language on a literal level and misses abstract meanings.\nTrouble with language use: He has trouble using language appropriately in social situations. He may also misunderstand common word meanings.\nUnusual use of words: He may use words in an unusual way or create her own words.\nNarrow range of interests and obsessions: The Aspie is intensely interested in a small number of activities and subjects and refuses to engage in other activities.\nSelf-stimulatory behavior: He may engage in stimming behavior such as hand flapping, rocking back and forth or twirling.\nStrict schedule: He prefers a rigid schedule and experiences anxiety when the schedule is interrupted.\nLimited food choices: The Aspie may choose and reject foods based upon smell or texture.\nOdors: She may react strongly to certain smells.\nSounds: She might be hypersensitive to different sounds.\nTouch: The Aspie may not want to be touched.\nDifficulty playing with others: The Aspie may not understand how to initiate play with his peers or how to play by common social rules. For example, he may take a ball from a group of children playing a game without asking to join the game first. He will not return the ball if they ask for the ball back because he does not understand the negative reaction.\nGeneral social skills: He wants to socialize with others,, but does not understand how to interact.\nInability to understand common social cues: The Aspie may not comprehend common social cues such as facial expressions, body language or gestures.\nInappropriate responses: He may behave or respond to social situations in an unusual or inappropriate manner. For example, he\/she may laugh at something sad.\nProblems with two-way conversation: He has trouble with initiating and maintaining a two-way conversation. He may appear to \"talk at\" someone rather than \"with\" them. Conversation topics may focus on an obsessive interest. He speaks inappropriately such as talking too loudly or softly.\nRelating to others: The Aspie does not understand other's emotions or social responses accurately in a group situation. He may not understand if an activity or conversation is boring or upsetting to another person.\nRigid range of interests for social interaction: He will only engage in a narrow range of activities or talk about certain subjects.\nThe Aspergers list above can help you recognize common symptoms of Aspergers. If you believe that your youngster or teenager has this disorder, contact your pediatrician for an Aspergers screening. Early intervention provides the best chance for your son or daughter to live a healthy and fulfilling life.\nMy child with Aspergers often convinces me that he is too sick to attend school. I end up calling-in for him at least a couple times a month. How can I tell if he's really sick \u2013 or if he just doesn't want to go to school that day?\nIs your Aspergers (high functioning autistic) child really too sick to attend school - or is he faking it?\nMany moms and dads have a hard time deciding if their children are well enough to go to school. After all, what well-intentioned mother or father hasn't sent a youngster off with tissues in hand only to get that mid-morning \"come get your son\" phone call? But making the right decision isn't as tough as you might think. It basically boils down to one question: Can your youngster still participate in school activities? After all, having a sore throat, cough, or mild congestion does not necessarily mean a student can't be active and participate in school activities.\nDetermining whether or not a youngster has a fever offers a fairly reliable way of judging whether or not he is truly too sick to go to school. Additionally, it is one of the few symptoms that can actually be quantified. States often have requirements regarding the exact temperature at which kids need to be sent home, especially in early childhood and child-care settings.\nThe time of day during which a youngster is experiencing a fever can also make a difference. Fevers usually run a bit higher in the evening than they do in the morning. So a high temperature in the evening may abate overnight. However, a high temperature in the morning will likely only get worse as the day progresses, so moms and dads should consider keeping children home in this case.\nThe child should probably stay home if it is the first day of the illness and the temperature is over 101. If it is the third day or later, and the youngster has been acting well during the day, but has a 101 temperature in the evening, he probably should go to school.\nA fever isn't the only symptom to track when it comes to judging a kid's ability to attend school. Vomiting, diarrhea, rashes and a host of other indicators can also mean the difference between a desk at school or the couch at home.\nVomiting and diarrhea can also be a tremendous source of discomfort for kids if severe or uncontrolled. In these cases, a day at home may be the best option. If the youngster is vomiting, it is inconvenient for the teacher and the other classmates. If mild and controllable, however, a bit of diarrhea may not be a big problem. In elementary age kids, diarrhea isn't as much of an issue if it doesn't interfere with their ability to remain in the classroom and if they aren't sick enough to potentially have accidents, have to run to the bathroom, or be in pain.\nRashes, particularly those that cannot be readily explained, may also be cause to keep your kid at home -- and perhaps even require a doctor's opinion.\nAnd kids with severe cases of conjunctivitis -- commonly known as pink eye -- should also be kept home from school. However, some doctors note that mild cases of this affliction may not warrant a day off.\nMany moms and dads may also wonder when it is safe for their youngster to go back to school after recovering from their illness. The answer to that one is a little bit trickier. In general, nobody would recommend that a youngster goes back to school unless he has been fever-free for 24 hours, and some would even say 48 hours. A full day of fever-free downtime is probably sufficient to safeguard the youngster's health, as well as that of his peers. It is fairly standard that kids are required to be fever-free for 24 hours before returning, which is a useful method of limiting the spread of infection during the febrile period when kids are thought to be most contagious. When the fever is gone for 24 hours, the contagiousness is greatly diminished.\nThe 24-hour rule may prove to be more than sufficient for vomiting. Vomiting is a temporary nuisance most of the time. So if the youngster feels OK and has not vomited since midnight, consider allowing him to go the next morning.\nIn most cases, however, the decision of whether or not to send a sick kid to school will not be clear-cut. In these cases, moms and dads must ask themselves certain questions to help them decide. Will the illness prevent the youngster's participation in normal school activities? Also, will the youngster's illness place an unusual burden on the staff? A third and very important question to consider is whether or not the illness that the youngster has poses a risk to other kids and adults.\nMoms and dads must also keep in mind that those complaints of abdominal pain could be from a food-borne illness -- or they could just be due to anxiety over the prospect of going to school. But in these situations, it may be better to err on the side of caution.\nMoms and dads have to be willing to trust their instincts. Even when their youngster is not having any objective signs of illness, if they think that the youngster is different from how she normally is, they need to trust their instincts and keep her at home until they figure out what's wrong.\nMoms and dads should also be aware of symptoms that suggest the youngster should be brought to a doctor. If the youngster cannot touch her chin to her chest, it could be a sign of meningitis, a serious infection that warrants immediate medical attention.\nDeciding whether or not to send your kid to school can sometimes be a tough call. After all, it's not always easy to distinguish simple theatrics from true illness. Worse yet, a diagnosis and decision must often be made in the few spare moments after breakfast (and before an angry call from your boss). Fortunately, there are a few rules of thumb that you can follow when determining whether a youngster is up to the task of a full day at school. And doctors say a mild case of sore throat or the sniffles is not necessarily a mandate to keep children at home.\nTrust your instincts. If your child has the sniffles but hasn't slowed down at home, chances are he's well enough for the classroom. On the other hand, he may need to take it easy at home if he's been coughing all night and needs to be woken up in the morning.\nAspergers (high functioning autism) affects behavior, personality, and the way a person interacts with others. As females with Aspergers become adults, they may feel isolated because they react differently to certain \"stressful\" situations. Their comments can seem insensitive and uncaring, when in reality, they simply may not fully understand the concept of empathy. These ladies often look for companionship with other adult females who have similar behavior patterns and outlook.\nThe symptoms of Aspergers in adult females are usually displayed in a more subtle manner, which often results in missed or incorrect professional diagnoses, a lack of access to special education services and provisions in school, and a greater chance of social and emotional problems in adulthood. Several distinct differences exist in regard to the ways that females and males with Aspergers behave.\nFemales with Aspergers are not often aggressive when they get frustrated. Instead, they tend to be withdrawn and can easily \"fly under the radar\" in classrooms and other social environments. These girls are also able to express their emotions in a calmer way than their male counterparts. Young females with Aspergers are often protected and nurtured by their \"non-Aspergers\" friends, who help them cope with difficult social situations. Acceptance from peers can sometimes mask the issues that these kids have so that they are not recognized by educators and parents, and as a result, they are less likely to suggest psychological and social evaluations for young female \"Aspies.\"\nThere are certain personality traits and symptoms that moms and dads, educators, and professionals can look for if they suspect that a young girl or woman has Aspergers. Females with the disorder often display obsessive tendencies in regard to animals, dolls, and other female-oriented interests. While \"non-Aspergers\" females will play with dolls by pretending that they are interacting socially, female Aspies may collect dolls and not use them to engage socially with other kids. Their fascination with certain subjects can lead to them lagging behind their peers in terms of maturity and age-appropriate behavior (e.g., a pre-teen with Aspergers may be fascinated with stuffed animals or cartoons long after other peers her age have outgrown these things).\nFemale Aspies may be mistakenly assumed to have a personality disorder because they mimic typical kids, but use phrases inappropriately. They tend to be bored with others their age and have difficulty empathizing with their peers' worries or problems. While their behaviors are more passive than those typical of males with Aspergers, people who pay close attention to female kids with social and emotional delays can ensure that proper diagnosis and treatment will take place. The younger a child is when she begins to receive the appropriate speech, occupational, and psychological services for the disorder, the greater likelihood she will have of living an independent and functional adult life.\n\u2022 Communication Difficulties\u2014 A girl with Aspergers finds that social communication does not come easily. She may struggle to find topics to talk about that will interest her peers. She often attempts to mimic the interests, behavior, and body language of others in an attempt to \"fit in.\" Many female Aspies become quite adept at this mimicking, causing them to elude diagnosis and treatment throughout life. A girl with Aspergers who does not mimic others appears shy and socially awkward. Her body language is different from her peers, and she seems oblivious to the body language of others. Her voice may lack inflection, and she may show no happiness at the good fortune of others.\n\u2022 Emotional Outbursts\u2014 It may be easier to identify males as having Aspergers, because they express their feelings and frustrations through emotional outbursts, which are more obvious to the observer. On the other hand, females with Aspergers may be more likely to internalize their emotions and feelings, and experience inward or passive signs of aggression. These certain gender-related behaviors might be part of the reason that fewer females are diagnosed with Aspergers.\n\u2022 Fantasy\u2014 Female Aspies are intrigued with fantasies that include magical kingdoms, princesses, and other fairy tale elements. It is possible that the princess fantasies are given little notice, because females in general are more prone to this type of imaginary fantasy and play; therefore, these fantasies are not used as criteria in diagnosing the disorder.\n\u2022 Highly Intelligent\u2014 Girls with Aspergers may be less talkative than other females their age. They are highly intelligent, but like their autistic counterparts, possess poor language skills. Communication and interaction with other kids may be difficult. Female Aspies may strive to learn as much as they can, even though social interaction is limited. Most \"typical\" kids who are socially active have no problem learning in a group setting, whereas Aspie girls may want to study and learn on their own.\n\u2022 Inflexibility\u2014 An girl with Aspergers may be inflexible about her daily routine. She may want to eat the same meal each day and avoid food that has touched other food on her plate. She may arrange her toys a certain way on the shelf, perform the same grooming ritual each day and become upset by any change in her schedule.\n\u2022 Obsessional Interests\u2014 Obsessional interests are another indicator of Aspergers. The child may talk incessantly about her topic of interest or spend the majority of her free time studying it or playing with it. A girl with Aspergers is more likely to have interests that are common to healthy females, whereas an Aspie male is more likely to have an unusual interest (e.g., a girl may be obsessed with horses, while a boy may be obsessed with AAA batteries). This highly focused interest can prove helpful or harmful (e.g., a strong interest in math can help a girl function well in school, while an interest in dolls may cause her to not focus on schoolwork and to eventually bore her friends).\n\u2022 Repetitive Behaviors\u2014 The girl with Aspergers may exhibit repetitive behaviors (e.g., hand-flapping, pacing, stomping, blinking, finger-tapping, etc.). These may become more obvious when she is stressed. Even when she is made aware of these behaviors, she may be unable to control them.\n\u2022 Socially Awkward\u2014 Inability to communicate and physical clumsiness will put young Aspies at a disadvantage from the very beginning. They may seem disinterested and aloof. Asperger kids have difficulty understanding slang and humor. They may seem out of place and will not make friends easily. As Asperger kids grow into teens, many find ways to adapt, and their differences may not seem as pronounced.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was searching for a Property and found this listing (MLS\u00ae #180015461). Please send me more information regarding 1015 Winters, Dayton, NV, 89403-6796. Thank you!\nI'd like to request a showing of 1015 Winters, Dayton, NV, 89403-6796 (MLS\u00ae #180015461). Thank you!\nListing information last updated on April 23rd, 2019 at 7:30pm PDT.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"August 25, 2015, 12:12 p.m.\nThis gorgeous night sky photo was taken in Borrego Springs during the Perseid meteor shower. You can see a couple trails of meteors as they zoomed through the sky during the photo exposure. The Perseid meteor shower provides a beautiful show each summer, and this year peaked in mid-August. Star gazers from all over headed out to the darkest skies they could find to watch.\nIf you missed the height of this shower, don't worry. There are others you can catch this year, and it's not too difficult to do.\nThe first thing you need to do is to look up when meteor showers are happening. For instance, Perseids meteor showers happens every year between mid-July and mid-August. You can check out StarDate for a chart of other showers during the year.\nOnce you know when you can see the most meteors, choose a night with cloud-free skies and travel out to an area well away from city lights. A mile or two outside of town won't be dark enough because the glow from the city will still dim the stars. If you want a great view, you want to travel way, way outside of the city. State and national parks are ideal.\nWhen at your location, lay out a blanket, pop open the thermos of hot chocolate, and enjoy the view. During some showers you will see the bright dash of a shooting star every few minutes.","meta":{"redpajama_set_name":"RedPajamaC4"}}